The performance of a battery depends heavily on the properties of its electrolyte. Total ionic conductivity is one part of this, but how the current is carried by the electrolyte has some important implications, especially when the battery is subjected to very high charge or discharge currents.

Let's consider the discharge of a Li-ion battery, containing an electrolyte with a simple salt such as LiPF_{6}, and which is completely dissociated in the solvent. Current is drawn from the cell; Li^{+} ions are extracted from the negative electrode and inserted into the positive electrode. Part of this current is carried by the transport of Li^{+} ions, and the rest is carried by the transport of PF_{6}^{-} in the opposite direction:

We call the fraction of the current carried by a specific species, e.g., Li^{+} or PF_{6}^{-}, the *transport number*, with the symbol *t _{+}* or

What does this mean? The transport number is effectively a ratio of the mobilities of the ions, which, following the Nernst-Einstein equation (more on this later) also relates the partial conductivities and the diffusion coefficients to each other:

$$\displaystyle t_+ = \frac{\mu_+}{\mu_+ + \mu_-} = \frac{\sigma_+}{\sigma_+ + \sigma_-} = \frac{D_+}{D_+ + D_-}$$

*t _{+}* is often smaller than

Back to the battery. Because only a *part* of the current goes into moving Li^{+} ions, Li^{+} is consumed at the interface with the positive electrode (or produced at the negative) faster than it is replenished by electrical *migration*. This creates a gradient in salt concentration in the electrolyte, between the electrodes of the battery. The gradient in concentration then drives diffusion of the salt, which makes up for the rest of the transport of Li^{+} which *isn't* supplied by migration.

The formation of this concentration gradient can ultimately limit the discharge (or charge) of the battery. If the concentration of salt at an electrode surface reaches zero, the ionic resistance becomes huge, and the battery stops. If the concentration of salt becomes too high, the salt can precipitate out altogether, and the resistance can again become huge. (These processes are, however, reversible, in time.)

The value of t_{+}, and the salt diffusion coefficient, determine how fast this concentration gradient forms, and in turn the maximum current that the *electrolyte* can sustain indefinitely (assuming nothing else is limiting). Both of these parameters are therefore important properties of any electrolyte, in addition to its total ionic conductivity. Ideally, *t _{+}* = 1 (and, correspondingly,

The most common experimental method for measuring *t _{+}* in a polymer electrolyte is the so-called Bruce-Vincent method, named for the work done by Colin Vincent and Peter Bruce on this subject, published in 1987 (here and here).

The method involves the polarisation of a *symmetrical* cell (i.e., an electrochemical cell with two Li metal electrodes) by a small potential difference, to induce a small concentration gradient, until the system reaches a steady state, with a concentration gradient that does not change further with time:

This method assumes - as I previously mentioned - that the ions in the electrolyte are perfectly dissociated. More specifically, that the electrolyte obeys the Nernst-Einstein equation, which relates the conductivity (and the electrical mobility) of an ion to its diffusion coefficient:

$$\displaystyle \sigma_i = \frac{\left|z_i\right|^2 F^2 c_i}{RT} D_i$$

The derivation of the following equations is long and rather complex (the paper detailing the derivation is 17 pages long), but I will summarise here briefly.

In this case, assuming a perfect system, the current that flows initially depends only on the conductance of the cell and the potential difference \(\Delta V\):

$$\displaystyle I_0 = \frac{\sigma}{k} \Delta V$$

where \(k\) is the cell constant, i.e., the ratio of the distance between the electrodes to the surface area. At steady state, the concentration gradient does not change with time. The migration of the anion is exactly balanced by its diffusion in the opposite direction (that is, the net flux of anions is zero - this has to be the case, because the electrode surfaces are "blocking" to the anions). Meanwhile, the current carried by Li^{+} is carried exactly 50:50 by migration and diffusion.

The end result of all this is that for **very small potential differences (< 10 mV)**, the current flow at steady state is given simply by:

$$\displaystyle I_{ss} = \frac{t_+ \sigma}{k} \Delta V$$

Or, indeed:

$$\displaystyle t_+ = \frac{I_{ss}}{I_0}$$

However! In real systems we have interfacial resistances resulting from surface layers and charge transfer kinetics, so we cannot apply this directly. These resistances may also change with time and concentration. We need to measure the interfacial resistance both initially and at steady state, and ideally the impedance spectra will look something like this:

In this case the series resistance *R _{s}* is the ionic resistance of the electrolyte, and

$$\displaystyle I_0 = \frac{\Delta V}{k/\sigma + R_{p,0}}$$

Correspondingly, the current at steady state is given by:

$$\displaystyle I_{ss} = \frac{\Delta V}{k/t_+ \sigma + R_{p,ss}}$$

The expression for the initial current can be rearranged so that:

$$\displaystyle \sigma = \frac{I_0 k}{\Delta V - I_0 R_{p,0}}$$

Substituting this equation into the expression for the steady state current gives:

$$\displaystyle I_{ss} = \frac{\Delta V}{\frac{\Delta V - I_0 R_{p,0}}{t_+ I_0} + R_{p,ss}}$$

And this ultimately rearranges to give:

$$\displaystyle t_+ = \frac{I_{ss}\left(\Delta V - I_0 R_{p,0}\right)}{I_0 \left(\Delta V - I_{ss} R_{p,ss}\right)}$$

This is the well-known Bruce-Vincent equation. Simple, right?!

Well, one important consideration, regardless of how carefully the experiment is set up, is that assumption of adherence to the Nernst-Einstein equation. This equation assumes that the ions do not interact with each other when they are dissolved, but this is only even approximately true in very dilute solution, for example concentrations < 0.01 M.

At typical salt concentrations used in batteries - 1 M, or maybe even higher - there are significant interactions between ions. Take a generic salt, LiX, for example. When we dissolve this into a solvent at a relatively high concentration, this dissociates into solvated Li^{+} and X^{-} ions, but some will remain as neutral [LiX] ion pairs.

We can also consider the potential formation of ion triplets such as [Li_{2}X]^{+} and [LiX_{2}]^{-}. Whether or not such species really exist is besides the point, but in considering these examples it is clear that the migration of the [Li_{2}X]^{+} with only 1 positive charge moves two Li^{+}; and the migration of [LiX_{2}]^{-} moves a Li^{+} in the opposite direction! These each have their own transport numbers. In this case, we should consider the concept of * transference*, distinct from

The *transference number* for lithium, for example, is defined as the number of moles of lithium transferred *by migration* per Faraday of charge. To avoid confusion with the transport number, we will use the symbol *T _{+}* (and the transference number of the anion, correspondingly, is

$$\displaystyle T_+ = t_{\text{Li}^+} + 2t_{[\text{Li}_2\text{X}]^+} - t_{[\text{LiX}_2]^-}$$

*T _{+}* therefore quantifies the net transference of all the Li

As with the transport numbers:

$$\displaystyle T_+ + T_- = 1$$

However, there are no bounds on individual transference numbers. From the equation above, it is theoretically possible to have *T _{+}* < 0 if the mobility of the [LiX

That's not all though: while this is all happening, neutral [LiX] ion pairs - which do not migrate, because they are neutral - are diffusing down the concentration gradient and also, in effect, transferring Li^{+}. However, this is not included in the definition of *T _{+}*, because the definition of

Ultimately, a system could have *T _{+}* < 0 and still operate, but diffusion of neutral ion pairs would have to be fast enough to both supply the current passed and compensate for the net effect of migration to transfer Li

Because of the above, the Bruce-Vincent method is only valid in very (and unrealistically) dilute systems. So how can we determine the *real* transference number?

One of the established and reliable methods for calculating the true transference numbers is the Hittorf method. In this case, a known amount of charge is passed through a cell, and the electrolyte is then divided into four or more sections (this is a bit easier with a polymer electrolyte!).

The reason it needs to be four or more is that there must be at least two reference sections where the concentration of the salt are identical to each other. During the passage of charge, migration transfers cations and anions to the electrode surface. If the passage of charge is stopped before a concentration *gradient* forms in the reference sections, then the transfer of cations and ions into the other sections is due only to migration, and not diffusion. No assumptions about the nature of the electrolyte need to be made!

In this case, there is a change in the amount of salt in section 1 in the diagram above according to:

$$\displaystyle T_- = \frac{-\Delta \text{moles}_\text{Li} F}{Q} $$

And therefore, if you can determine the concentration of salt in the sections, you can calculate *T _{-}* and then, accordingly,

Bruce, Hardgrave and Vincent did in fact use this method in 1992 to show that for a PEO polymer electrolyte containing LiClO_{4} salt, *T _{+}* was calculated to be 0.06 ± 0.05 by the Hittorf method, compared with ~0.2 by their own Bruce-Vincent method. This result showed clearly how the transport of neutral ion pairs and/or negatively charged triplets can cause an overestimate of the true transference number in the Bruce-Vincent method.

