Finite element modeling of carbon nanotube agglomerates in polymers

Accepted Manuscript Finite element modeling of carbon nanotube agglomerates in polymers Aggeliki Chanteli, Konstantinos I. Tserpes PII: DOI: Reference:

S0263-8223(15)00577-2 http://dx.doi.org/10.1016/j.compstruct.2015.07.033 COST 6612

To appear in:

Composite Structures

Please cite this article as: Chanteli, A., Tserpes, K.I., Finite element modeling of carbon nanotube agglomerates in polymers, Composite Structures (2015), doi: http://dx.doi.org/10.1016/j.compstruct.2015.07.033

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Finite element modeling of carbon nanotube agglomerates in polymers Aggeliki Chanteli, Konstantinos I. Tserpes* Laboratory of Technology and Strength of Materials Department of Mechanical Engineering & Aeronautics University of Patras, Patras 26 500, Greece

Abstract While the negative effect of carbon nanotube (CNT) agglomerates on the effective mechanical properties of CNT-reinforced polymers and composites has been well recognized through mechanical tests and microscope observations, little has been done on the modeling of CNT agglomerates and the understanding of the role of their specific characteristics. In the present work, a detailed finite element (FE) model of CNT agglomerates is proposed. The geometry modeled represents a cubic representative unit cell comprising the agglomerated CNTs and the surrounding matrix. The FE model of the unit cell was used to perform a parametric study on the effect of agglomerates’ geometrical characteristics on the homogenized elastic properties of the CNT/polymer material. The study shows that the waviness, the multitude and the topology of the agglomerated CNTs has a considerable effect which varies for the different material properties. Using the unit cells of the agglomerates, the Young’s modulus of the CNT/polypropylene material was predicted by employing a multiscale analysis. The characteristics of the unit cells used for the modeling of the CNT/polypropylene material were determined from processing and analysis of scanning electron microscopy images of the nanocomposites. The comparison between numerical and experimental results indicates that the model is capable of accurately predicting the Young’s modulus of materials at which the agglomeration effect is prominent, however, it tends to underestimate the Young’s modulus of materials at which agglomeration is minor.

Keywords: Carbon nanotubes; Agglomeration; Finite element analysis; Multiscale modeling; Polypropylene

*

Corresponding author. Tel.+302610969498, Fax: +302610997190, E-mail: [email protected]

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1. Introduction Reinforcement of polymers with a small amount of carbon nanotubes (CNTs) may improve significantly their mechanical, thermal and electrical properties thus making them very attractive for use in various technological applications. In the last decade, in view of realization of these applications, there has been made a noticeable progress in the manufacturing of CNT-reinforced polymers and composites [1]. However, the fundamental problem of poor CNT dispersion, which eventually leads to the formation of agglomerates, still remains unresolved. It has been well recognized from experiments and microscope observations that CNT agglomerates act as defects causing, in the worst scenario, a reduction in the mechanical properties of neat polymer [2-4], and in the best scenario, a smaller than the desired increase in the properties [5-12]. Besides recognition of the negative effect of CNT agglomerates, little has been done on the understanding of the role of agglomerates’ geometrical characteristics. Such a study could be conducted either by repeated mechanical tests and microscope interpretations or by modeling, with the latter being far more time- and costeffective. Several numerical and analytical models of CNT agglomerates in polymers and composites have been reported in the literature [13-18]. The reported analytical models [13-16] treat agglomerates as inclusions and employ the Mori-Tanaka micromechanics theory to obtain analytical relationships between the effective properties and the morphology of CNT-reinforced polymers. The only parameters considered in these models are the CNT volume fraction and the degree of agglomeration. On the other hand, most reported numerical models use representative unit cells at which the CNTs are stacked into different configurations which deviate from the actual configuration of CNT agglomerates e.g. [17,18]. The works that have modeled the exact structure of CNT agglomerates are very few: in [19] a two-scale model was developed to investigate the effect of the agglomerates on the stress-distribution in fiber-reinforced composites on the micro-level while in [20] the CNT agglomeration effect on piezoresistivity of polymer nanocomposites was studied using a multiscale model. From this short literature overview, it may be concluded that there is still a need for developing models representing the exact structure of CNT agglomerates and for conducting studies on the detailed effects of agglomerate’s geometrical characteristics on the mechanical properties of the CNT-reinforced polymers. In this context, a finite element (FE) model of CNT agglomerates was developed in the present work to meet two objectives. The first objective is to perform a parametric study on the effect of geometrical characteristics of the agglomerates on the homogenized elastic properties of the CNT/polymer material and the second objective is to examine whether the FE model could be used as a base of a multiscale methodology aiming to predict the elastic properties of CNT/polymer specimens. The

