Nucleation in systems with a metastable liquid–gas critical point is the prototypical example of a two-step nucleation process in which the appearance of the critical nucleus is preceded by the formation of a liquid-like density fluctuation. So far, the majority of studies on colloidal and protein crystallization have focused on one-component systems, and we are lacking a clear description of two-step nucleation processes in multicomponent systems, where critical fluctuations involve coupled density and concentration inhomogeneities. Here, we examine the nucleation process of a binary mixture of patchy particles designed to nucleate into a diamond lattice. By combining Gibbs-ensemble simulations and direct nucleation simulations over a wide range of thermodynamic conditions, we are able to pin down the role of the liquid–gas metastable phase diagram on the nucleation process. In particular, we show that the strongest enhancement of crystallization occurs at an azeotropic point with the same stoichiometric composition of the crystal.

1.
S.
Whitelam
and
R. L.
Jack
, “
The statistical mechanics of dynamic pathways to self-assembly
,”
Annu. Rev. Phys. Chem.
66
,
143
(
2015
).
2.
S. K.
Kumar
,
G.
Kumaraswamy
,
B. L. V.
Prasad
,
R.
Bandyopadhyaya
,
S.
Granick
,
O.
Gang
,
V. N.
Manoharan
,
D.
Frenkel
, and
N. A.
Kotov
, “
Nanoparticle assembly: A perspective and some unanswered questions
,”
Curr. Sci.
112
,
1635
(
2017
).
3.
N. C.
Seeman
and
H. F.
Sleiman
, “
DNA nanotechnology
,”
Nat. Rev. Mater.
3
(
1
),
17068
(
2017
).
4.
W.
Liu
,
M.
Tagawa
,
H. L.
Xin
,
T.
Wang
,
H.
Emamy
,
H.
Li
,
K. G.
Yager
,
F. W.
Starr
,
A. V.
Tkachenko
, and
O.
Gang
, “
Diamond family of nanoparticle superlattices
,”
Science
351
,
582
(
2016
).
5.
T.
Zhang
,
C.
Hartl
,
K.
Frank
,
A.
Heuer‐Jungemann
,
S.
Fischer
,
P. C.
Nickels
,
B.
Nickel
, and
T.
Liedl
, “
3D DNA origami crystals
,”
Adv. Mater.
30
,
1800273
(
2018
).
6.
D.
Kashchiev
,
P. G.
Vekilov
, and
A. B.
Kolomeisky
, “
Kinetics of two-step nucleation of crystals
,”
J. Chem. Phys.
122
,
244706
(
2005
).
7.
D.
Erdemir
,
A. Y.
Lee
, and
A. S.
Myerson
, “
Nucleation of crystals from solution: Classical and two-step models
,”
Acc. Chem. Res.
42
,
621
(
2009
).
8.
P. G.
Vekilov
, “
The two-step mechanism of nucleation of crystals in solution
,”
Nanoscale
2
,
2346
(
2010
).
9.
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
).
10.
R. P.
Sear
, “
The non-classical nucleation of crystals: Microscopic mechanisms and applications to molecular crystals, ice and calcium carbonate
,”
Int. Mater. Rev.
57
,
328
(
2012
).
11.
T. K.
Haxton
,
L. O.
Hedges
, and
S.
Whitelam
, “
Crystallization and arrest mechanisms of model colloids
,”
Soft Matter
11
,
9307
(
2015
).
12.
J.
Russo
and
H.
Tanaka
, “
Nonclassical pathways of crystallization in colloidal systems
,”
MRS Bull.
41
,
369
(
2016
).
13.
G. C.
Sosso
,
J.
Chen
,
S. J.
Cox
,
M.
Fitzner
,
P.
Pedevilla
,
A.
Zen
, and
A.
Michaelides
, “
Crystal nucleation in liquids: Open questions and future challenges in molecular dynamics simulations
,”
Chem. Rev.
116
,
7078
(
2016
).
14.
J. F.
Lutsko
, “
How crystals form: A theory of nucleation pathways
,”
Sci. Adv.
5
,
eaav7399
(
2019
).
15.
D.
James
,
S.
Beairsto
,
C.
Hartt
,
O.
Zavalov
,
I.
