Equilibrium gels of limited valence colloids

Percolation thresholds of disks with random nonoverlapping patches on four regular two-dimensional lattices

In percolation of patchy disks on lattices, each site is occupied by a disk, and neighboring disks are regarded as connected when their patches contact. Clusters of connected disks become larger as the patchy coverage of each disk χ increases. At the percolation threshold χ_{c}, an incipient cluster begins to span the whole lattice. For systems of disks with n symmetric patches on Archimedean lattices, a recent work [Wang et al., Phys. Rev. E 105, 034118 (2022)2470-004510.1103/PhysRevE.105.034118] found symmetric properties of χ_{c}(n), which are due to the coupling of the patches' symmetry and the lattice geometry. How does χ_{c} behave with increasing n if the patches are randomly distributed on the disks? We consider two typical random distributions of the patches, i.e., the equilibrium distribution and a distribution from random sequential adsorption. Combining Monte Carlo simulations and the critical polynomial method, we numerically determine χ_{c} for 106 models of different n on the square, honeycomb, triangular, and kagome lattices. The rules governing χ_{c}(n) are investigated in detail. They are quite different from those for disks with symmetric patches and could be useful for understanding similar systems.

Nucleic acid liquids

The confluence of recent discoveries of the roles of biomolecular liquids in living systems and modern abilities to precisely synthesize and modify nucleic acids (NAs) has led to a surge of interest in liquid phases of NAs. These phases can be formed primarily from NAs, as driven by base-pairing interactions, or from the electrostatic combination (coacervation) of negatively charged NAs and positively charged molecules. Generally, the use of sequence-engineered NAs provides the means to tune microsopic particle properties, and thus imbue specific, customizable behaviors into the resulting liquids. In this way, researchers have used NA liquids to tackle fundamental problems in the physics of finite valence soft materials, and to create liquids with novel structured and/or multi-functional properties. Here, we review this growing field, discussing the theoretical background of NA liquid phase separation, quantitative understanding of liquid material properties, and the broad and growing array of functional demonstrations in these materials. We close with a few comments discussing remaining open questions and challenges in the field.

Thermoresponsive oil-continuous gels based on double-interpenetrating colloidal-particle networks

Gels composed of multicomponent building blocks offer promising opportunities for the development of novel soft materials with unique and useful structures. While interpenetrating polymer networks have been extensively studied and applied in the creation of these gels, equivalent strategies utilizing colloidal particles have received limited scientific and technological attention. This study presents a novel class of thermo-responsive apolar double gels from interpenetrating networks of attractive colloidal silica and lipid particles. These double gels are easily assembled and suitable for the fabrication of 3D-printed edible soft constructs. Emphasis is focused on the rheological properties and structure emerging on the dilute regime (ϕ ≲ 0.1). Rheological investigations demonstrate that double gels exhibit greater stiffness and resilience to yielding compared to their single lipid gel counterparts. The scaling behavior of the oscillatory linear shear moduli and the critical strain for yielding with volume fraction remain comparable between single and double gels. Creep yielding in double gels exhibits two exponential decay regimes, suggesting the presence of thicker gel strands undergoing flow. Visualization and quantification of the quiescent microstructure confirms the existence of such denser aggregates devoid of larger clusters due to steric hindrance of interpenetrating networks in double gels. This is in stark contrast to lipid single gels where aggregates grow unrestrictedly into larger clusters. Our study constitutes the first demonstration on the assembly of apolar double gel networks as a promising avenue for the design of novel soft materials and foods with tailored structure and mechanics.

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.

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.

Predicting the size and morphology of nanoparticle clusters driven by biomolecular recognition

Nanoparticle aggregation is a driving principle of innovative materials and biosensing methodologies, improving transduction capabilities displayed by optical, electrical or magnetic measurements. This aggregation can be driven by the biomolecular recognition between target biomolecules (analytes) and receptors bound onto nanoparticle surface. Despite theoretical advances on modelling the entropic interaction in similar systems, predictions of the fractal morphologies of the nanoclusters of bioconjugated nanoparticles are lacking. The morphology of resulting nanoclusters is sensitive to the location, size, flexibility, average number of receptors per particle f̄, and the analyte-particle concentration ratio. Here we considered bioconjugated iron oxide nanoparticles (IONPs) where bonds are mediated by a divalent protein that binds two receptors attached onto different IONPs. We developed a protocol combining analytical expressions for receptors and linker distributions, and Brownian dynamics simulations for bond formation, and validated it against experiments. As more bonds become available (e.g., by adding analytes), the aggregation deviates from the ideal Bethe's lattice scenario due to multivalence, loop formation, and steric hindrance. Generalizing Bethe's lattice theory with a (not-integer) effective functionality feff leads to analytical expressions for the cluster size distributions in excellent agreement with simulations. At high analyte concentration steric impediment imposes an accessible limit value facc to feff, which is bounded by facc < feff < f̄. A transition to gel phase, is correctly captured by the derived theory. Our findings offer new insights into quantifying analyte amounts by assessing nanocluster size, and predicting nanoassembly morphologies accurately is a first step towards understanding variations of physical properties in clusters formed after biomolecular recognition.

