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45
УДК 539.199
Разработка новых моделей для модуля упругости металл-углеродных нанокомпозитов
Y. Zare, K.Y. Rhee
Университет Кёнхи, Йонъин, 446-701, Республика Корея
В статье предложены новые модели для оценки модуля упругости металл-углеродных нанокомпозитов на основе углеродных нанотрубок. Ряд моделей композитов (модели Hirsch, Hui-Shia, Halpin-Tsai для 3D наполнителей и модель Lewis-Nielsen) дают хорошие результаты в сравнении с экспериментальными данными для композитного материала из меди с многостенными углеродными нанотрубками. В соответствии с моделью Takayanagi для данного нанокомпозита необходимо учитывать влияние межфазных границ. В настоящей работе с использованием модели Ji проведена оценка толщины и модуля упругости межфазного слоя. Также при разработке новых моделей использованы модифицированное правило смесей и модель Kerner-Nielsen. Из предлагаемых моделей исключены многие неэффективные и трудно описываемые параметры. Значительно упрощенные модели демонстрируют высокую эффективность при расчете модуля упругости нанокомпозитов.
Ключевые слова: металл-углеродные нанокомпозиты на основе углеродных нанотрубок, модуль упругости при растяжении, моделирование
DOI 10.24411/1683-805X-2020-13007
Development of new models for tensile modulus of metal/carbon nanotubes nanocomposites
Y. Zare and K.Y. Rhee
Department of Mechanical Engineering, College of Engineering, Kyung Hee University, Yongin, 446-701, Republic of Korea
The current paper studies the development of new models for the prediction of tensile modulus in metal/carbon nano-tubes (CNTs) nanocomposites. Several composite models such as the Hirsch, Hui-Shia, developed Halpin-Tsai for 3D fillers and Lewis-Nielsen models can present appropriate predictions compared to experimental results of a Cu/multi-walled CNTs (MWCNTs) sample. Furthermore, Takayanagi model reveals that the effect of the interface phase should be considered for the present nanocomposite. Therefore, an appropriate evaluation of the thickness and modulus of the interphase is carried out by the Ji model. In addition, new models are developed based on the modified rule of mixtures and Kerner-Nielsen models. Many ineffective and difficultly characterized parameters are eliminated from the suggested models. The developed models demonstrate the best predictability together with significant simplicity for the calculation of tensile modulus in nanocomposites.
Keywords: metal/CNTs nanocomposites, tensile modulus, modeling
1. Introduction
There is a growing attentiveness on the nanostruc-tures such as nanocomposites in the recent years [15]. Carbon nanotubes (CNTs) formed by rolling a gra-phene sheet show much extraordinary mechanical, electrical and thermal properties [6-9]. Their unexpected stiffness, specific strength and specific modulus makes them an ideal reinforcement for composites.
CNTs reinforced polymer nanocomposites were extensively synthesized by surfactant assisted processing, repeated stirring, solution evaporation with high energy sonication and interfacial covalent functionalization [10, 11]. However, the investigation on ceramic and metal composites reinforced by CNTs has been comparatively limited due to the several factors: the diffi-
© Zare Y., Rhee K.Y., 2020
culty uniform dispersion of CNTs in metal; the poor interfacial reaction between CNTs and metal matrix, and the lack of suitable fabrication technique [10-14].
There have been little attempts to improve the me-tal/CNTs nanocomposites by modification of traditional powder metallurgy process and analysis of strengthening mechanism [15]. It is difficult to achieve a homogeneous distribution of CNTs in the composites, due to the much larger size of metal powder than that of CNTs particles. To overcome this problem, some methods such as nanoscale dispersion of CNTs by introducing into an elastomer precursor [16], molecular level mixing by a salt containing Cu ions [17] and in situ reduction [18] were suggested.
The excellent mechanical and physical properties of metal/CNTs nanocomposites have been extensively reported in the literature. Dong et al. [19] prepared the Cu/CNTs nanocomposites by hot pressing sintering which show an enhancement on fracture toughness, wear resistance and hardness. But, from a modeling point of view, no study has been carried out so far while the modeling helps to comprehensive analysis of nanocomposite behavior. In addition to provide more information about the materials, modeling removes the needs to more experiments performed for study of various parameters. The modeling can predict the properties of nanocomposite assuming different parameters without involving much cost, time and difficulty [20]. In this paper, the tensile modulus of metal/CNTs nanocomposites is analyzed. Several models are suggested for prediction of tensile modulus in this class of attractive nanocomposites. The developed models reveal the finest predictability and the highest simplicity for estimation of tensile modulus in metal/ CNTs nanocomposites.
