Isothermal equation of state of crystalline and glassy materials from optical measurements in diamond anvil cells

Elastic properties of materials define the structural and electronic response of the system to applied stress that strongly depends on the nature of interatomic interactions. This makes knowledge of the elastic properties of materials as a function of pressure and temperature indispensable in materials science. The equation of state (EOS) of a system defines the relationship between the thermodynamic variables, such as volume (V), pressure (P), and temperature (T), through the bulk modulus and the thermal expansion coefficients. At a constant temperature, pressure–volume relations of a solid can be described by different types of analytical EOSs^1,2 involving the isothermal bulk modulus $K0=−V⋅∂P∂V$] and its pressure derivatives (K′ = ∂K/∂P).

Investigating materials’ EOSs under pressure requires subjecting them to extreme conditions. The first studies of condensed matter under static compression in the gigapascal pressure range were done many decades ago^3,4 in large volume presses. The invention and technical development of the diamond anvil cell (DAC) technique significantly enlarged accessible thermodynamic space in high-pressure studies, and the DAC technique has evolved in the powerful and routine experimental method at in-house laboratories and synchrotron beamlines.⁵ X-ray diffraction (XRD) in DACs⁶ is the most common technique for deriving EOSs of crystalline materials through measuring the unit cell volume of a sample as a function of pressure, but it is not applicable to amorphous and glassy materials because of their topological and chemical disorder.⁷ This explains why EOSs of crystalline materials have been well studied, but so far little is known about P–V relations for non-crystalline matter (melts, metallic glasses, and other amorphous solids and nanocrystalline ceramics). These materials are currently in the focus of solid-state physics, chemistry, materials science, and geophysics research communities. In geosciences, glasses are considered as proxies of silicate melts whose properties are of great importance, as they control magmatic and volcanic activity and therefore play a central role in determining the chemical and physical evolution of the Earth throughout geologic time.^8,9 Studies of the compressional behavior, local structures, and densification mechanisms of silicate glasses at deep mantle conditions can shed light on the dynamics of the Earth’s interior, which is still insufficiently understood.¹⁰ 

Elastic properties of both crystalline and amorphous materials can be studied in situ in DACs by utilizing such methods as x-ray absorption,¹¹ Brillouin scattering (BS),¹² ultrasonic measurements (US),^6,13 impulsive stimulated scattering (ISS),¹⁴ and inelastic x-ray scattering (IXS),¹⁵ or by determining strain–stress relations using optical microscopy.^16,17 The sample density (and hence the EOS) can be obtained from high-pressure x-ray absorption measurements,^18–20 but this method is hard to apply in DACs and it works reasonably well only for materials containing heavy elements (i.e., good x-ray absorbers).

Determination of the strain–stress relations using optical microscopy^16,17 requires measurements of the dimensions of an object being observed under the high-resolution optical microscope. According to Abbe’s theory, the resolution limit of the optical system (assuming the absence of aberrations) is comparable with the size of the Airy disk.¹⁷ However, it is possible to achieve a much higher resolution by taking into account the spatial distribution of the intensity in the Airy disk itself.²¹ The technique for studying EOS through the sample length determination in DAC using an image shearing device²² was first presented by Scott and Jeanloz in 1984,¹⁷ who reported the precision of measurements of about 0.065 μm (for the samples with a linear size of around 100 μm) and validated the technique through determining the EOS of Au.¹⁷ The isothermal bulk modulus [K₀ = 156(35) GPa] determined from the optical length measurements was found to be comparable with that known from high-precision XRD studies [K₀ = 167(5) GPa],²³ and the high uncertainty was attributed to the limited pressure range of the investigations and small sizes of the samples used.¹⁷ Deriving the EOS of GeO₂ glass up to 12 GPa through the optical measurements was reported by Smith et al. in 1995.²⁴ In this work, the spacing between the lines deposited on the polished surface of the GeO₂ sample was determined using a magnified image and the precision was reported to be about 0.5 μm for the maximum line spacing of 100 μm. Such precision is significantly worse than the one reported by Scott and Jeanloz,¹⁷ but still reasonable for highly compressible glass samples.²⁴ One of the major problems of the approaches described above is the high uncertainty in the obtained sample dimensions: The observed sample length is strongly affected by the focus position of the sample, and the definition of the length relies on the subjective perception of the operator. In order to make the method reliable and accurate, the measurements have to be performed at the same focus position in each pressure point. Defining the focus point by eye is inaccurate.

