16128
J. Phys. Chem. 1995, 99, 16128-16135
Ostwald Ripening of KNO3 Grains in Acrylamide Gel Shigeo Sasaki" and Hiroshi Maeda Department of Chemistry, Faculty of Science, Kyushu University, 33 Hakozaki, Higashiku, Fukuoka 81 2, Japan Received: May 22, 1995; In Final Form: August 3, 1995@ Ostwald ripening of KNO3 grains in acrylamide gel was investigated for the grain's volume fractions of 0.05 and 0.10. The grains observed were rodlike at the initial stage but became spherical with ripening. The sizes and the number densities of the spherical grains were measured, and power relations for them against the ripening time t were obtained. At the low volume fraction (0.05) the exponents of the power relation with respect to the average size and the number density, respectively, were 0.365 and -0.66 for t = 20-800 h, while at the high volume fraction (0.10) they were 1.39 and -1.93 at the early stage t = 70-167.7 h but 0.34 and -0.88 at the late stage t = 167.7-900 h. The size distribution functions were unimodal for the low volume fraction and multimodal for the high volume fraction. The function at the low volume fraction for t = 20-50 h was found to be described by the reaction-controlled parabolic growth. The functions normalized by the average sizes became close to the Lifshitz-Slyozov-Wagner function as t increased. The present experimental results suggest that the concentration fluctuation at the initial period influences the growing behaviors of the grains for a long period.
1. Introduction Grain particles of the solute molecules precipitate from the supersaturated solution, evolve so as to minimize their surfaceto-volume ratio, and thereby minimize the surface free energy. With the lapse of time the number of grains is reduced and the grain size increases. This precipitation process is known as Ostwald ripening, which occurs in systems such as colloidal particles in solution,',* binary alloy^,^-^ and ionic glasses.6The precipitation of inorganic materials occurs also in biological bodies, which is termed as biomineralization?,s In the biological system the precipitation fields are the extracellular polymer matrices or highly concentrated polymer solutions, and the precipitation is highly organized to be tailored to the biological functions. For understanding the biomineralization mechanism it is important to clarify the growing behaviors of precipitated grains in the polymer matrices. Experimental studies on Ostwald ripening have been done extensively for the precipitates in metal alloys9 or in glassy electrolytes.'0,' I Theoretical descriptions of the ripening have been proposed for the condensing and evaporating systems by Lifshitz, Slyozov,'2 and WagnerI3 for the limit of zero volume fraction of the grain and by Marqusee and RossI4 for the case of the finite volume fraction. In the theories, the ripening rate is considered to be controlled by the diffusion of precipitating molecules in the solution phase. Jain and Hughes have taken into account in the theoryI5 the effect of the lateral diffusion of solute molecules over the grain boundary on the ripening, which results in a narrower size distribution. KahlweitI6has developed the theory to describe the case of reaction-controlled ripening. However, the experimental results have not always been satisfactorily explained by these theories based on the gross homogeneity of the system. Recently, various conditions of the system are recognized to influence the ripening behaviors. For example, elastic properties of the medium have been suggested to induce the narrower size distribution^^^-'^ by the computer simulations. Various types of Ostwald ripening may be possible, though many of their properties including biomineralizations have not been fully revealed. The purpose of this work is to examine growing behaviors of KN03 grains in gel matrices. It was found that KNO3 formed @
Abstract published in Aduunce ACS Abstracts, September 15, 1995.
0022-365419512099-16128$09.00/0
rodlike grains in the gel at the very beginning of ripening and that the rod grains become spherical with the lapse of time. The transformation rate from the rod to the sphere was found to depend on the degree of polymerization of cross-linked chains (D,) of the gel or the density of the network. The ripening behavior of the spherical grain was found to depend strongly on the degree of supersaturation of the solution A at the initial stage. Here A is defined by A = Ct, - CO-, where c b and CO, are the concentration in the bulk solution and the solubility of the bulk solid, respectively. Under the condition that D, = 100 and A = 2.5 wt %, the size distribution of the grains at the early times was described by the theory of KahlweitI6 for the reaction-controlled parabolic growth, and at the later times it became the closer to the theoretical prediction of Lifshitz, Slyozov,'2 and Wagner.I3 It was also found that the crystal formed in the gel was mechanically separated from the gel.
