Freely rotating rods

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]. 

Freely rotating rods consist of an assembly of rods linked in series to one another with no spatial orientation with respect to one another. The form factor is then written as [21] ... 

Unlike freely rotating rods, wormlike chains are characterized by a so-called persistence length Ip and the following... 

In a very crude sense, liquid crystalline polymers can be regarded as a freely-jointed-rod chain, shown in Figure 2.16. The freely-jointed-rod chain consists of a series of repeated segments of length lo- Each segment is able to rotate freely. It is assumed that the freely-jointed-rod chain is a replica of its small molecular mass liquid crystal counterpart in the liquid crystalline properties. 

For re > 20 and freely rotating particles, Simha (111) found that the Einstein constant for spheroids and rigid rods is given by... 

Yamakawa and Fuji verified the accuracy of the resulting eq 2.34 by Monte Carlo calculations of ( r - r ) for freely-rotating chains with veiy small complementary bond angles. Their finding was confirmed later by Norisuye et al. [23], who evaluated (t + c /4) ( r — r l ) near the rod limit up to the fifth power in t/2[Pg.148]

In 1944 Kramers [1] published a phase-space kinetic theory for the steady-state potential flow of monodisperse dilute polymer systems in which the polymer molecule is modeled as a freely jointed bead-rod chain. Subsequent scholars developed kinetic theories for shearing flows of monodisperse dilute polymer solutions Kirkwood [2] for freely rotating bead-rod chains with equilibnum-averaged hydrodynamic interaction. Rouse [3] and Zimm [4] for freely jointed bead-spring chains, and others. These theories were all formulated m the configuration space of a single polymer chain. 

Wang and Pecora [51] earlier presented an analytical solution applicable to a restricted rotational diffusion model, in which a rigid rod undergoing a rotational diffusion is allowed to change its direction only up to an angle o and not beyond. The solution contains two parameters, o and the diffusion coefficient D,. In Fig. 22 the solid lines are those fitted by the analytic equations using nearly identical numeric valu of o and D, for both Mi(t) and Mzft) in either the polyethylene or the freely-rotating chain model. Similarly in Fig. 23 the observed time correlation functions are fitted by analytical solutions for... 

The spectra of NF2 in Ar matrices at 4.2 K were compared for two different mole ratios of matrix-to-radical precursor, M/R = 300 and 1200 (deposition of room-temperature equilibrium mixtures of N2F4/NF2 and matrix gas onto a liquid helium cooled sapphire rod). The M/R = 300 sample gave an approximately axially symmetric g tensor indicating nearly free rotation of the NF2 radical about an axis perpendicular to the molecular plane (x axis, see below). The more diluted sample (M/R = 1200) unexpectedly exhibited the spectrum of a randomly oriented radical with three different principal values for the g tensor this may be due to a stronger crystalline field effect. The spectrum of NF2 in Kr at 4.2 K (M/R = 300) indicated freer rotation than in Ar with M/R = 300. At about 30 K the completely resolved isotropic triplet set of triplets of an almost freely rotating NF2 radical was observed in the Ar and Kr matrices [6, 7]. With CCI4 as the matrix material, somewhat distorted spectra were observed at 4.2 to 30 K [6]. 

It has not proved possible to develop general analytical hard core models for liquid crystals, just as for normal liquids. Instead, computer simulations have played an important role in extending our understanding of the phase behaviour of hard particles. It has been found that a system of hard ellipsoids can form a nematic phase for ratios L/D > 2.5 (rods) or L/D < 0.4 (discs). However, such a system cannot form a smectic phase, as can be shown by a scaling argument in statistical mechanical theory. However, simulations show that a smectic phase can be formed by a system of hard spherocylinders. The critical volume fractions for stability of a smectic A phase depend on whether the model is that of parallel spherocylinders or, more realistically, freely rotating spherocyHnders. 

Exercise. A rod-like molecule rotates freely in a plane, but is subjected to a Langevin force due to the surroundings ... 

We assume that the rods can rotate freely with respect to their origins in this case the operator g does not act on the variable 13. For this model, the operator g remains Hermitian and ip = ip. ... 

All the polymers studies so far contain alternating 1,3-trans cyclobutane and 1,4-arylene units in the main chain. As a matter of course, these polymers are stiff rods even when the o-bonds between the cyclobutane and aromatic rings in the chain rotate freely, as is obvious in poly-DSP (Fig. 17). 

A suspension of long rods with aspect ratio of 50 or more can only be considered dilute if its concentration is very low, less than 1 % by volume. The reason is that diluteness requires that the rods be able to rotate freely without being impeded by neighboring rods (Fig. 6-18a), and the volume that a single long rod can sweep out by rotation about its center of mass must be large, around. Thus, rod-rod interactions should be expected when the number concentration of rods, y, reaches a value proportional to 7. . Experimentally (Mori et al. 1982), found that the transition occurs at more than 30 times this estimate, apparently because a rod can easily dodge several other rods that invade its sphere of rotation. Thus... 

This arises from the loss in configurational entropy of the chains on the approach of a second particle. As a result of such an approach the volume available for the chains becomes restricted, which results in a loss of the number of configurations. This can be illustrated by considering a simple molecule, represented by a rod that rotates freely in a hemisphere across a surface (Figure 8.3). When the two surfaces are separated by an infinite distance, 00, the number of configurations of the rod is 2(oo), which is proportional to the volume of the hemisphere. When a second... 

This arises from the loss of configurational entropy of the chains, particularly at significant overlap. This is schematically illustrated in Figure 16.9 for a simple case where the chain is represented by one rod with one attachment point that rotates freely on the surface when the surfaces are separated by an infinite distance. Under these conditions, the rod has a number of configuration which is proportional to the volume... 

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