Adsorption under diffusion control

Adsorption under Diffusion Control Small Initial Perturbation Large Initial Perturbation Generalization for Ionic Surfactants Adsorption under Barrier Control... 

The comparison between Equations 5.42 and 5.43 shows that the Gibbs adsorption equation can be expressed either in terms of a, and a or in terms of o, E , and a. Note that Equations 5.42 and 5.44 are valid under equilibrium conditions, whereas Equation 5.43 can be used also for the description of dynamic surface tension (Section 5.2.2) in the case of surfactant adsorption under diffusion control, assuming local equilibrium between adsorptions E and subsurface concentrations of the respective species. 

The above eqnations show that in the case of adsorption under diffusion control the long-time asymptotics can be expressed in the form ... 

The formal transition in Equation 5.91 from mixed to diffusion control of adsorption is not trivial and demands application of scaling and asymptotic expansions. The criterion for occurrence of adsorption under diffusion control (presence of equilibrium between subsurface and surface) is... 

Valette-Hamelin approach,67 and other similar methods 24,63,74,218,225 (2) mass transfer under diffusion control with an assumption of homogeneous current distribution73 226 (3) adsorption of radioactive organic compounds or of H, O, or metal monolayers73,142,227 231 (4) voltammetry232,233 and (5) microscopy [optical, electron, scanning tunneling microscopy (STM), and atomic force microscopy (AFM)]234"236 as well as a number of ex situ methods.237 246... 

Formation and stripping of a cobalt adlayer on/from a polycrystalline Au electrode have been studied [469] applying electrochemical methods under underpotential conditions. The kinetics of deposition fitted a model of a simultaneous adsorption and diffusion-controlled two-dimensional instantaneous nucleation of cobalt on the electrode surface. 

The overall rate of surfactant adsorption is controlled by the slowest stage. If it is stage (1), we deal with diffusion control, whereas if stage (2) is slower, the adsorption occurs under barrier (kinetic) control. The next four subsections are dedicated to processes under diffusion control (which are the most frequently observed), whereas in Section 5.2.2.5 we consider adsorption under barrier control. 

Equation 5.70 is often used as a test to verify whether the adsorption process is under diffusion control data for o(0 are plotted vs. 1/Vt and we check if the plot comphes with a straight line moreover, the intercept of the line gives o. We recall that Equations 5.69 and 5.70 are valid in the case of a small initial perturbation alternative asymptotic expressions for the case of large initial perturbation are considered in the next subsection. 

Equation 5.88 predicts that the perturbation of surface tension, Ao(f) = o(f) — o,- relaxes exponentially. This is an important difference with the cases of adsorption under diffusion and electrodiffusion control, for which Ao(f) 1/x/f see Equations 5.70, 5.76, and 5.78. Thus, a test whether or not the adsorption occurs under purely barrier control is to plot data for ln[Aa(f)] vs. t and to check if the plot complies with a straight line. 

Equation 5.93 reflects the fact that in the diffusion regime the surface is always assumed to be equilibrated with the subsurface. In particular, if E, = 0, then we must have Cj = 0. In contrast, Equation 5.94 stems from the presence of barrier for time intervals shorter than the characteristic time of transfer, the removal of the surfactant from the interface (Tj = 0) cannot affect the subsurface layer (because of the barrier) and then Cij(O) = c. This purely theoretical consideration implies that the effect of barrier could show up at the short times of adsorption, whereas at the long times the adsorption will occur under diffusion control." The existence of barrier-affected adsorption regime at the short adsorption times could be confirmed or rejected by means of the fastest methods for measurement of dynamic surface tension. 

The determination of the real surface area of the electrocatalysts is an important factor for the calculation of the important parameters in the electrochemical reactors. It has been noticed that the real surface area determined by the electrochemical methods depends on the method used and on the experimental conditions. The STM and similar techniques are quite expensive for this single purpose. It is possible to determine the real surface area by means of different electrochemical methods in the aqueous and non-aqueous solutions in the presence of a non-adsorbing electrolyte. The values of the roughness factor using the methods based on the Gouy-Chapman theory are dependent on the diffuse layer thickness via the electrolyte concentration or the solvent dielectric constant. In general, the methods for the determination of the real area are based on either the mass transfer processes under diffusion control, or the adsorption processes at the surface or the measurements of the differential capacitance in the double layer region [56],... 

