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 measurements1 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 Vincent4 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 method4 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 Ma | Mz+Xzin polymer host | Mc, 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 Mz+ + ze(M) ⇄ M.

FIG. 1.

Illustration of the experiment setup; the dashed parts of potential ϕ and cation chemical potential μ+ cannot be measured, but their relation follows the Nernst equation. In this conceptualization, potential drops due to the interfacial/interphasial charge-transfer resistance at the steady-state under the anion-blocking condition are excluded.

FIG. 1.

Illustration of the experiment setup; the dashed parts of potential ϕ and cation chemical potential μ+ cannot be measured, but their relation follows the Nernst equation. In this conceptualization, potential drops due to the interfacial/interphasial charge-transfer resistance at the steady-state under the anion-blocking condition are excluded.

Close modal
To begin the derivation of t+,ss, we need to first introduce the driving forces, i.e., electrochemical potentials μ̃, for both cations and anions at the steady-state,
μ̃+,ss=μ+,ss+zFϕss,
(1)
μ̃,ss=μ,sszFϕss,
(2)
where μ+,ss and μ−,ss are the chemical potentials of cations and anions, respectively, and ϕss is the Galvani potential of the electrolyte solution.
According to the Onsager theory of ion transport,7 the flux and driving forces are connected through the Onsager coefficients and the reciprocal relation as follows:
J+,ss00=Ω++0Ω+0Ω+0Ω0μ̃+,ssμ̃,ss.
(3)
It is important to understand that the above relation is only valid in the solvent-fixed RF, which is the reason that the Onsager coefficients Ω+00 and Ω00 related to the solvent species can be omitted. The flux of anions is set to be zero in Eq. (3) by applying the anion-blocking condition at the steady-state,4,9 and this allows us to express the flux of cations just using the electrochemical potential of cations alone,
J+,ss0=Ω++0Ω+02/Ω0μ̃+,ss.
(4)
The initial current density due to the migration of both cations and anions is
itot=σV
(5)
=z2F2(Ω++0+Ω02Ω+0)V,
(6)
where ΔV = −dV is the applied potential and σ is the ionic conductivity. It is clear from Eq. (5) that the applied potential ΔV 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 iRct term.8,9
Then, the Bruce–Vincent transference t+,ss as the ratio between the steady-state current (density) i+,ss and the initial current (density) itot can be expressed as follows:
t+,ss=i+,ssitot=zFJ+,ss0itot
(7)
=Ω++0Ω+02/Ω0Ω++0+Ω02Ω+0μ̃+,ss(zFV).
(8)

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 Δμ̃+,ss and the applied potential ΔV.

Following the notations from Fawcett,10 the Galvani potential difference at the Ma |solution interface ΔMaSaϕ is defined as
ΔMaSaϕ=ϕMaϕSa=ΔG°Mz+/M°zF+RTzFlnaMaz+,
(9)
where ΔGMz+/M°° is the standard free energy for the reduction reaction of Mz+ and aMaz+ is the activity of Mz+ near the anode.
The same applies to the cathode side with the Galvani potential difference ΔMcScϕ as
ΔMcScϕ=ϕMcϕSc=ΔG°Mz+/M°zF+RTzFlnaMcz+.
(10)
Then, the potential difference between the two electrodes ΔMaMcϕ can be obtained by combining the two equations above,
ΔMaMcϕ=McΔScϕ+ScΔSaϕ+ΔMaSaϕ=RTzFlnaMaz+/aMcz++(ϕSaϕSc).
(11)
It is worth noting that ΔMaMcϕ does not account for any potential drop developed at the electrode–electrolyte interface/interphase due to the charge-transfer resistance, which allows us to apply the Nernst equation to the Galvani potential difference here.
By identifying ΔMaMcϕ=ΔV, Δμ+,ss=RTln(aMaz+/aMcz+), and Δϕss=ϕSaϕSc and applying the definition Eq. (1) of the electrochemical potential of cations, the above equation can be rewritten as follows:
zFΔV=Δμ+,ss+zFΔϕss
(12)
=Δμ̃+,ss.
(13)
Plugging Eq. (13) into Eq. (8), we get the following expression:
t+,ss=Ω++0Ω+02/Ω0Ω++0+Ω02Ω+0,
(14)
which is the main result of this work.
For the infinitely dilute solution, Ω+0 becomes zero and there are also no correlations among the same type of ions. Therefore, one recovers the apparent transference number tapp, which only depends on the self-diffusion coefficients Ds, i.e.,
limr0t+,ss=t+,app=D+sD+s+Ds.
(15)

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.

The point for discussion is whether the assumption regarding the relationship between the chemical potentials of cations and anions matters or not. For this purpose, consider the general case where μ+ = . The linear relation between the steady-state flux and the driving forces under the anion-blocking condition reads
J+,ss00=Mϕssμ+,ssμ,ss
(16)
with
M=Ω++0Ω+00Ω+0Ω00001zF10zF0101x.
(17)
Given that the only non-zero term on the left-hand side is J+,ss, the driving forces can be represented in terms of J+,ss, x, and Ω by inverting the linear relation, which leads to
ϕss=J+,ss0zFxΩ+0+Ω0(x+1)Ω++0Ω0Ω+02,μ,ss=zFΩ+0Ω0xΩ+0+Ω0ϕss,μ+,ss=xμ,ss.
(18)

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.

Importantly, the following two combinations of the driving forces are x-independent:
μ+,ss+zFϕss=Ω0Ω++0Ω0Ω+02J+,ss0,
(19)
μ+,ss+μ,ss=Ω+0Ω0Ω++0Ω0Ω+02J+,ss0.
(20)

This is certainly not a coincidence, as the left-hand side of Eq. (19) is related to Δμ̃+,ss, which corresponds to the applied potential ΔV, 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.

When taking the ratio between Eq. (19) and Eq. (20), one can relate the chemical potential change of the salt Δμsalt = Δμ+,ss + Δμ−,ss to the corresponding potential difference Δϕsalt defined as
Δϕsalt=RTzFlnγ±,amaγ±,cmc,
(21)
where γ± is the mean activity coefficient and m is the molal salt concentration. Then, this leads to the expression of Δϕsalt in terms of the applied potential and the Onsager coefficients as
Δϕsalt=RT2zFlnaaac=Ω0Ω+02Ω0ΔV,
(22)
where aa and ac are the salt activities near anode and cathode, respectively. This recovers the dilute limit (Ω+00), in which Δϕsalt 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 GROMACS14 package and the General AMBER Force Field (GAFF)15 at 157 °C because of a much higher glass transition temperature Tg 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

The Onsager coefficients under the barycentric RF (denoted as M) can be readily computed from the displacement correlations as a function of time t,
ΩijM=limtβ6VNA2tΔriM(t)ΔrjM(t),
(23)
where β is the inverse temperature, V is the system volume, NA is the Avogadro number, and ΔriM(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 
To compute 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 rule5,
Ωij0=k,l0Aik0MΩklMAjl0M=limtβ6VNA2tk0Aik0MΔrkM(t)l0Ajl0MΔrlM(t),
(24)
where Aij0M is the matrix converting the 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.

FIG. 2.

Experimental and simulation results of Bruce–Vincent transference numbers t+,ss and apparent transference numbers t+,app for the PEO–LiTFSI system. The experimental data were taken from Refs. 2 and 2022.

FIG. 2.

Experimental and simulation results of Bruce–Vincent transference numbers t+,ss and apparent transference numbers t+,app for the PEO–LiTFSI system. The experimental data were taken from Refs. 2 and 2022.

Close modal

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.

The authors have no conflicts to disclose.

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).

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

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