Frenkel’s model

Note that due to the sinusoidal function of the shear stress in Frenkel s model V Yb= p/(4dc)). The corresponding maximum value of the shear energy, W y is given by... 

The dependence of slip on the interaction strength of the surface and liquid was studied early on by Tolstoi [3], later revisited by Blake [12], a model that is linked to interfacial viscosity. Tolstoi modeled the surface using Frenkel s model for the bulk mobility of a liquid molecule [38],... 

The validity of Frenkel s model is limited to Newtonian flow and can only be used to predict the early stage of the coalescence process, when the diameter of the two spherical particles remains nearly unchanged. The inadequacy of a Newtonian model in describing the coalescence of polymers was also demonstrated in other studies, as reviewed by Mazur, and has led to the development of models as well as alternative methods for the characterization of the coalescence behavior of polymers for rotational molding applications.Based on theoretical and experimental analyses of the coalescence phenomenon, the material properties of primary interest in the evaluation of resin coalescence behavior in rotational molding have been identifled as the resin viscosity, surface tension, and elasticity. 

The emission parameter, is calculated, taking into account contribution of matrix frustration to the free energy of cluster distributions in binary alloys (Lepinoux 2006,2009). This effect is typical for concentrated alloys, when Frenkel s model of ideal cluster gas (Frenkel 1955) is not valid and it is necessary to consider the interaction between the clusters by means of so-called exclusion volume, i.e., the number of forbidden atomic sites (or volume normalized by the atomic volume) to a -mer by an n-mer. In our paper it is suggested to take into account the frustration effect empirically by the use of the thermodynamic free energy expression from CALPHAD (Andersson and Sundman 1987) with the correction suggested by Bonny et al. (2010). 

Flow fields for viscous sintering of spheres (schematic). Right-hand side, uniaxial contraction assumed in Frenkel s model. Left-hand side, form expected in real situations. From Uhlmann el al. (18]. Note added in proof Recent simulations using finite element analysis (A. Jagota and P.R. Dawson, to be published in J. Am. Ceram. Soc.) show that the flow field on the right is, indeed, correct. 

Relative density versus reduced time for Frenkel s model (dashed curve) and cylinder model (solid curve) for microstructures shown in Fig. 9. (Cylinder model used up to p = 0.94, MS model thereafter). From Scherer [30], reprinted by permission of the American Ceramic Society. 

Figure 9. Frenkel s model for early-stage sintering viscous flow [41].

The two spheres of the Frenkel s model use the concept of viscous flow of atoms that relates the vacancy diffusion coefficient D the volume of the atom or vacancy O, and vacancy concentration gradient per unit area of the material (dQ/dx), as shown in the following equation [41] ... 

Many aspects can be studied from the kinetic equations, as densification rate, determination of sintering mechanisms, and activation energy. The equation developed by Coble allows to estimate the sintering mechanisms for the initial stage, based on the two spheres Frenkel s model, as indicated in the Eq. 15 [43] ... 

J. Frenkel, T. Kontorova. J Phys USSR 7 137, 1939 E. Allroth, H. Muller-Krumbhaar. Phys Rev A 27 1515, 1983 S. Stoyanov, H. Miiller-Krumbhaar. Resonance-induced cluster mobility Dynamics of a finite Frenkel-Kontorova model. Surf Sci 159 49, 1985. 

The first section recalls the Frenkel-Davydov model in terms of a set of electromagnetically coupled point dipoles. A compact version of Tyablikov s quantum-mechanical solution is displayed and found equivalent to the usual semiclassical theory. The general solution is then applied to a 3D lattice. Ewald summation and nonanalyticity at the zone center are discussed.14 Separating short and long-range terms in the equations allows us to introduce Coulomb (dipolar) excitons and polaritons.15,16 Lastly, the finite extent of actual molecules is considered, and consequent modifications of the above theory qualitatively discussed.14-22... 

Dobmskin [57] proposed a model for the adsorption equilibria of multicomponent vapor mixtures based on the concept of TVFM and an adsorbed phase model in which the adsorbate-adsorbent interactions predominate over the lateral interaction between adsorbed molecules. The proportions of the component in the adsorbed phase are determined by a statistical distribution based on Frenkel s [70] mechanism and kinetic gas theory [71,72]. In Dobruskin s study, the equilibrium is viewed as a dynamic process in which the average molecular residence time T is the reciprocal of the rate constant for desorption, k. For adsorption of a binary mixture in an elementary volume dW, the ratio of the average times between two components is... 

The derivation of the sintering equations for Scherer s model is quite similar to that outlined earlier for Frenkel s initial stage model. The result is (21)... 

A feature that arises from our consideration of the analytical models is the difference in complexity between the sintering phenomena in polycrystalline materials and amorphous materials. The analysis of viscous sintering on the basis of Frenkel s energy balance concept appears relatively simple in principle. The idealization of the structure of amorphous materials leads to analytical solutions that describe the sintering behavior in a very satisfactory manner. 

Brown, O. M., Kivshar, J. S. (2004). The Frenkel-Kontorova model. Concepts, Methods, and Applications Springer, 519 p. 

Figure 4 shows the fit as a log of mobility plotted against the square root of the electric field. Plotting against s/f should yield a straight line according to the Poole-Frenkel phenomenological model for mobility [9, 11] ... 

The carrier mobility p is temperature- and field-dependent. Many theories have been developed to explain the temperature dependence, but no comprehensive model is yet available. It is still not clear whedier the charge carrier mobility follows a simple Arrhenius relationship (log p 1/7) as predicted by Gill [30] or if the more complex relationship log p 1/ 7 proposed by Busier et al. [35] is valid. The relationship between the mobility p and the electrical field strength E is equally unclear. Here Gill s model predicts a log pi E dependence which is consistent with a Pool-Frenkel formalism, whereas Bassler s calculations lead to a log pi E dependence. A detailed description of the different models and results obtained by fitting experimental mobility data to those models is beyond the scope of this chapter. It shall only be pointed out here that the main difficulty is the limited range of temperature and electric field in which carrier mobilities can be measured [36]. Additional experi-... 

The basic hypothesis is that along any polymer chain there is a Maxwellian distribution of energies with several low energy "cold spots." Inter- and intra-association of cold spots gives rise to a structure which he assumed disappeared at a temperature which increased linearly with log molecular weight, much as Tyin Fig. 5. Ueberreiter s model for structure is inherently similar to that of Frenkel depicted by us in Fig. 9 of ref. 1, as segment-segment association which melts out at T//(see Fig. 11). 

Braun, O.M. and Kivshar, Y.S. (2004) The Frenkel—Kontorova Model Concepts, Methods, Applications, Springer, Berlin. 

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