DOI: 10.1021/cg101213q
Crystal Nucleation Rates from Probability Distributions of Induction Times
2011, Vol. 11 256–261
Shanfeng Jiang and Joop H. ter Horst* Intensified Reaction & Separation Systems, Process & Energy Department, Delft University of Technology, Leeghwaterstraat 44, 2628CA Delft, The Netherlands. Received September 13, 2010; Revised Manuscript Received November 5, 2010
ABSTRACT: A novel method for the determination of stationary crystal nucleation rates in solutions has been developed. This method makes use of the stochastic nature of nucleation, which is reflected in the variation of the induction time in many measurements at a constant supersaturation. A probability distribution function was derived which describes, under the condition of constant supersaturation, the probability of detecting crystals as a function of time, stationary nucleation rate, sample volume, and a time needed to grow the formed nuclei to a detectable size. Cumulative probability distributions of the induction time at constant supersaturation were experimentally determined using at least 80 induction times per supersaturation in 1 mL stirred solutions. The nucleation rate was determined by the best fit of the derived equation to the experimentally obtained distribution. This method was successfully applied to measure the nucleation rates at different supersaturations of two model compounds, m-aminobenzoic acid (m-ABA) and L-histidine (L-His). The determined nucleation rates of m-ABA and L-His followed the trend expected from classical nucleation theory (CNT). The behavior indicated that, as expected, heterogeneous nucleation occurred. The relatively low kinetic parameter A for both compounds might indicate that the concentration of the active heterogeneous particles was relatively low. The novel method is a promising technique to determine nucleation rates in stirred solutions.
Introduction Nucleation is a key step in crystallization processes, since it controls crystal product quality aspects such as the kind of solid state and the crystal size distribution. Nucleation is the statistical process of appearance of nanoscopically small clusters of molecules of a new phase in a supersaturated old phase.1,2 There is an increasing demand for developing a reliable and relatively fast method to measure nucleation rates: such a method will enable scientists to validate nucleation theories and engineers to achieve control over the product quality in industrial crystallization processes. Microfluidic devices have shown great potential because of their ability to store, process, and control molecules in space and time.3 As recent promising results show,4-7 microfluidic devices will enable a systematic study into the fundamentals of the rate of nucleation of crystals, especially in combination with the double pulse method.8 Crystal nucleation rate measurement results from the double pulse method lead to the identification of a two-step mechanism for the nucleation of protein crystals from solution.9,10 Industrially applied crystallization processes fundamentally differ from those in small and stagnant solution volumes because it is believed that agitation of a supersaturated solution affects the nucleation rate.11 This may originate from effects of secondary nucleation12 or other phenomena. The agitation during nucleation thus might be an essential parameter in a nucleation study. It therefore is of importance to have available, next to methods for unstirred conditions, a nucleation rate measurement method under agitated conditions. *To whom correspondence should be addressed. E-mail: J.H.
[email protected]. Phone: þ31 15 278 6661. Internet: http://www.pe. tudelft.nl/. pubs.acs.org/crystal
Published on Web 11/19/2010
Here we report a novel method in which nucleation rates are determined from cumulative probability distributions of induction times in agitated solutions, closely related to industrial crystallization conditions. This new method is relatively easy to perform and less time-consuming compared to the double pulse method while the nucleation rate is straightforwardly accessible from the experimental results. This method makes use of the stochastic nature of nucleation, which is reflected in the induction time variation. First, an equation is derived describing the distribution as a function of the induction time. Then, the experimentally determined probability distributions of the induction time for two model compounds, m-ABA and L-His, are discussed. Finally, the nucleation rates are determined and discussed. Theory At a constant supersaturation, the appearance of a nucleus (often referred to as a critical nucleus) can be regarded as a random process. When the appearance of nuclei is independent, the probability Pm of forming m nuclei in a time interval is described by the Poisson distribution.13 Pm ¼
Nm expð - NÞ m!
