1. Introduction

The methods of reflection-absorption spectroscopy were developed for the infrared [1,2] (IRRAS) and ultraviolet-visible [3,4] (UV-Vis RAS) spectral ranges. Currently both techniques are used widely for investigations of films at gas/solid (see e.g. [5–8]) as well as gas/liquid interfaces (see e.g. [9–14]). The majority of studies deal with compounds of biological importance such as lipids, proteins, peptides and enzymes.

As a rule linear-polarized radiation is used, with the electric field parallel or perpendicular to the plane of incidence, known as $p$- or $s$-polarization, correspondingly. Based on general considerations one can expect that the analysis of $s$-polarized radiation leads to more clear and compact results. By now a lot of studies were performed for various systems at different angles of incidence. In some of them the angular dependence of the absorbance spectra has been studied [15–22] in order to obtain an additional knowledge interpretating data of multi-angle measurements.

In the quantitative description of experimental data on reflection-absorption spectroscopy, the authors mainly use two approaches. The first one is known model of two bulk phases with an anisotropic homogeneous film between, see, for example, [21]. This method can be generalized to a multi-layered surface that turns out to be relatively complex and opaque. Another approach is based on the molecular statistical theory of light propagation in non-absorbing dielectrics [15,23–25], quoted in a number of studies.

A common feature of both approaches to interpreting experimental data is that they begin with the consideration of an optical model under study, the determination of optical parameters of bulk phases and surface layer, and subsequent simulation of the experimental values of the reflection-absorption or reflectance spectra to obtain the tilt angle of the molecule or the orientation or dipole transition moment of corresponding band. This way is correct, but it makes simulation more complicated and hides considered physics behind the numerical calculations, since absorbance depends not only on the optical parameters of bulk phases and surface layer, but also on the boundary conditions.

In this work we perform simultaneously theoretical analysis and experimental study of the reflection-absorption spectra of a lipid film on the water surface. We propose an approach to the interpreting experimental data independent of a specific assumption about the surface profile; wherein the optical model itself appears at the final stage of the analysis. It is shown that the angular dependence of absorption spectra turns out to be universal, and can be reduced in case of s-polarization to invariant spectrum. The existence of this universal spectrum means that a multi-angular reflectance or reflectance-absorbance measurements do not provide any additional physical information about inhomogeneity compared to single-angle measurements. To demonstrate a method for interpreting the reflection-absorption data, a well-known system was chosen for the study. Namely we studied a monolayer of dipalmitoylphosphatidylcholine (DPPC) on the water surface with high surface density of lipid molecules for a number of different incidence angles. This work can be useful to a wide variety of researchers using the optical methods for studies of interfacial phenomena.

2. Theoretical analysis

We consider a flat boundary between two semi-infinite media. Let the plane electromagnetic wave incident from isotropic non-absorbing medium $A$ with permittivity $\tilde {\varepsilon }_A$ upon the interface with isotropic medium $B$ with permittivity $\tilde {\varepsilon }_B$, the sign “tilde” indicates that the respective parameters are the complex values. Let the medium A be diluted gas , undermining its bulk permittivity to be unit, $\tilde {\varepsilon }_A=1$; the coexisting condensed matter $B$ occupies the half-space $z> 0$, where $z$ is the Cartesian coordinate normal to the plane boundary. There exists between media a thin anisotropic surface layer, or a film, characterized by a profile of the permittivity tensor $\tilde {\varepsilon }(z)$. As was shown earlier [15] reflection coefficient of $s$-polarized radiation $\tilde {R}_s$ with account for the surface profile can be written as:

The factor $\tilde {I}_1$ in the brackets in (1) is integral surface permittivity excess describing contribution of the surface layer,

Equation (1) is valid up to the first order in parameter $k_0 L$, where $L$ is the characteristic thickness of a surface layer or film. Here we omit for brevity the dependence of $\tilde \varepsilon _B=\tilde \varepsilon _B(\omega )$, $\tilde {R}_s=\tilde {R}_s(\omega )$, $\tilde {R}^F_s=\tilde {R}^F_s(\omega )$ and $\tilde {I}_1=\tilde {I}_1(\omega )$ on the frequency $\omega$. The spectral permittivity dependence in the bulk phase as well in the surface layer leads to the frequency dependence of reflection coefficients, reflectance and absorbance.

