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40
УДК 539.32
Расчет модулей Юнга, сдвига и объемного сжатия полимерных нанокомпозитов при описании наночастиц с межфазными границами в рамках модели ядро-оболочка
Y. Zare1, K.Y. Rhee2
1 Исламский университет Азад, Тегеран, Иран 2 Университет Кёнхи, Сеул, 446-701, Южная Корея
Предложена простая методология для изучения влияния межфазных характеристик на модули Юнга, сдвига и объемного сжатия полимерныж нанокомпозитов, содержащих сферические наночастицы SiO2 и СаСОз. Каждая наночастица с прилегающей межфазной границей представлена в виде ядра с оболочкой. Для таких частиц рассчитаны модули сдвига и объемного сжатия. С помощью простыж уравнений предсказаны различные модули композита, содержащего полимерную матрицу и представленные частицы. Отсутствие межфазныж областей между полимерной матрицей и наночастицами приводит к заниженным значениям модулей нанокомпозитов, однако результаты расчетов хорошо согласуются с экспериментальными данными для соответствующих значений толщины и модулей межфазныж областей. Коэффициент Пуассона полимерной матрицы vm оказывает разное влияние на модули Юнга, сдвига и объемного сжатия нанокомпозитов. Изменение vm слабо влияет на модуль сдвига, но существенно изменяет модуль объемного сжатия. Межфазные области большой толщины и с большими значениями модулей оказывают положительное влияние на модули Юнга, сдвига и объемного сжатия нанокомпозитов. Согласно представленным результатам, свойства полимерной матрицы, наночастиц и межфазных границ обуславливают широкий разброс значений модуля объемного сжатия, однако незначительно влияют на модули Юнга и сдвига нанокомпозитов.
Ключевые слова: полимерные нанокомпозиты, модуль, межфазные свойства
DOI 10.24411/1683-805X-2019-12010
A core-shell structure for interphase regions surrounding nanoparticles to predict the shear, bulk and Young's moduli of polymer particulate
nanocomposites
Y. Zare1 and K.Y. Rhee2
1 Young Researchers and Elites Club, Science and Research Branch, Islamic Azad University, Tehran, Iran
2 Department of Mechanical Engineering, College of Engineering, Kyung Hee University, Yongin, 446-701, Republic of Korea
A simple methodology is suggested to examine the roles of interphase characteristics in the shear, bulk and Young's moduli of polymer nanocomposites containing spherical nanoparticles such as SiO2 and CaCO3. A core-shell structure is presumed for nanoparticles and surrounding interphase, and then the shear and bulk moduli of supposed particles are calculated. Subsequently, simple equations predict the different moduli of a composite containing a polymer matrix and supposed particles. The absence of interphase regions between the polymer matrix and nanoparticles underestimates the moduli of nanocomposites, but the calculations show good agreement with the experimental data assuming the proper levels for thickness and moduli of interphase regions. The Poisson ratio of polymer matrix vm differently affects the shear, bulk and Young's moduli of nanocomposites. The variation of vm unimportantly changes the shear modulus, while vm significantly affects the bulk modulus. Thick and high-modulus interphase regions cause the positive effects on the shear, bulk and Young's moduli of nanocomposites. According to the present results, the properties of the polymer matrix, nanoparticles and interphase introduce a wide variation in the bulk modulus, while cause moderate and slight changes in the Young's and shear moduli of nanocomposites, respectively.
Keywords: polymer particulate nanocomposites, modulus, interphase properties
© Zare Y., Rhee K.Y., 2019
1. Introduction
The properties of polymer nanocomposites depend on different parameters such as content, modulus, aspect ratio, arrangement, dispersion and agglomeration of nano-particles, as well as the strength of interfacial adhesion between polymer and nanofiller phases [1-9]. The researchers have tried to manipulate these parameters to obtain acceptable properties in polymer nanocomposites. Among the mentioned parameters, the interfacial adhesion/interaction between polymer matrix and nanofiller significantly affect the behavior of polymer nanocomposites, due to the significant surface area of nanoparticles and the strong interfacial adhesion at nanoscale [10-12]. However, the conventional models for composites such as Halpin-Tsai cannot consider the interfacial adhesion/interaction between polymer and nanoparticles and cannot present accurate predictions for Young's modulus of polymer nanocomposites [13].
