Self-similar approach

Step 4 Overall constitutive properties of the dilute and urridirectional SWCN/polymer composite are determined with Mori-Tanaka model with the mechanical properties of the effective fiber and the bulk polymer. The layer of polymer molecules that are near the polymer/nanotube interface (Figure 8.2) is included in the effective fiber, and it is assumed that the matrix polymer surrounding the effective fiber has mechanical properties equal to those of the bulk polymer. The self-similar approach was proposed by Pipes and Hubert [49] which consists of three major steps ... 

Many traditional simulation techniques (e.g., MC, MD, BD, LB, Ginzburg-Landau theory, micromechanics and FEM) have been employed, and some novel simulation techniques (e.g., DPD, equivalent-continuum and self-similar approaches) have been developed to study polymer nanocomposites. These techniques indeed represent approaches at various time and length scales from molecular scale (e.g., atoms), to microscale (e.g., coarse-grains, particles. 

The direct use of micromechanical models for nanocomposites is however doubtfid due to the significant scale difference between nanoparticles and macro-partides. As such, two methods have recently been proposed for modeling the mechanical behavior of polymer nanocomposites equivalent continuum approach and self-similar approach. In equivalent continuum approach, molecular dynamics (MD) simulation is first used to model the molecular interaction between nanopartide and polymer. Then, a homogeneous equivalent continuum reinforcing element (i.e., an effective nanopartide) is constmcted. Finally, micro-mechanical models are used to determine the effective bulk properties of a... 

Figure XVI-1 and the related discussion first appeared in 1960 [1], and since then a very useful mathematical approach to irregular surfaces has been applied to the matter of surface area measurement. Figure XVI-1 suggests that a coastline might appear similar under successive magnifications, and one now proceeds to assume that this similarity is exact. The result, as discussed in Section VII-4C and illustrated in Fig. VII-6, is a self-similar line, or in the present case, a self-similar surface. Equation VII-21 now applies and may be written in the form...

The chemical reactivity of a self-similar surface should vary with its fractional dimension. Consider a reactive molecule that is approaching a surface to make a hit. Taking Fig. VII-6d as an illustration, it is evident that such a molecule can see only a fraction of the surface. The rate of dissolving of quartz in HF, for example, is proportional to where Dr, the reactive... 

Similar approaches are applicable in the chemical industry. For example, maleic anhydride is manufactured by partial oxidation of benzene in a fixed catalyst bed tubular reactor. There is a potential for extremely high temperatures due to thermal runaway if feed ratios are not maintained within safe limits. Catalyst geometry, heat capacity, and partial catalyst deactivation have been used to create a self-regulatory mechanism to prevent excessive temperature (Raghaven, 1992). 

The calculations that have been carried out [56] indicate that the approximations discussed above lead to very good thermodynamic functions overall and a remarkably accurate critical point and coexistence curve. The critical density and temperature predicted by the theory agree with the simulation results to about 0.6%. Of course, dealing with the Yukawa potential allows certain analytical simplifications in implementing this approach. However, a similar approach can be applied to other similar potentials that consist of a hard core with an attractive tail. It should also be pointed out that the idea of using the requirement of self-consistency to yield a closed theory is pertinent not only to the realm of simple fluids, but also has proved to be a powerful tool in the study of a system of spins with continuous symmetry [57,58] and of a site-diluted or random-field Ising model [59,60]. 

Similar calculations were carried out for the single impurity systems, niobium in Cu, vanadium in Cu, cobalt in Cu, titanium in Cu and nickel in Cu. In each of these systems the scattering parameters for the impurity atom (Nb, V, Co, Ti or Ni) were obtained from a self consistent calculation of pure Nb, pure V, pure Co, pure Ti or pure Ni respectively, each one of the impurities assumed on an fee lattice with the pure Cu lattice constant. The intersection between the calculated variation of Q(A) versus A (for each impurity system) with the one describing the charge Qi versus the shift SVi according to eqn.(l) estimates the charge flow from or towards the impurity cell.The results are presented in Table 2 and are compared with those from Ref.lc. A similar approach was also found succesful for the case of a substitutional Cu impurity in a Ni host as shown in Table 2. 

A solution to this problem (Hansen and Ottino, 1996a) reveals that the cluster size distribution is bimodal, as expected, with c(x,t) for large x dependent upon the initial conditions (Fig. 35a). The distribution thus does not approach a self-similar form and the scaling results just given are not valid for this problem. This is a result of the non-homogeneous relative rate of breakup. 

Nealey and coworkers [75,76,146] took a similar approach and applied lithographically defined self-assembled monolayers as substrates to direct the orientation of block copolymer thin films. After EUV interferometic lithography on octadecyltrichlorosilane (OTS) or phenylethyltrichlorosilane (PETS) monolayers, PS-fr-PMMA block copolymers were deposited and annealed on the substrates. Due to the selective wetting of PS and PMMA on the unexposed and exposed regions, respectively, they were able to obtain large areas of perpendicular lamella when the commensurate condition was fulfilled. 

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