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PREDICTION OF RHEOLOGICAL PROPERTIES OF
STRUCTURED FLUIDS IN HOMOGENEOUS SHEAR BASED
ON A REALIZABLE MODEL FOR THE ORIENTATION DYAD

presented by

YoChan Kim

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2105 p:/ClRC/DaleDue.indd-p.1

PREDICTION OF RHEOLOGICAL PROPERTIES OF STRUCTURED FLUIDS IN
HOMOGENEOUS SHEAR BASED ON A REALIZABLE MODEL FOR THE
ORIENTATION DYAD
By

YoChan Kim

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Chemical Engineering and Materials Science

2006

ABSTRACT

PREDICTION OF RHEOLOGICAL PROPERTIES OF RIGID ROD FLUIDS IN
SIMPLE HOMOGENEOUS SHEAR FLOWS BASED ON A REALIZABLE MODEL
FOR THE ORIENTATION DYAD
By
YoChan Kim

Non-spherical particles dispersed in a fluid have a tendency to align in shear
flows because of particle-fluid drag. This phenomenon is opposed by rotary diffusion.
At high concentrations and in the absence of hydrodynamic couples, self-alignment can
also occur because excluded volume forces prevent the retum-to-isotropy of anisotropic
states by rotary Brownian motion. The balance between microhydrodynamic and
diffusive (i.c., Brownian and excluded volume) torques at the microscale has a direct
impact on the rheological properties of rigid rod fluids (particulate suspensions and liquid
crystalline polymers) at the continuum scale.

Over the past sixty years, important characteristics of the microstructure
associated with the foregoing alignment phenomenon have been quantified in terms of
the low order moments Of the orientation density function governed by the rotary

Smoluchowski equation. In this research, a closed model for the second order moment

< pp > (orientation dyad) has been identified based on the condition that in the absence

of an external field all realizable anisotropic states must relax to stable equilibrium states.
A key step in the development of the new closure is the use of an algebraic pre-closure

for the orientation tetrad < pppp > in terms of the orientation dyad < pp> that

preserves the six-fold symmetry and contraction properties of the original orientation

tetrad.

In the presence of a simple shear flow, the microstructure and the rheological
characteristics predicted for rigid-rod fluids agree with previous theoretical and

experimental results for a wide range of Péclet numbers. In addition to the Péclet number

(i.c., Pe :-:||Vg||/(6D;) ), the orientation director also depends on three other

dimensionless groups: a tumbling parameter, A; an excluded volume coefficient, U; and,
a dimensionless time t a 6 D: t. The rotary diffusion coefficient for dilute solutions, D; ,

is used to scale time. Unlike other closure models, the approach developed hereinafter
predicts that all two-dimensional and three-dirnensional realizable anisotropic states
relax to either a steady state (isotropic or anisotropic) or a periodic state, depending on

Pe, A, and U. The model predicts the existence of shear thinning and shear'thickening

phenomena, Newtonian plateau regions at low and high Péclet numbers, positive (and
negative) first normal stress differences, and negative (and positive) second normal stress
differences. For Pe = 0 , multiple equilibrium states exist for 4.72 S U S 5.00 . For
Pe > O and initial directors located in the flow-deformation plane, the predominant
feature for U < 25 is the existence of a unique nematic-like microstructure with a steady
alignment of the director that becomes completely aligned with the velocity as Fe —) 00.
For A < l and U > 25, tumbling and wagging of the director occur at low to moderate
values of the Péclet number. If the initial director has a component in the direction of the
vorticity, then director kayaking and director log-rolling may occur. The coexistence of
stable anisotropic states (or texture) predicted by the model may provide an explanation

of why micro defects occur during the processing of some structured fluids.

To My Family for Love, Support, and Patience

iv

ACKNOWLEDGEMENT

First of all, I would like to thank Professor Charles A. Petty for his support and
encouragement throughout my Ph.D. program. I truly believe that I would not have
completed my program without him. He has not only inspired me in many aspects from
an academic perspective, but has also given me personal guidance in my twenties. His
suggestion to participate in two internships helped me greatly. The NSF summer
program in Japan helped me broaden my world perspective. The summer internship at
Bechtel National, Inc. inspired me to become a better engineer.

In addition, I want to acknowledge Professors Peter W. Bates, Andre Bénard,
Krisnamurthy Jayaraman, and Michael E. Mackay for serving on my dissertation
committee. I would also like to recognize Professor Jun-ichi Takimoto fi'om Nagoya
University (now at Yamaguchi University) for giving me guidance during my research. I
also want to recognize my gradate school colleagues Dr. Steven Parks, Chinh Nguyen,
Hemant Kini, and Dilip Manda] for the inspiration of this research. I thank also Jon
Berkoe, Kelly Knight, Lin Lorraine, and Brigette Rosendall of Bechtel National, Inc for
their wonderful support.

I gratefully acknowledge financial support of this work by the National Science
Foundation and by Michigan State University.

Finally, I want to express a very special thanks to my wife Seiyoun Jung and my

parents for unlimited love and moral support.

TABLE OF CONTENTS

TABLE OF CONTENTS ................................................................................................... vi
LIST OF TABLES ............................................................................................................. ix
LIST OF FIGURES ............................................................................................................ x
NOTATION ..................................................................................................................... xvi
Chapter
1. INTRODUCTION
1.1 Motivation .......................................................................................................... l
1.2 Background ........................................................................................................ 4
1.3 Objective .......................................................................................................... 14
1.4 Outline .............................................................................................................. 16

2. SMOLUCHOWSKI EQUATION FOR RIGID ROD SUSPENSIONS

2.1
2.2
2.3
2.4
2.5

Introduction ...................................................................................................... 17
Jeffery’s Model for Rotary Convection: Tumbling Coefficient ...................... 18
Brownian Motion: Rotary Diffusivity ............................................................. 20
Excluded Volume Phenomena: Maier-Saupe Potential ................................... 21
Discussion ........................................................................................................ 23

3. MOMENTS OF THE ORIENTATION DISTRIBUTION

3.1
3.2
3.3
3.4
3.5

Introduction ...................................................................................................... 24
Orientation Dyad: Structure Tensor and the Order Parameter ......................... 26
Orientation Tetrad: Symmetry and Contraction Properties ............................. 28
Realizable Anisotropic States: Invariant Diagram ........................................... 29
Discussion ........................................................................................................ 32

4. EQUATIONS FOR THE MICROSTRUCTURE AND THE STRESS

4.1
4.2
4.3
4.4
4.5

Introduction ...................................................................................................... 33
Dynamic Equation for the Orientation Dyad ................................................... 33
Dynamic Equations for the Structure Invariants for Pe = O ............................. 34
Algebraic Equation for the Stress .................................................................... 35
Discussion ........................................................................................................ 36

5. CLOSURE FOR THE ORIENTATION TETRAD

5.1
5.2
5.3

Introduction ...................................................................................................... 38
Closure Models ................................................................................................ 40
Discussion ........................................................................................................ 44

6. REALIZABLE CLOSURE

6.1
6.2
6.3
6.4
6.5

Introduction ...................................................................................................... 45
Realizable Isotropic, Planar Isotropic, and Nematic States ............................. 46
Realizable Prolate and Oblate States ............................................................... 50
Realizable Planar Anisotropic Boundary ......................................................... 51
Discussion ........................................................................................................ 53

7. MICROSTRUCTURE IN THE ABSENCE OF AN EXTERNAL FIELD

7.1
7.2
7.3
7.4

Introduction ...................................................................................................... 56
Biphasic Phenomena ........................................................................................ 57
Relaxation to Isotropic and Anisotropic Steady States .................................... 62
Discussion ........................................................................................................ 65

8. MICROSTRUCTURE INDUCED BY HOMOGENEOUS SHEAR

8.1
8.2
8.3
8.4
8.5

8.6

Introduction ...................................................................................................... 72
Relaxation of Planar Anisotropic States for L/d = 00 ....................................... 73
Relaxation of Planar Anisotropic States for L/d g 12 ...................................... 76
Relaxation of Planar Anisotropic States for O S L/d < oo ................................. 93
The Effect of Tube Dilation on the Relaxation of Planar Anisotropic

States ................................................................................................................ 96
Discussion ...................................................................................................... 100

vii

9. VISCOSITY AND NORMAL STRESS DIFFERENCES

9.1 Introduction .................................................................................................... 102
9.2 Rheological Properties: L/d = 00 .................................................................... 103
9.3 Rheological Properties: L/d s 12 .................................................................. 117
9.4 Rheological Properties: 0 _<. L/d < oo .............................................................. 129
9.5 Rheological Properties: Effect of Tube Dilation ........................................... 133
9.6 Discussion ...................................................................................................... 143
10. CONCLUSIONS ..................................................................................................... 145
1 1. RECOMMENDATIONS ........................................................................................ 154
APPENDICES
APPENDDC A. Derivation of Moment Equations with Structure Tensor ...................... 161
APPENDIX B. Derivation of Moment Equations in Invariant Form ............................. 165
APPENDIX C. Normal Vectors in the Invariant Diagram ............................................. 169
APPENDIX D. Derivation of Various Closure Analysis in the Invariant Form ............ 172
APPENDIX E. Eigenvalues and Eigenvectors of the Orientation Dyad ........................ 182
APPENDIX F. Eigenvalues and Eigenvectors of the Orientation Dyad ........................ 189
APPENDD( G. Computational Code for Transient Calculations ................................... 189
LIST OF REFERENCES ................................................................................................. 199

viii

LIST OF TABLES

Table 7.1 Invariants and Order Parameter of the Equilibrium Structure Tensor on the
Prolate Line ........................................................................................ 63

Table 7.2 Invariants and Order Parameter of the Equilibrium Structure Tensor on the
Oblate Line ....................................................................................................... 64

Table 8.1 Invariants of the Equilibrium Anisotropic Tensor for Different Tumbling
Parameters (F SQ-model; FTD = l; U = 0) ........................................................ 77

Table 8.2 Effect of l on the Tumbling Period
(ESQ-model; FTD = 1; U = 27; Pe = 10) .......................................................... 98

Table 9.1 Viscous and Elastic Contributions to the Shear Viscosity and the First and
Second Normal Stress Differences at Selected Values of U and Pe for
L/d = 00 ........................................................................................................... 1 13

Table 9.2 Viscous and Elastic Contributions to the Shear Viscosity and the First and
Second Normal Stress Differences at Selected Values of U and Pe for
L/d g 12 .......................................................................................................... 124

Table 9.3 Legend for Figures 9.19 — 9.23 ....................................................................... 135

LIST OF FIGURES

Figure 1.1 Anisotropic Orientation States (Parks, 1997) ..................................................... 3

Figure 1.2 Molecular Structure of LPC (Larson, 1999) ....................................................... 6

Figure 1.3 Schematic of NMR Spectra for PBLG in DMF
(Abe and Yamazaki, 1987a; Robinson, 1966) ................................................... 8

Figure 3.1 Instantanous Orientation Vector Directed Along the Molecular Axis of a Rigid

Rod ................................................................................................................ 25
Figure 3.2 Three Orientation States for the Orientation Dyad .......................................... 27
Figure 6.1 Closure Coefficient C2 ( IIIb) for the Realizable FSQ-model ........................ 54

Figure 6.2 The Excluded Volume Potential for the Decoupling Approximation and the
FSQ-model ....................................................................................................... 55

Figure 7.1 Multiple Equilibrium States Predicted by the FSQ-model for C2 = constant
......................................................................................................................... 59

Figure 7.2 Multiple Equilibrium States Predicted by the FSQ-model for C2 = C2 (IIIb)

........................................................................................................................ 61

Figure 7 .3 Relaxation of the Microstructure due to Rotary Brownian Diffirsion
(U=0; i=1/(6D§)) ....................................................................................... 66

Figure 7.4 Effect of Excluded Volume on the Relaxation of the Microstructure
(F SQ-model ; FTD = l) ................................................................................... 67

Figure 7.5 The Effect of Tube Dilation on the Relaxation of the Microstructure
(F SQ-model; U = 3; initial conditions: 11b (0) = 2/9; IIIb (0) = 0). ............... 68

Figure 7.6 The Effect of U and Tube Dilation on the Characteristic Relaxation Time tc
(F SQ-model; (1(0) = 0.9). ................................................................................. 69

Figure 8.1 The Effect of U and Pe on the Steady State Director Angle for L/d = 00
(FSQ-model; FTD = 1; A = 1) .......................................................................... 74

Figure 8.2 The Effect of U and Pe on the Steady State Microstructure for L/d = 00
(FSQ-model; FTD = 1; 7» = 1) .......................................................................... 75

Figure 8.3 Instantaneous Microstructure for Director Tumbling for L/d s 12
(F SQ-model; FTD = l; I» = 0.987; U = 27, Pe = 10;
2<pp>(0)=§y_ey+gz§z) ............................................................................ 78

Figure 8.4 Microstructure for Director Tumbling in the Phase Plane for L/d s 12
(F SQ-model; FTD = 1; 7t = 0.987; U = 27, Pe = 10;
2<pp>(0)=gygy+ez§z) ............................................................................ 79

Figure 8.5 Instantaneous Microstructure for Director Wagging for L/d E 12
(F SQ-model; Fm = 1; X = 0.987; U = 27, Pe = 24;
2<pp>(0)=§ygy+e_zgz) ............................................................................ 81

Figure 8.6 Microstructure for Director Wagging in the Phase Plane for L/d s 12
(F SQ-model; FTD = 1; A = 0.987; U = 27, Pe = 24;
2 < 22 > (0) =§y9y +5225:z

......................................................................................................................... 82
Figure 8.7 Instantaneous Microstructure for Director Steady Alignment for L/d s 12

(F SQ-model; Fm = l; l = 0.987; U = 27, Fe = 95;

2<pp>(0)=ey_ey+gzgz) ............................................................................ 84

Figure 8.8 Microstructure for Director Steady Alignment in the Phase Plane for L/d a 12 A
(F SQ-model; Fm = 1; A. = 0.987; U = 27, Pe = 95;
2<pp>(0)=ey_e_y+gzgz ............................................................................ 85

Figure 8.9 Phase Diagram of U and Pe for L/d s 12
(FSQ-model; Fm =1;l=0.987; 2<_'p_p>(0)=§;ye_y +9292) .................... 86

Figure 8.10 Instantaneous Director Log—rolling for L/d 5 12
(F SQ-model; U = 27; Pe = 95; Fm = l; X = 0.987;
2<pp>(0)=_e_x§x+gz_ez) ............................................................................ 87

Figure 8.11 Director Log-rolling in the Phase Plane L/d E 12
(F SQ-model; U = 27; Pe = 95; Fm = 1; X = 0.987;
2<pp_>(0)=§x§x +2292) ............................................................................ 88

Figure 8.12 Phase Diagram of Shear-voriticity Plane Initial Condition L/d s 12
(FSQ-model;F1-D =1;7t=0.987; 2<pp>(0)=§xgx+§zgz) ................... 90

Figure 8.13 Instantaneous Director Kayaking L/d s 12
(F SQ-model; U = 27; Pe = 95; Fm = 1; 7L = 0.987) ...................................... 92

Figure 8.14 Microstructures for Director Tumbling and Steady State Alignment in the
Phase Plane

(FSQ-model; FTD = 1;)» = 0.5; U = 27, Pe = 95;
2<p_p>(0)=§y§y +£2.92) ............................................................................ 94

Figure 8.15 Microstructures for Director Tumbling and Steady State Alignment in the
Phase Plane

(FSQ-model; Fm =1;X=0;U=27,Pe=95;
2<pp>(0)=§y_e_y+§z_ez ............................................................................ 95

xii

Figure 8.16 Microstructure for Director Tumbling and Steady State Alignment in the
Phase Plane

(FSQ-model;F1-D =1;}t = 0; U= 0, Pe=95;
2<pp>(0)=§y§y+§z§z) ............................................................................ 97

Figure 8.17 The Effect of Tube Dilation on the Phase Diagram

(FSQ-model; Fm = 1; A= 0.987; 2<pp>(0)=e_y +£7.92 .................... 99

9y

Figure 9.1 The Effect of U and Pe on the Shear Viscosity for L/d = 00
(FSQ-model;Fm= 1; 1= 1) .......................................................................... 104

Figure 9.2 The Effect of U and Pe on the Viscous and Elastic Components of the Shear
Viscosity for L/d = 00

(FSQ-model; Fm = 1; A = 1). ........................................................................ 106

Figure 9.3 The Effect of U and Pe on the Brownian and Excluded Volume Contributions
to the Shear Elastic Component of Viscosity for L/d = 00
(FSQ-model;F1-D= 1; 7t= 1) .......................................................................... 108

Figure 9.4 Elastic Contributions to the Shear Stress
(FSQ-model; 7» =1; Fm =1) ........................................................................ 110

Figure 9.5 The Effect of U and Pe on the First Normal Stress Difference for L/d = 00
(FSQ-model; Fm = 1; k= 1). ..................................................................... 112

Figure 9.6 The Effect of U and Pe on the Second Normal Stress Difference
for L/d = 00

(FSQ-model;Fm= l;7\.= 1) .......................................................................... 115

Figure 9.7 The Contributions of Stress on the Time Averaged Normal Stress Differences
(FSQ-model; FTD =1;X=1;U=27). ........................................................ 116

Figure 9.8 The Effect of Tumbling Parameter on the Shear Viscosity for U = 0
(FSQ-model; Fm = 1). ................................................................................ 118

xiii

Figure 9.9 The Instantaneous Shear Viscosity for Director Tumbling
(FSQ-model; Fm = 1; k = 0.987; U = 27, Pe = 10). .................................... 119

Figure 9.10 The Effect of U and Pe on the Time Averaged Shear Viscosity for L/d 5 12
(FSQ-model; Fm = 1; x = 0.987) ................................................................. 121

Figure 9.11 The Contributions of Viscous and Elastic Stresses on the Time Averaged
Shear Viscosity for L/d E 12
(F SQ-model; FTD = 1; A = 0.987; U = 27). ................................................. 123

Figure 9.12 The Instantaneous First Normal Stress Difference for Director Tumbling
(FSQ-model; Fm = 1; A = 0.987; U = 27, Pe = 10). .................................... 125

Figure 9.13 The Effect of U and Pe on the Time Averaged First Normal Stress
Difference for L/d E 12

(FSQ-model; Fm = 1; A = 0.987). ............................................................... 126

Figure 9.14 The Instantaneous Second Normal Stress Difference for Director Tumbling
with L/d 5 12
(F SQ-model; Fm = 1; k = 0.987; U = 27, Pe = 10). .................................... 128

Figure 9.15 The Effect of U and Pe on the Time Averaged Second Normal Stress
Difference for L/d s 12

(FSQ-model; Fm = 1; A = 0.987) ................................................................. 130

Figure 9.16 The Effect of Tumbling Parameter on the Time Averaged Shear Viscosity
(FSQ-model; Fm =1;U=27,Pe=10). ...................................................... 131

Figure 9.17 The Effect of Tumbling Parameter on the Time Averaged First Normal Stress
Difference

(FSQ-model; Fm = 1; U = 27, Pe = 10). ...................................................... 132

Figure 9.18 The Effect of Tumbling Parameter on the Time Averaged Second Normal
Stress Difference

(FSQ-model; Fm = 1; U = 27, Pe = 10). ...................................................... 134

xiv

Figure 9.19 The Effect of Tube Dilation on the Time Averaged Shear Viscosity
(F SQ-model; it = 0.987; U = 27 see Table 9.3 for legend). .......................... 137

Figure 9.20 The Contributions of Stress on the Time Averaged Shear Viscosity
(FSQ-model; F196" in qu.(4.1) and (4.6); 7. = 0.987; U = 27). ................... 138

Figure 9.21 The Effect of Tube Dilation on the Time Averaged First Normal Stress
Difference

(F SQ-model; 7L = 0.987; U = 27 see Table 9.3 for legend) ............................ 139

Figure 9.22 The Effect of Tube Dilation on the Time Averaged Second Normal Stress
Difference

(F SQ-model; 7» = 0.987; U = 27 see Table 9.3 for legend). .......................... 141

Figure 9.23 The Contributions of Stress on the Time Averaged Normal Stress Differences
(FSQ-model; F193 ; 7. = 0.987; U = 27). ...................................................... 142

Figure 10.1 Validation of the FSQ-closure ...................................................................... 147

Figure 11.1 The Effect of Particle Aspect Ratio on the Dimensionless Friction
Coefficient ...................................................................................................... 156

Figure 11.2 Multiple Stable Steady States for Various Initial Orientation Dyads
(U = 27, Pe = 95, A = 0.987) .......................................................................... 158

XV

IIII>

"w

AijBkz

FSQ

I")

NOTATION
Second order tensor, or dyadic-valued operator
Second order tensor, or dyadic-valued operator

Index notation for tetradic-valued operator for dyadic valued

operators A and 2, equivalent to aibjcddl

Diameter of tube in tube dilation effect

Anisotropic structure tenser

Closure coefficient for FSQ-model

Number of rigid rods in unit volume

Rotary diffusion coefficient (units: 1/time)
Rotary diffusion coefficient in dilute solution

Diameter of rigid rod

Unit vector in Hb and H11, direction
Unit vector in the radius direction
Unit vector in the vorticity direction

Unit vector in the cross-flow direction
Unit vector in flow direction
Tube dilation effect

Tube dilation effect in Doi’s model
Fully symmetric quadratic

Invariant form of dynamic moment vector in the invariant space

xvi

Dimensional tumbling frequency

Dimensionless tumbling frequency
Dimensional wagging frequency
Dimensionless wagging frequency
Hinch and Lea] l closure

Hinch and Lea] 2 closure

Unit dyadic tensor
First invariant of 2

Index notation of unit dyadic tensor

Second invariant of 2
Third invariant of E

Boltzmann constant

Length of rigid rod

Number of species
Dimensionless first normal stress

Dimensional first normal stress

Dimensionless second normal stress

Dimensional second normal stress

Newtonian plateau

Normal vector

xvii

I'd

I'U'

Unit vector direct along rigid rod axis
Unit vector in rigid rod rotation
Neighboring rigid rod vector respect to p

Convective flux of rigid rod rotation

The first order moment
Components of unit vector direct along rigid rod axis

Component of unit vector direct along rigid rod axis, respect to

eigenvector x1, x2, and x3

Orientation dyad

The second order moment

Index notation of orientation dyad
Orientation tetrad

The fourth order moment

Péclet number

Three spherical coordinate axes

Dimensional strain rate tensor
Dimensionless strain rate tensor
Symmetry operator of arbitrary dyadic valued operator A and __B

Temperature

Dimensional time

xviii

Dimensionless time

tt Dimensional tumbling period
tt Dimensionless tumbling period
U Nematic potential coefficient
U Critical nematic potential coefficient of the phase transition
U1 Upper critical nematic potential coefficient for biphasic region
U2 Lower critical nematic potential coefficient for biphasic region
Q Fluid velocity
uz Scalar valued fluid velocity in flow direction
l Dimensionless vorticity tensor
)9, 2&2, x; Eigenvectors associated with 11,1, kpz, 21,3, respectively.
x, y, 2 Three Cartesian coordinate axes
z Arbitrary vector
Other Symbols
(1 Degree of orientation order parameter, structure parameter
)2 Motion of the suspension with velocity
6 Jaumann derivative
An Polymer contribution shear viscosity
A110 Zero-shear rate viscosity
At Change of time step
AUMs Maier-Saupe excluded volume potential

xix

AUo

AUMS

:3)

1'lE
TlS

nv

k
Ale~pzfl~p3

1b1,1b2,lbs,

0

"rt

123

I'd

Id

lie-l

Onsager nematic potential
Maier-Saupe nematic potential
Excluded volume potential

Angle between the projection of molecular axis on the plane [x1,

X3]
and x1 axis

Strain rate

Total shear viscosity
Elastic contribution of shear viscosity
Solvent contribution of shear viscosity

Solvent contribution of shear viscosity

Tumbling parameter

Three eigenvalues of < pp >

Three eigenvalues of lg)

Angle between rigid rod and y-direction

Total stress tensor

Elastic contribution of shear stress
Elastic contribution of stress tensor
Solvent contribution of stress tensor

Viscous contribution of stress tensor

Orientation density function

Q

Q)
lfi)

Subscript

MAC. 3

MS

Viscous drag coefficient

Gradient

dimensional velocity gradient
dimensionless velocity gradient
Ensemble average

Function of symmetry operator

Hand’s first order closure

Second order quadratic closure

Dimensional quantity

Structure tensor component
Convective contribution

Elastic contribution

Vector indices

Maier-Saupe

Body rotation

Three spherical coordinate axes
Solvent contribution

Tube Dilation

Viscous contribution

Three Cartesian coordinate axes

Material frame of reference

(1.13.7

Superscript
E

S

Vector indices

Elastic contribution

Solvent contribution

Viscous contribution

CHAPTER 1

INTRODUCTION

1.1 Motivation

The statistical theory of rigid rod suspensions provides a means for understanding
the microstructure and rheology of structured fluids (Doi and Edwards, 1978a, 1978b;
Doi, 1981). The microstructure may be either isotropic or anisotropic. Under extreme
conditions, a nematic phase may occur wherein the long axis of rod-like particles (or
molecules) align in the same direction. The tendency for some fluids to develop nematic-
like microstructures either spontaneously or under the influence of an external force field
has a significant and practical impact on the rheological, optical, and material properties
of structured fluids.

