1990 scherer theory of drying

5/22/2018 1990 Scherer Theory of Drying

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J . A m . Cerum. Soc. 73 [ l ] -14 1990)journal Theory of DryingGeorge 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 constantrate period CRP), when the pores arefull of liquid. It is during the CRP thatmost of the shrinkage occurs and thedrying stresses rise to a maximum . Weexamine the forces that produce shrink-age and the mechanisms responsible ortranspo rt of liqu id. By analyzing the in -terplay of fluid flow and s hrinkag e of thesolid ne twork, it is possible to calculatethe pressure distribution in the liquid inthe pores. The tension in the liquid isfound to be greatest near the drying sur-face, resulting in greater compressivestresses on the network in that region.This produces differential shrinkage ofthe solid, which is the cause of crackingduring drying. The p robab ility of fractureis related to the size of the body, the rateof evaporation, and the strength of thenetwork. A variety of strategies for avoid-ing fracture during drying are discussed.[Key words: drying, sh rinkage, cracking,models, gels.]

1 IntroductionREMOVALof liquid is particularly trouble-some in sol-gel processing, because gelstend to warp and crack during drying, andavoid ing fracture requires inconvenientlyslow dryin g rates. Howe ver, liquid trans-port processes are also of imp ortance inother 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 ofclays. Indeed, the princip les of flow in po-rous media are of such general interestthat they have been frequently redisco-vered over the past 60 years, and rele-vant literature is found in fields includingsoil science, food science, and polymermaterials science, as well as ce ramics . Inmost cases, liquid flows through a porousbody in response to a gradient in pres-sure; at the same time, the pressurecauses de formation of the solid network,and d ilatation of the p ores through wh ichthe liquid moves. In this review we ana-lyze the interaction between low of the li-qu id and dilatation of the solid in order topredict the stresses and strains that de-velop during drying. Special attention isgiven to the problems encountered in dry-ing gels, bu t the analysis is quite general.The driving forces for shrinkage of thesolid and the mechanisms for transport ofthe liquid are discussed n Section II.Thestages of dryin g are outlined in Section 111,and a mo del for calculation of dryingstresses is developed in Section IV. Thecause of c racking during d rying an d vari-ous strategiesfor avoiding fracture are de -scribed in Section V. These topics arediscussed in greater detail in a forthcom-ing book.'

I I Deformation and Flow1) Driving Forces for Shrinkage

The first stage of drying is illustrated inFig. 1(B): for every unit volume of liquidthat evaporates, the volume of the bodydecreases by one unit volume, so the li-quidlvapor interface (meniscus) remainsat the surface of the body. In gels, thisstage continues while the bo dy shrinks toas little as one-tenth of its orig ina l volume.The forces that pro duc e the shrinkage ofthe solid network are discussed below.A) CapillaryPressure: If evaporation

George W. Scherer has bee n a mem berof the Central Research Departm ent ofE. I. du Pont de Nemours & Co. since1985 . His work at Du Pont has dealt prrn-cipally with sol-gel proces sing, and es-pecially with drying. In collaboration withJeff Brinker of Sandia N ational Labs, hehas written a book entitled Sol-GelScience that will be published by Aca-dem ic Press in February. From 197 4 to1985 , Dr. Scherer was at Cornin g GlassWorks, 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. andM.S. egrees in 1972 and his Ph.D. inmaterials science in 1974 , all from MIT,where his thesis work was on crystalgrowth in glass.

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Vol. 73,No. 1Journal of the American Ceramic Society Schererof liquid from the pores were to exposethe solid phase, a solidlliquid interfacewould be replaced by a more energeticsolidhapor interface. To prevent such anincrease in the energy of the system, liquidtends to spread from the interior of thebody to cover that interface. Since the vol-

*The stress in the liquid, P. is positive when theliquid is in tension. The pressure, P i , follows the op-posite sign convention (Pi = - P ) , so tension isnegative pressure.

Fig. 1. Schematic illustrationof drying process: black network represents solid phase and shad-ed area is liquid filling pores. A) Before evaporation begins, the meniscus is flat. B)Capillarytension develops in liquid as it "stretches" to prevent exposure of the solid phase, and networkis drawn back into liquid. The network is initially so compliant that little stress is needed to keepit submerged, so the tension in the liquid is low and the radius of the meniscus is large.As thenetwork stiffens, the tension rises and, at the critical point (end of the constant rate period), theradius 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 curvedas indicated in Fig. 2. The tension 0 nthe liquid is related to the radius of curva-ture (r) of the meniscus by*where y ~ vs the liquidlvapor interfacialenergy (or surface tension). When the cen-ter of curvature s ir the vapor phase, theradius of curvature is negative and the li-quid is in tension PX).The maximum capillary tension PR)nthe liquid occurs when the radius of themeniscus is small enough to fit into. thepore; 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 andthe liquidhapor interface is flat F w ,P = 0). f 8= 0 the solid surface is coveredwith a liquid film. Of course, the pores ina real body are not cylindrical, but it canbe shownW that the maximum tension isrelated to the surface-to-volume ratio ofthe pore space, SplVp:

where ysv and y s ~ re the solidlvaporand solidliiquid interfacial energies,respectively. The specific surface area ofa porous body (interfacial area per gramof 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), andes is the density of the solid skeleton (theskeletal density). The quantity VplSp isalso known as the hydraulic radius.As weshall see, during most of the drying pro-cess the capillary tension is smaller thanthis maximum value.(6) Osmotic Pressure: Osmotic pres-sure (n) is produced by a concentrationgradient, as in the case of pure waterdiffusing through a semipermeable mem-brane to dilute a salt-richsolution on theother side. As indicated in Fig.3 pressuren must be exerted on the solution (or atension of - n must be exerted on thepure liquid) to prevent :he water from en-tering the solution. The pressure s a meas-ure of the difference in chemical potentialbetween the pure liquid and that in the so-lution. An analogous situation can arise ifthe pores of the drying body contain a so-lution of electrolyte: evaporation of solventincreases the salt concentration near thedrying surface,so liquid diffuses from theinterior to reduce the concentration gra-dient; the decrease in the volume of liquidin the interior causes tension in the liauid


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January 1990 Theory of Drying 5that remains there. If the pores are large,the diffusive flux is matched by counter-flow of liquid toward the interior, and nostress develops. However, if the pores aresmall enough to inhibit flow, diffusion awayfrom the interior can p roduce tension inthe liquid in that region; then the balanc-ing compression in the solid phase (whichin principle could approach fl can pro-duce shrinkage. In such a situation, thesolid network plays the role of a semi-permeab le mem brane, p ermitting trans-port in only one direct ion. Thisphenom enon could b e of imp ortance inclays and gels. Alkoxide-derived gelsgenerally contain a solution of liquids thatdiffer in volatility (viz., alcohol and w ater),so evaporation creates a com position gra-dient, and osmotic flow may result.(C) Disjoining Pressure: D isjoiningforces are short-range forces resultingfrom the presence of a solidlliquid inter-face. The most important examples aredou ble-layer repulsion between ch argedsurfaces and interactions caused by struc-ture created in the liquid by dispersionforces. Liquid molecules, especiallywater,43 tend to adop t a special structurein the vicinity of a solid surface. The inter-action with the surface is so strong that ad-sorbed layers=I nm thick resist freezing.6As evaporation occurs and solid surfacesare brought together, repulsive forces aris-ing from electrostatic repulsion, hydrationforces, and solvent structure resist contrac-tion of the gel. The pore liqu id will diffuseor flow from the swollen interior of the geltoward 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 inchemical potential in the liquid phase.Since these forces become importantwhen the separation between Surfaces issmall, they are most likely to be impor-tant near the end of dryin g of gels, whe nthe pore diameter may approach 2 nm.Macey7 argues that electrostatic rep ul-sion between particles of clay producestension in the liqu id that draws flow fromthe interior of a drying body. Even forclays, in which these phenomena are mostevident, it has be en argued8 that osmoticforces must be less important than capil-lary pressure, because moisture gradientspersist in clays for long periods whenevaporation is prevented. In addition, ithas been shown that the final shrinkageof kaolinite clay during drying is directlyrelated to the surface tension of the poreliquid.9 The swelling pressure of clays inwater is < I 0 MPa,lO which is com parableto the capillary pressure in pores with radii>14 nrn (according to Eq. (l),assumingyLv= 0.072 Jlm2 for water). In the case ofgels, the pores are generally smaller thanthat, so capillary forces are e xpected todominate.0) Mo isture Stress: Moisture stressor moisture potential y) s the partialspecific Gibbs free energy of liquid in aporous medium, and is given byir

5)=where eL and V,,, are the density and mo -lar volume of the liquid, Rs is the idealgas constant, T is the temperature, p v isthe vapor pressure of the liquid in the sys-tem, and po is the vapor pressure over aflat surface of the pure liquid. In soilscience12 it is conventional to define themoisture potential in terms of the equilibri-um height to which it would draw acolum n of water, so a factor of g (thegravitational acceleration) would be includ-ed in the denominator on the right side ofEq. (5). The m oisture potential is quite in-clusive, because the vapor pressure isdepressed by factors including capillarypressure, osmotic pressure, hydrationture potential subsumes all of the drivingforces discussed above, and can be o b-tained by m easuring the vapor p ressureof 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 thetotal potential to different transportprocesses. In soil science, it is customaryto assume that fluid flow is driven by thegradient in moisture potential, but it is donewith the understanding that factors otherthan cap illary pressure and gravitation arenegligible.12

e x ) k)

forces, and adsorption forces. Thus, mois- A) (B)Fig. 2 To prevent exposure of the solidPhase 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 ofthe impermeable membrane; pressure il would have to be exerted on the solution to preventthe influx of water.


