Journal of Nanomaterials

Volume 2015, Article ID 281308, 17 pages

http://dx.doi.org/10.1155/2015/281308

## Investigation of Theoretical Models for the Elastic Stiffness of Nanoparticle-Modified Polymer Composites

^{1}Norwegian Defence Research Establishment (FFI), P.O. Box 25, 2027 Kjeller, Norway^{2}Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, 0315 Oslo, Norway

Received 19 May 2015; Accepted 21 July 2015

Academic Editor: Abbas S. Milani

Copyright © 2015 T. Thorvaldsen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Mathematical models are investigated and suggested for the calculation of the elastic stiffness of polymer nanocomposites. Particular emphasis is placed on the effect on the elastic stiffness from agglomerates and the particle interphase properties. The multiphase Mori-Tanaka model and an interphase model are considered as two relevant models. These models only include and require the designation of a few system independent parameters with a clear physical meaning. Extensions of the models are also presented. The model calculations are compared to results from other models, as well as experimental data for different nanocomposites. For nanocomposites with spherical particles and with fiber-like particles, the suggested models are found to be the most flexible ones and are applicable to estimate the stiffness increase of nanocomposites for both low and high particle volume fractions. The suggested theoretical models can hence be considered as a general multiscale “model toolbox” for analysis of various nanocomposites.

#### 1. Introduction

Polymers are widely used in engineering applications, such as in adhesives and in fiber-reinforced composites, where it constitutes the continuous phase. Model analysis of lightweight materials is becoming more important in the design, development, and analysis of novel and complex structures. There is often a desire or need to improve the mechanical properties of the polymers, especially for advanced high-performance applications. The traditional approach has been to add particulate inclusions of microsize, such as inorganic or rubber particles, to the polymers. In recent years, however, a lot of effort has gone into investigating the effect of adding nanosized inclusions [1–3]. A range of nanosized inclusions with different shape and chemical composition has been investigated, such as nanoclay, nanosilica, carbon nanotubes (CNTs), and carbon nanofibers (CNFs). Since the required loading level of the nanoparticles in the polymer is much lower than that for inclusions of microsize, many of the intrinsic properties of the polymer will be retained after the addition of nanoparticles.

One mechanical property that is of significant interest is the nanocomposite elastic stiffness, that is, Young’s modulus of the nanocomposite. Most of the commonly used inclusions have a higher elastic modulus than the polymer itself and may therefore contribute to an increased elastic stiffness. The elastic stiffness of a nanocomposite material will, among others, depend on the type, orientation, and the volume fraction of the nanoparticles (assuming that the remaining constituents and the production process are kept constant).

The macroscopic nanocomposite material parameters are required in finite element (FE) modeling, among others. There may be a need for FE modeling of the nanocomposite itself or of a composite material where the nanomodified polymer constitutes the matrix of a continuous fiber-reinforced material. It would therefore be of interest to establish a set of mathematical models that can be used to calculate the macroscopic elastic stiffness of various nanoparticle-reinforced polymer composites.

For the models to be able to handle inclusions of various geometries, they must be flexible, while at the same time giving accurate predictions. In practice, the nanocomposite structure on the microlevel may be difficult to understand in detail. A detailed microstructural understanding may, however, not be required on a macroscopic level. Therefore, the mathematical models should have model parameters with physical meaning, without being directly system dependent, but they should still be sophisticated enough to give an accurate estimate and to predict the behavior observed in experimental testing.

The main aim of this work is to identify and establish flexible and system independent mathematical model tools for the prediction of the elastic stiffness of multiphase nanocomposites. The influence from the interphase around the nanoparticles and the influence from agglomerates are two effects that are of high interest to model for advanced and high-performance nanocomposites. Both effects will depend on the type of polymer, type of inclusion, loading level of the inclusion, and the processing conditions, and they thus have a practical interest for various nanoparticle/polymer systems. The main contribution from this work will therefore be more on defining a “model toolbox” rather than presenting new models for relevant nanocomposites. After the models are established and described, the model calculations are compared to experimental test data.

#### 2. Mathematical Models

##### 2.1. Nanoparticle/Polymer Characteristics

To actually increase, in this case, the elastic stiffness of a polymer material using particles of nanosize, several factors need to be considered and optimized. First of all, the nanoparticles are very small and therefore have a very large surface compared to larger particles; the surface area is inversely proportional to the particle size. The large surface area is, however, only available if the particles are well dispersed in the polymer. A good dispersion of the particles is believed to be a crucial requirement for obtaining a significant stiffness increase. If the particles form agglomerates, the inclusion phase will act more as defects/impurities in the homogenous polymer, which may instead reduce the bulk elastic stiffness [4, 5]. For nanocomposites with well-dispersed particles, significant increase of the stiffness (and also other properties, such as the toughness) is often explained by interphase effects. A precise description of the mechanical properties and behavior of the interphase is very challenging. Zhang et al. [6] describe these effects as the formation of a “three-dimensional physical network of interphase in a polymer matrix.” For the interphase, there may be other chemical and physical properties than in the bulk matrix, with differences in, among others, the polymer morphology and chain conformation [7], as well as the adhesive properties between the particle and the matrix. Several factors have been shown to affect the binding properties, such as the functionalization or surface treatment of the particles [8] and/or the choice of curing agent [9]. The stiffness of the composite is, furthermore, depending on the geometry and the orientation of the particles. Particles, where the size in one direction is significantly different from the others, such as for CNTs and CNFs, will introduce directional dependent changes in stiffness even for particles considered as isotropic. Also, composites with curved fiber-like particles (i.e., with waviness) have been reported to give a lower stiffness compared to composites with straight fiber-like particles [10].

