Introduction

Polymer nanocomposites have induced a lot of interest in each academic and industrial application [1,2,3,4,5,6,7]. Polyvinyl chloride (PVC) polymer has been studied intensively because of its attention-grabbing physical properties [8,9,10,11]. The optical properties of those materials may be altered through the addition of various nanoparticle ratios that could be used in optical fibers, optical waveguides and optical storage systems [1, 3, 6]. Optical properties of PVC compound films doped with numerous nanoparticles are investigated over the previous decades. Deshmukh et al. [12] considered the optical properties of PVC–PMMA thin films doped with 0.2, 0.4, 0.6, 0.8, and 1.0 wt% polyaniline (PANI). The optical energy gap (Eopt) increased with increasing PANI content except for the sample doped with 0.8 wt%, but the band-tailing decreased with increasing PANI concentration except for the samples doped with 0.6 and 1.0 wt%. The refractive index and high-frequency dielectric constant values indicated variety with PANI content. The ratio of free carriers to effective mass (N/m*) values are found to be within the order of 1021 cm−3/gm. Structure, optical and thermal properties of PVC/Cd0.5Zn0.5O nanocomposite films are explored [13]. The UV–Vis absorption spectra indicated that the prepared films are highly transparent and the transparency at higher slightly decreased with increasing Cd0.5Zn0.5O percentage from 0 to 0.5 wt% as a results of light scattering due to large aggregates. Optical properties of PVC-MWCNT nanocomposite films with totally different concentrations (0, 0.0005, 0.005, and 0.05 wt%) are contemplated [14]. The optical absorption increased with increasing MWCNT content. The direct optical energy gap decreased from 5.56 to 4.2 eV with increasing MWCNT percentage. The calculated refractive index, real and imaginary dielectric constants are found to increase with increasing MWCNT concentration. El Sayed and Morsi demonstrated that the direct band gaps and transparency of PVC/PbO nanocomposite films decreased with increasing PbO nanoparticles concentration [15]. The Urbach energies and refractive index of these nanocomposite films increased with increasing PbO nanoparticles content. Al-Taa’y et al. considered the impact of ZnO nanoparticles addition on the optical properties of polyvinyl chloride films [16]. The presence of ZnO causes an increase within the absorption and decrease within the nanocomposite films transparency. The extinction coefficient, refractive index, real and imaginary parts, optical conductivity, infinitely high-frequency dielectric constant, and average refractive index values increased with increasing ZnO content. The Urbach energy decreased with increasing ZnO concentration. The optical and dielectric properties of Cr2O3/PVC nanocomposite films have been explored [17]. The direct energy gap, single oscillator energy, and transparency decreased with increasing Cr2O3 concentration. However, Urbach energy, dispersion energy, refractive index at infinite wavelength, average interband oscillator wavelength, average oscillator strength, lattice dielectric constant and the ratio of carrier concentration to effective mass are found to increase with increasing Cr2O3 content. Structural, optical, and thermogravimetric analysis of Pb3O4/PVC nanocomposites films with totally different concentrations (0, 1, 2, 3, and 4 wt%) are contemplated [18]. The direct energy gap decreased from 5.05 to 4.34 eV with increasing Pb3O4 percentage. The presence of Pb3O4 nanoparticles enhanced the absorption and decreased the nanocomposite film transparency. Fermi energy, Urbach energy, and the solar material protection factor enhanced with increasing Pb3O4 wt%. This work aims to study the optical properties of PVC/Al2O3 nanocomposite films prepared by solution mixing and casting at room temperature. The linear optical parameters of these nanocomposites based on refractive index were calculated using Wemple–DiDomenico equation. The linear optical susceptibility (χ(1)), third-order nonlinear optical susceptibility (χ(3)), and the nonlinear refractive index (n2) were investigated. Also, the group velocity dispersion (GVD), dispersion coefficient for material dispersion (D), and third-order dispersion (TOD) are analyzed.

Experimental

In a common method, PVC/Al2O3 nanocomposite films with totally different concentrations (0, 2, 4, and 6 wt%) were prepared at room temperature. Two grams of PVC (extra pure powder with density 1.4 g/ml at 25 °C manufactured by Alpha Chemika, India) dissolved in 40 ml tetrahydrofuran (THF) and blended for 1 h on a magnetic stirrer, and then various ratios of Al2O3 nanoparticles are added to the clear solution with continuous stirring for one h. After that, the solution was thrown into glass Petri dish and left to dry in air to get the nanocomposite films.

XRD measurements of PVC/Al2O3 nanocomposite films were taken using XRD, Bruker, AXS D8 Advance, Germany, CuKa-radiation (k = 1.542). The size and morphology of Al2O3 nanoparticles were determined by transmission electron microscopy (TEM, JEOL 2100FX) worked at 200 kV accelerating voltage. Scanning electron microscopy (SEM) micrographs were recorded utilizing Quanta FEG250 SEM. The optical measurements of the synthesized PVC/Al2O3 nanocomposite films were taken utilizing JASCO UV–Vis–NIR double-beam spectrophotometer model V-570.

