Polymer chain, lattice-model

Iwata,L, Kurata,M. Brownian motion of lattice-model polymer chains. J. Chem. Phys. 50,4008 4013 (1969). 

Verdier,P.H., Stockmayer, W.H. Monte Carlo calculations on the dynamics of polymers in dilute solution. J. Chem. Phys. 36, 227-235 (1962). See also Verdier,P.H. Monte Carlo studies of lattice-model polymer chains. 1. Correlation functions in the statistical-bead model. J. Chem. Phys. 45,2118-2121 (1966). 

Another approach used to investigate dynamic behavior is to simulate Brownian motion using Monte Carlo techniques on a lattice-model polymer chain with the aid of a high speed digital computer (, lU, 15) In our lattice model the config-... 

It is noteworthy that for investigation of properties of real polymer systems with topological constraints it is not enough to be able to calculate the statistical characteristics of chains in the lattice of obstacles. It is also necessary to be able to compare any concrete physical system with the unique lattice of obstacles, which is a much more complicated problem than the first task. In this way, the model polymer chain in an array of obstacles is an intermediate between the microscopical and phenomenological approaches. The direct investigation of the PCAO-model was fulfilled in Refs. [16-25]. 

So far in this chapter, we have considered only systems in which T,V,N) are constant. In statistical mechanics, such a system is called the Canonical ensemble. Ensemble has two meanings. First, it refers to the collection of all the possible microstates, or snapshots, of the arrangements of the system. We have counted the number of arrangements W of particles on a lattice or configurations of a model polymer chain. The word ensemble describes the complete set of all such configurations. Ensemble is also sometimes used to refer to the constraints, as in the T, V,N) ensemble. The (T, V,N) ensemble is so prominent in statistical mechanics that it is called canonical. A system constrained by (T, p,N), is called the isobaric-isothermal ensemble. 

Monte Carlo Lattice Model for Chain Diffusion in Dense Polymer Systems and its Interlocking with Molecular Dynamics Simulations. 

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. 

Figure 12.5. (a) Lattice model showing a polymer chain of 200 beads , originally in a random configuration, after 10,000 Monte Carlo steps. The full model has 90% of lattice sites occupied by chains and 10% vacant, (b) Half of a lattice model eontaining two similar chain populations placed in contact. The left-hand side population is shown after 50,0000 Monte Carlo steps the short lines show the loeation of the original polymer interface (courtesy K. Anderson). 

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... 

FIQ. 1 Sketch of the BFM of polymer chains on the three-dimensional simple cubic lattice. Each repeat unit or effective monomer occupies eight lattice points. Elementary motions consist of random moves of the repeat unit by one lattice spacing in one lattice direction. These moves are accepted only if they satisfy the constraints that no lattice site is occupied more than once (excluded volume interaction) and that the bonds belong to a prescribed set of bonds. This set is chosen such that the model cannot lead to any moves where bonds should intersect, and thus it automatically satisfies entanglement constraints [51],... 

A particularly simple lattice model has been utilized by Harris and Rice [129] and subsequently by Stettin et al. [130] to simulate Langmuir mono-layers at the air/water interface chains on a cubic lattice which are confined to a plane at one end. Haas et al. have used the bond-fluctuation model, a more sophisticated chain model which is common in polymer simulations, to study the same system [131]. Amphiphiles are modeled as short chains of monomers which occupy a cube of eight sites on a cubic lattice and are connected by bonds of variable length [132], At high surface coverage, Haas et al. report various lattice artefacts. They conclude that the study... 

Lattice models have the advantage that a number of very clever Monte Carlo moves have been developed for lattice polymers, which do not always carry over to continuum models very easily. For example, Nelson et al. use an algorithm which attempts to move vacancies rather than monomers [120], and thus allows one to simulate the dense cores of micelles very efficiently. This concept cannot be applied to off-lattice models in a straightforward way. On the other hand, a number of problems cannot be treated adequately on a lattice, especially those related to molecular orientations and nematic order. For this reason, chain models in continuous space are attracting growing interest. 

Omstein [276] developed a model for a rigidly organized gel as a cubic lattice, where the lattice elements consist of the polyacrylamide chains and the intersections of the lattice elements represent the cross-links. Figure 7 shows the polymer chains arranged in a cubic lattice as in Omstein s model and several other uniform pore models for comparison. This model predicted r, the pore size, to be proportional to I/Vt, where T is the concentration of total monomer in the gel, and he found that for a 7.5% T gel the pore size was 5 nm. Although this may be more appropriate for regular media, such as zeolites, this model gives the same functional dependence on T as some other, more complex models. 

Mapping Atomistically Detailed Models of Flexible Polymer Chains in Melts to Coarse-Grained Lattice Descriptions ... 

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