J. Phys. Chem. 1987, 91, 6069-6073 in the infrared in the ObServed band frequencies, but changes are seen in bandwidths and intensities. Presumably these are associated with the disappearance of birefringence and a change in the mobility of the surfactant in the samples. Time Frame of the Nc+ to ND- Transitions. An interesting hysteresis effect was observed in the rate at which reorientation of the director occurs during a phase transition. Whereas the NDto Nc+ reorientation occurs very rapidly, going from Nc+ to NDrequires more than 5 h before the phase returns to equilibrium, as determined by monitoring the intensity of the 1398-cm-l absorption associated with the terminal methyl bending mode in the ND- phase (Figure 6). The same hysteresis is observed with samples in glass capillaries. This is consistent with the fact that the ND- phase is more ordered than the Nc+ phase. As the homeotropically aligned ND- phase converts to an Nc+ plane texture, the relaxation is energetically favorable, but it requires considerably more energy for an ND- texture to be reestablished. An experiment was also performed observing the time frame for the conversion of the homeotropically aligned Nc+ phase to the hoineotropically aligned ND- phase. This conversion, being
energetically more favorable, requires be achieved.
-
6069
1 h for equilibrium to
Conclusions Significant changes in frequency, intensity, and bandwidth in the infrared spectrum accompany the phase transitions from rodlike to disklike aggregates in SDS-DEC-D20. The Nc+ phase is more disordered, but there is a higher degree of hydration of the polar head groups of the surfactant. By monitoring the terminal methyl bending frequencies, the time frame for director reorientation and establishment of equilibrium can be determined. Reestablishing an ND- homeotropic texture requires about 5 h just below the phase-transition temperature, but the ND- to Nc+ textural change is energetically favorable and therefore rapid.
Acknowledgment. This work was supported by the National Science Foundation-Solid State Chemistry under Grant DMR84-04009. Registry No. SDS, 142-87-0; CloD2,0H,110510-78-6.
Crystal Formation (Nucleation) under Klnetlcally Controlled and Diffusion-Controlled Growth Conditionst Ingo H . Leubner Photographic Research Laboratories-Photographic Products Group, Eastman Kodak Company, Rochester, New York 14650 (Received: November 24, 1986; In Final Form: June 23, 1987)
A model of crystal formation for homogeneous nucleation of sparingly soluble compounds was derived which takes into account the balance between crystal growth and classical nucleation. The model quantitatively relates the number of stable crystals formed to the precipitation conditions and to the growth mechanism of the crystals. In the present work the general model was derived and solved for kinetically controlled growth conditions. The results were compared with those for a previously derived model under diffusion-controlled growth conditions. Both models show the same functional dependence of crystal number, Z,on addition rate, solubility,and temperature. Two significant differencesare as follows: (1) The diffusion-controlled model depends on the diffusion constant of the solutes in solution, D,while the kinetic model is dependent on the surface integration constant, Ki.(2) In the diffusion-controlled model, the crystal number is constant throughout the precipitation after a relatively short transition period, while in the kinetically controlled case the crystal number decreases as the crystal size, r, increases,so that Zr is constant. The latter is in agreement with published data for the initial stages of an AgI precipitation at 35 "C,indicating that in that case crystal growth was kinetically controlled. From the transition point from kinetically to diffusion-contralledgrowth the kinetic surface integration constant could be estimated at about 56 cm4/(s mol). For AgCl (cubic), AgBr (cubic and octahedral), and AgI (at 70 "C)the crystal numbers were constant after the first minute of the precipitation. This is in agreement with diffusioncontrolled growth processes. While the present nucleation model was confirmed for silver halide precipitations, it may also be applicable to other systems which rely on homogeneous nucleation.
