Stress from re-immersion of partially dried gel

N

ELSEVIER

]OURNA

L OF

Journal of Non-Crystalline Solids 212 (1997) 268-280

Stress from re-immersion of partially dried gel G e o r g e W. S c h e r e r * Department o['Cicil Engineering, Princeton Materials Institute, Princeton UniLersit),, Eng. Quad. E-319, Princeton, NJ 08544, USA Received 2 October 1996

Abstract When liquid evaporates from a gel, capillary tensile pressure (suction) develops in the liquid and corresponding compressive forces are imposed on the solid network, leading to contraction, At any point up to the time when shrinkage stops, the pores remain full of liquid. If, during that stage, the gel is immersed into liquid, the capillary pressure is suddenly relieved at the surface of the body, so the compressive forces on the network are eliminated and the network tends to expand elastically. The expansion of the surface creates compressive stresses near the surface and tensile stresses inside the body. These stresses can be great enough to cause cracking. In this paper, the stresses caused by re-immersion of a partially dried cylinder of gel are analyzed. The stresses are shown to be greatest at the axis of the cylinder, where fracture is observed to originate. By reducing the pressure at the surface slowly (as by exposing the gel to a humid atmosphere), the stresses can be kept low, so that the partially dried gel can be safely re-immersed.

1. Introduction In an earlier study, rods of silica gel were allowed to dry to a certain extent, then were placed into a liquid bath and subjected to bending in order to measure the elastic modulus and permeability of the gel at various degrees of shrinkage [1]. Recently those experiments were extended [2] to include gels that had almost reached the critical point of drying (CPD), where shrinkage stops and air invades the pores. Gels that were dried near to the CPD were found to crack when immersed in liquid, even though the pores of the gel were still full of liquid. As indicated in Fig. 1, the origin of the fracture was at the centerline of the rod of gel, rather than at the exterior surface (where cracks typically begin during drying); the rods broke into segments of roughly equal length ( ~ 1.5 cm). Qualitatively, the cause of this cracking is as follows. The tension (negative pressure) in the liquid is balanced by compressive forces on the network that cause contraction; elastic strain energy is stored in the network, as in a compressed spring. When the gel is immersed and the l i q u i d / v a p o r menisci at the exterior surface are eliminated, the capillary pressure disappears at the surface, and that region of the network tends to expand. The resulting stress is analogous to that produced in a rod of glass whose surface is suddenly heated: the surface tries to expand, but it is resisted by the cooler interior, so the surface goes into compression and the interior goes into tension; if there is a flaw in the interior of the body, fracture can occur. Similarly, the elastic

* Tel.: + 1-609 258 5680; fax: + 1-609 258 1563; e-mail: [email protected]. 0022-3093/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S 0 0 2 2 - 3 0 9 3 ( 9 7 ) 0 0 0 2 8 - 8

G. W. Scherer / Journal of Non-C~stalline Solids 212 (1997) 268-280

269

(a)

(b)

Fig. 1. Photo of partially dried rod of silica gel that cracked when immersedin water. (a) The crack originatesnear the axis of the cylinder, where the axial stress predominates, and grows perpendicular to the axis (creating the flat surface near the center). The radial stress increases with radius, so the crack turns to become parallel to the axis; steps form on the crack front as it bends, creatingthe radial lines. (b) The cylinder of gel breaks into fragments of equal length; most fragments have convex ends, as in (a), but one has the mating concave end. Diameter of rod is 0.41 cm; diameter of flat region on axis is ~ 0.06 cm.

expansion of the outer surface of the gel network imposes stress on the interior of the rod and can cause damage of the type shown in Fig. 1. Fracture of a gel is actually somewhat different from fracture of a glass under thermal stress, because the network of the gel is initially in compression. When the capillary pressure is released, the network near the surface springs back toward its stress-free length, so the interior of the gel is 'stretched' relative to its compressed state, but it is not driven into tension. However, as shown in Ref. [3], if the gel contains a flaw that is large compared to the mesh scale of the network, then tensile stresses appear at the tip of the flaw, and that can cause fracture. This explains why the rod typically breaks into a small number of pieces (corresponding to the frequency of large flaws) when immersed; if the intrinsic strength of the network were exceeded, the gel would shatter. In this paper, the stresses produced by immersion are analyzed. If the gel is placed directly into liquid, the stress is shown to be nearly half as large as the capillary pressure initially present in the pores. However, if the pressure is reduced gradually (for example, by exposing the gel to a humid atmosphere), then destructive stresses can be avoided. In any case, the highest stresses occur on the axis of the rod of gel. The results of the analysis are shown to be qualitatively consistent with experimental observations.

