1. Introduction

Nanostructures suggest novel applications due to the excellent properties of materials at nanoscale [1–4]. Therefore, the structure and interaction at nanoscale cause the significant effects on the properties. The dramatic enhancement in the mechanical properties of polymers can be attained by incorporation of a low weight percentage (wt %) of various nanofillers such as layered silicates [5, 6]. The large aspect ratio and stiffness of layered silicates may be the main reasons for the highly enhanced mechanical properties of polymer nanocomposites [7].

The Young’s modulus of polymer nanocomposites increases by addition of nanoparticles, because they usually have a much higher modulus than polymer matrices [8–10]. However, the yield strength of nanocomposites depends on the stress transferal between nanofiller and polymer matrix [11]. The stress applied to nanocomposites can be excellently transferred to nanoparticles in well-bonded nanoparticles to polymer matrix. In this condition, the yield strength of polymer nanocomposites noticeably improves in the tensile test. However, the yield strength reduces by adding of poorly bonded nanoparticles. As a result, the properties of interface/interphase between polymer matrix and nanoparticles cause main effects on the mechanical properties of polymer nanocomposites and discounting of interphase characteristics results in wrong prediction of nanocomposites performances [12, 13].

The interphase dimension and stiffness have been determined by micromechanical models for mechanical behavior such as Young’s modulus and tensile/yield strength [14, 15]. It was also reported that shape memory polymer nanocomposites with a strong adhesion at polymer–nanofiller interface show pronounced shape memory properties [16]. However, there is not a model which directly expresses the effects of interphase properties such as interphase fraction on the Young’s modulus and yield strength.

In this work, Ji and Pukanszky models are used to display the Young’s modulus and yield strength of polymer nanocomposites containing different filler geometries as a function of the volume fraction, thickness, tensile strength and modulus of interphase region. The influences of interphase properties on the Young’s modulus and yield strength of nanocomposites are discussed. Additionally, the mentioned equations are used to analyze the properties of interphase in various samples.

2. Background

Ji et al. [17] proposed a three-phase model for Young’s modulus of nanocomposites taking into account matrix, nanofiller and interphase between polymer and nanoparticles. The Ji model for nanocomposites containing layered (1), spherical (2) and cylindrical (3) nanoparticles is expressed attributed to geometry of nanofillers as

$$E = E_\text{m}\left( 1 - \alpha + \frac{\alpha - \beta}{1 - \alpha + \frac{\alpha(k - 1)}{\ln k}} + \frac{\beta}{1 - \alpha + \frac{1}{2}(\alpha - \beta)(k + 1) + \beta\frac{E_{\text{f}}}{E_\text{m}}} \right)^{- 1},\mspace{6mu}(1)$$
$$\alpha_{1} = \sqrt{(2\frac{t_\text{i}}{t} + 1)\varphi_{\text{f}1}},\mspace{6mu}(2)$$
$$\alpha_{2} = \sqrt{(\frac{r_{\text{i2}}}{r_{2}} + 1)^{3}\varphi_{\text{f}2}},\mspace{6mu}(3)$$
$$\alpha_{3} = \sqrt{(\frac{r_{\text{i3}}}{r_{3}} + 1)^{2}\varphi_{\text{f}3}},\mspace{6mu}(4)$$
$$\beta = \sqrt{\varphi_{\text{f}}},\mspace{6mu}(5)$$
$$k = \frac{E_\text{i}}{E_\text{m}},\mspace{6mu}(6)$$

where Em, Ef and Ei are the Young’s moduli of matrix, nanofiller and interphase, respectively, φf is volume fraction of nanofiller, r and t are the radius and thickness of nanofillers, respectively, and ri and ti are the thickness of interphase.

