Continuum theory, development

The Hamaker constant can be evaluated accmately using the continuum theory, developed by Lifshitz and coworkers [40]. A key property in this theory is the frequency dependence of the dielectric permittivity, e( ). If this spectrum were the same for particles and solvent, then A=0. Since the refractive index n is also related to t ( ), the van der Waals forces tend to be very weak when the particles and solvent have similar refractive indices. A few examples of values for for interactions across vacuum and across water, obtained using the continuum theory, are given in table C2.6.3. 

To discuss flow birefringence, we have to make use of some results of the continuum theory developed in chapter 3. In analogy with (3.1.38) we write for the isotropic phase< ... 

All the theory developed up to this point has been limited in the sense that translational motion (the continuum degree of freedom) has been restricted to one dimension. In this section we discuss the generalization of this to three dimensions for collision processes where space is isotropic (i.e., collisions in homogeneous phases, such as in a... 

The continuum theory of deformation of elastic solids is old and well developed [65T01, 74T01], and, in its linear version, is widely applied. Nonlinear theory is of much more recent origin. Most application of nonlinear theory has been to the behavior of highly deformable materials such as rubber or to the explanation of subtle effects observed by precise ultrasonic... 

This section provides an alternative measurement for a material parameter the one in the ensemble averaged sense to pave the way for usage of continuum theory from a hope that useful engineering predictions can be made. More details can be found in Ref. [15]. In fact, macroscopic flow equations developed from molecular dynamics simulations agree well with the continuum mechanics prediction (for instance. Ref. [16]). 

Baumgartner, L. P. Rumble III, D. (1988). Transport of stable isotopes I Development of a kinetic continuum theory for stable isotope transport. Contrib. Mineral. Petrol., 98, 417-30. 

Another approach to ET reactions originated in the work of Weiss and Libby, who suggested that activation energy for ET reactions in solution does not arise from the collisional-vibrational or electrostatic interaction of the solvent in the first and second solvation sphere, but rather from continuum solvent polarization fluctuation far out in solution. This approach, called continuum theory, was further developed by Kubo and Toyozawa, " Marcus, Platzmann and Erank, and by Levich and Dogo-nadze. ... 

Molecular modeling treatments of electron transfer kinetics for reactions involving bond breaking were developed much earlier than the continuum theories originated by Weiss in 1951. Gurney in 193l published a landmark paper (the foundation of quantum electrochemistry) on a molecular and quantum mechanical model of proton and electron transfer... 

With the acceptance of the atomic view of the world - accompanied by the necessity to explain reactions in extremely dilute gases (where the continuum theory fails) - the kinetic gas theory was developed. Using this it is possible not only to derive the ideal gas law in another manner but also to calculate many other quantities involved with the kinetics of gases - such as collision rates, mean free path lengths, monolayer formation time. 

Lifshitz (1955-60) developed a complete quantum electrodynamic (continuum) theory for the van der Waals interaction between macroscopic bodies. 

A major criticism of all this work is the treatment of the solvent as hydrodynamic continuum. To study the hydrodynamic repulsion of particles requires either molecular dynamics calculations or time-dependent liquid theories to be applied. It will be most interesting to see how the analysis using kinetic theory develops (see, for instance, Cukier et al. [454]). 

Two major developments since the mid-1950 s have greatly influenced the entire field of rheology. First, continuum theories have shown that, subject to very broad and physically plausible assumptions about mechanical response, many seemingly independent viscoelastic properties of a material are in fact related. These relationships have been confirmed experimentally and should hold... 

This part introduces variational principles relevant to the quantum mechanics of bound stationary states. Chapter 4 covers well-known variational theory that underlies modern computational methodology for electronic states of atoms and molecules. Extension to condensed matter is deferred until Part III, since continuum theory is part of the formal basis of the multiple scattering theory that has been developed for applications in this subfield. Chapter 5 develops the variational theory that underlies independent-electron models, now widely used to transcend the practical limitations of direct variational methods for large systems. This is extended in Chapter 6 to time-dependent variational theory in the context of independent-electron models, including linear-response theory and its relationship to excitation energies. 

The real world of compartmental systems has a strong stochastic component, so we will present a stochastic approach to compartmental modeling. In deterministic theory developed in Chapter 8, each compartment is treated as being both homogeneous and a continuum. But ... 

Derivations of conservation equations from the viewpoint of kinetic theory usually do not exhibit explicitly the diffusion terms, such as diffusion stresses, that appear on the right-hand sides of equations (49), (50), and (51), since it is unnecessary to introduce quantities such as afj specifically in these derivations. Kinetic-theory developments work directly with the left-hand sides of equations (49), (50), and (51). Transport coefficients (Appendix E) are defined only in terms of these kinetic-theory quantities because prescriptions for calculating the individual continua transports, afj and qf, are unduly complex. Moreover, measurement of diffusion stresses is feasible only by direct measurement of diffusion velocities, followed by use of equation (24). Therefore, it has not been fruitful to study the diffusion terms which, in a sense, may be viewed as artifacts of the continuum approach. 

Using the theory developed by Chapman-Enskog (see Ref. 14), a hierarchy of continuum fluid mechanics formulations may be derived from the Boltzmann equation as perturbations to the Maxwellian velocity distribution function. The first three equation sets are well known (1) the Euler equations, in which the velocity distribution is exactly the Maxwellian form (2) the Navier-Stokes equations, which represent a small deviation from Maxwellian and rely on linear expressions for viscosity and thermal conductivity and (3) the Burnett equations, which include second order derivatives for viscosity and thermal conductivity. 

In Sec. 2, we review the reactive empirical bond order potentials that we have developed over the last decade which possess these essential characteristics. Our model systems and computational techniques are also described. Simulations using the first of our models to exhibit classic detonation behavior are discussed in Sec. 3, where results involving both detonation and initiation are presented. In Sec. 4, the results of our molecular dynamics simulations are compared in detail with the predictions of continuum theory. Then, in Sec. 5, we review some simulations that raise... 

Willetts and Rice attempted to develop a continuum model (type II) to calculate the EFISH response of liquid acetonitrile. Their results were compared with accurate calculations and measurements of the hyperpolarizability of the isolated molecule and measured solution for two different cavity radii. One natural choice of cavity radius is based on the van der Waals radius of the compound and with this value of a the discrepancy between the gas and solution measurements is not resolved. A rather smaller value of a can be selected to give much better agreement. The arbitrariness of the choice of cavity radius places a serious limitation on the usefulness of the continuum theories and, in recent years, there has been a considerable effort to develop hyperpolarizability calculations based on discrete molecular simulations of liquid and solution structure. 

In microscale models the explicit chain nature has generally been integrated out completely. Polymers are often described by variants of models, which were primarily developed for small molecular weight materials. Examples include the Avrami model of crystallization,- and the director model for liquid crystal polymer texture. Polymeric characteristics appear via the values of certain constants, i.e. different Frank elastic constant for liquid crystal polymers rather than via explicit chain simulations. While models such as the liquid crystal director model are based on continuum theory, they typically capture spatiotemporal interactions, which demand modelling on a very fine scale to capture the essential effects. It is not always clearly defined over which range of scales this approach can be applied. 

Sometimes simple models derived from continuum theory promise better results, but MD or Monte-Carlo (MC) simulations are still the preferable approaches for condensed-matter investigations [3,4], These methods were developed in the early 1950s [3, 4], They include a model-inherent dynamical description in the case of MD, and large samples can be accounted for. These methods allow working scientists to treat more than one molecule and make use of so-called periodic boundary conditions which mimic images around the central cell in such a way that problems due to surface effects can be overcome. 

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