Introduction It is the objective of this paper to present a model of crystal formation which relates the number of stable crystals formed to the precipitation conditions and to the growth mechanism of the crystals. Homogeneous nucleation by sparingly soluble compounds is assumed. The model is based on a dynamic mass balance between crystal growth and (classical) nucleation and supplements the classical nucleation theory. It will be shown that predictions of the present model are in agreement with experimental results for silver halide precipitations. Crystallization of silver halides from solution is considered to occur in four different steps: nucleation, growth, Ostwald ripening, and recrystallization.'.* In modem precipitations of silver halides, silver nitrate and alkali halide solutions are added to an aqueous Paper presented at the 39th Annual Conference of the Society of PhotographicScientists and Engineers, May 18-22, 1986, Minneapolis, MN, and at the International Congress of Photographic Science, Sept 10-17, 1986, Cologne, Germany.
0022-3654/87/2091-6069$01.50/0
solution of gelatin under conditions where the temperature, excess halide or silver ion in solution, and reactant addition rates are tightly controlled. When the reactants are added at balanced addition rates in separate streams, the reaction is referred to as (controlled) double-jet precipitation. In double-jet precipitations, stable crystals are generally formed only during an initial nucleation step. After the nucleation period the number of crystals generally remains constant throughout the rest of the precipitation. Thus, nucleation is a critical process that determines the final crystal size and, indirectly, the photographic properties of the silver halides. Therefore, it is desirable to develop a model which quantitatively relates the crystal number to the precipitation conditions and variables. It will be shown that the classical nucleation theory predicts continuous nucleation throughout the precipitation and does not describe the formation of a stable crystal (1) James, T. H.,Ed. The Theory of the Photographic Process; Macmillan: New York, 1977; Chapters 1 and 3. (2) Berry, C. R. in ref 1, p 88 ff.
0 1987 American Chemical Society
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Leubner
The Journal of Physical Chemistry, Vol. 91, No. 23, 1987
Quasi Steady State (Dynamic Mass Balance)
Q
13
Time, t (non linear scale)
Log (Supersaturation ratio)
Figure 1. Relative nucleation rate, J / A (eq l), vs log (supersaturation ratio) as a function of surface energy, y. population for sparingly soluble materials like silver chloride, bromide, iodide, and their mixed-phase crystals. It is the intent of this paper to present an alternate nucleation model which overcomes the limitations of the classical nucleation theory. A limited model was previously presented for the case of nucleation under diffusion-controlled growth conditions of silver halides in controlled double-jet precipitations and was experimentally supported for the precipitations of silver bromide and silver c h l ~ r i d e . ~ -It~ is the purpose of this paper to present a general model and to extend this theory to nucleation under kinetically controlled growth conditions. The present model is based on the premises that (a) nucleation of crystals is an ongoing process during the precipitation (classical theory) and (b) that a constant number of stable crystals is determined by the kinetics and mechanisms of crystal growth and Ostwald ripening. Ostwald ripening is the process of dissolution of small crystals and reprecipitation on larger crystals. For the present model it is thus necessary to include the growth mechanism of the crystals. Under diffusion-controlled growth conditions, the diffusion of material to the crystal surface is significantly slower than the integration into the crystal surface. Reversely, under kinetically controlled growth conditions surface integration is the rate-limiting process. In the context of the present model, ‘diffusion-controlled and kinetically controlled nucleation” will refer to the underlying growth mechanism. To set the present model into perspective, previous nucleation models will be reviewed. Classical Nucleation Theory In a supersaturated solution, nucleation may begin spontaneously (homogeneous nucleation), may be triggered by foreign particles (hetergeneous nucleation), or may be furthered by break-up of existing particles (secondary nucleation). It is now generally accepted that silver halide nuclei are formed by homogeneous nucleation (spontaneously) in practical precipitations.’,’ According to the theory of homogeneous nucleation6the number of (spherical) nuclei formed per unit volume and unit time (nucleation rate, J) is J = A exp[-16ry3V:/3k3T3(ln SI2] (1) Here, A is a constant which has been estimated to be in the order of loz3to 1032.5(ref 7 and 8), y the energy of the crystal/solution interface per unit area (erg/cm’), V, the molecular volume of the crystal (cm3/molecule; molar volume/Avogadro’s number), k the (3) Leubner, I. H.; Jaganathan, R.; Wey, J. S . Photogr. Sci. Eng. 1980, 24, 268. (4) Leubner, I. H. In Colloids and Surfaces in Reprographic Technology, Hair, M., Croucher, C., EMS.; American Chemical Society: Washington, DC, 1982; ACS Symp. Ser. No. 200, p 81. ( 5 ) Leubner, I. H. J. ImagingSci. 1985, 29, 219. (6) Mullin, W. J. Crystallization, 2nd ed.; Butterworths: London, 1972; p 142. (7) Walton, A. G. Mikrochim. Acza 1963, 3, 422. (8) Nielsen, A. E. J. Phys. Chem. Solids, Suppl. 1 , Crystal Growth 1967, 419.
