Close-packed lattice model

In this review, we introduce another approach to study the multiscale structures of polymer materials based on a lattice model. We first show the development of a Helmholtz energy model of mixing for polymers based on close-packed lattice model by combining molecular simulation with statistical mechanics. Then, holes are introduced to account for the effect of pressure. Combined with WDA, this model of Helmholtz energy is further applied to develop a new lattice DFT to calculate the adsorption of polymers at solid-liquid interface. Finally, we develop a framework based on the strong segregation limit (SSL) theory to predict the morphologies of micro-phase separation of diblock copolymers confined in curved surfaces. 

To extend a close-packed lattice model Equation (15) to a lattice fluid model, we adopt a two-step process as shown in Figure 15 to establish an EOS (Hu et al., 1992). In the first step, pure chain molecules at close-packed lattice are mixed to form a close-packed mixture. In the second step, the close-packed mixture is mixed with N0 holes to form an expanded realistic system with volume V at temperature T and pressure p. According the two-step process, the Helmholtz energy of mixing can be expressed as... 

In Figure 1, a is plotted vs. ax for a face-centered lattice (a close-packed lattice) and for a simple cubic lattice (a loose-packed lattice). We notice that (1) the dependence of a on ax can be regarded as being practically the same for both lattices and that, (2) tx undergoes a rapid change around x = xc, which is the point at which a = 0 (Fig. 1). However, p/a does not attain the value it would have for the case of the unrestricted random walk model at x = xc, since at this point, p/a > 1 (Fig. 2), while for unrestricted chain pja = 1. Moreover, the dependence of p/a on ax is not the same for the two lattices while a as a function of ax is practically independent of the lattice. 

The crystal structure of a-Cr203 is made up by a hexagonal close-packed lattice of oxide ions (sequence ABAB ) Two-thirds of the octahedral sites are occupied by Cr3+ ions. Possible idealized surface structures, based on the (001), (100), and (101) planes and the creation of surface sites in the form of coordi-natively unsaturated cations and anions on dehydroxylation of the surface, have been discussed by Burwell et al. (21) and by Stone (144). The (001) face is the most likely crystal plane to predominate in the external surface of well-crystallized a-Cr203 (145). A possible surface model that maintains the overall as well as the local electrical neutrality, as proposed by Zecchina et al. (145) for the dehydroxylated (001) face, is shown in Fig. 2a. It can clearly be seen that equal numbers of four- and five-coordinate Cr3+ ions are to be expected on this idealized surface. Dissociative chemisorption of water would lead to the formation of surface OH groups, as shown in Fig. 2b, for a partially hydroxylated model surface. In fact, on adsorption of D20, Zecchina et al. (145) observed OD-stretching fundamental bands at 2700 and 2675 cm-1, which were narrow and isolated. As evidenced by the appearance of a H20 bending band at 1590... 

A close-packed lattice contains tetrahedral and octahedral interstitial holes (see Figure 5.5). Assuming a hard-sphere model for the atomic lattice, one can calculate that an atom of radius 0.41 times that of the atoms in the close-packed array can occupy an octahedral hole, while significantly smaller atoms may be accommodated in tetrahedral sites. 

The Finnis-Sinclair analytic functional form was introduced at about the same time as two other similar forms, the embedded-atom method > and the glue model." ° However, the derivation of the Finnis-Sinclair form from the second-moment approximation is very different from the interpretation of the other empirical forms, which are based on effective medium theory as discussed later. This difference in interpretation is often ignored, and all three methods tend to be put into a single class of potential energy function. In practice, the main difference between the methods lies in the systems to which they have been traditionally applied. In developing the embedded-atom method, for example, Baskes, Daw, and Foiles emphasized close-packed lattices rather than body-centered-cubic lattices. Given that angular interactions are usually ig-... 

Symmetry.—The Ising model, like the phenomenon of ferromagnetism which it was designed to simulate, has as an essential feature an exact symmetry. All the thermodynamic functions are symmetric or antisymmetric with respect to an axis of zero magnetic field. Transcription of this model to that of a one-component fluid leads to the highly artificial lattice-gas in which the critical density is exactly half the density po of the close-packed lattice. 

