Rouse modes time correlation function

Different equilibrium, hydrodynamic, and dynamic properties are subsequently obtained. Thus, the time-correlation function of the stress tensor (corresponding to any crossed-coordinates component of the stress tensor) is obtained as a sum over all the exponential decays of the Rouse modes. Similarly, M[rj] is shown to be proportional to the sum of all the Rouse relaxation times. In the ZK formulation [83], the connectivity matrix A is built to describe a uniform star chain. An (f-l)-fold degeneration is found in this case for the f-inde-pendent odd modes. Viscosity results from the ZK method have been described already in the present text. 

The time-correlation function of the Rouse modes is given by... 

We finish this section with the schematic drawing displayed in Fig. 6.4, meant to indicate how the time dependent fluctuations of the amplitude of a Rouse-mode could look-like. The interaction of a chain with its surroundings leads to excitations of this mode at random times. In-between, the mode amplitude decreases exponentially with a characteristic relaxation time as described by the equation of motion. These are the only parts in the time dependent curve which show a well-defined specific behavior the excitations occur irregularly during much shorter times. We may therefore anticipate that the shape of the time correlation function is solely determined by the repeated periods of exponential decay. Regarding the results of this section, we thus may formulate directly the time correlation function for the normal coordinate... 

Equation (6.73) relates G t) to the magnitudes and the time dependencies of the fluctuations of the Rouse-modes in thermal equilibrium. The time correlation functions are given by Eq. (6.58)... 

Now we employ the Rouse-model. As the end-to-end distance vector is essentially determined by the lowest order Rouse-modes, we can also represent the time correlation function in good approximation by... 

Figure 12. (a) Plots of the Rouse amplitude correlation functions for several modes versus time... 

Since l is proportional to and q is proportional to 1/L, i is proportional to. Substitution of Eq. (67) into Eq.(62) gives the Langevin equation for the Rouse modes of the chain within the approximations of preaveraging for hydrodynamic interactions and mode-mode decoupling for intersegment potential interactions. Equation (62) yields the following results for relaxation times and various dynamical correlation functions. 

Calculate the stress relaxation modulus of the Rouse model (Eq. 8.55) by showing that after a small step shear strain 7 at time t 0 the correlation function of normal modes decays as Xpx t)X y(t)) — i kTjkp) exp (- tjxp). 

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... 

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... 

Figure 13. Time-temperature superposition property for the Rouse mode correlation function, exemplified by the third mode, 33(f)j temperatures...

Rouse-modes with m < m do not exist. They become replaced by other relaxation processes, and the third term in Eq. (6.109) describes this contribution. The relaxation strength is identical to that of the replaced Rouse-modes, as this part remains unrelaxed after the decay of all modes with m>m. Writing the correlation function for the long-term part in the form (t/ra) implies the assumption that, similar to the Rouse-modes, also this part is controlled by a single characteristic time, the disentangling time ra, only. As introduced here, (f/ra) is a general normalized function... 

An analysis of the intramolecular dynamics in terms of the Rouse modes yields non-exponentially decaying autocorrelation functions of the mode amphmdes. At very short times, a fast decay is found, which turns into a slower exponential decay which is well fitted by Ap exp(-f/Tp), see Fig. 13. Within the accuracy of these calculations, the correlation functions exhibit universal behavior. Zimm theory predicts the dependence Tp for the relaxation times on the mode number for polymers with excluded-volume interactions [6]. With v = 0.62, the exponent a for the polymer of length Am = 40 is found to be in excellent agreement with the theoretical prediction. The exponent for the polymers with Am = 20 is slightly larger. 

Fig. 13 Correlation functions of the Rouse-mode amplitudes for various modes as a function of the scaled time tp for polymers with excluded volume interactions. The chain lengths are Am = 20 left) and Am = 40 right). The calculated correlations were fitted by Ap exp(—r/vp) and have been divided by Ap. The scaling exponents of the mode numbers are a = 1.93 (Am = 20) and a = 1.85 (Am = 40), respectively. From [73]...

This Rouse stress relaxation time is half of the end-to-end vector correlation time because stress relaxation is determined from a quadratic function of the amplitudes of normal modes (see Problem 8.36). 

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