MECHANICAL PROPERTY CHANGES IN THE CERAMIC
ELECTROLYTE LLZO DUE TO POROSITY, MICROCRACKING
AND LITHIUM INCORPORATION
By
Aaron Foster

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Materials Science and Engineering - Master of Science
2014

ABSTRACT
MECHANICAL PROPERTY CHANGES IN THE CERAMIC ELECTROLYTE LLZO DUE
TO POROSITY, MICROCRACKING AND LITHIUM INCORPORATION
By
Aaron Foster
The intent of this study was to determine the effects of lithium flux on the elastic
moduli of the solid electrolyte lithium lanthanum zirconium oxide (LLZO), with a goal of using
the elasticity measurements as a gauge of Li dendrite growth. Several LLZO billets were
prepared and analyzed using resonant ultrasound spectroscopy (RUS).
The specimens of LLZO had Young’s modulus, E, ranging from 90.7 GPa to 129.0
GPa, and the range for shear modulus, G, was 33.62 GPa to 50.65 GPa. Published values for
LLZO list E = 149.8 GPa and G = 59.6 GPa at 3% porosity [Ni 2012]. The cause for the
difference in elastic moduli values is uncertain. Porosity alone was determined not to account
for the difference in moduli among specimens. A microcrack damage model predicted that
surface microcracks induced by sanding would result in changes that are too small to be
measureable by RUS.
In order to predict whether Li dendrite growth would be detectable by elasticity
measurements, several models were considered for LLZO and Li forming a fiber/matrix
composite. In addition to fibers (dendrites), Li may also form a solid solution with LLZO, so
models based on Nordheim’s rule were also considered.

Dedicated to my grandfather
You were the man I will always strive to be

iii

ACKNOWLEDGMENTS

I would like to thank my advisor Dr. Case and Dr. Sakamoto for providing the
opportunity to be involved in this study, and to experience true scientific research first hand.
Thanks to Dr. Case for the guidance and invaluable expertise he offered over the course of this
work, as well as the extensive revision work and for helping me to find the motivation to press
on. Without his encouragement and mentoring, this work could not have been done. I would
like to thank Dr. Sakamoto and his students Asma Sharafi and Regina Garcia who provided the
samples and assisted me in this study.
I would also like to thank Robert Schmidt for showing me the ropes of graduate school,
research, RUS and our lab. Though he was a fellow graduate student, his knowledge and
guidance made him far more of a mentor than a peer.
The financial support for the research of Next Energy through the grant “Ceramic
electrolytes enabling solid state batteries” is gratefully acknowledged.
Lastly, I would to thank my parents who loved me unconditionally and set an example
to always work hard. I hope to make you proud.

iv

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................................ vii
LIST OF FIGURES ........................................................................................................................ x
1.0 Introduction ............................................................................................................................... 1
1.1 Need for Batteries ......................................................................................................... 1
1.2 Lithium-Ion Battery Components ................................................................................. 2
1.2.1 Intercalation Electrodes ................................................................................. 4
1.2.2 Solid Electrolytes ........................................................................................... 7
1.3 Lithium Deposition ....................................................................................................... 8
1.3.1 Lithium Deposition and Dendrite Formation in Electrolytes ........................ 8
1.3.2 Filaments ...................................................................................................... 11
1.3.3 Anodes as a Means to Control Lithium Deposition ..................................... 11
1.3.4 Electrolytes as a Means to Control Lithium Deposition .............................. 14
1.4 Major Developments in Lithium Ion Batteries ........................................................... 15
1.4.1 Lithium Lanthanum Zirconium Oxide (LLZO) ........................................... 18
1.5 Forced Damped Harmonic Oscillation ....................................................................... 21
1.5.1 Simple Harmonic Oscillator ........................................................................ 21
1.5.2 Damped Harmonic Oscillator ...................................................................... 24
1.5.3 Forced Damped Harmonic Oscillator .......................................................... 27
1.6 Mechanical Properties ................................................................................................. 32
1.6.1 Young’s Modulus of Solid Solutions as a Function of Composition .......... 33
1.6.2 Young’s Modulus of a Fiber Matrix Composite .......................................... 34
1.6.3 Effects of microcracking on Young’s modulus ........................................... 36
2.0 Experimental Procedure .......................................................................................................... 43
2.1 Labels .......................................................................................................................... 43
2.2 Specimen Preparation ................................................................................................. 44
2.2.1 Cutting.......................................................................................................... 44
2.2.1.1 Cutting Sample Billet into Several Specimens ................................. 44
2.2.1.2 Cutting Rectangular Parallelepipeds ................................................. 46
2.2.2 Sanding ........................................................................................................ 47
2.3 Resonant Ultrasound Spectroscopy Scan.................................................................... 50
2.3.1 Initial Measurement ..................................................................................... 50
2.3.2 Initial Prediction........................................................................................... 53
2.3.3 RUS Analysis ............................................................................................... 54
3.0 Results and Discussion ........................................................................................................... 59
3.1 Tables of Results ......................................................................................................... 59
3.1.1 Specimens Received 6-3-14 ......................................................................... 62
3.1.2 Specimens Received 6-20-14 ....................................................................... 63
3.1.3 Specimens Received 7-10-14 and 7-29-14 .................................................. 67
v

3.1.4 Specimens Received 9-23-14 ....................................................................... 70
3.1.4.1 Discussion of Processing .................................................................. 71
3.1.4.2 Discussion of Results ........................................................................ 73
3.1.4.3 Discussion of RUS Scans S-LLZO-92314 ....................................... 75
3.1.4.4 Further Processing of S-LLZO-92314-B .......................................... 79
3.2 Effects on Young’s modulus by Various Mechanisms ............................................... 84
3.2.1 Young’s Modulus as a Function of Porosity ............................................... 84
3.2.2 Effects on Young’s Modulus of Fiber Loading in Composite Materials..... 88
3.2.2.1 Models for Young’s Modulus of Fiber Containing Composites ...... 88
3.2.2.2 Plots for Young’s Modulus of Fiber Containing Composites .......... 96
3.2.2.3 Plot for Young’s Modulus of LLZO Containing Dendrites............ 119
3.2.3 Effects of Solid Solution on Elastic Modulus ............................................ 124
3.2.4 Microcracking ............................................................................................ 131
3.2.4.1 Method and Model .......................................................................... 133
3.2.4.2 Tables and Figures for Microcracking ............................................ 139
4.0 Summary and Conclusions ................................................................................................... 155
4.1 The Effect of Porosity on the Elastic Moduli of LLZO ............................................ 156
4.2 Fiber/Matrix Composite as a Model for the Change in Elastic Moduli of LLZO .... 157
4.3 Solid Solution as a Model for the Change in Elastic Moduli of LLZO .................... 160
4.4 The effect of Microcracking on the Elastic Moduli of LLZO .................................. 160
5.0 Future Work .......................................................................................................................... 163
REFERENCES ........................................................................................................................... 166

vi

LIST OF TABLES
Table 1.6.01 expressions for f, G and ε depending on alignment and crack geometry [Kim 1993a]
........................................................................................................................................... 39
Table 1.6.02 Equations for Sn based on crack geometry where ν is the Poisson’s ratio, c is the
idealized crack depth and a is half of the indentation mark diagonal length [Fan 2012] . 40
Table 2.1.1 Explanation of the specimen labels used in this report .............................................. 43
Table 2.3.1 Values of elastic moduli at select porosity values from Ni et al.[Ni 2012] ............... 54
Table 3.1.01 specimen label, dimensions, mass, density and date analyzed for specimens
analyzed over the course of this study. For the dimensions, if two dimensions are given,
they are height by diameter, if three dimensions are given, it is the three dimensions of a
parallelepiped. ................................................................................................................... 60
Table 3.1.02 specimen label, porosity (P), Young’s modulus (E), Poisson’s ratio, the RMS of the
RUS analysis using the CylModel or RPmodel software, and the number of peaks that
were matched in the analysis for various specimens analyzed over the course of this
study. ................................................................................................................................. 61
Table 3.1.03 specimen label, Young’s modulus (E), Poisson’s ratio (ν), the RMS of the RUS
analysis using the CylModel or RPmodel software, the number of peaks that were
matched, as well as mass, dimensions, density and porosity (P) of the initial
measurements for specimens S-LLZO-62014 .................................................................. 63
Table 3.1.04 specimen label, Young’s modulus (E), Poisson’s ratio (ν), the RMS of the RUS
analysis using the CylModel or RPmodel software, the number of peaks that were
matched, as well as mass dimensions density and porosity (P) of specimens S-LLZO62014-A and S-LLZO-62014-C ....................................................................................... 64
Table 3.1.05 specimen label, porosity (P), Young’s modulus (E), Poisson’s ratio, the RMS of the
RUS analysis using the CylModel or RPmodel software, and the number of peaks that
were matched in the analysis for samples S-LLZO-71014 and S-LLZO-72914 and
specimens from Ni et al. [Ni 2012] ................................................................................... 69
Table 3.1.06 specimen label, porosity (P), Young’s modulus (E), Poisson’s ratio, the RMS of the
RUS analysis using the CylModel or RPmodel software, and the number of peaks that
were matched in the analysis for S-LLZO-71014 before and after it was lithiated .......... 69
Table 3.1.07 Results from the RUS analysis. Dimensions, porosity, Young’s modulus, shear
modulus, Poisson’s ratio, RMS, peak count and unique peak count for the Samples A, B,
vii

C and B.2, each with before and after sanding results. P is porosity, E is Young’s
modulus, G is shear modulus and ν is the Poisson’s ratio ................................................ 71
Table 3.1.08 Thickness and changes in thickness for specimens A, B and C. For sample B, both
sides were from the inside, cut edges. The surfaces of B appeared identical and could not
be determined which side originally corresponded to sample A or sample C. ................. 73
Table 3.1.09 Porosity (P) Young’s modulus (E), shear modulus (G), Poisson’s ratio (ν) RMS,
peak count and geometry of sample S-LLZO-92314-B before being sanded as a disk and
as a rectangular parallelepiped (RP) ................................................................................. 81
Table 3.1.10 Porosity (P) Young’s modulus (E), shear modulus (G), Poisson’s ratio (ν) RMS,
peak count and geometry of S-LLZO-92314-B after being sanded as a disk and as a
rectangular parallelepiped (RP) ........................................................................................ 81
Table 3.2.01 Values of the exponential fit in Figure 3.1.01 according to Equation 3.1.01 .......... 85
Table 3.2.02 Table of Equations to determine Young’s modulus for a fiber/matrix composite .. 89
Table 3.2.03 values of x for the Curtis model based on ηl and ηo from the Krenchell expression
[Hull 1981; Sandler 2002] ................................................................................................ 93
Table 3.2.04 data and assumptions used to plot Figure 3.2.02 ..................................................... 96
Table 3.2.05 Data and assumptions used to plot Figure 3.2.03. This table is similar to table
3.2.04, but now considers aspect ratio, and therefore C to be constant. ........................... 98
Table 3.2.06 Data and assumptions used to plot Figure 3.2.04 .................................................... 99
Table 3.2.07 data and assumptions used to plot Figure 3.2.05 ................................................... 102
Table 3.2.08 data and assumptions used to plot Figure 3.2.06 ................................................... 104
Table 3.2.09 data and assumptions used to plot Figure 3.2.07 data originally from Dean and
Turner [Dean 1973]......................................................................................................... 107
Table 3.2.10 data and assumptions used to plot Figure 3.2.08 data originally from Dean and
Turner [Dean 1973]......................................................................................................... 109
Table 3.2.11 Slope, intercept and coefficient of determination, R2, of linear fit lines in Figure
3.2.09 (a) and (b) ............................................................................................................. 113

viii

Table 3.2.12 slope, intercept and linear coefficient of determination for Figure 3.2.10 ............ 114
Table 3.2.13 the Equation for each model in Table 3.2.02 as well as the partial derivative with
respect to volume fraction (Vf) ........................................................................................ 115
Table 3.2.14 approximate slope of models from Table 3.2.02, found by taking the derivative of
each Equation .................................................................................................................. 117
Table 3.2.15 data and assumptions used to plot Figure 3.2.11 ................................................... 120
Table 3.2.16 accompanying data for Figure 3.2.12 [Fan 2013a] ................................................ 127
Table 3.2.17 accompanying data for Figure 3.2.13 [Rao 2003] ................................................. 129
Table 3.2.18 lists the estimated values for properties of specimens of S-LLZO-92314-B.2 post
sanding after sanding with 600 grit sandpaper. The factor f is a function of the spatial
orientation of the microcracks, G is the microcrack geometry factor, r is the depth of the
microcrack layer divided by the depth of the microcrack free layer and ε is the crack
damage parameter [Case 1993] ....................................................................................... 140
Table 3.2.19 lists the estimated values for properties of specimens of S-LLZO-92314-B.2 post
sanding after sanding with 1200 grit sandpaper. f is a function of the spatial orientation of
the microcracks, G is the microcrack geometry factor, r is the depth of the microcrack
layer divided by the depth of the microcrack-free layer and ε is the crack damage
parameter......................................................................................................................... 141
Table 3.2.20 –Young’s modulus before and after sanding for specimens of S-LLZO-92314 with
the ΔE/E0, as measured by RUS ..................................................................................... 144

ix

LIST OF FIGURES
Figure 1.2.1 Diagram of the basic components of a lithium ion battery, not drawn to scale or
proportion. The anode on the left is composed of graphene layers with lithium
intercalated between the layers. The cathode on the right represents various compounds
that also intercalate lithium such as CoO2, TiS2 and many others. The electrolyte in the
middle can be a polymer, a non-aqueous liquid or a ceramic. (adapted from figure 3 from
Dunn et al.[Dunn 2011] and Figure 1 from Bruce et al.[Bruce 2008]) .............................. 4
Figure 1.2.2 A diagram of TiS2 intercalation cathode. The titanium occupies octahedral sites and
the material forms into layers in which the lithium ions can insert. Yellow atoms are
sulfur and the purple atoms are lithium. ............................................................................. 6
Figure 1.2.3 An illustration of lithium deposition in a lithium metal electrode. SEI stands for
Solid electrolyte interface. The top box is the starting point where the surface of the metal
anode is relatively smooth. If there is low current, the process moves to the middle left
image where lithium deposits under the SEI. If the current is high, the lithium deposits
under the SEI too rapidly, and the volume change causes the layer to break. There is
much less ionic resistance so more Li+ move towards that point causing the growth of the
protrusion. If the current stops, the SEI layer reforms around the protrusion. Adapted
from Aurbach [Aurbach 2002].......................................................................................... 10
Figure 1.5.01 Displacement of an object with simple harmonic motion. (a) δ=0 (b) same wave
with a phase shift of δ=45o, both graphs have tick marks at ω/2π but for (a) every other
tick mark is at a zero point and can’t be seen ................................................................... 24
Figure 1.5.02 Damped harmonic motion plotted versus time according to Equation 1.5.15. Time
is on the x axis and the displacement is on the y axis. The displacement experiences
exponential decay and the maximum amplitude at any point can be expressed as
A(t)=Aoe-bt/2m .................................................................................................................... 25
Figure 1.5.03 Amplitude with various drive frequencies assuming no dampening. The percentage
represents the drive frequency over the natural frequency, ω’’/ω. The value of x on the yaxis is arbitrary and will change based on the mass of an object and the other variables of
Equations 1.5.21 through 1.5.23. For clarity the dampening is set to 0 so they will
originate at the same point and be easier to compare. The important matter to note is that
as driving frequency approaches the natural frequency, the amplitude reaches a
maximum. Equation 1.5.21 shows the systems vibrations are dependent on the drive
frequency, and that is also shown here that the frequency of each curve is different and is
dependent on the drive frequency. .................................................................................... 30
Figure 1.5.04 A diagram of the amplitude with changing friction force based on Halliday
[Halliday 1966]. Maximum amplitude is on the y-axis and ω’’/ω is on the x-axis. Notice
x

when there is no friction the amplitude approaches infinity. As b increases, the maximum
amplitude and the resonance frequency moves to the left. ............................................... 31
Figure 1.6.01 Various cracks, (a) is a slit crack (b) is a modified slit crack geometry with the
mark from an indentation, (c) is a half ellipse geometry and (d) is a modified half ellipse
geometry. Lengths a and c for the Equations in Table 1.6.02 are shown in each figure
[Kim 1993a; Fan 2012] ..................................................................................................... 41
Figure 2.2.1 The LLZO billet being prepared for cutting. The left image shows the billet being
affixed to a glass slide with thermoplastic covering more than half of the specimen to
prevent chipping. Some minor chipping can be seen, as that was how the specimen was
received. The image on the right shows the glass slide mounted to the cutting arm of the
slow speed diamond saw. A cut that had been started can be seen on the left side of the
billet. ................................................................................................................................. 45
Figure 2.2.2 Notice the screw hole on the top side of the block in the left picture. In the picture
on the right, the block is remounted to the saw using that screw hole. In doing so, the
arrow will be rotated exactly 90o, so the cuts will be at exactly 90o to one another. ........ 47
Figure 2.2.3 The specimens to be sanded were placed between the two glass slides and were
sanded down to the thickness of the glass slide. In the image, this specimen is taped down
instead of being affixed with thermoplastic. Using thermoplastic to attach the specimens
to the aluminum plate would have required another round of acetone washing, so taping
the specimen was preferred. .............................................................................................. 48
Figure 2.3.1 An aluminum sample mounted to the three transducer setup. Notice in the image
how the transducers are on the edge of the specimen. The transducers are rounded so the
contact area is very small. Aluminum was not studied here, but the large specimen makes
a clear visual. .................................................................................................................... 51
Figure 2.3.2 RUS scan from an unlithiated LLZO specimen S-LLZO-62014-A. The actual scan
was taken from 3 kHz to 503 kHz. Notice the higher frequencies have much smaller
amplitudes, making them harder to identify and therefore less useful in the RUS analysis
of the elastic moduli. (AU stands for arbitrary units). ...................................................... 52
Figure 2.3.3 Same scan of S-LLZO-6-2014-A as Figure 2.3.2 but with the CylModel predicted
resonance peaks. The black line represents the RUS scan. The sharp peaks on the black
line represent a frequency from the driver transducer that resonates with the specimen.
The black crosses (+) represent the CylModel software’s calculation for the frequency at
which resonance is predicted to occur. The predicted frequencies are displayed here at an
amplitude of 3 so they can be distinguished clearly from the scan (AU stands for arbitrary
units). ................................................................................................................................ 55

xi

Figure 2.3.4 The RUS scan of S-LLZO-62014-A with CylModel predicted peaks as black
crosses (+) and peaks that were reported to the CylModel as red squares. The black line
represents the RUS scan. Notice the predicted peaks (+) have shifted from their location
in Figure 2.3.3. (AU stands for arbitrary units). ............................................................... 56
Figure 3.1.01 Porosity versus Young’s modulus, E, and shear modulus G, for LLZO specimens
S-LLZO-62014-A and S-LLZO-62014-C, cut from the billet S-LLZO-62014, compared
to elasticity data from the study by Ni et al. [Ni 2012]. Squares are the values from Ni
2012 and the circles are the data from this study. The red line represents the least-squares
fit of Equation 3.1.01 for shear modulus, G, and the black line represents the leastsquares fit of Equation 3.1.01 for Young’s modulus, E.................................................... 66
Figure 3.1.02 Porosity vs modulus for specimens received 7/10/14 and 7/29/14 as well as results
from Ni et al. [Ni 2012] and S-LLZO-62014 for comparison. Young’s modulus is in
black, and shear modulus is in red. The filled symbols are from literature and the open
symbols were tested in this study. E is Young’s modulus and G is shear modulus. ........ 68
Figure 3.1.03 An exaggerated representation of how the samples were cut and sanded, sample B
is in the center. Not drawn to scale or precise angle. ........................................................ 71
Figure 3.1.04 Young’s modulus, black symbols, and shear modulus, red symbols, versus porosity
for S-LLZO-92314. Each pair of data points shows a decrease in modulus and an increase
in porosity after sanding.................................................................................................... 74
Figure 3.1.05 RUS scans of (a) sample S-LLZO-62014 from previous work, (b) S-LLZO-92314A Post sanding from this study and (c) aluminum as a reference. .................................... 77
Figure 3.1.06 Young’s modulus, black symbols, and shear modulus, red symbols, versus porosity
for S-LLZO-92314. Each pair of data points shows a decrease in modulus and an increase
in porosity after sanding.................................................................................................... 80
Figure 3.1.07 Sample S-LLZO-92314-B.2 RUS scans before and after it was sanded. The post
sanding had a smoother surface because it was sanded with 1200 grit sandpaper, but
because it was a smaller specimen, comparable peaks between the two scans will appear
at lower frequencies. ......................................................................................................... 83
Figure 3.1.01 Young’s modulus (E) and shear modulus (G) as plotted versus porosity for Ni et al.
[Ni 2012] and S-LLZO-62014. The line is an exponential fit of Equation 3.1.01. The
black line and symbols are Young’s modulus and the red line and symbols are shear
modulus ............................................................................................................................. 85
Figure 3.2.01 Young’s modulus and shear modulus for all specimens in the study that could be
analyzed, plotted against porosity. Filled squares represent Ni et al. [Ni 2012] data and
opened symbols were experimentally determined during this study. Red symbols are
xii

shear modulus and black symbols are Young’s modulus. The black and red line
represents the least squares fit to Equations 3.2.01 and 3.2.02 respectively. ................... 87
Figure 3.2.02 Experimental data from Yeh [Yeh 2006] plotted using different C values, where C
is twice the ratio of length over width of a fiber. .............................................................. 95
Figure 3.2.03 Young’s modulus versus fiber volume. The experimental data is from [Yeh 2006]
ROM is the rule of mixtures model and IROM is the inverse rule of mixtures model. ... 97
Figure 3.2.04 Young’s modulus versus fiber volume fraction using data from Sandler [Sandler
2002]. ROM is the rule of mixtures model and IROM is the inverse rule of mixtures
model................................................................................................................................. 99
Figure 3.2.05 Young’s modulus versus fiber volume fraction using data from Manera [Manera
1977]. ROM is the rule of mixtures model and IROM is the inverse rule of mixtures
model............................................................................................................................... 102
Figure 3.2.06 Young’s modulus versus fiber volume fraction using data from Lee [Lee 1968].
ROM is the rule of mixtures model and IROM is the inverse rule of mixtures model. The
Christensen formula stops where Vf =0.20 because the model is no longer valid above
that................................................................................................................................... 104
Figure 3.2.07 Young’s modulus versus fiber volume fraction using data in Kriz [Kriz 1979] and
Dean and Turner [Dean 1973]. ROM is the rule of mixtures model and IROM is the
inverse rule of mixtures model. ...................................................................................... 106
Figure 3.2.08 Young’s modulus versus fiber volume fraction using data in Kriz[Kriz 1979] and
Dean and Turner [Dean 1973]. ROM is the rule of mixtures model and IROM is the
inverse rule of mixtures model. ...................................................................................... 108
Figure 3.2.09 Normalized data sets from Figures 3.2.03 through 3.2.08. (a) Longitudinally
oriented fibers data and their linear fit line are green, the transversely oriented data and
linear fit line are red and the randomly oriented fiber data and the corresponding linear fit
line are blue. (b) an expanded view of the plot in (a), the same legend applies to both . 112
Figure 3.2.10 Elastic modulus versus volume fraction for Kriz [Kriz 1979] according to Equation
3.2.19 data set when the red data point is not included in the linear fit (green line). ..... 114
Figure 3.2.11 Young’s modulus versus volume fraction in LLZO at 3% porosity. ROM is the
rule of mixtures model and IROM is the inverse rule of mixtures model. ..................... 120
Figure 3.2.12 The modulus of Cu10Zn2As4S13 doped with Cu12Sb4S13. Pure Cu12Sb4S13 at weight
fraction equal to 1 [Fan 2013a]. Together they form a solid solution, which follows
xiii

closely to the Nordheim’s rule. The solid red line represents least-squares fit of the Fan
2013a data to Nordheim’s rule, whose constants are determined from a polynomial fit of
the data. ROM is the rule of mixtures. The dashed lines represent the Nordheim’s rule if
only k changes................................................................................................................. 127
Figure 3.2.13 The modulus MnFe2O4 doped with various atomic fraction CdFe2O4[Rao 2003].
The solid red line represents a least-squares fit of the Rao 2003 data to Nordheim’s rule
whose constants are determined from a polynomial fit of the data. ROM is the rule of
mixtures. The dashed lines represent the Nordheim’s rule if only k changes. ............... 129
Figure 3.2.14 Palmqvist versus half-penny or modified half-ellipse radial-median crack system
from Vickers indentation (a) side view of Palmqvist (b) side view of half penny crack (c)
top view of indentation and cracks, which is the same whether the crack is Palmqvist or
half-penny. The small circles below 2a in (a) and (b) are the plastically deformed regions.
The larger circle in (b) represents the median crack and (c) shows that there are two
median cracks perpendicular to one another in a typically Vickers indentation [Kim
1993a; Fan 2012] ............................................................................................................ 132
Figure 3.2.15 The cut surface of sample of S-LLZO-92314-B after being cut with the low speed
diamond saw (viewed at 2000X magnification). No sanding was done to this specimen.
The image shows no clear sign of cracking or scratches that may produce microcracking.
......................................................................................................................................... 137
Figure 3.2.16 Sample of S-LLZO-92314-B after being sanded with 1200 grit SiC sandpaper at
600X magnification. The length and number of cracks were estimated from micrographs
such as this. ..................................................................................................................... 138
Figure 3.2.17 Sample of S-LLZO-92314-B after being sanded with 1200 grit SiC sandpaper at
2000X magnification. The width of the cracks could be estimated from micrographs such
as this. ............................................................................................................................. 138
Figure 3.2.18 ΔE/E0 versus crack damage parameter using Equation 3.2.22 (black line) and using
r and f values from Table 3.2.18. .................................................................................... 143
Figure 3.2.19 Change in Young’s modulus versus the crack length, L, the red circle is the value
of L calculated in Table 3.2.18. The green triangle is the crack length equal to the width
of specimen S-LLZO-92314-B.2. The plotted line is obtained from Equation 3.2.22 c. 145
Figure 3.2.20 Change in Young’s modulus as a function of crack depth according to Equation
3.2.22 c, black line. The red circle is the crack depth determined in Table 3.2.18 and the
blue square represents a 100 fold increase in crack depth, which would correspond to a
Young’s modulus change that would be detectable by RUS analysis ............................ 146

xiv

Figure 3.2.21 Change in Young’s modulus (black line) and crack length, c, in microns (red line)
as functions of Load, F, in mN. The red circles are the calculated values in Table 3.2.18
for each line and corresponds to the calculated load of 8.51 mN. The red line is a plot of
crack length c as a function of F, from Equation 3.2.26.The black line is a plot of ΔE/E0
from Equation 3.2.22 in conjunction with 3.2.26 ........................................................... 148
Figure 3.2.22 A semilog plot of the change in Young’s modulus as a function of fracture
toughness. The red circle represents the fracture toughness of 1.25 MPa *m1/2 from
Wolfenstine et al. [Wolfenstine 2013]. An increase in fracture toughness results in a
sharp decrease in change in Young’s modulus, since an increase in fracture toughness
corresponds to a decrease in crack length for a given load. The black line represents
Equation 3.2.22 c with varying c values, where c values are determined by Equation
3.2.26 with varying Kc values. ........................................................................................ 150
Figure 3.2.23 ΔE/E0 versus H. The red circle is the Hardness determined for LLZO by Ni et al.
[Ni 2012]. The plot was determined by solving F according to Equation 3.2.27 a, then
using F to solve for c according to Equation 3.2.26 and finally solving for 3.2.22 c ..... 152

xv

1.0 Introduction
Understanding and characterizing the elastic moduli of materials is critical to the
development of the products they make. The great interest in batteries today causes a great need
for research and development of the materials that compose them such as Lithium Lanthanum
Zirconium oxide (LLZO) which is used as a solid electrolyte. Resonance ultrasound
spectroscopy (RUS) is a nondestructive method for characterizing the elastic moduli of LLZO
so the same specimens could be further tested. In this study, RUS was used to analyze LLZO
and better understand its properties.
1.1 Need for Batteries
Improving battery technology is a critical need for modern technology and a major goal
for many researchers. The increasing cost of oil and increasing desire to use cleaner forms
energy is increasing the demand for electric vehicles (EV). The limiting factor in electric
vehicles is the energy supply, which is the battery. The specific energy of current batteries is
fairly low when compared to gasoline [Amine 2014]. State of the art lithium ion batteries have
specific energy values of 150- 200 Wh kg-1 which is only a fraction of gasoline, around 13,000
Wh kg-1 [Amine 2014]. The theoretical limit of intercalated batteries is around 900 Wh kg-1 and
even more exotic designs like Li/S have theoretical limits around 2600 Wh kg-1 [Amine 2014].
However, electric vehicles can be very efficient, with 80-90% of the energy in the battery going
to propulsion [Delucchi 2014]. For gasoline powered vehicles, only about 25% of the energy is
used for motion [Chau 2007]. Materials are being researched with the goal that future batteries
will make EVs competitive with internal combustion engine (ICE) vehicles.

1

Another major area of interest for batteries is load leveling [Whittingham 2012]. Solar
and wind energy have received a lot of attention because the energy can be harvested locally,
and without emissions [Amine 2014] . However the intermittent supply of energy from solar
and wind sources makes these technologies unsuitable to the current grid [Whittingham 2012].
If large scale batteries could be developed, solar and wind energy could be utilized to a much
greater extent [Goodenough 2010; Whittingham 2012; Amine 2014].
Furthermore, portable electronics have become ubiquitous in the modern world. The
increasing usage of cell phones, laptops, music players and other devices will also drive the
demand for battery performance, both in terms of power output and energy density.
1.2 Lithium-Ion Battery Components
A battery is a system with one or more electrochemical cells that convert chemical
energy to electrical energy [Huggins 2009]. Figure 1.2.1 shows the basic components of a
lithium ion battery. On the left is the negative electrode, or anode, which is where the electrons
leave the battery to power an external circuit as the battery is discharged [Dunn 2011]. As the
battery is discharged electrons will exit the cell to provide energy to an external circuit [Dunn
2011]. To balance the loss of the negatively charged electrons, positively charged ions leave the
anode and travel through the electrolyte. The lithium ions and the electrons arrive at the
positive electrode or cathode via different routes. The electrolyte can be a solid or a liquid but it
must not be an electrical conductor, or its electrical conductivity must be so low it is negligible.
For the electrochemical cell to function there must exist a chemical driving force
between the electrodes, which is the difference in the chemical potential [Huggins 2009]. In
lithium batteries the chemical driving force is due to the lithium ions inclination to move from

2

the anode to the cathode. This can be measured by the difference between the Gibbs free energy
of formation for the products and reactants:
ΔGrxno = Σ ΔGfo(products)- Σ ΔGfo (reactants)

1.2.1

where ΔGrxno is Gibbs free energy change per mole of reactant and ΔGfo is the Gibbs free
energy of formation [Huggins 2009] both expressed in units of joules per mole.
When the battery is not externally connected, there will be an internal electrostatic
driving force in the opposite direction as the chemical driving force that prevents or inhibits the
reaction [Huggins 2009]. The maximum work that the electrostatic force can exert can be
expressed by
Wmax=nFE

1.2.2

where n is the moles of electrons, F is Faraday’s constant and E is the cell voltage [Vincient
1984].
Because the electrostatic force and the chemical potential are balanced [Huggins 2009]
Wmax= -ΔGrxno

1.2.3

ΔGrxno =-nFE

1.2.4

and, therefore

The label for anode and cathode is reversed when the battery is charging [Huggins
2009]. To avoid confusion this thesis will only consider the condition when the battery is being
discharged. Components for the battery are discussed in more detail below.

3

Load
e-

-

e

-

+
Li+

Li

Li

Li

Li

Li

Li

Li

Li

Li+

Li+

Anode

electrolyte

Li

Li

Li

Li

Li

Li

Li

Li

Cathode

Figure 1.2.1 Diagram of the basic components of a lithium ion battery, not drawn to scale or
proportion. The anode on the left is composed of graphene layers with lithium intercalated
between the layers. The cathode on the right represents various compounds that also intercalate
lithium such as CoO2, TiS2 and many others. The electrolyte in the middle can be a polymer, a
non-aqueous liquid or a ceramic. (adapted from figure 3 from Dunn et al.[Dunn 2011] and Figure
1 from Bruce et al.[Bruce 2008])
1.2.1 Intercalation Electrodes
Until lithium ion batteries were created, most electrode materials consisted of a
homogeneous metal or alloy [Vincient 1984]. Lithium-ion batteries are different, they operate
as an insertion reaction electrode [Huggins 2009]. In this type of electrode, a host species has a
stable crystal structure that forms many parallel planes [Ebert 1976]. Another species can be
inserted to or removed from the voids between these planes without destroying the host
structure [Huggins 2009]. This is called an intercalation reaction. For a visual representation,
4

see Figure 1.2.1. The host structure may be altered by changes in the stacking order, volume or
bond distance [Ebert 1976].
Intercalation was precisely the advance that made rechargeable lithium batteries
possible. Lithium metal is highly reactive and has safety and structural issues when the battery
is recharged. Lithium in batteries could be made much more stable by placing the particles into
another compound. Rather than lithium ions interacting, the lithium particles are separated by
the host species. During charging or discharging, they become ionized and move across the
battery, thus the name lithium-ion batteries.
Many different crystal structures are utilized for intercalation electrodes. For the anode
side of a Lithium ion battery, graphite is commonly used, making a chemical composition of
LiC6 [Bruce 2008]. TiS2 was the first intercalation cathode material (Figure 1.2.2). Each sulfur
atom connects to three titanium atoms. Each titanium atom occupies an octahedral site and
connects to six sulfur atoms. These structures form into layers where Li-ions can be inserted.
LiCoO2 is a common cathode material that forms a similar structure.

