Introduction

A low content of carbon nanotubes (CNT)1,2,3,4,5,6,7,–8 or graphene9,10,11,12,13,14,15,16,17,–18 can make a polymer nanocomposite with significant stiffness, strength, heat distortion temperature (HDT) and barrier properties. The high levels of aspect ratio, specific surface area and stiffness of these nanoparticles are responsible for the substantial properties. The anti-static, electromagnetic shielding and electrical conductivity of polymer/CNT nanocomposites (PCNT) result in applications in biomedical devices, and in electronics, automotive and aerospace industries.19,20,21,22,23,–24 The PCNT are electrically conductive materials showing many benefits over conventional composites, such as noble thermal and mechanical performance, high electrical conductivity and low viscosity, allowing them to be simply molded at very low fractions of CNT. However, PCNT suffer from some problems, such as poor dispersion and waviness of the nanoparticles, which have motivate researchers to solve these deficiencies.25,26,–27

The general properties of polymer/CNT nanocomposites (PCNT) depend on many factors, such as the concentration, aspect ratio, strength and dispersion superiority of CNT, as well as the interphase properties between the polymer matrix and the CNT.28,29,30,–31 The interphase around the nanoparticles generally determines the molecular interaction at the nanoscale, which demonstrates the efficiency of the stress transfer from the polymer to the nanoparticles. Accordingly, a poor interphase eliminates the main advantages of nanoparticles in nanocomposites, such as excellent stiffness. However, the nanoscale manipulation limits the experimental measurement of the interphase dimension and strength. For this reason, theoretical models have been used to investigate the interphase characteristics. Previous studies found that the interphase with significant thickness and strength extensively controls the mechanical performance of nanocomposites.32,33,34,–35 An inadequate interfacial adhesion is commonly formed in many PCNT, which makes for a weak stress transfer from the polymer matrix to the nanoparticles. This occurrence is attributed to the incompatibility or lower compatibility between the polymer and the nanoparticles, as well as the aggregation/agglomeration and poor dispersion of the nanofiller or non-fitting processing parameters.36,37,38,–39 In nanocomposites, the main issue is the agglomeration of nanoparticles (hard layers) in the case of a lubricating matrix. Protective coatings for tribological applications fundamentally require the proper levels of hardness, toughness, and interfacial adhesion with the underlying substrate.40 To enhance the mechanical properties of these nanocoatings, the importance of these terms should be considered. The dispersion quality of nanoparticles affects the toughness, but the hardness, as the resistance to localized plastic deformation encouraged by mechanical indentation or abrasion, depends on many parameters, such as the elastic stiffness, strength, toughness, viscoelasticity, and shear modulus.

In this article, we focus on the interfacial adhesion between the polymer matrix and the nanoparticles, which mainly governs the tensile modulus and yield strength of nanocomposites, because the interface properties control the stress transferring from the matrix to the filler. Moreover, an adhesive interfacial layer between the substrate and crystalline/amorphous coating is essential for reducing the wear rate, which serves the dual purpose of enhancing the adhesion and stress relief.41 Laser cladding and laser annealing are also utilized to melt the materials, which can lead to a tremendous improvement in the adhesion of the coating materials.42

On the other hand, the formation of a connected network of nanotubes above the percolation threshold has been reported in PCNT.43,44 The percolation threshold is the smallest volume fraction of the nanoparticles, which form a continuous network in the polymer matrix.45,46 The percolation threshold meaningfully raises the electrical conductivity of the nanocomposite, which changes the insulating polymer matrix to a conductive sample. The percolation threshold is also effective for the mechanical properties of polymer nanocomposites.47,48 Favier et al.47 reported the high shear modulus of reinforced films with cellulose whiskers by percolation. Many theoretical approaches have been applied to consider and predict the effect of the percolation threshold on the mechanical behavior of polymer nanocomposites. The micromechanics models assessed the deformation energy stored in the tubes to determine the modulus.49,50 Chatterjee51 combined the Halpin–Tsai model assuming dispersed particles with the results from the networked filler and presented a model for the modulus of nanocomposites containing dispersed and networked nanofillers. However, this model needs some adjustable parameters for modeling, which is incorrect. Generally, the percolation threshold in the mechanical performances of polymer nanocomposite has been only briefly studied in the literature.

