Non-adsorbing Polymer Chains Research Articles

Depletion potential, correlation functions and demixing transition in model colloid-polymer mixtures

We describe a theoretical framework to calculate depletion potential between colloid particles induced by non-adsorbing ideal polymer chains (s-species) and correlation functions in a coarse-grained one-component system of colloids (c-species). A Padé approximant is used to express the depletion potential as a pair potential with many-body contributions subsumed in it. The depletion potential and correlation functions of c-species are calculated using a self-consistent procedure. Results for several values of size ratio q=σsσc (σs and σc are, respectively diameters of the polymer chain and a colloid particle) and packing fractions of s- and c-species are reported. The spinodal curve and critical point of demixing transition are determined for several values of q. Calculated values are compared with values found from other theories and simulations.

Multiphase Coexistences in Rod-Polymer Mixtures.

Using recently derived analytical equations of state for hard rod dispersions, we predict the phase behavior of athermal rod–polymer mixtures with free volume theory. The rods are modeled as hard spherocylinders, while the nonadsorbing polymer chains are described as penetrable hard spheres. It is demonstrated that all of the different types of phase states that are stable for pure colloidal rod dispersions can coexist with any combination of these phases if polymers are added, depending on the concentrations, rod aspect ratio, and polymer–rod size ratio. This includes novel two-, three-, and four-phase coexistences and isostructural coexistences between dilute and concentrated phases of the same kind, even for the more ordered (liquid) crystal phases. This work provides insight into the conditions at which particular multiphase coexistences are expected for well-defined model colloidal rod–polymer mixtures. We provide a quantitative map detailing the various types of isostructural coexistences, which confirms an early qualitative hypothesis by Bolhuis et al. (J. Chem. Phys.107, 19971551).

Phase behaviour of colloidal superballs mixed with non-adsorbing polymers

.Inspired by experimental work on colloidal cuboid-polymer dispersions (Rossi et al., Soft Matter, 7, 4139 (2011)) we have theoretically studied the phase behaviour of such mixtures. To that end, free volume theory (FVT) was applied to predict the phase behaviour of mixtures of superballs and non-adsorbing polymer chains in a common solvent. Closed expressions for the thermodynamic properties of a suspension of hard colloidal superballs have been derived, accounting for fluid (F), face-centred cubic (FCC) and simple cubic (SC) phase states. Even though the considered solid phases are approximate, the hard superballs phase diagram semi-quantitatively matches with more evolved methods. The theory developed for the cuboid-polymer mixture reveals a rich phase behaviour, which includes not only isostructural F1-F2 coexistence, but also SC1-SC2 coexistence, several triple coexistences, and even a quadruple-phase coexistence region (F1-F2-SC-FCC). The model proposed offers a tool to asses the stability of cuboid-polymer mixtures in terms of the colloid-to-polymer size ratio.Graphical abstract

Depletion-driven four-phase coexistences in discotic systems

ABSTRACTFree volume theory (FVT) is a versatile and tractable framework to predict the phase behaviour of mixtures of platelets and non-adsorbing polymer chains in a common solvent. Within FVT, three principal reference phases for the hard platelets are considered: isotropic (I), nematic (N) and columnar (C). We derive analytical expressions that enable us to systematically trace the different types of phase coexistences revealed upon adding depletants and confirm the predictive power of FVT by testing the calculated diagrams against phase stability scenarios from computer simulation. A wide range of multi-phase equilibria is revealed, involving two-phase isostructural transitions of all phase symmetries (INC) considered as well as the possible three-phase coexistences. Moreover, we identify the system parameters, relative disk shapes and colloid–polymer size ratios, at which four-phase equilibria are expected. These involve a remarkable coexistence of all three-phase states commonly encountered in discotics including isostructural coexistences I1–I2–N–C, I–N1–N2–C and I–N–C1–C2.

Colloidal gels tuned by oscillatory shear.

We examine the microstructural and mechanical changes which occur during oscillatory shear flow and reformation after flow cessation of an intermediate volume fraction colloidal gel using rheometry and Brownian Dynamics (BD) simulations. A model depletion colloid-polymer mixture is used, comprising a hard sphere colloidal suspension with the addition of non-adsorbing linear polymer chains. The results reveal three distinct regimes depending on the strain amplitude of oscillatory shear. Large shear strain amplitudes fully break the structure which results in a more homogenous and stronger gel after flow cessation. Intermediate strain amplitudes densify the clusters and lead to highly heterogeneous and weak gels. Shearing the gel to even lower strain amplitudes creates a less heterogonous stronger solid. These three regimes of shearing are connected to the microscopic shear-induced structural heterogeneity. A comparison with steady shear flow reveals that the latter does not produce structural heterogeneities as large as oscillatory shear. Therefore oscillatory shear is a much more efficient way of tuning the mechanical properties of colloidal gels. Moreover, colloidal gels presheared at large strain amplitudes exhibit a distinct nonlinear response characterized largely by a single yielding process while in those presheared at lower rates a two-step yielding process is promoted due to the creation of highly heterogeneous structures.

