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Abstract
The contribution of the surface layer to the reflection coefficients is shown to be determined by four surface integral values, which can be interpreted as real and imaginary parts of two complex permittivity excesses. The reflectance-absorbance spectra are determined by the spectra of these parameters. The spectra of the surface excess integrals cannot be found with the angular measurements of reflection-absorption spectra, which are determined by only three angular dependent terms. To determine these four surface excess integrals, it is necessary to involve the experimental data of spectroscopic ellipsometry or polarization-modulation infrared reflection absorption spectroscopy providing equivalent information about surface. In the case of weakly absorbing bulk medium, the real parts of the excesses can be neglected, permitting calculation of their imaginary parts using the angular dependence of the absorbance. The calculation of these parameters allows to check consistency of the data obtained. Measurements of the angular dependence of the absorbance of p-polarized radiation reflected from the DPPC monolayer upon distilled water were performed. The data obtained turned to be in good agreement with the proposed theoretical analysis.
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1. Introduction
The optical methods are known to be very effective for non-destructive investigation of surface layers and films on different substrates. Infrared reflection-absorption spectroscopy (IRRAS) [1–8], polarization modulation infrared reflection-absorption spectroscopy (PM-IRRAS) [9–17]), spectroscopic ellipsometry [8,18–21]), Brewster angle microscopy (BAM) [3,8,14] are highly effective to uncover the structure and events at different boundaries.
The theoretical background of these methods based on reflection coefficients dependent on surface layer characteristics. Interpretation of experimental results demands determination of surface layer correction to Fresnel reflection coefficients for two polarizations: s-polarisation, i.e. polarization in the plane perpendicular to the plane of incidence and p-polarization, in the plane of incidence. It has been detected that in comparence with s-polarization absorbance of p-polarized radiation demonstrate more pronounced bands for the same surface films. This fact makes it advisable in some experimental situations to perform experiments with p-polarization, despite to the lower signal-to-noise ratio due to the lower reflectivity.
There are two main approaches to the quantitative interpretation of the experimental data of the above methods. The first is based on the assumption that the surface layer can be modelled as a number of homogeneous isotropic or anisotropic layers between two homogeneous semi-infinite phases [22,23]. The cumbersome calculations can be simplified for a thin film approximation, i.e. for $\lambda \gg L$ , where $\lambda$ is the wave length and $L$ is the characteristic thickness of the surface layer [24–26].
Another approach is based on the molecular-statistical theory of light propagation in non-absorbing dielectrics, extended to the case of absorbing media [27–30], used e.g. [31–37]. It was developed for a thin anisotropic surface layer with an arbitrary permittivity profile. Nevertheless the interpretation of experimental data starts from construction of an optical model of the film.
In [38] this method was applied to analysis of the s-polarized radiation reflection and obtaining expressions for reflectance and reflectance-absorbance with account for the thin surface layer with arbitrary permittivity profile. It was shown that the contribution of the surface terms to both values is determined by only one parameter, the effective surface permittivity excess. This fact allows to calculate reflectance-absorbance spectrum at any angle of incidence on the base of experimental data measured at single angle. An optical model is used only at the final stage of interpreting this excess, and only one of its parameters can be calculated from experimental data.
The analysis of p-polarized radiation allows a quantitative analysis of the reflectance and reflectance-absorbance for this polarization, but also it permits also to obtain an additional knowledge about surface layer and allows to analyse ellipsometric and PM-IRRAS data, which are determined by the reflection coefficients for both polarizations. Presently we consider the polarized radiation reflection taking into account thin surface layer ($\lambda >> L$) with arbitrary permittivity profile. On the base of reflection coefficients derivation developed earlier [27–30] we calculate the reflectivities of p- and s- polarized radiation.
Surface corrections to the Fresnel reflection coefficients are known being angular dependent; it has been used repeatedly studying the surface parameters. Presently we succeed to prove experimentally and theoretically that the surface corrections to reflectances are wholly described in terms of four angular-independent integrals, real and imaginary parts of permittivity and surface permittivity reciprocal excess integrals. The surface contribution to the reflectance-absorbance of p-polarized radiation is determined by these four terms, while the contributions to the ellipsometric parameters and PM-IRAS data are determined by the difference between real and imaginary parts between these excesses.
