Size dependent transitions in grafted polymer brushes
Finding probabilities in order to solve for the most likely configuration of a grafted polymer chain is easily calculated by solving a random walk problem, starting form a given point (the surface). A property of the random walk is used to describe the partition function of a polymer in terms of a sum over possible loops configurations, which simplifies tremendously the problem of calculating the partition function of the grafted polymer. Instead of solving the non-interacting random walk that has been well studied in the theory of the probability, the focus is on the problem of self-interacting random walks, which cannot be solved exactly and has had various approximations suggested. The idea was to take the easily solved equations that give the solutions for the non-interacting random walks and add a new term with a specific given energy. This finds the probabilities of loop formation, and from there, the most likely configuration of interacting random walks. ^ The most likely configuration has also been calculated by Monte Carlo but in order to be accurate, it needs many sampling points, which means much computer time. Practically, they are limited to chains with less than 1000 segments, which are not close to the phase transitions. It is well known that the most accurate treatment planning procedures are based on Monte-Carlo calculations; however, their accuracy is limited by the computer time available. This method could be a better analytical approximation than Monte-Carlo calculations, and could drastically decrease the computing time requirement, and hence, more accurate TPS (Treatment Planning Systems) would be available to the Medical Physics community.^
Bosse, Courtney E, "Size dependent transitions in grafted polymer brushes" (2013). ETD Collection for University of Texas, El Paso. AAI1545152.