Contour length fluctuations dynamics

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... 

Having studied the static distribution of the contour length, we now examine the dynamics of the contour length fluctuation. As before, we use the Rouse model for the dynamics (see Fig. 6.7). Let s be the curvilinear coordinate of the n-th Rouse segment measured from a... 

The effect of the contour length fluctuation has been studied for slightly different models and it has been reported that the effect is less significant than in the case of the Rouse model. The discrepancy is perhaps due to the difference in the dynamics of the fluctuation, especially the short-time dynamics, which is quite important in the first passage time problem. 

The contour length fluctuation plays an essential role in the dynamics of branched polymers. Consider for example the star-shaped polymer shown in Fig. 6.11. Obviously simple reptation is not possible, but the polymer can change its conformation by utilizing the contour length fluctuation. 

Two different scenarios for chain-end dynamics have been suggested. Doi [142] introduced the so-called contour length fluctuation (CLF) as a modification of the tube/reptation model. Due to the stochastic nature of chain modes in the tube, the chain ends are fluctuating back and forth a length proportional to the square root of the chain length and, hence, are subject to tube constraints to a much lower degree than the central part of the chain. The chain-end blocks are therefore expected to have a molecular mass obeying... 

Statistical distribution of the contour length In the previous sections we regarded the primitive chain as an inexten-sible string of contour length L. In reality, the contour length of the primitive chain fluctuates with time, and the fluctuation sometimes plays an important role in various dynamical processes. 

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. 

These conclusions have been strengthened by an analysis of suitable correlation functions and structure factors [99]. These results show (Fig. 31) that a cylindrical bottle brush is a quasi-lD object and, as expected for any kind of ID system, from basic principles of statistical thermodynamics, statistical fluctuations destroy any kind of long-range order in one dimension [108]. Thus, for instance, in the lamellar structure there cannot be a strict periodicity of local composition along the z-axis, rather there are fluctuations in the size of the A-rich and B-rich domains as one proceeds along the z-axis, these fluctuations are expected to add up in a random fashion. However, in the molecular dynamics simulations of Erukhimovich et al. [99] no attempt could be made to study such effects quantitatively because the backbone contour length L was not very large in comparison with the domain size of an A-rich (or B-rich, respectively) domain. 

Fig. 11. Molecular dynamics (MD) simulations of dendronized polymers P12 (G3) (a) and P16/17 (G3) (b). P12 (G3) has 50 r.u s and the structure shown was obtained after 300 ps. End-to-end distance 9.1 nm, average diameter 4.4 ( 0.2) nm. The backbone atoms are kept in yellow, the terminal benzene rings in red, all other atoms in green. P16/17 (G3) has 40 r.u. s and a contour length of 33 nm. The backbone and the hexyl chain atoms are in yellow, the terminal benzene rings in red, and all other atoms in green. The diameter fluctuates between approximately 2 and 4 nm.

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