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PP ternary nanocomposite
Applied Clay Science 137 (2017) 176–182
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Applied Clay Science journal homepage: www.elsevier.com/locate/clay
Research paper
Development of Hashin-Shtrikman model to determine the roles and properties of interphases in clay/CaCO3/PP ternary nanocomposite Yasser Zare a, Kyong Yop Rhee b,⁎ a b
Young Researchers and Elites Club, Science and Research Branch, Islamic Azad University, Tehran, Iran Department of Mechanical Engineering, College of Engineering, Kyung Hee University, Yongin 446-701, Republic of Korea
a r t i c l e
i n f o
Article history: Received 17 May 2016 Received in revised form 19 December 2016 Accepted 21 December 2016 Available online xxxx Keywords: Ternary polymer nanocomposites Nanofiller Interphase Mechanical properties Modeling
a b s t r a c t The Hashin-Shtrikman model underpredicts the bulk, shear and Young's moduli of the prepared clay/CaCO3/PP ternary polymer nanocomposite (TPN) by ignoring the interphase between polymer matrix and nanoparticles. In this study, the Hashin-Shtrikman model was developed assuming the thickness and strength of interphases. Also, the thickness and strength of interphases in the ternary samples could be calculated by the developed Hashin-Shtrikman and Pukanszky models, respectively using the experimental results of mechanical properties. The predictions of the developed model showed good agreement with the experimental data at different Mt and CaCO3 contents assuming the interphase role which validate the current approach. According to the calculations, the strong and thick interphases were formed in the TPN at low nanofiller concentrations. This occurrence for the present samples was explained by the material and processing parameters. However, the thickness and strength of interphases weakened by increasing in clay content, probably due to the poor dispersion of nanoparticles and reduced interfacial area/adhesion at this condition. © 2016 Published by Elsevier B.V.
1. Introduction The development of polymer composites has continued in scientific and industrial areas, due to the significant properties achieved at low nanofiller contents such as high modulus, acceptable thermal stabilization, poor flammability and low permeability (Fasihi and Abolghasemi, 2012; Fernández et al., 2013; Huskić et al., 2013; Kalbasi et al., 2012; Mauroy et al., 2015; Monfared and Jalali-Arani, 2015; Salkhord and Sadeghi Ghari, 2015; Shabanian et al., 2015). The improvement of properties is mostly due to the excellent aspects of nanofillers such as very small size, high surface area and large modulus. For example, montmorillonite (Mt) as a known type of clay mineral shows a high modulus (about 180 GPa), and large specific surface area (750 m2 g− 1 at completely exfoliated state). Therefore, it can cause a large reinforcing effect in polymers (Mirabedini et al., 2012; Razavi-Nouri and Karami, 2014; Zare, 2016b, 2016e; Zare and Garmabi, 2015). However, the quality of filler dispersion and interfacial adhesion between polymer matrix and nanofiller play main roles in final properties of polymer composites. Although the effects of many material and processing parameters on the properties of polymer composites have been investigated for two decades, there is still a considerable attention in different communities to achieve desirable properties. The high-tech products require a wide ⁎ Corresponding author at: 1 Seocheon, Giheung, Yongin, Gyeonggi 449-701, Republic of Korea. E-mail address: [email protected] (K.Y. Rhee).
http://dx.doi.org/10.1016/j.clay.2016.12.033 0169-1317/© 2016 Published by Elsevier B.V.
knowledge of all factors affecting the properties of polymer composites. Also, polymer composites have shown a problem of toughness-stiffness optimization. Researchers have reported that TPN containing two nanofillers or polymers can offer much improved properties such as tensile modulus and impact strength (Chen et al., 2007). The mechanical properties of composites mostly depend on characteristics of components, composition, interfacial interaction between the constituents, etc. (Mauroy et al., 2015; Razavi et al., 2015; Zare, 2016c, 2016d). There are relatively few studies on the interphase properties of ternary samples in the literature. The modeling can be an acceptable technique to examine the interphase in polymer composites without any precise and practical method for evaluation of interfacial properties in composites. The theoretical analysis can provide valuable information which enables the achievement of anticipated properties (Pahlavanpour et al., 2013; Zare, 2016f, 2016g). Also, micromechanic models such as Halpin-Tsai, Guth, Paul, Cox, Kerner, etc. which consider two separate phases as matrix and filler cannot predict accurate values for mechanical properties (Zare and Garmabi, 2012, 2014). It seems that a strong interphase between polymer and nanofillers is formed which affects the properties of polymer composites. In other words, micromechanic models do not incorporate the role of interphase and thus, underpredict the mechanical properties of polymer composites. In this paper, Hashin-Shtrikman model for bulk and shear moduli of isotropic and quasi-homogeneous composites is developed for a TPN containing clay mineral and CaCO3 nanoparticles. Since experimental and theoretical results are spaced, the effects of interphase are assumed
Y. Zare, K.Y. Rhee / Applied Clay Science 137 (2017) 176–182
in this model and the obtained outputs are compared with experimental data. Additionally, the interphases properties are calculated by the developed Hashin-Shtrikman and Pukanszky models.