Saika-Voivod
,
R. K.
Bowles
, and
P. H.
Poole
, “
Phase transitions in fluctuations and their role in two-step nucleation
,”
J. Chem. Phys.
150
,
074501
(
2019
).
16.
C.
Desgranges
and
J.
Delhommelle
, “
Can ordered precursors promote the nucleation of solid solutions?
,”
Phys. Rev. Lett.
123
,
195701
(
2019
).
17.
D.
Kashchiev
, “
Classical nucleation theory approach to two-step nucleation of crystals
,”
J. Cryst. Growth
530
,
125300
(
2020
).
18.
S.
Lee
,
E. G.
Teich
,
M.
Engel
, and
S. C.
Glotzer
, “
Entropic colloidal crystallization pathways via fluid–fluid transitions and multidimensional prenucleation motifs
,”
Proc. Natl. Acad. Sci. U. S. A.
116
,
14843
(
2019
).
19.
P. R. t.
Wolde
and
D.
Frenkel
, “
Enhancement of protein crystal nucleation by critical density fluctuations
,”
Science
277
,
1975
(
1997
).
20.
P.
Tan
,
N.
Xu
, and
L.
Xu
, “
Visualizing kinetic pathways of homogeneous nucleation in colloidal crystallization
,”
Nat. Phys.
10
,
73
(
2014
).
21.
H.
Jiang
,
P. G.
Debenedetti
, and
A. Z.
Panagiotopoulos
, “
Nucleation in aqueous NaCl solutions shifts from 1-step to 2-step mechanism on crossing the spinodal
,”
J. Chem. Phys.
150
,
124502
(
2019
).
22.
D.
Gebauer
,
A.
Völkel
, and
H.
Cölfen
, “
Stable prenucleation calcium carbonate clusters
,”
Science
322
,
1819
(
2008
).
23.
E. M.
Pouget
,
P. H. H.
Bomans
,
J. A. C. M.
Goos
,
P. M.
Frederik
,
G.
de With
, and
N. A. J. M.
Sommerdijk
, “
The initial stages of template-controlled CaCO3 formation revealed by cryo-TEM
,”
Science
323
,
1455
(
2009
).
24.
D.
Fusco
and
P.
Charbonneau
, “
Crystallization of asymmetric patchy models for globular proteins in solution
,”
Phys. Rev. E
88
,
012721
(
2013
).
25.
S.
James
,
M. K.
Quinn
, and
J. J.
McManus
, “
The self assembly of proteins; probing patchy protein interactions
,”
Phys. Chem. Chem. Phys.
17
,
5413
(
2015
).
26.
J. J.
McManus
,
P.
Charbonneau
,
E.
Zaccarelli
, and
N.
Asherie
, “
The physics of protein self-assembly
,”
Curr. Opin. Colloid Interface Sci.
22
,
73
(
2016
).
27.
D.
Fusco
and
P.
Charbonneau
, “
Soft matter perspective on protein crystal assembly
,”
Colloids Surf., B
137
,
22
(
2016
).
28.
A.-Y.
Jee
,
K.
Lou
,
H.-S.
Jang
,
K. H.
Nagamanasa
, and
S.
Granick
, “
Nanoparticle puzzles and research opportunities that go beyond state of the art
,”
Faraday Discuss.
186
,
11
(
2016
).
29.
M.
Dijkstra
and
E.
Luijten
, “
From predictive modelling to machine learning and reverse engineering of colloidal self-assembly
,”
Nat. Mater.
20
,
762
(
2021
).
30.
S.
Whitelam
and
I.
Tamblyn
, “
Learning to grow: Control of material self-assembly using evolutionary reinforcement learning
,”
Phys. Rev. E
101
,
052604
(
2020
).
31.
A.
Bupathy
,
D.
Frenkel
, and
S.
Sastry
, “
Temperature protocols to guide selective self-assembly of competing structures
,”
Proc. Natl. Acad. Sci. U. S. A.
119
,
e2119315119
(
2022
).
32.
S.
Whitelam
and
I.
Tamblyn
, “
Neuroevolutionary learning of particles and protocols for self-assembly
,”
Phys. Rev. Lett.
127
,
018003
(
2021
).
33.
