Nucleation in systems with a metastable liquid-gas critical point is the prototypical example of a two-step nucleation process, in which the appearance of the critical nucleus is preceded by the formation of a liquid-like density fluctuation. So far, the majority of studies on colloidal and protein crystallization have focused on one-component systems, and we are lacking a clear description of two-step nucleation processes in multicomponent systems, where critical fluctuations involve coupled density and concentrations inhomogeneities. Here, we examine the nucleation process of a binary mixture of patchy particles designed to nucleate into a diamond lattice. By combining Gibbs-ensemble simulations and direct nucleation simulations over a wide range of thermodynamic conditions, we are able to pin down the role of the liquid-gas metastable phase diagram on the nucleation process. In particular, we show that the strongest enhancement of crystallization occurs at an azeotropic point with the same stoichiometric composition of the crystal.
We study a two-dimensional binary mixture of patchy colloids in sedimentation-diffusion-equilibrium using Monte Carlo simulation and Wertheim's theory. By tuning the buoyant masses of the colloids we can control the gravity-induced sequence of fluid stacks of differing density and percolation properties. We find complex stacking sequences with up to four layers and reentrant network formation, consistently in simulations and theoretically using only the bulk phase diagram as input. Our theory applies to general patchy colloidal mixtures and is relevant to understanding experiments under gravit…
We investigate the phase behaviour of 2D mixtures of bi-functional and three-functional patchy particles and 3D mixtures of bi-functional and tetra-functional patchy particles by means of Monte Carlo simulations and Wertheim theory. We start by computing the critical points of the pure systems and then we investigate how the critical parameters change upon lowering the temperature. We extend the Successive Umbrella Sampling method to mixtures to make it possible to extract information about the phase behaviour of the system at a fixed temperature for the whole range of densities and compositio…
In this work we develop the first theory to model self assembling mixtures of single patch colloids and colloids with spherically symmetric attractions. In the development of the theory we restrict the interactions such that there are short ranged attractions between patchy and spherically symmetric colloids, but patchy colloids do not attract patchy colloids and spherically symmetric colloids do not attract spherically symmetric colloids. This results in the temperature, density and composition dependent reversible self assembly of the mixture into colloidal star molecules. This type of mixtu…
We investigate the thermodynamics and percolation regimes of model binary mixtures of patchy colloidal particles. The particles of each species have three sites of two types, one of which promotes bonding of particles of the same species while the other promotes bonding of different species. We find up to four percolated structures at low temperatures and densities: two gels where only one species percolates, a mixed gel where particles of both species percolate but neither species percolates separately, and a bicontinuous gel where particles of both species percolate separately forming two in…
The addition of co-solutes to colloidal suspensions is often employed to induce tunable depletion interactions. In this work we investigate effective colloidal interactions arising from binary co-solute mixtures of hard spheres and patchy particles. By changing the relative concentration of the two species, we show that the resulting effective potential $V_\text{eff}$ continuously changes from the one obtained for a single-component hard sphere co-solute to that mediated by the single-component patchy particle co-solute. Interestingly, we find that, independently of the relative concentration …
We investigate percolation in binary and ternary mixtures of patchy colloidal particles theoretically and using Monte Carlo simulations. Each particle has three identical patches, with distinct species having different types of patch. Theoretically we assume tree-like clusters and calculate the bonding probabilities using Wertheim's first-order perturbation theory for association. For ternary mixtures we find up to eight fundamentally different percolated states. The states differ in terms of the species and pairs of species that have percolated. The strongest gel is a trigel or tricontinuous …
We explore the use of templated self-assembly to facilitate the formation of complex target structures made from patchy particles. First, we consider the templating of high-symmetry shell structures around a spherical core particle. We find that nucleation around the core particle can inhibit aggregate formation, a process which often hinders self-assembly. In particular, this new assembly pathway allows dodecahedral shells to form readily, whereas these structures never form in the absence of the template. Secondly, we consider the self-assembly of multi-shell structures, where the central ic…
The statistics of liquid-to-crystal nucleation are measured for both water and for clathrate-forming mixtures of tetrahydrofuran (THF) and water using an automatic lag time apparatus (ALTA). We measure the nucleation temperature using this apparatus in which a single sample is repeatedly cooled, nucleated and thawed. The effect of stirring on nucleation has been evaluated numerically and is discussed. We find that stirring of the solution makes no difference to the nucleation temperature of a given solution in a given tube.
