Nonspherical Brownian Particles Research Articles

Shear-induced particle migration in Couette and parallel-plate viscometers: NMR imaging and stress measurements

Couette and parallel plate viscometers are two commonly used flow geometries to characterize shear viscosity of concentrated suspensions. In Couette flow, it is well documented that prolonged shearing causes a decrease in the apparent viscosity of concentrated suspensions due to shear-induced particle migration from the annulus region to the stagnant region under the bob. In this study, the technique of nuclear magnetic resonance imaging (NMRI) was used to measure the evolution of suspension concentration profiles in Couette and parallel-plate flow devices upon shearing. Neutrally buoyant suspensions of nearly monodisperse, non-Brownian spherical particles at a volume fraction of 0.5 in a Newtonian fluid were used. The same flow cells and suspensions were also used in a rheometer to measure the changes in shear stress under identical experimental conditions such that a direct comparison can be made between the stress and concentration data. For Couette flow, the NMRI data correlated very well with the stress measurements and directly confirmed the Leighton–Acrivos [J. Fluid. Mech. 181, 415 (1987)] shear-induced migration theory. In torsional flow between parallel plates, no detectable change was found in particle concentration in the radial direction, but some decrease in the apparent viscosity was observed. These results provide some important clues for developing and evaluating more general descriptions of particle migration for nonrectilinear shear flows.

The approach to normality in the dispersion of sedimenting nonspherical Brownian particles

The dispersion process accompanying the sedimentation of nonspherical Brownian particles under the action of a constant external force field is studied with the goal of obtaining an asymptotic expansion of their distribution in the physical space in the long time limit (when an “equilibrium” orientational distribution has already been established). This expansion is obtained for homogeneous axisymmetric, centrosymmetric particles via an extension of Chatwin's aysmptotic scheme to the abstract, multidimensional phase-space of generalized Taylor dispersion theory and the recognition of the equivalence of the present problem and the generic formulation of the latter theory. The resulting expansion consists of a leading Gaussian distribution in physical space and correction terms proceeding in negative powers of t 1 2 . Thus the deviation from Gaussian behavior (essentially in the form of a residual “skewness” of the distribution) is fairly persistent. The coefficients of the expansion are evaluated explicitly (up to and including O( t −1) terms) for spheroidal particles, which enable an examination of the effects of particle shape and size, magnitude of external force, and the initial orientation, respectively, upon the departures from Gaussian asymptotic behavior. The results indicate that these are, in general, larger for prolate than for oblate spheroids of comparable aspect ratio and increase with increasing “Taylor effect” (i.e., increasing magnitude of external force, size of particles, and deviation of their shape from a spherical one). However, this increase is bounded. The leading effect of the initial orientation is a stationary shift of the centroid of the distribution relative to a frame of reference moving with the long-time average settling velocity. The observed trends in this (lateral and longitudinal) centroidal shift are explained in terms of the initial tendency of rod-like (disk-like) particles to slide along their axis (in their planes, perpendicularly to their axis).

On the formulation of boundary conditions for rigid nonspherical Brownian particles near solid walls: Applications to orientation-specific reactions with immobilized enzymes

Transport processes involving the motion of rigid nonspherical Brownian particles within porous media and, more generally, proximate to solid surfaces, arise in a multitude of natural and engineered contexts. Detailed modeling proceeds via solution of the microscale convective-diffusion equation in the six-dimensional configuration space composed of particle positions and orientations, subject to boundary conditions accounting for either impenetrability (inertness) or catalytic activity of the bounding surfaces. Owing to the geometrical complexity associated with even simple particle-wall combinations, the construction of such conditions has long been the source of ambiguities in the literature. We present a rigorous, general analysis of issues arising from particle-wall contact within the context of Brownian transport. Abstract configuration-space flux conditions are shown to possess a physically transparent interpretation in terms of rigid-body kinematics. As a specific application of our general theory, explicit asymptotic results are derived for a model problem pertaining to orientation-specific reactions of macromolecular substrates with immobilized enzymes. In the course of the analysis, a robust numerical technique is developed for the solution of configuration-space diffusion problems, the general utility of which transcends its application to the specific physical system considered here.

