Simulation lattice-based

So far, we have demonstrated that the MC simulation (lattice-based SRS model and off-lattice bead-spring model) results are in qualitative agreement with the experiments. A complementary approach is molecular dynamics (MD) simulation using the bead-spring model. Since MD study for PFPE is still the infant stage, we will discuss it only briefly. The equation of motion can be expressed in... 

Although this section has concentrated on MD, it should not be forgotten that lattice-based MC codes may be parallelized very efficiently for more infomration on parallel simulation methods see [220. 221. 222 and 223] and references therein. 

J. A. Kulkarni and A. N. Bens, Lattice-based Simulation of Chain Conformation in Semicrystalline Polymers with Application to Flow Induced Crystallization, J. Non-Newt. Fluid Meek, 82, 331-336 (1999). 

Using periodic boundary conditions (BCs), our simulations are based on a lattice construction set up earlier by Berg [135]. We have performed multicanonical MC simulations for the two models with the lattice sizes that correspond to the number of water molecules N = 128, 360, 576, 896, and 1,600. Combining the two fit results in the thermodynamic limit (N —> oo) leads to our final estimate... 

These simulation studies, based on either static lattice, MD or ab initio methods, have been able to provide deeper insight as to the fundamental solid state properties at the atomic level, particularly in the following key areas (a) defect chemistry and dopant-vacancy association, (b) mechanisms of oxygen ion migration, (c) structures and stability of surfaces (mainly (110) and (111)), (d) energetics of redox reactions... 

The master equation, however, can only be solved analytically for very simple systems such as the gas-phase reaction A—>B. The analysis of these systems typically requires numerical simulation of a lattice-based kinetic Monte Carlo model. The lattice gas model can then be used to formulate the respective transition probabilities in order to solve the master equationThe groups of both Zhdanov[ ° ° ] and Kreuzerl ° l have been instrumental in demonstrating the application of lattice gas models to solve adsorption and desorption processed from surfaces. Once a lattice model has been formulated there are three types of solution ... 

Commercial codes, e.g. PowerFLOW, which use lattice-based approaches are available, and this particular code was used in the present work. Based on discrete forms of the kinetic theory equations, this code employs an approach that is an extension of lattice gas and lattice Boltzmann methods in which particles exist at discrete locations in space, and are allowed to move in given directions at particular speeds over discrete time intervals. The particles reside on a cubic lattice composed of voxels, and move from one voxel to another at each time step. Solid surfaces are accommodated through the use of surface elements, and arbitrary surface shapes can be represented. Particle advection, and particle-particle and particle-surface interactions, are all considered at a microscopic level to simulate fluid behaviour in a way which ensures conservation of mass, momentum and energy, and which recovers solutions of the continuum flow... 

The MC and MD simulation approaches have become viable only after the introduction of fast computers. Starting from the pioneering works of Metropolis etal. [101] and Alder and Wainwright [102], the basic algorithms on which computer simulations are based were developed in the ensuing 20-30 years. They are now well established and described in standard textbooks [95,96], and able to provide a useful link between experiment and theory. Nowadays MC simulations are typically used for lattice and simple off-lattice models, while MD models are largely employed for atomistic systems (which are tricky to sample with MC) but also for coarse-grained models. 

Cell dynamics simulations are based on the time dependence of an order parameter, (i) (Eq. 1.23), which varies continuously with coordinate r. For example, this can be the concentration of one species in a binary blend. An equation is written for the time evolution of the order parameter, dir/dt, in terms of the gradient of a free energy that controls, for example, the tendency for local diffusional motions. The corresponding differential equation is solved on a lattice, i.e. the order parameter V (r) is discretized on a lattice, taking a value at lattice point i. This method is useful for modelling long time-scale dynamics such as those associated with phase separation processes. 

In the foregoing chapters, the simulation is based on the macroscopic point of view that the fluid is continuous medium and its physical properties, such as density, velocity, and pressure, are functions of time and space. Thus, the Navier—Stokes equation can be employed as modeling equation in the mathematical simulation. In this chapter, we turn to the mesoscopic point of view and use the lattice Boltzmann... 

For a given material, this input takes the form of an activation enthalpy versus shear stress curve and phonon-drag coefficients calculated for each pressure under consideration. Here full activation enthalpy curves have been calculated at selected pressures in Ta, Mo, and V, and phonon drag has been studied as a function of pressure and temperature in the case of Ta. These results have been fitted and modeled in suitable analytic forms to interface smoothly with the DD simulation codes. Detailed DD simulations have then been carried out in Ta and Mo as a function of pressure, temperature, and strain rate. Our DD simulations have been performed in part with the pioneering lattice-based serial code developed for bcc metals [21,22] but even more extensively with the general node-based Parallel Dislocation Simulator (ParaDiS) code recently developed at the Lawrence Livermore National Laboratory [27-30]. 

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