Tumbling coefficient

Small-step rotational diffusion is the model universally used for characterizing the overall molecular reorientation. If the molecule is of spherical symmetry (or approximately this is generally the case for molecules of important size), a single rotational diffusion coefficient is needed and the molecular tumbling is said isotropic. According to this model, correlation functions obey a diffusion type equation and we can write... 

In Eqs. (4.43)—(4.47), use is made of the well-known rclation(23)P = K/kBT, where k is the bending rigidity. Equation (4.45) differs from the corresponding BZ result by the inclusion of the D L / term. Normally, this is negligible on the fluorescence time scale, but for sufficiently short filaments it could make a significant contribution. It appears with the coefficient 1.0 instead of 2.0 because the correction term from the Euler-McLaurin summation formula actually cancels out one of the two D t terms in Eq. (4.43). Equation (4.43) or (4.45) is then inserted in Eq. (4.26) to obtain the BZ tumbling correlation function for the filament. 

The term p, V2u> reveals that vorticity (i.e., the strength of fluid rotation) can diffuse by molecular interactions throughout a flow field, with the viscosity being the diffusion coefficient. Quite often the source of vorticity is the fluid tumbling caused by the shearing action associated with a no-slip condition on a solid wall. This vorticity, once produced, is both convected and diffused throughout the flow. The relative strength of the convective and diffusive processes depends on the flow field and the viscosity. 

Nmr methods have unrivalled potential to explore interfaces, as this account has striven to show. We have been able to determine the mobility of hydrated sodium cations at the interface of the Ecca Gum BP montmorillonite, as 8.2 ns. We have been able to measure the translational mobility of water molecules at the interface, their diffusion coefficient is 1.6 10 15 m2.s. We have been able to determine also the rotational mobility of these water adsorbate molecules, it is associated to a reorientational correlation time of 1.6 ns. Furthermore, we could show the switch in preferred reorientation with the nature of the interlayer counterions, these water molecules at the interface tumbling about either the hydrogen bond to the anionic surface or around the electrostatic bond to the metallic cation they bear on their back. And we have been able to achieve the orientation of the Ecca Gum BP tactoids in the strong magnetic field of the nmr spectometer. 

Note that, in addition to translational motion, other kinds of motion will be subject to random thermal fluctuations as well. In particular, the tumbling of (macro)molecules in solution can be described in terms of rotational diffusion, for which equivalent rotational diffusion constants can be defined. By analogy with translational diffusion described above, the rotational diffusion coefficient (/>rot) and rotational friction coefficient (/rot) are related by the expression ... 

A, Cp, Q are the thermal conductivity, specific heat, and density of the product, respectively, n 0.33 for a disc dryer with rotating blades, trough dryer n 0.22 for tube dryers, t is the time between two product rearrangements [5.29]). A dimensionless method is applied to the heat transfer coefficient as a function of rearrangement frequency and layer thickness of the product in a rotary tube dryer, disc dryer and tumble dryer for the case of preheated free-flowing product [5.30]. 

AcAc, acetylacetonate EPR, electron paramagnetic resonance DPM, dipivaloylmethane Tc, Correlation time for molecular tumbling A/x, concentration of spins X (per unit volume) D, mutual translational self-diffusion coefficient of the molecules containing A and X a, distance of closest approach of A and X ye, magnetogyric ratio for the electron C, spin-rotation interaction constant (assumed to be isotropic) Ashielding anisotropy <7 <7j ) coo, Debye frequency 0d, the corresponding Debye temperature Fa, spin-phonon coupling constant. 

The meaning of the first mode, for instance, may become clearer by considering another superposition coefficient +2 for positive cosimn/N) and -2 for negative cosUmr/N). Then, qi is the vector that connects the centroid of the beads in the first half of the chain and the centroid of the beads in the second half of the chain (see Fig. 3.43a). As the chain makes an overall tumbling motion, qj changes its... 

A second requirement for this instability to occur is that the two Leslie viscosity coefficients tt2 and Oi are of opposite signs [276,312]. If the ratio between the two viscosities is positive, the director exhibits different dynamics it aligns with respect to the velocity at an angle 6I9 such that tan (6b) = a2/ 3- Note finally that, despite a complex microstructure, the classification in terms of flow-aligning and tumbling nematics, as defined for low molecular weight liquid-crystals, still applied to lyotropic systems. 

---