Steady-state diffusion layer, thickness

In convective systems, the situation is different, because the thickness of the nonsteady-state diffusion layer, is limited by that of the steady-state diffusion layer 5, which depends only on the prevailing hydrodynamic conditions (see Sect. 4.3). As the non-steady-state diffusion layer grows, convection contributes increasingly to the transport of the reacting species. Eventually, when becomes comparable to the steady-state diffusion layer thickness 5 the concentration profile no longer changes with time. 

Mass transport phenomena usually are effective on distance scales much larger than cell wall and double layer dimensions. Thicknesses of steady-state diffusion layers in mildly stirred systems are in the order of 10 5m. Thus one may generally adopt a picture where the local interphasial properties define the boundary conditions, while the actual mass transfer processes take place on a much larger spatial scale. 

The boundary layer thickness 5. For convective crystal dissolution, the steady-state boundary layer thickness increases slowly with increasing viscosity and decreasing density difference between the crystal and the fluid. It does not depend strongly on the crystal size. Typical boundary layer thickness is 10 to 100/rm. For diffusive crystal dissolution, the boundary layer thickness is proportional to square root of time. 

Figure 5.8 Concentration profiles near an electrode after the applieation of a potential step. The steady state is reached when the thickness of the non-steady-state diffusion layer, 5p(t), becomes equal to 5, the thickness of the steady-state diffusion layer.

This current is time-dependent too, but in the convective diffusion case, the current rapidly (within a few ms or so) reaches a stationary value. The speed with which this current plateau is reached arises from the establishment of a well-defined steady-state diffusion layer. The thickness d of this diffusion layer is given approximately by... 

It is worthwhile to note that Equations (A.22) and (A. 23) have genuine steady state solutions without the need to introduce a boundary layer. This is because the chemical reaction (A. 19) causes the formation of a steady state kinetic layer. Only within this boundary layer is R present in solution, and the concentration of 0 perturbed from its initial concentration. The thickness of the kinetic layer depends on k the larger k the thinner the layer. Certainly for high values of k, the kinetic layer will lie well within the normal steady state diffusion layer defined by natural convection. The time required to reach the steady state (form the kinetic layer) also depends inversely on k. 

If the electrode has a radius of the order 10 run, the diffusion layer is concentrated within a few tens of nanometres away from the electrode surface. This is in contrast to a microelectrode where the diffusion layer will be microns thick, or a macroelectrode where the steady-state diffusion layer is so large that the steady-state limit is not achieved on an experimental timescale. Instead, the diffusion layer continues to grow at a rate proportional to s/, throughout the course of an experiment. 

On the other hand, when the thickness of the diffusion layer becomes sufficiently thicker than the radius of the electrode, the diffusion layer spreads hemispherically around the electrode and the mode of diffusion becomes radiative. As a result, as a steady-state diffusion layer of the solute concentration that is inversely proportional to the distance from the electrode surface is formed, the current also becomes steady state. This steady-state current value (limiting current value) is expressed as follows ... 

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

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,... 

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... 

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. 

Here F is the Faraday constant C = concentration of dissolved O2, in air-saturated water C = 2.7 x 10-7 mol cm 3 (C will be appreciably less in relatively concentrated heated solutions) the diffusion coefficient D = 2 x 10-5 cm2/s t is the time (s) r is the radius (cm). Figure 16 shows various plots of zm(02) vs. log t for various values of the microdisk electrode radius r. For large values of r, the transport of O2 to the surface follows a linear type of profile for finite times in the absence of stirring. In the case of small values of r, however, steady-state type diffusion conditions apply at shorter times due to the nonplanar nature of the diffusion process involved. Thus, the partial current density for O2 reduction in electroless deposition will tend to be more governed by kinetic factors at small features, while it will tend to be determined by the diffusion layer thickness in the case of large features. 

