Long-Range Fluctuation-Induced Forces in Driven Electrolytes

Stochastic Density Functional Theory for Ions in a Polar Solvent

In recent years, the theoretical description of electrical noise and fluctuation-induced effects in electrolytes has gained renewed interest, enabled by stochastic field theories like stochastic density functional theory (SDFT). Such models, however, treat solvents implicitly, ignoring their generally polar nature. In the present study, starting from microscopic principles, we derive a fully explicit SDFT theory that applies to ions in a polar solvent. These equations are solved to compute observables like dynamic charge structure factors and dielectric susceptibilities. We unveil the relative importance of the different contributions (solvent, ions, cross terms) to the dynamics of electrolytes, which are key to understand the couplings between ions and the fluctuations of their microscopic environment.

Algebraic Depletion Interactions in Two-Temperature Mixtures

The phase separation that occurs in two-temperature mixtures, which are driven out of equilibrium at the local scale, has been thoroughly characterized, but much less is known about the depletion interactions that drive it. Using numerical simulations in dimension 2, we show that the depletion interactions extend beyond two particle diameters in dilute systems, as expected at equilibrium, and decay algebraically with an exponent -4. Solving for the N-particle distribution function in the stationary state, perturbatively in the interaction potential, we show that algebraic correlations with an exponent -2d arise from triplets of particles at different temperatures in spatial dimension d. Finally, simulations allow us to extend our results beyond the perturbative limit.

Correlation Decoupling of Casimir Interaction in an Electrolyte Driven by External Electric Fields

It is well established that the long-range van der Waals or thermal Casimir interaction between two semi-infinite dielectrics separated by a distance H is screened by an intervening electrolyte. Here we show how this interaction is modified when an electric field of strength E is applied parallel to the dielectric boundaries, leading to a nonequilibrium steady state with a current. The presence of the field induces a long-range thermal repulsive interaction, scaling just like the thermal Casimir interaction between dielectrics without the intervening electrolyte, i.e., as 1/H^{3}. At small E the effect is of order E^{2}, while at large fields it saturates to an E independent value. We explain the results in terms of a decoupling mechanism between the charge density fluctuations of cations and anions at large applied fields.

On analytical theories for conductivity and self-diffusion in concentrated electrolytes

Describing analytically the transport properties of electrolytes, such as their conductivity or the self-diffusion of the ions, has been a central challenge of chemical physics for almost a century. In recent years, this question has regained some interest in light of Stochastic Density Field Theory (SDFT) - an analytical framework that allows the approximate determination of density correlations in fluctuating systems. In spite of the success of this theory to describe dilute electrolytes, its extension to concentrated solutions raises a number of technical difficulties, and requires simplified descriptions of the short-range repulsion between the ions. In this article, we discuss recent approximations that were proposed to compute the conductivity of electrolytes, in particular truncations of Coulomb interactions at short distances. We extend them to another observable (the self-diffusion coefficient of the ions) and compare them to earlier analytical approaches, such as the mean spherical approximation and mode-coupling theory. We show how the treatment of hydrodynamic effects in SDFT can be improved, that the choice of the modified Coulomb interactions significantly affects the determination of the properties of the electrolytes, and that comparison with other theories provides a guide to extend SDFT approaches in this context.

Electrical noise in electrolytes: a theoretical perspective

Seemingly unrelated experiments such as electrolyte transport through nanotubes, nano-scale electrochemistry, NMR relaxometry and surface force balance measurements, all probe electrical fluctuations: of the electric current, the charge and polarization, the field gradient (for quadrupolar nuclei) and the coupled mass/charge densities. The fluctuations of such various observables arise from the same underlying microscopic dynamics of the ions and solvent molecules. In principle, the relevant length and time scales of these dynamics are encoded in the dynamic structure factors. However, modelling the latter for frequencies and wavevectors spanning many orders of magnitude remains a great challenge to interpret the experiments in terms of physical processes such as solvation dynamics, diffusion, electrostatic and hydrodynamic interactions between ions, interactions with solid surfaces, etc. Here, we highlight the central role of the charge-charge dynamic structure factor in the fluctuations of electrical observables in electrolytes and offer a unifying perspective over a variety of complementary experiments. We further analyze this quantity in the special case of an aqueous NaCl electrolyte, using simulations with explicit ions and an explicit or implicit solvent. We discuss the ability of the standard Poisson-Nernst-Planck theory to capture the simulation results, and how the predictions can be improved. We finally discuss the contributions of ions and water to the total charge fluctuations. This work illustrates an ongoing effort towards a comprehensive understanding of electrical fluctuations in bulk and confined electrolytes, in order to enable experimentalists to decipher the microscopic properties encoded in the measured electrical noise.

Ionic fluctuations in finite volumes: fractional noise and hyperuniformity

Observing finite regions of a bigger system is a common aim, from microscopy to molecular simulations. In the latter especially, there is ongoing interest in predicting thermodynamic properties from tracking fluctuations in finite observation volumes. However, kinetic properties have received little attention, especially not in ionic solutions, where electrostatic interactions play a decisive role. Here, we probe ionic fluctuations in finite volumes with Brownian dynamics and build an analytical framework that reproduces our simulation results and is broadly applicable to other systems with pairwise interactions. Particle number and charge correlations exhibit a rich phenomenology with time, characterized by a diversity of timescales. The noise spectrum of both quantities decays as 1/f3/2, where f is the frequency. This signature of fractional noise shows the universality of 1/f3/2 scalings when observing diffusing particles in finite domains. The hyperuniform behaviour of charge fluctuations, namely that correlations scale with the area of the observation volume, is preserved in time. Correlations even become proportional to the box perimeter at sufficiently long times. Our results pave the way to understand fluctuations in more complex systems, from nanopores to single-particle electrochemistry.

Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations

Using Brownian dynamics simulations, we investigate the effects of confinement, adsorption on surfaces, and ion-ion interactions on the response of confined electrolyte solutions to oscillating electric fields in the direction perpendicular to the confining walls. Nonequilibrium simulations allows to characterize the transitions between linear and nonlinear regimes when varying the magnitude and frequency of the applied field, but the linear response, characterized by the frequency-dependent conductivity, is more efficiently predicted from the equilibrium current fluctuations. To that end, we (rederive and) use the Green-Kubo relation appropriate for overdamped dynamics, which differs from the standard one for Newtonian or underdamped Langevin dynamics. This expression highlights the contributions of the underlying Brownian fluctuations and of the interactions of the particles between them and with external potentials. Although already known in the literature, this relation has rarely been used to date, beyond the static limit to determine the effective diffusion coefficient or the DC conductivity. The frequency-dependent conductivity always decays from a bulk-like behavior at high frequency to a vanishing conductivity at low frequency due to the confinement of the charge carriers by the walls. We discuss the characteristic features of the crossover between the two regimes, most importantly how the crossover frequency depends on the confining distance and the salt concentration, and the fact that adsorption on the walls may lead to significant changes both at high and low frequencies. Conversely, our results illustrate the possibility to obtain information on diffusion between walls, charge relaxation, and adsorption by analyzing the frequency-dependent conductivity.

Diffusion of a tracer in a dense mixture of soft particles connected to different thermostats

We study the dynamics of a tracer in a dense mixture of particles connected to different thermostats. Starting from the overdamped Langevin equations that describe the evolution of the system, we derive the expression of the self-diffusion coefficient of a tagged particle in the suspension, in the limit of soft interactions between the particles. Our derivation, which relies on the linearization of the Dean-Kawasaki equations obeyed by the density fields and on a path-integral representation of the dynamics of the tracer, extends previous derivations that held for tracers in contact with a single bath. Our analytical result is confronted to results from Brownian dynamics simulations. The agreement with numerical simulations is very good even for high densities. We show how the diffusivity of tracers can be affected by the activity of a dense environment of soft particles that may represent polymer coils-a result that could be of relevance in the interpretation of measurements of diffusivity in biological media. Finally, our analytical result is general and can be applied to the diffusion of tracers coupled to different types of fluctuating environments, provided that their evolution equations are linear and that the coupling between the tracer and the bath is weak.

