![Sodalite (a) unit cell volume, (b) bond lengths and (c) Si-O-Al bridging angle variations as a function of pressure. In panel (a), the results of Hazen and Sharp [14] are reported for a direct comparison.](https://www.researchgate.net/publication/364615378/figure/fig1/AS:11431281091372094@1666402701636/Sodalite-a-unit-cell-volume-b-bond-lengths-and-c-Si-O-Al-bridging-angle-variations_Q320.jpg)
![Sodalite framework at 22 GPa (left) and -6 GPa (right), as seen from the [111] and [100] directions. The wireframe structure is composed by Al (cyan), Si (blue) and O (red), whereas the Na and Cl atoms are shown as yellow and green spheres, respectively.](https://www.researchgate.net/publication/364615378/figure/fig2/AS:11431281091349521@1666402701710/Sodalite-framework-at-22-GPa-left-and-6-GPa-right-as-seen-from-the-111-and-100_Q320.jpg)
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Citation: Ulian, G.; Valdrè, G.
Structural and Elastic Behaviour of
Sodalite Na8(Al6Si6O24)Cl2at
High-Pressure by First-Principle
Simulations. Minerals 2022,12, 1323.
https://doi.org/
10.3390/min12101323
Academic Editor: Jordi Ibanez-Insa
Received: 6 October 2022
Accepted: 19 October 2022
Published: 20 October 2022
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Attribution (CC BY) license (https://
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4.0/).
minerals
Article
Structural and Elastic Behaviour of Sodalite Na8(Al6Si6O24)Cl2
at High-Pressure by First-Principle Simulations
Gianfranco Ulian * and Giovanni Valdrè*
Centro di Ricerche Interdisciplinari di Biomineralogia, Cristallografia e Biomateriali, Dipartimento di Scienze
Biologiche, Geologiche e Ambientali, Universitàdi Bologna “Alma Mater Studiorum”, Piazza di Porta San
Donato 1, 40126 Bologna, Italy
*Correspondence: gianfranco.ulian2@unibo.it (G.U.); giovanni.valdre@unibo.it (G.V.)
Abstract:
Sodalite Na
8
(Al
6
Si
6
O
24
)Cl
2
(space group
P4
3
n
) is an important mineral belonging to the
zeolite group, with several and manyfold fundamental and technological applications. Despite the
interest in this mineral from different disciplines, very little is known regarding its high-pressure
elastic properties. The present study aims at filling this knowledge gap, reporting the equation of state
and the elastic moduli of sodalite calculated in a wide pressure range, from
−
6 GPa to 22 GPa. The
results were obtained from Density Functional Theory simulations carried out with Gaussian-type
basis sets and the well-known hybrid functional B3LYP. The DFT-D3 a posteriori correction to include
the van der Waals interactions in the physical treatment of the mineral was also applied. The calculated
equation of state parameters at 0 GPa and absolute zero (0 K), i.e., K
0
= 70.15(7) GPa, K’ = 4.46(2) and
V
0
= 676.85(3) Å3 are in line with the properties derived from the stiffness tensor, and in agreement
with the few experimental data reported in the literature. Sodalite was found mechanically unstable
when compressed above 15.6 GPa.
Keywords:
sodalite; zeolite group; equation of state; stiffness tensor; seismic wave velocities;
DFT; B3LYP
1. Introduction
Zeolites are both natural and synthetic nanoporous crystalline aluminosilicates (frame-
work silicates) that are frequently used in important chemical industry applications, such
as catalysis, separation (“molecular sieves”), and ion exchange [
1
,
2
]. From the crystal
chemistry perspective, zeolites are made of a network of corner-sharing TO
4
tetrahedra,
where T = Si, Al, or other tetrahedrally coordinated atoms (see for instance [3]).
The present work focuses on sodalite, whose chemical formula is Na
8
(Al
6
Si
6
O
24
)Cl
2
and cubic structure (space group
P4
3
n
). As determined by different structural refine-
ments over the past century [
4
–
7
], this mineral is characterized by a
β
-cage that is made
of four-membered (Al,Si)O
4
rings on the (100) plane (see Figure 1a) linked together in
six-membered rings along the [111] direction (Figure 1b), leading to the (AlSiO
4
)
66–
frame-
work. The negative charge (–6) induced by the Al
3+
/Si
4+
substitutions is balanced by two
tetrahedral (Na
4
Cl)
3+
clusters. The structure is highly symmetric, with just five atoms in
the asymmetric unit, i.e., Cl (Wyckoff site 2a) in (0,0,0), Al (6d) in (1/4,0,1/2), Si (6c) in
(1/4,1/2,0), Na (8e) in (x,x,x) and O (24i) in (x,y,z).
In addition, sodalite and the hydrated variant known as hydrosodalite Na
6
(AlSiO
4
)
6·
8H
2
O [
8
]
are among the possible mineral phases synthesized during geopolymerisation [
9
]. This
process requires kaolinite or metakaolin precursors that provide SiO
2
and Al
2
O
3
in ratio
2:1 and a strong basic solution (pH > 12.5) of sodium hydroxide (NaOH) that hydrolyses
the silicon and aluminium oxides [
10
], as observed with several techniques, such as X-ray
diffraction and scanning electron microscopy [11–13].
Minerals 2022,12, 1323. https://doi.org/10.3390/min12101323 https://www.mdpi.com/journal/minerals
Minerals 2022,12, 1323 2 of 15
Minerals2022,12,xFORPEERREVIEW2of17
withseveraltechniques,suchasX‐raydiffractionandscanningelectronmicroscopy[11–
13].
Despitethediversifiedandmultidisciplinaryinterestinsodalite,totheauthor’s
knowledge,onlytwoexperimentalworksaddressedandinvestigatedtheelasticityofthis
mineral.HazenandSharp[14]reportedthehydrostaticcompressionbehaviourofa
naturalsodalitesamplewithcompositionNa7.99K0.01(Al5.98Fe0.04Si5.98)O23.99Cl1.96(SO4)0.02,
fromroomto2.6GPa,butlimitedtothemeasurementsoftheunitcellvolumeandthea
latticeparametervariations.Nootherworksreportedtheequationofthestateofsodalite
athigherpressures.ThesecondworkisthatofLietal.[15],whoprovidedthesecond‐
orderelasticmoduliofthemineralfromultrasoundmeasurements.Onthecontrary,there
havebeenseveraltheoreticalstudiesonsodalite,buttheyweremainlyfocusedonthe
stoichiometricSi12O24phase,whichwasselectedasaprototypeforframeworksilicates
becauseofitssimplicity[3,16].Theonlyavailableinvestigationontheelasticproperties
ofsodalitewithidealformulaNa8(Al6Si6O24)Cl2isthetheoreticaloneofWillamsandco‐
workers[17],whichwashoweverconductedwithclassicalmechanicsmethods(force
fields).Theknowledgeoftheelasticbehaviourofsodaliteisfundamentaltoassessingits
mechanicalstability,whichultimatelyaffectsthepossibleapplicationsofthiszeolite,for
exampleinbuildingmaterials(seegeopolymers),watertreatment,soilremediation,gas
separation,andcatalysis[18].Itisalsoworthrememberingthatsodaliteand,ingeneral,
naturalzeolitesalsooccurassecondarymineralsinseveralgeologicalenvironmentsin
theEarth’scrust.