In 2012, Amin et al.¹⁶ described a partially automatized algorithm for the determination of EOSs through high-resolution optical microscopy. The methodology relied on two-dimensional image acquisition and its subsequent analysis in order to quantify changes in the sample surface area. The authors applied the Canny edge detection algorithm²⁵ to define the sample boundaries and calculate its surface area. However, Amin et al.¹⁶ yielded higher experimental uncertainties if compared to the method of Scott and Jeanloz.¹⁷ The lower precision of the partially automatized measurements of Amin et al.¹⁶ could be explained by the problem with a subjective choice of the focus point or/and with applying Gaussian smoothing during the image processing with the Canny algorithm. Amin et al.¹⁶ chose samples of random shapes but applied smoothing algorithms, which blur out the corners and junctions, thus making it harder to detect their actual positions. Still, despite all obstacles, the method by Amin et al.¹⁶ performed reasonably well on several crystalline and amorphous compounds at pressures up to 12 GPa.

Here, we describe the methodology of the EOS determination based on optical studies of materials in DACs at pressures up to 30 GPa. The data analysis is fully automatized (i.e., the effect of the operator is negligible) and applicable to opaque crystalline and amorphous/glassy materials. Our methodology exploits high-resolution optical microscopy in DACs, image analysis, and statistical data treatment. A significant advantage of our approach is that experiments do not require access to synchrotron facilities or specialized x-ray sources. At the same time, our method can be easily applied in combination with x-ray imaging and diffraction.

In our experiments, we use the customized optical system based on a previously developed laser heating setup.²⁶ It is schematically shown in Fig. 1.

The image of the sample is formed by passing transmitted light through a DAC sample chamber and collected from the opposite side by a long working distance objective (Mitutoyo NIR infinity-corrected M Plan Apo NIR B 20×). To avoid the chromatic aberrations, the sample is illuminated by the monochromatic LED (Thorlabs M455L2, λ = 455 nm). The light passes through a set of lenses into a zoom objective (Nikon AF-S NIKKOR 28–300 mm f/3.5-5.6G ED VR Lens) and projects the sample image on the matrix of a high-resolution CMOS camera (EYE^© CMOS cameras, UI-3280CP, resolution 2456 × 2054 pixel², sensor size ∼8.4 × 7.0 mm², 12-bit depth). The setup provides variable magnification in the range of around 80–320 times. At maximum magnification, the size of an image projected on the camera sensor from the sample of 25 μm is around 8 mm, while a single pixel corresponds to ∼0.01 μm.

The pressure during the experiment is controlled by ruby fluorescence²⁷ using a 532 nm green laser (Laser Quantum, Inc. model gem 532) and an IsoPlane SCT 320 spectrometer equipped with a 1024 × 256 PI-MAX 4 camera (Princeton Instruments, Inc.).

In our experiments, we utilize the BX90-type large aperture DACs equipped with Boehler-Almax type diamonds (culet diameter of 250 μm). For each DAC, the sample chamber is formed by pre-indentation of a rhenium gasket to ∼30–35 μm thickness and drilling a hole of 120 μm in diameter in the center of the indentation. The pressure is measured by ruby fluorescence, and Ne is used as a pressure transmitting medium.

The geometrical shape of a sample plays an important role in data acquisition and analysis. Simple and symmetrical shapes, such as balls or rectangular plates, allow more measurements of the sample length to be made at every pressure point that improves the statistics and reduces the random error upon data analysis (this will be discussed in detail in Sec. II D).