On the foundations of generalized Taylor dispersion theory

Generalized Taylor dispersion theory extends the basic long-time, asymptotic scheme of Taylor and Aris greatly beyond the class of rectilinear duct and channel flow dispersion problems originally addressed by them. This feature has rendered it indispensable for studying flow and dispersion phenomena in porous media, chromatographic separation processes, heat transfer in cellular media, sedimentation of non-spherical Brownian particles, and transport of flexible clusters of interacting Brownian particles, to mention just a few examples of the broad class of non-unidirectional transport phenomena encompassed by this scheme. Moreover, generalized Taylor dispersion theory enjoys the attractive feature of conferring a unified paradigmatic structure upon the analysis of such apparently disparate physical problems. For each of the problems thus treated it provides an asymptotic, macroscale description of the original microscale transport process, being based upon a convective-diffusive ‘model’ problem characterized by a set of constant (position- and time-independent) phenomenological coefficients.The present contribution formally substantiates the scheme. This is accomplished by demonstrating that the coarse-grained (macroscale) transport ‘model’ equation leads to a solution which accords asymptotically with the leading-order behaviour of the comparable solution of the exact (microscale) convective–diffusive problem underlying the transport process. It is also shown, contrary to current belief, that no systematic improvement in the asymptotic order of approximation is possible through the incorporation of higher-order gradient terms into the model constitutive equation for the coarse-grained flux. Moreover, the inherent difference between the present rigorous asymptotic scheme and the dispersion models resulting from Gill–Subramanian moment-gradient expansions is illuminated, thereby conclusively resolving a long-standing puzzle in longitudinal dispersion theory.

Nonlinear ultrasonically induced birefringence in gold sols: Frequency-dependent diffusion.

Ultrasonically induced birefringence has been used to study coupled translational and rotational motions of large (100-nm) nonspherical colloidal gold particles in fluids. The effect is nonlinear in that the induced birefringence is quadratic in the ultrasonic amplitude. A theory is developed to explain the dependence of the observed signal on concentration, viscosity, temperature, particle size, and on the ultrasonic intensity and frequency. This enables us to determine the frequency dependence of the viscous drag on the translational motion of nonspherical Brownian particles.

Memory effects in the motion of nonspherical Brownian particles

A system of nonlinear generalized Langevin equations is stated to describe the motion of noninteracting nonspherical Brownian particles. The coupling of translational and rotational degrees of freedom is taken into account via anisotropic translational and rotational friction coefficients, a Magnus force term and a systematic torque. A kinetic equation is derived with time- dependent coefficient functions, which are explicitly calculated for the one-mode case. As an example the frequency dependence of the coefficient of flow birefringence is discussed.

Dynamic simulation of sheared suspensions. I. General method

A general method is presented for simulating the dynamical behavior of a suspension of particles which interact through both hydrodynamic and nonhydrodynamic forces. In the molecular-dynamics-like simulation there are two different procedures for computing the interactions among particles: a pairwise additivity of forces and a pairwise additivity of velocities. The pairwise additivity of forces is the preferred method as it preserves the hydrodynamic lubrication forces which prevent particles from overlapping. The two methods are compared in a simulation of a monolayer of identical rigid non-Brownian spherical particles in a simple shear flow. Periodic boundary conditions are used to model an infinite suspension. Both methods predict the presence of a shear induced anisotropic local structure whose form and strength depend on the concentration of particles, the nonhydrodynamic forces, and the shear rate. Increasing the particle concentration up to near the maximum fraction that can still flow results in a transition to a layered structure in which planes of particles slide relative to one another. The anisotropic local structure and transition to a layered structure predict a non-Newtonian suspension rheology.

