Hydrodynamic radius, layer thickness

For sufficiently large electrodes with a small vibration amplitude, aid < 1, a solution of the hydrodynamic problem is possible [58, 59]. As well as the periodic flow pattern, a steady secondary flow is induced as a consequence of the interaction of viscous and inertial effects in the boundary layer [13] as shown in Fig. 10.10. It is this flow which causes the enhancement of mass-transfer. The theory developed by Schlichting [13] and Jameson [58] applies when the time of oscillation, w l is small in comparison with the time taken for a species to diffuse across the hydrodynamic boundary layer (thickness SH= (v/a>)ln diffusion timescale 8h/D), i.e., when v/D t> 1. Re needs to be sufficiently high for the calculation to converge but sufficiently low such that the flow does not become turbulent. Experiment shows that, for large diameter wires (radius, r, — 1 cm), the condition is Re 2000. The solution Sh = 0.746Re1/2 Sc1/3(a/r)1/6, where Sh (the Sherwood number) = kmr/D and km is the mass-transfer coefficient,... 

Collision efficiency was calculated by the method proposed for the first time by Dukhin Derjaguin (1958). To calculate the integral in Eq. (10.25) it is necessary to know the distribution of the radial velocity of particles whose centre are located at a distance equal to their radius from the bubble surface. The latter is presented as superposition of the rate of particle sedimentation on a bubble surface and radial components of liquid velocity calculated for the position of particle centres. Such an approximation is possibly true for moderate Reynolds numbers until the boundary hydrodynamic layer arises. At a particle size commensurable with the hydrodynamic layer thickness, the differential of the radial liquid velocity at a distance equal to the particle diameter is a double liquid velocity which corresponds to the position of the particle centre. Such a situation radically differs from the situation at Reynolds numbers of the order of unity and less when the velocity in the hydrodynamic field of a bubble varies at a distance of the order ab ap. At a distance of the order of the particle diameter it varies by less than about 10%. Just for such conditions the identification of particle velocity and liquid local velocity was proposed and seems to be sufficiently exact. In situations of commensurability of the size of particle and hydrodynamic boundary layer thickness at strongly retarded surface such identification leads to an error and nothing is known about its magnitude. 

General hydrodynamic theory for liquid penetrant testing (PT) has been worked out in [1], Basic principles of the theory were described in details in [2,3], This theory enables, for example, to calculate the minimum crack s width that can be detected by prescribed product family (penetrant, excess penetrant remover and developer), when dry powder is used as the developer. One needs for that such characteristics as surface tension of penetrant a and some characteristics of developer s layer, thickness h, effective radius of pores and porosity TI. One more characteristic is the residual depth of defect s filling with penetrant before the application of a developer. The methods for experimental determination of these characteristics were worked out in [4]. 

Due to the extremely small size of the double layer (thickness ranges from 3 to 300 nm), the EOF originates close to or almost at the wall of the capillary. As a result, the EOF has a flat plug-like flow profile, compared to the parabolic profile of hydrodynamic flows (Figure 8). Flat profiles in capillaries are expected when the radius of the capillary is greater than seven times the double layer thickness and are favorable to avoid peak dispersion. Therefore, the flat profile of the EOF has a major contribution to the high separation efficiency of CE. 

Steric elution mode occurs when the particles are greater than 1 jm. Such large particles have negligible diffusion and they accumulate near the accumulation wall. The mean layer thickness is indeed directly proportional to D and inversely proportional to the field force F (see Equation 12.3). The condition is depicted in Figure 12.4b. The particles will reach the surface of the accumulation wall and stop. The particles of a given size will form a layer with the particle centers elevated by one radius above the wall the greater the particle dimension, the deeper the penetration into the center of the parabolic flow profile, and hence, larger particles will be displaced more rapidly by the channel flow than smaller ones. This behavior is exactly the inverse of the normal elution mode and it is referred to as inverted elution order. The above-described mechanism is, however, an oversimplified model since the particles most likely do not come into contact with the surface of the accumulation wall since, in proximity of the wall, other forces appear—of hydrodynamic nature, that is, related to the flow—which lift the particles and exert opposition to the particle s close approach to the wall. 

At the other extreme, if one wishes to use rotating microelectrodes, the minimum rotation rate may be dictated by the requirement that the diffusion layer thickness be small compared to the radius of the electrode. This becomes a limitation, however, only for r < 0.025 cm. Actually, the limitations imposed on the minimum radius of an RDE by the assumptions made in solving the hydrodynamic equations may be somewhat more severe, and under most circumstances one would be ill-advised to use an RDE having a radius of less than about 0.2 cm. 

