Recent developments in computational modelling of nucleation in phase transformations

Nucleation is one of the most common physical phenomena in physical,chemical,biological and materials sciences.Owing to the complex multiscale nature of various nucleation events and the difficulties in their direct experimental observation,development of effective computational methods and modeling approaches has become very important and is bringing new light to the study of this challenging subject.Our discussions in this manuscript provide a sampler of some newly developed numerical algorithms that are widely applicable to many nucleation and phase transformation problems.We first describe some recent progress on the design of efficient numerical methods for computing saddle points and minimum energy paths,and then illustrate their applications to the study of nucleation events associated with several different physical systems.

Heterogeneous Multiscale Methods: A Review

This paper gives a systematic introduction to HMM,the heterogeneous multiscale methods,including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for a particular problem.This is illustrated by examples from several application areas,including complex fluids,micro-fluidics,solids,interface problems,stochastic problems,and statistically self-similar problems.Emphasis is given to the technical tools,such as the various constrained molecular dynamics,that have been developed,in order to apply HMM to these problems.Examples of mathematical results on the error analysis of HMM are presented.The review ends with a discussion on some of the problems that have to be solved in order to make HMM a more powerful tool.

The graph limit of the minimizer of the Onsager-Machlup functional and its computation

The Onsager-Machlup(OM)functional is well known for characterizing the most probable transition path of a diffusion process with non-vanishing noise.However,it suffers from a notorious issue that the functional is unbounded below when the specified transition time T goes to infinity.This hinders the interpretation of the results obtained by minimizing the OM functional.We provide a new perspective on this issue.Under mild conditions,we show that although the infimum of the OM functional becomes unbounded when T goes to infinity,the sequence of minimizers does contain convergent subsequences on the space of curves.The graph limit of this minimizing subsequence is an extremal of the abbreviated action functional,which is related to the OM functional via the Maupertuis principle with an optimal energy.We further propose an energy-climbing geometric minimization algorithm(EGMA)which identifies the optimal energy and the graph limit of the transition path simultaneously.This algorithm is successfully applied to several typical examples in rare event studies.Some interesting comparisons with the Freidlin-Wentzell action functional are also made.

A Level Set Method for the Simulation of Moving Contact Lines in Three Dimensions

We propose an efficient numerical method for the simulation of the twophase flows with moving contact lines in three dimensions.The mathematical model consists of the incompressible Navier-Stokes equations for the two immiscible fluids with the standard interface conditions,the Navier slip condition along the solid wall,and a contact angle condition(Ren et al.(2010)[28]).In the numerical method,the governing equations for the fluid dynamics are coupledwith an advection equation for a level-set function.The latter models the dynamics of the fluid interface.Following the standard practice,the interface conditions are taken into account by introducing a singular force on the interface in themomentum equation.This results in a single set of governing equations in the whole fluid domain.Similarly,the contact angle condition is imposed by introducing a singular force,which acts in the normal direction of the contact line,into theNavier slip condition.The newboundary condition,which unifies the Navier slip condition and the contact angle condition,is imposed along the solid wall.The model is solved using the finite difference method.Numerical results are presented for the spreading of a droplet on both homogeneous and inhomogeneous solid walls,as well as the dynamics of a droplet on an inclined plate under gravity.

Sequential Multiscale Modeling Using Sparse Representation

The main obstacle in sequential multiscale modeling is the pre-computation of the constitutive relationwhich often involvesmany independent variables.The constitutive relation of a polymeric fluid is a function of six variables,even after making the simplifying assumption that stress depends only on the rate of strain.Precomputing such a function is usually considered too expensive.Consequently the value of sequential multiscale modeling is often limited to“parameter passing”.Here we demonstrate that sparse representations can be used to drastically reduce the computational cost for precomputing functions of many variables.This strategy dramatically increases the efficiency of sequential multiscale modeling,making it very competitive in many situations.