In this study, the percolation threshold (\( \phi_{\text{p}} \)) in PCNT is assumed by the aspect ratio of the CNT. Also, the incomplete interfacial adhesion is considered by the average normal stress, which suggests the effective aspect ratio and volume fraction of the CNT. After that, two known micromechanics models have been applied to study the effects of \( \phi_{\text{p}} \) and effective parameters on the tensile modulus and yield strength of the PCNT. The present paper reports that the levels of \( \phi_{\text{p}} \) and the interfacial adhesion have significant roles in the mechanical performances of PCNT.

Nomenclature

σf: tensile strength of CNTs, σ: normal stress, Lc: critical length of CNT essential for effective stress transferring, D: CNT diameter, l: CNT length, α: inverse aspect ratio, τ: interfacial shear strength, αeff: effective CNT aspect ratio, \( \phi_{\text{eff}} \): effective CNT volume fraction, \( \phi_{\text{f}} \): CNT volume fraction, \( \phi_{\text{p}} \): percolation threshold, Ec: Young’s modulus of nanocomposite, Em: matrix modulus, Ef: CNT modulus, σc: yield strength of nanocomposite, σm: strength of polymer matrix, s: interfacial stress transfer parameter.

Theoretical Views

A poor interfacial adhesion cannot bear the high interfacial shear stress during stress loading, which causes yielding or debonding at or near the interface. In this condition, the interfacial shear stress presents a low build-up of normal stress in the tube and a large distance is essential for normal stress to reach the tensile strength of the tubes (σf).52 As a result, a large portion of the tubes is not completely loaded due to imperfect interfacial adhesion, which decreases the strengthening effect of the nanoparticles.

Figure 1 shows the profiles of normal stress (σ) in a tube in two states. In the first case (\( L_{\text{c}} \le x \le l/2 \)), σ reaches σf before the full length of the tube debonds. However, the complete length of the tube is involved before reaching σ to σf when \( 0 \le x \le L_{\text{c}} \). In the second case. Lc is the critical length of the tube, which is essential for the effective transfer of stress from the matrix to the tube, i.e. Lc is the crucial distance for σ to reach σf.

Fig. 1
figure 1

The effective length of a tube (lef) assuming incomplete interfacial adhesion and the profiles of normal stress at two different cases of (1) x < 2Lc and (2) x > 2Lc

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Lc is expressed as:

$$ L_{\text{c}} = \frac{{\sigma_{\text{f}} D}}{2\tau } = \frac{{\sigma_{\text{f}} \alpha_{{}} l}}{2\tau } $$
(1)

where D and l are the diameter and length of the tube, respectively and α is the inverse aspect ratio, as α = D/l. Also, τ is the interfacial shear strength.

The average normal stress (\( \bar{\sigma } \)) equals σf when a tube is perfectly adhered to the matrix, but \( \bar{\sigma } \) is less than σf in the case of incomplete interfacial adhesion, which results in a smaller effective length of the tube than l52 as:

$$ \bar{\sigma }l = \sigma_{\text{f}} l_{\text{eff}} $$
(2)

As a result, the effective aspect ratio (αeff) and volume fraction (\( \phi_{\text{eff}} \)) of nanotubes are reduced by poor interfacial bonding, which decreases their reinforcing efficiency in PCNT.

αeff and \( \phi_{\text{eff}} \) in the case of l < 2Lc (case 1) were defined52 as:

$$ \alpha_{\text{eff}} = \alpha \frac{{4L_{\text{c}} }}{l} $$
(3)
$$ \phi_{\text{eff}} = \phi_{\text{f}} \left( {\frac{l}{{4L_{\text{c}} }}} \right) $$
(4)

where \( \phi_{\text{f}} \) is the volume fraction of the nanofiller in the sample. Moreover, when l > 2Lc (case 2), it was stated52 that:

$$ \alpha_{\text{eff}} = \alpha \left( {\frac{1}{{1 - \frac{{L_{\text{c}} }}{l}}}} \right) $$
(5)
$$ \phi_{\text{eff}} = \phi_{\text{f}} \left( {1 - \frac{{L_{\text{c}} }}{l}} \right) $$
(6)

