A Mathematical Model for Crystal Growth Rate Hysteresis Induced by

Morioka, 020-8551 Japan. John W. Mullin. Department of Chemical and Biochemical Engineering, University College London,. Torrington Place, London, WC1...
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A Mathematical Model for Crystal Growth Rate Hysteresis Induced by Impurity Noriaki Kubota,* Masaaki Yokota, Norihito Doki, Luis A. Guzman, and Shigeko Sasaki

CRYSTAL GROWTH & DESIGN 2003 VOL. 3, NO. 3 397-402

Department of Chemical Engineering, Iwate University, 4-3-5 Ueda, Morioka, 020-8551 Japan

John W. Mullin Department of Chemical and Biochemical Engineering, University College London, Torrington Place, London, WC1E, 7JE, U.K. Received September 30, 2002;

Revised Manuscript Received January 23, 2003

ABSTRACT: The growth rate of a crystal in the presence of impurity depends on the history of supercooling. This behavior is called growth rate hysteresis (GRH). In this paper, GRH is described by using a mathematical model. This mathematical model is devised by considering the pinning mechanism of Cabrera and Vermilyea, the twodimensional Gibbs-Thomson effect on step movement and slow adsorption of impurity species on a crystal surface. The model explains, reasonably but qualitatively, experimental literature data on GRH that its magnitude becomes large as the supercooling-changing rate R is decreased or the impurity concentration c is increased. The model also shows a possibility that the reverse effect of these two factors (R and c) on GRH may occur in the range of their small values and it predicts the GRH behavior over a wide range of experimental conditions. Limitations of the model are also discussed. Introduction Trace amounts of impurities present in the crystallizing systems can have dramatic effects on crystal growth.1-3 An interesting behavior, exhibited in the presence of impurities, is that the crystal growth rate depends on the history of supercooling. A crystal growing at a high supercooling can continue to grow at appropriate reduced growth rates, down to a low supercooling, if the supercooling is lowered continuously from the higher level. Yet, the crystal that has stopped growing at the low supercooling is unable to grow even when the supercooling is continuously raised to a high level. The supercooling must be further increased for the crystal to resume growth. This behavior is called crystal growth rate hysteresis (GRH). However, it is not widely known, and only a few cases4-10 have been reported. There is no completely satisfactory explanation, but in this study we try to describe the hysteresis behavior mathematically by considering the pinning mechanism of Cabrera and Vermilyea11 and the twodimensional Gibbs-Thomson effect on step movement, and by assuming slow adsorption of impurities, by which the step advancement velocity, and hence the crystal growth rate is reduced. Typical Experimental Data A typical example of GRH observed by Guzman et al.10 for a potassium sulfate crystal in the presence of chromium (III) is shown in Figure 1. A crystal was started to grow from a high supercooling of ∆To ≈ 9.5 °C, and the supercooling was lowered continuously at a constant rate of 0.2 °C/min down to a supercooling (∆Ts) * To whom correspondence [email protected].

should

be

addressed.

E-mail:

Figure 1. Face growth rate of a potassium sulfate crystal in the presence of chromium (III) impurity.10 Crystal GRH is seen in the presence of the impurity (see b, O), i.e., the crystal growth rate depends on the history of supercooling of the solution, while no GRH is observed for the corresponding pure solution (see 9, 0).

where the crystal ceases to grow. Then the direction of supercooling alteration was changed, and supercooling was increased at the same constant rate. Two different growth rates are seen at a supercooling-changing rate of 0.2 °C/min. The growth rate is reduced in both processes. The extent of the reduction is larger in the processes of increasing supercooling. In addition, the crystal does not grow, after it ceases to grow at ∆Ts, for a while, even when the supercooling is being increased. It begins to grow again at a supercooling of ∆Tr in the higher supercooling range. Finally, the growth rate returns to the original level at higher supercoolings. However, at a high impurity concentration of 2 ppm (see Figure 2), the crystal does not resume growth even when the supercooling is increased so that a closed loop of growth rate is not established. According to Guzman et al.,10 the hysteresis behavior was affected by impurity

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of a two-dimensional circular nucleus of the crystal and defined8,9,12,13 as

R)

Figure 2. Face growth rate of a potassium sulfate crystal in the presence of chromium (III) impurity.10 Impurity concentration is increased to 2 ppm from 0.5 ppm in the case shown in Figure 1. The GRH becomes larger. The growth rate does not return to the initial one even at the highest supercooling examined.

concentration and the rate of supercooling alteration. As the impurity concentration was increased, or the rate of supercooling alteration was decreased, the GRH became clearly seen or the distance between the supercoolings of ∆Tr and ∆Ts became large. It must be noted that no GRH was seen for the pure system (see lines indicated as “pure” in Figures 1 and 2).

