Rotational and translational self-diffusion in concentrated suspensions of permeable particles

In our recent work on concentrated suspensions of uniformly porous colloidal spheres with excluded volume interactions, a variety of short-time dynamic properties were calculated, except for the rotational self-diffusion coefficient. This missing quantity is included in the present paper. Using a precise hydrodynamic force multipole simulation method, the rotational self-diffusion coefficient is evaluated for concentrated suspensions of permeable particles. Results are presented for particle volume fractions up to 45% and for a wide range of permeability values. From the simulation results and earlier results for the first-order virial coefficient, we find that the rotational self-diffusion coefficient of permeable spheres can be scaled to the corresponding coefficient of impermeable particles of the same size. We also show that a similar scaling applies to the translational self-diffusion coefficient considered earlier. From the scaling relations, accurate analytic approximations for the rotational and translational self-diffusion coefficients in concentrated systems are obtained, useful to the experimental analysis of permeable-particle diffusion. The simulation results for rotational diffusion of permeable particles are used to show that a generalized Stokes-Einstein-Debye relation between rotational self-diffusion coefficient and high-frequency viscosity is not satisfied.     

Self-diffusion of rodlike and spherical particles in a matrix of charged colloidal spheres: a comparison between fluorescence recovery after photobleaching and fluorescence correlation spectroscopy

The fluorescence recovery after photobleaching (FRAP) method and the fluorescence correlation spectroscopy (FCS) have been applied on suspensions of highly charged colloidal spheres with a small content of rod-shaped tobacco mosaic virus (TMV) particles. Since these methods only determine the self-diffusion coefficient of the fluorescently labeled species, D(S) of the rods and the spheres could independently be measured. The ionic strength of the dispersion medium has been varied to measure self-diffusion of rods and spheres in dependence on the degree of order of the matrix spheres. In contrast to FRAP, which allows the determination of the long-time self-diffusion coefficient D(S) (L), FCS measures self-diffusion on a shorter time scale. Thus a comparison of the results that were obtained by FCS and FRAP, in combination with Brownian Dynamics simulations, gives insight into the time dependence of the self-diffusion coefficient of an interacting colloidal system. As the mean interparticle distance of the matrix is of the same order of magnitude as the length of a TMV rod, the rotational motion is influenced by the assembly of spheres around a TMV particle. Since FCS is sensitive both to translational and rotational motion, whereas FRAP, which probes the diffusion at much larger length scales, is only sensitive to the translational motion of TMV, the comparison of diffusion coefficients measured employing FRAP and FCS can give some insights in the rotational diffusion: the experimental data indicate a slowing down of the rotational motion of a TMV rod with increasing structural order of the matrix spheres.     

Reversible coagulation of colloidal suspension in shallow potential wells: Direct numerical simulation

Brownian dynamics computer simulations of aggregation in 2D colloidal suspensions are discussed. The simulations are based on the Langevin equations, pairwise interaction between colloidal particles and take into account Brownian, hydrodynamic and colloidal forces. The chosen mathematical model enables to predict the correct values of diffusion coefficient of freely moving particle, the mean value of kinetic energy for each particle in ensemble of interacting colloidal particles and residence times of colloidal particles inside the potential wells of different depths. The simulations allow monitoring formation and breakage of clusters in a suspension as well as time dependence of the mean cluster size. The article is published in the original.     

Short-time transport properties in dense suspensions: from neutral to charge-stabilized colloidal spheres

We present a detailed study of short-time dynamic properties in concentrated suspensions of charge-stabilized and of neutral colloidal spheres. The particles in many of these systems are subject to significant many-body hydrodynamic interactions. A recently developed accelerated Stokesian dynamics (ASD) simulation method is used to calculate hydrodynamic functions, wave-number-dependent collective diffusion coefficients, self-diffusion and sedimentation coefficients, and high-frequency limiting viscosities. The dynamic properties are discussed in dependence on the particle concentration and salt content. Our ASD simulation results are compared with existing theoretical predictions, notably those of the renormalized density fluctuation expansion method of Beenakker and Mazur [Physica A 126, 349 (1984)], and earlier simulation data on hard spheres. The range of applicability and the accuracy of various theoretical expressions for short-time properties are explored through comparison with the simulation data. We analyze, in particular, the validity of generalized Stokes-Einstein relations relating short-time diffusion properties to the high-frequency limiting viscosity, and we point to the distinctly different behavior of de-ionized charge-stabilized systems in comparison to hard spheres.     

