Long-time integration methods for mesoscopic models of pattern-forming systems

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Abstract

Spectral methods for simulation of a mesoscopic diffusion model of surface pattern formation are evaluated for long simulation times. Backwards-differencing time-integration, coupled with an underlying Newton–Krylov nonlinear solver (SUNDIALS-CVODE), is found to substantially accelerate simulations, without the typical requirement of preconditioning. Quasi-equilibrium simulations of patterned phases predicted by the model are shown to agree well with linear stability analysis. Simulation results of the effect of repulsive particle–particle interactions on pattern relaxation time and short/long-range order are discussed.

Introduction

Mesoscopic models [1], [2] are relevant to a broad set of physical phenomena [1], [2], [3]. Some examples include adsorption/desorption processes [2], transport and reaction through nanoporous films [4], non-Debye relaxation mechanisms [5], and heteroepitaxially-driven pattern formation [6], [7], [8], [9]. This range of applications highlights the versatility of mesoscopic modeling for different types of particle interactions, transition mechanisms, and dynamics on multiple time/length scales.

Mesoscopic models are derived through coarse-graining of microscopic lattice-models, and thus, they explicitly account for particle/particle interactions. These mesoscopic models can then be solved using less computationally intensive partial differential equation (PDE) techniques, compared to stochastic simulation of the underlying microscopic model, e.g., kinetic Monte Carlo [10].

The solution of mesoscopic models poses unique challenges to standard computational methods for PDEs. The standard approach involves local finite-element method techniques with spatial refinement and adaptivity [11], [12], [13]. These methods rely on the use of spatial discretization via localized basis functions. Past work has shown that global methods [14], such as the Fourier spectral method, provide distinct advantages over localized methods for mesoscopic models due to the presence of global interaction terms in the form a convolution:Jc=J(r-r)c(r)dr.These global terms present new challenges to the acceleration and scalability of computational methods for the simulation of mesoscopic models.

The overall objective of this work is to evaluate time-integration techniques suited to the simulation of mesoscopic models. The specific objectives of this work are:

  • 1.

    Perform a simulation study of the mesoscopic diffusion model for various time-integration methods using the Fourier spectral method for spatial discretization.

  • 2.

    Compare simulation results from the full nonlinear model to the predictions from past linear stability analysis [10].

  • 3.

    Predict the effect of repulsion strength on quasi-equilibrium pattern characteristics predicted via full simulation, including pattern quality, relaxation time, and short/long-range order.

The paper is organized as follows: Section 2 provides a concise introduction to a prototype experimental system (Pb/Cu heteroepitaxy) and the models used. In Section 3 we summarize the computational methods used and present the simulation conditions. In Section 4, the efficiency of our time-integration schemes is thoroughly analyzed and quasi-equilibrium simulation results are presented and evaluated. Finally, conclusions are drawn in Section 5.

Section snippets

Mesoscopic diffusion model

Focusing on pattern formation, a prototype experimental system is Pb/Cu (111) heteroepitaxy [6], [7], [8], [9]. Past work has modeled this material system using a mesoscopic diffusion model [3], [10] with particle/particle interactions consisting of a combination of short-range attractive and long-range repulsive components. This past work shows, through a linear analysis, that the this mesoscopic diffusion model predicts modulated phases [15] similar to experimental observations of the Pb/Cu

Methodology

Past numerical approaches to simulation of mesoscopic model PDEs have employed collocation methods, such as finite difference methods [2], [14], to approximate spatial derivatives and single-step explicit integration methods for solution of the resulting time-dependent ordinary differential equation (ODE) system. An alternate approach [14], using the Fourier spectral method, was shown to have a significant advantage in that convolution terms fg (Eq. (1)) are efficiently evaluated in the

Results and discussion

Simulations using the conditions described in Section 3 were performed for a range of repulsion strengths χ = (0.25, 0.50, 0.75) (see Eq. (9)). Simulations were performed at a fixed temperature and coverage of (Table 1) predicted by dominant mode analysis to be in a stable hexagonal or square pattern regime [10]. This past work [10] also predicted other stable patterned phases including stripe and inverse hexagonal at intermediate and high coverage values, respectively. Simulations of these higher

Conclusions

A simulation study of pattern formation via a mesoscopic diffusion model was conducted. Various time-integration schemes, based on global spectral collocation methods, were evaluated for pattern formation over a range of particle repulsion strengths. Quasi-equilibrium phase behavior was computed and validated with past linearized analysis. Furthermore, roughness, segregation length-scale and relaxation time were predicted.

Computational findings from this work show that the additional

Acknowledgements

This work was supported, in part, by the National Science Foundation (USA, CMMI-0835673) and the Natural Science and Engineering Research Council of Canada (NMA).

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