Assuming that the tubes comprise x < 2Lc and x > 2Lc region (Fig. 1), αeff and \( \phi_{\text{eff}} \) parameters are expressed for the tubes as:

$$ \alpha_{\text{eff}} = \frac{{2L_{\text{c}} }}{l}\left( {\alpha \frac{{4L_{\text{c}} }}{l}} \right) + \left( {\frac{{l - 2L_{\text{c}} }}{l}} \right)\alpha \left( {\frac{1}{{1 - \frac{{L_{\text{c}} }}{l}}}} \right) = \alpha \left( {\frac{{8L_{\text{c}}^{2} }}{{l^{2} }} + 1} \right) $$
(7)
$$ \phi_{\text{eff}} = \frac{{2L_{\text{c}} }}{l}\left( {\phi_{\text{f}} \frac{l}{{4L_{\text{c}} }}} \right) + \left( {\frac{{l - 2L_{\text{c}} }}{l}} \right)\phi_{\text{f}} \left( {1 - \frac{{L_{\text{c}} }}{l}} \right) = \phi_{\text{f}} \left[ {\frac{1}{2} + \left( {\frac{{l - 2L_{\text{c}} }}{{l^{2} }}} \right)(l - L_{\text{c}} )} \right] $$
(8)

When Lc is replaced from Eq. 1 into Eqs. 7 and 8, the effective factors are suggested as:

$$ \alpha_{\text{eff}} = \alpha \left( {\frac{{2\sigma_{\text{f}}^{2} \alpha^{2} }}{{\tau^{2} }} + 1} \right) $$
(9)
$$ \phi_{\text{eff}} = \phi \left[ {\frac{1}{2} + \left( {1 - \frac{{\sigma_{\text{f}} \alpha }}{\tau }} \right)\left( {1 - \frac{{\sigma_{\text{f}} \alpha }}{2\tau }} \right)} \right] $$
(10)

Chatterjee51 suggested an opposite linking between the volume fraction of the geometric percolation threshold (\( \phi_{\text{p}} \)) and the inverse aspect ratio of CNT (α) as:

$$ \phi_{\text{p}} \approx \alpha $$
(11)

Using the above equation, the effective parameters in Eqs. 7–10 can be expressed by \( \phi_{\text{p}} \), Lc and τ as:

$$ \alpha_{\text{eff}} = \phi_{\text{p}} \left( {\frac{{8L_{\text{c}}^{2} }}{{l^{2} }} + 1} \right) $$
(12)
$$ \alpha_{\text{eff}} = \phi_{\text{p}} \left( {\frac{{2\sigma_{\text{f}}^{2} \phi_{\text{p}}^{2} }}{{\tau^{2} }} + 1} \right) $$
(13)
$$ \phi_{\text{eff}} = \phi_{\text{f}} \left[ {\frac{1}{2} + \left( {1 - \frac{{\sigma_{\text{f}} \phi_{\text{p}} }}{\tau }} \right)\left( {1 - \frac{{\sigma_{\text{f}} \phi_{\text{p}} }}{2\tau }} \right)} \right] $$
(14)

To evaluate the effects of the \( \phi_{\text{p}} \), Lc and τ parameters on the tensile modulus and yield strength of P the CNT, two known micromechanics models are applied.

The Halpin–Tsai model53 is widely used for the calculation of the tensile modulus in PCNT, which is given by:

$$ E_{\text{R}} = \frac{{1 + 2\eta \phi_{\text{f}} /\alpha }}{{1 - \eta \phi_{\text{f}} }} $$
(15)
$$ \eta = (E_{\text{f}} /E_{\text{m}} - 1)/(E_{\text{f}} /E_{\text{m}} + 2/\alpha ) $$
(16)

where ER = Ec/Em, and Ec, Em and Ef are the Young’s moduli of the nanocomposite, the polymer matrix and the nanoparticles, respectively. Inserting the effective parameters into the Halpin–Tsai model results in:

$$ E_{\text{R}} = \frac{{1 + \frac{{2\eta \phi_{\text{f}} \left[ {\frac{1}{2} + \left( {\frac{{l - 2L_{\text{c}} }}{{l^{2} }}} \right)(l - L_{\text{c}} )} \right]}}{{\phi_{\text{p}} \left( {\frac{{8L_{\text{c}}^{2} }}{{l^{2} }} + 1} \right)}}}}{{1 - \eta \phi_{\text{f}} \left[ {\frac{1}{2} + \left( {\frac{{l - 2L_{\text{c}} }}{{l^{2} }}} \right)(l - L_{\text{c}} )} \right]}} $$
(17)
$$ \eta = (E_{\text{f}} /E_{\text{m}} - 1)/\left[ {E_{\text{f}} /E_{\text{m}} + \frac{2}{{\phi_{\text{p}} \left( {\frac{{8L_{\text{c}}^{2} }}{{l^{2} }} + 1} \right)}}} \right] $$
(18)

