Diffusive boundary exposure time

The theories vary in the assumptions and boundary conditions used to integrate Fick s law, but all predict the film mass transfer coefficient is proportional to some power of the molecular diffusion coefficient D", with n varying from 0.5 to 1. In the film theory, the concentration gradient is assumed to be at steady state and linear, (Figure 3-2) (Nernst, 1904 Lewis and Whitman, 1924). However, the time of exposure of a fluid to mass transfer may be so short that the steady state gradient of the film theory does not have time to develop. The penetration theory was proposed to account for a limited, but constant time that fluid elements are exposed to mass transfer at the surface (Higbie, 1935). The surface renewal theory brings in a modification to allow the time of exposure to vary (Danckwerts, 1951). 

Diffusion tends to equilibrate concentration differences between two reservoirs upon contact fluorine concentration profiles develop at the boundary of the two compartments as a function of time. Studies of the distribution of this trace element in archaeological samples such as bones, teeth or flints allow to gain some age information on the excavated objects of a burial site. The presented technique using beams of accelerated protons allows to measure fluorine diffusion profiles with an excellent space resolution. The surface exposure duration was deduced by the same method for Antarctic meteorites. 

The short time exposure SPME measurement described has an advantage associated with the fact that the rate of extraction is defined by diffusivity of analytes through the boundary layer of the sample matrix, and their corresponding diffusion coefficients, rather than distribution constants. 

It was observed that in some circumstances exposure to ambient fluid is initiated by the formation of a thin molecular layer that adheres to the exposed boundary of the polymer and that the diffusion process involves some time delay before proceeding at full capacity (e.g.. Long and Richman 1960). This observation motivated Long and Richman (1960) to employ a time-dependent boundary condition even under exposure to constant environment, namely... 

Equation (6.29) illuminates the effects of viscoelastic retardation and aging, as well as stress, on the diffusion process. In the absence of those effects, the boundary condition, which reads i9Ao/i9ot = translates into the familiar statement m x,t) = mo t), X on boundary and mo prescribed. Therefore, the term corresponds to a classical process of diffusion through a mechanically inert solid. On the other hand, (6.29) states that when the equilibrium boundary value is approached gradually with time (even for exposure to constant ambient vapor pressure), it is affected by age and depends quadratically on the applied stress. 

---