ð1Þ
where N is the average number of nuclei that form in the time interval. The probability P0 that no nuclei are formed within the time interval is, using eq 1 P0 ¼ expð - NÞ
ð2Þ
The probability Pg1 that at least 1 nucleus has formed in the time interval is thus Pg1 ¼ 1 - P0 ¼ 1 - expð - NÞ
ð3Þ
r 2010 American Chemical Society
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The average number of nuclei N formed in time interval tJ and volume V is related to the stationary nucleation rate J: N ¼ JVtJ
ð4Þ
The probability P*(tJ) that at least 1 nucleus has formed in time interval tJ is thus PðtJ Þ ¼ 1 - expð - JVtJ Þ
ð5Þ
This equation was used before to analyze nucleation data in small volumes (see ref 1, chapter 26 for a review). In order to be detected, the formed nuclei have to grow to appreciable sizes before they can be detected. This causes a delay tg, the growth time, between the time tJ of appearance of a nucleus and the time t of detection: tJ = t - tg. The probability P(t) to detect crystals at time t which were nucleated at an earlier time can thus be determined with the aid of eq 5: PðtÞ ¼ 1 - expð - JVðt - tg ÞÞ
ð6Þ
This cumulative distribution function is valid for times t g tg. The induction time is the period of time between the achievement of a constant supersaturation and the detection of crystals.14 Because of the stochastic nature of nucleation, the variation in the induction time measured under equal conditions can be described by eq 6. Then, the induction time probability P(t) at a certain supersaturation, temperature, and volume describes the chance that for certain conditions in an induction time measurement crystals are detected at time t. The probability distribution of the induction time can be determined from a large number of induction time measurements at constant supersaturation, temperature, and volume. For M isolated experiments, the probability P(t) to measure an induction time between zero and time t is defined as M þ ðtÞ ð7Þ M where Mþ(t) is the number of experiments in which crystals are detected at time t. The experimentally determined probability distribution P(t) of the induction time (eq 7) can be described by the cumulative probability distribution function (eq 6), leading to the determination of nucleation rate J and growth time tg. PðtÞ ¼
Experimental Section The two model compounds are m-aminobenzoic acid15 (m-ABA) and L-histidine16 (L-His), both of which have one amino group and one carboxylic acid group. m-ABA (TCI, chemical purity g99%), L-His (Fluka Chemie, chemical purity g99%), pure ethanol (chemical purity 100%), and ultrapure water were used. The solubility and induction time measurements were performed using the Crystal16 multiple-reactor setup (Avantium Pharma, Amsterdam) with accurate temperature control. It has 16 wells designed to hold 16 standard HPLC glass vials (1.8 mL) and measures the transmission of light through a sample in the vials in the wells. Because of a small constant temperature difference between the actual temperature in the well and the set temperature of the well, a recalibration of the Crystal16 set temperature was performed. Solubility Measurement. The solubility of m-ABA in 50 wt % water/ethanol mixtures and of L-His in water was measured as a function of temperature using the Crystal16 setup. Slurries of m-ABA or L-His with different concentrations were prepared by adding a known amount of crystalline material and 1 mL of solvent in the 16 vials containing a magnetic stirrer. The vials were placed in the setup at a stirring speed of 900 rpm. The heating and cooling rates were set to 0.5 °C/min. Upon increasing the temperature of a suspension, the light transmission through the sample reaches an upper limit at a certain temperature (clear point) when the suspension
Figure 1. Temperature-dependent solubilities of m-ABA (Δ) in water/ethanol mixtures (50 wt %) and L-His (0) in water. Each point is the average value of multiple measurements. The error bars are smaller than the symbols. The solid lines are the predicted solubilities using a fit of the measured data to the van ’t Hoff equation. turns into a clear solution. The clear point was taken as the saturation temperature. The saturation temperature was measured 4 to 5 times per sample by cycles of cooling and reheating. Induction Time Measurements. The induction times were measured at 25 °C in the Crystal16 setup. For m-ABA the mole fraction based supersaturation ratios S = x/x* used were S = 1.83, 1.87, 1.93, 1.96, 2.06, and 2.15, while for L-His S = 1.55, 1.60, 1.64, 1.69, 1.74, and 1.79 were chosen. Care was taken to keep variations in temperature and concentration during the induction time measurement minimal. For all measurements at one supersaturation ratio, a 50 mL solution was prepared by dissolving the corresponding amount of the model compound in the solvent. For m-ABA the solvent was a 50 wt % mixture of water and ethanol, while for L-His it was pure water. A bottle-top dispenser was used to dispense 1 mL of clear solution into each vial. The samples were stirred above their saturation temperature for at least 30 min to make sure that the crystals were dissolved. The stirring speed was controlled at 900 rpm. The clear solution was quickly cooled down to 25 °C with a rate of 5 °C/min. The moment the solution reached a temperature of 25 °C was taken as time zero, after which a constant temperature of 25 °C was maintained. At some point in time, the transmission decreased. The difference between the time at which the transmission started to decrease and time zero was taken as the induction time. Induction times up to 5 h were measured. Then, the sample was reheated with a rate of 1 °C/min and maintained above the saturation temperature to dissolve the crystals and to obtain a clear solution again. Then a subsequent induction time measurement was started. This coolhold-heat cycle was repeated 5 times to obtain a total of 80 (16 5) induction time measurements for each supersaturation ratio.
Results The temperature-dependent solubilities of m-ABA in water/ethanol mixtures (50 wt %) and L-His in water are shown in Figure 1. Each experimental point is the average of multiple saturation temperature measurements. The variation in the experimental values for a certain concentration is smaller than the corresponding symbol. The solubility of m-ABA in 50 wt % water/ethanol mixtures increases from 18.0 to 70.3 g/L-solvent upon increasing the temperature from 22 to 53 °C. The solubility of L-His in water increased from 49.6 to 87.7 g/L-solvent when the temperature increased from 33 to 64 °C. A fit of the van ’t Hoff equation to the data facilitated the interpolation of the solubility as well as the determination of the prevailing supersaturation ratio S = x/x* in a certain solution composition. Induction Time and Induction Time Probability Distributions. Induction time measurements of m-ABA and L-His
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Figure 2. (a, left) Induction time of 80 experimental data points for m-ABA solutions at the supersaturation ratio S = 1.96; (b, right) probability distribution of the induction times of these 80 points.
solutions were performed at a temperature of 25 °C for various supersaturation ratios. At each supersaturation ratio a total of M = 80 identical induction time measurements were performed in 1 mL samples. As expected, the induction times at one supersaturation show a large variation. For example, at a supersaturation ratio S = 1.96, the induction time t of m-ABA varied from 1.03 103 to 8.48 103 s (see Figure 2a); at S = 1.60, the induction time t of L-His varied from 75 to 10.4 103 s. These results make clear that for small volumes a single induction time measurement is insufficient to be the basis of scientifically sound conclusions. The large variation of induction times under equal supersaturation conditions reflects the stochastic nature of the nucleation process. At relatively small volume and low supersaturation, the probability that nuclei form in solution is low, leading to a large variation in induction time. With the total number of M = 80 induction times per supersaturation ratio, the probability distribution P(t) of the induction time can be determined using eq 7. As an example, for m-ABA at S = 1.96 and a time t = 5000 s (indicated by the dashed line in Figure 2a), in Mþ = 73 samples crystals were detected. Thus, the probability P(t) is positioned between the values 73/80 = 0.913 < P(t) < 74/80 = 0.925. Figure 2b shows the probability distribution determined from the induction times presented in Figure 2a. An experimental point in Figure 2b is taken exactly at the moment in time that in one of the 80 samples crystals were detected: at t = 4.74 103 s crystals were detected in the 73rd sample, giving a P(t) = 0.913. Starting at zero, the probability distribution rapidly increases from t = 1.03 103 s, where crystals were detected in the first sample. The probability distribution levels off when approaching P(t) = 1. At time t = 8.48 103 s, crystals were detected in the final sample. This shape is very well represented by the probability distribution function (eq 6). A fit of this function to the experimental data, represented by the curved line in Figure 2b, delivers a nucleation rate J = 0.63 103 ( 20 m-3 s-1 and a growth time tg = 1.17 103 ( 20 s. Nucleation Rate Determination. The experimental probability distributions of the induction time are shown in Figure 3 for m-ABA and in Figure 4 for L-His. The figures show that, at higher supersaturations, the probability P(t) approaches 1 in a shorter time, indicating a higher nucleation rate. As shown in Figure 3, at S = 2.15, the probability P(t) of m-ABA reaches 1 when t = 1.79 103 s, while, at S = 1.83, until t = 18 103 s the probability only reaches to
Figure 3. Experimentally obtained probability distribution P(t) of the induction time for m-ABA at supersaturation ratios S = 1.83 (0), 1.87 (þ), 1.93 (]), 1.96 ( ), 2.06 (O), and 2.15 (Δ) in 50 wt % water/ethanol mixtures calculated using eq 7. The solid lines are fits of eq 6 to the experimental data.
