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УДК 539.32
Evaluation and development of expanded equations based on Takayanagi model for tensile modulus of polymer nanocomposites assuming the formation of percolating networks
Y. Zare1, 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
In this study, the tensile modulus of polymer nanocomposites is analyzed by the development of expanded Takayanagi models considering the fractions of networked and dispersed nanoparticles above the percolation threshold. The tensile moduli of networked and dispersed phases are calculated by suitable models. This study focuses on "polymer-carbon nanotubes" nanocomposites, but the developed model can be applied for samples reinforcing with long fillers such as clay and graphene. The expanded Takayanagi model suggests two different forms which are evaluated by the experimental results of "polymer-carbon nanotubes" nanocomposites. Only one form shows the best results compared to the experimental data, whereas another form underestimates the modulus. The developed model (correct form) shows that the fraction of filler network meaningfully changes the reinforcement of nanocomposites. The network level and other correlated parameters with the percolation threshold can be calculated by comparing the experimental data with the developed model. The logical outputs confirm the correct development of Takayanagi model assuming the network and dispersion of nanoparticles in polymer nanocomposites.
Keywords: polymer nanocomposites, network structure, tensile modulus, percolation threshold
Оценка и построение развернутых уравнений на основе модели Такаянаги для модуля упругости при растяжении полимерных нанокомпозитов с учетом формирования перколяционных сетей
Y. Zare1, K.Y. Rhee2
1 Исламский университет Азад, Тегеран, Иран 2 Университет Кёнхи, Сеул, 02447, Республика Корея
В статье для анализа модуля упругости при растяжении полимерных нанокомпозитов построены расширенные модели Такаянаги с учетом доли наночастиц при сетчатом и дисперсном распределении, превышающей порог перколяции. С использованием соответствующих моделей рассчитаны модули растяжения для сетчатого и дисперсного распределения наночастиц. Исследованы нанокомпозиты «полимер - углеродные нанотрубки», однако предлагаемая модель также применима для образцов, наполненных глиной и графеном. Предложены два варианта расширенной модели Такаянаги, проведена их оценка на основе экспериментальных результатов для нанокомпозитов «полимер - углеродные нанотрубки». Один из вариантов модели показывает хорошее соответствие с экспериментальными данными, другой вариант дает заниженные значения модуля растяжения. Согласно корректному варианту доля наполнителя при сетчатом распределении существенно влияет на упрочнение нанокомпозитов. Размер ячеек сетки и другие параметры, связанные с порогом перколяции, могут быть рассчитаны с использованием экспериментальных данных. Показана корректность разработанной модели Такаянаги при сетчатом и дисперсном распределении в полимерных нанокомпозитах.
Ключевые слова: полимерные нанокомпозиты, сетчатая структура, модуль упругости при растяжении, порог перколяции
1. Introduction
Polymer nanocomposites contain a reinforcement which one of its dimensions changes in the range of 1-100 nm. The nanocomposites show excellent properties due to the nanometer filler which causes high specific surface area and aspect ratio [1-4]. Also, the strong interfacial interaction between polymer matrix and nanoparticles plays an
important role in nanocomposites behavior. The mechanical and conductivity properties of nanocomposites are well affected by the 3D network of nanoparticles formed at a filler concentration considered as percolation threshold [5, 6]. The percolation effect exceedingly depends to the degree of interaction between fillers. So, a percolated network with weakly interacting fillers behaves differently from
© Zare Y., Rhee K.Y., 2017
a strongly connected network of particles. Several methods for fabrication of conductive nanocomposites were developed. One of the best techniques is the filler prelocalization or segregated network model [7]. In this method, the filler is not consistently dispersed in the polymer matrix, but a three dimensional (3D) continuous network of filler-rich layers is shaped in the polymer matrix. This method can be performed by the hot compaction of mixtures of polymer and conductive fillers such as graphene, carbon nanotubes (CNT) and carbon black.
The prediction of mechanical properties in the nanocomposites with percolating filler is a challenge and the conventional models such as Halpin-Tsai cannot present accurate results. The connected structure is formed by the random and 3D arrangement of nanoparticles. Many models used to define the percolation have the functional form of power-law and depend on the minimum volume fraction of filler in which the connected structure is formed. These models fairly predict the critical conductivity in polymer nanocomposites. A similar effect of percolation on the electrical conductivity was also reported for mechanical properties referred to mechanical percolation [8, 9]. Also, the same power-law models were applied in modeling of mechanical response of composites. The mechanical behavior is generally controlled by the volume fraction of filler, even prior to the formation of a connected phase [10]. Therefore, the mechanical properties may depend on a combination of mechanisms, while the percolation threshold is critical.