Moreover, the Halpin–Tsai model can be presented as a function of \( \phi_{\text{p}} \) and τ by the effective parameters as:

$$ E_{\text{R}} = \frac{{1 + \frac{{2\eta \phi_{\text{f}} \left[ {\frac{1}{2} + \left( {1 - \frac{{\sigma_{\text{f}} \phi_{\text{p}} }}{\tau }} \right)\left( {1 - \frac{{\sigma_{\text{f}} \phi_{\text{p}} }}{2\tau }} \right)} \right]}}{{\phi_{p} \left( {\frac{{2\sigma_{\text{f}}^{2} \phi_{\text{p}}^{2} }}{{\tau^{2} }} + 1} \right)}}}}{{1 - \eta \phi_{\text{f}} \left[ {\frac{1}{2} + \left( {1 - \frac{{\sigma_{\text{f}} \phi_{\text{p}} }}{\tau }} \right)\left( {1 - \frac{{\sigma_{\text{f}} \phi_{\text{p}} }}{2\tau }} \right)} \right]}} $$
(19)
$$ \eta = (E_{\text{f}} /E_{\text{m}} - 1)/\left[ {E_{\text{f}} /E_{\text{m}} + \frac{2}{{\phi_{\text{p}} \left( {\frac{{2\sigma_{\text{f}}^{2} \phi_{\text{p}}^{2} }}{{\tau^{2} }} + 1} \right)}}} \right] $$
(20)

Additionally, the dependence of the yield strength of the PCNT to the material and interfacial properties can be displayed by the Callister model54 as:

$$ \sigma_{\text{R}} = 1 + \left( {\frac{s}{{\alpha \sigma_{\text{m}} }} - 1} \right)\phi_{\text{f}} $$
(21)

where σR is the relative yield strength as σR = σc/σm, and σc and σm are the yield strengths of the nanocomposite and the polymer matrix, respectively. Also, s is an interfacial stress transfer parameter, which demonstrates the quality of interfacial adhesion. Substituting the effective parameters in this model results in:

$$ \sigma_{\text{R}} = 1 + \left[ {\frac{s}{{\phi_{\text{p}} \left( {\frac{{8L_{\text{c}}^{2} }}{{l^{2} }} + 1} \right)\sigma_{\text{m}} }} - 1} \right]\left[ {\frac{1}{2} + \left( {\frac{{l - 2L_{\text{c}} }}{{l^{2} }}} \right)(l - L_{\text{c}} )} \right]\phi_{\text{f}} $$
(22)

Also, the Callister model (Eq. 21) can be presented by the \( \phi_{\text{p}} \) and τ parameters by replacing the effective parameters as:

$$ \sigma_{\text{R}} = 1 + \left[ {\frac{s}{{\phi_{\text{p}} \left( {\frac{{2\phi_{\text{p}}^{2} \phi_{\text{f}}^{2} }}{{\tau^{2} }} + 1} \right)\sigma_{\text{m}} }} - 1} \right]\left[ {\frac{1}{2} + \left( {1 - \frac{{\phi_{\text{p}} \sigma_{\text{f}} }}{\tau }} \right)\left( {1 - \frac{{\phi_{\text{p}} \sigma_{\text{f}} }}{2\tau }} \right)} \right]\phi_{\text{f}} $$
(23)

Results and Discussion

In this section, the micromechanics models are used to investigate the effects of the percolation threshold by \( \phi_{\text{p}} \) and the effective parameters (αeff and \( \phi_{\text{eff}} \)) by Lc and τ on the tensile modulus and yield strength of the PCNT. The software utilized to perform the simulations is MATLAB.

Figure 2 illustrates the 3D and contour plots of ER as a function of the \( \phi_{\text{p}} \) and Lc parameters according to Eqs. 17 and 18 at average values of \( \phi_{\text{f}} \) = 0.02, l = 5 μm, Em = 3 GPa and Ef = 1000 GPa. The worst modulus is observed at the highest levels of the \( \phi_{\text{p}} \) and Lc parameters. Also, the highest modulus is achieved by the smallest ranges of these parameters. Therefore, the values of the \( \phi_{\text{p}} \) and Lc parameters inversely affect the tensile modulus of the PCNT. The \( \phi_{\text{p}} \) parameter determines the volume fraction of nanoparticles in which the CNT forms a network. Clearly, a low value of \( \phi_{\text{p}} \) shows the formation of a CNT network in the polymer matrix at a low volume fraction of nanoparticles. In this status, the PCNT contains a network of CNT with a low fraction of nanoparticles, which reinforces the polymer matrix, well. As a result, a small level of \( \phi_{\text{p}} \) makes a stiff PCNT by the low concentration of CNT, and the higher levels of \( \phi_{\text{f}} \) strengthens the CNT network more. On the other hand, the Lc parameter shows the surface fraction of the nanotubes, which can effectively bear the load due to complete interfacial adhesion.

