Strength and texture of sodium chloride to 56 GPa Free

Sodium chloride, NaCl, is an archetypal ionic compound that has been the subject of extensive study. In high-pressure research, it is often employed as a pressure-transmitting medium for achieving near hydrostatic stress conditions under load and as a thermal insulator in simultaneous high pressure–temperature experiments.¹ NaCl is chosen for these purposes due to its high compressibility, low strength, and limited chemical reactivity as well as its general ease of use. NaCl has also been widely used as a secondary pressure standard^2–9 for both ambient and high temperature experiments. NaCl crystallizes in the cubic rock-salt structure (B1, $Fm3¯m$⁠) at ambient conditions and transforms into the CsCl structure (B2, $Pm3¯m$⁠) at a pressure of 25–32 GPa and room temperature.^9–16 

Due to its importance in high-pressure research, the elastic and mechanical properties of NaCl under compression are of strong interest. The presence of finite shear strength may generate a non-hydrostatic sample environment and affect the accuracy of pressure determination at high pressures.¹⁷ Most studies of the strength, elasticity, and texture of NaCl have focused on the B1 phase only; investigations of the strength and texture of the B2 phase are limited. Furthermore, existing studies are inconsistent with regard to the effect of pressure on the strength of the B1 phase. Some reports suggest that the strength of the B1 phase reaches a plateau at 2–8 GPa and then exhibits little further increase to the maximum experimental pressures.^18–20 Alternatively, a continuous increase in strength of the NaCl B1 phase to ∼30 GPa was also reported.^21–23 Furthermore, the variation of non-hydrostatic stress conditions under compression depending on sample conditions may facilitate the formation of experiment-specific textures that can account at least partially for the observed variability of the strength in NaCl. Texture information is important for understanding the mechanical response of polycrystalline materials and in the geological context is relevant for understanding tectonic deformation and seismic anisotropy.

Despite its simple cubic structure, the mechanical behavior of polycrystalline NaCl is complex. It exhibits high ductility and strong plastic anisotropy at high pressures,²⁴ making it challenging to model the high-pressure texture evolution in NaCl. A number of mechanical studies and simulations investigating texture development in NaCl B1 phase at low pressures and low temperatures have been reported.^24–28 The most active slip system of NaCl B1 phase was suggested to be $110⟨11¯0⟩$⁠.²⁶ Under compression, the poles of NaCl rotate along the compression direction and the maximum pole density was found to adopt a bimodal distribution, {111} and {100}, based on the experimental observations and the viscoplastic self-consistent theory (VPSC) predictions.^27,29

While there has been considerable progress in understanding the low-pressure deformation behavior of the B1 phase of NaCl, knowledge of the behavior of the B2 phase is almost entirely lacking. In this study, the high-pressure strength of NaCl was examined at pressures extending into the stability field of the B2 phase. Our experiments were conducted using a diamond anvil cell together with a synchrotron X-ray diffraction in a radial geometry. The data are interpreted using lattice strain theory to provide insight into the high-pressure behavior of NaCl across the B1-B2 transition and into the B2 phase to pressures at 56 GPa. We also record texture development and evaluate the slip systems of NaCl based on the VPSC model.^29–32 

NaCl powder (99.999% purity, Alfa Aesar) was mechanically ground to grain sizes of 1–2 μm. The sample was heated in an oven at 120 °C for 8 h immediately prior to loading to eliminate moisture. The samples were compressed in a diamond anvil cell with a pair of 300-μm culet diamonds. Two different types of gaskets were used in this study—a beryllium gasket with a 100-μm diameter hole (run 1) and a boron-epoxy gasket with a 60-μm diameter hole (run 2). In addition, a flake of gold (∼20 μm) was placed on top of the sample as a pressure standard and position marker. Pressure was determined using the equation of state of gold.³³ No pressure medium was used in the experiments. In run 1, in situ high-pressure X-ray diffraction was performed at beamline X17C at the National Synchrotron Light Source, Brookhaven National Laboratory. The incident X-ray beam was focused by a pair of Kirkpatrick-Baez mirrors to a size of 25 × 25 μm. Energy-dispersive X-ray diffraction (EDXD) patterns were collected in a radial geometry up to 44 GPa. At each pressure step, X-ray diffraction data were collected at ψ angles of 0°, 24°, 35°, 45°, 55°, 66°, and 90°, where ψ is the angle between the diffraction vector and the compression axis. In run 2, radial angle-dispersive X-ray diffraction (ADXD) was conducted at beamline 12.2.2 at the Advanced Light Source, Berkeley National Laboratory. The incident X-ray beam size was focused to 30 × 30 μm. Two-dimensional X-ray diffraction images were recorded up to 56 GPa. Each diffraction pattern was collected for 10–30 min. To minimize the uncertainty of the pressure determination, the sample was allowed to relax for at least 30–120 min after each compression step. The differential stresses of NaCl were investigated in both runs 1 and 2, whereas the texture of NaCl was evaluated only in run 2.

Lattice strain theory has been described in detail elsewhere.^34–40 The stress state of the sample in a diamond anvil cell can be expressed as a sum of the hydrostatic and differential stress components

where σ₁ and σ₃ are the stresses along the radial and axial directions, respectively, σ_p = (σ₁ + σ₁ + σ₃)/3 is the corresponding hydrostatic stress, D_ij is the deviatoric stress tensor, and t is the differential stress, which can be interpreted via the Von Mises yield criterion as a lower bound on the material's yield strength. Thus, the differential stress, t, can be expressed as

where τ is the shear strength and Y is the yield strength of the sample.

The measured lattice d-spacings, d_m, can be expressed as a function of ψ, the angle between the diamond cell loading axis and the normal to the corresponding diffracting plane

where d_p(hkl) is the d-spacing from the hydrostatic component of stress, and Q (hkl) is defined as

G_R (hkl) is the shear modulus for the crystal under the condition of constant stress across grain boundary (Reuss condition, α = 1). G_V is the shear modulus under iso-strain condition (Voigt limit, α = 0). The parameter α describes the relative variation of stress and strain across the grain boundaries.

The measurement of the linear relationship between d_m and 1–3cos²ψ from Eq. (3) allows one to obtain Q(hkl) and the hydrostatic d-spacing, d_p. The differential stress, t, can then be constrained using^34,35

where $⟨$Q(hkl)⟩ represents the average value over all observed reflections of Q(hkl) and G is the aggregate shear modulus for the polycrystalline sample. $f(x,α)$ is defined as

where symbol $⟨$⟩ denotes the average over all crystallographic reflections, and C_ij and S_ij are the single-crystal elastic moduli and compliances, respectively. In the cubic system, $⟨$Г(hkl)⟩ takes on the value of 1/5.^35,36

For texture investigation, diffraction patterns were analyzed using Rietveld refinement techniques as implemented in Material Analysis Using Diffraction (MAUD) program.^41,42 Debye rings were unrolled using the Fit2D software,^43,44 and diffraction intensity was integrated at a 5° increments. A moment pole stress model⁴³ was applied to calculate principal stresses from variations in peak positions. The elastic constants of NaCl reported by the Brillouin scattering measurements at high pressure were used for shear moduli inputs⁴⁵ in B1 phase. For the B2 cesium chloride phase, the elastic constants were computed using ab initio methods as described below. In the quantitative texture analysis, the orientation and pole densities are normalized and the observed densities are compared to those of a hypothetical sample with a random orientation distribution. The units are multiples of a random distribution (m.r.d.) with 1 m.r.d. indicating a fully random orientation. VPSC modeling³² was used to infer slip system activities of the NaCl B1 and B2 phases.

