Primitive chain length

An estimate for rrep may be found by considering the Brownian motion of the chain. At the transition from one isolated primitive tube to its successor, the chain must have executed enough excursions to travel the chain length Nb. If the distance associated with each excursion is u and the number of excursions is n, Eq. 7.47 also establishes the estimate by/n Nb. Therefore, the number of excursions, n, is approximately n = N2b2 ju2. If v is the excursion frequency,... 

At times t >Teq, the wriggling motion results merely in a fluctuation around the primitive path, so the ch moves coherently in a one-dimension diffusion process, keeping its arc length constant. The macroscopic diffusion coefficient of a reptating chain scales with chain length (molecular weight) as ... 

The primitive chain has a constant contour length L, so fluctuations of the contour length are neglected. 

The conformation of the primitive chain becomes Gaussian on a large length scale. This means that if the position of two points on the primitive chain are r(, t) and r(s, t), where 5 and s are the contour lengths measured from the chain end, then... 

The three parameters necessary for the characterization of the primitive chain, L, D, and a can be expressed in terms of the Rouse model parameters N, b, and Thus D is given by Eq. (11.14), while La is equal to Nb, the mean square end-to-end distance of the Rouse chain. As a result, the length of the primitive chain can be written as... 

L fluctuates in time as the chain (or snake) moves. A full description of chain dynamics requires knowledge of the probability distribution of the primitive path lengths. This problem has been solved exactly by Helfand and Pearson in 1983 for a lattice model of a chain in a regular array of... 

The contribution to the free energy is not strictly linear because for each primitive path length L, the entropy of not only the ends, but of the rest of the chain in an entanglement network needs to be considered. 

It is unknown how the fatty chain carbonic acids of the lipids originally formed. It is noteworthy that fatty acids with up to six and seven C-atoms were detectable in the Murchison meteorite. Thus, it might be possible that membranes that were even more primitive existed in the past. However, dicarboxylic acids of chain lengths up to 17 were detectable after hydrothermal treatment. It is also remarkable that isoprenoid derivatives - more precisely, phytanyl and biphytanyl, that is CIO and C20 chains - linked to each other by monoether or diether bridges are part of membranes of the Archaea (Fig. 6.2, bottom structure). 

The primitive path length Lm of a chain which has m path segments is... 

Doi and Edwards (1978, 1979, 1986). They started with the Rouse-segmented chain model for a polymer molecule. Because of the presence of neighboring molecules, there are many places along the chain where lateral motion is restricted, as shown in Fig. 21. To simplify the representation of these restrictions, Doi and Edwards assume that they are equivalent to placing the molecule of interest in the tube as shown in Fig. 22. This tube has a diameter d and length L. The mean field is represented by a three-dimensional cage. The primitive chain can move randomly forward or backward only along itself. For a monodisperse polymer, the linear viscoelasticity is characterized by... 

Doi and Edwards considered the primitive chain as a freely jointed chain with step length a. The positions of the joints (or links) can be labeled as... 

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... 

At the same time, the contour length of the primitive chain, L", recovers its equilibrium value, L. Thus,... 

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. 

While considering the thermal motions of the segments, the stretch-and-shrink motion of the primitive-chain contour length will help relax the tube stress at both ends of the tube. This effect occurs because when a chain moves out of the tube due to a stretching of the contour length following a... 

We first regard the fixed primitive-chain contom length L in Eqs. (8.3) and (8.51) as the time average of the fluctuating length L(t). In other words. 

To sort out such a complicated dynamic situation, we first assume that the primitive chain is nailed down at some central point of the chain, i.e. the reptational motion is frozen only the contour length fluctuation is allowed. This is equivalent to setting rg —> oo while allowing the contour length fluctuation 5L(t) to occur with a finite characteristic relaxation time Tb- In this hypothetical situation, the portion of the tube that still possesses tube stress tt fa tb is reduced to a shorter length Lq, because of the fluctuation SL(t). Then, tt tube length that still possesses tube stress can be defined by... 

Fig. 9.3 The Brownian motion of a primitive chain (a) with fixed contour length, and (b) with fluctuating contour length. The oblique lines denote the region that has not been reached by either end of the primitive chain. The length of the region

The effect of the primitive-chain contour length fluctuation was first considered by Doi. In an approach which was meant to be an approximation, Doi obtained the zero-shear viscosity and the steady-state compliance as... 

Appendix 9.A — Contour Length Fluctuations of the Primitive Chain... 

The time-correlation function 5L 0)5L t)) of Eq. (9.3) will be derived by considering the polymer chain as a Gaussian chain consisting of No segments each with the root mean square length b. Let 5 (t) be the contour position of the nth bead relative to a certain reference point on the primitive path. Then the contour length of the primitive chain at time t is given by... 

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