In a short survey the analogy between the theories on diffusion in solids and in solutions is shown. As a result, analogous theories are developed for the ceramic sintering process and the recrystallization in colloidal solutions. By a thermodynamic method a description is given of the available free energy, the diffusion flux and the driving force

Jonker, G.H.

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Electroanalytical Chemistry and Interforial Eiectrochemi.sf~y Elsevim sequoia S.A., Lam - Printed in The Netherlands 227 ON THE THEORY OF THE CERAMIC DENSIFICATION PROCESS AND OF RECRYSTALLIZATION IN COLLOIDAL SOLUTIONS* G. I-L JONKER Twente Urziuv-sizy of Technology, Depanmem of Elecmorechnology. Emchrde (The Netherlands) pazived 5th October 1971) In the theory of the ceramic densification process the starting point is that the material transport consists mainly of diffusion processes. At least three different contributions are however distingutshable : volume diffusion, grain boundary diffusion and surface difksionl. For volume diffusion it has been shown tbeorctica& that in ionic compounds an optimal delusion rate can bc obtained by appropriate controf of the crystal lattice defectz?. In these theories calculations are based upon equilibria and transport of the lattice defects rather than those of the real components. This has the advantage that the methods used are mathematically analogous to those applied to solutions of salts in water. lo demonstrate this we compare a porous crystal of Mg0 with AgCl crystal- lites in contact with 8 saturated solution- In MgO we distinguish two types of lattice vacancies V,, and &, with concentrations in accordance with the Schottky equili- brium: _ This is analogous to the solubility product of AgCl in water : (1) 12) Generally vacancies in ionic lattices release or attract electrons so that they may have’ electric charges . In the same way atomicvacancies in metals may be compared with a saturated soiution of a molecular compound. In both cases increased concentrations appear in equilibrium with curved surfaces, governed by the Kelvin relation: where q, denotes the equiiibrium concentration around a spherical pore in a metal or around a spherical crystal prith radius p, n, is the equilibrium concentration near a flat surface, y the specific surface free-energy and 17 the mokcular volume. The concen- trations are expressed in numbers per unit of volume- An analogous relation holds for .I_ E&k chon, 37 (1972) 22s G. H. JONKER ionic compounds but now for the values of K, or KS: K,?riETIC PART It is the object of this paper to discuss the theory of the diffusion in the dilute soIutions of vacancies or ions mentioned above. The starting point is Fick’s diffusion equation : wherej is the flux density in numbers of mobiie particles per second per unit of surface, D the diffusion coefficient and dnidx the concentration gradient_ For the diffusion of vacancies we use the same equation : L = - kc dn,,,/dx As j,_ = -j,* and d~,,/dx = - dn,/dx the values of D,,, and D,, are equal. In our problem the concentration gradient dn/dx originates from the difference in equilibrium concentrations around pores or crystals of different dimensions. In orde - to derive genend rules we consider the simple problem of a smaii spherical pore in a Iarse crystal with fiat surfaces or of a small spherical crystal in a solution with flat crystals at a iarg~ distance. For vacancy diffusion in metals Kuczynskij has derived that the total flux & from the surface of a spherical pore can be described by I (b = 4rC,CD,, 27C,,/kT (3 which resuit is independent of the radius p. In the case of charged vacancies in ionic compounds the diffusion equation has an extra electricat term as a result of the difference in mobiiities of the two types of vacancies : 3-c = -D+dn,jdx-n+b+dV/dx @a) j- = -D-dn-jdx+n_b_dV/&~ (6b) where the indices + and - denote the positively and nqatively charged vacancies and b, and b_ the mobilities For these coupled fluxes a general flux equation can be solved, using the reiation of Nenst and Einstein D/kT= b/q where q is the absolute value of the electric charge. _4ccording to Ready2 and Red-en3 this gives for the total flux from the surface of a pore : 4+ = #_ = 4x- 2 + + _L)-’ +Ttl;-) ( 3 + + again independent of the radius p_ With K,=n+n_ we then find that the vacancy fluxes are maximal if n, D, = n _ D _ _ The concentration ratio n, /n _ can be realized by appropriate doping. The same results hold, of course, for the ion fluxes from the surface of a small crystal. It is interesting to recognize in the factor 2(1/n+ D, + l/n _ D_)- I the relation of Nernst (1888) between the diffusion coefficient of a salt and those of the separate ions_ 1. Electroana,!