Short-time dynamics of permeable particles in concentrated suspensions

Structure and short-time diffusion of concentrated suspensions consisting of silicone-stabilised PMMA particles: a quantitative analysis taking polydispersity effects into account

We characterise structure and dynamics of concentrated suspensions of silicone-stabilised PMMA particles immersed in index-matching decalin-tetralin mixtures by means of static and quasielastic light scattering experiments. These particles can reproducibly be prepared via a comparatively easy route and are thus promising model systems with hard-sphere interaction. We demonstrate the hard-sphere behaviour of dense suspensions of these systems rigorously taking polydispersity effects into account. Structure factors S(Q) can in the entire range of volume fractions with liquid-like structure quantitatively be modelled using a multi-component Percus-Yevick ansatz regarding the particle size distribution and the form factor assuming a core-shell model with a scattering length density gradient in the PMMA core. Herewith, hydrodynamic functions H(Q) are in the whole accessible Q-range beyond the second maximum of H(Q) quantitatively modelled using a rescaled δγ-approach for all investigated volume fractions. With these data, previously provided characterisation of dilute systems is extended: the excellent agreement of structural and dynamic properties with theoretical predictions for hard spheres demonstrates the suitability of these particles as a model system for hard spheres.

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).

Dynamics of proteins in solution

The dynamics of proteins in solution includes a variety of processes, such as backbone and side-chain fluctuations, interdomain motions, as well as global rotational and translational (i.e. center of mass) diffusion. Since protein dynamics is related to protein function and essential transport processes, a detailed mechanistic understanding and monitoring of protein dynamics in solution is highly desirable. The hierarchical character of protein dynamics requires experimental tools addressing a broad range of time- and length scales. We discuss how different techniques contribute to a comprehensive picture of protein dynamics, and focus in particular on results from neutron spectroscopy. We outline the underlying principles and review available instrumentation as well as related analysis frameworks.

Short- and long-time diffusion and dynamic scaling in suspensions of charged colloidal particles

We report on a comprehensive theory-simulation-experimental study of collective and self-diffusion in concentrated suspensions of charge-stabilized colloidal spheres. In theory and simulation, the spheres are assumed to interact directly by a hard-core plus screened Coulomb effective pair potential. The intermediate scattering function, f_c(q, t), is calculated by elaborate accelerated Stokesian dynamics (ASD) simulations for Brownian systems where many-particle hydrodynamic interactions (HIs) are fully accounted for, using a novel extrapolation scheme to a macroscopically large system size valid for all correlation times. The study spans the correlation time range from the colloidal short-time to the long-time regime. Additionally, Brownian Dynamics (BD) simulation and mode-coupling theory (MCT) results of f_c(q, t) are generated where HIs are neglected. Using these results, the influence of HIs on collective and self-diffusion and the accuracy of the MCT method are quantified. It is shown that HIs enhance collective and self-diffusion at intermediate and long times. At short times self-diffusion, and for wavenumbers outside the structure factor peak region also collective diffusion, are slowed down by HIs. MCT significantly overestimates the slowing influence of dynamic particle caging. The dynamic scattering functions obtained in the ASD simulations are in overall good agreement with our dynamic light scattering (DLS) results for a concentration series of charged silica spheres in an organic solvent mixture, in the experimental time window and wavenumber range. From the simulation data for the time derivative of the width function associated with f_c(q, t), there is indication of long-time exponential decay of f_c(q, t), for wavenumbers around the location of the static structure factor principal peak. The experimental scattering functions in the probed time range are consistent with a time-wavenumber factorization scaling behavior of f_c(q, t) that was first reported by Segrè and Pusey [Phys. Rev. Lett. 77, 771 (1996)] for suspensions of hard spheres. Our BD simulation and MCT results predict a significant violation of exact factorization scaling which, however, is approximately restored according to the ASD results when HIs are accounted for, consistent with the experimental findings for f_c(q, t). Our study of collective diffusion is amended by simulation and theoretical results for the self-intermediate scattering function, f_s(q, t), and its non-Gaussian parameter α₂(t) and for the particle mean squared displacement W(t) and its time derivative. Since self-diffusion properties are not assessed in standard DLS measurements, a method to deduce W(t) approximately from f_c(q, t) is theoretically validated.