12
tl
-
t3
Time Figure 2. Supersaturation profile and crystal number vs precipitation time. Both axes are relative and nonlinear.
Boltzmann constant, T the absolute temperature (K), and S the supersaturation ratio. S is defined as S=
actual concentration = -C equilibrium concentration C,
(2)
For instance, for AgBr the silver ion concentration is the sum of all soluble silver ion species, e&, Ag’, AgBr, AgBr2-, etc.; e.g., at 70 OC and pAg, 8.0, the solubility is about 1.0 X 10” m ~ l / L . ~ If a 1.0 M AgNO, solution is introduced into this solution then the supersaturation ratio at this point is lo6. In Figure 1, J / A (eq l), the relative nucleation rate, is plotted as a function of log S and surface energy ( 7 ) . The value of A has not yet been determined for the silver halide systems. The following values were chosen for the parameters in eq 1: V, = 0.47 X cm3/molecule (value for AgBr), k = 1.38 X erg/(deg molecule), and T = 333 K (60 “C). Estimates of the surface energy vary from about 20 (ref 9) to about 140 erg/cm’ (ref 5). Thus the value of y was varied from 10 to 160 erg/cm’. The results show that for practical silver halide precipitations (S > lo5) the relative nucleation rate is approximately constant. Small changes in S (less than 1OX change) do not significantly change the relative nucleation ratio. The size of the primary crystal nuclei (AgX), was estimated to be n = 4 for AgBr and n = 8 for AgCl by using stopped-flow continuous precipitation The classical nucleation theory gives estimates of n = 4-10 for AgBr and supersaturation ratios of 20-273, and of n = 5-10 for AgCl (S = 58-179).” The primary nucleus sizes are significantly smaller than the crystal sizes obtained during silver halide precipitations. They are also significantly smaller than the critical nucleus sizes estimated for the steady-state stage of silver chloride and silver bromide precipitation~.~ Thus these primary nuclei will dissolve and reprecipitate onto existing larger crystals. It is apparent that the classical nucleation theory is not sufficient to predict observed variations in the nucleation of silver halides. Furthermore, this theory does not predict an end of nucleation. In practical systems apparent nucleation eventually ceases and a finite number of crystals is obtained. These continue to grow and apparent nucleation These difficulties are resolved when both growth and Ostwald ripening processes are included in the considerations of the nucleation step. The Dynamic Mass Balance Model of Nucleation Figure 2 is a sketch of the course of the precipitation of sparingly soluble salts via direct mixing of the reactants, e.g., double-jet (9) Wagner, C. Z . Electrochem. 1961, 65, 581. (10) Tanaka, T.; Iwasaki, M. J . Photogr. Sci. 1983, 31, 13. ( 1 1) Tanaka, T.; Iwasaki, M. J . Imaging Sci. 1985, 29, 86.
