Institute of Fundamental Technological Research

We investigate qualitatively a uniform non-Brownian sedimenting suspension in a stationary state. As a base of our analysis we take the BBGKY hierarchy derived from the Liouville equation. We then show that assumption of the plasma-like screening relations can cancel some long-range terms in the hierarchy but it does not provide integrable solutions for correlation functions. This suggests breaking the translational symmetry of the system. Therefore a non-uniform structure can develop to suppress velocity fluctuations and make the range of correlations finite.

non-Brownian sedimentation, stability, BBGKY hierarchy, hydrodynamic screening, correlation functions, low-Reynolds-number hydrodynamics

Inclusion of hydrodynamic interactions is essential for a quantitatively accurate Brownian dynamics simulation of colloidal suspensions or polymer solutions. We use the generalized Rotne–Prager–Yamakawa (GRPY) approximation, which takes into account all long-ranged terms in the hydrodynamic interactions, to derive the complete set of hydrodynamic matrices in different geometries: unbounded space, periodic boundary conditions of Lees–Edwards type, and vicinity of a free surface. The construction is carried out both for non-overlapping as well as for overlapping particles. We include the dipolar degrees of freedom, which allows one to use this formalism to simulate the dynamics of suspensions in a shear flow and to study the evolution of their rheological properties. Finally, we provide an open-source numerical package, which implements the GRPY algorithm in Lees–Edwards periodic boundary conditions.

Two main problems that arise in the context of hydrodynamic bead modeling are an inaccurate treatment of bead overlaps and the necessity of using volume corrections when calculating intrinsic viscosity. We present a formalism based on the generalized Rotne-Prager-Yamakawa approximation that successfully addresses both of these issues. The generalized Rotne-Prager-Yamakawa method is shown to be highly effective for the calculation of transport properties of rigid biomolecules represented as assemblies of spherical beads of different sizes, both overlapping and nonoverlapping. We test the method on simple molecular shapes as well as real protein structures and compare its performance with other computational approaches.

We develop a generalized Rotne–Prager–Yamakawa approximation for the dipolar components of the inverse friction matrix and use it for calculating the intrinsic viscosity of rigidly connected bead conglomerates. Such bead models are commonly used in the calculation of hydrodynamic properties of macromolecules. We consider both the case of non-overlapping constituent beads as well as overlapping beads of different sizes. We demonstrate the accuracy of the approximation in two test cases and show that it performs well even if the distances between the beads are small or if the beads overlap. Robust performance of this approximation in the case of overlapping beads stems from its correct limiting behaviour at a complete overlap, with one sphere fully immersed in the other. The generalized Rotne–Prager–Yamakawa approximation is thus well suited for evaluation of intrinsic viscosity, which is a key quantity in characterizing molecular conformations of biological macromolecules.

complex fluids, low-Reynolds-number flows, mathematical foundations

The exact analytical expressions for the time-dependent cross-correlations of the translational and rotational Brownian displacements of a particle with arbitrary shape were derived by us in [3, 4]. They are in this work applied to construct a method to analyze the Brownian motion of a particle of an arbitrary shape, and to extract accurately the self-diffusion matrix from the measurements of the crosscorrelations, which in turn allows to gain some information on the particle structure. As an example, we apply our new method to analyze the experimental results of D. J. Kraft et al. for the micrometer-sized aggregates of the beads [8]. We explicitly demonstrate that our procedure, based on the measurements of the time-dependent cross-correlations in the whole range of times, allows to determine the self-diffusion (or alternatively the friction matrix) with a much higher precision than the method based only on their initial slopes. Therefore, the analytical time-dependence of the cross-correlations serves as a useful tool to extract information about particle structure from trajectory measurements.

Brownian motion, Smoluchowski equation, hydrodynamic interactions, self-diffusion matrix, friction coefficients, cross-correlations of translational and rotational Brownian displacements

The analytical expressions for the time-dependent cross correlations of the translational and rotational Brownian displacements of a particle with arbitrary shape are derived. The reference center is arbitrary, and the reference frame is such that the rotational-rotational diffusion tensor is diagonal.

Rotational correlation time, Tensor methods, Brownian motion, Colloidal systems, Matrix equations

Hydrodynamic interactions with confining boundaries often lead to drastic changes in the diffusive behaviour of microparticles in suspensions. For axially symmetric particles, earlier numerical studies have suggested a simple form of the near-wall diffusion matrix which depends on the distance and orientation of the particle with respect to the wall, which is usually calculated numerically. In this work, we derive explicit analytical formulae for the dominant correction to the bulk diffusion tensor of an axially symmetric colloidal particle due to the presence of a nearby no-slip wall. The relative correction scales as powers of inverse wall-particle distance and its angular structure is represented by simple functions in sines and cosines of the particle’s inclination angle to the wall. We analyse the correction for translational and rotational motion, as well as the translation-rotation coupling. Our findings provide a simple approximation to the anisotropic diffusion tensor near a wall, which completes and corrects relations known from earlier numerical and theoretical findings.

