The Reptation Approach

The stress relaxation function GR(t) for a single chain is Rouse-like and given by 

We have expressed the relaxation behaviour in terms of the number of chains per unit volume. At this stage we are considering the polymer in an undiluted state. Suppose we now apply a step strain to the melt in the linear regime. Two different zones of behaviour can be seen relative to the time re. This is the time at which the tube constraints begin to affect the relaxation of the chain  

The applied strain is affine and the whole of the tube is deformed along with the polymer. As the strain is infinitesimally small the contour length is unaltered. At very short times / after the strain is applied, t re tr the stress is relaxed as a Rouse chain. At short times we can make an approximation and replace our sum by an integral  

Experimentally we know that the overlap between relaxation processes results in a smooth change in behaviour. This change in behaviour enables us to evaluate GN. We know that when t re we change from Rouse Gr(/) to reptation G(0 behaviour  

Now all we need do is substitute our expressions for the relaxation times to give  

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A comparison with the tube diameter derived from plateau moduli measurements on PEB-7 underlines this assertion. The coincidence of tube diameters determined macroscopically by application of the reptation model and direct microscopic results is far better than what could have been expected and strongly underlines the basic validity of the reptation approach. 

From the theoretical point of view, the first refinement of the reptation approach has been to introduce the collective dynamics of the chains in terms of the constraint release process[7, 8, 13]. Due to the motions of the surrounding chains, some constraints which constitute the tube may disappear during one reptation lime, and thus give more freedom to the test chain. Quantitative attempts have been made to take into account these additional... 

However, experiments (25) have indicated that the molecular weight exponent is close to 3.4. From the mc els that have been proposed to improve the reptation approach, it appears that in concentrated polymer solutions in good solvents, the solution viscosity is given as... 

MC simulations and semianalytical theories for diffusion of flexible polymers in random porous media, which have been summarized [35], indicate that the diffusion coefficient in random three-dimensional media follows the Rouse behavior (D N dependence) at short times, and approaches the reptation limit (D dependence) for long times. By contrast, the diffusion coefficient follows the reptation limit for a highly ordered media made from infinitely long rectangular rods connected at right angles in three-dimensional space (Uke a 3D grid). 

Polydispersity of the molecular weight is not so well described by the DE approach, even qualitatively. In reptation theory the blend of two compatible homopolymers A and B of different molecular weights is given by... 

Pokrovskii VN (2006) A justification of the reptation-tube dynamics of a linear macromolecule in the mesoscopic approach. Physica A 366 88-106 Pokrovskii VN (2008) The reptation and diffusive modes of motion of linear macromolecules. J Exper Theor Phys 106(3) 604-607... 

The double reptation approach allows us to visualize the blend of n different... 

Another important point is that, when approaching Me, the tube consistency becomes weaker or in other words, the constraint release scaling law is modified and the rubbery plateau disappears whereas the steady-state compliance J decreEises. A self-consistent approach should predict that aroimd Me, the reptation modes would be gradually replaced by Rouse modes in order to describe the non entangled - entangled transition. 

Theories of gel electrophoretic mobility are usually based on the reptation theory introduced for polymer melts by de Gennes, as well as Doi and Edwards [32,33,7]. Their approach succeeded in explaining the inverse relationship between mobility and chain length for short fragments of DNA [7d], By replacing the tube in the reptation model by open spaces and lakes connected by straights, Zimm explained the antiresonance phenomena observed in the field inversion experiments that occur when the time scale for the formation of the conformational change of the DNA coincides with the time scale for the field cycle [7c]. 

The linear viscoelastic behavior of the pure polymer and blends has already been described quantitatively by using models of molecular dynamics based on the reptation concept [12]. To describe the rheological behavior of the copolymers in this study, we have selected and extended the analytical approach of Be-nallal et al. [13], who describe the relaxation function G(t) of Hnear homopolymer melts as the sum of four independent relaxation processes [Eq. (1)]. Each term describes the relaxation domains extending from the lowest frequencies (Gc(t)) to the highest frequencies (Ghf( )), and is well defined for homopolymers in Ref [13]. 

Herman and Edwards [55] extended the Brochard and de Gennes approach [53] by considering in detail the stress accompanying the swelling of the polymer within the reptation model. They evaluated the contributions to the free energy and chemical potentials due to the deformation of the polymer due to swelling. The chemical potential of the solvent, pi, was obtained by taking two contributions into account. The first was the classical osmotic pressure... 

The reptation idea could account for the effect of polymer molecular weight and solvent concentration on the dissolution rate. However the key concentration identified in this approach [57,59] is one at the surface. This implies independence of the solvent concentration history. This may not be true as the disentanglement of polymer chains does not commence till the local solvent concentration is greater than a critical value at which the local glass transition temperature is lowered below the experimental temperature so that the glassy polymer changes into a gel. Also, all of the efforts discussed so far failed to take into account the effect of the viscoelastic properties of the polymer on the dissolution mechanism. 

The enhanced viscoelastic functions are attributable to additional relaxation processes that occur at low frequencies associated with deformation of the dispersed phase. Therefore, for cases such as mPE/LDPE, where partial miscibility at high LDPE content and the extremely different relaxatimi times of the phases in the blends rich in mPE are observed, a hybrid model including the double reptation approach for the matrix and the linear Palieme approach for the whole system could successfully explain the viscoelastic response of these blends (Peon et al. 2003). 

A different approach, based on the reptation concept, was initiated by the present author and was recently augmented by Edwards and Doi. As explained in the next section, it leads to tj — N , and there is no... 

The mutual steric restrictions of entangled chains at deformation are accotmted for in a tube model considering the reptation motion of network subchains. This approach was proposed by Edwards and Vilgis ° and Heinrich et al. Later the tube model was further developed (see References 42 and 43). 

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