How Self-Interaction Changes the Fractal Dimension

The conceptual random-walk polymers we have considered up to now are different from real polymers in an important way. A random-walk polymer typically intersects itself many times, while real polymers cannot intersect themselves. A real polymer is self-avoiding two monomers cannot be in the same place at the same time Thus instead of treating polymers as random walks, it would seem more realistic to treat them as self-avoiding random walks. The differences between random walks and self-avoiding random walks form an important subject in mathematical polymer physics. Our goal here is to give some account and some feeling for these differences. 

Strictly, the discarding probability is finite if the distance between the home particles of the two 

The actual properties of self-avoiding polymers have been studied by detailed calculations, by computer simulations, and by measurements on real polymer solutions. All these approaches confirm that these polymers, like random-walk polymers, are fractals. They have a D very close to 5/3, a value consistent with des Cloizeaux s inequality. 

If this cyclohexane is cooler than 36.6 degrees, the attraction becomes stronger than the compensating value. Polymers in such a solvent become progressively smaller than the random-walk size they have a D that is larger than 2. Such polymers attract one another in solution as well as attracting themselves. Thus it is difficult to isolate them and measure their D definitively. Such solvents are called poor solvents. 

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