Figure 4. Experimentally obtained probability distribution P(t) of the induction time for L-His at supersaturation ratios S = 1.55 (0), 1.60 (þ), 1.64 (]), 1.69 ( ), and 1.74 (O) in water calculated using eq 7. The solid lines are fits of eq 6 to the experimental data.
around P(t)=0.5, meaning that in only 50% of these 1 mL m-ABA solutions crystals were detected within that time. In Figure 4, at S=1.79, the probability P(t) of L-His reaches 1 at t=2.71 103 s, while, at S=1.55, it just approaches P(t)= 0.63 at t = 7.5 103 s. For both compounds and at all supersaturations the data is well represented by the characteristic shape of the distribution
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Table 1. Determined Nucleation Rate J and Growth Time tga m-ABA
L-His
S
J (m-3 s-1)
ΔJ (m-3 s-1)
tg (s)
Δ tg (s)
1.83 1.87 1.93 1.96 2.06 2.15 1.55 1.60 1.64 1.69 1.74
50 250 350 630 1.22 103 4.03 103 160 630 690 1.87 103 1.98 103
10 10 10 20 30 140 10 20 20 30 30
2900 1410 786 1170 637 294 195 89 31 30 39
150 60 30 20 10 10 30 20 20 10 10
a The values for ΔJ and Δtg reflect the quality of the fit of eq 6 to the experimental data.
function (eq 6). The determined nucleation rates J and growth times tg for both compounds are given in Table 1. The nucleation rates J of m-ABA increased from 50 to 4.03 103 m-3 s-1 when the supersaturation ratio S increased from 1.83 to 2.15, and that of L-His increased from 160 to 1.98 103 m-3 s-1 when the supersaturation ratio S increased from 1.55 to 1.74. The growth time tg seems slightly supersaturation ratio dependent, decreasing with an increasing supersaturation ratio. Nucleation Behavior. According to the classical nucleation theory (CNT), the dependence of the nucleation rate J on the supersaturation ratio can be described by1,2 B ð8Þ JðSÞ ¼ AS exp - 2 ln S with A the kinetic parameter and B the thermodynamic parameter for nucleation. The exponential part represents a free energy barrier for the formation of the nucleus while the pre-exponential part contains information on the kinetics of the barrier crossing. In conformity with eq 8, ln(J/S) should be a linear function of 1/ln2 S. This is shown in Figure 5 using the nucleation rate data from Table 1. The thermodynamic parameter B can be estimated from the slope of the best-fit straight line, while the kinetic parameter A can be derived from the intercept of that line, giving the value of ln A. Table 2 gives the obtained values of A and B for m-ABA and L-His. The thermodynamic parameter B for heterogeneous nucleation is expressed by1 B ¼
4 c3 v2 γef 3 27 k3 T 3
ð9Þ
with a shape factor c (e.g., c = (36π)1/3 for spheres, c = 6 for cubes1), the molecular volume v of the crystalline phase, and the effective interfacial energy γef. The effective interfacial energy γef = ψγ, with activity factor 0 < ψ < 1, accounts for the reducing effect of the heterogeneous particle on the nucleation work W* compared to homogeneous nucleation where γef = γ. Using the obtained values of the thermodynamic parameter B (Table 2), the values of the effective interfacial energy γef for heterogeneous nucleation were estimated based on eq 9 by assuming spherical nuclei. The molecular volumes were taken to be v = 151 10-30 m3 and 180 10-30 m3 for m-ABA and L-His, respectively. The effective interfacial energies γef for m-ABA in 50 wt % ethanol/water and of LHis in water at 25 °C are given in Table 2. To determine the activity factor ψ, the theoretical values for the interfacial energy γ were calculated using a relation
Figure 5. Plot of ln(J/S) as a function of 1/ln2 S for m-ABA (2) and L-His (9), which enables the determination of the kinetic parameter A and the thermodynamic parameter B of the nucleation rate equation (eq 8). The obtained values are shown in Table 2. Table 2. Determined Kinetic Parameter A, Thermodynamic Parameter B, Effective Interfacial Energy γef for Heterogeneous Nucleation, and Activity Factor ψ parameter
m-ABA
L-His
A (m-3 s-1) B γef (mJ/m2) ψ
0.87 106 3.6 8.7 0.27
36.3 103 1.1 5.1 0.22
between solubility and interfacial energy17 with a corrected shape factor of 0.514 for spherical nuclei.2 γ ¼ 0:514kT
1 1 ln v2=3 Na vc
ð10Þ
