Arrested states in colloidal fluids with competing interactions: A static replica study.

We present the first systematic application of the integral equation implementation of the replica method to the study of arrested states in fluids with microscopic competing interactions (short-range attractive and long-range repulsive, SALR), as exemplified by the prototype Lennard-Jones-Yukawa model. Using a wide set of potential parameters, we provide as many as 11 different phase diagrams on the density (ρ)-temperature (T) plane, embodying both the cluster-phase boundary, TC(ρ), and the locus below which arrest takes place, TD(ρ). We describe how the interplay between TC and TD-with the former falling on top of the other, or the other way around, depending on thermodynamic conditions and potential parameters-gives rise to a rich variety of non-ergodic states interspersed with ergodic ones, of which both the building blocks are clusters or single particles. In a few cases, we find that the TD locus does not extend all over the density range subtended by the TC envelope; under these conditions, the λ-line is within reach of the cluster fluid, with the ensuing possibility to develop ordered microphases. Whenever a comparison is possible, our predictions favorably agree with previous numerical results. Thereby, we demonstrate the reliability and effectiveness of our scheme to provide a unified theoretical framework for the study of arrested states in SALR fluids, irrespective of their nature.

Microphase versus macrophase separation in the square-well-linear fluid: A theoretical and computational study.

Due to the presence of competing interactions, the square-well-linear fluid can exhibit either liquid-vapor equilibrium (macrophase separation) or clustering (microphase separation). Here we address the issue of determining the boundary between these two regimes, i.e., the Lifshitz point, expressed in terms of a relationship between the parameters of the model. To this aim, we carry out Monte Carlo simulations to compute the structure factor of the fluid, whose behavior at low wave vectors accurately captures the tendency of the fluid to form aggregates or, alternatively, to phase separate. Specifically, for a number of different combinations of attraction and repulsion ranges, we make the system go across the Lifshitz point by increasing the strength of the repulsion. We use simulation results to benchmark the performance of two theories of fluids, namely, the hypernetted chain (HNC) equation and the analytically solvable random phase approximation (RPA); in particular, the RPA theory is applied with two different prescriptions as for the direct correlation function inside the core. Overall, the HNC theory proves to be an appropriate tool to characterize the fluid structure and the low-wave-vector behavior of the structure factor is consistent with the threshold between microphase and macrophase separation established through simulation. The structural predictions of the RPA theory turn out to be less accurate, but this theory offers the advantage of providing an analytical expression of the Lifshitz point. Compared to simulation, both RPA schemes predict a Lifshitz point that falls within the macrophase-separation region of parameters: in the best case, barriers roughly twice higher than predicted are required to attain clustering conditions.

Like aggregation from unlike attraction: stripes in symmetric mixtures of cross-attracting hard spheres.

Self-assembly of colloidal particles into striped phases is at once a process of relevant technological interest-just think about the possibility to realise photonic crystals with a dielectric structure modulated along a specific direction-and a challenging task, since striped patterns emerge in a variety of conditions, suggesting that the connection between the onset of stripes and the shape of the intermolecular potential is yet to be fully unravelled. Hereby, we devise an elementary mechanism for the formation of stripes in a basic model consisting of a symmetric binary mixture of hard spheres that interact via a square-well cross attraction. Such a model would mimic a colloid in which the interspecies affinity is of longer range and significantly stronger than the intraspecies interaction. For attraction ranges shorter enough than the particle size the mixture behaves like a compositionally-disordered simple fluid. Instead, for wider square-wells, we document by numerical simulations the existence of striped patterns in the solid phase, where layers of particles of one species are interspersed with layers of the other species; increasing the attraction range stabilises the stripes further, in that they also appear in the bulk liquid and become thicker in the crystal. Our results lead to the counterintuitive conclusion that a flat and sufficiently long-ranged unlike attraction promotes the aggregation of like particles into stripes. This finding opens a novel way for the synthesis of colloidal particles with interactions tailored at the development of stripe-modulated structures.

Competition between clustering and phase separation in binary mixtures containing SALR particles.

