Effect of microstructural features on the thermal conducting behavior of carbon nanofiber–reinforced styrene-based shape memory polymer composites

Abstract

This article presents a novel hierarchical micromechanics approach to carefully investigate the thermal conductivities of styrene-based shape memory polymer composites containing carbon nanofibers. The research is mainly focused on the simulation of carbon nanofiber/shape memory polymer interfacial thermal resistance and carbon nanofiber agglomeration as two critical microstructural features of carbon nanofiber–shape memory polymer composite materials. The computed results are compared with the available experimental measurements. It is found that both of those microstructural factors along with carbon nanofiber non-straight shape significantly affecting the thermal conducting behavior must be incorporated in the analysis to have a more realistic prediction. The thermal conductivity of carbon nanofiber–reinforced shape memory polymer composites reduces significantly due to the effects of carbon nanofiber/shape memory polymer interfacial resistance and carbon nanofiber agglomeration and waviness. It is suggested to uniformly disperse carbon nanofibers into the shape memory polymers and reduce interfacial resistance for improving the carbon nanofiber–styrene composite thermal properties. In addition, the present study reveals that the effective thermal conductivities of the shape memory polymer composites reinforced by aligned carbon nanofibers are greatly enhanced over those of the shape memory polymer composites containing randomly dispersed carbon nanofibers. The effects of percentage, waviness parameters, degree of agglomeration, material properties, length and diameter of carbon nanofibers as well as interfacial thermal resistance value on the thermal behavior of carbon nanofiber–reinforced styrene-based shape memory polymer composites are investigated.

Keywords

shape memory polymer carbon nanofiber thermal properties agglomeration interfacial resistance

1. Introduction

Shape memory materials (SMMs) have the capacity to recover from an initial temporary shape to their permanent shape under various types of external stimuli such as temperature, moisture, light, electricity, and magnetic fields (Hager et al., 2015; Liu et al., 2017; Wang and Wang, 2018). Among the active and deformable SMMs, shape memory polymers (SMPs) with advantages of more than 100% recoverable strain and low cost, have been used for a wide range of applications in daily life, like high-performance textiles, aerospace and civil engineering, actuators, electronic devices, and intelligent medical devices (Li et al., 2018; Mu et al., 2018; Qiao et al., 2017; Zhao et al., 2015). Styrene-based polymer developed by Cornerstone Research Group (CRG) Industries has received increasing attention due to its excellent shape memory properties (Beblo et al., 2010; Du et al., 2017; Tandon et al., 2009). This material is a two-part, fully formable thermoset SMP system. Also, in contrast to shape memory ceramics and shape memory metals, SMP materials show other natural advantages such as lower density, better biological and organic compatibility, and easier modification and processing (Liu et al., 2018; Meng and Hu, 2009). Furthermore, modern science offers possibilities to regulate the glass temperature via controlling the chemistry of the structure of SMP materials consistent with different applications. However, pure SMP materials possess low mechanical stiffness and strength and small recovery force which may lead to seriously limitations in engineering applications (Liu et al., 2004; Yu et al., 2011; Zhao et al., 2019).

Composite technology, which uses two or more materials mixed together to obtain the desired properties that cannot be achieved in any one of the constituents, has attracted increasing attention in various areas (Capela et al., 2019; Kundalwal and Ray, 2014a; Mouritz, 2019; Sharma and Lakkad, 2015). Overall, SMP composites show low mass density, adjustable glass transition temperature, and great damping capability making these smart composite materials appropriate for aerospace applications, such as trusses, hinges, antennas, and solar arrays (Chen et al., 2018; Dao et al., 2018). In this frame to satisfy different requirements, microscale fibers such as carbon fiber have been applied as reinforcements to improve the mechanical properties of SMP materials. For example, Li et al. (2019) measured the thermo-mechanical properties of unidirectional epoxy-based SMP composites with carbon fiber mass fractions of 16%, 23%, 30%, 37%. It was shown that both flexural modulus and strength of carbon fiber–SMP composites are temperature-dependent. Also, the carbon fiber–reinforced epoxy-based SMP system displayed good shape recovery capability, with recovery ratio of higher than 93% at 120°C. In another study, Herath et al. (2018) evaluated the thermo-mechanical properties, photothermal response and light activation of woven carbon fiber–reinforced epoxy-based SMP composites for large-scale engineering applications. They reported that the shape fixity and recovery ratio of the SMP composites are 100% and 86%, respectively. Guo et al. (2015) performed some experimental tests on the trans-l,4-polyisoprene (TPI) SMP composites reinforced with chopped carbon fiber weight fraction of 5%, 7%, 9%, 11%, and 13% to investigate the thermo-mechanical properties and shape memory behaviors. The largest Young’s modulus was observed for 7 wt.% chopped carbon fiber–reinforced SMP specimen.

