Model for Chain Dynamics

1 Model for Chain Dynamics In Sections 1.2 and 1.3, we learned about the conformation of the ideal chain and obtained the probability distribution of the conformation. The distribution tells how many chains in the system have a certain conformation at a given time. Each chain is moving and changing its conformation all the time. The probabihty distribution also gives the distribution of the period in which a given chain takes each conformation (ergodicity). 

The Rouse modeP is the simplest version of the bead-spring model that can treat the chain dynamics. The model assumes that the beads have no excluded volume (they are essentially a point) and that there are no hydrodynamic interactions 

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Orwoll, R. A., Stockmayer, W. H. Stochastic models for chain dynamics. Advan. Chem. Phys. 15,305-324 (1969). 

Stochastic Models for Chain Dynamics (Orwoll Stockmayer). 15 305... 

As quoted in Sect. 1.2, different theoretical models for chain dynamics generally lead to different analytical expression for the OACF, which can be compared with the experimental anisotropy. The knowledge of the statistical distribution of errors on each channel is an essential tool for this comparison. It provides objective criteria to decide whether a discrepancy between a model and a set of data is significant or not, and to compare different models. Among these criteria, the most well known one is the reduced which should be 1 for purely statistical deviations, and increases... 

The fact that the dielectric relaxations in Amorphous solid polymers was due to the motion of angularly correlated dipoles within a bulk amorphous medium comprising similar dipoles implied that a complete theoreticar description of dielectric relaxation for any given poljrmer would be an extremely formidable task. This was the daunting prospect in the late 1960 s and it appeared necessary to construct sophisticated models for chain dynamics, incorporating angular (dipole) correlations, in order to make progress. 

Originally the concern focussed on idiat kinds of local factors (torsional strain, steric effects) affected the cyclization of flexible organic molecules. Then emphasis shifted to the entropic factors affecting polymer cyclization (2). More recently one has begun to appreciate that many topical issues in polymer theory, particularly excluded volume effects and models for chain dynamics, find rather vivid expression in cyclization-related phencxnena. Thus studies of cyclization equilibria and dynamics provides new insights into polymer behavior and new approaches to testing predictions of contemporary theory. 

Within the Rouse model for polymer dynamics the viscosity of a melt can be calculated from the diffusion constant of the chains using the relation [22,29,30] ... 

A method for the detailed atomistic modelling of amorphous glassy polymers has been developed [50] and applied to atactic polypropylene. This method has been described in Sect. 5.4. The quasi-static modelling of chain dynamics [51] has been described in Sect. 5.4.2. 

The -alkanes have long been used as valuable models for the dynamics of polyethylene. The INS work confirms their utility but also highlights their limitations. Most work has neglected the effect of the finite length and the chain ends but both are present in the real compounds and modify the dynamics. 

Fenchenko studied free induction decays and transverse relaxation in entangled polymer melts. He considered both the effects of the dipolar interactions between spins in different polymer chains and within an isolated segment along s single chain. Sebastiao and co-workers presented a unifying model for molecular dynamics and NMR relaxation for chiral and non-chiral nematic liquid crystals. The model included molecular rotations/ reorientations, translational self-diffusion as well as collective motions. For the chiral nematic phase, an additional relaxation mechanism was proposed, associated with rotations induced by translational diffusion along the helical axis. The model was applied to interpret experimental data, to which we return below. 

Now we turn oirr attention to other models for imentangled dynamics that are different from the Rouse model in several ways. These models are divided into two categories single-chain models and midtichain models. 

Monte Carlo simulations with lattice models of chain dynamics have been concerned with segmental and with whole molecule relaxation. In each case the autocorrelation functions depend upon details of the lattice model. A cubic-lattice model, with random stifle bead motions has been used to study the relaxation of seven characteristics of instantaneous aspherical shape. Three decay functions are observed for iarge molecules with and without excluded volume considerations. The large difference in relaxation times and in their chain length dependence obtained here, and in earlier studies on including the excluded volume effect, is attributed to restrictions associated with the single bead mechanism, a view justified by simulations which obtained more flexible chains by... 

Sergi D, Scocchi G, Ortona A Coarse-graining MARTINI model for molecular-dynamics simulations of the wetting properties of graphitic surfaces with non-ionic, long-chain, and T-shaped surfactants, J Chem Phys 137 094904, 2012. 

These predictions of the simple phenomenological model are in accord with experimental dielectric data for amorphous solid polymers (4-7). The model does not specify detailed mechanisms for a and B processes, so, historically, the next stage was to develop such models. Many attempts were made and Table 1 summarizes a number of one-body models and their generalizations to include chain dynamics. Those for chain dynamics incorporate the basic models for one body motion e.g. the theory of Yamafuji and Ishida (22) is for coupled units each undergoing small-step rotational diffusion, while those of Jernigan (29) and Beevers and Williams (30) are for coupled units each undergoing motion in local (conformational) barrier systems. All the models in Table 1 exclude the short time effects associated with inertial factors and damped librations in a local potential. 

The starting point for molecular models for polymer dynamics based on the ideas introduced in Section 14.2.3 is the Rouse model for an isolated chain in a viscous medium, in which the chain is taken to behave as a sequence of m beads linked by Gaussian springs [Figure 14.9(a)] [13-16]. The chain interacts with the solvent via the beads, and the solvent is assumed to drain freely as the chain moves. Hence, Eq. (22) leads to Eqs. (45), where N is the number of links between adjacent beads, C is a friction coefficient per bead and r is the position of the ith bead. 

It had long been assumed that the solvent in a polymer solution provides a neutral hydrodynamic background, and that the properties of the solvent in a solution, such as viscosity, are the same as the properties found in the neat solvent. We know now that this simple assumption is incorrect Just as the solvent can alter properties of the polymer, so also do polymers alter the properties of the surrounding solvent Translational and rotational mobilities of solvent molecules may be reduced or increased by the presence of nearby polymer chains. Models for polymer dynamics that assume that the solvent has the same properties as the neat liquid are therefore unlikely to be entirely accurate. 

Comparison with models for colloid dynamics indicates that the drag coefficients for single-chain diffusion and for chain sedimentation are not the same at elevated matrix concentrations. Experiments testing this assertion for polymer solutions... 

In addition to linear chains, one may also examine systems having different chain topologies. Despite their appearance in predictions of some models for polymer dynamics, there do not appear to be substantial literature results on the viscosity... 

The form of equation (208) for E calM/pM ) is consistent with certain calculations based on the reptation model of chain dynamics with intermolecular entanglement constraints. In its simplest version, application of the reptation model gives... 

Monte Carlo Lattice Model for Chain Diffusion in Dense Polymer Systems and its Interlocking with Molecular Dynamics Simulations. 

Particular emphasis will be laid on the experimental verification of features of standard theories of polymer dynamics such as the Rouse model, the tube/reptation model [1] or the renormahzed Rouse models. Based on the special conditions under which predictions of these theories match experimental findings well, the application limits and the deficiencies of these models for more general scenarios become obvious and help to ameliorate the underlying model ansatz for chain dynamics. 

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. 

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