Photo-Crosslinkable Colloids: From Fluid Structure and Dynamics of Spheres to Suspensions of Ellipsoids

Recently-developed photo-crosslinkable PMMA (polymethylmethacrylate) colloidal spheres are a highly promising system for fundamental studies in colloidal physics and may have a wide range of future technological applications. We synthesize these colloids and characterize their size distribution. Their swelling in a density- and index- matching organic solvent system is demonstrated and we employ dynamic light scattering (DLS), as also the recently-developed confocal differential dynamic microscopy (ConDDM), to characterize the structure and the dynamics of a fluid bulk suspension of such colloids at different particle densities, detecting significant particle charging effects. We stretch these photo-crosslinkable spheres into ellipsoids. The fact that the ellipsoids are cross-linked allows them to be fluorescently stained, permitting a dense suspension of ellipsoids, a simple model of fluid matter, to be imaged by direct confocal microscopy.

Utilizing the Discrete Element Method for the Modeling of Viscosity in Concentrated Suspensions

The rheological behavior of concentrated suspensions is a complicated problem because it originates in the collective motion of particles and their interaction with the surrounding fluid. For this reason, it is difficult to accurately model the effect of various system parameters on the viscosity even for highly simplified systems. We model the viscosity of a hard-sphere suspension subjected to high shear rates using the dynamic discrete element method (DEM) in three spatial dimensions. The contact interaction between particles was described by the Hertz model of elastic spheres (soft-sphere model), and the interaction of particles with flow was accounted for by the two-way coupling approach. The hydrodynamic interaction between particles was described by the lubrication theory accounting for the slip on particle surfaces. The viscosity in a simple-shear model was evaluated from the force balance on the wall. The obtained results are in close agreement with literature data for systems with hard spheres. Namely, the viscosity is shown to be independent of shear rate and primary particle size for monodisperse suspensions. In accordance with theory and experimental data, the viscosity grows rapidly with particle volume fraction. We show that this rheological behavior is predominantly caused by the lubrication forces. A novel approach based on the slip of water on a particle surface was developed to overcome the divergent behavior of lubrication forces. This approach was qualitatively validated with literature data from AFM measurements using a colloidal probe. The model presented in this work represents a new, robust, and versatile approach to the modeling of viscosity in suspensions with the possibility to include various interaction models and study their effect on viscosity.

Short-time dynamics in dispersions with competing short-range attraction and long-range repulsion

Dynamic clustering of globular Brownian particles in dispersions exhibiting competing short-range attraction and long-range repulsion (SALR) such as low-salinity protein solutions has gained a lot of interest over the past few years. While the structure of the various cluster phases has been intensely explored, little is known about the dynamics of SALR systems. We present the first systematic theoretical study of short-time diffusion and rheological transport properties of two-Yukawa potential SALR systems in the single-particle dominated dispersed-fluid phase, using semi-analytic methods where the salient hydrodynamic interactions are accounted for. We show that the dynamics has unusual features compared to reference systems with pure repulsion or attraction. Results are discussed for the hydrodynamic function characterizing short-time diffusion that reveals an intermediate-range-order (cluster) peak, self-diffusion and sedimentation coefficients, and high-frequency viscosity. As important applications, we discuss the applicability of two generalized Stokes-Einstein relations, and assess the wavenumber range required for the determination of self-diffusion in a dynamic scattering experiment.

Near-wall dynamics of concentrated hard-sphere suspensions: comparison of evanescent wave DLS experiments, virial approximation and simulations

In this article we report on a study of the near-wall dynamics of suspended colloidal hard spheres over a broad range of volume fractions. We present a thorough comparison of experimental data with predictions based on a virial approximation and simulation results. We find that the virial approach describes the experimental data reasonably well up to a volume fraction of ϕ≈ 0.25 which provides us with a fast and non-costly tool for the analysis and prediction of evanescent wave DLS data. Based on this we propose a new method to assess the near-wall self-diffusion at elevated density. Here, we qualitatively confirm earlier results [Michailidou, et al., Phys. Rev. Lett., 2009, 102, 068302], which indicate that many-particle hydrodynamic interactions are diminished by the presence of the wall at increasing volume fractions as compared to bulk dynamics. Beyond this finding we show that this diminishment is different for the particle motion normal and parallel to the wall.

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.