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.

rownian motion of a particle with an arbitrary shape is investigated theoretically. Analytical expressions for the time-dependent cross-correlations of the Brownian translational and rotational displacements are derived from the Smoluchowski equation. The role of the particle mobility center is determined and discussed.

Tensor methods, Brownian motion, Matrix theory, Rotational correlation time, Eigenvalues

Systems of spherical particles moving in Stokes flow are studied for different particle internal structures and boundaries, including the Navier-slip model. It is shown that their hydrodynamic interactions are well described by treating them as solid spheres of smaller hydrodynamic radii, which can be determined from measured single-particle diffusion or intrinsic viscosity coefficients. Effective dynamics of suspensions made of such particles is quite accurately described by mobility coefficients of the solid particles with the hydrodynamic radii, averaged with the unchanged direct interactions between the particles.

hydrodynamic radius, Stokes equations

Short-time dynamics and high-frequency rheology for suspensions of non-overlapping core–shell particles with thin shells were analyzed. In the thin-shell limit, the single-particle scattering coefficients were derived and shown to define a unique effective radius. This result was used to justify theoretically (in the thin-shell limit) the accuracy of the annulus approximation with the inner radius equal to the effective hydrodynamic radius of the core–shell particle. The two-particle virial expansion of the translational and rotational self-diffusion, sedimentation and viscosity was performed. The virial coefficients were evaluated and shown to be accurately approximated by the effective annulus model, in contrast to the imprecise effective hard sphere model.

Stokes equations, Core–shell particles, Permeable medium

Short-time dynamic properties of concentrated suspensions of colloidal core-shell particles are studied using a precise force multipole method which accounts for many-particle hydrodynamic interactions. A core-shell particle is composed of a rigid, spherical dry core of radius a surrounded by a uniformly permeable shell of outer radius b and hydrodynamic penetration depth κ−1. The solvent flow inside the permeable shell is described by the Brinkman-Debye-Bueche equation, and outside the particles by the Stokes equation. The particles are assumed to interact non-hydrodynamically by a hard-sphere no-overlap potential of radius b. Numerical results are presented for the high-frequency shear viscosity, η∞, sedimentation coefficient, K, and the short-time translational and rotational self-diffusion coefficients, D t and D r. The simulation results cover the full three-parametric fluid-phase space of the composite particle model, with the volume fraction extending up to 0.45, and the whole range of values for κb, and a/b. Many-particle hydrodynamic interaction effects on the transport properties are explored, and the hydrodynamic influence of the core in concentrated systems is discussed. Our simulation results show that for thin or hardly permeable shells, the core-shell systems can be approximated neither by no-shell nor by no-core models. However, one of our findings is that for κ(b − a) ≳ 5, the core is practically not sensed any more by the weakly penetrating fluid. This result is explained using an asymptotic analysis of the scattering coefficients entering into the multipole method of solving the Stokes equations. We show that in most cases, the influence of the core grows only weakly with increasing concentration.

core-shell particles, suspension, diffusion, sedimentation, effective viscosity

A single Brownian particle of arbitrary shape is considered. The time-dependent translational mean square displacement W(t) of a reference point at this particle is evaluated from the Smoluchowski equation. It is shown that at times larger than the characteristic time scale of the rotational Brownian relaxation, the slope of W(t) becomes independent of the choice of a reference point. Moreover, it is proved that in the long-time limit, the slope of W(t) is determined uniquely by the trace of the translational-translational mobility matrix μtt evaluated with respect to the hydrodynamic center of mobility. The result is applicable to dynamic light scattering measurements, which indeed are performed in the long-time limit.

translational and rotational Brownian motion, mean square displacement, particle of arbitrary shape, mobility center

