Robust directed self-assembly of non-periodic nanoscale structures is a key process that would enable various technological breakthroughs. The dynamic evolution of directed self-assemblies towards structures with desired geometries is governed by the rugged potential energy surface of nanoscale systems, potentially leading the system to kinetic traps. To study such phenomena and to set the framework for the directed self-assembly of nanoparticles towards structures with desired geometries, the development of a dynamic model involving a master equation to simulate the directed self-assembly process is presented. The model describes the probability of each possible configuration of a fixed number of nanoparticles on a domain, including parametric sensitivities that can be used for optimization, as a function of time during self-assembly. An algorithm is presented that solves large-scale instances of the model with linear computational complexity. Case studies illustrate the influence of several degrees of freedom on directed self-assembly. A design approach that systematically decomposes the ergodicity of the system to direct self-assembly of a targeted configuration with high probability is illustrated. The prospects for extending such an approach to larger systems using coarse graining techniques are also discussed.

1.
N.
Stephanopoulos
,
E. O. P.
Solis
, and
G.
Stephanopoulos
,
AIChE J.
51
,
1858
(
2005
).
2.
K. J. M.
Bishop
,
C. E.
Wilmer
,
S.
Soh
, and
B. A.
Grzybowski
,
Small
5
,
1600
(
2009
).
3.
D. A.
Walker
,
K. P.
Browne
,
B.
Kowalczyk
, and
B. A.
Grzybowski
,
Angew. Chem., Int. Ed.
49
,
6760
(
2010
).
4.
D. A.
Walker
,
C. E.
Wilmer
,
B.
Kowalczyk
,
K. J. M.
Bishop
, and
B. A.
Grzybowski
,
Nano Lett.
10
,
2275
(
2010
).
5.
S. J.
Koh
,
Nanoscale Res. Lett.
2
,
519
(
2007
).
6.
B. D.
Gates
,
Q.
Xu
,
M.
Stewart
,
D.
Ryan
,
C. G.
Wilson
, and
G. M.
Whitesides
,
Chem. Rev.
105
,
1171
(
2005
).
7.
E. O.
Solis
,
P. I.
Barton
, and
G.
Stephanopoulos
,
Ind. Eng. Chem. Res.
49
,
7728
(
2010
).
8.
E. O.
Solis
,
P. I.
Barton
, and
G.
Stephanopoulos
,
Ind. Eng. Chem. Res.
49
,
7746
(
2010
).
9.
M. F.
Hagan
,
O. M.
Elrad
, and
R. L.
Jack
,
J. Chem. Phys.
135
,
104115
(
2011
).
10.
Y.
Min
,
M.
Akbulut
,
K.
Kristiansen
,
Y.
Golan
, and
J.
Israelachvili
,
Nature Mater.
7
,
527
(
2008
).
11.
S.
Whitelam
and
P. L.
Geissler
,
J. Chem. Phys.
127
,
154101
(
2007
).
12.
V.
Munoz
,
E. R.
Henry
,
J.
Hofrichter
, and
W. A.
Eaton
,
Proc. Natl. Acad. Sci. USA
95
,
5872
(
1998
).
13.
M.
Cieplak
,
M.
Henkel
,
J.
Karbowski
, and
J. R.
Banavar
,
Phys. Rev. Lett.
80
,
3654
(
1998
).
14.
S. B.
Ozkan
,
K. A.
Dill
, and
I.
Bahar
,
Protein Sci.
11
,
1958
(
2002
).
15.
S.
Kachalo
,
H.-M.
Lu
, and
J.
Liang
,
Phys. Rev. Lett.
96
,
058106
(
2006
).
16.
H.-X.
Zhou
,
J. Chem. Phys.
128
,
195104
(
2008
).
17.
D. T.
Gillespie
,
J. Phys. Chem.
81
,
2340
(
1977
).
18.
D. T.
Gillespie
,
Annu. Rev. Phys. Chem.
58
,
35
(
2007
).
19.
R.
Gunawan
,
Y.
Cao
,
L.
Petzold
, and
F. J.
Doyle
,
Biophys. J.
88
,
2530
(
2005
).
20.
M.
Rathinam
,
P. W.
Sheppard
, and
M.
Khammash
,
J. Chem. Phys.
132
,
034103
(
2010
).
21.
B.
Munsky
and
M.
Khammash
,
J. Chem. Phys.
124
,
044104
(
2006
).
22.
B.
Munsky
and
M.
Khammash
,
IEEE Trans. Autom. Control
53
,
201
(
2008
).
23.
J. E.
Tolsma
and
P. I.
Barton
,
SIAM J. Sci. Comput.
23
,
1861
(
2002
).
24.
H. C.
Kang
and
W. H.
Weinberg
,
Acc. Chem. Res.
25
,
253
(
1992
).
25.
H. C.
Kang
and
W. H.
Weinberg
,
Phys. Rev. B
41
,
2234
(
1990
).
26.
K. A.
Fichthorn
and
W. H.
Weinberg
,
Phys. Rev. Lett.
68
,
604
(
1992
).
27.
H. C.
Kang
and
W. H.
Weinberg
,
Surf. Sci.
299
,
755
(
1994
).
28.
H.
Huang
,
G. H.
Gilmer
, and
T. D.
de la Rubia
,
J. Appl. Phys.
84
,
3636
(
1998
).
29.
G. H.
Gilmer
,
H.
Huang
, and
C.
Roland
,
Comp. Mater. Sci.
12
,
354
(
1998
).
30.
G. H.
Gilmer
and
P.
Bennema
,
J. Appl. Phys.
43
,
1347
(
1972
).
31.
K. A.
Fichthorn
and
W. H.
Weinberg
,
J. Chem. Phys.
95
,
1090
(
1991
).
32.
P. N.
Brown
,
G. D.
Byrne
, and
A. C.
Hindmarsh
,
SIAM J. Sci. Stat. Comput.
10
,
1038
(
1989
).
33.
Y.
Saad
and
M. H.
Schultz
,
SIAM J. Sci. Stat. Comput.
7
,
856
(
1986
).
34.
I. S.
Duff
and
J. K.
Reid
,
ACM Trans. Math. Softw.
22
,
187
(
1996
).
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