The Question of Equilibrium Shapes

The first term on the right hand side represents the energy cost associated with the interface between the particle and the matrix, and for generality we have assumed (temporarily) that the interfacial energy is anisotropic, a fact that is revealed by the dependence of this energy on the normal n as y (n). The second term is the elastic energy of the medium by virtue of the presence of the second-phase particle. Note that both of these terms are functionals of the particle shape and our ambition is to 

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Equilibrium Shapes in Three Dimensions. The analyses described above have illustrated the fundamental competition between interfacial energies and elastic energies in governing the outcome of the question of equilibrium shapes. These calculations were introduced in the setting of reduced dimensionality with the... 

Equilibrium Shapes Revisited. In section 10.2.4, we described the examination of equilibrium shapes. Though the question of equilibrium shapes is one of terminal privileged states, nevertheless, the ideas introduced here which can in... 

Where are the nuclei . This is nob just a question of equilibrium shape as measured by n.m.r./ x-ray or neutron spectroscopy/ but also concerns what possible shapes the molecule can asscune as it interacts with its partner in general/ what flexibility it possesses. Flexibility is clearly a property of both small molecules and the protein binding sites. 

Hence, within this approximate model, the entire question of equilibrium shape has been reduced to the dependence of F2 on A which is gotten by adding the quadratic contributions to the energy from both the interfacial and elastic terms and is given by... 

Therefore, we first look at the question of how a crystal looks in thermodynamical equilibrium. Macroscopically, this is controlled by its anisotropic surface (free) energy and the shape can be calculated via the Wullf construction. 

McBain accounted for hysteresis by assuming that the pores contained a narrow opening and a wide body, the so-called bottle-neck shape. His model asserts that during adsorption the wide inner portion of the pore is filled at high relative pressures but cannot empty until the narrow neck of the pore first empties at lower relative pressures during desorption. Therefore, for bottle-neck pores the adsorption isotherm corresponds to the equilibrium condition. However, the model proposed by McBain ignores the question of how condensation into the wider inner portion of the pore can occur once the narrow neck has been filled at low relative pressures. 

Principle of corresponding states. The principle of corresponding states, originally introduced by van der Waals and applied since to model inter-molecular potentials, transport and equilibrium properties of fluids over a wide range of experimental conditions, was remarkably successful, albeit it is not exact in its original form. An interesting question is whether one could, perhaps, describe the diversity of spectral shapes illustrated above by some reduced profile, in terms of reduced variables. If all known rare-gas spectra are replotted in terms of reduced frequencies and absorption strengths,... 

In this research into the limits of stabihty of the shapes of equilibrium of revolution, we always have supposed the finite shape has two cross-sections perpendicular to the axis and equal in diameter. But it is clear that one could adopt other terminations and that then the limits of Stability would be different one could, for example, still take for the bases of the shape two cross-sections perpendicular to the axis, but give them, except in the case of the cylinder, unequal diameters. One saw ( 91) that Mr. Lindelof analytically treated, for this ratio, the question of the catenoid but I had arrived before at a remarkable result that I had made known in my 11th Series here when one takes the neck circle for one of the terminations, the catenoid does not have a limit of stability any more, i.e. the second base can as be far from the first as is wanted, without the shape tending to alter spontaneously. 

If thus one wanted to treat a priori, and only by calculation, the question of the limits of stability of equilibrium liquid shapes, the problem would consist in seeking, for each surface represented by the equation = C, the limits between... 

Since an actual crystal will be polyhedral in shape and may well expose faces of different surface tension, the question is what value of y and of r should be used. As noted in connection with Fig. VII-2, the Wulff theorem states that 7,/r,- is invariant for all faces of an equilibrium crystal. In Fig. VII-2, rio is the... 

In calculations of pore size from the Type IV isotherm by use of the Kelvin equation, the region of the isotherm involved is the hysteresis loop, since it is here that capillary condensation is occurring. Consequently there are two values of relative pressure for a given uptake, and the question presents itself as to what is the significance of each of the two values of r which would result from insertion of the two different values of relative pressure into Equation (3.20). Any answer to this question calls for a discussion of the origin of hysteresis, and this must be based on actual models of pore shape, since a purely thermodynamic approach cannot account for two positions of apparent equilibrium. 

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

A simple way to appreciate the shape of fullerene is to construct a physical model in which rigid planar trivalent nodal connectors represent the atoms and flexible plastic bars (tubes) of circular cross-section represent the bonds. From a mechanical point of view the model may be considered as a polyhedron-like space frame whose equilibrium shape is due to self-stress caused by deformation of bars. We suppose that the bars are equal and straight in the rest position and that they are inclined relative to each other at every node with angle of 120°. The material of the bars is assumed to be perfectly elastic and that Hooke s law is valid. All the external loads and influences are neglected and only self-stress is taken into account. Then we pose the question What is the shape of the model subject to these conditions To answer this question we apply the idea used for coated vesicles by Tarnai Gaspar (1989). 

Now, a question arises, Is there a way to quantitatively describe the phase boundaries in terms of P and T The phase rule predicts the existence of the phase boundaries, but does not give any clue on the shape (slope) of the boundaries. To answer the above question, we make use of the fact that at equilibrium the chemical potential of a substance is the same in all phases present. 

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