Tube length fluctuations

Note that the root-mean-square magnitude of these fluctuations is in perfect agreement with the value of tube length fluctuations derived above [Eq. (9.52)]. Even though this magnitude seems large, it is a small fraction of the contour length of the tube [Eq. (9.3)]  

At times longer than the Rouse time tr, all monomers move coherently with the chain. The chain diffuses along the tube, with a curvilinear diffusion coefficient given by the Rouse model Dg 

In entangled polymer melts this diffusion occurs along the contour of the tube, with the mean-square monomer displacement in space determined using Eq. (9.71)  

This curvilinear motion continues up to the reptation time trep where the chain has curvilinearly diffused the complete length of the tube, of order aNjN. At times longer than the reptation time (/ Trep) the mean-square displacement of a monomer is approximately the same as the centre of mass of the chain and is a simple diffusion with diffusion coefficient D [Eq. (9.12)]. 

There are four different regimes of monomer displacement in entangled linear polymer melts, shown in Fig. 9.20. The subdiffusive regime for the mean-square monomer displacement is a unique characteristic of Rouse motion of a chain confined to a tube, which has been found in both NMR experiments and computer simulations. 

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The refinement introduced by Doi [9] who considers tube length fluctuations is more relevant to experimental scaling laws as far as the viscosity/molecular weight dependence is concerned. Following that concept, Gc(t) can be cast into the integral form ... 

The Doi-Edwords equation [Eq. (9.20)] is the first attempt at a molecular model for viscoelasticity of entangled polymers. It ignores tube length fluctuation modes that relax some stress on shorter time scales. These modes significantly modify dynamics of entangled polymers, as described in Section 9.4.5. 

Thus, a typical tube length fluctuation is of the order of the root-mean-square end-to-end distance R of the chain and the confining tube has a wide range of typical lengths ... 

Displacements of monomers at the two ends of the tube are unrelated to each other on lime scales shorter than the Rouse time of the chain (r < Tr). These incoherent curvilinear displacements lead to tube length fluctuations [Eq. (9.70)] ... 

The last relation made use of the fact that (L) aNjNf. bN/y/N and Te ro77g. The tube length fluctuations grow and the stress relaxation modulus decreases up to the Rouse time of the whole chain [Eq. (8.17)]. Consequently, the stress relaxation modulus at the Rouse time of the chain is lower than Gg. 

The final result was obtained using Eq. (9.19) (tr/Tc (N/Nef) and fj, is a coefficient of order unity. The fraction N jN of the tube is vacated, and therefore relaxed, at the Rouse time of the chain by tube length fluctuations. The modulus at the relaxation time of the chain is also lower by the same factor ... 

Recall that Fig. 9.3 showed the linear viscoelastic response of a polybutadiene melt with MjM = 68. The squared term in brackets in Eq. (9.82) is the tube length fluctuation correction to the reptation time. With /i = 1.0 and NjN = 68, this correction is is 0.77. Hence, the Doi fluctuation model makes a very subtle correction to the terminal relaxation time of a typical linear polymer melt. However, this subtle correction imparts stronger molar mass dependences for relaxation time, diffusion coefficient, and viscosity. 

Tube length fluctuation modes significantly modify the rheological res-... 

Rouse modes of the chain, including tube length fluctuations and longitudinal Rouse modes, are active at times shorter than the Rouse... 

The results of models that include tube length fluctuation modes [Fig. 9.23(b)] are in much better agreement with the experimentally measured loss modulus G" (ic) of monodisperse melts than the prediction of the Doi Edwards reptation model [Eq. (9.83)]. Tube length fluctuation corrections predict that the loss peak broadens with decreasing molar mass because the fraction of the stress released by fluctuations is larger for shorter chains. 

Doi s estimate of the effect of tube length fluctuations [Eq. (9.84)] predicts a molar mass dependence that approximates over a reasonable... 

Single-chain motion of the P-mer within its confining tube by reptation and tube length fluctuations. 

Single-chain models, such as the Doi Edwards reptation model [Eq. (9.21)] or the Doi tube length fluctuation model, assume a linear contribution to... 

Edwards reptation model improves the dependence of the heights of the loss modulus peaks on the volume fractions (j)] and (ps of components. But this model lacks the higher frequency modes because it does not include tube length fluctuations. 

These tube length fluctuation modes (see Section 9.4.5) of the neighbouring chains affect the constraint release modes of a given chain. If entanglements between chains are assumed to be binary, there should be a duality between constraint release events and chain in a tube relaxation events. A release of an entanglement by reptation or tube length fluctuation of one chain in its tube leads to a release of the constraint on the second chain. If this duality is accepted, the distribution of constraint release rates can be determined self-consistently from the stress relaxation modulus of the tube model. 

These simple rules for connectivity, order, and motion allow the repton model to be analysed analytically and easily solved numerically. The time dependence of such motion is shown in Fig. 9.35(a) for the extremities of the repton chain. The repton model allows direct visualization of tube length fluctuations. 

The probability distribution function of the tube length L for a chain with N monomers is approximately Gaussian, with mean-square fluctuation of the order of the mean-square end-to-end distance of the chain. The tube length fluctuates in time, leading to stronger molar mass dependences of relaxation time, viscosity, and diffusion coefficient resembling experi-mental observations over some range of molar masses ... 

Reptation and tube length fluctuations of surrounding chains release some of the entanglement constraints they impose on a given chain and lead to Rouse-like motion of its tube, called constraint release. Constraint release modes are important for stress relaxation, especially in polydisperse entangled solutions and melts. 

Note that Eq. (9.144) is a probability distribution for a primitive path of an A -step walk to consist of K steps and is an approximate expression for tube length fluctuations. 

There is a conceptual difference between tube dilation and constraint release. The motion of the tube in constraint release does not affect singlechain modes inside the tube (reptation and tube length fluctuation). The tube diameter for these single chains in a tube processes is, on average, constant. In contrast, the tube diameter increases with time in the tube dilation process and dramatically affects the chain motion within the tube. 

The molecular theory of extensional viscosity of polymer melts is again based oti the standard tube model. It considers the linear viscoelastic factors such as reptation, tube length fluctuations, and thermal constraint release, as well as the nonlinear viscoelastic factors such as segment orientations, elastic contractimi along the tube, and convective constraint release (Marrucci and lannirubertok 2004). Thus, it predicts the extensional stress-strain curve of monodispersed linear polymers, as illustrated in Fig. 7.12. At the first stage, the extensional viscosity of polymer melts exhibits the Newtonian-fluid behavior, following Trouton s ratio... 

All these observed characteristics of viscoelasticity for star polymers are natural consequences of the tube model. As suggested by the sketch in Fig. 3.49, the presence of even one long branch would surely suppress reptation [53]. There is no longer any direction for the star to move freely into new positions and conformations, and accordingly relaxation and diffusion must occur by some other motion. The Pearson-Helfand theory for stars based on tube-length fluctuations alone [72]... 

The results of the repton model qualitatively agree with the conclusions of Doi, > but represent a more accurate treatment of tube-length fluctuations and are in better agreement with experiments at intermediate molecular weights. The two models probably coincide in the high-molecular-weight region where expansion used by Doi is... 

In the reptation model, the viscosity t) scales with N. The exponent 3 is slightly smaller than that found experimentally, eq. (4.19). There have been numerous models to explain this difference between the reptation model and experiment. Though the question is not completely settled, it is believed by many that the higher exponent is a finite size crossover effect due to the tube length fluctuations. DoP was the first to suggest this difference was a crossover effect and that for finite chain length, contour length fluctuations enhance the relaxation of the stress. Doi sug-... 

The 4/5 prefactor arises from tube length fluctuations, which allow for a better relaxation under deformation, while this is not the case in networks. 

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