Modelling stiffness of polymer/clay nanocomposites K. Hbaieb a, * , Q.X. Wang b , Y.H.J. Chia a , B. Cotterell a a Institute of Materials Research and Engineering (IMRE), Materials Science and Characterization Laboratory, 3 Research Link, Singapore 117602, Singapore b Institute of High Performance Computing (IHPC), 1 Science Park Road, #01-01 The Capricorn, Singapore Science Park II, Singapore 117528, Singapore Received 4 August 2006; received in revised form 27 November 2006; accepted 27 November 2006 Available online 8 January 2007 Abstract Aligned nanoclay particles can be distributed randomly in a polymer matrix even at high volume fractions, but randomly oriented particles cannot be randomly distributed at high volume fractions. Instead a nanocomposite where there are clusters of nearly aligned particles is obtained. The clusters of nearly aligned particles form an effective particle with lower aspect ratio. This phenomenon which produces a nanocomposite of less stiffness than might have been expected has implications for the processing of nanoclay polymer composites. It is shown by comparing two-dimensional to three-dimensional finite element studies that the two-dimensional model, often used because it is simpler, does not accurately predict the stiffness. The MorieTanaka model is shown to give a reasonably accurate prediction of the stiffness of clay nanocomposites whose volume fraction is less than about 5% for aligned particles but underestimates the stiffness at higher volume fractions. On the other hand for randomly oriented particles the MorieTanaka model overestimates the stiffness of clay nanocomposites. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Finite element; MorieTanaka; Stiffness 1. Introduction Polymer/clay nanocomposites are polymeric materials that are reinforced by nanoclay particles whose dimensions are in the sub-micron scale; the particles are composed of stacks of w1 nm thick mono-layers whose in-plane dimensions range from 100 nm to 1000 nm. The thickness of the stacks depends upon how well they are intercalated or exfoliated. For en- hanced functional properties of nanocomposites, full exfolia- tion is desired. The Toyota group [1e3] was the first to achieve successful exfoliation of clay in nylon 6 through in situ polymerization. They have shown that inserting as little as 4.7 wt% clay into nylon 6 doubles both elastic modulus and strength. However, it is the functional properties of nanocomposites that are the main driving force in nanocomposite development. Functional properties such as barrier [4e6], flammability resistance [7] and ablation performance [8] are all greatly improved by the addition of small volume fractions of nanoclay. To find applications for this new class of materials their mechan- ical properties have to be sufficient to ensure mechanical reliability. The established mechanics-based composite stiffness models, such as the MorieTanaka (MeT) [9e12] and the HalpineTsai [13e15], are only dependent on the volume fraction, aspect ratio of the particles and the elastic constants of both matrix and particles. The particle size will not affect the stiffness unless the particles affect the structure and stiffness of the ad- jacent polymer. Such an effect may be present if the polymer is semicrystalline, since the particles may affect the orientation of the lamellar crystallites to give a transcrystalline layer. However, even if there is a transcrystalline layer adjacent to the clay particles, Sheng et al. [16] have shown that the effect is slight. * Corresponding author. Tel.: þ65 6874 7168; fax: þ65 6774 4657. E-mail address: hb-kais@imre.a-star.edu.sg (K. Hbaieb). 0032-3861/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymer.2006.11.062 Polymer 48 (2007) 901e909 www.elsevier.com/locate/polymer