PHYS*7330 - Summer 2010
Problem Set 1
Due: Thursday, June 3rd .
Question 1. In order to study the effect of polydispersity, a frequently-used trick is to
blend monodisperse polymers together to prepare a well-controlled polydisperse sample. A
polymer sample is made by blending monodisperse 200 kDa, 500 kDa and 750 kDa polymers
together in a ratio of 1:2:2 by wieight. Calculate Mw , Mn and P I for the blend.
Question 2. Using the exponentially-decaying form for the orientational correlations along
the chain, show that the mean square end-to-end distance of a worm-like chain with persistence length ℓp and contour length LC satisfies
hR2 i = 2ℓp LC − 2ℓ2p 1 − e−LC /ℓp .
Show that this result reduces to what you would expect in the limits ℓp ≪ LC and ℓp ≫ LC .
Question 3. Calculate the radius of gyration of an ideal ring polymer of N segments
of Kuhn length b, and compare your result to the radius of gyration of a linear chain with
the same number of segments.
Question 4. The ideal polymer chain model we discussed in class has the drawback that
the chain can pass through itself, which is unrealistic. Furthermore, chain self-avoidance can
play an essential role in certain problems. Unfortunately. it is difficult to make analytical
progress on the self-avoiding chain problem – for one thing the interactions leading to selfavoidance are non-local. Here we resort to a numerical study.
Consider the self-avoiding walk on an effectively infinite two-dimensional square lattice.
Assume that the first step is “up”. The walks generated by the three other possible initial
directions only differ by a rotation of the whole lattice and do not have to be considered
explicitly. The second step can be in three rather than four possible directions because of the
constraint that the walk cannot return to the origin. To obtain unbiased results, we generate
a random number to choose one of the three directions. Sucessive steps are generated in
the same way. Unfortunately, the walk will very likely not continue indefinitely. To obtain
unbiased results, we must choose at random one of the three steps, even though one of these
steps might lead to self-intersection. If the next step does lead to a self-intersection (meaning
the polymer ends up on a lattice site that it has already visited), the entire walk must be
discarded to keep the statistics unbiased. Think a bit about why this is so, that is, why is it
not correct to simply choose another (allowed) direction and keep going.
(a) Write a program that implements this algorithm and record the fraction f (N) of successful attempts at constructing self-avoiding polymer chains with N total monomers. Represent
the lattice as an array, so that you can record the sites that have been already visited. Examine 100,000 walks to get a rough idea of what is going on, then increase to 10,000,000 or
more walks to get good results. What is the qualitative dependence of f (N) on N? What
is the maximum value of N that you can reasonably consider?
(b) Does the fraction you compute numerically for N = 4 agree with your expectation?
(c) Graph a few typical configurations of the self-avoiding random walk for a large N value.
~ and hR
~ 2 i. Examine as large a range of
(d) For an ensemble of random walks, compute hRi
~ 2 i ∼ N 3/2 .
walk lengths as possible. Compare with the ideal result, and the hypothesis hR
~ N). Plot this func(e) Compute the distribution function of the end-to-end vector, P (R,
tion as a surface plot. Compare with the exact result for the finite-N ideal chain in two
dimensions:
P
(2D)
~ N) =
(R,
!N
1 Z 2 i~q·R~ sin qb
dqe
,
(2π)2
qb
where q = |~q| and b is the Kuhn length.
You may wish to develop a code to simulate the ideal random walk first, and then build
in the self-avoidance.