Lattice polymer models

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). 

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]. 

A number of theoretical models have been proposed to describe the phase behavior of polymer—supercritical fluid systems, eg, the SAET and LEHB equations of state, and mean-field lattice gas models (67—69). Many examples of polymer—supercritical fluid systems are discussed ia the Hterature (1,3). 

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. 

This chapter is concerned with the application of liquid state methods to the behavior of polymers at surfaces. The focus is on computer simulation and liquid state theories for the structure of continuous-space or off-lattice models of polymers near surfaces. The first computer simulations of off-lattice models of polymers at surfaces appeared in the late 1980s, and the first theory was reported in 1991. Since then there have been many theoretical and simulation studies on a number of polymer models using a variety of techniques. This chapter does not address or discuss the considerable body of literature on the adsorption of a single chain to a surface, the scaling behavior of polymers confined to narrow spaces, or self-consistent field theories and simulations of lattice models of polymers. The interested reader is instead guided to review articles [9-11] and books [12-15] that cover these topics. 

Coarse-grained polymer models neglect the chemical detail of a specific polymer chain and include only excluded volume and topology (chain connectivity) as the properties determining universal behavior of polymers. They can be formulated for the continuum (off-lattice) as well as for a lattice. For all coarse-grained models, the repeat unit or monomer unit represents a section of a chemically realistic chain. MD techniques are employed to study dynamics with off-lattice models, whereas MC techniques are used for the lattice models and for efficient equilibration of the continuum models.36 2 A tutorial on coarse-grained modeling can be found in this book series.43... 

A basic polymer model is built by attaching successive units along the chain in neighboring sites of a geometrical lattice, with random orientations of the re-... 

Recently, some models (e.g., Halpin-Tsai, Mori- Tanaka, lattice spring model, and FEM) have been applied to estimate the thermo-mechanical properties [247, 248], Young s modulus[249], and reinforcement efficiency [247] of PNCs and the dependence of the materials modulus on the individual filler parameters (e.g., aspect ratio, shape, orientation, clustering) and on the modulus ratio of filler to polymer matrix. 

We have calculated the energy 0 in this way for some polymers and separation conditions (Table 2) and, using the lattice-like model and a slit-like pore, we have found the distribution coefficients, K 1, for these macromolecules as a function of N, D, 0 and 0f 65). It turned out that for such a crude model not only the calculated KJj 1 values were close to the experimental ones, but also, which is especially important, that the chemical nature of the macromolecule, the functional groups and the separation conditions (the mobile phase composition) were correctly accounted for. Two examples of such calculations are given in Figs. 8 and 9. 

In many production routes, and also during processing, polymer systems have to undergo pressure. Changes in the volume of a system by compression or expansion, however, cannot be dealt with in rigid-lattice-type models. Thus, non-combinatorial free volume ( equation of state ) contributions to AG have been advanced [23 - 29]. Detailed interaction functions have been suggested (but all of them are based on adjustable parameters, for blends, e.g., Mean-field lattice gas [30], SAFT [31], specific interactions [32]), and have been succesfully applied, for example, by Kennis et al. [33]. 

Equation of state for polymer systems based on lattice fluid model... 

The EOS based on the lattice fluid model has also be used to describe thermodynamic properties such as pVT behaviors, vapor pressures and liquid volumes, VLE and LLE of pure normal fluids, polymers and ionic... 

Figure 12 Free energy per unit area (a) and overall polymer density (b) as a function of the gallery height H for the Xsc= 84 system. Calculations are based on the compressible lattice SCFT model.

Figure 1.4 A 6 X 6 square lattice site model. The dots correspond to multifunctional monomers. (A) The encircled neighboring occupied sites are clusters (branched intermediate polymers). (B) The entire network of the polymer is shown as a cluster that percolates through the lattice from left to right.

Later Hildebrand [10] obtained the same result assuming that free volume available to the molecules per unit volume of liquid is the same for the polymer as for the solvent. The heat of mixing is defined as the difference between the total interaction energy in the mixture compared with that of pure components. Based on their lattice theory model, Flory [7,8,9] and Huggins [11,12] obtained the following expression for the heat of mixing ... 

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