Angular dependence of reflectance-absorbance of p-polarized radiation is determined by three angular dependent terms and hence only three free parameters can be found from the absorbance measurements. One parameter defines reflectance-absorbance of s-polarised light and allows to calculate reflectance or reflectance-absorbance of s-polarized radiation on the studying the angular dependence of p-polarized light. To determine real and imaginary parts of both excesses it is necessary simultaneously have data of reflectance-absorbance of p-polarized light and PM-IRRAS spectroscopy or spectroscopic ellipsometry for the system under investigation.
To demonstrate the method described we study the widely known monolayer of dipalmitoylphosphatidylcholine (DPPC) on the distilled water surface with high surface density of lipid molecules for a number of different incidence angles. We found remarkable agreement between these universal parameters and experimental data.
2. Four-parameter model
2.1 Thin-layer contribution to reflectance
We consider the plane boundary between two semi-infinite media, $A$ and $B$. Let the plane electromagnetic wave incidents from isotropic non-absorbing medium A with permittivity $\tilde {\epsilon }_A$ upon the boundary with isotropic medium B with permittivity $\tilde {\epsilon }_B$ (Fig. 1); the tilde $\tilde {}$ indicates that the respective parameters are the complex variables, containing the real and imagine parts due to absorbance. We take phase $A$ being a diluted gas with permittivity $\epsilon _A=1$, and phase $B$ being an isotropic condensed matter with tensor permittivity $\hat {\epsilon }_B$. The permittivity of the film, or the surface layer, at the boundary between media $A$ and $B$ we denote $\hat {\epsilon }(z)$ where $z$ is the cartesian coordinate normal to the plane boundary.
The reflection coefficients $\tilde {R}_s$ and $\tilde {R}_p$ of s and p-polarized radiation, respectively, taking into account the surface profile, in the first order of dimensionless thickness $L/\lambda$ can be written in the form [27,30]:
The factors $\tilde {I}_1$ and $\tilde {I}_2$ in Eqs. (1), (2) are the unique values describing the contribution of the surface layer profile to reflection coefficients,
We call the quantity $\tilde{I}_{1}$ as the integral surface permittivity excess and $\tilde{I}_{2}$ as the integral surface permittivity reciprocal excess. These integrals represent the contribution of the surface profile arising from the difference in the absorbing properties of the substance inside the surface layer and in the depth of the bulk medium.
Equations (1) and (2) are valid in the first order of thickness parameter $k_0 L$. We omit for brevity the argument $\omega$ indicating the permittivity dependence on frequency, where $\omega$ is angular frequency. Naturally, the spectral dependence of the permittivity both in the bulk phase and in the surface layer leads to a frequency dependence of the reflection coefficients and other quantities that depend on the permittivity.
Using Eqs. (1), (2), (3) and (4) we present the reflectance of s- and p-polarized radiation $r_s=\tilde{R}_s\tilde{R}^*_s$ and $r_p=\tilde{R}_p\tilde{R}^*_p$ as follows
$O_r={({\epsilon _B^{\prime }}^2+{\epsilon _B^{\prime \prime }}^2)^{-1}}\left (\epsilon _B^{\prime }(\epsilon _B^{\prime }+1)+{\epsilon _B^{\prime \prime }}^2\right )$ and $O_i=-{({\epsilon _B^{\prime }}^2+{\epsilon _B^{\prime \prime }}^2)^{-1}} {\epsilon _B^{\prime \prime }}$ are the bulk optical parameters. The quantities
Angular dependence of the p-polarized radiation reflectance is more complicated than that of the s-polarization and exhibits in numerator additional terms proportional to $\sin ^2\phi _0 \cos \phi _0$ and $\sin ^4\phi _0 \cos \phi _0$. For normal incidence the reflectances Eqs. (5) and (6) obviously coincide.
The $I_{eff}^{(s)}$ , the effective permittivity excess is determined by the excess of $\tilde \epsilon _{t}(z)$, tangential component of permittivity tensor only in the surface layer and does not depend on the angle of incidence. In contrast to this value $I_{eff}^{(p)}$ depends on the angle of incidence and determined according to Eq. (8) by real and imaginary parts of both integrals $\tilde {I}_1$ and $\tilde {I}_2$.