Further improvement of the effective properties of SMP composites is an object of interest for numerous research teams. More recently, the availability of nanoscale fillers has allowed the development of polymer composites (Boutaleb et al., 2009; Tsai et al., 2010; Yanase et al., 2013). SMP composite nanostructures are the important components where various types of nanofillers such as carbon nanofibers (CNFs) are utilized to gain excellent mechanical, thermal, and functional properties (Kai et al., 2016; Lu et al., 2014). Generally, decrease of the material dimensions from microscale to nanoscale leads to in several very interesting characteristics like great surface-to-volume ratio, surface functionalities, unique transport properties along the axis because of the confinement influence and higher mechanical properties in comparison with the bulk counterparts (Roduner, 2006; West and Halas, 2003). One characteristic of polymer composite systems filled with CNFs thought to be a vital aspect in their effective properties is the CNF specific surface area (Hammel et al., 2004; Jimenez and Jana, 2007). For the same reinforcement percentage, the number density of CNFs within the nanoscale filler–reinforced composites will be more than that of carbon fiber within the traditional microscale filler–reinforced composites. It is shown in Figure 1. Thus, due to very low size of CNFs, they present a greater surface area-to-volume ratio when compared to microscale carbon fibers which results the CNF/polymer interfacial region in the nanoscale filler–reinforced composite systems to be significantly higher than that of carbon fiber–reinforced composites. Several experimental works have been focused on the evaluation of effective properties of CNF–SMP composites. For example, Tang et al. (2013) prepared and measured the mechanical and electrical properties of specimens of polyester-based SMP composites with different volume fractions of CNFs. By a uniform dispersion of CNFs and good interfacial bonding between SMP and CNF, a significant improvement in the composite mechanical properties was revealed. Furthermore, adding CNFs could successfully convert the insulating SMP material into electrically conductive composites with a percolation threshold of 2.5 vol.%. It was found that a composite system reinforced with 11.6 vol.% CNFs reaches recovery ratio of 97% within 90 s subjected to a small triggered voltage of 20 V. Also, Dong et al. (2013) found that the CNF-reinforced epoxy-based SMP composite has a low glass transition temperature of 48°C which leads to a fast recovery rate at 65°C in 10 s. Also, the SMP composite filled with 1 wt.% CNFs showed a high shape recovery and fixity ratio of higher than 90% after several thermomechanical cycles. The strength and electrical conductivity of the epoxy SMP can be increased with adding CNFs. In another experimental work, Yu et al. (2014) evaluated elastic modulus, tensile strength, flexural strength, and shape recovery properties of the styrene-based SMP composites containing different weight fractions of CNFs in the range of 0%–2%. The isothermal static mechanical tests demonstrated that the mechanical properties of the SMP composite specimens increase by increasing CNF percentage. Moreover, they performed some experimental tests to measure the thermal conductivity of the styrene-based SMP composites filled with CNFs. In contrast to pure styrene SMP material, its composite filled with CNFs has higher thermal conductivity.

The study of thermal properties of CNF-reinforced SMP composites is of great significance in the field of smart nanoscale filler–reinforced composite design. Efficient thermal management is particularly imperative for microelectronic applications, because the enhancement in the heat elimination of such devices is directly related to their efficiency, prolongation of half-life time operating and the abatement of premature failures of equipment (Chen et al., 2016). To solve the heat dissipation problem of microelectronic devices, increasing the material thermal conductivity is still the most efficient technique. In this frame, thermal conductivity needs to be accurately investigated for application of CNF–SMP composites in various fields of engineering equipment (Chen et al., 2016; Zhang et al., 2015). Most of experimental works have been provided to assess the mechanical, electrical and shape memory properties of SMP composites filled with CNFs. However, there is a notable lack of data to characterize the effective thermal conductivity CNF-reinforced SMP composites in a meaningful and exhaustive way.

Fabrication of CNF-reinforced polymer composites is a difficult, time-consuming, and expensive process. For example, obtaining uniform dispersion of CNFs in polymeric materials still remains a challenge (Dong et al., 2013; Yu et al., 2014). The CNFs tend to agglomerate. A typical transmission electron microscopy (TEM) image of CNF agglomeration within the polyester-based SMP composites has been shown by Tang et al. (2013). In addition to the importance of producing a homogeneous microstructure, other essential characteristics of CNF–polymer composites are recognized to be the interfacial interactions between the CNF and surrounding polymer, non-straight shape and transversely isotropic behavior of CNF (Dong et al., 2013; Tang et al., 2013).

The micromechanical approach is a common method of investigating the effective properties of polymer composites reinforced with nanoscale fillers (Boutaleb et al., 2009; Yanase et al., 2013). These micromechanical models can predict the effective properties of heterogeneous materials from volume percent, material properties of constituents, and interactions between the phases. The micromechanical methods play a substantial role in development of nanofiller–polymer composites by providing the predictions to help understand, analyze and design such new material systems which can be attained readily and cost effectively.

The aim of the present work is to investigate the thermal conducting response of general CNF-based composites using a novel micromechanics approach. The main novelty of this work is considering the non-straight shape and agglomeration of CNFs and interfacial thermal resistance between the CNF and SMP matrix as three important parameters affecting the thermal properties of CNF–SMP composites into the micromechanical modeling. To validate our developed micromechanical model, comparisons are made between our numerical results and experimental data reported in literature. Also, the proposed micromechanical method is employed to investigate the influences of weight fraction, non-straight shape, length, diameter, degree of agglomeration and alignment of CNFs as well as CNF/SMP interfacial thermal resistance on the equivalent thermal conductivities of CNF-reinforced styrene-based SMP composite systems. This study not only provides a thermal evaluation of the CNF-reinforced styrene-based SMP composites for direct engineering applications, but also offers a valuable reference for different types of polymer composite systems filled with various carbon-based nanoscale reinforcements which could then be utilized to design and optimize new generations of heterogeneous material systems.

