How patchy can one get and still condense? The role of dissimilar patches in the interactions of colloidal particles (original) (raw)
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Physical Review E, 2009
We 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 spots͒. Two of the spots are of type A and one is of type B; the bonding interactions between each pair of spots have strengths ⑀ AA , ⑀ BB , and ⑀ 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 ͑⑀ AA 0, ⑀ AB = ⑀ BB =0͒, hyperbranched polymers ͑⑀ AB 0, ⑀ AA = ⑀ BB =0͒, and dimers ͑⑀ BB 0, ⑀ AA = ⑀ AB =0͒. 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 bonds͒; of hyperbranched polymers ͑AB clusters connected by AA bonds͒; or of dimers ͑BB clusters connected by AA bonds͒. Interestingly, there is no critical point when ⑀ AA vanishes despite the fact that AA bonds alone cannot drive condensation.
Remnants of the disappearing critical point(s) in patchy fluids with distinct interaction patches
The Journal of Chemical Physics, 2020
We 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.
Theory of Chain Association versus Liquid Condensation
Physical Review Letters, 1996
We combine the original van der Waals description for liquid condensation with the association theory of ideal particles into a simple association theory of nonideal chains. The theory shows that vapor-liquid coexistence becomes metastable if the tendency to form weakly interacting chains is sufficiently strong. Our findings qualitatively explain recent computer simulations on dipolar hard spheres. [S0031-9007(96) PACS numbers: 61.20. Gy, 64.70.Fx, 82.35.+t, 82.60.Hc In 1873 van der Waals argued that the existence of a dilute disordered phase, a vapor, and a condensed disordered phase, a liquid, can be explained by assuming longrange attraction and short-range repulsion between the constituent particles . By now it is well established that the prototype simple fluid, described by a repulsive core and an attractive isotropic 1͞r 6 pair potential (r the particle separation), indeed gives rise to vapor-liquid coexistence below a critical temperature . Also the prototype ionic fluid, consisting of charge carrying hard spheres, is now known to have a vapor-liquid critical point . Here the attraction is provided by the Coulombic 1͞r attractions (possibly screened) between oppositely charged particles.
The Journal of Chemical Physics, 2011
We study a model consisting of particles with dissimilar bonding sites ("patches"), which exhibits self-assembly into chains connected by Y-junctions, and investigate its phase behaviour by both simulations and theory. We show that, as the energy cost j of forming Y-junctions increases, the extent of the liquid-vapour coexistence region at lower temperatures and densities is reduced. The phase diagram thus acquires a characteristic "pinched" shape in which the liquid branch density decreases as the temperature is lowered. To our knowledge, this is the first model in which the predicted topological phase transition between a fluid composed of short chains and a fluid rich in Y-junctions is actually observed. Above a certain threshold for j , condensation ceases to exist because the entropy gain of forming Y-junctions can no longer offset their energy cost. We also show that the properties of these phase diagrams can be understood in terms of a temperature-dependent effective valence of the patchy particles.
2010
We calculate the network fluid regime and phase diagrams of binary mixtures of patchy colloids, using Wertheim's first order perturbation theory and a generalization of Flory-Stockmayer's theory of polymerization. The colloids are modelled as hard spheres with the same diameter and surface patches of the same type, AAA. The only difference between species is the number of their patches -or functionality-, fA(1)f_A^{(1)}fA(1) and fA(2)f_A^{(2)}fA(2) (with fA(2)>fA(1)f_A^{(2)}>f_A^{(1)}fA(2)>fA(1)). We have found that the difference in functionality is the key factor controlling the behaviour of the mixture in the network (percolated) fluid regime. In particular, when fA(2)ge2fA(1)f_A^{(2)}\ge2f_A^{(1)}fA(2)ge2fA(1) the entropy of bonding drives the phase separation of two network fluids which is absent in other mixtures. This changes drastically the critical properties of the system and drives a change in the topology of the phase diagram (from type I to type V) when fA(1)>2f_A^{(1)}>2fA(1)>2. The difference in functionality also determines the miscibility at high (osmotic) pressures. If fA(2)−fA(1)=1f_A^{(2)}-f_A^{(1)}=1fA(2)−fA(1)=1 the mixture is completely miscible at high pressures, while closed miscibility gaps at pressures above the highest critical pressure of the pure fluids are present if fA(2)−fA(1)>1f_A^{(2)}-f_A^{(1)}>1fA(2)−fA(1)>1. We argue that this phase behaviour is driven by a competition between the entropy of mixing and the entropy of bonding, as the latter dominates in the network fluid regime.
Chemical Physics, 2011
We investigate the effect of distinct bonding energies on the onset of criticality of low functionality fluid mixtures. We focus on mixtures of particles with two and three patches as this includes the mixture where "empty" fluids were originally reported. In addition to the number of patches, the species differ in the type of patches or bonding sites. For simplicity, we consider that the patches on each species are identical: one species has three patches of type A and the other has two patches of type B. We have found a rich phase behavior with closed miscibility gaps, liquid-liquid demixing, and negative azeotropes. Liquid-liquid demixing was found to pre-empt the "empty" fluid regime, of these mixtures, when the AB bonds are weaker than the AA or BB bonds. By contrast, mixtures in this class exhibit "empty" fluid behavior when the AB bonds are stronger than at least one of the other two. Mixtures with bonding energies ɛBB = ɛAB and ɛAA < ɛBB, were found to exhibit an unusual negative azeotrope.
