Discussion of the Question 09/02
POLYMER AT A WALL RADIATOR
The question was:
One of the simplest idealizations of a flexible polymer chain consists of replacing it by a random walk on a cubic lattice in three-dimensional space or on a square lattice in two-dimensional space. (This walk is allowed to self-intersect, and it has no "bending energy", i.e. each step is independent of the previous one.) An example of such "polymer" is depicted by a red line in Fig. (a). Assume that one end of the polymer is tied to a surface (denoted by brown color in the Figure). The surface will be considered adsorbing, i.e. every time the polymer (or the random walk) touches the surface its energy decreases by V. E.g., the energy of the configuration in Fig. (a) is -3V, because the walk touches the surface three times.
A very long polymer will be either localized near the surface as depicted in Fig. (c), or will be delocalized as in Fig. (b), depending on the temperature T. (Figs. (b) and (c) have different scale from Fig. (a) and therefore the lattice is not shown.)
Describe the temperature-dependence of the "localization" of the polymer.
(8/03) Y. Kantor: Nenad Vukmirovic, an undergraduate student at the University of Belgrade (Serbia, Yugoslavia) (e-mail nenadvuk@yubc.net) submitted (6/8/03) the following numerical solution of the problem. The numerical solution has several problems ((a) the method is not proper for low temperatures,- a true Monte Carlo technique would be a proper replacement of the numerical methods in the proposed solution; (b) the definition of the localization width through the maximal separation of the polymer from the wall is somewhat problematic, because it considers the extreme rather than a typical point). Nevertheless, the numerical solution provides us with a qualitative picture of the transition, and an order of magnitude estimate of the transition point - the critical T is of order of 0.1V/kB.
We are waiting for more accurate solutions.
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