Interfacial gradient energy

The stability of inverse micelles has been treated by Eicke (8,9) and by Muller (10) for nonaqueous systems, while Adamson (1) and later Levine (11) calculated the electric field gradient in an inverse micelle for a solution in equilibrium with an aqueous solution. Ruckenstein (5) later gave a more complete treatment of the stability of such systems taking both enthalpic (Van der Waals (VdW) interparticle potential, the first component of the interfacial free energy and the interparticle contribution of the repulsion energy from the compression of the diffuse part of the electric double layer) and entropic contributions into consideration. His calculations also were performed for the equilibrium between two liquid solutions—one aqueous, the other hydrocarbon. 

Here H is the Heaviside function and the sum is over sites in the (real space) lattice. In the interface region, p and Q are functions of z, with a ramped shape used. Working again in the square gradient approximation, Klupsch finds an interface of width 3.592a (where a is now the Lennard-Jones parameter) and an interfacial free energy of 0.968e/a. ... 

Presence of surfactant at the interface will also directly affect deformation. The surfactant allows formation of a /-gradient. This would affect the deformation mode of a drop, which has indeed been observed. Moreover, enlarging the interfacial area causes y to increase, as mentioned. This implies that the interfacial free energy increase includes two terms y dA + A dy. The first term is due to the deformation of the drop being counteracted by its Laplace pressure the second is due to surface enlargement being counteracted by the surface dilational modulus ESD. Making use of Eq. (10.20) for we obtain... 

Thus, the free energy cost of the interface can be expressed in terms of the gradient energy term (which of course balances the change in the bulk free energy due the presence of the interface). For the profile given by Eq. (2.33), this yields an interfacial tension... 

The simplest heterogeneous model considers only the interfacial gradients between sohd and fluid phases, imposing mass and energy balances for all phases involved and introdueing mass and heat transport from the gas bulk to catalyst surface and vice versa. Typical mass and energy balances equations are ... 

This suggests that under UHV conditions, for both copolymer systems (acrylates or polyurethanes), there is an increasing concentration gradient of the phos-phorylcholine groups from the surface to the bulk of the polymer. The copolymer system tends to lower Its Interfacial free energy by burying the PC group underneath the extreme surface. 

Within this contimiiim approach Calm and Flilliard [48] have studied the universal properties of interfaces. While their elegant scheme is applicable to arbitrary free-energy fiinctionals with a square gradient fomi we illustrate it here for the important special case of the Ginzburg-Landau fomi. For an ideally planar mterface the profile depends only on the distance z from the interfacial plane. In mean field approximation, the profile m(z) minimizes the free-energy fiinctional (B3.6.11). This yields the Euler-Lagrange equation... 

The kinetics of spinodal decomposition is complicated by the fact that the new phases which are formed must have different molar volumes from one another, and so tire interfacial energy plays a role in the rate of decomposition. Anotlrer important consideration is that the transformation must involve the appearance of concenuation gradients in the alloy, and drerefore the analysis above is incorrect if it is assumed that phase separation occurs to yield equilibrium phases of constant composition. An example of a binary alloy which shows this feature is the gold-nickel system, which begins to decompose below 810°C. 

When the two phases separate the distribution of the solvent molecules is inhomogeneous at the interface this gives rise to an additional contribution to the free energy, which Henderson and Schmickler treated in the square gradient approximation [36]. Using simple trial functions, they calculated the density profiles at the interface for a number of system parameters. The results show the same qualitative behavior as those obtained by Monte Carlo simulations for the lattice gas the lower the interfacial tension, the wider is the interfacial region in which the two solvents mix (see Table 3). 

It follows from Eqn. 4—13 that the electron level o u/av) in the electrode is a function of the chemical potential p.(M) of electrons in the electrode, the interfacial potential (the inner potential difference) between the electrode and the electrolyte solution, and the surface potential Xs/v of the electrolyte solution. It appears that the electron level cx (ii/a/v) in the electrode depends on the interfacial potential of the electrode and the chemical potential of electron in the electrode but does not depend upon the chemical potential of electron in the electrolyte solution. Equation 4-13 is valid when no electrostatic potential gradient exists in the electrolyte solution. In the presence of a potential gradient, an additional electrostatic energy has to be included in Eqn. 4-13. 

When acetic acid is diffusing from a 1.9 iV solution in water into benzene, spontaneous emulsion forms on the aqueous side of the interface, accompanied by a little interfacial turbulence. Results can be obtained with this system, however, if in analysing the refractive index gradient near the surface a correction is made for the spontaneous emulsion the rate of transfer is then in excellent agreement (57) with Eq. (20) (Fig. 6). Consequently there is no appreciable energy barrier due to re-solvation of the acetic acid molecules at the interface, nor does the spontaneous emulsion affect the transfer. With a monolayer of sodium lauryl... 

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