Mechanical properties of single-walled carbon nanotube reinforced polymer composites with varied interphase’s modulus and thickness: A finite element analysis study

Abstract

The role of the interphase on mechanical performance of glassy polymer/single-walled carbon nanotube composites has been investigated by finite element (FE) method. The matrix and the interphase are modeled using continuum elements and the nanotube is analyzed by Timoshenko beam elements. Stress distribution and mechanical property of the nanocomposites are quantified as a function of the interphase’s modulus and thickness. For composites that include an interphase, the predicted moduli are comparable to the values calculated by the rule of mixtures, but are much lower than those of an interphase-free composite. For composites consisting of only the CNT and the matrix, the predicted modulus values are in good agreement with those computed by the rule of mixtures and with theoretical data reported in the literature, although the predicted values are considerably higher than those of real polymer/CNT composites. Using Griffin’s fracture analysis for dissimilar materials, we have proved that fracture stress of the weak boundary layer in the CNT/polymer interphase is lower than that of the CNT/polymer interface or of the matrix. The weak boundary layer, which is always existing in a CNT/polymer’s three dimensional interphase, is proposed as the main reason for the large discrepancy between Young’s moduli predicted by the model observed in this study and reported in the literature and those measured experimentally from real-world polymer/SWCNT composites.

1. Introduction

Since their discovery in 1991 [1], carbon nanotubes (CNTs) have been explored intensively as a nanofiller for enhancing multifunctional properties of polymeric matrices, as demonstrated in numerous reviews [2–8]. CNTs can be thought of as rolled-up graphene sheets that form hollow cylindrical tubes [9–11]. Depending on how the graphene sheets are rolled-up, there are two main types of carbon nanotubes available today: (1) single-walled carbon nanotubes (SWCNTs) consisting of a seamless single cylinder having typical diameter of 1.4 nm, and (2) multi-walled carbon nanotubes (MWCNTs) consisting of concentric cylinders with an inter-tube spacing of 0.34 nm, and having a typical diameter between 2 nm and 30 nm. The lengths of both types of tube can be hundreds of micrometers or even several centimeters long. CNTs are suitable for a wide range of applications [10–12]. The combination of exceptional mechanical, thermal, and electrical properties along with low density and high aspect ratios makes CNTs ideal materials for polymer composites [9,11,13–16]. The strength of CNTs is about 30–100 times greater than that of steel, yet its density is only about one sixth of steel, and the elastic modulus is 1.2 TPa, which is comparable to that of diamond. The density of SWCNT is about 1.33–1.40 g/cm³, just one half that of aluminum, and the thermal conductivity of SWCNT is about 6000 W/m K at room temperature, nearly double that of diamond. And the electrical current carrying capability is estimated at 1.0 × 10⁹ amp/cm², which is three orders of magnitude higher than that of copper.

Despite significant progress witnessed in improving structural properties, there are several critical issues that need to be addressed before polymer/CNT composites can be reliably processed and extensively used [4,7,8,17]. These include (1) dispersion of CNTs in polymers, (2) interactions between polymer matrix and CNTs and the resulting interphase, (3) influence of CNTs and polymer properties and processing parameters on the performance of the resulting nanocomposites, and (4) development of advanced tools to effectively characterize and quantify dispersion, interphase, microstructure, and performance of the nanocomposites. Because these problems are complex and very challenging, simulation and modeling methods have been employed to provide manufacturing guidance and a better understanding of the processing-property relationships of polymer/CNT composites [17–23]. Various modeling techniques have been used to calculate the mechanical properties of CNTs and their polymer composites, including molecular mechanics and molecular dynamics simulations, continuum modeling, and multiscale modeling, as reviewed in Ref. [17].

Probably the most important requirement for CNT-reinforced composites is that the applied load must be effectively transferred from the matrix to the CNTs; that is, the interphase must be mechanically strong. Due to the small size and high aspect ratio of CNTs, the volume fraction of the interphase in a polymer/CNT composites is substantial and often plays an important role in the load transfer and its overall mechanical properties [24,25]. Issues associated with stress transfer and the interface/interphase in nanocomposites have been critically discussed [24–26], and Refs. [6,25] have reviewed some measurement and theoretical data on the interfacial shear strength of polymer/CNT composites. To provide a better understanding of its role in the mechanical performance of polymer/CNT composites, a brief description on the polymer/substrate interphase’s characteristics is presented here.

