TIME DEPENDENCE TORQUE

When the upper plate oscillates sinusoidally with frequency oi and angular amplitude yo (yo < 1), the time-dependent torque (T) is measured. The following expression is used to determine tj and rj from the measurements with yo 1 (Bird et al, 1987)... 

Fig. 1.1. Time-dependence of the components of angular momentum J, (Markovian process) and the torque M, (white noise) in the impact approximation.

This latter expression has been used to simplify KD(t)- Note that the time dependences of the linear and angular momentum autocorrelation functions depend only on interactions between a molecule and its surroundings. In the absence of torques and forces these functions are unity for all time and their memories are zero. There is some justification then for viewing these particular memory functions as representing a molecule s temporal memory of its interactions. However, in the case of the dipolar correlation function, this interpretation is not so readily apparent. That is, both the dipolar autocorrelation function and its memory will decay in the absence of external torques. This decay is only due to the fact that there is a distribution of rotational frequencies, co, for each molecule in the gas phase. In... 

Fig. 5.1. Time dependence of a typical classical trajectory in the photodissociation of C1CN — Cl 4- CN(j). (a) R(t) (solid line) is the Cl-CN separation and 7(t) (dashed line) is the orientation angle of CN with respect to the scattering vector R (for the definition of coordinates see Figure 3.1). (b) The angular momentum j(t) of CN (solid line) and the torque F1 = -dV/dj (dashed line).

Debye obtained his result by solving a forced diffusion equation Ci.e., with torque of the applied field included) for the distribution of dipole coordinate p - pcosS, with 6 the polar angle between the dipole axis and tSe field, and the same result for the model follows very simply from equation (3) using the time dependent distribution function in the absence of the field (5). The relaxation time is given by td = 1/2D, which for a molecular sphere of volume v rotating in fluid of viscosity n becomes... 

The velocity and angular velocity autocorrelation functions become more oscillatoiy as the externally tqiplied torque increases in strength (Fig 3). Moving-frame component acfs such as (o (f)o (0)>/ have different time dependences, that is,... 

In a liquid crystal most properties are best expressed relative to a director based coordinate system. This is not a problem in a macroscopic system where the director is virtually fixed. However, it can be a problem in a small system such as a simulation cell where the director is constantly diffusing on the unit sphere. Thus a director based frame is not an inertial frame. Correction terms should therefore be added to time dependent properties. Time correlation functions with slowly decaying tails might also be affected by the director reorientation. Transport coefficient obtained from them will consequently be incorrect. When NEMD-simulation algorithms are applied, the fictitious external field exerts a torque that constantly twists the director, which could make it impossible to reach a steady state. 

Fig. 6.6.7. Contact-time dependence of F signal intensity for a TORQUE experiment on a commercial sample of Viton, P(VDF-co-HFP). The constant time tsL + tcx = 3 ms was used throughout.

Physical origin of dielectric loss The foregoing conclusions correspond to a static description or cases for which the polarization can perfectly follow the oscillation of the electric field. Indeed, the electric field orientation depends on time with a frequency equal to 2.45 GHz (the electric field vector switches its orientation approximately every 10 s). The torque exercised by the electric field induces rotation of polar molecules, but they cannot always orient at this rate. The motion of the particles will not be sufficiently rapid to build up a time-dependent polarization P(t) that is in equilibrium with the electric field at any moment. This delay between electromagnetic stimulation and molecular response is the physical origin of the dielectric loss. 

Particles suspended in a liquid will experience a torque in a rotating E-field (Arnold and Zimmerman, 1982). A dipole is induced in the particle. Because the polarization process (redistribution of charges) is not immediate, the induced dipole will lag the external field and a frequency dependent torque will exist. It can be shown that the torque is dependent on a relaxation time constant identical to the time constant in the theory of P-dispersion (Schwan, 1985). Cell rotation is therefore a direct physical manifestation of dispersion. Theory predicts that the torque may have two maxima, usually of opposite sign (corotation and antirotation) and is given by (Zhou et al., 1995) ... 

G(t) is determined most easily with a thin-walled tube (see Figure 4.2). At t = 0 the tube is rotated through an angle d and the time dependence of the torque F(r) which keeps 6 constant is determined. Now the shear stress acting on a section (from Figure 4.2) is, from eqn 4.5,... 

Git) is measured 1 determining r(0 for fixed 6. Conversefy, in a design calculation, the time dependence of the torque in a tube twisted through an angle 6 may be obtained if Git) is known. 

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