Probabilistic Micromechanics Analysis of CNT Nanocomposites with Three-Dimensional Karhunen-Loève Expansion

Probabilistic Micromechanics Analysis of CNT Nanocomposites with Three-Dimensional Karhunen-Loève Expansion Fei-Yan Zhu 1, Sungwoo Jeong 2 and Gunjin Yun 3 1), 2, 3) Department of Mechanical & Aerospace Engineering, Seoul National University, Seoul 08826, Korea ABSTRACT A new method is presented for modeling spatially random distributions of effective modulus of CNT nanocomposites considering random waviness, dispersion (i.e. volume fraction) and interfacial stiffness within a 3D microscale RVE domain. For the random fields, 3D Karhunen-Loève expansion (KLE) was utilized with a transversely isotropic covariance kernel. Classical micromechanics models were adopted for computational efficiency with reasonable accuracy. Results were compared with numerical and experimental data from literatures for verification of the proposed method. Effective modulus was more sensitive to the waviness than the interfacial stiffness. 1. INTRODUCTION Carbon nanotubes (CNT) have Young s moduli of order 1TPa, and tensile strengths 20 times of those of high strength steel alloys, and thermal conductivities which are almost double of pure diamond (Collins and Avouris 2000). Since the CNT was discovered in 1991 by Iijima (Iijima 1991), a plethora of studies on CNT nano-fillers have been conducted as found in comprehensive review papers with different perspectives (De Volder, Tawfick et al. 2013, Gohardani, Elola et al. 2014, Rafiee and Moghadam 2014, Liew, Lei et al. 2015, Mittal, Dhand et al. 2015). CNT within a polymeric resin has distinct characteristics such as alignment, waviness, dispersion, bonding and interphase materials between CNT fibers and matrix, etc. that significantly influence physical properties of CNT-reinforced polymeric nanocomposite materials. The interphase materials are formed through either covalent or non-covalent functionalization of CNTs, which are used to prevent agglomeration of CNTs caused by van der Waals forces. CNTs alignment effects on enhancements of the mechanical and electrical properties were also studied experimentally (Khan, Pothnis et al. 2013). 1 Graduate student 2 Graduate student 3 Corresponding author, Associate Professor, Gwanak-gu Gwanak-ro 1 Building 301 Room 1308, Seoul, South Korea, 08826, Tel)+82-2-880-8302, Fax)+82-2-887-2662, Email)gunjin.yun@snu.ac.kr

Tremendous studies on effective properties of CNT-based nanocomposites and their modeling approaches are found in the literatures (Shokrieh and Rafiee 2010, Zare and Garmabi 2014). Improper modeling of agglomerations, waviness, interfacial debonding and dispersions of CNTs mainly causes discrepancies between experimental data and theoretical predictions of effective properties. Therefore, modeling efforts have been focused on those unique geometric features of CNTs. For example, Hammerand, et al. reported that effects of interphase regions on the effective properties are more significant than clustering effects (Hammerand, Seidel et al. 2007). Fisher et al. (Fisher, Bradshaw et al. 2002) reported that waviness and orientations can also reduce the effective stiffness significantly. Following the work by Fisher et al., Bradshaw et al. counted for waviness effects of CNTs and proposed a prediction method of aligned or randomly oriented CNT nanocomposites by utilizing finite elements and Mori-Tanaka micromechanics theory (Bradshaw, Fisher et al. 2003). Shao et al. recognized significant influences of the waviness and interfacial debonding effects on reinforcement efficiency of CNTs (Shao, Luo et al. 2009). Anumandla et al. proposed a closed-form effective stiffness model by incorporating waviness and random orientation of CNTs in matrix materials (Anumandla and Gibson 2006). Comprehensive micromechanics analysis of CNT nanocomposite materials were also studied by Seidel, et al. (Seidel and Lagoudas 2006). Shockrieh, et al., provided a review of existing modeling approaches for CNT nanocomposite materials under three classifications: atomistic, analytical continuum and numerical continuum approaches (Shokrieh and Rafiee 2010), stating advantages, limitations and drawbacks of each of the existing modeling methods. However, large variations of experimental observations could be explained through stochastic modeling approaches. According to our literature surveys, there are few studies on stochastic modeling of CNT nanocomposites. Recently, a probabilistic approach at the microscale was attempted focusing on electrical properties of CNT-embedded nanocomposites (Yang, Jang et al. 2017). Molecular dynamic simulations can simulate crosslinking densities and its probabilistic models were also used to obtain microscale CNT-reinforced matrix properties (Subramanian, Rai et al. 2015). This paper proposed a new stochastic method to compute random distributions of effective moduli of CNT-reinforced epoxy matrix taking into account the spatial randomness of CNTs waviness, interfacial stiffness and dispersion (i.e. volume fraction) at the microscale. For the purpose, a series expansion method, called Karhunen-Loève expansion (KLE), was utilized to randomize the waviness, interfacial modulus and volume fractions of CNT. First, KLE is generalized for three-dimensional cases. Random effective properties computed by the proposed are compared with numerical and experimental data from literatures. It was proved that the proposed method could provide statistical variations that are well matched with experimental data. By the proposed method, uncertainty at the nanoscale could reasonably propagate to the microscale. The proposed modeling method will be used for inverse design of CNT nanocomposite materials.