In polymer nanocomposites, the effect of mechanical percolation becomes inordinate due to the significant effect of interfacial region between polymer matrix and nano-filler [11, 12] resulted from the perturbation of the matrix in the presence of nanoparticles, adhesion between the phases, confinement of matrix between stiff regions or restriction of the mobility of flexible polymer chains which cause a limited increase in matrix stiffness. The local reinforcement is occurred in all composites, but the interfacial region in nanocomposites can create a third phase as interphase because of high surface area to volume ratio of na-noparticles [13, 14]. In the nanocomposites, the interphase may represent a volume fraction equal to or greater than that of the nanoparticles. As a result, the presence of interphase may suggest two levels of percolation [15]. The first level includes the interfacial region which assists the formation of a connected structure by particles and interface at a much lower volume fraction compared to the second level where the nanoparticles only percolate.
In this work, the tensile modulus of polymer nanocom-posites is estimated by the developed Takayanagi model above the percolation threshold considering the network and dispersion of nanoparticles in the polymer matrix. The moduli of regions containing networked and dispersed na-noparticles are predicted by different models and then, the
modulus of nanocomposite is calculated using the developed model. The Takayanagi model suggests two different forms which their predictions are evaluated by the experimental results of polymer/CNT nanocomposites (PCNT). Also, the effects of some parameters such as the network amount and percolation threshold on the tensile modulus are assessed. This paper focuses on PCNT, but the developed model can be used for the samples containing long fillers such as clay and graphene.
2. Modeling approach
Takayanagi et al. [16] proposed a model for tensile modulus of polymer composites based on the series and parallel models as:
"i -a+ P 1-1
E = Em
1 -a + PEf/ Em
(1)
where Em and Ef are the Young's moduli of matrix and filler, respectively. Also, a and P are functions of filler volume fraction tf.
Above the percolation threshold, a fraction of filler belongs to the infinite network, while other particles are dispersed in the polymer matrix. Loos and Manas-Zloczower [17] assumed the network and dispersion of nanoparticles in the polymer matrix above the percolation threshold and developed the Takayanagi model. They developed both forms of Takayanagi model as observed in Fig. 1.
The form I as Fig. 1, a is expressed by
(1 -f Em En +(tf -<U Ed En
Ei =-
(2)
tn (1 - tf )Em + tn (tf - tn )Ed + (1 - tn r E
where tn is the volume fraction of networked filler. Also, En and Ed are the moduli of regions containing networked and dispersed nanoparticles, respectively. When tn = 0 as the absence of network, form I reduces to parallel one or rule of mixtures model as
E = (1 -tf) Em +tf Ed (3)
which is correct according to Fig. 1, a.
1 - Л
1 - ф
Fig. 1. Schematic representation of Takayanagi model for a nano-composite above percolation containing dispersed and networked filler: form I (a) and II (b). 1 - X and At are the volume fractions of networked and dispersed phases, respectively (tf =1 - A + At)
Ф
Л
The developed form II (Fig. 1, b) is also given by
Eii =
= 4>n(1 - >f ) Ed En + t. (>f - >n ) Em En + (1 - )2 Ed Em
(1 ->f)Ed + (>f ->n)Em U
which reduces to series or inverse rule of mixtures model in the case of >n = 0 (see Fig. 1, b in absence of network phase) as
Eii =-
Ed Em
(1 -f Ed + < eJ' (5)
Although Loos and Manas-Zloczower [17] used the above models to calculate the modulus of nanocompo-sites above percolation threshold, they did not propose any accurate method for the prediction of moduli for networked and dispersed regions. Here, some suitable models are suggested to predict the moduli of these regions which can estimate the modulus of nanocomposite at the final step by the amount of networked particles and the volume fraction of percolation threshold expressed by aspect ratio of nano-particles.
A common version of classical model for tensile modulus of networked filler above percolation threshold includes a power-law form [10] as
E = EfOf -V', (6)
wherep is percolation exponent and <p is the volume fraction of percolation threshold. The volume portion of nano-particles which belong to the network part is given by f <f
>n =-
(7)
1 - (1 - f )<f
where f is the fraction of filler particles involved in the network. Applying <n in Eq. (6) results in
En = Ef (>n ->p)*
(8)
On the other hand, Paul [18] suggested a model for the composite containing dispersed particles assuming the homogeneous stress in two components as
E = Em
A = f
Em
1 + (A -1)f 1 + (A - 1)(f ->f)'
(9)
(10)
The volume fraction of dispersed nanoparticles in the nanocomposite above percolation threshold is calculated by
<f(1 - f)
>d =-
(11)
1 - f <f
which proposes the tensile modulus of the region containing matrix and dispersed nanoparticles by Paul model as
1 + (h -1)<2/3
v ^ (12)
Ed = Em
•1 + (h - 1)(<f -<d)
Additionally, Chatterjee [19] found a converse connection between the percolation threshold and aspect ratio of nanoparticles a as
1
a
(13)
where a = l/d, l and d are the length and diameter/thickness of nanoparticles, respectively. Using the above equation, the tensile modulus of nanocomposites can be calculated above the percolation threshold using the inverse aspect ratio of nanoparticles as the volume fraction of percolation <p.
