In this project, the simulation algorithm used is one that has been developed by Daan Frenkel and Berend Smit3 for studying polymer chains in a grand canonical ensemble. We have taken their algorithm and rewritten it in C for our own modification purposes in order to study the polymer system phase behavior using the Gibbs ensemble4.
In GCCBMC, polymers
are placed inside a box of length L, and the particles are allowed to move
around in the box or leave the box into a reservoir connected to the system.
In a grand canonical system, polymers are allowed to move in and out between
the reservoir and the box, but the thermodynamic variables 
(chemical potential), P (pressure), and T (temperature) are kept constant.
In this simulation,
the interactions between the polymers are described by using a Lennord-Jones
potential with the form
where i, j, denote
the polymers, epsion, sigma are parameters of the L-J potential.
Within a polymer, the
interactions between bonded monomers are described using a harmonic spring
potential of the form
where Kv is the spring
constant(in our simulations, we choose Kv=400) and it is chosen such
that the variance of the bond length is much smaller than the bond
length itself. L is the instantaneous bond length, and L0 is the
equilibrium
bond length. In this project L0 is chosen as the unit length and
defined as unity.
For systems using the grand canonical ensemble, there are four types of Monte Carlo moves, particle displacement, system volume change, particle exchange, and deformation of polymers. The deformation of polymers includes repation and configurational-bias polymer generation. In the following sections, the algorithm for each of these will be described in detail.
The following animation demonstrate the CBMC algorithm in action in a canonical system(NVT). The system has 10 chains, each consisting of 8 monomers. 20 displacement moves plus 10 configurational-bias moves are performed during each MC step. The animation shows 21 frames.
Deformation:Configurational-biasRepation
Volume Change: (Grand
Canonical) (Gibbs)
Chain Exchange: (Grand
Canonical)(Gibbs)