Figure1.SodaliteNa8(Al6Si6O24)Cl2polyhedralmodelseenalong(a)the[100]and(b)the[111]
directions.Thedashedbluelinesshowtheminerallattice.Colourcodeforatoms:blue—Si;cyan—
Al;red—O;yellow—Naandgreen—Cl.
Thepresentworkaimsatfillingtheseknowledgegaps,providingthereaderswith
anabinitioanalysisattheDensityFunctionalTheory(DFT)levelofthecrystalchemistry
andelasticpropertiesofsodalite.Regardingthemechanicalbehaviour,theequationof
stateofthemineraluptoabout22GPaisherereported,investigatingtheeffectsof
pressureontheinternalgeometry,i.e.,bonddistancesandangles,andpolyhedral
volumes.Then,ateachhydrostaticallycompressedstate,thesecond‐orderelasticmoduli
werecalculatedandanalysedtoobtainotherimportantproperties,suchasYoung’sand
shearmoduli,thecompressibility,Poisson’sratioandtheseismicwavevelocities.This
paperisorganisedasfollows:afterabriefdescriptionoftheemployedcomputational
parameters(Section2),theresultsrelatedtothecrystalchemistry,theequationofstate
andtheelasticmoduli(Section3)willbepresentedanddiscussedagainstthefew
Figure 1.
Sodalite Na
8
(Al
6
Si
6
O
24
)Cl
2
polyhedral model seen along (
a
) the [100] and (
b
) the [111]
directions. The dashed blue lines show the mineral lattice. Colour code for atoms: Blue—Si; cyan—Al;
red—O; yellow—Na and green—Cl.
Despite the diversified and multidisciplinary interest in sodalite, to the author’s knowl-
edge, only two experimental works addressed and investigated the elasticity of this mineral.
Hazen and Sharp [
14
] reported the hydrostatic compression behaviour of a natural sodalite
sample with composition Na
7.99
K
0.01
(Al
5.98
Fe
0.04
Si
5.98
)O
23.99
Cl
1.96
(SO
4
)
0.02
, from room to
2.6 GPa, but limited to the measurements of the unit cell volume and the alattice param-
eter variations. No other works reported the equation of the state of sodalite at higher
pressures. The second work is that of Li et al. [
15
], who provided the
second-order
elastic
moduli of the mineral from ultrasound measurements. On the contrary, there have been
several theoretical studies on sodalite, but they were mainly focused on the stoichiometric
Si
12
O
24
phase, which was selected as a prototype for framework silicates because of its
simplicity [3,16]
. The only available investigation on the elastic properties of sodalite with
ideal formula Na
8
(Al
6
Si
6
O
24
)Cl
2
is the theoretical one of Willams and co-workers [
17
],
which was however conducted with classical mechanics methods (force fields). The knowl-
edge of the elastic behaviour of sodalite is fundamental to assessing its mechanical stability,
which ultimately affects the possible applications of this zeolite, for example in building
materials (see geopolymers), water treatment, soil remediation, gas separation, and catal-
ysis [
18
]. It is also worth remembering that sodalite and, in general, natural zeolites also
occur as secondary minerals in several geological environments in the Earth’s crust.
The present work aims at filling these knowledge gaps, providing the readers with an
ab initio analysis at the Density Functional Theory (DFT) level of the crystal chemistry and
elastic properties of sodalite. Regarding the mechanical behaviour, the equation of state
of the mineral up to about 22 GPa is here reported, investigating the effects of pressure
on the internal geometry, i.e., bond distances and angles, and polyhedral volumes. Then,
at each hydrostatically compressed state, the second-order elastic moduli were calculated
and analysed to obtain other important properties, such as Young’s and shear moduli, the
compressibility, Poisson’s ratio and the seismic wave velocities. This paper is organised as
follows: after a brief description of the employed computational parameters (Section 2), the
results related to the crystal chemistry, the equation of state and the elastic moduli (Section 3)
will be presented and discussed against the few experimental and theoretical data available
in the literature. Finally, some general conclusions will be presented in Section 4.
Minerals 2022,12, 1323 3 of 15
2. Computational Methods
The ab initio (DFT) simulations performed in the present study were carried out with
the CRYSTAL17 code [
19
]. We selected the hybrid B3LYP functional [
20
,
21
] because it is
known to provide structural, vibrational and elastic data that are in very good agreement
with the experimental counterparts (see for instance [
22
–
25
]). The CRYSTAL code builds
the multi-electronic wave function using the so-called linear combination of atomic orbitals
(LCAO) approach, which employs Gaussian-type orbitals basis sets. Throughout the
simulations, an 88-31G* [
26
], an 85-11G* [
27
] an 8-411d11G [
28
] an 8-511G [
29
] and an
86-311G [
30
] basis sets were employed to describe Si, Al, O, Na and Cl, respectively.
They are double-
ζ
quality basis sets, i.e., they contain a double amount of Gaussian-type
functions that describe the atomic orbitals, which ensure high accuracy with affordable
computational costs. Furthermore, the basis sets for silicon, aluminium and oxygen were
previously adopted for the simulations of the crystal chemical and elastic properties of other
minerals, in particular talc [
31
], pyrophyllite [
32
,
33
], chlorite [
34
,
35
], topaz [
36
], brucite and
portlandite [37].
The exchange–correlation contribution to the total energy is calculated within CRYS-
TAL code by numerical integration of the electron density and its gradient, a task performed
on a pruned grid whose angular points are generated from the quadrature scheme of Gauss–
Legendre, whereas the Lebedev approach is adopted to obtain the radial points [
19
]. The
default grid of CRYSTAL17 was employed, which is made of 75 radial points and 974 an-
gular points and represents a good compromise between accuracy and cost of calculation.
