Force density functional theory in- and out-of-equilibrium.

When a fluid is subject to an external field, as is the case near an interface or under spatial confinement, then the density becomes spatially inhomogeneous. Although the one-body density provides much useful information, a higher level of resolution is provided by the two-body correlations. These give a statistical description of the internal microstructure of the fluid and enable calculation of the average interparticle force, which plays an essential role in determining both the equilibrium and dynamic properties of interacting fluids. We present a theoretical framework for the description of inhomogeneous (classical) many-body systems, based explicitly on the two-body correlation functions. By consideration of local Noether-invariance against spatial distortion of the system we demonstrate the fundamental status of the Yvon-Born-Green (YBG) equation as a local force-balance within the fluid. Using the inhomogeneous Ornstein-Zernike equation we show that the two-body correlations are density functionals and, thus, that the average interparticle force entering the YBG equation is also a functional of the one-body density. The force-based theory we develop provides an alternative to standard density functional theory for the study of inhomogeneous systems both in- and out-of-equilibrium. We compare force-based density profiles to the results of the standard potential-based (dynamical) density functional theory. In-equilibrium, we confirm both analytically and numerically that the standard approach yields profiles that are consistent with the compressibility pressure, whereas the force-density functional gives profiles consistent with the virial pressure. For both approaches we explicitly prove the hard-wall contact theorem that connects the value of the density profile at the hard-wall with the bulk pressure. The structure of the theory offers deep insights into the nature of correlation in dense and inhomogeneous systems.

Modeling the impedance response and steady state behaviour of porous CGO-based MIEC anodes.

Mixed ionic and electronic conducting (MIEC) materials recently gained much interest for use as anodes in solid oxide fuel cell (SOFC) applications. However, many processes in MIEC-based porous anodes are still poorly understood and the appropriate interpretation of corresponding electrochemical impedance spectroscopy (EIS) data is challenging. Therefore, a model which is capable to capture all relevant physico-chemical processes is a crucial prerequisite for systematic materials optimization. In this contribution we present a comprehensive model for MIEC-based anodes providing both the DC-behaviour and the EIS-spectra. The model enables one to distinguish between the impact of the chemical capacitance, the reaction resistance, the gas impedance and the charge transport resistance on the EIS-spectrum and therewith allows its appropriate interpretation for button cell conditions. Typical MIEC-features are studied with the model applied to gadolinium doped ceria (CGO) anodes with different microstructures. The results obtained for CGO anodes reveal the spatial distribution of the reaction zone and associated transport distances for the charge carriers and gas species. Moreover, parameter spaces for transport limited and surface reaction limited situations are depicted. By linking bulk material properties, microstructure effects and the cell design with the cell performance, we present a way towards a systematic materials optimization for MIEC-based anodes.

Pressure, surface tension, and curvature in active systems: A touch of equilibrium.

We explore the pressure of active particles on curved surfaces and its relation to other interfacial properties. We use both direct simulations of the active systems as well as simulations of an equilibrium system with effective (pair) interactions designed to capture the effects of activity. Comparing the active and effective passive systems in terms of their bulk pressure, we elaborate that the most useful theoretical route to this quantity is via the density profile at a flat wall. This is corroborated by extending the study to curved surfaces and establishing a connection to the particle adsorption and integrated surface excess pressure (surface tension). In the ideal-gas limit, the effect of curvature on the mechanical properties can be calculated analytically in the passive system with effective interactions and shows good (but not exact) agreement with simulations of the active models. It turns out that even the linear correction to the pressure is model specific and equals the planar adsorption in each case, which means that a known equilibrium sum rule can be extended to a regime at small but nonzero activity. In turn, the relation between the planar adsorption and the surface tension is reminiscent of the Gibbs adsorption theorem at an effective temperature. At finite densities, where particle interactions play a role, the presented effective-potential approximation captures the effect of density on the dependence of the pressure on curvature.

Flow induced crystallisation of penetrable particles.

