Adhesive Hard Sphere Research Articles

Comparative Study on the Models of Thermoreversible Gelation.

A critical survey on the various theoretical models of thermoreversible gelation, such as the droplet model of condensation, associated-particle model, site–bond percolation model, and adhesive hard sphere model, is presented, with a focus on the nature of the phase transition predicted by them. On the basis of the classical tree statistics of gelation, combined with a thermodynamic theory of associating polymer solutions, it is shown that, within the mean-field description, the thermoreversible gelation of polyfunctional molecules is a third-order phase transition analogous to the Bose–Einstein condensation of an ideal Bose gas. It is condensation without surface tension. The osmotic compressibility is continuous, but its derivative with respect to the concentration of the functional molecule reveals a discontinuity at the sol–gel transition point. The width of the discontinuity is directly related to the amplitude of the divergent term in the weight-average molecular weight of the cross-linked three-dimensional polymers. The solution remains homogeneous in the position space, but separates into two phases in the momentum space; particles with finite translational momentum (sol) and a network with zero translational momentum (gel) coexist in a spatially homogeneous state. Experimental methods used to detect the singularity at the sol–gel transition point are suggested.

Anomalous rheological aging of a model thermoreversible colloidal gel following a thermal quench.

We investigate the aging behavior in a well-studied model system comprised of a colloidal suspension of thermoreversible adhesive hard spheres (AHS) but thermally quenched below the gel transition to much larger depths than previously studied. The aging behavior in the model AHS system is monitored by small amplitude oscillatory shear rheology measurements conducted while rapidly quenching from the liquid state at 40 °C to a temperature below the gel temperature, and new, anomalous aging behaviors are observed. Shallow quenches lead to monotonic development of the elastic modulus with time, consistent with prior reports for the development of a homogeneous gel [Gordon et al., J. Rheol. 61, 23-34 (2017)]. However, for deeper quenches, a unique and new phenomenon is reported, namely, after an initial rise in the modulus, a reproducible drop in the modulus is observed, followed by a plateau in the modulus value. This drop can be gradual or sudden and the extent of the drop depends on the quench depth. After this drop in the modulus, AHS gel evolves toward a quench-path independent state over the experimental timescale. These effects of the extent of quenching on aging behavior are hypothesized to be a consequence of quenching into different underlying thermodynamic states of colloidal gels and the possible influence of the adhesive glass dynamical arrest for the deepest quenches. The research connects homogeneous gelation with heterogeneous gel formation due to phase separation and shows that the extent of quench can be used as an independent parameter to govern the rheological response of the arrested gel.

Convergence of Coupling-Parameter Expansion-Based Solutions to Ornstein–Zernike Equation in Liquid State Theory

The objective of this paper is to investigate the convergence of coupling-parameter expansion-based solutions to the Ornstein–Zernike equation in liquid state theory. The analytically solved Baxter’s adhesive hard sphere model is analyzed first by using coupling-parameter expansion. It was found that the expansion provides accurate approximations to solutions—including the liquid-vapor phase diagram—in most parts of the phase plane. However, it fails to converge in the region where the model has only complex solutions. Similar analysis and results are obtained using analytical solutions within the mean spherical approximation for the hardcore Yukawa potential. However, numerical results indicate that the expansion converges in all regions in this model. Next, the convergence of the expansion is analyzed for the Lennard-Jones potential by using an accurate density-dependent bridge function in the closure relation. Numerical results are presented which show convergence of correlation functions, compressibility versus density profiles, etc., in the single as well as two-phase regions. Computed liquid-vapor phase diagrams, using two independent schemes employing the converged profiles, compare excellently with simulation data. The results obtained for the generalized Lennard-Jones potential, with varying repulsive exponent, also compare well with the simulation data. Solution-spaces and the bifurcation of the solutions of the Ornstein–Zernike equation that are relevant to coupling-parameter expansion are also briefly discussed. All of these results taken together establish the coupling-parameter expansion as a practical tool for studying single component fluid phases modeled via general pair-potentials.

Comparison of Huggins Coefficients and Osmotic Second Virial Coefficients of Buffered Solutions of Monoclonal Antibodies.