The interface is defined as the two-dimensional boundary between the matrix and the substrate (e.g., CNTs, particles, fibers, or any planar surfaces), while the interphase (i.e., interfacial region) is the three-dimensional region that includes the interface plus a zone of finite thickness on both sides of the interface. A schematic presentation of a polymer/filler (substrate) interphase can be seen in Refs. [27,28]. The interphase boundaries are generally defined from the point in the matrix where the local properties start to deviate from the bulk properties in the direction of the polymer/substrate interface. Depending on the system, the interphase can extend from a few nanometers to a few hundred nanometers from the interface. For CNT/polymer composites, chemical composition, molecular orientation, and morphology at the CNT surface can be quite different from that found in the bulk. Chemical functionalization, polymer wrapping, and presence of dispersants/surfactants can substantially alter the original surface, resulting in a chemically and structurally different CNT surface. Likewise, the nanostructure, physical properties, and chemical reactivity of the polymer matrix near the interface may be different from the bulk due to diffusion, preferential adsorption, and molecular organization at the interface. The polymer/CNT interphase could also include surface contaminants, unreacted or partially-reacted low molecular mass molecules, and processing additives. It is also affected by processing conditions, which may cause chemical reactions, species diffusion, volumetric changes, and residual stresses. In addition, local internal stress may also develop in the interfacial region to counter the reduction in the entropy of the polymer chains due to their interactions with the CNT surface, as demonstrated experimentally [29–31] and computer simulation [32] for other particles. Each of these phenomena can vary in scale and can occur concurrently in the interphase. Therefore, the structure, properties, and volume fraction (i.e., thickness) of the interphase have a profound effect on the mechanical properties, thermal stability, and long-term performance of polymer/CNT composites.

Despite extensive experimental and theoretical studies on the mechanical properties of polymer/CNT composites [2–8,17], little quantitative information is available about the effect of interphase properties on mechanical performance of this advanced material. This is undoubtedly due to the complexity, undefined boundaries, and difficulty in the characterization of the buried, nanoscale size interphase in nanocomposites. Several recent modeling studies have examined the interphase mechanical properties on the mechanical behavior of polymer/CNT composites [49–51]. Spring elements were used to model the interphase in one study [49], in which it is unclear how disparities in the number of degrees of freedom for beam elements for CNT and solid elements for matrix are handled. In another study [50], the mechanical behavior of the nanocomposite was examined assuming both perfect bonding between matrix and CNT and considering an elastic interface region. The elastic interface layer was modeled using both a thin elastic material surrounding the CNT and a series of spring elements. Details of the implementation of interface layer are not explained and matrix strength considered are substantially greater than those that are normally reported in the literature. In yet another study [51], elastic (Young’s) modulus of CNT-reinforced cement paste using 3D and axisymmetric models was studied. The behavior of the CNT and the cement matrix was assumed to be fully elastic, while a cohesive surface framework was used to model the interface. It is unclear how appropriate it is to model the interfacial region by using the so called cohesive element or cohesive interaction between CNT and matrix, because there are many unknown parameters in this approach.

A review of these papers and others [17,23,52] suggests that a simple, multiscale modeling approach is needed to determine the effect of the interphase on the overall mechanical behavior of the nanocomposite. Therefore, the main objective of this study is to assess the role of the interphase on the overall mechanical property of polymer/CNT composites using finite element (FE) modeling. Specifically, the effects of both the thickness and modulus of the interphase on stress distribution, stress transfer, and Young’s modulus of an elastic amorphous polymer/SWCNT composite are quantified. The nanotube is modeled following the Timoshenko theory using linear finite strain beam elements, while the interphase and polymer matrix regions are analyzed by the continuum brick (hexahedral) elements. In addition, the model was modified to analyze interphase-free composites, and the results are compared with literature values. The current methodology takes into account all three major components in a nanocomposite, and their mechanical behaviors in the 3-dimensional model are expected to provide good predictive capability of the overall mechanical properties of the polymer/CNT composites.

2. Modeling

The approach used in this study is based on multiscale modeling where the size scale in the nanometer range of the CNT constituent is much smaller than the continuum scale in the micrometer range of the polymer matrix. The unique feature that distinguishes the present analysis from previous studies [22,23] is the introduction of an interphase between the reinforcing CNT and the matrix in the model to account for its contributions to the overall mechanical properties of the composite.

