# Multi-scale numerical simulations of thermal expansion properties of CNT-reinforced nanocomposites

- Alamusi
^{1}, - Ning Hu
^{1, 2}Email author, - Jianhui Qiu
^{3}, - Yuan Li
^{4}, - Christiana Chang
^{5}, - Satoshi Atobe
^{6}, - Hisao Fukunaga
^{6}, - Yaolu Liu
^{1}, - Huiming Ning
^{1}, - Liangke Wu
^{1}, - Jinhua Li
^{1}, - Weifeng Yuan
^{2}, - Tomonori Watanabe
^{1}, - Cheng Yan
^{7}and - Yajun Zhang
^{8}

**8**:15

**DOI: **10.1186/1556-276X-8-15

© Alamusi et al.; licensee Springer. 2013

**Received: **6 November 2012

**Accepted: **22 December 2012

**Published: **7 January 2013

## Abstract

In this work, the thermal expansion properties of carbon nanotube (CNT)-reinforced nanocomposites with CNT content ranging from 1 to 15 wt% were evaluated using a multi-scale numerical approach, in which the effects of two parameters, i.e., temperature and CNT content, were investigated extensively. For all CNT contents, the obtained results clearly revealed that within a wide low-temperature range (30°C ~ 62°C), thermal contraction is observed, while thermal expansion occurs in a high-temperature range (62°C ~ 120°C). It was found that at any specified CNT content, the thermal expansion properties vary with temperature - as temperature increases, the thermal expansion rate increases linearly. However, at a specified temperature, the absolute value of the thermal expansion rate decreases nonlinearly as the CNT content increases. Moreover, the results provided by the present multi-scale numerical model were in good agreement with those obtained from the corresponding theoretical analyses and experimental measurements in this work, which indicates that this multi-scale numerical approach provides a powerful tool to evaluate the thermal expansion properties of any type of CNT/polymer nanocomposites and therefore promotes the understanding on the thermal behaviors of CNT/polymer nanocomposites for their applications in temperature sensors, nanoelectronics devices, etc.

### Keywords

Polymer-matrix composites (PMC) Thermal properties Numerical analysis Carbon nanotube (CNT)## Background

As technology and modern industry has developed, reinforced composite materials, such as particle- or short-fiber-reinforced composites and long-fiber-reinforced or sandwich laminates, have been widely applied in the aerospace, construction, transportation, machinery, chemical, and other industries. In recent years, as a representative of new engineering materials, carbon nanotube (CNT) at nanoscale has shown superior mechanical, electrical, and thermal properties, as well as low density and high aspect ratio, which make it an ideal choice for composite reinforcement. CNT-reinforced nanocomposite is a multi-phase material, and its external macro-physical properties strongly depend on the properties of its constituents and complex internal microstructure. Experimental evaluation requires large amounts of material samples and a large testing work load, giving simulation of the physical properties of nanocomposites important engineering significance.

There has been extensive research on the mechanical, thermal, and electrical properties of CNT-reinforced nanocomposites. For instance, the thermal properties [1–3] and electrical properties of CNT-reinforced nanocomposites [4, 5] have been explored experimentally in some previous studies. Moreover, due to the complexity and variations of the CNT-reinforced composite microstructure, theoretical analyses and numerical simulation methods are common strategies to estimate composite physical properties. For instance, diffusion and thermal expansion coefficients of CNT-reinforced nanocomposites have been studied through micromechanics models without sufficient atomic scale information [6] or molecular dynamics (MD) models with very high computational cost and complexity [7].

In recent years, to deal with the remarkable scale difference in CNT-reinforced nanocomposites, multi-scale modeling has been widely used for predicting the mechanical properties [8], electrical properties [9], and thermal conductivity [10] of the CNT-reinforced nanocomposites. However, to the best knowledge of the present authors, there has been no report on the multi-scale modeling of thermal expansion properties of the CNT-reinforced nanocomposites to date. In this work, the thermal expansion properties of the CNT-reinforced nanocomposites, i.e., CNT/epoxy, were evaluated using a sequential multi-scale numerical model. The present study focused on the effects of two key parameters, i.e., temperature and CNT content, on the thermal expansion properties. Moreover, it was found that the results of the present multi-scale numerical model agree very well with those based on theoretical predictions and experimental measurements carried out in this work.

