Computational predictions of the polymorphic outcomes of a crystallization process, referred to as polymorph selection, can accelerate the process development for manufacturing solid products with targeted properties. Polymorph selection requires understanding the interplay between the thermodynamic and kinetic factors that drive nucleation. Moreover, post-nucleation events, such as crystal growth and polymorphic transformation, can affect the resulting crystal structures. Here, the nucleation kinetics of the Lennard-Jones (LJ) fluid from the melt is investigated with a focus on the competition between FCC and HCP crystal structures. Both molecular dynamics (MD) simulations and 2D free energy calculations reveal that polymorph selection occurs not during nucleation but when the cluster sizes exceed the critical cluster size. This result contrasts with the classical nucleation mechanism, where each polymorph is assumed to nucleate independently as an ideal bulk-like cluster, comprised only of its given structure. Using the 2D free energy surface and the MD simulation-derived diffusion coefficients, a structure-dependent nucleation rate is estimated, which agrees with the rate obtained from brute force MD simulations. Furthermore, a comprehensive population balance modeling (PBM) approach for polymorph selection is proposed. The PBM combines the calculated nucleation rate with post-nucleation kinetics while accounting for the structural changes of the clusters after nucleation. When applied to the LJ system, the PBM predicts with high accuracy the polymorphic distribution found in a population of crystals generated from MD simulations. Due to the non-classical nucleation mechanism of the LJ system, post-nucleation kinetic events are crucial in determining the structures of the grown crystals.
REFERENCES
1.
J.
Bernstein
, Polymorphism in Molecular Crystals
, 2nd ed. (International Union of Crystal
, 2020
), Vol. 30.2.
D. A.
Hegg
and M. B.
Baker
, “Nucleation in the atmosphere
,” Rep. Prog. Phys.
72
, 056801
(2009
).3.
H.
Nada
, “Pathways for the formation of ice polymorphs from water predicted by a metadynamics method
,” Sci. Rep.
10
, 4708
(2020
).4.
S. I.
Sanchez
, M. W.
Small
, E. S.
Bozin
, J.-G.
Wen
, J.-M.
Zuo
, and R. G.
Nuzzo
, “Metastability and structural polymorphism in noble metals: The role of composition and metal atom coordination in mono- and bimetallic nanoclusters
,” ACS Nano
7
, 1542
–1557
(2013
).5.
R.
Tycko
, “Amyloid polymorphism: Structural basis and neurobiological relevance
,” Neuron
86
, 632
–645
(2015
).6.
P.
Corradini
and G.
Guerra
, “Polymorphism in polymers
,” in Macromolecules: Synthesis, Order and Advanced Properties
(Springer
, 1992
), pp. 183
–217
, available at https://link.springer.com/book/10.1007/BFb0051632.7.
R.
Hilfiker
and M.
Von Raumer
, Polymorphism in the Pharmaceutical Industry: Solid Form and Drug Development
(John Wiley & Sons
, 2019
).8.
A. M.
Reilly
, R. I.
Cooper
, C. S.
Adjiman
, S.
Bhattacharya
, A. D.
Boese
, J. G.
Brandenburg
, P. J.
Bygrave
, R.
Bylsma
, J. E.
Campbell
, R.
Car
et al, “Report on the sixth blind test of organic crystal structure prediction methods
,” Acta Crystallogr., Sect. B: Struct. Sci., Cryst. Eng. Mater.
72
, 439
–459
(2016
).9.
C.
Stoica
, P.
Verwer
, H.
Meekes
, P. J. C. M.
van Hoof
, F. M.
Kaspersen
, and E.
Vlieg
, “Understanding the effect of a solvent on the crystal habit
,” Cryst. Growth Des.
4
, 765
–768
(2004
).10.
S.
Datta
and D. J. W.
Grant
, “Effect of supersaturation on the crystallization of phenylbutazone polymorphs
,” Cryst. Res. Technol.
40
, 233
–242
(2005
).11.
K. R.
Devi
and K.
Srinivasan
, “Towards a better understanding of the nucleation behavior of α and γ polymorphs of glycine from aqueous solution in the presence of selective additives by charge compensation mechanism
,” J. Cryst. Growth
401
, 227
–232
(2014
).12.
I. N.
Stranski
and D.
Totomanow
, “Keimbildungsgeschwindigkeit und ostwaldsche stufenregel
,” Z. Phys. Chem.
163A
, 399
–408
(1933
).13.
B.
