The Dynamics of Polymer Molecules

In the previous sections, theories were reviewed where the optical properties of polymer liquids were cast in terms of the microscopic properties of the constituent chains. The dynamics of polymer chains subject to external fields that orient and distort these complex liquids are considered in this section for a variety of systems ranging from dilute solutions to melts. Detailed descriptions of theories for the dynamics and structure of polymer fluids subject to flow are found in a number of books, including those by Bird et al. [62], Doi and Edwards [63], and Larson [64], 

An elastic spring tethers the two beads together and exerts a force on the beads tends to restore the system to its equilibrium coil-like shape. If the spring is taken to e a force constant, K (l R ), the spring force is 

The force constant can be determined by considering the force required to stretch 

The form of the distribution function will depend on the approximations that have been incorporated into the model. In its simplest form, where finite extensibility, hydrodynamic interaction and excluded volume have been neglected, the following Gaussian function describes the distribution of conformations, 

The Gaussian distribution is deficient in a number of ways. In particular, it suggests that the dumbbell can be stretched without limit, since there is a finite probability for any value of the end-to-end distance. The constraint that the end-to-end distance should not exceed the contour length NKa can be incorporated into the model and a discussion of that 

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Maclnnes DA (1977a) Internal viscosity in the dynamics of polymer molecules. J Polym Sci Polym Phys Ed 15(3) 465-476... 

Another largely unexplored area is the change of dynamics due to the influence of the surface. The dynamic behavior of a latex suspension as a model system for Brownian particles is determined by photon correlation spectroscopy in evanescent wave geometry [130] and reported to differ strongly from the bulk. Little information is available on surface motion and relaxation phenomena of polymers [10, 131]. The softening at the surface of polymer thin films is measured by a mechanical nano-indentation technique [132], where the applied force and the path during the penetration of a thin needle into the surface is carefully determined. Thus the structure, conformation and dynamics of polymer molecules at the free surface is still very much unexplored and only few specific examples have been reported in the literature. 

Molecularly motivated empiricisms, such as the solubility parameter concept, have been valuable in dealing with mixtures of weakly interacting small molecules where surface forces are small. However, they are completely inadequate for mixtures that involve macromolecules, associating entities like surfactants, and rod-like or plate-like species that can form ordered phases. New theories and models are needed to describe and understand these systems. This is an active research area where advances could lead to better understanding of the dynamics of polymers and colloids in solution, the rheological and mechanical properties of these solutions, and, more generally, the fluid mechaiucs of non-Newtonian liquids. 

When adsorbed (from ambient air), water molecules might act as plasticizers and alter the dynamics of polymers. Moreover, water has a strong dipole moment and, consequently, dielectric active relaxation processes, which could partially occlude significant parts of the dielectric spectra of interest. Special attention to this effect has to be paid when the dynamics of thin polymer films is investigated, for example in relation to phenomena like the glass transition, dewetting, pattern formation, surface mobility etc. 

In a real situation, the motion of the segments of a chain relative to the molecules of the solvent environment will exert a force in the liquid, and as a consequence the velocity distribution of the liquid medium in the vicinity of the moving segments will be altered. This effect, in turn, will affect the motion of the segments of the chain. To simplify the problem, the so-called free-draining approximation is often used. This approximation assumes that hydrodynamic interactions are negligible so that the velocity of the liquid medium is unaffected by the moving polymer molecules. This assumption was used in the model developed by Rouse (5) to describe the dynamics of polymers in dilute solutions. 

The latter asymptotic relation was first obtained by de Gennes [46] in 1967, and the factor 1/16 (= 0.0625) in eq 3.28 was previously estimated to be 0.055 by Akcasu and Gruol [41]. The factor 1/67t in eq 3.29 is about 15% smaller than 1/16, indicating that the error due to the preaveraging is rather serious. But, more importantly, these equations contain no parameter connected with local polymer structure. In order to obtain by QELS information about the local dynamics of polymer molecules we have to formulate the dynamic structure factor on a more realistic chain model. 

However, it has been shown that the dynamics of reorganization of a thin film upon thermal treatment not only is determined by the composition and organization of the film itself, but also depends on its local environment. Numerous experiments indicate that the mobility of polymer molecules in the vicinity of a surface or an interface is perturbed [16-18], and the extent to which they affect the mobility of polymer chains depends on the strength of their interactions with the surface/interface. 

After a critical discussion of novel algorithms for condensed poly mo phases(Leontidis et al.), the structural and equation of state properties of polymers (Theodorou et al.), as well as the dynamics of small molecules in bulk polymers (Gusev et al.) are presented. 

In the spirit of dumbbell models [9], we mimic the dynamics and the shape of polymer molecule by that of a particle at position r where r = 0 corresponds to its center of mass. A force F acting on this particle can be chosen such that the time average of coincides with the mean square radius of gyration of the polymer coil in equilibrium. Furthermore, the effect of the flow of the background fluid is taken into account via a friction force proportional to the difference between the velocity of the particle and the flow velocity v(r). For this model, reduced (dimensionless) variables are used, but they are denoted by the same symbols as the corresponding physical quantities. The reduced mass is put equal to 1. Then the equation of motion reads... 

This discrepancy comes from neglecting the hydrodynamic interactions between the solvent molecules and the polymer chains. Because of this discrepancy, the Rouse model is not suitable for describing the dynamics of polymer chains in dilute solution, but it is still very useful for describing the dynamics of undiluted polymer with M < M,. 

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