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development and implementation of the model were done on the basis of the CNT/polypropylene (PP) material.

2. Description of the model 2.1 Creation of RUC’s geometry The model represents a representative unit cell (RUC) consisting of a CNT agglomerate embedded into polymer matrix. In the model, no interface between the CNTs and the matrix was modeled (perfect bonding feature) based on the findings of [20] that the interface characteristics (stiffness and thickness) do not affect the elastic properties of CNT/polymers. The RUC is cubic and the length of its side is determined according to the CNTs volume fraction from




∙ = 

∙  ∙  4 

(1)

where  is the number of CNTs in the agglomerate,  and  are the CNT’s diameter and length,

respectively and  is the CNTs volume fraction. The CNTs have a solid circular cross-sectional area. The main challenge in the development of the model is the realistic representation of the agglomerate’s

geometry. In order to meet this challenge, the open source TexGen software [21] was used for developing the wavy form of CNTs. TexGen [21] has advanced capabilities in the modeling of wavy fiber-like structures as it is used for modeling the geometry of textile materials. Also, it has a direct link to FE codes. For the matrix, the PP thermoplastic material with a Young’s modulus of 1.64 GPa was used. The CNTs were modeled as isotropic materials with a Young’s modulus of 1 TPa, which belongs to pristine non-functionalized CNTs, and a Poisson’s ratio of 0.3.

2.2 Design parameters Several design parameters are involved in the modeling of CNT agglomerates. The most important are the diameter, length, waviness and orientation of CNTs as well as the density (number of CNTs) of the agglomerate. The FE model is fully parametric. For demonstration reasons, the model was developed on the basis of the CNT/PP material. The values of design parameters were determined from scanning electron microscopy (SEM) images presented in [10]. This was also necessary because one of the objectives of the present work is to use the RUC as a module of a two-scale model to evaluate the Young’s modulus of the CNT/PP material. The SEM images were imported into the image processing

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software ImageJ [22] and several measurements were taken. The average measured CNT’s diameter is 29 nm and the average length is 210 nm. Measurements on CNT’s waviness, multitude and orientation in the agglomerate were also taken; however, the range of these parameters that could be adopted is restricted by the modeling capacity. To this end, three different densities of the RUC were considered; namely 3, 5 and 10 CNTs representing a minimum, a medium and an extreme density. The CNT wavy shape is described by the following equation taken from [23]

 = cos 

2   

(2)

where  is the sinusoidal wavelength and z is the CNT’s axial direction. The amplitude  of the CNT

triangle. Using Eq.(2) the CNT waviness was determined by adjusting the parameter  in the waviness

wavy shape is estimated by assuming that the CNT orientation is described by the shape of an isosceles ratio  = /. Two waviness cases were considered in the present work: a moderate waviness ( =

10.5) and an extreme waviness ( = 82.26). In Fig.1, schematic representations of the two waviness cases are depicted. The effect of CNTs orientation was studied by means of the RUC with 5 CNTs

through the cases of 1 CNT, 2 CNTs and 3 CNTs at the loading axis, respectively. In the RUCs with 3 CNTs a uniform orientation was applied (1 CNT per axis) in all cases and in the RUC with 10 CNTs, a close to uniform orientation. The values of geometrical parameters are listed in Table 1. In Fig.2, typical geometries of the three RUCs created using TexGen [21] are shown. In order to obtain a more realistic shape of the agglomerates an extra tilt of 15° was assigned to each CNT.