A Brownian dynamics study on the self-diffusion of charged tracers in dilute polyelectrolyte solutions

Brownian dynamics simulations with hydrodynamic interactions are conducted to investigate the self-diffusion of charged tracer particles in a dilute solution of charged polymers, which are modeled by bead-spring chains. The Debye-Hückel approximation is used for the electrostatic interactions. The hydrodynamic interactions are implemented by the Ewald summation of the Rotne-Prager tensor. Our simulations find that the difference in short- and long-time diffusivities is very slight in uncharged short-chain solutions. For charged systems, to the contrary, the difference becomes considerable. The short-time diffusivity is found to increase with increasing chain length, while an opposite behavior is obtained for the long-time diffusivity. The former is attributed to the hydrodynamic screening among beads in a same chain due to the bead connectivity. The latter is explained by the memory effect arising from the electrostatic repulsion and chain length. The incorporation of hydrodynamic interactions improves the agreement between the simulation prediction and the experimental result.     

Short-time self-diffusion coefficient of a particle in a colloidal suspension bounded by a microchannel: virial expansions and simulation

Self-diffusion of colloidal particles confined to a cylindrical microchannel is considered theoretically and numerically. Virial expansion of the self-diffusion coefficient is performed. Two-body and three-body hydrodynamic interactions are evaluated with high precision using the multipole method. The multipole expansion algorithm is also used to perform numerical simulations of the self-diffusion coefficient, valid for all possible particle packing fractions. Comparison with earlier results shows that the widely used method of reflections is insufficient for calculations of hydrodynamic interactions even for small packing fractions and small particles radii, contrary to the prevalent opinion.     

Transport coefficients of aqueous dodecyltrimethylammonium bromide solutions: comparison between experiments, analytical calculations and numerical simulations

We study dynamical properties of ionic species in aqueous solutions of dodecyltrimethylammonium bromide, for several concentrations below and above the critical micellar concentration (cmc). New experimental determinations of the electrical conductivity are given which are compared to results obtained from an analytical transport theory; transport coefficients of ions in these solutions above the cmc are also computed from Brownian dynamics simulations. Analytical calculations as well as the simulation treat the solution within the framework of the continuous solvent model. Above the cmc, three ionic species are considered: the monomer surfactant, the micelle and the counterion. The analytical transport theory describes the structural properties of the electrolyte solution within the mean spherical approximation and assumes that the dominant forces which determine the deviations of transport processes from the ideal behavior (i.e., without any interactions between ions) are hydrodynamic interactions and electrostatic relaxation forces. In the simulations, both direct interactions and hydrodynamic interactions between solutes are taken into account. The interaction potential is modeled by pairwise repulsive 1/r(12) interactions and Coulomb interactions. The input parameters of the simulation (radii and self-diffusion coefficients of ions at infinite dilution) are partially obtained from the analytical transport theory which fits the experimental determinations of the electrical conductivity. Both the electrical conductivity of the solution and the self-diffusion coefficients of each species computed from Brownian dynamics are compared to available experimental data. In every case, the influence of hydrodynamic interactions (HIs) on the transport coefficients is investigated. It is shown that HIs are crucial to obtain agreement with experiments. In particular, the self-diffusion coefficient of the micelle, which is the largest and most charged species in the present system, is enhanced when HIs are included whereas the diffusion coefficients of the monomer and the counterion are roughly not influenced by HIs.     

Collective diffusion in charge-stabilized suspensions: concentration and salt effects

The authors present a joint experimental-theoretical study of collective diffusion properties in aqueous suspensions of charge-stabilized fluorinated latex spheres. Small-angle x-ray scattering and x-ray photon correlation spectroscopy have been used to explore the concentration and ionic-strength dependence of the static and short-time dynamic properties including the hydrodynamic function H(q), the wave-number-dependent collective diffusion coefficient D(q), and the intermediate scattering function over the entire accessible range. They show that all experimental data can be quantitatively described and explained by means of a recently developed accelerated Stokesian dynamics simulation method, in combination with a modified hydrodynamic many-body theory. In particular, the behavior of H(q) for de-ionized and dense suspensions can be attributed to the influence of many-body hydrodynamics, without any need for postulating hydrodynamic screening to be present, as it was done in earlier work. Upper and lower boundaries are provided for the peak height of the hydrodynamic function and for the short-time self-diffusion coefficient over the entire range of added salt concentrations.     

