For a porous polymer layer where diffusion is Fickian under some microscopic assumptions (Cunningham and Williams 1980; Neogi 1996), the diffusion equation Eq. (8.1) is still a valid phenomenological model as long as the diffusivity Dm is replaced by the effective diffusivity Dmeff. Hence, an estimate
Eq. (8.26) as a generalized form of Eq. (8.13) clearly illustrates how the fractal dimension d fr and the porosity parameters aV rp-o1re and hpore / h affect the
response time of a porous PSP. For aV rp-1re << 1 or hpore/h << 1, Eq. (8.26)
naturally approaches to the classical square-law estimate Eq. (8.13) for a homogenous polymer layer.
On the other hand, for aV r-1ore >> 1 and hpore /h ~ 1, another asymptotic estimate for rdiJf is a simple power-law as well
Tdiff ~ h2-dfr/Dm. (8.27)
The estimate Eq. (8.27) is asymptotically valid for a very porous polymer layer. The exponent in the power-law relation between the response time Tdiff and
thickness h deviates from 2 by the fractal dimension d fr due to the presence of
the fractal pores in the polymer layer. The relation Eq. (8.27) provides an explanation for the experimental finding that the exponent q in the power-law relation Tdiff hq is less than 2 for a porous PSP. In addition, this relation can
serve as a useful tool to extract the fractal dimension of the tube-like pores in a very porous polymer layer from measurements of the diffusion response time. For example, the fractal dimension dfr of a pore in the polymer Poly(TMSP) is
dfr = 1.71, while for GP197/BaSO4 mixture the fractal dimension dfr is close to one. In addition, based on the experimental results shown in Fig. 8.5, we know that the fractal dimension dfr for Poly(TMSP) linearly decreases with
temperature in a temperature range of 293.1-323.1 K. This implies that the geometric structure of a pore in Poly(TMSP) may be altered by a temperature
change. Note that the diffusivity Dm of oxygen mass transfer is also temperature – dependent, but it is independent of the coating thickness h. Therefore, the experimental results in Fig. 8.5 mainly reflect the temperature effect on the geometric structure of pores in the polymer rather than the diffusivity.
Table 8.1. Response times and luminescent lifetimes of PSPs