Monte Carlo Simulation of Lattice Polymer Models

Polymers are substances whose molecules have high molar masses and are composed of a large number of repeating units. They occur naturally, and are also produced in large quantities. The materials commonly called plastics are all synthetic polymers. Polymers consist of a large number of units called monomers, which are joined sequentially to form a long chain. Perhaps the simplest polymer is polyethylene (PE), which consists of many repeating CH2 units. It is produced by the addition polymerization of ethylene, CH2=CH2 (ethene). Other common synthetic polymers are polypropene (PP), made of polymerized CH2=CH-CH3 (propene), polystyrene (PS), made of polymerized CH2=CH-C6H5 (styrene), and polyvinyl chloride (PVC), which is polymerized CH2=CHCl (chloroethene). Naturally occurring polymers are for instance proteins. The length of a polymer can be expressed in the number of monomers, or alternatively in the number of bonds between monomers. Here, we adapt the latter choice, so that a polymer of length N consists of N +1 monomers, connected throughN bonds. The shape (configuration) of a polymer of length N is characterized by specifying for each monomer i = 0 . . . N its position ri. Alternatively, one can specify the bond directions ui ≡ ri − ri−1, in which i = 1 . . . N . The chemical bond between neighboring monomers is usually rather stiff, so that in good approximation the bond length equals some constant: