CHAPTER 4 MODELING OF MECHANICAL PROPERTIES OF POLYMER COMPOSITES

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1 CHAPTER 4 MODELING OF MECHANICAL PROPERTIES OF POLYMER COMPOSITES 4. Introduction Fillers added to polymer matrices as additives are generally intended for decreasing the cost (by increase in bulk) of the in addition to improving some of the physical and mechanical properties of the material. Filler, a chemically inert material can decrease properties like linear expansion coefficient, lubricancy, toughness and elongation percent, while properties like viscosity, thermal conductivity, stiffness, hardness, flame resistance and yield strength increase. Fillers can be organic and inorganic. Non-metallic materials like talc, calcium carbonate, metallic fibers and silica belong to inorganic category while natural fibers (organic origin), wood flour and synthetic fibers are organic fillers. Geometrically, fillers may be spherical, sheet, needle form and fibrous. Filler shape, type and its concentration can influence the mechanical properties of the polymer to which it is added. Particles with non-spherical shape lead to have more viscosity and manifest in several problems during composite shaping (Valavala and Odegard, 2005). Traditionally, composites were reinforced with micronsized inclusions. Recently, processing techniques have been developed to allow the size of inclusions to go down to nanoscale (Jordan, et al., 2005). Although experimental based research can ideally be used to determine structure-property relationships of nanostructured composites, experimental synthesis and characterization of nanostructured composites demands the use of sophisticated processing methods and testing equipment; which could result in exorbitant costs. To this end, computational modeling techniques for the determination of mechanical properties of nanocomposites have proven to be very effective (Liu & Chen, 2003; Sheng, et al., 2004; Chen & Liu, 2004). Computational modeling (CM) for predicting the mechanical properties of composites is highly desirable. Computational modeling of the mechanical properties of polymeric nano-composites for design and development of nanocomposite structures for engineering applications has been suggested by Valavala and Odegard, (2005). The modeling of polymer-based nano-composites has become an 8

2 important topic in recent times because of the need for the development of these materials for engineering applications. It was found that deal with modeling and offering appropriate formulations for the estimation of mechanical properties of polymeric composites. Different modeling theories and equations for describing the reinforcing mechanisms of polymer matrix by additives (like fillers) have been suggested that are discussed in this chapter. 4.2 Overview of the Modeling Method The importance of modeling in understanding of the behavior of matter is illustrated in Figure 4.. The earliest attempt to understanding material behavior is through observation via experiments. Careful measurements of observed data are subsequently used for the development of models that predict the observed behavior under the corresponding conditions. The models are necessary to develop the theory. The theory is then used to compare predicted behavior to experiments via simulation. This comparison serves to either validate the theory, or to provide a feedback loop to improve the theory using modeling data. Therefore, the development of a realistic theory of describing the structure and behavior of materials is highly dependent on accurate modeling and simulation techniques. Valavala and Odegard (2005) have presented a review of modeling techniques for predicting the mechanical behavior of polymeric nano-composites. It is proposed that mechanical properties of nano-structured materials can be determined by a select set of computational methods. Details Computational Chemistry and Computational Mechanics modeling are presented. Computational Chemistry techniques are primarily used to predict atomic structure using first-principles theory, while Computational Mechanics is used to predict the mechanical behavior of materials and engineering structures. To facilitate the development of nano-structured polyamide composite materials for this purpose, constitutive relationships must be developed that predict the bulk mechanical properties of the materials as a function of the molecular structure of the polyamide and reinforcement. These constitutive relationships can be used to influence the design of these materials before they are synthesized. Even though it has been shown that these materials have the potential to have excellent mechanical properties, the relatively 9