Taylor dispersion in systems of sedimenting nonspherical Brownian particles: III. Time-periodic forces

An Aris-type moment scheme is employed to calculate the time-average mean velocity vector and Taylor-Aris dispersion tensor for the Stokes sedimentation of small homogeneous ellipsoidal (and other orthotropic) particles which simultaneously undergo translational and rotational Brownian motions while settling in an otherwise quiescent viscous fluid under the influence of a time-periodic external force. This generalizes to time-periodic forces prior dispersion results previously restricted to time-independent forces. Each harmonic component of the force independently gives rise to its own Taylor dispersion contribution, in which appears a dimensionless coefficient quantifying the effects of frequency and particle geometry—namely size and shape. This parameter vanishes identically for spherical and other hydrodynamically isotropic particles (e.g., cubes, tetrahedra, octahedra, etc.), whose translational hydrodynamic resistances are orientation independent. Advantages attaching to an experimental “wet” particle-shape characterization device based upon the principles developed herein are pointed out. In this context a novel experimental device is proposed based upon preventing any net (i.e., time-average) gravitational sedimentation of the nonneutrally buoyant particles by continuous rotation of the suspension about a horizontal axis.

Transport and relaxation phenomena in suspensions of dipolar Brownian particles in an electric field

A system of noninteracting nonspherical Brownian particles with dipole moment parallel to the symmetry axis is studied in the presence of an external homogeneous electric field. A generalized Fokker-Planck equation is introduced which takes into account anisotropic translational and rotational friction as well as a translational-rotational coupling via a Magnus force term. From the corresponding kinetic equation for the reduced distribution function for rotationally overdamped particles a system of transport-relaxation equations for macroscopic variables is derived. They are used to treat the particle diffusion under the influence of the electric field and to calculate the spectral functions for infrared absorption and depolarized Rayleigh light scattering.

Nonspherical Brownian particles. Kinetic description and application to flow birefringence

Abstract For a system of noninteracting spheroidal Brownian particles a Fokker-Planck equation is derived which takes into account a hydrodynamical coupling between translational and orientational degrees of freedom. With the moment method, transport relaxation equations are obtained from which an expression for the coefficient of flow birefringence is inferred.

Taylor dispersion in systems of sedimenting nonspherical brownian particles. I. Homogeneous, centrosymmetric, axisymmetric particles

A recent theory of generalized Taylor dispersion phenomena is applied to the sedimentation of identical, nonneutrally buoyant, axisymmetric, Brownian particles settling in the Stokes-law regime. General expressions are derived for the mean (orientation-averaged) settling velocity U ̄ ∗ of the particles and the dispersion about this mean in the general case where the axisymmetric particles may lack fore-aft symmetry and where they may possess an inhomogeneous mass distribution. Specific calculations are carried out for the case where the particles possess a center of symmetry (as, for example, in the case of prolate and oblate spheroids) and are homogeneous. In such circumstances no gravitational couple acts upon the particles and they therefore possess no preferred orientations relative to the direction of the gravity field. The dispersion coefficients, D ̄ ‖ ∗ and D ̄ ⊥ ∗ , for motion of the particles parallel and perpendicular, respectively, to the direction of the gravity field are found to be of the forms D ∗ i = D + K iλ 2( U ∗) 2 d r with U ̄ ∗ the mean settling velocity of the particle, d r its rotary diffusion coefficient, and D ̄ = ( 1 3 ) × (D ‖ + 2D ⊥ ) the mean translational diffusion coefficient of the particle—wherein D ‖ and D ⊥ are the orientation-specific molecular diffusivities of the particle for translation parallel and perpendicular, respectively, to its symmetry axis. In addition, λ is the dimensionless parameter λ = |D ‖ − D ⊥ |/ D ̄ 1 2 , and κ ‖ = 2 135 and κ ⊥ = 1 90 . The forms of these transversely isotropic dispersivities agree with those recently obtained by Goren via a Langevin scheme, but the κ i values differ from his. A fundamental error in Goren's analysis is shown to be the source of the discrepancy. On the basis of these calculations, Smoluchowski's well-known equation for diffusion in an external field of force is shown to be generally inapplicable to nonspherical particles. Potential applications of the dispersion results are discussed.