Solution Of Nonadsorbing Polymer Research Articles

Flow induced by an oscillating sphere in probing complex viscosity of polymer solutions

A theoretical investigation is presented for a linear viscoelastic flow induced by an oscillatory colloidal particle in nonadsorbing polymer solutions. At small-amplitude oscillations, the polymer distribution is assumed to be at equilibrium and forms a depletion zone around the particle based on the mean-field approximation. The goal of the theoretical approach is to predict the apparent complex viscosity sensed by the particle and compare this with the actual viscosity of the bulk fluid. Due to the local inhomogeneity, substantial deviation between the apparent and true viscosity in the bulk needs to be corrected quantitatively. The resulting apparent complex viscosity or friction coefficient in the Fourier domain will help to interpret active and passive microrheological measurements of colloid-polymer mixtures that take polymer depletion into account.

Diffusiophoresis in nonadsorbing polymer solutions: The Asakura-Oosawa model and stratification in drying films.

A colloidal particle placed in an inhomogeneous solution of smaller nonadsorbing polymers will move towards regions of lower polymer concentration in order to reduce the free energy of the interface between the surface of the particle and the solution. This phenomenon is known as diffusiophoresis. Treating the polymer as penetrable hard spheres, as in the Asakura-Oosawa model, a simple analytic expression for the diffusiophoretic drift velocity can be obtained. In the context of drying films we show that diffusiophoresis by this mechanism can lead to stratification under easily accessible experimental conditions. By stratification we mean spontaneous formation of a layer of polymer on top of a layer of the colloid. Transposed to the case of binary colloidal mixtures, this offers an explanation for the stratification observed recently in these systems [A. Fortini etal., Phys. Rev. Lett. 116, 118301 (2016)PRLTAO0031-900710.1103/PhysRevLett.116.118301]. Our results emphasize the importance of treating solvent dynamics explicitly in these problems and caution against the neglect of hydrodynamic interactions or the use of implicit solvent models in which the absence of solvent backflow results in an unbalanced osmotic force that gives rise to large but unphysical effects.

Nonlinear electro-osmosis of dilute non-adsorbing polymer solutions with low ionic strength.

Nonlinear electro-osmotic behaviour of dilute non-adsorbing polymer solutions with low salinity is investigated using Brownian dynamics simulations and a kinetic theory. In the Brownian simulations, the hydrodynamic interaction between the polymers and a no-slip wall is considered using the Rotne-Prager approximation of the Blake tensor. In a plug flow under a sufficiently strong applied electric field, the polymer migrates toward the bulk, forming a depletion layer thicker than the equilibrium one. Consequently, the electro-osmotic mobility increases nonlinearly with increasing electric field and becomes saturated. This nonlinear mobility does not depend qualitatively on the details of the rheological properties of the polymer solution. Analytical calculations using the kinetic theory for the same system quantitatively reproduce the results of the Brownian dynamics simulation well.

Stochastic interactions of two Brownian hard spheres in the presence of depletants.

A quantitative analysis is presented for the stochastic interactions of a pair of Brownian hard spheres in non-adsorbing polymer solutions. The hard spheres are hypothetically trapped by optical tweezers and allowed for random motion near the trapped positions. The investigation focuses on the long-time correlated Brownian motion. The mobility tensor altered by the polymer depletion effect is computed by the boundary integral method, and the corresponding random displacement is determined by the fluctuation-dissipation theorem. From our computations it follows that the presence of depletion layers around the hard spheres has a significant effect on the hydrodynamic interactions and particle dynamics as compared to pure solvent and uniform polymer solution cases. The probability distribution functions of random walks of the two interacting hard spheres that are trapped clearly shift due to the polymer depletion effect. The results show that the reduction of the viscosity in the depletion layers around the spheres and the entropic force due to the overlapping of depletion zones have a significant influence on the correlated Brownian interactions.