5

Figure 1.2.2 A diagram of TiS2 intercalation cathode. The titanium occupies octahedral sites and
the material forms into layers in which the lithium ions can insert. Yellow atoms are sulfur and
the purple atoms are lithium.
Another popular crystal structure for a cathode material is the spinel structure named
after the naturally occurring mineral that bares this structure, MgAl2O4 [Barsoum 1997a]. It can
be expressed as the general formula AB2O4, where A has an oxidation state of +2 and B has an
oxidation state of +3. In the case of a normal spinel, A2+ occupies tetrahedral sites and B3+
occupies octahedral [Barsoum 1997a]. The structure has a face centered cubic stacking of the A
ions. Some examples of spinel cathodes are LiMn2O4 [Amine 2014], LiTi2O4 and LiTi2S4
[Goodenough 2010]. Spinel materials are structurally and thermally stable during lithium
extraction and spinel materials used for cathodes can have a voltage of over 3V against Li+/Li
anode [Goodenough 2010].
6

Similar to the spinel structure is the olivine structure. In 1997, the olivine material
LiFePO4 was developed for use as a cathode material [Nanjundaswamy 1997]. Olivines have a
hexagonal closed packed structure [Barsoum 1997a]. LiFePO4 is probably the most popular
olivine cathode because of the low cost of the materials; but Ni, Co and Mn can replace Fe
[Whittingham 2012]. In LiFePO4, phosphorus occupies tetrahedral sites and the iron occupies
octahedral sites. the crystal has 1D tunnels for Li deposition [Goodenough 2010]. A visual
representation of this material can be seen in Goodenough [Goodenough 2012].
1.2.2 Solid Electrolytes
Solid electrolytes have been known for several decades, and have seen a strong growth
in interest in recent years [Takada 2013]. Solid electrolytes can offer enhanced durability and
safety over liquid electrolytes [Fergus 2010], they are also more stable and are also less likely
to form side reactions when compared to liquid electrolytes [Takada 2013]. This is because
liquid electrolytes can act as a solvent for many compounds, whereas solid electrolytes
typically only conduct one type of species and are less reactive with other species [Takada
2013].
There are two main types of solid electrolytes: polymers and ceramics [Fergus 2010]. A
popular polymer is polyethylene oxide (PEO) which can have various salts dissolved into it
while the polymer is being processed [Thangadurai 2003]. Polymer electrolytes tend to be less
rigid and easier to produce [Fergus 2010]. Ceramic electrolytes have higher elastic moduli and
are more resilient to high temperatures and abusive environments [Fergus 2010]. The weakness
for ceramic electrolytes is their low ionic conductivity. The conduction of ions requires the
movement of point defects, which requires energy. Consequently, conductivity goes up in
ceramics as the temperature increases [Fergus 2010].
7

In 2003 Thangadurai et al. developed lithium ion conducting garnet materials of the
composition Li5La3M2O12 (where M could represent different metals) for use as a solid
electrolyte [Thangadurai 2003]. One metal that seemed particularly attractive was tantalum,
with an electrolyte composition of Li5La3Ta2O12 which could operate at room temperature. It
showed improved conductivity over other well-known Li ion conductors such as
Li2.88PO3.73N0.14 (LiPON) and lithium beta-alumina [Thangadurai 2003]. Li5La3Ta2O12 was also
shown to be stable against molten lithium metal, which attest to the stability of the material
[Thangadurai 2003].
1.3 Lithium Deposition
The phenomenon of lithium deposition has been extensively documented [Amine 2014;
Brandt 1994; Goodenough 2010; Goodenough 2012; Qi 2010; Kanno 1989; Li 2014]but not
fully understood. One idea that is clear and agreed upon, is that unregulated lithium deposition
can pose a major threat to the safety and cycle life of lithium and lithium ion batteries, and
addressing this issue will be critical to advancing battery technology. When a lithium battery is
cycled, lithium ions will pass back and forth between the anode and cathode through the
electrolyte. Ideally, when the ions come back to the anode they should redeposit in a uniform
layer, but this is not always observed.
1.3.1 Lithium Deposition and Dendrite Formation in Electrolytes
There have been numerous studies on dendrites in liquid electrolytes. In lithium metal
batteries, lithium deposition on the surface of the anode is intended but its morphology is
difficult to control [Li 2014]. Defects, such as cracks, pits or grain boundaries, can cause
uneven deposition below the Solid Electrolyte interphase (SEI) [Li 2014]. This can cause small
8

protrusions which are described as mossy or granular, but can stay confined within the SEI
layer [Li 2014]. If there is a sufficient current density or the protrusion is quite large, it can
grow beyond the plane of the SEI [Li 2014]. Subsequently, the protrusions can evolve into long
fiber or branch like structures called dendrites [Li 2014]. The exact mechanism as to how
dendrites grow is cause of some debate. Early models predicted that the dendrites would grow
from the tip as ions are deposited at the end [Barton 1962]. Later models predicted that
dendrites grow out from the surface of the anode [Yamaki 1998] Others predicted somewhat of
a hybrid theory: the dendrite initially grow from the base, but the lithium ions will eventually to
deposit on the tip of the dendrite [Nagy 1997].

9

Electrolyte
SEI

Electrolyte

Electrolyte
SEI

SEI

Lithium

Lithium

High Current

Low Current

Lithium

Electrolyte
SEI

Lithium
Figure 1.2.3 An illustration of lithium deposition in a lithium metal electrode. SEI stands for
Solid electrolyte interface. The top box is the starting point where the surface of the metal anode
is relatively smooth. If there is low current, the process moves to the middle left image where
lithium deposits under the SEI. If the current is high, the lithium deposits under the SEI too
rapidly, and the volume change causes the layer to break. There is much less ionic resistance so
more Li+ move towards that point causing the growth of the protrusion. If the current stops, the
SEI layer reforms around the protrusion. Adapted from Aurbach [Aurbach 2002]
Most researchers agree that when the electrolyte surface has been covered with
protrusions and the current is high enough, the ions will begin to deposit at the tip of the
dendrites and dendrite growth will continue with further cycling, see Figure 1.2.3 [Li 2014]. If
the dendrites grow until they extend from the anode to the cathode, the battery will be short
circuited resulting in a drop in battery voltage [Rosso 2006]. Even worse, the short circuit can
10

cause explosions or fires in the battery [Aurbach 2002]. The heating or explosion can occur
because the electrolyte is an electrical insulator, but Li metal is a very good electrical
conductor. When a dendrite connects the anode to the cathode, a surge of electricity can flow
through the dendrite, causing very rapid Joule heating [Wen 2012].
1.3.2 Filaments
A similar phenomenon similar to dendrite formation is Filament growth. The formation
of a filament is similar to the formation of a dendrite. A defect in the anode or in the SEI layer
leads to a high amount of lithium deposition at one point [Huggins 2009]. As the filament
grows, the SEI layer will be less thick at this protrusion than at the surface of the anode. This
causes diffusion to occur much easier along the protrusion [Huggins 2009], similar to dendrite
formation. The protective layer gets thinner along the protrusion as it gets further away from
the surface of the anode. Therefore, more lithium will deposit towards the tip [Huggins 2009].
During discharge, if the lithium is taken away from the protrusion at a uniform rate across the
surface, the protrusion will disconnect. Therefore, the filament is taken away from the reaction
process, so the lithium can be considered dead and the capacity of the battery is reduced
[Huggins 2009].
1.3.3 Anodes as a Means to Control Lithium Deposition
Perhaps the most effective strategy for stopping dendrite formation that is presently
available is the use of intercalation electrodes. When a lithium ion battery is recharged, the ions
diffuse through the electrolyte and then diffuse into the intercalated anode. If the lithium ions
are again intercalated, the ions are kept separate from one another in the anode, thus
eliminating the possibility they could form dendrites. However, if the ions arrive at the
11

anode/SEI layer faster than they diffuse into the anode, a surface layer of Li can be deposited
on the anode surface [Purushothaman 2006], then the battery would face the same issues as a
lithium metal battery. Overcharging can also cause lithium deposits to occur on the surface of a
graphite anode [Li 2014]. This is because the anode cannot intercalate any more of the lithium
trying to diffuse in so the ions in the SEI layer will instead be deposited on the surface [Li
2014]. Graphite is the most common of the intercalation anodes. If lithium particles can
interact, dendrite or deposition issues may ensue, but intercalation electrodes inhibit this.
Other strategies for limiting lithium dendrite growth include using metal electrodes. Al,
Sn or Si will form lithium alloys and will help to keep lithium particles from interacting and
generating dendrites, but most alloy-forming electrodes experience large volume changes
(320% and 260% for Si and Sn, respectively) which tends to pulverize the anode resulting in
irrevocable capacity loss[Zhang 2011].
Many methods have been used to try to overcome dendrite growth. In lithium ion
batteries, increasing the size of the anode relative to the cathode, or increasing the surface area
of the anode, can suppress the dendrite growth [Tang 2009]. To reduce charging time without
enabling dendrite growth, pulse charging has been utilized [Purushothaman 2006]. The pulse
charging concept is based on the idea that by quickly saturating the interface, but not exceeding
the saturation level, maximum lithium diffusion into the anode will occur. At a constant charge
rate, Purushothaman [Purushothaman 2006] observed that the time it takes to saturate the
interface could be up to an hour. By the same concept, dendrites can be avoided in lithium ion
batteries if they are charged over a long period of time so that the SEI is never saturated with
lithium [Li 2014]. This would be acceptable if the batteries were allowed long lengths of time
to recharge, and if the batteries were never exposed to frequent charging and discharging that

12

may lead to spikes in lithium at the SEI layer. Unfortunately, this would make most
applications of batteries (laptops, music players, electric vehicles) much less practical.
In Lithium metal batteries, the approach is somewhat different. Rather than trying to
prevent conditions that can cause lithium deposition on the surface, the focus is to manage
lithium deposition on the surface so that dendrites do not evolve [Li 2014]. It has been
observed that using a lower current densities will cause the lithium protrusions that form on the
surface of the metal anode to be mossy in nature rather than dendritic [Li 2014]. What exactly
defines “high” current density is not precise. Current densities as low as 0.2 mA cm-2 with a
lithium metal anode can generate protrusions [Dollé 2002], but they may only become mossy
up to current densities of 0.45 mA cm-2 [Orsini 1998]. Other sources report dendrite formation
at 0.5 mA cm-2 [Jeong 2008]. Experiments such as those done by Orsoni et al. observed clear
dendritic growth above 2.2 mA cm-2 in a system with a lithium metal anode [Orsini 1998]. For
many experiments that are deliberately avoiding dendrites, the current density is kept below 1
mA/cm-2 [Whittingham 1976; Kanno 1989; Yazami 1983]. But again, this is impractical for
most realistic battery applications.
Recent work also shows that defects in the subsurface of the lithium metal anode may
be the cause of dendrites [Harry 2014]. Using hard X-ray tomography, Harry found small
irregularities forming below the metal surface before protrusions started forming on the
surface. Harry et al.[Harry 2014] suggests that the structures are what dendrites on the surface
nucleate from. These structures form due to the presence of defects, most likely chemical
contamination [Harry 2014].

13

1.3.4 Electrolytes as a Means to Control Lithium Deposition
There are many methods for suppressing dendrite formation in electrolytes. In liquid
electrolytes, increasing concentration of the salt, such as LiPF6 or LiClO4, could accommodate
a thinner SEI layer which would then offer lower resistance to the ions so they would pass
through more rapidly [Jeong 2008]. This has been observed to increase the number of cycles
before the capacity drops to zero in a lithium metal battery with a liquid electrolyte [Li 2014].
Yet another method for suppressing dendrite growth with a metal lithium anode is the
application of mechanical pressure in the electrolyte [Li 2014]. Gireaud et al. showed that if the
lithium anode contacts a liquid electrolyte, then the cycling efficiency increased as the pressure
increased over a range from 0.7 kg cm-2 to 7 kg cm-2 [Gireaud 2006]. Gireaud et al. [Gireaud
2006] state that the development of dendrites was unavoidable and was caused by defects in the
microstructure of the lithium anode, such as cracks or stress lines.
Another method used to try to suppress dendrite growth was to increase the operating
temperature of the battery to help the ions diffuse through the SEI layer and prevent a buildup
of ions at the surface of the SEI [Mogi 2001; Li 2014]. Conversely, Park [Park 2008] found
increasing the temperature during recharging, while holding the current constant, would
actually enhance the dendrite formation. In the study by Park [Park 2008], it was observed that
as the temperature of the electrolyte material was increased, the resistance decreased, so in
order to keep the same current, the voltage also had to decrease. Li [Li 2014] suggests that the
contradiction in the findings may be a result of the SEI layer forming slower at lower
temperatures. Therefore, when the entire system is at an elevated temperature because of
ambient conditions, the SEI layer between the anode and electrolyte will form more rapidly
than at lower temperatures; but if the increase in temperatures is created by the resistance in the
14

electrolyte, the SEI will still be at a lower temperature to start with and will form slower than in
the case of elevated temperatures from an ambient conditions [Li 2014].
Many battery designs use solid and liquid electrolytes together [Johnson 1998;
Goodenough 2010]. The solid electrolytes can slow the process of dendrite growth [Fergus
2010; Miyashiro 2005]. However, Rosso et al. observed dendrite growth through a solid
polymer electrolyte that led to cell failure [Rosso 2006]. Dendrite growth in solid electrolytes
has been observed elsewhere, either at the anode/electrolyte interface or by penetrating the
electrolyte itself [Orsini 1998; Fergus 2010; Dollé 2002; Sakamoto 2013]. If the solid
electrolyte has a shear modulus of twice that of the lithium, surface roughening between the
anode and solid electrolyte interface can be suppressed [Monroe 2005]. Furthermore, Monroe
and Newman calculated that at a high enough shear modulus for the electrolyte, dendrites will
be prevented from penetrating into the electrolyte altogether [Monroe 2005].
1.4 Major Developments in Lithium Ion Batteries
There has been numerous challenges in creating lithium batteries, and there has been
even more efforts to overcome them. As early as the 1960’s there was research into secondary
lithium batteries, but they mostly focused on high temperature systems [Jasinski 1967]. The
first rechargeable lithium-ion batteries would come from the work of M.S. Whittingham while
working for Exxon [Fletcher 2011]. In 1976 he reported the first use of a lithium intercalating
material for electrochemical energy storage at ambient room temperatures [Whittingham 1976].
Titanium disulfide (TiS2) was used as a cathode, metallic lithium was used as the anode and a
liquid organic solution was used as the solvent [Whittingham 1976]. The experimental batteries
in this research were able to be cycled over a thousand times and had a theoretical density of
480Wh/kg, contemporary batteries were only able to achieve this charge at elevated
15

temperatures [Whittingham 1976]. A major drawback to this research was the low current
densities 1-10 mA cm-2 [Mizushima 1980]. Materials for the anodes, cathodes and electrolytes
would have to be further developed to achieve higher power output.
A group led by John. B Goodenough developed the LiCoO2 cathode, which was an
improvement over the previous cathode because the theoretical specific energy would be over 1
kWh kg-1[Mizushima 1980]. The voltage was also improved, over 4 volts in LiCoO2 versus
about 2 volts in titanium disulfide [Mizushima 1980]. However, the current density had to
remain low because of issues with the available electrolytes and the lithium metal anode
[Mizushima 1980]. At high current density or voltage, the liquid electrolyte could break down
[Mizushima 1980]; and lithium deposition was also a major concern. According to Mizushima,
using an anode other than lithium metal reduced the voltage that can be obtained from the
battery [Mizushima 1980]. The high voltage of the LiCoO2 cathode opened the door for
different anode materials to be used [Mizushima 1980].
The graphite electrode was first successfully used in 1983 by utilizing a solid
electrolyte, polyethylene oxide [Yazami 1983]. In this setup, the graphite electrode was used as
the cathode and lithium metal was used as the anode. To maintain high efficiency when cycled,
the cell was still limited to low current densities like the cells that first used TiS2 and LiCoO2.
By the late 1980’s many carbon based materials were being used for intercalation anodes
[Kanno 1989].
Sony made further improvements on this developments and in 1991 they were able to
introduce the first commercial lithium ion battery [Amine 2014]. This battery had a practical
specific energy of 90 Wh/kg, which was almost double that of Nickel Cadmium based batteries,

16

and at least 10% more than Nickel metal hydride, the two closest competing chemistries that
were commercially-available [Fletcher 2011].
A history that seems to run intertwined with batteries is the developments of the solid
electrolyte. In the 1960’s, researchers at Ford Scientific Laboratory made the discovery of an
alkali metal conducting material [Huggins 2009]. This was quite different from the batteries of
the time because it had liquid electrodes and a solid electrolyte [Kummer 1972]. Although this
was not a lithium based battery, it did prove that many ions could rapidly transport through
solids, including lithium [Yao 1967]. This helped to spark interest in solid electrolytes. Other
solid electrolytes would be developed that had faster lithium ionic conduction, such as Li3N
and LiTi2(PO4)3 [Takada 2013].
Polyethylene oxide (PEO) is a polymer solid electrolyte that is a very popular. Polymer
Solid electrolytes have the advantage of being flexible, easy to produce and many have good
ion conductivity, 10-4 to 10-3 S cm-1 [Chatani 1987]. Polymer solid electrolytes have increased
in ionic conductivity and generally have higher values than ceramic solid electrolytes. In 2003
Thangadurai and others developed a garnet structure as a fast ion conducting solid electrolyte
[Thangadurai 2003]. This material had a nominal structure of Li5La3M2O12, where M= Nb or
Ta. The solid electrolyte had an ionic conductivity comparable to other ceramic electrolytes of
the time, but it was still fairly low, on the order of 10-6 S cm-1, but it did have the advantage of
being remarkably stable against lithium, it appeared to be unreactive even when placed against
molten lithium [Thangadurai 2003].
Thangadurai and Weppner continued to work on the material and in 2007 they
published another paper with a reformulated version of the material, Li7La3M2O2 [Murugan
2007]. This material met many of the criteria needed for a usable solid electrolyte: high

17

temperature stability, low cost materials, high mechanical properties, stability against lithium
and negligible electronic conductivity [Murugan 2007]. The main advantage was its improved
ionic conductivity over previous ceramic electrolytes. At room temperature the ionic
conductivity was around 10-4 S cm-1 [Murugan 2007]. Work continues on these materials.
1.4.1 Lithium Lanthanum Zirconium Oxide (LLZO)
Further work would produce a new garnet material Li7La3Zr2O12 (LLZO) [Murugan
2007]. In the study, LLZO had a conductivity of approximately 6 X 10-6 S cm-1 at 27 ˚C, which
was more than an order of magnitude better than Li5La3Ta2O12 [Murugan 2007]. This finding
made LLZO the best ionic conductor of any Li5La3M2O12 garnet material that had yet been
synthesized [Murugan 2007]. LLZO is made from relatively abundant materials that are
environmentally benign and LLZO is chemically and thermally stable with most electrode
materials; it is also relatively easy to prepare and densify [Murugan 2007]. One drawback is
that when the LLZO is being processed, a different product may form, La2Zr2O7 also known as
pyrochlore, which has low conductivity and is sensitive to air [Murugan 2007]. Another
challenge for LLZO is to form the right phase. Two phases exist for LLZO: tetragonal phase,
which forms above 950 ˚C in pure LLZO and the more conductive cubic phase, which forms
above 1230 ˚C in pure LLZO, but could form as low as 1180 ˚C [Awaka 2011].
Cubic phase has more than an order of magnitude higher conductivity than the
tetragonal phase [Wolfenstine 2012a]. Meier and others investigated the ion conducting
mechanisms in LLZO [Meier 2014]. Meier found that the tetragonal phase has all of its lithium
sites fully occupied, in order to have lithium move through the electrolyte, many lithium ions
must move together, all at once [Meier 2014]. For the cubic phase, there are many more free
lithium sites in the LLZO electrolyte, and the motion of lithium ions is described as “single-ion
18

jumps”, meaning the ions can move individually to vacant lithium sites, which requires much
less energy [Meier 2014].
Several methods have been explored to generate and stabilize the cubic phase. Yttrium
had been used as an additive which achieved a conductivity of more than 8 X 10-4 S cm-1
[Murugan 2011]. Geiger el at. [Geiger 2011] used specimens with Aluminum added to the
LLZO material, because the addition of aluminum to LLZO may have occurred unintentionally
in previous studies of LLZO. Geiger suspected that in the first experiments to synthesize LLZO
as an electrolyte by Murugan and others [Murugan 2007], the aluminum oxide crucibles that
were used added small amounts of aluminum to the LLZO. Regardless of intent, Aluminum
addition to LLZO proved to be beneficial. Rangasamy et al. later explored the effects aluminum
had on the phase [Rangasamy 2012]. They were able to process a cubic phase sample with
composition of Li6.24La3Zr2Al0.24O11.98, pressed at 40 MPa at a temperature of 1000 ˚C. at 98%
density, the material had a conductivity of 4 X 10-4 S cm-1 [Rangasamy 2012]. The processing
temperature Rangasamy used, 1000 ˚C, was much below the 1180 ˚C threshold observed by
Awaka [Awaka 2011], and that was believed to be due to the Al additive.
To stabilize the cubic phase during calcination at 1000 ˚C, 0.204 moles of aluminum in
the LLZO was required, but an excess of 0.389 moles could cause a side product, LaAlO3, to
form [Rangasamy 2012]. Furthermore, when the Al concentration was held constant at 0.24
moles, Li concentration between 5.63 and 6.24 moles would form cubic phase, but above that,
the tetragonal phase would start to form [Rangasamy 2012].
To better understand the effects of Al substituted for Li in LLZO, Ga was used as the
substitute instead, with a formula of Li6.25La3Zr2Ga0.25O12 [Wolfenstine 2012a]. With Ga, the
LLZO material that was hot pressed at 1000 ˚C and 40 MPa appeared to form the cubic phase.

19

The conductivity was 3.5 X 10-4 S cm-1 and the density was approximately 91%; compared to
LLZO with Al substitution at a similar density, it had slightly better ionic conductivity
[Wolfenstine 2012a]. It was proposed that the larger ions of Ga tend to occupy lithium sites that
are less necessary for Li-conductivity, leaving Li-sites that are more involved in the lithium
transport process to be left vacant [Wolfenstine 2012a].
LLZO can have some variety in its composition, such as Al and Ga substitution
previously discussed. Substitutions can take the form Li7-3xM+3xLa3Zr2O12 where M+3 can be Al
[Rangasamy 2012], Ga[Wolfenstine 2012a] or Fe[Rettenwander 2013]. All of these cations
have a +3 charge and sit on a Li site [Thompson 2014]. To balance the charge, three Li ions
must be expelled from the lattice creating two Li vacancies. Other substitutions can take the
form Li7-yLa3Zr2-yM+5yO12 where M+5 can be Sb, Nb or Ta [Thompson 2014]. Because these
ions have a higher charge than the Zr they are replacing (M5+ versus Zr4+), it will be balanced
by creating a Li vacancy. In either method of substitution, 0.4-0.5 Lithium vacancies per
formula unit are required to stabilize the cubic structure in LLZO, which corresponds to x≈0.20
for M3+, direct substitution of lithium, and y=0.4-0.5 for M5+, substitution of Zr [Thompson
2014]. To verify that x=0.4 was needed to stabilize the cubic phase, Thompson et al. tested
specimens of the composition Li6.75La3Zr1.75Ta0.25O12, which should not have enough vacancies
to stabilize the cubic phase and would instead be in the tetragonal phase [Thompson 2014].
Thompson concluded that when there is not enough doping to create 0.4 to 0.5 Li vacancies,
both cubic and tetragonal phases will form in the material [Thompson 2014].
For all the advances and breakthroughs that have come, there is still much work to be
done and indeed interest and research continues to grow.

20

1.5 Forced Damped Harmonic Oscillation
In this study the elastic moduli will be determined from the resonant frequencies of disk
or rectangular parallelepiped specimens. The concepts of resonance can be related to a forced
damped harmonic oscillator. In order to discuss the forced damped harmonic oscillator, we
shall first briefly overview the harmonic oscillator and the damped harmonic oscillator.
To help visualize the concepts, an example of a simple harmonic oscillator would be a
pendulum in a vacuum. Only the force of gravity would be acting on this pendulum and there
would be no loss to the surroundings. A real world example of a pendulum is a person on a
swing which would be a damped harmonic oscillator because as the person swings, they are
losing energy to friction forces such as wind resistance and the moving parts. If another person
pushes the individual on the swing, the example now becomes a forced damped harmonic
oscillator. Even though the person on the swing is losing energy, more energy is being put into
the system by the person pushing.
The time it takes for a body in simple harmonic motion to complete one cycle or a
single round trip is called the period [Halliday 1966]. The frequency is the inverse of the
period, or how many cycles per unit time. The unit for frequency is cycles per second, or hertz
(Hz).
1.5.1 Simple Harmonic Oscillator
A harmonic oscillator can be modeled as a spring-mass system. A perfect spring has no
energy losses. When the mass-spring system of a perfect spring is displaced and then released,
the mass undergoes simple harmonic motion. The potential energy can be described by
Equation 1.5.01 [Halliday 1966]
21

1
𝑈(𝑥) = 𝑘𝑥 2
2

1.5.01

where x is the displacement and k is a constant dependent upon the material. The force acting
on a particle at any point can be described by
𝐹 = −𝑑𝑈/𝑑𝑥

1.5.02

Applying Equation 1.5.01 to Equation 1.5.02 it can be determined
𝐹 = −𝑘𝑥

1.5.03

This is also known as Hooke’s law [Halliday 1966]. Applying Equation 1.5.03 to the
well-known Newton’s Second Law F=ma and substituting a with d2x/dt2 will yield
−𝑘𝑥 = 𝑚

𝑑2𝑥
𝑑𝑡 2

or
𝑑2x
𝑘
=− 𝑥
2
𝑑𝑡
𝑚

1.5.04

According to Halliday the value of x in this Equation must be a function “whose second
derivative is the negative of the function itself” [Halliday 1966]. Cosine and sine functions
have this property. Therefore the general solution to Equation 1.5.04 can be expressed as
x= A cos(ωt + δ)

1.5.05

where A, ω and δ are constants. As a simple proof, take the second derivate of Equation 1.5.05
𝑑2x
= −ω2 A cos(ωt + δ)
𝑑𝑡 2

1.5.06

𝑑2x
= −ω2 𝑥
𝑑𝑡 2

1.5.07

apply Equation 1.5.05

dividing Equation 1.5.07 by Equation 1.5.04 gives
22

𝑘
= ω2
𝑚

1.5.07 a

With the derivation and proof aside, the constants should be discussed. δ is the phase
constant, which determines where the maximum is; for Equation 1.5.04, δ=0 and x=A at t=0
(see Figure 1.5.01). A is the amplitude of the motion and ω is the angular frequency [Halliday
1966].
From Halliday [Halliday 1966] The frequency, v, of the system is defined by the
Equation

𝑣=

1
𝑘
√
2𝜋 𝑚

which shows that the frequency of the spring is only dependent on the constant k and the mass.
Thus the displacement of the mass vs time is represented by a cosine function as depicted in
Figure 1.5.01.

23

Figure 1.5.01 Displacement of an object with simple harmonic motion. (a) δ=0 (b) same wave
with a phase shift of δ=45o, both graphs have tick marks at ω/2π but for (a) every other tick mark
is at a zero point and can’t be seen
1.5.2 Damped Harmonic Oscillator
For the damped harmonic oscillator the spring does have energy loss so that the time
versus displacement curve is different than for the harmonic oscillator [Ohanian 1985]. The
damped harmonic oscillator is a cosine function with amplitude that decrease exponentially
with increasing time as shown in Figure 1.5.02

24

Figure 1.5.02 Damped harmonic motion plotted versus time according to Equation 1.5.15. Time
is on the x axis and the displacement is on the y axis. The displacement experiences exponential
decay and the maximum amplitude at any point can be expressed as A(t)=Aoe-bt/2m
The expression for damped harmonic oscillators is similar to Equation 1.5.04, except
there will be an additional term to represent energy loss
𝑚

𝑑2x
𝑑𝑥
= −𝑘𝑥 − 𝑏
2
𝑑𝑡
𝑑𝑡

1.5.08

where b represents the friction force that is present.
In the case where b is larger than k/m, no oscillation takes place and x quickly goes to
zero [Reddick 1955]. Therefore, it would be more beneficial to explore the scenario where b is
small. To find the solution to Equation 1.5.08, the characteristic Equation should be considered
[Hermann 2014]
𝑚𝑟 2 + 𝑏𝑟 + 𝑘 = 0

25

1.5.09

where
𝑟=−

𝑏
√𝑏 2 − 4𝑚𝑘
±
2𝑚
2𝑚

or
𝑏
𝑘
𝑏 2
√
𝑟=−
±𝑖
− ( ) ≡ 𝛼 ± 𝑖𝛽
2𝑚
𝑚
2𝑚

1.5.10

Evidently, some of the roots of this characteristic Equation are imaginary. α and β can be
defined as

1.5.11

𝑘
𝑏 2
√
β=
− ( ) = ω′
𝑚
2𝑚

b
α=−
2m

1.5.12

Note that when b equals zero, ω' is simply ω. According to Tenenbaum [Tenenbaum
1985] second order, homogeneous differential Equations whose roots follow the form α ± iβ,
will have the general solution
x = ceαt cos(βt + δ)

1.5.13

where δ is an angle or the phase constant. Substituting 1.5.11 and 1.5.12 into 1.5.13 gives
bt

x = ce−2m cos(ω′ t + δ)

1.5.14

where c is a constant. Figure 1.5.02 shows the value of x will be the highest at the initial
amplitude, A0, at t=0 if there is no phase shift (δ=0). Applying these conditions to 1.5.14, the
constant c is shown to be A0. And the solution is
x = 𝐴0 e−bt/2m cos(ω′t + δ)

1.5.15

It is important to note that ω' is slightly smaller than ω. Using ω instead of ω’ to
simplify the Equation will introduce little error, and for simple calculations this is acceptable
[Ohanian 1985], but for maximum accuracy it is included here. A more detailed explanation
26

and following proofs can be found in Tenenbaum and other sources [Tenenbaum 1985;
Reddick 1955; Halliday 1966; Ohanian 1985; Hermann 2014].
1.5.3 Forced Damped Harmonic Oscillator
When energy is added at constant intervals to a damped harmonic oscillator system, it
becomes a forced damped harmonic oscillator. In the case of harmonic oscillators and damped
harmonic oscillators, the period is based on the natural oscillations of the system [Ohanian
1985]. But for a forced damped harmonic oscillator, the period is dependent on the force being
applied. Take for example a person on a swing. If they “pump” their legs at the same period as
the oscillation of the swing, they can increase how high they swing, or the amplitude, this is
called resonance [Ohanian 1985]. However if the person swinging were to “pump” out of phase
as the swing or at a different frequency as the swing, it could cause the swing to not swing as
high. The displacement will reach a maximum when the drive frequency is equal to the
resonant frequency. To find the displacement, start with Equation 1.5.08 and add a new term
for the driving frequency [Halliday 1966]
𝑚

𝑑2 x
𝑑𝑥
= −𝑏
− 𝑘𝑥 + 𝐹𝑚 cos(𝜔′′ 𝑡))
2
𝑑𝑡
𝑑𝑡

or
𝑑2x
𝑑𝑥
𝑚 2 +𝑏
+ 𝑘𝑥 = 𝐹𝑚 cos(𝜔′′ 𝑡)
𝑑𝑡
𝑑𝑡

1.5.16

where m is the mass, Fm is the maximum force of the driver, ω’’ is the frequency of the driver
2
with units of 1/time, b is the term for dampening with units of mass/time,𝑑 x⁄𝑑𝑡 2 is

acceleration, dx⁄𝑑𝑡 is velocity and t is time [Halliday 1966].