We augmented our experiments with 3-D periodic density-functional-theory (DFT) computations for the B1 and B2 phases using the Vienna ab-initio simulation package (VASP) software package.^46,47 Ion-electron interactions were described within the all-electron like projector augmented wave (PAW) formalism.^48,49 Electronic exchange and correlation effects were described within generalized gradient approximation (GGA).⁵⁰ Reference valence electron configurations for Na (3s¹; R_core = 2.2 a_B; a_B = Bohr radius) and Cl (3s²3p⁵; R_core = 1.9 a_B). All computations were performed using a common plane wave cutoff energy of E_cut = 600 eV and Monkhorst-Pack k-point grids⁵¹ of 4 × 4 × 4 for the B1 phase (conventional unit cell) and 8 × 8 × 8 for the B2 phase. Using these computational parameters, the B1 and B2 phases of NaCl were relaxed at pressures up to 50 GPa. In these static computations, the (cubic) cell parameter was relaxed until computed and target pressure deviated by less than 0.02 GPa and enthalpies were converged to better than 1 meV/atom. The three independent elastic constants of the B1 and B2 phases were determined in two steps: After the structures were optimized at fixed pressure values, the three independent elastic constants C₁₁, C₁₂, and C₄₄ of the NaCl B1 and B2 phases were determined by applying small positive and negative strains of magnitude 1%, following previous work.⁵² C₁₁ and C₁₂ were obtained from the deviatoric stress response to a strain tensor with ε₁₁ not equal to zero, and C₄₄ was obtained with ε₁₂ = ε₂₁ not equal zero (components of the strain tensor not listed were zero).

A representative unrolled diffraction image of NaCl B2 phase at 56 GPa from the ADXD is shown in Fig. 1. The calculated diffraction images obtained using MAUD program are in good agreement with the experimental observations. Sinusoidal variations in peak position as a function of ψ are attributed to lattice strains. Plots of d-spacings as a function of 1–3cos²ψ show a linear relation for all diffraction planes in both B1 and B2 phases. Figure 2 shows the d-spacings of B2 (100), (110), (200), and (221) as a function of 1–3cos²ψ at 32, 44, 51, and 56 GPa, respectively. The hydrostatic d-spacing, d_p(hkl), can be estimated from the intercept at 1–3cos²ψ = 0 or ψ = 54.7°. The corresponding hydrostatic pressure was determined from unit cell refinement of gold³² using Au 111 diffraction lines. The equations of state of the NaCl B1 and B2 phase were obtained from the previous studies.^2,33 The pressure difference between gold and NaCl was observed to be <2.5 GPa for the B1 phase and <1.5 GPa for the B2 phase.

Q(hkl) was directly obtained from the slope of the d-spacing as a function of 1–3cos²ψ and ⟨Q(hkl)⟩ represents the average of all the observed Q(hkl) for each pressure. In Fig. 3, ⟨Q(hkl)⟩ increases gradually to a value of about 0.0017 at 30 GPa and then drops sharply to 0.0006. This is followed by a steeper increase to 0.0035 at 56 GPa. For the NaCl B1 phase, the gradual increase in ⟨Q(hkl)⟩ with pressure implies an increase in differential stress in the sample that is almost equivalent to the rate of increase in the shear modulus with compression. That is, the ratio of t/G remains nearly constant under compression for this material. At about 30 GPa, a dramatic weakening is observed associated with the phase transition from the B1 to B2 phase. Across this transition, the cation coordination number increases from 6 to 8 accompanied by lengthening of Na-Cl bonds in the B2 phase. Above 30 GPa, the increase in the differential stress in the B2 phase exceeds the rate of increase in the shear modulus as shown in Fig. 3. Nevertheless, at pressure greater than 44 GPa, the increase in the ⟨Q(hkl)⟩ becomes monotonic, suggesting that the differential stress increases with a similar rate as the shear modulus of the B2 phase.

Based on Eq. (5), the differential stress can be evaluated under the assumptions of α = 0, 0.5, and 1, which corresponds to Voigt, Hill, and Reuss limits, respectively. The aggregate shear modulus and elastic constants required for this calculation were taken from Brillouin measurements⁴³ and our first-principles DFT computations for the B1 and B2 phases, respectively (Table I and Fig. 4). Our calculated elastic constants are in excellent agreement with the available experimental values and with the previous computations for the B1 phase (Fig. 4). The differential stresses supported by NaCl obtained from this study are compared to the previous reports^18–23 in Fig. 5 where the solid symbols denote the differential stresses of NaCl under the assumption of α = 0.5, whereas the upper and lower bounds (α = 0 and 1) are shown as solid curves. Based on Eqs. (6)–(10), when α is assumed as 0.5, the quantity f(⁠$x$, α) would be equal to 1.^35,36 The aggregate shear modulus was calculated from individual Cijs⁴⁵ using the Voigt-Reuss-Hill average.^53,54 The differential stress supported by the NaCl B1 phase increases weakly up to ∼28 GPa and then begins to decrease near 30 GPa. Specifically, the measured B1 stresses increase from 0.22(6) to 0.36(6) GPa at 29 GPa, just before the phase transition. In our studies, the B1 and B2 phase coexisted pressure ranges are from 28 to 32 GPa. At the transition, the differential stress decreases abruptly to 0.002(6) GPa. Above the phase transition, the differential stress supported by the B2 phase increases more strongly and eventually reaches 1.8(6) GPa at our maximum pressure of 56 GPa. It is important to note that the differential stress data demonstrate significant strength weakening across the B1-B2 phase transition. In fact, the strength weakening was also observed in the previous studies such as SiO₂ and Ta.^55,56 The volume change associated with the phase transition may relieve the differential stress and hence produces near hydrostatic conditions near the boundary.

