Critical Casimir Forces Research Articles

Nanoalignment by critical Casimir torques

The manipulation of microscopic objects requires precise and controllable forces and torques. Recent advances have led to the use of critical Casimir forces as a powerful tool, which can be finely tuned through the temperature of the environment and the chemical properties of the involved objects. For example, these forces have been used to self-organize ensembles of particles and to counteract stiction caused by Casimir-Liftshitz forces. However, until now, the potential of critical Casimir torques has been largely unexplored. Here, we demonstrate that critical Casimir torques can efficiently control the alignment of microscopic objects on nanopatterned substrates. We show experimentally and corroborate with theoretical calculations and Monte Carlo simulations that circular patterns on a substrate can stabilize the position and orientation of microscopic disks. By making the patterns elliptical, such microdisks can be subject to a torque which flips them upright while simultaneously allowing for more accurate control of the microdisk position. More complex patterns can selectively trap 2D-chiral particles and generate particle motion similar to non-equilibrium Brownian ratchets. These findings provide new opportunities for nanotechnological applications requiring precise positioning and orientation of microscopic objects.

Networks of Limited-Valency Patchy Particles.

Equilibrium gels provide physically attractive counterparts of nonequilibrium gels, allowing statistical understanding and design of the equilibrium gel structure. Here, we assemble two-dimensional equilibrium gels from limited-valency "patchy" colloidal particles and follow their evolution at the particle scale to elucidate cluster-size distributions and free energies. By finely adjusting the patch attraction with critical Casimir forces, we let a mixture of two-valent and pseudo-three-valent patchy particles approach the percolated network state through a set of equilibrium states. Comparing this equilibrium route with a deep quench, we find that both routes approach the percolated state via the same equilibrium states, revealing that the network topology is uniquely set by the particle bond angles, independent of the formation history. The limited-valency system follows percolation theory remarkably well, approaching the percolation point with the expected universal exponents.

Laser-induced heating for the experimental study of critical Casimir forces with optical trapping

Critical Casimir interactions represent a perfect example of bath-induced forces at mesoscales. These forces may have a relevant role in the living systems as well as a role in the design of nanomachines fueled by environmental fluctuations. Since the thermal fluctuations are enhanced in the vicinity of a demixing point of a second-order phase transition, we can modulate the magnitude and range of these Casimir-like forces by slight changes in the temperature. Here, we consider two optical trapped colloidal beads inside a binary mixture. The Casimir interaction is controlled by warming the mixture by laser-induced heating, whose local application ensures high reproducibility. Once this two-particle system is warmed, the critical behavior of different observables allows the system to become its self-thermometer. We use this experimental scheme for analyzing the energetics of a critical colloidal system under a non-equilibrium-driven protocol. We quantify how the injected work can be dissipated to the environment as heat or stored as free energy. Indeed, our system allows us to use the fluctuation theorems framework for analyzing the performance of this critically driven toy model. Our work paves the way for future experimental studies on the non-equilibrium features of bath-induced forces and the design of critically driven nanosystems.

Tunable critical Casimir forces counteract Casimir–Lifshitz attraction

In developing micro- and nanodevices, stiction between their parts, that is, static friction preventing surfaces in contact from moving, is a well-known problem. It is caused by the finite-temperature analogue of the quantum electrodynamical Casimir–Lifshitz forces, which are normally attractive. Repulsive Casimir–Lifshitz forces have been realized experimentally, but their reliance on specialized materials severely limits their applicability and prevents their dynamic control. Here we demonstrate that repulsive critical Casimir forces, which emerge in a critical binary liquid mixture upon approaching the critical temperature, can be used to counteract stiction due to Casimir–Lifshitz forces and actively control microscopic and nanoscopic objects with nanometre precision. Our experiment is conducted on a microscopic gold flake suspended above a flat gold-coated substrate immersed in a critical binary liquid mixture. This may stimulate the development of micro- and nanodevices by preventing stiction as well as by providing active control and precise tunability of the forces acting between their constituent parts.