The large specific surface area of nanoparticles and the strong interfacial adhesion cause the formation of a different phase between polymer matrix and filler phases—interphase [14-16]. The effect of the interphase on the mechanical properties of polymer nanocomposites is considerable, due to the extraordinary interfacial area at the poly-mer-nanoparticles interface. Many researchers have studied the interphase properties and their influences on the mechanical properties of polymer nanocomposites such as Young's modulus and tensile/yield strength by modeling methods [17-19]. The experimental results of Young's modulus and tensile strength can be used in these models to calculate the interphase properties.
The previous studies developed micromechanical models by direct adding another phase (interphase) to them. This assumption may load to error in calculating the modulus and strength, due to inappropriate considering the interphase surrounding nanoparticles. Moreover, the previous studies mainly focused on the interphase by modeling the Young's modulus and tensile/yield strength of polymer nanocomposites. Therefore, the effects of the interphase on the shear and bulk moduli of polymer nanocomposites have been disregarded in recent years.
2. Methods
In this paper, a core-shell structure is assumed for nanoparticles and interphase (Fig. 1) and its shear and bulk moduli are calculated by a proper model. Subsequently, the core-shell particles are supposed in the polymer matrix (Fig. 1) and the moduli of the composite are predicted by the same model. The predictions of the present method are compared with the experimental data of Young's modulus and the results are presented focusing on the interphase. The effects of several parameters attributed to the polymer matrix, nanoparticles and interphase on the various moduli of polymer nanocomposites are discussed.
Berryman [20] suggested a self-consistent method to estimate the effective elastic constants for inhomogeneous composites containing ellipsoidal inclusions using elastic-wave scattering theory. According to this method, the shear G and bulk K moduli of composites with spherical inclusions are presented as
Q%
G - Gm + (Gf - Gm)
K - Km + (Kf - Km)
G + F
q — _ m m
1 -% + Q%
P^f
1 -% + P%
P-
Fm -
Gf + Fm Km + 4/3 Gf Kf + 4/3 Gf '
Gm/6(9Km + 8Gm)
(1)
(2)
(3)
(4)
(5)
Km + 2Gm
where subscripts m and f show the matrix and filler, respectively, 9 is volume fraction.
To assume the interphase, the nanoparticles and surrounding interphase are supposed as core-shell particles at the first step. The G and K for these particles are calculated by the Berryman approach assuming the interphase as the matrix (Fig. 1) by
Gp = Gi + (Gf - Gi) x
x_Q%/(% + 9i)_
1 - %/(% + 9i) + Q%/(% + 9i)
(6)
Fig. 1. The core-shell structure of nanoparticles and interphase in the proposed method and the composite containing the core-shell particles (color online)
Kp = Ki + (Kf - Ki)X
X
2 =
1 -%/(%+Pf (%+9i)' G + F
Gf + Fi
p = Ki + 4/3 Gf Kf + 4/3 Gf
(7)
(8) (9)
F=
(10)
_ G^6(9Kj + SGj) Ki + 2Gi '
where the subscript i shows the interphase. The volume fraction of the interphase for composites containing spherical nanoparticles [21] can be calculated by
=%
r + r
-1
(11)
where r and ri are the radius of nanoparticles and interphase thickness, respectively. ri _ 0 results in ^ _ 0, indicating the absence of interphase in nanocomposite.
