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THESIS

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This is to certify that the
dissertation entitled
" Fluid Diffusion in Porous Silica"

presented by

LowellLMcCann

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in PhY'SiCS

Em! L OMW

Major professor T

 

Date February 10, 1998

 

MS U L! an Affirmative Action/Equal Opportunity Institution 0-12771

 

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Michigan State
University

 

 

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TO AVOID FINB return on or before date due.

 

DATE DUE

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JUL 2 7 2001

 

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1198 «mm-m4

 

FLUID DIFFUSION IN POROUS SILICA
By

Lowell I. McCann

A DISSERTATION
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1998

ABSTRACT
FLUID DIFFUSION IN POROUS SILICA
By

Lowell I. McCann

Fluid motion in porous media has received a great deal of theoretical and
experimental attention due to its importance in systems as diverse as ground water
aquifers, catalytic processes, and size separation schemes. Often, the motion of interest is
the random thermal motion of molecules in a fluid undergoing no net flow. This
diffusive motion is particularly important when the Size of the pores is nearly the same as
the size of the molecules. In this study, fluid diffusion is measured in several varieties of
porous silica whose pore structure is determined by the process by which it is made. The
samples in this study have porosities (ii), the ratio of the pore volume to the total sample
volume) that vary from 0.3 to 0.75 and average pore radii that range from approximately
15 to 120 A.

Determining the effect of the pore structure on the diffusion of a liquid in a porous
material is complicated by the chemical interactions between the diffusing molecules and
the pore surface. In this study, ions in a hydrophilic fluid are used to block the adsorption
of the diffusing dye molecules to the hydroxyl groups covering the Silica surface. This
technique is unlike typical surface treatments of silica in that it does not permanently alter
the pore geometry.

In this work, fluid diffusion is measured with a transient holographic grating

technique where interfering laser beams create a periodic refractive index modulation in

the fluid. The diffraction of a third laser off this grating is monitored to determine how
quickly the grating relaxes, thereby determining the diffusion coefficient of the molecules
in the fluid. Varying the grating periodicity controls the length scale of the diffusion
measurement from 1.2 to 100 pm which is much larger than the average pore sizes of the
samples. Therefore, over these large scales, we measure “normal” diffusion, where the
mean squared displacement of a diffusing particle varies linearly with time.

In one particular type of porous silica, manufactured to create a narrow
distribution of pore sizes in each sample, the normalized diffusion coefficient depends
upon 4) as D/D0 ~ ((1) - the)”, as (1) approaches a critical porosity (be. Here, D0 and D are the
diffusion coefficients of the free fluid and the fluid within the porous sample,
respectively. This result is compared with predictions of diffusion on a percolating
cluster of identical pores as well as with continuum models based on networks with a
distribution of pore sizes. While diffusion in these materials might be expected to behave
according to a continuum model of porous networks based on the aggregation of spherical
particles (the “Swiss—cheese” model), the behavior seen agrees with the prediction for
networks whose smallest bonds have a non-singular distribution of conductances. This
experiment is unique in that the materials chosen appear to produce a system that is close
enough to the percolation threshold to allow a measurement of the percolation exponents.
The diffusion coefficient in these samples is also shown to depend on the average pore
radius as D/Do ~ (Rp - Rc)°'49, a result which, while unpredicted, is shown to be consistent

with a previous study of fluid diffusion in silica.

To my parents.

iv

ACKNOWLEDGMENTS

I would like to thank my advisor, Brage Golding, for the many engaging
discussions about this research and for providing the opportunity to carry it out.

I also wish to thank Mike Dubson for all his help with my “other” thesis project,
and for sharing with me his boundless enthusiasm for Physics.

I thank my guidance committee members: Norman Birge, Tom Kaplan, Brad
Sherrill, and Dan Stump for their efforts to make sure I got it right.

The “shop guys”: Tom Palazzolo, Jim Muns, and Tom Hudson all have my
gratitude for their frequent help in the design and construction of a number of items over
the years. Baokang Bi and Reza Loloee have also been of great assistance on many
occasions.

I thank my fellow labmates: Jeeseong Hwang, Mike Jaeger, George Jeffers, J ung-
Uk Kim, Evstatin Krastev, Wenhao Wu, Ashraf Yussouff, Qifu Zhu, and particularly
Amy Bylsma Engebretson who deserves credit for my sanity after all these years in the
basement.

I have also benefited from my interactions with a number of faculty members
through the years: Bill Pratt, Phil Duxbury, Peter Schroeder, Carl Foiles, S.D. Mahanti,

and Jerry Cowen.

I am grateful to the succession of Julie Stone, Mary Curtis, and Sandy Teague in
Physics Stores for their help and pleasantness, even when I asked for something to be
rushed.

Finally, a special thank you to Stephanie Holland for taking on the MSU

bureaucracy a number of times to fix problems that she never caused, but always solved.

vi

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................... x
LIST OF FIGURES ............................................................................................................ xi
Chapter 1: INTRODUCTION ............................................................................................. 1
Chapter 2: BACKGROUND ............................................................................................... 4
I. Introduction .................................................................................................. 4

II. Diffusion and Brownian motion .................................................................. 4

HI. Theoretical models of diffusion in porous media ........................................ 9

a. Bundled tube models ........................................................................ 9

b. Discrete network models ................................................................ 12

c. Continuum network models ........................................................... 22

IV. Previous studies of diffusion in porous glass ............................................. 27

a. Techniques ..................................................................................... 27

b. Experiments ................................................................................... 34

V. Conclusions ................................................................................................ 42
Chapter 3: EXPERIMENTAL TECHNIQUES ................................................................ 47
I. Introduction and background ..................................................................... 47

H. Forced Rayleigh scattering theory .............................................................. 48

11]. Experimental setup ..................................................................................... 57

vii

IV. Detailed data acquisition procedure ........................................................... 63
Chapter 4: MATERIALS .................................................................................................. 67
I. Introduction ................................................................................................ 67
II. Porous Silica ............................................................................................... 67
a. Types of porous silica .................................................................... 67

b. Effect of heat on silica ................................................................... 69

III. Characterization of porous silica ............................................................... 71
a. Properties of porous silica .............................................................. 71

b. Common techniques for characterizing porous materials

............................................................................................ '723

c. Nitrogen adsorption ....................................................................... 75

IV. The manufacture and properties of Gelsil .................................................. 81
V. The manufacture and properties of gelled Ludox ...................................... 89
VI. The manufacture and properties of Vycor .................................................. 96
VII. Characteristics of the fluid and Methyl Red .............................................. 96
VIII. Surface chemistry of silica and its interactions with Methyl Red ............ 102
Chapter 5: DIFFUSION IN GELSIL POROUS SILICA ................................................ 113
I. Forced Rayleigh scattering decay curves ................................................. 113
H. Determination of the diffusion coefficient ............................................... 116
III. Temperature dependence of the Methyl Red lifetime .............................. 120
IV. Temperature dependence of the diffusion coefficient .............................. 121
V. Dependence of the diffusion coefficient on the porosity ......................... 125

viii

VI. Dependence of the diffusion coefficient on the pore size ........................ 127

VH. Conclusions .............................................................................................. 130
Chapter 6: DIFFUSION IN OTHER POROUS SILICAS .............................................. 138
I. Vycor ........................................................................................................ 138
II. Ludox glasses ........................................................................................... 140
III. Comparisons with the Gelsil data ............................................................ 143
IV. Conclusions .............................................................................................. 146
Chapter 7: SUMMARY AND FUTURE DIRECTIONS ............................................... 149
Appendix A: ABSORPTION MEASUREMENTS ........................................................ 153
Appendix B: DATA ACQUISITION AND ANALYSIS PROGRAMS ........................ 155

ix

LIST OF TABLES

Table 4.1: Character of the silica surface as a function of the temperature ...................... 69

Table 4.2: Properties of the Gelsil samples as determined from nitrogen
adsorption performed by Geltech, Inc.(GT) and Porous Materials,
Inc. (PMI). Rw is the Wheeler average pore radius, va and RSA are
the median pore radii as determined from the cumulative pore volume
and surface area, respectively ................................................................................ 82

Table 4.3: Properties of the HS-40 and AS-40 Ludox colloidal silica
suspensions ............................................................................................................ 89

Table 4.4: Properties of the Ludox porous glasses as determined from
nitrogen adsorption ................................................................................................ 93

Table 5.1: Activation energies of the thermal cis to trans isomerization of
Methyl Red in the free fluid and within the Gelsil samples ................................. 121

Table 5.2: Activation energies of D and Do in the free fluid and within the
Gelsil samples ...................................................................................................... 123

Table 5.3: Average normalized diffusion coefficients, D/Do, within the
Gelsil samples ...................................................................................................... 125

Table 6.1: The normalized diffusion coefficient in Vycor 7930 porous glass
compared to previous measurements. The study by Guo et al was made
with DLS using polystyrene (molecular weight 2500) as the tracer and
fluorobenzene as the fluid. Dozier et 01 made their measurements using
FRS with azobenzene as the tracer and (l) methanol/toluene or (2) l-
propanol/toluene as the fluid ................................................................................ 138

Table 6.2: Values of D/D0 in the seven gelled Ludox samples. All the data was
taken at 22.2 °C .................................................................................................... 141

LIST OF FIGURES

Figure 2.1: Schematic path of a random walker in (a) an unbounded fluid and
(b) in a fluid within a pore ....................................................................................... 7

Figure 2.2: The mean squared displacement (MSD) of a random walker over time
in three environments. Solid line: in an infinite cluster of pores, dotted line:
in a finite size cluster containing many pores, dashed line: in an isolated pore ...... 8

Figure 2.3: The tube geometry used in bundled tube models of porous media. rp
is the pore radius and rH is the hydrodynamic radius of the solute
molecule. The solute molecule is constrained to move along the tube axis
in this picture .......................................................................................................... 10

Figure 2.4: A two dimensional bond network. Top: at p = 1, bottom: at p = 0.37.
An infinite square lattice has a percolation threshold of pC = 0.5 .......................... 13

Figure 2.5: A schematic of the percolating cluster (also called infinite or sample-
spanning cluster). The solid lines are the bonds making up the backbone
of the cluster. The dashed lines are the Side branches ........................................... 15

Figure 2.6: The functional dependence of the percolation probability P, and the
conductivity 0' of a three dimensional network near pc .......................................... 16

Figure 2.7: Two dimensional Swiss-cheese model. The shaded spheres are the solid
phase and the white space are the pores. The solid lines are the open pores
and the dotted lines are the closed pores of the corresponding discrete network.
(After: S. Feng et al, Phys. Rev. B 35, 197 (1987).) .............................................. 23

Figure 2.8: Pulsed field gradient spin echo NMR pulse sequence. (After:
A. Mitzithras et al, J. Mol. Liq. 54, 273 (1992).) .................................................. 29

Figure 2.9: Dynamic Light Scattering experimental geometry. (After: K.S. Schmitz,
Dynamic Light Scattering by Macromolecules (Academic Press, New York,
1990).) .................................................................................................................... 31

Figure 2.10: Fluorescence recovery after photobleaching experimental setup.

(From: A. van Blaaderen et al, J. Chem. Phys. 96, 4591 (1992).) ......................... 33

xi

Figure 2.11: FRS decay rate for azobenzene in methanol/toluene within Vycor. The
squares (diamonds) are data from locations that do (do not) behave as yr oc qz.

(From: W.D. Dozier et al, Phys. Rev. Lett. 56, 197 (1986).) ................................ 35

Figure 2.12: FRS decay rate for azobenzene in l-propanol/toluene within Vycor.
(From: W.D. Dozier et al, Phys. Rev. Lett. 56, 197 (1986).) ................................ 36

Figure 2.13: Normalized diffusion coefficient, D/Do, of polystyrene in Vycor.
The x-axis is the ratio of the polymer hydrodynamic radius (m) to the
average pore radius (rp) of the glass. Shown are data for polystyrenes of
four different molecular weights. (From: Y. Guo et al, Phys. Rev. B 50,
3400 (1994).) ......................................................................................................... 38

Figure 2.14: Normalized diffusion coefficient of cyclohexane in sol-gel grown
silica as a function of (a) the pore radius and (b) the porosity. (Data from:
A. Mitzithras et al, J. Mol. Liq. 54, 273 (1992).) .................................................. 41

Figure 3.1: Schematic drawing of the interference region of a forced Rayleigh
scattering experiment ............................................................................................. 48

Figure 3.2: Intensity profile in the interference region of two crossed TEMOO laser
beams. The parameters for this Mathematics calculation are 4 mm beam
diameter, A = 488 nm, and 0 = 0.0005 rad. (After: A.E. Siegman,
J. Opt. Soc. Am. 67, 545 (1977).) .......................................................................... 49

Figure 3.3: (a) Sinusoidal intensity profile of the interference pattern. (b)
Corresponding profile of the change in concentration of the excited state
species. (c) Profile of the change in concentration of the ground state
species. The dotted lines illustrate the profiles after the interference pattern
is removed and the concentration grating is decaying through diffusion ............... 56

Figure 3.4: Experimental setup for the FRS measurements in this study. The
distance between the beam splitter and the sample cell is 2.3 m ........................... 58

Figure 3.5: Response of the photodetector to 632.8 nm light. Power measured
with Newport 1825-C meter and 818-SL detector ................................................. 61

Figure 3.6: Schematic drawing of the sample cell holder. The body and post are
made of copper separated by a thermoelectric heater/cooler used to control
the temperature of the cell. When in use, the opening (which reveals the
glass windows and the sample) is mostly covered with thin copper shim
stock to reduce heat loss from the window faces. The heater is 1 inch
square, and the opening in the cell body is 0.9 inches in diameter ........................ 62

xii

Figure 3.7: Schematic of the electronics used to control and monitor the FRS
measurements. The dotted line separates equipment that is on the optical table
(right hand side) from equipment on the overhead shelf. Italics indicates the
name of the connection on the equipment ............................................................. 63

Figure 4.1: The sol-gel process showing TEOS reacting with water to form silanol,
which then reacts to form the silica network ......................................................... 68

Figure 4.2: Schematic of the (a) hydrated and (b) dehydrated Silica surface,
showing the types of hyroxyl groups and bound water that can form. (After
R.K. Iler, The Chemistry of Silica (John Wiley and Sons, New York, 1979)
and K.K. Unger, Porous Silica (Elsevier, Amsterdam, 1979).) ............................. 70

Figure 4.3: (a) Nitrogen adsorption and desorption isotherms for Vycor porous glass.
(b) The pore size distributions calculated from (a) ................................................ 76

Figure 4.4: Top: Pore size distributions of the Gelsil porous silica samples calculated
from the desorption isotherms. Bottom: Pore size distributions of the Gelsil
porous silica samples calculated from the adsorption isotherms. Note the
discrepancy between the adsorption and desorption distributions, and with
the nominal average pore radius ............................................................................ 83

Figure 4.5: SEM image of the uncoated surface of the 28 A pore radius Gelsil
sample. The field of view is 2000 A x 2000 A ..................................................... 86

Figure 4.6: SEM image of the uncoated surface of the 91 A pore radius Gelsil
sample. The field of view is 6000 A x 4500 A ..................................................... 87

Figure 4.7: AFM topographic images of the surfaces of the 1.4, 2.8, and 9.1 nm
pore radius Gelsil samples (top to bottom). The images in the left column
and right column are (l um)2 and (200 nm)2, respectively .................................... 88

Figure 4.8: Raman spectra of HS-40 Ludox glasses heated to 600°C (top) and
850°C (bottom). The top spectra shows the broad Raman lines characteristic
of fused silica. The lines labeled in the bottom spectra are lines characteristic
of the crystalline silica phase of cristobalite. The spectra are offset and
magnified for clarity. Both were acquired using the 633 nm line of a HeNe
laser as the excitation. The insert shows the spectra of nonporous
amorphous silica .................................................................................................... 92

Figure 4.9: Pore size distribution of the AS-40 Ludox glasses calculated from the
desorption isotherms .............................................................................................. 94

Figure 4.10: Pore size distribution of the HS-40 Ludox glasses calculated from the
desorption isotherms .............................................................................................. 95

xiii

Figure 4.11: Pore size distribution of the Vycor porous glass calculated from the
desorption isotherm ................................................................................................ 97

Figure 4.12: Methyl Red molecule used as the tracer molecule in the FRS
experiments. The molecule is roughly planar with dimensions ~15 A x ~ 7 A.
(a) The alkaline and acid forms of Methyl Red, showing hydrogen bonding
to the azo bridge. (b) The trans and cis states of Methyl Red. The cis state
may be created through absorption of light, and will decay thermally back to
the trans state .......................................................................................................... 99

Figure 4.13: The absorption spectrum of Methyl Red in a mixture of 90% glycerol,
10% water, and 0.054 M NaOH ........................................................................... 100

Figure 4.14: The change in the absorbance of Methyl Red when illuminated with
488 nm light. Plotted is the absorbance with the pump beam off subtracted
from the absorbance with the pump beam on. A positive value means the
absorbance is larger when it is illuminated .......................................................... 101

Figure 4.15: The lifetime of the cis Methyl Red molecule in water with different
concentrations of NaOH. The “down triangle” point is the lifetime in the
90% glycerol/ 10% water solution used in the FRS measurements. The
lifetimes were determined by transient absorption at room temperature. The
dye was excited at 488 nm and the transmission of light at three other
wavelengths was monitored to determine the decay rate. The inset shows
one of these decays on a log scale ........................................................................ 103

Figure 4.16: (a) Silica surface reacted with hexamethyldisilazane to place
trimethylsilyl groups on the surface. (b) Silica surface with a N a+ counterion
(and its entourage of water molecules) above a charge site on the surface .......... 106

Figure 5.1: (Top): Normalized FRS decays of Methyl Red within the five Gelsil
samples (r, = 14.0, 16.4, 27.7, 35.5 and 91.5 A) at A = 1.225 mm and
10.0 °C plotted on a log scale. The Slope decreases with decreasing
pore size. (Middle and Bottom): FRS decays of Methyl Red in the free
fluid (middle) and within the 35.5 A pore radius Gelsil sample (bottom)
for three different values of A at 27.8 °C, plotted on a linear scale ..................... 114

Figure 5.2: The FRS decay rate (l/T) vs. q2 in the free fluid (top) and within the

16.4 A radius Gelsil sample (bottom) at four different temperatures. Also
shown as solid lines are the fits to Equation 5.2 .................................................. 117

xiv

Figure 5.3: The decay rate of the thermal isomerization of cis Methyl Red back to
trans Methyl Red (1/131) in the fluid and within all the Gelsil samples vs. l/I‘.
Each data set is fit to Equation 5.3 to determine an activation energy for
the process ........................................................................................................... 119

Figure 5.4: The diffusion coefficient in the free fluid and within the 35.5 A pore
radius Gelsil sample plotted vs. l/I‘. Both data sets are fit to Equation 5.4
to determine an activation energy ........................................................................ 122

Figure 5.5: D, Do, and D/D0 for the 14.0 A pore radius Gelsil sample as a function
of the temperature. D and D0 change by nearly and order of magnitude,
while D/Do is essentially constant ........................................................................ 123

Figure 5.6: The normalized diffusion coefficient, D/Do, for all the Gelsil samples
plotted against the temperature. The straight lines are the average values
of D/Do ................................................................................................................. 124

Figure 5.7: (a) D/Do plotted against the porosity, 4), of the Gelsil samples on a
linear scale. (b) D/Do plotted against 0 - 4),; on a log-log scale. The lines
shown are the fit to Equation 5.5. The value for (be used in (b) is that
determined from the fit ........................................................................................ 126

Figure 5.8: D/Do plotted against the average pore radius of the Gelsil samples.
The solid line is the fit to Equation 5.7 and the dotted line is the fit to the
Renkin model, Equation 5.8 ................................................................................. 128

Figure 5.9: D/D0 plotted against the pore radius for the data from Mitzithras et al,
for cyclohexane diffusing in sol-gel glasses. The line is a fit to Equation 5.7
of all but the top two data points. (After: A. Mitzithras et al, J. M01. Liq.
54, 273 (1992).) .................................................................................................... 129

Figure 5.10: Plot showing the range of the uncertainty of the continuum conductivity
exponent (II) from the fit to the Gelsil data and for two predictions of the.

continuum percolation model. 5: Gelsil data, with E = 1.94 i 0.32 .
I: Continuum model with CL S 0 , E = 2.0 i 0.2 . I: Continuum model

with (pg, n=2.44:gjgg .................................................................................. 134

Figure 6.1: The FRS decay rate (l/T) vs. q2 in the free fluid and within the Vycor
porous glass at 222°C. Also shown as solid lines are the fits using
Equation 5.2. Here D0 = 4.37 1- 0.01 x 10'8 cm2/s and
D = 4.4 :1: 0.5 x 10‘9 cmzls .................................................................................... 139

XV

Figure 6.2: The FRS decay rate (1/1:) vs. q2 in the free fluid and within the three
AS-40 Ludox porous glasses at 222°C. Also shown as solid lines are the
fits to Equation 5 .2 ............................................................................................... 142

Figure 6.3: The FRS decay rate (l/T) vs. q2 in the free fluid and within the four
HS-40 Ludox porous glasses at 222°C. Also shown as solid lines are the
fits to Equation 5 .2 ............................................................................................... 143

Figure 6.4: D/D0 in all the samples of this study as a function of (a) the porosity
and (b) the pore radius ......................................................................................... 144

Figure A. 1: Setup for transient absorption measurements. The dark shaded line
is the excitation laser, the light shaded line is the white light from the lamp,
and the PMT is the photomultiplier tube attached to the exit slit of the
monochromator .................................................................................................... 154

xvi

Chapter 1
INTRODUCTION

Porous materials could be defined as substances that “aren’t all there.” A solid
material is porous if it contains voids which are free of the solid phase and are
interconnected to provide a transport path through the material for a liquid or gas. By this
definition, most materials found in nature are porous and only a few materials like metals,
most crystals, and some plastics are nonporous. Most other solids, including rocks,
wood, soil, human skin, bones, and sponges are porous to some degree. The first, and
still most common use of porous media is as a filter to prevent objects larger than the
pores from being passed.

Porous materials play an important role in many fields and technologies. Among
these are oil recovery, groundwater motion, microbial transport, chromatography, and
catalysis. In chromatography, flows through porous gels or packings of porous particles
are used to separate molecules based on their size. The large surface area of porous
materials makes them very useful for chemical reactions which require the catalytic
action of a surface. The speed with which reactants are brought into contact with the
surface and the products removed determines the overall efficiency of the process.
Descriptions of these varied processes require an understanding of the motion of fluids
within the narrow passages of the pores. The fluid may flow by the pull of gravity or an
external pressure difference, but in small pores, the transport of fluids is dominated by
diffusion caused by the random thermal motions of the molecules. It is this motion which

is the focus of the present work.

2
In this study, the diffusion of a large number of probe molecules in porous silicas

is examined over length scales which are much larger than the sizes of the pores.
Diffusion on these macroscopic scales is different from diffusion on the scale of the pore
size and is affected by the average properties of the pore network. The main focus of this
work are materials created with a sol-gel process so that the average pore radius and the
porosity (the ratio of the pore volume to the total sample volume) are varied from sample
to sample in a controlled manner. In disordered porous media like rocks or sintered bead
packs, there is a wide distribution of pore sizes. These sol-gel glasses, however, are
prepared to have relatively narrow pore size distributions. This distribution of pore sizes,
in combination with the finite size of the diffusing molecules, causes the system to
approach the percolation threshold as the porosity and the average pore size are reduced.
Comparing our experimental results to continuum percolation models that account for
pore size distributions, we show that the diffusion is unlike that expected for porous
media made from the aggregation of spherical particles (“Swiss-cheese” models), where
the distribution of the conductances of the smallest pores contains a singularity as the
pores decrease in size. This leads us to believe that the pore structure of the sol-gel glass
is unlike that of these spherical particle models.

Another issue addressed in this work is the effect of chemical interactions
between the diffusing molecules and the surfaces of the pores. If the interaction is strong,
the diffusive behavior of the molecules in the pores is not Simply a result of the pore
structure. A number of techniques have been used in previous experiments to block the

adsorption of the diffusing molecules to the pore walls, but those techniques alter the pore

3
structure by effectively decreasing the pore size and even blocking off sections of the pore

network. In this work a different technique is employed that uses mobile ions attracted to
charge sites on the surface to block the adsorption.

The rest of this document is organized as follows. Chapter 2 introduces the topic
of diffusion and discusses several theoretical models describing diffusion in porous
media. Among the models investigated are percolation models of both discrete and
continuum networks. This chapter also contains a review of past fluid diffusion
experiments within porous silicas. The holographic technique used to measure diffusion
in this work is described in Chapter 3 along with Specific details regarding the equipment
and the measurements. Methods for characterizing porous materials are presented in
Chapter 4, along with the results of such measurements on the materials used in this
work. In addition, this chapter describes the structure and surface chemistry of the porous
silica samples as well as how that surface interacts with the fluid and the probe molecules
that are placed in the pores. Chapter 5 discusses the results of diffusion measurements in
specially prepared sol-gel grown glasses and compares the results to the different models
of diffusion in porous media. The results of diffusion measurements in other porous
silicas made with different processes are shown in Chapter 6 and are compared to the sol-
gel glasses. Finally, Chapter 7 summarizes the results of this study and lists potential

extensions of the research in the future.

Chapter 2
BACKGROUND

1. Introduction

Much of the interest in the flow of liquids in porous materials began with
experiments by H. Darcy in 1856 who studied the flow rate in a column of porous
material under the influence of external pressure and gravity.1 This rate was found to
depend upon the viscosity of the liquid and the permeability of the material. Later, many
workers tackled the problem of relating the permeability to the physical properties of a
material by modeling the structure of its pores.2 However, this type of forced motion is
not the only transport that occurs in fluids. Even for small (or nonexistent) pressure
differentials, fluids still move about due to the random thermal motion of the individual
molecules. This diffusive motion is responsible for the transport of liquids in small
structures where the pressure difference from one side to the other is extremely small.
Even in fluids at rest, diffusion produces transport through a liquid.
11. Diffusion and Brownian motion

Diffusive motion in fluids and solids has been widely studied3 since the discovery
of brownian motion4 and its later description with kinetic theory.5 In an isotropic system,
a single diffusing particle (or walker, if the particle is viewed in the context of a random
walk), released at time zero from the origin, will move such that the mean squared
displacement from the origin increases linearly with time. This relationship is easily
derived beginning with Fick’s First Law of Diffusion describing the flux of particles in a

concentration gradient:°

(2.1) I = —DVc(r, t)

where I is the particle flux through one cm2 in one second, D is the diffusion coefficient

in cmZ/S, and c('r°,t) is the number of particles in one cm3 at position 'r’ and time t.

Combining this with the equation of continuity:
(2.2) ——=—VoI,
we find:
(2.3) $- = DV2e(r, t) ,
dt

assuming that D is not a function of position (or a function of the particle concentration).
Solving this diffusion equation in one dimension by separation of variables, we find the

general solution to be:
(2.4) c(x, 0 =—- j C(v)exp(-72Dt>exp(iix)dv,

where C(y) is the Fourier transform of the initial concentration c(x,0). If we take c(x,0) to
be a delta function to describe a particle released from position x’ at time zero, then

(2.5) C(y) = (1/21t)

This produces the one dimensional probability distribution for finding a single diffusing

particle at position x and time t that was released from x = 0 at t = 0:

(2.6) c(x, t) =

 

1
e
ZJnDt
With this distribution, we can determine the mean squared displacement (MSD)

traveled by the diffusing particle in a time t. Because motion in all directions is equally

6
probable, the mean displacement (the first moment of the distribution) is zero, so the

MSD is the best measure of the distance traveled by the particle. Therefore, the MSD in

one dimension is:

(2.7) <x2> = Jx2c(x,t)dx = 2Dt.

In three dimensions,
(2.8) <r2> -_- 6Dt.
This result shows that classical diffusive behavior (i.e. Fickian diffusion) is characterized
by a MSD that is linear in time. It should be noted that in certain Situations7 the random
motion of a molecule may not obey Fick’s Law. This so-called anomalous diffusion is
characterized by <r2> ~ t° where 5 < l for “subdiffusion” and 5 > 1 for “superdiffusion”.
This can occur in situations such as the short-time motion on the percolating cluster of a
network of bonds8 or in fluids undergoing turbulent mixing.9

The solid phase of a porous material restricts the motion of molecules in a fluid
and causes a reduction in the effective diffusion coefficient from that found in the free
fluid. This effective diffusion coefficient accounts for the boundary conditions imposed
on the random walking particle by the pore walls. This effect is seen by imagining the
motion of a random walking molecule that starts its walk inside a pore (Figure 2.1).
Initially, it moves about free from interactions with the pore walls and will appear to
diffuse at the same rate as if it were in the bulk fluid. This changes once the walker
begins to interact with the pore walls. If the pore is isolated (i.e. there is no outlet from
the pore), the particle can never leave and therefore the measured effective diffusion

coefficient must drop to zero as time increases. If the pore is not isolated, the particle

"H

’

Figure 2.1: Schematic path of a random walker in (a) an unbounded fluid and
(b) in a fluid within a pore.

will eventually find its way through an opening and into another pore. In this case, the
particle will continue to diffuse through the porous material, but at a slower rate than it
would in the free fluid.