PP homopolymer as matrix (ZH500, MFI = 10 g/10 min, 230 °C, 2.16 kg) was received from Navid Zar Shimi Company, Iran. The modified Mt with a quaternary ammonium salt (Cloisite 20A) was purchased from Southern Clay Products with average thickness of 2 nm for each layer. Maleic anhydride grafted PP (PPgMA) (PB3150) with 0.5 wt% of maleic anhydride was supplied from Crompton Corp. and used as a compatibilizer between PP and Mt. A same content of PPgMA and Mt was mixed in all samples. Also, precipitated CaCO3 (Socal312) with average radius of 35 nm and a coated layer of stearic acid was provided from Solvay. After a dry mixing, melt compounding of samples was performed by a co-rotating twin screw extruder, Brabender TSE 20/40D (D = 20 mm, L/D = 40). Screw speed and feeding rate were kept at 250 rpm and 3 kg/h, respectively. The temperature profile was set from hopper to die at 210 to 230 °C. Also, the injection molding of extruded samples was performed using a MonoMat 80 injection molding machine at melt and mold temperatures of 245 and 80 °C, respectively. Tensile test was performed by Z050 (Zwick) according to ASTM D638 at crosshead speed of 50 mm/min. The reported values are the average measurement of at least 5 samples. 3. Results and discussion A linear elastic deformation was suggested by Hashin and Shtrikman (1961) assuming the isotropy and quasi-homogeneity of materials without any dependency to geometry of components. The lower bounds for bulk “K” and shear “G” moduli of composites were expressed as: φ f ðK f −Km Þ 3ðK f −Km Þ 1 þ ð1−φ f Þ 3Km þ 4Gm
K ¼ Km þ
G ¼ Gm þ
φ f ðG f −Gm Þ 6ðKm þ 2Gm ÞðG f −Gm Þ 1 þ ð1−φ f Þ 5Gm ð3Km þ 4Gm Þ
ð1Þ
ð2Þ
Also, the upper bounds of moduli were given by:
K ¼ Kf þ
G ¼ Gf þ
ð1−φ f ÞðKm −K f Þ 3ðKm −K f Þ 1 þ φf 3K f þ 4G f ð1−φ f ÞðGm −G f Þ 6ðK f þ 2G f ÞðGm −G f Þ 1 þ φf 5G f ð3K f þ 4G f Þ
ð3Þ
ð4Þ
where subscripts “m” and “f” indicate matrix and filler phases, respectively. “φf” is the volume fraction of nanofiller which for a TPN with two nanofillers is φf = φf1 + φf2, in which “φf1” and “φf2” are volume fractions of Mt and CaCO3, respectively. In this TPN, “Kf” and “Gf” at different nanofiller contents are expressed as: K f ¼ φ f1 K f1 þ φ f2 K f2 G f ¼ φ f1 G f1 þ φ f2 G f2
Young's modulus (E) of an isotropic solid can be determined (Dorigato et al., 2013) as: E¼
2. Materials and methods
ð5Þ ð6Þ
177
9KG 3K þ G
ð7Þ
Also, assuming matrix, filler and composite as isotropic elastic solids, “K” and “G” can be calculated (Dorigato et al., 2013) as: Kj ¼
Ej 3−6ν j
ð8Þ
Gj ¼
Ej 2 þ 2ν j
ð9Þ
where “j” index refers to matrix, filler or composite and “ν” is Poisson ratio. “ν” for a TPN is determined by: ν ¼ φ f1 ν f1 þ φ f2 ν f2 þ φm νm
ð10Þ
“E”, density (ρ) and “ν” for PP, Mt and CaCO3 from data sheets (Zare and Garmabi, 2012) and the calculations of “K” and “G” by Eqs. (8) and (9) are shown in Table 1. In addition, experimental “E” and yield strength (σ) by tensile test as well as the calculations of “ν”, “K” and “G” (Eqs. (8)–(10)) for all prepared TPN are reported in Table 2. These values of “E”, “K” and “G” are considered as experimental data and compared with model predictions. All moduli of prepared samples improve by addition of both Mt and CaCO3 nanofillers, but less moduli for sample reinforced with 6 wt% Mt and 20 wt% CaCO3 is measured, possibly due to some undesirable phenomena such as the poor dispersion of nanoparticles at high nanofiller contents. If only the weight percentages of nanofillers were important, the sample with 6 wt% Mt and 20 wt% CaCO3 displayed the highest moduli. However, the reported findings clearly indicate the significant role of dispersion quality of nanoparticles, interfacial adhesion between polymer matrix and nanofiller, etc. beside the reinforcing effects of nanoparticles in properties of composites. As a result, many parameters affect the moduli of polymer composites, which should be well considered in modeling of properties. In addition, “σ” data show a small variation at different nanofiller compositions. However, they give the worst levels at high nanofiller concentrations. Moreover, the values of “ν” for all samples are less than “ν” for neat PP matrix as 0.38, but, the calculated “ν” results change in a narrow range of 0.37–0.38 in all samples. Figs. 1–3 depict the experimental results of TPN and the lower and upper bounds of “K”, “G” and “E” by Eqs. (1)–(7). The large modulus of nanofillers causes a close predictions by upper and lower bounds models for polymer composites. However, the increase rate of moduli is higher by upper bound model compared to lower one. Also, high differences between experimental data and predictions are illustrated for all moduli of prepared samples. These discrepancies are not peculiar for polymer composites, because many effective parameters such as nanoparticles dispersion and interphase properties are not supposed in Hashin-Shtrikman model. The most improvement of mechanical properties in polymer composites is attributed to formation of strong interphases between polymer matrix and both nanofillers, which can properly transfer the load from matrix to nanoparticles. Likewise, the high dispersion and distribution of nanoparticles in polymer matrix result in a large interface Table 1 The characteristics of neat PP, MMT and CaCO3. Materials
E (GPa)
d (g/cm3)
ν
K (GPa)
G (GPa)
PP MMT CaCO3
2.17 178 26
0.91 1.77 2.71
0.38 0.27 0.31
3.01 129 22.81
0.79 70.1 9.92
178
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Table 2 The mechanical properties of ternary nanocomposites.