270

G. W. Scherer / Journal of Non-Cr3,stalline Solids 212 (1997) 268-280

2. Experimental procedure The gels used in this work were 2-step acid/base-catalyzed silica gels of the type known as B2 [4]. The preparation, aging, and drying procedure is described in Ref. [1]. The final stage of aging was performed in a solution of 95 vol.% w a t e r / 5 vol.% ethanol; the alcohol was used to prevent growth of microbes. A rod of gel was removed from the bath and allowed to dry until it had shrunk to the desired diameter, then it was immersed in water and subjected to a beam-bending experiment [5,6] to measure its permeability and viscoelastic properties. The initial diameter o f each gel (i.e., the inner diameter of the mold) was ~ 0.78 cm; when completely dried, the final diameter was ~ 0.36 cm. The gels remained completely clear until shrinkage stopped, then became opaque due to light scattering from the irregular drying front [7] that developed as the liquid/vapor interface entered the pores; thus, any sample with a diameter > 0.36 cm was fully saturated. Any sample that had shrunk to a diameter less than ~ 0.47 cm was subject to cracking when placed into water; the gel shown in Fig. 1 is an example. To make the rewetting process more gentle, a small quantity of water was placed into a large beaker and a smaller beaker was inverted inside of it to provi_de a platform for the gel. The gel rod was placed on the smaller beaker (above the water line), the large beaker was covered with plastic wrap, then it was placed on a hotplate and heated to ~ 65°C. When droplets condensed on the gel (within 2 h), the rod was pushed off the smaller beaker into the hot water, and the hotplate was turned off. Once the beaker had cooled (to avoid thermal shock to the gel), the gel could be transferred into the bath for the bending experiments.

3. Analysis The analysis is based on the theory developed by Biot [8] for deformation of porous bodies containing liquid. The relevance of the theory to gels has been discussed by Chandler and Johnson [9,10], and its application to stress development in gels is discussed in Ref. [11]. The theory assumes that liquid transport in the gel obeys Darcy's law [12,13]: D J = --VP,

(1)

'I~L

where J is the flux of liquid, ~TL is the viscosity of the liquid, D is the permeability (which is related to the pore size and porosity of the gel network); P is the stress in the liquid, so it is equal in magnitude to the pressure, but differs in sign (i.e., P is positive when there is tension in the liquid). The gel network is expected to behave in a purely elastic manner, because the viscoelastic relaxation time for silica gels is so long and the crack velocity is so high [3]. In cylindrical coordinates, the elastic constitutive equation has the form

1 eZ=E

P [~-

v(°'r+ ~)]

3K'

(2)

where e z is the axial strain, crz, o r, cr0 are the total stresses in the axial, radial, and circumferential directions, respectively. The total stresses represent the sum of the forces on the liquid and solid phases per unit area; the network stresses represent the force on the solid per unit area, and are related to the total stress by [1 l] 6"k=O-g--(1--p)P,

k=r,

O, z,

(3)

where p = Pb/Ps is the relative density (volume fraction of solids); Pb and p~ are the bulk density and skeletal density of the network, respectively. The factor of 1 - p appears because P is defined as the force on the liquid per unit area of liquid (rather than the total cross-sectional area). The properties E, v, and K = E/[3(1 - 2 v)] are Young's modulus, Poisson's ratio, and the bulk modulus of the drained network of the gel, respectively; that is, they reflect the response that the network would exhibit to applied stresses if the liquid were not present.

G. W. Scherer / Journal of Non-Crystalline Solids 212 (1997) 268-280 ,

1.4

(a)

1.2:

, ,,,,,,J

,

i ,,,,,,

i

__

__.<--<--.%

, llllJll

I

o. o

/--:°2, --

1

o~

271

I

I

- -0~s

0.8 0.6-

v cL

0.4 0.2 0 0 -3

I 0 -2

1 0 "I

1

0°

0.4-0"6 (b)l i i iI L I I I ~ I ~ i i illll1 ".1 i i i qlll

°

=

~'--

0.2

- ........

-0.2~

-

/ -0.4--

/

-O.6-0.8

..__.. ~ , 0 .3

.......

i

. . . . . . . .