Particle suspensions and liquid crystalline polymers may be either isotropic or
anisotropic, depending on the local environment. The microstructure is often
characterized statistically by low-order moments of the orientation distribution function,
referred to hereinafter as the orientation dyad (second order tensor) and the orientation

tetrad (fourth order tensor). The orientation dyad, < p p > , is symmetric and non-
negative (i.c., realizable). The orientation vector B has unit length and is aligned with

the principal axis of an axisymmetric ellipsoidal particle, cylindrical rod, or disk-like

particle. The eigenvalues of <22 > are real and non-negative (i.c.,
0 < 1p] < )‘pZ < 11,3 < 1). The “director” of the microstructure is defined as the

eigenvector associated with the largest eigenvalue of the orientation dyad. For an

isotropic material, the director has no preferred direction inasmuch as all the eigenvalues

of <pp> are the sameOtpl = A = 11,3 =1/3).

p2

Figure 1.1 illustrates the type of possible anisotropic states that can occur for
axisymmetric ellipsoidal suspensions (see Kini et al., 2003; Nguyen et al., 2001a; Parks
et al., 1998; Petty et al., 1999; Weispfennig et al., 1999). Each orientation state is

parameterized by two nontrivial invariants of the structure tensor l_) (IIb = hf; Q) and

1111, = egg-2) ). The b-operator is defined as the anisotropic component of the
orientation dyad (2=(<pp>—%l ). As noted on Figure 1.1, three-dimensional

anisotropic states for which 0 < )‘p1 = 11,2 < 11,3 < 1 have quadratic forms with prolate

surfaces (F -boundary) whereas three-dimensional anisotropic states for which

0 < hp] < lpz = 11,3 < 1 have quadratic forms with oblate surfaces (D-boundary). In

the absence of external hydrodynamic forces, all stable and unstable equilibrium states
are either on the prolate boundary or on the oblate boundary of the invariant diagram
(Doi and Edwards, 1986; Kini, 2003).

Two-dimensional planar anisotropic states are located on the B-boundary of
Figure 1.1. These states are associated with structure tensors with one eigenvalue equal

to zero and two unequal positive eigenvalues (lpl = 0, 11,2 ¢ 11,3 , 11,2 + Ap3= 1).

Two-dimensional planar isotropic states (Point C on Figure 1.1) have one zero eigenvalue

and two equal eigenvalues (lpl = 0, 11,2 = 11,3: 1/2). A fully-aligned microstructure

forms an ideal nematic phase with 11 = 0, 7&2 = 0, and 7.3 = 1 (Point A on Figure 1.1).

A

q
It’c?
lug

C. Planar Isotropic

\.

 

A. Nematic 2/3

2/9

B. Planar Anisotropic

F. Prolate

Set of Realizable Orientation States

' 2/9

 

 

 

 

 

 

 

 

 

 

 

 

 

. - 1
1/6 13 =< p p > ——l
—l/3 . 3
D. Oblate
HI = . b - b
E. Isotropic b g “(2 = =)
Invarinats of 2 Eigenvalues of < pp >
Orientation State _-
IIb IIIb M1 7~p2 7ups Notes
A Nematic 2/3 2/9 0 0 1
B. Planar Anisotropic 11b = 2/9 +ZIIIb 0 21,2 1 - 21,2 Ap2=[0,1/2]
c. Planar Isotropic 1/6 -l/36 0 1/2 1/2
D. Oblate 11b = 6(—IIIb/6)2/3 1—2xp2 rpz xpz rpz=[1/3,1/2]
E. Isotropic o 0 1/3 1/3 1/3
F. Prolate 11;, = «rub/6)”3 xpl xpl 1—2xp1 xp1=[o,1/3]

 

 

 

 

 

 

 

Figure 1.1 Anisotropic Orientation States (Parks, 1997)

Microstructures associated with axisymmetric suspensions must fall either on the

boundaries or within the bounded region of the (Hb , 1111, )-plane identified by Figure 1.1.

Microstructures outside this domain are unphysical because at least one of the
eigenvalues of the orientation dyad is negative.

The realizability domain defined by Figure 1.1 stems directly from the algebraic
properties of real, symmetric, non-negative operators and is a fundamental characteristic
of any second-order moment of a distribution function. Appendix A shows how the
boundaries depend on the invariants of the structure tensor. This model-independent
result places an important theoretical constraint on allowable models for the orientation
dyad. A practical consequence of Figure 1.1 is that it provides a means to identify a
closure model for the orientation tetrad in terms of the orientation dyad, i.e.,

< pppp > = S(< pp >). This closure is developed in CHAPTER 5 and 6 below.

1.2 Background

The primary objective of this research is to examine the influence of the low-order
moments of the orientation distribution (microstructure) on the equilibrium and
rheological properties of rigid-rod suspensions. Liquid crystalline polymers (LCPs), such
as poly (g-benzyl-L-glutamate) in m-cresol and hydroxypropylcellulose in water are often
represented as rigid-rod suspensions with a characteristic length L ~ 110 nm and a
characteristic diameter (1 ~ 1.16-1.75 nm (Bibe and Armstrong, 1988; Larson, 1999;
Walker and Wagner, 1994; Yousefi et al., 2003). The stiffness of LCPs stems from the

presence of aromatic rings in the backbone of the polymer or from the a-helix structure

due to hydrogen bonding (see Figure 1.2). The significant decrease in the shear viscosity
of thermotropic and lyotropic liquid crystalline polymers (LCPs) during processing
makes these materials commercially attractive. Specific end uses of LCPs exploit their
low elongation resistance to cutting, favorable thermal properties, high resistance to wear,
and high-strength, low-weight, and high-irnpact resistance (Collyer, 1992). The tensile
moduli of LCPs in the solid phase may vary between 1-100 GPa, depending on the
molecular orientation of the constituent polymers (Donald and Windle, 1992).
Applications of LCPs are numerous and range from reinforced bulletproof vest to optical
components in electrical devices (Collyer, 1992).

Liquid crystalline polymers are generally manufactured by a stepwise
polycondensation reaction in either a batch or a continuous process (J ansson, 1992). The
polymer is mixed with various additives and extruded as a filament. Over half of the
LCPs sold are reinforced with 30% — 40% glass fillers having polymeric sizing to
produce a strong interface between the fiber and the matrix material (Clarke et al., 1997).

Some LCPs are injection molded for special applications.

Equilibrium Microstructure

In the absence of an external field, a rigid rod suspension has an isotropic
microstructure (i.e., IIb= 0 and 1111, = 0) at low concentration. As the concentration
increases to a critical value, the microstructure undergoes a spontaneous transition to an

anisotropic nematic-like state. Several methods have been developed to study this

transition experimentally. For example, Robinson (1966) developed a birefiingence

H

o
+C O—(IJ—N ,R = -[—CH2—CH21—|Cl—?

CH2
R
\
poly (y—benzyl-L-glutamate), PBLG oc-helix l /
i d/2

 

 

3D view of PBLG Rigid Rod

Figure 1.2 Molecular Structure of LCP (Larson, 1999)

technique to study biphasic phenomena (i.e., coexistence of isotropic states and
anisotropic states at the same concentration) for poly-y-benzyl-L-glutamate (PBLG) in
dioxane solutions. This phenomenon was observed for PBLG aspect ratios from 10 to
100 and has been reported by other investigators using other methods (see Abe and
Yamazaki, 1989b; Kubo and Ogino, 1979; Murthy et al., 1976; Sartirana et al., 1987).
Abe and Yamazaki (1989a) developed a NMR technique to correlate the relative
orientation of the (rt-helical backbone of PBLG rigid-rod molecules by exploiting a
quadrupolar splitting phenomena related to the pendant side chain containing C-D and
N-D bonds. If the PBLG solution is isotropic, the NMR spectrum has only one resonance
peak. As the concentration increases and the microstructure approaches the biphasic
region, quadrupolar splitting occurs. The split increases as the fluid becomes more
anisotropic. In the biphasic region, the central peak corresponds to an isotropic
microstructure and the split signal corresponds to a nematic-like microstructure. As the
concentration increases firrther, the isotropic peak disappears while the peak-to-peak
distance in the split increases. Figure 1.3 illustrates the observations reported by Abe and
Yamazaki for PBLG in DMF (dimethylformamide) and in 1, 4-dioxane with aspect ratios

of32,121, and 185.

Non-Equilibrium Microstructure

In a time independent external field, director tumbling of LCP solutions may
occur at low shear rates, but direct measurements using optical methods are difficult.
However, optical measurements of this phenomenon for lower molecular weight liquid

crystal (LC) suspensions have been reported extensively (Bedford and Burghardt, 1996;

Goa .aomosom 352 .%§§> poo 23
”95 a 3mm é «€on Ezzuo ouoaooom 3 para

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5:26on
_ _ l noumbzooaoo awn; 3380 Z
295823 ,\ -
.7
. < < - m
9
22235 |< < < m 235
- ,M
oaobofl < 1
:osfifiooqoo Boa 88m oaobofl

 

 

 

 

 

 

Burghardt and Fuller, 1991; Fuller, 1995; Larson, 1999). Rheological measurements on
LCP solutions and melts have been used to study director tumbling. Erickson and others
(see p. 453-456 in Larson, 1999) related this phenomenon to a phenomenological
tumbling parameter A that couples the angular motion of the rigid rod to the strain rate of
the flow field (see Eq.(10-3) p. 448 in Larson, 1999). For homogeneous shear flows and |
X | > 1, the director attains a steady alignment relative to the flow direction in the shear
(or deformation) plane (i.e., flow/cross-flow plane). On the other hand, if | 7t | < 1, the
director rotates continuously in the shear plane (see p. 454 in Larson, 1999; Carlsson,
1982; Carlsson and Skarp, 1986). Because A is related to rheological Leslie-Ericksen
coefficients, tumbling phenomenon has been studied indirectly for more than thirty years
by measuring fluid properties of the suspensions (see Cladis and Torza, 1975; Gahwiller,
1973; Pieranski and Guyon, 1974; Skarp et al., 1981).

Skarp et al. (1981) (also see Carlsson and Skarp, 1986; Clark et al., 1981)
measured the Leslie-Ericksen coefficients for 4-n-octyl-4’-cyano-biphenyl (a
thermotropic liquid crystal) over a range of temperatures (35 — 40 °C). They used an
electromagnetic field to initially align the orientation director parallel and perpendicular
to the flow direction. After removing the magnetic field, the anisotropic microstructure
relaxes and the LE-coefficients were measured. The rheological data indicated that the
magnitude of the tumbling parameter was less than unity, which implies director
tumbling according to Ericson’s theory. This approach has also been applied to infer
director tumbling in LCP solutions subjected to homogeneous shear flows with limited

success (see Burghardt and Fuller, 1990; Larson, 1988).

Rheological Properties

The rheological characteristics of LCPs are important indicators of molecular
orientation because time-dependent molecular conformation is strongly coupled with the
flow (Walker et al., 1995). Some LCPs in simple shear flows show shear viscosity with
strain rate (or stress) response curves with three distinct characteristics: 1) a shear
thinning region at low strain rates (Region I); 2) a Newtonian plateau (Region II); and, 3)
an additional shear thinning region at high strain rates (Region 111) (Walker and Wagner,
1994; Walker et al., 1995; Larson, 1999). Region 11 occurs for a wide range of strain
rates because the molecular orientation and the conformation of the rigid-rod polymer
solution are maintained. Shear thinning occurs at higher strain rates because the flow
field distorts the microstructure by flow alignment. Clearly, flow alignment enhances the
relative motion between phases (i.e. translational diffusion) with the result that the shear
viscosity decreases (i.e., shear thinning).

An anomalous shear-thinning region at low strain rates has only been observed for
LCPs. However, not all LCPs show a Region I behavior. Walker and Wagner (1994)
have shown that (1,4-phenylene-2,6-benzobisthiazole) (PBZT') has a Region I response
only at relatively high concentrations. This phenomenon has not been firlly characterized
experimentally because of inaccurate shear stress measurements at low strain rates (see
Doraiswamy and Metzner, 1986; Larson, 1999; Walker et al., 1995). In addition, no clear
theoretical explanation for this phenomenon has been identified.

Another interesting characteristic of LCP solutions is the occurrence of a negative

first normal stress difference (N1) at intermediate strain rates (see Baek et al., 1993;

Chono et al., 1996; Kiss and Porter, 1980; Larson, 1999; Magda et al., 1991). At low

10

strain rates, N1 is positive; however, as the strain rate increases, N1 attains a maximum
value and then decreases to zero and becomes negative. At higher shear rates, the first
normal stress difference becomes positive again. This observation suggests that N1 is

sensitive to changes in the microstructure as the strain rate increases. Beck et a1. (1993)

have suggested that the transition from positive to negative values of N1 was associated

with the orientation director changing from a stable periodic tumbling state to a stable
periodic wagging state. This is consistent with the Leslie-Ericksen theory, which requires
director tumbling for negative first normal stress differences (also, see p. 449 in Larson,
1999; Burghardt and Fuller, 1990).

In addition to the negative first normal stress difference, direct oscillatory
response of the viscosity also indicates molecular tumbling phenomenon. When the rate
of shear is suddenly changed, the shear stress component of the deviatoric molecular
stress shows an oscillatory response including a reversal in the strain rate (Burghardt and
Fuller, 1991; Picken et al., 1991; Vermant et al., 1994; Walker et al., 1995). This
relaxation response has multiple overshoots and undershoots that can be imposed with
various shear rates before the steady state is reached against strain. Since this type of
response is independent of strain rate, it must be due to changes in the microstructure
rather than the flow properties. In addition, there is only one overshoot when the
microstructure relaxes to a steady flow alignment state. These experimental observations
support the hypothesis that multiple stress oscillations and director tumbling are

correlated (Burghardt and Fuller, 1991; Larson, 1999).

ll

Theoretical Studies

Liquid crystalline polymers have stiffness characteristics that are different from
other polymers. LCPs have been studied for many years. Although “industria ” LCPs
are not as rigid as “laboratory” LCPs, understanding the behavior of rigid rod suspensions
would nevertheless provide valuable information and insight related to processing LCPs.

Theoretical studies of LCP orientation phenomenon are primarily related to the
Smoluchowski’s (S-) equation. The S-equation governs the distribution of orientation
states. It is a partial differential equation that balances the accumulation of states subject
to rotary convection and rotary diffusion in orientation space (see CHAPTER 2 below).
The diffusive flow has two contributions: Brownian motion and the excluded volume
phenomenon. Brownian motion tends to mix the LCP molecules randomly whereas the
excluded volume effect tends to align the LCP molecules. The convective flux arises due
to the torque on the LCP molecules in a shear field.

There are several ways to study the S-equation. One approach is to develop a
solution using spherical harmonics (Chaubal and Leal, 1997, 1999; Larson, 1990).
Another approach is based on the method-of-moments. Developing an explicit expansion
for the density function is a complicated process and requires significant computational
resources. On the other hand, developing a solution based on low order moments of the
density function requires a closure approximation (Marrucci, 1996).

Doi (1981) developed a second order moment equation by integrating the S-
equation with Maier-Saupe potential for the excluded volume. Doi used a quadratic

closure approximation for the orientation tetrad (i.e., < p p p p > = < p p > < p p > ).

Unfortunately, these approaches do not retain the six-fold symmetry and six-fold

12

contraction properties of the fourth order moment (see CHAPTER 3 and the development

in CHAPTER 5).

Hand (1962) used a first order closure approximation for < p p p p > , which

satisfies six-fold symmetry and six-fold contraction. However, Hand’s closure is limited
to microstructures near the isotropic state (see E on Figure 1.1). Later, Hinch and Leal
(1976) introduced two closure approximations based on an anisotropic analysis of the
orientation tetrad near the isotropic and nematic states (see Figure 1.1). The HL-closures
predict phase transition from the isotropic to the nematic state, but predict unrealizable
behavior for some situations (Chaubal and Leal, 1999).

Later, Cintra and Tucker (1995) developed a new closure for the orientation tetrad
based on an orthotropic operator. The approach assumes that the symmetry directions of
the tetrad coincide with the orientation dyad. The closure coefficients of the tetrad are
obtained by fitting the moment equation with the “exact” solution based on a spherical
harmonic expansion of the orientation density function. The orthotropic closure is
applicable to simple geometries and can provide valuable bench mark information.

There are other closures that combine previous approximations (Tucker, 1988;
Larson, 1999). For example, Tucker (1988) has combined Hand’s closure approximation
at the isotropic state and the decoupling approximation at the nematic state. This
superposition of two asymptotic closures is similar to the strategy employed by Hinch
and Leak and is an example of a hybrid closure approximation. Larson (1990) used the
decoupling closure for the excluded volume potential and the HL-closure for the
convective flux in the same model. None of the foregoing closure approximations satisfy

the symmetry properties of the orientation tetrad and the realizable condition on the

13

second order moment. In addition, none have firlly predicted the microstructure and
rheological properties of LCPs.

Recent research at Michigan State University has developed a representation of
the orientation tetradic in terms of the orientation dyad that satisfies the six-fold

symmetry and the six-fold contraction properties associated with < p > (see Parks

et al., 1999; Parks and Petty, 1999a, 1999b; Petty et al., 1999; Imhoff, 2000; Nguyen,
2001; Kini, 2003; Mandal, 2004). This closure is incomplete and needs an appropriate
closure coefficient C2 (11b , HIb) so that the orientation dyad is realizable for all
conditions. Previous studies assumed that C 2 = 1/3 for all anisotropic states within the
invariant diagram (Figure 1.1). This condition must be true at the nematic state, but it is
not required elsewhere. However, it is noteworthy that C2 = 1/3 predicts biphasic
phenomena (Kini, 2003; Nguyen, 2001) and tumbling phenomena (Nguyen, 2001).
However, a “universal” value of C 2 = 1/3 causes unrealizable behavior for some

physically allowable initial conditions. This is unacceptable and a resolution of the
problem is developed in CHAPTER 6 below. Imhoff (2000) and Mandal (2004)

identified a value for C2 by using solutions of the S-equation. Their best fitted C2
value was 0.37, but this choice of C2 is also unacceptable because it yields unrealizable

results orientation dyad for certain initial conditions.

1.3 Objective

This research addresses a long-standing fundamental problem related to the

self-alignment and flow alignment of structured fluids. The approach, which builds on

14

the statistical theory developed earlier by Doi and many others (see, esp., Doi and
Edwards, 1986; Bird et al., 1987a,b; Larson, 1999) provides new insights and
understanding of the relationship between the microstructure and the phenomenological
properties of structured fluids at the continuum scale.

The objective of the research is to develop a closure for the orientation tetrad that
yields a realizable model for the orientation dyad. The new approach is used to predict
the microstructure and the rheological response of rigid rod suspensions to simple shear
flows. By using the S-equation based on Doi’s theory (1981), an equation for the second
order moment of the orientation density fimction can be developed that depends on the
fourth order moment (i.e., the orientation tetrad). Although the method-of-moments has
been employed for more than thirty years as a mean to study the microstructure of rigid
rod suspensions, understanding the equation has been limited by the absence of a
satisfactory closure model for the orientation tetrad. This is a significant theoretical
deficiency that hinders the interpretation of rheological anomalies associated with the
response of microstructure fluids to simple shear fields. To address this issue, this
research presumes that the orientation tetrad can be approximated by using an algebraic

closure, < pppp > = 3(< pp >) . The efficacy of this hypothesis will be evaluated for a

class of microstructured fluids (i.e., rigid rod suspensions) in homogeneous shear flows.
The aim of the research is to deve10p a realizable dynamic model for the orientation dyad

for a wide class of complex engineering flows.

15

1.4 Outline

The equilibrium relaxation of the orientation dyad in the absence of an external
field (CHAPTER 7) and the non-equilibrium relaxation of the orientation dyad in the
presence of an external field (CHAPTER 8) are addressed in this research. The S-
equation in orientation space (CHAPTER 2) is used to develop an ordinary differential

equation for the orientation dyad, < pp > (CHAPTER 4). The moment equation has

three physical contributions: rotary Brownian motion, excluded volume phenomenon,
and hydrodynamic interactions through particle/fluid torque. A Maier-Saupe potential is
used for the excluded volume effect and Jeffery’s model is used for the hydrodynamic
interactions (CHAPTER 2). In addition, the effect of tube dilation on the diffusive flux is
examined. Doi’s stress model is used to predict the viscosity and the normal stress

differences (CHAPTER 9).

Once the realizable closure approximation is obtained, the ordinary differential
equation is solved for the orientation dyad by using a fourth order Runge-Kutta algorithm
with a dimensionless time step less than 0.0003 (see APPENDIX G). The moment
equation has three independent dimensionless variables: U, Pe, and A. (see CHAPTER 2).
For A _<_ l, the model is used to study the microstructure and rheology of rigid rod
suspensions for a wide range of U and Pe (CHAPTER 7 and CHAPTER 8). Various
initial value problems are examined with and without homogeneous shear. In CHAPTER
3, two metrics of the microstructure are defined to evaluate the results: 1) the order

parameter a; and, the deviation of the director from the flow direction, 253 - g2 .

l6

CHAPTER 2

SMOLUCHOWSKI EQUATION FOR RIGID ROD SUSPENSIONS

2.1 Introduction

Smoluchowski’s equation (S-equation) governs the evolution of the orientation
density function ‘I’( p, t; Z (X, b) for a suspension of rigid rod particles (see p.50 in Doi
and Edwards, 1986). The fraction of particles with orientation vectors with angular
coordinates between (0, 4)) and (0 + A0, 4) + Ad>) is given by ‘I’(p, t; 2(3, D)sin9AOA¢.

In a frame of reference moving with the local velocity of the suspension, the S-equation

is a balance equation for orientation states and can be written as:

6‘? 6 .
(3)3 =-5E-(2‘I’). (2.1)

. 6 . . . . .
In the above equation, ‘6— rs a surface gradient operator on a sphere in orientation space
P

and the vector p is the angular velocity of the particle about its center of mass. The
vector X represents the spatial position of a material fluid element at some arbitrary
reference time; the spatial position of the material fluid element at time t is f; = )2 (X, t).

The vector zLX, B is the motion of the suspension with velocity fl (3, D defined by

A

it; t) Eta—i] . (2.2)
at X

The Operator on the left-hand-side of Eq.(2.l) represents the substantial (or material)

derivative of the orientation density function:

17

A

(2.3)

a:
at

 

 

 

E —T + —?
X at i at

The rotary flux of orientation states relative to a material frame of reference, p‘I’ ,

X.

can be separated into a rotary convective flux pC‘I’ and a rotary diffusive flux
lE—Ec )W;
E‘I’EEC‘I’+@-2c)‘l’- (2.4)

In this research, the rotary convective flux developed by Jeffery (1922) for ellipsoidal

particles suspended in a homogenous shear field will be used for pc‘l’ (see Section 2.2
below). Doi’s model for the rotary diffusive flux (see Section 2.3 below) will be used for

(p— p C )‘I’ . The rigid rods have the same density as the suspending fluid so gravity is

unimportant. The S-equation given by Eq.(2.1) above assumes that spatial diffusion of

the particles relative to the translational velocity is also unimportant.

2.2 Jeffery’s Model for Rotary Convection: Tumbling Coefficient

Jeffery’s model is used for the rotary convective flux (Jeffery, 1922).
Hydrodynamic drag causes the rotary motion of the suspended particles (Batchelor, 1976,

1982; Bibbo et al., 1985). A balance of angular momentum on an axisymmetric rigid rod

yields the following equation for p C (see Jeffery, 1922; Parks et al., 1999):

5p

re 5%», brawl-22H

III!»

'2 1, (2.5)

18

where 7» is a dimensionless tumbling parameter.

ll Cl»

and W are the rate-of-strain and
vorticity tensors, respectively:

§=§lf§+fitfl=§lv§+w2fl; (2.6a)
i=§l€7§-(rrfl=§[v§—(vyTl. (26»
For homogeneous shear, VQ = ye), 92 where

7: (6g):(vg)T =‘/2§:§T 423':ng =constant. (2.7)

In this research, A is only a function of the particle aspect ratio, L/d. For axisymmetric

 

particles (see Jeffery, 1922)

L
(592—1

._ 5 +1 . (2.2)
(g? +1

For large aspect ratio particles 71. i 1. For disk-like particles 7t -'—- 1. A typical tumbling
parameter for slender rod-like particles is about 0.7, which is equivalent to L/d s 2.38
(see p.280 Larson, 1999; Bretherton, 1962; Trevelyn and Mason, 1951). The rotational
period of a rigid rod can be related to the tumbling parameter (Jeffery, 1922; p.280
Larson, 1999). For )t = i 1, a prolate spheroidal particle and a disk-like oblate particle
have infinite rotation periods (i.e., they are not rotating). Experimental evidence for
particle rotation (or tumbling) in rigid rod suspensions has been given by Larson and
many others (see p.280 Larson, 1999; Anczurowski and Mason, 1967a, 1967b; Frattini

and Fuller, 1986).

19

2.3 Brownian Motion: Rotary Diffusivity

Rotary Brownian motion is an important phenomenon in particle/fluid
suspensions. This phenomenon has a direct impact on models for viscoelasticity,
diffusion, birefringence and dynamic light scattering.