2. PROBABILISTIC MICROMECHANICS MODEL FOR CNT NANOCOMPOSITES The concept of the proposed method is illustrated in Fig. F 1. Thee proposed method models CNT waviness, interfacial stiffness, and local volume fractionn based on various classical micromechanics theories. Spatially varying random fields in the microscale domain are modeled by the KLE method. The proposed method can probabilistically model heterogeneous distributions of the effective moduluss of CNT nanocomposites.. Fig. 1 Probabilistic prediction method for effective moduli of o 3D randomly oriented CNT nanocomposite, SEM imagee is from (Savvas, Stefanou et al. 2016) 2.1 Modeling of Random Fields Dispersion (i.e. volume fraction), waviness and interfacial properties of CNT nano-fillers show non-uniform distributions in the microscale space. Their infinite dimension in the probability space was reduced to finite f dimensions by threez to a dimensional KL expansion method. The KL expansion decomposes a spatial random field Hx, y, deterministic and a stochastic part as follows,, (1) where is the mean value, iss the normal variable,, are the eigenvalue and eigenvector of an analytical covariance kernel, respectively.. Solutions of the following Fredholm integral equation are,, (2) In the Fredholm integral equation,, is a covariance function that takes t an exponential forms:, Where,, and,, are randomly selected two points in the domains and L is the correlation length in i-th direction. Galerkin finite element method was used for discretization of the eigenfunction (Shang and Yun 2013).. (3)

2.2 Probabilistic Micromechanics Model for CNT Nanocomposites Two representative volume elements (RVE) of the CNT nanocomposites are shown in Fig. 1. The random waviness of CNT is defined as α, ω 2πA/L (4) Where is the position vector in the microscale domain and ω indicates the primitive randomness. The infinitesimal segment of CNT RVE was considered an off-axis transversely isotropic lamina as shown in Fig. 1. Five independent elastic properties (i.e.,,,,and ) of the off-axis lamina are obtained from Chamis s simplified micromechanics models (SMM) (Chamis 1984). In the CNT direction, effective modulus ( ) in the CNT direction and major Poisson ratio ( ) are determined by equal-strain rule of mixture (ROM). However, the transverse effective moduli (i.e. E 2 and E 3 ) are obtained by SMM as follows, ω, ω 1, ω1 / Where is the matrix modulus,, ω is the random local volume fraction of CNT bundle and is the transverse modulus of CNT. When the interfacial property is included, the transverse effective moduli is obtained as, ω, ω 1, ω, ω 1 1, ω 1, ω, ω 1, ω, ω 1 (5) (6) where is the diameter of outer interface, is the diameter of outer CNT, and, ω is the random interfacial elastic modulus. Random shear moduli (i.e. and ) are also obtained in the same form as Eq. (5) and (6) by properly replacing properties. These properties are used to compute random compliances of along the principal material axes. Following Hsiao and Daniel s approach (Hsiao and Daniel 1996), off-axis elastic properties averaged over one wavelength of the wavy CNT in RVE1 are obtained. For example, random E, G, ν are derived as E, ω 2 G, ω 2 ν, ω 2 (7)

where s are functions of waviness α (Hsiao and Daniel 1996). Other engineering properties can be found in (Hsiao and Daniel 1996). The second RVE2 has endcaps of polymer matrix (L m /2) at both ends of RVE1 to represent discontinuity of CNT. Quasi-isotropic elastic material properties for randomly oriented wavy CNT RVEs are obtained by the method of Christensen-Waals (Christensen and Waals 1972). They are derived as follows E, ω E, ω E 4ν 8ν 4K E 4ν 4ν 1K 6G G 32E 8ν 12ν 7K 2G G υ, ω E 4ν 16ν 6K 4G G 4E 16ν 24ν 14K 4G G In the case of RVE2 with matrix endcaps, the random quasi-isotropic elastic modulus ( ) was obtained by inverse ROM as follows 1, ω 1 1 In this paper, =40 nm and =20 nm were assumed. (8) (9) (10) 3. RESULTS AND DISCUSSIONS The proposed method analytically models random features of CNT s dispersion, waviness and interfacial stiffness, and produces statistical variations of effective elastic modulus. Results are compared both with existing numerical and experimental results in order to show verification and advantages of the proposed method. Table 1 and Table 2 include the assumed base material properties and statistical parameters of the three random fields, respectively. Table 1 Properties of base materials CNT(Shen and Li 2005, Yazdchi and Salehi 2011) Matrix Material (Anumandla and Gibson 2006) Interfacial Material α 0.1 1.9 GPa 1) 5.0 GPa 1.58 TPa 0.3 2) 1.7 nm 0.156 0.73 GPa 3) 1.36 nm 0.493 TPa (Masud, Tahreen et al. 2009) 2) (Yang, Yu et al. 2013) 10.57 GPa 3) (Shen and Li 2005) Table 2 Statistical parameters of random fields Volume fraction Waviness Interfacial Material 0.125 0.1 5.0 GPa σ 0.25 σ 0.1 σ 1.0 20 nm 20 nm 20 nm