The method of capillary Jlow measures the increase in resistance for solvent flow through a capillary (or a porous plug) due to an adsorbed polymer layer. This increase can be translated into a smaller effective capillary (or pore) radius through the Hagen-Polseuille law (1.6.4.18). The hydrodynamic radius d is supposed to be given by the difference between the "covered" and the "bare" radius. In such experiments the observed hydrodynamic thickness sometimes turns out to be flow-rate dependent. In such cases an extrapolation to zero flow rate needs to be carried out. 

A linear relationship exists between the ESA or CVP amplitude and the volume fraction of the suspended particles. At relatively high-volume fractions, hydrodynamic and electric double-layer interactions lead to a non-linear dependence of these two effects on volume fraction. Generally, non-linear behavior can be expected when the electric double-layer thickness is comparable to the interparticle spacing. In most aqueous systems, where the electric double layer is thin relative to the particle radius, the electro-acoustic signal will remain linear with respect to volume fraction up to 10% by volume. At volume-fractions that are even higher, particle-particle interactions lead to a reduction in the dynamic mobility. 

Relation (3.2.13) is valid in the region of laminar flow past the disk the laminar regime occurs until Re 104 to 105, depending on the roughness of the surface. For low Reynolds numbers (Re < 10), this relation is invalid, because the thickness of the hydrodynamic boundary layer becomes comparable with the disk radius and the boundary effects on the hydrodynamic flow and mass transfer become stronger. 

As a last remark on polymer adsorption, let s consider Fig. 6.3. If a polymer with a hydrodynamic radius Rg is present at low concentration, its configuration at the interface will be relatively flat with trains on the interface. After more polymer is added, the irreversible adsorption on the surface will produce an adsorbed layer thickness on the... 

The equilibrium thickness of the film greatly exceeded twice the ellipsomet-ric or hydrodynamic thickness of one adsorbed layer. This was attributed to the steric repulsion associated with isolated tails, in agreement with subsequent predictions of Scheutjens and Fleer (1979 1980 1982). Such isolated tails could be essentially free draining, contributing little to the hydrodynamic radius and being essentially invisible to ellipsometry. 

A major experimental complication for adsorption at a soUd/liquid interface is that such interfaces can be very variable due to differences in cleaning procedures. Additionally, because solid surfaces are rigid, the attainment of an equilibrium adsorbed conformation may be difficult. Furthermore, adsorption may be located in regions of high specificity and thus the experimental observation may correspond to a thick layer (i.e. the hydrodynamic radius) whereas the adsorbed amount suggests a much flatter conformation. 

The Dynamic Light Scattering (DLS) technique was used to measure radii of the PS latex spheres with and without adsorbed polymer brushes. We could then deduce the polymer brush hydrodynamic layer thickness by taking the difference of the radii. DLS measures the intensity autocorrelation as a function of delay time, which gives information on the diffusion constant of particles in a dilute solution. The translational diffusion coefficient, D, is related to the solution temperature T, particle radius r, and solvent viscosity ri by the Stokes-Einstein relation ... 

Several methods may be used to determine the adsorbed layer thickness, 8. Most of the methods depend on measuring the hydrodynamic radius of the particles with and without the adsorbed polymer layer. For example, one may measure the relative viscosity, of a dispersion with an adsorbed polymer layer. Assuming that the particles behave as hard spheres (when 8 is small compared with the particle radius R) of noninteracting units (low volume fraction of the disperse phase), can be related to the effective volume fraction, [Pg.355]

By carrying out the measurements in the presence and absence of polymer layers, one can obtain 8. In the presence of an adsorbed polymer layer, the hydrodynamic radius is the sum of the core radius and the adsorbed layer thickness, whereas in the absence of the polymer layer, 7 is simply the core radius. Again, as with the viscosity technique, this method cannot be directly applied to emulsions, which are polydisperse and with a radius that is large for significant Brownian motion. 

Colloidal suspensions can be classified as soft sphere systems because the repulsive intoactions occur at some characteristic distance from the particle surface. For electrostatic and stoic stabilization, this distance is the Debye length (1/ K) and the thickness of the adsorbed polymer layer, respectively. For stoically stabilized suspensions, the adsorbed polymer layer leads to an increase in the hydrodynamic radius of the particle. When the adsorbed layer is densely packed, the principles described above for hard sphere systems are applicable, provided that the volume fraction of particles/is replaced by an effective volume fraction /gy given by... 

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