The Casimir effect: From quantum to critical fluctuations

Strong Casimir-like Forces in Flocking Active Matter

Confining in space the equilibrium fluctuations of statistical systems with long-range correlations is known to result into effective forces on the boundaries. Here we demonstrate the occurrence of Casimir-like forces in the nonequilibrium context provided by flocking active matter. In particular, we consider a system of aligning self-propelled particles in two spatial dimensions that are transversally confined by reflecting or partially reflecting walls. We show that in the ordered flocking phase this confined active vectorial fluid is characterized by extensive boundary layers, as opposed to the finite ones usually observed in confined scalar active matter. Moreover, a finite-size, fluctuation-induced contribution to the pressure on the wall emerges, which decays slowly and algebraically upon increasing the distance between the walls. We explain our findings-which display a certain degree of universality-within a hydrodynamic description of the density and velocity fields.

Exact pretransition effects in kinetically constrained circuits: Dynamical fluctuations in the Floquet-East model

We study the dynamics of a classical circuit corresponding to a discrete-time version of the kinetically constrained East model. We show that this classical "Floquet-East" model displays pre-transition behavior which is a dynamical equivalent of the hydrophobic effect in water. For the deterministic version of the model, we prove exactly (i) a change in scaling with size in the probability of inactive space-time regions (akin to the "energy-entropy" crossover of the solvation free energy in water), (ii) a first-order phase transition in the dynamical large deviations, (iii) the existence of the optimal geometry for local phase separation to accommodate space-time solutes, and (iv) a dynamical analog of "hydrophobic collapse."

Critical Casimir effect in a disordered O(2)-symmetric model

The critical Casimir effect appears when critical fluctuations of an order parameter interact with classical boundaries. We investigate this effect in the setting of a Landau-Ginzburg model with continuous symmetry in the presence of quenched disorder. The quenched free energy is written as an asymptotic series of moments of the model's partition function. Our main result is that, in the presence of a strong disorder, Goldstone modes of the system contribute either with either an attractive or a repulsive force. This result was obtained using the distributional zeta-function method without relying on any particular ansatz in the functional space of the moments of the partition function.

Critical fluctuations in a confined driven-dissipative quantum condensate

Phase fluctuations determine the low-energy properties of quantum condensates. However, at the condensation threshold, both density and phase fluctuations are relevant. While strong emphasis has been given to the investigation of phase fluctuations, which dominate the physics of the quantum system away from the critical point, number fluctuations have been much less explored even in thermal equilibrium. In this work, we report experimental observation and theoretical description of fluctuations in a circularly confined nonequilibrium Bose-Einstein condensate of polaritons near the condensation threshold. We observe critical fluctuations, which combine the number fluctuations of a single-mode condensate state and competition between different states. The latter is analogous to mode hopping in photon lasers. Our theoretical analysis indicates that this phenomenon is of a quantum character, while classical noise of the pump is not sufficient to explain the experiments. The manifestation of a critical quantum state competition unlocks possibilities for the study of condensate formation while linking to practical realizations in photonic lasers.

Functionalized nanoparticles: Tailoring properties through surface energetics and coordination chemistry for advanced biomedical applications

Significant advances in nanoparticle-related research have been made in the past decade, and amelioration of properties is considered of utmost importance for improving nanoparticle bioavailability, specificity, and catalytic performance. Nanoparticle properties can be tuned through in-synthesis and post-synthesis functionalization operations, with thermodynamic and kinetic parameters playing a crucial role. In spite of robust functionalization techniques based on surface chemistry, scalable technologies have not been explored well. The coordination enhancement via surface functionalization through organic/inorganic/biomolecules material has attracted much attention with morphology modification and shape tuning, which are indispensable aspects in the colloidal phase during biomedical applications. It is envisioned that surface amelioration influences the anchoring properties of nano interfaces for the immobilization of functional groups and biomolecules. In this work, various nanostructure and anchoring methodologies have been discussed, aiming to exploit their full potential in precision engineering applications. Simultaneous discussions on emerging characterization strategies for functionalized assemblies have been made to gain insights into functionalization chemistry. An overview of current advances and prospects of functionalized nanoparticles has been presented, with an emphasis on controllable attributes such as size, shape, morphology, functionality, surface features, Debye and Casimir interactions.

Inducing oscillations of trapped particles in a near-critical Gaussian field

We study the nonequilibrium dynamics of two particles confined in two spatially separated harmonic potentials and linearly coupled to the same thermally fluctuating scalar field, a cartoon for optically trapped colloids in contact with a medium close to a continuous phase transition. When an external periodic driving is applied to one of these particles, a nonequilibrium periodic state is eventually reached in which their motion synchronizes thanks to the field-mediated effective interaction, a phenomenon already observed in experiments. We fully characterize the nonlinear response of the second particle as a function of the driving frequency, in particular far from the adiabatic regime in which the field can be assumed to relax instantaneously. We compare the perturbative, analytic solution to its adiabatic approximation, thus determining the limits of validity of the latter, and we qualitatively test our predictions against numerical simulations.

Boundary Criticality of the 3D O(N) Model: From Normal to Extraordinary

It was recently realized that the three-dimensional O(N) model possesses an extraordinary boundary universality class for a finite range of N≥2. For a given N, the existence and universal properties of this class are predicted to be controlled by certain amplitudes of the normal universality class, where one applies an explicit symmetry breaking field to the boundary. In this Letter, we study the normal universality class for N=2, 3 using Monte Carlo simulations on an improved lattice model and extract these universal amplitudes. Our results are in good agreement with direct Monte Carlo studies of the extraordinary universality class serving as a nontrivial quantitative check of the connection between the normal and extraordinary classes.