As suggested previously by Kubota et al.8,9 and Guzman et al.,10 the hysteresis is considered to be due to slow adsorption of impurity acting as a growth suppressor. Following this idea, we try to describe the hysteresis behavior mathematically by modifying the model, which was previously proposed by Guzman et al.10 and Kubota et al.,8,9,12,13 to explain the suppressing effect of impurity on crystal growth. Initially, we discuss the crystal growth rate where impurities adsorb instantaneously on the surface of a crystal. Although no hysteresis is expected in this case, by doing this we will show how crystal growth retardation can be described by the model taking into account the pinning mechanism of Cabrera and Vermilyea11 and the two-dimensional Gibbs-Thomson effect. Then, the growth rate in the case of slow adsorption of impurity, where the hysteresis of crystal growth rate appears, is discussed. Growth Rate in Case of Instantaneous Impurity Adsorption. According to the Kubota-Mullin model,8,9,12,13 the face growth rate of a crystal G in the presence of impurity can be described by

G ) 1 - Rθeq (for Rθeq e 1) Go

(1)

and G/G0 ) 0 for Rθeq > 1, where Go is the face growth rate of the corresponding pure system and θeq is the equilibrium coverage of active sites for impurity adsorption. Equation 1 was derived based on the pinning mechanism,11 by which steps are forced to curve, and the so-called two-dimensional GibbsThomson effect that curved steps proceed more slowly than straight steps due to an excess in the edge free energy. The parameter R is called the impurity effectiveness factor, which is related to the critical radius

(2)

where γ is the edge free energy of a step on the surface of the crystal, a is the size of a growth unit, k is the Boltzmann constant, and L is the average distance between active (or kink) sites, where impurities are to be adsorbed. According to eqs 1 and 2, the value of the impurity effectiveness factor is increased as supersaturation σ is decreased, and hence the growth rate G is reduced. When the supersaturation is reduced to a critical value σc, which corresponds to (1 - Rθeq) ) 0, the crystal ceases to grow. In a hysteresis experiment, supercooling was changed continuously at a constant rate by altering the temperature T of a given solution. For such a case, supersaturation can be related to the supercooling ∆T () Te - T) by assuming a linear solubility-temperature relation as

σ)

Te - T ∆T ) T T

(3)

where Te is the saturation temperature. From eqs 2 and 3, the impurity effectiveness factor R becomes

R)

Theory

γa kTσL

γa k∆TL

(4)

Substitution of eq 4 into eq 1 gives

G ) Go(∆T)-1 (∆T - ∆Tc) (for ∆T g ∆Tc)

(5)

where ∆Tc is a critical supercooling, below which G ) 0, written by

∆Tc )

γa θ kL eq

(6)

The critical supercooling can be written as a function of the impurity concentration, c, if the Langmuir adsorption isotherm is assumed to apply for θeq, the equilibrium coverage of active sites for impurity adsorption, as

∆Tc )

γa Kc kL 1 + Kc

(7)

where K is the Langmuir constant. Although, strictly speaking, the critical supercooling should be a function of temperature, we assume that it is constant, because the temperature change is not so large in the situation dealt with here. The face growth rate G in the presence of impurity can be written by the following equation in a nondimensional form, where Go is replaced by go(∆T) to emphasize the fact that it is a function of supercooling ∆T as

( ) (

go(∆T) ∆T G ) go(∆Tc) go(∆Tc) ∆Tc

-1

∆T -1 ∆Tc

)(

for

)

∆T g1 ∆Tc (8)

and, when ∆T/∆Tc < 1, G ) 0. Although any face growth rate function can be applied for go(∆T) in theory, we use

Mathematical Model for Crystal Growth Rate Hysteresis

Figure 3. Theoretical face growth rate of a crystal calculated using the model for the case of instantaneous adsorption of impurity. No GRH is observed even in the presence of impurity. The same growth rate is observed in both the supercooling-increasing and -decreasing processes.

