Effect of the Ion, Solvent, and Thermal Interaction Coefficients on Battery Voltage

In order to increase the adoption of batteries for sustainable transport and energy storage, improved charging and discharging capabilities of lithium-ion batteries are necessary. To achieve this, accurate data that describe the internal state of the cells are essential. Several models have been derived, and transport coefficients have been reported for use in these models. We report for the first time a complete set of transport coefficients to model the concentration and temperature polarization in a lithium-ion battery ternary electrolyte, allowing us to test common assumptions. We include effects due to gradients in chemical potentials and in temperature. We find that the voltage contributions due to salt and solvent polarization are of the same order of magnitude as the ohmic loss and must be taken into account for more accurate modeling and understanding of battery performance. We report new Soret and Seebeck coefficients and find thermal polarization to be significant in cases relevant to battery research. The analysis is suitable for electrochemical systems, in general.

as a basis S1 .Γ x ij is determined from the equilibrium simulations by evaluating the Kirkwood-Buff integrals S2,S3 : where g ij (r) is the pair distribution function.The radial distribution functions were calculated using the OCTP plugin S4 for LAMMPS with the finite-size correction method of van der Vegt et al.S5,S6 .For evaluating the integrals in Eq. ( 2), we have used the finite-size correction of Krüger et al.S7 and one example of the extrapolation is shown in Figure S1 for the 1 M LiPF 6 in EC:DEC system.Relations between G ij and Γ x ij for a ternary system are given by Liu et al.S3 and Krishna et al.S8 : where c i is the molar concentration of i and, and, For the evaluation of the thermodynamic factors, EC has been taken as the reference (component 3). S-3

Thermal coefficients
The heat of transfer can be computed from composition gradients once we have the thermodynamic factors, Γ x ij .In the stationary state for transport of salt, we obtain S-4 An equivalent expression can be found for D. In both cases, ∇T is directly measured and ∇µ i,T for i = L or D is determined from where R is the gas constant and x j is the mole fraction of (independent) components j = L or D. The gradient in the mole fraction ∇x j is determined in non-equilibrium simulations, and Γ x ij is determined from equilibrium simulations by evaluating equation ( 2).
The aim of the present investigation is to compute the electric potential gradient across the electrolyte from the last line in equation 2 in the main text.This can be done with knowledge of the conductivity L φφ /T and the transference coefficients t i = F (L iφ /L φφ ).
These coefficients can be obtained from equilibrium simulations, using fluctuation-dissipation theorems.The heat of transfer can be determined from non-equilibrium simulations by setting up a heat flux and measuring the resulting composition gradients.
The Peltier coefficient on the other hand is not directly obtainable from simulations.
It will here be determined from Seebeck coefficient measurements.By using the Onsager relations, we obtain the identity The expression applies to a subsystem as well as to the whole measuring cell.The single contributions to the Peltier heat was obtained for this cell from the entropy balance S9 : Here S * i are transported entropies of the lithium ion and the electron, respectively, S Li is the entropy of lithium.The heat of transfer and the transference coefficients were defined above.
The transported entropy of the electron is assumed to be small.The equilibrium structure of the electrolyte In Tables S2, S3 and S4, the coefficients are obtained using Eq. ( 13) in the barycentric (B) frame of reference, and converted to the EC-and DEC frames of reference.The conversion is shown in Ref. S1.As described by Liu et al.S3 , the barycentric Onsager coefficients, Λ ij , can be directly obtained in MD simulations from the particle displacements as a function of Table S3: Diffusion coefficients for the mixed component scenario of the isothermal electrolyte of 3:7 wt.% EC:DEC + 1 M LiPF 6 using the EC frame of reference.Transference coefficients, t, and transport numbers, τ , are dimensionless.

Figure S1 :
Figure S1: Calculation of G ij by linear extrapolation for the 1 M LiPF 6 in EC:DEC system.The dotted vertical line shows the start of the linear extrapolation and the extrapolated values are given in the legend (as the values following "ex.").

Figure S3 :
Figure S3: Kirkwood-Buff integrals as a function of inverse distance for the three replicates of the 1:1 EC:DEC electrolyte.The regions used for linear regression to find the intercepts of the curves are displayed by dotted vertical lines.

Figure S4 :
Figure S4: (a) Radial distribution functions and (b) coordination numbers of the central carbon atom of EC around Li.
Figure S2: Kirkwood-Buff integrals as a function of inverse distance for the four electrolytes studied.The regions used for linear regression to find the intercepts of the curves are displayed by dotted vertical lines.

Table S1 :
Thermodynamic factors (Γ x ij , L = LiPF 6 , D = DEC or DMC) used to calculate heats of transfer.

Table S4 :
Diffusion coefficients for the mixed component scenario of the isothermal electrolyte of 3:7 wt.% EC:DMC + 1 M LiPF 6 using the EC frame of reference.Transference coefficients, t, and transport numbers, τ , are dimensionless.

Table S5 :
Potential contributions to cell voltage in the isothermal caseA

Table S6 :
Diffusion coefficients for the mixed component scenario of the isothermal electrolyte of 1:1 wt.% EC:DEC + 1 M LiPF 6 using the EC frame of reference at 280 and 320 K. Transference coefficients, t, and transport numbers, τ , are dimensionless.Potential contributions to cell voltage in the bottom section.Mean values and standard deviations from two replicas.CoefficientValue (× 10 −11 m 2 /s) Value (× 10 −11 m 2 /s)