Effect of drying on properties of silica gel

]OURNAL OF

IlglltSWIII ELSEVIER

Journal of Non-Crystalline Solids 215 (1997) 155-168

Effect of drying on properties of silica gel George W. Scherer * Department of Cicil Engineering, Princeton Materials Inst.. Princeton Unicersit3', Eng. Quad. E-319. Princeton, NJ 08544, USA

Received 10 October 1996; revised 15 January 1997

Abstract Viscoelastic properties and permeability of a silica gel have been followed during the course of drying. The modulus and viscosity of the network rise by several orders of magnitude, and exhibit power-law dependence on the density of the gel. An aerogel was prepared from an undried gel, and its modulus was found by hydrostatic compression in a mercury porosimeter. Pore size distribution of the aerogel was determined by nitrogen sorption before and after compression. The bulk modulus of the aerogel was very close to that of the wet gel at low densities, but at higher densities the wet gels were more rigid that the compressed aerogels. This difference is attributed to aging of the wet gels in the pore liquid (water). The final shrinkage of the wet gels was greater than expected, based on the viscoelastic properties and the calculated capillary pressure, and the difference is attributed to accelerated viscoelastic relaxation of the network under high capillary stresses. The permeability decreases by almost four orders of magnitude as the gel contracts and pore size inferred from the permeability (rw) decreases in direct proportion to the pore volume. The pore size measured by nitrogen desorption (rBT) agrees closely with r w, as expected on theoretical grounds. For compliant aerogels, the nitrogen desorption process causes substantial compression of the get, because of the capillary pressure exerted by liquid nitrogen. In such cases, the true pore size and pore volume can be estimated, given knowledge of the elastic modulus of the network; the corrected value of rax agrees well with r w. At high densities r w was about 0.5 nm smaller than rBT, and this discrepancy is probably caused by neglect of the existence of a relatively immobile layer of water of that thickness on the surface of the network. © 1997 Elsevier Science B.V.

I. Introduction When liquid evaporates from the pores of a gel, concave l i q u i d / v a p o r menisci form at the exterior surface of the body, so that the liquid goes into tension and the solid network of the gel is compressed [1,2]. These stresses, which can exceed 100

* Tel.: + 1-609 258 5680; fax: +1-609 258 1563, e-mail: [email protected].

Mpa [3,4] cause a silica gel to decrease in volume by as much as a factor of 10 as it dries. However, the warping and cracking that often attend drying do not depend on the absolute value of the stress. If the stress were hydrostatic, then the body would contract isotopically and there would be no damage. It is the g r a d i e n t in capillary pressure within the pores that leads to differential contraction, and consequently to mechanical damage of the network [1,5]. The magnitude of the gradient depends on the permeability, D, of the network [1], and the permeability decreases

0022-3093/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PH S0022-3093(97)00079-3

156

G.W. Scherer/Journal of Non-Crystalline Solids 215 (1997) 155-168

rapidly as the body contracts [6], so the stresses are expected to increase as drying proceeds. To predict the maximum stresses, it is necessary to know how D depends on the volume of the gel, and that is the purpose of the present study. This is an extension of earlier work [6]. The measurements have been extended to the point where shrinkage of the gel is essentially complete. Moreover, we have used nitrogen desorption to determine the pore size distribution in xerogels and aerogels, so that the permeability can be related to the structure of the gel. We have also characterized the elastic and viscoelastic properties of the wet gels and the corresponding aerogels.

2. Experimental procedure The silica gels used in this study were of the type known as B2 [7], made from tetraethoxysilane (TEOS) in a two-step process: initial hydrolysis under acidic conditions with a substoichiometric amount of water, followed by gelation under basic conditions with a final water/alkoxide ratio of ~ 3.7. Details of the preparation and properties of this gel are available in Refs. [7-9]. The sol was cast into polystyrene pipette tubes (with inner diameter 7.77 mm) and left to gel at room temperature overnight; then the rods of gel were removed from the tubes and given two 24 h washings in a large excess of pure ethanol to remove partially hydrolysed alkoxide [8]. The rods were then given successive 24 h washings in ethanol/water solutions with volume ratios of 50/50, 25/75, and 5/95. The samples were stored in the 5 / 9 5 solution, rather than pure water, to avoid microbial growth during long aging. The new measurements reported here were done on samples from the same batch that was used previously [6], but which were stored in the 5 / 9 5 solution for 13 months longer. Based on the amount of alkoxide, and assuming a skeletal density for the gel of Ps = 2.0 g / c m 3, the solids content (or relative density, p) is estimated to be 0.065 at the time of gelation [9]. However, a significant fraction of the original silica (in the form of polysilicate oils) was removed in the first washing step, so the actual solids content was determined by measurement of the bulk and skeletal densities of

aerogels. An aerogel was made by exchanging a gel into acetone, then into liquid carbon dioxide, and supercritically drying at 35°C and 8.5 MPa pressure in a Polaron autoclave. The skeletal density of the aerogel was measured by helium pycnometry with a Micromeritics Accupync 1330. The bulk density was measured by mercury displacement in a home-made cell; the sample was heated to 150°C, then immediately transferred into the mercury to avoid adsorption of ambient moisture. The pore structure was examined by nitrogen sorption using a Micromeritics ASAP 2000. A mercury porosimeter (QuantaChrome Autoscan 33) was used to compress the aerogel to 227 MPa to determine the bulk modulus of the network. It has been established that the gels are so compliant that no mercury enters the pores, it merely compresses the structure [10]. Separate pieces of the aerogel were compressed to 1, 7, and 227 MPa, then removed from the porosimeter; the diameter of the cylindrical sample was measured with calipers and the density by mercury displacement. The pore structure was then examined by nitrogen sorption. For controlled drying, rods of gel were transferred onto a plastic mesh inside a glass tube with loose-fitting endcaps, and held at room temperature. When the partially dried samples had contracted to the desired diameter, they were transferred into a bath of pure water for three-point bending measurements; the bath and bending equipment were maintained at 2 8 _ I°C inside an oven. The apparatus and procedure are described in Ref. [11]. Briefly, the gel rod is supported at the ends on aluminum rollers, and a pushrod is dropped onto the gel to produce a sudden predetermined deflection; the load is measured as a function of time at constant deflection (typically < 1% strain). The original design used a pushrod with a radius of curvature of ~ 1 mm, but it has been established that significant indentation of the gel occurs at points of contact [12], so the end supports and the pushrod were both fabricated with radii of curvature of 6.35 mm. Each run lasted 20 h, during which time the baseline drift could be as much as 0.5 g; the initial load was chosen to be _> 20 g, of which > 15 g relaxed away, so the baseline drift amounts to < 3% of the measured change in load. The span/diameter ratio ranged from 8 to 14; previous work indicated that the measured modulus was rela-