No.
MMT (wt%)
CaCO3 (wt%)
E (GPa)
ν
K (GPa)
G (GPa)
1 2 3 4 5 6 7 8 9 10 11 12 13
0 2 2 2 2 4 4 4 4 6 6 6 6
0 2 8 14 20 2 8 14 20 2 8 14 20
2.17 2.41 2.49 2.57 2.7 2.48 2.57 2.65 2.95 2.6 2.71 2.92 2.52
0.38 0.3784 0.3768 0.3751 0.3732 0.3772 0.3755 0.3737 0.3718 0.3760 0.3743 0.3724 0.3703
3.01 3.3 3.37 3.43 3.55 3.37 3.44 3.5 3.83 3.49 3.59 3.81 3.24
0.79 0.87 0.9 0.93 0.98 0.9 0.93 0.96 1.08 0.94 0.99 1.06 0.92
area between polymer and nanofillers. Accordingly, dispersion quality as an important parameter which affects the mechanical properties of polymer composites can be presented by interphase properties. The formation of an interphase with dissimilar properties than matrix and nanofillers was shown from experimental and theoretical studies in the literature (Zare, 2015, 2016a, 2016h). Therefore, much attempt is made in present study to correctly model the interphases in the TPN. The volume fraction of interphase (φi) can be expressed for polymer composites containing layered and spherical nanoparticles (Yanovsky et al., 2013) as: φi1 ¼ φ f1
φi2 ¼ φ f2
2ti t
ð11Þ
r þ r 3 i
r
−1
ð12Þ
where “t” and “r” are the thickness and radius of layered and spherical
nanofillers, respectively. Also, “ri” and “ti” are the thickness of interphase in different composites. The total volume fraction of interphases is expressed as: φi ¼ φi1 þ φi2 ¼ aðφ f1 þ φ f2 Þ
ð13Þ
where “a” parameter is a parameter showing interphase properties. The effective volume fraction of nanoparticles (φ) assuming the role of interphase can be expressed as: φ ¼ φ f þ φi ¼ φ f þ aφ f ¼ ð1 þ aÞðφ f1 þ φ f2 Þ
ð14Þ
Now, the lower and upper results of “K”, “G” and “E” (Eqs. (1)–(7)) are calculated by new “φ” (Eq. (14)) and fitted to experimental data by an appropriate “a”. The calculations of “K”, “G” and “E” moduli are observed in Figs. 4–6, respectively. The predictions of lower bounds show a good agreement with experimental results of all moduli, while the upper results depict a high rate of increment for moduli of present samples. Accordingly, the lower bound results are chosen for determination of “a” values. The values of “E” and “G” well agree with the experimental data by “a” levels of 2.5, 2.1 and 2 at 2, 4 and 6 wt% of Mt, respectively. However, “K” calculations display a good consistency with experimental data by “a” values of 3, 2.4 and 2 at 2, 4 and 6 wt% of Mt, respectively. A larger level of “a” is required to predict “K” compared to “E” and “G”, demonstrating the important effects of interphase on “K”. Also, the successful modeling by Hashin-Shtrikman model results in consideration of TPN containing layered and spherical nanofillers as homogenously isotropic materials. In addition, the predictions show suitable consistently with the experimental data taking into account a proper interphase between polymer and nanoparticles, indicating the successful development of Hashin-Shtrikman model for TPN containing two different types of nanofillers. The average values of “a” parameter in different Mt contents are given in Table 3. They show the formation of different interphase in
Fig. 1. The plots of “K” according to Eqs. (1) and (3) for Mt contents of a) 2, b) 4, and c) 6 wt%.
Y. Zare, K.Y. Rhee / Applied Clay Science 137 (2017) 176–182
Fig. 2. The predicted results of “G” by Eqs. (2) and (4) in Mt contents of a) 2, b) 4, and c) 6 wt%.
Fig. 3. The calculations of “E” by Eqs. (1)–(7) in Mt concentrations of a) 2, b) 4, and c) 6 wt%.
179
180
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Fig. 4. The plots of “K” (Eqs. (1), (3) and (12)) assuming the effects of interphase in MT contents of a) 2, b) 4, and c) 6 wt%.
Fig. 5. The calculations of “G” by Eqs. (2), (4) and (12) assuming the interphase in Mt contents of a) 2, b) 4, and c) 6 wt%.
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181
Fig. 6. The results of “E” (Eqs. (1)–(7) and (12)) assuming the effects of interphase in Mt contents of a) 2, b) 4, and c) 6 wt%.
present composites. The most value of “a” is obtained at 2 wt% of Mt, while the worst level is found at 6 wt% of Mt. This evidence demonstrates the significant role of nanofiller composition in interphase properties. Possibly, good dispersion and distribution of nanoparticles at little nanofiller contents cause the formation of significant interphases in TPN. Using Eqs. (11) and (12), the thickness of interphases can be calculated at different Mt contents (Table 3). The maximum “ti” for PP-Mt is calculated as 2.75 nm at 2 wt% of Mt, but increasing in Mt decreases the value of “ti”. In addition, the average thickness of PP-CaCO3 interphase is reduced from 19.3 to 15.4 nm by increasing in Mt content. The aggregation and poor dispersion of nanoparticles as well as the reduced interfacial area and adhesion at high nanofiller concentrations may deteriorate the formation of thick interphases in TPN. Pukanszky (1990) established a model to describe the composition dependence of yield strength in composites as: σR ¼
1−φ f expðBφ f Þ 1 þ 2:5φ f
ð15Þ
where “σR” is relative yield strength as σ/σm, where “σ” and “σm” arethe yield strengths of composite and matrix, respectively. Also, “B” parameter which displays the extent of polymer-filler adhesion is expressed as:
σi B ¼ ð1 þ Ac ρ f ri Þ ln σm
where “Ac” is specific surface area of filler and “ρf” is density of filler. Also, “σi” is tensile strength of interphase. The Pukanszky model can be rewritten to: 1 þ 2:5φ f ¼ Bφ f ln ðσ Reduced Þ ¼ ln σ R 1−φ f
which “B” is obtained by a linear connection between ln(σReduced) and “φf”. By rearranging of Eq. (16), “σi1” can be expressed as: σ i1 ¼ σ m exp
B 1 þ Ac ρ f ti
ð18Þ
2
Ac ¼
A A 2l 2 ¼ ≅ ¼ m ρ f v ρ f l2 t ρ f t
ð19Þ
where “A”, “m”, “v” and “l” are surface area, mass, volume and length of Mt nanoparticles, respectively. Therefore, “σi1” can be calculated as: 0
Table 3 The characteristics of interphase in the TPN.
“Ac” for Mt nanoparticles is calculated as:
ð16Þ
ð17Þ
B σ i1 ¼ σ m exp@
1 B 1þ2
C ti A t
ð20Þ
Furthermore, “Ac” for spherical nanoparticles is expressed as:
No.
MT (wt%)
a
ti (nm)
ri (nm)
B
σi1 (MPa)
σi2 (MPa)
1 2 3
2 4 6
2.75 2.25 2
2.75 2.25 2
19.3 16.8 15.4
2.74 3.37 2.34
83.1 112.8 87.3
112.3 159.2 109.7
Ac ¼
A A 4πr2 3 ¼ ¼ ¼ 4 3 ρf r m ρf v ρ f πr 3
ð21Þ
182
Y. Zare, K.Y. Rhee / Applied Clay Science 137 (2017) 176–182
As a result, “σi2” at PP-CaCO3 interphase can be calculated as: 0
References
1
B B C σ i2 ¼ σ m exp@ rA 1þ3 i r
ð22Þ
Using the interphase thickness data reported in Table 3 and σm = 40 MPa, the values of interphase strengths are obtained by Eqs. (20) and (22) (Table 3). The best levels of “σi1” and “σi2” are obtained at average Mt concentration (4 wt%). It is expected, because “B” parameter shows the maximum level at 4 wt% Mt. Additionally, “σi1” and “σi2” are more than “σm” at all Mt contents revealing the high strength of formed interphases in TPN. As a result, the formation of different interphases in ternary samples should be well considered in experimental and theoretical work to understand the unexpected behavior of TPN at different nanofiller compositions. Apart from the theoretical approach suggested in the present work, the formation of strong interphases in present samples can be attributed to these factors: 1) the clay mineral with cation exchange reactions by alkyl ammonium encourages large compatibility between hydrophobic PP and organo-philic clay, 2) the polar groups of maleic anhydride in PPgMA interact with clay layers promoting the good dispersion of clay mineral in PP matrix, 3) hydrogen atoms of polymer backbone produce a strong interaction with clay mineral, 4) stearic acid coated on CaCO3 enhances the hydrophobicity of nanoparticles and improves the distribution of CaCO3 nanoparticle in PP matrix, 5) PPgMA can also be used as a compatibilizer to increase interfacial bonding between CaCO3 and PP matrix and 6) high shear stress in melt mixing process by twin screw extruder provides a good dispersion and distribution of nanoparticles in PP matrix without any distinct agglomeration and aggregation of both nanofillers. According to above explanation, assumption of strong and thick interphases between polymer and nanofillers phases is more logical from various aspects such as sample characteristics, mechanical properties, strong interaction at interface region and modeling procedure.
4. Conclusions The micromechanic models such as Hashin-Shtrikman underpredict the modulus of polymer composites, because the mechanical properties of composites depend on many parameters such as the level of filler dispersion and interfacial interaction. In this paper, Hashin-Shtrikman model which underpredicts the bulk, shear and Young's moduli of PP/ Mt/CaCO3 TPN was developed assuming the formation of interphases between polymer matrix and both nanofillers. The calculations of developed model showed a good agreement with the experimental results at different Mt and CaCO3 contents which validated the present modeling. The average values of “a” parameter changed from 2 to 3 for the present samples, similar to the calculated values of “B” indicating the correct definition of “a” in the current study. The thickness and strength of interphases were also calculated by the developed equations. The properties of both interphases worsened by increasing in Mt content, possibly due to poor dispersion of nanoparticles and reduced interfacial adhesion at high nanofiller concentrations. Conclusively, the material and processing conditions confirmed the proper assumption of strong and thick interphases between polymer and nanofillers phases in the present ternary samples.
