# Cahn and Hilliard Derivation of Interfacial Free Energy of a System

In the paper 'Free Energy of a Nonuniform System. I. Interfacial Free Energy' (https://doi.org/10.1063/1.1744102), the authors present an approximation to the free energy per molecule $$f$$ in a binary mixture of two chemical species, where $$c$$ denotes the concentration of a given component of the system.

In section 2, paragraph 2 of the paper, the authors say the following (find a picture of the relevant paragraph attached),

"Providing $$f$$ is a continuous function of these variables $$[c, \nabla c, \nabla^2c, ...]$$, it can be expanded in a Taylor series about $$f_0$$ the free energy per molecule of a solution of uniform composition $$c$$. . . leading terms in the expansion of $$f$$ are:

$$f(c, \nabla c, \nabla^2 c, ...) = f_0(c) + \sum_i [\partial f/\partial (\partial c/\partial x_i)]_0(\partial c/\partial x_i) + ...$$"

Now, my question is the following---since $$f$$ is not assumed to be a function of $$f_0$$, what sense does it make to say that the series presented above is a Taylor Series expansion of $$f$$ about $$f_0$$? In other words, how is this a mathematically defensible step to take?

I've never seen anything like this before, could someone help me parse this out?

Regarding the $$f_0$$: I think they mean $$$$f_0 = f(c,0,0,...)$$$$ which is exactly what they say: expanding $$f$$ around the free energy per molecule of a uniform composition. This also explains why you do not find a $$\left[\frac{\partial f}{\partial c}\right]_0=\left[\frac{\partial f}{\partial c}\right](c,0,0,...)$$ term on the right-hand side. You take the multidimensional series in regards to the vector of all derivatives of c $$$$\left[\frac{\partial c}{\partial x},\frac{\partial c}{\partial y},\frac{\partial^2 c}{\partial x^2},\frac{\partial^2 c}{\partial y^2},\frac{\partial^2 c}{\partial xy},...\right],$$$$ and not $$c$$ itself.

I think the reason for this formulation lies in the higher dimensional Taylor series:

Cahn and Hilliard want to develop a Taylor series around a uniform composition.

But also, they assume the local derivatives as independent variables. That's why the local free energy in (2.1) is written not only as a function of variable $$c$$, but also the local derivatives of $$c$$: $$\nabla^i c$$.

In order to form a more dimensional Taylor series, you then need to form the partial derivatives of the function of the local free energy. These are are dependent on the local derivatives $$\nabla^i c$$.

I hope this helped you out.

• Hello! I have edited your answer using MathJax (LaTeX) math typesetting. For future posts, you can refer to MathJax basic tutorial and quick reference. Thanks! Dec 14, 2021 at 15:15