Spherical boundary

This potential was developed to ensure that the molecules inside the sphere never escape and maintain a fully solvated system during molecular dynamics. Here, es, Rs, ew and Rw are the van der Waals constants for the solvent and the wall and rj is the distance between the molecule i and the center of the water sphere, Ro is the radius of the sphere. The quantities A, B and Rb are determined by imposing the condition that W and dW/dr, vanish at r, = Ro. The restraining potential W is set to zero for r, < R0. The van der Waals parameters Es, ew, Rs and Rw can also be specifically defined for different solvents. The constants Awaii and Cwan are computed using a well depth of es = ew = 0.1 kcal and the radius of Rs = Rw = 1.25 A. For the other set of simulations, especially for the hydride ion transfer, we applied periodic boundary conditions by using a spherical boundary shell of 10.0 A of TIP3P40 water to cover the edges of the protein. 

The Legendre functions of the second kind, Qn(cos 0), which are absent from problems involving spherical boundaries, enter into the expressions for potcntiul functions appropriate to the space between two coaxial cones. If 0 < a < Q < fi < n we must take a solution of the form (20.1). Suppose, for example, that rp = 0 on 0 = a, ami yj = iToc r" on Q = ( then we must have... 

The orbital angular-momentum quantum number, , defines the shape of the atomic orbital (for example, s-orbitals have a spherical boundary surface, while p-orbitals are represented by a two-lobed shaped boundary surface). can have integral values from 0 to (n - 1) for each value of n. The value of for a particular orbital is designated by the letters s, p, d and f, corresponding to values of 0, 1, 2 and 3 respectively (Table 1.2). 

The eigenfunctions of the free-electron Schrodinger equation with spherical boundary conditions can be written in separable form like that for the hydrogen atom, eqn (2.48), namely... 

The distinction between these three types of boundaries will become clearer once we deal with examples. In the following sections the mathematical tools will be derived that are necessary to describe transport across these boundaries. They will then be applied to real environmental boundaries. We will also distinguish between different geometrical shapes of the boundary. Flat boundaries are easier to describe mathematically than spherical boundaries. The latter will be used to describe the exchange between suspended particles or droplets, and the surrounding fluid (algal cells in water, fog droplets in air, etc.). Furthermore, boundaries can be simple (one layer ) or have a multiple structure. 

In this section we deal with mass fluxes across spherical boundaries like the surface of gas bubbles in liquids, droplets in air, suspended particles or algal cells in water. It is true that suspended solids are rarely shaped like ideal spheres. Nonetheless, the following discussion can serve as a conceptual starting point from which more complex structures can be analyzed. Obviously, such situations require the application of numerical models, yet some of the principles, like the existence of characteristic length and time scales, will remain the same. 

Solving for XF yields the integrated flux across the spherical boundary ... 

The first two terms on the right-hand side may be recognised as the diffusive currents of acceptors across the spherical boundaries at r - °° and r = R. Providing is taken sufficiently large, i.e. r>( >012,... 

To clarify the nature of the afterflow term, it is appropriate for underwater explosions to consider the source as a spherical boundary in the fluid containing gas initially at high pressure. The initial pressure in the pressure wave is determined by the initial gas pressure. This initial compression leaves behind it outward flowing water in an increasingly large sphere. 

If the compression is to be maintained in this volume, increasingly large displacements of water near the source are necessary, despite the weakening of the initial pressure at the front by spherical divergence. The pressure in the gas sphere, however, decreases as the volume determined by the spherical boundary increases, and the strength of this source must decrease. Outward accelerations of the water near the boundary will thus decrease, but as long as there is a pressure excess over hydrostatic, outward flow continues. 

Instead of drawing the s-orbital as a cloud, we usually draw its boundary surface, a surface that encloses the densest regions of the cloud. The electron is likely to be found only inside the boundary surface of the orbital. An s-orbital has a spherical boundary surface (Fig. 1.24), because the electron cloud is spherical. s-Orbitals with higher energies have spherical boundary surfaces of greater diameter. They also have a more complicated radial variation (Fig. 1.25). 

In 1920, Max Born, a Nobel Prize winner, published some work on the free energy of solvation of ions, AGgon [21]. He conceived the idea of approximating the solvent surrounding the ion as a dielectric continuum. Defining a spherical boundary between the ion and the continuum by an effective ion-radius, f lon, he got the simple result... 

A plausible estimate of the spatial extension of a hydrogenic orbital is the radius of a spherical boundary surface within which there is a high probability of finding the electron. In order to develop this criterion on a quantitative basis, it is useful to define a cumulative probability function FW(p) which gives the probability of finding the electron at a distance less than or equal to p from the nucleus ... 

Lorentz calculated EL in the following way. A spherical region within the dielectric, centred on the point X at which EL is required, is selected. The radius is chosen so that, as viewed from X, the region external to the spherical boundary can be regarded as a continuum, whereas within the boundary the discontinuous atomic nature of the dielectric must be taken into consideration. EL can then be written... 

In this section we consider the general solution of (7-99) in the spherical coordinate system (r, 9, coordinate system is particularly useful for flows in the vicinity of a spherical boundary, but we begin by simply deducing the most general solution of (7-99) that is consistent with the constraint of axisymmetry, namely,... 

Water and matrix are characterized by their respective dielectric constants, e and t. Typical value are e n. 80, e 2, so that e e. We first write the self energy of one cation in the vesicle. It expresses the repulsion by the electrical image due to the spherical boundary. As is well known (15), there is no exact analytical expression for it in the spherical geometry, but if we neglect terms in 1/e2 it can be written as... 

In this case, the force exerted by the fluid on any spherical boundary described by the equation R = const is given by... 

---