In this work, we revisit the classic problem of homogeneous nucleation of a liquid droplet in a supersaturated vapor phase. We consider this at different extents of the driving force, or equivalently the supersaturation, and calculate a reaction coordinate (RC) for nucleation as the driving force is varied. The RC is constructed as a linear combination of three order parameters, where one accounts for the number of liquidlike atoms and the other two for local density fluctuations. The RC is calculated from biased and unbiased molecular dynamics (MD) simulations using the spectral gap optimization approach “SGOOP” [P. Tiwary and B. J. Berne, Proc. Natl. Acad. Sci. U. S. A. 113, 2839 (2016)]. Our key finding is that as the supersaturation decreases, the RC ceases to simply be the number of liquidlike atoms, and instead, it becomes important to explicitly consider local density fluctuations that correlate with shape and density variations in the nucleus. All three order parameters are found to have similar barriers in their respective potentials of mean force; however, as the supersaturation decreases, the density fluctuations decorrelate slower and thus carry longer memory. Thus, at lower supersaturations, density fluctuations are non-Markovian and cannot be simply ignored from the RC by virtue of being noise. Finally, we use this optimized RC to calculate nucleation rates in the infrequent metadynamics framework and show that it leads to a more accurate estimate of the nucleation rate with four orders of magnitude acceleration relative to unbiased MD.

1.
T.
Bartels-Rausch
,
Nature
494
,
27
(
2013
).
2.
B. J.
Murray
,
T. W.
Wilson
,
S.
Dobbie
,
Z.
Cui
,
S. M.
Al-Jumur
,
O.
Möhler
,
M.
Schnaiter
,
R.
Wagner
,
S.
Benz
,
M.
Niemand
 et al,
Nat. Geosci.
3
,
233
(
2010
).
3.
P.
Mazur
,
Science
168
,
939
(
1970
).
4.
J. R.
Cox
,
L. A.
Ferris
, and
V. R.
Thalladi
,
Angew. Chem., Int. Ed.
46
,
4333
(
2007
).
5.
D.
Erdemir
,
A. Y.
Lee
, and
A. S.
Myerson
,
Curr. Opin. Drug Discovery Dev.
10
,
746
(
2007
).
6.
E. D.
Sloan
,
Nature
426
,
353
(
2003
).
7.
E.
Hammerschmidt
,
Ind. Eng. Chem.
26
,
851
(
1934
).
8.
J. D.
Harper
,
C. M.
Lieber
, and
P. T.
Lansbury
, Jr.
,
Chem. Biol.
4
,
951
(
1997
).
9.
S. I.
Cohen
,
S.
Linse
,
L. M.
Luheshi
,
E.
Hellstrand
,
D. A.
White
,
L.
Rajah
,
D. E.
Otzen
,
M.
Vendruscolo
,
C. M.
Dobson
, and
T. P.
Knowles
,
Proc. Natl. Acad. Sci. U. S. A.
110
,
9758
(
2013
).
10.
G. C.
Sosso
,
J.
Chen
,
S. J.
Cox
,
M.
Fitzner
,
P.
Pedevilla
,
A.
Zen
, and
A.
Michaelides
,
Chem. Rev.
116
,
7078
(
2016
).
11.
G.
Chkonia
,
J.
Wölk
,
R.
Strey
,
J.
Wedekind
, and
D.
Reguera
,
J. Chem. Phys.
130
,
064505
(
2009
).
12.
M.
Salvalaglio
,
P.
Tiwary
,
G. M.
Maggioni
,
M.
Mazzotti
, and
M.
Parrinello
,
J. Chem. Phys.
145
,
211925
(
2016
).
13.
G. M.
Torrie
and
J. P.
Valleau
,
J. Comput. Phys.
23
,
187
(
1977
).
14.
S.
Kumar
,
J. M.
Rosenberg
,
D.
Bouzida
,
R. H.
Swendsen
, and
P. A.
Kollman
,
J. Comput. Chem.
13
,
1011
(
1992
).
15.
A.
Laio
and
M.
Parrinello
,
Proc. Natl. Acad. Sci. U. S. A.
99
,
12562
(
2002
).
16.
A.
Barducci
,
G.
Bussi
, and
M.
