Simulation of interaction forces between nanoparticles in the presence of Lennard-Jones polymers: freely adsorbing homopolymer modifiers

The force between two nanoscale colloidal particles dispersed in a solution of freely adsorbing Lennard-Jones homopolymer modifiers is calculated using the expanded grand canonical Monte Carlo simulation method. We investigate the effect of polymer chain length (N), nanoparticle diameter (sigma(c)), and colloid-polymer interaction energy (epsilon(cp)) on polymer adsorption (Gamma) and polymer-induced forces (F(P)(r)) between nanoparticles in the full thermodynamic equilibrium condition. There is a strong correlation between polymer adsorption and the polymer-mediated nanoparticle forces. When the polymer adsorption is weak, as in the case of smaller diameters and short polymer chain lengths (sigma(c) = 5, N = 10), the polymers do not have any significant effect on the bare nanoparticle interactions. The adsorbed amount increases with increasing particle diameter, polymer chain length, and colloid-polymer interaction energy. In general, for strong polymer-particle adsorption the polymer-governed force profiles between nanoparticles show short-range repulsion and long-ranged attraction, suggesting that homopolymers would not be ideal for achieving stabilization in nanoparticle dispersions. The attraction is likely due to bridging, as well as polymer segment-segment interactions. The location and magnitude of attractive minimum in the force profile can be controlled by varying N and epsilon(cp). The results show partial agreement and some marked differences with previous theoretical and experimental studies of forces in the limit of flat walls in an adsorbing polymer solution. The difference could be attributed to incorporation of long-ranged colloid-polymer potential in our simulations and the influence of the curvature of the nanoparticles.     

Interactions between colloidal particles induced by polymer brushes grafted onto the substrate

We investigate the interaction energy between two colloidal particles on or immersed in nonadsorbing polymer brushes grafted onto the substrate as a function of the separation of the particles by the use of a self-consistent-field theory calculation. Depending on the colloidal size and the penetration depth, we demonstrate the existence of a repulsive energy barrier of several kBT, which can be interpreted by separating the interaction energy into three parts: colloid-polymer interfacial energy, entropic contribution due to "depletion zone" overlap of colloidal particles, and entropic elastic energy of grafted chains by the compression of particles. The existence of a repulsive barrier which is of entirely entropic origin can lead to kinetic stabilization of the mixture rather than depletion flocculation or phase separation. Therefore, the present result may suggest an approach for controlling the self-assembling behavior of colloids for the formation of target structures, by tuning the colloidal interaction on the grafting substrate under appropriate selection of colloidal size, effective gravity (influencing the penetration depth), and brush coverage density.     

Effects of colloid polydispersity on the phase behavior of colloid-polymer mixtures

We study theoretically the equilibrium phase behavior of a mixture of polydisperse hard-sphere colloids and monodisperse polymers, modeled using the Asakura-Oosawa model [S. Asakura and F. Oosawa, J. Chem. Phys. 22, 1255 (1954)] within the free volume approximation of H. N. W. Lekkerkerker, W. C. K. Poon, P. N. Pusey, A. Stroobants, and P. B. Warren [Europhys. Lett. 20, 559 (1992)]. We compute full phase diagrams in the plane of colloid and polymer volume fractions, using the moment free energy method. The intricate features of phase separation in pure polydisperse colloids combine with the appearance of polymer-induced gas-liquid coexistence to give a rich variety of phase diagram topologies as the polymer-colloid size ratio xi and the colloid polydispersity delta are varied. Quantitatively, we find that polydispersity disfavors fluid-solid against gas-liquid separation, causing a substantial lowering of the threshold value xi(c) above which stable two-phase gas-liquid coexistence appears. Phase splits involving two or more solids can occur already at low colloid concentration, where they may be kinetically accessible. We also analyze the strength of colloidal size fractionation. When a solid phase separates from a fluid, its polydispersity is reduced most strongly if the phase separation takes place at low colloid concentration and high polymer concentration, in agreement with experimental observations. For fractionation in gas-liquid coexistence we likewise find good agreement with experiment, as well as with perturbative theories for near-monodisperse systems.     

Density-functional study of interfacial properties of colloid-polymer mixtures

Interfacial properties of colloid-polymer mixtures are examined within an effective one-component representation, where the polymer degrees of freedom are traced out, leaving a fluid of colloidal particles interacting via polymer-induced depletion forces. Restriction is made to zero-, one-, and two-body effective potentials, and a free energy functional is used that treats colloid excluded volume correlations within Rosenfeld's fundamental measure theory, and depletion-induced attraction within first-order perturbation theory. This functional allows a consistent treatment of both ideal and interacting polymers. The theory is applied to surface properties near a hard wall, to the depletion interaction between two walls, and to the fluid-fluid interface of demixed colloid-polymer mixtures. The results of the present theory compare well with predictions of a fully two-component representation of mixtures of colloids and ideal polymers (the Asakura-Oosawa model) and allow a systematic investigation of the effects of polymer-polymer interactions on interfacial properties. In particular, the wall surface tension is found to be significantly larger for interacting than for ideal polymers, whereas the opposite trend is predicted for the fluid-fluid interfacial tension.     

Theoretical study of the three-phase contact line and its tension in adsorbed colloid-polymer mixtures

We perform a theoretical study of the three-phase contact line and the line tension in an adsorbed colloid-polymer mixture near a first-order wetting transition, employing an interface displacement model. We use a simple free-energy functional to describe a colloid-polymer mixture near a hard wall. The bulk phase behavior and the substrate-adsorbate interaction are modeled by the free-volume theory for ideal polymers. The large size of the colloidal particles and the suppression of the van der Waals interaction by optical matching of colloid and solvent justify the planar hard wall model for the substrate. Following the Fisher-Jin scheme, we derive from the free-energy functional an interface potential V(l) for these mixtures. For a particle diameter of 10-100 nm, the calculations indicate a line tension tau approximately 10(-12)-10(-13) N at room temperature. In view of the ultralow interfacial tension in colloid-polymer mixtures, gamma approximately 10(-7) Nm, this leads to a rather large characteristic length scale taugamma in the micrometer range for the three-phase contact zone width. In contrast with molecular fluids, this zone could be studied directly with optical techniques such as confocal scanning laser microscopy.     

Surface forces between telechelic brushes revisited: the origin of a weak attraction

Telechelic polymers are useful for surface protection and stabilization of colloidal dispersions by the formation of polymer brushes. A number of theoretical investigations have been reported on a weak attraction between two telechelic brushes when they are at the classical contact, i.e., when the surface separation is approximately equal to the summation of the brush thicknesses. While recent experiments have confirmed the weak attraction between telechelic brushes, its origin remains elusive because of conflicting approximations used in the previous theoretical calculations. In this paper, we have investigated the telechelic polymer-mediated surface forces by using a polymer density functional theory (PDFT) that accounts for both the surface-adhesive energy and segment-level interactions specifically. Within a single theoretical framework, the PDFT is able to capture both the depletion-induced attraction in the presence of weakly adhesive polymers and the steric repulsion between compressed polymer brushes. In comparison of the solvation forces between telechelic brushes with those between brushes formed by surfactant-like polymers and with those between two asymmetric surfaces mediated by telechelic polymers, we conclude that the weak attraction between telechelic brushes is primarily caused by the bridging effect. Although both the surfactant-like and telechelic polymers exhibit a similar scaling behavior for the brush thickness, a significant difference has been observed in terms of the brush microstructures, in particular, the segment densities near the edges of the polymer brushes.     

Phase stability of colloidal spheres mixed with supramolecular rod-like polymers

Mixtures of colloids and supramolecular polymers exhibit stimuli-responsive phase behavior. In previous work (Peters and Tuinier, Physica A 510, 233 [2018]) the polymers were treated as fully flexible chains, while in experimental systems supramolecular polymers may have a certain degree of stiffness. Here we predict that for stiff rod-like supramolecular polymers phase separation can occur at much lower polymer concentrations than for flexible supramolecular polymers. Additionally, it is demonstrated that colloid–polymer interactions can significantly influence the equilibrium polymer size distribution, however this does not strongly affect the phase behavior of the mixture. At the low polymer concentrations at which the system already gets unstable, the effect of excluded volume interactions between polymers chains themselves is small. Finally, for an experimental system it is predicted that a variety of re-entrant phase transitions may be observed within a realistic temperature range as illustrated by a specific example.     

Phase behavior in thin films of confined colloid-polymer mixtures

Using self-consistent-field and density-functional theories, we first investigate colloidal self-assembly of colloid-polymer films confined between two soft surfaces grafted by polymers. With increasing colloidal concentrations, the film undergoes a series of transitions from disordered liquid --> sparse square --> hexagonal (or mixed square-hexagonal) --> dense square --> cylindrical structures in a 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 regions of different in-layer ordering structures as the colloidal concentration is varied and layering transitions as the polymer-grafted density is decreased. Our results show a new control mechanism to stabilize the ordering of structures within the films.     

Phase behavior of charge stablized colloid dispersion with added water soluble polymers

Demixing and colloidal crystallization in the mixture of charge stabilized colloidal poly(methyl methacrylate) particles and soluble poly(ethylene oxide) were investigated by means of synchrotron small-angle X-ray scattering (SAXS) technique. Phase diagram of the mixture was obtained based on visual inspection and SAXS results. The phase behavior is determined as a function of the concentration of the polymer as well as the volume fraction of the colloidal particles. The system shows a one phase region when the concentration of the polymer is low, whereas a two-phase region is present when the concentration of the polymer is larger than a critical concentration at certain volume fraction of the colloids. Interestingly, a face centered cubic colloidal crystalline structure was formed under certain conditions, which has been rarely observed in experiments of colloid-polymer mixtures with competing interactions.     

Microstructure and depletion forces in polymer-colloid mixtures from an interfacial statistical associating fluid theory

By using a classical density functional theory (interfacial statistical associating fluid theory), we investigate the structure and effective forces in nonadsorbing polymer-colloid mixtures. The theory is tested under a wide range of conditions and performs very well in comparison to simulation data. A comprehensive study is conducted characterizing the role of polymer concentration, particle/polymer-segment size ratio, and polymer chain length on the structure, polymer induced depletion forces, and the colloid-colloid osmotic second virial coefficient. The theory correctly captures a depletion layer on two different length scales, one on the order of the segment diameter (semidilute regime) and the other on the order of the polymer radius of gyration (dilute regime). The particle/polymer-segment size ratio is demonstrated to play a significant role on the polymer structure near the particle surface at low polymer concentrations, but this effect diminishes at higher polymer concentrations. Results for the polymer-mediated mean force between colloidal particles show that increasing the concentration of the polymer solution encourages particle-particle attraction, while decreasing the range of depletion attraction. At intermediate to high concentrations, depletion attraction can be coupled to a midrange repulsion, especially for colloids in solutions of short chains. Colloid-colloid second virial coefficient calculations indicate that the net repulsion between colloids at low polymer densities gives way to net attraction at higher densities, in agreement with available simulation data. Furthermore, the results indicate a higher tendency toward colloidal aggregation for larger colloids in solutions of longer chains.     

Long-range attraction between surfaces: existence and amplitude?

The possible occurrence of a long-range attraction between hydrophobic surfaces is fundamental for understanding the kinetics of protein folding or self-assembling structures, such as biological membranes, the stability of emulsions and inorganic dispersions. Direct force measurements have revealed the presence in water of a long-range attraction between macroscopic and hydrophobic surfaces. Nevertheless, the existence of this so-called ‘hydrophobic force’ for smaller objects is still under discussion. For macroscopic surfaces, it appears that electrostatic contributions due to surface heterogeneities and gas effects and/or bubble bridging have to be taken into account, but do they define an intrinsic ‘hydrophobic force’? For charged colloidal particles, theoretical predictions of electrostatic attraction and a phase separation when the counterions are multivalent, are partially confirmed by experiments, and recent experimental evidences of an attraction in a confined geometry put an exciting challenge for theoreticians.     

Particle-induced phase separation in mixed polymer solutions

The influence of added colloidal particles on the phase separation of mixed aqueous polymer solutions is investigated. Two types of particles (polystyrene latex or silica) and different combinations of segregating polymers (dextran of varying molar mass combined with poly(ethylene oxide) (PEO) of varying molar mass, or Ucon, a copolymer of ethylene oxide and propylene oxide) were used. All systems displayed particle-induced instability effects, but the extent of the effect varied strongly between the various combinations and with the amount of added salt. Very large instability effects were seen in certain mixtures. Two mechanisms, both relying on the adsorption of at least one of the polymers to the particle surface, seem to operate. Close to the cloud-point curve of the particle-free polymer1/polymer2/water mixture, adsorption of PEO or Ucon to the particles gives rise to a capillary-induced phase separation. Close to the dextran/water axis of the phase diagram, the adsorbing polymer gives rise to a surface modification of the particles, which then interacts repulsively with the surrounding dextran solution.     

