Development of Chow model for tensile modulus of polymer nanocomposites assuming the interphase region and particle arrangement Текст научной статьи по специальности «Нанотехнологии»

УДК 539.3

Модификация модели Чоу для определения модуля упругости полимерных нанокомпозитов с учетом межфазного слоя и расположения частиц

Y. Zare1, K.Y. Rhee1, S.-J. Park2

1 Университет Кёнхи, Йонъин, 446-701, Республика Корея 2 Университет Инха, Инчхон, 22212, Республика Корея

В работе предложена упрощенная модель Чоу для определения модуля упругости при растяжении полимерных композитов с полностью упорядоченным расположением частиц, разработанная для описания полимерныж нанокомпозитов с учетом влияния межфазного слоя и расположения частиц. Проведено сравнение результатов, полученных на основе упрощенной модели, с экспериментальными данными для ряда образцов, содержащих сферические, слоистые и цилиндрические наночастицы. Полученные данные хорошо согласуются с результатами экспериментов, тогда как результаты расчетов на основе исходной модели Чоу не совпадают с экспериментальными данными. Модули нанокомпозитов, армированныж сферическими наночастицами, зависят только от свойств межфазного слоя. В случае малой объемной доли межфазного слоя коэффициент Пуассона матрицы не оказывает влияния на модули нанокомпозитов со слоистыми и цилиндрическими нанонаполнителями. Показано, что межфазный слой играет важную роль в механическом поведении полимерных нанокомпозитов.

Ключевые слова: полимерные нанокомпозиты, модель Чоу, модуль растяжения, межфазный слой

DOI 10.24411/1683-805X-2019-15008

Development of Chow model for tensile modulus of polymer nanocomposites assuming the interphase region and particle arrangement

Y. Zare1, K.Y. Rhee1, and S.-J. Park2

1 Department of Mechanical Engineering, College of Engineering, Kyung Hee University, Yongin, 446-701, Republic of Korea 2 Department of Chemistry, Inha University, Incheon, 22212, Republic of Korea

In this study, the conventional Chow model for the tensile modulus of polymer composites containing fully aligned particles is simplified and developed for polymer nanocomposites, taking into account the effects of interphase and particle arrangement. The results obtained from the developed model are compared to the experimental results of several samples containing spherical, layered, and cylindrical nanoparticles. The predictions of the developed model are in good agreement with the experimental data, whereas the calculations of the original model deviate. The moduli of nanocomposites reinforced with spherical nanoparticles depend on the properties of the interphase alone. Additionally, a low volume fraction of the interphase eliminates the effect of matrix Poisson's ratio on the moduli of nanocomposites containing layered and cylindrical nanofillers. These results demonstrate the important role of the interphase in the mechanical behavior of polymer nanocomposites.

Keywords: polymer nanocomposites, Chow model, tensile modulus, interphase

1. Introduction

There has been a growing interest in nanostructures especially nanocomposites in recent years due to their improved mechanical, thermal, physical, barrier, and flame properties [1-13]. It is known that structure and interaction at the nano-scale have a significant effect on the properties of polymer nanocomposites [14-27]. Accordingly,

the interface/interphase characteristics between polymer matrix and nanoparticles affect the mechanical performance of polymer nanocomposites and overlooking it would result in a wrong estimation of nanocomposite behavior [28, 29]. This fact has been well-established using conventional models, such as the Halpin-Tsai and Guth models, which

© Zare Y., Rhee K.Y., Park S.-J., 2019

underestimate the tensile modulus of polymer nanocom-posites [30]. Therefore, studying the interphase in polymer nanocomposites is essential to develop accurate models to predict nanocomposites behavior.

Various micromechanics models have predicted the properties of the interphase in nanocomposites such as volume fraction, thickness, modulus, and strength [31, 32]. It was reported that the significant thickness, modulus and strength of interphase in polymer nanocomposites cannot be neglected [33, 34]. Furthermore, some models were used to estimate the influence of interfacial interactions/adhesion on the properties of polymer nanocomposites [35, 36].

Chow [37] suggested a complex model for predicting the tensile modulus of composites containing fully aligned particles. He applied the shear and bulk moduli of the constituents, the Poisson's ratio of the matrix vm, and the aspect ratio of fillers p to predict the tensile modulus of a composite. However, this model cannot be used for polymeric nanocomposites as it disregards the effects of the interphase and nanoparticle dispersion. In this study, the Chow model was further developed by taking into account the roles of the interphase and nanoparticle arrangement. This model is also simplified for polymeric nanocomposites containing different filler geometries. Moreover, the developed equations are used to predict the volume fraction and thickness of the interphases of some samples reported in the literature. Three-dimensional (3D) and contour plots also explain the effects of interphase volume fraction and vm on the modulus of different nanocomposites.