With a relatively large set of properties, partly interrelated, that will affect and alter the elastic stiffness of a given nanoparticle/polymer system, a study purely based on experiments will soon become too complex, time consuming, and expensive. As mentioned above, the understanding and development of new nanocomposites will therefore benefit from having flexible and accurate mathematical models on macrolevel with only a few model parameters to vary. This set of models will then function as the multiscale mathematical “model toolbox” for understanding the behavior of the nanocomposites. It is, however, important to have a strong coupling to experimental test results for verification of the models.

##### 2.2. Modeling Approaches

Fisher and Brinson [10] refer to two main mathematical multiscale modeling approaches for nanocomposites, that is, the “top-down” and the “bottom-up” approach. In the “bottom-up” approach, one starts out with the atomistic structure of the nanoreinforcements and the matrix. Typical model techniques are quantum mechanics, molecular dynamics, and Monte-Carlo simulations [10, 11]. These simulations end up in fairly large systems to be solved for only a small part of the composite material, for example, a single CNT attached to a relatively short polymer chain. The output from these calculations should then be representative for the entire composite. Moreover, these simulations are very time consuming, even with large computing resources available.

The “top-down” approach, on the other hand, is based on continuum mechanics, where the polymer and the nano- and microinclusions are treated as continuum elements. The nanocomposite models often use short-fiber models and laminate theory [12] as a starting point. Equivalent models are then established for particles of smaller size. Additional effects, such as interphase effects, are often incorporated in the equivalent models, since these effects seem to be more significant when reducing the size of the particle. One main benefit of the “top-down” approach is that one is able to calculate for a larger part of the composite material (often referred to as a representative volume entity) to estimate the macroscopic properties of the materials.

Due to the size of the nanoparticle filler material, care must be taken when applying the continuum mechanics approach since the structure and interactions at the atomistic level are essential for a precise description of the mechanical properties. For this study, the “top-down” approach, employing continuum mechanics, is, however, assumed to be a valid approach.

##### 2.3. Analytical Models

Several analytical models for the elastic stiffness of nanocomposites are based on the pioneer work by Cox [13]. Rule of mixtures models specially designed for nanocomposites with nonspherical particle geometry have also been presented [14]. Moreover, to include a random orientation of the particles, models for weighting of the properties for the different orientations have been defined, for example [12]. Approaches employing laminate theory in describing the macroscopic properties of the composite have also been considered; see, for example, [15] and the references therein.

Most of these more “traditional” short-fiber based models assume a perfect dispersion of the particles in the matrix, that is, one inclusion phase in addition to the matrix phase, and a no-slip boundary condition at the particle-matrix interface. Moreover, the traditional models are often developed for a specific nanocomposite, that is, one type of nanoparticle with a given geometry, embedded in a given polymer system. The model parameters are then specific to the particular nanoparticle/polymer system and may not be transmissible to other nanocomposites. More flexible models with less system dependent parameters are therefore appropriate to establish.

In the following, a set of more flexible models for nanocomposites are described: the Mori-Tanaka approach and an interphase model. For comparison, the Halpin-Tsai equation and a slip/no-slip model are also included since these latter models are often applied in comparison with experimental data. It should also be mentioned at this point that expressions based on the Mori-Tanaka method have been established for specific nanocomposites and applied for comparison with experimental results [16–22], but, again, these expressions are not as flexible as the more general multiphase model.

###### 2.3.1. The Halpin-Tsai Equation

For aligned particles, the Halpin-Tsai equations, which are based on empirical data, are often applied [23–26]. For the special case of spherical particles, these equations are reported in the literature to give a reliable estimate for the stiffness properties of the nanocomposite and are often applied for comparison with experimental test results. The modified version of the Halpin-Tsai equations has also been presented, including an orientation factor that takes into account the randomness of discontinuous fibers.

The modified version of the Halpin-Tsai equation for the* longitudinal* elastic modulus can be expressed as [27, 28]where in the above expression is the elastic stiffness of the composite, and are the elastic stiffness of the inclusions and the matrix, respectively, and is the aspect ratio, being the length and being the diameter of the particle. In the original Halpin-Tsai equation, equals unity. For composites with a three-dimensional random orientation of fiber-like particles (where ), is used.

