# Higher-Order Model

In a micro-heterogeneous polymer matrix, the multiple-exponential luminescent emission decay can be observed in contrast to the single-exponential decay in a homogeneous medium (Carraway et al. 1991a; Sacksteder et al. 1993; Xu et al. 1994). This is associated with the fact that the host matrix has domains that vary with respect to their interaction with the luminescent probe molecules; as a result, the excited molecules decay at different rates, depending on their environments. Consider a paint system consisting of a number of independently emitting species with different single-exponential lifetimes тi (i = 1,2,3,-■■) and relative contributions. The multiple-exponential luminescent decay is described as

I(t) = ^ ai exp( -1/гi ), (6.9)

where a{ is the weighting constant for the ith component. The luminescent lifetime of each component obeys the Stern-Volmer relation

T0i / Ti = 1 + KSVip, (6.10)

where KSVi is the Stern-Volmer coefficient for the ith component. Hence, a

higher-order model is needed to describe the luminescent response of an inhomogeneous PSP to a time-varying excitation light. We consider a third-order model

a0 d3I/ dt3 + a1d2I / dt2 + a2dI / dt + a3I = E(t). (6.11)

With the initial conditions I(0) = I'(0) = I"(0) = 0, a solution for (6.11) is

I(t) = J E(u) ^ atexp[ -(t – u )/ti ] du. (6.12)

0 i=1

The lifetimes ti are related to the weighting constants at through the roots of the characteristic equation a0s3 + a1s2 + a2s + a3 = 0 . The weighted mean lifetime is usually expressed as < т > = ^ ai т{/ ^ ai. A general model for the nonexponential decay of luminescence was discussed by Ruyten (2004) and Ruyten and Sellers (2004) considering the continuous decay rate spectrum and excitation response function.

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