Studying the impact of the pre-exponential factor on templated nucleation

Traditionally, the enhancement of nucleation rates in the presence of heterogeneous surfaces in crystallisation processes has been attributed to the modi ﬁ cation of the interfacial energy of the system according to the classical nucleation theory. However, recent developments have shown that heterogeneous surfaces instead alter the pre-exponential factor of nucleation. In this work, the nucleation kinetics of glycine and diglycine in aqueous solutions have been explored in the presence and absence of a heterogeneous surface. Results from induction time experiments show that the presence of a heterogeneous surface increases the pre-exponential factor by 2-fold or more for both glycine and diglycine, while the interfacial energy remains unchanged for both species. This study suggests that the heterogeneous surface enhances the nucleation rate via hydrogen bond formation with both glycine and diglycine. This is veri ﬁ ed by hydrogen bond propensity calculations, molecular functionality analysis, and calculation of the time taken for a solute molecule to attach to the growing nucleus, which is an order of magnitude shorter than the estimated lifetime of the hydrogen bond. The e ﬀ ect of the heterosurface is of greater magnitude for diglycine than for glycine, which may be due to the heightened molecular complementarity between the hydrogen bond donor and acceptor sites on diglycine and the heterosurface.


Introduction
The universal nature of crystallization has been explored in every aspect of science, from organics to inorganics to bio-organics, and from atmospherics to minerals.Crystallisation is a two-step process, with nucleation being the initial step of forming a stable new phase from a supersaturated medium.Nucleation is subsequently followed by growth as well as ripening, agglomeration, aggregation, angles enhanced the nucleation rate, pointing towards lattice matching as the potential reason, this was only evident for the polymers with HBA properties.Polymers lacking HBA properties did not inuence the nucleation rate of aspirin.
The concept of hydrogen bonding complementarity in the template induced nucleation of pharmaceutical ingredients was further explored by Chadwick et al. 19,36 for the heterogeneous nucleation of acetaminophen (AAP) in the presence of graphite, d-mannitol, a-lactose monohydrate, L-histidine, poly(methylmethacrylate), and poly-n-butylmethacrylate.Interestingly, graphite was among the poorest in accelerating the heterogeneous nucleation of AAP.Later, Bueno et al. 22 studied the inuence the range of different heterosurfaces such as d-mannitol, silica, microcrystalline cellulose (MCC), carboxymethyl cellulose, a/blactose, and polycaprolactone (PCL) on the heterogeneous nucleation of fenobrate (FF).They found that FF nucleated slowest in the presence of the PCL heterosurface.This is because except for PCL, all the heterosurfaces studied possessed HBDs which could easily form hydrogen bonds with the HBAs of fenobrate.Therefore, this study shows the inuence of functional group complementarity in template induced nucleation systems.
Recently, Ouyang et al. 37 were able to successfully nucleate and stabilise the carbamazepine (CBMZ) metastable form (FII) at lower supersaturations, at which normally only the stable form of CBMZ (FIII) crystallises, when crystallised in the presence of a phenyl functionalised silica heterosurface.The possible mechanism for this observation was the stabilisation of CBMZ molecules on the heterosurface due to the aromatic-aromatic interaction between the surface of phenylfunctionalized silica and the CBMZ molecule.Later, Ouyang et al. 38 expanded the work of using functionalized silica (SiO 2 , SiO 2 -NH 2 , SiO 2 -COOH) as a heterosurface in the nucleation of vanillin.The functionalised silica was able to signicantly enhance the nucleation rate of vanillin, which is evident by the reduction in the induction time from 4 hours to 20 minutes.The heterosurface also increased the growth rate of the nucleated vanillin crystal.These increases in the nucleation and growth rates were potentially due to the formation of hydrogen bonds between the vanillin molecules and the SiO 2 -NH 2 and SiO 2 -COOH heterosurfaces.Recently Li et al. 39 discovered that the use of silica particles as a heterosurface improved the crystallisation rate of lysozyme by reducing the induction time even in the presence of an impurity protein, thaumatin.
A recent study by Verma et al. 14 studied the inuence of a heterosurface (microcrystalline cellulose (MCC)) on the nucleation kinetic parameters of seven different active pharmaceutical ingredients (APIs), namely carbamazepine (CBMZ), acetaminophen (AAP), caffeine (CAF), clozapine (CPB), risperidone (RIS), phenylbutazone (PBZ), and FF.Crystallisation of the latter ve APIs was accelerated in the presence of MCC compared to the former two, CBMZ and AAP.The maximum acceleration was observed for FF, which was 16 times faster than what was observed for homogeneous nucleation.This acceleration was possibly because of the formation of a stable hydrogen bond between the HBAs of the latter ve APIs with the HBD sites on MCC.This study conrmed that the functional group complementarity between the API and heterosurface is an important factor in improving the API nucleation rate.This conrmation was supported by the hypothesis that the lifetime of the single adsorbed molecule on the surface of a hetero substance is sufficiently long to allow for the adsorption of other API molecules or clusters to attach to the adsorbed molecule, thereby leading to the formation of stable nuclei that remain attached to the heterosurface.This hypothesis was recently proved by Cazade et al. 40 using Monte Carlo (MC) and molecular dynamics (MD) simulations.The MD simulation results for the adsorption of RIS onto MCC surfaces concluded that the hydrogen-bonded lifetime of 32 ns is comparatively longer than that of the RIS-RIS attachment lifetime in the solution.It is also an order of magnitude longer than the time to add a single molecule of RIS to a growing crystal at a moderate supersaturation.