The Journal of Chemical Physics, 2010
We 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 spots͒ 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-dependentfunctionality f and unique bonding probability p. 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-junctions͒ if ⑀ AB / ⑀ AA is small, or drive the formation of a hyperbranched polymer if ⑀ AB / ⑀ 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 coexistence͒ and the connectivity properties of the model ͑cluster size distributions and percolation locus͒.
Phase transition and crossover behavior of colloidal fluids under confinement
Chemical Physics Letters, 2010
We report a molecular simulation study on the non-monotonic behavior of critical temperature, T cp , of a confined Yukawa fluid. Close to the adhesive hard sphere (AHS) range of the surface-fluid interaction, T cp monotonically increases with increasing surface-fluid interaction range. Subsequently, after a certain threshold value, depending on the surface interaction well depth, T cp decreases monotonically with further increase in the surface interaction range. On the other hand, critical density and pressure show increasing monotonic trends with the surface interaction range. The crossover from 3D to 2D behavior for colloidal fluid in attractive pores is observed around a slit width of 14 molecular diameters for the studied system in this work.
Two types of dynamic crossovers in a network-forming liquid with tetrahedral symmetry
Liquids with tetrahedral symmetry of the first coordination shell often display anomalous thermodynamic and dynamic behaviors. The main reason for these anomalies is that pressurizing such liquids leads to the disordering of this local symmetry by the particles migrating from the second to the first coordination shell. This in some cases may lead to the increase of entropy upon pressurizing and consequently to the volume increase upon cooling, as well as increase of diffusivity upon pressurizing. Under certain circumstances, pressurizing or cooling these substances may lead to a first-order phase transition between two liquids with different local structures, entropies, energies and densities. The liquid-liquid first-order phase transition can end in a liquid-liquid critical point (LLCP). The Widom line, defined as the line of zero ordering field, emanates from the LLCP into the supercritical region. In the vicinity of the LLCP thermodynamic response functions have extrema along different loci that converge to the LLCP and can approximate the Widom line. In particular, the maxima of the specific heat are associated to continuous structural changes in the liquid and, in general, to dynamic crossovers. Here we present a model of a network-forming liquid with tetrahedral symmetry in which each response function has two loci of maxima as function of temperature at constant pressure. One locus has positive slope in the pressure-temperature (P-T) thermodynamic plane, and the other has negative slope. We show that for each locus there is a dynamic crossover in the diffusivity and that the two crossovers are qualitatively different. For the positively sloped locus, occurring at P above the pressure P C of the LLCP, the crossover is from low activation energy at high T to high activation energy at low T. For the negatively sloped locus with P b P C , the crossover is characterized by an increase of activation energy in a certain temperature interval but with similar activation energies at low and high T. Such a behavior has been proposed for water where an apparent glass transition, associated with the increase of the activation energy at high T, could be avoided if the activation energy would decrease in the region where experiments are difficult, the so called "no-man's-land". (S.V. Buldyrev).
Phase diagrams of binary mixtures of patchy colloids with distinct numbers of patches
2010
We calculate the network fluid regime and phase diagrams of binary mixtures of patchy colloids, using Wertheim's first order perturbation theory and a generalization of Flory-Stockmayer's theory of polymerization. The colloids are modelled as hard spheres with the same diameter and surface patches of the same type, AAA. The only difference between species is the number of their patches -or functionality-, fA(1)f_A^{(1)}fA(1) and fA(2)f_A^{(2)}fA(2) (with fA(2)>fA(1)f_A^{(2)}>f_A^{(1)}fA(2)>fA(1)). We have found that the difference in functionality is the key factor controlling the behaviour of the mixture in the network (percolated) fluid regime. In particular, when fA(2)ge2fA(1)f_A^{(2)}\ge2f_A^{(1)}fA(2)ge2fA(1) the entropy of bonding drives the phase separation of two network fluids which is absent in other mixtures. This changes drastically the critical properties of the system and drives a change in the topology of the phase diagram (from type I to type V) when fA(1)>2f_A^{(1)}>2fA(1)>2. The difference in functionality also determines the miscibility at high (osmotic) pressures. If fA(2)−fA(1)=1f_A^{(2)}-f_A^{(1)}=1fA(2)−fA(1)=1 the mixture is completely miscible at high pressures, while closed miscibility gaps at pressures above the highest critical pressure of the pure fluids are present if fA(2)−fA(1)>1f_A^{(2)}-f_A^{(1)}>1fA(2)−fA(1)>1. We argue that this phase behaviour is driven by a competition between the entropy of mixing and the entropy of bonding, as the latter dominates in the network fluid regime.