Table 1 reports the lattice parameters, tetrahedral (Tthick) and octahedral (Mthick) sheet thicknesses, interlayer distance (Ithick), and selected polyhedral properties of the 2M1 phlogopite polytype, obtained from DFT simulations with (B3LYP-D*) and without (B3LYP) the correction to include the effects of long-range interactions. Previous XRD refinements are also shown in Table 1 for a direct comparison.

The simulations at the DFT/B3LYP-D* level of theory are in good agreement with the experimental crystallographic data reported in literature, showing a small underestimation of the unit cell volume (ΔV =-1.0%), which is the result of a slight expansion of the tetrahedral and octahedral sheet thicknesses (ΔTthick=1.5% and ΔO_thick =1.8%, respectively) and a contraction of the interlayer distance (ΔIthick=-2.5%). These results for the 2M1-phlogopite are in agreement with previous simulations of the 1M polytype (Ulian and Valdrè, 2023a) and other phyllosilicates and layered minerals (Ulian et al., 2013; Ulian and Valdrè, 2015a, b, 2019), where the inclusion of long-range interactions in the physical treatment is of utmost importance to properly simulate the crystal-chemistry and properties of this kind of structure. When the DFT-D2 correction is not implemented in the simulation framework, a dramatic increase in the unit cell volume can be noted (+6% with respect to the DFT/B3LYP-D* results, +5% from the XRD data), which is the result of the increase of the lattice parameters a (+0.9%), b (+0.9 %), and c (+3.6 %) and the shrinking of the β angle (-2.1%). Considering the absence of any thermal effects in the Density Functional Theory simulations, the overestimation of the lattice constants and unit cell volume is a further sign of the relevant role played by van der Waals interactions in micas.

The compressional behaviour of phlogopite was modelled considering 10 different unit-cell volumes, both smaller (compressed, 7 volumes) and larger (expanded, 3 volumes), within 88 % and 108 % of the equilibrium geometry volume (Veq). Then, the internal coordinates and lattice parameters were optimized keeping the selected volume fixed during the procedure, constraining the space group of 1M-phlogopite (P2) and 2M1-phlogopite (P2/c), a procedure known as “symmetry preserving, variable cell-shape structure relaxation” that was proposed by Pfrommer et al. (1997). The equilibrium geometry of 1M-phlogopite was retrieved from the work of Ulian and Valdrè (2023a) that was obtained with the same computational settings here adopted for the 2M1 polytype.

The dependence of the total energy E of the mineral at the different volumes V, i.e., the E(V) curve, was described in terms of a volume-integrated 3rd-order Birch-Murnaghan equation of state, BM3 (Birch, 1947), as reported by Hebbache and Zemzemi (2004): 1EV=E0+916K0V0K′η2-13+η2-126-4η22η=V0V1/3 where η is a dimensionless parameter, K0 is the bulk modulus at 0 K, K′ its pressure first derivative and V0 the volume at zero pressure. The pressure values at each unit cell volume (reported in Tables S2 and S3 for the 1M and 2M1 polytypes, respectively) were calculated using the fitting parameters K0, K′ and V0 obtained from the previous equation and employed in the well-known P-V formulation of the BM3 (Birch, 1947): 3PV=P0+32K0η7-η51-344-K′η2-1 with P0 the reference pressure (0 GPa). In Table 2, the BM3 equation of state parameters for both 1M and 2M1 phlogopite are reported. A graphical representation of the normalised volume and axis lengths as a function of calculated pressure is presented in Fig. 2., alongside the experimental results from X-ray diffraction (Comodi et al., 2004; Pavese et al., 2003) and the theoretical calculations of Chheda et al. (2014) carried out at the LDA and GGA level of theory.