These situations are illustrated in Figure 2.2 where the MSD of a particle is
plotted as a function of time for several different environments. At short times the MSD
increases linearly in time with a slope proportional to Do, the diffusion coefficient for a
molecule in the unconstrained liquid. For longer times, the behavior depends upon the
particle’s environment. The dashed line illustrates diffusion in a single isolated pore,

where the MSD approaches a constant value related to the size of the pore. The slope in

2
<r >
A

N
V
ll
0
O
D
U)
B
E’,

 

 

Time

Figure 2.2: The mean squared displacement (MSD) of a random walker over time
in three environments. Solid line: in an infinite cluster of pores, dotted
line: in a finite size cluster containing many pores, dashed line: in an
isolated pore.

this region is zero and there is no diffusive motion. The solid line shows the behavior in
pores that are well connected to each other and diffusion is allowed, but at a slower rate
than in the free fluid (D < Do). The dotted line shows an intermediate case where the
particle is in a cluster of pores that is isolated from the rest of the porous network. Here,

the first two linear regions correspond to intrapore and interpore diffusion, respectively,

9
and the level of the constant region indicates the size of the cluster the particle is trapped

in.

These simple pictures demonstrate that the measured value of the diffusion
coefficient depends crucially on the environment where diffusion is occurring and the
time scale over which it is measured. For most technologically important situations
(groundwater motion and chromatography, for example) the diffusive motion of interest
occurs over distances large compared to the pores in the system. Therefore, the long-time
diffusion coefficient is of most interest in these situations.

Finally, it is important to note that the geometry of the porous media is not the
only characteristic that affects the diffusive motion of a fluid. As mentioned earlier,
porous materials are used in catalysis due to their large surface areas and the chemical
properties of the pore walls. Any chemical interaction between the walls and the fluid
will modify the molecular motion through the pores. Separating the geometric effects of
the pore sizes and shapes from the chemical effects of the surface is crucial for any
comparison to a model of the transport processes.
1H. Theoretical models of diffusion in porous media

III a. Bundled tube model

There are two common and distinct methods for modeling porous materials:
bundled tube models and network models.lo In bundled tube models, the porous medium
is viewed as a collection of nonintersecting capillary tubes whose diameter does not vary
along the length of each tube though it may differ from tube to tube. Hydrodynamic

models are a subset of these models and are concerned with the propagation of solute

10
molecules through channels of nearly the same size as the molecules,11 but with lengths

much larger than the diameter. The models are based on the assumption that the solute
molecule of interest is several times larger than the solvent molecules that surround it. In
this regime (rp >~ rH >> rsoivem, where rp is the radius of the channel, rH is the radius of the
solute molecule, and rsoivcm is the radius of the solvent molecule), the force on a single
solute molecule may be described by the hydrodynamic drag of the solvent on a similarly
shaped particle. For a spherical particle in an unconstrained fluid, this leads to an
expression for the diffusion coefficient given by the Stokes—Einstein equation:

(29) DSE = kBT/(67mfll),

where k3 is the Boltzmann constant, T is the temperature, I] is the solvent viscosity, rH is
the effective solute radius (sometimes called the hydrodynamic radius), and D55 is the
diffusion coefficient in a dilute solution. Placing the solute in a narrow pore (Figure 2.3),
the ratio of D, the diffusion coefficient in the confined fluid, to DSE may be calculated

over portions of the range 0 < rH/rp < l by considering the forces on the solute as the

 

 

 

Figure 2.3: The tube geometry used in bundled tube models of porous media. rp
is the pore radius and m is the hydrodynamic radius of the solute
molecule. The solute molecule is constrained to move along the tube axis
in this picture.

ll

solvent passes down the channel with a given velocity. The diffusive motion is recovered
by setting the solvent velocity to zero once the solution is found.

The most commonly used solution is that of the Renkin equation:

2 3 5
(2.10) D = [l — r—H] 1— 2.10445Ii + 2.089%] — 0.948['—"]
DSE rp rp rP rP

. . . r . .
which 15 valid for O _<. —”— < 0.4 and assumes the solute particle 18 centered on the tube

rp

 

axis. Regardless of the solution found in different regimes of this model,12 one feature is

always present:

 

(2.11) 11)) —>[ —%-J as :—”—91,
38 p p
with n > 1. This produces a gradual decrease in D/DSE as :1 —> 1 compared to the sharp
p

decrease seen if n < 1. As we will see later (Chapter 5), this is not the behavior found in
the disordered media examined in this work. The inapplicability of hydrodynamic
models to disordered porous media arises from the two oversimplifications assumed in
the model. First, is the assumption that 1'” >> rsolvcm. Frequently, rp >~ rH ~ rsolvcm for
small solute molecules, and so the hydrodynamic drag is not a realistic model of the force
acting on the solute. Second, and most important, is the description of the porous
medium as a collection of long, straight, independent channels. Disordered porous
materials are characterized by highly interconnected pores that are typically very irregular
in shape. For these reasons, it is difficult to view bundled tube models as valid

descriptions of disordered materials.

12
III b. Discrete network models

A more realistic model of disordered porous materials views the pore space as a
connected network13 where each bond in the network can be viewed as a pore. Each bond
can then be assigned a conductivity that is related to the size and shape of a corresponding
pore and is independent of all other pores. Just as in a real material, these bonds are
connected at nodes to form the full network. This level of detail makes it quite
complicated to solve for any transport property exactly and so, at least initially, we will
restrict the bonds to be identical.

Networks like these display percolation behavior.14 If bonds are randomly
removed from an initially complete network (e.g. a cubic lattice, where the bonds connect
nearest neighbor sites), eventually a bond is removed which severs the last path
connecting one side of the network to the other (Figure 2.4). The fraction (p) of the
original number of bonds that remain at this critical point defines the percolation
threshold, pc. A lattice filled with bonds such that p < pc is disconnected, while for p > pc
there is at least one sample-spanning path. This phenomenon is well known in systems
such as resistor networks and metal/insulator composites where an increase in the
fractional volume of metal can change the system from insulating to conducting.'5 In
porous media, the porosity (b (the ratio of the pore volume to the total volume), is often
viewed as being equivalent to p in a bond network. We should note that percolation is
not a true critical phenomenon because it does not involve a dynamic process.

One method for analyzing these complex networks is the effective medium

approximation (EMA).‘° Originally developed to calculate the dielectric constant of an

13

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.4: A two dimensional bond network. Top: at p = 1, bottom: at p = 0.37.
An infinite square lattice has a percolation threshold of pc = 0.5.

insulator with metal inclusions, EMA looks at a single bond and calculates the effect of
all the other bonds in the network on it. The rest of the network is divided into two
regions, the nearby bonds whose effect must be determined explicitly, and the distant
bonds which are replaced by a continuous effective field. Kirkpatrickl7 applied EMA to a
network of conducting bonds to make the first quantitative prediction of a percolation
threshold in a resistor network. EMA was used because it was the only analytical theory
capable of producing the necessary critical behavior. However, it tends to overestimate

the real value of the threshold. Extensions to three dimensions have shown that EMA

14
becomes even less accurate than for 2-dimensional nets. Several modifications to EMA18

have been made which improve the behavior near pC and make useful quantitative
predictions possible. However, EMA models require an accurate knowledge of the
distribution of pore Sizes (the distribution of the conductances of the individual bonds)
and of the connectivity of the network (how many bonds meet at each node) in order to
predict properties such as the total conductivity of the network. For disordered porous
media, this information is not typically well known, limiting the effectiveness of EMA.

A network near the percolation threshold behaves as a system near a critical point
and therefore its properties in that region behave as power laws of the occupation
probability p (recall that p is the ratio of the number of completed (occupied) bonds to the
total number of possible bonds). For example, near pc it is found that:

(2.12) P°<(p-p.)", §x(p-p.)‘”. andcoc(p-p.)“

where P is the fraction of all possible bonds that are occupied and connected to the
sample-spanning cluster, E is the correlation length of the system (proportional to the
average cluster radius when p < pc, and to the distance between locations on the network
that are connected by more than one path, when p > p), and o is the electrical
conductivity of a percolating network of resistive bonds. Critical exponents like [3 (=
0.41) and v (= 0.88), which depend on the connectivity of the network, can be found
through simulations, experiments, and scaling arguments and depend only upon the
dimensionality of the system, not the details of the network structure. This is one of the
strengths of percolation theory, it is a general statistical description with wide

applicability. However, not all percolation exponents behave in this universal fashion.

15

Specifically, transport exponents like u (s 2 for a simple uniform bond network), and
exponents of elastic properties do depend on the details of the network model, as we will
see later.

Before discussing the percolation predictions for diffusion, it is important to note
a few aspects of conduction on a percolating network. Because conduction requires a
continuous path across the network, it can only occur on the sample spanning cluster (also
referred to as the infinite or the percolating cluster). Any bonds which are part of isolated

clusters cannot contribute. The spanning cluster can be viewed as having two

 

 

 

 

 

 

 

 

--—- *1

 

 

 

 

---_r___ -__-
I
I
I

 

 

 

 

 

“WT ________

Figure 2.5: A schematic of the percolating cluster (also called infinite or sample-
spanning cluster). The solid lines are the bonds making up the backbone
of the cluster. The dashed lines are the side branches.

16

 

1.0 . . . . . u . u
0.41

P a (p - p)

0.6 - -

0.4 - 1

0.2 r -

 

 

 

0.0 ‘ f A A 1.0

Figure 2.6: The functional dependence of the percolation probability P, and the
conductivity 0' of a three dimensional network near pc.

components: a backbone and side branches (Figure 2.5). The backbone is made up of
those bonds that actually carry current, while the side branches (which are connected to
the backbone but lead nowhere) do not. This is the reason for the difference in the
percolation exponents of P and 0' (B = 0.41 and it = 2.0, respectively, in three

dimensions). P measures the number of total bonds in the spanning cluster while 0'

l7
depends only on the number of bonds in the backbone. Figure 2.6 illustrates the behavior

of P and 6 near the threshold.

A direct method for analyzing the properties of a fluid in a porous material is
through the use of computer simulations. A model of the pore structure is developed
(where the pore network is viewed as a network of bonds, with each pore corresponding
to an occupied bond, and the solid corresponding to the missing bonds) and random
walkers are released to explore the environment just as a diffusing molecule does. This
idea, first proposed by de Gennes,l9 has been used extensively to determine the transport
properties of percolation networks both near and far from the threshold, as well as to
model the signals produced in pulsed field gradient spin echo NMR diffusion
experiments.20 These simulations have provided an excellent means of examining the
properties of idealized networks.

Earlier in the chapter, it was shown that the diffusion coefficient measured in a
porous material depends upon the length scale that the measurement is made on. For the
same reason, a diffusion measurement on a network will also depend upon whether the
network is above or below the percolation threshold. If a random walker is released onto
the network and allowed to move along the occupied bonds, the MSD for very long times
will be:

(r2) oc D(p) t, for pc < p $1, and large t
(2.13) (r2) cc constant, for p < pc , and large t

(r2) oc t", for p = pc, and large t

18
Therefore, above pc, normal diffusion occurs (although the diffusion coefficient may be

reduced from its value at p = 1 when pC < p < 1). Below pc, the walker is confined to a
finite size cluster of bonds, and won’t diffuse any farther than some asymptotic distance
related to the average size of that cluster. Right at pc, these two regimes connect and

produce “anomalous” diffusion where n at 1. To find n, we first must determine the

behavior of (r2) below pc. The following discussion closely follows the derivation found

in Stauffer and Aharony.”

Here, the network is viewed as a collection of sites rather than bonds such that a
walker can move to a nearest neighbor site only if that site is “occupied.” Viewing the
lattice as a collection of sites rather than bonds changes only the predictions of the
percolation threshold-mot the values of the exponents.22 If a “blind” walker ( a walker
that cannot tell if a neighboring site is occupied or not) is placed at site i on the lattice,
where each site has 2 neighbors, the probability (Pi) that the walker is at site i evolves

with time as:

dP.
(2.14) P.(t+1)—P,<t) = 2mm) - mm] = 715’
i

where Yij is the probability per unit time of hopping from i to j (= 0 if site j is missing, =
1/z if site j is occupied). Looking only at a single finite cluster containing 8 occupied

sites, and for very long times so that the walker can visit each site many times, Pi will

reach an asymptotic value and 93:0. Equation 2.14 then implies that all sites are

equally probable, meaning that Pi(t —> oo) = 1. Therefore, the asymptotic distance that a
s

19
diffuser can reach on this finite cluster will simply be the average distance between two

points in the cluster, R3, the average cluster radius. For p < pc, all the clusters are finite,
so a walker placed at random on the network has a probability of nss of being on a cluster
of size s, where n, is the number of clusters of size 5 per lattice site. Therefore, the MSD
is

(2.15) (r2>(t = oo,p < p) = znsst.

it is

 

 

 

, Coclp—pci, and Rsocsx", wherex= and‘t = 2+

+7 [3+

If ns oc s’te'

CS I 1 B
Y

easily shown that
(2.16) Mo = «up < p.) cc (p. — p)”

Having determined the MSD below pc, we now turn to its behavior above pc. We
know that in this range (r2) oc D(p)t, and that D(p) decreases to zero as p approaches pc,
but have not determined how D(p) varies with p. Einstein23 showed that the diffusion
coefficient of a particle is proportional to its mobility, defined as the ratio of its velocity
to the applied force. For a conduction electron, the mobility is simply the current divided

by the applied voltage, which is the conductivity. The diffusion coefficient is then

proportional to the conductivity24 and therefore,

(2.17) DocO'oc(p—pc)” and,

(2.18) (r2)(t = 00.13 > 9.) cc (p- p.)“t.
[It may be tempting to say that D oc K , the permeability of a fluid flowing through

a channel, but it is an incorrect comparison. Because the forced flow of a fluid is

20
restricted by the boundary condition that the velocity is zero at the walls, a molecule at

the center of a channel travels faster than one near the walls (Hagen-Poiseuille flow). A
diffusing molecule, however, behaves more like a conduction electron whose mean free
path is essentially the same anywhere in a resistive wire.]

To reconcile these two results (Equations 2.16 and 2.18) at p = pc, we can

introduce a scaling function that depends on both t and p - pc:

(2.19) [(7) .. twp — pan] = t"P[Z].

For large t and p > pc, this function must behave as Eq. 2.18, therefore:

a g
(2.20) m oc tk[(p_pc)tx]2 oc (p_pc)2tk+xu/2

with k = (1/2)(1 - xii). For large t and p < pc, it must behave as Eq 2.16, so that

k

(2.21) p[z ——> oo] cc (—z)-;
this makes (r2) independent of t and implies that k = (x/2)(2v — B). Combining these

two values for k finds:

-13
(2.22) x=——1—— and k=—V—/2—.
2v+u-B 2v+u-B

Therefore, for long times and at p = pc, Equation 2.19 gives 11(3) oc tk as expected, where
k = 0.2 in three dimensions.
It is important to realize that the discussion above is concerned with the diffusion

of walkers that exist on all the clusters in network. When observed over sufficiently long

times, <r2> is constant on finite clusters and linearly proportional to time on the infinite

21
cluster, therefore only the walkers on the infinite cluster contribute to the measured

diffusion coefficient. In other words, on these time scales, only the fraction of walkers
that are on the infinite cluster can influence D. Since P is the fraction of all possible
bonds which are occupied and connected to the infinite cluster and p is the probability
that any given bond is occupied, P/p is the fraction of all occupied bonds that are part of
the spanning cluster. Because the walkers in this analysis are placed on occupied bonds at
random, P/p is the fraction of walkers on the spanning cluster. Therefore, because only
P/p walkers will contribute to the total diffusivity, we find that

(2.23) Dt oc <r2> cc (P/p)D't,

where D is the diffusion coefficient over the entire network and D' is the diffusion

coefficient on only the spanning cluster. This leads to:

(2.24) D' = 13% cc D(p - pc)'B

or,

(2.25) D' oc (p - pc)H3 for long times (and distances >> é).

As an aside, it has been shown that if walkers are released only on the percolating cluster,

but that the region explored is kept so that ,Krz) << {3, the diffusion is anomalous25 due

to the fractal nature of the percolation cluster over short length scales, with 11(3) oc tk' ,

V

—— = 0.26. Numerical simulations have verified this behavior for
2v+u-B

where k’ =

walkers on both the infinite cluster and the whole lattice.26

22
Equations 2.17 and 2.25 show that the diffusion percolation exponent near the

threshold depends upon whether the measurement is made over the entire network, or just
on the spanning cluster. If the measurement is made on the entire network, the
conductivity exponent will be recovered, but if only motion on the spanning cluster is
followed, that exponent will be reduced by B. In a porous material that is filled with fluid
while it is forming, the fluid will occupy all the pores, both those isolated and those
connected to the surface. Porous materials filled with fluid after forming will have fluid
only in those pores directly connected to the surface--which are the pores on the sample-
spanning cluster (neglecting a tiny volume of dead end pores on the surface). A
diffusion measurement made in these two materials should produce D and D',
respectively. Until now, no one has experimentally measured this behavior of D' in a
fluid system.

111 c. Continuum network models

Up to this point we have been discussing percolation models in which every bond
in the network is identical and the properties of the network arise solely from the number
of occupied bonds. This is not necessarily a realistic model of real porous materials
where the pores may vary considerably in size and shape. A more realistic model should
consider how the pore network forms. One of these models, the “Swiss-cheese” model,
creates a porous material by placing spheres at random into a volume, allowing them to
overlap. As more spheres are added or as the spheres grow in size, the volume not
occupied by spheres (the pore space) decreases until the percolation threshold is reached.

This pore space is highly disordered, with pores whose shapes and sizes vary widely. A

23
two dimensional representation of the “Swiss-cheese” model is shown in Figure 2.7,

where the shaded spheres make up the solid phase of the porous material, and the white
space is the pore structure. The black lines indicate the bonds of the discrete network that
the pore structure is mapped onto. In a three dimensional model, these discrete bonds are
located at the edges of Voronoi polyhedra that are constructed around the centers of the
spheres in the same manner as Wigner—Seitz cells are created. If the spheres are large
enough to overlap one of these bonds anywhere along its length, that bond is considered

to be absent from the discrete network (the dotted lines in Figure 2.7 are examples of

 

Figure 2.7: Two dimensional Swiss-cheese model. The shaded spheres are the
solid phase and the white space are the pores. The solid lines are the open
pores and the dotted lines are the closed pores of the corresponding
discrete network. (After: S. Feng et al, Phys. Rev. B 35, 197 (1987).).

24
these missing bonds). Each bond in this network corresponds to a narrowing in the pore

structure that has a characteristic width 8. The transport properties (e. g. the conductivity)
of each bond will vary with this width, producing a network where each bond is assigned
a unique conductance.

The percolation cluster of a network can be viewed as a group of strings that
connect to each other at nodes. Each string is made up of several “blobs” connected in
series by “links.” The blobs are regions of the network where more than one path exists
from one side of the blob to the other. The links, on the other hand, are singly connected
bonds--meaning that if any of the bonds in a link are broken, the whole string is broken.
A bond broken in a blob, however, will not cause the whole string to break. Therefore,
the “weakest” bond in the links of a string will determine the overall strength of the
string. Looking at this in terms of conductance, the bond with the lowest conductance
will determine the conductance of the whole string. So, in the “Swiss cheese” model, the
conductivity of the whole network near pc is determined by the bonds in the links with the
smallest conductances.

In the “Swiss cheese” model, if the pore space is considered to be an electrical
conductor and the spheres an insulator, a detailed analysis of the pore shapes produces an
analytical description of the conductance g (a function of the width 5) of each bond on the
network. This leads to a probability density function describing the distribution of the

conductances of the smallest bonds (pores):27

0c -%
(2.26) P(g) g ,asg—20,

25
where g is the conductivity of an individual bond. The difference between this model and

the networks discussed in the previous section is this distribution, because here each bond
is given a different “strength,” whereas all the bonds are identical in normal bond
percolation. It is found that the transport properties (such as the conductivity, the
permeability, or the elastic properties) of lattices with these distributions have different
percolation exponents than networks where every occupied bond is identical.28 However,
the exponents for properties that are determined solely by the connectivity of the network
(e.g. v and B), are unchanged.

A general continuum model of this type can be defined on a random, discrete
network such that:
(2.27) P(g) oc g“, as g —> 0
where P(g) is the distribution of the transport strengths g of the smallest bonds of the
network, and g is related to the “width” of the bond 8, through:
(2.28) g = 5’“.

Therefore, in this model, the distribution of conductances is singular as g —> 0.

If y and 0t are related by

(2.29) y = 13—

then the probability distribution of 5 behaves as:

(2.30) P(fi) —> constant, as 8 —> 0.

26

As the shapes of the pores in the network change, the values of y and or change.
This will alter the transport properties of the system, but not the pore size distribution,
which remains constant for small pores.

Feng et al.29 use both a scaling analysis of the structure of the percolating cluster
and a variational analysis of the conductance of the whole network to derive the upper
and lower bounds of the transport percolation exponents for this distribution of

conductances. For the conductivity exponent, It” , in this network, these bounds are:
(2.31) max(tt,+y,u) S E S u+y, wheny>0(or0<0tSl)

fi=tt, whenySO(or0tSO)
where u is the conductivity exponent found for networks of identical bonds and
(2.32) n, 2 1+ (d — 2)v.
Here, d is the dimensionality of the network, and v (= 0.88 for d = 3) is the percolation
exponent for the correlation length é. Therefore, for the three dimensional Swiss-cheese

model, where or = 1/3 and y = 1/2,

(2.33) 2.38 3 Emma... s 2.5.

Feng et al. argue that these bounds are valid even if there are correlations in the
occupation probabilities of the bonds, but that statistical correlations in the distribution of
strengths among occupied bonds will make these bounds inapplicable.

These predictions have been borne out in both numerical30 and experimental“
studies, but mostly in two dimensions. Feng et al. also show that this model predicts a
permeability exponent that is near values typically found in rocks, but they point out that

the measurements are made in the region where

27
(2.34) £¢_;9__) >1,

which is outside of the region where these asymptotic exponents should be found.

Therefore, we see that depending upon the characteristics of the porous material
and the location of the fluid that is diffusing, there are several different predictions for the
diffusive behavior near the percolation threshold. If the measurement is made in a fluid
trapped within all the pores, the diffusion coefficient should vanish as a power law with
an exponent u. If the measurement is made in the pores of the percolating cluster, this
exponent will be reduced to 11 - B. However, the value of it depends upon the geometry
of the pore structure as seen in the discussion of the Swiss Cheese model, giving at least
four possible values for the diffusion exponent. A measurement of this exponent might
provide information about the structure of the pore network.
IV. Previous studies of diffusion in porous glass

IV a. Techniques

Before the mid 1970’s, studies of diffusion in microporous glasses were difficult
to carry out in part because of the very slow diffusion that occurs in pores of molecular
size. Long times were required to measure diffusion over macroscopic distances in
standard diffusion cells, where a porous material is placed between two chambers of
solvent, with a solute added to one chamber. The concentration of that solute is then
monitored in the other chamber over time to measure the diffusion through the material.
Most importantly, for slow diffusion, any cracks in the material, or leaks around it, will

overwhelm the signal due to true diffusion through the pores.

28
Some of the first systematic studies of diffusion in microporous glasses were

made by Satterfield32 and Colton.33 In their experiments, porous glass beads were soaked
in a solvent containing a known amount of solute, then transferred to a container of the
pure solvent. The solute concentration in the solvent was then monitored to determine
the rate at which the solute escaped from the pores. These measurements were of
unsteady state diffusion, as the driving concentration gradient was continuously changing
as molecules left the beads. They compared their results to hydrodynamic models and
found some agreement with some solutes if an arbitrary “tortuosity” parameter, X, was

included to relate D/D0 to rH/rp,

D —__1. 111
(2.35) B—_Xf(/rp].

Here, f(rH/rp) is the result of a hydrodynamic model (like the Renkin Equation, Equation
2.10) of a sphere diffusing down a straight tube.

The last two decades has seen a dramatic increase in the number of techniques
available for the study of diffusion inside porous materials. Techniques such as pulsed
field gradient spin echo nuclear magnetic resonance (PFGSE), dynamic light scattering
(DLS), forced Rayleigh scattering (FRS), and fluorescence recovery after photobleaching
(FRAP) all emerged to allow the study of fluid diffusion in the interior of a porous
material without the need for any overall concentration gradient. By measuring over
relatively small length scales, the measurement of extremely slow diffusion became
possible on a reasonable graduate student time scale. Some of these methods are also
capable of measuring diffusion in a directional manner to determine if any anisotropy

exists in the fluid movement.

29
PFGSE is a variation of standard spin-echo NMR.”35 In a typical spin-echo

experiment, a uniform magnetic field in the 2 direction causes the protons of the
hydrogen nuclei to precess about the z axis with frequency (1)0, which depends upon the
magnitude of the field. A radio—frequency pulse applied along the y axis forces the
precessing “spin packets” into the x-y plane where they precess at different rates. After a
time 1:, a second, longer, RF pulse is applied which essentially reverses the precessional
motion of the spin packets. If all the packets maintain their phase coherence, they will all
reconverge at a time 't after the second pulse, producing the spin-echo signal.

In PFGSE, an additional inhomogeneous magnetic field is applied for a time 5 in
the 2 direction, with a magnitude g*z. This field is applied twice, once after the first RF

pulse, and then again (a time A later) after the second RF pulse (Figure 2.8). Because the

90° 1..., 180°

 

RF pulses

 

 

 

spin echo

   

magnetic
field pulses

 

 

 

V

A
D

attenuated I\ n
A

CChOCS (1116 U V“ V Unv UnU v
to diffusion

 

Figure 2.8: Pulsed field gradient spin echo NMR pulse sequence. (After:
A. Mitzithras et al, J. Mol. Liq. 54, 273 (1992).).

30
resonant frequency of the protons depends upon the magnetic field, this additional field

will increase 000 while it is applied. The second pulse (after the precession has been
reversed) corrects for the effect of the first pulse. However, if a proton has moved along
the z axis during the time A between the two field gradient pulses, it will experience a
different magnetic field during the second pulse than it did during the first pulse. In this
case, the resonant frequency of that proton during the two pulses will be different, and the
second pulse will not correct for the first one. Therefore, the amplitude of the spin-echo
will be reduced. For motion due to diffusion along the z axis, this decrease in the spin-
echo amplitude will be:

(2.36) I? = 6725250“.

where y is the gyromagnetic ratio for the proton, and D is the diffusion coefficient along
the z axis.

PFGSE experiments typically measure diffusion over times of a few milliseconds
which corresponds to length scales of a few microns. It is not easily used to measure
diffusion over large length scales. When using PFGSE to measure diffusion in a porous
material, magnetic impurities on the pore walls will interact with the diffusing particles,
and destroy the phase coherence. This relaxation of the magnetism is a significant
consideration in these experiments, and can lead to misinterpretations of the diffusion
coefficient.36

Dynamic Light Scattering measures diffusion by simply measuring the temporal

correlation of fluctuations in light scattered from molecules in a liquid.37 A laser beam of

wavevector K0 is incident on the sample (placed at the origin of the coordinate system,

31

see Figure 2.9), and the scattered light collected at position R at an angle of 0 from K0.

The electric field of this scattered light can then be written as:

_. _. a) 2112 . — - _.
(2,37) Escmed(R,t) = _-—092—R——Eoeux,.a_w,ua(K,t,),

where (no is the frequency of the incident light, 11 is the index of refraction of the

surrounding medium, K8 is the wavevector of the scattered light, a(K,t’) is the

polarizability of the scattering region, and K = K0 — Ks. If the frequency of the scattered

—.

light is the same as the incident light, then IKI = 2 K0

 

 

sin(g). Equation 2.37 shows that

the fluctuations of the scattered light arise from fluctuations of the polarizability in the
scattering volume which in turn arise from the motion of molecules in the solvent.

The autocorrelation function of the intensity of the scattered light is measured:

 

X Ex
KL incident light
K0
x
Z

 

scattered light

Figure 2.9: Dynamic Light Scattering experimental geometry. (After: K.S.
Schmitz, Dynamic Light Scattering by Macromolecules (Academic Press,
New York, 1990).).

32

_ . 1 V2 1 I_ 2 2
(2.38) (I(0)I(t))-— jg? j_%1(t)1(t +t)dt _(I(0)) +Q, <

 

E‘(0) - 13(1)]

 

E’a) - E<0>|).
where Qe is a constant. For a normal diffusion process, this can be shown to be:

(1(0)1(t)> _ 4x2. 2
(2.39) __(I(0))2 —1+A(e ),

where D is the diffusion coefficient, and A is a constant. Therefore, to determine D, the
decay of the correlation function is measured and fit to an exponential. Typically,
however, the measurement is made at several values of K to verify that D is independent
of K. For scattering at optical wavelengths, K ranges from approximately 5 x 104 to 4 x
105 cm", which corresponds to a range of length scales from about 0.1 pm to 1 tun.

FRS (discussed in detail in Chapter 3) measures diffusion by monitoring the decay
of a sinusoidal modulation of the optical properties in a fluid. Two coherent laser beams
cross each other in the liquid, creating a sinusoidal interference pattern to excite a tracer
molecule, giving rise to a corresponding modulation in the optical properties in the fluid
(e.g. the index of refraction). The diffraction of a third laser beam off this modulation is
used to monitor its creation and decay as the two exciting lasers are turned on and off.

The decay rate of the modulation is shown to be:

(2.40) l = W +i,
T

l
where D is the diffusion coefficient, q is the wavevector of the sinusoidal modulation, and
T] is the intrinsic lifetime of the excitation. Therefore, D and “t. are determined by

measuring the decay rate at several values of q (which is set by the angle between the two

33
exciting beams). q ranges from approximately 630 to 5 x 104 cm", corresponding to a

length scale of about 100 um down to 1 11m.

