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v_solv.

      The average molar frame of reference. In a multicomponent mixture, there is no excess of one component, and we use the average molar velocity:

(4.21)

      Here, xj = cj /c is the mole fraction of component j. This gives as flux of A:

(4.22)

      The average volume frame of reference is used when transport occurs in a closed volume. The average volume velocity is

(4.23)

      where Vj is the molar volume of component j. The flux of A becomes

(4.24)

      The barycentric (average center of mass) frame of reference. This frame of reference is used in the Navier–Stokes equation. The average mass velocity is

(4.25)

      where ρ is the mass density of the fluid, and ρj are the partial mass densities. The flux of A in mol/m²s in the barycentric frame of reference is therefore

(4.26)

      In the barycentric frame of reference, the dimension of the component fluxes used is normally kg/m²s [23]. The flux in kg/m²s is found by multiplication with M_A, the molar mass, M_AJ_A,bar ≡ ρ_A(v_A − v). This is the diffusion flux of component A. The sum of these diffusion fluxes (in kg/m²s) over all neutral components is equal to zero.

      Remark 4.2. In our analysis, we only consider electroneutral systems. The sum in Eq. (4.25) is therefore over neutral components. In charged systems, like those considered by de Groot and Mazur [23], the sum is over charged and uncharged components. For electroneutral systems, this results in an average mass velocity which differs from the one that we use, by a term proportional to the electric current density. We refer to the appendix at the end of this chapter for an example.

      In this book, we consider heterogeneous systems. As will be explained in the following two sections, the excess entropy production of a surface and a three-phase contact line should be calculated in a frame of reference which moves along with the surface or the contact line. We refer to these as the surface frame of reference and the contact line frame of reference. For practical reasons, the same frame of reference is usually taken for fluxes in the adjacent homogeneous phases. As will be explained in Chapter 12 for multi-component diffusion, one can use Gibbs–Duhem’s equation to obtain flux–force relations, which are independent of the frame of reference. For each example considered in this book, we shall indicate which frame of reference is used and why.

4.4.2Transformations between the frames of reference

      In an electroneutral system, the electric current density j is independent of the frame of reference. The measurable heat flux J′q is also independent of the frame of reference. The total heat flux, the mass fluxes and the entropy flux depend on the frame of reference. In the laboratory or the wall frame of reference, we denote these fluxes by Jq, Jk and Js. In any other frame of reference, they become

(4.27)

      All frames of reference defined above can be used for v_ref .

      We consider in this book systems that are in mechanical equilibrium. The hydrostatic pressure is then constant and there are no shear forces. The Gibbs–Duhem’s equation for constant pressure is

(4.28)

      By substituting J_q,ref, J_j,ref and J_s,ref into Eqs. (4.13)–(4.15), we find, using Gibbs–Duhem’s equation, that the term proportional to v_ref gives a zero contribution to the entropy production. This result is Prigogine’s theorem [23].

      Gibbs–Duhem’s equation gives a possibility to eliminate a thermodynamic force. Galilean invariance gives a possibility to eliminate a mass flux. One property is a consequence of the other, as we have seen above. De Groot and Mazur [23] used systematically the barycentric frame of reference, because they also treated hydrodynamic phenomena. In this frame of reference, the sum of the mass (diffusion) fluxes is zero, see above. The total heat flux J_q,bar equals the heat flux Jq used by de Groot and Mazur.

4.AAppendix: The first law and the heat flux

      The purpose of this appendix is to show how the description of de Groot and Mazur is compatible with the one used in this book. The analysis uses as starting point Chapter XIV in de Groot and Mazur [23]. We restrict ourselves to the case that the magnetic field and the magnetization are zero. The system is also electroneutral. As a final simplification, we shall assume the system to be in mechanical equilibrium. The last simplification leads to a form of the first law appropriate for the systems described in this book, Eq. (4.7). Vectors shall be indicated by bold letters in this appendix, meaning that they have an arbitrary direction, rather than being restricted to the x-direction as in most of the book.

      For a system in which there are no magnetic field or magnetization, the Maxwell equations (in SI units) can be written as

(4.29)

      where ε₀ is the dielectric constant of vacuum, E the electric field, j the electric current density and P the polarization density (in C/m²).

      For the molar density of component j, we have the following conservation equation:

(4.30)

      where vj and Jj are the velocity and the molar flux of component j. The potential energy density satisfies

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