Out-of-equilibrium behavior of Casimir-type fluctuation-induced forces for free classical fields

Momentum tunnelling between nanoscale liquid flows

The world of nanoscales in fluidics is the frontier where the continuum of fluid mechanics meets the atomic, and even quantum, nature of matter. While water dynamics remains largely classical under extreme confinement, several experiments have recently reported coupling between water transport and the electronic degrees of freedom of the confining materials. This avenue prompts us to reconsider nanoscale hydrodynamic flows under the perspective of interacting excitations, akin to condensed matter frameworks. Here we show, using a combination of many-body theory and molecular simulations, that the flow of a liquid can induce the flow of another liquid behind a separating wall, at odds with the prediction of continuum hydrodynamics. We further show that the range of this 'flow tunnelling' can be tuned through the solid's electronic excitations, with a maximum occurring when these are at resonance with the liquid's charge density fluctuations. Flow tunnelling is expected to play a role in global transport across nanoscale fluidic networks, such as lamellar graphene oxide or MXene membranes. It further suggests exploiting the electronic properties of the confining walls for manipulating liquids via their dielectric spectra, beyond the nature and characteristics of individual molecules.

Critical Casimir forces in soft matter

We review recent advances in the theoretical, numerical, and experimental studies of critical Casimir forces in soft matter, with particular emphasis on their relevance for the structures of colloidal suspensions and on their dynamics. Distinct from other interactions which act in soft matter, such as electrostatic and van der Waals forces, critical Casimir forces are effective interactions characterised by the possibility to control reversibly their strength via minute temperature changes, while their attractive or repulsive character is conveniently determined via surface treatments or by structuring the involved surfaces. These features make critical Casimir forces excellent candidates for controlling the equilibrium and dynamical properties of individual colloids or colloidal dispersions as well as for possible applications in micro-mechanical systems. In the past 25 years a number of theoretical and experimental studies have been devoted to investigating these forces primarily under thermal equilibrium conditions, while their dynamical and non-equilibrium behaviour is a largely unexplored subject open for future investigations.

Fluctuations of the critical Casimir force

The critical Casimir force (CCF) arises from confining fluctuations in a critical fluid and thus it is a fluctuating quantity itself. While the mean CCF is universal, its (static) variance has previously been found to depend on the microscopic details of the system which effectively set a large-momentum cutoff in the underlying field theory, rendering it potentially large. This raises the question how the properties of the force variance are reflected in experimentally observable quantities, such as the thickness of a wetting film or the position of a suspended colloidal particle. Here, based on a rigorous definition of the instantaneous force, we analyze static and dynamic correlations of the CCF for a conserved fluid in film geometry for various boundary conditions within the Gaussian approximation. We find that the dynamic correlation function of the CCF is independent of the momentum cutoff and decays algebraically in time. Within the Gaussian approximation, the associated exponent depends only on the dynamic universality class but not on the boundary conditions. We furthermore consider a fluid film, the thickness of which can fluctuate under the influence of the time-dependent CCF. The latter gives rise to an effective non-Markovian noise in the equation of motion of the film boundary and induces a distinct contribution to the position variance. Within the approximations used here, at short times, this contribution grows algebraically in time whereas, at long times, it saturates and contributes to the steady-state variance of the film thickness.

The role of dimensionality and geometry in quench-induced nonequilibrium forces

We present an analytical formalism, supported by numerical simulations, for studying forces that act on curved walls following temperature quenches of the surrounding ideal Brownian fluid. We show that, for curved surfaces, the post-quench forces initially evolve rapidly to an extremal value, whereafter they approach their steady state value algebraically in time. In contrast to the previously-studied case of flat boundaries (lines or planes), the algebraic decay for curved geometries depends on the dimension of the system. Specifically, steady-state values of the force are approached in time ast^-d/2ind-dimensional spherical (curved) geometries. For systems consisting of concentric circles or spheres, the exponent does not change for the force on the outer circle or sphere. However, the force exerted on the inner circles or sphere experiences an overshoot and, as a result, does not evolve to the steady state in a simple algebraic manner. The extremal value of the force also depends on the dimension of the system, and originates from curved boundaries and the fact that particles inside a sphere or circle are locally more confined, and diffuse less freely than particles outside the circle or sphere.

