Drying Theory

8/20/2019 Drying Theory

1/12

J . A m .

Cerum.

Soc.

73 [ l ]

-14 1990)

journal

Theory of Drying

George W. Scherer*

Central Research

and

Development ,

E.

I.du Pont de Nemours & Co.,

Wilmington, Delaware 19880-0356

This review exam ines the stage s of dry-

ing, with the emph asis on the constant

rate period

CRP),

when the pores are

full of liquid. It is during the CRP that

most of the shrinkage occurs and the

drying stresses rise to a maximum . We

examine the forces that produce shrink-

age and the mechanisms responsible or

transpo rt of liqu id. By analyzing the in -

terplay of fluid flow and s hrinkag e of the

solid ne twork, it is possible to calculate

the pressure distribution in the liquid in

the pores. The tension in the liquid is

found to be greatest near the drying sur-

face, resulting in greater compressive

stresses on the network in that region.

This produces differential shrinkage of

the solid, which is the cause of cracking

during drying. The p robab ility of fracture

is related to the size of the body, the rate

of evaporation, and the strength of the

network. A variety of strategies for avoid-

ing fracture during drying are discussed.

[Key words: drying, sh rinkage, cracking,

models, gels.]

1

Introduction

REMOVALof liquid is particularly trouble-

some in sol-gel processing, because gels

tend to warp and crack during drying, and

avoid ing fracture requires inconveniently

slow dryin g rates. Howe ver, liquid trans-

port processes are also of imp ortance in

other ce ramic-formingoperations, includ-

A. H.

Heuer-contributing editor

Manuscript

No.

198096. Rece ived September 25,

Mem ber, American Ceramic Society.

1989; approved October

17,

1989.

ing slip casting, tape casting, binder burn-

out, liquid-p hase sintering, and dryin g of

clays. Indeed, the princip les of flow in po-

rous media are of such general interest

that they have been frequently redisco-

vered over the past 60 years, and rele-

vant literature is found in fields including

soil science, food science, and polymer

materials science, as well as ce ramics . In

most cases, liquid flows through a porous

body in response to a gradient in pres-

sure; at the same time, the pressure

causes de formation of the solid network,

and d ilatation of the p ores through wh ich

the liquid moves. In this review we ana-

lyze the interaction between low of the li-

qu id and dilatation of the solid in order to

predict the stresses and strains that de-

velop during drying. Special attention is

given to the problems encountered in dry-

ing gels, bu t the analysis is quite general.

The driving forces for shrinkage of the

solid and the mechanisms for transport of

the liquid are discussed n Section II.The

stages of dryin g are outlined in Section 111,

and a mo del for calculation of drying

stresses is developed in Section IV. The

cause of c racking during d rying an d vari-

ous strategiesfor avoiding fracture are de -

scribed in Section

V.

These topics are

discussed in greater detail in a forthcom-

ing book.'

I I

Deformation and

Flow

1) Driving Forces for Shrinkage

The first stage of drying is illustrated in

Fig. 1(B): for every unit volume of liquid

that evaporates, the volume of the body

decreases by one unit volume, so the li-

quidlvapor interface (meniscus) remains

at the surface of the body. In gels, this

stage continues while the bo dy shrinks to

as little as one-tenth of its orig ina l volume.

The forces that pro duc e the shrinkage of

the solid network are discussed below.

A) CapillaryPressure: If evaporation

George W. Scherer has bee n a mem ber

of the Central Research Departm ent of

E. I. du Pont de Nemours & Co. since

1985 . His work at Du Pont has dealt prrn-

cipally with sol-gel proces sing, and es-

pecially with drying. In collaboration with

Jeff Brinker of Sandia N ational Labs, he

has written a book entitled Sol-Gel

Science

that will be published by Aca-

dem ic Press in February. From 197 4 to

1985 , Dr. Scherer was at Cornin g Glass

Works, where his research included op -

tical fiber fab rication, viscous sintering,

and viscoelastic stress analysis. The lat-

ter work was the su bject of his first bo ok,

Relaxation in Glass and Composites

(Wiley, 198 6). He receive d his

B.S.

and

M.S.

egrees in 1972 and his Ph.D. in

materials science in 1974 , all from

MIT,

where his thesis work was on crystal

growth in glass.

3


2/12

Vol. 73,No. 1

Journal

of

the American Ceramic Society Scherer

of

liquid from the pores were to expose

the solid phase, a solidlliquid interface

would be replaced by a more energetic

solidhapor interface. To prevent such an

increase in the energy of the system, liquid

tends to spread from the interior of the

body to cover that interface. Since the vol-

*The stress in the liquid, P. is positive when the

liquid is in tension. The pressure, P i , follows the op-

posite sign convention

(Pi =

- P ) ,

so

tension is

negative pressure.

Fig.

1.

Schematic illustration

of

drying process: black network represents solid phase and shad-

ed area is liquid filling pores. A) Before evaporation begins, the meniscus is flat.

B)

Capillary

tension develops in liquid as it "stretches" to prevent exposure

of

the solid phase, and network

is drawn back into liquid. The network is initially so compliant that

little

stress

is

needed

to

keep

it submerged, so the tension in the liquid is

low

and the radius of the meniscus is large.As the

network stiffens, the tension rises and, at the critical point (end

of

the constant rate period), the

radius of the meniscus drops

to

equal the pore radius. (C) During the falling rate period, the li-

quid recedes into the gel.

ume of liquid has been reduced by evapo-

ration, the meniscus must become curved

as indicated in Fig. 2. The tension 0 n

the liquid

is

related to the radius of curva-

ture

(r)

of the meniscus by*

where y ~ vs the liquidlvapor interfacial

energy (or surface tension). When the cen-

ter of curvature s

ir

the vapor phase, the

radius of curvature is negative and the

li-

quid is in tension

PX).

