Diffusion of spheres in isotropic and nematic networks of rods: electrostatic interactions and hydrodynamic screening

Translational diffusion of a small charged tracer sphere in isotropic and nematic suspensions of long and thin charged rods is investigated as a function of ionic strength and rod concentration. A theory for the diffusive properties of a small sphere is developed, where both (screened) hydrodynamic interactions and charge interactions between the tracer sphere and the rod network are analyzed. Hydrodynamic interactions are formulated in terms of the hydrodynamic screening length. As yet, there are no independent theoretical predictions for the hydrodynamic screening length for rod networks. Experimental tracer-diffusion data are presented for various ionic strengths as a function of the rod concentration, both in the isotropic and nematic states. Orientational order parameters are measured for the same ionic strengths as a function of the rod concentration. The hydrodynamic screening length is determined from these experimental data and scaling relations obtained from the above mentioned theory. For the isotropic networks, a master curve is found for the hydrodynamic screening length as a function of the rod concentration. For the nematic networks the screening length turns out to be a very sensitive function of the orientational order parameter.     

The short-time self-diffusion coefficient of a sphere in a suspension of rigid rods

The short-time self-diffusion coefficient of a sphere in a suspension of rigid rods is calculated in first order in the rod volume fraction phi. For low rod concentrations, the correction to the Einstein diffusion constant of the sphere due to the presence of rods is a linear function of phi with the slope alpha proportional to the equilibrium averaged mobility diminution trace of the sphere interacting with a single freely translating and rotating rod. The two-body hydrodynamic interactions are calculated using the so-called bead model in which the rod of aspect ratio p is replaced by a stiff linear chain of touching spheres. The interactions between spheres are calculated using the multipole method with the accuracy controlled by a multipole truncation order and limited only by the computational power. A remarkable accuracy is obtained already for the lowest truncation order, which enables calculations for very long rods, up to p=1000. Additionally, the bead model is checked by filling the rod with smaller spheres. This procedure shows that for longer rods the basic model provides reasonable results varying less than 5% from the model with filling. An analytical expression for alpha as a function of p is derived in the limit of very long rods. The higher order corrections depending on the applied model are computed numerically. An approximate expression is provided, valid for a wide range of aspect ratios.     

Diffusion of spheres in isotropic and nematic suspensions of rods

Diffusion of a small tracer sphere (apoferritin) in isotropic and nematic networks [of fd virus] is discussed. For a tracer sphere that is smaller than the mesh size of the network, screened hydrodynamic interactions between the sphere and the network determine its diffusion coefficient. A theory is developed for such interactions as well as their relation to the long-time self-diffusion coefficient. Fluorescence correlation spectroscopy measurements on mixtures of apoferritin and fd virus are presented. The long-time self-diffusion coefficient of apoferritin is measured as a function of the fd-virus concentration, both in the isotropic and nematic state, in directions parallel and perpendicular to the nematic director. The hydrodynamic screening length of the fd-virus network as a function of fd concentration is obtained by combining these experimental data with the theory. Surprisingly, the screening length increases with increasing concentration in nematic networks. This is due to the increase in the degree of alignment, which apparently leads to a strong increase of the screening length. Hydrodynamic screening is thus strongly diminished by alignment. A self-consistent calculation of the screening length does not work at higher concentrations, probably due to the strong variation of the typical incident flow fields over the contour of a rod.     

Structures and dynamics of thermosensitive microgel suspensions studied with three-dimensional cross-correlated light scattering

The structure factors, short- and long-time diffusion coefficients, and hydrodynamic interactions of concentrated poly(N-isopropylacryamide) microgel suspensions were measured with simultaneous static and dynamic three-dimensional cross-correlated light scattering. The data are interpreted through comparison to hard sphere theory. The structure factors are known to be described well by the hard sphere approximation. When the structure factor is fit to an effective hard sphere volume fraction and radius, the diffusion and hydrodynamic interactions are also well described by the hard sphere model. We demonstrate that one single hard sphere volume fraction is sufficient to describe the microgel structures, hydrodynamic interactions, and long- and short-time collective diffusion coefficients. This result is surprising because the particle form of the microgels at these temperatures is not rigid, but rather "fuzzy" spheres with dangling polymer chains.     