We consider, then, two media (1 for the cell-wall layer and 2 for the solution medium) where the diffusion coefficients of species i are /),yi and 2 (see Figure 3). For the planar case, pure semi-infinite diffusion cannot sustain a steady-state, so we consider that the bulk conditions of species i are restored at a certain distance <5,- (diffusion layer thickness) from the surface where c, = 0 [28,45], so that a steady-state is possible. Using just the diffusive term in the Nernst-Planck equation (10), it can be seen that the flux at any surface is ... 

Equation (14) also shows that for microorganisms with radii that are less than a few microns with a typical diffusion layer thickness > 10 pm, radial diffusion should predominate over linear diffusion [46], Under steady-state conditions, the area integrated cellular flux (mols-1), Q, for a small, spherical cell of surface = 4tt q, is given by ... 

One case is where the ions are traveling to the electrode by a process of diffusion. Then the steady-state diffusion problem can be looked at from the diffusion-layer point of view (Section 7.9). The variation of concentration with distance can be approximated to a linear variation, and the linear concentration gradient can be considered to occur over an effective distance of 8, die diffusion-layer thickness. Then the diffusion current is given by (Section 7.9.10). 

The steady state will be approached when the diffusion-layer thickness becomes much larger than the radius of the disk, roughly speaking, when (Dt)1/2/r 1, where D is the diffusion coefficient, t is the duration of the experiment, and r is the electrode radius. In the opposite extreme, where (Dt)l/2/ r 1, a normal peak-shaped response will be obtained that is characteristic of planar diffusion with little contribution of diffusion from the solution at the periphery of the disk. 

Reduction of the solution temperature allows transition from steady-state to peak-shaped response simply by way of the marked diminution of D at low temperatures. Figure 16.5 shows slow-scan cyclic voltammograms obtained at two microdisk electrodes as a function of solution temperature. Between -120 and -140°C there is a particularly clear transition for the 25-pm-diameter electrode as the diffusion-layer thickness becomes less than the disk radius. Also illustrated here is the immense decrease in the limiting currents that is seen over this range of temperatures due to the 100-fold decrease in D. 

For water-insoluble drugs, dissolution-controlled systems are an obvious choice for achieving sustained-release because of theirslow dissolution rate characteristics. Theoretically, the dissolution process at steady state can be described by the Noyes-Whitney equation as shown in Equation 22.7. The rate of dissolution of a compound is a function of surface area, saturation solubility, and diffusion layer thickness. Therefore, the rate of drug release can be manipulated by changing these parameter. 

At low frequencies, it approaches the macroelectrode behaviour. Indeed the steady-state solution shows that the microelectrode can be considered as a simple extension of the macroelectrode. In this quasi steady-state regime, the frequency is small enough to allow the concentration wave to propagate over the whole diffusion layer thickness. 

In these electrode processes, the use of macroelectrodes is recommended when the homogeneous kinetics is slow in order to achieve a commitment between the diffusive and chemical rates. When the chemical kinetics is very fast with respect to the mass transport and macroelectrodes are employed, the electrochemical response is insensitive to the homogeneous kinetics of the chemical reactions—except for first-order catalytic reactions and irreversible chemical reactions follow up the electron transfer—because the reaction layer becomes negligible compared with the diffusion layer. Under the above conditions, the equilibria behave as fully labile and it can be supposed that they are maintained at any point in the solution at any time and at any applied potential pulse. This means an independent of time (stationary) response cannot be obtained at planar electrodes except in the case of a first-order catalytic mechanism. Under these conditions, the use of microelectrodes is recommended to determine large rate constants. However, there is a range of microelectrode radii with which a kinetic-dependent stationary response is obtained beyond the upper limit, a transient response is recorded, whereas beyond the lower limit, the steady-state response is insensitive to the chemical kinetics because the kinetic contribution is masked by the diffusion mass transport. In the case of spherical microelectrodes, the lower limit corresponds to the situation where the reaction layer thickness does not exceed 80 % of the diffusion layer thickness. 

