**2.1 Hydrolysis and condensation**

390 Biomedical Science, Engineering and Technology

microelectronic industry, anti-corrosion and scratch resistant coatings, contact lenses or host materials for chemical sensors. In the recent years interest in those materials is connected to their possible applications as biomaterials (Gigant et al., 2002; Joshua et al., 2001; Klukowska et al., 2002; Mackenzie & Bescher, 1998; Matsuura et al., 2001; Spanhel et. al., 1995). One indirect advantage of including polymers is that it is possible to obtain synergistic effects that combine the best properties of polymers with the best properties of inorganic materials. These materials are considered as biphasic materials, where the organic and inorganic phase is mixed at the nm to sub-μm scales. Nevertheless, it is obvious that the properties of these materials are not just the sum of the individual contributions from both phases; the role of the inner interfaces could be predominant. The nature of the interface has recently been used to divide these materials into two distinct classes (Sanchez & Ribot, 1994). In class I, organic and inorganic compounds are embedded and only the weak bonds (hydrogen, van der Waals bonds) give the cohesion to the whole structure. In class II materials, the phases are linked together through strong chemical bonds (covalent or ionic-covalent bonds). Both class I and class II

The aim of the present chapter is to summarize the synthesis via sol-gel and the characterisation methods of amorphous and hybrid materials for biomedical applications. Therefore, the emphasis of our discussion will be focussed on the science, rather than on the technology, of sol-gel processing. The controlled release of pharmaceuticals such as anti-inflammatory agents and antibiotics from strong and biocompatible hosts has relevant applications: they include implantable therapeutic systems, filling materials for bone or teeth repair, which curtail inflammatory or infectious side effects of implant materials when coatings of biocompatible

Different types of colloids can be used to produce polymers or particles from which we can obtain a ceramic material: for example, sols (suspensions of solid particles in a liquid), aerosols (suspensions of particles in a gas) or emulsions (suspensions of liquid droplets in another liquid). The sol-gel chemistry is based on the hydrolysis and polycondensation of molecular precursors such as metal alkoxides M(OR)x, where M = Si, Sn, Ti, Zr, Al, Mo, V, W, Ce and so forth. The following sequence of reactivity is usually found as Si(OR)4 << Sn(OR)4 = Ti(OR)4 < Zr(OR)4 = Ce(OR)4 (Novak, B.M.,1993 ). Fig. 1 presents a schema of the procedures which one could follow within the scope of sol-gel processing. In the sol-gel process, the precursors for the preparation of a colloid consist of a metal or metalloid element surrounded by various

hybrids were prepared by sol-gel technique (Young, 2002).

**2. General processing methods** 

materials containing anti-inflammatory or antibiotic drugs are applied.

ligands. A list of the most commonly used alkoxy ligands is presented in Tab. 1.

methyl ●CH3 Methoxy ●OCH3 ethyl ●CH2CH3 Ethoxy ●OCH2CH3 n-propryl ●CH2CH2CH3 n-propoxy ●OCH2CH2CH3 Iso-propyl H3C(●C)HCH3 Iso-propyl H3C(●O)CHCH3 n-butyl ●CH2(●CH2)2CH3 n-butoxy ●O(CH2)3CH3 Sec-butyl H3C(●C)HCH2CH3 Sec- propoxy H3C(●O)CHCH2CH3 Iso- butyl ●CH2CH(CH3)2 Iso- propoxy ●OCH2CH(CH3)2 Tert-butyl ●C(CH3)3 Tert- propoxy ●OC(CH3)3

**Alkyl Alkoxy**

Table 1. Commonly used ligands in sol-gel process.

