Stochastic Nucleation of Polymorphs: Experimental Evidence and

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Stochastic nucleation of polymorphs: experimental evidence and mathematical modelling Giovanni Maria Maggioni, Leonard Bezinge, and Marco Mazzotti Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.7b01313 • Publication Date (Web): 08 Nov 2017 Downloaded from http://pubs.acs.org on November 9, 2017

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Crystal Growth & Design

Stochastic nucleation of polymorphs: experimental evidence and mathematical modelling Giovanni Maria Maggioni, Leonard Bezinge, and Marco Mazzotti∗ Separation Processes Laboratory, ETH Zurich, Zurich E-mail: [email protected] Phone: +41 44 632 24 56. Fax: +41 44 632 11 41

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November 7, 2017

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Abstract

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In this work, we investigate the primary nucleation of isonicotinamide, i.e. a com-

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pound exhibiting several different polymorphs, from ethanol at several supersaturations

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and at isothermal conditions. Experiments have been performed over a broad range of

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supersaturation and in two different volumes, collecting hundreds of data points. These

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data show that not only the detection times, but also the polymorphs observed in the

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system in different repetitions of the same experiments are statistically distributed.

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By using the wealth of experimental data produced and building upon previous

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theoretical work, we develop a simple stochastic model of primary nucleation from

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solution for a compound forming an arbitrary number of polymorphs. The model

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shows that, if the polymorphism of the compound cannot be monitored, the primary

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nucleation rate estimated from the experiments represents only an apparent nucleation

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rate. Furthermore, the first polymorph nucleating in the system is not expected to be

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a function of the system size, but only of its concentration and temperature.

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The model also allows one to perform a semi-quantitative analysis of the system,

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identifying the regions where each polymorph preferentially forms.

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1

Introduction

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The formation of a new crystal from a clear solution is an activated, rare event, hence primary

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nucleation is a stochastic process 1,2 and nucleation times measured in repeated experiments

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at the same conditions are distributed, e.g. according to a Poisson statistics 3–5 . Utilising

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experimental protocols that have been discussed in the literature 3,5–7 , empirical cumulative

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distributions of nucleation (or detection) times can be determined and used to estimate the

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associated nucleation rate. Note that new crystals are detected only after nuclei have evolved

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into fully grown crystals 8–11 .

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For substances that exhibit different polymorphs, there is the need of establishing which

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polymorph has been formed and of attributing the estimated nucleation rate to the appro-

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priate polymorph, which is challenging. The Ostwald’s rule of stages, stating that crystals of

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the least stable polymorph nucleate first and then possibly transform via a solution-mediated

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mechanism towards more stable forms, was believed to apply 12,13 , while it was understood

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that in a solution at a specific solute concentration the polymorph that forms first is that

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with the largest nucleation rate among those for which the solution is supersaturated, i.e.

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often, but not always, the least stable polymorph 14–16 . In a deterministic framework, both

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interpretations take for granted that at a given supersaturation and temperature there is

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one polymorph that, in repeated nucleation experiments, forms always first.

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Considering nucleation in a stochastic framework, as we should do, leads to questioning

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the conclusion above. If nucleation times are distributed, even if one polymorph has a

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shorter average nucleation time than another, it might well be that in a specific experiment

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the nucleation time of the former is longer than that of the latter; hence in that specific

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experiment, the generally slow-nucleating polymorph nucleates before the generally fast-


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nucleating polymorph does. Considering that the average nucleation time of a polymorph

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and the variance of the distribution of nucleation times scale with the reciprocal of the

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nucleation rate and of the system size, such effects are supposed to be more or less observable

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depending on the experimental conditions (supersaturation and temperature) and on the size

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of the crystalliser. Furthermore, it could also happen that the polymorph that forms first

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transforms into a more stable one before being detected (or before being collected at the end

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of the experiment for off-line analysis).

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Experimental evidence concerning the stochastic nature of nucleation of polymorphs has

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been reported in recent years 17–19 . Among others, Kulkarni et al. 20,21 and Caridi et al. 22

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studied the nucleation of isonicotinamide (INA), of which at least three polymorphs are

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known, in various solvents and solvent mixtures. They reported that when INA is in ethanol

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(EtOH), methanol (MeOH), and 2-propanol (IPA) only the most stable form (Form II, in

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their work) nucleated, whereas in nitrobenzene and nitromethane the Form IV and the Form

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I nucleated, respectively. Kulkarni and co-workers 8 studied also the crystallisation of INA

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in two solute mixtures: in the case of ethanol/nitrobenzene, they reported the formation of

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both Form II and Form I, while in the case of ethanol/nitromethane of both Form II and

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Form V. These authors observed that only one polymorph formed when crystallisation was

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performed in small volumes (3 mL). The polymorph forming in each individual experiment

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appeared to be random, while the occurrence frequency of the polymorphs depended on

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the initial concentration. On the contrary, in large volume experiments they observed solid

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mixtures of multiple polymorphs. In the same work 8 , Kulkarni et al. reported that also 4-

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hydroxyacetophenone in ethyl acetate yielded different polymorphs in different experiments.

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In this system, though, liquid-liquid phase separation was reported, which may admittedly

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have affected the nucleation process. Hansen et al. 17 investigated the crystallisation of a

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non-steroidal drug, piroxicam, in acetone and acetone/water mixtures with several additi-

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ves. They reported that piroxicam crystallised either Form I, or Form II, or a mono-hydrate

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crystal, in a random way and with an occurrence frequency depending on the initial solute



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concentration. To interpret their data, Kulkarni et al. 8 , as well as Hansen et al. 17 , postulated

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the stochastic formation from the clear solution of a first single nucleus, dictating the evo-

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lution of the system in small volumes. In larger systems, a more conventional poly-nuclear

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mechanism would act, resulting in mixtures of multiple solid forms. The formation of diffe-

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rent polymorphs at the same conditions randomly, rather than in the deterministic way pre-

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dicted by the classical theory, have also been recently reported for other systems 5,23,24 , such

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as p-ABA in water/EtOH mixtures 18,19 , L-glutamic acid in water 24 , and m-hydroxibenzoic

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acid in 1-propanol 25 . Furthermore, Cui et al. 26 have observed that either form α, or form

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β, or a mixture of the two crystallised on a functionalised substrate via contact secondary

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nucleation, which is an activated mechanism 27 similar to primary nucleation. It is also worth

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noting that Hansen et al. 28 in a recent study have further investigated the crystallisation of

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INA in different solvents (MeOH, acetonitrile, acetone, ethyl acetate, and dichloromethane)

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and have concluded that the polymorphism of INA depends not only on the solvent chosen,

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but also on the concentration and on the temperature at which crystallisation is carried out.

