AdsorptionControlled Model
Besides the collisioncontrolled quenching, an adsorbed oxygen molecule on a porous surface can also quench the luminescence; if this is the dominant mechanism, the oxygen quenching is controlled by adsorption and surface diffusion of the adsorbed oxygen on the porous surface. The oxygen concentration on a porous surface, [OJad, can be described by the fractional coverage of oxygen on the porous surface
Q [ O 2 ] ads
[O2] adsM
where [O2]adSM is the maximum oxygen concentration on the porous surface. The SternVolmer equation is then written as
I – _ 1 + kqT0[O2]adM Q, (2.38)
and accordingly the convenient form of the SternVolmer relation for aerodynamic applications is
Q
EaL _ A(T) + B(T)——, (2.39)
I Qref
The rate constant kq for the oxygen quenching, which is surfacediffusion – controlled, can be described by (Freeman and Doll 1983)
kq = 2nRAB *bd = К exP(Esdiff /RT), (2.41)
where Rab is the relative distance between an adsorbed oxygen and an adsorbed luminophore, and D is the diffusivity and the parameter AB is a ratio of the modified firstorder and secondorder Bessel functions of the second kind. Basically, kq is temperaturedependent due to the Arrhenius relation
D = D0 eXP(Esdiff /RT) .
To describe в, either the Langmuir isotherm or the Freundlich isotherm can be used (Carraway et al. 1991b). The Langmuir isotherm relates в to the partial pressure of oxygen pOq in the working gas by
b pO
в= °2 . (2.42)
1 + bPo2
The factor b in Eq. (2.42) is a function of temperature (Butt 1980)
The coefficient A[nngmut has the same temperature dependency as that for a conventional polymer PSP and that in the collisioncontrolled model, i. e.,
and the linearized form for Ar. is
Langmuir
Hence, Eq. (2.48) indicates that A[angmur is related to the temperature dependency of the nonradiative processes of the luminophore. On the other hand, B[amgmidr has the following temperature dependency
where E, — Esdiff + Eads. Rewriting Eq. (2.49) in an exponential form yields
and furthermore, linearization of Eq. (2.50) at T = T gives
Where El — E, – R Tref / 2 — Ediiff + E ads – RTref/ 2 . С^аГ^ the temperature
dependency of the coefficient B[angmiit, Eq. (2.51), is associated with both surface diffusion and adsorption; but it has the similar form to Eq. (2.23) for a conventional polymer layer. The reference SternVolmer coefficients ALangmuir, ref
and BLcngmuir, ref (their lengthy expressions are not given here) satisfy the
constraint Ahangmuir, ref + B Langmuir, ref = 1 .
The Freundlich isotherm can serve as another model for surface adsorption
0 — bp(Po2)r (2.52)
where the coefficient and exponent are
і RT
bf —~b= exp(Eads /RT) and у ——————– . (2.53)
■^TY EadsM
The exponent /is an empirical parameter that is temperaturedependent. For a known yref at a known reference temperature Trep EdisU is given by
































which is similar to B^ . After rewriting all the terms in Eq. (2.61) in an
exponential form, linearization at T = Tref yields
where
Similar to the Langmuirtype model, the coefficient BFrmndlich has the temperature dependency associated with surface diffusion and adsorption. However, the photophysical model Eq. (2.55) describes the nonlinear behavior of the Stern – Volmer plot for a porous PSP.
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