Concentration-Distance Curves

Figure 3.1 Concentration-distance curves for different periods of linear diffusion to a planar-electrode surface. Diffusion times (1) 10 s (2) 100 s (3) 1000 s (4) 10,000 s. Data for a diffusion coefficient D of 1 x 10-5 cm2 s 1.

The concentration-distance curve at time t serves to give dx/dc and... 

Figure 1.2. Concentration-distance profiles for diffusion of an electroactive substance to an electrode surface at different times. At zero time, a voltage large enough to cause the electrode reaction to occur is suddenly applied to the electrode. Note that with increasing time (tg > ta > ti), the concentration gradient—the slope of the concentration-distance curve —becomes less steep, and the diffusion layer thickness, 8, becomes larger.

The concentration-distance curves method is based on the measurement of the distribution of the diffusant concentration as a function of time. Light interference methods, as well as radiation adsorption or simply gravimetric methods, can be used for concentration measurements. Various sample geometries can be used, for example semiinflnite solid, two joint cylinders with the same or different material, and so on. The analysis is based on the solution of Pick s equation. 

Diffusion at the interface between PVC and poly- -caprolactone has been demonstrated using energy dispersive X-ray analysis (EDX) [85,86]. These polymers are compatible in the melt at all ratios, but phase separation takes place on cooling. The technique permits the measurement of the concentration-distance curve for PVC in the interfacial region. The results of work at 70°C gave a diffusion coefficient of 1.59 X 10 m s ... 

The zero plane may be chosen when the concentration-distance curve is plotted at a given time. Integration with respect to A then gives... 

The concentration-distance curves into the metal are, however, not always of a shape to which a simple solution of the Pick law is applicable. Pigs. 73 and 74 give different types of concentration-distance curve, for which the solution given above is manifestly not correct. In the former case a saturated carbon layer has established itself at the surface, and in the latter the surface has been to some extent decarburised by hydrogen present in the carburising atmosphere. An attempt has been made to treat diffusion problems of the type illustrated by Pig. 73(6), but a complete treatment of Pig. 74 is not available. However, Bramley and his co-workers neglected the initial parts of these unusual forms of concentration distance curve, and applied the simple solution of Pick s law only to the tail of the curves. The error involved in evaluating D is therefore smaller. 

Figure 18.8 Concentration-distance curves for molecules that begin at X = 0 and diffuse in one dimension along x. The numbers on the curves are the values of Df. Source J Crank, The Mathematics of Diffusion, 2nd edition, Clarendon Press, Oxford, 1993.

Sorption kinetics Permeation methods Concentration-distance curves... 

Tuominen, Y. (1971). Studies on some concentration-distance curves of the diffusion of Cs and Sr ions in columns composed of the thallus of Cladonia alpestris. Ann. Bot. Fenn. 8, 245. 

Models play a very important part in the study of the interphase. They give us ways to calculate the charge of the electrode as a function of electrode potential as well as complete potential-distance and concentration-distance curves. Models, however, are never ideal and those available for the interphase are quite crude. This must be borne in mind when using these models for quantitative calculations. 

Before correcting the faults of the Gouy-Chapman model by using another model, let us derive expressions for the potential-distance and concentration-distance curves. In order to do that, equation 4.15 should be integrated. 

Using the program PREMIX (Kee et al. 1985), premixed, stationary, laminar, one-dimensional methane-air flames were simulated. The cold mixture boundary conditions were p= atm and T = 298.15 K. The calculations were carried out for lean ( = 0.70), stoichiometric equivalence ratios. PREMIX calculated not only the concentration — distance curves, but also normed sensitivities dlnYJd and the sensitivities of the calculated concentrations with respect to the enthalpies of formation of species. 

This section is concerned with measurement techniques of the diffusivity and solubility from which the permeability can easily be calculated. In the following analysis we restrict ourselves to the measurement of constant values of D. Concentration- and position-dependent diffusivities are analyzed in Crank and Park (1968) and Crank (1975). Generally, the techniques are for permeability, steady-state and time lag techniques and for diffusivity, sorption and desorption kinetics and concentration-distance curves. For self-diffusivity in polymer melts the techniques are (Tirrell, 1984) nuclear magnetic resonance, neutron scattering, radioactive tracer, and infrared spectroscopy. 

FIGURE 4.17 Schematic of the concentration-distance curve used in the calculation of the diffusivity. 

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