Infrared spectra of the main optical properties of poly(methyl methacrylate) thin �lms

Infrared spectra are presented for the main optical properties of poly(methyl methacrylate) films. They are obtained from the complex permittivity using the Drude-Lorentz model for permittivity. This approach is original in obtaining the permittivity and then in obtaining the main optical properties of the polymer film. Classical dispersion analysis of the reflection-absorption spectrum of poly(methyl methacrylate) films cast on the metal mirror was used. Subsequently, the complex electric modulus, energy loss function, optical constants, absorption coefficient, penetration depth, dissipation factor, and complex optical conductivity were obtained from the complex permittivity.


Introduction
The optical properties of poly(methyl methacrylate) (PMMA) in the infrared spectrum are particularly important for polymer characterization.
Reflection and transmission techniques, such as Fourier transform infrared (FTIR) spectroscopy and Kramers-Kronig analysis, are often used to obtain infrared spectra of the optical constants n and k [1][2][3].Another method, classical dispersion analysis, was used successfully in this paper to process infrared reflection-absorption spectra of thin PMMA superficial films.This approach is original and allows us to easily obtain the main optical properties of superficial polymer films.
The measured spectrum is a transflection spectrum.The measured size is the transmittance.The transflection spectra thus obtained are similar to the transmission spectra [4,5].
Classical dispersion analysis uses the Drude-Lorentz model of independent oscillators, which describes complex relative permittivity by the equation: Each oscillator is represented by a Lorentzian with three adjustable parameters: the oscillator frequency ν0,j, the line-width γj and the plasma frequency νp,j.The parameter ε∞ is the contribution of higher-frequency oscillators.The parameter ε0 is the vacuum permittivity (= 8.854•10 -12 F/m) and ε∞ is the contribution of higher frequency oscillators [6].
The modeling process consists of introducing into the expression of permittivity described by the Drude-Lorentz equation the dispersion parameters ε∞, ν0, νp , γ corresponding to the independent oscillators of the different groups in the polymer structure and adjusting them until the transmittance Tmodel obtained from the modeled permittivity approximates the best Texperimental transmittance corresponding to the reflection-absorption spectrum of the polymer [4,5,7].
The relative permittivity r  of Eq. ( 1) is continuously modified by adjusting the parameters ε∞, ν0, νp, γ until a minimum value of the quantity 2  is obtained: The chi-square is the sum of the squares of the differences divided by the standard deviation between the experimental and model transmittance values.
In this equation, yj are the values of the experimental transmittance at different frequencies ν and f(xj, pj,….pM ) corresponds to the transmittance of the model at the same frequencies.The standard deviations are given by σi.The Levenberg-Marquardt algorithm, also known as the damped least squares method, was used [6].
The classical dispersion analysis based on the previously presented model (Drude-Lorentz) was used to obtain the complex permittivity.This quantity is the basis for the subsequent determination of the main optical properties based on the relationships between them, according to the scheme shown in Figure 1.
Fig. 1 Obtaining the main optical properties of the polymer film from the complex permittivity.This is obtained through the classic Drude-Lorentz dispersion analysis of the infrared reflection-absorption spectrum of the polymer Many values of other optical properties of the polymer film can be obtained from the complex relative permittivity , i ~2 1 r obtained by the classical dispersion analysis of the infrared (IR) reflection-absorption spectra of PMMA thin films.Relationships between them were used.All these quantities depend on the frequency of the radiation.Thus, the spectra of these quantities are obtained.
1 0 The complex electric modulus M ~ is another representation of the complex relative permittivity.It gives a clearer characterization of the dipolar relaxation mechanisms of synthesized polymeric sheets [8,9].The complex electric modulus M ~ is described by the following equation: The imaginary part of the complex electric modulus M2 was calculated in the RefFIT program [10] as a loss function Lf, or it was defined as a volume (bulk) energy loss function, VELF [11][12][13].The energy loss function (ELF) describes the energy lost while passing through the film.The ELF has two main sections, the surface energy loss function (SELF) and the volume energy loss function (VELF).ELF is the sum of the two quantities, both obtained from ε1 and ε2.The real part ε1 refers to the dispersion and ε2 to the absorption of light waves in the optical medium.The two terms of ELF are given by the relations: The complex refractive index n ~ is obtained from the permittivity value with the following equation: The refractive index n, which is the real part of the complex refractive index, describes the dispersive properties of the material.It can be calculated from the complex permittivity [14,15].
The extinction coefficient k is the imaginary part of the complex refractive index that describes the absorbing properties of the material and can be calculated from the complex permittivity with the equation: 3 0 The dissipation factor (or dielectric loss tangent) tan δ.The loss tangent, tan δ determines how well a material can absorb the electromagnetic field [9,13,21].It can be used to investigate the dielectric properties of the polymer in cases where its geometry is unknown [8].
. tan For many electrical engineering applications, the loss tangent, called the dissipation factor, is also frequently used to describe the amount of energy dissipated inside the polymer because of the alternative field.

