The expression for entropy production owing to diffusion used in Hackl et al. (2010 Proc. R. Soc. A 467, 1186–1196. (doi:10.1098/rspa.2010.0179)) has led to some discussion concerning its definition and comparability with terms found in the literature. Therefore, in this addendum, we introduce a slight modification which leads to a more consistent result that can be better interpreted in the light of existing literature.
We present, in this addendum, simplifications and corrections to Hackl et al. (2010) and the preceding paper by Hackl & Fischer (2008). The denomination of the individual quantities as well as the notation in this addendum are identical to those in Hackl et al. (2010).
2. Problem formulation
As central relations, we use the First Law of Thermodynamics as 2.1the Helmholtz free energy ψ as 2.2and the external entropy rate as 2.3(e.g. De Groot & Mazur 1969 ch. III, §2). Note that the second term in equation (2.3) now has a positive sign. The entropy production rate , 2.4follows as 2.5Using the so-called Coleman–Noll conditions, one gets 2.6The third term on the right-hand side of equation (2.6) can be rewritten as 2.7Furthermore, it holds that 2.8Elimination of the temperature-gradient term in equation (2.7) by equation (2.8) yields 2.9Replacement of the third term on the right-hand side of equation (2.6) by equation (2.9) results in 2.10Consider now that the total entropy production rate for a body B is defined as ‘surface terms’. Hence, we may apply the Gauß theorem to the last term of equation (2.10) and find that 2.11Obviously, only a surface term will appear after variation of the total entropy production rate with respect to ji. Here, we discuss solely the local field equations derived by variation of the entropy production rate. Therefore, we are allowed to neglect the last divergence term in equation (2.10) and write finally 2.12In the case of a closed system, the surface integral appearing in equation (2.11) disappears owing to ji⋅n=0.
Relation (2.12) agrees with the entropy production as derived by eqn (21) in De Groot & Mazur (1969 ch. III, §2), and was also used by Svoboda et al. (2005). However, in Hackl et al. (2010), the contribution of the diffusion process to the entropy production, , is formulated as . Such a relation can also be found in the open literature (e.g. Haase (1969 §4.8, eqn (4–8.15)), denoted there as ‘local entropy production’, or De Hoff (2006, table 3.2)). It is trivial to state that for T constant in space, and this is very often the case, at least locally, both formulations are identical.
In our case, we have for Pc 2.13with for i=2,…,n.
We would like to mention that a thermally driven diffusion, as was discussed in Svoboda et al. (2006), Mohanty et al. (2009) or Aouadi (2010), is only possible if, as in equation (2.13), thermodynamical forces of the form act on the fluxes ji, instead of (see equation (2.12)).
3. Principle of maximum dissipation
If the principle of maximum dissipation is applied to derive evolution equations for the thermodynamical fluxes, e.g. for the diffusive fluxes ji, one can always work with the dissipation Q and entropy production term P, which is equivalent to Q, Q=P; this is different from the approach in Hackl et al. (2010), where the terms or were used. Note that P must be equivalent in the case at hand to the negative rate of Gibbs energy, . Variation of the Lagrangian employing a Lagrange multiplier λ, with respect to the fluxes ji, i=2,…,n, yields 3.1The denominator N is derived in Hackl et al. (2010) as 3.2Equation (3.1) is valid for a coupled process. For a non-coupled process, we have 3.3Here, Qc is the dissipation owing to diffusion only, usually a quadratic functional in ji, i=2,…,n.
In this addendum, we have shown the immanent coupling of the diffusive fluxes with the corresponding chemical potentials and the temperature field. For a spatially constant temperature field, the coupling effect disappears. The modified thermodynamical forces derived here can be better related to the existing literature now and allow for thermally driven diffusion processes.
F.D.F. and J.S. gratefully acknowledge the financial support by the Austrian Federal Government (in particular from the Bundesministerium für Verkehr, Innovation und Technologie and the Bundesministerium für Wirtschaft und Arbeit) and the Styrian Provincial Government, represented by Österreischische Forschungsförderungsgesellschaft mbH and by Steirische Wirtschaftsförderungsgesellschaft mbH, within the research activities of the K2 Competence Centre on ‘Integrated Research in Materials, Processing and Product Engineering’, operated by the Materials Center Leoben Forschung GmbH in the framework of the Austrian COMET Competence Centre Programme as well as funding through DFG project AB.314/1 (FOR 741). J.S. gratefully acknowledges the financial support by the Research Plan of the Institute of Physics of Materials (project CEZ:AV0Z20410507). The cooperation of authors (F.D.F. and J.S.) is also supported in the frame of COST action P19.
- Received January 7, 2011.
- Accepted February 3, 2011.
- This journal is © 2011 The Royal Society