Bond fluctuation model

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:

${\displaystyle \mathbf {B} =\mathbf {P_{\pm }} \left({\begin{matrix}2\\0\\0\end{matrix}}\right)\cup \!\ \mathbf {P_{\pm }} \left({\begin{matrix}2\\1\\0\end{matrix}}\right)\cup \!\ \mathbf {P_{\pm }} \left({\begin{matrix}2\\1\\1\end{matrix}}\right)\cup \!\ \mathbf {P_{\pm }} \left({\begin{matrix}2\\2\\1\end{matrix}}\right)\cup \!\ \mathbf {P_{\pm }} \left({\begin{matrix}3\\0\\0\end{matrix}}\right)\cup \!\ \mathbf {P_{\pm }} \left({\begin{matrix}3\\1\\0\end{matrix}}\right)}$

The resulting bond lengths are ${\displaystyle 2,{\sqrt {5}},{\sqrt {6}},3}$ and ${\displaystyle {\sqrt {10}}}$.

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

${\displaystyle \mathbf {\Delta B} =\mathbf {P_{\pm }} \left(1,0,0\right)}$

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 ${\displaystyle 1,{\sqrt {2}},{\sqrt {3}}}$. In addition to the bond length constraint, polymers should not be allowed to cross. This is done most efficiently by the use of a secondary lattice which is twice as fine as the original lattice. The secondary lattice tracks the midpoints of the bonds in the system, and forbids the overlap of bond midpoints. This effectively leads to disallowing polymers from crossing each other.