Numerical study of metastability due to tunneling: The quantum string method

We generalize the string method, originally designed for the study of thermally activated rare events, to the calculation of quantum tunneling rates. This generalization is based on the formal analogy between quantum mechanics and statistical mechanics in the path-integral formalism. The quantum string method first locates the minimal action path (MAP), which is a smooth curve connecting two minima of the imaginary-time action in the space of imaginary-time trajectories. From the MAP, the saddle point of the action (called “the bounce”) associated with the exponential factor for barrier tunneling probability is obtained and the pre-exponential factor (the ratio of determinants) for the tunneling rate evaluated using stochastic simulation. The quantum string method is implemented to calculate the bounce and rate of tunneling for the Mueller potential in two dimensions. The quantum problem is much more difficult than the thermally activated barrier crossing problem for the same potential. The model calculations show the string method to be an efficient numerical tool for the study of barrier tunneling in higher dimension, from the determination of the saddle point to the computation of the pre-exponential factor.     

Multiscale modeling of the dynamics of solids at finite temperature

We develop a general multiscale method for coupling atomistic and continuum simulations using the framework of the heterogeneous multiscale method (HMM). Both the atomistic and the continuum models are formulated in the form of conservation laws of mass, momentum and energy. A macroscale solver, here the finite volume scheme, is used everywhere on a macrogrid; whenever necessary the macroscale fluxes are computed using the microscale model, which is in turn constrained by the local macrostate of the system, e.g. the deformation gradient tensor, the mean velocity and the local temperature. We discuss how these constraints can be imposed in the form of boundary conditions. When isolated defects are present, we develop an additional strategy for defect tracking. This method naturally decouples the atomistic time scales from the continuum time scale. Applications to shock propagation, thermal expansion, phase boundary and twin boundary dynamics are presented.     

Gauge finite element method for incompressible flows

A finite element method for computing viscous incompressible flows based on the gauge formulation introduced in [Weinan E, Liu J‐G. Gauge method for viscous incompressible flows. Journal of Computational Physics (submitted)] is presented. This formulation replaces the pressure by a gauge variable. This new gauge variable is a numerical tool and differs from the standard gauge variable that arises from decomposing a compressible velocity field. It has the advantage that an additional boundary condition can be assigned to the gauge variable, thus eliminating the issue of a pressure boundary condition associated with the original primitive variable formulation. The computational task is then reduced to solving standard heat and Poisson equations, which are approximated by straightforward, piecewise linear (or higher‐order) finite elements. This method can achieve high‐order accuracy at a cost comparable with that of solving standard heat and Poisson equations. It is naturally adapted to complex geometry and it is much simpler than traditional finite element methods for incompressible flows. Several numerical examples on both structured and unstructured grids are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Blowup of solutions of the unsteady Prandtl's equation

We prove that for certain classes of compactly supported C^∞ initial data, smooth solutions of the unsteady Prandtl's equation blow up in finite time. © 1997 John Wiley & Sons, Inc.     

Collapse transition in thin films of poly(methoxydiethylenglycol acrylate)

The thermal behavior of poly(methoxydiethylenglycol acrylate) (PMDEGA) is studied in thin hydrogel films on solid supports and is compared with the behavior in aqueous solution. The PMDEGA hydrogel film thickness is varied from 2 to 422?nm. Initially, these films are homogenous, as measured with optical microscopy, atomic force microscopy, X-ray reflectivity, and grazing-incidence small-angle X-ray scattering (GISAXS). However, they tend to de-wet when stored under ambient conditions. Along the surface normal, no long-ranged correlations between substrate and film surface are detected with GISAXS, due to the high mobility of the polymer at room temperature. The swelling of the hydrogel films as a function of the water vapor pressure and the temperature are probed for saturated water vapor pressures between 2,380 and 3,170?Pa. While the swelling capability is found to increase with water vapor pressure, swelling in dependence on the temperature revealed a collapse phase transition of a lower critical solution temperature type. The transition temperature decreases from 40.6?°C to 36.6?°C with increasing film thickness, but is independent of the thickness for very thin films below a thickness of 40?nm. The observed transition temperature range compares well with the cloud points observed in dilute (0.1?wt.%) and semi-dilute (5?wt.%) solution which decrease from 45?°C to 39?°C with increasing concentration.     

