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УДК 539.4
Моделирование модуля упругости вторично переработанных нанокомпозитов на основе полиэтилена и глины
Y. Zare1, K.-Y. Rhee2
1 Институт рака Мотамеда, Тегеран, 14155-4364, Иран 2 Университет Кёнхи, Йонъин, 446-701, Республика Корея
Проведено исследование модуля упругости вторично переработанных нанокомпозитов на основе полиэтилена и глины с использованием ряда традиционных моделей. Показано, что модели Paul и Takayanagi дают хорошие результаты по сравнению с экспериментальными данными. Модель Хирша дает более точные предсказания при значении x равном 0.1, показывая на соответствие модуля упругости нанокомпозитов обратному правилу в модели смесей. Сравнение экспериментальных и теоретических результатов моделей Гута подтверждает необходимость учета коэффициентов жесткости, таких как соотношение ширины и толщины силикатных слоев. Теоретические данные хорошо согласуются с экспериментальными данными при значении указанного соотношения равном 8 для силикатных слоев. При этом необходимо учитывать ориентацию и случайное объемное распределение пластинок наноглины в соответствии с моделью Halpin-Tsai. Для некоторых моделей выполнено упрощение и модификация с целью повышения точности прогнозирования. Модифицированные модели очень просты, поскольку используют для прогноза только модуль Юнга и объемную долю компонентов.
Ключевые слова: вторичная переработка, полимерные нанокомпозиты, механические свойства
DOI 10.24411/1683-805X-2020-15008
Modeling of tensile modulus for recycled PET/Clay nanocomposites
Y. Zare1 and K.-Y. Rhee2
1 Biomaterials and Tissue Engineering Research Group, Department of Interdisciplinary Technologies, Breast Cancer Research Center, Motamed Cancer Institute, ACECR, Tehran, 14155-4364, Iran 2 Department of Mechanical Engineering, College of Engineering, Kyung Hee University, Yongin, 446-701, Republic of Korea
The present study is devoted to tensile modulus of recycle PET/clay nanocomposites based on the various conventional models. Some models such as Paul and Takayanagi provide good results, when compared with the experimental data. Hirsch model predicts more accurate data by the x value of 0.1 revealing that the tensile modulus of current nanocomposites conforms to the inverse rule of mixture model. The comparison between experimental and theoretical results of Guth models confirm the necessity for taking account of stiffening factors such as aspect ratio of silicate layers. The theoretical data present a good agreement with experimental data considering the aspect ratio of 8 for silicate layers. In addition, the orientation and 3D random dispersion of the nanoclay platelets should be assumed according to Halpin-Tsai model. Further, some simplifications and modifications are carried out on the several models to enhance their predictability. The modified models are too simple, because they only require to Young's modulus and volume fraction of components for prediction.
Keywords: recycling, polymer nanocomposites, mechanical properties
1. Introduction
Every day, large quantities of waste polymers are produced creating more than 12% of the municipal solid waste stream [1]. The environmental, economic, and petroleum considerations have caused the scien-
tific community to increasingly deal with this problem. The efficient treatment of waste polymers is still a complicated challenge. The conventional methods such as combustion or burying underground cause the negative effect on the environment. Recycling process
© Zare Y., Rhee K.-Y., 2020
is one of the best ways to treat waste polymer causing an economic and ecologic benefits. The rate of plastic recycling for different polymers varies much, which an average rate of 7% was reported [2].
Recycling involves different difficulties including waste separation, finally limited application, and sensitivity of some polymers like polyethylene terephtha-late (PET) to heat, moisture and acidic environments [3, 4]. In addition, the main problem in recycling is the degradation of polymer structure in reprocessing which introduces much lower mechanical properties [5-7]. Therefore, many researchers have been interested in the recycling of the polymers, with the objectives of selecting the best amount and types of modifiers that could compensate the loss properties.