Dynamics of the critical Casimir force for a conserved order parameter after a critical quench

Fluctuation-induced forces occur generically when long-range correlations (e.g., in fluids) are confined by external bodies. In classical systems, such correlations require specific conditions, e.g., a medium close to a critical point. On the other hand, long-range correlations appear more commonly in certain nonequilibrium systems with conservation laws. Consequently, a variety of nonequilibrium fluctuation phenomena, including fluctuation-induced forces, have been discovered and explored recently. Here we address a long-standing problem of nonequilibrium critical Casimir forces emerging after a quench to the critical point in a confined fluid with order-parameter-conserving dynamics and non-symmetry-breaking boundary conditions. The interplay of inherent (critical) fluctuations and dynamical nonlocal effects (due to density conservation) gives rise to striking features, including correlation functions and forces exhibiting oscillatory time dependences. Complex transient regimes arise, depending on initial conditions and the geometry of the confinement. Our findings pave the way for exploring a wealth of nonequilibrium processes in critical fluids (e.g., fluctuation-mediated self-assembly or aggregation). In certain regimes, our results are applicable to active matter.

Nonequilibrium forces following quenches in active and thermal matter

Nonequilibrium systems with conserved quantities like density or momentum are known to exhibit long-ranged correlations. This, in turn, leads to long-ranged fluctuation-induced (Casimir) forces, predicted to arise in a variety of nonequilibrium settings. Here, we study such forces, which arise transiently between parallel plates or compact inclusions in a gas of particles, following a change ("quench") in temperature or activity of the medium. Analytical calculations, as well as numerical simulations of passive or active Brownian particles, indicate two distinct forces: (i) The immediate effect of the quench is adsorption or desorption of particles of the medium to the immersed objects, which in turn initiates a front of relaxing (mean) density. This leads to time-dependent density-induced forces. (ii) A long-term effect of the quench is that density fluctuations are modified, manifested as transient (long-ranged) (pair-)correlations that relax diffusively to their (short-ranged) steady-state limit. As a result, transient fluctuation-induced forces emerge. We discuss the properties of fluctuation-induced and density-induced forces as regards universality, relaxation as a function of time, and scaling with distance between objects. Their distinct signatures allow us to distinguish the two types of forces in simulation data. Our simulations also show that a quench of the effective temperature of an active medium gives rise to qualitatively similar effects to a temperature quench in a passive medium. Based on this insight, we propose several scenarios for the experimental observation of the forces described here.

Fluctuation spectra and force generation in nonequilibrium systems

Many biological systems are appropriately viewed as passive inclusions immersed in an active bath: from proteins on active membranes to microscopic swimmers confined by boundaries. The nonequilibrium forces exerted by the active bath on the inclusions or boundaries often regulate function, and such forces may also be exploited in artificial active materials. Nonetheless, the general phenomenology of these active forces remains elusive. We show that the fluctuation spectrum of the active medium, the partitioning of energy as a function of wavenumber, controls the phenomenology of force generation. We find that, for a narrow, unimodal spectrum, the force exerted by a nonequilibrium system on two embedded walls depends on the width and the position of the peak in the fluctuation spectrum, and oscillates between repulsion and attraction as a function of wall separation. We examine two apparently disparate examples: the Maritime Casimir effect and recent simulations of active Brownian particles. A key implication of our work is that important nonequilibrium interactions are encoded within the fluctuation spectrum. In this sense, the noise becomes the signal.

Viscosity of a sheared correlated (near-critical) model fluid in confinement

Second-order phase transitions are characterized by a divergence of the spatial correlation length of the order parameter fluctuations. For confined systems, this is known to lead to remarkable equilibrium physical phenomena, including finite-size effects and critical Casimir forces. We explore here some non-equilibrium aspects of these effects in the stationary state resulting from the action of external forces: by analyzing a model of a correlated fluid under shear, spatially confined by two parallel plates, we study the resulting viscosity within the setting of (Gaussian) Landau-Ginzburg theory. Specifically, we introduce a model in which the hydrodynamic velocity field (obeying the Stokes equation) is coupled to an order parameter with dissipative dynamics. The well-known Green-Kubo relation for bulk systems is generalized for confined systems. This is shown to result in a non-local Stokes equation for the fluid flow, due to the correlated fluctuations. The resulting effective shear viscosity shows universal as well as non-universal contributions, which we study in detail. In particular, the deviation from the bulk behavior is universal, depending on the ratio of the correlation length and the film thickness L. In addition, at the critical point the viscosity is proportional to [Formula: see text], where [Formula: see text] is a dynamic length scale. These findings are expected to be experimentally observable, especially for systems where the bulk viscosity is affected by critical fluctuations.

