Chain statistics freely rotating

The characteristic ratio changes from 1.3 to 2.8 with o changing from 0 to 1, when the virtual bond is used. On the other hand, when each bond of the phenylene group is taken into account individually, the two extreme values are 3.41 and 7.40. By assuming all the statistical weight factors to be unity, which corresponds to the freely-rotating chain, the characteristic ratio is 1.60 when the virtual bond is used, and 4.22 if it is not. 

The methods of conformational statistics, discussed so far, had as starting point the real polymer chain. The aim was to relate the dimensions of the coiled polymer molecule statistically to the mutual displaceability of the chain atoms. Nearly exact relationships are obtained for a large number of freely jointed or freely rotating elements. Under conditions of restricted movability, however, the statistical equations can generally not be solved and empirical factors like s, a and a are introduced. 

The freely-rotating chain of a polymer in dilute solution is represented by a statistical distribution of conformational structures of the chain. The resultant excimer fluorescence emission is a broad Gaussian band. The excimer fluorescence spectrum from a solid polymer results in a very broad spectrum which is characteristic of the distribution of conformations adopted by the polymer in the preparation process. This distribution depends upon the thermal history of the sample and the conditions used for the casting these dependencies were presented by Frank and co-workers [26]. The complexity of photophysical processes accompanying the excimer formation and the inherent complexity... 

The results of an illustrative calculation are depicted in Fig. 3.4. The chain has 0 = 112°, i = 3, and < > = 180° and 60°. The statistical weight matrix for all internal bonds is given by Eq. (3.9) with t = if/ = 1. When cr and freely rotating chain with the same bond angle. Imposition of a symmetric torsional potential that penalizes the g states, with tr = 0.4, increases the C . Introduction of a pair-wise interdependence, via cr = 0.4 and w = 0.1, produces a further increase in C . Obviously, the interdependence of the bonds can have a strong effect on the unperturbed dimensions of the chain. 

Real pol3oner chains have fixed bond lengths and possibly hindered rotation. These effects are taken into account by the concept of the equivalent freely joined chain [7] for example, a paraffin-type chain with unhindered rotation will have rms V2 times greater than the simple model. More sophisticated treatments are possible in terms of random walk statistics. 

The Debye s Gaussian chain is rather ideal as the statistical segment is considered negligible with respect to Rq. More realistic models include either larger statistical segments (freely rotating rods), and wormlike chains [19,20]. 

As discussed further in the following section, it can be shown that the statistical distribution of end-to-end distances for any real chain reduces to the Gaussian form if the number of rotatable links is sufficiently large. By suitably choosing n and / for the freely jointed random-link model, both rms and the fully extended length can be made equal to the corresponding values for the real chain. These values define the equivalent freely jointed random chain. For example, if it is assumed that in a real polyethylene chain (i) the bonds are fixed at the tetrahedral angle and (ii) there is free... 

Here v and m denote the volume and mass of the molecule or atom, respectively. The r.h.s of Equation 32 denotes the ground-state energy of a quantum mechanical particle enclosed in a potential well (particle in a box problem [Martin and Leonard, 1970]). This condition is not satisfied for liquid helium and liquid hydrogen, while liquid neon is a borderline case. For the theoretical description of their thermophysical properties, application of the Maxwell-Boltzmann statistics sometimes does not suffice. Another assumption states that the internal degrees of freedom of the molecules or atoms are the same in the gas phase and in the liquid phase. In other words, it is assumed that the molecules can rotate and vibrate freely in the liquid phase, too. Molecular rotation may be hindered in the case of long-chain hydrocarbons or silicone fluids with side groups but also for small, nonspherical molecules such as N2,02, CS2, and others, rotation around two axes is restricted due to steric hindrance. Polar molecules exhibit restricted rotation due to the effect of dipolar orientation. 

In the simplest model for polymer coils, the chain is supposed to consist of n volume-less links of length / which can rotate freely in space. This model is then called the freely jointed chain model. Since each link can adopt any orientation, the polymer coil effectively executes a random walk, as sketched in Fig. 2.2. This is similar to the Brownian motion of microscopic particles suspended in a fluid (Section 1.4). The effect of the random walk statistics is that the chain coils back, and even crosses itself, many times, leading to a dense clumped up structure. The statistics of random walks were worked out for Brownian motion by Einstein (Section 1.4), and we can simply use the same result for random polymer coils. It turns out that the mean-square end-to-end distance is... 

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