here a simple one to avoid mathematical complexity. If a second-order rate equation is assumed to apply, as go(∆T) ) kG(∆T)2, the face growth rate G in the presence of impurity can be written in a normalized form as

( )(

G ∆T ∆T ) -1 ∆Tc ∆Tc kG(∆Tc)2

)(

for

)

∆T g1 ∆Tc

(9)

and G ) 0 at ∆T/∆Tc < 1. Equation 9 is shown in Figure 3, where the face growth rate of a crystal in the corresponding pure solution is also indicated as a dotted line. Growth rate in the impure system is suppressed over a whole range of supercooling by the combination of the pinning mechanism and the two-dimensional Gibbs-Thomson effect. It does not show the hysteresis, namely, it is never affected by the supercooling history of solution, because the impurity adsorption is always at equilibrium. Growth Rate in Case of Slow Adsorption of Impurity. On the contrary to the above case of instantaneous adsorption, the hysteresis is expected to appear if adsorption of impurities proceeds slowly, because the amount of adsorbed impurities continuously increases and hence the growth suppression effect becomes stronger with time. If the Langmuir mechanism is assumed to apply for the adsorption process, the transient coverage of active sites by impurities can be described9,13,14 as

[

t τ

( )]

θ ) θeq 1 - exp -

Figure 4. Alteration of supercooling in both the experiments and calculations. Supercooling is lowered at a constant rate from ∆To until ∆Ts, where the crystal ceased to grow, and then it is increased at the same rate to the termination.

second-order rate equation is assumed to apply, as

{

t G ) 1 - Rθeq 1 - exp Go τ

( )]

[

(

( )) (12)

and G/kG(∆Tc)2 ) 0 when ∆T/∆Tc < 1 - exp(-t/τ). It should be mentioned that the time constant τ was implicitly assumed here not to depend on supercooling and temperature. As can be seen clearly from eq 12, growth rate G in the presence of impurity decreases with time at a given supercooling ∆T. Note that eqs 11 and 12 reduce to eqs 1 and 9, respectively, if the time constant approaches zero as τ f 0 (instantaneous adsorption). As discussed below, the GRH appears when the supercooling is altered continuously. Growth Rate Hysteresis. The time t for adsorption (see eqs 10-12) or the time period during which a crystal is exposed to an impure solution is related, as shown below, with the supercooling ∆T when it is changed as in the hysteresis experiment conducted by Guzman et al.10 See also Figure 4, where the supercooling in the hysteresis experiment is shown as a function of time. If a crystal is started to grow at a high supercooling of ∆To and the supercooling is lowered at a constant rate of R, the time t for adsorption, can be related with ∆T, for the supercooling-decreasing process (temperature-raising process), as

t)

(∆To - ∆T) R

(13)

until the supercooling reaches ∆Ts, where the crystal ceases to grow. For the reverse process, i.e., the supercooling-increasing process (temperature-lowering process), the exposure time, counted from the instance of the start of a hysteresis experiment at ∆To, can be written as

t)

(for Rθ [1 - exp(- τt)] e 1) (11)

( )]}

G ∆T ∆T t ) - 1 - exp τ kG(∆Tc)2 ∆Tc ∆Tc ∆T t for g 1 - exp ∆Tc τ

(10)

where τ is the time constant of net adsorption process, which is defined by τ ) (1/(k1c + k2); k1 and k2 are the rate constants for adsorption and desorption processes, respectively. Replacing θeq in eq 1 by θ, given by eq 10, we can obtain the following equation.

[

Crystal Growth & Design, Vol. 3, No. 3, 2003 399

(∆To - ∆Ts) + (∆T - ∆Ts) ∆T - 2∆Ts + ∆To ) R R (14)

eq

and G/G0 ) 0 for Rθeq[1 - exp(- t/τ)] > 1. Similarly as made in the discussion of the case of instantaneous adsorption, the normalized growth rates can be written as a function of supercooling ∆T and time t when the

when the supercooling is reversed to increase just at a supercooling of ∆Ts. The growth rate of a crystal under the conditions in which temperature is initially raised and later lowered both at a constant rate as in the GRH experiments (see