Residual stresses in colloidal gels.

A combination of experiments and Brownian Dynamics (BD) simulations is utilized to examine internal stresses in colloidal gels brought to rest from steady shear at different shear rates. A model colloidal gel with intermediate volume fraction is chosen where attractions between particles are introduced by adding non-adsorbing linear polymer chains. After flow cessation, the gel releases the stress in two distinct patterns: at high shear rates, where shear forces dominate over attractive forces, the shear-melted gel behaves as a liquid and releases stresses to zero after flow cessation. After low shear rates, though, stresses relax only partially, similar to the response of hard sphere glasses and jammed soft particles. The balance between shear and attractive forces which determines the intensity of structural distortion controls the amplitude of the residual stresses through a universal scaling. Stress decomposition to repulsive and attractive contributions in BD simulations reveals that internal stresses mainly originate from attractive forces. Moreover, analysis of particle dynamics indicates that internal stresses are associated with sub-diffusive particle displacements on average smaller than the attraction range as such short-range displacements are not sufficient to completely erase structural anisotropy caused during the course of shear.

Phase behaviour of colloids with short-range repulsions plus nonadsorbing polymer chains

The colloidal gas–liquid phase behaviour has been studied for aqueous mixtures of well-defined spherical particles with short-ranged repulsions mixed with relatively monodisperse poly(ethylene oxide) polymers. We show that our set of experimental phase diagrams are in quantitative agreement with theoretical predictions using generalized free volume theory [G. J. Fleer and R. Tuinier, Phys. Rev. E, 2007, 76, 041802]. The determination of the equilibrium composition of coexisting phases reveals qualitative deviations between the two, for which we propose a tentative explanation.

Hydrodynamic interaction of two colloids in nonadsorbing polymer solutions

The hydrodynamic interaction between two colloids mediated by non-adsorbing polymer chains is considered for spherical colloids moving along the symmetric line. We present a resistance analysis and determine the mobility function that incorporates the polymer depletion effect. The hydrodynamic friction is calculated by solving the Stokes equation with non-uniform viscosity using the vorticity-stream function method and a bispherical coordinate transformation. The numerical results show the effective viscosity asymptotically approaches the single sphere limit of Fan et al. (T.-H. Fan, B. Xie and R. Tuinier, Phys. Rev. E., 2007, 76, 051405) for a large interparticle distance. As the particles get closer the effective viscosity decreases and finally approaches the lubrication limit, where the friction equals that of two close-approached spheres in a pure solvent. The flow analysis shows that the circulation pattern, a characteristic for the presence of the depletion layer, expands upon approach of the particles. The pressure contribution to the total drag force is determined by a competition between the lubrication force and the apparent slip effect caused by overlap of the depletion layers. The friction of the single sphere limit is only attained in the limit of an extremely small colloid volume fractions. Our results help rationalize the long-time relative diffusivity of Brownian particles in polymer solutions and this is important for a better understanding of the depletion-induced demixing kinetics.

Scaling the Structure Factors of Protein Limit Colloid−Polymer Mixtures

The scaling of the phase boundaries and structure factors of protein limit colloid-polymer mixtures has been investigated through the addition of large nonadsorbing polymer chains to a solution of small microemulsion droplets. The colloid-polymer size ratio has been varied between 10 and 16 by changing the microemulsion droplet size; the phase boundaries were shown previously to observe theoretical scaling relations very well [Langmuir 2009, 25 (7), 3944-3952]. These thermodynamic scaling relations are now shown to also hold extremely well for the individual and cross-term partial structure factors. The structure factors for systems with different size ratios occupying the same point in scaled phase space show extremely good agreement. The properties and stability of these mixtures are governed by the polymer mesh.

Testing the Scaling Behavior of Microemulsion−Polymer Mixtures

The phase behavior and structural properties of "protein limit" mixtures of small (radius 20-30 A) water-in-oil microemulsion droplets (colloids) and large (radius 130-580 A) nonadsorbing polymer chains have been investigated. Accepted theoretical scaling relations for describing correlations have been applied and do not account fully for the observations; solvency effects may account for the deviations. The polymer/colloid size ratio has been varied from around 4 to 19 by using three different molecular weights of polyisoprene. Small-angle neutron scattering (SANS) has been used to determine partial structure factors (PSF) through contrast variation. The structure factors describing colloid-colloid interactions for the three polymers at fixed polymer concentration are shown to exhibit the same scaling behavior as the phase boundaries, provided that samples are sufficiently far from the demixing phase transition. The structure factors show a dramatic increase at low wavevectors on approaching the phase boundary, and behavior in this region does not obey expected scaling relations. By calculating effective polymer Flory-Huggins parameters, the effect of apparent solvent properties on adding microemulsion are shown to be less dramatic for the higher molecular weight polymers. This study extends previous work carried out on microemulsion-polymer mixtures.

