Kuhn model

The Kuhn model is presented in detail (and in parts mathematically justified) in the articles referred to above. It is unclear why Kuhn s ideas have not always received due attention in the literature in comparison with other theories. Without going into detail, some features will be mentioned briefly. 

Water sorption isotherms may be determined experimentally by gravimetric determination of the moisture content of a food product after it has reached equilibrium in sealed, evacuated desiccators containing saturated solutions of different salts. Data obtained in this manner may be compared with a number of theoretical models (including the Braunauer-Emmett-Teller model, the Kuhn model and the Gruggenheim-Andersson-De Boer model see Roos, 1997) to predict the sorption behaviour of foods. Examples of sorption isotherms predicted for skim milk by three such models are shown in Figure 7.12. 

Fig. 9.17 Models for a disordered polyacetylene chain (a) the Kuhn model and (b) the worm-like chain.

Fig. 30. Schematic representation of the shape of polydiacetylene chains. Top planar, fully conjugated chain, middle Kuhn model, bottom woim-like chain...

Fig. 7.1.8 A cross-link chain in a rubber network. Following the Kuhn model [Kuh3] it can be decomposed into Ne freely jointed but rigid segments. Their length and their number depends on the temperature and the stiffness of the chemical structure.

When all of these effects are taken into account, a characteristic ratio C may be introduced as a measure of the expansion of the actual end-to-end distance of the polymer chain, i o, Ifom that calculated from a Kuhn model ... 

Kuhn model, Equation (1.1). Data for several polymers in addition to polyethylene are given, including a rigid-rod aromatic nylon polymer, poly(p-phenylene terephthalamide) (Kevlar ), as well as the aliphatic nylon polymer poly(hexamethylene adipamide) (nylon-6,6). 

Assuming that the blobs have a Gaussian distribution and the Kuhn model of the blob chain applies then... 

Figure 32.2 The Kuhn model of a polymer, is the length of the chemical bond, bf is the length of a virtual, or Kuhn, bond directed along the chain axis, and (p is the angle between the chain axis and the chemical bond. 

These arguments suggest a strategy for treating real chains. The idea is to represent a real chain by an equivalent freely jointed chain. The equivalent chain has virtual bonds, or Kuhn segments, each of which represents more than one real chemical bond. The number of virtual bonds Nk and the length of each bond bK are determined by two requirements. First, the Kuhn model chain must have the same value of (r ) as the real chain, but it is freely jointed so its characteristic ratio equals one,... 

Abstract This introductory chapter provides a brief (textbook-like) survey of important facts concerning the conformational and dynamic behavior of polymer chains in dilute solutions. The effect of polymer-solvent interactions on the behavior of polymer solutions is reviewed. The physical meanings of the terms good, 9-, and poor thermodynamic quality of the solvent are discussed in detail. Basic assumptions of the Kuhn model, which describes the conformational behavior of ideal flexible chains, are outlined first. Then, the correction terms due to finite bond angles and excluded volume of structural units are introduced, and their role is discussed. Special attention is paid to the conformational behavior of polyelectrolytes. The pearl necklace model, which predicts the cascade of conformational transitions of quenched polymer chains (i.e., of those with fixed position of charges on the chain) in solvents with deteriorating solvent quality, is described and discussed in detail. The incomplete (up-to-date) knowledge of the behavior of annealed (i.e., weak) polyelectrolytes and some characteristics of semiflexible chains are addressed at the end of the chapter. 

In the part devoted to neutral polymers, we mentioned that semiflexible and stiff chains do not obey the behavior predicted by the Kuhn model. Restricted flexibility of the chain can be caused by the presence of stiff units with multiple bonds or bulky pendant groups, but it can be a result of external conditions or stimuli. In the preceding part, it was explained in detail that repulsive interactions together with entropic forces increase the stiffness of PE chains. Hence, a sudden pH change can be used as a stimulus affecting the stiffness of annealed PE chains. The properties of semiflexible polymers are usually treated at the level of the wormlike chain (WLC) model developed by Kratky and Porod [31]. The persistence length, /p, is an important parameter strongly related to the WLC model and has been used as the most common characteristic of chain flexibility—in both theoretical and experimental studies. It is used to describe orientational correlations between successive bond vectors in a polymer chain in terms of the normalized orientation correlation function, C(s) = (r,.r,+j). For the WRC model, this function decays exponentially ... 

Calculate from the Kuhn model equation the modulus at room temperature of natural rubber (p 970 kg m ) crosslinked with n molar fraction of organic peroxide. Assume that each peroxide molecule results in one crosslink. 

In addition to the above result for the size exponent, several quantities such as the probability distribution function for finding a particular end-to-end distance can be derived for the Kuhn model. In particular, the results become simple if the end-to-end distance is smaller than the chain contour length Nl.ln this limit, for example, the probability distribution function for the end-to-end distance is a Gaussian function. In view of this, a Kuhn model chain with large enough N is called a Gaussian chain. The major properties of a Gaussian chain are now summarized. 

Many polymer chains are not completely flexible under the usual experimental conditions of interest. In order to incorporate the local chain stiffness, the Kuhn model is modified slightly by introducing a bond angle 180-0 between the consecutive Kuhn steps, as sketched in Figure 2.12a. Obviously, this angle is a parameter to capture the backbone stiffness of the chain. Further, let us assume that the Kuhn steps are freely rotating, and now the model is called the Kratky-Porod or wormlike chain model. 

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