The thresholds controlling the accuracy of the calculation of the Coulomb and exchange
integrals were set to 10
−8
(Coulomb series) and 10
−16
(exchange series). The Hamilto-
nian matrix was diagonalized in 35 kpoints (reciprocal lattice points), corresponding to a
shrinking factor of 8 [
38
]. To properly include the effects of long-range interactions in the
simulations, typically neglected by both GGA and hybrid DFT functionals, we employed
the DFT-D3 scheme proposed by Grimme and co-workers [
39
], which adds the following
contribution term to the total energy of the system:
EDFT−D3=−1
2
N
∑
i=1
N
∑
j=1∑
g"C6ij
r6
ij,g
fdum p,6rij,g+C8i j
r8
ij,g
fdum p,8rij,g#(1)
The sums run over the atoms Nin the unit cell, with r
ij,g
the internuclear distance
between atom iin cell
g
= 0 (reference cell) and atom jin cell
g
, and C
nij
(n= 6, 8) terms
are the 6th- and 8th-order dispersion coefficients for atom pairs ij, which depends on the
geometry of the system. The damping function here adopted is the one proposed by Becke
and Johnson [40–42]:
fdum p,nrij,g=snrn
ij
rn
ij +fR0i jn(2)
with
R0ij =qC8ij/C6ij
and
fR0ij =α1R0ij +α2
,s
6
= 1, s
8
, whereas
α1
and
α2
are
adjustable parameters. The choice of including the long-range interactions is dictated by
the need for the correct treatment of all the forces acting on the system because the elastic
properties depend on their gradient (second derivatives of the energy with respect to the
deformation). In fact, we observed that these weak forces play a non-negligible role even
for those minerals and materials where long-range interactions are not predominant. We
recently demonstrated this behaviour for calcite [
43
] and aragonite [
44
], and we expect the
same for sodalite.
The lattice constants and the atomic coordinates were optimized within the same
run with a numerical gradient method and an analytical gradient approach, respectively.
The starting geometry used to create the sodalite model was taken from the experimental
XRD refinements of Hassan and collaborators [
4
]. The upgrade of the Hessian matrix was
performed by means of the well-known BFGS algorithm [
45
–
49
]. For the optimization
of both the equilibrium (0 GPa) and hydrostatically compressed/expanded unit cell, the
Minerals 2022,12, 1323 4 of 15
tolerances for the maximum allowed gradient and the maximum atomic displacement
for considering the geometry as converged have been set to 1
×
10
−5
hartree bohr
−1
and
4×10−5bohr, respectively [50,51].
Second-order elastic moduli were calculated according to the scheme proposed by
Perger and co-workers [
52
], which is implemented in the CRYSTAL code. The elastic
moduli are the components of the 4th-rank stiffness tensor
C
that, according to the Voigt’s
6×6 matrix representation, can be written as:
σv=Cvuηu(3)
where the indices v,urun from 1 to 6 (1 = xx, 2 = yy, 3 = zz, 4 = yz, 5 = xz and 6 = xy),
σv
and
ηu
are the components of the stress and pure strain second-rank tensors. The
interested reader can find more information on the theory [
53
], implementation in the
CRYSTAL code [
52
,
54
] and recent applications [
36
,
37
,
51
,
55
] in dedicated literature. Single-
crystal elastic properties, namely Young’s modulus (E), linear compressibility (
β
), shear
modulus (
µ
) and Poisson’s ratio (
υ
) were calculated from the elastic moduli using the
QUANTAS code [
56
], with well-known directional relations [
34
,
53
,
55
,
57
,
58
]. Voigt and
Reuss equations were employed to calculate the average elastic properties considering the
system as a polycrystalline aggregate as explained by Nye [53].
Graphical representations of the sodalite structure were made with the molecular
graphics program VESTA [59].
3. Results
3.1. Crystal Chemistry
The crystal structure data (unit cell parameter a, atomic distances, bond angles, poly-
hedral volumes and relevant atomic fractional coordinates) of sodalite obtained from
geometry optimization at equilibrium (0 K and 0 GPa) within the DFT/B3LYP-D3 approach
is reported in Table 1, alongside previous theoretical [
17
] and experimental crystal chemical
analyses [4–6].
Table 1.
Zero-pressure (equilibrium) structure of sodalite Na
8
(Al
6
Si
6
O
24
)Cl
2
, with athe lattice param-
eter and Vthe unit cell volume, and internal geometry (atomic distances, bond angles, polyhedral
volumes and relevant atomic fractional coordinates).
B3LYP-D3 ∆FF 1SR-XRPD 2ND 3XRD 4
a(Å) 8.78091 −1.19% 8.848 8.88696 8.882 8.8823
V(Å3)677.0477 −3.54% 692.684 701.875 700.700 700.771
Si–O (Å) (×4) 1.6298 0.60% 1.598 1.6201 1.620 1.620
Al–O (Å) (×4) 1.7437 0.10% 1.758 1.7419 1.740 1.742
Na–O (Å) (×3) 2.2750 −3.48% 2.316 2.357 2.354 2.353
Na–Cl (Å) (×4) 2.6486 −2.52% 2.749 2.717 2.735 2.716
O–Si–O (◦) (×4) 107.53 0.06% 108.66 107.466 107.66 107.7
(×2) 113.42 −0.12% 111.11 113.560 113.15 113.0
Mean 109.50 0.00% 109.48 109.50 109.49 109.47
O–Al–O (◦) (×4) 108.49 0.05% 109.88 108.442 108.62 108.7
(×2) 111.45 −0.09% 108.66 111.550 111.18 111.0
Mean 109.48 0.00% 109.47 109.48 109.47 109.47
Si–O–Al (◦) 133.90 −3.17% 137.49 138.288 138.24
VSiO4 (Å3)2.2208 2.18% 2.0951 2.1734 2.1730 -
VAlO4 (Å3)2.7207 0.38% 2.7840 2.7103 2.7033 -
Na1 (8e) x/a 0.17414 −2.11% 0.17937 0.1779 0.17780 0.1778
O1 (24i) x/a 0.13816 −1.13% 0.13416 0.13974 0.13925 0.1390
y/b 0.14813 −1.33% 0.14782 0.15013 0.14954 0.1494
z/c 0.42940 −2.18% 0.43525 0.43895 0.43851 0.4383
Notes:
∆
is the percentage difference between the B3LYP-D3 results and the experimental SR-XRPD refinements.
1—Force field (FF)
simulations [
17
];
2
—synchrotron radiation X-ray powder diffraction (SR-XRPD) measure-
ments [4]; 3—neutron diffraction (ND) determination; 4—X-ray diffraction.