A unit cell of the matrix/SWCNT composite shown in Fig. 1 has been chosen to perform the stress analysis calculation. The model consists of three components: an innermost region composing of a SWCNT, an interphase, and an outermost matrix region. The CNT is assumed to be continuous with its physical length identical to that of the matrix. An unmodified SWCNT is used as a model nanotube because its geometry is simple and its physical, mechanical, and chemical properties have been well characterized. The SWCNT is represented by an equivalent structural beam following the method of Ref. [23], and the interphase and matrix are modeled using hexahedral (brick) finite elements. For the analysis, we assume that the nanotube and the polymer matrix form “perfect” interfacial bonds, and the SWCNTs are uniform, straight, and continuous. Although the polymer/CNT interphase can be quite complex, as described earlier, the 2D interface is treated here as mainly due to van der Waals interactions, with minimum mechanical locking, no chemical reaction, and containing no physical defects. This assumption is valid for unmodified or unfunctionalized carbon nanotubes, because their relatively smooth surface provides little opportunity for entanglement or mechanical inter-locking, and the absence of reactive groups on their surface hinders the formation of hydrogen bonding or covalent bonding with the polymer matrices. Accordingly, the interphase is simulated by continuum brick elements having properties that are different from the polymer matrix. The three regions of the three-phase model are designated as: Phase 1 = SWCNT; Phase 2 = Interphase; and Phase 3 = Polymer matrix (Fig. 2).

Fig. 1.

A schematic showing SWCNT and the beam model in ANSYS.

Fig. 2.

A schematic of the cross-section of the model showing SWCNT, interphase, and the matrix regions (t_wall = 0.34 nm).

The nanotubes are modeled following the Timoshenko theory using finite strain beam elements (BEAM188 in ANSYS¹ [33]). This approach can handle linear, large rotation, and/or large strain non-linear behaviors, and is well-suited for analyzing slender to moderately thick beam structures. The analysis utilizes a linear (2-node) element in three dimensions (3D) and contains stress stiffness terms. It has six degrees of freedom (DOFs) at each node, consisting of translations in the x, y, and z directions and rotations about the x, y, and z directions. Both the interphase and the matrix regions are modeled using 3D 8-node (linear) structural solid elements (SOLID185 in ANSYS [33]). This solid element is defined by eight nodes having three DOFs at each node, e.g., translations in the nodal x, y, and z directions. In our assumption of perfect interfacial bonds between the CNTs and the matrix (and CNTs and interphase), the outer surface of the nanotube coincides with the inner surface of the polymer matrix (or interphase). For the purpose of modeling the atoms in the nanotube with nodes, it is assumed that the center of the atoms is located on the outer surface of the tube wall as opposed to being in the central region of the tube wall. This assumption is valid because the finite element nodes for the interphase are, in reality, a small distance away from these CNT atom positions. A multi-point constraint is used to connect the nanotube beam element nodes with the nodes for the interphase or matrix continuum brick elements. This will be explained later in more detail.

3. Computational details and data inputs

The finite element modeling has been performed using ANSYS software. The nanotube is modeled as space frame, and the interphase is surrounded by a rectangular (parallepiped) matrix region. The problem in implementing this type of multiscale model lies in the region where the 3D solid elements and 3D beam elements interface. A coupling needs to be imposed to handle this issue. This implementation must be handled with care because of the difference in the degrees of freedom of these two types of elements. For identifying the coupling between the CNT and the interphase, we define a “rigid” region by automatically generating constraint equations to relate nodes in the region. The CERIG command in ANSYS is used to generate a partial set of equations for the rigid region. The difference between CERIG and an actual element is that CERIG does not involve any element stiffness matrix. It simply represents a linear relationship between the DOFs (degrees of freedom) belonging to the master and slave nodes. The constraint equations generated between pairs of nodes belonging to two disparate regions are based on small deflection theory. This option is useful for transmitting the bending moment between elements having different DOFs at a node. We generate constraint equations needed for defining rigid lines in 3D space. In this case only three of the six equations are generated. A special label in the CERIG command allows generation of a partial set of rigid region equations. One disadvantage from the computational standpoint is that applying this command to a large number of slave nodes may result in constraint equations with a large number of coefficients. This could significantly increase the peak memory required during element assembly.

The 3-phase composite comprising three different regions is subjected to deformation in the longitudinal direction. No constraints were imposed in-plane. Table 1 gives dimensions and mechanical properties of the three regions that are used as inputs for the modeling. The outer radius of the nanotube is taken as 0.471 nm, the interphase starts at (R_nt) 0.471 nm, with the outer radius of the interphase (R_int) is taken as 2.355 nm or 8.007 nm (see Fig. 2). Together, these three regions constitute a representative volume element (RVE) or a 3-phase composite. Fig. 3 shows all three regions in ANSYS, and Fig. 4 displays only the CNT and the interphase regions. It should be mentioned that the dimensions of the three regions and the mechanical properties of the SWCNT and the matrix given in Table 1 are similar to those used in Ref. [23], mainly to validate the present model.

Table 1.

Dimensions and mechanical properties for the three regions with varied modulus and thickness of the interphase.

Quantity	Nanotube	Interphase		Matrix
Radius or thickness (nm)	0.471	2.355	8.007	27.355
Young’s modulus (GPa)	1000	Em, 0.75Em, 0.5Em, 0.25Em, 0.05Em	Em, 0.75Em, 0.5Em, 0.25Em, 0.05Em	2.41
Poisson ratio	0.4	0.45	0.45	0.45

Fig. 3.

A view of ANSYS sectional plot showing three different regions with separate colors. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4.

A sectional view of the ANSYS model of the nanotube and interphase only.

As a first case, a tensile iso-strain loading condition is assumed, and two maximum applied strains are considered: 0.103% and 0.414%. These strains are chosen to compare present results with those in Ref. [23]. The strain is applied as a displacement boundary condition in the z direction and is applied to all nodes on the top surface of the entire composite unit (i.e., the entire RVE). The nodes at the bottom region of the RVE are fixed. Sectional areas of the nanotube, interphase, and matrix are computed as follows (sectional areas represent areas obtained by cutting the body with the x–y plane) (see Fig. 2):

$$\begin{gathered} A_{nt}=\pi \left[R_{nt}^2-{\left(R_{nt}-0.34\right)}^2\right] \\ {A}_{int}=\pi \left[R_{int}^2-R_{nt}^2\right] \\ {A}_m=4a^2-\pi R_{int}^2 \end{gathered}$$ (1)

where A_nt, A_int, and A_m are the sectional areas of nanotube, interphase, and matrix respectively. R_nt, R_int, are outer radii of the nanotube and the interphase, respectively, and 2a is the width of the composite (see Fig. 2). Note that the wall thickness of the SWCNT is taken to be 0.34 nm, similar to that of Ref. [23]. The corresponding gross volume fractions of the three regions are obtained by:

$$\begin{gathered} V_{nt}=\frac{A_{nt}}{A_{nt}+A_{int}+A_m} \\ {V}_{int}=\frac{A_{\text{int}}}{A_{nt}+A_{int}+A_m} \\ {V}_m=1-V_{nt}-V_{int} \end{gathered}$$ (2)

For continuous fiber-reinforced composites, the axial Young’s modulus of the composite can be computed using the rule of mixtures:

$$E_c=E_{nt}{V}_{nt}+E_{int}{V}_{int}+E_m{V}_m$$ (3)

where E_nt, E_int, and E_m are the Young’s moduli, and V_nt, V_int, and V_m are the volume fractions, of CNT, interphase, and polymer matrix, respectively. The effective Young’s modulus of the composite can also be computed from the ratio of the sum of total forces in the matrix, interphase, and nanotube to the product of total sectional area and applied strain as shown in Eq. (4).

$$E_{eff}=\frac{F_{nt}+F_{int}+F_m}{\left(A_{nt}+A_{int}+A_m\right)*\epsilon }$$ (4)

where F_nt, F_int, and F_m are the total forces in nanotube, interphase, and matrix, respectively, and ε is the applied strain. The values for the total forces can be obtained from the ANSYS simulations, and the Young’s modulus computed using Eq. (4) can be compared with the value obtained with Eq. (3).

4. Results

4.1. 2-Phase polymer/CNT composite

For the purpose of comparing data obtained by our model with those obtained by Li and Chou [23] and also for providing a test case, the analysis presented in this section is for a 2-phase composite consisting of only the CNT and the matrix (i.e., no interphase). The following adjustments are made to the above equations to convert our 3-phase composite model to a 2-phase composite model.

$$\begin{gathered} A_{int}=0 \\ {A}_m=A_{int} \end{gathered}$$

Note that for the 2-phase composite, the region from “R_int” to “2a” as shown in Fig. 2 does not exist. Here, the interphase is replaced by the matrix. Therefore, we have only two cylindrical regions: (a) CNT: from (R_nt – t_wall) to R_nt and (b) Matrix: from R_nt to R_int. The cross sectional area for the matrix = $A_m=\pi \left[R_{int}^2-R_{nt}^2\right]$.

Using the dimensions and mechanical properties given in Table 1 and a thickness value of 2.355 nm for the matrix, the sectional areas of the nanotube and cylindrical matrix are 0.643 nm² and 16.73 nm², respectively, for the 2-phase composite. Note that the thickness value of 2.355 nm is employed here to duplicate that used in Ref. [23] so that the results can be directly compared. Applying these data, V_nt is computed to be 3.7% in the 2-phase material. Table 2 compares predictions from the present model with those reported in literature for a 2-phase polymer/CNT composite that is subjected to two different iso-strains in the axial/longitudinal (z) direction.

Table 2.

Computed values at two strains for a 2-phase composite consisting of only the polymer matrix and a SWCNT.

