Spherical Filler Particles

There are two basic theoretical approaches that have been followed to model the effect of spherical filler particles on a matrix. These two scenarios are shown schematically in Fig. 1. The first approach allows for the formation of an internal network structure within the matrix. This model considers the contribution of each phase separately, and uses percolation theory to arrive at an effective thermal conductivity for the composite [8-10]. The models which account for particle-particle interactions tend to maximize the effect of the dispersed phase. For example the simplest interactive model is the simple rule-of-mixtures  

Another series expansion approach that accounts for a wide range of k , and kf has the following form [15]  

In this equation is a multidimensional integral to the boimdary value teat conduction prolrfem around the particle n, weighted according to the size 

Hashin and Shtrikman produced the following lower bound equation to describe the effect of spherical filler particles on the thermal conductivity of a randomly dispersed, particle-in-matrix, two phase syston [16]  

Hamilton and Grosser produced the following lower bt und equation [17]  

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Non-spherical filler particles are also of considerable interest [50,69]. Prolate (needle-shaped) particles can be thought of as a bridge between the roughly spherical particles used to reinforce elastomers and the long fibers frequently... 

Figure 10 Deformation of spherical filler particles into prolate (needle-shaped) ellipsoids see text for details. 

Consider spherical filler particles (phase f) in a matrix (phase m). The probability of a positron hitting a filler particle is proportional to the volume fraction of filler vr. The probability of the positron thermalizing and annihilating in this filler particle can be written [21] as... 

In the case of non-spherical filler particles, it has been possible to simulate the anisotropic reinforcement obtained, for various types of particle orientation.68-70 127 135 Different types and degrees of particle agglomeration can also be investigated. 

Different concentration limits of the filler arise from the CCA concept [22]. With increasing filler concentration first an aggregation limit O is reached. For >+, the distance of neighboring filler particles becomes sufficiently small for the onset of flocculation and clusters with solid fraction A are formed. Dependent on the concentration of filler particles, this flocculation process leads to spatially separated clusters or, for 0>0, a through going filler network that can be considered as a space-filling configuration of fractal CCA-clusters. The different cases for spherical filler particles are shown schematically in Fig. 1. 

Here, Va is the solid volume and NA is the number of particles or primary aggregates of size d in the clusters of size . p is the solid fraction of primary aggregates considered in Sect. 3.2.2. For spherical filler particles it equals... 

The central point of the present survey is an attempt to show a complete analogy between the free volume of suspensions and that of molecular systems. It is characteristic that the limiting volume fraction of spherical filler particles leaves in the system another 25-40% of unoccupied volume. Precisely the same unoccupied volume exists in molecular systems if we liken them to a volume filled with spheres whose radii are calculated taking into account the Lennard-Jones potential. 

The shape factor ( d/hg) reflects the boundary condition s constraint on rubber flow during deformation, and can be considered as a measure of tightness for a junction. The shape factor, or the ratio d/hg, can be used to calculate the stored energy with a junction rubber between two spherical filler particles [86,87] ... 

The mechanical action of the immobilized rubber layer on spherical filler particles, that are assumed to form a CCA-filler network in a rubber matrix for

cp, is obtained if the mechanically effective solid fraction q>A (Eq. (71)), is applied in Eq. (69) instead of cpA and the space-filling condition [Pg.35]

Vacuole dilatation information itself is not simply interpreted. The data instead are best understood through models of microstruetural failure (1). Assuming a single size of spherical filler particles encompassed by elliptically shaped voids that form arbitrarily in strain, and once formed grow at a constant rate with further deformation, then one can readily separate vacuole growth from vacuole formation. Models such as the one described above have been substantiated by microscopic studies. The solution of such models (1) indicates that the first derivative of vacuole dilatation with respect to strain c, is directly proportional to the cumulative number of vacuoles per unit volume,n, that exist at any strain. The second derivative is then directly proportional to the instantaneous frequency distribution of vacuole formation. These two results can be expressed mathematically... 

Fig. 10. Modulus-concentration relationships for spherical filler particles...

In these equations, fa is the volume fraction of filler, and subscripts / and 0 refer to the filled and unfilled elastomers respectively. Note that equations (6-95) and (6-96) introduce a parameter m that accounts for the maximum packing fraction of the filler. For randomly placed spherical filler particles, m = 0.637. 