At the next step, the composite comprising the core-shell particles in the polymer matrix is considered and the moduli are predicted by the Berryman method assuming the core-shell particles as the filler phase (see Fig. 1) by
G _ Gm + (Gp - Gm)"--~
1
K = Km + ( Kp - Km )
+ 9i)+Q(% + 9i) P(% + 9i) 1 -(% +9i)+P (% +9i)'
(12) (13)
P =
Km + 4/3 Gp
Kp + 4/3Gp'
Q-
Gm + Fm
Gp + Fm'
(14)
(15)
The Young's modulus E can be generally determined for an isotropic solid as
e _-9KG_. (16)
3K + G
Assuming the matrix, filler, interphase and composite as isotropic elastic solids, K and G can be determined by
E
K
G=
3 - 6v
E
(17)
(18)
2 + 2v
where v is the Poisson ratio. The perfectly bonded interfaces of nanoparticles and interphase and matrix and interphase are supposed in the proposed method, and the imperfect bondings are revealed by the interphase properties.
3. Results and discussion
3.1. Evaluation of model by experimental results
Equation (16) calculates the Young's modulus in several samples assuming the absence and presence of interphase. The calculations are compared with the experimental data.
6.5-
O
5.0-
4.5
■ Experimental
-±-rx = 0
Proper interphase
■
■ Jtr
-▲
k^------
0.0 0.5 1.0 1.5 Si02, vol. %
2.0
4.5- ■ Experimental l£
4.0- -•- Proper interphase
<3 §3.5-
3.0-
2.51
2 3 4 CaCQ3, vol. %
3.5" 3.4
<3
PH
O3.3-
3.2-
3.1
■ Experimental
-A-r—0
-•-Proper interphase
■ ____
Ph
O
2.8 2.6 2.4 2.22.0
0
0.0 0.2 0.4 0.6 0.8 1.0 Si02, vol. %
■ Experimental \d_
rx = 0
-•- Proper interphase
4 6 8 CaCQ3, vol. °/c
10 12
Fig. 2. Comparison between experimental results and predictions of Young's modulus by the proposed method in the absence and presence of interphase for PEEK/SiO2 [22] (a), PS/SiO2 [23] (b), PA66/CaCO3 [13] (c) and PVC/CaCO3 [24] samples (d)
Figure 2 illustrates the experimental data and the predictions of Young's modulus for four samples including PEEK/SiO2 [22], PS/SiO2 [23], PA66/CaCO3 [13] and PVC/CaCO3 [24]. The (r, Em) values for these samples were reported as (6.5, 4.5), (8, 3.12), (19, 2.4) and (40, 2.12) (nm, GPa, respectively). The Ef is also considered as 80 and 26 GPa for SiO2 and CaCO3 nanoparticles, respectively. Moreover, v f is assumed as 0.17 [25] and 0.27 [13] for SiO2 and CaCO3. As observed in Fig. 2, the calculations in the absence of interphase r; = 0 are below the experimental data demonstrating that the absence of interphase underestimates the modulus. As a result, the interphase should be assumed in the nanocomposites, due to the high interfacial area and strong interfacial interaction between the polymer matrix and nanoparticles. The interphase occupies a large fraction of nanocomposites (even higher than nano-particles in some cases), which significantly affects the mechanical properties [26-28].
The present methodology for the shear and bulk moduli of nanocomposites accurately predicts the modulus for the reported samples. The calculations show good agreement with the experimental data by considering a proper interphase thickness and modulus as shown in Fig. 2. The highest value of r; should be smaller than the gyration radius of polymer macromolecules Rg, E; varies from the Young's modulus of polymer matrix Em to nanofiller modulus Ef. Thus, E; should be obtained between Em and Ef levels. The best values of (r;, E;) are calculated as (6, 22), (17, 65), (35, 25) and (3, 5) (nm, GPa) for PEEK/SiO2, PS/ SiO2, PA66/CaCO3 and PVC/CaCO3 samples, respectively. As observed, the interphase properties are obtained in the permitted ranges confirming the correctness of the proposed method assuming the interphase. The different values for the interphase properties in the reported samples indicate various levels of interfacial properties as a function of the miscibility between polymer matrix and nanoparticles, dispersion level of nanoparticles, etc. [29, 30].
3.2. Effects of component properties
The effects of component (matrix, nanofiller and interphase) properties on the predicted moduli are discussed.