Results and discussion

Figure 1 shows XRD patterns of pure PVC and PVC + 4 wt% Al2O3 films; it is discovered that pure PVC is partly crystallized and also the diffraction peaks related to rhombohedral Al2O3 crystal [JCPDS card number 43-1484] with positions 25.50°, 37.68°, 43.07°, and 52.50° noticed for PVC film doped with 4 wt% Al2O3.

Fig. 1
figure 1

XRD spectra of a pure PVC, b PVC doped with 4 wt% Al2O3

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The average Al2O3 nanoparticles size determined from TEM micrograph (Fig. 2a) is 10 nm.

Fig. 2
figure 2

a TEM micrograph of Al2O3 nanoparticles, b SEM image of pure PVC, c SEM image of PVC + 2 wt% Al2O3, d SEM image of PVC + 6 wt% Al2O3

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Scanning electron microscope (SEM) images are displayed in Fig. 2b–d that illustrated that Al2O3 nanoparticles are well distributed in PVC film surface.

The normalized optical absorption spectra for PVC compound films doped with 0, 2, 4, and 6 wt% Al2O3 nanoparticles are represented in Fig. 3. It is watched that the polymer films absorption increase with increasing Al2O3 nanoparticles content. The absorption band extended from 257 to 297 nm is appointed to π–π* electronic transition [18,19,20] and the increased absorbance at wavelengths below 257 nm is identified with C–Cl bond [18, 21].

Fig. 3
figure 3

Normalized optical absorption spectra for PVC/Al2O3 nanocomposite films

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The transmittance spectra and the first derivative of the optical transmittance for the nanocomposite films were plotted versus wavelength as shown in Fig. 4. The absorption band edges of those films were evaluated from the maximum peak position in Fig. 4b [22] and tabulated in Table 1. The addition of Al2O3 nanoparticles causes an increase in the maximum peak values from 269 to 316 nm as seen in Table 1. This leads to a decrease in the absorption band edge from 4.197 to 3.931 eV with the variation of Al2O3 percentage.

Fig. 4
figure 4

a The transmittance versus wavelength plot and b the first derivative of the optical transmittance versus wavelength plot for the nanocomposite films

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Table 1 Optical band gap (Eopt), Urbach energy (EU) and other optical parameters for the PVC/Al2O3 nanocomposite films
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The optical band gap (Eopt) values for allowed direct transitions are calculated utilizing the well-known relation [23,24,25];

$$\alpha h\nu = k(h\nu - E_{\text{opt}} )^{1/2}$$
(1)

where hν is the incident photon energy and k is a constant. The direct band gaps, Eopt, are dictated by extrapolating the linear portion of the curves in Fig. 5a to zero absorption and summarized in Table 1. The addition of Al2O3 nanofiller results in localized states formation within the band gap and lowers the optical band gap values that are less than the band gap of pure PVC film [15, 18].

Fig. 5
figure 5

a Plots of (αhυ)2 versus hυ for the PVC/Al2O3 nanocomposite films, b variation of Lnα with photon energy (hυ) for the prepared nanocomposite films

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The width of band tails (EU) for localized states within the forbidden band gap that is connected with the amorphous nature of the materials can be calculated via Urbach equation [26];

$${ \ln }\left( \alpha \right) = { \ln }\left( {\alpha_{0} } \right) + h\nu /E_{\text{U}}$$
(2)

where α0 is a constant. EU values were calculated from the inverse slope for straight lines of every curve in Fig. 5b.

The obtained values are found to increase with increasing Al2O3 wt% (Table 1), possibly due to the formation of imperfections and increased disorder within the nanocomposite [27, 28]. Likewise, it is seen that EU values for PVC polymer films doped with 2, 4, and 6 wt% Al2O3 nanoparticles are higher than that for pure PVC film [18].

The refractive index, n, was computed using values of reflectance calculated from [29, 30];

$$R = 1 - \sqrt {T*\exp (A)}$$
(3)

Through the subsequent equation [25, 31];

$$n = \left( {\frac{1 + R}{1 - R}} \right) + \left( {\frac{4R}{{\left( {1 - R} \right)^{2} }} - k^{2} } \right)^{{\frac{1}{2}}}$$
(4)

where k is the extinction coefficient (k = αλ/4π). The dependence of refractive index on wavelength is represented in Fig. 6.