In this study, metallic samples were prepared from titanium microspheres with a diameter of about ∼25 μm (purchased from Cospheric Microspheres, Inc.). The spheres were pressurized between two diamond anvils in a DAC to form a thin cylindrical plate with a thickness of about 5 μm. The obtained plates were given a square shape with an edge length of about 25–30 μm (Fig. 2) by milling using the FEI-QUANTA 3D Focused Ion Beam (FIB).

Adjustment of the camera and the sample illumination is the key to obtaining high-quality images and, hence, reproducible and precise measurements. Parameters such as the intensity and the angle of incidence of light, settings of the CCD camera, and the sample position within the field of view must be maintained constant during the whole set of measurements at different pressure points. To achieve stable illumination conditions during the whole experiment, the DAC was coupled with a membrane pressure controller and mounted on a three-axis motorized stage. Then, we manually adjusted the angle of incidence of light to achieve a flat field background in both horizontal and vertical planes on the entire image (Fig. 3).

Settings of the camera, which can affect the image representation [digital gain, automatic contrast/brightness/color adjustment, nonlinear LUT (Look-up table), white balance, black offset, infrared filter correction matrix], were disabled. At the last step, we adjusted the camera exposure time and the light intensity to use the complete dynamic range of the camera sensor.

If the length of an object is measured under an optical microscope, the result depends on the focal position of the object. To avoid ambiguities, the measurements should be made on the images in focus. In order to make the choice of the right image independent of the operator, we apply the focus-stacking technique by acquiring a set of optical images at different focal positions. At each pressure point, the images are collected upon continuous movement of the DAC along the optical axis of the system. First, the best focal position is determined by the eye on the sharpness of the image. Then, the DAC is moved in the range of about ±20 μm with a speed of ∼2 μm/s, and a series of sample images is taken with ∼100 ms exposure time (Fig. 4). Typically, the total number of images for each pressure point is up to 200. Although a single set data collection for one pressure point takes less than a minute, the stabilization of pressure upon the relaxation of the metallic membrane requires ∼5–10 min, and the total duration of an experiment with typically ∼20 pressure points takes ∼6 h. The algorithm for selecting the sharpest images is described in Sec. II D.

A fully automatic procedure (employing a custom Python script) has been developed for extracting the length of the sample from an acquired set of images at each pressure point. The procedure includes the analysis of image intensity profiles at the edges of the sample that results in unambiguous selections of in-focus images, whose intensity profiles are used for determining the exact positions of the sample edges (Fig. 5). This enables us to precisely measure the sample length at a given pressure point.

A digital image can be represented as a two-dimensional matrix of values of light intensities, where each element of the matrix corresponds to a pixel on the camera sensor. Considering the rows or columns of that matrix, one can make the image cross section intensity profiles (Fig. 5). The dotted line in Fig. 5(a) is the cross section line in the image of the square-shaped titanium plate. The intensity profile in Fig. 5(b) shows the intensity values taken from regularly spaced points along the line path in the image.

To describe the shape of the edges of the intensity profile, we used a parametric sigmoid function,

where x is the pixel coordinate, A_o is the lower asymptote, K is the upper asymptote, α is the growth rate (or steepness of the curve), and x₀ is the value of the sigmoids’ midpoint. Thereby, we define the sample length in pixels as a difference between the two sigmoids’ midpoints taken along a single image cross section (at the right and left edges of the sample) (⁠$x0right−x0left$⁠. Such parameterization allows defining the positions of the edges with subpixel accuracy.

If an object is not in focus, its edges look blur. It is obvious that the sharper the image, the steeper the edges of the cross section intensity profile (Fig. 6).

To select the images that are in-focus, we analyzed every image of the set and plotted the steepness parameter [i.e., the growth rate parameter α of the sigmoidal function, Eq. (1)] vs the image number, which indicates the DAC position along the optical axis. An example of such a plot is shown in Fig. 7.

Since both vertical and horizontal cross sections are used for in-focus image selection, the algorithm gives four, usually slightly different, focal positions. This can be attributed to slightly uneven illumination of different sample edges or minor inclination of the sample in the DAC. However, we found that for a set of 100–150 images (frames), the variations within up to ten frames do not introduce measurable errors in the length of the sample (see Sec. II D 4).