Higher-order multipole moments enhance the forces between particles at short distances and their neglect is extremely questionable, especially if fine effects are looked at, as for instance the ground-state properties of close-packed lattice structures [244,246-251] or the viscosity To go beyond the point dipole approximation Klingenberg and co-workers [ 173,252] developed an empirical force expression for the interaction between two dielectric spheres in a uniform external field from the munerical solution of Laplace s equation [253]. Recently, Yu and co-workers [254,255] proposed a computationally efficient (approximate) dipole-induced-dipole model based on a multiple image method which accounts partially for multipolar interactions. 

In the percolation model, the monomers are closely packed. The model describes polyfimctional polymerization in a melt. What happens if we dilute the reactants When we fix a reaction level p, we impose certain conditions on the monomers. They cannot be spread at random on the lattice because this would not give the correctp value. Thus, dilution leads... 

This equation gives the volume fraction in a close-packed lattice of spheres of diameter D when their center-to-center separation is D. The close agreement between results and expectations confirms the model, even in the case of aqueous latexes at concentrations so low that the particles are h diameters apart. 

To investigate the effect of the interaction of a matrix of holes with a multiple shock profile, a matrix of 10% air holes located on a hexagonal close-packed lattice in TATB was modeled. The spherical air holes had a diameter of 0.004 cm. The initial configuration is shown in Figure 3.39. The three-dimensional computational grid contained 16 by 22 by 36 cells each 0.001 cm on a side. The time step was 0.0002 /rsec. Figure 3.40 shows the density and mass fraction cross sections for a 40 kbar shock wave followed after 0.045 /rsec by a 290 kbar shock wave interacting with a matrix of 10% air holes of 0.004 cm diameter in TATB. 

Figure Bl.21.1. Atomic hard-ball models of low-Miller-index bulk-temiinated surfaces of simple metals with face-centred close-packed (fee), hexagonal close-packed (licp) and body-centred cubic (bcc) lattices (a) fee (lll)-(l X 1) (b)fcc(lO -(l X l) (c)fcc(110)-(l X 1) (d)hcp(0001)-(l x 1) (e) hcp(l0-10)-(l X 1), usually written as hcp(l010)-(l x 1) (f) bcc(l 10)-(1 x ]) (g) bcc(100)-(l x 1) and (li) bcc(l 11)-(1 x 1). The atomic spheres are drawn with radii that are smaller than touching-sphere radii, in order to give better depth views. The arrows are unit cell vectors. These figures were produced by the software program BALSAC [35]-...

In the CHS model only nearest neighbors interact, and the interactions between amphiphiles in the simplest version of the model are neglected. In the case of the oil-water symmetry only two parameters characterize the interactions b is the strength of the water-water (oil-oil) interaction, and c describes the interaction between water (oil) and an amphiphile. The interaction between amphiphiles and ordinary molecules is proportional to a scalar product between the orientation of the amphiphile and the distance between the particles. In Ref. 15 the CHS model is generalized, and M orientations of amphiphiles uniformly distributed over the sphere are considered, with M oo. Every lattice site is occupied either by an oil, water, or surfactant particle in an orientation ujf, there are thus 2 + M microscopic states at every lattice site. The microscopic density of the state i is p.(r) = 1(0) if the site r is (is not) occupied by the state i. We denote the sum and the difference of microscopic oil and water densities by and 2 respectively and the density of surfactant at a point r and an orientation by p (r) = p r,U(). The microscopic densities assume the values = 1,0, = 1,0 and 2 = ill 0- In close-packing case the total density of surfactant ps(r) is related to by p = Ylf Pi = 1 - i i. The Hamiltonian of this model has the following form [15]... 

Martensitic phase transformations are discussed for the last hundred years without loss of actuality. A concise definition of these structural phase transformations has been given by G.B. Olson stating that martensite is a diffusionless, lattice distortive, shear dominant transformation by nucleation and growth . In this work we present ab initio zero temperature calculations for two model systems, FeaNi and CuZn close in concentration to the martensitic region. Iron-nickel is a typical representative of the ferrous alloys with fee bet transition whereas the copper-zink alloy undergoes a transformation from the open to close packed structure. ... 

Among MC lattice models of the double layer, it is also worth mentioning the work of Nazmutdinov et al. (1988), who used a lattice model involving two mono-layers of water molecules on the surface of an electrode, forming a hexagonal close-packed array. The interaction of each water molecule in contact with the metal surface (assumed to be Hg) was taken from quantum-mechanical calculations. Information was obtained concerning the relative numbers of molecules with different numbers of hydrogen bonds, and it was concluded that the hypothesis of an icelike state of water in a monolayer on Hg is rather unlikely. 

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