Reptation model segmental motion

The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

A (rheology) [7]. Obviously the local freedom for segmental motion is larger than anticipated so far from rheology. The result casts some doubts on the determination of the plateau modulus from rheological data in terms of the reptation model. 

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. 

One disadvantage of the slithering snake model is that the time step cannot be controlled in one step the chain moves exactly one tube segment. This in particular leads to the unphysical oscillations in i,mid(t) at early time in Figure 17. In order to resolve the motion on smaller timescales (and more importantly to account for fluctuations of the chain inside the tube, see below), one has to distinguish between the tube and chain coordinates. Now we introduce the main set of variables of the tube model the 3D tube coordinates V),(t), fe = 0...Z as in the previous section plus the one-dimensional (ID) chain coordinates inside the tube Xj( ), i = 0...N. In total, we have 3(Z-r 1) -r (N-f 1) variables, and their equations of motion are coupled. The main idea of the tube theory is that the chain inside the tube moves independent of the tube coordinates, whereas the tube segments are deleted at the ends when the chain does not occupy them any more, and are created when the chain sticks out of the tube. In the pure reptation model, only the center-of-mass of X coordinates moves according to... 

In the following, we will discuss some microscopic dynamical models. We begin with the Rouse-model , which describes the dynamics of chains in a non-entangled polymer melt. The effects of entanglements on the motion can be accounted for by the reptation model , which we will treat subsequently. Finally, we shall be concerned with the motion of polymer chains in a solvent, when the hydrodynamic interaction between the segments of a chain plays a prominent role. 

The tube constraints of the reptation model are implicit in these equations of motion since the positions R- have a single index i the segments follow each other perfectly along the tube axis, and only the two end-segments can make the tube evolve. 

The basis of the previously published GR model for PET [6] is mechanical coupling of (i) perturbation of bond-stretehing potentials (linear elasticity relaxed by loeal segmental motion) and (ii) conformational entropy ehange (non linear elasticity relaxed by reptation), where E = S + Dialled description of the model ean be found in literature [4]. 

In interpreting the relaxation behavior of polydisperse systems by means of the tube model, one must consider that renewal of the tube occurs because the chain inside it moves thermally, either by reptation mode, by fluctuation of the tube length in time (breathing motion), or in both ways (13,14). Moreover, the tube wall can be renewed independently of the motion of the chain inside the tube because the segments of the chains of the wall are themselves moving. The relaxation mechanism associated with the renewal of the tube is called constraint release. 

The constraint release process for the P-mer can be modelled by Rouse motion of its tube, consisting of P/A e segments, where is the average number of monomers in an entanglement strand. The average lifetime of a topological constraint imposed on a probe P-mer by surrounding A -mers is the reptation time of the A -mers Trep(A ). The relaxation time of the tube... 

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 equations for the tube motion (eqn [60]) remain the same as in the pure reptation case the tube segments are aeated and destroyed at the ends. However, the number of tube segments Z becomes a random variable Z(t), as stressed by the model name contour length fluctuations (CLFs). These fluctuations relax stress and orientation faster than the reptation mode alone. To solve eqns [63] and [65], we should first subtract the equilibrium stretch from Xi coordinates y,=X( - i b /a) and then use eqn [23] to transform yi to the Rouse modes. These modes will again satisfy the same Omstein-Uhlenbeck equation [25]. 

Although Rgure 8.21 suggests small molecules, a similar model can be constructed for the motion of polymer chains, the main difference being that more than one hole may be required to be in the same locality, as cooperative motions are required (see reptation theory. Section 5.4).Thus, for a polymeric segment to move from its present position to an adjacent site, a critical void volume must first exist before the segment can jump. 

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