Surface melting of a colloidal glass

Despite their technological relevance, a full microscopic understanding of glasses is still lacking. This applies even more to their surfaces whose properties largely differ from that of the bulk material. Here, we experimentally investigate the surface of a two-dimensional glass as a function of the effective temperature. To yield a free surface, we use an attractive colloidal suspension of micron-sized particles interacting via tunable critical Casimir forces. Similar to crystals, we observe surface melting of the glass, i.e., the formation of a liquid film at the surface well below the glass temperature. Underneath, however, we find an unexpected region with bulk density but much faster particle dynamics. It results from connected clusters of highly mobile particles which are formed near the surface and deeply percolate into the underlying material. Because its thickness can reach several tens of particle diameters, this layer may elucidate the poorly understood properties of thin glassy films which find use in many technical applications.

Self-assembly of colloidal cube superstructures with critical Casimir attractions

The structure of self-assembled materials is determined by the shape and interactions of the building blocks. Here, we investigate the self-assembly of colloidal ‘superballs’, i.e. cubes with rounded corners, by temperature-tunable critical Casimir forces to obtain insight into the coupling of a cubic shape and short range attractions. The critical Casimir force is a completely reversible and controllable attraction that arises in a near-critical solvent mixture. Using confocal microscopy and particle tracking, we follow the self-assembly dynamics and structural transition in a quasi-2D system. At low attraction, we observe the formation of small clusters with square symmetry. When the attraction is increased, a transition to a rhombic Λ1-lattice is observed. We explain our findings by the change in contact area at faces and corners of the building blocks combined with the increase in attraction strength and range of the critical Casimir force. Our results show that the coupling between the rounded cubic shape and short-range attraction plays a crucial role for the superstructures that form and provide new insights for the active assembly control of micro and nanocubes.

Modeling ion channel clustering using critical phenomena

Recent superresolution microscopy experiments have shown that, even in unmyelinated axons, voltage gated channels are clustered together in a fashion that resembles the organization at the nodes of Ranvier. While it is known that localization of channels can be beneficial for increasing propagation speed and efficiency of action potentials, the mechanism underlying this clustering phenomenon is currently unclear. Here we investigate the role of lipid membrane phase separation and critical Casimir forces as microscopic determinants of pattern formation in cell membranes.

From spatial to temporal patterns: cooperativity across ion channels in supercritical membranes

Theoretically predicted by Fisher and de Gennes in 1978, critical Casimir forces are the thermodynamic analogues of the Casimir-Polder attraction forces due to quantum fluctuations of confined electromagnetic field in vacuum. In the context of lipid membranes close to a miscibility transition, critical Casimir interactions are thought to be responsible for clustering of membrane proteins. Here we explore the hypothesis that these effective statistical-field mediated interactions can explain the observed ion channel cooperativity that has been suggested to promote multistability of the membrane potential and thus “cellular memory”.

Revealing Polymerization Kinetics with Colloidal Dipatch Particles.

Limited-valency colloidal particles can self-assemble into polymeric structures analogous to molecules. While their structural equilibrium properties have attracted wide attention, insight into their dynamics has proven challenging. Here, we investigate the polymerization dynamics of semiflexible polymers in 2D by direct observation of assembling divalent particles, bonded by critical Casimir forces. The reversible critical Casimir force creates living polymerization conditions with tunable chain dissociation, association, and bending rigidity. We find that unlike dilute polymers that show exponential size distributions in excellent agreement with Flory theory, concentrated samples exhibit arrest of rotational and translational diffusion due to a continuous isotropic-to-nematic transition in 2D, slowing down the growth kinetics. These effects are circumvented by the addition of higher-valency particles, cross linking the polymers into networks. Ourresults connecting polymer flexibility, polymer interactions, and the peculiar isotropic-nematic transition in 2D offer insight into the polymerization processes of synthetic two-dimensional polymers and biopolymers at membranes and interfaces.

