Diffusion Timescale

Diffusion Timescale Подпись: (8.26)

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

Paint

Thickness

Qrm)

Life­

time

(ns)

Response

time

Comments

References

LPSF1 (pyrene)

2

5 ms

OPTROD formulation

Borovoy et al. (1995)

PSPL2 (pyrene)

20

0.2 s

OPTROD formulation

Fonov et al. (1998)

PSPL4 (pyrene) PSPF2 (pyrene) PF2B (Ru(dpp))

13

5

0.172 s 0.1-2.6 ms 0.48 s

OPTROD formultation OPTROD formulation McDonnell Douglas (MD)

Fonov et al. (1998) Fonov et al. (1998) Carroll et al. (1996b)

PF2B (Ru(dpp))

15

5

0.88 s

formulation MD formulation

Carroll et al. (1996b)

PF2B (Ru(dpp))

25

5

1.2 s

MD formulation

Carroll et al. (1996b)

PF2B (Ru(dpp))

35

5

2.4 s

MD formulation

Carroll et al. (1996b)

PtOEP/polymer

19

50

0.82 s

concentrated luminophore near

Carroll et al. (1996b)

PtOEP/GP197

22

50

1.4 s

outer surface of the binder

Carroll et al. (1996b)

PtOEP/GP197

26

50

1.6 s

Carroll et al. (1996b)

PtOEP/GP197

32

50

2.4 s

Carroll et al. (1996b)

Ru(dpp)/RTV

6

5

22.4 ms

Winslow et al. (1996)

Ru(dpp)/RTV

11

5

58.6 ms

Winslow et al. (1996)

Ru(dpp)/RTV

16

5

148 ms

Winslow et al. (1996)

Ru(dpp)/RTV

20

5

384 ms

Winslow et al. (1996)

Ru(dpp)/PDMS

4-5

5

3-6 ms

Hubner et al. (1997)

PtOEP/GP197

50

2.5 s

Baron et al. (1993)

PtOEP/copolymer

50

0.4 s

Baron et al. (1993)

H2TFPP/silica

1.5-10 ms

silica with a binder

Baron et al. (1993)

h2tfpp/tlc

luminophore/AA

25 ns 18-90 ns

depended on the luminophore

Baron et al. (1993) Mosharov et al. (1997)

Ru(dpp)/FIB and

5

<500 ns

and anodization processes approached the apparatus

Ponomarev &

alumina

PtTFPP/FIB and

50

<500 ns

response time approached the apparatus

Gouterman (1998) Ponomarev &

alumina

PtTFPP/porous

50

60 ns

response time

Gouterman (1998) Scroggin (1999)

ceramic

Ru(dpp)/AA

5

80 ns

Sakaue et al. (2001)

Ru(dpp)/TLC

5

70 ns

Sakaue et al. (2001)

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