We present micro-rheological measurments of the drag force on colloids pulled through a solution of lambda-DNA (used here as a monodisperse model polymer) with an optical tweezer. The experiments show a violation of the Stokes-Einstein relation based on the independently measured viscosity of the DNA solution: the drag force is larger than expected. We attribute this to the accumulation of DNA infront of the colloid and the reduced DNA density behind the colloid. This hypothesis is corroborated by a simple drift-diffusion model for the DNA molecules, which reproduces the experimental data surprisingly well, as well as by corresponding Brownian dynamics simulations.
We extend the Rouse model of polymer dynamics to situations of non-stationary chain growth. For a dragged polymer chain of length $N(t) = t^α$, we find two transitions in conformational dynamics. At $α= 1/2$, the propagation of tension and the average shape of the chain change qualitatively, while at $α= 1 $ the average center-of-mass motion stops. These transitions are due to a simple physical mechanism: a race duel between tension propagation and polymer growth. Therefore they should also appear for growing semi-flexible or stiff polymers. The generalized Rouse model inherits much of the ver…
We study the dynamics of a polymer that is pulled by a constant force through a viscoelastic medium. This is a model for a polymer being pulled through a cell by an external force, or for an active biopolymer moving due to a self generated force. Using a Rouse model with a memory dependent drag force, we find that the center of mass of the polymer follows a subballistic motion. We determine the time evolution of the length and the shape of the polymer. Through an analysis of the velocity of the monomers, we investigate how the tension propagates through the polymer. We discuss how polymers can…
Depletion-induced interactions between colloids in colloid-polymer mixtures depend in range and strength on size, shape, and concentration of depletants. Crowding by colloids in turn affects shapes of polymer coils, such as biopolymers in biological cells. By simulating hard-sphere colloids and random-walk polymers, modeled as fluctuating ellipsoids, we compute depletion-induced potentials and polymer shape distributions. Comparing results with exact density-functional theory calculations, molecular simulations, and experiments, we show that polymer shape fluctuations play an important role in…
We consider a simplified coarse-grained model for colloid-polymer mixtures, in which polymers are represented as monoatomic molecules interacting by means of pair potentials. We use it to study polymer-colloid segregation in the presence of a quenched matrix of colloidal hard spheres. We fix the polymer-to-colloid size ratio to 0.8 and consider matrices such that the fraction f of the volume that is not accessible to the colloids due to the matrix is equal to 40%. As in the Asakura-Oosawa-Vrij (AOV) case, we find that binodal curves in the polymer and colloid volume-fraction plane have a small…
The density profile and surface tension for the interface of phase-separated colloid-polymer mixtures have been studied in the framework of the square gradient approximation for both ideal and interacting polymers in good solvent. The calculations show that in the presence of polymer-polymer excluded volume interactions the interfaces have lower widths and surface tensions compared to the case of ideal polymers. These results are a direct consequence of the shorter range and smaller depth of the depletion potential between colloidal particles induced by interacting polymers.
Using video-microscopy imaging we demonstrate the existence of a short-ranged equilibrium attraction between heavy silica colloids diffusing on soft surfaces of cross-linked polymer gels. The inter-colloid potential can be tuned by changing the gel stiffness or by coating the colloids with a polymer layer. On sufficiently soft substrates, the interaction induced by the polymer matrix leads to large-scale colloidal aggregation. We correlate the in-plane interaction with a colloid-surface attraction.
