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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