Critical Casimir Forces Research Articles

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.

Ensemble dependence of critical Casimir forces in films with Dirichlet boundary conditions.

In a recent study [Phys. Rev. E 94, 022103 (2016)2470-004510.1103/PhysRevE.94.022103] it has been shown that, for a fluid film subject to critical adsorption, the resulting critical Casimir force (CCF) may significantly depend on the thermodynamic ensemble. Here we extend that study by considering fluid films within the so-called ordinary surface universality class. We focus on mean-field theory, within which the order parameter (OP) profile satisfies Dirichlet boundary conditions and produces a nontrivial CCF in the presence of external bulk fields or, respectively, a nonzero total order parameter within the film. Additionally, we study the influence of fluctuations by means of Monte Carlo simulations of the three-dimensional Ising model. We show that, in the canonical ensemble, i.e., when fixing the so-called total mass within the film, the CCF is repulsive for large absolute values of the total OP, instead of attractive as in the grand canonical ensemble. Based on the Landau-Ginzburg free energy, we furthermore obtain analytic expressions for the order parameter profiles and analyze the relation between the total mass in the film and the external bulk field.

Controlling Superstructure-Property Relationships via Critical Casimir Assembly of Quantum Dots.

The assembly of colloidal quantum dots (QDs) into dense superstructures holds great promise for the development of novel optoelectronic devices. Several assembly techniques have been explored; however, achieving direct and precise control over the interparticle potential that controls the assembly has proven to be challenging. Here, we exploit the application of critical Casimir forces to drive the growth of QDs into superstructures. We show that the exquisite temperature-dependence of the critical Casimir potential offers new opportunities to control the assembly process and morphology of the resulting QD superstructures. The direct assembly control allows us to elucidate the relation between structural, optical, and conductive properties of the critical Casimir-grown QD superstructures. We find that the choice of the temperature setting the interparticle potential plays a central role in maximizing charge percolation across QD thin-films. These results open up new directions for controlling the assembly of nanostructures and their optoelectronic properties.

Controlling the dynamics of colloidal particles by critical Casimir forces.

Critical Casimir forces can play an important role for applications in nano-science and nano-technology, owing to their piconewton strength, nanometric action range, fine tunability as a function of temperature, and exquisite dependence on the surface properties of the involved objects. Here, we investigate the effects of critical Casimir forces on the free dynamics of a pair of colloidal particles dispersed in the bulk of a near-critical binary liquid solvent, using blinking optical tweezers. In particular, we measure the time evolution of the distance between the two colloids to determine their relative diffusion and drift velocity. Furthermore, we show how critical Casimir forces change the dynamic properties of this two-colloid system by studying the temperature dependence of the distribution of the so-called first-passage time, i.e., of the time necessary for the particles to reach for the first time a certain separation, starting from an initially assigned one. These data are in good agreement with theoretical results obtained from Monte Carlo simulations and Langevin dynamics.

Measurement of the Curvature-Dependent Surface Tension in Nucleating Colloidal Liquids.

The curvature dependence of the surface tension is central to the nucleation of liquids, but remains difficult to access experimentally and predict theoretically. This curvature dependence arises from the curvature-dependent molecular structure, which, for small nuclei, can deviate significantly from that of the planar liquid interface. Simulations and density functional theory have been used to predict this curvature dependence, however with contradicting results. Here, we provide the first direct measurement of the curvature-dependent surface tension in nucleating colloidal liquids. We employ critical Casimir forces to finely adjust colloidal particle interactions and induce liquid nucleation, and image individual nuclei at the particle scale to measure their curvature-dependent surface tension directly from thermally excited surface distortions. Using continuum models, we elucidate the interplay between nucleus structure, particle pair potential, and surface tension. Our results reveal a 20% lower surface tension for nuclei of critical size compared to bulk liquids, leading to 3 orders of magnitude higher nucleation rates, thus highlighting the importance of surface tension curvature corrections for accurate prediction of nucleation rates.

Collective behavior of colloids due to critical Casimir interactions

If colloidal solute particles are suspended in a solvent close to its critical point, they act as cavities in a fluctuating medium and thereby restrict and modify the fluctuation spectrum in a way which depends on their relative configuration. As a result effective, so-called critical Casimir forces (CCFs) emerge between the colloids. The range and the amplitude of CCFs depend sensitively on the temperature and the composition of the solvent as well as on the boundary conditions of the order parameter of the solvent at the particle surfaces. These remarkable, moreover universal features of the CCFs provide the possibility for an active control over the assembly of colloids. This has triggered a recent surge of experimental and theoretical interest in these phenomena. We present an overview of current research activities in this area. Various experiments demonstrate the occurrence of thermally reversible self-assembly or aggregation or even equilibrium phase transitions of colloids in the mixed phase below the lower consolute points of binary solvents. We discuss the status of the theoretical description of these phenomena, in particular the validity of a description in terms of effective, one-component colloidal systems and the necessity of a full treatment of a ternary solvent-colloid mixture. We suggest perspectives on the directions towards which future research in this field might develop.

Nucleation of weakly attractive aggregates in microgravity

We investigate the incipient stages of the formation of aggregates of weakly attractive particles in microgravity. Using colloidal particles interacting with tunable critical Casimir forces, we drive the system increasingly out of equilibrium starting from near-equilibrium gas-solid coexistence, and follow the early stages of aggregation with near-field scattering. Contrary to measurements on ground that show a rapid exponential growth of aggregates, our microgravity measurements indicate that under pure diffusion-limited conditions, the formation of aggregates is still governed by a nucleation process over the investigated range of moderate attractive strength of a few kBT. By studying in detail the decrease of the critical radius with the increase of the attractive strength, we determine a surface tension of 115 nN/m, or , with σ the particle diameter, of the order of but larger than that for equilibrium colloidal liquids and crystals. These results highlight the importance of nucleation in the formation of non-equilibrium solid aggregates bound by weakly attractive forces.

Surface-induced nonequilibrium dynamics and critical Casimir forces for model B in film geometry

Using analytic and numerical approaches, we study the spatio-temporal evolution of a conserved order parameter of a fluid in film geometry, following an instantaneous quench to the critical temperature $T_c$ as well as to supercritical temperatures. The order parameter dynamics is chosen to be governed by model B within mean field theory and is subject to no-flux boundary conditions as well as to symmetric surface fields at the confining walls. The latter give rise to critical adsorption of the order parameter at both walls and provide the driving force for the non-trivial time evolution of the order parameter. During the dynamics, the order parameter is locally and globally conserved; thus, at thermal equilibrium, the system represents the canonical ensemble. We furthermore consider the dynamics of the nonequilibrium critical Casimir force, which we obtain based on the generalized force exerted by the order parameter field on the confining walls. We identify various asymptotic regimes concerning the time evolution of the order parameter and the critical Casimir force and we provide, within our approach, exact expressions of the corresponding dynamic scaling functions.

Investigating the effective interaction between silica colloidal particles near the critical point of a binary solvent by small angle neutron scattering.

An effective attractive potential can be introduced between colloidal particles dispersed in a binary solvent when the solvent condition approaches its demixing temperatures. Despite the debate of the physical origins of this effective attraction, it is widely termed as the critical Casimir force and is believed to be responsible for the colloidal stability in a wide range of particle concentration at both critical and near-critical solvent concentrations. Here, we study the effective attraction and equilibrium phase transition of charged spherical silica particles in the binary solvent of 2,6-lutidine and water as a function of the particle volume fraction and temperature at the critical solvent concentration. By analyzing our small angle neutron scattering (SANS) data, we found that at a relatively small particle volume fraction, the density fluctuation introduced attraction between silica particles can be satisfactorily explained by the function form commonly used for the critical Casimir interaction. However, at large silica particle volume fractions, an additional long range attraction has to be introduced to satisfactorily fit our SANS data and explain the large shift of the phase transition temperature. Therefore, while at relatively low volume fractions, the solvent introduced attraction may be dominated by the critical Casimir force, the physical mechanism of the effective attraction at large particle volume fractions seems to be different from the critical Casimir force. Furthermore, the range of this long range attraction is consistent with a recently proposed new theory, where the attraction can be introduced by the solvent capillary condensation between particles. We also demonstrate that the reduced second virial coefficient close to the particle phase transition is similar to the values of the binodal transition of the sticky hard sphere system.

