Classical continuum

In this chapter we provide an introductory overview of the imphcit solvent models commonly used in biomolecular simulations. A number of questions concerning the formulation and development of imphcit solvent models are addressed. In Section II, we begin by providing a rigorous fonmilation of imphcit solvent from statistical mechanics. In addition, the fundamental concept of the potential of mean force (PMF) is introduced. In Section III, a decomposition of the PMF in terms of nonpolar and electrostatic contributions is elaborated. Owing to its importance in biophysics. Section IV is devoted entirely to classical continuum electrostatics. For the sake of completeness, other computational... 

IV. CLASSICAL CONTINUUM ELECTROSTATICS A. The Poisson Equation for Macroscopic Media... 

The thermal healing has been studied most extensively for one-dimensional gratings. Above roughening, the gratings acquire, for small amplitude to wavelength ratios, a sinusoidal form, as predicted by the classical continuum theory of Mullins and confirmed by experimenf-s and Monte Carlo simulations. - The decay of the amplitude is, asymptotically, exponential in time. This is true for both evaporation dynamics and (experimentally more relevant) surface diffusion. 

In marked contrast, the classical continuum theory by mullins describes the sim-ulational data (profile shapes and amplitude decay) above roughening for wires even with small geometries surprisingly well, both for surface diffusion and evaporation-condensation The agreement may be a little bit fortuituous, because of a compensation of the competing effects of the anisotropic surface tension and anisotropic mobility, whereas continuum theory assumes isotropic quantities. In any event, the predicted decay laws with w= 1/4 for surface diffusion and w= 1/2 for evaporation kinetics are readily reproduced in the simulations. 

Above roughening (see Figure 6), the decay of the gratings is well described by the classical continuum theory for sufficiently small ratios amplitude/wavelenght, with g = 2 for evaporation, and 4 for surface diffusion. Deviations, observed otherwise, can be explained mostly by the anisotropy of the surface tension. ... 

Above roughening, the simulations confirm the classical continuum theory. The width of the bumps spreads with time as fwhere b= Ml for evaporation-condensation, and b= 1/4 for surface diffusion. In the latter case, the profile shows an oscillatory behavior away from the foot of the bump, as for pairs of steps and wires. ... 

So-called solvation/structural forces, or (in water) hydration forces, arise in the gap between a pair of particles or surfaces when solvent (water) molecules become ordered by the proximity of the surfaces. When such ordering occurs, there is a breakdown in the classical continuum theories of the van der Waals and electrostatic double-layer forces, with the consequence that the monotonic forces they conventionally predict are replaced (or accompanied) by exponentially decaying oscillatory forces with a periodicity roughly equal to the size of the confined species (Min et al, 2008). In practice, these confined species may be of widely variable structural and chemical types — ranging in size from small solvent molecules (like water) up to macromolecules and nanoparticles. 

The classical continuum approximation (13.80) becomes questionable for light nuclei, where inertial moments are reduced and quantum rotational spacings proportionally increased, and in this case the quantum sum over angular momentum states may be substituted. Furthermore, the treatment assumes sufficient free volume for unhindered rotations, and is therefore only appropriate at the lower-density conditions of gaseous reactions. 

When we come to examine the annals of classical hydrodynamics and electrodynamics, we find that the foundations of vector field theory have provided some key field structures whose role has repeatedly been acknowledged as instrumental in not only underpinning the structural edifice of classical continuum field physics, but in accounting for its empirical exhibits as well. 

It is necessary to drop completely the physical pictures of Schrodinger which aim at a revitalization of the classical continuum theory, to retain only the formalism and to fill that with new physical content. 

Another evident mechanism for energy transfer to activated ions may be by bimolecular collisions between water molecules and solvated ion reactants, for which the collision number is n(ri+ r2)2(87tkT/p )l/2> where n is the water molecule concentration, ri and r2 are the radii of the solvated ion and water molecule of reduced mass p. With ri, r2 = 3.4 and 1.4 A, this is 1.5 x 1013 s"1. The Soviet theoreticians believed that the appropriate frequency should be for water dipole librations, which they took to be equal 10n s 1. This in fact corresponds to a frequency much lower than that of the classical continuum in water.78 Under FC conditions, the net rate of formation of activated molecules (the rate of formation minus rate of deactivation) multiplied by the electron transmission coefficient under nonadiabatic transfer conditions, will determine the preexponential factor. If a one-electron redox reaction has an exchange current of 10 3 A/cm2 at 1.0 M concentration, the extreme values of the frequency factors (106 and 4.9 x 103 cm 2 s 1) correspond to activation energies of 62.6 and 49.4 kJ/mole respectively under equilibrium conditions for adiabatic FC electron transfer. 

It is important to be aware of clustering caused by electrostriction in order to understand reactions of ions in supercritical rare gases. If a classical continuum model is used to calculate clustering, the magnitude of the volume change due to electrostriction would be overestimated because such a model ignores the density build-up around the ion. Because of this density augmentation, the compressibility of the fluid near the ion is less and, since electrostriction is proportional to compressibility, the actual electrostriction will be less... 

Three broad classes of vibrational modes may contribute to the thermally averaged Franck-Condon factor the high-frequency (fast) modes hv > 1000 cm ) which are mainly intraligand vibrations, intermediate modes (1000 cm > hv > 100 cm ) that typically include the metal-ligand stretching vibrations and higher frequency solvent orientational-vibrational modes, and the low-frequency (slow) modes hv < 100 cm ) which are primary solvent modes but can include low-frequency intramolecular modes. At ordinary temperatures hv kT k hv hv and the low-frequency modes can be treated using classical (continuum) expressions. 

In this section the classical continuum theory of mixtures is reviewed [15]. In this concept the multiphase mixture is treated as a single homogeneous continuum. Thereby the balance principle can be applied to derive conservation laws for the macroscopic pseudo-fluid in analogy to the single phase formulation examined in chap 1. Approximate constitutive equations are postulated for the expected macroscopic behavior of the phases. 

Use of e, the variable for orientation, and of second derivatives with respect to 9. These practices lead to the shortening-rate equations (10.4) and (10.5), which show second derivatives of potential with respect to orientation. These are alternatives to the shortening-rate equations used in classical continuum mechanics. The latter are of course equally correct, but for present purposes less suitable in form than eqn. (10.5). 

Once the continuum hypothesis has been adopted, the usual macroscopic laws of classical continuum physics are invoked to provide a mathematical description of fluid motion and/or heat transfer in nonisothermal systems - namely, conservation of mass, conservation of linear and angular momentum (the basic principles of Newtonian mechanics), and conservation of energy (the first law of thermodynamics). Although the second law of thermodynamics does not contribute directly to the derivation of the governing equations, we shall see that it does provide constraints on the allowable forms for the so-called constitutive models that relate the velocity gradients in the fluid to the short-range forces that act across surfaces within the fluid. 

Once we adopt the continuum hypothesis and choose to describe fluid motions and heat transfer processes from a macroscopic point of view, we derive the governing equations by invoking the familiar conservation principles of classical continuum physics. These are conservation of mass and energy, plus Newton s second and third laws of classical mechanics. 

In classical, continuum theories of diffusion-reaction processes based on a Fickian parabolic partial differential equation of the form, Eq. (4.1), specification of the Laplacian operator is required. Although this specification is immediate for spaces of integral dimension, it is less straightforward for spaces of intermediate or fractal dimension [47,55,56]. As examples of problems in chemical kinetics where the relevance of an approach based on Eq. (4.1) is open to question, one can cite the avalanche of work reported over the past two decades on diffusion-reaction processes in microheterogeneous media, as exemplified by the compartmentalized systems such as zeolites, clays and organized molecular assemblies such as micelles and vesicles (see below). In these systems, the (local) dimension of the diffusion space is often not clearly defined. 

Due to the complexity of the formation of interphases, a completely satisfying microscopic interpretation of these effects cannot be given today, especially since the process of the interphase formation is not yet understood in detail. Therefore, a micromechanical model cannot be devised for calculating the global effective properties of a thin polymer film including the above-mentioned size effects governed by the interphases. On the other hand, a classical continuum-based model is not able to include any kind of size effect. An alternative to the above-mentioned classical continuum or the microscopical model is the formulation of an extended continuum mechanical model which, on the one hand, makes it possible to capture the size effect but, on the other hand, does not need all the complex details of the underlying microstmcture of the polymer network. 

Classical Continuum-Solvent Methods. The simplest and crudest continuum-solvent method is the solvent-accessible-surface area (SASA) model, which assumes that the standard free-energy of solvation AG°o y can be expressed as... 

The basic idea of the theory of electron-transfer concerning the dynamic role of the solvent, first suggested by LIBBI /150/, has been developed by MARCUS /40a/ for outer-sphere redox processes on the basis of a classical continuum model for solvent polarization. A quantum-mechanical treatment of the same model was done by LEVICH and DO-GONADZE /40b,143/, making use of the theory of non-adiabatic radiationless electron transfer in polar crystals. 

Figure 10.2 Schematic classification of classical continuum models.

The new features of the current work relate to the approach adopted in the modeling of the polymer matrix and the investigation of the CNT polymer interfacial properties as appose to the effective mechanical properties of the RVE. The idea behind the ABC technique is to incorporate atomistic interatomic potentials into a continuum framework. In this way, the interatomic potentials introduced in the model capture the underlying atomistic behavior of the different phases considered. Thus, the influence of the nanophase is taken into account via appropriate atomistic constitutive formulations. Consequently, these measures are fundamentally different from those in the classical continuum theory. For the sake of completeness, Wemik and Meguid provided a brief outline of the method detailed in their earlier work [133-134]. 

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