Drag and Torque on Clusters of N Arbitrary Spheres at Low Reynolds Number

Hydrodynamics of particle clusters suspended in viscous fluids is a subject of considerable theoretical and practical importance. Using a multipole expansion of the flow velocity in a series of spherical harmonics, Lamb's fundamental solution of the Stokes flow outside a single sphere is generalized in this work to the case of N nonoverlapping spheres of arbitrary size with slip boundary conditions. The expansion coefficients are found by transforming the boundary conditions to the Lamb form and by transforming the spherical coordinates and solid spherical harmonics centered at different spheres. The problem is reduced to the solution of the linear system of equations for the expansion coefficients, which is carried out numerically. Based on the developed theory, the relation between the hydrodynamic and gyration radius of fractal-like aggregates with different structure is established. In another application, an asymptotic slip-regime dependence of the aggregate hydrodynamic radius on the Knudsen number and the number of particles is found by performing calculations of drag forces acting on the gas-borne fractal-like and straight chain aggregates. A good agreement is shown in comparing predictions of the described theory with available experimental and theoretical results on motion of various small sphere clusters in viscous fluid. Copyright 2000 Academic Press.     

Flow properties of freshly prepared ettringite suspensions in water at 25 degrees C

The rheology of a complex, heterogeneous mineral colloid was rationalised using models devised for model rod systems. Mixing a calcium hydroxide slurry with an aluminium sulphate solution produces a suspension of rod-shaped ettringite particles. Ettringite rod suspensions exhibit non-Newtonian flow behaviour, which depends on the shape of the particles, their size distribution, concentration and surface properties as well as the suspension medium characteristics. We have measured the shear viscosity of suspensions of ettringite rods with a median aspect ratio, r(i) approximately 8, at 25 degrees C as a function of particle volume fraction, phi, in the range 0.0001-0.08. It was found that the viscosity of the suspensions increased with phi, and showed a marked change of slope at phi approximately 0.01, which we identified as the minimum overlap concentration phi(*). Above phi(*), the system is in the semi-dilute regime. At phi>phi(*), when Pe(rot)>1, hydrodynamic interactions between rods become increasingly significant, and we observe shear-thinning behaviour. The high effective hydrodynamic volume of rotating rods, resulting in much lower values of the maximum packing fraction, phi(c), than for spheres, dominates the rheological behaviour of ettringite suspensions.     

Critical flux of hard sphere suspensions in crossflow filtration: Hydrodynamic force bias Monte Carlo simulations

A Monte Carlo method is developed for crossflow membrane filtration to determine the critical flux of hard sphere suspensions. Brownian and shear-induced diffusion are incorporated into an effective hydrodynamic force exerted on the hard spheres in a concentrated shear flow. Effects of shear rate and particle size on the critical flux are investigated using hydrodynamic force bias Monte Carlo simulations, providing a baseline of the critical flux.     

Pair diffusion, hydrodynamic interactions, and available volume in dense fluids

We calculate the pair diffusion coefficient D(r) as a function of the distance r between two hard sphere particles in a dense monodisperse fluid. The distance-dependent pair diffusion coefficient describes the hydrodynamic interactions between particles in a fluid that are central to theories of polymer and colloid dynamics. We determine D(r) from the propagators (Green's functions) of particle pairs obtained from molecular dynamics simulations. At distances exceeding ~3 molecular diameters, the calculated pair diffusion coefficients are in excellent agreement with predictions from exact macroscopic hydrodynamic theory for large Brownian particles suspended in a solvent bath, as well as the Oseen approximation. However, the asymptotic 1/r distance dependence of D(r) associated with hydrodynamic effects emerges only after the pair distance dynamics has been followed for relatively long times, indicating non-negligible memory effects in the pair diffusion at short times. Deviations of the calculated D(r) from the hydrodynamic models at short distances r reflect the underlying many-body fluid structure, and are found to be correlated to differences in the local available volume. The procedure used here to determine the pair diffusion coefficients can also be used for single-particle diffusion in confinement with spherical symmetry.     

Thermodynamic perturbation theory for fused sphere hard chain fluids using nonadditive interactions

A model is developed for the equation of state of fused chains based on Wertheim thermodynamic perturbation theory and nonadditive size interactions. The model also assumes that the structure (represented by the radial distribution function) of the fused chain fluid is the same as that of the touching hard sphere chain fluid. The model is completely based on spherical additive and nonadditive size interactions. The model has the advantage of offering good agreement with simulation data while at the same time being independent of fitted parameters. The model is most accurate for short chains, small values of Delta (slightly fused spheres) and at intermediate (liquidlike) densities.     