3. Results and discussion
Both forms of the developed model by micromechanics models are applied for prediction of modulus in some reported samples. The theoretical predictions are compared with the experimental results of modulus. Moreover, the influences of some parameters such as f, p and a on the modulus of nanocomposites are discussed based on the developed model.
Figure 2 depicts the experimental relative modulus as nanocomposite modulus divided to modulus of neat matrix for poly(ethylene terephthalate) (PET)/multi wall carbon nanotubes (MWCNT) from Ref. [20], polyethylene (PE)/ MWCNT from [21], polypropylene (PP)/MWCNT from Ref. [22] and polyamide 6 (PA6)/MWCNT from Ref. [23] and the predicted results by average a = 200 or <p = 0.005 and Ef = 1000 GPa. The experimental data well follow the theoretical predictions of form II (Eq. (4)). However, the calculations of form I are smaller than the experimental data which fail to predict the modulus. In other words, the form I underestimates the modulus of present samples at all concentrations of nanofiller. As a result, only form II of the developed model can estimate the modulus of PCNT. In this study, we use the form II in all calculations and the developed model in the following text means the form II.
Moreover, it is found that thep value of 0.15 (Eq. (8)) shows the best predictions by the developed model. Assuming p = 0.15, the best values of f parameter are obtained as 0.07, 0.15, 0.15 and 0.85 for PET/MWCNT, PE/MWCNT, PP/MWCNT and PA6/MWCNT samples, respectively. These findings demonstrate that the lowest level of CNT network is shown in PET/MWCNT sample, while the best level of network is observed in PA6/MWCNT specimen. The experimental data of modulus in the reported samples confirm the outputs of the developed model for CNT network. For example, 2 wt% of MWCNT (<f = = 0.0142) in PET/MWCNT causes Er = 1.25, whereas 0.5 wt% of MWCNT (<f = 0.003) results in Er = 1.48 in PA6/MWCNT sample. A high level of stiffness in PCNT by a low amount of CNT is only obtained by the formation of a strong network in the nanocomposite. A network can bear a high level of stress by which a large level of stiffness is produced in the nanocomposite [10, 24]. Conclusively, the experimental results of modulus can exhibit the level of network in the samples by f parameter in the developed Takayanagi model. Also, the suitable results of form II re-
Fig. 2. The experimental relative modulus and the calculations by the developed model (both forms) for PET/MWCNT [20] (a), PE/MWCNT [21] (b), PP/MWCNT [22] (c) and PA6/MWCNT [23] (d) with different fraction volume of MWCNT
veal that the network of CNT is parallel to the dispersed CNT and polymer matrix regions in the PCNT. The dispersed CNT and polymer matrix parts are series along the stress direction (Fig. 1). Loos and Manas-Zloczower [17] also found that the form II is suitable for estimation of modulus in polymer composites containing CNT and cellulose nanowhisker.
Figure 3 displays the highest and the smallest levels of relative modulus calculated by the developed model at p = = 0.15, Em = 2 GPa, Ef = 1000 GPa, 4>p = 0.005 and different nanofiller concentrations. The highest value of stiffness is achieved by f= 1 indicating that all nanoparticles in the sample are involved in the network and no tube is separately dispersed in the polymer matrix. In this condition,
the modulus grows by incorporation of nanoparticles in the polymer matrix and reaches Er = 13 by = 0.04. It means that the modulus of PCNT increases as 1200% when 4 volume percent of CNT forms a network in the polymer matrix. As a result, the network of CNT significantly improves the stiffness of polymer matrix. However, when all CNT are dispersed in the polymer matrix ( f = 0), the modulus shows a negligible increment by addition of CNT signifying the undesirable role of network absence in PCNT. In fact, a high level of reinforcement is obtained by formation of a network [10, 25], due to storing much energy by elastic network in the polymer nanocomposites.