Phase behavior of a suspension of hard spherocylinders plus ideal polymer chains

We study isotropic-isotropic and isotropic-nematic phase transitions of fluid mixtures containing hard spherocylinders (HSC) and added non-adsorbing ideal polymer chains using scaled particle theory (SPT). First, we investigate isotropic-nematic (I -N phase coexistence using SPT in the absence of polymer. We compare the results obtained using a Gaussian form of the orientational distribution function (ODF) to minimize the free energy versus minimizing numerically. We find that formal numerical minimization gives results that are much closer to computer simulation results. In order to describe mixtures of HSC plus ideal chains we studied the depletion of ideal chains around a HSC. We analyze the density profiles of ideal chains near a hard cylinder and find the depletion thickness delta is a function of the ratio of the polymer's radius of gyration R(g) and the cylinder radius R(c). Our results are compared with a common approximation in which the depletion thickness is taken equal to the radius of gyration of the polymer chain. We incorporate the correct depletion thickness into SPT and find that for R (g)/R (c) < 1.56 using ideal chains gives phase transitions at smaller polymer concentrations, whereas for R (g)/R (c) > 1.56 , which is a common experimental situation, the phase transitions are found at larger polymer concentrations with respect to delta = R (g) . The differences are significant, especially for R (g) >> R (c) , so we can conclude it is essential to take into account the properties of ideal polymer chains and the resulting depletion near a cylinder. Finally, we present phase diagrams for rod-polymer mixtures which could be realized under experimental conditions.

Concentration and Solvency Effects on the Excess Amount and Surface Free Energy of a Colloidal Particle in a Solution of Nonadsorbing Polymer

Analytical expressions are derived for the polymer excess amount and the grand potential (surface free energy) of flat and spherical surfaces immersed in a solution of nonadsorbing polymer chains in the mean-field approximation. We start from a recent mean-field expression for the depletion thickness δ which takes into account not only the effect of the chain length N but also that of the polymer concentration φb and the solvency χ. Simple expressions are obtained for the interfacial properties at a colloidal surface, using both the adsorption method and the osmotic route. For a sphere of radius a, the excess amount can be separated into a planar contribution Γ = −φbδ and a curvature correction Γc = −(π2/12)φbδc2/a, where δc is a “curvature thickness” which is close to (but smaller than) δ. The grand potential has a planar contribution Ω = (2/9)φb/δ and a curvature part Ωc = (π/18)φb/a. We test the results against numerical lattice computations, taking care that the boundary conditions in the continuum and lattice models are the same. We find good agreement up to a polymer segment volume fraction of 10%, and even for more concentrated solutions our simple model is reasonable. For spherical geometry we propose a new equation for the segment concentration profile which excellently agrees with numerical lattice computations. The results can be used as a starting point for the pair interaction between colloidal particles in a solution containing nonadsorbing chains, which is discussed in the following paper.

Bending elasticity of a curved amphiphilic film decorated with anchored copolymers: asmall angle neutron scattering study

Microemulsion droplets (oil in water stabilized by a surfactant film) are progressivelydecorated with increasing amounts of poly(ethylene oxide) (PEO) chains anchored in thefilm by the short aliphatic chain grafted at one end of the PEO chain. The evolution of thebending elasticity of the surfactant film with increasing decoration is deduced from theevolution in size and polydispersity of the droplets as reflected by small angleneutron scattering. The optimum curvature radius decreases while the bendingrigidity modulus remains practically constant. The experimental results comparewell with the predictions of a model developed for the bending properties of acurved film decorated with non-adsorbing polymer chains, which takes into accountthe finite curvature of the film and the free diffusion of the chains on the film.