O’Malley
and I.
Snook
, “Crystal nucleation in the hard sphere system
,” Phys. Rev. Lett.
90
, 085702
(2003
).14.
S.
Auer
and D.
Frenkel
, “Numerical prediction of absolute crystallization rates in hard-sphere colloids
,” J. Chem. Phys.
120
, 3015
–3029
(2004
).15.
F.
Leoni
and J.
Russo
, “Nonclassical nucleation pathways in stacking-disordered crystals
,” Phys. Rev. X
11
, 031006
(2021
).16.
W.
Wöhler
and T.
Schilling
, “Hard sphere crystal nucleation rates: Reconciliation of simulation and experiment
,” Phys. Rev. Lett.
128
, 238001
(2022
).17.
W.
Gispen
and M.
Dijkstra
, “Kinetic phase diagram for nucleation and growth of competing crystal polymorphs in charged colloids
,” Phys. Rev. Lett.
129
, 098002
(2022
).18.
P. R.
Ten Wolde
, M. J.
Ruiz-Montero
, and D.
Frenkel
, “Numerical evidence for bcc ordering at the surface of a critical fcc nucleus
,” Phys. Rev. Lett.
75
, 2714
(1995
).19.
P. R.
ten Wolde
, M. J.
Ruiz‐Montero
, and D.
Frenkel
, “Numerical calculation of the rate of crystal nucleation in a Lennard-Jones system at moderate undercooling
,” J. Chem. Phys.
104
, 9932
–9947
(1996
).20.
D.
Moroni
, P. R.
ten Wolde
, and P. G.
Bolhuis
, “Interplay between structure and size in a critical crystal nucleus
,” Phys. Rev. Lett.
94
, 235703
(2005
).21.
H.
Wang
, H.
Gould
, and W.
Klein
, “Homogeneous and heterogeneous nucleation of Lennard-Jones liquids
,” Phys. Rev. E
76
, 031604
(2007
).22.
C.
Desgranges
and J.
Delhommelle
, “Controlling polymorphism during the crystallization of an atomic fluid
,” Phys. Rev. Lett.
98
, 235502
(2007
).23.
V. K.
de Souza
and D. J.
Wales
, “The potential energy landscape for crystallisation of a Lennard-Jones fluid
,” J. Stat. Mech.: Theory Exp.
2016
, 074001
.24.
W.
Ouyang
, B.
Sun
, Z.
Sun
, and S.
Xu
, “Entire crystallization process of Lennard-Jones liquids: A large-scale molecular dynamics study
,” J. Chem. Phys.
152
, 054903
(2020
).25.
S.
Auer
and D.
Frenkel
, “Quantitative prediction of crystal-nucleation rates for spherical colloids: A computational approach
,” Annu. Rev. Phys. Chem.
55
, 333
–361
(2004
).26.
W.
Lechner
, C.
Dellago
, and P. G.
Bolhuis
, “Role of the prestructured surface cloud in crystal nucleation
,” Phys. Rev. Lett.
106
, 085701
(2011
).27.
B.
Peters
, “Competing nucleation pathways in a mixture of oppositely charged colloids: Out-of-equilibrium nucleation revisited
,” J. Chem. Phys.
131
, 244103
(2009
).28.
L.
Lupi
, A.
Hudait
, B.
Peters
, M.
Grünwald
, R.
Gotchy Mullen
, A. H.
Nguyen
, and V.
Molinero
, “Role of stacking disorder in ice nucleation
,” Nature
551
, 218
–222
(2017
).29.
A.
Hudait
, S.
Qiu
, L.
Lupi
, and V.
Molinero
, “Free energy contributions and structural characterization of stacking disordered ices
,” Phys. Chem. Chem. Phys.
18
, 9544
–9553
(2016
).30.
T.
Mandal
and R. G.
Larson
, “Nucleation of urea from aqueous solution: Structure, critical size, and rate
,” J. Chem. Phys.
146
, 134501
(2017
).31.
M.
Salvalaglio
, M.
Mazzotti
, and M.
Parrinello
, “Urea homogeneous nucleation mechanism is solvent dependent
,” Faraday Discuss.
179
, 291
–307
(2015
).32.
M.
Li
, Y.
Chen
, H.
Tanaka
, and P.
Tan
, “Revealing roles of competing local structural orderings in crystallization of polymorphic systems
,” Sci. Adv.
6
, eaaw8938
(2020
).33.
P. R. t.
Wolde
and D.