FRAP is very closely related to FRS, as it also uses interfering lasers to create a
modulated pattern in the fluid, as illustrated in Figure 2.10.38 In FRAP, however, it is a
pattern of fluorescent dye. A fluorescent dye is dissolved in the fluid, and the interference
pattern is turned on at a very high intensity for a short time. This pulse of light bleaches
the dye in the bright bands of the interference pattern. (When dye is bleached, it no
longer fluoresces.) At the end of this pulse, the intensity of the interference pattern is
reduced to a level which will cause the dye to fluoresce, but not bleach it, and the pattern
itself is slightly oscillated by modulating the phase of one of the two interfering beams.

The intensity of the in-phase fluorescent emission from the sample is monitored over time

 

 

 

 

3
83“ ML]

 

 

 

we: CONTROLLER lk
=53? FIUER (L—As

'
'c’
"

/
man 91:20 mature: / I
mason J, ", m
cuss mean I

SAME

 

 

 

 

' {Mean

 

 

   
 

 

 

 

Figure 2.10: Fluorescence recovery after photobleaching experimental setup.
(From: A. van Blaaderen et al, J. Chem. Phys. 96, 4591 (1992).).

34
using phase sensitive detection.39 As the unbleached dye diffuses into the regions that

had been bleached with the initial pulse, the signal increases until the concentration

becomes spatially uniform. The rate of this fluorescence recovery is:
1 2

(2.41) — = Dq ,
1'

where D and q are the same as in a FRS experiment. The length scales involved in FRAP
are the same as FRS. The advantages of FRAP over FRS are that the bleaching is
permanent, so there is no intrinsic lifetime of the excitation, and the sample does not have
to be transparent. FRS needs transparent media because it relies on the diffraction of the
probe beam, and any scattering in the sample will affect the diffracted signal. However,
because the bleaching is permanent in FRAP, the signal level decreases with every pulse,
making signal averaging difficult, while the finite lifetime in an FRS experiment allows
an experiment to be repeated many times without having to change the fluid.

IV b. Experiments

One well-characterized commercial porous glass (Vycor 7930) has been used in
fluid diffusion studies on several occasions. Vycor is a borosilicate glass which is heat
treated to phase separate it into boron rich and boron poor regions. It is then placed in an
acid bath to selectively etch the boron rich regions-~leaving behind a tortuous network of
pores of highly uniform size (pore radius of ~ 25 A).40 Dozier et al. 4‘ used FRS to
measure the diffusion of azobenzene in alcohol-toluene mixtures inside the pores of
Vycor. Diffusion in glasses is affected by two main things: the geometry of the pore

network and the adsorption of the diffusing molecules to the pore surface. Dozier et al.

35
were interested in determining only the effect of the pore network structure on diffusion,

and so chemically blocked the adsorption of azobenzene to the walls.

The Vycor was first soaked in methanol so that it would attach to the binding sites
(hydroxyl groups) on the surface, and not allow the azobenzene to adsorb when it was
added to the fluid. However, with this treatment they found FRS signals that were
independent of the grating wavevector, q, indicating that the azobenzene was not
diffusing (Figure 2.11). Their second attempt was to derivatize the glass with l-propanol
in order to replace the hydroxyl groups. This was done by boiling the Vycor in 1-
propanol for 24 hours. With this treatment, they found normal diffusive behavior (an FRS

decay rate that was proportional to qz, see Figure 2.12) with a normalized diffusion

 

 

 

1.5 _ O .
O
O O
O
1.0 b
g I
i; I
F
0.5 -
D~4 x10-' cull/sec
I
0.0 I I 1 L I l I I 1 l

 

0 4 8 12 16 20
q3(cm - 2) x 1 0' '

Figure 2.11: FRS decay rate for azobenzene in methanol/toluene within Vycor.
The squares (diamonds) are data from locations that do (do not) behave as

% oc q2 . (From: W.D. Dozier et al, Phys. Rev. Lett. 56, 197 (1986).).

36

 

1.5 '-

1Ir (Hz)

0.5 i"

 

 

0 1 1 1 1 1 1 1
0 5 1O 15 20 25 30 35

q3(cm 4) x 10'

 

Figure 2.12: FRS decay rate for azobenzene in l-propanol/toluene within Vycor.
(From: W.D. Dozier et al, Phys. Rev. Lett. 56, 197 (1986).).

coefficient (D/Do) on the order of 0.01-0.03 for two different solvents. However, even in
their treated sample, they have a small number of quite scattered data points. In Figure
2.12, the slope of the fitted line is forced to be lower than the slope of the data because
the line (fitted with Equation 2.40) cannot intercept the y axis below the origin. This
leads to considerable uncertainty in their conclusions.

They compared their result with two separate models of diffusion in porous
materials. One was the bundled-tube model from which they calculated a value of the
“tortuosity”, T, of the pore network:

(2.42) D/D0 = (WT.

37
where q) = 0.28 is the porosity of their Vycor sample. However, as they pointed out, this

tortuosity parameter is not well defined and does not provide any true measurement of the
geometry of the pore network.42

At the time, the pore network of Vycor was suspected to be fractal in nature, so a
model relating the estimated fractal dimension and the diffusion coefficient was
compared to their measured result for D/Do. However, in a later publication they noted
that the pore network in Vycor had been shown to not be fractal, and therefore this
comparison was inappropriate.43 Finally, they found an order of magnitude difference
between their measured value of DID0 and that calculated from the Renkin equation (Eq
2.10). That is not surprising, however, since the Renkin equation is based on a model of
a single straight tube, and the pores in Vycor do not resemble straight tubes. Because
only one glass was examined in this study, any conclusions drawn about the dependence
of D/Do on the porosity or pore size is, at best, speculative.

At the University of Massachusetts, Guo et al.“, made DLS measurements of
polystyrene diffusing in the same Vycor glass. Again, adsorption of the diffusers to the
surface was a concern, so the hydroxyl groups on the surface of the silica were reacted
with hexamethyl disilazane to prevent hydrogen bonding of the polystyrenes. For four
polystyrenes, ranging in molecular weight from 2500 to 13,000, they found D/Do to range
from 0.1 to 0.01 (see Figure 2.13, where the normalized diffusion is plotted versus M; =
rH/rp, the ratio of the polystyrene hydrodynamic radius (m) to the pore radius in the Vycor
glass). These values are the same or greater than what Dozier et al. found for the

diffusion of the much smaller azobenzene molecule. Guo et al. attributed this difference

38
to possible residual unreacted hydroxyl groups in the earlier experiment, which would

also result in a reduced diffusion coefficient in the pores. The authors also stated that
they briefly examined the diffusion of azobenzene in the Vycor glass that had been
treated with hexamethyldisilazane, and found that the diffusion was occurring faster than
they could measure. This is very surprising, because the diffusion coefficient of
azobenzene and Methyl Red in solvents such as methanol/toluene, l-propanol/toluene,
benzene, ethanol, and 2-propanol is in the range of 1 x 10'6 to 1 x 10'5 cm2/s, which is

easily measurable.45 It is hard to explain how azobenzene in Vycor could diffuse faster

 

 

 

 

thanthat.
0.3 , j
o 0.2 " d
O
\ \
Q 0.1 - .
0. ' 1
8.0 0.5 1.0 1.5

AH

Figure 2.13: Normalized diffusion coefficient, D/Do, of polystyrene in Vycor.
The x-axis is the ratio of the polymer hydrodynamic radius (m) to the
average pore radius (rp) of the glass. Shown are data for polystyrenes of
four different molecular weights. (From: Y. Guo et al, Phys. Rev. B 50,
3400 (1994).).

39
One drawback to the surface treatments used in both these experiments is their

modification of the pore structure.46 The replacement of relatively small hydroxyl groups
with larger molecules in small channels, or at a small opening to another pore, can close
off portions of the pore network to the fluid. By restricting the fluid to the largest pores
in this way, the apparent diffusivity in the material will increase. Neither of these studies
examined the dependence of the diffusion on the pore size of the glass.

In an extensive series of DLS experiments,47 the group at the University of
Massachusetts examined the behavior of polystyrene diffusion in acid-etched glasses
similar to Vycor as a function of the polymer molecular weight, polymer concentration,
and the solvent species. The glasses studied in these experiments had pore radii ranging
from 75 to 1866 A, which are considerably larger than the pores in Vycor. They varied
the molecular weight of the polystyrene to change the hydrodynamic radius of the probe

molecule and compared their results in each glass to predictions of hydrodynamic models:

D__1_ r“
(2.43) _D——Xf(K/rp)’

o
where f(rH/rp) is one of several hydrodynamic models, such as the Renkin equation, X is
the “tortuosity”, an extra free parameter to take into account that the pores are connected
rather than single, isolated tubes, and 1c, another free parameter to account for the fact that
the linear polystyrene molecules are not the spheres that hydrodynamic models are
derived for.48 This analysis was able to fit their data for small values of rH/rp, but the use
of parameters such as the tortuosity makes it difficult to draw any useful conlusions about
the pore size dependence of diffusion in these large pore, long molecule systems. In these

studies, they only looked at how D/Do varied with r”, not with changes in rp.

40
Finally, the same group at the University of Massachusetts used FRS to measure

polystyrene diffusion in suspensions of silica in polystyrene. As expected, they found
that the diffusion coefficient decreased with increasing volume fraction of silica (up to
approximately 13% silica by volume), and showed that above 6% silica, where the
suspension eventually gels, the behavior deviated from theories of diffusion in
suspensions. In these gelling samples, they also tracked the diffusion coefficient over
time, and found a 5% reduction in D as the suspension gelled.

One systematic PFGSE study of the pore size dependence of diffusion has been
made with cyclohexane diffusing in porous silica powders and beads with pore radii
ranging from 245 - 30 A.49 Cyclohexane was chosen because it interacts only weakly
with the silica surface and therefore is not hindered by chemical bonding. As the pore
radius, RP, of the silica samples decreases, they found that D/D0 spans the range of
0.91 - 0.61. They fit their data to an empirical function (Figure 2.14a)

D -b(2r )‘
2.44 — = e "
( ) D

(with b = 71.3 and 8 = -1.02), which is based on no model and related their result to a
model based upon diffusion through stacks of thin membranes to describe l/b as an
“effective permeability.” Although their data shows the expected trend of D/Do
decreasing with rp, their data is limited to relatively large values of D/Do, and they make
no statement about how D/Do approaches zero. Also, they found no systematic
dependence of the diffusion on the limited range of porosities of their samples (Figure

2.14b).

41

 

 

 

 

0.0' . .

 

 

 

 

 

0 50 100 150 200 250
Pore radius (A)
1.0 . . , . , . ,
’ b
o 9 _ ) O o -
0.8 — _
_ 0
go 0.7 - o -
. o .
D 0.6 - -
. O 1
0.5 - ° -
0.4 - -
1- O '1
03 l I l 1 l . l I
0.60 0.65 0.70 0.75 0.80
Porosity

Figure 2.14: Normalized diffusion coefficient of cyclohexane in sol-gel grown
silica as a function of (a) the pore radius and (b) the porosity.
(Data from: A. Mitzithras et al, J. Mol. Liq. 54, 273 ( 1992).).

42

In other studies: PFGSE experiments by Kiirger50

examined the temperature
dependence of diffusion of several solvents in porous glasses although they did not draw
a conclusion as to the dependence of diffusion on the pore size or structure. Later,
D’Orazio51 used PFGSE to study the diffusion in partially filled pores of single silica
glass and found the behavior to be described by a variation of Archie’s Law.52 And, in a
FRAP experiment, where the length scale of the diffusion measurement is easily
controlled, Messager53 saw the crossover from intrapore (free fluid-like) diffusion to
interpore (hindered) diffusion in large porosity fumed silica gels.
V. Conclusions

With these past experiments as a base, the work described in this thesis was
initiated to examine the behavior of diffusion in porous glasses as a function of both their
porosity and average pore radius and use the results to distinguish between the models

that attempt to describe this behavior.

43
References

 

1 H. Darcy, Les fontaines publiques de la ville de Dijon, (Dalmont, Paris, 1856); A. E.
Scheidegger, The Physics of flow through porous media, 3rd ed., (University of Toronto
Press, Toronto, 1974).

2 A.E. Scheidegger, ibid., p. 140.

3 w. 1081, Difi‘usion in Solids, Liquids, Gases (Academic Press, New York, 1960).

4 R. Brown, Phil. Mag. (4) 1828, 161.

5 A. Einstein, Ann. D. Phys. 17, 549 (1905); A. Einstein, Investigations on the Theory of
the Brownian Movement, edited by R. Fiirth, translated by AD. Cowper (Dover, New
York 1956).

6 w. Jost, ibid., p.7.

7 M.A. Knackstedt, B.W. Ninharn, and M. Monduzzi, Phys. Rev. Lett. 75, 653 (1995); A.
Ott et al, Phys. Rev. Lett. 65, 2201 (1990); M. Sahimi, Rev. Mod. Phys. 65, 1393 (1993).

8 D. Stauffer and A. Aharony, Introduction to Percolation Theory, (Taylor and Francis,
London, 1994).

9 TH. Soloman, E.R. Weeks, and H.L. Swinney, Phys. Rev. Lett 71, 3975 (1993).

10 F.A.L. Dullien, Porous Media: Fluid Transport and Pore Structure (Academic Press,
New York, 1979), p. 42.

“ W.M. Deen, Amer. Inst. Chem. Eng. J. 33, 1409 (1987).
‘2 w.M. Deen, Amer. Inst. Chem. Eng. J. 33, 1409 (1987).
'3 Dullien, ibid., p. 44.

'4 D. Stauffer and A. Aharony, Introduction to Percolation Theory, (Taylor and Francis,
London, 1994); SR. Broadbent and J .M. Hammersley, Proc. Camb. Phil. Soc. 53, 629
(1957).

'5 B.I. Shklovskii and AL. Efros, Electronic Properties of Doped Semiconductors
(Springer-Verlag, Berlin, 1984); Electrical Transport and Optical Properties of
Inhomogeneous Media, edited by J .C. Garland and DB. Tanner (AIP Press, New York,
1978).

44

 

'6 M. Sahimi, Applications of Percolation Theory (Taylor and Francis, London, 1994); R.
Landauer in Electrical Transport and Optical Properties of Inhomogeneous Media,
edited by J .C. Garland and DB. Tanner (AIP Press, New York, 1978).

'7 S. Kirkpatrick, Rev. Mod. Phys. 45, 574 (1973).

'8 M. Sahimi, L.E. Scriven, and H.T. Davis, J. Phys. C. 17,1941 (1984).

‘9 P.G. de Gennes, La Rechereche 7, 919 (1976).

2" P.N. Sen et al, Phys. Rev. B 49, 215 (1994); DJ. Bergman et al, Phys. Rev. E 51, 3393
(1995); L.M. Schwartz et a1, Physica A 207, 28 (1994);L.M. Schwartz and J .R. Banavar,
Phys. Rev. B 39, 11965 (1989); A. Bhattacharya, S.D. Mahanti, and A. Chakrabarti, Phys.
Rev. B 53, 11495 (1996); M. Sahimi and V.L. Jue, Phys. Rev. Lett. 62, 629 (1989); P.
Argyrakis and R. Kopelman, Phys. Rev. B 29, 511 (1984).

21 D. Stauffer and A. Aharony, ibid., Chapter 6.

22 D. Stauffer and A. Aharony, ibid, p. 17.

23 A. Einstein, Investigations on the Theory of the Brownian Movement, edited by R.
Fiirth, translated by AD. Cowper (Dover, New York 1956).

24 P.G. de Gennes, La Rechereche 7, 919 (1976).
25 Y. Gefen, A. Arahony, and 8. Alexander, Phys. Rev. Lett. 50,77 (1983).

26 D. Ben Avraham and S. Havlin, J. Phys. A 15, L691 (1982); RB. Pandey et al, J. Stat.
Phys. 34, 427 (1984).

27 s. Feng, B.I. Halperin, and RN. Sen, Phys. Rev. B 35, 197 (1987)

2" P.M. Kogut and JP. Straley, J. Phys. C 12, 2151 (1979); A. Ben-Mizrahi and DJ.
Bergman, J. Phys. C 14, 909 (1981); J .P. Straley, J. Phys. C 15, 2333 (1982); J .P. Straley,
J. Phys. C 15, 2343 (1982); J. Petersen et al, Phys. Rev. B 39, 893 (1989).

29 S. Feng, B.I. Halperin, and RN. Sen, ibid.

30 RN. Sen, J .N. Roberts, and BI. Halperin, Phys. Rev. B 32, 3306 (1985); J .N. Roberts
and L.M. Schwartz, Phys. Rev. B 31, 5990 (1985).

3‘ C.J. Lobb and M.G. Forrester, Phys. Rev. B 35, 1899 (1987);L. Benguigui, Phys. Rev.
B 34, 8176 (1986).

45

 

32 C.N. Satterfield, C.K. Colton, and W.H. Pitcher, Jr., Amer. Inst. Chem. Eng. J. 19, 628
(1973).
33 C.K. Colton, C.N. Satterfield, and C.-J. Lai, Amer. Inst. Chem. Eng. J. 21, 289 (1975).

34 J. Karger, in Access in Nanoporous Materials, edited by T.J. Pinnavaia and M.F.
Thorpe (Plenum Press, NY, 1995), p. 175.

35 W.B. Mims, in Electron Paramagnetic Resonance, edited by S. Geschwind (Plenum
Press, NY, 1972), p. 263.

36 A. Bhattacharya, S.D. Mahanti, and A. Chakrabarti, Phys. Rev. B 53, 11495 (1996);
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143 (1992).

37 KS. Schmitz, An Introduction to Dynamic Light Scattering by Macromolecules
(Academic Press, Boston, 1990).

38 D. Axelrod et al, Biophys. J. 16. 1055 (1976)-

39 D. Chatenay et al, Phys. Rev. Lett. 54, 2253 (1985); A. Irnhof et al, J. Chem. Phys.
100, 2170 (1994); A. van Blaaderen et al, J. Chem. Phys. 96, 4591 (1992).

40 P. Levitz et al, J. Chem. Phys. 95, 6151 (1991).
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42 PAL. Dullien, Porous Media: Fluid Transport and Pore Structure (Academic, New
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43 J.M. Drake and J. Klafter, Phys. Today, May, 1990, p. 46.

44 Y. Guo, K.H. Langley, and FE. Karasz, Phys. Rev. B 50, 3400 ( 1994).

45 M. Terzima, K. Okamoto, and N. Hirota, J. Phys. Chem. 97, 5188 (1993).

46 J.M. Drake and J. Klafter, ibid.

47 Y. Guo, K.H. Langley, and FE. Karasz, Macromolecules 23, 2022 (1990); I. Teraoka
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48 MT. Bishop, K.H. Langley, and FE. Karasz, Macromolecules 22, 1220 (1989).

46

 

49 A.Mitzithras, F.M. Coveney, and J .H. Strange, J. Mol. Liq. 54, 273 (1992).

50 J. Karger et al, J. Amer. Chem. Soc. 66, 69 (1983); K. Fukuda et al, J. Phys. Soc. Jpn.
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53 R. Messager et al, Europhys. Lett. 10, 61 (1989).

Chapter 3
EXPERIMENTAL TECHNIQUES

I. Introduction and background

The measurement of diffusion in fluids can be accomplished in many ways, up to
and including the direct measurement of the position of individual particles. To make an
accurate measurement of the diffusivity, it is necessary to measure the motion of large
numbers of particles in order to determine the average behavior. Also, to find the long-
time diffusion in porous materials, it is necessary to measure on length scales that are
large compared to the pore size, but small enough to measure slow diffusion on
reasonable time scales. Forced Rayleigh scattering satisfies these requirements in
addition to providing a means of measuring diffusion over a known and variable length
scale. This technique was first applied to the measurement of thermal diffusion within
ruby and glycerol.1 Since that time it has been extended2 to measure the diffusion of
fluids, polymers, and electrons in a variety of media.3

The first application of forced Rayleigh scattering (FRS) to mass diffusion was
performed by Hervet et al,4 who advanced the technique with the addition of
photochromic tracer molecules. They also made use of the directional nature of FRS
measurements to verify the anisotropic diffusion in the nematic phase of a liquid crystal.
Numerous studies of diffusion in polymers with FRS have made significant advances in
the study of the glass transition in those systems.5 As discussed in Chapter 2, FRS has

also been used in diffusion studies within Vycor porous glass.

47

48
II. Forced Rayleigh scattering theory

FRS is a four-wave mixing holographic measurement. Two coherent laser beams
are interfered in the medium under study to produce a periodic excitation of the optical
properties of the medium. This modulation, in effect a diffraction grating, can then be
used to diffract another beam to monitor the strength of the excitation.

Figure 3.1 shows a schematic representation of the interference region of the two
excitation beams of vacuum wavelength 71” that intersect at an angle of 9 within the
sample. The fringe pattern formed by the interference of these two TEMoo beams

(illustrated in Figure 3.2) has a wavevector (q) determined by the momentum matching

     

2
’ x
diffracted beam

probe beam

Figure 3.1: Schematic drawing of the interference region of a forced Rayleigh
scattering experiment.

49

 

 

 

3?.
\_‘

—2

.7!

O

X (rim)

0.0005 rad. (After:

488 nm, and 6 =

545(1977))

25000

Intensity profile in the interference region of two crossed TEMOO
laser beams. The parameters for this Mathematica calculation are:

4 mm beam diameter, 7»

A.E. Siegman, J. Opt. Soc. Am. 67

Figure 3.2'

50

condition: 6] = k, — k2 . The corresponding grating wavelength is then simply:

(3.1) A = ism/(2 sin(0/2)),

where q = 2Jt/A. For light of a visible wavelength, A can vary from ~1 - 100 um.
Equation 3.1 is valid for beams interfering in vacuum or for beams that have passed into
the sample, providing that they are incident at an angle of 0/2 to the normal of the sample
face. If this condition is not met, then the indices of refraction of the sample and the air
must be taken into account using Snell’s Law.

If the two excitation beams are viewed as plane waves (a valid assumption if A <<
beam diameters), then the intensity of the interference pattern may be calculated as
follows. The wavevectors of the excitation beams are:

(3.2) fl = 2k, — xkx, and k, = 2k2 + ikx,
and therefore

X

(3.3) q=ixq=ix2k.

The amplitude of the electric field in the interference region is then:

with a corresponding intensity of

(3.5) I = geerO - E; = 34.04

 

_. 2 _. 2 —. _.
E1| +|E2| +2E,~E2cos(2kxx)]

or,

(3.6) I = II +12 + 2AIcos(2kxx)

where AI = geocEl E2. If EI = E2 Ily (s-polarized beams), then

51
(3.7) I = 2I,(1+ cos(qx)).

This shows that the intensity varies sinusoidally in the interference region with an
amplitude that varies from zero to four times the intensity of the individual beams. If the
polarizations of the two beams are not both s-polarized, then the interference pattern is no
longer described by Equation 3.7. If the beams are polarized perpendicular to each other,
there will be no interference pattern.

If the excitation beams interact with the material in the interference region, a
periodic excitation will form in that region. This excitation may be thermal, electronic, a
concentration fluctuation, or any other property the light may directly or indirectly couple
to. These material changes, which only occur in the bright fringes of the interference
pattern, produce a corresponding spatial modulation of the index of refraction: a
diffraction grating. The magnitude of this modulation may be as large as An = 103, but
changes on the order of 10"2 are detectable.

A third laser beam, the probe, is incident on the interference region at the Bragg
angle calculated from Equation 3.1,

(3.8) 9,, = 2sin"(xp,,b,/(2A)),

where 2.2.01” is the wavelength of the probe beam. If the index of refraction is modulated
at lpmbc, part of the probe will be diffracted at an angle of 0,, from the incident beam. The
intensity of the diffracted beam can then be used to measure the contrast of the grating.

The strength of the grating is usually measured by the diffraction efficiency,

P.
39 EM,
( ) 11 P

0

52
where Po and Pdiffmctcd are the powers of the incident and diffracted probe beams. For

holographic volume gratings such as those created in FRS, 11 may be written as:

-Kz
(3.10) n = 6W[Sin2{ip_1ct§§n(%/z_)] + sinhzflfigf‘fij],

where K is the absorption coefficient at the probe wavelength hp, AK is the modulation of
K, An is the modulation of the index of refraction, z is the thickness of the grating, and
0p/2 is the angle the probe makes to the grating.6 If AP is chosen so that K = 0, then An

can be determined from:

(3.11) mzm
TCZ

sin" (JR).
As an example, one of the weakest gratings produced in this study had P0 = 2.0 mW and
deffmctcd z 5.5 nW in a 3 mm thick sample at 02/2 = 7.83°. This implies an efficiency of
n = 2.8 x 10'6 and An = 1.1 x 107. This analysis is not necessary to the study of diffusion
using FRS, but it can be useful to understand the strength of the transient grating formed.
If the grating is thin (z is small, producing a grating that is more two dimensional
than three dimensional) all orders of the diffraction pattern will be found, not just the
order satisfying the Bragg condition. Thick gratings are advantageous because all the
diffracted light is concentrated into one diffraction spot and therefore produces a larger
signal.
For mass diffusion measurements in fluids, a tracer molecule is added to the

liquid. The tracer molecule changes its optical properties when it absorbs light from the

excitation beams. This excitation must have a lifetime long enough to allow the molecule

53
to diffuse a significant fraction of the grating wavelength before it decays. Because this

diffusion time in liquids is quite long compared to typical electronic excitations, the tracer
molecules used generally have metastable conformational states with lifetimes on the
order of seconds. Therefore, when the excitation beams are removed, the periodic
material excitation will decay through two processes: the intrinsic lifetime of the
molecule, and its diffusion through the fluid. If the molecule does not diffuse, the
transient diffraction grating will decay with a rate independent of the grating wavelength,
A. If diffusion is occurring, molecules that had been excited in the bright fringes will
move into regions that had not been excited. This motion will cause the grating to decay
with a rate that is dependent on both the diffusion coefficient and the grating wavelength.
Therefore, the diffusion coefficient may be extracted by monitoring the time dependence
of the diffracted beam intensity.

The concentration of the species excited by a FRS measurement can described by
the following 1-dimensional differential equation (when A << the width of the excitation
beams):

0cc(x, t) _ cc(x,t)

Ti

(3.12) = DVzce (x, t) + 0t I(x, t)cg(x, t) — [5 I(x, t)ec (x, t)

where ce(x,t) (cg(x,t)) is the concentration of the excited (ground state) species, D is the
diffusion coefficient, 1. is the lifetime of the excited species, I(x,t) is the intensity
distribution of the excitation pattern, and or (B) describes the rate at which the ground

(excited) state is converted to the excited (ground) state by the incident light of the

54

excitation beams. This can be simplified if we assume ce(x,t) << cg(x,t) E c(x,t), the

initial concentration of the ground state molecule, and that B << 01. Then,

cc(x, t)

T!

(3.13) flags! = DVzce(x, t) — + aI(x,t)c(x, t)

where, the intensity of the interference pattern for an excitation pulse that begins at t = 0
and ends at t = T is, from Equation 3.7:
(3.14) I(x,t) =10 (l + cos(qx)) (H(t) - H(t-T)),
given that H(t-T) = 0 for t < T, and 1 for t > T.
After the excitation pulse (when I(x,t) = 0), the solution is:
(3.15) e.,(x,t) = w e'“1 cos (qx)
where,
(3.16) l/t = W + 1m.
Therefore, the overall decay of the concentration pattern depends upon both the intrinsic
lifetime of the excitation (1'1) and the decay rate due to diffusion (qu).

Due to the direct relationship between the concentration modulation and the

amplitude of the diffracted beam (An = (cl—“AC and 1] oc An for small An), the amplitude
c

of the diffracted beam decays at the same rate as the amplitude of the concentration
grating. However, since the intensity of the beam is measured rather than its amplitude,
the measured time dependence of the diffracted beam is:7

(3.17) Idiffracteda) = (Ac-m: "l" B)2 + C2,

55
where A is the amplitude of the diffracted electric field, B is the amplitude of light

scattered from imperfections in the sample but remaining coherent with the diffracted
beam, and C is the amplitude of the incoherently scattered background light.

The decay rate of the transient grating (1/1) is measured over a range of grating
spacings (21dq) and the resulting data fit with a function of the form of Equation (3. 16) to
extract both the diffusion coefficient and the lifetime of the excited state. A decay rate
that does not depend quadratically on the grating wavector, indicates that anomalous
diffusion is occurring. This is true because the statement 1/1 or q2 is equivalent to the
result found earlier for a random walker: <r2> or t. Therefore, FRS may also be used to
determine if classical diffusion is occurring on a length scale given by A.

In general, however, the situation is more complicated than the decay of a single
grating. Because the excited state is formed from the ground state, the creation of a
periodic concentration profile of excited molecules requires that there is a corresponding
modulation of the ground state as well (Figure 3.3). In this case, there will be two decay
rates, one for each grating. These will be: l/te = ng2 + UT... and l/tg = ng2 + 1mg. The
intensity of the diffracted signal becomes:

(3.18) lament) = (Are'm + Are” + Br- + C2

which can lead to a variety of non-monotonic decay curves depending upon the values of
A., A2, D6, D3, tie, and 1.25.8 If the excited state decays directly back to the ground state
and not to a third state, relaxing molecules will reduce the modulation of both the excited
state grating and the ground state grating. Therefore tie = rig. If both species also diffuse

at the same rate (D6 = Dg), then the time dependence of the diffracted beam will again

56

21)

Intensity

 

 

b)

 

 

9)

  

 

I l
I I
I . I
I O C I ' D
I I I O I
I I I I I I
I I I I 2 g
I
I
I I I I | I
.l O. I .0 O. ' .' .
I I | I a | I o
O I . . I I
I. .I . '. .‘ ' . I.
..... : °o..- : '0...-
I I
I I
I I
I I
I l
I
I
I
I

Figure 3.3: (a) Sinusoidal intensity profile of the interference pattern.
(b) Corresponding profile of the change in concentration of the excited
state species. (c) Profile of the change in concentration of the ground state
species. The dotted lines illustrate the profiles after the interference
pattern is removed and the concentration grating is decaying through
diffusion.

