Surface conditions, continuum theory

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. 

None of these phenomena - magnetic field or surface orientation dependence, tube diameter dependence, and the shear rate dependence at low shear rates - are observed with Isotropic fluids under comparable conditions. They are all predicted very satisfactorily by the Lesiie-Ericksen (L-E) continuum theory of the mechanics of liquid crystals. It will not be attempted here to give more than a cursory account, copied from Reference 2, of the theory complete descriptions are given in the references cited in Reference 3, especially Leslie s review article. 

In this section I aim to describe in some detail continuum theory for nematics, and also to draw attention to some points of current interest, particularly surface conditions and surface terms. At this juncture it does seem premature to discuss new developments concerning smectics, polymers and lyotropics, although a brief discussion of an equilibrium theory for certain smectics seems appropriate, given that it relates to earlier work on this topic. Throughout, to encourage a wider readership, we endeavour to employ vector and matrix notation, avoiding use of Cartesian tensor notation. 

We shall always assume isothermal conditions and therefore ignore thermal effects. In these circumstances, as in any classically based continuum theory, conservation laws for mass, linear momentum and angular momentum must hold. The balance law for linear momentum, given below, is basically similar to that for an isotropic fluid, except that the resulting stress tensor (to be derived later) need not be symmetric. The balance law for angular momentum is also suitably augmented to include explicit external body and surface moments. 

The design of precision components with ultrasmooth surfaces, for example, in the field of microelectromechanical systems and nanotechnology, boosts the use of lubricating films of molecular thicknesses. Under these conditions, the validity of continuum theories to describe the hydrodynamics of the lubricant is questionable. This is a new lubrication regime, denoted as thin film lubrication. A review on thin film lubrication as the hmiting case of elastohydrodynamic lubrication is provided in Ref [1055]. We will focus here on the aspects of molecular thin films studied in the context of nanotribology (reviews are Refs [185, 1056]). 

A second similar consequence of the continuum hypothesis is an uncertainty in the boundary conditions to be used in conjunction with the resulting equations for motion and heat transfer. With the continuum hypothesis adopted, the conservation principles of classical physics, listed earlier, will be shown to provide a set of so-called field equations for molecular average variables such as the continuum point velocity u. To solve these equations, however, the values of these variables or their derivatives must be specified at the boundaries of the fluid domain. These boundaries may be solid surfaces, the phase boundary between a liquid and a gas, or the phase boundary between two liquids. In any case, when viewed on the molecular scale, the boundaries are seen to be regions of rapid but continuous variation in fluid properties such as number density. Thus, in a molecular theory, boundary conditions would not be necessary. When viewed with the much coarser resolution of the macroscopic or continuum description, on the other hand, these local variations of density (and other molecular variables) can be distinguished only as discontinuities, and the continuum (or molecular average) variables such as u appear to vary smoothly on the scale L, right up to the boundary where some boundary condition is applied. 

The continuum model, in which solvent is regarded as a dielectric continuum, has been used for a long time to study solvent effects [2]. Solvation energies can be primarily approximated by a reaction field owed to polarization of the dielectric continuum as solvent, and other contributions such as dispersion interactions, which must be explicitly considered for non-polar solvent systems, have usually been treated with an empirical quantity such as the macroscopic surface tension of the solvent. An obvious advantage of the method is its handiness, whilst its disadvantage is an artifact introduced at the boundary between the solute and solvent. Agreement between experiment and theory is considerably governed by the boundary conditions. 

In a staged multi-scale approach, the energetics and reaction rates obtained from these calculations can be used to develop coarse-grained models for simulating kinetics and thermodynamics of complex multi-step reactions on electrodes (for example see [25, 26, 27, 28, 29, 30]). Varying levels of complexity can be simulated on electrodes to introduce defects on electrode surfaces, composition of alloy electrodes, distribution of alloy electrode surfaces, particulate electrodes, etc. Monte Carlo methods can also be coupled with continuum transport/reaction models to correctly describe surfaces effects and provide accurate boundary conditions (for e.g. see Ref. [31]). In what follows, we briefly describe density functional theory calculations and kinetic Monte Carlo simulations to understand CO electro oxidation on Pt-based electrodes. 

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