27

Equation 1.5.16 is a nonhomogeneous second order differential Equation, therefore the
solution can be found using the method of undetermined coefficients [Tenenbaum 1985]. The
general solution will be the sum of the characteristic solution, xc, and a particular solution, xp.
The solution to the characteristic Equation can be determined as though it were a homogeneous
Equation and will be similar to Equation 1.5.15
𝑥𝑐 = 𝐴0 e−bt/2m cos(ω′′t + δ)

1.5.17

Then the particular solution can be found from the right side of Equation 1.5.16.which
will be a combination of all of its linearly independ derivatives: cos(ω’’t) and sin(ω’’t). In this
case, Fm is a constant and can be ignored
𝑥𝑝 = A sin(ω′′t ) + B cos(ω′′t)

1.5.18

To continue the method of undetermined coefficients xp’ and xp’’ need to be
determined
d𝑥𝑝
= ω′′ A cos(ω′′t ) − ω′′B sin(ω′′t)
dt

1.5.19

and
𝑑2 𝑥𝑝
= −ω′′2 A sin(ω′′t ) − ω′′2 B cos(ω′′t)
𝑑𝑡 2

1.5.20

From here Equations 1.5.18, 1.5.19 and 1.5.20 would be inserted into the original
Equation 1.5.16 to solve for the A and B variables and get an unambiguous Equation for xp.
Then the solution to x would simply be xc plus xp. The algebra for this is somewhat rigorous
and can be found in Reddick [Reddick 1955]. For brevity sake, we will skip to the solution
𝑥=

𝐹𝑚
sin( ω′′ t − δ)
𝐺

28

1.5.21

where
𝐺 = √𝑚2 (ω′′ 2 − ω2 )2 + 𝑏 2 ω′′

2

1.5.22

and
δ = cos −1 (

𝑏ω′′
)
𝐺

1.5.23

In Equation 1.5.21, x is a distance with units of length, and because the value of the sin
term is unitless, the Fm/G term also has units of length. In Equation 1.5.21 it is evident that the
system’s vibration is determined by the frequency of the driving force ω’’ and not the natural
frequency ω. From Equation 1.5.22 it can be seen that as ω’’ approaches ω, G becomes much
smaller and x, the amplitude, rises exponentially. The driving frequency that causes the
maximum amplitude is called the resonance frequency. As shown in Figure 1.5.04, as the value
of b gets smaller, the resonant frequency approaches the natural frequency. In Equation 1.5.22,
if b=0 the resonance frequency will be equal to the natural frequency, ω. For the simple spring
mass driver model of the forced damped harmonic oscillator, there is a single mass on a spring
and one driver, so there would be one resonant frequency. However for bulk solids have a
number of resonant frequencies. Figure 1.5.03 shows the effects different driving frequencies
have on a curve. Figure 1.5.04 shows the effects of friction on maximum amplitude.

29

Amplitude with Various Drive Frequencies

Amplitude

75%
90%
100%
150%

0

pi/2

pi

Time
Figure 1.5.03 Amplitude with various drive frequencies assuming no dampening. The percentage
represents the drive frequency over the natural frequency, ω’’/ω. The value of x on the y-axis is
arbitrary and will change based on the mass of an object and the other variables of Equations
1.5.21 through 1.5.23. For clarity the dampening is set to 0 so they will originate at the same
point and be easier to compare. The important matter to note is that as driving frequency
approaches the natural frequency, the amplitude reaches a maximum. Equation 1.5.21 shows the
systems vibrations are dependent on the drive frequency, and that is also shown here that the
frequency of each curve is different and is dependent on the drive frequency.

30

Forced Damped Harmonic
Oscillator with Varying Dampening Force
b=0
b=mw/4
b=mw/2
b=mw

Amplitude = Fm/G

3Fm/k

2Fm/k

Fm/k

0.0

0.5

1.0

1.5

2.0

Ratio of driving frequency to natural frequency
Figure 1.5.04 A diagram of the amplitude with changing friction force based on Halliday
[Halliday 1966]. Maximum amplitude is on the y-axis and ω’’/ω is on the x-axis. Notice when
there is no friction the amplitude approaches infinity. As b increases, the maximum amplitude
and the resonance frequency moves to the left.
In this study the resonant ultrasound spectroscopy (RUS) system uses a transducer to
send a sine wave through the material. The frequency of the transducer will increase and find
several frequencies that resonate. The resistance within the material will damp the wave. It also
acts as a forced damped harmonic oscillator by continuing to input energy to maintain the
wave. As the damping increases, the amplitude of the peak decreases. If the material is not
entirely homogeneous (such as a change in density or change in material phase) it will resonate
with less intensity and over a wider range of frequencies causing the peak to broaden, or
increases in width [Leisure 2004]. This is called internal friction. There are other sources of
31

internal friction such as point defects and [Ogi 2006]. RUS be discussed further in later
sections, but it is important to understand that the system acts as a forced damped harmonic
oscillator.
1.6 Mechanical Properties
How stress affects a material is determined by its mechanical properties. Mechanical
properties are inherent material properties that do not change based on the size of the specimen.
However, Specimens with the same chemical composition can have drastically different
mechanical properties. Many factors determine the mechanical properties of a particular
specimen including processing methods, temperature and defects.
RUS is often used to measure mechanical properties such as Young’s modulus, E, shear
modulus, G, and Poisson’s ratio, ν. Hooke’s law states the stress is proportionate to the strain
[Popplewell 1907]. Accordingly, Young’s modulus can be defined by Hooke’s law
E=

σ
ϵ

1.6.01

where σ is stress and ϵ is strain [Kingery 1976a] (this is an alternative form of Hooke’s Law as
stated in Equation 1.5.03).
Being that the strain is unitless, the Young’s modulus will have the same units as the
stress, namely Pascal (Pa), often on the order of GPa for metals and ceramics. The shear
modulus can be expressed in a similar way [Kingery 1976a]
G=

τ
γ

1.6.02

where G is the shear stress, τ is shear stress and γ is shear strain [Kingery 1976a].
Shear modulus for metals and ceramics can also have units of GPa, though of less
magnitude than the Young’s modulus. Porosity will decrease a materials modulus value. Higher
32

porosity will mean more empty space in a cross sectional area, which in turn means less
material to bear the stress, leading to a lower modulus value (this will be further discussed in
section 3.2.1).
Poisson’s ration, ν, can be defined by Equation 1.6.03 [Kingery 1976a]
ν=

Δd/d
Δl/l

1.6.03

where d is thickness and l is length.
This is the ratio of the thickness decrease to length increase. For elastic deformation,
values typically range from 0.2 to 0.3 for ceramics [Kingery 1976a]. Poisson’s ratio can be
related to Young’s modulus and shear modulus by the formula [Kingery 1976a]
ν=

E
−1
2G

1.6.04

A single crystal may have directional dependent values for E and G based on the
orientation of the crystal. In graphite, for example, when the stress is parallel to the graphene
sheets, they have high modulus values; but when the stress is perpendicular to the sheets, the
sheets can be pulled apart with little resistance resulting in a low modulus. In the same way as
graphite, any material, such as a composite material or a layered material, that has a different
structure in a different directions will also have different modulus values in the various
directions.
1.6.1 Young’s Modulus of Solid Solutions as a Function of Composition
As a crystal forms it may be exposed to a different type of atom than the ones that forms
its structure. If incorporating the secondary atom into the existing structure raises the energy,
the crystal will reject it or be unstable [Kingery 1976b]. But if incorporating the atom into the
crystal lowers the energy, it will be added to the [Kingery 1976b]. If more of the secondary
33

atoms are added, the crystal may change into a more order structure, or form two separate
crystals, the intermediate phase is called a solid solution [Kingery 1976b]. Perhaps the most
important characteristic of a solid solution is that the material will have a range of secondary
atoms it can absorb. It can be less than 1% like in the case CaCl2 or form a complete solid
solution over the entire range of composition such as the case of Si and Ge. When both
materials are completely soluble over the entire range of composition it is known as complete
solid solubility [Barsoum 1997b].
A substitutional solid solution is when the atom of one material replaces an atom of the
host structure, this is common in ceramics [Kingery 1976b]. Several factors determine whether
or not an atom can form a substitutional solid solution with another material including; size
factor, valance charge, chemical affinity and structure type [Kingery 1976b].
Another type of solid solution is the interstitial solid solution. Rather than the atoms
substituting in place of one of the host ions, the smaller added atoms sit on the interstitial sites
between the host atoms. This is most common with metallic bonds who then incorporate H, C,
N or B into interstitial sites [Kingery 1976b]. Solid Solutions will be discussed further in
section 3.2.5
1.6.2 Young’s Modulus of a Fiber Matrix Composite
Another type of material that may explain lithium deposition in LLZO is a composite
that contains fibers. Lithium-ion flux in a battery can cause dendrites to grow into the solid
electrolyte, in a form that resembles fibers [Dollé 2002; Rosso 2006]. This could change the
elastic moduli of the material. Kingery presents several models to analyze how changes in
composition can affect the elastic moduli [Kingery 1976a].

34

The simplest method to estimate the Young’s modulus in a composite material is the
rule of mixtures (ROM) and inverse rule of mixtures (IROM) for the upper and lower bounds,
respectively [Islam 2011]
𝐸𝐶𝑈 = 𝑉𝑚 𝐸𝑚 + 𝑉𝑟 𝐸𝑟 (Rule of mixtures)

1.6.05

and
1
𝑉𝑚
𝑉𝑟
=
+
𝐸𝐶𝐿
𝐸𝑚 𝐸𝑟

(Inverse Rule of mixtures)

1.6.06

ECU = Upper estimate for Young’s modulus of the composite material, ECL = Lower estimate for
Young’s modulus of the composite material, Vm = Volume of matrix, Em = Young’s modulus
of matrix, Vr = Volume of reinforcement, Er = Young’s modulus of reinforcement [Kingery
1976a].
The ROM model estimates the upper bound and assumes the fibers, or layers, are
aligned in the direction of the stress [Kingery 1976a]. The IROM model sets the lower bound
and assumes the two phases, as layers or fibers, are perpendicular to the applied stress [Kingery
1976a]. The ROM and IROM and also known as the Voigt and Ruess models, respectively. The
bounds set by these models are rather wide. Hashin and Shtrikman determined Equations with
narrower bounds which did not require any particular geometry for the phases, so it could be
more widely used [Kingery 1976a]. The Equations, where K2 > K1 and where G2 > G1, are
𝐾𝐶𝐿 = 𝐾1 +

𝐾𝐶𝑈 = 𝐾1 +

𝑉2
1
3𝑉1
𝐾2 − 𝐾1 + (3𝐾1 + 4𝐺1 )

𝑉1
1
3𝑉2
𝐾1 − 𝐾2 + (3𝐾2 + 4𝐺2 )

35

1.6.07

1.6.08

𝐺𝐶𝐿 = 𝐺1 +

𝑉2
6𝑉1 (𝐾1 + 2𝐺1 )
1
𝐺2 − 𝐺1 + 5𝐺1 (3𝐾1 + 4𝐺1 )

1.6.09

𝐺𝐶𝑈 = 𝐺2 +

𝑉1
6𝑉2 (𝐾2 + 2𝐺2 )
1
𝐺1 − 𝐺2 + 5𝐺2 (3𝐾2 + 4𝐺2 )

1.6.10

K = bulk modulus, G= shear modulus, V = Volume fraction. In Equation 1.6.07 through 1.6.10,
the subscripts 1 and 2 refer to the two materials, where the stronger material is labeled 2.
Bounds for the Young’s modulus using the bounds from the Hashin Shtrikman Equations
inserted into the relation
𝐸=

4𝐾𝐺
3𝐾 + 𝐺

1.6.11

An application of the ROM and IROM as well as other fiber/matrix composite models
will be utilized and discussed further in section 3.2.2.
1.6.3 Effects of microcracking on Young’s modulus
Microcracks are small cracks that cannot typically be seen by then naked eye and have a
detrimental effect on elastic moduli [Case 1993]. Microcrack can occur during polishing,
during machine operations, due to temperature change or accidental damage to the surface
[Barsoum 1997c]. Microcracks can be induced by small abrasive particles, like the sanding
medium, which can mimic a “sliding indentation” creating a crack on the specimen surface.
Microcracking can also be correlated to temperature. Microcracking is unavoidable when the
material is cooled below a critical microcrack initiation temperature [Zimmermann 2001; Case
1980]. One major cause can be buildup of stress at grain boundaries during temperature change
due to a mismatch in thermal expansion coefficient [Barsoum 1997c]. The critical microcrack
36

initiation temperature can be defined by Equation 1.6.12 [Case 1980]
𝑇𝑀𝐼

−14.4𝛾𝑓
1 1/2
=
∗ ( ) + 𝑇0
2
𝐸 ∗ ∆𝛼𝑚𝑎𝑥
𝑔𝑠

1.6.12

Where TMI is the temperature at which microcracks will begin to initiate upon cooling to (or
below) this temperature , gs is grain size, γf is the fracture energy, E is the Young’s modulus,
Δαmax is the maximum change in thermal expansion coefficient over the given temperature
range and T0 is temperature at which the internal stresses are no longer relieved [Case 1980].
Equation 1.6.12 shows the microcrack initiation temperature is a function of fracture
toughness, thermal expansion, grain size, and the total temperature change.
The extent to which the microcracks affect the elastic modulus properties is dependent
upon microcrack number density, orientation geometry and the size of the microcrack [Case
1993]. Models are available to characterize the effects that microcracking has on a material, as
discussed by Case and Kim [Case 1993; Kim 1993a; Kim 1993b]. The material was modeled as
a layered composite, where microcracks exist to a certain depth, but below this layer is an
unaltered, crack free region [Case 1993]. In a layer with microcracks, assuming homogeneous
distribution of the cracks, the change in Young’s modulus can be expressed by Equation 1.6.13
[Case 1993]
𝐸0 − 𝐸
= 𝑓𝐺𝑁𝑣
𝐸0

1.6.13

where E0 is the Young’s modulus of the material with no microcracks, E is the Young’s.13
modulus of the material with microcracking, f is a function of the spatial orientation of the
microcracks, Nv is the number of cracks per unit volume and G represents the crack geometry
which can be represented by the Equation
2 < 𝐴2 >
𝐺=
𝜋 <𝑝>
37

1.6.14

where A and p are the area and perimeter of the microcrack respectively [Case 1993].
Equation 1.6.13 is applicable for a single layer or a material with homogenous
microcrack distribution, but the Equation is inadequate for a material where the crack is limited
to a surface layer and a crack free layer exists. To predict the modulus of a multilayer of
material, Case and Kim [Case 1993]use a modified rule of mixtures (ROM) Equation
𝐸2𝑅𝑂𝑀 = 𝑉𝑙 𝐸𝑙 + 𝑉𝑚 𝐸𝑚

1.6.15

where E2ROM is the rule of mixtures for a material with 2 layers, a microcrack damaged layer
and a microcrack free layer Vl and Vm are the volume fraction of the microcrack layer and the
microcrack free layer, respectively, and El and Em are the Young’s modulus of the microcrack
layer and the microcrack free layer, respectively [Case 1993].
If there are cracks on both sides of the material it can be viewed as a 3 layered material
with a similar application of the ROM [Case 1993]
𝐸3𝑅𝑂𝑀 = 𝑉𝑙1 𝐸𝑙1 + 𝑉𝑚 𝐸𝑚 + 𝑉𝑙2 𝐸𝑙2

1.6.16

where l1 and l2 represent layer 1 and layer 2 respectively.
Equation 1.6.15 and 1.6.16 can be combined with Equation 1.6.13 to get Equation
1.6.17 and Equation 1.6.18 to get a general expression for changes due to the microcrack
containing layers [Case 1993]
𝐸𝑚 − 𝐸2𝑅𝑂𝑀
= 𝑓𝐺𝑙 𝑁𝑙 𝑉𝑙 = 𝑓𝐺𝑙 𝑁𝑣
𝐸𝑚
and

1.6.17
17

𝐸𝑚 − 𝐸3𝑅𝑂𝑀
= 𝑓𝑙1 𝐺𝑙1 𝑁𝑣1 𝑉𝑙1 + 𝑓𝑙2 𝐺𝑙2 𝑁𝑣2 𝑉𝑙2
𝐸𝑚
(for a more detailed derivation see Case and Kim [Case 1993])

1.6.18
.15212
31321

38

In the 3 layer model Equation 1.6.18, if the two layers microcrack containing layers
have the same cracks size, shape, orientation, geometry and number density, then Equation
1.6.18 reduces to Equation 1.6.17. With some understanding of the geometry of the cracks, f
and Gl can be calculated using the Equations in Table 1.6.01.
Table 1.6.01 expressions for f, G and ε depending on alignment and crack geometry [Kim 1993a]
Crack geometry
f(ν0)
G
ε
Orientation
3
3
2
πa /2
N<a >
Aligned
16 ∗ (1 − 𝜈0 )/3
Penny

Half
Penny

(1 − 𝜈02 ∗ (10 − 𝜈0 )
16 ∗
45(2 − 𝑣0 )
16 ∗ (1 − 𝜈02 )/3
(1 − 𝜈02 ∗ (10 − 𝜈0 )
16 ∗
45(2 − 𝑣0 )
𝜋02 ∗ (1 − 𝜈02 )/2

Slit

(1 − 𝜈0 )(5 − 4𝜈0 )
30
16 ∗ (1 − 𝜈02 )/3

Random
πa3/4

0.5N<a3>

Random
2c2l2/(c+l)

4N<c2l2>/(π(c+l))

16 ∗
HalfEllipse

(1 − 𝜈02 ∗ (10 − 𝜈0 )
45(2 − 𝑣0 )
16 ∗ (1 − 𝜈02 )/3

Aligned
Random

𝜋02 ∗

Ellipse

Aligned

π2bd2/4E(k)

Nπ<bd2>/2E(k)

Aligned
Random

π2bd2/8E(k)

Nπ<bd2>/4E(k)

Aligned

Random
(1 − 𝜈02 ∗ (10 − 𝜈0 )
45(2 − 𝜈0 )
ν0 is the Poisson’s ratio of the microcrack free layer, <a> is average radius of crack, c is half of
average crack size for slit, l is the depth of the slit crack, b is half of the major axis of crack, d is
half of the minor axis for the crack and E(k) is the complete elliptical integral of the second kind
where k = (b2-d2)1/2/2
16 ∗

A plot of (E0-E)/E0 versus Nv would be a straight line with a slope, Sn, which is defined
as the product of the crack orientation, f, and crack geometry, G. Sn can be defined by the
Equation 1.6.19 [Fan 2012]
2 < 𝐴2 >
𝑆𝑛 = 𝑓
𝜋 <𝑝>
39

1.6.19

which can then be used in Equation 1.6.13 to get Equation 1.6.20 [Kim 1993b]
𝐸0 − 𝐸
= 𝑆𝑛 ∗ 𝑁
𝐸0

1.6.20

Equation 1.6.20 applies to both 2 and 3 layer specimens if the outside layers of the 3
layer specimen have the same thickness, orientation and geometry [Kim 1993a]. Equations for
Sn can be found from Table 1.6.02 [Fan 2012]
Table 1.6.02 Equations for Sn based on crack geometry where ν is the Poisson’s ratio, c is the
idealized crack depth and a is half of the indentation mark diagonal length [Fan 2012]
Model
Equation
Unmodified slit
(1
− 𝑣02 ) 2(2𝑐 2 )2
2
𝑆𝑛 = 𝜋
geometry
2
4𝜋𝑐
2)
2
Modified slit crack
(1
−
𝑣
2(2𝑐
−
𝜋𝑎2 /2)2
0
2
𝑆
=
𝜋
𝑛
geometry
2
𝜋(4𝑐 + 𝜋𝑎)
2
Unmodified half
(1 − 𝑣0 ) 2(𝜋𝑐 2 /2)2
𝑆
=
16
𝑛
ellipse crack
3
𝜋2𝑐
geometry
modified half ellipse
(1 − 𝑣02 ) 2(𝜋𝑐 2 /2 − 𝜋𝑎2 /2)2
𝑆𝑛 = 16
crack geometry
3
𝜋(𝜋𝑐 + 𝜋𝑎)

To help understand Table 1.6.02, Figure 1.6.01 shows where c and a would be
measured on a crack and what each crack is labeled. Based on figures from Kim and Case and
Fan [Kim 1993a; Fan 2012].

40

2a
c

c
2c
(b)
(b)
2a

2c
(a)
(a)

c

c
2c
(d)
(d)

2c
(c)
(c)

Figure 1.6.01 Various cracks, (a) is a slit crack (b) is a modified slit crack geometry with the
mark from an indentation, (c) is a half ellipse geometry and (d) is a modified half ellipse
geometry. Lengths a and c for the Equations in Table 1.6.02 are shown in each figure [Kim
1993a; Fan 2012]
Similar to Equation 1.6.20 is Equation 1.6.21 which can to find a change in Poisson’s
ratio [Fan 2012]
𝑣0 − 𝑣
= 𝑆𝑣 ∗ 𝑁
𝑣0

1.6.21

where ν is the Poisson’s ratio of the specimen with microcracks, ν0 is the Poisson’s ratio of a
specimen free of microcracks. Sv can be defined as 4πc3/3 [Walsh 1965] for randomly aligned
microcracks.
Using Table 1.6.01 and Table 1.6.02, as well as Equations 1.6.13 to 1.6.21, require very
fine measurements and can become quite tedious. Furthermore, the flat face has a biaxial stress
state and the bulk material has a triaxial stress state, this would make it more difficult to get a
41

reliable number for Nv [Fan 2012]. A predicted S value, Sp, can be used in 1.6.20, which is
easier to measure and doesn’t require detailed measurements of the cracks. Sp can be found
from Equation 1.6.22
𝑆𝑝 =

𝑟2 + 3
𝑟
𝑓
2
(𝑟 + 1) (2 + 1)

1.6.22

where r is the depth of the microcrack layer divided by the depth of the microcrack free layer.
Microcracking can influence the elastic moduli values of a material. The effects of
microcracking on LLZO specimens in this study are discussed in section 3.2.4.

42

2.0 Experimental Procedure
The experimental procedure consisted of several different processes. The LLZO
material was fabricated by Professor Sakamoto’s group. In this project, specimens were
prepared for analysis by resonant ultrasound spectroscopy (RUS) by various means. RUS was
used to determine the elastic moduli of the specimens.
2.1 Labels
The First two specimens in table are from previously published work [Ni 2012].The rest
of the specimens that were analyzed in this study have the same labeling format. The first letter
stands for the Professor’s lab where the specimen was made. The second group of letters is the
abbreviated name of the material. The group of numbers indicates the date that the specimen
was received. Any specimens that share these first three parts of the label are cut from the same
billet. The last character is either the letter or number that designates individual specimens cut
from a single original billet. For example:
Table 2.1.1 Explanation of the specimen labels used in this report
S-LLZO-71014-A
S
LLZO
71014
Lab where specimen
specimen material:
date the specimen
was made:
Lithium Lanthanum
was received:
Dr. Sakamoto’s Lab
Zirconium Oxide
7/10/14

A
Individual
specimen label

Additionally some specimens were analyzed more than once after, typically after some
reprocessing. Therefore the specimen has an additional designation in their label such as “Presanding” or “Post Sanding”. The reprocessing, such as cutting a cylinder into a rectangular
parallelepiped or sanding a specimen, typically improved the RUS analysis result. If a label for

43

reprocessing is not listed on the specimen, it is the latest results from that specimen; and prior,
less accurate results are not included or do not exist.
2.2 Specimen Preparation
Samples need to be made into specimens that could be analyzed with a resonant
ultrasound spectroscopy RUS. For most of the samples in this project, a billet received from Dr.
Sakamoto’s group was cut it into disks one to two millimeters thick. Some of these samples
were sanded. Some samples were cut into rectangular parallelepipeds. All samples were
scanned using the RUS analysis. The RUS data was then analyzed using commercial software
to determine the elastic moduli.
2.2.1 Cutting
2.2.1.1 Cutting Sample Billet into Several Specimens
The billet was placed on a glass slide to prepare it for cutting. The slide and the sample
were slowly heated on a hot plate to prevent thermal shock, which may produce cracking. The
sample was affixed to the glass slide with enough thermal plastic to cover most of the edges in
order to prevent chipping during the cutting, see Figure 2.2.1. The heat setting on the hot plate
was adjusted so that the hotplate was typically just hot enough to soften the thermoplastic, but
not hot enough to liquefy the thermoplastic.

44

Figure 2.2.1 The LLZO billet being prepared for cutting. The left image shows the billet being
affixed to a glass slide with thermoplastic covering more than half of the specimen to prevent
chipping. Some minor chipping can be seen, as that was how the specimen was received. The
image on the right shows the glass slide mounted to the cutting arm of the slow speed diamond
saw. A cut that had been started can be seen on the left side of the billet.
The cuts of the as received billet were made using the low speed diamond blade saw
(Isomet Low Speed Saw, Buehler, Evanston, IL) at a speed setting between 2 and 4. The
micrometer mounted on the Buehler saw was used to move the cutting sample to ensure
opposite faces of the disks would be parallel. The next cut was also made at a setting between 2
and 4. Most individual cuts took from 15 to 45 minutes, depending on the rotations per minute
of the saw blade, the mass of the counter weight and the hardness of sample. After the cutting
was complete the specimens were removed from the thermoplastic by heating on the hot plate.
Again, the specimen was heated just enough to make the thermoplastic pliable, but not hot
45

enough to liquefy it which allowed the pieces of thermoplastic to be removed from the
specimen in complete sections and minimized the thermoplastic melting to the surface of the
specimen. The specimens were then washed with acetone to remove remaining residual
thermoplastic. In the early stages of this project, the acetone was applied directly to the sample,
but this was changed. Acetone was applied to Kim wipes and the samples were wiped
repeatedly until no visible signs of thermoplastic remained. The resulting specimens were
cylindrical disks.
2.2.1.2 Cutting Rectangular Parallelepipeds
Cutting the rectangular parallelepiped follows a similar procedure with some
modification. The specimen to be cut into a parallelepiped initially was cylindrical disk. The
disk was mounted to a glass slide and then the glass slide was mounted to an aluminum block.
The glass slide was placed between the sample and the aluminum block to prevent the saw
blade from cutting into the aluminum mounting block.
The block was mounted to the low speed diamond blade saw (Isomet Low Speed Saw,
Buehler, Evanston, IL) at two points 90o away from each other, that is to say, on adjacent faces
of the block which allows the block and samples to be rotated so the cuts in the specimen are
oriented at 90o from one another (see Figure 2.2.2).
When cutting a parallelepiped, the blade needs to cut deep into the glass so the curved
blade can cut through the flat material. Another reason for cutting deep into the glass slide is
that the blade has a beveled edge. So in order to not have any sort of ridge at the edge of the
specimen, the blade needs to cut sufficiently through the sample.

46

Figure 2.2.2 Notice the screw hole on the top side of the block in the left picture. In the picture
on the right, the block is remounted to the saw using that screw hole. In doing so, the arrow will
be rotated exactly 90o, so the cuts will be at exactly 90o to one another.
2.2.2 Sanding
Some of the cylinder samples were sanded. In some cases reducing the specimen
thickness improved the RUS scan. In other cases, sanding was needed because the low speed
diamond saw cut was not made adequately parallel to the existing surface. By sanding one face
of the specimen, the change in thickness across the specimen would be reduced. Different
methods of sanding were used as the study progressed.
At the beginning of the study, the specimens to be sanded were placed between two
glass microscope slides that were affixed to a smooth aluminum plate, see Figure 2.2.3. Initially
the specimens were secured in place using thermoplastic, enough to reach the thickness of the
glass slides but not to the top of the specimens. This way the thermoplastic would not be
sanded needlessly, until the sanding is nearly complete. After a few trials, the specimens were
taped down with double sided tape instead of using thermoplastic to glue them down between
the glass slides, see Figure 2.2.3. After the specimen was sanded to a uniform thickness, the
47

specimen would be removed from the tape. The specimen was then lightly sanded on the side
that contacted the tape to remove any adhesive residue. Using double sided tape avoided
applying a second coat of thermal plastic to the specimen and thus also eliminated the need for
the specimen to be washed with acetone a second time.
There was a slight difference in height between the two glass slides on either side of the
specimen, but it was very close and far more precise than sanding them free hand. It also
helped to control the angle at which the specimens were sanded so the edges of the specimen
wouldn’t be rounded. After sanding, the variation in thickness would usually decrease to less
than 0.06 mm. The specimens were sanded, individually or could be sanded in groups of 2 or 3,
see Figure 2.2.3.

Figure 2.2.3 The specimens to be sanded were placed between the two glass slides and were
sanded down to the thickness of the glass slide. In the image, this specimen is taped down
instead of being affixed with thermoplastic. Using thermoplastic to attach the specimens to the
aluminum plate would have required another round of acetone washing, so taping the specimen
was preferred.

48

The specimens were sanded using 400 grit sand paper. When they were sanded down to
the height of the glass slide, the specimens were then removed from the aluminum plate. If they
were secured with thermoplastic they would have to be cleaned with acetone wipes, similar to
the procedure after cutting. This method using the glass slides, as depicted in Figure 2.2.3, had
a sever limitation: all specimens would be sanded to the thickness of the glass slides, which
was about 1.16mm.
A later iteration of the sanding process involved using the lapping fixture from lab 2530
(Southbay Technologies model #150) to make parallel faces. If the specimen was a middle
section of the original billet, the specimen would have parallel faces; but if one of the two
circular faces of the specimen was one of the original billet faces, it would not be precisely
parallel to the cut face. In this circumstance the lapping fixture was used. The specimen was
affixed to the lapping fixture using double sided tape with the cut surface facing out, that way
the cut surface became parallel to the original billet surface which was at a 90˚ angle of the
radial surface of the disk. Once the surfaces were made parallel by sanding, the sample was
removed from the double sided tape. If further sanding was required the specimen was held
down against sandpaper by hand and lightly sanded in a circular fashion, then changing
directions. The specimen would be sanded an equally number of times clockwise as
counterclockwise. After a few changes of direction, the specimen would be rotated around and
sanded on a different area of the sand paper.
The sandpaper that was used was Silicon Carbide (SiC) with a grit number 600.
Typically, if more than 0.2mm needed to be removed (usually in the case where the specimen’s
faces were not parallel and were being sanded to make them parallel) lower grit number

49

sandpaper (400 or 240) would be used to start. A final polish of 600 would be used before the
specimen was analyzed using RUS. Later a final polish of 1200 would be utilized.
Changing the sanding procedure allowed less material to be wasted. The height of the
glass slides required every sample to be sanded to the same height. For example, In the later
sanding method, when a sample was cut with an average thickness of 1.40 mm but did not have
parallel faces, rather than sanding it with the glass slide fixture (see Figure 2.2.3) and losing
about 0.24 mm worth of material, it would only need to be sanded until it was parallel.
2.3 Resonant Ultrasound Spectroscopy Scan
The resonant ultrasound spectroscopy (RUS) scanning and analysis are the most
involved steps of the procedure, and accordingly have the most software to assist in the tasks.
Resonant Ultrasound Spectroscopy (RUS) is a non-destructive method for determining elastic
modulus of a specimen [Zadler 2004].
2.3.1 Initial Measurement
Resonant Ultrasound Spectroscopy (RUS) equipment (Quasar RUSpec, Quasar
International, Albuquerque, NM) was used for the measure the elastic moduli. A three
transducer setup was used, one drive transducer and two receivers. The transducers were
arranged so that they touched the corners of the specimen, this allows for the optimal transfer
of vibrations in and out of the specimen and minimal mechanical impendence [Migliori 1993].

50

Figure 2.3.1 An aluminum sample mounted to the three transducer setup. Notice in the image
how the transducers are on the edge of the specimen. The transducers are rounded so the contact
area is very small. Aluminum was not studied here, but the large specimen makes a clear visual.
One of the three transducers in the RUS apparatus is a driver transducer that induces
sinusoidal mechanical vibrations in the specimen. The driving frequency is swept through a
range of frequencies, where the upper and lower frequency limits are set prior to the RUS scan.
As the driving frequency is increased, the specimen will resonate at characteristic frequencies
that depend on the specimen density, composition, dimensions and porosity. When the
specimen vibrates at a resonant frequency, it creates a sharp rise in amplitude, or a peak, see
Figure 2.3.2. The set of characteristic resonance frequencies create a spectrum for this
particular specimen. The RUS experiment is controlled and recorded by the software (Quasar
Galaxy RI2000 and RPModel software, Quasar International). For each specimen, the
measurements were repeated several times. The specimen was rotated or flipped several times
to get different scans. Each time the specimen was moved, the transducer contacts the specimen
at a different set of points. Some points on the specimen will have little or no mechanical
51

vibration these are called antinodes. Antinodes cause the receiver transducers to detect lower
amplitude peaks or miss certain vibrational modes altogether. Taking multiple scans from
different transducer contact points can reveal low amplitude peaks or modes that do not show
up at different contact points. In light of the presence of antinodes in the specimen, it should be
noted that the corners of the specimen is the least likely point to contain the antinodes [Migliori
1993]. The frequency ranges was between 3 to 600 kHz. Most vibrational modes that were
detected were in the range of 30 to 400 kHz, above that range the vibrational modes were too
small in amplitude to differentiate from noise.

