Reptation motion of macromolecules

We now turn to di.scussion of the niacromolecule reptation motion model which has already been referred to above. 

De Gennes (1979) has empha.sized that, depending on the performance of an experiment, two different diffusion coefficients are observed in polymer systems, namely, the self-diffusion coefficient (Dj) and the interdiffusion one (D), the latter also called by him the cooperative diffusion coefficient (cf. subsections 2.4.3 and 3.2.1). 

The interdiffusion coefficient D is mea,sured when polymer is redistributed under a gradient of its concentration. Theoretical consideration of this process is ba.sed on the laws of unequilibriiim thermodynamics (sec subsections 2.4.3 and 3.3.1). If a labelled macro-molecule is present among their ensemble (see subsection 3.6.2) its motion is characterized by the self-diffusion coefficient (D2). Roth those quantities coincide in dilute solutions D D i Do, but the coefficient D rises with polymer concentration while D decreases significantly. 

It is apparently this circumstance that causes serious difficulties while interpreting data on dynamic light scattering in experiments like thos described in subsection 3.6.2. 

I litis, the time necessary to have the labelled chain unravelled depends on A. Estiiiialion of heat changes in the pipe itself (de Gennes, 1976a) leads to 

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The model described by equations (3.42)-(3.45) is valid for equilibrium situations. For chain in a flow, one ought to define displacements of the particles under flow and to consider the average values (3.44) to depend on the velocity gradient (Doi and Edwards 1986). McLeish and Milner (1999) considered mechanism of reptation motion of branched macromolecules of different architecture. 

One can see from the comparison of equations (5.10) and (5.19) that the reptation motion of the macromolecules is revealed at the condition... 

I lie experimental results concerning dynamic light scattering in polymer solutions indicate the existence of two diffusion modes of a maeromolecule. The first mode is concerned with the conventional self-diffusion coefficient of the polymer. To the second one is as cribed the reptation motion of the maeromolecule in an imaginary pipe of enlanglemonl points formed from other macromolecules. 

The general problem of polymer-polymer interdiffusion is studied and used to illustrate several unique aspects of the orientation and motion of macromolecules at interfaces. Specific new results are obtained for the short time evolution of the Interfacial concentration profile which is shown, following the reptation... 

These equations describe the reptation normal relaxation modes, which can be compared with the Rouse modes of the chain in a viscous liquid, described by equation (2.29). In contrast to equation (2.29) the stochastic forces (3.47) depend on the co-ordinates of particles, equation (3.48) describes anisotropic motion of beads along the contour of a macromolecule. 

There were different generalisations of the reptation-tube model, aimed to soften the borders of the tube and to take into account the underlying stochastic dynamics. It seems that the correct expansion of the Doi-Edwards model, including the underlying stochastic motion and specific movement of the chain along its contour - the reptation mobility as a particular mode of motion, is presented by equations (3.37), (3.39) and (3.41). In any case, the introduction of local anisotropy of mobility of a particle of chain, as described by these equations, allows one to get the same effects on the relaxation times and mobility of macromolecule, which are determined by the Doi-Edwards model. 

The realisation of a certain mode of motion of a macromolecule among other macromolecules depends on the lengths of both diffusing macromolecule and macromolecules of the environment. The solid line M divides the dilute blends into those, in which macromolecules of the additive can reptate, and those, where no reptation occurs. The dashed line marks the systems with macromolecules of equal lengths. 

The investigation of viscoelasticity of dilute blends confirms that the reptation dynamics does not determine correctly the terminal quantities characterising viscoelasticity of linear polymers. The reason for this, as has already been noted, that the reptation effect is an effect due to terms of order higher than the first in the equation of motion of the macromolecule, and it is actually the first-order terms that dominate the relaxation phenomena. Attempts to describe viscoelasticity without the leading linear terms lead to a distorted picture, so that one begins to understand the lack of success of the reptation model in the description of the viscoelasticity of polymers. Reptation is important and have to be included when one considers the non-linear effects in viscoelasticity. 

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... 

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). 

One of the most important and useful applications of the reptation concept concerns crack healing, which is primarily the result of the diffusion of macromolecules across the interface. This healing process was studied particularly by Kausch and co-workers [76]. The problem of healing is to correlate the macroscopic strength measurements to the microscopic description of motion. The difference between self-diffusion phenomena in the bulk polymer and healing is that the polymer chains in the former case move over... 

Work to further expand the reptation-tube model has been explored. Pokrovski (2008), for example, has shown that the underlying stochastic motion of a macromolecule leads to two modes of motion, namely, reptative and isotropically diffusive. There is a length of a macromolecule M = lOM where is the macromolecule length between adjacent entanglements above which macromolecules of a melt can be regarded as obstacles to motions of each other and the macromolecules reptate. The transition to the reptation mode of motion is determined by both topological restriction and the local anisotropy of the motion. 

The entanglement points of a polymer chain with the neighbouring chains arc represented as obstacles preventing the transversal motion of this macromolecule it can just diffusionally reptate along its longitudinal antis (see Figure 4.2). Such motion resembles a reptilian s creeping in a certain pipe, and is therefore named reptation. ... 

So the first-order equation of the dynamics of a macromolecule in very concentrated solutions and melts of polymers has the form (41) where the memory functions defined by relations (42) and (45). This linear equation does not include the reptation dynamics of a macromolecule introduced by de Gennes [8] as a special type of anisotropic motion a macromolecule moves along its contour like a snake. Unbounded lateral motion is assumed to be completely suppressed due to the entanglement of the tagged macromolecule with its many neighbouring coils which, it is assumed, effectively constitutes a tube of radius The reptation of a macromolecule is considered to be important to describe the dynamics of solutions and melts of polymer [9]. 

Localisation of a macromolecule in a tube was assumed by Edwards [7] and by de Gennes [8]. The latter introduced reptation motion for the macromolecule to explain the law of diffusion of very long macromolecules in entangled systems. We consider the reptation of the macromolecule on the basis of the Doi-Edwards model described in Sect. 3.3. 

In the polymer melt conformational rearrangements are associated with the motions of the macromolecule (reptation) curvilinearly along its contour in an environmental tube to which the molecule is confined. The tube can only be altered, topologically, by contractions and reelongations of the chain as its ends diffuse, eliminate and redefine, new tube sections. The time for the renewal of the entire tube, by this process, is proportional to the cube of the molecular weight. The rotation of the macromolecule in simple shear, as well, requires up and down motions along the tube. Hence, if the viscosity of the melt is governed by the same friction factor as in diffusion it should scale as... 

Entanglements of flexible polymer chains contribute to non-linear viscoelastic response. Motions hindered by entanglements are a contributor to dielectric and diffusion properties since they constrain chain dynamics. Macromolecular dynamics are theoretically described by the reptation model. Reptation includes fluctuations in chain contour length, entanglement release, tube dilation, and retraction of side chains as the molecules translate using segmental motions, through a theoretical tube. The reptation model shows favourable comparison with experimental data from viscoelastic and dielectric measurements. The model reveals much about chain dynamics, relaxation times and molecular structures of individual macromolecules. 

By now, sufficient information on the dynamics of macroinolecules in solution has bci ii accumulated by means of dynamic light scattering, this method being sensitive to the internal modes of a macromolecule s motion and to the process of its reptation among similar molcondensed state. At the same time, there is a lack of unambiguous evidence for association mode s" or the lifetime of as.sociates in the voluminous literature on dynamic light scattering from polymer solutions. 

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