2. Background
Hirsch [21] developed a model as
E = x( Em 9m + Ef 9f ) + (1 - x)-
Ef Em
Ef 9m + Em9f
where Em and Ef are the Young's modulus of matrix and filler, and 9m and 9 are the volume fraction of the matrix and filler, x and 1 - x represent the relative contribution of composite to the rule of mixtures and inverse rule of mixtures, respectively.
Cox and Kelly [22, 23] suggested a model as the shear lag model. The Cox model considers the transfer of tensile strength from matrix to reinforcement through the interfacial shear stresses. This model is shown as
E = Em 9m +nEf f (2)
^ = 1 -
tanh m
m
2G
m = a
Ef ln(r/R) '
(3)
(4)
where a is the aspect ratio of filler, G is the shear modulus of matrix, R and r are the filler radius and the centre-to-centre distance of the fillers, respectively.
In Ref. [24] the rule-of-mixtures model is developed and the model, nominated as modified rule of mixtures (MROM), is presented as
E = Em 9m +
MRF = 1 -
MRF9 Ef, ln(u +1)
1
u =-
a
9fG
Ef 9m
(5)
(6)
(7)
In Refs. [25, 26] the Halpin-Tsai model are implemented as
1 + ^9f
E = Em 1
Ef/Em -1 Ef/ Em + ^
(8)
(9)
= 2a. (10)
The tensile modulus of composites containing randomly oriented fillers can be calculated by laminate plate theory [27] as
E = 0.49E1 + 0.51E2, (11)
E1 and E2 are the tensile modulus of composite in the longitudinal and transverse directions, respectively. Both E1 and E2 are calculated from Eq. (8), but the aspect ratio of filler is assumed 1 for E2 (a = 1).
Nielsen and Landel [28] modified the Halpin-Tsai model as
E = E„ 1 + AB*f
(12)
(13)
(14)
(15) is the
maximum volumetric packing fraction of the filler: 9max = true volume of the filler/apparent volume occupied by the filler.
In addition, Lewis and Nielsen [29, 30] took account the 9max in the Halpin-Tsai model as
1 - PB9f A = (7 - 5v)/(8 - 10v), B = Ef/Em - 1 Ef/ Em + A'
P = 1 + 9f2(1 -9maxV9max ,
where v is the Poisson ratio of matrix and 9m
u
E = Em
^ = 1 + 9f
1 + ^<Pf 1 -fl<Pf
1 f max
f 2
rmax
(16)
(17)
Hui, Shia et al. [31, 32] developed a model assuming the perfect interfacial adhesion between polymer matrix and particles as
C = 9f +
E = En Em
1 -fL
1
—+ ■
-1
(1 - g)q2 - g/2
Ef - Em
q2 -1
A = (1 -f
+ 3(1 -f g = q,
3( q 2 + 0.25) g - 2q 2 q2 -1
(18)
(19)
(20) (21)
Ji et al. [33] suggested a three-phase model (matrix, filler and interphase) as
q-P
E = Em
(1 -q) +
1 -q + q( k - 1)/ln k
-1
+
P
E = Em
1-P+
1 -P + PEJ Ea
-1
(25)
(22)
1 -q + 1/2(q-p)(k + 1) + pEf/Em '
P = V%, (23)
q = V (2 tjt + 1)ff, (24)
where t and tx are the thickness of filler and interface, respectively, k is the ratio of the interphase modulus on the surface of filler Ei and the Young's modulus of matrix as k = Ej/Em. If the effects of interface are neglected (tx = 0), Ji model changes to the two-phase model of Takayanagi [34] as
p
3. Results and discussion
3.1. Analysis of models
The experimental results of tensile modulus of Cu/multiwalled CNTs (MWCNTs) nanocomposite were provided in Ref. [35]. The nanocomposite was fabricated by electroless deposition process. By increasing CNTs content in Cu matrix, the Young's modulus and hardness of samples increase in spite of the electrical conductivity which decreases. Figure 1 illustrates the experimental data of tensile modulus and the predictions of different models. The Hirsch model can give the most accurate data at x value of 0.3 indicating that the tensile modulus of Cu/MWCNTs nanocom-
Fig. 1. The prediction of tensile modulus by Hirsch, Hui-Shia, developed Halpin-Tsai model for 3D fillers, LewisNielsen and Takayanagi models
posite conform to the rule of mixtures (the first part of Hirsch equation) smaller than that to the inverse rule of mixtures (the second part of Hirsch model). However, the obtained level of x for the present nanocomposite is much larger compared to polymer nanocom-posites, where x value of 0.05 was found [20]. It confirms that the CNTs nanoparticles are more efficiently in the metal nanocomposites in comparison to the polymer nanocomposites.