As said before, for each pressure point, we collected a set of up to 200 images. Four in-focus images were selected as described above. In the case of rectangular shape samples for each of in-focus images, we made a cross section mesh of 50–100 lines with a line spacing of 5–20 pixels [Fig. 8(a)]. For each line, the length of the sample was determined independently. All measurements for vertical and horizontal lines were averaged independently, and we calculated the standard deviation for both vertical and horizontal dimensions.

For spherical shape samples, at each in-focus image, a cross section mesh was built around vertical and horizontal diameters [Fig. 8(b)]. The length of the sample was determined as an average of the largest horizontal and vertical dimensions with error corresponds to the standard deviation.

Independent of the shape, the final dimensions of the sample at each pressure point were the average values obtained for all four processed images (with the final random error being the outcome of error propagation for each individual image).

Filtering of the data is required in the presence of edge defects of the samples, which can induce significant errors in the selection of in-focus images and in the measurements of the sample length. A bulb-like defect on the top edge of the square-plate sample is seen in Fig. 9. In such a case, we manually excluded the defect region from the further analysis and performed the focus position analysis along with length measurements only on defect-free regions of the sample (Fig. 9).

Figure 10 presents an example of the uncertainty in the length measurements of the titanium square plate at different pressures. The average lengths of the sample, as measured at the first and the last pressure points, were 1217.2 and 1144.7 pixels, respectively. The maximum uncertainty in the sample length was less than three pixels that correspond to the value of relative uncertainty of ∼0.25% for the sample with a real size of 30 μm.

In this work, the determination of the sample volumetric strain is based on several assumptions: (1) Sample conditions in DAC are quasi-hydrostatic, (2) the sample does not have voids in the bulk, and (3) compression of the sample is isotropic. Then, the sample strain (f_e) can be described as the Eulerian finite strain,²⁸ 

where V is the sample volume at a given pressure and V_p is the sample volume at the reference pressure, and the following relationship is true:

where L is the average length at a given pressure and L_p is the average length at the reference pressure. To determine the bulk modulus (K₀) and it’s derivative (K′), the pressure–strain data, obtained by optical measurements, were fitted to the third order Birch–Murnaghan EOS,²⁹ 

To validate the technique described above, the EOS of a material obtained on the basis of optical microscopy measurements should be compared with the known EOS of the same material, previously determined using well-established methods. Titanium is a transition metal with the EOS well studied using x-ray diffraction,³⁰ so we chose Ti as the reference material. It is known that Ti undergoes the α-to-ω structural phase transition,³¹ which takes place between 2 and 12 GPa at 300 K, depending on the pressurization conditions.³¹ It results in a volume reduction of a few percent that provides a chance to test if the first-order phase transition can be detected using optical microscopy.

The sample was prepared by flattening of a Ti microsphere (Cospheric Microspheres, Inc.) and shaped using FIB to a square plate with an edge size of about 28 μm. After loading the DAC with Ne, the pressure was found to be ∼8 GPa. The sample was then studied in the pressure range of 8–30 GPa (Fig. 11).

The images were collected and processed as described above. The results of measurements, in the form of the dependence of the relative volume on pressure, are shown in Fig. 12. Some irregularities in the compressional behavior of the sample were observed at around 11 GPa, which may be related to the α-to-ω Ti phase transition.³⁰ However, the effect is hardly visible (Fig. 12, inset), and further experiments are needed to establish how sensitive the optical microscopy may be to detect phase transitions.

The data collected above 11 GPa are related to ω–Ti. The sample was pressurized with a step of 1.5–2 GPa from 11.1(1) to 30.5(1) GPa that resulted in 24 data points (Fig. 12). At each pressure point, a set of 100–150 images was collected by focus stacking. Four in-focus images were selected. The difference in their focal positions did not exceed 2 μm. Determination of the length was performed on each of the four images by making 100 cross section intensity profiles in the horizontal and vertical direction. Therefore, 200 lengths measurements were made for each image, thus 800 measurements for four images in total. The standard deviation of the average length lays within 3 pixels that corresponds to approximately 60 nm.