Phase Diagram of Solvophilic Nanodiscs in a Polymer Solution: Depletion Attraction

Entropic attraction between anisotropic nanoparticles in nonadsorbing polymer solutions is significantly greater than that between nanospheres. In this study, depletion attraction between two solvophilic nanodiscs and the aggregation behavior of a nanodisc suspension are explored by dissipative particle dynamics. The depletion force due to polymer addition is proportional to the product of the osmotic pressure and the area of the nanodisc, even for concentrated polymer solutions. For a suspension of nanodiscs, three possible equilibrium states are observed upon polymer addition: dispersion, pretransition, and phase separation. By varying nanodisc concentration (φD) and polymer concentration (φP), the phase diagram is obtained. Dispersion exists for small φD and φP. As φD exceeds critical aggregation concentration, finite sized clusters are mainly formed with columnar structure in the pretransition regime, similar to micellization associated with typical surfactants. The mean size of columnar clusters seems to grow with φD(1/2) and φP. For high enough φD and φP, finite sized clusters vanish and phase separation appears. The nanodisc-rich phase is formed by a bundle of columns or a pack of columns owing to face-to-side and side-to-side depletion attractions.

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.

Asymptotic analysis of tracer diffusivity in nonadsorbing polymer solutions

We present an asymptotic and scaling analysis of the long-time self-diffusivity of a Brownian spherical particle in dilute polymer solutions with nonadsorbing chains. The polymer depletion zone near the particle surface is described by a continuous polymer density profile. Hydrodynamics formulated by the modified Stokes equation with nonuniform viscosity is solved by a regular perturbation approximation using the Green function method. The asymptotes predict how polymer depletion alters the friction a spherical particle experiences during translational and rotational motion within a quiescent fluid. The analysis agrees very well with full numerical computation, which enables us to investigate the scaling law for the polymer-mediated retardation effect using a stretched exponential form that is commonly applied by experimentalists. The scaling exponents revealed are consistent with the nominal values from collected experiment observations.

Polymer depletion profiles around nonspherical colloidal particles

We study the effect of chain self-avoidance on the polymer density profiles that are induced by a single colloidal particle of nonspherical shape such as an ellipsoid, a dumbbell, or a lens in a solution of nonadsorbing polymers. For colloid sizes sigma much smaller than the size R(x) of the polymers, we observe a pronounced difference between ideal and self-avoiding chains. In the case of ideal polymers, the surfaces of constant density always have the same character as the surface of the particle, e.g., are oblate for an oblate ellipsoid. In the self-avoiding case, however, the character changes with increasing distance r from the particle, and an oblate particle induces prolate surfaces of constant density if sigma<<r<<R(x). For sigma<<r, R(x), the isotropic and anisotropic contributions to the densities factor into a dependence on the particle size and shape and a dependence on r, R(x). The latter is determined by distance distributions within a chain in the absence of the particle. For self-avoiding polymers in two spatial dimensions, exact density profiles are derived for sigma, r<<R(x), which explicitly show the above-mentioned change of the contours of constant density.

Effect of polydispersity on the depletion interaction in nonadsorbing polymer solutions

The effect of polymer polydispersity on the depletion interaction between two plates immersed in a nonadsorbing polymer solution is studied by self-consistent-field theory. The depletion potential is calculated numerically for the Schulz molecular weight distribution. The results show that as the two plates approach the polymers with different chain lengths are excluded from the gap gradually for conformational entropy penalty, and the range of the depletion potential increases and the depth of the potential decreases with increasing polydispersity. For the case of two large spheres the Derjaguin approximation is used to study the effect of polydispersity. The result shows that the polydispersity has little effect on the contact potential. However, it induces more broad interaction than the monodisperse polymers.

Dependence of depletion layer thickness on polymer concentration determined by Vincent's pragmatic and Donath's electrophoretic theories

On the interface of a solid surface and a solution of nonadsorbing polymer there exists a depletion layer (DL), where the concentration of polymer segments is lower. Donath's electrophoretic theory, based on the decreased viscosity in the DL region, allows computing DL thickness from the relative (with and without polymer) electrophoretic mobility, the bulk viscosity, and the ionic strength. The aim of this work is to check experimentally Donath's nonlinear electrophoretic (NLE) theory under the most favorable conditions—liposomes in solutions of low-molecular poly(ethylene glycol) (PEG). In order to determine DL thickness, the dependence of mobility on viscosity is chosen instead on ionic strength. The value obtained from NLE theory is compared with the DL thickness calculated by Vincent's pragmatic theory. The conformation-statistical parameters are calculated on the base of viscosimetric measurements of PEG solution. The results indicate a few shortcomings of NLE theory. The main one is that DL thickness does not depend on polymer concentration, a fact that is in discrepancy with the prediction of Vincent's theory. The conclusion is that NLE theory describes well the experimental dependence of the relative mobility on the bulk viscosity, but it is inapplicable to quantitative determination of DL thickness.

