The effect of explicit counterion binding on the transference number of polyelectrolyte solutions

Polyelectrolyte solutions have been proposed as a method to improve the efficiency of lithium-ion batteries by increasing the cation transference number because the polymer self-diffusion coefficient is much lower than that of the counterion. However, this is not necessarily true for the polymer mobility. In some cases, negative transference numbers have been reported, which implies that the lithium ions are transporting to the same electrode as the anion, behavior that is often attributed to a binding of counterions to the polyion. We use a simple model where we bind some counterions to the polymer via harmonic springs to investigate this phenomenon. We find that both the number of bound counterions and the strength of their binding alter the transference number, and, in some cases, the transference number is negative. We also investigate how the transference number depends on the Manning parameter, the ratio of the Bjerrum length to charge separation along the chain. By altering the Manning parameter, the transference number can almost be doubled, which suggests that charge spacing could be a way to increase the transference number of polyelectrolyte solutions.