1 0 "2

i

I

. . . . . . .

0 "1

i

1 0°

6

Fig. 2, Stresses generated when partially dried gel is plunged into bath: (a) stress in liquid (P); (b) total axial stress in gel (o'z). Both normalized by initial stress (P0) and plotted as function of reduced time, 0 = t/'c. At the surface, P(I, 0) = 0 and o-z(1, 0) < 0 for t > 0. Legend indicates radial positions, u = r / R .

The continuity equation is [11,13] -

k =

V"

VP

,

(4)

where e = G + ~0 + e.- is the volumetric strain. For a long cylindrical gel (i.e., long enough so that liquid flows principally in the radial direction, and flow through the ends is negligible), when the permeability is uniform within the body, Eq. (4) becomes [14]

.0.

.

=N

+

- - -

00

(5)

u = r / R , r is the radial coordinate, R is the radius of the cylinder, 13 = (1 + v)/[3(1 - v)], and 0 = t/"c is the reduced time. The hydrodynamic relaxation time r is defined by T/L R2 7=

(6)

DH'

where H = (1 - v ) E / [ ( 1 + u)(1 - 2u)] is the longitudinal modulus of the network. The mean pressure in the liquid is ( P) = 2 f'P( Jo

u, O ) u d u .

(7)

G. W. Scherer / Journal of Non-Crystalline Solids 212 (1997) 268-280

272

(a) 0.25 0.20

(b) - - 0 . 0 -- -- -0.2 .....

0.15

0.4

--

N

0.30-

n

0 . 2 5 -

- -0.6

.....

0.20

0.8

0.10

~°

0.15-

~.

0.10-

0.05

~

,

I

,

,

,

I

+

i

,

I

1 0 "2

,

~

I

~

,

1--o=10,) I- - -o,1o,,,

/ ",,~ 0.2

0o

1 0 "1

0.4

0.6

0.8

e

(d) i

,

i

I

,

,

i

J

i

i

i

I

L

- -

i

i

]

i

,

o=(0.6)1

0.20

k

0.10 N ,2

t.~ ~

I

" -,. o.o5

- - -

" ~

II J¢

0.00 -

0.15 L= ~

,

0.00 0 "3

0.25

,

0.05

0.00

(C)

0.35

O.

0.10

o=10.8) -%(o.8)

o

-0.05-

0.05

~

-0.10-

o"

-0.15

0.00 0.2

0.4

0.6 5

0.8

0,2

0.4

0.6

0.8

0

Fig. 3. (a) Total radial stress (orr) normalized by initial stress in liquid (P0), plotted as a function of reduced time, 0 = t/'c. At the surface, = 0. The legend indicates the radial positions, u = r/R. (b)-(d) Comparison of axial and radial stresses at indicated radial positions; by the time ~ peaks (0 = 0.1), o-r > o-z for u > 0.6.

We seek the solution of Eq. (5) for a gel that has been partially dried, then immersed in liquid. If the drying rate has not been extremely fast, then the pressure gradient within the gel is not large (indeed, a large pressure gradient would cause cracking of the gel), so it is reasonable to assume that the stress in the liquid in the partially dried gel is approximately uniform. Thus we impose the initial condition P ( u , 0) = P0.

(8)

Assuming that the gel is then plunged directly into the liquid, the boundary condition is P(I, 0)=0.

(9)

As in Ref. [14] 1, the solution is obtained by use of the Bessel transform [15]. The result is

P(u, =1~-=' =~=, 2J°(B'U)[exp(-B~O)-(1-~)~o +(1-[3)B2I'(0)

(10)

io(o)-=

(ll)

where

(e---Z> d0' Po

There is a typographical error in Ref. [14]: The minus sign on the right side of Eq. (31) is extraneous.

G. W. Scherer / Journal of Non-Crystalline Solids 212 (1997) 268-280 ,,,I,,,I

1.4

,,I,,,1:,,I

273

,,,I,,,1,,,

P(O) ] _ _ -p(0.2)/ ..... P(0.4)] ----P(O.6)j ..... P(O.8)[

(a)

1.2-

02 1 . 0 : 0.8=

--

0,6:

-P(1) [

/P0]

'-.:.%:N..

0.4: 0.2: 0.0:

,

0

0.6

,i,r,i

0.2

,,i,,,~,,,i

0.4

' ' ' ' , , , '

....

. . . . . . . . .

