Entanglement time

The simple probabihstic model was also confirmed by quantum-mechanical analysis [72]. Quantum-mechanical analysis gave expressions for ae and a2y that are very similar to each other. This similarity indicates that TPA of entangled and accidentally coincident photons is essentially identical. Differences between both expressions arise mainly due to the fact the expression for entangled photon TPA cross-section accounts for correlations between the photons it has a constant factor l/(TeAe). Moreover, it includes complex entanglement time-dependent siunmation coefficients, which may lead to periodic modiflation in the ae(Te) dependence, or entanglement-induced two-photon transparency. 

The entanglement time and area depend on the thickness of nonhnear crystal, the type of nonlinear interaction, and piunping conditions. Their chosen values are close to those used in [73]. Together, they yield the critical flux density of 0c = 3 x 10 cm. This results in the entangled photon absorption cross-section = 2.95 x 10 cm. The latter estimate falls between the values obtained earlier from quantum-mechanical calculations for Na (6.0 X 10-3° K2CsSb (2.6 x lO cm ) [73]. 

It was also assumed that internal times such as xe are proportional to the macroscopic viscosity xe = t]/ri0 (xe)0 (xe)0 or x0 is the mean entanglement time at equilibrium. The viscosity was then calculated from the expression tj = v/y2,... 

Theories based on the uniformly effective medium have the practical advantage that they can be extended quite easily to polydisperse systems (227). Viscosity master curves can be predicted from the molecular weight distribution, for example. The only new assumption is that the entanglement time at equilibrium for a chain of molecular weight M in a polydisperse system has the form suggested by the Rouse theory (15) ... 

This characteristic shear rate is equal to the reciprocal average entanglement time Tn at equilibrium. The solid curve is from Graessley s model, which reads... 

Multiphoton absorption cross sections as functions of entanglement times and path delayes reflect quantum correlations among the optical fields constituting an entangled state [93,94]. There may occur entanglement-induced multiphoton transparency—the value of absorption cross section drops to zero for some values of entanglement times and path delayes (see Fig. 22 for entangled three-photon absorption cross section). 

The dependence of multiphoton absorption cross sections on entanglement times and path delayes may, in principle, be used for the determination of spectrum and dipol moments of the absorbing material (for two-photon (three-photon) absorption, see Ref. 95 (Ref. 94). 

There are a number of important concepts which emerge in our discussion of viscosity. Most of these will come up again in subsequent chapters as we discuss other mechanical states of polymers. The important concepts include free volume, relaxation time, spectrum of relaxation times, entanglement, the friction factor, and reptation. Special attention should be paid to these terms as they are introduced. 

The tube is a construct which we might continue to sketch around an emerging chain as it diffuses out of the original sleeve. Instead, it is convenient to start with the tube initially in place and consider how long it takes for the molecule to escape. The initial entanglements which determine the contours of the tube comprise a set of constraints from which the molecule is relaxing, even if only to diffuse into another similar set. Accordingly, we identify this reptation time as a relaxation time r for the molecule. 

Chemical crosslinks and entanglements differ, however. The former is permanent, the latter transient. Given sufficient time, even the effects of entanglements can be overcome and stationary-state flow is achieved. An... 

In Section 4.2.2 the central role of atomic diffusion in many aspects of materials science was underlined. This is equally true for polymers, but the nature of diffusion is quite different in these materials, because polymer chains get mutually entangled and one chain cannot cross another. An important aspect of viscoelastic behavior of polymer melts is memory such a material can be deformed by hundreds of per cent and still recover its original shape almost completely if the stress is removed after a short time (Ferry 1980). This underlies the use of shrink-fit cling-film in supermarkets. On the other hand, because of diffusion, if the original stress is maintained for a long time, the memory of the original shape fades. 

There are three basic time scales in the reptation model [49]. The first time Te Ml, describes the Rouse relaxation time between entanglements of molecular weight Me and is a local characteristic of the wriggling motion. The second time Tro M, describes the propagation of wriggle motions along the contour of the chain and is related to the Rouse relaxation time of the whole chain. The important... 

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