It seems that the easiest way to recycle mixed plastic waste including blends and composites is the addition of other component to the wastes. The oriented polymers are typically ductile suffering a ductile-brittle transition for low temperatures and high strain speeds. To increase the properties of these materials, minimizing their plastic resistance was suggested [8]. The most effective method to toughen polymers includes the addition of cavitated rigid particles to attain an interparticle ligament dimension [9]. In the recent years, nano structures are an attractive topic [10-12] and a low content of nanofillers causes an appreciable enhancements of mechanical, thermal, optical and barrier properties, compared to conventional microparti-cles [13-17]. The nanofillers increase the interphase surface of the components and the surface area/volume ratio resulting in a high improvement of the overall performances. They can produce cavitations or debonding, transforming material into a cellular solid and increasing ductility and strength [18].
Polyethylene terephthalate as a low cost and high performance polymer, has been widely used in various applications such as textile, reinforcement of tires, food and beverage packaging. Food and beverage application of polyethylene terephthalate has shown enormous growth over the recent years. The international attempt has been made to reduce the environmental influence caused by the waste polyethylene terephthalate products through recycling process. The loss of melt strength causes an inconsistency of material after leaving the extruder die, which make impossible the production of sheets or perfect profiles. Reprocessing of polyethylene terephthalate waste with the lower intrinsic viscosity is not probable. These defects can be compensated by nanofiller addition.
Brazil is one of the main countries in polyethylene terephthalate recycling [19]. Large amounts of recy-
cled PET from municipal wastes are currently used in the same applications of virgin PET. The waste production of polyethylene terephthalate is in the highest quantity about 20-30% of total polymer wastes. From these points of view, the potential practical importance of recycled polyethylene terephthalate nano-composites and the possibility of significant enhancement of properties made the choice of this system very attractive. The potential applications of the recycled polymer/nanofiller systems can be noted as household applications and car components, building industry and other polymer products.
At the present, further development and optimization of nanocomposite behavior has been a major challenge in the scientific communities due to the potential aspect of these advanced materials.
The modeling of nanocomposite properties suggests the extensive information which reduces the number of experiments involving much cost and time. For that reason, the analysis of mechanical properties such as tensile modulus of nanocomposites has attracted much attention [20-23]. Although more studies have been performed for modeling of nanocom-posite properties, the behavior of recycled polymer nanocomposites have not been analyzed. Undoubtedly, the models would help the development of recycling process. In this paper, the tensile modulus of recycled PET/clay nanocomposite is studied based on the different models. Some modifications are carried out for predicting the tensile modulus of these materials.
2. Technical background
The simplest models for predicting the tensile modulus are "parallel" and "series" models [24-26]. In the "parallel" or "rule of mixtures" model, the equal strain is taken account in the matrix and filler phases. This model is expressed as
E = Em^m + Ef f (1)
where Em and Ef are the Young's modulus of matrix and filler, respectively, and ^m and ^f are the volume fraction of the matrix and filler. In most cases, rule of mixture suggests the upper limit of the tensile modulus.
The "series" or "inverse rule of mixtures" model assumes the same stress in the matrix and filler phases, represented as
1 <b
E Em
(2)
These models were modified to achieve accuracy in the modulus prediction.
Guth and Gold used the Smallwood-Einstein approach to taking account the interaction between filler particles [27, 28]. The Guth model is shown as
E = Em(1 + 2.5<f +14.14,2). (3)
Guth [29] later developed this equation by introducing a shape factor in order to account accelerated stiffening for rod-like filler particles as
E _ Em[1 + 0.67афг + 1.62(афг )2],
(4)
where a = wit is the aspect ratio of filler w and t are the width and thickness of the dispersed filler, respectively.