2. Experimental Section Gels were prepared by radical copolymerization in an aqueous solution of acrylamide (1.4 M) and N,K-methylenebis(acry1amide). The polymerization was initiated by ammonium peroxydisulfate and was carried out in an oven at 70 "C. The average degree of polymerization of cross-linked chains (D,) was defined as
D , = [Acrylamide]/(2[Bis(acrylamide)])
(2.1)
where [Acrylamide] and [Bis(acrylamide)] denote the concentrations of acrylamide and N,K-methylenebis(acrylamide), respectively. The gel synthesized in a plate form of 2 mm thickness was rinsed thoroughly with distilled water, cut into a rectangular piece (10 x 20 mm), and soaked in KN03 solution of a given concentration at 60 "C for more than 10 days. Polymer concentration of the gel on the monomer basis C,, was given by c , = Wd/w,M,, where wd, w, and Ma, respectively, were the weights of the gels in the fully water absorbing and dry states and the molecular weight of monomer. In the present study C, was about 0.1 M. All chemicals used were of reagent grade. For the experiment of Ostwald ripening, the gel was taken out from the KN03 solution and was sandwiched between two 0 1995 American Chemical Society
Ostwald Ripening of KNO3 Grains in Acrylamide Gel
J. Phys. Chem., Vol. 99, No. 43, 1995 16129
1 mm
Figure 1. Photograhs of the Ostwald ripening in experiment 1 at (a) t = 16.6 h, (b) t = 65.8 h, (c) t = 400.7 h, and (d) t = 808.5 h.
glass microscope slides with a 2 mm thick spacer of silicone rubber after carefully removing droplets of the solution on the gel surface by wiping with filter paper. The silicon grease was used for sealing the gel between the slides and the spacer. The setup operation of the gel on the slide was made above 40 "C. The gel was put onto the stage of an optical microscope for observation, where the temperature was controlled to be 25 f 0.5 "C. Observations were made from time to time and photomicrographs were taken with recording times (t) elapsing after setting the gel on the stage. The microscope system used was a Nikon SMZ-2T dissecting microscope (Nikon Ltd., Japan). The field of vision of the microscope was 4.1 x 2.9 mm, and the thickness of vision was 2 mm, within which a sphere of 20 pm diameter was recognizable, so that the volume of vision was about 24 mm3. The sizes of the grains in the photographs were measured with the aid of a computer in the following way. The grain boundaries, which were approximate circles and sometimes overlapped with each other in the photograph, were carefully traced onto transparent sheets. The traced figures were processed into the digital image data by using an image processor (Image Scanner GT-6000, Epson Ltd., Japan) and were displayed on a computer screen. A circle was drawn on the screen so as to overlap three arbitrary points of the trace of one grain on the circle, and the radius of the circle was evaluated by using the computer. The error in the radius determined was estimated to be about 5%, as judged from the measurement on the particles with known sizes. The number of grains was also counted when the radii were measured. For examining whether the gel networks penetrate into the ripened crystals or not, the gel was cut into pieces and some of pieces were observed with the diascopic Nomarski differential interference contrast optical microsope (OPTIPHOTO-2-NTF2 Nikon Ltd., Japan).
3. Theoretical Background The theories proposed by Lifshitz, S l y o ~ o v ,and ' ~ Wagner'3 and by KahlweitI6 are briefly reviewed. Ostwald ripening is caused by the diffusion flow of the solute molecules from a high-concentration region around the small grain to a lowconcentration region around the large grain. The diffusion diminishes the concentration in the former region and increases the concentration in the latter region with t. As a consequence of the diffusion, small grains decrease in size or dissolve completely. Concerning large grains, the solute molecules deposit on their surface and thereby they grow, contrary to the case of the small grains. The increase or decrease in the grain size occurs according to the situation that the concentration at the grain surface is higher or lower than the solubility of a particle of radius r. The solubility Cs is given by the GibbsThomson relation as follows.