0.6 0.8

1

i,,,

'=''''

1'.2 11.4

,,

.6

i,,,

]

~ ( b '

0.4

- - -o(o.2)]%(°)[ . . . . . az(0.4)/ ~ - - o=(0.6)J

~.o 0 . 2

. . . . . a(o.8)[

"-"

, _ .--"

---oz(1)

O" - 0 . 2 -0.45

/ J

/

-0.61 0

,,i,,,i,,,i 0.2 0.4

....... 0.6 0.8

t ~ , , i , , , i

1

1,2

,,,

1.4

1.6

9

Fig. 4. Stresses generated when stress in liquid at surface of partially dried gel is reduced to zero according to Eq. (18) with 3' = 10: (a) stress in liquid (P); (b) total axial stress in gel (o'=). Both normalized by initial stress (P0) and plotted as function of reduced time, 0 = t/r. Legend indicates radial positions, u = r/R.

The functions J0 and J= are Bessel functions of the first kind of order 0 and 1, respectively, and B, is a root of

Jo, Jo(Bn) = 0. The average stress in the liquid is given implicitly by P o = n~= l -fir,z, e x p ( - B : O ) + 4 ( l - fl ) ny" = ] In(O)"

(12)

It is k n o w n [16] that Y'.~= l(4/B.=) = 1. The axial stress in the gel resulting from P ( u , O) is given by [11]

% = C,(P-


(13)

where C, = (1 - 2 v ) / ( 1 - v). Eq. (13) represents the total stress, which is the sum of the forces on the solid and liquid c o m p o n e n t s of the gel [11]; this is the stress that causes p r o p a g a t i o n o f cracks [3]. W i t h Eqs. ( 1 0 ) - ( 1 2 ) , Eq. (13) b e c o m e s

cr

C,.

~ v"

2Jo( B , u )

[exp(-,,:o)

+ (1 -

8:,o(o)1

c, (P) /3 eo

(14)

The radial stress in the gel is given by [11] 1

g = 7[,,

o

o-:(u)udu.

(15)

G. W. Scherer / Journal of Non-Crystalline Solids 212 (1997) 268-280

274

i. ., .,.i.,.,. , J , , , i , , , i , , , i , , , i , , , J , , ,

1.2 1.0

o."

~,~

.

.

.

.

.

(a)

P(0) -- -- - P(0.2) P(0.4)

.

--

0,8

- - P(O.S)

-. .-. . .

\ *",;('~',~

\

¢D

0,6

0.

0.4 ~

-P(1)P(0"8)

+'+°

0.2 5 0.0 0

0=.2 0 . 4 OF.6 0 . 8

. . . . . . . . . . . . . . . . . .

1, . . . .1. + 1

...4. .

i.6

0.4 %(0)

0.3'

a.o

_

_

0.2-

.....

0.1

-.....

0

--

-0.1.

.0.2- \

- ~z(0.2)

Oz(0.4 ) - - % ( o . e) o" ( o . 8 ) -Oz(1

)

/

\/ -0.4

O' ' ;12' Oi 4 ' Oi" r ;i"+

' 1 I I '1 12' i14' 1. 6

6

Fig. 5. Stresses generated when stress in liquid at surface of partially dried gel is reduced to zero according to Eq. (18) with y = 3: (a) stress in liquid ( P ) ; (b) total axial stress in gel (O-z). Both normalized by initial stress ( P 0 ) and plotted as function of reduced time, 0 = t/'c. Legend indicates radial positions, u = r / R .

From Eq. (14) we find

2JI(B"U)[exp(-B~O)+(1-~)B~I"(O)]+C"B~JI(B.)

e--~°" r = ~-~C~'y=I

' }- - -~

P0

(16)

The circumferential (hoop) stress is [1 l] % = o-~ - ~ .

(17)

Suppose that the gel is not plunged directly into the liquid, but instead that the humidity is gradually raised so that the liquid/vapor menisci gradually flatten and the tension in the liquid is steadily reduced over a period of time. That procedure could be described by a boundary condition such as P(1, 0 ) =

Po(1-yO), 0<0<1/3,

0,

l/y<0

(18)

If Eq. (9) is replaced by Eq. (18), then in place of Eq. (10) we find

Po

f(u,O)-

P----~+ ~

=1B,J](B,)[exp(-B~O)+B~I~(O)],

(19)

G. W. Scherer / Journal of Non-Crystalline Solids 212 (1997) 268-280 ' .2 1

............... ~ ~~ . "

00:

a.o

o.5~

(a)

P(O)

~ -':'J~

" "'."','~ \':.q'N,

---'p(° 6)l ..... .(o5,,

--"P(') -P-',0~[

".",":"N

N",~,.N ",',7:-\ \',£%

oo 0

0.5

1

1.5

o.ts~---%(o2)]j 0.20

K3.°

~. o

~

- - -- - P ( O . 2 ) [ ..... P(0.4) ~

Q.