Effective temperatures and the breakdown of the Stokes-Einstein relation for particle suspensions

The short- and long-time breakdown of the classical Stokes-Einstein relation for colloidal suspensions at arbitrary volume fractions is explained here by examining the role that confinement and attractive interactions play in the intra- and inter-cage dynamics executed by the colloidal particles. We show that the measured short-time diffusion coefficient is larger than the one predicted by the classical Stokes-Einstein relation due to a non-equilibrated energy transfer between kinetic and configuration degrees of freedom. This transfer can be incorporated in an effective kinetic temperature that is higher than the temperature of the heat bath. We propose a Generalized Stokes-Einstein relation (GSER) in which the effective temperature replaces the temperature of the heat bath. This relation then allows to obtain the diffusion coefficient once the viscosity and the effective temperature are known. On the other hand, the temporary cluster formation induced by confinement and attractive interactions of hydrodynamic nature makes the long-time diffusion coefficient to be smaller than the corresponding one obtained from the classical Stokes-Einstein relation. Then, the use of the GSER allows to obtain an effective temperature that is smaller than the temperature of the heat bath. Additionally, we provide a simple expression based on a differential effective medium theory that allows to calculate the diffusion coefficient at short and long times. Comparison of our results with experiments and simulations for suspensions of hard and porous spheres shows an excellent agreement in all cases.

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.

Brownian motion and the hydrodynamic friction tensor for colloidal particles of complex shape

We synthesize colloidal particles with various anisotropic shapes and track their orientationally resolved Brownian trajectories using confocal microscopy. An analysis of appropriate short-time correlation functions provides direct access to the hydrodynamic friction tensor of the particles revealing nontrivial couplings between the translational and rotational degrees of freedom. The results are consistent with calculations of the hydrodynamic friction tensor in the low-Reynolds-number regime for the experimentally determined particle shapes.

Structure and dynamics of loosely cross-linked ionic microgel dispersions in the fluid regime

We report a comprehensive experimental-theoretical study of the temperature- and concentration-dependent swelling behavior of weakly cross-linked PNiPAm ionic microgel particles in the deionized fluid phase. The particles swell reversibly when the dispersion is cooled from the collapsed state to lower temperatures. While the collapsed state shows no dependence on the microgel number density, the swelling at lower T is more pronounced at lower concentrations. The static pair correlations and short-time diffusion functions, and the concentration and temperature dependence of the microgel radius and effective charge, are studied using static and dynamic light scattering in combination with state-of-the-art analytical theoretical schemes based on a Yukawa-type effective pair potential and a core-shell model. We show that only such a combined, simultaneous fit of static and dynamic scattering functions allows for an unambiguous determination of the microgel radius and effective charge.

Short-time rheology and diffusion in suspensions of Yukawa-type colloidal particles

A comprehensive study is presented on the short-time dynamics in suspensions of charged colloidal spheres. The explored parameter space covers the major part of the fluid-state regime, with colloid concentrations extending up to the freezing transition. The particles are assumed to interact directly by a hard-core plus screened Coulomb potential, and indirectly by solvent-mediated hydrodynamic interactions. By comparison with accurate accelerated Stokesian Dynamics (ASD) simulations of the hydrodynamic function H(q), and the high-frequency viscosity η(∞), we investigate the accuracy of two fast and easy-to-implement analytical schemes. The first scheme, referred to as the pairwise additive (PA) scheme, uses exact two-body hydrodynamic mobility tensors. It is in good agreement with the ASD simulations of H(q) and η(∞), for smaller volume fractions up to about 10% and 20%, respectively. The second scheme is a hybrid method combining the virtues of the δγ scheme by Beenakker and Mazur with those of the PA scheme. It leads to predictions in good agreement with the simulation data, for all considered concentrations, combining thus precision with computational efficiency. The hybrid method is used to test the accuracy of a generalized Stokes-Einstein (GSE) relation proposed by Kholodenko and Douglas, showing its severe violation in low salinity systems. For hard spheres, however, this GSE relation applies decently well.

Structural response of polyelectrolyte dendrimer towards molecular protonation: the inconsistency revealed by SANS and NMR

Polyamidoamine (PAMAM) dendrimers and their charged state in deuterium oxide have been investigated with proton pulsed field gradient diffusion nuclear magnetic resonance (PFG-NMR) and small-angle neutron scattering (SANS) techniques. NMR measurements suggest that significant variation of the hydrodynamic radius, calculated by the Stokes-Einstein relation with appropriate surface conditions, is observed upon increasing the molecular protonation. However, a comparative SANS experiment indicates little dependence of the dendrimer global size, in terms of its radius of gyration, on molecular protonation. The inconsistency indicates the necessity of incorporating the effect of molecular interface modification and molecular porosity provided by dressed counterions, when dynamical measurements are used for the determination of the structural characteristics of ionic soft colloids even in dilute suspensions.

