Attractive Hard Spheres Research Articles

Curvature Dependence and Tolman Length of Surface Tension of Nano-Droplet of Attractive Hard Sphere Liquid

Curvature Dependence and Tolman Length of Surface Tension of Nano-Droplet of Attractive Hard Sphere Liquid

From nuclei to micro-structure in colloidal crystallization: Investigating intermediate length scales by small angle laser light scattering.

Hard sphere suspensions are well recognized model systems of statistical physics and soft condensed matter. We here investigate the temporal evolution of the immediate environment of nucleating and growing crystals and/or their global scale distribution using time resolved Small Angle Light Scattering (SALS). Simultaneously performed Bragg scattering measurements provide an accurate temporal gauging of the sequence of events. We apply this approach to studies of re-crystallization in several different shear molten hard sphere and attractive hard sphere samples with the focus being on the diversity of observable signal shapes and their change in time. We demonstrate that depending on the preparation conditions different processes occur on length scales larger than the structural scale, which significantly influence both the crystallization kinetics and the final micro-structure. By careful analysis of the SALS signal evolution and by comparing different suggestions for small angle signal shapes to our data, we can for most cases identify the processes leading to the observed signals. These include form factor scattering from crystals surrounded by depletion zones and structure factor scattering from late stage inter-crystallite ordering. The large variety of different small angle signals thus in principle contains valuable information complementary to that gained from Bragg scattering or microscopy. Our comparison, however, also shows that further refinement and adaptation of the theoretical expressions to the sample specific boundary conditions is desired for a quantitative kinetic analysis of micro-structural evolution.

Diffusion-limited reactions in crowded environments: a local density approximation

In the real world, diffusion-limited reactions in chemistry and biology mostly occur in crowded environments, such as macromolecular complex formation in the cell. Despite the paramount importance of such phenomena, theoretical approaches still mainly rely on the Smoluchowski theory, only valid in the infinite dilution limit. In this paper we introduce a novel theoretical framework to describe the encounter rate and the stationary density profiles for encounters between an immobilized target and a fluid of interacting spherical particles, valid in the local density approximation. A comparison with numerical simulations performed for a fluid of hard spheres and square well attractive hard spheres confirms the accuracy of our treatment.

Scaled Particle Theory for the Coil–Globule Transition of an Isolated Polymer Chain

We have developed a scaled particle theory for the coil–globule transition (CGT) of a chain of attractive hard spheres in an attractive hard sphere solvent. The model predicts a lower critical solution temperature (LCST) related CGT that merges with a high temperature upper critical solution temperature (UCST) related CGT at a high pressure hypercritical point. The predicted hypercritical point is in good agreement with simulations of a bead–spring polymer chain in a symmetric solvent, without the use of any adjustable parameters. The model predicts that increasing bulkiness of monomers relative to solvent molecules, decreasing polymer cohesive energy relative to solvent cohesive energy, and reduced chain length stabilize the expanded coil state, in qualitative agreement with simulations. These trends in the UCST-CGT and LCST-CGT are found to emerge from incompressible-solvent effects and equation-of-state effects, respectively, consistent with the enthalpic origin of the UCST and the entropic origin of t...

Self-consistent phonon theory of the crystallization and elasticity of attractive hard spheres

We propose an Einstein-solid, self-consistent phonon theory for the crystal phase of hard spheres that interact via short-range attractions. The approach is first tested against the known behavior of hard spheres, and then applied to homogeneous particles that interact via short-range square well attractions and the Baxter adhesive hard sphere model. Given the crystal symmetry, packing fraction, and strength and range of attractive interactions, an effective harmonic potential experienced by a particle confined to its Wigner-Seitz cell and corresponding mean square vibrational amplitude are self-consistently calculated. The crystal free energy is then computed and, using separate information about the fluid phase free energy, phase diagrams constructed, including a first-order solid-solid phase transition and its associated critical point. The simple theory qualitatively captures all the many distinctive features of the phase diagram (critical and triple point, crystal-fluid re-entrancy, low-density coexistence curve) as a function of attraction range, and overall is in good semi-quantitative agreement with simulation. Knowledge of the particle localization length allows the crystal shear modulus to be estimated based on elementary ideas. Excellent predictions are obtained for the hard sphere crystal. Expanded and condensed face-centered cubic crystals are found to have qualitatively different elastic responses to varying attraction strength or temperature. As temperature increases, the expanded entropic solid stiffens, while the energy-controlled, fully-bonded dense solid softens.