The maximum capillary tension PR)n

the liquid occurs when the radius of the

meniscus is small enough to fit into. the

pore; for liquid in a cylindrical pore of ra-

dius

a ,

he minimum radius of the menis-

cus is

where

8

s the contact angle.

If 8 s go",

then the liquid does not wet the solid and

the liquidhapor interface is flat F

w ,

P =

0).

f

8=

0

the solid surface

is

covered

with a liquid film. Of course, the pores in

a real body are not cylindrical, but it can

be shownW that the maximum tension is

related to the surface-to-volume ratio of

the pore space,

SplVp:

where

ysv

and y s ~ re the solidlvapor

and solidliiquid interfacial energies,

respectively. The specific surface area of

a porous body (interfacial area per gram

of solid phase),S , s related to the surface-

to-volume ratio by3

where

e is

the relative density,

e =e ,

eb

is

the bulk density of the solid network

(not counting the mass

of

the liquid), and

es

is

the density of the solid skeleton (the

skeletal density). The quantity

VplSp is

also known as the hydraulic radius.As we

shall see, during most of the drying pro-

cess the capillary tension is smaller than

this maximum value.

(6) Osmotic Pressure: Osmotic pres-

sure

(n)

is

produced by a concentration

gradient, as in the case of pure water

diffusing through a semipermeable mem-

brane to dilute a salt-rich

solution

on the

other side. As indicated

in

Fig.

3

pressure

n

must be exerted on the solution (or a

tension of

- n

must be exerted on the

pure liquid)

to

prevent :he water from en-

tering the solution. The pressure s a meas-

ure of the difference in chemical potential

between the pure liquid and that in the so-

lution. An analogous situation can arise if

the pores of the drying body contain a so-

lution of electrolyte: evaporation of solvent

increases the salt concentration near the

drying surface,so liquid diffuses from the

interior

to

reduce the concentration gra-

dient; the decrease in the volume of liquid

in the interior causes tension in the liauid


3/12

January 1990

Theory of Drying 5

that remains there. If the pores are large,

the diffusive flux is matched by counter-

flow of liquid toward the interior, and no

stress develops. However, if the pores are

small enough to inhibit flow, diffusion away

from the interior can p roduce tension in

the liquid in that region; then the balanc-

ing compression in the solid phase (which

in principle could approach

fl

can pro-

duce shrinkage. In such a situation, the

solid network plays the role of a semi-

permeab le mem brane, p ermitting trans-

port in only one direct ion. This

phenom enon could b e of imp ortance in

clays and gels. Alkoxide-derived gels

generally contain a solution of liquids that

differ in volatility (viz., alcohol and w ater),

so

evaporation creates a com position gra-

dient, and osmotic flow may result.

(C) Disjoining Pressure: D isjoining

forces are short-range forces resulting

from the presence of a solidlliquid inter-

face. The most important examples are

dou ble-layer repulsion between ch arged

surfaces and interactions caused by struc-

ture created in the liquid by dispersion

forces. Liquid molecules, especially

water,43 tend to adop t a special structure

in the vicinity of a solid surface. The inter-

action with the surface is so strong that ad-

sorbed layers

=I

nm thick resist freezing.6

As

evaporation occurs and solid surfaces

are brought together, repulsive forces aris-

ing from electrostatic repulsion, hydration

forces, and solvent structure resist contrac-

tion of the gel. The pore liqu id will diffuse

or flow from the swollen interior of the gel

toward the exterior to allow the solid sur-

faces

to

move farther apart. The disjoin-

ing forces thus pro duce an osmotic flow,

where transport is driven by a gradient in

chemical potential in the liquid phase.

Since these forces become important

when the separation between Surfaces is

small, they are most likely to be impor-

tant near the end of dryin g of gels, whe n

the pore diameter may approach 2

nm.

Macey7 argues that electrostatic rep ul-

sion between particles of clay produces

tension in the liqu id that draws flow from

the interior of a drying body. Even for

clays, in which these phenomena are most

evident, it has be en argued8 that osmotic

forces must be less important than capil-

lary pressure, because moisture gradients

persist in clays for long periods when

evaporation is prevented. In addition, it

has been shown that the final shrinkage

of kaolinite clay during drying

is

directly

related to the surface tension of the pore

liquid.9 The swelling pressure of clays in

water is < I 0 MPa,lO which is com parable

to the capillary pressure in pores with radii

>14

nrn (according to Eq.

(l),

assuming

yLv=

0.072

Jlm2 for water). In the case of

gels, the pores are generally smaller than

that, so capillary forces are e xpected to

dominate.

0) Mo isture Stress: Moisture stress

or moisture potential y) s the partial

specific Gibbs free energy of liquid in a

porous medium, and is given byir

5)

=

where eL and V,,, are the density and mo -

lar volume of the liquid, Rs is the ideal

gas constant, T is the temperature, p v is

the vapor pressure of the liquid in the sys-

tem, and po is the vapor pressure over a

flat surface of the pure liquid. In soil

science12

it

is conventional to define the

moisture potential in terms of the equilibri-

um height to which it would draw a

colum n of water, so a factor of g (the

gravitational acceleration) would be includ-

ed in the denominator on the right side of

Eq.

(5).

The m oisture potential is quite in-

clusive, because the vapor pressure is

depressed by factors including capillary

pressure, osmotic pressure, hydration

ture potential subsumes all of the driving

forces discussed above, and can be o b-

tained by m easuring the vapor p ressure

of the liquid in the system. For that rea-

propriate potential driving shrinkage of On

the

solid

phase

shrinkage.

gels during drying. The difficulty in im-

plementing that suggestion is that capil-

lary pressure gradients produce flow,

while concentration gradients (that p ro-

duce osmotic pressure) cause diffusion,

so it is necessary to app ly portions of the

total potential to different transport

processes. In soil science, it is customary

to assume that fluid flow is driven by the

gradient in moisture potential, but it is done

with the understanding that factors other

than cap illary pressure and gravitation are

negligible.12

e x )

k)

forces, and adsorption forces. Thus, mois- A) (B)

Fig. 2

To

prevent exposure of the solid

Phase Ah the liquid must adopt a curved 11

son, Zarzycki13 recom men ds it as the ap-

quidhapor interface (B). Compressive forces

Fig.

