MODIFIED GUTH–GOLD EQUATION FOR CARBON BLACK–FILLED RUBBERS

ABSTRACT The modulus increase in rubbers filled with solid particles is investigated in detail here using an approach known widely as the Guth–Gold equation. The Guth–Gold equation for the modulus increase at small strains was reexamined using six different species of carbon black (Printex, super abrasion furnace, intermediate SAF, high abrasion furnace, fine thermal, and medium thermal carbon blacks) together with model experiments using steel rods and carbon nanotubes. The Guth–Gold equation is only applicable to such systems where the mutual interaction between particles is very weak and thus they behave independently of each other. In real carbon black–filled rubbers, however, carbon particles or aggregates are connected to each other to form network structures, which can even conduct electricity when the filler volume fraction exceeds the percolation threshold. In the real systems, the modulus increase due to the rigid filler deviates from the Guth–Gold equation even at a small volume fraction of the filler of 0.05–0.1, the deviation being significantly greater at higher volume fractions. The authors propose a modified Guth–Gold equation for carbon black–filled rubbers by adding a third power of the volume fraction of the blacks to the equation, which shows a good agreement with the experimental modulus increase (G/G0) for six species of carbon black–filled rubbers, where G and G0 are the modulus of the filled and unfilled rubbers, respectively; ϕeff is the effective volume fraction; and S is the Brunauer, Emmett, Teller surface area of the blacks. The modified Guth–Gold equation indicates that the specific surface volume ()3 closely relates to the bound rubber surrounding the carbon particles, and therefore this governs the reinforcing structures and the level of the reinforcement in carbon black–filled rubbers.