Figure 3 shows the effects of % and r on different moduli of polymer particulate nanocomposite by contour plots at Em = 3 GPa, Ef = 50 GPa, vm = 0.4, vf = 0.25, r; = 10 nm and E; = 30 GPa. The illustrations for moduli indicate that the % and r parameters have similar effects on the shear, bulk and Young's moduli of polymer nano-composites. The high concentration of nanofiller and the smallest level of nanoparticle radius yield the best moduli. On the other hand, the worst calculations are reported by a low concentration of large nanoparticles. The high content of nanoparticles in the nanocomposite causes a high stiffening effect, which significantly reinforces the polymer ma-
trix. Smaller nanoparticles introduce a larger interfacial area between polymer matrix and nanoparticles, which improves the mechanical involvement of polymer chains to nanoparticles and increases the reinforcement of the polymer matrix. Accordingly, the effects of the nanoparticle content and size on the predicted moduli are logical, which confirm the correctness of the proposed method.
Figure 4 depicts the effects of the vm and vf factors on the various moduli of polymer nanocomposite based on the suggested method at % = 0.02, r = 20 nm, Em = 3 GPa, Ef = 50 GPa, r = 10 nm and E; = 30 GPa.
0.01 0.02 0.03 0.04 9f
Fig. 3. Contour plots for the dependence of shear (a), bulk (b) and Young's moduli (c) (all in GPa unit) on 9f and r parameters at Em = 3 GPa, Ef = 50 GPa, vm = 0.4, vf = 0.25, r; = 10 nm and E; = 30 GPa (color online)
It is observed (Fig. 4) that the vf parameter does not affect the shear and bulk moduli and these moduli only depend on the vm parameter, where a high vm induces the low and high levels for G and K, respectively. Therefore, vm differently governs the shear and bulk moduli of nanocomposites at the same values of other parameters. However, the variation of vm from 0.3 to 0.45 insignificantly changes the shear modulus, while considerably affects the bulk modulus. As observed in Fig. 4, c, both vm and vf factors affect the values of Young's modulus based on the present method. The best level of Young's modulus
is obtained by low vm and high vf values. Moreover, vm > 0.37 causes a low Young's modulus at different vf. As a result, vm and vf dissimilarly control the values of Young's modulus in polymer particulate nanocomposites. However, the Young's modulus slightly varies from 3.41 to 3.44 GPa by variation of these parameters, demonstrating the negligible effects of vm and vf on E. So, vm and vf parameters only cause a high influence on the bulk modulus of nanocomposites, while the shear and Young's moduli remain approximately constant at different levels of these factors.
Ef, GPa
Ef, GPa
Ef, GPa
0.30
0.35
0.40
Fig. 4. Shear (a), bulk (b) and Young's moduli (c) of nanocomposites (all in GPa unit) as a function of vm and vf factors at 9f = 0.02, r = 20 nm, Em = 3 GPa, Ef = 50 GPa, r; = 10 nm and E; = 30 GPa (color online)
Fig. 5. Contour plots for the dependence of shear (a), bulk (b) and Young's moduli (c) of nanocomposite (all in GPa unit) on Em and Ef parameters at 9f = 0.02, r = 20 nm, vm = 0.4, vf = 0.25, r; = 10 nm and E; = 30 GPa (color online)
v
E-v GPa | 3025 . 2015105 1
m
V
1.50 1.45 1.40 1.35 1.30 1.25 1.20 1.15
Fig. 6. The roles of r and Ei in shear (a), bulk (b) and Young's moduli (c) of polymer nanocomposites (all in GPa unit) according to the suggested method at 9f = 0.02, r = 20 nm, Em = 3 GPa, Ef = 50 GPa, v m = 0.4 and vf = 0.25 (color online)
Figure 5 also shows the effects of the Young's moduli of polymer matrix and nanoparticles on the different moduli of nanocomposites at % = 0.02, r = 20 nm, v m = 0.4, vf = = 0.25, r = 10 nm and Ei = 30 GPa. It is shown that Ef insignificantly changes the different moduli, but Em mainly affects the moduli of nanocomposites. The high modulus of polymer matrix causes a high modulus for polymer nanocomposite. The possible reason for the unimportant effect of Ef on the moduli of nanocomposites may be its high level compared to Em. Although the Young's modu-
lus of nanoparticles is assumed to reach 200 GPa in Fig. 5, the Young's modulus of nanocomposites improves to 4.4 GPa at the best condition with Em = 4 GPa.