We investigate by Monte Carlo simulations a mixture of particles with competing interactions (hard-sphere two-Yukawa, HSTY) and hard spheres (HS), with same diameters σ and a square-well (SW) cross attraction. In a recent study [G. Munaò et al., J. Phys. Chem. B, 2022, 126, 2027-2039], we have analysed situations-in terms of relative concentration and attraction strength-where HS promote the formation of clusters involving particles of both species under thermodynamic conditions that would not allow for clustering of the pure HSTY fluid. Here, we focus on the role played by the range of cross attraction in determining the equilibrium structure of the mixture, starting from a homogeneous low-density state. When the width of the well exceeds approximately σ, clustering takes place in the system, with aggregates characterised by various sizes and shapes. Only for low HSTY concentrations (less than 10%) a single big cluster appears, anticipating the behaviour observed for a wider well, around 1.2σ. In the latter case, a spherical cluster encompassing almost all particles is the stable structure at equilibrium. We interpret this outcome as a macrophase, liquid-vapour separation where the spherical cluster is just the form taken at low density by the liquid phase inside the vapour phase: indeed, when the density takes larger values, periodic boundary conditions select liquid-vapour interfaces with other non-spherical shapes, similarly as found for a finite sample of simple fluid going through the liquid-vapour coexistence region. For still higher densities we document the existence of a solid phase characterized by the alternation of bilayers filled with particles of one species and bilayers of the other species, giving the solid a peculiar wafer structure.

Glass quantization of the Gaussian core model.

We use the replica method to study the dynamical glass transition of the Gaussian core model, a system of ultrasoft repulsive spheres interacting via a Gaussian potential, focusing on low temperatures and low-to-moderate densities. At constant temperature, an amorphous glassy state is entered upon a first compression but this glass melts as the density is further increased. In addition to this reentrant transition, a second, smooth transition is discovered between a continuous and a discretized glass. The properties of the former are continuous functions of temperatures, whereas the latter exhibits a succession of stripes, characterized by discontinuous jumps of the glassiness parameters. The glass physics of ultrasoft particles is hence richer than that of impenetrable particles for reasons that can be attributed to the ability of the former to create and break out-of-equilibrium clusters of overlapping particles.

Clustering in Mixtures of SALR Particles and Hard Spheres with Cross Attraction.

Self-assembling complex fluids are often modeled as particles with effective competing isotropic interactions, combining a short-range attraction (SA) followed by a longer-range repulsion (LR). For moderately low temperatures and densities, SALR particles form clusters in equilibrium, at least provided that the potential parameters are appropriate. Here we inquire into the possibility that cluster formation in SALR fluids might be pushed by a foreign species even under thermodynamic conditions that would not allow for clusterization of the pure system. To this aim, we study by Monte Carlo simulations a mixture of hard-sphere two-Yukawa particles and hard spheres, with a cross interaction modeled by a square-well attraction, and we investigate the conditions of clustering in terms of strength of attraction and relative concentration of the two species. We find that clusters can occur in the mixture for the same temperature and density where the pure SALR fluid is almost structureless. In particular, we single out a cross attraction such that clusters are formed with a SALR concentration as low as 5%. We also find a situation where nearly pure droplets of hard spheres are held together by a shell of SALR particles. Conversely, we show that clustering can be undermined in the mixture under conditions for which this process takes place in the parent SALR fluid. Using a simple criterion, based on the second virial coefficients of the attractive part of interaction potentials (the so-called "reference attractive fluids"), we are able to predict accurately whether clustering is favored (or hindered) in the mixture, as compared to the pure SALR fluid.

Local order and cluster formation in model fluids with competing interactions: a simulation and theoretical study.

In a preliminary study [Phys. Chem. Chem. Phys., 2017, 19, 15247], we have recently documented an elusive mechanism underlying the cluster formation in model fluids with microscopic competing interactions (hard-sphere two-Yukawa). This mechanism consists in a tiny rearrangement of a distant correlation peak in the local density profile. For weak attractions, this peak contributes to the shallow, long-wave oscillation typical of such fluids; as the attraction strengthens, such a portion progressively disengages from the long-range behaviour, and moving backwards takes on the character of a new shell of neighbours, falling beyond the existing ones at shorter distances. This "reversal of trend" - despite its tiny size, in comparison with the overall aspect of the density profile - is shown to precisely occur at the onset of clustering. The scope of the present study is twofold. In the first instance, we positively assess our preliminary finding. To this aim we have studied by Monte Carlo simulations different families of two-Yukawa fluids, under the same conditions investigated in the original paper, namely fixed temperature, high fluid-density and increasingly attractive strength. Apparently, the reversal of trend in spatial correlations sets as a sensitive criterion to identify the clustering threshold, complementing other common indicators, based on the modifications undergone by the low-wavevector peak in the structure factor. Secondly, we document the accuracy of the Hypernetted Chain theory in predicting the spatial rearrangement under scrutiny. This evidence paves the way to an extended investigation of the observed phenomenology by the complementary use of theoretical and simulation tools.