Kinetic trapping of a cobalt(ii) metallocage using a carbazole-containing expanded carbaporphyrinoid ligand

The meso-unsubstituted expanded porphyrinoid 3, incorporating two carbazole moieties, acts as an effective ligand for Co(ii) and permits the isolation and X-ray diffraction-based characterization of a 6 : 3 metal-to-ligand metallocage complex that converts spontaneously to the constituent 2 : 1 metal-to-ligand metalloring species in chloroform solution. The discrete metalloring is formed directly when the Co(ii) complex is crystallized from supersaturated solutions, whereas crystallization from more dilute solutions favors the metallocage. Studies with two other test cations, Pd(ii) and Zn(ii), revealed exclusive formation of the monomeric metalloring complexes with no evidence of higher order species being formed. Structural, electrochemical and UV-vis-NIR absorption spectral studies provide support for the conclusion that the Pd(ii) complex is less distorted and more effectively conjugated than its Co(ii) and Zn(ii) congeners, an inference further supported by TD-DFT calculations. The findings reported here underscore how expanded porphyrins can support coordination modes, including bimetallic complexes and self-assembled cage structures, that are not necessarily easy to access using more traditional ligand systems.

Carbazole containing expanded carbaporphyrinoid ligand supports the formation of 2 : 1 metal-to-ligand complexes with Pd, Co, and Zn. Solid-state studies also revealed formation of a 6 : 3 metal-to-ligand metallocage in the case of Co complexation.     

Convergence of a Stochastic Method for the Modeling of Polymeric Fluids

We present a convergence analysis of a stochastic method for numerical modeling of complex fluids using Brownian configuration fields (BCF) for shear flows. The analysis takes into account the special structure of the stochastic partial differential equations for shear flows. We establish the optimal rate of convergence. We also analyze the nature of the error by providing its leading order asymptotics.     

Minimum action method for the study of rare events

The least action principle from the Wentzell‐Freidlin theory of large deviations is exploited as a numerical tool for finding the optimal dynamical paths in spatially extended systems driven by a small noise. The action is discretized and a preconditioned BFGS method is used to optimize the discrete action. Applications are presented for thermally activated reversal in the Ginzburg‐Landau model in one and two dimensions, and for noise induced excursion events in the Brusselator taken as an example of non‐gradient system arising in chemistry. In the Ginzburg‐Landau model, the reversal proceeds via interesting nucleation events, followed by propagation of domain walls. The issue of nucleation versus propagation is discussed and the scaling for the number of nucleation events as a function of the reversal time and other material parameters is computed. Good agreement is found with the numerical results. In the Brusselator, whose deterministic dynamics has a single stable equilibrium state, the presence of noise is shown to induce large excursions by which the system cycles out of this equilibrium state. © 2004 Wiley Periodicals, Inc.     

Well-Posedness for the Dumbbell Model of Polymeric Fluids

The dumbbell model is a coupled hydrodynamic-kinetic model for polymeric fluids in which the configurations of the dumbbells are described by stochastic differential equations. We prove well-posedness of this model by deriving directly a priori estimates on the stochastic model. Our results can be used to analyze stochastic simulation methods such as the ones that are based on Brownian configuration fields.Supported by ONR grant N00014-01-1-0674 and National Science Foundation of China through a Class B Award for Distinguished Young Scholars 10128102.Partially supported by the special funds for Major State Research Projects G1999032804 and National Science Foundation of China for Distinguished Young Scholars 10225103.