275

....

.....

i

,

,

,

,

I

,

0.10

°z(O)

.....

0.05 0.00

.....

-0,05

~'"

.............

-0.15

~

-0.20

....

~

~ .... 0,5

"''f""

I

az(O,4)/ -

- Oz(O.6) i G (0.8)

--"az(1)

/ ~ .... 1

~ .... 1.5

6

Fig. 6. Stresses generated when stress in liquid at surfaces of partially dried gel is reduced to zero according to Eq. (18) with y = 1: (a) stress in liquid (P); (b) total axial stress in gel (%). Both normalized by initial stress (P0) and plotted as function of reduced time, 0 = t / r . Legend indicates radial positions, u = r/R.

where

~ 2Jo(B,u ) "

.

O
1 - ~,0+ ~, 2 B-~,/-~-7 [1 -exp(-8"~°)l' f ( u , O) -

n= 1

= 2Jo(B,u )

'y~, ~

n

lk

nl

(20)

{exp[-8;(o-1/'r)]- exp(-B20)},

1/y~

0

and in place of Eq. (12) we find

( P ) =g(O) + ( 1 - f l ) Po

4

~= --gT[exp(-B20) + B2l,(O)], =1

(21)

Bn

where

g(O)=

~ 4 f l ( l - y 0 ) + flyn__~ -Ta- [ 1 - e x p ( - B 2 0 ) ] , =1Bn ~ 4

0<0_
[3Yn~==]--L-7 {exp[-B~(0--B,

l/y<0

It is k n o w n [16] that ~ = (21).

iBn 4 =

l/T)]-

exp(--B20)},

(22)

1 / 3 2 . The stress in this case is given by Eq. (13), together with Eqs. (19) and

276

G.W. Scherer / Journal of Non-Crystalline Solids 212 (1997) 268-280

4. Sample calculations Fig. 2a shows how the stress in the liquid evolves at several radial locations after the partially dried gel is suddenly immersed in a bath; the calculations use the typical value [1,2,5] v = 0.2, so that /3 = 0.5 and C,. = 0.75. The peak stress on the centerline in P(0, 0 ) = 1.27P 0 at 0 = 0.064. The total axial stress in the gel is shown in Fig. 2b, which indicates that the surface goes into compression immediately (Crz(1,0) < 0), while the interior is in tension. The maximum tension at the centerline is o-z(0, 0) = 0.466 P0 at 0 = 0.095. The capillary pressure in a gel near the end of drying can be very large (P0 > 100 MPa), so the stresses generated during the quench into the bath are quite severe. The total radial stress, shown in Fig. 3a, is tensile everywhere; it drops monotonically to zero at the surface (u = 0). Near the axis of the cylinder (u < 0.4), ~ is less than half as large as o-z, so the crack tends to form at the centerline and grow perpendicular to the axis (i.e., normal to the maximum tensile stress). However, as the crack expands toward the exterior of the cylinder, the axial stress becomes compressive, while the radial stress has a significant positive value. This is indicated in Fig. 3 b - d , where o-z and ~ are compared at several radial positions. Where the radial stress dominates, the crack turns to become parallel to the axis, as indicated in the photo in Fig. 1. If the partially dried gel is subjected to a humid atmosphere such that the tension in the liquid at the surface is reduced to zero linearly over a period o f time equal to 0.1 0 (i.e., if y in Eq. (18) equals 10), then the stresses are slightly smaller. Fig. 4 shows the time dependence o f P and o-z in this case. Much greater stress reductions

1 ,2-~'

1.0

o.~

' ' I ....

[ ....

I ....

I ....

1~

I

(a)!

--

0.8

0.6

"%

P(O)

--

" P(0"2)

I

- - - .....

P(0.6) P(O.8)

j

4m

-PO)

o,.

o. 2

~'~k 0

1

2

3

4

5

6

0.06-~'

' ' i ....

i ....