G. W. Scherer / Journal of Non-Crystalline Solids 215 (1997) 155-168

tively insensitive to this ratio when it exceeded ~ 7 [13]. The sample diameter was measured with an optical microscope at the end of each run. Water (neutral pH) was chosen as the medium for the experiments, because it provides fast viscoelastic relaxation for silica gels [11]. In some cases where the partially dried samples had undergone substantial shrinkage, they cracked when re-immersed in liquid for the beam-bending measurement. Such samples could be safely rewet by suspending them over water that was heated to ~ 70°C; once beads of condensation appeared on the gel, it could be immersed in water, then cooled to room temperature and transferred into the bending apparatus. The stresses generated during rewetting are discussed elsewhere [14]. A few samples dried to the point that they cracked spontaneously, so they could not be used for property measurements. Several of these were allowed to dry completely, and the resulting xerogels were examined by nitrogen sorption. The stiffest samples could only be deflected by ~ 50 Ixm before the load cell reached its limit (50 g). To insure that the compliance of the load cell was not significant compared to the imposed deflection, a rigid bar (namely, an Allen wrench) with a diameter of 6.35 mm was put on the end supports in place of a sample and loaded. The displacement of the load cell was read from the display of the motor on the translation stage holding the cell; the displacement of the cell was carefully measured from the point of contact until the load cell read 50 g. Assuming that the metal bar did not bend under such small loads, the compliance of the cell was found to be 1 Ixm per 14.0 g load; this correction was applied in any case where the compression of the load cell exceeded 1% of the deflection of the sample. The load on the gel rod relaxes by two independent mechanisms: hydrodynamic relaxation caused by flow of liquid within the gel network, and viscoelastic relaxation of the network attributed to hydrolysis of siloxane bonds by the pore liquid [11]. As explained in Ref. [15], the kinetics of relaxation can be modeled as the product of the purely hydrodynamic effect and the purely viscoelastic effect:

W(t) - = , ~ ( t ) 0vE(t),

w(0)

(1)

157

where W(t) is the measured load at time t. The hydrodynamic relaxation function is [13] =

2(1 + v) 3 +

8(1-2

)

/3;2 exp( -/3~ ' / r b ) , n=l

(2) where v is Poisson's ratio for the network, /3. is the nth root of the Bessel function of the first kind of order 1, Jl(/3.) = 0. and the hydrodynamic relaxation time r b, is defined by l r b=2(1-v)

~L R2 OH - ( 1 - 2 v )

~lLR2 O--~'

(3)

where ~TL is the viscosity of the pore liquid, R is the radius of the gel rod, D is the permeability (defined by Darcy's law [1]), and H is the longitudinal modulus of the network, which is related to Young's modulus by H = (1 - v)E/[(1 + v)(l - 2v)]; G = E/[2(1 + v)] is the shear modulus. The viscoelastic relaxation function is represented by the stretched exponential, or Kohlrausch function [15,16],

I/IVE(t)

= exp[-( t/ryE)h],

(4)

where rVE is the modal viscoelastic relaxation time and b is a constant that is related to the breadth of the distribution of relaxation times. The average viscoelastic relaxation time (¥VE) is given by [16,17]

rvE=b-'rill ~blrvE"

(5)

where F is the gamma function. This quantity is related to the viscosity of the network (~7) by YvE = 3TI/E.

(6)

The load relaxation data were fit to Eqs. (1), (2) and (4) using a non-linear least-squares procedure described in Ref. [13]; the effect of indentation at points of contact was taken into account using Hertz

1 The factor of 2(1 -- v) was accidentally left out of the equation presented in Ref. [6]; worse, the same error occurred in the computer program used to calculate the permeability values in table 1 of that paper. Recalculated parameters for those experiments are presented in Section 3 of this paper.

G.W. Scherer/ Journal of Non-Crystalline Solids 215 (1997) 155-168

158

1

theory, as explained in Ref. [ 12]. The fitting parameters are u, r b, b, and TVE. From the initial load and deflection one can calculate the shear modulus, G, and with the fitted value of ~, one can then find E and H. The fitted value of ~'h can be used with Eq. (3) to find the permeability, D. Using the fitted values of b and ~'VE in Eqs. (5) and (6), one can calculate the viscosity of the network.

o-

~

_

-I---4__

I

I

I

)

0.6-

,.~ 0 . 4 0.2-

o

1

I

1 0°

0 "1

I

I

1 01

t

1 02

1 03

I

1 04

0s

t (s) 3. Results

Helium pycnometry measurements on four aerogel samples yielded a skeletal density of p~ = 2.01 + 0.03 g / c m 3. Fig. 1 shows the relationship between the bulk density (measured by mercury displacement) and the diameter (d) of the aerogel samples that were compressed to 0, 1, 7, and 227 MPa. Assuming that the gels contract isotopically when hydrostatically compressed, then the cube of the ratio of d to the diameter of the as-cast gel (d o = 7.77 mm) is related to the ratio of the volumes of the compressed and as-cast gels. Therefore, the extrapolation of the line in Fig. 1 to d i d o = 1 yields the initial bulk density (exclusive of liquid) of the original gel. In this way we find that the as-cast gel had a bulk density of Pb = 0.106 g / c m 3 and a solid fraction (or, relative density) of p---Pb/P~ = 0.053. Comparing this value to the solids fraction expected from the TEOS content of the sol (0.065), we find

1

[

I

I

I

0,8-

2" E u

J

0.6J 0.4-

J

s

J

J

~t3

J,'

f

o ~'

j, 0.2-

0

G

f

I

I

I

2

4

6

I 8

0

( d o / d )3 Fig. 1. Bulk density versus diameter of aerogels compressed to 0, 1, 7, and 227 MPa in a mercury porosimeter; d o = 7.77 m m = diameter of as-cast gel. Dashed line is least-squares fit to Pb = c(d o/d)3; fit yields c = 0.106 g / c m 3.

oA

0

.

