Lattice site representations

A fiirther step in coarse graining is accomplished by representing the amphiphiles not as chain molecules but as single site/bond entities on a lattice. The characteristic architecture of the amphiphile—the hydrophilic head and hydrophobic tail—is lost in this representation. Instead, the interaction between the different lattice sites, which represent the oil, the water and the amphiphile, have to be carefiilly constmcted in order to bring about the amphiphilic behaviour. 

Figure C2.3.2. Two-dimensional radial lattice representation of micelle stmcture using the approach of Dill and Flory [6], Each lattice site is considered to be equal in volume to tire otliers. Reproduced by pennission from [6],...

In tire simple version of tire lattice representation of proteins tire polypeptide chain is modelled as a sequence of connected beads. The beads are confined to tire sites of a suitable lattice. Most of tire studies have used tire cubic lattice. To satisfy tire excluded volume condition only one bead is allowed to occupy a lattice site. If all tire beads are identical we have a homopolymer model the characteristics of which on lattices have been extensively studied. 

A schematic representation of emergence is given in figure 12.6, which depicts the first three levels of a dynamical hierarchy and the rules or laws describing their behavior. The first, or lowest, level might be thought of as the level on which a CA system is usually defined. It consists of the lattice sites and values that define the microscopic dynamics. 

Figure 1. Schematic representation of the NAS1CON structure. The Si04 and P04 tetrahedra arc indicated by light blue, the ZrOfi octahedra by darkblue and the NaO, octahedra by green. The sodium ions are depicted by the red circles. The different radii represent the probability of lattice site occupation large radius 67 percent, small radius 1.1 percent. The Si/P ratio is 0.683 0.317. The a, b, and c axes are indicated.

As the number of lattice sites increases, the electrons experience additional correlations, so the representable region shrinks. That is, if TZi is the representable region for a lattice with i = A sites, then TZ4 dTZ dTZ% TZw d . This phenomenon is accurately tracked by the third-order estimates, and Fig. 3 shows that convergence to the limiting case where A oo is rapid. 

The experimental inaccessibility of the configurational entropy poses no problem for the LCT, apart from a consideration of whether to normalize the configurational entropy per lattice site or per monomer in order to provide a better representation of experiment within the AG model. Once the appropriate normalization of Sc has been identified, t can be calculated from Eq. (33) as a function of temperature T, molar mass Mmoi, pressure P, monomer structure, backbone and side group rigidities, and so on, provided that Ap is specified [54]. The direct determination of Ap from data for T > Ta is not possible for polymer systems because Ta generally exceeds the decomposition temperature for these systems. Section V reviews available information that enables specifying Ap for polymer melts. 

In the lattice representation of a polymer solution, each polymer segment or solvent molecule occupies one lattice site, while the system is regarded as a binary mixture of polymer and solvent. The Helmholtz energy of system can be expressed as... 

Since a lattice is just a set of points, we will need another entity to describe the real crystal. That is, it is required to locate a set of atoms named basis in the vicinity of the lattice sites. Therefore, a crystal will be a combination of a lattice and a basis of atoms. In Figure 1.3, a representation of the operation... 

The Flory-Huggins model uses a simple lattice representation for the polymer solution and calculates the total number of ways the lattice can be occupied by small molecules and by connected polymer segments. Each lattice site accounts for a solvent molecule or a polymer segment with the same volume as a solvent molecule. This analysis yields the following expression for A5m, the entropy of mixing A l moles of solvent with N2 moles of polymer. 

Fig. 4 Schematic representation of the BFS contributiions to the total energy of formation. The left hand side represents the reference atom (denoted by an arrow) in an alloy. The different terms on the right hand side indicate the strain energy (atoms in their actual positions but of the same atomic species as the reference atom), the chemical energy term (atoms in ideal lattice sites) and the reference chemical energy (same as before, but with the atoms retaining the original identity of the reference atoms).

Figure 10. Simulation of the EPR state preparation in an optical lattice with 25 sites, at three consecutive times. First row shows the joint probability distribution in x representation, the second one in p representation, (ol) and (a2) initially (t = 0), the atoms are cooled down to the external harmonic potential ground state, whereas the LIDDI is off. (61) and (62) at t = 1.4 x 10-4 s LIDDI and the repulsive linear potential (with the slope 0.04 Erec per lattice site) are on, whereas the harmonic potential is off. The diatoms are moving through the lattice very slowly in comparison to the single atoms, (cl) and (c2) at t = 2.16 x 10 4 s single atoms are ejected out of the lattice and discarded and the diatoms are separated out.

Figure 7. Schematic representation of the one dimensional Frenkel Kontorova Tomlinson model, a and b denote the lattice constant of the upper sohd and the substrate, respectively. The substrate is considered rigid, and its center of mass is kept fixed. In the shder, each atom is coupled with a spring of lateral stiffness to its ideal lattice site and with a spring of stiffness 2 to its neighbor. The PT model is obtained for 2 0, while the Frenkel Kontorova model corresponds to k = 0. We will drop the subscripts for these two cases since a single spring is relevant.

Magnetite exists in the spinel structure which can be represented by the formula (Fe " ") [Fe ,Fe " ]0, where the parentheses denote cations in tetrahedral lattice sites, and the brackets denote cations in octahedral lattice sites (J ). Figure 1 is a representation of the idealized spinel structure (note that the structure has been extended in the [001] direction for clarity). The oxygen anions form a cubic close-packed framework in which there are 2 tetrahedral vacancies and 1 octahedral vacancy per oxygen anion. From the above formula, it can be seen that one-eighth of the tetrahedral sites and one-half of the octahedral sites are occupied by iron cations. The ordered occupation of octahedral sites shown in Figure 1 facilitates electron hopping between ferrous and ferric cations at temperatures above 119 K( ). As a result, the oxidation state of these octahedral cations can be considered to be +2.5. 

Figure 30 Schematic representation of the zeolite ZSM-39 lattice framework. The three crystallographically inequivalent tetrahedral lattice sites are indicated by T, T2, and T3 (inside circles), and in each case the identities of the four nearest neighbors are shown.

The density n r) and the magnetization density m(r) are the key quantities in any (S)DFT calculation and require an accurate representation to allow for reliable total energy calculations. In the (R)FPLO scheme we need an adequate representation in terms of lattice sums (29). We split the expressions for n(r) and m(r) into on-site (net) contributions and overlap contributions, the latter arising from local orbitals situated at different lattice sites ... 

Fig. 5.2 Schematic representation of functional polymer chains configured on a cubic lattice. The darker cubes indicate a lattice site occupied by a functional end group, and the lighter cubes are occupied by polymer chain segments (a) illustrates a chain with a low-energy attractive end group, (b) depicts a nonfunctional polymer with neutral end groups. Reproduced with permission from [54]...

Figure 2. The nature of the statistical meehanical problem in the ease of a ciystalline solid. Uppermost is a representation of a perfect one-dimensional monatomie lattiee as would exist imder static conditions. Immediately beneath is an illustration of the effeet of temperature atoms no longer occupy their ideal lattice sites and the symmetry of the strueture at any instant is completely broken. The lowermost graph shows how the energy of the structure depends on the displacement of an atom the dependence is quadratic to first order, but higher order (anharmonie) terms may be important at conditions typicd of the Earth s interior.

The angle representation is a valid method when the change of the angle within the cell is less than 90°. Otherwise it must be handled carefully in the case where the liquid crystal director at two neighboring lattice sites are anti-parallel. The numerical calculation may produce a large elastic energy while the actual elastic energy is 0, because n and - n are equivalent. 

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