Diffusion, sedimentation, and rheology of concentrated suspensions of core-shell particles

Geometrical Influence on Particle Transport in Cross-Flow Ultrafiltration: Cylindrical and Flat Sheet Membranes

Cross-flow membrane ultrafiltration (UF) is used for the enrichment and purification of small colloidal particles and proteins. We explore the influence of different membrane geometries on the particle transport in, and the efficiency of, inside-out cross-flow UF. For this purpose, we generalize the accurate and numerically efficient modified boundary layer approximation (mBLA) method, developed in recent work by us for a hollow cylindrical membrane, to parallel flat sheet geometries with one or two solvent-permeable membrane sheets. Considering a reference dispersion of Brownian hard spheres where accurate expressions for its transport properties are available, the generalized mBLA method is used to analyze how particle transport and global UF process indicators are affected by varying operating parameters and the membrane geometry. We show that global process indicators including the mean permeate flux, the solvent recovery indicator, and the concentration factor are strongly dependent on the membrane geometry. A key finding is that irrespective of the many input parameters characterizing an UF experiment and its membrane geometry, the process indicators are determined by three independent dimensionless variables only. This finding can be very useful in the design, optimization, and scale-up of UF processes.

The effect of morphology and particle-wall interaction on colloidal near-wall dynamics

We investigated the near-wall Brownian dynamics of different types of colloidal particles with a typical size in the 100 nm range using evanescent wave dynamic light scattering (EWDLS). In detail we studied dilute suspensions of silica spheres and shells with a smooth surface and silica particles with controlled surface roughness. While the near wall dynamics of the particle with a smooth surface differ only slightly from the theoretical prediction for hard sphere colloids, the rough particles diffuse significantly slower. We analysed the experimental data by comparison with model calculations and suggest that the deviating dynamics of the rough particles are not due to increased hydrodynamic interaction with the wall. Rather, the particle roughness significantly changes their DLVO interaction with the wall, which in turn affects their diffusion.

Structure and dynamics in suspensions of soft core-shell colloids in the fluid regime

We report on a detailed experimental study of the structure and short-time dynamics in fluid-regime suspensions of soft core-shell spherical particles with different molecular weights of the chains forming the soft outer shell, and therefore different degrees of particle softness, using 3D dynamic light scattering (3D-DLS). Owing to the particle softness, the liquid-crystal coexistence regime is found to be broader than that of hard-sphere (HS) suspensions. Static light scattering in the dilute regime yields form factors that can be described using a spherical core-shell model and second virial coefficients A₂ > 0 indicative of purely repulsive interactions. The particle-particle interactions are longer ranged for all considered systems except those of the smaller molecular weight chain grafted particles which show a HS-like behavior. 3D-DLS experiments in the concentrated regime up to the liquid-crystal transition provide the short-time diffusion function, D(q), in a broad range of scattering wavenumbers, q, from which the structural (cage) and short-time self-diffusion coefficients D(q_m) and D_S = D(q ≫ q_m), respectively, are deduced as functions of the effective particle volume fraction, ϕ = c/c^*, where c^* is the overlap concentration, calculated using the hydrodynamic particle radius, R_H. The size of the nearest-neighbor cage of particles is characterized by 2π/q_m, with D(q) and the static structure factor S(q) attaining at q_m the smallest and largest values, respectively. Experimental data of D(q_m) and D_S are contrasted with analytic theoretical predictions based on a simplifying hydrodynamic radius model where the internal hydrodynamic structure of the core-shell particles is mapped on a single hydrodynamic radius parameter γ = R_H/R_eff, for constant direct interactions characterized by an (effective) hard-core radius R_eff. The particle softness is reflected, in particular, in the corresponding shape of the static structure factor, while the mean solvent (Darcy) permeability of the particles related to γ is reflected in the dynamic properties only. For grafted particles with longer polymer chains, D(q_m) and D_S are indicative of larger permeability values while particles with shorter chains are practically nonpermeable. The particle softness is also evident in the effective random close packing fraction estimated from the extrapolated zero-value limit of the cage diffusion coefficient D(q_m).