The Huggins coefficient kH is a well-known metric for quantifying the increase in solution viscosity arising from intermolecular interactions in relatively dilute macromolecular solutions, and there has been much interest in this solution property in connection with developing improved antibody therapeutics. While numerous kH measurements have been reported for select monoclonal antibodies (mAbs) solutions, there has been limited study of kH in terms of the fundamental molecular interactions that determine this property. In this paper, we compare measurements of the osmotic second virial coefficient B22, a common metric of intermolecular and interparticle interaction strength, to measurements of kH for model antibody solutions. This comparison is motivated by the seminal work of Russel for hard sphere particles having a short-range “sticky” interparticle interaction, and we also compare our data with known results for uncharged flexible polymers having variable excluded volume interactions because proteins are polypeptide chains. Our observations indicate that neither the adhesive hard sphere model, a common colloidal model of globular proteins, nor the familiar uncharged flexible polymer model, an excellent model of intrinsically disordered proteins, describes the dependence of kH of these antibodies on B22. Clearly, an improved understanding of protein and ion solvation by water as well as dipole–dipole and charge–dipole effects is required to understand the significance of kH from the standpoint of fundamental protein–protein interactions. Despite shortcomings in our theoretical understanding of kH for antibody solutions, this quantity provides a useful practical measure of the strength of interprotein interactions at elevated protein concentrations that is of direct significance for the development of antibody formulations that minimize the solution viscosity.

Evolution of structure and dynamics of thermo-reversible nanoparticle gels-A combined XPCS and rheology study.

A combined X-ray photon correlation spectroscopy and rheology study is carried out to capture the evolution of structure, fast particle-scale dynamics, and moduli (elastic and loss) at early times of gel formation near the fluid-gel boundary of a suspension of nanoparticles. The system is comprised of moderately concentrated suspensions of octadecyl silica in decalin (ϕ = 0.2) undergoing thermoreversible gelation. Near the gel boundary, the rate of gel formation is very sensitive to changes in attraction strength. However, we find that at different attraction strengths, the system goes through identical intermediate states of microscopic and macroscopic behavior, even though the absolute time needed to form a gel varies by orders of magnitude. We identify a single dimensionless time parameter, tw/tg, where tw is the wait time following the quench and tg is the rheologically determined gel time, that captures the similarity in gel formation at a range of attraction strengths. Following a temperature quench below the gel boundary, the system is initially fluidlike and forms diffusive clusters (∼8.5 times the particle diameter). After a lag-time, tL, clusters aggregate to form a network like structure which is characterized by the onset of mechanical rigidity and a rapid growth in microscopic relaxation times. At tg, the Baxter parameter obtained from adhesive hard sphere fits of the structure factor attains a constant value corresponding to the theoretical percolation boundary, thus demonstrating that gelation is percolation driven.

On the viscosity of adhesive hard sphere dispersions: Critical scaling and the role of rigid contacts

The viscosity of colloidal dispersions is a highly sensitive function of the microstructure of the dispersion and the nature of the interactions among the particles. While the viscosity of solutions of colloidal hard spheres is well understood, the viscosity in a broader array of colloidal solutions remains poorly studied. Here, we use immersed boundary simulations to investigate the viscosity of a prototypical colloidal solution, adhesive hard spheres, which under the right conditions can percolate to form sample spanning networks. We examine how rigidity of the particle–particle contacts in these dispersions can enhance the dispersion viscosity. By rigidly constraining a fraction of the adhered particles, we establish a reaction coordinate, the extent of rigidity, along which we can track the development of a gelled state characterized by a diverging viscosity. The fraction of rigid bonds is analogous to the extent of reaction for cross-linking in chemical gels, which suggests a close connection between chemical gelation with polymers and physical gelation with adhesive hard spheres. A critical gel point is identified when particles connected by rigid bonds begin to percolate the sample. In the vicinity of critical gel point, we observe a diverging viscosity and critical scaling behavior in agreement with previous studies on adhesive hard spheres. We use these results to formulate empirical equations for the viscosity across a broad range of volume fractions as a function of the extent of rigidity. Most importantly, through these simulations, we show that rigid constraints among the bonded particles are essential to producing the large viscosities measured experimentally in dispersions of adhesive hard spheres. Hydrodynamic interactions alone are insufficient for reproducing experimental observations.