###### 2.3.2. The Mori-Tanaka Method

To include more than one inclusion phase, for example, the combination of dispersed nanoparticles and voids/agglomerates, or a second type of particle with other elastic properties and/or a different geometric shape, more general multiphase models are established. The Mori-Tanaka method is one such attempt [29–31], which also has been reported to agree well with experimental results; see, for example, [10] and the references therein. Different model variants based on the Mori-Tanaka approach can be found in the literature; see, for example, [32] for a review.

The composite elastic stiffness in the Mori-Tanaka model for a multiphase composite with* unidirectionally aligned inclusions* can generally, following the derivation in [10], be expressed aswhereIn the above expressions, phase 0 is the continuous and homogeneous matrix phase, and phases 1 to are the inclusion phases. Moreover, is the volume fraction of the matrix phase, contains the stiffness properties of the matrix phase, is the volume fraction of the th inclusion phase, and contains the stiffness properties of the th inclusion phase. The quantity is the identity matrix and is the (second-order) Eshelby tensor for the th inclusion phase [30, 31]. For composites where the particles can be modeled as having a spheroidal shape (e.g., spherical, oblate, or prolate shape) and included in a homogeneous infinite matrix, the Eshelby tensor is constant. Expressions for the Eshelby tensor for relevant spheroidal inclusion geometries can be found in the literature, for example, [18, 22, 33].

For the special case of a two-phase composite, including the matrix (phase 0) and the nanoparticles (phase 1), the stiffness relation in (2) can be expressed aswhere now indicates the matrix phase and indicates the inclusion phase. Furthermore, is the volume fraction of the matrix, is the volume fraction of the inclusions, is the stiffness matrix of the matrix, is the stiffness matrix of the inclusions, is the identity matrix, andwhere is the (second-order) Eshelby tensor for the inclusion phase.

The Mori-Tanaka model for a multiphase composite with* randomly oriented inclusions* can in a similar way be expressed aswhere the curly brackets indicate orientational averaging of all possible orientations, as described and shown in [10] (and also in [34] for another orientation of the local axes of the inclusion compared to that used in [10]). In the same way as for aligned inclusions, the elastic stiffness may for a two-phase composite be written asFor the special case of a three-phase composite, the contribution from the second inclusion phase is added to the expressions for aligned and random oriented composites, respectively.

An extension of the above model, also taking into account the curvature (waviness) of the fiber-like particles, has been included by performing a finite element method model calculation for a single curved fiber-like particle surrounded by matrix [10, 35]. Curved fiber-like particles have a reduced stiffness contribution to the composite compared to straight fiber-like particles.

###### 2.3.3. Interphase Models

As for the rule of mixtures and short-fiber composite expressions described above, the Mori-Tanaka method assumes a no-slip condition at the interface between the particles and the surrounding matrix. Such an interface interaction condition may not be a correct description for all nanoparticle/polymer systems.

One approach for allowing a slippage at the interface is presented by Lewis and Nielsen [36] and McGee and McGullough [37]. In their model, the generalized Einstein coefficient () is altered for the “no-slip” and “slippage” conditions. This model parameter, however, has to be estimated for each nanoparticle/polymer system, which makes the model less flexible for estimating the elastic stiffness of different systems.

As an alternative approach, it can be assumed that the binding characteristics can be expressed in the form of an elastic stiffness for the interphase surrounding the particles, that is, a “second matrix phase.” The stiffness will typically vary (radially) through the interphase, with a smooth transition from the particle surface to the surrounding bulk polymer. As pointed out by Fornes and Paul [7], when employing a continuum mechanics approach, one should assume that each constituent material in the composite does not affect or alter the properties of the other constituents. The interphase should thus be defined as a geometrically well-defined region with given elastic properties. A high interphase elastic stiffness indicates good bonding and less flexibility to deform when loaded, whereas a lower inclusion phase elastic stiffness indicates a weaker bonding with more flexibility to deform when loaded.

Measurements of the interphase elastic properties have been done, for example [38]. Since it may be difficult to perform stiffness measurements of the interphase without including the stiffness of the particle itself, molecular model calculations have instead been applied for estimating the interphase properties, for example [39]. Furthermore, the interphase properties may vary as a function of the volume fraction of nanoparticles in the composite. This change in interphase properties may thus indicate a variation in the thickness of the interphase surrounding the particles. Alternatively, the variation of the properties in the interphase can be due to variation in elastic stiffness. Tuning the elastic stiffness and the interphase thickness can therefore give an adequate description of the interphase properties and behavior. The interphase properties are discussed further in Section 4.1.

An effective interphase model has been presented by Odegard et al. [39, 40]. In this case, a no-slip condition at the interface between the particle and the interphase and between the interphase and the bulk matrix has been assumed. However, the interphase itself introduces the wanted flexibility. Note that Odegard et al. refer to their model as an “effective interface model.” To the authors’ knowledge and understanding, the* interface* is the surface of the particle being in contact with its surroundings. The* interphase*, on the other hand, is the polymer matrix region surrounding the particle. A sketch is given in Figure 1 for a spherical particle with radius , surrounded by an interphase with thickness , . This latter terminology is also in accordance with Fisher and Brinson [10] and will be employed in this paper.