The volume fractions of interphase φi in different polymer nanocomposites are defined as

$$\varphi_{\text{i1}} = \varphi_{\text{f}1}\frac{2t_\text{i}}{t},\mspace{6mu}(7)$$
$$\varphi_{\text{i2}} = \varphi_{\text{f}2}\lbrack(\frac{r_{2} + r_{\text{i}2}}{r_{2}})^{3} - 1\rbrack,\mspace{6mu}(8)$$
$$\varphi_{\text{i3}} = \varphi_{\text{f}3}\lbrack(\frac{r_{3} + r_{\text{i}3}}{r_{3}})^{2} - 1\rbrack.\mspace{6mu}(9)$$

Therefore, all α parameters in Eqs. (2)–(4) can be related to φi as

$$\alpha = \sqrt{\varphi_\text{i} + \varphi_{\text{f}}}.\mspace{6mu}(10)$$

As a result, Ji model for all nanocomposites is given by φi as

$$E_\text{c} = E_\text{m}\lbrack 1 - \sqrt{\varphi_\text{i} + \varphi_{\text{f}}} + \frac{\sqrt{\varphi_\text{i} + \varphi_{\text{f}}} - \sqrt{\varphi_{\text{f}}}}{1 - \sqrt{\varphi_\text{i} + \varphi_{\text{f}}} + \frac{\sqrt{\varphi_\text{i} + \varphi_{\text{f}}}(k - 1)}{\ln k}}$$
$$+ \sqrt{\varphi}(1 - \sqrt{\varphi_\text{i} + \varphi_{\text{f}}} + \frac{1}{2}(\sqrt{\varphi_\text{i} + \varphi_{\text{f}}} - \sqrt{\varphi_{\text{f}}})(k + 1) + \sqrt{\varphi_{\text{f}}}\frac{E_{\text{f}}}{E_\text{m}})^{- 1}\rbrack^{- 1}.\mspace{6mu}(11)$$

According to Eqs. (7)–(9), φf can be expressed as a function of φi as

$$\varphi_{\text{f}1} = \frac{\varphi_{\text{i}1}t}{2t_\text{i}},\mspace{6mu}(12)$$
$$\varphi_{\text{f}2} = \frac{\varphi_{\text{i}2}}{\frac{r_{2} + r_{\text{i}2}}{r_{2}})^{3} - 1},\mspace{6mu}(13)$$
$$\varphi_{\text{f}3} = \frac{\varphi_{\text{i}3}}{\frac{((r_{3} + r_{\text{i}3})}{r_{3}})^{2} - 1}.\mspace{6mu}(14)$$

Accordingly, Ji model can be defined by the properties of interphase for polymer nanocomposites containing different nanoparticles as

