Lattices description

Mapping Atomistically Detailed Models of Flexible Polymer Chains in Melts to Coarse-Grained Lattice Descriptions ... 

In this section, the state of the art of the lattice description of real polymers in terms of the bond fluctuation model augmented with bond length and bond angle potentials has been discussed. It has been shown that the approach has both merits and weaknesses. 

The other class of motion only now being introduced into interpretive models is oscillatory motion. Anisotropic oscillatory motions of substituent groups have been considered by Chachaty (12) but not in conjunction with a lattice description of backbone motion. No attempt to develop a model based on oscillatory backbone rearrangements is known to these authors, and this avenue may be very important for the interpretation of concentrated solutions, rubbery or amorphous solids, and especially glassy polymers... 

Lattice description Primitive Face centered Hexagonal... 

Various continuum limits of the lattice description are taken (i.e., the number of Cl molecules Nm + , the number of C2 molecules Nm2 - °°, and the number of lattice sites M + ), such that the density and the other thermodynamic and molecular ordering variables can vary continuously for the system of molecules. (In these limits, the lattice statistics can treat molecules in which the number of segments per molecule is not an integer.)... 

Given the diversity of relevant applications, it is not surprising that the characterization of voids in disordered systems has an appreciable history, which can be traced back to primitive hole theories of the liquid state (Frenkel, 1955 Ono and Kondo, 1960). While the early theories offer an admittedly rudimentary lattice description of voids, recent computational advances permit an exact (and highly efficient) characterization of the continuum void geometry present in particle packings in two (Rintoul and Torquato, 1995) and three dimensions (Sastry et al., 1997a). 

We now consider a more quantitative model of the vibrational density of states which makes a remarkable linkage between continuum and discrete lattice descriptions. In particular, we undertake the Debye model in which the vibrational density of states is built in terms of an isotropic linear elastic reckoning of the phonon dispersions. Recall from above that in order to effect an accurate calculation of the true phonon dispersion relation, one must consider the dynamical matrix. Our approach here, on the other hand, is to produce a model representation of the phonon dispersions which is valid for long wavelengths and breaks down at... 

Structure Crystal system Space group Lattice description Lattice parameters ... 

The structure periodicity of composite incommensurate inclusion compounds generally requires four basis reciprocal lattice vectors for lattice description and a fourdimensional superspace group to characterize its symine-try. The use of the superspace approach in stmeture determination of ehaimel inclusion compounds is scarce but was applied recently in the study of urea inclusion compounds. " For a commensurate inclusion compound. 

Around 1920,1. Langmuir formalized the first nonlinear model to describe the surface tension dependence on surface concentration (see Ref. [1] for a detailed description). Langmuir s model is based on a lattice description of the... 

It is of interest to note that one may change the translation lattice of Fig. 5.3 by replacing the translation lattice vector c with the molecular helix lattice, keeping the translation symmetries a and This would lead to a match of the molecular helix symmetry with the crystal symmetry and even for irrational helices, a crystal stracture symmetry would be recognized. In fact, a whole set of new lattices can be generated replacing all three translation symmetry operations by helix symmetry operations [5]. Since a 1 1/1 hehx has a translational symmetry, this new space lattice description with helices would contain the traditional crystallography as a special case. 

Symmetric polymer blends do not exist in reality. A host erf asymmetries are present in real chemical alloys of interest These include attractive potential asymmetries (present even for isotopic blends) and specific interactions, molecular weight asymmetries and polydispersity, and single chain structural differences between the blend components (e.g., monomer shape and volume, backbone stiffness, and tacticity). Realistic accounting for most of these effects would seem to require an off-lattice description which includes local interchain density and concentration correlations, and compressibility effects [1, 2, 63, 66, 67, 80]. 

A first attempt to get couect microscopic properties from a discrete (lattice) description was done by Broadwdl for a gas in which the possible partide vdodties are restricted to a finite set. Then, Frisch a devised a cellular automaton... 

A variety of thermodynamic models qualitatively capture the central equilibrium and dynamic features of microemulsion behavior (41 5). These range from phenomenological models, which treat the oil-water interface as fluctuating membranes to lattice models which describes discrete surfactants interacting with oil and water. Membrane models range from simple cubic lattice descriptions to fluctuating film models that include curvature-dependent bending elasticity to prescribe lower limits on domain size. 

One possibility to motivate such an off-lattice description is to use the pairwise interactions in Eq. (5.49) in conjunction with a linear assignment function II and average the potential v(r, r ) over a uniform distribution of grid positions keeping the particle positions, r, r, fixed. 

An advantage of a soft, coarse-grained, off-lattice model is the ability to simultaneously and accurately calculate the pressure, p, and the chemical potential, p. Abandoning the lattice-description allows a precise calculation of the pressure, p, and simulations at constant pressure or tension. This is also possible in off-lattice models with harsh excluded volume interactions (e.g., a Lennard-Jones bead-spring model). The accurate calculation of the chemical potential by particle insertion methods. 

The Q-model was the first attempt at using a lattice description and is an inherently scalar model that can only describe forces in a single direction. A different lattice model, the "three-leg" model, that is tensorial in nature was proposed by Claudin et al. and is sketched in Figure 7.7. This microscopic tensoral model was constructed to justify their proposition that force transmission within a granular... 

Equations of state in the second category arise from molecular models and the development of a proper partition funetion (9). Most earlier works on this group are based on the lattice or quasi-lattice description of a fluid. Three types of models have played a dominant role in the past several decades. These are the cell, hole, and lattice fluid models. 

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