Example scan S-LLZO-62014-A RUS
14

Amplitude (AU)

12
10
8
6
4
2
0
50

100 150 200 250 300 350 400 450 500

Frequency (kHz)

Figure 2.3.2 RUS scan from an unlithiated LLZO specimen S-LLZO-62014-A. The actual scan
was taken from 3 kHz to 503 kHz. Notice the higher frequencies have much smaller amplitudes,
making them harder to identify and therefore less useful in the RUS analysis of the elastic
moduli. (AU stands for arbitrary units).
52

2.3.2 Initial Prediction
The spectrum is a bit like a finger print from which the properties of the Young’s
modulus (E), shear modulus (G) and Poisson’s ratio (ν) can be determined. To do so, the
Quasar Galaxy RI2000 and RPModel software are used with Microsoft Excel. An initial guess
for two of the three properties (E, G and ν) must be used. Using two of the three values (E, G
and ν), it is possible to calculate the elastic stiffness constants, C11, C12 and C44. Only C11
and C44 are independent for isotropic materials, where E, G and ν can be determined. The
initial guesses for the elastic moduli has to be relatively close to the actual properties to avoid
divergence and to minimize calculating time in the model [Zadler 2004]. Dimensions and mass
of the specimen being analyzed were also needed.
All of the LLZO specimens analyzed in this study are assumed to be cubic crystal
structures where it is assumed that there is no preferential orientation of grains (no texturing),
thus the specimens are assumed to be isotropic.
A search of literature can sometimes generate a sufficient starting point (an adequate
initial guess for the elastic moduli). In the case of LLZO, elastic modulus values obtained for
LLZO by Ni et al.[Ni 2012] provided a starting point for LLZO specimens included in this
study. In all cases, the elastic moduli are a function of specimen porosity, P, which can be
determined by finding the density of the specimen, ρ, and comparing it to the theoretical
density of 100% dense LLZO, ρtheo=5.107g/cm3 [Ni 2012]
𝑃=

ρtheo − ρ
ρtheo

Values determined by Ni et al. [Ni 2012] are available in Table 2.3.1.

53

2.3.1

Table 2.3.1 Values of elastic moduli at select porosity values from Ni et al.[Ni 2012]
Porosity
E
E (GPa)
G
G (GPa)
(GPa) uncertainty (GPa)
uncertainty
3.0%
149.8
± 0.4
59.6
± 0.1
LLZO-01a
6.0%
132.6
± 0.2
52.1
± 0.04
LLZO-02a

Table 2.3.1 leaves some uncertainty for samples that have porosity other than the
samples listed in Table 2.3.1. Initially whichever value was closer to the measured specimen
porosity was used as the starting point for a RUS analysis. As more results from analyzing
specimens were obtained, better starting estimates could be obtained, this will be discussed
further in Section 3.2.1.
2.3.3 RUS Analysis
Two different models can be used to analyze the specimens depending on their
dimensions. For thin cylinder or disk specimens, CylModel (Quasar International) is used, and
for a rectangular parallelepiped the RPModel (Quasar International) is used. Once the initial
guesses for elastic moduli and the dimensions and mass are entered into the Quasar Galaxy
software, the software will predict the set of resonant frequencies for the specimen, as shown in
Figure 2.3.3. The amplitude of the peaks are in arbitrary units [Fan 2013b]

54

S-LLZO-62014-A RUS scan with
CylModel predicted peaks

14
12

RUS scan
CylModel
Predicted
Peaks

Amplitude

10
8
6
4
2
0
50

100 150 200 250 300 350 400 450 500

Frequency (kHz)

Figure 2.3.3 Same scan of S-LLZO-6-2014-A as Figure 2.3.2 but with the CylModel predicted
resonance peaks. The black line represents the RUS scan. The sharp peaks on the black line
represent a frequency from the driver transducer that resonates with the specimen. The black
crosses (+) represent the CylModel software’s calculation for the frequency at which resonance
is predicted to occur. The predicted frequencies are displayed here at an amplitude of 3 so they
can be distinguished clearly from the scan (AU stands for arbitrary units).
The user then matches the predicted peaks to the RUS peaks, and then the software
recalculates. This is repeated several times until all the peaks that can be matched have been
entered. Figure 2.3.4 shows the same scan as Figure 2.3.2 and Figure 2.3.3 but with the peaks
that were reported to the CylModel marked with red squares.

55

14

S-LLZO-62014-A RUS scan with
CylModel predicted and matched peaks

Amplitude (AU)

12

RUS scan
Peaks
matched for
analysis
CylModel
Predicted
Peaks

10
8
6
4
2
0
50

100 150 200 250 300 350 400 450 500

Frequency (kHz)

Figure 2.3.4 The RUS scan of S-LLZO-62014-A with CylModel predicted peaks as black
crosses (+) and peaks that were reported to the CylModel as red squares. The black line
represents the RUS scan. Notice the predicted peaks (+) have shifted from their location in
Figure 2.3.3. (AU stands for arbitrary units).
Figure 2.3.4 is a RUS scan with indicators for the CylModel analysis. As previously
stated, the CylModel predicts where the resonance frequencies are, which are plotted with the
RUS scan in Figure 2.3.3 and Figure 2.3.4. The red squares in Figure 2.3.4 identifies the actual
frequency from the RUS scan that the predicted peak correlates to. As the user matches the
peaks, typically the lowest peaks are matched first. Lower frequencies tend to make more
distinct peaks, as seen in RUS scan in Figure 2.3.4. Matching the lower frequency predicted
peaks by the CylModel to the actual peaks in the RUS scan adjust the predicted peaks and helps
to identify matches between the prediction and the actual scan at higher frequencies. It can also
56

be observed in Figure 2.3.4 that higher frequencies (above about 250 kHz in this case) are
lower in amplitude, when the amplitude is very low, it is obscured by the background noise.
Sometimes there will be high frequency peaks that have high amplitudes but if several peaks
before it are not identifiable, it cannot be certain if the predicted peak and a RUS peak with
similar frequency are actually correlated. This emphasizes the necessity for matching peaks at
low frequencies and gradually working up to high frequencies.
After the user matches the predicted peak with the actual peak on the RUS scan, the
CylModel will recalculate and change the frequencies of the predicted plots. Initially a set of
values for E, G and ν were used to begin the RUS analysis. This set of elastic moduli values, in
conjunction with the mass and volume of the specimen, produced a set of predicted peaks. As
the predicted peaks were matched to the RUS peaks, the predicted peaks changed and so did
the predicted values for the elastic moduli. It can be observed that the predicted peaks in Figure
2.3.3 are different than the predicted peaks in Figure 2.3.4 due to the CylModel recalculating
after the predicted peaks and actual peaks are matched. As the matching of the predicted peaks
and RUS scan peaks improves, the root mean square error, RMS, will decrease. The RMS is a
measure of how well experimental RUS results can be modeled [Ulrich 2002].
A good fit will have many peaks matched and a low RMS. There are several sources to
help define what number of peaks and what value of RMS make a good fit for the RUS
analysis. Migliori et al. [Migliori 1997] suggests a good fit between a model and a RUS scan
should have an RMS less than 0.2% with 30 peaks fitted. Reports for inhomogeneous
specimens suggest that RMS values greater than 1% represent errors in the RUS analysis that
are too large to be regarded as valid data [Ulrich 2002]. One study by Ren et al.[Ren 2009]
reported RMS values of 0.23% to 0.99% and peak count of 18 to 34. However, there is no

57

standardly accepted value for RMS or peak count that will validate or invalidate RUS results
[Ulrich 2002].

58

3.0 Results and Discussion
Values for elastic moduli were obtained for various specimens of LLZO. The results
were also quite varied. Many factors could have contributed to the variation in elastic moduli,
which will be discussed here.
3.1 Tables of Results
Values for specimens analyzed over the course of this study are presented in Table
3.1.01 and Table 3.1.02.

59

Table 3.1.01 specimen label, dimensions, mass, density and date analyzed for specimens
analyzed over the course of this study. For the dimensions, if two dimensions are given, they are
height by diameter, if three dimensions are given, it is the three dimensions of a parallelepiped.
Specimen Label
dimensions
mass Density
Date
analyzed
(mm)
(g)
(g/cm³)
4.97
LLZO-01a [Ni 2012]
4.78
LLZO-02a [Ni 2012]
1.158 x 12.720
0.729
4.95
6/11/2014
S-LLZO-6314-1
1.175 x 12.683
0.711
4.79
6/12/2014
S-LLZO-6314-3
1.200 x 12.735
0.7059
4.62
7/04/2014
S-LLZO-62014-A
1.253 x 10.758 x 4.860 0.3124
4.77
7/04/2014
S-LLZO-62014-C
1.097 x 12.709
0.6587
4.73
7/16/2014
S-LLZO-71014-A
0.673 x 12.663
0.4114
4.85
7/15/2014
S-LLZO-71014-B
1.147 x 12.663
0.7155
4.95
7/16/2014
S-LLZO-71014-C
1.196 x 12.664
0.7547
5.01
7/16/2014
S-LLZO-71014-D
1.524 x 6.212 x 9.484 0.4464
4.97
7/31/2014
S-LLZO-72914-A
1.276 x 12.664
0.8112
5.05
7/31/2014
S-LLZO-72914-B
1.238 x 12.659
0.7896
5.07
7/31/2014
S-LLZO-72914-C
1.337 x 12.602
0.8336
5.00
10/3/2014
S-LLZO-92314-A pre sanding
0.7699
4.96
10/3/2014
S-LLZO-92314-A post sanding 1.241 x 12.625
1.495 x 12.613
0.9473
5.07
10/3/2014
S-LLZO-92314-B pre sanding
S-LLZO-92314-B post sanding
S-LLZO-92314-C pre sanding
S-LLZO-92314-C post sanding
S-LLZO-92314-B.2 pre sanding
S-LLZO-92314-B.2 post sanding

1.390 x 12.628
1.126 x 12.571
1.097 x 12.583
1.389 x 7.348 x 9.415
1.320 x 7.350 x 9.417

60

0.8721
0.7022
0.6840
0.4844
0.4606

5.01
5.03
5.01
5.04
5.04

10/3/2014
10/7/2014
10/7/2014
10/21/2014
10/21/2014

Table 3.1.02 specimen label, porosity (P), Young’s modulus (E), Poisson’s ratio, the RMS of the
RUS analysis using the CylModel or RPmodel software, and the number of peaks that were
matched in the analysis for various specimens analyzed over the course of this study.
Specimen Label
P
E
G
Poisson's
RMS peaks
ratio
(GPa) (GPa)
3.0% 149.8
59.60
0.257
N/A
LLZO-01a [Ni 2012]
6.0% 132.6
52.10
0.274
N/A
LLZO-02a [Ni 2012]
3.0% 107.4
40.68
0.320
0.435%
26
S-LLZO-6314-1
6.3% 95.4
36.35
0.312
0.215%
24
S-LLZO-6314-3
9.6% 116.5
45.07
0.293
0.690%
16
S-LLZO-62014-A
6.6% 129.0
50.65
0.274
0.262%
14
S-LLZO-62014-C
7.3% 114.5
44.02
0.300
0.661%
14
S-LLZO-71014-A
5.0% 117.2
45.00
0.302
0.829%
18
S-LLZO-71014-B
3.0% 118.9
46.22
0.286
0.734%
16
S-LLZO-71014-C
1.9% 116.0
44.24
0.311
0.399%
12
S-LLZO-71014-D
2.7% 101.0
38.36
0.317
0.578%
8
S-LLZO-72914-A
1.2% 99.7
37.52
0.329
0.667%
14
S-LLZO-72914-B
0.8% 96.3
35.97
0.339
0.552%
14
S-LLZO-72914-C
2.1% 92.2
34.31
0.343
0.939%
16
S-LLZO-92314- A pre sanding
2.9% 90.7
33.62
0.349
0.536%
18
S-LLZO-92314- A post sanding
0.7% 96.7
35.76
0.353
0.654%
18
S-LLZO-92314- B pre sanding
S-LLZO-92314- B post sanding
S-LLZO-92314- C pre sanding
S-LLZO-92314- C post sanding
S-LLZO-92314- B.2 pre sanding
S-LLZO-92314- B.2 post sanding

1.9%
1.6%
1.8%
1.3%
1.3%

94.0
99.1
97.4
97.1
94.5

61

34.91
37.51
36.21
35.88
34.94

0.346
0.322
0.346
0.353
0.352

0.610%
0.609%
0.740%
0.677%
0.324%

20
16
16
10
11

3.1.1 Specimens Received 6-3-14
The first values in Table 3.1.01 and Table 3.1.02 are the results from published work by
Ni et al.[Ni 2012]. Both tables show results for specimens that were analyzed over the course of
this study. Each set of specimens was compared to the values reported by Ni et al.
Samples from the billet received on 6/3/14 were labeled as S-LLZO-6314-A and SLLZO-6314-C. The results from the RUS analysis were reported in Table 3.1.01 and Table
3.1.02. The specimens were cut (according to section 2.2.1) and sanded using the glass slide
fixture (according to section 2.2.2). The specimens were sanded to reduce the variation in
height across the specimen, where the height variation was initially about 0.4 mm and was
reduced to about 0.04 mm or less for the two specimens via sanding. Both specimens were
disks, and although they were cut from the same billet, they had different densities. This
demonstrates that the processing used for LLZO specimens induced a density gradient axially
through the billet. The density gradient was most likely due to porosity which will be discussed
further in section 3.2.1) A density gradient was observed in each successive billet received and
analyzed. A low RMS and a good number of peaks were achieved from the RUS analysis (see
section 2.3.3 for discussion on what defines a “good fit”). A third specimen from the same
billet was analyzed, but the results were not included because the RMS (0.650%) was much
higher than either of the other specimens.
The results for the elastic moduli of specimens S-LLZO-6314-1 and S-LLZO-6314-3
were lower than the results reported in Ni et al. [Ni 2012] which had similar porosities (see
Table 3.1.02). S-LLZO-6314-1 had a porosity of 3%, E= 107.4 GPa and G= 40.68 GPa, where
Ni et al. [Ni 2012] reported E= 149.8 GPa and G= 59.6 GPa at the same porosity of 3%. SLLZO-6314-3 had E= 95.4 GPa and G= 36.35 GPa and LLZO-02a from Ni et al.[Ni 2012] had
62

E= 132.6 GPa and G= 52.10 GPa, and both specimens a porosity of approximately 6%. Some
of the possible causes for why the values for S-LLZO-6314-1 and S-LLZO-6314-3 were lower
than those reported in Ni are discussed in section 3.2.2, section 3.25 and section 3.2.4.
3.1.2 Specimens Received 6-20-14
The next specimens to be analyzed were S-LLZO-62014-A and S-LLZO-62014-C.
Initially the sample was cut into three specimens and analyzed. The initial results were
questionable because few peaks could be matched, and the RMS was higher than previous
specimens (see Table 3.1.03).
Table 3.1.03 specimen label, Young’s modulus (E), Poisson’s ratio (ν), the RMS of the RUS
analysis using the CylModel or RPmodel software, the number of peaks that were matched, as
well as mass, dimensions, density and porosity (P) of the initial measurements for specimens SLLZO-62014
E
G
ν
RMS
P
peak mass Dimensions density
count
(Units) (GPa) (GPa)
(g)
(mm)
(g/cm3)
A- disk 121.99 48.91
B- disk 104.69 43.85
C- disk 130.05 51.07

0.247
0.194
0.273

0.654%
1.202%
0.399%

11
13
14

1.03
0.89
0.77

12.75 x 1.75
12.71 x 1.50
12.67 x 1.27

4.62
4.70
4.85

9.6%
8.0%
5.1%

After the results in Table 3.1.03 were obtained the specimens were further processed in
an attempt to get better results from another RUS scan. Specimen S-LLZO-62014-A was
sanded from a thickness 1.75 mm to 1.20 mm. The thinner specimen attenuated the RUS
vibrations less, which allowed more peaks to be fitted, making the results more reliable.
Specimen S-LLZO-62014-C was cut from a disk 1.25 mm thick and 12.7 mm in diameter to a
rectangular parallelpiped 10.7 mm long and 4.86 mm wide, with the same thickness as before it
was cut, 1.27 mm. Making this specimen smaller reduced the material, which made the

63

specimen more uniform. This method also helped improve the RUS scan by reducing the RMS,
as shown in Table 3.1.04.
Table 3.1.04 specimen label, Young’s modulus (E), Poisson’s ratio (ν), the RMS of the RUS
analysis using the CylModel or RPmodel software, the number of peaks that were matched, as
well as mass dimensions density and porosity (P) of specimens S-LLZO-62014-A and S-LLZO62014-C
E
G
ν
RMS peak mass Dimensions density
P
count
(Units) (GPa) (GPa)
(g)
(mm)
(g/cm3)
S-LLZO116.52
62014-A
S-LLZO129.01
62014-C

45.07 0.293 0.690%

16

0.710

12.75 x 1.20

4.617

9.59%

50.65 0.274 0.260%

14

0.312

1.27 x 10.76 x
4.86

4.769

6.62%

The initial porosity of sample C, when cut into a disk, was 5.1% (Table 3.1.03), but
when it was cut into a parallelpiped the porosity was 6.6% (Table 3.1.04). This indicates there
is also a radial density gradient in the sample. Thus, from the density measurements, it was
apparent that there was a density gradient in both the axial and radial directions.
The RUS analysis was difficult since only a few resonant peaks were available for each
specimen. However, once the samples were further processed (made thinner by sanding or
being cut into a smaller rectangular parallelpiped) the RUS analysis improved, that is,
additional RUS peaks could be matched which in turn leads to more reliable determination of
the elastic modulus. Sanding sample A allowed 16 peaks to be fitted, as opposed to 11 before
the specimen was sanded. Sample C still had 14 peaks fitted after it was cut into a rectangular
parallelepiped, but the RMS dropped from 0.399% to 0.262%. The first five peaks for the
cylinder model for sample S-LLZO-62014-A were obvious, but over 250 kHz, the amplitude of
the resonance peaks dropped off. Some of the peaks over 250 kHz were still readable, but
matching the peaks of the model to the RUS scan was increasingly difficult above 250 kHz

64

because the amplitude of the peaks was very small and of a similar magnitude as the noise
floor. No peaks over 300 kHz could be matched for specimen S-LLZO-62014-A.
In general, for RUS analysis, Poisson’s ratio is much more sensitive to an erroneous
analysis than either the Young’s modulus or shear modulus. For specimens A and C, the values
for elastic modulus, E, and shear modulus, G, in Table 3.1.04 agree well with the results from
Ni et al. [Ni 2012]. In addition, the modulus versus porosity behavior of specimens A and C
agrees well with that of the two specimens from the Ni et al. [Ni 2012]. Sample C of this study
had a porosity of 6.6%, E=129.0 GPa and G= 50.7 GPa. In the study from Ni a specimen with a
porosity of 6.0% had E= 132.6 GPa and 52.1 GPa. The Poisson’s ratio for specimen C and the
6% porous specimen from et al. Ni et al. [Ni 2012]both had a Poisson’s ratio of 0.274. The
agreement between the Poisson’s ratio for specimen S-LLZO-62014-C of this study and the
Poisson’s ratio from the study in Ni et al. [Ni 2012] indicates that, at least for specimen C, the
results of the sample S-LLZO-62014 is in relatively good agreement with the Ni et al. [Ni
2012] study. Specimen A had a higher porosity, 9.59%, and lower elastic moduli values, and
even though there was no sample from Ni et al. [Ni 2012] with similar porosity to compare it
to, it did follow the trend of decreasing elastic moduli with increasing porosity, see Figure
3.1.01.
The four specimens in Figure 3.1.01 (specimens S-LLZO-62014-A and S-LLZO62014-C from this study and the two specimens from Ni et al. [Ni 2012]) each had a different
porosity. The Young’s modulus and shear modulus of the four specimens are plotted versus
porosity, (Figure 3.1.01.). A least squares fit of the four specimens is also plotted in Figure
3.1.01 (red line represents G and black line represents E) based on the relationship
R(P)=R0*exp(-B*P)

65

3.1.01

where R(P) is porosity dependent elastic modulus (either Young’s modulus or shear modulus),
R0 is the elastic modulus property of the material at theoretical 100% density, P is the porosity
of the specimen and B is a fitting parameter unique to the material.

Modulus versus Porosity for previous
200 sample S-LLZO-62014 and Ni 2012
E [Ni 2012]
G [Ni 2012]
E S-LLZO-62014
G S-LLZO-62014
Exponential fit of
Young's modulus
Exponential fit of
shear modulus

Modulus (GPa)

150

100

50

0.00

0.02

0.04

0.06

Porosity

0.08

0.10

Figure 3.1.01 Porosity versus Young’s modulus, E, and shear modulus G, for LLZO specimens
S-LLZO-62014-A and S-LLZO-62014-C, cut from the billet S-LLZO-62014, compared to
elasticity data from the study by Ni et al. [Ni 2012]. Squares are the values from Ni 2012 and the
circles are the data from this study. The red line represents the least-squares fit of Equation
3.1.01 for shear modulus, G, and the black line represents the least-squares fit of Equation 3.1.01
for Young’s modulus, E.
Analysis of S-LLZO-62014 yielded results for elastic moduli similar to values in
literature Ni et al. [Ni 2012]. The data set in Figure 3.1.01 allowed one to estimate the elastic
moduli for a given porosity, which will be discussed further in section 3.2.1.

66

The original intention was to cycle (and lithiated) a specimen after it was analyzed
using the RUS and then analyze the specimen again post-lithiation. Unfortunately, that would
not be possible for these specimens. The thermoplastic was washed off by setting the
specimens into acetone. Furthermore, the specimens were stored in a way that allowed it to be
exposed to air. Lithium is highly reactive and may react to moisture and carbon dioxide in the
air. This was corrected for future specimens, acetone was no longer applied directly to the
specimen and the specimens were stored in an argon filled glove box to prevent exposure to
open atmosphere. However because of the exposure already incurred, the specimens S-LLZO62014 were not lithiated and re-analyzed.
3.1.3 Specimens Received 7-10-14 and 7-29-14
Specimens received 7-10-14 and 7-29-14 underwent the same processing as one
another, which was slightly different than the processing for previous specimens (S-LLZO6314 and S-LLZO-62014). After each specimen was cut from the billet, they were washed with
acetone. However, to avoid the acetone getting into the pores of the specimen, the acetone was
only applied to the Kim wipes and no longer applied directly to the surface of the specimen.
The specimens were taped down using double sided tape when they were sanded, instead of
using thermoplastic to secure them in place. This allowed a second round of acetone washing to
be avoided. Sample S-LLZO-72914-A was cut into a rectangular parallelepiped; all other
specimens remained as cylindrical disks. S-LLZO-71014 was cut into 4 specimens, and SLLZO-72914 was cut into 3 specimens. The most important change to the procedure was the
specimens were stored in a glove box to minimize carbon dioxide and moisture exposure
whenever they were not being processed or analyzed.

67

The results of the analyses for samples S-LLZO-71014 and S-LLZO-72914 are
displayed in Table 3.1.01 and Table 3.1.02. The elastic modulus results versus porosity are
plotted in Figure 3.1.02 for S-LLZO-62014, S-LLZO-71014 and S-LLZO-72914, as well as Ni
et al. [Ni 2012] for comparison.

Porosity Vs Modulus S-LLZO-71014
and S-LLZO-72914 LLZO Specimens

Elastic Modulus (GPa)

150

E [Ni 2012]
G [Ni 2012]
E S-LLZO-62014
G S-LLZO-62014
G S-LLZO-72914
E S-LLZO-72914
G S-LLZO-71014
E S-LLZO-71014

120

90

60

30
0.00

0.02

0.04

0.06

0.08

0.10

Porosity
Figure 3.1.02 Porosity vs modulus for specimens received 7/10/14 and 7/29/14 as well as results
from Ni et al. [Ni 2012] and S-LLZO-62014 for comparison. Young’s modulus is in black, and
shear modulus is in red. The filled symbols are from literature and the open symbols were tested
in this study. E is Young’s modulus and G is shear modulus.
For specimens received 7/10/14 and 7/29/14, the modulus results were noticeably lower
than Ni et al. [Ni 2012], see Table 3.1.05. Each group of samples that were cut from a different
billet seemed to follow its own trend, but there is no clear overall trend between all the
specimens. At this point it was considered that some sort of defect may be forming in the
material. This could be visible defects such as cracking or chipping, or defects that were
68

invisible to the naked eye, such as impurities or microcracks. If impurities were present in the
specimen, it is possible that fewer peaks might be identified in the RUS scan. If the LLZO
samples from S-LLZO-71014 and S-LLZO-72914 contained multiple phases or impurities, the
elastic moduli of the specimen would be dependent on how much of each phase or impurity
was present, and the elastic moduli of the impurity phases themselves. Furthermore, if there
was an impurity that formed a solid solution with LLZO, it could a more complicated effect on
the elastic moduli. The effects of different phases, impurities and solid solutions will be
discussed further in sections 3.2.
Table 3.1.05 specimen label, porosity (P), Young’s modulus (E), Poisson’s ratio, the RMS of the
RUS analysis using the CylModel or RPmodel software, and the number of peaks that were
matched in the analysis for samples S-LLZO-71014 and S-LLZO-72914 and specimens from Ni
et al. [Ni 2012]
Specimen Label
P
E
G
Poisson's
RMS peaks
ratio
(GPa) (GPa)
3.0% 149.8
59.60
0.257
N/A
1
LLZO-01a [Ni 2012]
6.0% 132.6
52.10
0.274
N/A
2
LLZO-02a [Ni 2012]
7.3% 114.5
44.02
0.300
0.661%
14
3
S-LLZO-71014-A
5.0% 117.2
45.00
0.302
0.829%
18
4
S-LLZO-71014-B
3.0% 118.9
46.22
0.286
0.734%
16
5
S-LLZO-71014-C
1.9% 116.0
44.24
0.311
0.399%
12
6
S-LLZO-71014-D
2.7% 101.0
38.36
0.317
0.578%
8
7
S-LLZO-72914-A
1.2% 99.7
37.52
0.329
0.667%
14
8
S-LLZO-72914-B
0.8% 96.3
35.97
0.339
0.552%
14
9
S-LLZO-72914-C
Table 3.1.06 specimen label, porosity (P), Young’s modulus (E), Poisson’s ratio, the RMS of the
RUS analysis using the CylModel or RPmodel software, and the number of peaks that were
matched in the analysis for S-LLZO-71014 before and after it was lithiated
ν
RMS
peaks
Specimen Label dimensions
P
E
G
S-LLZO-71014C unlithiated
S-LLZO-71014C lithiated

(mm)
1.147 x
12.663
1.130 x
12.660

(GPa)

(GPa)

3.00%

118.90

46.22

0.286

0.734%

16

2.71%

122.64

47.63

0.287

0.775%

14

69

Specimen S-LLZO-71014-C was tested after being lithiated. This was the only
specimen that was tested after lithiation, and the results are displayed in Table 3.1.06. Fewer
peaks could be matched and the RMS was still high. As stated in section 2.3.3. Migliori et al.
[Migliori 1997] suggests a good fit between a model and a RUS scan should have an RMS less
than 0.2% with 30 peaks fitted. S-LLZO-72914-C before and after being lithiated did not meet
these standards. The results for S-LLZO-72914-C lithiated and unlithiated (Table 3.1.06) are
outside of the standard suggested by Migliori et al. [Migliori 1997]; The RMS of the specimen
is 0.734% which is greater than the suggested 0.2%; and the peak count is 14 (lithiated) and 16
(unlithiated) which is less than the suggested 30. However, LLZO has been inhomogeneous in
previous studies by Thompson et al.[Thompson 2014], in which case, the RMS is expected to
be higher [Ulrich 2002]. The RMS from S-LLZO-72914-C RUS analysis is within the range
reported by Ren et al. [Ren 2009] (0.23% to 0.99% ) for a study on alumina and
hydroxyapatite. The peaks count of S-LLZO-72914-C is close to the low end of the range of
peak counts in the same study by Ren et al.(18-34 peaks). With all this in consideration, the
results of S-LLZO-72914-C lithiated and unlithiated, are not “good”. Without more specimens
being analyzed using the RUS to determine elastic moduli before and after lithiation, it is
difficult to draw decisive conclusions from the data in Table 3.1.05.
3.1.4 Specimens Received 9-23-14
The LLZO billet received on 9-23-14 was cut into three specimens and analyzed using
resonance ultrasound spectroscopy (RUS) and CylModel software. The specimens were sanded
to improve geometry then scanned using RUS. Then the specimens were sanded and scanned
again to try to identify any impact sanding had on the specimens. The results are displayed
Table 3.1.07 and can be compared to previous work in Table 3.1.01.
70

The specimens had a low Young’s and shear modulus, 90 – 100 GPa and 33-38 GPa
respectively, when compared to the previous work by Ni et al.[Ni 2012] (see Table 3.1.01).
Table 3.1.07 Results from the RUS analysis. Dimensions, porosity, Young’s modulus, shear
modulus, Poisson’s ratio, RMS, peak count and unique peak count for the Samples A, B, C and
B.2, each with before and after sanding results. P is porosity, E is Young’s modulus, G is shear
modulus and ν is the Poisson’s ratio
Specimen Label dimensions
P
E
G
Poisson's RMS
Peak
ratio
Count
mm
sample A pre
sanding
sample A post
sanding
sample B pre
sanding
sample B post
sanding
Sample C pre
sanding
Sample C post
sanding
sample B.2 pre
sanding
sample B.2 post
sanding

(GPa) (GPa)

1.337 x 12.602

2.13%

92.17

34.31

0.343 0.939%

16

1.241 x 12.625

2.95%

90.72

33.62

0.349 0.536%

18

1.495 x 12.613

0.69%

96.74

35.76

0.353 0.654%

18

1.390 x 12.628

1.87%

93.95

34.91

0.346 0.610%

20

1.126 x 12.570

1.56%

99.14

37.51

0.322 0.609%

16

1.097 x 12.583
0.139 x 0.735 x
0.941
0.132 x 0.735 x
0.942

1.85%

97.45

36.21

0.346 0.740%

16

1.26%

97.13

35.88

0.353 0.677%

10

1.32%

94.49

34.94

0.352 0.324%

11

3.1.4.1 Discussion of Processing
There were several issues to mention in terms of
specimens preparation. Starting with the processing of the
specimen, the cutting of the specimen resulted in cuts that
were not parallel to the existing edges (see Figure 3.1.03).
This was corrected in samples A and C (the outside
specimens) by sanding down the interior edge, but could
71

Figure 3.1.03 An exaggerated
representation of how the
samples were cut and sanded,
sample B is in the center. Not
drawn to scale or precise angle.

not be corrected in sample B without significant effort. Part of this issue arises from the
“wobble” of the blade on the Buehler Low Speed Saw (Isomet Low Speed Saw, Buehler,
Evanston, IL). There is a slight “wobble” or bend in the blade so when the specimen was
aligned by eye, the blade may have been aligned to part of the blade that was not aligned with
the rest of the blade. It is visible but not a large amount, but with a 5 mm thick specimen, a
slight misalignment can be problematic.
User error cannot be dismissed either. If the glass slide slipped while it was being
secured to the cutting arm, the specimen may become misaligned with the cutting blade. The
wobble in the blade also makes the meticulous measurements for each cut less precise.
Furthermore the micrometer on the blade arm is ruled in inches, but typically all other
measurements for this study are metric so the conversation introduces opportunity for user
error. The difference in thickness from specimen to specimen was not intentional.
Specimen C had less material removed between “Pre-Sanding” and “Post Sanding”
RUS scans, as shown in Table 3.1.08. This was because the specimen was the thinnest to begin
with and if the same amount was removed as was removed from sample A or B, it may have
become too thin to re-sand and cycle.

72

Table 3.1.08 Thickness and changes in thickness for specimens A, B and C. For sample B, both
sides were from the inside, cut edges. The surfaces of B appeared identical and could not be
determined which side originally corresponded to sample A or sample C.
sand paper A
B
C
grit
mm
mm
mm
1.337
1.495
1.117
Pre-Sanding thickness
removed from outside
240
0.047
N/A
0.004
(original billet surface)
0.038/
removed from inside
240
0.024
0.043
0.006
(cut side)
removed final polishing
600
0.026
0.025
0.010
(both sides total)
0.096
0.105
0.019
total removed
1.241

Post Sanding thickness

1.390

1.097

Sample B had a chip missing and it was also not an ordinary circular cylinder shape, but
an elliptic cylinder (see Figure 3.1.03). The volume was calculated as a regular circular
cylinder so it may have had some error.
3.1.4.2 Discussion of Results
Each specimen increased in porosity after sanding as shown in Table 3.1.08. In Figure
3.1.04 each sample of S-LLZO-92314 has two data points for E and two data points for G and
it can be seen that each pair shows an increase in porosity and a decrease in modulus. This
shows each sample had a decreased in both density and elastic moduli after sanding.

