Equation of Macromolecular Dynamics

The set of stochastic equations given by (3.37) is equivalent (in the linear case) to equations (3.11) with the memory functions defined in Section 3.3, but, in contrast to equations (3.11), set (3.37) is written as a set of Markov stochastic equations. This enables us to determine the variables that describe the collective motion of the set of macromolecules. In this particular approximation, the interaction between neighbouring macromolecules ensures that the phase variables of the elementary motion are co-ordinates, velocities, and some other vector variables - the extra forces. This set of phase variables describes the dynamics of the entire set of entangled macromolecules. Note that the Markovian representation of the equation of macromolecular dynamics cannot be made for any arbitrary case, but only for some simple approximations of the memory functions. We are considering the case with a single relaxation time, but generalisation for a case with a few relaxation times is possible. 

Abstract The discussion of relaxation and diffusion of macromolecules in very concentrated solutions and melts of polymers showed that the basic equations of macromolecular dynamics reflect the linear behaviour of a macromolecule among the other macromolecules, so that one can proceed further. Considering the non-linear effects of viscoelasticity, one have to take into account the local anisotropy of mobility of every particle of the chains, introduced in the basic dynamic equations of a macromolecule in Chapter 3, and induced anisotropy of the surrounding, which will be introduced in this chapter. In the spirit of mesoscopic theory we assume that the anisotropy is connected with the averaged orientation of segments of macromolecules, so that the equation of dynamics of the macromolecule retains its form. Eventually, the non-linear relaxation equations for two sets of internal variables are formulated. The first set of variables describes the form of the macromolecular coil - the conformational variables, the second one describes the internal stresses connected mainly with the orientation of segments. 

According to speculations in Chapter 3 (see Section 3.2), the standard equation of macromolecular dynamics can be written in the form... 

We note once more that the Markovian representation of the equation of macromolecular dynamics cannot be made for any arbitrary case, but only for some simple approximations of the memory functions. The above system describes the situation when the medium is characterised by the only relaxation time, but generalisation for few relaxation times is possible. 

Of course, it would be best to introduce the anisotropy of motion in equation (41). However, we follow Curtiss and Bird [47] and write the equation of macromolecular dynamics instead of (41) in the form... 

Molecular theories, utilizing physically reasonable but approximate molecular models, can be used to specify the stress tensor expressions in nonlinear viscoelastic constitutive equations for polymer melts. These theories, called kinetic theories of polymers, are, of course, much more complex than, say, the kinetic theory of gases. Nevertheless, like the latter, they simplify the complicated physical realities of the substances involved, and we use approximate cartoon representations of macromolecular dynamics to describe the real response of these substances. Because of the relative simplicity of the models, a number of response parameters have to be chosen by trial and error to represent the real response. Unfortunately, such parameters are material specific, and we are unable to predict or specify from them the specific values of the corresponding parameters of other... 

Though the hypothetically correct equation for macromolecular dynamics must contain non-linear terms, it is useful to study the linear dynamics of a macromolecule based on Eq. (41). This equation ought to be regarded as the fundamental equation of the theory, and we can use normal co-ordinates (12) to get independent dynamic modes. 

Equation 15 gives AC(x) in terms of macromolecular dynamics and thus sets a basis for application of this approach. In terms of the dynamics of the chain molecule, the interpretation of AC(x) (Eq. 15) should be straightforward. This is because (a) the dynamics has been characterized by a single adjustable parameter, D, (b) the fluorescence intensities are well-defined functions of distance between the chromo-phores, and (c) the equilibrium distribution function, N Cr), can be obtained by independent experiments, as has been shown above. 

Robert D. Skeel, Jeffrey J. Biesiadecki, and Daniel Okunbor. Symplectic integration for macromolecular dynamics. In Proceedings of the International Conference Computation of Differential Equations and Dynamical Systems. World Scientific Publishing Co., 1992. in press. 