Tethered polymer chains: surface chemistry and their impact on colloidal and surface properties

In this review the grafting of polymer chains to solid supports or interfaces and the subsequent impact on colloidal properties is examined. We start by examining theoretical models for densely grafted polymers (brushes), experimental techniques for their preparation and the properties of the ensuing structures. Our aim is to present a broad overview of the state of the art in this field, rather than an in-depth study. In the second section the interactions of surfaces with tethered polymers with the surrounding environment and the impact on colloidal properties are considered. Various theoretical models for such interactions are discussed. We then review the properties of colloids with tethered polymer chains, interactions between planar brushes and nanocolloids, interactions between brushes and biocolloids and the impact of grafted polymers on wetting properties of surfaces, using the ideas presented in the first section. The review closes with an outlook to possible new directions of research.     

Polymer-induced phase separation and crystallization in immunoglobulin G solutions

We study the effects of the size of polymer additives and ionic strength on the phase behavior of a nonglobular protein-immunoglobulin G (IgG)-by using a simple four-site model to mimic the shape of IgG. The interaction potential between the protein molecules consists of a Derjaguin-Landau-Verwey-Overbeek-type colloidal potential and an Asakura-Oosawa depletion potential arising from the addition of polymer. Liquid-liquid equilibria and fluid-solid equilibria are calculated by using the Gibbs ensemble Monte Carlo technique and the Gibbs-Duhem integration (GDI) method, respectively. Absolute Helmholtz energy is also calculated to get an initial coexisting point as required by GDI. The results reveal a nonmonotonic dependence of the critical polymer concentration rho(PEG) (*) (i.e., the minimum polymer concentration needed to induce liquid-liquid phase separation) on the polymer-to-protein size ratio q (equivalently, the range of the polymer-induced depletion interaction potential). We have developed a simple equation for estimating the minimum amount of polymer needed to induce the liquid-liquid phase separation and show that rho(PEG) (*) approximately [q(1+q)(3)]. The results also show that the liquid-liquid phase separation is metastable for low-molecular weight polymers (q=0.2) but stable at large molecular weights (q=1.0), thereby indicating that small sizes of polymer are required for protein crystallization. The simulation results provide practical guidelines for the selection of polymer size and ionic strength for protein phase separation and crystallization.     

Coarse graining of star-polymer--colloid nanocomposites

We consider mixtures of self-avoiding multiarm star polymers with hard colloids that are smaller than the star polymer size. By employing computer simulations, and by extending previous theoretical approaches, developed for the opposite limit of small star polymers [A. Jusufi et al., J. Phys.: Condens. Matter 13, 6177 (2001)], we coarse-grain the mixture by deriving an effective cross-interaction between the unlike species. The excellent agreement between theory and simulation for all size ratios examined demonstrates that the theoretical approaches developed for the colloidal limit can be successfully modified to maintain their validity also for the present case of the protein limit, in contrast to the situation for mixtures of colloids and linear polymers. We further analyze, on the basis of the derived interactions, the non-additivity parameter of the mixture as a function of size ratio and star functionality and delineate the regions in which we expect mixing as opposed to demixing behavior. Our results are relevant for the study of star-colloid nanocomposites and pave the way for further investigations of the structure and thermodynamics of the same.     

Influence of added particles on the phase behavior of polymer solutions. Analysis by mean-field lattice theory

The influence of added colloidal particles on the phase stability of polymer solutions is investigated theoretically. The polymer has an affinity to the particle surface. A mean-field lattice theory based on the Flory-Huggins theory is used to calculate the phase behavior in solutions containing a single polymer component and particles. The particles are described in two different ways. The first approach considers the surface free energy associated with added solid particles and the mixing entropy of the particles. In the second approach, the particles are simply modeled as large polymers. Both ways of describing the added particles show that the added particles decrease the stability of the polymer solution when the polymer-particle attraction is strong. A higher particle concentration enhances the effect. Experiments where polystyrene latex particles are added at different concentrations to aqueous dispersions of ethyl(hydroxyethyl)cellulose (EHEC) support the theoretical findings.     