Fig. 2
figure 2

(a) 3D and (b) contour plots to show ER as a function of \( \phi_{\text{p}} \) and Lc parameters by Eqs. 17 and 18 at average values of \( \phi_{\text{f}} \) = 0.02, l = 5 µ, Em = 3 GPa and Ef = 1000 GPa

Full size image

As explained, a low Lc indicates the great efficiency of stress transfer between the polymer matrix and the nanoparticles, because \( \bar{\sigma } \) reaches the σf at x > 2Lc. However, a higher level of Lc shows the larger area of tubes, which poorly transfers the load from the matrix to the tubes and weakens the modulus. Accordingly, ER shows a reasonable relationship with the \( \phi_{\text{p}} \) and Lc parameters based on the Halpin–Tsai model.

Figure 3 displays the roles of the \( \phi_{\text{p}} \) and τ parameters on the tensile modulus of the PCNT according to the Halpin–Tsai model (Eqs. 19 and 20) at \( \phi_{\text{f}} \) = 0.02, Em = 3 GPa, Ef = 1000 GPa and σf = 1.7 GPa. It is obvious that the \( \phi_{\text{p}} \) parameter plays the main role in the modulus of the PCNT, while the τ factor cannot change the modulus in this condition. Also, an ER value of about 5 is observed at \( \phi_{\text{p}} \) = 0.01, while ER = 8.5 is achieved by \( \phi_{\text{p}} \) = 0.001. Both the upper and lower levels of ER are calculated at the different levels of τ, demonstrating the ineffective role of τ in the modulus. Accordingly, the modulus mainly depends on the \( \phi_{\text{p}} \) parameter and τ does not play a role. Moreover, an inverse relationship is observed between the modulus and the \( \phi_{\text{p}} \) parameter, where the highest modulus is observed at the lowest level of \( \phi_{\text{p}} \). This trend is reasonable, due to the network formation in the PCNT at \( \phi_{\text{p}} \). A low level of \( \phi_{\text{p}} \) shows the formation of the network by the low fraction of CNT, which induces a strong sample at different levels of CNT concentration. In addition, the ineffective role of τ as the interfacial shear strength between the polymer matrix and the nanoparticles may be due to the high level of Ef of 1000 GPa and low level of σf of 1.7 GPa. The roles of the τ parameter in the modulus and the strength of the polymer nanocomposites have been shown in previous articles,55,56 but its character may be eliminated when it is compared with \( \phi_{\text{p}} \) as a main parameter, which shows the networking of the CNT in the polymer nanocomposites. A CNT network can endure a high capacity of load and, thus, it can considerably reinforce the polymer matrix.

Fig. 3
figure 3

The effects of \( \phi_{\text{p}} \) and τ parameters on ER according to the Halpin–Tsai model (Eqs. 19 and 20) at \( \phi_{\text{f}} \) = 0.02, Em = 3 GPa, Ef = 1000 GPa and σf = 1.7 GPa: (a) 3D and (b) contour plots

Full size image

Figure 4 shows the influences of the \( \phi_{\text{p}} \) and Lc parameters on σR according to the Callister model (Eq. 22) at average values of \( \phi_{f} \) = 0.02, σm = 40 MPa, l = 5 μm and s = 5 MPa. The Callister model calculates σR = 1 at \( \phi_{\text{p}} \) > 0.006 and Lc > 2 μm, while the highest σR of about 1.9 is obtained by \( \phi_{\text{p}} \) = 0.002 and Lc = 1 μm. The low relative strength at high levels of \( \phi_{\text{p}} \) and Lc indicates the non-strengthening of the matrix by the addition of nanoparticles. This evidence is due to the undesirable effects of high values of \( \phi_{\text{p}} \) and Lc on the yield strength of the PCNT. A high \( \phi_{\text{p}} \) demonstrates the formation of the CNs network at large CNT concentrations, which means that the low fraction of CNT does not form a network and cannot significantly strengthen the polymer matrix. Furthermore, a high level of Lc indirectly indicates the poor interfacial adhesion between the polymer matrix and the CNT nanoparticles. A high Lc shows that the low surface area of the tubes can effectively transfer the stress from the polymer matrix to the nanoparticles, which produces a poor yield strength in the PCNT. According to this explanation, observing a low strength at the high levels of \( \phi_{\text{p}} \) and Lc is reasonable. However, a good strength is obtained by the low levels of the \( \phi_{\text{p}} \) and Lc factors in the PCNT. The small values of \( \phi_{\text{p}} \) and Lc demonstrate the formation of the CNT network at low CNT concentration and an efficient stress transfer from the polymer matrix to the nanoparticles at a high region of CNT, respectively. In this condition, the CNT network forms by the low volume fraction of the nanoparticles and, also, the CNT show a high level of interfacial bonding and stress transfer. Therefore, obtaining an acceptable strength in this condition is expected.

