Connecting Irreversible to Reversible Aggregation: Time and Temperature

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.

Correlated Ion Transport and the Gel Phase in Room Temperature Ionic Liquids

Here we present a theory of ion aggregation and gelation of room temperature ionic liquids (RTILs). Based on it, we investigate the effect of ion aggregation on correlated ion transport-ionic conductivity and transference numbers-obtaining closed-form expressions for these quantities. The theory depends on the maximum number of associations a cation and anion can form and the strength of their association. To validate the presented theory, we perform molecular dynamics simulations on several RTILs and a range of temperatures for one RTIL. The simulations indicate the formation of large clusters, even percolating through the system under certain circumstances, thus forming a gel, with the theory accurately describing the obtained cluster distributions in all cases. However, based on the strength and lifetime of associations in the simulated RTILs, we expect free ions to dominate ionic conductivity despite the presence of clusters, and we do not expect the percolating cluster to trigger structural arrest in the RTIL.

Exploiting limited valence patchy particles to understand autocatalytic kinetics

Autocatalysis, i.e., the speeding up of a reaction through the very same molecule which is produced, is common in chemistry, biophysics, and material science. Rate-equation-based approaches are often used to model the time dependence of products, but the key physical mechanisms behind the reaction cannot be properly recognized. Here, we develop a patchy particle model inspired by a bicomponent reactive mixture and endowed with adjustable autocatalytic ability. Such a coarse-grained model captures all general features of an autocatalytic aggregation process that takes place under controlled and realistic conditions, including crowded environments. Simulation reveals that a full understanding of the kinetics involves an unexpected effect that eludes the chemistry of the reaction, and which is crucially related to the presence of an activation barrier. The resulting analytical description can be exported to real systems, as confirmed by experimental data on epoxy-amine polymerizations, solving a long-standing issue in their mechanistic description.

Dynamics of Patchy Particles in and out of Equilibrium

We combine particle-based simulations, mean-field rate equations, and Wertheim's theory to study the dynamics of patchy particles in and out of equilibrium, at different temperatures and densities. We consider an initial random distribution of nonoverlapping three-patch particles, with no bonds, and analyze the time evolution of the breaking and bonding rates of a single bond. We find that the asymptotic (equilibrium) dynamics differs from the initial (out of equilibrium) one. These differences are expected to depend on the initial conditions, temperature, and density.

Dynamics of network fluids

Network fluids are structured fluids consisting of chains and branches. They are characterized by unusual physical properties, such as, exotic bulk phase diagrams, interfacial roughening and wetting transitions, and equilibrium and nonequilibrium gels. Here, we provide an overview of a selection of their equilibrium and dynamical properties. Recent research efforts towards bridging equilibrium and non-equilibrium studies are discussed, as well as several open questions.

Structure formation in nanocomposite hydrogels

We use molecular dynamics simulations to study structure formation in physically associating nanocomposite hydrogels. Nanofillers were modeled as rigid bodies of disk-like shapes and physical crosslinks were simulated by introducing a short-range attraction between the nanofillers and polymer chain ends. The structure, dynamics and mechanics of these polymer gels were studied as a function of nanofiller volume fraction. We observe the formation of a percolated network in the hydrogels, with an ordered local structure but disordered globally, as we increase the filler fraction. This locally ordered structure was a result of the anisotropy of the disk-like fillers. The dynamics of polymers showed significant caging effects in the gel state. Stress autocorrelation and elongation results were analyzed as a function of nano-filler concentrations. Comparisons with nanofillers of different shapes showed that disk-like nanofillers are more effective in strengthening the hydrogels than spherical nanofillers.