Polymer depletion interaction of small mesoscopic particles: Effects beyond leading order and anisotropic particles

We discuss the depletion interaction between a wall and a mesoscopic particle of ellipsoidal shape induced by long, flexible, nonadsorbing polymer chains. Both a force and a torque are exerted on the particle. We concentrate on the case in which the particle size is much smaller than typical polymer lengths, such as the radius of gyration Rg, where a rigid polymer approximation of the Asakura–Oosawa-type cannot be applied. Explicit analytical results are obtained for ideal polymers. For particle–wall distances z large compared to Rg an orientation of the ellipsoid perpendicular to the wall is favored. For z small compared to Rg (but z still large compared to the particle size), parallel orientation is favored. The perturbation of the polymer system due to the small particle is represented by a series of point-operators in the corresponding field theory, with next-to-next-to-leading anisotropic derivative-operators characterizing the particle orientation. For the interaction between a spherical particle and a wall the simple analytical results predicted by the proposed small particle expansion beyond leading order display an interesting structure which is confirmed by direct numerical computation.

Colloid-induced polymer compression

We consider a model mixture of hard colloidal spheres and nonadsorbing polymerchains in a theta solvent. The polymer component is modelled as a polydispersemixture of effective spheres, mutually noninteracting but excluded from thecolloids, with radii that are free to adjust to allow for colloid-inducedcompression. We investigate the bulk fluid demixing behaviour of this modelsystem using a geometry-based density functional theory that includesthe polymer size polydispersity and configurational free energy, obtainedfrom the exact radius-of-gyration distribution for an ideal (random-walk)chain. Free energies are computed by minimizing the free energy functionalwith respect to the polymer size distribution. With increasing colloidconcentration and polymer-to-colloid size ratio, colloidal confinement is found toincreasingly compress the polymers. Correspondingly, the demixing fluidbinodal shifts, compared to the incompressible-polymer binodal, to higherpolymer densities on the colloid-rich branch, stabilizing the mixed phase.

An Empirical Kinetic Model for the Gelation of a Concentrated Latex Suspension in the Presence of Nonadsorbing Polymer Chains

An empirical model has been proposed to describe the kinetic aspect of the gelation process of a concentrated latex mixture in the presence of nonadsorbing polymer chains. It was found to allow the identification of two predominant effects, a viscosity effect and an excluded volume one effect that balance during gelation, and to predict the polymer volume fraction for which the transition between these two predominant effects occurs in the dilute polymer concentration range.

Excluded-volume polymer-induced depletion interaction between parallel flat plates

The interaction between two parallel plates due to non-adsorbing polymer chains with excluded volume is calculated using the adsorption method. The adsorption is calculated from the profile of the polymer segment concentration between the plates, which is obtained from the product function of the concentration profile near a single wall, involving the correlation length. The renormalization group theory provides expressions for the osmotic pressure and consequently for the osmotic compressibility, chemical potential and correlation length of a polymer solution. Both the local polymer concentration profiles as well as the minimum of the interaction potential between the plates agree with recently published self-avoiding random walk computer simulations.

Density depletion profile and solvation free energy of a colloidal particle in a polymer solution

The solvation free energy and polymer density depletion profile of a single mesoscopic colloidal particle in a solution of free nonadsorbing polymer chains are investigated theoretically. Keeping both the particle to polymer size ratio and the degree of inter-chain overlap arbitrary, we see how the qualitatively different behavior evolves in the limits of small and large size ratios and of dilute and semidilute solutions. While most of our results are obtained within a mean-field approach, we also use a “renormalized tree approximation” to estimate the surface tension and the coefficient of spontaneous curvature in a Helfrich expansion for large particle to polymer size ratio. There is a weak maximum in the polymer density profile for arbitrary size ratio. For small size ratio the maximum can be explained in terms of a minimum in the bulk polymer density correlation function.

Macromolecular theory of solvation and structure in mixtures of colloids and polymers.

The structural and thermodynamic properties of mixtures of colloidal spheres and nonadsorbing polymer chains are studied within a general two-component macromolecular liquid state approach applicable for all size asymmetry ratios. The dilute limits, when one of the components is at infinite dilution but the other concentrated, are presented and compared to field theory and to models that replace polymer coils with spheres. Whereas the derived analytical results compare well, qualitatively and quantitatively, with mean-field scaling laws where available, important differences from "effective sphere" approaches are found for large polymer sizes or semidilute concentrations.

Rheological Study of the Gelation Kinetics of a Concentrated Latex Suspension in the Presence of Nonadsorbing Polymer Chains

The gelation kinetics of a concentrated latex suspension in the presence of two nonadsorbing sodium carboxymethylcellulose polymer chains with different molecular weight has been investigated. The experimental results indicate that the presence of the polymer chains generates two concomitant and opposite effects on the gelation time. Providing excluded volume effects are predominant, the gelation time was found to decrease. When a critical value of the polymer volume fraction in the bulk phase was reached, the gelation time increased with concentration due to viscosity effects. Moreover, the dynamic rheological measurements were shown to exhibit the influence of the excluded volume effects on the fractal dimension of the growing aggregates of the gelified networks.