2. Simulation of CNF–SMP composites

Schematic view of CNF–SMP composite microstructure and solution procedure divided into several parts are shown in Figure 2. Herein, a new two-scale homogenization approach is suggested to address the effect of CNF agglomeration on the thermal behaviors of SMP composites. The first homogenization scale (small-scale) deals with determining the thermal conductivity of spherical inclusion from CNF and SMP matrix. It in turn serves as reinforcement into the pure SMP matrix to obtain the final CNF-reinforced SMP composites on the large-scale. Note that CNFs are agglomerated within these spherical inclusions as presented in Figure 2(a) and (b). In the second homogenization scale (large-scale), the thermal conductivity of agglomerated CNF-reinforced SMP composites is computed from the homogenized spherical inclusion and SMP matrix properties. Simulation of non-straight shape of CNFs and CNF/SMP interfacial thermal resistance will be carried out in small-scale. Efforts for micromechanical modeling of CNF–SMP composites considering the critical microstructural features according to the strategy shown in Figure 2 are provided as follows.

2.1. Agglomeration of CNFs within the SMP composites

The existence of CNF agglomeration during the fabrication of polymer composites has been quite documented (Jimenez and Jana, 2007; Tang et al., 2013; Yu et al., 2014). So, it is required to reflect such an agglomerated state of CNFs into the micromechanical process. To this end, the whole SMP composite is divided into two domains (Barai and Weng, 2011). The first domain is the CNF-free SMP matrix indicated as phase $O$ with volume fraction $c_{O}$ . The second domain (small-scale) is adopted to be the spherical inclusions consisting of the CNFs and the rest of SMP matrix (Figure 2(b)) symbolized as phase $I$ with the volume fraction $c_{I}$ . In terms of this separation, it can be expressed

c_{I} + c_{O} = 1

(1)

The spherical inclusion (or phase $I$ ) is itself a composite material filled with homogeneous dispersion of CNFs. The volume fractions of SMP matrix and CNF within the inclusion have been specified by $c_{0}^{(I)}$ and $c_{1}^{(I)}$ . Based on this specification, it can be expressed

c_{1}^{(I)} + c_{0}^{(I)} = 1

(2)

Denoting the volume fractions of SMP matrix and CNF in the whole SMP composites by $c_{0}$ and $c_{1}$ results following relation

c_{1} = c_{1}^{(I)} \cdot c_{I}, c_{0} = c_{O} + c_{0}^{(I)} \cdot c_{I}

(3)

Degree of CNF agglomeration into the SMP composites is symbolized by $c_{I}$ . When all CNFs are uniformly dispersed within the SMP composite, it is expressed that $c_{I} = 1$ .

2.2. Large-scale problem

This section presents the Mori–Tanaka micromechanics model (Benveniste, 1987; Mori and Tanaka, 1973) for estimating the effective thermal properties of the particulate polymer matrix composites. For steady-state heat conduction problems, this model incorporates a single ellipsoidal heterogeneity embedded inside an infinite homogeneous polymer matrix domain subjected to a constant far-field heat flux (Benveniste, 1987; Mori and Tanaka, 1973). The Mori–Tanaka employs the continuum averaged heat flux vector $q$ and temperature gradient $\nabla T$ to estimate the equivalent thermal conductivities of the composite systems (Chen and Wang, 1996). The heat flow in a composite material can be expressed in terms of the far-field applied heat flux vector as follows

q = - \bar{K} \nabla T

(4)

where $\bar{K}$ refers to the effective second-rank thermal conductivity tensor. One fundamental assumption in deriving the thermal conducting formulations using the Mori–Tanaka model is that the resulting temperature gradient inside each heterogeneity is constant (Benveniste, 1987; Chen and Wang, 1996). The thermal conductivity tensor for a SMP composite containing spherical inclusions is given by

\begin{matrix} K_{LS}^{C} = K^{O} \cdot {I + c_{I} (S - I) \cdot {(A - S)}^{- 1}} \\ \cdot {I + c_{I} S \cdot {(A - S)}^{- 1}}^{- 1} \end{matrix}

(5)

in which

A = {(K^{O} - K^{I})}^{- 1} \cdot K^{O}

(6)

where $K^{O}$ and $K^{I}$ are the second-rank thermal conductivity tensors for SMP matrix and spherical inclusion (CNF-containing domain $I$ ), respectively. Also, $I$ is the second-rank identity tensor, and $S$ stands for the second-rank Eshelby tensor for heterogeneity. The components of Ehelby tensor for sphere shape have been presented by Hatta and Taya (1985).

To calculate the values of $K_{LS}^{C}$ , it is required to have $K^{I}$ . Following section will provide the effective thermal conductivity of inclusion $(K^{I})$ . Spherical inclusions are composed of the rest of SMP matrix and the randomly dispersed CNFs which it results in an isotropic behavior for phase $I$ ; that is, $K_{11}^{I} = K_{22}^{I} = K_{33}^{I}$ .

2.3. Small-scale problem

This section introduces four important steps to predict the thermal conductivity of inclusion. First, for development of any efficient homogenization scheme, the interfacial thermal resistance (i.e. the reciprocal of the interface thermal conductance) between the CNF and surrounding SMP needs to be carefully addressed since it has significantly effect on the heat transfer performance of the nanoscale filler–reinforced composite systems (Nan et al., 2004; Unnikrishnan et al., 2008; Xue, 2006). For instance, Wilson et al. (2002) informed that the magnitude of the interfacial thermal resistance between nanosized fillers and various matrixes can be in the range of 0.77 × 10⁻⁸–20 × 10⁻⁸ m² K/W. In another research, Huxtable et al. (2003) found that the interfacial thermal resistance between the carbon nanotube (CNT) and surrounding matrix can be about 8.3 × 10⁻⁸ m² K/W. Second step is related to predicting the effective thermal conductivities of the aligned straight CNF-reinforced SMP composites using a unit cell micromechanics technique. However, due to their great aspect ratio, small bending stiffness and processing induced effects (Hammel et al., 2004; Jimenez and Jana, 2007), the CNFs within polymer composites are not straight but rather have certain degree of waviness. Therefore, in the third step an efficient technique will be carried out to simulate the CNF waviness. Finally, in the fourth step, a homogenization process is performed mainly to evaluate the thermal conductivity of three-dimensional (3D) randomly oriented CNFs within the SMP composites. By applying this approach, the critical microstructural features, including CNF/SMP interfacial thermal resistance, waviness and random orientation of CNFs are taken into account in evaluating the thermal conductivity of CNF–SMP composite systems.