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Recently, the possibility of modelling these phenomena has been demonstrated using an

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approach based on the description of nucleation as a Poisson process 29,30 . By expanding

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on results that we have presented recently 29 , in this work the stochastic nucleation of poly-

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morphs is studied both experimentally (crystallising INA in ethanol) and theoretically. The

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paper is structured as follows: experimental methods and experimental results are reported

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in Sections 2 and 3; the stochastic model is developed, discussed, and utilised in Section 4;

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conclusions are drawn in Section 5.

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2

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2.1

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Isonicotinamide (INA), purity ≥99%, was purchased from Sigma-Aldrich; ethanol (EtOH)

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from Fluka (Ethanol Laboratory Reagent, absolute, ≤ 99.5%) and EMD Millipore (Ethanol

Experimental Materials


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Absolute For Analysis Emsure Acs, Iso, Reag. Ph. Eur.). Prior to crystallisation experi-

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ments, the solution has been filtered through 13 mm syringe filters (PTFE Hydrophobic,

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non-sterile, Pore Size 0.22 µm); INA was used as received. Different polymorphs of INA are

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known and their nomenclature in the literature 20,28 appears somewhat confusing. In this

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work, we have labelled the three polymorphs in order of decreasing stability at 298 K in

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EtOH as α, β, and γ. They correspond to the forms labelled as EHOWIH01, EHOWIH02,

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and EHOWIH04 in the CSD 20,28 , respectively. The solubilities of the polymorphs have been

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measured gravimetrically and their values are reported in Table 1. Note that, despite the

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overlap of the confidence intervals at 298.15 K, the form α is indeed the most stable, as we

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could independently verify in ad hoc experiments (see also Section 3.3). We have assumed

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the ratio of activity coefficients γ(T, c)/γ(T, c∗ ) ≈ 1, hence the supersaturation Sj for each

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polymorph has been expressed as:

Sj (T, c) =

c c∗j (T )

j = α, β, γ

(1)

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where c is the concentration of INA in EtOH and c∗j the solubility of the polymorph j,

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expressed in grams of solute per kilogram of solvent [g/kgs ]. Unless explicitly indicated

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otherwise, in the following, with the generic term “solubility” and “supersaturation”, we

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refer to those of the stable polymorph, i.e. c∗α and Sα . The crystallisation experiments have

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been performed in a range of supersaturations between Sα = 1.10 and Sα = 2.10.

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2.2

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Cooling crystallisation experiments of INA in EtOH were performed at a constant tempe-

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rature of 298.15 K; due to the accuracy of the thermocouples, this temperature as well as

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the other temperatures reported in the following, are accurate within 0.5 K. The crystalli-

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sation experiments were performed with an amount of solvent equal either to 1.40 g (small

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size experiments) or to 50 g (large size experiments), the former in a Crystal16 (Technobis

Methods



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Table 1: The values of the solubilities c∗j , of the three polymorphs of INA in EtOH, measured gravimetrically at different temperatures. T [K] 293.15 298.15 303.15 308.15 310.15 316.65

c∗α 79.02 ± 0.60 93.41 ± 1.00 138.4 ± 0.70 170.1 ± 0.30

c∗β c∗γ [g/kgs ] 80.69 ± 3.00 85.22 ± 2.20 95.75 ± 3.20 100.6 ± 2.20 115.1 ± 1.90 117.1 ± 4.05 134.4 ± 3.00 -

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Crystallization Systems, multiple reactor set-up with 16 wells, each equipped with an inde-

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pendent thermocouple 7 ), and the latter in an EasyMax 400 (Mettler Toledo). Both devices

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were equipped with glass reactors; for the Crystal16, standard HPLC glass screw topped

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vials were used as vessels. As discussed elsewhere, the reactors were properly sealed during

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the experiments to minimise solvent evaporation, since this would change the solute concen-

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tration, hence the supersaturation 7 . We have also monitored the evaporation by weighing

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the reactors (individual vials in Crystal16 and single glass reactor in EasyMax) before and

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after each experiment: if evaporation resulted in a change of supersaturation larger than

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2%, the experiment was discarded and repeated. The vials in the Crystal16 were stirred

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with magnetic bars at 700 rpm, while the reactor in the EasyMax with a 4-blade impeller at

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400 rpm, to guarantee that the system was well-mixed. The stock solution was kept for 60

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minutes at 333 K to ensure the complete dissolution of INA and immediately thereafter used

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to fill the vials. For the large scale experiments, the solution was prepared directly inside

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the glass reactor.

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Supersaturation was induced by cooling from different saturation points, and multiple

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cycles were performed for both small and large size experiments. A crystallisation cycle

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consisted of 4 stages: a heating ramp from 298.15 K to 333.15 K (2.0 and 2.5 minutes, for

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Crystal16 and EasyMax, respectively), a dissolution step at 333.15 K (60 minutes, both

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devices), a cooling ramp from 333.15 K to 298.15 K (2.0 and 2.5 minutes, for Crystal16 and 6 ACS Paragon Plus Environment

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EasyMax, respectively), and finally an isothermal step at 298.15 K (its duration varied from

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10 to 180 minutes, depending on the supersaturation and the system size). The differences

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in the heating and cooling stage depended on the different configuration and controller of

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the devices used.

