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УДК 539.3
Упрощенная методика расчета параметра и модуля межфазного слоя для многослойной межфазной границы в полимерном нанокомпозите с использованием модуля Юнга
Y. Zare, K.Y. Rhee
Университет Кёнхи, Йонъин, 446-701, Республика Корея
В работе предложена методика расчета параметра Y и модуля межфазного слоя для многослойной межфазной границы в полимерном нанокомпозите на основе простых моделей расчета модуля Юнга полимерных нанокомпозитов. Корректность разработанной модели проверена путем сравнения с экспериментальными результатами для нескольких образцов. Также изучено влияние параметра Y на модули Юнга межфазного слоя Ei и полимерного нанокомпозита Ec. Различные значения параметра Y, полученные для рассматриваемых образцов, указывают на различные диапазоны свойств межфазных слоев. Согласно результатам, достижение необходимого значения межфазного модуля в полимерных нанокомпозитах возможно при высоких значениях как параметра Y, так и модуля нанонаполнителя Ep.
Ключевые слова: полимерные нанокомпозиты, параметр межфазного слоя, модуль Юнга, микромеханические модели
DOI 10.24411/1683-805X-2019-15009
A simple technique for calculation of an interphase parameter and interphase modulus for multilayered interphase region in polymer nanocomposites via modeling of Young's modulus
Y. Zare and K.Y. Rhee
Department of Mechanical Engineering, College of Engineering, Kyung Hee University, Yongin, 446-701, Republic of Korea
In this work, a simple technique is suggested applying the simple models for Young's modulus of polymer nanocomposites to calculate the Y interphase parameter and interphase modulus for multilayered interphase in polymer nanocomposites. The developed model is compared to the experimental results of several samples to examine its correctness. The effects of Y on the Young's moduli of interphase Ei and polymer nanocomposite Ec are also studied. The different levels of Y parameter calculated for the reported samples demonstrate the various extents of interphase properties. The results show that the high levels of both Y parameter and nanofiller modulus Ep are required to achieve a desirable interphase modulus in polymer nanocomposites.
Keywords: polymer nanocomposites, interphase parameter, Young's modulus, micromechanical models
1. Introduction
The high performance of polymer nanocomposites has been widely used in various engineering applications. The nanocomposites demonstrate several benefits such as low weight, high mechanical properties, easy processing, etc. [1-19]. It is known that the development ofnanocomposites requires the understanding of nanostructures, the properties of constituents and the interaction between them. A considerable attempt has been concentrated on these issues in the previous studies to find the main parameters, which can give an optimized sample. It was stated that the morpho-
logy of the polymer nanocomposites plays a critical role in obtaining the multifunctional properties [20-27]. The conventional models for microcomposites are only a function of constituent properties such as volume fraction and modulus, while the mechanical properties of nanocomposites significantly depend on the interfacial interaction and the interphase characteristics between polymer matrix and na-noparticles [28, 29].
The interphase characteristics cannot be directly characterized from experiments, due to the small thickness of interphase. Accordingly, the modeling approaches are applied
© Zare Y., Rhee K.Y., 2019
to measure the properties of interphase. Ji et al. [30] proposed a simple model for Young's modulus of nanocompo-sites assuming matrix, nanofiller and interphase between polymer and nanoparticles. This model was productively applied to calculate the Young's modulus and thickness of interphase in polymer nanocomposites containing different nanofillers [30].
Several authors have modeled the interphase as a multi-layered phase, which includes different properties for each layer [31-34]. The different properties of interphase layers such as thickness and modulus and their influences on na-nocomposite behavior were discussed in literature. Shabana [31] investigated the effects of interphase thickness, number of layers, properties of each layer, progressive debond-ing damage, size and aspect ratio of nanoparticles on the effective thermomechanical properties of nanocomposites. However, the properties of multilayered interphase and its influences on the performance of nanocomposites were limi-tedly analyzed in the previous articles.
In this work, a simple method is developed by some micromechanical models for Young's modulus of nanocom-posites such as Ji model to calculate the interphase parameter and interphase modulus in polymer nanocomposites. This technique is originally developed for nanocomposites assuming interphase properties, whereas the existing models in this area were initially suggested for conventional composites. The developed equation simply correlates the Young's modulus of nanocomposites to Y interphase parameter by the properties of constituents. The Y parameter is calculated for some samples and its effects on the modulus of nanocomposites are studied.
