# Supplementary Material to Single-crystal elastic moduli, anisotropy and the B1-B2 phase transition of NaCl at high pressures

Feng Xu, Laurent Belliard, Chenhui Li, Philippe Djemia, Loic Becerra, Haijun Huang, Bernard Perrin, Andreas Zerr

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Feng Xu, Laurent Belliard, Chenhui Li, Philippe Djemia, Loic Becerra, et al., Supplementary Material to Single-crystal elastic moduli, anisotropy and the B1-B2 phase transition of NaCl at high pressures. Applied Physics Letters, 2024, 10.60893/figshare.apl.27174300.v1. hal-05377740

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<figure><img src="image_1.png" /></figure>
**Supplementary Material:**

"Single-crystal elastic moduli, anisotropy and the B1-B2 phase transition of NaCl at high pressures"

Feng Xu,<fnref n="1" /> Laurent Belliard,<fnref n="2" /> Chenhui Li,<fnref n="3" />,<fnref n="4" /> Philippe Djemia,<fnref n="3" /> Loïc Becerra,<fnref n="2" /> Haijun Huang,<fnref n="1" /> Bernard Perrin<fnref n="2" /> and Andreas Zerr<fnref n="3" />

Table of contents

1. Sample loading procedure and its polycrystallinity:

2. Estimation of pressure gradient and choice of the ruby-fluorescence pressure scale

3. Pressure range of the B1-B2 phase transition of NaCl

4. Orientation dependence of sound velocities in cubic crystals of NaCl-B1 and NaCl-B2

**1. Sample loading procedure and its polycrystallinity:**

Prior the loading in a DAC, the NaCl grains were dried in a vacuum oven at 400 °C over at least one night to evacuate any water. Then, the whole loading process was done by the operator wearing gloves and mask in order to prevent the sample hydrolysis during the loading and atmosphere in the preparation room was dry. Moreover, the loading was performed quickly which further excluded the potential sample hydrolysis. This loading procedure guaranteed purity of our NaCl samples.

In particular, we placed several grains of NaCl in the sample chamber which didn't fill its volume completely due to their regular cuboid-like shape and were pressed/crushed by squeezing between the anvils. Typically, the chamber could not be filled in one step and other grains were repeatedly added in the chamber (typically three or four times) until the chamber was filled and the excess of the material on the edges was removed in order to obtain a complete sample transparency. Consequently, our initial NaCl-B1 sample was a polycrystal, which was further plastically deformed upon compression, and contained small grains arbitrary orientated with respect to the DAC axis or any other direction. This followed from the 2D XRD patterns of similarly prepared samples recovered after measurements at much lower pressures and quasi-hydrostatic conditions. This could be concluded because the 2D XRD patterns showed continuous X-ray diffraction rings. Grain sizes required for such continuous XRD rings were estimated in an earlier work to be around 450 nm [1]. Also, it was reported that texturing of NaCl-B1 upon compression without any pressure medium is very weak [2]. Further support that our samples were polycrystals containing randomly oriented grains followed from the excellent agreement of our measured $V_{Lmax}$ and $V_{Lmin}$ of the B1 phase with the earlier lower pressure measurements (see Figure 2 in the main text) and with the BLS

<footnote n="1">¹School of Physics and Mechanics, Wuhan University of Technology, 430070 Wuhan, China</footnote>

<footnote n="2">²Institut des NanoSciences de Paris, CNRS UMR 7588, Sorbonne Université, 75005 Paris, France</footnote>

<footnote n="3">³Laboratoire des Sciences des Procédés et des Matériaux, CNRS UPR 3407, Université Sorbonne Paris Nord, Alliance Sorbonne-Paris-Cité, 93430 Villetaneuse, France</footnote>

<footnote n="4">⁴State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, 100083 Beijing, China</footnote>
result obtained for single crystals of NaCl-B1 at P=26 GPa [3]. Furthermore, even if orientations of grains of the B2 phase are partially controlled by orientation of the parent B1 grains (which is less probable for a diffusional phase transition), orientations of grains in the sample containing the B2 phase only had to be random if the parent grains of the B1 phase were randomly oriented. For this reason, application of the envelope method for a large number of independent spatially (in 3D) resolved V<sub>L</sub> measurements provided, in our work, reliable values for V<sub>L(100)</sub> and V<sub>L(111)</sub> for both B1 and B2 phases of NaCl.

