Time-correlation function chain

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... 

Using Eq. 1.2 the time correlation function of the end-to-end vector was calculated for the hexamer systems in slits of the same thickness as the interlayer gallery of PEO/fluorohectorite intercalated hybrids [38d Section 2.2], for the whole film (Fig. 14a) and for the adsorbed chains (Fig. 14b). The film as a whole exhibits a multimodal relaxation, including fast relaxing and much slower relaxing species. The fast, bulk-like, relaxation is due to the free chains and the slower part is due to the superposition of the slower modes of the adsorbed chains and the relaxation times involved for each mode depend on the number of adsorbed segments (Fig. 15). 

Fig. 14. End-to-end vector time correlation functions for various wall-polymer affinities (a) for the whole film (approximately same width as the intercalated PEO interlayer gap), (b) for the adsorbed chains independently of the number of contacts. Adopted from reference [38d].

The relaxation times of the end-to-end vector time correlation function when calculated for the whole film (through the end-to-end vector time correlation function) exhibits a multimodal relaxation. The fast relaxation corresponds to the free chains located in the middle of the film and the slower modes to the relaxation of the adsorbed chains. This is concluded both by the relaxation times as calculated for those classes of chains separately, as well as by the ratio of the correlation function amplitudes of the fast versus the slow modes, which is very close to the ratio of the number of free versus adsorbed chains. 

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).

Figure 9 shows the instantaneous conformation of a portion of a polymer chain with the unit vector m affixed to the central bond of the nine-bond sequence AB whose end-to-end vector is denoted by The instantaneous position of the chain-affixed coordinate system Axyz is denoted by R. The laboratory coordinate system is represented by Oxyz. The time correlation function involves the product of the external and internal correlation functions. These two are assumed to be independent. The internal autocorrelation function is given by... 

The limit in front of the ratio means that the time t has to be much longer than the longest relaxation time of the chain. The resulting diffusion coefficients obtained by Monte Carlo simulation of the Evans-Edwards model of entangled polymers are presented in Fig. 9.33(a). The diffusion coefficient decreases with the number of monomers in the chain. Another quantity that can be extracted from the Monte Carlo simulations of the Evans-Edwards model is the relaxation time of the chain. It can be defined as the characteristic decay time of the time correlation function of the end-to-end vector R[t)R 0)) exp( t/Trep). Figure 9.33(b) presents the results of such simulations. 

The time correlation function for the energy, over a chain of size l as... 

The first cumuleint F is generally calculated by assuming the hydro namic interaction eis described by Oseen (3) where no knowledge of the space-time correlation function is needed (4-6>. The purpose of the present contribution is an experimental test of theoretical relationships which are based on the Flory-Stockmayer (FS) breinching theory (7) of the solution properties from reui-domly crossllnked monodisperse primary chains. Most of the theoretical work emd peirt of the e q>erlmental work has been piA>llshed previously (8-13). We, therefore, bring here only a short outline of the theory and confine ourselves mainly to the discussion of the namlc properties. 

There are two forms of phenomenological equations for describing Brownian motion the Smoluchowski equation and the Langevin equation. These two equations, essentially the same, look very different in form. The Smoluchowski equation is derived from the generalization of the diffusion equation and has a clear relation to the thermodynamics of irreversible processes. In Chapters 6 and 7, its application to the elastic dumbbell model and the Rouse model to obtain the rheological constitutive equations will be discussed. In contrast, the Langevin equation, while having no direct relation to thermodynamics, can be applied to wider classes of stochastic processes. In this chapter, it will be used to obtain the time-correlation function of the end-to-end vector of a Rouse chain. 

In summary, we have used the Rouse chain model to obtain the diffusion constant of the center of mass and the time-correlation function of the end-to-end vector, which reflects the rotational motion of the whole polymer molecule. Since N is proportional to the molecular weight M, and K is independent of molecular weight, Eqs. (3.41) and (3.62) indicate that Dq and Tr depend on the molecular weight, respectively, as... 

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. 

The time-correlation function 5L 0)5L t)) of Eq. (9.3) will be derived by considering the polymer chain as a Gaussian chain consisting of No segments each with the root mean square length b. Let 5 (t) be the contour position of the nth bead relative to a certain reference point on the primitive path. Then the contour length of the primitive chain at time t is given 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. 

Through Eq. (16.8a) or (16.8b), the time-correlation function C t) = (R(0) R(t)) of the end-to-end vector and the relaxation modulus G t) calculated from Eqs. (3.59) and (7.58), respectively, may be compared with the corresponding simulation results for a Rouse chain. In the simulation of a time-correlation function, an equilibrium state is first established by running a sufficiently large number of Monte Carlo steps. Then, a time window is set up, within which the time-correlation function may be calculated as explained in the following ... 

Fig. 16.1 Comparison of the analytical solutions (solid lines) and the Monte Carlo simulations for the time-correlation functions of the end-to-end vector, C t) = (R(0) -R(t)), of the Rouse chains with two beads (o) and three beads ( ) (b = 10 and d = 0.4 are used in the simulations.).

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... 

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