Diffusion layer thicknesses

Tafel slope (Napieran loop) transfer coefficient diffusion layer thickness dielectric constant, relative electric field constant = 8.85 x 10 F cm overvoltage, polarization ohmic voltage drop, resistance polarization specific conductance, conductivity electrochemical potential of material X,... 

In case of a steady state with a constant diffusion layer thickness for all ions it follows [for simplicity the surfix (aq) is ommitted] ... 

Coagulation takes place over several minutes in a rapid mix tank and encompasses several mechanisms such as reducing diffuse layer thickness and charge neutralization. 

That is, the current decreases in proportion to the square root of time, with (nDQt)l/2 corresponding to the diffusion layer thickness. 

C o(0, /)/CR(0, /) = 1/10,n = 1). The decrease in Co(0, t) is coupled with an increase in the diffusion layer thickness, which dominates the change in the slope after Co(0, /) approaches zero. The net result is a peak-shaped voltammogram. Such... 

The resulting voltammogram thus has a sigmoidal (wave) shape. If the stirring rate (U) is increased, the diffusion layer thickness becomes thinner, according to... 

As the reaction proceeds, the diffusion layer extends into the bulk of the solution outside the double layer. When the diffusion-layer thickness increases much more than the autocorrelation distance of the asymmetrical nonequilibrium fluctuation, a steady state emerges. In contrast to Eq. (103), in this case the following condition holds,... 

The micro structured platelets, hold in a non-conducting housing, were realized by etching of metal foils and laser cutting techniques [69]. Owing to the small Nemst diffusion layer thickness, fast mass transfer between the electrodes is achievable. The electrode surface area normalized by cell volume amounts to 40 000 m m". This value clearly exceeds the specific surface areas of conventional mono- and bipolar cells of 10-100 m m. ... 

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 Eqs. (4.37) and (4.38) that the diffusion-layer thickness will increase without limits and the diffusion flux will decrease to zero when the electrolyte is not stirred (v = 0) or the electrode not rotated (co = 0). This implies that a steady electric cnrrent cannot flow in such cells. But this conclusion is at variance with the experimental data. 

Natural convection depends strongly on cell geometry. No convection can arise in capillaries or in the thin liquid layers found in narrow gaps between electrodes. The rates of natural convective flows and the associated diffusion-layer thicknesses depend on numerous factors and cannot be calculated in a general form. Very rough estimates show that the diffusion-layer thickness under a variety of conditions may be between 100 and 500 pm. 

The concentration change near the electrode surface gradually reaches solution layers farther away from the electrode. In these layer the rate of concentration change is the same as at the electrode, but there is a time lag. The concentration distributions found at different times are shown in Fig. 11.3. The diffusion-layer thickness 5. gradually increases with time it follows from Eqs. (11.3) and (11.6) that... 

From Eq. (11.13) we obtain an expression for the eflective transient diffusion-layer thickness ... 

The transient process continues until thickness 5. has attained the value of diffusion-layer thickness corresponding to the applicable experimental conditions. Hence, we obtain for the duration of this transitory process,... 

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

We see that the expression for the current consists of two terms. The first term depends on time and coincides completely with Eq. (11.14) for transient diffusion to a flat electrode. The second term is time invariant. The first term is predominant initially, at short times t, where diffusion follows the same laws as for a flat electrode. During this period the diffusion-layer thickness is still small compared to radius a. At longer times t the first term decreases and the relative importance of the current given by the second term increases. At very long times t, the current tends not to zero as in the case of linear diffusion without stirring (when is large) but to a constant value. For the characteristic time required to attain this steady state (i.e., the time when the second term becomes equal to the first), we can write... 

In a reversible process that occurs under diffusion control, the time-dependent drop of the faradaic current is due to the gradual increase in diffusion-layer thickness. According to Eq. (11.14), we have, for reactants. 

The concentration asymptotically approaches the value Cq with increasing distance X (i.e., the reaction zone has no distinct boundary). Conventionally, thickness 5,. is defined just like the diffusion-layer thickness 5 [i.e., by the condition that Cq/5,. = (dcldx) o for zero surface concentration. Using Eq. (13.41), we find that... 

Most successful is a rotating Pt wire microelectrode as illustrated in Fig. 3.75 as a consequence of the rotation, which should be of a constant speed, the steady state is quickly attained and the diffusion layer thickness appreciably reduced, thus raising the limiting current (proportional to the rotation speed to the 1/3 power above 200 rpm140 and 15-20-fold in comparison with a dme) and as a result considerably improving the sensitivity of the amperometric- titration. 

Equation (1) predicts that the rate of release can be constant only if the following parameters are constant (a) surface area, (b) diffusion coefficient, (c) diffusion layer thickness, and (d) concentration difference. These parameters, however, are not easily maintained constant, especially surface area. For spherical particles, the change in surface area can be related to the weight of the particle that is, under the assumption of sink conditions, Eq. (1) can be rewritten as the cube-root dissolution equation ... 

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,...

In Eqs. (40)-(42), cM and cb2 are experimentally measurable and the aqueous diffusion layer thickness can be estimated theoretically. Therefore, the only unknowns are the solute concentrations at the interfaces, csl and cs2. Their estimation is shown below. 

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. 

In Eq. (41), the concentration gradient is expressed as a difference between the surface concentration, Cm, and the bulk concentration, CHS), divided by the diffusion layer thickness, 8. If sink dissolution conditions are assumed (Cb 0) and the solid has a uniform density (p = m/V), then Eq. (42) can be derived. 

Fig. 2. Concentration profile of the reacting ion at an electrode. The so-called Nernst diffusion layer thickness is indicated by <5n. ...

---