The alkoxydes used as ligands can be organometallic compounds, where direct metalcarbon bonds are present, or also members of the family or metalloid atoms, the so called metal alkoxydes, among which the most widely known, as it has been extensively studied, is the silicon tetraethoxide (or tetrathoxy-silane, or tetraethyl orthosilicate, TEOS), Si(OC2H5)4. Silicate gels are most often synthesized by hydrolyzing monomeric, tetrafunctional alkoxide precursors employing a mineral acid (e.g., HCl) or base (e.g. NH3) as a catalyst. At the functional group level, the sol-gel process starts with the following reaction:

$$\begin{array}{rcl} \text{Hydrolysis} \\ \text{Si(OR)}\_{4} & + \text{H}\_{2}\text{O} & \rightarrow & \text{OH-Si(OR)}\_{3} + \text{ROH} \\ \text{Esterification} \end{array} \tag{1}$$

which can even be stopped while the metal is only partially hydrolyzed, Si(OR)4-n (OH)n. Then, two partially hydrolyzed molecules can link together in a condensation reaction, such as one of the following:

$$\begin{array}{c} \text{Alcohol} \\ \text{(OR)} \text{:} \text{Si-OR} + \text{OH-Si(OR)}\_3 \rightarrow \text{(OR)}\_3 \text{Si-O-Si(OR)}\_3 + \text{ROH} \\ \text{Alcohol} \end{array} \tag{2}$$

$$\begin{array}{c} \text{Water condensation} \\ \text{(OR)}\_3\text{Si-OH} + \text{OH-Si(OR)}\_3 \rightarrow \text{(OR)}\_3\text{Si-O-Si(OR)}\_3 + \text{H}\_2\text{O} \\ \text{Hydrolysis} \end{array} \tag{3}$$

where R is an alkyl group, CxH2x+1. The hydrolysis reaction (eq. 1) replaces alkoxide group (OR) with hydroxyl group (OH). Subsequent condensation reactions involving the silanol group produce siloxane bonds (Si-O-Si) and the by-products alcohol (ROH) (eq. 2) or water (eq. 3). Under most conditions, condensation starts (eqs. 2 and 3) before hydrolysis (eq. 1) is complete. A solvent such as an alcohol is normally used as a homogenizing agent, as water and alkoxysilanes are immiscible (Fig. 2). However, a gel can be prepared from silicon alkoxide-water mixtures without adding a solvent (Avnir & Kaufman, 1987), since the alcohol produced as the by-product of the hydrolysis reaction is sufficient to homogenize the initially phase separated system. It should be noted that the alcohol is not simply a solvent. As indicated by the reverse of eqs. 1 and 2, it can participate in esterification or alcoholysis reactions.

Fig. 2. TEOS , H2O, Synasol (95% EtOH, 5% water ) ternary-phase diagram at 25°C . For pure ethanol the miscibility line is slightly shifted to the right (Cologan & Setterstrom, 1946).

The H2O:Si molar ratio (r) in eq. 1 has been made to vary from less than one to over 50, and the concentration of acid or bases from less than 0.01 (Brinker et al., 1982) to 7M (Stober et al., 1968) depending on the desired end product. Typical gel-synthesis procedures used to produce bulk gels, films, fibres, and powders are listed in Tab 2. Hydrolysis occurs by the nucleophilic attack of the oxygen contained in water on the silicon atom as shown by the reaction of isotopically labelled water with TEOS that produces only unlabelled alcohol in both acid-base-catalyzed systems (Voronkov & et al., 1978). Hydrolysis is facilitated in the presence of homogenizing agents (alcohols, dioxane, THF, acetone, etc.) that are especially beneficial in promoting the hydrolysis of silanes containing bulk organic or alkoxy ligands. It should be emphasized, however, that the addition of solvents may promote esterification or depolymerization reactions according to the reverse of eqs. 1 and 2.

where R is an alkyl group, CxH2x+1. The hydrolysis reaction (eq. 1) replaces alkoxide group (OR) with hydroxyl group (OH). Subsequent condensation reactions involving the silanol group produce siloxane bonds (Si-O-Si) and the by-products alcohol (ROH) (eq. 2) or water (eq. 3). Under most conditions, condensation starts (eqs. 2 and 3) before hydrolysis (eq. 1) is complete. A solvent such as an alcohol is normally used as a homogenizing agent, as water and alkoxysilanes are immiscible (Fig. 2). However, a gel can be prepared from silicon alkoxide-water mixtures without adding a solvent (Avnir & Kaufman, 1987), since the alcohol produced as the by-product of the hydrolysis reaction is sufficient to homogenize the initially phase separated system. It should be noted that the alcohol is not simply a solvent. As indicated by the reverse of eqs. 1 and 2, it can participate in esterification or