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Figure 4) can be calculated as a function of supercooling ∆T by using eq 12 together with eqs 13 and 14. From calculated growth rates, the behavior of GRH is obtained. The results will be shown below. Results and Discussion Theoretical Crystal Growth Rate Behavior. Actual calculations were conducted by using the following equations obtained by the insertion of eqs 13 and 14 into eq 12, respectively, for the supercooling-lowering and -raising processes. For the supercooling-lowering process, the dimensionless growth rate was calculated by

G ) kG(∆Tc)2

{

((

[

)/ )]}

∆To ∆T Rτ ∆T ∆T - 1 - exp ∆Tc ∆Tc ∆Tc ∆Tc ∆Tc (for ∆Ts e ∆T e ∆To) (15)

Figure 5. Theoretical face growth of a crystal calculated using the model for the case of slow adsorption of impurity. Crystal growth was started at ∆To/∆Tc ) 0.5. GRH, i.e., the region of no growth is seen in the process of supercooling-increasing process. The degree of the hysteresis is increased as the dimensionless supercooling-changing rate Rτ/∆Tc is decreased.

where ∆Ts is the supercooling at which the crystal stops growing. It is obtained by solving the following equation for ∆T

((

[

)/ )]

∆To ∆T Rτ ∆T ) 1 - exp ∆Tc ∆Tc ∆Tc ∆Tc

(16)

And, for the supercooling-raising process, the dimensionless growth rate was calculated by

{

[

(( )/ )]}

∆T 2∆Ts G ∆T ∆T ) - 1 - exp + 2 ∆T ∆T ∆T ∆Tc kG(∆Tc) c c c ∆To Rτ (for ∆Tr e ∆T) (17) ∆Tc ∆Tc and, for ∆Ts e ∆T e ∆Tr,

G )0 kG(∆Tc)2

(18)

where ∆Tr is the supercooling at which the crystal restarts to grow. The value of ∆Tr is obtained by solving the following equation for ∆T

[

((

)/ )]

∆T ∆T 2∆Ts ∆To Rτ ) 1 - exp + ∆Tc ∆Tc ∆Tc ∆Tc ∆Tc

(19)

Typical results of calculations for a given value of the starting supercooling of ∆To/∆Tc ) 0.5 are shown in Figure 5, where theoretical face growth rates in dimensionless form are plotted as a function of dimensionless supercooling ∆T/∆Tc with dimensionless supercoolingchanging rate Rτ/∆Tc as a parameter. As a measure of the magnitude of the hysteresis GRH, we use here the dimensionless supercooling distance defined by (∆Tr ∆Ts)/∆Tc. This is the distance between the supercooling at which a crystal ceases to grow and the supercooling at which it starts to grow again. The hysteresis can be observed clearly in calculations as shown in Figure 5. The magnitude of hysteresis, dimensionless GRH, becomes larger as the value of the dimensionless supercooling-changing rate Rτ/∆Tc is decreased. This means

Figure 6. Theoretical face growth rate of a crystal calculated using the model for the case of slow adsorption of impurity. Crystal growth was started at ∆To/∆Tc ) 1. GRH, or the region of no growth in the process of supercooling-increasing process, is seen to decrease by the increase of ∆To/∆Tc from 0.5 (in Figure 5) to 1.

that the hysteresis becomes larger as the impurity concentration c is increased (see eq 7 for the relation between ∆Tc and c). The calculation results on the impurity concentration effect are qualitatively in agreement with experimental results (see Figures 1 and 2). The effect of the supercooling-changing rate R on GRH that it increased with a decrease in R also agreed with the experimental data10 (data not shown here). In Figure 6, theoretical growth rates for a higher starting supercooling of ∆To/∆Tc ) 1.0 are shown. The dimensionless GRH, (∆Tr - ∆Ts)/∆Tc, does not appear at higher supercooling-changing rates of Rτ/∆Tc of 1 and 1/3, while, at the lowest value examined of Rτ/∆Tc ) 0.1, it is observed. However, if the starting point of supercooling is increased to ∆To/∆Tc ) 1.5 (see Figure 7), no hysteresis can be seen even at the lowest supercooling-changing rate examined (Rτ/∆Tc ) 0.1). Thus, the GRH should depend on the dimensionless starting supercooling ∆To/∆Tc, from which point a crystal starts to be exposed to impure solution, and the supercooling-changing rate Rτ/∆Tc. This is because the exposure time of a crystal to the impure solutions is governed by these two factors. The effects of these two factors will be discussed more in detail later. The GRH is also expected to be affected by the system properties, because the system parameters γ, a, L, and

Mathematical Model for Crystal Growth Rate Hysteresis

Figure 7. Theoretical face growth rate of a crystal calculated using the model for the case of slow adsorption of impurity. Crystal growth was started at ∆To/∆Tc ) 1.5. GRH, or the region of no growth in the process of supercooling-increasing process, is not seen.