8

-

~

1 01

1 02

0.64-'

0.4I' 0.2-

o

1 0"1

Data Fit --

--

" YVE

1 0°

I

I

I

1 03

I

1 04

0s

t (s) Fig. 2. Normalized load on (a) aged gel and (b) partially dried gel

as function of time following deflection: measured W~ Wo (0), fit to Eq. (1) ( ), viscoelastic relaxation function from fit to Eq. (4) (---); W0 is load initially after deflection. Bar near 100 s in (a) represents maximum error in normalized load.

that almost 20% of the silica was washed out in the form of polysilicate oils. Three gels were dried completely and their final diameters were 3.55, 3.57, and 3.62 ram; according to Fig. 1, the corresponding bulk densities are 1.11, 1.09, and 1.05 g / c m 3. The BET surface areas of the latter two xerogels, measured by nitrogen adsorption, were 660 and 733 m 2 / g , respectively; the final pore diameters (BJH desorption average) were both 3.2 nm. The surface areas of the aerogel sample, and the aerogels compressed to 1, 7, and 227 MPa were 728, 734, 725, and 562 m 2 / g , respectively. Typical relaxation curves are shown in Fig. 2. The first part of the load decay (extending to ~ 20 s in Fig. 2a) results from hydrodynamic relaxation, as liquid flows within the gel, and between the gel and the bath. The subsequent relaxation is a viscoelastic response of the silica network to the applied strain. The sample in Fig. 2a (d = 6.98 mm) had been aged,

G. W. Scherer / Journal of Non-Crystalline Solids 215 (1997) 155-168 Table 1 Fitting parameters

Pb/P. 0.0732 0.0799 0.0799 0.0839 0,0928 0.0968 0.110 0.118 0.118 0.119 0.150 0.199 0.232 0.266 0.348 0.420 0.457

,t /&

1000~

rb (S)

V

122 132 211 184 187 149 126 104 105 100 102 64.7 47.3 47.7 37.0 35.8 40.0

0,156 6.54 21.4 0,171 6,24 19.2 0,150 4,75 16.2 0.155 5,73 14.8 0.117 6.48 12.6 0.133 7.37 13.3 0.130 10.6 10.1 0.108 18.8 6.73 0.0920 23.7 5.38 0.182 15.9 7.53 0.144 27.7 3.80 0.113 105 1.35 0.135 181 0.946 0.228 214 0.647 0.211 764 0.199 0.226 1660 0.0822 0.271 2470 0.0440

H (MPa)

D (nm 2)

YVE (Ms)

"0( 1012 Pa s)

0.0940 0.596 0.127 0.0903 0,213 0.244 0.195 0.151 0.197 0.232 0.189 0.0142 0.244 0.225 3.48 1.39 1.87

0.193 1.15 0.190 0.163 0.446 0.576 0.663 0.915 1.53 1.13 1.66 0.483 14.1 13.9 786 667 1230

but not dried, whereas the one in Fig. 2b had been dried to the point that its diameter was d = 4.15 mm. The drier sample shows faster hydrodynamic relaxation and slower viscoelastic relaxation. The fitting parameters for all of the runs are shown in Table 1; also included are recalculated values for the samples from Ref. [6] (correcting for the missing factor of 2(1 - ~,) and taking account of Hertzian indentation). The permeability was calculated from Eq. (3) using the fitted parameters, and assuming that the

o •

o

1

G (MPa)) E ( MPa

z

100

'

/

o

|

"0

o

/

t/'&

10

Pb ( g / c m 6 )

Fig. 4. Symbols represent shear modulus (G) and Young's modulus (E) found from beam-bending experiments; curves are powerlaw fits applied for Pb > 0.18 g / c m 3.

viscosity of the liquid inside the gel was identical to that of bulk water at 28°C (r/L = 8.3 x 10 -4 Pa s) [18]. Fig. 3 shows how D varies with the relative density of the network. The vertical bar near Pb = 0.6 g / c m 3 represents the maximum error in D, based on the estimated uncertainty in ~'b ( ~ 10%), V ( ~ 30%), R ( ~ 1%), and G ( ~ 5%). The density dependences of the shear modulus (G) and Young's modulus (E) are shown in Fig. 4. The maximum error in the modulus, which results from uncertainties in the dimensions of the sample and in the measured load, is estimated to be ~ 5% (although the variation from sample to sample is as great as 20%); the error bar would be smaller than the symbols in Fig. 4. The viscosity of the network varies strongly with density, as indicated in Fig. 5.

~ l J l ~ L I J t = = l L i b l J l =

1

014~

'

0

'

I

0,2

'

o

v

o

I 01~:

Cubic '

J

J J

jo

1 013:

o

o

°

w.

-- --

/

o

0.1

A

0.01

7"

1

~

---C-K

/00

P"

1 01s: o

c v

0.1

A O"

1 016;

E

159

'

~

I

'

'

I

0.4 0.6 Pb ( g / c m 3 )

'

'

'

I

'

'

'

0.8

Fig. 3. Permeability versus bulk density of partially dried gels: measured (O); calculated from Carman-Kozeny equation using measured surface area (C-K); calculated from window radius ( r . ) of cubic cell model, where r . of initial gel chosen to fit initial permeability. Vertical bar near Pb = 0.6 g / c m 3 represents maximum error ( ~ 30%) in D.

1 011~

!

1 0 TM

/

'

o

/

,

i

0.2

,

,

,

I

0.4 Pb

'

'

'

I

0.6

'

'

'

I

7 `

'

0.8

(g/cm3)

Fig. 5. Viscosity of gel network, calculated from Eq. (6) using values of b and YVE from fit, versus bulk density of gel; dashed curve is power-law fit yielding exponent of 4.5.

G. W. Scherer / Journal of Non-Crystalline Solids 215 (1997) 155-168

160

orption on the aerogel that had been compressed to 7 MPa in the porosimeter.

100a

10

" :E

4. Discussion

9.

4.1. Elasticity

1

0.1

....

I ....

0

I''

1

''

2

I'

'

' ' l '

3

'''

I'*'

4

'

5

6

VHg ( c m 3 / g )

Fig. 6. Mercury compression curves for aerogel; separate samples compressed to maximum pressures of 1, 7, and 227 MPa. Arrows show direction of pressure change. No mercury enters the pores of the aerogels, so VHg represents the change in the specific volume of the sample, not the amount of intrusion; P is the applied pressure of Hg.

The exponent b in Eq. (4) showed no trend with Pb; the values were clustered around b = 0.42 _+ 0.09. Since the stress does not relax to zero, the relaxation parameters are much less accurate than the modulus and permeability; we can only judge the error from the scatter, which indicates that T/ is only an orderof-magnitude estimate. Fig. 6 shows the pressure-volume curves for aerogel samples compressed to 1, 7, and 227 MPa in the mercury porosimeter; the low-pressure parts of the curves reproduce extremely well. Fig. 7 shows the pore size distribution measured by nitrogen des3.0

.........

J .........

~ , ,

2.5 ~.A~ 2.0

•

//"

.

.

.

.

.

.

.

.

0.8

.

""',

,' '~

-

'

0.7

•

0.50"6 ~"

1.5 >

o., 9 ;

1.o

0.3 ~

e e,' ~ IDe'

0.5

.........

