Modelling random structure methods

Random structure methods have proved useful in solving structures from X-ray powder diffraction patterns. The unit cell can usually be found from these patterns, but the normal single-crystal techniques for solving the structure cannot be used. A variation on this technique, the reverse Monte Carlo method, includes in the cost function the difference between the observed powder diffraction pattern and the powder pattern calculated from the model (McGreevy 1997). It is, however, always necessary to include some chemical information if the correct structure is to be found. Various constraints can be added to the cost function, such as target coordination numbers or the deviation between the bond valence sum and atomic valence (Adams and Swenson 2000b Swenson and Adams 2001). 

Stations and Inglesia [44] simulated for comparison the ordered surface of a crystal obtained by cutting the bulk material, the unrelaxed cut-off amorphous surface, as well as the latter relaxed. The last case was the random structure created by the Monte Carlo sphere packing method. They calculated the adsorption potential surface for some weakly bound adsorbates (N2, Ar, CH4) with the aim of judging the fidelity of the surface models by comparison with the available experimental data on the heats of adsorption and surface diffusivity. The adsorption energy profile in Fig. 5.12 gives an interesting look of the surface from the point of view of the problems discussed in this book the concrete data will be called for an analysis in later sections. 

The Calculation Results. The calculations were made for a two-component medium. Calculations were executed for a two-component 3D composite with random structure. First we shall consider a comparison of the outcome for the effective conductivity calculated by means of the iterative method with the calculation using formulas (240) obtained on the basis of the effective medium theory model. 

Chaotic fractal sets on rectangular lattices have been used to the define the effective conductivity of the composite material. The effective conductivity of the composite material is defined using the fractal random structure model of a composite and the iteration method of averaging. Comparison of the calculation with experimental data is also given. 

Molecular model-building (conformational search) methods fall into two general classes systematic and random. - Systematic methods search all possible combinations of torsional angles, whereas random methods usually involve a Monte Carlo (with Metropolis sampling ) or molecular dynamics trajectory. Both approaches attempt to search large areas of conformational space and eventually converge on the desired conformation or structure. Dis-... 

Although most of the amorphous materials modelled by MD have been prepared by melting either a crystalline or random structure and quenching the resulting melt to generate the appropriate glassy structure, other methods of preparation have also been used such as pressure induced amorphization, defect induced amorphization, and radiation induced amorphization. Examples will be considered below. 

In the present example the confl iration of the solid sites is build to model the mesoporous structure of a porous glass. Each sample of the glass material is obtained with the Gaussian random field method [30], During a calculation, we use periodic boundary conditions in all directions of space. An illustration of a Vycor glass sample obtained with the Gaussian random field is reported on Fig. 1. We use the same procedure for CPG. 

Figure 5 Optimization of the objective function in Modeller. Optimization of the objective function (curve) starts with a random or distorted model structure. The iteration number is indicated below each sample structure. The first approximately 2000 iterations coiTespond to the variable target function method [82] relying on the conjugate gradients technique. This approach first satisfies sequentially local restraints, then slowly introduces longer range restraints until the complete objective function IS optimized. In the remaining 4750 iterations, molecular dynamics with simulated annealing is used to refine the model [83]. CPU time needed to generate one model is about 2 mm for a 250 residue protein on a medium-sized workstation.

But a computer simulation is more than a few clever data structures. We need algorithms to manipulate our system. In some way, we have to invent ways to let the big computer in our hands do things with the model that is useful for our needs. There are a number of ways for such a time evolution of the system the most prominent is the Monte Carlo procedure that follows an appropriate random path through configuration space in order to investigate equilibrium properties. Then there is molecular dynamics, which follows classical mechanical trajectories. There is a variety of dissipative dynamical methods, such as Brownian dynamics. All these techniques operate on the fundamental degrees of freedom of what we define to be our model. This is the common feature of computer simulations as opposed to other numerical approaches. 

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