_ Ch.cm_ 37 (1972) CERAMIC DENSIRCATION AND COLLOID RECRYSTALLIZATION 229 THERMODYNAMIC PART Surprisingly the rather simple results of Ready and of Reynen have not been easily accepted. Their apparent result is that the maximum flux is the result of a maximum ofthe diffusion coefficient. On the other hand, the vacancy “solubility” has a maximum under stoichiometric conditions, where n, = n- and then the concen- tration gradients have a maximum too. These gradients are severely decreased if excess concentrations of one of the vacancies are added5. This -again is completely analogous to the case of the AgCl solubility if Ag+ or Cl’ ions are added. A confusion then results if dn/dx is considered to be the “driving force” of the diffusion. From a thermodynamic point of view this is not correct, however. According to the theory of irreversible processes one should use a general flux equation J =xX in such a way that the product of the flux J and the foraz X represents the dissipation rate F of the available free energy F. A simple example is Ohm’s law for electric currents : J =;AVwithfl=JAP Therefore a more useful description of the diffusion is. for exampIe, which is essentially identical with eqn. (6a). Here p denotes the chemical part and q5, the electrical part of the free energy f+ per particle, so that we can a!so write D+ df+=s j, =n, --- k7- dx (9) In the following we wish to show that the driving force of the diffusion can be derived without any kinetic model of the diffusion. Then also the problem whether the diffu- sion might be the result of a pressure gradient can be Ieft out of consideration. Starting with the general flux equation our program is : (i) to find the available free enera F (ii) to derive an equation for p (iii) to split th is equation in a flux J and a force X (iv) to find the relation between J and X. As a first example we consider a spherical pore with radius p in a metal. The total availabie free energy is; F = 43~~~7 (10) lf by some process the pore shrinks, then : 2=* dV ~~+-p-& if V is the volume and 0 the surface of the pore. As this expression does not take into account the nature of the shrinkage process, it might as well apply to viscous flow as to diffusion. If we assume a diffusion process then dV/dt = 6, cME where & represents the total vacancy flux from the surface of the pore. Then : J. Eieccroand Chenz, 37 (1972) 230 G. H. JONKER F = + Fwac 2yjp (11) As we are interested in the vacancy flux 4, the driving force will be : x = 2;*L’_,/p 02) and the flux equation : Q = Q $T,,,/p (1% The value of I cannot be derived from thermodynamics but needs a special kinetic model. It we thereforecompare eqn. (11) with eqns. (5) and (9) we find that z must be : RD z=4lr----~ kT where p appears as a geometrical factor_ The case of d-ion in an ionic compound MgO having two types cf vacancies vWg and V. can be solved in an analogous way, Lading to : q5=d2~(r,+G_)/p (1% Here we really find that the thermodynamic driving force is a constant, independent of the ratio n + jn-. The concentrations and diffusion coefficients appear in the coeffi- cient z only. According to eqn. (7), or by applying directly the Nemst equation we find : 1 -1 31’ - n-D_ P From eqr~ (II) we can also derive the local flux equation by remembering that this equation describes the dissipation rat- p of the total free energy F_ This can also be expressed as the sum of the local dissipations: -r=43 I’= J r=p 4x9 j, z dr where jr is the local fhux density at a distance r from the center of the pore (or crystal) znd J, the Iocal value of the free energy per particie. As the total flux 4 =47rz’j, is constant through each spherical surface it follows that: dfr - -= - - dr jr “;= Then we find for the local flux equation : and with eqns. f 15) and (16) a comparable expression for ionic compounds. APPLICATIONS (i) Ionic cmzducti~it~ It is interesting to J. ElecrrormnL Chetn, 35 (1972) compare the condition for the