O.G 0

0.2 0.1 ~......... 5

, .......... 10

0

When the pushrod makes contact with a rod of gel, some indentation of the surface occurs, in addition to flexure of the rod; if all of the movement of the crosshead is assumed to correspond to flexure, then the modulus is underestimated (since the measured load actually results from less than the apparent amount of flexure). The depth ( 8 ) of indentation depends in a complicated way on the diameter of the sample and pushrod, and the span, but it can be calculated using Hertz theory, as explained in Ref. [12]. The ratio of 6 to the displacement of the crosshead (A) decreases as the ratio of span to diameter increases. For softer gels, the indentation substantially reduces the actual deflection of the bar, so neglecting this effect would cause an error in G of > 10%; even the highest moduli in this series would be underestimated by ~ 5% if indentation were neglected.Fig. 4 indicates that the shear modulus varies with density according to a power-law:

G = Go(pb/PO) m,

Pb > 0.18 g / c m 3,

(7)

where P0 = 0.106 g / c m 3, G = 0.42 MPa, and m = 3.45; at lower densities, the modulus is constant, or shows a shallow minimum. This behavior has been seen in numerous studies of the mechanical properties of aerogels (reviewed in Refs. [10,19]), but the physical basis for it has not been established. Young's modulus shows analogous behavior, with E 0 = 0.90 MPa and m = 3.51. Poisson's ratio shows a great deal of scatter, but there seems to be a slight increase in u with the density of the gel. Data for resorcinol/formaldehyde gels examined by Gross et al. [20] show the same trend. The scatter results from the fact that u is obtained from the ratio of two large numbers,

5

3W(plateau)

r (nm) Fig. 7. Cumulative pore volume (V) and pore size distribution ( d V / d r ) obtained by nitrogen desorption for aerogel sample compressed to 7 MPa in mercury porosimeter (bulk density = 0.351 g/cm3).

=

2W(initial)

1,

(8)

where W(plateau) is the value of the load in the plateau region (just beyond t/'r b = 0.2); small errors

161

G. W. Scherer / Journal of Non-C~stalline Solids 215 (1997) 155-168

in the measured loads lead to large uncertainties in v. The mean value for the silica gel is v = 0.16 + 0.049. The bulk modulus (found from the values of E and v obtained by beam-bending) also shows a constant modulus at low densities, followed by a power-law increase with K 0 = 0.33 MPa and m = 3.77. The bulk modulus can also be obtained from the pressure-volume curve for the aerogel that was compressed in the mercury porosimeter, as explained in Ref. [10]. The bulk modulus of the aerogel can be approximated by

o

1 0L

-i

102_

v 1 01-

Bending -~ ~ C o m p r e s s i o n l j

o o ~ ..

j

+Y

1 0°

. . . . . .

0.1

Pb ( g / c m ~ )

Ko,

Po ~ Pb ~ fly

g 0 ( P b / / P y ) m'

Pb ~-~ Py

(9)

K

This means that the modulus is constant if the aerogel is subjected to small strains (namely, such that the density remains below a certain value, py); if the compression continues until the density exceeds py, then the mercury pressure is related to the sample volume by [ 10] PHg

=

K01n(PJPo),

P0 -< Pb -< Py I

Koln( py/PO) +---~ --~y]

1/ ,

Pb

>-

Py

(10) Fig. 8 shows a fit of the compression data for the aerogel to this expression. Using the parameters from

....

I ....

I , j j i i + , L , i , i r , I , , , , 8

,

•

=='~'*lo~, a,.

l

o°t,,1

/

. . . . . . . .

1

0.1

. . . .

[

1

. . . .

I

. . . .

I

2

3

VHg

(

. . . .

I

4

. . . .

I

5

. . . .

6

cm ~ / g )

Fig. 8. Pressure-volume curve from compression of aerogel in mercury porosimeter; symbols are measured values, curve is fit to Eq. (lO).

Fig. 9. Bulk modulus found from beam-bending (O) and from compression of aerogel in mercury porosimeter (curve)

the fit to find K from Eq. (9), we obtain the values shown in Fig. 9. There is good agreement, in the low-density region, with K found from beam-bending; as the density increases, the bending samples become stiffer than the compressed aerogel. The parameters obtained from the mercury compression experiment are K 0 = 2.4 MPa and m = 3.24 (when p > 0.194 g/cm3). When a hydrophilic aerogel is compressed to the point that it enters the power-law region, the compaction is found to be irreversible [10]; however, if the surface has been passivated (for example, with methyl [21] or methoxy groups [22]), the sample springs back to nearly its original density. Evidently, the irreversible compaction results from new siloxane bonds that are formed as the silicate chains in the aerogel are pushed together; if such bonds are prevented by the presence of organic groups, the compliant network springs back. The mercury compression experiment is performed in a few hours on a dry specimen, whereas the bending experiments are performed on a series of gels dried to various extends over a period of ~ 10-12 days. When the latter samples are subjected to large capillary pressures over such long times, chemical attack on the network by the water in the pores is to be expected. Just as water promotes syneresis (which results from condensation reactions) [23,24], it is likely to encourage formation of new crosslinks as the silicate chains in the gel network are pushed together in the shrink-

G. W. Scherer / Journal of Non-Crystalline Solids 215 (1997) 155-168

162

ing gel, and this effect is probably responsible for the difference in the curves in Fig. 9. 4.2. Viscoelasticity The viscosity of the silica network shows a very strong dependence on density, as indicated in Fig. 5, rising in power-law fashion with an exponent of m = 4.5. Viscoelastic relaxation in silica gels is attributed to attack by the pore liquid on the siloxane bonds that constitute the network, based on the fact that the rate of relaxation exhibits the same dependence as the rate of dissolution on the pH of the pore water [11,25,26]. Previous studies have shown that the viscosity increases with aging, as well as shrinkage, so both mechanisms contribute to the exponent [11,27]. In principle, viscous relaxation could result from random breaking and reformation of bonds; that is, there need not be specific attack on strained bonds, since strain would simply accumulate at sites where bond breaking occurs spontaneously. However, recent experiments indicate that attack does occur preferentially on strained siloxane bonds in the gel [28]. This suggests that the value of ~/ and the exponent m might vary with the magnitude of the applied stress. Unfortunately, only a limited range of stress can be applied in the beam-bending experiment, because of the fragility of the gel and the limited range of our load cell, so we have no information on the linearity of the relaxation.