For a system of Brownian particles interacting via a soft exponential potential we investigate the interaction between equilibrium crystallisation and spatially varying shear flow. For thermodynamic state points within the liquid part of the phase diagram, but close to the crystallisation phase boundary, we observe that imposing a Poiseuille flow can induce nonequilibrium crystalline ordering in regions of low shear gradient. The physical mechanism responsible for this phenomenon is shear-induced particle migration, which causes particles to drift preferentially towards the center of the flow channel, thus increasing the local density in the channel center. The method employed is classical dynamical density functional theory.

Erratum: Nonequilibrium Phase Behavior from Minimization of Free Power Dissipation [Phys. Rev. Lett. 117, 208003 (2016)].

This corrects the article DOI: 10.1103/PhysRevLett.117.208003.

Nonequilibrium Phase Behavior from Minimization of Free Power Dissipation.

We develop a general theory for describing phase coexistence between nonequilibrium steady states in Brownian systems, based on power functional theory [M. Schmidt and J. M. Brader, J. Chem. Phys. 138, 214101 (2013)]. We apply the framework to the special case of fluid-fluid phase separation of active soft sphere swimmers. The central object of the theory, the dissipated free power, is calculated via computer simulations and compared to a simple analytical approximation. The theory describes well the simulation data and predicts motility-induced phase separation due to avoidance of dissipative clusters.

Driven colloidal fluids: construction of dynamical density functional theories from exactly solvable limits.

The classical dynamical density functional theory (DDFT) provides an approximate extension of equilibrium DFT to treat nonequilibrium systems subject to Brownian dynamics. However, the method fails when applied to driven systems, such as sheared colloidal dispersions. The breakdown of DDFT can be traced back to an inadequate treatment of the flow-induced distortion of the pair correlation functions. By considering the distortion of the pair correlations to second order in the flow-rate we show how to systematically correct the DDFT for driven systems. As an application of our approach we consider Poiseuille flow. The theory predicts that the particles will accumulate in spatial regions where the local shear rate is small, an effect known as shear-induced migration. We compare these predictions to Brownian dynamics simulations with generally good agreement.

Particle conservation in dynamical density functional theory.

We present the exact adiabatic theory for the dynamics of the inhomogeneous density distribution of a classical fluid. Erroneous particle number fluctuations of dynamical density functional theory are absent, both for canonical and grand canonical initial conditions. We obtain the canonical free energy functional, which yields the adiabatic interparticle forces of overdamped Brownian motion. Using an exact and one of the most advanced approximate hard core free energy functionals, we obtain excellent agreement with simulations. The theory applies to finite systems in and out of equilibrium.

Power functional theory for the dynamic test particle limit.

For classical Brownian systems both in and out of equilibrium we extend the power functional formalism of Schmidt and Brader (2013 J. Chem. Phys. 138 214101) to mixtures of different types of particles. We apply the framework to develop an exact dynamical test particle theory for the self and distinct parts of the van Hove function, which characterize tagged and collective particle motion. The memory functions that induce non-Markovian dynamics are related to functional derivatives of the excess (over ideal) free power dissipation functional. The method offers an alternative to the recently found nonequilibrium Ornstein-Zernike relation for dynamic pair correlation functions.

Superadiabatic forces in Brownian many-body dynamics.

Theoretical approaches to nonequilibrium many-body dynamics generally rest upon an adiabatic assumption, whereby the true dynamics is represented as a sequence of equilibrium states. Going beyond this simple approximation is a notoriously difficult problem. For the case of classical Brownian many-body dynamics, we present a simulation method that allows us to isolate and precisely evaluate superadiabatic correlations and the resulting forces. Application of the method to a system of one-dimensional hard particles reveals the importance for the dynamics, as well as the complexity, of these nontrivial out-of-equilibrium contributions. Our findings help clarify the status of dynamical density functional theory and provide a rational basis for the development of improved theories.

Dynamic correlations in Brownian many-body systems.