2.3 FE model of RUC The geometries of the RUCs created in TexGen were imported into the ANSYS Workbench [24] to generate the FE models. In order to avoid meshing problems, which could arise due to the complex shape of the agglomerates, a voxel-based FE model was developed. Both the matrix and the CNTs were represented using the ANSYS SOLID185 element [24]. The mesh density used was determined from a convergence study conducted with regard to the accuracy of computed results, the accurate representation of CNTs’ shape and the required computational effort. Typical FE meshes of the RUC (half) and the CNT agglomerates are shown in Fig.3. The local coordinate system of the elements was rotated according to the orientation of the elements with regard to the global coordinate system of the RUC. To derive the elastic moduli, the RUC was loaded in axial tension at the three directions by applying a small normal displacement at one side and fully constraining the other side. Similarly, to derive the shear

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moduli, the three planes were loaded in shear by applying a shear displacement at one face and fully restraining the opposite face. The elastic and shear moduli were derived as the ratios of stress to strain while the Poisson’s ratio as the negative ratio between longitudinal and transverse strains. Aiming to obtain the homogenized engineering behavior of the RUC, which reflects the macroscopic behavior of the material, periodic boundary conditions were applied in the RUC.

3. Parametric study Using the RUC a parametric study on the effect of volume fraction, waviness, multitude and topology of the agglomerated CNTs on the effective elastic properties of the CNT/polymer material was conducted aiming to understand the role of specific characteristics of the agglomerates. The range of parameters considered in the study is listed in Table 1.

3.1 Effect of CNT volume fraction and multitude The effect of CNT volume fraction is not trivial for CNT agglomerates as it is for single straight CNTs due to the complex architecture of the agglomerates. On the other hand, it is important to ascertain how the multitude of CNTs in the agglomerate may affect the degree of reinforcement through the formation of different load paths between the CNTs. Fig.4 plots the variation of EX, GXY and νXY with regard to CNT volume fraction VCNT for agglomerates with different multitude. For VCNT=1%, a small increase in all elastic properties was predicted. With increasing the VCNT, EX and GXY increase while the variation of νXY depends on the multitude of CNTs; it increases for 3 CNTs and decreases for 5 and 10 CNTs. The predicted variations of EY and EZ are similar with EX, the variations of GXZ and GYZ are similar with GXY and the variations of νXZ and νYZ are similar to νXY. The increase in EX is reversely proportional to the multitude of CNTs. This is because the fewer the aligned CNTs are the largest is the load they carry as can be seen in Fig.5 for the cases of RUCs with 3 and 5 CNTs.

3.2 Effect of CNT waviness In Fig.6, the predicted values of EX for the single straight CNT, the single waved CNT of moderate waviness and the agglomerates of moderate and extreme waviness are compared. The comparison between the single CNTs shows clearly that the waviness reduces the degree of reinforcement offered by the CNT in the polymer. The reduction increases with increasing the wave amplitude (from moderate to extreme waviness). The opposite trend is observed for the agglomerates: the agglomerate of extreme waviness has a larger Young’s modulus than the agglomerate of moderate waviness. The reason for this unexpected finding is purely geometrical: while in the case of moderate waviness one out of three CNTs

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is aligned to the loading direction, in the case of extreme waviness the larger curling of CNTs leads to the alignment with the loading direction of parts of the other CNTs also, thus increasing the load-carrying capability of the agglomerate. For instance, in the case of 3 CNTs with extreme waviness, shown in Fig.3c, both the green CNT, placed at the x-direction, and the yellow CNT, placed at the y-direction, are contributing to the transfer of load which is applied at the x-axis. This phenomenon is also illustrated in Fig.7.

