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We argue in favour of developing a comprehensive dynamical theory for rationalizing, predicting, designing, and machine learning nonequilibrium phenomena that occur in soft matter. To give guidance for navigating the theoretical and practical challenges that lie ahead, we discuss and exemplify the limitations of dynamical density functional theory. Instead of the implied adiabatic sequence of equilibrium states that this approach provides as a makeshift for the true time evolution, we posit that the pending theoretical tasks lie in developing a systematic understanding of the dynamical functional relationships that govern the genuine nonequilibrium physics. While static density functional theory gives a comprehensive account of the equilibrium properties of many-body systems, we argue that power functional theory is the only present contender to shed similar insights into nonequilibrium dynamics, including the recognition and implementation of exact sum rules that result from the Noether theorem. As~a~demonstration of the power functional point of view, we consider an idealized steady sedimentation flow of the three-dimensional Lennard-Jones fluid and machine-learn the kinematic map from the mean motion to the internal force field. The trained model is capable of both predicting and designing the steady state dynamics universally for various target density modulations. This demonstrates the significant potential of using such techniques in nonequilibrium many-body physics and overcomes both the conceptual constraints of dynamical density functional theory as well as the limited availability of its analytical functional approximations., Comment: 21 pages, 5 figures. J. Phys.: Condens. Matter (invited Perspective)

Noether's Theorem is familiar to most physicists due its fundamental role in linking the existence of conservation laws to the underlying symmetries of a physical system. Typically the systems are described in the particle-based context of classical mechanics or on the basis of field theory. We have recently shown [Commun. Phys. $\textbf{4}$, 176 (2021)] that Noether's reasoning also applies to thermal systems, where fluctuations are paramount and one aims for a statistical mechanical description. Here we give a pedagogical introduction based on the canonical ensemble and apply it explicitly to ideal sedimentation. The relevant mathematical objects, such as the free energy, are viewed as functionals. This vantage point allows for systematic functional differentiation and the resulting identities express properties of both macroscopic average forces and molecularly resolved correlations in many-body systems, both in and out-of-equilibrium, and for active Brownian particles. To provide further background, we briefly describe the variational principles of classical density functional theory, of power functional theory, and of classical mechanics., Comment: 15 pages, 3 figures, J. Phys.: Condens. Matter (invited as Topical Review, revised version)

An account is given of density functional theory (DFT) for quenched–annealed fluid mixtures that are used to model fluids adsorbed in random porous matrices. The theory is based on the replica trick and allows the treatment of situations where the quenched random matrix as well as the annealed fluid are inhomogeneous on average. Applications of the framework include investigation of the adsorption properties of hard spheres and model colloid–polymer mixtures in bulk matrices and at matrix surfaces, and the influence of the quenched disorder on phase transitions like fluid demixing, isotropic–nematic ordering, and freezing. Particularly rich wetting behaviour was found for colloid–polymer mixtures adsorbed against a porous wall.

After a brief review of the theory of sedimentation equilibria of colloidal systems, we consider the specific case of binary systems of hard sphere colloids and non-interacting polymer coils, the latter of vanishing buoyancy mass. The density profiles of the two components are calculated within density functional theory and using Monte Carlo simulations. Under appropriate conditions the profiles exhibit discontinuities or steeply varying regions associated with the interface separating colloid-rich and colloid-poor phases. The position of the interface is shown to be very sensitive to the strength of the gravitational field and, more surprisingly, to the total height L of the suspension. Phase coexistence in the absence of gravity is shown to be entirely suppressed beyond a critical ratio of the height L over the gravitational length of the colloids.

We consider the Asakura-Oosawa-Vrij model of hard sphere colloids and ideal polymer coils in contact with a planar hard wall at (colloidal) liquid-gas coexistence. Using extensive numerical density functional calculations, the liquid-gas, wall-liquid and wall-gas interfacial free energies are calculated. The results are inserted into Young's equation to obtain the contact angle between the liquid-gas interface and the wall. As a function of polymer fugacity this angle exhibits discontinuities of slope (``kinks'') upon crossing first-order surface phase transitions located on the gas branch of the bulk binodal. Each kink corresponds to a transition from n-1 to $n$ colloid layers adsorbed at the wall, referred to as the n'th layering transition. The corresponding adsorption spinodal points from n-1 to n layers upon reducing the polymer fugacity along the bulk binodal were found in a previous study (J. M. Brader et al. J. Phys.: Cond. Matt., 14: L1, 2002; Mol. Phys., 101: 3349, 2003). Remarkably, we find desorption spinodal points from n to n-1 layers to be absent upon increasing polymer fugacity at bulk coexistence, and many branches (containing up to 7 colloid layers) remain metastable. Results for the first layering binodal and both spinodal branches off-bulk coexistence hint at a topology of the surface phase diagram consistent with these findings. Both the order of the transition to complete wetting and whether it is preceded by a finite or an infinite number of layering transitions remain open questions. We compare the locations of the first layering binodal line and of the second layering binodal point at bulk coexistence with recent computer simulation results by Dijkstra and van Roij (Phys. Rev. Lett., 89: 208303, 2002) and discuss our results for the contact angle in the light of recent experiments., submitted to J. Phys.: Cond. Matt. (CODEF proceedings)