Depletion interaction between two colloidal particles in a nonadsorbing polymer solution

The depletion effect between two spherical colloidal particles in nonadsorbing polymer solutions is investigated using the self-consistent field theory. The density distributions of polymer segments, the depleted amount and depletion potential are calculated numerically in bispherical coordinates. The effects of chain length, bulk concentration, and solvency are also investigated. In the dilute regime the depleted amount and the depletion potential decrease as the two spherical particles approach to each other. The depth of interaction increases and the width of interaction varies slightly with increasing bulk concentrations. In the semidilute regime, with increasing bulk concentrations the width of interaction decreases and the depth of interaction increases. No distinct repulsive potential is observed in semidilute regime. However, at high concentration the depleted amount exhibits a barrier. The width and the depth of depletion potential increase with increasing the chain length and the solvency. The contact potential is proportional to the polymer concentration and almost independent on the solvency. In addition, the effect of depletion interaction on colloidal stability is analyzed.

Colloidal brushes in complex solutions: Existence of a weak midrange attraction due to excluded-volume effects

Towards the prediction of thermodynamic properties for complex systems and the design of self-assembly techniques for the production of nanocomposite materials, we employ Monte Carlo simulations to calculate the effective interactions between colloidal brushes dissolved in nonadsorbing polymer solutions. For simplicity, all interactions are reduced to excluded-volume potentials. Our results indicate that (1) due to excluded-volume effects and the absence of attractive van der Waals forces in our calculations, the effective interactions between colloidal brushes in a continuum medium are always repulsive; and (2) that the short-ranged repulsion between the colloidal brushes may be coupled to a midrange attraction when nonadsorbing polymers are in solution. We prove that a depletion mechanism is responsible for the midranged attraction and we observe that the strength of the induced attraction is weaker compared to the depletion attraction computed for hard-sphere colloids in nonadsorbing polymer solutions. More interestingly, however, our results indicate that, even for the simple interaction potentials employed here, it is not possible to superimpose depletion attraction and steric repulsion to predict the correct pair potential between colloidal brushes in nonadsorbing polymer solutions.

Lubrication properties of non-adsorbing polymer solutions in soft elastohydrodynamic (EHD) contacts

The in-use performance and processing of many consumer products in the food, home and personal care industries are dependent on their tribological properties. A major component of these products is often a high molecular weight polymer, which is typically used to thicken aqueous systems. Polymer solutions tend to be non-Newtonian, and in particular their viscosity varies with shear rate, such that it is difficult to predict their friction or hydrodynamic film-forming behaviour. The present work relates the tribology of aqueous polymer solutions to their rheological properties in thin films in ‘soft’ contacts at high shear rates. The friction properties of three types of polymers in aqueous solution, polyethylene oxide, PEO; xanthan gum, XG; and guar gum, GG, have been studied as a function of polymer concentration over a wide range of entrainment speeds in a point contact formed between silicone rubber and steel. This has enabled the boundary lubrication and isoviscous-elastic lubrication properties of the solutions to be investigated using both hydrophilic and hydrophobic silicone surfaces. It is found that the friction vs. entrainment speed dependence follows the shape of a classical Stribeck curve. In general, a lower friction is observed with increasing polymer concentration in the mixed-regime. Using scaling factors for the entrainment speed, we have shown that this decrease in friction is likely to be due to viscous effects and that the scaling factors represent effective high shear rate viscosities. In the case of PEO and XG, and GG at low concentrations, a good correlation is found between this effective viscosity and the apparent viscosity measured at the highest shear rates attainable with the available rheometer. However, for GG at concentrations above 0.2%, the effective viscosity decreases with increasing polymer content. The three polymers do not significantly reduce friction in the boundary regime and in general give essentially the same response as water when an effective viscosity is taken into account. However, a slight increase in friction in comparison to pure water has been observed for XG and GG on hydrophobic surfaces. It is suspected that this may be due to a blocking of fluid entrainment, or possibly exclusion of polymer from the contact, due to the large hydrodynamic volume and rigid nature of the two biopolymers. Finally, for PEO solutions with full-film elastohydrodynamic conditions were reached, the measured friction coefficient of the film correlated quite well with the value calculated from the effective viscosity.