Fig. 2. TEOS , H2O, Synasol (95% EtOH, 5% water ) ternary-phase diagram at 25°C . For pure ethanol the miscibility line is slightly shifted to the right (Cologan & Setterstrom, 1946).

The H2O:Si molar ratio (r) in eq. 1 has been made to vary from less than one to over 50, and the concentration of acid or bases from less than 0.01 (Brinker et al., 1982) to 7M (Stober et al., 1968) depending on the desired end product. Typical gel-synthesis procedures used to produce bulk gels, films, fibres, and powders are listed in Tab 2. Hydrolysis occurs by the nucleophilic attack of the oxygen contained in water on the silicon atom as shown by the reaction of isotopically labelled water with TEOS that produces only unlabelled alcohol in both acid-base-catalyzed systems (Voronkov & et al., 1978). Hydrolysis is facilitated in the presence of homogenizing agents (alcohols, dioxane, THF, acetone, etc.) that are especially beneficial in promoting the hydrolysis of silanes containing bulk organic or alkoxy ligands. It should be emphasized, however, that the addition of solvents may promote esterification or depolymerization reactions according

alcoholysis reactions.

to the reverse of eqs. 1 and 2.


Table 2. Sol-gel Silicate compositions for bulk gels, fibres, film and powder.

The Hydrolysis is more rapid and complete when catalysts are employed (Voronkov et al., 1978). Although mineral acids or ammonia are most generally used in sol-gel processing, other known catalysts are acetic acid, KOH, amines , KF, HF, titanium alkoxides, and vanadium alkoxides and oxides (Voronkov et al , 1978). In the literature mineral acids are reported to be more effective catalysts than the equivalent base concentrations. However, neither the increasing acid of silanol groups with the extent of hydrolysis and condensation (Keefer, 1984) nor the generation of unhydrolyzed monomers via base-catalyzed alcoholic or hydrolytic depolymerization processes have generally been taken into account. Aelion et al., (1950a, 1950b) investigated the hydrolysis of TEOS under acid and basic conditions using several cosolvents: ethanol, methanol, and dioxane. The extent of hydrolysis (eq. 1) was determined by distillation of the ethanol by-product. Karl Fischer titration was used to follow the consumption of water by hydrolysis (eq.1) and its production by condensation (eq.3). Aelion et al. observed that the rate and extent of the hydrolysis reaction was mostly influenced by the strength and concentration of the acid or base catalyst. As under acid conditions, the hydrolysis of TEOS in base media was a function of the catalyst concentration (Aelion et al., 1950a, 1950b).

Steric (spatial) factors exert the greatest effect on the hydrolytic stability of organoxysilanes (Voronkov et al., 1978). Any complication of the alkoxy group delays the hydrolysis of alkoxysilanes, but the hydrolysis rate is lowered at most by the branched alkoxy group (Voronkov et al., 1978). The effects of alkyl length and the degree of branching observed by (Aelion et al., 1950a, 1950b) are illustrated in Tab. 3 for the hydrolysis of tetralkoxysilanes.


Table 3. Rate constant k for acid hydrolysis of tetralkoxysilanes (RO)4Si at 20°C

Fig. 3 compares the hydrolysis of TEOS and TMOS under acid and basic conditions. The delaying effect of the bulkier ethoxide group is clearly evident. According to (Voronkov et al., 1978) in the case of mixed alkoxides, (RO)x(R'O)4-xSi where R'O is a higher (larger) alkoxy group than RO, if the R'O has a normal (i.e. linear) structure, its retarding effect on the hydrolysis rate is manifest only when x= 0 or 1. If R'O is branched, its delaying effect is evident even when x= 2. The hydrolysis of the n-propoxide group was observed to be slower than the ethoxide group during the second hydrolysis step under both acid and basic conditions. This result suggests that a delaying effect of a higher, normal alkoxide group is realized regardless of the extent of substitution.