Mixtures of ideal polymers with hard spheres whose diameters are smaller than the radius of gyration of the polymer, exhibit extensive immiscibility. The interfacial tension between demixed phases of these mixtures is estimated, as is the barrier to nucleation. The barrier is found to scale linearly with the radius of the polymer, causing it to become large for large polymers. Thus for large polymers nucleation is suppressed and phase separation proceeds via spinodal decomposition, as it does in polymer blends.
We investigate the effect of distinct bonding energies on the onset of criticality of low functionality fluid mixtures. We focus on mixtures of particles with two and three patches as this includes the mixture where 'empty' fluids were originally reported. In addition to the number of patches, the species differ in the type of patches or bonding sites. For simplicity, we consider that the patches on each species are identical: one species has 3 patches of type $A$ and the other 2 patches of type $B$. We have found a rich phase behaviour with closed miscibility gaps, liquid-liquid demixing and …
Crystallization and vitrification of tetrahedral liquids are important both from a fundamental and a technological point of view. Here, we study via extensive umbrella sampling Monte Carlo computer simulations the nucleation barriers for a simple model for tetrahedral patchy particles in the regime where open tetrahedral crystal structures (namely cubic and hexagonal diamond and their stacking hybrids) are thermodynamically stable. We show that by changing the angular bond width, it is possible to move from a glass-forming model to a readily crystallizing model. From the shape of the barrier w…
We calculate the network fluid regime and phase diagrams of binary mixtures of patchy colloids, using Wertheim's first order perturbation theory and a generalization of Flory-Stockmayer's theory of polymerization. The colloids are modelled as hard spheres with the same diameter and surface patches of the same type, $A$. The only difference between species is the number of their patches -or functionality-, $f_A^{(1)}$ and $f_A^{(2)}$ (with $f_A^{(2)}>f_A^{(1)}$). We have found that the difference in functionality is the key factor controlling the behaviour of the mixture in the network (percola…
Asymmetric patchy particle models have recently been shown to describe the crystallization of small globular proteins with near quantitative accuracy. Here, we investigate how asymmetry in patch geometry and bond energy generally impact the phase diagram and nucleation dynamics of this family of soft matter models. We find the role of the geometry asymmetry to be weak, but the energy asymmetry to markedly interfere with the crystallization thermodynamics and kinetics. These results provide a rationale for the success and occasional failure of George and Wilson's proposal for protein crystalliz…
We present a theory to predict the structure and thermodynamics of mixtures of colloids of different diameters, building on our earlier work [J. Chem. Phys. 145, 074904 (2016)] that considered mixtures with all particles constrained to have the same size. The patchy, solvent particles have short-range directional interactions, while the solute particles have short-range isotropic interactions. The hard-sphere mixture without any association site forms the reference fluid. An important ingredient within the multi-body association theory is the description of clustering of the reference solvent …
Model patchy particles have been shown to be able to form a wide variety of structures, including symmetric clusters, complex crystals and even two-dimensional quasicrystals. Here, we investigate whether we can design patchy particles that form three-dimensional quasicrystals, in particular targeting a quasicrystal with dodecagonal symmetry that is made up of stacks of two-dimensional quasicrystalline layers. We obtain two designs that are able to form such a dodecagonal quasicrystal in annealing simulations. The first is a one-component system of 7-patch particles but with wide patches that a…
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 experim…
In this work we extend Wertheim's thermodynamic perturbation theory (TPT) to binary mixtures (species A and species B) of patchy colloids where each species has a single patch which can bond a maximum of twice (divalent). Colloids are treated as hard spheres with a directional conical asssociation site. We restrict the system such that only patches between unlike species share attractions; meaning there are AB attractions but no AA or BB attractions. The theory is derived in Wertheim's two density formalism for one site associating fluids. Since the patches are doubly bondable, associated chai…
A mixture of solvent particles with short-range, directional interactions and solute particles with short-range, isotropic interactions that can bond multiple times is of fundamental interest in understanding liquids and colloidal mixtures. Because of multi-body correlations predicting the structure and thermodynamics of such systems remains a challenge. Earlier Marshall and Chapman developed a theory wherein association effects due to interactions multiply the partition function for clustering of particles in a reference hard-sphere system. The multi-body effects are incorporated in the clust…
Effective colloid-colloid interactions can be tailored through the addition of a complex cosolute. Here we investigate the case of a cosolute made by self-assembling patchy particles. Depending on the valence, these particles can form either polymer chains or branched structures. We numerically calculate the effective potential $V_{eff}$ between two colloids immersed in a suspension of reversible patchy particles, exploring a wide region of the cosolute phase diagram and the role of valence. In addition to well-known excluded volume and depletion effects, we find that, under appropriate condit…
