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Correlation Among Directions in Curves Lacking Direction

Long polymer chains in solution show the marvellous property of self-similarity: the statistically fluctuating spatial configuration on the average is covariant under dilatations. This already holds for a trivial model, where the polymer configuration is identified with a simple random walk of n steps, which are to be interpreted as the segments of the polymer chain. In a more realistic description the polymer configuration is swollen by repulsive 'excluded volume' forces among the segments. In the limit of long chains we then are confronted with a nontrivial 'critical' system. The adequate theoretical tool is the renormalization group, which incorporates dilatation covariance, and also is basic to the modern theory of second order phase transitions, (and to quantum field theory, of course).
Simple heuristic 'scaling' considerations suggest that swelling combined with dilatation covariance implies correlations of infinite range among the spatial directions of the chain segments. In an infinitely long chain the direction correlation function, defined as thermodynamic average of the scalar product of two segment vectors, should decrease as a nontrivial power of the distance |j_{1} - j_{2}| of the two segments j_{1},j_{2} along the chain. For finite chain lengths n this power law should be multiplied by a 'scaling function', which depends only on ratios j_{1}/n and j_{2}/n.

Now a detailed theoretical analysis poses an interesting problem. To prove power law scaling, we have to 'renormalize' the chain. This implies that we shrink the segment size to zero, since otherwise the finite microscopic length of the segments breaks dilatation invariance. Taking this limit we replace the polymer configuration by a continuous space curve, and it is known that the generic configuration of a 'continuous chain' is nowhere differentiable. Thus local segment directions cannot be defined. We thus have to face the question whether despite this fact the ** correlations ** among segment directions survive the continuous chain limit.

In her Diploma-thesis A. Ostendorf analyzed this problem. She showed that indeed properly defined direction correlations can be 'renormalized', which means that they survive the continuous chain limit. They furthermore obey power law scaling as suggested by the handwaving scaling arguments. We have calculated the scaling function to first nontrivial order of perturbation theory. The figure compares our result to Monte Carlo data, taken for a chain of discrete segments. The agreement is remarkable in view of the fact that no fit parameter is involved. We conclude that direction correlations are bona fide macroscopic observables, even though the individual segment directions are not.

Besides being an intellectually appealing result this has consequences for the theory of polyelectrolytes. There the direction correlations and the associated 'persistence length' play a central role.

Publications

- L. Schäfer, A. Ostendorf, J. Hager,

*Scaling of the correlations among segment directions in a self-repelling polymer chain*, J.Phys.A: Mathematical and General 32 (1999) 7875-7899.