Surface step concentration profile

Studies have shown that it is also important to consider a continuous segment concentration profile from the solid surface to the edge of the brush layer, rather than a simple step concentration profile of height h as implied in the above scaling analysis. For example, the theoretical self-consistent field (SCF) analysis of Milner et al. [75] predicts that, given a SCF-calculated equilibrium brush height h, the polymer volume fraction (p(z) in the brush layer follows a parabolic distribution with respect to the distance z from the surface ... 

The Alexander model is based on two assumptions that enable simple expressions for these two terms (1) The concentration profile of the layer is step-like. That is, the monomer volume fraction within the layer, (p Na3/d2L, is constant, independent of position (2) The chains are uniformly stretched. That is, all chain ends are positioned on a single plane at a distance L from the surface. [In this paper, we use the symbol to mean approximately equal to or equal to within a numerical factor of order one we use to mean proportional to .] The first assumption simplifies the calculation of Fin, while the second yields a simple expression for Fel. 

Figure 14. Simple model demonstrating how adsorption and surface diffusion can co-Urnit overall reaction kinetics, as explained in the text, (a) A semi-infinite surface establishes a uniform surface coverage Cao of adsorbate A via equilibrium of surface diffusion and adsorption/desorption of A from/to the surrounding gas. (b) Concentration profile of adsorbed species following a step (drop) in surface coverage at the origin, (c) Surface flux of species at the origin (A 4i(t)) as a function of time. Points marked with a solid circle ( ) correspond to the concentration profiles in b. (d) Surface flux of species at the origin (A 4i(ft>)) resulting from a steady periodic sinusoidal oscillation at frequency 0) of the concentration at the origin.

Surface diffusion can be studied with a wide variety of methods using both macroscopic and microscopic techniques of great diversity.98 Basically three methods can be used. One measures the time dependence of the concentration profile of diffusing atoms, one the time correlation of the concentration fluctuations, or the fluctuations of the number of diffusion atoms within a specified area, and one the mean square displacement, or the second moment, of a diffusing atom. When macroscopic techniques are used to study surface diffusion, diffusion parameters are usually derived from the rate of change of the shape of a sharply structured microscopic object, or from the rate of advancement of a sharply defined boundary of an adsorption layer, produced either by using a shadowed deposition method or by fast pulsed-laser thermal desorption of an area covered with an adsorbed species. The derived diffusion parameters really describe the overall effect of many different atomic steps, such as the formation of adatoms from kink sites, ledge sites... 

Fig. 7.138. Adatom concentration profiles, c M/c, between two steps of growth, (a) for different cathodic (upper part of the plot) and anodic (lower part of the plot) overpotentials at X = 0.31 8. (b) For different surface diffusion (sd) rates 0.1 8step < < 3.1 8step atTo mV cathodic and anodic...

The simulation of other electrochemical experiments will require different electrode boundary conditions. The simulation of potential-step Nernstian behavior will require that the ratio of reactant and product concentrations at the electrode surface be a fixed function of electrode potential. In the simulation of voltammetry, this ratio is no longer fixed it is a function of time. Chrono-potentiometry may be simulated by fixing the slope of the concentration profile in the vicinity of the electrode surface according to the magnitude of the constant current passed. These other techniques are discussed later a model for diffusion-limited semi-infinite linear diffusion is developed immediately. 

Quantitative determinations of the thicknesses of a multiple - layered sample (for example, two polymer layers in intimate contact) by ATR spectroscopy has been shown to be possible. The attenuation effect on the evanescent wave by the layer in contact with the IRE surface must be taken into account (112). Extension of this idea of a step-type concentration profile for an adsorbed surfactant layer on an IRE surface was made (113). and equations relating the Gibbs surface excess to the absorbance in the infrared spectrum of a sufficiently thin adsorbed surfactant layer were developed. The addition of a thin layer of a viscous hydrocarbon liquid to the IRE surface was investigated as a model of a liquid-liquid interface (114) for studies of metal extraction ( Ni+2, Cu+2) by a hydrophobic chelating agent. The extraction of the metals from an aqueous buffer into the hydrocarbon layer was monitored kinetically by the appearance of bands unique to the complex formed. 