Fig. 4
figure 4

(a) 3D and (b) contour plots of σR as a function of \( \phi_{\text{p}} \) and Lc parameters according to Eq. 22 at average values of \( \phi_{\text{f}} \) = 0.02, σm = 40 MPa, l = 5 µ and s = 5 MPa

Full size image

Figure 5 also illustrates the roles of the \( \phi_{\text{p}} \) and τ parameters on the yield strength of the PCNT according to Callister model (Eq. 23) at \( \phi_{\text{f}} \) = 0.02, σm = 40 MPa, σf = 1.7 GPa and s = 5 MPa. The \( \phi_{\text{p}} \) parameter alone controls the level of strength in the PCNT based on this model. Also, the τ parameter does not change the values of σR. As a result, the yield strength of the PCNT mainly depends on the level of \( \phi_{\text{p}} \). In this condition, the poorest strength is calculated by \( \phi_{\text{p}} \) > 0.009 and the highest is obtained by \( \phi_{\text{p}} \) = 0.002. Thus, the \( \phi_{\text{p}} \) parameter indirectly affects the σR in the PCNT. As mentioned, a low level of \( \phi_{\text{p}} \) shows the formation of the network in the PCNT by a small amount of CNT. Therefore, the higher concentration of CNT than the percolation fraction creates a denser and stronger network in PCNT, which strengthens the PCNT more.

Fig. 5
figure 5

The relationship between the yield strength of the PCNT and the \( \phi_{\text{p}} \) and τ parameters according to the Callister model (Eq. 23) at \( \phi_{\text{f}} \) = 0.02, σm = 40 MPa, σf = 1.7 GPa and s = 5 MPa: (a) 3D and (b) contour plots

Full size image

The τ parameter does not play a role in the yield strength of PCNT, which is similar to its effect on the modulus (Fig. 3). This occurrence is opposite to the previous findings for the effect of τ on the strength of the polymer nanocomposites, which demonstrated a direct relationship between the strength and τ as interface/interphase properties.55,56 Possibly, the effect of the τ parameter on the strength of the PCNT is much less than that of the percolation threshold based on the Callister model. The influences of the τ factor on the mechanical behavior may change by coupling with another parameter or applying other models in this area, such as Pukanszky model. Undoubtedly, the interface/interphase parameters such as τ largely manage the level of the modulus and the strength in the PCNT, because they control the extent of stress transference between the polymer matrix and the nanoparticles.57,58 The results in Figs. 3 and 5 indicate that the effective properties may arise from the consideration of extended singular chains. The lowering of the percolation threshold may increase the yield strength of the chains, as the extended chains are capable of forming above the percolation threshold.

Conclusion

Two known models have been applied to evaluate the effects of the filler network and imperfect interfacial adhesion on the tensile modulus and yield strength of the PCNT at the same time. The highest levels of the modulus and strength are obtained by the smallest levels of the \( \phi_{\text{p}} \) and Lc parameters, while the high levels of these parameters decrease the mechanical performance. A low level of \( \phi_{\text{p}} \) makes a dense and strong network of CNT in the PCNT by the low concentration of CNT, which strengthens the sample. In addition, a low Lc shows the excellent efficiency of the stress transfer between the polymer matrix and the nanoparticles in a large portion of the CNT surface, which improves the mechanical behavior. However, the ineffective role of τ as the interfacial shear strength in the mechanical performance may be due to the high level of Ef of 1000 GPa and low level of σf of 1.7 GPa or its smaller effect compared to the percolation threshold. Undoubtedly, the interface/interphase properties significantly affect the extent of the stress transfer between the polymer matrix and the nanoparticles managing the level of the modulus and strength in the nanocomposites.