3.3 Effect of CNTs topology The effect of CNTs topology on the homogenized elastic properties of the RUC is illustrated in Fig.8. From the figure it is concluded that the effect on the elastic moduli is balanced between the different directions. The alignment of more CNTs in the x-direction increases the EX (Fig.8a) and decreases the EY and EZ (Fig.8b). On the other hand, the effect of CNTs topology on the shear moduli and Poisson’s ratio is minor as can be seen from Figs.7c and 7d, respectively. It is mentioned that these findings are for CNTs of moderate waviness. In the case of extreme waviness the trend, especially for the elastic moduli, is expected to be different due to the explanation given in the previous section.

4. MWCNT/PP material 4.1 Experimental In a previous study [10] the tensile behavior of the PlasticylTM PP2001 material has been characterized experimentally. This is a PP material reinforced by multi-walled CNTs (MWCNTs). Tensile tests were conducted according to the ASTM D638 standard [25] at room temperature using an electromechanical Tinius Olsen model H5KT machine. The applied displacement rate was 1 mm/min. Reference specimens and MWCNT/PP specimens of two different MWCNT weight ratios, namely 2 wt% and 5 wt%, were tested (six specimens for each material). Strains at the specimens were measured using strain gauges and the Digital Image Correlation system ARAMIS 5M. The results from the tensile tests were additionally evaluated by morphological inspection of the specimens using SEM. The experimental results show an increase in the average Young’s modulus and average maximum stress of the MWCNT/PP material with regard to the reference PP material (Table 2). The considerable scatter in the properties of reinforced specimens is attributed to the varying dispersion and size of MWCNT agglomerates. In the specimens with mechanical properties lower than the average values, SEM interpretation revealed poor MWCNT dispersion and formation of large agglomerates. Due to the prominent effect of MWCNT agglomerates, the MWCNT/PP material was chosen as an application case of the model developed herein.

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4.2 Modeling A two-scale modeling approach was implemented to predict the Young’s modulus of MWCNT/PP material. Initially, the RUC was used to compute the homogenized elastic properties of MWCNT agglomerates and then, the tension MWCNT/PP specimen was modeled and the computed elastic properties of the RUCs were assigned to the elements of the specimen’s FE model. Two different MWCNT/PP specimens were modeled: a 2 wt% specimen with a rather good MWCNT dispersion and formation of small agglomerates and a 5 wt% specimen with a poor dispersion and formation of large agglomerates.

4.2.1 Processing and analysis of SEM images The characteristics of the RUCs were selected from processing and analysis of SEM images performed using the ImageJ software [22]. Image analysis was based on the fact that the areas filled with MWCNTs are in general brighter in the SEM image than the unfilled areas (pure resin). The output from ImageJ [22] is a map of black and white areas in the form of a matrix. The original SEM images were reconstructed using a MATLAB routine developed for this purpose. The routine uses the user-defined parameter (, which acts as a comparing tool of the index of the ImageJ matrix, to define the distribution of black color at the image. For the reconstructed images, the percentages of black and white colors were made In Figs.8 and 9, the images reconstructed using different values of parameter ( are compared with

available.

specimens belong to the same material category. For the 2 wt% specimen, ( = 200 leads to a good

original SEM images taken from the two specimens. The specific SEM images are representative of all specimen, the two images correlate well for ( = 170. The adopted values of parameter ( were

correlation between the original SEM image (Fig.9a) and the reconstructed image (Fig.9d). For the 5 wt%

determined using a trial-and-error approach. The reconstructed images represent sufficiently the agglomeration state in the specimens. In Fig.9d, the uniformly dispersed small white spots represent the well dispersed MWCNT agglomerates of small density while the few large white spots represent the few agglomerates of large density. Similarly, in Fig.10d, the large white spots represent the MWCNT agglomerates of large density. From the analysis of reconstructed images it was revealed that the agglomerates of large density occupy the 1.72% of Fig.9d image and the 3.56% of Fig.10d image. Complementary to the information drawn from reconstructed images, the diameters of the agglomerates were measured from the SEM images using the ImageJ software [22]. Since it wasn’t computationally feasible to model the actual CNT agglomerates, the analogy of diameters between small and large agglomerates was used for selecting the appropriate RUCs. For the 2 wt% specimen the analogy is 0.3 and for the 5 wt% it is 0.5.