The theory of rotary diffusion in concentrated suspensions is well described by

the following hypothesis (see p.294, Doi and Edwards, 1986; Parks et al., 1999):

 

as! ea ALI
'—' =—<D > —-‘I’— . 2.3
2 EC R" [op 62(kBT)] ( )

In the above equation, AU is an excluded volume potential and < DR > represents an
average rotary diffusion coefficient. In general, < DR > depends on the phenomenon of
tube dilation (see p.360 in Doi and Edwards, 1986; Kuzuu and Doi, 1993, 1994; p.520 in
Larson, 1999), the particle aspect ratio L/d, the volume fraction of particles, and the
temperature. In this research, the influence of tube dilation on the microstructure and
rheology will be examined (see CHAPTER 8 and CHAPTER 9), but most of the

applications will assume that < DR > is given by

3ka

<DR>éD§a
1mg L3

 

(29)

In the above equation, k]; and 718 represent, respectively, the Boltzmann constant and the

solvent viscosity; T is the temperature. Dfi has units of 1/(time) and represents the

rotary diffusion coefficient for dilute suspensions of rigid rods (see p.334 in Doi and

Edwards, 1986; p.281 in Larson, 1999). For semi-dilute and concentrated suspensions,

Eq.(2.9) is multiplied by a tube dilation factor Fm (< DR > = Fm D; ), which depends

2O

on the invariants of the microstructure (see Figure 1.1). In this research, FTD = 1 for

most of the applications. For the tube dilation examples (see Sections 8.5 and 9.5), Doi’s

theory for FTD is used (see p.360 in Doi and Edwards, 1986):

1

Doi
11b )

(1-

3 (2.10)

2
Note that Eq.(2. 10) implies that < DR > —» 00 as the microstructure approaches a nematic

state (see Point A on Figure 1.1).

2.4 Excluded Volume Phenomena: Maier-Saupe Potential

The excluded volume potential introduced by Eq.(2.6) above accounts for the
interaction of a rigid rod particle with neighboring particles. The main physical idea is
that particles cannot occupy the same space at the same time. This phenomenon has
important consequences that partly explain the self-alignment and the flow-induced
alignment of particles. Doi (1981) and others have developed models for the
instantaneous excluded volume potential by minimizing the Onsager free energy for rigid
rod suspensions (11g et al., 1999; Onsager, 1949) with the result that (see p.359, Doi and

Edwards, 1986):

A

kAI'JI‘ =-Upp:<pp>+{higherorderterms}. (2.11)
B _- -_

 

The above equation stems from an expansion of the second virial coefficient of the

Onsager nematic potential. The lead term is the so—called Maier-Saupe potential:

A

AUMS
kBT

 

=-Upp:<p_p>. (2.12)

21

This model is used in the Doi theory for rigid rod suspensions and is also used

hereinafter. The average of Eq.(2.12) shows that < AUMS > is proportional to the

second invariant of the orientation dyad: < AU MS > = - ngT < p p > :< pp >. With

< p p > = 1+ 2 , the average excluded volume potential can be expressed as

l
3

(AUMS >=—UkBT(%+Hb). ‘ (2.13)

Thus, at the isotropic state (i.e., Point E on Figure 1.1), < AUMS >= —-;:UkBT ; and, at

the nematic state (i.e., Point A on Figure 1.1), < AUMS > = —UkBT .
The nematic coeffiCient U is dimensionless and, as indicated above, compares the

average excluded volume potential with kBT. For rigid rod suspensions (see p.66 in

Larson, 1999), U is proportional to the concentration of particles and the excluded

volume V}; = aoL. The parameter a0 is the diameter of a tube of length L that contains a
single rigid rod. In this research, a0 is assumed to be independent of the local
microstructure (i.e., 11b and 1111,). Tube dilation affects the nematic potential (and rotary

Brownian diffusion) through the F TD-factor introduced by Eq.(2.10).

22

2.5 Discussion

The S-equation with Jeffery’s model for rotary convection (see, Eq.(2.5)) and

Doi’s model for the excluded volume potential (see, Eq.(2.12)) can be written as

 

 

6‘1’ 6
(57); +Pe-a— K—l amt-.1221 L§ 21M=
' ‘ (2.14)
a 6 a
+FTD—- —+U‘I’—(pp<pp>
2 62 52 " '-
where t is a dimensionless time and Pe is Péclet number:
taongi, Pea 1 d“}. (2.15)
0 d
6DR y

The S-equation determines how the orientation density function changes with
time over the surface of a sphere in orientation (phase) space (see Edwards and Beris,

1989). The relaxation of ‘1’ (p , t) from an initial state is controlled by four physical

factors: 1) a rotational torque due to the antisymmetric component of the velocity
gradient; 2) a rotational torque due to the symmetric component of the velocity gradient; ,
3) a rotational torque due to Brownian motion; and, 4) a rotational torque due to the
excluded volume phenomenon. A direct numerical (or analytical) analysis of Eq.(2.14)
subject to arbitrary, but realizable, an initial condition has not been done. Some limited
results have been reported for isotropic initial conditions (see Doi, 1981; Hand, 1962;
Hinch and Leal, 1976; Petty et al.,1999; Tucker, 1988), but an understanding of Eq.(2.14)

has primarily resulted from a study of the low order moments of ‘I' (_p , t). The moment

method (see CHAPTER 3) will be used in this research.

23

CHAPTER 3

MOMENTS OF THE ORIENTATION DISTRIBUTION

3 .1 Introduction

In this research, a rigid rod is approximated as an axisymmetric ellipsoid. A single

orientation vector 2 defines the instantaneous orientation state of the rod in terms of the
angular variable 0 and d) (see Figure 3.1):

2:9r = sin(0)cos(¢t)ex + sin (0)sin((l>)§y + cos(0)§z. (3.1)
There is no distinction between the head and the tail of the rod. Therefore, the orientation
density function satisfies the symmetry condition ‘1’(p, t) = ‘1’ (-p, t). The vorticity
direction is ex; the cross-flow direction is gy; and, the flow direction is £2. The plane
that contains the flow direction and the cross-flow direction is called the shear plane (or
the deformation plane); the plane that contains the cross-flow direction and the vorticity
direction is called the cross-flow/vorticity plane; and, the plane that contains the vorticity
direction and the flow direction is called the vorticity/flow plane (see Figure 3.1).

Measuring the orientation of individual rigid rods in a suspension is not practical.

However, low-order moments of the density function ‘I’( p , t) can be measured. These

moments provide an objective means to understand the complex behavior of rigid rod

suspensions. For a suspension of axisymmetric particles, the first moment <p > is zero

because of symmetry:

24

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25

(0,4), t) sin(0)d0d4)= 0. (3.2)

”I”

All of the odd moments are zero. The zeroth moment (i.e., integration of the density

0:3

function over the sphere) is unity because the total number of particles is a constant.

3.2 Orientation Dyad: Structure Tensor and the Order Parameter

The second moment of the orientation density function is

I'd
I'D

22] I pp ‘I’(0,4),t)sin(0)d0d4). (3.3)
0 O

The second moment < pp > is also called the orientation dyad. < p _p_ > is a symmetric,
non-negative operator, and its trace is unity because

tr(<p_p>)a<_p-p>=1. (3.4)
This dyadic-valued operator defines the microstructure of rigid rod suspensions. Figure

3 .2 illustrates an isotropic (or three-dimensional random) orientation state for which

1 . .
<pp > = 3(gxgx +§y§y +§z§z) , isotropic state.

When the rigid rod particles are randomly distributed in a two-dimensional plane (see

Figure 3.2 and Figure 1.1), then
-1 l ' tr ' tat
<pp>——?:(_e_y§_y +_e_z_e_z), panarrso oprcs e.

When all of the rigid rods are pointing in one direction (see Figure 3.2), then

< p p > = g e nematic state.

2—2’

26

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ofiobofl 853
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27

The orientation dyad can be represented as the sum of an isotropic operator and an

anisotropic operator:

"H

<pp>= +2. (3-5)

wIH

A scalar-valued order parameter a is often used to define the orientation state or the
prolate (and oblate) boundary of Figure 1.1. This parameter defined in terms of the

second invariant of the structure tensor b:

3 1/2
{311,} . (3.6)

Note that:
or = 1 at the nematic state; or = 0 at the isotropic state; and, a = —1/2 at the planar

isotropic state (see Figure 1.1).

3.3 Orientation Tetrad: Symmetry and Contraction Properties

The fourth order moment of the orientation density function is:
21! 1!
_p p =j Jpp_ pp,‘I’(9 q), t) sm(0)d0d4) (3.7)
o o

This statistical property is called the orientation tetrad. Previous studies have used the
following closure hypothesis for the orientation tetrad (Hand, 1962; Hinch and Leal,

1976; Doi, 1981; Tucker, 1988; Petty et al., 1999):
<pppp> 3(< pp >) (3.8)
Eq.(3.8) is also used to support this research (see CHAPTER 5). Clearly, the orientation

tetrad has six-fold symmetry. For example, with p = a = b = g = d , it follows that

28

<pppp>_=. <pbgd>
E<h§9é>2<shaé>5<éhsa> (3-9)
E<agh§> E<adsh> E <_a.12<_19>-

IO
v V
II» Io‘
IO ID-

IO I19

1:3. 16:.
I!» I0‘

Io‘lo
V V

(3.10)

V V

III III

A A
A A

10" It”
IO-

Eqs.(3.9) and (3.10) are important properties that should be retained by any closure based

on Eq.(3.8) above.

3.4 Realizable Anisotropic States: Invariant Diagram

The orientation dyad is a symmetric and non-negative operator (see Parks et al., 1999).

The eigenvectors ii and the eigenvalues 7» pi of the orientation dyad < _p_p > are defined
by:

<pp>~xi=hpixi . (3.11)
The eigenvalues of < pp > are real and non-negative: 0 5 API 5 11,2 S 2,1,3 5 1. The

eigenvectors of < p p > are:

)‘1 r 5.1 = xxlgx +xyl§y +le§z
12 , x2 = xngx +xy2gy +x22§z (3.12)
B 2 $3 = xx3§x +xy3§y +xz3§z

The three invariants of < pp_ > are

29

Ip =I'I(<BE >)=}"pl + Apz '1' )‘p3

_ _ 2 2 2
11p —tr(<pp>-<pp>)—).pl+1.1)2+1.p3 (3.13)

_ _ 3 3 3
IIIp --t(<pp>-<pp>)—7.pl +1132 +1133.

The orientation director is defined as the eigenvector associated with the largest
eigenvalue of < pp > (i.e., x3 ).
The structure tensor p , defined by Eq.(3.5), has three eigenvalues: bbl , hbzl, and

21,3. The invariants of p are

1b =h’fg)=lb1+7»b2+’~b3=0

11b =u(b-p)=1§2+1§2+1§3 (3.14)
mb=tr< Bartlet“;-

IIU‘ II

The second and third invariants of p are non-zero. Figure 1.1 uses these invariants of the

microstucture to identify all possible realizable orientation (see Parks et al., 1999; esp.,
Lumley, 1978). The orientation states within this designated region are realizable

inasmuch as the eigenvalues of < pp > are real and non-negative.

The eigenvectors of < pp > and 1:) are the same. The eigenvalues are related by
1 . _
)‘bi =lpi ~§ , 1-1,2,3. (3.15)

For uniaxial alignment states (see point A of Figure 1.1),

1 l 2
Apl = O, 1.132: 0, and 11,3 =1 (xbl = -3, Abz =—§, lb3 :3). (3.163)

For planar isotopic states (see Point C of Figure 1.1),

A'pl = 0, 1.132: 1/2, and A133 =1/2 (hb1=-?13-, le =-;-, 71.133 =%). (3.161))

30

For isotopic states (see Point E of Figure 1.1),

1p, =1/3, 1P2 = 1/3, and 1p, =1/3 (ltbl =0, 11,2 :0, 1b, =0). (3.16c)

For planar anisotopic states (see Line B of Figure 1.1),

1 1 3
AP] = 0, A132: 1 -1133 , and A133 0‘13] =—§, )‘bZ =Ap2 —§, Ab3 =E-Ap2) (116(1)

For axisymmetric oblate states (see Line D of Figure 1.1),
_ _ 1 1

4 1 l
(1131:3—271132, M32 =7~p2 -§, 1133 =7~p2 -§)

(3.16e)

For axisymmetric oblate states (see Line D of Figure 1.1),

1
hp]: xpz = leand )‘p3 =1-21p1 (05113153)
1 1 4
0.14.17, ”b2 =0” '3’ x... 721.1) (3.160

The planar anisotopic boundary of Figure 1.1 follows by substituting the planar

anisotopic eigenvalues of 2 into Eq.(3.14):
11b = 2/9 + 2111b . (3.17)

The prolate boundary of Figure 1.1 follows by substituting the axisymmetric prolate

eigenvalues of p into Eq.(3.14):

2/3
111, = 6 (13151] . (3.18)

The oblate boundary of Figure 1.1 follows by substituting the axisymmetric oblate

eigenvalues of 2 into Eq.(3.14):

31

 

2/3
11b=6[ 1611b) . (3.19)

3.5 Discussion

This research proposes to examine a longstanding moment closure problem
related to the orientation tetrad of rigid rod suspensions, such as liquid crystalline
polymer (LCP). The proposed approach is based on an analysis of the low order
moments of the S-equation for the orientation density function. The moment equation for

< pp > is unclosed inasmuch as it depends explicitly on < p p p p >. In CHAPTER 4, the
moment equation for < pp > is presented. In CHAPTER 5 and CHAPTER 6, a
realizable closure for <p p p p > based on the hypothesis expressed by Eq.(3.8) is

developed that satisfies all the symmetry and contaction properties defined by Eq.(3.9)

and Eq.(3.10).

32

CHAPTER 4

EQUATIONS FOR THE MICROSTRUCTURE AND THE STRESS

4.1 Intoduction
In principle, the S-equation given by Eq.(2.14) can be solved numerically for any
initial condition, ‘I’(p, 0). The resulting solution can be used a posteriori to calculate the

low-order moments that appear explicitly in the stess equation. This approach does not
require an a priori closure model for moments. Although a direct numerical simulation
of the S-equation for relatively simple flows and initial conditions provides useful
predictions of statistical properties, this approach is not practical for complex flows or
complex initial conditions. The method-of-moments provides an alternative means to
study Eq.(2.14). Unfortunately, this approach is unclosed inasmuch as the dynamic

equation for < pp > depends explicitly on < p p p p >. However, once an appropriate
closure has been identified (and validated) the moment equation for < pp > can be used

to study the relaxation of the microstucture of rigid rod suspensions from arbitrary
anisotopic states. Thus, the main objective of this research is to develop an algebraic

closure for the orientation tetad in terms of the orientation dyad.

4.2 Dynamic Equation for the Orientation Dyad

An equation for < pp > follows directly from Eq.(2.14) by first multiplying the

equation by p p and then integrating over the unit sphere:

33

6<pp>
6t

 

>x+PellT-<22>+<22>_fl_l=

—FTD[(< pp> —%l) —U(< pp> - < pp> — < pppp>:< pp>)] (4.1)
+>~Pel§T° <22> +<22> -§-2 <2222>=§l
The terms on the lefi-hand-side of Eq.(4.1) is the Jaumann derivative of < pp > , which
represents the rate of change of < pp > relative to a tame rotating with an angular

velocity proportional to the vorticity (see Bird et al., 1987b). The first bracket on the
ri t-hand-side represents rotary diffusion due to Brownian motion and excluded volume
phenomenon. The second bracket on the right-hand-side accounts for rotary convection
due to fluid/particle drag. Clearly, Eq.(4.1) is unclosed due to the explicit appearance of

the orientation tetad < p p p p >. Note that both of the bracket terms, which represents

different physical phenomena, depend on < p p p p >.

4.3 Dynamic Equations for the Stucture Invariants for Pe = 0

For Pe = 0, Eq.(4.l) can be reduced to two coupled scalar equations for the

stucture tensor invariants IIb and 1111, (see Figure 1.1). The following equations for Pe

= 0 are derived in APPENDD( A and B:

 

dcllltb =—211b+2FTDU|:§-Ilb+IHb—p:<pppp>zp] (4.2)
din: =—3IHb +3FTDU[§-1Hb +-;—11b2 _2::-<pppp>(22)] (4.3)

34

The stucture tensor 2 (.=.< pp > —-13-£ ) appears explicitly in Eqs.(4.2) and (4.3). Once

closed, the above coupled nonlinear autonomous (i.e., the independent variable does not
appear explicitly in the differential equation) first-order, ordinary differential equations
can be integrated from any orientation state in the realizable region defined by Figure 1.1.
If U = 0, then the equations are closed and linear and can be integrated analytically (see

Section 7.3).

4.4 Algebraic Equation for the Stess

The deviatoric component of the total stess for a rigid rod suspension consists of

three contributions:
1 = is +i" +35 (4.4)
where is is the solvent contribution; iv is the viscous contribution; and, :E is the

elastic contribution (Baek and Magda, 1994). is depends on the solvent viscosity ns

and the stain rate of the flow (Newtonian fluid):

S = Znsg . (45)

111-1)

The microstucture of the suspension couples with the stain rate to produce an additional

viscous contribution to i :

v (4.6)

II ch
II (11>

= ch <2222>=

The viscosity coefficient cCR is given by

35

30k T
ccR =——;B—— (4.7)
6DR Fm

The viscous stess is due to the drag of the solvent on the rigid rod (see p.521 Larson,
1999)

Doi (1981) deve10ped an elastic stess based on the Onsager free energy with the

result that (see Han and Kim, 1993; Larson, 1988):
:E = 3ckBT[(<pp> —%l)—U(< pp>~<pp>-<pppp>:<pp>):l . (4.8)

The first term on the right-hand-side of Eq.(4.8) represents the stess induced by rotary
Brownian motion. The second term is the stess caused by the excluded volume

phenomenon. For homogeneous shear flows, the viscosity of the suspension is given by

A E .:.§z

n= ’ . (4.9)
1’

The first and second normal stess differences are defined as follows

Neel-3921,32,. (4.10)

1212:; '39 -§x-:-2x- (4.11)

Y

In a cone-and-plate viscometer, a positive N1 represents a force that pushes on the cone.

A negative N1 represents a force that causes the cone to push on the fluid.

4.5 Discussion

Eq.(4.l) provides the relaxation of the dyad. The solution governs depends on
three dimensionless groups: Pe, U, and A. These groups are independent and account for

different physical phenomena. A closure approximation for the orientation tetrad

36

< pp pp > in Eq.(4.1) is developed based on six-fold symmetry and six-fold contaction

properties (CHAPTER 5). In CHAPTER 6, Eqs.(4.2) and (4.3) are used to develop a
realizable closure for Eq.(4.1).

Doi’s elastic stess model defined by Eq.(4.8), is similar to but not as complete as
Ericksen, Leslie, and Parodi’s (ELP) stess. Doi (1981) noted that the ELP stess has a
limitation for predicting nonlinear viscoelasticity, which is important in rigid rod
suspensions (Doi, 1981). The Doi stess, even if it is not as general as the ELP stess,
separates the elastic and viscous contributions of the stess in both isotopic and nematic
phase tansition. However, Doi emphasizes that his model is incomplete because of
physical and mathematical assumptions. The influence of the microstucture (i.e.,

< pp > and < p pp p >) on Eqs.(4.6) and (4.8) is the focus of CHAPTER 9.

37

CHAPTER 5

CLOSURE FOR THE ORIENTATION TETRAD

5 .1 Intoduction
In this chapter, a closure stategy for the orientation tetad is intoduced based on

the hypothesis that
(5. 1)

<2222 > = 3<< 22 >>
Parks et a1. (1999) used the Cayley-Hamilton theorem of linear algebra (Frazer et al.,

1960) and developed a representation for 3(< pp >) in terms of the following six

independent tetadic Operators:
I ; l < p p >

;<p>

m—t
111—:
IT)

<22> <21)? ; <22>-<22> .1. :

They proposed that S(< pp >) could be written as a linear combination of two tetradic
operators with six-fold symmety and six-fold contaction (see Eqs.(3.9) and (3.10)):
<pppp>=C131(<pp>)+C232(<pp>). (5.2)
In the above representation, 31(< p p >) is first-order in < p p > and 32 (< p p >) is

second-order in < pp >. The scalar coefficients C1 and C2 are functions of the

eigenvalues of < p p > or, equivalently, the eigenvalues of the stucture tensor

The closure coefficients C1 and C2 are not independent because

p=<pp>- ;.

le

Eq.(5.2) must satisfy the six-fold contaction conditions (see Eq.(3.10):

38

tr<2222>=trlfi<<22>>l
= C1 t1‘[31(< B B >)]+C2 t[32(< E E >)]

=(C1+C2)<BB>

The above result implies that C1 + C2 = 1.

Parks, Petty, and Shao (see Parks et al., 1999) developed explicit expressions for

Sl(<pp >) and 32(<pp >) by using the following two six-fold symmetic tetadic

operators:

S[g g] =AijAkt +AikAj£ +AieAjk (5-3)
and

312, El=AijBkt + Aikle + AilBjk + BijAkl + BikAjt + BirAjk (5-4)

In the above equations, the operators A and B are symmetric and may be 1 , < pp > ,

or <pp>~<pp>. The operators Sl(<pp>) and 32(<pp>) are definedas follows:

1 5
31(<pp>)E--3—5—S[I {HESRBP l] (5.5)

2
32(<pp>):—:-3—5<pp>:<pp>S[ll]
+S[<pp><pp>] (5.6)
10
-—< >-< >1 .
3,122 22 .1

Eqs.(5.2), (5.5), and (5.6) define a preclosure for the orientation tetad

< >2

I'U
I'd
1'6
1'6

<2 pg>=[l—Cztnb.mb)]81(<gy>>+czmb.mb)32(<gg>). (5.7)

39

This representation satisfies all six-fold symmetry and six-fold contaction properties of

the exact orientation tetad. A closure for the second-order coefficient C2 (11b , 1111,) will

be identified in CHAPTER 6 based on the idea that solutions to Eq.(4.1) must be
realizable (i.e., <p p >2gz 2 0 , g arbitary) for all initial conditions <p p > (O) with
invariants IIb(0) and IIIb(0) on the invariant diagram defined by Figure 1.1. Realizable

solutions (unsteady, steady, or periodic) must be produced for any combination of the

physical property groups of 1., U, and Pe: —1 5 AS 1; 0 5 U < oo; 0 5 Pe < 00. The
development of C2(IIb, 1111,) in CHAPTER 6 is the primary accomplishment of this

research.

5.2 Closure Models

Hand’s Closure
Hand (1962) studied the microstucture and rheology of rigid-rod suspensions
near the isotopic state (see Figure 1.1) and assumed that Eq.(5.1) could be approximated

as
<£EBB>Hand=3M<BB>L (5.8)
Which is Eq.(5.7) with C2 = 0. As demonstated in CHAPTER 7 (Pe = 0), this yields

unrealizable orientation dyads for finite values of U and, thereby does not provide a basis

for understanding the self-alignment phenomenon of rigid-rod suspensions.

Doi ’s Closure (Decoupling Approximation)

40

Doi (1981) studied the microstucture and rheology of rigid rod suspensions near
the nematic state (see Figure 1.1) and assumed that Eq.(5.1) could be approximated as
<EEEB>Doi=<PP><EE>- (5.9)
Unfortunately, this closure does not satisfy the six-fold symmety and six-fold
contaction properties of the orientation tetad. Although Bq.(S .9) has been widely used
by many researches for more than fifty years, it misrepresents the fundamental symmety
characteristics of the orientation tetad and, thereby, the symmetry characteristics of the
orientation density function for rigid suspensions of ellipsoidal particles. This unphysical
property occurs for all realizable orientation states, including the nematic-like states near

Point A of Figure 1.1. Bq.(5.9) is not a good approximation for < p p p p > , comments to

the contary in the cmrent literature notwithstanding (Doi and Edwards, 1986; Chaubal,

1997; Larson, 1999).

Tucker '5' Hybrid Closure

Tucker (see Tucker, 1988) used the following hybrid closure for < p p p p > to

study the flow-induced alignment of rigid rod suspensions:

< EBEB>Tucker= 27(det <pp >)31(<pp >)
(5.10)

+[1—27(det<pp>)]<pp><pp> .
For isotopic states, Xpl = 21,2 = 711,3 = 1/3; therefore, 27(det<pp >) =1 and Eq.(5.10)

asymptotically approaches Hand’s closure for orientation states near the isotopic state

(see Point E, Figure 1.1). For the nematic state, 21,1 = 711,2 = 0 and 11,3 = 1; therefore,

27(det<pp >) = 0 and Eq.(5.10) asymptotically approaches Doi’s closure. For the

41

reason cited above, the hybrid closure also misrepresents the firndamental symmetry
characteristics of the orientation density function and, thereby, does not provide an

appropriate closure for the moment equation governing the behavior of < p p > .

Hinch and Leal ’3 Closure

Hinch and Leal (1976) developed the following closure for < p p p p > :< p p > ,

which appears in the moment equation for < p p > (see Eq.(4.1)):

<EBE£>HL3<BE>=CII+°2<22>+C3<EB>‘<EE> (511)

where

s——+-8—t[ pp> <pp>]-—6—t[<pp> <pp> <pp>]
15 15 -- 5 -- --
2 1

c2 =———t[<pp> <pp>]
5 5 -

c __3

3" 5

0 =2

4'5

Note that the above result is symmetric and that t[< p ppp >:< p p >] =< pp >:< p p > ,

as required by Eq.(4.1). The foregoing closure stems from the idea that for Pe = 0 and
U —) 0 (‘weak’ nematic stength), the microstucture asymptotically approached the
isotOpic state. And, for Pe = 0 and U —) oo (‘stong’ nematic stength), the
microstucture asymptotically approaches the nematic state. Unfortunately, as shown in

CHAPTER 7, the Hinch and Leal closure yields unrealizable predictions for Fe = 0 and

42

finite values of U and, thereby, does not provide a physically acceptable closure for

Eq.(4.1) (see APPENDIX D).