The influence of poor solvent quality on fluid demixing of a model mixture of colloids and nonadsorbing polymers is investigated using density functional theory. The colloidal particles are modelled as hard spheres and the polymer coils as effective interpenetrating spheres that have hard interactions with the colloids. The solvent is modelled as a two-component mixture of a primary solvent, regarded as a background theta-solvent for the polymer, and a cosolvent of point particles that are excluded from both colloids and polymers. Cosolvent exclusion favors overlap of polymers, mimicking the e…
In drying liquid films of polymer-colloid mixtures, the stratification in which polymers are placed on top of larger colloids is studied. It is often presumed that the formation of segregated polymer-colloid layers is solely due to the proportion in size at fast evaporation as in binary colloid mixtures. By comparing experiments with a theoretical model, we found that the transition in viscosity near the drying interface was another important parameter for controlling the formation of stratified layers in polymer-colloid mixtures. At high evaporation rates, increased polymer concentrations nea…
An extension of the Asakura-Oosawa-Vrij model of hard sphere colloids and non-adsorbing polymers, that takes polymer non-ideality into account through a repulsive stepfunction pair potential between polymers, is studied with grand canonical Monte Carlo simulations and density functional theory. Simulation results validate previous theoretical findings for the shift of the bulk fluid demixing binodal upon increasing strength of polymer-polymer repulsion, promoting the tendency to mix. For increasing strength of the polymer-polymer repulsion, simulation and theory consistently predict the interf…
We determine the depletion-induced phase-behavior of hard sphere colloids and interacting polymers by large-scale Monte Carlo simulations using very accurate coarse-graining techniques. A comparison with standard Asakura-Oosawa model theories and simulations shows that including excluded volume interactions between polymers leads to qualitative differences in the phase diagrams. These effects become increasingly important for larger relative polymer size. Our simulations results agree quantitatively with recent experiments.
Using self-consistent field and density-functional theories, we first investigate colloidal self-assembling of colloid/polymer films confined between two soft surfaces grafted by polymers. With the increase of colloidal concentrations, the film undergoes a series of transitions from disordered liquid $\to$ sparse square $\to$ hexagonal (or mixed square-hexagonal) $\to$ dense square $\to$ cylindric structures in plane, which results from the competition between the entropic elasticity of polymer brushes and the steric packing effect of colloidal particles. A phase diagram displays the stable re…
From studies via Molecular Dynamics simulations, we report results on structure and dynamics in mixtures of active colloids and passive polymers that are confined inside a spherical container with a repulsive boundary. Such systems mimic the presence of bacteria in the background of bio-polymers. All interactions in the fully passive case are chosen in such a way that in equilibrium coexistence between colloid-rich and polymer-rich phases occurs. For most part of the studies the chosen compositions give rise to Janus structure; one side of the sphere is occupied by the colloids and the rest by…
We show that the critical behavior of a colloid-polymer mixture inside a random porous matrix of quenched hard spheres belongs to the universality class of the random-field Ising model. We also demonstrate that random-field effects in colloid-polymer mixtures are surprisingly strong. This makes these systems attractive candidates to study random-field behavior experimentally.
We study fluid-fluid phase separation in a colloid-polymer mixture adsorbed in a colloidal porous matrix close to the θ-point. For this purpose we consider the Asakura-Oosawa model in the presence of a quenched matrix of colloidal hard spheres. We study the dependence of the demixing curve on the parameters that characterize the quenched matrix, fixing the polymer-to-colloid size ratio to 0.8. We find that, to a large extent, demixing curves depend only on a single parameter f, which represents the volume fraction which is unavailable to the colloids. We perform Monte Carlo simulations for vol…
We demonstrate a novel route of spatially organizing the colloid arrangements on the polymer-grafted substrate by use of self-consistent field and density functional theories. We find that grafting of polymers onto a substrate can effectively control spatial dispersions of deposited colloids as a result of the balance between colloidal settling force and entropically elastic force of brushes, and colloids can form unexpected ordered structures on a grafting substrate. The depositing process of colloidal particles onto the elastic "soft" substrate includes two steps: brush-mediated one-dimensio…
We use thermodynamic perturbation theory to calculate the free energies and resulting phase diagrams of binary systems of spherical colloidal particles and interacting polymer coils in good solvent within an effective one-component representation of such mixtures, whereby the colloidal particles interact via a polymer-induced depletion potential. MC simulations are used to test the convergence of the high temperature expansion of the free energy. The phase diagrams calculated for several polymer to colloid size ratios differ considerably from the results of similar calculations for mixtures of…
We study the stability of mixtures of highly screened repulsive charged spheres and non-adsorbing ideal polymer chains in a common solvent using free volume theory. The effective interaction between charged colloids in an aqueous salt solution is described by a screened-Coulomb pair potential, which supplements the pure hard-sphere interaction. The ideal polymer chains are treated as spheres that are excluded from the colloids by a hard-core interaction, whereas the interaction between two ideal chains is set to zero. In addition, we investigate the phase behaviour of charged colloid-polymer m…
