Correlation function CONTIN

The inset shows a unimodal distribution of relaxation times r = I 1 obtained by a CONTIN analysis. Besides CONTIN there is a number of alternative techniques [51] for the determination of the distribution from the correlation function. Detailed discussions of this topic have been given by Stock and Ray [52] and by Stepanek [50]. 

The function (t) can be analysed by the method of cumulants [57] or by inverse Laplace transformation. These methods provide the mean relaxation rate T of the distribution function G(T) (z-average). For the second analysis procedure mentioned above, the FORTRAN program CONTIN is available [97,98]. It is sometimes difficult to avoid the presence of spurious amounts of dust particles or high molecular weight impurities that give small contributions to the long time tail of the experimental correlation functions. With CONTIN it is possible to discriminate these artifacts from the relevant relaxation mode contributing to (t). 

Any attempt to explain our result of bond reorientation dynamics on the basis of superposition of rotational diffusion processes encounters a contradiction. If such an explanation was to be valid, the same spectrum of D had to be able to explain the shape of the observed Mi(t) and MaCO functions and the broad nature of the reorientation angle distribution W(6,t) at the same time. The spectrum g(x) of correlation time t or the equivalent spectrum g(D) of the rotational diffusion coefficient D can be evaluated from the correlation functions by means of a numerical procedure such as CONTIN [45]. When the correlation function can be represented by an analytical function, the spectrum can be obtained more conveniently by means of inverse Laplace transformation. In the case of a KWW function, with t and p characterizing the function as given in Eq. (12), g(D) can be calculated by [46]... 

In dynamic LLS, the Laplace inversion of each measured intensity-intensity time correlation function G q, t) in the self-beating mode can result in a line-width distribution G(L). G(7) can be converted into a translational diffusion coefficient distribution G(D) or further a hydrodynamic radius distribution /(Rh) via the Stokes-Einstein equation, Rh = (kBTI6nrio)/D, where kB, T and qo are the Boltzmann constant, the absolute temperature and the solvent viscosity, respectively. The time correlation functions were analyzed by both the cumulants and CONTIN analysis. 

The CONTIN method uses a regularization technique to seek smooth solutions, no matter whether the G(r) distribution is unimodal, multimodal, or broad. So the CONTIN method is appropriate for photocount correlation profile analysis without an a priori assumption on the form of the G(r) distribution. We used the CONTIN method, which was kindly provided by Dr. S.W. Provencher (European Molecular Biology Laboratory), mainly for correlation function profile analysis of unimodal and bimodal G(r) distributions. 

Another ternary solution SPS43 was a 4-arm star PS (Afu =3.0 x 10, M] /Mn 1.10) mixed with a similar molecular weight probe linear PMMA (Muj=3-3 X 10, Muj/Mfi 1.10). The results were very similar to those of LPS98. Figure 10 shows the measured correlation function and the CONTIN... 

Figure 10.18. Number of occurrences of Phanerozoic ironstones (upper diagram, data from Van Houten and Bhattacharyya, 1982) and odlitic limestones (lower diagram, data from Wilkinson et al., 1985) as a function of geologic age. The relative sea level curve is that of Hallam (1984). Minima in occurrences appear to correlate with times of sea level withdrawal from the continents and major cycles of orogenesis (Caledonian, Hercynian, and Alpine).

---