J. Am. Cerum. Soc.

, 73 [ l ] 3-14 (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 examines the stages of dry-
ing, with the emphasis 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 for
transport of liquid. By analyzing the in-
terplay of fluid flow and shrinkage of the
solid network, 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 probability 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, shrinkage, cracking,
models, gels.]
1. Introduction
REMOVAL of liquid is particularly trouble-
some in sol-gel processing, because gels
tend to warp and crack during drying, and
avoiding fracture requires inconveniently
slow drying rates. However, liquid trans-
port processes are also of importance in
other ceramic-forming operations, includ-
A. H. Heuer-contributing editor
Manuscript No. 198096. Received September 25,
"Member, American Ceramic Society.
1989; approved October 17, 1989.
ing slip casting, tape casting, binder burn-
out, liquid-phase sintering, and drying of
clays. Indeed, the principles 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 ceramics. In
most cases, liquid flows through a porous
body in response to a gradient in pres-
sure; at the same time, the pressure
causes deformation of the solid network,
and dilatation of the pores through which
the liquid moves. In this review we ana-
lyze the interaction between flow of the li-
quid 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, but the analysis is quite general.
The driving forces for shrinkage of the
solid and the mechanisms for transport of
the liquid are discussed in Section II. The
stages of drying are outlined in Section 111,
and a model for calculation of drying
stresses is developed in Section IV. The
cause of cracking during drying and 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 body shrinks to
as little as one-tenth of its original volume.
The forces that produce the shrinkage of
the solid network are discussed below.
(A) Capillary Pressure: If evaporation
George W. Scherer has been a member
of the Central Research Department of
E. I. du Pont de Nemours & Co. since
1985. His work at Du Pont has dealt prrn-
cipally with sol-gel processing, and es-
pecially with drying. In collaboration with
J eff Brinker of Sandia National Labs, he
has written a book entitled Sol-Gel
Science that will be published by Aca-
demic Press in February. From 1974 to
1985, Dr. Scherer was at Corning Glass
Works, where his research included op-
tical fiber fabrication, viscous sintering,
and viscoelastic stress analysis. The lat-
ter work was the subject of his first book,
Relaxation in Glass and Composites
(Wiley, 1986). He received his B.S. and
M.S. degrees 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
Vol. 73, No. 1
4
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, Pi , 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 isneeded 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, atthe 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 in
the liquid is related to the radius of curva-
ture (r) of the meniscus by*
where y ~v is the liquidlvapor interfacial
energy (or surface tension). When the cen-
ter of curvature is ir! the vapor phase, the
radius of curvature is negative and the li-
quid is in tension (PX).
The maximum capillary tension (PR) in
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, the minimum radius of the menis-
cus is
where 8 is the contact angle. If 8 is go",
then the liquid does not wet the solid and
the liquidhapor interface is flat (F - w,
P= 0). If 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 ys ~ are the solidlvapor
and solidliiquid interfacial energies,
respectively. The specific surface area of
a porous body (interfacial area per gram
of solid phase), S, is 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 is 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
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 produce 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-
permeable membrane, permitting trans-
port in only one direction. This
phenomenon could be of importance in
clays and gels. Alkoxide-derived gels
generally contain a solution of liquids that
differ in volatility (viz., alcohol and water),
so evaporation creates a composition gra-
dient, and osmotic flow may result.
(C) Disjoining Pressure: Disjoining
forces are short-range forces resulting
from the presence of a solidlliquid inter-
face. The most important examples are
dou ble-layer repulsion between charged
surfaces and interactions caused by struc-
ture created in the liquid by dispersion
forces. Liquid molecules, especially
water,43 tend to adopt 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 liquid 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 produce 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 drying of gels, when
the pore diameter may approach 2 nm.
Macey7 argues that electrostatic repul-
sion between particles of clay produces
tension in the liquid that draws flow from
the interior of a drying body. Even for
clays, in which these phenomena are most
evident, it has been 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 comparable
to the capillary pressure in pores with radii
>14 nrn (according to Eq. (l), assuming
yLv= 0.072 J lm2 for water). In the case of
gels, the pores are generally smaller than
that, so capillary forces are expected to
dominate.