The differential stress, t, of the B1 and B2 phases under assumption of α = 0.5 can be expressed by t_B1 = 0.0036P+ 0.2483 (at pressure to about 29 GPa) and t_B2 = 0.0673P − 1.8815, where t and P are both in GPa. The lower bound lines [Fig. 5(a)] are under assumption that α = 1, which corresponds to stress continuity (Reuss) conditions. In this case, the differential stress almost remains constant and reaches a maximum value of 0.20 GPa at 25 GPa for the B1 phase. The upper limits shown in Fig. 5(a) are the differential stresses in the Voigt limit (α = 0) reaching a maximum value of 0.8 GPa for the B1 phase. As pressure increases, the difference between lower and upper bound of differential stress becomes greater in part due to the increase in the f(⁠$x$⁠, α) value.^35,36

The differential stresses of the B1 phase at α = 0.5 measured here are generally lower than the previous reports.^18–23 The lower values may be attributed to one or more of the following: (1) different experimental setups—earlier studies used a Bridgman-anvil type of device,¹⁸ a modified Drickamer-type apparatus,²¹ or a large volume press;¹⁹ (2) different starting materials—a mixture of NaCl and MgO was investigated previously;^19,20 the presence of MgO could have influenced the strength behavior of NaCl at high pressures; (3) different techniques—Meade and Jeanloz²² used the pressure gradient method which may overestimate the strength.⁵⁷ It is important to point out that the differential stresses of NaCl B1 at the Voigt limit (α = 0) are close to several previous reports, indicating the potential variations of local stress environments in different experiments.

A large number of materials including various oxides, inert gases, etc., have been used in past years as insulating and pressure-transmitting media in diamond anvil cell and other high-pressure experiments.^58–63 A comparison of NaCl to several of these materials is shown in Fig. 5(b). Below 40 GPa, NaCl appears as a good pressure transmitting medium due to its differential stress is lower than or similar to other materials including even the rare gas solids (Ar, Ne, and He). Above 40 GPa, the rapid rise in the strength of the B2 phase implies that NaCl-B2 is less effective as a medium in the high-pressure region than some of the rare gas solids (He and Ne).

Figure 6 shows inverse pole figures of the B1 and B2 phases at different pressures. The B1 phase exhibits a weak texture at 10 GPa, with the poles to (100) planes tending to be parallel to the compression direction (Fig. 6). This texture is enhanced with compression as the maximum pole density reaches 2.17 m.r.d. at 30 GPa. After the phase transition, the B2 phase developed a maximum texture at poles to (110), reaching a maximum pole density of 2.25 m.r.d. at 56 GPa. A secondary (211) texture slowly develops over the examined pressure range. Molecular dynamic simulations of the B1 to B2 phase transition established an orientation relation, whereby the (100)_B1 and (110)_B1 planes parallel the (110)_B2 and (100)_B2 planes, respectively.⁶⁴ This means that the (100) planes in NaCl B1 phase become equivalent to the (110) planes in B2 phase. Therefore, the maximum density observed in B1 (100) and B2 (110) suggests a continuity of texture across the transition. In all cases, NaCl has relatively low pole density values, suggesting weak textural development with pressure.

