Measurement of the Optical Constants of Sand Samples Using Ellipsometry on Sand–Adhesive Composites

Materials and Methods

Samples were prepared by embedding particulate samples in Crystalbond 509, which is a clear polymerized solid mixture with low flow temperature (121 °C), typically used for sample mounting. A pure Crystalbond sample was formed by melting Crystalbond in a metal-bottomed Teflon beaker on a hot plate. For composite samples, Crystalbond and particulate material were batched into a Teflon beaker in a 50:50 ratio by weight. The beaker was placed on a hot plate, the Crystalbond was melted, and the sample was thoroughly mixed to form a composite. For all samples, standard optical polishing techniques were applied to finish the front surface, and the back surface was ground to a diffuse finish. Most samples were polished at room temperature, but small-grain samples such as the AZ road dust were cryogenically polished by freezing them in liquid nitrogen prior to and periodically throughout polishing to mitigate the disparate polishing rates of matrix and particles. Finished samples were 2–5 mm thick. Images of a pure Crystalbond sample, a composite Crystalbond/sand sample in the Teflon beaker, a polished composite sample, and a microscope image showing the cross-section of a polished sample are shown in Figure 1. Note that while the polish on the composite sample shown in Figure 1c is imperfect, and there is some reflected scatter, it is evident from the image that the reflection contains a strong specular component. The microscope image in Figure 1d shows sand grains of varying size and indicates that some grains are polished in cross-section.

OPEN IN VIEWER

Figure 1.

Images of (a) the pure Crystalbond sample, (b) a composite sample in the Teflon beaker, (c) a polished composite sample, and (d) a microscope image of the cross-section of a polished sample.

Several types of composite samples were prepared: (i) pure silica (SiO₂), 325 mesh (<44 µm) (Spectrum Chemical); (ii) AZ road dust, ISO 12103-1 A1 (Powder Technology, Inc.), ultrafine standard test dust used for evaluating filtering and other applications (<22 µm); and (iii) three different desert sand samples, labeled “Sand 1", “Sand 2", and “Sand 3” that were gathered from the ground at different desert locations.

The SE was performed in an infrared spectroscopic ellipsometer (Woollam, IR-VASE) for a wavelength range of 1.7–30 µm. The ellipsometric parameters Ψ and Δ were measured at 55°, 65°, and 75° incident angles. FT-IR measurements were performed in a Thermo Nicolet iS50R spectrometer with a deuterated triglycine sulfate detector and a KBr beamsplitter over a wavelength range of 2.5–25 μm. For FT-IR measurements, static dust and sand samples were prepared in a custom jig by sandwiching material of a given type between KBr windows, with light pressure holding the particles in place, so that the aperture was partially clear but partially obstructed by sand. FT-IR measurements were performed multiple times with a new sample of particles each time and averaged together to reduce uncertainty. Transmittance as a function of wavelength, T(λ), was calculated by referencing the measurement to the empty jig with no sand.

Ellipsometry

To carry out SE measurements, the polarization state of the incident beam is varied, and the ratio of reflection coefficients for p- and s-polarized light, R_p and R_s, respectively, are measured. Their ratio can be expressed as

Each material is then modeled with a Kramers–Kronig consistent general oscillator model—a model that treats the dielectric function as a summation of complex terms with defined functional forms. The constants in these oscillator terms are then fit to provide the best fit between the model and experimental data. This model includes a constant offset term, ε₁(∞), to account for the effect of resonances outside of the spectral range analyzed. It also includes a Sellmeier term, i.e., a zero-width Lorentz oscillator, or “pole”, with the position given by E₀ and magnitude given by A₀ to account for the absorption outside of the measured range. This term may be expressed as

Scattering and Absorption

With knowledge of n(λ) and k(λ), it is possible to calculate how a scattering particle will interact with incident light. Specifically, it is possible to calculate the scattering cross-section, C_sca, absorption cross-section, C_abs, and the extinction cross-section, where C_ext = C_sca + C_abs. These quantities, respectively, represent areas that are proportional to the amount of light scattered, absorbed, and extinguished by the scatterer. A rigorous model, based on Maxwell's equations, was developed, and data for n(λ) and k(λ), determined by SE, are input into this model. It is assumed that each particle is spherical. The values of C_sca, C_abs, and C_ext are determined as a function of particle diameter and λ.