2. Theoretical development
Ji et al. [30] proposed a model for Young's modulus of nanocomposites assuming the effects of matrix (m), nanoparticles (p) and interphase (i). The Ji model for nanocomposites including spherical (1), layered (2) and cylindrical (3) nanoparticles is stated as a-p
E =
1 -a + -
1 -a + a(m-1)/ln m
P
1 -a + (a-P)(m + 1)/2 + PEp/Em
(1)
If1 \3
a=v| r+1J 9f, (2)
a2 2 d + 1)9f, (3)
If t A2
a3 r + l) 9f, (4)
P=V97, (5)
E, m = ——, E Em (6)
where Em, Ep and E, are the Young's moduli of matrix, nanoparticles and interphase, respectively, Er is defined as Ec/Em in which Ec is the Young's modulus ofnanocom-posite, r and d are the radius and thickness of nanofiller, while t is the thickness of interphase, % is also the volume fraction of nanofiller.
The volume fractions of interphase 9, in different polymer nanocomposites [35] are defined as
-1 , (7)
9,1 = 9f
r +1
9,2 = 9f| —
9,3 = 9f
r
2t
d
r + t
-1
(8)
(9)
Assuming Eqs. (7)-(9) in Eqs. (2)-(4), all a parameters can be expressed as
a = V9, + 9f • (10)
As a result, Ji model for all types of nanocomposites can be presented as
Er = [1 -^9, +9f +
+ V9, + 9f -V%_+
1 ->/9, +9f +aM +9f (m -1)/ln m
+797(1 -V99+9+(V 9,+9f -V97) *
X (m +1)/2 + V97 Ep/ Em)-1]-1. (11)
The Young's modulus of polymer nanocomposites was related to 9, [35] as
Er = 1 + 11(9f +9,)1'7. (12)
This equation considers the role of interphase volume fraction beside the filler fraction, because both filler and interphase affect the modulus of nanocomposites at the same time. Since the interphase regions cover the nanoparticles, this assumption is true. Many experimental data of Young's modulus in various samples containing spherical nanopar-ticles, nanoclay and carbon nanotube (CNT) reported in [36-41] show good agreement with calculations of this model. Therefore, this model can be used for determination of 9, in nanocomposites.
By rearranging the latter equation, parameter a in Eq. (10) can be given by
a
= V9f +9, =
Er -1 11
0.294
(13)
expressing a by the experimental data of Young's modulus as Er(exp).
The interphase between polymer matrix and nanopar-ticles can be divided into n layers, in which the thermo-mechanical properties continuously change from nanopar-ticles surface to polymer matrix.
Assuming a similar thickness for all interphase layers, the thickness of kth layer is determined by
tk = t/n. (14)
Fig. 1. The schematic procedure for calculation of Y and Ei parameters
Also, x is expressed as the distance from a nanoparticle surface (x = 0) to polymer matrix (x = t). The distance x for central point of kth layer xk is expressed as
xk = ktk - h!2
(15)
In our previous work [42], it was shown that the Young's modulus of interphase layers can change by a power function as
Ek = Ep -(Ep - Em)(xk/t)Y, (16)
where Y is an interphase parameter demonstrating the interphase quality such as the interfacial area between polymer and nanoparticles correlated to dispersion of nanofiller, the interfacial interaction/adhesion, the thickness and strength of interphase regions surrounding nanoparticles. A higher Y indicates larger interfacial area between polymer and nanoparticles due to proper dispersion of nanoparticles in polymer matrix, stronger interfacial interaction/adhesion, thicker and stronger interphase. However, Y = 0 results in Ek = Em indicating the absence of interphase regions in the sample.
Assuming that the modulus of interphase area linearly changes from polymer matrix to nanoparticles, Ei as the average modulus of interphase can be considered as Ek at xk = t/ 2. So, the following equation can be derived for Ei
as
e = ep
(ep - em)0.5 . (17)
As a result, parameter m in Ji model (Eq. (6)) can be presented as
" ' (18)
m = [Ep - (Ep - Em)0.57]/e„
-p V p
Taking into account the suggested equations for a and m in Eqs. (13) and (18), Ji model can generally express the modulus of different nanocomposites as
which correlates Er to the properties of constituents and Y interphase parameter.
The schematic procedure for calculation of Y and Ei is illustrated in Fig. 1. To calculate the Y parameter, the experimental values of Er, Em, ep and % are applied into Eq. (19). When the prediction of this equation shows the highest agreement with the experimental Er, the most suitable value of Y is obtained. Otherwise, a different Y should be tested. After that the Y value can be applied into Eq. (17) to calculate the average level of Ei in the sample.
3. Results and discussion
Several nanocomposite samples from valid literature and their characteristics are reported in the table. The experimental Young's moduli of these samples are applied into Eq. (19) to calculate the suitable Y parameter and interphase modulus according to the procedure demonstrated in Fig. 1. The table shows the proper values of Y and ei for all samples. To confirm the values of Y parameter for the samples, the agreement between experimental relative modulus of nanocomposites er and the calculations of Eq. (19) assuming Y levels are illustrated.
