Conceptual Validation of Stochastic and Deterministic Methods To Estimate Crystal Nucleation Rates

Abstract

This work presents a generalized framework to assess the accuracy of methods to estimate primary and secondary nucleation rates from experimental data. The crystallization process of a well-studied model compound was simulated by means of a novel stochastic modeling methodology. Nucleation rates were estimated from the simulated data through multiple methods and were compared with the true values. For primary nucleation, no method considered in this work was able to estimate the rates accurately under general conditions. Two deterministic methods that are widely used in the literature were shown to overpredict rates in the presence of secondary nucleation. This behavior is shared by all methods that extract rates from deterministic process attributes, as they are insensitive to primary nucleation if secondary nucleation is sufficiently fast. Two stochastic methods were found to be accurate independent of whether secondary nucleation is present, but they underestimated rates in the case where a large number of primary nuclei are formed. We hence proposed a criterion to probe the accuracy of stochastic methods for arbitrary data sets, thus providing the theoretical foundations required for their rational use. Finally, we showed how both primary and secondary nucleation rates can be inferred from the same set of detection time data by combining deterministic and stochastic considerations.

Figure 1

Characterization of the reference system: Evolution of the system after the first nucleation event. (a) Total number of detected crystals, i.e., those with a size larger than L_det = 5 μm. (b) Total number of crystals, i.e., including those smaller than the detection threshold. Colors indicate the four supersaturation levels studied by Cedeno et al.:S = 1.40 (red), S = 1.33 (green), S = 1.26 (blue), and S = 1.20 (magenta). Solid lines indicate the simulation corresponding to the median detection time, while dashed lines show the profiles for the simulations corresponding to the 10% and 90% quantiles of the detection time distribution.

Figure 2

Characterization of the reference system: Distribution of the detection times (left) and of its components, i.e., of the first nucleation times (center) and of the growth times (right) for α = 10^–4. Total of 16 384 crystallization processes were simulated to generate the distributions. Colors indicate the four supersaturation levels studied by Cedeno et al.:S = 1.40 (red), S = 1.33 (green), S = 1.26 (blue), and S = 1.20 (magenta).

Figure 3

Characterization of the reference system: Number of primary nuclei formed in the 100 mL system. Total of 16 384 crystallization processes were simulated to generate the distributions. Dashed lines indicate the number of crystals formed by primary nucleation at the end of the process; solid lines show the number at the detection time for α = 10^–4. Colors indicate the four supersaturations studied: S = 1.40 (red), S = 1.33 (green), S = 1.26 (blue), and S = 1.20 (magenta).

Figure 4

Estimating primary nucleation rates by means of deterministic (methods 1 and 2) and stochastic methods (methods 4 and 5) for the supersaturation level of S = 1.40. Black line indicates the true value of the primary nucleation rate, i.e., the one used to generate the simulations. (a) Rate estimates for a process comprising primary nucleation and secondary nucleation, as presented in section 3. (b) Estimates for a simplified crystallization process in which only primary nucleation but no secondary nucleation occurs. (c and d) Mean number of crystals present at the detection time, which depends on the value of α. Note that all crystals are formed through primary nucleation if secondary nucleation is absent. Colors indicate the four values of the detection threshold α (methods 2, 4, and 5): 10^–2 (dark blue), 10^–3 (light blue), 10^–4 (red, reference value), and 10^–5 (yellow). For method 1, they denote the value of the detection threshold L_det: 5 μm (red, reference value) and 0 (dark red).

Figure 5

Effect of the primary nucleation rate on process behavior. Six sets of increasingly fast primary nucleation are shown for two supersaturation levels: S = 1.40 (red) and S = 1.20 (magenta). Slowest primary nucleation kinetics (solid lines) correspond to the reference values, while faster kinetics have been evaluated using larger values of the pre-exponential parameter A_PN: 10² times larger (dashed lines), 10⁴ times larger (dotted lines), 10⁶ times larger (dashed–dotted lines), 10⁸ times larger (solid lines), and 10¹⁰ times larger (dashed lines).

Figure 6

(a and b) Secondary nucleation rates obtained through the deterministic method 3 (cf. eq 18). (c and d) Primary nucleation rates obtained through the stochastic method 5. Colors indicate the four supersaturations studied: S = 1.40 (red), S = 1.33 (green), S = 1.26 (blue), and S = 1.20 (magenta). Dashed lines denote the true values of the nucleation frequency. (a and c) Plotted versus crystallizer volume. (b and d) Plotted versus mean number of primary nuclei formed.

Figure 7

Scatter plot comprising all process conditions studied in this work, i.e., nine volumes, four supersaturation levels, and with/without secondary nucleation. For each condition, both the value of the standard deviation ratio and the relative accuracy are shown. Relative accuracy was computed as the ratio of the estimated value for the nucleation rate and the true value. Red line indicates the diagonal.

Figure 8

(a and b) Standard deviation ratio of first nucleation time and detection time for nine volumes and four supersaturation levels. Simulations were carried out both in the presence and in the absence of secondary nucleation. (c and d) Coefficient of variation for the same data sets. (a and c) Volume-based plots. (b and d) Number of nuclei-based plots. Solid black line at a value of one indicates the theoretical value corresponding to the single-nucleus assumption (SNM).