Bond Angle Restrictions

Influence of Bond Angle Restrictions.—In all real polymer chains the direction assumed by a given bond is strongly influenced by the direction of its predecessor in the chain. The orientation of other nearby bonds (second, third, and possibly fourth neighbors) may also exert an appreciable influence, but the orientation of the immediate predecessor usually is of greatest importance. The exact nature of these restrictions on the direction assumed by a given bond 

Kuhn has shown how a real polymer chain may be approximated by an equivalent freely jointed chain. Instead of taking the individual bonds as statistical elements, one may for this purpose choose sequences of m bonds each. In Fig. 79, arbitrarily chosen statistical elements consisting of five bonds are indicated, the displacement vectors for these elements being shown by the dashed lines. The direction assumed by a statistical element will be nearly independent of the direction of the preceding element, provided the number m of bonds per 

The introduction of vectors of constant displacement length to represent the individual elements, which actually vary in length, is rendered more plausible by inquiry into the effect of incorporating this artifice in the treatment of the freely jointed chain. In this case V = m H. Upon substitution of this expression together with n nlm in Eq. (17), the previous expression for / , Eq. (6), is recovered. Hence the calculated distribution is unaff ected by an arbitrary subdivision of the chain in this manner. We conclude that the value chosen for m in the reduction of the real chain to an equivalent freely jointed chain likewise is inconsequential (within the limits on m stated above). 

The foregoing discussion of equivalent chains requires merely that its root-mean-square end-to-end distance shall equal that of the real chain. In order to define completely the equivalent chain, its contour lengths may also be required to coincide with that of the real chain. 

Thus the fully extended length of the equivalent chain will be set equal to that of the real chain with all valence angle B and t ) restrictions removed. Then n and V (and therefore m) for the equivalent chain are completely defined by 

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C b2 is the effective bond length , b. For the case of the bond angle restriction on the random walk ... 

Figure 2 (a) Sketch of the variables determining the rotation isomeric state model of a statistical coil 9 = bond angle, = bond torsion angle [3], (b) Representation of one of the possible conformations of a two-dimensional statistical coil (33 monomers) with bond-angle restriction ( — 90° <0 <90°) [3],... 

For a freely jointed chain. Coo = 1 but for real chains, bond angle restrictions lead to values of Coo in the range 5-10. For polystyrene, for example. Coo 9.6 at 140°C. A tabulation of Coo for various polymers can be found by looking ahead to Table 3-3, as well as to Fetters et al. (1994 see also Flory 1969). 

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... 

Thus for polymers such as linear polyethylene, bond angle restrictions cause the RMS end-to-end distance to increase by a factor of - /2 from that of the freely-jointed chain. 

As an example, we consider a polymethylene chain (Figure 6-16). Its fully extended length and mean square end-to-end distance are, respectively, (considering only bond angle restrictions)... 

The freely rotating state is a hypothetical state of the chain in which the bond angle restrictions are operative but in which there are no steric hindrances to internal rotation. The Stockmayer-Kurato ratio, a, reflects such rotational isomerism preferences. That is, it is a measure of the effect of steric hindrance to the average chain dimension. It is given by... 

Influence of bond angle restrictions and rotational characteristics on the odd-even effect of thermodynamic quantities... 

Influence of Bond Angle Restrictions and Rotational Characteristics on the Odd-Even Effect of Thermodynamic Quantities... 

Characteristic Ratio A measure of the expansion of a polymer chain due to steric interactions and valence-bond angle restrictions. It is defined as... 

Chain with Bond-Angle Restrictions. Although the chain in the aforementioned treatment was assumed to be freely jointed, a real polymer chain has fixed bond angles 9. Therefore, the second term in equation (9) is no longer zero. The scalar product of two vectors Iflj is cos 0. It can be shown (9,73,74) that... 

Flory P J, Hoeve C A J and Ciferri A (1959) Influence of bond angle restrictions on polymer elasticity, J Polym Sci Polym Symp 34 337-347. 

The latter quantity a represents the effect of long-range interactions which can be described as an osmotic swelling of the chain by the solvent-polymer interactions, while the imperturbed dimension (r ) q represents the effeet of short-range interactions such as bond angle restrictions and steric hindrances to internal rotation. The steric hindrances are also influenced by the torques exerted on the chain by solvent molecules, but the effect is rather small in many cases (11). 

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