Calculated interface width

FIGURE 9.5 Calculated interface width. For parameter values, see Figure 9.2. 

Though a powerfiil technique, Neutron Reflectivity has a number of drawbacks. Two are experimental the necessity to go to a neutron source and, because of the extreme grazing angles, a requirement that the sample be optically flat over at least a 5-cm diameter. Two drawbacks are concerned with data interpretation the reflec-tivity-versus-angle data does not directly give a a depth profile this must be obtained by calculation for an assumed model where layer thickness and interface width are parameters (cf., XRF and VASE determination of film thicknesses. Chapters 6 and 7). The second problem is that roughness at an interface produces the same effect on specular reflection as true interdiffiision. 

To demonstrate the systematics of how an oscillatory time variation in the reflectivity is possible during dissolution, we assume that the surface is characterized by occupation factors described by an error function profile (Fig. 28A). These occupation factors can be thought of as blocks of orthoclase, so that occupation factors <1 represent a partially filled layer that is locally stoichiometric. The position of the error function moves continuously as the surface dissolves. For simplicity, we first assume that the interface width does not change during dissolution. The reflectivity is calculated as a function of the error function position. At the anti-Bragg condition, neighboring terraces are out of phase. The phase factor for each layer, n, varies as ( <9 ) = (-1) , so that interfacial structure factor becomes ... 

Molecular dynamics simulations have been used to test the validity of the CW theory down to distances comparable to 4b- Equation [14] predicts a specific dependence of the interface width on the temperature. Simulations at different temperatures can be used to determine (C ) (by fitting the density profile to Eq. [13]). This, combined with surface tension calculations (see below), can be used to verify that V(C ) s proportional to - T/y. Figure 3 shows this plot generated using the data published in the very recent million particles simulation of the Lennard-Jones liquid/vapor interface. As can be seen, the relation in Eq. [14] holds quite well at low T. Another simple approach is to obtain 4b from the bulk radial distribution function (g(4b) 1) and confirm the validity of Eq. [14] using the independently calculated surface tension and (C ), as has been done for several liquid/liquid interfaces. Alternatively, if several simulations with different surface areas are performed, Eq. [14] suggests that a plot of straight line with a slope of... 

The interfacial capacity is then obtained by calculating the profiles for various potential drops A0 and subsequent differentiation. Figure 7 shows several examples of capacity-potential characteristics for several widths of the interface. Obviously, the wider the interface, the higher the capacity. In all cases investigated it was higher than that calculated from the Verwey-Niessen model, in which ... 

The capacitance C of the SCR is usually much smaller than that of the double layer in the electrolyte and dominates the AC behavior of the whole system. The capacitance for an electrode of interface area A and an SCR of width W can be calculated according to... 

The thickness and width of the solid bed as a function of the helical downstream position z are calculated from the melting velocities at the interfaces over a small Az increment. The calculation is progressed down the transition section until the value approaches zero. The balances for the solid in the x and y directions for an increment in the z direction are as follows ... 

The details of the interactions between ions and surfaces are not known, because they involve large-scale ab initio quantum mechanical calculations. A simple manner to account for the change in the free energy when the ions approach the interface is via a potential well (or a potential barrio ) with a depth (height) A W(x) of the order of kT and with a width ir of the order of a few Angstroms [10]. The results are not qualitatively affected by the shape of the interaction potentials, a square or a triangular well providing similar results as a power-law interaction [10]. 

FIG. 2 A topographical plot of the electrostatic potential adjacent to and outside of a layer of charged hemicylindrical micelles adsorbed to a solid interface. The results were obtained using the boundary integral method (or boundary element method, BEM). All values shown are dimensionless. The half-width of a unit cell was taken to be b = 5 k -1, while the maximum extent into the electrolyte calculated was L = 10k-1. The radius of the cylinder s circular cross-section was assumed to be K —1. The potential is given in dimensionless units, y = efhp. 

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