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2.3

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The polymorphism of the crystals produced in the experiments was assessed off-line, ex situ

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through XRD measurements (experiments in Crystal16 and EasyMax) and on-line, in situ

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through Raman spectroscopy (RA 400 Raman spectrometer from Mettler Toledo, for the

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experiments in the EasyMax). XRD and Raman spectra of the pure polymorphs are shown

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in Figure 1a and 1b, respectively.

Measurement protocols

(a)

(b)

Figure 1: Characterisation of the polymorphs. (a) The XRD spectra of the powder of α, β, and γ; the bands in blue, magenta, and yellow, highlight the characteristic peaks of the three polymorphs, respectively. (b) The Raman spectra of pure α, β, and γ in ethanol, in blue, red, and yellow, respectively.

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Three characteristic peaks for each polymorph, i.e. peaks which are exhibited by that 7 ACS Paragon Plus Environment


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polymorph but not by the others, have been highlighted in Figure 1a as vertical bands in blue

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(2θ = 17.8, 26.0, 31.0), red (2θ = 14.5, 23.1, 32.2), and yellow (2θ = 23.9, 26.9, 37.0), for α, β,

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and γ, respectively. Note that the peak at 2θ = 23.4, though very prominent for polymorphs

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α and γ, is no selected because of its ambiguous attribution. For the experiments in the

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Crystal16, we measured the XRD spectra only of the crystals produced during the very last

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cycle of a series of experiments using the same solution.

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The Raman spectra of the pure forms in EtOH are reported in Figure 1b and are consis-

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tent with those given by Kulkarni et al. 20 . The three polymorphs can be clearly distinguished

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in two intervals of the Raman spectrum, namely 980 − 1010 cm-1 and 1200 − 1230 cm-1 , as

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observed in Figure 1b, where the characteristic peaks of α, β, and γ are highlighted by the

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blue, red, and yellow bars. Raman spectra were collected in the range of Raman shift from

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100 to 3600 cm-1 , with a resolution of 1 cm-1 and averaged over 10 scans using an exposure

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time of 6 s, corresponding to one measurement per minute. The raw data from Raman

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measurements have been treated according to standard procedures for spectroscopic data

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pre- and post-processing: the signal has been smoothed with a Savitzky-Golay filter 31 and

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a linear baseline correction 32 has been applied to the spectra. In the EasyMax, at least nine

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cycles for each supersaturation Sα have been performed, thus obtaining a number of repeti-

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tions comparable with those (sixteen) obtained from the last cycle of the experiments in the

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Crystal16. We used Raman spectroscopy also to verify that the system attained unbiased

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initial conditions during the dissolution step after each crystallisation step. The complete

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disappearance of the peaks associated to the solid forms of INA within 5 minutes from the

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beginning of the dissolution stage and the reproducibility of the Raman spectra of the clear

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solution at high temperature in different cycles indicated that the initial conditions of the

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system were the same at the beginning of each crystallisation experiment.

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Several XRD spectra measured from samples of small scale experiments were markedly

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different from those of pure polymorphs. Since solvates of INA have been reported for

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crystallisation from water 33 , we have taken care of excluding the presence of solvates or


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solvent inclusions by means of two different tests. The first test consisted in repeating

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the XRD measurements on the same samples after 5 days, during which the crystals were

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kept in a dry environment: all spectra were unchanged. The second test was a thermal

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gravimetric measurement, with temperature up to 500 K, i.e. well beyond the melting

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temperature of any possible solvent, of all known polymorphs and of any solvates. The

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thermal gravimetric analysis showed no loss of material, hence indicating that no ethanol

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was trapped in the solids and also pointing out that the spectra exhibiting characteristic

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peaks of two or three polymorphs were originated by a solid mixture of pure crystals (i.e., a

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mixture of polymorphs).

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Detection times, tD , were also recorded during all experiments (see Figure 2). For the

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Crystal16 experiments, the detection time was defined as the difference in time between the

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attainment of the isothermal temperature and the decrease of light transmissivity below a

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given threshold. Following the same protocol discussed in detail elsewhere 7 , we retained only

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the detection times in the small scale experiments whose temperature was within a band of

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±0.5 K from the set value of 298.15 K.

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For the EasyMax experiments, the detection time was defined as the difference in time

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between the attainment of the saturation point and the time when a characteristic peak of

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Ethanol (at wavelength 883 cm-1 ) decreased below a threshold value with respect to the peak

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maximum in the dissolved state before cooling (see Fig. 2). This unconventional choice of

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defining the detection condition with respect to a property of the liquid phase, rather than

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to one of the solid, has been based on the fact that the intensity of this Ethanol peak was

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always quite strong and clearly weaker upon the formation of crystals. On the contrary, the

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peaks associated to the solid forms of INA were initially quite weak at low supersaturations,

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becoming more visible only upon significant crystal growth. For this reason, we opted to

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define detection with respect to the Ethanol peak at 883 cm-1 . We note that nucleation (and

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detection) can occur also during the cooling ramp, as soon as Sj > 1: this can be more

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likely observed in systems at high supersaturations and/or of large size, since the average



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nucleation time is inversely proportional to the system size and the nucleation rate, the latter

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increasing with increasing Sα 4,10 . Indeed, in the large size experiments detection occurred

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always during the cooling ramp for all experiments with Sα > 1.40. A direct quantitative

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comparison between the values of the detection times measured in the Crystal16 and those

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measured in the EasyMax is not possible, since different detection techniques and detection

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conditions were applied in the two devices.

Figure 2: An illustration of detection times based on the temperature profile (top) and on the Raman intensity I (bottom), measured for a peak typical of ethanol, for the case with S = 1.30. T ∗ indicates the saturation temperature of the most stable form.