For classical Brownian systems driven out of equilibrium, we derive inhomogeneous two-time correlation functions from functional differentiation of the one-body density and current with respect to external fields. In order to allow for appropriate freedom upon building the derivatives, we formally supplement the Smoluchowski dynamics by a source term, which vanishes at the physical solution. These techniques are applied to obtain a complete set of dynamic Ornstein-Zernike equations, which serve for the development of approximation schemes. The rules of functional calculus lead naturally to non-Markovian equations of motion for the two-time correlators. Memory functions are identified as functional derivatives of a unique space- and time-nonlocal dissipation power functional.

Nonequilibrium Ornstein-Zernike relation for Brownian many-body dynamics.

We derive a dynamic Ornstein-Zernike equation for classical fluids undergoing overdamped Brownian motion and driven out of equilibrium. Inhomogeneous two-time correlation functions are obtained from functional differentiation of the one-body density and current with respect to an appropriately chosen external field. Functional calculus leads naturally to non-Markovian equations of motion for the two-time correlators. Memory functions are identified as functional derivatives of a space- and time-nonlocal power dissipation functional. We propose an excess (over ideal gas) dissipation functional that both generates mode-coupling theory for the two-body correlations and extends dynamical density functional theory for the one-body fields, thus unifying the two approaches.

Power functional theory for Brownian dynamics.

Classical density functional theory (DFT) provides an exact variational framework for determining the equilibrium properties of inhomogeneous fluids. We report a generalization of DFT to treat the non-equilibrium dynamics of classical many-body systems subject to Brownian dynamics. Our approach is based upon a dynamical functional consisting of reversible free energy changes and irreversible power dissipation. Minimization of this "free power" functional with respect to the microscopic one-body current yields a closed equation of motion. In the equilibrium limit the theory recovers the standard variational principle of DFT. The adiabatic dynamical density functional theory is obtained when approximating the power dissipation functional by that of an ideal gas. Approximations to the excess (over ideal) power dissipation yield numerically tractable equations of motion beyond the adiabatic approximation, opening the door to the systematic study of systems far from equilibrium.

Effect of mixing and spatial dimension on the glass transition.

We study the influence of composition changes on the glass transition of binary hard disk and hard sphere mixtures in the framework of mode coupling theory. We derive a general expression for the slope of a glass transition line. Applied to the binary mixture in the low concentration limits, this method allows a fast prediction of some properties of the glass transition lines. The glass transition diagram we find for binary hard disks strongly resembles the random close packing diagram. Compared to three dimensions from previous studies, the extension of the glass regime due to mixing is much more pronounced in two dimensions where plasticization only sets in at larger size disparities. For small size disparities we find a stabilization of the glass phase quadratic in the deviation of the size ratio from unity.

Glass rheology: From mode-coupling theory to a dynamical yield criterion.

The mode coupling theory (MCT) of glasses, while offering an incomplete description of glass transition physics, represents the only established route to first-principles prediction of rheological behavior in nonergodic materials such as colloidal glasses. However, the constitutive equations derivable from MCT are somewhat intractable, hindering their practical use and also their interpretation. Here, we present a schematic (single-mode) MCT model which incorporates the tensorial structure of the full theory. Using it, we calculate the dynamic yield surface for a large class of flows.

Nonlocal effects in nonisothermal hydrodynamics from the perspective of beyond-equilibrium thermodynamics.

We examine the origins of nonlocality in a nonisothermal hydrodynamic formulation of a one-component fluid of particles that exhibit long-range correlations, e.g., due to a spherically symmetric, long-range interaction potential. In order to furnish the continuum modeling with physical understanding of the microscopic interactions and dynamics, we make use of systematic coarse graining from the microscopic to the continuum level. We thus arrive at a thermodynamically admissible and closed set of evolution equations for the densities of momentum, mass, and internal energy. From the consideration of an illustrative special case, the following main conclusions emerge. There are two different source terms in the momentum balance. The first is a body force, which in special circumstances can be related to the functional derivative of a nonlocal Helmholtz free energy density with respect to the mass density. The second source term is proportional to the temperature gradient, multiplied by the nonlocal entropy density. These two source terms combine into a pressure gradient only in the absence of long-range effects. In the irreversible contributions to the time evolution, the nonlocal contributions arise since the self-correlations of the stress tensor and heat flux, respectively, are nonlocal as a result of the microscopic nonlocal correlations. Finally, we point out specific points that warrant further discussions.

