Читать книгу Non-equilibrium Thermodynamics of Heterogeneous Systems - Signe Kjelstrup - Страница 21

In Sec. 3.1, we considered a heterogeneous system in global equilibrium. The temperature, the chemical potentials, the pressure and the displacement field were constant throughout the system, Eqs. (3.17) and (3.18). In Sec. 3.3, we defined excess densities using expressions that can also be used in non-equilibrium systems. In order to use the thermodynamic identities from Sec. 3.1 for the excess densities, we need to cast the Gibbs equation and its equivalent forms into a local form.

For the homogeneous phases, we introduce the extensive variables per unit of volume. In the i-phase, we then have uⁱ = Uⁱ/Vⁱ, sⁱ = Sⁱ/Vⁱ, cjⁱ = N_jⁱ/Vⁱ and the polarization density Pⁱ. By dividing Eq. (3.12) by Vⁱ, the internal energy density becomes

(3.25)

By replacing Uⁱ by uⁱVⁱ etc. in Eq. (3.11), differentiating and using Eq. (3.25), we obtain

(3.26)

Gibbs–Duhem’s equation becomes

(3.27)

All quantities in these equations refer now to a local position in space.

All expressions for phase i are also true for phase o. (Replace all the super- and subscripts i by o.) Other thermodynamic relations can also be defined. We give below the i-phase internal energy, Gibbs energy, Helmholtz energy, internal energy density, Gibbs energy density and the Helmholtz energy density:

(3.28)

For the surface, the local variables are given per unit of surface area. These are the excess internal energy density u^s = U^s/Ω, the adsorptions γ_j = N_j^s/Ω, the excess entropy density, s^s = S^s/Ω, and the surface polarization, P^s. When we introduce these variables into Eq. (3.14), and use Eq. (3.15), we obtain the Gibbs equation for the surface:

(3.29)

The surface excess internal energy density is

(3.30)

and Gibbs–Duhem’s equation becomes

(3.31)

We give below the surface internal energy, surface internal energy density, Gibbs equation, Gibbs–Duhem’s equation, surface Gibbs energy density and the surface Helmholtz energy density:

(3.32)

For the contact line, local variables are given per unit of length. These are the excess internal energy density u^c = U^c/L, the adsorptions Γ_j^c = N_j^c/L and the excess entropy density, s^c = S^c/L. When we introduce these variables into Eq. (3.19), and use Eq. (3.20), we obtain the Gibbs equation for the line:

(3.33)

The contact line excess internal energy density is

(3.34)

and Gibbs–Duhem’s equation becomes

(3.35)

Thermodynamic relations for the contact line are given below. We give the line internal energy, line internal energy density, Gibbs equation, Gibbs-Duhem’s equation, contact line Gibbs energy density and the contact line Helmholtz energy density:

(3.36)