Model, oriented-lattice

In a fundamental paper [265] Ruland develops an advanced method for the analysis of scattering patterns showing moderate anisotropy. The deduction is based on a 3D model and the concept of highly oriented lattices. The addition of distortion terms makes sure that the theory is applicable to distorted structures and their scattering. 

A theoretical treatment has recently been carried out by the author in collaboration with Matheson along the lines discussed above with appeal only to the spatial requirements of hard rods as represented in the lattice model, orientation-dependent interactions being appropriately ignored. The two transitions, one conformational and the other a cooperative intermolecular transition, are found to be mutually affected each promotes the other as expected. The coil-helix conformational transition is markedly sharpened so that it becomes virtually discrete, and hence may be represented as a transition of first-order. These deductions follow from the steric interactions of hard rods alone intermolecular attractive forces, either orientation-dependent or isotropic, are not required. 

Orthogonal plytyps. In the Trigonal model, the lattice is hP (o = 0) in the true structure for orthorhombic polytypes the lattice is normally oC but pseudo-/ / (co 0). For subfamily B and mixed-rotation polytypes the limiting symmetry is hP and there is only one independent orientation of the w.r.l. Twinning is either by complete merohedry or by pseudo-merohedry and does not modify the geometry of the diffraction pattern. 

Quantum dimers provide models for anti-ferromagnets. Using a combination of continuous-time lattice diffusion Monte Carlo [56] and RQMC, the square lattice quantum dimer model for lattice sizes up to 48 x 48 sites was investigated to estimate the location of the columnar to plaquette phase transition [57], The former phase has no parallel dimers, while the latter one has sets of plaquettes with parallel dimers continuously changing orientation. The author found significant finite-size corrections to scaling for the plaquette phase and liquid phase. 

In this chapter we summarize our recent studies of phase behavior in aqueous/polymer systems. We begin by reviewing the thermodynamic framework for describing phase (swelling) equilibria in gel/solvent systems. We then introduce into this framework an oriented-lattice model for describing polymer/solvent mixing effects In aqueous systems [11] this model describes correctly lower critical-solution temperature... 

Figure 1. Oriented-lattice model for polymer/solvent mixture [11]. Each segment may possess up to three, energetically different types of contact sites thus, interaction energies between neighboring segments are orientation-dependent.

The physical premise of the oriented-lattice model is simple however, extensive computations are required to reduce this model to analytical expressions for describing phase behavior in aqueous/polymer systems. Prange et al [11] present details of the model derivation. For our purposes here, we are Interested primarily in the resulting expression for the polymer/solving mixing contribution in equation (2) ... 

For applying the oriented-lattice theory (equation 3) to polyelectrolyte gels, we need expressions for the elastic and ionic contributions to the gel-swelling pressure (i.e, 7tg and TCjQri). Prange et al [11] used the affine network model for providing a simple... 

This work discusses new models and techniques for Improving our understanding and description of phase behavior in aqueous/polymer systems. We have considered only some of the potentially important classes of aqueous/polymer systems this work provides a framework on which to base further studies in this area. Important classes of hydrogels not considered here are pH-sensitive gels [4, 38], and gels in mixed-solvent solutions [4], A satisfactory theoretical description has not yet been obtained for the large volume transitions observed In these systems. It Is likely that volume transitions induced by changing solvent composition (i.e., acetone/water ratio) may be explained by the oriented-lattice framework extended to ternary systems. 

There is, of course, a mass of rather direct evidence on orientation at the liquid-vapor interface, much of which is at least implicit in this chapter and in Chapter IV. The methods of statistical mechanics are applicable to the calculation of surface orientation of assymmetric molecules, usually by introducing an angular dependence to the inter-molecular potential function (see Refs. 67, 68, 77 as examples). Widom has applied a mean-held approximation to a lattice model to predict the tendency of AB molecules to adsorb and orient perpendicular to the interface between phases of AA and BB [78]. In the case of water, a molecular dynamics calculation concluded that the surface dipole density corresponded to a tendency for surface-OH groups to point toward the vapor phase [79]. 

Face-centered cubic crystals of rare gases are a useful model system due to the simplicity of their interactions. Lattice sites are occupied by atoms interacting via a simple van der Waals potential with no orientation effects. The principal problem is to calculate the net energy of interaction across a plane, such as the one indicated by the dotted line in Fig. VII-4. In other words, as was the case with diamond, the surface energy at 0 K is essentially the excess potential energy of the molecules near the surface. 

Fabbri U and Zannoni C 1986 A Monte Carlo investigation of the Lebwohl-Lasher lattice model in the vicinity of its orientational phase transition Mol. Phys. 58 763-88... 

For structures with a high curvature (e.g., small micelles) or situations where orientational interactions become important (e.g., the gel phase of a membrane) lattice-based models might be inappropriate. Off-lattice models for amphiphiles, which are quite similar to their counterparts in polymeric systems, have been used to study the self-assembly into micelles [ ], or to explore the phase behaviour of Langmuir monolayers [ ] and bilayers. In those systems, various phases with a nematic ordering of the hydrophobic tails occur. 

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. 

After this short intermezzo, we turn back to introduce the last class of lattice models for amphiphiles, the vector models. Like the three-component model, they are based on the three state Ising model for ternary fluids however, they extend it in such a way that they account for the orientations of the amphiphiles explicitly amphiphiles (sites with 5 = 0) are given an additional degree of freedom a vector with length unity, which is sometimes constrained to point in one of the nearest neighbor directions, and sometimes completely free. It is set to zero on sites which are not occupied by amphiphiles. A possible interaction term which accounts for the peculiarity of the amphiphiles reads... 

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]... 

Finally, we assume that the fields 4>, p, and u vary slowly on the length scale of the lattice constant (the size of the molecules) and introduce continuous approximation for the thermodynamical-potential density. In the lattice model the only interactions between the amphiphiles are the steric repulsions provided by the lattice structure. The lattice structure does not allow for changes of the orientation of surfactant for distances smaller than the lattice constant. To assure similar property within the mesoscopic description, we add to the grand-thermodynamical potential a term propor-tional to (V u) - -(V x u) [15], so that the correlation length for the orientational order is equal to the size of the molecules. 

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