Free energy curvature

In these symmetrical cases (Figs 4.2 and 4.3), the binodal points are given by the two minima. The binodal curve is the equilibrium curve differentiating stable one-phase systems from heterogeneous systems. In the concentration range between the binodal and spinodal points, the free energy curvature is positive and the solution is metastable towards composition flutuations. Nucleation and growth are associated with the phase separation and only one of the components forms a continuous phase. 

A very important thermodynamic relationship is that giving the effect of surface curvature on the molar free energy of a substance. This is perhaps best understood in terms of the pressure drop AP across an interface, as given by Young and Laplace in Eq. II-7. From thermodynamics, the effect of a change in mechanical pressure at constant temperature on the molar h ee energy of a substance is... 

Here, r is positive and there is thus an increased vapor pressure. In the case of water, P/ is about 1.001 if r is 10" cm, 1.011 if r is 10" cm, and 1.114 if r is 10 cm or 100 A. The effect has been verified experimentally for several liquids [20], down to radii of the order of 0.1 m, and indirect measurements have verified the Kelvin equation for R values down to about 30 A [19]. The phenomenon provides a ready explanation for the ability of vapors to supersaturate. The formation of a new liquid phase begins with small clusters that may grow or aggregate into droplets. In the absence of dust or other foreign surfaces, there will be an activation energy for the formation of these small clusters corresponding to the increased free energy due to the curvature of the surface (see Section IX-2). 

Here X is tire reorganization energy associated witli the curvature of tire reactant and product free energy wells and tlieir displacement witli respect to one another. Assuming a stmctureless polarizable medium, Marcus computed the solvent or outer-sphere component of tire reorganization energy to be... 

A real foam has further degrees of freedom available for estabHshing local mechanical equiHbrium the films and Plateau borders may curve. In fact, curvature can be readily seen in the borders of Figure 1. In order to maintain such curvature, there must be a pressure difference between adjacent bubbles given by Laplace s law according to the surface free energy of the film and the principle radii of curvature of the film AP = ) Note that the... 

A = 2k/ /3 for the case of cyhnders. In order to avoid this problem, Gompper and Kroll [241] have recently argued that a more appropriate discretization of the bending free energy should be based directly on the square of the local mean curvature ... 

The second possible cause of nonlinearity is a change in mechanism. Within a reaction series any change in mechanism must be such as to provide a smaller free energy of activation for the reaction (otherwise the mechanism would not change). If a substituent effect can produce a change in mechanism, the result must therefore be curvature that is concave upward. Figure 7-2 is a per plot for the S l solvolyses... 

As a result of the high surface free energy of water, the vapor pressure of a water droplet increases with decreasing radius of curvature, r, as deduced by Kelvin ... 

The use of Eq. (5-10) to evaluate the reaction rate is characterised by the calculation of Hessians for a large number of points along the MEP which are required to locate the free energy maximum and also to evaluate the curvature required for evaluation of the transmission coefficient. In view of the associated computational expense, high-level electronic structure calculations are not feasible and alternative strategies, one of which is to use a semi-empirical method, are usually employed [81]. 

If the medium is linear, the reactant and product state parabolas have the same curvature. In that case, one can show that the free energies to impose the constraints at the beginning and to remove them later exactly cancel, and we obtain the useful relations [56-58]... 

We have seen that the free energy curves for the reactant and product states have the same curvature, so that the relaxation free energy is the same in the reactant and product states /L4plxjd = AA C. This equality reflects the fact that the dielectric susceptibility a (12.24) does not depend on the perturbing field or charge, and is the same in the reactant and product states. We then obtain... 

SPT provides a conceptual basis relating the nonpolar free energy contribution to the solvent-exposed surface area. An attractive approximation is to ignore curvature effects and write... 

Fig. 4 Free energy reaction coordinate profiles that illustrate a change in the relative kinetic barriers for partitioning of carbocations between nucleophilic addition of solvent and deprotonation resulting from a change in the curvature of the potential energy surface for the nucleophile addition reaction. This would correspond to an increase in the intrinsic barrier for the thermoneutral carbocation-nucleophile addition reaction.

We deemed it necessary to confirm the CV results by the alternate method using convolutive potential sweep voltammetry, which requires no assumptions as to the form of the free energy relationship and is ideally suited for an independent analysis of curvature revealed in Figure 7. In convolutive linear sweep voltammetry, the heterogeneous rate constant ke is obtained from the cur-... 

Afi = QKy where y is the surface free energy/cm2 and K is the surface curvature... 

More recently, Smith et al. have developed another model based on spontaneous curvature.163 Their analysis is motivated by a remarkable experimental study of the elastic properties of individual helical ribbons formed in model biles. As mentioned in Section 5.2, they measure the change in pitch angle and radius for helical ribbons stretched between a rigid rod and a movable cantilever. They find that the results are inconsistent with the following set of three assumptions (a) The helix is in equilibrium, so that the number of helical turns between the contacts is free to relax, (b) The tilt direction is uniform, as will be discussed below in Section 6.3. (c) The free energy is given by the chiral model of Eq. (5). For that reason, they eliminate assumption (c) and consider an alternative model in which the curvature is favored not by a chiral asymmetry but by an asymmetry between the two sides of the bilayer membrane, that is, by a spontaneous curvature of the bilayer. With this assumption, they are able to explain the measurements of elastic properties. 

The first term in this free energy is the standard isotropic curvature rigidity. The second term is a chiral term with coefficient Xhp, which can exist only in chiral membranes and is prohibited by reflection symmetry in nonchiral membranes. This term favors curvature in a direction 45° from m. Thus, it gives an intrinsic bending force in any membrane with both chirality and tilt order. 

It is also possible that a membrane might have an even lower symmetry than a chiral smectic-C liquid crystal in particular, it might lose the twofold rotational symmetry. This would occur if the molecular tilt defines one orientation in the membrane plane and the direction of one-dimensional chains defines another orientation. In that case, the free energy would take a form similar to Eq. (5) but with additional elastic constants favoring curvature. The argument for tubule formation presented above would still apply, but it would become more mathematically complex because of the extra elastic constants. As an approximation, we can suppose that there is one principal direction of elastic anisotropy, with some slight perturbations about the ideal twofold symmetry. In that approximation, we can use the results presented above, with 4) representing the orientation of the principal elastic anisotropy. 

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