Fully Symmetric Quadratic (FSQ-) and Orthotropic Closure

Petty and Bénard together with their students (see Imhoff, 2000; Imhoff et al.,
2000; Kim et al., 2001, 2002, 2003, 2004, 2005; Kini 2003; Kini et al., 2003, 2004;
Mandal et al., 2003, 2004; Nguyen, 2001; Nguyen et al., 2001a, 2001b; Parks et al., 1999;

Parks and Petty, 1999a, 1990b; Petty et al., 1999) have used Eq.(5.7) as a closure for the
orientation tetrad by assuming that C2 is a universal constant. However, realizability at
the nematic state (Point A or Figure 1.1) and at the planar isotopic state (Point C or

Figure 1.1) requires C2 = 1/3 and C2 = 1/2, respectively (see CHAPTER 6). Clearly, C2
must depend or the local orientation state characterized by 11b and 1111,. Therefore, as

demonstrated in CHAPTER 7, Eq.(5.7) with C2 = constant may yield unrealizable

predictions for Pe = 0 and U < co and, thereby, does not provide a physically acceptable

closure for Eq.(4.1).

Cinta and Tucker (1995) also developed a closure for < p p p p > that satisfies

all six-fold symmety and six-fold contaction properties of the exact orientation tetad.
They related their closure coefficients to the local properties of the microstucture by
“fitting” model predictions with results based on a direct numerical simulation of the
S-equation for homogeneous shear flows and for homogeneous extensional flows. The

general realizability of the resulting orthotropic closure has not been determined.

43

5.3 Discussion

Hand’s closure satisfies six-fold symmetry and six-fold contaction properties, but
it is not realizable for finite values of U. Doi’s (decoupling) closure predicts realizable
solutions to Eq.(4.1) (see APPENDIX D), but does not satisfy six-fold symmety and six-
fold contaction properties. Fully symmetric quadratic (F SQ) closure and the orthotropic

closure satisfy six-fold symmety, and six-fold contaction. In CHAPTER 6, the FSQ-

closure coefficient C2 is related to the local microstucture so that < pp > is also

realizable for all rigid rod suspensions subject to simple homogeneous shear.

CHAPTER 6

REALIZABLE CLOSURE

6.1 Intoduction

In this chapter, a realizable closure model for the orientation dyad < pp > is

identified for the relaxation of anisotopic microstuctures in the absence of an external

field (i.e., Pe = 0). A closure for the orientation tetad < p p pp > , defined by Eq.(5.7),

will be completed by developing an equation for C2(IIb, 1111,) based on the condition that

all initial orientation states on the boundary of the realizable region (see Figure 1.1) must
remain either on the boundary or be attracted by states within the realizable region.

Clearly, this condition requires

(1

'11

p- go. (6.1)

all boundaries of Figure 1.1

D:
H

In the above inequality, the vector _11 is an outward pointing unit vector perpendicular to
the local tangent of the realizable boundary. The components of the vector E are the

invariants of the stucture tensor:

 

The vectors g1 and m are orthogonal unit vectors on the two-dimensional invariant
plane (i.e., 911 “9111 =0). For Pe = 0, FTD = 1, and <pppp> defined by Eq.(5.7), it

follows directly from Eq.(4.1) that (see Appendix D):

45

dIIb 7 3 54 2
——=—211 +2U —II +—III -—-II C 6.2b
dt b [35 b 7 b 35 b 2] ( )

dIIIb
dt

= —31111, + 3U[ 315111, +3-an -3311bmbcz] . (6.26)

 

14 35

Eqs.(6.2b) and (6.2c) govern the relaxation of all orientation states. The objective of this

chapter is to relate the second-order closure coefficient C2 to the local invariants IIb and

III), by using Ineq.(6.1).

6.2 Realizable Isotopic, Planar Isotopic, and Nematic States
The excluded volume term in Eq.(4.1) is zero at the isotopic, the planar isotopic,
and the nematic states. This can be seen by evaluating < p ppp >:< pp > at these three

states.

An eigenvector representation for the orientation dyad < p p > is

3 3 3
<pp>(t)=Z?~pi(t)zsi(t)ri(t)=ZZ<pipj >(t)§i 9,- (6.3)

Eq.(6.3) is a representation of < pp > using the fixed mutually orthogonal base vectors

g1, 92, and Q3. The instantaneous orientation vector p can be expressed as

3 3
g = 2140);, = 213101310). (6.4)
i=1 k=l
which implies that
3 3 3 3
_2 = 22p.(t)p,(t)s.9, = 225. (0'5: (081mm). (6.5)
i=1 3:1 k=1£=1

Therefore, the average of Eq.(6.5) shows that

46

~ (t)~ (0) O, ifk¢t (66)
< = .
pk p‘ 7.1,, ifk=t
Note that <p>=0_ implies that <pi >=0 and <pp >=0. Also,
tr<pp>=Z<pipi >=X<iik p”) >=Zxk =1 (6.7)
i

k k

Eqs.(6.3) and (6.5) can be used to represent the components of < p p p p >:< pp >

in terms of the eigenvectors of < p p > with the result that

<BBBB>KBE> E< 2222K B>

< (2251133 Kifij )(ZZEkF! £185! ):(Z)"pm 35me ) >

I'd

(6.8)
Eq.(6.8) holds for all orientation states (see Figure 1.1).

Isotropic States

For an isotopic state, < p p > =
1° + 1° + 1.0 - 1 Thus Eq (6 8) reduces to
l 2 3 — 3 o , o o

[<2222>=<22>]L.mpic

222031 15ij 13m >33?

mij

922611311me pm)>?£? a? (6.9)

=§ZZ<I51 151' >33?
1 J

lulu-t

<pp> =11.

isotopic -

belt—-

47

Eq.(6.9) implies that the excluded volume effect in Eq.(4.1) is zero at the isotropic state.

This conclusion holds for the FSQ-closure for any value of C2(O, 0). Thus, the isotropic

state is a fixed point (i.e., steady state) of Eq.(4.1) for Fe = 0 and U 2 0.

Nematic States
For a nematic state, <pp>=x§x§ and A? :43;- =O,k‘; =1 (see Figure 1.1).

Thus, Eq.(6.9) reduces to

~ ... (6.10)
=< 22133133 >
= ZZ< 155133153 >883).
I 1'
At the nematic state,
~ ~ ~ ~ 1, if i = . = 3

< PinP3P3 > = 0 otherwise (6.11)
Therefore,
[<BEEE >:<BE 1| - =£§§§5 (6.12)

which shows that the excluded volume effect in Eq.(4.1) is zero at the nematic state:

[<22>'<22>-<2222>=<22>]| =(z§z§)~(§§x§)—(L§z§)=g. (6.13)

nematic

This conclusion holds for the FSQ-closure provided C2(2/9, 2/3) = 1/3. Unlike the

isotropic state, the nematic state is not a fixed point (i.e., steady state) of Eq.(4.1)

inasmuch as the Brownian motion term is non-zero:

48

 

1 2 1
[QP‘gl] =§l§£§ 73%? +L‘2’L‘2’) (6.14)
nematic
Planar Isotropic States
For a planar isotropic state, <pp >=§ngg + xgxg) and 1‘; = 0, A3 = 1‘; =%

(see Figure 1.1). Thus, Eq.(6.9) reduces to

 

[< 2222 >:< BE >]Iplanar isotropic
1 ~ ~ ~ ~
=§ZZZ< Pi Pjpm Pm >505?
m i j
1 ~ ~ ~ ~
=§ZZ< Pi 15(me pm)>§?§§’ (6'15)
1 J m
1 ~ ~ 0 o
=EZZ< pi pj >25 Zfij
1 J
_1 _1 o o o o
_§<pp> -z(lz§2+§353)'
planarisotropic

Eq.(6.15) implies that the excluded volume effect in Eq.(4.1) is zero at the planar

isotropic state:

[<22>'<22>-<2222>=<22>]|

planar isotropic

(6.16)
(8353 +§§a§)-(x‘2’2s2 +x§z§)-%(§§l‘2’ +2313) =2

This conclusion holds for the FSQ-closure provided C2(1/6, —1/36) = 1/2. This result

shows that the planar isotropic state is not a fixed point of Eq.(4.1) because the Brownian

motion term is not zero

[<pp>-ll]
_... 3:

(533.3 +£33.54). (6.17)

 

planar isotropic

6.3 Realizable Prolate and Oblate States
The prolate boundary is defined by (see Figure 1.1):

III 2/3
11b = 6 ('61) , o s 1111, 5 2/9. (6.18)

The outward pointing normal vector on the prolate boundary is given by

 

 

Ep =+D¥I§H ++n¥n§nl (6.193)
where
up a — 1 (6.1%)
11 2/3
4 6
1+— —
J 41111))
1/3
2(3)
3 III
nfn .=. + b (6.19c)

4 6 2/3
1+— _—
9 111,,

Note that 111, pp =(ni’1)2 + (n‘l’n)2 =1. On the prolate boundary, the components of

dE/dt are defined by Eqs.(6.2b) and (6.20) with 111, and HIb related by Eq.(6.18). It is

noteworthy that Ineq.(6. 1) on the prolate boundary (F -line on Figure 1.1) reduces to

1:1

which implies that initial orientation states on the prolate boundary remain on the prolate

z 0 , (6.20)
prolate boundary

 

boundary for all time without any additional conditions on C2(Ilb, 1111,).

A similar result holds for the oblate boundary (see Figure 1.1):

50

=0. (6.21)

1:1

The analog of Eq. (6.18) for the oblate boundary (see Figure 1.1) is

 

oblate boundary

 

— — 5 111b s o (6.22)

_mb 2/3 1
’ 36

Hb=6(

The outward pointing normal vector on the oblate boundary is

 

E0 = +nfign + +ncfilgln (6.233)
where
o _ 1
1111 = - 2/3 (6.23b)
4 6
1+— ——
J 9( 1111))
1/
2[__6_] 3
3 111
° — b (6.23c)

 

Note that no 3110 = (n a)? + (ni’n)2 =1. Eq.(6.21) implies that initial orientation states
on the oblate boundary remain on the oblate boundary for all time without any additional

conditions on C2(Ilb, 1111,).

6.4 Realizable Planar Anisotropic Boundary

The planar anisotropic boundary is defined by (see Figure 1.1):

2 1 2
H =ZIII +—- , —— SIII _<_—. 6.24
b b 9 36 b 9 ( )

51

The outward pointing normal vector on the planar anisotropic boundary is given by

npa = +n {la 911 + “11131:; 9.111 (6.25a)
where
pa 1
nII = — (6.25b)
5
11pa = ——2— (6.25c)
III \f5—

Note that npa ~npa =(ng’la)2 +(n11’fi)2 =1. On the planar anisotropic boundary, the

components of dE/dt are defined by Eqs.(6.2b) and (6.2c) with IIb and IIIb related by

Eq.(6.24) above. Ineq.(6.1) on the planar anisotrOpic boundary has two contributions

(see APPENDD( F):
npa

E - 51—:- = 1111): £1 + III d mb
(1 t . . d t d t
planar amsotropic pa

=2121:1111){mum-8:511:+18<22<1+H1>1><o mm. (6.26)

9453

 

 

 

 

p8

 

Note that (—2 + 9 mb) 5 0 for -1/36 5 mb 5 8/36. Clearly, a necessary condition for

Ineq.(6.26) to be satisfied for all U 2 O is

84-45111},
18(1+9111b)'

 

2 2 (6.27)

Ineq.(6.27) is a significant finding and represents one of the primary results of this
research. The conclusion here is that Eq.(4.1) and the FSQ-closure (see Eq.(5.7))
produces realizable microstructures for 0 S U < co and Pe = 0. This theory will be tested

in CHAPTERS 7, 8, and 9 for a wide range of conditions.

52

6.5 Discussion

In this research,

C2 = 8 + 45111}, (628)
18(1 +9111b)

 

for —1 S 71 5 +1, 0 _<_ U < 00, and O S Pe < 00 for all realizable state defined by Figure 1.1.
The analysis in this chapter shows that Eq.(6.28) is consistent with the idea that planar

anisotropic states are attracted to 3D anisotropic states (see Figure 1.1). The results of

CHAPTER 8 (Pe > 0) show that this important feature of the FSQ-closure is satisfied by

a wide class of planar anisotropic states. Figure 6.1 illustrates the behavior of C2(IIIb)

for —1/36 S III}, S 8/36. A major hypothesis for the theoretical results developed in

CHAPTER 7, 8, and 9 is that the FSQ-closure coefficient defined by Eq.(6.28) above
applies for all anisotropic states.
Although the Doi-closure (see Bq.(5.9)) does not satisfy all the six-fold

symmetry and six-fold contraction properties of < p 2 pp > , Eq.(4.1) nevertheless yields
a realizable orientation dyad for Fe = 0 and < pppp > = < pp > <pp >. Figure 6.2

shows that how the excluded volume (EV) terms that appear in Eqs.(6.2b) and (6.2c) vary
over the planar anisotropic boundary of Figure 6.1. Note that the EV-terms are zero for
the Doi-closure and the F SQ-closure at the planar isotropic state and the nematic state.
The Doi EV-term in Eq.(6.2b) is significantly larger than the FSQ EV-term in Eq.(6.2b).
This indicates that the Doi-closure has a higher tendency for self-alignment. This
conjecture is confirmed by the equilibrium calculations presented in CHAPTER 7

inasmuch as the Doi theory predicts biphasic phenomenon at smaller values of U.

53

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54

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55

CHAPTER 7
MICROSTURCTURE IN THE ABSENCE OF AN EXTERNAL FIELD

7.1 Introduction

In the absence of an external field, the relaxation of < pp > is governed Eq.(4.1)
with Fe = 0:

d<pp>

1
d-t— =FTD[—(<EE>-‘3‘l)+U(<EB>-<Bp>—<pp>z<pppp>)). (7.1)

 

The Fm -factor in the above equation accounts for the tube dilation phenomenon

(see Eq. (2.6)). The first term on the right-hand-side of Eq.(7.1) represents rotary
Brownian diffusion. The second term accounts for the excluded volume phenomenon.
The dimensionless group U measures the relative importance of self-alignment and rotary
Brownian motion.

Experimental studies for rigid rod suspensions, such as lyotropic liquid crystalline
polymers, show that a transition fi'om an isotropic state to an anisotropic state occurs at
some critical concentration (Abe and Yamazaki 1989a, 1989b; Farhoudi and Rey, 1993;
Kubo and Ogino, 1979; Murthy et al.,l976; Orwoll and Vold, 1971; Robinson, 1966;
Sartirana et al., 1987; Srinivasarao and Barry, 1991). The equilibrium orientation state is

isotropic for dilute solutions (U << Uc) and anisotropic for concentrated solutions
(U >> Uc ). Uc is a critical value of the nematic coefficient that depends on the

concentration of the dispersed phase. The objective of this chapter is to determine the
effect of U on the steady state solutions of Eq. (7.1) for the FSQ-closure developed in

CHAPTER 5 and CHAPTER 6.

56

7.2 Biphasic Phenomena

The asymptotic solutions of Eq.(7.l) are steady equilibrium states for all U.
Clearly, the Fm-factor does not affect the steady state solutions. In APPENDIX B and
APPENDD( D, the following equations for the second and third invariants of the

structure tensor 2 are derived from Eq.(7.1):

 

d—HL- = -2FTDIIb + ZFTDU —7—Hb +3111}, —fiIIb2C2 :| (723)
35 7 35
dmb = -3FTDIIIb +3FmU[ 375-1111; +finb2 -%Hbmbcz]- (731’)

The steady state solutions to Eq. (7.1) have two equal eigenvalues
(see APPENDIX D). This means that the equilibrium states in the absence of an external
field are either prolate states or oblate states (see Figure 1.1). Thus, application of

Eq.(3.6) implies that the steady state solutions to Eqs.(7.2a) and (7.2b) can be represented

2

in terms of the order parameter or. defined by 11b = i— (a) . Since the equilibrium

solutions of Eq.(7.2a) and Eq.(7.2b) are either prolate or oblate states, Eq.(3.6) can be
represented in terms of a and 1111,: 111b = g (:t (1)3 . The positive sign is for prolate

states and the negative sign is for oblate states (see Figure 1.1). It follows directly from

Eq. (7.2a) that the order parameter is determined by the following algebraic equation:

0: l—U[l-+-l-a--3-§a2C2]. (7.3)
35 7 35

Eq. (7.3) has three solutions:

57

a=0

 

l/7i‘fil/7)2-(144/35)C2(-113-—%). (7.4)

(72/35)c2

 

a:

The solution a = 0 corresponds to the isotropic state. This steady state may be stable or
unstable, depending on the value of U. If the steady state is on the prolate boundary of
Figure 1.1, then the positive sign of Eq.(7.4) applies. For oblate steady states, the

negative sign applies. Previous application of the FSQ-closure assumed that C2 was

constant and independent of the microstructure (Imhoff, 2000; Imhoff et al., 2000; Kim et
al., 2001, 2002, 2003, 2004, 2005; Kini 2003; Kini et al., 2003, 2004; Mandal et al., 2003,
2004; Nguyen, 2001; Nguyen et al., 2001a, 2001b; Parks et al., 1999; Parks and Petty,

1999a, 1990b; Petty et al., 1999).

Figure 7.1 shows how U influences the steady state order parameter a for C2 =
constant. For U < U1, all steady states are isotropic (a = 0). For U1 < U < U2 , three

steady states exist: two stable and one unstable. The unstable state is on the prolate

boundary. The region U1 < U <U2 is called the biphasic region. Regardless of the C 2
value, U2 = 5 for the FSQ-closure, but the value of U1 depends on C2 (see Figure 7.1).
For C2 < 1/2, the oblate solutions become unrealizable for large finite value of U. For
C2 < 1/3, the prolate solutions become unrealizable for finite values of U. The
orientation state is always realizable if C2 > 1/2, but it can not cover all possible
orientation states inasmuch as the order parameter a S 0.78 for C2 5 1/2. Nevertheless,

this result provides the possibility that the FSQ-closure coefficient C2 can be fitted to the

58

.3838 M NO Sm _oucEAvmm 05 3 @8235 83% gunfincm 032:2 fin oSwE

 

 

 

D
N _
2: S D a F
p n _r O®.OI
033:;
........... .I/l I.I..I.I.I.I.I.I.I.I.I.l
; h . 98-
.k/ n u
.l/ n u
3me §85> can on< 35m 2290 / /. m m I 5&8 3%an r ONd-
808598 0 M M
N: III . D
m: I 33.». 030.50% \ m
and Iul 3.. .. ON C We
ONO I0I a. 0.4..
0 8m 2a 9a a . (I
eduaolol s _ . .25 U
, .. cod
:\,
mud I \ u omd
\
mad I I I c .
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII .. co _.

 

 

59

experimental results with C2 = C2 (111, ,IIIb) (see Abe and Yamazaki, 1986 for the

experimental data and CHAPTER 6 for the FSQ-closure coefficient).
Analogous to Eq. (7.4), the order parameters for the decoupling-closure and the

HLl-closure are (Chaubal et al., 1995):

a = 0, -1— 1:31 ’1 — —§— for decoupling-closure (7.5a)
4 4 3U

a: 0, %i%‘/49—2—:J—0 for HLl-closure. (7.5b)

Figure 7.2 shows the comparison between decoupling-closure, HLl-closure, and the
F SQ-closure developed in CHAPTER 6. The phase transition from the isotropic state to
the anisotropic state appears in both closure approximations. The figure also shows the
biphasic region for the decoupling-closure, HLl-closure, and FSQ-closure. The biphasic
region has two stable states and one unstable state that can coexist for the same U. The
isotropic state is a steady state solution to Eq. (7.1) for all three closures. The
equilibrium orientation state is the anisotropic state if the initial condition on a is above
the unstable state. When the initial condition on a is below the unstable state, the
orientation state relaxes to the isotropic state. The locations of the unstable states were
determined by solving Eq.(7.l) as an initial value problem.

Although the qualitative trends of each closure are similar, the quantitative

differences are significant. The decoupling approximation predicts the existence of

prolate states for which a(U) —> g as U —-> 00 (see Figure 7.2). However, the biphasic

transition point is lower than the other models (U2 = 3). The HLl—closure, however, has

60

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033283:

61

the same U2 as the FSQ-closure, but the HLl-closure predicts unrealizable oblate states
for U < 00. The FSQ-closure with C2 (1111,) defined by (6.12) satisfies both prolate and

oblate realizability conditions. Therefore, the decoupling approximation and the FSQ-
closure are realizable for all equilibrium steady states. Tables 7.1 and 7.2 show how U

influences the order parameter and the invariants for the equilibrium states.

7.3 Relaxation to Isotropic and Anisotropic Steady States

When U and PC are zero and Fm = 1, Eq.(4.1) reduces to
Nguyen (2001) discussed various aspects of this equation. An analytical solution to
Eq.(7.6a) is

<pp>(t)=§l+(<pp>(O)—%l)exp(—t). (7.6)

Eq. (7.6a) becomes an isotropic state (< pp > = 1/3) as t —> 00. The structure tensor
corresponding to < p p > (t), defined by Eq.(7.6b), is

gm= 2(0)exp(—t) (7.7)

The two invariants, IIb and 1111, of g are

111, = tr(b Q) = Kb (0) exp(—2 t) , and (7.8a)
1111, = tr(g 111:3) = mb (0) exp(—3 t). (7.8b)
Eqs.(7.8a) and (7.8b) imply that

62

111.

Table 7.1: Invariants and Order Parameter of the Equilibrium
Structure Tensor on the Prolate Line

mb

 

0.0221

0.0013

 

0.0682

0.0073

 

0.1640

0.0271

 

0.231 1

0.0454

 

0.2828

0.0614

 

0.3239

0.0752

 

0.3572

0.0872

 

 

0.5511

 

0.1670

 

 

. = (3/2 IIb)1/2

63

 

Table 7.2: Invariants and Order Parameter of the Equilibrium
Structure Tensor on the Oblate Line

Hb IIIb
0.0000 0.0000
-0.2744 0.0162
-0.3200 0.0352
-0.3510 0.0509

-0.3728 0.0637
-0.3 890 0.0741

a0 = — (3/211b)“2

 

 

 

 

 

 

 

 

 

 

 

3/2
M] (7,.)

III. (t)=III.<0)[
IIb(0)
Figure 7.3 shows that all planar anisotropic states relax to isotropic states on a time scale

of order 4tc if U = O and PC = 0. If U > 0, the relaxation trajectories in the invariant

plane depend on the closure approximation and the excluded volume potential model.
When the initial state is planar isotropic (see Figure 1.1), solutions to Eqs.(7.2a) and
(7.2b) remain on the oblate line. All other initial conditions relax to the prolate boundary.

Figure 7.4 shows that transient solutions for U = 0 and U = 3. Both solutions
relax to the isotropic state. However, the transient solution for U = 3 is attracted towards
the nematic state before it reaches the isotropic steady state. When the rigid rod
suspension is concentrated, the orientation state is anisotropic state (U > U 2 = 5).

Tube dilation does not affect the steady state solutions. However, as illustrated by
Figures 7.5 and 7.6, tube dilation makes the orientation state relax faster to the
equilibrium state. In addition, the relaxation time increases as the orientation state is

closer to U1 , and it decreases for higher U (see Figures 7.5 and 7.6).

7.4 Discussion

The microstructures of rigid-rod particle suspensions predicted by the FSQ-
closure are realizable for all values of U. For U > U1, the orientation structure parameter
a increases as U increases. For U —> 00, a -> 1, which is the nematic state (see Figure

1.1). These predictions are consistent with other closure approximations and

experimental observations.

The decoupling approximation with the Maier-Saupe potential has U1 = 8/3 with

65

0.7 -

0.6 -

  
  
 

0.5 - planar anisotropic states

 

 

 

 

 

0.4 -
Kb
0.3 -
(a) relaxation of planar
0'2 ' anisotropic states
0.1 note: time increment
between points = 0.09t
0'0 0.00 0.05 0.10 0.15 0.20 0.25
KL,

1.

0.9

0'8 (b) relaxation of the nematic state

0.7

0.6
< Pi P j > 0'5”

0.4

0.3

0'21. ——<p1p1>&<p2p2 >

0.1

o 1 2 3 3, 5 6 7

= E/Ec
Figure 7.3 Relaxation of the Microstructure due to Rotary
Brownian Diffusion (U = 0; E =1/(6D0R) ).

66

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.25
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. . . . . 85

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5N

:

fl cud
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o E QN z .. mNd

83:8 3:5

67

0.25

 

0.20
Fm =1
11b 0.15 .‘ . 3 _2
l ------- F1196" = (17111,) , p. 360 Doi & Edward (1986)
0.10 -

0.05 .

 

 

 

 

 

III],

 

 

 

 

Figure 7.5 The Effect of Tube Dilation on the Relaxation of the
Microstructure (FSQ-model; U = 3; initial conditions:
IIb(O) = 2/9; IIIb(O) = O).

68

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0
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4-«n

D

 

   

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f 8.0

 

. co...

    

‘.
.-

_..o

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mm venue—e J

 

T :2on 33.35

.. coo?

69

a = 1/4. The second critical U for the decoupling approximation is U 2 = 3 with a = 0
and 1/2. The HLl-closure predicts that U1 = 240/49 with a = 1/8, and U2 = 5 with a =
0 and 1/4. The FSQ-closure predicts that U1 = 4.73 with O. = 0.167, and U2 = 5 with a
= 0 and 0.321 (see Table 7.1). de Genne (1974) estimated a value for U2 (5 4.55) from
the orientation density function. Chaubal et al. (1995) also calculated the order parameter
curve based on the density function.