Study on Effects of Polydispersity on the Form Factor and Depletion Interaction in Colloid‐Polymer Mixtures with Laser Distribution

AbstractSummary: We investigate the influence of the polydispersity of colloidal particles and polymer, respectively, on the form factor and depletion interaction by using the laser distribution for parallel plates immersed in a solution of nonadsorbing polymers. For colloidal particles, we show that the measurement of scattering intensity of a polymer‐colloid mixture can be described qualitatively in terms of the parameters K and M of the laser distribution. The form factor of the monodisperse system varies considerably, but generally stays close to the average value that corresponds to the form factor of polydisperse system. By using the laser distribution in the chain length of polymers, we present the complex polymer segment density of polymer chain between particles with the product‐function approximation comprehensively and accurately. In particular, we find that for small values of parameter M, the change of values of parameter K has no effect on depletion interaction.Predicted form factor for polydisperse core‐shell particles.magnified imagePredicted form factor for polydisperse core‐shell particles.

The Depletion Attraction between Pairs of Colloid Particles in Polymer Solution

NVT Monte Carlo simulations were used to assess the effective interaction between pairs of colloid particles dissolved in non-adsorbing polymer solutions. The polymers were represented as freely-jointed-hard-sphere chains composed of 10, 20, or 30 segments. The size of the interacting colloid particles was similar to or smaller than the radius of gyration of the polymers. Results show a short-range colloid–colloid depletion attraction. At low polymer concentration, this attraction slowly decays to zero at increasing separations. At higher polymer concentration, the depletion attraction is coupled to a mid-range repulsion, especially for solutions of short, stiff polymers. From the simulated forces, osmotic second virial coefficients were computed for colloids as a function of polymer concentration. The calculated osmotic second virial coefficients exhibit a non-monotonic dependence on polymer concentration, in qualitative agreement with experimental results. The simulated colloid–colloid potentials of mean force were used, within a perturbation theory, to calculate fluid–fluid and fluid–solid coexistence curves. The colloids are treated as a pseudo one-component system, and the polymers in solution are considered only through the effective pair potential between the dissolved colloids. When long flexible polymers are dissolved in solution, the phase diagram for small colloid particles shows a fluid–fluid coexistence curve at low colloid packing fraction, and a fluid–solid coexistence curve at higher packing fraction. As the size of the colloid particles increases, the molecular weight of the polymer decreases, or the polymer concentration in solution increases, the fluid–fluid coexistence curve becomes metastable with respect to the fluid–solid coexistence curve.

The importance of the interfacial stabilising layer on the macroscopic flow properties of suspensions dispersed in non-adsorbing polymer solution

In this paper, the rheological properties, microscopic appearance and macroscopic sedimentation behaviour of 147- and 482-nm polystyrene latices in HEC solutions, bearing different adsorbed poly(ethyleneoxide)–poly(propyleneoxide)–poly(ethyleneoxide) (PEO–PPO–PEO) copolymers are presented and compared with previous results with a 67-nm latex (Langmuir 13 (1997) 2922). The ratio of the steric layer thickness to the particle radius varies from 1:5 to 1:40 for the three latex sizes covering a range of particle softness. The 147-nm latex showed gas, liquid and solid phases, including three phase coexistence with increasing concentrations of HEC. The solid phase was a viscoelastic gel that sedimented slowly and showed initially slow, then faster sedimentation rates for HEC concentrations close to the fluid–gel phase boundary. These properties depended on the adsorbed PEO–PPO–PEO copolymers; the fluid–gel phase boundary moved to lower HEC concentrations with increasing PEO chain length. Oscillatory shear measurements were sensitive to the floc network structure and showed the transition from the fluid to gel phases. The values for the elastic modulus in the gel region were independent of the PEO chain length in the stabiliser implying the presence of similar floc structures with each of the different PEO–PPO–PEO copolymers. Stress relaxation measurements gave long relaxation times (>10 3 s) and showed that the suspensions were viscoelastic fluids with high zero shear viscosities. Higher values were obtained with longer PEO chain lengths. Extrapolated yield stress values showed stronger flocculation with increasing PEO chain length in the adsorbed stabilising layer. There was good correlation between the extrapolated yield stress and the sedimentation velocity, indicating that the collapse of the floc network is governed by the strength of the interparticle bonds (for the same floc structure) and also with the long-time behaviour from the stress relaxation measurements. With the 482-nm latex there was less effect of the adsorbed copolymer both for the rheology and sedimentation behaviour, with the particles behaving more like hard spheres. The transition between hard and soft sphere behaviour, interpreted here as the ratio of the steric layer thickness to the particle radius at which the adsorbed stabilising layer starts to have an effect on the rheology and phase behaviour is estimated to be approximately 1:20.

