Dielectric response and viscosity due to dipolar interactions

Abstract

The dielectric response and viscosity are two fundamental properties of liquids that are usually treated separately. Here we show that in highly polar liquids the viscosity can be predicted directly from the dielectric function. We employ a stochastic field theory for thermal dipole-field dynamics coupled to hydrodynamic flow, and derive a very general Kubo relation for the response of an observable to the flow. We then use this to derive a Green-Kubo formula for the viscosity operator in terms of the correlation function for the body force, rather than the usual stress tensor formulation, and from this we derive the contribution to the viscosity due to dipolar interactions. In strongly polar liquids like water we show that viscous dissipation arising from these thermal van der Waals interactions is the dominant dissipative mechanism, leading to a direct connection between dielectric relaxation and viscosity. The theory also predicts the emergence of a second relaxation time in the dielectric response even when only a single microscopic relaxation mechanism is present. This additional timescale contributes to the intrinsic Debye relaxation and provides a natural explanation for the widespread empirical observation that many liquids require two relaxation times to fit their dielectric spectra. By establishing a predictive link between dielectric properties and viscosity, our results revisit classical ideas of liquid dynamics originating with Debye and suggest a practical route for identifying promising solvents for electrochemical energy storage.

Figures (5)

• Figure 1: (a) Relation between the bare polarizability and static dielectric constant, according to the Debye theory [Eq. (\ref{['Debye']}), dashed line] and the present work [Eq. (\ref{['chi_epsilon']}), solid line]. Only for $\epsilon_s/\epsilon_0-1\ll 1$ do the two theories converge. For large $\epsilon_s/\epsilon_0$ we get $\chi\simeq(3/2)\epsilon_s$ compared to Debye's $\chi\simeq\epsilon_s$. (b) Real and imaginary permittivities as a function of frequency for the static value $\epsilon_s=5\epsilon_0$. Dashed lines: Debye's theory [Eq. (\ref{['Debye']})]. Solid lines: the present work [Eq. (\ref{['epsilon']})]. The polarizability and permittivities are scaled by $\epsilon_0$. The frequency is scaled by $\tau_D^{-1}=\kappa/\chi$.
• Figure 2: Ratio of the two relaxation times as a function of the static permittivity (scaled by $\epsilon_0$), as obtained from Eqs. (\ref{['linktau']}) and (\ref{['chi_epsilon']}).
• Figure 3: Viscosity of water as a function of temperature. The solid line shows the theoretical viscosity obtained from Eq. (\ref{['Deta_intro']}) using measured values of the relaxation time $\tau_D(T)$ and static permittivity $\epsilon_s(T)$, and a value for the cutoff length $a$ fitted from the data at room temperature, $T=298$ K. Experimental data are taken from Refs. Kaatze89Kaatze07. The dashed line is obtained from an empirical interpolation formula for $\eta(T)$ of water Viswanath89.
• Figure 4: Viscosity of pentanol isomers as a function of temperature. (a) 3-pentanol; (b) 2-pentanol; (c) 1-pentanol; (d) isopentylalcohol; (e) tert-pentanol. Dots show the theoretical viscosity obtained from Eq. (\ref{['etafinal']}) using measured values of Debye's relaxation time $\tau_D$ and the static permittivity $\epsilon_s$, and a value for the cutoff length $a$ fitted from the data at room temperature. Solid lines are obtained from known empirical formulas for the measured viscosities. Experimental data are taken from Ref. Kaatze10.
• Figure 5: Second Debye-like relaxation time as a function of temperature for water. Dots show the measured values Ronne99. The solid line shows the theoretical result, Eq. (\ref{['linktau']}), using the known Debye relaxation time $\tau_D(T)$ and static permittivity $\epsilon_s(T)$ for water as a function of temperature Kaatze07. The inset shows $\tau_D(T)$ for comparison.