Thus in linear approximation in parameter $L/\lambda$ the contribution of the surface layer profile into the reflection coefficients (1), (2) and reflectancies (5), (6) are defined by unique four parameters - real and imaginary parts of surface excess integrals $\tilde {I}_1$ and $\tilde {I}_2$.
2.2 Surface excess contribution into ellipsometric parameters
Equations (1)–4) yield directly the ellipsometric parameters $\Delta$ and $\Psi$ (see [22])
Equation (11) shows that the surface term in ellipsometrical parameters $\Psi$ and $\Delta$ are defined by two quantities, namely real and imaginary parts of the difference of integral surface excesses $\tilde {I}_1-\tilde {I}_2$. The spectrum of ellipsometrical parameters is determined by the spectra of these two quantities and by the bulk permittivity spectrum. The sign of $\imath \Delta$ is determined by the choice of the wave in the form $E(r,t)=E_{0}exp(i(\mathbf {kr}-\omega t))$ , where - $\mathbf {k}$ is the wavevector, $\mathbf {r}$ is cartesian coordinate, and t is time.
For a non-absorbing system and isotropic thin surface layer this difference term is simplified to the Drude integral [28,39])
The expressions for ellipsometric parameters $\Delta$ and $\Psi$ may be derived from Eq. (11) after some simple but cumbersome algebraic calculations. Presently we don’t discuss spectra of ellipsometrical parameters. The only important fact is that both parameters depend on two integral surface excesses ${I}_{1r}-{I}_{2r}$ and ${I}_{1i}-{I}_{2i}$
2.3 Surface excess in PM-IRRAS signal
The expression for PM-IRRAS signal can be written as [40–42]
where - C is a constant accounting for the different amplification of the two parts during the two-channel electronic processing, $J_0$ and $J_2$ are the zero- and second-order Bessel functions of the maximum dephasing $\delta _0$, introduced by the photoelastic modulator.As seen the PM-IRRAS signal is determined by ratio $r_p/r_s$ i.e. as it follows from (11) the surface contribution to this signal is defined, by the difference of surface excesses integrals ${I}_{1r}-{I}_{2r}$ and ${I}_{1i}-{I}_{2i}$, quite similarly to ellipsometric parameters.
2.4 Surface excesses in reflectance-absorbance
Defining the reflection-absorbance as the logarithm of the reflected intensity related to its Fresnel’s value, $a_s= -\log _{10}\frac {r_s}{r^F_s}$ and $a_p= -\log _{10}\frac {r_p}{r^F_p}$ we obtain, in the main thickness order
The angular independent surface contribution $I_{eff}^{(s)}$ can be immediately derived from the IRRAS measurement data with s-polarized light [30]. Studying the angular dependence of spectrum $a_p$ we succeed to obtain an additional knowledge on the surface profiles.
Functions $I_{eff}^{(s)}$ and $I_{eff}^{(p)}$ describe the surface contribution to the reflectance-absorbance for $s$- and $p$- polarizations; the IRRAS multi-angle measurements with s-polarized radiation confirm angular invariance of parameter $I_{eff}^{(s)}$ [38]. The spectrum of $a_p$ (Eq. (15)) is determined by the spectra of the bulk phase permittivity and $I_{eff}^{(p)}$. According to Eq. (8) the spectrum of $I_{eff}^{(p)}$ is defined by three spectra - $I_{eff}^{(s)}$, ${I}_{s2}$ and ${I}_{s4}$ , which depend on the characteristics of the surface layer. The first angular-independent term in Eq. (8) is $I_{eff}^{(s)}$ depending on $\epsilon _t(z)$ only; the other terms depend on both components of the surface permittivity tensor. Thus the surface contribution to the reflectance and reflectance-absorbance is defined, according to Eqs. (8), (9) and (10), by four integrals $I_{1r}, I_{1i}, I_{2r}, I_{2i}$ and their spectra are defined by the spectra of this integrals. This means, in particular, that it is not possible to determine the spectra of these four integrals or their values for any specific wavelength based on multi-angle measurements of reflectance-absorbance.
