The packing fraction of the oxygen sublattice: Its impact on the heat of mixing

The packing fraction of the oxygen sublattice: Its impact on the heat of mixing | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article The packing fraction of the oxygen sublattice: Its impact on the heat of mixing Artur Benisek, Edgar Dachs This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3950395/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The heat of mixing of some petrological relevant substitutions (i.e., Mg-Al, Si-Al, Mg-Ti, Mg-Ca, and Mg-Fe) was investigated systematically in silicates, titanates, tungstates, carbonates, oxides, hydroxides, and sulphates by density functional theory calculations (e.g., melilite, chlorite, biotite, brucite, cordierite, amphibole, talc, pseudobrookite, pyroxene, olivine, wadsleyite, ilmenite, MgWO 4 , ringwoodite (spinel), perovskite, pyrope-grossular, magnesite-calcite, MgO-CaO, anhydrous and different hydrated MgSO 4 ). A specific substitution is characterised by different microscopic interaction energies in different minerals, e.g., the octahedral Mg-Al exchange on a single crystallographic site in pyroxene has a microscopic interaction energy that is more than twice compared to that in biotite. A comparative investigation of the heat of mixing using microscopic interaction energies on a single crystallographic site has the advantage that they are not influenced by cation ordering. They could be successfully correlated with the stiffnesses of the minerals, which in turn were scaled to the oxygen packing fraction, a parameter that is easily available for poorly investigated minerals. With this information, the interaction energies of a certain substitution can be transferred from minerals where they are well-known to mineral groups where they are less- or unknown. Using the cross-site terms and the microscopic interaction energies, the macroscopic interaction energies of the coupled substitution, e.g., Mg+Si = Al+Al, of biotite and pyroxene were calculated, which are, however, affected by cation ordering and different degrees of local charge balance, for which appropriate models are necessary. Physical Chemistry interaction energy excess enthalpy of mixing stiffness oxygen packing density density functional theory Figures Figure 1 Figure 2 Figure 3 Figure 4 1. Introduction To describe the heat of mixing ( Δ H mix ) more accurate, a decomposition of the macroscopic interaction energies into interaction energies between elements on different crystallographic sites has been proposed, termed the micro-f approach (Powell et al. 2014 ). It was used by these authors to extend these micro-f parameters from minerals where good data exist to minerals where no or little data exist without considering the physical differences between them. Such an extension is, however, questionable. The different polyhedron sizes of the substituted and substituent cations generate local strain heterogeneities and hence strain energy in the solid solution giving rise to the heat of mixing/disordering (e.g., Boffa Ballaran et al. 1998, Tarantino et al. 2003 , Carpenter 2003 ). Strain was found to play an important role in generating heat of mixing in many studies (e.g., Davies and Navrotsky 1983 , Christian 1975, Geiger 2001 , Urusov 2001 ). Furthermore, it is to be expected that the generated strain energy depends on the stiffness of the mineral (e.g., Urusov 2001 , Tarantino et al. 2003 ). Interaction energy differences between minerals might thus be correlated to differences in mineral elasticity. In biotite, for example, the energetic effect of the Tschermak´s substitution is lower than half of that in pyroxene (Dachs and Benisek 2019 , Benisek et al. 2007 ), although the difference in end-member volumes is similar in both minerals (Dachs and Benisek 2019 , Etzel et al. 2007 ). However, a systematic investigation of Δ H mix versus mineral stiffness is difficult because cation ordering has also a significant influence on the interaction energies, especially when coupled substitutions are considered, where the degree of local charge balances complicates the situation (Vinograd et al. 2007 , Benisek and Dachs 2020 ). The decomposition of the macroscopic interaction energies into interaction energies between elements on a single crystallographic site (microscopic interaction energies), however, allows a comparison excluding the effect of cation ordering and local charge balances. We present here results from density functional calculations on microscopic interaction energies for a number of petrological relevant substitutions and their dependence on mineral elasticity. 