Using molecular simulation, we study the nucleation of liquid droplets from binary mixtures and determine the free energy of nucleation along entropic pathways. To this aim, we develop the $μ_1 μ_2 VT-S$ method, based on the grand-canonical ensemble modeling the binary mixture, and use the entropy of the system $S$ as the reaction coordinate to drive the formation of the liquid droplet. This approach builds on the advantages of the grand-canonical ensemble, which allows for the direct calculation of the entropy of the system and lets the composition of the system free to vary throughout the nu…
We study the phase diagram of a binary mixture of patchy particles which has been designed to form a reversible gel. For this we perform Monte Carlo and molecular dynamics simulations to investigate the thermodynamics of such a system and compare our numerical results with predictions based on the analytical parameter-free Wertheim theory. We explore a wide range of the temperature-density-composition space that defines the three-dimensional phase diagram of the system. As a result, we delimit the region of thermodynamic stability of the fluid. We find that for a large region of the phase diag…
We systematically study the design of simple patchy sphere models that reversibly self-assemble into monodisperse icosahedral clusters. We find that the optimal patch width is a compromise between structural specificity (the patches must be narrow enough to energetically select the desired clusters) and kinetic accessibility (they must be sufficiently wide to avoid kinetic traps). Similarly, for good yields the temperature must be low enough for the clusters to be thermodynamically stable, but the clusters must also have enough thermal energy to allow incorrectly formed bonds to be broken. Ord…
Differences in activities in colloidal particles are sufficient to drive phase separation between active and passive (or less active) particles, even if they have only excluded volume interactions. In this paper, we study the phase separation kinetics and propose a theory of phase separation of colloidal mixtures in the diffusive limit. Our model considers a mixture of diffusing particles coupled to different thermostats, it thus has a non-equilibrium nature due to the temperature differences. However, we show that indeed the system recovers an effective equilibrium thermodynamics in the dilut…
Explicit simulations of fluid mixtures of highly size-dispersed particles are constrained by numerical challenges associated with identifying pair-interaction neighbors. Recent algorithmic developments have ameliorated these difficulties to an extent, permitting more efficient simulations of systems with many large and small particles of disperse sizes. We leverage these capabilities to perform molecular dynamics simulations of binary sphere mixtures with elastically stiff particles approaching the hard sphere limit and particle size ratios of up to 50, approaching the colloidal limit. The sys…
Two-component mixtures of Bose-Einstein condensates are shown to support solitary wave complexes that move with a constant velocity preserving their form. I obtain the families of such solitary wave complexes in two-dimensional two-component mixture condensates. These solutions are classified according to the structure of the wavefunction in each component. I show that these complexes nucleate from the surface of the disk when it moves supercritically, therefore, suggesting a mechanism by which these waves can be obtained in condensates by a moving laser beam. The condition for such a nucleati…
Recent simulations have studied the formation of patterns in a binary mixture of immiscible surfactants absorbed onto the surface of a spherical nanoparticle. The resulting patterns (Janus, spots and stripes) were in good agreement with experimental results. We perform dissipative particle dynamics (DPD) simulations to study the patterns obtained by adding a third surfactant to the monolayer as a guide towards increasing the richness and diversity of patchy particles synthesized this way. We predict a variety of new patterns that can be produced through different combinations of simple design …
Equilibrium gels of colloidal particles can be realized through the introduction of a second species, a linker that mediates the bonds between the colloids. A gel forming binary mixture whose linkers can self-assemble into linear chains while still promoting the aggregation of particles is considered in this work. The particles are patchy particles with $f_C$ patches of type $C$ and the linkers are patchy particles with $2$ patches of type $A$ and $f_B$ patches of type B. The bonds between patches of type $A$ ($AA$ bonds) promote the formation of linear chains of linkers. Two different ways (m…
Patchy particles are a class of colloids with functionalized surfaces. Through surface functionalization, the strength and directionality of the colloidal interactions are tunable allowing control over coordination of the particle. Exquisite equilibrium phase diagrams of mixtures of coordination two and three have been reported. However, the kinetics of self-organization and the feasibility of the predicted structures are still largely unexplored. Here, we study the irreversible aggregation of these mixtures on a substrate, for different fractions of two-patch particles. Two mechanisms of mass…
Colloidal patchy particles with divalent attractive interaction can self-assemble into linear polymer chains. Their equilibrium properties in 2D and 3D are well described by Wertheim's thermodynamic perturbation theory which predicts a well-defined exponentially decaying equilibrium chain length distribution. In experimental realizations, due to gravity, particles sediment to the bottom of the suspension forming a monolayer of particles with a gravitational height smaller than the particle diameter. In accordance with experiments, an anomalously high monomer concentration is observed in simula…
Selective removal of heavy metals from water is a complex topic. We present a theoretical computational approach, supported by experimental evidences, to design a functionalized nanomaterial that is able to selectively capture metallic ions from water in a self-assembling process. A theoretical model is used to map an experimental mixture of Ag nanoparticles and either Co$^{2+}$ or Ni$^{2+}$ onto an additive highly asymmetric attractive Lennard Jones binary mixture. Extensive NVT (constant number of particles, volume, and temperature) Monte Carlo simulations are performed to derive a set of pa…
We study low-salt, binary aqueous suspensions of charged colloidal spheres of size ratio Phi = 0.57, number densities below the eutectic number density n_E, and number fractions of p = 1.00-0.40. The typical phase obtained by solidification from a homogeneous shear-melt is a substitutional alloy of body centred cubic structure. In strictly gas-tight vials, the polycrystalline solid is stable against melting and further phase transformation for extended times. For comparison, we prepare the same samples also by slow and mechanically undisturbed deionization in commercial slit cells. These cells…
We discuss the interplay between a slow continuous drift of temperature, which induces continuous phase separation, and the non-linear diffusion term in the $ϕ^4$-model for phase separation of a binary mixture. This leads to a bound for the stability of diffusive demixing. It is argued that these findings are not specific to the $ϕ^4$ model, but that they always apply up to slight modifications of the bound. In practice stable diffusive demixing can only be achieved when special precautions are taken in experiments on real mixtures. Therefore, the recent observations on complex dynamical behav…
Unlike fluids at thermal equilibrium, biomolecular mixtures in living systems can sustain nonequilibrium steady states, in which active processes modify the conformational states of the constituent molecules. Despite qualitative similarities between liquid--liquid phase separation in these systems, the extent to which the phase-separation kinetics differ remains unclear. Here we show that inhomogeneous chemical reactions can alter the nucleation kinetics of liquid--liquid phase separation in a manner that is consistent with classical nucleation theory, but can only be rationalized by introduci…
The thermodynamics of strongly anisotropic crystalline surfaces is analogous to that of a binary mixture exhibiting phase separation. On a metastable planar surface, formation of stable orientations requires a nucleation process, in which the energy associated with the presence of corners must be considered. In this context, a nucleation event corresponds to the formation of a critical shape for the crystalline surface before the system enters the growth regime. We first derive the Euler-Lagrange equation for crystal surface nucleation, in two dimensions, and show that the saddle-point conditi…
This mini-review is concerned with two-dimensional colloidal mixtures exposed to various kinds of external fields. By a magnetic field perpendicular to the plane, dipole moments are induced in paramagnetic particles which give rise to repulsive interactions leading to complex crystalline alloys in the composition-asymmetry diagram. A quench in the magnetic field induces complex crystal nucleation scenarios. If exposed to a gravitational field, these mixtures exhibit a brazil--nut effect and show a boundary layering which is explained in terms of a depletion bubble picture. The latter persists …
We report a molecular dynamics study of crystallization in highly asymmetric binary hard-sphere mixtures, in which the large spheres can form a crystal phase while the small ones remain disordered during the crystallization process of the large spheres. By taking advantage of assisting crystal nucleation with a patterned substrate, direct evidence is presented that there is a close link between the diffusive redistribution of the small spheres and the crystal formation of the large spheres. Although the addition of a second component with large size disparity will not alter the crystal structu…
We perform particle resolved experimental studies on the heterogeneous crystallisation process of two compo- nent mixtures of hard spheres. The components have a size ratio of 0.39. We compared these with molecular dynamics simulations of homogenous nucleation. We find for both experiments and simulations that the final assemblies are interstitial solid solutions, where the large particles form crystalline close-packed lattices, whereas the small particles occupy random interstitial sites. This interstitial solution resembles that found at equilibrium when the size ratios are 0.3 [Filion et al…
We present a theory of nonionic solutes in a mixture solvent composed of water-like and alcohol-like species. First, we show relationship among the solvation chemical potential, the partial volumes $v_i$, the Kirkwood-Buff integrals, the second osmotic virial coefficient, and the Gibbs transfer free energy. We examine how the solute density $n_3$ is coupled to the solvent densities $n_1$ and $n_2$ in thermodynamics. In the limit of small compressibility, we show that the space-filling condition $\sum_i v_i n_i=1$ nearly holds for inhomogeneous densities $n_i$, where the concentration fluctuati…