Minerals 2022,12, 1323 5 of 15
As expected from static ab initio simulations carried out at 0 K and without including
zero-point effects, the unit cell volume is about 3.5% smaller than that found from the
refinements of the structural data collected from synchrotron radiation X-ray powder
diffraction (SR-XRPD) of Hassan et al. [
4
]. By analysing the internal geometry, it is possible
to note the interplay between the Si–O and Al–O bond lengths, slightly larger than the
experimental ones (less than 1%), and the Na–O and Na–Cl distances, which are instead
underestimated by about 3%. The mean O–Si–O and O–Al–O bond angles are in excellent
agreement with the experimental ones, and very close to the ideal tetrahedral angle (109.47
◦
)
whereas the bridging Si–O–Al angle is slightly underestimated (about
−
3%). The bridging
angle is the one that showed the greatest variation as a function of temperature, a behaviour
that was modelled with a parabolic function in the range of 28–1000
◦
C in the work of
Hassan and collaborators [
4
]. By extrapolating at 0 K (
−
273.15
◦
C) with the functional form
proposed by the cited authors, we obtained a Si–O–Al angle of 137.47
◦
, which is still larger
than that at the B3LYP-D3 level. This could be due to the absence of any thermal effects,
including zero-point energy, i.e., the ground-state vibrational contribution to the total
internal energy, and the associated isotropic displacement parameters that were not taken
into account in the present investigation. The volumes of the SiO
4
and AlO
4
tetrahedra
are also in line with the XRD refinements. Our results are also in good agreement with the
statistical data collected for hundreds of zeolites, including sodalite-like structures reported
by Baur and Fischer [
60
]. The authors calculated a mean T–O bond distance of 1.617(7) Å
(1.629 Å by applying a correction for bond distances) for SOD zeolites and observed
O–Si–O
and O–Al–O angles in the ranges 101.0
◦
–117.7
◦
and 94.6
◦
–123.7
◦
, respectively. Our
simulations on stoichiometric sodalite provided both distances and angular values that fall
in these ranges.
The present results at the DFT/B3LYP-D3 level of theory are in line with the force
field (classical mechanics) simulations of Willams et al. [
17
], and provide a better mean
absolute deviation of the structural features of sodalite (1.28%) in comparison to the cited
work (1.49%). Unfortunately, despite recent ab initio studies on sodalite performed in the
last decade [
61
–
63
], none of them reported the crystal chemistry of the optimized zeolite
structure for a proper comparison, focusing on other properties of the mineral (electronic
band structure and optical properties, spectroscopic models of localized defects in the
structure, reactivity towards sulphur).
3.2. Equation of State
The equation of the state of sodalite was obtained with a two-step process. In the first
one, it was performed a volume-constrained geometry optimization on a series of mineral
unit cells with larger (expansion) and smaller (compression) volumes. Here, eleven models
between 0.82
·
V
eq
and 1.12
·
V
eq
, with V
eq
the equilibrium unit cell volume, were simulated
using the internal routines of CRYSTAL [
54
]. The geometry optimization results at different
hydrostatic compression/expansion states are reported in Table 2.
Then, in the second step of the procedure, the unit cell internal energy as a function
of volume, U(V), was fitted to a volume-integrated 3rd-order Birch–Murnaghan (BM3)
formulation [64], as proposed by Hebbache and Zemzemi [65]:
U(V) = U0+9
16K0V0K0X2−13+X2−126−4X2 (4)
with X= (V
0
/V)
−1/3
. In this equation, the fitting parameters are U
0
(internal energy),
K
0
(bulk modulus), K’ (pressure derivative of the bulk modulus) and V
0
(unit cell vol-
ume), with the subscript zero meaning they were obtained at 0 GPa. This operation was
performed using the QUANTAS code [
56
], obtaining K
0
= 70.15(7) GPa,
K’ = 4.46(2)
and
V0= 676.85(3) Å3
. This result is in quite good agreement with that reported by Hazen
and Sharp [
14
], with second-order Birch–Murnaghan equation of state fitting parame-
ters K
0
= 52(8) GPa and K’ = 4 (fixed). The difference between the fitting results could
be due to several reasons, the first one the absence of thermal effects in the simulations,
Minerals 2022,12, 1323 6 of 15
as previously introduced during the discussion of the sodalite crystal chemistry. Then,
the composition of the experimental sample and the theoretical model is slightly dif-
ferent, because the former has some percentages of other elements [chemical formula
Na
7.99
K
0.01
(Al
5.98
Fe
0.04
Si
5.98
)O
23.99
Cl
1.96
(SO
4
)
0.02
, as obtained from microprobe analysis],
whereas the present DFT investigation considered an ideal, stoichiometric sodalite. Finally,
the present study considered a very wide pressure range, up to about 22 GPa, whereas the high-
pressure XRD refinements were carried out only up to 2.6 GPa, as can be noted from Figure 2a.
Table 2.
Unit cell volume V, lattice parameter a, differences in internal energy (
∆
U) and enthalpy
(
∆
H) with respect to the equilibrium geometry (0 GPa), mean bond lengths and angles, polyhedron
volume of SiO
4
and AlO
4
tetrahedra and position of the sodium and oxygen atoms as a function
of pressure.
P(GPa) 22.4 17.2 12.8 9.0 5.7 2.9 0.5 0.0 −1.6 −3.4 −4.9 −6.3
a(Å) 8.21885 8.30886 8.39887 8.48889 8.57890 8.66891 8.75892 8.78091 8.84893 8.93895 9.02896 9.11897
V(Å3)555.1791 573.6204 592.4656 611.7191 631.3853 651.4685 671.9731 677.0477 692.9035 714.2641 736.0593 758.2934
∆U(Ha) 0.25216 0.17133 0.10779 0.06053 0.02783 0.00821 0.00029 0.00000 0.00293 0.01512 0.03580 0.06407
∆H(Ha) 3.10532 2.43655 1.84532 1.32244 0.85921 0.44818 0.08293 0.00000 −0.24208 −0.53150 −0.78961 −1.02002
Si–O (Å) (×4) 1.593 1.600 1.607 1.613 1.618 1.624 1.629 1.630 1.633 1.638 1.642 1.646
Al–O (Å) (×4) 1.692 1.702 1.711 1.719 1.727 1.735 1.742 1.744 1.749 1.756 1.762 1.768
