Institute of Fundamental Technological Research

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

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

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

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