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Response to: On methods of determination of homogeneous nucleation rates of protein crystals
Colloids and Surfaces A: Physicochem. Eng. Aspects 215 (2003) 137 /140 www.elsevier.com/locate/colsurfa
Response to: On methods of determination of homogeneous nucleation rates of protein crystals C.F. Zukoski *, A.M. Kulkarni, N.M. Dixit Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, 600 S Mathews Avenue, Urbana, IL 61801, USA Received 24 May 2002; received in revised form 10 September 2002; accepted 11 September 2002
Abstract In their letter, Galkin and Vekilov suggest that our analysis [Dixit et al., Colloids Surf. A 190, 47 (2001)] of their experimental technique [Galkin and Vekilov, J. Phys. Chem. B 103, 10965 (1999); J. Am. Chem. Soc. 122, 156 (2000)] for estimating protein crystal nucleation rates contains inconsistencies. Here, we examine their criticisms and find that our analysis does not contain inconsistencies. We conclude by restating our claim in [Dixit et al., Colloids Surf. A 190, 47 (2001)] that Galkin and Vekilov underestimate protein crystal nucleation rates by several orders of magnitude because of the way their experimental data is interpreted. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Protein crystals; Nanoparticle; Nucleation rates
The rates of nanoparticle crystal nucleation are poorly understood. This is clear in the controversy that exists in methods to measure these rates and the range of values reported for nucleation rates determined by different methods on the same system [1]. The best studied system involves experimental and theoretical determination of hard sphere crystal nucleation kinetics. Even here, however, interpretation of the time dependence of angle dependent light scattering requires careful analysis to extract nucleation rates from measurable properties [4].
* Corresponding author. Tel.: 1-217-333-5076 E-mail address: [email protected] (C.F. Zukoski).
In our paper ‘Comparison of Experimental Estimates and Model Predictions of Protein Crystal Nucleation Rates’ [1], we discuss inconsistencies seen in predictions and measurements of protein crystal nucleation kinetics. Three sets of nucleation rate data were analyzed: nucleation rates determined from heats of reaction [5,6], a two step temperature quench attempting to separate nucleation and growth [2,3], and measurements of induction time to detectable crystals in a single temperature quench measurement [7]. Using very similar conditions, the three different methods result in nucleation rates that differ by several orders of magnitude. With this comparison in hand we attempted to determine the cause of the poor agreement. We applied a kinetic model for nucleation to understand the physical processes
0927-7757/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 0 2 ) 0 0 4 2 4 - 7
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C.F. Zukoski et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 215 (2003) 137 /140
involved [8]. Our modeling approach was based on that of Narasimhan and Ruckenstein [9]. Due to the methods used and the origin of heterogeneous nucleation the heat of reaction measurements [5,6] represent upper bounds on nucleation rates. The induction time measurements [7] suffer from the fact that the induction time, tind, is a measure of the time after supersaturating the suspension before the light scattered by the suspension increases rapidly. Scattered light intensity is a function of particle size and number density. Without a complete model capable of predicting cluster number density as a function of time, induction times can only be used as a measure of nucleation rate by making several approximations. Never-the-less, using these approximations, one predicts that ln(tind) should be a linear function of [ln(s)] 2, where s is the supersaturation, with the slope of the line being proportional to the cluster/solution interfacial tension [7]. The link between tind and s was supported by the experiments while the interfacial tension was of the order expected from the strength of attraction of the particles. However the prefactor B (where tind /B exp (A /[ln(s)]2) is related in poorly understood ways to the prefactor for nucleation. Thus we expect the induction times to give only qualitative estimates of nucleation rates. The nucleation rates estimated from the induction time measurements [7] (and also from the heat of reaction measurements [5,6]) were several orders of magnitude higher than the estimates of Galkin and Vekilov [2,3]. In our paper [1] we suggest that the discrepancies are partly due to the technique employed by Galkin and Vekilov, which, as outlined below, we believe underestimates crystal nucleation rates. In their letter, Galkin and Vekilov offer different interpretations. In particular they are concerned with the fact that the induction time data referred to in the paper (as described by Kulkarni and Zukoski [7]) are subject to heterogeneous nucleation that will result in anomalously high nucleation rates. Heterogeneous nucleation can be important for many conditions (in particular at low supersaturations). The limits where one moves from heterogeneous to homogeneous nucleation for the system investigated in both studies have