73

Porosity Versus Modulus in Specimen S-LLZO-92314
100

Elastic Modulus (GPa)

E Sample A
E Sample B
E Sample C
G Sample A
G Sample B
G Sample C

80

60

40
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035

Porosity
Figure 3.1.04 Young’s modulus, black symbols, and shear modulus, red symbols, versus porosity
for S-LLZO-92314. Each pair of data points shows a decrease in modulus and an increase in
porosity after sanding.
There was a density gradient in the material, as shown in Table 3.1.07, which was
denser at the side sample C was cut from. However, this is not likely the reason for the
decreasing elastic moduli between sanding stages because both sides of each specimen were
sanded, so the specimen should still have the same average density.
It is possible a reaction was occurring with the air that could be forming a reacted layer
on the surface. If exposure to the laboratory atmosphere caused the decrease in modulus and
increase porosity, the reaction rate would be an important factor. There was some variability to
how long each of the specimens was in open air; therefore the impact of exposure would be a
function of time. Without further analysis it is hard to determine what in particular caused the
74

decreased in modulus and increase in porosity. However, it can clearly be observed that in this
sample, sanding and scanning the specimen using the RUS in open air corresponds with a
decrease in modulus and an increase in porosity.
The most important matter to be discussed is the low modulus. The specimen is quite
dense, which is good, but this specimen had the lowest modulus of any LLZO sample tested so
far (see Table 3.1.01 of all the LLZO specimens tested so far). As to what is causing the low
modulus there are many possible factors. There is some possibility that multiple phases or side
products are forming that could reduce the elastic moduli, such as tetragonal LLZO, LaAlO3 or
Al2O3 [Thompson 2014; Geiger 2011; Rangasamy 2012], but authors often mention great care
is taken to avoid multiple phases and further analysis is typically done to make sure no major
side product have formed [Wolfenstine 2012b; Geiger 2011; Rangasamy 2012; Shimonishi
2011]. If multiple phases are present, and the material is heterogeneous, this would hinder the
RUS scan making it harder to analyze. Solid solution systems can have a significant impact on
elastic moduli properties, see section 3.2.5. These are possible explanations not just for SLLZO-92314 but for the rest of the LLZO specimens that were analyzed in this study.
Therefore more discussion will be devoted to these topics in section 3.2.
3.1.4.3 Discussion of RUS Scans S-LLZO-92314
Figure 3.1.05 shows three RUS scans, (a) the scan of sample S-LLZO-62014-A from
early in the study with high elastic modulus, (b) is the scan of the specimen S-LLZO-92314-A
Post sanding and (c) is a scan of an aluminum disk, as a reference. The frequency on the x-axis
refers to the driving frequency. When the driving frequency is equal to a natural frequency in
the material, it will resonate and cause a rise in amplitude, measured on the y-axis. The red
squares are where the CylModel predicted the resonance frequencies would occur. Comparing
75

(a) to (b) in Figure 3.1.05, it can be observed that the first three major peaks are distinct in both
scans and occur at higher frequencies in (a). Sample S-LLZO-62014-A in figure (a) had a
higher Young’s modulus, 116.5 GPa, and a higher porosity, 9.6%, than Sample S-LLZO92314-A post sanding in figure (b). Increasing porosity is known to cause a decrease in
Young’s modulus as seen in Figure 3.1.06, but even with the higher porosity in S-LLZO62014-A, it was still higher in modulus than S-LLZO-92314-A.
The materials in (a) and (b) of Figure 3.1.05 have similar composition so they should
have similar scans. However, S-LLZO-62014 in (a) of Figure 3.1.05 and S-LLZO-92314 in (b)
of Figure 3.1.05 are composed of the same precursors but underwent different processing
techniques (S-LLZO-62014 used hydrogel and S-LLZO-92314 used solid state synthesis)
therefore it is uncertain if the specimens are exactly the same or if the samples formed different
impurities or different phases as one another. The specimens may be slightly different because
a material with a higher Young’s modulus should have peaks occurring at higher frequencies.
Also, making a specimen thinner, by sanding for example, will make the peaks occur at lower
frequencies without significantly altering the Young’s modulus (as previously discussed,
sanding the S-LLZO-92314 sample increased porosity and decreased Young’s modulus which
can be observed in Figure 3.1.01 and Figure 3.1.02, but there is no indication that sanding in
which the porosity is not changed should decrease the Young’s modulus). S-LLZO-62014-A
was only slightly thinner than S-LLZO-92314-A post sanding, therefore Figure 3.1.05 (a) and
(c) should be similar but they are noticeably different. There is a difference in porosity, but if
that only affects elastic moduli, it should only shift the RUS frequencies of the resonant peaks
scan, not add or remove resonant frequencies, and therefore peaks, altogether.

76

S-LLZO-62014 RUS scan

10

RUS scan
CylModel
Predicted
Peak

Amplitude

8
6
4
2
0
50

100

(a)

150

200

Frequency (kHz)

250

300

S-LLZO-92314 Post Sanding RUS scan
10

RUS scan
CylModel
Predicted
Peak

Amplitude

8
6
4
2
0
50

(b)

100

150

200

250

300

Frequency (kHz)

Figure 3.1.05 RUS scans of (a) sample S-LLZO-62014 from previous work, (b) S-LLZO-92314A Post sanding from this study and (c) aluminum as a reference.

77

Figure 3.1.05 (cont’d)

Aluminum RUS scan
50

RUS scan
CylModel
Predicted
Peak

Amplitude

40
30
20
10
0

(c)

50

100

150

200

250

300

Frequency (kHz)

An important observation of Figure 3.1.05 is the unusual satellite peaks indicated by the
red arrows in Figure 3.1.05 (b). Ideally, a RUS scan has sharp, distinct peaks that occur at the
predicted frequencies; Figure 3.1.05 (c) is a good example of an ideal RUS scan of an
aluminum disk. Several of the peaks in Figure 3.1.05 (b) occur close to where the model
predicts they would, but there are several small, wide peaks that are unaccounted for by the
model. Some of these small peaks occur near larger peaks like the first and third peak in Figure
3.1.05 (b). This may indicate a layer of another phase, possibly a reaction with the open air, but
this is not seen as clearly in samples B and C. The satellite peaks may be an indicator of defects
in the case of cylindrical specimens [ASTM standard E 2001-98 1999], which S-LLZO-92314A was. For axisymmetric specimens, such as a cylindrical disc, some of the vibrational modes
are degenerate, which means that two or more vibrational modes have the same frequency.
However, if there is a defect in a axisymmetric specimen the redundant frequency may begin to
78

split and the RUS scan will show two close peaks where the model will predict one peak (for
two distinct modes) [ASTM standard E 2001-98 1999].
3.1.4.4 Further Processing of S-LLZO-92314-B
Specimen S-LLZO-92314-B was cut and sanded again, this time as a rectangular
parallelpiped, to provide a rough means of verifying the data in Figure 3.1.04 and Table 3.1.06.
Analyzing a different geometry of the same specimen prevents the user from using the same
pattern to match the peaks of the scan to the peaks of the model. Specimen S-LLZO-92314-B
was also scanned using RUS after it was sanded a second time. The results of the RUS analysis
seem to agree with the RUS analysis of the other specimens from the S-LLZO-92314 billet.
The results are shown in Table 3.1.07 (last two items) and Figure 3.1.06 (S-LLZO-92314-B.2 is
represented by X symbols).

79

Porosity Vs Modulus in Specimen S-LLZO-92314
E Sample A
E Sample B
E Sample C
G Sample A
G Sample B
G Sample C
E Sample B.2
G Sample B.2

Elastic Modulus (GPa)

100

80

60

40
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035

Porosity
Figure 3.1.06 Young’s modulus, black symbols, and shear modulus, red symbols, versus porosity
for S-LLZO-92314. Each pair of data points shows a decrease in modulus and an increase in
porosity after sanding.
The results of the RUS analysis on the rectangular parallelepiped (RP) were fairly
consistent with the previous analysis. Table 3.1.09 and Table 3.1.10 show the difference
between sample S-LLZO-92314 as a disk and as a RP, and compares before and after sanding
specimens.

80

Table 3.1.09 Porosity (P) Young’s modulus (E), shear modulus (G), Poisson’s ratio (ν) RMS,
peak count and geometry of sample S-LLZO-92314-B before being sanded as a disk and as a
rectangular parallelepiped (RP)
P
E(GPa) G(GPa)
ν
RMS peaks Geometry
0.007 96.74
35.76
0.353 0.65%
18
Disk
sample B pre sanding
35.88
0.353 0.68%
10
RP
sample B.2 pre sanding 0.013 97.13
Percent difference

0.4%

0.3%

0.0%

Table 3.1.10 Porosity (P) Young’s modulus (E), shear modulus (G), Poisson’s ratio (ν) RMS,
peak count and geometry of S-LLZO-92314-B after being sanded as a disk and as a rectangular
parallelepiped (RP)
P
E(GPa) G(GPa)
ν
RMS peaks Geometry
0.019 93.95
34.91
0.346 0.61%
20
Disk
sample B post sanding
34.94
0.352 0.32%
11
RP
sample B.2 post sanding 0.013 94.49
Percent difference

0.6%

0.1%

1.7%

Table 3.1.09 shows less than 1 percent difference between the E, G and ν for the disk
geometry and RP geometry of S-LLZO-92314-B before it was sanded. Table 3.1.10 shows for
both the disk and the rectangular parallelepiped (RP) geometry, the elastic modulus dropped
from the pre-sanding values in Table 3.1.09 after the specimen was sanded, and almost to the
same values. Table 3.1.10 shows the post sanding E and G values for the RP and disk geometry
were less than 1% different and the ν was only 1.7% different.
After sanding, the elastic moduli dropped and the porosity increased. The porosity of SLLZO-92314-B.2 was constant at 1.3% before and after sanding. This value was between the
before and after results of specimen S-LLZO-92314-B that it was cut from (7% and 1.9%
porosity respectively). This may be due to the fact that the dimensions could be measured more
accurately on a rectangular sample compared to the semi-oval shape of S-LLZO-92314-B (see
Figure 3.1.03).
Higher grit sandpaper, which has a smaller grit size, was used to polish the surface of
sample B.2, 1200 for sample B.2 versus 600 for samples A, B and C. The result of the RUS
81

analysis was not significantly different between sample B.2 and samples A, B and C (see
Figure 3.1.06). Sanding the surface with higher grit (smaller grit size) sandpaper seemed to
make higher frequency peaks identifiable, in the range of 200 to 270 kHz. This allowed the
lowest frequency peak with the highest individual RMS to be removed from the model to
improve the RMS of the overall model. Therefore, the analysis of S-LLZO-92314-B.2 that was
sanded with 1200 grit sand paper, had the lowest RMS of any analysis of an S-LLZO-92314
sample, see Table 3.1.07. Sanding with a higher grit sandpaper to produce more identifiable
peaks, as was done with sample S-LLZO-92314-B.2, could possibly be a means of improving
RUS scans and would be worth exploring.
After the 1200 grit sanding, the 3 peaks with the highest frequency could be identified,
but before sanding, the same peaks could not be identified (see Figure 3.1.07). The specimen
was sanded with 600 and then 1200 grit sandpaper for the post sanding, whereas the pre
sanding specimen was only sanded with 600 grit sandpaper. In Figure 3.1.07, the post sanding
scan of S-LLZO-92314-B.2 has similar peaks as the pre sanding scan, except shifted to the left;
this is because the sanding made the specimen smaller and therefore the resonance peaks would
occur at lower frequencies. It is important to note that after the large peaks around 200 kHz in
both scans of Figure 3.1.07, the amplitude drops. Figure 3.1.07 shows a few peaks over 200
kHz for the post sanding scan. Therefore, using 1200 grit sandpaper, instead of 600 grit
sandpaper, allowed three more peaks to be identified.

82

4

S-LLZO-92314-B.2 RUS
scans before and after sanding
Pre Sanding
Post Sanding

Amplitude

3

2

1

0
150

200

250

Frequency (kHz)
Figure 3.1.07 Sample S-LLZO-92314-B.2 RUS scans before and after it was sanded. The post
sanding had a smoother surface because it was sanded with 1200 grit sandpaper, but because it
was a smaller specimen, comparable peaks between the two scans will appear at lower
frequencies.
Though the results of the RUS analysis of sample S-LLZO-92314-B.2 agree with the
results of the RUS analysis of S-LLZO-92314-A, B and C, They are both lower in elastic
moduli than what is expected for LLZO, namely the results found in Ni et al. [Ni 2012]. Ni
reports E and G of 149.8 GPa and 59.6 GPa, respectively, for a sample with 3% porosity [Ni
2012]. There are many possible explanations as to why the current results are lower than the
results reported in Ni et al. [Ni 2012]. Each will be discussed in detail in section 3.2.

83

3.2 Effects on Young’s modulus by Various Mechanisms
3.2.1 Young’s Modulus as a Function of Porosity
As stated in section 2.3.1, a starting guess for the RUS analysis could be obtained from
Ni et al. [Ni 2012], but a closer estimate could be made with more experimental data. Ni et al.
[Ni 2012] reports Young’s and shear modulus of LLZO specimens at 3% and 6% porosity,
which are listed in Table 2.3.1, but this leaves some uncertainty for samples that have porosity
other than the samples listed in Table 2.3.1. As more results from analyzing specimens were
obtained, a starting guess would be based on Equation 3.1.01 from Barsoum [Schmidt 2014;
Rice 1998; Barsoum 1997c]:
R(P)=R0*exp(-B*P)

3.1.01

where R(P) is porosity dependent elastic modulus property, R0 is an elastic modulus property of
the material at theoretical 100% density, P is the porosity of the specimen and B is a fitting
parameter unique to the material [Schmidt 2014]. Estimates for R0 and B could be found from
the data available in Ni et al. [Ni 2012], and values that were obtained from sample S-LLZO62014.

84

Modulus Versus Porosity for Sample
S-LLZO-62014 and [Ni 2012]

Elastic Modulus (GPa)

200

160

E [Ni 2012]
G [Ni 2012]
E S-LLZO-62014
G S-LLZO-62014
Exponential fit of
Young's modulus
Exponential fit of
shear modulus

120

80

40
0.00

0.02

0.04

0.06

0.08

0.10

Porosity
Figure 3.1.01 Young’s modulus (E) and shear modulus (G) as plotted versus porosity for Ni et al.
[Ni 2012] and S-LLZO-62014. The line is an exponential fit of Equation 3.1.01. The black line
and symbols are Young’s modulus and the red line and symbols are shear modulus
Table 3.2.01 Values of the exponential fit in Figure 3.1.01 according to Equation 3.1.01
A0
B
R2
Young’s
3.816
0.997
modulus 167.16
shear
67.40
4.236
0.998
modulus

The values listed in Table 3.2.01 and can be substituted into Equation 3.1.01 in place of
A0 and B to find initial guesses for the porosity dependent Young’s modulus and shear, which
yield

85

E(P)=167.16*exp(-3.816*P)

3.2.01

G(P)=67.40*exp(-4.236*P)

3.2.02

The porosity was easily measured, so one or both of these Equations were used to
obtain starting values for the RUS analysis. The exponential fit in Figure 3.1.01 and Table
3.1.01, show a strong relationship between samples cut from a billet received on 6-20-14 (and
labeled S-LLZO-62014-A and S-LLZO-62014-C) and the values published in Ni et al. [Ni
2012]. Therefore, Equations 3.2.01 and 3.2.02 were used in subsequent RUS analysis to obtain
starting points. Unfortunately the elastic moduli of specimens S-LLZO-71014, S-LLZO-72914
and S-LLZO-92314 were much lower and did not fit the relationship of Equation 3.2.01 and
Equation 3.2.02, shown in Figure 3.2.01.

86

Elastic Modulus (GPa)

Porosity Vs Modulus Various LLZO Specimens
E Ni 2012
G Ni 2012
G S-LLZO-72914
E S-LLZO-72914
G S-LLZO-71014
E S-LLZO-71014
E S-LLZO-62014
G S-LLZO-62014
E S-LLZO-60314
G S-LLZO-60314
E S-LLZO-92314
G S-LLZO-92314
fit of equation
3.2.01
fit of equation
3.2.02

120

80

40
0.00

0.02

0.04

0.06

0.08

0.10

Porosity
Figure 3.2.01 Young’s modulus and shear modulus for all specimens in the study that could be
analyzed, plotted against porosity. Filled squares represent Ni et al. [Ni 2012] data and opened
symbols were experimentally determined during this study. Red symbols are shear modulus and
black symbols are Young’s modulus. The black and red line represents the least squares fit to
Equations 3.2.01 and 3.2.02 respectively.
As shown in Figure 3.2.01, most of the specimens did not follow the relationships of
Equations 3.2.01 and Equation 3.2.02 shown in figure 3.1.01. This limited the utility of the
Equations for estimating initial guesses for the RUS analysis. Fortunately, when the first three
peaks of the RUS scan were matched to the model the CylModel software recalculated the
predicted peaks, which shifted the model down in frequency so that the model was in line with
the RUS scan (see section 2.3.3 for more details).
Clearly the porosity can have a strong effect on the elastic moduli of the LLZO
specimens.
87

3.2.2 Effects on Young’s Modulus of Fiber Loading in Composite Materials
3.2.2.1 Models for Young’s Modulus of Fiber Containing Composites
Dendrite formation inside a solid electrolyte may be fiber like. There are many models
available to analyze a fiber/matrix composite. Most of the models in the literature were
constructed with the intention of analyzing a matrix with reinforcement fibers that would
enhance the elastic moduli. The scenario of lithium dendrites forming in LLZO is the opposite;
the fibers have lower elastic modulus values and diminish elastic modulus values in the
composite. Several models of elastic modulus for isotropic matrices with fiber inclusions, or a
fiber/matrix composite, will be discussed here.

88

Table 3.2.02 Table of Equations to determine Young’s modulus for a fiber/matrix composite
Name
Equation
Assumptions
Source
Rule of
Assumes fibers are
[Kingery
𝐸𝐶𝑈 = 𝑉𝑚 𝐸𝑚 + 𝑉𝑓 𝐸𝑓
mixtures
parallel to applied stress, 1976a]
This is an upper bound
for an estimate
𝑉𝑓
1
𝑉𝑚
=
+
𝐸𝐶𝐿
𝐸𝑚 𝐸𝑓

Inverse Rule
of mixtures

Christensen

Manera

𝐸𝐶 =

Modified
Halpin-Tsai
with
constant
shape factor

𝑉𝑓
𝐸 + (1 + (1 + 𝜈𝑚 )𝑉𝑓 )𝐸𝑚
6 𝑓

𝐸𝐶 = 𝑉𝑓 (

16
8
𝐸𝑓 + 2𝐸𝑚 ) + 𝐸𝑚
45
9

𝐸𝐶 = 𝑥 ∗ 𝑉𝑓 𝐸𝑓 + 𝑉𝑚 𝐸𝑚

Curtis
Pan

Assumes fibers are
[Kingery
perpendicular to applied 1976a]
stress, This is a lower
bound for an estimate

𝐸𝐶 = 𝐸𝑓

𝑉𝑓
𝑉𝑓
+ 𝐸𝑚 (1 − )
2𝜋
2𝜋

1 + 𝐶𝜂𝑉𝑓
𝐸𝐶 =
𝐸
1 − 𝜂𝑉𝑓 𝑚

𝜂=

𝐸𝑓
𝛼𝐸 −1
𝑚

Vf <0.2
Gf>Gm
Poisson’s ratio must be
known
Fibers are much longer
than they are thick,
2 GPa<Em<4 GPa,
0.1<Vf<0.4
x =1 if the fibers are
perfectly aligned
Assume: all fibers are
same length, uniform
fiber distribution, both
materials behave
elastically, fiber-fiber
interaction is negligible
Assumes random fiber
orientation

[Christensen
1972]

[Manera
1977]

[Curtis
1978]
[Pan 1996]

[Yeh 2006;
Tai 2008]

𝐸𝑓
𝛼𝐸 +𝐶
𝑚

2𝜆
𝑑
Ec= Young’s modulus of the fiber/matrix composite, Em= Young’s modulus of the matrix, Ef=
𝐶=

Young’s modulus of the fiber, Vm= Volume fraction of the matrix, Vf= Volume fraction of the
fiber, x= orientation constant( x =1 when the fibers are parallel to the applied stress), νm=
Poisson’s ratio of the matrix, λ= length of fiber, d=thickness of fiber and α=1/3 if fibers are
longer than the thickness of the test specimen or 1/6 if the fibers are shorter than the thickness
of the test specimen
89

The rule of mixtures (ROM) and the inverse rule of mixtures (IROM) have been
discussed in section 1.6.2. Together they serve as upper and lower bounds, respectively, of the
Young’s modulus of a fiber/matrix composite. Shear modulus can be found using the same
Equations by replacing Ef and Em with Gf and Gm, respectively.
Christensen developed his model to analyze commodity plastics that were enhanced
with low cost fibers [Christensen 1972]. Through a detailed derivation with several
assumptions, the Christensen Equation is derived
𝐸𝐶 =

𝑉𝑓
𝐸 + (1 + (1 + 𝜈𝑚 )𝑉𝑓 )𝐸𝑚
6 𝑓

3.2.03

where Vf is the volume fraction of the fiber, Ef is the Young’s modulus of the fiber, Vm is the
volume fraction of the matrix, Em is the Young’s modulus of the matrix and Ec is the Young’s
modulus of the composite. The Christensen Equation only applies when Vf is less than 0.2 and
if the shear modulus of the fiber is greater than the shear modulus of the matrix [Christensen
1972].
To obtain Equation 3.2.03, Christensen assumes the shear modulus of the fiber is
greater than the shear modulus of the matrix. In the case of the present study, lithium has a
much lower shear modulus than LLZO, so the Christensen model is not applicable to the
system with LLZO as the matrix and Li fibers (dendrites). Furthermore, the model is always
increasing even if the fiber being added has significantly lower elastic modulus values.
The Manera model was developed to be simple but still very accurate [Manera 1977].
16
8
𝐸𝐶 = 𝑉𝑓 ( 𝐸𝑓 + 2𝐸𝑚 ) + 𝐸𝑚
45
9

90

3.2.04

where Vf is the volume fraction of the fiber, Ef is the Young’s modulus of the fiber, Vm is the
volume fraction of the matrix, Em is the Young’s modulus of the matrix and Ec is the Young’s
modulus of the composite.
The Manera model has several limitations as well. First it is assumed that the material in
question has randomly oriented fibers, and that a material with fibers randomly oriented in
three dimensions acts similar to a block made of layers with fibers randomly oriented in twodimensions. Next, the fibers are much longer than they are thick. Lastly, the Young’s modulus
of the matrix is between 2 and 4 GPa and the volume fraction of the fiber must be between 0.1
and 0.4. The assumptions of Young’s modulus of the matrix below 4 GPa also makes this
model unsuitable for the present study.
The Curtis model can be explained as the rule of mixtures with an orientation constant
added to the fiber term, as shown in Equation 3.2.05
𝐸𝐶 = 𝑥 ∗ 𝑉𝑓 𝐸𝑓 + 𝑉𝑚 𝐸𝑚

3.2.05

where Vf is the volume fraction of the fiber, Ef is the Young’s modulus of the fiber, Vm is the
volume fraction of the matrix, Em is the Young’s modulus of the matrix, x is the orientation
constant and Ec is the Young’s modulus of the composite.
In their study, Curtis et al. physically tested carbon fiber in thermoplastic materials with
destructive tests. As the stress is increased, cracks will start to form at the tips of the fibers, first
in the fibers that aligned in the direction of the stress, then to more randomly aligned fibers
[Curtis 1978]. As these cracks form, the fibers contribute less to the load bearing and other
fibers, without cracks, will have to bear more of the applied stress [Curtis 1978]. The cracks
will also grow as stress continues to be applied, but as the cracks progress further away from

91

the tip of the fiber, there will be less force acting to propagate the cracks, and the cracks will
eventually reach a maximum [Curtis 1978].
Curtis et al. found that if the fibers are being added to reinforce the material, longer
fibers will enhance the mechanical properties more than short fibers [Curtis 1978]. When all
the fibers are parallel to the applied stress, x =1, which reduces the Equation to the rule of
mixtures. Thus, the modulus of the composite reaches a maximum when all the fibers are
parallel to the applied force [Curtis 1978]. x can be experimentally determined after results are
obtained, but Curtis does not offer a first principles method for solving for x.
Sandler [Sandler 2002] offers a similar expression to Equation 3.2.05 called the
Krenchel expression
𝐸𝐶 = 𝜂𝑙 𝜂𝑜 ∗ 𝑉𝑓 𝐸𝑓 + 𝑉𝑚 𝐸𝑚

3.2.06

where ηl and ηo are the length correction factor and the orientation efficiency factor, and can be
represented by the Equations [Hull 1981]

3.2.07

2𝐺𝑚
𝜂𝑙 = (
)
R
𝐸𝑓 𝑟 2 ln ( r )

and

𝜂𝑜 = ∑ ∆ 𝑎𝑓 𝑐𝑜𝑠 cos 4 (𝜑) /𝑎𝑓

3.2.08

where Gm is the shear modulus of the matrix, Ef so the Young’s modulus of the fiber, r is the
radius of the fiber, R is half of the inter fiber spacing, φ is the angle between the fiber and the
direction of the applied load, af is the cross sectional area of a fiber and Δaf is the total cross
section area of a group of fibers with the same radius.
The complexity of the Krenchell Equation quickly diminishes the simplicity of the
Curtis model. However, values for ηl and ηo have been determined for certain cases. If it can be

92

assumed that ηl*ηo in Equation 3.2.06 is equal to x in Equation 3.2.05, Table 3.2.03 can be used
[Hull 1981; Sandler 2002]
Table 3.2.03 values of x for the Curtis model based on ηl and ηo from the Krenchell expression
[Hull 1981; Sandler 2002]

ηl

ηo

fiber orientation
aligned
random 2D
random 3D

value of ηo
1
0.375
0.2

aspect ratio
value of ηl
values for x

1000
1

100
0.9

10
0.2

1
0.375
0.2

0.9
0.3375
0.18

0.2
0.075
0.04

Table 3.2.03 shows that x in the Curtis model (Equation 3.2.05) is a function of fiber
orientation and aspect ratio of the fiber.
The Pan model is also similar to the rule of mixtures [Pan 1996].
𝐸𝐶 = 𝐸𝑓

𝑉𝑓
𝑉𝑓
+ 𝐸𝑚 (1 − )
2𝜋
2𝜋

3.2.09

where Vf is the volume fraction of the fiber, Ef is the Young’s modulus of the fiber, Vm is the
volume fraction of the matrix, Em is the Young’s modulus of the matrix and Ec is the Young’s
modulus of the composite.
The assumptions for this form of the Pan model are: all fibers are uniform length; there
is a uniform fiber distribution; fiber to fiber interaction is negligible; and both the fiber and the
matrix materials must behave elastically. This model is applicable for Poisson’s ratio by
replacing Ec with νc, Em with νm and Ef with νf. To find shear modulus of an isotropic material
using the Pan model, Young’s modulus and Poisson’s ratio must first be found, then Equation
3.2.10 can be used

93

𝐺𝐶 =

𝐸𝑐
2(1 + ν𝑐 )

3.2.10

where Gc is the shear modulus of the composite, Ec is the Young’s modulus of the composite and
νc is the Poisson’s ratio of the composite phase.
There are several forms of the Halpin-Tsai Equation that can be used to predict the
elastic moduli, a particularly useful form is the modified Halpin-Tsai model with a constant
shape factor [Yeh 2006; Tai 2008]

𝐸𝐶 =

1 + 𝐶𝜂𝑉𝑓
𝐸
1 − 𝜂𝑉𝑓 𝑚

𝜂=

𝐸𝑓
𝛼𝐸 −1
𝑚

𝐸𝑓
𝛼𝐸 +𝐶
𝑚

𝐶=

2𝜆
𝑑

3.2.11

where EC is the Young’s modulus of the composite, Em is the Young’s modulus of the original
matrix, Ef the is Young’s modulus of the fiber phase, Vf is the volume fraction of the fiber
phase, Vm is the volume fraction of the original matrix phase, λ is the length of the fiber, d is
the thickness or diameter of the fiber.
Despite how the complicated the structure of Equation 3.2.11 is, it is actual a rather
general Equation. As C becomes very large, Equation 3.2.11 reduces to the rule of mixtures
(Equation 1.6.05) and if C approaches zero, Equation 3.2.11 reduces to the inverse rule of
mixtures (Equation 1.6.06) [Islam 2011]. Still, the modified Halpin-Tsai model with a constant
shape factor Equation is useful for an estimate of elastic moduli of a fiber loaded material.
The α term in Equation 3.2.11 will have one of two values: 1/3 if the fibers are longer
than the specimen is thick or 1/6 if the fibers length fits within the thickness of the specimen.
The addition of the α term is one of the features that makes the modified Halpin-Tsai model
different from the Halpin-Tsai model, however the main difference is the shape factor [Tai
2008]. Equation 3.2.11 has a constant shape factor, C. The modified Halpin-Tsai could have an
94

exponential shape factor, ξ, which has fitting parameters based on experimental data. However
the fitting parameters are unique to the set of data being analyzed, rather than being based on
first principles. Therefore ξ for the modified Halpin-Tsai model will not be used here. Instead
the constant shape factor will be used, because it can be physically measured from the length
and width of the fibers. Measuring the length and width of the fibers is not an easy task, but it is
possible. The constant shape factor, C, will cause less and less change to the elastic moduli as it
increases, see Figure 3.2.02.

160

Young's Modulus vs Volume Fraction for Data
From Yeh (Yeh 2006) at Various Values of C

Elastic Modulus (GPa)

140
120
C= 1
C= 5
C= 20
C= 100
C= 500
C= 1000
C= 10000
C= 100000

100
80
60
40
20
0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Volume Fraction

Figure 3.2.02 Experimental data from Yeh [Yeh 2006] plotted using different C values, where C
is twice the ratio of length over width of a fiber.
In Equation 3.2.11 as the ratio of the length to width of a fiber increases, so does the
value of the constant shape factor, C. Figure 3.2.02 plots Young’s modulus versus volume
95

fraction for Yeh [Yeh 2006] using Equation 3.2.11. When all other factors are held constant, if
C increases, the relationship becomes more linear, as shown in Figure 3.2.02. For C greater
than 500, the slope is almost exactly the same [Yeh 2006], especially at volume fractions below
0.10.
Figure 3.2.02 is plotted according to Equation 3.2.11 and using data from Table 3.2.04.
Each plot in Figure 3.2.02 uses the same values except the value of C.
Table 3.2.04 data and assumptions used to plot Figure 3.2.02
Parameter
Value/assumption used
composite description
phenolic epoxy reinforced with multi-walled
carbon nanotubes [Yeh 2006]
Young’s modulus of matrix (Em) (GPa) 5.13 [Yeh 2006]
Young’s modulus of fiber (Ef) (GPa)
953 [Yeh 2006]
Matrix Possion’s ratio (νm)
N/A
Fiber alignment
Random orientation [Yeh 2006]
3.2.2.2 Plots for Young’s Modulus of Fiber Containing Composites
The models in Table 3.2.02 were chosen because they are applicable to the analysis of
fiber/matrix composite systems, and therefore may be applicable to dendrites in LLZO. Some
of the discussed models are displayed in Figure 3.2.03. The data was taken from Yeh [Yeh
2006]. The matrix was a phenolic epoxy with a Young’s modulus of 5.13 GPa, and the fibers
were multi-walled carbon nanotubes with an estimated Young’s modulus of 953 GPa. It was
assumed that the fibers were dispersed and oriented in all directions [Yeh 2006]. The mean
length of the fibers was 17.57 micrometers and the mean diameter was 23.63 nanometers [Yeh
2006], thus a length to width ratio of over 744. A summary of the data used in Figure 3.2.03 is
available in Table 3.2.05.
The values on the x-axis were originally given in weight percent, but could be
converted to volume fraction using the formula provide in Yeh [Yeh 2006]
96

𝑉𝑓 =

𝜌𝑓 𝑊𝑓
𝜌𝑓 − 𝜌𝑓 𝑊𝑓 + 𝜌𝑚 𝑊𝑓

3.2.13

where Wf is the weight fraction of the fiber, Vf is the volume fraction of the fiber and ρf is the
density of the fiber and ρm is the density of the matrix. Values for these variables can be found
in Yeh [Yeh 2006].

Young's Modulus (GPa)

12

Young's Modulus Versus Volume
Fraction for [Yeh 2006] Data

10
ROM
IROM
Curtis
Pan
Halpin-Tsai
Experimental
Data (Yeh 2006)

8

6

0.00

0.01

0.02

0.03

0.04

Volume Fraction
Figure 3.2.03 Young’s modulus versus fiber volume. The experimental data is from [Yeh 2006]
ROM is the rule of mixtures model and IROM is the inverse rule of mixtures model.

97

Table 3.2.05 Data and assumptions used to plot Figure 3.2.03. This table is similar to table
3.2.04, but now considers aspect ratio, and therefore C to be constant.
Parameter
Value/assumption used
composite description
phenolic epoxy reinforced with multiwalled carbon nanotubes [Yeh 2006]
Young’s modulus of matrix (Em) (GPa) 5.13 [Yeh 2006]
Young’s modulus of fiber (Ef) (GPa)
953 [Yeh 2006]
Matrix Possion’s ratio (νm)
N/A
Fiber alignment
Random orientation [Yeh 2006]
Fiber aspect ratio
Average ~744 [Yeh 2006]

The Curtis model requires a value for the orientation constant. Using Equation 3.2.14
𝑥 = (𝐸𝑐 − 𝐸𝑚 𝑉𝑚 )/(𝐸𝑓 𝑉𝑓 )

3.2.14

x was calculated for each data point. When x= 1, all fibers are perfectly aligned in the direction
of the applied stress. The average x for the data points in Figure 3.2.03 was 0.15, meaning there
was little alignment of the fibers. The average value of x was used in the Curtis model.
The modified Halpin-Tsai model in Figure 3.2.03 uses a C value of 1,487. The value of
C is dependent on the ratio of length to width of the fiber, and C values of more than 500 do
very little to alter the value of Ec [Yeh 2006]. α is known to be 1/6 for this data set [Yeh 2006].
Figure 3.2.2 shows a data set from Sandler [Sandler 2002] with the same models to
compare the data.

