Automated Growth Rate Measurement of the Facet Surfaces of Single Crystals of the β-Form of l-Glutamic Acid Using Machine Learning Image Processing

Precision measurement of the growth rate of individual single crystal facets (hkl) represents an important component in the design of industrial crystallization processes. Current approaches for crystal growth measurement using optical microscopy are labor intensive and prone to error. An automated process using state-of-the-art computer vision and machine learning to segment and measure the crystal images is presented. The accuracies and efficiencies of the new crystal sizing approach are evaluated against existing manual and semi-automatic methods, demonstrating equivalent accuracy but over a much shorter time, thereby enabling a more complete kinematic analysis of the overall crystallization process. This is applied to measure in situ the crystal growth rates and through this determining the associated kinetic mechanisms for the crystallization of β-form l-glutamic acid from the solution phase. Growth on the {101} capping faces is consistent with a Birth and Spread mechanism, in agreement with the literature, while the growth rate of the {021} prismatic faces, previously not available in the literature, is consistent with a Burton–Cabrera–Frank screw dislocation mechanism. At a typical supersaturation of σ = 0.78, the growth rate of the {101} capping faces (3.2 × 10–8 m s–1) is found to be 17 times that of the {021} prismatic faces (1.9 × 10–9 m s–1). Both capping and prismatic faces are found to have dead zones in their growth kinetic profiles, with the capping faces (σc = 0.23) being about half that of the prismatic faces (σc = 0.46). The importance of this overall approach as an integral component of the digital design of industrial crystallization processes is highlighted.

This supplementary material supports the main manuscript by providing further details of the following: Section S1 discusses about the linear fitting methods with Figure S1 presenting a typical linear fitting for determining the face-based growth rate and Table S1 listing the goodness-of-fitting (R 2 ) for all experimental runs.Figure S2 shows a typical sequence of images of -form l-glutamic acid (LGA) crystals growing from water with time at relative supersaturations of 0.28 and 1.05.Section S3 describes a method to calculate normal distances between the paired prismatic {021} faces with Figure S3 showing the schematic of the method.Section S4 presents the repeated experimental data at σ = 0.78 and 1.05 to examine their repeatability, with Figures S4 & S5 for experimental data and Tables S2&S3 for their growth rates.Figure S6 presents the lengths of facet growth as a function of growth time for all nine experiments (nine different supersaturations).

S1. Linear Fitting for Determining Growth Rate
Figure S1.Determination of growth rate from relative length (to the first image) against relative time.The linear fitting is applied, and the gradient of the fitted line represents the growth rate.The red dotted line fits all data within the time range.The decrease of growth rate over time would cause poor fitting results, hence the R 2 of fitting being relatively lower.
The decrease of growth rate is likely caused by the decrease of supersaturation within the cuvette during crystal growth.It is preferred to only use the data at the beginning of the growth, hence the initial supersaturation still being valid.For example, the green dotted line fits only the first 100 min of data with very decent fitting results (R 2  1).Therefore, the gradient of the green line is used as the growth rate.Over all sets of the experimental data, the data from the first 60 -810 min (about 20 -180 data points) as shown in Table S1 are included for the linear fitting.As shown in Table S1, all of the linear fittings for the {101} capping faces have very high R 2 (> 0.99), whilst most of the goodness of fittings for the {021} prismatic faces are  2 = 0.9994  2 = 0.9601 reasonably satisfactory with some runs at lower supersaturations being found to have lower R 2 .This most possibly results from the very low growth rate of the prismatic faces at low supersaturations, hence highly scattered data measured, even more at the lowest  (= 0.28) there is no growth rate for the prismatic faces at all as evidenced in literature 1 .

S3. Calculation of Normal Distance between Paired {021}
As shown in Figure S3 (SI), the normal distance between the paired ( 021) and (0-2-1) faces were calculated based on the projected width, w, and the angle, , between ( 021) and (02-1) faces by the following equation (S1): Figure S3.Schematic of facet normal distance calculation of the prismatic {021} faces.
Table S2.Facet growth rates measured from the repeated experiments at  = 0.78.The colour of each run matches the corresponding colour as shown in Figure S4.Run #6 is an outlier which was not used for calculating the mean growth rate.

Figure S2 .
Figure S2.Typical sequence of images of beta-formLGA crystals growing in water with time at relative supersaturations of 0.28 and 1.05.

Figure S4 .
Figure S4.Lengths of capping and prismatic faces vs time at  = 0.78 from six repeated experiments.Note that the experimental run (coloured as red) can be treated as an outlier due to its much faster growth (about 2 times of growth rate compared with other five runs).

Figure S5 .Figure S6 .
Figure S5.Lengths of capping and prismatic faces vs time at  = 1.05 from six repeated experiments.

Table S1 .
The R 2 of linear growth rate fitting with the time for the early stage and the corresponding data points.Note that the run #6 (σ = 0.78) is an outlier.