Frenkel
, “Enhancement of protein crystal nucleation by critical density fluctuations
,” Science
277
, 1975
–1978
(1997
).34.
G. I.
Tóth
, T.
Pusztai
, G.
Tegze
, G.
Tóth
, and L.
Gránásy
, “Amorphous nucleation precursor in highly nonequilibrium fluids
,” Phys. Rev. Lett.
107
, 175702
(2011
).35.
R. K. R.
Addula
and S. N.
Punnathanam
, “Molecular theory of nucleation from dilute phases: Formulation and application to Lennard-Jones vapor
,” Phys. Rev. Lett.
126
, 146001
(2021
).36.
N. D.
Loh
, S.
Sen
, M.
Bosman
, S. F.
Tan
, J.
Zhong
, C. A.
Nijhuis
, P.
Král
, P.
Matsudaira
, and U.
Mirsaidov
, “Multistep nucleation of nanocrystals in aqueous solution
,” Nat. Chem.
9
, 77
–82
(2017
).37.
P. S.
Bulutoglu
, S.
Wang
, M.
Boukerche
, N. K.
Nere
, D. S.
Corti
, and D.
Ramkrishna
, “An investigation of the kinetics and thermodynamics of NaCl nucleation through composite clusters
,” PNAS Nexus
1
(2
), 1
–11
(2022
).38.
S.
Menon
, G.
Díaz Leines
, R.
Drautz
, and J.
Rogal
, “Role of pre-ordered liquid in the selection mechanism of crystal polymorphs during nucleation
,” J. Chem. Phys.
153
, 104508
(2020
).39.
Y.-C.
Hu
and H.
Tanaka
, “Revealing the role of liquid preordering in crystallisation of supercooled liquids
,” Nat. Commun.
13
, 4519
(2022
).40.
C.
Desgranges
and J.
Delhommelle
, “Can ordered precursors promote the nucleation of solid solutions?
,” Phys. Rev. Lett.
123
, 195701
(2019
).41.
P.
Tan
, N.
Xu
, and L.
Xu
, “Visualizing kinetic pathways of homogeneous nucleation in colloidal crystallization
,” Nat. Phys.
10
, 73
–79
(2014
).42.
J. R.
Espinosa
, C.
Vega
, C.
Valeriani
, and E.
Sanz
, “Seeding approach to crystal nucleation
,” J. Chem. Phys.
144
, 034501
(2016
).43.
C.
Parks
, A.
Koswara
, F.
DeVilbiss
, H.-H.
Tung
, N. K.
Nere
, S.
Bordawekar
, Z. K.
Nagy
, and D.
Ramkrishna
, “Solubility curves and nucleation rates from molecular dynamics for polymorph prediction–moving beyond lattice energy minimization
,” Phys. Chem. Chem. Phys.
19
, 5285
–5295
(2017
).44.
I.
Sanchez-Burgos
, E.
Sanz
, C.
Vega
, and J. R.
Espinosa
, “Fcc vs. hcp competition in colloidal hard-sphere nucleation: On their relative stability, interfacial free energy and nucleation rate
,” Phys. Chem. Chem. Phys.
23
, 19611
–19626
(2021
).45.
M.
Salvalaglio
, C.
Perego
, F.
Giberti
, M.
Mazzotti
, and M.
Parrinello
, “Molecular-dynamics simulations of urea nucleation from aqueous solution
,” Proc. Natl. Acad. Sci. U. S. A.
112
, E6
–E14
(2015
).46.
C.
Liu
, F.
Cao
, S. A.
Kulkarni
, G. P. F.
Wood
, and E. E.
Santiso
, “Understanding polymorph selection of sulfamerazine in solution
,” Cryst. Growth Des.
19
, 6925
–6934
(2019
).47.
J. F. B.
Black
, P. T.
Cardew
, A. J.
Cruz-Cabeza
, R. J.
Davey
, S. E.
Gilks
, and R. A.
Sullivan
, “Crystal nucleation and growth in a polymorphic system: Ostwald’s rule, p-aminobenzoic acid and nucleation transition states
,” CrystEngComm
20
, 768
–776
(2018
).48.
P.
Cui
, Q.
Yin
, Y.
Guo
, and J.
Gong
, “Polymorphic crystallization and transformation of candesartan cilexetil
,” Ind. Eng. Chem. Res.
51
, 12910
–12916
(2012
).49.
L.