## 2. Estimation of pressure gradient and choice of the ruby-fluorescence pressure scale

Applying a model of pressure gradients in a sample compressed in a DAC (see the classical book [4]), we calculated the upper limit of the pressure differences, δP, between locations of our TDBS measurements and the ruby grains used for the pressure measurements: δP = σ<sub>y</sub> d/h, where d≤30 µm is the maximal distance between locations of the TDBS measurements and the ruby grains, h≥14 µm is thickness of the whole sample (thickness of the NaCl layer was ~7 µm, at least, while that of the transducers ~7 µm [5]). Using σ<sub>y</sub>~0.5 GPa, yield stress of NaCl at the highest pressure of our measurements [2], we obtained δP≤1.1 GPa or ≤2.7% of the pressure value.

In should be noted that purely hydrostatic conditions exist at room temperature only to pressures around 10 GPa, above which all frequently used pressure transmitting fluids solidify thus leading to nonhydrostatic loading conditions, as is the case for any solid [6]. Thus, hydrostatic conditions are not accessible above this pressure not to mention at much higher pressures such as that achieved in our work, namely 41 GPa. As mentioned above, we used pressure dependence of the ruby fluorescence line and estimated the pressure difference between the used ruby grains and location of our TDBS measurements, in the most extreme case, to be ≤1.1 GPa. This quasi-hydrostatic condition is actually as good as that when solid argon or neon is used as a pressure transmitting medium (PTM) [7,8], and even better than commonly used PTM such as silicon oil and methanol-ethanol mixture [6]. The suitability of NaCl as a quasi-hydrostatic PTM follows from its low yield strength, σ<sub>y</sub>, under compression, which is very similar to that of the softest PTM (solid argon, neon or helium) within the pressure range of our experiments [2,7,8]. At 40 GPa, σ<sub>y</sub> of NaCl-B2 is just ~10% higher than that of neon, and even smaller than that of argon [2]. It has been also reported that NaCl is as soft as solid helium even at 304 GPa [9]. Consequently, the pressure in this work was determined using the quasi-hydrostatic scale proposed in [10] which provided slightly higher pressures when compared with that developed by [11]: At the maximal pressure of our measurements, 41 GPa according to the scale of [10], the scale of [11] would provide 40.0 GPa. This difference is below the uncertainty due to the pressure gradient estimated above. For completeness, the non-hydrostatic ruby fluorescence calibration of [12] would provide, at the maximal pressure of our work, P=39 GPa. It should be also emphasized, that pressure gradients, defining pressure uncertainty for samples compressed in a DAC, depend not only on σ<sub>y</sub> of the PTM but also on the sample thickness (see above).

## 3. Pressure range of coexistence of the B1 and B2 phases of NaCl
The pressure-induced B1-B2 phase transition in NaCl has already been studied using XRD techniques in several works which, however, can be subdivided in a group of the rather outdated measurements based on in-house X-ray sources with large X-ray spots covering almost whole sample area, and the more recent measurements based on high-resolution synchrotron XRD measurements. Basset et al. were the first who reported observation of this phase transition occurring rapidly and reversibly at 30 GPa, suggesting that the transition hysteresis is very narrow [13]. However, some subsequent XRD studies revealed that the B1 and B2 NaCl phases could coexist in a certain pressure region: Piermarini et al. observed the presence of the both phases at 29.1 GPa [14] and stated a rapid and reversible phase transition. Liu et al. observed the B1 phase from 30 to 31.1-33.4 GPa but the pressure was not determined reliably [15]. Heinz et al. stated that the B1 phase was not detectable above 32 GPa [16]. An anomalous result reported [17] who stated not only a very high pressure of disappearance of the B1 phase on compression of 37 GPa but also the outranging pressure of the transition onset of 34 GPa which indicates a systematic error in the pressure measurement of an unknown origin. In our opinion, the above publications are less reliable because large X-ray spots and, accordingly, illuminated sample areas could mimic, in the presence of lateral pressure gradients, extended pressure regions where the B1 and B2 phases of NaCl coexist. As above, we would like emphasize that pressure gradients depend not only on the yield strength of the sample or PTM but also on the sample or PTM thickness. Unfortunately, even the initial sample thickness is rarely stated in high-pressure publications, in contrast to our work.

