The BFM (bond fluctuation model or bond fluctuation method) is a lattice model for simulating the conformation and dynamics of polymer systems. There are two versions of the BFM used: The earlier version was first introduced by I. Carmesin and Kurt Kremer in 1988,[1] and the later version by J. Scott Shaffer in 1994.[2] Conversion between models is possible.[3]
In this model the monomers are represented by cubes on a regular cubic lattice with each cube occupying eight lattice positions. Each lattice position can only be occupied by one monomer in order to model excluded volume. The monomers are connected by a bond vector, which is taken from a set of typically 108 allowed vectors. There are different definitions for this vector set. One example for a bond vector set is made up from the six base vectors below using permutation and sign variation of the three vector components in each direction:
B=
P\pm |
\left(\begin{matrix}2\ 0\ 0\end{matrix}\right)\cup
P\pm |
\left(\begin{matrix}2\ 1\ 0\end{matrix}\right)\cup
P\pm |
\left(\begin{matrix}2\ 1\ 1\end{matrix}\right)\cup
P\pm |
\left(\begin{matrix}2\ 2\ 1\end{matrix}\right)\cup
P\pm |
\left(\begin{matrix}3\ 0\ 0\end{matrix}\right)\cup
P\pm |
\left(\begin{matrix}3\ 1\ 0\end{matrix}\right)
The resulting bond lengths are
2,\sqrt{5},\sqrt{6},3
\sqrt{10}
The combination of bond vector set and monomer shape in this model ensures that polymer chains cannot cross each other, without explicit test of the local topology.
The basic movement of a monomer cube takes place along the lattice axes
\DeltaB=
P\pm |
\left(1,0,0\right)
so that each of the possible bond vectors can be realized.[4]
As in the case of the Carmesin-Kremer BFM, the Shaffer BFM is also constructed on a simple-cubic lattice. However, the lattice points, or vertices of each cube are the sites that can be occupied by a monomer. Each lattice point can be occupied by one monomer only. Successive monomers along a polymer backbone are connected by bond vectors. The allowed bond vectors must be one of: (a) A cube edge (b) A face diagonal or (c) A solid diagonal. The resulting bond lengths are
1,\sqrt{2},\sqrt{3}
In both versions of the BFM, a single attempt to move one monomer consists of the following steps which are standard for Monte Carlo methods:
\DeltaB\inP\pm(1,0,0)
The conditions to perform a move can be subdivided into mandatory and optionalones.
If the move leads to an energetic difference
\DeltaU
pM
pM=
-\DeltaU/kBT | |
e |
is compared to a random number r from the interval [0, 1). If the Metropolis rate is smaller than r the move is rejected, otherwise it is accepted.
The number of Monte Carlo steps of the total system is defined as:
\#MCS=
\#attempts | |
\#monomers |