Chemical potential surfaces

Once the assumption of mechanical equilibrium has been incorporated, the free energy expression can be reinterpreted. Instead of viewing the free energy as a functional of both strain and surface shape, it can instead be viewed as a functional of surface shape alone (Shenoy and Freund 2002). In principle, the strain is known in terms of surface shape through the solution of the boundary value problem represented by (8.131) and (8.132). The goal in this section is to write the time rate of change of free energy in the form 

Following the reasoning outlined in Section 8.2.1, the time rate of change of free energy iF t) is 

The quantity q can be interpreted as a distributed surface torque or moment that tends to rotate the surface toward an orientation with lower surface energy. 

If the relations (8.127) and (8.128) are incorporated into (8.134), and if integration by parts is invoked to isolate a factor in each term, it is found that 

The terms obtained in the course of integration by parts have zero net value because of the assumed workless nature of the boundary constraints. The quantity appearing in (8.136) is the shear strain introduced in (8.128) 

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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. 

For the reasons we have just been discussing, we shall focus attention on the uptake of hydrogen by samples hydrogenated by exposure to plasma products for times of the order of an hour at 300°C and shall analyze the data on the assumption that the surface chemical potential / for given external and surface conditions is roughly independent of donor or acceptor doping. However, our conclusions will be tentative, since presently available data are limited and both the assumption of local equilibration and that of constant surface p need further checking. 

Such control of surface functionahties, surface chemical potential, and surface dipole is not possible in mixed SAMs of cofunctionalized saturated -alkanethiols, since dipolar interaction of surface functionahties wiU result in surface reorganization [95, 99-101]. 

In Eq. (24) the gradient of the surface chemical potential, g , is assumed to be the driving force. The surface phase concentrations, qi and qsat, and the surface diffusiv-ities, D j and D , must be known in order to predict the surface fluxes, J ... 

The first term on the r.h.s. is a kind of surface chemical potential. It is assumed that this term can be written as a standard term and a configurational part RTln. where = n°/ Zj n°. Hence,... 

This result is an extension of equation (28.3) the surface chemical potential of any constituent of a system is thus equ to its chemical potential in the bulk phases at equilibrium. 

If the system is in equilibrium, at constant temperature, pressure and surface area, the surface chemical potential of any constituent must always be equal to its chemical potential in the bulk phase, i.e., the solution, by equation (28.21). It is thus permissible to write equation (28.26) in the form... 

The reference state is taken to be the chemical potential and vapor pressure of the bulk lubricant m(oo) = 0 and P°(oo) is the vapor pressure of the bulk liquid. The surface chemical potential is approximated by the unretarded atom-slab dispersive interaction energy ... 

SURFACE CHEMICAL POTENTIALS FOR SPECIES ADSORBED ON HOMOGENEOUS SURFACES... 

Finally, we should make the following clarification in what concerns the surface chemical potential of the solvent clusters appearing in Eq. (10). The various solvent clusters do not have different surface coverages, since the following relationship is valid ... 

Therefore, the surface chemical potential of the i-th solvent cluster may be expressed as... 

In section 3 the contribution to the surface chemical potentials arising from the field of the potential drop Aif was derived from the combination of classical thermodynamics with the electrostatic theory of dielectrics. This contribution may be alternatively calculated as follows. Suppose that in general an adsorbed layer with M lattice sites, area A and thickness 1 consists of N neutral molecules of the i-th species, i 1, 2,. .., N. If... 

Each surfactant adsorption isotherm (that of Langmuir, Volmer, Frumkin, etc.), and the related expressions for the surface tension and surface chemical potential, can be derived from an expression for the surface free energy, F, which corresponds to a given physical model. This derivation helps us obtain (or identify) the self-consistent system of equations, referring to a given model, which is to be applied to interpret a set of experimental data. Combination of equations corresponding to different models (say, Langmuir adsorption isotherm with Frumkin surface tension isotherm) is incorrect and must be avoided. 

In equilibrium, the surface chemical potential is equal to the bulk (juf = / , ), and the chemical potential change on addition of any material is independent of the a, /3, interface phases. 

Equation (55) also suggests the introduction of the name surface-chemical potential for [i thus underlining the facts that fi enters into the fundamental equation for a surface phase in the same manner as enters into the corresponding equation for a bulk phase and that generally /t. In these respects is analogous to the electrochemical potential. However, the counterpart of Eq. (55) in the thermodynamics of electrochemical systems is the well-known relation... 

On the other hand, well-defined adsorption isotherms of proteins have been reported. Figure 1 shows one example, that for chymotrypsin in pure water at 20 C. The attainment of steady surface pressure values, which Increase with increasing protein concentration in solution indicating true equalization of bulk and surface chemical potentials, argues in favor of a reversible adsorption process. In addition, desorption from protein monolayers has been measured. How to rationalize these apparently conflicting results therefore presents an intriguing challenge. 

For any surface morphologies to appear, a surface reconstruction mechanism is required that would transform initially planar surface into a structured one. In epitaxially grown solid films such mechanism is surface diffusion in which atoms jump along the surface from one site to another, driven by the gradient of the surface chemical potential. The surface flux of atoms, j, is given by the analog of the Pick s law js = —MVsPs, where ps is the sur-... 

Vn is the normal velocity of the surface caused by the atoms redistribution and ps is the surface density of atoms. The surface chemical potential is typically determined by the film elastic energy and surface energy, and as such it is a function of the him local thickness as well as its slope, curvature, and may be higher spahal derivahves (see below). For very thin hlms (a few atomic layers) wetting interactions between the him and the substrate can also become important. These interachons are somewhat similar to wetting interactions between a liquid him and a solid substrate. They are responsible for the presence of an ultra-thin wetting layer of the him material between the islands resulting from the him instabihty and depend on the him thickness and its slope. Naturally, this dependence decays rapidly with the increase of the him thickness. 

The surface chemical potential is given by the equilibrium relationship with the gas phase ... 

The surface chemical potential gradient can be expressed in terms of the surface concentration as follows ... 

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