The Journal of Physical Chemistry, Vol. 91, No. 23, 1987 6071
Crystal Formation precipitation of silver halide crystals. Here, the supersaturation ratio, S, is plotted as a function of time into the precipitation where the time scale is arbitrary and nonlinear. Initially, the system is at equilibrium. At the start of the precipitation S increases rapidly and exceeds the critical value S, for spontaneous nucleation at the time tl. As soon as crystal nuclei are formed, growth of the nuclei will compete with nucleation and Swill increase at a lower rate until a maximum is reached. After that, S will begin to decrease and drop below S, at the time t2. The time span (tl to t2), when S is greater than S,, is usually referred to as the “nucleation region”. After the time t2 a transient period occurs where S decreases to a quasi-steady state where the supersaturation ratio is determined by an equilibrium between reactant addition and crystal growth. It has been estimated that in silver halide s and that steady-state t3 precipitations t, is about 10” to is reached after about 1-2 min.’S2 This nucleation model was supported experimentally.’* Our initial analysis indicates that the actual supersaturation at the silver nitrate entrance point is about lo5 to lo7, using the solubility data of ref 3. At the same time, the critical supersaturation ratio for renucleation was determined to be less than 2.5 Renucleation is the formation of new stable crystals after the initial phase of formation of stable crystals has ended. It has been suggested that at the introduction point of the reactants a high supersaturation region exists where “transient nuclei” are formed continuously, which quickly dissolve during the growth phase and act as source material for the growth of existing larger crystal^.'^^'^ This continuous formation of transient nuclei is not explicitly included in the present model which was derived to relate the number of stable nuclei to the precipitation conditions. It could be shown that the ratio of critical nucleus to actual crystal size (r*/r) is dependent on the precipitation conditions and generally is larger than 0.5 for common precipitation condition^.^ The critical radius (critical nucleus size), r*, represents a crystal size above which a crystal grows spontaneously and below which it dissolves. Crystals formed in the “reaction zone” where the silver nitrate enters the reactor would be well below the critical nucleus size and thus dissolve under the quasi-steady-state conditions. The dynamic mass balance model of crystal nucleation considers a dynamic mass balance for the quasi-steady-state regione3 Because S does not decrease substantially and was determined to be a small number, the bulk concentration is negligibly small and can be neglected relative to the total added mass of reagents. Thus, for a given population of crystals at time t > t3, the dynamic mass balance can be written as R=
c(:
-$4nr2Gn
dr
(3)
The following variables have not been defined previously: D, diffusion constant; g, maximum growth rate; Ki, the rate constant for the surface integration step; E , the ratio of relative resistance of bulk diffusion to surface integration; and R,, the gas constant. Successive introduction of eq 4-7 into eq 3 gives
Equation 8 describes the nucleation for the diffusion- and kinetically controlled crystal growth conditions. The crystal number is implicit in the integrals. Since eq 8 is difficult to evaluate, we will concentrate on the two extreme cases, first diffusion-controlled and then kinetically controlled growth conditions during the nucleation phase. Diffusion-Controlled Growth/Nucleation For diffusion-controlled growth, e.g., AgCl and AgBr, E is relatively large. If we introduce the approximation of eq 9 and the values for the integrals (eq 10 and 11) then we obtain eq 12. (r
+ l/e)
Z=
=r
(9)
L m nd r
Zi; = X - n r d r RR, T
z = 8nyVmC,D(F/r* - 1 .O)
(12)
The crystal size 7 on the left side of eq 1 1 represents the average number weighted crystal size, and Z is the total crystal number. This equation describes well the nucleation of AgBr and AgCl in controlled double-jet precipitation^.^-' Kinetically Controlled Crowth/Nucleation Under kinetically controlled growth conditions, the diffusion of material to the crystal surface is significantly faster than its integration into the surface, Le., D >> Ki. In this case, e becomes smaller than 1 and we can introduce the approximation (r
+ l/e)
= l/t
(13)
Insertion of this approximation into eq 8 leads to
R=
8nyCSDV,e 1 R,T -r*( L m r 2 n d r - L - r n dr]
(14)
Here, R is the molar addition rate of material (mol/s), r is the The first integral represents the total surface area of the system radius of a spherical crystal, G is the linear growth rate (drldt), and can be expressed by the total crystal number Z and the and n is the crystal population density (ndr represents the number area-weighted crystal size (radius), r, (eq 15). The second integral of crystals in the system having a size in the range r, r dr). V , is the molar volume of the crystal (cm3/mol). Zr: = r2n d r For the growth of silver halide crystals under normal double-jet conditions, the linear growth rate G can be represented by an was defined in eq 1 1 , where p is the number-weighted average expression which includes bulk diffusion, surface integration, and the Gibbs-Thomson effect (also known as Ostwald ripeni~~g).’~.’~ crystal size (radius). Inserting the values from eq 1 1 and 15 into 14, substituting eq 7 for t, and solving for Z , we obtain (4) RR,T Z= (16) 8nyC,Ki(r:/r* - i;)
+
Lm
Equation 16 correlates the number of stable crystals formed, Z, with the precipitation conditions under kinetically controlled growth conditions. To evaluate this equation, both the numberand area-weighted crystal sizes must be determined. For monodisperse emulsions, which are generally obtained in controlled double-jet precipitations, we can use the following approximation: (12) Jagannathan, R.; Wey, J. S. J . Crystal Growth 1985, 73, 226. (13) Berry, C. R. Photogr. Sci. Eng. 1964, 20, 1. (14) Margolis, G.; Gutoff, E. B. AIChE J. 1974, 20, 467.
