Non-equilibrium Thermodynamics of Heterogeneous Systems. Signe Kjelstrup
23).
We extend Gibbs’ formulation for the surface to the contact line. Equilibrium relations for the three-phase contact line can be formulated for excess line densities. The line, described in this manner, is also an autonomous thermodynamic system.
3.1Two homogeneous phases separated by a surface in global equilibrium
Consider two phases of a multi-component system in equilibrium with each other. The system is polarizable in an electric field. The last property is relevant for electrochemical systems. In systems that transport heat and mass only, this property is not relevant. The homogeneous phases i and o have their internal energies Ui and Uo entropies Si and So mole numbers Nji and Njo of the components j (see for instance [100]), and their polarizations in the direction normal to the surface, Piand Po. For the total system, we have
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(3.1) |
The internal energy for the total system U minus the sum of the two bulk values gives the internal energy of the interfacial region Us [65, 96]. For the entropy, the mole numbers and the polarization, we similarly have
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(3.2) |
The surface consists of layer(s) of molecules or atoms, with negligible volume, so
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(3.3) |
The surface area Ω is an extensive variable of the surface, comparable to the volume as a variable of a homogeneous phase. The surface tension γ plays a similar role for the surface as the pressures pi and po do for the homogeneous phases.
The polarizations of the homogeneous phases can be expressed in terms of the polarization densities (polarization per unit of volume) Pi and Po by
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(3.4) |
The polarization of the surface can similarly be expressed as the polarization per unit of surface area Ps:
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(3.5) |
The polarization density is particular for both homogeneous phases and the surface. The system may contain polarizable molecules (apolar), polar molecules and free charges. An electric field leads to polarization of the apolar molecules and to orientation of the polar molecules. Furthermore, it leads to a redistribution of free charges. In all cases, polarization occurs in a manner that keeps homogeneous phases and surfaces electroneutral. For a double layer, we define a surface polarization. For a conductor, we define a polarization density. The divergence of the polarization of a conductor equals the charge distribution, but the induced charge distribution in an electric field integrates to zero net charge.
In this book, we restrict ourselves to surfaces that are planar and contact lines that are straight. From the second Maxwell equation, and the condition of electroneutrality, we have
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(3.6) |
where D is the displacement field normal to the surface. The normal component of the displacement field across an electroneutral surface is continuous, and has accordingly no contribution for the surface; Ds = 0. The normal component of the displacement field is therefore constant throughout a heterogeneous system. Its value can be controlled from the outside by putting the system between a parallel plate capacitor. In this book, we do not consider electric potential gradients along the surface. Assuming the system furthermore to be invariant for translations along the surface, rotations around a normal on the surface and reflection in planes normal to the surface, it follows that electric fields, displacement fields and polarizations are normal to the surface. The displacement field is related to the electric fields in the two phases by
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(3.7) |
where εi and εo are the dielectric constants of the phases i and o, respectively. Furthermore, ε0 is the dielectric constant of vacuum. A similar relation can be written for the electric field of the surface
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(3.8) |
For more precise definitions of the interfacial densities, we refer to the following section and to Sec. 5.2. For an in-depth discussion of the excess polarization and electric field, we refer to the book by Bedeaux and Vlieger, Optical Properties of Surfaces [8, 9].
Remark 3.1.The surface polarization is equal to minus ε0 times the excess electric field of the surface. This follows from the following argument: The displacement field is the sum of ε0 times the electric field plus the polarization density. The displacement field is continuous through the surface. The sum of the electric field times ε0 plus the surface polarization density must therefore be zero.
For heterogeneous, polarizable systems, we can then write the internal energy as a total differential of the extensive variables S, Nj, V, Ω, Pi, Po and Ps:
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(3.9) |
where Deq is the displacement field that derives from a reversible transformation. This is the Gibbs equation [65], extended with terms due to polarization. By integrating with constant intensive variables, the energy of the system becomes
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(3.10) |
Since we are dealing with state functions, this result is generally valid. The contributions to the internal energy from the polarization are DeqPi/εi and DeqPo/εo in the homogeneous phases and DeqPs/ε0 [6] for the surface.
Remark 3.2.The contributions to the total polarization, i.e. from polarizable molecules, from polar molecules and from free charges, have very different relaxation times. They should therefore be counted as separate contributions Pα = Σk Pkα, where α equals i, o or s. The corresponding equilibrium displacement fields, Deq,k, found from