207

208

3

Experimental results

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We have performed two sets of nucleation experiments, which provide different, complemen-

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tary pieces of information. The Crystal16 allows, on the one hand, to gather a large number

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of detection time data, but cannot monitor which polymorph is associated to such events

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and the XRD measurements can be obtained only from the samples collected after the final

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cycle of a series. On the other hand, the EasyMax has a much lower productivity in terms

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of number of experiments, but it allows to measure online the time-resolved Raman spectra. 10 ACS Paragon Plus Environment

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3.1

Detection times

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Figures 3a (Crystal16) and 3b (EasyMax) illustrate, for each experimental supersaturation

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(horizontal axis) the detection times measured in tens or hundreds of repeated experiments

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(vertical axis). The detection times are clearly distributed and their spread reflects the

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stochastic nature of primary nucleation, which has already been reported for several sys-

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tems 3,4,9,34–36 . In Figure 3a, tD varies from a minimum of about 1 min to a maximum of

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about 5 hours at Sα = 1.20, and from a minimum of about 1 minute to a maximum of about

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30 minutes at Sα = 2.10. In Figure 3b, for the same supersaturation values, the maximum

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tD decreases to about 10 and 4 minutes, at Sα = 1.20 and Sα = 2.10, respectively. This

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sharp decrease of the broadness of the distribution observed in the data when increasing

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the supersaturation, from Sα = 1.20 to Sα = 2.08, or the system size, from 1.40 to 50 g

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of solvent, is also consistent with the statistics associated to the measurement of stochastic

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processes 3,4 .

(a)

(b)

Figure 3: The detection times measured from the experiments in the Crystal16 (Figure 3a) and in the EasyMax (Figure 3b) at different supersaturation levels. M is the mass of ethanol used in the experiments.

228



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Since nucleation is stochastic, the detection time tD is a stochastic variable (as illustrated in Figure 3), distributed according to an intrinsic underlying cumulative probability function, F (tD ). For each supersaturation, we have performed N experiments and measured n detection times (n ≤ N , since not all runs produce crystals during the duration of the experimental observation), with which we form the ordered vector tD : tD = [tD,1 , tD,2 , ..., tD,n ]T : 0 ≤ tD,1 ≤ tD,2 ≤ ... ≤ tD,n < +∞

(2)

through which the following empirical cumulative distribution function (eCDF) FN∗ (tD ) is defined as: FN∗ (tD )

n 1 X = 1[t ,+∞) (tD ) N j=1 D,j

(3)

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where 1[a,+∞) is the indicator function on the interval [a, +∞), i.e. equal to 1 if a ≤ tD < ∞,

230

and 0 otherwise. The vector tD constitutes a discrete, finite sample of the stochastic variable

231

tD , whereas FN∗ (tD ) is an estimate of F (tD ).

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3.2

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Figure 4a (Crystal16) and Figure 4b (EasyMax) demonstrate that, in repeated experiments

234

carried out at exactly the same conditions from identical initial solutions, different polymor-

235

phs are obtained first.

Polymorphic form

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This is illustrated in Figure 4a through XRD spectra of the final crystals obtained in

237

four different experiments at the same supersaturation: for each experiment, we have indi-

238

cated the polymorph to which the crystal powder has been attributed. The XRD spectrum

239

of experiment E4 clearly belongs to the α polymorph. The characteristic peaks of the β

240

polymorph are present in the XRD spectra of experiments E1, E2, and E3, although with

241

different intensities. Experiments E1 and E3 also present two peaks characteristic of γ, while

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E2 two characteristic peaks of α, i.e. experiments E1, E2, and E3 seem to be mixtures of

243

different polymorphs. However, it is evident that the solid products obtained in different 12 ACS Paragon Plus Environment

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experiments are different, even though they have been produced at the same conditions and

245

from the same stock solution.

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The samples exhibiting mixed mixed XRD spectra, such as those of E1, E2, and E3

247

in Figure 4a, were analysed again after 3 and 6 days; the spectra did not change, further

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confirming that the product crystals did not contain any trace of solvent. As mentioned in

249

Section 2.3, we have also verified by thermal gravimetric analysis that the extra peaks could

250

not be related to solvates. This analysis is also confirmed by observing the XRD spectra

251

of the powder obtained from experiments at other supersaturations: examples of such XRD

252

spectra at four different supersaturations are reported in Figures 10 and 11, in Appendix ??.

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Figure 4b shows at each supersaturation the relative occurrence Q of the three poly-

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morphs (i.e., the number of experiments in which a specific polymorph was observed, as

255

a percentage of all experiments performed at that specific supersaturation level), as obtai-

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ned from the Raman spectra both at detection (large vertical bars) and at the end of the

257

experiment (thin vertical bars). The Raman spectra of the experiments performed at four su-

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persaturations, used to construct Figure 4b, are reported in Figures 12 and 13, in Appendix

259

??.

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The mixed-colour striped bars in Figure 4b indicate the experiments in which two different

261

polymorphs were observed together. Depending on the supersaturation, between 10% (e.g.

262

Sα = 1.10 and 1.20) and 50% (e.g. Sα = 1.50) of the experiments exhibit mixed Raman

263

spectra, which are associate to the presence of multiple polymorphs. The β and γ forms

264

appear typically together for Sα > 1.50, whereas the α form appears with β for the lowest

265

supersaturation, Sα = 1.10.

266

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Two remarks are worth making. First, the distribution of relative occurrences changes

268

markedly from one supersaturation to the next; we will provide a justification thereof in

269

Section 4, based on the different supersaturation dependence of the nucleation rate of the



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(a)

Page 14 of 31

(b)

Figure 4: Evidence of polymorph formation in Crystal16 (Figure 4a) and in EasyMax (Figure 4b). (a) XRD spectra at Sα = 2.00 obtained from four different repetitions (labelled as E1, E2, E3, E4) of the same typical nucleation experiment. Depending on the vial analysed, either one of the three forms (or possibly a mixture) is observed. The colours of the vertical bands associated to the peaks typical of each polymorph, as in Figure 1: blue, red, and yellow refer to α, β, and γ, respectively. (b) Relative occurrence of the three polymorphs in the experiments carried out at different supersaturation. Blue, red, and yellow bars refer to α, β, and γ polymorphs. Mixed-colour striped bars in Figure 4b indicate the experiments in which two different polymorphs were observed together. The thicker bars indicate the relative occurrence at detection (see the detection times in Table ??, Appendix ??), while the thinner bars with lighter colours indicate the relative occurrence at the end of the experiment. 270

different polymorphs. Secondly, the distribution of relative occurrences changes from the

271

time of detection to the end of the experiment, because of polymorph transformation, as

272

discussed in Section 4.

