Curvilinear diffusion

Fig.4. By curvilinear diffusion a chain evacuates its original tube and creates new tube segments from an isotropic distribution. Forgotten portions of the original tube are shown shaded...

The backbone of the polymer moves by reptation along the contour of its tube, with curvilinear diffusion dominated by the branch point friction (br- An H-polymer is the simplest comb polymer with q = 4 branches per molecule. For any trifunctional comb polymer g>4) the... 

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... 

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)]. 

The primary mode of motion of a linear chain along its confining tube is reptation, first proposed by de Gennes. Reptation is a snake-like diffusion of a chain, as a whole, along the contour of its tube, with a Rouse curvilinear diffusion coefficient. The relaxation time of the melt is the time it takes the chain to reptate out of its original tube, called the reptation time Trep. The reptation time and the viscosity of entangled polymers are... 

Consider a segment 5 of the tube of an entangled A-mer at time t = 0. Assume that the chain moves along its tube by simple diffusion with curvilinear diffusion coefficient D. ... 

Here, we shall relate the translational diffusion motion of the center of mass of a polymer chain to the curvilinear diffusion constant. Experimentally, the translational diffusion constant can readily be measured. 

Then R — Rn-i = o,. Assume that in a time interval At, the primitive chain jumps forward or backward with equal probability one step of length a. Then the curvilinear diffusion constant can be defined by... 

According to the Doi-Edwards theory, after time t = Teq following a step deformation at t = 0, the stress relaxation is described by Eqs. (8.52)-(8.56). In obtaining these equations, it is assumed that the primitive-chain contour length is fixed at its equilibrium value at all times. And the curvilinear diffusion of the primitive chain relaxes momentarily the orientational anisotropy (as expressed in terms of the unit vector u(s,t) = 5R(s,t)/9s), or the stress anisotropy, on the portion of the tube that is reached by either of the two chain ends. The theory based on these assumptions, namely, the Doi-Edwards theory, is called the pure reptational chain model. In reality, the primitive-chain contour length should not be fixed, but rather fluctuates (stretches and shrinks) because of thermal (Brownian) motions of the segments. 

The fluctuation forces also cause the curvilinear diffusion along the primitive path. However, because of the constraint effect of the tube, the positive and negative fluctuation forces perpendicular to the primitive path cancel each other out when averaged over a time period longer than required for a segment to travel over an entanglement distance, and thus make no net contribution to the curvihnear diffusion and the translational diffusion of the center of mass. Thus, in the study of the cmvilinear diffusion, Eq. (9.A.3 ) should be used, instead of Eq. (9.A.3). 

Reptation of the primitive chain along the centroid of a curved tube is a curvilinear diffusion process. It can be shown [6] that the diffusion coefiicient Dp for this process is defined by... 

In the semidilute solution, the hydrodynamic interactions are shielded over the distance beyond the correlation length, just as the excluded volume is shielded. We can therefore approximate the dynamics of the test chain by a Rouse model, although the motion is constrained to the space within the tube. In the Rouse model, the chain as a whole receives the friction of N, where is the friction coefficient per bead. When the motion is limited to the curvilinear path of the primitive chain, the friction is the same. Because the test chain makes a Rouse motion within the tube, only the motion along the tube survives over time, leading to the translation of the primitive chain along its own contour. The one-dimensional diffusion coefficient for the motion of the primitive chain is called the curvilinear diffusion coefficient. It is therefore equal to Dq of the Rouse chain (Eq. 3.160) and given by... 

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... 

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