Protein self-diffusion in crowded solutions

Macromolecular crowding in biological media is an essential factor for cellular function. The interplay of intermolecular interactions at multiple time and length scales governs a fine-tuned system of reaction and transport processes, including particularly protein diffusion as a limiting or driving factor. Using quasielastic neutron backscattering, we probe the protein self-diffusion in crowded aqueous solutions of bovine serum albumin on nanosecond time and nanometer length scales employing the same protein as crowding agent. The measured diffusion coefficient D(ϕ) strongly decreases with increasing protein volume fraction ϕ explored within 7% ≤ ϕ ≤ 30%. With an ellipsoidal protein model and an analytical framework involving colloid diffusion theory, we separate the rotational D(r)(ϕ) and translational D(t)(ϕ) contributions to D(ϕ). The resulting D(t)(ϕ) is described by short-time self-diffusion of effective spheres. Protein self-diffusion at biological volume fractions is found to be slowed down to 20% of the dilute limit solely due to hydrodynamic interactions.

Short-time diffusion of charge-stabilized colloidal particles: generic features

Analytical theory and Stokesian dynamics simulations are used in conjunction with dynamic light scattering to investigate the role of hydrodynamic interactions in short-time diffusion in suspensions of charge-stabilized colloidal particles. The particles are modeled as solvent-impermeable charged spheres, repelling each otherviaa screened Coulomb potential. Numerical results for self-diffusion and sedimentation coefficients, as well as hydrodynamic and short-time diffusion functions, are compared with experimental data for a wide range of volume fractions. The theoretical predictions for the generic behavior of short-time properties obtained from this model are shown to be in full accord with experimental data. In addition, the effects of microion kinetics, nonzero particle porosity and residual attractive forces on the form of the hydrodynamic function are estimated. This serves to rule out possible causes for the strikingly small hydrodynamic function values determined in certain synchrotron radiation experiments.

High-frequency viscosity and generalized Stokes-Einstein relations in dense suspensions of porous particles

We study the high-frequency limiting shear viscosity, η∞, of colloidal suspensions of uncharged porous particles. An individual particle is modeled as a uniformly porous sphere with the internal solvent flow described by the Debye-Bueche-Brinkman equation. A precise hydrodynamic multipole method with a full account of many-particle hydrodynamic interactions encoded in the HYDROMULTIPOLE program extended to porous particles, is used to calculate η∞ as a function of porosity and concentration. The second-order virial expansion for η∞ is derived, and its range of applicability assessed. The simulation results are used to test the validity of generalized Stokes-Einstein relations between η∞ and various short-time diffusion coefficients, and to quantify the accuracy of a simplifying cell model calculation of η∞. An easy-to-use generalized Saitô formula for η∞ is presented which provides a good description of its porosity and concentration dependence.

Generic behavior of the hydrodynamic function of charged colloidal suspensions

We discuss the generic behavior of the hydrodynamic function H(q) and diffusion function D(q) characterizing the short-time diffusion in suspensions of charge-stabilized colloidal spheres, by covering the whole fluid regime. Special focus is given to the behavior of these functions at the freezing transition specified by the Hansen-Verlet freezing rule. Results are presented in dependence on scattering wavenumber q, effective particle charge, volume fraction, salt concentration, and particle size, by considering both the low-charge and high-charge branch solutions of static structure factors. The existence of two charge branches leads to the prediction of a re-entrant melting-freezing-melting transition for increasing particle concentration at very low salinity. A universal limiting contour line is derived for the principal peak height value of H(q), independent of particle charge and diameter, and concentration and salinity, which separates the fluid from the fluid-solid coexistence region. This line is only weakly dependent on the value of the structure factor peak height entering the Hansen-Verlet rule. A dynamic freezing criterion is derived in terms of the short-time cage diffusion coefficient, a quantity easily measurable in a scattering experiment. The higher-dimensional parameter scans underlying this study make use of the fast and highly efficient deltagamma-scheme in conjunction with the analytic rescaled mean spherical approximation input for the static structure factor. Our results constitute a comprehensive database useful to researchers performing dynamic scattering experiments on charge-stabilized dispersions.

Dynamics of permeable particles in concentrated suspensions

We calculate short-time diffusion properties of suspensions of porous colloidal particles as a function of their permeability, for the full fluid-phase concentration range. The particles are modeled as spheres of uniform permeability with excluded volume interactions. Using a precise multipole method encoded in the HYDROMULTIPOLE program, results are presented for the hydrodynamic function, H(q) , sedimentation coefficient, and self-diffusion coefficients with a full account of many-body hydrodynamic interactions. While self-diffusion and sedimentation are strongly permeability dependent, the wave-number dependence of the hydrodynamic function can be reduced by appropriate shifting and scaling, to a single master curve, independent of permeability. Generic features of the permeable sphere model are discussed.