Bond fluctuation model
Encyclopedia
The BFM 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 Carmesin and Kremer
in 1988, and the later version by Shaffer in 1994 . Conversion between models is possible.
s 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:
The resulting bond lengths are
and 
.
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
so that each of the possible bond vectors can be realized.
. 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 mid-points of the bonds in the system, and forbids the overlap of bond midpoints. This effectively leads to disallowing polymers from crossing each other.
s:
The conditions to perform a move can be subdivided into mandatory and optional
ones.
for example due to an electric field or an adsorbing force to the walls. In this case a Metropolis algorithm
is applied: The Metroplis rate
which is defined as
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:
Lattice model (physics)
In physics, a lattice model is a physical model that is defined on a lattice, as opposed to the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice. Currently, lattice models are...
for simulating
Simulation
Simulation is the imitation of some real thing available, state of affairs, or process. The act of simulating something generally entails representing certain key characteristics or behaviours of a selected physical or abstract system....
the conformation and dynamics of polymer
Polymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...
systems. There are two versions of the BFM used: The earlier version was first introduced by Carmesin and Kremer
Kremer
- People :* Honorable I. Raymond Kremer, Court of Common Pleas Judge* Andrea Kremer, ESPN and HBO reporter focusing on sports* Gerard de Kremer, Gerardus Mercator* Gidon Kremer, a celebrated Latvian violinist and conductor....
in 1988, and the later version by Shaffer in 1994 . Conversion between models is possible.
Carmesin and Kremer version
In this model the monomers are represented by cubeCube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
s 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
Excluded volume
The concept of excluded volume was introduced by Werner Kuhn in 1934 and applied to polymer molecules shortly thereafter by Paul Flory.- In liquid state theory :...
. 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
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
and sign variation of the three vector components in each direction:
The resulting bond lengths are
and .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
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
.
The basic movement of a monomer cube takes place along the lattice axes
so that each of the possible bond vectors can be realized.
Shaffer's version
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. 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 mid-points of the bonds in the system, and forbids the overlap of bond midpoints. This effectively leads to disallowing polymers from crossing each other.Monte Carlo step
In both versions of the BFM, a single attempt to move one monomer consists of the following steps which are standard for Monte Carlo methodMonte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
s:
- Select a monomer m and a direction
randomly - Check list of conditions (see below)
- If all conditions are fulfilled, perform move
The conditions to perform a move can be subdivided into mandatory and optional
ones.
Mandatory conditions for Carmesin–Kremer BFM
- Four lattice sites next to monomer m in the direction d are empty
- The move does not lead to bonds that are not contained in the bond vector set.
Mandatory conditions for Shaffer BFM
- The lattice site to which the chosen monomer is going to be moved is empty
- The move does not lead to bonds that are not contained in the bond vector set.
- The move does not lead to overlapping of bond mid-points.
Optional conditions
If the move leads to an energetic difference for example due to an electric field or an adsorbing force to the walls. In this case a Metropolis algorithmMetropolis-Hastings algorithm
In mathematics and physics, the Metropolis–Hastings algorithm is a Markov chain Monte Carlo method for obtaining a sequence of random samples from a probability distribution for which direct sampling is difficult...
is applied: The Metroplis rate
which is defined asis compared to a random number r from the interval
The number of Monte Carlo steps of the total system is defined as:
External links
- JBFM – a Java appletJava appletA Java applet is an applet delivered to users in the form of Java bytecode. Java applets can run in a Web browser using a Java Virtual Machine , or in Sun's AppletViewer, a stand-alone tool for testing applets...
from the Leibniz Institute of Polymer Research Dresden (GermanyGermanyGermany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
) for simulating polymers with the BFM