Confining a fluid membrane vesicle of toroidal topology in an adhesive hard sphere

We discuss how the equilibrium shapes of a confined toroidal fluid membrane vesicle change when an adhesion between membrane and confining sphere is taken into account. The case without adhesion was studied in Ref. [1]. Different types of solution were found and assembled in a phase diagram as a function of area and reduced volume of the membrane. Depending on the degree of confinement the vesicle is either free, in contact along a circle (contact-circle solutions) or on a surface (contact-area solutions). All solutions without adhesion are up-down symmetric. When the container is adhesive, the phase diagram is altered and new kinds of solution without up-down symmetry are found. For increasing values of adhesion the region of contact-circle solutions shrinks until it vanishes completely from the phase diagram.

The rheology and microstructure of an aging thermoreversible colloidal gel

Colloidal gels exhibit time-dependent bulk properties. However, the processes and mechanisms by which aging occurs are poorly understood, which complicates the prediction of macroscopic behavior in these systems. Using a model, thermoreversible, adhesive hard sphere system consisting of octadecyl-coated silica nanoparticles dispersed in tetradecane, rheological aging is quantitatively related to structural aging. By simultaneously measuring the bulk properties and gel microstructure using rheometry and small angle neutron scattering, respectively, we show a one-to-one correspondence between the time-dependent storage modulus and the microstructure, and further, that this correspondence is independent of the gel's thermal and shear history. At the working volume fraction, the gel is homogeneous and, unlike phase-separated gels, aging behavior is not due to heterogeneous coarsening. Instead, the results presented here are consistent with homogeneous, local particle rearrangements as the mechanism of rheolog...

Erratum to: The fragile-to-strong dynamical crossover and the system viscoelasticity in attractive glass forming colloids

The dynamical arrest phenomena of an adhesive hard-sphere (AHS) colloid, L64-D 2O system has been studied by using calorimetry and the complex shear modulus. This system is characterized by a rich temperature (T) and volume fraction (ϕ) phase diagram with a percolation line (PT). According to the mode-coupling theory (MCT), a cusp-like singularity and two glassy phases, one attractive (AG) and one repulsive (RG), are supposed to coexist in the phase diagram. The MCT scaling laws used to study the shear viscosity with ϕ and T as control parameters propose the existence of fragile-to-strong dynamic crossover (FSDC) analogous to that observed in molecular supercooled liquid glass formers. The measured critical values of the control parameters, coincident with the PT line, where the clustering process generates the AG phase, define the FSDC locus. This is in agreement with the extended mode-coupling theory that takes into account both cage and inter-cluster hopping effects. In this work, we demonstrate, by considering the frequency dependence of the complex moduli, that there is the onset of a system viscoelasticity as an effect of the clustering accompanying the FSDC. We will show as the measured frequency-dependent complex moduli satisfy the scaling relations predicted by the scalar elasticity percolation theory and well account for the system evolution toward the glass transition process.

The fragile-to-strong dynamical crossover and the system viscoelasticity in attractive glass forming colloids

The dynamical arrest phenomena of an adhesive hard-sphere (AHS) colloid, L64-D 2O system has been studied by using calorimetry and the complex shear modulus. This system is characterized by a rich temperature (T) and volume fraction (ϕ) phase diagram with a percolation line (PT). According to the mode-coupling theory (MCT), a cusp-like singularity and two glassy phases, one attractive (AG) and one repulsive (RG), are supposed to coexist in the phase diagram. The MCT scaling laws used to study the shear viscosity with ϕ and T as control parameters propose the existence of fragile-to-strong dynamic crossover (FSDC) analogous to that observed in molecular supercooled liquid glass formers. The measured critical values of the control parameters, coincident with the PT line, where the clustering process generates the AG phase, define the FSDC locus. This is in agreement with the extended mode-coupling theory that takes into account both cage and inter-cluster hopping effects. In this work, we demonstrate, by considering the frequency dependence of the complex moduli, that there is the onset of a system viscoelasticity as an effect of the clustering accompanying the FSDC. We will show as the measured frequency-dependent complex moduli satisfy the scaling relations predicted by the scalar elasticity percolation theory and well account for the system evolution toward the glass transition process.