Nonequilibrium relaxation of a trapped particle in a near-critical Gaussian field

We study the nonequilibrium relaxational dynamics of a probe particle linearly coupled to a thermally fluctuating scalar field and subject to a harmonic potential, which provides a cartoon for an optically trapped colloid immersed in a fluid close to its bulk critical point. The average position of the particle initially displaced from the position of mechanical equilibrium is shown to feature long-time algebraic tails as the critical point of the field is approached, the universal exponents of which are determined in arbitrary spatial dimensions. As expected, this behavior cannot be captured by adiabatic approaches which assumes fast field relaxation. The predictions of the analytic, perturbative approach are qualitatively confirmed by numerical simulations.

Controlled deposition of nanoparticles with critical Casimir forces

Nanocrystal assembly represents the key fabrication step to develop next-generation optoelectronic devices with properties defined from the bottom-up. Despite numerous efforts, our limited understanding of nanoscale interactions has so far delayed the establishment of assembly conditions leading to reproducible superstructure morphologies, therefore hampering integration with large-scale, industrial processes. In this work, we demonstrate the deposition of a layer of semiconductor nanocrystals on a flat and unpatterned silicon substrate as mediated by the interplay of critical Casimir attraction and electrostatic repulsion. We show experimentally and rationalize with Monte Carlo and molecular dynamics simulations how this assembly process can be biased towards the formation of 2D layers or 3D islands and how the morphology of the deposited superstructure can be tuned from crystalline to amorphous. Our findings demonstrate the potential of the critical Casimir interaction to direct the growth of future artificial solids based on nanocrystals as the ultimate building blocks.

A temperature-dependent critical Casimir patchy particle model benchmarked onto experiment

Synthetic colloidal patchy particles immersed in a binary liquid mixture can self-assemble via critical Casimir interactions into various superstructures, such as chains and networks. Up to now, there are no quantitatively accurate potential models that can simulate and predict this experimentally observed behavior precisely. Here, we develop a protocol to establish such a model based on a combination of theoretical Casimir potentials and angular switching functions. Using Monte Carlo simulations, we optimize several material-specific parameters in the model to match the experimental chain length distribution and persistence length. Our approach gives a systematic way to obtain accurate potentials for critical Casimir induced patchy particle interactions and can be used in large-scale simulations.

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.

Long-Range Fluctuation-Induced Forces in Driven Electrolytes

We study the stochastic dynamics of an electrolyte driven by a uniform external electric field and show that it exhibits generic scale invariance despite the presence of Debye screening. The resulting long-range correlations give rise to a Casimir-like fluctuation-induced force between neutral boundaries that confine the ions; this force is controlled by the external electric field, and it can be both attractive and repulsive with similar boundary conditions, unlike other long-range fluctuation-induced forces. This work highlights the importance of nonequilibrium correlations in electrolytes and shows how they can be used to tune interactions between uncharged biological or synthetic structures at large separations.

Boundary Critical Behavior of the Three-Dimensional Heisenberg Universality Class

We study the boundary critical behavior of the three-dimensional Heisenberg universality class, in the presence of a bidimensional surface. By means of high-precision Monte Carlo simulations of an improved lattice model, where leading bulk scaling corrections are suppressed, we prove the existence of a special phase transition, with unusual exponents, and of an extraordinary phase with logarithmically decaying correlations. These findings contrast with naïve arguments on the bulk-surface phase diagram, and allow us to explain some recent puzzling results on the boundary critical behavior of quantum spin models.

Exact Critical Casimir Amplitude of Anisotropic Systems from Conformal Field Theory and Self-Similarity of Finite-Size Scaling Functions in d≥2 Dimensions

The exact critical Casimir amplitude is derived for anisotropic systems within the d=2 Ising universality class by combining conformal field theory with anisotropic φ^{4} theory. Explicit results are presented for the general anisotropic scalar φ^{4} model and for the fully anisotropic triangular-lattice Ising model in finite rectangular and infinite strip geometries with periodic boundary conditions. These results demonstrate the validity of multiparameter universality for confined anisotropic systems and the nonuniversality of the critical Casimir amplitude. We find an unexpected complex form of self-similarity of the anisotropy effects near the instability where weak anisotropy breaks down. This can be traced back to the property of modular invariance of isotropic conformal field theory for d=2. More generally, for d>2 we predict the existence of self-similar structures of the finite-size scaling functions of O(n)-symmetric systems with planar anisotropies and periodic boundary conditions both in the critical region for n≥1 as well as in the Goldstone-dominated low-temperature region for n≥2.

Bridging transitions and capillary forces for colloids in a slit

Capillary bridges can form between colloids immersed in a two-phase fluid, e.g., in a binary liquid mixture, if the surface of the colloids prefers the species other than the one favored in the bulk liquid. Here, we study the formation of liquid bridges induced by confining colloids to a slit, with the slit walls having a preference opposite to the one of the colloid surface. Using mean field theory, we show that there is a line of first-order phase transitions between the bridge and the no-bridge states, which ends at a critical point. By decreasing the slit width, this critical point is shifted toward smaller separations between the colloids. However, at very small separations and far from criticality, we observe only a minor influence of the slit width on the location of the transition. Monte Carlo simulations of the Ising model, which mimics incompressible binary liquid mixtures, confirm the occurrence of the bridging transitions, as manifested by the appearance of "spinodal" regions where both bridge and no-bridge configurations are stable or metastable. Interestingly, we find that there is no such spinodal region in the case of small colloids, but we observe a sharpening of the transition when the colloid size increases. In addition, we demonstrate that the capillary force acting between the colloids can depend sensitively on the slit width and varies drastically with temperature, thus achieving strengths orders of magnitude higher than at criticality of the fluid.