Rotational friction of dipolar colloids measured by driven torsional oscillations

Despite its prominent role in the dynamics of soft materials, rotational friction remains a quantity that is difficult to determine for many micron-sized objects. Here, we demonstrate how the Stokes coefficient of rotational friction can be obtained from the driven torsional oscillations of single particles in a highly viscous environment. The idea is that the oscillation amplitude of a dipolar particle under combined static and oscillating fields provides a measure for the Stokes friction. From numerical studies we derive a semi-empirical analytic expression for the amplitude of the oscillation, which cannot be calculated analytically from the equation of motion. We additionally demonstrate that this expression can be used to experimentally determine the rotational friction coefficient of single particles. Here, we record the amplitudes of a field-driven dipolar Janus microsphere with optical microscopy. The presented method distinguishes itself in its experimental and conceptual simplicity. The magnetic torque leaves the local environment unchanged, which contrasts with other approaches where, for example, additional mechanical (frictional) or thermal contributions have to be regarded.

Generalization of Clausius-Mossotti approximation in application to short-time transport properties of suspensions

In 1983, Felderhof, Ford, and Cohen gave microscopic explanation of the famous Clausius-Mossotti formula for the dielectric constant of nonpolar dielectric. They based their considerations on the cluster expansion of the dielectric constant, which relates this macroscopic property with the microscopic characteristics of the system. In this article, we analyze the cluster expansion of Felderhof, Ford, and Cohen by performing its resummation (renormalization). Our analysis leads to the ring expansion for the macroscopic characteristic of the system, which is an expression alternative to the cluster expansion. Using similarity of structures of the cluster expansion and the ring expansion, we generalize (renormalize) the Clausius-Mossotti approximation. We apply our renormalized Clausius-Mossotti approximation to the case of the short-time transport properties of suspensions, calculating the effective viscosity and the hydrodynamic function with the translational self-diffusion and the collective diffusion coefficient. We perform calculations for monodisperse hard-sphere suspensions in equilibrium with volume fraction up to 45%. To assess the renormalized Clausius-Mossotti approximation, it is compared with numerical simulations and the Beenakker-Mazur method. The results of our renormalized Clausius-Mossotti approximation lead to comparable or much less error (with respect to the numerical simulations) than the Beenakker-Mazur method for the volume fractions below ϕ≈30% (apart from a small range of wave vectors in hydrodynamic function). For volume fractions above ϕ≈30%, the Beenakker-Mazur method gives in most cases lower error than the renormalized Clausius-Mossotti approximation.

Ultrafiltration modeling of non-ionic microgels

Membrane ultrafiltration (UF) is a pressure driven process allowing for the separation and enrichment of protein solutions and dispersions of nanosized microgel particles. The permeate flux and the near-membrane concentration-polarization (CP) layer in this process is determined by advective-diffusive dispersion transport and the interplay of applied and osmotic transmembrane pressure contributions. The UF performance is thus strongly dependent on the membrane properties, the hydrodynamic structure of the Brownian particles, their direct and hydrodynamic interactions, and the boundary conditions. We present a macroscopic description of cross-flow UF of non-ionic microgels modeled as solvent-permeable spheres. Our filtration model involves recently derived semi-analytic expressions for the concentration-dependent collective diffusion coefficient and viscosity of permeable particle dispersions [Riest et al., Soft Matter, 2015, 11, 2821]. These expressions have been well tested against computer simulation and experimental results. We analyze the CP layer properties and the permeate flux at different operating conditions and discuss various filtration process efficiency and cost indicators. Our results show that the proper specification of the concentration-dependent transport coefficients is important for reliable filtration process predictions. We also show that the solvent permeability of microgels is an essential ingredient to the UF modeling. The particle permeability lowers the particle concentration at the membrane surface, thus increasing the permeate flux.