Paul assumed that consistent stress is applied at the matrix-filler boundary [30]. Paul model is given by following equations:
1 + (m -1)f
E Em
(5)
1 + (m - 1)(<2/3 -<f)' m = Ef/ Em. (6)
Counto assumed the perfect adhesion in the matrix-filler interface and suggested the following model [31]. Counto model is presented in equation
1 _ 1-Ф1
12
1
E Em (1 -фГ)/фГЕт + Ef
(7)
Hirsch combined the series and parallel models and presented Eq. (8) [32], in which x and (1 - x) represent the relative contributions of composite conforming to the upper and lower limits of modulus, respectively:
E _ X(EmФm + Ef ф^ + (1 - x)-
Ef Em
(8)
Ef 4m + Em4f
Riley developed the rule of mixtures model by introducing a modulus reduction factor (MRF) [33, 34]. He took account the aspect ratio of filler and the shear modulus of matrix G in MRF factor. Riley model, nominated as modified rule of mixtures (MROM) is presented as
E = Em4m + MRF4f Ef, (9)
ln(u +1)
MRF _ 1 --
u _
1 fl
(10)
(11)
а V Ef К
Halpin and Tsai [35-37] introduced a mathematical model as
1 +
E - Em
1 '
(12)
"Л _
Ef/Em -1
(13)
Ef/ Em
^ = 2a. (14)
For unidirectional oriented fillers, in the direction of the fiber or in the plane of the platelet, it can be stated that the modulus of composite is calculated from Eq. (12).
The calculation of the modulus of a composite with randomly oriented fillers can be carried out by laminate plate theory [36, 38]. For a 2D in-plane random material, the modulus of the composite can be predicted as E2Drand = 0.375E1 + 0.625E2. (15)
Here Ei and E2 are the elastic moduli of the composite in the longitudinal and transverse directions, respectively coinciding with upper and lower bound values. Both Ei and E2 are calculated from Eq. (12), but for E2, the aspect ratio is assumed to be 1 (a = 1).
For a fully random 3D isotropic material [39], the modulus for fibre and platelet composites can be presented as
E3D rand fibre = 0.184^ + 0.8^, (16)
E3D rand platelet = 0.49^ + 0.5^. (17)
Halpin-Tsai model was later modified by Kerner and Nielsen in 1990's [40]. They eliminated the effect of filler aspect ratio and consider two parameters: Af as a function of the Poisson ratio of matrix vm, and ф!^ as the maximum volumetric packing fraction of the filler (true volume of the filler/apparent volume occupied by the filler).
The Kerner-Nielsen model is given by following equations:
1 + ABфf
E _ Em
A _
1 -
7 - 5Vm
B _
- 10v m
_ Ef/Em - 1
Ef/ Em + Af '
p _ 1 + ф21-^max.
(18)
(19)
(20)
(21)
Lewis and Nielsen [41, 42] also considered the maximum volumetric packing fraction of the filler and modified the Halpin-Tsai model into the following equations:
1 +
E - Em
ф_ 1 + 1
1-ФЧ^ '
1 -ф™
(22)
(23)
u
max
Kerner proposed a model [43-45] which was initially developed to predict the shear modulus of composites. He supposes the Poisson ratio of matrix Vm as a key parameter. The model is expressed as 15(1 -v) ^
Table 1. The experimental data of tensile modulus
E = Em
1 +-
(24)
8 - 10v 1
Hui and Shia [46, 47] developed a model for composites assuming the perfect interfacial adhesion between the polymer matrix and platelets, given by equations
3
E - Em
1
1
.C + C + A
_1
C = ^f +
Ef _ Em
+ 3(1 -f
(1 - g ) a2 - g 12
a2 -1
A = (1 -f
g = a, 3(a2 + 0.25)g - 2a2 a2 -1 '
(25)
(26)
(27)
(28)
E - Em
1 -a + -
1 -a + a(k-1)/ln k
ß
1 - a + (a - ß)(k + 1)/2 + ßEf/ Em
-1
(29)
(30)
(31)
where t and ti are the thickness filler and interface, respectively. Parameter k is the ratio of the interphase modulus on the surface of the platelet Ei and the Young's modulus of the matrix as k = Ei/Em.