2.5 Weakly absorbing bulk phases
For a surface layer at a low absorbing bulk phase the excess integrals (7) – (10) are simplified significantly. Evaluating the permittivity parameters of homogeneous substrate $B$ with inequality $\epsilon _B^{\prime }-1 \gg \epsilon _B^{\prime \prime }$ we obtain Eqs. (16), (17) ${I}_{s2}\approx -(\epsilon _B^{\prime }-1)((\epsilon _B^{\prime }+2)I_{1i}+\epsilon _B^{\prime } I_{2i})/\epsilon _B^{\prime }$, ${I}_{s4}\approx ({\epsilon _B^{\prime }}^2-1){\epsilon _B^{\prime }}^{-2}(\epsilon _B^{\prime }I_{2i}+I_{1i})$ where $I_{1i}\approx \int _{0}^{\infty }dz\epsilon ^{\prime \prime }_t(z)$ and $I_{2i}\approx \int _{0}^{\infty }dz\epsilon ^{\prime }_B \epsilon ^{\prime \prime }_n(z)/({{\epsilon ^{\prime }_n(z)}^2 + {\epsilon ^{\prime \prime }_n(z)}^2})$.
It permits to rewrite expressions for the surface contributions to the reflectance-absorbance for $s$- and $p$- polarizations $I_{eff}^{(s)}$ and $I_{eff}^{(p)}$ as follows
Thus reflectance and reflectance-absorbance of surface layers for the low bulk absorbance are determined only by imaginary parts of the surface permittivity and surface permittivity reciprocal excesses, which can be obtained from experiment.
3. Experimental setup and materials
Fourier transform infra-red (FTIR) spectrometer Nicolet 8700 (Thermo Scientific) equipped with table optical module (TOM) was used for studying of the IRRAS angular dependencies. The optical scheme and parameters of the setup for measuring the angular dependencies of the reflection-absorption spectra of linearly polarized light were published earlier [38]. MCT-D detector was used for the registration of IR single beam spectra in the 1000 - 4000 cm$^{-1}$ spectral range with resolution of 2 cm$^{-1}$. All spectra were averaged over 1024 scans. Spectrometer and TOM were purged with nitrogen. Wire grid polarizer was used to select the s- or p-polarized radiation.
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 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
As shown above the reflectance (5), (6) and reflection-absorbance spectra (14), (15) are determined by spectra of the permittivity excess (3) and the permittivity reciprocal excess (4). Taking into account that these excesses are complex variables we may state that in the first approximation in thickness that the surface inputs to the IRRAS and PM-IRRAS spectra as well as ellipsometrical parameters spectra are defined by for spectra $I_{1r},I_{1i} ,I_{2r},I_{2i}$, the experimental data for any frequency may be defined by the magnitudes of these four surface integrals. That means that for thin surface layers it is possible to determine from experimental data only four values and hence optical model should not contain more than four free parameters.
Ellipsometric parameters $\Delta$ and $\Psi$ as well as PM-IRRAS parameter $S$ depend on the ratio ${R}_p/{R}_s$. The surface term to this ratio is defined by differences of integral surface excesses $I_{1r}-I_{2r}$ and $I_{1i}-I_{2i}$. In particular that means that spectra-ellipsometry and PM-IRRAS can produce equivalent information about surface characteristics.
The obtained expression $I_{eff}^{(p)}$ demonstrates that the angular independent term presents the effective surface permittivity excess. There is obviously fulfilled the equivalence $\tilde {R}_p =\tilde {R}_s$ in the limit $\phi _0=0$. This fact allows to determine effective surface permittivity excess from experiments with p-polarised radiation, i.e. theoretically measurements with s-polarized radiation do not provide additional information as compared to data obtained from p-polarization. Practically, however, s-polarization measurements seem more preferable since they deliver more accurate experimental data because for this polarization the signal-to-noise ratio is significantly higher due to larger reflectivity.
