Particle motion driven by solute gradients with application to autonomous motion: continuum and colloidal perspectives

Diffusiophoresis, the motion of a particle in response to an externally imposed concentration gradient of a solute species, is analysed from both the traditional coarse-grained macroscopic (i.e. continuum) perspective and from a fine-grained micromechanical level in which the particle and the solute are treated on the same footing as Brownian particles dispersed in a solvent. It is shown that although the two approaches agree when the solute is much smaller in size than the phoretic particle and is present at very dilute concentrations, the micromechanical colloidal perspective relaxes these restrictions and applies to any size ratio and any concentration of solute. The different descriptions also provide different mechanical analyses of phoretic motion. At the continuum level the macroscopic hydrodynamic stress and interactive force with the solute sum to give zero total force, a condition for phoretic motion. At the colloidal level, the particle's motion is shown to have two contributions: (i) a ‘back-flow’ contribution composed of the motion of the particle due to the solute chemical potential gradient force acting on it and a compensating fluid motion driven by the long-range hydrodynamic velocity disturbance caused by the chemical potential gradient force acting on all the solute particles and (ii) an indirect contribution arising from the mutual interparticle and Brownian forces on the solute and phoretic particle, that contribution being non-zero because the distribution of solute about the phoretic particle is driven out of equilibrium by the chemical potential gradient of the solute. At the colloidal level the forces acting on the phoretic particle – both the statistical or ‘thermodynamic’ chemical potential gradient and Brownian forces and the interparticle force – are balanced by the Stokes drag of the solvent to give the net phoretic velocity

Response to "Comment on 'The rheological behavior of concentrated colloidal dispersions'"

A reply to the comments of Cichocki and Felderhof concerning my paper on the rheological behavior of concentrated colloidal dispersions [J. Chem. Phys. 99, 567 (1993)] is given. Differences exist between my scaling predictions as maximum packing is approached for the reduced dynamic viscosity and the dilute analysis of Cichocki and Felderhof

The rheological behavior of concentrated colloidal dispersions

A simple model for the rheological behavior of concentrated colloidal dispersions is developed. For a suspension of Brownian hard spheres there are two contributions to the macroscopic stress: a hydrodynamic and a Brownian stress. For small departures from equilibrium, the hydrodynamic contribution is purely dissipative and gives the high-frequency dynamic viscosity. The Brownian contribution has both dissipative and elastic parts and is responsible for the viscoelastic behavior of colloidal dispersions. An evolution equation for the pair-distribution function is developed and from it a simple scaling relation is derived for the viscoelastic response. The Brownian stress is shown to be proportional to the equilibrium radial-distribution function at contact, g(2;phi), divided by the short-time self-diffusivity, D0(s)(phi), both evaluated at the volume fraction phi of interest. This scaling predicts that the Brownian stress diverges at random close packing, phi(m), with an exponent of -2, that is, eta0' approximately eta(1 - phi/phi(m))-2, where eta0' is the steady shear viscosity of the dispersion and eta is the viscosity of the suspending fluid. Both the scaling law and the predicted magnitude are in excellent accord with experiment. For viscoelastic response, the theory predicts that the proper time scale is a2/D0s, where a is the particle radius, and, when appropriately scaled, the form of the viscoelastic response is a universal function for all volume fractions, again in agreement with experiment. In the presence of interparticle forces there is an additional contribution to the stress analogous to the Brownian stress. When the length scale characterizing the interparticle forces is comparable to the particle size, the theory predicts that there is only a quantitative contribution from the interparticle forces to the stress; the qualitative behavior, particularly the singular scaling of the viscosity and the form of the viscoelastic response, remains unchanged from the Brownian case. For strongly repulsive interparticle forces characterized by a length scale b (much greater than a), however, the theory predicts that the viscosity diverges at the random close packing volume fraction, phi(bm), based on the length scale b, with a weaker exponent of -1. The viscoelastic response now occurs on the time scale b2/D0s(phi), but is of the same form as for Brownian dispersions