A direct comparison between the different experimental and computational approaches is however possible on the phlogopite-1M, which is better depicted in Fig. S1 (Supplement) considering the absolute values of the cell volume and lattice vectors. In general, compared to the high-pressure XRD refinements, the crystal structure from the DFT/B3LYP-D* approach well agrees with the results of Pavese et al. (2003) at low pressure (P<5 GPa), however there is a slight overestimation of the unit cell volume at high pressure (see Figs. 2a and S1a). Analysing the lattice vectors, it can be noted that the a and b lengths are very close to the experimental ones at each pressure state, whereas the c-axis shows a stiffer behaviour and an overestimation that increases with pressure. By considering the results of Comodi et al. (2004), the present B3LYP-D* data are slightly underestimated in terms of unit cell volume, which is due to the lower values of the a- and b-axes, whereas the evolution of the clattice parameters is correctly described by the proposed computational approach (see Fig. S2b–d). The stiffer behaviour of both polytypes at higher pressures elucidated by the V/V0 trend is related to the static conditions of the simulations, i.e., the zero-point and thermal effects were neglected. It is suggested that the use of the quasi-harmonic approximation, which is however beyond the scope of the present work, could reduce the mismatch between the theoretical and experimental data.

It is interesting to note that if the long-range interactions are not included in the theoretical framework, the B3LYP functional leads to a general overestimation of the unit cell volume and lattice parameters. The a and b vectors are closer to the results of Comodi and collaborators (2004) than those of Pavese et al. (2003), however, the c lattice parameter is severely overestimated when compared to both XRD results. All these considerations are reflected in the PV fitting using the BM3 equation of state, where the B3LYP-D* approach resulted in a small overestimation of the bulk modulus K0 when compared to the experimental data (see Table 2), and the theoretical K' and V0 values between the ranges obtained from the XRD refinements. Conversely, the B3LYP data shows a large underestimation of the bulk modulus of about 18 GPa, and an overestimation of the cell volume.

Compared to previous theoretical results, the crystal structure of the phlogopite 1M polytype and the related PV fitting obtained from the uncorrected B3LYP functional are almost identical to the GGA values of Chheda et al. (2014). Instead, the B3LYP-D* approach, which includes the modified DFT-D2 correction, provided crystallographic results that are between those calculated using the LDA (lower bound in Fig. S1) and the GGA functionals (upper bound). Interestingly, the LDA approach led to a correct description of the c lattice parameter with pressure, whereas the generalised-gradient approximation functional resulted in the same overestimation noted with the B3LYP approach. Thus, GGA and hybrid B3LYP functionals provide an adequate description, i.e., within the experimental information, of the a and b lattice parameters, where strong ionic/covalent bonds are responsible of the TOT structure; though, both approaches fail in properly describing the c-axis behaviour. This is a further confirmation of the hypothesis that the forces acting along the [001] direction of micas are not only electrostatic/ionic, but there is also a contribution from weak van der Waals interactions (Ulian and Valdrè, 2015c, 2023a), as demonstrated using the DFT-D2 correction. The authors are aware that the B3LYP-D* approach, whose DFT-D2 parameters were empirically recalibrated on molecular crystals (Civalleri et al., 2008), is just one of the possible methods to properly treat long-range interactions, and other more ab initio methods could be employed, e.g., the vdW functionals developed of Dion et al. (2004). However, the proposed theoretical framework was already suitable to provide new insights into this fundamental topic regarding the bonding nature of mica phyllosilicates.

The axial compressibility, M0, was obtained using a linearized form of the third-order Birch-Murnaghan equation of state, as described by Angel et al. (2014), which was also employed to re-calculate the experimental compressibility reported by Pavese et al. (2003) and Comodi et al. (2004) including the uncertainties on pressure and lattice parameters. The results are shown in Table 3, alongside the theoretical results of Chheda and collaborators (2014). The results are consistent for all the simulations, as the bulk modulus K0=M0-1a+M0-1c+M0-1c calculated from the axial compressibility is within 2.64 % (1M) and 1.67 % (2M1) from the K0 value obtained from the equation of state fit. In general, it can be noted that the K0 values are almost the same between the two polytypes. However, the 1M and 2M1 phlogopite models showed a slightly different behaviour in terms of axial compression. In fact, while the M0(a) and M0(b) values are the same between the two phlogopite models, the M0(c) modulus is significantly higher in the 2M1 polytype, with a stiffness increase of about 10 % with respect to the 1M one. Similar results were obtained by Chheda et al. (2014) from DFT simulations and by Pavese et al. (2003) from XRD experiments. Also, the recalculated values of Comodi et al. (2004) with the linearized formulation lead to a bulk modulus within 0.57 % the value obtained from the PV EoS fitting.