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6 Journal of the American Ceramic Society Scherer VOl. 73, No. 12) Transport Processes

A) Darcy's Law: Fluid flow throug hporous media obeys Darcy's la~,14~15which states that the flux of liquid, J , isproportional o the grad ient in pressure inthe liquid, VP

The flux is in units of volume per area ofthe porous body (not the area occupiedby the liquid) per time, PL is the force perunit area of the liquid, q~ is the viscosityof the liquid, and D is called the p ermea-bility and has units of area. Positive fluxmoves in the direction of increasinglynegative pressure (i.e., he flow is towardregions of greater tension in the liquid).Equation (6) s an empirical equation der-ived from observation of flow of waterthrough soi1,16 but it is analogous toPoiseuille's aw for flow of liquid througha straight circular pipe. This analogy hasgiven rise to man y models for the perme -ability of porous media based onrepresentationsof the pores by arrays oftubes, many of which are discussed in theexcellent texts by Scheideggerl4 and D ul-lien; l5 van Brake117 offers a critic al reviewof 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 ofthe relative density and specific surfacearea:(7)

The factor of 5 is an empirical correctionfor the noncircular cross section and non-linear path of a ctual pores . This equationis reasonably successful for many typesof granular m aterials, but it often fails, andshould be applied with caution.The p roportionality of the flux to thepressure gradient is obeyed by manymaterials, including those with po ressmaller than 10 nm, as in porousVycort718 and alkoxide -derived gels.19Even in unsaturated bodies (i.e., where thepores contain both liquid and gas),Da rcy's law is obeyed1420 as long as theliquid phase is funicular (i.e., interconnect-ed); if the liquid is pendular (i.e., isolatedin pockets), t can only b e transported bydiffusion of the vapor. The permeability ofunsa turated materials is a strong functionof liquid content and shows considerablehysteresis as the liquid content is raisedand lowered.In gels the pores are so small that alarge portion of the liquid may be in struc-tured layers within =1 nm of a solid sur-face, so the effective viscosity may begreater 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 giveincorrect results. Spe ctrosco pic methodsindicate an increase in viscosity by a fac-tor of -3, so the effect of solvent structureon the flux in gels can be substantial.(B) Diffusion: According to Fick'slaw, the diffusive flux Jo) s proportionalto the concentration gradient (VC):27

where D, is the chem ical diffusion coeffi-cient, C is the concentration, and p is thechem ical potential. As noted above, diffu-sion can contribute to the shrinkage of gelsin special cases (e.g., when the gel is im-merse d in a salt solution) and may be im -portant during evaporative drying, if aconcentration gradient develops in thepore s by p referential evaporation of onecomponent 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 eorganic polymers28 or clay29), but it is notclear whether transport occurs by diffusionor flow. One can com pare the fluxes giv-en by Eqs. (6) and (8) by converting thechemical potential gradient to pressure-volume work, then relating the diffusioncoefficient to the viscosity by use of theStokes-Einstein equation.1 The conclusionis that flow is faster than diffusion when-ever the pore diameter is more han a fewtimes the diameter of the liquid molecule.

Howe ver, this conclusion a pplies only tosituations such as flow within a clay bod y(where tension in the liquid is produ cedby disjo ining orces), where there is a gra-dient in concentration of solid phase. Flowcannot reduce a concentration gradient inthe liquid phase. For example, if a gel isimmersed n a salt solution, flow of the so-lution into the pores does not affect thedifference in salt concentration betweenthe bath and the original pore liquid; thatcan be achieved only by diffusion. Simi-larly, if evaporation creates a concentra-tion gradient in the p ore liquid, flow fromthe interior of the gel cannot eliminate it;only interdiffusion within the po res can doI l l Stages of Drying

The stages of drying were clearlydiscussed in the classic work ofShewmd30-3260 years ago. Several textsprov ide qualitative desc riptions of thephenom enology and detailed discussionof the technology of drying,%-% he scien-tific aspects are discussed in several verygood reviews (e.g., Refs. 8 and 36) andin the series of b ooks called Advance s inDr~ing.37~38

so.

demonstrated. using nuclear m agnetic ( 1 ) Constant Rate Periodre 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 theconstant rate period (CRP), because therate of evaporatlon per unit area of the dry-ing surface is independent of time 7,8 TheCorningGlass Works Corning NY


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January 1990 Theory of Drying 7evaporation rate is close to that from anopen dish of liquid, as indicated, for ex-ample, by Dwivedis data for drying of alu-mina ge1.39 The rate of evaporation, V , , sproportional o the difference between pvand the ambient vapor pressure, pA:

(9)where k s a factor that depends on thetemperature, draft, and geometry of thesystem. The vapor pressure of the liquidis related to the capillary tension (P) byPv Po exp( --PVm

%ITFrom Eqs. ( l) , (9), and (10) we see thatevaporation will continue as long as

The fact that the evaporation rate is simi-lar to that of bulk liquid indicates that thevapor pressure reduction is insignificantduring the CRP. However, in some gelsthe pores are so small that a significantreduction in pv could occur; moveover,the composition of the liquid in the porescould change with time if the initial liquidis a solution. The latter factors have beenproposedto explain the absence of a CRPfor an alkoxide-derived silica gel.4WIt seems reasonable to conclude thatthe surface of the body must be coveredwith a film of liquid during the CRP, be-cause the proportion of the surface co-vered by menisci shrinks faster than thetotal area, so the rate would decrease asthe body shrank if evaporation occurredonly from the menisci. However, Suzukiand MaedaQ proved that the evaporationrate can remain constant even when drypatches form on the surface of the body.There is a stagnant (or slowly flowing)boundary layer of vapor over the dryingsurface, and if the breadth of the drypatches is small compared to the thick-ness of the layer, diffusion parallel to thesurface homogenizes the boundary layerat 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 ofnanometers. Therefore, transport of vaporacross the boundary layer obeys Eq. (9),and the rate of evaporation per unit areaof surface is constant, whether or not thereare small dry patches.Evaporation causes cooling of a bodyof liquid, but the reduced temperatureleads to a lower rate of evaporation, andthis feedback process equilibrates whenthe drying surface reaches the wet bulbtemperature (T,). As indicated by Eq. (9),V , ncreases as PA decreases, so T,decreases with the ambient humidity. Theexterior surface of a drying body is at thewet bulb temperature during the CRP.3The surface temperature rises only afterthe rate of evaporation decreases (in thefalling rate period discussed in Section Ill