Quantity	Applied strain 0.103%		Applied strain 0.414%
	Model prediction	Values from Ref. [23]	Model prediction	Values from Ref.[23]
Sectional area of matrix (nm2)	16.726	19.074	16.726	19.074
Sectional area of CNT (nm2)	0.643	0.674	0.643	0.674
Total force in composite, nN	0.682	0.736	2.739	2.953
Effective Young’s modulus, GPa (Eeff, Eq. (4))	38.121	36.022	38.093	36.122
Young’s modulus from rule of mixtures, GPa (Ec, Eq. (3))	39.342	36.463	39.342	36.463

Table 2 shows that the Young’s moduli of the composite computed from the model, E_eff, for both strains agree well with those calculated using the rule of mixtures, E_c. This observation is similar to that reported previously [23]. Table 2 also shows that E_eff for the 2-phase composite is essentially independent of the applied strain. For these strain levels, the material is still within the linear elastic regime as explained in more detail later. It may also be noted that the effective Young’s modulus computed with Eq. (4) is somewhat lower than the Young’s modulus computed with Eq. (3), which agrees with the trend seen in values reported in ref [23].

4.2. 3-Phase polymer/CNT composites

It is obvious that the properties of the interphase and the exact thickness of this region are not precisely known in real polymer matrix/SWCNT composites. In order to investigate the effects of the thickness and mechanical property of the interphase on the overall composite’s Young modulus, FE computation is performed with the 3-phase composite (i.e., matrix, interphase, and CNT) model using the data given in Table 1. Only a 0.414% iso-strain in the axial direction is used in this case. For each interphase thickness, the Young’s modulus of the interphase, E_int, is arbitrarily varied to provide the following values: E_m, 0.75E_m, 0.5E_m, 0.25E_m, and 0.05E_m, where E_m is the matrix Young’s modulus. It should be noted that when the interphase thickness is increased from 2.355 nm to 8.007 nm (Table 1), the volume fraction of the matrix in the composite decreases from 99.4% to 93.3%. This is because the total cross-sectional areas of the composite for both cases of the interphase remain the same. Table 3 presents E values computed using the present model (Eq. (4)) and the rule of mixtures (Eq. (3)) for the 3-phase polymer/SWCNT composites with varied interphase’s thickness and modulus.

Table 3.

Computed Young’s moduli of the composite using the present model and rule of mixtures for 3-phase composite having varied interphase’s moduli and thicknesses.

Eint (GPa)	Interphase thickness 2.355 nm		Interphase thickness 8.007 nm
	Eeff (beam model) (GPa), Eq. (4)	Ec (mixture), (GPa), Eq. (3)	Eeff (beam model) (GPa), Eq. (4)	Ec (mixture), (GPa), Eq. (3)
2.41 (Em)	3.663	2.624	3.661	2.624
1.8075 (3Em/4)	3.660	2.621	3.610	2.584
1.205 (Em/2)	3.652	2.618	3.548	2.543
0.6025 (Em/4)	3.647	2.614	3.476	2.503
0.1205 (Em/20)	3.630	2.611	3.364	2.471

Table 3 shows that values of E_eff calculated by the present model and E_c obtained by the rule of mixtures for the 3-phase composite are substantially lower than those for the 2-phase composite (compare Table 2 with Table 3). This is mainly due to the fact that the volume fraction of SWCNT in the 3-phase composite is much lower than the volume fraction of SWCNT for the 2-phase material (e.g. 0.02% volume fraction of SWCNT in 3-phase compared to 3.7% for 2-phase, when interphase thickness is 2.355 nm). Table 3 also shows that the computed E_eff values from the model are slightly higher than those obtained from the rule of mixtures. The results suggest that the mechanical property of this type of composites is possibly controlled by local properties such as defects and stress concentrations.

The effect of E_int on the rate of change of E_eff is illustrated in Fig. 5 for two interphase thicknesses. As expected, for both cases the overall modulus as computed from the model decreases with a decrease in E_int. Further, the rate of decrease is more pronounced for the composite having a thicker interphase than that for same composite with a thinner interphase. This is mainly because the volume fraction of the interphase in the composite is 6.7% when the interphase thickness is 8.007 nm. This is much larger than the volume fraction of the interphase of 0.6% when the interphase thickness is 2.355 nm in the composite.

Fig. 5.

Dependence of composite modulus on Young’s modulus of the interphase for the 3-phase composite using dimensions in Table 1. (a) For interphase thickness of 2.355 nm, and (b) for interphase thickness of 8.007 nm. Lines are for eye’s guide.

The modeling technique using in this study also allows us to observe other complex phenomenon, such stress distributions, in a CNT/polymer matrix composite. One example is illustrated in Fig. 6, which displays the distributions of von Mises stress in half section of a 2-phase composite consisting of only the SWCNT and the matrix. An applied strain of 0.414% is used for this analysis, and the units for the stress are in N/nm². It can be seen that the maximum von Mises stress is found to be approximately 12.7 MPa, and the maximum shear stress appears to occur in the region near the matrix/CNT interface. The yield strength of such real composites has been reported to be in the range of (50–60) MPa (see Fig. 5 in Ref. [7]). Since the maximum von Mises stress is less than the yield strength, no yielding is expected to take place under this loading condition.

