Problem 1. (100 points) Nanoparticle self-assembly
Interactions between nanoparticles can lead to self-assembled arrays that resemble the structure of
crystals. In this problem, you will use Metropolis Monte Carlo simulations to describe the distribution of
a set of non-charged nanoparticles. We will approximate the interaction between two particles using a
Lenard-Jones potential:
π(π ππ) = 4π [( π
π ππ ) 12 β (
π
π ππ ) 6 ] π€ππ‘β π ππ= |ππ β ππ|
where ra and rb are the positions of particle a and b respectively, and the values for Ο and Ξ΅ are provided
below. You may assume that the sample at equilibrium follows a Boltzmann distribution with probability
of finding the assemble in a state q, π(π) β exp(βππ‘ππ‘(π)/(πππ)), where
ππ‘ππ‘(π) = β β π(π ππ)
π
π=π+1
π
π=1
and q is the state vector containing the position of all the nanoparticles in the system. For simplicity, we
will consider the self-assembly of a group of N spherical nanoparticles of radius R in two dimensions, and
the state vector has the following structure, q = [x1 y1 x2 y2 β¦. xN yN]T.
i. (25 points) Write a function that computes the value of Utot for a given state vector q.
ii. (40 points) Write a function that performs a Monte Carlo routine with an N_MC Monte Carlo
steps, a constant temperature T, and an N number of nanoparticles in the initial state q_0. For
each MC step, you should attempt moves of one nanoparticle at a time for all the particles. The
function should output a matrix containing the values of the state vector at each MC step in
different columns, and a vector containing Utot for each MC step.
iii. (35 points) Use your function to generate histograms containing the distribution of
nanoparticle distances, Rab, for a set of 16 nanoparticles. Assume that the particles are initially
arranged in a square grid of 4×4 nanoparticles, with center-to-center distance between the
particles equal to 2R. Generate graphs for T = 100 K, 500 K, 1000 K. Use a minimum of 100,000
MC steps. Discuss your results.
Physical parameters: R = 5 nm, Ο = 5 nm, Ξ΅/kb = 5000 K.