Verma et al. 14 also reported that the heterogeneous nucleation of APIs in the presence of MCC had minimal inuence on the interfacial energy of APIs in the nucleating solution.This is contradictory to the reported literature claiming that heterosurfaces decrease the interfacial energy, thereby reducing the nucleation energy barrier for nucleation, resulting in an increased nucleation rate. 2,41nstead, they claimed that the increase in the nucleation rate in the presence of MCC was due to the increase in the pre-exponential factor (A) in the presence of MCC.The value of A increased more than 2-fold for the API, exhibiting an increase in the nucleation rate in the presence of MCC.This signies that the rate of subcritical size cluster formation is at least 2-fold greater in the presence of MCC for most of the APIs, thereby promoting heterogeneous nucleation.This is consistent with their previous ndings on the nucleation of clozapine-methanol solvate in the presence of MCC, conrming that the pre-exponential factor is the major contributing factor to heterogeneous nucleation. 42lycine and diglycine were selected as molecules of interest for this work.The reason for this is that glycine is the simplest and most studied amino acid, consisting of a non-heavy hydrogen side chain, which eliminates the need to consider the effect of the side chain of the amino acid when analysing the effect of the heterosurface.Glycine exhibits a total of six polymorphs (a, b, g, d, 3, and z), 43 but the latter three only form under extreme pressures. 43At atmospheric conditions, only a, b, and g will form. 43The g polymorph is the thermodynamically stable form of glycine, while aand b-glycine exist as metastable and unstable polymorphs, respectively. 44However, the nucleation of g-glycine is kinetically hindered when compared to that of a-glycine, meaning that under agitation at atmospheric conditions, a-glycine is the initially forming polymorph.The solution-mediated phase transformation of glycine occurs slowly, taking around 20 hours to begin. 45There have been numerous studies into the effects of process conditions on the polymorphic outcome of glycine crystallisation, which have identied factors such as humidity, 46 pH, 47 the usage of microdroplets, 48 and the addition of precipitants such as sodium chloride. 45It is possible to distinguish between the a and g forms visuallya-glycine forms as rod-like monoclinic crystals belonging to the space group P2 1 /n, 49 whereas g-glycine forms trigonal pyramidal crystals belonging to P3 1 or P3 2 . 50ecently, Vesga et al. 51 showed that for the cooling crystallisation of aqueous solutions of glycine, agitation was a critical factor in determining the polymorphic outcomethe presence of agitation led to the preferential formation of a-glycine, while g-glycine was able to nucleate under quiescent conditions. 51They also showed that the degree of supersaturation inuenced the polymorphic outcome, with higher concentrations leading to the co-existence of both a-glycine and gglycine.
Diglycine was selected for a similar reason as it is the simplest dipeptide, consisting of two glycine residues.There have been comparatively fewer studies on the crystallisation of diglycine, but it has been shown to exhibit three polymorphs, a, b, and g. 52 However, the b and g polymorphs are notoriously elusive, requiring many recrystallisations to form. 52As well as this, the a-polymorph is the thermodynamically stable form, meaning that diglycine does not suffer from the same polymorphic issues as glycine. 53As a result, the a-polymorph is the most extensively studied polymorph as it forms most readily.The a-polymorph forms plate-like crystals belonging to the space group P2 1 /c. 54his work further explores the hypothesis that the heterosurface (glass beads) will enhance the nucleation rates of the amino acid (glycine) and dipeptide (diglycine) through heterogeneous nucleation.Previous work from the authors revealed that functional group complementarity between the heterosurface and the nucleating solute encouraged hydrogen bond formation between them, resulting in the heterogeneous nucleation of the solute.However, their work was only restricted to small organic drug molecules.The aim of this work is to advance the hydrogen bond complementarity and hydrogen bond lifetime hypotheses to small biomolecules, such as amino acids and dipeptides.The goal is to crystallise glycine and diglycine in the absence and presence of silica at different supersaturations to calculate heterogeneous nucleation parameters and to use the classical nucleation theory (CNT) to deduce the kinetic parameters of nucleation for each.The possible knowledge of hydrogen bonding behaviour between heterosurfaces and the crystallising solute, alongside appropriate crystallisation conditions, will result in the generation of crystals with the desired size, morphology, and polymorphic outcome.