M. C.
Rechtsman
,
F. H.
Stillinger
, and
S.
Torquato
, “
Optimized interactions for targeted self-assembly: Application to a honeycomb lattice
,”
Phys. Rev. Lett.
95
,
228301
(
2005
).
34.
É.
Marcotte
,
F. H.
Stillinger
, and
S.
Torquato
, “
Optimized monotonic convex pair potentials stabilize low-coordinated crystals
,”
Soft Matter
7
,
2332
(
2011
).
35.
É.
Marcotte
,
F. H.
Stillinger
, and
S.
Torquato
, “
Communication: Designed diamond ground state via optimized isotropic monotonic pair potentials
,”
J. Chem. Phys.
138
,
061101
(
2013
).
36.
G.
Zhang
,
F.
Stillinger
, and
S.
Torquato
, “
Probing the limitations of isotropic pair potentials to produce ground-state structural extremes via inverse statistical mechanics
,”
Phys. Rev. E
88
,
042309
(
2013
).
37.
M. Z.
Miskin
,
G.
Khaira
,
J. J.
de Pablo
, and
H. M.
Jaeger
, “
Turning statistical physics models into materials design engines
,”
Proc. Natl. Acad. Sci. U. S. A.
113
,
34
(
2016
).
38.
B. A.
Lindquist
,
R. B.
Jadrich
, and
T. M.
Truskett
, “
Communication: Inverse design for self-assembly via on-the-fly optimization
,”
J. Chem. Phys.
145
,
111101
(
2016
).
39.
D.
Chen
,
G.
Zhang
, and
S.
Torquato
, “
Inverse design of colloidal crystals via optimized patchy interactions
,”
J. Phys. Chem. B
122
,
8462
(
2018
).
40.
R.
Kumar
,
G. M.
Coli
,
M.
Dijkstra
, and
S.
Sastry
, “
Inverse design of charged colloidal particle interactions for self assembly into specified crystal structures
,”
J. Chem. Phys.
151
,
084109
(
2019
).
41.
É.
Ducrot
,
M.
He
,
G.-R.
Yi
, and
D. J.
Pine
, “
Colloidal alloys with preassembled clusters and spheres
,”
Nat. Mater.
16
,
652
(
2017
).
42.
D. R.
Nelson
, “
Toward a tetravalent chemistry of colloids
,”
Nano Lett.
2
,
1125
(
2002
).
43.
V. N.
Manoharan
,
M. T.
Elsesser
, and
D. J.
Pine
, “
Dense packing and symmetry in small clusters of microspheres
,”
Science
301
,
483
(
2003
).
44.
Z.
Zhang
,
A. S.
Keys
,
T.
Chen
, and
S. C.
Glotzer
, “
Self-assembly of patchy particles into diamond structures through molecular mimicry
,”
Langmuir
21
,
11547
(
2005
).
45.
F.
Romano
,
J.
Russo
, and
H.
Tanaka
, “
Influence of patch-size variability on the crystallization of tetrahedral patchy particles
,”
Phys. Rev. Lett.
113
,
138303
(
2014
).
46.
J. D.
Halverson
and
A. V.
Tkachenko
, “
DNA-programmed mesoscopic architecture
,”
Phys. Rev. E
87
,
062310
(
2013
).
47.
F.
Romano
and
F.
Sciortino
, “
Patterning symmetry in the rational design of colloidal crystals
,”
Nat. Commun.
3
,
975
(
2012
).
48.
D. F.
Tracey
,
E. G.
Noya
, and
J. P. K.
Doye
, “
Programming patchy particles to form complex periodic structures
,”
J. Chem. Phys.
151
,
224506
(
2019
).
49.
F.
Romano
,
J.
Russo
,
L.
Kroc
, and
P.
Šulc
, “
Designing patchy interactions to self-assemble arbitrary structures
,”
Phys. Rev. Lett.
125
,
118003
(
2020
).
50.
J.
Russo
,
F.
Romano
,
L.
Kroc
,
F.
Sciortino
,
L.
Rovigatti
, and
P.
Šulc
, “
SAT-assembly: A new approach for designing self-assembling systems
,”
J. Phys.: Condens. Matter
34
,
354002
(
2022
).
51.
L.
Rovigatti
,
J.
