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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?

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Regarding the $f_0$: I think they mean \begin{equation} f_0 = f(c,0,0,...) \end{equation} 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 \begin{equation} \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], \end{equation} and not $c$ itself.

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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.

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