Lattice Boltzmann simulations of nonequilibrium fluctuations in a nonideal binary mixture

In recent years the lattice Boltzmann (LB) methodology has been fruitfully extended to include the effects of thermal fluctuations. So far, all studied cases pertain to equilibrium fluctuations, i.e., fluctuations with respect to an equilibrium background state. In this paper we take a step further and present results of fluctuating LB simulations of a binary mixture confined between two parallel walls in the presence of a constant concentration gradient in the wall-to-wall direction. This is a paradigmatic setup for the study of nonequilibrium (NE) fluctuations, i.e., fluctuations with respect to a nonequilibrium state. We analyze the dependence of the structure factors for the hydrodynamical fields on the wave vector q in both the directions parallel and perpendicular to the walls, highlighting the long-range (∼|q|^{-4}) nature of correlations in the NE framework. Results at the small scales (high wave numbers) quantitatively agree with the predictions of fluctuating hydrodynamics without fitting parameters. At larger scales (low wave numbers), however, results show finite-size effects induced by confinement and call for further studies aimed at controlling boundary conditions in the fluctuating LB framework as well as compressibility effects. Moreover, in the presence of a nonideal equation of state of the mixture, we also observe that the (spatially homogeneous) average pressure changes, due to a genuinely new contribution triggered by NE fluctuations. These NE pressure effects are studied at changing the system size and the concentration gradient. Taken all together, we argue that the results of this article are useful and instrumental to boost the applicability of the fluctuating LB methodology in the framework of NE fluctuations, possibly in conjunction with experiments.

Crossover from low-temperature to high-temperature fluctuations: Universal and nonuniversal Casimir forces of isotropic and anisotropic systems

We study the crossover from low-temperature to high-temperature fluctuations including Goldstone-dominated and critical fluctuations in confined isotropic and weakly anisotropic O(n)-symmetric systems on the basis of a finite-size renormalization-group approach at fixed dimension d introduced previously [V. Dohm, Phys. Rev. Lett. 110, 107207 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.107207]. Our theory is formulated within the φ^{4} lattice model in a d-dimensional block geometry with periodic boundary conditions. We calculate the finite-size scaling functions F^{ex} and X of the excess free-energy density and the thermodynamic Casimir force, respectively, for 1≤n≤∞, 2<d<4. Exact results are derived for n→∞. Applications are given for L_{∥}^{d-1}×L slab geometry with an aspect ratio ρ=L/L_{∥}>0 and for film geometry (ρ=0). Good overall agreement is found with Monte Carlo (MC) data for isotropic spin models with n=1,2,3. For ρ=0, the low-temperature limits of F^{ex} and X vanish for n=1, whereas they are finite for n≥2. For ρ>0 and n=1, we find a finite low-temperature limit of F^{ex}, which deviates from that of the Ising model. We attribute this deviation to the nonuniversal difference between the φ^{4} model with continuous variables and the Ising model with discrete variables. For n≥2 and ρ>0, a logarithmic divergence of F^{ex} in the low-temperature limit is predicted, in excellent agreement with MC data. For 2≤n≤∞ and ρ<ρ_{0}=0.8567 the Goldstone modes generate a negative low-temperature Casimir force that vanishes for ρ=ρ_{0} and becomes positive for ρ>ρ_{0}. For anisotropic systems a unified hypothesis of multiparameter universality is introduced for both bulk and confined systems. The dependence of their scaling functions on d(d+1)/2-1 microscopic anisotropy parameters implies a substantial reduction of the predictive power of the theory for anisotropic systems as compared to isotropic systems. An exact representation is derived for the nonuniversal large-distance behavior of the bulk correlation function of anisotropic systems and quantitative predictions are made. The validity of multiparameter universality is proven analytically for the d=2,n=1 universality class. A nonuniversal anisotropy-dependent minimum of the Casimir force scaling function X is found. Both the sign and magnitude of X and the shift of the film critical temperature are affected by the lattice anisotropy.

Underscreening in ionic liquids: a first principles analysis

An attempt is made to understand the underscreening effect, observed in concentrated electrolyte solutions or melts, on the basis of simple, admittedly crude models involving charged (for the ions) and neutral (for the solvent molecules) hard spheres. The thermodynamic and structural properties of these 'primitive' and 'semi-primitive' models are calculated within mean spherical approximation, which provides the basic input required to determine the partial density response functions. The screening length [Formula: see text], which is unambiguously defined in terms of the wave-number-dependent response functions, exhibits a cross-over from a low density, Debye-like regime, to a regime where [Formula: see text] increases with density beyond a critical density at which the Debye length [Formula: see text] becomes comparable to the ion diameter. In this high density regime the ratio [Formula: see text] increases according to a power law, in qualitative agreement with experimental measurements, albeit at a much slower rate.

Casimir amplitudes in topological quantum phase transitions

Topological phase transitions constitute a new class of quantum critical phenomena. They cannot be described within the usual framework of the Landau theory since, in general, the different phases cannot be distinguished by an order parameter, neither can they be related to different symmetries. In most cases, however, one can identify a diverging length at these topological transitions. This allows us to describe them using a scaling approach and to introduce a set of critical exponents that characterize their universality class. Here we consider some relevant models of quantum topological transitions associated with well-defined critical exponents that are related by a quantum hyperscaling relation. We extend to these models a finite-size scaling approach based on techniques for calculating the Casimir force in electromagnetism. This procedure allows us to obtain universal Casimir amplitudes at their quantum critical points. Our results verify the validity of finite-size scaling in these systems and confirm the values of the critical exponents obtained previously.

