Equations of Relaxation

Abstract The discussion of relaxation and diffusion of macromolecules in very concentrated solutions and melts of polymers showed that the basic equations of macromolecular dynamics reflect the linear behaviour of a macromolecule among the other macromolecules, so that one can proceed further. Considering the non-linear effects of viscoelasticity, one have to take into account the local anisotropy of mobility of every particle of the chains, introduced in the basic dynamic equations of a macromolecule in Chapter 3, and induced anisotropy of the surrounding, which will be introduced in this chapter. In the spirit of mesoscopic theory we assume that the anisotropy is connected with the averaged orientation of segments of macromolecules, so that the equation of dynamics of the macromolecule retains its form. Eventually, the non-linear relaxation equations for two sets of internal variables are formulated. The first set of variables describes the form of the macromolecular coil - the conformational variables, the second one describes the internal stresses connected mainly with the orientation of segments. 

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We use the last equation from (7.4) and equations (7.14) to obtain the equation of relaxation for above-defined quantities. After the procedure, which is quite similar to that used in Section 7.2, we write down... 

Now one can return to the first equation from the set (7.33) and, also using equation (7.21), obtain the equation of relaxation of the orientational variables... 

The quantitative application of IMC for the rate of enthalpy relaxation of a pure amorphous APIs and excipients are explicitly documented but unfortunately few examples are available hitherto for ASD (Caron et al. 2010). Exothermic heat flow is detected in TAM below Tg as a consequence of relaxation. During IMC measurement, the early data points are often excluded during data analysis to avoid noise resulting from sample positioning (Kawakami and Pikal 2005). However, IMC records more temporal data points during enthalpy relaxation and hence yields relaxation parameters from a single run when compared to DSC. Thus, the power-time profiles obtained in TAM can be directly treated with the power equations of relaxation models viz., KWW (Eq. 14.3) or MSE equation with respect to time. The derivative form of the MSE equation can describe the experimental relaxation data measured by IMC more consistent, especially those recorded at lower annealing temperature (Kawakami and... 

When this is used in conjunction with Eq. (5.3), it represents the solution to the master equation of relaxation. If the spin-lattice couplings fluctuate slowly, treatment of spin relaxation in the slow motion regime requires solving the stochastic Liouville equation [5.6]. 

In the presence of some fomi of relaxation the equations of motion must be supplemented by a temi involving a relaxation superoperator—superoperator because it maps one operator into another operator. The literature on the correct fomi of such a superoperator is large, contradictory and incomplete. In brief, the extant theories can be divided into two kinds, those without memory relaxation (Markovian) Tp and those with memory... 

The fimdamental kinetic master equations for collisional energy redistribution follow the rules of the kinetic equations for all elementary reactions. Indeed an energy transfer process by inelastic collision, equation (A3.13.5). can be considered as a somewhat special reaction . The kinetic differential equations for these processes have been discussed in the general context of chapter A3.4 on gas kmetics. We discuss here some special aspects related to collisional energy transfer in reactive systems. The general master equation for relaxation and reaction is of the type [H, 12 and 13, 15, 25, 40, 4T ] ... 

Tabor M, Levine R D, Ben-Shaul A and Steinfeld J I 1979 Microscopic and macroscopic analysis of non-linear master equations vibrational relaxation of diatomic molecules Mol. Phys. 37 141-58... 

The concept of relaxation time was introduced to the vocabulary of NMR in 1946 by Bloch in his famous equations of motion for nuclear magnetization vector M [1] ... 

We begm tliis section by looking at the Solomon equations, which are the simplest fomuilation of the essential aspects of relaxation as studied by NMR spectroscopy of today. A more general Redfield theory is introduced in the next section, followed by the discussion of the coimections between the relaxation and molecular motions and of physical mechanisms behind the nuclear relaxation. 

Levitt M H and Di Bari L 1994 The homogeneous master equation and the manipulation of relaxation networks Bull. Magn. Reson. 16 94-114... 

A good introductory textbook, includes a nice and detailed presentation of relaxation theory at the level of Solomon equations. 

The classical description of magnetic resonance suffices for understanding the most important concepts of magnetic resonance imaging. The description is based upon the Bloch equation, which, in the absence of relaxation, may be written as... 

Relaxation or chemical exchange can be easily added in Liouville space, by including a Redfield matrix, R, for relaxation, or a kinetic matrix, K, to describe exchange. The equation of motion for a general spin system becomes equation (B2.4.28). 

Davis, M. E., McCammon, J. A. Solving the finite difference linearized Poisson-Boltzmann equation A comparison of relaxation and conjugate gradients methods.. J. Comp. Chem. 10 (1989) 386-394. 

It is interesting to note that the Voigt model is useless to describe a relaxation experiment. In the latter a constant strain was introduced instantaneously. Only an infinite force could deform the viscous component of the Voigt model instantaneously. By constrast, the Maxwell model can be used to describe a creep experiment. Equation (3.56) is the fundamental differential equation of the Maxwell model. Applied to a creep experiment, da/dt = 0 and the equation becomes... 

One-equation models relax the assumption that production and dissipation of turbulence are equal at all points of the flow field. Some effects of the upstream turbulence are incorporated by introducing a transport equation for the turbulence kinetic energy k (20) given by... 

In an ideal fluid, the stresses are isotropic. There is no strength, so there are no shear stresses the normal stress and lateral stresses are equal and are identical to the pressure. On the other hand, a solid with strength can support shear stresses. However, when the applied stress greatly exceeds the yield stress of a solid, its behavior can be approximated by that of a fluid because the fractional deviations from stress isotropy are small. Under these conditions, the solid is considered to be hydrodynamic. In the absence of rate-dependent behavior such as viscous relaxation or heat conduction, the equation of state of an isotropic fluid or hydrodynamic solid can be expressed in terms of specific internal energy as a function of pressure and specific volume E(P, V). A familiar equation of state is that for an ideal gas... 

In the presence of a field H, rotating at the precessional frequency the nuclear system can absorb energy, following which nuclear relaxation occurs. Thus, the equation of motion must include both the precessional and the relaxation contributions ... 

The Fourier transform of a pure Lorentzian line shape, such as the function equation (4-60b), is a simple exponential function of time, the rate constant being l/Tj. This is the basis of relaxation time measurements by pulse NMR. There is one more critical piece of information, which is that in the NMR spectrometer only magnetization in the xy plane is detected. Experimental design for both Ti and T2 measurements must accommodate to this requirement. 

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