Motion and Stokes Law

For a spherical particle of radius Vp, density Pp and mass mp, if the particle motion obeys Stokes law, then fp = nprp. From equation (3.1.4), the magnitude of the vertically downward force on this particle is... 

Eig. 5. Target efficiency of spheres, cylinders, and ribbons. The curves apply for conditions where Stokes law holds for the motion of the particle (see also N j ia Table 5). Langmuir and Blodgett have presented similar relationships for cases where Stokes law is not vaUd (149,150). Intercepts for ribbon or... 

Stokes diameter is defined as the diameter of a sphere having the same density and the same velocity as the particle in a fluid of the same density and viscosity settling under laminar flow conditions. Correction for deviation from Stokes law may be necessary at the large end of the size range. Sedimentation methods are limited to sizes above a [Lm due to the onset of thermal diffusion (Brownian motion) at smaller sizes. 

For the present we consider the case of very small frictional effects due to the beads i.e., the Stokes law radius a is small. We assume that the effects are so small that the motion of the surrounding medium is only very slightly disturbed by the movement of the polymer molecule relative to the medium. The frictional effects due to the polymer molecule are then comparatively easy to treat, for the velocity of the medium everywhere is approximately the same as though the polymer molecule were not present. The solvent streams through the molecule almost (but not entirely) unperturbed by it hence the term free-draining is appropriate for this case. The velocity difference we require in Eq. (11) is simply defined by the motion of the molecule on the one hand and the unperturbed flow of the medium on the other. 

Increasing the radius of the suspended particles, Brownian motion becomes less important and sedimentation becomes more dominant. These larger particles therefore settle gradually under gravitational forces. The basic equation describing the sedimentation of spherical, monodisperse particles in a suspension is Stokes law. It states that the velocity of sedimentation, v, can be calculated as follows ... 

For very dilute solutions, the motion of the ionic atmosphere in the direction of the coordinates can be represented by the movement of a sphere with a radius equal to the Debye length Lu = k 1 (see Eq. 1.3.15) through a medium of viscosity t] under the influence of an electric force ZieExy where Ex is the electric field strength and zf is the charge of the ion that the ionic atmosphere surrounds. Under these conditions, the velocity of the ionic atmosphere can be expressed in terms of the Stokes law (2.6.2) by the equation... 

The velocity of motion of the particles depends on their dimensions and shape, on the interaction (e.g. association) between the solvent molecules and finally on the interaction between particles of the dissolved substance and solvent molecules. Consider the simplest case, where the molecule of the dissolved substance is much larger than the solvent molecule, is spherical and the interaction between the solute molecules and the solvent is negligible. Then the motion of the particles of the solute can be considered as the motion of spherical particles with radius rf through a viscous medium with viscosity coefficient rj. The velocity v is then described by the Stokes law ... 

In a hydraulic jig, a mixture of two solids is separated into its components by subjecting an aqueous slurry of the material to a pulsating motion, and allowing the particles to settle for a series of short time intervals such that their terminal falling velocities are not attained. Materials of densities 1800 and 2500 kg/m3 whose particle size ranges from 0.3 mm to 3 mm diameter are to be separated. It may be assumed that the particles are approximately spherical and that Stokes Law is applicable. Calculate approximately the maximum time interval for which the particles may be allowed to settle so that no particle of the less dense material falls a greater distance than any particle of the denser material. The viscosity of water is 1 mN s/m2. 

The ions are regarded as rigid balls moving in a liquid bath. It is assumed that the macroscopic laws of motion in a viscous medium hold, and that the electrostatic interaction is determined by the theory of continuous dielectrics. This assumption implies that the moving particles are large compared to the molecular structure of the liquid. The most successful results of continuous theories can be found in any textbook of physical chemistry Stokes , law for viscous motion, Einstein s derivation of the dependence of viscosity on the concentration... 

Information on particle size may be obtained from the sedimentation of particles in dilute suspensions. The use of pipette techniques can be rather tedious and care is required to ensure that measurements are sufficiently precise. Instruments such as X-ray or photo-sedimentometers serve to automate this method in a non-intrusive manner. The attenuation of a narrow collimated beam of radiation passing horizontally through a sample of suspension is related to the mass of solid material in the path of the beam. This attenuation can be monitored at a fixed height in the suspension, or can be monitored as the beam is raised at a known rate. This latter procedure serves to reduce the time required to obtain sufficient data from which the particle size distribution may be calculated. This technique is limited to the analysis of particles whose settling behaviour follows Stokes law, as discussed in Section 3.3.4, and to conditions where any diffusive motion of particles is negligible. 

It may be noted that the equations of motion for the two directions (X and F) are coupled, with x and y appearing in each of the equations. General solutions are therefore not possible, except as will be seen later for motion in the Stokes law region. 

The critical nucleus of a new phase (Gibbs) is an activated complex (a transitory state) of a system. The motion of the system across the transitory state is the result of fluctuations and has the character of Brownian motion, in accordance with Kramers theory, and in contrast to the inertial motion in Eyring s theory of chemical reactions. The relationship between the rate (probability) of the direct and reverse processes—the growth and the decrease of the nucleus—is determined from the condition of steadiness of the equilibrium distribution, which leads to an equation of the Fourier-Fick type (heat conduction or diffusion) in a rod of variable cross-section or in a stream of variable velocity. The magnitude of the diffusion coefficient is established by comparison with the macroscopic kinetics of the change of nuclei, which does not consider fluctuations (cf. Einstein s application of Stokes law to diffusion). The steady rate of nucleus formation is calculated (the number of nuclei per cubic centimeter per second for a given supersaturation). For condensation of a vapor, the results do not differ from those of Volmer. 

For instance, molar conductivities cannot conceivably be negative. Given this, one may try to look for an ion with very small conductivity, or one may apply Stokes law for motion in a viscous fluid and assume a proportionality between reciprocal ionic radii and the ionic conductivity. It turns out that the Stokes radius of Na+... 

Settling of particles less than 0.5 pm is slowed by Brownian motion (random motion of small particles from thermal effects) in the water. Conversely, large sand-sized particles are not affected by viscous forces and typically generate a frontal pressure or wake as they sink. Thus, Stokes law can only apply to particles with Reynolds numbers (Re) that are less than unity. The particle Reynolds number according to Allen (1985) is defined as follows ... 

By accelerating deposition with a slowly rotating ultracentrifuge, Stokes law [Eq. (4.1)] is modified in uniform circular motion to an equilibrium distribution along the axis of rotation (0 to x), as a function of solute mass (mt). At higher centrifugal velocities, sedimentation succeeds the equilibrium distribution. Sedimentation equilibrium and sedimentation velocity provide a means to determine M. 

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