2. Simplification and development of Chow model

Chow [37] predicted the longitudinal tensile modulus for ellipsoidal particles embedded in matrix with major axis aligned along the stress direction as

(f Km -1)Ai + 2(Gf/Gm - 1)B,

Er = 1 + f

2 A3 B, + AjB3

Az = 1 + f Gm -1)(1 -<Pf)Pz, B2 = 1 + (Kf/Km -1)(1 -Pf)az, a1 = 4n Q/3 - 2(2n-1 ) R, a3 = 4n Q/3 + 4(I + n )R,

P. =

P3 =

Q= R=

( 4n 4n-3I ^

3 1-p 4n (4n - 3I ) p

Q - 4(I - 2n)R,

2 ^

1-P2

Q + (4n-1 )R,

1

8n1-Vm

1 1 - 2v,

8n 1 - Vm

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8) (9)

where Er is relative modulus as Ec/ Em, Ec and Em are the tensile modulus of composite and matrix, respectively, K and G are bulk and shear moduli and subscripts m and f show matrix and filler phases, respectively. The subscript z can be replaced by 1 or 3, % and p are also the volume fraction and aspect ratio of particles and vm is Poisson ratio of matrix.

Kf >> Km and Gf >> Gm in polymer nanocomposites and thus Kf/Km -1 = Kf/Km and Gf/Gm -1 = Gf/Gm. In addition, p for layered and cylindrical nanoparticles is commonly more than 100. As a result, parameter I and Eqs. (2)-(7) for layered and cylindrical nanoparticles p >> 1 can be expressed as

2nP [P(P2 - 1)V2 - cosh-1 p] = 2n, (10)

I =

(P2 -1)3/2

a = -

1 1

a3 =

P =

P3 =

21-Vm 1 + 3(1-2Vm) 2(1 -Vm) '

1_1_

21-Vm, 1 -2Vm

41-v

1 1

A1 = 1 + -^(1 -pf)- ,

1 GmK 21-Vm

B = 1+■K^(1-Pf )1- 1 ,

Km 21-Vm

A3 = 1 + -GL(1 -Pf)l z2^,

3 Gm 41 - Vm

B3 S 1 + f -pf)1 + 3(1 - 2Vm) . 3 Km ^ 2(1 - Vm)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

By replacing of Az and Bz parameters into Eq. (1) and performing some simplifications, the Chow model is simplified to

3(1 -Vm)

Er = 1 + Pf

2 - 4v

(19)

The simplified model gives the longitudinal tensile modulus for composites containing ellipsoidal particles. An arrangement factor n0 was introduced, which is equally to 1 for fully aligned fillers, 3/8 for randomly in-plane two-dimensional and 1/5 for randomly 3D arrangement [38]. Assuming the average n0 = 0.5 for fully dispersed fillers in three-dimensions, the Chow model is given by

3(1 -Vm)

Er = 1 + 0.5pf

(20)

2 - 4v

m

This model can simply calculate the tensile modulus for conventional composites p >> 1. For polymer nanocomposites, an interphase is formed between polymer and

nanoparticles. Assuming the interphase in polymer nano-composites p >> 1, the Chow model is developed to

Er = 1 + 0.5q)f3(1 Vm) + 0.5^ 3(1 Vm)

2 - 4v„

2 - 4v„

(21)

v m v m

where ^ is volume fraction of interphase. The above equations are expressed for nanocomposites containing layered or cylindrical fillers p >> 1.

For polymer composites containing spherical fillers p = 1, the parameters of Chow model are given by 4n

' - 311 + v

a = a = a = —

ß =ß3 =ß =

31-Vm' 2 4 - 5v n 15 1 -Vm

A = A3 = 1 + (1 -f

2_ Gf 4 - 5vr 15 G 1 -v

51 = 53 = 1 + (1 -%)1^f 1+ V

(22)

(23)

(24)

(25)

(26)

3 Km 1-Vm

Using these parameters in Eq. (1), the Chow model for conventional polymer composites p = 1 can be simplified to

9(1 -Vm)

Er = 1 + f

(27)

(4 - 5V m)(1 + V m)

Assuming the arrangement of nanoparticles and the role of interphase in polymer nanocomposites reinforced with spherical nanoparticles p = 1, the Chow model is expressed as

9(1 -Vm) ,

Er = 1 + 0.5% + 0.5q>i

(4 - 5v m)(1 + v m) 9(1 -Vm)

(4 - 5v m)(1 + v m)'

(28)

which results in Er = 1, when % = qtj = 0.

