ISSN:
1089-7690
Source:
AIP Digital Archive
Topics:
Physics
,
Chemistry and Pharmacology
Notes:
We provide a mathematical formalism for a self-consistent mean field treatment of long chain molecules. The formalism is applied to the case of a neutral polymer under the excluded volume interaction. Upon scaling the problem in the N→∞ limit we find the natural scaling length RN, of the polymer, which is made up of (N+1) monomers or beads, is RN∼N3/5, the well known Flory result. The asymptotics of the problem is dominated by the neighborhood of the turning point, so that a uniformly valid Green's function solution of the differential equations is necessary. In the neighborhood of a point y* the scaled polymer density fN(x), is found to decay sharply. If we let x denote the scaled distance from one end of the chain to a point in space we obtain, for y*−x(approximately-greater-than)O(N−2/15), a closed form expression for the polymer density viz., fN(x)∼{1/2x2[fN(x)−fN(y*)]1/2} while for x−y*(approximately-greater-than)O(N−2/15) the density is shown to be, to leading order, zero. Although our results imply the rate of decay of the density at y* is O(N1/5) we are unable to verify this explicitly by calculating fN′(y*). We believe this is due to the inability of the WKB theory to correctly approximate solutions in regions of rapid variation. We suggest remedies for this, so that a complete self-consistent solution may be obtained. © 1996 American Institute of Physics.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1063/1.471616
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