(0) Moisture Stress: Moisture stress
or moisture potential (y) is the partial
specific Gibbs free energy of liquid in a
porous medium, and is given byir
(5)
y = ~
where eL and V,,, are the density and mo-
lar volume of the liquid, Rs is the ideal
gas constant, T is the temperature, pv 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
column 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 moisture 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 ob-
tained by measuring the vapor pressure
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 pro-
duce osmotic pressure) cause diffusion,
so it is necessary to apply 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 capillary pressure and gravitation are
negligi ble.12
( e x ) 1' 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 recommends 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.
6
Journal of the American Ceramic Society-Scherer
VOl. 73, No. 1
(2) Transport Processes
(A) Darcy's Law: Fluid flow through
porous media obeys Darcy's la~,14~15
which states that the flux of liquid, J, is
proportional to the gradient 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 permea-
bility and has units of area. Positive flux
moves in the direction of increasingly
negative pressure (i.e., the flow is toward
regions of greater tension in the liquid).
Equation (6) is an empirical equation der-
ived from observation of flow of water
through soi1,16 but it is analogous to
Poiseuille's law for flow of liquid through
a straight circular pipe. This analogy has
given rise to many models for the perme-
ability of porous media based on
representations of the pores by arrays of
tubes, many of which are discussed in the
excellent texts by Scheideggerl4 and Dul-
lien;l5 van Brake117 offers a critical 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 actual pores. This equation
is reasonably successful for many types
of granular materials, but it often fails, and
should be applied with caution.
The proportionality of the flux to the
pressure gradient is obeyed by many
materials, including those with pores
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),
Darcy'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), it can only be transported by
diffusion of the vapor. The permeability of
unsaturated 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. Spectroscopic 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) is proportional
to the concentration gradient (VC):27
where D, is the chemical diffusion coeffi-
cient, C is the concentration, and p is the
chemical potential. As noted above, diffu-
sion can contribute to the shrinkage of gels
in special cases (e.g., when the gel is im-
mersed in a salt solution) and may be im-
portant during evaporative drying, if a
concentration gradient develops in the
pores by preferential evaporation of one
component of the pore liquid.
In some cases, a gradient in concen-
tration of the solid phase can produce os-
motic transport (as in the swelling of some
organic polymers28 or clay29), but it is not
clear whether transport occurs by diffusion
or flow. One can compare 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 than a few
times the diameter of the liquid molecule.
However, this conclusion applies only to
situations such as flow within a clay body
(where tension in the liquid is produced
by disjoining forces), 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 in 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 pore liquid, flow from
the interior of the gel cannot eliminate it;
only interdiffusion within the pores can do
Il l . Stages of Drying
The stages of drying were clearly
discussed in the classic work of
Shewmd30-32 60 years ago. Several texts
provide qualitative descriptions of the
phenomenology and detailed discussion
of the technology of drying,%-% The scien-
tific aspects are discussed in several very
good reviews (e.g., Refs. 8 and 36) and
in the series of books called Advances in
Dr~ing.37~38
so.
demonstrated. using nuclear magnetic (1) Constant Rate Period
resonancgl or optical s pectro~copy, ~~~~~
The first stage of drying 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
January 1990
Theory of Drying 7
evaporation rate is close to that from an
open dish of liquid, as indicated, for ex-
ample, by Dwivedis data for drying of alu-
mina ge1.39The rate of evaporation, V, , is
proportional to the difference between pv
and the ambient vapor pressure, pA:
(9)
where k is 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 , increases 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. If 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 ismuch 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 et d. 43
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, in 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 . The impor-
tance of contact angle is nicely illustrated
by the work of Mitsyuk et a1.45They pre-
pared aqueous silica gels from sodium sili-
cate, then soaked them in various alcohols
8 Jour
Fi g. 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 in 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~s36~47.48 of the many theories that
have been proposed to 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., the liquidlvapor interface) is 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 for the stability of
the drying front: the capillary pressure is
SO low in advanced regions of the front
that the radius of the meniscus is too 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 comparable to the evaporation
ratel the funicular condition is preserved.