We report molecular dynamics simulations of a system of repulsive, polymer-tethered colloidal particles. We use an explicit polymer model to explore how the length and the behavior of the polymer (ideal or self-avoiding) affect the ability of the particles to organize into ordered structures when the system is compressed to moderate volume fractions. We find a variety of different phases whose origin can be explained in terms of the configurational entropy of polymers and colloids. Finally, we discuss and compare our results to those obtained for similar systems using simplified coarse-grained…
Large-scale Monte Carlo simulations of a phase-separating colloid-polymer mixture are performed and compared to recent experiments. The approach is based on effective interaction potentials in which the central monomers of self-avoiding polymer chains are used as effective coordinates. By incorporating polymer nonideality together with soft colloid-polymer repulsion, the predicted binodal is in excellent agreement with recent experiments. In addition, the interfacial tension as well as the capillary length are in quantitative agreement with experimental results obtained at a number of points i…
Colloid-polymer mixtures may undergo either fluid-fluid phase separation or gelation. This depends on the depth of the quench (polymer concentration) and polymer-colloid size ratio. We present a real-space study of dynamics in phase separating colloid-polymer mixtures with medium- to long-range attractions (polymer-colloid size ratio q_R=0.45-0.89, with the aim of understanding the mechanism of gelation as the range of the attraction is changed. In contrast to previous studies of short-range attractive systems, where gelation occurs shortly after crossing the equilibrium phase boundary, we fin…
We discuss structural correlations in mixtures of free polymer and colloidal particles based on a microscopic, 2-component liquid state integral equation theory. Whereas in the case of polymers much smaller than the spherical particles the relevant polymer degree of freedom is the center of mass, for polymers larger than the (nano-) particles conformational rearrangements need to be considered. They have the important consequence that the polymer depletion layer exhibits two widely different length scales, one of the order of the particle radius, the other of the order of the polymer radius or…
Understanding the phase behaviour of mixtures continues to pose challenges, even for systems that might be considered "simple". Here we consider a very simple mixture of two colloidal and one non-adsorbing polymer species which can be simplified even further to a size-asymmetrical binary mixture, in which the effective colloid-colloid interactions depend on the polymer concentration. We show that this basic system exhibits surprisingly rich phase behaviour. In particular, we enquire whether such a system features only a liquid-vapor phase separation (as in one-component colloid-polymer mixture…
We study phase separation and transient gelation in a mixture consisting of polydisperse colloids and non-adsorbing polymers, where the ratio of the average size of the polymer to that of the colloid is approximately 0.063. Unlike what has been reported previously for mixtures with somewhat lower colloid polydispersity, the addition of polymers does not expand the fluid-solid coexistence region. Instead, we find a region of fluid-solid coexistence which has an approximately constant width but an unexpected re-entrant shape. We detect the presence of a metastable gas-liquid binodal, which gives…
The non-equilibrium distribution of colloids in a polymer solution under a temperature gradient is studied experimentally. A slight increase of local temperature by a focused laser drives the colloids towards the hot region, resulting in the trapping of the colloids irrespective of their own thermophoretic properties. An amplification of the trapped colloid density with the polymer concentration is measured, and is quantitatively explained by hydrodynamic theory. The origin of the attraction is a migration of colloids driven by a non-uniform polymer distribution sustained by the polymer's ther…
Using Gibbs ensemble Monte Carlo simulations and density functional theory we investigate the fluid-fluid demixing transition in inhomogeneous colloid-polymer mixtures confined between two parallel plates with separation distances between one and ten colloid diameters covering the complete range from quasi two-dimensional to bulk-like behavior. We use the Asakura-Oosawa-Vrij model in which colloid-colloid and colloid-polymer interactions are hard-sphere like, whilst the pair potential between polymers vanishes. Two different types of confinement induced by a pair of parallel walls are consider…
A new potential of mean force is proposed for colloidal dispersions, which is obtained from coarse grained, pair interactions between colloidal particles formed by the explicit grouping of particles that are themselves groups of atoms and molecules. Using numerical simulations, we start by constructing colloidal particles made up of 250 mesoscopic particles joined by springs and interacting with each other through short range forces. Afterward we proceed to model several colloidal concentrations and obtain the colloidal particles pair correlation function, from which we derive the potential of…
A grand canonical Monte Carlo method for the simulation of a simple colloid-polymer mixture called the AO model will be described. The phase separation known to occur in this model is driven by entropy. The phase diagram of the unmixing transition, the surface tension and thecritical point will be determined. To appear in: "Computer Simulation Studies in Condensed Matter Physics XVIII, Eds. D.P. Landau, S.P. Lewis, and H.B. Schuettler (Springer Verlag, Heidelberg, Berlin, 2004).
