Chain model, freely jointed valence angle

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... 

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... 

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... 

Problem 2.7 For a linear molecule of polyethylene of molecular weight 1.4x10 what would be the RMS end-to-end distance according to the valence angle model as compared to that according to the freely-jointed chain model and the end-to-end distance of a fully extended molecule. Comment on the values obtained, indicating which one is a more realistic estimate of chain dimensions. 

This means that directions of neighboring bonds are completely uncorrelated. The freely jointed chain model corresponds to a chain with fixed bond lengths and with unconstrained, free to adjust valence angles and with free torsional rotations. The mean square end-to-end vector in the unperturbed state (denoted by subscript 0) for the freely jointed chain is... 

As discussed in previous chapters, the choice of the bond / o rf( r/ ) and short-range —r l) potential varies from simulation to simulation. Off-lattice models, for example, have used the harmonic-spring potential, the FENE (finitely extendable, nonlinear elastic) potential, the rigid bond with fixed valence angles, and the freely-jointed chain model to represent the bonding interaction between adjacent monomers. For the short-range... 

The simplest modification to the freely jointed chain model allows for a fixed valence angle, 6, between successive bonds (Fig. 2.3), although it is assumed that free rotation about these bonds is still possible (i.e. 0 can take any value). This is called the valence angle model. Here... 

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... 

Calculation of the End-to-End Distance. The characteristic dimensions of a chain with perfectly flexible joints can be described by means of the random walk model. In this case, the N-step trajectory of a random walker—who can freely return to a site previously visited—is assimilated to a chain of length NL the only element taken into consideration is thus the bond length (L). The valence (0) and rotation (macromolecular chain is to calculate the end-to-end distance (r) connecting its two chain ends by vector analysis. 

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