Simulation and Theory of Antibody Binding to Crowded Antigen-Covered Surfaces

In this paper we introduce a fully flexible coarse-grained model of immunoglobulin G (IgG) antibodies parametrized directly on cryo-EM data and simulate the binding dynamics of many IgGs to antigens adsorbed on a surface at increasing densities. Moreover, we work out a theoretical model that allows to explain all the features observed in the simulations. Our combined computational and theoretical framework is in excellent agreement with surface-plasmon resonance data and allows us to establish a number of important results. (i) Internal flexibility is key to maximize bivalent binding, flexible IgGs being able to explore the surface with their second arm in search for an available hapten. This is made clear by the strongly reduced ability to bind with both arms displayed by artificial IgGs designed to rigidly keep a prescribed shape. (ii) The large size of IgGs is instrumental to keep neighboring molecules at a certain distance (surface repulsion), which essentially makes antigens within reach of the second Fab always unoccupied on average. (iii) One needs to account independently for the thermodynamic and geometric factors that regulate the binding equilibrium. The key geometrical parameters, besides excluded-volume repulsion, describe the screening of free haptens by neighboring bound antibodies. We prove that the thermodynamic parameters govern the low-antigen-concentration regime, while the surface screening and repulsion only affect the binding at high hapten densities. Importantly, we prove that screening effects are concealed in relative measures, such as the fraction of bivalently bound antibodies. Overall, our model provides a valuable, accurate theoretical paradigm beyond existing frameworks to interpret experimental profiles of antibodies binding to multi-valent surfaces of different sorts in many contexts.

Antibodies are the main working horses of the human immune system. Remarkably, no matter the size or the shape of the pathological intruders, these extremely flexible three-lobe molecules are able to form a complex, thus eliciting an immune response. What makes antibodies so effective? To answer this and other questions, we have developed a simplified computational scheme to simulate the dynamics of many antibodies interacting with each other and with antigens. Coarse-grained models are a great opportunity, as they give access to a true multi-scale approach to biologically relevant problems. In this work, our innovative method allowed us to simulate the binding process of many antibodies to surface-adsorbed antigens. This led us to elucidate and quantify many important physical aspects of their biological function in agreement with experiments, such as the role of their flexibility and crowding effects at the hapten-covered surface, which were shown to finely regulate the avidity.

Generic model for tunable colloidal aggregation in multidirectional fields

Based on Brownian Dynamics computer simulations in two dimensions we investigate aggregation scenarios of colloidal particles with directional interactions induced by multiple external fields. To this end we propose a model which allows continuous change in the particle interactions from point-dipole-like to patchy-like (with four patches). We show that, as a result of this change, the non-equilibrium aggregation occurring at low densities and temperatures transforms from conventional diffusion-limited cluster aggregation (DLCA) to slippery DLCA involving rotating bonds; this is accompanied by a pronounced change of the underlying lattice structure of the aggregates from square-like to hexagonal ordering. Increasing the temperature we find a transformation to a fluid phase, consistent with results of a simple mean-field density functional theory.

Adsorbed films of three-patch colloids: continuous and discontinuous transitions between thick and thin films

We investigate numerically the role of spatial arrangement of the patches on the irreversible adsorption of patchy colloids on a substrate. We consider spherical three-patch colloids and study the dependence of the kinetics on the opening angle between patches. We show that growth is suppressed below and above minimum and maximum opening angles, revealing two absorbing phase transitions between thick and thin film regimes. While the transition at the minimum angle is continuous, in the directed percolation class, that at the maximum angle is clearly discontinuous. For intermediate values of the opening angle, a rough colloidal network in the Kardar-Parisi-Zhang universality class grows indefinitely. The nature of the transitions was analyzed in detail by considering bond flexibility, defined as the dispersion of the angle between the bond and the center of the patch. For the range of flexibilities considered we always observe two phase transitions. However, the range of opening angles where growth is sustained increases with flexibility. At a tricritical flexibility, the discontinuous transition becomes continuous. The practical implications of our findings and the relation to other nonequilibrium transitions are discussed.

Structure and rheology of colloidal particle gels: insight from computer simulation