Pure Crystalbond

Spectroscopic ellipsometry was first performed on the pure Crystalbond sample. The data for all angles were fit simultaneously with a general oscillator model with 19 Gaussian oscillators. The best fit provided values of ε₁(∞) = 1.41, E₀ = 9.61 eV, and A₀ = 89.0. The values for the Gaussian oscillator fit parameters are provided in Table I. Values of E_0,j and σ _j are shown in units of µm as well as eV for convenience.

OPEN IN VIEWER

Table I.

Fit parameters for Gaussian terms for Crystalbond.

Oscillator index, j	E0,j (eV)	E0,j (µm)	Aj	σ j (eV)	σ j (µm)
1	0.0919	13.49	0.640	1.55 × 10−3	0.23
2	0.0933	13.29	0.304	9.70 × 10−3	1.38
3	0.1087	11.41	0.150	1.10 × 10−2	1.15
4	0.1216	10.20	0.070	8.57 × 10−3	0.72
5	0.1261	9.836	0.068	2.21 × 10−3	0.17
6	0.1296	9.566	0.474	3.19 × 10−3	0.24
7	0.1330	9.322	0.889	2.61 × 10−3	0.18
8	0.1400	8.857	0.922	5.60 × 10−3	0.35
9	0.1575	7.874	1.664	7.02 × 10−3	0.35
10	0.1664	7.452	0.088	2.58 × 10−3	0.12
11	0.1699	7.299	0.222	2.08 × 10−3	0.09
12	0.1739	7.130	0.111	3.45 × 10−3	0.14
13	0.1793	6.916	0.206	2.96 × 10−3	0.11
14	0.1845	6.720	0.064	3.42 × 10−3	0.12
15	0.1958	6.333	0.116	1.13 × 10−3	0.04
16	0.1993	6.222	0.094	1.38 × 10−3	0.04
17	0.2032	6.101	0.028	8.02 × 10−2	2.41
18	0.2141	5.791	1.570	3.79 × 10−3	0.10
19	0.2185	5.673	0.073	1.40 × 10−2	0.36

Plots of measured and modeled Ψ and Δ are provided in Figure S1 (Supplemental Material). The resulting values of n(λ) and k(λ) are not of high importance here, since the main role of the Crystalbond is as a matrix material, but they are provided in Figure S2 (Supplemental Material), for reference.

Silica in Crystalbond

To test the effectiveness of this method, a composite sample of pure SiO₂ powder embedded in Crystalbond was evaluated. SE was carried out, and the data were fit, treating the sample as a Bruggeman effective medium composed of Crystalbond, with fixed fit parameters described above, and a material with unknown properties, representing silica. The ratio of silica to Crystalbond is treated as a fit parameter in order to account for the amount of silica versus Crystalbond being probed. A best fit was obtained using a model with 5 Gaussian oscillators and a 37.4% SiO₂ fraction. Plots of measured and modeled Ψ and Δ are shown in Figure S3 (Supplemental Material). A good fit was obtained between the measured data and the model. Figure 2, adapted from Frantz et al., ¹⁵ shows the values for n(λ) and k(λ) obtained from this sample compared to literature data for thin film SiO₂. ²² Consistent with the literature data, the measurements performed here show a large absorption peak near 9 µm and a smaller one near 12.5 µm, with corresponding redshifted peaks in n(λ), as well as a minimum in n(λ), with an uplift in the middle of the feature, near 8.5 µm. There are some differences, however the features measured here in both n(λ) and k(λ) are sharper, and the positions of the peaks are shifted. A Christiansen wavelength of λ_C = 7.69 µm is observed, slightly larger than the value of λ_C = 7.36 µm reported by Kischkat et al. ²² The vertical solid and dotted lines show the positions of λ_C measured here and for the data in Kischkat et al. ²² respectively. (Note that a second wavelength at which n = 1 is present, at λ = 9 µm, but this feature is of less interest because it occurs in a region of high absorption, and only the feature at the shorter wavelength is typically referred to as the “Christiansen feature” in this context). These variations are not completely unexpected since different samples of silica have different properties based on variations in stoichiometry and bonding. The sample measured here has a much greater surface area than a thin film which could lead to some differences. Generally, the model predicts realistic values for n(λ) and k(λ) indicating the method can be used effectively.

OPEN IN VIEWER

Figure 2.

The n(λ) and k(λ) for the Crystalbond/silica composite sample and comparison to literature values. ²² The vertical solid and dotted lines show the positions of λ_C measured here and for the data in Kischkat et al., ²² respectively.

