Polymer–filler interaction in nanocomposites: New interface area function to investigate swelling behavior and Young’s modulus Mithun Bhattacharya, Anil K. Bhowmick * Rubber Technology Center, Indian Institute of Technology, Kharagpur, West Bengal 721302, India article info Article history: Received 6 March 2008 Received in revised form 1 September 2008 Accepted 2 September 2008 Available online 12 September 2008 Keywords: Nanocomposites Polymer–filler interaction Interface area function abstract Polymer–filler interaction for nanocomposites was quantified by introducing Interface Area Function (IAF), to account for the nanofiller characteristics comprising of the specific surface area, correlation length and the filler volume fraction. IAF supplants the immeasurable filler characteristic terms, rendering tractability to the equation derived by considering the restraining forces acting on a nanofiller- elliptical platelet-embedded in polymer matrix. However, neglecting such terms reduces the same to Kraus’s equation. Recognition of the due importance of such filler characteristics, by introduction of IAF, resulted in better fitment of swelling data and also conformance with the trend predicted by Zisman’s interpretation of surface energy. Experimental values of Young’s modulus of natural rubber and styrene– butadiene rubber nanocomposites and those predicted by Guth–Gold and Halpin–Tsai equations for composites conform post-introduction of IAF, with mere 5–20% deviations. The accurate fitment of the resulting constitutive equations indicates suitable integration of the shape and aggregate effects. Ó 2008 Elsevier Ltd. All rights reserved. 1. Introduction The degree of crosslink density in vulcanized elastomeric gum compounds can be estimated by applying the Flory–Rehner network theory [1]. But their extrapolation into the domain of filled compounds presents certain theoretical and practical complica- tions, prime among them being the restriction to deformation in proportion with the sample dimension. These restrictive forces arise from the presence of the rigid inclusions engrafted in the rubber matrix. Bueche [2], Kraus [3] and Boonstra and Dannenberg [4] have separately studied this problem to come up with the understanding that although strong theoretical reasons cannot be assigned, the apparent crosslink density from equilibrium swelling data accounts for the polymer–filler interaction. This has established that swelling experimentation on filled compounds yields a measure of the physical crosslinking in elastomeric compounds. The observations of Lorenz and Parks [5] have led to the conclusion that the swelling of bulk rubber is essentially the same in both the filled and the unfilled compounds and the effect of the inclusion is predominantly felt in the interfacial region only. Within this region, the restriction to swelling is maximum due to adher- ence of rubber to filler by means of adsorption. On moving radially outwards from the filler, which is the epicenter of the restraining forces, the effect subsides and beyond a certain imaginary sphere of influence, the rubber swells to the same extent as the bulk of the gum compound. This swelling model was effectively utilized by Kraus [6] to quantize the effect on swelling exerted by bonded spherical particles in rubber matrices. Although the swelling in the case of polymer nanocomposites (PNCs) agrees in principle with this, it undervalues the importance of the extended interface offered by the change in filler shape and aggregate characteristics. Literature search indicates that the suit- ability of the Kraus plot to platelet-like nanofillers has not been investigated. In the present work, we evaluate the same and attempt to extend it to the domain of PNCs by incorporating certain modifications. The extra terms that evolve on application of ‘‘the swelling of the rubber shell’’ approach to the platelet filler are represented by an interface area function (IAF). This function maps the shape and aggregate effects, characteristic of nanofillers, to accurately determine the polymer–filler interaction in PNCs. Since the polymer–filler interaction has direct consequence on the modulus, the derived function is subjected to validation by intro- ducing the function in established models for determination of composite modulus. Some earlier reports dealing with the micromechanics of the intercalated or exfoliated polymer–clay nanocomposites [7–9] have attempted to understand the reinforcing mechanism of polymer- layered silicate nanocomposites. In our earlier communications [10–15], on clay and silica nanocomposites we have attempted to correlate the reinforcement with swelling behavior in terms of the volume fraction of rubbers. It appears that there is a gap in the * Corresponding author. Tel.: þ91 3222 283180; fax: þ91 3222 220312. E-mail address: anilkb@rtc.iitkgp.ernet.in (A.K. Bhowmick). Contents lists available at ScienceDirect Polymer journal homepage: www.elsevier.com/locate/polymer 0032-3861/$ – see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymer.2008.09.002 Polymer 49 (2008) 4808–4818