(3.1) where y and V', respectively, are the interfacial energy per unit area of the grain and the molar volume of KNO3 in the grain. According to Kahlweit,Ih the growth kinetics for a grain of radius r in the surrounding solution of concentration C is generally described as
where k and 1 are positive integer numbers, and K is a positive constant. Here 1 denotes the reaction order in the deposition of molecules on the grain surface or the dissolution from the grain. It should be noted that the case k = 1, I = 1, and K = 4nD
UJUJ 1
J. Phys. Chem., Vol. 99, NO.43, 1995 16131
Ostwald Ripening of KNO3 Grains in Acrylamide Gel
Figure 3. Photographs of the Ostwald ripening in experiment 3 at (a) t = 126.1 h, (b) t = 139.6 h, (c) t = 163.8 h, and (d) t = 188.5 h. The rodlike grain pointed out by the arrow becomes spherical with elapsing time.
controlled growth as follows (3.13) These theories predict that the number densities and the average sizes of the grains, respectively, decay as t-l or r 3 / 4 and grow as t1I3 or P 9 . 4. Results
F(u) =
b
The constants in eqs 3.8 and 3.9 stand for factors for the normalization. The mass conservation law combined with the solution of eq 3.8 gives the following power relations of the grain number density, N,and the average radius (r) with respect to f .
N = AQt-’
(r)=a,(
2yV2C,D RT
(3.10)
($)
113
t)
;a, *
(1
+ 0.746) (3.1 1)
.
where Q and A are the initial degree of oversaturation and a constant, respectively. The same type of asymptotic equations for N and (r) have been obtained for the case of the solution of eq 3.8 by KahlweitI6 as follows
Within several tens of minutes after setting up the gel on the stage, the transparent gel became opaque white. The opaque formation started from the edge of the gel plate and spread over the gel in a few minutes. After the elapse of a few hours tiny grains were observed in the gel under a microscope, although their sizes were too small to be accurately measured. The gel became less opaque at t more than 10 h, the transparency of the gel was gradually recovered with the lapse of time, and the sizes of grains became measurably large. Figures 1-3 show different stages of the ripening of grains in the gels as time elapses. The KNO3 concentrations of the solutions (CKNO~) used in the experiments shown in Figures 1, 2, and 3 were 30, 32, and 30 wt %, respectively. The solubility of KNO3 at 25 “Cis 27.5 wt The experiments shown in Figures 1, 2, and 3 respectively are referred to as experiment 1, experiment 2, and experiment 3. The D, values of the gels used in experiments 1 and 2 were 100, and the D, value of the gel used in experiment 3 was 400. As shown in Figures 1, 2, and 3, the shape of the crystals ripened in experiments 1 and 2 is spherical, while the shape in experiment 3 is rodlike. Rodlike grains were also formed in the supersaturated solutions. It is noticeable in Figure
Sasaki and Maeda
16132 .IPhys. . Chem., Vol. 99, No. 43, 1995
1: t= 16.6 Hours
Id c
5 B z
n
100
0
io3
Time(t)iHours
100
b
I 103
Figure 4. Power relations of the average sizes (0)and the number (0)of grains against t for (a) experiment 1 and (b) experiment 2. The solid lines are obtained by a least squares fit to the data. TABLE 1: Power Relations of the Average Size and the Number of Grains in a Gel Volume of 0.238 cm3 against t (s) exoeriment no.: time sizelcm number 1: t = 16.6-809 h 2: t=71.9-168 h 2: t = 168-864 h
(2.8 x 10-'2t)037 (2.5 x 10-7t)'39 (9.9 x 10-13t)034
300
1: t= 71.9 Hours 2: k167.7 Hours 3:t=360.6 Hours 4:1 4 6 4 . 1 Hours
0
Time(t)/Hours
~~
200
Radius/ P m
-
(2.5 10-'0t)-066 (2.0 x 10-8t)-' 93 (1.4 x 10-9t)-o88
