A thermodynamics question
how can I determine if the Helmholtz Free Energy can depend on the 
gradient of temperature? I think the answer depends on violating the 
second law.

http://arxiv.org/PS_cache/astro-ph/pdf/0008/0008399.pdf

A method is presented for the calculation of the gradient of the free
energy with respect to all the internal and external degrees of
freedom of a periodic crystal. This gradient can be used in
conjunction with a static-energy Hessian for efficient geometrical
optimization of systems with large unit cells. The free energy is
calculated using lattice statics and lattice dynamics in the
quasiharmonic approximation, and its derivatives by means of
first-order perturbation theory. In the present application of the
method, particles are assumed to interact via arbitrary short-ranged
spherically-symmetric pair potentials and long-ranged Coulomb forces,
and polarizability effects are accounted for by use of the shell
model. The method can be used directly as the basis for a computer
program which makes efficient use of both storage and CPU time,
especially for large unit cells. Detailed expressions for all the
lattice sums are presented.

The inclusion of anisotropic surface free energy and anisotropic
linear interface kinetics in phase-field models is studied for the
solidification of a pure material. The formulation is described for a
two-dimensional system with a smooth crystal-melt interface and for a
surface free energy that varies smoothly with orientation, in which
case a quite general dependence of the surface free energy and kinetic
coefficient on orientation can be treated; it is assumed that the
anisotropy is mild enough that missing orientations do not occur. The
method of matched asymptotic expansions is used to recover the
appropriate anisotropic form of the Gibbs-Thomson equation in the
sharp-interface limit in which the width of the diffuse interface is
thin compared to its local radius of curvature. It is found that the
surface free energy and the thickness of the diffuse interface have
the same anisotropy, whereas the kinetic coefficient has an anisotropy
characterized by the product of the interface thickness with the
intrinsic mobility of the phase field.

thanks, but I'm asking a much simpler question. 
can I include the gradient of temperature in the Helmholtz Free Energy
without violating the second law? I have a feeling this is a textbook 
type problem. It's not my area, and I don't know how to proceed.

No. The Helmholtz Free Energy can not depend on the gradient of
Temperature. (I think).

http://tinyurl.com/byr6s

The author discusses the Helmholtz F.E. then looks at the constraints
place upon it by the Second Law of Thermodynamics, and comes to the
conclusion that for a thermoelastic system, the Helmholtz is
independant of the Temperature Gradient.

(thermoelastic is a system where the only work exchanged with the
surroundings is that of pressure forces -> http://tinyurl.com/d6v7g


Now. I'm not completely sure if this is right, as it's kinda out of my
league (not to mention the fact that I hate thermodynamics and
Statistical Mechanics...)

to theeldest: 
ok, I looked at that textbook you suggsted http://tinyurl.com/byr6s and I 
think my answer is in there, so I'm willing to cough up the dough. I think 
you have to post it as the answer first. 
FYI, the question came up in a course I'm taking thru distance learning on
continuum mechanics, specifically on thermoelasticity. It's a brand new 
area for me, and I also hate thermodynamics. 
thanks again
sanjoser

Oh. I'm not an Answerer. I was just browsing around here while I was
looking for help on my Thermodynamics homework, and thought I could
help out. (we had just gotten done with Helmholtz and a bunch of
associated derivations).

to theeldest
well then, how can I do something for you? I'm actually doing my
continuum mechanics homework. I took your suggestion and worked out
a pretty good answer to the problem I was doing. Is there something
I can help you do? I know a fair bit of stuff.
best
sanjoser