Materials
Glycine (Gly, >99%, suitable for electrophoresis) and diglycine (Digly, >99% by titration) were supplied by Sigma-Aldrich and used without further purication.Deionized water was generated by a PURELAB Chorus 1 water purication system (ELGA LabWater (High Wycombe, UK)).Glass beads with an average diameter of 50 mm were supplied by Sigmund Linder GmbH (Warmensteinach, Germany).The molecular structures of the heterosurface and the two organic compounds are presented in Fig. 1.
Fig. 1 Chemical structures of the heterosurface (glass beads) and the two organic compounds (glycine and diglycine) used in this study with the number of hydrogen bond donor (HBD) and hydrogen bond acceptor (HBA) groups in each structure shown.

Determination of the metastable zone width (MSZW) for the glycine and diglycine solutions in water in the absence of silica
The MSZW for both glycine and diglycine were determined using the following procedure.A known amount of glycine/diglycine was added to a lidded bottle containing 200 mL of deionised water.The solution was placed in a water bath controlled by a Grant GP200 circulator (Royston, UK) at 45 C for an hour to allow the solids to dissolve.Aer this, the solutions were checked visually to ensure the solids had dissolved and the solution was ltered via a Sartorius 0.22 mm pore size PTFE syringe lter (Göttingen, Germany) and transferred to a 250 mL glass bottle submerged in a water bath connected to a JULABO F32-GB thermoregulator (Seelbach, Germany).The thermoregulator was programmed to follow a set heating prole.Firstly, the system was held at 30 C (50 C for diglycine) for one hour to allow the system to equilibrate and to ensure solid dissolution.The temperature was then decreased to 25 C (45 C for diglycine), and the system was held for a further 30 minutes to ensure thermal equilibrium.From here, a cooling ramp of 0.1 C min À1 was imposed until the system reached 5 C. At this point, the system was maintained at 5 C for 1 hour.The onset of nucleation was determined via a Nikon D90 camera tted with an AF-S 18-105 mm lens, which took regularly timed images of the solution, and the metastable zone widths were calculated from these images.