The HLl- and F SQ-closures agree better with the transition point than the
decoupling approximation. However, the anisotropic transition fits the decoupling
approximation better (i.e., the curve is much steeper in both the exact and Chaubal et al.
solutions). The I-ILl-closure becomes unrealizable on the oblate boundary. Using other
types of excluded volume potential model, qualitative difference with Chaubal et al. can

be modified, but U2 is much higher than other approximations (see Ilg et al., 1999).
The computational results are based on initial condition for < pp_ > with only
diagonal components in the planar anisotropic state. When U = 0 and Pe =0, Brownian

motion is the only driving force that makes the orientation state random. The nematic

potential coefficient U influences the anisotropic orientation state as well as the
equilibrium isotopic state. For U > U2 , all the steady states are either on the prolate or

on the oblate boundaries (see Figure 1.1). In addition, relaxation experiments provide a
means to determine the rotary diffusion coefficient DR Previous studies have estimated

DR values by fitting the first normal stress difference N1 with computational results.

Beak and Magda (1993) have reported that the DR coefficient for PBLG solutions is

about 2 s'l. With DR ~ 2s‘1 , the time scale for relaxation to an equilibrium state in the

70

biphasic region may be as long as 10 — 100$ inasmuch as tc 5 10 to 100 (see Figure 7.6).

Figure 7.6 may be used to design an experiment to estimate the rotary diffusion

coefficient by measuring the relaxation time of rigid rod suspensions are a wide range of

concentations inasmuch as to ~1/(6Dfi) .

71

CHAPTER 8

MICROSTUCTURE INDUCED BY HOMOGENOUS SHEAR

8.1 Intoduction

The relaxation of < pp > (t) in the presence of a homogeneous shear field
(7 = constant) will be examined in this chapter by solving Eq.(4.1) for Pe > 0. The
velocity gradient for the rigid rod suspension is

Vfi=tsysz- (8.1)

~ . 1
§=§M =§(§y§z +§z_e.y) (8.2)
W-Wl'-1 83
=—= 7’5 sygz-gzsy). (.)

The objective of the chapter is to explore the effect of the tumbling parameter 7L
on the microstructure for a wide range of U and Pe. Appendix B shows how the director

of < pp > can be calculated in terms of the components of < pp > . The motion of the

director relative to the fixed flow direction (3 = quZ) will be examined for L/d = 00

(Section 8.2), L/d =12 (Section 8.3), and 0 _<_ L/d < 00 (Section 8.4). In Section 8.5, the

influence of tube dilation on the microstucture will be discussed. The results of this

chapter are based on the F SQ-closure for < p p p p > defined by Eqs. (5.11) and (6.12).

Eq. (4.1) is integrated by using a fourth-order Runge-Kutta algorithm (see APPENDIX

G).

72

8.2 Relaxation of Planar Anisotopic States for L/d = 00

For L/d = 00, 7. = 1 (see Eq.(2.4)). For this case, Eq. (4.1) with E = 1, Fm = l, and
8 Q defined by Eq.(8.l) above predicts that all realizable orientation states (see Figure
1.1) relax to steady states for U 2 0 and Fe > 0 (see CHAPTER 7 for Fe = 0). If the initial

condition for the director (i.e., the eigenvector associated with the largest eigenvalue of

< pp > ) is in the deformation plane, then the steady state is unique.

Figure 8.1 shows the effect of U and Pe on the steady state angle between the

flow direction ( u =uzgz) and the director (cost) = x3 o_e_z ). The director angle

decreases monotonically as U increases for fixed Pe, and as Fe increases for fixed U. The
results show that hydrodynamic coupling (l-parameter) and the excluded volume
phenomenon (U-parameter) cause anisotropic steady states.

For U S 3 and Pe —> 0, Figure 8.2 indicates that the microstucture is isotopic

(i.e., IIb = 0 and UL, = 0). As Pe increases, anisotopic steady states occur. For fixed

values of Pe, the microstucture becomes more nematic-like as U increases. This occurs
because the excluded volume term in Eq.(4.1) mitigates the return-to-isotropy due to
rotary Brownian motion. Thus, the flow-induced alignment mechanism and the
excluded-volume effect simultaneously promote nematic-like microstuctures. The
excluded volume mechanism makes the microstucture more prolate axisymmetric (see

CHAPTER 7) whereas the hydrodynamic coupling of < pp > with g tends to make the

microstructure more isotopic.

73

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75

8.3 Relaxation of Planar Anisotropic States for L/d E 12

The tumbling parameter k in Eq.(4.1) plays a significant role in the relaxation of
the microstucture. This is anticipated by Eq.(2.2), which predicts that a single rigid rod
in a steady shear flow will tumble continuously if | A | < 1 (see p.449 , Larson 1999). For
U = 0 and l 5 Fe _<_1,000, Table 8.1 compares the invariants of the structure tensor for [M
= 00 (A. = 1) and L/d = 12 (2. = 0.987). For 7» = 1, the steady state invariants approach the
nematic state (see Figure 1.1) for large values of Pe. However, for Fe = 1,000 and

1. = 0.987, the microstucture of the suspension is less nematic (11b = 0.48, 1111, = 0.13).

For 7t = 0.987 and U > 27, Eq.(4.1) predicts director tumbling for Pe a 10. For
this case, Figures 8.3 and 8.4 show that an initial planar isotopic state (see Figure 1.1)

relaxes to a periodic state characterized by director tumbling with a frequency of
ft 5 6D°R1rl15. The initial condition for < pp_ > (0) is planar isotopic:
2<22><0)=@y2y +2.2.). (8.4)
The eigenvectors of < pp > (0) are in the deformation (or shear) plane (see Figure 3.1).

The director tumbles 180' because there is no distinction between the head and the tail of
a rigid rod.

Director tumbling occurs for two reason: 1) the excluded volume phenomenon
mitigates rotary Brownian motion and, thereby, reduces the intrinsic diffusive torque on
the microstucture; and, 2) the torque on the microstructure due to particle coupling with
the shear rate is weakened by a reduced tumbling parameter (i.e., A = 0.987 < 1). As a

consequence, the torque due to the antisymmetric component of the velocity gradient can

76

Table 8.1 Invariants of the Equilibrium Anisotropic Tensor for

Different Tumbling Parameters
(FSQ-model; Fm = 1; U = 0)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7. 1 7. = 0.987

P6 111, 1111, 11b 111b
1 0.0412 0.0017 0.0402 0.0016
5 0.1754 0.0249 0.1686 0.0232
10 0.2588 0.0488 0.2458 0.0447
50 0.4477 0.1202 0.4062 0.1029
100 0.5099 0.1475 0.4475 0.1201
500 0.6020 0.1905 0.4785 0.1335
1000 0.6240 0.2011 0.4801 0.1342
00 0.6667 0.2222 0.4805 0.1344

 

77

 

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79

sustain the phenomenon of director tumbling (see Eq.(4.1)). The dimensionless tumbling
frequency ft aft/6D?l depends on U and Pe. For U > 27, Eq.(4.1) predicts that ft

increases as Pe increases. This conclusion is consistent with previous theoretical and
experimental studies related to director tumbling (see p.280 and p. 463, Larson, 1999).
Figure 8.3 shows that the director rotates slowly when it is nearly aligned with the
flow direction and rapidly rotates as it crosses the vorticity/cross-flow plane (see Figure
3.1). The temporal response of the eigenvalue associated with the director is also shown
in Figure 8.3. It is noteworthy that 13 (t) is a minimum during the rapid tumbling phase
of the motion. The invariants of the structure tensor for this example are shown on the
invariant diagram, Figure 8.4. Clearly, Eq.(4.1) together with the FSQ—closure yields a

realizable orientation dyad for director tumbling. Note also that the initial planar
isotropic orientation state rapidly locks onto the tumbling orbit for ts 0.09/6Dfi .

For A. = 0.987 and U = 27, Eq.(4.1) predicts that director wagging will occur for
Pe = 24. This phenomenon is illustrated by Figure 8.5, which shows the angle between

the fixed flow direction and the director. For this case, the wagging frequency
?w z 6D‘fin/l 1. Note that the director eigenvalue 13 is a minimum as the director wags

about the flow director. A planar isotropic initial condition is also specified for this

calculation and, as illustrated by Figure 8.6, the orientation state locks onto the periodic
wagging state within t5 0.09/6Dfi. Eq.(4.1) predicts that f 5 f‘w/6Da —-> 00 for Pe E

30 and U a 27. As the Péclet number increases, the microstructure approaches the fully

aligned state (Point A, Figure 1.1). The coupling between the strain rate dyadic and the

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82

orientation dyad weakens as (11b,IIIb) —> Point A. Thus, for U = 27 and Pe > 30, the

diffusive torque balances the hydrodynamic torque to produce a steady state
microstructure. Figures 8.7 and 8.8 illustrate this phenomenon for 7t. = 0.987, U = 27, and
Pe = 95. A planar isotropic initial condition with the director in the shear plane was also

specified for the calculation (2 < pp_ > (O) = ey e + 92 e2 ). The director angle relaxes

y
rapidly fi'om its initial condition of 45' and fluctuates around the flow director.
Eventually the director attains a steady state with a small negative offset from the flow
direction. Figure 8.8 shows that the invariants of the structure tensor are close to the
prolate axisymmetric boundary. It is noteworthy that for U = 27, Pe = 95, and 7t. = 1
(rather than 0.987), the director also attains a steady state, but with a small positive offset
fi'om the flow direction (see results on Figure 8.1).

Figure 8.9 gives a phase diagram for A = 0.987. The diagram was constructed by
integrating Eq.(4.1), F”) = l, and the FSQ-closure from a planar isotropic initial

condition (2 < p_p > (0) = eye + 9292 ). The asymptotic state depends on U and Pe.

Y
Three possible states were found: 1) steady states; 2) periodic tumbling states; and 3)
periodic wagging states. The boundaries shown on Figure 8.9 were determined within
(APe, AU) = (1, 1). For example, at Pe = 10, steady alignment occurs for U = 25 and
director tumbling occurs for U = 26. For U = 27, Fe = 95, and 7L = 0.987, Figures 8.10
and 8.11 show how a planar isotropic state with an initial director colinear with the
vorticity (see Figure 3.1) relaxes to its steady state. The initial condition for the
orientation dyad is 2 < pp > (O) = ex ex + 9292. The eigenvalues and eigenvectors for

this initial condition are

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XI (0) = 0, 2&1 (0) = an
8 (OH/2 3.2 <0)=sz (8.5)
13 (0)=1/2 233 (0) =gx.

The director 53 is initially aligned with the vorticity V xu =3 = wxex and as
indicated by Figure 8.10, remains colinear with _w_ for the entire relaxation process. The
steady state invariants of the structure tensor are Kb (00) = 0.492 and 111-, (00) = 0.138 (see

Figure 8.11). It is noteworthy that this steady state is different from the one that deve10ps
from a planar isotropic state with an initial director in the shear plane (see Figure 8.7 and
8.8). Thus, if U = 27., Fe = 95, and A = 0.987, Eq.(4.1) predicts the existence of two
steady states:

1) 11b (00) = 0.546, H11, (00) = 0.165 (see Figure 8.8); and,

2) 11b (00) = 0.492, Int, (00) = 0.138 (see Figure 8.10).

Figure 8.10 shows that g1 '92 = $2 -§y = 0.052 for t —+ 00. The rocking motion of the

two eigenvectors _igl and 2&2 around the director g3 (see Figure 8.10) has been termed
log-rolling (see p.450 Larson, 1999). The three eigenvalues associated with the steady
state orientation dyad < pp > (oo) are ll (00) = 0.014, 282 (00) = 0.083, and 13(00) =
0.903. Thus, a relaxation process with the initial director colinear with the vorticity (i.e.,
3 3 (0) ~ v_v) produces a final microstructure which is less nematic than a process with the

initial director colinear with the cross flow direction (i.e., £3 (0) ~ 3x31). Figure 8.12

gives a phase diagram for k = 0.987 for a planar isotropic initial condition with the

director in colinear with the vorticity (see Figure 3.1): 2 < pp > (0) = ex ex + gzgz. The

89

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90

asymptotic states depend on U and Pe. Three possible states were found: 1) log-
rolling/steady alignment; 2) periodic tumbling; and, 3) periodic wagging.

Figure 8.13 shows the relaxation of an anisotropic microstructure with the initial
director located in the vorticity/flow plane. The initial condition for the orientation dyad

is
<pp>(0)=-5—e 2 +—2-§ s +-5-e 9 +i(§ 2 +§ 9,.) (8.6)

The dimensionless groups U, Pe, and 78 are the same as the relaxation process illustrated
by Figures 8.8 and 8.10 (i.e., U = 27, Pe = 95, and 7t. = 0.987). The initial condition for

the eigenvalues and eigenvectors for < p p > (0) are

2x-(0)=1/6, 51(0)=-§y,
12(0)=2/6, 8 (0)= J5 (Ex—£2) (8.7)
x3(0)=3/6, 8 (0)= J5 (8+8).

As indicated by Figure 8.13, the director x3 relaxes to a final steady state ( x3 (00) = g2 )

by executing a complex three dimensional motion wherein the “cross-flow” component

of £3 (i.e., x3 - ey) first increases to a maximum and then decreases to a steady state
(i.e., x3 (00) - 9y = 0 ) by a damped oscillation through the vorticity/cross-flow plane. The
“vorticity” component of 53 (i.e., x3 ex) shows a monotonic decrease from its initial
condition (33 (0)-§x = «[2- ) to its final steady state (x3 (ao)-gx < 0). This relaxation

process, which has been termed director kayaking, produces a microstructure with

invariants 11b (00) = 0.546, and Int, (00) = 0.165 (see p537, in Larson, 1999). Steady state

component of < pp > (oo) are

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< pp > (oo) = 0.031 exex — 0.001 exey — 0.001exez
— 0.001eyex + 0.032 eyey - 0.004 eyez . (8.8)
— 0.001ezex - 0.004 62 ey — 0.937 ezez

The eigenvalues and eigenvectors associated with < pp > (00) are

9.1 (8) = 0.031, x1 (00) = 0.998 g, + 0.065 9y + 0.052 g,
12(8) = 0.032, 252(00) = 0.065 9,, - 0.098 _e_y — 0.004 g2 (8.9)
9.380) = 0.937, 103(00): —0.0019x — 0.004 gy + 0.997 g,

8.4 Relaxation of Planar Anisotropic States for 0 _<. L/d < 00

Figure 8.14 shows relaxation of a planar anisotropic state with
(< _p_p > (0) =§gy§y +§gz§z) for A. = 0.5, U = 27, Pe = 95, and Fm = 1. The director
and its eigenvalue are periodic. One eigenvalue relaxes to a steady state. The other two
eigenvalues fluctuate and the director tumbles in the deformation plane with a period
it i 0.08/6D°R.

Figure 8.15 shows director tumbling for A = 0, U = 27, FTD = 1, and Pe = 95.
The director tumbles 180' with period of it i 0.07/6Dfi. However, for this case the
invariants of the structure tensor and the eigenvalues of < p p > relax to steady state
values with a microstructure on the prolate boundary (11b (00) = 0.551, 111-, (00) = 0.167).

For this case (A = 0), asymptotic solutions to Eq.(4. 1) split into two contributions: 1) solid
body rotation; and, 2) a steady state microstructure that exactly balances the Brownian

torque and the torque due to the excluded volume potential. It is noteworthy that Ill) (00)

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and IIIb (00) are the same as shown in Table 7.1, for U = 27, Pe = 95, and A. = 0 and for as
well as U = 27, Pe = O, A =1. For A = O, U = O, and Fm =1, Eq.(4.1) reduces to

6< >
RE +Pe[lT-<pp>+<pp>-1]=

I-< >. 8.10
at X = ‘32 ( )

.1.
3

For Pe = 95, Figure 8.16 shows the relaxation of the invariants of the structure tensor

llc"

( < pp > — g l) to the isotropic state. The director tumbles with a period

tt i 0.07/ 6D§ , which is the same dynamic response as k = 0, U = 27, and Fe = 95 (see

Figure 8.15). Figures 8.15 and 8.16 support the idea that the asymptotic solutions to
Eq.(4.1) for A = 0 splits into a solid body rotation and a steady state axisymmetric prolate
microstructure on the F-boundary of Figure 1.1.

For U = 27, Fe = 10, and Fm = 1, Table 8.2 shows that the dimensionless

tumbling period decreases as 7L decreases. Larson noted that the tumbling period tt ~ L/d.

Table 8.2 shows the relation of tt ~ (L/d)2/3 for O < k < 0.91, but as 7t —> 1, tt —-> 00.

8.5 The Effect of Tube Dilation on the Relaxation of Planar Anisotropic
States

Figure 8.17 shows the effect of tube dilation on the microstructure for A = 0.987.

The phase transition between steady alignment and tumbling for fixed Pe is the same as

Figure 8.9 (i.e., U = 26), but the tumbling and wagging regions are extended to larger

values of Pe. This occurs because the diffusive flux becomes larger inasmuch as

FTB‘ —+ 00 as 11}, -—> 3/2.

96

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97

Table 8.2 The Effect of K on the Tumbling Period
(FSQ-model; Fm = 1; U = 27; Fe = 10)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A. tt 5 6D§Et
L 2
(—) -1
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0.987 14.883 (EV +1
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0.970 3.312 0.4 .
0.950 2.286
0.930 1.858 1 o -
0.920 1.720 _o_4 .
0.910 1.610
0.900 1.520 '0-8 ' ___________________________________
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99

8.6 Discussion

The microstructure of the orientation dyad, which is induced by a homogeneous shear,
has four independent variables: U; Pe; l; and, the initial state of the orientation dyad.
Figures 8.2 and 8.15 show that the nematic potential U influences the axisymmetric
orientation state (the prolate state) regardless of director tumbling. For
Pe > 0 and A = 1, the microstructure is anisotropic and inside the invariant diagram. For
large values of U and Pe, the orientation states are near the nematic state.

The tumbling parameter k influences the periodic orientation state and its period.
For k = 1, the orientation states are steady states. For A. < 1, periodic orientation states
occur. Table 8.2 shows that reducing 7» reduces the tumbling period. Eq.(4.1) has four
physical features that determine the orientation state: vorticity, Brownian motion,
nematic, and strain contributions. The vorticity contribution tends to have periodic
rotation, but the strain contribution hinders its motion. For U < 26, the nematic
contribution is not large enough to reduce the strain contribution (steady alignment). For
U > 26 with J. < 1, the director tumbling occurs at low value of Pe because the vorticity
contribution becomes larger than the strain rate term. However, for large values of Pe,
the strain rate contribution regains its strength (steady alignment) (see Figure 8.9). If the
strain rate contribution is reduced by 1 (Figure 8.14 and 8.15), director tumbling occurs
for even larger values of Pe.

The HLl-closure approximation predicts tumbling phenomenon with k = 1.
According to Chaubal (1995), the decoupling approximation used by Doi and others can
also predict tumbling phenomenon (Chaubal and Leal, 1997, 1998; Chaubal et al. 1995)

provided the flow field is modified. The results developed in this chapter shows that

100

Doi’s theory predicts director tumbling for homogenous shear provided 71. < 1
(see Figure 11.3).

The log-rolling and kayaking phenomena have been predicted by other closure
models (Chaubal et al. 1995; Faraoni et al. 1999; Larson and Ottinger 1991). FSQ-
closure also predicts director tumbling, log-rolling, and kayaking. As demonstrated in

this chapter, the realizable F SQ-closure also predicts log-rolling, and kayaking by the

director as well as the existence of multiple steady states for Pe > 0.

101

CHAPTER 9

VISCOSITY AND NORMAL STRESS DIFFERENCES

9.1 Introduction

For homogenous shear flow, the deviatoric stress has three nontrivial normal

components and two non tr1v1al shear components: ;zgxgx , ;zeygy, ;zgzgz , __rzgygz

= izgz 52),. In this chapter, the effect of the tumbling parameter it, the excluded volume

coefficient U, and the Péclet number Pe on the viscosity, the first normal stress
difference, and the second normal stress difference will be developed by using Doi’s
theory for the stress (see Eqs.(4.8)) and the FSQ-closure for the orientation tetrad (see

Eqs.(5.8) and (6.12)). The dimensional rheological properties are defined as follows:

 

viscosity
1:9 9
05‘ iy. (90
Y

zgz — Eygy) , (9.2)

1:12 5::(§y§y_§z§z)' (93)

The objective is to access the impact of the realizable FSQ-closure on the rheological

properties of rigid rod suspensions.

102

9.2 Rheological Properties: L/d = 00

Shear Viscosity
Figure 9.1 shows the effect of the excluded volume coefficient U and the Péclet

number, Pe = y/(6DR ) , on the steady state shear viscosity due to the viscous and elastic

components of Doi’s stress for the realizable FSQ-closure (see Eq.(4.8)):

_ A .. 6DR
Tl Tls-(Tl Tls) 3ckBT

 

=F(U,Pe) , osU<oo , OSPe<oo. (9.4)

The theory predicts the existence of a Newtonian plateau for low shear rates (i.e.,
y < 60 DR) and a shear-thinning region for high strain rates (i.e., 73> 600 DR ). For large

Pe, Figure 9.1 shows that the shear viscosity becomes independent of U for large Pe:

Pe —900 -3/5

n—nsEF(U,Pe)——)O.18Pe (9.5)

PBLG in m-cresol solution and PBZT in methane sulfonic acid solution, which are
lyotropic LCPs, also show shear thinning phenomena at large strain rates with

n 5 —1/3 and n .2. —3/4, resp. (see, p.286 and p.510 in Larson, 1999).

For low values of Pe, the shear viscosity becomes independent of Fe

(i.e., Newtonian plateau) but still depends on U. Figure 9.1 shows that

lim F(U,Pe)=o.01+——O‘—25——.
96—30 1+1.17U

(9.6)
For dilute suspensions, DR is independent of concentration (i.e., DR —> D3) and

U << 1. Therefore, Eq.(9.4) and Eq.(9.6) predict that Afio at c, which is consistent with

experiments and other theoretical predictions for rigid rod suspensions and LCPs

(see p.281 in Larson, 1999).

103

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As the concentration increases, U becomes large and DR at: c“2 (see p.287 and p.

520 in Larson, 1999). Under these conditions, Eq.(9.4) indicates that Afio at c3 / U for

"concentrated" suspensions. For "high" concentrations of PBLG in m-cresol (i.e., > 0.5

wt% ), Mead and Larson (see p. 290 in Larson, 1999) observed that the zero-shear rate
viscosity increases with concentration as Afio 0: c3. Thus, Eq.(9.6) and the foregoing

experimental observation imply that U becomes independent of concentration for semi-
dilute and concentrated suspensions. For dilute suspensions, U at c (see p.66 in Larson,
1999)

Figure 9.2 shows how the viscous and elastic contributions to the shear viscosity

(i.e., Afi :— fi —fi s: fiv + fig) depend on Pe for U = 0 and U = 27. For the stress model

used herein (see Eqs.(4.6) and (4.8) ), fiv and ma are defined as follows:

 

 

 

 

fi 5‘7": " . (9-7)
V Y Ilg
and
fi _fi _3ckBT[<pypz>—U(<pyp>-<ppz>—<pypzpp>:<pp>)] (98)
E=—.‘- '* - -
1 |l§||

In the above equations, cQR represents the viscous drag coefficient between the rod and

the suspending fluid per unit volume of mixture and has units of viscosity (force-

time/area). The parameter 3ckBT has units of energy per unit volume of mixture, and

c CR is related to 3 ckBT and the rotary diffusion coefficient DR as follows

3ckBT
c = . 9.9
CR 615, ( >

 

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. 50.0

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Bachelor developed Eq.(9.9) for dilute suspensions of slender rods (see p. 284 in Larson,

1999). This parameter is used to scale the viscosity coefficients: TlV =fiV/cCR and

n5 =fiE/CCR. For Pe > 50, Figure 9.2a and 9.2b show that the shear viscosity is
primarily due to the viscous stress; however, for Fe << 5, the elastic stress becomes more

important although the viscous contribution remains significant (Tlv is approximately

25% ofAn at Pe = 0.1).

For U = 0, the elastic stress is due to Brownian motion. For U > 0, the excluded
volume phenomenon mitigates rotary Brownian motion. With U = 27, Figure 9.3 shows
that the elastic stress does not contribute significantly to the viscosity because rotary
Brownian motion is balanced by counter diffusion due to the excluded volume potential.

