Projected surface free energy

Interaction increases quadratically with the step density (b) Relationship between the projected surface free energy and the crystal shape as a function of temperature. 

This is called the projected surface free energy, because it is the surface free energy projected along the surface normal direction, z. One of the most important meanings of the projected surface free energy is that it is equivalent to the crystal shape through the Legendre transformation. The conversion of a polar coordinates (fr 9) into a... 

This is defined as the surface free energy per unit area projected onto the low index facet plane. The use of the projected free energy allows a direct analogy with the thermodynamics of a liquid-vapor system. See, e.g., Williams et al. for a clear discussion. 

Jayaprakash et al. (98, 99) presented an interacting TLK model, focusing on the effect of long- and short-range interactions between distinct facets and those between facets and curved surfaces. In this model the 2D statistics of steps is reduced to the 1D quantum fermion system, and the variation of the surface free energy (per projected area in the low-index plane) with angle 9 and temperature T is described by the equation... 

G. Electrostatic Contribution to Adhesion. Surface free energies describe adhesion phenomena once molecular contact has been achieved between the adhering phases. For a cell to come from a distance into molecular contact with a surface requires consideration of long-range forces which influence approach. Two factors are of prime importance to the attachment of microbial cells to solid surfaces 1) electrostatic interactions and 2) fine surface projections. 

Fig. 39a, b. Contour plot of relative surface free energy of Cu measured at T = 1200 K. Data are shown as a stereographic projection in the unit triangle of low-index orientations. All values are normalized to unity at the (100) orientation, (a) Clean Cu surfaces (b) oxygen covered Cu surfaces at a O2 partial pressure of lO mbar [70Hon2]. 

In the case of a rough surface, the expression for the free energy of a hquid layer, per unit projected area, can be represented as... 

Perhaps the most important conclusion arising from a study of such models is that the projected free energy density of a uniform vicinal surface with slope s is given by the familiar Gruber-Mullins " expression ... 

The l.h.s. simply evaluates H in this limit and the r.h.s. is the surface area LyNsW of the flat reference plane times the projected free energy density J s) for a uniform system with Slope s = Mw. Thus we find... 

If, at a given temperature and for optimized values of all other independent variables than A, for example, we consider the projection of the free energy surfaces on the G,X plane, we could, in principle, calculate the (G,X)T, etc., curve for each phase and so derive the boundary compositions of coexisting phases. 

Fig. 9. Three dimensional projection of the 6D free energy surface, showing the local minima (black crosses) and the E2 and Sjv2 reaction channels. The gray arrows depict the lowest free energy paths for the two reactions. After [41]...

In this section several examples will be presented that are representative of ongoing research projects. Even though the results are still preliminary, the projects selected further illustrate the utility of MD simulations in soil chemistry. Topics include the calculation of free energy changes associated with cation exchange and the adsorption of proteins on clay mineral surfaces. 

Mackor (1951) was perhaps the first to endeavour to calculate the free energy of repulsion between sterically stabilized particles. This work was instigated after van der Waarden (1950 1951) had shown experimentally that aromatic molecules with long-chain aliphatic substituents could have a profound effect on the stability of carbon black particles dispersed in a paraffin (see Section 2.4.2). For this reason, Mackor adopted a model in which he assumed that the aromatic nuclei were adsorbed onto the carbon black particles in a flat configuration, thus anchoring the alkyl chains to the surface. These chains were assumed to project into the dispersion medium and were modelled as rigid rods, of length L, flexibly attached to the particle surfaces by ball joints. 

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