Diffuse double layer potential profile

Kg. 17.2 Schematic representation of transmembrane potential profile. Em, transmembrane potential difference Eo DL, exterior diffuse double layer potential difference Ei DL, interior diffuse double layer potential difference ED potential difference due to membrane molecular dipoles EDi = EDo, symmetric membrane potential EDiff diffusion potential difference. 

Figure 28. A schematic transmembrane potential profile E = transmembrane potential iJ/q = the outer diffuse double layer potential = the inner diffuse double layer potential = polarization potential due to membrane molecular dipoles iJ/do d — asymmetrical polarization potentials = diffusion potential.

A general mathematical formulation and a detailed analysis of the dynamic behavior of this mass-transport induced N-NDR oscillations were given by Koper and Sluyters [8, 65]. The concentration of the electroactive species at the electrode decreases owing to the electron-transfer reaction and increases due to diffusion. For the mathematical description of diffusion, Koper and Sluyters [65] invoke a linear diffusion layer approximation, that is, it is assumed that there is a diffusion layer of constant thickness, and the concentration profile across the diffusion layer adjusts instantaneously to a linear profile. Thus, they arrive at the following dimensionless set of equations for the double layer potential, [Pg.117]

Figure 8.1 sketches the basic mechanism underlying this type of stationary pattern formation [5] The concentration c of the activator (red line) is locally slightly perturbed from its steady-state value Css (s) This initiates the self-enhanced production of the activator that now slowly diffuses into neighboring regions (b). As a consequence, the inhibitor, that is, the double-layer potential grows (blue dashed line) that propagates more rapidly out of the activator spot (c). Since the inhibitor consumes the activator, the concentration of the latter will always be suppressed outside the spot (d). In this way, a stationary concentration profile may develop surrounded by a halo of increased inhibitor < dl-... 

Figure 21. A schematic diagram of the Stern adsorption layer (top) and the average potential profile of the Stern layer and Gouy-Chapman diffuse double layer.

In this case, besides a molecular condenser due to the dipole array, 4770-1 = e"( A(0) - iA(/)), similar to that of the Stern layer, a diffuse layer potential due to electrolytes and the charges of dipoles may be formed in the aqueous solution which is slightly different from the double layer potential discussed above. Depending upon the magnitude of the dipoles /t and their orientation at the membrane surface, the contribution of such polarization potentials to the interfacial potential as well as to the transmembrane potential could be considerable. In addition, it is possible that molecular (nonspecific or specific) adsorption of ions or water molecules occurs. This would further complicate the profile of the diffuse layer potential. 

Any modification in the double layer potential affects the concentration profile of ions that reside within the double layer structure. In the case of an enzyme-substrate reaction, in which a nonequilibrium contribution to the potential arises due to differences in substrate and product diffusion coefficient, as seen from Eq. (16), variation in the equilibrium concentration profiles of not only charged reactant and charged product but also variations in the concentration profiles of nonreacting ionic species present is possible.The concentration profiles of all ionic species adjust according to changes in the reaction rate, In order to... 

Without actually studying the whole mathematical equation of the potential profile in the double layer, one can remember that when the potential difference between the terminals of the diffuse double layer is moderate, then the variations inside the latter are close to an exponential function ... 

Revil et al. [5] explain that the influence of temperature on the zeta potential is a result of changes in silanol equilibrium, adsorption equilibria, and diffuse double-layer thickness. Their analysis and data are geared towards the geophysical conununity, but the linear profiles they show describing change in zeta potential with temperature is consistent with Venditti et al. [3]. A linear zeta potential—temperature profile is again reported in a more recent paper by Revil et al. [6]. 

Regardless of whether the transfer or transport approach is followed, the flux equation can, in principle, be solved for any given potential profile in the diffuse layers. Usually, the potential profile is quite well described using the Poisson-Boltzmann equation, which in the case of a z z supporting electrolyte can be solved analytically. More advanced theories may also be applied [81, 82]. In a recent theoretical study based on the idea of determining the permeabilities of the diffuse double layers and the interface independently, different potential profiles were analyzed. The Poisson-Boltzmann potential profile, a stepwise linear potential profile, and a potential step profile were compared. Interestingly, even a very simple stepwise potential profile is hardly distinguishable from the linear potential profile and the relatively complicated Poisson-Boltzmann... 

Overbeek and Booth [284] have extended the Henry model to include the effects of double-layer distortion by the relaxation effect. Since the double-layer charge is opposite to the particle charge, the fluid in the layer tends to move in the direction opposite to the particle. This distorts the symmetry of the flow and concentration profiles around the particle. Diffusion and electrical conductance tend to restore this symmetry however, it takes time for this to occur. This is known as the relaxation effect. The relaxation effect is not significant for zeta-potentials of less than 25 mV i.e., the Overbeek and Booth equations reduce to the Henry equation for zeta-potentials less than 25 mV [284]. For an electrophoretic mobility of approximately 10 X 10 " cm A -sec, the corresponding zeta potential is 20 mV at 25°C. Mobilities of up to 20 X 10 " cmW-s, i.e., zeta-potentials of 40 mV, are not uncommon for proteins at temperatures of 20-30°C, and thus relaxation may be important for some proteins. 

Since the separation between the tip and the surface is such that their respective double layers do not overlap, the nanostmcturing process can be described simply through the diffusion of the ions toward the surface. Thus, the concentration profiles of the diffusing ions dehne effective Nemst potential prohles that can be employed to predict the regions where the oversaturation conditions will contribute to metal nucleation and growth. 

The dependence of dx on qM is central in a model, proposed by Price and Halley,93 for the metal surface in the double layer which is related to that discussed above. The positively charged ion background profile p+(z) is assumed uniform, with a value equal to the bulk density pb, from z = -oo to z = 0, with the electronic density profile n(z) more diffuse. In contrast to the previous model30 which emphasizes penetration by the conduction electrons of the region of solvent, this model93 supposes that the density profile n(z) is zero for z > dx, where z > dx defines the region of the electrolyte. Then the potential at dx is given by... 

As suggested before, the role of the interphasial double layer is insignificant in many transport processes that are involved with the supply of components from the bulk of the medium towards the biosurface. The thickness of the electric double layer is so small compared with that of the diffusion layer 8 that the very local deformation of the concentration profiles does not really alter the flux. Hence, in most analyses of diffusive mass transport one does not find any electric double layer terms. For the kinetics of the interphasial processes, this is completely different. Rate constants for chemical reactions or permeation steps are usually heavily dependent on the local conditions. Like in electrochemical processes, two elements are of great importance the local electric field which affects rates of transfer of charged species (the actual potential comes into play in the case of redox reactions), and the local activities... 

The properties of the surface layers have a strong effect on the deposition process. The driving force of the electrochemical reaction is the potential difference over the electrochemical double layer. Adsorption of species can change this potential. For example, the additives used in electrodeposition adsorb in the Helmholtz layer. They can change the local potential difference, block active deposition sites, and so on. The thickness of the diffusion layer affects the mass-transfer rate to the electrode. The diffusion layer becomes thinner with increasing flow rate. When the diffusion layer is thicker than the electrode surface profile, local mass-transfer rates are not equal along the electrode surface. This means that under mass-transfer control, metal deposition on electrode surface peaks is faster than in the valleys and a rough deposit will result. 

The potential profile across the double layer is modified, including the value of the diffuse-layer potential, (/ i, for a given metal/solution p.d., A<. ... 

The thickness of the diffuse layer depends on the total ionic concentration in the solution for concentrations greater than 10 M, the thickness is less than 100 A. The potential profile across the double-layer region is shown in Figure 1.2.4. 

Figure 13.3.6 (a) A view of the differential capacitance in the Gouy-Chapman-Stem (GCS) model as a series network of Helmholtz-layer and diffuse-layer capacitances. (b) Potential profile through the solution side of the double layer, according to GCS theory. Calculated from (13.3.23) for 10 M 1 1 electrolyte in water at 25°C. 

At frequencies below 63 Hz, the double-layer capacitance began to dominate the overall impedance of the membrane electrode. The electric potential profile of a bilayer membrane consists of a hydrocarbon core layer and an electrical double layer (49). The dipolar potential, which originates from the lipid bilayer head-group zone and the incorporated protein, partially controls transmembrane ion transport. The model equivalent circuit presented here accounts for the response as a function of frequency of both the hydrocarbon core layer and the double layer at the membrane-water interface. The value of Cdl from the best curve fit for the membrane-coated electrode is lower than that for the bare PtO interface. For the membrane-coated electrode, the model gives a polarization resistance, of 80 kfl compared with 5 kfl for the bare PtO electrode. Formation of the lipid membrane creates a dipolar potential at the interface that results in higher Rdl. The incorporated rhodopsin may also extend the double layer, which makes the layer more diffuse and, therefore, decreases C. ... 

This formula for the electroosmotic velocity past a plane charged surface is known as the Helmholtz-Smoluchowski equation. Note that within this picture, where the double layer thickness is very small compared with the characteristic length, say alX t> 100, the fluid moves as in plug flow. Thus the velocity slips at the wall that is, it goes from U to zero discontinuously. For a finite-thickness diffuse layer the actual velocity profile has a behavior similar to that shown in Fig. 6.5.1, where the velocity drops continuously across the layer to zero at the wall. The constant electroosmotic velocity therefore represents the velocity at the edge of the diffuse layer. A typical zeta potential is about 0.1 V. Thus for = 10 V m" with viscosity that of water, the electroosmotic velocity U 10 " ms, a very small value. 

A schematic representation of the inner region of the double layer model is shown in Fig. 1. Figure lb describes the distribution of counterions and the potential profile /(a ) from a positively charged surface. The potential decay is caused by the presence of counterions in the solution side (mobile phase) of the double layer. The inner Helmholtz plane (IHP) or the inner Stem plane (ISP) is the plane through the centers of ions that are chemically adsorbed (if any) on the solid surface. The outer Helmholtz plane (OHP) or the outer Stem plane (OSP) is the plane of closest approach of hydrated ions (which do not adsorb chemically) in the diffuse layer. Therefore, the plane that corresponds to x = 0 in Eq. (4) coincides with the OHP in the GCSG model. The doublelayer charge and potential are defined in such a way that ao and /o, op and Tp, and <5d and /rf are the charge densities and mean potentials of the surface plane, the Stem layer (IHP), and the diffuse layer, respectively (Fig. 1). 

Refer to section 2.2.3 and to figures 2.15 and 2.16 in section 1.2.4.2 for the qualitative description of potential profiles in an electrochemical cell, including the different contributions to the voltage across an electrochemical system with a current flowing. In addition, remember that the description of the concentration or potential profiles is on the spatial scale of the diffusion layers and not on the double-layers scale (see section 3.3.1). In particular, while the potential profile is continuous on the scale of the double-layer dimensions, it shows discontinuities on the scale of the diffusion layers at the interfaces. 

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