Q-limit

For non-interacting, incompressible polymer systems the dynamic structure factors of Eq. (3) may be significantly simplified. The sums, which in Eq. (3) have to be carried out over all atoms or in the small Q limit over all monomers and solvent molecules in the sample, may be restricted to only one average chain yielding so-called form factors. With the exception of semi-dilute solutions in the following, we will always use this restriction. Thus, S(Q, t) and Sinc(Q, t) will be understood as dynamic structure factors of single chains. Under these circumstances the normalized, so-called macroscopic coherent cross section (scattering per unit volume) follows as... 

Further note that for t=0 Eq. 3.24 does not resemble the Debye function but yields its high Q-limiting behaviour i.e. it is only valid for QR >1. In that regime the form of Dr immediately reveals that the intra-chain relaxation increases in contrast to normal diffusion ocQ, Finally, Fig. 3.2 illustrates the time development of the structure factor. 

It is therefore clear that particle number conservation considerations are not sufficient to determine S q, q) at very small but finite q. In the case of broken translational symmetry, as certainly occurs in the vicinity of a surface, the perfect screening of density fluctuation matrix elements, which is characteristic of homogeneous systems, does not hold due to nonconservation of momentum, and the small q limit of S(q, q) is nonuniversal even in the zero temperature case. 

Physically, ds, measures the distance over which the probability of finding a diffusing atom decays as one moves away from the step edge. In the small q limit, the eigenvalues of M are given by,... 

Thus the second step (described by h Qc/)) fluctuates exactly as given by Eq. (26), while in the small q limit, we find. 

II that the second-order, high-momentum CGE, which will be referred as the correct large-q limit (CLQL) in later sections, is given by... 

E(Q), F(Q) and P(Q) are the correlations between chain extremities, the form factor and the structure factor respectively and < > represents an average over all possible chain conformations. Note that these expressions are normalized such that they become unity at the zero Q limit. 

This scattering function goes to the square of the total number of monomers in the gel, (nNbnB)2, at the zero Q limit as it should. 

A. Time Dependence of Mean-Squared Displacement Consider the evaluation of Eq. (47) in the low q limit, that is,... 

Recall from Section 2.8.4 that the form factor for an ideal chain is the Debye function [Eq. (2.160)]. The high q limit of the Debye function is... 

As shown by Friend Abbott (1986), rotating massive stars have an enhanced radiation driven stellar wind mass loss well before they actually hit the Q-limit, with an enhancement factor... 

Formally, M —> oo for H —> 1 in Eq. (5.33). However, the mass loss remains finite due to the effect that it imposes a loss of angular momentum which results in a (non-magnetic) spin-down of the star In rigidly rotating stars, the specific angular momentum increases from the center to the surface due to mass loss, it is continuously transported from the core to the envelope where it is blown off. Therefore, the mass loss rate (due to Eq. 5.33) of the considered stars at the Q-limit, Mq, is self-regulated by the condition... 

These main sequence models at the H-limit can certainly not be related to LBVs, but perhaps rather to B[e] stars the material of stars at the Q-limit which has excess angular momentum can not remain on the star but it is also not clear whether it can be pushed to infinity some of it may form a ring or disk close to the star. However, as the mass loss rate at the H-limit depends on the evolutionary speed of the star in the HR diagram, one can expect values of the order 10 5 M0 yr-1 x th/tkh for the expansion phase after core H-exhaustion, when the star hits the fl-limit again due to the helium opacity peak. Here, th is the H-burning time scale and tkh the thermal time scale of the star, such that values of Mq of the order of 10 3 — 10 2 M0 yr-1 can be expected, which is the order of magnitude observed for LBVs. 

Figure 20. Upper panel Evolution of the equatorial rotation velocity with time during the core hydrogen burning phase of four 60 M sequences with different initial rotation rates (see at t = 0). The evolution of the critical rotation velocity (Eq. 5.31) is displayed for the sequence with Vrot.init. = 100 kms-1 by the triangles. It is very similar for the other sequences. For Vcrit — ( rot, the stars evolve at the Q-limit. Lower panel Evolution of the stellar mass with time for the same 60M sequences. The initial equatorial rotation velocities are given as labels. For comparison, the evolution of a non-rotating star is shown in addition.

Of course pressure fluctuations occur in a real binary mixture. Our neglect of these fluctuations simplified the equations considerably. Nevertheless, the full set of equations should be analyzed. This has been done by Mountain and Deutch (1969). The full treatment of these equations gives in the limit of small q precisely the same central component that we have just calculated so that our preceding discussion is valid. The full treatment also gives Brillouin doublets. The full spectrum in the small q limit is... 

We have succeeded in showing that the phenomenological equation for A naturally arises from microscopic consideration in the small q limit. In the process we have obtained a formal microscopic definition of the transport coefficient A. It should be noted that in this derivation we did not postulate a linear constitutive relation between / and A. 

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