Interactions in protein mixtures. Part I: Second virial coefficients from osmometry

The interaction of proteins (β-lactoglobulin, Bovine Serum Albumin (BSA), gelatins and whey protein isolate (WPI)) in solution was quantified by measuring their second virial coefficient using membrane osmometry. At neutral pH below 20–40 mM ionic strength, electrostatic repulsion dominated the interaction. At higher ionic strength, BSA, WPI and whey protein aggregates (WPA) were well approximated as hard spheres. On the other hand β-lactoglobulin behaves as an adhesive hard sphere for which the stickiness parameter τ (known from the Baxter model for sticky hard spheres) and depth and width of the adhesive part of the interaction potential were calculated.

Gel transition in adhesive hard-sphere colloidal dispersions: the role of gravitational effects.

The role of gravity in gelation of adhesive hard spheres is studied and a critical criterion developed for homogeneous gelation within the gas-liquid binodal. We hypothesize that gelation by Brownian diffusion competes with phase separation enhanced by gravitational settling. This competition is characterized by the gravitational Péclet number Pe(g), which is a function of particle size, volume fraction, and gravitational acceleration. Through a systematic variation of the parameters, we observe the critical Pe(g) of ∼ 0.01 can predict the stability of gels composed of adhesive hard spheres.

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.

Self-Assembled Calixarene Derivative as a Supramolecular Polymer

A light scattering study shows that the HCl-promoted self-assembly of an amino-calix[5]arene monomer precursor UC5-NH(2), bearing an ancillary ureido binding site, efficiently yields supramolecular polymers with a concentration-independent average mass. These polymers adopt a random-coil conformation and interact among themselves via the adhesive hard sphere pair potential. The findings also indicate the formation of polymer clusters, coexisting in low concentration with the polymers and resulting from the adhesive interpolymer interactions. Unlike polymers, the size of these clusters depends on the monomer concentration.

Investigating the evolution of the phase behavior of AOT-based w/o microemulsions in dodecane as a function of droplet volume fraction

AOT-based water in oil (w/o) microemulsions are one of the most extensively studied reverse micellar systems because of their rich phase behavior and their ability to form in the absence of any co-surfactant. The aggregation characteristics and interaction of the microemulsion droplets in these systems are known to be governed by AOT–oil compatibility and water to AOT molar ratio (w). In this manuscript by using Dynamic Light Scattering (DLS) and viscometry techniques, we show that droplet volume fraction too plays an important role in shaping the phase behavior of these microemulsions in dodecane. The phase separation characteristics and the evolution of the viscosity and the hydrodynamic radius of the microemulsion droplets on approaching the cloud points have thus been found to undergo complete transformation as one goes from low to high droplet volume fraction even at a fixed ‘w’. Modeling of the DLS data attributes this to the weakening of inter droplet attractive interaction caused by the growing dominance of the excluded volume effect with increase in droplet volume fraction. In the literature, the inter droplet attractive interaction driven phase separation in these microemulsions is explained based on gas–liquid type phase transition, conceptualized in the framework of Baxter adhesive hard sphere theory. The modeling of our viscosity data, however, does not support such proposition as the characteristic stickiness parameter (τ−1) of the microemulsion droplets in this system remains much lower than the critical value (τc-1≈10.25) required to enforce such phase transition.