Fig. 6
figure 6

Calculated refractive index as a function of wavelength for the PVC/Al2O3 films

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From investigation of this figure, it is ascertained that the refractive index for the current polymer nanocomposite films increases with increasing Al2O3 nanoparticles content and higher than that of pure PVC film. Also, the obtained refractive index values are higher than that in the literature [6, 14,15,16]. The increase in the refractive index values of these nanocomposite films may be due to the condensation of smaller ceramic molecules into larger clusters [32]. The high values of refractive index (typically >  1.65) for the PVC/Al2O3 nanocomposite films make it suitable in improving the performance of optical and photovoltaic devices in many technologies like solar cells [32, 33], Bragg gratings [34], photonic crystals [35] and waveguide-based optical circuits [36].

The refractive index dispersion of the prepared PVC/Al2O3 films is expressed by the formula [37];

$$\left( {n^{2} - 1} \right)^{ - 1} = \frac{{E_{ 0} }}{{E_{\text{d}} }} - \left( {\frac{1}{{E_{ 0} E_{\text{d}} }}} \right)\left( {h\nu } \right)^{2}$$
(5)

where n is that the refractive index, hν is the incident photon energy, E0 is that the average excitation energy for electronic transitions and Ed is the dispersion energy that measure the interband optical transitions strength and is identified with the changes within the material structural order and also the effective oscillator energy. E0 and id values were calculated from the slope and intercept on the vertical axis of (n2 − 1)−1 versus (hν)2 plots (Fig. 7) and displayed in Table 1.

Fig. 7
figure 7

(n2 − 1)−1 versus (hv)2 plots for the PVC/Al2O3 films

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The static refractive index n0 (at zero photon energy) is calculated by extrapolating Eq. 4 to h → 0, that is obtained from the subsequent relation [37]:

$$n_{0} = \sqrt {\left( {1 + \frac{{E_{\text{d}} }}{{E_{0} }}} \right)}$$
(6)

The static dielectric constant is computed from the static refractive index using \(\varepsilon_{s} = n_{0}^{2}\) [38, 39] and also optical oscillator strengths (f) for optical transitions are defined as absorption of a photon by the electron between the initial state and the final state that is correlated with E0 and Ed as f  = E0Ed [30, 40]. The got values of n0, εs and f are recorded in Table 1 which increased with increasing Al2O3 wt%.

The moments of optical spectrum M−1 and M−3 for the PVC/Al2O3 films were computed from the following relations [41];

$$E_{0}^{2} = \frac{{M_{ - 1} }}{{M_{ - 3} }}\;{\text{and}}\;E_{d}^{2} = \frac{{M_{ - 1}^{3} }}{{M_{ - 3} }}$$
(7)

The M−1 and M−3 moments for the PVC/Al2O3 films were obtained and are found to increase with the addition of Al2O3 nanoparticles in Table 1.

The linear optical susceptibility χ(1) for PVC/Al2O3 nanocomposite films could be computed from the relation [42, 43];

$$\chi^{(1)} = E_{\text{d}} /4\pi E_{0}$$
(8)

The calculated χ(1) values are given in Table 2 which are found to increase with increasing Al2O3 content. Additionally, the third-order nonlinear optical susceptibility χ(3) is estimated utilizing the following formula [42, 44];

$$\chi^{(3)} = 6.82 \times 10^{ - 15} \left( {E_{d} /E_{0} } \right)^{4} \quad ({\text{e}}.{\text{s}}.{\text{u}})$$
(9)

The nonlinear refractive index for the prepared PVC/Al2O3 nanocomposite films may be written as [45,46,47]:

$$n_{2} = \frac{{12\pi \chi^{(3)} }}{{n_{0} }}$$
(10)

The obtained χ(3) and n2 values are increased with increasing Al2O3 concentration (see Table 2).

Table 2 Values of S0, λ0, N/m*, ωp, n∞, ε∞, χ(1), χ(3), n2 and for the PVC/Al2O3 nanocomposites
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The average oscillator wavelength λ0 and oscillator strength S0 parameter values for the considered nanocomposite samples can be acquired from the linear of (n2 − 1)−1 versus λ−2 (Fig. 8) by utilizing the single oscillator demonstrate as [39];

$$\left( {n^{2} - 1} \right)^{ - 1} = \left( {\frac{1}{{S_{0} \lambda_{0}^{2} }}} \right) - \left( {\frac{1}{{S_{0} }}} \right)\lambda^{ - 2}$$
(11)

The obtained λ0 and S0 values are given in Table 2. It is clear that the average oscillator wavelength λ0 and oscillator strength S0 parameter values increased with the steady increase in Al2O3 nanoparticles wt%.

