Brownian motion dilute suspensions

Consider a dilute suspension of spherical particles A in a stationary liquid B. If the spheres are sufficiently small, yet large with respect to the molecules of stationary liquid, the collisions between the spheres and the liquid molecules B lead to a random motion of the spheres. This motion is called the Brownian motion. Dilute diffusion of suspended spherical colloid particles is related to the temperature and the friction coefficient by... 

For dilute suspensions, particle-particle interactions can be neglected. The extent of transfer of particles by the gradient in the particle phase density or volume fraction of particles is proportional to the diffusivity of particles Dp. Here Dp accounts for the random motion of particles in the flow field induced by various factors, including the diffusivity of the fluid whether laminar or turbulent, the wake of the particles in their relative motion to the fluid, the Brownian motion of particles, the particle-wall interaction, and the perturbation of the flow field by the particles. 

Problem 2-25. Complex fluids. Consider a suspension consisting of a Newtonian suspending fluid and micrometer-sized particles in which Brownian motion is a factor. The particles are spherical at equilibrium but are made of a Hookean elastic (rubberlike) material. Assume that the suspension is dilute (the motion of each particle is independent of the fact that other particles are present). Discuss whether this material can be described at a continuum level as a Newtonian fluid. Does this depend on the magnitude of the elastic modulus On the shear rate ... 

Suspensions, even in Newtonian liquids, may show elasticity. Hinch and Leal [1972] derived relations expressing the particle stresses in dilute suspensions with small Peclet number, Pe = y/D 1 (D is the rotary diffusion coef-hcient) and small aspect ratio. The origin of elastic effect lies in the anisometry of particles or their aggregates. Rotation of asymmetric entities provides a mechanism for energy storage. Brownian motion for its recovery. Eor suspensions of spheres, this mechanism does not exist. 

In this section we examine the flow of a suspension of particles, particularly the apparent viscosity coefficient of the suspension. Our interest is in calculating the convective mass flux of a suspension as distinct from the diffusive flux of Brownian motion. As previously, we shall assume a very dilute suspension in which each particle behaves as if it were in a liquid of infinite extent. To simplify the calculation, we neglect Brownian motion, although, as we discuss later, in the very dilute limit considered and for spherical particles it has no effect on the suspension viscosity. 

If the particles are not spherical, even in the very dilute limit where the translational Brownian motion would still be unimportant, rotational Brownian motion would come into play. This is a consequence of the fact that the rotational motion imparts to the particles a random orientation distribution, whereas in shear-dominated flows nonspherical particles tend toward preferred orientations. Since the excess energy dissipation by an individual anisotropic particle depends on its orientation with respect to the flow field, the suspension viscosity must be affected by the relative importance of rotational Brownian forces to viscous forces, although it should still vary linearly with particle volume fraction. 

We begin by examining the rate of collision of suspended spherical particles in a static fluid due to Brownian motion. This theory was first put forward by the great Polish physicist M. Von Smoluchowski, to whom we have often referred. In consequence of the equivalence between diffusion and Brownian motion, we consider the relative motion between the particles as a diffusion process. The particles are assumed to be in sufficiently dilute concentration that only binary encounters need be considered. To further simplify the calculation, we consider the suspension to be made up of only two different-sized spherical particles, one of radius a, and the other of radius... 

Consider an extremely diluted suspension consisting of a viscous liquid and small spherical particles suspended in it. Small volume concentration of particles allows to neglect their interactions and to assume that each particle behaves as if it were surrounded by an infinite volume of pure hquid. It is obvious that with the increase of volume concentration of particles their mutual influence will become more and more prominent, and at a certain point it will become impossible to neglect it. Further on, for simplicity, the Brownian motion will be neglected. Besides, we assume that particles are small, so it is possible to neglect the influence of gravity and consider their motion as inertialess. It means that the velocity of particle motion is equal to the velocity of the flow, in other words, the particles are freely suspended in the liquid. 