(15) Wey, J. S.;Strong, R. W. Photogr. Sci. Eng. 1977, 21, 248. (16) Wey, J. S.; Strong, R. W. Photogr. Sci. Eng. 1979, 23, 344.
6072
The Journal of Physical Chemistry, Vof. 91, No. 23, 1987
Leubner
and eq 16 can be further simplified to
RR,T Z=
8.rryCsKir(r/r*- 1.0)
Discussion and Experimental Results It is now possible to compare the equations for diffusion- and kinetically controlled nucleation (eq 12 and 18) and to relate the predictions to experimental results. A comparison of eq 12 and 18 shows that the crystal number has the same functional dependency on addition rate, R , solubility, C,, temperature, T , and r/r* independent of growth mechanisms. Thus, these functionalities cannot be used to discern between the growth mechanisms. However, two significant differences are apparent, which are due to the different growth mechanisms: 1. In the kinetically controlled model (eq 18) the surface integration constant, Ki, replaces the diffusion constant D in eq 12. Thus, the nucleation is no longer dependent on diffusion processes. While the diffusion constant, D,in eq 12 could be approximated by the diffusion constant in gelatin,5~15~16 the determination of Ki is more difficult and must be determined for each substrate. It is a function of the crystal composition, its surface properties (e.g., adsorbed species), growth species, (e.g., Ag', AgX, AgX2-, etc. (X = monovalent ligand)), and temperature. The experimental determination of r / r * , which was relatively straightforward for the diffusion-controlled nucleation of AgCl and AgBr,5 becomes thus quite difficult for the kinetically controlled nucleation system. 2. The other significant difference between eq 12 and 18 is the presence of the crystal size, r, in the divisor of eq 18 (the quotient r/r* is common to both systems). This predicts that in the kinetically controlled nucleation and growth systems the number of stable crystals, Z , should decrease as the average crystal size increases (2 0: 1/ r ) . The dependence of Z on l / r reflects a dependence on the surface/volume ratio of the crystal population. As long as crystal growth is kinetically controlled, material is transported to the surface but also away from it. This leads to a decrease in crystal size for a fraction of the crystals. When the size of a crystal decreases below the critical crystal radius r*, it will completely dissolve, resulting in an overall decrease in crystal number. When the total surface area has increased above a critical value where diffusion to the crystal surface becomes the limiting growth factor, dissolution of crystals will cease. At the critical surface area kinetic and diffusion-controlled growth are in equilibrium. From a determination of this transition point the value of the kinetic growth constant, Ki, may be estimated (eq 12 = eq 18):
where rc is the crystal size at the transition from kinetically to diffusion-controlled crystal growth. Reevaluation of eq 18 shows that for fixed precipitation conditions, addition rate, solubility, temperature, and r/r*, the product Zr should be constant as long as kinetically controlled growth processes dominate. It was determined previously that r/r* is a constant for a given precipitation condition for AgCl and AgBr.5 The present theoretical derivations thus predict the following: In precipitations where nucleation and growth are controlled by diffusion processes the number of crystals remains constant after the initial nucleation and transition period. In precipitations that are kinetically controlled, the product Zr will be constant. Transition from kinetically to diffusion-controlled growth will result in change from declining to constant crystal population. What is the experimental evidence? It was reported that for AgCl and AgBr the number of crystals becomes constant after about 1 m h 2 In this study, the crystal numbers of AgCI, AgBr (cubic and octahedral crystal forms), and AgI were followed under controlled precipitation conditions for 20 min (Figure 3). The results show that the crystal number did not change after the first
AgBr (oct.)