A particle gel is a network of aggregated colloidal particles with soft solid-like mechanical properties. Its structural and rheological properties, and the kinetics of its formation, are dependent on the sizes and shapes of the constituent particles, the volume fraction of the particles, and the nature of the interactions between the particles before, during and after gelation. Particle gels may be permanent or transient depending on whether the colloidal forces between the aggregating particles lead to irreversible bonding or weak reversible interactions. With short-range reversible interactions, network formation is typically associated with phase separation or kinetic arrest due to particle crowding. Much existing knowledge has been derived from computer simulations of idealized model systems containing spherical particles interacting with well-defined pair potentials. The status of current progress is reviewed here by summarizing the underlying methodology and key findings from a range of simulation approaches: Monte Carlo, molecular dynamics, Brownian dynamics, Stokesian dynamics, dissipative particle dynamics, multiparticle collision dynamics, and fluid particle dynamics. Then it is described how the technique of Brownian dynamics simulation, in particular, has provided detailed insight into how different kinds of bonding and weak reversible interactions can affect the aggregate fractal structure, the percolation behaviour, and the small-deformation rheological properties of network-forming colloidal systems. A significant ongoing development has been the establishment and testing of efficient algorithms that are able to capture the subtle dynamic structuring effects that arise from effects of interparticle hydrodynamic interactions. This has led to an appreciation recently of the potentially important role of these particle-particle hydrodynamic effects in controlling the evolving morphology of simulated colloidal aggregates and in defining the location of the sol-gel phase boundary.

Self-assembly of multicomponent structures in and out of equilibrium

Theories of phase change and self-assembly often invoke the idea of a "quasiequilibrium," a regime in which the nonequilibrium association of building blocks results nonetheless in a structure whose properties are determined solely by an underlying free energy landscape. Here we study a prototypical example of multicomponent self-assembly, a one-dimensional fiber grown from red and blue blocks. We find that if the equilibrium structure possesses compositional correlations different from those characteristic of random mixing, then it cannot be generated without error at any finite growth rate: there is no quasiequilibrium regime. However, by exploiting dynamic scaling, structures characteristic of equilibrium at one point in phase space can be generated, without error, arbitrarily far from equilibrium. Our results, supported by mean-field theory in higher dimensions, thus suggest a "nonperturbative" strategy for multicomponent self-assembly in which the target structure is, by design, not the equilibrium one.

Electrostatics and aggregation: how charge can turn a crystal into a gel

The crystallization of proteins or colloids is often hindered by the appearance of aggregates of low fractal dimension called gels. Here we study the effect of electrostatics upon crystal and gel formation using an analytic model of hard spheres bearing point charges and short range attractive interactions. We find that the chief electrostatic free energy cost of forming assemblies comes from the entropic loss of counterions that render assemblies charge-neutral. Because there exists more accessible volume for these counterions around an open gel than a dense crystal, there exists an electrostatic entropic driving force favoring the gel over the crystal. This driving force increases with increasing sphere charge, but can be counteracted by increasing counterion concentration. We show that these effects cannot be fully captured by pairwise-additive macroion interactions of the kind often used in simulations, and we show where on the phase diagram to go in order to suppress gel formation.

Primitive models of patchy colloidal particles. A review

In this article I will review some recent studies of the phase behavior and of the self-assembly of patchy colloidal particles. These studies have been based on simple primitive models for colloid–colloid interactions, effectively extending to soft matter the seminal work of I. Nezbeda on associated fluids. I will discuss the possibilities offered by the study of the self-assembly of particles with limited valence in deepening our understanding of the onset of the liquid state, of the differences between gels and glasses and of the possible connection between physical and chemical gels. A review with 55 references.

Gel formation through reversible and irreversible aggregation

We study the kinetics of formation of branched loopless structures in mixtures of particles with different shapes and functionalities. These systems are treated with the appropriate Smoluchowski rate equations, including condensation and fragmentation terms, and it is shown that it is possible to provide a parameter-free description of the assembly process, including the limit of irreversible aggregation at low temperatures. Using dynamics simulations we provide evidence of a connection between physical and chemical gelation in low-valence particle systems, and the possibility of relating ageing time with temperature.

Reversible gels of patchy particles: role of the valence

We simulate a binary mixture of colloidal patchy particles with two and three patches, respectively, for several relative concentrations and hence relative average valences. For these limited-valence systems, it is possible to reach low temperatures, where the lifetime of the patch-patch interactions becomes longer than the observation time without encountering phase separation in a colloid-poor (gas) and a colloid rich (liquid) phase. The resulting arrested state is a fully connected long-lived network where particles with three patches provide the branching points connecting chains of two-patch particles. We investigate the effect of the valence on the structural and dynamic properties of the resulting gel and attempt to provide a theoretical description of the formation and of the resulting gel structure based on a combination of the Wertheim theory for associated liquids and the Flory-Stockmayer approach for modeling chemical gelation.