Distribution function Kirkwood-Fuoss

In order to obtain better agreement with experimental results, the concept of a distribution of correlation times was introduced in nuclear magnetic relaxation. Different distribution functions, G(i c), such as Gaussian, and functions proposed by Yager, Kirkwood and Fuoss, Cole and Cole, and Davidson and Cole (asymmetric distribution) are introduced into the Eq. (13), giving a general expression for... 

Treating the problem as one of rotary Brownian movement, Kirkwood and Fuoss were able to calculate the distribution functions and F (r) of Eqs. 22 and 29. In their case F(r) was a symmetrical function and they identified the average relaxation time with the value corresponding to the maximum in the loss curve. Unfortunately their theory is incompatible with existing experimental data on dilute solutions, since it specifies that should be proportional to the degree of polymerization. 

Two other distribution functions are due to Kirkwood and Fuoss [1941] and Davidson and Cole [1951] (see also Davidson [1961]). 

It was soon realized that a distribution of exponential correlation times is required to characterize backbone motion for a successful Interpretation of both carbon-13 Ti and NOE values in many polymers (, lO). A correlation function corresponding to a distribution of exponential correlation times can be generated in two ways. First, a convenient mathematical form can serve as the basis for generating and adjusting a distribution of correlation times. Functions used earlier for the analysis of dielectric relaxation such as the Cole-Cole (U.) and Fuoss-Kirkwood (l2) descriptions can be applied to the interpretation of carbon-13 relaxation. Probably the most proficient of the mathematical form models is the log-X distribution introduced by Schaefer (lO). These models are able to account for carbon-13 Ti and NOE data although some authors have questioned the physical insight provided by the fitting parameters (], 13) ... 

Density functions used earlier to interpret the relaxation data of polymers were the Cole-Cole function,70 the Fuoss-Kirkwood function,71 and the log 2) function.72 These functions, particularly the skewed log ( 2) distribution, were accounted for by 13C T, and n.O.e. data of some polymers, but the physical significance of the adjustable parameters has been questioned by some authors.68... 

The inversion of transform (5.3.2) and the determination of L (t ), when the analytical dependence Y j is known, have been considered previouslyThe results were based on the well-known inversion equations of Kirkwood and Fuoss (see also Ref. ) establishing a relationship between L(t ) and the corresponding dynamic compliance function Z(ico). Indeed, the reduced complex dynamic compliance corresponding to a distribution L j) is given by... 

---