Carbon black effective volume fraction

By relating the endpoint of crushed DBF absorption to the void space within and between equivalent spheres of aggregates, and assuming the spheres to be packed at random, Wang et al. obtained the following equation for the effective volume fraction of carbon black ... 

Considering a highly irregular shape and a presence of occluded rubber in cavities of the carbon black aggregates, Medalia [16] introduced effective volume to represent filler concentration. His effective volume fraction, V, replaces the real volume fraction, v, in Equation 8.8. The effective volume fraction, V, is not an adjustable parameter, but calculated from the DBP absorption [17] (see also Chapter 9). The shear storage modulus, G, was measured at 25 °C and 0.25 Hz, with 20 phr of carbon black loading where G was practically independent of the strain amplitude. With 12 carbon blacks of varying particle size and structure, the calculated, G, from the equation. 

It is shown below that p fulfills a scaling relation which involves the size and mass fractal dimension of the primary aggregates. Due to significant deviations of the solid fraction p from 1, the filler volume fraction of carbon black in rubber composites has to be treated as an effective one in most applications, i.e., 0eff=0/0p (compare [22]). 

Fast extrusion furnace black with a particle size of 360 A, was used to verify different theoretical concepts of percolation which by definition predicts a rapid change in conductance when volume fraction of conductive particles attains a critical value. Figure 15.38 shows the effect of a carbon black addition to polychloroprene. Up to 30 phr carbon black, the conductivity of poly chloroprene is almost constant and then it increases linearly as concentration of carbon black increases. The following equation applies o = o (P - P Jwhere c is constant, P is concentration of conducting particles, Pc is percolation threshold, and P is exponent which accounts for cluster size."" When data from the Figure 15.38 are replotted as in Figure 15.39 it is evident that the percolation law is valid. 

It was concluded that this large effect was due to a network of carbon black aggregates. This, of course, was the same as the conclusion of Payne who found that even when the rubber is replaced by an equal volume of oil, the very-low-strain modulus remains about the same [84]. Kraus [85] has proposed a theoretical treatment of the carbon-particle interaction network in which he concludes that the modulus should be proportional to 4>, where (j) is the volume fraction of carbon black. This has been confirmed by Arai and Ferry [81]. 

Interparticle contact is of critical importance to the behavior of lithium batteries. Most lithium-ion electrodes contain 2 to 15 wt% conductive filler, such as carbon black, in order to maintain contact among aU the particles of active material and in order to reduce ohmic losses in the electrodes. Presently, there are few models available for predicting contact resistance, and the effect of the weight fraction of conductive filler on the overall electronic conductivity of the composite electrode must be determined experimentally. Doyle et al. [35] demonstrate how the fuU-cell-sandwich model can be used to determine what minimum value of effective electronic conductivity is needed to make solid-phase ohmic resistance negligible. Then, one need only measure the effective conductivity of the composite electrode as a function of filler content, and one need not run separate experiments on complete cells to determine the optimum filler content. Modeling techniques for predicting effective electroitic conductivities of composite electrodes are under development, and hold promise to aid in optimizing filler shape and volume fraction [85]. 

At moderate strains, sufficient for dismption of the filler network, some modulus enhancement is still observed, due to strain amplification of polymer chains near the carbon black [73-75], This hydrodynamic effect is analogous to the viscosity increase of liquids upon addition of solid particles. The effect in carbon black can be described using the Einstein equation however, quantitative agreement with experimental results requires that a higher than actual volume fraction of filler be used in the calculation, presumably to account for occluded rubber [66,75]. Such... 

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