Dynamics of suspensions of hydrodynamically structured particles: analytic theory and applications to experiments

We present an easy-to-use analytic toolbox for the calculation of short-time transport properties of concentrated suspensions of spherical colloidal particles with internal hydrodynamic structure, and direct interactions described by a hard-core or soft Hertz pair potential. The considered dynamic properties include self-diffusion and sedimentation coefficients, the wavenumber-dependent diffusion function determined in dynamic scattering experiments, and the high-frequency shear viscosity. The toolbox is based on the hydrodynamic radius model (HRM) wherein the internal particle structure is mapped on a hydrodynamic radius parameter for unchanged direct interactions, and on an existing simulation data base for solvent-permeable and spherical annulus particles. Useful scaling relations for the diffusion function and self-diffusion coefficient, known to be valid for hard-core interaction, are shown to apply also for soft pair potentials. We further discuss extensions of the toolbox to long-time transport properties including the low-shear zero-frequency viscosity and the long-time self-diffusion coefficient. The versatility of the toolbox is demonstrated by the analysis of a previous light scattering study of suspensions of non-ionic PNiPAM microgels [Eckert et al., J. Chem. Phys., 2008, 129, 124902] in which a detailed theoretical analysis of the dynamic data was left as an open task. By the comparison with Hertz potential based calculations, we show that the experimental data are consistently and accurately described using the Verlet-Weis corrected Percus-Yevick structure factor as input, and for a solvent penetration length equal to three percent of the excluded volume radius. This small amount of solvent permeability of the microgel particles has a significant dynamic effect at larger concentrations.

A general approach for predicting the filtration of soft and permeable colloids: the milk example

Membrane filtration operations (ultra-, microfiltration) are now extensively used for concentrating or separating an ever-growing variety of colloidal dispersions. However, the phenomena that determine the efficiency of these operations are not yet fully understood. This is especially the case when dealing with colloids that are soft, deformable, and permeable. In this paper, we propose a methodology for building a model that is able to predict the performance (flux, concentration profiles) of the filtration of such objects in relation with the operating conditions. This is done by focusing on the case of milk filtration, all experiments being performed with dispersions of milk casein micelles, which are sort of ″natural″ colloidal microgels. Using this example, we develop the general idea that a filtration model can always be built for a given colloidal dispersion as long as this dispersion has been characterized in terms of osmotic pressure Π and hydraulic permeability k. For soft and permeable colloids, the major issue is that the permeability k cannot be assessed in a trivial way like in the case for hard-sphere colloids. To get around this difficulty, we follow two distinct approaches to actually measure k: a direct approach, involving osmotic stress experiments, and a reverse-calculation approach, that consists of estimating k through well-controlled filtration experiments. The resulting filtration model is then validated against experimental measurements obtained from combined milk filtration/SAXS experiments. We also give precise examples of how the model can be used, as well as a brief discussion on the possible universality of the approach presented here.

Responsive emulsions stabilized by stimuli-sensitive microgels: emulsions with special non-Pickering properties

Recent studies revealing the unique properties of microgel-stabilized responsive emulsions are discussed, and microgels are compared to classical rigid-particle Pickering stabilizers. Microgels are strongly swollen, lyophilic particles that become deformed at the oil-water interface and protrude only a little into the oil phase. Temperature- and pH-sensitive microgels allow us to prepare temperature- and pH-sensitive emulsions and thus enable us to prepare and break emulsions on demand. Although such emulsions are sensitive to pH, the stabilization of droplets is not due to electrostatic repulsion, instead the viscoelastic properties of the interface seem to dominate droplet stability. Being soft and porous, microgels behave distinctly differently from rigid particles at the interface: they are deformed and strongly flattened especially in the case of oil-in-water emulsions. The microgels are located mainly on the water side of the interface for both oil-in-water and water-in-oil emulsions. In contrast to rigid, solid particles, the behavior of microgels at oil-water interfaces does not depend only on the interfacial tension but also on the balance among the interfacial tension, swelling, elasticity, and deformability of the microgel, which needs to be considered. It is obvious that microgels as soft, porous particles are significantly different from classical rigid colloidal stabilizers in Pickering emulsions and we suggest avoiding the term Pickering emulsion when swollen microgels are employed. Microgel-stabilized emulsions require the development of new theoretical models to understand their properties. They open the door to new sophisticated applications.