Curvilinear length of the tube

Fig. 3.12 (a) A pom-pom with three arms at each branch point (q = 3). At short times the polymer chains are confined to the Doi-Edwards tuhe. Sc is the dimensionless length of branch point retraction into the tube X is the stretch ratio where L is the curvilinear length of the crossbar and Lq is the curvilinear equilibrium length, (b) Relaxation process of a long-chain-branched molecule such as LDPE. At a given flow rate e the molecule contains an unrelaxed core of relaxation times t > g 1 connected to an outer fuzz of relaxed material of relaxation t < g 1, behaving as solvent. [Reprinted by permission from N. J. Inkson et al., J. Rheol., 43(4), 873 (1999).]... 

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)]. 

As in Sect. 2.1, Dj is the curvilinear centre-of-mass diffusion constant of the chain, and is given in terms of the monomeric friction constant by the Einstein relation Dj =kT/Nl. L is as before the length of the primitive path, or tube length of the chain, which is Finally, we need the initial condition on p(s,t), which... 

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)] ... 

In order for one chain to move a distance of the order of the tube diameter a, the other chain must move out of its way by the curvilinear distance of the order of the tube length (L). 

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. 

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... 

This is the natural law for spatial motions of one labeled monomer in an entangled melt after a time t, the chain carrying this monomer has moved along its own tube by a curvilinear length ... 

Unfortunately, the slope-ratio method of calculation has only a very restricted use in the computation of microbiological assay values. It is the exception rather than the rule that the mean response to a tube assay is linearly related to the dose, i. e., that there is a strictly linear portion of the standard curve. In the vast majority of assays, there is no linear portion of the standard curve which is curvilinear throughout its length. 

The curvilinear diffusion constant of the chain in its tube is given by D= g7V5, where =(L/6)5q is the friction coefficient of the whole chain and 6=3.4 A is the contour length of one bp. The time it takes to disengage completely from a tube of length Na by this diffusion process is2 ... 

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