Using density functional theory and Monte Carlo simulations we investigate the Asakura-Oosawa-Vrij mixture of hard sphere colloids and non-adsorbing ideal polymers under selective confinement of the colloids to a planar slab geometry. This is a model for confinement of colloid-polymer mixtures by either two parallel walls with a semi-permeable polymer coating or through the use of laser tweezers. We find that such a pore favours the colloidal gas over the colloidal liquid phase and induces capillary evaporation. A treatment based on the Kelvin equation gives a good account of the location of the capillary binodal for large slit widths. The colloid density profile is found to exhibit a minimum (maximum) at contact with the wall for large (small) slit widths.

The fundamental measure density functional theory for hard spheres is generalized to binary mixtures of arbitrary positive and moderate negative non-additivity between unlike components. In bulk the theory predicts fluid-fluid phase separation into phases with different chemical compositions. The location of the accompanying critical point agrees well with previous results from simulations over a broad range of non-additivities and both for symmetric and highly asymmetric size ratios. Results for partial pair correlation functions show good agreement with simulation data., 8 pages with 4 figures

Using the Asakura–Oosawa–Vrij model for mixtures of hard sphere colloids and non-adsorbing polymer coils, an analytic formula for the interfacial free energy of the fluid mixture in contact with a hard wall is obtained within a scaledparticle treatment. The results compare well with explicit density functional calculations for the binary mixture. We also give expressions for the wall tension of the mixture when polymers interact via a simple stepfunction pair potential, and for the case of contact with a polymer-coated wall, which we take to be hard for the colloids but penetrable for the polymers. On the gas side of the fluid–fluid demixing binodal we confirm the wetting transition at the hard wall and predict complete drying of the polymer-coated wall on the liquid side of the binodal.

We consider the free interface between demixed fluid phases in a mixture of hard spheres and vanishingly thin hard rods using Monte Carlo simulations and density functional theory. Both approaches treat the full binary mixture and hence include all rod-induced many-body depletion interactions between spheres. The agreement between theoretical and simulation results for density and orientation order profiles across the interface is remarkable, even for states not far from the critical point. The simulation results confirm the previously predicted preferred vertical (parallel) alignment of rod orientation to th ei nterface plane at the sphere-rich (sphere-poor) side. This ordering should be experimentally observable in phase-separated colloidal rod–sphere mixtures.

We investigate the fluid-fluid demixing phase transition of the Asakura-Oosawa model colloid-polymer mixture confined between two smooth parallel hard walls using density functional theory and computer simulations. Comparing fluid density profiles for statepoints away from colloidal gas-liquid coexistence yields good agreement of the theoretical results with simulation data. Theoretical and simulation results predict consistently a shift of the demixing binodal and the critical point towards higher polymer reservoir packing fraction and towards higher colloid fugacities upon decreasing the plate separation distance. This implies capillary condensation of the colloid liquid phase, which should be experimentally observable inside slitlike micropores in contact with a bulk colloidal gas.

We investigate the freezing transition in a two-dimensional lattice model of annealed hard squares that are subject to the influence of randomly placed quenched particles of the same size. The latter model is a porous medium. By combining two recent density functional approaches we arrive at a theory for quenched-annealed lattice fluids that treats the quenched particles on the level of their one-body density distribution. We show that this approach yields thermodynamics that compare well with results from treating matrix realizations explicitly and performing subsequent averaging over the disorder. The freezing transition from a fluid to a columnar phase is found to be continuous. On increasing matrix density it shifts towards close packing and vanishes beyond a threshold matrix density.