Revisiting the replica theory of the liquid to ideal glass transition.

The replica theory of the "Random First Order Transition" (RFOT) from a supercooled liquid to an "ideal" glass of a system of "soft spheres" is revisited. Following the seminal work of Mézard and Parisi [J. Chem. Phys. 111, 1076 (1999)], the number m of weakly interacting replicas of the system is varied continuously from m = 2 to m < 1. Relevant order parameters and the free energy of the liquid and glass phases are calculated using the hypernetted chain (HNC) approximation for the pair correlation functions. The scenario observed for all m confirms the existence of two glass branches G1 and G2. The latter has the lowest free energy for all m > 1, while the former has a lower free energy for m < 1 but is shown to be unstable against spinodal decomposition for any nonzero value of the attractive inter-replica coupling. The critical temperature Tcr of the RFOT turns out to depend on m, which may be a by-product of the approximation inherent in the HNC closure. The RFOT is predicted to be weakly first order, characterized by a small jump in density between the coexisting liquid and G2 phases for all m > 1. Estimating Tcr in the limit m → 1 requires a proper extrapolation of high resolution HNC calculations. The present protocol explores the behavior of the free energy of the ideal glass phase below Tcr as a function of m.

A semianalytical "reverse" approach to link structure and microscopic interactions in two-Yukawa competing fluids.

We study theoretically a prototype hard-sphere two-Yukawa model with competing interactions, under thermodynamic conditions associated with the formation of clusters. We adopt the analytically solvable random phase approximation and show that this theory predicts reasonably well the structure of the fluid-in comparison with exact Monte Carlo results-within a unique parameterization of the direct correlation function inside the hard core of particles. In particular, the theory follows correctly the development, in the structure factor, of a local peak at low wavevectors, as peculiarly associated with the onset of aggregation. We then model the direct correlation function in the same wavevector regime by a Gaussian function, so as to systematically investigate, in a "reverse" scheme, how varying the properties of the local peak modifies the original underlying competing interaction. We show that large variations in the height of the peak are generally associated with comparatively smaller variations in the height of the microscopic repulsive barrier; moreover, the shrinking and shifting towards lower wavevectors of the peak may be interpreted in terms of the displacement of the barrier, producing a substantial enlargement of the range of both the attractive and repulsive contributions to the interaction potential. Finally, we document the way the repulsive barrier tends to vanish as the two-Yukawa fluid approaches a "simple fluid" behavior, heralding the onset of a liquid-vapor phase separation.

Tiny changes in local order identify the cluster formation threshold in model fluids with competing interactions.

We use Monte Carlo simulations to carry out a thorough analysis of structural correlations arising in a relatively dense fluid of rigid spherical particles with prototype competing interactions (short-range attractive and long-range repulsive two-Yukawa model). As the attraction strength increases, we show that the local density of the fluid displays a tiny reversal of trend within specific ranges of interparticle distances, whereupon it decreases first and increases afterwards, passing through a local minimum. Particles involved in this trend display, accordingly, distinct behaviours: for a sufficiently weak attraction, they seem to contribute to the long-wave oscillations typically heralding the formation of patterns in such fluids; for a stronger attraction, after the reversal of the local density has occurred, they form an outer shell of neighbours stabilizing the existing aggregation seeds. Following the increment of attraction, precisely in correspondence of the local density reversal, the local peak developed in the structure factor at small wavevectors markedly rises, signalling-in agreement with recent structural criteria-the onset of a clustered state. A detailed cluster analysis of microscopic configurations fully validates this picture.

Coexistence of low and high overlap phases in a supercooled liquid: An integral equation investigation.

