Barrier Properties of Polymer Nanocomposites † Lakshmi N. Sridhar* Department of Chemical Engineering, UniVersity of Puerto Rico, Mayaguez, Puerto Rico 00681-9046 Rakesh K. Gupta and Mohit Bhardwaj Department of Chemical Engineering, West Virginia UniVersity, P.O. Box 6102, Morgantown, West Virginia 26506 A computationally efficient strategy, based on a resistance-in-series and resistance-in-parallel approach, is developed for obtaining diffusivities when nanometer-sized flakes are introduced in a polymer matrix. The mass-transfer resistance is properly defined, and several examples are presented to demonstrate the reliability of this technique. One is also able to determine the optimum configuration to minimize the diffusivity by maximizing the resistance in the polymer matrix. The extent of diffusivity reduction with variables such as the filler loading level and filler geometry is explored and explained. Comparison is also made with a two- dimensional model where the Laplace differential equation is solved using a finite difference method, and agreement with experimental results demonstrates the effectiveness of this technique. Introduction Polymer nanocomposites (PNCs) formed by dispersing a few weight percent of nanometer-sized fillers, such as carbon nanotubes and montmorillonite (clay), in either thermoplastic or thermosetting polymers are now commercially available. Compared to neat polymers, PNCs have a tendency to have higher tensile and flexural moduli, improved barrier properties, and enhanced flame resistance. In terms of barrier properties, a reduction in the diffusion coefficient is thought to result simply from the increase in path length that is encountered by a diffusing molecule, because of the presence of an enormous number of (passive) barrier particles during mass transfer. This principle has been utilized to develop better tennis balls, improved packaging for juice and beer, and protective coatings for fuel and chemical tanks. 1 Here, we examine diffusion through PNCs in detail, and we seek to quantitatively relate the reduction in diffusivity to the morphology of the nano- composite. Diffusion through a heterogeneous, two-phase medium can be expected to be dependent on the properties of the individual phases, and the amount, shape, size, size distribution, and orientation of the dispersed phase, and early work on diffusion through arrays of spheres, cylinders, and ellipsoids has been summarized by Crank. 2 Barrer 3 studied diffusion through two types of heterogeneous media: (i) laminates, where layers of different properties are sandwiched together, and (ii) particulate composites, where discrete particles of one phase are dispersed in a continuum of another. Although the mass-transfer resistance has often not been explicitly defined in the literature, the series- parallel formulation to obtain the effective diffusivity has been proposed for both situations (i) and (ii) by several authors. In addition, Fidelle and Kirk 4 and Bell and Crank, 5 among other researchers, numerically solved the two-dimensional Laplace equation over a domain appropriate to case (ii) and compared the results with the expressions predicted by the series-parallel and parallel-series formulations. They found that the numerical results were bounded by the two analytical expressions. However, this approach does not seem to have been attempted for the situation where the filler is of nanometer size. Quantitatively predicting the reduction in diffusivity of a diffusing molecule in a polymer containing an impenetrable filler is an important problem that is of both fundamental and practical importance. On the fundamental side, Nielsen 6 considered two- dimensional diffusion through a polymer that contained infinitely long, rectangular-cross-section plates that were uniformly dispersed in the polymer but were placed normal to the direction of mass transfer. By calculating the maximum possible tortuosity factor, he determined that the largest possible ratio of the diffusivity of a molecule through the neat polymer (D 0 ) to that of the same molecule through the filled polymer (D) was given as where φ is the volume fraction of filler and R is the aspect ratio of the rectangle that forms the barrier cross section (the latter parameter is defined as R) w/(2t); see Figure 1). More recently, Cussler and co-workers 7-12 have studied the diffusion problem through flake-filled membranes and developed an extensive theory for predicting the changes in diffusivity, again as a function of the loading level and aspect ratio of the filler particles. Their basic equation is and the predictions of this equation can differ significantly from † This paper is submitted to the special issue of Industrial and Engineering Chemistry Research honoring Dr. Warren Seider on the occasion of his 65th birthday. This particular section is Dr. Sridhar’s tribute to Dr. Seider. Dr. Sridhar wishes to state that “...Dr. Seider has been a role model and a source of tremendous inspiration to me, not only as an awesome academician but also as a wonderful human being. In an atmosphere where grant chasing seems of primary importance, Dr. Seider has demonstrated the importance of teaching and imparting knowledge to both undergraduate and graduate students. This attitude is reflected in his textbook Process Design Principles, which enables undergraduate students to learn the principles of process design without too much aid from an instructor. Despite his academic achievements and a hectic schedule, Dr. Seider has always taken time to advise younger faculty like me on various academic and nonacademic issues. I therefore feel tremendously honored to be able to submit an article for this special issue.” * To whom all correspondence should be addressed. Tel: (787) 832- 4040. Fax: (787) 265-3818. E-mail: Sridhar_lakshmi@hotmail.com. D 0 D ) 1 + Rφ 2 (1) D 0 D ) 1 + R 2 φ 2 1 - φ (2) 8282 Ind. Eng. Chem. Res. 2006, 45, 8282-8289 10.1021/ie0510223 CCC: $33.50 © 2006 American Chemical Society Published on Web 02/18/2006