Na–O (Å) (×3) 2.047 2.081 2.116 2.152 2.189 2.227 2.265 2.275 2.306 2.348 2.392 2.438
Na–Cl (Å) (×4) 2.307 2.355 2.405 2.458 2.514 2.571 2.633 2.649 2.697 2.764 2.835 2.910
O–Si–O (◦) (×4) 107.51 107.53 107.52 107.52 107.52 107.52 107.53 107.53 107.54 107.55 107.57 107.59
(×2) 113.46 113.44 113.45 113.46 113.45 113.44 113.43 113.42 113.41 113.38 113.34 113.30
Mean 109.50 109.50 109.50 109.50 109.50 109.50 109.50 109.50 109.50 109.50 109.49 109.49
O–Al–O (◦) (×4) 108.50 108.51 108.50 108.50 108.50 108.49 108.49 108.49 108.49 108.50 108.50 108.51
(×2) 111.43 111.41 111.42 111.43 111.44 111.44 111.45 111.45 111.44 111.43 111.42 111.41
Mean 109.48 109.48 109.48 109.48 109.48 109.48 109.48 109.48 109.48 109.48 109.48 109.48
Si–O–Al (◦) 124.34 125.62 127.01 128.49 130.06 131.71 133.46 133.90 135.30 137.25 139.34 141.56
VSiO4 (Å3)2.0743 2.1018 2.1273 2.1513 2.1739 2.1956 2.216 2.2208 2.2354 2.2539 2.2714 2.2879
VAlO4 (Å3)2.4868 2.5300 2.5698 2.6078 2.6442 2.6792 2.7127 2.7207 2.7454 2.7769 2.8069 2.8361
VNa4Cl (Å3)6.3022 6.6986 7.1392 7.6251 8.1519 8.7258 9.3685 9.5350 10.0683 10.8319 11.6879 12.6480
Na (8e)x/a 0.16207 0.16361 0.16533 0.16720 0.16917 0.17126 0.17356 0.17414 0.17597 0.17850 0.18125 0.18425
O (24i)x/a 0.13402 0.13458 0.13524 0.13592 0.13660 0.13729 0.13799 0.13816 0.13868 0.13937 0.14006 0.14075
y/b 0.14366 0.14432 0.14505 0.14578 0.14651 0.14723 0.14796 0.14813 0.14867 0.14937 0.15008 0.15077
z/c 0.40887 0.41163 0.41464 0.41783 0.42118 0.42472 0.42845 0.42940 0.43239 0.43657 0.44107 0.44591
Minerals2022,12,xFORPEERREVIEW6of17
withX=(V0/V)–1/3.Inthisequation,thefittingparametersareU0(internalenergy),K0(bulk
modulus),K’(pressurederivativeofthebulkmodulus)andV0(unitcellvolume),withthe
subscriptzeromeaningtheywereobtainedat0GPa.Thisoperationwasperformedusing
theQUANTAScode[56],obtainingK0=70.15(7)GPa,K’=4.46(2)andV0=676.85(3)Å3.
ThisresultisinquitegoodagreementwiththatreportedbyHazenandSharp[14],with
second‐orderBirch–MurnaghanequationofstatefittingparametersK0=52(8)GPaandK’
=4(fixed).Thedifferencebetweenthefittingresultscouldbeduetoseveralreasons,the
firstonetheabsenceofthermaleffectsinthesimulations,aspreviouslyintroducedduring
thediscussionofthesodalitecrystalchemistry.Then,thecompositionoftheexperimental
sampleandthetheoreticalmodelisslightlydifferent,becausetheformerhassomeper‐
centagesofotherelements[chemicalformulaNa7.99K0.01(Al5.98Fe0.04Si5.98)O23.99Cl1.96(SO4)0.02,
asobtainedfrommicroprobeanalysis],whereasthepresentDFTinvestigationconsidered
anideal,stoichiometricsodalite.Finally,thepresentstudyconsideredaverywidepres‐
surerange,uptoabout22GPa,whereasthehigh‐pressureXRDrefinementswerecarried
outonlyupto2.6GPa,ascanbenotedfromFigure2a.
Figure2.Sodalite(a)unitcellvolume,(b)bondlengthsand(c)Si–O–Albridginganglevariations
asafunctionofpressure.Inpanel(a),theresultsofHazenandSharp[14]arereportedforadirect
comparison.
ThebulkmoduluscalculatedattheDFTlevelisalsoinlinewiththeadiabaticbulk
modulus(KS=55.3GPa)calculatedfromtheelasticmodulimeasuredbyultrasonicmeth‐
odsbyLietal.[15].Totheauthors’knowledge,noothertheoreticalandexperimental
dataontheequationofstatewerereportedinliteratureregardingsodalitewiththeideal
chemicalformula[Na8(Al6Si6O24)Cl2]herereported.Albeitthedifferentcagestructure,the
elasticbehaviourofsodaliteisinlinewiththatofcancrinite‐groupzeolitesasrecently
Figure 2.
Sodalite (
a
) unit cell volume, (
b
) bond lengths and (
c
) Si–O–Al bridging angle variations
as a function of pressure. In panel (
a
), the results of Hazen and Sharp [
14
] are reported for a
direct comparison.
Minerals 2022,12, 1323 7 of 15
The bulk modulus calculated at the DFT level is also in line with the adiabatic bulk
modulus (K
S
= 55.3 GPa) calculated from the elastic moduli measured by ultrasonic methods
by Li et al. [
15
]. To the authors’ knowledge, no other theoretical and experimental data on
the equation of state were reported in literature regarding sodalite with the ideal chemical
formula [Na
8
(Al
6
Si
6
O
24
)Cl
2
] here reported. Albeit the different cage structure, the elastic
behaviour of sodalite is in line with that of cancrinite-group zeolites as recently reviewed
by Chukanov and co-workers [66], whose bulk moduli fall in the range 30–48 GPa.
An inspection of the variation of the internal geometry (Figure 2b,c and Table 2)
provides more insights into the compression mechanism. The structural features that are
affected the most by hydrostatic pressure are the Si–O–Al bridging angles (–7.1% at about
22 GPa with respect to the equilibrium geometry), the Na–O (–10.0%) and
Na–Cl (–12.9%)
bond distances and the Na
4
Cl polyhedral volume (–33.9%). Conversely, the SiO
4
and AlO
4
bond lengths are just slightly affected by compression (up to about
−
3%), whereas the
O–Si–O
and O–Al–O angles remain constant. A graphical representation of the sodalite
structure at the extremes of the compression regime investigated here is reported in Figure 3.
All these observations agree with the general behaviour of framework silicates, such as
zeolites and feldspathoids. Indeed, these minerals undergo a small distortion of the
framework on the T–O–T angles, with more or less rigid T–O bonds, and high compression
of the cation sites [14].
Minerals2022,12,xFORPEERREVIEW7of17
reviewedbyChukanovandco‐workers[66],whosebulkmodulifallintherange30–48
GPa.
Aninspectionofthevariationoftheinternalgeometry(Figure2b,candTable2)pro‐
videsmoreinsightsintothecompressionmechanism.Thestructuralfeaturesthatareaf‐
fectedthemostbyhydrostaticpressurearetheSi–O–Albridgingangles(–7.1%atabout
22GPawithrespecttotheequilibriumgeometry),theNa–O(–10.0%)andNa–Cl(–12.9%)
bonddistancesandtheNa4Clpolyhedralvolume(–33.9%).Conversely,theSiO4andAlO4
bondlengthsarejustslightlyaffectedbycompression(uptoabout–3%),whereastheO–
Si–OandO–Al–Oanglesremainconstant.Agraphicalrepresentationofthesodalitestruc‐
tureattheextremesofthecompressionregimeinvestigatedhereisreportedinFigure3.
