Diffusion effective thickness

The quantity 1 /k is thus the distance at which the potential has reached the 1 je fraction of its value at the surface and coincides with the center of action of the space charge. The plane at a = l//c is therefore taken as the effective thickness of the diffuse double layer. As an example, 1/x = 30 A in the case of 0.01 M uni-univalent electrolyte at 25°C. 

The rate of dissolving of a solid is determined by the rate of diffusion through a boundary layer of solution. Derive the equation for the net rate of dissolving. Take Co to be the saturation concentration and rf to be the effective thickness of the diffusion layer denote diffusion coefficient by . 

Cg = the concentration of the saturated solution in contact with the particles, D = a diffusion coefficient (approximated by the Hquid-phase diffusivity), M = the mass of solute transferred in time t, and S = the effective thickness of the liquid film surrounding the particles. For a batch process where the total volume H of solution is assumed to remain constant, dM = V dc and... 

The concept of surface concentration Cg j requires closer definition. At the surface itself the ionic concentrations will change not only as a result of the reaction but also because of the electric double layer present at the surface. Surface concentration is understood to be the concentration at a distance from the surface small compared to diffusion-layer thickness, yet so large that the effects of the EDL are no fonger felt. This condition usually is met at points about 1 nm from the surface. 

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. 

It follows that convection of the hqnid has a twofold influence It levels the concentrations in the bnlk liquid, and it influences the diffusional transport by governing the diffusion-layer thickness. Shght convection is sufficient for the first effect, but the second effect is related in a qnantitative way to the convective flow velocity The higher this velocity is, the thinner will be the diffusion layer and the larger the concentration gradients and diffusional fluxes. 

Let CO be the angular velocity of rotation this is equal to Inf where/is the disk frequency or number of revolutions per second. The distance r of any point from the center of the disk is identical with the distance from the flow stagnation point. The hnear velocity of any point on the electrode is cor. We see when substituting these quantities into Eq. (4.34) that the effects of the changes in distance and hnear vefocity mutuaUy cancel, so that the resulting diffusion-layer thickness is independent of distance. 

It follows from this equation that the effective diffusion-layer thickness can be described as... 

SPAN module. It was mentioned at the beginning that the special polyacrylonitrile fibers of SPAN have a wall thickness of 30 gm, which is considerably thicker than the 8 gm wall thickness of the SMC modules [19]. As a consequence, the presence of stronger capillary effects from the special porous fiber material of the SPAN module would be a reasonable conclusion. Furthermore, the texture of the special polyacrylonitrile fibers is expected to have better surface properties, supporting the permeation of molecules as compared with synthetically modified cellulose. In conclusion, both convection and diffusion effectively contribute to the filtration efficiency in a SPAN module, whereas for the SMC membrane, diffusion is the driving force for molecular exchange, the efficiency of which is also considerable and benefits from the large surface-to-volume ratio. 

Figure 5.37 depicts the stationary distribution of the electroactive substance (the reaction layer) for kc—> oo. The thickness of the reaction layer is defined in an analogous way as the effective diffusion layer thickness (Fig. 2.12). It equals the distance [i of the intersection of the tangent drawn to the concentration curve in the point x = 0 with the line c = cA/K,...

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. 

Daq = aqueous diffusion coefficient, cm2/sec hAEL = effective thickness of the ABL, cm... 

Free convection at vertical cylinders differs from that along vertical plates because of the influence of curvature, but the effect is important only if the diffusion layer thickness is appreciable compared with the diameter. This can be expressed as... 

The metal ion in electroless solutions may be significantly complexed as discussed earlier. Not all of the metal ion species in solution will be active for electroless deposition, possibly only the uncomplexed, or aquo-ions hexaquo in the case of Ni2+, and perhaps the ML or M2L2 type complexes. Hence, the concentration of active metal ions may be much less than the overall concentration of metal ions. This raises the possibility that diffusion of metal ions active for the reduction reaction could be a significant factor in the electroless reaction in cases where the patterned elements undergoing deposition are smaller than the linear, or planar, diffusion layer thickness of these ions. In such instances, due to nonlinear diffusion, there is more efficient mass transport of metal ion to the smaller features than to large area (relative to the diffusion layer thickness) features. Thus, neglecting for the moment the opposite effects of additives and dissolved 02, the deposit thickness will tend to be greater on the smaller features, and deposit composition may be nonuniform in the case of alloy deposition. 

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],... 

Figure 10.13. Variations in the effective thickness of diffusion layer 8 over an electrode with a microprofile.

The two major causes of uneven current distribution are diffusion and ohmic resistance. Nonuniformity due to diffusion originates from variations in the effective thickness of the diffusion layer 8 over the electrode surface as shown in Figure 10.13. It is seen that 8 is larger at recesses than at peaks. Thus, if the mass-transport process controls the rate of deposition, the current density at peaks ip is larger than that at recesses since the rate of mass transport by convective diffusion is given by... 

It should be understood that even for micro surface features the potential is uniform and the ohmic resistance through the bath to peaks and valleys is about the same. Also, electrode potential against SCE will be uniform. What is different is that over micro patterns the boundary of the diffusion layer does not quite follow the pattern contour (Fig. 12.3). Rather, it thus lies farther from depth or vias than from bump peaks. The effective thickness, 8N, of the diffusion layer shows greater variations. This variation of 8N over a micro profile therefore produces a variation in the amount of concentration polarization locally. Since the potential is virtually uniform, differences in the local rate of metal deposition result if it is controlled by the diffusion rate of either the depositing ions or the inhibiting addition (leveling) agents. 

The diffusion flux J, in mol/cm, is proportional to the concentration gradient and inversely proportional to the diffusion layer s effective thickness 5 (also called the Nemst thickness). The proportionality constant D is the diffusion constant hence,... 

As can be expected, with bath agitation the effective thickness 8 will diminish hence the diffusion rate will increase. Typical values of 8 are about 0.2 mm without agitation, and with a rotating-disk electrode it can be made as small as 0.02 mm. Of course, this thickness will vary from species to species. 

While Eq. (10) is mathematically similar to Eq. (9), it is dramatically different when H is significantly different from unity. It is evident from Eqs. (9) and (10) that k em does depend on the diffusion coefficient of the solute in the solvent within the pores and effective thickness, S, which is a membrane property. It does not depend on flow properties of the adjacent solutions. The flow properties affect only and kout-... 

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