Transport theory for highly correlated electrolytes with non-local species interactions

Kurzfassung

The principal research focus of this thesis lies on highly correlated battery electrolytes in the bulk, and near electrified interfaces. The bulk regime can be characterized as a mesoscopic continuum, which spans over the length scale of a few hundred micrometers. In contrast, the electrochemical double layer (EDL) is a microscopic effect. It is constituted by a charged electrolyte region adjacent to an electrode, and decays towards the electroneutral bulk region with increasing distance from the electrode, typically over some nanometers. As consequence, our research focus spans over various length scales. In this work, we derive a holistic continuum transport theory for highly correlated electrolytes which captures mesoscopic transport effects of bulk electrolyte, i.e. migration, diffusion and convection, and the formation of the EDL in ionic liquids (ILs) near electrified interfaces. To address this goal, we use the framework of rational thermodynamics (RT), which combines elements from non-equilibrium thermodynamics, mechanics and electromagnetic theory to describe a wide class of materials. RT has a rigorous physics-based foundation which is constituted by mutually coupled universal balancing laws, and the second axiom of thermodynamics. This description takes account for the strong correlations between arbitrary many charged or uncharged electrolyte species, and ensures that the description for the evolution of the system is thermodynamically consistent. The method of Coleman and Noll allows a concise description of the system in the form of constitutive equations via thermodynamic derivatives of the Helmholtz free energy, which is the focal quantity of our constitutive modeling. We obtain a consistent description for the thermodynamic fluxes via using an Onsager approach. This coupling between the fluxes and forces closes our flux-explicit transport theory. Our manuscript is split into two main parts. In the first part, we present a detailed derivation of our continuum transport theory for the bulk electrolyte. Here, we treat the electrolyte as a continuum at liquid state, and neglect the particle nature of the constituents. This implies that we do not account for microscopic interactions explicitly, but use an averaged description based on macroscopic energy contributions. As consequence, for the bulk, it suffices to focus on modeling the Helmholtz free energy density of the system. In addition to the balancing laws for mass, momentum, energy and charge, and to the second axiom of thermodynamics, we make use of volume being an extensiveproperty and account for the volume-filling property of liquid electrolytes. We use the resulting constraints and identify the independent set of species, fluxes and transport parameters. This simplifies our description and rationalizes the transport theory. Altogether, for an electrolyte mixture composed of N species, we obtain a system of equations which consists of one transport equation for the charges, and N-2 transport equations for the species concentrations. These transport equations are supplemented by the Poisson equation and a heat equation. Because convection plays an important role in electrolyte solutions with high amount of salt, we derive an equation for the convection velocity as function of volume fluxes and local volume productions due to chemical reactions. The set of independent transport parameters follows from the Onsager matrix, and is determined by symmetry arguments, flux constraints and thermodynamic consistency. We clarify the ongoing debate regarding the sign and magnitude of transport parameters via a rational discussion of the frame dependence, and derive transformation rules between different reference frames. Our consistent coupling of thermodynamics, mechanics and electromagnetic theory yields a constitutive equation for the forces, which accounts for electrostatic forces, Lorentz forces in charged electrolyte regions, forces stemming from volume penalties due to non-equal molar volumes of the species, dissipative friction forces due to the viscosity of the electrolyte, and entropic forces due to concentration gradients. These forces can be supplemented by non-ideal interactions via modification of the thermodynamic factor. We validate our bulk description for highly correlated electrolytes using numerical methods, and apply it to a zinc ion battery which is based on an electrolyte composed of an IL-mixture with water and salt. A comparison of the simulation results for charging and discharging the battery with experimental results shows that both are quantitatively in very good agreement. In the second part of this manuscript, we focus on the description of the electrochemical double layer (EDL) of binary ILs and IL / salt mixtures. Typically, the EDL spans over some nanometers, and thus constitutes a system at length scales comparable to the size of the molecules, and the effective range of particle interactions. As consequence, our continuum assumption of a structureless bulk liquid must be relaxed, and we must account for non-local correlations between hardcore particles in our EDL description. For this purpose, we generalize our constitutive approach to modeling the free energy of the system as a functional, with contributions stemming from non-local interactions. Although this does not affect the main structure of our transport theory, it yields constitutive equations in the form of functional derivatives and there appear additional contributions in the transport equations in the form of integrals. In principle, the resulting framework can be used to incorporate a variety of different interactions. Here, we focus on the effect of hardcore particles via modeling a short-ranged repulsive interaction potential. In a first step, we apply this framework to binary ILs next to electrified interfaces. In this case, the system is completely described by the Poisson equation and one transport equation for the charge. We show that short-ranged interactions can be approximated systematically by expanding the interaction integral in higher order gradients of the species concentrations. This gradient description has the advantage that it is susceptible to an analytic investigation of the EDL in stationary state, which parametrizes the EDL description and rationalizes the appearance of higher order derivative operators in modified Poisson equations, as recently proposed in this context in the literature. Our analytic analysis of the stationary EDL shows that the charge distribution in the EDL, i.e. the shape and width of the long-ranging screening profile, is completely determined by three competing energy scales. These energy scales describe the electrostatic forces between ions, the molecular repulsion between all molecules, and the thermal motion. Depending upon the relative magnitude of the three energy scales, the EDL profile of the charge density can have three different shapes. For negligible molecular repulsion, the screening profile is determined by the competition between charge ordering (due to electrostatics) and thermal disordering, and decays exponentially. However, in the case where the repulsion between molecules is comparable with the thermal energy and the Coulomb interactions, the EDL spans over some ion diameters and is characterized by a nanostructured electrolyte region with charge oscillations ("overscreening"). Finally, once the molecular repulsion becomes dominant, the bulk electrolyte undergoes a phase transition into ionic layers, and the EDL spans over the complete electrolyte region. Eventually, upon further increase of molecular repulsion, the layered structure phase separates into pure ionic layers. We confirm the instability onset of the stationary electrolyte via a linear stability analysis of our dynamical description with respect to the electroneutral bulk state. Depending upon the magnitude of the electrode polarization, the charge profile saturates near the interface ("crowding"). However, in contrast to the effect of overscreening, this is a "bulk effect" which happens independently from molecular repulsion, and results from the assumption of finite molar volumes. The two characteristic parameters of this crowding effect, i.e. the width of the saturation layer and the maximal charge density, are both predicted by our framework. Our description allows the complete analytical reconstruction of the charge distribution in the EDL. In particular, it predicts the saturation width, the damping parameter, the oscillation frequency and the exact phase boundaries between the three screening phases as function of the energy scales. We validate our description by comparing the EDL forces as obtained from our theory with experimental results obtained from AFM measurements. We also we apply our EDL formalism to a ternary electrolyte mixture composed of a neat IL with a minor salt. Via an analytic discussion of the stationary state, we predict the critical amount of salt additive, which is necessary to perturb the interface screening by the IL ions, and validate our theoretical prediction using experimental results. Altogether, our theoretical description yields a rigorous multiscale methodology from atomistic quantum chemistry calculations to phenomenological continuum models. We identify the interaction contribution appearing in the chemical potential with the pair correlation function used in atomistic frameworks and liquid state theory. Also, we rationalize phenomenological continuum EDL models proposed in the literature, e.g. the BSK approach, which are comprised in our framework as limiting cases. Furthermore, macroscopic thermodynamic descriptions for ion correlations in non-ideal electrolytes, e.g. the Flory Huggins approach, can be obtained from our functional approach via the method of coarse graining.