Yu
, “Survival of the fittest polymorph: How fast nucleater can lose to fast grower
,” CrystEngComm
9
, 847
–851
(2007
).50.
Y.
Gui
, C.
Huang
, C.
Shi
, T.
Stelzer
, G. G. Z.
Zhang
, and L.
Yu
, “Polymorphic selectivity in crystal nucleation
,” J. Chem. Phys.
156
, 144504
(2022
).51.
M.
Kobari
, N.
Kubota
, and I.
Hirasawa
, “A population balance model for solvent-mediated polymorphic transformation in unseeded solutions
,” CrystEngComm
16
, 6049
–6058
(2014
).52.
D.
Ramkrishna
, Population Balances: Theory and Applications to Particulate Systems in Engineering
(Elsevier
, 2000
).53.
T.
Rosenbaum
, L.
Tan
, M.
Dummeldinger
, N.
Mitchell
, and J.
Engstrom
, “Population balance modeling to predict particle size distribution upon scale-up of a combined antisolvent and cooling crystallization of an active pharmaceutical ingredient
,” Org. Process Res. Dev.
23
, 2666
–2677
(2019
).54.
B. J.
McCoy
, “A new population balance model for crystal size distributions: Reversible, size-dependent growth and dissolution
,” J. Colloid Interface Sci.
240
, 139
–149
(2001
).55.
B.
Szilagyi
, W.-L.
Wu
, A.
Eren
, J.
Mackey
, S.
Kshirsagar
, E.
Szilagyi
, I.
Ostergaard
, H.
Qu
, K.
Sinha
, L.
Mlinar
et al, “Cross-pharma collaboration for the development of a simulation tool for the model-based digital design of pharmaceutical crystallization processes (CrySiV)
,” Cryst. Growth Des.
21
, 6448
–6464
(2021
).56.
B.
Szilágyi
, A.
Eren
, J. L.
Quon
, C. D.
Papageorgiou
, and Z. K.
Nagy
, “Digital design of the crystallization of an active pharmaceutical ingredient using a population balance model with a novel size dependent growth rate expression. From development of a digital twin to in silico optimization and experimental validation
,” Cryst. Growth Des.
22
, 497
–512
(2021
).57.
M.
Lin
, Y.
Wu
, and S.
Rohani
, “Identifying the polymorphic outcome of hypothetical polymorphs in batch and continuous crystallizers by numerical simulation
,” Cryst. Growth Des.
20
, 7312
–7319
(2020
).58.
G. M.
Maggioni
and M.
Mazzotti
, “A stochastic population balance equation model for nucleation and growth of crystals with multiple polymorphs
,” Cryst. Growth Des.
19
, 4698
–4709
(2019
).59.
W.
Tang
, Y.
Quan
, J.
Gong
, J.
Wang
, Q.
Yin
, and T.
Li
, “Form selection of concomitant polymorphs: A case study informed by crystallization kinetics modeling
,” AIChE J.
67
, e17129
(2021
).60.
R.
Schneider
, J.
Kerkhoff
, A.
Danzer
, A.
Mattusch
, A.
Ohmann
, M.
Thommes
, and G.
Sadowski
, “The interplay of dissolution, solution crystallization and solid-state transformation of amorphous indomethacin in aqueous solution
,” Int. J. Pharm.: X
2
, 100063
(2020
).61.
A.
Burrows
, S.
Cooper
, and P.
Schwerdtfeger
, “Instability of the body-centered cubic lattice within the sticky hard sphere and Lennard-Jones model obtained from exact lattice summations
,” Phys. Rev. E
104
, 035306
(2021
).62.
A.
Travesset
, “Phase diagram of power law and Lennard-Jones systems: Crystal phases
,” J. Chem. Phys.
141
, 164501
(2014
).63.
J. S.
Langer
, “Statistical theory of the decay of metastable states
,” Ann. Phys.
54
, 258
–275
(1969
).64.
S.
Plimpton
, “Fast parallel algorithms for short-range molecular dynamics
,” J. Comput. Phys.
117
, 1
–19
(1995
).65.
D. J.
Evans
and B. L.
Holian
, “The Nose–Hoover thermostat
,” J. Chem. Phys.
83
, 4069
–4074
(1985
).66.
L.
Filion
, M.
Hermes
, R.
Ni
, and M.
Dijkstra
, “Crystal nucleation of hard spheres using molecular dynamics, umbrella sampling, and forward flux sampling: A comparison of simulation techniques
,” J. Chem. Phys.