For low values of Pe, both the viscous and the elastic components of the viscosity
are approximately independent of the strain rate (i.e., Newtonian plateau, NP); therefore,

Eqs.(9.7) and (9.8) imply that in this region

 

fi
<p:p:>..=—:,':P ”(llélll W
[< pypz > —U (< pyp> - < ppz> — < pypzpp>z< 22>):le oc ll§||~ (9.11)

For U = 0 and large Pe, Eq.(9.7), Eq.(9.8), and Figure 9.2a imply that

lim <p§p§ > at: ”gm—3’5 (9.12)
Pe—wo -

l/5

lim < pypz > at H S ||- (9.13)
Pe—No -
For U = 27 and large Pe, Eq.(9.7) and Figure 9.2b imply that

lim <p§p§ >°c|I§||’3’5, (9.14)
Pe—mo

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108

which is the same as Eq.(9.12). It also follows from Figure 9.2b and Eq.(9.5) that

. —/
P119133 [<pypz>—U(<pyp>-<p_pz>—<pypzpp_>:<pp>)]ocl|§|| 15. (9.15)
00

The effect of U and Pe on the elastic contribution of i : gzg is shown in Figure

Y
9.4. If U E O for dilute and semi-dilute solution, then Doi’s shear stress for the elastic

contribution can be approximated by the Brownian motion contribution of stress (see

p.308 and p.338, Doi and Edward, 1986; and, Smyth et al., 1995). Thus, with U = 0,

Eq.(9.8) implies that
“352 |U=o = 3nk13T < Psz > (9-16)

Figure 9.4 shows that the general trend of total elastic contribution, predicted by
Doi’s shear stress depends on U and Pe. Smyth et al. (1995) used a birefringence method

to estimate the elastic contribution to the shear stress for semi-dilute solutions of xanthan

gum (L = 1440 nm and L/d 660) in fructose solvent (r13 = 0.483 Pa-s). They used

Eq.(9.16) to relate the stress to the microstructure. For limited range of shear rates, their

1/3

experimental data indicated that $32 at (y) . Figure 9.4 shows that these experiments

are consistent with the Doi theory with a realizable FSQ-closure provided U = 5 and 0.2
< Pe < 2 (i.e., 1.2 Dfi <7 < 12 D‘fi ). The strain rates in the Smyth/Mackay experiments

covered the range ls-1 < ‘y < 20s_1. Therefore a combination of Doi’s theory and the

experimental observations gives the following estimate D‘fi s 1.3 s_1. For U = 0, the

elastic stress makes a significant contribution to the total stress for Pe < 100.

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110

Normal Stress Differences

Figure 9.5 shows that N is positive for 10’1 S Pe S 103 and 0 S U S 35. For large

3/5

values of Pe, Nl oc Pe . For small values of Pe, Nl oc Pe. These results are consistent

with the experimental measurements of Zirnsak et al. (see p. 295 in Larson, 1999) for
glass fibers suspended in Newtonian fluids and with experimental measurements of Kim

and Han (see p. 513 in Larson, 1999) for thermotropic polyesters, OQO (phenylsulfonyl)
10. By contrast, at low Péclet numbers, Nl cc Fe2 for isotropic viscoelastic fluids
(see p. 450 in Larson, 1999).

For a fixed value of Pe, Figure 9.5 also shows that the first normal stress

difference decreases as the excluded volume coefficient U increases. This occurs because

counter diffusion due to the excluded volume effect balances rotary Brownian motion for

sufficiently large values of U (see Table 9.1). For U —> 00 at low Pe, N1 is primarily

determined by the viscous stress. This prediction is qualitatively supported by the

thermotropic polyester experiments mentioned of Kim and Han (1993) above inasmuch

as N1 increases with an increase in molecular weight. In an earlier study of PBLG

solutions (lyotropic LCP), Robinson (1965) observed that the critical concentration
corresponding to a transition from an isotropic to a nematic state decreases with an
increase in molecular weight. Therefore, if U decreases as the molecular weight
increases, it follows that Figure 9.5 is qualitatively consistent with trends observed for

both thermotropic and lyotropic LCPs.

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112

Table 9.1 Viscous and Elastic Contributions to the Shear Viscosity and

the First and Second Normal Stress Differences at Selected
Values of U and Fe for L/d = no

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Elastic
Property Pe U Total Viscous

. Excluded

Total Browman Vl
oume

0 0.0761 0.0374 0.0387 0.0387 0
5 15 0.0295 0.0272 0.0023 0.0180 —0.0157
n-ns 27 0.0122 0.0109 0.0013 0.0208 —0.0195

0 0.0019 0.0019 0.0000+ 0.0000+ 0
1,000 15 0.0028 0.0028 0.0000+ 0.0000+ 0.0000+
27 0.0017 0.0017 0.0000+ o.0000+ 0.0000+

0 0.8928 0.4011 0.4972 0.4972 0
5 15 0.5027 0.3965 0.1062 0.8337 —0.7274
27 0.3619 0.3160 0.0459 0.9040 —0.8581

N1 0 28.487 27.515 0.9717 0.9717 0
1,000 15 27.675 26.840 0.8351 0.9756 —0.1405
27 27.016 26.267 0.7488 0.9780 —0.2292

0 —0.0420 0.0420 —0.0840 —0.0840 0
5 15 —0.0159 0.0159 -0.0318 0.0037 —0.0355
27 —0.0071 0.0071 —0.0142 0.0037 —0.0179

N2 0 —0.0055 0.0055 —0.0110 -0.0110 0
1,000 15 —0.0294 0.0294 —0.0588 —0.0076 —0.0512
27 —0.0394 0.0394 —0.0788 —0.0059 —0.0729

 

 

 

 

 

 

113

 

Unlike N1, Figure 9.6 shows thath < 0 for 10‘1 3 Fe :103 and 0 s U s 35. As

expected, the magnitude of the second normal stress difference is significantly smaller
than the first normal stress difference (|N2| .<_|N1|/200). The calculations support the

conclusion that INzl/INIIAO for Pe—>O and Pe—)oo . Note that for PeleO ,

|N2| SlNll/IOO for U = 35 and decreases to lNzlslNll/ZSO for U = 0. For a fixed

value of U, the second normal stress difference increases in magnitude as Pe increases,

but reaches a maximum for an intermediate value of Pe. For large values of U (i.e., U >

15), max |N2| occurs at Pe 5 200. For Pe E 10 and U > 15, |N2| decreases as U

ll

increases; however, for Pe _ 500, |N2| increases as U increases. Coincidently, for

10 3 Fe 5 500 and U > 15, the shear viscosity decreases significantly as Pe increases

whereas the first and second normal stresses increase as Pe increases.
Figure 9.7 shows the contribution of normal stress differences with FTD = 1, 7s =

0.987, and U = 27. For Pe > 10, the viscous contribution to N1 becomes dominant,
though the elastic contribution is increasing (see Table 9.1). On the other hand, the

elastic contribution to N2 is more important than the viscous contribution. Table 9.1
shows that the elastic contribution of N2 is nearly twice as larger as the viscous

contribution

114

 

 

 

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116

9.3 Rheological Properties: L/d s 12

Shear Viscosity

Figure 9.8 shows the effect of the tumbling parameter )1 on n — n5 for U = 0. This

rigid rod contribution to the viscosity for k = 0.987 (L/d 5 12) is the same as X = l for Pe
<100. However, for Pe > 100, a Newtonian plateau occurs for 71 = 0.987 and the shear
thinning region continues for it = 1. Because Pe is high (see Eq. (4.1)), the diffusive flux
becomes negligible, but it still makes a relatively significant contribution because the
convective flux (hydrodynamic interaction part) is mitigated by 71. For example, Table
8.1 shows that the invariants of the structure tensor are nearly constant with 71 = 0.987 at
high value of Pe. Hypothetically, if the microstructure could be forced to align more for
k = 0.987, another shear thinning state may appear for larger Pe (Region 111) (see Larson,
1999 p. 509 — 511; Walker and Wagner, 1994; Walker et al., 1995).

In the tumbling region (high U and low Pe), the shear viscosity has a periodic
behavior because the microstructure does (see CHAPTER 8). Figure 9.9 shows an
example of the instantaneous viscosity for director tumbling (I. = 0.987, U = 27, Pe = 10,
and Fm = 1). The dimensionless frequency of the dynamic viscosity humbling is 0.21.
Note that it has two local maximums and one local minimum per period. The shear
viscosity shows instantaneous thickening when the director is not aligned with the flow
direction. A period of shear thinning follows when the director rotates towards the cross

flow direction (i.e., u x E ), but the flow and rigid rod resistance is not significant. The

resistance to director rotation becomes significant when the director passes through the

117

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119

cross-flow/vorticity plane, which trigger another shear thickening episode occurs (see
Figure 8.3). As the director rotates away from the cross flow direction, another shear
thinning episode occurs. It is noteworthy that this prediction agrees qualitatively with
experimental results reported by Gu and Jarnieson for thermotropic liquid crystals, 8CP at
36.6 °C (see p.465 in Larson 1999; Chaubal and Leal, 1999).

The time averaged shear viscosity can be obtained by averaging the instantaneous
shear viscosity over several time periods. Figure 9.10 shows the effect of U and Fe on
the time averaged shear viscosity for A. = 0.987. As U increases, the Newtonian plateau
region is extended to higher values of Pe because the tumbling region is extended for
A. = 0.987 (see Figure 8.9). Shear thinning occurs in the wagging region. In the steady
alignment region (high values of Pe), the shear viscosity is independent of Pe. In
addition, shear viscosity becomes independent of U and Fe for 71 = 0.987 at high Pe
because the orientation state becomes nearly constant (see Table 8.1).

For some LCP experimental studies, the shear viscosity shows a shear thinning
phenomenon at low strain rates (Region I), a Newtonian plateau region at medium strain
rates (Region II), and a shear thinning region at high strain rates (Region III) (Walker and
Wagner 1994, Walker et a1. 1995, and see Larson 1999 p. 509 — 511). Region I may be
due to layers of different orientation states (texture affect) in LCP solutions (Marrucci,
1991), Region II is due to the director tumbling phenomenon, and Region II] is due to the
shear aligning phenomenon (see Larson 1999, p. 509 — 511). Figure 9.10 shows that
Region II coincides with director tumbling and Region III with the director wagging

region (see Figure 8.9). Note, however, that additional Newtonia plateau occurs in the

120

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121

steady alignment region at high Pe’clet numbers (see CHAPTER 11 for further
discussion). For U = 27, Figure 9.11 shows that the elastic contribution to the shear
viscosity is relatively small for all Péclet number. Indeed, the elastic contribution to the
rigid rod suspension stress is almost negligible for Pe > 20, and the viscous contribution
becomes independent of Pe (see Table 9.2). This shows that the Newtonian plateau
region at high Fe is determined by the viscous contribution of the stress and that the local

microstructure is insensitive to further increases in Pe.

Normal Stress Differences

Figure 9.12 shows the instantaneous first normal stress difference (N 1) for

director tumbling. The frequency of tumbling is approximately 6 Dfi rt/15 when U = 27,
Fe = 10, A = 0.987, and FTD = 1. When the director tumbles, N1 changes sign. N1

increases when the initial director is off-aligned from the flow direction (t s 44.3), and
decreases when the director rotates towards the cross-flow direction (compare with
Figure 8.3). N1 increases again when the director completes the rotation from the cross-
flow direction to the flow direction. In a cone-and-plate viscometer, the plate is pushed

apart when the director rotates away from flow direction because this causes an increase
in N1. On the other hand, when the director aligns with the flow direction the plates push

back on the fluid.

Figure 9.13 shows the effect of U and Pe on the time averaged first normal stress

difference for k = 0.987 and FTD = 1. When U is relatively small (steady alignment

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Table 9.2 Viscous and Elastic Contributions to the Shear Viscosity and
the First and Second Normal Stress Differences at Selected
Values of U and Fe for L/d -_'=. 12

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Elastic
Property Pe U Total Viscous
. Excluded
Total Browman

Volume

0 0.0809 0.0560 0.0249 0.0249 0
5 15 0.0251 0.0238 0.0011 0.0126 -0.0115
11 _ TI 8 27 0.0228 0.0214 0.0014 -0.0004 0.0018

0 0.0099 0.0099 0.0000+ 0.0000+ 0
1,000 15 0.0123 0.0123 0.0000+ 0.0000+ 0.0000+
27 0.0132 0.0132 0.0000+ 0.0000+ 0.0000+

0 0.8728 0.3813 0.4915 0.4915 0
5 15 0.3586 0.2803 0.0190 0.8223 -0.8033
27 -0.0128 -0.0111 -0.0017 0.8699 -0.8716

N1 0 5.5053 4.6326 0.8727 0.8727 0
1,000 15 0.2563 1.9266 0.3297 0.8952 -O.5655
27 -0.3244 -0.2789 -0.0455 0.9046 -0.9501

0 -0.0470 0.0399 -0.0869 -0.0869 0
5 15 -0.0140 0.0102 -0.0242 -0.0245 0.0003
27 0.0003 -0.0001 0.0004 0.0196 -0.0192

N2 0 -0.0579 -0.0074 -0.0505 0.0505 0
1,000 15 -0.0537 0.0275 -0.0813 -0.0135 -0.0678
27 0.0094 -0.0056 0.0150 0.0014 0.0136

 

 

 

 

 

 

 

 

 

 

 

 

 

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region), N1 is always positive. However, N1 is negative in the tumbling region and

remains negative for higher values of Pe. Figure 8.7 shows that the director angle for
high U and Pe is negative. The negative angle indicates that the director is pointing

downward so that the plate is pushed towards the local microstate. Larson (1999)

explains that N1 is positive when the director is tumbling, and becomes negative when it

is wagging. He suggests that the negative N1 in both the computational results and

experimental results supports the idea of director tumbling for LCPs. Both results also

show increase in N1 at high strain rates. Figure 9.13, however, shows that Nl,avg
predicted by the Doi stress is independent of Pe at high values of Pe. However, N1,avg
still depends on U. N1 remains negative as long as the director tumbles. A dependence

of Pe only occurs when 7» = 1 (see Figure 9.1). As mentioned previously, a negative N1,
which is independent of Pe, is caused by a constant orientation state.
Figure 9.14 shows the instantaneous second normal stress difference (N 2) for

director tumbling. The dimensionless frequency of tumbling is 0.21 when U = 27,

Pe = 10, and FTD = 1. Analogous to the instantaneous N1 analysis, the sign of N2 can be

related to the microstructure of the rigid rod suspension. N2 decreases to negative values

when the director is close to the cross flow direction. However, when the director passes

the cross flow direction (t E 44.3), it changes to positive values. Then, N2 slowly

decreases to zero as the director approaches the flow direction (see Figure 8.3).

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Figure 9.15 shows the effect of U and Fe on the time averaged N2. When U is relatively
small (steady alignment region), N2 is always negative. For U > 27 (where the director

tumbles), N2 is always positive. At the high value of Pe, N2 becomes independent of Pe,

but depends on U. As previously noted by Beak and Magda (1993), Beak et al. (1993),

and Larson (see p.534, 1999), the results summarized by Figure 9.15 show that 1) N1 and
N2 have opposite signs; 2) N2 has a local minimum where N1 has a local maximum; and,

3) [NZ] 5 |N1| /20 (see Table 9.2).

9.4 Rheological Properties: 0 S L/d < 00

Figure 9.16 shows the effect of the tumbling parameter A on the shear viscosity
with Fm = 1. In this research, the tumbling parameter is related to the aspect ratio of a

rigid rod by Eq. (2.3), which shows that 7t —-> 1 as L/d —-> co; and, A —> —1 as L/d —+ 0. If
L/d = 1, then 2. = 0. The suspension viscosity 1]an — 113 2 A11an has a maximum at k =

0.100, which corresponds to L/d = 1.106. Note that the shear viscosity is not symmetric
about A. = 0. As I k | —> 0, the frequency of tumbling increases, which causes the
viscosity to increase (see Table 8.2).

Figure 9.17 shows the effect of A on the first normal stress difference with

Fm = 1. There are two local positive maxima: one at 7» = 0.750 and N1 = 0.1084; and,

another at A = -0.965 and N1 = —0.0168. A local minimum is located at A = —0.600 and

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N1 = —0.070. N1 is negative for both positive and negative values of 2.. However, N1 is
positive for A = :1 (A = 1, N1 = 0.699; and, A = —1, N1 = 0.531). According to Larson
(1999 p. 294), N1 oc (L/d)21n(L/d) for rigid rods (2. > 0). This relationship can be
approximated by power law relationship, N1 oc (L/d)7'4. Figure 9.17 shows qualitative

agreement with this theory until A s 0.7. N1 decreases with the tumbling parameter as l
—-> 1.
Figure 9.18 shows the effect of 2. on the second normal stress difference with

FTD = 1. Notice that there are two local minima: X = 0.700, N2 = —0.025; and,
X = —0.920, N2 = —0.007. A local maximum occurs at k = —0.500 (N2 = 0.012). N2
increases to positive values as X —* 21:1; however, N2 is negative for X = :tl (A = 1,

N2 = —0.0127; and, A = —1, N2 = —0.5688). The sign of N2 is always opposite NLbut the

magnitude varies with A.

9.5 Rheological Properties: Effect of Tube Dilation

Shear Viscosity
In this section, the effect of tube dilation on the rheological properties will be

examined. This phenomenon directly impacts the rotary diffusion coefficient in Eqs.(4.1)

and (4.6). The tube dilation coefficient FTD depends on the local microstructure through

. —2
. the second invariant of the structure tensor: F196” = (I —%Ilb) . Table 9.3 defines how

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. -2
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legend Eq.(4.1) Eq.(4.6)

_a_ No No

 

 

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135

the FTD-factor is applied in the results presented hereinafter. If no tube dilation is

considered, then this is designated as “No” in the table. If tube dilation is included, then
the affected equations are identified as “Yes.” Figure 9.19 shows the effect of tube
dilation on the time averaged shear viscosity for 2. = 0.987 and U = 27. With tube
dilation, the tumbling region is extended to larger values of Pe (cf. Figures 8.9 and 8.17).
Tube dilation enhances the diffusive flux in the moment equation so the director has more
freedom to rotate. Because of director tumbling, the Newtonian plateau is also extended.
When the tube dilation is included in Eq.(4.1) and Eq.(4.6), shear thickening occurs near
the tumbling and wagging transition region. Larson notes that shear thickening occurs
because the increase in interparticle spacing makes it harder for the solution to deform
(see p.273, Larson, 1999). In Figure 9.19, the shear thickening phenomenon may be
explained as an interparticle spacing effect. However, Larson also mentions that there is
no known direct relationship between tube dilation and shear thickening. Figure 9.20

shows the contribution of stress components on the time averaged shear viscosity when

Fm = E??? in Eq. (4.1) and Eq.(4.6). Shear viscosity is nearly identical with the viscous

contribution of the stress, which is consistent with 3. = l and A = 0.987 study without tube

dilation.

Normal Stress Differences

Figure 9.21 shows the effect of tube dilation on the time averaged of the first

normal stress difference with k = 0.987 and U = 27. Note that N] is independent of Pe

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for Pe >> 2,000. In addition, N1 with and without tube dilation have similar trends in the

tumbling, wagging, and steady alignment regions. The inclusion of tube dilation in

Eq.(4.1) causes the tumbling at high value of Pe. This influences prediction of the

minimum of N1 and its magnitude of N1. If tube dilation is included in Eq.(4.1) and
Eq.(4.6), N] has a local minimum in the wagging region. Tube dilation affects the

magnitude of N1 at the local minimum significantly. Figure 9.22 shows the effect of tube
dilation on the time averaged second normal stress difference with k = 0.987 and U = 27.
Though N2 values are independent of large Pe value for all tube dilation cases (see Table
9.3), they have different trends related to tumbling, wagging, and steady alignment. With

the tube dilation effect, signs of N2 values are not always opposite to N1. When BB? is

only applied to Eq.(4.1), N2 is negative for low value of Pe. In the director wagging
region, N2 changes from negative to positive. In the steady alignment region, N2 remains
positive and becomes independent of Pe. When FPSi is applied to both Eq.(4.1) and
Eq.(4.6), N2 is positive for low value of Pe and decreases to negative in the wagging

region. In the steady alignment region, N2 remains negative and becomes independent of

Pe. None of these results have been observed experimentally.

Figure 9.23 shows the contribution of stress on the time averaged normal stress

differences with EB? only in Eq.(4.1), A = 0.987, and U = 27. The viscous contribution

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to N1 determines the total N1, while the sum of Brownian and the excluded volume

contribution to the elastic component of N1 are very small.

9.6 Discussion

Predictions of rheological properties, N1, N2, and A1] are coupled with the

microstructure of the orientation states. These orientation states can be categorized by a

tumbling parameter A. For )- = 1, all solutions are steady state. A11 has a Newtonian
plateau and a shear thinning region, N1 is always positive, and N2 is always negative

(see Figures 9.1, 9.5, and 9.6).

For )- 3 0.987, a steady alignment region at small values of U (U < 27) is predicted.
An, N1, and N2 have similar trends as the case with 7t. = 1, until the orientation state
becomes independent of Pe at large values of Pe. Based on previous experimental and

theoretical studies, negative N1 is caused by tumbling phenomenon, which is predicted

by this research. For )- 5 0.987, N] becomes independent of Pe for large values of Pe.

The effect of tube dilation in the moment equation adds additional diffusive flux in
the orientation state, which gives an extended tumbling region. This phenomenon
broadens the Newtonian plateau region. When the tube dilation effect is included in the
moment equation and in the viscous contribution of the stress, shear thickening occurs
near the tumbling/wagging transition region. Moreover, the magnitudes of the normal

' stress differences are increased significantly.

143

In general, An and N1 are mainly determined by the viscous contribution to the

stress, while the elastic contribution, which includes rotary Brownian motion and the
excluded volume effect, is very small. The second normal stress difference, however, is
determined mainly by the excluded volume contribution. These results offer useful

insights on how to modify the stress model.

144

CHAPTER 10
CONCLUSIONS

In this research, the Smoluchowski equation (S-equation) for the orientation
density fimction was used to study the self-alignment and the flow-induced alignment of
semi-dilute and concentrated suspensions of ellipsoidal particles. A Maier—Saupe
excluded volume potential and Jefi‘ery’s model for rotary convection in a homogeneous
shear field were used to close the S-equation. Doi’s model for the rotary diffusion
coefficient accounts for the influence of tube dilation on highly aligned suspensions.

Low-order moments of the orientation density fimction were used to quantify the
relaxation of the microstructure from initial anisotropic states. An unclosed moment

equation for the orientation dyad < pp > was derived from the S-equation. An algebraic
closure for the orientation tetrad < pppp > , which appears in the moment equation for

< pp > , was developed based on the hypothesis that all planar anisotropic states are

attracted by three dimensional anisotropic states.

The objective of this chapter is to summarize the salient conclusions of this
research. The following four complementary topics, which relate to the self-alignment
and the flow-induced alignment of rigid rod suspensions, will be addressed: 1) closure for
the orientation tetrad; 2) equilibrium states; 3) non-equilibrium states; and, 4) rheology of

rigid-rod suspensions.

145

Closure for the Orientation T etrad
This research has developed a new algebraic closure approximation (FSQ-
closure) for the orientation tetrad < pppp > in terms of the orientation dyad < pp > with
the following three characteristics:
0 the closure retains the six-fold symmetry properties of the exact orientation tetrad;
o the closure retains the six-fold contraction properties of the exact orientation
tetrad; and,

o the second-order FSQ-closure coefficient, defined by

 

2 = 8*“ mb , —is IIIb s—8— (10.13)
18(1+9 1111,) 36 36

1111- Etr[2°2'2] (10.1b)

ga<pp>—%l , (10.10)

has the feature that all planar anisotropic states are attracted by three dimensional
anisotropic states (see Figure 1.1).

The FSQ-closure for < pppp > is

< E 222 > = [1 — c2(1nb)]31(< £2 >)+c2( IIIb)32(< 22 >). (10.2)
The tetradic operators 310 and 32 (-) are defined by Eqs.(5.5) and (5.6), respectively.
Eqs.(10.1a) and (10.2) are significant discoveries that will have an immediate impact on
the further development and understanding of the flow (and light scattering) properties of
suspensions, liquid crystalline polymers, and other fluids with microstructure.

Figure 10.1 compares the steady state solutions of the moment equation (see

~ Eq.(4.1)) with statistical properties calculated from solutions of the S-equation deve10ped

146

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147

by Cintra and Tucker (1994). Cintra and Tucker solved Eq.(2.14) numerically subject to
a uniform initial condition (isotropic) with 7» = l, U = O, FTD/Pe = 9C1, and t* = tPe. The

orientation dyad was calculated directly from the orientation density function. No

closure approximation was required for < p_ppp >. It is noteworthy that solutions to
Eq.(4.1) with the realizable FSQ-closure for < pppp > predicts qualitatively the same
steady states as the direct numerical simulation for a wide variation in the dimensionless

diffusion coefficient C1. Figure 10.1a shows a comparison of the F SQ-closure

predictions of < pp > : ezez with the results of Cintra and Tucker (1994). Figure 10.1b
compares the relaxation of the microstructure (i.e., IIb and 1111,) for the two approaches.

The results show that the FSQ-closure with C2(IIIb) defined by Eq.(6.28) provides a

qualitatively consistent prediction of the exact statistical properties. Note that for

C1 —> 0, the steady state approaches the nematic state (see Point A on Figure 1.1); and, as
C1 —-> co, the steady state approaches the isotropic state. Earlier application of the FSQ-
closure with C2(IIIb) = constant (see Parks et al., 1999; Nguyen et al., 2001; Kini et al.,
2003) predicted unrealizable behavior for small values of C1. For example, if C2 = 1/3,
unrealizable steady states occur for C1 < 0.01. Clearly, the discovery of the closure

coefficient C2(IIIb) defined by Eq.(6.28) is a major accomplishment of this research.

148

Equilibrium States

The steady state (equilibrium) solutions to Eq.(4.1) with Pe = 0 are either
isotropic, prolate anisotropic, or oblate anisotropic. All initial planar anisotropic
microstructures are attracted (relax) to the prolate anisotropic boundary of Figure 1.1. If
the initial state is oblate anisotropic, then the equilibrium state is oblate anisotropic.
However, if an infinitesimal disturbance produces a microstructure within the invariant
domain (see Figure 1.1) near the oblate line, the relaxed equilibrium state will be prolate
anisotropic, not oblate. Thus, oblate equilibrium states are stable to disturbance that are

oblate; however, they are unstable to arbitrary realizable disturbance.
Figure 7.2 shows how the steady state order parameter or (a (3/211b)1/2) depends
on the excluded volume coefficient U. The isotropic state (a = 0) is a solution to Eq.(4. 1)

for all U, 0 S U < 00. For U < U1, or = 0 is the only steady state solution.