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.

Revealing pseudorotation and ring-opening reactions in colloidal organic molecules

Colloids have a rich history of being used as ‘big atoms’ mimicking real atoms to study crystallization, gelation and the glass transition of condensed matter. Emulating the dynamics of molecules, however, has remained elusive. Recent advances in colloid chemistry allow patchy particles to be synthesized with accurate control over shape, functionality and coordination number. Here, we show that colloidal alkanes, specifically colloidal cyclopentane, assembled from tetrameric patchy particles by critical Casimir forces undergo the same chemical transformations as their atomic counterparts, allowing their dynamics to be studied in real time. We directly observe transitions between chair and twist conformations in colloidal cyclopentane, and we elucidate the interplay of bond bending strain and entropy in the molecular transition states and ring-opening reactions. These results open the door to investigate complex molecular kinetics and molecular reactions in the high-temperature classical limit, in which the colloidal analogue becomes a good model.

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.

Concentration and size effects on the size-selective particle purification method using the critical Casimir force

Critical Casimir force (CCF) is a solvent fluctuation introduced interaction between particles dispersed in a binary solvent. Recently, it has been demonstrated that the CCF induced attraction between particles can trigger particle size-sensitive aggregation, and has thus been used as an efficient way to purify nanoparticles by size. Here, combining small angle neutron scattering and dynamic light scattering, we investigate the effects of size and concentration on this particle size separation method. Increasing the particle concentration does not significantly affect the purification method, but the solvent composition needs to be adjusted for an optimized efficiency. This purification method is further demonstrated to work also very efficiently for systems with particle size ranging from 15 nm to about 50 nm with a very large size polydispersity. These results indicate that for both short-ranged and long-ranged attraction relative to the particle diameter, the CCF introduced particle aggregation is always size sensitive. This implies that particle aggregation is strongly affected by size polydispersity for many colloidal systems. We further propose a method to use light scattering to help identify the temperature range within which this particle purification method can work efficiently instead of using neutron scattering.

Revealing viscoelastic bending relaxation dynamics of isolated semiflexible colloidal polymers.

The viscoelastic properties of filaments and biopolymers play a crucial role in soft and biological materials from biopolymer networks to novel synthetic metamaterials. Colloidal particles with specific valency allow mimicking polymers and more complex molecular structures at the colloidal scale, offering direct observation of their internal degrees of freedom. Here, we elucidate the time-dependent viscoelastic response in the bending of isolated semi-flexible colloidal polymers, assembled from dipatch colloidal particles by reversible critical Casimir forces. By tuning the patch-patch interaction strength, we adjust the polymers' viscoelastic properties, and follow spontaneous bending modes and their relaxation directly on the particle level. We find that the elastic response is well described by that of a semiflexible rod with persistence length of order 1000 μm, tunable by the critical Casimir interaction strength. We identify the viscous relaxation on longer timescales to be due to internal friction, leading to a wavelength-independent relaxation time similar to single biopolymers, but in the colloidal case arising from the contact mechanics of the bonded patches. These tunable mechanical properties of assembled colloidal filaments open the door to "colloidal architectures", rationally designed (network) structures with desired topology and mechanical properties.

Statics and dynamics of a finite two-dimensional colloidal system with competing attractive critical Casimir and repulsive magnetic dipole interactions.