Sign change in the net force in sphere-plate and sphere-sphere systems immersed in nonpolar critical fluid due to the interplay between the critical Casimir and dispersion van der Waals forces

We study systems in which both long-ranged van der Waals and critical Casimir interactions are present. The latter arise as an effective force between bodies when immersed in a near-critical medium, say a nonpolar one-component fluid or a binary liquid mixture. They are due to the fact that the presence of the bodies modifies the order parameter profile of the medium between them as well as the spectrum of its allowed fluctuations. We study the interplay between these forces, as well as the total force (TF) between a spherical colloid particle and a thick planar slab and between two spherical colloid particles. We do that using general scaling arguments and mean-field-type calculations utilizing the Derjaguin and the surface integration approaches. They both are based on data of the forces between two parallel slabs separated at a distance L from each other, confining the fluctuating fluid medium characterized by its temperature T and chemical potential μ. The surfaces of the colloid particles and the slab are coated by thin layers exerting strong preference to the liquid phase of the fluid, or one of the components of the mixture, modeled by strong adsorbing local surface potentials, ensuring the so-called (+,+) boundary conditions. On the other hand, the core region of the slab and the particles influence the fluid by long-ranged competing dispersion potentials. We demonstrate that for a suitable set of colloids-fluid, slab-fluid, and fluid-fluid coupling parameters, the competition between the effects due to the coatings and the core regions of the objects involved result, when one changes T, μ, or L, in sign change of the Casimir force (CF) and the TF acting between the colloid and the slab, as well as between the colloids. This can be used for governing the behavior of objects, say colloidal particles, at small distances, say in colloid suspensions for preventing flocculation. It can also provide a strategy for solving problems with handling, feeding, trapping, and fixing of microparts in nanotechnology. Data for specific substances in support of the experimental feasibility of the theoretically predicted behavior of the CF and TF have been also presented.

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.

Statistical field theory with constraints: Application to critical Casimir forces in the canonical ensemble

The effect of imposing a constraint on a fluctuating scalar order parameter field in a system of finite volume is studied within statistical field theory. The canonical ensemble, corresponding to a fixed total integrated order parameter (e.g., the total number of particles), is obtained as a special case of the theory. A perturbative expansion is developed which allows one to systematically determine the constraint-induced finite-volume corrections to the free energy and to correlation functions. In particular, we focus on the Landau-Ginzburg model in a film geometry (i.e., in a rectangular parallelepiped with a small aspect ratio) with periodic, Dirichlet, or Neumann boundary conditions in the transverse direction and periodic boundary conditions in the remaining, lateral directions. Within the expansion in terms of ε=4-d, where d is the spatial dimension of the bulk, the finite-size contribution to the free energy of the confined system and the associated critical Casimir force are calculated to leading order in ε and are compared to the corresponding expressions for an unconstrained (grand canonical) system. The constraint restricts the fluctuations within the system and it accordingly modifies the residual finite-size free energy. The resulting critical Casimir force is shown to depend on whether it is defined by assuming a fixed transverse area or a fixed total volume. In the former case, the constraint is typically found to significantly enhance the attractive character of the force as compared to the grand canonical case. In contrast to the grand canonical Casimir force, which, for supercritical temperatures, vanishes in the limit of thick films, in the canonical case with fixed transverse area the critical Casimir force attains for thick films a negative value for all boundary conditions studied here. Typically, the dependence of the critical Casimir force both on the temperaturelike and on the fieldlike scaling variables is different in the two ensembles.

Fluctuation-induced forces in confined ideal and imperfect Bose gases

Fluctuation-induced ("Casimir") forces caused by thermal and quantum fluctuations are investigated for ideal and imperfect Bose gases confined to d-dimensional films of size ∞^{d-1}×D under periodic (P), antiperiodic (A), Dirichlet-Dirichlet (DD), Neumann-Neumann (NN), and Robin (R) boundary conditions (BCs). The full scaling functions Υ_{d}^{BC}(x_{λ}=D/λ_{th},x_{ξ}=D/ξ) of the residual reduced grand potential per area φ_{res,d}^{BC}(T,μ,D)=D^{-(d-1)}Υ_{d}^{BC}(x_{λ},x_{ξ}) are determined for the ideal gas case with these BCs, where λ_{th} and ξ are the thermal de Broglie wavelength and the bulk correlation length, respectively. The associated limiting scaling functions Θ_{d}^{BC}(x_{ξ})≡Υ_{d}^{BC}(∞,x_{ξ}) describing the critical behavior at the bulk condensation transition are shown to agree with those previously determined from a massive free O(2) theory for BC=P,A,DD,DN,NN. For d=3, they are expressed in closed analytical form in terms of polylogarithms. The analogous scaling functions Υ_{d}^{BC}(x_{λ},x_{ξ},c_{1}D,c_{2}D) and Θ_{d}^{R}(x_{ξ},c_{1}D,c_{2}D) under the RBCs (∂_{z}-c_{1})ϕ|_{z=0}=(∂_{z}+c_{2})ϕ|_{z=D}=0 with c_{1}≥0 and c_{2}≥0 are also determined. The corresponding scaling functions Υ_{∞,d}^{P}(x_{λ},x_{ξ}) and Θ_{∞,d}^{P}(x_{ξ}) for the imperfect Bose gas are shown to agree with those of the interacting Bose gas with n internal degrees of freedom in the limit n→∞. Hence, for d=3, Θ_{∞,d}^{P}(x_{ξ}) is known exactly in closed analytic form. To account for the breakdown of translation invariance in the direction perpendicular to the boundary planes implied by free BCs such as DDBCs, a modified imperfect Bose gas model is introduced that corresponds to the limit n→∞ of this interacting Bose gas. Numerically and analytically exact results for the scaling function Θ_{∞,3}^{DD}(x_{ξ}) therefore follow from those of the O(2n)ϕ^{4} model for n→∞.

