The Compensation Effect in the Vogel–Tammann–Fulcher (VTF) Equation for Polymer-Based Electrolytes

Abstract

Single-ion conducting polymer electrolytes have been proposed to significantly enhance lithium ion battery performance by eliminating concentration gradients within the cell. Such electrolytes have universally suffered from poor conductivity at low to moderate temperatures. In an attempt to improve conductivity, numerous studies have sought to better understand the fundamental interplay of ion content and segmental motion, with typical analyses relying on a fit of temperature-dependent conductivity data using the Vogel–Tammann–Fulcher (VTF) equation to assist in separating these effects. In this study, we leverage the large accessible composition window of a newly synthesized, single ion conducting polysulfone–poly­(ethylene glycol) (PSf-co-PEG) miscible random copolymer to more completely understand the interrelationship of glass transition temperature, ion content, and the polymer’s Li+ conductivity. It is demonstrated here that choice of fitting procedure and Vogel temperature plays a crucial role in the observed trends, and importantly, after optimization of the data fitting procedure, a strong positive correlation was observed between the VTF equation prefactor and apparent activation energy for polymers in this electrolyte class. This relationship, known as the compensation effect (among other names) for the related Arrhenius-type behavior of activated processes such as chemical kinetics and diffusion, is shown here to exist in several other polymer electrolyte classes. Given conductivity’s inverse exponential dependence on the apparent activation energy, maximum conductivity within an electrolyte class is achieved in samples where the activation energy is small. For a system in which the compensation effect exists, decreasing activation energy also decreases the prefactor, highlighting the limiting nature of the compensation effect and the importance of escaping from it. Blending of small molecules is shown to break the apparent trend within the PSf-co-PEG system, suggesting a clear route to high transference number, high conductivity electrolytes.

Poly(p-phenylene)s tethered with oligo(ethylene oxide): synthesis by Yamamoto polymerization and properties as solid polymer electrolytes

“…Furthermore, the difference between T 0 and T g increased with the salt concentration, as has previously been observed. 48 For P( pP-EO 3 )-y (y = 40 and 30) this difference was 35°C and increased to 48 and 72°C for P( pP-EO 3 )-20 and P( pP-EO 3 )-10, respectively. For P( pP-EO 4 )-40, T g − T 0 was estimated to be 23°C and increased to ∼35°C upon further increase in salt concentration.…”

“…As is typically observed, increasing ion concentration led to an increase in E a , which is typically due to an increase in the level of coordination between lithium cations and EO impeding chain motions. 48 Furthermore, the P( pP-EO 4 ) membranes tended to have a lower E a than P( pP-EO 3 ), due to the longer and more flexible side chains. The Vogel temperature, T 0 , can be considered the thermodynamical glass transition temperature (free from kinetic effects) at which the polymer can only adopt one conformation.…”

“…The apparent activation energy, E a , is evaluated from conductivity data and is typically considered to include the activation energy for segmental mobility and ionic dissociation. 48 The difference in E a between the electrolyte membranes had limited statistical significance, but some trends could be discerned. As is typically observed, increasing ion concentration led to an increase in E a , which is typically due to an increase in the level of coordination between lithium cations and EO impeding chain motions.…”

“…The parameter T 0 is often taken as T g −50 K. However, apparent trends of the three parameters (E a , T 0 , and σ 0 ) can be heavily dependent on fitting method and choice of T 0 . 48 Consequently, in this study T 0 was included in the fitting through non-linear regression of the natural logarithm of the VTF equation (eqn (1)). The values of the fitted parameters are shown in Table 2, and conductivities calculated with the VTFequation are indicated in Fig.…”

Synergy of Single‐ion Conductive and Thermo‐responsive Copolymer Hydrogels Achieving Anti‐Arrhenius Ionic Conductivity

“…The equations can be expressed as σ = σ 0 exp(− E a / kT ) or σ = σ 0 exp[− E a / k ( T − T 0 )], where σ 0 is the pre‐expotential factor, E a the activation energy, k the Boltzman constant. T 0 is referred to as the Vogel temperature and is equal to the glass transition in ideal glasses . As a new class of ion‐conducting materials, plastic crystals display plastic phase transition that usually results in much enhanced ionic conductivity versus temperature .…”