Self-avoiding walk continued

The parameter /r tunes the stiffness of the potential. It is chosen such that the repulsive part of the Leimard-Jones potential makes a crossing of bonds highly improbable (e.g., k= 30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Leimard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly fomi liquid crystalline phases. 

Fisher concluded that the above values of the loop class exponent c intjply a continuous phase transition in d = 2 tuid d = 3. In [89] it is shown that the phase transition condition is satisfied in both d = 2 tuid in d = 3 for the case when the double stranded segments are treated as straight paths. Note that in d = 2, self-avoiding walks as paths will result in no phase transition. 

For long flexible polymer chains it has been customary for a Imig time [1, 2] to reduce the theoretical description to the basic aspects such as chain connectivity and to excluded volume interactions between monomers, features that are already present when a macromolecule is described by a self-avoiding walk (SAW) on a lattice [3]. The first MC algorithms for SAW on cubic lattices were proposed in 1955 [164], and the further development of algorithms for the simulation of this simple model has continued to be an active area of research [77, 96, 165 169]. Dynamic MC algorithms for multichain systems on the lattice have also been extended to the simulation of symmetric binary blends [15, 16] comprehensive reviews of this work can be found in the literature [6, 81, 82]. It mms out, however, that for the simulation both of polymer blends [6, 9, 21, 82, 170, 171] and of solutions of semiflexible polymers [121 123], the bond fluctuation model [76, 79, 80] has a number of advantages, and hence we shall focus attention only on this lattice model. 

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