Phase diagrams of binary mixtures of patchy colloids with distinct numbers and types of patches: The empty fluid regime

Machine Learning of a Density Functional for Anisotropic Patchy Particles

Anisotropic patchy particles have become an archetypical statistical model system for associating fluids. Here, we formulate an approach to the Kern-Frenkel model via the classical density functional theory to describe the positionally and orientationally resolved equilibrium density distributions in flat wall geometries. The density functional is split into a reference part for the orientationally averaged density and an orientational part in mean-field approximation. To bring the orientational part into a kernel form suitable for machine learning (ML) techniques, an expansion into orientational invariants and the proper incorporation of single-particle symmetries are formulated. The mean-field kernel is constructed via ML on the basis of hard wall simulation data. The results are compared to the well-known random-phase approximation, which strongly underestimates the orientational correlations close to the wall. Successes and shortcomings of the mean-field treatment of the orientational part are highlighted and perspectives are given for attaining a full-density functional via ML.

Engineering Azeotropy to Optimize the Self-Assembly of Colloidal Mixtures

The goal of inverse self-assembly is to design interparticle interactions capable of assembling the units into a desired target structure. The effective assembly of complex structures often requires the use of multiple components, each new component increasing the thermodynamic degrees of freedom and, hence, the complexity of the self-assembly pathway. In this work we explore the possibility to use azeotropy, i.e., a special thermodynamic condition where the system behaves effectively as a one-component system, as a way to control the self-assembly of an arbitrary number of components. Exploiting the mass-balance equations, we show how to select patchy particle systems that exhibit azeotropic points along the desired self-assembly pathway. As an example we map the phase diagram of a binary mixture that, by design, fully assembles into cubic (and only cubic) diamond crystal via an azeotropic point. The ability to explicitly include azeotropic points in artificial designs reveals effective pathways for the self-assembly of complex structures.

Comparative Study on the Models of Thermoreversible Gelation

A critical survey on the various theoretical models of thermoreversible gelation, such as the droplet model of condensation, associated-particle model, site–bond percolation model, and adhesive hard sphere model, is presented, with a focus on the nature of the phase transition predicted by them. On the basis of the classical tree statistics of gelation, combined with a thermodynamic theory of associating polymer solutions, it is shown that, within the mean-field description, the thermoreversible gelation of polyfunctional molecules is a third-order phase transition analogous to the Bose–Einstein condensation of an ideal Bose gas. It is condensation without surface tension. The osmotic compressibility is continuous, but its derivative with respect to the concentration of the functional molecule reveals a discontinuity at the sol–gel transition point. The width of the discontinuity is directly related to the amplitude of the divergent term in the weight-average molecular weight of the cross-linked three-dimensional polymers. The solution remains homogeneous in the position space, but separates into two phases in the momentum space; particles with finite translational momentum (sol) and a network with zero translational momentum (gel) coexist in a spatially homogeneous state. Experimental methods used to detect the singularity at the sol–gel transition point are suggested.

Empty liquid state and re-entrant phase behavior of the patchy colloids confined in porous media

Patchy colloids with three and four equivalent patches, confined in an attractive random porous medium, undergo re-entrant gas–liquid phase separation with the liquid phase density approaching zero at low temperatures. The (bonding) colloid–colloid interaction causes the liquid–gas phase separation, which is modulated by the presence of the randomly distributed hard-sphere obstacles, attracting the colloids via Yukawa potential. Due to this interaction, a layer of mutually bonded colloids around the obstacles is formed. The network becomes nonuniform, with colloid particles locally centered on the obstacles. Features described in this article may open possibilities to produce equilibrium gels with predefined nonuniform distribution of particles and indicate how complicated the phase behavior of biological macromolecules in a crowded environment may be.

Building up DNA, bit by bit: a simple description of chain assembly

We simulate the assembly of DNA copolymers from two types of short duplexes (short double strands with a single-stranded overhang at each end), as described by the oxDNA model. We find that the statistics of chain lengths can be well reproduced by a simple theory that treats the association of particles into ideal (i.e., non-interacting) clusters as a reversible chemical reaction. The reaction constants can be predicted either from SantaLucia's theory or from Wertheim's thermodynamic perturbation theory of association for spherical patchy particles. Our results suggest that theories incorporating very limited molecular detail may be useful for predicting the broad equilibrium features of copolymerisation.