2.3.1. CNF/SMP interfacial thermal resistance

Due to the high surface area-to-volume ratio of nanoscale carbon fillers, there is a huge interface between the CNFs and polymer matrix in composites which significantly affects the thermal conducting response of composite materials (Huxtable et al., 2003; Nan et al., 2004; Xue, 2006). The interfacial thermal resistance is classified as a heat flow barrier related to a weak contact at the interface, varieties in phonon spectra according to the atomic arrangements and densities of the two phases. The interfacial thermal resistance has been recognized as the Kapitza resistance $R_{K}$ which can be expressed by the heat flux $Q$ , and the temperature drop $δ T$ , across the interface as

Q \approx \frac{δ T}{R_{K}}

(7)

In the micromechanical methods, the interfacial thermal resistance between the polymer and the nanoscale fillers can be usually taken as interfacial region influence (Bryning et al., 2005; Kundalwal and Ray, 2014b). The CNF has been covered with a very thin interfacial thermal barrier layer as presented in Figure 2(f). Herein, the CNF and surrounding interfacial region are integrated as an equivalent nanofiber (ENF). This ENF is indicated in Figure 2(e). So, a homogenization process by the rule of mixture (ROM) is performed to obtain the equivalent thermal conductivities of the ENF along the axial $(K_{A}^{NF})$ and transverse $(K_{T}^{NF})$ directions. In this process, the CNF and interfacial layer have the role of reinforcement and matrix, respectively. Thus, the values $K_{A}^{NF}$ and $K_{T}^{NF}$ can be estimated via the ROM for a simple series model of the barrier layer/CNF/barrier layer as follows

K_{A}^{NF} = \frac{L_{CNF} K^{O} K_{A}^{CNF}}{L_{CNF} K^{O} + 2 r_{K} K_{A}^{CNF}}

(8)

K_{T}^{NF} = \frac{d_{CNF} K^{O} K_{T}^{CNF}}{d_{CNF} K^{O} + 2 r_{K} K_{T}^{CNF}}

(9)

where $L_{CNF}$ and $d_{CNF}$ stand for the length and diameter of CNFs, respectively. Also, $r_{K}$ refers to the so-called Kapitza radius expressed as

r_{K} = R_{K} K^{O}

(10)

In other words, the interfacial thermal property has been concentrated on a surface of zero thickness and described via Kapitza radius. When no interfacial thermal resistance exists between the polymer matrix and CNF, the heat flux will be entirely transferred from the polymer matrix to the CNFs. In this situation $r_{K} = 0$ . But, because of the interaction between the CNF and polymer resin making high interfacial region, the heat flux cannot be completely transferred from the matrix to the CNFs which results $r_{K} \neq 0$ .

2.3.2. Unit cell method

This section presents the micromechanics model based on the method of cell (MOC) approach (Kundalwal et al., 2014) to estimate the effective thermal properties of the aligned straight CNF-reinforced SMP composites. This composite system can be viewed such that the ENF is the reinforcement and the matrix phase is the SMP as shown in Figure 2(e). To implement a unit cell micromechanical model for unidirectional composite system, a representative volume element (RVE) is constructed such that all effective characteristics and global behavior of the composite are similar to those of the RVE (Hassanzadeh-Aghdam and Ansari, 2019; Kundalwal and Ray, 2012). The RVE of the SMP composites with four sub-cells which one of them has been occupied by the ENF and three other sub-cells are SMP matrix is displayed in Figure 3. It is assumed that ENFs are aligned along the x₃-direction. The composite system is regarded to be consisted of cells which make doubly periodic arrays along the x₁ and x₂ directions. Notation $β γ$ is used to denote each sub-cell of the RVE, with $β$ and $γ$ signifying the location of the sub-cell along the $x_{1}$ and $x_{2}$ directions, respectively. As shown in Figure 3, $h$ , $b$ , and $l$ are the lengths of the RVE in the $x_{1}$ , $x_{2}$ , and $x_{3}$ directions, respectively. Also, $h_{β}$ , $b_{γ}$ , and $l$ denote the lengths of the sub-cell $β γ$ in the $x_{1}$ , $x_{2}$ , and $x_{3}$ directions, respectively.

To extract the micromechanical relations of MOC, four local coordinate systems, including ${\bar{x}}_{1}^{β}$ , ${\bar{x}}_{2}^{β}$ , and $x_{3}$ have been presented. All these coordinate systems possess origins located at the centroid of each sub-cell. According to the MOC approach, the deviation of the temperature from a reference temperature can be extended in the following form

Δ Θ^{β γ} = Δ T + {\bar{x}}_{1}^{β} ξ_{1}^{β γ} + {\bar{x}}_{2}^{γ} ξ_{2}^{β γ}

(11)

where $ξ_{1}^{β γ}$ and $ξ_{2}^{β γ}$ refer to the linear dependence of the temperature on the local coordinates.