Figures 2 and 3 exhibit the difference between experimental and theoretical results of er calculated by Eq. (19) for all samples at the suitable values of Y reported in the table. As observed, the theoretical values of er acceptably agree with the experimental results. As a result, the presented procedure for calculation of Y parameter by experimental data of Young's modulus gives accurate outputs. In other words, the proper agreements between the experimental data and the predictions of Eq. (19) confirm the correctness and uniqueness of Y parameter for the samples.
ET = [1 -((ET -1)/1 If294 +...
((Er -1)/1 1)0294 -V^f
1 - ((Er -1)/11)0 294 +
((er -1)/11) [(ep - (ep - em )0.5 )jEm -1]
ln[(ep - (ep - em)0.57)/em]
_>£_
- + ...
1-1
1 - ((er -1)/11)°'294 + 1/2[((er - 1)/11)0 294][(ep - (ep - em )0.57 )/em + 1] + Jqf e^en
(19)
No. Sample [Ref.] Em' GPa Ep, GPa Y (Eq. (19)) Ei, GPa (Eq. (17))
1 iPP1 /PPgMA2/SiO2 [40] 1.420 80 0.250 13.9
2 PA63/OMt4 [43] 1.500 178 0.600 61.5
3 PLA5/OMt [44] 1.700 178 0.180 22.4
4 PP/OMt [45] 0.780 178 0.080 24.9
5 Epoxy/MWCNT6 [41] 1.900 1000 0.050 35.9
6 PI7/MWCNT-COOH [46] 0.910 1000 0.007 5.70
7 PET8/MWCNT [47] 1.550 1000 0.005 5.00
8 PU9/MWCNT [48] 0.076 1000 0.003 2.80
Table
The characteristics of samples and their interphase properties
iPP1 /PPgMA2/SiO2 [40]
PA63/OMt4 [43]
PLA5/OMt [44]
PP/OMt [45]
Epoxy/MWCNT6 [41]
PI7/MWCNT-COOH [46]
PET8/MWCNT [47]
PU9/MWCNT [48]
Em, GPa
1.420
1.500
1.700
0.780
1.900
0.910
1.550
0.076
Ep, GPa
80
178
178
178
1000
1000
1000
1000
Y (Eq. (19))
0.250
0.600
0.180
0.080
0.050
0.007
0.005
0.003
Ei, GPa (Eq. (17))
13.9
61.5
22.4
24.9
35.9
5.70
5.00
2.80
1—isotactic polypropylene, 2—maleic anhydride grafted polypropylene, 3—polyamide 6, 4—organically modified montmorillonite, 5—poly (lactic acid), 6—multiwalled carbon nanotubes, 7—polyimide, 8— poly (ethylene terephthalate), 9—polyurethane.
The different values of Y are obtained for the reported samples based on the properties of interphase formed between polymer matrix and nanoparticles. The different values of Y demonstrate the various levels of interphase properties such as thickness, strength and modulus. The properties of interphase depend on several parameters such as interfacial area, dispersion quality, compatibility between polymer matrix and nanofiller and interfacial interaction/adhesion [49, 50]. The fine dispersion of nanoparticles and proper treatment, modification and functionalization of nanofiller or polymer chains can promote the interfacial interaction/adhesion and thus, enhance the interphase properties.
The highest level of Y parameter is calculated as 0.6 for PA6/OMt [43]. The morphological images showed the intercalation and some exfoliation of organically-modified layered silicates in PA6 matrix. The mechanical properties of this nanocomposite demonstrate that the clay nanoparticles stiffen the polymer matrix. Accordingly, good dispersion of nanoparticles providing large interfacial area between polymer and nanoparticles in addition to the strong interaction/bonding between polymer matrix and nanoparticles introduce a strong interphase, which obtains a high Y. So, the significant level of Y parameter calculated for this sample by the developed model is reasonable.