2. Experimental methods Calculations using the density functional theory (DFT) Quantum-mechanical calculations were based on the DFT plane wave pseudopotential approach implemented in the CASTEP code (Clark et al. 2005) included in the Materials Studio software from Biovia®. The calculations used the local density approximation (LDA) for the exchange-correlation functional (Ceperley and Alder 1980 ) and norm-conserving pseudopotentials to describe the core-valence interactions. Studying the Mg-Fe substitution, the LDA + U (2.5 eV) method and a ultrasoft potential was used. For the k-point sampling of the investigated unit cells, a Monkhorst-Pack grid (spacing of 0.04 Å − 1 ) was used (Monkhorst and Pack 1976 ) and convergence was tested by performing calculations using a denser k-point grid. The structural relaxation was calculated by applying the BFGS algorithm (Pfrommer et al. 1997 ), where the convergence threshold for the force on an atom was 0.01 eV Å − 1 . The excess internal energy of mixing ( Δ E mix ) was calculated using the single defect method, which is a particularly effective way to calculate it using DFT methods (e.g., Sluiter and Kawazoe 2002 ). The obtained value represents Δ E mix of the disordered state (Vinograd and Sluiter 2006 , Vinograd and Winkler 2010). The difference between Δ E mix and Δ H mix is given by the volume term P Δ V mix , where P is the pressure and Δ V mix is the excess volume of mixing, which is typically less than 1 J/bar/mol. At a pressure of 0 Pa, where the results of this study are compiled, P Δ V mix is zero and Δ E mix = Δ H mix . Unit cells with element substitutions involving different charges, i.e., the Mg-Al substitution, were generated by charging the unit cell itself. The energy calculations of charged unit cells can be corrected (Makov and Payne 1995 ), which was not feasible during this study because this correction is not implemented in the Materials Studio´s CASTEP code. However, as shown below, the results were verified by transforming the microscopic interaction energies into macroscopic ones, where no charged unit cells exist, and compared with independently derived DFT-calculated macroscopic interaction parameters. Generally, the symmetry constraints of a unit cell that incorporates a single defect are destroyed generating a cell with P1 symmetry. This local structural situation cannot be observed by X-ray diffraction because this technique averages over hundreds of unit cells and does not yield the correct crystal system of a given single unit cell. However, in a real crystal, a given unit cell cannot change its symmetry independently from their neighbouring cells. The unit cell with a defect will probably have a symmetry that is lowered compared to the end-member unit cell, but it will be higher than P1 symmetry. These considerations were used to generate the crystal structure of the unit cell that contains a single defect. The investigation focused on the octahedral Mg-Al substitution. The studied minerals are partly characterised by different octahedral sites. If the preferred site for Al is not known, the interaction energies were calculated for each octahedral site. In the case that one site is characterised by a significant lower interaction energy, ordering on this site was assumed (e.g., Al on M1 in olivine). On the other hand, if the energies are similar for different sites, they were combined into one single crystallographic site (e.g., Al on M1 + M2 in wadsleyite). Table 1 presents detailed information on the end-member choice and in consequence on the crystallographic site where mixing was assumed to take place (given in the supplementary material). 3. Results To express the elastic properties of a mineral, the bulk modulus or compressibility is the mostly used parameter, however, for rare minerals it is not well known. During the search for another parameter, the question raised, which structural units determine the stiffness of a mineral. For minerals containing an oxygen sublattice, its packing fraction is likely to be a good proxy for the elastic properties since it dominates the occupancy of the unit cell volume. The oxygen packing fraction (OPF) is a parameter that has the great advantage that it is easily available for poor investigated minerals because OPF can be defined as OPF = n O V O /( V UC ), 1) where n O is the number of oxygens in the unit cell, V O is the volume of the oxygen anion calculated from the data