Fuoss-Kirkwood function

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... 

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) ... 

There were several attempts to generalize the Debye function like the Cole/Cole formula (Cole and Cole 1941) (symmetric broadened relaxation function), the Cole/Davidson equation (Davidson and Cole 1950, 1951), or the Fuoss/Kirkwood model (asymmetric broadened relaxation function) (Fuoss and Kirkwood 1941). The most general formula is the model function of Havriliak and Negami (HN function) (Havriliak and Negami 1966,1967 Havriliak 1997) which reads... 

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... 

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... 

Fuoss and Kirkwood have obtained equations identical with Eqs. 31 and 32 without introducing, explicitly, the exponential decay function. Like Debye they reasoned as if the problem were mainly one of diffusion by Brownian motion under the influence of an external force. Treating this problem as a Sturm-Liouville equation, they developed /i, into a complete set of orthogonal functions fx- A relaxation time tx is associated with each of these functions. 

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. 

Kirkwood and Fuoss [1941] first showed that G(t) could be recovered by integration from a set of e" values. A general treatment has been given by Macdonald and Brachman [1956], who provided a usefnl set of relations between the varions functions used to describe networks and systems as well as between responses to various types of input. 

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

The well-known Cole-Cole plot (0single line in this a, jS plane see Figure 29 (on page 264). The Kirkwood-Fuoss as well as the Gaussian functions are essentially coincident with the Cole-Cole function. The Cole-Davidson function Os]8 l.O is also a single line in this plane and forms one of the bounds that is perpendicular to the Cole-Cole line. The stretched exponential is represented by a hyperbolic-like line in the a, )3 plane and takes the form ... 

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