Two different approaches have appeared in the literature in order to derive transport equations for coupled diffusion and stress in lithium host materials. The first approach is taken from solid mechanics and is described in detail, for example, in the work of Larché and Cahn [1] or in the reference text by Gurtin et al. [2]. Other examples of this approach can be found in Li [3] and Zhang et al. [4]. The second approach has been used by electrochemical researchers including Christensen and Newman [5] and Rengenathan et al. [6] to derive transport equations for lithium host materials, based on the Stefan-Maxwell equations as described in Hirschfelder et al. [7] and discussed in Bird and Klingenberg [8] for the treatment of gases and liquids. One of the motivations for this paper was the need to reconcile the different flux relations that result from these two approaches. Gases and liquids do not sustain a gradient in pressure or stress under equilibrium conditions in the absence of other body forces, such as gravity.  In contrast, solids can maintain a stress gradient under equilibrium conditions.  Hence, the most basic thermodynamic underpinnings in the treatment of a solid must differ from that of a liquid or gas, and this fact led us to employ the methods reviewed in [1,2] (the first approach). The second approach results in a version of the Stefan-Maxwell equations in which driving forces for diffusion are given by gradients of species concentrations, pressure, and temperature. The rigorous basis for these equations has led to their wide application to transport problems involving liquids and gases, but their application to solids is made difficult by the fact that the driving forces do not explicitly consider stress as it appears in solids. In [5] and [6] the pressure gradient is replaced by the hydrostatic-stress gradient in a solid. It is at this point in the derivation of flux relations where differences in the two approaches appear.

The derivation of transport relations starts with the second law of thermodynamics, which leads to the formulation of a positive definite transport matrix relating fluxes and driving forces. The choice of which transport matrix to use must then be made based on the specific problem in question. Our approach is to modify the driving forces in the Stefan-Maxwell equations so as to accommodate stress gradients in the solid state. This is done using a generalized Gibbs-Duhem equation that includes gradients of stress within a solid. The transport matrix resulting from the Stefan-Maxwell equations is shown to be positive definite if all of the binary diffusion coefficients have positive values. These two steps demonstrate that the resulting Stefan-Maxwell equations are consistent with the thermodynamics as described in Gurtin et al. [2].

We consider substitutional and interstitial alloys. One of the purposes of this work is to present a rigorous derivation of a simple form of such models, thereby providing a basis on which further complications can be added. In particular, the model considered here ignores creation or annihilation of lattice sites, so that the total amount of all species (which is the same as the number of lattice sites) is conserved. Phase changes are also not considered. Plasticity effects will not be considered and we restrict our attention here only to the case of small strain theory, which is viewed as a preliminary step to the more general finite strain theory. The models considered will allow for the host species to be either mobile or immobile, and some comparisons of these two theories are given by numerical simulation.  The model is also assumed to be isothermal. Complete details are given in [9].

 

1. F.C. Larché and J.W. Cahn, Acta metall., 30, 1835 (1982).
2. M.E. Gurtin, E. Fried and L. Anand, The Mechanics and Thermodynamics of Continua, Cambridge University Press, New York, 2010.
3. J. C-M. Li, Metallurgical Transactions A, 9A, 1353 (1978).
4. X. Zhang, Wei Shyy, and A. M. Sastry, J. Electrochem. Soc., 154, A910 (2007).
5. J. Christensen and J. Newman, J. Solid State Electrochem., 10, 293 (2006).
6. S. Renganathan, G. Sikha, S. Santhanagopalan, and R. E. White, J. Electrochem. Soc., 157, A155 (2010).
7. J. 0. Hirschfelder, C. F.  Curtiss, and  R. B. Bird, Molecular Theory of  Gases and Liquids, Wiley, New  York, 1954.
8. R. B. Bird and D. J. Klingenberg, Advances in Water Resources, 62, 238 (2013).
9. D. R. Baker, M. W. Verbrugge and A.F. Bower, J. Solid State Electrochem., DOI 10.1007/s10008-015-3012-7 (2015).