Drying gels

324

Journal of Non-Crystalline Solids 99 (1988) 324-358 North-Holland, Amsterdam

DRYING GELS VI. Viscoelastic plate George W. SCHERER E.L DuPont de Nemours & Co., Central R&D Dept., Experimental Station 356/384, Wilmington, DE 19898, USA Received 13 July 1987

The model of drying presented in earlier parts of this series is reviewed. In those papers, the gel network was assumed to be either purely elastic or purely viscous. The latter solution seems most appropriate for alkoxide-derived gels, which exhibit considerable irreversible shrinkage during drying, but it does not describe the rise in stress as evaporation begins. In this paper, an analysis is presented of the stresses and strains that develop during drying of a plate of gel in which the solid phase is viscoelastic. Initially the pressure in the liquid phase rises as predicted by the elastic analysis (Part V of this series), then evolves into the distribution predicted by the viscous analysis (Parts I and II of this series). The viscous solution is shown to apply at times greater than the viscoelastic relaxation time, which is small compared to the drying time of alkoxide-derived gels. Therefore, the simple equations provided by the viscous solution are suitable for prediction of the stress development and shrinkage during drying of gels.

1. Introduction In earlier segments of this series, drying of gels was modelled on the assumption th,at the solid phase is either purely viscous [1-5] * or purely elastic [6]. The analysis of the viscous gel provides a steady state solution with no explicit time dependence. The solution for the elastic case [6] indicates that the stress rises rapidly to a maximum (similar in magnitude to that predicted for a viscous gel), then decays to zero. In the present paper, a viscoelastic analysis is presented that illustrates the transition from elastic to viscous behavior. It is shown that the simple viscous solution is sufficient for prediction of the maximum stresses in most cases. Before presenting the new analysis, we re-examine the implications, and shortcomings of the model developed to date. The model is summarized in fig. 1. The gel consists of a solid network enclosing a continuous liquid phase; initially the liquid/vapor interface (meniscus) is flat. Evaporation of liquid from the pores of a gel exposes the solid phase. Since the solid/vapor interfacial energy (Ysv) is larger than the solid/liquid interfacial energy (YSL), the liquid tends to spread from the interior of the gel to cover the exposed solid. As the liquid stretches to cover * Errors in Part I are corrected in ref. [5]. 0022-3093/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

G. W. Scherer / Drying gels V1

a)

Initial

325

condition

Liquid/vapor

meniscus

flat

Pore liquid Solid phase

b)

Evaporation Pressure

creates

in liquid

meniscus with at exterior:

radius

PE =

Evaporation ww ~

Shrinkage

c)

r

2 (Ysv- YSL) r

~

r

Dry region forms

Empty pores Minimum radius of curvature PE = PR Fig. 1. Schematic illustration of drying process: (a) Initial condition (i.e., before evaporation begins); (b) meniscus develops curvature and pressure rises as evaporation removes pore liquid, but meniscus remains at surface of gel; (c) when gel is too stiff to flow under redistribution pressure, PR, meniscus enters gel (surface region becomes dry).

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G. IV.. Scherer / Drying gels VI

the solid, tensile stress develops in the liquid and compressive stress is imposed on the solid phase. The meniscus begins to develop curvature (fig. lb); the pressure in the liquid at the exterior surface (PE) is related to the radius of curvature of the meniscus (r) by PE = 2(Ysv- 7SL)/r- Initially, the gel is so compliant that the compressive stress causes viscoelastic deformation of the solid, drawing it under the surface of the liquid. Thus, the liquid does not flow to the surface, but the solid "flows" into the liquid. The tension in the liquid tends to do two things: pull in the walls of the pores (causing contraction of the gel network) and induce flow of liquid from the interior of the body. The small pore size gives rise to considerable resistance to flow (low permeability) in the gel, so the tension in the liquid cannot readily draw liquid from the interior. Therefore, the tension in the liquid and the resulting contraction of the gel network are isolated near the exterior surface. The drying stress arises because the shrinkage of the gel is faster near the exterior than in the interior. This is analogous to the development of stress during cooling of a plate of glass, when the cooler exterior contracts relative to the interior, causing tensile stress in the surface region. There would be no stress in the gel if the permeability were high, because in that case there would be no pressure gradient in the liquid, and the gel would contract uniformly. It is the spatial variation in strain rate that causes stress. As the gel shrinks it becomes more viscous (stiffer), because the structural units (particles or polymeric clusters) become more tightly packed, and because chemical reactions continue to form bridging bonds. The meniscus deepens and the pressure in the liquid rises to a level just sufficient to compress the solid network at the rate at which pore liquid is evaporating; the liquid/vapor interface remains at the exterior surface of the body (fig. lb). As the gel stiffens, the pressure in the liquid rises, but it cannot increase beyond [5] PR = (YSV-- 7se)Sp/Vp, where Sp and Vp are the surface area and volume of the (liquid-filled) pore space; PR is called the redistribution (or capillary) pressure. Eventually the gel becomes too stiff to contract under the influence of PR, and further evaporation drives the liquid/vapor meniscus into the gel (fig. lc). This occurs when shrinkage is nearly complete. As shown in section 2.1 (below) for viscous gels, and in ref. [6] for elastic gels, the stress has a maximum when the interface enters the gel, and this is consistent with the observation that gels tend to crack just as shrinkage stops. As the pores continue to empty by evaporation, the dry region expands because it is released from the pressure in the liquid, but the saturated region continues to be compressed by that pressure. This also produces differential strain, resulting in high tensile stress in the wet region as it becomes very thin. This picture of the drying process is consistent with the observations of Simpkins et al. [7], who used an acoustic method to detect cracking in a drying gel. They found two burst of acoustic activity (presumably cracking): first, when the meniscus entered the gel, then when drying was almost complete. The model developed in this series makes it possible to quantify the stresses and strains that develop during drying. The next section provides a review and

G.W. Scherer / D~ing gels VI

327

reinterpretation of the results obtained so far, and illustrates the need for the viscoelastic treatments presented in sections 3 and 4. A differential equation is derived to describe the pressure in the pores in a viscoelastic plate. A relatively simple solution is derived in section 3 by assuming that the viscosity of the gel is constant: a more realistic, but more complex solution is presented in section 4 for a gel whose viscosity increases with time (and becomes infinite). The implications of the analysis are discussed in section 5. A table of notation is provided at the end of the paper.

2. Limiting cases 2.1. Viscous plate

The flow of liquid through a porous medium (or, equivalently, the collapse of the gel network into the liquid) obeys Darcy's law, J= ~vP,

(11

where J [units of volume/ (area × time)] is the flux of liquid, D (units of area) is the permeability, Or, is the viscosity of the pore liquid, and ~TP is the pressure gradient. The pressure, P, is positive when tensile (as for stresses), so the flux is positive (flows toward) regions in which the liquid is in tension. In a viscous gel, the pressure in the liquid P, causes a constant volumetric strain rate, ~, given by [1] - I ? / V = - 3~ s - P / K o ,

(2)

where V is the volume of the gel (solid plus liquid) and K~ is the volume viscosity of the gel network; ~s is the syneresis strain rate (i.e., the rate of spontaneous contraction when there is no pressure in the liquid). In Part I [1] eqs. (1) and (2) were used to obtain a differential equation describing the pressure distribution in the pore liquid. Darcy's law was used to predict the rate of accumulation of liquid in a given region (which depends on ~72P), and the rate of change in liquid volume was set equal to the rate of change in pore volume found from eq. (2). The result is [1]

v,2e_

e=----5--

(3)

Since the gel contracts more rapidly near the exterior surface, the structure-dependent properties D and K o will vary with position; moreover, since the gel shrinks there is a moving boundary. A precise solution of eq. (3) must therefore be obtained by numerical method. In refs. [1-4] these complications were ignored so that a simple analytical solution could be obtained. Such solutions are believed to include the essential features of the true pressure distribution, and will be reasonably accurate in many cases. For a plate of

G. IV.. Scherer / Drying gels VI

328

thickness 2 L with faces at z = L and z = - L , if we neglect syneresis, the result is [1,2]

P ( z ) = PEI-I(a, z),

(4)

where Pz is the pressure at the exterior surface, P ( L ) = PE. The function Fl(a, z) is called the permeability factor, and for a plate is given by *

H ( a, z) = cosh( a z / L )/cosh( a),

(5)

where 2

a - ( L ~ L / D K G)

1/2

.