Here, Na is Avogadro’s number and c* is the molar solubility. For m-ABA, the theoretical value of the interfacial energy γ is 32.1 mJ/m2 with c* = 0.15 mol/L-solvent in 50 wt % ethanol/water at 25 °C. Comparing this to the experimentally obtained effective interfacial energy γef = 8.7 mJ/m2 shows that the activity factor (ψ = γef/γ) is 0.27. For L-His, the theoretical value of the interfacial energy γ is 23.4 mJ/m2 with c* = 0.27 mol/L-solvent in water at 25 °C, giving an activity factor ψ of 0.22. The determined values for the activity factor are typical values observed for heterogeneous nucleation1 and indicate that heterogeneous nucleation is the dominant nucleation mechanism in our experiments. The obtained kinetic parameter A for m-ABA and L-His was respectively 0.87 106 and 36.3 103 m-3 s-1. Discussion Statistical Significance. Due to the strongly nonlinear behavior of the nucleation rate with respect to the supersaturation, we expect the fluctuation in the supersaturation to be the most significant experimental error. This variation in supersaturation in turn is a function of the temperature fluctuations through the temperature dependence of the solubility. Therefore, care has been taken to avoid fluctuations in temperature during the induction time measurements. Our measurements were performed with a temperature accuracy of 0.1 °C. In the case of m-ABA, a fluctuation of 0.1 °C in the temperature would result in a supersaturation fluctuation of less than ΔS = 0.01 at the used supersaturations. At a supersaturation of S = 1.92, 1.93, and 1.94 and using the nucleation parameters from Table 2, we calculate a nucleation rate of respectively J=359, 406, and 459 m-3 s-1. At a
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supersaturation of S = 1.93 and a structural deviation in the supersaturation, we would therefore expect to measure a nucleation rate between 359 and 459 m-3 s-1. However, we expect the temperature fluctuations to be randomly distributed over our experiments: some solutions experience larger supersaturations while some experience smaller supersaturations. This would lead to slightly decreased induction times at low probabilities and increased induction times at higher probabilities. We estimate our experimental nucleation rates, when sampled with a sufficient number of experiments, might be accurate well within 15%. In order to check whether M=80 experiments per supersaturation is a sufficient number, we performed a statistical analysis. By generating a series of random numbers, with a set nucleation rate, growth time, and volume, eq 6 was used to numerically simulate a series of induction times. These induction times were converted to a probability distribution of induction times. Using this probability distribution, a nucleation rate can be back-determined and compared to the set nucleation rate. This numerically simulated nucleation rate is equivalent to our experimental nucleation rate but does not show the deviations due to experimental fluctuations in, e.g., temperature discussed previously. We estimate from this analysis that for M = 80 induction time measurements per supersaturation there is about an 80% chance that we measured a nucleation rate within 20% of the actual nucleation rate. This chance can be increased when performing many more induction time measurements per supersaturation. The analysis of the fluctuations in supersaturation and the statistical analysis give us good confidence that this method gives sufficiently reliable nucleation rate data. Crystallization Mechanism. The induction time was determined by using the change in the transmission of light through the solution. When a crystal suspension forms in the solution, a decrease in the transmission is observed. The time period between the achievement of constant supersaturation and the moment the transmission starts to decrease is taken as the induction time. A considerable volume of crystals has to be present in the suspension before a decrease in transmission will be detected. The determined nucleation rates can be used to calculate the number of nuclei formed during the measurement. Using the parameters in Table 2 for a supersaturation of S = 1.93, a nucleation rate of J = 406 m-3 s-1 is calculated. The number of nuclei at time t = 2500 s in 1 mL can be determined by eq 4 to be around N = 1: on average, there is only 1 nucleus present in a 1 mL solution. This seems to contradict the considerable volume of crystals needed to decrease the transmission. A likely explanation for this apparent contradiction is given by the crystallization mechanism during the experiment: we believe this mechanism to involve subsequently the nucleation of a single parent crystal, the growth of this crystal to its attrition size, the attrition18 of the single parent crystal to form secondary nuclei, and the growth of the attrition fragments to fill the detectable crystal volume. This crystallization mechanism was also proposed for sodium chlorate crystallization.19 Sodium chlorate is an achiral compound which crystallizes in chiral forms. Crystallization in unstirred solutions results in a 50:50 mixture of crystals of both chiral forms. Crystallization in stirred solutions, however, resulted in crystals of a single handedness. This chiral symmetry breaking is the result of the attrition of a single parent crystal which has grown out from a nucleus formed earlier.
