Continuous space models

Results of recent theoretical and computer simulation studies of phase transitions in monolayer films of Lennard-Jones particles deposited on crystalline solids are discussed. DiflFerent approaches based on lattice gas and continuous space models of adsorbed films are considered. Some new results of Monte Carlo simulation study for melting and ordering in monolayer films formed on the (100) face of an fee crystal are presented and confronted with theoretical predictions. In particular, it is demonstrated that the inner structure of solid films and the mechanism of melting transition depend strongly on the effects due to the periodic variation of the gas - solid potential. 

Reduced continuous-space models were also employed in studies of the various aspects of protein [27,139] or polypeptide [49,140,141] folding. 

Configurational-bias methods trace their ancestry to biased sampling for lattice polymer configurations proposed by Rosenbluth and Rosenbluth [85]. Development of configurational-bias methods for canonical and grand canonical simulations and for continuous-space models took place in the early 1990s [86-90] and dramatically expanded the range of intermolecular potential models that can be studied by the methods described in the previous sections. 

A somewhat similar assumption of exact knowledge of secondary structure, as employed in the aforementioned continuous space models, was also used in an early lattice Monte Carlo model due to Skolnick and Kolinski [67] and Godzik et al. [192], However, there, the preferred local geometry was encoded... 

The static conformations on scales larger than the persistence length or the average spacing between nonbonded nearest neighbors do not depend on the specific model. For shorter distances differences occur and the lattice models are somewhat less realistic than the continuous space models. The MD simulations of Kremer and Grest are described very well by the PRISM (polymer reference interaction site model) model of Curro and Schweizer. ... 

The block diagram of the system is shown in Figure 9.10. Continuous state-space model From equations (9.77)-(9.81)... 

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. 

The simplest atomistic model for the formation of a crystal in continuous space requires the definition of some effective attractive potential between any two atoms, which is defined independently of the other atoms in the cluster or crystal. The most frequently studied potential is the Lennard-Jones potential... 

A suitable approach to the equilibration of an amorphous polymer system at bulk density becomes much more likely when the fully atomistic model in continuous space is replaced by an equivalent coarse-grained model on a lattice with sufficient conformational flexibility. Different strategies, which seek results at different levels of detail, can be employed to create an appropriate coarse-grained model. Section 4 (Doruker, Mattice) describes an approach which attempts to retain a connection with the covalent bonds in the polymer. The rotational isomeric state (RIS) [35,36] model for the chain is mapped into... 

The most austere representation of a polymer backbone considers continuous space curves with a persistence in their tangent direction. The Porod-Kratky model [99,100] for a chain molecule incorporates the concept of constant curvature c0 everywhere on the chain skeleton c0 being dependent on the chemical structure of the polymer. It is frequently referred to as the wormlike chain, and detailed studies of this model have already appeared in the literature [101-103], In his model, Santos accounts for the polymer-like behavior of stream lines by enforcing this property of constant curvature. 

This exercise underscores one more time that there is no unique way to define state variables. Since our objective here is to understand the association between transfer function and state space models, we will continue our introduction with the ss2tf () and tf2ss o functions. 

Continuous Transfer functions in polynomial or pole-zero form state-space models transport delay... 

Export processes are often more complicated than the expression given in Equation 7, for many chemicals can escape across the air/water interface (volatilize) or, in rapidly depositing environments, be buried for indeterminate periods in deep sediment beds. Still, the majority of environmental models are simply variations on the mass-balance theme expressed by Equation 7. Some codes solve Equation 7 directly for relatively large control volumes, that is, they operate on "compartment" or "box" models of the environment. Models of aquatic systems can also be phrased in terms of continuous space, as opposed to the "compartment" approach of discrete spatial zones. In this case, the partial differential equations (which arise, for example, by taking the limit of Equation 7 as the control volume goes to zero) can be solved by finite difference or finite element numerical integration techniques. 

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. 

As the dynamics of the system are removed from the model, it is no longer necessary to allow the molecules to live in a continuous space. Instead, the use of lattices - discrete sets of coordinates on to which the molecules are restricted - is popular. Digital computers are of course much more efficient with discrete space than with continuum space. The use of a lattice implies that one removes all properties that occur on shorter length scales than the lattice spacing from the model. This is no problem if the main interest is in phenomena that are larger than this length scale. 

The presence of a low-viscosity interfacial layer makes the determination of the boundary condition even more difficult because the location of a slip plane becomes blurred. Transitional layers have been discussed in the previous section, but this is an approximate picture, since it stiU requires the definition of boundary conditions between the interfacial layers. A more accurate picture, at least from a mesoscopic standpoint, would include a continuous gradient of material properties, in the form of a viscoelastic transition from the sohd surface to the purely viscous liquid. Due to limitations of time and space, models of transitional gradient layers will be left for a future article. 

Some of the mathematical tools, such as the linear one-box model, are both fairly simple and nonetheless sufficient for handling a great variety of situations. More complex systems require the use of multibox models. In some cases, continuous time-space models are needed. The mathematics of the latter involve partial differential equations and quickly lead beyond the scope of this book. In this chapter, a few important concepts were discussed which allow the reader both to make some approximative calculations and to critically analyze the results from computer models in which time-space processes are employed. 

Most of these tools use the finite difference method (at least for one-dimensional models) in which the continuous space coordinate is divided into a number of boxes. So we are back to the box-model technique. To demonstrate the procedure, in Box 23.4 we show how the partial differential Eqs. 23-44 and 23-45 are transformed into discrete (box) equations. 

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