Solute boundary conditions

Boundary Conditions Solution of the Problem of the Reduced Ionic Cloud. For equipotential conditions the total potential at the surface of the... 

For problems with relatively simple boundary and initial conditions, solutions can probably be found in a library. However, it can be difficult to find a closed-form solution for problems with highly specific and complicated boundary conditions. In such cases, numerical methods could be employed. For simple boundary conditions, solutions to the diffusion equation in the form of Eq. 4.18 have a few standard forms, which may be summarized briefly. 

If bz(r) was a scalar function Z z(r), (3.57) would be equivalent to the Schrodinger equation for a three-dimensional free particle, yielding, for periodic boundary conditions, solutions of the form... 

We will have many occasions in future chapters to solve (4.4.12) under a variety of boundary conditions. Solutions of this equation yield concentration profiles, Cq x, t). 

Figure 7.10 Illustration of Stick Boundary Condition. Solute biological macromolecule shown as single grey sphere moves with an average velocity through water (shown as light blue spheres). Water molecules in immediate hydration layer move at the same average velocity due to tight hydration interactions. Under Slip Boundary Conditions, water molecules do not possess hydration interactions and therefore do not move with the biological macromolecule at all.

In a string under tension, the ends are restricted from moving (that s how we make the tension). The conditions of non-movement at the ends are called boundary conditions. The boundary condition solution to an ideal string of length L terminated at both ends is ... 

Distortions and defects can be interpreted in terms of the continuum theory through equations derived from the expressions of the elastic energy and the imposed boundary conditions. Solutions are known in certain simple situations. Oseen [35] has found configurations, named disinclinations by Frank [33], or disclinations today, which are solutions of this problem for planar samples in which the director n is confined to... 

In addition to the spatial boundary conditions, solution of the model also requires a set of initial conditions, i.e., a specification of the initial state of the system. As with the one-dimensional system, we assume that the concentration of each species is initially uniform and equal to its bulk value across all space ... 

The equations are transcendental for the general case, and their solution has been discussed in several contexts [32-35]. One important issue is the treatment of the boundary condition at the surface as d is changed. Traditionally, the constant surface potential condition is used where po is constant however, it is equally plausible that ag is constant due to the behavior of charged sites on the surface. 

For many-electron systems such as atoms and molecules, it is obviously important that approximate wavefiinctions obey the same boundary conditions and symmetry properties as the exact solutions. Therefore, they should be antisynnnetric with respect to interchange of each pair of electrons. Such states can always be constmcted as linear combinations of products such as... 

The solutions of such partial differential equations require infomiation on the spatial boundary conditions and initial conditions. Suppose we have an infinite system in which the concentration flucPiations vanish at the infinite boundary. If, at t = 0 we have a flucPiation at origin 5C(f,0) = AC (f), then the diflfiision equation... 

The solution of Laplace s equation, (A3.3.71), with these boundary conditions is, for [Pg.749]

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as... 

Often in numerical calculations we detennine solutions g (R) that solve the Scln-odinger equations but do not satisfy the asymptotic boundary condition in (A3.11.65). To solve for S, we rewrite equation (A3.11.65) and its derivative with respect to R in the more general fomi ... 

A is a coelTicient matrix that is designed to transfomi between solutions that obey arbitrary boundary conditions and those which obey the desired boundary conditions. A and S can be regarded as unknowns in equation (A3.11.72) and equation (A3.11.73). This leads to the following expression for S ... 

A fiill solution of tlie nonlinear radiation follows from the Maxwell equations. The general case of radiation from a second-order nonlinear material of finite thickness was solved by Bloembergen and Pershan in 1962 [40]. That problem reduces to the present one if we let the interfacial thickness approach zero. Other equivalent solutions involved tlie application of the boundary conditions for a polarization sheet [14] or the... 

Pick s second law of difflision enables predictions of concentration changes of electroactive material close to the electrode surface and solutions, with initial and boundary conditions appropriate to a particular experiment, provide the basis of the theory of instrumental methods such as, for example, potential-step and cyclic voltanunetry. 

Here the distortion (diagonal) and back coupling matrix elements in the two-level equations (section B2.2.8.4) are ignored so that = exp(ik.-R) remains an imdistorted plane wave. The asymptotic solution for ij-when compared with the asymptotic boundary condition then provides the Bom elastic ( =f) or inelastic scattering amplitudes... 

Numerical solution of this set of close-coupled equations is feasible only for a limited number of close target states. For each N, several sets of independent solutions F.. of the resulting close-coupled equations are detennined subject to F.. = 0 at r = 0 and to the reactance A-matrix asymptotic boundary conditions,... 

If tire diffusion coefficient is independent of tire concentration, equation (C2.1.22) reduces to tire usual fonn of Pick s second law. Analytical solutions to diffusion equations for several types of boundary conditions have been derived [M]- In tlie particular situation of a steady state, tire flux is constant. Using Henry s law (c = kp) to relate tire concentration on both sides of tire membrane to tire partial pressure, tire constant flux can be written as... 

An advantage of Eq. (90) for computational purposes is that the solutions are subject to single-valued boundary conditions. It is also readily verified that inclusion of an additional factor qjj the right-hand side of Eq. (89) adds a... 

This justifies the use of the simpler language over the one. The solution of the Poisson equation and the boundary conditions used are explained in detail elsewhere [55]. Here, we will present some selected results. 

The diabatic LHSFs are not allowed to diverge anywhere on the half-sphere of fixed radius p. This boundary condition furnishes the quantum numhers n - and each of which is 2D since the reference Hamiltonian hj has two angular degrees of freedom. The superscripts n(, Q in Eq. (95), with n refering to the union of and indicate that the number of linearly independent solutions of Eqs. (94) is equal to the number of diabatic LHSFs used in the expansions of Eq. (95). 

Another subject with important potential application is discussed in Section XIV. There we suggested employing the curl equations (which any Bohr-Oppenheimer-Huang system has to obey for the for the relevant sub-Hilbert space), instead of ab initio calculations, to derive the non-adiabatic coupling terms [113,114]. Whereas these equations yield an analytic solution for any two-state system (the abelian case) they become much more elaborate due to the nonlinear terms that are unavoidable for any realistic system that contains more than two states (the non-abelian case). The solution of these equations is subject to boundary conditions that can be supplied either by ab initio calculations or perturbation theory. 

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