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4.2.2 FE modeling of the MWCNT/PP specimen A FE model of the tensile specimen was developed using the ANSYS commercial FE code [24]. The MWCNT/PP material was represented using the 3D ANSYS SOLID185 element [24]. A typical FE mesh of the specimen is shown in Fig.11. To simulate the tensile loading, boundary conditions were applied at the parts of the specimen captured by the tabs. One side was fully built in, while at the other side an axial displacement was applied. For the elements of the specimen’s FE model, an isotropic material behavior was assumed. The Young’s modulus of the elements was computed from the RUCs of the CNT agglomerates. Using the data derived from section 4.2.1 and the available RUCs it was decided to model the 2 wt% specimen using the RUCs of 3 and 5 CNTs in the analogy 98.28% and 1.72%, respectively, and the 5 wt% specimen using the RUCs of 5 and 10 CNTs in the analogy of 96.44% and 3.56%, respectively. These RUCs had a uniform or a close-to uniform topology and the CNTs inside them had a moderate waviness. The elements which had properties of the denser RUCs (5 CNTs for the 2 wt5 specimen and 10 CNTs for the 5 wt% specimen) were selected using a random numbers algorithm (Fig.12). The use of RUCs, whose characteristics are defined from the processing and image analysis of SEM images, to derive the Young’s modulus of the specimen’s elements is based on the assumption that the reinforcement characteristics identified from SEM images apply to the entire specimen. A very important issue in the modeling of the RUCs is the conversion of MWCNT weight ratio to volume fraction. In the conversion, the basic parameters are the number of CNT walls and the density of CNTs. The PlasticylTM PP2001 material contains the NC7000 MWCNTs for which the manufacturer reports 7 walls, an outer diameter of 9.5 nm and a density ranging from 1.3 to 2.0 gr/cm3. However, as already mentioned in section 2.2, our measurements showed an average CNT diameter of 29 nm. This density is a function of diameter [26-29], in the present work the density * was derived from the

finding indicates that MWCNTs of more than 7 walls have been eventually produced. Given the nanotube

following relation taken from [29]

*+,

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4000 / = . = − 32 454 6 7 < / 01 1315 01

(3)

9:;

where 01 is the outer diameter of MWCNTs equals to 29 nm, 454 is the interlayer distance equals to

0.34 nm and /(= 16) is the number of walls derived using the 01 and /. Eq.(3) gives a density of 1.38 gr/cm3 which converts 2 wt% to a volume fraction of 1.3% and 5 wt% to a volume fraction of 3.3%.

These MWCNT volume fractions were applied to the respective RUCs. In the RUCs, for the Young’s modulus of the matrix the average experimental value of neat PP (1.64 GPa) was used.

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4.3 Prediction of Young’s modulus The predicted values of Young’s modulus are compared with the average experimental values for the two reinforced materials and the neat PP in Fig.13. In the histograms, the standard deviations of the experimental values are also included. While for the 5 wt% material the model agrees very well with the test, for the 2 wt% there is a considerable deviation which still lies within the standard deviation of the mean experimental value. This finding suggest that while the model is capable of accurately predicting the Young’s modulus of materials at which the agglomeration effect is prominent , i.e. the 5 wt% material, it underestimates the Young’s modulus of materials at which the agglomeration is minor, i.e. the 2 wt% material. For modeling materials with a good dispersion of CNTs a more realistic approach could be the use of representative volume elements with single wavy randomly oriented CNTs.

5. Conclusions In the present work, a detailed FE model of CNT agglomerates was developed. The modeled geometry belongs to a cubic RUC comprising the agglomerated wavy CNTs and the surrounding matrix. The parametric studies conducted using the RUC show that with increasing the multitude of CNTs in the agglomerate the elastic properties decrease. On the contrary, with increasing the waviness of CNTs the elastic properties increase since crimping of CNTs leads to the alignment of larger parts of the CNTs. As for the effect of CNTs topology, it is balanced between the different directions; the largest elastic modulus appears in the direction with the more aligned CNTs. The shear moduli and Poisson’s ratios are rather independent of CNTs topology. In addition to the parametric studies, a two-scale model was developed based on the RUC for predicting the Young’s modulus of MWCNT/PP material. For this study, the geometrical characteristics of the RUCs were determined from processing and analysis of SEM images of the nanocomposite. The comparison between numerical and experimental results reveals that the model is capable of accurately predicting the Young’s modulus of materials at which the agglomeration effect is prominent, however, it tends to underestimate the Young’s modulus of materials at which agglomeration is minor.