$$E = E_\text{m}\lbrack 1 - \sqrt{\varphi_{\text{i}1} + \frac{\varphi_{\text{i}1}t}{2t_\text{i}}} + (\sqrt{\varphi_{\text{i}1} + \frac{\varphi_{\text{i}1}t}{2t_\text{i}}} - \sqrt{\frac{\varphi_{\text{i}1}t}{2t_\text{i}}})(1 - \sqrt{\varphi_{\text{i}1} + \frac{\varphi_{\text{i}1}t}{2t_\text{i}}} + \frac{\sqrt{\varphi_{\text{i}1} + \frac{\varphi_{\text{i}1}t}{2t_\text{i}}}(k - 1)}{\ln k})^{- 1}$$
$$+ (\sqrt{\frac{\varphi_{\text{i}1}t}{(2t_\text{i})}}(1 - \sqrt{\varphi_{\text{i}1} + \frac{\varphi_{\text{i}1}t}{(2t_\text{i})}} + \frac{1}{2}(\sqrt{\varphi_{\text{i}1} + \frac{\varphi_{\text{i}1}t}{(2t_\text{i})}} - \sqrt{\frac{\varphi_{\text{i}1}t}{(2t_\text{i})}})(k + 1) + \sqrt{\frac{\varphi_{\text{i}1}t}{(2t_\text{i})}}\frac{E_{\text{f}}}{E_\text{m}})^{- 1}\rbrack^{- 1},\mspace{6mu}(15)$$
$$E = E_\text{m}\lbrack(1 - \sqrt{\varphi_{\text{i}2} + \varphi_{\text{i}2}(\frac{(r_{2} + r_{\text{i}2})^{3}}{r_{2}^{3}} - 1)^{- 1}} + (\sqrt{\varphi_{\text{i2}} + \varphi_{\text{i}2}(\frac{(r_{2} + r_{\text{i}2})^{3}}{r_{2}^{3}} - 1)^{- 1}} - \sqrt{\varphi_{\text{i}2}(\frac{(r_{2} + r_{\text{i}2})^{3}}{r_{2}^{3}} - 1)^{- 1}})$$
$$\times \ {{(1-\sqrt{{{\varphi }_{\text{i2}}}+{{\varphi }_{\text{i}2}}{{({{{({{r}_{2}}+{{r}_{\text{i}2}})}^{3}}}/{r_{2}^{3}}\;-1)}^{-1}}}+\sqrt{{{\varphi }_{\text{i2}}}+{{\varphi }_{\text{i}2}}{{({{{({{r}_{2}}+{{r}_{\text{i}2}})}^{3}}}/{r_{2}^{3}}\;-1)}^{-1}}}(k-1){{\ln }^{-1}}k)}^{-1}}$$
$$+ \sqrt{\varphi_{\text{i}2}(\frac{(r_{2} + r_{\text{i}2})^{3}}{r_{2}^{3}} - 1)^{- 1}}(1 - \sqrt{\varphi_{\text{i}2} + \varphi_{\text{i}2}(\frac{(r_{2} + r_{\text{i}2})^{3}}{r_{2}^{3}} - 1)^{- 1}} + \frac{1}{2}(\sqrt{\varphi_{\text{i2}} + \varphi_{\text{i}2}(\frac{(r_{2} + r_{\text{i}2})^{3}}{r_{2}^{3}} - 1)^{- 1}}$$
$$- \sqrt{\varphi_{\text{i}2}(\frac{(r_{2} + r_{\text{i}2})^{3}}{r_{2}^{3}} - 1)^{- 1}})(k + 1) + \sqrt{\varphi_{\text{i}2}(\frac{(r_{2} + r_{\text{i}2})^{3}}{r_{2}^{3}} - 1)^{- 1}}\frac{E_{\text{f}}}{E_\text{m}})^{- 1}\rbrack^{- 1},\mspace{6mu}(16)$$
$$E = E_\text{m}\lbrack(1 - \sqrt{\varphi_{\text{i}3} + \varphi_{\text{i}3}(\frac{(r_{3} + r_{\text{i}3})^{2}}{r_{3}^{2}} - 1)^{- 1}} + (\sqrt{\varphi_{\text{i}3} + \varphi_{\text{i}3}(\frac{(r_{3} + r_{\text{i}3})^{2}}{r_{3}^{2}} - 1)^{- 1}} - \sqrt{\varphi_{\text{i}3}(\frac{(r_{3} + r_{\text{i}3})^{2}}{r_{3}^{2}} - 1)^{- 1}})$$
$$\times ((1 - \sqrt{\varphi_{\text{i}3} + \varphi_{\text{i}3}(\frac{(r_{3} + r_{\text{i}3})^{2}}{r_{3}^{2}} - 1)^{- 1}} + \sqrt{\varphi_{\text{i}3} + \varphi_{\text{i}3}(\frac{(r_{3} + r_{\text{i}3})^{2}}{r_{3}^{2}} - 1)^{- 1}}(k - 1)\ln^{- 1}k)^{- 1}$$
$$+ \sqrt{\varphi_{\text{i}3}(\frac{(r_{3} + r_{\text{i}3})^{2}}{r_{3}^{2}} - 1)^{- 1}}(1 - \sqrt{\varphi_{\text{i}3} + \varphi_{\text{i}3}(\frac{(r_{3} + r_{\text{i}3})^{2}}{r_{3}^{2}} - 1)^{- 1}} + \frac{1}{2}(\sqrt{\varphi_{\text{i}3} + \varphi_{\text{i}3}(\frac{(r_{3} + r_{\text{i}3})^{2}}{r_{3}^{2}} - 1)^{- 1}}$$
$$- \sqrt{\varphi_{\text{i}3}(\frac{(r_{3} + r_{\text{i}3})^{2}}{r_{3}^{2}} - 1)^{- 1}}(k + 1)) + \sqrt{\varphi_{\text{i}3}(\frac{(r_{3} + r_{\text{i}3})^{2}}{r_{3}^{2}} - 1)^{- 1}}\frac{E_{\text{f}}}{E_\text{m}})^{- 1}\rbrack^{- 1},\mspace{6mu}(17)$$

which display the effects of interphase properties on the Young’s modulus of polymer nanocomposites.