98

Young's Modulus Versus Volume
Fraction for [Sandler 2002] Data

Young's Modulus (GPa)

8

ROM
IROM
Curtis
Pan
Halpin-Tsai
Experimental
Data [Sandler 2002]

6

4
0.00

0.05

0.10

Volume Fraction
Figure 3.2.04 Young’s modulus versus fiber volume fraction using data from Sandler [Sandler
2002]. ROM is the rule of mixtures model and IROM is the inverse rule of mixtures model.
Table 3.2.06 Data and assumptions used to plot Figure 3.2.04
Parameter
Value/assumption used
Composite description
poly(ether ether ketone) with carbon
naonfiber reinforcement [Sandler
2002]
Young’s modulus of matrix (Em) (GPa) 4.0 [Sandler 2002]
Young’s modulus of fiber (Ef) (GPa)
100 [Sandler 2002]
Matrix Possion’s ratio (νm)
N/A
Fiber alignment
Mostly ordered with some random
orientation [Sandler 2002]
Fiber aspect ratio
500 (not explicitly stated, fiber aspect
ratio started around 103 but may have
been shortened during processing
[Sandler 2002])

99

In the Sandler study, the Young’s modulus of the matrix with no fiber was 4.0 ±0.1
GPa, but the Young’s modulus of the fiber was uncertain but estimated to be in the range 19 to
280 GPa but most likely around 100 GPa [Sandler 2002], so for the models in Figure 3.2.04 the
Young’s modulus was assumed to be 100 GPa. The fibers were mostly aligned in the direction
of the tensile stress. In the injection moulding process most fibers are aligned in the direction of
the extruder except in the center of the specimen where there is some misalignment [Sandler
2002]. However, Sandler did not estimate the degree of alignment of the fibers [Sandler 2002].
The lack of a specified degree of alignment creates some uncertainty for the Curtis model
where an orientation factor is needed. Because no in-depth description of fiber orientation is
offered in Sandler [Sandler 2002], an orientation factor of 0.5 was chosen. This would imply a
moderate to high degree of alignment (see Table 3.2.03).
The aspect ratio of the as received fiber was around 103 [Sandler 2002].The effective
load bearing aspect ratio could be as low 10, [Sandler 2002] but this could be due to low
modulus of the fiber material or misoriented fibers. This leaves a lot of room for interpretation
of what to use for the C value in the modified Halpin-Tsai model. Because carbon nanotubes
have very high elastic modulus and are very small, it was assumed they would experience
minimum degradation during the processing of the composite material. So a high value was
chosen and because values over 500 do little to change the model, 500 was used for the value
of C in the modified Halpin-Tsai model [Yeh 2006].
With the assumptions being stated, a discussion of the models in Figure 3.2.04 can
begin. Again, the ROM and the IROM form upper and lower bounds for the other models in
Figure 3.2.04. The best fit seems to be from the Pan model which doesn’t require any
assumptions or additional measurements beyond modulus and volume fraction of the matrix

100

and fiber material. The modified Halpin-Tsai model is lower than the experimental data. α and
C could be changed so that the model would better fit the experimental data, but increasing α
would imply the fibers are longer than the thickness of the test specimens, which was not the
case [Sandler 2002], and increasing C even a hundred fold will only have negligible changes.
The modified Halpin-Tsai model may still be correct if the Young’s modulus of the fiber is
higher than 100 GPa which was used in the prediction, but Sandler’s estimate was around 100
GPa so that is what was used [Sandler 2002]. Lastly, the Curtis model is much too high. If C is
set to 0.20, the line will be much closer to the experimental data.
A third data set is available from figure 5 in Manera [Manera 1977]. This data set is
plotted in Figure 3.2.05 with several models. Table 3.2.07 list details of the composite and its
components.

101

Young's Modulus, E (GPa)

Young's Modulus Versus Volume Fraction for [Manera 1977] Data
15
ROM
IROM
Curtis
Pan
Halpin-Tsai
Manera
Experimental
Data [Manera
1977]

10

5

0.00

0.20

0.40

Volume Fraction
Figure 3.2.05 Young’s modulus versus fiber volume fraction using data from Manera [Manera
1977]. ROM is the rule of mixtures model and IROM is the inverse rule of mixtures model.
Table 3.2.07 data and assumptions used to plot Figure 3.2.05
Parameter
Value/assumption used
composite description
Polyester resin with fiber glass
reinforcement
Young’s modulus of matrix (Em) (GPa) 2.25 [Manera 1977]
Young’s modulus of fiber (Ef) (GPa)
73 [Manera 1977]
Matrix Possion’s ratio (νm)
0.4 [Manera 1977]
Fiber alignment
Random orientation [Manera 1977]
Fiber aspect ratio
Assumed to be greater than 300, with
lengths ranging from 10 to 50 cm and
widths ranging from 5 to 30 μm
[Manera 1977]

102

The Young’s modulus of the fiber is much higher than the Young’s modulus of the
matrix without any fiber added. The fibers are randomly oriented and assumed to be much
longer than they are wide.
The ROM and IROM again provide upper and lower bounds for the data and the models. The
Curtis model has the best fit, but this is by design. The Curtis model requires a value for the
orientation constant which could, again, be found using Equation 3.2.14
𝑥 = (𝐸𝑐 − 𝐸𝑚 𝑉𝑚 )/(𝐸𝑓 𝑉𝑓 )

3.2.14

Equation 3.2.14 results in an average x of 0.40. This value was used to plot the Curtis
model in Figure 3.2.05 and that is why the Curtis model fits the best.
The Manera model has strict specifications for when the model can be applied. The data
set plotted in Figure 3.2.05 fits these specifications, namely Em must be between 2 and 4 GPa
and the volume fraction of fiber, Vf, must be between 0.1 and 0.4. The Manera model is plotted
in Figure 3.2.05 extends outside of these boundaries and it can be seen that the Manera model
at Vf=0 does not equal the Young’s modulus of the matrix. However, it can be clearly seen in
Figure 3.2.05 the model fits fairly close to the data where the specifications are met.
The modified Halpin-Tsai model is also fairly close to the data set in Figure 3.2.05. One
variable that needed to be assumed was the orientation factor, α.The orientation factor will be
1/3 when the fiber is longer than the thickness of the specimen and 1/6 when the fiber is shorter
than the thickness of the specimen [Yeh 2006]. The fiber length is stated as 1 to 5 cm [Manera
1977]. The dimensions of the specimen for torsion testing were 3 cm diameter with a 3 mm
thickness [Manera 1977]. If the specimens that were used to test elastic moduli were close to
the 3 mm thickness, then α would be 1/3. Using α=1/3 makes the modified Halpin-Tsai model
closer than if α=1/6 is used. It is ultimately an assumption that α=1/3 rather than 1/6, but it

103

seems to be a reasonable assumption given the fibers are greater than 1 cm in length and the
specimen is 3 mm thick [Manera 1977].

Young's Modulus Versus Volume
Fraction for [Christensen 1972] Data

Young's Modulus, E (GPa)

15
ROM
IROM
Curtis
Pan
Halpin-Tsai
Christensen
Experimental
Data
[Christensen
1972]

10

5

0.00

0.20

Volume Fraction
Figure 3.2.06 Young’s modulus versus fiber volume fraction using data from Lee [Lee 1968].
ROM is the rule of mixtures model and IROM is the inverse rule of mixtures model. The
Christensen formula stops where Vf =0.20 because the model is no longer valid above that
Table 3.2.08 data and assumptions used to plot Figure 3.2.06
Parameter
Value/assumption used
composite description
polystyrene with fiber glass addition
Young’s modulus of matrix (Em) (GPa) 3.2 [Lee 1968]
Young’s modulus of fiber (Ef) (GPa)
72.4 [Lee 1968]
Matrix Poisson’s ratio (νm)
0.32 [Lee 1968]
Fiber alignment
Random orientation [Lee 1968]
Fiber aspect ratio
675, Length of 1/4 inches and diameter
of 0.00037 inches

104

Figure 3.2.06 shows a data set from Christensen [Christensen 1972], that was
experimentally determined by Lee [Lee 1968]. The study done by Lee [Lee 1968] involved
reinforcing polymer materials with fiber glass and the composite was tested for tensile and
flexural strength. The composite’s fiber would have a random orientation, which wouldn’t be
ideal for strength properties, but the material could be mass produced cheaper and faster than
materials that had fibers all aligned in one direction [Lee 1968].
As in the previous figures, the ROM and IROM form the upper and lower bounds,
respectively. The Curtis model’s orientation factor, x, was chosen to be 0.36 because the
average x calculated from the 4 non-zero data points in Figure 3.2.06 was 0.36. The
Christensen model is included in Figure 3.2.06. The Christensen model has requirements that
have made it unsuitable for the data sets in Figure 3.2.03 through Figure 3.2.05; namely the
Poisson’s ratio of the unaltered matrix must be known and the Vf must be less than 0.20.
Because the volume fraction must be less than 0.20, the line plotting the Christensen model in
Figure 3.2.06 ends at 0.20. For the modified Halpin-Tsai model, the aspect ratio was
approximately 675 [Lee 1968]. The samples that were tested in the study were 3.12 mm thick,
and because the fibers were twice this length, the orientation factor, α, for the modified HalpinTsai model was 1/3. The Pan model does not seem to fit well to this data. The Christensen,
modified Halpin-Tsai and the Curtis model all fit rather well to the data set.
Two data sets were available from Kriz et al.[Kriz 1979]. The data was experimentally
determined by Dean and Turner using ultrasonic transmission techniques [Dean 1973].

105

Young's Modulus (GPa)

Young's Modulus Versus Volume Fraction
for Transverse Fibers from [Kriz 1979] Data
50
40
30

ROM
IROM
Curtis
Pan
Halpin-Tsai
Experimental
Data [Kriz 1979]

20
10
0
0.0

0.2

0.4

0.6

0.8

1.0

Volume Fraction
Figure 3.2.07 Young’s modulus versus fiber volume fraction using data in Kriz [Kriz 1979] and
Dean and Turner [Dean 1973]. ROM is the rule of mixtures model and IROM is the inverse rule
of mixtures model.
The materials that were tested were a composite of epoxy with carbon fiber aligned
uniformly in one direction of the material ranging from Vf equal to 0.3 to 0.7 Vf that were cut
into specimens with dimensions of 10 mm by 40 mm by 2.5 mm [Dean 1973]. The first data set
is plotted in Figure 3.2.07 with some details of the materials listed in Table 3.2.09.

106

Table 3.2.09 data and assumptions used to plot Figure 3.2.07 data originally from Dean and
Turner [Dean 1973]
Parameter
Value/assumption used
composite description
Isotropic epoxy resin with graphite fibers (Ciba
LY558 epoxy resin with modmor carbon
fiber)[Dean 1973]
Young’s modulus of matrix (Em) (GPa)
5.35 [Kriz 1979]
Young’s modulus of fiber (Ef) (GPa)
233 [Kriz 1979]
Matrix Possion’s ratio (νm)
0.354 [Kriz 1979]
Fiber alignment
Measured transverse to fiber orientation [Kriz
1979]
Fiber aspect ratio
[Kriz 1979] not specified, assumed to be much
longer than the diameter

The second data set from Kriz is plotted in Figure 3.2.08 with further details of the
materials listed in Table 3.2.10. The main difference of the two data sets in the orientation of
the fibers; Figure 3.2.07 is measured in the longitudinal direction of the fiber and Figure 3.2.08
is measured in the transverse direction of the fiber.
Figure 3.2.07 has fibers oriented in the transverse direction [Kriz 1979]. The Young’s
modulus of the composite is noticeably lower in the transverse direction (Figure 3.2.07) than in
the longitudinal direction (Figure 3.2.08). In Figure 3.2.07 and Figure 3.2.08, the ROM and
IROM form upper and lower bounds for the models, except the Curtis model in Figure 3.2.07,
but not bounds for the data. The ROM, IROM, Pan and modified Halpin-Tsai models are the
same for Figure 3.2.07 and Figure 3.2.08. The aspect ratio of the fiber was not given in Dean
and Turner[Dean 1973] or Kriz [Kriz 1979], but because the aspect ratio of fibers discussed so
far have been large, it was assumed the aspect ratio of these fibers was also very large. The
orientation factor,α, for the modified Halpin-Tsai model was assumed to be 1/6 implying the
fibers were shorter than the thickness of the specimen[Yeh 2006] which was given as 2.5 mm
[Dean 1973]. The only model that can explicitly include more than one fiber orientation is the
Curtis model. In Figure 3.2.07 the Curtis model is given the minimum value of x it can have
107

without having a negative slope for Ec versus Vf. This minimum value of x is Em/Ef, which
reduces the Equation to Ec= Em. In Figure 3.2.08 the Curtis model, is given a value of 0.95,
because x is equal to one when the fibers are all aligned in one direction like in this data set, but
then it would simply be equal to the ROM, so it was set to be slightly less so both models
would be visible in the figure.

Young's Modulus Versus Volume Fraction
for Longitudinal Fiber from [Kriz 1979] Data

Young's Modulus (GPa)

200

150

ROM
IROM
Curtis
Pan
Halpin-Tsai
Experimental
Data [Kriz 1979]

100

50

0
0.0

0.2

0.4

0.6

0.8

1.0

Volume Fraction
Figure 3.2.08 Young’s modulus versus fiber volume fraction using data in Kriz[Kriz 1979] and
Dean and Turner [Dean 1973]. ROM is the rule of mixtures model and IROM is the inverse rule
of mixtures model.

108

Table 3.2.10 data and assumptions used to plot Figure 3.2.08 data originally from Dean and
Turner [Dean 1973]
Parameter
Value/assumption used
composite description
Isotropic epoxy resin with graphite fibers
(Ciba LY558 epoxy resin with modmor
carbon fiber)[Dean 1973]
Young’s modulus of matrix (Em) (GPa)
5.35 [Kriz 1979]
Young’s modulus of fiber (Ef) (GPa)
233 [Kriz 1979]
Matrix Possion’s ratio (νm)
0.354 [Kriz 1979]
Fiber alignment
measured along length of fiber [Kriz 1979]
Fiber aspect ratio
[Kriz 1979] not specified, assumed to be
much longer than the diameter

As seen in Figure 3.2.08 the ROM can be rather accurate for a composite material with
fiber orientation in the longitudinal direction, but as seen in Figure 3.2.07 the IROM is not as
close of an approximation for fiber composite materials in the transverse direction. The closest
model for the data set in Figure 3.2.08 is the ROM model because the actual experiment fits
very closely to the assumption that the fibers are aligned in the direction of the test. The closest
model for Figure 3.2.07 is more difficult. The IROM model has an exponential curve. The data
set does not seem to follow the exponential curve and looks more linear. According to Equation
1.6.06
1
𝑉𝑚
𝑉𝑟
=
+
𝐸𝐶𝐿
𝐸𝑚 𝐸𝑟

1.6.06

In Equation 1.6.06, if the Vf is equal to 1, the material should be completely made out
of the fiber material and no matrix material. Therefore Ec at Vf equal to 1 should be equal to
Em. The data set in Figure 3.2.08 does not seem to follow Equation 1.6.06 and even appears to
have a linear trend.
The data sets in Figure 3.2.07 and Figure 3.2.08 are the extremes of fiber orientation
and show the limitations of the models. The exact aspect ratio and orientation factor for the
modified Halpin-Tsai Equation are not critical to the discussion because the model is intended
109

for random oriented fibers [Yeh 2006], making it is unsuitable for data sets and experiments
involving uniformly transverse or longitudinal alignment. So even if α and the aspect ratio were
changed the modified Halpin-Tsai model would still be inaccurate in Figure 3.2.07 and Figure
3.2.08. The Pan model is also meant for composites with random fiber orientation so the model
has similar results as the modified Halpin-Tsai and is also unsuitable for uniformly aligned
fiber composites [Pan 1996].
The various data sets in Figure 3.2.03 through Figure 3.2.08 are quite different from one
another, but it is worth comparing them in order to compare the effect fiber orientation has on
the Young’s modulus of a composite. Each data set for Figure 3.2.03 through Figure 3.2.08 has
been roughly normalized using Equation 3.2.15 and plotted versus the volume fraction, Vf, in
Figure 3.2.09.
𝐸𝑁 =

(𝐸𝐶 − 𝐸𝑚 )
(𝐸𝑓 − 𝐸𝑚 )

3.2.15

where EN is the normalized Young’s modulus of the composite, which is unitless.
To obtain this normalization, first consider the rule of mixtures,
𝐸𝐶 = 𝑉𝑚 𝐸𝑚 + 𝑉𝑓 𝐸𝑓 (Rule of mixtures)

1.6.05

Which can written as
𝐸𝐶 = (1 − 𝑉𝑓 )𝐸𝑚 + 𝑉𝑓 𝐸𝑓 (Rule of mixtures)

1.6.05 a

The slope of the changes in Young’s modulus of the composite versus volume fraction
of fibers will be the partial derivative of Equation 1.6.05 a, with respect to volume fraction, Vf,
𝜕𝐸𝐶
𝜕
𝜕
𝜕
(𝐸𝑚 ) −
=
(𝑉𝑓 ∗ 𝐸𝑚 ) +
(𝑉 ∗ 𝐸 ) = −𝐸𝑚 + 𝐸𝑓
∂𝑉𝑓
∂𝑉𝑓
∂𝑉𝑓
∂𝑉𝑓 𝑓 𝑓

3.2.16

Derivatives of the various models are listed in Table 3.2.13. A linear Equation has the
form y=mx+b where m is the slope and b is the y-intercept. As seen in Figure 3.2.03 through
110

Figure 3.2.08,each model besides the IROM shows a linear relationship between the volume
fraction and Young’s modulus of the composite. In the case of the rule of mixtures model, the
linear relationship can be expressed as
𝐸𝐶 = (𝐸𝑓 − 𝐸𝑚 )𝑉𝑓 + 𝐸𝑚
when Vf=0, Ec equals Em, therefore Em is the y-intercept. The Equation can then be rearanged
by subtracting the intercept from both sides, which is the translation of the coordinate axis
𝐸𝐶 − 𝐸𝑚 = (𝐸𝑓 − 𝐸𝑚 )𝑉𝑓
Then divide both sides by the slope (Ef-Em)
𝐸𝐶 − 𝐸𝑚
= 𝑉𝑓
𝐸𝑓 − 𝐸𝑚
Which is equal to the normalization factor
𝐸𝐶 − 𝐸𝑚
= 𝐸𝑁 = 𝑉𝑓
𝐸𝑓 − 𝐸𝑚

3.2.18

These results seem to suggest that the normalization facor will be equal to the volume
fraction for all specimens. However, Figure 3.2.09, which shows three distinct slopes for each
of the three orientations, random, transverse and longitudinal. Therefore, Vf in Equation 3.2.18
would be multiplied by an orientation dependent factor, M, and Equation 3.2.18 can be more
accurately expressed as
𝐸𝐶 − 𝐸𝑚
= 𝐸𝑁 = 𝑉𝑓 ∗ 𝑀
𝐸𝑓 − 𝐸𝑚

3.2.19

the orientation dependent factor, M, would be equal to the slope of each line Figure 3.2.09, see
Table 3.2.11.

111

Normalized Young's
Modulus of Composite

Normalized Young's Modulus Versus Volume Fraction

0.6

0.4

0.2

0.0
0.0

0.2

0.4

Volume Fraction
(a)

0.6

[Yeh 2006]
[Sandler 2002]
[Manera 1977]
[Christensen
1972]
[Kriz 1979]
Transverse
[Kriz 1979]
Linear fit of
randomly
oriented fiber
Linear fit of
transversely
oriented fiber
Linear fit of
longitudinal
oriented fiber

Normalized Young's Modulus Versus Volume Fraction

Normalized Young's
Modulus of Composite

0.15

0.10

0.05

0.00
0.0

0.1

0.2

Volume Fraction
(b)

0.3

Figure 3.2.09 Normalized data sets from Figures 3.2.03 through 3.2.08. (a) Longitudinally
oriented fibers data and their linear fit line are green, the transversely oriented data and linear fit
line are red and the randomly oriented fiber data and the corresponding linear fit line are blue. (b)
an expanded view of the plot in (a), the same legend applies to both

112

Table 3.2.11 Slope, intercept and coefficient of determination, R2, of linear fit lines in Figure
3.2.09 (a) and (b)
Intercept
Fiber Alignment
Slope
R2
-0.006
Random Orientation
0.388
0.980
-0.002
Transverse
0.040
0.985
0.192
Longitudinal
0.683
0.884

The roughly normalized data sets are plotted against their volume fraction of fiber in
Figure 3.2.09. The data sets from Yeh, Sandler, Manera and Christensen [Yeh 2006; Sandler
2002; Manera 1977; Christensen 1972] all contain randomly oriented fiber and had Vf values
below 0.40. A linear fit of the randomly oriented fiber data sets was plotted in blue and most
points fall fairly close to this line, R2=0.98. A linear fit was also plotted for the longitudinal and
transverse data sets as well, which are green and red respectively. The linear fit for the
longitudinally aligned fibers has a slope of 0.683. The slope for transverse oriented fibers is
nearly flat, and the slope of randomly oriented fiber is between the two. This shows that the
maximum Young’s modulus enhancement will come from fibers oriented in the longitudinal
direction.
The y-intercept, as shown in Table 3.2.11, is not equal to zero. For the randomly
oriented fiber and transverly oriented fiber, it is quite close. For longitudinally oriented fiber, it
is is noticably high. This may just be attributed to the fact that there are few data points
available for the data set. If the lowest volume fraction data point in the Kriz longitudinal data
set [Kriz 1979] was removed, the plot would have a slope closer to one and an intercept closer
to zero, which may indicate that the data point is an outlier, see Figure 3.2.10.

113

Normalized Young's
Modulus of Composite

Normalized Young's Modulus Versus Volume Fraction

0.6
[Kriz 1979]
longitudinal
Linear fit of
longitudinal
oriented fiber

0.4

0.2

0.0
0.0

0.2

0.4

0.6

(a)
Volume Fraction
Figure 3.2.10 Elastic modulus versus volume fraction for Kriz [Kriz 1979] according to Equation
3.2.19 data set when the red data point is not included in the linear fit (green line).
Table 3.2.12 slope, intercept and linear coefficient of determination for Figure 3.2.10
Intercept
Fiber Alignment
Slope
R2
0.048
Longitudinal
0.909
0.945

However, a data point cannot be ignored without good cause, so the linear fit shown in
Figure 3.2.09 and Table 3.2.11 should be considered a more valid analysis of the data set in
Kriz [Kriz 1979] than Figure 3.2.10 and Table 3.2.12. If there was much more data, a more
exact slope and intercept could be determined for the longitudinally aligned fiber in Figure
3.2.09. However, it can still be observed that the longitudinally oriented fiber will have a
greater slope than randomly oriented or transversely oriented fibers.

114

The normalization factor, Equation 3.2.15, used for Figure 3.2.09 and Table 3.2.11 is
based on the rule of mixtures, but similar results could be obtained with the other models. The
derivative, and therefore the slope, of each model is displayed in Table 3.2.13.
Table 3.2.13 the Equation for each model in Table 3.2.02 as well as the partial derivative with
respect to volume fraction (Vf)
Name
Equation
Derivative
𝜕𝐸𝐶𝑈
Rule of
𝐸𝐶𝑈 = 𝑉𝑚 𝐸𝑚 + 𝑉𝑓 𝐸𝑓
= 𝐸𝑓 − 𝐸𝑚
mixtures
𝜕𝑉𝑓
𝑉𝑓
1
𝑉𝑚
=
+
𝐸𝐶𝐿
𝐸𝑚 𝐸𝑓

Inverse
Rule of
mixtures
Christensen

𝐸𝐶 =

Manera

Curtis

Pan

𝐸𝐶 =

𝑓 𝑚

𝑉𝑓
𝐸 + (1 + (1 + 𝜈𝑚 )𝑉𝑓 )𝐸𝑚
6 𝑓

𝑚 𝑓

𝜕𝐸𝐶 𝐸𝑓
= + 𝐸𝑚 (1 + 𝜈𝑚 )
𝜕𝑉𝑓
6

16
8
𝐸𝐶 = 𝑉𝑓 ( 𝐸𝑓 + 2𝐸𝑚 ) + 𝐸𝑚
45
9

𝜕𝐸𝐶 16
=
𝐸 + 2𝐸𝑚
𝜕𝑉𝑓 45 𝑓

𝐸𝐶 = 𝑥 ∗ 𝑉𝑓 𝐸𝑓 + 𝑉𝑚 𝐸𝑚

𝜕𝐸𝐶
= 𝑥𝐸𝑓 − 𝐸𝑚
𝜕𝑉𝑓

𝐸𝐶 = 𝐸𝑓

Halpin-Tsai
with
constant
shape factor

𝜕(𝐸𝐶𝐿 ) 𝐸𝑓 ∗ 𝐸𝑚 (𝐸𝑓 − 𝐸𝑚 )
=
2
𝜕𝑉𝑓
(𝐸 𝑉 − 𝐸 𝑉 )

𝑉𝑓
𝑉𝑓
+ 𝐸𝑚 (1 − )
2𝜋
2𝜋

𝜕𝐸𝐶
1
=
(𝐸 − 𝐸𝑚 )
𝜕𝑉𝑓 2𝜋 𝑓

𝐸𝑓
𝛼𝐸 −1

𝜕𝐸𝐶
𝜂(1 + 𝐶)
= 𝐸𝑚
2
𝜕𝑉𝑓
(1 − 𝜂𝑉 )

1 + 𝐶𝜂𝑉𝑓
𝐸
1 − 𝜂𝑉𝑓 𝑚

𝜂=

𝑚

𝐸𝑓
𝛼𝐸 +𝐶
𝑚

𝐶=

2𝜆
𝑑

𝑓

In a linear Equation using the Curtis or Pan model they would take a similar form as
Equation 3.2.17, except with some constants attached to Em and Ef, as shown in Equation 3.2.20
𝐸𝐶 = (𝐴 ∗ 𝐸𝑓 − 𝐵 ∗ 𝐸𝑚 )𝑉𝑓 + 𝐸𝑚

3.2.20

where A and B are constants dependent on the coefficients of Em and Ef from the derivative in
Table 3.2.13.
115

For each of the two models, A and B will have different values causing M, the
orientation dependent factor, to be slightly different, but they will still be similar. If Ef is much
larger than Em, the slope will be positive, that is to say, an increasing volume fraction will
create a higher Young’s modulus composite. If Em is much larger than Ef, the slope will be
negative or the composite will decrease in elastic modulus as volume fraction increase. The
slope of the various models when either Em or Ef is much larger than the other is displayed in
Table 3.2.14.

116

Table 3.2.14 approximate slope of models from Table 3.2.02, found by taking the derivative of
each Equation
Name
The slope of Ec with increasing Vf if The slope of Ec with increasing Vf if
Ef>> Em
Em>> Ef
𝜕𝐸𝐶𝑈
𝜕𝐸𝐶𝑈
Rule of
≈ 𝐸𝑓
≈ −𝐸𝑚
Mixtures
𝜕𝑉𝑓
𝜕𝑉𝑓
Inverse Rule
of Mixtures

𝜕(𝐸𝐶𝐿 ) 𝐸𝑚
≈ 2
𝜕𝑉𝑓
𝑉𝑓

𝜕(𝐸𝐶𝐿 ) −𝐸𝑓
≈ 2
𝜕𝑉𝑓
𝑉𝑓

Christensen

𝜕𝐸𝐶 𝐸𝑓
≈
𝜕𝑉𝑓
6

𝜕𝐸𝐶
≈ 𝐸𝑚 (1 + 𝜈𝑚 )
𝜕𝑉𝑓

𝜕𝐸𝐶 16
≈
𝐸
𝜕𝑉𝑓 45 𝑓

𝜕𝐸𝐶
≈ 2𝐸𝑚
𝜕𝑉𝑓

Curtis

𝜕𝐸𝐶
≈ 𝑥𝐸𝑓
𝜕𝑉𝑓

𝜕𝐸𝐶
≈ −𝐸𝑚
𝜕𝑉𝑓

Pan

𝜕𝐸𝐶 𝐸𝑓
≈
𝜕𝑉𝑓 2𝜋

𝜕𝐸𝐶
𝐸𝑚
≈−
𝜕𝑉𝑓
2𝜋

Manera

Halpin-Tsai
with constant
shape factor

𝜂 = positive
∂EC
the Equation for
does not
∂Vf
significantly reduce, but it will be
positive

𝜂 = 𝑛egative, becomes very
small as 𝐶 increases
𝜕𝐸𝐶
≈ −𝐸𝑚
𝜕𝑉𝑓

The Manera and Christensen models do not work very well when Em>> Ef. In the
derivatives of the Manera and Christensen models (Table 3.2.13) Em and Ef of are summed
together, meaning regardless of which is bigger, the slope will be positive and therefore
Young’s modulus will continue to increase as volume fraction increases. But the Young’s
modulus should decrease with increasing volume fraction if the fiber has a smaller Young’s
modulus than the matrix. Both of these models have restrictions for what volume fraction they

117

can apply to, and they seemingly have the restriction of only applying to systems that increase
with fiber addition.
This leaves the inverse rule of mixtures (IROM) and the modified Halpin-Tsai model.
Both of these Equations have Vf 2 in the denominator of their derivative (Table 3.2.13), making
them the only two Equations that are not linear. In the IROM model, Em and Ef are both in the
denominator and the difference between the two is squared, so the greater the difference, the
greater the bowing. The numerator of the IROM has Ef –Em, so like the ROM, Curtis and Pan
models, if Ef is much larger than Em, the slope will be positive, and if Em is much larger than Ef,
the slope will be negative.
As for the modified Halpin-Tsai model, the complexity helps give it robustness. The
derivative, according to Table 3.2.13 is positive unless one of its terms is negative. The only
term that could be negative is η. If Em is significantly larger than Ef, the denominator of η
becomes negative, and therefore the slope is negative. The slope will be zero when Ef /Em= 1/α.
If Ef is greater than Em/α, η will be positive; and therefore the slope of Ec versus Vf will also be
positive.
The reason for examining these models is to better understand the elastic moduli of
LLZO and lithium at varying composition. Unlike the data sets previously discussed, if lithium
dendrites act as fibers in LLZO, the matrix has a higher Young’s modulus than the fiber being
added. In fact, the Em is higher than Ef by an order of magnitude [Ni 2012], see Figure 3.2.16.
Table 3.2.15 shows what should happen when Em is significantly higher in a fiber composite.
The rule of mixture, the Curtis model and the modified Halpin-Tsai model all show the same
relationship: the slope of the Young’s modulus of the composite versus the volume fraction will
be equal to -Em. This is an interesting observation because the rule of mixtures is meant for the

118

case of longitudinally oriented fibers and the Curtis and modified Halpin-Tsai models are for
randomly oriented fiber, but they yield the same slope. Therefore, in a composite where Em>>
Ef, as long as the fibers are either randomly oriented or oriented in the longitudinal direction,
they will have a slope of -Em. The pan model gives a slope of -Em/2π, which is about 1/6 of the
estimate made by the ROM, Curtis model and modified Halpin-Tsai model. The inverse rule of
mixtures(IROM) predicts the slope will be much smaller but still negative. However, it is
reasonable for the IROM to be much smaller because IROM is for the case of transversely
aligned fibers, and is therefore used as a lower bounds for estimates of a composite material
[Kingery 1976a]. Lastly, the Christensen and Manera models both predict a positive slope for
Ec versus Vf even in the case of Em>> Ef, see Table 3.2.14. The Christensen models are probably
not applicable for the LLZO dendrite composite and this will be discussed more in section
3.2.2.3
3.2.2.3 Plot for Young’s Modulus of LLZO Containing Dendrites
Figure 3.2.11 shows the discussed models applied to LLZO with lithium dendrites. The
fiber/matrix composite is based on a sample from Ni et al. [Ni 2012] that had 3% porosity and
Young’s modulus of 149.8 GPa. The Young’s modulus for lithium dendrites was assumed to be
10.35 GPa. This is the average of the Hashin and Shtrikman bounds at 25˚ C, which are 12.2
and 8.5 GPa, respectively [Trivisonno 1961]. The Poisson’s ratio could also be found from the
average of the Hashin and Shtrikman bounds found in Trivisonno and Smith [Trivisonno 1961].

