Rouse model fluctuations

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. 

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. 

Finally, we will briefly discuss the properties of polymer blends under shear flow. In small molecule mixtures, shear flow is known to produce an anisotropy of critical fluctuations and anisotropic spinodal decomposition [244, 245], In polymer mixtures, the shear has the additional effect of orienting and stretching the coils, thus making the single-chain structure factor anisotropic. In the framework of the Rouse model these effects have been incorporated into the RPA description of polymer blends [246, 247]. Assuming a velocity field v = yyex, where x, y, z are cartesian coordinates, y the shear rate, and ex is a unit vector in x direction, the single chain structure factor becomes [246, 247]... 

Based on the fluctuation-dissipation theorem, the equilibrium-simulated Gs t) is predicted to be equivalent to the step strain-simulated Gs t) in the linear region. In Fig. 16.2, the equilibrium-simulated Gs(t) curves for two-bead, five-bead and ten-bead Rouse chains are also shown. These equilibrium-simulated Gs(t) results are in perfect agreement with the step strain-simulated results and the Rouse theoretical curves, illustrating the fluctuation-dissipation theorem as applied to the Rouse model and confirming the validity of the Monte Carlo simulations. 

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... 

Fig. 6.7. A Rouse model describing the contour length fluctuation. The point O denotes the origin of the curvilinear coordinate r..

Precise calculation of requires the first passage problem in multidimensional phase space. A variational calculation for the Rouse model shows that X is larger that 1.47. Hence the effect of the contour length fluctuation is significant even if Z is as large as 100. 

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. 

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... 

Fig. 11. Resolved normal mode relaxation of PI-3 sample. The lines represent the behaviour predicted by the Rouse model including for the actual sample polydispiersity (soUd line) and without it (dotted line). Triangles correspond to the difference between the experimental losses and the Rouse model predictions. The inset shows schematically how the presence of configuration defects allows fluctuations of the whole chain dipole moment without variation of the end-to-end vector. Reprinted with permsision from Riedel et al, Macromolecules 42, 8492-8499 Copyright 2009 American Chemical Society...

These should he compared to the dependencies on molar mass predicted hy the Rouse model (Eq. 2.34). As indicated in Fig. 2.21, experiments reveal that for entangled polymers the zero-shear viscosity scales with The subtle difference in exponent compared to the predictions of reptation theory indicates additional effects not considered in the original model. These include the release of the entanglement constraints and fluctuation-driven stretchings and contractions of the chain along the tube. 

Coo/ and Ne, Me these scales are fixed. This procedure was used for the data of Table 4.1. A direct and simple test is to compare for the bond fluctuation simulation from Table 4.1 with Fig. 4.9. The agreement and thus the consistency of different simulations is excellent. From this mapping, one finds that the crossover time varies considerably from one polymer to another as one would expect. Some typical values for are 5.5 x 10 s for PS at 485 K compared to 3.2 x 10 s for PTHF at a comparable temperature, 500 K. For PDMS at 273 K, = 4.1 x 10 s while it decreases to 1.7 X 10 s at 373 K. This sheds some hght on the long-standing discussion about whether neutron spin-echo scattering could be used to observe the predicted plateaus in S q, t) or not. The first spin-echo experiments were for PDMS and PTHF. From the estimates of Ref. 54 of for the temperatures of the experiments it became clear that the neutron spin-echo experiments on PTHF should have seen a deviation from the Rouse model which they did. However for PDMS this was not the case, since the times were beyond the resolution of the experiment, which was around 10 seconds at the temperature used. In both cases the q-range was sufficient in spite of some early concerns. Later experiments on p p 124,135,191 pj 135 pg (pEB 2)i24,i35 well defined cross-... 

So far in this review, we have confined our attention to dense melts, where we found good agreement to the reptation model. For short times, however, not all the data fit to the Rouse model perfectly. One way to examine this in more detail is to study crossover from solution to melt in the free draining limit as a function of density. Experimentally this certainly is not possible, because of the effects of hydrodynamics, which influence the dynamics very strongly. The bond fluctuation algorithm was used because at the relatively low densities of interest the MD is not as suitable. ... 

Often, the bond is simply considered as a harmonic spring and, therefore, a harmonic approximation is used to model bond fluctuations (Rouse model). 

The frequency dependence of the spin-lattice relaxation time predicted by the Rouse model was verified with polymer melts for M fluctuation time determined from the Ti minimum condition. Melts with M Mc did not show any such behavior in the short-time limit although predicted so by all three model theories. On the other hand, dilution of the polymer by a low-molecular solvent increases Me and diminishes the entanglement effect. Under such conditions, Rouse-Hke behavior was observed. 

Although this athermal bond fluctuation model is clearly not yet a model for any specific polymeric material, it is nevertheless a useful starting point from which a more detailed chemical description can be built. This fact already becomes apparent, when we study suitably rescaled quantities, such that, on this level, a comparison with experiment is already possible. As an example, we can consider the crossover of the self-diffusion constant from Rouse-like behavior for short chains to entangled behavior for longer chains. 

Fig. 5.3. Log-log plot of the self-diffusion constant D of polymer melts vs. chain length N. D is normalized by the diffusion constant of the Rouse limit, DRoUse> which is reached for short chain lengths. N is normalized by Ne, which is estimated from the kink in the log-log plot of the mean-square displacement of inner monomers vs. time [gi (t) vs. t]. Molecular dynamics results [177] and experimental data on PE [178] are compared with the MC results [40] for the athermal bond fluctuation model. From [40]...

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