Diffusive front

Soott S K and Showalter K 1992 Simple and oomplex propagating reaotion-diffusion fronts J. Phys. Chem. 96 8702-11... 

Florvath D and Showalter K 1995 Instabilities in propagating reaction-diffusion fronts of the iodate-arsenous acid reaction J. Chem. Rhys. 102 2471-8... 

The concentration distributions found at different times after the start of current flow are shown in Fig. 11.4. It is a typical feature of the solution obtained that the variable parameters x and t do not appear independently but always as the ratio Like Eq. 11.15), this indicates that the diffusion front advances in proportion to the square root of time. This behavior arises because as the diffusion front advances toward the bulk solution, the concentration gradients become flatter and thus diflusion slows down. 

K. Tucci and R. Kapral, Mesoscopic multiparticle collision dynamics of reaction-diffusion fronts, J. Phys. Chem. B 109, 21300 (2005). 

Limitations of the Shrinking Core Model. The assumptions of this model may not match reality precisely. For example, reaction may occur along a diffuse front rather than along a sharp interface between ash and fresh solid, thus giving behavior intermediate between the shrinking core and the continuous reaction models. This problem is considered by Wen (1968), and Ishida and Wen (1971). 

The diffusion profile is a smooth profile. Even if the initial concentration distribution is not smooth, diffusion smoothes out any initial discontinuities. Therefore, there is no well-defined diffusion front (except for some special cases). 

Because the diffusion distance is proportional to the square root of time, instead of the first power of time, diffusion rate is a less well-defined concept. The rate of the diffusion front moving into the diffusion medium would be dx/dt, which is not a constant, but is proportional to Hence, there is no fixed... 

The solution of Example 7.3 will be compared with an analytical solution of a diffusive front moving at velocity U, with D = 1/2U Az. First, we must derive the analytical solution. This problem is similar to Example 2.10, with these exceptions (1) convection must be added through a moving coordinate system, similar to that described in developing equation (2.36), and (2) a diffusion gradient will develop in both the +z-and -z-directions. 

Gray, P., Showalter, K., and Scott, S. K. (1987). Propagating reaction-diffusion fronts with cubic autocatalysis the effects of reversibility. J. Chim. Phys., 84, 1329-33. 

Saul, A. and Showalter, K. (1985). Propagating reaction-diffusion fronts. In Oscillations and traveling waves in chemical systems, (ed. R. J. Field and M. Burger), ch. 11, pp. 419-39. Wiley, New York. 

Gas-solid chromatography is best described by this theory. Here one finds diffuse front and rear boundaries with definite tailing of the rear boundary. Mathematical descriptions of systems of this type can become very complex however, with proper assumptions mathematical treatments do fairly represent the experimental data. The bands (zones) are similar to those shown in Figures 1.16 and 1.17. 

Kuge and Yoshikawa (3) related a change in the gas chromatographic peak shape to the beginning of multilayer adsorption on the surface of the solid. For small adsorbate volumes, the peak shape is symmetrical. As the adsorbate volume is increased, a sharp front, diffuse tail, and a defect at the front of the peak top is observed (Figure 11.2). It then acquires a diffuse front and sharp tail. This point corresponds to the B point of the BET Type II adsorption isotherm at which the relative surface area may be calculated. 

Both treatments assume that the dissolution of drug is rapid compared with the diffusion of drug, and both predict release of drug that is linear with Vf. This is a consequence of the increased diffusional distance and decreased diffusional area at the diffusion front as drug release proceeds. Figure 4.1 illustrates the amount released Qt versus square-root-time -Jt plots for two cases both dispersed (loading greater than saturation,... 

The diffusion front between the dissolved and undissolved drug in the gel (or swelled) phase... 

P. Colombo, R. Bettini, G. Massimg, et al. Drug diffusion front movements important in drug release control from swellable matrix tablet. J. Pharm. Sci. 84 991—997, 1995. 

We extend this analysis to the multilayer situation to describe particle activation energies for ID diffusion on a 3D cubic lattice. The constant source row boundary condition is extended to a thickness of up to five layers for this analysis. The diffusion front for each layer is defined using the seawater methodology of Sapoval et al. [161]. 

For front particles at the average position of the diffusion front, the mass M of the front particles scales with radius r as... 

The fractal nature of monolayer diffusion fronts has been investigated both computationally [161] and analytically [164,165]. Findings for the case of a monolayer diffusing from a constant source row indicate a Df of exactly 1.75 for a self-avoiding random walk (H = 0) [165], Sapoval et al. [161] found that Df decreases below 1.75 when the equivalent of J/kBT in Eq. (1.26) exceeds a critical value of 1.76. Surely the minimum for Df is 1.0 as J/kBT —> oo. 

B. Sapoval, M. Rosso, and J. F. Gouyet, The fractal nature of a diffusion front and the relation to percolation, J. Phys. Lett. 46(4), L149-L156 (1985). 

Bunde and J. F. Gouyet, On scaling relations in growth-models for percolating clusters and diffusion fronts, J. Phys. A—Math Gen. 18(6), L285-L287 (1985). 

The shape of the profile alters with time, until it becomes completely flat and the inner concentration of fluorine corresponds to the one measured on the periosteal surface. This equation is used to describe diffusion processes if there is a concentration gradient only along one axis, i.e. if diffusion is one-dimensional. If diffusion and environmental conditions are constant (e.g. a constant supply of fluorine due to invariant soil humidity), the profile shape and its penetration depth carry the information on exposure time t. The profile will be more developed if the sample has been exposed to this environmental system for a longer time (Fig. 6). This fact leads to the idea of a mathematical evaluation of the diffusion length Dt (the parameter that describes how far the diffusion front has penetrated into the material), which allows the calculation of the burial time t, i.e. the age of the archaeological sample, if the diffusion constant is known [80],... 

Fig. 8. Fluorine scans obtained by PIGE Fluorine-containing soil water enters the tooth mainly through the nerve canal into the pulpa. The cementum also readily takes up fluorine which slowly diffuses into the dentine, while the enamel crown forms a barrier. Fluorine enters a long bone as well from the periosteal surface as from the marrow cavity. The thickness of the bone wall does not influence the shape of the diffusion front itself, but limits the time window where age determination is possible, as the profile becomes flat much faster. (Human molar, Seeberg BE, Switzerland, 3750 bc, and human tibia, grave 132, Buren a. A. BE, Switzerland, medieval).

To gain quantitative information on the profile characteristics, the profile shape must be evaluated mathematically. The parameter Dt (D, diffusion constant t, exposure time) that describes the depth of the diffusion front that penetrated into the sample was determined by fitting the data with an error function (erf). The resulting curve describes the result of an undisturbed diffusion process. If the exposure time t is known, e.g. by radiocarbon dating, the diffusion constant D, a material constant, can be derived from this data. 

Matrix Diffusion. Historically, the most popular diffusion-controlled delivery system has been the matrix system, such as tablet and granules, where the drug is uniformly dissolved or dispersed, because of its low cost and ease of fabrication. However, the inherent drawback of the matrix system is its first-order release behavior with continuously diminishing release rate. This is a result of the increasing diffusional resistance and decreasing area at the penetrating diffusion front as matrix diffusion proceeds. 

The center of the circular sector lies outside the cylinder, thereby producing a slit for drug release from the drug containing matrix in the cavity. The release profiles from this system also show a substantial constant rate region (19,20). It is clear that, in both systems, the increase in diffusional distance and consequently the decrease in diffusion rate have been balanced by the increase in area at the diffusion front thereby giving rise to a near constant rate region. 

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