An overview of recent developments and applications of a specific density functional approach that originates from Rosenfeld's fundamental measure theory for hard spheres is given. Model systems that were treated include penetrable spheres that interact with a step function pair potential, the Widom–Rowlinson model, the Asakura–Oosawa colloid–polymer mixture, ternary mixtures of spheres, needles, and globular polymers, hard-body amphiphilic mixtures, fluids in porous media, and random sequential adsorption that describes non-equilibrium processes such as colloidal deposition and random car parking. In these systems various physical phenomena were studied, such as correlations in liquids, freezing and demixing phase behaviour, the properties of fluid interfaces with and without orientational order, and wetting and layering phenomena at walls.

We consider a model mixture of hard colloidal spheres and non-adsorbing polymer chains in a theta solvent. The polymer component is modelled as a polydisperse mixture of effective spheres, mutually noninteracting but excluded from the colloids, with radii that are free to adjust to allow for colloid-induced compression. We investigate the bulk fluid demixing behaviour of this model system using a geometry-based density-functional theory that includes the polymer size polydispersity and configurational free energy, obtained from the exact radius-of-gyration distribution for an ideal (random-walk) chain. Free energies are computed by minimizing the free energy functional with respect to the polymer size distribution. With increasing colloid concentration and polymer-to-colloid size ratio, colloidal confinement is found to increasingly compress the polymers. Correspondingly, the demixing fluid binodal shifts, compared to the incompressible-polymer binodal, to higher polymer densities on the colloid-rich branch, stabilizing the mixed phase., 14 pages, 4 figures

We treat the non-equilibrium process of random sequential adsorption of hard particles onto a solid substrate by means of a geometry-based density functional theory. As a prerequisite we solve the zero-dimensional case exactly and use it to construct density functionals in higher dimensions, permitting the treatment of adsorption onto arbitrary spatially inhomogeneous substrates. As applications we study the influence of a hard boundary of the adsorption region in the one-dimensional car-parking problem and for colloidal deposition on a two-dimensional solid substrate. Comparing to our computer simulation results, we find that the respective density functionals correctly predict the oscillatory density profiles near the boundary, with amplitudes that are considerably smaller than in the corresponding equilibrium models.

We test the accuracy of a recently proposed density functional (DF) for a fluid in contact with a porous matrix. The DF was constructed in the spirit of Rosenfeld’s fundamental measure concept and was derived for general mixtures of hard core and ideal particles. The required double average over fluid and matrix configurations is performed explicitly. As an application we consider am odel mixture where colloids and matrix particles are represented by hard spheres and polymers by ideal spheres. Integrating over the degrees of freedom of the polymers leads to a binary colloid–matrix system with effective Asakura– Oosawa pair potentials, which we treat with an integral-equation theory. We fin dt hat partial pair correlation functions from both theories are in good agreement with our computer simulation results, and that the theoretical results for the demixing binodals compare well, provided the polymer-to-colloid size ratio, and hence the effect of many-body interactions neglected in the effective model, is not too large. Consistently, we find that hard (ideal) matrix–polymer interactions induce capillary condensatio n( evaporation) of the colloidal liquid phase.

We describe a density functional theory for mixtures of hard sphere (HS) colloids and ideal polymers, the Asakura–Oosawa model. The geometrybased fundamental measures approach which is used to construct the functional ensures the correct behaviour in the limit of low density of both species and in the zero-dimensional limit of a cavity which can contain at most one HS. Dimensional crossover is discussed in detail. Emphasis is placed on the properties of homogeneous (bulk) fluid phases. We show that the present functional yields the same free energy and, therefore, the same fluid–fluid demixing transition as that given by a different approach, namely the freevolume theory. The pair direct correlation functions c (2) ij (r ) of the bul km ixture are given analytically. We investigate the partial structure factors S ij (k) and the asymptotic decay, r →∞ ,o f the to tal pair correlation functions h ij (r ) obtained from the Ornstein–Zernike route. The locus in the phase diagram of the crossover from monotonic to oscillatory decay of correlations is calculated for several size ratios q = R p/Rc ,w hereR p is the radius o ft hepolymer sphere and Rc that of the colloid. We determine the (mean-field) behaviour of the partial structure factors on approaching the fluid–fluid critical (consolute) point.

A recent density functional theory is used to investigate the free interface between demixed fluid phases in a model colloid-polymer mixture. Both the colloid and polymer density profiles oscillate on the colloid-rich side of the interface, provided the polymer reservoir packing fraction ηpr is sufficiently high. Results for the surface tension are in reasonable agreement with experiment. When the mixture is adsorbed against a hard wall, entropic depletion effects give rise to a wetting transition whereby the colloid-rich phase wets completely. Prior to complete wetting we find three layering transitions, the first of which extends far into the single-phase region. This pattern of surface phase transitions is very different from that observed for simple one-component fluids at planar substrates.

