Single Crystal Elastic Properties of Hemimorphite, a Novel Hydrous Silicate

: Hemimorphite, with the chemical formula Zn 4 Si 2 O 7 (OH) 2 · H 2 O, contains two di ﬀ erent types of structurally bound hydrogen: molecular water and hydroxyl. The elastic properties of single-crystal hemimorphite have been determined by Brillouin spectroscopy at ambient conditions, yielding tight constraints on all nine single-crystal elastic moduli (C ij ). The Voigt–Reuss–Hill (VRH) averaged isotropic aggregate elastic moduli are K S (VRH) = 74(3) GPa and µ (VRH) = 27(2) GPa, for the adiabatic bulk modulus and shear modulus, respectively. The average of the Hashin–Shtrickman (HS) bounds are Ks (HS) = 74.2(7) GPa and and µ (HS) = 26.5(6) GPa. Hemimorphite displays a high degree of velocity anisotropy. As a result, di ﬀ erences between upper and lower bounds on aggregate properties are large and the main source of uncertainty in Ks and µ . The HS average P wave velocity is V P = 5.61(4) km / s, and the HS S-wave velocity is V S = 2.77(3) km / s. The high degree of elastic anisotropy among the on-diagonal longitudinal and pure shear moduli of hemimorphite are largely explained by its distinctive crystal structure.


Introduction
The influence of hydrogen, or "water", on the elastic properties of minerals is a topic of great current interest for possibly identifying hydrous phases in the deep crust and mantle from seismic models for these regions. Adding hydrogen bonds into crystal structures is widely assumed to decrease the density, and elastic moduli of silicate minerals by increasing their volume, at least at ambient conditions. However, the effect of water on the elastic properties of minerals can depend greatly on details of the crystal structure into which hydrogen is incorporated, and on whether hydrogen is present in the form of hydroxyl OH − or as molecular water. Hydration of wadsleyite, ringwoodite and garnet can change elastic moduli significantly [1,2], in contrast to the olivine-chondrodite group [3], which varies less, with a similar water content. These differences are considered to be caused by different incorporation mechanisms of hydrogen into their structures. To investigate the influence of water on the direction-dependent compressibility, or elastic anisotropy, of minerals, it is necessary to measure the single-crystal elastic moduli matrix, C ij . Hemimorphite, Zn 4 Si 2 O 7 (OH) 2 ·H 2 O, a hydrated alteration product of willemite, Zn 2 SiO 4 , is interesting, in that it contains both molecular H 2 O and OH − hydroxyl bonded into its structure. In nature hemimorphite is an economic ore mineral in supergene non-sulfide zinc deposits. Some significant deposits are dominated by hemimorphite, such as Shaimerden supergene deposit in Kazakhstan, the Cho Dien district in Vietnam, and the Skorpion deposit in Namibia [4].
The content of structurally bound H 2 O of hemimorphite is 7.18 wt% [5]. It has two phase transitions below room temperature related to ordered versus dynamically disordered H 2 O and OH − in the crystal structure at 20 and 90 K, respectively [6]. At elevated temperatures, hemimorphite dehydrates in two stages. It loses molecular H 2 O without breakdown of the structure at 550 • C, then undergoes a phase transition to β-Zn 2 SiO 4 , involving release of the OH − group between 725-760 • C. If heated above 960 • C, it transforms into willemite (α-Zn 2 SiO 4 ) [7].
The structure of orthorhombic hemimorphite, with space group Imm2, is composed of a three-dimensional framework, in which two ZnO 3 (OH) tetrahedra and one SiO 4 tetrahedron form three-membered rings that are corner-shared to form corrugated sheets parallel to (010) [8,9] (Figure 1). SiO 4 tetrahedra in one layer are linked to SiO 4 tetrahedra in an adjacent layer, while the ZnO 3 (OH) tetrahedra in adjacent layers also only link to each other, completing the framework. Large open channels run parallel to [001], giving hemimorphite a structural similarity to zeolites. The molecular water, orientated in the (010) plane, is near the center of large cavities of an eight-membered ring and bonded to hydroxyl groups of the ZnO 3 (OH) tetrahedra by hydrogen bonds. The hydrogen bonds formed by H of hydroxyl group (H35) are stronger than those formed by H of molecular water (H53) [8]. The presence of both hydroxyl and molecular water, corrugated sheets in a tetrahedral framework, and large zeolitic cavities, make hemimorphite a rather novel silicate mineral. The objective of this project was to measure the elastic properties of hemimorphite, in order to investigate its structure-property relations, and as a first step toward understanding how H 2 O and OHinfluence sound velocities and elasticity.
Minerals 2020, 10, x FOR PEER REVIEW 2 of 9 The structure of orthorhombic hemimorphite, with space group Imm2, is composed of a threedimensional framework, in which two ZnO3(OH) tetrahedra and one SiO4 tetrahedron form threemembered rings that are corner-shared to form corrugated sheets parallel to (010) [8,9] (Figure 1). SiO4 tetrahedra in one layer are linked to SiO4 tetrahedra in an adjacent layer, while the ZnO3(OH) tetrahedra in adjacent layers also only link to each other, completing the framework. Large open channels run parallel to [001], giving hemimorphite a structural similarity to zeolites. The molecular water, orientated in the (010) plane, is near the center of large cavities of an eight-membered ring and bonded to hydroxyl groups of the ZnO3(OH) tetrahedra by hydrogen bonds. The hydrogen bonds formed by H of hydroxyl group (H35) are stronger than those formed by H of molecular water (H53) [8]. The presence of both hydroxyl and molecular water, corrugated sheets in a tetrahedral framework, and large zeolitic cavities, make hemimorphite a rather novel silicate mineral. The objective of this project was to measure the elastic properties of hemimorphite, in order to investigate its structure-property relations, and as a first step toward understanding how H2O and OHinfluence sound velocities and elasticity.