been studied [10] and the measurements reported in our experiments are in the homogeneous nucleation region. Further, the reproducibility of the induction time experiments, their consistency with model predictions for the interfacial tension, and comparisons to other data where the relative rates of heterogeneous and homogeneous nucleation were studied [10] lead us to conclude that heterogeneous nucleation did not dominate the induction time data of [7]. That heterogeneous nucleation may be important cannot be ruled out without further modeling and experimental studies. However, given the previous studies in this field and the controls we are able to run, we believe heterogeneous nucleation does not dominate the induction time kinetics we report. The kinetic model for nucleation was applied to provide a consistent treatment of cluster size distribution and the effects of changes in temperature (supersaturation) on rates of nucleation. The purpose of presenting the model was to present a framework to describe the effects of step changes in supersaturation on nucleation kinetics. We explicitly state that the model is incorrect in an absolute sense and thus emphasize the qualitative arguments in our analysis of the experimental techniques rather than focusing on quantitative comparisons. As Galkin and Vekilov have rightly pointed out, quantitative comparisons indeed result in large discrepancies between experiments and model predictions. One reason for the failure of the model lies in its oversimplified representation of protein interactions as short-ranged centrosymmetric attractions. In a more recent application of the model [11], we show that accounting for the inherent anisotropy in protein interactions leads to dramatic differences in predicted nucleation rates. In addition, the predictions allowed quantitative comparisons with the solid /fluid interfacial tensions estimated from the induction time data of [7]. Thus, within the framework of aggregation and dissociation processes and the thermodynamics of phase changes, the model is expected to capture the qualitative features of the experimental system. Ignoring the merit of the qualitative arguments purely on the basis of the quantitative discrepancies between models and experiments, as Galkin and Vekilov
C.F. Zukoski et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 215 (2003) 137 /140
repeatedly do in their letter, does not address the physical interpretation captured by the model. Of interest in the response of Galkin and Vekilov [2,3] is our analysis of their extensive and extremely careful measurements of the number of crystals formed in a two step temperature quench measurement. While developing an analysis of this approach we found that classical nucleation and growth models and a literature on the two step method suggest difficulties in separating nucleation and growth in the two step method [1,12]. Our analysis indicates that irrespective of the comparisons with other measures, the interpretation of the two step method employed in [2,3] underestimates nucleation rates. The two step method involves quenching a suspension to a temperature T1 for a time period t1 and then raising the temperature to T2 for a time period t2. The number of crystals at the end of t2 is determined. Looking at how this number changes with t1, the nucleation rate is backed out. The authors [2,3] argue that T2 is sufficiently close to the melting temperature at the concentration of interest that no nucleation is observed for several months at T2. Of particular concern in our analysis is the opportunity for clusters formed at T1 to melt upon raising the temperature to T2. Here we attempt to restate our argument in a few lines. We define the critical cluster size at T1 as R* (T1) and that at T2 as R* (T2). Because T2 is chosen very close to the melting temperature at the concentration of interest, we must assume R* (T2)/R* (T1). Thus upon raising the temperature one predicts the particles at size R *(T1) will be unstable and melt. On the other hand, if there is a steady rate of nucleation and some clusters can grow to a size of R *(T2) or larger at a time t1, the solution will contain a cluster size distribution containing clusters with sizes smaller than R *(T2). When the temperature is raised to T2, these clusters will have a tendency to melt rather than grow. Experimental evidence was provided in our paper that such melting may indeed occur [1]. In their original paper and in their response to our paper, Galkin and Vekilov largely ignore the question of why crystals formed at high supersaturations (T1), where the critical size is small,
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and do not grow sufficiently during t1 cannot melt when the temperature is raised rapidly to just below the melting temperature. (We note that our use of the term ‘melting’ is consistent with our description of protein solutions as one component systems, where the protein molecules interact via an effective interaction potential.) In their letter, Galkin and Vekilov seem to suggest that the melting of crystals we predict is due to the ‘solution depletion to below the solubility at T2’, and attempt to prove that their system is not depleted upon increasing the temperature to T2. We note that we expect crystals to melt not because the solution is depleted to below the solubility (in which case, ‘all’ and not ‘some’ of the crystals would melt) but because some of the crystals nucleated at T1 will be smaller than the critical cluster size at T2 at the end of the interval t1. The light scattering experiments reported in our paper [1] do indeed show a reduction in the intensity of light scattered from suspensions exposed to a two-step process as proposed by Galkin and Vekilov, indicating that freshly nucleated crystals (independent of their being heterogeneously or homogeneously nucleated) melted upon rapidly reducing the supersaturation. Analysis of the time for the signal to reach a maximum would require a detailed study of the rates of nucleation and growth (shrinkage) coupled with how light scattered is related to particle number density and size distributions. We did not carry out such a kinetic analysis for our paper. However, it is clear that the number and/or size of the clusters in the solution decrease upon raising the temperature to T2. In their response, Galkin and Vekilov further argue that even if clusters melt, the number that do so would be independent of the quench time t1. As a result, experimental nucleation rates, determined as the slope of the number of crystals counted at t2 versus the quench time t1, remain unaffected by crystals melting. We believe that this argument needs some careful thought. Our approach is based on the idea that as clusters grow, the monomer concentration and thus the driving force for nucleation is decreased. As in our paper [1], we define tn as the time characterizing the formation of a cluster of size
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R *(T1) in a unit volume of suspension (note that this definition involves no inconsistencies in the units), and tg(R ) as the time required for a cluster to grow from R *(T1) to a particular size, R (/ R *(T1)). The number of crystals that melt, Nmelt, is then estimated to equal tg(R* (T2))/tn. Since R *(T2 )/R *(T1), it follows that tg(R *(T2)) /tn. Also, to see crystals at T2, the quench time must be longer than the time for clusters to grow to R *(T2), so that t1 /tg(R *(T2)). As a result, t1 / tn. This implies that in the quench time t1, the number of crystals that nucleate, which is approximated by t1/tn, is large. This causes s to reduce to a lower value than at the beginning of the crystallization experiment. (Galkin and Vekilov argue that s does not change much in their experiments by observing that no crystals bigger than 2 mm were present in their suspensions before and after raising the temperature to T2. However, the critical cluster consists of a few protein molecules and thus is on the order of a few nanometers in size. As a result, a large number of post critical clusters could exist in the suspensions and lower s and still be smaller than 2 mm and therefore remain undetected.) As t1 is increased, t1/tn also increases, causing s to reduce to an even lower value. This changes tn and tg(R *(T2)) (continuously) and hence Nmelt. We note that in the experiments of Galkin and Vekilov, t1 is increased by an order of magnitude (10 min /8 h) [2,3]. As a result, Nmelt could conceivably change significantly with t1, thereby affecting the measured nucleation rates. In a supersaturated solution, clusters develop as the result of association and dissociation processes giving rise to a cluster size distribution. The critical cluster is that cluster which does not have a tendency to grow or shrink. Measuring the flux of clusters through this critical size is extremely difficult whether for proteins or for smaller molecules. Interpreting experiments requires that
they be rigorously linked back to the properties the measurement techniques probe. Light scattering and crystal counting probe different averages of the cluster size distribution [4]. However, assuming that the number of clusters at t2 is a measure of the stable clusters in solution at t1 requires an investigation of the kinetics of cluster growth and how the cluster size distribution is altered when the supersaturation is suddenly lowered. One of the goals of our paper is to suggest that more detailed investigations of growth kinetics and the supersaturation dependence of the critical cluster size be carried out. Given that there is at least a little evidence for the shrinkage of crystals in the two step process and that two step methods have been investigated in molecular systems and are interpreted differently [1,12], we believe such studies are warranted.
References [1] N.M. Dixit, A.M. Kulkarni, C.F. Zukoski, Colloids Surf. A 190 (2001) 47. [2] O. Galkin, P.G. Vekilov, J. Phys. Chem. B 103 (1999) 10965. [3] O. Galkin, P.G. Vekilov, J. Am. Chem. Soc. 122 (2000) 156. [4] N.M. Dixit, C.F. Zukoski, Phys. Rev. E, in press. [5] P.A. Darcy, J.M. Wiencek, Acta. Cryst. D 54 (1998) 1387. [6] P.A. Darcy, J.M. Wiencek, J. Crystal Growth 196 (1999) 243. [7] A.M. Kulkarni, C.F. Zukoski, J. Cryst. Growth 232 (2001) 156. [8] N.M. Dixit, C.F. Zukoski, J. Colloid Interface Sci. 228 (2000) 359. [9] G. Narasimhan, E. Ruckenstein, J. Colloid Interface Sci. 128 (1989) 549. [10] T.E. Paxton, A. Sambanis, R.W. Rousseau, Langmuir 17 (2001) 3076. [11] N.M. Dixit, C.F. Zukoski, J. Chem. Phys, accepted. [12] P.E. Wagner, R. Strey, J. Phys. Chem. 85 (1981) 2694.