Hydrodynamic properties of fibrinogen molecules were theoretically calculated. Their shape was approximated by the bead model, considering the presence of flexible side chains of various length and orientation relative to the main body of the molecule. Using the bead model, and the precise many-multipole method of solving the Stokes equations, the mobility coefficients for the fibrinogen molecule were calculated for arbitrary orientations of the arms whose length was varied between 12 and 18 nm. Orientation averaged hydrodynamic radii and intrinsic viscosities were also calculated by considering interactions between the side arms and the core of the fibrinogen molecule. Whereas the hydrodynamic radii changed little with the interaction magnitude, the intrinsic viscosity exhibited considerable variation from 30 to 60 for attractive and repulsive interactions, respectively. These theoretical results were used for the interpretation of experimental data derived from sedimentation and diffusion coefficient measurements as well as dynamic viscosity measurements. Optimum dimensions of the fibrinogen molecule derived in this way were the following: the contour length 84.7 nm, the side arm length 18 nm, and the total volume 470 nm3, which gives 16% hydration (by volume). Our calculations enabled one to distinguish various conformational states of the fibrinogen molecule, especially the expanded conformation, prevailing for pH < 4 and lower ionic strength, characterized by high intrinsic viscosity of 50 and the hydrodynamic radius of 10.6 nm. On the other hand, for the physiological condition, that is, pH = 7.4 and the ionic strength of 0.15 M NaCl, the semi-collapsed conformation dominates. It is characterized by the average angle equal to = 55, intrinsic viscosity of 35, and the hydrodynamic radius of 10 nm. Additionally, the interaction energy between the arms and the body of the molecule was predicted to be 4 kT units, confirming that they are oppositely charged than the central nodule. Results obtained in our work confirm an essential role of the side chains responsible for a highly anisotropic charge distribution in the fibrinogen molecule. These finding can be exploited to explain anomalous adsorption of fibrinogen on various surfaces.

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, permeable particles, concentrated suspensions

For suspensions of permeable particles, the short-time translational and rotational self-diffusion coefficients, and collective diffusion and sedimentation coefficients are evaluated theoretically. An individual particle is modeled as a uniformly permeable sphere of a given permeability, with the internal solvent flow described by the Debye-Bueche-Brinkman equation. The particles are assumed to interact non-hydrodynamically by their excluded volumes. The virial expansion of the transport properties in powers of the volume fraction is performed up to the two-particle level. The first-order virial coefficients corresponding to two-body hydrodynamic interactions are evaluated with very high accuracy by the series expansion in inverse powers of the inter-particle distance. Results are obtained and discussed for a wide range of the ratio, x, of the particle radius to the hydrodynamic screening length inside a permeable sphere. It is shown that for x≥10, the virial coefficients of the transport properties are well-approximated by the hydrodynamic radius (annulus) model developed by us earlier for the effective viscosity of porous-particle suspensions.

Stokes equations, hydrodynamic interactions, diffusion, sedimentation, permeable particles, suspesnion, virial expansion

The motion of a millimeter size spherocylinder particle settling in a very viscous oil in a closed container is measured by laser interferometry, with the goal to model the motion of a particle of this shape in a fluid at microscales. The container is a cylinder with vertical axis and closed at both ends by horizontal plates. The displacement of the particle along the container axis is recorded with a resolution of the order of 100 nm, that is much smaller than the particle–wall separation when in the lubrication regime. The particle friction coefficient, measured as a function of the particle–wall distance, is then used to test the theoretical predictions of an accurate hydrodynamic analysis. The Stokes flow problem is solved by using the hydromultipole method, that is in general appropriate for spheres but is extended here to a non-spherical particle by using a compound of overlapping spheres. The lateral wall effect is negligible but the two parallel horizontal end plane walls are accurately taken into account. The result of the theoretical model is in good quantitative agreement with experiment for the whole settling motion of the spherocylinder, that is for any position between the walls.

Stokes flows, Suspensions, Sedimentation

We study short-time diffusion properties of colloidal suspensions of neutral permeable particles. An individual particle is modeled as a solvent-permeable sphere of interaction radius a and uniform permeability k, with the fluid flow inside the particle described by the Debye–Bueche–Brinkman equation, and outside by the Stokes equation. Using a precise multipole method and the corresponding numerical code HYDROMULTIPOLE that account for higher-order hydrodynamic multipole moments, numerical results are presented for the hydrodynamic function, H(q), the short-time self-diffusion coefficient, Ds, the sedimentation coefficient K, the collective diffusion coefficient, Dc, and the principal peak value H(qm), associated with the short-time cage diffusion coefficient, as functions of porosity and volume fraction. Our results cover the full fluid phase regime. Generic features of the permeable sphere model are discussed. An approximate method by Pusey to determine Ds is shown to agree well with our accurate results. It is found that for a given volume fraction, the wavenumber dependence of a reduced hydrodynamic function can be estimated by a single master curve, independent of the particle permeability, given by the hard-sphere model. The reduced form is obtained by an appropriate shift and rescaling of H(q), parametrized by the self-diffusion and sedimentation coefficients. To improve precision, another reduced hydrodynamic function, hm(q), is also constructed, now with the self-diffusion coefficient and the peak value, H(qm), of the hydrodynamic function as the parameters. For wavenumbers qa > 2, this function is permeability independent to an excellent accuracy. The hydrodynamic function of permeable particles is thus well represented in its q-dependence by a permeability-independent master curve, and three coefficients, Ds, K, and H(qm), that do depend on the permeability. The master curve and its coefficients are evaluated as functions of concentration and permeability.