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Research paper
Development of Hashin-Shtrikman model to determine the roles and properties of interphases in clay/CaCO3/PP ternary nanocomposite Yasser Zare a, Kyong Yop Rhee b,⁎ a b
Young Researchers and Elites Club, Science and Research Branch, Islamic Azad University, Tehran, Iran Department of Mechanical Engineering, College of Engineering, Kyung Hee University, Yongin 446-701, Republic of Korea
a r t i c l e
i n f o
Article history: Received 17 May 2016 Received in revised form 19 December 2016 Accepted 21 December 2016 Available online xxxx Keywords: Ternary polymer nanocomposites Nanofiller Interphase Mechanical properties Modeling
a b s t r a c t The Hashin-Shtrikman model underpredicts the bulk, shear and Young's moduli of the prepared clay/CaCO3/PP ternary polymer nanocomposite (TPN) by ignoring the interphase between polymer matrix and nanoparticles. In this study, the Hashin-Shtrikman model was developed assuming the thickness and strength of interphases. Also, the thickness and strength of interphases in the ternary samples could be calculated by the developed Hashin-Shtrikman and Pukanszky models, respectively using the experimental results of mechanical properties. The predictions of the developed model showed good agreement with the experimental data at different Mt and CaCO3 contents assuming the interphase role which validate the current approach. According to the calculations, the strong and thick interphases were formed in the TPN at low nanofiller concentrations. This occurrence for the present samples was explained by the material and processing parameters. However, the thickness and strength of interphases weakened by increasing in clay content, probably due to the poor dispersion of nanoparticles and reduced interfacial area/adhesion at this condition. © 2016 Published by Elsevier B.V.
1. Introduction The development of polymer composites has continued in scientific and industrial areas, due to the significant properties achieved at low nanofiller contents such as high modulus, acceptable thermal stabilization, poor flammability and low permeability (Fasihi and Abolghasemi, 2012; Fernández et al., 2013; Huskić et al., 2013; Kalbasi et al., 2012; Mauroy et al., 2015; Monfared and Jalali-Arani, 2015; Salkhord and Sadeghi Ghari, 2015; Shabanian et al., 2015). The improvement of properties is mostly due to the excellent aspects of nanofillers such as very small size, high surface area and large modulus. For example, montmorillonite (Mt) as a known type of clay mineral shows a high modulus (about 180 GPa), and large specific surface area (750 m2 g− 1 at completely exfoliated state). Therefore, it can cause a large reinforcing effect in polymers (Mirabedini et al., 2012; Razavi-Nouri and Karami, 2014; Zare, 2016b, 2016e; Zare and Garmabi, 2015). However, the quality of filler dispersion and interfacial adhesion between polymer matrix and nanofiller play main roles in final properties of polymer composites. Although the effects of many material and processing parameters on the properties of polymer composites have been investigated for two decades, there is still a considerable attention in different communities to achieve desirable properties. The high-tech products require a wide ⁎ Corresponding author at: 1 Seocheon, Giheung, Yongin, Gyeonggi 449-701, Republic of Korea. E-mail address: [email protected] (K.Y. Rhee).
http://dx.doi.org/10.1016/j.clay.2016.12.033 0169-1317/© 2016 Published by Elsevier B.V.
knowledge of all factors affecting the properties of polymer composites. Also, polymer composites have shown a problem of toughness-stiffness optimization. Researchers have reported that TPN containing two nanofillers or polymers can offer much improved properties such as tensile modulus and impact strength (Chen et al., 2007). The mechanical properties of composites mostly depend on characteristics of components, composition, interfacial interaction between the constituents, etc. (Mauroy et al., 2015; Razavi et al., 2015; Zare, 2016c, 2016d). There are relatively few studies on the interphase properties of ternary samples in the literature. The modeling can be an acceptable technique to examine the interphase in polymer composites without any precise and practical method for evaluation of interfacial properties in composites. The theoretical analysis can provide valuable information which enables the achievement of anticipated properties (Pahlavanpour et al., 2013; Zare, 2016f, 2016g). Also, micromechanic models such as Halpin-Tsai, Guth, Paul, Cox, Kerner, etc. which consider two separate phases as matrix and filler cannot predict accurate values for mechanical properties (Zare and Garmabi, 2012, 2014). It seems that a strong interphase between polymer and nanofillers is formed which affects the properties of polymer composites. In other words, micromechanic models do not incorporate the role of interphase and thus, underpredict the mechanical properties of polymer composites. In this paper, Hashin-Shtrikman model for bulk and shear moduli of isotropic and quasi-homogeneous composites is developed for a TPN containing clay mineral and CaCO3 nanoparticles. Since experimental and theoretical results are spaced, the effects of interphase are assumed
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in this model and the obtained outputs are compared with experimental data. Additionally, the interphases properties are calculated by the developed Hashin-Shtrikman and Pukanszky models.
