Composite sphere model

The Kerner equation, a three phase model, is applicable to more than one type of inclusion, Honig (14,15) has extended the Hashin composite spheres model to include more than one inclusion type. Starting with a dynamic theory and going to the quasi-static limit, Chaban ( 6) obtains for elastic inclusions in an elastic material... 

Transient Stresses During Sintering 11.4.2.1 Composite Sphere Model... 

Figure 11.18 The composite sphere model. A composite containing spherical inclusions is conceptually divided into composite spheres (cross-sectional view). The core of each sphere is an inclusion and the outer radius of the cladding of the sphere is chosen such that each sphere has the same volume fraction of inclusion as the whole body.

Scherer also considered a self-consistent model in which a microscopic region of the matrix is regarded as an island of sintering material in a continuum (the composite) that contracts at a slower rate (34). The mismatch in sintering rates causes stresses that influence the densification rate of each region. It is found that the equations for the self-consistent model differ from those of the composite sphere model only in that the shear viscosity of the matrix Gm is replaced by the shear viscosity of the composite Gc. Taking Eq. (11.33), the corresponding equation for the self-consistent model is therefore... 

Using Eqs. (11.34) and (11.37), the ratio AGJ3K can be calculated in terms of the relative density of the matrix and this ratio can be substituted into Eqs. (11.28), (11.29), and (11.35) to calculate the stresses and densification rates for the composite sphere model. Alternatively, for the self-consistent model, Gc can be found from Eq. (11.36), and the same procedure repeated to determine the stresses and strain rates. Figure 11.20 shows the predicted values for tjt as a function of the relative density of the matrix for the composite sphere and self-consistent models. For v less than 20 vol%, the predictions for the two models are almost identical, but they deviate significantly for much higher values of V,-. When v is less than —10-15 vol%, the predicted values of tJtT are not... 

Figure 11.20 Comparison of the predictions of Scherer s theory of sintering with rigid inclusions with the predictions of the rule of mixtures. The linear strain rate of the composite normalized by the strain rate from the rule of mixtures [Eq. (11.7)] is plotted versus the relative density of the matrix for the indicated volume fraction of inclusions Vf. The dashed curves represent the composite sphere model [Eq. (11.33)], and solid curves, the self-consistent model [Eq. (11.35)]. (From Ref. 33.)...

The change of electronic conductivity G(r) over diameter of such two-sphere model composition as element in a system of contacting particles is shown in a Figure 10.6b. The transfer of electron across this composition consists of three stages electron tunneling over the interspace — Rq is replaced by the M/SC conductivity across a particle with subsequent electron tunneling over the further interspace R — Rq. The probability of electron tunneling falls down exponentially with increase in distance from the surface of particle. 

Fig. 10.6. Percolation cluster model of tunnel current in composite film containing M/SC nanoparticles (a) two-sphere model of spherical M/SC nanoparticle of radius Rq surrounded by outer sphere (radius Rd) that is defined by a degree of electron delocalization extending the nanoparticle and characterizes electron tunneling (see text) (b) the distribution of conductivity G(r) over the two-sphere particle (c) two-dimensional pattern of cluster from overlapping two-sphere particles (overlapping areas of outer spheres are shown).

Equations (32) and (34), which are based on a hard-sphere model, are in agreement in predicting no dependence of the diffusion constant on gas composition. However, for real molecules, a slight composition dependence should exist, which depends on the form of the intermolecular potential. ... 

The types of surface moieties stabilizing the latex also are important. The binders used in waterborne coatings are not the hard-sphere, model polymer colloids used in adsorption studies. They are soft (low glass transition temperature), deformable moieties that are stabilized by grafted polymer fragments [e.g., (hydroxyethyl)cellulose (16) or poly(vinyl alcohol)] or by terpolymerized acid monomers extended from the surface of the colloid (IT). Such stabilizers produce a far less hydrophobic surface than is generally depicted in colloid texts. This situation is particularly true if the composition of the latex is predominately methacrylate or vinyl acetate, as they are in most U.S. commercial products. 

Predicted composite Young s modulus vs. particle volume fraction, for (a) voids (b) rigid spheres, for the random sphere model of Fig. 4.8a, with a matrix Poisson s ratio of 0.25 (Segurado, J., Llorca, J. et al.J. Mech. Phys. Solids, 50, 2107, 2002) Elsevier. 

In addition to tire standard model systems described above, more exotic particles have been prepared witli certain unusual properties, of which we will mention a few. For instance, using seeded growtli teclmiques, particles have been developed witli a silica shell which surrounds a core of a different composition, such as particles witli magnetic [12], fluorescent [13] or gold cores [14]. Anotlier example is tliat of spheres of polytetrafluoroetliylene (PTFE), which are optically anisotropic because tire core is crystalline [15]. 

The composites with the conducting fibers may also be considered as the structurized systems in their way. The fiber with diameter d and length 1 may be imagined as a chain of contacting spheres with diameter d and chain length 1. Thus, comparing the composites with dispersed and fiber fillers, we may say that N = 1/d particles of the dispersed filler are as if combined in a chain. From this qualitative analysis it follows that the lower the percolation threshold for the fiber composites the larger must be the value of 1/d. This conclusion is confirmed both by the calculations for model systems [27] and by the experimental data [8, 15]. So, for 1/d 103 the value of the threshold concentration can be reduced to between 0.1 and 0.3 per cent of the volume. 

Figures 1 a and 1 b represent the two-phase and the three-phase models respectively in the representative volume element of the composite. In the modified model three concentric spheres were considered with each phase maintaining a constant volume 4). The novel element in this model is the introduction of the third intermediate hybrid phase, lying between the two principal phases.

Thus, in the three-layer model, with the intermediate layer having variable physical properties (and perhaps also chemical), subscripts f, i, m and c denote quantities corresponding to the filler, mesophase, matrix and composite respectively. It is easy to establish for the representative volume element (RVE) of a particulate composite, consisting of a cluster of three concentric spheres, that the following relations hold ... 

A better approach for the Rosen-Hashin models is to adopt models, whose representative volume element consists of three phases, which are either concentric spheres for the particulates, or co-axial cylinders for the fiber-composites, with each phase maintaining its constant volume fraction 4). 

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