.
AgBr (cub.)
-
* *
-
6
A
v
1
I
AgCl
4)
I
Figure 3. log(crysta1number) vs precipitation time (min) for AgC1, AgBr (cubic and octahedral), and AgI at 70 OC (present work). TABLE I: Crystal Size and Number vs Precipitation Time
AgCl"
AgBr(cubic)"
time, zx min size size 1 0.063 4.14 0.044 2 0.104 1.84 0.070 4 0.131 1.84 0.084 6 0.141 2.21 0.094 8 0.153 2.31 0.105 12 0.173 2.40 0.119 16 0.196 2.20 0.126 20 0.215 2.08 0.137 silver halide AgCl AgBr AgBr
AgBr(oct)"
zx
AgIb
Z X
size
10-l~
Z X
size
10-l~
13.60 0.058 12.60 6.76 0.091 6.53
7.82 0.109
7.60 0.035 77.30
8.37 0.118 8.01 0.129
8.99 0.037 98.10
9.17 0.044 8.26 0.146 9.48 0.047 9.27 0.153 11.0 0.059 9.02 0.167 10.60 0.063
Precipitation Conditions' addition rate, temp, "C mol/min morphology cubic 70 0.040 70 0.040 cubic octahedral 70 0.040 70 0.020 undefined
77.70 95.70 64.50 66.20
pAg
5.85 6.95 8.75
9.50 AgI 'Size = edge length (pm), Z = crystal number. bSize = equivalent diameter (pm), Z = relative crystal number (see text). CInitialreactor volume: 2.0 L, 2.5% deionized bone gelatin, pH 5.60. TABLE 11: Effect of Precipitation Time on the Crystal Size and Number of an AgI Precipitation" moles of AgI formed r, nm Z , X lo-' Zr X 0.255 45 26.8 1.2 0.860 81 12.6 1.1 1.37 107 11.0 1.2 2.04 129 9.4 1.2 2.91 147 9.5 1.4 4.0 166 9.3 1.5
'Data obtained from ref 13; size ( r ) is a crystal linear dimension (CLD, see ref 13); Z, is a relative crystal number; precipitation conditions: pAg 8.0, 35
OC.
2 min of precipitation (Table I). The crystal number at the first minute of precipitation is significantly larger than during the steady state. This indicates that the transient region extends beyond the first minute of precipitation. Small crystals are prone to Ostwald ripening, which may result in apparent larger crystal sizes after sampling. To minimize this effect, a known growth restrainer (2-phenylmercaptotetrazole)was added to the silver halide samples (Table I). These results support the hypothesis that the nucleation and growth of AgCl, AgBr, and AgI are dominated by diffusion-controlled processes under these precipitation conditions (70
"C). A decrease in crystal number over an extended time of precipitation was reported by Daubendiek for AgI under different
Crystal Formation
R
Figure 4. log(re1ative crystal number) vs silver ion addition for AgI (data from ref 13).