PP homopolymer as matrix (ZH500, MFI = 10 g/10 min, 230 °C, 2.16 kg) was received from Navid Zar Shimi Company, Iran. The modified Mt with a quaternary ammonium salt (Cloisite 20A) was purchased from Southern Clay Products with average thickness of 2 nm for each layer. Maleic anhydride grafted PP (PPgMA) (PB3150) with 0.5 wt% of maleic anhydride was supplied from Crompton Corp. and used as a compatibilizer between PP and Mt. A same content of PPgMA and Mt was mixed in all samples. Also, precipitated CaCO3 (Socal312) with average radius of 35 nm and a coated layer of stearic acid was provided from Solvay. After a dry mixing, melt compounding of samples was performed by a co-rotating twin screw extruder, Brabender TSE 20/40D (D = 20 mm, L/D = 40). Screw speed and feeding rate were kept at 250 rpm and 3 kg/h, respectively. The temperature profile was set from hopper to die at 210 to 230 °C. Also, the injection molding of extruded samples was performed using a MonoMat 80 injection molding machine at melt and mold temperatures of 245 and 80 °C, respectively. Tensile test was performed by Z050 (Zwick) according to ASTM D638 at crosshead speed of 50 mm/min. The reported values are the average measurement of at least 5 samples. 3. Results and discussion A linear elastic deformation was suggested by Hashin and Shtrikman (1961) assuming the isotropy and quasi-homogeneity of materials without any dependency to geometry of components. The lower bounds for bulk “K” and shear “G” moduli of composites were expressed as: φ f ðK f −Km Þ 3ðK f −Km Þ 1 þ ð1−φ f Þ 3Km þ 4Gm
K ¼ Km þ
G ¼ Gm þ
φ f ðG f −Gm Þ 6ðKm þ 2Gm ÞðG f −Gm Þ 1 þ ð1−φ f Þ 5Gm ð3Km þ 4Gm Þ
ð1Þ
ð2Þ
Also, the upper bounds of moduli were given by:
K ¼ Kf þ
G ¼ Gf þ
ð1−φ f ÞðKm −K f Þ 3ðKm −K f Þ 1 þ φf 3K f þ 4G f ð1−φ f ÞðGm −G f Þ 6ðK f þ 2G f ÞðGm −G f Þ 1 þ φf 5G f ð3K f þ 4G f Þ
ð3Þ
ð4Þ
where subscripts “m” and “f” indicate matrix and filler phases, respectively. “φf” is the volume fraction of nanofiller which for a TPN with two nanofillers is φf = φf1 + φf2, in which “φf1” and “φf2” are volume fractions of Mt and CaCO3, respectively. In this TPN, “Kf” and “Gf” at different nanofiller contents are expressed as: K f ¼ φ f1 K f1 þ φ f2 K f2 G f ¼ φ f1 G f1 þ φ f2 G f2
Young's modulus (E) of an isotropic solid can be determined (Dorigato et al., 2013) as: E¼
2. Materials and methods
ð5Þ ð6Þ
177
9KG 3K þ G
ð7Þ
Also, assuming matrix, filler and composite as isotropic elastic solids, “K” and “G” can be calculated (Dorigato et al., 2013) as: Kj ¼
Ej 3−6ν j
ð8Þ
Gj ¼
Ej 2 þ 2ν j
ð9Þ
where “j” index refers to matrix, filler or composite and “ν” is Poisson ratio. “ν” for a TPN is determined by: ν ¼ φ f1 ν f1 þ φ f2 ν f2 þ φm νm
ð10Þ
“E”, density (ρ) and “ν” for PP, Mt and CaCO3 from data sheets (Zare and Garmabi, 2012) and the calculations of “K” and “G” by Eqs. (8) and (9) are shown in Table 1. In addition, experimental “E” and yield strength (σ) by tensile test as well as the calculations of “ν”, “K” and “G” (Eqs. (8)–(10)) for all prepared TPN are reported in Table 2. These values of “E”, “K” and “G” are considered as experimental data and compared with model predictions. All moduli of prepared samples improve by addition of both Mt and CaCO3 nanofillers, but less moduli for sample reinforced with 6 wt% Mt and 20 wt% CaCO3 is measured, possibly due to some undesirable phenomena such as the poor dispersion of nanoparticles at high nanofiller contents. If only the weight percentages of nanofillers were important, the sample with 6 wt% Mt and 20 wt% CaCO3 displayed the highest moduli. However, the reported findings clearly indicate the significant role of dispersion quality of nanoparticles, interfacial adhesion between polymer matrix and nanofiller, etc. beside the reinforcing effects of nanoparticles in properties of composites. As a result, many parameters affect the moduli of polymer composites, which should be well considered in modeling of properties. In addition, “σ” data show a small variation at different nanofiller compositions. However, they give the worst levels at high nanofiller concentrations. Moreover, the values of “ν” for all samples are less than “ν” for neat PP matrix as 0.38, but, the calculated “ν” results change in a narrow range of 0.37–0.38 in all samples. Figs. 1–3 depict the experimental results of TPN and the lower and upper bounds of “K”, “G” and “E” by Eqs. (1)–(7). The large modulus of nanofillers causes a close predictions by upper and lower bounds models for polymer composites. However, the increase rate of moduli is higher by upper bound model compared to lower one. Also, high differences between experimental data and predictions are illustrated for all moduli of prepared samples. These discrepancies are not peculiar for polymer composites, because many effective parameters such as nanoparticles dispersion and interphase properties are not supposed in Hashin-Shtrikman model. The most improvement of mechanical properties in polymer composites is attributed to formation of strong interphases between polymer matrix and both nanofillers, which can properly transfer the load from matrix to nanoparticles. Likewise, the high dispersion and distribution of nanoparticles in polymer matrix result in a large interface Table 1 The characteristics of neat PP, MMT and CaCO3. Materials
E (GPa)
d (g/cm3)
ν
K (GPa)
G (GPa)
PP MMT CaCO3
2.17 178 26
0.91 1.77 2.71
0.38 0.27 0.31
3.01 129 22.81
0.79 70.1 9.92
178
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Table 2 The mechanical properties of ternary nanocomposites.