Protein Association in Solution: Statistical Mechanical Modeling

Protein molecules associate in solution, often in clusters beyond pairwise, leading to liquid phase separations and high viscosities. It is often impractical to study these multi-protein systems by atomistic computer simulations, particularly in multi-component solvents. Instead, their forces and states can be studied by liquid state statistical mechanics. However, past such approaches, such as the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, were limited to modeling proteins as spheres, and contained no microscopic structure-property relations. Recently, this limitation has been partly overcome by bringing the powerful Wertheim theory of associating molecules to bear on protein association equilibria. Here, we review these developments.

Aggregation of amphiphilic nanocubes in equilibrium and under shear

We investigate the self-assembly of amphiphilic nanocubes into finite-sized aggregates in equilibrium and under shear, using molecular dynamics (MD) simulations and kinetic Monte Carlo (KMC) calculations. These patchy nanoparticles combine both interaction and shape anisotropy, making them valuable models for studying folded proteins and DNA-functionalized nanoparticles. The nanocubes can self-assemble into various finite-sized aggregates ranging from rods to self-avoiding random walks, depending on the number and placement of the hydrophobic faces. Our study focuses on suspensions containing multi- and one-patch cubes, with their ratio systematically varied. When the binding energy is comparable to the thermal energy, the aggregates consist of only few cubes that spontaneously associate/dissociate. However, highly stable aggregates emerge when the binding energy exceeds the thermal energy. Generally, the mean aggregation number of the self-assembled clusters increases with the number of hydrophobic faces and decreases with increasing fraction of one-patch cubes. In sheared suspensions, the more frequent collisions between nanocube clusters lead to faster aggregation dynamics but also to smaller terminal steady-state mean cluster sizes. The results from the MD and KMC simulations are in excellent agreement for all investigated two-patch cases, whereas the three-patch cubes form systematically smaller clusters in the MD simulations compared to the KMC calculations due to finite-size effects and slow aggregation kinetics. By analyzing the rate kernels, we are able to identify the primary mechanisms responsible for (shear-induced) cluster growth and breakup. This understanding allows us to tune nanoparticle and process parameters to achieve desired cluster sizes and shapes.

Colloidal Clusters and Networks Formed by Oppositely Charged Nanoparticles with Varying Stiffnesses

Colloidal clusters and gels are ubiquitous in science and technology. Particle softness has a strong effect on interparticle interactions; however, our understanding of the role of this factor in the formation of colloidal clusters and gels is only beginning to evolve. Here, we report the results of experimental and simulation studies of the impact of particle softness on the assembly of clusters and networks from mixtures of oppositely charged polymer nanoparticles (NPs). Experiments were performed below or above the polymer glass transition temperature, at which the interaction potential and adhesive forces between the NPs were significantly varied. Hard NPs assembled in fractal clusters that subsequently organized in a kinetically arrested colloidal gel, while soft NPs formed dense precipitating aggregates, due to the NP deformation and the decreased interparticle distance. Importantly, interactions of hard and soft NPs led to the formation of discrete precipitating NP aggregates at a relatively low volume fraction of soft NPs. A phenomenological model was developed for interactions of oppositely charged NPs with varying softnesses. The experimental results were in agreement with molecular dynamics simulations based on the model. This work provides insight on interparticle interactions before, during, and after the formation of hard-hard, hard-soft, and soft-soft contacts and has impact for numerous applications of reversible colloidal gels, including their use as inks for additive manufacturing.

Patchy colloidal gels under the influence of gravity

In this contribution, gravitational effects in gel-forming patchy colloidal systems are studied. We focus on how the gel structure is modified by gravity. Through Monte Carlo computer simulations of gel-like states recently identified by the rigidity percolation criterion [J. A. S. Gallegos et al., Phys. Rev. E 104, 064606 (2021)], the influence of the gravitational field, characterized by the gravitational Péclet number, Pe, on patchy colloids is studied in terms of the patchy coverage, χ. Our findings point out that there exists a threshold Péclet number, Pe_g, that depends on χ above which the gravitational field enhances the particle bonding and, in consequence, promotes the aggregation or clustering of particles; the smaller the χ value, the higher the Pe_g. Interestingly, when χ ∼ 1 (near the isotropic limit), our results are consistent with an experimentally determined threshold Pe value where gravity affects the gel formation in short-range attractive colloids. In addition, our results show that the cluster size distribution and the density profile undergo variations that lead to changes in the percolating cluster, i.e., gravity is able to modify the structure of the gel-like states. These changes have an important impact on the structural rigidity of the patchy colloidal dispersion; the percolating cluster goes from a uniform spatially network to a heterogeneous percolated structure, where an interesting structural scenario emerges, namely, depending on the Pe value, the new heterogeneous gel-like states can coexist with both diluted and dense phases or they simply reach a crystalline-like state. In the isotropic case, the increase in the Pe number can shift the critical temperature to higher temperatures; however, when Pe > 0.01, the binodal disappears and the particles fully sediment at the bottom of the sample cell. Furthermore, gravity moves the rigidity percolation threshold to lower densities. Finally, we also note that within the values of the Péclet number here explored, the cluster morphology is barely altered.

Metal-Organic Framework-Based Colloidal Particle Synthesis, Assembly, and Application

Metal-organic frameworks (MOFs) assembled from metal nodes and organic ligands have received significant attention over the past two decades for their fascinating porous properties and broad applications. Colloidal MOFs (CMOFs) not only inherit the intrinsic properties of MOFs, but can also serve as building blocks for self-assembly to make functional materials. Compared to bulk MOFs, the colloidal size of CMOFs facilitates further manipulation of CMOF particles in a single or collective state in a liquid medium. The resulting crystalline order obtained by self-assembly in position and orientation can effectively improve performance. In this review, we summarize the latest developments of CMOFs in synthesis strategies, self-assembly methods, and related applications. Finally, we discuss future challenges and opportunities of CMOFs in synthesis and assembly, by which we hope that CMOFs can be further developed into new areas for a wider range of applications.

Density-tunable pathway complexity in a minimalistic self-assembly model

An open challenge in self-assembly is learning how to design systems that can be conditionally guided towards different target structures depending on externally-controlled conditions. Using a theoretical and numerical approach, here we discuss a minimalistic self-assembly model that can be steered towards different types of ordered constructs at the equilibrium by solely tuning a facile selection parameter, namely the density of building blocks. Metadynamics and Langevin dynamics simulations allow us to explore the behavior of the system in and out of equilibrium conditions. We show that the density-driven tunability is encoded in the pathway complexity of the system, and specifically in the competition between two different main self-assembly routes. A comprehensive set of simulations provides insight into key factors allowing to make one self-assembling pathway prevailing on the other (or vice versa), determining the selection of the final self-assembled products. We formulate and validate a practical criterion for checking whether a specific molecular design is predisposed for such density-driven tunability of the products, thus offering a new, broader perspective to realize and harness this facile extrinsic control of conditional self-assembly.

Entropy-Driven Thermo-gelling Vitrimer

Thermo-gelling polymers have been envisioned as promising smart biomaterials but limited by their weak mechanical and thermodynamic stabilities. Here, we propose a new thermo-gelling vitrimer, which remains at a liquid state because of the addition of protector molecules preventing the crosslinking, and with increasing temperature, an entropy-driven crosslinking occurs to induce the sol-gel transition. Moreover, we find that the activation barrier in the metathesis reaction of vitrimers plays an important role, and experimentally, one can use catalysts to tune the activation barrier to drive the vitrimer to form an equilibrium gel at high temperature, which is not subject to any thermodynamic instability. We formulate a mean-field theory to describe the entropy-driven crosslinking of the vitrimer, which agrees quantitatively with computer simulations and paves the way for the design and fabrication of novel vitrimers for biomedical applications.

Engineering DNA-based synthetic condensates with programmable material properties, compositions, and functionalities

Biomolecular condensates participate in diverse cellular processes, ranging from gene regulation to stress survival. Bottom-up engineering of synthetic condensates advances our understanding of the organizing principle of condensates. It also enables the synthesis of artificial systems with novel functions. However, building synthetic condensates with a predictable organization and function remains challenging. Here, we use DNA as a building block to create synthetic condensates that are assembled through phase separation. The programmability of intermolecular interactions between DNA molecules enables the control over various condensate properties including assembly, composition, and function. Similar to the way intracellular condensates are organized, DNA clients are selectively partitioned into cognate condensates. We demonstrate that the synthetic condensates can accelerate DNA strand displacement reactions and logic gate operation by concentrating specific reaction components. We envision that the DNA-based condensates could help the realization of the high-order functions required to build more life-like artificial systems.

Designing Enhanced Entropy Binding in Single-Chain Nanoparticles

Single-chain nanoparticles (SCNPs) are a new class of bio- and soft-matter polymeric objects in which a fraction of the monomers are able to form equivalently intra- or interpolymer bonds. Here we numerically show that a fully entropic gas-liquid phase separation can take place in SCNP systems. Control over the discontinuous (first-order) change-from a phase of independent diluted (fully-bonded) polymers to a phase in which polymers entropically bind to each other to form a (fully-bonded) polymer network-can be achieved by a judicious design of the patterns of reactive monomers along the polymer chain. Such a sensitivity arises from a delicate balance between the distinct entropic contributions controlling the binding.