57
be described by a single exponential, just as in the case of a single grating. If DC at Dg

(for example, if one species is more attracted to binding sites on the walls of a pore), then
the time dependence will no longer decay as a single exponential.9 Because the analysis
of such signals is difficult to interpret unambiguously,lo usually only data that exhibit
single exponential behavior are considered reliable. However, there are instances where
the source of the nonexponential decay is well understood and where the non-exponential
signals are sufficiently distinctive to allow analysis with Equation 3.18.
III. Experimental setup

The experimental setup used in most of the FRS experiments in this work is
shown in Figure 3.4. The excitation laser is a Coherent Innova I-400-15 Ar+ laser
operating at 488 nm with an internal cavity etalon to produce a coherence length of at
least several meters (when operated in ModeTrack mode to lase in only one cavity mode).
The laser typically operates at 3.5 - 4.1 W of output power, corresponding to a tube
current of 32.8 A. The diameter of the vertically polarized beam at the output of the laser
is 2 mm, but widens to 3 mm by time it reaches the sample cell. The beam is attenuated
by a sampling optic which reduces the power to ~100 mW. This is reduced further with a
partially aluminized variable attenuator down to the final power of ~20 mW. The beam
next passes into an assembly of optics making up the ferroelectric liquid crystal shutter. A
vertically oriented Glan-laser polarizer first “cleans up” the polarization of the beam. The
polarization is then rotated by a quartz half-wave waveplate, folowed by a rotation to
either vertical or horizontal polarization by the Displaytech (PV 050 AC) liquid crystal

rotator, which acts as a half-wave waveplate where the rotation depends upon the voltage

58

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ar" laser . sampling optic
I variable attenuator
2 power U
I stabilizer
I detector [ ]
l: :1
I shutter
, 1::
I mirrors [ ]
I beam
I splitter
I
Y (8‘. — . U‘l selection
detector 66 iris and lens mirror
interference
filter

Figure 3.4: Experimental setup for the FRS measurements in this study. The
distance between the beam splitter and the sample cell is 2.3 m.
applied across the liquid crystal. Finally, the beam exits through another Glan-laser
polarizer which is oriented to pass only vertically polarized light. Depending upon the
voltage applied to the bi-stable liquid crystal element, the excitation beam either passes
through this final polarizer or is extinguished. This shutter has an extinction ratio of at
least 1000:1 (the ratio of the power transmitted in the “open” state to that transmitted in
the “closed” state) but is typically 5000 - 10000:1 with proper alignment and temperature
stability. The advantage of this shutter over a mechanical shutter is its response speed. A
mechanical shutter can open and close in 2-3 ms while the liquid crystal can switch state

(90% open - 10% open) in 35 us with much less variability from shot to shot.

59
The excitation beam next encounters a beam sampling optic which sends a portion

of the beam to a photodiode used to trigger the oscilloscope that acquires the signal. The
remaining beam is split by a cube beamsplitter--half is sent directly to the sample cell
(excitation beam 1), and half is passed to the selection mirror (excitation beam 2). By
rotating the selection mirror, beam 2 can be sent to one of the prealigned mirrors which
direct it to intersect beam 1 inside the sample cell. The plane formed by the two
intersecting excitation beams defines the direction of the interference pattern. Therefore,
if the fringes of the interference pattern are to be vertical, the plane of the excitation
beams must be parallel to the table surface.

As mentioned earlier, for the periodicity of the grating to be calculated simply, the
normal vector to the sample cell face must bisect the angle between the excitation beams.
So, to select the grating spacing, A, only the selection mirror and the sample cell need be
rotated. To determine 0cxc and A, the distance between the excitation beams is measured

at a known distance from the intersection point. For example, at the smallest grating

_ 16.7cm
39.4cm

 

wavelength, A = 1.225 mm, tan(0m) , where the uncertainty in A is i2.0%

(from the uncertainty in the measured distances).

The probe beam is generated by a Uniphase 1137P 7 mW, linearly polarized
HeNe laser. The beam power is controlled and stabilized externally by a Cambridge
Research and Instrumentation power stabilizer to produce a beam of typically 2.0 - 3.0
mW that is stable to better than 0.5% over a period of several hours. The angle of
incidence of the probe on the sample cell is changed by translating and rotating its own

selection mirror. The approximate angle and location of the mirror is known by

60

calculating the Bragg angle from A, but is determined precisely by maximizing the
intensity of the diffracted beam. The part of the beam transmitted through the cell is
stopped in a beam block, while the diffracted portion is directed to the'detector by a
mirror. The probe beam is 2 mm in diameter when it is incident on the sample cell.

The detector assembly consists of an iris diaphragm, a lens, a 632.9 nm bandpass
interference filter (with a 2.6 nm bandwidth, Oriel part number 52730), and a photodiode.
The iris and the filter serve to block out all but the diffracted beam, while the lens is used
to direct the light onto the photodiode so that the active area of the diode is filled (but not
overfilled) by the beam. The peak transmittance of the interference filter is 50%,
meaning that half the light at the center of the passed band is transmitted. The angular
alignment of this filter is a significant source of error in the measurement of the absolute
power of the diffracted beam because the transmittance of the interference filter is
dependent on the angle of incidence of the beam. For a misalignment of 5 degrees, the
central wavelength of this 2.6 nm bandwidth filter will shift by 0.3%, which can cause up
to a 50% change in the transmitted intensity.“ However, because FRS diffusion
measurements only depend upon the rate of change of the diffracted signal and not its
absolute magnitude, this effect is only important when trying to make a quantitative
measurement of the diffraction efficiency. The detector is a Centronic OSD15-5T silicon
photodiode with a 3.8 mm x 3.8 mm active area and a native response time of ~12 ns.
The photocurrent produced by the diode flows through a 11.1 k!) resistance to produce

the measured voltage. The response of this detector is measured to be 4.1 V/mW at 633

61

 

0.35 ' I ' ] T I r l f

0.30

I
0
1

0.25

l
0
1

(V)
<1.

... 0.20

0.15

Diode outpu

0.10

  
 

0.05
V = (4.060 t 0.008) * (power in mW) + (5.4 t 2.0 x 104)

 

 

0.00 1 l 1 1 n l n l n
0.00 0.02 0.04 0.06 0.08 0.10
Incident Power at 632.8 nm (mW)

 

Figure 3.5: Response of the photodetector to 632.8 nm light. Power measured
with Newport 1825-C meter and 818-SL detector.

nm (Figure 3.5). The linear response region of this detector extends to ~50 11W, while the
diffracted signals in this experiment range from ~5 nW to ~ 211W.

The sample cell (Figure 3.6) is a homemade copper cell designed for 27 mm outer
diameter optical windows (Edmund Scientific #A2199). The sample and fluid are placed
between two windows separated by a Viton o-ring whose thickness is chosen to be
slightly larger than the sample. The glass windows are then pressed together in the
copper cell--compressing the o—ring until the windows touch both sides of the sample and

hold it in place. This is done so that no fluid is between the sample and the window

62
Pt thermometer embedded
leads for Pt thermometer 1n COP?“ body

  
     

screws used to
close cell

thermoelectric
heater/cooler

copper post

Figure 3.6: Schematic drawing of the sample cell holder. The body and post are
made of copper separated by a thermoelectric heater/cooler used to control
the temperature of the cell. When in use, the opening (which reveals the
glass windows and the sample) is mostly covered with thin copper shim
stock to reduce heat loss from the window faces. The heater is 1 inch
square, and the opening in the cell body is 0.9 inches in diameter.

which could provide an additional diffusion signal. Soldered between the top of the cell
(containing the windows and the sample) and the supporting post is a Melcor (CPl.0-7l-
06TI‘) thermoelectric heater/cooler used to control the temperature of the cell between 10
- 50 °C. An Omega Pt thermometer (#W2103) embedded in the top copper piece allows
an Omega CN76000 temperature controller to maintain the temperature within :1: 0.2°C.

Figure 3.7 is a schematic of the electronics used to control the experiment and

acquire the data. The signal from the detector is amplified by a Stanford Research

63

 

 

temperature Pt
controller thermometer

HP6284A
power supply

 

 

thermoelec.
heater/cooler

 

 

   

 

 

 

 

 

 

 

 

-—-——q-————-———- ---

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

LeC rigg er photodiode
roy
9409A channel A
Oscrlloscope SR560
——. photodiode
PC RS-232 preamp :
I
I
I
RS-2 2 l .

3 Vincent Assoc. Displaytech , liq. crystal
shutter driver E shutter
controller :

I
external :
I

pulse output ,
input

Figure 3.7: Schematic of the electronics used to control and monitor the FRS
measurements. The dotted line separates equipment that is on the optical
table (right hand side) from equipment on the overhead shelf. Italics
indicates the name of the connection on the equipment.

Systems low noise preamplifier before being collected with a digital oscilloscope. The
oscilloscope allowed many shots to be averaged together to improve the signal to noise
ratio of the decay signal before it was transferred to the PC for storage and analysis.
Except for the temperature controller and some of the features of the shutter controller,
the entire experiment is controlled through a PC. (The control program and the file
structure of the data files are shown in Appendix B)

IV. Detailed data acquisition procedure

After the sample has been loaded into the sample cell along with the fluid, and the

lasers turned on and allowed to warm up for at least 20 minutes, the excitation beams are

64

aligned to intersect in the fluid next to the sample in the sample cell. The distance from
the table top to each of the two excitation beams is measured to ensure that the beams are
parallel to the table top, and at the same height. The excitation beams are aligned to
overlap in the sample cell (there is enough scattering and fluorescence from the liquid to
see the beam locations as they pass through the cell) by rotating the selection mirror that
directs beam 2. The sample cell is then rotated so that the reflection of beam 1 off the
front face retraces the path that beam 2 follows, and vice versa. This guarantees that the
normal vector of the sample cell face bisects the angle between the excitation beams so
that equation (3.1) may be used. The angle between beams 1 and 2 (0) is determined by
measuring the distance between the beams (x) at a known distance (y) along beam 1 from
the point of intersection in the sample. The angle is then found from tan(0) = x/y, and the
grating spacing A is found from equation (3.1). The power in each beam is then
measured just in front of the sample cell. This power is typically 6 - 7 mW in each beam,

producing an intensity of ~ 8 mW/mm2 at the peaks of the interference pattern (from

2
P = M1 and Equation 3.7, where P is the power in a TEMoo beam, w is its radius, and

I0 is the maximum intensity in the center of the beam).

The HeNe probe beam is set at the same height off the tabletop as the excitation
beams, and aligned at approximately the Bragg angle by knowing A and hpmbe, and using
equation (3.8). The final alignment is achieved with small adjustments to the mirror to
maximize the power in the diffracted beam.

The detector assembly is then aligned so that the diffracted beam is sent to the

photodiode through the iris diaphragm, lens, and interference filter.

65
Once the optics are all aligned, the delay between pulses of the excitation beams is

adjusted to be longer than the time required for the grating to relax, and at least as long as
the lifetime of the dye molecule. Normally, the delay is 10 to 20 seconds. The duration
of the pulses is 20 ms to reduce heating of the liquid by the excitation beams. Next, the
gain of the preamplifier is chosen to provide the maximum amplification without
overloading it. This gain ranges from 5000 to 50,000 depending on the power in the
diffracted beam. The appropriate timebase, vertical scale, and trigger time of the
oscilloscope are then adjusted so the entire diffracted signal (before, during, and after the
excitation pulse) is captured on the screen. Finally, the oscilloscope is allowed to average
a number of these events (anywhere from 10 to 200 or more), to improve the signal to
noise ratio. When the acquisition is stopped, the data is saved for later analysis.12

The next step is to measure the diffracted signal from within the sample.
Therefore, the sample cell is translated so that the beams will intersect within it, rather
than the fluid next to it. The excitation beams are then checked to verify that they overlap
within the sample, the probe beam alignment is adjusted, if necessary, to maximize the
diffracted signal, and the detector assembly realigned. The electronics are adjusted as

before, and the diffracted signal from the sample is measured. Once this is complete, the

angle between the excitation beams is changed and the process is repeated.

66
References

 

‘ H. Eichler, G. Salje, and H. Stahl, J. Appl. Phys. 44, 5383 (1973).

2 H.J. Eichler, P. Gunter, and D.W. Pohl, Laser-Induced Dynamic Gratings (Springer-
Verlag, Berlin, 1986).

3 M.Terazima, K. Okamoto, and N. Hirota, J. Phys. Chem. 97, 5188 (1993); J .A. Wesson
et al, Macromolecules 17, 782 (1984);T.P. Lodge, J .A. Lee, and TS Frick, J. Polym. Sci.
B 28, 2607 (1990).

4 H. Hervet, w. Urbach, and F. Rondelez, J. Chem. Phys. 68, 2725 (1978).

5 D. Ehlich and H. Sillescu, Macromolecules 23, 1600 (1990).

6 H. Kogelnick, Bell Sys. Tech. J. 48, 2909 (1969); H.J. Eichler, P. Giinter, and D.W.
Pohl, ibid., p. 104.

7 L. Leger, H. Hervet, and F. Rondelez, Macromolecules 14, 1732 (1981).

8 H. Sillescu and D. Ehlich, in Lasers in Polymer Science and Technology: Applications,
edited by Fouassier and Rabek (CRC Press, 1990).

9 J.L. Xia and CH. Wang, J. Chem. Phys. 88, 5211 (1988); J.A. Wesson et al,
Macromolecules 17, 782 (1984).

'0 S. Park et al, J. Phys. Chem. 95, 7121 (1991); DR. Spiegel, M.B. Sprinkle, and T.
Chang, J. Chem. Phys. 104, 4920 (1996), S. Park, H. Yu, and T. Chang, Macromolecules
26, 3086 (1993).

H For light at an angle of incidence of 0, this filter’s center wavelength is
he = XOJI — 0.690 sin2(0) , where 3.0: 632.9 nm, the center wavelength at normal
incidence. Oriel Corporation, Optics and Filters Catalog, p. 2-31 to 2-37.

 

‘2 Microcal Origin v. 4.1 is used to fit the data. The nonlinear least squares curve fitting

routine uses the Levenberg-Marquardt algorithm to find the fit that produces the smallest
2

x value.

Chapter 4
MATERIALS

I. Introduction

Finding porous materials is not difficult. More materials absorb liquids or gases
than are impervious to them. Everything from wood, biological cells, plastics, concrete,
stone, ceramics, and even some forms of metals are porous. The transport of fluids
through each of these materials is crucial to their formation, their technological
usefulness, or (in the case of living porous materials) their existence. Restricting our
focus to materials that are manufactured for their porous nature, we still find a
tremendous variety of forms: sintered metals, disordered porous glasses, crystalline media
like zeolites,l layered materials,2 and self-assembled materials.3 These manufactured
materials are of great interest to the catalysis and separation (sieving) community where
the ability to design a material for specific processes is crucial.
II. Porous silica

II a. Types of porous silica

This work focuses on fluid diffusion in porous silica, chosen partly because it is
transparent to visible light, and partly because of the wide range of ways that exist to
prepare it. The first commercial porous glass was Vycor 7930,4 a borosilicate glass that is
heat treated to phase separate it into two interpenetrating boron-rich and silica-rich
networks. The boron rich phase is then dissolved in acid, leaving behind a silica rich
porous glass with a very narrow pore size distribution (20 - 30 A pore radius). Glasses

may also be produced through the gelation of colloidal silica sols5 or aggregates of flame-

67

68

H

l
0

—O—70

l
R—O—Si—O—R +4H20 —-> H—o—81—o—H +4R0H

o o

| |

R H
H H H H
| I l |
o o o o

| |
H—O—Si—O—H ——> H—o—si—o—Si—o—Si—o—H + H20

1
o o o o
l l l l
H H H H

Figure 4.1: The sol-gel process showing TEOS reacting with water to form
silanol, which then reacts to form the silica network.

hydrolyzed silica.6 The most common way of making porous silica today is through the
polymerization of hydrolyzed metal alkoxides such as tetraethylorthosilicate (TEOS) in
an alcohol/water mixture. This sol-gel process is shown in Figure 4.1: the TEOS reacts
with water to produce silanol and an organic alcohol (here, ethanol). The silanol
molecules react with each other, releasing water and producing a network of silica. The
network formed is not a regular lattice as the bonds form during random interactions
between molecules. This network is said to have gelled once the network spans the entire
body of the container (therefore, the gel point is a percolation threshold7). In practice,
this point is defined as the moment when tilting the sample produces no flow.

The water and alcohol evolved in the gelling process must be removed from the
network in order for the pore structure to be accessed. This drying step is the most

critical phase of the process. As the liquid/vapor interface progresses into the porous

69
material, the surface tension on the meniscus produces large stresses on the network.8

These stresses, which are greatest in small pores, can easily crack the gel. There are
several ways to produce uncracked dried gels:9 dry the gel under tightly controlled
pressure and humidity to produce highly porous materials such as aerogel,10 add a
surfactant to lower the surface energy at the drying interface, remove the small pores, dry
the gel under very high humidity (to increase the gelling time and allow the network to
respond to the stresses), or produce highly homogeneous gels that allow the stress to be
balanced over the entire network. Once the network is successfully dried, it may be
heated to remove the remaining adsorbed water, to modify the surface chemistry, or to
reduce the porosity.

II b. The effect of heat on silica

The surface of silica in air is covered by one or more layers of adsorbed water
(water hydrogen bonded to the hydroxyl groups on the surface). As silica is heated, the
surface passes through several regimes of dehydration (Table 4.1, Figure 4.2) eventually

reaching a hydrophobic state and, if heated further, shrinking to become fully dense silica.

Table 4.1: Character of the silica surface as a function of the temperature.

 

 

 

 

 

 

 

 

 

Temperature (C) surface character

< 170 hydrated: bound water and hydroxyl groups

170 - 400 reversible dehydration: no more water, vicinal groups begin

condensing

400 - 800 irreversible dehydration: shrinkage closes offiiores
> 800 only isolated hydroxyl groups exist--hydrophobic surface
> 850 isolated hydroxyl groups react-—closing network

1000 - 1700 densification

 

 

70

(a) hydrated
H H
vicinal isolated \ / hydrogen-bonded
.2 water
717-...‘1 7177‘?
(I ‘1’ 7O\S (1715783 ‘7? ‘P
- "Si Surface
077'770771707777779777177707717707I77o7177o717770717 77\
O O
O O O O

(b) partially dehydrated

717 71 717
‘1) 70 \S 77. 8270 x 51'- Oxs 77..
\ IS7\ 7777677717770 77777077177077770771770 7177071777 5777777777
0

U'i—o—z:

‘0

77/
000

O O O O

0

Figure 4.2: Schematic of the (a) hydrated and (b) dehydrated silica surface,
showing the types of hyroxyl groups and bound water that can form.
(After R.K. Iler, The Chemistry of Silica (John Wiley and Sons, New
York, 1979) and K.K. Unger, Porous Silica (Elsevier, Amsterdam,
1979).).
Figure 4.2a schematically shows the surface structures on a hydrated silica surface:
vicinal hydroxyl groups, isolated hydroxyl groups, and hydrogen bonded water. As the
temperature is raised above 170°C, the adsorbed water is removed and vicinal hydroxyl
groups begin condensing to form siloxane and water. Up to ~400°C, the dehydration is
fully reversible and exposure to water will produce a hydrated surface equivalent to the
original surface. Above ~400°C, shrinking of the network begins to close off pores. By

~800°C only isolated hydroxyl groups exist on the surface, and at ~850°C the motion of

silicon atoms at the surface is significant enough to allow isolated hydroxyl groups to

71
react and begin densifying the network. The silica will fully densify (i.e. lose its porosity

and reach the density of fused silica, 2.2 g/cm3) at temperatures greater than 1000°C.ll
This heating process to transform a wet silica gel into a fully dense silica at
relatively low temperatures is a significant advance in the manufacture of preformed
silica objects.‘2 Previously, very pure fused silica could only be produced in a high
temperature (> 1600°C) process. With sol-gel processes, glass can be made in pre-shaped
molds (which significantly reduces the time required for grinding and polishing)--but only
if the optic survives the drying step without cracking. Another difficulty in glass
formation is the transformation of silica to a crystalline form before it fully densifies. It is
found that the presence of alkali ions in the network promotes this crystallization at
temperatures as low as 700°C. Lithium causes quartz to form, while sodium produces

cristobalite.‘3

This is a concern in the production of glasses from charge-stabilized
colloids where alkali counter ions from the suspension are incorporated into the network
upon gelation, and means that the creation of a hydrophobic amorphous silica surface
through heat treatment is impossible to achieve.
III. Characterization of porous silica

III a. Properties of porous silica

Understanding the pore structure in porous materials is crucial to understanding
the transport processes that take place within them. Therefore, the determination of the
pore structure is of great importance, although it is a task that has proven very difficult to

accurately accomplish. A number of techniques are employed to characterize the

structure--each with its own strengths and weaknesses.

72

The properties of interest of a porous material are the surface area, pore volume,
average pore size, pore size distribution, and connectivity. The surface area of a porous
material is all the surface that is accessible from the exterior, including the surface area of
the pores. A large surface area is advantageous for catalysts, because the chemical
reactions take place on the surface and a larger surface area allows more reactions to
occur simultaneously. The pore volume is the volume of the material not occupied by the
rigid network. The porosity (the ratio of the pore volume to the total sample volume) is
the characteristic most often reported as it is easily measured by either comparing the
weight of the material with empty pores to the weight with the pores full of a liquid of
known density or by measuring the volume of liquid displaced by the addition of the
material. Note that with these measurements, isolated pores are not detected. The
porosity characterizes how much of the material is empty and how much is solid. The
pore size distribution is, as the name implies, the distribution of the pore sizes throughout
the material. The distribution is defined by the amount of the pore volume that results
from pores of a given size. Therefore, a narrow pore distribution implies that most of the
available pore volume occurs in pores of nearly the same size. The connectivity of the
pore network is the average number of passages that lead out of one pore to a neighboring
pores (e.g. a perfect cubic pore network would have a connectivity of 6). The
connectivity is perhaps the most difficult of all the parameters to determine14 as it cannot

easily be probed externally.

73
III b. Common techniques for characterizing porous materials:

The most straightforward technique for determining these properties'5 is to
repeatedly make cross sections of the material. After each slice the amount of the
exposed area that is solid and the amount that is empty is measured. This has several
obvious disadvantages including the destruction of the sample and the large amount of
time it takes to make the measurement. Nevertheless, variations of this technique are
used-—particularly for materials with nanometer-sized features, where the sample is
thinned down for measurement in a transmission electron microsc0pe.

A very powerful technique that has been applied to the analysis of the structure of
porous rocks16 is that of X-ray microtomography, where (just like in medical CT scans) a
three dimensional image of the sample interior is built up by analyzing the absorption of
X-rays as they pass through a sheet of the sample. While it is a useful nondestructive
technique, it is limited to a resolution of about 1 micron and requires the use of
synchrotron X-ray sources.

Small angle X-ray scattering (SAXS) and small angle neutron scattering (SANS)
have been used successfully to determine average properties of porous materials17 such as
the correlation length of the network, which can be related to the average pore size.
Structural properties on the order of a few Angstroms to about 2000 A may be probed
with these techniques, and they have been useful in the determination of the fractal nature
of some porous networks. However, SAXS can make no determination of the pore

volume or surface area of a sample.

74
Nuclear magnetic resonance imaging (MRI) can also be used to image the internal

pore structure of a porous material78 as well as measure the motion of fluids within the
pores.19 MRI is limited in spatial resolution to fractions of a millimeter and so is (like X-
ray tomography) not useful in the characterization of microporous materials.

One of the oldest techniques for the determination of the pore size distribution is
mercury intrusion.20 Because mercury does not wet the pore walls, it must be forced into
the pores with externally applied pressure. At 1 atmosphere, mercury will fill pores
connected to the surface if their pore radius is greater than 7 um. Filling pores of 100 A
and 20 A radius requires pressures of 700 atm. and 3500 atm., respectively.21 The
measurement is very simple: a known quantity of mercury is brought into contact with
the surface of the material and the pressure increased to fill the pores. As the pores fill,
the amount of mercury removed from the “bath” at each pressure step is measured to
determine the uptake curve. In order to relate this to the pore size, a model of the pore
structure must be assumed.22 Normally, a bundled tube model is used, where the
diameter of each tube is constant along its length, but each tube may have a different
diameter. This is the largest source of error in the determination of the pore size
distribution--if the pores do not have a constant diameter, a large pore in the interior will
fill only after the smaller pores between it and the surface do. This means that the amount
of mercury taken up at a given pressure does not correspond to the total volume occupied
by pores of only one size. Therefore, the measured distribution really gives information
about only the smallest pores that the mercury passes through, as they are the limiting

connections. Mercury intrusion works best for large pore samples, but is capable of

75
measuring distributions below 40 A. This technique has two main drawbacks: the use of

a toxic liquid (that may remain in the pores after the measurement), and the use of high
pressure which may destroy the network of pores.

III e. Nitrogen adsorption

By far the most common method for determining the internal structure of a porous
material is the isotherm analysis of the adsorption of a gas, typically nitrogen.23 This
condensation technique is capable of determining the pore volume, specific surface area
(surface area per gram of adsorbent), and pore size distribution of a material without
subjecting it to high pressures or contaminating it. It does not, however, produce a
microscopic image of the pores and its conclusions about the pore distribution are
affected by the model used to represent the pore shape. Therefore, while these
measurements are highly useful in providing an understanding of the pore structure, they
are not without serious ambiguities.

Nitrogen adsorption measurements are made by placing the sample (the
adsorbent) into a cell of known volume that is cooled to the temperature of liquid

24 Attached to this cell through a valve is another chamber of known volume

nitrogen.
(the dosing volume) in which the pressure is monitored. After evacuating both chambers,
and closing off the sample cell, the dosing volume is filled with a known quantity of
nitrogen gas. The valve between the chambers is then opened, allowing the gas to
condense inside the pores of the sample. The amount that condenses is determined from

the change in the pressure after the system has equilibrated. This cycle is repeated until

the saturation pressure (p0) has been reached and all the pores are filled. The procedure is

76

 

 

180 ' 1 ' l v I u I
:7 160 . desorption -
U)
:5 140 -
an .
"g 120 -
g 100 -
7E7 .
U 80 -
B .
6o -
'2 .

40 -
0.020

 

(I --I—- desorption

I

p
O
p—L
til

—0-- adsorption

 

.9

8

LII
l

 

d(pore volume)/d(pore diameter) (cm3/ g)
o
S
o

 

 

- 0‘0—0 0.0/0 ‘
I {'3‘ Xi‘l \o\o
0.000 , 1 . I 1M

0.0 0.2 0.4 0.6 0.8 1.0

 

Figure 4.3: (a) Nitrogen adsorption and desorption isotherms for Vycor porous
glass. (b) The pore size distributions calculated from (a).

77
then reversed to measure the amount desorbed as the pressure is reduced (Figure 4.33).

Once the isotherms (the total amount of gas condensed in the sample at each
pressure step) are measured, they are analyzed to determine the surface area, pore volume,
and pore distribution. The surface area is typically determined through a method first
described by Brunauer, Emmett, and Teller:25 the BET model. The BET surface area is
found from the amount of gas adsorbed on the surface after the first monolayer has
formed and before any pores are completely filled. The model assumes that all the
adsorption sites are equivalent (that no one site is favored), that no interaction exists
between molecules adsorbed in the same layer (i.e. an adsorbed molecule interacts only
with the molecules directly below it and directly above it), and that an infinite number of
layers will exist on the surface when p = p0. These are significant assumptions that are
generally violated in any adsorption experiment. The result of the model is the BET

equation relating the mass of the adsorbed gas to the relative pressure:

(4.1) p = 7 +°"l—P—,
x(1).-p) xmc ch p.

where p is the pressure of the gas, po is the saturation vapor pressure, x is the ratio of the
mass of adsorbed gas at p to the total mass of the adsorbent, xm is the ratio of the mass of

one monolayer of adsorbed gas to the mass of the adsorbent, and c is a constant.

Therefore, if the isotherm data are plotted as —p——vs.—Ii, a straight line should be

x(Po - P) Po
found and xm and c can then be determined from the slope (s) and intercept (i) of that line
through the relations: xm = 1/(s+i) and c=(s+i)/i. This linear region in the adsorption

isotherm is typically in the range 0.05 < p/p0 < 0.35 for silica. The specific surface area

78
(S, the surface area per gram of adsorbent) of the sample is then determined from the

mass adsorbed in the first monolayer by:
in
(4.2) S = M NAAm,

where M is the molecular weight of the adsorbate, N A is Avogadro’s number, and Am is
the area covered by one adsorbate molecule (= 16.2 A2 for nitrogen).