The imposition of the continuity conditions of the temperature at the sub-cell interfaces on an average basis results the following relations

{\begin{matrix} h_{1} ξ_{1}^{1 γ} + h_{2} ξ_{1}^{2 γ} = h \frac{\partial T}{\partial x_{1}} \\ b_{1} ξ_{2}^{β 1} + b_{2} ξ_{2}^{β 2} = b \frac{\partial T}{\partial x_{2}} \end{matrix}

(12)

For the average heat flux in the sub-cell $β γ$ $({\bar{q}}_{i}^{β γ})$ , based on the Fourier’s law, one can arrive following relations

{\bar{q}}_{1}^{β γ} = - K_{11}^{β γ} \frac{\partial T}{\partial x_{1}}, {\bar{q}}_{2}^{β γ} = - K_{22}^{β γ} \frac{\partial T}{\partial x_{2}}, {\bar{q}}_{3}^{β γ} = - K_{33}^{β γ} \frac{\partial T}{\partial x_{3}}

(13)

where $K_{ii}^{β γ}$ denotes the thermal conductivity of the sub-cell $β γ$ . The average heat flux of the RVE of the composites can be calculated as

q_{i} = \frac{1}{V} \sum_{γ = 1}^{2} \sum_{β = 1}^{2} V_{β γ} {\bar{q}}_{i}^{β γ}

(14)

where $V$ is the total volume of the RVE and equals to

V = hbl

(15)

and the volume of each sub-cell $(V_{β γ})$ is

V_{β γ} = h_{β} b_{γ} l

(16)

The imposition of the continuity conditions of the heat flux at the sub-cell interfaces results

{\bar{q}}_{1}^{1 γ} = {\bar{q}}_{1}^{2 γ}, {\bar{q}}_{2}^{β 1} = {\bar{q}}_{2}^{β 2}

(17)

It is possible to connect the average heat flux components to the temperature gradients by means of the thermal conductivity coefficients as follows

{\bar{q}}_{1} = - K_{11}^{NC} \frac{\partial T}{\partial x_{1}}, {\bar{q}}_{2} = - K_{22}^{NC} \frac{\partial T}{\partial x_{2}}, {\bar{q}}_{3} = - K_{33}^{NC} \frac{\partial T}{\partial x_{3}}

(18)

where $K_{ii}^{NC}$ is the composite thermal conductivity. Finally, eliminating the microvariables $(ξ_{1}^{β γ}, ξ_{2}^{β γ})$ in conjunction with continuity conditions, the equivalent thermal conductivities of aligned straight CNF-reinforced SMP composites will be determined as

{\begin{matrix} K_{11}^{NC} = \frac{K^{O} {K_{T}^{NF} [h (V_{11} + V_{21}) + h_{2} (V_{12} + V_{22})] + K^{O} h_{1} (V_{12} + V_{22})}}{hbl (K^{O} h_{1} + K_{T}^{NF} h_{2})} \\ K_{22}^{NC} = \frac{K^{O} {K_{T}^{NF} [b (V_{11} + V_{21}) + b_{2} (V_{12} + V_{22})] + K^{O} b_{1} (V_{12} + V_{22})}}{hbl (K^{O} b_{1} + K_{T}^{NF} b_{2})} \\ K_{33}^{NC} = \frac{K_{A}^{NF} V_{11} + K^{O} (V_{12} + V_{21} + V_{22})}{hbl} \end{matrix}

(19)

So, the related second-rank thermal conductivity tensor can be expressed as

K^{NC} = [\begin{matrix} K_{11}^{NC} & 0 & 0 \\ 0 & K_{22}^{NC} & 0 \\ 0 & 0 & K_{33}^{NC} \end{matrix}]

(20)

2.3.3. Simulation of CNF waviness

In this study, the analytical approach developed by Hsiao and Daniel (1996) is suggested to investigate the effects of CNF waviness. It is assumed that the waviness shape of CNFs to be sinusoidal (Dastgerdi et al., 2013; Hsiao and Daniel, 1996) as follows

y = A \sin (\frac{n π}{L} z)

(21)

where $A$ and $L$ represent the amplitude of the CNF wave and the linear distance between the CNF ends, respectively. Herein, the ratio of $A / L$ has been called waviness factor. Also, n refers to the number of waves of the CNF. Figure 2(c) illustrates the RVE of a SMP composite reinforced with aligned wavy CNFs. This RVE is divided into infinitesimally thin slices of thickness $dz$ . Averaging the thermal conductivities of those slices over the RVE length leads to obtaining the effective thermal conductivities of the aligned wavy CNF-reinforced SMP composites. Each slice may be assumed to be as an off-axis unidirectional lamina which results that its thermal conductivities are determined by way of transforming the thermal conductivities of the corresponding specially ply. So, by applying the successive coordinate transformation, the thermal conductivities of the SMP composites filled with aligned wavy CNFs at any point in the ply where the CNF is inclined at the angle $θ$ with regard to the z-direction are attained as

{\begin{matrix} {\bar{K}}_{11} = K_{11}^{NC} \\ {\bar{K}}_{22} = K_{22}^{NC} p^{2} + K_{33}^{NC} q^{2} \\ {\bar{K}}_{33} = K_{22}^{NC} q^{2} + K_{33}^{NC} p^{2} \end{matrix}

(22)

in which

{\begin{matrix} p = \cos θ = {[1 + {\frac{n π A}{L} \cos (\frac{n π z}{L})}^{2}]}^{- \frac{1}{2}} \\ q = \sin θ = \frac{n π A}{L} \cos (\frac{n π z}{L}) {[1 + {\frac{n π A}{L} \cos (\frac{n π z}{L})}^{2}]}^{- \frac{1}{2}} \end{matrix}

