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The Basis for Convexification

In this section we show that the functions defined in Eq. 2 must be convex in the kind of minimizations that are representative of thermodynamic equilibrium. It is apparent from equation (5) that convexity applies to the of any chemical system. If any two chemical systems are combined, the masses of their individual components are added; this is equivalent to a vector addition of the as and are added in equation (5). But the equilibrium free energy of the combined system can not be greater than the sum of the equilibrium free energies of the separated systems. If the two systems remain unmixed, the resultant free energy would be the sum of the free energies of the parts; any relaxation towards equilibrium can only lead to a reduction in free energy. Thus with the use of equation (5), the convexity of is a simple consequence of thermodynamics.

We next show with reasoning that is quite similar that convexity also applies to . The convexified is the lowest free energy that a surface with a planar perimeter with orientation and spanning an area can achieve, allowing facetting to all other orientations. Note that adding two area vectors, , , gives a another area vector, say , which lies in the plane spanned by and . This allows a simple construction for the addition of area vectors. Let the area vectors be represented by rectangles of area, , , and . Since the normals to these three rectangles lie in a plane, the three rectangles form a `tent' (or, triangular prism) and we will take the rectangle representing the summed area as the `tent floor.' The proof that is convex parallels the proof that is convex. Consider the area . Its energy cannot exceed the energy of the combined areas and ; if this were not true would spontaneously form a tent. This must hold for all possible configuration of `tent sides.' But clearly can be less than the tent energy. Thus is convex at equilibrium since for all . Note that the magnitude of area itself is a convex function; the combined area cannot exceed the sum of the separate areas.

The consequences are also similar. If is a convex function, then all orientations are stable with respect to facetting. Since , formation of a tent or corrugation of a surface represented by into any a configuration represented by two other vectors that sum to cannot decrease the energy of the original structure. Conversely, if is not convex at , then there must be a corrugated structure which is composed of alternating pieces and which has a lower surface energy. The same construction can be applied to the formation of corners by considering a partition into three or more orientations.

It is important to note that this convexity criterion comes from thermodynamics on a very general and fundamental level. The inequality (5) applies to and , and not to the molar free energy or the surface free energy per unit area . The inequality (5) should not and does not apply to . Note that when convexified for equilibrium curves up instead of down. For a binary solution, comparing with the sum of and does not make any sense, since mass is not conserved. Although the solute species is conserved, the mass of the solvent is not.

The inequality (4) applies to and , and to any planar submanifold (a lower dimensional planar cut, including any straight line section) of the extended functions of or dimensional variables. When and are taken as end points of vectors, the end point of the vector is always on the connecting straight line segment. Thus is convex from below on any straight line section; this includes the hyperplane (for which ) of molar free energies . Thus convexity applies to . The widely used graphical convexification methods for are thus validated.

Applying the inequality (5) for is valid; but applying (4) makes little sense for . When and are taken on the unit sphere, that is as end points of unit vectors, the end point of the vector is always on the connecting chord; the inequality, while correct, applies to a vector that is not a unit vector, one that is in the interior of the unit sphere, and thus not to . Thus whether the scaled functions , , and when restricted to some submanifold are also convex depends on the somewhat arbitrary choice of how a unit of the argument is measured; in other words, whether the choice of the operation which is implied by the operator is linear.

Another use of the inequality (4) is to note that the surfaces defined by level sets of any convex function have to be convex when the function is a positive constant and concave when the function is negative. Because is the usual length of the area vector and plots as distance from the origin , we can convert the equation () for the level surfaces for into radial plots of the reciprocal of . Setting the constant to the equation for the level surface becomes . Since is positive, the radial plot of the reciprocal of has to be convex at equilibrium.

Note that is not the usual length of a vector, and is not the radial distance to the level set of . As a result the convexity of an inverse plot of has as little significance as the convexity of a radial plot of . We next look into the definitions of the metrics of these quantities, to understand the basis for these differences and to look for alternate definitions.



Next: Metrics Up: ConvexificationCommon Tangents, Previous: ConvexificationCommon Tangents,


wcraig@ctcms.nist.gov
Wed Mar 8 09:54:09 EST 1995