Let us summarize the obtained results. From the expression for the viscosity of an infinite diluted suspension, it follows that the viscosity factor does not depend on the size distribution of particles. The physical explanation of this fact is that in an infinite diluted suspension W 1), particles are spaced far apart (in comparison with the particle size), and the mutual influence of particles may be ignored. Besides, under the condition a/h 1, we can neglect the interaction of particles with the walls. It is also possible to show that in an infinite diluted suspension containing spherical particles. Brownian motion of particles does not influence the viscosity of the suspension. However, if the shape of particles is not spherical, then Brownian motion can influence the viscosity of the suspension. It is explained by the primary orientation of non-spherical particles in the flow. For example, thin elongated cylinders in a shear flow have the preferential orientation parallel to the flow velocity, in spite of random fluctuations in their orientation caused by Brownian rotational motion. 

Consider a problem on definition of collision frequency of small spherical particles executing Brownian motion in a quiescent liquid. In Section 8.2, Brownian motion was considered as diffusion with a effective diffusion factor. It was supposed that suspension is sufficiently diluted, so it is possible to consider only the pair interactions of particles. To simplify the problem, consider a bi-disperse system of particles, that is, a suspension consisting of particles of two types particles of radius ai and particles of radius a2. In this formulation, the problem was first considered by Smolukhowski [59]. 

In a QELS experiment, a monochromatic beam of light from a laser is focused on to a dilute suspension of particles and the scattering intensity is measured at some angle 0 by a detector. The phase and the polarization of the scattered light depend on the position and orientation of each scatterer. Because molecules or particles in solution are in constant Brownian motion, scattered light will result that is spectrally broadened by the Doppler effect. The key parameter determined by QELS is the diffusion coefficient, D, or particle di sivity which can be related to particle diameter, d, via the Stokes-Einstein equation ... 

Motion of a fiber in flow is described by Jeffery s model [3]. It is assumed that the fiber is a single rigid ellipsoidal partide suspended in a viscous fluid, the flow is a creeping flow of a Newtonian and incompressible fluid, and Brownian motion and inertia terms of the fiber are neglected. Jeffery s model was used for prediction of fiber orientation in the early period of injection molding CAE. Since it is, however, for dilute suspension, the model is replaced with the Folgar-Tucker model for concentrated suspension. 

Free-disperse systems comprise dilute emulsions, sols, and suspensions in which the participation of particles in thermal Brownian motion plays a dominant role over the cohesive forces between them. In these systems, we are particularly interested in the stability resisting the transition from the free-disperse state to the connected-disperse state via aggregation, flocculation, or sedimentation (Figure 4.2). 

Einstein [63-65] was the pioneer in the study of the viscosity of dilute suspensions of neutrally buoyant rigid spheres without Brownian motion in a Newtonian hquid. He proposed the following relationship between the relative viscosity of the suspension and the volume fraction of the suspended particles... 

The theory of the viscosity of dilute ellipsoidal suspensions without Brownian motion was developed by Jef y [109]. He observed tt at Einstein s equation (4.1) could be used to estimate the viscosity by appropriately defining the Einstein constant a. Jeffrey [109] has tabulated the values of ag for prolate and oblate spheroids. He found that ag depended on file ratio of the semi-axes of the ellipsoid of rotation. 

Consider a dilute dispersion of uniform spherical polymer particles as shown in Fig. 19. These spheres experience Brownian motion and therefore diffuse in all directions, causing collisions between the particles. If an adhesion bond forms between the surface molecules, then a collision has a chance of creating a doublet, that is. two particles adhering together at the single molecular bond which forms at the point of contact. If the adhesive bond is weaker than kT, then thermal collisions can break this bond in a period of time. The spheres will then separate and move apart. Thus there is a dynamic equilibrium between joining and separation, giving a certain number of doublets in the suspension at equilibrium. 

The hydrodynamic forces are always proportional to the viscosity of the medium. Therefore, suspension viscosities are scaled with the viscosity of the suspending medium, meaning that relative viscosities are used. As for dilute systems, the balance between Brownian motion and flow can be expressed by a Peclet number. Here the translational diffusivity D, has to be used, but that does not change the functionality (for spheres. Dr is proportional to D,). A dimensionless number is obtained by taking the ratio of the time scales for diffusion (D,) and convective motion (y). This is again a Peclet number ... 

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