Extended Wertheim theory predicts the anomalous chain length distributions of divalent patchy particles under extreme confinement

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 experi- mental 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 simulations which is not well understood. To account for this observation, we interpret the polymerization as taking place in a highly confined quasi-2D plane and extend the Wertheim thermodynamic perturbation theory by defining addition reactions constants as functions of the chain length. We derive the theory, test it on simple square well potentials, and apply it to the experimental case of synthetic colloidal patchy particles immersed in a binary liquid mixture that are described by an accurate effective critical Casimir patchy particle potential. The important interaction parameters entering the theory are explicitly computed using the integral method in combination with Monte Carlo sampling. Without any adjustable parameter, the predictions of the chain length distribution are in excellent agreement with explicit simulations of self-assembling particles. We discuss generality of the approach, and its application range.

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.

Percolation thresholds of randomly rotating patchy particles on Archimedean lattices

We study the percolation of randomly rotating patchy particles on 11 Archimedean lattices in two dimensions. Each vertex of the lattice is occupied by a particle, and in each model the patch size and number are monodisperse. When there are more than one patches on the surface of a particle, they are symmetrically decorated. As the proportion χ of the particle surface covered by the patches increases, the clusters connected by the patches grow and the system percolates at the threshold χ_{c}. We combine Monte Carlo simulations and the critical polynomial method to give precise estimates of χ_{c} for disks with one to six patches and spheres with one to two patches on the 11 lattices. For one-patch particles, we find that the order of χ_{c} values for particles on different lattices is the same as that of threshold values p_{c} for site percolation on these lattices, which implies that χ_{c} for one-patch particles mainly depends on the geometry of lattices. For particles with more patches, symmetry becomes very important in determining χ_{c}. With the estimates of χ_{c} for disks with one to six patches, using analyses related to symmetry, we are able to give precise values of χ_{c} for disks with an arbitrary number of patches on all 11 lattices. The following rules are found for patchy disks on each of these lattices: (1) as the number of patches n increases, values of χ_{c} repeat in a periodic way, with the period n_{0} determined by the symmetry of the lattice; (2) when mod(n,n_{0})=0, the minimum threshold value χ_{min} appears, and the model is equivalent to site percolation with χ_{min}=p_{c}; and (3) disks with mod(n,n_{0})=m and n_{0}-m (m<n_{0}/2) share the same χ_{c} value. The results can be useful references for studying the connectivity of patchy particles on two-dimensional lattices at finite temperatures.

The physics of empty liquids: from patchy particles to water

Empty liquids represent a wide class of materials whose constituents arrange in a random network through reversible bonds. Many key insights on the physical properties of empty liquids have originated almost independently from the study of colloidal patchy particles on one side, and a large body of theoretical and experimental research on water on the other side. Patchy particles represent a family of coarse-grained potentials that allows for a precise control of both the geometric and the energetic aspects of bonding, while water has arguably the most complex phase diagram of any pure substance, and a puzzling amorphous phase behavior. It was only recently that the exchange of ideas from both fields has made it possible to solve long-standing problems and shed new light on the behavior of empty liquids. Here we highlight the connections between patchy particles and water, focusing on the modelling principles that make an empty liquid behave like water, including the factors that control the appearance of thermodynamic and dynamic anomalies, the possibility of liquid-liquid phase transitions, and the crystallization of open crystalline structures.

Extended law of corresponding states: square-well oblates

The vapour-liquid coexistence collapse in the reduced temperature,T_r=T/T_c, reduced density,ρ_r=ρ/ρ_c, plane is known as a principle of corresponding states, and Noro and Frenkel have extended it for pair potentials of variable range. Here, we provide a theoretical basis supporting this extension, and show that it can also be applied to short-range pair potentials where both repulsive and attractive parts can be anisotropic. We observe that the binodals of oblate hard ellipsoids for a given aspect ratio (κ= 1/3) with varying short-range square-well interactions collapse into a single master curve in theΔB2*-ρ_rplane, whereΔB2*=(B2(T)-B2(Tc))/v0,B₂is the second virial coefficient, andv₀is the volume of the hard body. This finding is confirmed by both REMC simulation and second virial perturbation theory for varying square-well shells, mimicking uniform, equator, and pole attractions. Our simulation results reveal that the extended law of corresponding states is not related to the local structure of the fluid.