(6)

The parameter a determines the shape of the pressure distribution: the larger it is, the greater the difference in P through the thickness of the plate. (This means that the analytical solution applies only for small a, say a < 3, since the z-dependence of the properties become large when a is large.) This is such an important parameter that it is worth examining its physical significance in some detail. Suppose we want to find the time rj in which liquid flows a distance L (i.e., J = L/'rj) under a pressure gradient of ~TP = P / L . From eq. (1),

rs = L2nL/DP.

(7)

At the same time, that pressure causes a strain rate in the gel of

i

=

P / K G = 1/r,,

(8)

where ,, is the time to achieve a volumetric strain of unity. From eqs. (6)-(8), .

.1/2

a = (rj/r,)

,

(9)

so a is related to the ratio of the characteristic times for flow of pore liquid (rj) and for viscous deformation of the network (r,); that is, it depends on the ratio of the time required to drain the pores and the time to collapse them. If a is large, the pressure in the liquid causes relatively more contraction than flow, so that shrinkage is concentrated near the exterior surface; since liquid is not extracted from the interior, the pressure does not rise there, and the pressure drop occurs in a small region near the exterior. In extreme cases this could produce a relatively dense skin on the gel. Whenever a is large the exterior of the gel densifies faster, creating spatial variation in structure-dependent properties, such as D and K o. Therefore, the analytical solution becomes increasingly inaccurate with time. Another way of looking at a is to define a characteristic breadth, LE, of the pressure gradient at the exterior surface, ZE

-

dz z=L

=

a

tanh(a),

(10)

* T h e equation in ref. [1] looks different, because the surfaces of the plate were taken to be z = 0 and 2 L , rather than z = + L as in this paper.

G.IV. Scherer / Drying gels V1

329

where the gradient is found from eqs. (4) and (5); then L/L E = a tanh(a) = a,

(lla,b)

a > 2.

Thus 1 / a represents the fraction of the thickness of the plate over which most of the pressure drops occurs: if a is large, the pressure drop occurs near the external surface; if a is very small, the pressure is relatively uniform through the plate. It was shown in Part II [2] that the stress at the surface of a plate drying by evaporation from both sides is

where ox is the stress in the plane of the plate, VE is the evaporation rate [units of v o l u m e / (area X time)l, CN-- (1 -- 2 N ) / ( 1 - N ) = 1, and N is the Poisson's ratio of the gel network (exclusive of liquid). The hyperbolic cotangent is approximated by c o t h ( a ) ~ 1 / a + a / 3 when a is small ( < 1) and coth(a) = 1 when a is large ( > 2), so eq. (12) can be written as * I}'ELTIL/3D O'x =

•

V E( K01 LID )

a <1 1/2

,

(13a,b)

ot >> 1.

This shows that the stress is proportional to the evaporation rate and inversely related to the permeability of the network. When a is small, the stress is proportional to the thickness of the plate (2L) but when a is large, the pressure drop does not extend through the plate (only through the thickness LE) and the stress is independent of the thickness of the plate. When a is small, the stress is independent of the viscosity of the network, K o, even though eq. (12) suggests otherwise, because a decreases when K o increases and the changes exactly balance. This is no longer true when a is large: the stress rises in proportion to K~/2. This is important because the viscosity of the gel increases as it shrinks, rising abruptly as shrinkage stops. Eq. (13b) thus indicates that the end of contraction of the gel and the entrance of the liquid/vapor interface into the gel are accompanied by a rise in stress. This explains why cracking is frequently observed in alkoxide-derived gels at the time when shrinkage stops [8]. Recall that a changes during contraction of the gel, because L and D decrease and K G increases; the stress depends on the current value of a, so the stress will increase as the meniscus enters only if a is still large at that time. The value of a when the meniscus enters the gel ( at time tR) is given by eq. (71) of Part I [1] (again neglecting the syneresis strain rate): t a n h ( a ) / a = KGfYE/PR L

(t = tR).

(14)

* Eq. (13b) represents a range of a in which the accuracy of the analytical solution is questionable, so this relationship must be regarded as serniquantitative.

G.W. Scherer / Drying gels VI

330

1.0

0.8

meniscus enters .u 0 . 6 Q.=

.>= ~o 0.4

0.2

[meniscus at surface] 0

2

4

6

8

10

O~(tR) Fig. 2. Plot of eq. (14), showing relationship between ~ = Ko("E/PRL and the value of c~ at the time t R when the meniscus enters the gel. For example, if a = 2, then the meniscus will enter if > 0.48, but will remain at the exterior surface of the gel otherwise.

1.0

0.8

J

¸

nO.

z (J

0.6'

0.4.

0.2,

0 0

I 2

I 4

6

~(tR) or

rt

I 8

I 10

Fig. 3. Plot of eq. (15) showing stress at surface of plate of gel at the time t R when the meniscus enters the gel. The same curve applies to a viscous gel [characterized by the parameter a defined in 0% (6)1 and to an elastic gel [characterized by the parameter # defined in eq. (20)]. The larger a or #, the greater the stress, which asymptotically approaches a m a x i m u m value of PR.

G. V¢~Scherer / Drying gels VI

331

This relationship is illustrated in fig. 2: for values of a to the right of the curve, the meniscus will enter the gel; otherwise it remains at the exterior surface. The higher the evaporation rate (I?E), the smaller the value of a that will permit entry. From eqs. (12) and (14) we find that the stress in the plate at that time is

a~(L)/CNPR = 1 -- [a c o t h ( a ) ] - '

(t = tR)

(15)

SO oX approaches the redistribution pressure, PR, when a is large, as shown in fig. 3. Let us define (16)

JR =-- D P R / ~ L L

so that JR is the flux produced by a pressure gradient of PR/L, according to Darcy's law [eq. (1)]. From eqs. (6), (14) and (16) we find that this flux is JR = l?E[a t a n h ( a ) ] - 1

= l?E/a

(a > 2)

(17a,b)

so that at the time that the meniscus enters the gel, a ~ I?E/J R and eq. (15) can be written as

o~(L)/CNPR = 1 --JR/I?E

(a >_2).

(18)

This means that the stress depends on the relative magnitudes of the evaporation rate and the liquid flux that can be sustained with a pressure gradient extending to the midplane of the plate. If that flux is smaller than the imposed evaporation rate, a steeper pressure gradient will develop, causing high stress.

2.2. Elastic plate The elastic analysis presented in Part V [6] applies for rigid porous materials such as porous VYCOR *, or for gels dried so rapidly that no viscous deformation can take place. The differential equation governing the pressure distribution in the liquid in the gel is

where Kp is the elastic bulk modulus of the gel network (exclusive of liquid); the liquid is assumed to be incompressible (i.e., the bulk modulus of the liquid, K L --* ~c). The magnitude of the drying stress is found to depend on the parameter

tt-- IZEL.qL/DP R = ("E/JR.

(20)

Comparison with eq. (17) reveals t h a t / , has the value that a has in a viscous gel at the time (tR) when the meniscus enters the gel. In contrast to the viscous solution, which has no explicit time dependence, in the elastic gel the stress is predicted to rise with time following the start of evaporation of pore liquid.