119

Young's Modulus of Composite LLZO
with Lithium Dendrites at 3% Porosity
Young's Modulus (GPa)

180

ROM
IROM
Curtis
Pan
Halpin-Tsai
Christensen

170
160
150
140
130
120
110
0.00

0.05

0.10

0.15

0.20

Volume Fraction

Figure 3.2.11 Young’s modulus versus volume fraction in LLZO at 3% porosity. ROM is the
rule of mixtures model and IROM is the inverse rule of mixtures model.
Table 3.2.15 data and assumptions used to plot Figure 3.2.11
Parameter
Value/assumption used
composite description
LLZO at 3% porosity with dendrites
Young’s modulus of matrix (Em) (GPa) 149.8 [Ni 2012]
Young’s modulus of fiber (Ef) (GPa)
10.35 [Trivisonno 1961]
Matrix Possion’s ratio (νm)
0.257 [Ni 2012]
Fiber alignment
Dendrites would align mostly in the
direction between the electrodes, not
exclusively [Rosso 2006; Dollé 2002]
Fiber aspect ratio
Assumed to be 500

Figure 3.2.11 shows the previously discussed models plotted for Lithium Lanthanum
Zirconium Oxide (LLZO). For the terms in the models, the values could be found in the
literature. At 25˚ C, LLZO has a Young’s modulus of 10.35 GPa [Trivisonno 1961], this will
120

decrease as the temperature rises, and it is likely that the inside of a battery will increase in
heat, but it serves as a start point. LLZO has been measured at various porosities [Ni 2012]. A
fully dense LLZO specimen is desirable, but it is easier to make LLZO with some porosity so
3% porosity LLZO was chosen which has been measured to have a Young’s modulus of 149.8
GPa and a Poisson’s ratio of 0.257 [Ni 2012].
Some assumptions had to be made for the models in Figure 3.2.11. For the Modified
Halpin-Tsai Equation, α was assumed to be 1/6 because it is likely the dendrites are shorter
than the thickness of the electrolyte [Yeh 2006]. If the dendrites were longer than the thickness
of the electrolyte, they would bridge the electrodes which would lead to catastrophic failure.
The aspect ratio was assumed to be very large so the C term in the modified Halpin-Tsai
Equation was assumed to be 500. This is a reasonable assumption because the dendrites are
likely to be thin and grow in length faster than they grow in thickness[Rosso 2006; Dollé
2002], however the dendrites will start short so the aspect ratio will be somewhat dependent on
how long the dendrites has been growing. The Curtis model presented some uncertainties. The
orientation parameter x is equal to 1 when all the fibers are aligned in the direction parallel to
the stress. For the case of LLZO, x is equal to 1 when all dendrites extend directly along the
path the ions travel from the anode to the cathode. This is the ideal scenario, but in reality,
lithium ions are more likely to travel a nonlinear path. Brissot et al. [Brissot 1999] observed
dendrites growing through a polymer solid electrolyte by in-situ microscopy, and the images
reveal a “zig-zag” path or a mostly linear path through the electrolyte. Therefore x was assumed
to be 0.5. If x =1 it will be the same value as the ROM.

The validity of the models in Figure 3.2.11 is yet to be proven, but should still be
discussed. It is evident that the ROM and IROM models do not provide the bounds in Figure
121

3.2.11. The ROM model assumes all fibers are oriented longitudinally to the applied stress, the
IROM model assumes the fibers are oriented transversely to the applied stress. The IROM
model is the lower bound and experiences exponential decay which is hard to observe in Figure
3.2.11 because it drops so quickly. The Pan model is noticeably higher than the ROM model.
This warrants a closer look at the Equation
𝐸𝐶 = 𝐸𝑓

𝑉𝑓
𝑉𝑓
+ 𝐸𝑚 (1 − )
2𝜋
2𝜋

When Vf =0, Ec = Em and this boundary condition holds true. However, when Vf = 1, the
modulus of the matrix should no longer contribute to the modulus of the composite, but in the
Pan model it does. In fact, more than 80% of the matrix modulus value is still added to the fiber
modulus at Vf = 1. In light of this, the Pan model seems better suited for a material where the
added fiber has a higher modulus than the original matrix. Therefore, the ROM and IROM are
likely to be the bounds for the data, or at least similar to Figure 3.2.07 and Figure 3.2.08, the
ROM and IROM would be near the edge of the true bounds for the data.
The Christensen model makes an even more unreasonable model for the LLZO/dendrite
system plotted in Figure 3.2.11. The Christensen model anticipates the composite Young’s
modulus, Ec, to increase as fiber is added. Increasing Young’s modulus was the anticipated
results in the data sets previously discussed, and indeed it was the reason the fiber was added in
the first place [Yeh 2006; Sandler 2002; Manera 1977; Christensen 1972; Kriz 1979].
However, dendrites are not intentionally added, and the new inclusion has a lower Young’s
modulus than the existing matrix (see Table 3.2.15). This identifies the unintended assumption
of the Christensen model: increasing Vf increases Ec.
𝐸𝐶 =

𝑉𝑓
𝐸 + (1 + (1 + 𝜈𝑚 )𝑉𝑓 )𝐸𝑚
6 𝑓

122

3.2.03

What further limits the Christensen model, Equation 3.2.03, is that it is only valid for Vf
less than 0.20 [Christensen 1972]. This is a reasonable limitation, considering when Vf is equal
to 1, Ec will include twice the value of Em. Therefore, the Christensen model is unsuited for
modeling the LLZO/dendrite composite.
In conclusion of these models, Figure 3.2.07 and Figure 3.2.08 show that the ROM and
IROM provide good bounds for fiber containing composites, though in the most extreme
situations the data points maybe outside of these bounds, they should still be fairly close. As
seen in Figure 3.2.05, the Manera model is good when the required conditions are met, but this
is limiting because the Em and the Vf must both be within a certain range. The Christensen
model is also useful when its conditions are met, which are less restrictive, but the model is
also only suitable for composites whose fibers have a higher Young’s modulus than the original
matrix, which is not the case for LLZO. In the plots in section 3.2.2.2, the Pan model seemed to
be a fairly useful to model randomly oriented fiber containing composites. However in Figure
3.2.11, it was shown that the model is clearly not intended for composites whose matrix has a
higher Young’s modulus than the fibers, which again is the case for LLZO and the Pan model
is thus unsuitable. The Curtis model may or may not be close depending on the orientation
factor x. Unfortunately it cannot be empirically determined from the material and ultimately
comes down to being decided almost arbitrarily by the person using the Equation. Table 3.2.03
is based on studies by Hull et al. and Sandler [Hull 1981; Sandler 2002] and lists several x
values based on specific aspect ratios and fiber orientations, but this is only helpful when the
fiber/matrix composite has the one of the specific aspect ratios and fiber orientations. In the
plots in section 3.2.2.2, it was more useful to use the Curtis model to calculate the orientation
factor, x, from the experimental data which could give an idea of the overall orientation of the

123

fibers. Lastly is the modified Halpin-Tsai model, which is shown in Figure 3.2.03 through
Figure 3.2.08 to be most applicable to randomly oriented fiber composites. Even so it still
makes a reasonable model for the LLZO composite.
In conclusion, if lithium in LLZO forms dendrites, it can be modeled as a fiber/ matrix
composite. Most models available for fiber/matrix composite have been used to model a system
where adding fiber increases the elastic moduli of the composite. LLZO with lithium dendrites
would be the opposite scenario, where dendrite addition decreases the elastic moduli of the
composite. The modified Halpin-Tsai Equation was shown to be a good model for a
fiber/matrix composite system, whether the fiber addition increases or decreases the
composite’s elastic moduli. And in either case, the ROM and IROM will provide upper and
lower bounds for the composite, respectively. The ROM, Curtis model and modified HalpinTsai models all have approximately the same derivative with respect to the volume fraction of
fibers. Therefore, for the case that Em >> Ef such as in the case of LLZO and lithium
composite. This would mean that whether the fibers are aligned longitudinally or randomly
oriented, they will follow a similar relationship for Young’s modulus of the composite versus
volume fraction of the fiber. Furthermore, when several data sets from the literature, that had
different fiber and different matrix materials, were normalized (Figure 3.2.09) it could be seen
that the normalized modulus depends on the fiber orientation. And therefore, the fiber
orientation is major factor in the elastic moduli of the material.
3.2.3 Effects of Solid Solution on Elastic Modulus
As discussed in section 1.6.1, a solid solution is where two or more elements can mix
completely over a certain composition range. This may be the case with lithium in LLZO. The

124

effects on the elastic moduli from the a solid solution of lithium in LLZO can be described by
the Nordheim’s rule
𝐴 = 𝐴0 + (𝐴1 − 𝐴0 )𝑋 + 𝑘𝑋(1 − 𝑋)

3.2.21

where X is the atomic fraction of the solute atom, A can be a a varity of properties, A1
is the value of A when X=1, the pure solute, and A0 is the value of A when X= 0, the pure
solvent. When the bowing parameter k =0, the Nordheim’s rule is the rule of mixtures. The
parameter k is the bowing parameter, which is a measure of the departure from linearity and
has the same units as variable A, A0 and A1 [Kasap 2007]. X can also be expressed in terms of
weight fraction [Fan 2013a].
The Nordheim’s relation is often simply refered to as a parabolic function when
discussing elastic moduli or hardness [Darrow 1969; Schenk 1998b; Schenk 1998a;
Shchennikov 2003; Ren 2007; Ravinder 2001; Fan 2013b]and refered to by name when
discussing thermal conductivity [Terada 1997]or electrical resistivity [Mori 1996; Jen 1991]
For the elastic moduli in particular, a rough gauge of the dependence of elastic moduli
on composition is the ratio of the maximum and minimum values of Young’s modulus
(Emax/Emin) or bulk modulus, Bmax/Bmin over a given range of composition [Ren 2007; Fan
2013a]. Furthermore, the composition dependence of a number of solid solution systems
follows Equation 3.2.21 and can exhibit large changes in elastic moduli as a function of X.
Experimental studies of solid solution systems show, for example, Emax/Emin = 2.3 for
Ni0.7Zn0.3GdxFe2−xO4 (0.2 < x < 1) [Rao 2003], Emax/Emin = 2.4 and Bmax/Bmin = 1.9 for Mn1xCdxFe2O4

(0 < x < 1) [Ravinder 2001]. Also, Bmax/Bmin values range from 1.4 to 2.8 for

theoretical bulk modulus studies of solid solution systems including (Mg1-xBex)3N2 [Mokhtari
2004], BNxP1−x and BNxAs1−x [Hassan 2005],for 0 < x < 1 in each case. For a solid solution of

125

the naturally-occurring thermoelectric mineral tetrahedrite Cu10Zn2As4S13 and the synthetic
compound Cu12Sb4S13 Fan et al.[Fan 2013a] found Emax/Emin and Bmax/Bmin values of 1.9 and 1.7,
respectively.
The Young’s modulus of a solid solutions for Cu10Zn2As4S13 and Cu12Sb4S13 is plotted
in Figure 3.2.12 from data in Fan et al.[Fan 2013a]. It is important to note that the Nordheim’s
rule is an empirical relationship because the bowing parameter k is not determined by first
principles. For example, Kasap [Kasap 2007] list various experimentally determined values for
k for the resistivity of metal alloys.
Figure 3.2.12 plots both the rule of mixtures and Nordheim’s rule for data from Fan et
al. For the Nordheim’s rule, k was found from a polynomial fit. Equation 3.2.21 can be
rewritten more explicitly as a polynomial
𝐴 = 𝐴0 + ((𝐴1 − 𝐴0 ) + 𝑘)𝑋 − 𝑘𝑋 2

126

3.2.21 a

Young's Modulus Versus Volume Fraction
in a Solid Solution [Fan 2013]
[Fan 2013]
fitted Nordheim
ROM
Nordheim (increased k)
Nordheim (decreased k)
Nordheim (negative k)

Young's Modulus (GPa)

70

60

50

40

30
0.0

0.2

0.4

0.6

0.8

1.0

Atomic Fraction
Figure 3.2.12 The modulus of Cu10Zn2As4S13 doped with Cu12Sb4S13. Pure Cu12Sb4S13 at weight
fraction equal to 1 [Fan 2013a]. Together they form a solid solution, which follows closely to the
Nordheim’s rule. The solid red line represents least-squares fit of the Fan 2013a data to
Nordheim’s rule, whose constants are determined from a polynomial fit of the data. ROM is the
rule of mixtures. The dashed lines represent the Nordheim’s rule if only k changes.
Table 3.2.16 accompanying data for Figure 3.2.12 [Fan 2013a]
𝐸 = 𝐸0 + (𝐸1 − 𝐸0 + 𝑘)𝑋 − 𝑘𝑋 2
Material at x=0 Cu9.7Zn1.9Fe0.4As4S13 natural
[Fan 2013a]
Material at x=1 Cu12Sb4S13
synthetic
[Fan 2013a]
Value
E0
(GPa)
57.597
E1-E0+k (GPa)
25.447
k
(GPa)
48.977
2
R
0.964

Error
1.847
8.034
7.580

In Figure 3.2.12, the data is fitted to a polynomial in red, which takes the form
a+bx+cx2 where a is equal to the E0 of Equation 3.2.21, b is equal to E1-E0+k and c will be
127

equal to k, the bowing parameter. These values are listed in Table 3.2.16 with the
accompanying error, as well as the materials that compose the solid solution. As mentioned
previously, k must be determined experimentally and fitted from data. Once k was determined,
Nordheim’s rule was recalculated and replotted with various k values. The orange dashed line
in Figure 3.2.12 labeled “Nordheim (increased k)” has a k value twice the calculated k value,
that is to say k=97.954 GPa instead of 48.977 GPa. The blue dashed line in Figure 3.2.12
labeled “Nordheim (decreased k)” has a k value half the calculated k value, that is to say
k=24.489 GPa instead of 48.977 GPa. The green line represents the k value multiplied by
negative 1. The red line was the Nordheim relation that fit the data, but the other lines are
added to elucidate how the bowing parameter can affect the modulus of a material. This bowing
parameter is the most unique aspect of the Nordheim’s relation. It demonstrates the fact that in
a solid solution, the elastic moduli do not necessarily follow a linear relationship. Furthermore,
in Figure 3.2.12 the Young’s modulus of the doped material increased above the Young’s
modulus of both pure materials, a phenomenon that would not be expected by the composite
models in section 3.2.2.
To further illustrate the changes in Young’s modulus due to solid solution composition
change, a second data set from Rao [Rao 2003] has been plotted in Figure 3.2.13. Table 3.2.17
has data for the fitted line as well as the materials that compose the solid solution material.
Again, the k value for the orange dashed line is double the fitted k value, see Table 3.2.17, and
the k blue dashed line has a k value half that of the fitted k value. The green line represents
Nordheim’s rule with –k.

128

Young's modulus versus volume fraction
in a solid solution [Rao 2003]

Young's Modulus (GPa)

300

250

200
[Rao 2003]
fitted Nordheim
ROM
Nordheim (increased k)
Nordheim (decreased k)
Nordheim (negative k)

150

100
0.0

0.2

0.4

0.6

0.8

1.0

Atomic Fraction
Figure 3.2.13 The modulus MnFe2O4 doped with various atomic fraction CdFe2O4[Rao 2003].
The solid red line represents a least-squares fit of the Rao 2003 data to Nordheim’s rule whose
constants are determined from a polynomial fit of the data. ROM is the rule of mixtures. The
dashed lines represent the Nordheim’s rule if only k changes.
Table 3.2.17 accompanying data for Figure 3.2.13 [Rao 2003]
𝐸 = 𝐸0 + (𝐸1 − 𝐸0 + 𝑘)𝑋 − 𝑘𝑋 2
Material at x=0
Material at x=1

MnFe2O4
CdFe2O4

[Rao 2003]
[Rao 2003]
Value

E0
E1-E0+k
k
R2

Error
135.857
-1.214
-162.5
0.978

129

8.572
40.314
38.697

In Figure 3.2.13, the modulus values obtained from Nordheim’s rule are less than those
obtained from the linear relationship. Therefore the solid solution may also result in lower
Young’s modulus than would be expected from the rule of mixtures or the other models used to
for composites in section 3.2.2. The fitted polynomial results in Young’s modulus between the
two pure substances.
In both Figure 3.2.12 and 3.2.13, the dashed lines are based on Equation 3.2.21 a
making them terminate at the first and last data point, but because the k value had to be
calculated from the data, a polynomial fit had to be used. The E0 and E1 values calculated in the
polynomial are slightly different than the E0 and E1 values that are plotted. For example,
Rao[Rao 2003] provides E0 as 131 GPa and E1 as 304 GPa , but the values determined by the
polynomial fit in Figure 3.2.13 were 135.857 GPa and 297.143 GPa. The error for each variable
in the polynomial for Figure 3.2.13 (see Table 3.2.17) is much higher than the polynomial fit of
the data in Figure 3.2.12 (see Table 3.2.16). The R2 value for the polynomial fit in Figure
3.2.13 is of 0.978, so the polynomial is still a good fit.
No bowing parameter, k, is available from literature for LLZO with lithium, as no
experiment has been done to determine elastic moduli of LLZO as a function of the
concentration of any solute. Therefore, a curve cannot be made to generate the effects of
lithium forming a solid solution with LLZO. However, it can be stated that as the LLZO
specimens are doped (to enhance sinterability or electrochemical properties, for example) the
modulus of the LLZO specimens can also change. For solid solutions, the moduli may even
increase above the higher moduli material or decrease below the lower moduli material.

130

3.2.4 Microcracking
Microcracking can have a strong effect on the elastic moduli of a material. The extent to
which microcracking affects the elastic moduli is a function of crack width, crack depth, crack
length, force applied to generate a crack, temperature and the crack density [Case 1993; Kim
1993b; Fan 2012]. During this study, it was noticed that the elastic modulus for specimens of
LLZO were decreasing after sanding (see Table 3.1.02 for values of Young’s modulus, shear
modulus and Poisson’s ratio). Sanding may induce microcracking which can be detrimental to
the elastic modulus of LLZO.
In the work by Case and Kim [Fan 2012; Case 1993]and by Fan et al. [Fan 2012], the
microcrack damage layer was produced in either alumina [Fan 2012; Case 1993] or
hydroxyapatite specimens Fan et al. [Fan 2012] via Vickers indentation. Thus, the number of
cracks, crack length and crack depth could be measured and controlled. In contrast, if we apply
these models to the microcrack damage layer produced by sanding, it is not straightforward to
determine the number of cracks, crack length, crack depth and the applied load. The crack
pattern from sanding is difficult to control, the loads on particular grit are not known, and the
damage from sanding is much closer to that of a “sliding indentation” rather than a Vickers
indentation. First, we shall discuss how the damage due to a sliding indentation differs from
that of a Vickers indentation and then we shall discuss how the microcrack damage due the
sliding indentation was estimated.
Vickers indentation is used to study brittle fracture and how cracks propagate in a
material, and can be used to determine the hardness of the material[Bower 1994]. The Vickers
hardness test use a pyramid shape “sharp” indenter that causes plastic deformation at the point
of contact [Bower 1994]. Vickers indenters make a diamond shaped impression and the cracks
131

will propagate from the corners forming two median cracks [Swain 1979], see Figure 3.2.14
(c). The median cracks propagate from the plastically deformed region. Two types of cracks
can occur from a Vickers indenter, half-penny (or semi-ellipse) or Palmqvist, see Figure 3.2.14
[Ponton 1989]. Systems that develop Palmqvist cracks at low loads may develop into halfpenny cracks at increased loads, thus the half-penny system is considered a fully developed
crack [Kim 1993a]. The depth of the half-penny crack, c, will be equal to half the distance of
the surface diagonal crack.

2a

2a

(a)
(a)

2c
(b)
(b)

2a

2c
(c)
(c)

Figure 3.2.14 Palmqvist versus half-penny or modified half-ellipse radial-median crack system
from Vickers indentation (a) side view of Palmqvist (b) side view of half penny crack (c) top
view of indentation and cracks, which is the same whether the crack is Palmqvist or half-penny.
The small circles below 2a in (a) and (b) are the plastically deformed regions. The larger circle in
(b) represents the median crack and (c) shows that there are two median cracks perpendicular to
one another in a typically Vickers indentation [Kim 1993a; Fan 2012]
Sliding indentation differs from Vickers indentation because of a tangential or shearing
force is introduced perpendicular to the indentation force [Sharp 1993]. At the point of contact,
sliding indentation is initially the same as in Vickers indentation, a plastic region of
deformation is made, and diagonal cracks form either as Palmqvist or half-penny crack [Swain
1979]. As the indenter begins to slide and form a scratch on the materials surface, one of the
cracks will continue to propagate in the direction of the sliding, whereas the other median crack
will stay immobile at the initial site of contact [Swain 1979]. The curved edges of a half-penny
crack will only be at the beginning and end of the crack, so over far enough an area, the crack
132

can be considered a slit crack (see Figure 1.6.01). The median crack below the scratch may
move to the left and the right as the crack forms because the crack propagates mainly along the
grain boundaries [Zarudi 1996]. Zarudi et al. found the area immediately connected to the
scratch was plastically deformed and the median crack formed below the plastically deformed
region of the scratch [Zarudi 1996]. Therefore there is a set of subsurface cracks generated by
sliding indenters or by sanding/machining that extend into the specimen (from the plastically
deformed surface) that are not visible by observing the specimen surface.
It is important to note Lateral cracks may also form in systems where loads are high
enough that friction forces cause stick and slip to occur [Swain 1979]. These lateral cracks
extend nearly parallel to the surface and can cause chipping or spalling near stationary cracks,
or chipping in the case of surface abrasion. Also, Swain observed that at very low loads (<0.05
N) in a study using various glass samples and sapphire, no surface or subsurface cracks were
detected and the track of the scratch was exclusively plastically deformed [Swain 1979]. This is
believed to be the mechanism for sanding where scratches plastically deform the material and
the plastically deformed region is removed without causing cracks [Swain 1979]. If the
calculated loads are very low for the present study of LLZO, there is the possibility that
microcracking does not form at all, but if this is the case, the calculated change due to
microcracking would be very small as well. This is consistent with a crack “pop-in” threshold
load that has been observed for Vickers indentation cracks.
3.2.4.1 Method and Model
To determine the change in Young’s modulus for LLZO due to sanding, the specimen
was first sanded using two different grades of sandpaper (600 and 1200), as these were used as
the final polish for specimens during the study. Then the specimens were analyzed using
133

scanning electron microscopy (SEM). In order to avoid surface charging in the SEM, the
specimens had to be coated with platinum, since the uncoated LLZO surface is non-conductive.
The micrographs were used to view the scratches. As stated by Swain and Zarudi et al. [Swain
1979; Zarudi 1996], the microcracks form beneath the scratches. The number of scratches,
length of the scratch and width of the scratch were directly observed from the micrographs. The
number of scratches determines the number of cracks, and the length of the scratches would be
the length of the cracks. The width of the scratch could help to estimate the applied load as
described below.
For the subsurface cracks due to abrasion, the depth of the crack is difficult to estimate.
Subsurface microcracks below the scratch do not have the same relationship with surface
cracks that a typical Vickers indentation has, (see Figure 3.2.14) which allow the crack depth to
be measure. However, an understanding of the material’s fracture toughness and hardness can
be used to estimate the load and crack depth based on the micrograph measurements of the
crack width.
A model for the change in Young’s modulus due to a surface damage layer has been
given by Case and Kim and by Fan et al. [Fan 2012; Case 1993]
∆𝐸
𝑟2 + 3
𝑟
=
∗𝑓∗ε
2
𝐸0 (𝑟 + 1) (r + 1)

3.2.22

where E0 is the Young’s modulus of the material with no microcracks, f is a function of the
spatial orientation of the microcracks, ∆E is the change in Young’s modulus due to
microcracking, r is the depth of the microcrack layer divided by the depth of the microcrack
free layer and ε is the crack damage parameter [Fan 2012].
For simplicity sake, R can be defined as

134

𝑟2 + 3
𝑟
𝑅=
(𝑟 + 1)2 (r + 1)

3.2.23

Which reduces Equation 3.2.22
∆𝐸
=R∗𝑓∗ε
𝐸0

3.2.22 a

f can be determined based on the type of crack that forms. For a slit crack

𝑓 = 𝜋 2 (1 − 𝜈02 )/2

3.2.24

where ν is the Poisson’s ratio.
The crack damage parameter ε is defined by the Equation [Budiansky 1975; Hoenig
1979; Fan 2012]
ε = 𝑁𝑣 ∗ 𝐺 =

2𝑁𝑣 < 𝐴2 >
𝜋 <𝑃>

3.2.25

where Nv is the crack number density, G is the microcrack geometry factor (not the shear
modulus) and A and P are the area and perimeter of the microcrack respectively [Case 1993].
In order to use the technique used by Case et al. [Case 1993] to find the damage due to
surface abrasion rather than Vickers indentation cracks, we need to estimate the applied load on
the abrasive particle, the number of cracks, and the crack dimensions. Also, instead of a
standard half-penny crack generated by Vickers indentation, we shall assume that the sandinginduced cracks are long slot cracks.
Equation 3.2.25 shows ε is a function of area and perimeter which will be a function of
the crack depth; r is also a function of crack depth. The crack depth, c, can be determined from
Equation 3.2.26 from Lawn and Cook [Lawn 2011; Schmidt 2014]
𝐾𝐶 =

(𝐸/𝐻)0.5 𝐹
3
𝑐2

135

3.2.26

which can be rearranged to solve for c
2

𝐸 0.5 𝐹 3
𝑐 = (0.016 ∗ ( )
)
𝐻
𝐾𝐶

3.2.26 a

where Kc is the fracture toughness, E is the Young’s modulus, F is the load on the microcrack
and c is the crack depth [Schmidt 2014].
Assuming the width of the abrasion scratch is approximated by the indentation width
2a, the load, F, can be estimate using Equation 3.2.27 from Wachtman [Wachtman 2009]
1.8544𝐹
(2𝑎)2

𝐻=

3.2.27

which can be rearranged to
𝐻(2𝑎)2
𝐹=
1.8544

3.2.27 a

where H is the hardness, F is the load and 2a is the width of the scratch.
For the abrasion induced cracks, the mean crack area, A, and mean crack perimeter, P,
are given by
3.2.28

𝐴 =𝑐∗L

P = 2𝑐 + 2L

3.2.29

assuming a rectangular crack of length L and depth c for the subsurface crack that is beneath
the surface scratch and is induced by sanding.
After the specimens were processed, micrographs of the surfaces were analyzed using
SEM. one specimen that was not sanded was analyzed under SEM, but it showed no clear signs
of microcracking (Figure 3.2.15). The specimen was cut from the same larger sample, SLLZO-92314-B, that the other specimens were cut from.

136

Figure 3.2.15 The cut surface of sample of S-LLZO-92314-B after being cut with the low speed
diamond saw (viewed at 2000X magnification). No sanding was done to this specimen. The
image shows no clear sign of cracking or scratches that may produce microcracking.
Figure 3.2.15 shows the cut surface of sample of S-LLZO-92314-B. No signs of
scratches that would cause microcracking are visible.
Figure 3.2.16 and Figure 3.2.17 are micrographs of a different piece cut from S-LLZO92314-B that were sanded using 1200 grit SiC sandpaper. Figure 3.2.16 shows magnification of
600X which was used to estimate crack number density, Nv, and crack length. Figure 3.2.17
shows the same area at a magnification of 1200X, which was used to estimate the width of the
scratches.

137

Figure 3.2.16 Sample of S-LLZO-92314-B after being sanded with 1200 grit SiC sandpaper at
600X magnification. The length and number of cracks were estimated from micrographs such as
this.

Figure 3.2.17 Sample of S-LLZO-92314-B after being sanded with 1200 grit SiC sandpaper at
2000X magnification. The width of the cracks could be estimated from micrographs such as this.

138

Figure 3.2.16 and Figure 3.2.17 show scratches from the sandpaper, which resembles
sliding indentations. Microcracks form under the surface of the scratches, so the length of the
microcrack can be assumed to be approximately the length of the scratch [Swain 1979]. From
the micrographs of S-LLZO-92314-B, the average length of the scratch (and therefore of the
cracks) was estimated as 263 μm and the width of the scratch was estimated to be 1.6 μm, see
Table 3.2.18. Using this information, the impact of microcracking could be estimated for
sample S-LLZO-92314-B.2. Values for S-LLZO-92314-B.2 post sanding were used because
this specimen had the lowest RMS and was cut from S-LLZO-92314-B using the same low
speed diamond saw procedure as was used to cut the LLZO pieces that were analyzed using the
SEM. Furthermore, S-LLZO-92314-B.2 was sanded and thus could potentially have
microcrack damage due to the sanding.
3.2.4.2 Tables and Figures for Microcracking
Table 3.2.18 lists the properties that had to be identified in order to estimate the change
in Young’s modulus due to microcracking for S-LLZO-92314-B.2. The thickness, Young’s
modulus and Poisson’s ratio for S-LLZO-92314-B.2 are listed in Table 3.1.02. Kc and H are
available in from Wolfenstine et al. [Wolfenstine 2013] and Ni et al. [Ni 2012], respectively.
Measurements of H for specimens of the LLZO specimens with porosities of 3% and 6% were
6.3 GPa and 5.2 GPa respectively [Ni 2012]. S-LLZO-92314-B.2 post sanding had a porosity
of 1.3%, and the Young’s modulus was more than 30% less than 3% porous LLZO specimen in
the study by Ni et al. [Ni 2012]. Nevertheless, the hardness values from Ni et al. [Ni 2012]
were used to approximate H in this study. Also, the fracture toughness, Kc, determined by
Wolfenstine et al. [Wolfenstine 2013] were used as an approximation here. Length and width
of the crack as well as the number of cracks per volume were estimated by inspecting the SEM
139

micrographs. All of the other values (f, depth of crack, load per crack, average area of a crack,
average perimeter per crack, crack damage parameter and r) were calculated.
Table 3.2.18 lists the estimated values for properties of specimens of S-LLZO-92314-B.2 post
sanding after sanding with 600 grit sandpaper. The factor f is a function of the spatial orientation
of the microcracks, G is the microcrack geometry factor, r is the depth of the microcrack layer
divided by the depth of the microcrack free layer and ε is the crack damage parameter [Case
1993]
Source
Property
Description
value
units
Table 3.1.02
T
Sample thickness
1320 μm
Table 3.1.02
E
Young’s modulus
94.5 GPa
Table 3.1.02
ν
Poisson's ratio
0.352
1/2
Fracture toughness
[Wolfenstine 2013]
Kc (average)
1.25 MPa *m
[Ni 2012]
H
Hardness
6.3 GPa
Equation 3.2.24
f (slit)
3.2
micrograph
L
scratch length
263 μm
micrograph
2a
scratch width
1.6 μm
micrograph
a
Half of the scratch width
0.79 μm
Equation 3.2.24
F
Load
8.5 mN
Equation 3.2.26
c
Depth
0.56 μm
2
Equation 3.2.28
A
Area
149 μm
Equation 3.2.29
P
Perimeter
528 μm
3
Micrographs
Nv (average)
Crack number density
468 cracks/mm
3
Equation 1.6.14
G
26.6 μm
=c/(T-c)
r
0.00043
-5
Equation 3.2.25
ε
1.2 X 10
-8
Equation 3.2.22
ΔE/E0
5.1 X 10
Table 3.2.18 includes an estimate of the ΔE/E0 in S-LLZO-92314-B.2 due to
microcracks after sanding with 600 grit sandpaper. Specimen S-LLZO-92314-B.2 was sanded
with 600 grit sandpaper and then sanded again with 1200 grit sandpaper. Table 3.2.19 includes
the same properties and calculations that are displayed in Table 3.2.18 except with 1200 grit
sandpaper being used instead of 600 grit. Calculations for Table 3.2.18 and Table 3.2.19 use the
Young’s modulus, specimen thickness and Poisson’s ratio for S-LLZO-92314-B.2 post

140

sanding, thus the calculated difference in ΔE/E0 due to microcracking between Table 3.2.18
and Table 3.2.19 are due to the change in sandpaper grit.
Table 3.2.19 lists the estimated values for properties of specimens of S-LLZO-92314-B.2 post
sanding after sanding with 1200 grit sandpaper. f is a function of the spatial orientation of the
microcracks, G is the microcrack geometry factor, r is the depth of the microcrack layer divided
by the depth of the microcrack-free layer and ε is the crack damage parameter
Property
Description
value
units
Source
T
Sample thickness
1320 μm
Table 3.1.02
E
Young’s modulus
94.5 GPa
Table 3.1.02
ν
Poisson's ratio
0.352
Table 3.1.02
1/2
Kc (average)
Fracture toughness
1.25 MPa *m
[Wolfenstine 2013]
H
Hardness
6.3 GPa
[Ni 2012]
f (slit)
3.2
Equation 3.2.24
L
scratch length
174 μm
micrograph
2a
scratch width
0.72 μm
micrograph
a
Half of the scratch width
0.36 μm
micrograph
F
Load
1.74 mN
Equation 3.2.24
c
Depth
0.20 μm
Equation 3.2.26
2
A
Area
34 μm
Equation 3.2.28
P
Perimeter
349 μm
Equation 3.2.29
3
Nv (average)
Crack number density
540 cracks/mm Micrographs
G
2.1 μm3
Equation 1.6.14
r
0.00015
=c/(T-c)
-6
ε
1.1 X 10
Equation 3.2.25
-9
ΔE/E0
1.6 X 10
Equation 3.2.22

Table 3.2.18 and Table 3.2.19 include several different values for the selected crack
damage model parameters. First, the scratch length and scratch width is larger in Table 3.2.18,
also the scratch number is lower in Table 3.2.18. This is understandable as the 600 grit
sandpaper that was used for Table 3.2.18 has a large particle size and fewer particles per unit
area. Therefore the load at each sliding indentation would be higher in the 600 grit sandpaper
than in the 1200 grit sandpaper because the same force is distributed over a smaller area (for
the 600 grit), or fewer contact points, which accounts for the fewer microcracks. This would in

141

turn cause the scratches to penetrate deeper and wider, which accounts for the larger value for
2a in the Table 3.2.18 for sanding with 600 grit sandpaper.
Table 3.2.18 and Table 3.2.19 show ΔE/E0 is very small. Therefore the model used here
predicts microcracking induced by sanding the specimen surface is not likely to have a
significant effect on the Young’s modulus of the S-LLZO-92314-B.2. Since many assumptions
had to be made to determine the value of ΔE/E0 in Table 3.2.18 and Table 3.2.19, the value
could be larger or smaller. Plotting Equation 3.2.22 can be quite useful to explore how
changing the values of the parameters in Table 3.2.18 can affect the influence microcracking
has on the Young’s modulus. The crack damage parameter, ε is a function of Nv, and G. Figure
3.2.18 shows ΔE/E0 as a function of ε.