of Shannon (1976), and V UC is unit cell volume. The correlation between OPF and bulk modulus for different minerals is shown in Fig. 1 demonstrating that OPF increases from 51 to 71 % during a bulk-modulus increase from 300 to 2500 kbar. The data are from the compilation of Holland and Powell (2011) and range from tridymite with low bulk modulus over all mineral groups that contain oxygen to stishovite with a high bulk modulus (including 95 minerals). Δ H mix is defined as the difference between the enthalpy of a solid solution ( H AB ) at a given composition and the enthalpy of its mechanical mixture ( X A H A + X B H B ), i.e., Δ H mix = H AB – ( X A H A + X B H B ), 2) where X A , X B and H A, H B represent the mole fractions and the enthalpy of the A and B component, respectively. Δ H mix of different substitutions in different minerals was investigated by DFT using the single defect method (Sluiter and Kawazoe 2002). The data were then described by a symmetric Margules model, Δ H mix = X A X B w H , 3) where w H is the interaction energy between elements on a single crystallographic site (micro w H ). This parameter is listed in Tab. 1 (given in the supplementary material) and plotted against OPF of the Mg-end members in Fig. 2. The Mg-Al substitution was investigated in 16 minerals (ordered by increasing w H ): melilite, chlorite, biotite, cordierite, amphibole, talc, pseudobrookite, pyroxene, sulphate (4 x hydrated), olivine, wadsleyite, sulphate (1 x hydrated), ilmenite, Mg-wolframite, anhydrous sulphate, and ringwoodite that has the spinel structure. As can be seen from Fig. 2, a linear positive correlation of w H versus OPF was found (fit parameters are given in the supplementary material). The Mg-Al substitution in biotite, as an example, has an energetic effect that is less than 1/2 of that in pyroxene, and less than 1/6 of that in spinel. This behaviour is expected to be a consequence of the different oxygen packing fractions, which are 53.5 % in biotite, 59.1 % in pyroxene, and 65 % in spinel. In addition to the Mg-Al substitution, the Si-Al substitution has been investigated in 8 minerals. Here, two trends were observed. One, where the tetrahedra are connected with each other (chain and sheet silicates) investigated in 4 minerals (ordered by increasing w H ): biotite, amphibole, pyroxene, perovskite. The second trend is from island and double island silicates defined by 4 minerals, i.e., melilite, olivine, wadsleyite, ringwoodite. This group is characterised by a much flatter positive correlation. MgSiO 3 perovskite, with w H SiAl /OPF = 577.4/77.1, formed an outlier and was not included in the fit. The interaction energies of the Mg-Ca substitution were investigated in 7 minerals (ordered by increasing OPF): MgO-CaO solid solution, amphibole, olivine, pyroxene, carbonate, garnet, and perovskite). This substitution is characterised by a flat w H /OPF correlation, whereas the Mg-Ti 4+ substitution shows a steep slope that is defined by 4 minerals (ordered by increasing w H ): biotite, pyroxene, olivine, ringwoodite. Finally, 7 minerals were studied for the Mg-Fe substitution finding ideal to slightly negative D H mix behaviour (ordered by increasing OPF): biotite, brucite, pyroxene, olivine, spinel, garnet, and perovskite. Using the interaction energies of the separate Mg-Al and Si-Al substitutions (micro w ´s, i.e., w H MgAl and w H SiAl ) in combination with the underlying cross-site interaction ( w H cross ), the macroscopic interaction energy for the Tschermak´s substitution ( W H TS ) can be calculated (e.g., Powell et al., 2014): W H TS = ¼ (4 w H MgAl + w H SiAl – 2 w H cross ), 4) where the cross-site interaction is given by the energies of the reciprocal reaction: w H cross = H (Mg M Si T ) + H (Al M Al T ) – H (Mg M Al T ) – H (Al M Si T ). 5) Such calculations were performed for the Mg-Al biotite in order to verify the DFT results of charged unit cells. Using eq (4), w H MgAl = 82.5 kJ/mol, w H SiAl = 95.6 kJ/mol, and w H cross of 175.1 kJ/mol, the macroscopic interaction energy of W H TS = 18.8 kJ/mol is obtained. It represents the interaction energy with ordering of Al on M1 and T1 sites, but without strict local-charge balance (without short-range ordering). To compare it with DFT values obtained without using charged unit cells, a Mg-biotite (KMg 3 [(OH) 2 AlSi 3 O 10 ], phlogopite), a Tschermak substituted biotite (KMg 2 Al[(OH) 2 Al 2 Si 2 O 10 ], eastonite) and intermediate solid-solution biotites were investigated. Using 10 different super cells for the