Based on classical density functional theory (DFT), we investigate the demixing phase transition of a two-dimensional, binary Heisenberg fluid mixture. The particles in the mixture are modeled as Gaussian soft spheres, where one component is characterized by an additional classical spin-spin interaction of Heisenberg type. Within the DFT we treat the particle interactions using a mean-field approximation. For certain magnetic coupling strengths we calculate phase diagrams in the density-concentration plane. For sufficiently large coupling strengths and densities, we find a demixing phase trans…
A full equilibrium treatment of molecular aggregation is presented for prototypes of 1D and 3D aggregates, with and without nucleation. By skipping complex kinetic parameters like aggregate size-dependent diffusion, the equilibrium treatment allows to predict directly time-independent quantities such as critical concentrations. The relationships between the macroscopic equilibrium constants for the different paths are first established by statistical corrections and so as to comply with the detailed balance constraints imposed by nucleation, and the composition of the mixture resulting from ho…
We consider the demixing of a binary fluid mixture, under gravity, which is steadily driven into a two phase region by slowly ramping the temperature. We assume, as a first approximation, that the system remains spatially isothermal, and examine the interplay of two competing nonlinearities. One of these arises because the supersaturation is greatest far from the meniscus, creating inversion of the density which can lead to fluid motion; although isothermal, this is somewhat like the Benard problem (a single-phase fluid heated from below). The other is the intrinsic diffusive instability which…
Micro and nanodroplets have many important applications such as in drug delivery, liquid-liquid extraction, nanomaterial synthesis and cosmetics. A commonly used method to generate a large number of micro or nanodroplets in one simple step is solvent exchange (also called nanoprecipitation), in which a good solvent of the droplet phase is displaced by a poor one, generating an oversaturation pulse that leads to droplet nucleation. Despite its crucial importance, the droplet growth resulting from the oversaturation pulse in this ternary system is still poorly understood. We experimentally and t…
Demixing of binary fluids subjected to slow temperature ramps shows repeated waves of nucleation which arise as a consequence of the competition between generation of supersaturation by the temperature ramp and relaxation of supersaturation by diffusive transport and flow. Here, we use an advection-reaction-diffusion model to study the oscillations in the weak- and strong-diffusion regime. There is a sharp transition between the two regimes, which can only be understood based on the probability distribution function of the composition rather than in terms of the average composition. We argue t…
We present theory and simulation of simultaneous chemical demixing and phase ordering in a polymer-liquid crystal mixture in conditions where isotropic-isotropic phase separation is metastable with respect to isotropic-nematic phase transition. In the case the mechanism is nucleation and growth, it is found that mesophase growth proceeds by a transient metastable phase that surround the ordered phase, and whose lifetime is a function of the ratio of diffusional to orientational mobilities. In the case of spinodal decomposition, different dynamic regimes are observed depending on the mobility r…
Phase separation has recently emerged as an important organizational principle in the dense and heterogeneous environment within living cells. Here, we use a synthetic system to show that compressive stresses in a polymer network suppress phase separation of the solvent that swells it. These stresses create a barrier to droplet nucleation that leads to robust, stabilized mixtures well beyond the liquid-liquid phase separation boundary. Network stresses not only alter the stability of mixtures, but they also have a dramatic effect on the ripening of droplets. Gradients in network stresses can d…
The interplay between phase separation and wetting of multicomponent mixtures is ubiquitous in nature and technology and recently gained significant attention across scientific disciplines, due to the discovery of biomolecular condensates. It is well understood that sessile droplets, undergoing phase separation in a static wetting configuration, exhibit microdroplet nucleation at their contact lines, forming an oil ring during later stages. However, very little is known about the dynamic counterpart, when phase separation occurs in a non-equilibrium wetting configuration, i.e., spreading dropl…
A method of simulating the drying process of a soft matter solution with an implicit solvent model by moving the liquid-vapor interface is applied to various solution films and droplets. For a solution of a polymer and nanoparticles, we observe "polymer-on-top" stratification, similar to that found previously with an explicit solvent model. Furthermore, "polymer-on-top" is found even when the nanoparticle size is smaller than the radius of gyration of the polymer chains. For a suspension droplet of a bidisperse mixture of nanoparticles, we show that core-shell clusters of nanoparticles can be …