Alltheseobservationsagreewiththegeneralbehaviourofframeworksilicates,suchas
zeolitesandfeldspathoids.Indeed,thesemineralsundergoasmalldistortionoftheframe‐
workontheT–O–Tangles,withmoreorlessrigidT–Obonds,andhighcompressionof
thecationsites[14].
Figure3.Sodaliteframeworkat22GPa(left)and–6GPa(right),asseenfromthe[111]and[100]
directions.ThewireframestructureiscomposedbyAl(cyan),Si(blue)andO(red),whereastheNa
andClatomsareshownasyellowandgreenspheres,respectively.
3.3.ElasticModuliandtheirVariationwithPressure
Cubiccrystalshavethreeindependentelasticmoduli(C11,C44andC12)thatcanbe
representedina6×6matrixCusingVoigt’snotation[53]:
11 12 12
11 12
11
44
44
44
CCC
CC
C
C
C
C
C
(5)
wherethedotsindicatetheCijvalueiszero.Forsymmetryreasons,twolatticedefor‐
mationsaresufficienttoobtainalltheindependentelasticmoduli:
Figure 3.
Sodalite framework at 22 GPa (
left
) and
−
6 GPa (
right
), as seen from the [111] and [100]
directions. The wireframe structure is composed by Al (cyan), Si (blue) and O (red), whereas the Na
and Cl atoms are shown as yellow and green spheres, respectively.
3.3. Elastic Moduli and their Variation with Pressure
Cubic crystals have three independent elastic moduli (C
11
,C
44
and C
12
) that can be
represented in a 6 ×6 matrix Cusing Voigt’s notation [53]:
C=
C11 C12 C12 · · ·
C11 C12 · · ·
C11 · · ·
C44 · ·
C44 ·
C44
(5)
Minerals 2022,12, 1323 8 of 15
where the dots indicate the C
ij
value is zero. For symmetry reasons, two lattice deformations
are sufficient to obtain all the independent elastic moduli:
ε1=δ
100
000
000
(6)
and
ε4=δ
000
001
010
(7)
with
ε1
and
ε4
being uniaxial and biaxial (shear) strains, respectively. The factor
δ
controls
the amount of applied strain, which was varied between
±
0.015 with a step of 0.005, hence
seven configurations for each lattice strain were simulated.
The calculated elastic moduli of sodalite in equilibrium conditions (0 K and 0 GPa)
and by varying pressure are reported in Table 3, together with the polycrystalline prop-
erties calculated with the Voigt, Reuss and Hill averaging schemes. According to the
Born
criteria [67]
, a cubic crystalline structure is stable when the following necessary and
sufficient conditions are met:
C11 −C12 > 0, C11 + 2C12 > 0, C44 > 0. (8)
Table 3. Elastic moduli Cij (GPa), density ρ(kg m−3), bulk modulus (K, GPa), linear compressibility
(
β
, TPa
−1
), Young’s modulus (E, GPa), shear modulus (
µ
, GPa), Poisson’s ratio (
υ
), and average
longitudinal and shear wavevelocities (v
P
and v
S
, respectively), of sodalite as a function of pressure P(GPa).
P22.4 * 17.2 * 12.8 9.0 5.7 2.9 0.5 0.0 −1.6 −3.4 −4.9 −6.3
C11 146.00 138.78 130.19 121.50 114.29 107.26 101.19 99.91 95.88 89.45 83.98 78.95
C44 5.94 16.91 23.57 28.35 32.05 35.02 37.49 37.99 39.39 40.58 41.09 41.00
C12 168.76 143.79 121.38 101.79 85.16 70.39 57.85 55.16 47.18 36.85 28.52 21.38
ρ2894 2801 2712 2626 2544 2466 2391 2373 2319 2249 2183 2119
K- - 124.32 108.36 94.87 82.68 72.30 70.08 63.41 54.38 47.01 40.57
β- - 2.68 3.08 3.51 4.03 4.61 4.76 5.26 6.13 7.09 8.22
EV- - 45.76 59.05 69.08 76.41 81.74 82.74 85.18 86.18 85.55 83.55
ER- - 25.21 46.28 60.36 69.99 76.77 78.07 81.27 83.04 83.05 81.64
EVRH - - 35.58 52.71 64.74 73.22 79.26 80.42 83.23 84.62 84.30 82.60
µV- - 15.90 20.95 25.05 28.39 31.16 31.75 33.38 34.87 35.74 36.12
µR- - 8.60 16.20 21.65 25.75 29.01 29.70 31.59 33.34 34.45 35.05
µVRH - - 12.25 18.57 23.35 28.07 30.09 30.72 32.48 34.10 35.09 35.58
υV- - 0.439 0.409 0.379 0.346 0.312 0.303 0.276 0.236 0.197 0.157
υR- - 0.466 0.429 0.394 0.359 0.323 0.314 0.286 0.245 0.206 0.165
υVRH - - 0.452 0.419 0.386 0.352 0.317 0.309 0.281 0.241 0.201 0.161
vS- - 2.125 2.660 3.030 3.313 3.547 3.598 3.743 3.894 4.010 4.098
vP- - 7.202 7.120 7.038 6.940 6.857 6.841 6.784 6.663 6.555 6.445
Note: the bulk modulus values are the same between the three polycrystalline averaging schemes
(KV=KR=KVRH)
.
The elastic properties were not calculated for the mechanically unstable sodalite models (marked with an asterisk).
The second and third criteria are satisfied throughout the pressure conditions explored
in the present work. However, the first condition is not met at 17.2 GPa and 22.4 GPa
(see Figure 4a) and, by fitting the data with a second-order polynomial, it was observed that
the maximum allowed compression is about 15.6 GPa (black square in Figure 4a). Hence,
it is expected that sodalite is not mechanically stable above this pressure threshold and it
could undergo a phase transition or decomposition, according to our symmetry-constrained
simulations at absolute zero.
The value of the bulk modulus at equilibrium geometry, K
R
=K
V
=K
VRH
= 70.08 GPa,
can be exploited to assess the quality of the simulation approach by comparing it to
Minerals 2022,12, 1323 9 of 15
the K
0
equation of state parameter. Since their difference is extremely low (0.1%), the
computational methods here employed are consistent and physically valid.
Further positive assessment of the simulated stiffness components comes from the
comparison with the experimental ones determined by Li and co-workers [
15
] with ultra-
sound techniques (C
11
= 88.52 GPa C
44
= 36.46 GPa and C
12
= 38.72 GPa). In detail, the
DFT/B3LYP-D3 results at zero temperature and pressure differ by about +13%, +4% and
+42% for the C
11
,C
44
and C
12
moduli, respectively, an observation in line with the equation
of state previously discussed.