Pukanszky [18] suggested an equation based on the formation of interphase in composites, where the yield strength is determined as a function of filler content. Pukanszky model is presented as

$$\sigma_{\text{r}} = \frac{1 - \varphi_{\text{f}}}{1 + 2.5\varphi_{\text{f}}}\exp(B\varphi_{\text{f}}),\mspace{6mu}(18)$$

where σr is relative yield strength as σc/σm, σc and σm are yield strengths of composite and matrix, respectively, B is an interfacial parameter which assumes the capability of stress transfer between matrix and filler. This model was well applied for different polymer nanocomposites in the recent studies [19, 20]. Therefore, it is applied in this work to analyze the effects of interphase on yield strength of dissimilar nanocomposites. Parameter B depends to interphase characteristics as

$$B = (1 + A_\text{c}\rho_{\text{f}}r_\text{i})\ln\left( \frac{\sigma_\text{i}}{\sigma_\text{m}} \right),\mspace{6mu}(19)$$

where Ac is the specific surface area of filler, ρf is density of filler, and σi is the tensile strength of interphase. To calculate parameter B, Pukanszky model can be rewritten as

$$\ln\left( \sigma_{\text{r}}\frac{1 + 2.5\varphi_{\text{f}}}{1 - \varphi_{\text{f}}} \right) = B\varphi_{\text{f}},\mspace{6mu}(20)$$

where the linear plot of ln (σr(1 + 2.5φf)/(1 – φf)) against φf shows the slope of B. Using Eqs. (12)–(14), Pukanszky model can be expressed as a function of interphase properties for polymer nanocomposites as

$$\sigma_{\text{r}1} = \frac{1 - \frac{t\varphi_{\text{i}1}}{(2t_\text{i})}}{1 + 2.5\frac{t\varphi_{\text{i}1}}{(2t_\text{i})}}\exp(B_{1}\frac{t\varphi_{\text{i}1}}{2t_\text{i}}),\mspace{6mu}(21)$$
$$\sigma_{\text{r}2} = \frac{1 - {\varphi_\text{i}}_{2}(\frac{(r_{2} + r_{\text{i}2})^{3}}{r_{2}^{3}} - 1)^{- 1}}{1 + 2.5{\varphi_\text{i}}_{2}(\frac{(r_{2} + r_{\text{i}2})^{3}}{r_{2}^{3}} - 1)^{- 1}}\exp\lbrack B_{2}{\varphi_\text{i}}_{2}(\frac{(r_{2} + r_{\text{i}2})^{3}}{r_{2}^{3}} - 1)^{- 1}\rbrack,\mspace{6mu}(22)$$
$$\sigma_{\text{r}3} = \frac{\frac{(r_{3} + r_{\text{i}3})^{2}}{r_{3}^{2}} - 1 - {\varphi_\text{i}}_{3}}{\frac{(r_{3} + r_{\text{i}3})^{2}}{r_{3}^{2}} - 1 + 2.5{\varphi_\text{i}}_{3}}\exp\lbrack B_{3}{\varphi_\text{i}}_{3}(\frac{(r_{3} + r_{\text{i}3})^{2}}{r_{3}^{2}} - 1)^{- 1}\rbrack.\mspace{6mu}(23)$$

Additionally, Ac can be defined for layered, spherical and cylindrical nanoparticles as

$$A_{\text{c}1} = \frac{A_{1}}{m_{1}} = \frac{A_{1}}{\rho_{\text{f}1}v_{1}} \cong \frac{2l^{2}}{\rho_{\text{f}1}l^{2}t} = \frac{2}{\rho_{\text{f}1}t},\mspace{6mu}(24)$$
$$A_{\text{c}2} = \frac{A_{2}}{m_{2}} = \frac{A_{2}}{\rho_{\text{f}2}v_{2}} = \frac{4\pi r_{2}^{2}}{\rho_{\text{f}2}\frac{4}{3}\pi r_{2}^{3}} = \frac{3}{\rho_{\text{f}2}r_{2}},\mspace{6mu}(25)$$
$$A_{\text{c}3} = \frac{A_{3}}{m_{3}} = \frac{A_{3}}{\rho_{\text{f3}}v_{3}} = \frac{2\pi r_{3}l}{\rho_{\text{f3}}\pi r_{3}^{2}l} = \frac{2}{\rho_{f3}r_{3}},\mspace{6mu}(26)$$