In our opinion, more recent publications based on the X-ray synchrotron radiation [2,18] provided reliable information on the extension of the two-phase region due to small sizes of the probing X-ray spots. Ono et al. didn't observe the B1 phase at $P \ge 32$ GPa [18] which agrees with the most recent XRD measurements of Mi et al. who reported co-existence of the two phases between 28 and 32 GPa [2]. Basing on the synchrotron data, we consider as evident that above 32 GPa only the B2 phase was present in our measurements, while at pressures below 28 GPa our sample contained the B1 phase only. Therefore, we fixed the coexistence pressure range to 28-32 GPa.

In order to answer potential concerns due to the anomalous report of Sato-Sorensen et al. stating that the B1 phase could be present above 32 GPa, we also treated our data for the B2 phase in a shorter pressure range between 36-41 GPa. Figure S1 shows that the 2^nd^ order polynomial fits of the extreme $f_B$ data points in the latter pressure region is almost identical to that for the pressure ranges from 32 to 41 GPa. This indicates that the following processing of $V_L$ values and elastic moduli of the B1 phase in the pressure range from 32 to 41 GPa is reliable.
<figure><img src="image_2.png" /><figcaption>FIG. S1. Second order polynomial fits of the extreme $f_B$ data points for the B2 phase. Black lines and red lines are fits for the pressure ranges between 32-41 GPa and between 36-41 GPa, respectively.</figcaption></figure>

4. Orientation dependence of sound velocities in cubic crystals of NaCl-B1 and NaCl-B2

Dependence of sound velocity on direction of propagation in a single crystal follows from the solution of the Christoffel equation for a particular set of single crystal elastic moduli, also at high pressures, $C_{ij}(P)$. A general solution for longitudinal sound velocities, $V_L$, in single crystals of any symmetry and a more detailed consideration for cubic crystals can be found, for example, in the book of Zhang & Chen [19]. If the $C_{ij}(P)$ values are known, then the 3D distribution of $V_L$ in a single crystal can be visualized using an open-source code "AnisoVis" [20] providing solutions of the Christoffel equation.

It was shown in the literature that extremes of Young's modulus (max and min or inverse) in any cubic crystal correspond to the crystallographic directions (111) and (100) [21-23]. In most cases, this is also the case for the $V_L$ in cubic crystals [8,24,25]. All published DFT calculations of $C_{ij}(P)$ of NaCl-B1 and NaCl-B2 [2,26-28] confirm the above statements: They all show that (100) is the fastest and (111) is the slowest directions of propagation of the longitudinal acoustic wave in single crystals of the two NaCl phases at any pressure. This follows from the calculated directional distributions of $V_L$ in both phases using the $C_{ij}(P)$ predicted in the four works [2,26-28], see figures in Table S2 at the end of this section. Thus, consensus of all DFT calculations based on various functionals further supports our finding that $A<1$ holds for both B1 and B2 phases of NaCl and, accordingly, longitudinal sound waves propagate at the fastest along (100) and at the slowest along (111).

In order to meet potential interests of the readers, we summarize here some relevant mathematical formulas for elastic moduli and sound velocities in cubic single crystals including: single-crystal elasticity matrix, longitudinal and transverse sound velocities along the main crystallographic directions, as well as their relations with the aggregate elastic
moduli.

For cubic crystals, the elasticity matrix has a simple form:

$$ \begin{bmatrix} C_{11} & C_{12} & C_{12} \\ C_{12} & C_{11} & C_{12} \\ C_{12} & C_{12} & C_{11} \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \begin{bmatrix} C_{44} & 0 & 0 \\ 0 & C_{44} & 0 \\ 0 & 0 & C_{44} \end{bmatrix} \quad (S1) $$

Deformation of a cubic crystal can be described using the moduli $C_{11}$, $C_{12}$ and $C_{44}$ or other three independent moduli such as bulk modulus $B$, and two principal shear moduli $C_{44}$ and $C'=(C_{11}-C_{12})/2$. The former shear modulus stands for the simple shear along one of the symmetric directions, and the latter represents the tetragonal shear deformation.