Fig. 3. Relative water concentration versus time during acid- or base-catalzed hydrolysys of □: TMOS with HCl ; X: TEOS and TMOS with NH3. ∆:TEOS with HCl (Shih et al., 1987)

The substitution of one alkyl group with alkoxy groups increases the electron density on the silicon. Conversely, hydrolysis (substitution of OH for OR ) or condensation (substitution of OSi for OR or OH) decreases the electron density on the silicon Fig. 4. Inductive effects are evident from investigations on the hydrolysis of methylethoxysilanes (Schimdt et al., 1984), (CH3)x(C2H5O)4-xSi where x varies from 0 to 3. Fig. 5 shows that under acidic (HCl) conditions, the hydrolysis rate increases with the degree of substitution x, of electronproviding alkyl group, whereas under basic (NH3) conditions the reverse trend is clearly observed.

Fig. 4. Inductive effects of substituents attached to silicon,R, OR, OH or OSi (Brinker, 1988)

Fig. 5 also shows the accelerating effect of methoxide substitution on the hydrolysis rate (TMOS versus TEOS). The acceleration and retardation of hydrolysis with increasing x under acid and basic conditions respectively, suggest that the hydrolysis mechanism is sensitive to inductive effects and is apparently unaffected by the extent of alkyl substitution. Because increased stability of the transition state will increase the reaction rate, the inductive effects are evident for positively and negatively charged transition states or intermediates under acid and basic conditions respectively. This reasoning leads to the hypothesis that

Fig. 3. Relative water concentration versus time during acid- or base-catalzed hydrolysys of □: TMOS with HCl ; X: TEOS and TMOS with NH3. ∆:TEOS with HCl (Shih et al., 1987)

The substitution of one alkyl group with alkoxy groups increases the electron density on the silicon. Conversely, hydrolysis (substitution of OH for OR ) or condensation (substitution of OSi for OR or OH) decreases the electron density on the silicon Fig. 4. Inductive effects are evident from investigations on the hydrolysis of methylethoxysilanes (Schimdt et al., 1984), (CH3)x(C2H5O)4-xSi where x varies from 0 to 3. Fig. 5 shows that under acidic (HCl) conditions, the hydrolysis rate increases with the degree of substitution x, of electronproviding alkyl group, whereas under basic (NH3) conditions the reverse trend is clearly

Fig. 4. Inductive effects of substituents attached to silicon,R, OR, OH or OSi (Brinker, 1988) Fig. 5 also shows the accelerating effect of methoxide substitution on the hydrolysis rate (TMOS versus TEOS). The acceleration and retardation of hydrolysis with increasing x under acid and basic conditions respectively, suggest that the hydrolysis mechanism is sensitive to inductive effects and is apparently unaffected by the extent of alkyl substitution. Because increased stability of the transition state will increase the reaction rate, the inductive effects are evident for positively and negatively charged transition states or intermediates under acid and basic conditions respectively. This reasoning leads to the hypothesis that

observed.

under acid conditions, the hydrolysis rate decreases with each subsequent hydrolysis step (electron withdrawing), whereas under basic conditions the increased electron-withdrawing capabilities of OH (and OSi) compared to OR may establish a condition in which each subsequent hydrolysis step occurs more quickly as hydrolysis and condensation proceed.