--

-

"cw = 1 . 0 x 1 0 6 p b 1"16

4.3. Permeability The permeability values shown in Fig. 3 were calculated assuming that the viscosity of the water in the gel was the same as in bulk liquid, and that is not necessarily true. In this section we will compare those values of D to standard models, then return to the question of the viscosity of the pore liquid. The hydraulic radius, r h, is defined by 2Vp rh =

0

o °

D = (30 O0~ ~ 0

(s) 1 o5-

~

~0

~0 ~ 0

0

(ll) '

( 1 - Pb/P~) rh

( 1 - pb/p~) 3

4Kh

Kh PbS2

,

(12)

0

~

1 04 0.1

Pb ( g / c m ~

S

where Vp is the specific volume, Vp = 1 / p ~ - 1/p~, and S is the specific surface area of the porous body. The Carman-Kozeny equation relates the permeability to r h [31]:

1 07--

Table 1 indicates that the viscosity is greater for denser samples, even though they sustain higher loads, and this seems to contradict the suggestion that viscoelastic relaxation is accelerated by stress. However, Gross [29] points out that the connectivity of the network is greater in the denser gels, so the stress concentration on the load-bearing skeleton is greater in the low-density gels that exhibit lower viscosity. This effect will be discussed in a future publication [30]. Since the elastic modulus also rises strongly, the average viscoelastic relaxation time given by Eq. (6) increases roughly linearly with density, as indicated in Fig. 10; YVE varies from ~ 1 day for the virgin samples to almost a month for the driest samples. There is no trend in the exponent b, that controls the shape of the viscoelastic relaxation function.

)

Fig. 10. Average viscoelastic relaxation time (s), calculated using Eq. (5) with values of rVE and b obtained from fit to beam-bending experiment; dashed curve is fit to power-law, yielding exponent of 1.16.

where Kh is called the Kozeny 'constant', but is actually a function of density [31,32]. The curve labeled ' C - K ' in Fig. 3 was calculated from Eq. (12) using the measured surface area of ~ 730 m Z / g for the aerogel, and values for Kn calculated by Happel and Brenner for the case of flow through a random array of cylindrical obstacles (representing the gel network). The agreement is not very good at low densities. If Eq. (12) is used to calculate Kh from D,

G. W. Scherer / Journal of Non-Co'stalline Solids 215 (1997) 155-168 15

, L ~ , L J , , J I , ,

I

. . . .

o

I

x:h ( from D ) lCh(H&B)

_ _ -Kw(H

lO:

& B)

©

5 0

0

0

O

0

0 ....

011 . . . .

01.2 ' ' 0 1 3

' ' 014

' 0.5

Pb / P,

Fig. 11. Carman-Kozeny parameter calculated from Eq. (12) using measured values of S and D (labeled 'from D') compared to Kh predicted by Happel and Brenner [31] for flow through random array of cylinders (H&B); Kh applies when the characteristic pore size controlling the permeability is assumed to be the hydraulic radius. Also shown is Kw from Eq. (14), also based on calculations by H&B [32]; this is the Carman-Kozeny parameter that applies when the characteristic pore size is assumed to be the window (pore entry) radius.

then the trend in Kh with density is opposite to the prediction, as shown in Fig. 11. The permeability can also be written in terms of the radii of the 'windows', or pore entries, in the network: (1

D =

-

Pbl P~) r~ 4Kw

= 1.0 + 6.0500.5 - 8.60 0 + 6.56015,

m 2 / g , which is less than half of the measured surface area. The actual gel has a substantial distribution of pore sizes, as indicated in Fig. 7, and the smaller pores contribute the most to the surface area, while the larger pores carry most of the flow. Therefore, if one chooses a uniform microstructure with a pore size to match D (as we have done in the cubic cell model), then the surface area is underestimated (because the contribution of the small pores is neglected). On the other hand, if one assumes a uniform structure with a pore size related to S (as in Eq. (12)), the permeability is underestimated; the quality of the prediction improves at higher densities (see Fig. 3), where the larger pores have collapsed and the flow actually passes through smaller pores. Katz and Thompson [34,35] argue that the permeability is controlled by a characteristic pore size called the 'breakthrough radius', rBT, which is the radius of the largest pore that connects a percolating network of pores with radii >__rST. We can use Eq. (13) to calculate raT from the measured permeability:

rw = rBT =

4KwD Pb/Ps

(15)

1 --

(13) 20-

In this case, Happel and Brenner's calculations show that r w varies weakly with Pb, compared to Kh, as indicated in Fig. 11. For a network consisting of a random array of cylinders, Kw is given approximately by [32] K w

163

(14)

where p = Pb/P~ is the relative density. The curve labeled 'Cubic' in Fig. 3 was calculated using Eqs. (13) and (14), with rw = 13.6 nm; this value was chosen to give a good fit to D at Pb = 0.154 g / c m 3 (the density of the aerogel). The calculated values are seriously in error at the higher densities. The surface area of a cubic array of cylinders can be calculated from equations given in Ref. [33]; given r w = 13.6 nm and Pb = 0.154 g / c m 3, we find that the cylinder radius is a = 2.79 nm and S = 334

- c- - r w ( permeability

• A

E

)

rat ( d r i e d ) rat ( compressed )

15-

/

f

c

O/

a•~

O

A~ : °

.~,u

%

10zx9 /

I I

/©

v Z3

J

5

,,A

c~o"m 0

,,,I

....

I

I ....

I ....

2 3 Pore volume

I ....

i ....

4 5 (cm 3/g)

I ....

6

I

7

Fig. 12. Breakthrough radii (raT) obtained from peak of nitrogen desorption curves for fully dried xerogel ('dried') and aerogels compressed in mercury porosimeter ('compressed'); window radii calculated from measured permeability using Eq. (15) with K,~ given by Eq. (14): abscissa is pore volume, Vp. Dashed line is fit forced through origin. Arrows indicate the magnitude of the compression predicted to occur during nitrogen desorption (see text).

164

G.W. Scherer/Journal of Non-CrystallineSolids 215 (1997) 155-168

The results are plotted in Fig. 12 against the pore volume, Vp: Vp

1

1

Pb

Ps

(16)

A linear fit of r w versus Vp indicates that r w = 0 when Vp = 0.07 cm3/g; that is, the extrapolation indicates that the pores close when the relative density reaches p -- 0.88. The dashed line in Fig. 12 is a fit forced through the origin. A linear dependence of pore size on density or pore volume has been found in several other studies of silica gel [36-38]. In the next section, we compare r W with the breakthrough radii found from nitrogen desorption.

Desorption is a drying process where the pore fluid is liquid nitrogen. The peak of the pore size distribution obtained from the desorption branch is related to the break-through radius, rBT [39--41]. The maximum capillary pressure exerted on the network during drying is determined by rBv [4], as is the permeability [34]. Nitrogen desorption has been applied to measure rBT for several dried gels, as well as the aerogel before and after compression by mercury; the results are shown in Fig. 12. For the denser gels, rBT agrees well with r w from the permeability measurements, but the aerogels with Vp > 5 g / c m 3 fall well below the line. The latter discrepancy is an artifact caused by stress exerted on the network by the liquid nitrogen. Capillary pressure exerted by the liquid nitrogen causes substantial compression of compliant aerogels, so that the pore volume and pore size found by desorption are seriously underestimated [42]. The capillary pressure (Pc) exerted on the gel during the measurement can be found from the partial pressure of nitrogen, since [43]

t=

bH

) 1/3

In(P/Po)

'

(19)

where b n is a constant; for nitrogen, From Eqs. (17) and (19),

(bHRT) t=

bn ~

0.221.