Facile self-assembly of colloidal diamond from tetrahedral patchy particles via ring selection

The self-assembly of colloidal diamond–a classic example of an open crystal with the low coordination number of four and much sought after due to its applications in visible light management–from designer spherical colloidal particles has proved challenging over the years. The formation of the diamond lattice from tetrahedral patchy particles is hampered by the propensity to form competing open periodic structures for narrow patches or dynamically arrested states for wider patches, leaving a narrow window in design space where diamond crystals may be realized. Our two-component system of designer tetrahedral patchy particles supports a significantly wider range for patch sizes for programmed self-assembly, thus facilitating experimental fabrication, and offers fundamental insight into crystallization into open lattices.

Diamond-structured crystals, particularly those with cubic symmetry, have long been attractive targets for the programmed self-assembly of colloidal particles, due to their applications as photonic crystals that can control the flow of visible light. While spherical particles decorated with four patches in a tetrahedral arrangement—tetrahedral patchy particles—should be an ideal building block for this endeavor, their self-assembly into colloidal diamond has proved elusive. The kinetics of self-assembly pose a major challenge, with competition from an amorphous glassy phase, as well as clathrate crystals, leaving a narrow widow of patch widths where tetrahedral patchy particles can self-assemble into diamond crystals. Here we demonstrate that a two-component system of tetrahedral patchy particles, where bonding is allowed only between particles of different types to select even-member rings, undergoes crystallization into diamond crystals over a significantly wider range of patch widths conducive for experimental fabrication. We show that the crystallization in the two-component system is both thermodynamically and kinetically enhanced, as compared to the one-component system. Although our bottom-up route does not lead to the selection of the cubic polytype exclusively, we find that the cubicity of the self-assembled crystals increases with increasing patch width. Our designer system not only promises a scalable bottom-up route for colloidal diamond but also offers fundamental insight into crystallization into open lattices.

Phase behavior of a binary mixture of patchy colloids: Effect of particle size and gravity

We study theoretically the effect of size difference and that of gravity in the phase behavior of a binary mixture of patchy particles. The species, 2A and 3B, have two A and three B patches, respectively, and only bonds between patches A and B (AB bonds) are allowed. This model describes colloidal systems where the aggregation of particles (3B) is mediated and controlled by a second species, the linkers (2A) to which they bind strongly. Thermodynamic calculations are performed using Wertheim's perturbation theory with a hard sphere reference term that accounts for the difference in the size of the two species. Percolation lines are determined employing a generalized Flory-Stockmayer theory, and the effects of gravity are included through a local density approximation. The bulk phase diagrams are calculated, and all the stacking sequences generated in the presence of gravity are determined and classified in a stacking diagram. The relative size of the particles can be used to control the phase behavior of the mixture. An increase in the size of particles 3B, relative to the size of the linkers 2A, is found to promote mixing while keeping the percolating structures and, in certain cases, leads to changes in the stacking sequence under gravity.

Crystal structures in binary hard-sphere colloid-droplet mixtures with patchy cross interactions

A binary mixture of droplets and patchy colloids, where patches are arranged in tetrahedral symmetry, is studied with Metropolis Monte Carlo simulations. The colloidal patches attract droplets, while both the colloid-colloid and the droplet-droplet interactions are hard sphere like. We find stable crystal structures with atomic analogs ZnS, CaF_{2}, and fcc or hcp (face centered cubic or hexagonal close packed) of the droplets coexisting with a dispersed fluid of the colloids. The simulated crystal structures agree well with those predicted by close-packing calculations for an intermediate range of droplet-colloid size ratios. A discrepancy between the simulations and theoretical predictions occurs at low and high size ratios. The results of the simulations for mixtures with anisotropic colloid-droplet interactions reveal a richer phase diagram, with ZnS-gas and ZnS-fluid coexistence, as compared to the isotropic case. For the example of a square planar patch arrangement, we find a particular crystal structure, consisting of two interpenetrating fcc or hcp lattices with right bond angles. Such a structure has no known atomic analog. Our study of generic models of anisotropic colloid-droplet mixtures could provide a promising way towards the fabrication of novel and complex colloidal structures.

