# Простое моделирование модуля упругости высокопрочных тройных нанокомпозитов

УДК 539.32

Простое моделирование модуля упругости высокопрочных

тройных нанокомпозитов

у. /аге1, к.-у. мее2

Проведен анализ модуля упругости при растяжении высокопрочного тройного нанокомпозита, содержащего термопласт, эластомер и наноглину. На основе данных, полученных с помощью простых моделей, показано влияние различных параметров на модуль упругости нанокомпозитов, таких как размер и форма нанонаполните-ля и характеристики межфазного слоя. В связи с этим в модели На1рт-Т8а1 учитывается влияние межфазной границы в предположении, что наноглина имеет трехмерную (3Б) и случайную ориентацию. Также для рассматриваемой системы разработан ряд моделей, включая модели ОиШ, МЯОМ, На1рт-Т8а1, Кегпег-№екеп и Кегпег.

БО! 10.24411/1683-805Х-2020-14010

Simple modeling of tensile modulus for toughened ternary nanocomposites

Y. Zare1 and K.-Y. Rhee2

The present paper analyzes the tensile modulus in a toughened ternary nanocomposite containing a thermoplastic, an elastomer and nanoclay. The unsuitable predictions by simple models reveal the effect of various parameters on the modulus of nanocomposites such as the size and shape of nanofiller and the characteristics of interphase section. Therefore, the effect of interphase is taken into account and three dimensional (3D) and random orientation of nanoclay are assumed in Halpin-Tsai model. In addition, many models including Guth, MROM, Halpin-Tsai, Kerner-Nielsen and Kerner are developed for the present system.

There is a growing attentiveness on the nanostruc-tures such as nanocomposites in recent years [1-5]. In the recent two decades, one of the main reasons for development of polymer nanocomposites is the significant enhancement of most of their properties at very low filler content, compared to the conventional microcomposites [6-8]. The studies on the preparation, characterization and application of different polymer-layered silicate nanocomposites have shown that the silicate layers improve the overall properties

of polymers [9-11]. However, the addition of silicate layers typically sacrifices the toughness of nanocom-posites. This problem has been a major challenge for some applications, which require an optimum balance of stiffness and toughness. Two scenarios can be suggested for toughness-stiffness optimization: the addition of a rubbery phase to increase the toughness and the incorporation of filler to compensate the loss of stiffness. The elastomer toughened PP or other thermoplastic olefin (TPO) containing talc filler are widely used in the automotive exterior and interior

© Zare Y., Rhee K.-Y., 2020

parts. Nevertheless, the utilization of much filler loading, about 20 wt % leads to some processing and performance defects.

The remarkably enhanced mechanical properties along with additional advantages such as low weight, low cost and easy processing can be obtained when nanoclay are dispersed in the blends of elastomer and thermoplastic polymers. The localization of nanoclay layers in the ternary systems has been reported to play a significant role in the morphology and the final properties of nanocomposites. In the ternary nano-composites, the improvement of toughness and stiffness is attributed to the reduction of dispersed phase size or the formation of a mixed nanostructure [12]. When nanoclay is located in the continuous phase, a reduction in the domain size of dispersed phase occurs and inversely, an increment in the dispersed phase size is shown, when the nanoclay is placed in the dispersed phase [13].

The addition of nanoclay to blends may lead to higher stiffness and lower toughness while the best balance is achieved when the nanoclay layers are located in the continuous phase [14]. On the contrary, nanoclay can increase both stiffness and toughness of a blends compared to the neat ones [15]. Some papers also reported the enhancement of toughness when the nanoclay layers were located in the interphase or in the elastomeric disperse phase. These obvious contradictories indicate to the important effect of nanoclay location on the mechanical properties.

Most researchers investigated only the experimental processing and characterization aspects of ternary nanocomposites. From a modeling point of view, the tensile modulus of nanocomposites has been studied in the recent years [16, 17]; but this key issue for ternary systems has not been investigated, up till now. The modeling offers the faster, easier and cheaper prediction of behavior, which can help the researchers in academia and industries to develop the most desirable products for different applications. The current paper studies the prediction of tensile modulus in the toughened ternary nanocomposites through development of different models.

The simplest models for estimation of tensile modulus are parallel and series models [18, 19]. In the parallel or rule of mixtures model, the equivalent strain is taken into account in the matrix and filler phases. This model is expressed as

E _ Em K + Ef f (1)

where Em and Ef are the tensile modulus of matrix and nanofiller, respectively and and are the volume fractions of matrix and nanofiller, in that order. In a three phase system containing two matrix phases, the rule of mixtures is given by

E _ Em1^m1 + Em2^m2 + Ef fc . (2)

In the present work, Em1 and Em2 indicate to the Young&s modulus of thermoplastic and elastomer phases, respectively, ^m1 and ^m2 are the volume fractions of thermoplastic and elastomer components.