Structural precursor to freezing: an integral equation study.

Recent simulation studies have drawn attention to the shoulder which forms in the second peak of the radial distribution function of hard spheres at densities close to freezing and which is associated with local crystalline ordering in the dense fluid. We address this structural precursor to freezing using an inhomogeneous integral equation theory capable of describing local packing constraints to a high level of accuracy. The addition of a short-range attractive interaction leads to a well known broadening of the fluid-solid coexistence region as a function of attraction strength. The appearance of a shoulder in our calculated radial distribution functions is found to be consistent with the broadened coexistence region for a simple model potential, thus demonstrating that the shoulder is not exclusively a high density packing effect.

Learning real-world stimuli in a neural network with spike-driven synaptic dynamics.

We present a model of spike-driven synaptic plasticity inspired by experimental observations and motivated by the desire to build an electronic hardware device that can learn to classify complex stimuli in a semisupervised fashion. During training, patterns of activity are sequentially imposed on the input neurons, and an additional instructor signal drives the output neurons toward the desired activity. The network is made of integrate-and-fire neurons with constant leak and a floor. The synapses are bistable, and they are modified by the arrival of presynaptic spikes. The sign of the change is determined by both the depolarization and the state of a variable that integrates the postsynaptic action potentials. Following the training phase, the instructor signal is removed, and the output neurons are driven purely by the activity of the input neurons weighted by the plastic synapses. In the absence of stimulation, the synapses preserve their internal state indefinitely. Memories are also very robust to the disruptive action of spontaneous activity. A network of 2000 input neurons is shown to be able to classify correctly a large number (thousands) of highly overlapping patterns (300 classes of preprocessed Latex characters, 30 patterns per class, and a subset of the NIST characters data set) and to generalize with performances that are better than or comparable to those of artificial neural networks. Finally we show that the synaptic dynamics is compatible with many of the experimental observations on the induction of long-term modifications (spike-timing-dependent plasticity and its dependence on both the postsynaptic depolarization and the frequency of pre- and postsynaptic neurons).

Entropic interfaces in hard-core model amphiphilic mixtures.

We investigate bulk and interfacial properties of a recently proposed hard-body model for a ternary mixture of amphiphilic particles, spheres and needles using density functional theory. The simple model amphiphiles are formed by bonding a vanishingly thin needle tail radially to a hard-sphere head group. Such particles provide a natural amphiphile when added to a binary mixture of spheres and needles. As all interactions are hard, we seek to find whether amphiphilic effects can be driven by entropy without the need to invoke attractive interactions. In order to assess the amphiphilic character of the model we first examine the spatial and orientational distribution of the amphiphiles at the free interface between demixed needle-rich and amphiphile-rich fluid phases of the binary amphiphile-needle subsystem. We then consider the free interface between sphere-rich and needle-rich phases upon adding amphiphiles with low concentration to the demixed system. In both cases the orientational distribution of the particles in the interface provides strong evidence that amphiphilic properties can arise purely from geometrical packing effects.

Colloidal rod-sphere mixtures: fluid-fluid interfaces and the Onsager limit.

Using a geometry-based density functional theory we investigate the free interface between demixed bulk fluid phases of a colloidal mixture of hard spheres and vanishingly thin needles. Results are presented for the spatial and orientational density distributions of the particles, as well as for the interface tension. Density profiles display oscillations on the sphere-rich side of the interface provided the sphere liquid phase is on the oscillatory side of the Fisher-Widom line in the bulk phase diagram. Needles tend to align parallel (perpendicular) to the interface on the needle-rich (sphere-rich) side displaying biaxial (uniaxial) order. Furthermore, we generalize the theory to the Onsager limit for interacting rods, and give explicit expressions for the functional in simple geometries.