# Collision-Controlled Model

When the rate constant kq for the oxygen quenching and the oxygen concentration

[O2] are considered in a collision-controlled reaction, the Stern-Volmer relation is called the collision-controlled model to distinguish from the diffusion-controlled relation (or adsorption-controlled model). The rate of collision of the oxygen molecules on a porous surface is [O2]c /4, where c* is the average speed of the molecules. According to the theory of ideal gas, one knows

where pOq is the partial pressure of oxygen, T is the absolute temperature in

Kelvin, Mm is the molar mass, R is the universal gas constant, and N0 is the Avogadro’s number.

The rate of the oxygen quenching is modeled by a product of an effective contact area and the collision rate

Hence, the rate of the oxygen quenching is proportional to the partial pressure of oxygen or air pressure, but is inversely proportional to the square root of temperature. The Stern-Volmer relation for the luminescent lifetime then becomes

1 Geff No Фо2

-_ ka + , =P

T pKMmRT

For aerodynamic applications, the Stern-Volmer relation for the collision – controlled quenching process can be written as

where the coefficients at the reference conditions are defined as

Acollision, ref 1 /(1 + Z) , Bcollision, ref Z /(1 + Z) ,

OeffN 0Фо2Р ref

karef-l2nMmR Tref

Although Eq. (2.33) has the same form as that for a conventional polymer binder, the Stern-Volmer coefficients AcoUiion and Bcoliisinn have different physical meanings. The coefficient Bmllso>n has weaker temperature dependency that is inversely proportional to the square root of temperature. In contrast, the temperature dependency of Acoliiiion has the same form as that for a conventional polymer binder; linearization of Eq. (2.34) at T = Tref leads to

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