0 The absorption coefficient α
The absorption index is obtained from the extinction coefficient.It allows us to measure how much light is absorbed by a material.[11][12][13][16][17][18][19][20] In these equations, λ is the wavelength of the radiation, ω is the angular frequency, and c is the speed of light in vacuum.

0 Penetration depth δp and skin depth δe.
The penetration depth is defined as the depth at which the intensity or power of the field inside the material decays to 1/e of its surface value.It can be defined as the reciprocal of the absorption coefficient .The penetration depth depends on wavelength, material properties, and experimental conditions [22].The skin depth δe is the depth at which the intensity of the electric field drops to 1/e of its value at the surface.
Skin depth is sometimes called the penetration depth.6 0 Complex optical conductivity  ~ Complex optical conductivity measures the response of the material to light and is related to complex relative permittivity r  [11-13, 19, 23-25].Complex permittivity describes how much electrons displace when the field is applied, whereas complex conductivity describes how fast electrons move under the influence of the applied field.Complex optical conductivity can provide details of the structural and optoelectronic properties of the material.
The imaginary part of the permittivity ε2 that determines the absorption of the wave inside the medium contributes to the real part of the optical conductivity , and the real part of the permittivity ε1 that expresses the dispersive properties of the medium contributes to the imaginary part of the optical conductivity . ; In this equation ω is the angular frequency (rad/s).
There are subtleties in the calculus of optical conductivity.Some authors [25][26][27] calculated the real part of the optical conductivity in the CGS system of units with the equation: To obtain the permittivity value in the International System of Units (SI), the previous equation must be multiplied by 4πε0 [28].

Experimental
Poly (methyl methacrylate) PMMA films were obtained by casting benzene polymer solutions on metal mirrors [5].Benzene was considered a good solvent for PMMA [29].Montedison, Italy, Montedison, Italy atactic poly (methyl methacrylate) granules (Mw ~ 97 kg/mol by GPC, Mn ~ 46 kg/mol, Tg = 105 0 C, Tm = 150 0 C) were used, without further purification.The concentration of the solution used was approximately 3.3 g/L.The sample with the metallic surface on which the polymer solution was deposited was coated with a Griffin glass beaker to allow very slow evaporation of the solvent [5].Very slow evaporation of the solvent from the polymer solution, in a solventvapor saturated environment, occurs under the spout of the Griffin beaker.The middle part of the film was used in the tests to ensure uniform thickness.This polymer film casting was used because it allows one to obtain surface films of controlled and uniform thickness on the entire metal mirror surface [7].The spin coating technique [30][31][32][33] or the dip coating technique [34,35] commonly used to obtain thin surface films can induce a certain orientation of the polymer chain during the casting of the polymer film.
The solid substrate used as the reflective mirror was the very well-polished surface of a steel sample.
The surface film deposited on the metal mirror was lightly heated to a temperature of 80 degrees for 30 minutes to completely remove the residual solvent.The presence of residual solvent can cause changes in the appearance of the polymer spectrum [36].
The IR reflection-absorption spectrum of the PMMA thin film was recorded at a 20 0 incidence angle within the spectral range of 500-4000 cm -1 using the specular reflectance device of the Carl Zeiss Jena UR-20 spectrophotometer.The experimental error bar for the measured infrared reflection-absorption (IRRA) spectrum was 0.01.
The theoretical transmission spectrum of PMMA corresponding to the Drude-Lorentz model of the dielectric function was obtained using the classical dispersion analysis of the experimental IRRA spectrum of PMMA and the RefFIT computing program [10].The model parameters were continuously adjusted by fitting the theoretical spectrum to the experimentally measured spectrum.The processed spectrum contains a set of 3501 experimental points, in digitized form.The experimental error bar for the experimental IRRA spectrum was 0.01.In the dielectric function model, 83 internal parameters were included.These parameters correspond to a set of 27 damped harmonic oscillators, the 'high-frequency dielectric constant', and the thickness of the superficial film.A value of 10.6 m for the superficial film was obtained by fitting.The thickness of the polymer film can be considered to be 5.3m, because the incident radiation passes through the polymer film twice, after reflection on the metal mirror.The thickness is in agreement with the concentration of the polymer in the benzene solution.

Results and discussions
The reflection-absorption spectrum (RAS) of the superficial PMMA film deposited on the metallic steel mirror was processed with the RefFIT calculation program that uses the classical dispersion analysis based on the Drude-Lorentz model, according to relation (1) [6]. Figure 2 shows the experimental reflection-absorption spectrum of the superficial PMMA film deposited on the metallic mirror and the model spectrum based on the Drude-Lorentz equation, which best approximates the experimental spectrum.The IR spectra of the real and imaginary parts of the complex relative permittivity and complex electric modulus are similar.Fig. 4 Infrared spectra of the real part M1 and the imaginary part M2 of the complex electric modulus of the superficial poly(methyl methacrylate) film cast on a metallic mirror The two components of the energy loss function: surface energy loss function (SELF) and volume energy loss function (VELF), defined by Eq. ( 4), for the PMMA film cast on the metallic mirror, are presented in Figure 5.
The VELF described by El-Nahass et al. [11] is the same as the imaginary part M2 of the complex electric modulus and the loss function Lf defined by Kuzmenko [10].
Fig. 5 Infrared spectra of the energy loss functions and the ratio of the two components volume energy loss function (VELF) and surface energy loss function (SELF) for the poly(methyl methacrylate) film cast on the metallic mirror SELF describes the inelastic reflection of electrons on the flat surface of a solid without accounting for spatial dispersion.SELF represents the energy loss by the free charge carriers when traversing the surface of the material [13].
The ratio of the two components of the energy loss function VELF/SELF is shown in Figure 6.In the spectral regions where the superficial film shows high absorption, it is observed that high values of this ratio are obtained.
The refractive index n and the extinction coefficient k spectra are obtained from the complex relative permittivity, based on the relations (5-7).
Figure 6 shows infrared spectra of the refractive index n and the extinction coefficient k of the PMMA film cast on a metal mirror.Fig. 6 Infrared refractive index n and extinction coefficient k spectra of the poly(methyl methacrylate) film cast on a metal mirror Figure 7 shows the IR spectrum of the dissipation factor tan δ of the PMMA film cast on the metal mirror.Fig. 7 Infrared spectrum of the dissipation factor tan δ of the poly(methyl methacrylate) film cast on the metal mirror This spectrum is similar to that of the imaginary part ε2 of the permittivity.The absorption coefficient α expressed by relation ( 8) is proportional to the extinction coefficient k and the frequency of radiation.For this reason, the absorption bands from high frequencies are expected to be more intense than those in the extinction coefficient spectrum.The change in the appearance of the absorption band from 3400 cm -1 is better highlighted.Figure 8 shows the IR spectrum of the absorption coefficient α.Fig. 8 Infrared spectrum of the absorption coefficient of the poly(methyl methacrylate) thin film cast on a metalic mirror, obtained by classical disperssion analysis of the reflection-absorption spectrum with the RefFIT program Figure 9 shows the IR spectrum of the penetration depth through the PMMA film cast on the metallic mirror.Fig. 9 Infrared spectrum of the penetration depth δp of the radiation through the superficial poly(methyl methacrylate) film, cast on a metallic mirror It can be seen that in the spectral regions where the film has higher absorption, the penetration depth of radiation through the superficial film decreases if the IR spectrum of the absorption coefficient α is compared with that of the penetration depth δp.The penetration depth is greater than the thickness of the superficial film (5.3 μm) in almost all spectral regions.Radiation refracted at the air-film interface can reach the polymer-metal interface and is reflected on the metallic mirror (with a high reflectance coefficient R23=0.894)[4].
The infrared spectrum of the real part σ1 and the imaginary part σ2 of the optical conductivity for the PMMA film cast on the metal mirror are presented in Figure 10.The optical properties of PMMA (ε2, M2, VELF, SELF, k, α, tan δ and σ1) which depend directly on the absorbing properties of the polymer, present similar IR spectra.The optical properties (ε1, M1 and n) that depend on the dispersion of the radiation in the medium through which it passes also have similar IR spectra.