273

274

Figure 5 illustrates the distribution of detection times as observed in the same experiments, by plotting the corresponding empirical cumulative distribution functions (eCDFs).

275

Figure 5a shows the eCDFs (as black lines) of the data obtained in the Crystal16 at

276

Sα = 1.30, 1.58 and 1.91, as computed using Eq. (3); for each supersaturation level, the

277

experiments for which the XRD spectra were also measured are represented with a coloured

278

symbol, in blue, red, or yellow to indicate that polymorphs α, β, or γ, respectively, has

279

nucleated first. It is apparent that all polymorphs have distributed detection times, and

280

any among the three can form first. Figure 5b shows the eCDFs of the data measured in 14 ACS Paragon Plus Environment

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281

the EasyMax experiments at Sα = 1.30 and 2.10, after analysing the Raman spectra at

282

detection. The black eCDF is that of the detection times, when ignoring which specific

283

polymorph was detected, i.e. the same type of curve plotted in Figure 5a (but with a

284

much smaller number of data points). The coloured eCDFs are those of the individual

285

polymorphs, where the horizontal coordinate, i.e. the detection time, is the same as that of

286

the corresponding point on the black eCDF, whereas the vertical coordinate, corresponding

287

to the cumulative probability, attains a smaller value because it is calculated on the data

288

points of that polymorph only.

(a)

(b)

Figure 5: (a) The empirical cumulative distribution functions for experiments at S = 1.30, 1.58, and 1.91 in the Crystal16; the symbols represents the samples for which the XRD spectra have been measured. The blue, red, and yellow symbols refer to α, β, and γ, respectively. (b) The empirical cumulative distribution functions for S = 1.30 and 2.10 obtained from identifying the polymorph nucleating with time-resolved Raman spectroscopy in the EasyMax, where a much smaller number of points was available. The coloured empirical cumulative distributions represent the frequency of the individual polymorph (see also Section 4.1).

289



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290

3.3

Solid phase transformation

291

Figure 4b provides evidence that new crystals, after forming in a certain polymorphic form,

292

whose relative occurrence depends on supersaturation, may transform into a more stable

293

form, possibly through a solvent mediated transformation that has been well documented

294

by several other studies 18,20 , including a few from our group 37,38 . While Figure 4b provi-

295

des information about the polymorphic form at two points in time, namely detection and

296

experiment end, and proves that less stable forms may evolve into more stable ones, time

297

resolved Raman spectra, as those shown in Figure 6, allow to monitor the dynamics of the

298

whole transformation with great clarity.

299

More specifically, Figure 6a shows the transformation from polymorph γ to α occurring

300

at Sα =1.30, and Figure 6b shows that from polymorph γ to β occurring at Sα = 1.90; in

301

both cases seven spectra are shown, namely from detection to 30 min after that, with each

302

spectra taken five minutes after the previous.

303

A few remarks are worth making. First, the transformation in Figure 6a is possibly not

304

yet completed, as the characteristic peak of polymorph γ is still visible in the last spectra

305

recorded. This indicates that at these conditions the γ to α transformation takes more

306

than thirty minutes. Secondly, in the case of Figure 6b the transformation of form β to

307

form α is not visible, but it indeed occurred as we could observe after six hours during

308

which the suspension of β crystals had been left undisturbed. Incidentally, we note that the

309

transformation from the β to the α form also proves that α is the most stable form observed

310

in our experiments. Finally, as we observe with reference to the experiment in Figure 6b

311

that the transformation from the β to the α form may take very long, we also conclude

312

that when the most stable polymorph α is present at detection (see measurements at low

313

supersaturation levels in Figure 4b), it has most likely formed via primary nucleation and

314

not through solid phase transformation from another metastable polymorph (which would

315

have taken too long to occur in the short time before detection).

316


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Figure 6: The transformation of γ polymorph, monitored over 30 minutes. (a) A direct transformation of γ into α at low supersaturations. (b) A transformation from γ into β, at high supersaturations. Typically, the β form is then stable for several hours, but transforms ultimately into the α form. 317

3.4

Discussion of experimental results

318

The experiments in the Crystal16 and in the EasyMax provide complementary information

319

and allow to identify the main features of the nucleation process, particularly the effect of the

320

system volume and of the solute concentration. The experimental data clearly show not only

321

that detection times are statistically distributed (see Section 3.1, Figure 3), but also that

322

the type of polymorph nucleating in each experiment is itself a stochastic property (Section

323

3.2, Figure 4). Moreover, the data indicate that the relative occurrence of the different

324

polymorphs depends strongly on the solute concentration (see Figure 5b). Therefore, a

325

proper understanding of this non-deterministic behaviour requires analysing the experiments

326

in the context of statistical models of nucleation 3,34,35,39 .

327

Stochastic models of nucleation for non-polymorphic systems describe the nucleation

328

time, tN , i.e. the time when the first primary nucleus forms, with S > 1, as a a stochastic

329

quantity. Even though in some cases crystal growth has been found to be itself stochas-

330

tic 40–42 , primary nucleation is considered to be the major source of stochasticity during

331

crystallisation from a clear solution, whereas growth can be considered to be deterministic,

332

and can be modelled accordingly. Unfortunately, nucleation times tN cannot be measured 17 ACS Paragon Plus Environment


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333

directly, because the nuclei are too small for being observed 3,4,9–11,35 . Consequently, one

334

measures tD and, using a model, retrieves tN , from which then the nucleation kinetics can

335

be estimated. Since nucleation times are distributed 3,9,34 , also detection times are distribu-

336

ted. A thorough and comprehensive discussion about the estimation of tN from tD and the

337

necessary assumptions behind it can be found elsewhere 3,6,9–11 .