Fig. 6.

von Mises equivalent stress in the 2-phase polymer/SWCNT composite (stress units are in N/nm²; maximum value of 0.127E—10 corresponds to 12.7 MPa).

5. Discussion

The computed modulus value obtained by this study (see Table 2 for the case of 2-phase polymer/CNT composite) and also by Li and Chou [23] for a polymer matrix/SWCNT composite is nearly 15 times that of the polymer matrix (using the 2.41 GPa value for the matrix given in Table 1, a typical value for a glassy styrene polymer). A study by Haggenmueller et al. [34] using the Halpin–Tsai composite theory also reported that the experimental elastic modulus of polymer/SWCNT was an order of magnitude smaller than the predicted value. However, a survey of experimental elastic modulus values tabulated in several reviews [5,6,35] for nanocomposites comprising of unmodified or non-functionalized SWCNTs dispersed randomly in a variety of polymers at between 1% and 2.5% mass loadings showed only a moderate enhancement of this mechanical property, from 1.1 to 2.5 times that of the matrix. The discrepancy between theoretical and experimental elastic moduli (i.e. low nanofiller reinforcement potential) for the polymer/CNT composites is similar to that observed for the interfacial strength for polymer/substrate systems. For these materials, the calculated interfacial strengths between a polymer and a substrate, even at a separation of 1 nm, are still substantially higher than their experimental values [36–38].

This section presents a plausible mechanism for explaining the disagreement between theoretical and experimental Young’s modulus for real polymer matrix/CNT composites. The discussion is limited to only unmodified CNTs or CNTs that had been functionalized with non-or low-polar groups mainly to improve the chemical compatibility between the matrix and the CNTs. This discussion does not cover CNTs bearing reactive groups that can form strong hydrogen, electrostatic, or covalent bonds with the polymer matrix. At present, three reasons have been cited for the low reinforcement potential of CNTs in polymers: inefficient load transfer between the matrix and the CNTs [6,7], low (cohesive) strength of the matrix [35], and slippage of individual CNTs in bundles [39,40]. The third reason (i.e., individual CNT slippage in a bundle) is still a subject of debate. For example, a modeling and experimental study by Li et al. [35] has demonstrated that both bundle intercalation and inter-bundle bridging do not contribute to the enhancement of mechanical properties of CNTs and that only the outermost individual CNTs of the bundle carry the load. Although these reasons have been proposed, the exact mechanism for the low reinforcement potential of CNTs in polymer nanocomposites has not been resolved. We believe that one major reason for the low CNT reinforcement potential is due to a weak boundary layer (WBL) near the matrix/CNT interface, i.e., due to the presence of a WBL in the interphase or in the interfacial region. This proposed mechanism satisfactorily explains both the inefficient load transfer and the low cohesive strength of the matrix.

The mode of failure (i.e., adhesive failure at the interface or cohesive failure in the matrix) during fracturing of a polymer/CNT composite can provide essential data to support the WBL mechanism. For unmodified CNTs where little polar force is present, the interfacial interactions between a matrix and a CNT, which is expressed as the work of adhesion, W_a, is well known to be due mainly to their dispersion force components; that is, $W_a=2{\left({\gamma }_p^d{\gamma }_{nt}^d\right)}^{1/2}$ where W_a is the work of adhesion, ${\gamma }_p^d$ and ${\gamma }_{nt}^d$ are the dispersion force components of the polymer and CNTs, respectively. Although the unit bond energy of the dispersion force is small as compared to other bond types, its abundance and universal presence in any two bodies make the interactions by this force very large, about 1500 MPa for a typical polymer/substrate system (including polymer/CNT), which is approximately two orders of magnitude greater than the actual value experimentally obtained [37]. Based on this theoretically predicted value, it has been suggested that the dispersion forces alone are sufficient to provide strong interfacial bonds, and that, as long as the polymer matrix wets the substrate, a strong interface between a polymer and a substrate is formed. Therefore, the discrepancy between theoretical and experimental interfacial strength of a polymer/substrate system is generally attributed to the interphase, where defects such as voids, residual stresses, disordered structure, are generated during processing [36–38]. These physical defects cause the composite to rupture below its theoretical value.