Polymer-enforced crystallization of a eutectic binary hard sphere mixture

We prepared a buoyancy matched binary mixture of polydisperse polystyrene microgel spheres of size ratio Γ = 0.785 at a volume fraction of Φ = 0.567 just below the kinetic glass transition. In line with theoretical expectations, a eutectic phase behaviour was observed, but only a minor fraction of the samples crystallized at all. By adding a short non-adsorbing polymer we enforce inter-species fractionation into coexisting pure component crystals, which in turn also shows signs of intra-species fractionation. We show that in formerly inaccessible regions of the phase diagram binary hard sphere physics are made observable using attractive hard spheres.

The Free-Energy Landscape of Clusters of Attractive Hard Spheres

The study of clusters has provided a tangible link between local geometry and bulk condensed matter, but experiments have not yet systematically explored the thermodynamics of the smallest clusters. Here we present experimental measurements of the structures and free energies of colloidal clusters in which the particles act as hard spheres with short-range attractions. We found that highly symmetric clusters are strongly suppressed by rotational entropy, whereas the most stable clusters have anharmonic vibrational modes or extra bonds. Many of these clusters are subsets of close-packed lattices. As the number of particles increases from 6 to 10, we observe the emergence of a complex free-energy landscape with a small number of ground states and many local minima.

Crystallization and dynamical arrest of attractive hard spheres

Crystallization of hard spheres interacting with a square well potential was investigated by numerical simulations using so-called Brownian cluster dynamics. The phase diagram was determined over a broad range of volume fractions. The crystallization rate was studied as a function of the interaction strength expressed in terms of the second virial coefficient. For volume fractions below about 0.3 the rate was found to increase abruptly with increasing attraction at the binodal of the metastable liquid-liquid phase separation. The rate increased until a maximum was reached after which it decreased with a power law dependence on the second virial coefficient. Above a critical percolation concentration, a transient system spanning network of connected particles was formed. Crystals were formed initially as part of the network, but eventually crystallization led to the breakup of the network. The lifetime of the transient gels increased very rapidly over a small range of interaction energies. Weak attraction destabilized the so-called repulsive crystals formed in pure hard sphere systems and shifted the coexistence line to higher volume fractions. Stronger attraction led to the formation of a denser, so-called attractive, crystalline phase. Nucleation of attractive crystals in the repulsive crystalline phase was observed close to the transition.

The critical endpoint in phase diagrams of attractive hard spheres

We calculate the critical endpoint (cep) in phase diagrams of attractive hard spheres (HS), for both a Yukawa attraction and a polymer depletion potential, using approximate analytical theories. The cep corresponds to the minimum attraction range and the maximum contact potential for which liquid can exist. For both sytems we find essentially the same result: in the cep the attraction range is about one-third of the particle radius, and the attraction strength is around 2 kT. This same cep is found from simulations on particles interacting through a Lennard–Jones-type potential. It thus seems that the cep is essentially independent of the details of the pair potential for any system with a smooth attractive pair potential.

Are thermoresponsive microgels model systems for concentrated colloidal suspensions? A rheology and small-angle neutron scattering study.