However, as the distance from the exteri-
or to the drying front increases, the capil-
lary pressure gradient decreases and
therefore so does the flux. Eventually (if the
body is thick enough) it becomes so slow
that the liquid near the outside of the body
is isolated in pockets (i.e., enters the pen-
dular condition), so flow to the surface
stops and liquid is removed from 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 the
ambient temperature and the rate of
evaporation becomes less sensitive to ex-
ternal conditions (temperature, humidity,
draft rate, etc.). As indicated in 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 relie~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 to the aver-
aging implicit in Darcy's law, where the
permeability, D, "smears out" the ge-
ometrical details of the microstructure. Use
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/.,55-56 who 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 interface had been. This would
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 Darcys 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 J ust 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, the 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~7~12 or 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~SCOUS, ~~ or viscoelastic.66-70 For
the sake of discussion, we will employ the
simpler elastic analysis. When the network
isassumed to be elastic, Eq. (13) has the
mathematical form 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 this case, a pore is a space not occupied by
the solid phase, which may be occupied by liquid
andlor gas During the 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.
(1 5) 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. (1 5) that results when the permea-
bility and elastic properties vary with the
porosity (and therefore with position in 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<O<OR.
Beyond time OR the tension at the sur-
face cannot increase, but P(z,O) rises in
the interior until the tension in the liquid
becomes uniform (Fig. 6(E)); as the pres-
sure gradient decays, the differential
strain disappears and the stress drops
toward zero (Fig. 6(F)). It can be
shown63,69 that the stress is related to the
tension in the liquid by
a,= P-(P> (1 8)
where (P) is the average tension in the
plate. If the pressure is hydrostatic,
P=(P), the network is uniformly com-
pressed; there is no differential strain and
no stress. In the limiting case of extremely
rapid evaporation, P(L,OR) reaches PR
while the tension is still zero throughout
the bulk of the liquid. Then (R=O and
Eq. (18) indicates that the stress at the ex-
terior surface has its maximum possible
value, ux(L)=PR. Under less drastic con-
ditions, as illustrated in Fig. 6, the maxi-
mum stress occurs while the pressure
10 Journal of the American Ceramic Society-Scherer Vol. 73, No. 1
distribution is parabolic, in which case Eq.
(1 8) becomes61
The stress increases in proportion to the
thickness of the plate and the rate of
evaporation, and in inverse proportion to
the permeability; that is, the stress is 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 sphere, 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 volume that increases as P and
r 3, respectively. Since these results are
derived from Eq. (13), they are valid only
as long as the pores remain filled with li-
quid. At some point the network will stop
shrinking and the meniscus will retreat
into the gel; then Eq. (19) will apply only
within the saturated pores inside the ge1.63
(3) Diffusion
If the pores contain a solution of liquids
with intrinsic diffusion coefficients D, and
D2, then diffusion contributes to the trans-
port and the diffusion term must be ad-
ded to the flow term. Then Eq. (13)
becomes72
h = -v. - VP +(I -@)V.
K i
Note that diffusion has no influence if 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 concentration gra-
dient (since interdiffusion of liquids is
rapid), so diffusion can extract liquid from
the interior of the body 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 expla-
nation for the phenomenon of cracking
during drying. Any suitable theory should
a'ccount for the common 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-derived gels.39174
We now examine two models of fracture
during drying, a macroscopic model
(described in Section IV) that attributes
cracking to stresses produced by a pres-
sure gradient in the liquid phase, and a
microscopic model that explains crack-
ing as a result of the distribution of pore
sizes.
The stress that causes fracture is not
the macroscopic stress, a, , that acts on
the network. Rather, it is the stress con-
centrated at the tip of a flaw of length c
which is proportional to75
Fracture occurs when o,>K\,, where KI,
is a material property called the critical
stress intensity.76 It is reasonable to as-
sume that the flaw size distribution is in-
dependent 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 . , it does not offer any explanation for
the common observation that slowly
dried gels crack at the critical point. The
stress is predicted to rise continually until
that moment, but there is no sudden jump
predicted for a, at time f3R that would en-
hance the likelihood of cracking.
The microscopic model for fracture is
based on the idea77178 illustrated in Fig.
7. After the critical point, liquid is 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 appears
to account quite clearly for the occur-
rence of cracking at the critical point.
However, the flaws produced 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 by supposing that the flaws per-
colate through the structure until they
achieve the critical length. A more impor-
tant problem with this mechanism is that
it does not explain the importance of dry-
ing rate or body size. The local stresses
result from the local heterogeneity of the
microstructure, so fracture should be in-
evitable when the pore size distribution
is wide.