Figure 4 demonstrates the effect of p parameter as the percolation exponent on the prediction of modulus by the
0.00 0.01 0.02 0.03 (|>f
Fig. 3. The highest and the lowest levels of modulus predicted by developed Takayanagi model (form II) at p = 0.15, Em = 2 GPa, Ef = 1000 GPa and = 0.005 for different nanofiller (MWCNT) concentrations
= 0.15 p = 0.30 -a- p = 0.50 -a-;? = 0.80
0.00 0.01 0.02 0.03 фг
Fig. 4. The effect ofp parameter on the prediction of modulus by developed model at f = 0.5, Em = 2 GPa, Ef = 1000 GPa and 0p = 0.005 for different nanofiller (MWCNT) concentrations
developed model at f = 0.5, Em = 2 GPa, Ef = 1000 GPa and <p = 0.005. As observed, the modulus of PCNT significantly improves when p has a small level. Similarly, a low level ofp creates a major increment of modulus started at <f = 0.01. However, a high value ofp produces a poor modulus which unimportantly changes at different concentrations of CNT. Accordingly, the best modulus is obtained by the lowest level of p which shows the inverse relation between the modulus level and p factor as percolation exponent. As mentioned, the p value of 0.15 shows acceptable results for modulus of PCNT by the developed Takaya-nagi model. The p parameter depends to several parameters such as the density and strength/stiffness of CNT network in polymer nanocomposites. In this regard, a dense and strong CNT causes a low level forp which produces a high stiffness in nanocomposite. The estimation of percolation threshold for mechanical properties is generally derived from the similar theories developed for electrical conductivity [10].
The electrical and mechanical properties can be studied in the same samples and the measurements of conductivity can be used to determine the threshold value used in the models which predict the mechanical properties [8, 9]. The micromechanics models in this area were frequently suggested based on a power-law form in which the conductivity term is replaced by the elastic modulus. Therefore, the p term in Eq. (8) is similar to the t parameter in the common model for electrical conductivity of polymer nano-composites [10] as
a = CT0(<f -<h)',
(14)
where a and a0 are the conductivity of nanocomposite and nanofiller, respectively and t is the critical exponent of conductivity. The nature and range of p and t parameters depend to the characteristics of the percolating network.
Figure 5 exhibits the roles off and a parameters on the calculated relative modulus at <f = 0.02, p = 0.15, Em = = 2 GPa and Ef = 1000 GPa by contour plot. The worst level of modulus is observed at low ranges of f, whereas the best modulus is observed at the highest levels off and a parameters. A low level off shows the absence or low content of CNT network in the PCNT which cannot considerably reinforce the polymer matrix. Therefore, observing a poor modulus at a low f is expected. However, a high level of f displays the considerable involvement of CNT in the network which produces a dense and strong network in the polymer matrix. In other words, a high f shows that a high fraction of CNT makes a strong network in the polymer matrix. In this condition, obtaining a high reinforcement by CNT is reasonable.
On the other hand, a high level of a creates a low percolation threshold in the nanocomposite. The percolation threshold shows a volume fraction of CNT in which the modulus rapidly increases due to formation of network in the polymer matrix. So, a low percolation threshold by the
a 500
400
300
Fig. 5. Contour plot for the effects of f and a parameter on the predicted modulus by developed model at <f = 0.02, p = 0.15, Em = 2 GPa and Ef = 1000 GPa
high aspect ratio of CNT results in a greatly improved modulus in PCNT. Additionally, a high level of a introduces a considerable amount of interfacial area/interaction between polymer chains and nanoparticles which can excellently transfer the stress from polymer matrix to nanopar-ticles and improve the modulus [26, 27].
It was shown that the achievement of high modulus is not possible without obtaining a high a in polymer nano-composites. In fact, a high aspect ratio is a main advantage of layered and cylindrical nanofillers which significantly affects the mechanical, thermal and flammability of nano-composites [28, 29]. As a result, achieving a high level of modulus by the large aspect ratio of nanoparticles is not a peculiar trend in the mechanical performance of polymer nanocomposites. According to above description, the logical effects of f and a parameters on the modulus calculations confirm the correctness of the developed model.
4. Conclusions
A modeling method based on expanded Takayanagi model was developed which can calculate the tensile modulus of polymer nanocomposites above the percolation threshold by the fractions of networked and dispersed nanoparticles in the polymer matrix. The moduli of networked and dispersed regions were calculated by power-law and Paul models. Also, both different forms of Takayanagi model were evaluated by experimental measurements of modulus.
One form of Takayanagi model demonstrates the acceptable results compared to experimental results, but another one underestimates the modulus. The suitable form shows that the network of CNT is parallel to the series order of dispersed CNT and polymer matrix regions in the PCNT. Also, the modulus of nanocomposite considerably increases by decrease in p value as percolation exponent in the power-law model illustrating an inverse relation between p value and modulus level. The best level of modulus is also obtained by the highest levels of f and a parameters.
The modulus shows a negligible improvement at all filler concentrations in absence of network. A high level off expresses the high fraction of CNT network in the sample. Similarly, a high level of a shows the low percolation threshold and high interfacial area/interaction between polymer matrix and nanoparticles. As a result, suggesting a significant modulus at the high levels of f and a parameters by the developed model demonstrates the correct development of Takayanagi model by micromechanics theories.
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Поступила в редакцию 11.02.2017 г.
Сведения об авторах
Yasser Zare, Dr., Islamic Azad University, Iran, y.zare@aut.ac.ir Kyong Yop Rhee, Prof., Kyung Hee University, Korea, rheeky@khu.ac.kr
