Chain with fixed bond angles

FIGURE 8-28 Schematic diagram depicting on the left a real" chain with fixed bond angles and hindered bond rotation and on the right a freely jointed chain with unrestricted bond angles and bond rotations. 

Chains with Fixed Bond Angles and Independent Potentials for Internal Bond Rotation 69... 

In the first part of this article the review of various theoretical models for polymer chains is given. The models of freely jointed chains, freely rotating chains (including wormlike chains), and chains with fixed bond angles and independent rotational potentials and with interdependent potentials, including rotatimial isomeric state approximation, are presented. 

CHAINS WITH FIXED BOND ANGLES AND INDEPENDENT POTENTIALS FOR INTERNAL BOND ROTATION... 

A chain with fixed bond angle and free internal rotation. The angle 0 between consecutive bonds is fixed for this model 0 - const the internal torsional rotation is completely free. In this case K(l) = cos 0,K n) = (cos 0)". The latter result can be obtained, for example, by calculating (m2), (m3), and so on for a fixed orientation of the first bond ntj. A strongly stretched chain adopts the trans-conformation with Tmax = L = IbN co for even N, that is, Z b = ZbCO. On using eqns [3], [4], and [8] one can find the persistence and the Kuhn lengths ... 

Schematic of a freely jointed chain, (b) Freely rotating chain with fixed bond angle. These models do not incorporate excluded volume effects in their description... 

For a free-rotation chain with fixed bond angle 0 and bond length /, the cross terms in equation (2) do not vanish. After some algebra, there is obtained, for large N... 

FIGURE 3 Freely rotating chain with fixed bond length a and bond angle 0, free rotation around single bond is characterized by angle [Pg.31]

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... 

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... 

Figure 2.8 shows part of a polyethylene glycol chain and illustrates the type of problem encountered with real chains. First, there are fixed bond angles (in this case the tetrahedral angle), and second, there is the physical volume occupied by each atom so that the walk cannot for example cross itself. Both of these constraints mean that the random... 

Here the states for bond i - 1 index the rows of this matrix (in the order t, g from top to bottom), and those of the bond i index the columns (in the same order but from left to right). In the formulation of this matrix, it is implied that the interdependence of rotational states does not extend beyond first neighbors. It is also assumed that all bond lengths and bond angles are fixed, and therefore the conformation of a chain with n bonds may be specified by assigning a rotational state to each rotatable bond. 

FIGURE 1.5 Diagrammatic representation of the cones or revolution available to the third and fourth bonds of a simple carbon chain with a fixed bond angle 0. 

The carbon-carbon bonds constitute the backbone of a polyolefin chain. In principle, each carbon atom contains four bonds aligning along the tetrahedron sp hybrid orbits with the bond angles fixed at 109°287 In other words, the connection of... 

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