G.W. Scherer / Drying gels VI

332

The maximum stress appears in the plate as the meniscus enters, and its value is [61

t~?ax(t)/CvPR

=

1 - t,0 R.

(21)

where C,-= ( 1 - 2 v ) / ( 1 - v)---1, v is the Poisson's ratio of the network, OR - t R / z is the reduced (dimensionless) time at which the meniscus enters, and r is defined by

r -- L2•L/DKp,

(22)

which is the time to flow a distance L under a pressure gradient of K p / L , according to Darcy's law [eq. (1)]. If the evaporation rate is slow enough so that ~t < 1.5, then/~ and OR are related by [6] --- 3 / ( 1 + 30R)

(23)

SO that eq. (21) is identical to eq. (13a). When /~ is larger (# > 1.5), /* = ( ~r/40R ),/2

(24)

SO eq. (21) becomes °max(L) = 1 - rr -- 1 - --rrJR C, PR 41" 412E '

(25)

which is the same (within the accuracy of the approximations used) as eq. (18). Thus the elastic and viscous analyses give the same predictions for the maximum stresses during drying and fig. 3 applies to elastic gels as well as viscous gels, if the abscissa is changed from a to /*. Eq. (25) explains the fracture of fresh gels that are suddenly exposed to the atmosphere: the rapid evaporation causes tensile stress at the exterior surface that approaches PR (which may amount to several megapascals), so cracks tend to run perpendicularly to the exterior surface. The elastic solution [6] could apply to an alkoxide-derived gel only if drying were sudden (in which case ox ~ Pp, and fracture is almost inevitable), or at the end of drying. In the latter case, when the viscosity diverges and the gel becomes rigid, the liquid has a pressure distribution established while the gel was viscoelastic. Therefore the stress development at that stage may not obey eq. (21). In the next section, a viscoelastic analysis is presented that illustrates the transition from elastic to viscous behavior. It is shown that the preceding analyses provide upper bounds on the actual stresses, but that the viscous solution is generally a reasonable approximation.

3. Viscoelastic analysis with constant properties 3.1. Derivation of pressure distribution We now consider a simple viscoelastic model of a gel in which both the shear and dilatational relaxation moduli can be represented by Maxwell

G. I,E Scherer / Drying gels V1

333

Kp KG(t) Fig. 4. Maxwell element, consisting of spring with modulus Kp and dashpot with viscosity K~, used to represent the viscoelastic response of the gel to hydrostatic stress. In section 3, K G is assumed to be constant, while in section 4 it is allowed to vary with time.

elements (see fig. 4). For a viscoelastic material the dilatational (volumetric) strain, c, can be written as [9] c ( t ) = 3%(t)

=foJ2(t_t,)_~dt,+o(O)J2(t) , ' ~o

(26)

where % ( 0 is the syneresis strain at time t, o is the dilatational (hydrostatic) stress, o = ox + Oy+ oz, and J2(t) is the dilatational compliance. If the properties of the gel are described by the model shown in fig. 4 with Kp and K G constant, then Jz(t) = 1/3Kp + t/3K~ (27) so that the response to a constant stress consists of an instantaneous elastic strain (% = o/3Kp) and a viscous strain that increases in proportion to time (~v = ot/3K~). In the present case, o is the mean stress on a cross-section of the gel network, which is equal and opposite to the mean stress in the liquid, P; that is, o(t)= - 3 P . Using eq. (27) and taking the time derivative of eq. (26) leads to i ( t ) = 34s(t )

/5(t) gp

P(t)

(28)

KG "

The superscript dot indicates the partial derivative with respect to time throughout this paper. We now proceed as in the viscous [1] and elastic [6] cases. Let the (liquid-filled) pore volume in a region of the gel be Vp and the volume of the incorporated liquid be VL. Assuming that no bubbles are nucleated, the change in pore volume must equal the change in liquid volume, so

l?p/Vp =

~'L/VL.

(29)

It was shown in Part I that

l?p/Vp = ~/(1 - y ) ,

(30)

G. IV.. Seherer / Drying gels VI

334

where y is the relative density of the network; if p is the bulk density of the gel network (exclusive of liquid) and Ps is the skeletal density, then y -- O/Os. It was also shown in Part I that D

L/VL

--y) V2P.

(3 0

Substituting eqs. (28), (30), and (31) into eq. (29) leads to the differential equation that governs the pressure distribution in the pores of a viscoelastic gel:

v2e -

( ~L / P

-g2S !

3BLi"

BE 0P

+ - - 5 - = DKp at

(32)

By solving eq. (32) we obtain the pressure distribution in the liquid in the gel, and that result can be substituted into eq. (28) to find the strain rate at any point in the gel. Before proceeding with the solution of eq. (32), let us examine the viscoelastic model more closely. The Maxwell model shown in fig. 4 gives the simplest possible viscoelastic response; it would be more realistic to use several such models in parallel, so that eq. (27) would include a delayed elastic component consisting of one or several exponential terms [9]. The latter could be important for describing the response of a body to cyclical loading and unloading, because "cross-over" effects may occur in such cases [ref. 9, p. 46]. However, for the slowly increasing load imposed by the capillary pressure on the gel, the simpler model is sufficient to illustrate the principal effects of viscoelasticity. A seemingly more important limitation is the assumption that the viscosity is constant (i.e., the viscosity, K G, of the dashpot in fig. 4 is not dependent on time, which is to say that it does not vary with the relative density of the gel). This assumption is relaxed in section 4, but it is shown to be of relatively little practical importance.

3.2. Solution for constant flux The pressure distribution in the liquid is found by solving eq. (32) subject to the appropriate boundary conditions. We now examine a plate of thickness 2L with faces at z = L and z = - L ; it is assumed that the external conditions (temperature, humidity, draft) are arranged to achieve a constant rate of evaporation, I)'E, from each surface. Therefore, the boundary condition is /';'E = D d_ff_PI BL dz z=L

(33)

As explained in Appendix A, this is equivalent to the boundary condition used in the viscous case [1], although it looks different. This condition can apply only until the pressure in the liquid reaches its maximum value, PR at time tR. At that point the meniscus enters the gel, and the appropriate boundary

335

G. IV. Scherer / Drying gels VI

condition becomes P(L, t)= PR. In this paper we consider only the time period t < t R, during which the stress rises to its maximum value. If the z- and t-dependence of K C, D, and ~/L are ignored, eq. (32) can be solved using the Laplace transform, as shown in Appendix B. The result is

P(z,

q)) = 3KoCh(1 -- e -~) +

(KGl)Eet)fo°H(z L

~') dqS',

(34)

where

H(z, q,)-

~ (exp[-cn(z)/q,] +exp[-G(z)/q,]}

(35)

n=O

and the position-dependent functions are c,,(z) = (,~/2)2(2n + 1 - z / L ) 2

(36)

d,(z)

(37)

and = (c~/2)2(2n + 1 +

z/L) 2

The reduced time, qS, is defined by

O-~Kpt/KG = cot.

(38)

The viscoelastic dilatational relaxation time for the gel is % KG/Kp, so the reciprocal of that time, co = 1/%, is the relaxation frequency. The linear free strain rate of the gel, ~f = ~//3, is obtained by substituting eq. (34) into eq. (28), with the result, =

(39)

ir = - (('E/3L)F(z, ~,), where

F(z, q~)-a(H(z, q~)+ fo*H(z,

q/) dq~').