(23)

where $K_{11}^{NC},$ $K_{22}^{NC}$ , and $K_{33}^{NC}$ were calculated by the MOC approach; that is, equation (19). As the value of $θ$ changes over the length of the CNF, the values of ${\bar{K}}_{11}$ , ${\bar{K}}_{22}$ , and ${\bar{K}}_{33}$ of the ply vary along the its length. Thus, the homogenized thermal conductivities of aligned wavy CNF-reinforced SMP composites can be evaluated by the following integral equation

{\begin{matrix} {\bar{K}}_{11}^{NC} \\ {\bar{K}}_{22}^{NC} \\ {\bar{K}}_{33}^{NC} \end{matrix}} = \frac{1}{L} \int_{0}^{L} {\begin{matrix} {\bar{K}}_{11} \\ {\bar{K}}_{22} \\ {\bar{K}}_{33} \end{matrix}} dz

(24)

2.3.4. SMP composites filled with randomly oriented CNFs

Alignment of CNFs within the polymer matrixes is a very difficult task and the distribution of CNFs is random. In other words, CNFs can be oriented into the polymer matrixes in all possible directions. Figure 2(b) shows a spherical inclusion in which randomly oriented CNFs are embedded into SMP matrix. Accordingly, orientation averaging has been conducted to study the effect of randomly oriented CNFs on the equivalent thermal conductivity of spherical inclusion $(K^{I})$ . Applying the successive transformation for a second-rank tensor from global-to-local coordinates, it can be written as

K_{ij}^{l} = α_{im} α_{jn} K_{mn}^{g}, (i, j, m, n = 1, 2, 3)

(25)

where $K^{l}$ and $K^{g}$ refer to the second-rank thermal conductivity tensors in terms of local and global coordinates, respectively. It should be noted that the components of $K^{g}$ are equal to those of ${\bar{K}}^{NC}$ determined from equation (24). Also, for 3D randomly oriented condition, the general form of the global-to-local transformation tensor is given by

α = [\begin{matrix} \cos θ \sin \emptyset & \sin θ \sin \emptyset & \cos \emptyset \\ - \cos \emptyset \cos θ & - \cos \emptyset \sin θ & \sin \emptyset \\ \sin θ & - \cos θ & 0 \end{matrix}]

(26)

Finally, the homogenized second-rank thermal conductivity tensor for a SMP composites filled with randomly oriented CNFs can be calculated by the following integral equation

K_{R} = \frac{1}{2 π} \int_{0}^{π} \int_{0}^{π} K^{l} \sin θ d θ d \emptyset

(27)

It can be noticed that the non-zero components of second-rank thermal conductivity tensor of inclusion are equal to the non-zero components of $K_{R}$ .

3. Results and discussion

In this section, first, the predictions by the proposed micromechanics modeling approach are compared with existing experimental data (Yu et al., 2014) on the CNF-reinforced styrene-based SMP composites. The thermal conductivities of this composite have been experimentally characterized by Yu et al. (2014). In this validation example, the thermal conductivity and density $(ρ^{O})$ of styrene matrix are 0.17 W/mK and 0.92 g/cm³ (Yu et al., 2014). The thermal conductivity of pure styrene matrix was measured experimentally by Yu et al. (2014). Also, the density of styrene shape memory resin is reported by Yu et al. (2014). The axial and transverse thermal conductivities of the transversely isotropic CNFs are selected to be 500 and 30 W/mK (McIvor et al., 1990; Mahanta et al., 2010), respectively. Mahanta et al. (2010) experimentally measured the thermal conductivities of the CNFs. The equivalent axial thermal conductivities of the CNFs were in the range of 429–589 W/mK. Finally, they concluded that the axial thermal conductivity of the CNF can be about 500 W/mK (Mahanta et al., 2010). The values of dimeter and length of CNF are reported from Chengdu Organic Chemicals Co. Ltd, Chinese Academy of Sciences (Yu et al., 2014). It has been stated that the nanofibers have an average diameter of 100 nm and an average aspect ratio of 50 (Yu et al., 2014). So, the average length of nanofiber is calculated to be 5 µm. In the case of $R_{k}$ , Wilson et al. (2002) stated that the magnitude of the interfacial thermal resistance between nanoscale particles and different matrixes can be in the range of 0.77 × 10⁻⁸–20 × 10⁻⁸ m² K/W. Also, the CNT/matrix interfacial thermal resistance evaluated by Huxtable et al. (2003) is about 8.3 × 10⁻⁸ m² K/W. Numerous studies have reported that the interfacial thermal resistance between the CNT and polymer matrix may be about 8.3 × 10⁻⁸ m² K/W (Bryning et al., 2005; Chen et al., 2016; Huxtable et al., 2003; Kundalwal and Ray, 2014b; Nan et al., 2004). Selection of waviness parameters, including waviness factor $A / L$ , number of waves $n$ has some complexity. These parameters are usually evaluated based on the experimental observation of micrographs. According to the previous studies on the elastic modulus of CNT-reinforced polymer composites, when the values of $A / L$ and $n$ are 0.5 and 2, respectively, very good agreement has been resulted between the theoretical predictions and experimental measurements (Dastgerdi et al., 2013; Pan et al., 2016; Yanase et al., 2013; Yang et al., 2012). Therefore, the values of waviness factor, number of waves, density, length and diameter of CNFs, and interfacial thermal resistance are considered to be A/L = 0.5, n = 2, ρ_CNF = 1.7 g/cm³, L_CNF = 5 µm, d_CNF = 100 nm and R_k = 8.3 × 10⁻⁸ m² K/W, respectively.