Russo
,
F.
Romano
,
M.
Matthies
,
L.
Kroc
, and
P.
Šulc
, “
A simple solution to the problem of self-assembling cubic diamond crystals
,”
Nanoscale
14
,
14268
(
2022
).
52.
C.
Desgranges
and
J.
Delhommelle
, “
Unraveling the coupling between demixing and crystallization in mixtures
,”
J. Am. Chem. Soc.
136
,
8145
(
2014
).
53.
Y.
Wang
,
A.
Lomakin
,
J. J.
McManus
,
O.
Ogun
, and
G. B.
Benedek
, “
Phase behavior of mixtures of human lens proteins gamma D and beta B1
,”
Proc. Natl. Acad. Sci. U. S. A.
107
,
13282
(
2010
).
54.
Y.
Wang
,
A.
Lomakin
,
R. F.
Latypov
, and
G. B.
Benedek
, “
Phase separation in solutions of monoclonal antibodies and the effect of human serum albumin
,”
Proc. Natl. Acad. Sci. U. S. A.
108
,
16606
(
2011
).
55.
M.
Heidenreich
,
J. M.
Georgeson
,
E.
Locatelli
,
L.
Rovigatti
,
S. K.
Nandi
,
A.
Steinberg
,
Y.
Nadav
,
E.
Shimoni
,
S. A.
Safran
,
J. P. K.
Doye
, and
E. D.
Levy
, “
Designer protein assemblies with tunable phase diagrams in living cells
,”
Nat. Chem. Biol.
16
,
939
(
2020
).
56.
G.
van Anders
,
N. K.
Ahmed
,
R.
Smith
,
M.
Engel
, and
S. C.
Glotzer
, “
Entropically patchy particles: Engineering valence through shape entropy
,”
ACS Nano
8
,
931
(
2013
).
57.
Z.
Zhang
and
S. C.
Glotzer
, “
Self-assembly of patchy particles
,”
Nano Lett.
4
,
1407
(
2004
).
58.
A. B.
Pawar
and
I.
Kretzschmar
, “
Fabrication, assembly, and application of patchy particles
,”
Macromol. Rapid Commun.
31
,
150
(
2010
).
59.
E.
Bianchi
,
R.
Blaak
, and
C. N.
Likos
, “
Patchy colloids: State of the art and perspectives
,”
Phys. Chem. Chem. Phys.
13
,
6397
(
2011
).
60.
F.
Romano
and
F.
Sciortino
, “
Colloidal self-assembly: Patchy from the bottom up
,”
Nat. Mater.
10
,
171
(
2011
).
61.
K.
Suzuki
,
K.
Hosokawa
, and
M.
Maeda
, “
Controlling the number and positions of oligonucleotides on gold nanoparticle surfaces
,”
J. Am. Chem. Soc.
131
,
7518
(
2009
).
62.
J.-W.
Kim
,
J.-H.
Kim
, and
R.
Deaton
, “
DNA-linked nanoparticle building blocks for programmable matter
,”
Angew. Chem., Int. Ed.
50
,
9185
(
2011
).
63.
Y.
Wang
,
Y.
Wang
,
D. R.
Breed
,
V. N.
Manoharan
,
L.
Feng
,
A. D.
Hollingsworth
,
M.
Weck
, and
D. J.
Pine
, “
Colloids with valence and specific directional bonding
,”
Nature
491
,
51
(
2012
).
64.
L.
Feng
,
R.
Dreyfus
,
R.
Sha
,
N. C.
Seeman
, and
P. M.
Chaikin
, “
DNA patchy particles
,”
Adv. Mater.
25
,
2779
(
2013
).
65.
P. W. K.
Rothemund
, “
Folding DNA to create nanoscale shapes and patterns
,”
Nature
440
,
297
(
2006
).
66.
Y.
Tian
,
J. R.
Lhermitte
,
L.
Bai
,
T.
Vo
,
H. L.
Xin
,
H.
Li
,
R.
Li
,
M.
Fukuto
,
K. G.
Yager
,
J. S.
Kahn
et al, “
Ordered three-dimensional nanomaterials using DNA-prescribed and valence-controlled material voxels
,”
Nat. Mater.
19
,
789
(
2020
).
