Conformation of a polymer

It is unrealistic to expect that any single conformer of a polymer will adequately represent the overall size and shape of the polymer. The low-energy conformer for each polymer strand shown here is merely meant to allow identification of the polymer in terms of its components. 

The rheological behaviour of polymeric solutions is strongly influenced by the conformation of the polymer. In principle one has to deal with three different conformations, namely (1) random coil polymers (2) semi-flexible rod-like macromolecules and (2) rigid rods. It is easily understood that the hydrody-namically effective volume increases in the sequence mentioned, i.e. molecules with an equal degree of polymerisation exhibit drastically larger viscosities in a rod-like conformation than as statistical coil molecules. An experimental parameter, easily determined, for the conformation of a polymer is the exponent a of the Mark-Houwink relationship [25,26]. In the case of coiled polymers a is between 0.5 and 0.9,semi-flexible rods exhibit values between 1 and 1.3, whereas for an ideal rod the intrinsic viscosity is found to be proportional to M2. 

The conformation of a polymer in solution is the consequence of a competition between solute intra- and intermolecular forces, solvent intramolecular forces, and solute-solvent intermolecular forces. Addition of a good solvent to a dry polymer causes polymer swelling and disaggregation as solvent molecules adsorb to sites which had previously been occupied by polymer intra- and intermolecular interaction. As swelling proceeds, individual chains are brought into bulk solution until an equilibrium solubility is attained. 

We have added a companion option to PBUILD, PRANDOM which eases considerably the problem of finding good conformations of a polymer segment. PRANDOM automatically selects all of the polymer backbone and/or side chain bonds and will randomly select rotations for each bond. In a few minutes, one can not only build a polymer fragment, but also set up a Monte-Carlo search of its conformational space. However, even this cannot solve the problems for large models (pentamer or larger), again due to the number of bonds to be rotated. 

Several experimental parameters have been used to describe the conformation of a polymer adsorbed at the solid-solution interface these include the thickness of the adsorbed layer (photon correlation spectroscopy(J ) (p.c.s.), small angle neutron scattering (2) (s.a.n.s.), ellipsometry (3) and force-distance measurements between adsorbed layers (A), and the surface bound fraction (e.s.r. (5), n.m.r. ( 6), calorimetry (7) and i.r. (8)). However, it is very difficult to describe the adsorbed layer with a single parameter and ideally the segment density profile of the adsorbed chain is required. Recently s.a.n.s. (9) has been used to obtain segment density profiles for polyethylene oxide (PEO) and partially hydrolysed polyvinyl alcohol adsorbed on polystyrene latex. For PEO, two types of system were examined one where the chains were terminally-anchored and the other where the polymer was physically adsorbed from solution. The profiles for these two... 

Equivalence Principle. The conformation of a polymer chain in the crystalline state is defined by a succession of equivalent structural units which occupy geometrically (not necessarily crystallographically) equivalent positions with respect to the chain axis. The chain axis is parallel to a crystallographic axis of the crystal. 

Principle of Minimum Internal Conformational Energy. The conformation of a polymer chain in a crystal approaches one of the minima of the internal conformational energy, which would be taken by an isolated chain subjected to the restrictions imposed by the equivalence principle. 

An enhancement of stability due to the contracted conformation of a polymer-ligand chain may also be deduced from Table 5. The (34 value of a PAA-Cu chelate is larger in the system with higher ionic strength. The interpretation is that the local concentration of ligands is increased by the contraction of the PAA chain on the addition of the neutral salts°). 

The same types of polyrotaxanes were also prepared by a different method, Method 2 (Figure 6). In this method, a preformed polymer is used and the cyclic is threaded onto the polymer in a melt or in solution. A solution of 28 and polystyrene in THF under reflux afforded a polyrotaxane with an min value of 5.0X1CT4, much lower than those via Method 1 [69]. Threading 28 on to poly (butylene sebacate) afforded poly(ester rotaxane) 33 of Type 4 [70]. Although a laige excess of cyclic was used, 33 only had a min value of 0.0017. This value is 100-fold lower than that for the corresponding polyrotaxane prepared by Method 1 [19]. A possible reason is that the concentration of chain ends is very low and the random coiled-chain conformation of a polymer disfavors threading. 

Another important characteristic for describing the mean conformation of a polymer chain is the characteristic ratio, C00, defined as ... 