Jiang and ter Horst
This crystallization mechanism also explains the need for the growth time tg in eq 6. If the time for attrition and the outgrowth of the attrition fragments is considered to be sufficiently small, the growth time is just the time needed for the nucleus to grow out to its attrition size. In essence, if this crystallization mechanism is valid, the nucleation and growth stage are effectively split, similar to the case of the double pulse method.8 The advantage of this method is that, to determine the nucleation rate at a certain supersaturation, only one series of experiments is needed compared to the several series of experiments in the case of the double pulse method. We aim for a validation of this crystallization mechanism as well as its generality of its occurrence in future work. A further advantage of the new method is that, due to the stirring during the measurements, the method relates more directly to industrial crystallization conditions. To what extend this method can be used in the design and scale up of industrial crystallization processes will be investigated in future work. Nucleation Parameters. The obtained thermodynamic parameter B values fall within the expected range and show that heterogeneous nucleation rather than primary nucleation occurs. The effective interfacial energies γef are the same order of magnitude as those of L-asparagines (6.1 mJ/ m2) and lovastatin (1.57 mJ/m2), which were determined from measured nucleation rates.20,21 With the aid of the thermodynamic parameter B or the effective interfacial energy γef and the supersaturation S, the size of the nucleus can be estimated.1 For both systems, nucleus sizes of 10 to 35 molecules were determined. The obtained kinetic parameters A for m-ABA and L-his, however, were respectively 0.87 106 and 36.3 103 m-3 s-1, which is relatively low. In other solution nucleation rate measurements, such relatively low values were also found. For instance, the kinetic parameter A of lysozyme22 was between 107 and 109 m-3 s-1 while that of potassium nitrate23 was about 3 107 m-3 s-1. Since the pre-exponential factor AS in eq 8 is the product of the Zeldovich factor, the concentration of heterogeneous nucleation sites C0, and the attachment frequency f*, the low value of A indicates that either the concentration of heterogeneous nucleation sites C0 or the attachment frequency f* would be much smaller than expected. Currently, not much is known about the heterogeneous particles onto which heterogeneous nucleation generally takes place. Investigating template assisted nucleation with well-defined template particles24 might help to elucidate heterogeneous nucleation mechanisms and the size of the kinetic parameter A. It should be noted that in our case nucleation might also take place on the glass or stirrer wall or at the solution-air interface. A low attachment frequency f* might result from a strong solvation shell around the solute molecules or an energetically costly conformational change upon attaching a molecule to a cluster. Conclusions A novel experimental method of measuring the nucleation rate from probability distributions of induction times was developed. This method makes use of the stochastic nature of nucleation, which is reflected by the variation in induction times at constant supersaturation. The method is applicable to study nucleation kinetics in solution of soluble substances with temperature-dependent solubility. It was successfully tested on two model systems, m-aminobenzoic acid (m-ABA)
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in water/ethanol (50 wt %) mixtures and L-histidine (L-His) in water. The induction times were measured over the range of supersaturation ratios 1.83-2.15 for m-ABA and 1.55-1.79 for L-His, giving nucleation rates of respectively 50 to 4.03 103 m-3 s-1 and 160 to 1.98 103 m-3 s-1. The stationary nucleation rate J was determined by fitting the experimentally obtained probability distribution to the proposed probability distribution function using the nucleation rate and a growth time as fitting parameters. The results indicate that the nucleation of m-ABA and L-His in a 1-mL solution occurred via the subsequent heterogeneous nucleation of a single parent crystal, the growth of this crystal to its attrition size, the generation of attrition fragments from the single parent crystal, and the growth of these fragments to fill the detectable volume. This novel method seems a promising technique to determine nucleation kinetics in stirred solutions. Acknowledgment. We thank Peter Jansens, Dimo Kashchiev, and Roger Davey for the stimulating discussions during the preparation of this work.
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