## Methods

*ε*of the present MWCNT and epoxy from 30°C to 120°C are shown in Figure 3. As shown in [14], the axial thermal expansion rate of MWCNT is dominated by MWCNT's inner walls. We modeled MWCNT's six innermost walls [14] to obtain the approximate axial thermal expansion rate of the present MWCNT in Figure 3.

- 1.
The CNT content of CNT/epoxy nanocomposites ranged from 1 to 15 wt%.

- 2.
The length and diameters of the outmost and innermost walls of CNT were set as 5 μm, 50 nm, and 5.4 nm, respectively, which are in accordance with the experimental measurement using a transmission electron microscope [9, 15]. The properties of MWCNT used in the present experiments are shown in Table 1.

**Properties of MWCNT**

Property | Value |
---|---|

Fiber diameter (nm) | Average 50 |

Aspect ratio (−) | >100 |

Purity (%) | >99.5 |

- 3.
We only considered the axial thermal expansion/contraction of MWCNT, and the radial thermal expansion/contraction was neglected since they are very small as identified in [14]. Therefore, CNT thermal expansion properties were orthotropic. Other properties of CNT were assumed to be isotropic, as well as those of epoxy. The detailed material properties in simulations are listed in Table 2.

- 4.
For the uni-directional model, simulations were conducted using a quarter of the cross section of a cylinder representative volume element (RVE) containing a CNT, i.e., an axisymmetrical model (see Figure 1). Under thermal loading, some forces along the radial direction were imposed on the nodes of the outmost lateral surface of the RVE and adjusted through an iterative procedure so that all points on the outmost lateral surface moved at the same distance in the radial direction to simulate the periodic conditions [16]. The length of the polymer was two times longer than that of the CNT in Figure 1, implying that the short CNTs are distributed evenly in both longitudinal and lateral directions in a matrix so that the RVE is the same for any CNT [16].

- 5.
For the multi-directional model, there were randomly distributed 100 CNTs per model (see Figure 2). This model was built up under plane-strain conditions. The boundary conditions were applied at the two external edges which is similar to those for the uni-directional model above. In order to reflect the 3D characteristics of real nanocomposites, the volume fraction should be converted to the half of the real one [12, 13]. Note that the number of the CNTs in this model, i.e., 100, was determined by some trial computations, such as testing of models containing 10, 25, and 50 CNTs. It was found that 100 is the minimum number, which can yield isotropic, convergent, and stable results. This number is just the same with that of holes for modeling the effective mechanical properties of a porous plate [17].

## Results and discussion

### Uni-directional models

*ε*| becoming gradually smaller and finally converging to a stable value when the CNT content reaches 10 wt%. Note that the thermal expansion rate is negative at 30°C.

### Multi-directional models

*ε*| becomes smaller gradually. However, unlike the uni-directional nanocomposites (Figure 5), the thermal expansion rate of the multi-directional nanocomposites still decreases proportionally to the CNT content even when the CNT content is over 10 wt%.

### Verification

To verify the effectiveness of the above multi-scale numerical simulations, the following theoretical prediction and experimental measurements were carried out.

#### Theoretical prediction

The following assumptions are made to derive conventional micromechanics models for the coefficient of thermal expansion (CTE). Note that the CTE, which is generally understood as a constant and temperature-independent, is different from the thermal expansion rate used here. Following the terminology of conventional micromechanics models, we still use CTE in this section. The two-phase composite consisting of matrix and short fiber is of perfect interfaces at phase boundaries. Therefore, it is impossible for the two components, i.e., the matrix and short fiber, to separate at their interfaces when the composite is loaded or heated. Additionally, only macro-composites are considered, namely the scale of the reinforcement is large compared to that of the atom size or grain size so that composite properties can be modeled by continuum methods. This assumption may be reasonable here since the present MWCNT is comparatively large in diameter. Finally, the composite properties are an appropriate average of those of the components.

*f*(

*φ*), which is independent of dimension, can be given by [18]

*f*(

*φ*) = 1, and therefore, the CTE of the nanocomposites is

*f*(

*φ*) = 1/

*n*, where

*n*represents the number of different orientations of the MWCNTs in the matrix. If

*n*is the number of possible orientations, the CTE of the nanocomposites is

In the above equations, the nomenclatures for the parameters are as follows:

α, CTE

V, volume fraction

E, Young's modulus

ν, Poisson's ratio

and the subscripts are as follows:

c, nanocomposite

m, the matrix

f, the reinforcement phase (MWCNT here)

Note that Poisson's ratio of the nanocomposites, *v*_{c} in Equation 3, was directly obtained from the rule of mixture and the data in Table 2. For 1 ~ 5 wt% addition of CNTs, *v*_{c} ranges from 0.338 (1 wt%) to 0.333 (5 wt%).