The structure of concentrated temperature-sensitive poly(N-isopropylacrylamide) (PNiPAM) microgel suspensions has been investigated employing rheology and small-angle neutron scattering (SANS). A previously described model expression for the particle form factor P(inho)(q) is extended by a model hard sphere structure factor S(q), and the average radial density profiles phi(r) are calculated from the amplitude of the form factor A(q) and the structure factor S(q). By this procedure, a direct real space description of the spatial ordering in the neighborhood of a single particle is obtained. The overall particle size and the correlation length xi of the concentration fluctuations of the internal polymer network decrease with concentration, revealing the increasing compression of the spheres. Thus, the particle form factor P(inho)(q) of the swollen PNiPAM microgels depends on concentration. The particle-particle interaction potential does not change significantly between 25 and 32 degrees C. Even approximately 1 K below the lower critical solution temperature (LCST), the experimental scattering intensity distributions I(q)/c are described very well by the hard sphere structure factor when an equivalent hard sphere particle size R(HS) and volume fraction eta(HS) are used. Microgels with different degrees of cross-linking and particle size resemble true hard sphere behavior up to effective volume fractions of phi(eff) < 0.35. At higher effective volume fractions phi(eff) > 0.35 strong deviations from true hard spheres are observed. Interpenetration of the outer, less cross-linked regions of the soft spheres as well as particle compression occurred at higher concentrations. In agreement with this, the equilibrium colloidal phase behavior and rheology also has some features of soft sphere systems. At temperatures well above the LCST, the interaction potential becomes strongly attractive and the collapsed microgel spheres form aggregates consisting of flocculated particles without significant long-range order. Hence, an attractive interaction potential in concentrated suspensions of PNiPAM microgels leads to distinctively different structures as compared to attractive hard sphere colloids. When the peculiar structural properties of the PNiPAM microgels are considered, they can be used as model systems in colloidal science.

Global phase diagrams of the van der Waals–Dieterici and the BMCSL–Dieterici equations of state

Global phase diagrams of a binary mixture of attractive hard spheres described by the van der Waals–Dieterici and Boublik–Mansoori–Carnahan–Starling–Leland–Dieterici equations of state (with a square root combining rule for cross-attraction) are calculated in dependence on the ratio of the hard-sphere diameters and the attraction strength.

Mapping of theoretical equations of state for molecular fluids on a biquadratic equation

A biquadratic equation of state for the attractive hard sphere chain fluid with dipolar interactions is developed. The background of the development is the mapping of molecular model equations onto a simplified mathematical form. In course of this mapping procedure known properties such as virial coefficients and physical limits are accounted for. For the hard sphere reference model a simple theoretical equation is developed, which is almost as accurate as the Carnahan–Starling (CS) equation and includes the nearly correct high density limit. The perturbation of this reference model due to chain formation and dipolar forces is accounted for by variations of the coefficients in the reference model. These coefficients are constant values for the hard sphere reference fluid and become dependent on the chain length parameter or the reduced dipole moment for the chain fluid or the dipolar fluid, respectively. By simple mathematical manipulations one can recover the usual equation of state form which consists of a sum of additive terms for each molecular property.

Interaction between macroparticles in a binary fluid mixture: anomalous effects due to the bulk composition

The interaction between macroparticles in a binary fluid mixture has been analysed using the reference hypernetted-chain (RHNC) theory. The mixture is composed of particles interacting through an attractive potential (component 1) and hard spheres (component 2). Particles of different components interact through a hard-sphere potential. Significantly high affinity is assumed between the macroparticle surface and component 1. Two cases have been studied: case 1 where the bulk mixture unavoidably separates into two immiscible phases in the regime x 1p < x 1 < x 1Q (x 1 is the mole fraction of component 1, and x 1p and x 1Q denote values at the spinodal points); and case 2 where the bulk mixture is miscible over the entire composition range. As x 1 increases from x 1 = 0 or x 1 decreases from x 1 = 1 in case 1, the macroparticle interaction always shifts to the more attractive side. As x 1 → x 1R;1—0 or x 1 → x 1S;1 + 0 (x 1R;1 and x 1S;1 are threshold values, and x 1R;1 < x 1P and x 1S;1 > x 1Q), the range of the interaction exhibits sudden growth to a scale that is far longer than molecular dimensions, and the interaction becomes strongly attractive. This behaviour is ascribed to a surface-induced phase transition, the partial wetting of one of the components accompanied by the partial drying of the other. Even in case 2, similar behaviour has been observed as x 1 → x 1R;2—0 or x 1 → x 1S;2 + 0 (x 1R;2 and x 1S;2 are threshold values). Over the regime x 1R;2 < x 1 < x 1S;2 the RHNC theory possesses no solutions. The interaction between unlike macroparticles, whose surfaces have high affinity only with component 1 and only with component 2, respectively, has been analysed also. Although the interaction between like macroparticles is strongly attractive except for very small separations, the interaction between unlike macroparticles is strongly repulsive for all separations.