Minerals2022,12,xFORPEERREVIEW9of17
GPa(seeFigure4a)and,byfittingthedatawithasecond‐orderpolynomial,itwasob‐
servedthatthemaximumallowedcompressionisabout15.6GPa(blacksquareinFigure
4a).Hence,itisexpectedthatsodaliteisnotmechanicallystableabovethispressure
thresholdanditcouldundergoaphasetransitionordecomposition,accordingtoour
symmetry‐constrainedsimulationsatabsolutezero.
Thevalueofthebulkmodulusatequilibriumgeometry,KR=KV=KVRH=70.08GPa,
canbeexploitedtoassessthequalityofthesimulationapproachbycomparingittotheK0
equationofstateparameter.Sincetheirdifferenceisextremelylow(0.1%),thecomputa‐
tionalmethodshereemployedareconsistentandphysicallyvalid.
Furtherpositiveassessmentofthesimulatedstiffnesscomponentscomesfromthe
comparisonwiththeexperimentalonesdeterminedbyLiandco‐workers[15]withultra‐
soundtechniques(C11=88.52GPaC44=36.46GPaandC12=38.72GPa).Indetail,the
DFT/B3LYP‐D3resultsatzerotemperatureandpressuredifferbyabout+13%,+4%and
+42%fortheC11,C44andC12moduli,respectively,anobservationinlinewiththeequation
ofstatepreviouslydiscussed.
Figure4.(a)EvolutionofthemechanicalstabilitycriterionC11−C12>0,withtheblacksquareshow‐
ingtheoccurrenceofinstabilityabove15.6GPa.(b)Elasticmoduliofsodaliteasafunctionofpres‐
sure,withinthestabilityregion.Thelinesare2nd‐orderpolynomialfitofthedata(seetextforde‐
tails).
Thegraphshowingtheevolutionoftheelasticmoduliwithpressureispresentedin
Figure4b,withtheuniaxialcomponentsC11andC12increasingwithP,whereasaninverse
trendisobservedfortheshearcomponentC44.Thefunctionaldescriptionofthevariation
ofthestiffnessmatrixwithpressurewasobtainedforeachCij(P)curvebymeansofafinite
strainfitofthetype:
Cij(P)=Cij0+Cij′P+Cij″P2(9)
whereCij0istheelasticmodulusvalueatzeropressure(inGPaunits),Cij′isthepressure
derivativeofCij(dCij/dP)isdimensionlessandij
C isthepressuresecondderivativeof
theelasticmodulus(d2Cij/d2P,inGPa–1units).Theregressionofthedataresultedina0=
99.59GPa,a1=2.9102anda2=–4.4238∙10–2GPa–1forC11(R2=0.9988);a0=37.86GPa,a1=–
0.9051anda2=–1.6801∙10–2GPa–1forC44(R2=0.9997);anda0=55.19GPa,a1=5.3134and
a2=–1.1938∙10–2GPa–1forC12(R2=0.9999).Hence,therateofstiffeningoftheoff‐diagonal
C12modulus,i.e.,C12′,ishigherthanthatoftheC11termbyabout1.8times,explainingthe
mechanicalinstabilitypreviouslydiscussed.
Finally,thewavevelocitieswerecalculatedforsingle‐crystalsodalitebysolvingthe
Christoffel’sequationinthecaseamonochromaticplanewavewithwavevectorq,which
isthepropagationdirection[68]:
Figure 4.
(
a
) Evolution of the mechanical stability criterion C
11 −
C
12
> 0, with the black square
showing the occurrence of instability above 15.6 GPa. (
b
) Elastic moduli of sodalite as a function
of pressure, within the stability region. The lines are 2nd-order polynomial fit of the data (see text
for details).
The graph showing the evolution of the elastic moduli with pressure is presented in
Figure 4b, with the uniaxial components C
11
and C
12
increasing with P, whereas an inverse
trend is observed for the shear component C
44
. The functional description of the variation
of the stiffness matrix with pressure was obtained for each C
ij
(P) curve by means of a finite
strain fit of the type:
Cij(P) = Cij 0+Cij
0P+Cij
00 P2(9)
where C
ij0
is the elastic modulus value at zero pressure (in GPa units), C
ij0
is the pres-
sure derivative of C
ij
(dC
ij
/dP) is dimensionless and
C00
ij
is the pressure second deriva-
tive of the elastic modulus (d
2
C
ij
/d
2
P, in GPa
−1
units). The regression of the data re-
sulted in
a0= 99.59 GPa
,a
1
= 2.9102 and a
2
=
−
4.4238
·
10
−2
GPa
−1
for C
11
(R
2
= 0.9988);
a0= 37.86 GPa,
a
1
=
−
0.9051 and a
2
=
−
1.6801
·
10
−2
GPa
−1
for C
44
(R
2
= 0.9997); and
a0= 55.19 GPa
,a
1
= 5.3134 and a
2
=
−
1.1938
·
10
−2
GPa
−1
for C
12
(R
2
= 0.9999). Hence, the
rate of stiffening of the off-diagonal C
12
modulus, i.e., C
120
, is higher than that of the C
11
term by about 1.8 times, explaining the mechanical instability previously discussed.
Finally, the wave velocities were calculated for single-crystal sodalite by solving the
Christoffel’s equation in the case a monochromatic plane wave with wave vector
q
, which
is the propagation direction [68]:
∑
ij hMi j −ρv2
pδij isj=0, (10)
where Mij is a component of the Christoffel matrix M:
Mij =∑
kl
qkCikl j ql, (11)
Minerals 2022,12, 1323 10 of 15
C
ijkl
are the elastic moduli (in 4th-rank notation),
ρ
is the crystal density, v
p
is the
wave velocity (known as phase velocity),
δij
is the Kronecker’s delta function and
s
is the
polarization. Since the velocities are independent of the wavelength,
q
is assumed as a
dimensionless unit vector denoting only the direction of travel of a monochromatic plane
wave. The solutions of Christoffel’s equation above reported are one primary (P-wave,
longitudinal, v
P
) and two secondary (S-wave, transverse, v
S
) acoustic wave velocities, with
the eigenvectors describing the polarization directions. However, since sound waves are
never purely monochromatic, it is more realistic to calculate the so-called group velocities,
v
g
, which consider the sound as a wave packet with a small spread in wavelength and
direction of travel, according to the formula [69]:
vg=
→
∇vp(12)
The gradient of the phase velocities is a derivative to the components of
q
, which is
calculated in the reciprocal space (assumed dimensionless). It is worth noting that, while
vpis a scalar function of q,vgis a vector that is not necessarily parallel to q, and the angle
between the two velocities (phase and group) is given by
vp=vgcos ψ(13)
with
ψ
being called the power flow angle. The power flow angle varies as a function of the
direction because
q
and
vg
are not parallel, thus there are some directions showing concen-
tration of the energy flux, whereas others are characterized by its dispersion. This effect,
also known as the phonon focusing effect, is quantified by the enhancement factor A[70]:
A=∆Θp
∆Θg, (14)
where
∆Θp
and
∆Θg
are defined as the solid angles that are crossed phase wave vectors
np
and group wave vectors
ng
, respectively.
np
and
ng
are the normalized vectors of
the phase and group velocities, respectively. More details can be found in the dedicated
literature [69–71].
The seismic velocity results, i.e., phase velocity v
p
, group velocity v
g
, enhancement
factor Aand power flow angle
ψ
for sodalite at 0 GPa are reported in Figure 5, as upper
hemisphere (Z > 0) Lambert equal-area projections on the XY plane. The cubic symmetry of
the mineral is immediately recognizable from the patterns in each panel and for longitudinal
and transverse acoustic waves. The enhancement factor is different according to the type
of wave. For the P-mode, A is lowest along the Cartesian [100], [010] and [001] directions
(Cartesian X, Y and Z) and the highest along the [111] direction. While the distribution
of the enhancement factor is simple for P-waves, when considering the transverse modes
there are complex patterns, with the lowest A value along the [111] direction.
By increasing pressure, the phase velocities change their minimum–maximum value
ranges, but their directional distribution is almost unvaried, as shown in Figure 6relatively
to the mineral compressed at 12.8 GPa. However, there are striking differences in the group
velocities v
g
, especially for the slow and fast S-waves (Figure 6b), maintaining the cubic
symmetry. These variations are due to the power flow angle values increasing up to about
70
◦
(slow waves) and 45
◦
(fast waves) at the maximum pressure investigated (12.8 GPa),
whereas they were much lower (30
◦
and 20
◦
, respectively), at 0 GPa. Conversely, primary
waves are less affected by pressure effects. The enhancement factor Aclearly shows higher
variations of the power flow angle, as reported in the logarithmic plots in Figures 5c and 6c.
Minerals 2022,12, 1323 11 of 15
Minerals2022,12,xFORPEERREVIEW12of17
Figure5.Analysisoftheseismicwavevelocities(Lambertequal‐areaupperhemisphereprojections)
ofsodaliteat0GPa,showing(a)thephasevelocitiesvP(km/s),(b)thegroupvelocitiesvP(km/s),(c)
theenhancementfactorAand(d)thepowerflowangle(PF,°).
Figure 5.
Analysis of the seismic wave velocities (Lambert equal-area upper hemisphere projections)
of sodalite at 0 GPa, showing (
a
) the phase velocities v
P
(km/s), (
b
) the group velocities v
P
(km/s),
(c) the enhancement factor Aand (d) the power flow angle (PF, ◦).
Minerals 2022,12, 1323 12 of 15
Minerals2022,12,xFORPEERREVIEW13of17
Figure6.Analysisoftheseismicwavevelocities(Lambertequal‐areaupperhemisphereprojections)
ofsodaliteat12.8GPa,showing(a)thephasevelocitiesvP(km/s),(b)thegroupvelocitiesvP(km/s),
(c)theenhancementfactorAand(d)thepowerflowangle(PF,°).
4.Conclusions
Inthepresentwork,weprovidedforthefirsttimeanabinitioinvestigationofthe
crystalchemicalandelasticpropertiesofsodalite[Na8(Al6Si6O24)Cl2,spacegroup
𝑃4
3𝑛],
azeolitemineralwithimportantfundamental(geologicalandminero‐petrographic)and
technological(e.g.,catalysisandseparation,buildingandconstructionmaterials)applica‐
tions.Theknowledgeoftheelasticbehaviourofthismineralunderpressureisimportant
Figure 6.
Analysis of the seismic wave velocities (Lambert equal-area upper hemisphere projections)
of sodalite at 12.8 GPa, showing (
a
) the phase velocities v
P
(km/s), (
b
) the group velocities v
P
(km/s),
(c) the enhancement factor Aand (d) the power flow angle (PF, ◦).
4. Conclusions
In the present work, we provided for the first time an ab initio investigation of the
crystal chemical and elastic properties of sodalite
[Na8(Al6Si6O24)Cl2, space group P43n]
,
a zeolite mineral with important fundamental (geological and minero-petrographic) and
technological (e.g., catalysis and separation, building and construction materials) applica-
tions. The knowledge of the elastic behaviour of this mineral under pressure is important
Minerals 2022,12, 1323 13 of 15
to assess its mechanical stability and guide the possible use of the material in specific
applications. The study was conducted at the Density Functional Theory level, using the
well-known hybrid functional B3LYP corrected with the DFT-D3 scheme to include the
contributions arising from van der Waals interactions.
Albeit being at absolute zero (0 K), the third-order Birch–Murnaghan equation of the
state of the mineral calculated from the U(V) curves up to about 22 GPa is in line with the
few experimental studies reported in the literature. The analysis of the internal geometry
showed that the compression mechanism is controlled by both the Na–O and Na–Cl
distances, and hence on the respective tetrahedral volumes, and the T–O–T bridging angle.
Conversely, the aluminosilicate framework, i.e., the Si–O and Al–O bonds, was the least
affected by pressure. No sign of abrupt structural variations in the considered hydrostatic
pressure range suggested a possible phase transition or decomposition. However, it must
be recalled that the simulations were conducted by constraining the symmetry, thus the
unit cell and its atoms were free to relax within the symmetry operations of the
P4
3
n
space
group. In fact, from the elastic point of view, the necessary and sufficient Born stability
criteria were not satisfied above 15.6 GPa, meaning that sodalite is not mechanically stable
above this pressure threshold. In future, high-pressure simulations and thermodynamic
analyses will be performed to check if other possible, more energetically favourable crystal
structures exist.
Finally, the second-order elastic moduli of sodalite and their functional variation with
pressure were reported, information of utmost relevance for both geophysical studies
and applications of the zeolite. The variation of the elastic moduli is well described by a
finite-strain fit using second-order polynomial functions. In addition, single-crystal and
polycrystalline elastic properties derived from the stiffness tensor were calculated as a
reference for future studies on this mineral phase.
Author Contributions:
Conceptualization, G.U. and G.V.; methodology, G.U.; validation, G.U. and
G.V.; formal analysis, G.U.; investigation, G.U. and G.V.; data curation, G.U.; writing—review and
editing, G.U. and G.V.; visualization, G.U.; supervision, G.V. All authors have read and agreed to the
published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data are available within the present article.
Acknowledgments:
The authors wish to thank the University of Bologna for supporting the present research.
Conflicts of Interest: The authors declare no conflict of interest.
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