Figure 8.7 Phase diagram of the conformation of a polymer chain, deduced from theoretical calculations (multicanonical Monte Carlo method Noguchi and Yoshikawa, 1998).

Figure 1.10 Random conformation of a polymer chain s carbon-carbon backbone.

As discussed earlier, solid polymers can be distinguished into amorphous and the semicrystalline categories. Amorphous solid polymers are either in the glassy state, or - with chain cross linking - in the rubbery state. The usual model of the macromolecule in the amorphous state is the "random coil". Also in polymer melts the "random coil" is the usual model. The fact, however, that melts of semi-crystalline molecules, although very viscous, show rapid crystallisation when cooled, might be an indication that the conformation of a polymer molecule in such a melt is more nearly an irregularly folded molecule than it is a completely random coil. 

Schick and Harvey (49) summarize an interesting investigation of the effect of the choice of solvent on the conformation of a polymer adsorbed at the solution interface with Spheron 6 carbon black. A noteworthy conclusion concerns the occurrence of extended and looped configurations of the adsorbed polymer molecules formed from good or poor solvents, respectively. 

Depending on its structure, the polsmier, will have different conformations in a mixture with itself than in a mixture with a solvent. The conformation of a polymer is typically coiled. The rms end-to-end distance, (r ), is one meastme of the diameter of the coil. The rms... 

The average conformation of a polymer chain with an end-to-end distance r may be quantified by defining a general partition function Z r)... 

As we have already mentioned, the conformation of a polymer chain is determined by the position taken in space by their atoms that can be interchanged by simple rotation about single bonds (4). There are flexible polymers that can adopt a large number of conformations, and rigid chains for which only a limited number of conformations are accessible. On the other hand, flexible polymers in the crystalline state adopt fixed conformations, whereas in solution or in the molten state they adopt a wide range of conformations. To illustrate what the conformational change consists of, we refer to the molecule of -butane, shown in Figure 1.7. It can be seen that two extreme conformations can occur the one known as cis, in which... 

The most drastic approximations leading to Eq. (1) are credited by the following arguments [39] a chain conformation of a polymer coil in dense melt sys-... 

The surface segregation from the mixture of chemically identical polymers with chain length disparity is predicted by another model [198-200]. It represents the spatial conformation of a polymer coil as a random walk reflected by an external interface. The associated loss in system configurations is minimized when shorter chains are adsorbed at the surface. Preferential surface seg-... 

The restrictions imposed by short range steric interactions upon the conformations of a polymer molecule occur at a local level in short sequences of chain segments. There is, however, an interdependence of local chain conformations, that is, the conformation about a given chain segment is dependent upon the conformations about the segments to which it is directly connected (see Problem 2.1). Such interdependent steric restrictions affect the local chain conformations all along a polymer chain and have a significant effect upon chain dimensions. 

The conformation of a polymer in its crystal will generally be that with the lowest energy consistent with regular placement of structural units in the unit cell. Helical conformations occur frequently in polymer crystals. Helices are characterized by a number fj where / is the number of monomer units per j number of complete turns of the helix. Thus, polyethylene could be characterized as a li helix in its unit cell with an -trans conformation. The arrangement of the molecules in the polyethylene crystal structure is illustrated in Fig. 2.8. 

The deficiencies of the Flory-Huggins theory result from the limitations both of the model and of the assumptions employed in its derivation. Thus, the use of a single type of lattice for pure solvent, pure polymer and their mixtures is clearly unrealistic since it requires that there is no volume change upon mixing. The method used in the model to calculate the total number of possible conformations of a polymer molecule in the lattice is also unrealistic since it does not exclude self-intersections of the chain. Moreover, the use of a mean-field approximation to facilitate this calculation, whereby it is assumed that the segments of the previously added polymer molecules are distributed uniformly in the lattice, is satisfactory only when the volume fraction (f>2 of polymer is high, as in relatively concentrated polymer solutions. 

Figure 6-14. Conformation of a polymer chain with one end fixed at the origin of a Cartesian coordinate system.

In this Section, the general characteristics of the conformation of a polymer molecule in solution are considered. The general model for a linear polymer molecule in solution is based on a randomly coiled, flexible chain, the average form of which possesses spherical symmetry. The distribution of chain ends about the center of this sphere is further supposed to be Gaussian. Since the total number of conformations which the macromolecule may adopt is exceedingly large, only an average dimension can... 

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