Crystal structure data of both 1M and 2M1 polytypes are reported in Tables S2 and S3, respectively, providing further insights into the pressure effects on the phlogopite internal geometry.

The TO₄ tetrahedra showed a stiff behaviour, with zero-pressure bulk moduli equal to 282(3) and 285(2) GPa for the 1M and 2M1 polytypes, respectively, whereas the octahedral MO₆ units were more compressible, presenting a bulk modulus of 143(1) GPa for both phlogopite polytypes (Fig. 3a). This means that the TOT layer of the two minerals presented similar elastic properties. As also explained in previous works on phyllosilicates (Comodi et al., 2004; Chheda et al., 2014; Ulian and Valdrè, 2015c), the variation of the tetrahedral rotation angle αr (Donnay et al., 1964b, a) is the main mechanism that accommodated the compression of the mineral. The αr value is greater than 0 for both phlogopite polytypes at equilibrium, which means that the TO₄ mesh is not hexagonal (ideal) but forms a di-trigonal ring (see Fig. 1), and αr increases with pressure as reported in Fig. 3b. This trend is in line with the simulations performed by Chheda and collaborators (2014), and with the general compressional behaviour of trioctahedral micas reported by several authors (Comodi and Zanazzi, 1995; Gatta et al., 2015, 2011; Pavese et al., 1999, 2003; Pawley et al., 2002). In addition, it is interesting to note that the octahedral flattening angle ψ became smaller by increasing pressure in the 1M polytype, whereas a direct proportionality was observed for the 2M1 one.

To provide a complete analysis of the elastic behaviour of phlogopite polytypes, the second-order elastic tensor of the two models was also calculated. The approach relies on stress-strain relationships based on total energy E(V, ε) calculations through a Taylor expansion in terms of the strain components truncated at the second order: 4EV,ε=EV0+V∑ασαεα+V2∑αβCαβεαεβ+… where ε is the strain, σ is the stress, C is the 6×6 matrix representation of the elastic tensor (Voigt's notation), V is the volume of the strained unit cell and V0 is the volume at equilibrium. In this expression, α, β=1, 2, 3, … 6. The adiabatic second-order elastic moduli are related to the second derivatives of the total energy E on the strain according to: 5Cαβ=1V∂2E∂εα∂εβ0 The Cαβ values were calculated by imposing a certain amount of strain ε along the crystallographic direction corresponding to the component of the dynamical matrix. This procedure is automated in the CRYSTAL code (Perger et al., 2009), and in the present simulations the default values that control the calculation of the elastic moduli were employed. To calculate the elastic tensor, the crystal was oriented with the b and c crystallographic axes parallel to the y and z Cartesian axes, respectively. Within this crystal orientation and considering the monoclinic lattice, the 6×6 matrix C of the elastic moduli in Voigt's notation is given by: 6C=C11C12C130C150C22C230C250C330C350C440C46C550C66, whose values are reported in Table 4, considering both B3LYP and B3LYP-D* approaches. As expected, the inclusion of long-range interactions leads to a stiffer behaviour of phlogopite, especially for the diagonal terms Cαα.