3.1 Model Verification for 3D Randomly Oriented CNT Nanocomposites Effective moduli (E wavy ) of a wavy CNT reinforcement are a back-calculated from f rule of mixture (ROM) equation as 1 where E is from Eq. (7)(a).( E was compared with w valuess from a literature where E was computed from a finitee element model (Fisher, Bradshaw et al. 2002). As shown in Fig. 2(a), E from this paper are very closee to literature values.. Effective moduli (E, ω) dramatically changee with waviness and volume fraction of CNT as shown in Fig. 2(b). Experimental results from a literature(andrews, Jacques et al. 2002) were also included for comparisons. These results well verify the proposed method. (11) (a) (b) Fig. 2 (a) Comparisons off effective moduli of wavy CNT with results from literature (Fisher, Bradshaw et al. 2002) E m =1 GPa and Poisson ratio is 0.3and (b) comparisons of effective moduli E 3D-RVE1 (Eq.( (8)(a))) with experimental results (Andrews, Jacques et al. 2002) E ratio =E CNT /E m 3.2 Effect of Interfacial Stiffness and Waviness on Effective Stiffness According to simulations, E, ω was larger than, ω for a range of volume fraction. As the volume fraction increases, the difference also increased (Fig. 3). Therefore, matrix end caps in the micromechanicss model reduce the effectivee stiffness, which is physically reasonable.

Fig. 3 Comparison of E 3D-RVE E1 and E 3D- of the random waviness and interfacial stiffness, two combinations of random fields weree simulated by a Monte Carlo sampling method, that is, E v and αvv. Probabilistic ranges in Fig. 4(a) and (b) -RVE2 vs. CNT volumee fraction In order to compare the level of influences indicate effects of random waviness and random interfacial stiffness at each volume fraction, respectively. In case of αv, experimental data d are included within the range. As increasing the volume fraction, mean value of the effective stiffness gets closer to experimental data. However, in case of E v, the effect of interfacial stiffnesss was observed as marginal. Therefore, the effective modulus of 3D randomly r oriented CNT nanocomposite is more sensitive to random waviness than random interfacial stiffness. In this simulation, we assume no specific s correlation between interfacial stiffnesss and waviness. (a) Fig. 4 (b) (a) Random waviness and volume fraction and (b) random interfacial stiffness and volume fraction effects on effective modulus As illustrated in Fig. 5, isosurfaces of E indicate truly random effectivee modulus due to random nanoscale CNT dispersion, waviness and interfacial stiffness. The assumed small correlation length seeminglyy caused the heterogeneous effectivee modulus. It is still challenging g to estimate the correlation length experimentally.

4. CONCLUSIONSS Probabilistic effective modulus of 3D randomly oriented CNT nanocomposites was predicted through a series of classical micromechanics models, 3DD Karhunen-Loève expansion and Monte Carlo sampling method. Morphologic cal and physical properties of CNT, for example, waviness, dispersion and interface stiffness dramatically change effectivee modulus. The proposed method has advantages s of computational efficiency giving proper accuracy. Analysing effective modulus and comparingc g it with experiment data prove that the proposed method iss credible and reliable. ACKNOWLEDGMENT This work was supported by Creative-Pioneering Researchers Program through Seoul National University (SNU). Authors aree grateful for their supports. REFERENCES Andrews, R., D. Jacques, M. Minot and T. Rantell (2002). "Fabrication off carbon multiwall nanotube/polymer composites by shear mixing." Macromolecular Materials and Engineering 287(6): 395-403. Anumandla, V. and R. F. Gibson (2006). "A comprehensive closed form micromechanics model for estimating the elastic modulus of nanotube-reinforced composites." Composites Part a-applied Science and Manufactur ring 37(12): 2178-2185. Bradshaw, R. D., F. T. Fisher and L. C. Brinsonn (2003). " Fiber waviness in nanotube- dilute reinforced polymer composites-ii: modeling via numerical approximation of the strain concentration tensor." Composites Sciencee and Technology 63(11): 1705-1722. (a) Effective Modulus of RVE1 (E 3D-RVE1 =2.8) (b) Effective Modulus of RVE2 (E 3D-RVE2 2=2.8) Fig. 5 Random distributions of effective modulus of 3D randomly oriented CNT- reinforced matrix within microscale RVE

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