Models for a polymer chain

The simplest, and most primitive, model for a polymer molecule is the random flight chain, also termed the freely jointed chain. In this model, the bonds are represented by volumeless lines in space and there are no restrictions on the valency angles or on the rotations about bonds. 

The scalar product of r tvith itself gives. Following this prescription, we have 

There are two different types of terms involved in those sununations. First, there are terms of the type U.lj (k j) that equal cost, where t is the angle between the two vectors. Each of these terms, on averaging, yields / cost which must be zero, since the angle t is random. The other terms, and there are n of them, are characterized by A =y so that U.l =cosO, which averages to P. Accordingly 

Note that equation (4.8) predicts that the dimensions of a freely jointed chain vary as Af, where AT=molecular weight. For real chains, this prediction is strictly only correct under certain specific conditions in 0-solvents, in the melt and, perhaps, in the so-called concentrated regime (see Section 4.6). 

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From such a walk we can observe certain features of the general model for a polymer chain. The chain structure differs from conventional structures in that it does not display an obvious surface and incorporates a significant fraction of solvent within the structure. We can notice ... 

Earlier we developed a model for a polymer chain by using a long strand of beads. We could also choose other one-dimensional chain models, such as spaghetti noodles or strands of Easter basket grass. Based on their diam-... 

The first modification to the freely jointed chain model is the introduction of bond angle restrictions while retaining the concept of free rotation about bonds. This is called the valence angle model. For a polymer chain with all backbone bond angles equal to 9, this leads to Eq. (2.5) for the mean square end-to-end distance... 

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 classical model for a polymer chain with self-excluded volume is the SAW. Let be the appropriate self-consistent field (SCF) for a SAW of N steps but of any size R, By analogy with the Flory argument, we are tempted to assume that for a SAW is proportional to the mean density of occupied sites,, where the volume in p (i ) has been replaced by its mean value ATva because we are considering the total ensemble of iV-step SAWs and not just the subset of size R, The SCF for a SAW must satisfy certain properties. As will be shown below, this choice for is not self-consistent for < 4. 

Figure 2. A depiction of the bead-and-spring (Rouse Zimm) model for a polymer chain. The chain is represented by n beads and (n-1) springs of length b.

The simplest hydrodynamic model for a polymer chain in solution assumes that the solvent drains freely into the polymer chain. In this model, first studied by Rouse, the motions of different monomers are independent, i.e. the friction force on a given monomer depends solely on the velocity of that monomer and not on that of the other monomers. 

The mapping procedure provides an explicit connection between an atomistic model of a polymer chain and the corresponding coarse-grained model. For PE, we use an united atom description for the CH2 groups, resulting in a potential of the following type [146,182,183] ... 

In his paper Domb presents a detailed analysis of the statistical properties of self-avoiding walks on lattices.1 These walks serve as models for linear polymer chains with hard-core intramolecular interactions associated with the exclusion of multiple occupancies of the lattice sites by the chain so-called chains with excluded volume. 

Among other approaches, a theory for intermolecular interactions in dilute block copolymer solutions was presented by Kimura and Kurata (1981). They considered the association of diblock and triblock copolymers in solvents of varying quality. The second and third virial coefficients were determined using a mean field potential based on the segmental distribution function for a polymer chain in solution. A model for micellization of block copolymers in solution, based on the thermodynamics of associating multicomponent mixtures, was presented by Gao and Eisenberg (1993). The polydispersity of the block copolymer and its influence on micellization was a particular focus of this work. For block copolymers below the cmc, a collapsed spherical conformation was assumed. Interactions of the collapsed spheres were then described by the Hamaker equation, with an interaction energy proportional to the radius of the spheres. 

As the size and shape of a polymer chain are of considerable interest to the polymer scientist, it is useful to know the factors that govern these properties. We shall, however, confine ourselves to the models of the random coil for a polymer chain, as this is usually believed to be most appropriate for synthetic polymers other models —rods, discs, spheres, spheroids—are also postulated, but these need not concern us at this level. 

A similar model can also be constructed for the motion of polymer chains, the main difference being that more than one hole will now be required to be in the same locality, as cooperative motions of a host of consecutive chain atoms are required for the movement of polymer chain segments. Thus, for a polymer chain segment to move from one position to an adjacent site, a critical void volume must first exist before the segment can jump. That empty spaces exist can be inferred from the fact that when a sample of polystyrene is dissolved in benzene there is a contraction in the total volume. This indicates that the polymer can occupy less volume when surrounded by benzene molecules and there must have been unused space in the polymer matrix to allow this increase in packing eflSciency to occur. On this basis, the observed specific volume of a sample, v, can be described as a sum of the volume actually occupied by the polymer molecules, iiq, and the free volume, Vf, in the system [see Fig. 2.27(a)],... 

The simplest mathematical model of a polymer chain is the freely jointed chain. It has n links, each of length I, joined in a linear sequence with no restrictions on the angles between successive bonds. The length of the chain along its backbone is known as the contour length and is given by nl. However, for linear flexible chains, it is more usual, and more realistic, to consider the dimensions of the molecular coil in terms of the distance between the two chain ends, that is the end-to-end distance r [Fig. A2.1(a)]. 

In considering the type of packing possible in the polymer structure, it is clear that the hydrogen bonding seen for the A-urethanes is likely to occur for the polymer, but not that for the B-urethanes. Another feature of the MeMMe structure is that the molecules have crystallized end-to-end see the shaded mole-cules in fig, lb. A model for the polymer chain was derived (9)... 

Fig. 3. Schematic illustration of the Flory-Huggins lattice model for a polymer mixture. Lattice sites taken by (effective) monomers are indicated by full dots lattice sites taken by vacancies are denoted by empty circles. Chains of type A are indicated by thick bonds between the monomers, and B chains by wavy bonds. Nearest neighbor nonbonded interactions between monomers of the same kind (eAA or eBb) are shown as full straight lines and those between monomers of a different kind (eAB) by broken lines. Interactions between monomers and vacancies (or solvent molecules, respectively), eAV and ebv, could be introduced as well but will be assumed here to be zero throughout...

Doi and Edwards (1978, 1979, 1986). They started with the Rouse-segmented chain model for a polymer molecule. Because of the presence of neighboring molecules, there are many places along the chain where lateral motion is restricted, as shown in Fig. 21. To simplify the representation of these restrictions, Doi and Edwards assume that they are equivalent to placing the molecule of interest in the tube as shown in Fig. 22. This tube has a diameter d and length L. The mean field is represented by a three-dimensional cage. The primitive chain can move randomly forward or backward only along itself. For a monodisperse polymer, the linear viscoelasticity is characterized by... 

In the present case of chain-growth polymerization, a Green s function, Gp(r, t), can be calculated for each addition of a monomer to the reactive group at the end of a polymer of degree n. Within the framework of the Rouse model, wherein a polymer chain is represented as a set of n beads connected by harmonic potentials, the propagator has the Gaussian form [10,66],... 

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