The series or inverse rule of mixtures model assumes the same stress in the matrix and filler phases, represented as

For a three phase system, this model is developed to Eq. (4) as

E Eml Em2 Ef

These models were later modified to achieve more accuracy for modulus prediction. The rule of mixtures model developed in Refs. [20, 21] by introducing a modulus reduction factor (MRF) nominated as modified rule of mixtures (MROM) as

E _ Em^m + MRFfc Ef, (5)

ln(u +1)

MRF = l-l

where a is the aspect ratio of filler defined as a = w/t, which w and t are the width and thickness of the dispersed filler, respectively, G is the shear modulus of matrix.

The MROM model for a ternary system is expressed as

E = EmiKl + Em2^m2 + MRF, <f Ef, (8) ln(ut + l)

MRFt = l-l <f(Gl + G2)

a V 2Ef(Kl + ^m2) ,

where G1 and G2 are the shear moduli of thermoplastic and elastomer constituents, respectively.

Guth and Gold [22, 23] took account the interaction between filler particles using the Smallwood-Einstein approach and proposed a model as

E = Em(l + 2.5<f + l4.lf

In a three phase nanocomposite, the Guth model is shown as

[1 + 0.67aфf + 1.62(aфf )2]. (14)

In Refs. [25, 26] a mathematical Halpin and Tsai model has been introduced as

E = E„

n= Er - Er+, (16)

V Em z V Em z

= 2a. (17)

For a ternary system, the Halpin-Tsai model is presented as

E = - m1

E = E„

& 1 - PBфf 7 - 5v

- lll^f-E

p=l+ф2

(1 + 2.5(f +14.1(2). (12)

In Ref. [24] a shape factor was introduced the Guth to his model in order to account the accelerated stiffening for rod-like fillers as

E = Em[1 + 0.67a(f + 1.62(a(f )2]. (13)

The developed Guth model can be expressed for ternary systems as

For ternary systems, the Nielsen model is shown as E = Eml + Em2 1 + 0.5(A + A)Bt(f , (25)

S - 10v1&

(26) (27) (2S)

where vi and v2 are the Poisson ratio of thermoplastic and elastomer, respectively.

Lewis and Nielsen [30, 31] also considered the фтзх and modified the Halpin-Tsai model into the following equation:

E = E„

ф= 1 + ф

E = 0.49E1 + 0.51E2, (20)

E1 and E2 are the tensile moduli of the composite in the longitudinal and transverse directions, respectively. Both E1 and E2 are calculated from Eq. (26), but the aspect ratio is assumed 1 (a = 1) for calculation of E2.

Halpin-Tsai model was afterward modified by Nielsen and Landel [29] in 1990&s. They considered the Af as a function of the Poisson ratio of matrix v and фщж as the maximum volumetric packing fraction of the filler ^max = true volume of the filler/apparent volume occupied by the filler). The Kerner-Nielsen model is presented as

The Lewis-Nielsen model for a three phase system is presented as

E = Em1 + Em2 1 + % (31)

Kerner proposed a model in which v was assumed as a key parameter [32, 33]

E = E„

The Kerner model for the current ternary nano-composite is presented as

"1 + 15(1 - (vi +v 2)/2) <f !. (33) 8 - 5(v1 +v 2) 1

Ji et al. [34] suggested a model, which consider the interphase phase in addition to the matrix and filler phases as

E = Em1 + Em2

E = E„

a_7 (2 tj t + 1)^, (36)

where t and ti are the thickness of filler and interphase, 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.