DNA Origami as an Experimental Tool to Probe Critical Casimir Forces in Membranes

DNA origami are nanoscale assemblies of DNA designed to fold into two or three dimensional 10-100 nm sized structures designed by the user. Furthermore, it is possible to assemble larger DNA objects by encoding interactions between origami. Past work has taken this approach to assemble nanotracks or nanogrids in solution or have used coupling to homogeneous bilayers to facilitate two dimensional assembly over large areas. In membranes near a miscibility phase transition, theoretical investigations predict that inclusions feel effective potentials based on their preference for a specific local membrane environment. Similar critical Casimir forces have been shown experimentally to facilitate phase transitions in colloidal systems in three dimensions. In this poster, we will present ongoing experimental work aimed at exploring if and how coupling origami to membranes near a miscibility transition facilitates their assembly into extended structures.

On the effects from the simultaneous occurrence of the critical Casimir and dispersion forces between conical colloid particle and a thick plate immersed in nonpolar critical fluid

Here we study the interplay between the van der Waals (vdWF) and critical Casimir forces (CCF), as well as the total force (TF) between a conical colloid particle and a thick planar slab. We do that using general scaling arguments and mean-field type calculations utilizing the so-called “surface integration approach”, a generalization of the well known Derjaguin approximation. Its usage in the present research, requires knowledge on the forces between two parallel slabs, confining in between some fluctuating fluid medium characterized by its temperature T and chemical potential μ. The surfaces of the colloid particle and the slab are assumed coated by thin layers exerting strong preference to the liquid phase of a simple fluid, or one of the components of a binary 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 particle, influence the fluid by long-ranged competing dispersion potentials. We demonstrate that for a suitable set of colloid-fluid, slab-fluid, and fluid-fluid coupling parameters the competition between the effects due to the coatings and the core regions of the objects, result, when one changes T or μ, in sign change of the Casimir force (CF) and the TF acting between the colloid and the slab. Such an effect can provide a strategy for solving problems with handling, feeding, trapping and fixing of microparts in nanotechnology.

Tuning Patchy Bonds Induced by Critical Casimir Forces.

Experimental control of patchy interactions promises new routes for the assembly of complex colloidal structures, but remains challenging. Here, we investigate the role of patch width in the assembly of patchy colloidal particles assembled by critical Casimir forces. The particles are composed of a hydrophobic dumbbell with an equatorial hydrophilic polymer shell, and are synthesized to have well-defined patch-to-shell area ratios. Patch-to-patch binding is achieved in near-critical binary solvents, in which the particle interaction strength and range are controlled by the temperature-dependent solvent correlation length. Upon decreasing the patch-to-shell area ratio, we observe a pronounced change of the bonding morphology towards directed single-bonded configurations, as clearly reflected in the formation of chain-like structures. Computer simulations using an effective critical Casimir pair potential for the patches show that the morphology change results from the geometric exclusion of the increasingly thick hydrophilic particle shells. These results highlight the experimental control of patchy interactions through the engineering of the building blocks on the way towards rationally designed colloidal superstructures.

Liquid bridging of cylindrical colloids in near-critical solvents.

Within mean field theory, we investigate the bridging transition between a pair of parallel cylindrical colloids immersed in a binary liquid mixture as a solvent that is close to its critical consolute point Tc. We determine the universal scaling functions of the effective potential and of the force between the colloids. For a solvent that is at the critical concentration and close to Tc, we find that the critical Casimir force is the dominant interaction at close separations. This agrees very well with the corresponding Derjaguin approximation for the effective interaction between the two cylinders, while capillary forces originating from the extension of the liquid bridge turn out to be more important at large separations. In addition, we are able to infer from the wetting characteristics of the individual colloids the first-order transition of the liquid bridge connecting two colloidal particles to the ruptured state. While specific to cylindrical colloids, the results presented here also provide an outline for identifying critical Casimir forces acting on bridged colloidal particles as such and for analyzing the bridging transition between them.

