Thermal Conductivity of Ionic Liquids: Recent Challenges Facing Theory and Experiment

Abstract

Recently ionic liquids have been considered as prospective substances for application as heat transfer fluids, which requires accurate knowledge of their thermal conductivity as a function of thermodynamic parameters of state. Here we present a review of the state of the art for datasets of experimental data existing to date as well as methods developed for predicting this transport property. Special attention is focused on the usage of other thermodynamic quantities, which are more accessible, for such predictive calculations supplied with analysis of physical premises of their interplay with the thermal conductivity instead of phenomenological correlations. We formulate several sets of open problems in this field of study supplied with some preliminary illustrative estimations. As supplementary material, we describe how an automatised access to the comprehensive collection of the respective experimental data, Ionic Liquids Database – ILThermo (v2.0), can be implemented in a programmable way.

Appendix: Basic Assumptions Made During the Derivations of Classic Dependencies for the Thermal Conductivity of Simple Systems

Einstein [86], considering the thermal molecular motions in solids, proposed a simple formula for the frequency $\nu$ of particles’ oscillations

$$\begin{aligned} \nu \sim \sqrt{\frac{\delta }{m\kappa _S}} \end{aligned}$$

(22)

based on arguments of dimensionality and the consideration of this process as dependent on the particle’s mass m, the inter-particle spatial separation $\delta$ and the inter-particle forces, which can be estimated by the compressibility coefficient $\kappa _S$ (note that Einstein did not specify the kind of compressibility but it should be the adiabatic compressibility when considering the oscillations within the frame of acoustic modes as well as in the context of the thermal conductivity). Estimation of the inter-particle distances from the density $\rho \sim m/\delta ^3$ and the particle’s mass $m=M/N_A$ expressed from the molar weight M and Avogadro’s number $N_A$ leads to the formula

$$\begin{aligned} \nu \sim \left( \frac{N_A\rho }{M}\right) ^{1/3}u, \end{aligned}$$

(23)

where the thermodynamic definition of the speed of sound $u=1/\sqrt{\rho \kappa _S}$ is used.

For estimation of the proportionality constant, Einstein considered [75] the simplified 3D lattice system of a central particle placed in its centre and interacting with 26 nearest neighbours. However, for the simplification, instead of pair-wise distances between points distributed in a cubic lattice, the isotropic interaction of the central particle with particles distributed on a spherical surface was considered. Consequently, the inter-particle distance $\delta$ was introduced in such a way that it is equal to the radius of the mentioned sphere with the volume coordinated with volume of cubic-regularly packed particles derived from the macroscopic density:

$$\begin{aligned} \frac{4\pi }{3}\delta ^3=8\frac{M}{\rho N_a}. \end{aligned}$$

(24)

It is worth noting that such an approximation is even more suitable for the consideration of liquid rather than solids because of macroscopic isotropy of the former accompanied with the local order defined by the radial distribution function, i.e. by the consideration of the isotropic interaction of a central particle and its nearest neighbours forming the first coordination shell.

Respectively, considering small displacements x of the central particle as governed by elastic forces $f=-(26/3)k_{\kappa }\Delta$ (i.e. Hooke’s law), where the angle averaging procedure was applied, and $k_{\kappa }$ is the coefficient of linear elasticity, the frequency has the same form as known for the mass-spring harmonic oscillator

$$\begin{aligned} \nu =\frac{1}{2\pi }\sqrt{\frac{26}{3}k_{\kappa }\frac{N_a}{M}}. \end{aligned}$$

(25)

The coefficient of linear elasticity can be related to the adiabatic compressibility when the sound mode of oscillations equalising the change of elastic potential energy and the thermodynamic work per unit volume under the displacement $\delta$, for which Eq. A3 was used, Einstein obtained

$$\begin{aligned} \nu =\frac{6^{1/2}}{2\pi }\left( \frac{\pi }{6}\right) ^{1/3} \left( \frac{\rho N_A}{M} \right) ^{1/3}\frac{1}{\sqrt{\kappa _S\rho }}. \end{aligned}$$

(26)

Einstein also attracts attention [86] to the need of comparing his formula with the frequency which follows from the Lindemann criterion [87], where this quantity is expressed via the melting temperature, molar weight and inter-particle separation, which also can be estimated from the density.

