Rigid dumbbells

At low energies, the rotational and vibrational motions of molecules can be considered separately. The simplest model for rotational energy levels is the rigid dumbbell with quantized angular momentum. It has a series of rotational levels having energy... 

For rigid materials, measuring modulus and strength in flexure is almost as commonly practised as tensile tests. Its popularity is largely due to the fact that a strip is easier to produce than a dumbbell and there are no gripping problems. It can also be argued... 

In the idealized case for rotation of a diatomic molecule, one assumes the molecule is analogous to a dumbbell with the atoms held at a fixed distance r from each other that is, it is a rigid rotor. The simultaneous vibration of the molecule is ignored, as is the increase of internuclear distance at high rotational energies arising from the centrifugal force on the two atoms. 

Rotaxane 316+ was specifically designed36 to achieve photoinduced ring shuttling in solution,37 but it also behaves as an electrochemically driven molecular shuttle. This compound has a modular structure its ring component is the electron donor macrocycle 2, whereas its dumbbell component is made of several covalently linked units. They are a Ru(II) polypyridine complex (P2+), ap-terpheny 1-type rigid spacer... 

Completely rigid models appear to provide rather peculiar short time response. The stress relaxation modulus for rigid dumbbells is (102) ... 

Bird and co-workers have recently examined a number of low-order models which mimic finite extensibility (102, 338). The rigid dumbbell consists of two... 

Internal viscosity (Section 4) provides another possible source of shear-rate dependence. For sufficiently rapid disturbances, a spring-bead model with internal viscosity acts like a rigid body for sufficiently slow disturbances it is flexible and indefinitely extensible. The analytical difficulties for coupled, non-linear spring-bead systems are equally severe in linear spring-bead systems with internal viscosity. Even the elastic dumbbell with internal viscosity has only been solved exactly in the limit of small e (559), where e is the ratio of internal friction coefficient to molecular (external) friction coefficient Co n. For this case, the viscosity decreases with shear rate. 

Values of p22 — P33 = N2 appear to be negative and approximately 10-30% of Nj in magnitude (82). The conventional bead-spring models yield N2=0. Indeed, N2 in steady shear flow is identically zero for all free draining models, regardless of the force-distance law in the connectors (102a). Thus, finite extensibility and, by inference at least, internal viscosity do not in themselves provide non-zero N2 values. Bird and Warner (354) have recently analyzed the rigid dumbbell model with intramolecular hydrodynamic interaction, the latter represented by the Oseen approximation. In this case N2 turns out to be non-zero but positive. 

Bird,R.B., Warner,H.R.,Jr. Hydrodynamic interaction effects in rigid dumbbell suspensions. I. Kinetic theory. Trans. Soc. Rheol. 15,741-750 (1971). 

In the treatment of a rigid dumbbell, where the whole time-correlation functions (TCF) can be solved exactly, Stockmayer and Burchard21 disclosed the origin for the discrepancy between theory and experiments. They recognized that all measurements of the TCF can be carried out down only to a limiting minimum delay time. With common instruments, this lower limit lies at about 100 ns but the lowest time is often much higher under conditions such that the TCF should have decayed to e"2 at channel 8Q220). These experimental condition imply that only an apparent first cumulant is determined defined by... 

The probability distribution function describing the configuration of a collection of rigid dumbbells will have the form, (u, t), and will only depend on the unit vector U and time. The conservation of this probability is expressed as... 

The two contributions to the rate of rotation, li, of the rod are convection and Brownian diffusion. Unlike the elastic dumbbell, where the springs were allowed to deform by the flow, the fixed separation of the beads in the rigid dumbbell must be maintained. For that reason, the vector u can rotate, but it cannot stretch. This constraint is satisfied by ensur-... 

The convection-diffusion equation for y (u, f) will be of the same form as the rigid dumbbell model of section 7.1.6.2 except that the diffusivity must be replaced by Dr(u, i) to give... 

In the limit of an infinite aspect ratio (p -> °°), this result tends to the dynamics of the rigid dumbbell shown in equation (7.61). 

This result is referred to as the Giesekus expression [62,86] and can be used to develop the form of the stress tensor for the rigid dumbbell model. Equation (7.63) for the rate of change of the second-moment tensor for this model is used to give the following result ... 

In one such TTF/DNP-based bistable [2]rotaxane 54+ (Fig. 8.6a), a long, rigid p-ter-phenyl spacer was employed between the TTF and DNP recognition sites and two tetraarylmethane stoppers were employed [22a] on both ends of the dumbbell component for the solution-phase switching studies. UV-Vis spectroscopy (Fig. 8.6b), which reveals the precise location of the CBPQT4+ ring on the dumbbell... 

Figure 6.19. The rigid dumbbell model of a diatomic molecule.

The simplest model of a rotating diatomic molecule is a rigid rotor or dumbbell model in which the two atoms of mass and m2 are considered to be joined by a rigid, weightless rod. The allowed energy levels for a rigid rotor may be shown by quantum mechanics to be... 

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