Fractal Properties of Branched Polymers

Abstract

It has been known for a long time that amorphous polymers¹ behave as what is called nowadays fractals² This was found for linear polymers and also for branched structures³ in the vicinity of the sol-gel transition. The latter was studied some years ago, and it was realized very early that the distribution of masses is very broad, and becomes infinitely polydisperse near the gelation threshold. More recently, excluded volume effects were taken into account^4–6 ‚ and it was shown that each of the branched polymers in a sol is a fractal. In the following, we wish to show some consequences of such polydispersity on the (average) fractal behavior. As we shall see, although the study of branched macromolecules seems to be analogous to that of linear chains, it differs from it in a basic way. For instance, the measured dimension D_eff of a dilute solution is not the actual fractal dimension D of every polymer, but is a function of both D and of the polydispersity index τ, to be introduced below. This implies that a single measurement does not necessarily give the actual dimension of fractal objects, and one has to check for possible effects of polydispersity. The origin of such effects is that the mass distribution function cannot be reduced to a single average mass, but to at least two masses, namely N_w and N_z that diverge in different ways near the gelation threshold. In what follows, we will discuss some consequences of this for the static properties. Another important consequence, for the rheological characteristics of these systems is that in the same way as they cannot be reduced to a single mass, they also have a distribution of relaxation times that is very broad^{7 8} and may be described by at least two times. As we shall see, there is still discussion about the divergences of these times near the sol-gel transition, and there might be more than one universality class for the dynamics in the reaction bath. In what follows, we will remind briefly the main properties of the polymers and of the distribution of molecular weights in section 2. Then we will discuss the static properties of dilute solutions. Section 4 is devoted to the semi-dilute solutions. Finally, the distribution of relaxation times and its consequences for the rheological properties both in the reaction bath and in dilute solutions will be discussed in section 5.