Arizona Road Dust in Crystalbond

Next, this method was applied to a composite Crystalbond/AZ road dust sample with the aim of testing the approach on a standard material with a more complex absorption spectrum than SiO₂. The inset of Figure 3a shows a microscope image of the cross-section of the polished composite sample. As expected, the dust particles appear to be very fine, with most appearing to measure several microns in diameter. The sample was again treated as a Bruggeman effective medium, and the Ψ and Δ data were fit with a general oscillator model with six Gaussian oscillators. The ratio of road dust to Crystalbond is treated as a fit parameter, and the best fit was obtained with a 51.6% road dust fraction. Plots of measured and modeled Ψ and Δ are shown in Figure S4 (Supplemental Material), and again, a good fit was obtained between measured data and the model, indicating that the proposed method can be applied to materials with more complex optical properties. It is important to note that the AZ road dust, unlike the SiO₂ powder, is not a pure material and is instead a mixture of silicate dust particles with a variety of impurities. Thus, the values of the optical constants obtained here are best understood as effective values that treat the road dust itself as an effective medium.

OPEN IN VIEWER

Figure 3.

Results for AZ road dust: (a) n(λ) and k(λ) for the Crystalbond and AZ road dust composite sample, where the position of λ_C is indicated by the vertical line, and the inset shows a microscope image of the cross-section of the polished composite sample; (b) FT-IR measurement results for T(λ).

The plots in Figure 3a, adapted from Frantz et al. ¹⁵ show the values for the optical constants obtained from this sample. Additional absorptions, in comparison to the pure SiO₂ sample, can be observed at wavelengths near 9.7 and 11.6 µm with associated features in the refractive index. A Christiansen wavelength of λ_C = 8.20 µm is evident, indicated by the vertical line in the figure.

To test whether these values of n(λ) and k(λ) are reasonable, FT-IR measurements for the AZ road dust were carried out for static sand samples that were mounted between KBr windows, as described in the Experimental section above. The measurement was repeated six times, and the results were averaged. Figure 3b shows T(λ), and because only some fraction of the incident beam encounters sand, i.e., a fraction that varies from one measurement to the next depending on how sand is loaded between the windows. Therefore, the measurement provides information on the position and relative magnitude of features in T(λ) rather than a quantitative determination of sample transmittance. From the plot, a peak at λ = 7.54 µm is observed, corresponding to the Christiansen feature at λ_C = 8.20 µm, but apparently blueshifted due to the high absorption beginning near λ = 8 µm. A decrease in T(λ) between λ = 8–12 µm is also evident, corresponding to the broad, high absorption features in k(λ). Finally, a decrease in T(λ) between λ = 12–13 µm is observed, resulting from higher scatter due to the increased value of n(λ) in this range. Given these observations, overall, the measured values of T(λ) correspond well with the measured values of n(λ) and k(λ).

Desert Sand in Crystalbond

Finally, this method was applied to three composite Crystalbond/desert sand samples, labeled Sand 1, Sand 2, and Sand 3. Unlike the AZ road dust sample, these samples contain a wide variety of particle sizes, ranging from several microns to >100 µm in diameter. Thus, the beam from the ellipsometer, which is several millimeters in diameter, encounters a mixture of particle sizes on each sample.

As above, each sample was treated as a Bruggeman effective medium, and the Ψ and Δ data were fit with general oscillator models. For Sand 1, a best fit was obtained with a 21.9% sand fraction as a fit parameter with eight Gaussian oscillators; for Sand 2, a best fit was obtained with a 25.8% sand fraction with eight Gaussian oscillators; and for Sand 3, a best fit was obtained with a 29.2% sand fraction with eight Gaussian oscillators. Plots of measured and modeled Ψ and Δ are shown in Figure S5 (Supplemental Material). These samples were rougher than the other composite samples after polishing, resulting in less light in the specular reflection from the samples, and therefore more noise in experimental values for Ψ and Δ. Nevertheless, it was possible to fit the data well in each case and extract values for the optical constants.