(2)). For alkoxide-derived gels the vaporpressure 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 bythe solid phase, which therefore goes intocompression. f the network is compliant,as it is in alkoxide-derived gels, the com-pressive forces cause it to contract into theliquid and the meniscus remains at the ex-terior surface, as indicated in Fig. 1 B). Ina gel, it does not take much force to sub-merge the solid phase, so the capillarytension is low and the radius of the menis-cus is much larger than the pore radius.As drying proceeds, the network becomesincreasingly stiff, because new bonds areforming and the porosity is decreasing; themeniscus deepens and the tension in theliquid rises correspondingly. Once the ra-dius of the meniscus becomes equal tothe radius of the pores in the gel, the li-quid exerts the maximum possible force.That marks the end of the CRP: beyondthat point the tension in the liquid cannotovercome further stiffening of the network,so the meniscus recedes into the pores,leaving air-filled pores near the outside ofthe gel (Fig. 1 C)). Thus, during the CRP,the shrinkage of the gel is equal to the vol-ume of liquid evaporated; the meniscusremains at the exterior surface, but rdecreases continuously. This behavior isillustrated by the data of Kawaguchi etd . 4 3for alkoxide-derived gels; equivalentresults have been reported for particulategels made from fumed silica.44The end of the CRP is called the criticalpoint (or leatherhard point, n clay technol-ogy), and it is at this point that shrinkagevirtually stops.At the critical point, the ra-dius of curvature of the meniscus is smallenough to enter the pores, so the capil-lary tension is found from Eqs. (3)and (4):

For an alkoxide-derived gel with S-300 to800 m*/g, eb-0.4 to 1.6 glcm3, Q-0.2 to0.6, and yLv cos (0)-0.02 to 0.07 J/m*,this is an enormous pressure: P p 3 to200 MPa The amount of shrinkage thatprecedes the critical point depends on themagnitude of the maximum capillarystress, PR. Since PR increases with the in-terfacial energy (yLV) and with decreasingpore size, it is not surprising to find thatthe porosity of a dried body is greater (be-cause less shrinkage has occurred) whensurfactants are added to the liquid. For ex-ample, Kingery and Franc19 found a line-ar proportionality between YLV and drieddensity for clay bodies mixed with surfac-tants. It is important to recognize, howev-er, that the pressure depends on thecontact angle, and the surfactant could in-crease 0 while reducing y ~ v . he impor-tance of contact angle is nicely illustratedby the work of Mitsyuk et a1.45 They pre-pared aqueous silica gels from sodium sili-cate, then soaked them in various alcohols


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8 Jour

Fig . 4. After the critical point, the li-quidlvapor meniscus retreats into the pores ofthe body. In the first falling rate period, liquidis in the funicular state, so transport by fluidflow is possible. There is also some diffusionin the vapor phase.

Fig. 5. During the second falling rate peri-od evaporation occurs inside the body, at theboundary between the funicular (continuousliquid) 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 gelswere dried, the final porosity was foundto be linearly related to the heat of wet-ting. The heat of wetting is related* to thequantity ysv-ysL=yLv cos (8); in thiscase, yLv is nearly the same for all the al-cohols, so the variation in capillary stressis caused by 8.(2) First Falling Rate PeriodWhen shrinkage stops, further evapo-ration drives the meniscus into the body,as illustrated n Fig. 1(C); as air enters thepores, the surface may begin to lose itstranslucency.@ In the first falling rate peri-od (FRPI), the rate of evaporationdecreases and the temperature of the sur-face rises above the wet bulb temperature.Most of the evaporation is still occurringat the exterior surface, so the surface re-mains below the ambient temperature,and the rate of evaporation is sensitive tothe ambient temperature and vapor pres-sure.83 The liquid in the pores near thesurface remains in the funicular condition,so there are contiguous pathways alongwhich flow can cccur (Fig.4). At the sametime, some liquid evaporates within the un-saturated pores and the vapor is transport-ed by diffusion. Analysis of this situationinvolves coupled equations for flow of heatand liquid and diffusion of vapor, withtransport coefficients that are generallydependent on temperature and con-,centration. There are several goodrevie~s36~47.48f the many theories thathave been proposed o descibe the FRP1.'The most complete and rigorous treatmentis by Whitaker.49,50Shaw51r52 performed an elegant seriesof experiments showing that the dryingfront (i.e., he liquidlvapor interface) s frac-tally rough on the scale of the pores, butstable on a much larger scale. It is thepressure gradient in the unsaturated re-gion that is responsible or the stability ofthe drying front: the capillary pressure isSO low in advanced regions of the frontthat the radius of the meniscus istoo largeto pass through the pores. Since the ir-regularity in the drying front is on the scaleof the pores, it is very small compared totlhe 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 dryingfront is visible as a sharp line between thetranslucent saturated region and theopaque dry region. No doubt this linewould be rough if observed in the SEM,but it is quite smooth on a macroscopicscale.3) Second Falling Rate PeriodAs the meniscus recedes into the body,the exterior does not become completelydry right away, because liquid continuesto flow to the outside; as long as the fluxof liquid is comparableto the evaporationratel the funicular condition is preserved.However, as the distance from the exteri-or to the drying front increases, he capil-