3 that the lengths of some rods decrease, their diameters increase with the lapse of time, and their shapes become more spherical. It should be noted that even in experiments 1 and 2 very small rodlike structures (a few tens of micrometers) were observed at the initial stage. This implies that the rodlike grains were made in the opaque gel of experiments 1 and 2 at the very early stage. As shown in Figures l a and 2a, at the early stage grain sizes were very small and number densities of grains were very high. The larger grains were recognized after about t = 9 h in experiment 2; before that only densely distributed small grains were observed. It is noticeable in Figure 2a that larger grains are surrounded by many densely distributed smaller grains. With the lapse of time those larger grains grew, while smaller grains disappeared and the number of grains decreased. After the elapse of more than 300 h the grains shown in Figures lc,d and 2c,d were obtained. Most of the grains were large. These are characteristic features of Ostwald ripening. The volume fractions at t more than 300 h were measured to be about 0.05 and 0.10 for experiments 1 and 2, respectively. The average sizes and the observed numbers of grains in experiments 1 and 2 are shown, respectively, in parts a and b of Figure 4 as functions oft. The power relations against t are shown to hold for both the sizes and the numbers in Figure 4. It is noticeable that the exponents observed in experiment 2 change values at about t = 168 h, as shown in Figure 4b. This might be related to the fact that most grains of radius smaller than about 30 pm which were densely distributed around the larger grains, as shown in Figure 2a, disappeared before t = 168 h. The numerically described power relations are shown in Table 1. The grain size distributions observed in experiment 1 and experiment 2 at various t are shown in parts a and b of Figure
loo
200
300
Radius/ v m
Figure 5. Changes of the grain size distributions with time observed for experiment 1 (a) and experiment 2 (b).
5 respectively. The size distributions shown in Figure 5b seem multimodal, while those in Figure 5a appear to be unimodal. The size distributions normalized by the averaged sizes are shown in Figure 6a,b. The LSW the modified LSW functioni4of eq 3.8, and the distribution functions of eq 3.9 for the cases of the reaction-controlled linear growth ( k = 2 and 1 = 1) and the reaction-controlled parabolic growth (k = 2 and 1 = 2)16 are also shown in Figure 6a,b for comparison. With the lapse of time the profiles of distribution became close to that of the LSW function or the modified LSW function, as shown in Figure 6a,b. The distributions at the early stage in experiment 1 is very close to the distribution function for the reaction-controlled parabolic growth as shown in Figure 6a. The distribution shown in Figure 6b deviates from any of the theoretical distribution functions mentioned above. Figure 7 shows a crystal grain allowed to ripen for 7 days in the gel with D, = 100 under the condition of CKNO, = 30 wt % and being exposed on the surface of the gel by cutting the gel into pieces. Before cutting no exposed crystals were observed on the surface of the gel. On the surface of the exposed side of the grain no pieces of the gel were observed, as shown in Figure 7 . From this observation it can be concluded that no penetration of the gel network into the grain crystal occurs. If the network penetrated into the grain, small pieces of the gel should be distributed over the surface of the exposed grain. 5. Discussions The rodlike grains as shown in Figure 3 were observed at the early stage of ripening. Generally, crystals grow in a manner to increase the area of the crystal plane whose interfacial energy is the lowest among possible crystal planes. The appearance of the rodlike grains is similar to the crystal growth in the solution phase, and this means that the interfacial energies of various planes in the gel are similar to those in the solution. That is, at the early stage of growth the interaction between the gel chain and the grain surface is considered to be too weak to affect the shape of the grain. As the grains ripen, however, the gel chains in the vicinity of the grain surface are condensed, deformed, and/or scissored. The interfacial energy is considered to be ascribed to the stress in the gel, which is caused by the
J. Phys. Chem., Vol. 99, No. 43, 1995 16133
Ostwald Ripening of KNO3 Grains in Acrylamide Gel
a
2 1
0 2 x
' = I 9
1 0 " 2
U
L
1
0
i
P
-:
0. Inm Figure 7. A grain exposed on the surface of the gel by cutting it into pieces. The grain is partly buried in the gel (the bottom part).