Measured n(λ) and k(λ) are shown in Figure 4. The insets in the figure show microscope images of cross-sections of the polished composite samples. Sand 1 and Sand 2 appear to have generally similar values for optical constants, but there are some subtle differences. For example, the absorption features between ∼8 and 10 µm are higher for Sand 2 than for Sand 1. The associated Christiansen wavelength is shifted as well; for Sand 1, λ_C = 7.86 µm, and Sand 2, λ_C = 7.67 µm. Additionally, n for Sand 1 remains below unity over ∼0.4 µm for Sand 1 but over a larger value of 0.7 µm for Sand 2. Sand 3 has significantly different optical constants with, for example, a stronger and sharper absorption feature between ∼8 and 9 µm, a higher peak value of n, and a value of λ_C = 7.54 µm. Interestingly, unlike the pure SiO₂ or the AZ road dust, all three sands also have a secondary Christiansen feature, at wavelengths of λ_C = 6.38 µm for Sand 1, λ_C = 6.30 µm for Sand 2, and λ_C = 6.29 µm for Sand 3. All Christiansen wavelengths are indicated by vertical lines in the figure.

OPEN IN VIEWER

Figure 4.

The n(λ) and k(λ) for the Crystalbond and desert sand composite samples for (a) Sand 1, (b) Sand (2), and (c) Sand 3. The insets show microscope images of cross-sections of polished composite samples. All Christiansen wavelengths are indicated by vertical lines.

In order to fully evaluate the effects of scattering and absorption on transmittance through a sand sample, n(λ) and k(λ) are entered into the rigorous scattering model, and C_sca, C_abs, and C_ext are calculated. The plots on the left of Figure 5 show heat maps of the magnitude of C_ext for each sand sample as a function of both particle diameter for a range of 5–30 µm and wavelength from λ = 3–13 µm. The plots on the right show C_ext for specific values of particle diameter. Heat maps for C_sca and C_abs are included in Figure S6 (Supplemental Material). For Sands 1 and 2, C_ext appears to be similar across particle diameter and λ with subtle differences such as the magnitudes of the maxima and minima. For Sand 1, C_ext has a minimum at λ = 7.85 µm, and Sand 2 has one at λ = 7.64 µm, near the values of λ_C determined above. Both also have a sharp increase in C_ext immediately on the short wavelength side of the Christiansen feature. The origin of this peak is less obvious from the n(λ) and k(λ) data alone but arises from a sharp increase in scattering, as evident in Figure S6 (Supplemental Material). For Sand 3, C_ext is similar in general but with several differences. The minimum value is at λ = 7.56 µm, corresponding to its shorter λ_C, and it has a higher peak near λ = 9 µm corresponding to its high absorption. All of the sand samples show a secondary band of low C_ext sat λ ≈ 6.3 µm corresponding to the secondary Christiansen feature.

OPEN IN VIEWER

Figure 5.

C_ext as a function of particle diameter and λ. Plots on the left show heat maps for (a) Sand 1, (c) Sand 2, and (e) Sand 3; plots on the right show calculations for fixed particle diameters for (b) Sand 1, (d) Sand 2, and (f) Sand 3. All values for C_ext are in units of µm².

To test these predictions, FT-IR measurements for each sand sample were carried out as described above. For each sample, the measurement was repeated three times, and the results were averaged. Each plot was then normalized based on T at the shortest wavelength and the peak value near λ = 8 µm. Figure 6 shows T(λ) for each sand sample. As above, the measurement provides information on the position and relative magnitude of features in T(λ) rather than a quantitative determination of sample transmittance. The scattering model accurately predicts a Christiansen feature for each sample. The FT-IR data show this feature at slightly longer wavelengths in comparison to the model, with measured values near λ = 8 µm. The reason for this shift is unclear but may be related to the fact that the model assumes spherical particles whereas the real sand particles are highly irregular. The model correctly predicts very similar transmission spectra for Sands 1 and 2 despite the fact that they were gathered from different locations. It also accurately predicts the dip in T beyond λ = 8 µm including the fact that Sand 3 has the largest decrease. Finally, it accurately predicts higher T at the secondary Christiansen feature near λ = 6.3 µm with decreased T between the two sets of Christiansen features.

OPEN IN VIEWER

Figure 6.

T(λ) for each sand sample.

Current work focuses on improving the FT-IR characterization of sand samples. The jig for static sand samples is being replaced with a setup to allow for dynamic, falling sand. This change will help provide more precise values of λ_C and more quantitative measurements of T(λ) that can be more readily compared to predictions. Additionally, data on particle size and sand sample composition is being collected to give a fuller picture of how these factors relate to the measurement of n(λ) and k(λ) and how they relate to sand sample transmittance.