lary pressure gradient decreases andtherefore so does the flux. Eventually (if thebody is thick enough) it becomesso slowthat the liquid near the outside of the bodyis isolated n pockets (i.e., enters the pen-dular condition), so flow to the surfacestops and liquid s removed rom the bodyonly by diffusion of its vapor. At this stage,drying is said to enter the second fallingrate period (FRP2), where evaporation oc-curs inside the body (see Fig. 5 .31 Thetemperature of the surface approaches heambient temperature and the rate ofevaporation becomes ess sensitive to ex-ternal conditions (temperature, humidity,draft rate, etc.). As indicated n Fig.5, thedrying front is drained by flow of funicularliquid which evaporates at the boundaryof the funicularlpendular regions. In thependular region, vapor is in equilibriumwith isolated pockets of liquid and ad-sorbed films, and the principal transportprocess is expected to be diffusion ofvapor.As the saturated region recedes into thebody, the body expands slightly as the to-tal stress on the network is re lie~ed.32~43~~At the same time, differential strain buildsup because the solid network is beingcompressed more in the saturated regionthan near the drying surface. This cancause warping in a plate dried from oneside, as faster contraction of the wet sidemakes the plate convex toward the dry-ing side.% The fact that the warping is per-manent (i.e., does not spring back whendrying is complete) indicates that the un-saturated region retains some viscosity orplasticity during FRP2. As the saturated re-gion becomes thinner, its contraction ismore effectively prevented by the largerunsaturated region, and this raises the ten-sion in the network in the saturated region.This phenomenon probably accounts forthe observation by Simpkins et a/. thatcracks in drying gels often originated nearthe nondrying surface.Whitaker499 developed an analysis ofheat and mass transfer during drying ofrigid 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 comparedto the pore size, but small compared tothe sample. This is analogous o the aver-aging implicit in Darcy's law, where thepermeability, D, "smears out" the ge-ometrical details of the microstructure.U s eof Whitaker's model requires knowledgeof a large number of physical properties(permeability, thermal conductivity, diff u-sivity of vapor), and the analysis must beperformed numerically. A successful testof the model was performed by Wei eta/.,55-56who studied the drying of poroussandstone.

IV. Drying StressIf evaporation of liquid from a porousbody exposed the solid network, a solid/vapor interface would appear where asolid/liquid nterface had been. This would


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January 1990 Theory of Drying 9raise the energy of the system, becauseysv>vsL, so liquid tends to flow from theinterior to prevent exposure of the solid.As it stretches toward the exterior, the li-quid goes into tension, and this has twoconsequences: 1) liquid tends to flowfrom the interior along the pressure gra-dient, according to Darcys law; (2) thetension is balanced by compressive stressin the network that causes shrinkage. Thelower the permeability, the more difficultit is to draw liquid from the inside of thebody, and therefore the greater the pres-sure gradient that develops. As the pres-sure gradient increases, so does thevariation in free strain rate, with the sur-face tending to contract faster than the in-terior. It is the differential strain (i.e., thespatial variation in strain (for an elastic ma-terial) or strain rate (for a viscous materi-al)) that produces stress. This is analogousto the development of thermal stresses inresponse to a temperature gradient, anobservation that has been exploited by anumber of authors.57-60 Just as calculationof thermal stresses requires knowledge ofthe temperature distribution, prediction ofdrying stresses depends on calculation ofthe pressure distribution, which we nowexplore.( 1 ) Pressure Distribution

If we consider an isolated region of aporous body, the rate of change of the vol-ume of liquid in that region depends onthe divergence of the flux (i.e., the differ-ence between the flux entering and theflux leaving). During the CRP, when thepores are full of liquid, he change in liquidcontent must be equal to the change inpore volume,S which is. related to thevolumetric strain rate, E . Setting thesechanges equal, we obtain the equation forcontinuity (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 havedone this by using empirical (nonlinearelastic) equation~7~12r by assuming elas-tic behavior with the solid and liquidphases compressible57P or incompress-ible,63 a or allowing the network to bepurely V~S CO US ,~ ~r viscoelastic.66-70 Forthe sake of discussion, we will employ thesimpler elastic analysis. When the networkis assumed to be elastic, Eq. 13) has themathematical orm of the diffusion equa-tion. For the CRP it is appropriate to in-troduce the boundary condition that theflux at the exterior surface is constant:

$In his case, a pore is a space not occupied bythe solid phase, which may be occupied by liquidandlor 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 ofthe drying plate, u =z /L is the coordinatenormal to the drying surface, the dimen-sionless time is defined as O = flr and

where Kp and GPare the bulk and shearmoduli of the solid network (i.e., theproperties that would be measured withthe liquid drained away). By solving Eq.15) we obtain the pressure distributionin the drying body; the stresses andstrains follow from the constitutive equa-tions.1 612) Stress DistributionPhilip12 discusses at length themethods for solving the nonlinear versionof Eq. 15) that results when the permea-bility and elastic properties vary with theporosity (and therefore with position n thebody). In the simple case where theproperties are constant and the shrinkageduring drying is negligible, an analyticalsolution is readily obtained;63 typicalresults are shown in Fig. 6. The tensionP in the liquid rises until at the critical point

(time 0) it reaches the maximum valueat the exterior surface, P(L,OR)= PR, asshown in Fig. 6(A). If the evaporation rateis not too fast, the distribution through theplate becomes roughly parabolic at amuch earlier stage (when O=eR/3 in Fig.6(C)) and, since the stress depends onthe shape of the pressure distribution,a,is approximately constant (Figs. 6(B) and(D)) during the time interval OR/3


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10 Journal of the American Ceramic Society Scherer Vol. 73, No. 1distribution s parabolic, in which case Eq.(18) becomes61

The stress increases in prop ortion to thethickness of the plate and the rate ofevaporation, and in inverse proportion tothe perm eability; that is, the stress s in-creased by those factors that steepen thepressure gradient. The reason that gelsare so much more difficult to dry thanconventional ceramics is that the perme-ability of gels is low, as a result of theirvery small pore size. Comparing thestress at the surface of a drying plate,cylinder, and sphe re, it is found71 that thetension decreases in the ratio plate/cylinderlsphere =; /a/ ; The lower stressreflects the shallower pressure gradientsin the cylinder and sphere, where the li-quid flowing from the interior passesthrough a volum e that increases as P andr 3 respectively. Since these results arederived from Eq. (13),hey are valid onlyas long as the pores rem ain filled with li-quid. At s ome point the netw ork will stopshrinking and the meniscus will retreatinto the gel; then Eq. (19) will apply on lywithin the saturated po res nsid e he ge1.633) Diffusion

If the po res conta in a solution of liqu idswith intrinsic diffusion coefficientsD , andD2, then diffusion contributes to the trans-port and the diffusion term m ust be a d-ded to the flow term. Then Eq. (13)becomes72h = v. P + I -@)V.iNote that d iffusion has no influence f theintrinsic diffusion coefficients of the twoliquids are equal, because the diffusivevolume fluxes are then equal and oppo-site (i.e., diffusion produces no volumeflow). It has been shown72 that dryingstresses can be reduced considerablywhen the diffusion term is significant. Thereason is that a substantial flux can beproduced by a shallow co ncen tration gra-dient (since interdiffusion of liquids israpid ), so diffusion can e xtract liquid fromthe interior of the bod y almost as fast asit evaporates from the surface. Conse-quently, the pressure distribution is flat-ter, the differential strain is reduced, andthe drying stresses are smaller whendiffusion occurs.

V. Fracture( 1) Models of Fracture During DryingThere is no generally accepted ex pla-nation for the phenom enon of crackingduring drying. Any suitable he ory shoulda'ccount for the com mon observationsthat cracking is more likely if the body isthick or the drying rate is high, and thatcracks generally appear at the critical

point (i.e., when shrinkage stops and thevapor/liquid interface moves into thebody of the gel). The tendency for slow-ly dried bodies to crack at the critical pointhas been noted for clay,35 particulategels, l73 and alkoxide -deriv edgels.39174We no w examine two mod els of fractureduring drying, a macroscopic model(described in Section IV) that attributescracking to stresses produced b y a pres-sure gradien t in the liquid ph ase, and amicroscopic mod el that explains crack-ing as a result of the distribution of p oresizes.The stress that causes fracture is notthe m acrosc opic stress, a that acts onthe network. R ather, it is the stress con-centrated at the tip of a flaw of length cwhich is proportional to75Fracture occurs w hen o,>K\,, where KI,is a material property called the criticalstress intensity.76 It is reasonable to as-sume that th e flaw size distribution is in-depen dent of the size and drying rate ofthe gel, so the tendency to fracture willincrease with the stress, ax. Although Eq.(19) accounts qualitatively for the ob-served dependence of cracking on L andV . , t does not offer any explanation forthe common observation that slowlydried ge ls crack at the critical point. Thestress is p redic ted to rise continu ally untilthat moment, but there is no sudden umppredicted for a at time f3R that would en-hance the likelihood of cracking .The microscopic model for fracture isbased on the idea77178 llustra ted in Fig .7. After the critical point, liquid s removedfirst from the largest pores; then the ten-sion in the neighboring small pores isclaimed to deform the pore wall andcause cracking. This mechanism app earsto account quite clearly for the occur-rence of cracking at the critical point.How ever, he flaws produ ced in this wayhave lengths on the order of the spacebetween pores, which is typically 1 to 5nm in alkoxide-derived gels, and suchflaws should be subcritical (i.e., non-propagating). This difficulty could beavoided b y suppo sing that the flaws per-colate through the structure until theyachieve the critical length. A more im por-tant prob lem with this mechanism is thatit does not explain the impo rtance of dry-ing rate or body size. The local stressesresult from the local hetero geneity of themicrostructure,so fracture should be in-evitable when the pore size distributionis wide.Another version of this mode l wou ld at-tribute the flaws to the irregu larity of thedrying front, illustrated schematically inFig. 8. The width of the drying front, w ,is 2 or 3 orders of m agnitude larger thanthe po re size, but the drying front is quitesmooth on the scale of the thickness ofthe sample. The crack might be expect-ed to have a length similar to w , so thestress intensity would be p roportional to

ac= a x e (21)