A coarse-graining strategy, previously developed for polymer solutions, is extended here to mixtures of linear polymers and hard-sphere colloids. In this approach groups of monomers are mapped onto a single pseudoatom (a blob) and the effective blob-blob interactions are obtained by requiring the model to reproduce some large-scale structural properties in the zero-density limit. We show that an accurate parametrization of the polymer-colloid interactions is obtained by simply introducing pair potentials between blobs and colloids. For the coarse-grained model in which polymers are modelled as…
We determine the phase diagram of mixtures of spherical colloids and neutral nonadsorbing polymers in the thermal crossover region between the $θ$ point and the good-solvent regime. We use the generalized free-volume theory (GFVT), which turns out to be quite accurate as long as $q = R_g/R_c\lesssim 1$ ($R_g$ is the radius of gyration of the polymer and $R_c$ is the colloid radius). Close to the $θ$ point the phase diagram is not very sensitive to solvent quality, while, close to the good-solvent region, changes of the solvent quality modify significantly the position of the critical point and…
Mixtures of ideal polymers with hard spheres whose diameters are smaller than the radius of gyration of the polymer, exhibit extensive immiscibility. The interfacial tension between demixed phases of these mixtures is estimated, as is the barrier to nucleation. The barrier is found to scale linearly with the radius of the polymer, causing it to become large for large polymers. Thus for large polymers nucleation is suppressed and phase separation proceeds via spinodal decomposition, as it does in polymer blends.
We examine the demixing transition in star polymer-colloid mixtures for star arm numbers f=2,6,16,32 and different star-colloid size ratios. Theoretically, we solve the thermodynamically self-consistent Rogers-Young integral equations for binary mixtures using three effective pair potentials obtained from direct molecular computer simulations. The numerical results show a spinodal instability. The demixing binodals are approximately calculated, and found to be consistent with experimental observations.
Using monomer-resolved Molecular Dynamics simulations and theoretical arguments based on the radial dependence of the osmotic pressure in the interior of a star, we systematically investigate the effective interactions between hard, colloidal particles and star polymers in a good solvent. The relevant parameters are the size ratio q between the stars and the colloids, as well as the number of polymeric arms f (functionality) attached to the common center of the star. By covering a wide range of q's ranging from zero (star against a flat wall) up to about 0.75, we establish analytical forms for…
We derive the short distance interaction of star polymers in a colloidal solution. We calculate the corresponding force between two stars with arbitrary numbers of legs f_1 and f_2. We show that a simple scaling theory originally derived for high f_1, f_2 nicely matches with the results of elaborated renormalization group analysis for f_1 + f_2 =< 6 generalizing and confirming a previous conjecture based only on scaling results for f_1=f_2=1,2.
Low-density "equilibrium" gels that consist of a percolated, kinetically arrested network of colloidal particles and are resilient to aging can be fabricated by restricting the number of effective bonds that form between the colloids. Valence-restricted patchy particles have long served as one archetypal example of such materials, but equilibrium gels can also be realized through a synthetically simpler and scalable strategy that introduces a secondary linker, such as a small ditopic molecule, to mediate the bonds between the colloids. Here, we consider the case where the ditopic linker molecu…
The study of competitive and cooperative adsorption of functionalized molecules such as polymers, rheology modifiers and surfactant molecules on colloidal particles immersed in a solvent is undertaken using coarse grained, dissipative particle dynamics simulations. The results show that a complex picture emerges from the simulations, one where dispersants and surfactants adsorb cooperatively up to certain concentrations, on colloidal particles, but as the surfactant concentration increases it leads to dispersant desorption. The presence of rheology modifying agents in the colloidal dispersion …
We investigated the viscoelastic properties of colloid-polymer mixtures at intermediate colloid volume fraction and varying polymer concentrations, thereby tuning the attractive interactions. Within the examined range of polymer concentrations, the samples ranged from fluids to gels. Already in the liquid phase the viscoelastic properties significantly changed when approaching the gelation boundary, indicating the formation of clusters and transient networks. This is supported by an increasing correlation length of the density fluctuations, observed by static light scattering and microscopy. A…
We present further evidence that gelation is an arrested phase separation in attractive colloid-polymer mixtures, based on a method combining confocal microscopy experiments with numerical simulations recently established in {\bf Nature 453, 499 (2008)}. Our results are independent of the form of the interparticle attractive potential, and therefore should apply broadly to any attractive particle system with short-ranged, isotropic attractions. We also give additional characterization of the gel states in terms of their structure, inhomogeneous character and local density.