Materials and Methods
Our samples of hemimorphite are from the Mapimí mine, Durango, Mexico. The sample contains a "spray" of transparent euhedral crystals with clearly defined faces. Most crystals exhibit tabular planes parallel to (010), and striations parallel to [001]. From single-crystal X-ray diffraction using a four-circle diffractometer, the average lattice parameters obtained on three single crystals are: a = 8.37(2) Å, b = 10.722(4) Å, and c = 5.118(7) Å, with unit cell volume V 0 = 459(1) Å 3 . The lattice parameters are consistent with the values reported in earlier crystal structure determinations by Hill et al. [8], McDonald and Cruickshank [9], Takeuchi et al. [10] and Cooper and Gibbs [11].
Scanning electron microscopy-energy dispersive spectroscopy (SEM-EDS) was used to obtain the major element chemical composition of hemimorphite. The results show that our samples are iron-free, nearly pure zinc silicates. The calculated density ρ = 3.48 (3) g/cm 3 was derived from the chemical formula Zn 4 Si 2 O 7 (OH) 2 ·H 2 O with Z = 2 and the unit cell volume given by X-ray diffraction.
Three near-principal sections of hemimorphite were polished with parallel faces in the a-b, b-c, and a-c crystallographic planes. The orientations of the polished surfaces were obtained by specular goniometry measurements and the X-ray orientation matrices for each crystal. The accuracy of sample orientation was within 0.5 degrees. For the Brillouin scattering measurements, a single-frequency diode pumped solid-state laser of wavelength 532 nm was used as a light source. All measurements were performed using a 90 • platelet symmetric scattering geometry [12] at ambient conditions. The scattered light was analyzed by a piezoelectrically-scanned tandem Fabry-Perot interferometer [13].
For each principal section, acoustic velocities were determined in 26 distinct crystallographic directions by changing the chi angle on the three-circle Eulerian cradle in 15 degrees increments. Measurements were made in a total of 78 distinct crystallographic directions, yielding 156 acoustic mode velocities (78 longitudinal and 78 single shear modes).