Fig. 5. Relative silane concentration versus time during acid- and base-catalyzed hydrolysis of different silanes in ethanol (volume ratio to EtOH=1:1). ●: (CH3)3SiOC2H5. ∇:(CH3)2Si(OC2H5)2. □: (CH3)2Si(OC2H5)3. ○:Si(OC2H5)4. ∆:Si(OCH3)4. (Shih et al., 1987)

From the standpoint of organically modified alkoxysilanes, RxSi(OR)4-x, the inductive effects indicate that acid-catalyzed conditions are preferable (Schimdt et al., 1984), since acids are effective in promoting hydrolysis both when x=0 and x>0. As indicated in Tab. 2, the hydrolysis reaction has been performed with r values ranging from <1 to over 25 depending on the desired polysilicate product, for example, fibers, bulk gel or colloidal particles. From eq. 1, an increased value of r is expected to promote the hydrolysis reaction. (Aelion et al., 1950a, 1950b) found the acid-caltayzed hydrolysis of TEOS to be first-order in [H2O]; however, they observed an apparent zero-order dependence of the water concentration under base-catalyzed conditions. As explained, this is probably due to the production of monomers by siloxane bond hydrolysis and redistribution reactions.

Solvents are usually added to prevent liquid-liquid phase separation during the initial stages of the hydrolysis reaction and to control the concentrations of silicate and water that influence the gelation kinetics. More recently, the effects of solvents have been studied primarily in the context of drying control chemical additives (DCCA) used as cosolvents with alcohol in order to facilitate rapid drying of monolithic gels without cracking (Hench et al., 1986). Solvents can be classified as polar or nonpolar and as protic or aprotic . The dipole moment of a solvent determines the length over which the charge of one species can be "felt" by surrounding species. The lower the dipole moment, the larger this length becomes. This is important in electrostatically stabilized systems and when considering the distance over which a charged catalytic species, for example an OH- nucleophile or H3O+ electrophile, is attracted to or repelled from potential reaction sites, depending on their charge. The availability of labile protons determines whether anions or cations are solvated more strongly through hydrogen bonding. Because hydrolysis is catalyzed either by hydroxyl (pH>7) or hydronium ions (pH<7), solvent molecules that hydrogen bonds to hydroxyl or hydronium ions reduce the catalytic activity under basic or acid conditions respectively. Therefore, aprotic solvents that do not form a hydrogen bond to hydroxyl ions have the effect of making hydroxyl ions more nucleophilic, whereas protic solvents make hydronium ions more electrophilic (Morrison & Boyd, 1966).

Hydrogen bonding may also influence the hydrolysis mechanism, hydrogen bonding with the solvent can sufficiently activate weak leaving group to realize a bimolecolar, nucleophilic (SN2-Si) reaction mechanism (Voronkov et al., 1978). The availability of labile protons also influences the extent of the reverse reactions, reesterification (reverse eq. 1) or siloxane bond alcoholysis or hydrolysis (reverse of eqs. 2 and 3). Aprotic solvents do not participate in reverse reactions such as reesterification or hydrolysis, because they lack a sufficiently electrophilic proton and are unable to be deprotonated to form sufficiently strong nucleophiles (OH- or OR- ) necessary for reaction 4. Therefore compared to alcohol or water, aprotic solvents such as THF or dioxane are considerably more "inert" (they do not formally take part in sol-gel processing reactions), they may influence reaction kinetics by increasing the strength of nucleophiles or decreasing the strength of electrophiles.

$$\begin{array}{c|c|c|c} \text{---} & \text{---} & \text{---} & \text{---} \\ & \text{---} & \text{---} & \text{---} & \text{---} \\ & \text{~-} & \text{---} & \text{---} & \text{---} \\ \text{~-} & \text{---} & \text{---} & \text{---} & \text{---} \\ \end{array}$$

### **2.2 Gelation**

Clusters resulting from the hydrolysis and condensation reactions eventually collide and link together into a gel, which is often defined as "strong" or "weak" according to whether the bonds connecting the solid phase are permanent or reversible; however, as noted by (Flory, 1974), the difference between weak and strong ones is a matter of time scale. Even covalent siloxane bonds in silica gel can be cleaved, allowing the gel to exhibit a slow and irreversible (viscous) deformation. Thus the chemical reactions that bring about gelation continue long beyond the gelation point, permitting flow and producing gradual changes in the structure and properties of the gel. The outline of the aging of a gel is as follows:


The simplest picture of gelation is that clusters grow by condensation of polymers or aggregation of particles until the clusters collide; then, links form between the clusters to produce a single giant cluster that is called gel. When the gel forms, many clusters will be present in the sol phase, entangled in but not attached to the spanning cluster; with time, they progressively become connected to the network and the stiffness of the gel will increase. The gel appears when the last link is formed between two large clusters to create the spanning cluster. This bond is no different from innumerable others that form before and after the gel point, except that it is responsible for the onset of elasticity by creating a continuous solid network. The sudden change in rheological behaviour is generally used to identify the gel point in a crude way.

The classic theory explains the theory developed by Flory (1953) and Stockmayer (1945) to account for the gel point and the molecular-weight distribution in the sol. The most important deficency of this model is that it neglects the formation of closed loops within the growing clusters, and this leads to unrealistic predictions about the geometry of the polymers. The percolation theory offers a description that does not exclude the formation of closed loops and so does not predict a divergent density for large clusters. The disadvantage of the theory is that it generally does not lead to analytical solutions for such properties as the percolation threshold or the size distribution of polymers. However, these features can be determined with great accuracy from computer simulations, and the results are often quite different from the predictions of the classical theory. Excellent reviews of percolation theory and its relation to gelation have been written by Zallen (1983 ) and Stauffer et al. (1982); and the kinetic models are based on Smoluchowski' s analysis of the growth and aggregation of clusters.

The Smoluchowski equation describes the rate at which the number, ns, of clusters of size s changes with time t, during an aggregation process :

$$\frac{dn\_s}{dt} = \frac{1}{2} \Sigma\_{l+j=s} K \left( l, f \right) n\_l n\_j - n\_s \sum\_{f=1}^{\infty} K \left( s, f \right) n\_f \tag{5}$$

The coagulation kernel, K(i,j) is the rate coefficient for aggregation of a cluster of size i with another and of size j. The first term in eq. 5 gives the rate of creation of size s by aggregation of two smaller clusters, and the second term gives the rate at which clusters of size s are eliminated by further aggregation. For this equation to apply, the sol must be so diluted that collisions between more than two clusters can be neglected, and the clusters must be free to diffuse so that the collisions occur at random. Further, since K depends only on i and j, ignoring the range of structures that could be present in a cluster of a given size, this is a mean-field analysis that replaces structural details with averages.

### **2.3 Drying**

396 Biomedical Science, Engineering and Technology

is important in electrostatically stabilized systems and when considering the distance over which a charged catalytic species, for example an OH- nucleophile or H3O+ electrophile, is attracted to or repelled from potential reaction sites, depending on their charge. The availability of labile protons determines whether anions or cations are solvated more strongly through hydrogen bonding. Because hydrolysis is catalyzed either by hydroxyl (pH>7) or hydronium ions (pH<7), solvent molecules that hydrogen bonds to hydroxyl or hydronium ions reduce the catalytic activity under basic or acid conditions respectively. Therefore, aprotic solvents that do not form a hydrogen bond to hydroxyl ions have the effect of making hydroxyl ions more nucleophilic, whereas protic solvents make hydronium

Hydrogen bonding may also influence the hydrolysis mechanism, hydrogen bonding with the solvent can sufficiently activate weak leaving group to realize a bimolecolar, nucleophilic (SN2-Si) reaction mechanism (Voronkov et al., 1978). The availability of labile protons also influences the extent of the reverse reactions, reesterification (reverse eq. 1) or siloxane bond alcoholysis or hydrolysis (reverse of eqs. 2 and 3). Aprotic solvents do not participate in reverse reactions such as reesterification or hydrolysis, because they lack a sufficiently electrophilic proton and are unable to be deprotonated to form sufficiently

water, aprotic solvents such as THF or dioxane are considerably more "inert" (they do not formally take part in sol-gel processing reactions), they may influence reaction kinetics by

Clusters resulting from the hydrolysis and condensation reactions eventually collide and link together into a gel, which is often defined as "strong" or "weak" according to whether the bonds connecting the solid phase are permanent or reversible; however, as noted by (Flory, 1974), the difference between weak and strong ones is a matter of time scale. Even covalent siloxane bonds in silica gel can be cleaved, allowing the gel to exhibit a slow and irreversible (viscous) deformation. Thus the chemical reactions that bring about gelation continue long beyond the gelation point, permitting flow and producing gradual changes in the structure and properties of the gel. The outline of the aging of a gel

Si ⇋ <sup>+</sup> **. .** 

The simplest picture of gelation is that clusters grow by condensation of polymers or aggregation of particles until the clusters collide; then, links form between the clusters to produce a single giant cluster that is called gel. When the gel forms, many clusters will be present in the sol phase, entangled in but not attached to the spanning cluster; with time,

increasing the strength of nucleophiles or decreasing the strength of electrophiles.

) necessary for reaction 4. Therefore compared to alcohol or

HO

**. .** 

<sup>O</sup> **. .** Si OR Si **. .** 

(4)

ions more electrophilic (Morrison & Boyd, 1966).

strong nucleophiles (OH- or OR-

**. .** R O

**. .** 

**. .** Si <sup>O</sup> **. .** 

H

**2.2 Gelation** 

is as follows:

• Phenomenology • Classical theory • Percolation theory • Kinetic model

The drying of a porous material is a process which can be divided into several stages. At first the body shrinks by an amount equal to that volume of the evaporated liquid and the liquid-vapor interface remains at the exterior surface of the body. The second stage begins when the body becomes too stiff to shrink and the liquid recedes into the interior, leaving air-filled pores near the surface. Even as air invades the pores, a continuous liquid film supports flow to the exterior, so evaporation continues to occur from the surface of the body. Eventually, the liquid becomes isolated into pockets and drying can proceed only by evaporation of the liquid within the body and diffusion of the vapor to the outside. In the specialized literature the factors affecting stress development are discussed and various strategies to avoid warping and cracking are described. The outline is as follows:


The first stage of drying is called the constant rate period (CRP), because the rate of evaporation per unit area of the drying surface is uniform (Fortes & Okos, 1980; Macey, 1942; Moore, 1961). The evaporation rate is close to that of an open dish of liquid, as indicated by the data for the drying of alumina gel (Dwivedi, 1986), shown in Fig. 6. The rate may differ slightly, depending on the texture of the surface. For example, as sand beds dry, the water conforms to the shapes of the particles, so the wet area is larger than the planar one pertaining to the surface of the body, and the rate of evaporation is correspondingly higher (Ceaglske & Hougen, 1937). The distribution of a spreading liquid is illustrated schematically in Fig.6. The chemical potential, µ, of the liquid in the adsorbed film is equal to the one under the concave meniscus, otherwise liquid would flow from one to the other to balance the potential. The chemical potential µ is lower than bulk liquid because of disjoining and capillary forces, therefore the vapour pressure (pv) decreases according to:

$$\frac{p\_v}{p\_o} = \exp\left(\Delta\mu/R\_gT\right) \tag{6}$$

where p0 is the vapour pressure of bulk liquid, Rg is the ideal gas constant, T is the temperature and Δµ is the increment of the chemical potential. The rate of evaporation, VE, is proportional to the difference between pv and the ambient vapour pressure, pA:

$$N\_E = k \left( p\_v - p\_A \right) \tag{7}$$

where k is a coefficient that depends on the design of the drying chamber, draft rate, etc. It appears reasonable to conclude that the surface of the body must be covered with a film of liquid (as in Fig. 6a), because the rate would decrease as the body shrinks if evaporation occurrs only from the menisci, Fig. 6b.

Fig. 6. Distribution of liquid at the surface of a drying porous body, when liquid is (a) spreading (contact angle θ=0°) or (b) wetting, but not spreading (90°> θ >0°). The chemical potential of the liquid in the adsorbed film is equal to that under the meniscus.