In order to evaluate the slip system activities in the NaCl B1 and B2 phases under high pressure, we compare the experimental textures with those simulated from VPSC.^29–32 Plausible slip systems were identified by applying different critical resolved shear stresses (CRSSs) to possible slip systems, and comparing the VPSC results with the experimentally obtained textures of the B1 and B2 phases. Both B1 and B2 phases were individually simulated with six different models [Table II and Fig. 7(a)]. For the B1 phase, simulations were performed with 2000 grains compressed to 40% equivalent strain in 1% increments. The simulations for the B2 phase were performed incrementally in 30 steps to a total of 15% strain. A stress exponent of n = 8 was assumed in both cases as suggested in a previous study.²⁴ Our simulated texture results show that model 1 and model 7 are in good agreement with the experimental observation for the B1 and B2 phases, respectively. In this case, deformation is strongly dominated by the {110}⟨$11¯0⟩$ slip system (99% of the total plastic activity) for the B1 phase and {112}⟨$11¯0⟩$ slip system (100% of the total plastic activity) for the B2 phase. The results are also in agreement with the previous studies^25,26,30 on the NaCl B1 phase at room pressure and/or high temperature.

The B1-B2 phase transition occurs in many types of AB compounds. In magnesium oxide, an important component within terrestrial planet interiors, the B1-B2 transition has been observed experimentally under dynamic laser-ramp compression at ultrahigh-pressures near 600 GPa.⁶⁵ It has been suggested that this transition in MgO may affect the rheology and lower the viscosity in the deep mantles of large super-Earth extra-solar planets with potentially significant effects on their mantle dynamics. In Karato's analysis,⁶⁶ low-pressure, high-temperature creep data for alkali halides were used to infer that B2 structured materials are softer than their B1 counterparts. However, this was based on a comparison of NaCl in the B1 structure with CsCl in the B2 structure, that is, the materials differed in both structure and composition. Our study enables us to directly compare NaCl in both the B1 and B2 structures. At low-temperature, high-pressure conditions, we find instead that the B2 phase exhibits a stronger pressure dependence of strength than the B1 phase. Further studies to develop a better systematic understanding of the strength behavior of B1 and B2 structured materials over a wide range of conditions are warranted. Until then, any application of relatively low P-T data on alkali halides to behavior of oxides under the ultrahigh P-T conditions of large extra-solar planets may be premature.

Radial X-ray diffraction experiments combined with lattice strain theory for strength and texture determination on NaCl were conducted at pressure of 56 GPa. The marked decrease in $⟨$Q(hkl)$⟩$ across the B1 to B2 transition boundary indicating strong stress relaxation accompanies the phase transition as a result of the associated volume reduction. In general, the differential stresses show a weak increase with pressure in the low-pressure phase, confirming that NaCl-B1 is a good pressure-transmitting medium below 30 GPa. The B2 phase exhibits a stronger increase in strength with compression compared with B1 and sustains higher differential stresses but, however, it may still be useful as a pressure transmitting medium. Our differential stress results also show that the previous studies are consistent with α = 0.5–1. NaCl B1 and B2 phases exhibit texture with the poles to (100) and (110), respectively, along compression direction. This is consistent with the orientation relations showing that the {100} planes in NaCl B1 phase become equivalent to the {110} planes in B2 phase. The VPSC simulation suggests a {110}⟨$11¯0⟩$ slip system for B1 phase. For the B2 phase, the simulation of 15% strain with 1 GPa CRSS on {112}⟨$11¯0⟩$ gives a texture profile consistent with the experimental results.

We are grateful for experimental assistance from Zhiqiang Chen and Susannah Dorfman at X17C, National Synchrotron Light Source and Nobumichi Tamura, Jinyuan Yan, and Martin Kunz at 12.2.2 and 12.3.2, Advanced Light Source. We thank Jay Bass for providing access to Brillouin Scattering data of NaCl B1 phase. We appreciate the constructive comments from Haini Dong and Eloisa Zepeda-Alarcon. This work was financially supported by NSERC and NSF and support to A.K. from the Carnegie/DOE alliance center NNSA Grant No. DE-NA-00006. This research was also partially supported by COMPRES, the Consortium for Materials Properties Research in Earth Sciences under NSF Cooperative Agreement No. EAR 1606856.