Manipulation and amplification of the Casimir force through surface fields using helicity

We present both exact and numerical results for the behavior of the Casimir force in O(n) systems with a finite extension L in one direction when the system is subjected to surface fields that induce helicity in the order parameter. We show that for such systems, the Casimir force in certain temperature ranges is of the order of L^{-2}, both above and below the critical temperature, T_{c}, of the bulk system. An example of such a system would be one with chemically modulated bounding surfaces, in which the modulation couples directly to the system's order parameter. We demonstrate that, depending on the parameters of the system, the Casimir force can be either attractive or repulsive. The exact calculations presented are for the one-dimensional XY and Heisenberg models under twisted boundary conditions resulting from finite surface fields that differ in direction by a specified angle, and the three-dimensional Gaussian model with surface fields in the form of plane waves that are shifted in phase with respect to each other. Additionally, we present exact and numerical results for the mean-field version of the three-dimensional O(2) model with finite surface fields on the bounding surfaces. We find that all significant results are consistent with the expectations of finite-size scaling.

Tricritical Casimir forces and order parameter profiles in wetting films of ^{3}He-^{4}He mixtures

Tricritical Casimir forces in ^{3}He-^{4}He wetting films are studied, within mean field theory, in terms of a suitable lattice gas model for binary liquid mixtures with short-ranged surface fields. The proposed model takes into account the continuous rotational symmetry O(2) of the superfluid degrees of freedom associated with ^{4}He and it allows, inter alia, for the occurrence of a vapor phase. As a result, the model facilitates the formation of wetting films, which provides a strengthened theoretical framework to describe available experimental data for tricritical Casimir forces acting in ^{3}He-^{4}He wetting films.

Work, work fluctuations, and the work distribution in a thermal nonequilibrium steady state

Long-ranged correlations generically exist in nonequilibrium fluid systems. In the case of a nonequilibrium steady state caused by a temperature gradient, the correlations are especially long-ranged and strong. The anomalous light scattering predicted to exist in these systems is well-confirmed by numerous experiments. Recently, the Casimir force or pressure due to these fluctuations or correlations has been discussed in great detail. In this paper, the notion of a Casimir work is introduced, and an alternative way to measure the nonequilibrium Casimir force is suggested. In particular, the nonequilibrium Casimir force is related to nonequilibrium heat, and not, as in equilibrium, to a volume derivative of an average energy. The nonequilibrium work fluctuations are determined and shown to be very anomalous compared to equilibrium work fluctuations. The nonequilibrium work distribution is also computed, and it is contrasted with work distributions in systems with short-range correlations. Again, there is a striking difference in the two cases. Formal theories of work and work distributions in nonequilibrium steady states are not explicit enough to illustrate any of these interesting features.

Casimir interaction of rodlike particles in a two-dimensional critical system

We consider the fluctuation-induced interaction of two thin, rodlike particles, or "needles," immersed in a two-dimensional critical fluid of Ising symmetry right at the critical point. Conformally mapping the plane containing the needles onto a simpler geometry in which the stress tensor is known, we analyze the force and torque between needles of arbitrary length, separation, and orientation. For infinite and semi-infinite needles we utilize the mapping of the plane bounded by the needles onto the half plane, and for two needles of finite length we use the mapping onto an annulus. For semi-infinite and infinite needles the force is expressed in terms of elementary functions, and we also obtain analytical results for the force and torque between needles of finite length with separation much greater than their length. Evaluating formulas in our approach numerically for several needle geometries and surface universality classes, we study the full crossover from small to large values of the separation to length ratio. In these two limits the numerical results agree with results for infinitely long needles and with predictions of the small-particle operator expansion, respectively.

Direct calculation of the critical Casimir force in a binary fluid

We show that critical Casimir effects can be accessed through direct simulation of a model binary fluid passing through the demixing transition. We work in the semi-grand-canonical ensemble, in slab geometry, in which the Casimir force appears as the excess of the generalized pressure, P_{⊥}-nμ. The excesses of the perpendicular pressure, P_{⊥}, and of nμ, are individually of much larger amplitude. A critical pressure anisotropy is observed between forces parallel and perpendicular to the confinement direction, which collapses onto a universal scaling function closely related to that of the critical Casimir force.

Failure of Local Thermal Equilibrium in Quantum Friction

Recent progress in manipulating atomic and condensed matter systems has instigated a surge of interest in nonequilibrium physics, including many-body dynamics of trapped ultracold atoms and ions, near-field radiative heat transfer, and quantum friction. Under most circumstances the complexity of such nonequilibrium systems requires a number of approximations to make theoretical descriptions tractable. In particular, it is often assumed that spatially separated components of a system thermalize with their immediate surroundings, although the global state of the system is out of equilibrium. This powerful assumption reduces the complexity of nonequilibrium systems to the local application of well-founded equilibrium concepts. While this technique appears to be consistent for the description of some phenomena, we show that it fails for quantum friction by underestimating by approximately 80% the magnitude of the drag force. Our results show that the correlations among the components of driven, but steady-state, quantum systems invalidate the assumption of local thermal equilibrium, calling for a critical reexamination of this approach for describing the physics of nonequilibrium systems.