Another version of this model would at-
tribute the flaws to the irregularity of the
drying front, illustrated schematically in
Fig. 8. The width of the drying front, w,
is 2 or 3 orders of magnitude larger than
the pore 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 proportional to
ac =a x e (21)
January 1990
PRwlQ, However, Shawsz has shown
that
w a pp ) - 1 ~ a (VE)-1/2
(22)
which means that the drying front be-
comes smoother (w decreases) as the
drying rate increases. Thus
(23)
oc =pRwl/2 a (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,<PR) and is
probably responsible for the creation
of critical flaws, the probability of frac-
ture can be reduced by decreasing
OLV cos (0) or by increasing the pore
size. Surfactants can be added to the
pore liquid to reduce the interfacial ener-
gy and thereby decrease the capillary
stress. It has been demonstrated for
particulate44 and alkoxide-derived77 gels
that cracking is reduced by surfactants,
though it is not necessarily eliminated.
The shrinkage at the critical point is
reduced by surfactants;9177 this will in-
crease the permeability of the gel and,
since Eq. (19) indicates that oxal ID, will
contribute to reduction of the stress dur-
ing the CRP. Since larger pores will de-
mand higher firing temperatures, there is
a tradeoff between easy drying and easy
sintering.
Aging before drying helps to strength-
en the network and thereby reduce the
risk of fracture.77 This has been demon-
strated by direct measurements of the
modulus of rupture79180 and stress inten-
sity factor81 of gels subjected to various
periods of aging in their own pore liquid.
The process is accelerated in an aggres-
sive chemical environment where the rate
of condensation of M-OH groups, as well
as dissolution and reprecipitation of the
solid phase, is accelerated. For exam-
ple, Mizuno et al.82found that they could
dry silica gels up to 5 times faster after
u, =uXwl/2 a (VE)3/4
Theory of Drying 11
The lower bulk density of the aged gels
indicates that they were coarser, so the
capillary pressure was lower and the
permeability higher, and the gels were
probably stiffer and stronger
Drying control chemical additives
(DCCA)78 are reported to allow faster dry-
ing when they are included in the liquid
phase One example is formamide
(NH2CHO), which is used to replace
about half of the solvent ordinarily used
when making silica gels from alkoxide
The resulting gel is found to be harder (as
determined by indentation tests83) and to
have a larger and more uniform pore size,
Fig. 6. Drying behavior of flat plate: (A) normalized pressure (P/PR) in liquid versus reduced
time (0 =th) at several locations in the plate (exterior surface at z/L =1 and midplane at ziL =0);
(B) normalized stress (ox/PR) versus reduced time at same locations as in (A); (C) pressure distri-
bution at several times during CRP (KO,); (D) normalized stress distribution at same times as
in (C); (E) normalized pressure distribution at several times during FRP (@>OR); (F) normalized
stress distribution at same times as in (E). From Ref. 63.
soaking f& 24 h in 4N HCI or 2N NH3.
12 Journal of the American Ceramic Society-Scherer Vol. 73, No. 1
Constant rate period
Fig. 7. Illustration of microscopic model:
during the constant rate period, meniscus has
same radius of curvature for pores of 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., liquid-filled) and un-
saturated regions. Flaw of length c is subjected
to stress over width w 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 (=150C). 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
J lmz), but not vapors of methanol (0.023
J lm2) 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, the 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 group9192 wanted to make
rnonolithic gels to be sintered into 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 for silica, the high tem-
peratures and pressures make the
process expensive and dangerous. A
convenient alternative is 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 It 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 seem to 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 C02 is not
miscible with water (Kistler88 notes that li-
quidlliquid interfaces formed 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 is widely used in the prepa-
ration of foods,47 but does not permit the
preparation of monolithic gels. The rea
son is that the growing crystals reject the
gel network, pushing it 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 fractures appear in the
gel. If a silica sol is frozen, flakes of silica
gel (sometimes called lepidoidal silica) are
produced;99 if 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-
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 strengthening of 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 involves 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.
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I I
Temperature Tc
Fig. 9. Schematic phase diagram, indicat-
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Table I: Critical Points of Selected Solvents*
Substance Formula T, ("C) P, ( MW
31.1 7.36
19.7 2.97
Carbon dioxide CO2
Freon 11 6 CF3CF3
Methanol CH30H 240 7.93
Ethanol C~HSOH 243 6.36
Water H20 374 22.0
'Data collected by Tewari, Hunt, and Lofltus (Ref. 95)
14 Journal of the American Ceramic Society-Scherer
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