We continue our theoretical study of a recently proposed two-dimensional colloidal system with attractive critical Casimir and repulsive magnetic dipole forces that can be tuned easily and independently from each other via the temperature and the strength of an external magnetic field, respectively [K. Marolt, M. Zimmermann, and R. Roth, Phys. Rev. E 100, 052602 (2019)2470-004510.1103/PhysRevE.100.052602]. Using this freedom, it is possible to construct a competing interaction potential that causes microphase separation featuring spatially inhomogeneous cluster, stripe, and bubble phases in the bulk, i.e., in an infinite system without an external potential. In the present work, we demonstrate by means of density functional theory that microphase separation can also occur in finite geometries. In a square cell with a side length of 20 or 30 colloid diameters, we observe the emergence of highly structured cluster and ring phases at intermediate bulk densities in addition to almost uniform fluid phases for lower and higher bulk densities. We then employ dynamic density functional theory to determine how the system reacts when the temperature and the magnetic field are altered over time, and we show how to induce a transition from the liquid to the cluster/ring phase and also from the cluster directly to the ring phase. We find that often a slowly varying and nontrivial path in parameter space is required to reach a stable state, whereas abrupt changes are prone to lead to metastable configurations.

Casimir force

Casimir force in quantum electrodynamics is the representation of zero point energy of vacuum. Depending on the type of fluctuation medium, generalized Casimir force covers a wide spectrum of topics in physics, such as, quantum, critical, Goldstone mode, and non-equilibrium Casimir force. In general, long range correlated fluctuations and constraints are two conditions for generating the Casimir force. In this paper, through a survey of the development of Casimir physics, we discuss several types of Casimir forces and several regularization methods. We end the paper with an outlook for the further development of Casimir physics in the future.

Microphase separation in a two-dimensional colloidal system with competing attractive critical Casimir and repulsive magnetic dipole interactions.

We propose and study theoretically a colloidal system in two dimensions with attractive critical Casimir and repulsive magnetic dipole forces, wherein the strength of attraction and repulsion can be easily and independently tuned by adjusting the temperature and an external magnetic field, respectively. We expect this setup to be experimentally accessible and are confident that it can serve to deepen our understanding of the mechanisms behind microphase separation due to competing interactions. We develop a density functional theory for our model and present first results of our calculations in the form of a phase diagram for fixed temperature, but varying magnetic fields and bulk densities. For certain values of these parameters, we are able to confirm the existence of thermodynamically stable inhomogeneous density profiles in the bulk, such as parallel lamellar stripes, as well as clusters and voids on a hexagonal lattice.

Stochastic buckling of self-assembled colloidal structures

Although buckling is a prime route to achieve functionalization and synthesis of single colloids, buckling of colloidal structures---made up of multiple colloids---remains poorly studied. Here, we investigate the buckling of the simplest form of a colloidal structure, a colloidal chain that is self-assembled through critical Casimir forces. We demonstrate that the mechanical instability of such a chain is strikingly reminiscent of that of classical Euler buckling but with thermal fluctuations and plastic effects playing a significant role. Namely, we find that fluctuations tend to diverge close to the onset of buckling and that plasticity controls the buckling dynamics at large deformations. Our work provides insight into the effect of geometrical, thermal and plastic interactions on the nonlinear mechanics of self-assembled structures, of relevance for the rheology of complex and living matter and the rational design of colloidal architectures.

Physical origin of the expansion of polymer coils in a binary solvent in the vicinity of its demixing critical point

Critical fluctuations are known to induce a collapse of polymer chains in a mixed solvent upon approaching its liquid–liquid critical point, as originally predicted by Brochard and de Gennes. Recently, we have found that closer to the critical point this collapse is followed by a reswelling of the polymer coils well beyond the original dimensions, a phenomenon not predicted by the theory of Brochard and de Gennes. We submit that upon approaching the critical temperature more closely, the correlation length of the critical fluctuations inside the polymer coils can no longer further increase due to the finite size of the coils, resulting in the appearance of large critical Casimir forces that cause a significant expansion of the polymer coils. Eventually, micro-phase separation inside the coils will appear and the coils will reshrink. This entire process takes place while the bulk solution is still in the one-phase region.

An effective method for calculating the critical Casimir force between two parallel plates for the Ising universality class

An effective method for calculating the critical Casimir force between two parallel plates for the Ising universality class