133
, 244115
(2010
).67.
V.
Ramasubramani
, B. D.
Dice
, E. S.
Harper
, M. P.
Spellings
, J. A.
Anderson
, and S. C.
Glotzer
, “freud: A software suite for high throughput analysis of particle simulation data
,” Comput. Phys. Commun.
254
, 107275
(2020
).68.
W.
Mickel
, S. C.
Kapfer
, G. E.
Schröder-Turk
, and K.
Mecke
, “Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter
,” J. Chem. Phys.
138
, 044501
(2013
).69.
W.
Lechner
and C.
Dellago
, “Accurate determination of crystal structures based on averaged local bond order parameters
,” J. Chem. Phys.
129
, 114707
(2008
).70.
J.
Russo
and H.
Tanaka
, “Selection mechanism of polymorphs in the crystal nucleation of the Gaussian core model
,” Soft Matter
8
, 4206
–4215
(2012
).71.
V.
Agarwal
and B.
Peters
, “Solute precipitate nucleation: A review of theory and simulation advances
,” Adv. Chem. Phys.
155
, 97
–160
(2014
).72.
L.
Maibaum
, “Comment on ‘Elucidating the mechanism of nucleation near the gas-liquid spinodal
,'” Phys. Rev. Lett.
101
, 019601
(2008
).73.
J.
Guo
, A.
Haji-Akbari
, and J. C.
Palmer
, “Hybrid Monte Carlo with LAMMPS
,” J. Theor. Comput. Chem.
17
, 1840002
(2018
).74.
M.
Tuckerman
, Statistical Mechanics: Theory and Molecular Simulation
(Oxford University Press
, 2010
).75.
S. E. M.
Lundrigan
and I.
Saika-Voivod
, “Test of classical nucleation theory and mean first-passage time formalism on crystallization in the Lennard-Jones liquid
,” J. Chem. Phys.
131
, 104503
(2009
).76.
A. L.
Ferguson
, “BayesWHAM: A Bayesian approach for free energy estimation, reweighting, and uncertainty quantification in the weighted histogram analysis method
,” J. Comput. Chem.
38
, 1583
–1605
(2017
).77.
J.
Wedekind
, R.
Strey
, and D.
Reguera
, “New method to analyze simulations of activated processes
,” J. Chem. Phys.
126
, 134103
(2007
).78.
J.
Wedekind
and D.
Reguera
, “Kinetic reconstruction of the free-energy landscape
,” J. Phys. Chem. B
112
, 11060
–11063
(2008
).79.
A. F.
Voter
, “A method for accelerating the molecular dynamics simulation of infrequent events
,” J. Chem. Phys.
106
, 4665
–4677
(1997
).80.
L. V.
Woodcock
, “Entropy difference between the face-centred cubic and hexagonal close-packed crystal structures
,” Nature
385
, 141
–143
(1997
).81.
E.
Simone
, B.
Szilagyi
, and Z. K.
Nagy
, “Systematic model identification and optimization-based active polymorphic control of crystallization processes
,” Chem. Eng. Sci.
174
, 374
–386
(2017
).82.
H. A.
Kramers
, “Brownian motion in a field of force and the diffusion model of chemical reactions
,” Physica
7
, 284
–304
(1940
).83.
V.
Špillar
and D.
DolejŠ
, “Calculation of time-dependent nucleation and growth rates from quantitative textural data: Inversion of crystal size distribution
,” J. Petrol.
54
, 913
–931
(2013
).84.
J. Q.
Broughton
, G. H.
Gilmer
, and K. A.
Jackson
, “Crystallization rates of a Lennard-Jones liquid
,” Phys. Rev. Lett.
49
, 1496
(1982
).85.
J.
Anwar
and D.
Zahn
, “Polymorphic phase transitions: Macroscopic theory and molecular simulation
,” Adv. Drug Delivery Rev.
117
, 47
–70
(2017
).86.
D. R.
Jones
and M. F.
Ashby
, Engineering Materials 2: An Introduction to Microstructures and Processing
(Butterworth-Heinemann
, 2012
).87.
J. A.
Nelder
and R.
Mead
, “A simplex method for function minimization
,” Comput. J.
7
, 308
–313
(1965
).88.
T.
Mandal
, R. L.
Marson
, and R. G.
Larson
, “Coarse-grained modeling of crystal growth and polymorphism of a model pharmaceutical molecule
,” Soft Matter
12
, 8246
–8255
(2016
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
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