(121f) A Master-Equation Approach to Simulate Directed Self Assembly

Robust
fabrication of nanoscale structures is of great importance to enable
technological breakthroughs in various fields such as molecular computing,
diagnostics, and molecular factories [1]. Top-down methods such as e-beam
lithography can be used to produce non-periodic nanoscale structures with a
resolution in the range of 15 nm [2]. Bottom-up methods driven by self assembly
can be used to fabricate nanoscale structures with unmatched resolution at the
molecular scale. Fabrication of nanoscale structures via self assembly has been
demonstrated particularly for periodic structures. A key challenge is also to
fabricate robustly non-periodic nanoscale structure via directed self assembly.
Top-down methods such as e-beam lithography can be used to fabricate
nano-electrodes, which in turn can direct the self-assembly process by placing
time-varying controls in the form of electrostatic charges on a scaffold [3][4].
Static charge can also be placed on a scaffold by using micro-contact printing
[5]. Electrostatic forces are particularly relevant for directing self assembly
due to the tuneable direction and strength of these forces [6]. Furthermore,
nanoparticles can be specifically functionalized to manipulate interactions
between nanoparticles. The often competing forces at the nanoscale have to be
engineered such that the self assembly will be directed towards a desired
nanostructure.

            Solis
et al. [7][8] addressed the influence of local point charges on directed self
assembly. The point charges interact with nanoparticles to direct their self
assembly into a desired configuration. First, Solis et al. [7] developed a
method to stabilize a desired configuration. Secondly, Solis et al. [8] developed
a method to arrive at a desired configuration with maximum probability from an
unknown initial structure. A sequence of pseudostatic problems approximates the
overall dynamic problem. Although such an approach is an important step forward
as it protects the system from favoring undesired configurations, it fails to
predict the influence of kinetic traps during the course of a specific
pseudostatic phase. Self-assembled systems are prone to kinetic traps [6],
which can dynamically arrest the system in local minima of the potential energy
surface.

            The
aim of this contribution is to demonstrate the application of master equations
to simulate directed self assembly of nanoparticles including kinetic trapping.
Master equations describe time-varying continuous-time Markov processes. The
states represent the probabilities that the system will be in a certain
configuration at a given time. Models based on master equations have been
successfully applied to simulate kinetic traps in related fields such as the
folding kinetics of proteins. A generic model framework will be presented based
on the Ising lattice model to describe self assembly. The number of states in these
models grows exponentially with the size of the system. Therefore, particular
emphasis has been paid to the design of efficient computational algorithms. The
variable-coefficient ODE solver DVODPK [9] with pre-conditioned iterative
solver GMRES is used to solve the resulting sparse and stiff system of ODEs.
The complete algorithm runs in linear computational complexity, which makes the
algorithm favorable for large systems.

            The
effect of the various degrees of freedom to direct self assembly will be illustrated
for several case studies. First, the capability of the model to simulate
kinetic trapping of a subset of state space will be illustrated. Second, the
influence of temperature with respect to kinetic trapping will be illustrated.
Third, the influence of the relative strengths between various inter-particle
interaction forces is illustrated. Fourth, several degrees of freedom will be
combined to direct the self assembly of a system systematically towards a
desired configuration by reducing the ergodicity of the system in a specific
manner. The strategy is to first direct the desired number of nanoparticles to
a desired part of the domain. Subsequently, the ergodicity of the system is
broken by restricting transitions that involve the movement of particles
between the desired parts of the domain. Finally, the nanoparticles are
directed to their precise location. A strategy that utilizes ergodicity
breaking offers the prospect for approaching larger systems. The exponential
growth of the model inevitably requires coarse graining techniques to be able
to approach larger systems mathematically. The systematic decomposition of
system ergodicity allows for the application of such coarse graining techniques
in a natural way.

            The
master equations are solved explicitly for each state variable to yield the
time evolution of the entire probability distribution. Following this approach,
the dynamic trajectories of parametric sensitivities can in principle be
calculated, which are useful for dynamic optimization. An interesting direction
for future research is to apply rigorous dynamic optimization to reveal optimal
and robust fabrication routes towards non-periodic nanoscale structures via
self assembly.

References

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