#### Experimental measurements

The thermal expansion properties of the MWCNT/epoxy nanocomposites were measured using a TMA equipment (TMA-50, Shimadzu Co., Kyoto, Japan). The TMA measurement methodology is described as follows: a rectangular sample (3 cm wide, 3 cm long) was cut from the nanocomposites at a point 3 cm from the parallel portion of the tensile test specimen (according to JIS K 7197 [22]). Specimens were heated from 30°C to 120°C at a scanning rate of 5°C/min in air for continuous measurements. The thermal expansion properties of pure epoxy were similarly measured for the same specimen size and test conditions. Note that the highest test temperature, i.e., 120°C, is close to the glass transition point of bisphenol-F epoxy resin, which usually ranges from 120°C to 130°C, depending on fabrication conditions. In our tests, it was found that even at 120°C, the obtained thermal expansion rates were still normal and a molten or rubber-like state in epoxy was not identified.

#### Comparison

## Conclusions

In this work, the thermal expansion properties of CNT/epoxy nanocomposites with CNT content ranging from 1 to 15 wt% were investigated using a multi-scale numerical technique in which the effects of two parameters, temperature and CNT content, were investigated extensively. For all CNT contents, the obtained results clearly revealed that within a wide low-temperature range (30°C ~ 62°C), the nanocomposites undergo thermal contraction, and thermal expansion appears in a high-temperature range (62°C ~ 120°C). It was found that at any CNT content, the thermal expansion properties vary with the temperature. As temperature increases, the thermal expansion rate increases linearly. However, at a specified temperature, the absolute value of the thermal expansion rate decreases nonlinearly as the CNT content increases. Moreover, the results provided by the present multi-scale numerical model are verified with those obtained from a micromechanics-based theoretical model and from experimental measurement. Therefore, this multi-scale numerical approach is effective to evaluate the thermal expansion properties of any type of CNT/polymer nanocomposites.

## Declarations

### Acknowledgements

The authors are grateful to be partly supported by the Grand-in-Aid for Scientific Research (no. 22360044) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.