338

4

339

4.1

340

Primary nucleation has successfully been modelled as a Poisson process 3,4,9–11,21,35,36,43 , which

341

is the mathematical formalism describing the occurrence of rare events 44 , e.g. activated

342

phenomena such as chemical reactions and nucleation 2 . It is often assumed that only the

343

first nucleus forms stochastically at the time tN , whereas the later nuclei – primary and

344

secondary alike – form deterministically 3,4,6,10,11,35 . For the sake of simplicity, but without

345

loss of generality, we consider here only unseeded isothermal crystallisation 4,10,35 . Note that

346

the primary nucleation rate J is constant in an isothermal, closed system with no crystal and

347

no chemical reactions, since the solution composition does not change until the first nucleus

348

forms. We also emphasise that assuming the first nucleation event as the only source of

349

stochasticity is a rather strong hypothesis, whose validity for non-polymorphic systems we

350

discussed in another work 10 . This hypothesis, however, allows to build a model conceptually

351

simple, computationally affordable, and semi-quantitatively accurate.

Modelling stochastic nucleation of polymorphs Mathematical model

Let us now extend the stochastic model of nucleation to a system with an arbitrary number of polymorphs (α, ..., π). Such a system can be thought of as occupying discrete states: an empty state with no nuclei (which is also the initial state of the system) and a state where a nucleus of polymorph j has formed (j = α, ..., π); hence, the presence of a nucleus of polymorph j represents a possible state different from the empty state. The (random) appearance of the first nucleus causes the irreversible transition of the system 18 ACS Paragon Plus Environment

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from the original empty state to the new state. The functions P0 (t) and Pj (t) represent the probabilities that, at time t, the system is still in the initial state and that it is in the state j, respectively. Note that the system must be in one of the possible states, therefore:

P0 (t) +

π X

Pj (t) = 1

(4)

j=α

and that the probability P (t) that a nucleus of any polymorph has formed, irrespectively of which one, is by definition of P0 (t) simply:

P (t) = 1 − P0 (t) =

π X

Pj (t)

(5)

j=α

352

The evolution of P0 (t) and Pj (t) can be described in terms of the rates of transition between

353

the possible states of the system. The rate of change of the probability Pj (t) (state with

354

one nucleus of polymorph j) is given by the probability P0 (t) (state without nuclei) times

355

the transition rate λj , while the rate of change of P0 is minus the product of P0 (t) times an

356

ˆ overall transition rate λ:  dP   ˆ 0  0 = −λP dt dP    j = λj P0 dt

357

(6) ∀j = α, ..., π

(7)

The corresponding initial conditions are:

 P0 (0) = 1

(8) ∀j = α, ..., π

P (0) = 0 j

(9)

By summing term by term all Eqs. (6) and (7), and employing the conservation of total probability given by Eq. (4), one obtains:

ˆ= λ

π X

λj

j=α


(10)


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358

359

360

Page 20 of 31

which is a physically sensible result. Based on physical arguments 35 , λj is the product of the primary nucleation rate of the polymorph j, Jj , and the system size, V , i.e. λj = Jj V ˆ = V Pπ Jj = V J, ˆ where Jˆ = Pπ Jj represents an apparent (j = α, ..., π), hence λ j=α j=α

361

nucleation rate and corresponds to what an observer incapable of distinguishing the different

362

polymorphs would measure. Solving Eqs. (6) and (7), using also Eq. (5), yields: ˆ P0 (t) = exp −λt ˆ P (t) = 1 − exp −λt Pj (t) =

λj P (t) ˆ λ

(11) (12)

∀j = α, ..., π

(13)

363

Note that Pj (t) is the so-called joint probability, i.e. the probability that at time t a nucleus

364

has formed and it is of polymorph j. Dividing Pj by the probability P that a nucleus of any

365

type has formed, yields the occurrence probability ξj :

ξj =

Jj Pj (t) λj = = ˆ P (t) Jˆ λ

∀j = α, ..., π

(14)

366

which is the conditional probability, namely the probability that, at any time, the nucleus

367

which has formed belongs to polymorph j, given that a (first) nucleus of any type has formed.

368

Note also that, for a homogeneous system such as the one considered here, ξj is a function

369

only of the nucleation kinetics.

370

Figure 7 illustrates the results obtained above for the case with three polymorphs, namely

371

j = α, β, γ. The black solid line represents the cumulative probability function given by Eq.

372

(12), while the blue, magenta, and yellow line, for form α, β, and γ, respectively represent

373

ˆ the cumulative probability functions given by Eq. (13). The probability P , associated to J,

374

behaves identically to the cumulative probability of a standard isothermal Poisson process

375

and, as expected, asymptotically approaches 1 for t → ∞. On the contrary, t → ∞ the

376

probabilities Pj (t) do not approach 1, but the asymptotic value ξj , thus reflecting the fact


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377

that a polymorph can or cannot form. Figure 7 also shows clearly that, as long as the

378

value of Jˆ is constant (see Eq. (14)), the occurrence probability of the polymorphs does not

379

depend on the nucleation time itself, as indicated by Eq. (14): to change such probability,

380

one must alter the nucleation kinetics of the polymorphs, for instance by operating at a

381

different supersaturation, or at a different temperature.

Figure 7: The cumulative probabilities for a system with three polymorphs in a system with V = 1.8 mL. The black, blue, red, and yellow lines refer to the probability of forming any polymorph, α, β, γ, respectively, and are calculated for the nucleation rates Jˆ = 490, Jα = 32, Jβ = 48, Jγ = 410 [#/s/m3 ]. The values of the nucleation rates have been computed using Eqs. (15) and (16) with the parameters reported in Table 2, at T = 298.15 K and Sα = 1.50.