The pore volume is easily found from nitrogen adsorption because at the
saturation pressure, the pores are completely filled with a known mass of liquid nitrogen.
However, just as in mercury intrusion experiments, the pore distribution cannot be
determined without making a model of the pore structure. At a pressure p < p0, the pores
are partly filled by liquid, with nitrogen condensing in the smallest pores first. It can be
shown that a capillary pore will fill according to:26

fl=——XY_—
(4'3) dS RTln(%0)cos(7777

where v is the volume of the capillary, S is the specific surface area of the capillary walls,
V is the molar volume of the liquid, g is the surface tension of the liquid, and 4) is the
contact angle between the walls and the liquid. For liquid nitrogen, V = 34.6 cm3/mole
and 'y = 8.85 dyne/cm. Because of the difficulty in determining the contact angle in a
porous material, the fluid is assumed to wet the pore walls (4) = 0°).

The relationship describing the pore size at which condensation occurs for a given
pressure must arise from a physical model of the pore geometry. Because the true pore

geometry is nearly always unknown, the pores are frequently modeled as being perfectly

79
cylindrical. In that case, the radius of the pore (the Kelvin radius, rk) where condensation

occurs at pressure p is:

L1 = __VY_
(4.4) 2 RTln(%o) cos((p) .

Given this (and correcting for the amount of liquid being adsorbed on the pore walls at
the same pressure), the sorption isotherms can be converted into a pore distribution by
calculating the amount of liquid nitrogen added to the pores at each pressure step27--
which is essentially taking the derivative of the isotherm. Figure 4.3b shows the pore
distributions produced from the adsorption and desorption isotherms shown in Figure
4.3a.

This procedure is complicated by the hysteresis observed between the adsorption
and desorption isotherms in porous media28 (Figure 4.3a). The hysteresis is a result of
pores which vary in size along their length. As p/p0 increases during the adsorption
phase, vapor will condense in the narrower sections of the pore first--followed by the
wider regions. However, during the desorption, wide areas which are separated from the
surface by a narrow region must wait until that narrow region empties (at a lower value of
p/Po) before it can vaporize (similar to the situation in mercury intrusion experiments).
This will cause the measured adsorbed volume at a given p/p0 to be larger in the
desorption isotherm than the adsorption isotherm. This discrepancy produces two very
different calculated pore distributions (Figure 4.3b): a narrow distribution that provides
more information about the size of the “flow-limiting” pore necks (desorption), and a
broader distribution that some view as a more accurate representation of the true

distribution of pores (adsorption).

”Q

(70

ii

80
This hysteresis in the isotherms vividly shows the problems of using a cylindrical

pore model in the determination of the pore radius. If the pores were really just straight
cylinders, there would be no hysteresis in the isotherms. In reality, very few porous
materials have cylindrical pores, and certainly disordered porous materials like porous
glasses do not fit the model. For this reason, the pore radii determined from the isotherm
analysis cannot be accepted without question. The shape of the hysteresis curves have
been used as a way to determine the geometry of the pores,” although it is not a direct
measure of the pore shape. The Gelsil samples used in this work all have hysteresis
curves that classify them as being made of cylindrical pores.

Due to these difficulties, an estimate of the average pore radius is used that is
based on the measured surface area and pore volume (two quantities that are relatively

well known).3° This estimate, known as the Wheeler radius or the hydraulic radius, is:

 

v
(4.5) Rw=28 P.

BET
where Vp and Saar are the pore volume per gram adsorbent and the BET surface area per
gram adsorbent, respectively. This estimate can vary by as much as 20% from the peak

37 However, the Wheeler

value of the pore distribution determined from the isotherms.
radius is simple to determine, is based on the same assumption of cylindrical pores, and
has been found useful in comparing materials of similar compositions.32 For these
reasons, Rw is frequently used to characterize the pore size—-particularly in conjunction

with isotherm analysis to gauge the width of the distribution, and the possible geometry

of the pores.

81
IV. The manufacture and properties of Gelsil

The main emphasis of this study is focused on fluid diffusion in a set of
commercial sol-gel glasses made by GelTech, Inc.33 These porous glasses are a
byproduct of the development of fully densified sol-gel glasses for the manufacture of
pre-shaped optical elements.7777 This process requires forming gels that do not crack upon
drying or heating. In these materials, this is achieved with the addition of a “drying

control chemical additive”35

which serves as a surfactant to reduce the capillary stresses
found at the liquid/vapor interface during drying. One effect of these additives is to
produce extremely narrow pore distributions during the gelation phase, which also helps
diminish the drying stresses caused by unequal evaporation rates from pores of different
sizes. After drying, these porous optics are heated to drive off the bound water and, if the
heating is stopped at ~650°C, a porous glass of small (< 100 A radius) and very uniform
pore size is recovered. The drying control additive is chosen to also be easily removed
from the glass during the heating process and so not contaminate the end product. Due to
their small pores and narrow pore size distribution, these porous glasses (of porosity >
0.4) are optically transparent. If the glass is heated up to 1150°C, it will shrink to fully
dense silica. One thing to note is that the porosity of these glasses is much larger than the
porosities of a Vycor-like glass of the same pore size. This means that in the sol-gel
glasses, the walls of the rigid network are thinner or there are more connections between
the pores.

According to the manufacturer, the surfaces of the porous glass (Gelsil) is that of a

dehydrated silica surface--siloxane bonds with both vicinal and isolated hydroxyl groups.

82

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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83

 

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Adsorption Isotherrn
de/de (cc/g/A)

0.005

 

 

 

0.000

 

Pore Radius (A)

Figure 4.4: Top: Pore size distributions of the Gelsil porous silica samples
calculated from the desorption isotherms. Bottom: Pore size distributions
of the Gelsil porous silica samples calculated from the adsorption
isotherms. Note the discrepancy between the adsorption and desorption
distributions, and with the nominal average pore radius.

hf

si.

02

C3

th.

C21

84
The pores are characterized by nitrogen adsorption and the average pore radius

(calculated by R.) is reported. By varying the initial concentrations in the sol and
changing the temperature treatment, the average pore radius can be varied from ~15 -
100A. The Gelsil was used as it was received from the manufacturer with no additional
heat treatments applied.

To verify the manufacturer’s values for the pore size and porosities of these
7 silicas, the samples underwent additional nitrogen adsorption analysis by a third party.3°
Even though the measurements were made approximately nine months after the samples
were made and first analyzed, the agreement (Table 4.2) between the results is quite
good. Analysis of the adsorption and desorption isotherms (Figure 4.4) confirms the
narrow pore distributions claimed by the manufacturer and indicates the pores are roughly
cylindrical (based on the shape of the hysteresis curves). As discussed earlier, while these
distributions provide some information about the pore size, they are not a true
representation of the pore geometry. In order to gain some additional insight into the
structure, preliminary analysis with Scanning Electron Microscopy (SEM) and Atomic
Force Microscopy (AFM) has been carried out.

It should be noted that the adsorption analysis performed by the manufacturer was
carried out on a larger amount of the material than was available for the later analysis.
The accuracy of the measurements increase with a larger amount of adsorbent because of
the larger total pore volume. This, and the fact that the manufacturer’s analysis was

carried out shortly before the materials were used in the diffusion studies gives us greater

85
confidence in those results. The additional analysis was performed to verify the

manufacturer’s results and gain some knowledge of the pore distribution.

The SEM analysis was carried out by Dr. Baokang Bi on a Hitachi S-4700
microscope on the uncoated Gelsil surface. Figure 4.5 shows a 200 x 200 nm image of
the 2.8 nm average pore radius Gelsil glass, while Figure 4.6 shows a 600 x 450 nm
image of the 9.1 nm pore radius glass. Here, the bright regions correspond to the
protruding silica, and the dark regions to the space between them. Assuming that the
pores exist at the boundaries between the particles in these images, the feature sizes in
both images are consistent with the pore radii obtained from nitrogen adsorption.

The AFM analysis was performed on a Digital Instruments Dimension 3100 SPM
operating in Tapping Mode. Figure 4.7 shows the surfaces of the 1.4, 2.8, and 9.1 nm
pore radius samples. All show colloidal structures as seen in the SEM images. It is
important to realize that in all probe microscopies, the topographic image obtained is a
convolution of the surface features with the shape of the probing tip. Therefore, small
lateral features on rough surfaces can appear larger than they actually are.

Both SEM and AFM indicate the colloidal-like growth that is expected in sol-gel
glasses37 where the colloidal gel particles grow first, then combine to form the continuous
network of the glass. The pores in these samples may occur in the gaps between the
colloidal particles, similar to the “Swiss-cheese” model discussed in Chapter 2. In
Chapter 5, we will compare predictions of the Swiss-cheese model to the results of

diffusion in these Gelsil samples.

 

Figure 4.5: SEM image of the uncoated surface of the 28 A pore radius Gelsil
sample. The field of view is 2000 A x 2000 A.

87

 

Figure 4.6: SEM image of the uncoated surface of the 91 A pore radius Gelsil
sample. The field of View is 6000 A x 4500 A

 

Figure 4.7: AFM topographic images of the surfaces of the 1.4, 2.8, and 9.1 nm
pore radius Gelsil samples (top to bottom). The images in the left column
and right colurrm are (l um)2 and (200 nm)2, respectively.

89

Table 4.3: Properties of the HS-40 and AS-40 Ludox colloidal
silica suspensions.

AS-40

counter ion 0
' le diameter nm 22
wt% silica 40
H . 9.1
surface area m/

with

 

V. The manufacture and properties of gelled Ludox

The gelled Ludox samples were made by the author from Ludox AS-40 and HS-
40 colloidal silica38 (Table 4.3). The silica particles are kept in a suspension of high pH
water to produce a negative charge on the particle surface, (Si-O-H + O-H‘ —> Si-O' +
H20) to cause the spheres to repel when they are close, thereby preventing aggregation
and settling. The positively charged counter ions surrounding each particle screen the
surface charges and eliminate any long range coulomb interactions between particles. For
Ludox to gel, the particles must be allowed to interact with each other. This may be
accomplished in several ways: one can reduce the surface charge (by reducing the pH of
the solution), increase the concentration of the silica (to force the spheres closer together),
increase the salt concentration (Cl' ions force the positively charged counter ions closer to
the surface of the spheres, reducing the length over which the repelling force exists), or
increase the temperature (to increase the number of interactions between particles).
When the particles interact, the spheres can stick together through hydrogen bonds

formed by the surface hydroxyl groups and adsorbed water. This bonding becomes

90
permanent with the formation of Si-O-Si bonds between particles. Because hydroxyl ions

are a catalyst for Si-O-Si bond formation, as well as the cause of the negative surface
charge on the spheres, their concentration strongly effects the stability of the particles in
the sol. At high pH, the surface charge causes the stability, while at low pH the lack of
OH' ions increases the stability by not causing the formation of Si-O-Si bonds between
particles when they do interact. The sol ends up being least stable for a pH in the range of
5-6.

The porous silica samples prepared from Ludox for this study were gelled through
the addition of a dissolved salt (NaCl for HS-40, NH4C1 for AS-40). 5 ml of the
appropriate salt solution was added to 20 ml of the colloidal silica while stirring to
produce the desired molarity of salt in the final mixture. (The 0.290M NH4C1 in AS-40
mixture used an addition of 15 ml salt water to 40 ml of AS-40.) The solution was then
poured into four polystyrene cuvettes which were sealed with parafilm and a plastic cap
and allowed to sit undisturbed until they were gelled. The remaining solution was sealed
in a vial to gel as well. One additional HS-40 sample was created by allowing it to gel
without any salt being added. It was sealed in a cuvette and sat undisturbed for nearly
two years. As expected, the solutions with the largest salt concentration gelled the
quickest.

After gelling, the network begins to shrink as additional bonds form, drawing the
particles together and expelling water from the pores. Eventually, the gel pulls away
from the walls of the cuvette and becomes loose. At this point, it was removed to a

plastic sample box for further drying. To provide more thorough drying and perhaps

91
additional densification, the samples (not including the 0 M HS-40 sample) were heated

in air to 600°C or 900°C for six hours, which produced weight losses of 5 - 10%. As
discussed earlier, these temperatures will remove water and hydroxyl groups from the
silica surface and, for samples heated to 900°C, a surface that is at least partly
hydrophobic surface should result. This surface will have a greatly reduced number of
hydroxyl groups which should reduce the binding of molecules to the surface. The HS-40
samples, however, cannot be heated above ~700°C due to the presence of sodium in the
network. Alkali ions in silica are known to cause crystallization at relatively low
temperatures:39 Li produces quartz while Na produces cristobalite. This crystallization
was seen in an HS-40 sample heated to 850°C in air. A Kaiser Optical Systems, Inc.
Raman Microscope was used to identify the crystalline cristobalite phase formed in this
piece as shown in Figure 4.8.40 Therefore, these samples were only heated up to 600°C to
dehydrate them. The presence of Na in the network is not entirely undesirable, though,
because of the ability of the ion to block molecules from adsorbing to the surface of silica
(this will be discussed in more detail later).

Nitrogen adsorption was used to characterize the pore size and distribution of
these samples.41 Table 4.4 lists the results of these measurements and Figures 4.9 and
4.10 show the pore size distributions determined from the desorption isotherms. The
three AS-40 Ludox samples do not have a very large range of pore radii or porosity. The
0.149 M and 0.088 M samples appear to be nearly identical in terms of their surface area
and porosity, which leads to essentially the same Wheeler average pore radius. Their

pore size distributions are slightly different (Figure 4.9) and indicate that the 0.149 M

92

 

 

 

 

 

 

 

    

  

0.290M NaCl HS-40 Ludox heated to 600°C

231

Raman Intensity (arb. units)

 

 

 

H840 Ludox heated to 850:7C

 

 

0 500 1 000 1 500

Raman Shift (cm77)

Figure 4.8: Raman spectra of HS-40 Ludox glasses heated to 600 °C (top) and
850 °C (bottom). The top spectra shows the broad Raman lines
characteristic of fused silica. The lines labeled in the bottom spectra are
lines characteristic of the crystalline silica phase of cristobalite. The
spectra are offset and magnified for clarity. Both were acquired using the
633 nm line of a HeNe laser as the excitation. The insert shows the

spectra of nonporous amorphous silica.

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3, —o— 0.149 M NH4C1

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pore radius (A)

Figure 4.9: Pore size distribution of the A840 Ludox glasses calculated from
the desorption isotherms.

sample has a smaller median pore radius. The 0.290 M sample, which gelled much faster
than the others, has a larger average pore radius and a wider pore distribution. This is
expected for a gel that formed quickly and did not have time to create a more compact
structure.

Three of the four HS-40 Ludox samples follow the same progression: as the
concentration of NaCl was reduced in the 0.290 M, 0.088M, and 0 M samples, the gel
time increased, the average pore radius decreased, and the width of the pore size

distribution decreased (Figure 4.10). This implies that as the sol is given more time to gel

95

 

 
  
  
     
 

   

  

 

 

    
   

A
°< 1 1 T r r
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5 0.010 - :~ -
v .’ —A— 0.149 M NaCl

12 .’ . —<>— 0.088 M NaCl

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10 100 1000
pore radius (A)

Figure 4.10: Pore size distribution of the HS-40 Ludox glasses calculated from
the desorption isotherms.

it does form a more compact and uniform network. The 0.149M sample, on the other
hand, is unlike any of the other samples. It has unusually low surface area and pore
volume, and has no peak in the pore size distribution. This sample also has a mottled
appearance, with some regions appearing whiter than others. This suggests that some
regions have been sealed off from the rest of the networ --leading to both the mottled
appearance and the lowered surface area and pore volume. Fluid diffusion in this sample

also does not conform to the other samples.

96
Comparing these results to the Gelsil results (Table 4.2 and Figure 4.4) we see

that these samples are very different. The average pore sizes are quite large, but with
porosities that are much less than the sol-gel produced Gelsil. The pore size distributions
of the Gelsil glasses are significantly narrower than those of the Ludox, showing that their
special preparation does create a porous network with pores of more uniform size.
VI. The manufacture and properties of Vycor

The Vycor 7930 sample was a gift from Coming42 in the form of a 1/8” diameter
rod. Vycor is produced from a borosilicate glass which is carefully heat treated to phase
separate the Boron—rich region from the silica—rich regions, and then acid etched to
remove the boron-rich phase. The rod was sectioned with a low-speed diamond
impregnated saw to produce a section 0.065” thick for the FRS measurements. Another
length was removed for nitrogen adsorption analysis. Figure 4.11 shows the pore size
distribution for this glass which has an average pore radius of Rw = 24.5 A and a porosity
of 0.359. These values are very similar to the values found for the Vycor used in
previous diffusion studies (Chapter 2). This glass is believed43 to have a pore structure
very different from that found in sol-gel produced glasses (as indicated by the difference
in the porosities between it and Gelsil with a similar pore size), making it a very good
comparison for the results found in the Gelsil glasses.
VII. Characteristics of the fluid and Methyl Red

The fluid used throughout this study was a mixture of 90% (by volume) glycerol
(HOCHzCH(OH)CHzOH) and 10% deionized water. This fluid is transparent to visible

light and has an index of refraction (n) at room temperature close to that of silica (~l.46).

97

 

 

 

 

 

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EODOOM‘ "Mu—PP “4*“-
10 100 1000

pore radius (A)

Figure 4.11: Pore size distribution of the Vycor porous glass calculated from the
desorption isotherm.

In an optical experiment such as FRS, light scattered by variations in n causes a
background signal which must be compensated for in the analysis of the detected signal.
By closely matching the refractive index of the sample to the fluid in its pores, this
scattered light may be eliminated. A second reason for choosing this fluid is that it
mimics the surface of the silica. The tracer molecule added to this fluid is surrounded by
-OH groups regardless of whether it is near or far from the pore walls. This means that
the tracer molecule does not have a stronger affinity for the walls than the fluid, a

situation which does occur if an anhydrous fluid is used.

98
Sodium hydroxide (NaOH) was added to the fluid for two purposes: to improve

the solubility of the tracer dye, and to supply Na+ ions to help block adsorption of the dye
to the silica surface. The two batches of fluid made during this study had concentrations
of 0.014M and 0.054M NaOH.

To this mixture of glycerol, water, and NaOH is added the tracer molecule,
Methyl Red (2-[4-(dimethylamino)phenyl-azo]benzoic acid). (see Figure 4.12) Methyl
Red is an azobenzene-based dye which was first used in an FRS experiment by Hervet et
al.44 to study mass diffusion in a liquid crystal and has since become one of the most
common tracers in use in these experiments.45 Methyl Red is a roughly planar molecule
with lateral dimensions of ~7 by 15 A. Commonly used as a pH indicator, the color of
Methyl Red changes from red to yellow as the pH changes from 4.4 to 6.0. This color
change is the result of a H+ ion bonding with one of the nitrogen atoms making up the azo
bridge between the phenyl rings.46 (Figure 4.12a) As we will see later, this color change
is a useful method of examining the interaction between Methyl Red and the surface of
silica.

Like many members of the azobenzene-based family of dyes, Methyl Red is
photochromic, meaning that it changes its color when it absorbs light. This color change
occurs as the molecule undergoes isomerization--a conformational change of the
molecule structure. As light is absorbed and the molecule is lifted to an electronic excited
state, half of the double azo bond is broken, allowing the molecule to twist along that
axis. When the electronic excitation decays, the azo bond is reformed, and the molecule

may be left in the metastable cis conflgurational state (the cis isomer).47 (Figure 4.12b)

99
COOH COOH
a
) (I ©(
N<CH3)2 H ElN(CH3>2
alkaline (yellow) acid (red)

b)

(ICOOH hv

kT
E (trans) Z (cis)

Figure 4.12: Methyl Red molecule used as the tracer molecule in the FRS
experiments. The molecule is roughly planar with dimensions
~15 A x ~ 7 A. (a) The alkaline and acid forms of Methyl Red, showing
hydrogen bonding to the azo bridge. (b) The trans and cis states of Methyl
Red. The cis state may be created through absorption of light, and will
decay thermally back to the trans state.

The two states of the dye are chemically identical, but differ in their electronic and optical
properties.48 Most important are the optical changes: the absorption bands change and the
index of refraction changes. These changes are what make Methyl Red useful for FRS
diffusion measurements. If the excitation beams are within the absorption band of the
ground (trans) state, the interference pattern will be transferred into an identical pattern of
cis state molecules, producing the desired modulation of the refractive index.

Figure 4.13 shows the ground state absorption spectrum (see Appendix A) of
Methyl Red in the glycerol/water fluid. It is important to note that in the FRS

experiments, the excitation laser (488 nm) lies within the absorption band, while the

100

 

 

 

 

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1.2 - 1
l- 1
1.0 - -
L.
0.8 P _
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.8
<
(14 - -
*- 1
0.2 — ..
0.0 1 l 1 l 1 l n I I A |

 

200 300 400 500 600 700 800 900
Wavelength (nm)

Figure 4.13: The absorption spectrum of Methyl Red in a mixture of 90%
glycerol, 10% water, and 0.054 M N aOH.

probe laser (633 nm) lies well outside. When the dye is illuminated with the excitation
laser, the absorption bands shift slightly. Figure 4.14 shows the difference in the
absorbance between the illuminated fluid and the unilluminated fluid, indicating the
difference in the absorbance of the cis and trans states of Methyl Red. We see that the
absorption does not change at the probe wavelength. This indicates that the grating
probed by the probe laser is an index of refraction grating, not an absorption grating. This

is the conclusion reached in other experiments as well.49

101

 

 

 

 

 

 

 

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<t '{LZO b _
)-
_0'25 n l L l l l 1 l A I I I n l j I

100 200 300 400 500 600 700 800 900

Wavelength (nm)

Figure 4.14: The change in the absorbance of Methyl Red when illuminated with
488 nm light. Plotted is the absorbance with the pump beam off
subtracted from the absorbance with the pump beam on. A positive value
means the absorbance is larger when it is illuminated.

The metastable cis state may relax to the trans state through thermal excitation at
room temperature. The lifetime of this process is quite long compared to electronic
excitations (on the order of seconds). This is necessary for studies of slow diffusion,
because the excitation must exist long enough for the molecules to diffuse far enough to
effect the refractive index modulation. The lifetime of cis Methyl Red in water has been

shown50 to depend upon the concentration of the OH' ion in the solution. To verify this, a

102
room temperature transient absorption experiment was performed on three solutions of

Methyl Red in water with varying concentrations of NaOH. Appendix A discusses the
experimental setup used in this and other absorption experiments during this study.
Figure 4.15 shows the lifetime of the cis state in 2.6mM Methyl Red in water with 0.006
to 0.14 M NaOH, excited by a 2 5 long, 245 mW, 488nm laser pulse. Plotted are the
results obtained at three different wavelengths of the absorbed light, which show good
agreement over this range. Also shown is the lifetime measurement at 515 nm of the
2.64mM Methyl Red in 90% glycerol, 10% water (0.054 M NaOH) fluid. This result is
quite consistent with the data of Methyl Red in water, and with the lifetime obtained from
the analysis of FRS signals in this fluid.

Studies of other azobenzene derivatives that were incorporated into sol-gel glasses
(incorporated directly into the silica network, not placed in a fluid in the pores), have

1 Molecules that are

shown that the lifetime of the cis state is affected in two ways.5
hydrogen bonded to the surface show an increased lifetime, while molecules that can only
reach the cis state in a strained configuration (because of the small volume available to
it), show a faster relaxation to the trans state. This suggests that when the Methyl Red
molecules in these experiments are placed in small pores and are not bound to the surface,
the lifetime should be reduced from its value in the free, unbounded fluid.
VIII. Surface chemistry of silica and its interactions with Methyl Red

As discussed earlier, the hydrated surface of silica is covered with hydroxyl

groups and adsorbed water. The surface hydroxyl groups are the sites that provide the

acidic and chemically active character of silica.52 One method for reducing the activity of

103

 

 

  
  

 

 

 

 

   

 

 

 

20 ' 1 g . l . ' 1 r l ' l ' I ' I
*\ Transmission at 5l5 nm after
1;, a 25, 488 nm pulse in 2.64mM
' S Wethyl Red in 90% glycerol/water ‘
Ewe .
E
15 " 2 -'
2
17: E
v ' E 10 . ‘
Q) m
a a
”-4 l—
6 10 _
u-q 2 ‘
S
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5
m .
m
6
5 - —El— 1 = 5 l5 nm -
—O— l = 375 nm
—A— 9. = 235 nm _
—V— A = 5 l5 nm (in glycerol/water)
A l A I A l A l A I A I A I

 

 

 

o
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
Molarity NaOH

Figure 4.15: The lifetime of the cis Methyl Red molecule in water with different
concentrations of NaOH. The “down triangle” point is the lifetime in the
90% glycerol] 10% water solution used in the FRS measurements. The
lifetimes were determined by transient absorption at room temperature.
The dye was excited at 488 nm and the transmission of light at three other
wavelengths was monitored to determine the decay rate. The inset shows
one of these decays on a log scale.

the surface (i.e. reducing the adsorption of molecules from solution onto the surface) is to
remove the hydroxyl groups through dehydration. This will produce a hydrophobic
surface that has no hydroxyl groups and 53 will not adsorb molecules readily. Placing a
completely dehydrated surface into water will not rehydrate it unless the water is at a very

high pH (9 or greater), or the temperature is raised significantly above 100°C.

104
Methyl Red will readily adsorb to the surface hydroxyl groups on silica and in

turn become bright red in color.54 This is expected for this dye as it enters an acidic
environment (see the earlier discussion on the pH properties of Methyl Red). The
adsorption of Methyl Red has been used to measure the number of active hydroxyl sites
on the silica surface, and in one study,55 it was shown that as the number of hydroxyl sites
decreases on a dehydrated silica surface, the adsorption of Methyl Red is reduced. It was
also found that the presence of adsorbed water reduces the adsorption of Methyl Red56
because the water interacts preferentially with the hydroxyl groups,57 producing a less
active surface in an aqueous solution. This demonstrates that the solvent used to carry
Methyl Red will determine how strongly it interacts with the surface of the silica. If an
anhydrous solvent like toluene or benzene is used, the dye will be preferentially pulled
out of solution to adsorb on the surface.

To prevent molecules from adsorbing to the surface, the hydroxyl groups must be
either removed or access to them must be blocked. One way to remove them is the
aforementioned dehydration of silica. However, this treatment can cause the structure of
the silica to change by closing pores. Another common technique is to use esterification--
in other words, replace the hydroxyl groups with another chemical group that is not as
active. This has been shown to be an effective method to block adsorption of Methyl
Red.58 In fact, it is not necessary to replace every hydroxyl group to produce the desired
hydrophobic surface, as a “bushy” molecule can cover nearby hydroxyl groups and

thereby block access to them as well.59 A molecule does not need to be very large to

105
produce this steric blocking effect, because molecules as small as a methyl group (CH3)

are necessary if all the hydroxyl groups on a silica surface are to be removed.

A variety of chemicals have been used to esterify the surface of silica,60 but a few
are quite common and have also been used in previous studies of diffusion within porous
silica: propanol,"l dimethyldichlorosilane,62 and hexamethyldisilazane.63 Figure 4.16a
schematically shows the silica surface after reacting with hexamethyldisilazane to leave
trimethylsilyl groups on the surface.

There are problems with the use of esterification to stop adsorption on the silica
surface. Not all the hydroxyl groups are replaced, and while most of those remaining may
be sterically blocked, not all will be. This will allow some adsorption to occur,
particularly when an anhydrous solvent is used. More importantly, the addition of these
molecules on the surface can dramatically alter the structure of the pores, particularly
when the pores are very small. As the surface is covered, the effective pore radius will be
reduced, which can cause pores that had been accessible to a diffusing particle through
narrow openings to become inaccessible. As an aside, it should be noted that by
“removing” small pores the measured diffusion coefficient may actually increase, because
the diffusing particles are then allowed to move only in the relatively unrestrictive larger
pores. Therefore, the surface treatment used can alter the fluid diffusion in the pores.

A third way of reducing adsorption to hydroxyl sites is well known, but not as
commonly used, because it does not produce a permanent hydrophobic surface. For pH >
2, negatively charged Si-O' sites will attract counterions from the surrounding fluid. Na”

counterions in an aqueous solution are surrounded by six water molecules which increase

106

(a) treated with hexamethyldisilazane

... ... '... ... ... '... ... ... '... . S rf

\ IT\ IT\ \ IT\ I \ I|’\ IT\ IT\ " a“
O O O O O O O O O O O
(b) pH above neutrality

Ii‘ i‘
‘P

'- —? (P (P Surface
\OI I \OI1KO}I\O}1\O}1\01 0

Figure 4.16: (a) Silica surface reacted with hexamethyldisilazane to place
trimethylsilyl groups on the surface. (b) Silica surface with a N a‘
counterion (and its entourage of water molecules) above a charge site on
the surface.

the effective size of the ion. While the ion is not physically bound to the charge site on
the surface, it will remain very close to it and shield it from other molecules in the
solution. Due to the size of the water “cloud” surrounding it, neighboring hydroxyl sites
are also covered (Figure 4.16b).(’4 The effect of the counterion is the same as the attached

molecule on an esterified surface, except for two key features: the surface is not

107
hydrophobic and the sodium ion is still mobile. This means that the pore structure is left

unchanged, and the likelihood of a pore being completely blocked off is greatly reduced.

If Methyl Red is within a porous silica containing sufficient sodium and OH’, it
will not change its color, indicating that the molecule has not been adsorbed onto the
hydroxyl sites and is not experiencing a very low pH. Without Na+ ions, the bright red
(low pH) color is found, and no FRS signal can be detected, indicating the molecules are
bound at the azo bridge. If the porous silica is made with sodium incorporated into it
(like in the HS-40 Ludox samples), no additional sodium ions are required to prevent
adsorption.