Determination of the induction time of glycine and diglycine in the absence of glass beads at different supersaturations
All induction time experiments in the absence of glass bead particles were carried out via the methodology outlined below (see Fig. 2).A known amount of glycine/ diglycine was added to a lidded bottle containing 100 mL of deionised water.The solution was placed in a water bath controlled by a Grant GP200 circulator (Royston, UK) at 45 C for an hour to allow the solids to dissolve.Aer this, the solutions were checked visually to ensure that the solids had dissolved and 50 mL of solution was then ltered via a syringe lter and transferred to a lidded 100 mL bottle.The bottle was then placed back in the water bath at 45 C for an hour to further ensure that no crystallisation or precipitation had occurred.The concentration of the ltrate was veried via a Thermo Scientic NanoDrop One C UV-Vis spectrophotometer (Waltham, Massachusetts).The bottle was then placed in a second water bath controlled by a JULABO F32-GB thermoregulator, which was set at a lower temperature to generate supersaturation.The onset of nucleation was determined via a Nikon D90 camera tted with an AF-S 18-105 mm lens, which took regularly timed images of the solution.The images were then analysed to determine the onset of nucleation.Aer nucleation had occurred, the solution was removed from the bath and the entire procedure was repeated at least three times to obtain an average value for the induction time, alongside an indication of the standard deviation.This was then repeated for two more different crystallisation temperatures, allowing for induction time measurements for three different supersaturations for both glycine and diglycine.The conditions for each experiment are given in Table 1.

Crystallisation of glycine and diglycine in the presence of glass beads at different supersaturations
The procedure of determining the induction time for the presence of glass beads is similar to that outlined in Section 2.3 (see Fig. 2).However, there are two key differences.Firstly, once the solutions were ltered, a known mass of glass beads corresponding to 50% of the maximum theoretical yield of glycine and diglycine (calculated as the difference in concentration between the initial supersaturation and the solubility) was added to the bottles containing ltered solution.Secondly, it was not possible to monitor crystallisation via visual observation methodologies, as the presence of glass beads rendered the solutions cloudy from the offset.Instead, 150 mL of the solution was collected via a syringe into an Eppendorf tube, and then ltered using 200 nm PTFE membrane syringe lters.From the supernatant, 100 mL was then diluted in 500 mL of deionised water in a second Eppendorf tube, to prevent any further crystallisation from occurring and to bring the concentration to within the limits of calibration.The concentration of the diluted sample was then characterised via a Thermo Scientic NanoDrop One C UV-Vis spectrophotometer (Waltham, Massachusetts) by collecting absorbance data at 220 nm (glycine) and 230 nm (diglycine), and the concentration of the original sample was back-calculated via a linear correlation between concentration and absorbance.This correlation was calculated by dissolving known quantities of glycine and diglycine at concentrations well below the solubility limit and measuring their absorbance at 220 nm (glycine) and 230 nm (diglycine) to form a linear calibration curve.From this, it was possible to plot desupersaturation proles for each experiment, from which the induction time could be estimated.These experiments were again repeated a minimum of three times for each of the conditions described in Table 1.
2.5.2.Microscopy imaging.Visual analysis of the systems was performed using an Olympus CX-41 microscope (Essex, UK) at magnications of 5Â and 20Â.The microscope was connected to a GT Vision GXCAM HiChrome Met display (Suffolk, UK), which allowed for digital imaging and postprocessing of the images taken.A sample of around 50 mL was taken at the end of each experiment via a syringe and dispensed onto a clear glass slide, on top of which a cover slip was placed to prevent solvent evaporation.Images were obtained to help verify the polymorphic outcome of each crystallisation experiment.

Hydrogen bond propensity calculations
To nd the most probable HBAs and HBDs in the glycine and diglycine, hydrogen bond propensities were calculated 57 using the program Mercury 3.10.1 as follows.
The training dataset for the statistical models was composed of molecules extracted from the Cambridge Structural Database (CSD) that contain all the functional groups present in the target APIs.A logistic regression was then applied to the training dataset that allowed the predictions in the form of H-bond propensities upon consideration of the environmental variables for the functional groups (e.g., aromaticity, steric density) of the target API.The H-bond propensity for a donor-acceptor pair can take a value between 0 and 1, where 0 indicates no likelihood of H-bond formation and 1 indicates that a hydrogen bond will always be found.

