Relaxation Phenomena in Poled Electro-Optic Polymers
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been probed by time and frequency dependent studies in order to determine the characteristic times associated with the decay. In addition, temperature dependent studies of the characteristic times helps elucidate the mechanism as well as estimate the energies which characterize the relaxation. Many of the studies are carried out near and above the glass transition temperature. Others, more relevant to the actual behavior of the polymers in devices, were carried out deep in the glassy state. In this paper, we describe measurements of the activated reorientation of molecules which is more apparent on short time scales. These measurements connect well with dielectric measurements and suggest a figure of merit for the degree of coupling between the nonlinear optical chromophore and the polymer host. Before describing the experiments, we now discuss the stretched exponential with which we fit our activated reorientation data. THE STRETCHED EXPONENTIAL A great deal of literature exists on relaxation phenomena in polymers in the glassy state. These include mechanical and dielectric relaxation studies which are carried out both in the time and frequency domain. An early study of mechanical relaxation was carried out by Kohlrausch in the last century, and later applied to many polymer phenomena by Williams and Watt.[17] They found that relaxation behavior followed a stretched exponential decay:
x(t) = xoexp[-(a(T)t)1l
(1)
where x(t) is the quantity of interest being studied. In the case of mechanical relaxation it would be the normalized strain. The quantity a(T) is the temperature dependent relaxation rate or inverse relaxation time, a = 1/r. The exponent f is quantity describing the dispersive nature of the relaxation. If Q = 1, for example, a simple exponential decay results. For • 1. Though, originally, the stretched exponential was an ema stretched exponential 0 < < pirical model, statistical models leading to the stretched exponential have been proposed.[18] The concept underlying the stretched exponential is related to behavior exhibiting fractal time.[19] This means that the first moment of the distribution, < t >, is infinite; that is, no mean time characterizes the physical process. In this case, the relaxation process appears similar when viewed on any time scale. In order to generate fractal time (and the stretched exponential), one requires Bernoulli scaling where an order of magnitude longer intermittence is accompanied by an order of magnitude less probability. With b describing the intermittence and a the probability, a probability distribution reflecting this property can be written as,[19] 1-a 00 (2) 1a'b"exp(- bt) 1(t) = a
n=1
which is finite for b < a < 1. This distribution decays asymptotically to 40(t) -. t-1-6 where # = In a/ In b is the width or dispersion parameter of the stretched exponential. The probability distribution can also be related to a weighted distribution of Poisson processes, ¢(t) = foJA exp(- At)p(A)dA
(3)
where p is the weighting function. If A varies as bn and p as a , then
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