Conclusions
The significance of this work is the simple way to determine the complex electrical permittivity and then obtain the main optical properties of PMMA.Classical dispersion analysis and the Drude-Lorentz model were used to process the IRRA spectra of the polymer.The IR spectra of the real and imaginary parts of the permittivity were the basis for the subsequent obtaining of the spectra of the main optical properties of PMMA.
From the spectra obtained and presented, it can be observed that the IR spectra of some quantities presented have a similar appearance.The extinction coefficient k, the absorption coefficient α, the real part of the conductivity σ1, the imaginary part of the complex electric modulus M2, the surface energy loss function SELF and the dielectric loss tangent tan δ have IR spectra with an appearance similar to that of the imaginary part of the permittivity ε2.For this reason, Jitian and Bratu [4] only presented the change in the IR spectrum aspect of the extinction coefficient k during UV irradiation of the PMMA film.The refractive index n and the real part of the complex electric modulus M1 have IR spectra with an appearance similar to that of the real part of the permittivity ε1.For this reason, it is sufficient to follow only the IR spectrum of the refractive index n.The optical properties (ε1, M1 and n) that depend on the dispersion of the radiation in the medium through which it passes also have similar IR spectra.Most of the articles present only the IR spectra of the optical constants n and k and their changes during different structural transformations of the polymer.
It is important to know the spectral regions where the penetration depth is less than the thickness of the surface film.If the thickness of the superficial film is greater than δp, there is no transflection through the superficial film.In these spectral regions, specular reflection occurs and the appearance of the spectrum can be distorted, providing false information.
In the spectral regions where the surface film shows high absorption, large values of the VELF/SELF ratio are observed.

Fig. 2
Fig. 2 Comparison of the experimental infrared reflection-absorption spectrum Texperimental and the transmission model spectrum Tmodel of the poly(methyl methacrylate) film obtained by classical dispersion analysis with the RefFIT program The complex relative permittivity r  and thickness of the superficial film of PMMA deposited from a benzene solution on the surface of a steel mirror were obtained by fitting the Drude-Lorentz model of relative permittivity with that of the experimental reflection-absorption spectrum.A good fit of the model built with the experimental spectrum is observed.The IR spectra of the real and imaginary parts of the complex relative permittivity of a PMMA film deposited from solutions on the metallic steel mirror are shown in Figure3.A value of 5.3 μm for the superficial film was obtained, by fitting.

Fig. 3
Fig.3Infrared spectra of the real part ε1 and respective the imaginary part ε2 of the permittivity for the superficial poly(methyl methacrylate) film cast on a metal mirror

Figure 4
Figure 4 presents the real part and the imaginary part of the complex electric modulus , iM M 1 M ~2 1 r + = =  expressed in relation (3).The IR spectra of the real and imaginary parts of the complex relative permittivity and complex electric modulus are similar.

Fig. 10
Fig. 10 Infrared spectra of the real part σ1 and the imaginary part σ2 of the optical conductivity for the superficial poly(methyl methacrylate) film cast on a metallic mirrorThe real part of the conductivity contains the same information as the imaginary part of the dielectric function.Their IR spectra are similar for this reason.The optical conductivity σopt is less significant for simple polymers with dielectric properties.It is used mainly in the case of conductive polymers, called organic metal polymers, or in the case of semiconductor materials to detect the allowed inter-band optical transitions.