TABLE 2: y and y2Kr Values Estimated from the Power Relation Shown in Table 1" experiment no.: time
yldyn cm-l"
y2K&g2cm4 s-' mol-''
I : t = 16.6-809 h 2: t = 168-864 h
6.79 x IO2 2.38 x IO2
8.8 x IOX 3.1 x IOx
It is assumed in the estimation that C h = 3.18 mol/cm-3, V' = 47.9 cm3 mol-', and D = 1.5 x IO-' cm2 s-'. "The evaluation was made by using eq 3.1 1. The evaluation was made by using eq 3.13.
0
2
1
3
rl
Figure 6. Size distributions normalized by the avergae for (a) experiment 1 at t = 16.55.65.8.400.7. and 808.5 h from the top to the bottom, respectively, and (b) experiment 2 at t = 7 1.88, 167.7, 360.6, and 864.1 h from the top to the bottom, respectively (thick bar). For a comparison the distribution functions of eq 3.9 for the reactioncontrolled parabolic growth law, for the reaction-controlled linear growth law, eq 3.7 for the LSW theory, and eq 3.8 for the modified LSW theory at 4 = 0.05 and 0.1, respectively, are drawn by the thick solid, thick broken, thin dotted, thin broken, and thin dash-dot lines in each figure.
deformation of the gel network and condensing gel chains and/ or to the energy of scissoring the chains. To decrease the interfacial energy, the grains change shape from rodlike to spherical, which is the shape for minimizing the surface area per unit volume of the grain. The interfacial energy increases with the chain density in the gel, and hence it is considered to be greater for smaller D,. Therefore, the stability of the rodlike grains increases with D,. The rodlike grains in the gel of D, = 400 might be much more stabilized and were observed for a much longer period than those in the gel of D, = 100. In the present experiment the power laws as shown in Figure 4a,b are observed. The numerical data of the power law are shown in Table 1. The observed exponents of the size growth for experiment I in the whole range of t as well as for experiment 2 at the late stage are very close to 113. The exponents for the number densities at t mentioned above are smaller than 1. These exponents, 113 and 1, are predicted by the LSW theory'2.13(eq 3.10) or the Kahlweit theory (eq 3.13) for the case k = 1 = 2.16 The discrepancy in the exponent for
the number densities may be due to the fact that the coarsegrained concentration of the solute is spatially inhomogeneous and the growth of grains causes interference because of the low spacing between their surface, as shown in Figures 1 and 2. The fact that the exponent for the size growth coincides with the theoretical prediction suggests that the growth behavior with t i l 3 is not much influenced by both the inhomogeneity and the interference. Such a small effect of the inhomogeneity might be suggested by eqs 3.1 I and 3.13, which show no concentration dependency of the growth of (r). Making use of eq 3.1 1 or 3.13, we can estimate the value of y or y'Kr from the power relations of the size shown in Table 1 by assuming C h = 3.18 mol ~ m - V ~ ,= 47.9 cm3 mol-' (=partial molar volume of the crystal), and D = 1.5 x cm2 s-I (=measured D value of KNO3 in acrylamide gel2'). The estimated results are shown in Table 2. Hohmann and Kahlweit22have reported that NaNOB grains in the solution grow according to the reaction-controlled parabolic growth law and that a value of y2Kr = 7.5 x IO6 g2 cm4 s - ~mol-' is obtained from the power relation of the grain size against the time. The values observed in the present experiment are about 2 orders of magnitude greater than this. This means that the growing rate of the grain in the gel is much lower than that in the solution. If the ripening kinetics, Kr, of KNO3 grain is not so much different from that of NaNO3 grain, the y value in the gel should be much greater than that in the solution. In the case of grain growth in the solution, y is the surface tension of the grain and has been considered to be about 5 x 10' dyn cm-'.22 The y values estimated from the results shown in Table 2 are 5-10 times the surface tension. This indicates that the interfacial energy is ascribed not only to the surface tension of KNO3 grain only but to the stress in the gel caused by the grain ripening, as mentioned above. The interfacial energy U required for increasing the grain size from