Results
From the measured Brillouin frequency shifts, ∆ν B , velocity v in a given crystallographic direction i is derived from the equation for symmetric platelet geometry [12]: where λ 0 is the wavelength of the incident laser light, and θ is the external angel between the incident and scattered light. The single-crystal elastic moduli tensor of orthorhombic hemimorphite contains nine independent non-zero elastic moduli C ij . On-diagonal moduli were well constrained by the acoustic velocities in the directions very close to the crystallographic axes. A linearized inversion method of Weidner and Carleton [14], was used to obtain a least-squares best-fit model of the elastic moduli C ij derived from the velocities. Figure 2 and Table 1 show a comparison between model and observed data of phonon velocities. The root-mean-square (RMS) residual in velocity for the final best-fit C ij model is 32.4 m/s. Hemimorphite exhibits considerable longitudinal and shear velocity anisotropy ( Figure 2 and Table 2).     Using the inversion method of Brown [15], we obtained virtually identical results for the C ij and their 1σ uncertainties. Table 3 shows the resulting C ij values for hemimorphite. We note the large differences among the moduli C 11 , C 22 , and C 33 , with C 11 40% smaller than C 22, are a result of the large acoustic anisotropy of hemimorphite. From the single-crystal moduli, the Voigt, Reuss, and Hashin-Shtrikman bounds on the adiabatic bulk modulus K S , and shear modulus µ of an isotropic aggregate were calculated, along with the Hill (Voigt-Reuss-Hill (VRH)) averages ( Table 4). The Voigt and Reuss bounds on the bulk modulus and shear modulus differ by factors of 7% and 17%, respectively. The differences are due to the moderately strong anisotropy for both longitudinal waves and shear waves. We note that the Hill average is within the range of the Hashin-Shtrikman bounds.  3 3.48 (3) From the values of Ks (VRH) and µ (VRH), the sound velocities appropriate to an isotropic polycrystalline aggregate were calculated to be V P,aggr = √ K+ 4 3 µ ρ = 5.6(1) km/s; V S, aggr = √ µ ρ = 2.8(1) km/s ( Table 4). The longitudinal and shear velocity anisotropy are defined as (V P ,maximum − V P ,minimum)/V P ,(VRH) = 26% and (V S ,maximum − V S ,minimum)/V S ,(VRH) = 47%), respectively.
Seryotkin et al. [16] studied the structural changes of hemimorphite at pressures up to 4.2 GPa by X-ray diffraction and the diamond-anvil cell. The purpose of their study was to search for a high-pressure phase transition, which they found near 2.5 GPa. We fitted their data on hemimorphite to a 2 nd -order Birch Murnaghan equation of state and obtain K T = 70(6) GPa (K T ' assumed to be 4), in agreement with our result of Ks = 74 (3). Their experiments also show that the a-axis of hemimorphite is most compressible, while the b-axis is most rigid, in agreement with our result C 22 > C 33 > C 11 . For the purpose of comparing our results and static compression results, we have ignored the difference between the adiabatic and isothermal bulk modulus, which is typically~1%.
Interestingly, we note that our value of Ks for hemimorphite is in excellent agreement with the value of K T = 72(2) GPa for bertrandite, Be 4 Si 2 O 7 (OH) 2 , measured by Hazen and Au [17]. Bertrandite is topologically identical to hemimorphite, but differs chemically and does not contain molecular water in the large open cavities of the structure. The fact that bertrandite and hemimorphite have the same bulk modulus, despite having significant chemical differences, likely indicates a dominant influence of crystal-structural topology on the physical properties of these materials. However, it is also possible that the substitution of Zn for Be, and the presence of H 2 O in hemimorphite, have opposite but equal effects on the bulk modulus that nearly cancel. There is some evidence to support this possibility. The Zn-O bond is longer than the Be-O bond and the bulk modulus of the BeO 4 tetrahedron is significantly larger than that of the ZnO 4 tetrahedron [18]. On the other hand, Cooper and Gibbs [11] found that upon dehydrating H 2 O from the structure, the large cavities contract. This may suggest that H 2 O provides support for the structure and stiffens it. It would be interesting to test this possibility on dehydrated hemimorphite, which would shed light on the effect of molecular water on this structure.