The previous researches have reported that some phenomena such as aggregation/agglomeration of nanoparticles, weak interfacial adhesion and unsuitable condition for production of samples may make a poor improvement in the Young's modulus of nanocomposites [31-33]. Accordingly, the insignificant effect of Ef on the moduli of nanocomposites is logical based on the mentioned explanations. However, the variation of the shear modulus by Em and Ef parameters is negligible (Fig. 5, a), while the highest variation is observed in the bulk modulus (Fig. 5, b). Hence, Em and Ef parameters cause high differences in the bulk and Young's moduli of polymer nanocomposites.
Figure 6 illustrates the influences of the interphase properties including r and Ei on the shear, bulk and Young's moduli of polymer nanocomposites according to the suggested method at % = 0.02, r = 20 nm, Em = 3 GPa, Ef = 50 GPa, vm = 0.4 and vf = 0.25. The r and Ei parameters show relatively similar effects on the moduli of polymer nanocomposites. The best moduli are achieved by the highest levels of r and E;, whereas their small levels decrease the moduli. So, the interphase thickness and modulus have positive roles in the moduli of polymer nano-composites. These evidences are reasonable, because the interphase reinforces the polymer nanocomposites beside the nanoparticles. Clearly, a more-content and stiffer interphase produces a stiffer nanocomposite, because the interphase is commonly stiffer than the polymer matrix. The literature reports also indicated that the thickness and modulus of interphase positively improve the moduli of polymer nanocomposites [34-36]. The rational outputs of the proposed methodology for the effects of the interphase properties on the moduli prove the correctness of the currently proposed method. However, the most effects of ri and Ei are observed on the bulk modulus (Fig. 6, b), while the shear modulus (Fig. 6, a) is slightly affected by these parameters. As a result, the bulk modulus considerably relates to the interphase properties, while the shear modulus shows very low dependence on them. Generally, the bulk modulus demonstrates a wide variation by the properties of polymer matrix, nanoparticles and interphase based on the current approach. Furthermore, the shear modulus slightly changes depending on the properties of nanocomposite components.
4. Conclusions
In this paper, a core-shell structure was assumed for nanoparticles and interphase, and its shear and bulk moduli were calculated by the Berryman model. These core-shell particles were supposed as filler in the polymer matrix (see Fig. 1), and the moduli of the composite were predicted by the same model. The calculations of the proposed method
in the absence of interphase were below the experimental data demonstrating that the absence of interphase underestimates the moduli. Nevertheless, the Young's modulus calculations showed fine agreement with the experimental data by proper levels of the interphase properties. The best moduli of polymer nanocomposite were obtained by the high concentration of the smallest nanoparticles. Also, vm differently governed the shear and bulk moduli of nano-composites at the same values of other parameters. However, the variation of vm insignificantly changed the shear modulus, while considerably affected the bulk modulus. The Young's modulus of nanoparticles does not change all moduli, and only Em changes the moduli of nanocom-posites. The variation of shear modulus by the Em and Ef parameters was negligible, while the highest variation was observed in the bulk modulus. The r and E; parameters also represented relatively similar influences on the moduli of polymer nanocomposites. The best moduli were logically achieved by the highest levels of r; and E;, because the interphase reinforces the polymer nanocomposite in addition to nanoparticles. According to the present method, the bulk modulus demonstrated a wide variation by the properties of polymer matrix, nanoparticles and interphase, while the Young's and shear moduli showed moderate and slight changes by the properties of nanocomposite components, respectively.
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Received 19.12.2018, revised 18.02.2019, accepted 18.02.2019
Сведения об авторах
Yasser Zare, PhD, Dr., Islamic Azad University, Iran, [email protected]
Kyong Yop Rhee, PhD, Prof., Kyung Hee University, Republic of Korea, [email protected]