IRRAS spectra for s-polarized radiation are defined by the spectra of the bulk phase permittivity and the effective surface permittivity excess $I_{eff}^{(s)}$, i.e a combination of integrals $I_{1r}$ and $I_{1i}$. The spectra of p-polarised light allow to obtain more knowledge about surface characteristics. The surface input in this case is determined by the magnitude $I_{eff}^{(p)}$ dependent on the angle of the incidence. The angular dependence of the absorbance allow to extract three independent combinations of four parameters $I_{1r},I_{1i} ,I_{2r},I_{2i}$, i.e. themselves multi-angular measurements do not allow to determine these integrals within method outlined. To determine all for integrals it is necessary to have results of complementary PM-IRRAS or ellipsometric measurements.
As mentioned above, it is impossible to determine the above integrals based on the experimental absorption spectra of p-polarized radiation. Nevertheless, the applicability of the described approach can be verified, i.e., check whether the angular dependence of the absorption spectra is described by Eq. (15) and, in particular, whether the value of $I_{eff}^{(s)}$ calculated in this article is close to that determined earlier in experiments with s-polarized radiation [38].
In Fig. 2 there are presented the absorbance spectra of $I_{eff}^{(p)}$ within the wavenumber interval $1150 - 4000\:cm^{-1}$ for angles of incidence $\phi _0 = 20{^\circ }$, $\phi _0 = 30{^\circ }$ , $\phi _0 = 40{^\circ }$; $I_{eff}^{(p)}$ values are given in nm. To calculate $I_{eff}^{(p)}$ from experimental absorbance data we use the values of permittivity of the water from [43].
Fig. 2. p-polarized radiation reflectance-absorbance on wavenumber dependence, angle of incidence $20{^\circ }$ (solid line), $30{^\circ }$ (dash line), $40{^\circ }$ (dot line). DPPC film on the water surface, area per molecule 0.49 $nm^2$
Spectra in Fig. 2 demonstrate the decline the signal/noise ratio approaching Brewster angle due to reduced reflectivity. It is convenient to distinguish intense bands for analysis in the regions near 2850 $cm^{-1}$ and 2920 $cm^{-1}$. Therefore, further consideration will be limited to the interval of wave numbers 2700-3100 $cm^{-1}$ ( see inset in Fig. 2), corresponding $I_{eff}^{(p)}$ spectra are presented in Fig. 3.
Fig. 3. $I_{eff}^{(p)}$dependence on wavenumber, angle of incidence $20{^\circ }$ (solid line), $30{^\circ }$ (dash line), $40{^\circ }$ (dot line). DPPC film on the water surface, area per molecule 0.49 $nm^2$
The positions and maxima of the methylene bands were determined for $I_{eff}^{(p)}$ spectra obtained from 43 experimental reflection-absorption spectra measured at 13 angles of incidence in the range of incidence angles of $20{^\circ }$-$50{^\circ }$. The average positions of bands maxima for methylene asymmetric and symmetric stretching vibrations were determined as 2919.3 $\pm$ 1.1 $\:cm^{-1}$ and 2850.2 $\pm$ 0.7 $\:cm^{-1}$ , respectively. These values are in agreement with results [38] 2918.8 $\pm$ 0.4 and 2850.5 $\pm$ 0.4 $\:cm^{-1}$ obtained for s-polarized radiation [2,35,44].
The $I_{eff}^{(p)}$ maxima were fitted by using fore order approximation of $\sin ^2\phi _0$. Results are presented in Table 1 and in Fig. 4 and Fig. 5.
Fig. 4. Dependence of $I_{eff}^{(p)}$ value maxima on the $\sin ^2\phi _0$ for methylene symmetric band. Solid line - fitting by square function. DPPC film on the water surface, area per molecule 0.49 $nm^2$
Fig. 5. Dependence of $I_{eff}^{(p)}$ value maxima on the $\sin ^2\phi _0$ for methylene asymmetric band. Solid line - fitting by square function. DPPC film on the water surface, area per molecule 0.49 $nm^2$
Fitting results are $I_{eff}^{(p)}=1.07-2.2\sin ^2\phi _0+0.79\sin ^4\phi _0$ for symmetric and $I_{eff}^{(p)}=1.61-3.48\sin ^2\phi _0+1.54\sin ^4\phi _0$ for asymmetric methylene bands. The constant terms are in good agreement with values obtained from experiments with s-polarized light [38] - 1.08 and 1.60 respectively.