The maximal binding rate constant is thus of the order of 10, 2 x 10, and 5 X 10" mol 1 sec for typical antibodies, for DNA strands, and for conventional organic molecules respectively. It is seen that it is easier to observe the activation-controUed adsorption kinetics with large biomolecules than with conventional organic molecules, explaining why, in the latter case, the results are so scarce or so uncertain and also why the adsorption kinetics are considered to be under diffusion control in most circumstances. 

Plots of log( — di/dt)v vs. t were found to be linear up to about lOmin. The slopes obtained were essentially independent of potential and numerically in agreement with values calculated using values of the adsorption coefficient " " K and of D determined independently. Thus, the variation of current with time at constant potential is due to slow attainment of adsorption equilibrium under diffusion control. The break in the curves after about 10 min is indicative of a different cause for the time effects at longer times, which probably involves slow, activation-controlled adsorption on a small fraction of the electrode surface. 

In summary, four distinct kinetic regimes of adsorption from micellar solutions exist, called AB, BC, CD, and DE see Figures 4.8 and 4.10. In regime AB, the fast micellar process governs the adsorption kinetics. In regime BC, the adsorption occurs under diffusion control because the... 

However, there remains a possibility that the appearance of two pairs of voltammetric peaks may arise from particle adsorption onto the electrode surface. As in solid films, the ferrocene moieties might exhibit different energetic states and accessibility to counterions because of spatial effects. This hypothesis is discounted by results from two additional experiments. First, the cathodic and anodic current density of the redox peaks was found to be linearly proportional to the square root of potential scan rates, suggesting that the charge-transfer processes were under diffusion control. Second, after the electrochanical measurements in the Ru=CH-Fc particle solution, the Au electrode was taken out and rinsed with a copious amount of DMF and then immersed into a same electrolyte solution without the nanoparticles. Only featureless voltammetric responses were observed, as shown in Figure 3.13 (long dashed curve). In short, both measurements signify minimal surface adsorption of the particles. 

As mentioned earlier, below we focus om attention on the kinetics of surfactant adsorption. First, we introduce the basic equations. Next, we consider the two alternative cases of surfactant adsorption under diffusion and barrier control. Special attention is paid to the adsorption of ionic surfactants, whose molecules are involved in long-range electrostatic interactions. Finally, we consider the adsorption from micellar surfactant solutions, which is accompanied by micelle diffusion, assembly, or disintegration. 

Due to these complicated transport mechanisms and often occurring irreversibilify, particle adsorption is a highly path-dependent process characterized by a wide spectrum of time scale ranging from seconds for concentrate colloid suspensions to days for dilute suspensions of larger particles adsorbing under diffusion-controlled transport [9,10,16]. Therefore, in contrast to... 

This equation describes the nonstationary adsorption of particles at planar interfaces under diffusion-controlled transport conditions. 

Equations [2.1] and [2.2] also describe the fact that, under diffusion-controlled conditions, increases with decreasing the membrane thickness (5). Equation [2.2] is valid only for a very low value of n (i.e., 0.5), which corresponds to the a phase at relatively low pressure. At high pressures, H-H interactions within the palladium bulk are not negligible and Equation [2.2] is considered no longer valid. For instance, at 160°C, the H2 adsorption isotherm (hydrogen loading, H/Pd, as a function of p ) is linear at low pressures, but starts to curve as the miscibility gap is approached. 

Also, under continuous CO oxidation conditions, alkaline media exhibit a much higher activity than acidic media. Markovic and co-workers observed a shift of about 150 mV of the main oxidation wave, and a pre-wave corresponding to CO oxidation at potentials as low as 0.2-0.3 V [Markovic et al., 2002]. Remarkably, the hysteresis that is so prominently observed in the diffusion-controlled CO oxidation wave in acidic media (see Fig. 6.9), is no longer present in alkaline media. Markovic and co-workers also attribute the high activity of alkaline media to a pH-dependent adsorption of OH ds at defect/step sites. 

In contrast, physical adsorption is a very rapid process, so the rate is always controlled by mass transfer resistance rather than by the intrinsic adsorption kinetics. However, under cerlaiii conditions the combination of a diffusion-controlled process with an adsorption equilibrium constant that varies according to equation 1 can give the appearance of activated adsorption. 

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