In general, acoustic sound velocity along an arbitrary direction in a single crystal is expressed by:

$$ V = \sqrt{\frac{C_{eff}}{\rho}} \quad (S2) $$

where $C_{eff}$ is the effective elastic modulus depending on the acoustic wave type and propagation direction $\langle hkl \rangle$. In the case of a cubic crystal, we have the expressions shown in Table S1:

<table><caption>**Table S1:** $C_{eff}$ along specific orientations in a cubic crystal, and for different types of waves.</caption><thead><tr><th>Wave type</th><th>⟨100⟩</th><th>⟨110⟩</th><th>⟨111⟩</th></tr></thead><tbody><tr><td>Longitudinal</td><td>C<sub>11</sub></td><td>(C<sub>11</sub>+C<sub>12</sub>+2C<sub>44</sub>)/2</td><td>(C<sub>11</sub>+2C<sub>12</sub>+4C<sub>44</sub>)/3</td></tr><tr><td>Fast transverse</td><td>C<sub>44</sub></td><td>C<sub>44</sub></td><td>(C<sub>11</sub>-C<sub>12</sub>+C<sub>44</sub>)/3</td></tr><tr><td>Slow transverse</td><td>C<sub>44</sub></td><td>(C<sub>11</sub>-C<sub>12</sub>)/2</td><td>(C<sub>11</sub>-C<sub>12</sub>+C<sub>44</sub>)/3</td></tr></tbody></table>

For polycrystalline aggregates, their elastic moduli can be expressed through the combinations of the $C_{ij}$ values. The aggregate bulk modulus $B$ of a cubic solid is given by:

$$ B = \frac{C_{11}+2C_{12}}{3} \quad (S3) $$

The aggregate shear modulus $G$ is usually calculated using the Voigt-Reuss-Hill approximation, where the Voigt and Reuss approximations represent the extremes of $G$:

$$ G_{Hill} = \frac{1}{2} (G_{Voigt} + G_{Reuss}) \quad (S4) $$

For cubic solids, the Hill approximation leads to:

$$ G_{Hill} = \frac{1}{2} \left( \frac{C_{11}-C_{12}+3C_{44}}{5} + \frac{5(C_{11}-C_{12})C_{44}}{4C_{44}+3(C_{11}-C_{12})} \right) \quad (S5) $$

The corresponding aggregate longitudinal and transverse sound velocities, $V_{L(agg)}$ and $V_{T(agg)}$ respectively, are
$$V_{\text{L(agg)}} = \sqrt{\frac{3B+4G}{3\rho}} \qquad (\text{S6})$$

$$V_{\text{T(agg)}} = \sqrt{\frac{G}{\rho}} \qquad (\text{S7})$$
<table><caption>**Table S2:** 3D distributions of $V_L$ in both NaCl phases calculated using the code "AnisoVis" and $C_{ij}(P)$ predicted by different DTF calculations. In each figure, "a", "b", and "c" indicate the directions [100], [010] and [001], respectively.</caption><thead><tr><th rowspan="2">C<sub>ij</sub>(P) by<br>DFT</th><th colspan="2">3D distribution of V<sub>L</sub> by AnisoVis</th></tr><tr><th>@10.5 GPa</th><th>@25.5 GPa</th></tr></thead><tbody><tr><td>[Liu_2010]<br>B1 phase by<br>GGA-PBE</td><td><img src="image_1.png" alt="P-wave phase velocity, km s<sup>-1</sup><br>min=5.49, max=7.53, VRH<sub>average</sub> = 6.68"></td><td><img src="image_2.png" alt="P-wave phase velocity, km s<sup>-1</sup><br>min=6.51, max=9.46, VRH<sub>average</sub> = 8.29"></td></tr><tr><td>[Liu_2010]<br>B2 phase by<br>GGA-PBE</td><td><img src="image_3.png" alt="P-wave phase velocity, km s<sup>-1</sup><br>min=7.79, max=8.87, VRH<sub>average</sub> = 8.36"></td><td><img src="image_4.png" alt="P-wave phase velocity, km s<sup>-1</sup><br>min=8.11, max=9.12, VRH<sub>average</sub> = 8.64"></td></tr><tr><td>[Mi_2018]<br>B1 phase by<br>GGA</td><td><img src="image_5.png" alt="P-wave phase velocity, km s<sup>-1</sup><br>min=5.60, max=7.35, VRH<sub>average</sub> = 6.60"></td><td><img src="image_6.png" alt="P-wave phase velocity, km s<sup>-1</sup><br>min=6.62, max=9.36, VRH<sub>average</sub> = 8.25"></td></tr><tr><td>[Mi_2018]</td><td>@35 GPa</td><td>@40 GPa</td></tr></tbody></table>
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