No.
MMT (wt%)
CaCO3 (wt%)
E (GPa)
ν
K (GPa)
G (GPa)
1 2 3 4 5 6 7 8 9 10 11 12 13
0 2 2 2 2 4 4 4 4 6 6 6 6
0 2 8 14 20 2 8 14 20 2 8 14 20
2.17 2.41 2.49 2.57 2.7 2.48 2.57 2.65 2.95 2.6 2.71 2.92 2.52
0.38 0.3784 0.3768 0.3751 0.3732 0.3772 0.3755 0.3737 0.3718 0.3760 0.3743 0.3724 0.3703
3.01 3.3 3.37 3.43 3.55 3.37 3.44 3.5 3.83 3.49 3.59 3.81 3.24
0.79 0.87 0.9 0.93 0.98 0.9 0.93 0.96 1.08 0.94 0.99 1.06 0.92
area between polymer and nanofillers. Accordingly, dispersion quality as an important parameter which affects the mechanical properties of polymer composites can be presented by interphase properties. The formation of an interphase with dissimilar properties than matrix and nanofillers was shown from experimental and theoretical studies in the literature (Zare, 2015, 2016a, 2016h). Therefore, much attempt is made in present study to correctly model the interphases in the TPN. The volume fraction of interphase (φi) can be expressed for polymer composites containing layered and spherical nanoparticles (Yanovsky et al., 2013) as: φi1 ¼ φ f1
φi2 ¼ φ f2
2ti t
ð11Þ
r þ r 3 i
r
−1
ð12Þ
where “t” and “r” are the thickness and radius of layered and spherical
nanofillers, respectively. Also, “ri” and “ti” are the thickness of interphase in different composites. The total volume fraction of interphases is expressed as: φi ¼ φi1 þ φi2 ¼ aðφ f1 þ φ f2 Þ
ð13Þ
where “a” parameter is a parameter showing interphase properties. The effective volume fraction of nanoparticles (φ) assuming the role of interphase can be expressed as: φ ¼ φ f þ φi ¼ φ f þ aφ f ¼ ð1 þ aÞðφ f1 þ φ f2 Þ
ð14Þ
Now, the lower and upper results of “K”, “G” and “E” (Eqs. (1)–(7)) are calculated by new “φ” (Eq. (14)) and fitted to experimental data by an appropriate “a”. The calculations of “K”, “G” and “E” moduli are observed in Figs. 4–6, respectively. The predictions of lower bounds show a good agreement with experimental results of all moduli, while the upper results depict a high rate of increment for moduli of present samples. Accordingly, the lower bound results are chosen for determination of “a” values. The values of “E” and “G” well agree with the experimental data by “a” levels of 2.5, 2.1 and 2 at 2, 4 and 6 wt% of Mt, respectively. However, “K” calculations display a good consistency with experimental data by “a” values of 3, 2.4 and 2 at 2, 4 and 6 wt% of Mt, respectively. A larger level of “a” is required to predict “K” compared to “E” and “G”, demonstrating the important effects of interphase on “K”. Also, the successful modeling by Hashin-Shtrikman model results in consideration of TPN containing layered and spherical nanofillers as homogenously isotropic materials. In addition, the predictions show suitable consistently with the experimental data taking into account a proper interphase between polymer and nanoparticles, indicating the successful development of Hashin-Shtrikman model for TPN containing two different types of nanofillers. The average values of “a” parameter in different Mt contents are given in Table 3. They show the formation of different interphase in
Fig. 1. The plots of “K” according to Eqs. (1) and (3) for Mt contents of a) 2, b) 4, and c) 6 wt%.
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Fig. 2. The predicted results of “G” by Eqs. (2) and (4) in Mt contents of a) 2, b) 4, and c) 6 wt%.
Fig. 3. The calculations of “E” by Eqs. (1)–(7) in Mt concentrations of a) 2, b) 4, and c) 6 wt%.
179
180
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Fig. 4. The plots of “K” (Eqs. (1), (3) and (12)) assuming the effects of interphase in MT contents of a) 2, b) 4, and c) 6 wt%.
Fig. 5. The calculations of “G” by Eqs. (2), (4) and (12) assuming the interphase in Mt contents of a) 2, b) 4, and c) 6 wt%.