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Figure 9. Dimensionless GRH, ∆Tr/∆Tc and ∆Ts/∆Tc as a function of dimensionless supercooling-changing rate Rτ/∆Tc.

Figure 10. Dimensionless GRH as a function of dimensionless supercooling-changing rate with different dimensionless starting supercoolings ∆To/∆Tc. Figure 8. Face growth rate of an ammonium sulfate crystal in the presence of aluminum (III) impurity. Crystal growth rate hysteresis is seen in the presence of the impurity (see b, O), i.e., the crystal growth rate depends on the history of supercooling of the solution, while no growth rate hysteresis is observed for the pure solution.

K are included in ∆Tc as seen in eq 7, and τ is included in eqs 11 and 12. Therefore, a different growth pattern can be obtained for a different combination of crystal and impurity. As an example, the growth behavior of an ammonium sulfate crystal in the presence of Al(III)15 is shown in Figure 8. The GRH is also seen clearly in this case. However, the magnitude of the hysteresis is different from that of the above potassium sulfatechromium (III) system, and the growth rate pattern is also different. Effects of ∆To/∆Tc and Rτ/∆Tc. The effects of these two factors on the dimensionless GRH and the other two dimensionless supercoolings ∆Ts/∆Tc and ∆Tr/∆Tc, at which the crystal stops growing and resumes growth respectively, are shown in Figure 9 for ∆To/∆Tc ) 1.0. The values of ∆Ts/∆Tc and ∆Tr/∆Tc, and hence the dimensionless GRH, are largely affected by the dimensionless supercooling-changing rate Rτ/∆Tc. In the range of high values of Rτ/∆Tc, the dimensionless supercoolings ∆Ts/∆Tc and ∆Tr/∆Tc exhibit the same value; therefore, no GRH is observed. As Rτ/∆Tc is decreased, the GRH increases, passing through a maximum and finally reaching a constant value of 0.5. This theoretical calculation indicates that the GRH has a possibility to decrease as the dimensionless supercooling-changing rate Rτ/∆Tc is decreased in the low value range. It must

be noted, however, that this decrease in GRH was not observed in actual experimental data obtained for the potassium sulfate-chromium (III) system.10 Figure 10 shows calculation results of the dimensionless GRH obtained for various values of the dimensionless starting-supercooling ∆To/∆Tc. The value of the dimensionless GRH becomes small as ∆To/∆Tc is increased and it exhibits the maximum at the starting supercoolings other than ∆To/∆Tc ) 1.5. From Figure 10, we can theoretically predict that the GRH depends on the value of the dimensionless supercooling-starting point and the value of dimensionless supercoolingchanging rate, and it has a maximum. In the above calculations, we assumed a second-order growth rate equation for mathematical simplicity. However, the generality of the theory is not lost by this assumption, because the GRH behavior is basically governed by the term of {∆T/∆Tc - [1 - exp(-t/τ)]} in eq 12, which appears always regardless of the rate expression of crystal growth for the corresponding pure system. Limitations of the Present Theory. As shown above, the present theory was able to describe, though qualitatively, the experimental behavior of the GRH observed for the system of potassium sulfate crystalchromium (III) impurity. However, this theory cannot explain, even in qualitative manner, the experimental growth behavior that a crystal, in the supercoolingincreasing process, finally returns the original growth rate at high supercoolings (see Figure 1). In other words, the model could not simulate the case that is observed