There are conceptually similar methods currently under development based on electrophoretic NMR, which also allow for determination of the true transference number, but these approaches are not yet accessible for most researchers. Also, this method (at the time of writing) has yet to be applied to solid polymer electrolytes.

John Newman and co-workers have also been developing an electrochemical method based on concentrated solution theory which also does not require any prior assumptions about the electrolyte. They have shown that under the same DC polarisation conditions as in the Bruce-Vincent experiment, that for a concentrated electrolyte:

$$\displaystyle \frac{I_{ss}}{I_{0}} = \frac{1}{1 + Ne}$$

where \(Ne\), the dimensionless "Newman number" is a more complicated term containing the true transference number:

$$\displaystyle Ne = a \frac{\sigma RT \left(1 - T_+ \right)^2}{F^2 Dc} \left(1 + \frac{d \ln \gamma_\pm}{d \ln m}\right)$$

To use this method, several things need to be measured: \(\frac{I_0}{I_{ss}}\), as in the Bruce-Vincent method; the ionic conductivity, \(\sigma\); the salt diffusion coefficient, \(D\), which can be obtained from the voltage of the cell as it relaxes after the DC polarisation; and the "thermodynamic factor", \(\left(1 + \frac{d \ln \gamma_\pm}{d \ln m}\right)\), which quantifies the concentration dependence of the activity of the ions, and can be obtained from measurements on a concentration cell.

This is a lot of things to measure, and there are different errors associated with all of the measurements; so the overall determination of *T _{+}* is subject to significant experimental error. Nonetheless, Pesko

The Bruce-Vincent method is so convenient, and has become so widely used in the polymer electrolyte research area, that it has effectively become a standard technique used in most reports on new materials. Because of this, there are probably a lot of researchers - maybe even most - using it without fully understanding it.

This is especially relevant now, where a lot of research effort is directed towards, for example, electrolytes with very high salt concentration - where we are a long way from ideality and transference numbers are very high (or so it would appear)! This is also relevant in systems based on ionic liquids, where there are *only* ions in the electrolyte, and Li^{+} transference has to compete with the transference of other positively charged species.

In these cases, clearly the Bruce-Vincent method does not give the transference number, *T _{+}*, or indeed the transport number,

Bruce and Vincent themselves recognised this was the case very shortly after publishing their method, and suggested in such cases that the result of their method should instead be termed the "limiting current fraction", *F _{+}*. I would tend to agree - it surely doesn't make sense to report some quantity you have measured as being

Is *F _{+}* a useful quantity, and what does it mean? Well, this debate is likely to continue long into the future. For what it's worth, I think it can be a helpful value to allow comparisons between different materials (you would expect in most cases that a higher

But, *F _{+}*, as we saw in the expression for the Newman number before, is a mash-up of several other different properties, and so therefore it doesn't mean very much in itself. For modelling purposes, it's hopeless - the real

The problem is, it's not easy to get. As appealing as the assumption-free Newman approach looks, I'm cautious: it's not that much more work, but the errors can get pretty large, the *T _{+}* values don't always seem to match up with the Bruce-Vincent method in the very dilute case where you expect they should agree, and I would like to see the results backed up by other methods such as the Hittorf cell or electrophoretic NMR to demonstrate its robustness.

I reckon the Bruce-Vincent method is here to stay, but its users could probably do the field a favour by not calling the result of the experiment something that it isn't.

Hopefully this article has been helpful in making sense of the minefield of transference and transport in battery electrolytes. If I would pick out any specific points as particularly important, I would say:

There is a difference between transport,

*t*, and transference,*T*. And they are often mixed up, so it's a good idea to know how to spot this. Electrochemistry textbooks like Bard & Faulkner discuss transport, but not transference; polymer electrolyte research papers often discuss transference, not transport, but both use small*t*to denote this. It's confusing, and I suggest using*T*for transference to be clear about this!Know what the assumptions in the Bruce-Vincent method are - no ion association (Nernst-Einstein equation is valid), and only applies to very small DC polarisations (< 10 mV).

If you know that you have a system which behaves non-ideally (that would be essentially all of them), then consider using

*F*instead of_{+}*T*to report the results of a Bruce-Vincent experiment._{+}Whoever invents a simple, convenient, accurate method for measuring true transference numbers of both solid and liquid electrolytes is going to be a very well cited scientist indeed.

If it was helpful, or unclear, or if I've got something wrong (it's very possible), then please let me know in the comments below, and I'll do my best to improve this.