# Free energy Functional and its Relation to Pressure and Chemical Potential

I am reading a paper and trying to follow along some of the mathematical steps and reasoning; however, I can't quite understand one aspect. The authors give an expression for the free energy functional of the system: $$\mathcal{F} = \mathcal{F}[h,\psi,\nabla h,\nabla\psi]$$ which depends on the films thickness and the effective solute layer thickness. Later on the authors use this to develop coupled evolution equations for $$h$$ and $$\psi$$ derived within the framework of gradient dynamics. What I don't quite understand is that in these coupled equations the authors define the total pressure, $$p$$, and the chemical potential, $$\mu$$, as: $$p = \frac{\delta\mathcal{F}}{\delta h} \quad\text{and}\quad \mu = \frac{\delta\mathcal{F}}{\delta \psi}$$

Specifically, the units don't even seem to match up. The fundamental units of energy are $$\frac{ML^2}{T^2}$$ whereas for pressure they are $$\frac{M}{LT^2}$$. Given that $$[h]=L$$ how could the variational derivative of an energy functional possibly represent the total pressure?

• For clarity and readability, please define your variables and identify the paper so that the context can be understood. Commented Jul 6, 2022 at 15:18

In that case $$F\sim M/T^{2}$$ and $$p\sim F/h\sim M/LT^2$$ as expected