382

It is worth highlighting three points. First, according to Eq.(12) all polymorphs contri-

383

bute to the total nucleation probability, P (t), through their individual intensities λj , hence

384

ultimately through Jj . Second, the polymorphs associated with higher nucleation rates are

385

more likely to nucleate, but in general they do not necessarily nucleate first in each in-

386

dividual repetition of an experiment: all polymorphs have non-zero (albeit possibly very

387

small) probability to nucleate, unless their supersaturation is below 1. Third, the system

388

size V influences the nucleation probability P , i.e. the nucleation time, but does not affect

389

the occurrence probability ξj . Thus, which polymorph nucleates depends on the conditions



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Page 22 of 31

390

of the system, but not on the system size. Finally, we can use the model to investigate

391

the influence of the operating parameters, namely the solute concentration and the system

392

temperature.

393

4.2

394

In this section, we focus on studying the effect of the solute concentration on the system; since

395

experiments were isothermal, we equivalently look at the influence of the supersaturation,

396

Sα . The primary nucleation rate of the individual polymorphs is assumed to follow the

397

expression of the Classical Nucleation Theory 1,45,46 :

Comparison between the model results and the experiments

Bj Jj (c, T ) = Sj (c, T )Aj exp − 3 2 T ln Sj (c, T )

j = α, β, γ

(15)

398

At constant temperature T , the pseudo-rate Jˆ is then a function of the actual solute con-

399

centration c (through the supersaturation, Eq. (1)) and of all individual nucleation rates of

400

α, β, and γ:

ˆ = Jα (c) + Jβ (c) + Jγ (c) J(c)

(16)

401

Equations (15) and (16) show that such model needs six parameters to describe the INA/EtOH

402

isothermal system, namely the pair (Aj , Bj ) for each polymorph, with j = α, β, γ. Deter-

403

mining these six parameters is not trivial; two points are worth mentioning. First, when

404

no information about the specific polymorph forming is available, such as during detection

405

time experiments in the Crystal16, the empirical cumulative distribution function obtained

406

from experiments allows to estimate only the value of Jˆ at the chosen supersaturation level.

407

Second, a proper estimation of the parameters would require the characterisation not only

408

of crystal growth, secondary nucleation, and dissolution, but also of the detection technique

409

itself 11,47 : such studies go beyond the scope of this contribution. For these reasons, even

410

though we have selected the parameters (see Table 2) to run our simulations based on a 22 ACS Paragon Plus Environment

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411

fitting procedure, we consider such values appropriate only for semi-quantitative studies.

412

The selection procedure is detailed in Appendix ??. Table 2: The parameters of the nucleation rate Jj selected and used in the simulations. Parameters / Form Aj [# /s/m3 ] Bj [K3 ]

α 23 475

β 17,715 2.37×107

γ 392 8.85×105

413

Let us now compare the simulations results and the experimental data. Eq. (15) indicates

414

that one can modify the occurrence probability of the polymorphs (Eq. (14)) by varying the

415

supersaturation, hence the nucleation rate; using the parameters reported in Table 2, one

416

can compute the values of ξα , ξβ , and ξγ as a function of Sα . These values are represented

417

in Figure 8 as the blue, red, and yellow solid line, for ξα , ξβ , and ξγ , respectively.

Figure 8: The occurrence probabilities ξα , ξβ , and ξγ , in blue, red, and yellow, plotted as function of Sα , while the stacked bars represents the rescaled relative occurrences Q measured in the large scale experiments and reported also in Figure 4b. 418

419

Figure 4b, on the other hand, reports the (percentage) occurrence frequency Q of form

420

α, β, and γ, at different values of Sα : one can interpret the occurrence frequency of each 23 ACS Paragon Plus Environment


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421

polymorph as an estimate of its occurrence probability. By normalising Q from Figure 4b,

422

one can then compare the values of Q with those of ξj , at each Sα , as shown in Figure 8.

423

One can readily see that the model results and the experimental measurements exhibit a

424

very similar behaviour.

425

Thus concluding, these results suggest that the model presented in Section 4.1 is not only

426

plausible, but also consistent with the data, further indicating that it indeed describes the

427

main, fundamental physical features of the system.

428

5

429

First, we make a few remarks about the deterministic, kinetic interpretation of Ostwald’s

430

rule of stages developed by Davey, Garside, Cardew and others 12–16,48 , mentioned in Section

431

1. Even though the deterministic theory cannot explain the experiments of Section 3 of INA

432

in EtOH, a deep connection exists between such theory and the stochastic model. Davey,

433

Garside and others interpreted the primary nucleation rate as a measure of how much a

434

certain polymorph j is favoured against the others in a deterministic framework. This

435

“bias” towards a specific polymorph would promote the formation of pre-critical clusters of

436

a specific configuration and would eventually lead to the formation of the polymorph with

437

the highest nucleation rate at the given conditions. In the stochastic framework, the “bias” of

438

the deterministic theory can be interpreted as the probability of the system to form a certain

439

polymorph (see Eq.(14)). In fact, the intervals of Sα where the stochastic model predicts

440

that a polymorph j is more likely to appear than the others correspond to the intervals

441

where the deterministic theory predicts that the same polymorph j nucleates always first.

442

Thus, the deterministic model can be seen as a special case of the stochastic model, in the

443

sense that its prediction corresponds to the most-likely outcome predicted by the stochastic

444

approach. Second, we note that when Jj at the given conditions is such that ξj ≈ 1, then the

445

system exhibits a pseudo-deterministic behaviour, i.e. the experiments yield almost always

Discussion and conclusions


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446

the polymorph j only. Finally, it is worth noting that, through a qualitative analysis not

447

reported here, we can show that the region where the α polymorph nucleates with higher

448

probability should become larger when T decreases; this conclusion further supports the

449

conjecture that the difference between our findings and those of Kulkarni et al. 20 can be

450

due to the different temperatures at which these authors have conducted their experiments,

451

namely 298.15 K in our case and 278.15 K in theirs.

