428 Nanoscale Res Lett (2009) 4:420–430 Fig. 6 Tensile stress-elongation plot of 4 phr of different SBS gels ﬁlled NR samples thereby increasing the overall tensile strength of gel ﬁlled rubber matrix. In order to understand the reinforcement mechanism of these nanogels in neat elastomer matrix, tensile properties of gel ﬁlled systems were analyzed in detail with the help of various particulate reinforcement models. Normally, intro- duction of particulate ﬁllers in a rubber matrix leads to an increase in modulus of the composite material. This is due to the fact that modulus of inorganic particles is usually much higher than that of the polymer matrices; as a result the composite modulus is easily enhanced by adding particles to matrix. Many empirical or semi-empirical equations have been proposed to predict the modulus of particulate– polymer composites. Smallwood [28] introduced, for the ﬁrst time, the following equation, using an analogy to the Einstein viscosity equation, viz., Ec ¼ Emð1 þ 2:5UÞ where Ec and Em are Young’s modulus of composite and matrix, respectively and U is the volume fraction of the 4 ð Þ Fig. 7 Comparison between experimental and theoretical Young’s modulus values for SB1 gel ﬁlled NR and NS1 gel ﬁlled SBR systems as determined by a Guth model and b Kerner model 123 5 ð Þ 6 ð Þ ﬁllers. The constant 2.5 is applicable for spherically shaped particles. Later, Guth [29] modiﬁed the above equation by taking into account the polymer–ﬁller interaction, they proposed the following equation, Ec ¼ Em 1 þ 2:5U þ 14:1U2 where the linear term is the stiffening effect of individual particles and the second power term is the contribution of particle–particle interaction. Another deﬁnitive equation for determining the modulus of a composite that contains spherical particulate inclusions in a matrix was proposed by Kerner [30] and is given below: EC=Em ¼ 1 þ ð1 / 15ð1 /Þ ð8 tmÞ 10tmÞ where tm is the matrix Poisson ratio taken as 0.5 here. The equation is based on the assumption that the Young’s modulus of the particulate inclusions (Ef) is greater than that of the matrix (i.e. Ef Em). In the present case, the Young’s moduli of the nanogel ﬁlled elastomers are compared with the calculated theoret- ical values following the Guth and Kerner reinforcement models. These are presented in Fig. 7a–b. It is apparent that the nano SBR gel ﬁlled NR systems show reasonable ﬁtting with both the models, particularly with Guth model. How- ever, in the case of NR gel ﬁlled SBR systems, the experimental data deviate considerably from their calculated counterpart. This anomaly can be explained by taking the Young’s moduli of the gels into consideration. In all par- ticulate reinforcement theories, it is assumed that there is a great difference in the respective Young’s modulus values of particulate ﬁller and neat matrix. However, in the case of present systems, sulfur crosslinked nanogels have been used which are partially deformable and their moduli are mar- ginally higher than that of the virgin polymer. Because of the relatively large difference in modulus values between SB1 nanogels (3.42 MPa) and neat NR (0.7 MPa), SB1 gel ﬁlled NR systems show better matching with theoretical values.