Diffusion coefficient reptation theory

Diffusion of flexible macromolecules in solutions and gel media has also been studied extensively [35,97]. The Zimm model for diffusion of flexible chains in polymer melts predicts that the diffusion coefficient of a flexible polymer in solution depends on polymer length to the 1/2 power, D N. This theoretical result has also been confirmed by experimental data [97,122]. The reptation theory for diffusion of flexible polymers in highly restricted environments predicts a dependence D [97,122,127]. Results of various... 

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). 

The rod is visualised as being constrained to a tube in a similar fashion to entanglements constraining a polymer in reptation theory. So for a finite concentration our diffusion coefficient and rotary Peclet number changes ... 

Further, reptation theory asserts26) that the molecular-weight dependence of the diffusion coefficient in an entangled gel should have the form... 

Reptation theory has been developed further by Doi and Edwards (1986) and is being applied to both viscoelastic and solution behavior. It has been shown that for a chain moving in the melt, over time-scales that greatly exceed the lifetime of the tube X, a reptation self-diffusion coefficient D p, can be measured, which is inversely proportional to rP-, i.e., the diffusion law is... 

In an excellent review article, Tirrell [2] summarized and discussed most theoretical and experimental contributions made up to 1984 to polymer self-diffusion in concentrated solutions and melts. Although his conclusion seemed to lean toward the reptation theory, the data then available were apparently not sufficient to support it with sheer certainty. Over the past few years further data on self-diffusion and tracer diffusion coefficients (see Section 1.3 for the latter) have become available and various ideas for interpreting them have been set out. Nonetheless, there is yet no established agreement as to the long timescale Brownian motion of polymer chains in concentrated systems. Some prefer reptation and others advocate essentially isotropic motion. Unfortunately, we are unable to see the chain motion directly. In what follows, we review current challenges to this controversial problem by referring to the experimental data which the author believes are of basic importance. 

The crucial assumption of the Doi-Edwards theory is that the primitive chain reptates in a tube fixed in space. However, in monodisperse solutions, all chains are wriggling simultaneously, so that the tube around each chain is never fixed but successively renewed by different chains. Hence, the Doi-Edwards theory is not self-consistent. This fact has given rise to recent measurements of the tracer diffusion coefficient as a function of the molecular weight and concentration of the matrix component. 

Finally, we discuss the dependence of the diffusion coefficient D on the molecular weight N. This dependence is often used to evaluate the tube theory, which originally predicted D N scaling resulting from pure reptation motion (no CLF or CR). This is easy to obtain by a simple argument that at long timescales the center-of-mass diffusion might be approximated by random jumps of size Ar with frequency of reptation time Tdo The diffusion coefficient of this simple jump... 

On the other hand, the reptation theory proposed by de Gennes<4) assumes that a flexible chain is diffusing in a fixed three-dimensional mesh of obstacles that the chain cannot cross (Figure 2). Thus, the chain would be topologically constrained to move by a curvilinear, or snake-like, motion alone. This motion has been termed reptation (from reptile ). One can visualize that the flexible chain is reptating by a Brownian diffusion within a tube surrounded by obstacles, but motions proceed perpendicular to the axis if the tube is blocked. For a chain made from N monomers of size a, the coefficient of the curvilinear diffusion, along the tube is... 

Fig. 6 Diffusion coefficient of labeled PS in toluene solutions as a function of polymer concentration and its prediction according to reptation and scaling theory (adapted with permission from Liu...

The Doi-Edwards theory assumes that reptation is the dominant mechanism for conformational relaxation of highly entangled linear chains. Each molecule has the dynamics of a Rouse chain, but its motions are now restricted spatially by a tube of uncrossable constraints, illustrated by the sketch in Fig. 3.38. The tube has a diameter corresponding to the mesh size, and each chain diffuses along its own tube at a rate that is governed by the Rouse diffusion coefficient (Eq. (3.37)). If the liquid is deformed, the tubes are distorted as in Fig. 3.39, and the resulting distortion of chain conformations produces a stress. The subsequent relaxation of stress with time corresponds precisely to the progressive movement of chains out of the distorted tubes and into random conformations by reptation. The theory contains two experimental parameters, the unattached mer diffusion coefficient T>o... 

Reptation theory predicts that the self-diffusion coefficient varies with the molecular weight M according to... 

The next question is how the diffusion coefficient is related to the radical chain lengths and system viscosity. According to de Geimes chain reptation theory ... 

The transition from one regime to the other occurs precisely at Me, the critical molar mass above which entanglements occur. Through the image of an imaginary tube and the resistance it exerts on the motion of the test polymer, the reptation theory accounts for the experimental facts more suitably than the Rouse theory indeed, the viscosity of the polymer increases as a third power of M (qo h-instead of ho h- M) and its diffusion coefficient decreases as the inverse square of M D2 M instead of >2 h- M ). 

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