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.

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 , where 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.

Switching Colloidal Superstructures by Critical Casimir Forces.

Recent breakthroughs in colloidal synthesis promise the bottom-up assembly of superstructures on nano- and micrometer length scales, offering molecular analogues on the colloidal scale. However, a structural control similar to that in supramolecular chemistry remains very challenging. Here, colloidal superstructures are built and controlled using critical Casimir forces on patchy colloidal particles. These solvent-mediated forces offer direct analogues of molecular bonds, allowing patch-to-patch binding with exquisite temperature control of bond strength and stiffness. Particles with two patches are shown to form linear chains undergoing morphological changes with temperature, resembling a polymer collapse under poor-solvent conditions. This reversible temperature switching carries over to particles with higher valency, exhibiting a variety of patch-to-patch bonded structures. Using Monte Carlo simulations, it is shown that the collapse results from the growing interaction range favoring close-packed configurations. These results offer new opportunities for the active control of complex structures at the nano and micrometer scale, paving the way to novel temperature-switchable materials.

From 2D to 3D: Critical Casimir interactions and phase behavior of colloidal hard spheres in a near-critical solvent.

Colloids dispersed in a binary solvent mixture experience long-ranged solvent-mediated interactions (critical Casimir forces) upon approaching the critical demixing point of the solvent mixture. The range of the interaction is set by the bulk correlation length of the solvent mixture, which diverges upon approaching the critical point. This presents a great opportunity to realize the reversible self-assembly of colloids by tuning the proximity to the critical point of the solvent. Here, we develop a rejection-free geometric cluster algorithm to study the full ternary mixture of colloidal hard spheres suspended in an explicit three-dimensional lattice model for the solvent mixture using extensive Monte Carlo simulations. The phase diagram displays stable colloidal gas, liquid, and crystal phases, as well as broad gas-liquid and gas-crystal phase coexistence, and pronounced fractionation of the solvent in the coexisting colloid phases. The topology of the phase diagram in our three-dimensional study shows striking resemblance to that of our previous studies carried out in two dimensions.

Conformal field theory of critical Casimir forces between surfaces with alternating boundary conditions in two dimensions

Systems as diverse as binary mixtures and inclusions in biological membranes, and many more, can be described effectively by interacting spins. When the critical fluctuations in these systems are constrained by boundary conditions, critical Casimir forces (CCF) emerge. Here we analyze CCF between boundaries with alternating boundary conditions in two dimensions, employing conformal field theory (CFT). After presenting the concept of boundary changing operators, we specifically consider two different boundary configurations for a strip of critical Ising spins: (I) alternating equi-sized domains of up and down spins on both sides of the strip, with a possible lateral shift, and (II) alternating domains of up and down spins of different size on one side and homogeneously fixed spins on the other side of the strip. Asymptotic results for the CCF at small and large distances are derived. We introduce a novel modified Szegö formula for determinants of real antisymmetric block Toeplitz matrices to obtain the exact CCF and the corresponding scaling functions at all distances. We demonstrate the existence of a surface renormalization group flow between universal force amplitudes of different magnitude and sign. The Casimir force can vanish at a stable equilibrium position that can be controlled by parameters of the boundary conditions. Lateral Casimir forces assume a universal simple cosine form at large separations.

Energy Transfer between Colloids via Critical Interactions

We report the observation of a temperature-controlled synchronization of two Brownian-particles in a binary mixture close to the critical point of the demixing transition. The two beads are trapped by two optical tweezers whose distance is periodically modulated. We notice that the motion synchronization of the two beads appears when the critical temperature is approached. In contrast, when the fluid is far from its critical temperature, the displacements of the two beads are uncorrelated. Small changes in temperature can radically change the global dynamics of the system. We show that the synchronisation is induced by the critical Casimir forces. Finally, we present the measure of the energy transfers inside the system produced by the critical interaction.