The pair structure, free energy, and configurational overlap order parameter Q of an annealed system of two weakly coupled replicas of a supercooled "soft sphere" fluid are determined by solving the hypernetted-chain (HNC) and self-consistent Rogers-Young (RY) integral equations over a wide range of thermodynamic conditions ρ (number-density), T (temperature), and inter-replicas couplings ε12. Analysis of the resulting effective (or Landau) potential W(ρ,T; Q) and of its derivative with respect to Q confirms the existence of a "precursor transition" between weak and strong overlap phases below a critical temperature Tc well above the temperature To of the "ideal glass" transition observed in the limit ε12→0. The precursor transition is signalled by a loss of convexity of the potential W(Q) and by a concomitant discontinuity of the order parameter Q just below Tc, which crosses over to a mean-field-like van der Waals loop at lower temperatures. The HNC and RY equations lead to the same phase transition scenario, with quantitative differences in the predicted temperatures Tc and To.

Hypernetted-chain investigation of the random first-order transition of a Lennard-Jones liquid to an ideal glass.

The structural and thermodynamic behavior of a deeply supercooled Lennard-Jones liquid, and its random first-order transition (RFOT) to an ideal glass is investigated, using a system of two weakly coupled replicas and the hypernetted chain integral equation for the pair structure of this symmetric binary system. A systematic search in the density-temperature plane points to the existence of two glass branches below a density-dependent threshold temperature. The branch of lower free energy exhibits a rapid growth of the structural overlap order parameter upon cooling and may be identified with the ideal glass phase conjectured by several authors for both spin and structural glasses. The RFOT, signaled by a sharp discontinuity of the order parameter, is predicted to be weakly first order from a thermodynamic viewpoint. The transition temperature T(cr) increases rapidly with density and approximately obeys a scaling relation valid for a reference system of particles interacting via a purely repulsive 1/r(18) potential.

An investigation of the liquid to glass transition using integral equations for the pair structure of coupled replicae.

Extensive numerical solutions of the hypernetted-chain (HNC) and Rogers-Young (RY) integral equations are presented for the pair structure of a system of two coupled replicae (1 and 2) of a "soft-sphere" fluid of atoms interacting via an inverse-12 pair potential. In the limit of vanishing inter-replica coupling ɛ12, both integral equations predict the existence of three branches of solutions: (1) A high temperature liquid branch (L), which extends to a supercooled regime upon cooling when the two replicae are kept at ɛ12 = 0 throughout; upon separating the configurational and vibrational contributions to the free energy and entropy of the L branch, the Kauzmann temperature is located where the configurational entropy vanishes. (2) Starting with an initial finite coupling ɛ12, two "glass" branches G1 and G2 are found below some critical temperature, which are characterized by a strong remnant spatial inter-replica correlation upon taking the limit ɛ12 → 0. Branch G2 is characterized by an increasing overlap order parameter upon cooling, and may hence be identified with the hypothetical "ideal glass" phase. Branch G1 exhibits the opposite trend of increasing order parameter upon heating; its free energy lies consistently below that of the L branch and above that of the G2 branch. The free energies of the L and G2 branches are found to intersect at an alleged "random first-order transition" (RFOT) characterized by weak discontinuities of the volume and entropy. The Kauzmann and RFOT temperatures predicted by RY differ significantly from their HNC counterparts.

A theoretical study of structure and thermodynamics of fluids with long-range competing interactions exhibiting pattern formation.

We study the structure and phase behavior of a model fluid with competing short-range attraction and long-range repulsion, constituted by hard spheres interacting by means of two opposite Kac potentials. We use, to this purpose, a thermodynamically self-consistent integral equation approach developed by one of the authors [J.-M. Bomont and J.-L. Bretonnet, J. Chem. Phys. 119, 2188 (2003)], which proven accurate in predicting the properties of other competing fluids. We choose the potential parameters in such a way that, upon appropriate thermodynamic conditions, the fluid displays microphase separation terminating, at sufficiently low temperatures, with a phase transition into an ordered-pattern fluid. The propensity toward the pattern formation is indicated by long-wavelength, slowly decaying oscillations in the pair correlation function, and by the presence of a sharp peak in the structure factor S(q) at a small but finite wavevector q(c). The limits of stability of the micro-separated phase are identified by a drastic, diverging-like, increase of S(q(c)) as the temperature drops. The behavior of S(q) in the disordered-pattern phase suggests that different morphologies of the ordered patterns should be expected, depending on the ratio between the strengths of competing interactions. The structural predictions are confirmed, at the thermodynamic level, by the change of sign observed in the "residual multi-particle entropy," according to the one-phase ordering criterion developed by Giaquinta and Giunta [Physica A 187, 145 (1992)], and by the trend shown by the chemical potential. Our self-consistent approach succeeds in describing the thermodynamic regime where the phase transition occurs, whereas, as reported in the literature, other sophisticated schemes within the same theoretical framework generally fail; reasons of this outcome and putative remedies are discussed.