(40)

The quantity ~r is called a free strain rate, because it represents the strain rate that would result from the pressure in the pore liquid, if the neighboring portions of the gel did not interfere. That is, the actual contraction rate of the gel is influenced by the fact that any local region of the network is bound to other regions that are contracting at different rates; it is this difference in local contraction rates that generates drying stresses. It is shown in Appendix C that eq. (39) reduces to the elastic solution [6] at very short times (cot << 1) and to the viscous solution [1,2] at long times (cot ---, ~ ) . It was shown in Part II [2] that the strain rate in the plane of a drying plate, G, is equal to the volume-weighted average of the free strain rate in the plate; the same is true in the viscoelastic case, as shown in Appendix D:

1 rL

G

G = 2-L J-L~f dz = - 3--L"

(41)

336

G. ~ Scherer / Drying gels I,'1

The viscoelastic stress analysis, presented in Appendix D, assumes that the shear relaxation behavior of the gel network can also be modelled by a Maxwell element. The result is

Ox(Z, t ) = 3CvKpfot('x-'f)exp(- fto~' d t " ) d t ' , at'

(42)

where w' is the biaxial stress relaxation time. To minimize the number of parameters, it is assumed that o~' and the dilatational relaxation time, ~, are equal. This is a very good approximation, as explained in Appendix D. Eq. (42) indicates that the stress results from the difference between the actual contraction rate of the plate, i x, and the local free strain rate, if. Substituting from eqs. (39)-(41) into eq. (42), with the assumption ~ = o~', we find

Ox(Z, q~)= C~PR~k~(z, q~),

(43)

where

~,(z, q~) = af~'H(z, ao

~') d q , ' - (1 - e - * )

(44)

and

q~= KGI?E/ PR L = i~/et2.

(45)

The physical meaning of ~k is easy to see. If the plate has area A and evaporation is occurring from both faces, then the rate of volume decrease by evaporation is 2AI?E; since the volume of the plate is 2AL, the volumetric strain rate is ("E/L, if the meniscus remains at the exterior surface of the plate. According to eq. (2), the mean pressure that must be applied to force the gel network to contract at that rate is ff = KGI?E/L, SO ~ = P/PR. Obviously, the mean pressure cannot be greater than PR, SO when a is small (so that the pressure is uniform) the meniscus must enter before ~b > 1; when a is large, the pressure near the surface is much greater than the mean pressure in the liquid, so the meniscus must enter when ~p << 1. This relationship between a and ~p is illustrated in fig. 2 for a viscous gel. The situation is a bit more complicated for a viscoelastic gel, because some of the strain is accommodated elastically, so a higher value of + may be required to drive the meniscus into the gel (as explained below).

3.3. Sample calculations In this section we evaluate eqs. (40) and (44) to show the variation with time of the free strain rate and the stress in a drying plate of viscoelastic gel. Fig. 5a shows F(z, q0, a quantity proportional to the free strain rate, for a gel with et = 3. Initially the strain rate is very high at the exterior surface (z = L), because the gel responds elastically to the redistribution pressure; at this point the strain is still small, so the stress is small. As indicated in Appendix C,

G.W. Scherer / Drying gels VI

337

12-

0 = 0.03

/

10-

8" A -O-

I (x=3

0.1

6-

N" ,..,*

L.

/ 0.3

4.

2oo

0

a

0

012

0,4

0.6

0.8

1.

z/L 2.5 =oo

2.

1 1.5

0.3 0.1

"~" 0.5

0.03

O. -0.5

b

"1'o

o12

o14

ols

o18

1.

z/L Fig. 5. (a) The function F(z, ~,), defined in eq. (40), is proportional to the free strain rate as shown in eq. (39). It is plotted as a function of position in the plate, where z = L at the surface and z = 0 at the midplane of the plate; a = 3. Curves are shown for several values of the reduced time ~,, which is defined in eq. (38); for ¢ > 1, the curves approach the shape [Fv(z ), defined in eq. (46), labelled oo] predicted from the viscous solution. (b) The function %(z, (~), defined in eq. (44), is proportional to the stress, as shown in eq. (43). It is plotted as a function of position in the plate, as in fig. 5a; for ¢, > 1 , the curves approach the shape [-Yv(z), defined in eq. (47), labelled oo] predicted from the viscous solution.

G.W. Scherer / Dryinggels 111

338

before the reduced time (q~ = ~0t = t/¢v) reaches unity, F(z, cO) approaches the form, Fv(z), predicted by the viscous analysis (eq. (73) of ref. [1]) *:

/~(Z, I~)),~1)

a cosh(ctz/L) sinh(a)

"-~/'v(Z).

(46)

As shown in Appendix C, F(z, cO) becomes identical to Fv(Z ) as q~--* ~ . Similarly, as shown in fig. 5b, the stress distribution approaches the value, ~;v(z), given by eq. (23) of Part II [2]:

Z(z, ep)~

a cosh(az/L) _ 1 = Zv(Z).

(47)

sinh(a) Thus, the viscoelastic relaxation occurs within a period on the order of K G / K p, which is not a very long time. For example, preliminary data on alkoxide-derived silica gels [10] indicate that the viscosity shortly after gelation is < 1011 Pa s and the elastic modulus is a few megapascals, so the viscous solution would apply after a few hours of drying, whereas the total drying time is more than a week for a gel several millimeters thick. Qualitatively the same result is obtained, regardless of the value of a. Of course, if the rate of evaporation is very high, the liquid/vapor interface will enter the gel before viscoelastic relaxation is complete, so the stresses would be more accurately predicted by the viscoelastic solution than by the purely viscous solution. This situation will exist if the time at which the meniscus enters, tR, is not much larger than the viscoelastic relaxation time, %. The pressure in the liquid at the surface of the plate is P(L, ~ R ) = PR when the meniscus enters the gel, where q~R -= ~°tR. Recognizing this, we can use eq. (34) to obtain a relation between q~R, a, and the evaporation rate parameter ~ [defined in eq. (45)]. Neglecting the syneresis strain rate, the result is q~= (af0q'RH(L , ~') dq~') -1

(48)

This relation is illustrated in fig. 6a for several values of a. For a given evaporation rate (~k), the larger is a, the earlier the meniscus enters the gel. That is true because the liquid does not flow as easily from the interior when a is large, so evaporation rapidly dries the surface region. For a given value of a, the faster the evaporation rate the earlier the meniscus enters. We have seen that when 4~> 1 the pressure in the liquid approaches the distribution predicted by the viscous solution; therefore when q~R > 1, q, approaches the value given by eq. (14). The stress at the surface of the plate at time ~R is found from eqs. (43) and (48) to be

ox( L, dpR)/C~P R = N( L, dpR)/a fq'RH( L, ~b') dq~', Jo

(49)

* The equation in ref. [1] looks different, because the surfaces of the plate were taken to be z = 0 and 2L, rather than z = + L as in this paper.

G.W. Scherer / Drying gels VI

339

4

3.

~=0.1

2

a

I

o

I

0.5

1.0

I

I

1.5

I

2.0

i

2.5

3.0

~R

ii -'-"-

1.0-

I

J

/

0.8-

/

/

/

0,6

%(L, %)

cv~ 0.4

IIII / I

i! /

0.2

0.3

/

/

/

0.1

1

2

3

4

5

Fig. 6. (a) Plot of magnitude of ~p-= K G V E / P R L at reduced time CR -= wtR when meniscus enters gel, for several value of the parameter a defined in eq. (6). The lower the evaporation rate, the later the meniscus enters; the larger a, the easier it is to dry the surface region, so the meniscus enters sooner. (b) Stress at surface of plate, at moment when meniscus enters, versus tp for several values of a. Dashed lines represent prediction of viscous solution, eq. (12).

where Z is given by eq. (44). From eqs. (48) and (49) we can determine the relation between ~ and the stress when the meniscus enters; the result is illustrated in fig. 6b. It is shown in fig. 5 that when ¢ > 1 the stress distribution approaches that given by the viscous solution [1], and at earlier times the stress is smaller. This is also indicated in fig. 6b, where the dashed

340

G. W. Soberer / Drying gels VI

lines represents the stresses predicted by the viscous solution [2], eq. (12); i.e., the slopes of the lines equal a coth(a) - 1. When ~k is small (slow evaporation), before the meniscus enters the gel, the stress distribution approaches that given by the viscous solution. For larger ~b (faster evaporation) the stress is lower according to the viscoelastic solution, because some of the differential strain is accommodated elastically. For a > 1, numerical evaluations indicate that the deviation from the viscous solution occurs when ~k> 1 / a . According to eq. (12) this means that deviations from the viscous solution occur when ° x ( L ) > coth(a) - 1 / a = 1 - 1 / a C~PR

( a > 2),

(50)

which is a very large stress when a > 2. Therefore, the viscoelastic effects will be important only when the evaporation rate is so fast that the stress approaches the capillary pressure. These calculations are rather artificial, because neither the viscosity nor the elastic modulus is allowed to increase with time. In the next section, the analysis is generalized to allow for an increase in viscosity. As we might expect from the results of this section, the transition to viscous behavior happens so early in drying that the increase in viscosity is not important. The casual reader is advised to skip to the discussion in section 5.