Fluctuation-induced forces between rings threaded around a polymer chain under tension

We characterize the fluctuation properties of a polymer chain under external tension and the fluctuation-induced forces between two ring molecules threaded around the chain. The problem is relevant in the context of fluctuation-induced forces in soft-matter systems, features of liquid interfaces, and to describe the properties of polyrotaxanes and slide-ring materials. We perform molecular-dynamics simulations of the Kremer-Grest bead-spring model for the polymer and a simple ring-molecule model in the canonical ensemble. We study transverse fluctuations of the stretched chain as a function of chain stretching and in the presence of ring-shaped threaded molecules. The fluctuation spectra of the chains are analyzed in equilibrium at constant temperature, and the differences in the presence of two-ring molecules are compared. For the rings located at fixed distances, we find an attractive fluctuation-induced force between the rings, proportional to the temperature and decaying with the ring distance. We characterize this force as a function of ring distance, chain stretching, and ring radius, and we measure the differences between the free chain spectrum and the fluctuations of the chain constrained by the rings. We also compare the dependence and range of the force found in the simulations with theoretical models coming from different fields.

Critical adsorption and critical Casimir forces in the canonical ensemble

Critical properties of a liquid film between two planar walls are investigated in the canonical ensemble, within which the total number of fluid particles, rather than their chemical potential, is kept constant. The effect of this constraint is analyzed within mean-field theory (MFT) based on a Ginzburg-Landau free-energy functional as well as via Monte Carlo simulations of the three-dimensional Ising model with fixed total magnetization. Within MFT and for finite adsorption strengths at the walls, the thermodynamic properties of the film in the canonical ensemble can be mapped exactly onto a grand canonical ensemble in which the corresponding chemical potential plays the role of the Lagrange multiplier associated with the constraint. However, due to a nonintegrable divergence of the mean-field order parameter profile near a wall, the limit of infinitely strong adsorption turns out to be not well-defined within MFT, because it would necessarily violate the constraint. The critical Casimir force (CCF) acting on the two planar walls of the film is generally found to behave differently in the canonical and grand canonical ensembles. For instance, the canonical CCF in the presence of equal preferential adsorption at the two walls is found to have the opposite sign and a slower decay behavior as a function of the film thickness compared to its grand canonical counterpart. We derive the stress tensor in the canonical ensemble and find that it has the same expression as in the grand canonical case, but with the chemical potential playing the role of the Lagrange multiplier associated with the constraint. The different behavior of the CCF in the two ensembles is rationalized within MFT by showing that, for a prescribed value of the thermodynamic control parameter of the film, i.e., density or chemical potential, the film pressures are identical in the two ensembles, while the corresponding bulk pressures are not.

Finite-size scaling of the magnetization probability density for the critical Ising model in slab geometry

The magnetization probability density in d  =  2 and 3 dimensional Ising models in slab geometry of volume [Formula: see text] is computed through Monte-Carlo simulation at the critical temperature and zero magnetic field. The finite-size scaling of this distribution and its dependence on the system aspect-ratio [Formula: see text]and boundary conditions are discussed. In the limiting case [Formula: see text] of a macroscopically large slab ([Formula: see text]) the distribution is found to scale as a Gaussian function for all tested system sizes and boundary conditions.

Nonadditivity of critical Casimir forces

In soft condensed matter physics, effective interactions often emerge due to the spatial confinement of fluctuating fields. For instance, microscopic particles dissolved in a binary liquid mixture are subject to critical Casimir forces whenever their surfaces confine the thermal fluctuations of the order parameter of the solvent close to its critical demixing point. These forces are theoretically predicted to be nonadditive on the scale set by the bulk correlation length of the fluctuations. Here we provide direct experimental evidence of this fact by reporting the measurement of the associated many-body forces. We consider three colloidal particles in optical traps and observe that the critical Casimir force exerted on one of them by the other two differs from the sum of the forces they exert separately. This three-body effect depends sensitively on the distance from the critical point and on the chemical functionalisation of the colloid surfaces.

The critical Casimir force, rising from fluctuating field confined between surfaces, is predicted to be nonadditive, but there is no experimental verification to date. Here the authors provide data support by quantifying the forces between three interacting colloidal particles using holographic traps.

Physical origin of nonequilibrium fluctuation-induced forces in fluids

Long-range thermal fluctuations appear in fluids in nonequilibrium states leading to fluctuation-induced Casimir-like forces. Two distinct mechanisms have been identified for the origin of the long-range nonequilibrium fluctuations in fluids subjected to a temperature or concentration gradient. One is a coupling between the heat or mass-diffusion mode with a viscous mode in fluids subjected to a temperature or concentration gradient. Another one is the spatial inhomogeneity of thermal noise in the presence of a gradient. We show that in fluids fluctuation-induced forces arising from mode coupling are several orders of magnitude larger than those from inhomogeneous noise.

Critical and near-critical phase behavior and interplay between the thermodynamic Casimir and van der Waals forces in a confined nonpolar fluid medium with competing surface and substrate potentials