Breakdown of the law of rectilinear diameter and related surprises in the liquid-vapor coexistence in systems of patchy particles

The phase diagram of molecular or colloidal systems depends strongly on the range and angular dependence of the interactions between the constituent particles. For instance, it is well known that the critical density of particles with "patchy" interactions shifts to lower values as the number of patches is decreased [see Bianchi et al. Phys. Rev. Lett. 97, 168301 (2006)]. Here, we present simulations that show that the phase behavior of patchy particles is even more interesting than had been appreciated. In particular, we find that, upon cooling below the critical point, the width of the liquid-vapor coexistence region of a system of particles with tetrahedrally arranged patches first increases, then decreases, and finally increases again. In other words, this system exhibits a doubly re-entrant liquid-vapor transition. As a consequence, the system exhibits a very large deviation from the law of rectilinear diameter, which assumes that the critical density can be obtained by linear extrapolation of the averages of the densities of the coexisting liquid and vapor phases. We argue that the unusual behavior of this system has the same origin as the density maximum in liquid water and is not captured by the Wertheim theory. The Wertheim theory also cannot account for our observation that the phase diagram of particles with three patches depends strongly on the geometrical distribution of the patches and on the degree to which their position on the particle surface is rigidly constrained. However, the phase diagram is less sensitive to small angular spreads in the patch locations. We argue that the phase behavior reported in this paper should be observable in experiments on patchy colloids and may be relevant for the liquid-liquid equilibrium in solutions of properly functionalized dendrimers.

Optimal number of linkers per monomer in linker-mediated aggregation

We study the dynamics of diffusion-limited irreversible aggregation of monomers, where bonds are mediated by linkers. We combine kinetic Monte Carlo simulations of a lattice model with a mean-field theory to study the dynamics when the diffusion of aggregates is negligible and only monomers diffuse. We find two values of the number of linkers per monomer which maximize the size of the largest aggregate. We explain the existence of the two maxima based on the distribution of linkers per monomer. This observation is well described by a simple mean-field model. We also show that a relevant parameter is the ratio of the diffusion coefficients of monomers and linkers. In particular, when this ratio is close to ten, the two maxima merge at a single maximum.

A cluster size distribution theory to study the thermodynamics and phase behavior of multi-bonding single site solutes in patchy colloidal mixtures

We study binary mixtures of multi-bonding single site solute particles in a solvent comprising patchy colloid particles. The particles in the mixture interact by very short-ranged attraction and hard-sphere repulsion. The attractive patch on the solute can bond with multiple solvent particles, whereas the patch on the solvent is restricted to bond only once. From a quasi-chemical analysis of association, in the hard-sphere reference we develop an accurate multi-body correlation information for the distribution of solvent particles over the patch region of the solute. We use this information within Wertheim's multi-density formalism to develop a cluster size distribution theory that is capable of capturing the physics of multi-body association for any geometry of association sites on the solute. We use this general framework to study a mixture containing Janus solutes and one- or two-patch solvent particles over a range of concentration of the solute and association strengths. We find that a mixture of two-patch solvent (with both patches of the same kind) and multi-bonding solutes with different patch geometries can have a vapor-liquid equilibrium, although the pure components themselves cannot phase separate. The liquid state occurs at very low densities, forming a so-called empty liquid. For the relative association strengths studied in this work, we observe that the vapor-liquid coexistence curve broadens as the concentration of the patchy solvent particles in the liquid phase is increased. The pressure-composition phase equilibrium curves show negative azeotropes for these mixtures. We also observe that, for these mixtures, as the size of the patch on the solute particles is decreased, the critical temperature and the critical packing fraction decreases.