3. Water diffuses into the salt solution to equilibrate the concentration on either side

of

the impermeable membrane; pressure il would have to be exerted

on

the solution

to

prevent

the influx of water.


4/12

6

Journal

of


VOl. 73, No.

1

2)

Transport

Processes

A) Darcy's Law: Fluid flow throug h

porous media obeys Darcy's la~,14~15

which states that the flux of liquid, J , is

proportional o the grad ient in pressure in

the liquid,

VP

The flux is in units of volume per area of

the porous body (not the area occupied

by the liquid) per time,

PL is

the force per

unit area of the liquid, q~ is the viscosity

of the liquid, and D is called the p ermea-

bility and has units of area. Positive flux

moves in the direction of increasingly

negative pressure (i.e., he flow is toward

regions of greater tension in the liquid).

Equation

(6)

s an empirical equation der-

ived from observation of flow of water

through soi1,16 but it is analogous to

Poiseuille's aw for flow of liquid through

a straight circular pipe. This analogy has

given rise to man y models for the perme -

ability

of

porous media based on

representationsof the pores by arrays of

tubes, many of which are discussed in the

excellent texts by Scheideggerl4 and D ul-

lien; l5 van Brake117 offers a critic al review

of over

300

such models. The most popu -

lar model, because of its simplicity and ac-

curacy, is the Carman-Kozeny equation,

which gives the permeability in terms of

the relative density and specific surface

area:

(7)

The factor of 5 is an empirical correction

for the noncircular cross section and non-

linear path of a ctual pores . This equation

is reasonably successful for many types

of granular m aterials, but it often fails, and

should be applied with caution.

The p roportionality of the flux to the

pressure gradient is obeyed by many

materials, including those with po res

smaller than 10 nm, as in porous

Vycort718 and alkoxide -derived gels.19

Even in unsaturated bodies (i.e., where the

pores contain both liquid and gas),

Da rcy's law is obeyed1420 as long as the

liquid

phase is funicular (i.e., interconnect-

ed); if the liquid is pendular (i.e., isolated

in pockets), t can only b e transported by

diffusion of the vapor. The permeability of

unsa turated materials is a strong function

of liquid content and shows considerable

hysteresis as the liquid content is raised

and lowered.

In gels the pores are

so

small that a

large portion of the liquid may be in struc-

tured layers within =1 nm of a solid sur-

face, so the effective viscosity may be

greater than in the bulk liquid. The re-

duced mobility in such lavers can be

unfortunately, attempts at direct measure-

ment of the viscosity near solid sur-

faces243 have been shown26 to give

incorrect results. Spe ctrosco pic methods

indicate an increase in viscosity by a fac-

tor of -3, so the effect of solvent structure

on the flux in gels can be substantial.

(B) Diffusion: According to Fick's

law, the diffusive flux

Jo)

s proportional

to the concentration gradient (VC):27

where D, is the chem ical diffusion coeffi-

cient, C is the concentration, and p is the

chem ical potential. As noted above, diffu-

sion can contribute to the shrinkage of gels

in special cases (e.g., when the gel is im-

merse d in a salt solution) and may be im -

portant during evaporative drying, if a

concentration gradient develops in the

pore s by p referential evaporation of one

component of the pore liquid.

In some cases, a gradient in concen-

tration of the solid phase can p roduce

os-

motic transport (as in the sw elling of som e

organic polymers28 or clay29), but it is not

clear whether transport occurs by diffusion

or flow. One can com pare the fluxes giv-

en by Eqs. (6) and

(8)

by converting the

chemical potential gradient to pressure-

volume work, then relating the diffusion

coefficient to the viscosity by use of the

Stokes-Einstein equation.1 The conclusion

is that flow is faster than diffusion when-

ever the pore diameter is more han a few

times the diameter of the liquid molecule.

Howe ver, this conclusion a pplies only

to

situations such as flow within a clay bod y

(where tension in the liquid is produ ced

by disjo ining orces), where there is a gra-

dient in concentration of solid phase. Flow

cannot reduce a concentration gradient in

the liquid phase. For example, if a gel is

immersed n a salt solution, flow of the

so-

lution into the pores does not affect the

difference in salt concentration between

the bath and the original pore liquid; that

can be achieved only by diffusion. Simi-

larly,

if

evaporation creates a concentra-

tion gradient in the p ore liquid, flow from

the interior of the gel cannot eliminate it;

only interdiffusion within the po res can do

I l l

Stages of

Drying

The stages of drying were clearly

discussed in the classic work of

Shewmd30-32

60

years ago. Several texts

prov ide qualitative desc riptions of the

phenom enology and detailed discussion

of the technology of drying,%-% he scien-

tific aspects are discussed in several very

good reviews (e.g., Refs. 8 and 36) and

in the series of b ooks called Advance s in

Dr~ing.37~38

so.

demonstrated. using nuclear m agnetic

( 1 )

Constant

Rate

Period

re so na nc gl or optical s p e c t r o ~ c o p y , ~ ~ ~ ~ ~

he first stage of dryin g is called the

constant rate period (CRP), because the

rate of evaporatlon per unit area

of

the dry-

ing surface is independent of time 7,8 The

Corning

Glass

Works Corning

NY


5/12

January 1990

Theory

of

Drying

7

evaporation rate is close to that from an

open dish of liquid, as indicated, for ex-

ample, by Dwivedi’s data for drying of alu-

mina ge1.39 The rate of evaporation, V , , s

proportional o the difference between

pv

and the ambient vapor pressure, pA:

(9)

where k s a factor that depends on the

temperature, draft, and geometry of the

system. The vapor pressure of the liquid

is related to the capillary tension (P) by

Pv Po exp(

--PVm

%IT

From Eqs. ( l) , (9), and (10) we see that

evaporation will continue as long as

The fact that the evaporation rate is simi-

lar to that of bulk liquid indicates that the

vapor pressure reduction is insignificant

during the CRP. However, in some gels

the pores are so small that a significant

reduction in pv could occur; moveover,

the composition of the liquid in the pores

could change with time if the initial liquid

is a solution. The latter factors have been

proposed

to

explain the absence of a CRP

for an alkoxide-derived silica gel.4W

It seems reasonable to conclude that

the surface of the body must be covered

with a film of liquid during the CRP, be-

cause the proportion of the surface co-

vered by menisci shrinks faster than the

total area,

so

the rate would decrease as

the body shrank if evaporation occurred

only from the menisci. However, Suzuki

and MaedaQ proved that the evaporation

rate can remain constant even when dry

patches form on the surface of the body.

There is a stagnant (or slowly flowing)

boundary layer of vapor over the drying

surface, and

if

the breadth of the dry

patches is small compared

to

the thick-

ness of the layer, diffusion parallel

to

the

surface homogenizes the boundary layer

at the equilibrium concentration of vapor.

This would certainly be expected in gels,

where the expanse of dry solid phase be-

tween menisci would be on the order

of

nanometers. Therefore, transport of vapor

across the boundary layer obeys Eq. (9),

and the rate of evaporation per unit area

of surface is constant, whether or not there

are small dry patches.

Evaporation causes cooling of a body

of liquid, but the reduced temperature

leads to a lower rate of evaporation, and

this feedback process equilibrates when

the drying surface reaches the wet bulb

temperature

(T,).

As indicated by Eq.

(9),

V ,

ncreases as PA decreases,

so

T,

decreases with the ambient humidity. The

exterior surface of a drying body is at the

wet bulb temperature during the CRP.3’

The surface temperature rises only after

the rate of evaporation decreases (in the

falling rate period discussed in Section Ill

(2)).

For alkoxide-derived gels the vapor

pressure must be kept high to avoid rap-

id drying, so the temperature of the sam-

ple remains near ambient.

The tension in the liquid is supported by

the solid phase, which therefore goes into

compression. f the network is compliant,

as it is in alkoxide-derived gels, the com-

pressive forces cause it to contract into the

liquid and the meniscus remains at the ex-

terior surface, as indicated in Fig. 1 B). In

a gel, it does not take much force to sub-

merge the solid phase,

so

the capillary

tension is low and the radius of the menis-

cus is much larger than the pore radius.

As drying proceeds, the network becomes

increasingly stiff, because new bonds are

forming and the porosity is decreasing; the

meniscus deepens and the tension in the

liquid rises correspondingly. Once the ra-

dius of the meniscus becomes equal to

the radius of the pores in the gel, the li-

quid exerts the maximum possible force.

That marks the end of the CRP: beyond

that point the tension in the liquid cannot

overcome further stiffening of the network,

so

the meniscus recedes into the pores,

leaving air-filled pores near the outside of

the gel (Fig. 1 C)). Thus, during the CRP,

the shrinkage of the gel is equal

to

the vol-

ume of liquid evaporated; the meniscus

remains at the exterior surface, but r

decreases continuously. This behavior is

illustrated by the data of Kawaguchi etd . 4 3

for alkoxide-derived gels; equivalent

results have been reported for particulate

gels made from fumed silica.44

The end of the CRP is called the critical

point (or leatherhard point, n clay technol-

ogy), and it is at this point that shrinkage

virtually stops.At the critical point, the ra-

dius of curvature of the meniscus is small

enough to enter the pores, so the capil-

lary tension is found from Eqs. (3)and (4):

For an alkoxide-derived gel with S-300 to

800

m*/g,

eb-0.4

to 1.6 glcm3, Q-0.2 to

0.6, and yLv cos

(0)-0.02

to 0.07 J/m*,

this is an enormous pressure: P p 3 to

200

MPa The amount of shrinkage that

precedes the critical point depends on the

magnitude of the maximum capillary

stress,

PR.

Since PR increases with the in-

terfacial energy

(yLV)

and with decreasing

pore size, it is not surprising to find that

the porosity of a dried body is greater (be-

cause less shrinkage has occurred) when

surfactants are added to the liquid. For ex-

ample, Kingery and Franc19 found a line-

ar proportionality between

YLV

and dried

density for clay bodies mixed with surfac-

tants. It is important to recognize, howev-

er, that the pressure depends on the

contact angle, and the surfactant could in-

crease

0

while reducing

y ~ v .

he impor-

tance of contact angle is nicely illustrated

by the work of Mitsyuk et a1.45 They pre-

pared aqueous silica gels from sodium sili-

cate, then soaked them in various alcohols


6/12

8 Jour

Fig . 4. After

the critical point, the li-

quidlvapor meniscus retreats into the pores

of

the body. In the first falling rate period, liquid

is

in the funicular state, so transport by fluid

flow

is

possible. There

is also

some diffusion

in the vapor phase.

Fig.

5.

During the second falling rate peri-

od evaporation occurs inside the body,

at

the

boundary between the funicular (continuous

liquid) and pendular (isolated pockets of liquid)

regions. Transpqrt in the pendular region oc-

curs by diffusion.of vapor.

wal of the American Ceramic Society Scherer

Vol. 73, No.

1

(methanol, ethanol, 1 propanot, 1 butanol)

to replace the pore liquid. When the gels

were dried, the final porosity was found

to be linearly related to the heat of wet-

ting. The heat of wetting is related*

to

the

quantity ysv-ysL=yLv cos

(8); in

this

case, yLv is nearly the same for all the al-

cohols, so the variation in capillary stress

is caused by

8.