A clear indication of this is the behavior of a piece of untreated silica placed in a
solution of Methyl Red, glycerol, water, Na+, and OH“. The initially white surface turns
red as the fluid diffuses into the pores and Methyl Red is bound to the surface. However,
as time goes on, the Na+ and OH' ions diffuse into the pores--slower than the dye because
they interact strongly with the pore walls. As they appear in the pores, the pH increases,
and the Na+ ions block the hydroxyl binding sites, freeing any Methyl Red bound to the
surface and turning color back to the original orange color of the solution. If the silica
had been made with Na already in the network, the color of the solution never turns red as
it diffuses in.

To incorporate Na into the pores before the addition of Methyl Red, the Gelsil
samples and the Vycor are first soaked in a solution of water and sodium hydroxide, with
the pH kept below ~9 to avoid the dissolution of the silica. The sample must be left in

the solution for many days, in order to allow the ions time to diffuse into the pores. The

108

time required depends upon the pore size of the sample, because the diffusion is slower
through small pores. It may be necessary to periodically refresh the solution to add more
ions. The sample is then removed from the liquid and placed in the solution of Methyl
Red and glycerol. If the sample begins to turn red, it is removed, then rinsed and soaked
in water and NaOH. Once in the Methyl Red fluid, the sample is allowed to sit for
several days to equilibrate. It is sometimes heated to reduce the viscosity of the fluid and
speed its infusion into the pores. It should be noted that in all these samples, the fluid is
seen to enter the silica uniformly from all sides. This implies that the pores form a well
connected network.

In conclusion, we use a hydrophilic fluid along with added Na’r ions in order to
block the adsorption of our Methyl Red tracer molecule to the surface. By simply
monitoring the color of the fluid solution in the silica, we can determine if the tracer is
bound to the surface. With this system, we can probe only the effect of the pore geometry
on the diffusive behavior of the fluid, not the effect of the chemical interactions between

the tracer and the walls.

l 09
References

 

l D.W. Breck, Zeolite Molecular Sieves: Structure, Chemistry and Use (John Wiley and
Sons, New York, 1974).

2 RM. Barrer, Zeolites and Clay Minerals as Sorbent and Molecular Sieves (Academic
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3 CT. Kresge et al, Nature 359, 710 (1992).
4 R.K. Iler, The Chemistry ofSilica (John Wiley and Sons, NY 1979), p. 551.
5 R.K. Iler, ibid., Chapter 4.

6 EM. Rabinovich, in Sol-Gel Optics: Processing and Applications, edited by LC. Klein
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7 R. Zallen, The Physics ofAmorphous Solids (John Wiley and Sons, Inc., New York,
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110

 

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23 SJ. Gregg and K.S.W. Sing, ibid.

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3‘ K.K. Unger, ibid., p. 37-40.
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40 RS. Darling, I.-M. Chou, and R]. Bodnar, Science 276, 91 (1997); SR. Elliott,
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47 H. Rau, in Photochromism: Molecules and Systems, edited by H. Diirr and H. Bouas-
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48 H. Zollinger, Azo and Diazo Chemistry (Interscience, New York, 1990), p. 59; M. Irie,
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49 M. Terazima, K. Okamoto, and N. Hirota, J. Phys. Chem. 97,5188 (1993).

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52 R.K. Iler, ibid., p. 182.

53 R.K. Iler, ibid., p. 661,663.

112

 

54 I. Shapiro and I.M. Kolthoff, J. Amer. Chem. Soc. 72, 776 (1950); W.K. Lowen and
EC. Broge, J. Phys. Chem. 65, 16 (1961).

55 I. Shapiro and I.M. Kolthoff, ibid.

5" I. Shapiro and I.M. Kolthoff, ibid; K.K. Unger, ibid., p. 210.
57 K.K. Unger, ibid., p.130.

5“ W.K. Lowen and EC. Broge, ibid.

59 R.K. Iler, ibid., p. 689.

6" Cabot Corp., PO. Box 188, Tuscola, IL 61953. Technical data sheets TSD-
121,130,131.

6‘ W.D. Dozier, J.M. Drake, and J. Klafter, Phys. Rev. Lett. 56, 197 (1986).
62 R. Messager et al, Europhys. Lett. 10, 61 ( 1989).

63 C.K. Colton, C.N. Satterfield, and OJ. Lai, Amer. Inst. Chem. Eng. J. 21, 289 (1975);
Y. Guo et al , Phys. Rev. A 46, 3335 (1992).

6“ R.K. Iler, ibid., p. 657-8.

Chapter 5
DIFFUSION IN GELSIL POROUS SILICA

I. Forced Rayleigh scattering decay curves

Gelsil samples that were infused with Methyl Red, glycerol, and water were
transferred along with some of the fluid to the temperature controlled sample cell and
allowed to equilibrate for at least 12 hours before any data were taken. In order to press
the windows against the sample on both sides, the sample cell was screwed closed to
compress the viton o-ring between the windows. The distance between the windows
varied from sample to sample because of the different sizes of each cylindrical sample (5
- 6 mm in diameter and 2 - 3 mm thick). This was done to prevent a layer of free fluid
between the sample and the window from producing an additional diffracted signal. FRS
signals were then acquired over a range of temperatures of 10 - 38.8 0C and grating
periods of A z 1.2 - 7.5 11m. These grating wavelengths are several orders of magnitude
larger than the sizes of the pores in the silica samples. In this fluid, maximum diffraction
efficiencies (see Chapter 3) of n z 1 x 10’6 to 5 x 104 were found, which correspond to
modulations of the index of refraction of An 2 1 x 10'7 to 2 x 106. These values of n
were calculated from the largest diffracted signal during an excitation pulse and are
therefore not steady-state values of the diffraction efficiency.

The decay signals behaved as single exponentials both in the free fluid and the
Gelsil, with only a few exceptions. The top plot in Figure 5.1 shows the decays in the
five Gelsil samples at A = 1.225 um and 10.0 0C on a log scale to demonstrate the single

exponential behavior. In this plot, the steepest slope corresponds to the largest pore

113

114

 

 

 

 

 

 

 

 

 

Normalized Diffracted Intensity
C

 

 

 

 

 

 

 

 

 

 

 

 

0.01 ,-

5 - I I I I I f q
A l
3
'E 4 r- 5
‘3
1‘5. 3- -
A? t
8
E 2 ' "
.8 t (a) A = 6.052 nm
‘3 1_ (h)It=3.551um _
E t (c) A = 1.224 um

O . l . r . r 1 ~

0.0 0. l 0.2 0.3 0.4 0.5

 

4 T ‘r T T I I V I ‘7

 

 

 

 

 

 

 

 

 

7.2‘

'E A
3.

A? .
é (a) A = 6.052 urn

- (b) A = 3.551 run

(2 (c) A = 1.224 11m

'5 -~ we"- ea; 2-; -

0.00 0.05 0.10 0.15
Time (s)

Figure 5.1: (Top): Normalized FRS decays of Methyl Red within the five Gelsil
samples (rp = 14.0, 16.4, 27.7, 35.5 and 91.5 A) at A = 1.225 um and
10.0 °C plotted on a log scale. The slope decreases with decreasing pore
size. (Middle and Bottom): FRS decays of Methyl Red in the free fluid
(middle) and within the 35.5 A pore radius Gelsil sample (bottom) for
three different values of A at 27.8 0C, plotted on a linear scale.

115
radius sample, and the slope of the decay curves decreases as the pore size decreases.

The inset plot is the data from the smallest pore size sample, shown separately because its
slope is much less than the others. Shown in the lower two plots of Figure 5.1 are
examples of the decay curves found in both the fluid and the 35 A sample for three
different values of A, demonstrating how the decays change with A. In two of the
samples, the data taken at the largest values of A deviated from single exponential
behavior. As discussed previously, these deviations may occur if the cis and trans states
of the Methyl Red molecule diffuse at different rates, if some of the Methyl Red diffuses
slower than the rest because it is interacting with the surface, or if another species is
diffusing as well. For the 27.7 A pore radius sample, the non-exponential behavior was
attributed to the sample cracking under the pressure of the windows and the thermal
cycling of the sample cell. In previous samples that had broken apart, the slow movement
of pieces containing a population grating of Methyl Red was seen to produce an
additional slow decay similar to that found in the 27.7 A sample. Knowing this, the

signals were fit to a double exponential decay equation of the form of Equation 3.18:

(5.1) I = (Ae‘b‘ + Ce’d‘ + F)2 +6

in order to extract the relatively fast decay due to the diffusion of fluid from the
additional, relatively slow decay from the motion of the silica pieces. These results were

then combined with the single exponential decays found at smaller A to determine the

diffusion coefficient.

The 14.0 A pore radius sample showed non-single exponential behavior at the

very largest values of A, but did not deviate enough to reliably fit to Equation 5.1. In this

116
case, the three points showing this behavior were discarded. In some of the gelled Ludox

and Vycor samples discussed in the next chapter, the same behavior was seen. In those
cases, resoaking the samples in N aOH and water to supply more Na+ in the pores resulted
in decays showing single exponential behavior, but with a normalized diffusion
coefficient that was unchanged from what was found before the samples were resoaked.
This shows that it is possible to analyze non-single exponential decays to determine
meaningful diffusion coefficients.
11. Determination of the diffusion coefficient

Once the decay rates were found, they were plotted against the square of the

grating wavevector (qz). If a straight line results, then it verifies that normal, Fickian

diffusion is occurring, with (r2) oc time. The lines are fit to Equation 3.16:

(5.2) l = qu +i
1:

to determine the associated diffusion coefficient and dye lifetime. All the samples in this

study show this normal diffusive behavior. Figure 5.2 is a plot of 1 vs. q2 for both the
1

free fluid and the fluid within the 16.4 A pore radius sample at four temperatures. There
are several things to note in these plots. First, the difference in the slope between the
fluid and the silica data, which indicates the difference in the diffusion coefficient in the
two environments. Second, the slope increases dramatically as the temperature rises.
This is due to the decreased viscosity of the fluid mixture (causing a corresponding
increase in the diffusion coefficient) at higher temperatures. Third, there is a change in

the y-intercept of the data as the temperature changes. The intercept corresponds to the

1/1: (l/s)

1/17 (Us)

117

 

 

 

 

 

5m ' 1* Y I v I v 1 v I

400 - ~
v 37.8 °C
A 27.8 0C

300 " 0 178°C -
1:1 10.0“C as

200 P d

100 [- . 7 -

O .._.,.—- “'3 1 . 1 1 I 1

 

 

 

0-0 5.0x108 1.0x109 1.5x109 20x109 2.5x109 3.0x109

 

 

 

 

 

2 -2
q (cm )
l I I ' I ' r ' I
v 37.8“C
150 ' A 278°C '
o 17.8°C
:1 100°C
4v- M
100 - W _

M—T"
50W q

 

 

 

0 l L A l I L L

0-0 5.0x108 1.0x109 1.5x109 2.0X109 2.5x109 3.0x109

 

q2 (cm'z)

Figure 5.2: The FRS decay rate (llt) vs. q2 in the free fluid (top) and within the

16.4 A radius Gelsil sample (bottom) at four different temperatures. Also
shown as solid lines are the fits to Equation 5.2.

118
decay rate of the dye back to the ground state, so an increasing intercept implies a

decreasing lifetime of the excited state of the dye. Therefore, as the temperature
increases, the dye lifetime decreases, as expected for a thermally activated process.
Fourth, the dye lifetime is much shorter in the silica than in the free fluid. This will be
discussed in the next section.

Having very slow diffusion and short dye lifetimes means that the molecules may

not diffuse very far before they decay to the ground state. That distance can be estimated

from <r2> = 6Dt, where t is set equal to Tlire. which (along with D) is determined from the

1 vs. q2 data. This distance can then be compared to N2, the distance between the peak
I

and the valley of the transient concentration grating created in the fluid. The smallest

ratio of 1l<r2> to N2 found in these data is ~5%, which occurred in the 14.0 A sample at

the highest temperature, 388°C, and the largest grating spacing A = 7.42 pm, which
means that an average particle traveled approximately 1800 A, or the length of over 60
pores. Even in this extreme case, the diffusive behavior is easily measured (and is seen to
be normal diffusion), because the grating decay rate depends quadratically on q. This
points out the need to measure the decay rate at several values of A in order to prove the
existence of normal diffusive behavior and to separate D from “cure. The temptation to
save a great deal of time and only measure the decay rate at one value of A will almost

certainly lead to an incorrect conclusion about the diffusive behavior. Note that even

when <r2> < N2, the grating modulation is still reduced by diffusion because the

119
grating has a sinusoidal profile, and a diffuser does not have to travel the entire A/2

distance to change the concentration profile.

 

 

Sou- \\
P \‘ d
A
m 10 r \ 1
v ,. ct
K— r _ ink I
v—i )- _ \\ ll
45 _ \ x-.. _
5 1 \3\ -
:11
>5 D 14.0 A \ 2
g o 16.4 A . 4-- g
A A
G 0.1 __ V :22? in free fluid . 4
E o 91.5 A E -
)-

 

 

 

 

 

1 a l A l A l

3.2x10'3 3.3x10‘3 3.4x10'3 3.5x10‘3 3.6x10‘3
1/T(1/K)

 

Figure 5.3: The decay rate of the thermal isomerization of cis Methyl Red back to
trans Methyl Red (1/11) in the fluid and Within all the Gelsil samples vs.
1/T. Each data set is fit to Equation 5.3 to determine an activation
energy for the process.

120
III. Temperature dependence of the Methyl Red lifetime

Since the decay rate of cis Methyl Red back to trans Methyl Red may carry some

information regarding the environment the molecule is in (Chapter 4), the activation

energy of this process is of interest. Figure 5.3 plots the log of the decay rate —1— against
TI

——1— for the data taken in every Gelsil sample and the free fluid just outside each sample.

Each data set is fit with:

 

(5.3) i=Aexp[— E” J
I, kBT

where A is a constant, Ea is the activation energy, and k3 is the Boltzmann constant. The
most striking aspect of this plot is the clear separation between the data in the Gelsil and
the data in the free fluid, showing an order of magnitude difference in the decay rate
between the two locations.

Table 5.1 lists the activation energies determined from the fits in Figure 5.3.
While there is no believable correlation between the pore size and the activation energy,
there is a real reduction in the activation energy when the Methyl Red is in the pores,
from the value in the free fluid. These two results, that the Methyl Red has a higher
decay rate and a lower activation energy in the pores may be due to two effects: the pH of
the fluid in the pores, and the small volume of the pores themselves. As discussed in
Chapter 4 (VII), the lifetime of Methyl Red is dependent upon the concentration of the
OH' ion. Because OH' ions react with surface hydroxyl groups on silica (producing H20

and Si—O', which attracts the Na+ ions), the region near the silica surface has a pH less

121

Table 5.1: Activation energies of the thermal cis to trans isomerization of
Methyl Red in the free fluid and within the Gelsil samples.

 

 

 

 

 

 

Gelsil pore radius Activation energy in free fluid Activation energy in Gelsil
(A) (W) (eV)
14.0 0.77 1- 0.08 0.552 1 0.006
16.4 0.67 i 0.02 0.44 1 0.01
27.7 0.68 t 0.03 0.54 :1: 0.02
35.5 0.72 :t 0.02 0.467 t 0.007
91.5 0.77 :t 0.05 0.42 :t 0.01

 

 

 

 

 

than the bulk fluid. As the pores get smaller, the Methyl Red in the pores is always close
to a wall, and is therefore always in an environment with a lower pH level. Also, other
work has shown (Chapter 4 VI) that the decay rate of the cis state of azobenzenes
increases if the molecules are located in regions where the cis state can only form in a
strained configuration.1 The same work indicates that the decay rate will decrease if the
molecule is bound to the surface through hydrogen bonds. These results for the decay
rate and the activation energy of Methyl Red in the Gelsil imply that the molecule is not
bound to the surface of the silica, but is in a region of lower pH.
IV. Temperature dependence of the diffusion coefficient

As we saw in Figure 5.2, the diffusion coefficient varies with temperature both in

the free fluid, Do, and in fluid within the sample, D. Figure 5.4 plots the log of D and Do

as a function of :1; for one of the Gelsil samples. Over this small range of temperatures,

the data can be fit to an exponential

122

 

El free fluid
\- 0 fluid in 35.5 A pore Gelsil

p—A
9.
q
I"
j. LLLLI

10'8 3' \§\ 1

Diffusion Coefficent (cmzls)

 

 

 

0.0032 0.0033 0.0034 0.0035 0.0036
1 / Temperature (l/K)
Figure 5.4: The diffusion coefficient in the free fluid and within the 35.5 A pore

radius Gelsil sample plotted vs. 1/T. Both data sets are fit to Equation 5.4
to determine an activation energy.

 

(5.4) D=Aexp[— E“ ]
kBT

to determine an activation energy (Ea) for the diffusive process. Table 5.2 gives the
results of these fits which show no clear trend. The small range and small number of

temperatures examined make it difficult to draw any conclusion about these results. We

do see, however, that the diffusion coefficient is highly temperature dependent.

123

Table 5.2: Activation energies of D and Do in the free fluid and
within the Gelsil samples.

 

Gelsil pore radius

Activation energy in free fluid

Activation energy in Gelsil

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(A) E. (eV> E, (eV)
14.0 0.491 :1: 0.007 0.501 :1: 0.117
16.4 0.489 3: 0.008 0.475 1 0.045
27.7 0.515 :1: 0.008 0.449 :1: 0.033
35.5 0.493 i 0.010 0.551 :1: 0.033
91.0 0.448 t 0.009 0.457 :1: 0.009
‘7 f I 1 , . , , , r.— 0.010
10 I. * t
A t ' “ " D. - 0.008
(g b I- .— o D (in 14.0 A pore Gelsil)
E —’ 0 D/DO 'l
3 10’8 :-
E? E __ E - 0.006
.2 .
g l) A g
8 9 - 0.004 °
U 10' -
S __ t
.5. _ i
g _ ~ 0.002
G I
10-10 :- i
A l A 4 A 1 1 l 1 1 omo
10 15 20 25 30 35 40
Temperature (C)

Figure 5.5: D, Do, and D/Do for the 14.0 A pore radius Gelsil sample as a

function of the temperature. D and Do change by nearly and order of

magnitude, while D/D0 is essentially constant.

 

124
Figure 5.5 plots the diffusion coefficients in both the free fluid and in the silica

against temperature for one of the Gelsil samples. Also plotted is the normalized
diffusion coefficient D/Do. Even though D and Do vary considerably over this range,
their ratio does not. Figure 5.6 demonstrates that this normalized diffusion coefficient is
temperature independent for all the Gelsil samples. This indicates that D/Do is insensitive
to changes in the properties of the fluid, but is highly sensitive to the pore structure of the

silica. This is important, because it means that the normalized diffusion coefficient is

 

 

 

 

 

 

 

 

 

1 : I I I I I I I I
' $ at 9 1|—
3 g E “' l
E 1
: I i -I— 1 .
9° I n 91.0 A Gelsil
Q o 35.5 A Gelsil
A 27.7 A Gelsil
v 16.4 A Gelsil
0.01 .- 0 14.0 A Gelsil -,
I T T § 4.
F j i i -t
r l l A 41 A J A l A L r l

 

10 15 20 25 30 35 40

Temperature (C)

Figure 5.6: The normalized diffusion coefficient, D/Do, for all the Gelsil
samples plotted against the temperature. The straight lines are the
average values of D/Do.

125

Table 5.3: Average normalized diffusion coefficients, D/Do,
within the Gelsil samples.

 

 

 

 

 

 

Gelsil pore radius (A) D/Do
14.0 0.0045 x 0.0008
16.4 0.073 1- 0.006
27.7 0.20 i 0.02
35.5 0.29 i 0.02
91.0 0.53 t 0.01

 

 

 

 

unaffected by temperature dependent factors such as the dye lifetime. For the rest of the
analysis, the average value of D/Do is used with its uncertainty representing the standard
deviation over the entire temperature range. (See Table 5.3) The average value of D/Do
for each Gelsil sample is shown in Figure 5.6 as a straight line.
V. Dependence of the diffusion coefficient on the porosity
Figure 5.7a Shows D/Do as a function of the porosity of the Gelsil silica. The first
thing to notice about this figure is that D/Do decreases in a nonlinear fashion as the
porosity decreases to a finite value, just as expected for a percolation system. The values
for porosity used here are those determined by the nitrogen adsorption experiments.
Because the behavior appears to be percolation-like, a power-law is fit to the data

with the form

D m
(5.5) D—-A(¢—¢.) .

0

with each point weighted by the uncertainty in the value of D/Do. The uncertainty in (11 is

difficult to determine and its absolute value is assumed to be the same for all values of q).

126

 

0.6 J ' 1 ' I fl I ' I ' fl ' I
0.5

I
l

0.4

I

°0.3
0.2

D/D

 

 

 

 

0.0 l . 1 . 1 r I A 1 . 1 A A
0.45 0.50 0.55 0.60 0.65 0.70 0.75
Porosity (0)

 

0.1:

D/D

0.01 E

 

 

 

 

0.01 l ””101
‘1’"?

Figure 5.7: (a) D/D0 plotted against the porosity, in, of the Gelsil samples on a
linear scale. (b) D/Do plotted against (1) - (be on a log-log scale. The lines
shown are the fit to Equation 5.5. The value for (be used in (b) is that
determined from the fit.

127
The results of the fit are m = 1.53 i 0.32, the = 0.45 i 0.01, and A = 3.7 :1: 1.4. To view

this behavior more clearly, Figure 5.7b plots D/D0 against (1) - (be on a log scale along with

the power law fit. The range of the data is such that m < 0.3 , suggesting that the data

may lie within an asymptotic region near (be Where percolation exponents might apply.2 It
also suggests that the mean field results of the Effective Medium Approximation may not
apply in this region.3
VI. Dependence of the diffusion coefficient on the pore size

Figure 5.8 plots D/Do against the average pore radius of the Gelsil silica samples.
The striking aspect of the plot is the sharp cutoff seen as the pore size approaches the size
of the Methyl Red molecule. Also shown in the figure are two fits to the data, one is a
power law (solid line),
(5.6) D/Do = B(rp - RC)n
(with n = 0.49 i 0.06, Rc = 15.0 .4: 0.7 A, and B = 0.06 d: 0.02), the other is the Renkin

equation (Equation 2.10, dashed line),

2 3 5
(5.7) —D—-=F —-r-"— l—2.1044r—”+2.089 51 —0.948 L" .
D0 rp rp rP rp

Both are fit by weighting each point with the uncertainty in D/Do, and the Renkin
equation has an additional free parameter (F) to adjust the amplitude. As with 0, the
uncertainty in the average pore radius is difficult to estimate, in part because the value
depends upon the model used to determine it. Therefore, while the uncertainty in rp is

significant, it is assumed to be the same for all values of rp. The fit of the Renkin

128

 

 

 

 

 

06 ' I I I I l ' I
0.4 -
Q
D
0.2 -
0.0 4‘ l r 1 r 1 A 1 A
0 20 40 60 80 100

r, (A)

Figure 5.8: D/D0 plotted against the average pore radius of the Gelsil samples.
The solid line is the fit to Equation 5.6 and the dotted line is the fit to the
Renkin model, Equation 5.7.

equation gives: I“ = 9.5 1 1.1 A and F = 0.9 :1: 0.1, where r“ should be the effective radius
of the solute molecule (here Methyl Red).

The one previous study that can be compared to these results is that of Mitzithras
et al4 (discussed in Chapter 2 and shown in Figure 5.9). Their data show remarkably
similar behavior with a power law fit of the form of Equation 5.6 over the same range as

the Gelsil data (11,.)l = 0.54 :t 0.08, Rcm = 2.9 i 8.1 A, and Bm = 0.06 i 0.03). The only

129
significant difference is in the critical pore size Rem, which should be smaller in their

data, because they measured the diffusion of cyclohexane, a smaller molecule than
Methyl Red. This agreement is somewhat surprising, given the very different behavior
the Mitzithras data Show as a function of porosity (Figure 2.14b). It may be that the
structure of the glasses is quite different, but that the average pore size is a very good

measure of the diffusive properties of sol-gel glasses.

 

 

 

 

1.0 . . . . r .
0.9; D :1 .
0.8} _
0.7L —

QC 0.6; -
0.5; -
0.4L 4
0.31 -
0'20 I 50 l 100 I 150 I 200 I 250

Pore radius (A)

Figure 5.9: D/Do plotted against the pore radius for the data from Mitzithras et al,
for cyclohexane diffusing in sol-gel glasses. The line is a fit to Equation
5.6 all but the top two data points. (Data from: A. Mitzithras et al, J. Mol.
Liq. 54, 273 (1992).).

l 30
VII. Conclusions

There are several different regimes that might be relevant to these measurements.
First, we might have expected to find behavior similar to that of Archie’s Law (0' cc (1)” )
found in rocks, where the percolation threshold is very near (1) = 0. This does not describe
the behavior in these glasses at all. We might also have expected that the reduced
diffusion coefficient is due simply to the narrowing pores, and not necessarily related to
some pores becoming inaccessible. In this case, we might expect hydrodynamic models,
which are based on this idea, to be appropriate.

The Renkin equation, which is based on the hydrodynamic drag on a large solute
molecule flowing down the center of a long cylindrical pore, does not reproduce the
behavior found in these glasses (Figure 5.8). This is not surprising, given the disordered
structure of the glasses and the fact that the Methyl Red molecule is only slightly larger
than the solvent molecules--violating one of the basic assumptions of the model. The
effective radius it predicts is consistent with Methyl Red, but given the poor description
of the data by the Renkin Equation, should be viewed with caution. This fit shows

r

0 P

v
behavior that is consistent with all the hydrodynamic models, where DI)— —->[ —r—“] , as

rH -—> rp, with v > 1. This leads to a gradual decrease in D/Do as rH —> rp, which

qualitatively disagrees with this data at small values of rp. It may be that this model is
appropriate for comparison to the data at large rp only.
As mentioned earlier, the data in Figure 5.7 suggest that a percolation model

might be appropriate to describe itS behavior. However, it is important to realize how this

131
system is approaching the percolation threshold. For a network of uniform bonds, the

threshold is approached by removing bonds to reduce (l) (or p). In this case, the bonds that
remain are unchanged. The situation is different in the Gelsil materials, where a
reduction in (b is accompanied by a corresponding reduction in the pore size. How, then,
does this type of system approach a percolation threshold? First, consider the case of a
material with a distribution of pore sizes about some large average pore radius. If the
diffusing molecule is small relative to that average pore size, it will have access to nearly
all the interconnected pores and will only be slowed by the hydrodynamic interactions of
the fluid and the finite pore size. If the average pore radius is reduced, the lower end of
the pore size distribution will begin to overlap significantly with the size of the diffuser.
At this point, the smallest pores will become inaccessible to the molecule--effectively
removing them from the network. Therefore, in a real material, it is the finite size of the
diffusing particle and the distribution of pore sizes which cause the system to approach a
percolation threshold as rp and 0 are reduced. In contrast, a network of identical pores
will either be completely accessible or totally inaccessible to a molecule, depending upon
the relative sizes of the molecule and the pores, or if adsorbed molecules block some
pores. In that case, there will be no percolation threshold, as the reduction in the
diffusion coefficient will be due entirely to the hydrodynamic interactions of the fluid in
the pores.

Although we cannot show with certainty that these materials are within the
asymptotic region of a percolation threshold, we expect that this system must approach a

percolation threshold due to the reasons given above. The behavior of the diffusion

132

coefficient in this range of 4) appears to agree with this assumption and so we will assume
these systems are near the threshold in order to compare our results with percolation
models.

Although uniform bond percolation models describe systems very different from
these glasses, continuum percolation models are based on materials with nontrivial
relationships between the porosity and the pore radii, and we might expect them to
describe the behavior we find. As discussed in Chapter 2 (IIIc), continuum percolation
models assign each bond in the network a conductance based upon the size of the pore
associated with it. In these models, it is shown that the smallest pores are responsible for
the overall transport through the network near the percolation threshold. The continuum
model described in Chapter 2, Where the bonds on a random, discrete network have a
distribution of conductances g given by P(g)oc g'“ (for g-—> 0), gives rise to the

following limits on the continuum conductivity exponent (II) near the percolation

threshold (Equation 2.31):
(5.8) max(p.l +y,u) S E S u+y, wheny >0(or0<oc S 1)

5:11, whenySO(orocS0)

where y=—1—g—, gzfiy”, 5 is the pore width, and u, E l+(d—2)v which becomes

11, = 1.88 in three dimensions (d = 3, v = 0.88). Also, recall that u is the conductivity

exponent found in networks with uniform bonds, with an uncertain value ranging from

approximately 1.855 to 2.0.6

133
We also note that the exponent recovered in a diffusion measurement depends

upon how the fluid is distributed in the pores. If the measurement is made with fluid in

every cluster of the network, the conductivity exponent, It", should be found. If the
measurement only probes fluid on the infinite cluster, the exponent should be H— B.

Because the fluid is placed into these porous silicas after it is manufactured (therefore,
after the pore network is formed), it can only reach pores which are connected to the
outside surface of the silica. This means that only those pores on the sample spanning
cluster are filled with fluid. Any isolated pores cannot be filled. The only exceptions are
the negligible volume of pores which make contact to the outside surface but do not
connect to any other pores. Therefore, in these glasses percolation theory predicts the

diffusion exponent to be H - B.