where A, m, v and l are the surface area, mass, volume and length of nanoparticles, respectively. As a result, B can be expressed for polymer nanocomposites as

$$B_{1} = \left( 1 + \frac{\varphi_{\text{i}1}}{\varphi_{\text{f1}}} \right)\ln\frac{\sigma_{\text{i}1}}{\sigma_\text{m}},\mspace{6mu}(27)$$
$$B_{2} = \left\lbrack 3\left( \frac{\varphi_{\text{i}2}}{\varphi_{\text{f2}}} + 1 \right)^{\frac{1}{3}} - 2 \right\rbrack\ln\frac{\sigma_{\text{i2}}}{\sigma_\text{m}},\mspace{6mu}(28)$$
$$B_{3} = \left\lbrack 2\left( \frac{\varphi_{\text{i}3}}{\varphi_{\text{f3}}} + 1 \right)^{\frac{1}{2}} - 1 \right\rbrack\ln\frac{\sigma_{\text{i3}}}{\sigma_\text{m}}.\mspace{6mu}(29)$$

3. Results and discussion

3.1. The Analysis of Experimental Data from the Literature

In this section, the mentioned models are utilized to determine the properties of interphase in several samples from the literature. In addition, the effects of interphase characteristics on the modulus and strength of polymer nanocomposites are plotted. Table 1 shows different samples from the literature as well as the properties of neat polymer and nanofiller. The experimental tensile moduli of samples are applied to Ji model (Eqs. (1)–(6)) and the average levels of ti or ri and Ei are calculated. The interphase thickness cannot be more than the gyration radius of polymer chains and Ei changes between the moduli of polymer matrix and nanofiller. Therefore, suitable ri or ti and Ei are chosen from the explained ranges and finally, the average values of interphase properties are given in Table 1. The presented data show the significant thickness of interphase in the reported samples. Additionally, a high interphase modulus is calculated in all reported samples, which is more than the stiffness of polymer matrix. Accordingly, the interphase can play an important role in the performances of polymer nanocomposites.

Table 1. The samples and calculated interphase properties
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Moreover, the experimental yield strength of samples is fitted to Pukanszky model (Eq. (18)) to determine the B parameter. Parameter B is applied into Eq. (19) to measure σi values. Table 1 gives the values of B and σi data. The diverse B data show the dissimilar levels of interfacial adhesion in the samples. Additionally, different σi data are obtained for reported samples. The highest σi is obtained for sample no. 3 as 4467 MPa and the least level is found for sample no. 1 as 80 MPa. The σi results are much higher than σm demonstrating the significant strength of formed interphase in the reported samples.

The dimension and level of interphase are attributed to some parameters such as interfacial area and compatibility between polymer matrix and nanofiller which control the interfacial interaction. Some procedures such as treatment, modification and functionalization of nanofillers can encourage the compatibility and interfacial adhesion between the components of nanocomposites. Furthermore, Ac data are reported in Table 1 using Eqs. (24)–(26). The nanoclay produces the highest level of Ac among the nanoparticles. This occurrence causes the highest level of interfacial area between polymer matrix and nanoclay layers, which finally creates the highest level of reinforcement in polymer nanocomposite. In fact, the high value of Ac is the significant advantage of nanofiller, which makes the unexpected behavior in polymer nanocomposites. According to Eqs. (24)–(26), Ac is inversely related to r and t and the smallest nanoparticles create the highest level of Ac as shown in Table 1.

Figure 1 illustrates the volume fraction φi in some reported samples by Eqs. (7)–(9). It is observed that the interphase occupies a large volume in polymer nanocomposites which is more than the nanofiller volume in some samples. Volume fraction φi increases with nanofiller content in all samples. The high level of interphase confirms the significant influence of this phase beside matrix and nanofiller phases. As a result, assuming the interphase is compulsory for estimation of mechanical properties in polymer nanocomposites. In addition, φi is directly related to the thickness of interphase (according to Eqs. (7)–(9)), i.e. φi increases when the thickness of interphase enlarges.