3 high costs of development and manufacturing of nano-polymer composites has been prohibitive. A lower cost approach is the use of particles in the polymer. Figure 4. The schematic of developing theory and validation of experimental data through simulation. Vollenberg and Heikens (989) explained that if there is a strong interaction between the polymer and the particle, the polymer layer in the immediate proximity of the particle will have a higher density. For most systems, density is proportional to elastic modulus, so the region directly surrounding the inclusions will be a region of high modulus. The polymer right outside this high modulus region will have a lower density due to the polymer chains that are moved towards the particle. For large particles, the size of the low density region will be relatively large, and the contribution of the high modulus filler will be diminished. For nano-particles, the number of particles for a given volume fraction is much larger, thus the particles will be much closer to one another. If the particles are densely packed, the boundary layer of polymer at the interface will comprise a large percentage of the matrix and can create a system where there is no space for a low modulus region to form. This results in the elastic moduli of composites with smaller particle (nano) size being greater than the moduli of composites with larger inclusions (Jordan, et al., 2005). The small inter-particle distance in nano-composites was used as another parameter to explain the changes in the elastic modulus and strength of these materials compared to the composites with micron-sized particles. 20

4 The same parameter also plays a role in the glass transition temperature changes observed in nano-composites versus composites with micron-sized reinforcement. Ash, et al. (2002) found that for their system the glass transition temperature was constant until around 0.5% weight fraction of particles, and then had a sharp drop, and then it remained constant for weight fractions above %. When there is little or no interfacial interaction between the filler and matrix and the inter-particle distance is small enough, the polymer between two particles acts as a thin film, and for thin films, the glass transition temperature decreases with the drop of film thickness. The distance between particles in a composite with the filler weight fraction below 0.5% is relatively large, and in such case the polymer between each particle is not considered to belong to the thin film regime. As the filler concentration increases, the inter-particle distance and the resulting thickness of the film decrease. This theory, however, does not explain why the glass transition temperature levels off rather than continues to drop as a function of increasing weight fraction of the filler. The behavior of polymeric nano-composite systems are shown in the Table 4.. From the viewpoint of fundamental laws, reinforcing effect of nano-particles is dependent on Aspect Ratio and also on the interactions between fillers particles and polymeric matrix (Jordan, et al., 2005). Figure 4.2 shows typical tensile stress displacement curves for PVC and its nano-composites. Pure PVC is brittle, however, when CaCO 3 nano-particles are added in the PVC matrix, the composites show ductile behaviors, such as stress whitening and necking. Their Young s modulus, tensile yield strength and elongation at break are calculated and plotted in Figures 4.3 to 4.5. In Figure 4.3, Young s modulus of the PVC/CaCO 3 nano-composites is observed to increase with the loading of CaCO 3 nano-particles up to 5 wt% and then decrease marginally at 7.5 wt%. These results confirm that the CaCO 2 nano-particles do stiffen PVC. The yield strength of the nano-composites is plotted in Figure 4.5 and is broadly independent of nano-particle loading (Xie, et al., 2004). 2

5 Table 4. Behavior of polymer nano-composite (Jordan, et al., 2005). Attribute Crystalline State Amorphus State Elastic modulus Increase w/volume fraction Increase or no change with decrease of size Increase w/volume fraction Increase w/decrease size Greater increase than for good interaction Yield stress/strain Ultimate stress/strain Density/volume Strain-to failure T g Crystallinity Viscoalasticity Increase w/volume fraction Increase w/decrease size Decrease with addition of particles Increase w/ decrease size No unfilled result for change in V f Lower than pure for small volume fraction Increased volume as size decreases N/A Decrease with addition of particles Decrease with addition of particles Decrease with addition of particles N/A No major effect No major effect Increase w/volume fraction Increase w/decrease size N/A Increase w/volume fraction Increase w/decrease size Increase w/volume fraction Increase w/decrease size N/A Decrease with addition of particles Nano>micro after 20% weight Decrease with addition of particles Increased volume as size decreases N/A increase with addition of particles increase with addition of particles Increase w/decrease size Level until 0%, drops off level -0 N/A N/A Increase w/volume fraction nano less Decrease with addition of particles Figure 4.2 Tensile stress - displacement curves for PVC and nano-composites 22

6 Figure 4.3 Variation of Young s modulus of nano-composite with nano-particle loading. Figure 4.4 Variation of tensile yield strength of nano-composite with nano-particle loading. 4.3 Theoretical Developments 4.3. Guth s Equation and Nicolais-Narkis Theory The modulus and yield strength of particle-filled composites can be predicted by Guth s equation (Equation 4.) and Nicolais-Narkis theory (Equation 4.2), respectively (Bliznakov, et al., 2000): E C = E m φ f + 4. φ f 2 (4.) 23

7 σ yc = σ ym.2 φ f 2/3 (4.2) Where, E and σ y are Young s modulus and yield strength, respectively; subscripts m, f, and c denote matrix, filler and composite, and Φ f is volume fraction of particles. It is quite obvious, that the experimental modulus (except at 7.5 wt% loading) and yield strength values are larger than the predicted values, as shown in Figures 4.3 and 4.4. These results indicate the limitations of the theories when applied to nano-composites. The strong interaction between CaCO 3 nano-particles and PVC matrix caused by the large interfacial areas has led to much higher elongations-at-break, with a maximum at 5 wt% nanoparticles, as shown in Figure 4.5 (Xie, et al., 2004). Figure 4.5 Variation of Charpy notched impact energy of nano-composites with nanoparticle loading (Xie, et al., 2004). The specific surface area of filler gives important information about the filler reinforcement properties. In his review paper on filler-elastomeric interactions, Wang (998) notes that the specific surface area of carbon black directly affects dynamic mechanical properties. When the surface area for silica increases, there is a higher compound viscosity which requires more energy (torque) for mixing and also contributes to the buildup of heat in the mixture. If too much heat is generated the compound will cure prematurely. At the 24

8 same time, as the filler surface areas increases the tensile and tear strength of the rubber increases. Much is still unknown concerning the mechanisms of how these properties directly relate to reinforcement and what the exact mechanisms of reinforcement are (Jordan, et al., 2005) The Mixture Rule The field of composite material behavior can be studied from two perspectives: micromechanics and macromechanics. The goal of most micromechanics approaches is to determine the elastic moduli or stiffness of a composite material in terms of the elastic properties of the constituent materials. Most of the analytical models presented presume the idealization that there is perfect adhesion between the phases and that the particles are spherical and evenly dispersed. Some of the earlier attempts in modeling composites were performed by Einstein and Guth. Guth and Smallwood extended Einstein s theory to explain rubber reinforcement. Both of these attempts have proved to be applicable, but only at low concentrations of particulate. Thus, the focus will be on the newer works, separated into two approaches, defined as either a mechanics of materials or an elasticity approach. In the mechanics of materials approach some simplifying assumptions are made, the most significant of which is that the strain in the matrix is equal to the strain in the particulate. With this assumption the most simplistic of all methods of predicting the moduli of a composite, known as the rule of mixtures, can be obtained (Haghighat, et al., 2005): E c = E f φ f + E p φ p (4.3) E c, E f, and E p, are the elastic modulus of compound, filler and polymer matrix, respectively. φ f is the volume fraction of filler and φ p is the volume fraction of polymer matrix. The mixing law represents the linear relationship of elastic modulus in which the effects of size, shape and particle distribution has been neglected. Generally, the law of mixtures has been considered as the upper limit of elastic modulus. The absolute lower bound on elastic modulus can be obtained, assuming equal stress in the matrix and particulate (Haghighat, et al., 2005; Zhang, et al., 2003): 25

9 E c = φ f + φ (4.4) p E f E p The upper and lower limits on elastic moduli represent the most widely used relationships derived through material mechanics approach. Although other expressions have also been proposed using various assumptions, the elasticity approach has received maximum attention (Haghighat, et al., 2005). Two of the most important models which are now applied for the nano-composites are proposed by Hashin and Halpin-Tsai model (Haghighat, et al., 2005). Although, the calculated values resulting from these models do not accurately predict the mechanical properties, these may be considered as an estimation of elastic modulus for nanocomposite samples. In the following, the related equations for these models have been discussed Hashin-Shtrikman Model The composite spheres model, introduced by Hashin, consists of a graduation of sizes of spherical particles fixed in a continuous matrix phase. In line with this model, Hashin and Shtrikman developed the bounds for the shear and bulk modulus (Hashin, 962; Wang and Pyrz, 2004; Haghighat, et al., 2005). The resulting bounds on the Young s modulus are as follows: Lower bound for the Bulk moduli, K l : K l = K + φ 2 (4.5) 3φ + K 2 K 3K + 4G The upper bound on K u is, K u = K 2 + φ (4.6) 3φ + 2 K K 2 3K 2 + 4G 2 Similarly, the lower bound of the Shear moduli, G l : G l = G + φ 2 G 2 G + 6(K + 2G )φ 5G (3K + 4G ) (4.7) 26

10 and the corresponding upper bound: G u = G 2 + φ G G 2 + 6(K 2 + 2G 2 )φ 2 5G 2 (3K 2 + 4G 2 ) (4.8) The resulting bounds on the Young s modulus are following equations: Lower bound: E l = 9Kl G l 3K l + G l = 9 k + 3 k + φ 2 k 2 k + k 2 k + φ 2 3φ G + 3k + 4G 3φ + G + 3k + 4G φ 2 G 2 G + 6(k + 2G )φ 5G (3k + 4G ) φ 2 G 2 G + 6(k + 2G )φ 5G (3k + 4G ) (4.9) Upper bound: E u = 9Ku G u 3K u + G u = 9 k k 2 + k k 2 + k k 2 + φ φ 3φ G k 2 + 4G 2 3φ + G k 2 + 4G 2 φ G G 2 + 6(k + 2G )φ 5G 2 (3k 2 + 4G 2 ) φ G G 2 + 6(k + 2G )φ 5G 2 (3k 2 + 4G 2 ) (4.0) In the above equation, subscripts and 2 refer to the polymer and the filler respectively. It may be worth mentioning that the above equations are applicable when K <K 2 and G <G 2 (Wang and Pyrz, 2004; Haghighat, et al., 2005). 27

11 4.3.4 Halpin-Tsai Model The semi-empirical equations suggested by Halpin-Tsai are yet another way for predicting the composite properties, equations 4. and 4.2: Where E = E ( + ζηφ 2 ) ηφ 2 (4.) η = E 2 E E 2 E + ζ (4.2) These equations result from simplification and approximation of micromechanical complex models. The importance of this method lies in its simplicity and its ability for generalization. The only problem in this method is related to the parameter of ζ, which is best determined experimentally. Generally, this parameter for particulate composites has been approximated to have a value of 2 for matter properties (Haghighat, et al., 2005). Some other mechanical relationships for the Young s modulus and Poisson s ratio of composites have been given by Budiansky (965) as follows: E = υ = 9KG 3K + G 3K 2G 6K + 2G (4.3) (4.4) and for each constituent K n = E n 3 6υ n (4.5) G n = E n 2 + 2υ n (4.6) Where, n = or 2 (denoting continuous and filler phases respectively). 28