For U1 < U < U2, three equilibrium solutions exist: two are stable (isotropic and

prolate anisotropic) and one is unstable (prolate anisotropic). The existence of this so-
called biphasic region has been confirmed by experiments and by other theories (see

CHAPTER 1).
For U > U2 = 5, Eq.(4.1) still predicts the existence of three equilibrium states for

the same value of U: l) a stable prolate anisotropic state; 2) an unstable isotropic state;
and, 3) a conditionally stable oblate state. For U —> co, the prolate state approaches the
nematic state (Point A on Figure 1.1) and the oblate state approaches the planar isotropic

state (Point C on Figure 1.1)

149

Non-equilibrium States
The class of transient solutions to Eq.(4.1) with the FSQ-closure is sensitive to the

tumbling parameter )- and to the initial conditions for < _p_p > (0). For 7L = l, Pe > 0, and

U > 0, all planar anisotropic states with the initial director in the flow/cross-flow plane
(see Figure 8.8) relax to three-dimensional anisotropic states. None of these states are on

the prolate boundary. Figure 8.2 shows the distribution of the steady state invariants.

The locus of steady states (111,, 1111,)SS is parameterized by U and Pe. For fixed Pe, the

steady states approach the prolate boundary as U increases. Also, for fixed U, the steady
states approach the nematic state as Pe increases. Increases in U and Pe have
qualitatively similar effect on director alignment. Thus, Figure 8.1 suggests that an
initially isotropic material could be processed to attain the same three-dimensional
anisotropic microstructure by following different paths in the plane variations in U and
Pe.

For A. < 1, Pe > 0, and U > 0, the asymptotic solutions to Eq.(4.1) with the F SQ-
closure may be either steady or periodic. Periodic solutions require a relatively large
value of the excluded volume coefficient (e.g., U > 27). If the initial director is in the
flow/cross-flow plane and if U > 27, then the theory predicts director tumbling for
Pe < 20 and director wagging as Pe increases. For large Pe and U > 27, the director
relaxes to a steady state alignment relative to the flow direction. Figure 8.9 summarizes
the physical nature of the asymptotic states for different combinations of U and Pe. This
phase diagram or microstructure map, provides the insight needed to attain (or to avoid)
specific microstructures.

Table 8.1 summarizes an unanticipated result related to the alignment

150

phenomenon for large values of Pe. Note that for A. < l, the steady state invariants
approach a highly aligned state independent of Pe; whereas, for 2» = 1, the steady state
invariants continue to approach the nematic state (see Point A on Figure 1.1).
Apparently, the rotational torque due to the vorticity (~ Pe) is unable to counter the strong
rotational torque due to diffusion because the complementary rotational torque due to the
strain rate (~lPe) is weaker with A. < 1.

For high Péclet numbers (Pe > 90) and for high excluded volume coefiicients
(U > 27), the asymptotic response of the director for 7s. < 1 depends on the initial

condition < pp > (O). For example, if the initial director is in the flow/cross-flow plane,

then the director relaxes to a steady state, which has a negative offset from the flow
direction (see Figure 8.7; it = 0.987; U = 27; Pe = 95). However, if the initial director is
in the vorticity/flow plane, director log-rolling occurs (see Figure 8.10; A = 0.987; U =
27; Pe = 95) with a positive offset from the flow direction. On the other hand, if the

initial director is in the vorticity/flow plane and the initial condition of < pp > (0) has an

off-diagonal component, then director kayaking occurs (see Figure 8.13; l = 0.987; U =
27; Pe = 95) with a negative offset from the flow direction. This type of behavior has
been reported by others based on direct simulations of the S-equation, but the
comprehensive set of results given in CHAPTER 8 has not been previously developed
using low-order moments to the S-equation primarily because a realizable theory for

< p p > was unavailable heretofore.

151

Rheologt of Rigid Rod Suspensions
The microstructure of a rigid rod suspension has a significant impact on the

viscous and elastic components of the deviatoric stress (see Eqs.(4.5) and (4.8)) inasmuch

as

.v .~

; °C<BEBB>-§ - (10.3)
. 1

;E at [(<pp_> —§;)—U(<pp>.<p.p>—<p_p_p_p>:<pp>):| . (10.4)

The tumbling parameter A, which controls the behavior of < pp > and, thereby,
< pppp > , also impacts the rheology of the suspensions. For X = 1, the shear viscosity

has a Newtonian-like behavior for low Pe and a shear thinning behavior at high Pe (see
Figure 9.1). As indicated by Figure 9.2, the viscous component of the suspension
viscosity determines the shear thinning behavior at high Pe. At low Pe, the elastic and
viscous components are equally important.

Figure 9.4 shows how U and Pe influence the shear component of the elastic
stress for it = 1(see Eq.(10.4) above; and, Eq.(4.8)). It is not surprising that for Pe > 1,

the contribution of the elastic stress decreases inasmuch as the excluded volume

contribution to 152 diminishes as the director aligns with the flow. Note that the stress
. - - V
model (Eq.(4.7)) wrth the FSQ-closure for < pppp > (see Figure 9.4) predicts that ‘Cyz
1/3 E 1/3
at Pe for U = 0 and Pe > 10. However, for U = 5 and 0.2 < Pe < 2, ryz oc Pe also.

E

Thus, in a flow/ stop experiment for which iyz

can be measured (see Smyth et al., 1995),

. it is important to cover a wide range of U (CC C) and Pe (cc 7) to compare with appropriate

152

theoretical models. For 0.05 wt% xanthan gum in a fructose solution, Smyth et al. (1995)

. . / -l . — . . .
observed that 152 at y] 3 for 1 s < y < 20 s 1. A companson of 121118 experimental

result with the microstructure theory developed herein indicates that DR ~ 1.3 s.1 if U E

5. This estimate is consistent with the rotary diffusion coefficients for PBLG solutions
(Baek and Magda, 1993; also see Figure 9.4).

As anticipated by the microstructure results in Table 8.1, the shear viscosity
predicted for it < 1 has a Newtonian plateau for Pe —+ co (i.e., it = 0.987, U = 0, and Pe
> 500). Once again, this phenomenon implies that shear thinning is limited by the
weaker rotational torque due to particle coupling with the strain rate.

Finally, Figure 9.19 shows that tube dilation causes shear thickening. This
unanticipated result occurs provided tube dilation afi‘ects rotational diffusion in the
moment equation and hydrodynamic drag in the viscous stress equation. This interesting
(and perhaps doubtful) result requires additional research (see p.337 Doi and Edwards

1986)

153

CHAPTER 11

RECOMMENDATIONS

The F SQ-closure for the orientation tetrad needs additional development and
validation to become a practical closure for Eq.(4.1). Although the traditional models for
the rotary convective flux and the rotary dtfiitsive flux used to close the Smoluchowski
equation could be improved, the purpose of this chapter is to identify specific problems

that would support the use of Eqs.(4.1) and (4.4).

Suspension Theory
In the approach adopted in this research, the tumbling parameter A (see Eq.(2.5)),

the excluded volume coefficient U (see Eq.(2.12)), and the Péclet number Pe (see

Eq.(2.15)) were treated as independent dimensionless groups. In general, 7», U, <DR>,
and CR depend on the aspect ratio L/d, the volume fraction of the dispersed phase av, and

the local microstructure (11b and 1111,):

A=Fl(-I§,(1v , 11b, HIb)

U =Fu(%-, (1v , 11b, IIIb)

 

<DR>=FU(£,avaHbaIHb)
Dfi d

PEEL—Fol: a 11 111)
kBT gd’ V’ b’ b'

154

A systematic parametric study of Eq.(4.1) and (4.4) with the F SQ-closure should be

developed for suspensions of prolate ellipsoids for which 1 S L/d < 00. An explicit
dependence of L/d and av on the above groups should be used in the parametric study.
The tube dilation model of Doi was used in this research to determine the effect

of the microstructure on <DR >/D§ and DfiQR lkBT. Shear thickening occurs

because tube dilation was included in both factors. The influence of tube dilation on the
tumbling parameter A should also be developed, if appropriate. An explicit dependence

of L/d on the tumbling parameter was used in this research to study oblate and prolate
ellipsoidal suspensions. However, the effect of L/d on <DR> and CR were not studied.
Figure 11.1 shows the variation of the rotary drag coefficient with L/d for prolate
ellipsoidal suspensions (see p.292, Doi and Edwards, 1986; and Jeffery, 1922). This

theory, and its generalization to oblate ellipsoidal suspensions, should be used in the

proposed parametric study of Eqs.(4.1) and (4.4).

F SQ-closure coefficient
The FSQ-closure was developed based on the Cayley-Hamilton theorem, which

led to the representation of < pppp > given by Eq.(5.7). The second order closure

coefficient was selected so that planar anisotropic states are attracted by three-
dimensional anisotropic states for all values of 2», U, and Pe. The results in CHAPTER 7,

8, and 9 validated that this approach produces realizable orientation states for simple

shear flows. However, the universality of C2(IIIb) as a second-order closure coefficient

155

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D I I o
. S
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7 S0 1 NAE-cv +050 TN 05: _
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- 0.0

156

needs to be tested for other flows. C2(IIIb) also needs to be validated experimentally for

simple shear flows.
As demonstrated in CHAPTER 7 as well as in CHAPTER 8, the relaxation of the
microstructure to an anisotropic state has a dynamic signature that is sensitive to the

closure model for < pppp > . For example, the initial trajectory from a planar

anisotropic state depends on the closure assumption and could be tested directly (see
Figure 7.4). Stress relaxation experiments and/or optical experiments that directly
measure the relaxation of the director could be designed to test the ad hoc assumption of

replacing Ineq.(6.27) with Eq.(6.28).

Texture (Pe > 0)
Figure 11.2 shows that Eq.(4.1) with an FSQ-closure has multiple steady states
for the same parameter set (7L, U, Pe). Note that for A. = 0.987, U = 27, and Pe = 95, two

stable steady state solutions to Eq.(4. 1) occur with distinct microstructures:
(11b, IIIb)1 = (0.4923, 0.1381) and (111,, 1111,); = (0.5463, 0.1648). This result was"

discovered serendipitously by numerically integrating Eq.(4.1) from two different initial
conditions (see Figure. 11.2). The first microstucture was obtained by the relaxation of
an initial director in the flow/vorticity plane, whereas the second microstructure resulted
fiom the relaxation of an initial director in the flow/cross-flow plane. The existence of
multiple steady states for Fe = 0 has been known for many years (see biphasic region in
CHAPTER 7). The existence of multiple steady states for Fe > 0 have not been

systematically studied or discussed in the literature. Larson (p.515, 1999) and Marrucci

157

Sad n a .3 u on .R u a
mega aouflECO 3:5 30?; 8m mouflm >3on 038m 03252 N: 23me

 

E
wwd :3 No 2.6 3.0 Cd 9.0 mto 36 2.6
. . . . . . . . _ m3
0
. md
9%; .838 u E: 55
II /
NmNm+sm$n§A§vN .39
a:
$25 .838 u 2: ea .2
as? xmxmustmmvm
. mad

 

. 5.0

158

(1999) have previously suggested that multiple steady state phenomenon may be the
underlying cause for the existence of small strains associated with defects in the
microstructure (texture). For very small strain rates, the viscosity would be high, but
would decrease significantly for marginal increases in the strain rate inasmuch as the
multiple steady states would be eliminated. Marrucci has hypothesized that this
phenomenon may explain the Region I behavior observed for some liquid crystalline
polymers. Thus, the results illustrated by Figure 11.2 together with the possibility that
multiple steady states for Fe > 0 may relate to Region I behavior supports the suggestion
that a comprehensive study of the possible multiple steady state solutions to Eq.(4.1) with
the FSQ-closure should be conducted for 0 < U < co and 0 < Pe < 00. This could provide
significant new insights and understanding on the origins of texture and other phenomena

associated with structured fluids.

159

APPENDICES

160

APPENDIX A

Derivation of Moment Equations with Structure Tensor

 

161

The objective of APPENDIX A is to find the moment equation in terms of the

structure tensor 2. From Eq.(4.1), the derivative of dynamic equation with the structure

tensor 2 for Fm = 1 is starting from:

 

 

 

d<pp> b U(

-- =—+< >< >—< >< >

5, PP 22 22 2222) (M)
+1Pe(S-<pp>+<pp>§—2<pppp>§)

Where

b-< >-lI

= 22 3=’

<pp>Pe= Y , y= 2SzS,

__ 6DR ==

S=_I-§, §:Vu+Vu ,

= Y: = 2

and

8<pp> 1 a . .r

—— s +fi-V< +W-< >+< >-W A.2
5t 6DR[[6—t - HEB == (>92 22 = ( )

Once the orientation dyad < pp > is substituted with 2 from left hand side of Eq.(A.1)

becomes:

II €>
A
I'd
I'd
V
+
A
I'd
I'd
V
Ilé’
||
Ila
l
H
+
O‘
\_/
+
/'_\
be I H
llr—a
+
"c"
\_/
HS»
.3
ll
llé’
"0‘
+
llc‘
"2

(A3)

Once the orientation dyad < p p > is substituted with 2 , right hand side of the first line in

Eq.(A.1) becomes:

162

III/J
A
I'd
I’U
V
+
/\
I'd
I'd

Combining Eqs.(A.4) and (A5),

3?

Note]:
62 1
5t — 6DR
1
s—tr
6DR

since =\i=’:<pp>=<pp>:\iIT =0

And

5t

+XPe(§:<pp>+<pp>:

= —t1-(2)

Therefore, t = O, 3(2) =0 and t > 0, 121-(b) = 0.

A

Ilc‘
"6*
. .4;

U)
110‘
+
IIO"
110‘
l
m—a
l
ooh—a
llo‘

n6:
1
Ilo‘
x
In:
m
u:
re
v

b 1
= -2+U(§2+2-2—22< pp_pp >)+XPe(§§+=S__-I_J+§'§-2 <PPPP >3_S_)

 

(i:<pp>+<pp>:_:)

 

 

5 —tr
6BR Dt

6b
tr[—=—]=-tr(2)+U(<pp>z<-pp> —<22>2<1_)E(B'B) >)

_2<(E.E)

um
I'd

(A.5)

(A.6)

(A.7)

(A.8)

I“

Note2:

T T
5b 6b . .
[ =) --1— [—3] +<pp>Tle+lz<pp>T

E 6DR at
--———-——l —a—bT+< >T-WT+W'< >T
6DR 6t= 39 ‘= =' 22 (A9)
a) _—
=—q—)—tb =U(<pp>T<pp>T—<pp><pppp>)
+2.Pe(<pp>T:§T +§T :<p_p>T —2<p_ppp>:§)
And
2T=2, “(2)20
S181, tr<§=0

bT satisfies the same equation as _b_. If 2T = Eat t = 0, then 2T =2 for t > 0.

Therefore, "(2) and the symmetry of g are “conserved.”

164

APPENDIX B

Derivation of Moment Equations in the Invariant Form

165

The objective of APPENDIX B is to find the moment equations in terms of the invariants
IIb and 1111, of the structure tensor 2.
General Second Invariant Dynamic Equation

The second invariant dynamic equation is derived by taking trace of the dot

product of the structure tensor from Eq.(A.6):

+UJ + APeK]-= b
—+UJ+>.PeK] (3.1)

+
IO‘
lé

’-1
IICIT

The trace of the ordinary derivative is:

u[d—(2~2)]=%[trbb tr2-( 2)]=‘”‘—-b—

dt

= —2IIb + 2U[ J. b ]+ 2m: (3.2)

l—_‘
M
ucr
I-———l

Note that

0

(E-2)=2=E=(2-2)

because b is traceless tensor.

Since gT=-g,(g-g7):g=—(g-g)z g: (31; 2) £50. Therefore,the

vorticity does not influence dynamic equation of second invariant. The second and third

terms in the second line of Eq. (3.2) is

166

2:242 2+2 2):2—2 <2222>2
3; (2.3)
=§Hb+IHb—g:<pppp>2
and
22-3-2 2+2 2) 2+(2-2) 2—22 <2222>2
=§§=2+§ (2 2)+§=(2'2)—2§<EEBB> 2 (3.4)
=2§=E2+2 1.2-2 <2222>]=2§;

Combining Eqs.(B.3) and (BA) into Eq.(B.2), the dynamic equation for the second
invariant becomes:

c111b

dt =—211b+2U[;:g]+2xPe[;zg]

(13.5)

ll'-—1
H

=—211b+2U[ ;Hb+IHb—2:<p_ppp>:g ]+4kPe[§:

General Third Invariant Dynamic Equation

The third invariant dynamic equation is derived by taking trace of another dot

product from Eq.(B.1):

__‘Lb b-b)=i‘2 b-b+b d_2.b+h 2.92

dt "" dt=a== _ t: ""dt
=—32-2 2+Uu-2-2+2 2 2+2-2 J]+>~PeK-2-2+2 22+2 2 2]
—[W 2+2 229-222 W 2+2 2?) 2+2-2- W 2+2-.W__T)

(3.6)

1 2

,where J=§2+2 2—2 <pppp> and §=§§+§ 2+1; __S_-2<_pppp> §

Notethat

167

And since 2“=—2, (22.“ )=(2-2)=-(2-2)=(2-2H222 )=

"2

'50.

Therefore, the vorticity does not influence dynamic equation of second invariant. The

second and third terms in the second line of Eq. (B.6)

 

d t V (13.7)

(B.8)
=—111b+ IIbZ—b <2222>Qz 2)
and
2 (2 2)=ZS b 2)+s (2 2 2)+2 (2°--2)—22:<2222>=(22)
=22 [—2 2+2 2 2—<2222> (2 2)] (39)

Notice that tr(2 - 2 2 - __1;): -;—11b2 by using Caley-Hamilton theorem.

Combining Eqs.(B.8) and (B9) into Eq.(B.6), the dynamic equation for the second
invariant becomes:

dIII
t" =-3111b +3U[%Illb +-12—11b2 —b:<2222>:(2.2 )]

 

1 (13.10)
+6>~Pe§:[g22+222-<2222>=<2~2)]

168

APPENDIX C

Normal Vectors in the Invariant Diagram

169

The objective of APPENDD( C is to find the outward pointing normal vectors in

the boundary of the invariant diagram. Normal vector _n of the invariant diagram is

defined as:
E = 1111911 +En1§111 (C-l)
In order to be realizable for the orientation state, the dot product of invariant dynamic

equations (dIIb/dt and dIIIb/dt) and the normal vector must be less than zero:

 

 

n-d—£50 orn-hSO (C.2a)
dt
where d—-f- = _de 9H +——dmb gm, 2 = di ’dt (C.2b)
dt d t d t ,/(d§ /d t)-(d§ /d t)

Normal vector on the planar anisotropic line

The relationship of 111J and 1m, is

IIb = 2111b + 2/9 (C.3)
Therefore slope on the planar anisotropic line is 2.

Based on Eq.(C.3) the normalized vector of the planar anisotropic line is:

2 1

Then, the normal vector becomes

1 2
Eplanar anisotropic = ‘J‘ggll _ TEEIII (C-S)

Normal vector on the prolate line

From Eq.(3.18), the slope of the prolate line is

(4/6)(6/111b)“3 (06)

Then, normalized base vector becomes:

170

(4/6X6/Inb)“3 1
= /3 911+ /3 E111
J1+(4/9)(6/111b)2 J1+(4/9)(6/111b)2

Eprolate

Therefore, the normal vector on the prolate line becomes:

—1 (4/6)(6/111b)“3
/3 E11+ ,3 $2111
J1+(4/9)(6/111b)2 J1+(4/9)(6/111b)2

Qprolate =

Normal vector at oblate line
From Eq.(3.17), the normalized base vector becomes:

(4/6X—6/mb)“3 e + -1
-—II
J1+(4/9X—6/IIIb)2/3 fi+(4/9)(—6/mb)2’3

 

 

 

Hoblate =

 

2111

Therefore, the normal vector becomes:

—1 + -(4/6)(—6/mb)“3

E blt = $11 9111
0 ac J1+(4/9X—6/IIIb)2/3 ,/1+(4/9)(-6/1111,)2’3

171

(C7)

(C8)

(C9)

(010)

APPENDD( D

Derivation of Various Closure Analysis in the Invariant Form

172

The objective of APPENDIX D is to find the moment equations in terms of the

invariants 11b and 1111, of the structure tensor ‘9 for various closure approximations. With

the invariants of moment equations, the realizability of orientation dyad < p p > is

determined.
Realizability of Decoupling Closure
Decoupling approximation provides realizable orientation dyad without the

external field (Pe = 0). The moment equation can be represented with invariants of Q,

then the moment equation can be represented with scalar value of 111, and 1111,.

The invariant form of decoupling closure double dot with dyad 2 from Eq.(A.6) is:

"O"

<2222>=2=<_2><22> (DI)

Based on general the invariant dynamic equations (see Eq. (B5) and Eq. (B.10)), each

the double dot product of decoupling closure becomes:

2 <gp><22> >=b (D.2)
gz<pp><pp>:(=b.-g)
Since <pp>:b==(2+ 5:1}: b= b: g=llb (D.3a)

and<pp>:(2_b_)= [b+-;—I):(=b-g)=tr(Q-=b_-g)+-;-tr(l=)-=g) IIIb+;IIb, (D.3b)

the invariant form from Eq.(D.2) becomes:

IIO"

<EE><PB>32=Hb2 (D.4a)

and

173

HO"

:<pp><_p_p>:(b~b)=IIbIIIb+-:1;Ilb2 (D.4b)

Therefore, the invariant forms of moment equations for the decoupling closure, when Pe

 

 

= O are:

d 11" = —2111, +2U[ 1111, +111b «nbz ] (11561)
d t 3

and

d mb -_- —3111b +3U[%1Hb +24le -111be ] (D'Sb)

Realizability at planar anisotropic line
Substituting 11., and III}, relationship at planar anisotropic line (111, = 21111) + 2/9) into

Eq.(D.5a) and Eq.(D.5b), the invariant form of moment equations become:

 

dH
—b— : 4111b —3+U[—8mb2 +3111}, +1] (D-6a)
dt 9 9 81
dIIIb 2 7 2

=-3111 +U —4III +—III +— . D-6b
(it b i b 9 b 81] ( )

Using normal vector analysis (see Bq.(C.2)),

 

df dIIb dIIIb
o—:-—= . _ + $0 D07
9 '[dt 911 dt9111) ( )

Therefore dot product becomes:

— +
d d
t J? t J3 t (D.8)

 

Since the maximum III}, value is 2/9, the dot product is always negative.

Realizability prolate and oblate line

174

Using the same method from Eq.(D.6), the invariant form of moment equations on the

prolate line become:

 

2 2 4
(111 III “ III ‘ 111 ‘
—9—=—12 —‘l 3 +2U 2 J 3’+1111,-3 ——‘-’- 3 (D-9a)
dt 6 6 6
2
dIII m ‘ III ‘
dtb =—3lIIb +3U 31-1111, +6[—€b-)3 —6IIIb [-6—bj3 (ng)

Afier taking dot product with normal vector from Eq.(C.10) to Eq.(D.9a) and Eq.(D.9b),
Eq.(D.7) is zero. Analogous to the prolate line, the oblate line provides the same result.

Therefore, decoupling closure in prolate line and oblate line are realizable.

Realizability of Hand’s Closure
From section 6.2 mention, Hand’s closure provides realizable orientation dyad on
the prolate and the oblate line. Based on general the invariant moment equations (see Eq.
(B5) and Eq. (B.10)), each the double dot products of Hand’s closure are:
2K PinPkPl >Hand=2 = bij [- 3%(13'121 +Iik1j1 + Iilek)

1 (p.10)
+ 7(Pipj11d + PiPkIjl +PiP11jk + IiijPl +IikPjP1 + Iiipjpkl] bij

1
2K P1P ijP1 >Hand=(2'2) =bij [‘Efiijlkl + 111:1 ji + 1111 jk)
1 (D.11)
+ 7(Pin1k1 + PiPkIjl + PiPlek + IiijPl +11kPjP1 + 1111’ij )](biybjy)
The right hand side from Eq. (D.11) becomes

2 4
—II +—III D.12
15 b 7 b ( )

175

—__-.. .—. _p

Substituting Eq.(D.]Z) into Eq. (B.5), the second invariant moment equation becomes:

c111,,

 

= —2Fm11b + ZFTDUI: 373-11}, +21%] (D.13)

The right hand side from Eq. (D.11) becomes

3 2 2
—II +_m D.14
7 b 15 b ( )

Substituting Eq.(D. 14) into Eq. (8.10), the third invariant moment equation becomes:

dIIIb
dt

 

= —3FTDIIIb +3FTDU[ %mb +li411b2] (D.15)

Realizability at planar anisotropic line
Substituting 11., and 1111, relationship at planar anisotropic line (111, = 2111b + 2/9) into

Eq.(D.]3) and Eq.(D.]S), the invariant form of moment equations become:

 

c111,, 4 2 29
——=—4HI -—-+U -+——III -16
(it b 9 [9 7 b] (D )
dIII

1’ =--3111b +U[-—6IIIb2 +3111b +327] (D.17)

Using normal vector analysis (see Eq.(C.2)),

 

df dIIb (1111b '
n--—==n- —e + e SO .18
_ dt ( -11 dt -111] (D )

T d t
Therefore dot product becomes:

_p_-ii:— 2(- 2 +9mb )(-105 + 2(8 + 4SIIIb)U) (D1813)
d t 9453

 

Therefore, Hand’s closure is not realizable at a certain U and 111;, (where 2(8+ 45111b )U

is more than 105).