intermediate biotites (50:50 composition with randomly distributed Al on M1 and T1 with sizes up to 16 formula units), a W H TS of 19.0 +/- 3.0 kJ/mol was obtained, which is in good agreement with the W H TS calculated via micro w ´s (18.8 kJ/mol). This value is slightly lower than an experimentally derived value of 25.4 kJ/mol (Dachs and Benisek 2019) because of different ordering schemes. The same calculations were performed for the diopside – CaTs solid solution, ( w H MgAl = 213.6 kJ/mol, w H SiAl = 285.6 kJ/mol, and a w H cross = 422.0 kJ/mol) yielding a W H TS of 74.0 kJ/mol. The uncharged cells delivered 74.8 +/- 3.7 kJ/mol, also in good agreement with W H TS obtained via micro w ´s. This value is larger than the experimentally derived value of ~ 24 kJ/mol (Benisek et al. 2007), if a symmetric fit is applied to their data. The difference comes mainly from the fact that the experimental data were derived using a partly disordered CaTs end-member (Benisek and Dachs 2020). 4. Discussion Davies and Navrotsky ( 1983 ) correlated the interaction parameter ( W ) with a normalised volume difference ( Δ V norm ) as a quantity to include the strain, i.e., Δ V norm = (V 2 – V 1 ) / V 2 , 6) where V 2 and V 1 are the end-member volumes. The investigated minerals were also studied in this respect by the DFT methods, and the corresponding results are shown in Fig. 3 and compiled in Table 1 (given in the supplementary material), demonstrating good correlations, if the different substitutions are separated. The w H / Δ V norm and w H /OPF correlations yielded similar results, i.e., the Mg-Ti 4+ has a steep and the Mg-Ca substitution has a very flat slope. They have also similar R-squared values (see supplementary material). However, the Si-Al substitution in MgSiO 3 perovskite is no longer an outlier, as is the case for the w H /OPF correlation. From these points of view, there would be no need to postulate a relationship between w H and OPF or between w H and another parameter describing the mineral elasticity. However, Δ V norm may also incorporate information on the packing density of the oxygen sublattice. To test this idea, Δ V norm was plotted against the oxygen packing fraction (Fig. 4 ) demonstrating a good Δ V norm /OPF correlation. Δ V norm , as used in the study of Davies and Navrotsky ( 1983 ) to parameterise the strain, contains thus also information about the packing density of the oxygen sublattice. For minerals with the same formula but different volumes (olivine versus ringwoodite), the Δ V norm /OPF correlation is easy to explain, because the volume of the ringwoodite is small compared to that of olivine increasing Δ V norm from olivine to ringwoodite (from 0.258 to 0.361) as it is the case with their oxygen packing fractions (from 59.1 to 65.0%). However, for minerals with different formulae, the Δ V norm /OPF correlation is not intuitively understandable. Δ V norm decreases generally with an increase of the number of atoms in the formula unit. It may be that the increase of the number of different elements and different polyhedra in the structure increases the probability that the mineral becomes less dense. However, the data of melilite should also be mentioned in this context. Melilite´s formula has neither a large nor a small number of elements. Nevertheless, the Mg-Al substitution in melilite is characterised by a very low strain ( Δ V norm = 0.042), which is certainly a consequence of its open packed structure (OPF = 49.9%). The oxygen packing fraction may, therefore, be the dominant factor for determining Δ V norm . The coordination number of the substituted cation does not always have the same value within a certain substitution and is quite heterogenous for the Mg-Ca substitution. The Ca site in different minerals increases from 6 to 12 with a simultaneous increase of OPF. These circumstances may result in a flattening of the w H MgCa / Δ V norm and w H MgCa /OPF correlations. In MgSiO 3 perovskite, as another example, Si(Al) has not tetrahedral but octahedral coordination. This might be a reason for producing the perovskite outlier in the w H /OPF correlation. On the other hand, melilite has a tetrahedral Mg(Al)-site and plots on the Mg-Al line, where all other minerals have Mg(Al) in octahedral coordination. Melilite´s w H MgAl is slightly negative, which is a surprising observation, because cation substitutions are thought to produce zero (substitutions without local strain heterogeneities) or positive Δ H mix in the case that local strain heterogeneities exist (Tarantino et al., 2003 ). However, the generated strain for the Mg-Al substitution in melilite is small ( Δ V norm = 0.042). 