We have only rules of thumb with which to predict how a material will crystallize, chief among which is Ostwald's rule of stages. It states that the first phase to appear upon transformation of a parent phase is the one closest to it in free energy. Although sometimes upheld, the rule is without theoretical foundation and is not universally obeyed, highlighting the need for microscopic understanding of crystallization controls. Here we study in detail the crystallization pathways of a prototypical model of patchy particles. The range of crystallization pathways it exhibits is richer than can be predicted by Ostwald's rule, but a combination of simulation and analytic theory reveals clearly how these pathways are selected by microscopic parameters. Our results suggest strategies for controlling self-assembly pathways in simulation and experiment.
REFERENCES
1.
T. H.
Zhang
and X. Y.
Liu
, Angew. Chem. Int. Ed.
48
, 1308
(2009
).2.
S. Y.
Chung
, Y. M.
Kim
, J. G.
Kim
, and Y. J.
Kim
, Nat. Phys.
5
, 68
(2008
).3.
S.
Chung
, S. H.
Shin
, C. R.
Bertozzi
, and J. J.
De Yoreo
, Proc. Natl. Acad. Sci.
U.S.A. 107
, 16536
(2010
).4.
W.
Ostwald
, Z. Phys. Chem.
22
, 289
(1897
).5.
T.
Threlfall
, Org. Process Res. Dev.
7
, 1017
(2003
).6.
R. E.
Cech
, Trans. Am. Inst. Min., Metall. Pet. Eng.
585
(1957
).7.
L. E.
Fox
, S. C.
Wofsy
, D. R.
Worsnop
, and M. S.
Zahniser
, Science
267
, 351
(1995
).8.
P. R.
Wolde
and D.
Frenkel
, Phys. Chem. Chem. Phys.
1
, 2191
(1999
).9.
J. M.
Leyssale
, J.
Delhommelle
, and C.
Millot
, Chem. Phys. Lett.
375
, 612
(2003
).10.
P.
Cardew
, R.
Davey
, and A.
Ruddick
, J. Chem. Soc., Faraday Trans. 2
80
, 659
(1984
).11.
12.
J. M.
Leyssale
, J.
Delhommelle
, and C.
Millot
, J. Chem. Phys.
122
, 184518
(2005
).13.
D.
Sanders
, H.
Larralde
, and F.
Leyvraz
, Phys. Rev. B
75
, 132101
(2007
).14.
E.
Sanz
, C.
Valeriani
, D.
Frenkel
, and M.
Dijkstra
, Phys. Rev. Lett.
99
, 55501
(2007
).15.
I. N.
Stranski
and D.
Totomanov
, Z. Phys. Chem. Abt. A
163
, 399
(1933
).16.
O.
Galkin
and P. G.
Vekilov
, Proc. Nat. Acad. Sci. U.S.A.
97
, 6277
(2000
).17.
A.
Radha
, T.
Forbes
, C.
Killian
, P.
Gilbert
, and A.
Navrotsky
, Proc. Natl. Acad. Sci. U.S.A.
107
, 16438
(2010
).18.
W.
Dai
, S.
Kumar
, and F.
Starr
, Soft Matter
6
, 6130
(2010
).19.
An object of radius a in physiological aqueous solution possesses a range of attraction Δ that is characteristically a nanometer. The fact that Δ does not scale with a is important to the thermodynamics of particle association (Ref. 24), and is likely to be important dynamically. From Stokes' law, the characteristic time for two particles to explore the extent of their range of attraction is Δ2a, and the characteristic time for particles to explore their orientational degrees of freedom (e.g., thinking of the present model, to move from parallel to antiparallel registry) scales as a3. The ratio of these characteristic times scales as a2, and we therefore expect a large change of this ratio upon moving from a small protein of a ≈ 1 nm to a patchy nanoparticle of a ≈ 100 nm.
20.
G.
Torrie
and J.
Valleau
, J. Comput. Phys.
23
, 187
(1977
).21.
P. G.
Bolhuis
, D.
Chandler
, C.
Dellago
, and P. L.
Geissler
, Annu. Rev. Phys. Chem.
53
, 291
(2002
).22.
A.
Pan
and D.
Chandler
, J. Phys. Chem. B
108
, 19681
(2004
).23.
R.
Allen
, P.
Warren
, and P.
Ten Wolde
, Phys. Rev. Lett.
94
, 18104
(2005
).24.
P. R.
Wolde
and D.
Frenkel
, Science
277
, 1975
(1997
).25.
Y. C.
Shen
and D. W.
Oxtoby
, Phys. Rev. Lett.
77
, 3585
(1996
).26.
J.
Lutsko
and G.
Nicolis
, Phys. Rev. Lett.
96
, 046102
(2006
).27.
N.
Duff
and B.
Peters
, J. Chem. Phys.
131
, 184101
(2009
).28.
C.
Desgranges
and J.
Delhommelle
, Phys. Rev. Lett.
98
, 235502
(2007
).29.
F.
Zhang
, G.
Zocher
, A.
Sauter
, T.
Stehle
, and F.
Schreiber
, J. Appl. Crystallogr.
44
, 755
(2011
).30.
R. J.
Allen
, D.
Frenkel
, and P. R.
ten Wolde
, J. Chem. Phys.
124
, 024102
(2006
).31.
B.
Peters
and B. L.
Trout
, J. Chem. Phys.
125
, 054108
(2006
).32.
B.
Peters
, G. T.
Beckham
, and B. L.
Trout
, J. Chem. Phys.
127
, 034109
(2007
).33.
J.
Geng
and J.
Selinger
, Phys. Rev. E
80
, 11707
(2009
).34.
S.
Whitelam
, J. Chem. Phys.
132
, 194901
(2010
).35.
N.
Asherie
, A.
Lomakin
, and G. B.
Benedek
, Phys. Rev. Lett.
77
, 4832
(1996
).36.
See supplementary material at http://dx.doi.org/10.1063/1.3655358 for movie of nucleation pathway for fast (r = 99) rotation rate at point B on the phase diagram; slow (r = 0.01) rotation rate at point B on the phase diagram; fast (r = 99) rotation rate at point C on the phase diagram; and for movies showing representative nucleation pathways for fast (r = 99) and slow (r = 0.01) rotation rate at points B and C on the phase diagram of Fig. 1.
© 2011 American Institute of Physics.
2011
American Institute of Physics
You do not currently have access to this content.