Figure 4 demonstrates the comparison between the two sets of results for the CNF-reinforced styrene-based SMP composites. Also, the effects of important microstructural features, including interfacial thermal resistance, non-straight shape, directional behavior and dispersion type of CNFs on the thermal conducting behavior of styrene-based SMP composites. In the case of isotropic CNF, its transverse thermal conductivity is considered to be equal to that axial thermal conductivity; that is, $K_{A}^{CNF} = K_{T}^{CNF} = 500 W / mK$ . It can be observed from Figure 4 that:

Without interfacial thermal resistance, the predictions are very far from the experimental data. Also, the thermal conductivity of SMP composites without interfacial thermal resistance is higher than that of SMP composites with interfacial thermal resistance. So, eliminating the interfacial resistance causes an improvement in the thermal properties.

The predictions with straight CNFs are higher than the experimental data. Also, the thermal conductivity of SMP composites filled with straight CNFs is higher than that of SMP composites filled with wavy CNFs. Thus, using the straight CNFs increases heat dissipation from the CNF–polymer composite structures.

The SMP composite thermal properties depend on the directional behavior of CNFs. The thermal conductivity of SMP composites considering isotropic CNFs is higher than that of SMP composites considering transversely isotropic CNFs.

When CNFs are uniformly dispersed, (i) the interfacial thermal resistance, (ii) waviness shape, and (iii) transversely isotropic behavior of CNFs must be included in the simulation to have good predictions.

At high content of CNF, that is, when the CNF weight fraction is higher than 1%, the discrepancy between the experiment and predictions considering uniform dispersion of CNFs is noticeable. This is due to the agglomeration of CNFs within the SMP composites as reported by Yu et al. (2014). Thus, besides the interfacial thermal resistance, waviness shape and transversely isotropic behavior of CNFs, the agglomerated state of CNFs $(c_{I} = 0.5)$ must be included in the simulation to get more realistic predictions.

The CNF agglomeration significantly decreases the composite thermal conductivity.

The SMP composite thermal conductivity increases with the increase of CNF content.

Figure 5 shows the effect of agglomeration degree of CNFs on the micromechanical predictions. Three values are selected for agglomeration degree, including c_I = 0.4, 0.5 and 0.7. It is found that the value of c_I = 0.5 can generate acceptable predictions as compared to the experimental data.

The influence of interfacial thermal resistance between the CNF and SMP in the range of 0–8.3 × 10⁻⁸ m² K/W on the thermal conducting behavior of styrene-based composites is shown in Figure 6. The thermal conductivity CNF–SMP composite significantly increases with the decrease of interfacial thermal resistance, and its maximum value is when R_k = 0. When the CNF weight fraction is 2%, compared with the case of R_k = 8.3 × 10⁻⁸ m² K/W, the thermal conductivity of the SMP composite without interfacial thermal resistance increases by 130.4%.

The effects of waviness factor (A/L) and number of waves (n) of CNFs on the thermal conductivity of styrene-based composites are explored and the results are summarized in Figure 7(a) and (b), respectively. Note that $A / L = 0$ or n = 0 is corresponding to straight CNFs. The increase of $A / L$ or $n$ leads to a reduction in thermal conductivity, thus, use of straight CNFs improves the heat dissipation. For example, the thermal conductivities of 2 wt.% CNF–SMP composite are 0.316 W/mK for wavy CNFs $(A / L = 0.5)$ and 0.392 W/mK for straight CNFs, respectively.

The effect of CNF transverse thermal conductivity on the styrene-based composite thermal conductivity is investigated in Figure 8. The thermal conductivity improves with increasing the CNF transverse thermal conductivity up to $K_{T}^{CNF} = 100 W / mK$ . The change of CNF transverse thermal conductivity slightly affects the composite thermal properties when $K_{T}^{CNF} > 100 W / mK$ .

High aspect ratio of CNFs and other factors induced during the fabrication of CNF-polymer composite systems lead to the agglomeration of CNFs (Pan et al., 2016; Yang et al., 2012). Figure 9 shows the thermal conductivity of the SMP composites as a function of CNF weight fraction for different values of $c_{I}$ . Note that c_I = 1 is corresponding to a uniform dispersion of CNFs into the SMP matrix. The thermal conductivities of SMP composites with CNF agglomeration are lower than those of styrene-based SMP composites with uniform dispersion of CNFs. The increase of $c_{I}$ enhances the heat dissipation from SMP composites. So, producing a SMP composite with the uniform dispersion of CNFs is an efficient way for obtaining maximum level of thermal properties.

Figure 10(a) and (b) presents the variation of the effective thermal conductivity of styrene-based SMP composites with length and diameter of CNFs, respectively. The values of $L_{CNF}$ and $d_{CNF}$ are selected to be in the range of 1–1000 µm and 1–1000 nm, respectively. The thermal conductivity of the SMP composites improves with increasing the length and diameter of CNFs. This is due to the fact that increasing the size of CNFs can decrease the effect of interfacial thermal resistance on the final composite thermal conductivity. This statement is extracted from equations (7) and (8). There is a threshold value for $L_{CNF}$ after which further increase of CNF length does not affect the thermal behavior of SMP composites. Similar trend is observed for the effect of CNF diameter on the composite thermal response.

The effect of CNF alignment on thermal conductivity of the styrene-based composites is shown in Figure 11. The thermal conductivities are extracted in the presence and the absence of interfacial thermal resistance. The thermal conductivities of aligned CNF-reinforced composites are significantly higher than those of randomly dispersed CNF-reinforced composites. Thus, alignment of CNFs into the composite systems along the thermal loading direction is a suitable candidate for heat transfer problems.

