Rouse model simulation

Like the dynamic structure factor for local reptation it develops a plateau region, the height of which depends on Qd. Figure 20 displays S(Q,t) as a function of the Rouse variable Q2/ 2X/Wt for different values of Qd. Clear deviations from the dynamic structure factor of the Rouse model can be seen even for Qd = 7. This aspect agrees well with computer simulations by Kremer et al. [54, 55] who found such deviations in the Q-regime 2.9 V Qd < 6.7. 

D. Richter, Phys. Rev. Lett., 80, 2346 (1998). Chain Motion in an Unentangled Polymer Melt A Critical Test of the Rouse Model by Molecular Dynamics Simulations and Neutron Spin Echo Spectroscopy. 

Molecular-Dynamics Simulation of a Glassy Polymer Melt Rouse Model and Cage Effect. 

In order to learn more about the Rouse model and its limits a detailed quantitative comparison was recently performed of molecular dynamics (MD) computer simulations on a 100 C-atom PE chain with NSE experiments on PE chains of similar molecular weight [52]. Both the experiment and the simulation were carried out at T=509 K. Simulations were imdertaken,both for an explicit (EA) as well as for an united (l/A) atom model. In the latter the H-atoms are not explicitly taken into account but reinserted when calculating the dynamic structure factor. The potential parameters for the MD-simulation were either based on quantum chemical calculations or taken from literature. No adjusting... 

Figure 3.10 compares the same experimental data, with a best fit to the Rouse model (Eq. 3.19). Here a good description is observed for small Q-values (Q<0.14 A 0, while at higher Q important deviations appear. Similarly, the simulations cannot be fitted in detail with a Rouse structure factor. Recently this result was confirmed by an atomistic computer simulation on PE molecules of different lengths. Again, at high Q the Rouse model predicts a too-fast decay forSpair(Q,0 [53]. 

Having obtained a very good agreement between experiment and simulation, the simulations containing the complete information about the atomic trajectories may be further exploited in order to rationalize the origin of the discrepancies with the Rouse model. A number of deviations from Rouse behaviour evolve. 

While the Rouse model predicts a linear time evolution of the mean-square centre of mass coordinate (Eq. 3.14), within the time window of the simulation t<9 ns) a sublinear diffusion in form of a stretched exponential with a stretching exponent of (3=0,83 is found. A detailed inspection of the time-dependent mean-squared amplitudes reveals that the sublinear diffusion mainly originates from motions at short times t[Pg.39]

An example from recent works is the study of the dynamics of unentangled PEO chains in 35% PEO/65% PMMA by quasielastic neutron scattering (Niedzwiedz et al., 2007), and in 25% PEO/75% PMMA by molecular dynamics simulations (Brodeck et al., 2010). The Rouse model has the mean-square displacement, (r (r)>, of a chain segment, which increases proportionally to the square root of time according to... 

The fractional exponent, finiT), of the stretched correlation function of the p = 1 mode had been obtained by simulations at various temperatures. Arrese-Igor et al. showed they can obtain the (A// ) -dependence of the Rouse time xr in the blend from the (A/p) -dependence of of the Rouse model by raising it to the power of /fiR T). The operation is expressed explicitly by... 

In Chapters 3, 6 and 7, the two equivalent descriptions of Brownian motion the Langevin and Smoluchowski equations for an entanglement-free system have been studied in the cases where analytic solutions are obtainable the time-correlation function of the end-to-end vector of a Rouse chain and the constitutive equation of the Rouse model. When the Brownian motion of a more complicated model is to be studied, where an analytical solution cannot be obtained, the Monte Carlo simulation becomes a useful tool. Unlike the Monte Carlo simulation that is employed to calculate static properties using the Metropolis criterion, the simulation based on the Langevin equation can be used to calculate both static and dynamic quantities. 

The Basic Monte Carlo Simulation Scheme as Applied to the Rouse Model... 

Both the Rouse theory and the Rouse-model Monte Carlo simulation are a mean-field representation, meaning that the stress relaxation is the sum of contributions from all the chains in a unit volume, each represented by its statistically averaged time dependence (Chapters 6 and 7). Thus, simulations as explained above are performed on a singe chain. 

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. 

Analysis of the Rouse Modes. In order to test the applicability of the Rouse model we calculated the basic quantities, the Rouse modes, and compared the simulation results with the theoretical predictions. The Rouse modes are defined as the cosine transforms of the position vectors, Vn, to the monomers. For the discrete polymer model under consideration they can be written as (66)... 

Figure 13 tests another prediction of the Rouse model, the time-temperature superposition property. Again, a representative example is shown, t.e., the correlation function of the third Rouse mode. As the theory anticipates, it is indeed possible to superimpose the simulation data, obtained at different temperatures, onto a common master curve by rescaling the time axis. The required scaling time, T3, is defined by the condition pp(r3) = 0.4. The choice of this condition is arbitrary. Since the Rouse model predicts that the correlation function satisfies equation (10) for all times, any other value of pp(t) could have been used to define T3. This scaling behavior is in accordance with the theory. However, contrary to the theory, the correlation functions do not decay as a simple exponential, but as... 

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