Rouse model tube 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. 

Early-time motion, for segments s such that UgM(s)activated exploration of the original tube by the free end. In the absence of topological constraints along the contour, the end monomer moves by the classical non-Fickian diffusion of a Rouse chain, with spatial displacement f, but confined to the single dimension of the chain contour variable s. We therefore expect the early-time result for r(s) to scale as s. When all prefactors are calculated from the Rouse model [2] for Gaussian chains with local friction we find the form... 

The reptation ideas discussed above will now be combined with the relaxation ideas discussed in Chapter 8 to describe the stress relaxation modiihis G t) for an entangled polymer melt. On length scales smaller than the tube diameter a, topological interactions are unimportant and the dynamics are similar to those in unentangled polymer melts and are described by the Rouse model. The entanglement strand of monomers relaxes by Rouse motion with relaxation time Tg [Eq. (9.10)] ... 

On length scales larger than the correlation length but smaller than the tube diameter a, hydrodynamic interactions are screened, and topological interactions are unimportant. Polymer motion on these length scales is described by the Rouse model. The relaxation time Tg of an entanglement strand of monomers is that of a Rouse chain of N jg correlation volumes [Eg. (8.76)] ... 

Constraint release has a limited effect on the diffusion coefficient it is important only for the diffusion of very long chains in a matrix of much shorter chains and can be neglected in monodisperse solutions and melts. The effect of constraint release on stress relaxation is much more important than on the diffusion and cannot be neglected even for monodisperse systems. Constraint release can be described by Rouse motion of the tube. The stress relaxation modulus for the Rouse model decays as the reciprocal square root of time [Eq. (8.47)] ... 

Tube length fluctuations modify the rheological response of entangled polymers. Reptation dynamics adds a regime to the mean-square monomer displacement that was not present in the free Rouse model. This extra regime is a characteristic signature of Rouse motion of a chain confined to a tube. 

The word reptation was created by De Gennes in 1971 (see De Gennes23). The term tube model is used to describe complete theories that incorporate Rouse and reptation motions within a tube-like constraint of the surrounding polymer chains. 

The next important question now is how to calculate all observables of the previous section from the tube coordinates Vj(t). In the slithering snake model, we will not distinguish the chain from the tube and assume that N = Z and Ri = Vj. This means that R can be thought of as the centers of the blobs of size Ne = N/Z, and one understands that the results of such a model are not valid at timescales treptation motion is associated with the center-of-mass motion of the chain inside the tube, it is reasonable to assume that the jump time t scales linearly with the molecular weight, that is, t=Ztss. Thus, the parameters of the model areZ, a, and z s- Since the last two parameters are just imits of space and time (similar to b and zq in the Rouse model), Z is the only nontrivial parameter (analogous to N in the Rouse model). 

For t = 0 or i=N, this equation should be modified. The Rouse model used no tension boundary condition, equivalent to setting Xn+ 1 = Xn and x i = Xq. However, these conditions for the ID motion inside the tube will lead to an escape from the tube after the Rouse time since the tube length will fluctuate in the range from 0 to i/Nb, with 0 being the most probable value. To prevent this, it was argued that a constant tension condition... 

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]. 

The first discrepancy at large q can be suppressed within the framework of the reptation model, but with a realistic value of the tube diameter corresponding to Eq. (7.24). The result will be the disappearance of the plateau in the calculated curve q S(q). A finite value of D is in fact the result of a Rouse motion during the period (0, T, where T. cx M is the Rouse time of a chain of mass M. Thus it resembles the Rouse model at this short T. At longer t the resemblance remains but the time decay is now slower and ind mdmt of q at large q, which is different from the Rouse model. 

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... 

In contrast to D, the prediction of other viscoelastic properties, such as the friction coefficient f or the zero-shear rate viscosity i/o, requires that the atomistic MD data be mapped upon a mesoscopic theoretical model. For unentangled polymer melts, such a model is the Rouse model, wherein a chain is envisioned as a set of Brownian particles connected by harmonic springs [25,28]. For entangled polymer melts, a better model that describes more accurately their dynamics is the tube or reptation model [26]. According to this model, the motion of an individual chain is restricted by the surrounding chains within a tube defined by the overall chain contour or primitive path. During the lifetime of this tube, any lateral motion of the chain is quenched. 

In this section experiments on the diffusion of d-PS and d-PMMA chains in PS and PMMA hosts, respectively, are discussed in light of the Doi-Edwards model. In this model the motion of a labelled chain in an entangled melt is imagined to occur along an average trajectory called the primitive path which is defined as the center line of the "tube". The dynamical modes of this chain in its "tube" are assumed to be described by the Rouse model therefore the diffusion coefficient of the chain along the primitive path is denoted by ... 

While the Rouse model considers only intramolecular motions, the tube model deals with intermolecular interactions due to entanglement couplings and neglects intramolecular motions. The neglect of intramolecular motions that may occur on the timescale shorter than the timescale of reptation motions was thought to be responsible for the 3.0-power dependence of jjq on M. 

The better model, called reptation, imagines the polymer chain confined within a curved tube (deGennes, 1979). Within this tube, the Rouse model governs the chain dynamics, but the polymer diffusion is governed by the time required to escape from the tube. Because motion in the tube is one-dimensional, this escape time x is given by... 

In the simple reptation model, there is a delay in relaxation (the rubbery plateau) between te and the reptation time of the chain trep [Eq. (9.11)]. By restricting the chain s Rouse motions to the tube, the time the chain takes to diffuse a distance of order of its size is longer than its Rouse time by a factor of 6 N/N. This slowing arises because the chain must move along the confining tube. The reptation time of the chain trep — 0.2 s is measured experimentally as the reciprocal of the frequency at which G = G" in Fig. 9.3 at low frequency (see Problem 9.8). In practice, this time is determined experimentally and tq, Te and Tr are determined from Trep-... 

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