Generation of Kinetic "Phase Diagrams" for Self-Assembled Nanopattern Formation in Heteroepitaxy Via Hierarchical Multiscale Modeling

Abhijit Chatterjee, Theoretical Division, Los Alamos National Laboratory, T-12 MS B268, Los Alamos, NM 87545 and Dionisios G. Vlachos, University of Delaware, Department of Chemical Engineering, 150 Academy St., 325 Colubrn Lab, Newark, DE 19716.

Kinetic Monte Carlo (KMC) is an ideal tool for studying dynamics and steady state of noise-controlled phenomena. However, KMC can typically handle only short length and time scales. The growing interest in the design and control of nano- and micro-scale devices necessitates the extension of KMC to larger scales, with the objective of capturing accurately and efficiently both short and large-scale phenomena. In this poster we present two complementary multiscale techniques to study self-assembled nanopattern formation over realistic scales (length scales ranging from Angstroms to microns, and time scales ranging from nanoseconds to seconds). The experimental system [1] we model exhibits nanodisc, labyrinth, and “inverted nanodisc” formation resulting from an interplay between competing short ranged attractive and long ranged repulsive elastic interactions. The first multiscale technique comprises of partial differential equations, called mesoscopic equations, derived via coarse-graining the underlying master equation by passing to the continuum limit. Mesoscopic equations are directly linked to the microscopic physics but because of the lack of noise terms, they can be viewed as low hierarchical tools in modeling self-assembly. In comparison to computationally intensive stochastic simulations, these continuum models render the generation of the dynamic phase diagram of patterns shapes, and scaling laws for pattern feature size, shape, and growth times in terms of the interaction potential parameters, substrate temperature, film thickness and material properties a rather easy task. The second approach we propose for studying pattern formation is the newly developed coarse-grained Monte Carlo (CGMC) method of our group [2-4]. CGMC is a stochastic, multiscale modeling tool obtained by spatially coarse-graining the same underlying master equation used for the continuum mesoscopic equations. It has been found that CGMC accurately captures the dynamics, steady state and noise. CGMC can be thought of as an ideal high level tool in the hierarchy of models for investigating the role of thermal fluctuations in nucleation of materials' patterns and defects. However, it is computationally more expensive and not easily amenable to nonlinear analysis techniques. In this poster we propose to use the two methods symbiotically. Based on the phase diagrams obtained from analysis of mesoscopic equations, the effect of noise on pattern nucleation and shape of patterns is efficiently explored using CGMC and compared to experimental data [1].

References [1] Plass, R., Last, J.A., et al. (2001), "Self-assembled domain patterns", Nature, 412, 875. [2] M. Katsoulakis, A. J. Majda, and D. G. Vlachos (2003) "Coarse-grained stochastic processes for microscopic lattice systems", Proc. Natl. Acad. Sci. 100(3), 782-787. [3] M. A. Katsoulakis and D. G. Vlachos (2003) "Coarse-grained stochastic processes and kinetic Monte Carlo simulators for the diffusion of interacting particles", J. Chem. Phys. 119(18), 9412-9428. [4] Chatterjee, A., Vlachos, D.G., & Katsoulakis, M.A. (2004). Spatially adaptive lattice coarse-grained monte carlo simulations for diffusion of interacting molecules. Journal of Chemical Physics, 121(22), 11420-11431.