Diffusion across a thin film

Membrane transport represents a major application of mass transport theory in the pharmaceutical sciences [4], Since convection is not generally involved, we will use Fick s first and second laws to find flux and concentration across membranes in this section. We begin with the discussion of steady diffusion across a thin film and a membrane with or without aqueous diffusion resistance, followed by steady diffusion across the skin, and conclude this section with unsteady membrane diffusion and membrane diffusion with reaction. 

Figure 2 illustrates steady diffusion across a thin film of thickness h. The solutions on both sides of the film are dilute, so the diffusion coefficient can be considered constant. The solute molecules diffuse from the well-mixed higher concentration Ci to the well-mixed lower concentration c2. The concentrations on both sides of the film are kept constant. After sufficient time, a steady state is reached in which... 

Steady diffusion across a thin film is mathematically straightforward but physically subtle. Dissolution film theory, suggested initially by Nernst and Brunner, is essentially based on steady diffusion across a thin film. 

It may be appropriate here to introduce film theory. As mentioned in reference to the steady diffusion across a thin film, we often hypothesize a film called an unstirred layer to account for the aqueous diffusion resistance to mass transfer. Film theory is valuable not only because of its simplicity but also because of its practical utility. However, the thickness of the film is often difficult to determine. In the following, we try to answer the question, What does the thickness of the film represent ... 

Steady diffusion across a thin film with a fixed boundary... 

The classic steady-state diffusion problem is diffusion across a thin film at constant pressure and temperature and with no convection in the direction of diffusion (z direction), as shown in Figure 15-1. At steady state, there is no accumulation in the film and the concentration profile does not change with time. Over a segment of the film of thickness Az, the mass balance is input = output, which can be written in this case as = Ja,z+Az> leads to... 

This emphasis on dilute solutions is found in the historical development of the basic laws involved, as described in Section 2.1. Sections 2.2 and 2.3 of this chapter focus on two simple cases of diffusion steady-state diffusion across a thin film and unsteady-state diffusion into an infinite slab. This focus is a logical choice because these two cases are so common. For example, diffusion across thin films is basic to membrane transport, and diffusion in slabs is important in the strength of welds and in the decay of teeth. These two cases are the two extremes in nature, and they bracket the behavior observed experimentally. In Section 2.4 and Section 2.5, these ideas are extended to other examples that demonstrate mathematical ideas useful for other situations. 

Steady diffusion across a thin film is illustrated schematically in Fig. 2.2-1. On each side of the film is a well-mixed solution of one solute, species 1. Both these solutions are dilute. The solute diffuses from the fixed higher concentration, located at z 0 on the left-hand side of the film, into the fixed, less concentrated solution, located at z / on the right-hand side. 

Fig. 2.2-1. Diffusion across a thin film. This is the simplest diffusion problem, basic to perhaps 80% of what follows. Note that the concentration profile is independent of the diffusion coefficient.

This example is a mainstay of the analysis of diffusion. It is a good mathematical introduction of spherical coordinates, and it gives a result which is much like that for steady diffusion across a thin film. After all, Eq. 2.4-25 is the complete parallel of Eq. 2.2-10, but with the sphere radius i o replacing the film thickness /. Thus most teachers repeat this example as gospel. 

However, we must also see a different and broader blueprint based on physics, not mathematics. This blueprint includes the two limiting cases of diffusion across a thin film and diffusion in a semi-infinite slab. Most diffusion problems fall between these two limits. The first, the thin film, is a steady-state problem, mathematically easy and sometimes physically subtle. The second, the unsteady-state problem of the thick slab, is harder to calculate mathematically and is the limit at short times. 

If the concentration difference for diffusion across a thin film is doubled, what happens to the flux ... 

The second characteristic of this result is its strong parallel with the diffusion across a thin film and diffusion away from a dissolving sphere. For diffusion across a thin film of thickness / from a solution at cjo to a pure solvent with cj = 0, we found in Equation 2.2-10 that... 

Example 7.3-2 Steady-state multicomponent diffusion across a thin film In steady-state binary diffusion, we found that the solute s eoncentration varied linearly across a thin film. Will solute concentrations vary linearly in the multieomponent ease What will the flux be Solution By comparison with Eq. 2.2-9, we see that... 

The sections of this chapter are different attempts to give a prediction of Equation 9.0-1. In Sections 9.1 and 9.2, we discuss the film theory, based on diffusion across a thin film and the penetration and surface-renewal theories, based on diffusion into a semi-infinite slab. In Section 9.3, we discuss why these theories do not predict Equation 9.0-1, and how this disagreement may be resolved. In Section 9.4, we talk about... 

---