Isotropic–nematic phase separation and demixing in mixtures of spherical nanoparticles with length‐polydisperse nanorods

An extension of Onsager theory is developed to simulate isotropic–nematic phase separation in a mixture of spheres with length‐polydisperse system of rods. This work is motivated by recent experimental data on nanorod liquid crystals. Prior theoretical investigations indicate that both polydispersity and the presence of spheres should increase the biphasic–nematic phase transition, that is, the nematic cloud point. Results indicate that the phase diagrams undergo drastic changes depending upon both particle geometry and rod length polydispersity. The key geometric factor is the ratio between the sphere diameter and the rod diameter. In general, length fractionation is enhanced by the addition of spheres, which may be experimentally advantageous for separating short nanorods from a polydisperse population. Simulation results also indicate that the nematic cloud and shadow curves may cross one another because of the scarcity of spheres in the shadow phase. In general, these results do indicate that the nematic cloud point increases as a function of sphere loading; however, in certain areas of phase space, this relationship is nonmonotonic such that the nematic cloud point may actually decrease with the addition of spheres. This work has application to a wide range of nanoparticle systems, including mixtures of spherical nanoparticles with nanorods or nanotubes. Additionally, a number of nonspherical particles and structures may behave as spheres, including crumpled graphene and tightly coiled polymers. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012     

Brownian dynamics simulations of the self- and collective rotational diffusion coefficients of rigid long thin rods

Recently a microscopic theory for the dynamics of suspensions of long thin rigid rods was presented, confirming and expanding the well-known theory by Doi and Edwards [The Theory of Polymer Dynamics (Clarendon, Oxford, 1986)] and Kuzuu [J. Phys. Soc. Jpn. 52, 3486 (1983)]. Here this theory is put to the test by comparing it against computer simulations. A Brownian dynamics simulation program was developed to follow the dynamics of the rods, with a length over a diameter ratio of 60, on the Smoluchowski time scale. The model accounts for excluded volume interactions between rods, but neglects hydrodynamic interactions. The self-rotational diffusion coefficients D(r)(phi) of the rods were calculated by standard methods and by a new, more efficient method based on calculating average restoring torques. Collective decay of orientational order was calculated by means of equilibrium and nonequilibrium simulations. Our results show that, for the currently accessible volume fractions, the decay times in both cases are virtually identical. Moreover, the observed decay of diffusion coefficients with volume fraction is much quicker than predicted by the theory, which is attributed to an oversimplification of dynamic correlations in the theory.     

Interactions between colloidal particles in polymer solutions: A density functional theory study

We present a density functional theory study of colloidal interactions in a concentrated polymer solution. The colloids are modeled as hard spheres and polymers are modeled as freely jointed tangent hard sphere chains. Our theoretical results for the polymer-mediated mean force between two dilute colloids are compared with recent simulation data for this model. Theory is shown to be in good agreement with simulation. We compute the colloid-colloid potential of mean force and the second virial coefficient, and analyze the behavior of these quantities as a function of the polymer solution density, the polymer chain length, and the colloid/polymer bead size ratio.     

Molecular alignment of rigid rods in nonrigid spherical pores

We have investigated the orientation ordering of two shish-kebab chains confined by spherically harmonic potentials through Monte Carlo simulations and asymptotic analysis. The rigid rod is modeled as shish-kebab chains consisting of tangent hard spheres aligned in the same axis, and the harmonic potential is chosen to model nonrigid cavities. We first show that the interactions between a rod and the spherically harmonic potential are independent of chain orientation, indicating that the alignment of two confined rods arises from the excluded volume interactions alone. In the strong fields, the order parameter of two confined rods converges to different values, depending on the parity of chain length. From asymptotic order parameters, we find that the rods of odd-number beads rotate more freely even under the limiting strong confinement. However, the two rods of even-number beads are essentially trapped in a configuration of perpendicular alignment through intercalation of their central grooves. We attribute the dependence of the parity of chain length to the different locations of the center-of-mass in a rod for these two cases. Furthermore, we compare the shish-kebab chains with different rod models in the simulations, and utilize these models to explore the effect of the local rod smoothness on molecular alignment. Our findings suggest that increasing local rod smoothness enhances the rotational degree of freedom for confined rods, and the effect of local rod roughness emerges under strong enough applied potentials.     