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January 1990PRwlQ, However, Shawsz has shownthatw a p p ) - 1 ~ (VE)-1/2 (22)which means that the drying front be-comes smoother (w decreases) as thedrying rate increases. Thus

(23)c = pRwl/2a VE)-1/4which means that the stress intensitydecreases as the evaporation rateincreases, in contradiction to the ex-perimental evidence. Further, no depen-dence of stress on sample size isexpected according to this mechanism.On the other hand, if these flaws are act-ed upon by the stress predicted by themacroscopic mechanism, then the stressintensity is

(241which increases almost in proportion tothe drying rate. Thus, the flaws generat-ed by the irregular drying front, togetherwith the macroscopic stress, may explainthe appearance of cracking at the criti-cal point. The macroscopic nature of thestress also explains the observation thata drying body will often break into onlytwo or three pieces; if the stresses werelocal, failure should always result in a verylarge number of fragments.2) Avoiding FractureSince the capillary pressure sets thelimit on the drying stress (o,OR); (F) normalizedstress distribution at same times as in (E). From Ref. 63.

soaking f& 24 h in 4N HCI or 2N NH3.


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2 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 hassame radiusof curvature for poresof all sizes;after the critical point, the largest pores areemptied first.The capillary tension compress-ing the smaller pores causes localstressesthatcrack 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 lengthc is subjectedto stress over width of drying front.

and all of these features help to reducecracking. The coarser structure may bea result of the higher pH produced byhydrolysis of formamide.84 Unfortunate-ly, the additive is difficult to remove uponheating, so bloating and cracking result.The original claims85 of rapid processing(-48 h) for centimeter-thick pieces of gelprocessed with formamide have not beenrepeated nor reproduced, but promisingresults have been reported for dimethyl-formamide (DMF).86,87 That additiveyields gels with larger pores, and they areeven larger after aging at elevated tem-peratures (=150C). Gels made with DMFdo not crack at drying rates that destroygels made with formamide, or withoutany DCCA. Interestingly, the dried gelcracks when exposed to vapors of water(yLv=0.072 J/m2) or formamide (0.058Jlmz), but not vapors of methanol(0.023Jlm2) or DMF (0.036J/m2), so the lowersurface tension of the additive may be im-portant. To the extent that these additivesare effective, their success can be at-tributed to coarsening of the microstruc-ture (which increases D and decreasesPR) and strengthening of the network.They may also provide a medium throughwhich the more volatile components(water and alcohol) can diffuse, therebyallowing diffusion to reduce the pressuredifferential within the body.72Since shrinkage and cracking areproduced by capillary forces, KistlePreasoned that those problems could beavoided by removing the liquid from thepores 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 noliquidhapor interface, and no capillarypressure. In the process of supercritical(or hypercritical) drying, a sol or wet gelis placed into an autoclave and heatedalong a path such as the one indicatediin Fig. 9. The pressure and temperature,are increased in such a way that thephase boundary is not crossed; once thecritical point is passed, he solvent is vent-led at a constant temperature (>T,). Theresulting gel, called an aerogel, has avolume similar to that of the original sol.This process makes it possible toproduce monolithic gels as large as thevolume of the autoclave. Table I containsvalues of T, and Pc for some relevant li-quids. Two groups succeeded at aboutthe same time in making large monolithicgels by supercritical drying. In one case89the aerogel itself was the objective: theLOW density of the silica gel was requiredfor a Cherenkov radiation detector.90 Theother group9192 wanted to makernonolithic gels to be sintered nto denseglasses or ceramics, and found that largecrack-free bodies could be made withinvvide ranges of concentration of reac-tants. Although supercritical drying givesvery good results or silica, the high tem-peratures and pressures make the

process expensive and dangerous. Aconvenient alternative s to exchange thepore liquid for a substance with a muchlower critical point. As shown in Table I,carbon dioxide has T, = 31C andPc=7.4 MPa, so the process can beperformed near ambient temperatures.Supercritical drying following COP ex-change has become a standard tech-nique for preparing biological samples forTEM examination.93 t was apparently firstapplied for producing monolithic silicagels by Woignier,94 and was indepen-dently developed by Tewari ef al.95 formaking large windows. For some materi-als, supercritical treatment in alcoholcauses dissolution, so a milder processis essential. Brinker et al.96 used CO2 ex-change to make aerogels of lithiumborate compositions that would dissolvein alcohol. This would seemto be an idealway of making aerogels, but it does havesome disadvantages. Long times can berequired to achieve complete solvent ex-change, especiaily because C 0 2 is notmiscible with water (Kistler88 notes that li-quidlliquid interfaces ormed by rmmisci-ble liquids could produce capillarycompression of the gel). It may be neces-sary to exchange first with a mutual sol-vent such as amyl acetate,96 then to flushfor hours with liquid C02. Moreover, be-cause of the low density of the driedbody, sintering of monolithic crystallineaerogels to full density is impractical.Another way of avoiding the presenceof the liquidlvapor interface is to freezethe pore liquid and sublime the resultingsolid under vacuum. This process offreeze-drying s widely used in the prepa-ration of foods,47 but does not permit thepreparation of monolithic gels. The re ason is that the growing crystals reject thegel network, pushing t out of the way untilit is stretched to the breaking point. It isthis phenomenon that allows gels to beused as hosts for crystal growth:97198 thegel is so effectively excluded that the crys-tals nucleated in the pore liquid are notcontaminated with the gel phase; thecrystals can grow up to a size of a fewmillimeters before the strain is so greatthat macroscopic ractures appear in thegel. If a silica sol is frozen, flakes of silicagel (sometimes called lepidoidal silica) areproduced;99 f freezing is done unidirec-tionally, fibers of gel are obtained.OO*JAttempts to freeze-dry gels typically resultin flakes (e.g., Ref. 102) or in translucentbodies with large pores that are the fos-sils of the crystals.

