Transference number is a key design parameter for electrolyte materials used in electrochemical energy storage systems. However, the determination of the true transference number from experiments is rather demanding. On the other hand, the Bruce–Vincent method is widely used in the lab to approximately measure transference numbers of polymer electrolytes, which becomes exact in the limit of infinite dilution. Therefore, theoretical formulations to treat the Bruce–Vincent transference number and the true transference number on an equal footing are clearly needed. Here, we show how the Bruce–Vincent transference number for concentrated electrolyte solutions can be derived in terms of the Onsager coefficients, without involving any extrathermodynamic assumptions. By demonstrating it for the case of poly(ethylene oxide)–lithium bis(trifluoromethane)sulfonimide system, this work opens the door to calibrating molecular dynamics (MD) simulations via reproducing the Bruce–Vincent transference number and using MD simulations as a predictive tool for determining the true transference number.

Transference number, defined as the fraction of current due to the migration of certain ionic species, is an essential design parameter for the energy storage application of electrolyte materials. While important progress has been made in the quest for determining the true transference number, notably with the combinations of concentration cell and steady-state measurements^{1} or electrophoretic nuclear magnetic resonance (NMR),^{2} its determination in polymer electrolytes remains difficult in practice.

The usage of steady-state currents for the determination of transport coefficients dates back to experiments of Wagner on metal oxides and sulfides.^{3} It is Bruce and Vincent^{4} who first derived the equivalence of such measurements to determine the transference number of Li ions in polymer electrolytes. The method should give the true transference number in the limit of infinite dilution; however, it becomes approximated in concentrated solutions where the ion–ion correlations become non-negligible. Nevertheless, the Bruce–Vincent method^{4} remains the most widely used one in the lab to gauge the transference number because of its simplicity.

Computations of the true transference number $t+0$ from molecular dynamics (MD) simulations have come on the scene recently, which allow for a direct comparison between the theory and experiment.^{5,6} However, due to the challenge in measuring the true transference and the high accessibility of the Bruce–Vincent transference number *t*_{+,ss} from experiments, it would be desirable to also obtain the Bruce–Vincent transference number directly from MD simulations.

In this communication, we derive *t*_{+,ss} with the Onsager equations of ion transport and apply the method to the PEO–LiTFSI [poly(ethylene oxide)–lithium bis(trifluoromethane)sulfonimide] system using MD simulations. By clarifying the extrathermodynamic assumptions used previously, we show that *t*_{+,ss} has a well-defined connection with the Onsager coefficients and can be computed from MD simulations accordingly with a proper consideration of the reference frame (RF). Comparing the experiment and simulation for the PEO–LiTFSI system, we observe a consistent trend of its reduction with respect to its dilute limit, determined by the self-diffusion coefficients of ions.

The generic cell under consideration here is M_{a} | M^{z+}X^{z−}in polymer host | M_{c}, as shown in Fig. 1, where *a* and *c* denote the anode and the anode, respectively. The two electrodes are separated by a distance *d*, and the electrode reaction for cations is M^{z+} + *ze*(M) ⇄ M.

*t*

_{+,ss}, we need to first introduce the driving forces, i.e., electrochemical potentials $\mu \u0303$, for both cations and anions at the steady-state,

*μ*

_{+,ss}and

*μ*

_{−,ss}are the chemical potentials of cations and anions, respectively, and

*ϕ*

_{ss}is the Galvani potential of the electrolyte solution.

^{7}the flux and driving forces are connected through the Onsager coefficients and the reciprocal relation as follows:

^{4,9}and this allows us to express the flux of cations just using the electrochemical potential of cations alone,

*V*= −

*d*∇

*V*is the applied potential and

*σ*is the ionic conductivity. It is clear from Eq. (5) that the applied potential Δ

*V*in this context stands for the one after excluding any potential drop due to the charge-transfer resistance at the interface. A similar procedure is also used in the experiment by subtracting the

*iR*

_{ct}term.

^{8,9}

*t*

_{+,ss}as the ratio between the steady-state current (density)

**i**

_{+,ss}and the initial current (density)

**i**

_{tot}can be expressed as follows:

Therefore, the key step for obtaining *t*_{+,ss} is to establish the relationship between the change in the electrochemical potential of cations at the steady-state $\Delta \mu \u0303+,ss$ and the applied potential Δ*V*.

^{10}the Galvani potential difference at the M

_{a}|solution interface $\Delta MaSa\varphi $ is defined as

^{z+}and $aMaz+$ is the activity of M

^{z+}near the anode.

*D*

^{s}, i.e.,

Before showing how to compute the Onsager coefficients from MD simulations and taking care of their RF-dependence, it is necessary to make a connection of Eq. (14) with previous studies,^{1,11} where similar results were either implied or indicated but with seemingly rather different assumptions.

