Rouse motion

Rouse motion has been best documented for PDMS [38-44], a polymer with little entanglement constraints, high flexibility and low monomeric friction. For this polymer NSE experiments were carried out at T = 100 °C to study both the self- and pair-correlation function. 

Thus, with only two parameters, the values of which are both close to expectations, the Hess model allows a complete description of all experimental spectra. In the complex crossover regime from Rouse motion to entanglement controlled behavior, this very good agreement confirms the significant success of this theory. 

Local reptation regime For times t > xe we have to consider curvilinear Rouse motion along the spatially fixed tube. The segment displacement described by Eq. (18) (n = m) must now take the curvilinear coordinates s along the tube into consideration. We have to distinguish two different time regimes. For (t < xR), the second part of Eq. (19) dominates - when the Rouse modes... 

Fig. 19a-c. Schematic representation of a reptating chain in different time regimes a Short-time unrestricted Rouse motion b equilibration of density fluctuations along tha chain c creep motion of a chain out of its tube. 

A quantitative analysis of scattering data, originating from the crossover regime between short-time Rouse motion and local reptation, needs explicit consideration of the initial Rouse motion neglected by de Gennes. Ronca [50] proposed an effective medium approach, where he describes the time-dependent... 

The simplest case of comb polymer is the H-shaped structure in which two side arms of equal length are grafted onto each end of a linear cross-bar [6]. In this case the backbones may reptate, but the reptation time is proportional to the square of Mj, rather than the cube, because the drag is dominated by the dumb-bell-like frictional branch points at the chain ends [45,46]. In this case the dependence on is not a signature of Rouse motion - the relaxation spectrum itself exhibits a characteristic reptation form. The dynamic structure factor would also point to entangled rather than free motion. 

Fig. 3.13 Schematic sketch of the different time regimes of reptation a unrestricted Rouse motion for b local reptation, i.e. Rouse relaxation along the confining tube, and...

The two situations are displayed in Fig.3.13b and c. The first process, where the chain performs Rouse motion along the tube, is called local reptation the creeplike diffusion along the tube which eventually leads to a complete tube renewal is also termed pure reptation. 

In generalized Rouse models, the effect of topological hindrance is described by a memory function. In the border line case of long chains the dynamic structure factor can be explicitly calculated in the time domain of the NSE experiment. A simple analytic expression for the case of local confinement evolves from a treatment of Ronca [63]. In the transition regime from unrestricted Rouse motion to confinement effects he finds ... 

Rubber-like models take entanglements as local stress points acting as temporary cross finks. De Cloizeaux [66] has proposed such a model, where he considers infinite chains with spatially fixed entanglement points at intermediate times. Under the condition of fixed entanglements, which are distributed according to a Poisson distribution, the chains perform Rouse motion. This rubber-like model is closest to the idea of a temporary network. The resulting dynamic structure factor has the form ... 

Inserting the Rouse rate W(, 3Q9 K)=(7 0.7)xl0 AVns (Table 3.2) obtained from single chain structure factor measurement into Eq. 3.18 the solid line is obtained. It quantitatively corroborates the correctness of the Rouse description at short times. The data also reveal clearly a transition to a law, though Eq. 3.36 would predict the dotted line. The discrepancy explains itself in considering the non-Gaussian character of the curve-linear Rouse motion (Eq. 3.38). Fixing and d to the values obtained from the single chain struc-... 

Again, at high Q the RPA predicts that the dynamics of arm A is identical to the Rouse motion of an A polymer in an A homopolymer melt. At low Q, Ai(Q) turns into a breathing mode with a non-vanishing relaxation rate at Q=0, as the collective mode A3(Q). 

Fig. 6.15 Dynamic structure factor from the junction-labelled triblock copolymer for different Q-values. T=433 K filled circles Q=0.20 A y filled squares Q=0.18 A open triangles down Q=0.14 A open triangles up Q=0.114 A open circles Q=0.08 A open squares Q=0.05 A T=473 K filled circles Q=0.20 A open triangles up Q=0.10 A open circles Q=0.08 A open squares Q=0.05 A"k The solid lines are result of the fit with the complete structure factor for surface undulations and Rouse motion. (Reprinted with permission from [284]. Copyright 2002 EDP Sciences)...

One can try to locate a critical polymerisation index above which the data are no longer compatible with a Rouse-like dynamics, Ng = 500, lager than the Ng= 100 value determined from the diffusion measurements in a frozen matrix. This is an illustration of the fact that the two processes. Rouse motion and entangled motion are in competition the slowest process is the one which is indeed observed.When the matrix chains are mobile, the entangled dynamics becomes more rapid than pure reptation, and the Rouse motion can dominate the dynamics for larger molecular weights than when the matrix chains are immobile. 

Comparison of Eqs (8.16) and (8.25) reveals that the Zimm time is shorter than the Rouse time in dilute solution. In principle, a chain in dilute solution could move a distance of order of its size by Rouse motion, by Zimm motion, or some combination of the two. The chain could simply move its monomers by Rouse motion through the solvent without dragging any of the solvent molecules with it, or it could drag all of the solvent in its pervaded volume with it, thereby moving by Zimm motion. In dilute solution, Zimm motion has less frictional resistance than Rouse motion, and therefore, the faster process is Zimm motion. The chain effectively moves as though it were a solid particle with volume of order of its per-... 

The Rouse model is the simplest molecular model of polymer dynamics. The chain is mapped onto a system of beads connected by springs. There are no hydrodynamic interactions between beads. The surrounding medium only affects the motion of the chain through the friction coefficient of the beads. In polymer melts, hydrodynamic interactions are screened by the presence of other chains. Unentangled chains in a polymer melt relax by Rouse motion, with monomer friction coefficient C- The friction coefficient of the whole chain is NQ, making tha diffusion coefficient inversely proportional to chain length ... 

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)] ... 

At the Rouse time of an entanglement strand Te, the chain finds out that its motion is topologically hindered by surrounding chains. Free Rouse motion of the chain is no longer possible on time scales t > Te. The value of the stress relaxation modulus at le is the plateau modulus G, which is kT per entanglement strand [Eq. (9.5)] ... 

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-... 

This time dependence is slower than for unrestricted Rouse motion [Eq. (8.58)] because displacement along the contour of the tube leads to a smaller displacement in space [Eq. (9.71)]. At the Rouse time of the chain,... 

There are four different regimes of monomer displacement in entangled linear polymer melts, shown in Fig. 9.20. The subdiffusive regime for the mean-square monomer displacement is a unique characteristic of Rouse motion of a chain confined to a tube, which has been found in both NMR experiments and computer simulations. 

Constraint release as a Rouse motion of the tube confining the P-mer. 

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... 

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