Effects of gas interparticle interaction on dissipative wake-mediated forces

We examine how the short-range repulsive interaction in a gas of Brownian particles affects behavior of the nonequilibrium depletion forces between obstacles embedded into the gas flow. It is shown that for an ensemble of small and widely separated obstacles the dissipative wake-mediated interaction belongs to the type of induced dipole-dipole interaction governed by an anisotropic screened Coulomb-like potential. For closely located obstacles, formation of a common density perturbation "coat" around them leads to enhancement of dissipative interaction, manifested by characteristic peaks in its dependence on both the bath fraction and the external driving field. Moreover, additional screening of the gas flow due to nonlinear blockade effect gives rise to generation of a pronounced step-like profile of gas density distribution around the obstacles. This can lead to additional enhancement of dissipative interaction between obstacles. The possibility of the dissipative pairing effect and dissipative interaction switching provoked by wake inversion is briefly discussed. All the results are obtained within the classical lattice-gas model.

Transient Casimir Forces from Quenches in Thermal and Active Matter

We compute fluctuation-induced (Casimir) forces for classical systems after a temperature quench. Using a generic coarse-grained model for fluctuations of a conserved density, we find that transient forces arise even if the initial and final states are force free. In setups reminiscent of Casimir (planar walls) and van der Waals (small inclusions) interactions, we find comparable exact universal expressions for the force. Dynamical details only scale the time axis of transient force curves. We propose that such quenches can be achieved, for instance, in experiments on active matter, employing tunable activity or interaction protocols.

Hydrodynamic fluctuation-induced forces in confined fluids

We study thermal, fluctuation-induced hydrodynamic interaction forces in a classical, compressible, viscous fluid confined between two rigid, planar walls with no-slip boundary conditions. We calculate hydrodynamic fluctuations using the linearized, stochastic Navier-Stokes formalism of Landau and Lifshitz. The mean fluctuation-induced force acting on the fluid boundaries vanishes in this system, so we evaluate the two-point, time-dependent force correlations. The equal-time correlation function of the forces acting on a single wall gives the force variance, which we show to be finite and independent of the plate separation at large inter-plate distances. The equal-time, cross-plate force correlation, on the other hand, decays with the inverse inter-plate distance and is independent of the fluid viscosity at large distances; it turns out to be negative over the whole range of plate separations, indicating that the two bounding plates are subjected to counter-phase correlations. We show that the time-dependent force correlations exhibit damped temporal oscillations for small plate separations and a more irregular oscillatory behavior at large separations. The long-range hydrodynamic correlations reported here represent a "secondary Casimir effect", because the mean fluctuation-induced force, which represents the primary Casimir effect, is absent.

Relaxation of the thermal Casimir force between net neutral plates containing Brownian charges

We investigate the dynamics of thermal Casimir interactions between plates described within a living conductor model, with embedded mobile anions and cations, whose density field obeys a stochastic partial differential equation which can be derived starting from the Langevin equations of the individual particles. This model describes the thermal Casimir interaction in the same way that the fluctuating dipole model describes van der Waals interactions. The model is analytically solved in a Debye-Hückel-like approximation. We identify several limiting dynamical regimes where the time dependence of the thermal Casimir interactions can be obtained explicitly. Most notably we find a regime with diffusive scaling, even though the charges are confined to the plates and do not diffuse into the intervening space, which makes the diffusive scaling difficult to anticipate and quite unexpected on physical grounds.

Nonequilibrium critical Casimir effect in binary fluids

Colloids immersed in a critical binary liquid mixture are subject to critical Casimir forces (CCFs) because they confine its concentration fluctuations and influence the latter via effective surface fields. To date, CCFs have been primarily studied in thermodynamic equilibrium. However, due to the critical slowing down, the order parameter around a particle can easily be perturbed by any motion of the colloid or by solvent flow. This leads to significant but largely unexplored changes in the CCF. Here we study the drag force on a single colloidal particle moving in a near-critical fluid mixture and the relative motion of two colloids due to the CCF acting on them. In order to account for the kinetic couplings among the order parameter field, the solvent velocity field, and the particle motion, we use a fluid particle dynamics method. These studies extend the understanding of CCFs from thermal equilibrium to nonequilibrium processes, which are relevant to current experiments, and show the emergence of significant effects near the critical point.