Fig. 1.
figure 1

φi as a function of φf in PBT/nanoclay [21] (no. 1) (a), LLDPE/SiO2 [23] (no. 3) (b) and epoxy/MWCNT [24] (no. 4) samples (c).

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3.2. The Roles of Interphase Properties According to the Models

Figure 2 demonstrates the effects of φi and Ei on modulus of polymer nanocomposites containing spherical nanoparticles by Eq. (16). The Young’s modulus more depends to φi than Ei.

Fig. 2.
figure 2

Roles of φi and Ei in the Young’s modulus of polymer nanocomposites containing spherical nanoparticles by Eq. (16) in r2 = 15 nm, ri2 = 10 nm and Em = 2 GPa: (a) 3D and (b) contour plots (color online).

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The low φi generally results in a low modulus at all Ei levels. However, the best modulus is obtained by high levels of φi and Ei. It means that φi and Ei have optimistic effects on the modulus of nanocomposites, where the effect of Ei becomes important at high values of φi indicating the important role of φi in Young’s modulus of polymer nanocomposites.

Figure 3 exhibits the influences of φi and σi on parameter B in polymer nanocomposites containing spherical nanoparticles (Eq. (28)). Parameter B shows comparatively same levels at all φi values depended to the level of σi. In other words, σi more expressively affects the level of parameter B compared with φi.

Fig. 3.
figure 3

3D (a) and contour plots (b) to show the effects of φi and σi on parameter B in polymer nanocomposites containing spherical nanoparticles (Eq. (28)) in φf3 = 0.1 and σm = 50 MPa (color online).

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Parameter B shows negative values when σi is lower than σm at all φi. The best level of B is obtained in the highest values of φi and σi. As a result, B is more depended to σi value in polymer nanocomposites (especially at low σi), while formation of a high-volume interphase improves the magnitude of B interfacial parameter.

Figure 4 also shows the roles of nanoparticle radius and interphase thickness on the interphase fraction φi2 of nanocomposites containing spherical nanoparticles by Eq. (8) in φi2 = 0.02. The high values of nanoparticle size decrease the φi2, but the small radius of nanoparticles increases it. On the other hand, a thick interphase grows the level of φi2, while the thinner one produces a smaller φi2. Therefore, small nanoparticles and thick interphase show beneficial effects on the φi2.

Fig. 4.
figure 4

Volume fraction φi2 as a function of r2 and ri2 (Eq. (8)) in φf2 = 0.02: 3D (a) and contour plots (b) (color online).

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Since a higher volume fraction of interphase causes a better level for tensile modulus and strength (Figs. 2 and 3), it is concluded that small particles and thick interphase introduce the high mechanical properties in nanocomposites. Likewise, large particles and thin interphase result in low levels for mechanical performances. The small nanoparticles induce the high specific surface area between polymer and nanoparticles. A good interphase is gained by the high level of interfacial interaction/adhesion in nanocomposites [20, 26].

A micromechanical model was also proposed by Boutaleb et al. [27] to calculate the modulus and yield stress in polymer/SiO2 nanocomposites. It considers the interphase as the perturbed region of polymer matrix around the nanoparticles. The predicted effects of nanoparticle radius, interphase thickness and modulus on the modulus and yield strength of nanocomposites in that work are similar to those suggested in the present study. All these remarks confirm the progressive roles of interphase properties in the mechanical behavior of nanocomposites.

4. Conclusions

Ji and Pukanszky models were used to show the Young’s modulus and yield strength of different polymer nanocomposites as a function of interphase volume fraction, thickness, modulus and strength. The interphase properties of different samples were also studied based on Young’s modulus and yield strength. The Young’s modulus more depends on φf and φi than Ei. A low φi generally results in a low modulus at all Ei values. The poorest Young’s modulus is found by the thinnest interphase. Besides, small nanoparticles and thick interphase increase the yield strength of nanocomposites. Parameter σi more expressively affects the level of parameter B compared to φi. Nevertheless, the highest values of φi and σi produce the highest level of interfacial adhesion expressed by parameter B. Since a high level of B increases the yield strength of nanocomposites, the high values of φi and σi play positive roles in the yield strength.