Polymer chains in a ©-solvent

Swollen polymers fall, in a sense, between the case of a few small polymers in a polymer matrix and a polymer chain in a solvent. The amount of... 

These models are designed to reproduce the random movement of flexible polymer chains in a solvent or melt in a more or less realistic way. Simulational results which reproduce in simple cases the so-called Rouse [49] or Zimm [50] dynamics, depending on whether hydrodynamic interactions in the system are neglected or not, appear appropriate for studying diffusion, relaxation, and transport properties in general. In all dynamic models the monomers perform small displacements per unit time while the connectivity of the chains is preserved during the simulation. 

In 1981 Bruns and Bansal [94] used a Lennard-Jones (LJ) model of polymer and solvent and analyzed the structural properties of the chain. In contrast to previous reports [92,93], a significant solvent effect was observed by Bruns and Bansal [94], The study of solvent effect on the static properties of polymer was succeeded by Khalatur et al. [95] on the static properties of a 16-bead polymer chain in a solvent. All potentials used were of LJ type. There are many other reports in the literature to understand the equilibrium size and shape of polymer as a function of solvent quality [96,97,98], Most of these studies are exploratory rather than quantitative, probably due to the computational expense, since the large relaxation times of the polymer chains as well as the large system sizes imply powerful computer resources. However, with the availability of high-performance computers, the problem has been addressed in earnest [99],... 

In the dumbbell model, a polymer chain in a solvent is pictured as two massless spheres of equal size connected by a frictionless spring. The spheres experience a hydrodynamic drag proportional to their size, characterized by the Flory radius. Assume that the displacement of the spring generated by the thermal energy is also characterized by the Flory radius. Write the equation of motion for the dumbbell and show that the characteristic relaxation time for the chain deformation is that given by Eq. (9.2.1). 

Figure 15.8. A schematic illustration of the mixing of a polymer chain in a solvent. Eaeh monomer unit (black) of the polymer or one solvent molecule (gray) occupies one lattice site. The figure in general illustrates polymer conformation in solution.

The assumption of unperturbed chain statistics. Implicit in Flory-Huggins theory is the assumption that the long-range chain statistics of polymer chains are ideal random walks. This is not to be expected a polymer chain in a solvent collapses as conditions are changed to bring about phase separation between the polymer and the solvent (Grosberg and Khokhlov 1994). One would expect a polymer chain in a mixture to do the same as the conditions for phase separation were approached (Sariban and Binder 1987). 

Monomers on a polymer chain in a solvent interact with each other through the effective long-range force. The temperature blob model predicts a crossover from a random coil to a compact globule. On the basis of the mean-field free energy, this section studies the possibility of a sharp CG transition [16,20]. 

For polymer chains in a -solvent the scaling exponent y takes its mean-field value y — 1. The polymer concentration derivative of the reduced osmotic pressure follows from (4.29) as... 

Fig. 4.13 Comparison of experimental gas-liquid coexistence binodals (data) compared to GFVT (curves). Left panel, spherical colloids mixed with polymer chains in a -solvent for q = 0.84 (open triangles, [20]), 1.4 (stars, [21]) and 2.2 (crosses, [21]). Right panel, colloidal spheres plus polymers in a good solvent for q = 0.67 (open squares, [20]), 0.86 (inverse filled triangle, [54]) and 1.4 (pluses, [20])...

Fig. 1. Schematic representation of the lattice model for the determination of the combinatorial entropy of mixing (a) mixture of small molecules (b) polymer chain in a solvent and (c) polymer/polymer mixture.

EDMD and thermodynamic perturbation theory. Donev et developed a novd stochastic event-driven molecular dynamics (SEDMD) algorithm for simulating polymer chains in a solvent. This hybrid algorithm combines EDMD with the direa simulation Monte Carlo (DSMC) method. The chain beads are hard spheres tethered by square-wells and interact with the surrounding solvent with hard-core potentials. EDMD is used for the simulation of the polymer and solvent, but the solvent-solvent interaction is determined stochastically using DSMC. 

The subject of Chapter 4 (originally the third lecture) is the problem of the reduction of turbulent losses by polymer chains in a solvent. The topic appears to be especially intricate in view of the coupling between the hydrodynamic aspects of turbulent flow and the viscoelastic behavior of chains in strongly perturbed conformations. The lecture followed closely a paper Towards a scaling theory of drag reduction published in 1986 by Professor de Gennes in Physica which is here reprinted by permission of the publisher. Because this lecture is the most tentative (and difficult) part in the series we have put it in the last chapter. 

In Section II we look more closely at the computational aspects of DPD, before focusing attention on the specific application to polymer systems. Section III describes the matching of simulation parameters to the properties of real polymer systems, with an emphasis on the relation between the conservative force field and the common Flory-Huggins / parameter for mixtures. The dynamics of individual polymer chains in a solvent and in a melt are discussed in Section IV, and the ordering dynamics of quenched block copolymer systems is described in Section V. A summary and conclusions are given in Section VI. 

In the following, we will discuss some microscopic dynamical models. We begin with the Rouse-model , which describes the dynamics of chains in a non-entangled polymer melt. The effects of entanglements on the motion can be accounted for by the reptation model , which we will treat subsequently. Finally, we shall be concerned with the motion of polymer chains in a solvent, when the hydrodynamic interaction between the segments of a chain plays a prominent role. 

The result indicates that the ratio Cr/< r should be independent of the choice of the sequence. This is true if the friction coefficient Cr is proportional to the number of monomer units in the sequence. Strictly speaking, the latter property constitutes a basic requirement for the validity of the Rouse-model The friction coefficient of a sequence has to be proportional to the number of monomer units. In fact, this is not trivial and clear from the very beginning. It seems to be correct in a melt because, as we shall see, here the Rouse-model works quite satisfactorily, if compared with experimental results. On the other hand, the assumption is definitely wrong for isolated polymer chains in a solvent where hydrodynamic interactions strongly affect the motion we shall be concerned with this point in a subsequent section. 

To this point, we have focused on the behavior of polymer chains in a solvent, concentrating on the dilute case, in which the individual chains do not strongly interact with each other. In this section, we now turn to more concentrated states in polymeric materials and their mechanical properties. Polymers take a variety of forms in everyday objects, but there are really only two significant states that are relevant to our understanding of everyday materials the glassy state and the melt state. In this section, we are focusing on polymer materials containing very little or no solvent. 

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