Using (1) and (2) we present the reflectance of $s$-polarized radiation $r_s=\tilde {R}_s\tilde {R}^*_s$ as follows:

According to (4) the spectrum of ${I}_{\textrm {eff}}$ is determined by the absorptive properties of the surface layer and absorptive properties of the bulk medium. Thus the absorbance, as it follows from (6), does not exhibit any angular dependence except the linear decrease with the cosine of the incidence angle. It means that for the $s$-polarization multi-angular measurements do not provide any additional physical or chemical knowledge compared with the single-angle measurements. All the information about thin surface layer which can be obtained from reflection absorption measurement with $s$-polarized radiation is contained in the effective permittivity excess only. The experimental value ${I}_{\textrm {eff}}$ may be related to an optical model of a surface layer described by Eq. (4) .

3. Experimental setup and materials

The angular dependencies of the IRRAS were measured using the Fourier Transform Infra-Red (FTIR) spectrometer Nicolet 8700 (Thermo Scientific) equipped with Table Optical Module (TOM). Registration of IR single beam spectra was performed with the MCT-D detector. In all spectra measurements, 1024 scans were accumulated with resolution of 2 $cm^{-1}$. Spectrometer and TOM were purged with nitrogen. Wire grid polarizer was used to select the $s$-polarized radiation. The single beam spectra reflected from the pure water surface prior film spreading were used as reference spectra. The optical scheme of the TOM is shown in Fig. 1. This module has been modified for automatic detection of the angular dependence of $s$- and $p$-polarized IR reflection/absorption spectra.

 

Fig. 1. Optical scheme of the TOM module optimized for measuring the angular dependence of IR reflection/absorption spectra.

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The non-polarized beam from the FT-IR spectrometer was focused using a $BaF_{2}$ lens with a focal length of 750 mm. The use of such a lens allows obtaining a small variation in angle for each experiment and the IR light localization on the sample surface in a spot diameter about 8 mm. The size of the sample cell is $100 \times 150$ mm, thus as a first approximation, the surface of the liquid sample can be considered flat in the focal spot even with the meniscus at the edges of cell. A linear wire grid polarizer mounted on a motorized rotary translator was used to form the $s$-polarization of IR light. The polarizer positioning accuracy was about $0.1^\circ$. The angle of incidence ${\phi }_0$ was set by the rotation angle ${\theta }_1$ of the uncoated gold mirror 1. The spatial position of the focal region on the surface of the sample was kept constant due to the motorized linear positioning of $X_1$ of mirror 1. The angle of rotation ${\theta }_2$ and the position $X_2$ of the uncoated gold mirror 2 were set in such a way as to direct IR radiation into the detector.

Dipalmitoylphosphatidylcholine (DPPC) (Sigma-Aldrich, 99% purity) was used without further purification. Chloroform (Sigma-Aldrich) was purified by distillation. All water used was triply distilled. The last two distillations were implemented in an apparatus made entirely of glass. The solution of DPPC, 17 $\mu$L, in chloroform were spread drop-wise with a micro-liter syringe onto the water surface to get the surface area /molecule equal 0.49 $nm^2$. The temperature during measurement was $23\pm 0.5^\circ$C. Measurements began 1.5 hours after DPPC film spreading.

4. Results and discussion

We have measured the reflectance spectra at the air-liquid boundary for two samples of pure water and for three samples with DPPC monolayer at the interface, varying the angle of incidence in the $20^\circ$–$50^\circ$ deg interval. In Fig. 2 there are presented the spectra of absorbance and effective permittivity excess ${I}_{\textrm {eff}}$ within the wavenumber interval $1250-4000$ $cm^{-1}$, for angle of incidence $\phi _0=25 ^{\circ }$; the ${I}_{\textrm {eff}}$ data are given in $nm$. We restrict considerations with the frequency range $2700-3100$ cm$^{-1}$ containing the methylene asymmetric and symmetric stretching bands (gray area on the Fig. 2). This range is convenient for analysis firstly due to the good signal-to-noise ratio, and secondly for it is located far from the maxima of the water bands, in order to exclude discussion of their changes after film deposition.