In the context of this continuum model, our result of m = 1.53 i 0.32, implies

that a diffusion measurement on the infinite cluster will produce E— B = 1.53 i 0.32 , or
H = 1.94 i 0.32 (recall that B = 0.41). Comparing this to Equation 5.8, and assuming that
u = 2.0, we would say that CL S O (y S O) in these silicas. This suggests that the

distribution of the conductances of the smallest pores does not have a singularity at g = 0.
We should also compare these results to the exponent expected from the “Swiss-

cheese” model discussed in Chapter 2. In this specific continuum model, where the solid

phase of the porous network is made of overlapping solid spherical particles, Feng et al.

showed that the distribution of the conductances of the smallest pores obey Equation 5.8

with OL =% (y = é). This implies that the conductivity exponent, E , should fall

134
1.94 2.0 2.44

   
     

OLSO oc=1l3

Gelsil data

\

     

1.0 1.5 2.0 2.5 3.0
Conductivity Exponent (II)

Figure 5.10: Plot showing the range of the uncertainty of the continuum
conductivity exponent (H) from the fit to the Gelsil data and for two

predictions of the continuum percolation model. E: Gelsil data, with
E = 1.94 i 0.32 . I: Continuum model with on S O , E = 2.0 i 0.2 .

I: Continuum model with CL = l , E = 2.44 :33.
3 .

between the values 2.38 and 2.5. Figure 5.10 illustrates the value and uncertainty of the

continuum conductivity exponent (E) found from our data and compares it to the values

and estimated uncertainties of the predictions of the continuum models for the cases:
CL S 0 and on =% (the "Swiss-cheese" result). In this figure, the width of each box
signifies the area within the uncertainty of the value of E. The “nominal” value of each

is shown as a vertical line. The box corresponding to our data illustrates E = 1.94 i 0.32 .

For these boxes, the value of u is assumed to be 2.0 1- 0.2, which is an arbitrary

135

uncertainty based on the range of values found in the literature. Therefore, the on = 1/3

box (the “Swiss-cheese” result) has a lower limit of u, +y = 2.38 (because it is larger
than the value of u, see Equation 5.8), and an upper limit of u+ y + 0.2 = 2.7 , with 0.2

added to account for the uncertainty in u. This plot illustrates how our experimental
result compares to these two realizations of the continuum model. It appears that our
result is more consistent with continuum models with CL S 0 than with the “Swiss-
cheese” model. Given the uncertainties, though, we cannot say that the “Swiss-cheese”
model is inconsistent with our data (particularly if the “two-sigma” errors are considered,
which would show a large overlap between the model and the data). However, the
difference between our conductivity exponent and that expected for on = 1/3 suggests that
these glasses may not have a pore structure like that described by this model, where the
pores exist in the gaps between the particles making up the network.

As seen in Chapter 4, the sol-gel process that creates these glasses does appear to
form a network of colloidal particles. However, the large porosities found in these
glasses suggest that the particles themselves are porous, and a network of porous particles
with gaps between the particles would have a bimodal pore size distribution
corresponding to the inter- and intra-particle pores, whereas the pore size distributions for
the Gelsil glasses have only a single peak. This suggests that as the particles formed a
continuous network during gelation, the gaps between the particles filled in with porous
silica. All of this strengthens the conclusion that the Gelsil pore network is not like that

of the “Swiss-cheese” model.

136

Finally, if the diffusion coefficient is viewed as simply a function of the average
pore radius of this silica we find that, as expected, the diffusion coefficient drops to zero

as the pore size approaches the size of the diffusing molecule. But it does so as a power

law with an exponent of 2 V2 which has not been previously predicted. The comparison

of these results in Gelsil with the results found by Mitzithras suggests that this power law

behavior may be common to other sol-gel glasses.

1 37
References

 

' M. Ueda et al, Chem. Mater. 6, 1771 (1994).

2 s. Feng, B.I. Halperin, and RN. Sen, Phys. Rev. B 35, 197 (1987).

3 M. Sahimi et al, Phys. Rev. B 28, 307 (1983), Figure 3.

4 A. Mitzithras, F.M. Coveney, and J.H. Strange, J. Mol. Liq. 54, 273 (1992).
5 AB. Harris, Phys. Rev. B 28, 2614 (1983).

6 D. Stauffer and A. Aharony, Introduction to Percolation Theory, (Taylor and Francis,
London, 1994), p. 52.

“I

Chapter 6
DIFFUSION IN OTHER POROUS SILICAS

I. Vycor

Following procedures similar to those used for Gelsil glasses, Vycor 7930 porous
glass was first soaked in water and NaOH before placing it in the Methyl
Red/glycerol/water mixture. If the glass was not soaked in NaOH long enough, a red
stain could be seen in the glass and a double exponential decay was found for all values
of A. The signal became a single exponential decay by simply allowing the glass to soak
in the solution of NaOH and water for a longer period of time to place more Na+ ions in
the pores and remove the red color. The FRS decay rates from the sample, plotted against
qz, are shown in Figure 6.1.

The normalized diffusion coefficient at 22.2 °C in this glass is DID0 = O. 10 i 0.01,
where D = 4.4 :1: 0.5 x 10'9 cmzls and D, = 4.37 a 0.02 x 10'8 cmz/s. Table 6.1 compares

this to the results of other diffusion measurements in the same glass.

Table 6.1: The normalized diffusion coefficient in Vycor 7930 porous
glass compared to previous measurements. The study by Guo
et al.1 was made with DLS using polystyrene (molecular weight 2500)
as the tracer and fluorobenzene as the fluid. Dozier et al.2 made
their measurements using FRS with azobenzene as the tracer and
(l) methanol/toluene or (2) l-propanol/toluene as the fluid.

 

 

 

 

 

 

 

 

 

Porosity Pore Radius (A) D/Dn
This study 0.36 24.5 A 0.10 :1: 0.01
Guo et al 0.28 20 A 0.105 :t 0.006
Dozier et al (fluid 1) 0.28 30 A 0.010 :r 0.003
Dozier et al (fluid 2) 0.28 30 A 0.026 :r 0.007

 

 

138

5 IA“:

14‘!
"w

139

 

 

120

 

 

 

OO
O

UT (US)

.5
O

 

 

 

 

0-0 5.0x108 1.0x109 1.5x109 2.0x109 25x109 3.0x109

q2 (cm’z)

Figure 6.1: The FRS decay rate (l/T) vs. q2 in the free fluid and within the Vycor

porous glass at 22.2 0C. Also shown as solid lines are the fits using

Equation 5.2. Here D0 = 4.37 :1: 0.01 x 10‘8 cmzls and D = 4.4 :1: 0.5

x 10'9 cmzls.
Our results are very similar to those of Guo et al., but differ considerably from that found
by Dozier et al. Given that Methyl Red has a molecular weight of 270, and is similar in
size and shape to azobenzene (molecular weight 182), these results are rather surprising.
The polystyrene used in the DLS study has a hydrodynamic radius (calculated from
Equation 2.9) of 13 A while Methyl Red in the glycerol/water solution used in this study

has a calculated radius of roughly 4 A (assuming a viscosity of ~200 centipoise, T =

140
295.5 K, and D0 = 2.4 x 10'8 cm2/s in the free fluid). By comparison, azobenzene in the

alcohol/toluene fluids used by Dozier et al., has a hydrodynamic radius of approximately
3 A?

Guo et al. speculated that the difference between their results and those of Dozier
et al. may have been due to residual binding sites that remained on the surface of the
Vycor that Dozier et al. treated in boiling l-propanol. These remaining sites would then
reduce the observed diffusion coefficient, resulting in the surprising low values found in
that experiment. Guo et al. also state that they briefly examined the diffusion of
azobenzene in their hexamethyldisilazane treated Vycor glass, and found that the
diffusion was occurring faster than they could measure. This is very surprising, because
the diffusion coefficient of azobenzene and Methyl Red in solvents such as
methanol/toluene, 1-propanol/toluene, benzene, ethanol, and 2-propanol is in the range of
1 x 10'6 to 1 x 10'5 cmZ/s, which is easily measurable.4 It is hard to explain why
azobenzene in Vycor would diffuse faster than that.

II. Ludox glasses

As a result of the heat treatment of the AS-40 glasses to remove most of the
hydroxyl sites on the surface and the “native” Na in the HS-40 glasses, it was unnecessary
to presoak these samples in the NaOH/water solution. When placed in the Methyl
Red/glycerol/water mixture, the fluid entering the pores did not turn red in the HS-40
glasses, while in the AS-4O glasses the fluid turned faintly red before sufficient Na+ ions
entered the pores from the bulk fluid and blocked the remaining binding sites on the

surface to return the fluid to its original color.

141
Non-single exponential decays of the diffracted intensity found in the 0.088M

NH4C1 gel appeared to be due to free fluid trapped between the curved surface of the
sample and the window because simply translating the beams to a region closer to the
sample edge (where the sample had to touch the window) produced single exponential
behavior. The non-single exponential signal in the 0.149M NH4C1 sample disappeared
after the sample was soaked in the Methyl Red/glycerol/water solution for several more
days, to allow time for more Na+ to enter the pores. However, in both samples, the
normalized diffusion coefficient, D/Do, found from the single exponential decays was the
same as that found from the non-single exponential decays. (The values of D/D0 in the
double exponential and single exponential cases were, respectively, 0.259 and 0.258 :1:
0.004 in the 0.088M sample and 0.26 and 0.265 t 0.011 in the 0.149M sample.) This is a
strong verification that the correct diffusive behavior can be recovered even when an
additional signal is present in the FRS experiment.

Figures 6.2 and 6.3 show the FRS decay rates vs. q2 in the free fluid and within
the seven Ludox samples taken at 222°C. Table 6.2 lists the values of D/Do derived from

this data. D/Do appears to depend strongly on the salt concentration used in the

Table 6.2: Values of D/Do in the seven gelled Ludox samples. All the

 

 

 

 

 

data was taken at 22.2 °C.
Molarity of salt when gelled AS-40 Ludox HS-40 Ludox
D/Dn D/DL
0.290 M 0.46 i 0.01 0.457 :1: 0.003
0.149 M 0.265 :1: 0.011 0.537 :1: 0.007
0.088 M 0.258 :1: 0.004 0.318 1 0.002
0 M ---- 0.197 t 0.003

 

 

 

 

 

142

 

 

 

 

80 ' U ' I ' I ' I I I
I fluid
:1 0.290M NH4C1
O fluid
60 - 0 0.149MNH,C1 -
A fluid
A 0.088MNI-14Cl

 

4,". / j

‘;“-‘V' “V

UT (Us)

I
l

 

 

' 6’1":
1 l I 1 l I l

0 = " . ‘ ‘ ‘
0-0 5.0x108 1.0x109 1.5x109 2.0x109 2.5x109 3.0x109

£12 (cm'z)

Figure 6.2: The FRS decay rate (1/1:) vs. q2 in the free fluid and within the three
AS-40 Ludox porous glasses at 22.2 0C. Also shown as solid lines are the
fits to Equation 5.2.

liquid to make the initial gel, which also determined the time it took to gel the silica.
However, the 0.149 M NaCl HS-40 sample does not follow this trend, just as it showed
very different behavior in the nitrogen adsorption results in Chapter 4. In the following
section we will examine how these data depend upon the porosity and pore size of these

glasses and how they compare to the Gelsil samples discussed in the previous chapter.

143

 

 

80 ..,.,

fluid
0.290 M NaCl HS-40
fluid
0.149 M NaCl HS-4O -
fluid
0.088 M NaCl its—40 ,
fluid 1
O M NaCl HS-40

60

OODDOODI

 

 

4:.

O
L

..,

 

1/1: (l/s)

N
O
1
L

..
0".
'1’.)

l I l l 1

x108 1.0x109 1.5x109 2.0x109 25x109 3.0x109

7;"
' L

O :.‘.
0.0 5.0

 

 

<12 (cm'z)

Figure 6.3: The FRS decay rate ( 1/1) vs. q2 in the free fluid and within the four
HS-40 Ludox porous glasses at 22.2 °C. Also shown as solid lines are the
fits to Equation 5.2.

III. Comparison with the Gelsil data

 

Figure 6.4 plots D/D0 vs. porosity and pore radius for the Gelsil data along with
the Vycor and Ludox samples. These plots graphically illustrate the difference between
the sol-gel grown Gelsil glasses and the others. Gelsil has a much higher porosity than
the other glasses, even though the average pore radius is the same or less. This indicates

that the geometry of the pore networks must be very different.

1.0
0.9
0.8

0.7

0.6

0.2
0.1
0.0

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3

D/D

0.2

0.1

0.0 a r n n l
O 20 4O 6O 80 100 1 20

Figure 6.4: D/Do in all the samples of this study as a function of (a) the porosity

0.4 i
0.3

144

 

 

Gelsil data "
Vycor ‘
Ludox HS-40 "
Ludox AS-4O

 

 

I
QDDO

 

I
I A

D 1
- It 4
b

 

 

.1

 

I a I r I n I .‘u I A I a L 1 I a I n I
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Porosity

 

I j I I I ' I ' I T

'I'I'I‘I
IIII

' I
I

1.1
it

i=1

 

 

 

Pore Radius (A)

and (b) the pore radius.

 

145
First comparing Vycor to the Gelsil, we see from Figure 6.4a that the Vycor point

does not fall along the curve formed by the Gelsil data as a function of porosity, but that
in Figure 6.4b, the point is close to the Gelsil curve as a function of pore radius. Taken
by itself, this might indicate that for two different pore geometries (both with narrow pore
size distributions), the pore radius is a better predictor of the diffusion coefficient than the
porosity. However, with only one point, that is purely speculation.

Looking at the Ludox data, we see that the 0.149 M NaCl HS-40 (D/D0 = 0.54, 0
= 0.12, and Rp = 58 A) sample does not behave like any of the other samples, most likely
for the reasons discussed in Chapter 4. That data point will be ignored for the rest of this
discussion. The Ludox samples group themselves with respect to the Gelsil data in a
fashion that seems to follow the relative sizes of the particles that make up the glass. The
progression of the data is ¢AS-40 < ¢HS-4O < ¢Ge1sfl, and Rp A340 > Rp “3.40 > Rp Gelsil. From
Table 4.3, we know that HS-40 is made of 12 nm diameter silica particles and A840 is
made from 22 nm particles, while the Gelsil is made from initially molecular—sized silica
particles, though they grow to larger sizes before the continuous network forms (Figure
4.7). It is no surprise, then, that the AS-40 glasses have the lowest porosities and the
largest pore sizes of these glasses, since they are made from larger particles. Also, the
AS-40 glasses were subjected to a higher temperature than the other glasses, so some of
the small pores may have been closed off, which would lead to an increase in the average
pore size and a decrease in the porosity. This also indicates that the particles making up
the Gelsil glasses are likely to be porous in order for those samples to have such large

porosities.

 

146
The normalized diffusion coefficient in the AS-40 glasses decreases with porosity

and pore radius, but the range of the data is too small to determine if it behaves with the
same functionality as the Gelsil data. The porosity dependence of the diffusion
coefficient in the HS-4O samples is similar to the AS-40 data, but again covers too small a
range and has too much scatter to quantitatively compare it with the Gelsil. The pore size
dependence, however, appears to follow the shape of the Gelsil data quite closely. If the
data are fit to a curve of the form of Equation 5 .7:
(5.7) D/D0 = 0.06*(rp - RC)n
where the prefactor is arbitrarily set to 0.06, because that was the value found with the fits
to the Gelsil and Mitzithras data (Chapter 5), the result is n = 0.48 t 0.02 and Rc = 33 i 5
A. This larger value of Rc might be due to the apparently wider pore size distributions of
the Ludox glasses. As discussed previously, the percolation threshold is approached
because pores smaller than the diffuser’s size are effectively removed from the network.
For a given diffuser and average pore radius, a network with a wide distribution will have
more pores excluded than a network with a narrow distribution. Therefore, a network
with a wider distribution will be “closer” to the percolation threshold and the “critical”
average pore radius (RC) will then be larger. However, this result from the Ludox data is
based on too few data points to be considered anything but qualitative.
IV. Conclusions

These comparisons between the four different types of glasses show that while the
normalized diffusion coefficient depends upon both the porosity and the pore size, neither

of these parameters alone can be used to predict the diffusive behavior of a fluid in a pore

147
network. These glasses seem to group themselves in the following manner. Vycor , with

its small porosity, small pore size, and narrow pore size distribution must have a pore
structure unlike the others. The Ludox glasses, with small porosities, large pore sizes,
and relatively large pore size distributions may be good candidates for comparison to
“Swiss-cheese”-like models because they are made from presumably solid colloidal
particles. The sol-gel Gelsil glasses have large porosities, easily varied pore sizes, but
with relatively narrow pore size distributions which make them distinctly different from
the Ludox glasses, even though both are made from colloidal silica particles. The large
porosities of the Gelsil indicate that the particles in those glasses are actually porous,
which would also help explain why the pore distribution in so narrow.

The results in these different glasses make it clear that the details of the pore
structure affect the diffusive behavior of fluids in the pores. Because the pore size
distribution appears to be responsible for the percolation behavior in these systems, it
seems likely that a detailed knowledge of the pore size distribution of a glass, along with
its porosity and average pore size, is necessary to describe the diffusion of a fluid whose

molecular size is close to the size of the pores.

148

References

 

1 Y. Guo, K.H. Langley, and FE. Karasz, Phys. Rev. B 50, 3400 (1994).
2 W.D. Dozier, J.M. Drake, and J. Klafter, Phys. Rev. Lett. 56, 197 (1986).
3 Y. Guo, K.H. Langley, and FE. Karasz, ibid.

4 M. Terzima, K. Okamoto, and N. Hirota, J. Phys. Chem. 97,5188 (1993).

Chapter 7
SUMMARY AND FUTURE DIRECTIONS
In this work, fluid diffusion within porous silica has been studied to determine
how parameters such as the porosity and average pore size affect the diffusion. We have
used a holographic optical technique to measure diffusion on length scales much larger
that the size of the pores in order to examine the “normal” diffusion through the entire
network rather than the anomalous diffusion found on the length scale of the pore size.

In one particular set of porous glasses, we have found that the normalized

diffusion coefficient varies with the porosity of the glass as: D2 acct—(1)6)” and with

O

. D .
the average pore radius as D- °‘ (RP — R.)° 49. As expected, these results do not agree

0
with hydrodynamic models of media with uniform nonintersecting pores, but we have
shown that these results are consistent with the prediction of continuum models of
percolation for diffusion on the percolating cluster of a network. This is the first time this
power-law behavior has been found in an experimental study of fluid diffusion. These
glasses appear not to behave in accordance with the “Swiss-cheese” model, where the
distribution of conductances for the smallest pores contains a singularity at zero
conductance, although our data cannot exclude this model completely. The difference
between these sol-gel glasses and the “Swiss-cheese” model can be seen by their
relatively large porosities, which indicate that the particles making up the network are

porous, unlike the solid particles that are assumed in the model.

149

 

150
Comparisons with porous silicas made through different processes verifies the

importance of the pore size distribution on the fluid diffusion. These systems approach
the percolation threshold because the diffusing molecule cannot pass through pores that
are smaller than it is. By comparing glasses with relatively narrow pore distributions to
glasses with wider pore distributions we find that, as expected, the latter are closer to the
percolation threshold and therefore have smaller diffusion coefficients than glasses with
narrower distributions--leading to a larger value for RC.

The importance of the width of the pore size distribution on the transport
properties in porous materials makes this a ripe area for further investigation. The
creation of materials with controlled pore size distributions should make it possible to
probe the crossover from “Swiss-cheese” behavior to that found in this work. Such a
study may also provide insight into whether knowledge of the width of the pore
distribution, along with the porosity and the average pore size, can provide a prediction of
the diffusive behavior in pore networks. Also, a study of the conductivity and
permeability of the Gelsil glasses should be carried out in order to compare those results
with the diffusion measurements in this study.

An additional line of research could systematically vary the size and shape of the
probe molecule to determine how that parameter affects the percolation thresholds (Dc and
RC. This may be somewhat difficult to achieve, given the restrictions on the properties of
a probe molecule in these experiments. A related study could use molecules of different

size to modify the surface structure of the pores. This can provide a way to vary the pore

W73.“ .7 ~—

 

151
size and porosity in a controlled fashion, although it will also alter the connectivity of the

pore network.

Finally, the color change of Methyl Red could be used to perform a very simple
measurement of the diffusion of the Na+ ion through a porous material. As seen in this
study, Methyl Red turns red in the pores of an untreated silica, so if a sample were filled
with Methyl Red in water, and then a known quantity of Na” were added to the
surrounding fluid, the boundary of the color change could be monitored to measure the
diffusion of the ions into the pores. In fact, with this dye and two immiscible fluids, it
would be possible to monitor the phenomenon of “viscous fingering” as one fluid

containing the dye displaces the other in the pore network.l

 

1 52
References

 

l U. Oxaal et al, Nature 329, 32 (1987).

 

Appendices

 

Appendix A

 

Appendix A
ABSORPTION MEASUREMENTS

The absorbance measurements made in this study were carried out with an Acton
Research Corporation SpectraPro 300i 300mm focal length scanning monochromator
with an attached SC-447 sample chamber. Figure Al is a schematic drawing of the
sample chamber showing how it is set up for a transient absorption measurement. An
absorbance spectra is made by illuminating the sample (in a 0.2 or 0.5 mm path length
cell) with light from a 30W tungsten halogen light. The transmitted light is imaged onto
the entrance slit of the monochromator which is scanned over the wavelength range of
interest. The light is detected by a photomultiplier tube (PMT) attached to the exit slit of
the monochromator, and the data is collected on a computer which is interfaced to both
the detector and the monochromator. The spectral output of the lamp, the response of the
monochromator, and the reflection off the sample cell faces are corrected for by taking a
spectra of the same sample cell filled with only glycerol. The absorbance (A) is
calculated from Lambert-Beer’s Law: A0») = log( [JD/MO») ) where 100‘.) is the incident
light intensity at wavelength A and 1,10») is the transmitted light intensity at 2». Here, 100»)
is the intensity through the glycerol-filled cell.

For the transient absorption measurements, an excitation source is needed to
excite the dye in the cell. This is accomplished with the 488nm line of an Ar+ laser. In
addition to passing through a shutter to control the length of the excitation pulse, the
beam is expanded with a lens to assure that it illuminates all of the sample that is probed

by the white light plus the surrounding region. This is necessary to prevent molecules not

153

 

154

    
  

excitation laser

 

monochromator

 

 

 

 

 

PMT ::-=;:':éiiif.ii if.L.;.,:.;',:.jjQ,j§:5-.§.». -- 30W lamp

\ / sample {11
exit and entrance slits

Figure A. 1: Setup for transient absorption measurements. The dark shaded line
is the excitation laser, the light shaded line is the white light from the
lamp, and the PMT is the photomultiplier tube attached to the exit slit of
the monochromator.

 

 

 

 

 

 

 

 

 

 

 

 

excited by the laser from entering the probed region and affecting the lifetime
measurement. In practice, the beam is expanded to cover the entire sample. For these
measurements, the monochromator remains set at one wavelength and the intensity of the
transmitted light is monitored over time. After the excitation pulse ends, the time

dependence of the transmitted intensity can be used to determine the dye lifetime.

Appendix B

Appendix B
DATA ACQUISITION AND ANALYSIS PROGRAMS

The program used to control the shutter controller, the Stanford Research Systems
preamplifier, and the digital oscilloscope is listed on the following pages. It was written
in C++ in Borland C++, v. 4.5.

The format of the data files is as follows (shown on the following page). An
ASCII header containing the equipment settings occupies the first 24 lines. The next five
lines contain the values needed to convert the raw data (in integer form) to the real
number values in the form of x(t), y(V). The rest of the values are integers corresponding
to the voltage level of the oscilloscope at each point. The header itself describes the
conversion in detail. One irregularity is the value of the “# of averages” field. Because
the LeCroy 9400A oscilloscope does not output this value, the value entered is the
maximum number of averages allowed during that data acquisition.

If the data is analyzed with Microcal Origin, the following script file can be used

to convert the data file into an Origin worksheet:

offsetall=%H_a[l]; {*These 5 lines copy the first five*}
prefix=%H_a[2]; {*values used to convert the remaining*}
tpoint=%H_a[3]; {*integer data to useful numbers.*}

delay=%H_a[4];
numpoints=%H_a[5];

mark-d %H_a -b 1 -e 5; {*deletes the just copied 5 values*}

%H_bz prefix*( (%H_a-32768)/8l92 -offsetall); {*creates the voltage*}
{*data in column B*}

%H_a=data(0,numpoints-1,1); {*Creates column A, then converts*}

%H_a=(%H_a*tpoint)-delay; {*it into the correct time values.*}

155

 

1 56
Data file header:

Data taken with LeCroy 9400A, and SR560 preamp.

c:\users\lowell\data\gelsil\gsn2 1607.txt
Data taken on: November 21, 1996
Number of points saved: 2500
scopetimebase: .05 s/DIV
# of averages: 200
trigger level: 0.08 V
scope delay: 10 % fullscale
scopevertscale: .9 V/DIV
scopeoffset: -2.8 V
trigchannel: EX
preampgain: 20000
preampfilter mode: 2
High pass freq: 300 Hz
low pass freq: 30000 Hz
To convert data to volts:

y(V) = prefix*{ (data[i]-32768)/8l92 -offsetall}
To convert x=0,1,2,3... to time scale:

x(t) = (x*tpoint)-delay
(lst data value:) y scale offset (offsetall):
(2nd data value:)y scale multiplier (prefix):
(3rd data value:)tirne per point (tpoint):
(4th data value:)trigger time delay (delay):
(5th data value:)number of points (numpoints):
-2.8
0.900901
0.0002
0.05
2500
4198
3864
4134
4108
4326
3992
4210
4070

r___

 

157

//////////////////////////////////////l///////////////////////////////////////
//

// LeCroy94.cpp

// For the control of LeCroy 9400A oscilloscope, SRS 560 Low Noise
// preamplifier and Uniblitz T132 shutter driver.
//////I////////////////////////////////////////////////////////////////////////

// 9/6/97 Added lines to correctly import scope delay time when the

// ScopeDelayTime value is set <0.

// 12/12/96 Removed Dual grid display. Fix (again) the saving routine
// so that the origin script in lec94tmp.otw will work.

// lO/21/96 "Totally" working: save only integer values of data-~must
// must convert externally.

// 10/19/96 Created from LeCroy2.cpp

// Change to control LeCroy 9400A scope

// 9/19/96 Modified: don't save last point (buff-l)

// label all saved parameters

// only save y, not x,y pair

// 9/7/96 Modified: change x[],y[], to x,y in savedata()

// change "abort" message box in getdata()

// change max memory to 2500000, but only can seem
// to get 25000 without a GPF

// 8/26/96 Modified from scope02, Now controls 9310AM scope.

// 7/13/96 Modified--got the SR560 command (and sr560.h) working.
// 7/11/96 Modified from scope2. Add functions for HP54600b and

// SR560 Low Noise preamp. Remove SR850 functions.
/// 4/22/96: add dialog box for changing filename before saving

// Needs no .def, .rc, or any special library (besides gpib)
// Modified:4/20/96:reliable preamble extraction

// add .ini support

// Modified: 4/18/96—4/19/96: Add SR850 controls.

//

//////////l/////////////////////////////////////////////////////

// Modified from scopel.cpp, Now use Edit boxes.

// Control of HP54600b Oscope

// Opening and closing the shutter using buttons in main window.

// writtenz4/ 1 1/96

//////I/////////////////////////////////////////////////////////

”includes for windows stuff

#include <owl\applicat.h>

#include <owl/framewin.h>

#include <owl\edit.h>

#include <owl\inputdia.h> //For input dialog boxes

// Also need ...bc45\include\owl\inputdia.rh ?
// ...bc45\include\owl\inputdia.rc ?

158
// IDD_INPUTDIALOG ?
// added to the scopel.rc in the project
// 4/18/95-removed all input dialog boxes in favor of Edit boxes.

