Hydrodynamic diffusion layer, thickness

Diffusion in a convective flow is called convective diffusion. The layer within which diffnsional transport is effective (the diffnsion iayer) does not coincide with the hydrodynamic bonndary layer. It is an important theoretical problem to calcnlate the diffnsion-layer thickness 5. Since the transition from convection to diffnsion is gradnal, the concept of diffusion-layer thickness is somewhat vagne. In practice, this thickness is defined so that Acjl8 = (dCj/ff) Q. This calcniated distance 5 (or the valne of k ) can then be used to And the relation between cnrrent density and concentration difference. 

Rigorous calibration is a requirement for the use of the side-by-side membrane diffusion cell for its intended purpose. The diffusion layer thickness, h, is dependent on hydrodynamic conditions, the system geometry, the spatial configuration of the stirrer apparatus relative to the plane of diffusion, the viscosity of the medium, and temperature. Failure to understand the effects of these factors on the mass transport rate confounds the interpretation of the data resulting from the mass transport experiments. 

The hydrodynamic boundary layer has an inner part where the vertical velocity increases to a maximum determined by a balance of viscous and buoyancy forces. In fluids of high Schmidt number, the concentration diffusion layer thickness is of the same order of magnitude as this inner part of the hydrodynamic boundary layer. In the outer part of the hydrodynamic boundary layer, where the vertical velocity decays, the buoyancy force is unimportant. The profile of the vertical velocity component near the electrode can be shown to be parabolic. 

Figure 5. Exact (numerical solution, continuous line) and linearised (equation (24), dotted line) velocity profile (i.e. vy of the fluid at different distances x from the surface) at y = 10-5 m in the case of laminar flow parallel to an active plane (Section 4.1). Parameters Dt = 10 9m2 s-1, v = 10-3ms-1, and v = 10-6m2s-1. The hydrodynamic boundary layer thickness (<50 = 5 x 10 4 m), equation (26), where 99% of v is reached is shown with a horizontal double arrow line. For comparison, the normalised concentration profile of species i, ct/ithe linear profile of the diffusion layer approach (continuous line) and its thickness (<5, = 3 x 10 5m, equation (34)) have been added. Notice that the linearisation of the exact velocity profile requires that <5,

Figure 8. Variation of the hydrodynamic boundary layer thickness (So, equation (26), continuous line), the diffusion layer thickness (<5,-, equation (34), dotted line) and the ensuing local flux (/, equation (32), dashed line) with respect to the distance from the leading edge (y) in the case of laminar flow parallel to an active plane (the surface is a sink for species i). Parameters /), = 10-9nrs, v= 10 3ms, c = lmolm-3, and v — 10-6 m2 s 1. Notice that c5, o (as required for the derivation of the flux equation (32)), and that the flux decreases when <5, increases...

If instead of semi-infinite diffusion, some distance (5m acts as an effective diffusion layer thickness (Nernst layer approximation), then a modified expression of equation (63) applies where ro is substituted by 1 / (1 /Vo + 1 /<5m ) (see equation (38) above). For some hydrodynamic regimes, which for simplicity, are not dealt with here, the diffusion coefficient might need to be powered to some exponent [57,58],... 

Buccal dosage forms can be of the reservoir or the matrix type. Formulations of the reservoir type are surrounded by a polymeric membrane, which controls the release rate. Reservoir systems present a constant release profile provided (1) that the polymeric membrane is rate limiting, and (2) that an excess amoimt of drug is present in the reservoir. Condition (1) may be achieved with a thicker membrane (i.e., rate controlling) and lower diffusivity in which case the rate of drug release is directly proportional to the polymer solubility and membrane diffusivity, and inversely proportional to membrane thickness. Condition (2) may be achieved, if the intrinsic thermodynamic activity of the drug is very low and the device has a thick hydrodynamic diffusion layer. In this case the release rate of the drug is directly proportional to solution solubility and solution diffusivity, and inversely proportional to the thickness of the hydrodynamic diffusion layer. 

The parameters that determine the release rate of a drug from a delivery device include polymer solubility, polymer diffusivity, thickness of the polymer diffiisional path, and the aqueous solubility, partition coefficient, and aqueous diffusivity of the drug. Finally, the thickness of the hydrodynamic diffusion layer, the amount of drug loaded into the matrix, and the smface area of the device all affect the release rate. 

Hydrodynamic electrodes permit the control of the diffusion layer thickness by imposing convection. This thickess can also be modulated. Implicit functions link the current, potential and convection modulation. For the rotating disc electrode... 

The last of these is the impedance which has been considered throughout this chapter. We now consider forced convection. For low frequencies the diffusion layer thickness due to the a.c. perturbation is similar to that of the d.c. diffusion layer in these cases convection effects will be apparent in the impedance expressions. For the rotating disc electrode these frequencies are lower than 40 Hz33. For higher frequencies where the two diffusion layers are of quite different thicknesses, the advantage of hydrodynamic electrodes is that transport is well defined with time, as occurs with linear sweep voltammetry. 

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,... 

The diffusion layer thickness is controlled by the hydrodynamics (fluid flow). Although more details on mass transfer effects are discussed in Chapter 5, it is worthwhile to point out here that the diffusion-limited current density is independent of the substrate material. 

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. 

Fig. 3 Release of drug from various shapes of pol5mer membrane permeation-controlled drug-delivery systems (A) sphere-type, (B) cylinder-type, and (C) sheet-type. In (D), the drug concentration gradients across the rate-controlling polymeric membrane and hydrodynamic diffusion layer exist in series. Both the polymer membrane, which is either porous or non-porous, and the diffusion layer have a controlled thickness and h, respectively).

Helmholtz layer contains the second water molecule layer. From the Helmholtz double layer toward the bulk electrolyte are the diffusion layer and the hydrodynamic layer. In the diffusion layer, the concentration of species changes from that of the bulk electrolyte to that of the electrode surface. The diffusion layer does not move, but its thickness will decrease with increasing bulk electrolyte flow rate to allow higher reaction rates. The diffusion layer thickness is inversely proportional to the square root of the flow rate. The hydrodynamic layer or Prandtl layer has the same composition as the bulk electrolyte, but the flow of the electrolyte decreases from that of the bulk electrolyte to the stationary diffusion layer. 

In short, the accurate selection of experimental conditions can considerably reduce the incidence of these problems on the results of measurements. In particular, the type of electrode, the deposition potential and the hydrodynamic conditions at the electrode surface (which influences the diffusion layer thickness) are the most important. 

For HDI applications, the dimension of surface features such as lines or vias are approximately equivalent (25 to 100pm) to the diffusion layer thickness (as shown in Figure 6). However, in this case the width of the HDI is also on the order of the diffusion layer thickness and the contour of the lines or vias are inaccessible to the diffusion layer. We designate this a special case - a hydrodynamically inaccessible mlcroproflle . Consequently, the optimum MREF waveform for HDI plating should consist of a short... 

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