Several sources of information are available [53,54] to suggest that the interface between a polymer and a CNT is strong and that fracture generally does not occur at the interface but in a weak boundary layer in the interphase. Wagner and Vaia [25] have extensively reviewed the interface aspect of CNTs formed with various polymers, including a single CNT pulled from the matrix. This review also provided ample evidence to indicate that (1) the wetting of CNTs by polymers is adequate, as evidenced by transmission electron microscopy (TEM) [25] and Wilhelmy force balance methods [41]; (2) the stress transfer, adhesion, and interfacial shear strength of polymer and CNTs are high, as evidenced by Raman spectroscopy [42], TEM [43], AFM pull-out measurement of single SWCNT from polymer [44], and molecular simulations [44] and (3) the locus of failure is not at the polymer interface but away from it [25]. Other convincing evidence to indicate a strong polymer/CNT interfacial strength is from SEM imaging of fractured epoxy/CNT composite surfaces. For example, the results of Li et al. [35,45] showed clearly that the pull-out SWCNT bundles are covered with a sheath of epoxy polymer and the size of the circular pull-out holes correlates well with the sheathed CNT bundles. The presence of a matrix layer on the pull-out CNT bundles again indicates strongly that, if properly prepared, the locus of fracture in a polymer/CNT composite is in the matrix region near but not at the interface. Our SEM result, as presented in Fig. 7, also shows that the pull-out holes from an epoxy/MWCNT composite are uniformly circular and their diameter is larger than that of the nanotube or nanotube bundles. This observation suggests that the matrix layer near the MWCNT surface or near the MWCNT bundle surface is mechanically weak and was fractured during testing. This sample was prepared by cryo-fracturing (after immersing in liquid nitrogen for 1 h followed by breaking with a tweezers) of a 125 μm thick glassy amine-cured epoxy/0.72% mass fraction MWCNT composite. The glass transition temperature of this epoxy polymer was 90 °C, which behaves as an elastic material at ambient conditions. The nearly circular geometry of the pull-out holes in Fig. 7 suggests that the outer circumference of this mechanically-weak matrix layer corresponding to the axial fractured plane has a similar structure. If the microstructure of this layer is similar to that of the bulk matrix, the fractured holes are expected to have an irregular shape because the propagation during fracturing is random in an isotropic amorphous polymer such as the amine-cured epoxy.

Fig. 7.

SEM image of a cryo-fractured epoxy/MWCNT composite, showing clearly the pull-out CNTs and the circular interfacial space between CNT and the matrix.

The microscopic results of Fig. 7 together with those shown in Refs. [35,46] clearly indicate that for dispersion force dominated polymers, such as epoxy and poly(methyl methacrylate) the fracture of a polymer/CNT composite occurs cohesively in the matrix region near the interface. We believe that such fracture is a result of a weak boundary layer (as defined in Fig. 8) formed near the polymer/CNT interface (i.e., in the interphase). As indicated earlier, in addition to normal polymer chains, the interphase may contain defects, such as voids, internal stress (resulting from solidification and curing processes), impurities, and unreacted and partially-polymerized molecules. Therefore, the WBL in the interphase region would serve as local stress concentration sites. Under applied stress, the stress in this WBL exceeds its strength and fracture will occur. It is not known what the thickness of this WBL is or whether it has the same outer boundary as that of the interphase.

Fig. 8.

A cross-section schematic of a polymer/SWCNT composite, showing the interface, interphase, and weak boundary layer (not to scale).

To better understand quantitatively the effect of the WBL on the fracture characteristics of a polymer matrix/CNT composite, hence the load transfer, an analysis of fracture for a system of dissimilar materials (e.g., a polymer and a substrate) is presented here. Griffith’s criterion for cohesive fracture of a homogeneous material is given [47]:

$${\sigma }_f={(EG/\pi a)}^{1/2}$$ (5)

where σ_f is the fracture stress, E is the material’s elastic modulus, G is the fracture energy, which is the energy required to create one unit interfacial area of the crack, and a is the half length of the central crack. G is the sum of work of cohesion, W_c, and plastic work (also known as viscoelastic dissipation), W_p:

$$G=W_c+W_p$$ (6)

For ideally brittle materials where plastic yielding does not occur, G = W_c; but for most polymers, W_p is many orders of magnitudes greater than W_c. The Griffith’s fracture analysis is valid for atomic-size flaws, which cannot be measured in reality because small flaws are blunted by local plastic flow. This analysis is equally applicable for determining fracture stress of a polymer/substrate system. In this case, two dissimilar fractured surfaces are created (for a brief review of fracture mechanics application in adhesive bondings, see Ref. [37].