Communication: thermodynamic signatures of cluster formation in fluids with competing interactions.

Convergent theoretical evidence, based on self-consistent integral equations for the pair structure and on Monte Carlo simulations, is presented for the existence of small simultaneous jump discontinuities of several thermodynamic and structural properties of systems of colloidal particles with competing short-range attractive and long-range repulsive interactions, under physical conditions close to the onset of particle clustering. The discontinuities thus provide a signature of the transition from a homogeneous fluid phase to a locally inhomogeneous cluster phase.

Crystallization limits of the two-term Yukawa potentials based on the entropy criterion.

We examine the fluid-solid transition for the potential with two Yukawa terms (one attractive and the other repulsive) and a hard core by exploration of the parameter space of (K(1), Z(1), and Z(2)), i.e., the parameters of interaction strength and interaction ranges, respectively. We apply the single-phase crystallization rule of Giaquinta and Giunta (1992) by searching for the conditions where the residual entropy reaches zero. To obtain accurate entropy properties, we adopt the self-consistent closure theory of the zero-separation genre. This closure gives accurate thermodynamic properties. The Ornstein-Zernike equation is solved to obtain the correlation functions. The structure factor S(q) is examined with respect to its cluster-cluster peak, whose value is another indication of phase transition according to Hansen and Verlet (1969). We discover that the parameter Z(1) (which determines the range of attractive forces) is important in crystal formation, so long as sufficient attraction (parameter K(1)) is present. If the range of attraction is too narrow, strength alone is not adequate to satisfy the Giaquinta rule or to solidify at given concentration and temperature. The control of the range of repulsion rests with the Z(2)-parameter. Its variations can bring about a high peak in S(q) at zero wave number (i.e., at q=0). Implications for the crystallization of protein and colloidal solutions are discussed.

Approximative "one particle" bridge function B(1)(r) for the theory of simple fluids.

New properties for the one particle bridge function B(1)(r), which are necessary to the calculation of the excess chemical potential betamue), are derived for the hard sphere fluid. The method, which only requires the knowledge of the bridge function B(2)(r), is based on an investigation of the correlation function dependence on the Kirkwood charging parameter. In this framework, the unavoidable question of topological homotopy is addressed. As far as B(2)(r) is considered as exact, this work provides useful information on B(1)(r) in the well identified dynamical regimes of the hard sphere fluid. Signatures of the transitions between these regimes are identified on the trends of B(1)(r). This approach provides self-consistent results for betamue) that agree very well with simulation data.

A consistent calculation of the chemical potential for dense simple fluids.

A general method to calculate the excess chemical potential betamuex, that is based on the Kirkwood coupling parameter's dependence of the correlation functions, is presented. The expression for the one particle bridge function B(1)r is derived for simple fluids with spherical interactions. Only the knowledge of the bridge function B(2)r is required. The accuracy of our approach is illustrated for a dense hard sphere fluid. As far as B(2)r is considered as exact, B(1)r is found to be, at high densities, the normalized bridge function -B(2)rB(2)(r=0). This expression ensures a consistent calculation of the excess chemical potential by satisfying implicitly the Gibbs-Duhem constraint. Only the pressure-consistency condition is necessary to calculate the structural and thermodynamic properties of the fluid.

An effective pair potential for thermodynamics and structural properties of liquid mercury.

The properties of liquid mercury are investigated by using an empirical effective pair potential. Its parameters are determined with the aid of Monte Carlo simulation along the liquid branch of the liquid-vapor coexistence curve. The complexity of the electronic structure of dense metal mercury supposes a state dependence of the interatomic interactions, while no more state dependence is found in the metal-nonmetal transition region. It is shown that the use of this effective potential leads to an accurate description of the structural and thermodynamic properties of the expanded liquid mercury. Then, the melting and freezing phenomena are investigated with that potential. Sharp melting and freezing temperatures are observed at 234 and 169 K, respectively. This large hysteresis loop between freezing and melting is consistent with the experiments for the bulk mercury.