1/3 (20)

PcVm

r=

b.Rr] 'j3 I

(21)

Pc

Eq. (21) has been solved using the parameters appropriate for nitrogen (Vm = 34.6 cm3/g and y = 8.85 ergs/cm 2 at T = 77 K [43]), and the result is very accurately approximated by Pc "~ - 3 0 / r 1 ' 8 , (22) where Pc is in megapascals and r is in nanometers. As the relative pressure of nitrogen vapor decreases during desorption, the capillary pressure increases in magnitude, and the network is compressed; as the network contracts, its modulus rises according to Eq. (9). The network will stop contracting when the following condition is satisfied [38,42]:

(23) The first and third terms on the left are small and offsetting, so the condition can be simplified to

(17)

(24)

where Vm is the molar volume of liquid nitrogen. Pc is also related to the pore radius (r) by Laplace's equation:

where the second equality follows from Eq. (22). Fig. 12 indicates that the pore size varies approximately in proportion to the pore volume, so

Pc =

Vm ] ~'~00 '

(

Eqs. (18) and (20) can be used to relate the pore size to the capillary pressure:

4.4. Sorption

Pc =

where y is the liquid/vapor interracial energy of liquid nitrogen, 0 is the contact angle (taken to be zero), and t is the thickness of the layer of nitrogen that remains adsorbed on the pore surface when the meniscus enters. The layer thickness can be estimated from the Halsey equation [43]:

2y cos 0 - - , r-t

(18)

rw(pb) = r w ( o 0 )

Ob

Ps /

Po

'

G. W. Scherer / Journal of Non-Ct3,stalline Solids 215 (1997) 155-168

Inserting Eq. (25) into Eq. (24) leads to an expression that can be solved for the bulk density at the point when shrinkage stops. For our silica gels, Ko/m = 0.744 MPa, fly = 0.194 g / c m 3, Ps = 2.01 g / c m 3, and the permeability data indicate that r w = 13.6 nm when P0 = 0.154 g / c m 3. With these values, Eqs. (24) and (25) lead to an equation for Pb whose solution is Pb = 0.289 g/cm3; the corresponding pore size from Eq. (25) is r w = 6.74 nm. This means that any aerogel with a density less than 0.289 g / c m 3 will contract during nitrogen desorption until its density reaches that value, and its pore radius is reduced to ~ 6.7 nm. That value compares well with the pore radii (5.5 and 5.8 nm) found for the two lightest aerogels (namely, the two outliers in Fig. 12); the arrows in the figure indicate the magnitude of the predicted compression. Thus, we conclude that the original pore sizes of those two samples actually lay near the dashed line (so that raw = rw), and the desorption measurement caused them to contract by about a factor of two. Shrinkage of that magnitude has been directly observed during desorption of nitrogen from a similar silica aerogel [42].

4.5. Viscosity of pore liquid Confinement in small pores influences the structure and mobility of a liquid, and that could have important implications for the permeability of our gels. The fit to the beam-bending experiment yields the hydrodynamic relaxation time ~'b and the elastic modulus, which permits calculation of the quantity D/~IL. The permeability values given in Fig. 3 were obtained by assuming that r/L in the gel is equal to its value in bulk water; if that assumption is erroneous, then D is inaccurate, and comparisons to models are meaningless. Attempts have been made to measure the viscosity of water in a very small gap between mica sheets in a surface force apparatus, but careful consideration of the experimental details indicated that those results were unreliable [44]. The mobility of liquids in small pores can also be explored using optical spectroscopy [45], neutron diffraction [46,47], and nuclear magnetic resonance [48-50]. If the molecule interacts weakly with the pore wall, then there is little effect of confinement on mobility [45,50,51], but if the liquid molecules are

165

attracted to the wall, then the mobility of a few molecular layers is strongly reduced. In the present case, the water in the pores will interact strongly with the silanol groups on the surface of the silica gel. In a study of water in biological systems and clays, water was found to be strongly bound in a layer ~ 0.5 nm thick [48]. In a variety of oxides including silica, there is a layer of water ~ 0.9 nm thick that does not freeze at temperatures well below the normal freezing point [52]; however, the unfrozen layer remains mobile even at temperatures as low as 200 K [53]. Tracer diffusion studies show that the diffusion coefficient of water is ~ 5.5 times smaller in a gel with an average pore radius of 1.45 nm than in bulk water [54]. On the other hand, NMR relaxation time analysis indicates that the mobility of water in a porous glass with pore radius 1.75 nm is reduced only in a monolayer ( ~ 0.3 nm) of water on the solid surface [55]. Another NMR study of water in porous glass indicates that two layers of water ( ~ 0.5 nm) show reduced mobility [56]. If the actual viscosity of the liquid in our gels is higher than we have assumed, then the true magnitude of D is higher than the values given in Fig. 3, and the pore sizes in Fig. 12 are underestimated. However, if the viscosity of the pore liquid were 5.5 times higher than r/L for bulk water (i.e., if the effect on ~/L were as large as that reported for diffusivity of water [54]), then the pore radius would be 5.51/2 = 2.3 times higher than we have calculated; for the densest of our gels, this would raise r w from 1.0-1.3 nm to 2.3-3.0 nm. In fact, as shown in Fig. 13, the nitrogen desorption data indicate that raT = 1.6 nm

c- - r w ( p e r m e a b i l i t y ) • raT ( d r i e d ) A r.T ( c o m p r e s s e d )

-

2.5A

E

A

/

1.5

/

/

2-

/

©

//

o

J n-

/

1

©

J /

.= o O.

/-

0.5

J j

0

J

,,I,,,I,,, 0.2

I,,,I,,,I 0.4

Pore v o l u m e

0.6

0.8

(cm 3 / g )

Fig. 13. High-density region from Fig. 12.

1

166

G.w. Scherer/ Journal of Non-Crystalline Solids 215 (1997) 155-168

for samples of that density, so we conclude that the viscosity is not as strongly affected by confinement as the diffusivity. This difference might arise because the pressure gradient encourages the flow to pass through a percolating network of relatively large pores, while diffusion has no such bias; consequently, diffusing molecules will spend more time in micropores where their mobility is severely restricted, while molecules caught up in a hydrodynamic flow will tend to avoid those pores. Alternatively, tracer molecules might spend a significant amount of time in the relatively immobile surface layer (whose presence is indicated by NMR), whereas molecules moving under the influence of a pressure gradient do not.

radii of ~ 1.65 nm measured on the xerogels, Eq. (26) predicts Pc = - 124 MPa, so the maximum stress on the network would be ~ = (1 - p b / p ~ ) P c = --67--72 MPa; in contrast, the stress imposed on those gels during the beam-bending experiment was < 1 MPa. The viscoelastic response of silica gel is believed to result from chemical attack by the pore liquid on the siloxane network [11,13,28], and the rate of hydrolysis of siloxane bonds depends strongly on the strain in the bond [59]. Therefore it is likely that the rate of relaxation of silica gels is greater during the later stages of shrinkage, when the capillary stresses are quite high.