Location of the gel-like boundary in patchy colloidal dispersions: Rigidity percolation, structure, and particle dynamics

During the past decade, there has been a hot debate about the physical mechanisms that determine when a colloidal dispersion approaches the gel transition. However, there is still no consensus on a possible unique route that leads to the conditions for the formation of a gel-like state. Based on gel states identified in experiments, Valadez-Pérez et al. [Phys. Rev. E 88, 060302(R) (2013)PLEEE81539-375510.1103/PhysRevE.88.060302] proposed rigidity percolation as the precursor of colloidal gelation in adhesive hard-sphere dispersions with coordination number 〈n_{b}〉 equal to 2.4. Although this criterion was originally established to describe mechanical transitions in network-forming molecular materials with highly directional interactions, it worked well to explain gel formation in colloidal suspensions with isotropic short-range attractive forces. Recently, this idea has also been used to account for the dynamical arrest experimentally observed in attractive spherocylinders. Then, by assuming that rigidity percolation also drives gelation in spherical colloids interacting with short-ranged and highly directional potentials, we locate the thermodynamic states where gelation seems to occur in dispersions made up of patchy colloids. To check whether the criterion 〈n_{b}〉=2.4 also holds in patchy colloidal systems, we apply the so-called bond-bending analysis to determine the fraction of floppy modes at some percolating clusters. This analysis confirms that the condition 〈n_{b}〉=2.4 is a good approximation to determine those percolating clusters that are either mechanically stable or rigid. Furthermore, our results point out that not all combinations of patches and coverages lead to a gel-like state. Additionally, we systematically study the structure and the cluster size distribution along those thermodynamic states identified as gels. We show that for high coverage values, the structure is very similar for systems that have the same coverage regardless the number or the position of the patches on the particle surface. Finally, by using dynamic Monte Carlo computer simulations, we calculate both the mean-square displacement and the intermediate scattering function at and in the neighborhood of the gel-like states.

The flexibility-based modulation of DNA nanostar phase separation

Phase separation of biomolecules plays key roles in physiological compartmentalization as well as pathological aggregation. A deeper understanding of biomolecular phase separation requires dissection of a relation between intermolecular interactions and resulting phase behaviors. DNA nanostars, multivalent DNA assemblies of which sticky ends define attractive interactions, represent an ideal system to probe this fundamental relation governing phase separation processes. Here, we use DNA nanostars to systematically study how structural flexibility exhibited by interacting species impacts their phase behaviors. We design multiple nanostars with a varying degree of flexibility using single-stranded gaps of different lengths in the arm of each nanostar unit. We find that structural flexibility drastically alters the phase diagram of DNA nanostars in such a way that the phase separation of more flexible structures is strongly inhibited. This result is not due to self-inhibition from the loss of valency but rather ascribed to a generic flexibility-driven change in the thermodynamics of the system. Our work provides not only potential regulatory mechanisms cells may exploit to dynamically control intracellular phase separation but also a route to build synthetic systems of which assembly can be controlled in a signal dependent manner.

Sol/gel transition of oil/water microemulsions controlled by surface grafted triblock copolymer dodecyl–PEO227–dodecyl: molecular dynamics simulations with experimentally validated interaction potential

We studied a large range of identical spherical oil/water microemulsion (O/W-MI) volume fractions. The O/W-MIs are stabilized by cetylpyridinium chloride ionic surfactant (CpCl) and octanol cosurfactant and dispersed in salt water. We grafted different numbers of dodecyl–(polyEthylene oxide)₂₂₇–dodecyl triblock copolymer that we note (n(D–PEO₂₂₇–D)), where n varies from 0 to 12. We accomplished the grafting process by replacing a small amount of CpCl and octanol with the appropriate n(D–PEO₂₂₇–D). The aim is to determine the interaction/structure relationship of the covered microemulsions. Precisely, we are interested in a quantitative investigation of the influence of volume fraction Φ, temperature (T), and n(D–PEO₂₂₇–D) on the microemulsion sol/gel transition. To this end, we first study the uncoated microemulsion structure depending only on Φ. Second, we determine the coated microemulsions structure as a function of n(D–PEO₂₂₇–D) for different Φ. Third, we examine the effect of temperature on the uncoated and coated microemulsion. We show that the sol/gel transition is controlled by the three main parameters, Φ, T, and n(D–PEO₂₂₇–D). Accordingly, the uncoated microemulsion sol/gel transition, at ambient temperature, occurred for Φ ≃ 33.65%. By increasing Φ, the O/W-MIs show a glass state, which occurs, along with the gel state, at Φ ≃ 37% and arises clearly at Φ ≃ 60%. The coated O/W-MI sol/gel transition is found to be linearly dependent on n(D–PEO₂₂₇–D) and takes place for Φ ≃ 26.5% for n(D–PEO₂₂₇–D) = 12. Ordinarily, the decrease in temperature leads to gel formation of microemulsions for low Φ. Additionally, in this work, we found that the gelation temperature increases linearly with n(D–PEO₂₂₇–D). Thus, the parameter n(D–PEO₂₂₇–D) can control the sol/gel transition of the O/W-MIs at ambient temperature and moderate Φ.

We studied the sol/gel transition of oil/water microemulsion (O/W-MI) covered with dodecyl–(polyEthylene oxide)₂₂₇–dodecyl triblock copolymer.