Herein, another methodology is proposed to investigate the effect of CNF agglomeration on the effective thermal conductivity of SMP composites. This method has been previously used to model the CNT agglomeration on the elastic properties of polymer composites (Pan et al., 2016; Yang et al., 2012). A number of CNTs are assumed to be uniformly distributed in the SMP matrix and remaining CNTs appear in agglomerated state in spherical inclusions, as depicted in Figure 12. The volume fraction of CNFs within the spherical inclusion is different from that in other domains of the SMP matrix. The proposed model shows that two kinds of reinforcing phases exist, (i) CNFs and (ii) spherical inclusions. Thus, a multi-step procedure is used to analyze the thermal conductivity of the CNF–SMP composites. In this model, the whole volume of CNFs $V_{r}$ in the total SMP composite consists of two parts as follows

V_{r} = V_{r}^{inclusion} + V_{r}^{m}

(28)

in which $V_{r}^{inclusion}$ and $V_{r}^{m}$ are the volumes of CNFs into the spherical inclusion and SMP matrix outside the spherical inclusion, respectively. To characterize the CNF agglomeration, two parameters are presented as follows

ξ = \frac{V_{inclusion}}{V}, ζ = \frac{V_{r}^{inclusion}}{V_{r}} 0 \leq ξ, ζ \leq 1

(29)

in which $V$ and $V_{inclusion}$ are the volumes of composite and agglomerated spherical inclusions. Also, $ξ$ signifies the volume fraction of spherical inclusions with respect to the total volume of composite system $V$ , and $ζ$ indicates the volume ratio of the CNFs within the spherical inclusions over the total CNF $V_{r}$ . When $ξ$ = 1, a uniform dispersion of CNFs in the whole composite system occurs. As the value of $ξ$ decreases, the CNF agglomeration degree can be more severe. All CNFs are located in spherical inclusions when $ζ = 1$ . When $ξ = ζ$ , it means a uniform dispersion of CNFs without agglomeration. For a composite system, the volume fraction of CNFs is denoted by

v_{r} = \frac{V_{r}}{V}

(30)

Considering $v_{r}^{inclusion}$ and $v_{r}^{m}$ as the volume fractions of CNF in the agglomerated spherical inclusion and CNFs in the matrix outside the spherical inclusions, respectively, following relation can be derived

v_{r}^{inclusion} = \frac{V_{r}^{inclusion}}{V_{inclusion}} = \frac{ζ}{ξ} v_{r}, v_{r}^{m} = \frac{V_{r}^{m}}{V - V_{inclusion}} = \frac{(1 - ζ)}{(1 - ξ)} v_{r}

(31)

To compute the thermal conductivity of SMP composites containing CNF agglomeration, a three-step micromechanical procedure is employed. First, by defining the parameters $ξ$ and $ζ$ , the thermal conductivity of spherical inclusion consisting of CNFs and SMP matrix is obtained. Second, similar process is used to obtain the thermal conductivity of an equivalent matrix consisting of the rest of SMP material reinforced by the rest CNFs outside the spherical inclusion. In these two steps, both effects of CNF waviness and interfacial resistance are taken into account in the micromechanical modeling by the MOC. Finally, in the third step, the thermal conductivity of the CNF–SMP composites is predicted by the Mori–Tanaka micromechanical model using the properties of equivalent matrix and spherical inclusions.

Figure 13 shows the variations of thermal conductivity of the styrene-based SMP composites with $ζ$ under $ξ = 0.5$ . Two different values, including 1% and 2% are selected for the CNF volume fraction. It can be observed that with increasing the value of $ζ$ from 0.5 to 1, the effective thermal conductivity of the SMP composites the composite decreases. The maximum level of thermal conductivity will be obtained when the value of $ζ$ becomes equal to that of $ξ$ ; that is, $ζ = ξ = 0.5$ . This condition (i.e. $ζ = ξ$ ) is corresponding to uniform dispersion of CNFs into the composites. However, in the extreme case of agglomeration where all CNFs are concentrated in spherical inclusions, that is $ζ = 1$ , the SMP composite thermal conductivity is minimum.

Figure 14 illustrates the variation of styrene-based SMP composite thermal conductivity with CNF volume fraction for the different values of $ζ$ . In this analysis, the value of $ξ$ is 0.5. It is found that the effective thermal conductivity increases with the decrease of agglomeration parameter $ζ$ .

4. Conclusion

The current work was directed to develop a new hierarchical micromechanics approach comprising the Mori–Tanaka method and a unit-cell model for estimating the thermal conductivities of general CNF-based composites. The major contribution of this work was to include the critical microstructural parameters such as waviness, directional behavior and agglomerated state of CNFs as well as CNF/SMP interfacial thermal resistance into the micromechanical analysis. Generally, an excellent agreement was found between the predictions of the micromechanical model considering important microstructural features and experimental data. Due to the presence of interface thermal resistance, CNFs induced no significant improvement in SMP composite thermal conductivities. One efficient way to enhance the thermal conductivity was found to be eliminating or at least decreasing interfacial thermal resistance. Agglomeration of CNFs reduced the effective thermal properties of SMP composite, whereas their uniform distribution maximized the thermal conductivity. Hence, another way to increase the thermal conductivity was suggested to be uniform dispersion of CNFs into the SMP composites. Also, the thermal properties of the SMP composite were enhanced by reinforcement with straight CNFs. The results indicated that the SMP composite thermal conductivity increases with increasing (i) weight fraction, (ii) length, and (iii) diameter of CNFs. Another outcome from this works is that the effective thermal conductivity of the CNF–SMP composite can be greatly enhanced by the alignment of CNFs along the thermal load.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Lilin Jiang

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