We study the structure of colloidal fluids with reference to colloid-polymer mixtures. We compare the one component description of the Asakura-Oosawa (AO) idealisation of colloid-polymer mixtures with the full two-component model. We also consider the Morse potential, a variable range interaction, for which the ground state clusters are known. Mapping the state points between these systems, we find that the pair structure of the full AO model is equally well described by the Morse potential or the one component AO approach. We employ a recently developed method to identify in the bulk fluid th…
An extended theoretical study of interface potentials in adsorbed colloid-polymer mixtures is performed. To describe the colloid-polymer mixture near a hard wall, a simple Cahn-Nakanishi-Fisher free-energy functional is used. The bulk phase behavior and the substrate-adsorbate interaction are modelled by the free-volume theory for ideal polymers with polymer-to-colloid size ratios q=0.6 and q=1. The interface potentials are constructed with help from a Fisher-Jin crossing constraint. By manipulating the crossing density, a complete interface potential can be obtained from natural, single-cross…
The surface tension, the adsorption, and the depletion thickness of polymers close to a single nonadsorbing colloidal sphere are computed by means of Monte Carlo simulations. We consider polymers under good-solvent conditions and in the thermal crossover region between good-solvent and $θ$ behavior. In the dilute regime we consider a wide range of values of $q$, from $q = 0$ (planar surface) up to $q\approx 30$-50, while in the semidilute regime, for $ρ_p/ρ_p^*\le 4$ ($ρ_p$ is the polymer concentration and $ρ_p^*$ is its value at overlap), we only consider $q = 0,0.5,1$ and 2. The results are …
Understanding the dynamics of soft colloids, such as star polymers, dendrimers, and microgels, is of scientific and practical importance. It is known that the excluded volume effect plays a key role in colloidal dynamics. Here, we propose a condition of compressibility equivalence that provides a simple method to experimentally evaluate the excluded volume of soft colloids from a thermodynamic view. We apply this condition to survey the dynamics of a series of star polymer dispersions. It is found that as the concentration increases, the slowing of the long-time self-diffusivity of the star po…
We discuss the reliability of integral-equation methods based on several commonly used closure relations in determining the phase diagram of coarse-grained models of soft-matter systems characterized by mutually interacting soft and hard-core particles. Specifically, we consider a set of potentials appropriate to describe a system of hard-sphere colloids and linear homopolymers in good solvent, and investigate the behavior when the soft particles are smaller than the colloids, which is the regime of validity of the coarse-grained models. Using computer-simulation results as a benchmark, we fin…
Colloidal suspensions with free polymer coils which are larger than the colloidal particles are considered. The polymer-colloid interaction is modeled by an extension of the Asakura-Oosawa model. Phase separation occurs into dilute and dense fluid phases of colloidal particles when polymer is added. The critical density of this transition tends to zero as the size of the polymer coils diverges.