I ....

i .... o=(0)

0.042

~. . . . . . . . . . . . . . .

-.....

0.02 ¢D

0

. . . . . .....

~

--

-Oz(0.2

)

Oz(0.4

)

- - Oz(0.6

)

0,(0.8) -oz(l )

tO" - 0 . 0 2 2 \

0o4 \ -0.06

~ 0

/ , , .... 1

, .... 2

, .... 3

, .... 4

5

e

Fig. 7. Stresses generated when stress in liquid at surface of partially dried gel is reduced to zero according to Eq. (18) with 3' = 0.3: (a) stress in liquid (P); (b) total axial stress in gel (o'z). Both normalized by initial stress (P0) and plotted as function of reduced time, 0 = t/'r. Legend indicates radial positions, u = r / R .

G. W. Scherer / Journal of Non-Cm'stalline Solids 212 (1997) 268-280

1.2......... I............. Ja)~ -I/

277

P(O)

| - - -- - P(0.2)

0.65

[--

=¢~

0.65

m--

Q.

0.4 ~

::i

0.2

-.(,)

/F

~

0

,

2'

0.020

'

,

;''

I .......

6''

8'''1'0'

i ~

:,,,

I

. . . . . . . . . . .

112

,

b)

0.0015 100'!-~-_ii--_i!i--.iii--_ii_!iii!! (

W

- - PlO.6)

~. . . . . P(O.8)

%(0) ---a(0.2)[ .....

o.oo5

_

0.000

.....

-0. -0.001015-

1

~ (0.4)/

_ - o=(o.6) /

%(0.8)]

-0.005

°

-0.020

~

J I

i 2

~ r

i ' ' 6

4

, 8

i 10

,-

0

Fig. 8. Stresses generated when stress in liquid at surface of partially dried gel is reduced to zero according to Eq. (18) with 3' = 0.1: (a) stress in liquid ( P ) ; (b) total axial stress in gel (o-=). Both normalized by initial stress (P0) and plotted as function of reduced time, 0 = t/'r. Legend indicates radial positions, u = r / R .

are achieved if P(1, 0) is reduced more slowly, as indicated in Figs. 5-8. The peak stress, and the time ( 0 m a x ) at which the peak stress occurs, are shown as functions of the depressurization rate ( y ) in Fig. 9.

5. Discussion The solid network in a drying gel is compressed by the capillary pressure in the liquid, but cracking requires tensile stresses. If a flaw is present in the gel, then the total stress (~rz) acts on the faces of the flaw to pull it 0.5

~ r l l l l , l , , l l l , l I J l , l r , i

0.4

2.5

-2

/ )

0.3

-1.5

3~ v

N

0.2

1

0.1

0.5 0

0 2

4

6

8

10

12

1/7

Fig. 9. Maximum value of the total axial stress (left ordinate) and reduced time at which maximum occurs as functions of reciprocal of depressurization rate 3'.

G. W. Scherer / Journal of Non-Crystalline Solids 212 (1997) 268-280

278

0o4

~

i

J ii

0.2-

/

Q

.~.

o

~'

grcl Po /

-0.2 ,/ -0.4

/

/

E

o Z

%

/

/

/

/ Po

J

-0.6 1 0~

1 0"2

1 '0"1

1 0°

O Fig. 10. Axial stress on the network ( ~ ) at the centerline when the gel is plunged into a bath, calculated using Eq. (3) with values from Fig. 2, and assuming p = 0.4 (as for gel in Fig. 1), and corresponding stress at tip of flaw (6-c ) located on centerline; O"c calculated from Eq. (23)

assuming c / r¢ = 1.

open, and this generates tensile n e t w o r k stresses ( ~ ) at the tip of the flaw [3] that can lead to crack growth. Thus, cracking is possible only in the presence of flaws; moreover, the flaw must be free of liquid, since otherwise the tension in the liquid will oppose the total stress and prevent opening of the flaw. One expects to find flaws at the surface of a gel (owing to irregularities in the mold or damage from handling), but it is somewhat surprising that there should be voids (i.e., liquid-free flaws) in the interior of the gel. However, we shall see that the stresses are so high that very small flaws would suffice to allow cracking. Voids could result from entrained air bubbles or gas exsolved during solvent exchange (a phenomenon observed in this lab when the solvents have not been de-aired), or they could be formed by cavitation [17]. The stress on the solid network is related to the total stress by Eq. (3). For the gel plunged into liquid, Fig. 10 shows that this stress is negative (compressive), even though the total stress is highly tensile. The stress at the tip of a flaw (free of liquid) in a gel is given by [3] ~ = 2Bo-:

-(1 -o)P,

(23)

where c is the length of the flaw, r c is the radius of curvature of the tip of the flaw (expected to be comparable to the pore size in the gel), and B is a constant with a value near unity whose exact magnitude depends on the shape of the flaw [18]. Fig. 10 shows that ~c is tensile and reaches its maximum near 0m,x for any flaw with c / r ~ > 1; in fact, Eq. (23) applies only when c >> r c (and a flaw is only identifiable as such when that condition is met). Thus, whenever there is a void that is large compared to the pore size, Eq. (23) indicates that the crack-tip stress will be tensile. The sample calculations show that the total stress is greatest on the centerline of the gel rod, and this accounts for the location of the fracture origin in Fig. 1. We now proceed to calculate the magnitude of the stress. Using a combination of permeability and nitrogen desorption measurements, it has been established [2] that the pore radii (%) in the gels used in this study vary in proportion to the specific pore volume (Vp) according to rp ~

2.27 Vp ~ 2.27

, Pb

(24)

Ps

where rp is in nanometers and Vp is in cm3/g. The maximum stress in the liquid is given by Laplace's equation, P

2yLv Fp-- ~'

(25)

G. W. Scherer / Journal of Non-Crystalline Solids 212 (1997) 268-280

279

where TLV is the liquid/vapor interfacial energy and 6 is the thickness of the adsorbed layer of liquid that remains on the pore wall as the meniscus enters. Using the known values of YLV and 6 for water, it can be shown [2] that Eq. (25) is very accurately approximated by 216 P --~ - - 1.1 Fp

(26)

'

where P is in MPa. For the gel in Fig. 1 (diameter 0.407 cm), Pb 0.74 g / c m 3 and p~ = 2.01 g / c m 3, so Eq. (24) indicates that rp = 1.9 nm and Eq. (26) yields P = 105 MPa; if the gel were plunged into water, the stress in the liquid at the centerline would rise to P = 1.27(105)= 134 MPa. For gels that were allowed to dry completely (to a final diameter of ~ 0.36 cm), nitrogen desorption indicates a final pore radius of about 1.65 nm [2], which would correspond to a stress of P = 124 MPa, according to Eq. (26); suddenly rewetting such a gel would generate a centerline stress in the liquid of P = 157 MPa. Water cavitates (by homogeneous nucleation of a bubble of water vapor) at a stress of ~ 140 MPa [17], so it is plausible that cavitation would occur in any flaw on the centerline of a gel with a diameter < 0.41 cm, leading to crack growth. For a gel with a diameter of 0.47 cm, which does not crack when immersed, Eqs. (24) and (26) yield rp = 3.6 nm and P = 53 MPa, so that the peak centerline stress would be P = 68 MPa, which is well below the cavitation threshold; in this case, a pre-existing flaw in the network would remain full of liquid upon immersion, so crack growth would not occur (unless a bubble were already present). In the preceding calculation we assume that the radius of curvature of the meniscus is small enough to enter the pore, but that is not expected to be true until shrinkage stops, and the gel in Fig. 1 has not reached that point. Another way to estimate the stress in the liquid is to calculate the stress that would have to be applied to the network to compress it to a given size. The bulk modulus of the gel is given by [2] =

K = Ko(pb/Py) m ,

(27)

P b >~ Py"

For the gels of the present study the constants are K 0 = 3.19 MPa, p~. = 0.194 g / c m 3, m = 3.77 [2]. The stress required to compress the network to any density beyond p~ is approximately [19]

m

.

When the gel is being slowly dried, the total stress is near zero [11], so Eq. (3) reduces to 5z=-(l-p)P•

K° ( pb ] m m- p y ]

'

(29)

where the second equality follows from Eq. (28). Thus, the stress that must develop in the liquid in order to compress the gel to a density of Pb is

P "~ m(1 - p) For a gel rod with a diameter of 0.47 cm, Pb = 0.484 g / c m 3 and Eq. (30) yields P = 36 MPa, which is less than the 53 MPa calculated from Eq. (26); this means that the menisci would have radii larger than the pore size at this stage of shrinkage. For a rod with a diameter of 0.407 cm (as in Fig. 1), Pb -----0.74 g / c m 3 and Eq. (30) yields P ~ 213 MPa; this is not possible, since the calculation based on Eq. (26) indicated that the menisci will enter the pores of this gel when the pressure reaches 105 MPa. The contradiction between Eqs. (26) and (30) indicates that the network undergoes viscoelastic relaxation under the high capillary pressure exerted during drying, so the elastic analysis overestimates the actual stresses.