Friction-Controlled Entropy-Stability Competition in Granular Systems

Using cyclic shear to drive a two-dimensional granular system, we determine the structural characteristics for different interparticle friction coefficients. These characteristics are the result of a competition between mechanical stability and entropy, with the latter's effect increasing with friction. We show that a parameter-free maximum-entropy argument alone predicts an exponential cell order distribution, with excellent agreement with the experimental observation. We show that friction only tunes the mean cell order and, consequently, the exponential decay rate and the packing fraction. We further show that cells, which can be very large in such systems, are short-lived, implying that our systems are liquidlike rather than glassy.

Programmable Anisotropy and Percolation in Supramolecular Patchy Particle Gels

Patchy particle interactions are predicted to facilitate the controlled self-assembly and arrest of particles into phase-stable and morphologically tunable "equilibrium" gels, which avoids the arrested phase separation and subsequent aging that is typically observed in traditional particle gels with isotropic interactions. Despite these promising traits of patchy particle interactions, such tunable equilibrium gels have yet to be realized in the laboratory due to experimental limitations associated with synthesizing patchy particles in high yield. Here, we introduce a supramolecular metal-coordination platform consisting of metallic nanoparticles linked by telechelic polymer chains, which validates the predictions associated with patchy particle interactions and facilitates the design of equilibrium particle hydrogels through limited valency interactions. We demonstrate that the interaction valency and self-assembly of the particles can be effectively controlled by adjusting the relative concentration of polymeric linkers to nanoparticles, which enables the gelation of patchy particle hydrogels with programmable local anisotropy, morphology, and low mechanical percolation thresholds. Moreover, by crowding the local environment around the patchy particles with competing interactions, we introduce an independent method to control the self-assembly of the nanoparticles, thereby enabling the design of highly anisotropic particle hydrogels with substantially reduced percolation thresholds. We thus establish a canonical platform that facilitates multifaceted control of the self-assembly of the patchy nanoparticles en route to the design of patchy particle gels with tunable valencies, morphologies, and percolation thresholds. These advances lay important foundations for further fundamental studies of patchy particle systems and for designing tunable gel materials that address a wide range of engineering applications.

Gelling without Structuring: A SAXS Study of the Interactions among DNA Nanostars

We evaluate, by means of synchrotron small-angle X-ray scattering, the shape and mutual interactions of DNA tetravalent nanostars as a function of temperature in both the gas-like state and across the gel transition. To this end, we calculate the form factor from coarse-grained molecular dynamics simulations with a novel method that includes hydration effects; we approximate the radial interaction of DNA nanostars as a hard-sphere potential complemented by a repulsive and an attractive Yukawa term; and we predict the structure factors by exploiting the perturbative random phase approximation of the Percus-Yevick equation. Our approach enables us to fit all the data by selecting the particle radius and the width and amplitude of the attractive potential as free parameters. We determine the evolution of the structure factor across gelation and detect subtle changes of the effective interparticle interactions, that we associate to the temperature and concentration dependence of the particle size. Despite the approximations, the approach here adopted offers new detailed insights into the structure and interparticle interactions of this fascinating system.

Enhanced protein adsorption upon bulk phase separation

In all areas related to protein adsorption, from medicine to biotechnology to heterogeneous nucleation, the question about its dominant forces and control arises. In this study, we used ellipsometry and quartz-crystal microbalance with dissipation (QCM-D), as well as density-functional theory (DFT) to obtain insight into the mechanism behind a wetting transition of a protein solution. We established that using multivalent ions in a net negatively charged globular protein solution (BSA) can either cause simple adsorption on a negatively charged interface, or a (diverging) wetting layer when approaching liquid-liquid phase separation (LLPS) by changing protein concentration (cp) or temperature (T). We observed that the water to protein ratio in the wetting layer is substantially larger compared to simple adsorption. In the corresponding theoretical model, we treated the proteins as limited-valence (patchy) particles and identified a wetting transition for this complex system. This wetting is driven by a bulk instability introduced by metastable LLPS exposed to an ion-activated attractive substrate.

Multivalent Patchy Colloids for Quantitative 3D Self-Assembly Studies

We report methods to synthesize sub-micron- and micron-sized patchy silica particles with fluorescently labeled hemispherical titania protrusions, as well as routes to efficiently characterize these particles and self-assemble these particles into non-close-packed structures. The synthesis methods expand upon earlier work in the literature, in which silica particles packed in a colloidal crystal were surface-patterned with a silane coupling agent. Here, hemispherical amorphous titania protrusions were successfully labeled with fluorescent dyes, allowing for imaging by confocal microscopy and super-resolution techniques. Confocal microscopy was exploited to experimentally determine the numbers of protrusions per particle over large numbers of particles for good statistical significance, and these distributions were compared to simulations predicting the number of patches as a function of core particle polydispersity and maximum separation between the particle surfaces. We self-assembled these patchy particles into open percolating gel networks by exploiting solvophobic attractions between the protrusions.

