Freely orienting chain

The effect of tetrahedral bond angles on the radial distribution function enters through the parameter b, the value of which decreased from 3/2nl2 in the freely orienting chain to 3/4nl2 in the chain with tetrahedral bond angles. 

The mean-square end-to-end distance of a freely orienting chain is deduced from equations (7) and (10) (73-76) as follows ... 

One common postulate or model used is that of the freely orienting chain where it is stipulated that two bonds may assume with equal probability any orientation whatsoever with respect to each other. Consequently the average projection of one bond on another will be zero and the average value of ai 82 or of any other pair will be zero so that eqn (3.4) reduces to ... 

There are two obvious objections to the application of the freely orienting chain molecule when applied to real polymer molecules. 

Exact Treatment for the Freely Jointed Chain (or Equivalent Chain).4 >5— Consider one of the bonds of a freely jointed chain acted upon by a tensile force r in the x direction. Letting xpi represent the angle between the bond and the o -axis, its component on the x-axis is Xi = l cos pi. The orientation energy of the bond is —rXi, and the probability that its x component has a value between Xi and Xi- -dxi therefore is proportional to... 

Considering the large variation of / for the poly[2]catenand 51b, it is expected that little correlation will exist between the spatial orientation of neighboring monomer segments and that it will represent the closest synthetic equivalent of the freely jointed chain model [63]. In this model, a real polymer chain is replaced by an equivalent chain consisting of N rectilinear segments of length Z, the spatial orientations of which are mutually independent (Scheme 24) [63]. 

In Sect. 3, we will consider the orientational ordering in the solution of semiflexible macromolecules. In general, semiflexible macromolecules can have different flexibility distributions along the chain contour compare, for example, the freely-jointed chain of the long thin rods (Fig. 1 b) and the persistent chain, which is homogeneous along the contour (Fig. lc). We will see what properties of the liquid-crystalline transition do depend on the flexibility distribution along the drain contour and what properties are universal from this point of view. 

The extended or contour length of the freely oriented macromolecule will heal. Its rms end-to-end distance will be from the above equation. It can be seen that the ratio of the average end-to-end separation to the extended length is ct Since a will be of the order of a few hundred even for moderately sized macromolecules, d will be on the average much smaller than the chain end separation in the fully extended conformation. 

To find the anisotropy of the whole molecule in molecular coordinates h, the value of cos d should be averaged over all chain elements and all its conformations corresponding to a given value of h. To solve this problem Kuhn and Griin ) have considered the statistics of the intramolecular orientational distribution of segments with respect to h in a freely jointed chain and have shown that for the most probable distribution the value of cos d averaged over all chain segments can be expressed as a unique function of the h/L ratio... 

Different conformations in the freely jointed chain model correspond to different sets of orientations of bond vectors Tj in space [see Fig. 2.14(a)]. The orientation of each bond vector F] can be defined by the two angles of the spherical coordinate system 9 and [Fig. 2.14(b)]. Therefore, the sum... 

Figure 6-15. Radial distribution function of a chain of 104 freely orienting segments each of length 0.25 nm.

The case of the freely-jointed chain is trivial, /iso is, of course, the same as that given by Eq. (8.2.26). The other terms are vastly simplified by the fact that the orientation of a segment is uncorrelated with its center-or-mass position and that orientations of different segments are uncorrelated. Thus in Eqs. (8.9,5) and (8.9.6) all terms in the summations proportional to P2 are zero unless / = j. Note also that the / term in lyy is zero. 

The model of freely connected chain, introduced by Kuhn and Mark, has a great significance for polymers conformational properties description [1], In this model a polymer chain is replaced by an equivalent one, consisting of rod-like segments, spatial orientation of which is mutually independent. The freely connected chain statistics application possibility to real macromolecules is based on the fact, that any real chain at sufficient length acquires freely connected chain properties. 

In the simplest picture, the polymer molecule can be considered as a chain consisting of N segments with length b, each of which is free of any constraint to orient in an arbitrary direction. In other words, the orientation of a segment is totally independent of other segments in the chain and is random. Such a chain is referred to as the freely jointed chain model. 

Let us discuss the conformation of a freely rotating chain a little bit more in detail. A part of the chain is depicted in Figure 3, quantity a symbolizes the length of the C-C bond. For the end-to-end distance, we get Equation (5). Without fixed orientation between two successive bonds, it is ( n m ) = 0 for /w. However... 

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