It is intuitively obvious that at longer times, when the diffusion layer thickness far exceeds the radius of a disk or hemisphere (for small P), or of the width of a band or the hemicylinder, currents at flat electrodes (disk, band) must resemble those at round electrodes (hemisphere, hemicylinder). Some relations between these have been established. Oldham found [427] that the steady-state currents at a microdisk and microhemisphere are the same if their diameters along the surfaces are the same. This means that for a microdisk of radius a, the steady-state current is the same as that at a microhemisphere of radius 2a/ir. At band or hemicyclindrical electrodes, there is... 

If one considers a steady-state diffusion across a layer of thickness Ax,... 

Diffusion time (diffusion time constant) — This parameter appears in numerous problems of - diffusion, diffusion-migration, or convective diffusion (- diffusion, subentry -> convective diffusion) of an electroactive species inside solution or a solid phase and means a characteristic time interval for the process to approach an equilibrium or a steady state after a perturbation, e.g., a stepwise change of the electrode potential. For onedimensional transport across a uniform layer of thickness L the diffusion time constant, iq, is of the order of L2/D (D, -> diffusion coefficient of the rate-determining species). For spherical diffusion (inside a spherical volume or in the solution to the surface of a spherical electrode) r spherical diffusion). The same expression is valid for hemispherical diffusion in a half-space (occupied by a solution or another conducting medium) to the surface of a disk electrode, R being the disk radius (-> diffusion, subentry -> hemispherical diffusion). For the relaxation of the concentration profile after an electrical perturbation (e.g., a potential step) Tj = L /D LD being - diffusion layer thickness in steady-state conditions. All these expressions can be derived from the qualitative estimate of the thickness of the nonstationary layer... 

The current-potential relationship of the totally - irreversible electrode reaction Ox + ne - Red in the techniques mentioned above is I = IiKexp(-af)/ (1+ Kexp(-asteady-state voltammetry, a. is a - transfer coefficient, ks is -> standard rate constant, t is a drop life-time, S is a -> diffusion layer thickness, and

logarithmic analysis of this wave is also a straight line E = Eff + 2.303 x (RT/anF) logzc + 2.303 x (RT/anF) log [(fi, - I) /I -The slope of this line is 0.059/a V. It can be used for the determination of transfer coefficients, if the number of electrons is known. The half-wave potential depends on the drop life-time, or the rotation rate, or the microelectrode radius, and this relationship can be used for the determination of the standard rate constant, if the formal potential is known. 

Unlike macroelectrodes which operate under transient, semi-infinite linear diffusion conditions at all times, UMEs can operate in three diffusion regimes as shown in the Figure for an inlaid disk UME following a potential step to a diffusion-limited potential (i.e., the Cottrell experiment). At short times, where the diffusion-layer thickness is small compared to the diameter of the inlaid disc (left), the current follows the - Cottrell equation and semi-infinite linear diffusion applies. At long times, where the diffusion-layer thickness is large compared to the diameter of the inlaid disk (right), hemispherical diffusion dominates and the current approaches a steady-state value. 

If (95) is used to estimate values for the diffusion layer thickness obtained for sonovoltammetry in acetonitrile, values of the order of a few micrometres are obtained - much smaller than encountered in conventional voltammetry under silent (stationary) conditions unless either potential scan rates of hundreds of mVs, or more, are employed or alternatively steady-state measurements are made with microelectrodes with one or more dimensions of the micrometre scale (Compton et al., 1996b). 

Under the present conditions of negligible diffusion, flame propagation in the solid is associated with an excess enthalpy per unit area given by PsCps(T — Tq) dx just ahead of the reaction sheet. This excess provides a local reservoir of heated reactant in which a flame may propagate at an increased velocity. If = KI(Ps ps) denotes the thermal diffusivity of the solid, then for the steady-state solution, the thickness of the heated layer of reactant is on the order of where is the steady-state flame... 

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