For the weak boundary layer (discussed above) and the polymer/CNT interface, Eqs. (5) and (6) can be expressed as:

WBL fracture stress:

$${\sigma }_{wl}={\left(E_{wl}{G}_{wl}/\pi a\right)}^{1/2}$$ (7)

WBL fracture energy:

$$G_{wl}=W_{\alpha wl}+W_{pwl}$$ (8)

Interface fracture stress:

$${\sigma }_i={\left(E_i{G}_i/\pi a\right)}^{1/2}$$ (9)

Interface fracture energy:

$$G_i=W_{ai}+{\text{W}}_{pi}$$ (10)

where W_cwl and W_pwl are the work of cohesion and plastic work of the WBL, respectively; W_ai and W_pi are the work of adhesion and plastic work of the interface, respectively; E_wl and E_i are the elastic modulus of the WBL and the interface, respectively. For the polymer/CNT interface, E_i is given:

$$E_i=E_{w1}{E}_{nt}/\left({\varphi }_p{E}_p+{\varphi }_{nt}{E}_{nt}\right)$$ (11)

where ϕ_wl and ϕ_nt are fractional lengths of the WBL and CNT, respectively. For a system where the dispersion force dominates, W_ai of the interface is between the two phases [28]; that is,

$$W_{cnt}>W_{ai}>W_{wt}$$ (12)

where W_cnt is the work of cohesion of the CNT.

It is clear from Eqs. (5)–(12) that the real strength of a polymer/CNT system is determined not only by the strength but also the flaw size and viscoelastic properties (mostly plastic yielding) of the two materials. A survey of data on the elongation at break of polymer/CNT composites given in the review of Ref. [6] indicates that this property generally decreases with CNT incorporation, suggesting that the plastic yielding (W_p) of the fractured zone is lower than that of the matrix. Further, because E of the CNT is much greater than that of the matrix, from Eq. [11] E_i > E_wl. Similarly, because W_ai > W_cwl, G_i > G_wl (Eqs. (8) and (10)). Inputting these factors in Eqs. (5), (7), and (9), it becomes clear that the fracture stress of the WBL (σ_wl) is lower than that of both the polymer/CNT interface and of the matrix. Therefore, fracture likely occurs in this weak layer, which provides a good explanation for the low Young’s modulus value of real-world polymer/CNT composites.

This fracture analysis is consistent with the weak boundary layer theory of Bikerman [48], who asserted that “in a proper joint, true interfacial failure practically never occurs. What is taken for interfacial failure is actually separation of a weak boundary layer; that is, a thin layer greater than atomic dimensions with mechanical strength much weaker than the bulk phase”. One strong scientific argument for this theory is that the attraction between two unlike molecules is always intermediate between the two molecules; that is, the interaction at the interface is stronger than that of the polymer matrix but weaker than that of the substrate. This reason is similar to that given in Eq. (11) for E and in Eq. (12) for W_a. Another rationale to support that fracture is likely to occur in the WBL is that the probability for a flaw to happen and travel along the tortuous interface is infinitely small [48]. Although the generality of the WBL theory has been disputed, its presence in many polymer/substrate systems is factual, as extensively reviewed and validated by Wu [37].

The postulation of a WBL near the interface responsible for the discrepancy between experimental and theoretical E values of real polymer/CNT composites is consistent with the greatly enhanced E of polymer/CNT composites prepared by in situ polymerization (for a review on this subject, see Ref. [6]). In addition to providing high-strength covalent bondings with the CNT surface, this processing technique likely provides optimum conditions for forming orderly, long chain molecular mass near the CNT surface, which increases the mechanical properties of the interfacial layer and most likely gets rid of the WBL. The strength enhancement by chemical modifications of the interfacial region has also been demonstrated in many studies for polymer/flat substrate systems, as reviewed in [37]. These studies showed that, by plasma treatment to activate the surface reactivity, the adhesive strengths, which were low due to the presence of a WBL, were reported to increase substantially. This enhancement was attributed to cross-linking across the interfacial layer that extends tens of nanometer into the matrix.

6. Concluding remarks

A stress analysis of a single-walled carbon nanotube/polymer matrix composite has been conducted using the finite element method. Atomistic molecular structural mechanics modeling was used to model the carbon nanotube and continuum mechanics approach was applied to the polymer matrix and the CNT/matrix interphase region. Effects of changes in both Young’s modulus and thickness of the interphase on composite mechanical property were analyzed for the 3-phase composites. Further, stress distribution in polymer/SWCNT system was also studied. The elastic moduli of the composite computed using the present approach are in good agreement with those reported in the literature. However, the predicted values are considerably higher than those measured experimentally from real-world polymer/SWCNT composites, and this discrepancy has been attributed to the presence of a weak boundary layer formed in the interphase zone. Using Griffin’s fracture analysis for dissimilar materials, we have demonstrated that fracture stress of the weak boundary layer in a CNT/polymer interphase is lower than that of the CNT/polymer interface or of the matrix. The present work can be extended further to study the fracture and failure of polymer matrix/carbon nanotube composites through the use of multiscale modeling.