Critical Casimir interactions between colloids around the critical point of binary solvents

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.

Phases of surface-confined trivalent colloidal particles

Patchy colloids promise the design and modelling of complex materials, but the realization of equilibrium patchy particle structures remains challenging. Here, we assemble pseudo-trivalent particles and elucidate their phase behaviour when confined to a plane. We observe the honeycomb phase, as well as more complex amorphous network and triangular phases. Structural analysis performed on the three condensed phases reveals their shared structural motifs. Using a combined experimental and simulation approach, we elucidate the energetics of these phases and construct the phase diagram of this system, using order parameters to determine the phase coexistence lines. Our results reveal the rich phase behaviour that a relatively simple patchy particle system can display, and open the door to a larger joined simulation and experimental exploration of the full patchy-particle phase space.

Visualizing defect dynamics by assembling the colloidal graphene lattice

Graphene has been under intense scientific interest because of its remarkable optical, mechanical and electronic properties. Its honeycomb structure makes it an archetypical two-dimensional material exhibiting a photonic and phononic band gap with topologically protected states. Here, we assemble colloidal graphene, the analogue of atomic graphene using pseudo-trivalent patchy particles, allowing particle-scale insight into crystal growth and defect dynamics. We directly observe the formation and healing of common defects, like grain boundaries and vacancies using confocal microscopy. We identify a pentagonal defect motif that is kinetically favoured in the early stages of growth, and acts as seed for more extended defects in the later stages. We determine the conformational energy of the crystal from the bond saturation and bond angle distortions, and follow its evolution through the energy landscape upon defect rearrangement and healing. These direct observations reveal that the origins of the most common defects lie in the early stages of graphene assembly, where pentagons are kinetically favoured over the equilibrium hexagons of the honeycomb lattice, subsequently stabilized during further growth. Our results open the door to the assembly of complex 2D colloidal materials and investigation of their dynamical, mechanical and optical properties.

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.

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.

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. Our results 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.

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.

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.

Debye vs. Casimir: controlling the structure of charged nanoparticles deposited on a substrate

Fine-tuning the interactions between particles can allow one to steer their collective behaviour and structure. A convenient way to achieve this is to use solvent criticality to control attraction, via critical Casimir forces, and to control repulsion via the Debye screening of electrostatic interactions. Herein, we develop a multiscale simulation framework and a method for controlled deposition of quantum dots to investigate how these interactions affect the structure of charged nanoparticles deposited on a substrate, altogether immersed in a binary liquid mixture intermixed with salt. We consider nanoparticles and substrates favouring the same component of the mixture and find that the critical Casimir interactions between the nanoparticles become drastically reduced at the substrate. In particular, the interactions can become a few k_BT weaker and their decay length a few orders of magnitude smaller than in the bulk. At off-critical compositions, the decay length increases upon approaching criticality, as expected, but the interaction strength decreases. With molecular dynamics simulations and experiments, we reveal that the nanoparticles can self-assemble into crystalline clusters which form superstructures resembling cluster fluids and spinodal morphology. The simulations additionally predict the formation of fractal-like nanoparticle gels and bicontinuous phases. Our results demonstrate that charged nanoparticles in a salty binary liquid mixture provide exciting opportunities to study the formation of complex structures experimentally and theoretically, which may lead to applications in optoelectronics and photonics.

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.

Nonequilibrium master kinetic equation modeling of colloidal gelation

We present a detailed study of the kinetic cluster growth process during gelation of weakly attractive colloidal particles by means of experiments on critical Casimir attractive colloidal systems, simulations, and analytical theory. In the experiments and simulations, we follow the mean coordination number of the particles during the growth of clusters to identify an attractive-strength independent cluster evolution as a function of mean coordination number. We relate this cluster evolution to the kinetic attachment and detachment rates of particles and particle clusters. We find that single-particle detachment dominates in the relevant weak attractive-strength regime, while association rates are almost independent of the cluster size. Using the limit of single-particle dissociation and size-independent association rates, we solve the master kinetic equation of cluster growth analytically to predict power-law cluster mass distributions with exponents -3/2 and -5/2 before and after gelation, respectively, which are consistent with the experimental and simulation data. These results suggest that the observed critical Casimir-induced gelation is a second-order nonequilibrium phase transition (with broken detailed balance). Consistent with this scenario, the size of the largest cluster is observed to diverge with power-law exponent according to three-dimensional percolation on approaching the critical mean coordination number.

Nonequilibrium continuous phase transition in colloidal gelation with short-range attraction

The dynamical arrest of attractive colloidal particles into out-of-equilibrium structures, known as gelation, is central to biophysics, materials science, nanotechnology, and food and cosmetic applications, but a complete understanding is lacking. In particular, for intermediate particle density and attraction, the structure formation process remains unclear. Here, we show that the gelation of short-range attractive particles is governed by a nonequilibrium percolation process. We combine experiments on critical Casimir colloidal suspensions, numerical simulations, and analytical modeling with a master kinetic equation to show that cluster sizes and correlation lengths diverge with exponents  ~1.6 and 0.8, respectively, consistent with percolation theory, while detailed balance in the particle attachment and detachment processes is broken. Cluster masses exhibit power-law distributions with exponents  -3/2 and  -5/2 before and after percolation, as predicted by solutions to the master kinetic equation. These results revealing a nonequilibrium continuous phase transition unify the structural arrest and yielding into related frameworks.

Effective pair interaction of patchy particles in critical fluids

We study the critical Casimir interaction between two spherical colloids immersed in a binary liquid mixture close to its critical demixing point. The surface of each colloid prefers one species of the mixture with the exception of a circular patch of arbitrary size, where the other species is preferred. For such objects, we calculate, within the Derjaguin approximation, the scaling function describing the critical Casimir potential, and we use it to derive the scaling functions for all components of the forces and torques acting on both colloids. The results are compared with available experimental data. Moreover, the general relation between the scaling function for the potential and the scaling functions for the force and the torque is derived.

Rigidity and auxeticity transitions in networks with strong bond-bending interactions

A widely studied model for gels or biopolymeric fibrous materials are networks with central force interactions, such as Hookean springs. Less commonly studied are materials whose mechanics are dominated by non-central force interactions such as bond-bending potentials. Inspired by recent experimental advancements in designing colloidal gels with tunable interactions, we study the micro- and macroscopic elasticity of two-dimensional planar graphs with strong bond-bending potentials, in addition to weak central forces. We introduce a theoretical framework that allows us to directly investigate the limit in which the ratio of characteristic central-force to bending stiffnesses vanishes. In this limit we show that a generic isostatic point exists at [Formula: see text], coinciding with the isostatic point of frames with central-force interactions in two dimensions. We further demonstrate the emergence of a stiffening transition when the coordination is increased towards the isostatic point, which shares similarities with the strain-induced stiffening transition observed in biopolymeric fibrous materials, and coincides with an auxeticity transition above which the material's Poisson's ratio approaches -1 when bond-bending interactions dominate.

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.

Probing interface localization-delocalization transitions by colloids

Interface localization-delocalization transitions (ILDT) occur in two-phase fluids confined in a slit with competing preferences of the walls for the two fluid phases. At low temperatures the interface between the two phases is localized at one of the walls. Upon increasing temperature it unbinds. Although intensively studied theoretically and computationally, such transitions have not yet been observed experimentally due to severe challenges in resolving fine details of the fluid structure. Here, using mean field theory and Monte Carlo simulations of the Ising model, we propose to detect these ILDT by using colloids. We show that the finite-size and fluctuation induced force acting on a colloid confined in such a system experiences a vivid change if, upon lowering the temperature, the interface localizes at one of the walls. This change can serve as a more easily accessible experimental indicator of the transition.

Nonadditive interactions and phase transitions in strongly confined colloidal systems

The behaviour of colloids can be controlled effectively by tuning the solvent-mediated interactions among them. An extensively studied example is the temperature-induced aggregation of suspended colloids close to the consolute point of their binary solvent. Here, using mean field theory and Monte Carlo simulations, we study the behaviour of colloids confined to a narrow slit containing a nearly-critical binary liquid mixture. We found that the effective interactions in this system are highly non-additive. In particular, the effective interactions among the colloids can be a few times stronger than the corresponding sum of the effective pair potentials. Inter alia, this non-additivity manifests itself in the phase behaviour of confined colloids, which depends sensitively on the slit width and temperature. In addition, we demonstrate the possibility of a first-order bridging transition between colloids confined to a slit and suspended in a phase-separated fluid well below the critical point of the solvent and at its critical composition in the bulk. This transition is accompanied by a remarkably large hysteresis loop, in which the force between the colloids varies by two orders of magnitude.

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.

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.