Dipole relaxation times

The second step of the evolution towards equilibrium is the Zeeman dipole-dipole relaxation. Hartmann and Anderson estimated this time using the hypothesis that p at any time is of the form (22). As a consequence of the shortness of the dipole-dipole relaxation time we may assume that the dipole-dipole system always remains in equilibrium we are thus led to treat the evolution of the Zeeman system as the Brownian motion of a collective coordinate in the dipole-dipole heat bath. We assume that the diagonal elements of p have the form... 

It follows that the spin-spin relaxation time (exactly the Zeeman, dipole-dipole relaxation time) is not r12 but... 

Another transport property of interfacial water which can be studied by MO techniques is the dipole relaxation time. This property is computed from the dipole moment correlation function, which measures the rate at which dipole moment autocorrelation is lost due to rotational motions in time (63). Larger values for the dipole relaxation time indicate slower rotational motions of the dipole... 

Monte Carlo and Molecular Dynamics simulations of water near hydrophobic surfaces have yielded a wealth of information about the structure, thermodynamics and transport properties of interfacial water. In particular, they have demonstrated the presence of molecular layering and density oscillations which extend many Angstroms away from the surfaces. These oscillations have recently been verified experimentally. Ordered dipolar orientations and reduced dipole relaxation times are observed in most of the simulations, indicating that interfacial water is not a uniform dielectric continuum. Reduced dipole relaxation times near the surfaces indicate that interfacial water experiences hindered rotation. The majority of simulation results indicate that water near hydrophobic surfaces exhibits fewer hydrogen bonds than water near the midplane. 

In order to quantify diffiisional effects on curing reactions, kinetic models are proposed in the literature [7,54,88,95,99,127-133]. Special techniques, such as dielectric permittivity, dielectric loss factor, ionic conductivity, and dipole relaxation time, are employed because spectroscopic techniques (e.g., FT i.r. or n.m.r.) are ineffective because of the insolubility of the reaction mixture at high conversions. A simple model, Equation 2.23, is presented by Chem and Poehlein [3], where a diffiisional factor,//, is introduced in the phenomenological equation, Equation 2.1. 

The connection between anisotropic molecular motion and nuclear relaxation was derived by Woessner as early as 1962 [161]. Accordingly, the dipole-dipole relaxation time of a carbon nucleus is a function of the diagonal components R, R2, and R3 of the rotational diffusion tensor and the cosines X, p, and v of the angles assumed by the C —H bonds relative to the principal axes of this tensor ... 

Plotting the logarithm of the dipole-dipole relaxation time Tj (DD) versus the reciprocal temperature therefore gives an activation energy AE for molecular reorientation, which is of the order of 8.4 kJ/ mol for most of the molecules hitherto studied. In the case of 4,4 -dimethylbiphenyl a value of 16.8 kJ/ mol was found from the temperature dependence of Ti for C-2 and C-3 (Fig. 3.21) [169], For 7) of the methyl carbon atom the Arrhenius plot is curved at lower temperatures, since the internal rotation of this group is then probably faster than the overall motion of the molecule. 

We can say that such a static device is a U( ) unipolar, set rotational axis, sampling device and the fast polarization (and rotation) modulated beam is a multipolar, multirotation axis, SU(2) beam. The reader may ask how many situations are there in which a sampling device, at set unvarying polarization, samples at a slower rate than the modulation rate of a radiated beam The answer is that there is an infinite number, because from the point of the view of the writer, nature is set up to be that way [26], For example, the period of modulation can be faster than the electronic or vibrational or dipole relaxation times of any atom or molecule. In other words, pulses or wavepackets (which, in temporal length, constitute the sampling of a continuous wave, continuously polarization and rotation modulated, but sampled only over a temporal length between arrival and departure time at the instantaneous polarization of the sampler of set polarization and rotation—in this case an electronic or vibrational state or dipole) have an internal modulation at a rate greater than that of the relaxation or absorption time of the electronic or vibrational state. 

The implication of all these results is a prediction that the temperature dependence of the dipole relaxation time at fixed conversion will obey the WLF Equation and... 

Fig. 25. Shito s test plot of the Williams-Landel-Ferry equation for the dipole relaxation time in an anhydride-cured epoxy. (Reprinted from Ref.50) with permission of John Wiley and Sons, Inc.)...