Acknowledgments The research leading to these results has gratefully received funding from the European Union Seventh Framework Programme (FP7/2007- 2013) under Grant Agreement no 284562.

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Captions to Figures: Fig.1:

Schematic representation of a. moderate and b. extreme CNT waviness and definition of geometrical parameters. Fig.2: Typical geometries of the RUCs (CNT agglomerate and matrix) created using TexGen [21]. a. 3 CNTs with a uniform orientation, b. 5 CNTs with a close to uniform orientation and c. 10 CNTs with a close to uniform orientation. All CNTs in the figure have a moderate waviness. Fig.3: Typical FE mesh of a. half the RUC with 5 CNTs of moderate waviness, b. the agglomerate with 5 CNTs of moderate waviness and c. the agglomerate with 3 CNTs of extreme waviness Fig.4: Variation of elastic properties with regard to CNT volume fraction and CNTs multitude. CNTs have a moderate waviness and a uniform or a close-to uniform topology. Fig.5: Contour of ?@ stress at a. Agglomerate with 3 CNTs and b. Agglomerate with 5 CNTs. Loading is the x-axis. The agglomerates belong to RUCs with the same CNT volume fraction. Fig.6: Effect of CNT waviness on longitudinal modulus EX. The agglomerate has 3 uniformly oriented CNTs and the CNT volume fraction is 2%. Fig.7: Contour of ?@ stress at a. RUC with 3 CNTs of moderate waviness and b. RUC with 3 CNTs of extreme waviness. Loading is applied at the x-axis. The RUCs have the same CNT volume fraction. Fig.8: Effect of CNTs topology on the elastic properties of the RUC (VCNT = 3.3%, moderate waviness). Fig.9: a. SEM image from a MWCNT/PP specimen of 2 wt%. b., c. and d. reconstruction of SEM image using β=120, β=150 and β=200, respectively. Fig.10 a. SEM image from a MWCNT/PP specimen of 5 wt%. b., c. and d. reconstruction of SEM image using β=120, β=150 and β=170, respectively. Fig.11: Typical FE mesh of the tension specimen. Fig.12: a. Assignment of properties of the RUC with 5 CNTs at the elements of the tension specimen (blue elements), b. Assignment of properties of the RUC with 10 CNTs at the elements of the tension specimen (red elements) Fig.13: Comparison between predicted and experimental Young’s moduli for a. the 2 wt% MWCNT/PP specimens and b. the 2 wt% MWCNT/PP specimens. The standard deviation of the average experimental values is also included.

Captions to Tables: Table 1: Table 2:

Range of parameters measured from SEM images of the CNT/PP material. This range was considered in the parametric study Experimental Young’s moduli

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Fig.1

y

z

a

a

b

13

Fig.2

a

b

c

14

Fig.3

a

b

c

15

Fig.4

a

b

c

16

Fig.5

a

b

17

Fig.6

18

Fig.7

a

b

19

Fig.8

a

b

c

d

20

Fig.9

a

b

c

d

21

Fig.10

a

c

b

22

d

Fig.11

23

Fig.12

a

b

24

Fig.13

a

b

25

Table 1 Parameter CNT volume fraction CNT waviness CNT multitude CNT topology

Range 1-5% Extreme, Moderate 3, 5 and 10 1-3 CNTs in the X-direction

Table 2 Specimen

Average Young’s modulus (standard deviation)

PP MWCNT/PP 2wt% MWCNT/PP 5wt%

1641.63 (85.746) 1974.73 (146.66) 2058.55 (109.25)

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