The long-time self-diffusivity in concentrated colloidal dispersions

The long-time self-diffusivity in concentrated colloidal dispersions is determined from a consideration of the temporal decay of density fluctuations. For hydrodynamically interacting Brownian particles the long-time self-diffusivity, D^s_∞, is shown to be expressible as the product of the hydrodynamically determined short-time self-diffusivity, D^s_0(φ), and a contribution that depends on the distortion of the equilibrium structure caused by a diffusing particle. An argument is advanced to show that as maximum packing is approached the long-time self-diffusivity scales as D^s_∞(φ)~ D^s_0(φ)/g(2;φ), where g(2;φ) is the value of the equilibrium radial-distribution function at contact and φ is the volume fraction of interest. This result predicts that the longtime self-diffusivity vanishes quadratically at random close packing, φ_m ≈ 0.63, i.e. D^s_∞D_0(1-φ/φ_m)^2 as φ → φ_m, where D_0 = kT/6πηα is the diffusivity of a single isolated particle of radius α in a fluid of viscosity η. This scaling occurs because Ds_0(φ) vanishes linearly at random close packing and the radial-distribution function at contact diverges as (1 -φ/φ_m)^(−1). A model is developed to determine the structural deformation for the entire range of volume fractions, and for hard spheres the longtime self-diffusivity can be represented by: D^s_∞(φ) = D^s_∞(φ)/[1 + 2φg(2;φ)]. This formula is in good agreement with experiment. For particles that interact through hard-spherelike repulsive interparticle forces characterized by a length b(> α), the same formula applies with the short-time self-diffusivity still determined by hydrodynamic interactions at the true or ‘hydrodynamic’ volume fraction φ, but the structural deformation and equilibrium radial-distribution function are now determined by the ‘thermodynamic’ volume fraction φ_b based on the length b. When b » α, the long-time self-diffusivity vanishes linearly at random close packing based on the ‘thermodynamic’ volume fraction φ_(bm). This change in behaviour occurs because the true or ‘hydrodynamic’ volume fraction is so low that the short-time self-diffusivity is given by its infinite-dilution value D_0. It is also shown that the temporal transition from short- to long-time diffusive behaviour is inversely proportional to the dynamic viscosity and is a universal function for all volume fractions when time is nondimensionalized by α^2/D^s_∞(φ)

Microstructure of strongly sheared suspensions and its impact on rheology and diffusion

The effects of Brownian motion alone and in combination with an interparticle force of hard-sphere type upon the particle configuration in a strongly sheared suspension are analysed. In the limit Pe[rightward arrow][infty infinity] under the influence of hydrodynamic interactions alone, the pair-distribution function of a dilute suspension of spheres has symmetry properties that yield a Newtonian constitutive behaviour and a zero self-diffusivity. Here, Pe=[gamma][ogonek]a2/2D is the Péclet number with [gamma][ogonek] the shear rate, a the particle radius, and D the diffusivity of an isolated particle. Brownian diffusion at large Pe gives rise to an O(aPe[minus sign]1) thin boundary layer at contact in which the effects of Brownian diffusion and advection balance, and the pair-distribution function is asymmetric within the boundary layer with a contact value of O(Pe0.78) in pure-straining motion; non-Newtonian effects, which scale as the product of the contact value and the O(a3Pe[minus sign]1) layer volume, vanish as Pe[minus sign]0.22 as Pe[rightward arrow][infty infinity]

Self-diffusion in sheared suspensions

Self-diffusion in a suspension of spherical particles in steady linear shear flow is investigated by following the time evolution of the correlation of number density fluctuations. Expressions are presented for the evaluation of the self-diffusivity in a suspension which is either raacroscopically quiescent or in linear flow at arbitrary Peclet number Pe = ẏa^2/2D, where ẏ is the shear rate, a is the particle radius, and D = k_BT/6πηa is the diffusion coefficient of an isolated particle. Here, k_B is Boltzmann's constant, T is the absolute temperature, and η is the viscosity of the suspending fluid. The short-time self-diffusion tensor is given by k_BT times the microstructural average of the hydrodynamic mobility of a particle, and depends on the volume fraction ø = 4/3πa^3n and Pe only when hydrodynamic interactions are considered. As a tagged particle moves through the suspension, it perturbs the average microstructure, and the long-time self-diffusion tensor, D_∞^s, is given by the sum of D_0^s and the correlation of the flux of a tagged particle with this perturbation. In a flowing suspension both D_0^s and D_∞^s are anisotropic, in general, with the anisotropy of D_0^s due solely to that of the steady microstructure. The influence of flow upon D_∞^s is more involved, having three parts: the first is due to the non-equilibrium microstructure, the second is due to the perturbation to the microstructure caused by the motion of a tagged particle, and the third is by providing a mechanism for diffusion that is absent in a quiescent suspension through correlation of hydrodynamic velocity fluctuations. The self-diffusivity in a simply sheared suspension of identical hard spheres is determined to O(φPe^(3/2)) for Pe « 1 and ø « 1, both with and without hydro-dynamic interactions between the particles. The leading dependence upon flow of D_0^s is 0.22DøPeÊ, where Ê is the rate-of-strain tensor made dimensionless with ẏ. Regardless of whether or not the particles interact hydrodynamically, flow influences D_∞^s at O(øPe) and O(øPe^(3/2)). In the absence of hydrodynamics, the leading correction is proportional to øPeDÊ. The correction of O(øPe^(3/2)), which results from a singular advection-diffusion problem, is proportional, in the absence of hydrodynamic interactions, to øPe^(3/2)DI; when hydrodynamics are included, the correction is given by two terms, one proportional to Ê, and the second a non-isotropic tensor. At high ø a scaling theory based on the approach of Brady (1994) is used to approximate D_∞^s. For weak flows the long-time self-diffusivity factors into the product of the long-time self-diffusivity in the absence of flow and a non-dimensional function of Pe = ẏa^2/2D^s_0(φ)$. At small Pe the dependence on Pe is the same as at low ø