Discussion
The compressibility of minerals depends on the compressibility of the constituent cation polyhedra and their linkages to each other via angle-bending forces [17,18]. In their high-pressure X-ray structural study of bertrandite, Be 4 Si 2 O 7 (OH) 2 , which is structurally identical to hemimorphite Hazen and Au [17], concluded that polyhedral rotation and angle bending were the main compression mechanisms, as opposed to compression of tetrahedra. This is supported by the fact that the bulk modulus of bertrandite, 70(3) GPa, is much smaller than the polyhedral bulk moduli of the BeO 4 and SiO 4 tetrahedra (~200 GPa for both [17]). As noted above, our result for the bulk modulus of hemimorphite is identical to K T measured by Hazen and Au [17], providing additional support for this interpretation.
The C ij tensor of hemimorphite shows that strong anisotropy is a distinctive property of this mineral. We believe that this high degree of anisotropy is largely due to the topology of the hemimorphite crystal structure. C 11 and C 33 are the compressional moduli acting parallel to the corrugated sheets of tetrahedra. C 11 is a measure of the stiffness in the a direction, normal to the corrugations in the tetrahedral sheets ( Figure 3). Strain in this direction can be accommodated by a high degree of angle bending of the corrugations in an accordion-like fashion, and with little or no strain of the tetrahedra themselves. Thus, the value of C 11 is the lowest of the three compressional moduli.
The strongest elements in the hemimorphite crystal structure are the Si 2 O 7 groups that bond across the apical oxygens in adjacent (010) layers. The structure cannot be compressed along [010] without some strain being accommodated by the Si tetrahedra, which are the strongest polyhedra [18]. The Si 2 O 7 groups are essentially rigid pylons that stiffen the structure in the [010] direction. Thus, we attribute C 22 being the largest longitudinal modulus of hemimorphite as being largely due to the Si 2 O 7 groups. This situation is reminiscent of the olivine structure, in which SiO 4 tetrahedra are contained in columns along the a direction, yielding C 11 as the largest longitudinal modulus [19].
C 55 is the largest of the on-diagonal pure-shear moduli. The C 55 modulus corresponds to shear within the (010) plane, parallel to the tetrahedral sheets. These corrugated sheets contain only 3-membered rings of Zn and Si tetrahedra. There is virtually no rotational freedom or angle-bending for 3-membered rings in response to a shear stress within their tetrahedral basal planes. Strain within the tetrahedral sheets must be accommodated by either shearing of the tetrahedra themselves, or perhaps the tilting of tetrahedra out of the basal plane. This gives the structure high shear rigidity within the plane of a tetrahedral sheet. In support of this interpretation, we look to mica, which has tetrahedral sheets within the a-b plane, parallel to (001). The shear modulus corresponding to this tetrahedral sheet is C 66 = 72 GPa [20]. In comparison, the other shear moduli are far smaller, with C 44 = 16.5 GPa and C 55 = 19.5 GPa. In the cases of both hemimorphite and muscovite, the polyhedral topology, in particular tetrahedral sheets, dictates which of the pure shear moduli is largest. We suggest that the tetrahedral sheets in hemimorphite are less rigid than those in muscovite, for two reasons. Firstly, in hemimorphite, only 1/3 of the tetrahedra are SiO 4 , the strongest tetrahedral unit [18], whereas in muscovite, all the tetrahedra are SiO 4 in composition. Secondly, in muscovite the sheets are planar, whereas in hemimorphite the sheets are corrugated and exhibit some tilting of tetrahedra out of the (010) plane. Both the structural and chemical heterogeneity of hemimorphite make the rigidity of the tetrahedral sheets less than those in muscovite. Figure 3. Schematic illustration of a polyhedral model of mineral elasticity. Hydrogens in molecular water and hydroxyls are omitted to emphasize linkages among Si and Zn tetrahedra into corrugated sheets of 3-membered rings parallel to (010), and the bridging between sheets. Stiffness parallel to [010], normal to the sheets, depends on the strongest structural element in this direction, which are Si2O7 and Zn2O6(OH)2 groups that bridge the sheets along b. Within the plane of the sheets, the moduli depend on the weakest structural elements [19]. C11, measuring stiffness in the a direction, depends, in part, on the stiffness of relatively weak 6-membered and 8-membered rings, which can accommodate strain along [100].

Conclusions
The nine single-crystal elastic moduli of hemimorphite have been measured at ambient conditions. The relative magnitudes of the longitudinal elastic moduli, C22 > C33 > C11, can be qualitatively explained by the topology of the hemimorphite crystal structure. Strong Si2O7 groups greatly stiffen the structure along [010]. For the pure shear moduli, C55 is the largest, due to the lack of rotational freedom of Zn and Si within three-membered rings in sheets parallel to (010). The elastic character of hemimorphite is largely determined by the topology of the crystal structure.   . Schematic illustration of a polyhedral model of mineral elasticity. Hydrogens in molecular water and hydroxyls are omitted to emphasize linkages among Si and Zn tetrahedra into corrugated sheets of 3-membered rings parallel to (010), and the bridging between sheets. Stiffness parallel to [010], normal to the sheets, depends on the strongest structural element in this direction, which are Si 2 O 7 and Zn 2 O 6 (OH) 2 groups that bridge the sheets along b. Within the plane of the sheets, the moduli depend on the weakest structural elements [19]. C 11, measuring stiffness in the a direction, depends, in part, on the stiffness of relatively weak 6-membered and 8-membered rings, which can accommodate strain along [100].

Conclusions
The nine single-crystal elastic moduli of hemimorphite have been measured at ambient conditions. The relative magnitudes of the longitudinal elastic moduli, C 22 > C 33 > C 11 , can be qualitatively explained by the topology of the hemimorphite crystal structure. Strong Si 2 O 7 groups greatly stiffen the structure along [010]. For the pure shear moduli, C 55 is the largest, due to the lack of rotational freedom of Zn and Si within three-membered rings in sheets parallel to (010). The elastic character of hemimorphite is largely determined by the topology of the crystal structure.