The Diffusion Layer Model

Although the diffusion layer model is the most commonly used, various alterations have been proposed. The current views of the diffusion layer model are based on the so-called effective diffusion boundary layer, the structure of which is heavily dependent on the hydrodynamic conditions, fn this context, Levich [102] developed the convection-diffusion theory and showed that the transfer of the solid to the solution is controlled by a combination of liquid flow and diffusion. In other words, both diffusion and convection contribute to the transfer of drug from the solid surface into the bulk solution, ft should be emphasized that this observation applies even under moderate conditions of stirring. 

Noyes and Whitney published [103] in 1897 the first quantitative study of a dissolution process. Using water as a dissolution medium, they rotated cylinders of benzoic acid and lead chloride and analyzed the resulting solutions at various time points. They found that the rate c(t) of change of concentration c (t) of dissolved species was proportional to the difference between the saturation solubility cs of the species and the concentration existing at any time t. Using 

Although it was not stated in the original article of Noyes and Whitney, it should be pointed out that the validity of the previous equation relies on the assumption that the amount used, qo, is greater than or equal to the amount required to saturate the dissolution medium, qs. Later on, (5.1) was modified [102,104] and expressed in terms of the dissolved amount of drug q(t) at time t while the effective surface area A of the solid was taken into account  

The integrated form of (5.2) gives the cumulative mass dissolved at time t  

The limit t — oo defines the total drug amount, qs = csV, that could be eventually dissolved in the volume V assuming that the amount used qo is greater than qs. Thus, we can define the accumulated fraction of the drug 

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Fig. 15 Two of the simplest theories for the dissolution of solids (A) the interfacial barrier model, and (B) the diffusion layer model, in the simple form of Nemst [105] and Brunner [106] (dashed trace) and in the more exact form of Levich [104] (solid trace). c is the concentration of the dissolving solid, cs is the solubility, cb is the concentration in the bulk solution, and x is the distance from the solid-liquid interface of thickness h or 8, depending on how it is defined. (Reproduced with permission of the copyright owner, John Wiley and Sons, Inc., from Ref. 1, p. 478.)...

The diffusion layer model satisfactorily accounts for the dissolution rates of most pharmaceutical solids. Equation (43) has even been used to predict the dissolution rates of drugs in powder form by assuming approximate values of D (e.g., 10 5 cm2/sec), and h (e.g., 50 pm) and by deriving a mean value of A from the mean particle size of the powder [107,108]. However, as the particles dissolve, the wetted surface area, A, decreases in proportion to the 2/3 power of the volume of the powder. With this assumption, integration of Eq. (38) leads to the following relation, known as the Hixon-Crowell [109] cube root law ... 

To be useful in modeling electrolyte sorption, a theory needs to describe hydrolysis and the mineral surface, account for electrical charge there, and provide for mass balance on the sorbing sites. In addition, an internally consistent and sufficiently broad database of sorption reactions should accompany the theory. Of the approaches available, a class known as surface complexation models (e.g., Adamson, 1976 Stumm, 1992) reflect such an ideal most closely. This class includes the double layer model (also known as the diffuse layer model) and the triple layer model (e.g., Westall and Hohl, 1980 Sverjensky, 1993). 

After the initial jubilation that the diffuse-layer model has overcome the weakness of constant capacity with change of potential of the parallel-plate model, one has to... 

In many cases, a less exact treatment suffices, e.g. to obtain the mean concentrations mentioned above. With the diffusion layer model, eqns. (144) are replaced by... 

Fortunately, in most cases, the salt form under serious consideration exhibits a faster dissolution rate than the corresponding parent drug at an equivalent pH. This dissolution phenomenon can be explained in light of the parameters that govern the dissolution rate, as found in the diffusion layer model of Brunner (1904) ... 

Two of the simplest theories to explain the dissolution rate of solutes are the interfacial barrier model and the diffusion-layer model (Figures 17.1 and 17.2). Both of these theories make the following two assumptions ... 

Early studies in this field of research formulated two main models for the interpretation of the dissolution mechanism the diffusion layer model and the... 

Figure 5.2 Schematic representation of the dissolution mechanisms according to (A) the diffusion layer model, and (B) the interfacial barrier model.

Since the fundamental rate equation of the diffusion layer model has the typical form of a first-order rate process (5.1), using (5.4) and (5.14), the MDT is found equal to the reciprocal of the rate constant k ... 

In the interfacial barrier model of dissolution it is assumed that the reaction at the solid-liquid interface is not rapid due to the high free energy of activation requirement and therefore the reaction becomes the rate-limiting step for the dissolution process (Figure 5.1), thus, drug dissolution is considered as a reaction-limited process for the interfacial barrier model. Although the diffusion layer model enjoys widespread acceptance since it provides a rather simplistic interpretation of dissolution with a well-defined mathematical description, the interfacial barrier model is not widely used because of the lack of a physically-based mathematical description. 

Figure 10,18 Schematic plot of surface species and charge (a) and potential ) relationships versus distance from the surface (at the zero plane) used in the constant capacitance (CC) and the diffuse-layer (DL) models. The capacitance, C is held constant in the CC model. The potential is the same at the zero and d planes in the diffuse-layer model i/fj). Reprinted from Adv. Colloid Interface Sci. 12, J. C. Westall and H. Hohl, A comparison of electrostatic models for the oxide/solution interface, pp. 265-294, Copyright 1980 with kind permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.

With the diffuse-layer model and the sweep option in MINTEQA2, calculate the adsorption of zinc onto HFO at 0.5-pH-unit increments between pH 4 and 8.5 and determine corresponding surface speciation. Comment on the relationship of surface speciation to the pHpzNpc and on the behavior of surface charge and surface potential around the pHpzNPc- Finally, compute and plot for zinc adsorption from pH 4 to 8.5. Assume the same system conditions as in the previous problem, but with a total added zinc concentration of 10 M. System conditions include I = 0.01 M, Sa = 600 mVg, Cj = 8.9 x 10 g/L, and F = 2 x 10" mol active sites/L. 

TABLE 10.10. Adsorption of Zn by HFO computed in Example 10.2 using the sweep option in MINTEQA2 with the diffuse-layer model... 

In this problem we use the diffuse-layer model to compute the adsorption of orthophosphate species by goethite between pH 3 and 10. Intrinsic constants for the adsorption reactions are available in the MINTF.QA2 file,/ o-d/w.f/6r. Similar calculations have been discussed and performed by Hohl etal. (1980). (a) Assume 0.6 g/L of goethite in suspension, with a surface-site density of 1.35 x 10 moles of sites per... 

The diffuse layer model was originally introduced for surface charging of mercury by Gouy and Chapman in the early 1900s. Combination of... 

This is a big advantage of the diffuse layer model, namely the (To(pH) can be predicted for the ionic strengths, for which the experimental data are not available, without resorting to empirical interpolation or extrapolation. 

Similarly as with the diffuse layer model, for the data sets presented in Figs. 5.58 and 5.59 better fit can be obtained taking K (reaction (5.23)) somewhat higher than the PZC, and for the data sets presented in Figs. 5.57 and 5.60 better fit can be obtained taking K (reaction (5.23)) somewhat lower than the PZC. With the data sets... 

The Stern model was also tested for silica. With the Kosmulski. Ldbbus. and Szekeres experimental data the goodness of fit with physically realistic C values was satisfactory, but worse than for diffuse layer model (cf. Figs. 5,47. 5.48 and 5.51). Since C is the additional adjustable parameter with respect to the diffuse layer model, application of Stern model to these systems is not justified. Somewhat improved with respect to the diffuse layer model, but still rather poor modeling results were obtained for the Milonjic and Sidorova experimental data. The best-fit... 

Figures 5.66-5.68 (and Fig. 5.42) compare the best fit model curves for alumina (four sets of experimental data) calculated by means of the diffuse layer model combined with the 1-pK model on the one hand and the 2-pK model on the other. The best-fit in the 1-pK model are presented in Table 5.12. The best-fit parameters of the 2-pK model are presented in Table 5.17. 

In view of different PZC in these four diffuse layer models, the K values in Table 5.23 are not fully correlated with used in the diffuse layer model. Model uptake curves calculated for different diffuse layer models for 10 mol dm inert electrolyte (only one such curve is shown in Fig, 5.120) are practically identical. 

Figure 5.132 presents the ionic strength effect on the model uptake curves calculated for one proton released per one adsorbed Pb, using the diffuse layer model Kosmulski, for model parameters cf. Table 5.13). The model curves are significantly steeper, and the ionic strength effect is less significant than in the analogous Pb adsorption model (inner sphere, one proton released) combined with TLM (Fig. 5.126). The calculated stability constant of the surface complex is higher by three orders of magnitude for the diffuse layer model (Table 5.28) than for TLM (Table 5.27).

Two best-fit Ns values in Table 5.29 are significantly higher than the highest Ns reported in Tables 5.1 and 5.3, and their physical sense is questionable. The log K (=SiOPb ) calculated for the best-fit Stern model (Table 5.29) are lower by about 0.2 than the corresponding log K for the diffuse layer model (Table 5.28), but higher by 2-3 than the log A)(=SiOPb ) calculated for the best-fit TLM (Table 5.27). The calculated uptake curves presented in Fig. 5.132 (diffuse layer model) are practically identical as corresponding curves calculated for the Stem model (Kosmulski. model... 

Diffuse Layer Model The diffuse layer model of the oxide-solution interface (Huang and Stumm, 1973 Dzombak and Morel, 1990) contains the following assumptions ... 

In the diffuse layer model the surface reactions include Eqs. (6.6) and (6.7) for protonation and dissociation of the surface functional groups. In the two-site version of the model, surface complexation with metals is considered to occur on at most two types of sites a small set of high-affinity strong sites, S OH, and a large set of low-affinity weak sites, S OH, analogous to Eq. (6.8) (Dzombak and Morel, 1990) ... 

Although the diffuse layer model has the capability to consider bidentate metal complexes, such species are not usually considered. 

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