VI. ConclusionsAlthough the principles of drying havebeen recognized for decades, the meansof calculating drying stresses and strainshave been developed relatively recently.The stresses result from a gradient in thepressure in the liquid in the pores of thedrying body. The stress increases withthe drying rate and the size of the body,and is inversely related to the permeabil-


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January 1990ity of the structure. It is the latter factor thatmakes gels so much harder to dry thanconventional ceramics: their small poresresult in very low permeability. Fracturemay result from the action of dryingstresses on preexisting flaws, but in manycases seems to result from flaws gener-ated by the irregularity of the drying frontas it enters the body at the critical point.Unfortunately, many of the physicalproperties needed to predict failure (e.g.,permeability and critical stress intensityof wet bodies) have not yet beenmeasured.A variety of strategies have been de-vised to avoid fracture during drying.These include strengtheningof the solidnetwork by aging or chemical additives,increasing permeability by increasingpore size, and reducing capiliary pres-sure by increasing pore size, reducing in-terfacial energies, or drying undersupercritical conditions. Each of these ap-proaches nvolves some tradeoff (for ex-ample, in processing time or sinteringtemperature), so the best method mustbe chosen by regarding the process asa whole.

Theory of Dryingand Pore Structure. Academic Press, New York,1979.16H. Darcy. Les Fontaines Publiques de la Ville deDiion. Libraire des Corps lmperiaux des Ponts etChaussees et des Mines, Paris, France, 1856.17J. van Brakel, "Pore Space Models for Trans-port Phenomena in Porous Media-Review andEvaluation with Special Emphasis on Capillary LiquidTransport," Powder Techno/., 11, 205-36 (1975)lap. Debye and R. L. Cleland, "Flow of LiquidHydrocarbons in Porous VYCOR," J. Appl. Phys.,30 [6] 843-49 (1959).19G. W. Scherer and R. M. Swiatek, "Measurementof Permeability: II. Silica Gels"; to be published inNon-Cryst. Solids.20K. K. Watson, "An Instantaneous Profile Methodfor Determining he Hydraulic Conductivity of Unsalu-rated Porous Materials," Water Resources Res., 2[4] 709-15 (1966).21K. J. Packer, "The Dynamics of Water in Heter-ogeneous Systems." Pbilos. Trans. R. Soc. London,Ser. 8 278. 59-87 (1977).22J. Warnock, D. D. Awschalom, and M. W. Shafer,"Orientational Behavior of Molecular Liquids in Res-tricted Geometries," P h y s Rev. 6, 34 [I ] 475-78(1986).

23M. W. Shafer, G. D. Awschalom. J. Warnock, andG Ruben, "The Chemistry of and Physics withPorous Sol-Gel Glasses," J . Appl. Pbys., 61 [12]5438-46 (1987).24D. Y. C. Chan and R. G. Horn, "The Drainageof Thin Liquid Films between Solid Surfaces," J.Cbem. Pbys.. 83 [l o] 5311-24 (1985).25J. N. Israelachvili, "Measurement of the Viscosi-ty of Liquids in Very Thin Films," J . Colloid lnterface

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Eng. Cbem., 21 [I] 12-16 (1929).3 T K. Sherwood, "The Drying of Solids-11," lnd.Eng. Chem , 21 [lo] 976-80 (1929).32T. K. Sherwood, "The Drying of Solids-Il l,'' lnd.

Eng. Cbem.. 22 (21 132-36 (1930).33R.W. Ford, Ceramics Drying. Pergamon Press,New York, 1986.W . B. Keey, Drying, Principles and Practice. Per-gamon Press, New York, 1972"F. H. Clews, Heavy Clay Technology. Academ-ic Press, New York. 1969.36M. Fortes and M. R . Okos, "Drying Theories:Their Bases and Limitations as Applied to Foods and

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13

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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 vaporphases are the same and the surface tensionis zero.

Table I: Critical Points of Selected SolventsSubstance Formula T ("C) P, ( M W31.1 7.3619.7 2.97Carbon dioxide CO2Freon 116 CF3CF3Methanol CH30H 240 7.93Ethanol C~H SO H 243 6.36Water H20 374 22.0

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1990 scherer theory of drying

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