Theories of Dissolution

Dissolution is generally defined as a process by which a solid substance is sol-ubilized into the solvent to yield a solution. This process is fundamentally controlled by the affinity between the solid substance and the solvent and consists of two consecutive steps. The first step involves the liberation of molecules from the solid phase to the liquid layer near the solid surface (an interfacial reaction between the solid surface and the solvent). It is followed by the transport of solutes from the solid-liquid interface into the bulk solution. The dissolution of solid substance is generally modeled based upon the relative significance of these two transport steps.

The diffusion layer model proposed originally by Nernst and Brunner (Brunner, 1904; Nernst, 1904) is widely used to describe the dissolution of pure solid substances. In this model, it is assumed that a diffusion layer (or a stagnant liquid film layer) of the thickness h is surrounding the surface of a dissolving particle. The reaction at the solid-liquid interface is assumed to be instantaneous. Thus, equilibrium exists at the interface, and hence the concentration of the surface is the saturated solubility of the substance (Cs). Once the solute molecules diffuse through the film layer and reach the liquid film-solvent interface, rapid mixing takes place, resulting in a uniform bulk concentration (C). Based upon this description (see Fig. 3.2a), the dissolution rate is determined entirely by Brownian motion diffusion of the molecules in the diffusion layer.

To model the diffusion process through the liquid film, Fick's first law, which relates flux of a solute to its concentration gradient, can be applied:

dx where J is the amount of solute passing through a unit area perpendicular to the surface per unit time. D is the diffusion coefficient, and dC/dx is the concentration gradient, which represents a driving force for diffusion. At steady state, (3.1) becomes

h where C is the bulk concentration, Cs is the saturation concentration, and h is the thickness of the stagnant diffusion layer. Based on (3.2), the dissolution rate, which is proportional to the flux of solutes across the diffusion layer, can be described by dC

dt a. Diffusion Layer Model b. Interfacial Barrier Model c. Danckwerts' Model

Particle surface

Particle surface

Particle surface

Diffusion Layer

Film ^ Boundary

C

h cs

Film ^ Boundary

C

Cs \

Figure 3.2. Schematic illustration of (a) the diffusion layer model, (b) the interfacial barrier model, and (c) the Danckwerts model

dC _ SD ~dt ~ ~Vh where S is the total surface area of particles, and V is the volume of dissolution medium. The term Cs — C represents the concentration gradient within the stagnant diffusion layer with thickness h. This equation is known as the Nernst-Brunner equation (Brunner and Tolloczko, 1900; Nernst, 1904).

In addition to film theory, two other theories were also used to describe the dissolution process. These theories include the interfacial barrier model (Higuchi, 1961) and the Danckwerts model (Danckwerts, 1951). In contrast to the film model, the interfacial barrier model assumes that the reaction at the solid surface is significantly slower than the diffusion across the interface. Therefore, no equilibrium exists at the surface, and the liberation of solutes at the solid-liquid interface controls the overall rate of the transport process. This model is illustrated in Fig. 3.2b. Based on this model, the dissolution rate is given by

where G is the dissolution rate per unit area, and ki is the interfacial transport coefficient.

Under the assumption that the solid surface reaction is instantaneous, the Danckwerts model suggests that the transport of solute is achieved by the macroscopic packets that reach the solid surface, absorb solutes at the surface, and deliver them to the bulk solution. This transport phenomenon is depicted in Fig. 3.2c. The dissolution rate is expressed as

dt where m is the mass of dissolved substances and y is the interfacial tension.

These three models have been employed alone or in combination to describe the mechanism of dissolution. Nevertheless, the diffusion layer model is the simplest and most commonly used to describe the dissolution process of a pure substance among these three models.

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