Parrinello
,
Phys. Rev. Lett.
100
,
020603
(
2008
).
17.
T. S.
van Erp
,
D.
Moroni
, and
P. G.
Bolhuis
,
J. Chem. Phys.
118
,
7762
(
2003
).
18.
D.
Moroni
,
T. S.
van Erp
, and
P. G.
Bolhuis
,
Physica A
340
,
395
(
2004
).
19.
R. J.
Allen
,
D.
Frenkel
, and
P. R.
ten Wolde
,
J. Chem. Phys.
124
,
024102
(
2006
).
20.
R. J.
Allen
,
C.
Valeriani
, and
P. R.
ten Wolde
,
J. Phys.: Condens. Matter
21
,
463102
(
2009
).
21.
A.
Berezhkovskii
and
A.
Szabo
,
J. Chem. Phys.
122
,
014503
(
2005
).
22.
P.
Tiwary
and
B. J.
Berne
,
Proc. Natl. Acad. Sci. U. S. A.
113
,
2839
(
2016
).
23.
O.
Valsson
,
P.
Tiwary
, and
M.
Parrinello
,
Annu. Rev. Phys. Chem.
67
,
159
(
2016
).
24.
R. S.
DeFever
and
S.
Sarupria
,
J. Chem. Phys.
150
,
024103
(
2019
).
25.
C.
Dellago
,
P. G.
Bolhuis
, and
D.
Chandler
,
J. Chem. Phys.
108
,
9236
(
1998
).
26.
P.
Bolhuis
 et al,
Faraday Discuss.
110
,
421
(
1998
).
27.
J.
Kuipers
and
G.
Barkema
,
Phys. Rev. E
79
,
062101
(
2009
).
28.
I.
Ford
,
Proc. Inst. Mech. Eng., Part C
218
,
883
(
2004
).
29.
P.
Tiwary
and
B. J.
Berne
,
J. Chem. Phys.
147
,
152701
(
2017
).
30.
P.
Tiwary
and
B. J.
Berne
,
J. Chem. Phys.
145
,
054113
(
2016
).
31.
Z.
Smith
,
D.
Pramanik
,
S.-T.
Tsai
, and
P.
Tiwary
,
J. Chem. Phys.
149
,
234105
(
2018
).
32.
M.
Volmer
and
A.
Weber
,
Z. Phys. Chem.
119
,
277
(
1926
).
33.
L.
Farkas
,
Z. Phys. Chem.
125
,
236
(
1927
).
34.
R.
Becker
and
W.
Döring
,
Ann. Phys.
416
,
719
(
1935
).
35.
Y. B.
Zeldovich
,
Acta Physicochem., USSR
18
,
1
(
1943
).
36.
H. J.
Maris
,
C. R. Phys.
7
,
946
(
2006
).
37.
V. I.
Kalikmanov
, “
Classical nucleation theory
,” in
Nucleation Theory
(
Springer Netherlands
,
Dordrecht
,
2013
), pp.
17
41
.
38.
A.
Berezhkovskii
and
A.
Szabo
,
J. Chem. Phys.
135
,
074108
(
2011
).
39.
G.
van der Zwan
and
J. T.
Hynes
,
J. Chem. Phys.
77
,
1295
(
1982
).
40.
J. T.
Hynes
,
Annu. Rev. Phys. Chem.
36
,
573
(
1985
).
41.
B.
Peters
,
P. G.
Bolhuis
,
R. G.
Mullen
, and
J.-E.
Shea
,
J. Chem. Phys.
138
,
054106
(
2013
).
42.
B.
Peters
,
J. Chem. Phys.
135
,
044107
(
2011
).
43.
P.
Tiwary
and
M.
Parrinello
,
Phys. Rev. Lett.
111
,
230602
(
2013
).
44.
M.
Rao
,
B.
Berne
, and
M.
Kalos
,
J. Chem. Phys.
68
,
1325
(
1978
).
45.
H.
Wang
,
H.
Gould
, and
W.
Klein
,
Phys. Rev. E
76
,
031604
(
2007
).
46.
V.
Kalikmanov
,
J.
Wölk
, and
T.
Kraska
,
J. Chem. Phys.
128
,
124506
(
2008
).