Very few experimental data were reported in the scientific literature because suitable single crystals of phlogopite are rare, and many of these specimens are required for a complete characterisation of the elastic tensor of low-symmetry phases. Nevertheless, the present simulations are in good agreement with the experimental results of Alexandrov and Ryzhova (1961) and Aleksandrov et al. (1974), where the authors measured the elastic moduli from two crystals, one from the Slyudyanka and one from the Aldan region, near Lake Baikal (Russia). It is worth noting that only 5 independent elastic tensor components related to the hexagonal symmetry were reported instead of the 13 moduli required for the actual monoclinic symmetry of phlogopite. Also, two pairs of elastic moduli were equal, i.e., C11= C22 and C44=C55, and the off-diagonal components C15, C25, C35 and C46 were zero due to symmetry constraints. The present theoretical results at the B3LYP-D* level of theory indeed showed that C11≈C22 and C44≈C55, suggesting the near-isotropic elastic behaviour of the basal plane. Vaughan and Guggenheim (1986) measured the the full monoclinic elastic tensor for the dioctahedral equivalent of phlogopite, i.e., muscovite KAl₂Si₃AlO₁₀(OH)₂, finding a similar elastic behaviour with a small, but not insignificant, deviation from the hexagonal symmetry. Within this computational framework, the calculated Cαα moduli of phlogopite are overestimated by about 31 GPa (α=1, 2, 3) and 13 GPa (α=4, 5, 6) on average, but the trends of the diagonal components C11≈C22>C33 and C44≈C55<C66 are the same as those experimentally obtained from elastic wave propagation experiments (Aleksandrov et al., 1974; Alexandrov and Ryzhova, 1961). The uncorrected B3LYP functional resulted smaller absolute deviations from the experimental data (ca. 4 and 2 GPa for the uniaxial and biaxial tensor components, respectively) and the same C11≈C22>C33 relationship; however, the C44 modulus was smaller than the C55 one, and also much smaller than the one calculated at the B3LYP-D* level. Regarding the off-diagonal components, both approaches resulted in C12>C13, although the uncorrected functional showed C13>C23, while the inclusion of the van der Waals interactions provided C13≈C23, which is in better agreement with the experimental evidence.

The different performance of the B3LYP and B3LYP-D* approaches in calculating the diagonal and off-diagonal components of the stiffness tensor depends on the underlying physics of the elastic response. The diagonal elastic constants are dominated by the response of the strongly bonded T–O–T layers. When dispersion is neglected (B3LYP), long-range interactions are underestimated, leading to an overexpanded interlayer region. In Gaussian basis-set calculations, this partly compensates the overestimation of forces caused by Pulay stress (as discussed below), producing an accidental error cancellation and a seemingly better agreement for the diagonal terms. In contrast, the off-diagonal terms describe shear-normal coupling and interlayer-intralayer stress transfer, which are very sensitive to the correct description of weak interactions. Without dispersion (B3LYP), the layers are too decoupled, and the coupling terms Cαβ (α≠β) are underestimated or show incorrect relative magnitudes. Including dispersion (B3LYP-D*) restores realistic interlayer distances and mechanical coupling, improving the prediction of the off-diagonal elastic constants. This further confirms that the proper treatment of long-range interactions is relevant for the fundamental analysis of the elastic properties of layered silicates, especially when using Gaussian-type orbitals basis sets.

The present results at the B3LYP(-D*) level of theory are also in line with the simulations performed by Chheda and collaborators (2014) using LDA and GGA functionals and plane-waves basis sets, albeit some notable differences are present (see Table 4). Considering the cited work, it is known that the local density approximation generally underestimates lattice vectors and overestimates the forces, resulting in a stiffer behaviour of simulated phases. Conversely, GGA functionals are generally underbinding, hence the calculated elastic properties can be slightly underestimated. Then, hybrid functionals like B3LYP, which include a small fraction of the exact Hartree-Fock exchange to the total energy of the system, are expected to provide results that are between the LDA/GGA extremes if, and only if, other computational parameters (basis set type and quality, k-point sampling, converge criteria, etc.) are the same. Another source of discrepancy between the present elastic results and those of Chheda et al. (2014) is indeed the basis set type. Localised Gaussian-type orbitals generally provide slightly larger Cαβ values than those obtained with plane-waves basis sets in density functional theory simulation because of the Pulay forces that arise from the Hellmann-Feynman theorem and the use of atom-centred GTOs. As previously explained (Ulian et al., 2021; Ulian and Valdrè, 2018), the Pulay forces are associated with the basis set superposition error (BSSE), which is due to the incompleteness of GTOs and that artificially increases the calculated stress/force within a solid phase. The authors are aware that an approach like the geometrical counterpoise (gCP) method (Brandenburg et al., 2013; Kruse and Grimme, 2012) devised for molecule and molecular crystals could at least partially remove this BSSE-related error. However, the current implementation is limited to some general-purpose GTO basis sets that do not include the one employed in the present work, and a thorough analysis of its suitability for inorganic crystals and minerals is still ongoing (Ulian and Valdrè, 2024b). Besides the Pulay stress, the general overestimation of the theoretical results, both the present one and those of Chheda et al. (2014), could be also due to the athermal conditions, i.e., the DFT simulations were carried out at 0 K without zero-point energy and any thermal contribution, while the elastic wave propagation measurements are typically carried out at room temperature (about 300 K). Regarding the present work, the use of gCP and including temperature effect, e.g., with the quasi-harmonic approximation, can provide results that are in more agreement with the experimental findings.