If the effects of interface are neglected (ti = 0), Ji et al. model is reduced to the two phase model of Taka-yanagi [49] as
p
E - Em
1 -ß +
1 -ß + ßEf/ E,
(32)
m J
3. Results and discussion
3.1. Evaluation of models
The experimental results of tensile modulus for recycled PET/Cloisite 25A nanocomposite was selected
No. Nanoclay, wt % Tensile modulus, GPa
1 0 2.55 ± 0.08
2 1 2.8 ± 0.1
3 3 3 ± 0.16
4 5 3.34 ± 0.2
Ji et al. [48] proposed a three-phase model (matrix, filler and interphase) and considered the filler size and content, interface size and the Young's modulus of phases. The aspect ratio of filler was assumed to be much higher than 10 (a >>10).
The Ji model is expressed as
a-p
from [50] for modeling. The experimentally measured data of tensile modulus are shown in Table 1.
Figure 1 shows the theoretical predictions of tensile modulus by rule of mixture, inverse rule of mixture, MROM, Paul and Counto models. As shown, the rule of mixture calculates the much higher tensile modulus than that of experimental results. In the most cases, this model overestimates the modulus. The possible reason can be the high modulus of nanofillers causing the deviation of predicted data from experimentally measured ones.
Inverse rule of mixtures and MROM models cannot give any accurate predictions while the MROM suggests the lowest calculations of tensile modulus. For MROM model, the various aspect ratios of nano-clay have not any effect on the theoretical results. This phenomena has been reported in the ternary nanocomposite [51], but Borse et al. [33] have found that the MROM model was fitted to the experimental data of PA6/clay nanocomposite for various nanoclay types with the aspect ratio of 7-130. The predicted tensile modulus by Paul and Counto models demonstrates that the Counto has better prediction ability than that of Paul model while the Paul give higher predictions compared to Counto equation.
Figure 2 illustrates the theoretical results of Hirsch model. As expected based on the series and parallel models in Fig. 1, the comparison between values of x
CLh <
o<
3 o '
£4-
• Experimental —■—Rule of mixtures
—•—Paul >
—t— Counto
—*— Inverse rule
of mixturesys
—»—MROM/ ,1t:-- " T -—
0 1 2 3 4 5 6 Nanoclay, wt %
Fig. 1. Experimental and theoretical tensile modulus by rule of mixtures, inverse rule of mixtures, MROM, Paul and Counto models
+
4.5
ed PH
O
2.5J
• Experimental
—■— Hirsch
2
■—' 7*
4.5
Fig. 3. The predicted tensile modulus by Guth and developed Guth models. a is the aspect ratio of nanoclay layers, a = 5 (1), 8 (2), 12 (3)
Ph
o
3
-a o
o 3, £ jj
"cfl
a eS
5-
0 1 2 3 4 5 Nanoclay, wt %
Fig. 2. Experimental tensile modulus and theoretical data by Hirsch model. x demonstrates the relative contribution of modulus conforming to the rule of mixtures model: x = 0.05 (1), 0.10 (2), 0.15 (3), 0.20 (4)
and 1 - x should be much higher. The x value of 0.1 provides more precise data especially in the higher nanoclay contents. It reveals that the tensile modulus of recycled PET/clay nanocomposite follow the rule of mixtures and inverse rule of mixture about 10% and 90%, respectively. The high difference between the x and 1 - x can be due to the much high Young's modulus of nanoclay layers (about 180 GPa) which is much more than microparticles.
Figure 3 demonstrates the predictions of Guth and developed Guth models. The Guth model underestimates the tensile modulus confirming the stiffening effect of nanofiller presented here as aspect ratio. The nanofiller geometry depends on the constituent nature as well as the processing method of nanocomposites which affect the filler size applying the shear stress. The developed Guth model can predict the tensile modulus with nanoclay aspect ratio of 8 (a = 8). The size of silicate layers agrees with the obtained results about the morphology of recycled PET/clay nanocomposite. As indicated in Pegoretti work [50], the
2.5
• Experimental
—■— Halpin-Tsai
—*— 2D in-plane random
—*— 3D random fibre
— ■- 3D random platelet >
^ . 1
2 3 4 Nanoclay, wt %
Fig. 4. The tensile modulus predictions by Halpin-Tsai and related models for 2D in-plane and 3D random fillers
randomly dispersed nanoclay layers are shown in the STEM micrographs of recycled PET/clay nanocomposite. The level of exfoliation is correlated with aspect ratio of nanofillers [52]. In these samples, the lower aspect ratio of silicate layers represent the higher level of exfoliated structure of nanoclay in the recycled PET matrix.