optimal diffusion rate with CERAMIC DENSIFICATION AND COLLOID RECRYSTALLIZ4TION 231 ihat for the extreme value of ionic conduction Q : G = n,qb, +n_qb_ where q denotes the electric chaqe and b, and b_ the mobiiities of the mobile particles, respectively. As according to Nemst-Einstein D/kT = b/q, we can also write a=q2!kT (n+D+ +n_D_). In this case the condition n+D, =n_D_ corresponds to a minimum of the conductivity. (ii) The ceramic sinrering process In many oxides the diffusion is mainly a vacancy process. In such cases there is an enormous difference m diffusion constants of the metal ion vacancies D, and the oxygen ion vacancies Do_ Do has the lower value, probably as a result of the large size of the oxygen ions. Therefore the sintering rate can be improved by increasing the oxygen vacancy concentration by slight reduction or by appropriate additions. Many experimental results are given in ref. 3. An example is given in Fig. 1 for the sintering experiments on Al,OJ. i i ! I I I 54321012345 ,a-3 dope_ - _ __-__ x10-3 Fig. 1. Porosittes ofceramlc samples of AI,Oa after 6 h 5nng at 1850°C tn air m dependence on dtITerent dope conct~s. (after Reynen3) Fig Z Extmction coefficients (proporttonal to mean particle dtameter) of freshly prepared A&k sots containing 20 mol AgBr l- I. after 2 h aging (after Honker and Kruyt’). C = recrystalhzation maximum, F=zero point flocculation maximum- (iii) Recrystallization in fresh colloidal solutions of “insoluble” sairs For equally charged ions (with exception of H+ and OH- ions) in water the difference in ionic radius is mainly compensated by the thickness of the hydration layer so that the differences in diflksion constants are small in comparison with those of ions in crystals. Therefore the mmimum of the diffusion constant of a salt will genera!ly be quite near the stoichiometric composition of the solution, and as a result the maximum rate of recrystallization will be found in colloidaI solutions where no excess of one of the components is added. In many cases the zero-point of charge is found at a different composition of the solution, having a clear excess of one of the components. i-his is especially well known for the silver halides in water6_ As a result, under favourable circumstances, two separate types of particle growth may be dis- J. Electroanal. Chem_, 37 (1972) 232 G. H. JONKER tinguished, one by recrystallization and one by flocculation. as is shown in Fig. 2. As, moreover, at higher excess concentrations flocculation of colloids will also occur, rather complicated patterns of particle growth will appear in a series of colloidal solutions prepz-ed with a variation in the excess_ concentration of the components. This has led to confusion in the literature as regards the properties of freshly prepared colloidal solutions of silver halides ‘S In work described in ref. 8 the recrystallization _ phenomenon codd be distinguished from flocculation by using protecting hydro- philic colloids Moreover electron microscope pictures have later shown that true single crystals are formed in this area of compositions In contradistinction to the explanations used in refs 7 and 8 our present impression is that the theory of diffusion rates offers the simplest explanation. SUMMARY In a short survey the analogy between the theories on diffusion in solids and in solutions is shown As a result, analogous theories are developed for the ceramic Grit&g process and the recrystallization in colloidal solutions_ By a thermodynamic method a description is given of the available free energy, the diffusion flux and the driving force_ REFERENCES 1 D- L. Johnson, J. AppL Phj-s_. 40 (X%9) 192 2 D. w. RCXky, J. Appf. Phys., 3? (: 366) 2309 ; 3. Amer. Ccrum. Sot., 47 (1966) 366. 3 P- J. L R R-en_ Rmctimv of Suiids, 6. John Why and Sons. Nca York, 1969. p_ 99. 4 G. C_ Kucqnski. Acra Met. 4 (1956) 58. 5 lI L Jobson, Smtmng und Related Pknomena, Gordon and Breach, New York. 1967. p_ 369; Z_ Ph_m_ Chem &i-Q=., 19 I (1942) 270. 6 H. R Kruyt. Colloid Sciencq Elxwier. Amsterdam, 195z. p_ 231. 7 a TciblL 3. Ph,r chmr_. 57 (1953) Ml. 8 G. H. Joriker and H. R lquyt, Discuss. Fm~uy Sot_. 1s (1954) 170. J. Ekcr7omraL Ch7x_. 37 (1972) 

On the theory of the ceramic densification process an of recrystallization in colloidal solutions

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