(2)

First Falling

Rate

Period

When shrinkage stops, further evapo-

ration drives the meniscus into the body,

as illustrated n Fig. 1(C); as air enters the

pores, the surface may begin

to

lose its

translucency.@ In the first falling rate peri-

od (FRPI), the rate of evaporation

decreases and the temperature of the sur-

face rises above the wet bulb temperature.

Most of the evaporation is still occurring

at the exterior surface, so the surface re-

mains below the ambient temperature,

and the rate of evaporation is sensitive to

the ambient temperature and vapor pres-

sure.83 The liquid in the pores near the

surface remains in the funicular condition,

so

there are contiguous pathways along

which flow can cccur (Fig.

4).

At the same

time, some liquid evaporates within the un-

saturated pores and the vapor is transport-

ed by diffusion. Analysis of this situation

involves coupled equations for flow

of

heat

and liquid and diffusion of vapor, with

transport coefficients that are generally

dependent on temperature and con-

,centration. There are several good

revie~s36~47.48f the many theories that

have been proposed o descibe the FRP1.

'The most complete and rigorous treatment

is by Whitaker.49,50

Shaw51r52 performed an elegant series

of experiments showing that the drying

front (i.e., he liquidlvapor interface) s frac-

tally rough on the scale of the pores, but

stable on a much larger scale. It is the

pressure gradient in the unsaturated re-

gion that is responsible or the stability of

the drying front: the capillary pressure is

SO

low in advanced regions of the front

that the radius of the meniscus istoo large

to pass through the pores. Since the ir-

regularity in the drying front is on the scale

of the pores, it is very small compared to

tlhe dimensions of the body. Even in a par-

ticulate gel with 60-nm pores.53 if a par-

tially dried gel is broken in half, the drying

front is visible as a sharp line between the

translucent saturated region and the

opaque dry region. No doubt this line

would be rough if observed in the

SEM,

but it is quite smooth on a macroscopic

scale.

3) Second Falling Rate Period

As the meniscus recedes into the body,

the exterior does not become completely

dry right away, because liquid continues

to

flow

to

the outside; as long as the flux

of liquid is comparableto the evaporation

ratel the funicular condition is preserved.

However, as the distance from the exteri-

or

to the drying front increases, he capil-

lary pressure gradient decreases and

therefore so does the flux. Eventually (if the

body is thick enough) it becomesso slow

that the liquid near the outside of the body

is isolated n pockets (i.e., enters the pen-

dular condition), so flow to the surface

stops and liquid s removed rom the body

only by diffusion of its vapor. At this stage,

drying is said to enter the second falling

rate period (FRP2), where evaporation

oc-

curs inside the body (see Fig. 5 .31 The

temperature of the surface approaches he

ambient temperature and the rate of

evaporation becomes ess sensitive to ex-

ternal conditions (temperature, humidity,

draft rate, etc.). As indicated n Fig.5, the

drying front is drained by flow of funicular

liquid which evaporates at the boundary

of the funicularlpendular regions. In the

pendular region, vapor is in equilibrium

with isolated pockets of liquid and ad-

sorbed films, and the principal transport

process is expected to be diffusion of

vapor.

As

the saturated region recedes into the

body, the body expands slightly as the to-

tal

stress on the network is re lie~ed.32~43~~

At the same time, differential strain builds

up because the solid network is being

compressed more in the saturated region

than near the drying surface. This can

cause warping in a plate dried from one

side, as faster contraction of the wet side

makes the plate convex toward the dry-

ing side.% The fact that the warping is per-

manent (i.e., does not spring back when

drying is complete) indicates that the un-

saturated region retains some viscosity or

plasticity during FRP2. As the saturated re-

gion becomes thinner, its contraction is

more effectively prevented by the larger

unsaturated region, and this raises the ten-

sion in the network in the saturated region.

This phenomenon probably accounts for

the observation by Simpkins et

a/.

that

cracks in drying gels often originated near

the nondrying surface.

Whitaker499 developed an analysis of

heat and mass transfer during drying of

rigid materials that offers the most com-

plete description

of

the falling rate periods.

He uses transport coefficients that are

lo-

cal averages for regions large compared

to the pore size, but small compared to

the sample. This is analogous

o

the aver-

aging implicit in Darcy's law, where the

permeability,

D,

"smears out" the ge-

ometrical details of the microstructure.

U s e

of Whitaker's model requires knowledge

of a large number of physical properties

(permeability, thermal conductivity, diff u-

sivity of vapor), and the analysis must be

performed numerically.

A

successful test

of the model was performed by Wei et

a/.,55-56who studied the drying of porous

sandstone.

IV.

Drying

Stress

If evaporation of liquid from a porous

body exposed the solid network, a solid/

vapor interface would appear where a

solid/liquid nterface had been. This would


7/12

January 1990

Theory of Drying

9

raise the energy of the system, because

ysv>vsL,

so liquid tends to flow from the

interior to prevent exposure of the solid.

As it stretches toward the exterior, the

li-

quid goes into tension, and this has two

consequences:

1)

liquid tends

to

flow

from the interior along the pressure gra-

dient, according to Darcy’s law; (2) the

tension is balanced by compressive stress

in the network that causes shrinkage. The

lower the permeability, the more difficult

it

is to draw liquid from the inside of the

body, and therefore the greater the pres-

sure gradient that develops.

As

the pres-

sure gradient increases, so does the

variation in free strain rate, with the sur-

face tending to contract faster than the in-

terior. It is the differential strain (i.e., the

spatial variation in strain (for an elastic ma-

terial) or strain rate (for a viscous materi-

al)) that produces stress. This is analogous

to

the development of thermal stresses in

response to a temperature gradient, an

observation that has been exploited by a

number of authors.57-60 Just as calculation

of thermal stresses requires knowledge of

the temperature distribution, prediction of

drying stresses depends on calculation of

the pressure distribution, which we now

explore.

( 1 ) Pressure Distribution

If we consider an isolated region of a

porous body, the rate

of

change of the vol-

ume of liquid in that region depends on

the divergence of the flux (i.e., the differ-

ence between the flux entering and the

flux leaving). During the CRP, when the

pores are full of liquid, he change in liquid

content must be equal to the change in

pore volume,S which is. related

to

the

volumetric strain rate,

E .

Setting these

changes equal, we obtain the equation for

continuity (conservation of matter):el

We need

to

express in terms of the ten-

sion in the liquid using a constitutive equa-

tion for the network. Various authors have

done this by using empirical (nonlinear

elastic) equation~7~12r by assuming elas-

tic behavior with the solid and liquid

phases compressible57P or incompress-

ible,63 a or allowing the network to be

purely V~S CO US ,~ ~r viscoelastic.66-70 For

the sake of discussion, we will employ the

simpler elastic analysis. When the network

is

assumed

to

be elastic, Eq.

13)

has the

mathematical orm of the diffusion equa-

tion. For the CRP it

is

appropriate to in-

troduce the boundary condition that the

flux at the exterior surface is constant:

$In his case, a pore is a space not occupied by

the solid phase, which may be occupied by liquid

andlor gas Duringthe CRP, there are no gas pock-

ets, so a pore is full

of

liquid

where

V

is the constant evaporation rate.

For

an elastic network Eq. 13) becomes

In this equatior,

L

is the half-thickness of

the drying plate, u =z /L is the coordinate

normal to the drying surface, the dimen-

sionless time is defined as

O = flr

and

where

Kp

and

GP

are the bulk and shear

moduli of the solid network (i.e., the

properties that would be measured with

the liquid drained away). By solving Eq.

15)

we obtain the pressure distribution

in the drying body; the stresses and

strains follow from the constitutive equa-

tions.1 61

2) Stress Distribution

Philip12 discusses at length the

methods for solving the nonlinear version

of

Eq.

15)

that results when the permea-

bility and elastic properties vary with the

porosity (and therefore with position n the

body). In the simple case where the

properties are constant and the shrinkage

during drying is negligible, an analytical

solution is readily obtained;63 typical

results are shown in Fig. 6. The tension

P in the liquid rises until at the critical point

(time

0)

it

reaches the maximum value

at the exterior surface,

P(L,OR)= PR,

as

shown in Fig. 6(A). If the evaporation rate

is not too fast, the distribution through the

plate becomes roughly parabolic at a

much earlier stage (when O=eR/3 in Fig.

6(C)) and, since the stress depends on

the shape of the pressure distribution,a,

is approximately constant (Figs. 6(B) and

(D)) during the time interval OR/3


8/12

10

Journal of the American Ceramic Society Scherer

Vol.

73,

No. 1

distribution s parabolic, in which case Eq.

(18) becomes61

The stress increases in prop ortion

to

the

thickness of the plate and the rate of

evaporation, and in inverse proportion to

the perm eability; that is, the stress

s

in-

creased by those factors that steepen the

pressure gradient. The reason that gels

are so much more difficult to dry than

conventional ceramics is that the perme-

ability of gels is low, as a result of their

very small pore size. Comparing the

stress at the surface of a drying plate,

cylinder, and sphe re, it is found71 that the

tension decreases in the ratio plate/

cylinderlsphere =; /a/ ; The lower stress

reflects the shallower pressure gradients

in the cylinder and sphere, where the

li-

quid flowing from the interior passes

through a volum e that increases as P and

r 3

respectively. Since these results are

derived from Eq.

(13),

hey are valid only

as long as the pores rem ain filled with li-

quid. At s ome point the netw ork will stop

shrinking and the meniscus will retreat

into the gel; then Eq. (19) will apply on ly

within the saturated po res nsid e he ge1.63

3) Diffusion

If

the po res conta in a solution of liqu ids

with intrinsic diffusion coefficientsD , and

D2,

then diffusion contributes to the trans-

port and the diffusion term m ust be a d-

ded

to

the flow term. Then Eq.

(13)

becomes72

h =

v.

P

+ I

-@)V.

i

Note that d iffusion has no influence f the

intrinsic diffusion coefficients of the two

liquids are equal, because the diffusive

volume fluxes are then equal and oppo-

site (i.e., diffusion produces no volume

flow). It has been shown72 that drying

stresses can be reduced considerably

when the diffusion term is significant. The

reason is that a substantial flux can be

produced by

a

shallow co ncen tration gra-

dient (since interdiffusion of liquids is

rapid ), so diffusion can e xtract liquid from

the interior of the bod y almost as fast as

it evaporates from the surface. Conse-

quently, the pressure distribution is flat-

ter, the differential strain is reduced, and

the drying stresses are smaller when

diffusion occurs.

V. Fracture

( 1)

Models of Fracture During Drying

There is no generally accepted ex pla-

nation for the phenom enon

of

cracking

during drying. Any suitable he ory should

a'ccount for the com mon observations

that cracking is more likely

if

the body is

thick or the drying rate is high, and that

cracks generally appear at the critical

point (i.e., when shrinkage stops and the

vapor/liquid interface moves into the

body

of

the gel). The tendency for slow-

ly dried bodies

to

crack at the critical point

has been noted for clay,35 particulate

gels, l73 and alkoxide -deriv edgels.39174

We no w examine two mod els of fracture

during drying, a macroscopic model

(described in Section IV) that attributes

cracking

to

stresses produced b y a pres-

sure gradien t in the liquid ph ase, and a

microscopic mod el that explains crack-

ing as a result of the distribution of p ore

sizes.

The stress that causes fracture is not

the m acrosc opic stress,

a

that acts on

the network. R ather, it is the stress con-

centrated at the tip of a flaw of length c

which is proportional to75

Fracture occurs w hen o,>K\,, where KI,

is a material property called the critical

stress intensity.76 It is reasonable to as-

sume that th e flaw size distribution is in-

depen dent of the size and drying rate of

the gel,

so

the tendency

to

fracture will

increase with the stress,

ax.

Although Eq.

(19) accounts qualitatively for the ob-

served dependence of cracking on L and

V . , t

does not offer any explanation for

the common observation that slowly

dried ge ls crack at the critical point. The

stress is p redic ted to rise continu ally until

that moment, but there is no sudden ump

predicted for a at time f3R that would en-

hance the likelihood of cracking .

The microscopic model for fracture is

based on the idea77178 llustra ted in Fig .

7. After the critical point, liquid s removed

first from the largest pores; then the ten-

sion in the neighboring small pores is

claimed to deform the pore wall and

cause cracking. This mechanism app ears

to

account quite clearly for the occur-

rence of cracking at the critical point.

How ever, he flaws produ ced in this way

have lengths on the order of the space

between pores, which is typically

1

to

5

nm in alkoxide-derived gels, and such

flaws should be subcritical (i.e., non-

propagating). This difficulty could be

avoided b y suppo sing that the flaws per-

colate through the structure until they

achieve the critical length. A more im por-

tant prob lem with this mechanism is that

it does not explain the impo rtance of dry-

ing rate or body size. The local stresses

result from the local hetero geneity of the

microstructure,

so

fracture should be in-

evitable when the pore size distribution

is wide.

Another version of this mode l wou ld at-

tribute the flaws to the irregu larity of the

drying front, illustrated schematically in

Fig. 8. The width

of

the drying front,

w ,

is 2 or 3 orders of m agnitude larger than

the po re size, but the drying front is quite

smooth on the scale of the thickness of

the sample. The crack might be expect-

ed to have a length similar

to

w ,

so

the

stress intensity would be p roportional to

ac= a x e (21)


9/12

January

1990

PRwlQ, However, Shawsz has shown

that

w a p p ) - 1 ~ (VE)-1/2

(22)

which means that the drying front be-

comes smoother (w decreases) as the

drying rate increases. Thus

(23)

c

=

pRwl/2a VE)-1/4

which means that the stress intensity

decreases as the evaporation rate

increases, in contradiction

to

the ex-

perimental evidence. Further, no depen-

dence of stress on sample size is

expected according to this mechanism.

On the other hand, if these flaws are act-

ed upon by the stress predicted by the

macroscopic mechanism, then the stress

intensity is

(241

which increases almost in proportion

to

the drying rate. Thus, the flaws generat-

ed by the irregular drying front, together

with the macroscopic stress, may explain

the appearance of cracking at the criti-

cal point. The macroscopic nature of the

stress also explains the observation that

a drying body will often break into only

two or three pieces;

if

the stresses were

local, failure should always result in a very

large number of fragments.

2)

Avoiding

Fracture

Since the capillary pressure sets the

limit on the drying stress

(o,OR);

(F) normalized

stress distribution at same times as in

(E).

From Ref.

63.

soaking f&

24

h in

4N

HCI or

2N

NH3.


10/12

2

Journal

of


Vol.

73,

No. 1

Constant rate period

Fig.

7.

Illustration of microscopic model:

during the constant rate period, meniscus has

same radiusof curvature

for

poresof all sizes;

after

the

critical point, the largest pores are

emptied first.The capillary tension compress-

ing the smaller pores causes local

stresses

that

crack the network. After Ref. 77.

Fig. 8. Drying front is fractally rough bound-

ary between saturated (i.e., iquid-filled) and un-

saturated regions. Flaw of length

c is

subjected

to stress over width of drying front.

and all of these features help to reduce

cracking. The coarser structure may be

a result of the higher pH produced by

hydrolysis of formamide.84 Unfortunate-

ly, the additive is difficult to remove upon

heating, so bloating and cracking result.

The original claims85 of rapid processing

(-48

h) for centimeter-thick pieces of gel

processed with formamide have not been

repeated nor reproduced, but promising

results have been reported for dimethyl-

formamide (DMF).86,87 That additive

yields gels with larger pores, and they are

even larger after aging at elevated tem-

peratures (=150°C). Gels made with DMF

do not crack at drying rates that destroy

gels made with formamide, or without

any DCCA. Interestingly, the dried gel

cracks when exposed to vapors of water

(yLv=0.072 J/m2) or formamide

(0.058

Jlmz), but not vapors of methanol

(0.023

Jlm2) or DMF

(0.036

J/m2), so the lower

surface tension of the additive may be im-

portant. To the extent that these additives

are effective, their success can be at-

tributed to coarsening of the microstruc-

ture (which increases

D

and decreases

PR) and strengthening of the network.

They may also provide a medium through

which the more volatile components

(water and alcohol) can diffuse, thereby

allowing diffusion to reduce the pressure

differential within the body.72

Since shrinkage and cracking are

produced by capillary forces, KistleP

reasoned that those problems could be

avoided by removing the liquid from the

pores above the critical temperature

(T,)

and critical pressure

(Pc)

of the liquid.

Under such conditions there is no distinc-

tion between the liquid and vapor phases:

the densities become equal, there is no

liquidhapor interface, and no capillary

pressure. In the process of supercritical

(or hypercritical) drying, a

sol

or wet gel

is placed into an autoclave and heated

along a path such as the one indicated

iin Fig.

9.

The pressure and temperature

,are increased in such a way that the

phase boundary is not crossed; once the

critical point is passed, he solvent is vent-

led

at a constant temperature

(>T,).

The

resulting gel, called an aerogel, has a

volume similar

to

that of the original

sol.

This process makes it possible to

produce monolithic gels as large as the

volume of the autoclave. Table I contains

values of T, and

Pc

for some relevant li-

quids. Two groups succeeded at about

the same time in making large monolithic

gels by supercritical drying. In one case89

the aerogel itself was the objective: the

LOW

density of the silica gel was required

for a Cherenkov radiation detector.90 The

other group91’92 wanted to make

rnonolithic gels

to

be sintered nto dense

glasses or ceramics, and found that large

crack-free bodies could be made within

vvide ranges of concentration of reac-

tants. Although supercritical drying gives

very good results or silica, the high tem-

peratures and pressures make the

process expensive and dangerous. A

convenient alternative s to exchange the

pore liquid for a substance with a much

lower critical point. As shown in Table I,

carbon dioxide has T, = 31“C and

Pc=7.4 MPa,

so

the process can be

performed near ambient temperatures.