Percolation of functionalized colloids on patterned substrates

We study the percolation properties for a system of functionalized colloids on patterned substrates via Monte Carlo simulations. The colloidal particles are modeled as hard disks with three equally-distributed attractive patches on their perimeter. We describe the patterns on the substrate as circular potential wells of radius Rp arranged in a regular square or hexagonal lattice. We find a nonmonotonic behavior of the percolation threshold (packing fraction) as a function of Rp. For attractive wells, the percolation threshold is higher than the one for clean (non-patterned) substrates if the circular wells are non-overlapping and can only be lower if the wells overlap. For repulsive wells we find the opposite behavior. In addition, at high packing fractions the formation of both structural and bond defects suppress percolation. As a result, the percolation diagram is reentrant with the non-percolated state occurring at very low and intermediate densities.

In-situ Observation of Hierarchical Self-Assembly Driven by Bicontinuous Gelation in Mixed Nanodisc Dispersions

The search for new functional soft materials with precise and reconfigurable structures at the nano and meso-scale is a major challenge as well as objective of the current science. Patchy colloids of different shapes and functionalities are considered important new building blocks of a bottom-up approach towards rational design of new soft materials largely governed by anisotropic interactions. Herein, we investigate the self-assembly, growth of hierarchical microstructures and aging dynamics of 2D nano-platelets of two different aspect ratios (Laponite ~25 and Montmorillonite ~250) which form gels with different porosity that is achieved by tuning their mixing ratios. Qualitative in situ real-space studies are carried out, including fluorescent confocal microscopy imaging of the bicontinuous gelation process or final states, which provides dynamic visualization of the self-organization. The bicontinuous gels exhibit a foam-like morphology having pores of a few micrometers in size that can be tuned by varying the mixing ratio of nanoplatelets. It is shown that this new class of clay gels has unique and tunable physical properties that will find potential applications in the development of low cost lithium ion batteries, nanocomposites and nuclear waste management.

Phase behavior and bulk structural properties of a microphase former with anisotropic competing interactions: A density functional theory study

Using classical density functional theory, we investigate systems exhibiting interactions where a short-range anisotropic attractive force competes with a long-range spherically symmetric repulsive force. The former is modelled within Wertheim's first-order perturbation theory for patchy particles, and the repulsive part is assumed to be a Yukawa potential which is taken into account via a mean-field approximation. From previous studies of systems with spherically symmetric competing interactions, it is well known that such systems can exhibit stable bulk cluster phases (microphase separation) provided that the repulsion is sufficiently weak compared to the attraction. For the present model system, we find rich phase diagrams including both reentrant clustering and liquid-gas binodals. In particular, the model predicts inhomogeneous bulk phases at extremely low packing fractions, which cannot be observed in systems with isotropic competing interactions.

A density functional theory for association of fluid molecules with a functionalized surface: fluid-wall single and double bonding

In this manuscript we extend Wertheim's two-density formalism beyond its first order to model a system of fluid molecules with a single association site close to a planar hard wall with association sites on its surface in a density functional theory framework. The association sites of the fluid molecules are small enough that they can form only one bond, while the wall association sites are large enough to bond with more than one fluid molecule. The effects of temperature and of bulk fluid and wall site densities on the fluid density profile, extent of association, and competition between single and double bonding of fluid segments at the wall sites versus distance from the wall are presented. The theory predictions are compared with new Monte Carlo simulation results and they are in good agreement. The theory captures the surface coverage over wide ranges of temperature and bulk density by introducing the effect of steric hindrance in fluid association at a wall site.