We study, using general scaling arguments and mean-field type calculations, the behavior of the critical Casimir force and its interplay with the van der Waals force acting between two parallel slabs separated at a distance L from each other, confining some fluctuating fluid medium, say a nonpolar one-component fluid or a binary liquid mixture. The surfaces of the slabs are coated by thin layers exerting strong preference to the liquid phase of the fluid, or one of the components of the mixture, modeled by strong adsorbing local surface potentials ensuring the so-called (+,+) boundary conditions. The slabs, on the other hand, influence the fluid by long-range competing dispersion potentials, which represent irrelevant interactions in renormalization-group sense. Under such conditions, one usually expects attractive Casimir force governed by universal scaling function, pertinent to the extraordinary surface universality class of Ising type systems, to which the dispersion potentials provide only corrections to scaling. We demonstrate, however, that below a given threshold thickness of the system L(crit) for a suitable set of slabs-fluid and fluid-fluid coupling parameters the competition between the effects due to the coatings and the slabs can result in sign change of the Casimir force acting between the surfaces confining the fluid when one changes the temperature T, the chemical potential of the fluid μ, or L. The last implies that by choosing specific materials for the slabs, coatings, and the fluid for L≲L(crit) one can realize repulsive Casimir force with nonuniversal behavior which, upon increasing L, gradually turns into an attractive one described by a universal scaling function, depending only on the relevant scaling fields related to the temperature and the excess chemical potential, for L≫L(crit). We present arguments and relevant data for specific substances in support of the experimental feasibility of the predicted behavior of the force. It can be of interest, e.g., for designing nanodevices and for governing behavior of objects, say colloidal particles, at small distances. We formulate the corresponding criterion for determination of L(crit). The universality is regained for L≫L(crit). We also show that for systems with L≲L(crit), the capillary condensation phase diagram suffers modifications which one does not observe in systems with purely short-ranged interactions.

Line contribution to the critical Casimir force between a homogeneous and a chemically stepped surface

Recent experimental realizations of the critical Casimir effect have been implemented by monitoring colloidal particles immersed in a binary liquid mixture near demixing and exposed to a chemically structured substrate. In particular, critical Casimir forces have been measured for surfaces consisting of stripes with periodically alternating adsorption preferences, forming chemical steps between them. Motivated by these experiments, we analyze the contribution of such chemical steps to the critical Casimir force for the film geometry and within the Ising universality class. By means of Monte Carlo simulations, mean-field theory and finite-size scaling analysis we determine the universal scaling function associated with the contribution to the critical Casimir force due to individual, isolated chemical steps facing a surface with homogeneous adsorption preference or with Dirichlet boundary condition. In line with previous findings, these results allow one to compute the critical Casimir force for the film geometry and in the presence of arbitrarily shaped, but wide stripes. In this latter limit the force decomposes into a sum of the contributions due to the two homogeneous parts of the surface and due to the chemical steps between the stripes. We assess this decomposition by comparing the resulting sum with actual simulation data for the critical Casimir force in the presence of a chemically striped substrate.

Critical Casimir forces between planar and crenellated surfaces

We study critical Casimir forces between planar walls and geometrically structured substrates within mean-field theory. As substrate structures, crenellated surfaces consisting of periodic arrays of rectangular crenels and merlons are considered. Within the widely used proximity force approximation, both the top surfaces of the merlons and the bottom surfaces of the crenels contribute to the critical Casimir force. However, for such systems the full, numerically determined critical Casimir forces deviate significantly from the pairwise addition formalism underlying the proximity force approximation. A first-order correction to the proximity force approximation is presented in terms of a step contribution arising from the critical Casimir interaction between a planar substrate and the right-angled steps of the merlons consisting of their upper and lower edges as well as their sidewalls.

Inverse-scattering-theory approach to the exact n→∞ solutions of O(n) ϕ⁴ models on films and semi-infinite systems bounded by free surfaces

The O(n) ϕ(4) model on a strip bounded by a pair of planar free surfaces at separation L can be solved exactly in the large-n limit in terms of the eigenvalues and eigenfunctions of a self-consistent one-dimensional Schrödinger equation. The scaling limit of a continuum version of this model is considered. It is shown that the self-consistent potential can be eliminated in favor of scattering data by means of appropriately extended methods of inverse scattering theory. The scattering data (Jost function) associated with the self-consistent potential are determined for the L=∞ semi-infinite case in the scaling regime for all values of the temperature scaling field t=(T-T(c))/T(c) above and below the bulk critical temperature T(c). These results are used in conjunction with semiclassical and boundary-operator expansions and a trace formula to derive exact analytical results for a number of quantities such as two-point functions, universal amplitudes of two excess surface quantities, the universal amplitude difference associated with the thermal singularity of the surface free energy, and potential coefficients. The asymptotic behaviors of the scaled eigenenergies and eigenfunctions of the self-consistent Schrödinger equation as function of x=t(L/ξ(+))(1/ν) are determined for x→-∞. In addition, the asymptotic x→-∞ forms of the universal finite-size scaling functions Θ(x) and ϑ(x) of the residual free energy and the Casimir force are computed exactly to order 1/x, including their x(-1)ln|x| anomalies.

Three-body critical Casimir forces

Within mean-field theory we calculate universal scaling functions associated with critical Casimir forces for a system consisting of three parallel cylindrical colloids immersed in a near-critical binary liquid mixture. For several geometrical arrangements and boundary conditions at the surfaces of the colloids we study the force between two colloidal particles in the direction normal to their axes, analyzing the influence of the presence of a third particle on that force. Upon changing temperature or the relative positions of the particles we observe interesting features such as a change of sign of this force caused by the presence of the third particle. We determine the three-body component of the forces acting on one of the colloids by subtracting the pairwise forces from the total force. The three-body contribution to the total critical Casimir force turns out to be more pronounced for small surface-to-surface distances between the colloids as well as for temperatures close to criticality. Moreover, we compare our results with similar ones for other physical systems such as three atoms interacting via van der Waals forces.

