Boundary conditions interfaces

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

Equation (A3.3.73) is referred to as the Gibbs-Thomson boundary condition, equation (A3.3.74) detemiines p on the interfaces in temis of the curvature, and between the interfaces p satisfies Laplace s equation, equation (A3.3.71). Now, since ] = -Vp, an mterface moves due to the imbalance between the current flowing into and out of it. The interface velocity is therefore given by... 

HyperChem uses th e ril 31 water m odel for solvation. You can place th e solute in a box of T1P3P water m oleeules an d impose periodic boun dary eon dition s. You may then turn off the boundary conditions for specific geometry optimi/.aiion or molecular dynamics calculations. However, th is produces undesirable edge effects at the solvent-vacuum interface. 

In finite boundary conditions the solute molecule is surrounded by a finite layer of explicit solvent. The missing bulk solvent is modeled by some form of boundary potential at the vacuum/solvent interface. A host of such potentials have been proposed, from the simple spherical half-harmonic potential, which models a hydrophobic container [22], to stochastic boundary conditions [23], which surround the finite system with shells of particles obeying simplified dynamics, and finally to the Beglov and Roux spherical solvent boundary potential [24], which approximates the exact potential of mean force due to the bulk solvent by a superposition of physically motivated tenns. 

Establishing the interface design parameters is easy enough, but forcing designers to establish acceptable tolerance on interface boundary conditions is difficult. Operating parameters need tolerance just as much as manufactured dimensions. 

Computer simulations of bulk liquids are usually performed by employing periodic boundary conditions in all three directions of space, in order to eliminate artificial surface effects due to the small number of molecules. Most simulations of interfaces employ parallel planar interfaces. In such simulations, periodic boundary conditions in three dimensions can still be used. The two phases of interest occupy different parts of the simulation cell and two equivalent interfaces are formed. The simulation cell consists of an infinite stack of alternating phases. Care needs to be taken that the two phases are thick enough to allow the neglect of interaction between an interface and its images. An alternative is to use periodic boundary conditions in two dimensions only. The first approach allows the use of readily available programs for three-dimensional lattice sums if, for typical systems, the distance between equivalent interfaces is at least equal to three to five times the width of the cell parallel to the interfaces. The second approach prevents possible interactions between interfaces and their periodic images. 

Simulations of water in synthetic and biological membranes are often performed by modeling the pore as an approximately cylindrical tube of infinite length (thus employing periodic boundary conditions in one direction only). Such a system contains one (curved) interface between the aqueous phase and the pore surface. If the entrance region of the channel is important, or if the pore is to be simulated in equilibrium with a bulk-like phase, a scheme like the one in Fig. 2 can be used. In such a system there are two planar interfaces (with a hole representing the channel entrance) in addition to the curved interface of interest. Periodic boundary conditions can be applied again in all three directions of space. 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

It is not an easy task to develop computer codes which correctly treat the advancement of a folding interface as a boundary condition to a diffusion or flow field. In addition, the interface between a solid and a liquid, for example, is usually is not absolutely sharp on an atomic scale, but varies over a few lattice constants [32,33]. In these cases, it is sometimes convenient to treat the interface as having a finite non-zero thickness. An order parameter is then introduced, which for example varies from the value zero on one side of the interface to the value one on the other, representing a smooth transition from liquid to solid across the interface. This is called the phase-field... 

The interface free energy per unit area fi,u is taken to be that of a planar interface between coexisting phases. Considering a solution v /(z) that minimizes Eq. (5) subject to the boundary conditions vj/(z - oo) = - v /coex, v /(z + oo) = + vj/ oex one finds the excess free energy of a planar interface ... 

The situation is envisaged in which the total energy of a box of atoms can be calculated, and one wants to obtain the excess energy of an interface which has been constructed within the box. The simplest situation is if a static calculation has been made and the atomic positions are relaxed to the structure of minimum energy. However, free energy calculations are also feasible. Periodic boundary conditions parallel to the interface are employed, and perhaps also three dimensional periodicity, which implies that two boundaries per box are necessary. These technicalities as well as the method for calculating energies will not be discussed further here. 

The boundary conditions are given by specifying the panicle currents at the boundaries. Holes can be injected into the polymer by thermionic emission and tunneling [32]. Holes in the polymer at the contact interface can also fall bach into the metal, a process usually called interlace recombination. Interface recombination is the time-reversed process of thermionic emission. At thermodynamic equilibrium the rates for these two time-reversed processes are the same by detailed balance. Thus, there are three current components to the hole current at a contact thermionic emission, a backflowing interface recombination current that is the time-reversed process of thermionic emission, and tunneling. Specifically, lake the contact at Jt=0 as the hole injecting contact and consider the hole current density at this contact. 

Finally, the boundary condition (276) refers to the continuity of the flux J = — ClXd /dr) at the interface. 

The boundary conditions should account for continuity of stresses and displacements at the respective two interfaces and would be expressed as follows ... 

Because the boundary condition at y = oo Cr = 0) is unaltered by the fact that the concentration at the interface is a function of time, equation 10.106 is still applicable, although the evaluation of the constant 2 is more complicated because (C )y=o is no longer constant. 

The solution of this equation has been discussed by DANCKWERTSt28), and here a solution will be obtained using the Laplace transform method for a semi-infinite liquid initially free of solute. On the assumption that the liquid is in contact with pure solute gas, the concentration Cm at the liquid interface will be constant and equal to the saturation value. The boundary conditions will be those applicable to the penetration theory, that is ... 

Then the diffusion equation for the fluctuation of the metal ion concentration is given by Eq. (68), and the mass balance at the film/solution interface is expressed by Eq. (69). These fluctuation equations are also solved with the same boundary condition as shown in Eq. (70). 

In the previous sections, we have seen how computer simulations have contributed to our understanding of the microscopic structure of liquid crystals. By applying periodic boundary conditions preferably at constant pressure, a bulk fluid can be simulated free from any surface interactions. However, the surface properties of liquid crystals are significant in technological applications such as electro-optic displays. Liquid crystals also show a number of interesting features at surfaces which are not seen in the bulk phase and are of fundamental interest. In this final section, we describe recent simulations designed to study the interfacial properties of liquid crystals at various types of interface. First, however, it is appropriate to introduce some necessary terminology. 

The heat transfer model, energy and material balance equations plus boundary condition and initial conditions are shown in Figure 4. The energy balance partial differential equation (PDE) (Equation 10) assumes two dimensional axial conduction. Figure 5 illustrates the rectangular cross-section of the composite part. Convective boundary conditions are implemented at the interface between the walls and the polymer matrix. 

Examples of the electrophoretic mobility, m as functions of the molecular properties, e.g., solute size, charge, and shape, and solution conditions, were discussed in a previous section. For the two-phase system considered in Figure 30, the flux and equilibrium boundary conditions at the interface between the a and p phases are given by... 

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