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- Criticality of colloids with three distinct interaction patches: As simple as A, B, C?Publication . Tavares, Jose; Teixeira, PauloWe systematically study the phase behavior of a simple model of associating fluids which consists of hard spherical particles with three short-ranged attractive sites on their surfaces (sticky spots or patches), of types A, B, and C, that can form bonds with energy ij (i,j = A,B,C). We consider realizations of the model with one, two, or three nonzero ij. Using Wertheim’s first order perturbation theory of association, we establish the minimum requirements on the bond energies for the model to exhibit a liquid-vapor critical point, and investigate the nature of criticality in each case. As a preliminary, we rigorously show that, within this theory, particles with M identical sites do not condense if M < 3, a result that was previously conjectured, but never proved.
- Phase behaviour of pure and mixed patchy colloids - theory and simulationPublication . Teixeira, Paulo; Tavares, JoseWe review the phase behaviour of pure and mixed patchy colloids, as revealed (mostly) by theory and computer simulation. These experimentally-realisable systems are excellent models for investigating the general problem of the interplay between (equilibrium) phase transitions and self-assembly in soft condensed matter. We focus on how liquid vapour condensation can be pre-empted by the formation of different types of aggregates, in particular rings, which we argue is relevant to the criticality of empty fluids and network fluids, and possibly also of dipolar fluids. In this connection we also discuss percolation and gelation in pure and mixed patchy colloids. Finally, we describe the rich phase behaviour of (mostly binary) patchy colloid mixtures.
- How patchy can one get and still condense? The role of dissimilar patches in the interactions of colloidal particlesPublication . Tavares, Jose; Teixeira, Paulo; Gama, MargaridaWe investigate the influence of strong directional, or bonding, interactions on the phase diagram of complex fluids, and in particular on the liquid-vapour critical point. To this end we revisit a simple model and theory for associating fluids which consist of spherical particles having a hard-core repulsion, complemented by three short-ranged attractive sites on the surface (sticky spots). Two of the spots are of type A and one is of type B; the interactions between each pair of spots have strengths [image omitted], [image omitted] and [image omitted]. The theory is applied over the whole range of bonding strengths and results are interpreted in terms of the equilibrium cluster structures of the coexisting phases. In systems where unlike sites do not interact (i.e. where [image omitted]), the critical point exists all the way to [image omitted]. By contrast, when [image omitted], there is no critical point below a certain finite value of [image omitted]. These somewhat surprising results are rationalised in terms of the different network structures of the two systems: two long AA chains are linked by one BB bond (X-junction) in the former case, and by one AB bond (Y-junction) in the latter. The vapour-liquid transition may then be viewed as the condensation of these junctions and we find that X-junctions condense for any attractive [image omitted] (i.e. for any fraction of BB bonds), whereas condensation of the Y-junctions requires that [image omitted] be above a finite threshold (i.e. there must be a finite fraction of AB bonds).
- Percolation of colloids with distinct interaction sitesPublication . Tavares, Jose; Teixeira, Paulo; Gama, MargaridaWe generalize the Flory-Stockmayer theory of percolation to a model of associating (patchy) colloids, which consists of hard spherical particles, having on their surfaces f short-ranged-attractive sites of m different types. These sites can form bonds between particles and thus promote self-assembly. It is shown that the percolation threshold is given in terms of the eigenvalues of a m x m matrix, which describes the recursive relations for the number of bonded particles on the ith level of a cluster with no loops; percolation occurs when the largest of these eigenvalues equals unity. Expressions for the probability that a particle is not bonded to the giant cluster, for the average cluster size and the average size of a cluster to which a randomly chosen particle belongs, are also derived. Explicit results for these quantities are computed for the case f = 3 and m = 2. We show how these structural properties are related to the thermodynamics of the associating system by regarding bond formation as a (equilibrium) chemical reaction. This solution of the percolation problem, combined with Wertheim's thermodynamic first-order perturbation theory, allows the investigation of the interplay between phase behavior and cluster formation for general models of patchy colloids.
- Building up DNA, bit by bit: a simple description of chain assemblyPublication . Foffi, Riccardo; SCIORTINO, Francesco; Tavares, Jose; Teixeira, PauloWe 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.
- Interplay between self-assembly and condensation in models with asymmetric patchesPublication . Tavares, Jose; Teixeira, Paulo
- Criticality of colloids with distinct interaction patches: The limits of linear chains, hyperbranched polymers, and dimersPublication . Tavares, Jose; Teixeira, Paulo; Gama, MargaridaWe use a simple model of associating fluids which consists of spherical particles having a hard-core repulsion, complemented by three short-ranged attractive sites on the surface (sticky spots). Two of the spots are of type A and one is of type B; the bonding interactions between each pair of spots have strengths epsilon(AA), epsilon(BB), and epsilon(AB). The theory is applied over the whole range of bonding strengths and the results are interpreted in terms of the equilibrium cluster structures of the phases. In addition to our numerical results, we derive asymptotic expansions for the free energy in the limits for which there is no liquid-vapor critical point: linear chains (epsilon(AA)not equal 0, epsilon(AB)=epsilon(BB)=0), hyperbranched polymers (epsilon(AB)not equal 0, epsilon(AA)=epsilon(BB)=0), and dimers (epsilon(BB)not equal 0, epsilon(AA)=epsilon(AB)=0). These expansions also allow us to calculate the structure of the critical fluid by perturbing around the above limits, yielding three different types of condensation: of linear chains (AA clusters connected by a few AB or BB bonds); of hyperbranched polymers (AB clusters connected by AA bonds); or of dimers (BB clusters connected by AA bonds). Interestingly, there is no critical point when epsilon(AA) vanishes despite the fact that AA bonds alone cannot drive condensation.