Determination of the MSZW
Table 1 summarizes the key process parameters used during the various crystallizations of glycine and diglycine from their respective saturated aqueous solutions in the presence of the heterosurface, glass beads.The MSZW determined during this study for Digly is included in Table 1, along with the previously reported MSZW for Gly. 55Bonnin-Paris et al. 55 showed that the MSZW of glycine aqueous solution is approximately 9 C at a cooling rate of 0.1 C min À1 .Recently, Ramakers et al. 58 studied the inuence of the addition of antisolvents such as ethanol, methanol, and dimethyl formamide to the aqueous glycine solution on the MSZW of glycine.The MSZW of glycine is narrower in aqueous solution but wider when the initial glycine solution has a greater antisolvent mass fraction.The MSZW of Digly (T sat ¼ 43 C) was determined to be 17.5 C at a cooling rate of 0.1 C min À1 .Table 1 further indicates that most of the crystallization temperatures used to generate the desired supersaturations reside inside or within 0.5 C of the MSZW based on the data presented, thereby favouring heterogeneous nucleation in the presence of glass beads at the applied supersaturations.

Induction time of glycine and diglycine in the absence and presence of glass beads
Table 2 summarises the induction time of glycine and diglycine in both the presence and absence of glass beads.The extent of the change in induction in the absence and the presence of the heterosurface between both compounds varied considerably.As such, the glass beads display a positive yet discriminating inuence over the nucleation of Gly and Digly.Table 2 also summarises the nucleation rate for both Gly and Digly in the absence and presence of glass beads (J* and J, respectively), calculated according to eqn (1), previously reported by Mealey et al., 59 along with the corresponding nucleation rate ratios (J/J*).
where J or J* is the nucleation rate (nuclei m À3 s À1 ), t ind is the induction time (s), and V is the volume of the crystallization solution (m 3 ).When comparing the nucleation behaviour of Gly and Digly, the nucleation rate ratio clearly indicates that glass beads have a pronounced effect on the nucleation of Digly compare to Gly.One of the possible reasons for this change in the nucleation rate in presence of glass beads is the abundant presence of hydrogen bond accepting oxygen atoms on the silibead molecules, resulting in hydrogen bond formation between the heterosurface and the crystallising solutes.Both Gly and Digly are rich in hydrogen bond donors (HBDs), which present complementarity to the HBA of glass beads, thereby favouring the formation of hydrogen bonds.These hydrogen bonds facilitate the formation of clusters of Gly and Digly on the surface of the glass beads, resulting in the faster nucleation.Similar results were observed by Verma et al. 40 for the nucleation of various APIs in the absence and presence of microcrystalline cellulose (MCC).It was observed that the APIs with only HBAs could easily form hydrogen bonds with the HBD sites of MCC, which resulted in the stable cluster of API molecules on the surface of MCC, therefore enhancing the nucleation rate of the APIs possessing only HBAs.Fig. 3 presents the desupersaturation prole of Gly and Digly in the presence of glass beads.Irrespective of the supersaturation, the rates of the desupersaturation of Gly and Digly aqueous solutions in the presence of the heterosurface are similar, as observed by the slope of the desupersaturation proles.The standard deviations are slightly bigger due to the stochastic nature of the solution, as the volume is only 50 mL.
Despite similar molecular functionalities on Gly and Diglyamide and carboxylic groupsthe extent of nucleation is different in each case.For example, the nucleation rate of Gly increased by more than 1.5 times at the lowest S of 1.17 in the presence of glass beads, whereas the corresponding increase in the case of Digly at its lowest S of 1.30 is more than 3-fold.This change in the nucleation rate of crystallising solute in the presence of the heterosurface, as shown in Table 2, is a clear indication that heterogeneous nucleation can be inuenced by the interplay of supersaturation and the presence or absence of a heterosurface.