47.
P. R.
ten Wolde
and
D.
Frenkel
,
Science
277
,
1975
(
1997
).
48.
F.
Trudu
,
D.
Donadio
, and
M.
Parrinello
,
Phys. Rev. Lett.
97
,
105701
(
2006
).
49.
D.
Moroni
,
P. R.
ten Wolde
, and
P. G.
Bolhuis
,
Phys. Rev. Lett.
94
,
235703
(
2005
).
50.
S. G.
Kwon
,
G.
Krylova
,
P. J.
Phillips
,
R. F.
Klie
,
S.
Chattopadhyay
,
T.
Shibata
,
E. E.
Bunel
,
Y.
Liu
,
V. B.
Prakapenka
,
B.
Lee
 et al,
Nat. Mater.
14
,
215
(
2015
).
51.
J.
Zhou
,
Y.
Yang
,
Y.
Yang
,
D. S.
Kim
,
A.
Yuan
,
X.
Tian
,
C.
Ophus
,
F.
Sun
,
A. K.
Schmid
,
M.
Nathanson
 et al,
Nature
570
,
500
(
2019
).
52.
P. R.
ten Wolde
and
D.
Frenkel
,
J. Chem. Phys.
109
,
9901
(
1998
).
53.
G. A.
Tribello
,
F.
Giberti
,
G. C.
Sosso
,
M.
Salvalaglio
, and
M.
Parrinello
,
J. Chem. Theory Comput.
13
,
1317
(
2017
).
54.
G. A.
Tribello
,
M.
Ceriotti
, and
M.
Parrinello
,
Proc. Natl. Acad. Sci. U. S. A.
107
,
17509
(
2010
).
55.
S. M.
Kathmann
,
G. K.
Schenter
, and
B. C.
Garrett
,
J. Chem. Phys.
120
,
9133
(
2004
).
56.
S. I.
Resnick
,
Adventures in Stochastic Processes
(
Springer Science & Business Media
,
2013
).
57.
M.
Salvalaglio
,
P.
Tiwary
, and
M.
Parrinello
,
J. Chem. Theor. Comput.
10
,
1420
(
2014
).
58.
G.
Bussi
,
D.
Donadio
, and
M.
Parrinello
,
J. Chem. Phys.
126
,
014101
(
2007
).
59.
E.
Lindahl
,
B.
Hess
, and
D.
van der Spoel
,
Mol. Model. Annu.
7
,
306
(
2001
).
60.
C.
Camilloni
,
M.
Bonomi
, and
G.
Bussi
,
Nat. Methods
16
,
670
(
2019
).
61.
P. D.
Dixit
,
A.
Jain
,
G.
Stock
, and
K. A.
Dill
,
J. Chem. Theor. Comput.
11
,
5464
(
2015
).
62.
E. T.
Jaynes
,
Annu. Rev. Phys. Chem.
31
,
579
(
1980
).
63.
S.
Pressé
,
K.
Ghosh
,
J.
Lee
, and
K. A.
Dill
,
Rev. Mod. Phys.
85
,
1115
(
2013
).
64.
P. D.
Dixit
,
J.
Wagoner
,
C.
Weistuch
,
S.
Pressé
,
K.
Ghosh
, and
K. A.
Dill
,
J. Chem. Phys.
148
,
010901
(
2018
).
65.
P.
Tiwary
and
M.
Parrinello
,
J. Phys. Chem. B
119
,
736
(
2014
).
66.
D.
Bicout
and
A.
Szabo
,
J. Chem. Phys.
109
,
2325
(
1998
).
67.
R.
Casasnovas
,
V.
Limongelli
,
P.
Tiwary
,
P.
Carloni
, and
M.
Parrinello
,
J. Am. Chem. Soc.
139
,
4780
(
2017
).
68.
B.
Peters
and
B. L.
Trout
,
J. Chem. Phys.
125
,
054108
(
2006
).
69.
R. B.
Best
and
G.
Hummer
,
Proc. Natl. Acad. Sci. U. S. A.
102
,
6732
(
2005
).
70.
K.
Binder
,
Phys. Rev. A
29
,
341
(
1984
).

Supplementary Material

You do not currently have access to this content.