The effect of rotational and translational energy exchange on tracer diffusion in rough hard sphere fluids

A study is presented of tracer diffusion in a rough hard sphere fluid. Unlike smooth hard spheres, collisions between rough hard spheres can exchange rotational and translational energy and momentum. It is expected that as tracer particles become larger, their diffusion constants will tend toward the Stokes-Einstein hydrodynamic result. It has already been shown that in this limit, smooth hard spheres adopt "slip" boundary conditions. The current results show that rough hard spheres adopt boundary conditions proportional to the degree of translational-rotational energy exchange. Spheres for which this exchange is the largest adopt "stick" boundary conditions while those with more intermediate exchange adopt values between the "slip" and "stick" limits. This dependence is found to be almost linear. As well, changes in the diffusion constants as a function of this exchange are examined and it is found that the dependence is stronger than that suggested by the low-density, Boltzmann result. Compared with smooth hard spheres, real molecules undergo inelastic collisions and have attractive wells. Rough hard spheres model the effect of inelasticity and show that even without the presence of attractive forces, the boundary conditions for large particles can deviate from "slip" and approach "stick."     

Creeping motion of an assemblage of composite spheres relative to a fluid

The sedimentation of a homogeneous distribution of spherical composite particles and the fluid flow through a bed of these particles are investigated theoretically. Each composite particle is composed of a spherical solid core and a surrounding porous shell. In the fluid-permeable porous shell, idealized hydrodynamic frictional segments are assumed to distribute uniformly. The effect of interactions among the particles is taken into explicit account by employing a fundamental cell-model representation which is known to provide good predictions for the motion of a swarm of nonporous spheres within a fluid. In the limit of a small Reynolds number, the Stokes and Brinkman equations are solved for the flow field in a unit cell, and the drag force exerted by the fluid on the particle is obtained in a closed form. For a distribution of composite spheres, the normalized mobility of the particles decreases or the particle interactions increase monotonically with a decrease in the permeability of their porous shells. The effect of particle interactions on the creeping motion of composite spheres relative to a fluid can be quite significant in some situations. In the limiting cases, the analytical solutions describing the drag force or mobility for a suspension of composite spheres reduce to those for suspensions of solid spheres and of porous spheres. The hydrodynamic behavior for composite spheres may be approximated by that for permeable spheres when the porous layer is sufficiently thick, depending on the permeability.     

Effect of attractions on the structure of polymer solutions confined between surfaces: a density functional approach

A density functional theory is presented to study the effect of attractions on the structure of polymer solutions confined between surfaces. The polymer molecules have been modeled as a pearl necklace of freely jointed hard spheres and the solvent as hard spheres, both having Yukawa-type attractions and the mixture being confined between attractive Yukawa-type surfaces. The present theory treats the ideal gas free energy functional exactly and uses weighted density approximation for the hard chain and hard sphere contributions to the excess free energy functional. The attractive interactions are calculated using the direct correlation function obtained from the polymer reference interaction site model theory along with the mean spherical approximation closure. The theoretical predictions on the density profiles of the polymer and the solvent molecules are found to agree quite well with the Monte Carlo simulation results for varying densities, chain lengths, wall separations, and different sets of interaction potentials.     

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.     

Effect of attractive interactions on the structure of polymer melts confined between surfaces: a density-functional approach

A density-functional theory is presented to study the structure of polymers, having attractive interactions, confined between attractive surfaces. The theory treats the ideal-gas free-energy functional exactly and uses weighted density approximation for the hard-chain contribution to the excess free-energy functional. The bulk interactions of freely jointed hard spheres are obtained from generalized Flory equation of state and the attractive interactions are calculated using the direct correlation function obtained from the polymer reference interaction site model theory along with the mean spherical approximation closure. The theoretical predictions are found to be in quite good agreement with the Monte Carlo simulation results for varying densities, chain lengths, and different interaction potentials. The results confirm important implications of using different approximations for the hard-sphere and attractive interactions.     

Depletion potentials in colloidal mixtures of hard spheres and rods

The depletion potential between a hard sphere and a planar hard wall, or two hard spheres, imposed by suspended rigid spherocylindrical rods is computed by the acceptance ratio method through the application of Monte Carlo simulation. The accurate results and ideal-gas approximation results of the depletion potential are determined with the acceptance ratio method in our simulations. For comparison, the depletion potentials are also studied by using both the density functional theory and Derjaguin approximations. The density profile as a function of positions and orientations of rods, used in the density functional theory, is calculated by Monte Carlo simulation. The potential obtained by the acceptance ratio method is in good agreement with that of density functional theory under the ideal-gas approximation. The comparison between our results and those of other theories suggests that the acceptance ratio method is the only efficient method used to compute the depletion potential induced by nonspherical colloids with the volume fraction beyond the ideal-gas approximation.