Relaxation motion

With the boundary conditions that the chain ends are free of forces, Eq. (13) is readily solved by cos-Fourier transformation, resulting in a spectrum of normal modes. Such solutions are similar, e.g. to the transverse vibrational modes of a linear chain except that relaxation motions are involved here instead of periodic vibrations. 

The results obtained show that the dipole-relaxational motions in protein molecules are really very retarded as compared to such motions in the environment of aromatic molecules dissolved in liquid solvents (where they occur on a time scale of tens of picoseconds).(82) Dipole-relaxational motions on the nanosecond time scale have been observed for a variety of proteins. For example, such motions were recorded for apohemoglobin and bovine serum albumin0 04 105) labeled with the fluorescent probe 2,6-TNS. 

Numerical calculations inspired in the ZK method for stars have also been applied for the description of the dynamics of model dendrimers. La Ferla [232] used a freely-rotating model, including a topology-dependence stiffness parameter and preaveraged HI. With this model, he obtained a complex analytical expression for the mean size. Cai and Chen [233] used a Gaussian model without HI and performed a detailed analysis of the relaxation motions. They investigated the diffusion of the center of mass, the relaxation of the center of mass position relative to the core monomer, and also the rotational and internal modes. 

From this brief description it is quite apparent that the qualitative elements of the Marcus treatment for an electron transfer process are identical to the CM model. In CM terms the reaction involves the avoided crossing of reactant (Fe2+ + Fe3+) with product (Fe3+ + Fe2+) configurations, with the reaction co-ordinate just being the distortion-relaxation motion of the solvation sphere. Thus in CM terms any electron transfer reaction involves the avoided crossing of the DA (donor-acceptor) and D+A" configurations, and for such reactions at least, based on the equivalence with Marcus theory, the CM model has a solid foundation. 

As has long been known, every derivation of the bulk properties of matter from its atomic properties by statistical methods encounters essential difficulties of principle. Their effect is that in all but the simplest cases (i.e., equilibrium) the development does not take the form of a deductive science. This contrasts with the usual situation in physics e.g., Newtonian or relativistic mechanics, electromagnetism, quantum theory, etc. The present paper, after focusing on this difficulty, seeks a way out by exploring the properties of a special class of statistical kinetics to be called relaxed motion and to be defined by methods of generalized information theory. 

On the basis of these ideas, we are led to the study of relaxed motion, according to the following ... 

Definition Under the conditions set forth above regarding the existence of the relaxation time x, a tangential macroscopic path (Eq. (2)) shall be said to constitute a mode of x-relaxed motion initiating at (p,, E) if P,l9,... 

It cannot be too strongly emphasized that from the point of view of the deductive logical structure of the present theory, the definition of x-relaxed motion is the basic starting point all the preceding qualitative discussion has been for motivation, and could in pure logic have been omitted. 

Molecular Statistics, Irreversible, Relaxed Motion in (Koopman). 15 37... 

This type of molecular motion seems to occur less frequently than the preceding ones. The existing results indicate that it is probably more characteristic of polyacrylates127 than of polymethacrylates149. Fragmentary evidence of this relaxation motion obtained up to now is presented in Sect. 5.3. 

The absorption of vapor by the surface layer of a polymer film will necessitate some rearrangement of the polymer molecules, and it is reasonable to consider that the more active the segmental motion of polymer chains becomes, the more rapidly the surface layer takes up penetrant to the equilibrium concentration. This implies that the surface concentration gradually approaches an equilibrium value at a finite rate which may depend upon the rate of relaxation motions of the polymer molecules. Crank and Park (1951) expressed this situation by the equation ... 

Examination of the Long-Richman solution to Eq. (1) indicates that the quantity defined previously, i. e., the time for the inflection point on the sigmoid second stage portion, is very nearly equal to 1/2/8, provided D/L2 is not too small compared with /8. If, as has been discussed in section 3.3, it is plausible to interpret / as being related to the rate of relaxation motions of polymer molecules, then the second stage portion should shift to the short time region as the initial concentration of the experiment becomes higher, since as the solid contains more diluent it is more plasticized and thus the chain relaxation becomes more rapid [Fujita and Kishimoto (1958)]. This expectation is borne out by the data shown in Fig. 9. 

For both linear and star polymers, the above-described theories assume the motion of a single molecule in a frozen system. In polymers melts, it has been shown, essentially from the study of binary blends, that a self-consistent treatment of the relaxation is required. This leads to the concepts of "constraint release" whereby a loss of segmental orientation is permitted by the motion of surrounding species. Retraction (for linear and star polymers) as well as reptation may induce constraint release [16,17,18]. In the homopol5mier case, the main effect is to decrease the relaxation times by roughly a factor of 1.5 (xb) or 2 (xq). In the case of star polymers, the factor v is also decreased [15]. These effects are extensively discussed in other chapters of this book especially for binary mixtures. In our work, we have assumed that their influence would be of second order compared to the relaxation processes themselves. However, they may contribute to an unexpected relaxation of parts of macromolecules which are assumed not to be reached by relaxation motions (central parts of linear chains or branch point in star polymers). 

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