Note that recently there have been a number of works devoted to the prediction of the melting temperature of ionic liquids [88, 89]. Thus, it is worth investigating, how the frequencies determined from the extrapolation of experimental thermal conductivities down to the condition of solidifications are quantitatively related to the Lindemann criterion and does it have potential for predicting a reference point for the usage in complex with regressions like Eqs. 2 and  3.

While Einstein was interested in the deriving a formula for the frequency of the acoustic mode in solids, especially within frames of developing the theory of heat capacity, this molecular oscillation approach was applied to the thermal conductivity in liquids by Osida [46] who consider the thermal transport as originated from the exchange of energy during collisions of an molecule oscillating with the frequency $\nu$ with surrounding ones. Assuming the mean energy of such a molecule equal to $3k_BT$ (it combines translational and rotational motions), where $k_B$ is Boltzmann’s constant and the temperature, respectively, and integrating over the angle between the direction of oscillatory motion and the direction of the heat flow as well as the mean distance $\delta$ between the planes normal to the heat flow (separated by the typical distance between molecules), he obtained the expression

$$\begin{aligned} \lambda =\frac{4k_B\nu }{\delta }. \end{aligned}$$

(27)

Osida estimated the frequency from the Lindemann criterion for a number of molecular liquids and took inter-particle distance in the simplest form $d=\left( M/(\rho N_A)\right) ^{1/3}$ declaring a reasonable correspondence to experimental data. He noted also that Eq. 27 can be transformed to the form used by Bridgman using the connection between $\nu$ and the adiabatic compressibility derived by Einstein, and, respectively, the speed of sound, Eq. 23 but with the different numerical factor because Bridgman [45] considered more simplified physical picture of arranging of molecules in a simple cubic lattice with an average distance $\delta$ between centres of molecules, and the average energy of a molecule $2k_BT$ (one-dimensional motion, a half of the energy is considered as potential and half as kinetic); the speed of sound was used simply to define a time step of the one-dimensional heat flow transfer parallel to the lattice’s edges $\delta t^{-1}=u/\delta$, and $\delta ^2$ gives the area of surface crossed by the thermal flow that resulted in $\lambda =2k_BTu\delta ^{-2}$.

Another detailed exploration of the vibrational theory of thermal compressibility was carried out by Horrocks and McLaughlin [37]. The general line of derivation is practically the same in Osida’s work and even within a more simplified geometric picture: the linear heat flow normal to planes corresponding the lattice arrangement of molecules. However, the isochoric heat capacity was introduced ab initio in a general form as governing local changes of thermal energy as well as the probability ${{\mathcal {P}}}^*$ that energy is transferred when two molecules collide. As a result, the formula

$$\begin{aligned} \lambda =2{\mathcal {P}}^*C_Vn_S\delta \nu , \end{aligned}$$

(28)

was obtained, where $n_S$ is the number of molecules per unit area of a reference plane and the factor 2 takes into account that a molecule crosses the reference plane twice in every complete vibration. Under assumption of the cubic lattice, Eq. 28 reduces to the Osida’s form but with another numerical factor:

$$\begin{aligned} \lambda =2{\mathcal {P}}^*C_V\frac{\nu }{\delta }. \end{aligned}$$

(29)

More physically important feature during its derivation in the work [37] was the comparative estimation of the vibrational and convective contributions in molecular motions revealed the prevalence of the former at not so high temperatures that is in line with the modern phonon theory of liquid thermodynamics [64].

However, what is more significant, McLaughlin et al. proposed how to eliminate indefinite factors operating with isobaric [38] and isothermal processes [39] obtaining Eqs. 5 and 6. It should be pointed out that their calculations were carried out under assumption that ${\mathcal {P}}^*$, the factor connecting the density and the average inter-particle distance and the isochoric heat capacity $C_V$ are constant.