Maxwell equation molecular dynamics

The basic principles are described in many textbooks [24, 26]. They are thus only sketchily presented here. In a conventional classical molecular dynamics calculation, a system of particles is placed within a cell of fixed volume, most frequently cubic in size. A set of velocities is also assigned, usually drawn from a Maxwell-Boltzmann distribution appropriate to the temperature of interest and selected in a way so as to make the net linear momentum zero. The subsequent trajectories of the particles are then calculated using the Newton equations of motion. Employing the finite difference method, this set of differential equations is transformed into a set of algebraic equations, which are solved by computer. The particles are assumed to interact through some prescribed force law. The dispersion, dipole-dipole, and polarization forces are typically included whenever possible, they are taken from the literature. 

All these different mechanisms of mass transport through a porous medium can be studied experimentally and theoretically through classical models (Darcy s law, Knudsen diffusion, molecular dynamics, Stefan-Maxwell equations, dusty-gas model etc.) which can be coupled or not with the interactions or even reactions between the solid structure and the fluid elements. Another method for the analysis of the species motion inside a porous structure can be based on the observation that the motion occurs as a result of two or more elementary evolutions that are randomly connected. This is the stochastic way for the analysis of species motion inside a porous body. Some examples that will be analysed here by the stochastic method are the result of the particularisations of the cases presented with the development of stochastic models in Sections 4.4 and 4.5. 

All chemical process involves the motion of atoms within a molecule. Molecular dynamics (MD), in the broadest sense, is concerned with molecules in motion. It combines the energy calculations from molecular mechanics with equations of motion. Generally, an appropriate starting structure is selected (normally an energy minimized structure). Each atom in the system is then assigned a random velocity that is consistent with the Maxwell-Boltzmann distribution for the temperature of interest. The MM formalism is used to calculate the forces on all the atoms. Once the atom positions are known, the forces, velocities at time t, and the position of the atoms at some new time t + 5t can be predicted. More details about the method can be found in Ref.. ... 

In classical molecular dynamics simulations, atoms are generally considered to be points which interact with other atoms by some predehned potential form. The forms of the potential can be, for example, Lennard-Jones potentials or Coulomb potentials. The atoms are given velocities in random directions with magnitudes selected from a Maxwell-Boltzman distribution, and then they are allowed to propagate via Newton s equations of motion according to a finite-difference approximation. See the following references for much more detailed discussions Allen and Tildesley (1987) and Frenkel... 

This picture is developed to a high level of sophistication within the classical theory of the electromagnetic field, where dynamics is described by the Maxwell equations. Some basics of this theory are described in Appendix 3 A. Here we briefly outline some of the important results of this theory that are needed to understand the nature of the interaction between a radiation field and a molecular system. 

A molecular dynamics approach can also be used to predict mixed gas diffusivities in microporous materials, at the expense of computation cost (e.g., Qureshi and Wei, 1990 Chitra and Yashonath, 1995 Trout et al., 1997 Snurr and Karger, 1997). The empirical correlation of Vignes (1966) for binary diffusivities in liquid solutions and also metallic alloys has been used extensively for calculating binary diffusivities, using the Maxwell-Stefan formalism for flux equations (e.g., Krishna, 1990). 

This is the constitutive equation or rheological equation of state for the elastic dumbbell suspensions. It is identical to the upper-convected Maxwell model, eq. 4.3.7. The molecular dynamics have led to a proper (frame-indifferent) time derivative and to a definition... 

For the transport of gas mixtures, the generalised Maxwell-Stefan equation (Krishna and WesseUngh, 1997) has been widely adopted to describe multi-component diffusion. Although quantitative descriptions of gas diffusion in various microporous or mesoporous ceramic membranes based on statistical mechanics theory (Oyama et al., 2004) or molecular dynamic simulation (Krishna, 2009) have been reported, the prediction of mixed gas permeation in porous ceramic membranes remains a challenging task, due to the difficulty in generating an accurate description of the porous network of the membrane. 

These equations are often used in terms of complex variables such as the complex dynamic modulus, E = E + E", where E is called the storage modulus and is related to the amount of energy stored by the viscoelastic sample. E" is termed the loss modulus, which is a measure of the energy dissipated because of the internal friction of the polymer chains, commonly as heat due to the sinusoidal stress or strain applied to the material. The ratio between E lE" is called tan 5 and is a measure of the damping of the material. The Maxwell mechanical model provides a useful representation of the expected behavior of a polymer however, because of the large distribution of molecular weights in the polymer chains, it is necessary to combine several Maxwell elements in parallel to obtain a representation that better approximates the true polymer viscoelastic behavior. Thus, the combination of Maxwell elements in parallel at a fixed strain will produce a time-dependent stress that is the sum of all the elements ... 

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