Zero-Temperature Equilibrium Methods

ATOMISTIC/CONTINUUM COUPLING Zero-Temperature Equilibrium Methods 

Atomistic/Continuum Coupling to Simulate Static Loads A simple case of atomistic/continuum coupling is the methodology devised by Kwon and Jung to study the atomic behavior of materials under static loads. In this case, the coupling to a continuum description is 

This method has been tested by studying atomic rearrangements for two-dimensional (2D) systems with a dislocation under shear, tensile, and compressive forces. The atomistic domain was modeled with the Morse potential. A 2D system was chosen only for computational convenience nothing in the algorithm is limited to such a dimensionality. 

The finite-element combined with atomistic modeling (FEAt) method,developed by Kohlhoff et al. in 1988, is one of the oldest atomistic/ continuum coupling methods, and it was inspirational in the development of several other coupling schemes (e.g., the one used in the CADD methodology discussed below). 

Sizewise, for three-dimensional problems zone II is actually a surface (a line for a 2D problem), while zone III should be sufficiently wide to provide a complete set of neighbors to those atoms in III that interact with atoms in II. This means that the width of zone III should be at least equal to the cutoff distance of the interatomic potential that is being used in the calculations. However, for density-dependent potentials or potentials containing three-body terms, the zone HI width should be twice that distance. 

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In the atomistic part of the procedure, particles are used as FEM nodes, even though, as will become clearer in the following, an AFEM element is quite different from the standard FEM definition. As for any zero-temperature equilibrium method, the objective is to determine the state of lowest energy, i.e., the system configuration such that... 

At the conclusion of a geometric optimization calculation, we have the equilibrium positions of all the atomic nuclei, as well as the overall electron density distributed in space (x, y, z). Many important properties, especially for an isolated single molecule at absolute zero temperature, can be obtained by solving the quantum mechanical or the molecular mechanical equations. Only the former method can produce electronic properties, such as electron distributions and dipole moments, but both methods can produce structural and energy properties. 

There have been several other methods proposed for the statistical mechanical modeling of chemical reactions. We review these techniques and explain their relationship to RCMC in this section. These simulation efforts are distinct from the many quantum mechanical studies of chemical reactions. The goal of the statistical mechanical simulations is to find the equilibrium concentration of reactants and products for chemically reactive fluid systems, taking into account temperature, pressure, and solvent effects. The goals of the quantum mechanics computations are typically to find transition states, reaction barrier heights, and reaction pathways within chemical accuracy. The quantum studies are usually performed at absolute zero temperature in the gas phase. Quantum mechanical methods are confined to the study of very small systems, so are inappropriate for the assessment of solvent effects, for example. 

The equilibrium constant for the reverse, esterification, reaction has been measured by Berthelot and P6an de St. Gilles and found to be 3.96, corresponding to 66.57 per cent esterification. The forward reaction would thus reach equilibrium at 33.43 per cent hydrolysis with K equal to 0.253. The equilibrium position was shown to be independent of the temperature. A calculation of the heat of reaction by the method of bond energies gives a value of zero, since the bonds broken are of the same type as the bonds formed. From the van t Hoff equation (see later section) the condition for a zero temperature coefficient of equilibrium is that 6H be zero. The heat of reaction for the hydrolysis of ethyl acetate, therefore, is n ligible. At a temperature of 60°C and a pressure of 5,000 atm, the equilibrium position remained at approximately 33 per cent hydrolysis. This is to be expected, since an equal number of molecules appears in reactants and products. 

A thermodynamic equilibrium method is used to determine the equilibrium ratio of concentrations of the low-molecular-mass substances in materials in contact. The equilibrium state is usually determined with the help of kinetic curves of mass change in bodies in contact versus time. The analysis of this data permits estimation of distribution coefficients between the studied materials. A long experiment duration (usually a few months at room temperature) is a serious hindrance. If data must be collected at sub-zero temperatures, the experiment may take several years. 

Both the Anderson and the Kondo (or Coqblin-Schrieffer) model have been solved exactly for thermodynamic properties such as the 4f-electron valence, specific heat, static magnetic and charge susceptibilities, and the magnetization as a function of temperature and magnetic field B by means of the Bethe ansatz (see Schlottmann 1989, and references therein). This method also allows one to calculate the zero-temperature resistivity as a function of B. Non-equilibrium properties, such as the finite temperature resistivity, thermopower, heat conductivity or dynamic susceptibility, could be calculated in a self-consistent approximation (the non-crossing approximation), which works well and is based on an /N expansion where N is the degeneracy of the 4f level. 

Extensions of the QC Method Because of its versatility, the QC method has been widely applied and, naturally, extended as well. While its original formulation was for zero-temperature static problems only, several groups have modified it to allow for finite-temperature investigations of equilibrium properties as well. A detailed discussion of some of these methodologies is presented in the discussion of finite-temperature methods below. Also, Dupoy et al. have extended it to include a finite-temperature alternative to molecular dynamics (see below). Lastly, the quasi-continuum method has also been coupled to a DFT description of the system in the OFDFT-QC (orbital-free DFT-QC) methodology discussed below. 

Shenoy et al. in 1999 proposed the quasi-continuum Monte Carlo (QCMC) method as a way to extend the quasi-continuum method to the study of equilibrium properties of materials at finite temperature. The objective of this treatment is to construct a computationally manageable expression for a temperature-dependent effective energy for a system maintained at fixed temperature. Such an energy would then be used instead of the zero-temperature effective energy (e.g., Eq. [10]) in a Monte Carlo formulation of the QC method. 

Side-draw Temperature. The method of calculating side-draw temperatures is much the same as the calculation of the top temperature except. that complications arise because of the presence of the low-boiling materials that pass the draw plate. Furthermore, the equilibrium condensation curve, and particularly the point on this curve that denotes complete condensation, is the basis for computing these temperatures. At the zero per cent point on the flash curve the side-draw product can be completely condensed at 760 mm pressure if no lighter products or steam are present at the plate. 

The values of Km and T2d from Eq.(36) can be obtained from the transfer function of the linearized model at the equilibrium point, applying conventional methods from the linear control theory (see [1]). In order to investigate the self-oscillating behavior, one can determine the linearized system at the equilibrium point, and the corresponding complex eigenvalues with zero real part, when the parameters Km and of the PI controller are varied. For example, taking into account Eq.(34), the Jacobian matrix of the linearized system at dimensionless set point temperature xs is the following ... 

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