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I have a problem understanding how to deal with the chemical potential when two thermodynamic systems are in contact. For simplicity, let us consider $T=0\ K$. We have a system that is composed of two parts - I and II - where I occupies a space of fixed volume and II occupies the remaining space (essentially going to infinity in all directions). Both parts are initially filled with a crystal that is made up of a finite number of atomic types $i$. This is an equilibrium configuration (1). Now, I apply a long-range displacement field that originates in part I. If the boundary between parts I and II is permeable for the matter, some atoms move across this boundary. Here, part II can be thought of as a reservoir for the matter. The system eventually settles into another equilibrium (2), where the number of atoms $n_i$ in the part I is different from the above, i.e. there is nonzero

$$\Delta n_i=n_i^{(2)}-n_i^{(1)}\ .$$

The free energy difference between the two equilibrium states of part I is then

$$\Delta F \equiv F^{(2)}-F^{(1)} = V\sum_{ij}\sigma_{ij}\Delta\epsilon_{ij}+\sum_i\mu_i\Delta n_i\ .$$

In each of the two configurations (1) and (2), the parts I and II are in contact equilibrium, which means that $\mu_i^{I(1)}=\mu_i^{II(1)}$ and $\mu_i^{I(2)}=\mu_i^{II(2)}$, where the number in the bracket refers to the configuration. So, we have two chemical potentials for each $i$, one corresponding to configuration (1) and the other (generally different) for the configuration (2). Moreover, the particles move into I from the immediate neighborhood of the boundary I-II, which implies that $\mu_i$ should correspond to the atoms in the part II that are close to this boundary. However, how am I to understand a single $\mu_i$ in the equation above?

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I think someone made a mistake on the indices of the summations. The first summation is the elastic deformation energy, using the applied stress tensor $\sigma_{ij}$ and resulting strains $\Delta\epsilon_{ij}$. This applies across the entire volume that your strain field covers. This applies to the volume $V$ which is area I. The $i$ and $j$ in this term refer to the components of the stress/strain tensors (i.e. $x,y,z$), not the chemical constituents. (Apparently volume II is not impacted by this long range strain field - that is, well, difficult to arrange in real life). The second term is a simple summation across the constituents. So to make things clearer I would certainly have used different summation indices in the first term.

Consider a simple one-component system, say a silicon crystal. You magically reach in and apply a stress field in a confined volume, doing elastic work to deform that volume, but keeping the number of atoms in it constant. Now what happens? The atoms in that volume have had work performed on them during the elastic deformation. If an atom hops out of the volume, it gains that energy back by elastic relaxation (remember, volume II is not affected by this stress field). The Helmholtz free energy of the volume will decrease because of the loss of that atom. This will continue until the volume has lost enough atoms, at the chemical potential $\mu$ set by your 'infinite bath' of atoms, to balance the elastic deformation in your defined volume.

Thermodynamic equilibrium requires that the chemical potentials of each constituent are constant, else there is a driving force to move the system into a different configuration.

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