Modeling phase transitions in mixtures of β-γ lens crystallins

We analyze the experimentally determined phase diagram of a γD-βB1 crystallin mixture. Proteins are described as dumbbells decorated with attractive sites to allow inter-particle interaction. We use thermodynamic perturbation theory to calculate the free energy of such mixtures and, by applying equilibrium conditions, also the compositions and concentrations of the co-existing phases. Initially we fit the Tcloudversus packing fraction η measurements for a pure (x2 = 0) γD solution in 0.1 M phosphate buffer at pH = 7.0. Another piece of experimental data, used to fix the model parameters, is the isotherm x2vs. η at T = 268.5 K, at the same pH and salt content. We use the conventional Lorentz-Berthelot mixing rules to describe cross interactions. This enables us to determine: (i) model parameters for pure βB1 crystallin protein and to calculate; (ii) complete equilibrium surface (Tcloud-x2-η) for the crystallin mixtures. (iii) We present the results for several isotherms, including the tie-lines, as also the temperature-packing fraction curves. Good agreement with the available experimental data is obtained. An interesting result of these calculations is evidence of the coexistence of three phases. This domain appears for the region of temperatures just out of the experimental range studied so far. The input parameters, leading good description of experimental data, revealed a large difference between the numbers of the attractive sites for γD and βB1 proteins. This interesting result may be related to the fact that γD has a more than nine times smaller quadrupole moment than its partner in the mixture.

A density functional theory for colloids with two multiple bonding associating sites

Wertheim's multi-density formalism is extended for patchy colloidal fluids with two multiple bonding patches. The theory is developed as a density functional theory to predict the properties of an associating inhomogeneous fluid. The equation of state developed for this fluid depends on the size of the patch, and includes formation of cyclic, branched and linear clusters of associated species. The theory predicts the density profile and the fractions of colloids in different bonding states versus the distance from one wall as a function of bulk density and temperature. The predictions from our theory are compared with previous results for a confined fluid with four single bonding association sites. Also, comparison between the present theory and Monte Carlo simulation indicates a good agreement.

Temperature (de)activated patchy colloidal particles

We present a new model of patchy particles in which the interaction sites can be activated or deactivated by varying the temperature of the system. We study the thermodynamics of the system by means of Wertheim's first order perturbation theory, and use Flory-Stockmayer theory of polymerization to analyse the percolation threshold. We find a very rich phase behaviour including lower critical points and reentrant percolation.

Colloidal Recycling: Reconfiguration of Random Aggregates into Patchy Particles

The key ingredients to the successful bottom-up construction of complex materials are believed to be colloids with anisotropic shapes and directional, or patchy, interactions. We present an approach for creating such anisotropic patchy particles based on reconfiguring randomly shaped aggregates of colloidal spheres. While colloidal aggregates are often undesirable in colloidal dispersions due to their random shapes, we exploit them as a starting point to synthesize patchy particles. By a deliberate destabilization of the colloidal particles, diffusion-limited aggregation is induced which partitions the particles into randomly shaped aggregates with controlled size distribution. We achieve a reconfiguration of the aggregates into uniform structures by swelling the polymer spheres with an apolar solvent. The swelling lowers the attractive van der Waals forces, lubricates the contact area between the spheres, and drives the reorganization through minimization of the interfacial energy of the swollen polymer network. This reorganization process yields patchy particles whose patch arrangement is uniform for up to five patches. For particles with more patches, we find that the patch orientation depends on the degree of phase separation between the spheres and the monomer. This enables the synthesis of patchy particles with unprecedented patch arrangements. We demonstrate the broad applicability of this recycling strategy for making patchy particles as well as clusters of spheres by varying the swelling ratio, swelling solvent, surfactant concentration, and swelling time.

Thermodynamic perturbation theory for self-assembling mixtures of divalent single patch colloids

In this work we extend Wertheim's thermodynamic perturbation theory (TPT) to binary mixtures (species A and species B) of patchy colloids were each species has a single patch which can bond a maximum of twice (divalent). Colloids are treated as hard spheres with a directional conical association 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 chains, of all chain lengths, as well as 4-mer rings consisting of two species A and two species B colloids are accounted for. With the restriction of only AB attractions, triatomic rings of doubly bonded colloids, which dominate in the corresponding pure component case, cannot form. The theory is shown to be in good agreement with Monte Carlo simulation data for the structure and thermodynamics of these patchy colloid mixtures as a function of temperature, density, patch size and composition. It is shown that 4-mer rings dominate at low temperature, inhibiting the polymerization of the mixture into long chains. Mixtures of this type have been recently synthesized by researchers. This work provides the first theory capable of accurately modeling these mixtures.