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.