In Section 4, we have examined, from a fundamental point of view, how temperature and cure affect the dielectric properties of thermosetting resins. The principal conclusions of that study were (1) that conductivity (or its reciprocal, resistivity) is perhaps the most useful overall probe of cure state, (2) that dipolar relaxations are associated with the glass transition (i.e., with vitrification), (3) that correlations between viscosity and both resistivity and dipole relaxation time are expected early in cure, but will disappear as gelation is approached, and (4) that the relaxed permittivity follows chemical changes during cure but is cumbersome to use quantitatively. 

By its nature the TRMC technique is not applicable to polar liquids for which the dipole relaxation times usually lie in the picosecond time regime, i.e. the same regime as the oscillation period of microwaves. 

Both the dipole-relaxation time and the ionic conductivity are related to the glass-transition temperature Fg. As a material is heated through its glass-transition temperature, static dipoles gain mobility and start to oscillate in an electric field. This causes an increase in permittivity and a loss-factor peak is noted. Obviously this motion is affected by frequency (lower frequencies have greater effects). This effect is shown in Figure 3.62 (Prime, 1997a), which shows the peaks in permittivity and loss factor at Tg. 

FIG. 2 Plots of dipole-dipole relaxation times Tinn and Tiqd versus correlation time at spectrometer frequencies of 200 MHz (dashed line) and 500 MHz. (a) Homonuclear case two silicon nuclei separated by 324 ppm in a Si — O — Si substructure (b) heteronuclear case a silicon and a proton separated by 258 ppm in a Si — O — H — substructure. (Bond lengths from Ref. 70 T, T 2 calculations according to equations provided in Ref 71.)... 

Fig. 7.13 The Debye (a) and Cole-Cole (b) diagrams for calculations of characteristic dipole relaxation times...

We can see that relaxation time Tm is independent of the sample dimensions and includes only material parameters, namely, specific COTiductivity a and dielectric constant (real part s = s ). This time is called space charge relaxation time. It is the same Maxwell dielectric relaxatimi time we met in Section 7.2.1. Note that time Tm has no relation to the dispersion frequency of ionic conductivity (Ti) neither to Debye dipole relaxation time. 

Here a is the bulk ionic or dc conductivity is the angular frequency (27rf) r is the dipole relaxation time is the relaxed dielectric constant or low frequency/high temperature dielectric constant (relative permittivity due to induced plus static dipoles) is the unrelaxed dielectric constant or high frequency/low temperature dielectric constant (relative permittivity due to induced dipoles only) o is the permitivity of free space E p is the electrode polarization term for permittivity and E"-p is the electrode polarization term for loss factor. The value of E p and E"p is usually unity, except when ionic conduction is very high (75). 

Many polymeric materials consist of dipoles (chemical bonds which have an unbalanced distribution of charge in a molecule) and traces of ionic impurities. If a polymer containing polar groups is heated so that an immobile dipole becomes mobile, an increase in permittivity is observed as the dipole starts to oscillate in the alternating electric field. This effect is referred to as a dipole transition and has a characteristic relaxation time (t) associated with it (76). When exposed to an electric field, the dipoles tend to orient parallel to the field direction and the ions move toward the electrodes, where they form layers. The dipole relaxation time... 

All the features just listed are eminently reasonable and together with other specific conclusions too numerous to be given here provide a substantial basis for the assertion (69) that MD calculations on an isolated system can yield reliably the complex permittivity of the system - for of course the model used. The necessarily more elaborate 3D computations for a single dipole strength give qualitatively similar results. The derived value of dipole relaxation time... 

Fig. 2.3 Behaviour of the longitudinal (a) and transverse (b) nuclear dipole-dipole relaxation times T (DD) and T2(DD) for different field strengths (2.114, 4.23, 6.34, 8.46 and 11.74 T)...

The molecular dynamics, such as flexibility and inter-intramolecular interactions, and the electrical properties of polar molecules like the poly(alkylene oxide)s can be investigated by measurement of simple and complex dielectric phenomena. From dipole relaxation times, the time and temperature dependence of polymer flexibility and mobility or viscoelasticity in bulk and in solution, which is important to flow characteristics and utility, can be analyzed and studied. It is well known that complex mechanical moduli are analogous to dielectric phenomena. 

---