#include <owl/button.h>

#include <owl/static.h>

#include <string.h>

#include <stdio.h> //need for sprintf function ?
#include <iostream.h>

#include <fstream.h>

”#include <math.h>

//#include <ctype.h>

#include "c:\bc45\include\classlib\date.h"

#include "shutter.h" //include function for sending commands to shutter

#include <windecl.h> //include GPIB commands

#include "SR560.h" //include function for sending COM2 commands to
// SR560 Low Noise preamp

”definitions for whole program
#define IDB_OPEN 64
#define IDB_CLOSE 65
#define IDB_TRIGGER 66
#define IDB_RESET 67

#define IDB_SCOPERUN 101
#define IDB_SCOPECLEAR 102
#define IDB_SCOPESTOP 103
#define IDB_SCOPEDELAY 104
#define IDB_SCOPEGETDATA 105
#define IDB_SCOPERESET 106
#define IDB_SCOPEUPDATE 107
#define IDB_SR56OUPDATE 108

#define IDE_FILEN AME 201

#define IDE_SCOPETIMEBASE 202
#define IDE_SCOPEAVER 203
#define IDE_SCOPETRIG 204

#define IDE_SCOPEDELAYTIME 205
#define IDE_SCOPEVERTSCALE 206
#define IDE_ERRORBOX 207

#define IDE_SCOPEOFFSET 208
#define IDE_SCOPETRIGCHAN 209
#define IDE_SCOPENUMPOINTS 210

#define IDE_SR56OGAIN 301

' Ann-“Q?
‘I

 

159

#define IDE_SR56OFILTERMODE 302
#define IDE_SR56OHIFREQFILTER 303
#define IDE_SR56OLOFREQFILTER 304

int shutter(const int&);
void sr560 (const char*);
int lec9400a=0;

class TShutWin : public TWindow
{
public:
TShutWin(TWindow* parent=0);
virtual BOOL CanClose();

int savedata(int*,long int,unsigned int*,char* );

void send(char *cmd);

protected:
void CmOpen();
void CmClose();
void CmTriggerO;
void CmReset();
void CmScopeGetDataO;
void CmScopeReset();
void CmScopeStop();
void CmScopeClear();
void CmScopeRun();
void CmScopeUpdateO;
void Cer560Update();

private:
TEdit *filename;
TEdit *ScopeTimeBase;
TEdit *ScopeAver;
TEdit *ScopeTrig;
TEdit *ScopeDelayTime;
TEdit *ScopeVertScale;
TEdit *ErrorBox;
TEdit *ScopeOffset;
TEdit *ScopeTrigChan;
TEdit *ScopeNumPoints;
TEdit *Sr560Gain;
TEdit *Sr560FilterMode;
TEdit *Sr560HiFrequlter;

 

1 60
TEdit *Sr560LoFrequlter;

char* get_info(TEdit* edit);

DECLARE_RESPONSE_TABLE(TShutWin);

DEFINE_RESPONSE_TABLE 1 (TS hutWin,TWindow)
EV_BN_CLIC KED(IDB_OPEN,CmOpen),
EV_BN_CLICKED(IDB_CLOSE,CmClose),
EV_BN_CLICKED(IDB_TRIGGER,CmTrigger),
EV_BN_CLICKED(IDB_RESET,CmReset),
EV_BN_CLICKED(IDB_SCOPEGETDATA,CmScopeGetData),
EV_BN_CLICKED(IDB_SCOPERESET,CmScopeReset),
EV_BN_CLICKED(IDB_SCOPESTOP,CmScopeStop),
EV_BN_CLICKED(IDB_SCOPECLEAR,CmScopeClear),
EV_BN_CLICKED(IDB_SCOPERUN,CmScopeRun),
EV_BN_CLICKED(IDB_SCOPEUPDATE,CmScopeUpdate),
EV_BN_CLICKED(IDB_SR560UPDATE,Cer560Update),

END_RESPONSE_TABLE;

TShutWin: :TShutWin(TWindow* parent)
{//Constructor
Init(parent,0,0);

”Uniblitz text and buttons

int x=600,y=650, space=90, w=80,hfl0;
new TStatic(this,-1,"Uniblitz T132 Shutter Driver/Timer Controller",
x,y-20,400,20);
new TButton(this,IDB_OPEN,"Open",x,y,w,h);
new TButton(this,IDB_CLOSE,"Close",x+space,y,w,h);
new TButton(this,IDB_TRIGGER,"Trigger",x+2*space,y,w,h);
new TButton(this,IDB_RESET,"Reset",x+3*space,y,w,h);

//1ec9400a scope text, buttons and edit boxes

new TStatic(this,-1,"LeCroy 9400A Oscilloscope",140,20,400,20);

 

161

x=70,y=70, space=50, w=100,h=40;

new TButton(this,IDB_SCOPERESET,"Reset",x,y,w,h);

new TButton(this,IDB_SCOPESTOP,"Stop",x,y+1*space,w,h);

new TButton(this,IDB_SCOPECLEAR,"Clear",x,y+2*space,w,h);

new TButton(this,IDB_SCOPERUN,"Run",x,y+3*space,w,h);

new TButton(this,IDB_SCOPEGETDATA,"Get Data",20,305,120, 120);

x=220,y=65,w=200,h=20,space=50;

new TStatic(this,-1,"Filename for data:",x,y,w,h);
new TStatic(this,-l,"Time Base

(s/DIV):(50,100,200,500ns... 100s)",x,y+space,w+300,h); i
new TStatic(this,- 1 ,"#of Averages (10,20,50,.. 1M): ",x,y+2*space,w,h);
new TStatic(this,- 1,"Trigger level (V): ",x,y+3*space,w,h); .
new TStatic(this,-1,"Delay time (% >0, 3 <0):",x,y+4*space,w,h); !
new TStatic(this,-l,"Channel 1 (V/DIV):",x,y+5*space,w,h); i...
new TStatic(this,-1,"Vertical Offset (V): ",x,y+6*space,w,h);
new TStatic(this,- l ,"Trigger channel (C 1 ,C2,EX,LINE):",x,y+7*space,w+300,h);
new TStatic(this,-l,"# of points:

(50,100,250,500...25000)",x,y+8*space,w+300,h);

y+=20,h=30;
if (NULL != (filename = new TEdit(this,IDE_FILENAME,"",
x,y,w+200,h)));
if (NULL 1: (ScopeTimeBase = new
TEdit(this,IDE_SCOPETIMEBASE,"",x,y+space,w,h)));
if (NULL 1: (ScopeAver = new TEdit(this,IDE_SCOPEAVER,"",
x,y+2*space,w,h)));
if (NULL != (ScopeTrig = new TEdit(this,IDE_SCOPETRIG,"",
x,y+3*space,w,h)));
if (NULL 1: (ScopeDelayTime = new
TEdit(this,IDE_SCOPEDELAYI‘IME,"",x,y+4*space,w,h)));
if (NULL 1: (ScopeVertScale = new
TEdit(this,IDE_SCOPEVERTSCALE,"",x,y+5*space,w,h)));
if (NULL 1: (ScopeOffset = new TEdit(this,IDE_SCOPEOFFSET,"",
x,y+6*space,w,h)));
if (NULL 1: (ScopeTrigChan = new
TEdit(this,IDE_SCOPETRIGCHAN,"",x,y+7*space,w,h)));
if (NULL 1: (ScopeNumPoints = new
TEdit(this,IDE_SCOPENUMPOINTS,"",x,y+8*space,w,h)));

 

new TButton(this,IDB_SCOPEUPDATE,"Update
Scope",x,y+9*space, 150,40);

162
//SR560 text and edit boxes

new TStatic(this,—1,"SR560 Low-Noise Preamp",690,20,200,20);

x=690,y=65,w=200,h=20,space=50;
new TStatic(this,— 1 ,"Gain (1,2,5,10,..,50000):",x,y,w,h);
new TStatic(this,- 1,"Filter mode (Ozbypass, 1 :lowpass(6dB),",x,y+space,w+200,h);
new TStatic(this,-l,"2:low(12dB),3:hi(6dB),4:hi(12dB),5:band )",
x,y+20+space,w+200,h);
new TStatic(this,- 1 ,"High pass
freq.(0.03,0.l,0.3,..,10000):",x,y+20+2*space,w+200,h);

new TStatic(this,-l,"Low Pass '
req.(0.03,0.1,0.3 ,...,1000000):",x,y+20+3 *space,w+200,h); I”

y+=20,h=30; 1

if (NULL != (Sr56OGain = new TEdit(this,IDE_SR560GAlN,"",x,y,w,h))); r

if (NULL 1: (Sr560FilterMode = new TEdit(this,IDE_SR560FILTERMODE,"", "e.
x,y+20+space,w,h)));

if (NULL != (Sr560HiFrequlter = new
TEdit(this,H)E_SR560HIFREQFILTER,"",x,y+20+2*space,w,h)));

if (NULL 1: (Sr560LoFrequlter = new
TEdit(this,H)E_SR560LOFREQFILTER,"",x,y+20+3*space,w,h)));

new TButton(this,IDB__SR560UPDATE,"Update Sr560",x,y+5*space, 150,40);

”Message text box

new TStatic(this,-1,"Messages:",50,670,200,30);

if (NULL 1: (ErrorBox = new
TEdit(this,IDE_ERRORBOX,"",50,700,600,30))); .—

lec9400a=ibdev(0,4,0,T10$, 1 ,0);

 

//Input .ini parameters
ifstream inFile;
inFile.open( "c :\\users\\lowell\\programs\\lecroy94.ini " ,ios: :in);

if (linFile)
{ ErrorBox->SetCaption("Error opening lecroy94.ini file");
}

char tempword[40]=" ";

inFile>>tempword;

163

filename->SetCaption(tempword);

inFile>>tempword;

ScopeTimeBase->SetCaption(tempword);

inFile>>tempword;

ScopeAver->SetCaption(tempword);

inFile>>tempword;

Sc0peTrig->SetCaption(tempword);

inFile>>tempword;

ScopeDelayTime->SetCaption(tempword);

inFile>>tempword;

ScopeVertScale->SetCaption(tempword);

inFile>>tempword; '
ScopeOffset->SetCaption(tempword);

inFile>>tempword; F1
ScopeTrigChan->SetCaption(tempword);
inFile>>tempword; I
Sr56OGain->SetCaption(tempword); "‘1
inFile>>tempword;

Sr560FilterMode->SetCaption(tempword);

inFile>>tempword;

Sr560HiFrequlter->SetCaption(tempword);

inFile>>tempword;

Sr560LoFrequlter->SetCaption(tempword);

inFile>>tempword;

ScopeNumPoints->SetCaption(tempword);

inFile.close();

MessageBox("Must update instrument parameters before starting!",
"Warning!");

}

char*
TShutWin: :get_info(TEdit* edit)
{ char *str,*str2;
int size;
if (edit)
{ str = new char[size=edit~>GetWindowTextLength()+l];
if (str)
{ edit->GetWindowText(str,size);
str2=str;

 

}

delete str;

164
return str2;

}

void
TShutWin::CmOpen()
{ shutter(IDB_OPEN);

}

void
TShutWin::CmClose()
{ shutter(IDB_CLOSE);

}

void
TShutWin::CmTrigger()
{ shutter(IDB_TRIGGER);

}

void
TShutWin::CmReset()
{ shutter(IDB_RESET);

}

void
TS hutWin: :Cer560Update()
{ // Must use "\n" before and after every command!!!
sr560("\nLALL\n*RST\nCPLGl\nDYNRO\nINVTO\nSRCEO\nUCALO\n");

char temp[20],updateinfo[200];

strcpy(temp," ");
strcpy(updateinfo," ");

strcpy(temp,get_info(Sr560Gain));

switch (atoi(temp))

{
case 1:strcpy(temp,"\nGAIN 0\n");break;
case 2:strcpy(temp,"\nGAIN 1\n");break;
case 5zstrcpy(temp,"\nGAIl\I 2\n");break;
case 10:strcpy(temp,"\nGAIN 3\n");break;
case 20:strcpy(temp,"\nGAIl\I 4\n");break;
case 50:strcpy(temp,"\nGAIN 5\n");break;
case 100:strcpy(temp,"\nGAII\l 6\n");break;
case 200:strcpy(temp,"\nGAIN 7\n");break;
case 500:strcpy(temp,"\nGAIN 8\n");break;
case 1000:strcpy(temp,"\nGAIN 9\n");break;

165

case 2000:strcpy(temp,"\nGAm 10\n");break;
case 5000:strcpy(temp,"\nGAIl\I 11\n");break;
case 10000:strcpy(temp,"\nGAIN 12\n");break;
case 20000:strcpy(temp,"\nGAIN 13\n");break;
case 50000:strcpy(temp,"\nGAIN l4\n");break;
default:
ErrorBox->SetCaption("Gain must be 1,2,5,10,...,50000!");
return;
} //end switch
strcat(updateinfo,temp);

strcpy(temp,get_info(Sr560FilterMode)); 1
strcat(updateinfo,"\nFLTM ");

strcat(updateinfo,temp); f ]

strcat(updateinfo,"\n"); .f. Y.
1'"

strcpy(temp,get_info(SrS60HiFrequlter)); lie.

double hi=atof(temp);

if(hi==(double)0.03) { strcpy(temp,"\nHFRQ 0\n"); }

else if (hi==(double)0. l){strcpy(temp,"\nHFRQ 1\n");}

else if (hi==(double)0.3){strcpy(temp,"\nHFRQ 2\n");}

else if (hi==(double) 1) { strcpy(temp,"\nHFRQ 3\n");}

else if (hi==(double)3) { strcpy(temp,"\nHFRQ 4\n");}

else if (hi==(double) 10) { strcpy(temp,"\nHFRQ 5\n"); }

else if (hi==(double)30) { strcpy(temp,"\nHFRQ 6\n");}

else if (hi==(double) 100) { strcpy(temp,"\nHFRQ 7\n"); }

else if (hi==(double)300){ strcpy(temp,"\nHFRQ 8\n");}

else if ( '==(double)1000){ strcpy(temp,"\nHFRQ 9\n");}

else if (hi==(double)3000){strcpy(temp,"\nHFRQ 10\n");}

else if (hi==(double)10000){strcpy(temp,"\nHFRQ 11\n");}

else { ErrorBox->SetCaption("High freq. filter must be: 0.03,0.1,0.3,...,10000");
retum;}

strcat(updateinfo,temp);

strcpy(temp,get_info(Sr560LoFrequlter));

double lo=atof(temp);

if(lo==(double)0.03) { strcpy(temp,"\nLFRQ 0 "); }

else if (lo==(double)0. 1){ strcpy(temp,"\nLFRQ 1\n"); }
else if (lo==(double)0.3) { strcpy(temp,"\nLFRQ 2\n");}
else if (lo==(double)1){ strcpy(temp,"\nLFRQ 3\n"); }
else if (lo==(double)3) { strcpy(temp,"\nLFRQ 4\n");}
else if (lo==(double) 10) { strcpy(temp,"\nLFRQ 5\n"); }
else if (lo==(double)30) { strcpy(temp,"\nLFRQ 6\n");}
else if (lo=(double)100){ strcpy(temp,"\nLFRQ 7\n");}
else if (lo==(double)300) { strcpy(temp,"\nLFRQ 8\n");}

 

166

else if (lo==(double)1000){strcpy(temp,"\nLFRQ 9\n"); }

else if (lo==(double)3000) { strcpy(temp,"\nLFRQ 10\n");}

else if (lo==(double) 10000) { strcpy(temp,"\nLFRQ 1 1\n");}

else if (lo==(double)30000) { strcpy(temp,"\nLFRQ 12\n"); }

else if (lo==(double)100000.0){ strcpy(temp,"\nLFRQ 13\n"); }

else if (lo==(double)300000.0) { strcpy(temp,"\nLFRQ l4\n");}

else if (lo==(double)1000000.0){strcpy(temp,"\nLFRQ 15\n"); }

else { ErrorBox->SetCaption("Low freq. filter must be: 0.03,0.1,0.3,...,1000000");
retum;}

strcat(updateinfo,temp);

sr560(updateinfo); l

} //end Sr560Update f

void
TShutWin::CmScopeRun()
{ send("TRIG_MODE NORM\n");

}

void
TShutWin::CmScopeClear()
{ send("CLEAR_SWEEPS\n");

}

void
TShutWin::CmScopeStop()
{ send("STOP\n");

}

void
TShutWin::CmScopeReset()
{ send("*RST\n");

}

 

void

TShutWin::CmScopeUpdateO

{ ErrorBox->SetCaption(" ");

// ibclr(lec9400a);

// send("*rst;\n");

// send("*rst;INE 256;*SRE l;\n");
send("TRACE_CHANNEL_1 ON;TRACE_CHANNEL_2 OFF;\n");
send("TRACE_FUNCTION_E ON;\n");
send("SELECT FUNCTION_E;VERT_POSITION 0;\N");

//

//

//

167

//removed VERT_POSITION -9 when no longer dual grid
send( "TRIG_COUPLIN G DC;TRIG_MODE NORM;TRIG_SLOPE NEG;\n");
send("CHANNEL_1_COUPLING D 1M;\n");
send("DUAL__GRID ON;\n");
send("AVERAGE_RESET;\n");

ErrorBox->SetCaption("here");

char temp[20]= "\0",temp2[20]="\0",updateinfo[400]="\0";
strcpy(temp," ");

strcpy(updateinfo, " ");

strcpy(temp,get_info(ScopeTrigChan));
strcpy(temp2,get_info(ScopeTrig));
strcat(updateinfo,"TRIG_SOURCE ");
strcat(updateinfo, temp);
strcat(updateinfo, " ;;")

strcat(updateinfo, "TRIG_LEVEL ");
strcat(updateinfo, temp2);
strcat(updateinfo, " ;;")

strcpy(temp,get_info(ScopeTimeBase));
strcat(updateinfo,"TIME/DIV ");
strcat(updateinfo, temp),
strcat(updateinfo, " ,";)

strcpy(temp,get_info(ScopeAver));
strcpy(temp2,get_info(ScopeNumPoints));
strcat(updateinfo,"REDEFIl\lE, AVERAGE, SUMMED, 25000, ");
strcat(updateinfo,temp2); //MAXPTS
strcat(updateinfo,"CHANNEL_1, ");

strcat(updateinfo,temp); //MAXSWEEPS

strcat(updateinfo,";");

strcpy(temp,get_info(ScopeDelayTime));
strcat(updateinfo,"TRIG_DELAY ");
strcat(updateinfo, temp),
strcat(updateinfo, " ;;")

strcpy(temp,get_info(ScopeVertScale));
strcat(updateinfo,"CHANNEL_1_VOLT/DIV ");
strcat(updateinfo, temp);

strcat(updateinfo, " ;;")

strcpy(temp,get_info(ScopeOffset));

168

strcat(updateinfo,"CHANNEL_l_OFFSET ");
strcat(updateinfo,temp);

strcat(updateinfo,";\n");
send(updateinfo);

void
TShutWin::CmScopeGetData()
{ // This block of code is useful for seeing double variables

// char text[20];

// sprintf(text,"%g",preamble[5]);

// if(IDCANCEL==MessageBox(text,"like it?", 1))
// return;

// char text[30]; //general text variable

ErrorBox->SetCaption("BEFORE CRASH? ");
ibclr(lec9400a);

CmScopeUpdate();

Cer560Update();

if (IDNO==MessageBox("Do you want to start an acquisition?"," ",4))
{ ibclr(lec9400a);
return;

}

// Start shutter
CmTrigger();
////////////////

if(lDNOzzMessageBoxC'Do you want to retrieve the current data?\nNo will
Abort and end the acquisition.","Accept Data",4))
{ CmReset(); //Stop Shutter
// ibclr(lec9400a);
return;

// else {
// ibwait(lec9400a,2048);

// }

 

 

169

char pre[200];

strcpy(pre," ");

ErrorBox->SetCaption("here 2");
send("COMM_FORMAT,L,WORD,UNSIGNED_SHORT;\N");
send("READ FUNCTION_E.DESC;\N");

ibrd(lec9400a,pre, 154);

n n,

char * token: , ,
char *ptr=pre;

int preamble[25];
ptr=strtok(pre,token);
int x=0;

char *temp;

while (ptr)

{ temp=ptr;
preamble[x]=atoi(temp);
x++;
if (x>24) break;
strcpy(temp," ");
ptr=strtok(NULL,token);

}

// Added to correctly retrieve the scope delay time when it is

//

negative.
double signdelay;
signdelay = atof(get_info(ScopeDelayTime));
if(signdelay<0.0)
{ preamble [20]=abs(preamble[20]-255);
preamble[21]=abs(preamble[2l]-255);
preamble[22]=abs(preamble[22]-255);
preamble[23]=abs(preamble[23]-256);

}

// Stop Shutter

CmReset();

////////////////

//

Pull out the data

 

 

 

//
//

170

long int buffer = 25000;

unsigned int stf1,stf2;

stf 1:0;

stf2=0;

int unsigned *data = new unsigned int[buffer];

send("COMM_FORMAT,A,WORD;\N");
send("READ FUNCTION_E.DATA;\N");
//read out the first four (useless) bytes

ibrd(lec9400a,&stf1,l);
ibrd(lec9400a,&stf1,1);
ibrd(lec9400a,&stf1,1);
ibrd(lec9400a,&stf1, 1);
stfl =0;

for (int qq=0;qq<buffer;qq++)
{ ibrd(lec9400a,&sth , 1);
ibrd(lec9400a,&stf2, 1 );
data[qq]=stf1*256+stf2;
sprintf(text," %u" ,dataqu});
MessageBox(text,"data",1);

}

ErrorBox->SetCaption("Data retrieved");
char file1[60];

BOOL goodfile=FALSE;
while (lgoodfile)
{ strcpy(filel,get_info(filename));
if(IDYES=MessageBox(file1,"Save to this file?",4))
{ savedata(preamble,buffer,data,file 1 );
char text[50];
strcpy(text," ");
strcat(text,"Data saved to file: ");
ErrorBox->SetCaption(strcat(text,file 1));
goodfile=TRUE;

else if(IDYES==MessageBox("Use a different filename?",
"choosing No will abandon data",4))

{ char t[60];

strcpy(t," ");
TInputDialog *filechangeDlg;

filechangeDlg=new TInputDialog(this,"Change File",

 

 

 

171

"New Filename--give full path:",
t,sizeof(t));

if (filechangeDlg->Execute( ==IDOK)

{ filename->SetCaption(t);

}

else
{ break;

1
} //end while

delete [] data;

} //end ScopeGetData

class TWinShutApp : public TApplication
{
public:
TWinShutApp():TApplication()
{ nCmdShow=SW_SHOWMAXIMIZED;
}

void InitMainWindowO;
} ;

BOOL TShutWin::CmClose()
{ int result=MessageBox("Update .ini file?","Close lecroy94",3);
if(result==IDCANCEL)
{return FALSE;}
else if(result==IDYES)
{ ibonl(lec9400a,0);

ofstream outFile;
outFile.open( "c:\\users\\lowell\\programs\\lecroy94.ini" ,ios: :out);
if (loutFile)
{ ErrorBox->SetCaption("Error opening lecroy94.ini file");

}
outFile<<get_info(filename)<<"\n";
outFile<<get_info(ScopeTimeBase)<<"\n";
outFile<<get_info(ScopeAver)<<"\n";
outFile<<get_info(ScopeTrig)<<"\n";
outFile<<get_info(ScopeDelayTime)<<"\n";

172

outFile<<get_info(ScopeVertScale)<<"\n";
outFile<<get_info(ScopeOffset)<<"\n";
outFile<<get_info(ScopeTrigChan)<<"\n";
outFile<<get_info(Sr560Gain)<<"\n";
outFile<<get_info(Sr560FilterMode)<<"\n";
outFile<<get_info(Sr560HiFrequlter)<<"\n";
outFile<<get_info(Sr560LoFrequlter)<<"\n";
outFile<<get_info(ScopeNumPoints)<<"\n";
outFile.close();

}

else
{ ibonl(lec9400a,0);

}
return TRUE;

}

void
TWinShutApp: :InitMainWindow()
{

// Set the main window and its menu
SetMainWindow(new TFrameWindow(0,"LeCroy94",new TShutWin));
GetMainWindow()—>AssignMenu("start");

} ;

int

OwlMain(int/*argc */,char*/*argv*/ [D
{

};

return TWinShutApp().Run();

void
TShutWin::send(char *cmd)
{ ibwrt(lec9400a,cmd,strlen(cmd));

}

int TShutWin::savedata(int *header,long int buff, unsigned int *data, char* file )
{ //I think that this only passes a pointer--not something the size of

// the array.
// double x,y;

// Put in a MessageBox to ask if want to change the number of data points here?

Ah __.__ p
A m . . 4‘ I!
r 1“
a
o

 

 

 

//

173

long int npoints;

int step;

npoints = atoi(get_info(ScopeNumPoints));
step = 25000/npoints;

TDate today=TDate();
ofstream outFile;
outFile.open(file,ios::out);
if (loutFile)
{ ErrorBox->SetCaption("Error opening output file.");
return 1;
}
outFile<<"Data taken with LeCroy 9400A, and SR560 preamp."<<"\n";
outFile<<file<<"\n"<<"Data taken on: "<<today<<"\n"<<
"Number of points saved: "<<get_info(ScopeNumPoints)<<"\n";

save state of instruments
outFile<<"scopetimebase: "<<get_info(ScopeTimeBase)<<" s/DIV\n";
outFile<<"# of averages: "<<get_info(ScopeAver)<<"\n";
outFile<<"trigger level: "<<get_info(ScopeTrig)<<" V\n";
outFile<<"scope delay: "<<get_info(ScopeDelayTime)<<" % fullscale if >0,
seconds if <0\n";
outFile<<"scopevertscale: "<<get_info(ScopeVertScale)<<" V/DIV\n";
outFile<<"scopeoffset: "<<get_info(ScopeOffset)<<" V\n";
outFile<<"trigchannel: "<<get_info(ScopeTrigChan)<<"\n";
outFile<<"preampgain: "<<get_info(Sr560Gain)<<"\n";
outFile<<"preampfilter mode: "<<get_info(Sr560FilterMode)<<"\n";
outFile<<"High pass freq: "<<get_info(Sr560HiFrequlter)<<" Hz\n";
outFile<<"low pass freq: "<<get_info(Sr560LoFrequlter)<<" Hz\n";

double gain;

switch (header[2])

{
case 22: gain=.005;break;
case 23: gain=.010;break;
case 24: gain=.020;break;
case 25: gain=.050;break;
case 26:gain=.l;break;
case 27:gain=.2;break;
case 28:gain=.5;break;
case 29: gain=1.;break;
case 30:gain=2.;break;
case 31: gain=5.;break;

 

174

default:
ErrorBox->SetCaption("switching error gain");
return;
} //end switch

// so convert data by: if in WORD format (2-byte, unsigned int):

// offset = 256.0*preamble[6] + preamble[7]

// gain = gain from above

// vgain = preamble[3]

// V(t) = (gain*200/(vgain+80))*((data—32768)/8192 - (offset-200)/25)

double offsetall = (256.0*header[6] + header[7]-200.0)/25.0;
double prefix: (gain*200.0/((float)(header[3] +80.0)));

I"?

// Figure out time per point:
char text[30]; we.

float tpoint=0.0;

switch (header[12])

{
case 16:tpoint=0.00000001;break;
case 17:tpoint=0.00000002;break;
case 18:tpoint=0.00000004;break;
case 19:tpoint=0.00000008;break;
case 20:tpoint=0.0000002;break;
case 21:tpoint=0.0000004;break;
case 22:tpoint=0.0000008;break;
case 23:tpoint=0.000002;break;
case 24:tpoint=0.000004;break;
case 25:tpoint=0.000008;break;
case 26:tpoint=0.00002;break;
case 27:tpoint=0.00004;break;
case 28:tpoint=0.00008;break;
case 29:tpoint=0.0002;break;
case 30:tpoint=0.0004;break;
case 3 1 :tpoint=0.0008;break;
case 32:tpoint=0.002;break;
case 33:tpoint=0.004;break;
case 34:tpoint=0.008;break;
case 35:tpoint=0.02;break;
case 36:tpoint=0.04;break;

default:

 

 

ErrorBox->SetCaption("switching error tpoint");
return;

 

175
} //end switch

tpoint = tpoint*step;

float tbase=0.0;

switch (header[11])

{
case 8:tbase=0.00000005;break;
case 9:tbase=0.0000001;break;
case 10:tbase=0.0000002;break;
case ll:tbase=0.0000005;break;
case 12:tbase=0.000001 ;break;
case 13:tbase=0.000002;break;
case 14:tbase=0.000005;break;
case 15:tbase=0.00001;break;
case l6:tbase=0.00002;break;
case 17:tbase=0.00005;break;
case 18:tbase=0.0001;break;
case 19:tbase=0.0002;break;
case 20:tbase=0.0005;break;
case 21 :tbase=0.001;break;
case 22:tbase=0.002;break;
case 23:tbase=0.005;break;
case 24:tbase=0.01;break;
case 25:tbase=0.02;break;
case 26:tbase=0.05;break;
case 27:tbase=0. 1 ;break;
case 28:tbase=0.2;break;
case 29:tbase=0.5;break;
case 30:tbase=1.;break;
case 31:tbase=2.;break;
case 32:tbase=5.;break;
case 33:tbase=10.;break;
case 34:tbase=20.;break;
case 35 :tbase=50.;break;
case 36:tbase=100.;break;

default:
ErrorBox->SetCaption("switching error tbase");
// return;
} //end switch

// calculate the delay, first combine the 4 bytes, then convert to time
double signdelay;
signdelay = atof(get_info(ScopeDelayTime));

‘x'wd

 

176
int sign=1;
if(signdelay<0.0)
{ sign=-l;}
float delay;
delay = l.*header[23] +256.*header[22] + 65536.*header[21] +
16777216.*header[20];
delay = sign*delay*0.02*tbase;

// So, to calc the x values, x=(x*tpoint) -delay

outFile<<"To convert data to volts:\n y(V) = prefix*{ (data[i]-32768)/8192 -
offsetall }\n";
outFile<<"To convert x=0,1,2,3... to time scale:\n x(t) = (x*tpoint)-delay\n";

outFile<<"(lst data value): y scale offset (offsetall):\n";
outFile<<"(2nd data value): y scale multiplier (prefix):\n"; q
outFile<<"(3rd data value): time per point (tpoint):\n"; 1m ..
outFile<<"(4th data value): trigger time delay (delay):\n"; ‘
outFile<<"(5th data value): number of points (npoints):\n";

outFile<<offsetall<<"\n"<<prefix<<"\n"<<tpoint<<”\n"<<delay<<"\n"<<npoints<
("\n H;

for ( int i=0;i<(buff-1);i=i+step)
{ //x=(i*tpoint)-delay;
// y= prefix*( ((double)data[i]-32768.)/8192. - offsetall);
// outFile<<x<<","<<y<<"\n";
outFile<<data[i]<<"\n";

}

outFile.close();
return 0;

 

 

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