Spin models in three dimensions: Adaptive lattice spacing

Aiming at the study of critical phenomena in the presence of boundaries with a nontrivial shape we discuss how lattices with an adaptive lattice spacing can be implemented. Since the parameters of the Hamiltonian transform nontrivially under changes of the length scale, adapting the lattice spacing is much more difficult than in the case of the numerical solution of partial differential equations, where this method is common practice. Here we shall focus on the universality class of the three-dimensional Ising model. Our starting point is the improved Blume-Capel model on the simple cubic lattice. In our approach, the system is composed of sectors with lattice spacing a,2a,4a,.... We work out how parts of the lattice with lattice spacing a and 2a, respectively, can be coupled in a consistent way. Here, we restrict ourself to the case where the boundary between the sectors is perpendicular to one of the lattice axes. Based on the theory of defect planes one expects that it is sufficient to tune the coupling between these two regions. To this end we perform a finite-size scaling study. However, first numerical results show that slowly decaying corrections remain. It turns out that these corrections can be removed by adjusting the strength of the couplings within the boundary layers. As a benchmark, we simulate films with strongly symmetry breaking boundary conditions. We determine the magnetization profile and the thermodynamic Casimir force. For our largest thickness L0=64.5, we find that results obtained for the homogeneous system are nicely reproduced.

Critical Casimir force in the presence of random local adsorption preference

We study the critical Casimir force for a film geometry in the Ising universality class. We employ a homogeneous adsorption preference on one of the confining surfaces, while the opposing surface exhibits quenched random disorder, leading to a random local adsorption preference. Disorder is characterized by a parameter p, which measures, on average, the portion of the surface that prefers one component, so that p=0,1 correspond to homogeneous adsorption preference. By means of Monte Carlo simulations of an improved Hamiltonian and finite-size scaling analysis, we determine the critical Casimir force. We show that by tuning the disorder parameter p, the system exhibits a crossover between an attractive and a repulsive force. At p=1/2, disorder allows to effectively realize Dirichlet boundary conditions, which are generically not accessible in classical fluids. Our results are relevant for the experimental realizations of the critical Casimir force in binary liquid mixtures.

Crossover aspects in Ising strips under the influence of variable surface fields and a grain boundary

I use an exact variational formulation of Mikheev and Fisher to study the critical Ising strip with a grain boundary and confining surfaces characterized by arbitrary and different surface magnetic fields. Energy density profiles that serve as order parameters of the system within the used method show strong nonmonotonous behavior in the vicinity of confining surfaces. I consider short-distance expansion of energy density profiles. New universal amplitudes associated with distant-wall corrections exhibit nontrivial crossover behaviors from positive to negative values as the field's variables are continuously varied. Casimir amplitudes calculated in a self-contained manner are characterized by complex manifolds which may comprise two disconnected positive wings separated by an area of negative values. I define and determine the generalized de Gennes-Fisher amplitude, which strongly suggests that a stress tensor is present within one of the distant-wall corrections. This is an unanticipated result given that a similar discovery for standard extraordinary (E) and ordinary (O) surface universality classes was based on the conformal invariance symmetry, which is broken under the present boundary conditions. A grain boundary essentially influences both energy density profiles and Casimir amplitudes, besides surface fields with a number of accompanying features that are examined in detail. We present closed analytic forms of energy density profiles for standard (N) and (O) BCs: ɛNN(x,g),ɛOO(x,g),ɛNO(x,g) [(N) is the normal surface universality class, g is the grain boundary's strength, x is the relative distance x:=z/L with L as a film width], thus enabling us to deduce their important symmetry properties and to have a detailed insight into their general behaviors.

Pronounced minimum of the thermodynamic Casimir forces of O(n) symmetric film systems: analytic theory

Thermodynamic Casimir forces of film systems in the O(n) universality classes with Dirichlet boundary conditions are studied below bulk criticality. Substantial progress is achieved in resolving the long-standing problem of describing analytically the pronounced minimum of the scaling function observed experimentally in ^{4}He films (n=2) by Garcia and Chan [Phys. Rev. Lett. 83, 1187 (1999)] and in Monte Carlo simulations for the three-dimensional Ising model (n=1) by O. Vasilyev et al. [Europhys. Lett. 80, 60009 (2007)]. Our finite-size renormalization-group approach describes the film systems as the limit of finite-slab systems with vanishing aspect ratio. This yields excellent agreement with the depth and the position of the minimum for n=1 and semiquantitative agreement with the minimum for n=2. Our theory also predicts a pronounced minimum for the n=3 Heisenberg universality class.

Casimir effect in active matter systems

We numerically examine run-and-tumble active matter particles in Casimir geometries composed of two finite parallel walls. We find that there is an attractive force between the two walls of a magnitude that increases with increasing run length. The attraction exhibits an unusual exponential dependence on the wall separation, and it arises due to a depletion of swimmers in the region between the walls by a combination of the motion of the particles along the walls and a geometric shadowing effect. This attraction is robust as long as the wall length is comparable to or smaller than the swimmer run length, and is only slightly reduced by the inclusion of steric interactions between swimmers. We also examine other geometries and find regimes in which there is a crossover from attraction to repulsion between the walls as a function of wall separation and wall length.

Critical Casimir interactions between spherical particles in the presence of bulk ordering fields

The spatial suppression of order parameter fluctuations in a critical media produces critical Casimir forces acting on confining surfaces. This scenario is realized in a critical binary mixture near the demixing transition point that corresponds to the second-order phase transition of the Ising universality class. Due to these critical interactions similar colloids, immersed in a critical binary mixture near the consolute point, exhibit attraction. The numerical method for computation of the interaction potential between two spherical particles using Monte Carlo simulations for the Ising model is proposed. This method is based on the integration of the local magnetization over the applied local magnetic field. For the stronger interaction the concentration of the component of the mixture that does not wet colloidal particles should be larger than the critical concentration. The strongest amplitude of the interactions is observed below the critical point.