The Hui-Shia model is well fitted to the experimental data at the aspect ratio q of 10. This outstanding agreement shows a perfect interfacial adhesion between the metal matrix and CNTs particles as claimed by Hui and Shia [31, 32]. The developed Halpin-Tsai model for 3D fillers also gives more ingenuous predictions. In addition, the theoretically calculated results by Lewis-Nielsen model demonstrate a good conformity with the experimental data at fmax values of higher than 0.65. It demonstrates that the true volume of the CNTs filler is similar to the apparent volume occupied by the filler in the current nanocomposite. The optimum parameters of the models were obtained from the fitting of experimental data to the models. Certainly, the vitiation of these parameters causes the underestimation or overprediction of the models compared to experimental data. Figure 1 generally shows good agreement of the studied models with the experimental data. Actually, this is the best predictions of the models assuming experimental data. Some deviations are observed at different CNT concentrations, due to the undesirable terms such aggregation/agglomeration or curvature of CNTs.
The underestimation of Takayanagi model is more evident in Fig. 2 indicative to the fact that the interphase should be taken into account. In other words, the interphase plays a key role in the mechanical pro-
2 6 10 14 18 k
Fig. 2. The values of t/t{ as a function of k parameter according to the Ji model
0 40 80 120 160 r/R
Fig. 3. Determination of aspect ratio of CNTs a and r/R parameters based on Cox model; r is centre-to-centre distance of fillers and R is filler radius
perties of metal/CNTs nanocomposites. Therefore, the Ji model is more helpful to calculate the tensile modulus. The Ji model requires to appropriate values of interphase thickness ti and k ratio for modulus calculation. However, the careful role as well as the characterization procedure of k value has not been exactly determined and it was so arbitrarily chosen in the previous works [33, 36].
As indicated before, k is the ratio of interphase modulus on the surface of the filler Ei and the Young's modulus of matrix k=Ei/Em supposing the linear dependence of the modulus on space variable from the matrix to the surface of particles. The possible values of the interphase modulus are Ei(min) = Em and Ei(max) = Ef and so, the k parameter can be varied in the range of 1 to 20 for the present nanocomposite. In addition, the t/ti value shows the reduced thickness of the dispersed CNTs particles in the metal matrix. The smaller value of t/ti impart the little size of broken particles along with the high level of interphase thickness which both means the higher level of dispersion of nanoparticles in the matrix. In this work, the k values are selected from the range of 1 to 20 and the t/ti is calculated from fitting process to the experimental data as shown in Fig. 2. As observed, t/ti increases from 0.45 to 1.8 at different k values. The average value of t/ti is obtained as 1.1 which demonstrates that the thickness of interphase is relatively equivalent to the thickness of CNTs particles. Certainly, further studies by microscopic characterizations are required for perfect determination of these parameters.
In the Cox model, two parameters including the aspect ratio of filler a and the ratio of centre-to-centre distance of the fillers r and filler radius R as r/R should be determined. These values were accurately characterized in the fiber-reinforced composites [37], but the exact measurement of these factors in the particulate nanocomposites is more difficult due to the
different size and uncertain dispersion of broken filler in the matrix. Figure 3 shows the values of aspect ratio a as a function of r/R based on the fitting procedure. The a is obtained from 6 to 17 at r/R values of 2 to 200 while the growth rate of a diminish at higher values of r/R. The Cox model suggests the average a of 11.5 which confirms the previously obtained a by developed Halpin-Tsai and Hui-Shia models.
3.2. Suggestion of new models
Figure 4 shows the poor capability of MROM model for prediction of tensile modulus in Cu/CNTs nanocomposite. Therefore, this model is developed for metal nanocomposites.
Table 1 shows the calculation of modulus by MROM model at different values of aspect ratio of CNTs a and shear modulus of metal matrix G. Clearly, the variation of modulus at a large range of a from 10 to 1000 and G at most different values of 20500 GPa is insignificant. It demonstrates that these parameters have not play a key role in the calculation of modulus.
■ Experimental
■•■MROM
-o- Modified MROM
Kerner-Nielsen
-o- Developed Kerner-Nielsen
■
0 4 8 12 16 20
CNTs, vol %
Fig. 4. The theoretical results of tensile modulus by MROM, Kerner-Nielsen and new developed models
Table 1. The variation of predicted modulus by MROM model at different aspect ratio of CNTs q and shear modulus of matrix G
ф£, vol % a G, GPa E, GPa
5 10 20 49.4809
5 10 100 49.5805
5 10 500 49.8012
5 100 20 49.4081
5 100 100 49.4181
5 100 500 49.4405
5 1000 20 49.4008
5 1000 100 49.4018
5 1000 500 49.4041
20 10 20 42.3038
20 10 100 43.1647
20 10 500 45.0543
20 100 20 41.6707
20 100 100 41.7579
20 100 500 41.9527
20 1000 20 41.6071
20 1000 100 41.6158
20 1000 500 41.6353
Therefore, q and G can be removed from the MROM model (Eqs. (5)-(7)) and the new model using trial and error procedure can be suggested as
E = Em9m + MRFm9f Ef, (26)
ln(um +1)
MRFm = 1.37 —
9f
lEf Ф„
(27)
(28)
Table 2. The effect of Poisson's ratio v of the matrix and maximum volumetric packing fraction fmax of the filler on B and P parameters by Eqs. (14), (15) in Kerner-Nielsen model
Ф& vol % V фшгх B, Eq. (14) P, Eq. (15)
5 0.2 0.05 0.9011 1.0475
5 0.2 0.95 0.9011 1.0001
5 0.5 0.05 0.8794 1.0475
5 0.5 0.95 0.8794 1.0001
20 0.2 0.05 0.9011 1.7600
20 0.2 0.95 0.9011 1.0021
20 0.5 0.05 0.8794 1.7600
20 0.5 0.95 0.8794 1.0021
Young's modulus of MWCNTs (about 1000 GPa) compared to the modulus of matrix (52 GPa). As a result, the Kerner-Nielsen model can be more simplified by eliminating the A, B and P parameters (Eqs. (13)-(15)). Eventually, the developed KernerNielsen model using fitting procedure is presented as
1 + 4ff
E = Em
1-ф
(29)
The correlation between experimental data and predicted results by MROM model is illustrated in Fig. 4. The new proposed model introduces good predictability together with more simplicity for metal nanocomposites.
As shown in Fig. 4, the Kerner-Nielsen model cannot correctly predict the tensile modulus of current nanocomposite. Some parameters in this model do not considerably affect the calculated modulus. Table 2 illustrates the variation of B and P parameters (Eqs. (14), (15)) at different values of Poisson ratio of matrix v and possible levels of maximum volumetric packing fraction of the filler 9max. As observed, v and 9max do not significantly change B and P due to the higher
Figure 4 shows the obtained results by developed Kerner-Nielsen model in which the improvement of predictability of Kerner-Nielsen model is evidently observed. It is worth noting that the simplicity of new proposed models is much excellent. The proposed models do not need to difficulty characterized parameters such as the aspect ratio of filler a and maximum volumetric packing fraction of the filler фшзх, etc. The simple models outstandingly facilitate the prediction process of nanocomposite properties.
4. Conclusions
This paper studied the estimation of tensile modulus in metal/CNTs nanocomposites. Hirsch model predicts the tensile modulus in x value of 0.3 for Cu/MWCNTs sample. Moreover, Hui-Shia and developed Halpin-Tsai models for 3D fillers show a good conformity with experimental results of Cu/MWCNTs sample assuming the aspect ratio of 10. Lewis-Nielsen model also gives a good agreement with the experimental data at фшзх values of higher than 0.65. Several new models based on MROM and Kerner-Nielsen models are suggested for prediction of tensile modulus. In these models, many ineffective parameters such as the aspect ratio of filler and maximum volumetric packing fraction of the filler are removed. The proposed models exhibit the outstanding
m
Um =
predictability as well as noteworthy simplicity compared to the conventional models, which facilitate the
prediction of properties.
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Received 09.04.2020, revised 09.04.2020, accepted 13.05.2020
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
