Model of Polymer Chain

The bond fluctuation model (BFM) [51] has proved to be a very efficient computational method for Monte Carlo simulations of linear polymers during the last decade. This is a coarse-grained model of polymer chains, in which an effective monomer consists of an elementary cube whose eight sites on a hypothetical cubic lattice are blocked for further occupation (see... 

Fig. 21 Mean-square displacement vs. evolution time for 16-mers with an occupation density of 0.9375 in a 32-sized cubic lattice. The triangles are for four middle chain units, the circles are for the mass center, and the crosses are for the chain units relative to the center of mass. The lines with slopes of 1.0 and 0.5 indicate the scaling expected according to the Rouse model of polymer chains [56]... 

The probability distribution Ptri of the end-to-end distance is studied for the RIS model of polymer chains. A Monte-Carlo investigation provided reliable numerical data for Pfrl, which was then compared with results from two related analytical studies. 

The probability distribution of the end-to-end vector for the RIS model of polymer chains is discussed in terms of the characteristic function. For PE the characteristic function calculated from the RIS model is found to be in good agreement with the much simpler worm model. 

The dynamic RIS model of polymer chains is applied to the interpretation of nuclear magnetic relaxation measurements of local chain dynamics. According to the proposed model, the relaxation times Tlc and T1H may be related to the chemical structure of a specific polymer. 

A self-avoiding walk on a lattice is a random walk subject to the condition that no lattice site may be visited more than once in the walk. Self-avoiding walks were first introduced as models of polymer chains which took into account in a realistic manner the excluded volume effect1 (i.e., the fact that no element of space can be occupied more than once by the polymer chain). Although the mathematical problem of... 

Priss LS, Popov VF (1971) Relaxation spectrum of one-dimensional model of polymer chains. J Macromol Sci B 5(2) 461-472... 

Kloczkowski A and Kolinski A, "Theoretical Models of Polymer Chains", In Mark JE (Ed), "Physical Properties of Polymers Handbook", 2nd Ed, Springer-Verlag, Berlin, 2007, Chap. 5. 

Usually one is interested in the geometry and thermodynamic properties of various models of polymer chains. In many cases, the effect of intramolecular long-range interactions on the shape and dimensions of the polymer chain are of interest. 

Fig. 16. Moves used to equilibrate coil configurations for the self-avoiding walk model of polymer chains on the simple cubic lattice (upper party end rotations, kinkjump motions and crankshaft rotations f 107]. From time to time these local moves alternate with a move (lower pan) where one attempts to replace an A-chain by a B-chain in an identical coil configuration, or vice versa. In the transition probability of this move, the chemical potential difference Ap as well as the energy change SjF enter. From Binder [2S8]...

The reason why the random flight model has proved so popular theoretically stems from its simplicity, which offers hope for the development of analytic solutions. The problem can usually be cast in the form of a diffusionlike or a Schrodinger-wave-equation-like differential equation, the solutions of which are reasonably well explored. A tendency has developed in recent times to apply extremely sophisticated mathematical procedures to what are really very primitive models for polymer chains (see, e.g. Levine et al., 1978). Whether the ends merit the means in such instances cannot yet be assessed objectively. A strategy that might be more productive in terms of the development of a practical theory for steric stabilization is to aim for a simpler mathematical description of more complex models of polymer chains. It should also be borne in mind in developing ab initio theories that a simple model that may well suffice in polymer solution thermodynamics may be quite inadequate for the simulation of the conformational properties of polymers. Polymer solution thermodynamics seem to be relatively insensitive to molecular architecture per se whereas the conformation of a polymer chain is extremely sensitive to it. 

Fig. 2.5 A worm-like polymer chain (a) and a freely jointed polymer chain (/>) — two simple and most common models of polymer chain flexibility.

We introduce first the (lattice) Self-Avoiding Walk (SAW) model of polymer chains, their critical statistics and the criteria indicating effects of lattice disoder on the critical behaviour. Prominent indications for the effect of disorder on the SAW statistics are then discussed. Next, some mean field and scaling arguments are discussed for the SAW statistics in disordered medium percolating lattice in particular. 

The problems of the description of the observable properties corresponding to realistic models of polymer chains often appear to be insurmountable. They are, however, minute in comparison to those encountered in the characterization of the properties of polymers in bulk, e.g., rubber elasticity, crystallization upon stretching. - - The statistical mechanical description of polymers in bulk would be difficult enough if the system could be considered to be just a liquid of the constituent atoms or groups thereof (the monomers). But even this approximation is untenable, since the connectivity of polymer chains and networks is of prime importance in determining the properties of polymers in bulk. ... 

Models of polymer chain extension were first used to compare the effect of the glycosidic linkage geometry of simple polysaccharide chains, eg cellulose and amylose (43). Both polymers are 1,4-linked glucans the only difference is in the anomeric configuration on the C-1 atom of the monomeric unit, a for amylose and for cellulose. The calculated data show a remarkable pseudohelical chain... 

Models of Polymer Chains In the Bulk Amorphous State... 

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