12 4.3.5 Micro-mechanics Models It is well known that most composite materials are anisotropic/transversely isotropic. Many composite analyses are performed using a macroscopic approach where the properties of the composites are homogenized to produce an anisotropic, yet homogeneous continuum before the analysis is conducted. However, the micromechanical approach to analyzing composites considers the filler and the matrix separately and applies the loading and the boundary conditions at the individual filler and matrix level. The overall properties of the composite are developed by relating the average stresses and strains. Thus the micro-mechanical approach is expected to provide much more detail into the true interactions between the filler and matrix, potentially leading to a more accurate model of the composite behavior (Gardner, 994). Two continuum-based micro-mechanic models are briefly described in the following sub-sections Mori-Tanaka Model The Mori-Tanaka approach, which is based on the Eshelby Tensor, can be used to predict the elastic properties of two-phase composites (matrix and effective particle phases) as a function of the effective particle volume fraction and geometry and they may be perfectly bonded to each other (Mori & Tanaka, 973; Benveniste, 987). The Mori- Tanaka approach has been used to accurately predict overall properties of composites when the reinforcements are on the micrometer-scale level, or higher (Odegard, et al., 2005). At these higher length scales, the assumption of the existence of two phases is apparently acceptable. However, for nanometer-sized reinforcement, it has been shown that the molecular structure of the polymer matrix is significantly perturbed at the reinforcement/polymer interface, and this perturbed region is on a length scale that is the same at that of the nanometer-sized reinforcement. Therefore, at the nanometer level, the reinforcement and adjacent polymer region is not accurately described as consisting of just two phases, thus the Mori-Tanaka model is not expected to perform well for nanostructured reinforcements Effective Interface Model Because of the aforementioned drawbacks to the Mori-Tanaka approach, another modeling approach was developed. The effective interface model can be used to predict 29

13 the elastic properties of a composite with effective particles that have an interface of the same spherical shape as the effective particle. The effective interface has a finite size and models the region immediately surrounding the spherical reinforcement, which is commonly referred to as an interphase or an interaction zone (Dunn & Ledbetter, 995). Odegard, et al. (2005) reported that unlike the Mori-Tanaka model, the effective interface model should be applicable to both nanometer-sized and larger-sized reinforcement. 4.4 Final Remarks on Models A critical analysis of the brief literature on the modeling of the mechanical properties of polymer composites presented above suggests that there is no general appropriate and correct model applicable for all nano-composites for estimating their properties. The conclusions from these researches and studies may be summarized as follows:. Considering the dimensions of polymer chains and their crystalline assemblies, it can be said that all polymers have structure on the nanometer size scale and, further, that the mechanical properties of polymers are governed by the interactions of these nanostructures with one another. Therefore, to influence the interactions that govern the mechanical properties of polymers, specific nano-scale reinforcement is efficient and beneficial. 2. Where the particle/matrix interface has a high energy, a very stiff network can be formed. In such situations, the stiffness of the matrix is much lower than the network stiffness and the mechanical behavior of the composite material can be rather well predicted on the basis of a structural modeling. 3. From a very general point of view, the elastic modulus of a filled polymer is affected by (i) the elastic properties of its constitutive phases (i.e., modulus and Poisson s ratio), (ii) the volume fraction of filler (iii) the morphology (i.e., shape, aspect ratio, and distribution of the filler into the polymeric matrix), and (iv) the interactions between fillers. Various models are proposed in the literature to understand the complex interplay between these parameters and to predict of the elastic modulus of polymer composites. 4. The available theoretical models may be used to predict elastic modulus as a first approximation or initial estimation. 30

14 5. Micromechanical approach to analyzing composites considers the filler and the matrix separately and applies the loading and the boundary conditions at the individual filler and matrix level. 4.5 Summary To facilitate the development of nano-structured composites, constitutive relationships are developed that predict the bulk mechanical properties of the materials as a function of the molecular structure of the polyamide and reinforcement. These constitutive relationships can be used to influence the design of these materials before they are synthesized. In this study the following theoretical models obtained from literature were used to predict Young s modulus of micro and nano-composite samples synthesized.. The Rule of Mixtures 2. Nicolais-Narkis Theory and the Guth s equation 3. The composite sphere model of Hashin-Shtrikman. 4. The semi-empirical micromechanical model of Halpin-Tsai. Results obtained from these computations and their predictive capability is discussed later in Chapter-6. 3