4.6. Drying shrinkage

This study provides very detailed information about the properties of a particular type (B2) of silica gel, and it is not clear to what extent the trends we observe can be expected to apply to other gels. For example, the power-law dependence of modulus on density seems to apply to a wide variety of silica gels and resorcinol-formaldehyde gels, but the exponent has been found to vary widely from gel to gel, and one recent study [20] indicates that the power-law is only an approximation that applies over a limited range of density. Unfortunately, the physical basis for the power-law behavior is not understood, so the connection between the exponent and gel structure cannot be predicted. In the present study, the viscosity exhibits a higher exponent that the modulus, so the viscoelastic relaxation time increases with p, but that might not be true for other gels. Moreover, since the viscosity results from a chemical reaction, the exponent might change with temperature, so that the trend in relaxation time with density might reverse at higher or lower temperatures. It has been observed [60,61] that when aerogels are subjected to mechanical compression, the larger pores collapse preferentially. Therefore, a broad distribution would be narrowed by elimination of the larger pores, whereas a monodisperse distribution would remain monodisperse during compression. It seems that the dependence of the average pore size on density must reflect the initial pore size distribution, so that the linear relation between r w and Vp observed for B2 might be unique. Surprisingly, the

The analysis presented in Section 4.4 can be adapted to predict the final density of a gel dried from water. The adsorbed layer thickness for water on silica has been determined [57], and the results are well approximated by Eq. (19) with b H = 0.125 nm 3, so (with Vm= 18 cm3/g and 3,=0.072 ergs/cm z at 298 K), Eq. (22) is replaced by Pc -" - 2 1 6 / r 1 ~ 1 ,

(26)

where Pc is the capillary pressure in water in MPa and r is the pore radius in nanometers. Eq. (24) is replaced by

m ~py

where m = 3.77; if we adopt py = 0.194 g / c m 3 (as for the aerogel), then K 0 = 3.19 MPa. Solving Eq. (27) for the final density after drying yields Pb = 0.68 g / c m , which is much less than the measured value ( ~ 1.05 g/cm3); even allowing for viscoelastic relaxation does not bring the calculated density close to the measured value, since the viscosity of the network is very high. This method of calculating the density after drying has been shown to work well for similar silica gels, when the pore liquid was inert (i.e., where no viscoelastic relaxation occurred) [58]. Therefore, we interpret the large discrepancy in water as evidence of enhanced relaxation at high stress (i.e., non-linear viscoelasticity). Given the desorption

4. 7. Generality o f results

G. W. Scherer / Journal of Non-Crystalline Solids 215 (1997) 155-168

167

same linearity has been observed for resorcinol-formaldehyde gels [62], as well as for a quite different family of silica gels [63].

for providing experimental facilities and helpful suggestions. I also thank Professor Jean Phalippou for helpful comments on the manuscript.

5. Conclusions

References

The properties of B2-type silica gels change dramatically during drying. The modulus, viscosity, and viscoelastic relaxation time of the network exhibit power-law dependence on density. The viscosity is attributed to attack on siloxane bonds by the pore liquid, so one would expect the viscosity to decrease as the applied stress increases. Our measurements permit only a small range of stress to be explored, but indirect evidence (namely, the amount of drying shrinkage) indicates that the viscosity of the silica network is much lower under the large capillary stresses generated during drying. For relative densities up to ~ 0.3, the permeability decreases with a power-law exponent of ~ 2.8, but at higher densities, the decrease is faster than a power-law; that is to be expected, since the pore size will go to zero at a finite density. If the viscosity in the pores is assumed to equal that of the bulk liquid, then the pore size inferred from D(rw) is quite close to the breakthrough radius (rBT) found from nitrogen desorption (as expected from theory). At the highest densities, r w is a few Angstroms smaller than rBT, which probably corresponds to the presence of a layer of that thickness with low mobility; however, the data indicate that most of the liquid in the gel has the viscosity of the bulk liquid. The implications of these results for drying stress and fracture will be the subject of a future publication.

[1] C.J. Brinker, G.W. Scherer, Sol-Gel Science (Academic Press, New York, 1990) ch. 7&8. [2] G.W. Scherer, in: Drying "92, Part A, ed. A.S. Mujumdar (Elsevier, Amsterdam, 1992) p. 92. [3] J.H.L. Voncken, C. Lijzenga, K.P. Kumar, K. Keizer. A.J. Burggraaf, B.C. Bonekamp, J. Mater. Sci. 27 (1992) 472. [4] G.W. Scherer, D.M. Smith, J. Non-Cryst. Solids 189 (1995) 197. [5] G.W. Scherer, J. Non-Cryst. Solids 144 (1992) 210. [6] G.W. Scherer, in: Better Ceramics Through Chemistry VI, ed. A.K. Cheetham, C.J. Brinker, M.L. Mecartney and C. Sanchez (Materials Research Society, Pittsburgh, PA, 1994) p. 209. [7] C.J. Brinker, K.D. Keefer, D.W. Schaefer, R.A. Assink, B.D. Kay, C.S. Ashley, J. Non-Cryst. Solids 63 (1984) 45. [8] G.W. Scherer, J. Non-Cryst. Solids 109 (1989) 183. [9] G.W. Scherer, R.M. Swiatek, J. Non-Cryst. Solids 113 (1989) 119. [10] G.W. Scherer, D.M. Smith, X. Qiu, J.M. Anderson, J. NonCryst. Solids 186 (1995) 316. [11] G.W. Scherer, J. Sol-Gel Sci. Tech. 2 (1994) 199. [12] G.W. Scherer, J. Non-Cryst. Solids 201 (1996) 1. [13] G.W. Scherer, J. Non-Cryst. Solids 142 (1992) 18. [14] G.W. Scherer, J. Non-Cryst. Solids 212 (1997) 268. [15] G.W. Scherer, J. Sol-Gel Sci. Tech. 1 (1994) 169. [16] G.W. Scherer, Relaxation in Glass and Composites (Wiley, New York, 1986: reprinted by Krieger, Malabar, FL, 1992) pp. 41-44, 91-92, 133. [17] J. DeBast, P. Gilard, Phys. Chem. Glasses 4 (4) (1963) 117. [18] R.C. Weast, M.J. Astle. eds., CRC Handbook of Chemistry and Physics. 62nd Ed. (CRC, Boca Raton, FL, 1981) p. F-42. [19] R.W. Pekala, L.W. Hrubesh, T.M. Tillotson, C.T. Alviso, J.F. Poco, J.D. LeMay, in: Mechanical Properties of Porous and Cellular Materials, ed. K. Sieradzki, D.J. Green and L.J. Gibson (Materials Research Society, Pittsburgh, PA, 1991) p. 197. [20] J. Gross, G.W. Scherer, C.T. Alviso, R.W. Pekala, J. NonCryst. Solids 211 (1997) 132. [21] G.W. Scherer, D.M. Smith, J. DiGregorio, J.M. Anderson, 'Bulk modulus of gels by mercury compression', J. NonCryst. Solids, to be published. [22] L. Duffours, T. Woignier, J. Phalippou, J. Non-Cryst. Solids 186 (1995) 321. [23] Z.Z. Vysotskii, V.I. Calinskaya, V.I. Kolychev, V.V. Strelko, D.N. Strazhesko, in: Adsorption and Adsorbents, No. 1, ed. D.N. Strazhesko (Wiley, New York, 1973) p. 72. [24] C.J. Brinker, G.W. Scherer, Sol-Gel Science (Academic Press, New York, 1990) ch. 6. [25] G.W. Scherer, Faraday Discuss. 101 (1995) 225.

