Free energy per unit area

The total free energy of the system is then made up of the molar free energy times the total number of moles of the liquid plus G, the surface free energy per unit area, times the total surface area. Thus... 

The interfacial free energy per unit area is given by the adhesion force Fo/lrR", estimate the Hamaker constant responsible for the adhesion force in the crossed-cylinder geometry illustrated in the inset to Fig. VI-6. 

Referring to Section V-2, the double-layer system associated with a surface whose potential is some value j/o requires for its formation a free energy per unit area or a t of... 

In this relationship. S is alkane solubility, A is the cavity surface area and a is the hydrophobic free energy per unit area. Extensive fitting of this equation [24] yields a value of 88 kJ mol A for the proportionality constant a. This value corresponds to an unfavourable free energy of about 3.6 kJ mol for the transfer of a CH2 group to aqueous solution. 

As in the qualitative discussion above, let 7 be the Gibbs free energy per unit area of the interface between the crystal and the surrounding hquid. This is undoubtedly different for the edges of the plate than for its faces, but we... 

Here,. Ai(X) is the partial SASA of atom i (which depends on the solute configuration X), and Yi is an atomic free energy per unit area associated with atom i. We refer to those models as full SASA. Because it is so simple, this approach is widely used in computations on biomolecules [96-98]. Variations of the solvent-exposed area models are the shell model of Scheraga [99,100], the excluded-volume model of Colonna-Cesari and Sander [101,102], and the Gaussian model of Lazaridis and Karplus [103]. Full SASA models have been used for investigating the thermal denaturation of proteins [103] and to examine protein-protein association [104]. 

The interface free energy per unit area fi,u is taken to be that of a planar interface between coexisting phases. Considering a solution v /(z) that minimizes Eq. (5) subject to the boundary conditions vj/(z - oo) = - v /coex, v /(z + oo) = + vj/ oex one finds the excess free energy of a planar interface ... 

FIG. 15 Free energy per unit area as a function of surface separation for five different values of x-The parameters are Ng = 25, p = 0.04, N = 100, and Xsurf = 0. The cartoon on the left shows the reference state, where the grafted chains form a melt between the surfaces. In the cartoon on the right, the surfaces are separated by polymers that have localized between the interfaces. (From Ref. 100.)... 

Let e > 0 be the energy gain per segment in the nucleus (in units of kT) and a be the lateral surface free energy per unit area. The free energy Fm>/t per chain in the nucleus of Fig. 11 is given by... 

There is a discussion in the literature about the effect of undulation entropy on the equilibrium membrane tension [14,15], Formally, undulations are included in the surface tension, and thus we need not worry about this. However, if in some model the two are artificially decoupled, one may allow for a very small (positive) surface tension as the equilibrium structure. In other words, the entropy (per unit area) from undulations should compensate for the tension (excess free energy per unit area). 

The total change in free energy per unit area in charging up the interface is obtained by integrating Equation 14... 

The interaction free energy per unit area between two planar double layers V(L) is just the difference between the change in free energy per unit area in charging up two surfaces at a distance L apart and that at infinite separation... 

On the other hand, the electrical free energy per unit area of double layer (second term) is high and positive even for relatively low surface potential. The contribution of this term could be tens of m Nm. This requires to have a... 

Comparison of the Different Techniques—Free Energy per Unit Area... 

In Fig. 2.4, we compare the free energy levels that are accessible using SFA, TFB, and MCT techniques. For TFB and MCT, the accessible range is narrower but these two techniques offer the advantage to reach lower levels of free energy per unit area compared to SFA. 

Thus, the tension per unit length in a single surface, or surface tension y, is numerically equal to the surface energy per unit area. Then Gs, the surface free energy per unit area is... 

Consider a finite crystal of thickness L, along the ith axis, as sketched in Figure 1.7. Let c, be the surface free energy per unit area of surfaces perpendicular to the ith axis. 

Consider the cylinder sketched in Figure 1.9 with radius R and length L, where lateral surface free energies per unit area. 

FIGURE 15.10 Plots of the Gibbs free energy per unit area, AG/A, as a function of the distance between two oppositely charged planar surfaces, L, with the ionic strength as a parameter. The curves are calculated from Equation 15.63 with e=80, c =-0.16C/m, and Op = 0.03C/m. ... 

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. 

The interface between two fluids is in reality a thin layer, typically a few molecular dimensions thick. The thickness is not well defined since physical properties vary continuously from the values of one bulk phase to that of the other. In practice, however, the interface is generally treated as if it were infinitesimally thin, i.e., as if there were a sharp discontinuity between two bulk phases (LI). Of special importance is the surface or interfacial tension, a, which is best viewed as the surface free energy per unit area at constant temperature. Many workers have used other properties, such as surface viscosity (see Chapter 3) to describe the interface. 

Here Vij denotes the distance between atoms i and j and g(i) the type of the amino acid i. The Leonard-Jones parameters Vij,Rij for potential depths and equilibrium distance) depend on the type of the atom pair and were adjusted to satisfy constraints derived from as a set of 138 proteins of the PDB database [18, 17, 19]. The non-trivial electrostatic interactions in proteins are represented via group-specific dielectric constants ig(i),g(j) depending on the amino-acid to which atom i belongs). The partial charges qi and the dielectric constants were derived in a potential-of-mean-force approach [20]. Interactions with the solvent were first fit in a minimal solvent accessible surface model [21] parameterized by free energies per unit area (7j to reproduce the enthalpies of solvation of the Gly-X-Gly family of peptides [22]. Ai corresponds to the area of atom i that is in contact with a ficticious solvent. Hydrogen bonds are described via dipole-dipole interactions included in the electrostatic terms... 

Reduced Surface Tension. Just as surfactants self-organize in the bulk solution as a result of their hydrophilic and hydrophobic segments, they also preferentially adsorb and organize at the solution—vapor interface. In the case of aqueous surfactant solutions, the hydrophobic tails protrude into the vapor and leave only (he hydrophilic head groups in contact with the solution. The favorable energetics of the arrangement can be seen by the reduction in Ihe interracial free energy per unit area, nr surface tension, it. 

The partial derivative of the Gibbs free energy per unit area at constant temperature and pressure is defined as the interfacial coefficient of the free energy or the interfacial tension (y), a key concept in surface and interface science ... 

To consider the elastic increase of the surface area the surface stress T is introduced. The change in the Gibbs free energy per unit area is given by the reversible work required to expand the surface against the surface tension 7 and the surface stress T... 

Forces of surface tension act on the surface of the particles in the liquid phase. It can be expressed as the work for the formation of a unit of interphase surface under constant thermodynamic parameters of the state (temperature, pressure, chemical potentials of the components). This process is reversible and isothermal. The surface tension forces can be regarded also as free energy per unit area, i.e. specific free energy (Gs). Then, the free energy per unit weight of particles would be... 

Its value was calculated to be 6>o = 68.18°. At 6>o < 90° (as is in this case), the process called limited wetting or wetting with impregnation. Therefore, the change of the free energy per unit area can be found by the expression... 

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