The temporal evolution of the concentration profiles of the adspecies with allowance for their interaction seems to have been studied for the first time by Bowker and King [158]. Initially, the distribution of the adspecies density has been given up in the form of a step (this technique is often applied to surface diffusion studies). Consideration has been given to the concentration profiles in the case of attraction and repulsion of the adspecies to conclude that they can be used to estimate the lateral interactions. The applicability of the model to the description of diffusion in the O/W (110) system [159] is discussed. 

For the monofunctional APTS, the observed deposition profile is identical to that reported by Vandenberg et al., using liquid/solid ellipsometry.15 The rapid, first equilibrium is followed by an additional adsorption. The equilibrium situation is related to the localized adsorption of silane molecules on the surface hydroxyls. Thus forming a monolayered coating on the surface. The extent of the additional adsorption was found to be not very reproducible and not strictly correlated to the initial silane concentration. At any time it showed not to have the step-wise profile, as found for the bifunctional aminosilane. Therefore, this last adsorption appears to be of a nonspecific type. We will discuss this further below. 

For PIGE measurements, transverse bone sections are cut with a diamond saw and polished with SiC paper, and then placed directly in front of the external proton beam. It is not necessary to coat the sample surface with a conductive layer as the charges are dissipated in air and helium. Step width of the concentration profiles is determined by precisely recorded sample translation in front of the beam. The above experimental conditions were used for F analysis in archaeological bone materials in the applications described in this chapter. 

Using a self-consistent field theory, Misra et al. [10] examined the effect of the brush charge, electrolyte concentration, and surface charge density on the brush thickness. They extended the self-consistent field polymer brush theory suggested by Milner et al. [11] to the case of a polyelectrolyte brush. The theory involves a parabolic monomer concentration profile rather than the step-function suggested by Alexander [12] and de Gennes [13,14], The repulsion force... 

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... 

Fig. UK Evolution of the concentration profile with time near an electrode surface, just after the potential has been stepped to the limiting current region.

Let us consider, for example, the simple nernstian reduction reaction in Eq. (221) and a solution containing initially only the reactant R. Before any electrochemical perturbation the electrode rest potential Ej is made largely positive to E . At time zero the potential is stepped to a value E2, sufficiently negative to E , so that the concentration of R is close to zero at the electrode surface. After a time 6, the electrode potential is stepped back to El, so that the concentration of P at the electrode surface becomes zero. When this potentiostatic perturbation, represented in Fig. 21a, is applied in a steady-state method, the R and P concentration profiles are linear and depend only on the electrode potential but not on time, as shown in Fig. 20a (for k 0). Yet when the same perturbation is applied in transient methods, the concentration profiles are curved and time dependent, as evidenced in Fig. 21b. Thus it is seen from this figure that a step duration at Ei, much longer than the step duration 0 at E2, is needed for the initial concentration profiles to be restored. This hysterisis corresponds to the propagation of the diffusion perturbation within the solution, which then keeps a memory of the past perturbation. This information is stored via the structuring of the concentrations in the space near the electrode as a function of the elapsed time. 

Two different techniques have been employed for the precipitation of membranes from a polymer casting solution. In the first method, the precipitant is introduced from the vapor phase. In this case the precipitation is slow, and a more or less homogeneous structure is obtained without a dense skin on the top or bottom side of the polymer film. This structure can be understood when the concentration profiles of the polymer, the precipitant and the solvent during the precipitation process are considered. The significant feature in the vapor-phase precipitation process is the fact that the rate-limiting step for precipitant transport into the cast polymer solution is the slow diffusion in the vapor phase adjacent to the film surface. This leads to uniform and flat concentration profiles in the film. The concentration profiles of the precipitant at various times in the polymer film are shown schematically in Figure 13. 

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