142

2.0x10

E/E0

1.5x10

Change in Young's Modulus with
Increasing Crack Damge Parameter
-6

-6

1.0x10

-6

5.0x10

-7

E/E0
calculated in

Table 3.2.18

0.0
0.0

1.0x10

-4

2.0x10

-4

3.0x10

-4

4.0x10

Crack Damage Parameter, 

-4

5.0x10

-4

Figure 3.2.18 ΔE/E0 versus crack damage parameter using Equation 3.2.22 (black line) and using
r and f values from Table 3.2.18.
Figure 3.2.18 shows the relationship between ΔE/E0 and the crack damage parameter, ε.
For specimen S-LLZO-92314-B.2 post sanding using 600 grit sandpaper, ε is 1.1 X 10-5 which
results in ΔE/E0 equal to 4.3 X 10-8, as shown in Table 3.2.18. From Equation 3.2.22, the slope
of the line is 3.87 X 10-3 which is a function of r and f as shown in Equation 3. 2-021
∆𝐸
𝑟2 + 3
𝑟
=
∗ 𝑓 ∗ ε = 3.87 ∗ 10−3 ∗ ε
2
𝐸0 (𝑟 + 1) (r + 1)

3.2.30

where f is a function of the spatial orientation of the microcracks and r is the depth of the
microcrack layer divided by the depth of the microcrack free layer.
It is important to compare the predictions from Figure 3.2.18, Table 3.2.18 and Table
3.2.19 with the E values of S-LLZO-92314 measured by RUS, because S-LLZO-92314 is a
143

specimen that was sanded and may have a reduced E due to microcracking. Table 3.2.20 shows
the change in E before and after sanding as measured by RUS.
Table 3.2.20 Young’s modulus before and after sanding for specimens of S-LLZO-92314 with
the ΔE/E0, as measured by RUS
Young’s modulus (GPa)
Specimen Label
before sanding after sanding ΔE/E0
92.2
90.7
1.6%
S-LLZO-92314-A
96.7
94.0
2.8%
S-LLZO-92314-B
99.1
97.4
1.7%
S-LLZO-92314-C
97.1
94.5
2.7%
S-LLZO-92314-B.2
96.3
94.2
2.2%
S-LLZO-92314 average

The average ΔE/E0 for specimens S-LLZO-92314 after sanding was 2.2%. According to
Equation 3.2.30, in order for microcracking to be responsible for a 2.2% ΔE/E0, ε would have
to be 5.69. This is larger than the calculated value in Table 3.2.18 by a factor of 5.1 X 105.
Thus, if the difference between the “before” and “after” sanding Young’s modulus values are
due to microcracking in the specimen, then the measured change in E due to microcracking is
much larger than the change predicted from Equation 3.2.22. If the change in Young’s modulus
in Table 3.2.20 is due to microcracking, ε (a function of Nv and G) would have to be larger than
was calculated. So either Nv or G is larger than estimated, or the change in Young’s modulus in
Figure 3.2.20 is not due to microcracking. Increasing Nv shows the same relationship as ε, so
the number of cracks would have to be several orders of magnitude larger than was calculated
to explain ΔE/E0 in Figure 3.2.20. G is dependent on several parameters, as shown in Equation
3.2.26 and Equation 3.2.27. Plots based on crack length, crack depth, load, fracture toughness
and hardness will be explored in order to understand the relationship between ΔE/E0 and each
of these parameters.

144

Change in Young's Modulus With
-6
Increasing Crack Length
2.0x10

E/E0

1.5x10

-6

1.0x10

-6

5.0x10

-7

E/E0

crack length
calculated in
Table 3.2.18
width of
S-LLZO92314-B.2

0.0
0

2000

4000

6000

8000

10000

L (m)
Figure 3.2.19 Change in Young’s modulus versus the crack length, L, the red circle is the value
of L calculated in Table 3.2.18. The green triangle is the crack length equal to the width of
specimen S-LLZO-92314-B.2. The plotted line is obtained from Equation 3.2.22 c.
Figure 3.2.19 shows ΔE/E0 versus crack length. The relation between ∆E/E0 and L,
Equation 3.2.25 can be incorporated into Equation 3.2.22 to give
∆𝐸
= R ∗ 𝑓 ∗ 𝑁𝑣 ∗ 𝐺
𝐸0

3.2.22 b

and incorporate Equation 3.2.28 and Equation 3.2.29 to obtain
∆𝐸
2𝑁𝑣
2𝑁𝑣
< 𝑐 2 𝐿2 >
= (𝑓 ∗
) R ∗ 𝐺 = (𝑓 ∗
)R ∗
𝐸0
𝜋
𝜋
< 2𝑐 + 2𝐿 >
where R is a function of r (see Equation 3.2.23) which is in turn a function c

145

3.2.22 c

When L is much larger than c and all parameters besides L are held constant, ∆E/E0 will
be proportionate to L. ΔE/E0 is still very small, on the order of 10-6, even when the cracks run
the entire width of a specimen (the width of specimen S-LLZO-92314-B.2 is used as an
example in figure3.2.19. Therefore increasing the scratch length, and there by increasing the
crack length, will not have a significant impact on ΔE/E0.
The crack depth is a critical property of microcracks. Figure 3.2.20 shows the
relationship of ΔE/E0 and c.

Change in Young's Modulus With
Increasing Crack Depth

0.05
0.04

E/E0

0.03
0.02
dE/E0
crack depth
calculated in Table 3.2.18

0.01
0.00
0

10

20

30

40

50

60

c (m)
Figure 3.2.20 Change in Young’s modulus as a function of crack depth according to Equation
3.2.22 c, black line. The red circle is the crack depth determined in Table 3.2.18 and the blue
square represents a 100 fold increase in crack depth, which would correspond to a Young’s
modulus change that would be detectable by RUS analysis

146

Figure 3.2.20 shows the relationship between ΔE/E0 and the crack depth, c, using
Equation 3.2.22 c. The red circle represents the calculated value of c as shown in Table 3.2.18,
which is 5.1 X 10-8. If c was actually two orders of magnitude larger than was estimated, then
ΔE/E0= 3.86%. A Young’s modulus change on the order of 4% would be detectable by the
RUS analysis. To explain the power law relationship in Figure 3.2.20, Equation 3.2.22 c will be
examined
∆𝐸
2𝑁𝑣
2𝑁𝑣
< 𝑐 2 𝐿2 >
= (𝑓 ∗
) R ∗ 𝐺 = (𝑓 ∗
)R ∗
𝐸0
𝜋
𝜋
< 2𝑐 + 2𝐿 >

3.2.22 c

where R is a function of r (see Equation 3.2.23).
In Equation 3.2.22 c, we consider the crack depth to change and other factors such
as f and Nv are held constant. R can be approximated as a linear function of c for the range
shown in Figure 3.2.20. The crack length L is much larger than c, therefore <2c+2L> will be
dominated by the L term, which is also a constant. Thus G will be proportional to c2, making G
a power law function of the second order. In turn, the product of G and R will be a power law
function.
Figure 3.2.20 shows crack depth can have a noticeable impact on the Young’s modulus,
albeit the crack depth would have to be 100 fold higher than currently estimated in order to
explain the decrease in E from sanding. A 100 fold increase in c correlated to a nearly a 106
fold increase in ΔE/E0. Thus it is important to consider which factors cause c to rise. Crack
depth is a function of the applied load, hardness and fracture toughness, so their relationship to
c and ΔE/E0 is plotted, using Equation 3.2.27, Equation 3.2.26 and finally Equation 3.2.22.

147

Change in Young's Modulus With
Increasing Load Per Crack
5
3.0x10

-5

2.0x10

-5

1.0x10

-5

3

2

E/E0

c (m)

4

dE/Eo
vs Load
c vs
Load
calculated
in table
Table 3.2.18

1

0
0

40

80

120

160

0.0
200

F (mN)
Figure 3.2.21 Change in Young’s modulus (black line) and crack length, c, in microns (red line)
as functions of Load, F, in mN. The red circles are the calculated values in Table 3.2.18 for each
line and corresponds to the calculated load of 8.51 mN. The red line is a plot of crack length c as
a function of F, from Equation 3.2.26.The black line is a plot of ΔE/E0 from Equation 3.2.22 in
conjunction with 3.2.26
Figure 3.2.21 shows the F dependence of both ΔE/E0 (black line) and ΔE/E0 versus c
(red line). The calculated value for the load is 8.51 mN, as shown in Table 3.2.18, which
corresponds to ΔE/E0=5.1 X 10-8. Figure 3.2.21 shows F, like c, has a power law relationship
with ΔE/E0, which is reasonable because c is a function of F. From Equation 3.2.26 it can be
seen that c is proportionate to F2/3.
2

𝐸 0.5 𝐹 3
𝑐 = (0.016 ∗ ( )
)
𝐻
𝐾𝐶

148

3.2.26

Thus the red line for c versus load, in Figure 3.2.21 is equal to F2/3 multiplied by some
constant, which can be determined from Equation 3.2.26 as
2

2

𝑐 = 0.13531 𝑁 −3 𝑚 ∗ 𝐹 3

3.2.31

The calculated crack depth in Table 3.2.18 is 0.56 microns. Figure 3.2.20 shows c
would have to increase by 1 or 2 orders of magnitude in order for ΔE/E0 to be measurable for
the RUS analysis. Based on the relationship stated in Equation 3.2.31 the load would have to
increase by well over two orders of magnitude. For example, the average change after sanding
for specimens s-LLZO-92314 was 2.2%, as shown in Figure 3.2.20, to get this ΔE/E0 from
sanding, c would have to be 46 microns, and in order to get this c value from changing the load
alone, F would have to be 6.3 N. that is 6.3 N per scratch or contact point by the sand paper,
which is significantly larger than the 8.5 mN load calculated in Table 3.2.18.
Values from literature were used to calculate the effects of microcracking on LLZO, see
Table 3.2.18. However it has been seen in this study that LLZO can have drastically different
elastic moduli values (see Table 3.1.02 for Young’s modulus and shear modulus of various
LLZO specimens from this study) compared to the Young’s modulus values available in
literature [Ni 2012]. Therefore, the LLZO specimens used in this study may have different Kc
and H values than what is available in literature, Kc from Wolfenstine et al.[Wolfenstine 2013]
and H from Ni et al.[Ni 2012]. Figure 3.2.22 plots ΔE/E0 versus Kc and Figure 3.2.23 plots
ΔE/E0 versus H.

149

-3

10

-4

10

-5

E/E0

10

10

-6

10

-7

10

-8

10

-9

Change in Young's Modulus With
Changing Fracture Toughness
E/E0

[Wolfenstine 2013]

1

2

3

4

5

6

7

8

9

10

1/2

Fracture Toughness (MPa m )
Figure 3.2.22 A semilog plot of the change in Young’s modulus as a function of fracture
toughness. The red circle represents the fracture toughness of 1.25 MPa *m1/2 from Wolfenstine
et al. [Wolfenstine 2013]. An increase in fracture toughness results in a sharp decrease in change
in Young’s modulus, since an increase in fracture toughness corresponds to a decrease in crack
length for a given load. The black line represents Equation 3.2.22 c with varying c values, where
c values are determined by Equation 3.2.26 with varying Kc values.
Figure 3.2.22 shows Young’s modulus as a function of fracture toughness for S-LLZO92314-B.2. Figure 3.2.22 follows Equation 3.2.22 c
∆𝐸
2𝑁𝑣
2𝑁𝑣
< 𝑐 2 𝐿2 >
= (𝑓 ∗
) R ∗ 𝐺 = (𝑓 ∗
)R ∗
𝐸0
𝜋
𝜋
< 2𝑐 + 2𝐿 >

3.2.22 c

Where c is defined by Equation 3.2.26
2

𝐸 0.5 𝐹 3
𝑐 = (0.016 ∗ ( )
)
𝐻
𝐾𝐶

150

3.2.26

The initial fracture toughness value was experimentally determined by Wolfenstine et
al. [Wolfenstine 2013] based on measurements that ranged from 1.63 to 0.86 MPa *m1/2, which
would correspond to ΔE/E0= 1.1 X 10-7 and ΔE/E0= 3.0 X 10-8, when all other factors held
constant. The red circle in Figure 3.2.22 is the average value, 1.25 MPa *m1/2. As shown in
Figure 3.2.22, as the fracture toughness increases the relative change in E decreases, again with
other factors held constant. Of course, it is unlikely that the fracture toughness of LLZO could
be significantly increased without perhaps and addition of fibers or particulate phases which
might comprise LLZO’s performance as an electrolyte. However, Figure 3.2.22 does illustrate
as the higher the fracture toughness for any electrolyte material increases the resulting drop in
Young’s modulus due to sanding should decrease.

151

Change in Young's Modulus With
-6
Changing Hardness
1.0x10
-7

6.0x10

-7

4.0x10

-7

E/E0

8.0x10

2.0x10

E/E0

Hardness value
of LLZO determined
by Ni [Ni 2012]

-7

0.0
0

20

40

60

80

100

120

Hardness (GPa)
Figure 3.2.23 ΔE/E0 versus H. The red circle is the Hardness determined for LLZO by Ni et al.
[Ni 2012]. The plot was determined by solving F according to Equation 3.2.27 a, then using F to
solve for c according to Equation 3.2.26 and finally solving for 3.2.22 c
Figure 3.2.23 shows ΔE/E0 versus hardness, H. H values for LLZO were experimentally
determined by Ni et al. [Ni 2012]. The value used in the calculations in Table 3.2.18 and for
Figure 3.2.23 is 6.3 GPa (measured for an LLZO specimen with 3% porosity). The calculations
done in Table 3.2.18 were for S-LLZO-92314-B.2 post sanding, which had a porosity of 1.3%.
It was assumed the value from Ni would apply to S-LLZO-92314-B.2, so Figure 3.2.23 is
particularly useful to determine the value of ΔE/E0 with different values of H. From Figure
3.2.23, even if the hardness was more than an order of magnitude larger, ΔE/E0 would still be
quite small and this change in modulus would still not be measureable by RUS.

152

By incorporating Equation 3.2.27 into Equation 3.2.26, it is found that c is proportional
to H1/3. As previously stated, as long as L is much greater than c, G will be proportional to c2.
ΔE/E0 is proportional to G, and thus ΔE/E0 will be proportional to H2/3. However, at low values,
such as the H values in Figure 3.2.23, the Equation is effectively linear.
In conclusion, if sufficient microcracking present, noticeable change in elastic moduli
of the material will occur. SEM micrographs were used to estimate the length and width as well
as frequency of the surface scratches in order to estimate the dimensions of subsurface
microcracks. Values for hardness and fracture toughness are available in literature [Ni 2012;
Wolfenstine 2013]. Calculations for microcracking for specimen S-LLZO-92314-B.2 are
shown in Table 3.2.18 and Table 3.2.19 for 600 and 1200 grit sandpaper respectively. From the
model discussion here, the change in Young’s modulus due to microcracking (ΔE/E0) was
estimated to be 5.1 X 10-8 for the specimen that was sanded with 600 grit sandpaper and 1.6 X
10-9 for specimens sanded with 1200 grit sandpaper. These values would indicate
microcracking has an extremely low impact on elastic moduli. However it was observed that
sanding damage has a lesser effect on ΔE/E0 for sandpaper with a high grit number. Swain
[Swain 1979] found that below 50mN of force per crack, no microcracking was observed in the
various glass materials used in his study. It is uncertain if such a crack initiation threshold
exists in LLZO, but using higher grit number sand paper as opposed to lower grit number sand
paper can help to reduce the accumulation of microcrack damage.
With the information that was available, several assumptions had to be made in order to
make rough estimates of the change in Young’s modulus as a function of sanding-induced
microcracks. The plots in Figure 3.2.19 through Figure 3.2.23 show microcracking as a
function of crack length, L; crack depth, c; load per crack, F; Hardness, H; and fracture

153

toughness Kc. These figures explore how each of these parameters affects the ΔE/E0. Although
the crack depth can have a strong influence on the ΔE/E0, ΔE/E0 is so low that the crack depth
would have to increase significantly (more than an order of magnitude) before ΔE/E0 would be
measureable by RUS.

154

4.0 Summary and Conclusions
The original intent of this study was to understand how the electrochemical cycling of
Lithium Lanthanum Zirconium Oxide (LLZO) affects the Young’s modulus, shear modulus
and Poisson’s ratio of the material. Any change in elastic moduli would likely be due lithium
deposition, possibly as dendrites. The only published study of elastic moduli of LLZO was
done by Ni et al. [Ni 2012]. Ni et al. used specimens of LLZO that were hot pressed at 1000˚C
under a pressure of 40 MPa [Ni 2012]. The LLZO billets were then cut into rectangular
parallelepipeds and analyzed using resonant ultrasound spectroscopy (RUS). Ni et al. found
high elastic modulus values for LLZO using RUS; the Young’s modulus, E, and shear
modulus, G, of LLZO were 149.8 GPa and 59.6GPa at 3% porosity [Ni 2012]. Ni et al. also
found LLZO had E= 132.6 GPa and G= 52.1 GPa at 6% porosity [Ni 2012]. The results from
Ni et al. [Ni 2012] and the results from this study are listed in Table 3.1.02 which list the
Young’s modulus, shear modulus, Poisson’s ratio, porosity and the RMS and peak count from
the resonant ultrasound spectroscopy (RUS) analysis.
Table 3.1.02 shows that the measured elastic moduli in this study were noticeably
lower than the results from Ni et al.[Ni 2012]. Each sample in this study was analyzed using
RUS. Only specimens S-LLZO-62014 had results comparable to the LLZO specimens from Ni
et al.[Ni 2012], see Figure 3.1.01 and figure 3.1.02. Sample S-LLZO-62014 was the second
sample received in this study, so the procedure was not entirely established. Though the
specimens had elastic moduli values that were comparable to published work, [Ni 2012], they
were exposed to air for more than a few hours (they were stored in air overnight for multiple
nights) and they were placed in acetone in order to wash off the thermoplastic, so the
specimens were not electrochemically cycled. There were concerns that the prolonged exposure
155

to the acetone and the moisture in the air may have caused a reaction. The experimental
procedure was changed so the specimens were stored in a glove box and the specimen was
exposed less to acetone (see section 2.2.1.1 for more on the washing procedure).
Unfortunately, the billets received after S-LLZO-62014, would have noticeably lower
elastic modulus compared to Ni et al. [Ni 2012]. Table 3.1.02 shows specimens S-LLZO-71014
had E values between 114 and 119 GPa, S-LLZO-72914 had E values from 96.3 to 101 GPa
and S-LLZO-92314 had E vales from 90 to 100 GPa. The shear modulus was also noticeable
lower than the samples from Ni et al. [Ni 2012]. From that point, much of the thesis work was
focused on exploring possible causes for the discrepancies in moduli.
4.1 The Effect of Porosity on the Elastic Moduli of LLZO
Porosity is known to have an effect on the elastic moduli of LLZO. Figure 3.1.01 shows
the Young’s modulus versus the porosity of LLZO specimens from Ni et al. [Ni 2012] and the
specimens S-LLZO-62014 (Section 3.2.1)
E(P)=167.16*exp(-3.816*P)

3.2.01

G(P)=67.40*exp(-4.236*P)

3.2.02

where E is Young’s modulus, G is shear modulus P is the porosity.
The R2 value for the exponential fit of the lines were 0.995 for E and 0.997 for G, see
Table 3.1.01. Although there were only four data points to fit, Equations 3.2.01 and 3.2.02 fit
extremely well. Unfortunately the rest of the specimens did not follow this pattern. Figure
3.2.02 shows all specimens from all billets received during this study and shows virtually no
pattern between elastic moduli and porosity for the specimens as a whole group. However,
Figure 3.1.04 shows a pattern of decreasing elastic modulus with increasing porosity for
specimen S-LLZO-92314. The same pattern, of increasing porosity and decreasing elastic
156

moduli , is seen in Figure 3.1.01 for specimens S-LLZO-62014 and LLZO specimens from Ni
et al. [Ni 2012]. Figure 3.1.01 and Figure 3.1.04 show that within a set of samples cut from the
same billet, increasing porosity will cause decreasing modulus.
4.2 Fiber/Matrix Composite as a Model for the Change in Elastic Moduli of LLZO
Section 3.2.2 discusses how the elastic moduli could be affected by lithium deposition
forming a fiber composite in the matrix of LLZO. Several models are explored, including the
rule of mixtures(ROM), inverse rule of mixtures (IROM),modified Halpin-Tsai, Pan, Curtis,
Manera and Christensen (Table 3.2.02 lists the Equations used to model the lithium LLZO
composite system, as well as their source and conditions or assumptions inherent to the model).
The models used in this discussion, have mainly, if not exclusively, applied to systems where
the E and G values of the fiber were much higher than the E and G values of the matrix, and the
fiber would serve to reinforce the matrix.
Several data sets [Yeh 2006; Sandler 2002; Manera 1977; Christensen 1972] with
random fiber orientation were used to compare the model. These data sets are plotted in Figure
3.2.03 through Figure 3.2.06. The maximum fiber contribution to modulus of the composite is
achieved when the fibers are oriented in the direction of the applied force [Kingery 1976a], like
in Figure 3.2.08, so the experimental data follows most closely to the ROM model. On the
other hand, the lowest fiber contribution of the fiber to the modulus is when the fibers are
aligned perpendicular to the application of stress, like in Figure 3.2.07 and thus, the
experimentally data follows closely to the IROM. The analysis of the models in section 3.2.2
helps to determine the utility of each model.
Each data sets had different materials which had different Young’s modulus values for
each fiber/matrix system. Each data set was normalized using Equation 3.2.15. The normalized
157

data was plotted in Figure 3.2.09 which plots the normalized Young’s modulus versus the
volume fraction of fiber for all the data sets, using Equation 3.2.19
𝐸𝐶 − 𝐸𝑚
= 𝐸𝑁 = 𝑉𝑓 ∗ 𝑀
𝐸𝑓 − 𝐸𝑚
where EN is the normalized Young’s modulus of the composite, which is unitless. EC is the
Young’s modulus of the composite, Ef is the Young’s modulus of the fiber and Em is the
Young’s modulus of the matrix. M is a fiber-orientation dependent factor.
Figure 3.2.09 shows the data sets from Figure 3.2.03 through Figure 3.2.08 plotted
according to Equation 3.2.19, would separate into 3 groups with different slopes, M, based on
the orientation of the fiber. The slope, intercept and R2 values are listed in Table 3.2.11. This is
rather significant because it shows the data sets’ major difference is the orientation of the
fibers, but otherwise the data sets can be normalized (using Equation 3.2.19) to fit a line
according to its orientation regardless of the fiber or matrix material. Therefore the
normalization might be used as a guide to analyzing the orientation of the fibers in a particular
data set, which would be applicable to a wide range of composite materials.
Figure 3.2.11 shows several of the models in Table 3.2.02 plotting Ec versus Vf for a
matrix of LLZO with lithium dendrites as the fiber. Values had to be assumed for the E of
LLZO, E of the lithium dendrites, Poisson’s ratio of the matrix, the alignment of the
fiber/dendrites and the aspect ratio of the dendrites. The Young’s modulus value used for
lithium metal was 10.65 GPa [Trivisonno 1961]. For LLZO, the Young’s modulus was 149.8
GPa and the Poisson’s ratio was 0.257 [Ni 2012]. The ROM and IROM typically serve as upper
and lower bounds for a composite system when plotting the Young’s modulus versus the
volume fraction of the fiber; however the Pan and Christensen models were above these
boundaries. The modified Halpin-Tsai and the Curtis model follow closely to ROM model and
158

can be considered good models to predict the relationship between Ec and Vf for a fiber/matrix
composite consisting of dendrites in LLZO.
The Pan model gives a slightly decreasing slope for E versus Vf. When Vf =0 Ec is
equal to Em which is true in Figure 3.2.11. However, as Vf approaches 1, the modulus of the
matrix should contribute very little to the modulus of the composite, but in the Pan model more
than 80% of the matrix modulus value is still added to the composite modulus at Vf = 1.
Therefore, the Pan model seems better suited for a material where the added fiber reinforces the
material and increases the E of the composite.
The Christensen model makes an even less reasonable model for the LLZO/dendrite
composite shown in Figure 3.2.11 since it predicts an increase in Young’s modulus of the
composite as the volume fraction of fibers increases, which is only valid if the Young’s
modulus of the fiber is greater than that of the matrix, but this is not the case for the
LLZO/dendrite composite, see section 3.2.2.3. Furthermore, the Christensen model is only
applicable when the shear modulus of the fiber is greater than the shear modulus of the matrix,
and this is not the case for the LLZO/dendrite composite, see Table 3.2.02.
The modified Halpin-Tsai makes a reasonable model for the LLZO/dendrite composite
in Figure 3.2.11. The Curtis model does as well, but the orientation factor, x, is not determined
by first principles and has to be determined by modeling the data. Since the Pan model and the
Christensen model cannot be considered appropriate models for Li fibers in LLZO, the ROM
and IROM once again serve as the upper and lower bounds of the models for the
LLZO/dendrite system.
The Manera model was not used because the model assumes that the Young’s modulus
for the matrix, in this case LLZO, has to be between 2 and 4 GPa. LLZO was over 90 GPa in

159

all samples during this study (see table 3.1.02). Thus the LLZO matrix was much above the
threshold of the Manera model.
A composite system of dendrite fibers in an LLZO matrix is a reasonable result of
lithium deposition in a LLZO electrolyte. If the change in volume fraction of lithium in LLZO
can be determined, then the changing modulus of LLZO upon cycling can be used to test the
models included in this thesis.
4.3 Solid Solution as a Model for the Change in Elastic Moduli of LLZO
Lithium deposition within the LLZO could also act as a solid solution as discussed in
section 3.2.3. A solid solution follows the Nordheim rule as shown in Equation 3.2.21 a
𝐴 = 𝐴0 + (𝐴1 − 𝐴0 + 𝑘)𝑋 − 𝑘𝑋 2

3.2.21 a

where X is the atomic fraction of the solute atom, A can be a a varity of properties, A1 is the
value of A when X=1 ,the pure solute, k is the bowing parameter, A0 is the value of A when
X= 0, the pure solvent.
Nordheim’s rule is effective for analyzing a solid solution, unfortunately no bowing
parameter, k, is available for LLZO with lithium so Nordheim’s rule cannot be used to predict
the magnitude of changes in the Young’s modulus. However, if LLZO specimens can be shown
to form a solid solution with lithium after electrochemical cycling, the bowing parameter of
lithium in LLZO could be determined for the first time. More importantly the model of Li
deposition in LLZO could be determined.
4.4 The effect of Microcracking on the Elastic Moduli of LLZO
Microcracking can be detrimental to the elastic moduli of a material. There are
multiples ways for microcracking to occur but of particular interest to this study is
160

microcracking due to sanding, which is a very common part of processing specimens.
Specimens S-LLZO-92314 were analyzed using RUS, then sanded and analyzed again using
RUS. The results for the before and after sanding RUS analysis are listed in Table 3.1.07. To
analyze the effects of microcracking, a model modified from the model in Kim and Case [Kim
1993a; Case 1993; Kim 1993b] was used. This model considers the sanded layer as a
microcrack damaged layer, and the rest of the material as a crack free layer. Kim and Case
[Kim 1993b] induced a microcrack layer using Vickers indentation in alumina. Below the
plastically deformed zone of the scratches microcracks will form [Zarudi 1996]. These cracks,
unlike Vickers indentation cracks cannot be seen on the surface [Swain 1979; Zarudi 1996].
Sanding with 1200 grit number sandpaper caused a 1.6 X 10-9 relative change in
Young’s modulus, ΔE/E0, and sanding with 600 grit number sand paper produced a 5.1 X 10-8
relative change Young’s modulus. Several estimates had to be made in order to make these
calculations. First, measuring the length and width of the scratches is a tedious task that is
subject to error as is counting the number of scratches. Also, the values in literature were
assumed to be accurate for the particular specimens, but the Young’s modulus, E, for the
specimens in this study does not match the published values for E, so Kc and H may have
different values for the specimens in this study versus the values available in literature. Due to
the assumptions that were made, section 3.2.4.2 discusses some of the variables in the
calculations for ΔE/E0 due to microcracking, including load, crack length, crack depth, hardness
and fracture toughness. The depth of the crack has a strong impact on the change in Young’s
modulus due to sanding. However, it would still take more than an order of magnitude increase
in the crack depth to cause the change in Young’s modulus that was seen between sanding
stages, see Figure 3.2.20. Therefore either (1) microcracking due to sanding is not the cause of

161

the change in modulus between sanding stage or (2) the model used to analyze the changes in
modulus due to sanding is inadequate.

162

5.0 Future Work
Much work can still be done in this study. Table 3.1.07 shows there is a noticeable drop
in Young’s modulus and shear modulus after the specimen is sanded. Logically it would seem
the specimen experiences some degradation due to the processing and the most likely
explanation would be microcracking induced by the sanding. Section 3.2.4 discusses
microcracking and shows that the model considered, which is applicable to a surface damage
layers induced by Vickers indentation cracks, does not explain the changes observed in the
LLZO after sanding. Thus, either the model, as applied to the sanding damage, is inadequate to
describe the damage or another mechanism other than microcracking is responsible for the
observed changes in modulus after sanding the LLZO specimens. Other than microcracking,
there may be a reaction with the air that causes a drop in elastic modulus. The specimen is only
exposed for a few hours at the most, and the time between initial RUS scanning and a second
RUS scanning was even shorter. This would mean the reaction with air for LLZO is quite rapid,
or at least fast enough to have noticeable effects within an hour or few hours of exposure to
open air. Therefore, in the future more study of microcracking can be done to more thoroughly
determine whether or not microcracking is the mechanisms causing a drop in elastic moduli.
According to the model presented in Section 3.2.4, it is highly unlikely microcracks are
causing a measureable change in Young’s modulus of the LLZO samples after sanding.
However many assumptions had to be made to make these calculations. A key issue is that it is
very difficult to determine the crack depth and crack length of the subsurface cracks induced by
sanding the specimen surface. Also, only one specimen for each sandpaper grit was analyzed
using SEM. More specimens could be inspected to get a better estimate of the scratch lengths,
scratch widths and scratch number density, which are all used in the calculation for ΔE/E0.
163

Also, a direct determination of the subsurface crack depths and crack lengths would be very
helpful.
This study serves as a good basis, or even a guide to continue the work. Simple
mistakes during the study include washing the specimens with acetone directly and storing the
specimens in air. These practices are completely acceptable for other materials, but are
inappropriate for LLZO, and were corrected for specimens after S-LLZO-62014. Other
incremental changes in the procedure included using a lapper to sand the specimens to a desired
height (which limited the amount of material wasted) and cutting parallelepipeds using the
aluminum block fixture discussed in section 2.2.1.2, which helped to ensure perpendicular cuts
in the specimen.
RUS is a very useful tool to nondestructively determine the elastic moduli of a material.
Defects such as chipping, cracks and impurities can reduce the elastic moduli of the specimens
and make them difficult to analyze. Also, it was suspected that the LLZO specimens in this
work may have had other phases present. What these phases were was beyond the scope of this
study, but it might explain some of the difficulty with the RUS analysis. RUS alone cannot
determine if other phases are present, but the analysis could be affected if there are.
Several models for the case of LLZO and lithium forming a fiber/matrix composite
have been considered. The modified Halpin-Tsai model would be effective for modeling the
composite. Also a model for a solid solution has been discussed. As data becomes available,
these models could be useful to help understand how the lithium is being deposited.
Most importantly, the intended experiment should still be done. The intended work was
to take specimens of LLZO, analyze the specimens using resonant ultrasound spectroscopy
(RUS) to determine the elastic moduli of the specimens, cycle the specimens in a glove box

164

free of moisture, and then analyze them again with RUS to find the elastic moduli. Changes in
elastic moduli would most likely be due to lithium deposition. Unfortunately, few specimens
could be produced for this study, and even worse was that the RUS results for the LLZO
specimens were different from billet to billet. When specimens can be produced with consistent
elastic moduli values, the intended study should take place. If this work continues, the number
of variables should be reduced, for instance: if all the work could be done in a glove box, or a
moisture free environment, the risk of the materials interacting with the atmosphere would be
reduced. Other methods could also be utilized to reduce the number of variables in play during
the experimental procedure, but these would have to be determined by the researchers who
continue the study. Ultimately, reducing the variables and having consistent specimens, should
allow for further progress in this study.

165

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