Table 1. The values of the three integral parameters $I_{eff}^{(p)}, {I}_{s2}, {I}_{s4}$ obtained for the symmetric $\omega _{1}$ and antisymmetric $\omega _{2}$ methylene bands.
We note, that $I^{(s)}_{eff}$ obtained from experiments with $p$-polarisation light is in a complete agreement with the data from $s$-polarised measurements. It confirms the validity of four-parameter model, developed presently.
The coefficients linear in $\sin ^2\phi _0$ are determined with an accuracy slightly better than $10{\%}$ for symmetric and asymmetric bands. The low accuracy of the coefficients proportional to $\sin ^4\phi _0$ is a consequence of using a relatively narrow range of incidence angles in this work. It emphasizes the feasibility of making measurements over a wider range of angles of incidence to improve accuracy of polynomial coefficients. These coefficients permit to calculate directly parameters $I_{1i}$ and $I_{2i}$. The real and imaginary parts of water permittivity are taken $\epsilon _B^{\prime }\approx 1.92, \epsilon _B^{\prime \prime }\approx 0.02$ and $\epsilon _B^{\prime }\approx 1.96, \epsilon _B^{\prime \prime }\approx 0.04$ for maxima wavenumbers of methylene symmetric and asymmetric bands, correspondingly. Using Eqs. (16) and (17) it’s possible to obtain from fitting data for methylene symmetric $I_{1i}\approx 1.07$, $I_{2i}\approx 0.00$ or $I_{2i}\approx 0.0$ and asymmetric bands $I_{1i}\approx 1.61$, $I_{2i}\approx 0.24$ or $I_{2i}\approx 0.21$, correspondingly. Two values of $I_{2i}$ are presented because it is possible to calculate $I_{2i}$ from both $I_{s2}$ and $I_{s4}$.
Experimental values of $I_{1i}$ and $I_{2i}$ show that $\epsilon ^{\prime \prime }_t(z) \gg \epsilon ^{\prime \prime }_n(z)$. This allows us to conclude that hydrocarbon tails orientation is close to the normal to the surface. This fact is rather obvious from general point of view for high film density and was admitted in a number of works.
5. Conclusion
We have shown that surface layer contribution to any reflection and reflection-absorption spectrum in the long-wavelength approximation can be described by a combination of four spectra that have a clear physical meaning, i.e. spectra of the real and the imaginary parts of the integral surface permittivity excess (Eq. (3)) and the integral surface permittivity reciprocal excess (Eq. (4)). All information about the surface layer can be obtained from the values of these permittivity surface excesses. From the formal point of view, the mathematical expressions for these quantities are quite simple. To obtain information about the surface layer, there is only one task - to establish a relationship between the components of the permittivity tensor in the surface layer and the parameters characterizing the surface.
It was shown that the angular dependence of IRRAS spectra of p-polarized light measured at the interface between two media with an arbitrary surface inhomogeneity is a superposition of three independent spectra with different angular dependencies. They are determined by the above spectra of surface excesses, as well as by the real and imaginary parts of the bulk permittivity. This means that only three independent spectra can be determined from experiment, or only three independent values for any particular wavelength. Therefore, every model containing more than 3 parameters cannot be solved from the angular dependence of the IRRAS data and, as a result, will include dependent parameters. Theoretically, absorbance of s-polarized radiation can be calculated from multi angular absorbance data for p-polarized light. The only condition that must be met is to achieve a sufficiently high measurement accuracy.
Additional data can be obtained from PM-IRRAS and spectra-ellipsometric measurements. The spectra obtained by both methods are determined by the same two parameters - the differences between the real and imaginary parts of the surface excesses. This fact means that these methods provide equivalent information about surface layer. In theory, reflectance-absorption measurements with additional data from one of these methods allow one to determine all four spectra associated with surface excesses.
In the case of films on weakly absorbing substrates, the experimental data can be described by two parameters, the imaginary parts of the surface excesses, which can be easily calculated from the IRRAS experimental data. This conclusion was confirmed by the obtained experimental data.
Funding
Russian Science Foundation (22-23-00235).
Acknowledgment
The experimental measurements were performed at the Center for Optical and Laser Materials Research, Research Park of Saint-Petersburg University.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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