Dynamical Arrest, Percolation, Gelation, and Glass Formation in Model Nanoparticle Dispersions with Thermoreversible Adhesive Interactions

We report an experimental study of the dynamical arrest transition for a model system consisting of octadecyl coated silica suspended in n-tetradecane from dilute to concentrated conditions spanning the state diagram. The dispersion's interparticle potential is tuned by temperature affecting the brush conformation leading to a thermoreversible model system. The critical temperature for dynamical arrest, T*, is determined as a function of dispersion volume fraction by small-amplitude dynamic oscillatory shear rheology. We corroborate this transition temperature by measuring a power-law decay of the autocorrelation function and a loss of ergodicity via fiber-optic quasi-elastic light scattering. The structure at T* is measured using small-angle neutron scattering. The scattering intensity is fit to extract the interparticle pair-potential using the Ornstein-Zernike equation with the Percus-Yevick closure approximation, assuming a square-well interaction potential with a short-range interaction (1% of particle diameter). (1) The strength of attraction is characterized using the Baxter temperature (2) and mapped onto the adhesive hard sphere state diagram. The experiments show a continuous dynamical arrest transition line that follows the predicted dynamical percolation line until ϕ ≈ 0.41 where it subtends the predictions toward the mode coupling theory attractive-driven glass line. An alternative analysis of the phase transition through the reduced second virial coefficient B(2)* shows a change in the functional dependence of B(2)* on particle concentration around ϕ ≈ 0.36. We propose this signifies the location of a gel-to-glass transition. The results presented herein differ from those observed for depletion flocculated dispersion of micrometer-sized particles in polymer solutions, where dynamical arrest is a consequence of multicomponent phase separation, suggesting dynamical arrest is sensitive to the physical mechanism of attraction.

On the impossibility of defining adhesive hard spheres as sticky limit of a hard-sphere-Yukawa potential

For fluids of molecules with short-ranged hard-sphere-Yukawa (HSY) interactions, it is proven that the Noro-Frenkel "extended law of corresponding states" cannot be applied down to the vanishing attraction range, since the exact HSY second virial coefficient diverges in such a limit. It is also shown that, besides Baxter's original approach, a fully correct alternative definition of "adhesive hard spheres" can be obtained by taking the vanishing-range-limit (sticky limit) not of a Yukawa tail, as is commonly done, but of a slightly different potential with a logarithmic-Yukawa attraction.

Dynamical Arrest Transition in Nanoparticle Dispersions with Short-Range Interactions

We measure the dynamical arrest transition in a model, thermoreversible, adhesive hard sphere dispersion. At low volume fractions ϕ, below the critical point, gelation occurs within the gas-liquid phase boundary. For ϕ slightly below and above the critical concentration, the phase boundary follows the predicted percolation transition. At high ϕ, it melds into the predicted attractive-driven glass transition. Our results demonstrate that for ϕ above ∼20% physical gelation is an extension of the attractive-driven glass line and occurs without competition for macroscopic phase separation.

TODGA based w/o microemulsion in dodecane: An insight into the micellar aggregation characteristics by dynamic light scattering and viscometry

N,N,N′,N′-tetraoctyl diglycolamide abbreviated as TODGA, is one of the most promising extractant for actinide partitioning from high level nuclear waste. It forms reverse micelles in non polar solvents on equilibration with aqueous HNO 3 solutions. This reverse micellar system undergoes phase separation into dilute and concentrated reverse micellar solutions at high aqueous acid concentration. Small angle neutron scattering (SANS) studies reported in the literature explained this phenomenon based on gas–liquid type phase transition in the framework of Baxter adhesive hard sphere theory in the presence of a strong inter-micellar attractive interaction. The present investigation attempts to throw further light on this system by carrying out systematic dynamic light scattering (DLS) and viscometry studies, and their modeling on the TODGA reverse micellar solutions in the dodecane medium. The variation of the diffusion coefficient with the micellar volume fraction observed from the DLS studies is suggestive of the presence of an attractive interaction between the TODGA reverse micelles, which weakens at the high micellar volume fraction due to the increased dominance of the excluded volume effect. It is suggested that this weakened interaction is responsible for the absence of phase separation in this system at high TODGA concentration. The results thus highlight the importance of the presence of an attractive interaction between the TODGA micelles in determining the observed phase separation in the TODGA reverse micellar systems. The modeling of the DLS and viscosity data, however, suggest that the characteristic stickiness parameter of this system could be smaller than the critical value required for inducing a gas–liquid type phase transition.

Phase transition and crossover behavior of colloidal fluids under confinement

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.