The volume fraction of interphase in polymer nanocomposites containing layered (1), cylindrical (2) and spherical (3) nanofillers are calculated [39] by

It-

^12 = f

<Pi3 = f

2

r +

j

3

-1

(29)

(30)

(31)

where r and t are radius and thickness of nanofillers, respectively, tj and r are the thickness of interphase around nanoparticles. From above equations, t{ and r{ are obtained for polymer nanocomposites containing dissimilar nano-fillers as

t. t i

2^f1

ri2 = r2

ri3 = r3

<Pf2

+1 -1

31 + 1 - 1 <Pf3

(33)

(34)

which can calculate the thickness of interphase by ^ and filler properties.

Er

Experimental Original model Developed model

0.00 0.02 0.04

Nanoclay, vol. fraction

1.6-

Experimental Original model Developed model

"0

0.00 0.01 0.02 0.03

MWCNT, vol. fraction

Er -r

Experimental Original model Developed model

T3

0.00

0.01

Si02, vol. fraction

0.02

Fig. 1. The experimental moduli and the calculations of original and developed Chow models for PBT/nanoclay [40] (a), epoxy/ MWCNT [41] (b) and PEEK/SiO2 [42] samples (c)

3. Results and discussion

The original and developed Chow models are compared with the experimental observations of different samples reported in the literature, including poly(butylene tereph-thalate) (PBT)/nanoclay [40], epoxy/multiwalled carbon nanotubes (MWCNT) [41], and poly(ether ether ketone) (PEEK)/SiO2 [42]. Assuming v m for poly(butylene tereph-thalate), epoxy, and PEEK to be 0.38, 0.4, and 0.38, the predictions made by the original and developed models at a suitable 9 are shown in Fig. 1.

The results indicate that the original Chow model cannot accurately calculate the tensile modulus of polymer nanocomposites as a result of disregarding the interphase. However, the developed model, which takes into account the role and properties of the interphase, can accurately predict the tensile modulus of the samples under consideration. In fact, the high modulus of polymer nanocomposites, which exceeds the predicted value by the conventional Chow model, is related to the strong interfacial interactions/adhesion at the interphase between the polymer matrix and nanoparticles. Conventional models, such as Chow, Halpin-Tsai, and Guth, however, disregard the interphase and hence fail to calculate the mechanical properties of nanocomposites accurately.

Figure 2 illustrates the 9 obtained by comparing the experimental data with the calculations of the developed Chow model at different % for the tested samples. It can be observed that 9 grows as 9f increases in all the samples. This phenomenon is predictable because a large number of well-dispersed nanoparticles (large f generally induces a high value of 9 in polymer nanocomposites. However, poor dispersion and aggregation/agglomeration of nanoparticles in the polymer matrix may reduce Accordingly, 9 shows a low rate of increase at high 9f values, as shown in Fig. 2, due to the aggregation/agglomeration of nanoparticles, which lead to a small interfacial area/fraction in polymer nanocomposites [43]. Therefore, it can be inferred that using a high volume of nanoparticles to produce a large fraction of the interphase is not logical, because nanoparticles may show some undesirable phenomena at high f such as aggregation/agglomeration, which would adversely affect the properties of the nanocomposite. In addition, the interphase volume fraction is higher than the volume fraction of nanoparticles in nanocomposites, which demonstrates the influence of interphase on the behavior of the nanocomposites. Further, it can be understood that the dimensions and strength of the interphase significantly control the mechanical performance of a polymer nanocomposite. Therefore, it is important to control interphase properties by improving the level of interfacial interaction or adhesion between polymer matrix and nanoparticles [44, 45]. It can be stated that a strong interphase produces a stiff nanocomposite, while a poor interphase may result in a weaker sample than the neat matrix.

Fig. 2. Volume fraction of interphase 9 as a function of 9f in the samples: PBT/nanoclay from [40] (a), epoxy/MWCNT from [41] (b) and PEEK/SiO2 from [42] (c)

The previous researchers have also investigated the effects of interphase on the mechanical behavior of polymer nanocomposites [32, 33].

Figure 3 depicts the interphase thickness (t{ and r{) at different 9f values (calculated using Eqs. (32)-(34) for the reported samples. The thickness of nanoclay and the radii of MWCNT and SiO2 nanoparticles are 2, 15, and 6.5 nm, respectively, according to the reported data.

In general, interphase thickness decreases with an increase in f demonstrating that a higher loading of nanoparticles leads to a thinner interphase in polymer nanocomposites. The volume fraction of an interphase may be in-

0.00 0.02 0.04

Nanoclay, vol. fraction

Fig. 3. The interphase thickness (Eqs. (32)-(34)) in the samples: PBT/nanoclay from [40] (a), epoxy/MWCNT from [41] (b) and PEEK/SiO2 from [42] (c)

creased by the addition of nanoparticles in the polymer matrix (Fig. 2), at which point its thickness would reduce. Therefore, a thick interphase is obtained by incorporating a smaller weight percentage of nanofillers in the polymer matrix. High concentrations of nanoparticles generally lead to aggregation/agglomeration, which decreases the interfacial area in nanocomposites [46]. Moreover, the numerous defects observed at high cp f induce stress concentration in the samples. Accordingly, a thin interphase at high pf is expected due to a reduction in the interfacial area/interaction. Additionally, the interphase thickness is higher than the thickness of clay platelets and nanoparticle radii. It was

0.40

0.35

0.30

0.3 ^

Fig. 4. The relative modulus as a function of and Vm for nanocomposites containing nanoparticles with p >> 1 by Eq. (21) at pf = 0.02: 3D (a) and contour plots (b) (color online)

indicated in previous studies that interphase thickness should be smaller than the gyration radius of polymer mac-romolecules, which is about 40 nm [47]. The interphase thicknesses calculated using Eqs. (32)-(34) for different samples are smaller than 40 nm, which confirms the precision and accuracy of the developed model.

Equation (21) is applied at p f = 0.02 to explain the effects of p j and Vm on the moduli of different samples; these values were predicted by the developed Chow model. Figure 4 shows the 3D and contour plots of nanocomposites containing layered or cylindrical nanoparticles p >> 1. It is found that low values of pj and V m yield poor tensile moduli. However, simultaneous increments in p j and Vm enhance the tensile modulus. As a result, p j and Vm have positive effects on the modulus of polymer nanocomposites containing layered or cylindrical nanoparticles. Moreover, the effect of V m on tensile modulus at a low pj is negligible. A low Pj ( Pj < 0.05 in this condition) significantly reduces the tensile modulus of nanocomposites at different values of Vm. On the other hand, the effect of Vm on the modulus is more important at higher pj values. These observations indicate that the volume fraction of the interphase exerts a greater effect on the properties of polymer nanocomposites compared to other parameters, such as V m. Hence, it should be noted that the compatibility of a poly-

0.3 0.0

0.40

0.3 5

0.30

« L

1.30 1.25 -1.20 1.15 1.10 1.05

0.3

Fig. 5. The relative modulus as a function of and vm for nanocomposites containing spherical nanoparticles (p = 1) by Eq. (28) at 9f = 0.02: 3D (a) and contour designs (b) (color online)

mer matrix with nanoparticles should be enhanced during the production of polymer nanocomposites because the mechanical properties of a nanocomposite rely on its interphase.

Figure 5 shows the relative modulus as a function of

and vm at 9f = 0.02 for nanocomposites containing spherical nanofillers (p = 1, Eq. (28)). It is clear that tensile modulus is more correlated to 9 than to vm, similar values of the modulus are obtained at similar 9 and different vm levels. In other words, 9 exerts the greatest influence on the tensile modulus of polymer nanocomposites, while vm is less effective. Therefore, according to the developed Chow model, the interphase can eliminate the effect of vm on the tensile modulus of polymer nanocomposites reinforced with spherical nanofillers. As mentioned previously, the interfacial properties should be paid more attention than the material conditions when manufacturing polymer nano-composites.

Interfacial properties can be improved using several techniques, such as compatibilization, functionalization, treatment, and modification of the polymer matrix and nanoparticles. For example, maleic anhydride-grafted polypropylene (PP-g-MA) was used as a compatibilizer to increase interfacial adhesion between a polypropylene (PP) matrix and nanoclay, which promotes the intercalation/exfoliation of clay platelets in the PP matrix [48]. Further, the surfaces of spherical nanoparticles, such as CaCO3 and SiO2, were modified to improve their compatibility with polymer matrices and produce strong interfacial bonding [42, 48].

4. Conclusions

The Chow model was simplified and developed for polymer nanocomposites containing dissimilar nanofillers taking into consideration the influences of interphase and filler orientation. The developed equations were used to calculate the volume fraction and thickness of the interphase in several reported nanocomposites. The values pre-

dicted by the developed model are in good agreement with the experimental data. The original Chow model underestimates the modulus of nanocomposites as a result of disregarding the interphase, while the developed model considers the role of the interphase and accurately predicts the tensile modulus. Therefore, the high moduli of polymer nanocomposites, which are much larger than the values predicted by conventional models, are related to the formation of interphase regions. Simultaneous increments in and vm improve the tensile moduli of nanocomposites containing layered and cylindrical nanoparticles p >> 1. Moreover, 9 exerts a greater effect (compared to vm) on the tensile modulus of nanocomposites reinforced with spherical nanoparticles (p = 1). Therefore, a good interphase is necessary in polymer nanocomposites to achieve a high modulus.

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Received July 15, 2019, revised August, 12, 2019, accepted August, 14, 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 Soo-Jin Park, Inha University, Republic of Korea, sjpark@inha.ac.kr