Figure 4 represents the modeling of tensile modulus by Halpin-Tsai equation and other related models considering the orientation and dimension of nano-layers. The Halpin-Tsai overestimates the modulus as found in other studies [24, 53, 54]. The overprediction of Halpin-Tsai model could be associated to the lower contribution of the plate-like nanoclay (two dimensional filler) to tensile modulus in comparison to the rod-like (one dimensional filler) [51]. However, the models assuming the orientation and dimension of silicate layers show better agreement with experimentally measured data. It is known that the nanoclay layers are dispersed in the matrix as in tercalated or exfoliated morphology [55-57]. In the most cases like the current work, partially intercalated/exfoliated platelets are formed in the samples having the 3D random structure. According to modeling results, the experimental data coincide with the 3D random platelet model.
Fig. 5. The theoretical tensile modulus by Kerner-Nielsen, Lewis-Nielsen and Kerner models
The theoretical results of Kerner-Nielsen, LewisNielsen and Kerner models can be observed in the Fig. 5. The calculated predictions using Lewis-Nielsen are well fitted to the experimental data but Kerner-Nielsen model show some error for prediction. It was found that the $max values from 0.05 to 0.95 do not change the predicted data by both Kerner-Nielsen and Lewis-Nielsen models. As a result, the data are presented independent of $max. The predicted data by Kerner model do not show any reasonable agreement with the experimental results. The theoretical results of Kerner-Nielsen and Kerner models are much similarly while they consider different parameters for prediction.
Figure 6 represents the modeling of tensile modulus through Hui-Shia and Takayanagi equations. The theoretical values of Hui-Shia model are better fitted to the experimental data assuming the aspect ratio of 12. According to the modeling results, the aspect ratio between 8 and 12 can be chosen for the nanoclay layers in the recycled PET/clay nanocomposite. The prediction capability of the Takayanagi model is significant which reveals that the interface phase is not taken account in Ji et al. model, i.e. ti = 0, as indicated in background. The experimental tensile modulus of PA6/polypropylene-grafted-maleic anhydride/nanoclay ternary nanocomposite are also fitted to the theoretical data assuming ti = 0 [58]. It was indicated that Ji's model can be applicable for higher aspect ratio of filler much more than 10. Possibly, this factor affects the results of Ji's model. However, judging about the interface phase in the recycled PET/clay nanocomposite needs to further research in future using the microscopic observations.
3.2. Development of MROM model
In the MROM model, it was indicated that the aspect ratio of silicate layers has not any influence on the results. Also, the various values of shear modulus of matrix have not any significant role in the predicted
Table 2. MROM results at different values of shear modulus of matrix G and aspect ratio of filler a
3.5-
cö PH
a
C/T J3
3
o -
B
JD
"co G
£
2.5-
• Experimental
...a— Hui-Shia (a = 8)
—■—Hui-Shia (a = 12)
— -Takayanagi
*****
1 2 3 4 5 Nanoclay, wt %
Fig. 6. The predicted tensile modulus by Hui-Shia and Takayanagi models
Nanoclay, wt % G, GPa a E, GPa
1 1 10 2.5312
1 1 1000 2.5308
1 3 10 2.5315
1 3 1000 2.5308
5 1 10 2.4580
5 1 1000 2.5308
5 3 10 2.4617
5 3 1000 2.4531
results. Table 2 shows the variation of tensile modulus in the long range of nanoclay content, G and a according to MROM model. As shown, the difference of results is more negligible.
Therefore, MROM model can be developed as follows for recycled PET/clay nanocomposite:
E = Em $ m + MRF^f Ef, (33)
ln(u +1)
MRF = 1.12 —
u =
'Ef ^n
(34)
(35)
The predicted data by developed MROM model are illustrated in Fig. 7. The developed model can calculate more exact data.
