Deformable Lattice Model

Now, we obtained discrete representation of adsorption-induced deformation model. It is named deformable lattice model, since the cell (lattice) is deformable. 

The constitutive equations (2.7) and (2.12) of derived adsorption-induced deformation model and deformable lattice model are regarded as dynamical systems. Its definition[178] is as ... 

At first, discrete model, deformable lattice model, were proposed from adsorption-induced deformation model, utihzing cell dynamic scheme. Both adsorption model and cellular model are ubiquitously used in the field of physics. 

The mean sizes of windows, dw, and contacting cross sections, Dpc can be measured during analysis of the electron microscopy images as the relation of the first statistical moment to the zero one the sizes of dw can also be measured by adsorption methods (see Section 9.3). The direct interrelation between dw and, for example, Z)pc, is determined in view of a used model (e.g., in the framework of a model of isotropic deforming lattice of particles). Besides, also possible are correlations type of dwi dCi that relate the possible size of a cavity dCj to corresponding sizes of windows dWi from the cavity to the neighboring cavities. 

Figure 8 also shows values of f that have been calculated by two other methods. In the first, Jaswal (19) has used lattice-vibration eigen-frequencies and eigenvectors which have been calculated in the first Brillouin zone using the deformation-dipole model for the lattice. This... 

Lattice Model Carlo simulations of a block copolymer confined between parallel hard walls by Kikuchi and Binder (1993,1994) revealed a complex interplay between film thickness and lamellar period. In the case of commensurate length-scales (f an integral multiple of d), parallel ordering of lamellae was observed. On the other hand, tilted or deformed lamellar structures, or even coexistence of lamellae in different orientations, were found in the case of large incommensurability. Even at temperatures above the bulk ODT, weak order was observed parallel to the surface and the transition from surface-induced order to bulk ordering was found to be gradual. The latter observations are in agreement with the experimental work of Russell and co-workers (Anastasiadis et al. 1989 Menelle et al. 1992) and Foster et al. (1992). 

In the same paper Picken et al [13] also describe a more general two-dimensional lattice model to deal with this additional aspect. In this model only splay and bend deformations are allowed to occur. The longitudinal relaxation (i.e. the band spacing) will be determined mainly by the bend elastic constant K, as used in the one-dimensional model, and the lateral relaxation is driven by the splay modulus K, They find the result of this model to be not completely satisfactory.. .. it is clear that from the present model the aspect ratio of the bands Lj /L l is expected to be a constant (and equal to (iC /K ) ). This is not in agreement with the experimental results where the lateral correlation length seems to depend on the applied preshear rate, (and the longitudinal correlation length does not). They speculate that Possibly, the applied shear rate influences the details of the initial texture that is formed upon cessation of flow. ... 

Brillouin spectroscopy can also be used to study the change of sound velocity with deformation. Anders et al. reported the longitudinal sound velocity in stretched poly (urethane) and poly (diethylsiloxane) (PDFS, Figure 60.8) networks [15]. They used the lattice-model to determine the force constants [11]. The samples showed different deformation-dependent behavior of the force constants. For the... 

The lattice models of polymers reach their limits when one wants to study phenomena related to hydrodynamic flow. Although study of how chains in polymer brushes are deformed by shear flow has been attempted, by modeling the effect of this simply by assuming a smaller monomer jump rate against the velocity field rather than along it [61], the validity of such nonequilibrium Monte Carlo procedures is still uncertain. However, for problems regarding chain configurations in equilibrium, thermodynamics of polymers with various chemical architectures, and even the diffusive relaxation in melts, lattice models still find useful applications. 

There are three ways to simulate reaction-diffusion system. The traditional method is to solve partial differential equation directly. Another way is to divide system into cells, which is called cell dynamic scheme (CDS). Typical models are cellular automata (CA)[176] and coupled map lattice (CML)[177]. In cellular automata model, each value of the cell (lattice) is digital. On the other hand, in coupled map lattice model, each value of the lattice (cell) is continuous. CA model is microscopic while CML model is mesoscopic. The advantage of the CML is compatibility with the physical phenomena by smaller number of cells and numerical stability. Therefore, the model based on CML is developed. Each cell has continuum state and the time step is discrete. Generally, each cell is static and not deformable. Deformable cell (lattice) is supposed in order to represent deformation process of the gel. Each cell deforms based on the internal state, which is determined by the reaction between the cell and the environment. 

