Optional Project 1B  (Total of 4 Quizzes possible)

1) Freely Jointed Chain: This project involves an alternative method to that of Project 1 (Flory Method) of calculating a random walk using two random angles between 0 and pi and 0 and 2pi for the bond direction.  The intent is to compare the speed of calculation with the method of project 1, compare the persistence length and compare the mean end-to-end distance and RMS end-to-end distance with the results of Project 1.  (1 Quiz)  Do this for a chain of 100,000 steps and averaging 100 of such chains. (1 Quiz)

2) Freely Rotating Chain: This appears to be similar to the Freely Jointed Chain except that the angle from 0 to 180 is fixed at a bond angle such as 71.5 for a carbon bond in polyethylene (180-109). This should result in a persistence length of sqrt(2) according to Flory's second book. This differs from the first problem since you will need to perform reference frame transformations as described in the following link (http://local.wasp.uwa.edu.au/~pbourke/geometry/rotate/). Example code using this reference frame transformation are available at http://local.wasp.uwa.edu.au/~pbourke/geometry/rotate/example.c, http://local.wasp.uwa.edu.au/~pbourke/geometry/rotate/source.c, and at http://local.wasp.uwa.edu.au/~pbourke/geometry/rotate/PointRotate.py. (Thanks to Drew Gilpin for finding these links.) (1 Quiz) (Local Links to these A, B, C, D)

3) Rotational Isomeric State Model (RISM) Chain: For this simulation you use the program of part 2) and restrict the rotation to 3 angles related to trans, gauche+ and gauche-. One of the three angles is chosen at random then the random choice is either accepted or declined depending on a Boltzman probability exp(-E/kT) where E/kT is about 1 for trans at 298 K and about 0.6 for gauche+ and - at 298 K. (There should be a lower probability of acceptance for gauche than for trans.) You should get a much larger persistence for this case and you should be able to observe coil expansion and contraction with temperature changes. These results should be reported. (1 Quiz)

4) Wignall reported in 1985 that the radius of gyration and chain scaling for polyethylene in the melt and in the semi-crystalline state does not vary.  This seemingly unusual result can be verified in a simulation by first simulating a chain of 100,000 steps averaging 100 of such chains (part 1 above) and then considering a random condition where A) the chain does not change position for a number of steps equal to the lamellar thickness in step units (30 or so) followed by further chain steps.  B)  A more realistic approach is to make the chain display a directed growth for the lamellar thickness (30 steps) with random onset associated with the degree of crystallinity.  C)  Finally, a fixed probability of adjacent re-entry can be incorporated in the simulation of the chain using a reversed persistence of 30 steps (returning to 0 propagation) for cases where adjacent re-entry occurs.  (1 Quiz)