Rouse theory simulations

Sikorsky and Romiszowski [172,173] have recently presented a dynamic MC study of a three-arm star chain on a simple cubic lattice. The quadratic displacement of single beads was analyzed in this investigation. It essentially agrees with the predictions of the Rouse theory [21], with an initial t scale, followed by a broad crossover and a subsequent t dependence. The center of masses displacement yields the self-diffusion coefficient, compatible with the Rouse behavior, Eqs. (27) and (36). The time-correlation function of the end-to-end vector follows the expected dependence with chain length in the EV regime without HI consistent with the simulation model, i.e., the relaxation time is proportional to l i+2v The same scaling law is obtained for the correlation of the angle formed by two arms. Therefore, the model seems to reproduce adequately the main features for the dynamics of star chains, as expected from the Rouse theory. A sim-... 

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. 

Fig. 16.3 Comparison of the Rouse theory (solid line) and the results of Gs(t) (o at A = 1 and A at A = 2) and G i(f) ( at A = 1 and A at A = 2) obtained from simulations on the five-bead Rouse chain following the application of a step shear strain A.

An imderstanding of the influence of polydispersity on chain dynamics in the melt was achieved by Baschnagel and co-workers (177) in a simulation using BFM. These dynamic Monte Carlo simulations showed that long chains move more rapidly in the presence of short chains, and the short ones move more slowly in the presence of the long ones. The net effect is that the dsniamics of a polydis-perse melt is close to Rouse theory predictions, ie, the chains act as if they are not entangled An indirect approach to reptation dynamics was described by Byutner... 

Vladkov, M. and Barrat, J.-L. (2006) linear and nonlinear viscoelasticity of unentangled polymer melts molecular dynamics and Rouse mode analysis. Macromol. Theory Simul, 15, 252-262. 

Apart from the introductory section, the article is subdivided into four major sections NMR Methods Modeling of Chain Dynamics and Predictions for NMR Measurands Experimental Studies of Bulk Melts, Networks, and Concentrated Solutions and Chain Dynamics in Pores. First, the NMR techniques of interest in this context will be described. Second, the three fundamental polymer dynamics theories, namely the Rouse model, the tube/reptation model, and the renormalized Rouse theories are considered. The immense experimental NMR data available in the literature will be classified and described in the next section, where reference will be made to the model theories wherever possible. Finally, recent experiments, analytical treatments, and Monte Carlo simulations of polymer chains confined in pores mimicking the basic premiss of the tube/reptation model are discussed. 

MC simulations and semianalytical theories for diffusion of flexible polymers in random porous media, which have been summarized [35], indicate that the diffusion coefficient in random three-dimensional media follows the Rouse behavior (D N dependence) at short times, and approaches the reptation limit (D dependence) for long times. By contrast, the diffusion coefficient follows the reptation limit for a highly ordered media made from infinitely long rectangular rods connected at right angles in three-dimensional space (Uke a 3D grid). 

Comparison between Simulation and Theory of the Rouse Model... 

These results are exactly confirmed by the simulations cases (l)-(3) are illustrated in the following The relaxation time in units of the time step as expressed by Eq. (16.8b) allows one to compare the simulated Gs t) curve with that calculated from Eq. (16.14). In Fig. 16.2, such a comparison is made for two-bead, five-bead and ten-bead Rouse chains. The perfect agreements between the simulated and theoretical line shapes without any shift along both the modulus and time-step axes confirm the predicted N and p dependences of the relaxation times of the Rouse normal modes. As predicted by the theory, no nonlinear effect can be observed between the... 

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... 

Fig. 13 Global structure factor versus wave vector for different times for a quench from the mixed state at /N = 0.314 to X-N = 5. Lines represent Monte Carlo results, symbols dynamic SCF theory results, (a) Compares dynamic SCF theory using a local Onsager coefficient with Monte Carlo simulations. Local dynamics obviously overestimates the growth rate and shifts the wavevector that corresponds to maximal growth rate to larger values, (b) Compares dynamic SCF theory using anon-local Onsager coefficient that mimics Rouses dynamics with Monte Carlo results showing better agreement. From [29]...

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