6 8 10 12 14 16
Si02, wt %
4 6 OMt, wt %
2 3
OMt, wt % OMt, wt %
Fig. 2. The difference between experimental and theoretical Er calculated by Eq. (19) for samples 1 (a), 2 (b), 3 (c) and 4 (d) in the table
1.6-
0 1 2 3 4 5 MWCNT, wt %
1.15H
i.ioh
1.05 H
1.00
0.0 0.5 1.0 1.5 2.0
MWCNT, wt %
-1-1-1-n-
0 1 2 3 4 5 MWCNT, wt %
0.0 0.5 1.0 1.5
MWCNT, wt %
Fig. 3. The experimental and theoretical E (Eq. (19)) for samples 5 (a), 6 (b), 7 (c) and 8 (d) in the table
According to the Y values exhibited in the table, the samples with very high Ep such as samples 5-8 show lower levels of Y compared to others, while the morphological and mechanical properties of these samples demonstrate acceptable results. For example, raw and carboxylic func-tionalized MWCNT were incorporated to polyimide (PI) to study the effects of functionalized MWCNT on the overall properties of nanocomposite [46]. The presence of hydrogen bonds between MWCNT-COOH and PI chains caused a strong interaction between functionalized MWCNT and PI matrix, which greatly enhanced the dispersion as well as interfacial adhesion. The overall mechanical performance of this sample was improved, but large difference between Em and Ep introduces a low Y for this sample. Accordingly, a low level of Y is calculated for these na-nocomposites, because their Young's modulus did not significantly improve assuming Ep extent. The table also reports the values of Ei calculated by Eq. (17) for each sample. Various Ei values are obtained for the reported samples. Some samples such as sample 1 with high Y show low Ei and also, some samples with large ep dis-
play poor Ei. These evidences indicate that the Ei values are correlated to both Y and Ep parameters.
Figure 4 illustrates the effect of Y value on Er calculated by Eq. (19) at constant values of other parameters (the characteristics of sample 3 are used). When the effects of interphase is negligible (Y = 0.01), very low level of E
Fig. 4. The impacts of different Y values on Er calculated by Eq.(19) based on the characteristics of sample 3 in the table
Ei, GPa 150 r-
Ep, GPa 200
Ep, GPa
500
400
300
140
120
100
0 0.1
Fig. 5. The roles of Yand Ep parameters in E, values (Eq. (17)): 3D (a) and contour plots (b) (color online)
is calculated. However, a stronger interphase causes higher Er in the nanocomposite. It is observed that a greater level of Y results in a better Er, which indicates the important role of Y interphase parameter in the modulus of polymer nanocomposites. The interphase is a bridge in polymer nanocomposites, which transfers the stress from matrix to nanoparticles. A poor bridge undoubtedly breaks by slight stress before the failure of sample.
Modulus E; as a function of parameters Y and Ep (Eq. (17)) is exhibited in Fig. 5 by 3D and contour plots (Em = 2 GPa). E; increases by simultaneous increment of parameters Y and Ep. It means that a great level of one parameter at lowlevel of another shows a low E;. The highest level of E; is obtained by the highest levels of Y and Ep, whereas the slightest values of E; are found at the smallest Y and Ep. As a result, both Y and Ep parameters have important effects on E;. A high level of E; produces a strong stiffness in polymer nanocomposites, because the mechanical properties of polymer nanocomposites are a function of their constituents as matrix, nanofiller and interphase. In fact, the positive effects of Y and Ep on E; demonstrate that the formation of a strong interphase is necessary to improve the mechanical properties of nanocom-posite. A strong interphase can adhere the nanoparticles to matrix, which transfers the loaded stress from matrix to nanoparticles [51]. So, its role in the modulus of nanocom-posite is important.
It should be indicated that the interfacial/interphase properties in polymer nanocomposites can be calculated by modeling the properties. The former models were basically proposed for two-phase microcomposites, while the micro-mechanical models applied in the suggested method correctly consider the interphase in polymer nanocomposites. Accordingly, the suggested technique in the present article
can express very careful results for interphase level by Y parameter.
4. Conclusions
An interphase parameter Y was assumed for multilayer-ed interphase regions in polymer nanocomposites. The parameter Y and the average interphase modulus were simply calculated by a developed technique for Young's modulus of nanocomposite. The developed technique applied some micromechanical models, which appropriately consider the interphase regions in polymer nanocomposites. Many experimental data of Young's modulus from several nanocom-posites confirm the accuracy of the developed technique. The different calculations of Y for the reported samples are attributed to the dissimilar properties of interphase regions. A higher Y commonly produces a stronger interphase, which more improves the modulus of nanocomposite. The interphase modulus increases by simultaneous increment of Y and filler modulus. So, the interphase modulus depends on Y parameter and filler modulus indicating that only the excellent stiffness of nanofiller cannot grow the mechanical properties of nanocomposite, but a strong interphase is required to transfer more stress from polymer matrix to nano-particles.
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Rece,ved September 06, 2019, rev,sed September 18, 2019, accepted September 25, 2019
Сведения об авторах
Yasser Zare, PhD, Dr., Kyung Hee University, Republic of Korea, y.zare@aut.ac.ir Kyong Yop Rhee, PhD, Prof., Kyung Hee University, Republic of Korea, rheeky@khu.ac.kr