Solid state characterisation of isolated solids
Fig. 4 presents the powder X-ray diffraction (PXRD) patterns of the isolated solids aer the full desupersaturation of Gly and Digly aqueous solutions in the presence of glass beads at the desired supersaturations.The PXRD patterns conrm the presence of the metastable a-form for Gly and the stable a-form for Digly in the presence of glass beads.
The solids isolated aer the homogeneous and heterogeneous nucleation of Gly and Digly were observed under an inverted light microscope.Crystals (100 mm) of the metastable a-form of Gly were observed in the presence and absence of glass beads, while crystals (100 mm) of the stable a-form of Digly were crystallised in the absence and presence of the heterosurface, as seen in Fig. 5.Most of the Gly and Digly crystals seem to be associated with the silibead micro particles, supporting the arguments made in Section 3.2.Similar nding of drug crystals attached to the heterosurface have previously been reported extensively in the literature. 14,21,22,42g. 4 Powder X-ray diffraction patterns of the composite solids isolated after the complete desupersaturation of both glycine (left) and diglycine (right) in the presence of glass beads at the indicated supersaturations, along with the glass beads and glycine and diglycine patterns.

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Effect of glass beads on the kinetic parameters of glycine and diglycine
The energy barrier associated with this desupersaturation is called the free energy barrier to nucleation ðDG * c Þ, which is mainly dependent on the temperature, supersaturation, and the interfacial energy at the crystal-solution interface.An equation for DG * c , which is based upon the classical nucleation theory, is shown in eqn (2).
Here, g is the interfacial energy, v m is the molecular volume, N a is Avogadro's constant, k is the Boltzmann constant, T is temperature (in Kelvin), and S is the supersaturation.
According to eqn (2), DG * c is inversely proportional to the supersaturation (S), therefore at low S, the DG * c is relatively large, and thus the possibility of the conversion of the solution to crystals is a rare event.A sufficiently large S is required to cross this energy barrier for nucleation to occur.The formation of nuclei has been postulated to occur either through a single step process of exchanging monomer units with other growing clusters in a structured way, or via a two-step process, whereby an unstructured aggregation of solute molecules is followed by the dened arrangement to form a crystal. 60he knowledge of supersaturation and the induction time of Gly and Digly in the absence and presence of a heterosurface were used to calculate the interfacial energy (g* and g) and the pre-exponential factor (A* and A) in the absence and presence of glass beads, respectively, for each compound.The kinetic and the thermodynamic parameters were derived using eqn (3) published by Kashchiev et al. 61 and used by the authors in the past. 14,42S) ¼ AS exp(ÀB/ln 2 S) where J(S) ¼ nucleation rate (nuclei per m 3 per s) at a given supersaturation, S, A ¼ pre-exponential factor (nuclei per m per s), S ¼ supersaturation, and where v 0 ¼ molecular volume (m 3 ) (for Gly: 9.94 Â 10 À29 m 3 and Digly: 1.45 Â 10 À28 m 3 ), 62 g ef ¼ interfacial energy of the cluster/solution interface for heterogeneous nucleation (J m À2 ), k ¼ Boltzmann constant (J K À1 ) (1.38 Â 10 À23 J K À1 ) and T ¼ crystallization temperature (K) Eqn ( 3) can be re-written as follows: Best t linear equations, as presented in Fig. 6(A) and (B), were used to calculate g* and g, and A* and A for Gly and Digly crystallization systems in the absence and presence of glass beads, respectively, plotted using eqn (5).According to previously published literature, an interface "promotes nucleation by lowering the interfacial energy of aggregates", 32,63 but interestingly the data presented in Fig. 6(C) suggests that the presence of a heterosurface does not appreciably inuence the interfacial energy of either the Gly or Digly crystallisation systems.Surprisingly, the major contributing factor to the heterogeneous nucleation of Gly and Digly is the pre-exponential factor (A*).As reported earlier, we interpret A as the number of clusters of a size less than the critical radius which form per cubic metre of supersaturated solution each second. 14

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These results are in support of the previous ndings from the group, where the presence of MCC in the crystallisation system resulted in a minimal change to the interfacial energy of the crystallising solute but instead resulted in an increase of the pre-exponential factor by 2-fold or more. 14,42Similar results were obtained by Heffernan et al., 64 where the presence of structurally similar impurities (demethoxycurcumin (DMC) and bisdemethoxycurcumin (BDMC)) in the curcumin solution did not inuence the interfacial energy of the crystallising solute, but instead changed the pre-exponential factor. 64Another report by Bodnár et al. 65 also presents similar ndings, where the presence of docusate sodium (DOSS) in a mefenamic acid solution in dimethylacetamide (DMA)-water mixtures resulted in an approximately 50% increase in the pre-exponential factor, while the interfacial energy remained uninuenced.Hence, the nucleation of organics in the presence of heterosurfaces in either a dissolved or undissolved state is inuenced by the change in the pre-exponential factor, while the interfacial energy remains uninuential.

Analysis of the intermolecular interactions
The presence of hydrogen bond donor (HBD) or acceptor (HBA) sites on the organic molecules allow them to form inter-or intra-molecular hydrogen bonds.This hydrogen bonding network has been proven to inuence the crystallisation kinetics, 14,22 as well as providing thermodynamic stability to the growing crystals. 66The crystallising molecules in this study, glycine and diglycine, possess 1 and 2 HBDs, respectively, and 2 and 3 HBAs, respectively, as shown in Fig. 1 and Table 3.To quantify the contribution of HBAs and HBDs to the formation of the crystal structures of Gly and Digly, the logit hydrogen-bonding propensity (LHP) model was used, based on the statistical analysis of hydrogen bonds in the Cambridge Structural Database (CSD). 57The hydrogen bond propensity (p) can be dened as a probability measure of an intermolecular hydrogen bond forming between two molecules within a crystal structure, with the assumption that the strongest possible donor-acceptor pairs will form this bond. 57The values of p for Gly (0.98) and Digly (0.87) are reported in Table 3, suggesting that Gly has a slightly higher affinity to form hydrogen bonds with itself compared to Digly.This result is consistent with other work on glycine polymorphism in aqueous solutions, which has shown that a-glycine is constructed from zwitterionic dimers, 47 and that the presence of glycine dimers in solution (as a result of hydrogen bonding) is conducive to the preferential formation of a-glycine over g- glycine. 67Comparing the values of p with the nucleation rate ratio (J/J*), suggests that the solute with a larger value of p resulted in a lower nucleation rate ratio, and vice versa.Both Gly and Digly can form hydrogen bonds with the glass beads due to hydrogen bonding complementarity between the crystallising solute and the heterosurface.Considering the cumulative number of HBDs and HBAs on Gly and Digly, and the value of p, suggests that the amide (-NH 2 ) and carboxylic groups (-C]O) of Digly have a higher propensity to form hydrogen bonds with the oxygen molecules (-O-) of the glass beads compared to Gly.This, therefore, results in the comparatively faster nucleation rate of Digly to that of Gly in the presence of glass beads.Heterogeneous nucleation is governed heavily by the lifetimes of the hydrogen bond interactions, as reported in past literature. 14,22,40,42Previous literature has reported that the average lifetime of single molecules attached to a surface via hydrogen bonding ranges from <10 ns to <70 ns.A previous paper from the authors suggests that the lifetime of the hydrogen bond between a risperidone molecule and the surface of microcrystalline cellulose is about 30 ns, as computed by molecular dynamics (MD) simulations. 40Table 3 also presents the time required (t m ) to add a single molecule to growing crystals of either Gly or Digly.These values were calculated from the growth phases of Fig. 3 for each of the APIs using eqn (6). 14 where t g is the time required for a single crystal to grow to a certain size (s) (obtained from the growth phases of Gly and Digly in Fig. 3), M w is the molecular mass of the crystallising solute (g mol À1 ), r solid is the density of the crystallising solute (g m À3 ) (for Gly: 1.3 Â 10 6 g m À3 and Digly: 1.5 Â 10 6 g m À3 ), 62 V p is the volume of the solute particle (m 3 ) calculated using the particle diameter from the microscope images (particle diameter for Gly: 100 mm and Digly: 100 mm), and N A is Avogadro's constant (6.023Â 10 23 mol À1 ).
The times required to add a single molecule to a growing crystal of Gly and Digly are 0.66 ps and 2.01 ps, respectively, which are much smaller than the average lifetime of the hydrogen bond (30 ns) between a molecule of the crystallising solute and the heterosurface.This also conrms that Gly is a fastnucleating molecule compared to Digly.Alternatively, the adsorbed solute molecule can exist attached to a surface for a time scale which will allow the attachment of multiple solute molecules from the solution phase and facilitate the formation of stable nuclei and eventually fully grown crystals.However, in the solution phase, solute-solute interactions are much shorter-lived, increasing the difficulty of nucleus formation, as reported previously. 22Therefore, the modest enhancement in the nucleation rate of Gly could be explained using the "lifetime effect", where once a Gly molecule adheres to the silibead surface via hydrogen bond formation, it remains adhered for a long enough period for other Gly molecules or clusters to attach to this silibead-bound Gly molecule.This would result in stable nuclei which subsequently grow into a well-formed crystal.Therefore, the heterogeneous nucleation observed for Gly in the presence of glass beads is postulated to be due to the "lifetime effect", whereas for Digly it is the combination of the "lifetime effect" and molecular complementarity.

Conclusions
The homogeneous and heterogeneous nucleation of glycine and diglycine in the absence and presence of glass beads were performed from aqueous solutions.It was observed that there was a modest increase in the nucleation rate of Gly compared to Digly.This conrms that glass beads displayed a positive yet discriminating effect on the heterogeneous nucleation of Gly and Digly.PXRD and microscope images conrmed the presence of the a-forms of both Gly and Digly aer both the homogeneous and heterogeneous crystallisation processes.Experimental data suggests that the presence of glass beads did not dramatically inuence the interfacial energies of Gly and Digly, but instead increased the preexponential factor by a factor of at least 2. Furthermore, molecular functionality analysis, hydrogen bond propensity calculations, and the time required to add a single molecule to the growing crystal of Gly and Digly all suggest that the heterogeneous nucleation of Gly is the consequence of the hydrogen bond "lifetime effect", whereas for Digly it is a combination of this effect as well as molecular complementarity.

Fig. 2
Fig. 2 Schematic representation of the experimental set-up.

Fig. 5
Fig. 5 Microscope images of the solids isolated after the homogeneous and heterogeneous nucleation of glycine and diglycine in the absence and presence of glass beads.
Fig.6(C) suggests that the presence of glass beads coincides with an increase in the preexponential (A*) factor of ca.2-fold for the crystallization of Gly and Digly.

Fig. 6 (
Fig.6 (A and B) Plots of ln(J/S) or ln(J*/S) against (1/(T 3 ln 2 S)) Â 10 6 , which illustrate the dependence of the nucleation rate on supersaturation for the crystallization of glycine (A) and diglycine (B) in the absence (blue diamond) and presence (red square) of glass beads at different supersaturations; the lines shown are the best linear fits and include the respective line equations and R 2 values.(C) Interfacial energies and pre-exponential factors for the crystallization of glycine and diglycine from aqueous solutions in the absence and presence of glass beads, and the related nucleation rate ratios at the lowest supersaturations.

Table 1
Complete list of crystallisation temperatures and associated supersaturations, along with the solubility and metastable zone width information used in this study, n $ 3; MSZW: metastable zone width

Table 2
Summary of the induction times, the nucleation rates, and the nucleation rate ratios for the crystallization of glycine and diglycine from supersaturated aqueous solutions in the absence and presence of glass beads at various supersaturations

Table 3
Summary of the number of hydrogen bond donor and acceptor groups of glycine and diglycine with their hydrogen bond propensity (p), the time required to add one molecule to a growing crystal (picoseconds), and nucleation rate ratio