For practical applications, it is commonly preferred the use of the polycrystalline averages, namely the bulk (K) and shear (μ) moduli, calculated with the Voigt–Reuss–Hill approach (Hill, 1952; Nye, 1957) according to the formulas: 7KV=1/9C11+C22+C33+2C12+C13+C238KR=S11+S22+S33+2S12+S13+S23-19μV=C11+C22+C33+3C44+C55+C66-C12+C13+C231510μR=154S11+S22+S33-S12+S13+S23+3S44+S55+S66 where Sij=Cij-1 are the component of the compliance tensor, i.e., the inverse of elastic moduli tensor C, and the subscripts V and R indicate the Voigt (upper) and Reuss (lower) bound of the isotropic elastic properties. The Young's modulus E and the Poisson's ratio υ were calculated from the following relations: 11E=9Kμ3K+μ and 12υ=3K-2μ23K+μ The polycrystalline mechanical properties were reported in Table 5. The KR values obtained for both phlogopite 1M and 2M1 polytypes were 58.8 and 59.4 GPa, respectively, which are consistent with those calculated through the BM3 fitting procedure (57.9 and 58.3 GPa, respectively).

The elastic anisotropy of phlogopite was described using the universal anisotropic index, AU, as proposed by Ranganathan and Ostoja-Starzewski (2008): 13AU=5μVμR+KVKR-6≥0. For all but isotropic systems, for which AU=0, the ratio between the Voigt and Reuss moduli is not equal to unity, thus AU>0. The percentage anisotropy in compression (AK) and shear (Aμ) was also calculated according to the following formulas (Ranganathan and Ostoja-Starzewski, 2008): 14AK=100KV-KRKV+KRAμ=100μV-μRμV+μR The anisotropies obtained via these expressions were reported in Table 5, showing that the AU values of the two phlogopite polytypes are very similar (about 3.8 at the B3LYP-D* case), and between the values calculated from the theoretical results of Chheda and collaborators (2014) using the LDA (2.9) and GGA (4.4) functionals. There is a slight discrepancy between the different DFT approaches and the experimental results (AU=11.8), which is related to the high anisotropy of the shear moduli. It is suggested that the high μV/μR ratio is probably due to the very small values of the C44=C55 elastic tensor components that were experimentally measured by Aleksandrov et al. (1974) with elastic wave propagation methods. Similar figures were found for the bulk and shear anisotropies, with AK≈14% and Aμ≈26% calculated at the B3LYP-D* level of theory, intermediate values between those obtained using LDA and GGA functionals by Chheda and co-workers (2014). Conversely, the uncorrected B3LYP functional led to an anomalously high anisotropy index (35.2) due to the very large difference between the Voigt and Reuss shear moduli (Aμ≈77%).

To further assess the validity of the linear model used to calculate the axial compressibilities, the axial moduli M(a), M(b) and M(c) were calculated from the elastic compliance tensor components according to the expressions reported by Mookherjee et al. (2016): 15Ma=βa-1=S11+S12+S13-1Mb=βc-1=S12+S22+S23-1Mc=βc-1=S13+S23+S33-1 For instance, the calculated moduli were M(a)=344.3, M(b)=365.2 and M(c)=88.0 GPa for phlogopite-1M, and M(a)=360.5, M(b)=353.9 and M(c)=89.1 GPa for the 2M1 polytype, values consistent with those reported in Table 3 obtained from the equation of state fit and that provide another measure of the anisotropy of the mineral, with a ratio M(a):M(b):M(c)≈4:4:1 for both polytypes.