An important consequence of the memory-lattice model is that high moduli can be accommodated provided that the memory term in the potential function does not vanish. In terms of the model, a memory effect is present if is not equal to unity. This is evident in Eq(14d), low fluctuations corresponding to small Af lead to large values ofa A, the free energy of deformation, and corresponding large values of the retractive force. The coupling of the modulus to junction fluctuations does not appear in either the phantom network or fixed junction models. It arises here only because the minimum in the total potential is not exactly centered on the lattice. 

The memory-lattice model has the same structure here as for the isotropic material. The potential is changed. Consider a chain in an uncrosslinked polymer which is at a minimum potential energy before stretching at R = Ro. The sample is stretched so that Ro changes to Rc where X = Xo. The sample is frozen, crosslinked at this elongation, and then reheated to the rubbery state. The potential associated with an arbitrary deformation is given by... 

Hence, the comparison of the Eqs. (4.21), (4.46) and (4.47) shows that polymer behavior at deformation is defined by change exactly, if this parameter is considered as probabilistic measure. Let us remind, that such definition exists actually within the frameworks of lattice models, where tiiis parameter is connected with the ratio of free volume microvoids number and lattice nodes number N (N jN) [49]. The similar definition is given and for Pj in the Eq. (4.46) [86]. 

An interesting extension of the shell model is the deformable shell model or breathing shell model introduced by SCHRODER [4.16]. This model allows for radial deformations of the shells in the course of lattice vibrations which leads to three-body interactions and, correspondingly, the model does not predict the Cauchy relations [4.46]. 

It should also be mentioned that other lattice dynamical model exist which, in many respects, are equivalent to the shell model an important one is the deformation dipole model put forward by HARDY [4.47]. 

Hulbert [77] discusses the consequences of the relatively large concentrations of lattice imperfections, including, perhaps, metastable phases and structural deformations, which may be present at the commencement of reaction but later diminish in concentration and importance. If it is assumed [475] that the rate of defect removal is inversely proportional to time (the Tammann treatment) and this effect is incorporated in the Valensi [470]—Carter [474] approach it is found that eqn. (12) is modified by replacement of t by In t. This equation is obeyed [77] by many spinel formation reactions. Zuravlev et al. [476] introduced the postulate that the rate of interface advance under diffusion control was also proportional to the amount of unreacted substance present and, assuming a contracting sphere (radius r) model... 

The first section involves a general description of the mechanics and geometry of indentation with regard to prevailing mechanisms. The experimental details of the hardness measurement are outlined. The tendency of polymers to creep under the indenter during hardness measurement is commented. Hardness predicitions of model polymer lattices are discussed. The deformation mechanism of lamellar structures are discussed in the light of current models of plastic deformation. Calculations... 

Palladium hydride is a unique model system for fundamental studies of electrochemical intercalation. It is precisely in work on cold fusion that a balanced materials science approach based on the concepts of crystal chemistry, crystallography, and solid-state chemistry was developed in order to characterize the intercalation products. Very striking examples were obtained in attempts to understand the nature of the sporadic manifestations of nuclear reactions, true or imaginary. In the case of palladium, the elfects of intercalation on the state of grain boundaries, the orientation of the crystals, reversible and irreversible deformations of the lattice, and the like have been demonstrated. 

Contributions to pressure drop have also been studied by lattice Boltzmann simulations. Zeiser et al. (2002) postulated that dissipation of energy was due to shear forces and deformational strain. The latter mechanism is usually missed by capillary-based models of pressure drop, such as the Ergun equation, but may be significant in packed beds at low Re. For a bed of spheres with N — 3, they found that the dissipation caused by deformation was about 50% of that... 

The linear model can be extended to include more distant neighbours and to three dimensions. Let us consider an elastic lattice wave with wave vector q. The collective vibrational modes of the lattice are illustrated in Figure 8.6. The formation of small local deformations (strain) in the direction of the incoming wave gives rise to stresses in the same direction (upper part of Figure 8.6) but also perpendicular (lower part of Figure 8.6) to the incoming wave because of the elasticity of the material. The cohesive forces between the atoms then transport the deformation of the lattice to the... 

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