Fig. 8
figure 8

(n2 − 1)−1 versus (λ)−2 plots for the PVC/Al2O3 nanocomposite films

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The real part of the dielectric constant ε1 can be analyzed to obtain the high-frequency dielectric constant ε∞ as indicated by the following relation [48,49,50,51];

$$\varepsilon_{1} = n^{2} - k^{2} = \varepsilon_{\infty } - \left( {\frac{{e^{2} }}{{4\pi^{2} c^{2} \varepsilon_{0} }}} \right)\left( {\frac{N}{{m^{*} }}} \right){{\lambda }}^{2}$$
(12)

where λ is the wavelength, e is that the charge of the electron, N is that the free charge-carrier concentration, ε0 is the permittivity of the free space, m* the effective mass of the charge carriers (kg), and c is that the velocity of light in vacuum. The high-frequency dielectric constant ε∞ and (N/m*) can be calculated from the intercept and the slope of the linear portion in ε1 versus λ2 plots as appeared in Fig. 9. Moreover, the long wavelength refractive index n∞, is calculated utilizing \(\varepsilon_{\infty } = n_{\infty }^{2}\) and furthermore the plasma frequency ωp is calculated from the relation; \(\omega_{p}^{2} = \frac{{e^{2} N/m^{*} }}{{\varepsilon_{0} }}\).

Fig. 9
figure 9

Plots of ε1 versus λ2 for the prepared PVC/Al2O3 nanocomposite films

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It is clear that the ratio of free carriers to effective mass (N/m*) increase with increasing Al2O3 nanoparticles content as indicated in Table 2. Likewise, the plasma frequency (ωp), n∞, and ε∞ are increased with increasing Al2O3 nanoparticles wt% for the prepared PVC/Al2O3 nanocomposite films.

The optical conductivity (σopt) comes because of the movement of the charge carriers by alternating electric field of the incident electromagnetic waves which is given by the equation below [52];

$$\sigma_{\text{opt}} = \frac{nc\alpha }{4\pi }$$
(13)

where n is that the refractive index, c is that the speed of light in vacuum, and α is the absorption coefficient. The calculated optical conductivity as a function of incident photon energy is displayed in Fig. 10.

Fig. 10
figure 10

The dependence of optical conductivity on the incident photon energy for the PVC/Al2O3 nanocomposite films

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The ascertained increase in polymer nanocomposite films optical conductivity with increasing Al2O3 nanoparticles content could also be a direct result of the formation of new levels within the band gap that facilitate crossing of the electrons from the valence band to those local levels to the conduction band, consequently the band gap decreases and also the conductivity increase [53].

Dispersion in PVC/Al2O3 nanocomposite films is defined by the group velocity dispersion (GVD) which is related to the second derivative of refractive index with respect to the incident light wavelength by [54];

$${\text{GVD}} = \frac{{\lambda^{3} }}{{2\pi c^{2} }}\left( {\frac{{d^{2} n}}{{d\lambda^{2} }}} \right)$$
(14)

GVD obtained values using second derivative of refractive index with respect to wavelength data plotted in Fig. 11 are given in Table 3, which causes a short pulse of light to spread in time as a result of different frequency components of the pulse traveling at different velocities. The dispersion coefficient for material dispersion (D) is related to GVD by [55];

$$D = - \frac{\lambda }{c}\left( {\frac{{d^{2} n}}{{d\lambda^{2} }}} \right) = - \frac{2\pi c}{{\lambda^{2} }}{\text{GVD}}$$
(15)

The medium has positive dispersion, if D is less than zero. If D is greater than zero, the medium has negative dispersion. The third-order dispersion (TOD) is defined as the frequency dependence on GVD which is given by the following formula [54];

$${\text{TOD}} = - \left( {\frac{{3\lambda^{4} }}{{4\pi c^{3} }}\frac{{d^{2} n}}{{d\lambda^{2} }} + \frac{{\lambda^{5} }}{{4\pi c^{3} }}\frac{{d^{3} n}}{{d\lambda^{3} }}} \right)$$
(16)

The prepared polymer nanocomposite samples have high GVD as well as TOD, which increased with increasing Al2O3 nanoparticles concentration as seen in Table 3. Also, these samples have positive dispersion, as D values are less than zero.

Fig. 11
figure 11

Second derivative of refractive index with respect to wavelength for the PVC/Al2O3 nanocomposite films

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Table 3 GVD, D, and TOD values for the PVC/Al2O3 nanocomposite films
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Conclusion

PVC/Al2O3 nanocomposite films have been prepared successfully with the well-known solution casting route at room temperature. The optical parameters such as optical energy gap, Urbach energy, refractive index, dielectric constant, average oscillator wavelength, oscillator strength, bond strength, and optical conductivity were contemplated as a function of Al2O3 content and increase with increasing Al2O3. The third-order nonlinear optical susceptibility (χ(3)) and the nonlinear refractive index (n2) are found to increase with increasing Al2O3 percentage that makes these films accommodating in optical devices technology. The group velocity dispersion, dispersion coefficient for material dispersion, and third-order dispersion were calculated and are found to increase with increasing Al2O3 percentage, which makes these films useful in laser pulse broadening.