We consider the Asakura-Oosawa model of hard sphere colloids and ideal polymers in contact with a porous matrix modeled by immobilized configurations of hard spheres. For this ternary mixture a fundamental measure density functional theory is employed, where the matrix particles are quenched and the colloids and polymers are annealed, i.e. allowed to equilibrate. We study capillary condensation of the mixture in a tiny sample of matrix as well as demixing and the fluid-fluid interface inside a bulk matrix. Density profiles normal to the interface and surface tensions are calculated and compare…
Microcapsules allow for the controlled containment, transport, and release of cargoes ranging from pharmaceuticals to fragrances. Given the interest from a variety of industries in microcapsules and other core-shell structures, a multitude of fabrication strategies exist. Here, we report on a method relying on a mixture of temperature-responsive microgel particles, poly(N-isopropylacrylamide) (pNIPAM), and a polymer which undergo fluid-fluid phase separation. At room temperature this mixture separates into colloid-rich (liquid) and colloid-poor (gas) fluids. By heating the sample above a criti…
As a generic model for liquid-vapour type transitions in random porous media, the Asakura-Oosawa model for colloid-polymer mixtures is studied in a matrix of quenched spheres using extensive Monte Carlo (MC) simulations. Since such systems at criticality, as well as in the two-phase region, exhibit lack of self-averaging, the analysis of MC data via finite size scaling requires special care. After presenting the necessary theoretical background and the resulting subtleties of finite size scaling in random-field Ising-type systems, we present data on the order parameter distribution (and its mo…
We consider a model mixture of hard colloidal spheres and non-adsorbing polymer chains in a theta solvent. The polymer component is modelled as a polydisperse mixture of effective spheres, mutually noninteracting but excluded from the colloids, with radii that are free to adjust to allow for colloid-induced compression. We investigate the bulk fluid demixing behaviour of this model system using a geometry-based density-functional theory that includes the polymer size polydispersity and configurational free energy, obtained from the exact radius-of-gyration distribution for an ideal (random-wal…
We investigate the influence of confinement on phase separation in colloid-polymer mixtures. To describe the particle interactions, the colloid-polymer model of Asakura and Oosawa [J. Chem. Phys. 22, 1255 (1954)] is used. Grand canonical Monte Carlo simulations are then applied to this model confined between two parallel hard walls, separated by a distance D=5 colloid diameters. We focus on the critical regime of the phase separation and look for signs of crossover from three-dimensional (3D) Ising to two-dimensional (2D) Ising universality. To extract the critical behavior, finite size scalin…
The structure and the statistical fluctuations of interfaces between coexisting phases in the Asakura-Oosawa (AO) model for a colloid--polymer mixture are analyzed by extensive Monte Carlo simulations. We make use of a recently developed grand canonical cluster move with an additional constraint stabilizing the existence of two interfaces in the (rectangular) box that is simulated. Choosing very large systems, of size LxLxD with L=60 and D=120, measured in units of the colloid radius, the spectrum of capillary wave-type interfacial excitations is analyzed in detail. The local position of the i…
We report on the formation of colloidal complexes resulting from the electrostatic self-assembly of polyelectrolyte-neutral block copolymers and oppositely charged surfactants. The copolymers investigated are asymmetric and characterized by a large neutral block. Using light, neutron and x-ray scattering experiments, we have shown that the colloidal complexes exhibit a core-shell microstructure. The core is described as a dense and disordered micro-phase of micelles connected by the polyelectrolyte blocks, whereas the shell is a diffuse brush made from the neutral chains. Colloidal complexes w…
Using computer simulations, we study the dynamic arrest in a schematic model of colloid-polymer mixtures combining short-ranged attractions with long-ranged repulsions. The arrested gel is a dilute rigid network of colloidal particles bonded due to the strong attractions. Without repulsions, the gel forms at the spinodal through arrested phase separation. In the ergodic suspension at sufficiently high densities, colloidal clusters form temporary networks that percolate space. Recently [Nat. Commun. 7, 11817 (2016)], it has been proposed that the transition of these networks to directed percola…
We computed the phase-separation behavior and effective interactions of colloid-polymer mixtures in the "protein limit", where the polymer radius of gyration is much larger than the colloid radius. For ideal polymers, the critical colloidal packing fraction tends to zero, whereas for interacting polymers in a good solvent the behavior is governed by a universal binodal, implying a constant critical colloid packing fraction. In both systems the depletion interaction is not well described by effective pair potentials but requires the incorporation of many-body contributions.
The depletion interactions between two colloidal plates or between two colloidal spheres, induced by interacting polymers in a good solvent, are calculated theoretically and by computer simulations. A simple analytical theory is shown to be quantitatively accurate for case of two plates. A related depletion potential is derived for two spheres; it also agrees very well with direct computer simulations. Theories based on ideal polymers show important deviations with increasing polymer concentration: They overestimate the range of the depletion potential between two plates or two spheres at all …
We investigate the structure of colloid-polymer mixtures by calculating the structure factors for the Asakura-Oosawa model in the PY approximation. We discuss the role of potential range, polymer concentration and polymer-polymer interactions on the colloid-colloid structure. Our results compare reasonably well with the recent experiments of Moussaïd et. al. for small wavenumber $k$, but we find that the Hansen-Verlet freezing criterion is violated when the liquid phase becomes marginal.