Basu, D. Banerjee, A.N. Misra, A. (1992). Comparative Rheological Studies on Jute-Fibre-and Glass-Fibre-Filled Polypropylene Composite Melts. Journal of Applied Polymer Science, Vol.46, No.ll, pp. 1999-2009 ISSN 0021-8995 Bigg, D.M (1982). Rheological Analysis of Highly Loaded Polymeric Composites Filled with Non-Agglomerating Spherical Filler Particles. Polymer Engineering and Science, Vol.22, No.8, p>p. 512-518 ISSN 0032-3888... 

Hence this equation is a natural generalization of the Einstein-Smallwood reinforcement law. For rigid and spherical filler particles at low volume firaction, the Einstein-Smallwood formula is recovered, since in this case the intrinsic modulus [/a] = 5/2 (the intrinsic modulus [/a] follows from the solution of a single-particle problem). Exact analytical results can be obtained for the most relevant cases, such as uniform soft spheres, which describe the softening of the material in a proper way, as well as in the case of soft cores and hard shells [5]. 

In one application, a filled PDMS network was modeled as a composite of cross-linked polymer chains and spherical filler particles arranged on a cubic lattice. The filler particles increase the non-Gaussian behavior of the chains and increase the moduli. It is interesting to note that composites with such structural regularity have actually been produced and mechanical properties have been reported. -... 

Table 1. Composition of the simulated dense systems (Afp = number of polymer chains of 100 units Nf = number of randomly distributed spherical filler particles = diameter of the particles ifi = volume fraction of filler)...

Spherical filler particles have received more attention than non-spherical and asymmetric particles. In the following, the effect of concentration and dimensions of the particles are presented under the sub-sections of different shapes of particles. 

Bigg, D.M. (1982) Rheological analysis of highly loaded pol5mneric composites filled with non-agglomerating spherical filler particles, Polym. Engg Sci., 22,512-18. 

Spherical filler particles, usually hollow, used to reduce weight in a product without much adverse effect on mechanical properties. 

Most of the equations assume spherical filler particles, but more recent work has taken their shape into account. Models have also been proposed for strength. 

Abstract This chapter deals with the non-linear viscoelastic behaviour of rubber-rubber blend composites and nanocomposites with fillers of different particle size. The dynamic viscoelastic behaviour of the composites has been discussed with reference to the filler geometry, distribution, size and loading. The filler characteristics such as particle size, geometry, specific surface area and the surface structural features are found to be the key parameters influencing the Payne effect. Non-Unear decrease of storage modulus with increasing strain has been observed for the unfilled vulcanizates. The addition of spherical or near-spherical filler particles always increase the level of both the linear and the non-linear viscoelastic properties. However, the addition of high-aspect-ratio, fiber-like fillers increase the elasticity as well as the viscosity. 

Bigg, D. M., Rheological analysis of highly loaded polymeric composites filled with nonagglomerating spherical filler particles, Polym. Eng. ScL, 22, 512-518 (1982). 

By virtue of the strong anisotropy of the shape of particles of Na -montmorillonite mentioned above, for theoretical estimation of the degree of reinforcement of nanocomposites filled by them the models of Halpin-Tsai and Mori-Tanaka are used [19]. For the case of isotropic (spherical) filler particles EJE estimation can be carried out according to the equation, obtained in paper [35] ... 

The dependence of measured retardation with distance (r ) is shown in Fig. 9 for the systems NR-benzene and NR-o-dichlorobenzene. These data were obtained using particles of ca. lOOy radius. The theoretical curves represent the results given in Table 1. Since the physical constants used in the theory were obtained independently of the inhomogeneous swelling experiments, the agreement in Fig. 9 is quantitative and contains no adjustable parameters. It is concluded that the inhomogeneous swelling theory [7] describes correctly the deformation and stress fields about an isolated, spherical filler particle. 

In recent calculations Stemstein3 has worked out the problem of swelling of rubber containing spherical filler particles subjected to the boundary conditions that the tangential strain is zero at the surface of the particle. He has obtained expressions... 

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