Dissolution diffusion layer thickness

Equation (1) predicts that the rate of release can be constant only if the following parameters are constant (a) surface area, (b) diffusion coefficient, (c) diffusion layer thickness, and (d) concentration difference. These parameters, however, are not easily maintained constant, especially surface area. For spherical particles, the change in surface area can be related to the weight of the particle that is, under the assumption of sink conditions, Eq. (1) can be rewritten as the cube-root dissolution equation ... 

In Eq. (41), the concentration gradient is expressed as a difference between the surface concentration, Cm, and the bulk concentration, CHS), divided by the diffusion layer thickness, 8. If sink dissolution conditions are assumed (Cb 0) and the solid has a uniform density (p = m/V), then Eq. (42) can be derived. 

FIGURE 17.2 Dissolution pro Jes of ne (open circles) and coarse ( lied circles) hydrocortisone (Lu et al., 1993). Simulated curves were drawn using spherical geometry without (a) and with (b) a time-dependent diffusion-layer thickness and cylindrical geometry without (c) and with (d) a time-dependent diffusion-layer thickness. Error bars represent 95% 6I=(= 3). (Reprinted from Lu, A. T. K., Frisella, M. E., and Johnson, K. C. (1993pharm. Res., 10 1308-1314. Copyright 1993. With permission from Kluwer Academic Plenum Publishers.)... 

For water-insoluble drugs, dissolution-controlled systems are an obvious choice for achieving sustained-release because of theirslow dissolution rate characteristics. Theoretically, the dissolution process at steady state can be described by the Noyes-Whitney equation as shown in Equation 22.7. The rate of dissolution of a compound is a function of surface area, saturation solubility, and diffusion layer thickness. Therefore, the rate of drug release can be manipulated by changing these parameter. 

A drug powder has uniform particle sizes of 500 pm diameter and 75 mg weight. If the solubility of the drug in water = 3 mg/mL, the density of the drug = 1.0 g/mL, and the diffusion coefficient of the drug in water = 5 x 10 6 cm2/sec, calculate the diffusion layer thickness when it takes 0.3 hours for the complete dissolution. 

Although these approaches demonstrate the important role of the drug material s surface and its morphology on dictating the dissolution profile, they still suffer from limitations regarding the shape and size distribution of particles as well as the assumptions on the constancy of the diffusion layer thickness S and the drug s diffusivity V throughout the process implied in (5.5), (5.6), (5.7), and... 

In reality, the parameters 6 and 2 cannot be considered constant during the entire course of the dissolution process when poly-disperse powders are used and/or an initial phase of poor deaggregation of granules or poor wetting of formulation is encountered. In addition, the diffusion layer thickness appears to depend on particle size. For all aforementioned reasons, (5.5), (5.6), (5.7), and (5.8) have been proven adequate in modeling dissolution data only when the presuppositions of constancy of terms in (5.3) are fulfilled. 

In the context of EMM, the diffusion layer thickness at the anode may be more compared to that in conventional ECM due to absence of high flow rate of the electrolyte. Hence the effect of Warburg impedance is more prominent during anodic dissolution in the microscopic domain. 

Nevertheless, the basic reason should still be found in transport phenomena similar to those leading to amplification of the surface profile in the opposite process of cathodic deposition the diffusion flux of products of anodic dissolution away from the surface is larger at elevated points than at recesses, since the effective diffusion layer thickness is smaller at the former than at the latter, and there is also an increased diffusion flux in the convex diffusion field at the peaks, leading to conditions close to those of spherical diffusion. However, diffusion away from the electrode in anodic processes is known not to lead to any limiting current, unless the solubility limit of the reaction product is reached or a very steep viscosity increase follows increase in concentration. Hence, a more likely cause could be found in the limitations of diffusion toward the electrode of an agent complexing the reaction product. 

Of the many theories proposed for studying the dissolution of solids, the simple diffusion model mostly suffices for pharmaceutical applications. The diffusion layer model assumes a thin stagnant diffusion layer (thickness = K) at the interface of the dissolving solid (x = 0) and the dissolution medium x = h), referred to as bulk (Figure 7.5). ... 

Sample agitation The higher is the extraction rate, the higher is the sensitivity, since the agitation reduces the diffusion layer thickness. However, the agitation rate has an upper limit due to the drop dislodgement and dissolution. The use of small stir bars is recommended. 

In the proposed mechanism of film formation, both iron and chromium form ions before undergoing hydrolysis and crystallization on the surface of the stainless steel. This mechanism can account for the existence of a critical velocity, since ions and hydrolyzed particles can diffuse into the circulating. stream before forming the film if the diffusion layer is too thin. Since the diffusion-layer thickness depends on the flow rate or turbulence, the process of film formation is, in essence, in competition with diffusion and turbulence. The critical velocity, then, is that flow rate (or turbulence) at which the diffusion layer is reduced to such an extent that most of the corrosion products get into the main circulating stream before they can form on the surface of the stainless steel. Under conditions where the rates of dissolution and hydrolysis are fast, the critical velocity would be expected to be relatively high. 

Although it is possible to control the dissolution rate of a drug by controlling its particle size and solubility, the pharmaceutical manufacturer has very little, if any, control over the D/h term in the Nernst-Brunner equation, Eq. (1). In deriving the equation it was assumed that h, the thickness of the stationary diffusion layer, was independent of particle size. In fact, this is not necessarily true. The diffusion layer probably increases as particle size increases. Furthermore, h decreases as the stirring rate increases. In vivo, as GI motility increases or decreases, h would be expected to decrease or increase. In deriving the Nernst-Brunner equation, it was also assumed that all the particles were... 

It can be seen that the dissolution rate constant kD is equivalent to the diffusion coefficient divided by the thickness of the diffusion layer (D/h). 

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 theory, illustrated in Fig. 15B, is the most useful and best-known model for transport-controlled dissolution. The dissolution rate here is controlled by the rate of diffusion of solute molecules across a diffusion layer of thickness h, so that kT kR in Eq. (40), which simplifies to kx = kT. With increasing distance, x, from the surface of the solid, the concentration, c, decreases from cs at x = 0 to cb at x = h. In general, c is a nonlinear function of x, and the concentration gradient dddx becomes less steep as x increases. The hyrodynamics of the dissolution process has been fully discussed by Levich [104]. In a stirred solution, the flow velocity of the liquid dissolution medium increases from zero at x = 0 to the bulk value at x = h. 

The dissolution rate of a solid from a rotating disc is governed by the controlled hydrodynamics of the system, and it has been treated theoretically by Levich [104]. This theory considers only forced convection due to rotation and ignores natural convection, which may occur at low speeds of rotation. Figure 16 shows the solvent flow held near the surface of the rotating disc. The apparent thickness, h, of the diffusion layer next to the surface of the disc is given by... 

The rate of agitation, stirring, or flow of solvent, if the dissolution is transport-controlled, but not when the dissolution is reaction-con-trolled. Increasing the agitation rate corresponds to an increased hydrodynamic flow rate and to an increased Reynolds number [104, 117] and results in a reduction in the thickness of the diffusion layer in Eqs. (43), (45), (46), (49), and (50) for transport control. Therefore, an increased agitation rate will increase the dissolution rate, if the dissolution is transport-controlled (Eqs. (41 16,49,51,52), but will have no effect if the dissolution is reaction-controlled. Turbulent flow (which occurs at Reynolds numbers exceeding 1000 to 2000 and which is a chaotic phenomenon) may cause irreproducible and/or unpredictable dissolution rates [104,117] and should therefore be avoided. 

---