Non-equilibrium Casimir force between vibrating plates

We study the fluctuation-induced, time-dependent force between two plates confining a correlated fluid which is driven out of equilibrium mechanically by harmonic vibrations of one of the plates. For a purely relaxational dynamics of the fluid we calculate the fluctuation-induced force generated by the vibrating plate on the plate at rest. The time-dependence of this force is characterized by a positive lag time with respect to the driving. We obtain two distinctive contributions to the force, one generated by diffusion of stress in the fluid and another related to resonant dissipation in the cavity. The relation to the dynamic Casimir effect of the electromagnetic field and possible experiments to measure the time-dependent Casimir force are discussed.

Out-of-equilibrium relaxation of the thermal Casimir effect in a model polarizable material

Relaxation of the thermal Casimir or van der Waals force (the high temperature limit of the Casimir force) for a model dielectric medium is investigated. We start with a model of interacting polarization fields with a dynamics that leads to a frequency dependent dielectric constant of the Debye form. In the static limit, the usual zero frequency Matsubara mode component of the Casimir force is recovered. We then consider the out-of-equilibrium relaxation of the van der Waals force to its equilibrium value when two initially uncorrelated dielectric bodies are brought into sudden proximity. For the interaction between dielectric slabs, it is found that the spatial dependence of the out-of-equilibrium force is the same as the equilibrium one, but it has a time dependent amplitude, or Hamaker coefficient, which increases in time to its equilibrium value. The final relaxation of the force to its equilibrium value is exponential in systems with a single or finite number of polarization field relaxation times. However, in systems, such as those described by the Havriliak-Negami dielectric constant with a broad distribution of relaxation times, we observe a much slower power law decay to the equilibrium value.

Thermal Casimir drag in fluctuating classical fields

A uniformly moving inclusion which locally suppresses the fluctuations of a classical thermally excited field is shown to experience a drag force that depends on the dynamics of the field. It is shown that in a number of cases the linear friction coefficient is dominated by short distance fluctuations and takes a very simple form. Examples where this drag can occur are for stiff objects, such as proteins, nonspecifically bound to more flexible ones such as polymers and membranes.

Forces exerted by a correlated fluid on embedded inclusions

We investigate the forces exerted on embedded inclusions by a fluid medium with long-range correlations, described by an effective scalar field theory. Such forces are the basis for the medium-mediated Casimir-like force. To study these forces beyond thermal average, it is necessary to define them in each microstate of the medium. Two different definitions of these forces are currently used in the literature. We study the assumptions underlying them. We show that only the definition that uses the stress tensor of the medium gives the sought-after force exerted by the medium on an embedded inclusion. If a second inclusion is embedded in the medium, the thermal average of this force gives the usual Casimir-like force between the two inclusions. The other definition can be used in the different physical case of an object that interacts with the medium without being embedded in it. We show in a simple example that the two definitions yield different results for the variance of the Casimir-like force.

Dynamical approach to the Casimir effect

Casimir forces can appear between intrusions placed in different media driven by several fluctuation mechanisms, either in equilibrium or out of it. Herein, we develop a general formalism to obtain such forces from the dynamical equations of the fluctuating medium, the statistical properties of the driving noise, and the boundary conditions of the intrusions (which simulate the interaction between the intrusions and the medium). As a result, an explicit formula for the Casimir force over the intrusions is derived. This formalism contains the thermal Casimir effect as a particular limit and generalizes the study of the Casimir effect to such systems through their dynamical equations, with no appeal to their Hamiltonian, if any exists. In particular, we study the Casimir force between two infinite parallel plates with Dirichlet or Neumann boundary conditions, immersed in several media with finite correlation lengths (reaction-diffusion system, liquid crystals, and two coupled fields with non-Hermitian evolution equations). The driving Gaussian noises have vanishing or finite spatial or temporal correlation lengths; in the first case, equilibrium is reobtained and finite correlations produce nonequilibrium dynamics. The results obtained show that, generally, nonequilibrium dynamics leads to Casimir forces, whereas Casimir forces are obtained in equilibrium dynamics if the stress tensor is anisotropic.