For ternary system, the Ji model is represented as

E _ Em1 + Em2

+-T-;- .(37)

If the effects of interphase are neglected ti = 0, the Ji model is reduced to the two phase model of Ta-kayanagi [35] as

nary systems as

E = Em1 + Em2

Nanoclay, wt % 1 3 5 7

Experimental 1.051 1.192 1.312 1.484

Rule of mixtures 1.921 3.661 5.367 7.039

Inverse rule of mixtures 0.026 0.026 0.026 0.026

MROM 1.033 1.023 1.014 1.005

Takayanagi model 0.808 0.855 0.889 0.919

The toughened ternary nanocomposite based on the blends of PP and ethylene-octene based elastomer (EOR) and organoclay is studied from Lee et al. work [15]. The table illustrates the experimental modulus of samples containing 30 wt % of elastomer and the predictions from several models. As observed, the rule of mixtures overestimates the modulus, even as others such as the inverse rule of mixtures, MROM and Takayanagi models underestimate the data. In addition, the MROM model does not present the considerably dissimilar predictions in different aspect ratio of nanoclay a as found in another ternary nanocomposite

The experimental data of tensile modulus and the predicted results by the models

Fig. 1. The values of t/ti as a function of k according to the Ji model

[36]. The observed results in the table demonstrate a most disagreement with the experimental data may be due to the much larger difference between the Young&s modulus of nanoclay (about 178 GPa) and the matrix moduli (maximum 1.5 GPa). On the other hand, it confirms that the tensile modulus of current ternary nanocomposite depends on the further effective factors such as the filler size and shape, the properties of interphase, etc.

As shown in the table, the Takayanagi model is not capable of prediction of tensile modulus indicating that the interphase phase should be taken into account for the present system. In other words, the interphase plays a key role in the mechanical behavior of toughened ternary nanocomposites. However, the Ji model, which assumes the interphase can suggest much superior result when the suitable values of interphase thickness ti and k are determine. In the previous works [34, 37], the k values were arbitrarily chosen and its role has not been obviously shown, so far. As mentioned before, k is the ratio of the interphase modulus on the surface of the platelet Ei and the Young&s modulus of matrix k = Ei/Em assuming the linear dependence of modulus from matrix to surface of particles. The possible values of interphase modulus are Ei(min) = Em and Ei(max) = Ef. As a result, the k parameter can be varied in the range of Ef/Em = 1 to 120 for the present nanocomposite. On other hand, the reduced thickness of dispersed clay platelets t/ti can specify the degree of exfoliation. An inferior t/ti value shows the small size of broken silicate layers in the matrix together with the high extent of interphase thickness, which both indicates to the higher dispersion of silicate layers, i.e. the greater degree of exfoliation.

In the present work, the values of k are selected from the determined range 1-120 and the t/ti is obtained from the fitting process to the experimental data. Figure 1 shows the values of t/ti as a function of

■ Experimental

—•— Halpin-Tsai

~ " 3D random platelet

—■--■ Kerner-Nielsen ^^

—o— Lewis-Nielsen/^

—cd™ Kerner m ------- ■

Fig. 2. The predicted tensile modulus by the Halpin-Tsai, developed model for 3D random platelets, Kerner-Nielsen, Lewis-Nielsen and Kerner models

k parameter. For k values between 1 and 50, the t/ti increases to about 0.2 and then, remain almost constant for larger k values. The results of t/ti at the possible values of k are smaller than 1, which evidently demonstrate that the interphase thickness is usually higher than that of silicate layers emphasizing the good exfoliation and stiffening effect of nanoparticles. Clearly, further research by microscopic observations can be more helpful for perfect determination of these parameters.

Figure 2 shows the calculated results by the developed Halpin-Tsai model for 3D random platelets, Kerner-Nielsen, Lewis-Nielsen and Kerner models. The overestimation of the Halpin-Tsai model was also reported in other studies [24, 38], which can be due tothe lower contribution of 2D filler to tensile modulus in comparison to 1D fillers [36]. On the contrary, when the orientation and dimension of silicate layers are considered a = 40, a better agreement with experimentally measured data is obtained. As well known, in most cases such as the current samples, 3D nanoclay platelets are partially intercalated/exfoliated in the matrix. Lewis-Nielsen model presents the more accurate predictions despite the similarly calculated results by Kerner-Nielsen and Kerner models, which show some disagreement with experimental data. It is worth noting that all possible values of (max in the range of 0.01 to 0.99 have not much influence on the predicted data by both Kerner-Nielsen and LewisNielsen models and their predictions are therefore observed in Fig. 2 independent of (max.

Figure 3 shows the calculated modulus by the Guth and developed Guth models. The Guth model offers

Fig. 3. The theoretical results of tensile modulus from the Guth, developed Guth and modified Guth models

the much lower predictions whereas the developed Guth model, which considers the aspect ratio of nano-clay overestimates the tensile modulus especially at greater nanoclay contents. Accordingly, a MRF can be introduced to the developed Guth model due to the lower contribution of 2D fillers to tensile modulus compared to 1D ones.