Supercritical drying following COP ex-

change has become a standard tech-

nique for preparing biological samples for

TEM

examination.93 t was apparently first

applied for producing monolithic silica

gels by Woignier,94 and was indepen-

dently developed by Tewari

ef

al.95 for

making large windows. For some materi-

als, supercritical treatment in alcohol

causes dissolution,

so

a milder process

is essential. Brinker et al.96 used CO2 ex-

change

to

make aerogels of lithium

borate compositions that would dissolve

in alcohol. This would seemto be an ideal

way of making aerogels, but it does have

some disadvantages. Long times can be

required to achieve complete solvent ex-

change, especiaily because

C 0 2

is not

miscible with water (Kistler88 notes that li-

quidlliquid interfaces ormed by rmmisci-

ble liquids could produce capillary

compression of the gel). It may be neces-

sary to exchange first with a mutual

sol-

vent such as amyl acetate,96 then

to

flush

for hours with liquid C02. Moreover, be-

cause of the low density of the dried

body, sintering of monolithic crystalline

aerogels to full density is impractical.

Another way

of

avoiding the presence

of the liquidlvapor interface is

to

freeze

the pore liquid and sublime the resulting

solid under vacuum. This process

of

freeze-drying s widely used in the prepa-

ration of foods,47 but does not permit the

preparation of monolithic gels. The re a

son is that the growing crystals reject the

gel network, pushing t out of the way until

it is stretched to the breaking point. It is

this phenomenon that allows gels to be

used as hosts for crystal growth:97198 the

gel is so effectively excluded that the crys-

tals nucleated in the pore liquid are not

contaminated with the gel phase; the

crystals can grow up to a size of a few

millimeters before the strain is

so

great

that macroscopic ractures appear in the

gel. If a silica sol is frozen, flakes of silica

gel (sometimes called lepidoidal silica) are

produced;99 f freezing is done unidirec-

tionally, fibers of gel are obtained.’OO*”J’

Attempts to freeze-dry gels typically result

in flakes (e.g., Ref. 102) or in translucent

bodies with large pores that are the “fos-

sils” of the crystals.

VI. Conclusions

Although the principles of drying have

been recognized for decades, the means

of calculating drying stresses and strains

have been developed relatively recently.

The stresses result from a gradient in the

pressure in the liquid in the pores of the

drying body. The stress increases with

the drying rate and the size of the body,

and is inversely related to the permeabil-


11/12

January 1990

ity of the structure. It is the latter factor that

makes gels so much harder to dry than

conventional ceramics: their small pores

result in very low permeability. Fracture

may result from the action

of

drying

stresses on preexisting flaws, but in many

cases seems to result from flaws gener-

ated by the irregularity of the drying front

as it enters the body at the critical point.

Unfortunately, many of the physical

properties needed to predict failure (e.g.,

permeability and critical stress intensity

of wet bodies) have not yet been

measured.

A

variety of strategies have been de-

vised

to

avoid fracture during drying.

These include strengtheningof the solid

network by aging or chemical additives,

increasing permeability by increasing

pore size, and reducing capiliary pres-

sure by increasing pore size, reducing in-

terfacial energies, or drying under

supercritical conditions. Each of these ap-

proaches nvolves some tradeoff (for ex-

ample, in processing time or sintering

temperature),

so

the best method must

be chosen by regarding the process as

a whole.

Theory of Drying

and Pore Structure. Academic Press, New York,

1979.

16H. Darcy. Les Fontaines Publiques de la Ville de

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Chaussees et des Mines, Paris, France, 1856.

17J. van Brakel, "Pore Space Models for Trans-

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19G. W. Scherer and R. M. Swiatek, "Measurement

of Permeability: II. Silica Gels"; to be published in

Non-Cryst. Solids.

20K. K. Watson, "An Instantaneous Profile Method

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2

[4] 709-15 (1966).

21K. J. Packer, "The Dynamics of Water in Heter-

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22J. Warnock, D. D. Awschalom, and M. W. Shafer,

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23M. W. Shafer,

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24D. Y. C. Chan and

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G. Horn, "The Drainage

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25J. N. Israelachvili, "Measurement of the Viscosi-

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Colloid lnterface

26J. van Alsten. S. Granick. and J.

N.

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27J. Crank, Mathematics of Diffusion. Clarendon

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2*T, Tanaka and

D.

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29H. van Olphen, Introduction

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Clay Colloid

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30T. K. Sherwood, "The Drying of Solids-I,"

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3 T K. Sherwood, "The Drying of Solids-11," lnd.

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32T. K. Sherwood, "The Drying of Solids-Il l,'' lnd.

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33R.W. Ford, Ceramics Drying. Pergamon Press,

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W . B. Keey, Drying, Principles and Practice. Per-

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"F. H. Clews, Heavy Clay Technology. Academ-

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13

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(1977).

517-39 (1961).

I

I

Temperature Tc

Fig. 9. Schematic phase diagram, indicat-

ing temperature/pressure path followed dur-

ing supercritical drying. At the critical point

(T,,P,)

the densities

of

the liquid and vapor

phases are the same and the surface tension

is zero.

Table I: Critical Points of Selected Solvents

Substance Formula T ("C) P, ( M W

31.1 7.36

19.7

2.97

Carbon dioxide CO2

Freon

11

6 CF3CF3

Methanol CH30H 240 7.93

Ethanol C~H SO H 243 6.36

Water H20 374

22.0

'Data collected by Tewari, Hunt, and Lofltus (Ref.

95)


12/12

14

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