Using Patchy Particles to Prevent Local Rearrangements in Models of Non-equilibrium Colloidal Gels

Simple models based on isotropic interparticle attractions often fail to capture experimentally observed structures of colloidal gels formed through spinodal decomposition and subsequent arrest: the resulting gels are typically denser and less branched than their experimental counterparts. Here, we simulate gels formed from soft particles with directional attractions ("patchy particles"), designed to inhibit lateral particle rearrangement after aggregation. We directly compare simulated structures with experimental colloidal gels made using soft attractive microgel particles, by employing a "skeletonization" method that reconstructs the three-dimensional backbone from experiment or simulation. We show that including directional attractions with sufficient valency leads to strongly branched structures compared to isotropic models. Furthermore, combining isotropic and directional attractions provides additional control over aggregation kinetics and gel structure. Our results show that the inhibition of lateral particle rearrangements strongly affects the gel topology and is an important effect to consider in computational models of colloidal gels.

Arrested dynamics of the dipolar hard sphere model

We report the combined results of molecular dynamics simulations and theoretical calculations concerning various dynamical arrest transitions in a model system representing a dipolar fluid, namely, N (soft core) rigid spheres interacting through a truncated dipole-dipole potential. By exploring different regimes of concentration and temperature, we find three distinct scenarios for the slowing down of the dynamics of the translational and orientational degrees of freedom: at low (η = 0.2) and intermediate (η = 0.4) volume fractions, both dynamics are strongly coupled and become simultaneously arrested upon cooling. At high concentrations (η≥ 0.6), the translational dynamics shows the features of an ordinary glass transition, either by compressing or cooling down the system, but with the orientations remaining ergodic, thus indicating the existence of partially arrested states. In this density regime, but at lower temperatures, the relaxation of the orientational dynamics also freezes. The physical scenario provided by the simulations is discussed and compared against results obtained with the self-consistent generalized Langevin equation theory, and both provide a consistent description of the dynamical arrest transitions in the system. Our results are summarized in an arrested states diagram which qualitatively organizes the simulation data and provides a generic picture of the glass transitions of a dipolar fluid.

Length-Dependence and Spatial Structure of DNA Partitioning into a DNA Liquid

Cells can spatially and temporally control biochemistry using liquid-liquid phase separation to form membrane-less organelles. Synthetic biomolecular liquids offer a means to study the mechanisms of this process, as well as offering a route to the creation of functional biomimetic materials. With these goals in mind, we here examine the partitioning of long double-stranded DNA linkers into a liquid composed of small DNA particles ("nanostars") whose phase separation is driven by base pairing. We find that linker partitioning is length-dependent because of a confinement penalty of inserting long strands within the liquid's characteristic mesh size. We quantify this entropic-confinement effect using a simple partitioning theory and show that its magnitude is consistent with classic Odijk pictures of confined worm-like chains. Linker partitioning can also lead to inhomogeneous structures: long linkers excluded from the liquid interior tend to preferentially accumulate on the surface of liquid droplets (i.e., acting as surfactants), while linkers forced at high concentrations into the liquid undergo a secondary phase separation, forming metastable droplet-in-droplet structures. Altogether, our work demonstrates the ability to rationally engineer the composition and structure of a model biomolecular liquid.

A Colloid Approach to Self-Assembling Antibodies

Concentrated solutions of monoclonal antibodies have attracted considerable attention due to their importance in pharmaceutical formulations; yet, their tendency to aggregate and the resulting high viscosity pose considerable problems. Here we tackle this problem by a soft condensed matter physics approach, which combines a variety of experimental measurements with a patchy colloid model, amenable of analytical solution. We thus report results of structural antibodies and dynamic properties obtained through scattering methods and microrheological experiments. We model the data using a colloid-inspired approach, explicitly taking into account both the anisotropic shape of the molecule and its charge distribution. Our simple patchy model is able to disentangle self-assembly and intermolecular interactions and to quantitatively describe the concentration-dependence of the osmotic compressibility, collective diffusion coefficient, and zero shear viscosity. Our results offer new insights on the key problem of antibody formulations, providing a theoretical and experimental framework for a quantitative assessment of the effects of additional excipients or chemical modifications and a prediction of the resulting viscosity.

Increasing valence pushes DNA nanostar networks to the isostatic point

The classic picture of soft material mechanics is that of rubber elasticity, in which material modulus is related to the entropic elasticity of flexible polymeric linkers. The rubber model, however, largely ignores the role of valence (i.e., the number of network chains emanating from a junction). Recent work predicts that valence, and particularly the Maxwell isostatic point, plays a key role in determining the mechanics of semiflexible polymer networks. Here, we report a series of experiments confirming the prominent role of valence in determining the mechanics of a model system. The system is based on DNA nanostars (DNAns): multiarmed, self-assembled nanostructures that form thermoreversible equilibrium gels through base pair-controlled cross-linking. We measure the linear and nonlinear elastic properties of these gels as a function of DNAns arm number, f, and concentration [DNAns]. We find that, as f increases from three to six, the gel's high-frequency plateau modulus strongly increases, and its dependence on [DNAns] transitions from nonlinear to linear. Additionally, higher-valence gels exhibit less strain hardening, indicating that they have less configurational freedom. Minimal strain hardening and linear dependence of shear modulus on concentration at high f are consistent with predictions for isostatic systems. Evident strain hardening and nonlinear concentration dependence of shear modulus suggest that the low-f networks are subisostatic and have a transient, potentially fractal percolated structure. Overall, our observations indicate that network elasticity is sensitive both to entropic elasticity of network chains and to junction valence, with an apparent isostatic point [Formula: see text] in agreement with the Maxwell prediction.