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Crystal Growth & Design, Vol. 3, No. 3, 2003

in Figure 1 at high supercoolings. At the present stage, two reasons are considered for this drawback in the theory. First, we assumed a constant value of the time constant τ for the adsorption process to avoid mathematical difficulties. However, this assumption is probably not justified, because, according to Kubota et al.,9 the time constant τ seemed to be increasing with an increase in supersaturation. The second reason appears to be more serious for the present model, because it may not describe the reality occurring in the supersaturation-raising process. Actually, it was observed under an optical microscope in our laboratory that, as the supercooling was increased, the crystal began to grow suddenly at ∆Tr in a catastrophic manner, and a new macrostep appeared suddenly somewhere, probably at a point where population density of adsorbed impurities was low, and proceeded to cover the whole crystal surface. The generation of such a macrostep followed by the surface covering was repeatedly observed at random during the supercooling-raising process, and finally the growth rate reached the original one at the starting point. Thus, the actual growth process was quite different from the stable growth assumed implicitly in the present model. Because of this gap between the model and the reality, it is considered that a satisfactory result was not obtained for the supercoolingraising process. However, it is impossible to establish a mathematical model describing a process consisting of a series of catastrophic events. Notation a c G Go go(∆T) k kG k1 k2 K L R t T Te

size of growth unit or area per growth unit appearing on the surface, m2 impurity concentration, mg/dm3 or ppm face growth rate of crystal in the presence of impurity, m/s or µm/min face growth rate of crystal in pure solution, m/s or µm/min face growth rate of crystal in pure solution ∆T, m/s or µm/min Boltzmann constant, J/K growth rate constant in Go ) kG(∆T)2 adsorption constant, (mg/dm3)-1 s-1 or (ppm)-1 s-1 desorption constant, s-1 Langmuir constant () k1/k2), (mg/dm3)-1 or (ppm)-1 average distance between the active sites on the step, m supercooling-changing rate, °C/s time for adsorption, s temperature, °C saturation or equilibrium temperature of a solution, °C

Kubota et al. To ∆T ∆Tc ∆Tr ∆Ts (∆Tr - ∆Ts)/ ∆Tc R γ θ θeq σ τ

temperature when a crystal starts to grow, °C supercooling () Te - T), °C critical supercooling when equilibrium adsorption is established, below which a crystal never grows (see eq 7), °C supercooling at which a crystal begins to grow in the supercooling-increasing process, °C supercooling at which a crystal ceases to grow in the supercooling-decreasing process, °C a degree of the magnitude of the growth rate hysteresis impurity effectiveness factor defined by eq 2 (see also eq 4) edge free energy of a step on the surface of crystal, J/m coverage of active sites of crystal surface by adsorbed impurities θ at equilibrium relative supersaturation time constant of impurity adsorption () 1/(k1c + k2), s

References (1) Mullin, J. W. Crystallization, 4th ed.; Butterworth-Heinemann, Oxford, 2001; pp 254-260. (2) Meenan, P. A.; Anderson, S. R.; Klug, D. L. In Handbook of Industrial Crystallization; Myerson, A. S., Ed.; ButterworthHeinemann, Boston, 2002; pp 67-100. (3) Sangwal, K. Prog. Cryst. Growth Charact. 1998, 36, 163248. (4) Dugua, J.; Simon B. J. Cryst. Growth 1978, 44, 280-286. (5) Kubota, N.; Uchiyama, I.; Nakai, K.; Shimizu, K.; Mullin, J. W. Ind. Eng. Chem. Res. 1988, 27, 930-934. (6) Simon, B.; Grass, A.; Boistelle, R. J. Cryst. Growth 1974, 26, 90-96. (7) Punin, Yu O.; Artamonova, O. I. Kristallografiya 1989, 34, 1262-1266. (8) Kubota, N.; Yokota, M.; Mullin, J. W. J. Cryst. Growth 1997, 182, 86-94. (9) Kubota, N.; Yokota, M.; Mullin, J. W. J. Cryst. Growth 2000, 212, 480-488. (10) Guzman, L. A.; Kubota, N.; Yokota, M.; Sato, A.; Ando, K. Cryst. Growth Des. 2001, 1, 225-229. (11) Cabrera, N.; Vermilyea, D. A. In Growth and Perfection of Crystals; Doremus, R. H., et al., Eds; Wiley: New York, 1958; pp 393-410. (12) Kubota, N.; Mullin, J. W. J. Cryst. Growth 1995, 152, 203208. (13) Kubota, N. Cryst. Res. Technol. 2001, 36, 749-769. (14) Guzman, L. A.; Maeda, K.; Hirota, S.; Yokota, M.; Kubota, N. J. Cryst. Growth 1997, 181, 272-280. (15) Kubota, N.; Yokota, M.; Doki, N.; Sasaki, S.; Hatakeyama, T. Proc. of the 67th Annual Meeting of the Society of Chemical Engineers (CD-ROM), Japan, 2001, No. 1302.

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