Density-induced coupling effects on the dispersivity of a flexible chain particle

A model is introduced to investigate the transport properties of an inhomogeneously dense flexible chain particle. The specific model used is a sedimenting non-neutrally buoyant inhomogenously weighted flexible Brownian dumbbell, and it is shown that density inhomogeneity gives rise to a novel coupling effect between the "shape-fluctuation" and "size-fluctuation" dispersion mechanisms. The previously reported shape-fluctuation dispersion term stems from the dumbbell's nonspherical shape and the ensuing anisotropic mobility tensor, while the already investigated size fluctuation term is the result of the dependence of the overall dumbbell translational mobility on the separation distance between the constitutive spheres. Because the density of the constitutive spheres is unequal, the external force simultaneously reorients and deforms the flexible dumbbell, and it is this mutual dependence between dumbbell orientation and size that induces the coupling. Numerical results are presented for the case of a tethered dumbbell composed of two spheres, identical in size but differing in density. The "weak-field" limit is addressed, where the externally applied torque and particle deformation forces are dominated by the thermal fluctuations associated with rotational and deformation Brownian motion. This numerical solution, obtained by including a large number of higher order hydrodynamic interactions (120 terms), describes the Brownian particle's long-time transport without resorting to ad hoc approximations, such as preaveraging the hydrodynamic force or incorporating only first-order hydrodynamic interaction effects (such as employing the Burgers-Oseen tensor). Separate analytical solutions, based on these respective approximations, are also presented and it is concluded that in the limit of "long tethers," where the ratio of tether length to sphere size is greater than seven, no more than 15% error is introduced by neglecting higher-order hydrodynamic interactions. Similarly, the preaveraging approximation introduces no more than a few percent error in the limit of "almost-rigid" dumbbells, where the ratio of tether length to sphere size is less than three. For tethers of "intermediate" length, the full numerical solution must be employed.     

Diffusion of spheres in crowded suspensions of rods

Translational tracer diffusion of spherical macromolecules in crowded suspensions of rodlike colloids is investigated. Experiments are done using several kinds of spherical tracers in fd-virus suspensions. A wide range of size ratios L/2a of the length L of the rods and the diameter 2a of the tracer sphere is covered by combining several experimental methods: fluorescence correlation spectroscopy for small tracer spheres, dynamic light scattering for intermediate sized spheres, and video microscopy for large spheres. Fluorescence correlation spectroscopy is shown to measure long-time diffusion only for relatively small tracer spheres. Scaling of diffusion coefficients with a/xi, predicted for static networks, is not found for our dynamical network of rods (with xi the mesh size of the network). Self-diffusion of tracer spheres in the dynamical network of freely suspended rods is thus fundamentally different as compared to cross-linked networks. A theory is developed for the rod-concentration dependence of the translational diffusion coefficient at low rod concentrations for freely suspended rods. The proposed theory is based on a variational solution of the appropriate Smoluchowski equation without hydrodynamic interactions. The theory can, in principle, be further developed to describe diffusion through dynamical networks at higher rod concentrations with the inclusion of hydrodynamic interactions. Quantitative agreement with the experiments is found for large tracer spheres, and qualitative agreement for smaller spheres. This is probably due to the increasing importance of hydrodynamic interactions as compared to direct interactions as the size of the tracer sphere decreases.     

Brownian dynamics studies of dilute dispersions

The method of brownian dynamics of used to study the non-equilibrium properties of very dilute colloids electrostatically stabilised in dilute aqueous electrolyte. It is assumed that the colloid is a monodisperse system of structureless spherical particles embedded in a hydrodynamic continuum. Although the particles are interacting electrostatically through a screened Coulomb potential, the dilution is such that effects arising from coupling of hydrodynamic flow can be ignore. Studies of the self-diffusion coefficient and van Howe functions show that after an initial period, during which the particles move essentially independently, the flow properties of the colloids are significantly different from those expected on the basis of free brownian motion.     

Dynamic light scattering,a probe of brownian particle dynamics

The principles and techniques of dynamic light scattering (DLS) are outlined and its application to the study of suspensions of interacting colloidal particles is discussed. We show how, under appropriate conditions, DLS can measure long-time collective and self-diffusion coefficients as well as study short-time motions (characterized by the cumulants). These theoretical considerations are illustrated by experimental data. Finally, we discuss the relevance of certain characteristic timescales to theories of the diffusion of interacting particles.     

Cooperative motion of spheres arranged in periodic grids between two parallel walls

In this article, we analyze the collective motion of a two-dimensional periodic array of spheres in a slit-pore confined by two parallel planar walls. We determine the friction coefficient of the spheres when all particles move with the same velocity along a particular direction and cooperate with each other in their motion. In order to solve this many-body problem, we use Stokesian dynamics algorithm and resolve multiparticle hydrodynamic interactions in wall-bounded geometry. Apart from particle-particle interactions, we also recognize that the aforementioned collective motion of all particles creates a cumulative effect on the fluid medium. This effect is manifested as either a net induced flow for a periodic pressure field or an additional pressure gradient for quiescent fluid. In our analysis, we focus on both periodic pressure and no-flow conditions. For both cases, the hydrodynamic friction on the translating particles is calculated using our multiparticle Stokesian dynamics simulation. The simulation for the no-flow condition is relatively straightforward-we only need to compute the multiparticle hydrodynamic interactions in quiescent fluid. However, for the periodic pressure condition, the net induced flow dragged by the particles has to be evaluated also. We express this net induced flow in terms of an additional pressure-driven velocity field. We present the hydrodynamic friction as a function of the dimensions of the two-dimensional periodic lattice. For closely packed arrays, the results show a considerable reduction in friction coefficients that usually increase with interparticle distance. Hence, our work renders the theoretical justification for other recent findings that indicate the importance of interparticle mutual cooperation.     