Global phase diagrams of model and real binary fluid mixtures Part II. Non-Lorentz–Berthelot mixtures of attractive hard spheres

The global phase diagram of a binary mixture of attractive hard spheres described by the Boublik–Mansoori–Carnahan–Starling–Leland equation of state with a van der Waals mean field attractive term is investigated in its dependence on the ratio of the hard-sphere diameters and the strength of the attraction. Two values of the energetic combining rule parameter k12, determining the deviation from the geometric mean rule, are considered. In addition to phenomena reported recently for k12=1, the shield region of quadruple points is investigated for k12=0.8, along with its associated azeotropic phenomena. For k12=1.15, the mixture exhibits a rich set of low-temperature azeotropic phenomena: cusps on the p–T projections of azeotropic lines, exchange of branches of azeotropic lines, zero-temperature limited azeotropy, and double critical azeotropic points. Some of these azeotropic phenomena are described here for the first time.

Statistical associating fluid theory for chains of attractive hard-spheres: Optimized equations of state

New equations of state for chain molecules are proposed. The equations of state are physically sound and have a simple mathematical structure. The equations combine the recently developed simplified hard-sphere equations with the first order thermodynamic perturbation theory for chain molecules. The resulting equations of state show almost identical thermodynamic properties compared to the original equation of state for chains of attractive hard spheres. The advantage of the equations proposed here is the simple mathematical structure. They can be written as fourth order or cubic polynomial in the molar volume for which the molar volume can be calculated analytically for given temperature and pressure. This work shows that physically based model equations of state and simple equation structures can be incorporated. Furthermore the equations presented here enlarge the applicability of the statistical associating fluid theory for chain molecules.

The global phase behaviour of binary mixtures of chain molecules: theory and application

In this work the global phase behaviour of binary fluid mixtures of chain molecules is investigated. The underlying model employed for the investigation is based on our recently developed equation of state for chain molecules. This equation is based on the statistical associating fluid theory for chains of attractive hard spheres. It is an optimized equation of state for chain molecules which can be rearranged into a fourth order polynom in the molar volume. With this equation of state the global phase diagram for several mixtures of spherical molecules with non-spherical molecules has been investigated. The non-spherical molecules range from small non-spherical over short chain molecules to long chain polymer molecules. Special attention is paid to mixtures of small spherical molecules with long chain molecules. Such mixtures are employed to model polymer–solvent mixtures. Based on the calculated global phase diagram for polymer solutions, several processes such as co- or anti-solvent processes and polymerisations are discussed. These processes are represented by paths in the global phase diagram.

Global phase diagrams of model and real binary fluid mixtures: Lorentz–Berthelot mixture of attractive hard spheres

The phase behavior and global phase diagram of binary mixtures of attractive hard spheres described by a hard-sphere mixture equation of state with a mean field term and the Lorentz–Berthelot combining rule are examined in detail in dependence on the ratio of (i) the hard-sphere diameters and (ii) the strength of the mean field attraction. It is shown that in addition to the usual phenomena, the studied mixture exhibits also the closed liquid–liquid immiscibility loop (Type VI and VII behavior) and a variety of new azeotropic phenomena. Topology of phase diagrams is discussed in detail with emphasis on the boundary states and is compared with that based on the one-fluid van der Waals equation approach.