Realizability prolate and oblate line

176

The invariant forms of moment equation of Hand’s closure on the prolate line become:

_2_ 2

 

 

(111 In 111 ‘
b —12 ——b 3+2U 9 —‘l 3 +9111}, (D.19)
dt 6 5 6 7
4
dIII "
b =—3IIIb+3U ;mb +29%? (p.20)

1/ 3
Multiplying Eq.(D.]9) and Eq. (D.20) with _%[%] (see Eq.(C.lO)), Eq.(D.7)
b

becomes zero. Therefore, 11 3% = 0 at the prolate line. Analogous to the prolate line,

the oblate line provides the same result. Therefore, Hand’s closure on the prolate line

and the oblate line are realizable.

Realizability of HLl Closure
HLl closure provides realizable orientation dyad on the prolate and the oblate line
for Fe = 0. Based on general the invariant moment equations (see Eq. (BS) and Eq.

(B.10)), each the double dot products of in 111.1 closure are:

l

b:< :=b —b. 6< >-b-< >-< >< >:b

= 2222> = 5-< <22 .. 22 22 22 2 (D21)
+21< 22> =-2 22<22><22>=2>=2

b:< >: ——b: 6< >b< ->< >< >:b

= <2222 £22)= < 22> 2 <>22 22 22 ._. (D22)
+21<252>=2-21< 22> - 2<2>=2>=(2-2)

Eq. (D21) becomes

1 2 2

§(4IIIb +3111, + 2111, ) (D23)

177

 

Substituting Eq.(D.22) into Eq. (8.5), the second invariant moment equation becomes:

 

d H
b = -2FT'D Hb + ZFTD U[: -1- 1111) + l 11b --Z 111,2] (13.24)
(1 t 5 5 5
Eq.(D.22) becomes
%(§mb + 2111,1111; +9113) (D25)

Substituting Eq.(D.25) into Eq. (B.lO), the third invariant moment equation becomes:

(1111,,
dt

 

= —3FTDIIIb +3FTDU[ £112, €111,111}, «Lg-611$] (D26)

Realizability at planar anisotropic line
Substituting 11b and 1111> relationship at planar anisotropic line (11), = 2111b + 2/9) into

Eq.(D.24) and Eq.(D.26), the invariant form of moment equations become:

 

dIIb 4 3
——=—4111 ——+U— —2+9111 5+72111 .27
dt b 9 405[( bx 1.)] (D )
dIII

b =-3111b +U[(—2+9IIIb)(l+9OIIIb)]. (D28)

Using normal vector analysis (see Eq.(C.2)),

 

dm" 2m] 5 0 . (D.29a)

Therefore dot product becomes:

. d_§ _ 2(— 2 + 9111b )(45 + 2(—2 + 9111b )U)

_n — (D.30b)
d t 405J§

 

Therefore, HLl closure is not realizable at a certain U and IIIb (where 2(—2+9IIIb )U is

more than 105).

178

Realizability prolate line

The invariant forms of moment equation of HM on the prolate line become:

 

 

2 2 4
(111 111 ‘ " III "
b =—12 —b— 3 +2U 9 E’- 3 +1111b £3 ——b- 3 (D31)
dt 6 5 6 5 5 6
4 5
(1111b 1 6 1111, 3' 72 1111, 3
=—3111 +3U —111 +— — —— — .32
dt b 5 b 5L 6) 5i 6) (D )

1/ 3
Multiplying Eq.(D.3 1) and .Eq. (D32) with -%[-III—:] (see Eq.(C.lO)), Eq.(D.7)

becomes zero. Therefore, 11 g—gt = 0 at prolate line. Analogous to the prolate line, the

oblate line provides the same result. Therefore, decoupling closure on the prolate line and
the oblate line are realizable.
Realizability of FSQ Closure
FSQ-model provides realizable orientation dyad on the planar anisotropic line if

8+45111b
18(1+9111b)'

 

2 In addition, FSQ-model provides realizable orientation dyad on the

prolate and the oblate line, regardless of C2 without the external field The first term of
FSQ-model is Hand’s closure and it is already proven in Eq. (D.18), Eq. (D.19), and
(D20). Based on general the invariant moment equations (see Eq. (B5) and Eq. (3.10)),

each the double dot products of in second term of FSQ closure are:

179

 

 

"O"

a, 2
2J2(<EE >):2=C2(llb,111b)bij [-3-§<p°pfi >< pupB >(Iij1kl +Iik1jl + Iilek)
+ <13in ><PkPl >+<Pipk ><PjP1 >+<PiP1><Pij >)
2
_7(< PiPY ><13:yl’i'>llcl+<pipr >< Pka >Ijl+<Pin ><PYPI >111

+ Iij <1)ka ><pr1 >+Iik <pjp7 ><pyp1 >+111 <13ij ><Pyl’k >)]bij

(D.33)

IIO‘

:32(<pp>):(2-2)
2
=C2(nbalnb)bij [§<Paps ><P¢1PB >(Iij1kl +12;le +Iu1jk)
+ (<19in ><PkPl >+ <PiPk ><PjPl >+<PiP1><Pij >)
2
—7(< piPy >< 13ij >Ik1+<PiPY XPYPk >Ijl+<PiP7 >< P791 >11k

+Iij <1)ku ><pypi >+Iik <1)ij >< p721 >+In <1)ij >< PyPk >)](biybjy)

(D.34)
Eq. (D.33) becomes
4 1 2 10 2 4 2( 2 )
— II +— +— II +—II +—III =-—7II +8111 +30111 D.35
35("3)21]"7"7"105b b b ()

Substituting Eq.(D.35) into Eq. (B.5), the second invariant moment equation becomes:

de 7 3 54 2
—=—2F II +2F U —II +—III ——II C .36
(it TD b TD [35 b 7 b 35 b 2] (D )
Eq. (D.34) becomes

4 1 2 3 2 10 3 2 2 54

— II +— +— III +—II +—II III =—II +—III +—II III .37
35(b3lzlib7b7bb7b15b35bb(D)

Substituting Eq.(D.37) into Eq. (B.5), the third invariant moment equation becomes:

dIIIb
dt

7 l 2 54
=—3F III +3F U —III +—II -—II III C D.38
TD b TD [35 b 14 b 35 b b 2] ( )

180

Realizability prolate line

The invariant forms of dynamic equation of FSQ-model on the prolate line become:

 

2 2 4
(111 111 ‘ 111 “ 111 ‘
——b =—12 —b 3 +2U 9 —b 3 +3112, -—1944 —b 3C2 (D39)
dt 6 5 6 7 35 6
4 5
(1111}, 1 18 111,, 3 1944 111,, 3
=—3111 +3U —111 +—— — -— — c .40
dt b 5 b 7( 6 i 35 i 6 j 2 (D )

1/3
Multiplying Eq.(D.39) and .Eq. (D.40) with "it—T6) (see Eq.(C.10)), Eq.(D.7)
b

becomes zero. Therefore, 11% = 0 at prolate line. Analogous to the prolate line, the

oblate line provides the same result. Therefore, FSQ-model closure on the prolate line

and the oblate line are realizable.

181

APPENDIX E

 

Eigenvalues and Eigenvectors of the Orientation Dyad

 

182

The objective of APPENDIX E is to find the eigenvalues and eigenvectors when

the dyadic valued operator has zero components in pxpy, pxpz, pypx, and psz. Thus, the

eigenvalues and the eigenvectors can be solved algebraically. If the dyadic valued
operator is given as:
<EB>’Z‘-i = Ai Ki (E-l)

, then it can be expanded to:

Pxpx 0 0 xix xix
0 Pypy Pypz xiy = A'i xiy a (E2)
0 pzpy Pzpz xiz xiz

where ZSi is the eigenvector, related to the eigenvalues Xi

Therefore,

Pxpx xix = Ai xix (E33)
PyPy Xiy + Psz Xiy = Xi xiy (5313)
mm Xiz + Psz xiz = 1i Xiz (E30)

Based on Eqs.(E.3b) and (E.3c), x1 x, xzy, x22, 1(3),, and X32 are only non-zero component

of eigenvector if all of non-zero components of the dyad from Eq.(E.2) and all of the
eigenvalues are different each other.

Using Eqs.(E.3b) and (E.3c),
[(pzpz -?~i) (Pypy ->~l)-pzpy Pszin2 = 0 (E4)
For i = 1, Eq.(E.3a) becomes:
PxPx xlx = 11 x1x (55)

Therefore, 11 is the same as pxpx.

183

The first eigenvector related to 2.1 becomes:

1
x]: xlx 0 (E6)
0

For i = 2, Eqs.(E.3a) and (E4) becomes:

pypzx22
X2 = — (E73)
y (pypy "'12)

 

[(prz —>»2> (pypy —82)—pzpy pyszxay = 0 (271»
Since X32 is non-zero component, the inside bracket in Eq.(E.7b) can be solved using

quadratic equation.

 

_ (pypy +pzpz)-\/(pypy +Psz)2 -4(pypy pzpz -pypz pzpy)

 

 

7»2 2 (E8)
From Eq.(E.7a), the second eigenvector related to lg becomes:
P0P
£2 2 - (WP: :12) x” (E9)
1
For i = 3, Eqs.(E.3b) and (E4) becomes:
x22 ___ _ Pzpyx22 (£103)
(13sz - 13)
[(22192 -?~3) (pypy -?~3)-pzpy pyszX3z = 0 (31%)

Since X32 is non-zero component, the inside bracket in Eq.(E.lOb) can be solved using

quadratic equation.

184

 

2
_ (Pypy +pzpz)+J(pypy +2222) -4(pypy pzpz -pypz Pzpy)

 

2 El]
3 2 ( )
From Eq.(E.7a), the third eigenvector related to 23 becomes:
(
0 i
x 3 = 1 x 22 (E12)
_ Pzpy
\ (1)sz _ 13 ) )

 

 

The eigenvectors are orthogonal to each other, the dot product of each eigenvector itself

becomes:
2121=2222=23°23=1 (E13)
Therefore,
XIX =1 (E.l4a)
P P 2
xZ-xz = 1+ _X_i_ xgy =1 (E.l4b)
pypy '32
P P 2
953 33 = 1+[—z—y——] x§z :1 (2.146)
Psz"x3

Solving Eqs.(E.14b) and (E.l4c),

x3, __. 1 (E.lSa)

PP 2
1+ ___2L£L_.
Pypy"12

and

185

 

 

x2 = (E.le)

z p p 2
pzpz-M

 

 

 

 

l
0
O
22 = - pypz 1 (E.l6b)
(Pypy "'12) p P 2
1 1+ —y-z_._
pypy ->~2
( N
0
53 = 1 1 (E.l6c)
-_..EL— 1 Pzpy 2
+ _—
( (Psz ’96)} pzpz —7L3

186

APPENDD( F

Realizability in the Planar Anisotropic Line

187

The objective of APPENDIX F is to derive the inequality equation in Ineq.(6.27).

In order to find the closure coefficient C2, Ineq.(6.1) is applied to the planar anisotropic

line. The outward pointing normal vectors in the prolate line are

n}? :1}? (RI)
nffi =72§ (F.2)

In the planar anisotropic line Eqs.(6.2b) and (6.2c) become

dIIb 4 4 58 16 96 423 2
_=———4111 +U—+—III — —+—111 +—111 c M
dt 9 b [45 35 b (105 35 b 35 b) 2] ( )
dIIlb 2 83 6 2 36 324 2
——=—3111 +U——+—III +-111 — —111 —-—m c F.4
dt b [189 105 b 7 b (35 b 35 b) 2] ( )

Multiplying Bq.(F .1) with Eq.(F.3) and Bq.(F .2) with Eq.(F .4), Ineq.(6.1) becomes

dF 4 2

 

 

n-—==-—+—m
‘dt 96 J? b
64 8 12 2
+U + H1 --———--111 (F.5)
[9454/3 105J§ 743 b
16 24 216 2
—( + 111 ——111 c ]so
105J§ 3545 b 353 b) 2

After arrange Eq.(F .5) the inequality becomes

2(—2 + 9111,, ){105 + 2U[—8 -45111b +18C2(l+ 1111,)]}< 0

9453

 

(R6)

188

APPENDIX G

Computational Code for Transient Calculations

189

The objective of APPENDIX G is to demonstrate the computation scheme using

flow chart and list the computational code for MATLAB 12.

 

[start]

1

time step (dt), number iteration (stop). shape factor (lam), U, Pe, solving order (order),
< p p > (a_real), Identity tensor (1D), velocity gradient (Du)

 

 

 

 

 

symmetric Tensor (s), vorticity tensor (w)

l

Dummy <32) (a)

l

n = number iteration (n) = stop yes

1110

yes
r = solving order (n) = 4?
i no

anisotropic tensor (b), eigen value of (lamda),
eigen vector of anisotopic tensor (1amdab)

l

second invariant (iid). third invariant (iiid),
C3 (C2). C1 (C1), difi‘usion coefficient (Drbar)

l

<2£>'<££> (a), 2'2 ('bb) ‘2? § (a5) <2£>'V“(aDu),
<22>§ <22> (aSa). <22>°V° <22>caDua). ”kw-9 (traceas).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

”(<pp>-V” (traceDu), <pp>c<pp> <pp >(aaa) b- b b’Cbbb)

l _

 

 

 

 

 

 

 

 

 

 

 

<2222>=< 22>(ppppddpp).< 132>1§ (ppppdds).
§<pp>+ <pp>S’(saas), §<pgpp> +<_p>-§’(waaw)
i <> 0 o

190

 

     

moment equation (dk)

 

 

 

1 yes
third Runge-Kutta
iteration (dk3)

fourth Runge-Kutta
iteration (dk4)
stress tensor (tau), stress components (W, X, Y. Z)
<32 > tensor (a), <p_ g >components (A, B, C, D, F), physical time (time)

/ print the results /<—-—

 

 

     

 

 

 

 

 

191

%clf
clear all

%time step
dt=0.003;

%number of time step
stop=20000;

%tumbling parameter
lam=0.987;

%nematic potential
U=27;

%Péclet number
Pe=30;

%1st order Euler order=1
%2nd order R-K order=2
%4th order R-K order=4
order=1;

%Constant Diffusion Coefficient Dr=1
%Doi Tube Dilation Dr=2

Dr=l;

dbar=l;

%Decoupling approximation closure=1
%Hybrid approximation closure=2
%FSQ approximation closure=3
closure=3;

%Doi stress st=l
%Ottinger stress st=2
st=1;

%Constant C2 const=1
%function of IIIb const=2
const=2;

c2const=l/3;

%Initial Condition
a_real(1,1)=0;a_real(1,2)=O;a_real(1,3)=0;
a_real(2,1)=O;a_real(2,2)=1/3;a_real(2,3)=0;
a_real(3,1)=0;a_real(3,2)=0;a_real(3,3)=2/3;

%Unit tensor;

ID(1,1)=1;ID(1,2)=0;ID(1,3)=0;
ID(2,l)=O;ID(2,2)=1;ID(2,3)=O;
ID(3,1)=0;ID(3,2)=0;ID(3,3)=1;

%Velocity gradient for simple shear flow;

192

 

Du(l,l)=O;Du(1,2)=O;Du(1,3)=0;
Du(2,l)=O;Du(2,2)=O;Du(2,3)=l;
Du(3,l)=O;Du(3,2)=O;Du(3,3)=0;

for i=l:3
for j=l:3
s(i,j)=.5*(Du(i,j)+ou(j.i));
W(i,j)=.5*(DU(i,j)-DU(j,i));
end
end
a_test=a_real;
a=a_real;
for n=l:stop

for r=lzorder
b=a-l/3*ID;

%Determine eigenvalues;

lamda1=a(1,l);

lamda2=(a(2,2)+a(3,3)-((a(2,2)+a(3,3))“2-
4*(a(2,2)*a(3,3)-a(2,3)*a(3,2)))“(1/2))/2;

lamda3=(a(2,2)+a(3,3)+((a(2,2)+a(3,3))“2-
4*(a(2,2)*a(3,3)-a(2,3)*a(3,2)))“(1/2))/2;

lamdab1=lamdal-l/3;

lamdab2=lamda2-l/3;

lamdab3=lamda3-l/3;

%determine C2
iiid=lamdab1“3+lamdab2“3+lamdab3“3;
iid=lamdabl“2+lamdab2“2+lamdab3“2;

if const==
C2=c2¢onst;
end
if const==2
C2=(8+45*iiid)/(18*(1+9*iiid));
end

C1=l-C2;

C3=0;
C4=1-C1-C2-C3;
c2(n)=C2;

%Tube Dilation
if Dr==
Drbar=dbar;
end

if Dr==

193

 

if st==l
Drbar=(l-3/2*iid)“(-2);

end

if st==2
Drbar=(l-3/2*iid)“(-l);

end

end

%Determine S[I(a*a)];
aa=a*a;
bb=(a-ID/3)*(a-ID/3);
aS=a*s;

aDu=a*Du;

aSa=aS*a;

aDua=aDu*a;
traceas=trace(aS);
traceaDu=trace<aDu);
aaa=aa*a;
bbb=bb*(a-ID/3);

%Determine aza (Ila) and alpha(2,1);
IIa=aa(l,1)+aa(2,2)+aa(3,3);
IIb=bb(1,1)+bb(2,2)+bb(3,3);

%Determine alpha(3,l);
IIIa=aaa(l,1)+aaa(2,2)+aaa(3,3);
IIIb=bbb(1,1)+bbb(2,2)+bbb(3,3);

if closure==
%Decoupling
ppppddpp=IIa*a;
ppppddS=traceas*a;
end

if closure==
%Hybrid
ppppddpp=(l-27*det(a))*IIa*a+27*det(a)*((-
1/35+l/7*IIa)*ID+3/35*a+4/7*aa);
ppppddS=(1—27*det(a))*traceas*a+27*det(a)*(-
2/35*s+1/7*(2*a*s+2*s*a+traceas*ID));
end

if closure==
%<pppp>=<pp>
ppppddpp=cl*((—
l/35+1/7*IIa)*ID+3/35*a+4/7*aa)+c2*(2/35*IIa*ID-
2/7*IIIa*ID+6/7*aaa-2/7*aa+39/35*IIa*a);

194

%<pppp>:S
ppppddS=Cl*(-
2/35*s+1/7*(2*a*s+2*s*a+traceas*ID))+C2*(4/35*s*IIa+a*traceas+2*a
Sa-2/7*(trace(aa*s)*ID+2*aa*s+2*s*aa));

ppppddDu=Cl*(-
2/35*Du+1/7*(2*a*Du+2*Du*a+traceaDu*ID))+C2*(4/35*Du*IIa+a*tracea
Du+2*aDua-2/7*(trace(aa*Du)*ID+2*aa*Du+2*Du*aa));

PPPPddSC2=(4/35*s*IIa+a*traceas+2*aSa—
2/7*(trace(aa*s)*ID+2*aa*s+2*s*aa));

end

saas=s*a+a*s;

waaw=w*a+a*(W');

if st==
ot=l;

end

if st==
ot=(6*(l-3/2*iid))“(-1/2);

end

%Moment Equation
dk=Drbar*((ID/3-a)+ot*(U*(aa-ppppddpp)))+lam*Pe*(saas-
2*ppppddS)-Pe*waaw;

% lst order Eular
if order==
a=a+dk*dt;
end

%4th order Runge-Kuttar
if order==4

if r==
dk1=dk;
a=a_real+dk1*dt*0.5;
end
if r==2
dk2=dk;
a=a_real+dk2*dt*0.5;
end
if r==
dk3=dk;
a=a_real+dk3*dt;
end
if r==4
dk4=dk;
a=a_real+(1/6)*(dkl+2*dk2+2*dk3+dk4)*dt;
end
end

195

 

%Second order Runge-Kuttar
if orde ==
if r==
a(i,j)=a_real(i,j)+dkl(i,j)*dt;
end
if r==
dk2(i,j)=dk(i,j);
a(i,j)=a_real(i,j)+(1/2)*(dkl(i,j)+dk2(i,j))*dt;
end
end
end

%Decomposition of Moment Equation
test1(n)=U*(aa(2,3)-ppppddpp(2,3));
test2(n)=U*(aa(2,2)-ppppddpp(2,2));
test3(n)=U*(aa(3,3)-ppppddpp(3,3));
test4(n)=Pe*(saas(2,3)-2*ppppddS(2,3)-waaw(2,3));
test5(n)=Pe*(saas(2,2)-2*ppppddS(2,2)-waaw(2,2));
test6(n)=Pe*(saas(3,3)-2*ppppddS(3,3)-waaw(3,3));
test7(n)=ID(2,3)/3-a(2,3);
test8(n)=ID(2,2)/3-a(2,2);
test9(n)=ID(3,3)/3-a(3,3);
test10(n)=dk(2,3);
test11(n)=dk(2,2);
test12(n)=dk(3,3);
test13(n)=Pe*(saas(2,3)-2*ppppddS(2,3));
testl4(n)=Pe*(saas(2,2)-2*ppppddS(2,2));
tetslS(n)=Pe*(saas(3,3)-2*ppppdds(3,3));
test16(n)=-Pe*waaw(2,3);
testl7(n)=-Pe*waaw(2,2);
test18(n)=-Pe*waaw(3,3);
test19(n)=(dk(2,2)-(ID(2,2)/3—a(2;2))-(U*(aa(2,2)-

ppppddpp(2,2)))-(-Pe*waaw(2,2)))/(Pe*(saas(2,2)-2*ppppddS(2,2)));
test20 (n)=(dk(3, 3) - (ID(3,3) /3-a(3,3) ) - (U* (aa(3,3)-

ppppddpp(3,3)))-(-Pe*waaw(3,3)))/(Pe*(saas(3,3)-2*ppppddS(3,3)));
test21(n)=(dk(2,3)-(ID(2,3)/3-a(2,3))-(U*(aa(2,3)-
ppppddpp(2,3)))-(—Pe*waaw(2,3)))/(Pe*(saas(2,3)-2*ppppddS(2,3)));

if st==1
tau=b-U*(aa-ppppddpp)+Pe*ppppddS/Drbar;
end
if st==2
tau=b-U/((l-IIa)“(l/2))/3*(aa-ppppddpp)+Pe*ppppddS/Drbar;
end

%Decomposition of Stress
N1btest(n)=b(3,3)-b(2,2);
Nlntest(n)=-U*(aa(3,3)-ppppddpp(3,3))+U*(aa(2,2)-
ppppddpp<2.2));
Nlhtest(n)=Pe*ppppddS(3,3)/Drbar-Pe*ppppddS(2,2)/Drbar;
N2btest(n)=b(2,2)-b(1,l);

196

N2ntest(n)=-U*(aa(2,2)-ppppddpp(2,2))+U*(aa(1,1)-
ppppddpp(1.l));
N2htest(n)=Pe*ppppddS(2,2)/Drbar—Pe*ppppddS(l,l)/Drbar;
shearb(n)=b(2,3)/Pe;
shearn(n)=-U*(aa(2,3)—ppppddpp(2,3))/Pe;
shearh(n)=Pe*ppppddS(2,3)/Drbar/Pe;

All(n)=all;
A22(n)=a22;
A33(n)=a33;
A23(n)=a23;
a_real=a;

%Components of <pp> and Stress
W(n)=tau(2,3);
X(n)=tau(2,2);
Ytn)=tau(1,l);
Z(n)=taU(3,3);
A(n)=a(111);

B(n)=a(2,2);
C(n)=a(3.3);
D(n)=a(2,3);
F(n)=a(3,2);

%Time step
time(n)=(n-l)*dt;

%First Normal Stress
Nl=Z-X;

%Second Normal Stress
N2=X-Y;
end

%Eigenvalue and Eigenvector

eigenvalue_1=A11;
eigenvalue_2=(A22+A33—sqrt((A22+A33).“2-4.*(A22.*A33-A23.“2)))/2;
eigenvalue_3=(A22+A33+sqrt((A22+A33).“2-4.*(A22.*A33-A23.“2)))/2;
eigenvector_32=1./sqrt(1+(A23./(B-eigenvalue_2)).AZ);
eigenvector_23=l./sqrt(1+(A23./(A33-eigenvalue_3)).“2);
eigenvector_33=(-A23./(A33—eigenvalue_3))./sqrt(l+(A23./(A33-
eigenvalue_3)).“2);
eigenvector_22=(-A23./(A22-eigenvalue_2))./sqrt(1+(A23./(A22—
eigenvalue_2)).“2);

eigenvalueb_1=eigenvalue_1-1/3;
eigenvalueb_2=eigenvalue_2-1/3;

eigenvalueb_3=eigenvalue_3-l/3;

%Shear Viscosity
shear=W/Pe;

197

%Invariants
II=eigenvalueb_l.“2+eigenvalueb_2.“2+eigenvalueb_3.“2;
III=eigenvalueb_l.“3+eigenvalueb_2.“3+eigenvalueb_3.“3;

%Invariant Domain
xIII=[-l/36:0.000l:2/9];
yII=2/9+2*XIII;

nyI=[O:0.000l:1/6];
xxIII=-6*(nyI/6).“(3/2);

ynyI=[O:0.000l:2/3];
xxxIII=6*(ynyI/6)-“(3/2);

plot(XIII,yII,xxIII,nyI,xxxIII,ynyI,III,II,'+')

198

 

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Abe, A. and T. Yamazaki, 1989, “Orientational Order of the a—Helical Poly(y-benzyl L-
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