Acknowledgements This work was done while the author was an employee of the DuPont Co. Their support of this work is appreciated. I thank Raj Patel (DuPont Co.) for his help with sample preparation. I am deeply indebted to Julie Anderson (University of New Mexico, Albuquerque) for preparing the aerogel, and performing the mercury compression and nitrogen sorption experiments, and to Professor Doug Smith

168

G.W. Scherer / Journal of Non-Crystalline Solids 215 (1997) 155-168

[26] G.W. Scherer, Faraday Discuss. 101 (1995) 287. [27] G.W. Scherer, S.A. Pardenek, R.M. Swiatek, J. Non-Cryst. Solids 107 (1988) 14. [28] K.G. Sharp, G.W. Scherer, J. Sol-Gel Sci. Tech. 8 (1997) 132. [29] J. Gross, private communication. [30] J. Gross, G.W. Scherer, to be published. [31] J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics (Martinus Nijhoff, Dordrecht, 1986). [32] G.W. Scherer, J. Sol-Gel Sci. Tech. 1 (1994) 285. [33] G.W. Scherer, J. Am. Ceram. Soc. 74 (7) (1991) 1523. [34] A.J. Katz, A.H. Thompson, Phys. Rev. B34 (1986) 8179. [35] A.J. Katz, A.H. Thompson, J. Geophys. Res. 92 (B1) (1987) 599. [36] S. Wallace, C.J. Brinker, D.M. Smith, in: Better Ceramics Through Chemistry VI (Materials Research Society, Pittsburgh, PA, 1995) pp. 241-246. [37] G. Reichenauer, C. Stumpf, J. Fricke, J. Non-Cryst. Solids 186 (1995) 334. [38] D.M. Smith, G.W. Scherer, J.M. Anderson, J. Non-Cryst. Solids 188 (1995) 191. [39] H. Liu, L. Zhang, N.A. Seaton, Chem. Eng. Sci. 47 (1992) 4393. [40] H. Liu, L. Zhang, N.A. Seaton, J. Colloid Interf. Sci. 156 (1993) 285. [41] H. Liu, L. Zhang, N.A. Seaton, J. Colloid Interf. Sci. 162 (1994) 265, erratum to Ref. [40]. [42] G.W. Scherer, D.M. Smith, D. Stein, J. Non-Cryst. Solids 186 (1995) 309. [43] S. Lowell, J.E. Shields, Powder Surface Area and Porosity (Chapman and Hall, New York, 1984). [44] J. van Alsten, S. Granick, J.N. Israelachvili, J. Colloid Interf. Sci. 125 (1988) 739. [45] J. Wamock, D.D. Awschalom, M.W. Shafer, Phys. Rev. B34 (1986) 475. [46] M.-C. Bellissent-Funel, J. Lal, L. Bosio, J. Chem. Phys. 985 (1993) 4246.

[47] D.C. Steytler, J.C. Dore, C.J. Wright, Molec. Phys. 48 (1983) 1031. [48] K.J. Packer, Phil. Trans. R. Soc. (London) B278 (1977) 59. [49] W.P. Halperin, F. D'Orazio, S. Bhattacharja, J.C. Tarczon, in: Molecular Dynamics in Restricted Geometries, ed. J. Klafter and J.M. Drake (Wiley, New York, 1989) p. 311. [50] G. Liu, M. Mackowiak, Y. Li, J. Jonas, Chem. Phys. 149 (1990) 165. [51] G. Liu, M. Mackowiak, Y. Li, J. Jonas, J. Chem. Phys. 94 (1991) 239. [52] M. Brun, A. Lallemand, J.F. Quinson, C. Eyraud, Thermochim. Acta 21 (1977) 59. [53] P.G. Hall, R.T. Williams, R.C.T. Slade, J. Chem. Soc. Faraday Trans. 1 81 (1985) 847. [54] N. Koone, Y. Shao, T.W. Zerda, J. Phys. Chem. 99 (1995) 16976. [55] F. D'Orazio, S. Bhattacharja, W.P. Halperin, K. Eguchi, T. Mizusaki, Phys. Rev. B42 (1990) 9810. [56] E. Almagor, G. Belfort, J. Colloid Interf. Sci. 66 (1978) 146. [57] J. Hagymassy Jr., S. Brunauer, R.Sh. Mikhail, J. Colloid Interf. Sci. 29 (1969) 485. [58] G.W. Scherer, S. H~ereid, E. Nilson, M.A. Einarsrud, J. Non-Cryst. Solids 202 (1996) 42. [59] T.A. Michalske, B.C. Bunker, J. Appl. Phys. 56 (1984) 2686. [60] G.A. Nicolaon, S.J. Teichner, Bull. Soc. Chim. 9 (1968) 3555. [61] R. Pirard, S. Blacher, F. Brouers, J.P. Pirard, in: Mater. Res. Symp. Proc., Vol. 371 (Materials Research Society, Pittsburgh, PA, 1995) p. 517-528. [62] G.W. Scherer, C. Alviso, R. Pekala, J. Gross, in: Microporous and Macroporous Materials, Mater. Res. Soc. Symp. Proc., Vol. 431, ed. R.F. Lobo, J.S. Beck, S.L. Suib, D.R. Corbin, M.E. Davis, L.E. Iton and S.I. Zones (Materials Research Society, Pittsburgh, PA, 1996) p. 497. [63] M.-A. Einarsrud, E. Nilsen, G.W. Scherer, 'Effect of precursor and hydrolysis conditions on drying shrinkage', submitted to J. Non-Cryst. Solids.