(393y) Approximating the Solution to the Master Equation to Simulate Directed Self Assembly of Nanostructures

Reliable self assembly of nanoscale systems into a
structure of desired geometry will enable various future technological
applications such as nanoelectronic circuits and molecular factories [1].
Significant progress has already been made on the self assembly of periodic
nanostructures (for example, self-assembled monolayers [2]). However, less
progress has been achieved in fabricating systems with non-periodic
nanostructures. Self assembly, being spontaneous, fast and having high spatial
resolution, provides a choice for fabricating non-periodic nanostructures.
Various interaction forces (electrostatic, Van der Waals, hydrogen bonding,
etc.) play a role in self assembly at the nanometer scale [3]. The
nanoparticles can interact with each other and with an external force field to
form a desired structure [1]. Such a force field can be the result of nanoscale
electrodes (or actuators) that are placed on a scaffold, which can be
fabricated using various top-down methods such as lithography [4]. The key question
is to design the nanoparticles and utilize the external fields to guide self
assembly towards a desired structure rapidly and with high probability.

Lakerveld et al. [5] demonstrated efficient
algorithms to solve simultaneously a very large number of master equations that
describe the probability of finding the system in a certain configuration at a
given time during the course of directed self assembly including parametric
sensitivities. However, instances of their model eventually become
prohibitively large upon increasing the size of the physical domain as each
possible configuration is being considered explicitly.

The aim of this contribution is to demonstrate the
application of the Finite State Projection (FSP) method [6] to simulate
directed self assembly. The FSP method is based on simulation of only a subset
of the total configuration space (called a projection space). In this work, the
projection space is reduced at equal intervals of time. The projection space is
also adjusted if the error (caused by simulating only a subset of the total
configuration space) exceeds a bound. This adaptation of the projection space
is guided by the probability and flux values, and hence it brings the error
back within the bound. The adjustment of the projection space is continued
until the maximum allowable size of the projection space is reached. This
method helps in identifying the time scale over which the system can be
simulated in detail and this will depend on the computational resources and the
user specified error bounds. It also provides critical information about the
dominant configurations and the transition flux between those configurations. A
couple of case studies are chosen to demonstrate the method and useful
information is revealed about them.

References

[1] N.
Stephanopoulos, E. O. P. Solis, and G. Stephanopoulos, ?Nanoscale process
systems engineering: Toward molecular factories, synthetic cells, and adaptive
devices,? AIChE Journal, vol. 51, no. 7, pp. 1858-1869, 2005.

[2] J. C. Love, L.
A. Estroff, J. K. Kriebel, R. G. Nuzzo, and G. M. Whitesides, ?Self-assembled monolayers
of thiolates on metals as a form of nanotechnology,? Chemical Reviews, vol. 105, no. 4, pp. 1103-1170, 2005.

[3] K. J. M.
Bishop, C. E. Wilmer, S. Soh, and B. A. Grzybowski, ?Nanoscale forces and their
uses in self-assembly,? Small, vol. 5, no. 14, pp. 1600-1630, 2009.

[4] B. D. Gates, Q.
Xu, M. Stewart, D. Ryan, C. G. Willson, and G. M. Whitesides, ?New approaches
to nanofabrication: molding, printing, and other techniques,? Chemical Reviews, vol. 105, no. 4, pp. 1171-1196, 2005.

[5] R. Lakerveld,
G. Stephanopoulos, and P. I. Barton, ?A master-equation approach to simulate kinetic
traps during directed self assembly,? The
Journal of Chemical Physics, In
press, 2012.

[6] B. Munsky and
M. Khammash, ?The finite state projection algorithm for the solution of the chemical
master equation,? The Journal of
Chemical Physics, vol. 124, no. 4,
p. 044104, 2006.