Possibilities of fluid-solid and solid-solid phase transformations in colloidal suspensions and star polymer solutions are reviewed. We start from given interparticle pair potentials and predict the corresponding phase diagrams using computer simulations and density functional theory. When possible, the results are compared with experimental data. In particular, we discuss a cascade of freezing transitions for confined colloids and re-entrant melting and anisotropic solid phases for star polymer solutions.

We develop a density functional theory for a system of penetrable spheres that interact with a constant pair potential energy if their separation distance is smaller than their diameter. As sufficient input, just the geometrical properties of the particles and the exactly known statistical behaviour of the system under strong confinement are needed. The theory predicts the bulk fluid properties in good agreement with computer simulations and better than liquid integral theories, as well as the freezing transition to a multiply occupied face-centred-cubic lattice. It becomes exact in the limits of strong confinement and high temperature, and coincides with a successful hard-sphere theory for zero temperature.

A modified geometrically based free-energy functional for hard spheres is proposed which gives reliable results even for situations of extreme confinements that reduce the effective dimensionality D. It is accurate for hard spheres between narrow plates .D D 2/, inside narrow cylindrical pores .D D 1/, and is exact in the 0D limit (a cavity that cannot hold more than one particle). This functional also predicts the hard-sphere fluid-solid transition in excellent agreement with the simulations.

Using fundamental density functional theory we calculate the partial radial distribution functions, g(ij)(r), of a binary non-additive hard sphere mixture using either Percus' test particle approach or inversion of the analytic structure factor obtained via the Ornstein-Zernike route. We find good agreement between the theoretical results and Monte Carlo simulation data for both positive and moderate negative non-additivities. We investigate the asymptotic, [Formula: see text], decay of the g(ij)(r) and show that this agrees with the analytic analysis of the contributions to the partial structure factors in the plane of complex wavevectors. We find the test particle density profiles to be free of unphysical artefacts, contrary to earlier reports.

We study interfacial properties of the phase-separated two-dimensional Ising model. The interface between coexisting phases is stabilized by two parallel walls with opposing surface fields. A driving field parallel to the walls is applied which (i) either acts locally at the walls or (ii) varies linearly with distance across the strip. Using computer simulations with Kawasaki dynamics, we found (Smith et al 2008 Phys. Rev. Lett. 101 067203) that the system reaches a steady state with a sharper magnetization profile, reduced interfacial width, and faster decay of correlations along the interface, as compared to the equilibrium case. Here we present new results for the bond energy profile, providing further evidence for the picture wherein shear acts as effective confinement in this system. As a prerequisite for understanding the driven system, we investigate the pronounced differences between Kawasaki (spin-exchange) and Glauber (spin-flip) dynamics in the confined equilibrium system.

Experimental and computational details are presented for an investigation of the transient time evolution of colloidal dispersions confined in a horizontal slit pore and under the influence of gravity (Royall et al 2007 Phys. Rev. Lett. 98 188304). We demonstrate that the interparticle interactions can be well described by those of effective hard spheres by comparing experimental results for the pair distribution function obtained in the homogeneous part of the settling system to the theoretical result for hard spheres in equilibrium. Using an effective hard sphere diameter that is 10% larger than that obtained by static light scattering takes account of the (screened) electrostatic repulsion between particles. As a simple computational model, we use Brownian dynamics computer simulations with hard sphere pair interactions and investigate the time evolution of the one-body density profile during sedimentation. We show that an 'intrinsic clock', that ticks only when trial moves are accepted, facilitates high accuracy of the time evolution of the density profile, even when using relatively large integration time steps for the Langevin equations of motion.

We use density functional theory to study the capillary phase behaviour of a discotic system of colloidal platelets that are confined in a planar slit pore. The model plates have circular shape, continuous orientations and vanishing thickness; they interact via hard-core repulsion with each other and with the walls which induces homeotropic wall anchoring of the nematic director. We find that the isotropic–nematic capillary binodal is shifted to lower values of the chemical potential as compared to bulk isotropic–nematic coexistence. Capillary isotropic–nematic coexistence vanishes below a critical wall separation distance which is significantly larger than it is in a reference system of thin hard (Onsager) rods confined between two parallel hard walls that act on the particle centres.