Random molecular rotations

In liquid samples (and in gases), the individual molecules rotate randomly (above cryogenic temperatures) with frequencies large compared to those of nuclear magnetic effects. Then each parameter matrix present can be taken as reduced to a single parameter (times the 3x3 unit matrix), and of course the sample becomes effectively isotropic. 

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Because of the orientational freedom, plastic crystals usually crystallize in cubic structures (Table 4.2). It is significant that cubic structures are adopted even when the molecular symmetry is incompatible with the cubic crystal symmetry. For example, t-butyl chloride in the plastic crystalline state has a fee structure even though the isolated molecule has a three-fold rotation axis which is incompatible with the cubic structure. Such apparent discrepancies between the lattice symmetry and molecular symmetry provide clear indications of the rotational disorder in the plastic crystalline state. It should, however, be remarked that molecular rotation in plastic crystals is rarely free rather it appears that there is more than one minimum potential energy configuration which allows the molecules to tumble rapidly from one orientation to another, the different orientations being random in the plastic crystal. 

Relaxation is strongly dependent on molecular motions. The overall random molecular tumbling, which is expressed in the rotational correlation time Xc, governs the overall relaxation process. Larger molecules have slower tumbling motions that lead to higher Xc values. However, local dynamics and independent domains can modulate the relaxation parameters, which account for differences in their flexibility and mobility. 

The amorphous or molten polymer is a conglomeration of badly packed interlacing chains, and the extra empty space caused by this random molecular arrangement is called the free volume, which essentially consists of all holes in the matrix. When sufficient thermal energy is available, the vibrations can cause a segment to jump into a hole by cooperative bond rotation and a series of such jumps will enable the polymer chain eventually to change its position. 

NMR spectroscopy in isotropic solution is characterised by the fact that free and random molecular reorientation averages many parameters that have a directional dependence, e.g. dipolar-coupling constants, chemical shifts, etc. Some parameters, such as dipolar-coupling constants, average to zero in isotropic solution so once the ability of a molecule to rotate and move freely and isotropically is removed, its NMR spectra immediately become more complex. 

There are two proper explanations, one based on physical intuition and the other based on the principle of material objectivity. The latter is discussed in many books on continuum mechanics.19 Here, we content ourselves with the intuitive physical explanation. The basis of this is that contributions to the deviatoric stress cannot arise from rigid-body motions -whether solid-body translation or rotation. Only if adjacent fluid elements are in relative (nonrigid-body) motion can random molecular motions lead to a net transport of momentum. We shall see in the next paragraph that the rate-of-strain tensor relates to the rate of change of the length of a line element connecting two material points of the fluid (that is, to relative displacements of the material points), whereas the antisymmetric part of Vu, known as the vorticity tensor 12, is related to its rate of (rigid-body) rotation. Thus it follows that t must depend explicitly on E, but not on 12 ... 

The fields which are responsible for relaxation are generated within the sample, often due to interactions of spins with one another or with their environment in some way. They are made time varying by the random motions (rotations, in particular) which result from the thermal agitation of the molecules and the collisions between them. Thus we will see that NMR relaxation rate constants are particularly sensitive to molecular motion. 

A nuclear electric quadrupole moment arises from a non-spherical distribution of electrical charges in the nucleus. Because the energy of an electric quadrupole moment depends on its orientation with respect to an EFG, a randomly varying electric field gradient associated, for example, with molecular rotation having a finite amplitude of its spectral density at the Larmor speed can induce effective relaxation of the quad-rupolar nucleus. In liquids, it is possible to write down... 

The heating effect in a microwave oven is mainly due to induction of molecular rotations in water molecules in the sample. The rotating water molecules quickly collide with neightxiuring molecules to generate the more random molecular motions that we know of as heat. Most other nonconducting materials are transparent to microwaves at this frequency, so the heating effect can be distributed throughout. 

The renaturation process of globular proteins takes place within 10 and 10" seconds. Internal molecular rotations are known to be in the order of 10 seconds Taking into account only three energetically favorable conformational states for each amino acid, a relatively small protein of 150 residues can potentially adopt more than 10 conformations. A comparison of these figures illustrates in an impressive manner that the time available to a protein for the folding process is by far not sufficient to find the energetically most favorable conformation by a random search mechanism. 

The rate of change of the total energy A on the moving fluid element consists of the kinetic energy due to its translational motion and of the internal energy due to random molecular motion inside the moving fluid element The random motion of molecules is a combination of translational, rotational, vibrational, and electronic... 

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