4. Viscoelastic analysis with increasing viscosity In the ensuing analysis we allow the bulk viscosity, K~, to increase with time, as it would during drying. However, we must assume (unrealistically) that the elastic bulk modulus, K e, remains constant, because otherwise the response would be nonlinear (i.e., the Boltzmann superposition principle - the convolution integral in eq. (26) - would not apply). This is not a very serious weakness in the analysis, because we are principally interested in the time dependence of the stress development, which is dominated by the viscosity. 4.1. Solution for constant flux We seek a solution to eq. (32) that allows for K~ to be a function of time. This means that the Laplace transform is not applicable, but, as shown in Appendix E, the result is readily obtained by use of the finite Fourier transform. Two reduced time scales are required: O(t) - t/'r,

(51)

where ~" is defined by eq. (22), and ep(t) - fottO(t ") d t ' =

o ~ K~ ] d t ' ,

(52)

341

G. t4/, Scherer / Drying gels VI

where the viscosity is now a function of time, K G = KG(t ). The pressure in the liquid is

+2

---L--

n=l

or, neglecting the syneresis strain rate and using eqs. (38) and (45),

P(z,t) PR

o~+ I o ( t ) + 2 ~ ( - 1 1 " c o s ( - - - ~ ) I n ( t )

(54)

n=l

where the integral In(t) (n = 0, 1, 2 .... ), is

In( t ) - fo'eXp( - f;w( t") dt"

n2erz( t- t') )

= f0'exp(-[,t,(t)-,t,(t')]-n2~r2[O(t)-O(t')])

dr'.

(55a,b)

The linear free strain rate of the gel is found from eqs. (28) and (54):

if = - ( I ? j 3 L ) [ ' ( z ,

t),

(56)

where P(z,t)-=l+2

(-1)"cosi---~--

1-

I,(t)

.

(57)

n=l

The stress in the plate is given by eq. (42) with the mean strain rate of the plate given by eq. (41) and the free strain rate by eq. (56). To evaluate the stress we must make some assumption about the variation in K G with time. Let us assume that K~ diverges according to

KG(t ) = KG(0)/(1 -- t/ts),

(58)

SO that ts is the time at which the viscosity becomes infinite and shrinkage stops• Then the relaxation frequency is

w( t ) = Kp(1 - t / t s ) / K G ( O ) = o~0(1 -- tits)

(59)

so the initial value is ~(0) -- o~0. The reduced time from eq. (52) becomes ~(t)=q~o(/)(1

~b°(t))2q~s,

(60)

where q~o(t) - ~0ot and q~s- Wots; t~ is also a function of time, since it depends on KG,

a ( t ) = Cto[1 - qJo(t)/eps] a/2

(61)

and we have defined its initial value as cto. Although a goes to zero at t~, the

G. W. Scherer / Drying gels VI

342

rising viscosity slows shrinkage and allows the meniscus to enter the gel at t < ts, while a is finite. The reduced time, ~, stops increasing and relaxation stops when the viscosity diverges; the maximum value of ~ is ~(ts) = ~Oots/2,

(62)

which is half of the value it would have had at time t~ if K~ had remained constant. Substituting eq. (60) into eq. (54) leads to P(z, %)

PR

=

go(q,o) + 2 __ ( - 1 ) " cos -7--- g.(~o)

(63)

n=]

where g.(~o) =- foC~'°exp{ - [ u2 + 2D.(q~o) u] } du

(64)

and D.(~0)

- C. - C ~ 0 ( t ) ,

c.=

1+

,~o2 ,

C - (2q~s)-1/2.

(65a,b,c)

From eqs. (57) and (60),

co (7)[1

(66,

n=l

Finally, the stress is found by using eqs. (41), (56), and (66) in eq. (42), with the result [ ncrz ~ , ( - 1 ) " cosl --~--) gn ( % ) ,

o x ( z , t)

(67)

n=l

where ~0 -

Ko (0) ~E/PRL,

(68)

which is the initial value of the function defined in eq. (45). 4.2. Sample calculation

Fig. 7 shows F(z, ~), a quantity proportional to the free strain rate that is given by eq. (66). It is assumed the a 0 = 3 and that the viscosity diverges when t s = 3/~00 = 3Tv. The viscous result (labelled Viscous) is given by eq. (46) with the value of a corresponding to q~o= 1; from eq. (61), a = 3 (~)l/e = 2.45. As in fig. 5a, the strain rate distribution approaches the value given by the viscous solution when q% ~ 1. It is evident that the increase in viscosity will not

G. W. Scherer / Drying gels V1

343

~0 0.03

10

~.

6

I~-

4

a

-2

0.2

0.4

0.6

0.8

1.0

z/L

2.s

%

2.

Viscous

/ 1.0

1.5

0.3

1. I~

=

0.1

0.5

0.03

0 -0.5 b

.1.

0.2

014

016

018

1'.0

z/L Fig. 7. (a) The function F(z, ¢), defined in eq. (57), is proportional to the free strain rate, as shown in eq. (56). It is plotted as a function of position in the plate, where z = L at the surface and z = 0 at the midplane of the plate; a = 3 initially, then decreases according to eq, (61.). The viscosity K c diverges at reduced time q's according to eq. (58); the reduced time obeys eq. (60); in this case, es = 3. Curves are shown for several values of ¢0 --- ~0t; for ¢0 > 1, the curves approach the shape [F v (z), defined in eq. (46), labelled Viscous] predicted from the viscous solution; F v(z) is calculated using the value a = 2.45 that applies at ¢0 = 1. (b) The function Y,(z, ¢), defined in eq. (67), is proportional to the stress. It is plotted as a function of position in the plate, as in fig. 7a; for ¢ 0 > 1 , the curves approach the shape [~v(Z), defined in eq. (47), labelled Viscous] predicted from the viscous solution. The parameters used in the calculation are the same as in fig. 7a.

G. 14/.. Scherer / Drying gels VI

344

3.

2.5

Viscous

1.49

~ / /

2.

1.12

1.5 1.

N"

v

0.5 0 -0.5 ,1. -1.5 0

0.2

0.4

0.6

0.8

1.0

z/L

3,

(~0

/

Viscous ~

2.97

2.5

2.

1.49

1.5 ~-

1.

0~0= 3

I~,1 0.5 0 -0.5 -1. b

-1.5 0

0.2

0.4

,

,

,

0.6

0.8

1.0

z/L Fig. 8. The function ~.(z, ~) is shown at times 25% and 99% of the way from ~0 = 1 to ~s; the curve labelled Viscous is calculated from eq. (47) using the value of a given by eq. (61) for ~0 = 0-99~s; results are shown for Os = 1.5 (a), 3 (b), and 10 (c).

prevent the development of the distribution characteristic of a viscous gel [Fv(z)], unless a substantial increase occurs before t = 1/600. Similarly, as shown in fig. 7b, the stress distribution approaches that given by the viscous

G. W. Scherer / D~ing gels V1

345

3.

