Continuum statics based models

Finally, it deserves to be mentioned that considerable numbers of models of static friction based on continuum mechanics and asperity contact were proposed in the literature. For instance, the friction at individual asperity was calculated, and the total force of friction was then obtained through a statistical sum-up [35]. In the majority of such models, however, the friction on individual asperity was estimated in terms of a phenomenal shear stress without involving the origin of friction. 

From Eq, (1) it is clear that a model of crystal polarization that is adequate for the description of the piezoelectric and pyroelectric properties of the P-phase of PVDF must include an accurate description of both the dipole moment of the repeat unit and the unit cell volume as functions of temperature and applied mechanical stress or strain. The dipole moment of the repeat unit includes contributions from the intrinsic polarity of chemical bonds (primarily carbon-fluorine) owing to differences in electron affinity, induced dipole moments owing to atomic and electronic polarizability, and attenuation owing to the thermal oscillations of the dipole. Previous modeling efforts have emphasized the importance of one more of these effects electronic polarizability based on continuum dielectric theory" or Lorentz field sums of dipole lattices" static, atomic level modeling of the intrinsic bond polarity" atomic level modeling of bond polarity and electronic and atomic polarizability in the absence of thermal motion. " The unit cell volume is responsive to the effects of temperature and stress and therefore requires a model based on an expression of the free energy of the crystal. 

Now that we have settled on a model, one needs to choose the appropriate algorithm. Three methods have been used to study polymers in the continuum Monte Carlo, molecular dynamics, and Brownian dynamics. Because the distance between beads is not fixed in the bead-spring model, one can use a very simple set of moves in a Monte Carlo simulation, namely choose a monomer at random and attempt to displace it a random amount in a random direction. The move is then accepted or rejected based on a Boltzmann weight. Although this method works very well for static and dynamic properties in equilibrium, it is not appropriate for studying polymers in a shear flow. This is because the method is purely stochastic and the velocity of a mer is undefined. In a molecular dynamics simulation one can follow the dynamics of each mer since one simply solves Newton s equations of motion for mer i,... 

The possible role of solvent dynamics in influencing reaction in solutions has recently received considerable scrutiny. We cannot exhaustively review the impressive number of recent methodological and applicative contributions in this field, which have been supported and stimulated by new experimental evidence based on innovative techniques, and by the increasing reliability of molecular dynamics and MC simulations. Following the approach used in the previous Section to treat the static description of the solvent, we shall focus our review about dynamical aspects almost entirely on methods in which the continuum model plays a key role. 

The above examples illustrate that continuum models such as the Kirkwood model are reasonably successful in describing the static permittivity, provided one has an independent means of estimating the correlation parameter Unfortunately, these estimates are available for only a few polar solvents, so that gK must be considered an independent parameter. The version of Kirkwood s theory presented here only considers orientational polarization. When distortional polarization, that is, the effect of molecular polarizability, is included, interpretation of experimental results is less clear. Since the approach taken here involves continuum concepts, it is necessarily limited. In the following section, a simple model based on a molecular description of a polar liquid is presented. 

Recently, a new category of methods, the cavity model, has been proposed to account for the solvent effect. Molecules or supermolecules are embedded in a cavity surrounded by a dielectric continuum, the solvent being represented by its static dielectric constant. The molecules (supermolecules) polarize the continuum. As a consequence this creates an electrostatic potential in the cavity. This reaction potential interacts with the molecules (supermolecules). This effect can be taken into account through an interaction operator. The usual SCF scheme is modified into a SCRF (self consistent reaction field) scheme, and similar modifications can be implemented beyond the SCF level. Several studies based on this category of methods have been published on protonated hydrates. They account for the solvent effect on the filling of the first solvation shell (53, 69), the charges (69, 76) and the energy barrier to proton transfer (53, 76). 

The continuum mechanics modeled by JAS3D are based on two fundamental governing equations. The kinematics is based on the conservation of momentum equation, which can be solved either for quasi-static or dynamic conditions (a quasistatic procedure was used for these analyses). The stress-strain relationships are posed in terms of the conventional Cauchy stress. JAS3D includes at least 30 different material models. 

There are three ways to simulate reaction-diffusion system. The traditional method is to solve partial differential equation directly. Another way is to divide system into cells, which is called cell dynamic scheme (CDS). Typical models are cellular automata (CA)[176] and coupled map lattice (CML)[177]. In cellular automata model, each value of the cell (lattice) is digital. On the other hand, in coupled map lattice model, each value of the lattice (cell) is continuous. CA model is microscopic while CML model is mesoscopic. The advantage of the CML is compatibility with the physical phenomena by smaller number of cells and numerical stability. Therefore, the model based on CML is developed. Each cell has continuum state and the time step is discrete. Generally, each cell is static and not deformable. Deformable cell (lattice) is supposed in order to represent deformation process of the gel. Each cell deforms based on the internal state, which is determined by the reaction between the cell and the environment. 

The generalized mode approach has been adapted to ab initio molecular dynamics combined with a polarizable continuum model [66] to include the effect of a bulk solvent [63]. In this context a simple procedure for the vibrational analysis of a generic molecular property was also proposed. Moreover, comparison to Hessian-based perturbative approaches [67] to treat anharmonic frequencies validated the method with very promising results. A similar static versus dynamic comparison has been performed for isolated systems [68]. 

Static modeling and energy minimization studies have been used to relate IR data and the orientation of molecules in membranes. A number of studies have used continuum approaches to mimic membrane properties. These have defined the membrane and water as regions having different dielectric constants. A recent study used a three-phase model including a hydrogen bonding potential. Lattice-based approaches have been used to study the influence of solutes on the phase transition of membranes and solute placement as a function of lipid surface area. Lattice-based approaches... 

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