5996
J. Phys. Chem. B 2009, 113, 5996–6000
Temperature Dependence of Ion Transport: The Compensated Arrhenius Equation Matt Petrowsky and Roger Frech* Department of Chemistry and Biochemistry, UniVersity of Oklahoma, Norman, Oklahoma 73019 ReceiVed: NoVember 17, 2008; ReVised Manuscript ReceiVed: February 12, 2009
The temperature-dependent conductivity originating in a thermally activated process is often described by a simple Arrhenius expression. However, this expression provides a poor description of the data for organic liquid electrolytes and amorphous polymer electrolytes. Here, we write the temperature dependence of the conductivity as an Arrhenius expression and show that the experimentally observed non-Arrhenius behavior is due to the temperature dependence of the dielectric constant contained in the exponential prefactor. Scaling the experimentally measured conductivities to conductivities at a chosen reference temperature leads to a “compensated” Arrhenius equation that provides an excellent description of temperature-dependent conductivities. A plot of the prefactors as a function of the solvent dielectric constant results in a single master curve for each family of solvents. These data suggest that ion transport in these and related systems is governed by a single activated process differing only in the activation energy for each family of solvents. Connection is made to the shift factor used to describe electrical and mechanical relaxation in a wide range of phenomena, suggesting that this scaling procedure might have broad applications. Introduction Ion transport in organic liquid electrolytes has been studied for well over a century, yet the temperature dependence of the conductivity in these (and other) electrolytes is poorly understood at the molecular level. In large part, this problem arises because of the interdependence of the temperature with other factors that play a role in transport of ions, particularly the dielectric constant. The conductivity of a thermally activated ion-transport process in an amorphous phase below the glass transition temperature (Tg) is usually described by a simple Arrhenius expression
σ ) σoe-Ea/RT
(1)
where the exponential prefactor σo is constant (or exhibits a weak temperature dependence) and the quantity Ea is the activation energy for the thermally activated process. However, the temperature dependence of the ionic conductivity, σ(T ) in amorphous polymer electrolytes above Tg and organic liquid electrolytes is decidedly non-Arrhenius, that is, a plot of the logarithm of the conductivity versus the reciprocal temperature is markedly curved. In these electrolyte systems, conductivity data are fit to an empirical equation such as the Williams-Landel-Ferry (WLF) equation,1 the Vogel-Tamman-Fulcher (VTF) equation,2-4 or other phenomenological equations.5,6 Unfortunately, calculation of the fitting parameters in these equations does not lead to physical insight into the fundamental aspects of ion transport. Here, we take a novel approach to the temperature dependence of organic liquid electrolytes by a bold hypothesis: we write the conductivity σ as an Arrhenius expression and assume that the non-Arrhenius behavior of the experimental data results from the temperature dependence of the exponential prefactor σo, that is
σ(T, ε) ) σo(T, ε)e-Ea/RT
(2)
where Ea is an activation energy. Because the conductivity of an electrolyte solution is dependent on the dielectric constant ε, an implicit dependence of σ on ε is also included in eq 2. It is well-known that the dielectric constant depends on the temperature;7 therefore, we further hypothesize that the temperature dependence of σo is given completely by the temperature dependence of the dielectric constant, that is, eq 2 can be rewritten as
σ(T, ε) ) σo[ε(T)]e-Ea/RT
(3)
It is then possible to cancel the exponential prefactor in eq 3 by scaling the temperature-dependent conductivities of a particular solution of interest, σ(T,ε), to reference conductivities at a chosen reference temperature Tr, σ(Tr,ε). This procedure is illustrated below for dilute electrolyte solutions (0.0055 M) of tetrabutylammonium trifluoromethanesulfonate (TbaTf) dissolved in various linear alcohols or ketones. The choice of TbaTf as the salt minimizes cation-anion interactions because a close approach of the anion to the nitrogen atom of the cation is frustrated by the bulky butyl groups. Consequently, there is no formation of contact ion pairs or more highly associated species. This conclusion is confirmed by vibrational spectroscopic measurements of the internal modes of the anion that are especially sensitive to cation-anion interactions.8-10 Therefore, when TbaTf is used as the solute, all species that exist in solution are “free” ions, and consequently, the temperature- or concentration-dependent conductivity can be interpreted in terms of changes in ionic mobility and not ionic association. Both the temperature and concentration dependence of conductivities in liquid and polymeric organic electrolytes have been interpreted in terms of the formation of ionically associated species such as ion pairs and more highly associated ionic aggregates.11,12 However, it has been pointed out that the association constants obtained from these analyses do not necessarily correspond to formation of contact ion pairs or complexes; rather, these constants should be viewed as a
10.1021/jp810095g CCC: $40.75 2009 American Chemical Society Published on Web 04/01/2009
Temperature Dependence of Ion Transport
Figure 1. Conductivity of 0.0055 M TbaTf-alcohol solutions vs dielectric constant of the solvents at ([) 25 and (9) 50 °C. 1, octanol; 2, hexanol; 3, butanol; 4, propanol; 5, ethanol.
way of representing the effects of Coulombic forces at short interionic distances.13,14 Experimental Methods All solvents (g99% pure) were obtained from Aldrich and used as received. Salts were obtained from Aldrich (purity g 99%) and dried under vacuum at 85 °C for 24 h. The samples were prepared by dissolving weighed amounts of salt into the appropriate amount of solvent and then stirring for 24 h. The sample preparations (excluding aqueous KCl to determine the cell constant) and all measurements were carried out in a glovebox (
