Atomistic lattice simulations

In this chapter we have selected a number of case studies to show how atomistic lattice simulations can be used to investigate the structural and defect chemistry of a wide range of high-7 cuprates. Space limitations have necessarily restricted the number of examples to a small range of compounds, chiefly from the La Cu-O, Nd-Cu-O, Y Ba Cu O and Sr-Cu-O systems. 

The placement and motion of potassium ions as well as the distortion of the polymer chains have been addressed by Corish el al. (see [116] and references therein), using atomistic lattice simulations. 

Note added in proof A powerful atomistic lattice simulation program GULP has recently been made available. Details may be obtained on application to Julian Gale, Dept, of Chemistry, Imperial College, South Kensington, London SW7 2AY, UK. 

N. L. Allan and W. C. Mackrodt,/. Chem. Soc., Faraday Trans., 86,1227 (1990). Structural and Energetic Aspects of Defects in High-T Oxides from Atomistic Lattice Simulations. 

Fig. 2. Steps in developing the atomistic pore from the mesoscaJe model, (a) snapshot of lattice simulation box (b) isolated pore (c) pore surface modeled as b-spline (d) silica block (e) mesopores carved out (f) model material after carving out the micropores and pore surface relaxation. White particles are the hydrogen atoms on the pore surface.

Before we can discuss in detail the simulation of adsorption and diffusion in zeolites using atomistic simulation we must ensure that the methods and potentials are appropriate for modelling zeolites. The work of Jackson and Catlow reviewed in the previous section shows the success of this approach. Perhaps the most critical test is to apply lattice dynamics and model the effect of temperature as any instability will cause the calculation to fail. Thus we performed free energy minimization calculations on a range of zeolites to test the methodology and applicability to zeolites. As noted in Section 2.2, the extension of the static lattice simulation technique to include the effects of pressure and temperature leading to the calculations of thermodynamic properties of crystals and the theoretical background to this technique have been outlined by Parker and Price [21], and this forms the basis of the computer code PARAPOCS [92] used for the calculations. 

In this review, we have chosen to focus mainly on the atomistic chain simulations, and will describe in detail equilibrium and dynamic properties of polymer melts at interfaces predicted by these simulations. The ability of the simulations to provide insight into the equilibrium and dynamic behavior of the interfaces as seen experimentally will be discussed. A brief description of systems of bead chains will also be presented, as these models also take into account the off-lattice continuum nature of the interface that... 

Although the folding of short proteins has been simulated at the atomic level of detail [159,160], a simplified protein representation is often applied. Simplifications include using one or a few interaction centers per residue [161] as well as a lattice representation of a protein [162]. Some methods are hierarchical in that they begin with a simplified lattice representation and end up with an atomistic detailed molecular dynamics simulation [163]. 

The current understanding of the protein folding process has benefited much from studies that focus on computer simulations of simplified lattice models. These studies try to construct as simple a model as possible that will capture some of the more important properties of the real polypeptide chain. Once such a model is defined it can be explored and studied at a level of detail that is hard to achieve with more realistic (and thus more complex) atomistic models. 

Very recently, people who engage in computer simulation of crystals that contain dislocations have begun attempts to bridge the continuum/atomistic divide, now that extremely powerful computers have become available. It is now possible to model a variety of aspects of dislocation mechanics in terms of the atomic structure of the lattice around dislocations, instead of simply treating them as lines with macroscopic properties (Schiotz et al. 1998, Gumbsch 1998). What this amounts to is linking computational methods across different length scales (Bulatov et al. 1996). We will return to this briefly in Chapter 12. 

Fig. 4.1. Schematic representation of three numbered steps in a MC simulation on a high coordination lattice (solid arrows) that replace a simulation of the fully atomistic system in continuous space (single dashed line)...

MD simulations of melts of C44H90, based on classic techniques in continuous space, have been reported recently using united atom [146] and fully atomistic [145] representations of the chain. Time in the conventional MD simulations is expressed in seconds, whereas time in the simulation of the coarse-grained chains on the 2nnd lattice is expressed in MC steps. Nevertheless, a few comparisons are possible via the longest relaxation time, rr, deduced from the decorrelation of the end-to-end vector ... 

Mapping Atomistically Detailed Models of Flexible Polymer Chains in Melts to Coarse-Grained Lattice Descriptions Monte Carlo Simulation of the Bond Fluctuation Model... 

A rather crude, but nevertheless efficient and successful, approach is the bond fluctuation model with potentials constructed from atomistic input (Sect. 5). Despite the lattice structure, it has been demonstrated that a rather reasonable description of many static and dynamic properties of dense polymer melts (polyethylene, polycarbonate) can be obtained. If the effective potentials are known, the implementation of the simulation method is rather straightforward, and also the simulation data analysis presents no particular problems. Indeed, a wealth of results has already been obtained, as briefly reviewed in this section. However, even this conceptually rather simple approach of coarse-graining (which historically was also the first to be tried out among the methods described in this article) suffers from severe bottlenecks - the construction of the effective potential is neither unique nor easy, and still suffers from the important defect that it lacks an intermolecular part, thus allowing only simulations at a given constant density. 

The side chain separation varies in a range of 1 nm or slightly above. The network of aqueous domains exhibits a percolation threshold at a volume fraction of 10%, which is in line with the value determined from conductivity studies. This value is similar to the theoretical percolation threshold for bond percolation on a face-centered cubic lattice. It indicates a highly interconnected network of water nanochannels. Notably, this percolation threshold is markedly smaller, and thus more realistic, than those found in atomistic simulations, which were not able to reproduce experimental values. 

Embedded atom potentials have been extensively used for performing atomistic simulations of point, line and planar defects in metals and alloys (e.g. Vitek and Srolovitz 1989). The pair potential ( ), atomic charge density pBtom(r), and embedding function F(p) are usually fitted to reproduce the known equilibrium atomic volume, elastic moduli, and ground state structure of the perfect defect-free lattice. However, the prediction of ground state structure, especially the competition between the common metallic structure types fee, bcc, and hep, requires a more careful treatment of the pair potential contribution ( ) than that provided by the semiempirical embedded atom potential. This is considered in the next chapter. 

Single crystals of /S-A1203 are essentially two dimensional conductors. The conducting plane has hexagonal symmetry (honeycomb lattice). This characteristic feature made -alumina a useful model substance for testing atomistic transport theory, for example with the aid of computer simulations. Low dimensionality and high symmetry reduce the computing time of the simulations considerably (e.g., for the calculation of correlation factors of solid solutions). 

Different scale features give different scale properties. At the smallest level, the lattice parameter is a key length scale parameter for atomistic simulations. Since atomic rearrangement is intimately related to various types of dislocations, Orowan [88], Taylor [89], Polyani [90], and Nabarro [91] developed a relationship for dislocations that related stress to the inverse of a length scale parameter, the... 

Some conclusions about the shape of highly branched macromolecules based on successive perfect generations of the controlled condensation of A-R-B2 are accessible from simulations on a diamond lattice/24 This type of simulation has the disadvantage that it is not atomistic, and therefore does not accurately describe in detail any particular macromolecule. But the advantages, namely the ability to convincingly demonstrate the attainment of equilibrated structures and the likelihood that the overall trends are reflec-... 

Abbreviations A(hkl), surface area of the simulation cell in atomistic approaches (m2) am. lattice parameter of an AM oxide (A) , fivefold coordinated, n-valent anion , four-... 

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