Polymer Lattices

The first Hamiltonian was used in the early simulations on two-dimensional glass-forming lattice polymers [42] the second one is now most frequently used in two and three dimensions [4]. Just to illustrate the effect of such an energy function, which is given by the bond length, Fig. 10 shows two different states of a two-dimensional polymer melt and, in part. 

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. 

A Dynamic Monte Carlo Simulations of Lattice Polymers. 27... 

Combining all contributions to the partition function of the disordered state of a lattice polymer solution, we obtain... 

Fig. 9 Snapshot of a single crystal of lattice polymers viewed from the chain direction. The bonds are drawn as solid cylinders. The viewing angle is large for better observation of folds. The chain length is 512 units and the thickness of the crystallite is about 12 units. The dissolved chains are not shown for clarity [57]...

Panagiotopoulos, A. Z. Wong, V. Floriano, M. A., Phase equilibria of lattice polymers from histogram reweighting Monte Carlo simulations, Macromolecules 1998, 31, 912-918... 

Lattice parameter determination, diffractometers in, 26 428 Lattice polymers, fullerene, 12 250, 251 Lattice-type inclusion compounds,... 

Version of the Bond Fluctuation Method for Lattice Polymers Erratum Notice, ibid., 71, 343 (1992). 

In this case, the equation of the isotherm of the adsorption of the oligomer on a onedimensional lattice (polymer) under conditions... 

The HP model is a coarse-grained (lattice or off-lattice) polymer model that abstracts from real polymers in two important ways (i) Instead of modeling the positions of all atoms of the polymer, it models only the backbone structure of the polymer, i.e., one position for each monomeric unit, (ii) Usually, only the hydrophobic interaction between the monomeric units is modeled, therefore the model distinguishes only two kinds of monomeric units, namely hydrophobic (H) and hydrophilic (or polar, P). 

Dynamic Monte Carlo simulations were first used by Verdier and Stockmayer (5) for lattice polymers. An alternative dynamical Monte Carlo method has been developed by Ceperley, Kalos and Lebowitz (6) and applied to the study of single, three dimensional polymers. In addition to the dynamic Monte Carlo studies, molecular dynamics methods have been used. Ryckaert and Bellemans (7) and Weber (8) have studied liquid n-butane. Solvent effects have been probed by Bishop, Kalos and Frisch (9), Rapaport (10), and Rebertus, Berne and Chandler (11). Multichain systems have been simulated by Curro (12), De Vos and Bellemans (13), Wall et al (14), Okamoto (15), Kranbu ehl and Schardt (16), and Mandel (17). Curro s study was the only one without a lattice but no dynamic properties were calculated because the standard Metropolis method was employed. De Vos and Belleman, Okamoto, and Kranbuehl and Schardt studies included dynamics by using the technique of Verdier and Stockmayer. Wall et al and Mandel introduced a novel mechanism for speeding relaxation to equilibrium but no dynamical properties were studied. These investigations indicated that the chain contracted and the chain dynamic processes slowed down in the presence of other polymers. 

D. Y. Yoon, Polym. Prepr. ACS, Div. Polym. Chem., 30(2j, 71 (1989). Chain Packing and Phase Transition of Cubic-Lattice Polymer Systems with Orientation-Dependent Interactions. 

Simple, exact models [51] consist of simple lattice polymers or heteropolymers where each amino acid in the hypothetical protein is represented by a bead occupying a single lattice point. Because of their simplicity, these models have generated considerable attention [57,58]. 

A measurement of compactness for lattice polymers is based on a similar idea.213,216 in this case, the parameter Q is defined as the ratio N/Nq, where N/ is the number of nearest-neighbor (nonbonded) contacts in the polymer and No its expected value for a compact global minimum. Another compactness parameter, based on residue contacts, is also used to classify folding features.223 Note that these relative descriptors explicitly use the molecular connectivity that is, they characterize molecular ID models. 

J.H. Jang and W. L. Mattice, The effect of solid wall interactions on an amorphous polyethylene thin film, using a Monte Carlo simulation on a high coordination lattice. Polymer 40 4685 (1999). 

Lattice Polymer chain Cell dimension (A) Cell angles Reference... 

Knots can of course be addressed also for toy lattice proteins discussed in section 10.10. The shortest cubic lattice knot has 24 monomers (Figure C11.7 a), but it is not space filling. The shortest space filling (open ended) lattice polymer to have knot is 36-mer. Its conformation with a trefoil knot is shown in the Figure C11.7. By the way, if the sequence of monomer species in it is properly selected (see Section 10.6), it folds in virtuo, of course) quite successfully, and not much slower that the corresponding chain without a knot — which opens even wider the question as to why real proteins are statistically less likely to have knots than random. 

In addition to carbon, many other elements can form chain structures with themselves or with other elements. Polymers that do not contain carbon atoms in the main chain are called inorganic polymers. According to the kinds of elements in the main chain, they are classed as isochains or heterochains, and, depending on the kind of linkage in the chains, they are called linear chains, ladder polymers, parquet polymers, or lattice polymers (see also Chapter 2). 

The oxidation and thermal stability can be increased by various measures. For one, the individual chains can be suitably substituted so that chemical attack on the main chain is sterically impeded or made electronically more difficult. In ladder, parquet, and lattice polymers, on the other hand, a simultaneous attack on neighboring chains is statistically improbable and additionally prevented by the close packing of the chains. 

Let us now put this polymer in a random medium. In the lattice model of Fig. 1, each site has an independent random energy and the total energy of the lattice polymer is the sum of the energies of the sites visited. In continuum, the Hamiltonian can be written as... 

The development of molecular simulations of a simple lattice-polymer model has allowed us to survey the topography of free-energy landscapes for singlechain melting and crystallization [43,44]. Thus, a quantitative thermodynamic description to the phase transitions of a single macromolecule can be verified [45]. 

Alkali-soluble resins n. These are generally lower molecular weight (than conventional lattices) polymers containing about 5-15% carboxyl groups which require amine and/ or cosolvent to solubilizer them. These systems are generally dispersions of micelles rather than true solutions. Abbreviation for ASR. 

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