Permeation through a functional barrier

The starting point for modeling permeation (migration) to the liquid is the second case (ii). This is because it represents the well-studied diffusion of a solute from a polymer of limited volume, VP, into a stirred solution of limited volume, VL. A suitable equation for all of these cases is Eq. (7-51), where cP0 = cPe. 

Let us consider the laminate system for situation (ii) with a 1 and a very short contact time t = t. This means the initial solute concentration in the vicinity of x=d at t=0 is Cp=cPe and cl.i = 0 (Fig. 7-14a). This illustration is the case of a system with diffusion between two semi-infinite media (Crank, 1975) for which Eq. (7-51) reduces to Eq. (7-54). 

A more realistic situation for diffusion in a laminate is illustrated in Fig. 7-14b, which shows the solute concentration profile in the barrier layer after a short contact time t=tj. In this illustration the concentration profile of the solute just reaches the polymer/liquid interface and cL.t = 0. If we now consider a similar case with a semi-infinite polymer system with the initial solute concentration (cP e) at the distance x xQ = a+b/2 and cP=0 at x x0 and t=0 (Fig.7-14c), then the possible concentration profiles for the three different times, tctj, t=t, and t tj can be illustrated in Fig. 7-14d. If we assume a mass transfer through the interface A at x=x at t=t, in Fig. 7-14d, then mpt/A = 0.5cpepp(d-X ), which corresponds to mP, /A= cPepp(x0-a) = cPeppb/2 in Fig. 7-14c. If we combine this result with Eq. (7-54) for t=t, then we obtain the time 

If we allow diffusion to continue until t=t2 t, then under the same assumptions of a semi-infinite system, the mass transfer during At = t2—ti is 

Dp is the diffusion coefficient of the solute at some temperature (T ) for time t, for example, the extrusion temperature of the laminate, where the diffusion of the solute into the barrier layer is most significant. Dp is the diffusion coefficient of the solute in the polymer at the temperature during the contact with liquid L. By using the relative time 0r instead of 0 and the general valid Eq. (7-51) instead of Eq. (7-54), a final equation for the migration of the solute from the core layer P through the barrier layer B after the contact time t can be written similar to the form of Eq. (7-51)  

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In chapter 7, section 7.2.8 an example of permeation through a functional barrier is described. Three-layered coextruded PET films were produced in which the core layer (P) was contaminated with chlorobenzene and the outer barrier layers (B) were made with virgin material. During the coextrusion process a partial contamination of the virgin layer occurred. The symmetrical structure of this film leads to a simplified treatment of it as a two layer laminate with the thickness d = a + b = 160 + 40 = 200 pm. For the modeling of this problem with numerical mathematics all parameters given in Section 7.2.8 are used. 

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