452

Thus summarising, in this work we have presented nucleation experiments of isonicoti-

453

namide in ethanol where all polymorphs, and not only the stable one (α, form EHOWIH01),

454

have been crystallised from clear solutions. The production of the metastable β and γ forms

455

(form EHOWIH02 and EHOWIH04, respectively) has been consistently obtained in a broad

456

range of initial concentrations, and has been shown to occur also at two different system

457

sizes. The formation of the polymorphs β and γ is in contrast with one hypothesis made in

458

the literature about crystallisation of INA in EtOH 20,22 , but is consistent with the behaviour

459

of INA observed in other solvents 28 . Even more interestingly, our experiments exhibited a

460

stochastic nature in terms not only of detection times, but also of the polymorph forming

461

first. On the basis of the theoretical results concerning the stochastic nature of primary

462

nucleation, we have developed (Section 4) a simple model which naturally extends previous

463

work on stochastic primary nucleation 4,9–11,35 to a polymorphic system. Such model descri-

464

bes in a coherent manner the stochasticity of nucleation/detection times, as well as that of

465

the type of polymorph nucleating, as observed in the experiments.



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466

467

468

469

470

References (1) Kashchiev, D. Nucleation, Basic theory with applications; Butterworth Heinemann, 2000. (2) Davey, R. J.; Schroeder, S. L. M.; ter Horst, J. H. Angew. Chemie Int. Ed. 2013, 52, 2166–2179.

471

(3) Jiang, S.; ter Horst, J. H. Cryst. Growth Des. 2011, 11, 256–261.

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(4) Kadam, S. S.; Kulkarni, S. A.; Coloma Ribera, R.; Stankiewicz, A. I.; ter Horst, J. H.;

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Kramer, H. J. Chem. Eng. Sci. 2012, 72, 10–19.

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(5) Little, L. J.; Sear, R. P.; Keddie, J. L. Cryst. Growth Des. 2015, 15, 5345–5354.

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(6) Xiao, Y.; Tang, S. K.; Hao, H.; Davey, R. J.; Vetter, T. Cryst. Growth Des. 2017, 17,

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2852–2863. (7) Maggioni, G. M.; Bosetti, L.; dos Santos, E.; Mazzotti, M. Crys. Growth Des. 2017, In press.

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(8) Kulkarni, S. A.; Meekes, H.; ter Horst, J. H. Cryst. Growth Des. 2014, 14, 1493–1499.

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(9) Sullivan, R. A.; Davey, R. J.; Sadiq, G.; Dent, G.; Back, K. R.; ter Horst, J. H.;

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Toroz, D.; Hammond, R. B. Cryst. Growth Des. 2014, 14, 2689–2696.

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(10) Maggioni, G. M.; Mazzotti, M. Faraday Discuss. 2015, 179, 359–382.

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(11) Maggioni, G. M.; Mazzotti, M. Cryst. Growth Des. 2017, 17, 3625–3635.

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(12) Ostwald, W. Zeitschrift f¨ ur Phys. Chemie 1897, 22, 289–302.

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(13) N´ yvlt, J. Cryst. Res. Technol. 1995, 30, 443–449.

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(14) Cardew, P. T.; Davey, R. J.; Ruddick, A. J. J. Chem. Soc. Faraday 1984, 80, 659–668.


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(15) Davey, R.; Cardew, P.; McEwan, D.; Sadler, D. J. Cryst. Growth 1986, 79, 648–653.

488

(16) Davey, R.; Garside, J. From molecules to crystallizers; 2000; pp 1–52.

489

(17) Hansen, T. B.; Qu, H. Cryst. Growth Des. 2015, 15, 4694–4700.

490

(18) Black, J. F. B.; Davey, R. J.; Gowers, R. J.; Yeoh, A. CrystEngComm 2015, 17, 5139–

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5142.

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(19) Garg, R. K.; Sarkar, D. J. Cryst. Growth 2016, 454, 180–185.

493

(20) Kulkarni, S. A.; McGarrity, E. S.; Meekes, H.; ter Horst, J. H. Chem. Commun. 2012,

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499

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48, 4983. (21) Kulkarni, S. A.; Kadam, S. S.; Meekes, H.; Stankiewicz, A. I.; ter Horst, J. H. Cryst. Growth Des. 2013, 13, 2435–2440. (22) Caridi, A.; Kulkarni, S. A.; Di Profio, G.; Curcio, E.; Ter Horst, J. H. Cryst. Growth Des. 2014, 14, 1135–1141. (23) Bhamidi, V.; Lee, S. H.; He, G.; Chow, P. S.; Tan, R. B. H.; Zukoski, C. F.; Kenis, P. J. A. Cryst. Growth Des. 2015, 15, 3299–3306.

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(24) Jiang, N.; Wang, Z.; Dang, L.; Wei, H. J. Cryst. Growth 2016, 446, 68–73.

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(25) Liu, J.; Sv¨ard, M.; Rasmuson, ˚ A. C. Cryst. Growth Des. 2014, 14, 5521–5531.

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(26) Cui, Y.; Stojakovic, J.; Kijima, H.; Myerson, A. S. Cryst. Growth Des. 2016, 16, 6131–

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6138.

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(27) Agrawal, S. G.; Paterson, A. H. J. Chem. Eng. Commun. 2015, 202, 698–706.

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(28) Hansen, T. B.; Taris, A.; Rong, B.-G.; Grosso, M.; Qu, H. J. Cryst. Growth 2016, 450,

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508

509

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Supporting Information

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Supporting Information to this paper is available online. It contains two sections: in the first

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one (Appendix A), we have detailed how the parameters for the simulations in Section 4 have

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been selected; in the second one (Appendix B), we have reported additional experimental

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data mentioned in Section 2, namely further XRD spectra and further Raman spectra, and

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the table reporting the detection times associated to each Raman spectrum measured.


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For Table of Contents Use Only

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Stochastic nucleation of polymorphs: experimental evidence and mathematical

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modelling

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Giovanni Maria Maggioni, Leonard Bezinge, and Marco Mazzotti

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Based on several experiments, and building upon previous theoretical work, we develop

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a stochastic model of primary nucleation from solution for a compound with an arbitrary

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number of polymorphs. Analysing the behaviour predicted by the model, we show that

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the nucleation probability depends on an apparent nucleation rate, given by the sum of the

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individual rates of all polymorphs at the conditions at which the system is operated.


Stochastic Nucleation of Polymorphs: Experimental Evidence and

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