Polymers valence angle model

The simplest modification to the freely jointed chain model is the introduction of bond angle restrictions while still allowing free rotation about the bonds. This is known as the valence angle model and for a polymer chain with backbone bond angles all equal to 6, it leads to Eq. (2.5) for the mean square end-to-end distance... 

The valence angle model, though more realistic than the freely jointed model, still underestimates the true dimensions of polymer molecules, because it ignores restrictions upon bond rotation arising from short-range steric interactions. Such restrictions are, however, more difficult to quantify theoretically. The usual procedure is to assume that the conformations of each sequence of three backbone bonds are restricted to the rotational... 

One parameter that is commonly used to specify the dimension of a linear polymer molecule is the root-mean-square (rms) end-to-end length. The simplest, and also the most primitive, model for a polymer molecule is the random flight chain, also termed the freely jointed chain. In this model, the bonds are represented by volumeless lines in space, and there are no restrictions on the valency angles or on the rotations about the bonds. The rms end-to-end length, can be... 

The random flight chain has the simplest mathematical properties, but, unfortunately, also has the smallest degree of strnctnral similarity to real polymers. A more complex model considering the fixed valency angle (x) and the restrictions of free rotation has been proposed ... 

This conclusion is not quite original. As early as the 1930s, Kuhn [71] and Fuoss and Kirkwood [72] suggested that conformational transitions in certain local regions of a macromolecule should be accompanied by deformation of valence angles and bonds in other regions. Similar theoretical models of polymer deformation were considered by Schatzki [73], Boyer [74], Pechhold [75, 76] Gotlib et al. [77-79], and Robertson [80]. 

We begin by considering the liquid state conformation of a typical synthetic polymer, such as a polymer chain consisting of n bonds of identical length / joined at fixed valence angles 9. Linear polyethylene or vinyl polymers (-CH2 — —), fit this model... 

The following three mechanisms of polymer chain flexibility are the best known in polymer physics freely jointed persistence, and rotational isomeric mechanisms [27]. The freely jointed mechanism corresponds to the simplest freely jointed model of a semiflexible polymer chain in which the chain is in the form of a sequence of hinged, long, rigid rods of length I and diameter d, with I d (Fig. 1.2a). In the persistence mechanism, the flexibility is due to gradual accumulation of the effect of small vibrations of valence angles, bonds, etc. A... 

The rotational isomeric mechanism of flexibility is basically characteristic of carbon-carbon chains [28, 29]. The simplest model of the polymer chain which illustrates the basic features of the rotational isomeric mechanism of flexibility is shown in Fig. 1.2c [30]. In this model, the polymer chain is represented as a sequence of units of length a and characteristic thickness d, each of which can be in either the trans position (step forward) or in the gauche position (kink in valence angle Yq)- The sites of the kinks are not fixed, as they can appear with the same probability p in each stq> (p Kuhn segment of this model of the chain I by the relation [31]... 

Experimental information on the valence levels comes essentially from photoemission XPS and UPS measure densities of states (DOSs) convoluted with absorption cross sections, and these DOS values can be compared with those computed from VEH valence-band structures [195]. This has now been done for several CPs and the agreement is good. It would be more instructive to compare the actual band structure to angle-resolved (ARUPS) measurements, but this has never been done. What comes nearest is an ARUPS study of a series of long alkanes taken as models for polyethylene, a nonconjugated polymer [196]. 

While the freely jointed chain is a simple model from which to begin predictions of chain dimensions, it is physically unrealistic. Since each carbon atom in a real polymer chain is tetrahedral with fixed valence bond angles of 109.5°, the links are subject to bond angle restrictions. Moreover, the links do not rotate freely because, as we have seen earlier, there are energy differences between diflferent conformations (cf. Fig. 2.3). Both of these effects cause to be larger than that predicted by the freely jointed... 

Theoretical attempts to relate dimensions of polymers to chemical structure were pioneered by Flory (2). Statistical macromolecular size in solution can be modeled from first principles by considering the number and length of bonds along with valence bond angles and conformational restrictions. Excluded volume, segmental interactions, specific intramolecular interactions, and chain solvation contribute to dimensions. 

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