67.
W.
Bol
, “
Monte Carlo simulations of fluid systems of waterlike molecules
,”
Mol. Phys.
45
,
605
(
1982
).
68.
N.
Kern
and
D.
Frenkel
, “
Fluid–fluid coexistence in colloidal systems with short-ranged strongly directional attraction
,”
J. Chem. Phys.
118
,
9882
(
2003
).
69.
L.
Rovigatti
,
J.
Russo
, and
F.
Romano
, “
How to simulate patchy particles
,”
Eur. Phys. J. E
41
,
59
(
2018
).
70.
C.
Beneduce
,
F.
Sciortino
,
P.
Sulc
, and
J.
Russo
, “
How to include azeotropy in the design of self-assembling patchy particles systems
,” arXiv:2208.09856 (
2022
).
71.
M.
Rubinstein
,
R. H.
Colby
et al,
Polymer Physics
(
Oxford University Press
,
New York
,
2003
), Vol. 23.
72.
H.
Tanaka
,
H.
Tong
,
R.
Shi
, and
J.
Russo
, “
Revealing key structural features hidden in liquids and glasses
,”
Nat. Rev. Phys.
1
,
333
(
2019
).
73.
W.
Lechner
and
C.
Dellago
, “
Accurate determination of crystal structures based on averaged local bond order parameters
,”
J. Chem. Phys.
129
,
114707
(
2008
).
74.
D.
Frenkel
and
B.
Smit
,
Understanding Molecular Simulation: From Algorithms to Applications
(
Elsevier
,
2001
), Vol. 1.
75.
L.
Xu
,
S. V.
Buldyrev
,
H. E.
Stanley
, and
G.
Franzese
, “
Homogeneous crystal nucleation near a metastable fluid-fluid phase transition
,”
Phys. Rev. Lett.
109
,
095702
(
2012
).
76.
A.
Stukowski
, “
Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool
,”
Modell. Simul. Mater. Sci. Eng.
18
,
015012
(
2010
).
77.
I.
Chakraborty
,
D. J. G.
Pearce
,
R. W.
Verweij
,
S. C.
Matysik
,
L.
Giomi
, and
D. J.
Kraft
, “
Self-assembly dynamics of reconfigurable colloidal molecules
,”
ACS Nano
16
,
2471
(
2022
).
78.
Y.
Xiong
,
S.
Yang
,
Y.
Tian
,
A.
Michelson
,
S.
Xiang
,
H.
Xin
, and
O.
Gang
, “
Three-dimensional patterning of nanoparticles by molecular stamping
,”
ACS Nano
14
,
6823
(
2020
).
79.
J.
Russo
,
F.
Leoni
,
F.
Martelli
, and
F.
Sciortino
, “
The physics of empty liquids: From patchy particles to water
,”
Rep. Prog. Phys.
85
,
016601
(
2021
).
80.
E.
Bianchi
,
J.
Largo
,
P.
Tartaglia
,
E.
Zaccarelli
, and
F.
Sciortino
, “
Phase diagram of patchy colloids: Towards empty liquids
,”
Phys. Rev. Lett.
97
,
168301
(
2006
).
81.
F.
Romano
,
E.
Sanz
, and
F.
Sciortino
, “
Crystallization of tetrahedral patchy particles in silico
,”
J. Chem. Phys.
134
,
174502
(
2011
).
82.
I.
Altan
and
P.
Charbonneau
, “
Obtaining soft matter models of proteins and their phase behavior
,” in
Protein Self-Assembly
(
Springer
,
2019
), pp.
209
228
.
83.
N.
Gnan
,
F.
Sciortino
, and
E.
Zaccarelli
, “
Patchy particle models to understand protein phase behavior
,” in
Protein Self-Assembly
(
Springer
,
2019
), pp.
187
208
.
84.
J. J.
McManus
,
Protein Self-Assembly
(
Springer
,
2019
).
85.
T. V.
Hvozd
,
Y. V.
Kalyuzhnyi
,
V.
Vlachy
, and
P. T.
Cummings
, “
Empty liquid state and re-entrant phase behavior of the patchy colloids confined in porous media
,”
J. Chem. Phys.
156
,
161102
(
2022
).
You do not currently have access to this content.