280

G. W. Scherer / Journal of Non-Crystalline Solids 212 (1997) 268-280

On the basis of these calculations we cannot say precisely how much stress is present in the gel in Fig. 1. However, it is clear that P is no greater than 36 MPa in a sample with diameter 0.47 cm (which does not crack on immersion), and that is not high enough to cause cavitation. On the other hand, the pore size of a fully dried gel indicates a final stress of P = 124 MPa, and that would generate more than enough pressure for cavitation. Thus, it is plausible that all of the gels that do crack on sudden immersion (viz., those with diameters < 0.41 cm) have voids created by cavitation. For the gels studied in Ref. [2], the hydrodynamic relaxation time of nearly-dried gels (i.e., those subject to cracking on immersion) was in the range ~-= 30 s. The sample calculations show that the stress is substantially reduced if P is reduced over a period of time much longer than ~- (i.e., if 0max > 3), so a conservative procedure would be to reduce the stress at the surface over a period of 1 0 0 = 6 - 8 min. The method used in our experiments was much slower than that.

6. Conclusions Rapid rewetting of a partially dried rod of gel (whose pores remain full of liquid) causes the tension in the liquid to rise, with the m a x i m u m increase occurring on the centerline of the rod. If the capillary tension in the liquid is P0 before the gel is immersed, then the total stress ~ is predicted to approach P o / 2 (the exact value being weakly dependent on Poisson's ratio, v). If there is a void in the gel that can act as a flaw, then cracking can occur; in all cases where cracking occurred in the present study, the stress was great enough to generate voids by cavitation in the liquid. Calculations show that the stresses are virtually eliminated by gradually reducing the capillary stress in the liquid over a period of time > 3~-, where ~- is the hydrodynamic relaxation time of the gel. This prediction was qualitatively confirmed experimentally, as gels that were exposed to a gradually increasing humidity could be rewet without cracking.

References [1] 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. [2] G.W. Scherer, J. Non-Cryst. Solids, in press. [3] G.W. Scherer, J. Non-Cryst. Solids 144 (1992) 210. [4] C.J. Brinker, K.D. Keefer, D.W. Schaefer, R.A. Assink, B.D. Kay and C.S. Ashley, J. Non-Cryst. Solids 63 (1984) 45. [5] G.W. Scherer, J. Non-Cryst. Solids 142 (1992) 18. [6] G.W. Scherer, J. Sol-Gel Sci. Tech. 1 (1994) 169. [7] C.J. Brinker and G.W. Scherer, Sol-Gel Science (Academic Press, New York, 1990) ch. 8. [8] M.A. Biot, J. Appl. Phys. 12 (1941) 155. [9] R.N. Chandler and D.L. Johnson, J. Appl. Phys. 52 (1981) 3391. [10] D.L. Johnson, J. Chem. Phys. 77 (1982) 1531. [11] G.W. Scherer, J. Non-Cryst. Solids 109 (1989) 171. [12] J. Happel and H. Brenner, Low ReynoldsNumber Hydrodynamics(MartinusNijhoff, Dordrecht, 1986). [13] C.J. Brinker and G.W. Scherer, Sol-Gel Science (Academic Press, New York, 1990) ch. 7. [14] G.W. Scherer, H. Hdach and J. Phalippou, J. Non-Cryst. Solids 130 (1991) 157; correction 136 (1991) 269. [15] F.B. Hildebrand,AdvancedCalculus for Applications(Prentice-Hall,EnglewoodCliffs, NJ, 1962) pp. 226-231. [16] G.N. Watson, A Treatise on the Theory of Bessel Functions(Cambridge, New York, 1945) p. 502. [17] G.W. Scherer and D.M. Smith, J. Non-Cryst. Solids 189 (1995) 197. [18] H. Tada, The Stress Analysis of Cracks Handbook, 2rid Ed. (Paris Productions, St. Louis, MO, 1985). [19] D.M. Smith, G.W. Scherer and J.M. Anderson,J. Non-Cryst. Solids 188 (1995) 191.