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.     

Electrostatic relaxation and hydrodynamic interactions for self-diffusion of ions in electrolyte solutions

The concentration dependence of self-diffusion of ions in solutions at large concentrations has remained an interesting yet unsolved problem. Here we develop a self-consistent microscopic approach based on the ideas of mode-coupling theory. It allows us to calculate both contributions which influence the friction of a moving ion: the ion atmosphere relaxation and hydrodynamic interactions. The resulting theory provides an excellent agreement with known experimental results over a wide concentration range. Interestingly, the mode-coupling self-consistent calculation of friction reveal a nonlinear coupling between the hydrodynamic interactions and the ion atmosphere relaxation which enhances ion diffusion by reducing friction, particularly at intermediate ion concentrations. This rather striking result has its origin in the similar time scales of the relaxation of the ion atmosphere relaxation and the hydrodynamic term, which are essentially given by the Debye relaxation time. The results are also in agreement with computer simulations, with and without hydrodynamic interactions.     

Stabilization of magnetorheological suspensions by polyacrylic acid polymers

This work is devoted to the synthesis and stabilization of magnetorheological suspensions constituted by monodisperse micrometer-sized magnetite spheres in aqueous media. The electrical double-layer characteristics of the solid/liquid interface were studied in the absence and presence of adsorbed layers of high molecular weight polyacrylic acids (PAA; Carbopol). Since the Carbopol-covered particles can be thought of as "soft" colloids, Ohshima's theory was used to gain information of the surface potential and the charge density of the polymer layer. The effect of the pH of the solution on the double-layer characteristics is related to the different conformations of the adsorbed molecules provoked by the dissociation of the acrylic groups present in polymer molecules. The stability of the suspensions was experimentally studied for different pH and polymer concentrations, and in the absence or presence of a weak magnetic field applied. The stability of the suspensions was explained using the classical DLVO theory of colloidal stability extended to account for hydration, steric, and magnetic interactions between particles. Diagrams of potential energy vs interparticle distance show the predominant effect of steric, hydrophilic/hydrophobic, and magnetic interactions on the whole stability of the system. The best conditions to obtain stable suspensions were found when strong steric and hydrophilic repulsions hinder the coagulation between polymer-covered particles, simultaneously avoiding sedimentation by the thickening effect of the polymer solution. When a not too high molecular weight PAA was employed in a low concentration, the task of a long-time antisettling effect compatible with the desired magnetic response of the fluid was achieved.     

Colloidal dynamics: influence of diffusion, inertia and colloidal forces on cluster formation

Computer simulations of colloidal suspensions are discussed. The simulations are based on the Langevin equations, pairwise interaction between colloidal particles and take into account Brownian, hydrodynamic and colloidal forces. Comparison of two models, one taking into account inertial term in Langevin equation and another based on diffusional approximation proposed in [D.L. Ermak, J.A. McCammon, J. Chem. Phys. 69 (1978) 1352], has shown that both models enable the prediction of the correct values of the diffusion coefficient and residence time of particle in a doublet and are therefore suitable to study the dynamics of formation and breakage of clusters in colloidal suspensions. It is shown that the appropriate selection of the time step and taking into account inertia of particles provides also the correct value of the average kinetic energy of each particle during the simulations, what allows to use the model based on full Langevin equations as a reference model to verify the validity of the numerical scheme for simulation using diffusion approximation.     

Tracer diffusion in colloidal suspensions under dilute and crowded conditions with hydrodynamic interactions

We consider tracer diffusion in colloidal suspensions under solid loading conditions, where hydrodynamic interactions play an important role. To this end, we carry out computer simulations based on the hybrid stochastic rotation dynamics-molecular dynamics (SRD-MD) technique. Many details of the simulation method are discussed in detail. In particular, our choices for the SRD-MD parameters and for the different scales are adapted to simulating colloidal suspensions under realistic conditions. Our simulation data are compared with published theoretical, experimental and numerical results and compared to Brownian dynamics simulation data. We demonstrate that our SRD-MD simulations reproduce many features of the hydrodynamics in colloidal fluids under finite loading. In particular, finite-size effects and the diffusive behavior of colloids for a range of volume fractions of the suspension show that hydrodynamic interactions are correctly included within the SRD-MD technique.