Stokes equations, hydrodynamic interactions, self-diffusion, sedimentation, permeable particles, suspension

We determine the high-frequency limiting shear viscosity in colloidal suspensions of rigid, uniformly porous spheres of radius a as a function of volume fraction and inverse porosity parameter x. Our study covers the complete fluid-state regime. The flow inside the spheres is modeled by the Debye–Bueche–Brinkman equation using the boundary condition that fluid velocity and stress change continuously across the sphere surfaces. The many-sphere hydrodynamic interactions in concentrated systems are fully accounted for by a precise hydrodynamic multipole method encoded in our HYDROMULTIPOLE program extended to porous particles. A truncated virial expansion is used to derive an accurate and easy-to-use generalized Saitô formula for. The simulation data are used to test the performance of two simplifying effective particle models. The first model describes the effective particle as a nonporous sphere characterized by a single effective radius dependent on x. In the more refined second model, the porous spheres are modeled as spherical annulus particles with an inner hydrodynamic radius as a function of x, defining the nonporous dry core and characterizing hydrodynamic interactions, and an outer excluded volume radius a characterizing the unchanged direct interactions. Only the second model is in a satisfactory agreement with the simulation data.

Stokes flow, permeable particles, effective viscosity, lubrication, concentrated suspensions

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 Saitˆo formula for η∞ is presented which provides a good description of its porosity and concentration dependence.

Stokes flow, hydrodynamic interactions, permeable particles, dense suspensions, effective viscosity

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.Rychlewski

Stokes equations, hydrodynamic interactions, permeable particles, concentrated suspensions, self-diffusion, hydrodynamic function, collective diffusion

A fluid flow through a nonisotropic porous medium with an axial symmetry is considered. The Green tensors for the corresponding nonisotropic Debye–Büche–Brinkman equations are calculated in terms of single integrals. Short-distance and far-field limiting behavior is discussed. The exact solution for the Green tensors is found explicitly in the limiting case of an infinite shielding length along the symmetry axis of the system.

Tensor methods, Porous media, Fluid equations, Integral equations, Exact solutions

Self-diffusion of a sphere in a network of rods is analyzed theoretically.Hydrodynamic interactions are taken into account according to the model of Dhont et al. [J. Chem. Phys.122, 044905 (2005); Dhont et al., J. Chem. Phys.124, 044907 (2006); Dhont et al., J. Chem. Phys.126, 214501 (2007)] based on the Debye–Bueche–Brinkman equation. The hydrodynamic screening length of the effective medium is assumed to be much larger than the sphere radius and the rod thickness. The self-diffusion coefficient, given by Dhont et al. in terms of four-dimensional integrals, is in this work expressed in terms of a single integral only and therefore evaluated numerically with a high precision. Moreover, simple expressions for the self-diffusion coefficient are derived and shown to be independent of the rod length. They can be useful for experimental verification of the model.

Hydrodynamics, Friction, Hydrological modeling, Self diffusion, Tensor methods

Hydrodynamic interactions between spheres immersed in a low-Reynolds-number fluid flow close to a flat free surface or hard wall are investigated. The spheres may have different or equal radii, and may be separated from the boundary or at contact with the free surface. A simple and useful expression is derived for the propagator (Green operator) connecting centers of two spheres. In the derivation, the method of images and the displacement theorems are used. Symmetry of the displacement operators is explicitly shown. The significance of these results in efficient Stokesian and Brownian dynamics simulations is outlined. An example of an application is shown.

Free surface, Hydrodynamics, Friction, Fluid flows, Mirrors

General expressions for the frequency-dependent Brownian contribution to the intrinsic viscosity of arbitrary-shaped particles have been derived from the Smoluchowski equation.

intrinsic viscosity, Brownian motion, particle of arbitrary shape

Short-time dynamic properties of concentrated suspensions of colloidal core-shell particles have been recently studied [1] using a precise force multipole method which accounts for many-particle hydrodynamic interactions (HIs). A core-shell particle is composed of a rigid, spherical dry core of radius a surrounded by an uniformly permeable shell of outer radius b and hydrodynamic penetration depth κ-1. The solvent flow inside the permeable shell is described by the Brinkman-Debye-Bueche equation, and outside the particles by the Stokes equation. The particles are assumed to interact non-hydrodynamically by a hard-sphere no-overlap potential of radius b. Numerical results are presented for the high-frequency shear viscosity, sedimentation coefficient and the short-time translational and rotational self-diffusion coefficients. The simulation results cover the full three-parametric fluid-phase space of the composite particle model, with the volume fraction extending up to 0.45, and the whole range of values for κb, and a/b. Many-particle hydrodynamic interaction effects on the transport properties are explored, and the hydrodynamic influence of the core in concentrated systems is discussed.

Stokes equations, Brinkman-Debye-Bueche equations, permeable particles, translational and rotational self-diffusion, sedimentation, effective viscosity