Elsevier

Polymer

Volume 48, Issue 3, 26 January 2007, Pages 901-909
Polymer

Modelling stiffness of polymer/clay nanocomposites

https://doi.org/10.1016/j.polymer.2006.11.062Get rights and content

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 Mori–Tanaka 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 Mori–Tanaka model overestimates the stiffness of clay nanocomposites.

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 ∼1 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 enhanced functional properties of nanocomposites, full exfoliation is desired.

The Toyota group [1], [2], [3] 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 [4], [5], [6], 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 mechanical properties have to be sufficient to ensure mechanical reliability.

The established mechanics-based composite stiffness models, such as the Mori–Tanaka (M–T) [9], [10], [11], [12] and the Halpin–Tsai [13], [14], [15], 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 adjacent 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.

Finite element analyses of composites containing high aspect ratio plate-like particles, although accurate, are not suitable as a general method for calculating the stiffness because of their complexity. The M–T model has a good theoretical basis since it is based on the equivalent inclusion model of Eshelby [17], [18] and is generally agreed to be superior to the Halpin–Tsai model particularly for composites with high aspect ratio particles [19]. However, the M–T model, though an improvement on the dilute particle concentration model of Eshelby, becomes less accurate at high particle volume fractions where there is considerable particle interaction. It is the purpose of this paper to explore the limits of the M–T model by comparing its stiffness predictions with finite element analyses.

A stiffness model does not only depend on the accuracy and robustness of the technique used, but also on the accuracy of the elastic constants. However, these constants are not known very precisely. Even in reasonably well-exfoliated clay nanocomposites the platelets can be made up of a number of intercalated silicate sheets and the spacing of the silicate sheets affects the Young's modulus and weight/volume relationship of the effective particle. The Young's modulus is a continuum parameter; when the clay exists in a single sheet, its stiffness in terms of the force–strain relation can be estimated from its structure, but the assignment of an equivalent thickness to the sheet so that the concept of stress–strain relationship can be used is not straightforward. The distribution of the aspect ratio of the clay particles is usually wide and so the appropriate aspect ratio for any model is uncertain. Moreover, the distribution of clay particles is far from uniform. The clay particles are usually not fully dispersed so that in typical epoxy/clay nanocomposites there are clusters of high particle concentration dispersed in a matrix of low particle concentration. The stiffness of clustered composite is less than that of the same volume fraction of particles that are uniformly dispersed [20]. Thus the accuracy of the modelling of the stiffness of the nanocomposite is crucially dependent on the properties of the effective particle. Since it is difficult to accurately determine the properties of the effective particle, the apparent accuracy obtained by using a finite element analysis is largely illusory. Thus it is preferable to use an analytical model, such as the M–T model, provided it is reasonably accurate.

Three-dimensional finite element models (FEMs) of nanocomposites containing plate-like particles are difficult especially if they are randomly oriented and many researchers such as Sheng et al. [16] have used plane strain two-dimensional FEMs. However, it will be shown in this paper that two-dimensional FEMs predict a Young's modulus that differs significantly from that obtained with three-dimensional FEMs and should not be used as a basis for deciding whether analytical models are sufficiently accurate to be used for nanocomposites. Gusev [21] has used a three finite element based approach to model a composite reinforced by fibers of different shape, size and distribution. A range of composite material properties, mainly those that are governed by Laplace's equation such as dielectric constant, but also including the elastic constants were modelled [21]. In the elastic example examined, the finite element results were compared with the Halpin–Tsai [13], [14], [15] model (here almost identical to the M–T predictions). For a particle volume fraction of 3%, the Halpin–Tsai considerably overestimated the stiffness for aspect ratios greater than about 20.

Section snippets

Finite element model

Both two-dimensional and three-dimensional finite element models are presented for aligned and randomly oriented clay particles which are randomly distributed. To avoid overlong computational times the representative volume element (RVE) must be reasonably small. A periodic RVE is often used, where the particles that are cut by any of the edges (or faces) of the RVE are continued from the opposite edges (or faces) with the same orientations. The parts of particles that intersect the boundary

Mori–Tanaka model

The M–T model, based on the equivalent inclusion of Eshelby [17], models the clay particle as an oblate spheroid, whereas in the finite element model we have assumed that the particles have constant thickness. Steif and Hoysan [23] have defined an elastic reinforcement factor by:Ep/Em=1+λvp,where Ep is the particle elastic modulus, Em is the matrix elastic modulus and vp is the particle Poisson's ratio. They compared the elastic reinforcement factor, λ of cylinders and ellipsoids albeit for a

Aligned particles

Finite element simulations are carried out for the aligned particles for 2D and 3D configurations. Fig. 5, Fig. 6 show the finite element results of the composite elastic modulus, Ec, normalized by the matrix elastic modulus, Em, as a function of the volume fraction of clay particles and the predictions of the M–T model for particle aspect ratios of 100 and 50, respectively. As shown in Fig. 6, the 2D FEM results are consistently lower than those for 3D, hence testing the accuracy of the M–T

Discussion

A 3D configuration should be used to model any property of a composite. However, some researchers have used 2D models to approximate the behaviour of composites. We have shown by performing both 2D and 3D FEMs that the 2D FEM cannot be used to accurately predict the stiffness of a nanoclay polymer composite. As we have demonstrated that 2D results are consistently lower than 3D results, we believe that the better agreement of 2D FEM results with experimental results as compared to M–T

Conclusions

Two-dimensional models do not predict the elastic modulus of real polymer/clay nanocomposites accurately.

If the particles are aligned the M–T model will accurately predict the elastic modulus up to volume fractions of about 5% but will underestimate the elastic modulus at higher volume fractions.

Fully exfoliated randomly oriented polymer/clay nanocomposites cannot be processed at high volume fractions but clusters of particles with nearly the same alignment form. The elastic modulus of such

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