2.5

9.90

Viscous . . ~ . ~ /

3.23 1.5

C

0.2

0.4

I

0

0.6

0.8

1.0

z/L Fig. 8 (continued).

5.

4

3

~o

= 0.1 2

3 o0

~

~ 0 5~

' 1.0

1 5~

2 0i

' 2.5

' 3.0

CO0 t R

Fig. 9. Relation between the initial value of g'o, defined in eq. (68), and the reduced time ~o = ~°otR at the moment when the meniscus enters the gel; calculated for ~ = 3 and several choices for the initial value of ~.

G. 14~ Scherer / Drying gels V1

346

solution [eq. (47)] when q~0= 1. Fig. 8 shows the development of stress for gels in which the viscosity diverges at reduced times of ~s = 1.5, 3, and 10. Only in fig. 8a is there any significant deviation from the solution for the purely viscous gel, but this requires a much more rapid increase in viscosity than is ever observed. That is, all of the examples in fig. 8 require the gel to become rigid in a period of hours, whereas alkoxide-derived gels are observed to shrink at a substantial rate for days or weeks. Therefore the viscous solution will generally be quite accurate for prediction of the drying stresses The maximum stress appears when the meniscus enters the gel at time tR, when the reduced time equals q~R-¢00tR. Since P(L, ~ R ) = PR, eq. (63) provides a relation between q~0 [proportional to the evaporation rate, eq. (68)] and fiR: ~o = C g0(*R) + 2

(69)

This result is plotted in fig. 9 for several values of a, and is seen to be similar to fig. 6a, which applies when the viscosity is constant. The curves in fig. 9 drift downward at long times because a(t) is decreasing according to eq. (61).

5. Discussion and conclusions

The elastic analysis in Part V [6] describes the drying of a rigid porous body that is initially fully saturated and stress-free. The stress is predicted to increase to a maximum by the time (t R) that the liquid/vapor meniscus enters the body, and to decrease rapidly thereafter. The stress rises again to a maximum as the saturated region shrinks and disappears. (Under some circumstances the liquid/vapor interface will become fractally rough after it enters the interior of the body [11], so the saturated region will have a complex shape. The stress distribution in that situation has not been determined.) The magnitude of the stress depends on the value of the parameter/t [defined in eq. (20)], and the maximum stress is given by eq. (13a) i f / t < 1.5 and by eq. (25) if # is larger. If the gel is viscoelastic, the stress rises as predicted by the elastic solution [6], and evolves into the distribution predicted by the viscous solution [2] within a period of time about equal to the viscoelastic relaxation time, %. If the evaporation rate is small enough [as defined by eq. (48) and fig. 6a] so that the meniscus does not enter the body before %, then the subsequent evolution of stress and strain are adequately described by the simple viscous solution. In alkoxide-derived gels, the relaxation time is expected to be short compared to the drying time, so the viscous solution will describe almost the entire drying process. In this case, the stress depends on the parameter a [defined in eq. (6)], and is given by eq. (13). If a is small, the stress is independent of the viscosity of the gel, and does not change when the viscosity rises and the meniscus

347

G.W. Scherer / Drying gels VI

enters the gel. If a is large, as expected for alkoxide-derived gels, then the stress rises with the viscosity, and reaches the value given by eq. (15) and shown in fig. 3. If the meniscus enters the viscoelastic gel before %, then the maximum stress will be less than predicted by the purely viscous or elastic solutions (see fig. 6b); however, this only occurs at high evaporation rates when the stress is almost equal to PR.

Appendix A. Equation boundary conditions The boundary condition used in eq. (70) of Part I [1] was

vF =

dz.

(A.1)

This equation means that the rate of volumetric contraction of gel is equal to the rate at which liquid volume is removed by evaporation. Using eq. (2) of the text, eq. (A.1) becomes

j:(') °d.

12E = -

3~ S - ~

dz.

(1.2)

For this to be equivalent to eq. (33) of the text, it must be true that nk dz

= - 3isL +

-~

(A.3)

dz,

where the first term of eq. (A.2) has been integrated assuming that the syneresis strain rate is constant. Integrating eq. (3) of the text once, recognizing that V'2P = d 2 P / d z 2 in this case, leads to d P z=L dz

d P ,=0 dz

=

3i,L~IL ~

~L f L p + O K G Jo - dz.

(A.4)

Symmetry requires that there be no flux across the midplane of the plate (i.e., J = 0 when z = 0), so the second term on the left side of eq. (A.4) is zero. Therefore eqs. (A.3) and (A.4) are the same, and eq. (A.1) is equivalent to eq. (33) of the text. To make the same comparison for a viscoelastic material, we substitute eq. (28) of the text into eq. (A.1) and set the result equal to eq. (33): ~L dz

=-3isL+

~

dz+

~

dz.

(A.5)

The identical relation is found by integrating eq. (32) of the text once, so eq. (A.5) is true, and eqs. (33) and (A.1) are equivalent for viscoelastic materials.

Appendix B. Solution of eq. (32) using the Laplace transform The Laplace transform can be used to solve eq. (32) only if the coefficients of P are not functions of time, so we must assume that D and K~ are

G. V~ Scherer / Drying gels V1

348

constants. The solution is more complicated than that obtained using the Fourier transform (discussed in Appendix E), but the sums converge more rapidly, so this solution is more convenient for numerical evaluation. The properties of the Laplace transform are discussed in ref. [9], p. 306, and a table of transforms is provided in ref. [12], p. 412. The Laplace transform with respect to time t, is defined by OO

[e(z, t)] = P ( z , s) = f0 e-Ste(z' t) dt,

(B.1)

where the constant s is called the transform parameter. The transform of the time derivative is [12, p. 413, formula 3]

~ aP] =/3(z,s)_ sP(z, O)=/3(z,s).

(B.2)

the second equality follows from the fact that the pressure in the liquid is initially zero. Applying eqs. (B.1) and (B.2) to eq. (32) of the text leads to

d2/3 A(s)P= -h(s),

(B.3)

dz 2

where

-

(s +

(B.4)

and, assuming that ~s is constant,

h (s) = 3is*lL/DS.

(8.5)

The solution of eq. (B.3) is h

/3(z, s)= -~ +c,(s) Cv~- + c2(s) e -z¢~-.

(B.6)

The plate is assumed to dry by evaporation from both sides, so P(L, t)= P(-L, t), which means that c 1 = c 2. Imposing the boundary condition given by eq. (33) of the text leads to

q(s) = ~'ECX s ~

+~

(eL~-e -L~) .

(8.7)

Using eqs. (8.6) and (B.7), the transformed solution can be written as

+/32,

(a.8)

where

3K,is (1 t 3 1 = s ( - s ~ o ) =3K°/s s

1 ) s+~0 "

(B.9)

G. W. Scherer / Drying gels VI

349

Inverting the transform in eq. (B.9) gives the first term on the right side of eq. (34) of the text. The other term is the inverse of

sf'2 = ( KGI)JL )(avr~ )G(z, s + oa),

(B.10)

where

~(z, s+oa)=(s+oa)-l/2( eNg +e z¢7 ) eL~--e

LO- "

(B.11)

This can be put into a more convenient form by expanding the denominator: (eeO - _ e-L¢~-)-I = e-LgX(1 + e - 2 L O + e-4CO- + . . . ) = ~ exp[--(2n + 1)LfA-].

(B.12a,b)

n=0

The parameter s appears in eq. (B.11) only as s + ~, so we can use the relation [12, p. 413 formula 11] G(z, s + oa) = £ [e-~'G(z, t)],

(B.13)

which means that we only have to find the inverse of (~(z, s). From eqs. (B.11) and (B.12),

G(z, s) = s -'/2 E ( e-~"¢~ + e-n"¢~),

(B.14)

n=O

where

A, = a(Zn + 1 - z/L)/v/-d, B. = a(2n + 1 + z / L ) / v ~ .

(B.15a,b)

The inverse of eq. (B.14) is [12, p. 419, formula 84]

G(z, t)=(rrt) -1/2 ~ [exp(-A2J4t)+exp(-B2/4t)]

(B.16)

n=O

so the inverse of eq. (B.10) is

P2( z, t ) = :

--KLVE)(av/-d)f0'e-'~/G(z , t' ) dt'

KG~E

,')d;

(B.17a,b)

where the function H is defined by eq. (35) of the text. Appendix C. Limiting case of the viscoelastic solution The viscous limit is approached as q~~ ~ , in which case H ( z , q~) -+ O. Then eq. (40) of the text becomes lim F ( z , cO) = a

/0

H ( z , q,') d , ' .

(C.1)

G. W. Scherer / Drying gels VI

350

From the definition of H, eq. (35), it is evident that eq. (C.1) is a sum of integrals of the form

I = f°°~ -'/2 exp[-fl/ep - ~1 d~,

(C.2)

J0

which can be evaluated [13, p. 340, integral 9 with v = ½, 2, = 1; definition of K1/2 on p. 967] to obtain

I = 2ff/4K,/2(2Vcfi) = f~-e-2V~.

(C.3)

Thus eq. (C.1) can be written as lira F(z, q,) = a E ( e-2 ~v~"+ e-2V~") ¢--*~

(C.4)

n=0

or, using eqs. (36) and (37) of the text, lim F(z, ¢p) = a(e az/L + e -~z/L) ~ e -~(2n+l). ~'--' ~

(C.5)

n=0

Recognizing that the sum in eq. (C.5) is the expansion of 2 sinh(a), we find that lim F(z, q~) = F,,(z),

(C.6)

a cosh(az/L) sinh(a)

(C.7)

where rv(z) =

is the solution given by the viscous solution [2, eq. (16)]. The elastic limit is obtained as ~ ~ 0 so that e -~ ~ 1. In that case we find from eq. (34) that lira P( L, 8) ep"-~0

PR

n=0

=~l°(~rO) -x/z 1 + 2 Y~. e -"2/° dO. ao

n=l

(C.8a,b)

J

When the meniscus enter the gel at reduced time OR, the pressure in the liquid at the surface of the plate equals the redistribution pressure, P(L, OR) = PR, and eq. (C.8) furnishes the following relation between/~ and 0R: /~=--~-¢'~

+

°RO-1/2

e -n2/°

dO

.

(C.9)

n=l

Numerical evaluation reveals that eq. (C.9) is indistinguishable from eq. (20) of Part V [6], so the viscoelastic result indeed reduces to the purely elastic solution of short times.

G.W. Scherer /Drying gels VI

351

Appendix D. Viscoelastic stress analysis

We begin with the assumption that both the dilatational and shear response of the gel can be modelled by Maxwell elements (shown in fig. 3), so that the dilatational compliance is

J2(t)

1

t

= - ~ p + 3K--"~,

(D.1)

and the shear compliance is 1 t J , ( , ) = -~p + 2-~o,

(D.2)

where Kp and Gp are the elastic bulk and shear moduli of the gel network, respectively, and Ko and Ga are the bulk and shear viscosities of the gel. The moduli can be written in terms of the Young's modulus (E) and Poisson's ratio (v) of the network as E Gp 2(1 + v ) ' E Kp - 3(1 - 2 v ) '

(D.3a,b)

and we can write analogous relations for the viscosities in terms of the uniaxial viscosity ( F ) and Poisson's ratio (N) of the viscous network: F F G~ 2(1 + N ) ' Kc 3(1 - 2 N ) " (D.4a,b) The quantities N and v differ in that the Poisson's ratio of the viscous network (N) approaches ½ as the porosity goes to zero, while v approaches the elastic Poisson's ratio, which is about half as large. The Laplace transform of the constitutive equation of a viscoelastic material can be written as [9, p. 78]

where 3 s(2,~ + ,~)

(D.6)

"~-.--~= .

(D.7)

and

2Jl + Jz

The tilde indicates that the Laplace transform has been applied with respect to time (see Appendix B). When eqs. (D.1) and (D.2) are used in eqs. (D.6) and (D.7), eq. (D.5) can be written in terms of the moduli and viscosities; inverting the transform leads to

~x=ct+F-1[ox-N(o~.+o~)]+E-1[dx-v(6v+dz)].

(D.8)

G. IV. Scherer / Drying gels 111

352

In the plate, symmetry requires that % = oy; since there is no constraint normal to the plate, % = 0, so eq. (D.8) reduces to ]--P

,

(D.9) There can be no net force normal to the free edges of the plate, so y

a~ dz = -~

OX dz = O.

(D.10)

By integrating eq. (D.9) and using eq. (D.10), we find that the strain rate of the plate is the average of the free strain rate:

lfoL

ix = ~

i, dz.

(D.11)

The linear free strain rate is found from eq. (28) of the text,

P i, = i / 3 = g s

P

3Kp

3K G "

(D.12)

The easiest way to evaluate eq. (D.11) is to take the Laplace transform of eq. (D.12), S~'f

is s

s/5 3Kp

P 3KG

is s

( ^~. ] ( s + co) \ 31~p ]

(D.13)

and of eq. (D.11), ix

s=Z

1

f0Ls•f dz.

(D.14)

Then, using eqs. (B.8)-(B.10), we find that (D.15)

i x = - I?E/3L.

Eq. (D.9) can be written as dx + W'Ox = (1E-~_ ~ )( i x -

,f ) = 3C~Kp( ix - if ),

(D.16)

CNKG .

(D.17)

where co'=-

T-77_v

The solution of eq. (D.16) is eq. (42) of the text. The biaxial relaxation frequency, co', differs from the dilatational relaxation time, co, by the factor C J C u. Given the high porosity of the gel network, both v and N are likely to be < 0.1, so C, and C u will both be = 1 as will their quotient. Therefore we can approximate co' by to with a high degree of accuracy, and eq. (D.16) reduces to Ox = C , ( P - P ) ,

as in the elastic case [6].

(D.18)

G.W. Scherer / Drying gels VI

353

Appendix E. Solution of eq. (32) using the finite fourier transform Instead of using the Laplace transform to remove time the Fourier transform to remove z as a variable. Let us that the interval of interest changes from 0 < z < L to finite Fourier cosine transform is defined by [12, p. 405]

as a variable, we use define x = ~rz/L, so 0 < x < ~r. Then the *

¢t

~.[P(x,

t)]

- f t . ( t ) - f P(x, t)

cos(nx) dx

(E.1)

go

and the transform of the second derivative is [ OP2 ] =

~ [--~x2 ]

-n2Z -

~-

x=0

" oP

+(-1)-~x

~=,"

(E.2)

Since symmetry requires that there be no flux across the midplane of the plate (x = 0), the second term on the right side of eq. (E.2) must be zero. The third term is determined by eq. (33) of the text: ~-~ x=,~

I?E~LL D~r

(E.3)

Thus the transform of eq. (32) is a---~-+ 1 +

~

/3,=h,

(E.4)

where h = 3Ko s

(E.5)

+

and 6=

It 0

n =0 n:g0.

(E.6)

The solution of eq. (E.4) is

P, =f-' foe°hfdO',

(E.7)

where

+

2] ao)_- ex. + : 20,

The inverse transform of the pressure is [12, p.405] P ( x , t) = --/3°+ _2 ~ /3 cos(nx).

(E.9)

n=l

Substituting eqs. (E.7) and (E.8) into eq. (E.9) leads to eq. (53) of the text. * Note that there is an error in eq. (2) of this reference: the minus sign ( - ) before the sum should be a plus sign ( + ) .