After introducing the MRF to the developed Guth model, the proposed model for ternary nanocompo-site can be given by

Em1 + Em

x [1 + 0.67MRFa(f +1.62(MRFa(f )2 ]. (40)

The MRF value of 0.52 shows the best conformity between the experimental and the predicted data as demonstrated in Fig. 3. It shows that the incorporation of MRF significantly improve the predicting ability of the Guth model.

In the MROM model, the different values of aspect ratio of silicate layers (a = 1-1000) together with the shear moduli of PP and EOR elastomer do not extensively change the predicted results. As a result, a, G1 and G2 can be removed from the MROM model and the modified model for ternary nanocomposites is shown as

E = Em1( m + Em2 (m2 + MRFm (f Ef, (41)

ln(«m +1)

MRFm = 1.06-(f

Ef(( m1 +(m2)

The developed MROM model suggests the much accurate predictions as illustrated in Fig. 4.

As shown in Fig. 2, the Halpin-Tsai model overestimates the tensile modulus of toughened ternary nanocomposite. The accuracy of this model can be improved by addition of MRF such as the modified Guth equation. When MRF is introduced to the Halpin-Tsai model, the modified model is presented as

Em1 + Em2 1 + MRF^ ^f 2 1

The calculations of the modified Halpin-Tsai model are observed in Fig. 4 assuming the MRF value of 0.52. The suggested model can predict the tensile modulus of toughened ternary nanocomposite better than the conventional Halpin-Tsai model.

As observed in Fig. 2, the Kerner-Nielsen model cannot estimate the tensile modulus of current ternary nanocomposite. It was also indicated that the possibly different values of ^max do not significantly affect the predictions. In addition, Bt parameter (Eq. (28)) exhibit an extremely small variation at different Pois-son&s ratio of matrices v1 and v2, may be due to the much larger level of Young&s modulus of nanoclay (about 178 GPa) compared to the matrix moduli (maximum 1.5 GPa). So, P, A1, A2 and Bt parameters (Eqs. (24)-(28)) can be eliminated from the KernerNielsen model and the modified equation using the fitting procedure can be presented as

E = Em1 + Em2 1 + 32^f 2 1 -*f &

The modified model shows a good ability for predicting the tensile modulus as illustrated in Fig. 4.

Fig. 4. The predicted modulus by the modified MROM, Halpin-Tsai, Kerner-Nielsen and Kerner models

According to Fig. 2, the Kerner predictions are not appropriately fitted to the experimental data. The different values of Poisson&s ratio of matrices v1 and v2 in the possibly long range of 0.33 to 0.5 do not play a key role in the calculations. Therefore, the Poisson ratio factor can be disregarded in this model and, the modified Kerner model is eventually developed to the modified Kerner-Nielsen presented in Eq. (45). The simplicity of suggested models is exceedingly excellent for analysis of tensile modulus in the toughened ternary nanocomposites. The conventional models need to various factors such as the aspect ratio of filler a, the Poisson ratio of matrix v and the maximum volumetric packing fraction of the filler ^max involving much difficulty, cost and time while the suggested models in the present work require more simple parameters for prediction.

The tensile modulus of toughened ternary nano-composites containing a thermoplastic and an elastomer was analyzed. Much disagreement was found between the experimental data and the predicted results by rule of mixtures, inverse rule of mixtures, MROM models. The possible reason is the large difference between the Young&s modulus of nanoclay (about 178 GPa) and the matrix modulus (maximum 1.5 GPa). The Takayanagi predictions corroborated that the effect of interphase phase should be taken into account for the mechanical properties of ternary nanocomposites. Therefore, the Ji model was employed for modulus analysis. The Halpin-Tsai model for 3D random platelets and Lewis-Nielsen models also provided the accurate calculations. Furthermore, some models such as the Guth, MROM, Halpin-Tsai, Kerner-Nielsen and Kerner models were modified for the toughened ternary nanocomposite as following: The MRF factor of 0.52 was introduced to the developed Guth and Halpin-Tsai models; the MROM, KernerNielsen and Kerner models were simplified by ignoring the ineffective parameters. The suggested models demonstrated good predictability and significant simplicity.

References

platelet-reinforced PP nanocomposites // Comptes Rendus Mecanique. - 2007. - V. 335. - P. 702-707.

Received 09.06.2020, revised 06.07.2020, accepted 10.07.2020

Сведения об авторах

Yasser Zare, PhD, Dr., Motamed Cancer Institute, Iran, y.zare@aut.ac.ir

Kyong Yop Rhee, PhD, Prof., Kyung Hee University, Republic of Korea, rheeky@khu.ac.kr