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181
Fig. 6. The results of “E” (Eqs. (1)–(7) and (12)) assuming the effects of interphase in Mt contents of a) 2, b) 4, and c) 6 wt%.
present composites. The most value of “a” is obtained at 2 wt% of Mt, while the worst level is found at 6 wt% of Mt. This evidence demonstrates the significant role of nanofiller composition in interphase properties. Possibly, good dispersion and distribution of nanoparticles at little nanofiller contents cause the formation of significant interphases in TPN. Using Eqs. (11) and (12), the thickness of interphases can be calculated at different Mt contents (Table 3). The maximum “ti” for PP-Mt is calculated as 2.75 nm at 2 wt% of Mt, but increasing in Mt decreases the value of “ti”. In addition, the average thickness of PP-CaCO3 interphase is reduced from 19.3 to 15.4 nm by increasing in Mt content. The aggregation and poor dispersion of nanoparticles as well as the reduced interfacial area and adhesion at high nanofiller concentrations may deteriorate the formation of thick interphases in TPN. Pukanszky (1990) established a model to describe the composition dependence of yield strength in composites as: σR ¼
1−φ f expðBφ f Þ 1 þ 2:5φ f
ð15Þ
where “σR” is relative yield strength as σ/σm, where “σ” and “σm” arethe yield strengths of composite and matrix, respectively. Also, “B” parameter which displays the extent of polymer-filler adhesion is expressed as:
σi B ¼ ð1 þ Ac ρ f ri Þ ln σm
where “Ac” is specific surface area of filler and “ρf” is density of filler. Also, “σi” is tensile strength of interphase. The Pukanszky model can be rewritten to: 1 þ 2:5φ f ¼ Bφ f ln ðσ Reduced Þ ¼ ln σ R 1−φ f
which “B” is obtained by a linear connection between ln(σReduced) and “φf”. By rearranging of Eq. (16), “σi1” can be expressed as: σ i1 ¼ σ m exp
B 1 þ Ac ρ f ti
ð18Þ
2
Ac ¼
A A 2l 2 ¼ ≅ ¼ m ρ f v ρ f l2 t ρ f t
ð19Þ
where “A”, “m”, “v” and “l” are surface area, mass, volume and length of Mt nanoparticles, respectively. Therefore, “σi1” can be calculated as: 0
Table 3 The characteristics of interphase in the TPN.
“Ac” for Mt nanoparticles is calculated as:
ð16Þ
ð17Þ
B σ i1 ¼ σ m exp@
1 B 1þ2
C ti A t
ð20Þ
Furthermore, “Ac” for spherical nanoparticles is expressed as:
No.
MT (wt%)
a
ti (nm)
ri (nm)
B
σi1 (MPa)
σi2 (MPa)
1 2 3
2 4 6
2.75 2.25 2
2.75 2.25 2
19.3 16.8 15.4
2.74 3.37 2.34
83.1 112.8 87.3
112.3 159.2 109.7
Ac ¼
A A 4πr2 3 ¼ ¼ ¼ 4 3 ρf r m ρf v ρ f πr 3
ð21Þ
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As a result, “σi2” at PP-CaCO3 interphase can be calculated as: 0
References
1
B B C σ i2 ¼ σ m exp@ rA 1þ3 i r
ð22Þ
Using the interphase thickness data reported in Table 3 and σm = 40 MPa, the values of interphase strengths are obtained by Eqs. (20) and (22) (Table 3). The best levels of “σi1” and “σi2” are obtained at average Mt concentration (4 wt%). It is expected, because “B” parameter shows the maximum level at 4 wt% Mt. Additionally, “σi1” and “σi2” are more than “σm” at all Mt contents revealing the high strength of formed interphases in TPN. As a result, the formation of different interphases in ternary samples should be well considered in experimental and theoretical work to understand the unexpected behavior of TPN at different nanofiller compositions. Apart from the theoretical approach suggested in the present work, the formation of strong interphases in present samples can be attributed to these factors: 1) the clay mineral with cation exchange reactions by alkyl ammonium encourages large compatibility between hydrophobic PP and organo-philic clay, 2) the polar groups of maleic anhydride in PPgMA interact with clay layers promoting the good dispersion of clay mineral in PP matrix, 3) hydrogen atoms of polymer backbone produce a strong interaction with clay mineral, 4) stearic acid coated on CaCO3 enhances the hydrophobicity of nanoparticles and improves the distribution of CaCO3 nanoparticle in PP matrix, 5) PPgMA can also be used as a compatibilizer to increase interfacial bonding between CaCO3 and PP matrix and 6) high shear stress in melt mixing process by twin screw extruder provides a good dispersion and distribution of nanoparticles in PP matrix without any distinct agglomeration and aggregation of both nanofillers. According to above explanation, assumption of strong and thick interphases between polymer and nanofillers phases is more logical from various aspects such as sample characteristics, mechanical properties, strong interaction at interface region and modeling procedure.
4. Conclusions The micromechanic models such as Hashin-Shtrikman underpredict the modulus of polymer composites, because the mechanical properties of composites depend on many parameters such as the level of filler dispersion and interfacial interaction. In this paper, Hashin-Shtrikman model which underpredicts the bulk, shear and Young's moduli of PP/ Mt/CaCO3 TPN was developed assuming the formation of interphases between polymer matrix and both nanofillers. The calculations of developed model showed a good agreement with the experimental results at different Mt and CaCO3 contents which validated the present modeling. The average values of “a” parameter changed from 2 to 3 for the present samples, similar to the calculated values of “B” indicating the correct definition of “a” in the current study. The thickness and strength of interphases were also calculated by the developed equations. The properties of both interphases worsened by increasing in Mt content, possibly due to poor dispersion of nanoparticles and reduced interfacial adhesion at high nanofiller concentrations. Conclusively, the material and processing conditions confirmed the proper assumption of strong and thick interphases between polymer and nanofillers phases in the present ternary samples.
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