*μ*

_{+}=

*xμ*

_{−}. The linear relation between the steady-state flux and the driving forces under the anion-blocking condition reads

**J**

_{+,ss}, the driving forces can be represented in terms of

**J**

_{+,ss},

*x*, and Ω by inverting the linear relation, which leads to

Equation (18) seems complicated, but one can verify that they reduce to the same set of equations reported in the literature under specific choices of *x*, namely, *x* = 0 as in Ref. 1 and *x* = 1 as in Ref. 11.

*x*-independent:

This is certainly not a coincidence, as the left-hand side of Eq. (19) is related to $\Delta \mu \u0303+,ss$, which corresponds to the applied potential Δ*V*, and that of Eq. (20) is proportional to the chemical potential of the salt. In both cases, these are the quantities that can be measured in experiments.

*μ*

_{salt}= Δ

*μ*

_{+,ss}+ Δ

*μ*

_{−,ss}to the corresponding potential difference Δ

*ϕ*

_{salt}defined as

*γ*

_{±}is the mean activity coefficient and

*m*is the molal salt concentration. Then, this leads to the expression of Δ

*ϕ*

_{salt}in terms of the applied potential and the Onsager coefficients as

*a*

_{a}and

*a*

_{c}are the salt activities near anode and cathode, respectively. This recovers the dilute limit $(\Omega +\u22120\u21920)$, in which Δ

*ϕ*

_{salt}is equal to half of the applied potential at the steady-state.

^{4}

Therefore, what we reveal here is yet another example of the Gibbs–Guggenheim principle,^{12,13} stating that chemical potentials of individual ions are a mathematical construct and cannot be measured experimentally without extrathermodynamic assumptions (e.g., the choice of *x* in this case). Nevertheless, what matters is that *t*_{+,ss} is a well-defined quantity and its derivation in terms of the Onsager coefficients does not need to involve any of these extrathermodynamic assumptions.

Now, we are ready to apply Eq. (14) to the PEO–LiTFSI system by using MD simulations. The simulations were performed using the GROMACS^{14} package and the General AMBER Force Field (GAFF)^{15} at 157 °C because of a much higher glass transition temperature *T*_{g} found in the simulation system.^{16} Further details regarding the simulation setup and the force field parameterization can be found in previous studies.^{16,17}

*t*,

*β*is the inverse temperature,

*V*is the system volume,

*N*

_{A}is the Avogadro number, and $\Delta riM(t)$ is the total displacement of species

*i*over a time interval

*t*. The long-time limit is estimated by fitting the correlation as a linear function over the interval 10–20 ns, which was shown to reach the diffusion regime in a previous study.

^{5}

*t*

_{+,ss}, it is necessary to convert the Onsager coefficients from the barycentric RF to the solvent-fixed RF (denoted as 0) using the transformation rule

^{5}

^{,}

*n*− 1 independent fluxes in an

*n*-component system from the barycentric RF to the solvent-fixed one. Equation (24) shows how this transformation relates to the correlations of ions. The detailed derivation of such a relation from constraints of the fluxes and driving forces can be found elsewhere.

^{18,19}An important implication from these works is that when the response relation as shown here is applied in other RFs, it also entails a corresponding RF transformation of these driving forces.

The results of the computed *t*_{+,ss} for the PEO–LiTFSI system in comparison with experiments are shown in Fig. 2, together with those of *t*_{+,app}. It is found that *t*_{+,ss} is always positive as expected since both the diagonal Onsager coefficients and the determinant are positive; the same is true for *t*_{+,app} and the two quantities approach each other at the dilute condition. In addition, the relation *t*_{+,ss} < *t*_{+,app} seems to hold for the entire range of concentration, in both the simulation and experiment.

Given that the measurements of Bruce–Vincent transference numbers are accessible in most labs and *t*_{+,ss} does reflect a defined facet of the ion–ion correlations, this work provides a way to calibrate the MD simulations of polymer electrolytes and concentrated electrolytes alike by reproducing *t*_{+,ss}. Through a quantitative comparison and calibration, this will allow us to use MD simulations as a predictive tool to obtain the true transference number $t+0$ for new types of polymer platforms beyond PEO.

This work was supported by the Swedish Research Council (VR), Grant No. 2019-05012. The authors thank the funding from the Swedish National Strategic e-Science program eSSENCE, STandUP for Energy, and BASE (Batteries Sweden). The simulations were performed on the resources provided by the National Academic Infrastructure for Supercomputing in Sweden (NAISS) at PDC.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Yunqi Shao**: Data curation (lead); Formal analysis (lead); Investigation (equal); Writing – original draft (equal). **Chao Zhang**: Conceptualization (lead); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Supervision (lead); Writing – original draft (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

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