3.3. Modification of Kerner-Nielsen model
For increasing the capability of prediction by Kerner-Nielsen model, some modifications are suggested in this model. It was indicated that the effect of various $max on the tensile modulus of current nanocomposite is insignificant. Table 3 demonstrates the P values by Eq. (21) for various $max. In addition, A and B from Eqs. (19), (20) show a very small variation in the different Poisson ratio of matrix as observed in
Fig. 7. The experimental tensile modulus and calculations of modified Guth, Kerner-Nielsen and Kerner models
Table 3. The variation of P in different maximum volumetric packing fraction of filler ^max and nanofiller contents
Table 5. Kerner predictions at various values of Poisson ratio of matrix Vm
Nanoclay, wt % фш^ P Nanoclay, wt % Vm E, GPa
1 0.05 1.0011 1 0.33 2.5914
3 0.05 1.009S 1 0.4 2.5935
5 0.05 1.0275 1 0.5 2.59S4
1 0.5 1.0001 5 0.33 2.7656
3 0.5 1.0005 5 0.4 2.7769
5 0.5 1.0014 5 0.5 2.S021
1 0.95 1.0000
3 0.95 1.0000 4. Conclusions
5 0.95 1.0001 The tensile modulus of recycle PET/clay nano
Table 4. This occurrence can be due to the much higher Young's modulus of nanoclay compared to the matrix modulus (fm = 178/2.55 = 69.8) in Eq. (20). As a result, the P, A and B parameters can be eliminated from the Kerner-Nielsen model.
After supposing P, A and B as 1, the modified Kerner-Nielsen model was fitted to the experimental results using trial and error method. The modified Kerner-Nielsen equation is presented as
1 +
E - Em
1 —
(36)
The predictions of modified Kerner-Nielsen are observed in Fig. 7. The suggested model has a considerable suitability with the experimental results.
3.4. Modification of Kerner model
Table 5 shows that the variation of theoretical modulus by MROM model at different nanoclay content and Poisson ratio of matrix is not considerable. By ignoring the Poisson ratio factor in this model, the Kerner model would be expressed as the modified Kerner-Nielsen model in Eq. (36) for exact predicting of tensile modulus. Importantly, the modified KernerNielsen model can predict the tensile modulus of recycled PET nanocomposite using the volume fraction of nanoclay and the matrix modulus. The modifications cause the simplicity in modeling which needed to difficulty measured factors, previously.
Table 4. The variation of A and B at different Poisson ratio of matrix Vm
Vm 0.33 0.40 0.45 0.50
A 1.13S3 1.2500 1.3571 1.1500
B 0.9699 0.96S3 0.9669 0.9649
composite was evaluated using the various conventional models. Some models such as Paul, LewisNielsen, and Takayanagi suggest good results compared with the experimental data. The Hirsch model provides too accurate data by the x value of 0.1 showing that the tensile modulus of nanocomposites conforms to the inverse rule of mixture more, about 90% and rule of mixtures about 10%. The theoretical results show that Halpin-Tsai model for 3D random platelet fillers and developed Guth model produce accurate data considering the aspect ratio of 8 for silicate layers. Additionally, the findings show that some parameters such as a, fmax, Vm and G have small effect on the theoretical results of MROM, Kerner-Nielsen and Kerner models. Therefore, for simplicity and presenting some models for recycled PET nanocompo-site, various modifications for these models were carried out to improve their predictions. The developed models calculate the tensile modulus of recycled PET nanocomposite using the volume fraction and Young's modulus of components, too simpler than the conventional models involving shear modulus and Poisson ratio of matrix or filler geometry.
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Received 20.08.2020, revised 11.09.2020, accepted 14.09.2020
Сведения об авторах
Yasser Zare, PhD, Dr., Motamed Cancer Institute, Iran, [email protected]
Kyong Yop Rhee, PhD, Prof., Kyung Hee University, Republic of Korea, [email protected]