Electric-field-induced oscillations in ionic fluids: a unified formulation of modified Poisson-Nernst-Planck models and its relevance to correlation function analysis

We theoretically investigate an electric-field-driven system of charged spheres as a primitive model of concentrated electrolytes under an applied electric field. First, we provide a unified formulation for the stochastic charge and density dynamics of the electric-field-driven primitive model using the stochastic density functional theory (DFT). The stochastic DFT integrates the four frameworks (the equilibrium and dynamic DFTs, the liquid state theory and the field-theoretic approach), which allows us to justify in a unified manner various modifications previously made for the Poisson-Nernst-Planck model. Next, we consider stationary density-density and charge-charge correlation functions of the primitive model with a static electric field. We predict an electric-field-induced synchronization between emergences of density and charge oscillations. We are mainly concerned with the emergence of stripe states formed by segregation bands transverse to the external field, thereby demonstrating the following: (i) the electric-field-induced crossover occurs prior to the conventional Kirkwood crossover without an applied electric field, and (ii) the ion concentration dependence of the decay lengths at the onset of oscillations bears a similarity to the underscreening behavior found by recent simulation and theoretical studies on equilibrium electrolytes. Also, the 2D inverse Fourier transform of the correlation function illustrates the existence of stripe states beyond the electric-field-induced Kirkwood crossover.

Stochastic Density Functional Theory on Lane Formation in Electric-Field-Driven Ionic Mixtures: Flow-Kernel-Based Formulation

Simulation and experimental studies have demonstrated non-equilibrium ordering in driven colloidal suspensions: with increasing driving force, a uniform colloidal mixture transforms into a locally demixed state characterized by the lane formation or the emergence of strongly anisotropic stripe-like domains. Theoretically, we have found that a linear stability analysis of density dynamics can explain the non-equilibrium ordering by adding a non-trivial advection term. This advection arises from fluctuating flows due to non-Coulombic interactions associated with oppositely driven migrations. Recent studies based on the dynamical density functional theory (DFT) without multiplicative noise have introduced the flow kernel for providing a general description of the fluctuating velocity. Here, we assess and extend the above deterministic DFT by treating electric-field-driven binary ionic mixtures as the primitive model. First, we develop the stochastic DFT with multiplicative noise for the laning phenomena. The stochastic DFT considering the fluctuating flows allows us to determine correlation functions in a steady state. In particular, asymptotic analysis on the stationary charge-charge correlation function reveals that the above dispersion relation for linear stability analysis is equivalent to the pole equation for determining the oscillatory wavelength of charge–charge correlations. Next, the appearance of stripe-like domains is demonstrated not only by using the pole equation but also by performing the 2D inverse Fourier transform of the charge–charge correlation function without the premise of anisotropic homogeneity in the electric field direction.

Non-hyperuniform metastable states around a disordered hyperuniform state of densely packed spheres: stochastic density functional theory at strong coupling

The disordered and hyperuniform structures of densely packed spheres near and at jamming are characterized by vanishing of long-wavelength density fluctuations, or equivalently by long-range power-law decay of the direct correlation function (DCF). We focus on previous simulation results that exhibit the degradation of hyperuniformity in jammed structures while maintaining the long-range nature of the DCF to a certain length scale. Here we demonstrate that the field-theoretic formulation of stochastic density functional theory is relevant to explore the degradation mechanism. The strong-coupling expansion method of stochastic density functional theory is developed to obtain the metastable chemical potential considering the intermittent fluctuations in dense packings. The metastable chemical potential yields the analytical form of the metastable DCF that has a short-range cutoff inside the sphere while retaining the long-range power-law behavior. It is confirmed that the metastable DCF provides the zero-wavevector limit of the structure factor in quantitative agreement with the previous simulation results of degraded hyperuniformity. We can also predict the emergence of soft modes localized at the particle scale by plugging this metastable DCF into the linearized Dean-Kawasaki equation, a stochastic density functional equation.