Moscow State University, Russia
1 Basic concepts of polymer physics
1.1 Fundamentals of the physical viewpoint in polymer science
Polymer chains of different chemical structure have, of course, different properties. How- ever, there are many common properties characteristic of large classes of polymer systems.
For example, all rubbers (cross-linked polymer networks, see below) exhibit the property of high elasticity, all polymer melts are vdscoelastic, all polyelectrolyte gels absorb a large amount of water, etc. Such properties can be described on a molecular level by taking into account only the general polymeric nature of constituent molecules, rather than the details of their chemical structure. It is these properties that are studied using polymer physics. For a more complete introduction, and many further references, see Grosberg and Khokhlov 1994; Grosberg and Kliokhlov 1997.
What are the main factors governing the general physical behaviour of polymer sys- tems? Three of them should be mentioned in the first place.
-CH,-CH$H,XH,- pd~@Ylene)
-CH,<H-CHrFH- poly(v1nyI chloride)
Figure 1. Common polymer chains
CI CI
First of all, polymers are long molecular chains. In Figure 1 three of the most common polymer chains with carbon backbones are shown. One can see that small atomic groups (monomer units) are connected in linear chains by covalent chemical bonds. The chain structure of constituent molecules is the first fundamental feature of polymer systems.
50 Alexei Khokhlov
In particular, this means that monomer units do not have the freedom of independent translational motion, and therefore polymers do not possess the entropy associated with this motion (the so-called translational entropy). This is sometimes expressed as follows:
polymer systems are poor in entropy.
Second, the number of monomer units in the chain N , is large: N >> 1 (otherwise we have an ‘oligomer’, not a polymer). For macromolecules synthesised in the chemical laboratory, normally N = 102-104. For biological macromolecules the values of N can be much larger, for example, the longest polymer chains are those of DNA molecules:
N N 109-1010. Such large objects can be seen by a normal optical microscope (if DNA is labelled with fluorescence dyes), since the linear size of DNA coil turn out to be larger than the wavelength of light.
Figure 2. Polymer chains are generally flexible, they normally take the configuration of the coal (right), not of the rigid rod (left).
Third, polymer chains are generally fiedble (see Figure 2), they normally take the conformation of a random coil, rather than that of a rigid rod. We will discuss in detail the notion of polymer chain flexibility in Section 1.2.
In summary, their chain structure, the large number of monomer units in each chain, and chain flexibility are the three main factors responsible for the special properties of polymer systems.
1.2 Flexibility mechanisms of a polymer chain
1.2.1 Rotational-isomeric flexibility mechanism
Let us consider the simplest polyethylene chain (Figure 1) and let us ask ourselves for which conformation do we have the absolute energetic minimum? Such a conformation corresponds to a straight line and is shown schematically in Figure 3. For this confor- mation all the monomer units are in the so-called trans position. This would be the equilibrium conformation at T = 0.
Figure 3. T h e rectilinear (all trams) conformation of a polyethylene chain.
Polymer physics: from basic concepts to modern developments 51
At T
#
0, due to the thermal motion, deviations from the minimum-energy confor- mation are possible. According to the Boltzmann law, the'probability of realisation of a conformation with the excess energy U over the minimum-energy conformation isWhat are the possible conformational deviations from the structure shown in Figure 3?
For a carbon backbone the valency angle y (see Figure 4) should be normally considered as fixed (for different chains 50"
<
7<
80"). However, rotation with fixed y by changing the angle of internal rotation cp (see Figure 4), is possible. Any value cp#
0 gives rise to deviations from the rectilinear conformation, i.e. to chain flexibility, though usually there are only two or three preferred values of cp corresponding to different rotational isomers.This kind of flexibility is called the rotational-isomeric flexibility mechanism.
Figure 4. The valency angle y and angle of internal rotation cp for a carbon backbone.
1.2.2 Persistent flexibility mechanism
Another flexibility mechanism can be realised when rotational isomers are not allowed, e.g. in a-helical polypeptides or DNA double helix. The conformations of these chains are stabilised by hydrogen bonds and internal rotation is impossible. In this case small thermal vibrations around the equilibrium conformation play the most important role.
Via their accumulation over large distances along the chain, these vibrations give rise to the deviations from the rectilinear conformation, i.e. to the chain flexibility. This is a persistent flexibility mechanism; it is analogous to the flexibility of a homogeneous elastic filament.
1.2.3 Freely-jointed flexibility mechanism
Another mechanism of flexibility is realised in the so-called freely-jointed model of a poly- mer chain. In this model the flexibility is located in freely-rotating junction points: p in Figure 4 takes any value. This mechanism is not very characteristic of real chains, but it is frequently used for model theoretical calculations.
52 Alexei Khokhlov
Figure 5 . %ical confomation of a polymer coal of freely-jointed segments.
1.3
In Figure 5 a typical conformation of a polymeric coil is presented. It was constructed on the computer for the freely-jointed model by allowing each subsequent segment to be oriented in an arbitrary direction with respect to the previous one. n o m this picture one can draw the following conclusions: