The goal of the present investigation was to extend the recently proposed inverse modelling approach for determining concentration-dependent transport properties (salt diffusivity and Li⁺ cation transference number) to concentrated lithium salt solutions in solvents with relatively low permittivity. We demonstrated in our previous studies that NMR concentration profiles can be used for estimating the concentration dependent transport properties of LiTFSI solution in propylene carbonate (PC) under minimal assumption, including that of negligible ion pairing (simplified transport model) [1], which is valid for solvents with relative high permittivity [2] (ε_r ≈ 65 for PC). Herein we consider the case where ion paring effects cannot be neglected, due to the low permittivity of the solvent, as is the case for concentrated solutions of Li salts in mixed organic carbonate solvents (ε_r≈ 26 for EC:DMC 1:1).

The data for the present study was obtained in the form of NMR intensity images of the ¹⁹F nuclei in the PF₆^- anions of LiPF₆ dissolved in a binary mixture of ethylene carbonate (EC) and dimethyl carbonate (DMC), recorded under galvanostatic conditions. The NMR images were then post-processed to obtain lithium concentration profiles. The formation of non-negligible amounts of ion pairs in the present work significantly affects the estimation of transport properties of the system. As a result, the simplified transport model produces thermodynamically inconsistent results, manifested as negative transference numbers obtained from the inverse modelling analysis (Fig. 1). To remedy this situation, we considered an extended transport model which assumes that oppositely charged ions combine to form neutral ion pairs and which tracks the concentration of neutral ion pairs in addition to the concentration of free ions. However, due to the presence of a reaction term describing the formation and disintegration of ion pairs, the resulting PDE system is extremely stiff. Using methods of asymptotic analysis, corresponding to the limit of fast reaction rates, we have reduced the modeling problem to a more simple system describing the evolution of the total lithium concentration, which is computationally tractable. As this reduced system may not satisfy all boundary conditions of the original extended system, an additional material property called boundary transference number was introduced and its optimal dependence on the total concentration was identified via inverse analysis. The unknown material properties are concentration dependent (except for the equilibrium constant K) and are reconstructed with minimal assumptions using methods of variational optimization, to minimize the least-square error between the experimentally determined and simulated concentration values. The optimization problem is solved using a gradient-based method with a careful treatment of uncertainties resulting from the presence of noise in the experimental data. A key element of our computational approach is identification of the sensitivity of the PDE system solutions to perturbations of the constitutive relations, which is achieved through a novel adjoint-based approach. As shown in Fig. 2, the reconstruction produces thermodynamically consistent results for the Li⁺transference number under conditions when ion pairing effects cannot be neglected.

References:

1. A.K. Sethurajan, S.A. Krachkovskiy, I. C. Halalay, G.R. Goward, B. Protas, J. Phys. Chem. B, 119, 12238 (2015).
2. Y. Marcus and G. Hefter, Chem. Rev., 106, 4585 (2006).

Figure 1. The simplified diffusion-migration model equations and inverse modelling results for LiPF₆/EC:DMC 1:1 v/v. The assumption of negligible ion pairing yields nonphysical values for the Li⁺ transference number. Note also that the cost functional is above the 95% confidence bound. (D = salt diffusion coefficient, t⁺ = cation transference number; units for the ion concentration care mM).

Figure 2. Extended diffusion-migration model (including ion pairing) equations and results of the inverse modeling analysis for LiPF₆/EC:DMC 1:1 v/v. (D_i = salt diffusion coefficient for free ions, D₀ = salt diffusion coefficient for ion pairs, t⁺ = cation transference number, r⁺ = boundary transference number for cations, c_t = c_i + c₀; units for the salt concentration c_t are mM).