precipitation conditions (35 OC).” The data are plotted in Figure 4 and listed in Table 11, where the product Zr was calculated from the published data. Throughout the first half of the precipitation, Zr is a constant in agreement with the derivation of eq 18. After that time the number of crystals becomes constant. This is in agreement with a transition from kinetically to diffusion-controlled growth processes. The different results for precipitations at 70 and 35 O C indicate that a change in nucleation mechanism may occur between these temperatures. One can hypothesize that at the lower temperature the kinetics of surface integration has slowed to a degree where crystal growth becomes kinetically controlled throughout much of the precipitation. Once the total surface area of the system has increased to a level where the kinetics of material incorporation becomes faster than the speed with which the material can be transported to the surface, the crystal growth becomes diffusion controlled. We can now use eq 19 to estimate the magnitude of the kinetic growth constant for the system. We estimate from Table I1 that r, = 65 nm (CLD/2). For 35 O C the diffusion constant was cm*/s (ref 5) and the molar volume estimated to be 0.88 X of AgI (hexagonal structure) is 41.4 cm3/mol. These data give an estimate for Kiof about 56 cm4/(s mol).
Conclusions A general theory of crystal nucleation was derived which is based on dynamic mass balance and growth mechanisms which include bulk diffusion, the kinetic surface integration, and the Gibbs-Thomson effect (Ostwald ripening). The validity of the diffusion-controlled model was confirmed for AgCl and AgBr.3-3 A distinction between the diffusion- and kinetically controlled nucleation and growth processes was made by using the history of the crystal population throughout the precipitation. Under diffusion-controlled conditions, the number of crystals remains essentially constant after an initial transition period of less than (17) Daubendiek, R. L. Proc. Int. Congr. Photogr. Sci. 1978, 141.
The Journal of Physical Chemistry, Vol. 91, No. 23, 1987 6073 1 min. This was observed for precipitations of AgCl, cubic and octahedral AgBr, and AgI under the present precipitation conditions 70 OC. For kinetically controlled conditions, the theory predicts that the number of crystals decreases with the progress of the precipitations and that the product of crystal number and crystal size, Zr, is constant. A decrease of crystal number with Precipitation time was observed for the initial stages of an AgI precipitation (at 35 OC).” In agreement with the theory, a constant value of Zr was calculated for the first half of the precipitation. At this point, a transition from kinetically to diffusion-controlled growth is indicated by a transition from decreasing to constant crystal numbers. From this the kinetic integration constant Kiwas estimated to be about 56 cm4/(s mol). The theory of kinetically controlled nucleation and growth may also apply in the initial transient region of nucleation (tz to t3, Figure 2) where the crystal number decreases after nucleation has ceased. Additional experiments in this time region are needed to confirm this hypothesis. The present model could be extended to quantify the effect of Ostwald ripening agentsi8and crystal growth restrainersig during the nucleation phase on the crystal number. The present theoretical treatment thus explains previous experimental results and points to interesting directions for future research.
Acknowledgment. I thank R. A. Chasman, R. L. Daubendiek, and J. S . Wey for helpful discussions, A. Kocher and J. Mitchell for editing this manuscript, and C. E. Flowers for technical assistance.
Glossary A
cs
D G J Ki
R
Psc T
vm
v,
Z €
g Y
k n
r F ra
r8
preexponential factor (eq 1 ) solubility, mol/cm3 diffusion constant, cm2/s linear growth rate nucleation rate surface integration rate constant molar addition rate, mol/s gas constant supersaturation ratio (eq 2) critical supersaturation ratio absolute temperature, K molar volume, cm’/mol molecular volume, cm3/molecule total crystal number defined in eq 2 maximum growth rate surface energy, erg/” Boltzmann constant, erg/(deg molecule) crystal population density crystal size average crystal size (number weighted) average crystal size (area weighted) critical crystal size
(18) (a) Leubner, I. H. Presented at the 39th Annual Conference of the Society of Photographic Scientists and Engineers, May 18-22, 1986, Minneapolis, MN, and at the International Congress of Photographic Science, Sept 10-17, 1986, Cologne, Germany. (b) Leubner, I. H. J . Imaging Sri., 1987, 31, 145. (19) (a) Leubner, I. H., see ref 18a. (b) Leubner, I. H., J . CrysralGrowth, to be published.
