Fillers dispersion model

These tests were performed using a universal Zwick tester, model 1435, coimected to a computer with appropriate software. Particle size was measured in paraffin oil using Zetasizemano S90 (Malvern Instmments) analyzer. Zeta potential of filler dispersion in water was studied by the means of Zetasizer 2000 (Malvern Instruments) apparams. Rheological properties of filler suspensions in paraffin oil were determined by viscometer RM500 (Rheometric Scientific). Dibutylphtalate (DBP) absorption was measured by means of an Absorptometer C (Brabender). The modification process was carried out in Brabender Measuring Mixer N50 at following parameters temperature 125°C rotor speed 40 RPM duration 0,5 h. 

The analysis of DMA results shows that theoretical models of a composite with a hard filler dispersed in a soft matrix do not account for the observed increase in the modulus. The experimental moduli in Fig. 9 are much higher compared with the theory of the Kemer-Nielsen (11) model (curve 1) (eq.l). 

It should be mentioned that in general, hard phase clusters can be non-spherical, as discussed in various earlier papers. In this case, the modulus increase could strongly depend on the aspect ratio the effect of the aspect ratio can be modeled through the micromechanical models of Halpin and Tsai [55] or Mori and Tanaka [56]. However, as we already commented above, below the spherical-to-cylindrical transition, most of the hard phase nano-domains have an aspect ratio close to 1. Above the spherical-to-cylindiical transition that is in our model associated with percolation threshold, most of the cylinders participate in the formation of the percolated hard phase, while the soft phase primarily contains hard phase islands with smaller aspect ratios. Therefore, in our analysis we assume that all the fillers dispersed within the soft phase are spherical (or have aspect ratios close to one). 

Permeation of gases through composite systems is a complex event and is influenced by a number of system parameters (Mai et al., 2004 Shields, 2008 Robeson, 2003 Bhardwaj, 2001) like the aspect ratio of the filler, extent of filler dispersion, orientation of the fillers and interparticle dispersion distance, filler loading fraction by volume, the density and crystallinity of the matrix, and the affinity between diffusion gases and the composite system. However, for modeling the permeabUity behavior, much simpler approaches have been used successfully. 

Using finite element techniqnes, a mathematical model was developed for the two-dimensional analysis of non-isothermal and transient flow and mixing of a generalised Newtonian fluid with an inert filler. The model could incorporate no-slip, partial-slip or perfect-slip wall conditions using a universally applicable numerical technique. The model was used to simulate the convection of carbon black with flowing rubber in the dispersive section of a tangential rotor (Banbury) mixer. The Carreau equation was used to model the rheological behaviour of the fluid in this example. 31 refs. 

Typical nonidealities such as polydispersity in filler size and conductivity, filler waviness and entanglements, and impurities impact the measured electrical properties of polymer nanocomposites. Most analytical and simulation studies of these nonidealities have been conducted for highly simplified systems, so that the extent to which these factors can modify composite properties, particularly within the context of more dominant factors such as filler dispersion and network stmcture, is unclear. To clarify the importance of these effects, theoretical analysis or modeling of more complex systems is required. Conducting parallel experiments in model systems can enhance the efficacy of such studies. 

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. 

Monte Carlo computer simulations were also carried out on filled networks [50,61-63] in an attempt to obtain a better molecular interpretation of how such dispersed fillers reinforce elastomeric materials. The approach taken enabled estimation of the effect of the excluded volume of the filler particles on the network chains and on the elastic properties of the networks. In the first step, distribution functions for the end-to-end vectors of the chains were obtained by applying Monte Carlo methods to rotational isomeric state representations of the chains [64], Conformations of chains that overlapped with any filler particle during the simulation were rejected. The resulting perturbed distributions were then used in the three-chain elasticity model [16] to obtain the desired stress-strain isotherms in elongation. 

This behavior can be understood if a superimposed kinetic aggregation process of primary carbon black aggregates in the rubber matrix is considered that alters the local structure of the percolation network. A corresponding model for the percolation behavior of carbon black filled rubbers that includes kinetic aggregation effects is developed in [22], where the filler concentrations and c are replaced by effective concentrations. In a simplified approach, not considering dispersion effects, the effective filler concentration is given by ... 

The CCA-model considers the filler network as a result of kinetically cluster-cluster-aggregation, where the size of the fractal network heterogeneity is given by a space-filling condition for the filler clusters [60,63,64,92]. We will summarize the basic assumptions of this approach and extend it by adding additional considerations as well as experimental results. Thereby, we will apply the CCA-model to rubber composites filled with carbon black as well as polymeric filler particles (microgels) of spherical shape and almost mono-disperse size distribution that allow for a better understanding of the mechanisms of rubber reinforcement. 

Quemada (1978a, 1978b) examined the rheology and modelling of concentrated dispersions and described simple viscosity models that incorporate the effects of shear rate and concentration of filler and separate effects of Brownian motion (or aggregation at low shear) and particle orientation and deformation (at high shear). The ratio of structure-build-up and -breakdown rates is an important parameter that is influenced by the ratio of the shear rate to the particle diffusion. A simple form of viscosity relation is given here ... 

Bicerano et al. (1999) provide a simplified scaling viscosity model for particle dispersions that states the importance of the shear conditions, the viscosity profile of the dispersing fluid, the particle volume fraction and the morphology of the filler in terms of its aspect ratio, the length of the longest axis and the minimum radius of curvature induced by flexibility. 

Viscoelastic stress analysis of two component systems shows that a broadening of the dispersion zone is to be expected 166,167), even if the disperse phase (filler) is purely elastic 166) and it is not necessary to ascribe different molecular properties to the continuous phase. The simplest way to visualize this mechanical interaction is by the use of phenomenological mechanically equivalent models. The model of Takayanagi (/68) is illustrated in Fig. 16. The elastic solution for this model is easily derived from elementary considerations. By the correspondence principle of viscoelastic stress analysis 169), the viscoelastic solution is obtained simply by substituting complex moduli in place of purely elastic moduli... 

Closed-form expressions from composite theory are also useful in correlating and predicting the transport properties (dielectric constant, electrical conductivity, magnetic susceptibility, thermal conductivity, gas diffusivity and gas permeability) of multiphase materials. The models lor these properties often utilize mathematical treatments [54,55] which are similar to those used for the thermoelastic properties, once the appropriate mathematical analogies [56,57] are made. Such analogies and the resulting composite models have been pursued quite extensively for both particulate-reinforced and fiber-reinforced composites where the filler phase consists of discrete entities dispersed within a continuous polymeric matrix. 

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