The straight solid line - the consequence of equations (3.52) and (3.53) at — 0 -depicts the analytical result for the Rouse dynamics. The solid curves represents the displacement for a macromolecule of length M = 25Me (% = 0.04) with corresponding (according to relations (3.25) and (3.29)) values of parameters B = 429 and ijj = 8.27 and the values of parameters of local anisotropy ae = 0 and ae = 0.3. For the isotropic situation (ae = 0), the curve can be calculated analytically according to equation (5.5), but for the parameter of local anisotropy ae = 0.3, the displacement ought to be calculated numerically. Internal resistance (parameters E and cq) does not affect mobility of macromolecular coil. Adapted from Pokrovskii (2006). 

Let us remind that equation (9.24), describing the relaxation of macro-molecular conformation, can be considered only as an assumed results of accurate derivation of the relaxation equation from the macromolecular dynamics. 

The Eq. (182) at the condition /g=const=0.270 nm and C =const=10 is reduced to a purely fractal form, that is, to the Eq. (8) with 5=0.349. Let us note essential distinctions of the Eqs. (180) and (8). Firstly, if the first from the indicated equations takes into accoimt object mass only, then the second one uses elements number N of macromolecule, that is, takes into account dynamics of molecular structure change. Secondly, the Eq. (8) takes into account real structural state of macromolecule with the aid of its fractal dimension The indicated above factors appreciation defines correct description by the equation (8) the dependence of macromolecular coil gyration radius R on molecular weight MIT of polynner [235]. 

In the simplest case the macromolecular coil can be considered to be perfectly drained and without internal friction [2]. These assumptions gives the simplest linear form of the dynamic equation... 

The Oid(7) and dependences obtained for PATAC are fairly well fitted by equation 15 with hcOpi, = 0.18 eV and Eq. 14 with E = 0.061 eV, respectively (Fig. 6). The break in the 03d(7) curve can be attributed to a change in the conformation of the system. The energy necessary for the charge-carrier activation hopping between polymer chains is close to the energy of activation of the polymer chain librations (see below). This fact is evidence of the interference of charge transfer and macromolecular dynamics processes in the polymer. 

In the first problem class mentioned above (hereinafter called class A), a collection of particles (atoms and/or molecules) is taken to represent a small region of a macroscopic system. In the MD approach, the computer simulation of a laboratory experiment is performed in which the "exact" dynamics of the system is followed as the particles interact according to the laws of classical mechanics. Used extensively to study the bulk physical properties of classical fluids, such MD simulations can yield information about transport processes and the approach to equilibrium (See Ref. 9 for a review) in addition to the equation of state and other properties of the system at thermodynamic equilibrium (2., for example). Current activities in this class of microscopic simulations is well documented in the program of this Symposium. Indeed, the state-of-the-art in theoretical model-building, algorithm development, and computer hardware is reflected in applications to relatively complex systems of atomic, molecular, and even macromolecular constituents. From the practical point of view, simulations of this type are limited to small numbers of particles (hundreds or thousands) with not-too-complicated inter-particle force laws (spherical syrmetry and pairwise additivity are typically invoked) for short times (of order lO" to 10 second in liquids and dense gases). 

Introduction of the local anisotropy of mobility allows us to specify the matrixes of the extra forces of external and internal resistance and to formulate dynamic equations, which will be discussed in Section 3.4. One can expect that, as a result of the introduction of the local anisotropy, mobility of a particle along the axis of a macromolecule appears to be bigger than mobility in the perpendicular direction, so that the entire macromolecule can move more easily along its contour. The local anisotropy hinders also change of the form of the macromolecular coil, and, by this way, plays a role similar to the role of the term with internal resistance in linear version of the model. 

Local motions which occur in macromolecular systems can be probed from the diffusion process of small molecules in concentrated polymeric solutions. The translational diffusion is detected from NMR over a time scale which may vary from about 1 to 100 ms. Such a time interval corresponds to a very large number of elementary collisions and a long random path consequently, details about mechanisms of molecular jump are not disclosed from this NMR approach. However, the dynamical behaviour of small solvent molecules, immersed in a polymer melt and observed over a long time interval, permits the determination of characteristic parameters of the diffusion process. Applying the Langevin s equation, the self-diffusion coefficient Ds is defined as... 

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