Directional single-crystal elastic properties (Young's modulus E, linear compressibility β= K⁻¹, shear modulus μ and Poisson's ratio υ) were calculated in the Cartesian space using the QUANTAS code (Gaillac et al., 2016; Ulian and Valdrè, 2022). The interested reader can find in the cited works all the formulations employed to calculate the spatial dependence of the cited properties. Differently from the other elastic properties reported above, where no significant variation between the 1M and 2M1 phlogopite polytypes was evinced, the directional properties showed some effect due to the different TOT stacking (see Fig. S1). For example, the shape of the Young's modulus on the (xy) plane (Fig. S1a) is slightly more compressed along NE-SW and NO-SE directions in the polytype-1M, while it appeared more isotropic in phlogopite-2M1. The shear moduli on the (xz) plane of the latter polytype (see Fig. S1c) is slightly canted, with the oblate maxima direction almost parallel to the east-west direction.

Finally, the phase (seismic) velocities were calculated by solving Christoffel's equation (Musgrave, 1970) using the routines implemented in the QUANTAS code (Jaeken and Cottenier, 2016; Ulian and Valdrè, 2022). The results are shown in Fig. 4, where the compressional wave velocity (vP, P-wave, primary), the anisotropy of the shear velocities (S-wave, secondary), and the polarization of the fast shear wave (vS1) are reported for both phlogopite polytypes. From the stereographic projections, particularly the S-wave anisotropy, it is immediately visible an east-west mirror plane normal to the b-axis that is related to the monoclinic crystal system of the mineral. This figure is in very good agreement with the theoretical analysis of Chheda et al. (2014), with some degree of deviation of the absolute values of the seismic velocities and anisotropy due to the stiffer elastic moduli obtained with the combination of the Gaussian-type orbitals basis sets and hybrid B3LYP-D* functional.

Table 6 reports the maximum and minimum vP values, the percentage of vP anisotropy (AvP) and the range of S-wave anisotropy (AvS), as calculated from the present GTO/B3LYP(-D*) simulations for both phlogopite polytypes, together with previous experimental (Aleksandrov et al., 1974) and theoretical (Chheda et al., 2014) data related to the phlogopite 1M. The results are in general good agreement, with a slight overestimation of the maximum P-wave velocity and a smaller minimum primary wave velocity obtained from the uncorrected B3LYP functional. The shear wave anisotropy at the B3LYP-D* level of theory is within the values obtained by Chheda and collaborators (2014), with the maximum AvS value being underestimated. This latter datum is instead more than double when calculated from the B3LYP results, and very overestimated compared to the experimental value reported by Aleksandrov et al. (1974).

As mentioned in the introduction, the knowledge of the elastic properties could help explain and interpret some geophysical observations reported in previous analyses. For instance, a large delay time (δt) of about 1–2 s was measured in certain subduction zones, like the Ryukyu and Calabrian arcs, in addition to the trench-parallel shear wave splitting (Long and Silver, 2008). Chheda and collaborators (2014) suggested that peridotitic rocks with olivine as the major component would require a very large thickness ranging from 100 to 150 km to justify this large δt value. However, phyllosilicates like talc (Mainprice et al., 2008; Ulian et al., 2014), muscovite (Ulian and Valdrè, 2015a), antigorite (Mookherjee and Capitani, 2011) and phlogopite (Pavese et al., 2003; Chheda et al., 2014) present significantly higher anisotropic behaviours than olivine. Thus, their presence in the subduction layer would require smaller thickness (∼10–15 km) to explain both the trench parallel shear wave polarization anisotropy and the large delay time (Chheda et al., 2014). Indeed, according to previous investigations (Trønnes, 2002), phlogopite has a wide PT stability, thus it can be found in different settings, e.g., in the upper mantle and subduction zone, the latter being also characterised by hydrating conditions that could further stabilise this trioctahedral mica and other hydrous phases.