Nanoscale colloids in a freely adsorbing polymer solution: a Monte Carlo simulation study

A key issue in nanoscale materials and chemical processing is the need for thermodynamic and kinetic models covering colloid-polymer systems over the mesoscopic length scale (approximately 1-100 nm). We have applied Monte Carlo simulations to attractive nanoscale colloid-polymer mixtures toward developing a molecular basis for models of these complex systems. The expanded ensemble Monte Carlo simulation method is applied to calculate colloid chemical potentials (micro(c)) and polymer adsorption (gamma) in the presence of freely adsorbing Lennard-Jones (LJ) homopolymers (surface modifiers). gamma and micro(c) are studied as a function of nanoparticle diameter (sigma(c)), modifier chain length (n) and concentration, and colloid-polymer attractive strength over 0.3 < Rg/sigma(c) < 6 (Rg is the polymer radius of gyration). In the attractive regime, nanocolloid chemical potential decreases and adsorbed amount increases as sigma(c), or n is increased. The scaling of gamma with n from the simulations agrees with the theory of Aubouy and Raphael (Macromolecules 1998, 31, 4357) in the extreme limits of Rg/sigma(c). When Rg/sigma(c) is large, the "colloid" approaches a molecular size and interacts only locally with a few polymer segments and gamma approximately n. When Rg/sigma(c) is small, the system approaches the conventional colloid-polymer size regime where multiple chains interact with a single particle, and gamma approximately sigma(c)2, independent of n. In contrast, adsorption in the mesoscopic range of Rg/sigma(c) investigated here is represented well by a power law gamma approximately n(p), with 0 < p < 1 depending on concentration and LJ attractive strength. Likewise, the chemical potential from our results is fitted well with micro(c) approximately n(q)sigma(c)3, where the cubic term results from the sigma(c) dependence of particle surface area (approximately sigma(c)2) and LJ attractive magnitude (approximately sigma(c)). The q-exponent for micro(c) (micro(c) approximately n(q)) varies with composition and LJ attractive strength but is always very close to the power exponent for gamma (gamma approximately n(p)). This result leads to the conclusion that in attractive systems, polymer adsorption (and thus polymer-colloid attraction) dominates the micro(c) dependence on n, providing a molecular interpretation of the effect of adsorbed organic layers on nanoparticle stability and self-assembly.     

Two-stage phase separation in ternary colloid-polymer mixtures

Whilst binary colloid-polymer mixtures have been studied in detail over the past few decades, here the first results are presented on a ternary mixture involving two particle sizes. Novel and unusual phase separation kinetics are found, with a liquid phase separating from an aggregate phase.     

Particle Shape Dependence of Rheological Behavior for Colloid-polymer Mixtures

The effect of particle shape on the rheological behavior of small particle-large polymer chain mixture solutions has been investigated with two model colloidal silica dispersions, one of which is ellipsoidal（BINDZIL20/440） and the other is spherical（TM40）. It was found that BINDZIL20/440 series showed shear-thickening at lower shear rates and had a lower upper limit in PEO concentration to demonstrate shear-thickening phenomena. The particle shape was identified as the major factor accounting for these differences. This work enables one to control the rheological behavior of colloid-polymer mixture through simply changing particle geometry instead of performing surface modifications, which could be especially useful in cases where only certain chemicals are allowed, for example in vivo applications.     

Interaction forces between silica surfaces in aqueous solutions of cationic polymeric flocculants: effect of polymer charge

Three cationic polymers with molecular weights and charge density of 3.0 x 10(5) g/mol and 10% (D 6010), 1.1 x 10(5) g/mol and 40% (D6040), and 1.2 x 10(5) g/mol and 100% (D6099) were investigated in aqueous NaCl solutions in the presence of silica. The atomic force microscope (AFM) colloidal probe technique was used to determine silica interparticle interaction forces, which were compared to macroscopic information on the strength of interactions such as compressive yield stress measurements. It was found that in 30 mM NaCl solution the 10% charged polymer produced steric repulsion upon approach and long-range adhesion with multiple pull off events upon retraction at the optimum flocculation concentration. This suggests that the polymer was adsorbed in a conformation where segments extend from the surface, resulting in bridging flocculation. The 40 and 100% charged polymers produced attraction upon approach and strong adhesion with snap out from contact upon separation at optimum polymer dosages. This suggests that these polymers are adsorbed with flat conformations and is typical of charge neutralization or patch attraction. The attractions for 40 and 100% charged polymers measured with the AFM are significantly larger than for the 10% charged polymer. The polymer dose that produced the optimum flocculation and the maximum compressive yield stress typically corresponded to the polymer concentration that produced the maximum adhesion for each polymer. It was found that the magnitude of the adhesive force was more significant in determining the compressive yield stresses of the silica particle sediments than the aggregate size and structure.