Stokesian Dynamics

Particles suspended or dispersed in a fluid medium occur in a wide variety of natural and man-made settings, e.g. slurries, composite materials, ceramics, colloids, polymers, proteins, etc. The central theoretical and practical problem is to understand and predict the macroscopic equilibrium and transport properties of these multiphase materials from their microstructural mechanics. The macroscopic properties might be the sedimentation or aggregation rate, self-diffusion coefficient, thermal conductivity, or rheology of a suspension of particles. The microstructural mechanics entails the Brownian, interparticle, external, and hydrodynamic forces acting on the particles, as well as their spatial and temporal distribution, which is commonly referred to as the microstructure. If the distribution of particles were given, as well as the location and motion of any boundaries and the physical properties of the particles and suspending fluid, one would simply have to solve (in principle, not necessarily in practice) a well-posed boundary-value problem to determine the behavior of the material. Averaging this solution over a large volume or over many different configurations, the macroscopic or averaged properties could be determined. The two key steps in this approach, the solution of the many-body problem and the determination of the microstructure, are formidable but essential tasks for understanding suspension behavior. This article discusses a new, molecular-dynamics-like approach, which we have named Stokesian dynamics, for dynamically simulating the behavior of many particles suspended or dispersed in a fluid medium. Particles in suspension may interact through both hydrodynamic and nonhydrodynamic forces, where the latter may be any type of Brownian, colloidal, interparticle, or external force. The simulation method is capable of predicting both static (i.e. configuration-specific) and dynamic microstructural properties, as well as macroscopic properties in either dilute or concentrated systems. Applications of Stokesian dynamics are widespread; problems of sedimentation, flocculation, diffusion, polymer rheology, and transport in porous media all fall within its domain. Stokesian dynamics is designed to provide the same theoretical and computational basis for multiphase, dispersed systems as does molecular dynamics for statistical theories of matter. This review focuses on the simulation method, not on the areas in which Stokesian dynamics can be used. For a discussion of some of these many different areas, the reader is referred to the excellent reviews and proceedings of topical conferences that have appeared (e.g. Batchelor 1976a, Dickinson 1983, Faraday Discussions 1983, 1987, Family & Landau 1984). Before embarking on a description of Stokesian dynamics, we pause here to discuss some of the relevant theoretical literature on suspensions, and dynamic simulation in general, in order to put Stokesian dynamics in perspective

O-H...O, C-H...O and C-H...[pi]arene intermolecular interactions in (2S)-2-(1-oxo-1H-2,3-dihydroisoindol-2-yl)pentanoic acid and (2S)-3-methyl-2-(1-oxo-1H-2,3-dihydroisoindol-2-yl)butanoic acid

In the first of the title compounds, (2S)-2-(1-oxo-1H-2,3-dihydroisoindol-2-yl)pentanoic acid, C₁₃H₁₅NO₃, prepared from L-norvaline, a hydrogen-bonded network is formed in the solid state through O-H...O=C, C-H...O=C and C-H...πarene intermolecular interactions, with shortest O...O, C...O and C...centroid distances of 2.582 (13), 3.231 (11) and 3.466 (3) Å, respectively. In the L-valine derivative, (2S)-3-methyl-2-(1-oxo-1H-2,3-dihydroisoindol-2-yl)butanoic acid, C₁₃H₁₅NO₃, O-H...O=C and Carene-H...O=C intermolecular interactions generate a cyclic R²₂(9) motif through cooperativity, with shortest O...O and C...O distances of 2.634 (3) and 3.529 (5) Å, respectively. Methylene C-H...O=Cindole interactions complete the hydrogen bonding, with C...O distances ranging from 3.283 (4) to 3.477 (4) Å

Shear-induced self-diffusion in non-colloidal suspensions

Self-diffusion in a monodisperse suspension of non-Brownian particles in simple shear flow is studied using accelerated Stokesian dynamics (ASD) simulation. The availability of a much faster computational algorithm allows the study of large systems (typically of 1000 particles) and the extraction of accurate results for the complete shear-induced self-diffusivity tensor. The finite, and often large, autocorrelation time requires the mean-square displacements to be followed for very long times, which is now possible with ASD. The self-diffusivities compare favourably with the available experimental measurements when allowance is made for the finite strains sampled in the experiments. The relationship between the mean-square displacements and the diffusivities appearing in a Fokker–Planck equation when advection couples to diffusion is discussed