Over the compression from 11.1(1) to 30.5(1) GPa, a decrease in the average sample length of 48 pixels (∼1 μm) was observed, while the sample volume contracted by about 10%. The pressure–relative volume data (Fig. 12) were fitted to the third order BM EOS with the parameters $K0=1082.6GPa$ and K′ = 3.5(3), in good agreement with those obtained from XRD data of Dewaele et al. [$K0=1073GPa$⁠, K′ = 3.55(3)].³⁰ 

Glassy carbon (GC) is one of the carbon forms, which consists of two-dimensional structural elements and does not exhibit “dangling” bonds.³² GC possesses very high hardness, high-temperature stability,^33,34 high resistance to chemical attack,³⁵ and impermeability for gases and liquids that have found a wide range of scientific and technological applications. However, the EOS of GC has never been studied.

Here, two glassy carbon spheres (microspheres type-I, purchased from Alpha Aesar, Inc.) with the initial diameters of about 12–14 μm were loaded into the sample chamber (Fig. 13) and pressurized in a Ne pressure transmitting medium up to 30 GPa.

The starting pressure (after Ne loading) was 6.9 (1) GPa. The sample volume at this pressure was chosen as the reference for further strain calculations. Like in the experiment with Ti, we performed the length measurements at each pressure step with an interval of 1.5–2 GPa (18 data points in total). A set of 100–150 images was obtained by the focus stacking at each pressure point, and four in-focus images were selected. The sample in-focus positions were found within the range of 2 μm.

For the GC-1 sphere (the smaller one), we observed a decrease in the average sample size (the diameter of the sphere) of ∼81 pixels (∼1.1 μm) with a volume contraction of about 20% upon compression from 6.9 to 30 GPa (Fig. 14). A similar compressional behavior was observed for the GC-2 sphere (the larger one) up to 23 GPa (Fig. 14), but beyond its volume stayed unchanged within the error of size measurements. Considering the larger initial diameter of the GC-2 sphere, it might be a result of its bridging between the diamond anvils; therefore, we excluded the corresponding data points from the further analysis. The fit of the experimental data to the third order BM EOS gave the following parameters: $K0=282GPa$ and K′ = 5.5(5) (Fig. 14). As expected, the compressibility of glassy carbon appeared to be relatively high. According to the obtained EOS, the total volume contraction (relative to the ambient pressure) was about 31%. Thus, the compressional behavior of glassy carbon was found to be very smooth, in agreement with the previous studies based on Raman spectroscopy,³⁶ which gave no evidence of a significant change in the type of chemical bonding in GC up to 60 GPa.

In this work, we have developed the technique to determine the EOS of opaque crystalline and amorphous solids using high-resolution optical microscopy in DACs. Our methodology is based on acquiring sample images at variable pressure and determining changes in the sample linear dimensions upon compression with a very high precision down to 60 nm. The analysis of images is automatized that grants the results to be independent of the operator.

This method was validated by studies of the EOS of ω-Ti up to 30 GPa. Our results agree well with the literature data obtained for ω-Ti on the basis of synchrotron XRD.³⁰ The EOS of glassy carbon reported in this work was determined for the first time.

The significant advantage of the presented method is that experiments do not require access to synchrotron facilities or specialized x-ray sources (but it can be easily coupled with x-ray imaging and diffraction). We are convinced that our work opens the way for wide investigations of opaque glasses, amorphous alloys, and crystalline materials at high pressures.

N.D. and L.D. thank the Federal Ministry of Education and Research, Germany (BMBF, Grant No. 05K19WC1), and the Deutsche Forschungsgemeinschaft (DFG Project Nos. DU 954-11/1, DU 393-9/2, and DU 393-13/1) for financial support. N.D. thanks the Swedish Government Strategic Research Area in Materials Science on Functional Materials at Linköping University (Faculty Grant SFO-Mat-LiU No. 2009 00971).

The data that support the findings of this study are available from the corresponding author upon reasonable request.