- Phase diagrams of particles with dissimilar patches: X-junctions and Y-junctionsPublication . Tavares, Jose; Teixeira, PauloWe use Wertheim's first-order perturbation theory to investigate the phase behaviour and the structure of coexisting fluid phases for a model of patchy particles with dissimilar patches (two patches of type A and f(B) patches of type B). A patch of type alpha = {A, B} can bond to a patch of type beta = {A, B} in a volume nu(alpha beta), thereby decreasing the internal energy by epsilon(alpha beta). We analyse the range of model parameters where AB bonds, or Y-junctions, are energetically disfavoured (epsilon(AB) < epsilon(AA)/2) but entropically favoured (nu(AB) >> nu(alpha alpha)), and BB bonds, or X-junctions, are energetically favoured (epsilon(BB) > 0). We show that, for low values of epsilon(BB)/epsilon(AA), the phase diagram has three different regions: (i) close to the critical temperature a low-density liquid composed of long chains and rich in Y-junctions coexists with a vapour of chains; (ii) at intermediate temperatures there is coexistence between a vapour of short chains and a liquid of very long chains with X-and Y-junctions; (iii) at low temperatures an ideal gas coexists with a high-density liquid with all possible AA and BB bonds formed. It is also shown that in region (i) the liquid binodal is reentrant (its density decreases with decreasing temperature) for the lower values of epsilon(BB)/epsilon(AA). The existence of these three regions is a consequence of the competition between the formation of X- and Y-junctions: X-junctions are energetically favoured and thus dominate at low temperatures, whereas Y-junctions are entropically favoured and dominate at higher temperatures.
- Remnants of the disappearing critical point(s) in patchy fluids with distinct interaction patchesPublication . Tavares, Jose; Teixeira, PauloWe investigate the disappearance of the critical points of a model consisting of particles decorated with two patches of type A and a variable number (n) of patches of type B (2AnB patchy particles), in which only AA and AB bonds can form. This has been shown to exhibit a very rich phase behavior including one, two, or no liquid-vapor critical points, depending on two parameters: the ratio of the volumes available to each type of bond and the ratio of the bond strengths. We apply Wertheim's theory in the limit of strong AA bonds to a lattice version of the model [Almarza et al., J. Chem. Phys. 137, 244902 (2012)] and show that the critical point does not always vanish at zero density and temperature, in contrast with results for particles decorated with only one type of patch. We uncover two remnants of the critical points-the lines of maximum and ideal compressibility-that survive even when no critical points are present.
- Equilibrium self-assembly of colloids with distinct interaction sites: Thermodynamics, percolation, and cluster distribution functionsPublication . Tavares, Jose; Teixeira, Paulo; Gama, Margarida; SCIORTINO, FrancescoWe calculate the equilibrium thermodynamic properties, percolation threshold, and cluster distribution functions for a model of associating colloids, which consists of hard spherical particles having on their surfaces three short-ranged attractive sites (sticky spots) of two different types, A and B. The thermodynamic properties are calculated using Wertheim's perturbation theory of associating fluids. This also allows us to find the onset of self-assembly, which can be quantified by the maxima of the specific heat at constant volume. The percolation threshold is derived, under the no-loop assumption, for the correlated bond model: In all cases it is two percolated phases that become identical at a critical point, when one exists. Finally, the cluster size distributions are calculated by mapping the model onto an effective model, characterized by a-state-dependent-functionality (f) over bar and unique bonding probability (p) over bar. The mapping is based on the asymptotic limit of the cluster distributions functions of the generic model and the effective parameters are defined through the requirement that the equilibrium cluster distributions of the true and effective models have the same number-averaged and weight-averaged sizes at all densities and temperatures. We also study the model numerically in the case where BB interactions are missing. In this limit, AB bonds either provide branching between A-chains (Y-junctions) if epsilon(AB)/epsilon(AA) is small, or drive the formation of a hyperbranched polymer if epsilon(AB)/epsilon(AA) is large. We find that the theoretical predictions describe quite accurately the numerical data, especially in the region where Y-junctions are present. There is fairly good agreement between theoretical and numerical results both for the thermodynamic (number of bonds and phase coexistence) and the connectivity properties of the model (cluster size distributions and percolation locus).