 

Fig. 2. (a) Effective surface permittivity excess ${I}_{\textrm {eff}}$ dependent on wavenumber, angle of incidence $25 ^{\circ }$ degrees, (b) reflectance - absorbance as function of wavenumber, angle of incidence $\phi _0= 25 ^{\circ }$ degrees; DPPC film on the water surface, area per molecule - 0.49 $nm^{2}$

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To calculate the effective permittivity excess from experimental absorbance data we use the known values of permittivity of the water [26]. We demonstrate the angular dependence of the maxima of the methylene bands instead of spectrum of effective surface permittivity excess measureed at diffrent angles.

Since spectra of effective surface permittivity excess measured for different angles are reduced to a single spectrum with high accuracy we demonstrate not these spectra but the angular dependence of the maxima of the methylene bands. The position of bands maxima for methylene asymmetric and symmetric stretching vibrations were determined as 2918.8$\pm$0.4 $cm^{-1}$ and 2850.5$\pm$0.4 $cm^{-1}$, respectively. These values are in good agreement with the values obtained earlier for films with high 2D densities where alkyl chains are in all-trans conformation [18].

The experimental values of ${I}_{\textrm {eff}}$ maxima for different angles of incidence for both methylene bands are presented in Fig. 3. As it is seen in Fig. 3 the experimental data for ${I}_{\textrm {eff}}$ exhibit no visible dependence on the angle of incidence, in correspondence with the theory. The average maxima values are 1.60$\pm$0.03 nm and 1.08$\pm$0.04 nm for asymmetric and symmetric stretching vibrations, respectively. Dashed lines correspond to standard deviation of ${I}_{\textrm {eff}}$ maxima for asymmetric stretching vibrations and double standard deviation of ${I}_{\textrm {eff}}$ maxima for symmetric stretching vibrations, which demonstrates fair reliability of obtained experimental results.

 

Fig. 3. Effective permittivity excess ${I}_{\textrm {eff}}$ via the incidence angles, maxima at $2850.5$ $cm^{-1}$ - circles (blue solid line corresponds to the mean value, dashed lines - deviation 8%) and $2918.8$ $cm^{-1}$ - triangles (red solid line corresponds to the mean value, dashed lines - deviation 4%). DPPC film on the water surface, area per molecule 0.49 $nm^2$

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Thus in case of $s$-polarized radiation an information on characteristics and structure of a surface layer or film can be obtained by interpretation of ${I}_{\textrm {eff}}$ spectrum only according to (4). This spectrum is determined by two spectra - the permittivity spectrum of the bulk $\tilde \varepsilon _B$ and profile of the permittivity tensor component $\tilde \varepsilon _{t}(z)$. The results obtained do not contradict to the homogeneous layer model [27].

The absorbance (6) and effective permittivity excess (4) can be useful comparing published experimental data obtained using different incidence angles. It’s worth to note that in case of $s$-polarization measurements it is advisable to perform measurements nearer to normal incidence increasing the signal-noise ratio. The fact that the surface layer contribution, namely the effective surface permittivity excess, is invariant with respect to the incidence angle suggests a method of alignment check. Since parameter ${I}_{\textrm {eff}}$ should be constant and independent on the incidence angle, any significant dependence indicates a hardware setup problem.

5. Conclusion

We have proposed a new simple method for interpreting reflection-absorption data obtained with s-polarized radiation. We have shown that the angular dependence of the absorption spectra turns out to be universal and in the particular case of the s-polarization can be reduced to a single invariant spectrum. This spectrum demonstrates a clear physical meaning of the excess of the surface dielectric constant, and the optical model of the surface layer appears only for its interpretation. The existence of this single spectrum leads to the conclusion that multi-angle measurements of reflectivity or absorbance do not provide more additional physical information about the inhomogeneity in comparison with one-angle measurements of a thin surface layer. The fact that the surface layer contribution, namely effective surface permittivity excess, is invariant with respect to the angle of incidence may help checking an alignment of rather complicated optical systems applied for reflectance-absorbance measurements or to compare experimental data measured at different angles of incidence.

Our derivation does not require any assumption on the surface profile. The additional advantage of the defined invariant is that this integral parameter, $I_{\textrm {eff}}$, itself can be easily calculated in terms of any specific surface layer profile.