Mean-Field Equation for the Depletion Thickness

We derive a general equation for the depletion thickness δ next to a flat wall in a solution of nonadsorbing polymer, which is easily extended to spherical geometry. This equation has the simple form δ-2 = δ0-2 + ξ-2. Here, δ0 is the value of δ in the limit of infinite dilution, which depends only on the chain length: δ0 = 2R/ , where R is the radius of gyration of the polymer. The parameter ξ is a correlation length in solution, which depends on the polymer concentration φb and the solvency χ, but not on the chain length. We use a mean-field form of ξ = ξ(χ,φb) which provides a smooth crossover from good to ϑ solvency conditions. We show that the depletion thickness is actually a generalized bulk solution correlation length which does depend on chain length. In all cases the profile for a flat wall is given by ρ = φ/φb = tanh2(z/δ). The extension to a sphere of radius a is also very simple: ρs = [z/a + tanh(z/δ)]2/[z/a + 1]2. These analytical results are compared to numerical self-consistent-field comp...

Interaction potential between two spheres mediated by excluded volume polymers.

The interaction between two spheres in a solution of nonadsorbing polymers, with excluded volume interaction, is calculated from the depletion layers around the spheres using the generalized Gibbs adsorption equation. By combining the bulk correlation length with the curvature-dependent interfacial tension between a sphere and the surrounding polymer solution [Hanke, Eisenriegler, and Dietrich, Phys. Rev. E 59, 6853 (1999)], the depletion layer thickness around a sphere is obtained. The resulting contact potential agrees with a scaling prediction of de Gennes in the semidilute regime.

Coil size oscillatory packing in polymer solutions near a surface

The theory developed by Scheutjens and Fleer to describe polymer adsorption and depletion is used to calculate the density profile of nonadsorbing polymers near a surface. The theory predicts damped oscillations in the segment density profile with a wavelength of about the coil size. As a consequence, the interaction energy between two surfaces immersed in a solution of nonadsorbing polymers is an oscillatory function of the separation distance, too. The decay length of the oscillations is proportional to the coil size and independent of the polymer concentration. The oscillations are associated with a liquid-like layering of polymer coils near the surface. An increase in concentration or chain length causes a decrease in the amplitude of the oscillations, because the stronger interpenetration of the coils suppresses inhomogeneities. In dilute solutions no oscillations are observed, because the decay length of the oscillations is smaller than the depletion correlation length, in analogy with the Fisher–Widom criterion in simple fluids. For polymer solutions it is found that the Fisher–Widom condition coincides with the overlap concentration. Our results show that the blob size is not the only relevant length scale above the overlap concentration. The size of the individual coils plays a role, too.

Hydrodynamic slippage inferred from thin film drainage measurements in a solution of nonadsorbing polymer

Thin film drainage measurements are presented for submicron films of an “ideal elastic” or Boger fluid, which is a high molecular weight polymer solution in a high viscosity solvent. The measurements are made in a surface force apparatus, with the fluid being squeezed between two mica surfaces in a crossed cylinder geometry and the film thickness measured as a function of time to study its drainage behavior. No equilibrium surface forces are detected in this system, indicating that the polymer is nonadsorbing. The effect of fluid elasticity is predicted to make drainage more rapid in a Boger fluid than for the equivalent Newtonian fluid. Qualitatively this is what is observed for films less than 600 nm thick, but the drainage is even more rapid than predicted for the elastic fluid. To account for this, it is suggested that slippage is occurring at the fluid–solid interfaces, and the data is analyzed in terms of a simple slip model. The slip length required to fit the data is in the range 30–50 nm.