Response to: On methods of determination of homogeneous nucleation rates of protein crystals C.F. Zukoski *, A.M. Kulkarni, N.M. Dixit Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, 600 S Mathews Avenue, Urbana, IL 61801, USA Received 24 May 2002; received in revised form 10 September 2002; accepted 11 September 2002
Abstract In their letter, Galkin and Vekilov suggest that our analysis [Dixit et al., Colloids Surf. A 190, 47 (2001)] of their experimental technique [Galkin and Vekilov, J. Phys. Chem. B 103, 10965 (1999); J. Am. Chem. Soc. 122, 156 (2000)] for estimating protein crystal nucleation rates contains inconsistencies. Here, we examine their criticisms and find that our analysis does not contain inconsistencies. We conclude by restating our claim in [Dixit et al., Colloids Surf. A 190, 47 (2001)] that Galkin and Vekilov underestimate protein crystal nucleation rates by several orders of magnitude because of the way their experimental data is interpreted. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Protein crystals; Nanoparticle; Nucleation rates
The rates of nanoparticle crystal nucleation are poorly understood. This is clear in the controversy that exists in methods to measure these rates and the range of values reported for nucleation rates determined by different methods on the same system [1]. The best studied system involves experimental and theoretical determination of hard sphere crystal nucleation kinetics. Even here, however, interpretation of the time dependence of angle dependent light scattering requires careful analysis to extract nucleation rates from measurable properties [4].
* Corresponding author. Tel.: 1-217-333-5076 E-mail address: [email protected] (C.F. Zukoski).
In our paper ‘Comparison of Experimental Estimates and Model Predictions of Protein Crystal Nucleation Rates’ [1], we discuss inconsistencies seen in predictions and measurements of protein crystal nucleation kinetics. Three sets of nucleation rate data were analyzed: nucleation rates determined from heats of reaction [5,6], a two step temperature quench attempting to separate nucleation and growth [2,3], and measurements of induction time to detectable crystals in a single temperature quench measurement [7]. Using very similar conditions, the three different methods result in nucleation rates that differ by several orders of magnitude. With this comparison in hand we attempted to determine the cause of the poor agreement. We applied a kinetic model for nucleation to understand the physical processes
0927-7757/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 0 2 ) 0 0 4 2 4 - 7
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C.F. Zukoski et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 215 (2003) 137 /140
involved [8]. Our modeling approach was based on that of Narasimhan and Ruckenstein [9]. Due to the methods used and the origin of heterogeneous nucleation the heat of reaction measurements [5,6] represent upper bounds on nucleation rates. The induction time measurements [7] suffer from the fact that the induction time, tind, is a measure of the time after supersaturating the suspension before the light scattered by the suspension increases rapidly. Scattered light intensity is a function of particle size and number density. Without a complete model capable of predicting cluster number density as a function of time, induction times can only be used as a measure of nucleation rate by making several approximations. Never-the-less, using these approximations, one predicts that ln(tind) should be a linear function of [ln(s)] 2, where s is the supersaturation, with the slope of the line being proportional to the cluster/solution interfacial tension [7]. The link between tind and s was supported by the experiments while the interfacial tension was of the order expected from the strength of attraction of the particles. However the prefactor B (where tind /B exp (A /[ln(s)]2) is related in poorly understood ways to the prefactor for nucleation. Thus we expect the induction times to give only qualitative estimates of nucleation rates. The nucleation rates estimated from the induction time measurements [7] (and also from the heat of reaction measurements [5,6]) were several orders of magnitude higher than the estimates of Galkin and Vekilov [2,3]. In our paper [1] we suggest that the discrepancies are partly due to the technique employed by Galkin and Vekilov, which, as outlined below, we believe underestimates crystal nucleation rates. In their letter, Galkin and Vekilov offer different interpretations. In particular they are concerned with the fact that the induction time data referred to in the paper (as described by Kulkarni and Zukoski [7]) are subject to heterogeneous nucleation that will result in anomalously high nucleation rates. Heterogeneous nucleation can be important for many conditions (in particular at low supersaturations). The limits where one moves from heterogeneous to homogeneous nucleation for the system investigated in both studies have
been studied [10] and the measurements reported in our experiments are in the homogeneous nucleation region. Further, the reproducibility of the induction time experiments, their consistency with model predictions for the interfacial tension, and comparisons to other data where the relative rates of heterogeneous and homogeneous nucleation were studied [10] lead us to conclude that heterogeneous nucleation did not dominate the induction time data of [7]. That heterogeneous nucleation may be important cannot be ruled out without further modeling and experimental studies. However, given the previous studies in this field and the controls we are able to run, we believe heterogeneous nucleation does not dominate the induction time kinetics we report. The kinetic model for nucleation was applied to provide a consistent treatment of cluster size distribution and the effects of changes in temperature (supersaturation) on rates of nucleation. The purpose of presenting the model was to present a framework to describe the effects of step changes in supersaturation on nucleation kinetics. We explicitly state that the model is incorrect in an absolute sense and thus emphasize the qualitative arguments in our analysis of the experimental techniques rather than focusing on quantitative comparisons. As Galkin and Vekilov have rightly pointed out, quantitative comparisons indeed result in large discrepancies between experiments and model predictions. One reason for the failure of the model lies in its oversimplified representation of protein interactions as short-ranged centrosymmetric attractions. In a more recent application of the model [11], we show that accounting for the inherent anisotropy in protein interactions leads to dramatic differences in predicted nucleation rates. In addition, the predictions allowed quantitative comparisons with the solid /fluid interfacial tensions estimated from the induction time data of [7]. Thus, within the framework of aggregation and dissociation processes and the thermodynamics of phase changes, the model is expected to capture the qualitative features of the experimental system. Ignoring the merit of the qualitative arguments purely on the basis of the quantitative discrepancies between models and experiments, as Galkin and Vekilov
C.F. Zukoski et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 215 (2003) 137 /140
repeatedly do in their letter, does not address the physical interpretation captured by the model. Of interest in the response of Galkin and Vekilov [2,3] is our analysis of their extensive and extremely careful measurements of the number of crystals formed in a two step temperature quench measurement. While developing an analysis of this approach we found that classical nucleation and growth models and a literature on the two step method suggest difficulties in separating nucleation and growth in the two step method [1,12]. Our analysis indicates that irrespective of the comparisons with other measures, the interpretation of the two step method employed in [2,3] underestimates nucleation rates. The two step method involves quenching a suspension to a temperature T1 for a time period t1 and then raising the temperature to T2 for a time period t2. The number of crystals at the end of t2 is determined. Looking at how this number changes with t1, the nucleation rate is backed out. The authors [2,3] argue that T2 is sufficiently close to the melting temperature at the concentration of interest that no nucleation is observed for several months at T2. Of particular concern in our analysis is the opportunity for clusters formed at T1 to melt upon raising the temperature to T2. Here we attempt to restate our argument in a few lines. We define the critical cluster size at T1 as R* (T1) and that at T2 as R* (T2). Because T2 is chosen very close to the melting temperature at the concentration of interest, we must assume R* (T2)/R* (T1). Thus upon raising the temperature one predicts the particles at size R *(T1) will be unstable and melt. On the other hand, if there is a steady rate of nucleation and some clusters can grow to a size of R *(T2) or larger at a time t1, the solution will contain a cluster size distribution containing clusters with sizes smaller than R *(T2). When the temperature is raised to T2, these clusters will have a tendency to melt rather than grow. Experimental evidence was provided in our paper that such melting may indeed occur [1]. In their original paper and in their response to our paper, Galkin and Vekilov largely ignore the question of why crystals formed at high supersaturations (T1), where the critical size is small,
139
and do not grow sufficiently during t1 cannot melt when the temperature is raised rapidly to just below the melting temperature. (We note that our use of the term ‘melting’ is consistent with our description of protein solutions as one component systems, where the protein molecules interact via an effective interaction potential.) In their letter, Galkin and Vekilov seem to suggest that the melting of crystals we predict is due to the ‘solution depletion to below the solubility at T2’, and attempt to prove that their system is not depleted upon increasing the temperature to T2. We note that we expect crystals to melt not because the solution is depleted to below the solubility (in which case, ‘all’ and not ‘some’ of the crystals would melt) but because some of the crystals nucleated at T1 will be smaller than the critical cluster size at T2 at the end of the interval t1. The light scattering experiments reported in our paper [1] do indeed show a reduction in the intensity of light scattered from suspensions exposed to a two-step process as proposed by Galkin and Vekilov, indicating that freshly nucleated crystals (independent of their being heterogeneously or homogeneously nucleated) melted upon rapidly reducing the supersaturation. Analysis of the time for the signal to reach a maximum would require a detailed study of the rates of nucleation and growth (shrinkage) coupled with how light scattered is related to particle number density and size distributions. We did not carry out such a kinetic analysis for our paper. However, it is clear that the number and/or size of the clusters in the solution decrease upon raising the temperature to T2. In their response, Galkin and Vekilov further argue that even if clusters melt, the number that do so would be independent of the quench time t1. As a result, experimental nucleation rates, determined as the slope of the number of crystals counted at t2 versus the quench time t1, remain unaffected by crystals melting. We believe that this argument needs some careful thought. Our approach is based on the idea that as clusters grow, the monomer concentration and thus the driving force for nucleation is decreased. As in our paper [1], we define tn as the time characterizing the formation of a cluster of size
140
C.F. Zukoski et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 215 (2003) 137 /140
R *(T1) in a unit volume of suspension (note that this definition involves no inconsistencies in the units), and tg(R ) as the time required for a cluster to grow from R *(T1) to a particular size, R (/ R *(T1)). The number of crystals that melt, Nmelt, is then estimated to equal tg(R* (T2))/tn. Since R *(T2 )/R *(T1), it follows that tg(R *(T2)) /tn. Also, to see crystals at T2, the quench time must be longer than the time for clusters to grow to R *(T2), so that t1 /tg(R *(T2)). As a result, t1 / tn. This implies that in the quench time t1, the number of crystals that nucleate, which is approximated by t1/tn, is large. This causes s to reduce to a lower value than at the beginning of the crystallization experiment. (Galkin and Vekilov argue that s does not change much in their experiments by observing that no crystals bigger than 2 mm were present in their suspensions before and after raising the temperature to T2. However, the critical cluster consists of a few protein molecules and thus is on the order of a few nanometers in size. As a result, a large number of post critical clusters could exist in the suspensions and lower s and still be smaller than 2 mm and therefore remain undetected.) As t1 is increased, t1/tn also increases, causing s to reduce to an even lower value. This changes tn and tg(R *(T2)) (continuously) and hence Nmelt. We note that in the experiments of Galkin and Vekilov, t1 is increased by an order of magnitude (10 min /8 h) [2,3]. As a result, Nmelt could conceivably change significantly with t1, thereby affecting the measured nucleation rates. In a supersaturated solution, clusters develop as the result of association and dissociation processes giving rise to a cluster size distribution. The critical cluster is that cluster which does not have a tendency to grow or shrink. Measuring the flux of clusters through this critical size is extremely difficult whether for proteins or for smaller molecules. Interpreting experiments requires that
they be rigorously linked back to the properties the measurement techniques probe. Light scattering and crystal counting probe different averages of the cluster size distribution [4]. However, assuming that the number of clusters at t2 is a measure of the stable clusters in solution at t1 requires an investigation of the kinetics of cluster growth and how the cluster size distribution is altered when the supersaturation is suddenly lowered. One of the goals of our paper is to suggest that more detailed investigations of growth kinetics and the supersaturation dependence of the critical cluster size be carried out. Given that there is at least a little evidence for the shrinkage of crystals in the two step process and that two step methods have been investigated in molecular systems and are interpreted differently [1,12], we believe such studies are warranted.
References [1] N.M. Dixit, A.M. Kulkarni, C.F. Zukoski, Colloids Surf. A 190 (2001) 47. [2] O. Galkin, P.G. Vekilov, J. Phys. Chem. B 103 (1999) 10965. [3] O. Galkin, P.G. Vekilov, J. Am. Chem. Soc. 122 (2000) 156. [4] N.M. Dixit, C.F. Zukoski, Phys. Rev. E, in press. [5] P.A. Darcy, J.M. Wiencek, Acta. Cryst. D 54 (1998) 1387. [6] P.A. Darcy, J.M. Wiencek, J. Crystal Growth 196 (1999) 243. [7] A.M. Kulkarni, C.F. Zukoski, J. Cryst. Growth 232 (2001) 156. [8] N.M. Dixit, C.F. Zukoski, J. Colloid Interface Sci. 228 (2000) 359. [9] G. Narasimhan, E. Ruckenstein, J. Colloid Interface Sci. 128 (1989) 549. [10] T.E. Paxton, A. Sambanis, R.W. Rousseau, Langmuir 17 (2001) 3076. [11] N.M. Dixit, C.F. Zukoski, J. Chem. Phys, accepted. [12] P.E. Wagner, R. Strey, J. Phys. Chem. 85 (1981) 2694.