Patchy Particle Models to Understand Protein Phase Behavior

In this chapter we describe numerical procedures to evaluate the phase behavior of coarse-grained models for globular proteins. Specifically we focus on models based on hard spheres complemented with "patchy-like" anisotropic interactions that mimic the attractive regions on the surface of the proteins. We introduce the basic elements of grand canonical Monte Carlo simulations for these types of models in which rotational and translational moves need to be accounted for. We describe the techniques for the estimation of the fluid-fluid critical point, coexistence curve, and fluid-crystal boundaries. We also discuss an efficient method for the evaluation of the fluid-fluid phase diagram: the successive umbrella sampling technique. Finally we briefly describe how to exploit the same tools for the calculation of the phase behavior of protein binary mixtures.

Microrheology of DNA hydrogel gelling and melting on cooling

We present systematic characterisation by means of dynamic light scattering and particle tracking techniques of the viscosity and of the linear viscoelastic moduli, G'(ω) and G''(ω), for two different DNA hydrogels. These thermoreversible systems are composed of tetravalent DNA-made nanostars whose sticky sequence is designed to provide controlled interparticle bonding. While the first system forms a gel on cooling, the second one has been programmed to behave as a re-entrant gel, turning again to a fluid solution at low temperature. The frequency-dependent viscous and storage moduli and the viscosity reveal the different viscoelastic behavior of the two DNA hydrogels. Our results show how little variations in the design of the DNA sequences allow tuning of the mechanical response of these biocompatible all-DNA materials.

The Proof Is in the Pidan: Generalizing Proteins as Patchy Particles

The Chinese century egg, or pidan, is a traditional preparation of duck eggs that can be stored for months at room temperature without degradation. Raw eggs are soaked in a strong alkaline and salt solution, and the albumin gradually forms a stable, transparent gel. Here, we show that pidan gels belong to the class of materials formed from "patchy particles". We found that the β-sheet structure of ovalbumin, the major protein constituent of egg white, is preserved during gelation, while α-helical regions undergo a degree of unfolding into unstructured random coils that may form attractive patches. Upon dilution in additional strong base, the phase behavior of pidan gels is consistent with patchy-particle thermodynamics. This protein gel is also physically and structurally similar to the protein gels that form the squid lens. Both systems exhibit patchy thermodynamics, and the constituent proteins share physical features including a structured, charged core, and polar, unstructured "arms" that form attractive patches. Our work provides a path toward rational design of proteins for precisely structured, volume-spanning materials.

Binding branched and linear DNA structures: From isolated clusters to fully bonded gels

The proper design of DNA sequences allows for the formation of well-defined supramolecular units with controlled interactions via a consecution of self-assembling processes. Here, we benefit from the controlled DNA self-assembly to experimentally realize particles with well-defined valence, namely, tetravalent nanostars (A) and bivalent chains (B). We specifically focus on the case in which A particles can only bind to B particles, via appropriately designed sticky-end sequences. Hence AA and BB bonds are not allowed. Such a binary mixture system reproduces with DNA-based particles the physics of poly-functional condensation, with an exquisite control over the bonding process, tuned by the ratio, r, between B and A units and by the temperature, T. We report dynamic light scattering experiments in a window of Ts ranging from 10 °C to 55 °C and an interval of r around the percolation transition to quantify the decay of the density correlation for the different cases. At low T, when all possible bonds are formed, the system behaves as a fully bonded network, as a percolating gel, and as a cluster fluid depending on the selected r.

Speeding up Monte Carlo simulation of patchy hard cylinders

The hard cylinder model decorated with attractive patches proved to be very useful recently in studying physical properties of several colloidal systems. Phase diagram, elastic constants and cholesteric properties obtained from computer simulations based on a simple hard cylinder model have been all successfully and quantitatively compared to experimental results. Key to these simulations is an efficient algorithm to check the overlap between hard cylinders. Here, we propose two algorithms to check the hard cylinder overlap and we assess their efficiency through a comparison with an existing method available in the literature and with the well-established algorithm for simulating hard spherocylinders. In addition, we discuss a couple of optimizations for performing computer simulations of patchy anisotropic particles and we estimate the speed-up which they can provide in the case of patchy hard cylinders.

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.