16134 J. Phys. Chem., Vol. 99, No. 43, 1995
Sasaki and Maeda
+
r to r d r is approximately given by U % E 4 n P d r (=8nyr dr) for deformation of the gel chains and U !z (Cp,,,/Dp)E,4n$ d r (=8nyr dr) for scissoring the gel chains, where E and Es, respectively, are the elastic modulus of the gel and the energy for scissoring a covalent bond constituting the chain. Correspondingly, y can be described as y = rEl2 and y = (Cpm/ 2Dp)E,r. The y value due to the deformation is estimated as 1 x lo2 dyn cm-' for the case where r = cm, and the reported value of E (=3 x 10" x Cpm3.O7) = 3 x 10"dyn cm-2.23 This is too small to explain the observed y values. If E is more than double the reported value,23the observed y value can be explained in terms of the deformation of the chains. This might be possible since the elastic force increases with the concentration of the chain and the chains in the vicinity of the grain surface are concentrated with growing of the grain. When the energy for scissoring a chain is assumed compatible with the thermal energy at the degradation temperature of vinyl polymers (T 600 K24), E, = 5 x 10'O erg mol-', and thereby y = 5 x lo3 dyn cm-I. This is too large compared with the observed y values. However, this is overestimated since scissoring all chains, which is assumed in this estimation, in the space for the grain to ripen is not necessary. One per 10 chains in the space is considered enough to be scissored to make space for growing the grain. Thus, the observed large y values can be ascribed either to the stress in the gel caused partly by the deformation or to the scissoring of chains, or both. It is noticeable that in experiment 2 the power relation in the early stage differs from that at the late stage. At the early stage the concentration in the regions around the small grains may be a multiple of that at the regions around the large grains, as indicated by eq 3.3, and thereby the flow rate of the diffusion may be very high. The small grains dissolve, the number of grains decays very rapidly, and the average size appears to grow at a high rate. After dissolution of most of the small grains, the regional concentration inhomogeneously may decrease and the growth of grains may take place under a rather homogeneous concentration background. The supersaturation degree of the solution A of experiment 2 was about twice as much as that of experiment 1. At the initial stage, grain sizes are smaller in experiment 2 than in experiment 1. The large A value might induce spontaneous concentration fluctuations in space because of its instability. The large concentration variation results in the large variation of the IC - C,I value and thereby induces the wide size distribution. An extremely wide distribution was observed at t = 71.9 h in experiment 2, as shown in Figure 6b. The corresponding photograph shown in Figure 2a suggests that the correlation length of the fluctuation inferred from the average distance between the larger particles is a few hundred micrometers and that the ratio of the large IC - C,l value to the small one inferred from the grain sizes is about 10. The number of modes in the size distribution could be a measure of the magnitude of concentration fluctuation. In this respect the size distributions shown in parts a and b of Figure 6, respectively, suggest that the spatial concentration variation in experiment 1 was relatively small and that the magnitude of the variation in experiment 2 decreased with t . The multimodal or wide size distributions have been also reported for other systems such as manganese aggregates in the NaCl crystal' or silver particle formation in photosensitive glass.25 In some cases wide distributions have been phenomenologically interpreted as the sum of several narrow distribut i o n ~ . 'The ~ observed distributions shown in Figure 6 cannot be ascribed to the effect of volume fraction, which makes the distribution slightly wider, as has been observed in the precipita-
-
Time(t)iHours
Figure 3. Growth of individual grains in experiment 2. The average size of the grain is shown as a function o f t (solid line).
tion in an alloy of Ni-Co-A1.26 The present results suggest that the concentration fluctuation at the initial stage and the interferrence among the grains both influence the ripening in several aspects. After an elapse of time much longer than 800 h the concentration may become homogeneous as a result of the diffusion, and the distribution might be comparable with the theoretical predictions of Marqusee and Ross.I4 For the purpose of examining the growing or diminishing behaviors of individual grains, the sizes of selected grains in experiment 2 have been measured as functions of t and are shown in Figure 8. For the largest grain, Figure 8 shows that the power relation is almost the same as that for the average size at the later stage. Figure 8 also shows that the grain whose growth stops decreases in size and finally disappears. It was found that the grains of smaller size than the average size, which were surrounded by much smaller grains, grew for a while. After dissolution of these much smaller grains, the smaller grain dissolved very rapidly and disappeared, as shown in Figure 8. From the present study, the effects of the elastic deformation of the network, the scission of the chain, and the inhomogeneous diffusion of the solute in the gel on the growth kinetics are partly elucidated, although they cannot be clearly identified. But it is obvious that the existence of the gel chains makes the grain spherical. The interaction between the polymer chains in the gel and the molecules on the crystal facet may diminish the surface energy density differences among the facets; thereby, the crystal habit observed in the solution has been suppressed in the gel. Further investigations are necessary for clarifying the role of gel chains in the growth behaviors. References and Notes (1) Dunning, W. J. In Parricle Growth in Suspensions; Smith, A. L., Ed.; Academic Press: New York, 1973. (2) Kahlweit, M. Adv. Colloid Inteface Sci. 1975, 5, 1. (3) Fischmeisher. H.; Grimvall, G. In Sintering and Related Phenomena; Kuczynski, G. C., Ed.; Plenum: New York, 1973. (4) Greenwood, G. W. The Mechanism of Phase Transfunnations in Crystalline Solids; The Institute of Metals: London, 1969. ( 5 ) Gunton, J. D.; San Miguel, M.; Sahn, P. S. In Phase Transitions and Critical Phenomena; Domb, C . , Ed.; Academic Press: New York, 1983. (6) Jain, S. C.: Hughes, A. E. J. Mater. Sci. 1978, 13, 1611. (7) Mann, S. Nature 1988, 3.32, 119. (8) Dameron, C. T.; Reese, R. N.; Mehra, R. K.; Kortan, A. R.; Carroll, P. J.; Steigerwald, M. L.; Bruns, L. E.; Winge, D. R. Nature 1989, 338, 596. (9) Chellman, D. J.; Ardell, A. J. Acta Metall. 1974, 22, 577. (10) Jain, S. C.; Arora, N. D. J . Phys. Chem. Solids 1974, 35, 1231. (11) Kirk, D. L.; Kahn, A. R.; Pratt, P. L. J . Phys. D 1975, 8, 2013. (12) Lifshitz, I. M.; Slyozov, V. V. Phys. Chem. Solids 1961, 19, 35. (13) Wagner, C. Z. Electrochem. 1961, 65, 581. (14) Marqusee, J. A.; Ross, J. J. Chem. Phys. 1984, 80, 531. (15) Jain, S. C.; Hughes, A. E. J. Mater. Sci. 1978, 13, 1611. (16) Hanitzsch, E.; Kahlweit, M. In Indusfrial Crystallization; The
Ostwald Ripening of KNO3 Grains in Acrylamide Gel Institution of Chemical Engineers: London, 1969; p 130. Kahlweit, M. In Physical Chemistry; Eyring, H., Henderson, D., Jost, W., Eds.; Academic Press: New York, 1970; Vol. 10, p 719. (17) Enomoto, Y.; Kawasaki, K.; Tokuyama, M. Acta Mefall. 1987,35, 915. (18) Kawasaki, K.; Enomoto, Y. Physica A 1988, 150, 463. (19) Enomoto, Y.; Kawasaki, K. Acta Metall. 1989, 37, 1399. (20) Stephen, H.; Stephen, T. Solubility of Inorganic and Organic Compounds; Pergamon Press: Oxford, 1963. (21) Sasaki, S.; Maeda, H. Submitted to J. Phys. Chem.
J. Phys. Chem., Vol. 99, No. 43, 1995 16135 (22) Hohmann, H. H.; Kahlweit, M. Eer. Bunsen-Ges. Phys. Chem. 1972, 76, 933. (23) Geisler, E.; Hecht, A. M. J . Phys. (Paris) 1978, 39, 955. (24) Brandrup, J.; Immergut, E. H. Polymer Handbook, 3rd ed.; John Wiley & Sons Inc.: New York, 1989; Chapter 2, p 365. (25) Kreibig, U. J . Phys. F 1974, 4,999. (26) Davies, C. K. L.; Nash, P.; Stevens, R. N. Acta Metall. 1980, 28, 179. JP95 1401H
