Non-equilibrium Thermodynamics of Heterogeneous Systems. Signe Kjelstrup
the normal thermodynamic relations, like the first and second laws are valid [65].
Remark 3.5.Excess densities for the surface can be much larger than the typical value of the integrated quantity times the thickness of the surface of discontinuity. The surface tension of a liquid–vapor is for instance a factor between 103 to 104 larger than the typical pressure of 1 bar times 3 Angstrom. This implies that contributions from the surface play a much more important role than what the volume of the surface of discontinuity would suggest.
Van der Waals constructed a continuous model for the liquid–vapor interface at equilibrium, around the end of the 19th century [107]. In this model, one adds to the Helmholtz energy density per unit of volume a contribution proportional to the square of the gradient of the density. It is therefore often called the square gradient model. The model was reinvented by Cahn and Hilliard in 1958 [108]. It has become an important model for studies of the equilibrium structure of the interface [96, 105]. The model gives a density profile through the closed surface and enables one to calculate, for instance, the surface tension. An extension of his model to non-equilibrium systems [109–111] is discussed in Chapter 23. It is found in the context of this model [110] that the surface, as described by excess densities, is in local equilibrium.
Exercise 3.3.1.The variation in density of A across a vapor–liquid interface between a = 0 and b = 5 nm is given by cA(x) = Cx3 + 50 mol/m3. The vapor density is cgA = 50 mol/m3 and the liquid density is clA = 50000 mol/m3. Determine the value of C and the position of the equimolar surface.
•Solution: The density function at b gives
C (5 × 10−9)3 + 50 = 5 × 104 ⇒ C = 4 × 1029 mol/m6
For the equimolar surface, we have
This results in the position d = 3.75 nm. The equimolar surface is close to the liquid side.
3.4Local thermodynamic identities
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 = Nji/Vi and the polarization density Pi. By dividing Eq. (3.12) by Vi, the internal energy density becomes
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(3.25) |
By replacing Ui by uiVi etc. in Eq. (3.11), differentiating and using Eq. (3.25), we obtain
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(3.26) |
Gibbs–Duhem’s equation becomes
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(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:
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(3.28) |
For the surface, the local variables are given per unit of surface area. These are the excess internal energy density us = Us/Ω, the adsorptions γj = Njs/Ω, the excess entropy density, ss = Ss/Ω, and the surface polarization, Ps. When we introduce these variables into Eq. (3.14), and use Eq. (3.15), we obtain the Gibbs equation for the surface:
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(3.29) |
The surface excess internal energy density is
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(3.30) |
and Gibbs–Duhem’s equation becomes
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(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:
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(3.32) |
For the contact line, local variables are given per unit of length. These are the excess internal energy density uc = Uc/L, the adsorptions Γjc = Njc/L and the excess entropy density, sc = Sc/L. When we introduce these variables into Eq. (3.19), and use Eq. (3.20), we obtain the Gibbs equation for the line:
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(3.33) |
The contact line excess internal energy density is
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(3.34) |
and Gibbs–Duhem’s equation becomes
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(3.35) |
Thermodynamic relations