Chain dynamics 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. 

Fig. 18. Stress time-correlation function of EV linear chains and stars with different functionalities. Comparison of Brownian dynamics (crosses) and generalized Zimm calculations from MC averages (solid lines). Reprinted with permission from [89]. Copyright (1996) American Institute of Physics...

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-... 

The spectrum of scattered light contains dynamical information related to translational and internal motions of polymer chains. In the self-beating mode, the intensity-intensity time correlation function can be expressed (ID) as... 

Fig. 35. Effect of water on parvalbumin dynamics. Time constants (tI) were determined from time correlation functions for the vector between the two outermost nonhydrogen atoms in each side chain, ordered by residue type. (Top) In vacuo simulation of parvalbumin (bottom) simulation with waters. From Ahlstroem etal. (1987).

Hatada, et al. have reported C-13 NMR experiments on a dilute solution of polyisoprene in methylene ch1oride(21). It is difficult to make a rigorous comparison with these measurements for two reasons 1) The NMR measurements were made at a single frequency and temperature. Hence no information about the shape of the correlation function is available. 2) The NMR experiment senses reorientation of a C-H vector which is perpendicular to the chain backbone. Our optical experiments measure the correlation function for a vector parallel to the chain backbone. In order to make at least a rough comparison with the NMR data, we make use of the Brownian dynamics simulations of Weber and He1fand(22). These simulations for a polyethylene-like chain compared the correlation functions for vectors both parallel and perpendicular to the chain backbone. Their results indicate that the correlation function for the perpendicular bond decays 4 times faster than the correlation function for the parallel bond. If we assume that the correlation function for a C-H vector in polyisoprene has the shape obtained in the computer simulation, and further assume that the relationship between parallel and perpendicular vectors revealed by the simulations is valid for polyisoprene, a comparison can be made between the optical and NMR experiments. 

In Chapter 3, we used the Rouse model for a polymer chain to study the diffusion motion and the time-correlation function of the end-to-end vector. The Rouse model was first developed to describe polymer viscoelastic behavior in a dilute solution. In spite of its original intention, the theory successfully interprets the viscoelastic behavior of the entanglement-free poljuner melt or blend-solution system. The Rouse theory, developed on the Gaussian chain model, effectively simplifies the complexity associated with the large number of intra-molecular degrees of freedom and describes the slow dynamic viscoelastic behavior — slower than the motion of a single Rouse segment. 

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. 

We now study the dynamics of the primitive chain and show that certain time correlation functions can be calculated by a straightforward method. For example, consider the time correlation function of the end-to-end vector P(r) lf(L, r) -lf(0, t). Figure 6.4 explains the prin-dple of calculating tlfis correlation function. At r = 0, the chain is trapped in a certain tube. As time passes, the primitive chain reptates and at a certain later time (Fig. 6Ad), the part of the chain CD remains in the original tube while the parts AC and DB are in a new tube. To calculate... 

Stochastic equation for reptation dynamics Although the above probabilistic description is quite useful in understanding the essence of reptation dynamics, it becomes progressively more difficult to proceed with the calculation for other types of time correlation function. For example, it is not easy to calculate the mean square displacement of a primitive chain segment (R(s, t)-R(s, 0)) ) by this method. In this section we shall describe a convenient method" for calculating general time correlation functions. 

The anisotropy of segmental motion exhibited in Fig. 19 may arise, as noted above, either from the intramolecular or from the intermoleoilar ccmstraint to the rotational motion. The anisotropy d orioitational condadon decay was indeed noted already by Weber and Helfand [47] in their Brownian dynamics simulation of polyethylene of infinite chain length. Their orioitational time-correlation function of the chord vector ( = 0°) decayed much more slowly than those of either the bisector vector ( = 0°, = 0°) or the out-of-plane vector ( = 0°, = 90°). What they modeled was a phantom chain having no... 

The MC method was first applied to polymer chain dynamics by Verdier and Stockmayer, using a bead model on a simple cubic lattice. Beads are moved from site to site on the lattice, in a way which satisfies certain criteria e.g. chain connectivity, excluded volume effects), and both the equilibrium average chain dimensions, and (by sampling to obtain the time correlation function) the relaxation behaviour of chains may be studied. One of the results is that excluded volume effects slowed the relaxation times significantly. Deutch and Boots have criticized the original model, attributing this unexpected result to unrealistic... 

Other than static properties, hydrogen bond (HB) dynamics was calculated between PA-66 chains. Time correlation functions namely the continuous HB time correlation function S t) and intermittent HB time correlation function C t) as defined in the equation 7 was calculated from... 

Chu et al. have studied the structure and dynamics of a polymer solution composed of PS(polystyrene), PMMA(poly(methyl methacrylate)), TOL(toluene), and CNA(a-chloronaphthalene). According to the photocount intensity-intensity time correlation function measurements of the PMMA probe in the PS/MS(mixed solvent) isorefractive matrix, one surprising feature was that at least two dominant characteristic modes appeared even at small scattering angles. While the slow mode could be identified with the translational motion of the center of mass of the PMMA probe chain, it remained unconfirmed that the fast mode might be related to a coupling of PMMA motion with the cooperative motion of the isorefractive PS/MS(matrix). 

For a chain of elements with three states, closer to the polymer model, the correlation functions for an analogous dynamic model can not be exactly determined. One can develop a theory which ignores all the terms that do not have a diffusional character. One must take into account that there are two types of deviations from equilibrium that can diffuse. We will not present more detail but merely mention that the resulting formulas can be found elsewhere, and shown to fit well to various conformational state time correlation functions determined from the simulations. ... 

The dynamic behavior of macromolecules can be explained by adopting the time correlation functions of the equilibrium fluctuations of the chain end-to-end vectors, CRR(t) (for a precise definition of correlation... 

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. 

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