354

(7.145. Scherer / Drying gels VI

We now show that this solution reduces to the correct results at short and long times. If K~ is constant, eq. (53) becomes p = 3K~4s(1 - e-~) + ~

1-e-*+2a

2~

( - 1 ) " cos(n~rz/L)

n=l

n 2 ~ 2 -1- 0¢2

× (1 - exp[-(n:~r 2 + a2)0] ).

"'~

(E.10)

The elastic limit is obtained when K G ~ oo, in which case a, w and q~--*0. Noting that K~ (1 - e-*) ~ KGwt = Kpt

(E.11)

Ko a2 = L2~L/D

(E.12)

and and recognizing that [12, p. 402] ~r---~, = , 2~ ( _ 1).n_Z cos(~_z) = 12(L)2

6'1

(E.13)

eq. (E.10) becomes p---~-=/t 0 + ~ ( L )

6

~r2 =l

~

expt--ff-)exp(-n2~r20) ' (E.14)

which is the same as the elastic solution given in eq. (16) of Part V [6]. The viscous limit is obtained when Kp ---, oo, in which case ~o and q, ~ oo. Then (since a20 = q, ~ oo) eq. (E.10) becomes

From eqs. (68) and (71) of Part I [1] we expect the viscous solution to be P = 3 K j s + (KGI;'JL)Fv(z),

(E.16)

where Fv(Z ) = a cosh(az/L) sinh(a)

(E.17)

If we define 2 L i , = ~ f ° c o s h ( ~ ) c o s ( ~ -~) d z = 2 ( - sinh(a)n27r 1)"a2 + a 2

(E.18)

2 c i o = - [ f ° cosh(--~)dz = (2)sinh(a),

(E.19)

and

G. IV. Scherer / Drying gels VI

355

then the Fourier expansion of F~(z) is [12, p. 399] F~(z)=

t o/[ sinh-(a)

½i°+ ~ i ,

cos(~

,

(E.20)

n=l

which is the same as the quantity in braces in eq. (E.15), so the viscoelastic solution does reduce to the viscous solution at long times. Eq. (56) and (57) reveal that all of the z-dependence of the free strain rate is contained in the cosine terms, which go to zero when averaged over the interval 0 < z _< L. Therefore when eq. (56) is substituted into eq. (D.11) the result is eq. (41).

Notation

Variable

cl(s), c2(s) c,(z) d,(z) f g g. h

Defining equation (B.6, (B.7) (36) (37) (E.8) (E.17) (64) (E.5)

h(s)

(B.5)

s

(B.1)

t tR

t~ y

(58) (30)

A(s) A.,B. C

(B.4) (B.15) (65) (65) (12) (21) (1) (65) (D.3) (D.4), (D.8) (Kll) (D.4) (35) (C.2)

G CN

G D

Do E F

d co H(z, ep) I

Explanation

Laplace transform parameter time time at which liquid/vapor interface enters gel time at which viscosity diverges relative density of gel network, exclusive of pore liquid, y = P/Ps

CN = (1 - 2 N ) / ( 1 - N) C~ = (1 - 2v)/(1 - ~) permeability Young's modulus uniaxial viscosity of gel network shear viscosity of gel network

356 in

io

I.(t) J

JR

G. W. Soberer / Drying gels VI

(E.18) (E.19) (55) (1) (16)

n = 0 , 1,2 .... liquid flux flux produced by pressure gradient

PR/L

J,(t) 4(0 Kp K1/2

(D.2) (26), (D.1) (2), (D.4b) (19), (D.3b) (C.3)

L LE

(lo)

half thickness of plate effective width of pressure gradient in liquid

(D.6) (D.4), (D.8) (D.7)

Poisson's ratio of viscous network

KG

./g N .A/" P PE

(E.1) (4)

shear compliance dilatational compliance bulk viscosity of gel network elastic bulk modulus of gel network

pressure in liquid in pores of gel finite Fourier cosine transform of P pressure in liquid at surface, PE =

P(L, t) redistribution pressure (maximum tensile pressure attainable in liquid), Prt

PR

= (Vsv - VsL)Sp/V.

/31

(B.8), (B.9) (B.8), (B.10)

V

(2)

v~

(29)

a(t)

(6), (9), (11) (61) (C.3) (E.6)

Oto 6 YLV

YSL YSV Jr 4s ix "OL 0

(z) (39), (56), (D.5) (41), (D.5), (D.8) (1) (51)

surface area of pore space in gel volume of gel rate of evaporation of pore liquid volume of liquid in region of gel volume of pore space in gel initial value of a(t)

liquid/vapor interracial energy solid/liquid interfacial energy solid/vapor interracial energy volumetric strain rate free strain rate of gel syneresis strain rate strain rate in x-direction (in plane of plate) viscosity of liquid in pores of gel reduced time in elastic solution

G.W. Scherer / Drying gels VI

OR

(20) ?)

(21), (D.3)

o os %,%,o:

(12)

r,

(22) (8)

rj

(7)

T

357

reduced time t R / r when liquid/vapor interface enters gel evaporation parameter in elastic solution elastic Poisson's ratio of gel network bulk density of gel network, exclusive of pore liquid skeletal density of gel network components of stress in the x, y, z-directions (mean stress on cross-section of gel, not stress concentrated in solid phase) characteristic time in elastic solution time to achieve volumetric strain of unity under pressure P time to flow distance L under gradient P/L

dilatational relaxation time,

rv

rv = Ko/K

q, % ePR

(38), (52), (60)

q's

(60)

q, ~o

(45) (68)

(60) (48)

60

(38)

600 l0 t

(59) (42), (D.17)

V' Vr2

dilatational relaxation frequency; timedependent if K G is, as in eq. (59) initial value of 60(t) biaxial relaxation frequency gradient operator Laplacian operator; for plate, V"2= d2/dz

F ( z, 'i') C(z)

P(z, ep) Fl(a, z)

2(z, ~,) Zv(z) 2(z, ep) superscripts: dot (") tilde .( - ) hat ( )

1, = 1/60

reduced time in viscoelastic analysis reduced time defined in terms of 60o reduced time 60tR when liquid/vapor interface enters gel reduced time 600t~ when viscosity diverges

2

(40) (46), (C.7) (57) (5) (44) (47) (67)

permeability factor

(B.1)

partial derivative with respect to time Laplace transform Fourier transform

(E.1)

358

G.W. Scherer / Drying gels VI

if]

(E.1)

9~ [f]

(B.1)

f i n i t e F o u r i e r c o s i n e t r a n s f o r m of f L a p l a c e t r a n s f o r m of f

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

G.W. Scherer, J. Non-Cryst. Solids 87 (1986) 199. G.W. Scherer, J. Non-Cryst. Solids 89 (1987) 217. G.W. Scherer, J. Non-Cryst. Solids 91 (1987) 83. G.W. Scherer, J. Non-Cryst. Solids 91 (1987) 101. G.W. Scherer, J. Non-Cryst. Solids 92 (1987) 375. G.W. Scherer, J. Non-Cryst. Solids 92 (1987) 122. P.G. Simpkins, D.W. Johnson Jr and D.A. Fleming, Am. Ceram. Soc. Annual Meeting, Pittsburgh, 1987, abstract 120-(3-87. P. Anderson and L.C. Klein, J. Non-Cryst. Solids 93 (1987) 415. G.W. Scherer, Relaxation in Glass and Composites (Wiley-Interscience, New York, 1986) p. 32. S. Pardenek, J.W. Fleming G.W. Scherer and R.M. Swiatek, to be published. T. Shaw, in: Better Ceramics Through Chemistry II, Materials Research Society Symposia Proc., Vol. 73, 1986, p. 215. W.H. Beyer, ed. CRC Standard Mathematical Tables, 27th ed. (CRC Press, Boca Raton, Florida, 1981). I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 1965).