Surface boundary conditions

Juffer, A.H., Berendsen, H.J.C. Dynamic surface boundary conditions A simple boundary model for molecular dynamics simulations. Mol. Phys. 79 (1993) 623-644. 

A convenient concept for introducing the surface boundary condition into the mathematical formulation of migration theory is that of what may be called a diffusional offset length d. Suppose that the external and surface conditions are describable by a set of parameters X, which we do not need to specify in detail we also allow the surface conditions to depend on the internal hydrogen concentration just beneath the surface. If the hydrogen complexes that are continually forming in the crystal are sufficiently immobile, the balance between inflow and outflow across the surface will depend only on X and on the concentration no(0) of H0 just beneath the surface. (If mobile H+ or H are present, the statement just... 

Fig. 9. Typical profile of the distribution n0(x) of mobile H° (a convenient measure for the density of all monatomic hydrogen if H+ and/or H are equilibrated with H0) at an arbitrary time during an experiment on a slab-shaped crystal specimen. The surface boundary conditions at the entrance and exit surfaces are defined by the diffusion offset lenghts dem and, respectively.

Under what conditions can experiments yield data relevant to the goal we have just described Two conditions have to be fulfilled the various dissolved hydrogen species have to have had time to get equilibrated with each other before the surface boundary conditions have changed appreciably, and the surface chemical potential /x must be, if not known, at least reproducible in experiments involving different bulk dopings. At the present writing, there have been no experiments that are entirely beyond question in either of these respects, but several experiments, which we shall presently discuss, can plausibly be argued to satisfy both criteria. 

Depending on the catalyst, electron transfer at the electrode is not necessarily fast. The Nemst law used as electrode surface boundary condition may thus have to be replaced by an equation depicting the electron transfer kinetic law (Section 1.4.2) ... 

The surface boundary condition for the diffusion of fuel is the same as that for pure evaporation [Eq. (6.75)] and takes the form... 

Since there is no oxidizer leaving the surface, the surface boundary condition for diffusion of oxidizer is... 

The interfacial boundary condition may be written in terms of the convective heat transfer of sensible heat, latent heat transfer due to evaporation, and, if the surface temperature is high enough, radiant heat transfer. Mathematically, the surface boundary condition is... 

If the surface temperature does not differ greatly from the surrounding temperature, the highly nonlinear surface boundary condition may be simplified by linearizing the expression for the radiation flux and the Clausius-Clapeyron equation to yield the approximation... 

Thermal/structural response models are related to field models in that they numerically solve the conservation of energy equation, though only in solid elements. Finite difference and finite element schemes are most often employed. A solid region is divided into elements in much the same way that the field models divide a compartment into regions. Several types of surface boundary conditions are available adiabatic, convection/radiation, constant flux, or constant temperature. Many ofthese models allow for temperature and spatially dependent material properties. 

Numerical solutions of the flow around and inside fluid spheres are again based on the finite difference forms of Eqs. (5-1) and (5-2) (BIO, H6, L5, L9). The necessity of solving for both internal and external flows introduces complications not present for rigid spheres. The boundary conditions are those described in Chapter 3 for the Hadamard-Rybczynski solution i.e., the internal and external tangential fluid velocities and shear stresses are matched at R = 1 (r = a), while Eq. (5-6) applies as R- co. Most reported results refer to the limits in which k is either very small (BIO, H5, H7, L7) or large (L9). For intermediate /c, solution is more difficult because of the coupling between internal and external flows required by the surface boundary conditions, and only limited results have been published (Al, R7). Details of the numerical techniques themselves are available (L5, R7). 

Here the last term is determined by the usual surface boundary condition... 

Develop and discuss a set of boundary conditions to solve the Graetz problem. Take particular care with the effects of surface reaction, balancing heterogeneous reaction with mass diffusion from the fluid. A second Damkohler number should emerge in the surface boundary condition,... 

Stagnation Surface The stagnation-surface boundary conditions are common to all the subcases. They are... 

Put in nondimensional form, the surface boundary condition is written as... 

Rate Expressions. A major difficulty in CVD reactor modeling is the choice of appropriate rate expressions, Rp for the gas-phase-species balance and the surface boundary conditions. As described previously (see Nucleation and Growth Modes), most of CVD chemical kinetics is unknown. Therefore, rate parameters may have to be estimated from experimental growth data as part of the reactor-modeling effort. 

As discussed earlier, not only does deposition of POM to the sediment surface typically vary over time in estuaries, but there are also issues of POM actively growing at the sediment-water interface, such as benthic microalgae. This is an important issue because the rate of supply of POM is assumed to be equal to the depositional flux from the water column (Berner, 1980). More details on diagenetic models addressing the surface boundary condition constraints as well as POM lability can be found in Aller (1982) and Rice and Rhoads (1989). When examining biological mixing as a one-dimensional... 

The well-known Princeton model with a vertical -coordinate, a curvilinear horizontal grid adapted to the coastline, a turbulent closure of the order of 2.5 was used for the studies of the BSGC in [58]. Eighteen levels were specified over the vertical and the horizontal spacing was about 10 km. Similarly to [48], various combinations of the surface boundary conditions were specified. The model started with the wintertime climatic temperature analysis salinity fields [11] and three years later reached a quasi-stationary regime in the upper 200-m layer. 

Assume initially that the excess pressure 8 is determinable by some function 8 = f(x) for t = 0 at any depth to the impervious stratum. Let the depth of this stratum be L/2. If the excess of pressure at the surface is simply harmonic, 8 = 0O sin cot for x = 0 and any value of t, we have the upper surface boundary condition. The value of 60 is of course the maximum oscillation from the mean value 760 mm Hg. At L/2 no flow can take place. This boundary can be accomplished by the following mathematical artifice. Instead of assuming the impervious layer to exist at L/2, assume that it exists at a depth L. At the depth L, let there be a forced oscillation of the same magnitude as that at the surface, so that the region then encompassed will be symmetrical about L/2, and no flow will take place across a plane parallel to the surface at this point. The boundary conditions to be satisfied by Eq (14-20) will then be ... 

The mean field potential for this system, a solution of the linear Poisson-Boltzmann equation, Eq. (32), will appropriately have the same periodic structure as the surface boundary condition. Thus, we expect that if/ will have the Fourier series,... 

Mathematically, this problem bears some resemblance to those considered above. The governing partial differential equation is still Eq. (6), and on the surfaces boundary conditions of constant potential, constant charge density or linear regulation [i.e., Eq. (45)] must be imposed. However, a further constraint arises from the need to satisfy mechanical equilibrium at the interface, and it is this new condition that provides the mathematical relation needed to calculate the interface shape. The equation is the normal component of the surface stress balance, and it is given by [12]... 

Here, the vector constants V and 1 are to be determined so as to satisfy the respective pair of dyadic and triadic particle-surface boundary conditions... 

The formulation of a proper surface boundary condition is a delicate matter, as noted by DiMarzio (1965) and de Gennes (1969). Lattice models simply require that P(i, s) = 0 for layers i < 0, a form proven correct by DiMarzio (1965). In continuum models, chains intersecting the surface undergo both reflection and adsorption, the relative amount of each depending on the energy of contact at the surface. The result is a mixed boundary condition expressed by de Gennes (1969) as... 

Equation (5-118) makes full allowance for multiple reflections in an enclosure of any degree of complexity. To apply Eq. (5-118) for design or simulation purposes, the gas temperatures must be known and surface boundary conditions must be specified for each and every surface zone in the form of either E, or Q,. In application of Eq. (5-118), physically impossible values of E, may well result if physically unrealistic values of Q, are specified. ... 

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