## Authors’ Affiliations

## References

- Haggenmueller R, Guthy C, Lukes JR, Fischer JE, Winey KI: Single wall carbon nanotube/polyethylene nanocomposites: thermal and electrical conductivity.
*Macromolecules*2007, 40: 2417–2421. 10.1021/ma0615046View ArticleGoogle Scholar - Biercuk MJ, Llaguno MC, Radosavljevic M, Hyun JK, Johnson AT, Fischer JE: Carbon nanotube composites for thermal management.
*Appl Phys Lett*2002, 80: 2767–2769. 10.1063/1.1469696View ArticleGoogle Scholar - Ruoff RS, Lorents DC: Mechanical and thermal properties of carbon nanotubes.
*Carbon*1995, 33: 925–930. 10.1016/0008-6223(95)00021-5View ArticleGoogle Scholar - Hu N, Karube Y, Cheng Y, Masuda Z, Fukunaga H: Tunneling effect in a polymer/carbon nanotube nanocomposite strain sensor.
*Acta Mater*2008, 56: 2929–2936. 10.1016/j.actamat.2008.02.030View ArticleGoogle Scholar - Hu N, Karube Y, Arai M, Watanabe T, Yan C, Li Y, Liu Y, Fukunaga H: Investigation on sensitivity of a polymer/carbon nanotube composite strain sensor.
*Carbon*2010, 48: 680–687. 10.1016/j.carbon.2009.10.012View ArticleGoogle Scholar - Seidel GD, Stephens SN: Analytical and computational micromechanics analysis of the effects of interphase regions on the effective coefficient of thermal expansion of carbon nanotube-polymer nanocomposites. In
*Proceedings of the 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference: April 12–15 2010; Orlando*. Reston: AIAA; 2010:2010–2809.Google Scholar - Wei C: Thermal expansion and diffusion coefficients of carbon nanotube-polymer composites.
*Nano Lett*2002, 2: 647–650. 10.1021/nl025554+View ArticleGoogle Scholar - Hu N, Fukunaga H, Lu C, Kameyama M, Yan B: Prediction of elastic properties of carbon nanotube-reinforced composites.
*Proc R Soc Lond A Math Phys Sci*2005, 461: 1685–1710. 10.1098/rspa.2004.1422View ArticleGoogle Scholar - Hu B, Hu N, Li Y, Akagi K, Yuan W, Watanabe T, Cai Y: Multi-scale numerical simulations on piezoresistivity of CNT/polymer nanocomposites.
*Nanoscale Res Lett*2012, 7: 402. 10.1186/1556-276X-7-402View ArticleGoogle Scholar - Clancy TC, Frankland SJV, Hinkley JA, Gates TS: Multiscale modeling of thermal conductivity of polymer/carbon nanocomposites.
*Int J Therm Sci*2010, 49: 1555–1560. 10.1016/j.ijthermalsci.2010.05.007View ArticleGoogle Scholar - Park C, Wilkinson J, Banda S, Ounaies Z, Wise KE, Sauti G, Lillehei PT, Harrison JS: Aligned single wall carbon nanotube polymer composites using an electric field.
*J Polym Sci, Part B: Polym Phys*2006, 44: 1751–1762. 10.1002/polb.20823View ArticleGoogle Scholar - Okabe T, Motani T, Nishikawa M, Hashimoto M: Numerical simulation of microscopic damage and strength of fiber-reinforced plastic composites.
*Adv Compos Mater*2012, 21: 147–163. 10.1080/09243046.2012.688495View ArticleGoogle Scholar - Huang H, Talreja R: Numerical simulation of matrix micro-cracking in short fiber reinforced polymer composites: initiation and propagation.
*Compos Sci Technol*2006, 66: 2743–2757. 10.1016/j.compscitech.2006.03.013View ArticleGoogle Scholar - Alamusi , Hu N, Jia B, Arai M, Yan C, Li J, Liu Y, Atobe S, Fukunaga H: Prediction of thermal expansion properties of carbon nanotubes using molecular dynamics simulations.
*Comput Mater Sci*2012, 54: 249–254.View ArticleGoogle Scholar - Yamamoto G, Liu S, Hu N, Hashida T, Liu Y, Yan C, Li Y, Cui H, Ning H, Wu L: Prediction of pull-out force of multi-walled carbon nanotube (MWCNT) in sword-in-sheath mode.
*Comput Mater Sci*2012, 60: 607–612.View ArticleGoogle Scholar - Hu N, Fukunaga H, Lu C, Kameyama M, Yan B: Prediction of elastic properties of carbon nanotube-reinforced composites.
*Proc R Soc A*2005, 461: 1685–1710. 10.1098/rspa.2004.1422View ArticleGoogle Scholar - Hu N, Wang B, Tan GW, Yao ZH, Yuan W: Effective elastic properties of 2-D solids with circular holes: numerical simulations.
*Compos Sci Technol*2000, 60: 1811–1823. 10.1016/S0266-3538(00)00054-3View ArticleGoogle Scholar - Wang YQ, Zhang MD, Zhou BL, Shi CX: A theoretical model of composite thermal expansion.
*Mater Sci Prog*1989, 3: 442–446.Google Scholar - Hu N, Masuda Z, Yamamoto G, Fukunaga H, Hashida T, Qiu J: Effect of fabrication process on electrical properties of polymer/multi-wall carbon nanotube nanocomposites.
*Composites: Part A*2008, 39: 893–903. 10.1016/j.compositesa.2008.01.002View ArticleGoogle Scholar - Hu N, Li Y, Nakamura T, Katsumata T, Koshikawa T, Arai M: Reinforcement effects of MWCNT and VGCF in bulk composites and interlayer of CFRP laminates.
*Composites: Part B*2012, 2012(43):3–9.View ArticleGoogle Scholar - Li Y, Hu N, Kojima T, Itoi T, Watanabe T, Nakamura T, Takizawa N, Inoue T, Cui H, Atobe S, Fukunaga H: Experimental study on mechanical properties of epoxy/MWCNT nanocomposites - effects of acid treatment, pressured curing, and liquid rubber.
*ASME J Nanotechnol Eng Med*2012, 3: 011004. 10.1115/1.4007018View ArticleGoogle Scholar - Japanese Industrial Standards Committee:
*JIS K 7197–1991: Testing Method for Linear Thermal Expansion Coefficient of Plastics by Thermomechanical Analysis*. Tokyo; 1991.Google Scholar

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