5. Conclusions It was shown in this study that the interaction energies of some substitutions strongly depend on the oxygen packing fraction, i.e., the Mg-Al and Mg-Ti exchanges and that of the Si-Al in chain and sheet silicates. This dependence can be understood as follows: The denser the oxygen packing, the stronger the impact of the local strain heterogeneities – caused by a substituted cation – will be on the whole unit cell. The strain heterogeneities of the Si-Al substitutions in island silicates may be absorbed by the less rigid neighbouring polyhedra producing the flat w H /OPF correlation. On the other hand, the w H MgCa /OPF correlation may be superimposed by the increasing coordination numbers with increasing OPF. There is also a w H / Δ V norm correlation, which is, however, not easily applicable to substitutions involving cations with different charges because the volume of the charged end members is only accessible by DFT methods. The calculated w H parameters represent the interaction energies between elements on a single crystallographic site. However, the degree of local charge balances and the degree of cation ordering influence the macroscopic interaction energies as demonstrated by DFT calculations (Dachs and Benisek 2019 , Benisek and Dachs 2020 ). Other theoretical studies have also shown that Δ H mix increases with temperature because of such effects, as demonstrated for garnets, diopside-jadeite solid solution, carbonites, and halides (e.g., Vinograd and Sluiter 2006 , Vinograd et al. 2007 , Vinograd et al., 2009 , Vinograd and Winkler 2010). If one end-member is involved in the disordering processes as is the case with Tschermak´s substituted pyroxene and melilite, Δ H mix can decrease with temperature (Sack and Ghiorso 2017 , Sack 2021 ). The use of OPF-dependent interaction energies for substitutions that are characterised by large strains and the formulation of appropriate models for cation ordering should further contribute to the reliability of future mixing models. Declarations Acknowledgement This work was supported by grants from the Austrian Science Fund (FWF), project number P33904, which is gratefully acknowledged. We thank E. Forsthofer for maintaining the Materials Studio software at the Salzburg University. Author Contributions ED initiated the study. AB had the idea, worked out the conception and design, did the calculations and drafted the manuscript. Both authors contributed to the analysis of data, read, and approved the final manuscript. Conflict of interest The authors declare no competing interests. Data availability All available data are given in the text and supplementary material. 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Vinograd VL, Gale JD, Winkler B (2007) Thermodynamics of mixing in diopside-jadeite, CaMgSi 2 O 6 -NaAlSi 2 O 6 , solid solution from static lattice energy calculations. Phys Chem Minerals 34: 713-725. Vinograd VL, Sluiter MHF, Winkler B (2009) Subsolidus phase relations in the CaCO 3 -MgCO 3 system predicted from the excess enthalpies of supercell structures with single and double defects. Phys Rev B 79: 104201. Vinograd VL, Winkler B (2010) An efficient cluster expansion method for binary solid solutions: application to the halite-sylvite, NaCl-KCl, system. Rev Mineral Geochem 71: 413-436. Additional Declarations The authors declare no competing interests. Supplementary Files TablesSupplementaryMaterial.pdf Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3950395","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":272378197,"identity":"a68e8308-f9d5-43c9-b137-cd7b1a8c8de9","order_by":0,"name":"Artur Benisek","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABBUlEQVRIie2RsWrDMBCGfyGwF9GsESnOK8gYWrz0WRzyEoaWxiWQyQ/Q0NJnSJbMAkGyiHp1wYsWd223lC5VSFoyVMRjofqm47hP+o8DPJ6/iwQoQIoc5x2m6bGiwb77/dOKhdzNOii9h2o9yNFEl2FozPypYaKS5G2Lq1uX0n8eB1yjTdIpS+LlqmWizigvMXYH0zTgBdRooVjAzUpZBRjYvE5leFAmCxW+fphHtQtGP4GJUxEHJRMKF2RZWEVmgf1FOZVY0yQtRBvbYAm/X7eM16NZWoqNfeh3Ik3MS5E3Q1FtzHt500RnlVL1Nr/uOdffx/upJHYXPe6cRHYf9Xg8nn/DF+nCVMkETaGSAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0002-1570-5850","institution":"Universtity of Salzburg","correspondingAuthor":true,"prefix":"","firstName":"Artur","middleName":"","lastName":"Benisek","suffix":""},{"id":272378198,"identity":"508b2edf-ef9b-484c-b60e-e2cb22300836","order_by":1,"name":"Edgar Dachs","email":"","orcid":"","institution":"Universtity of Salzburg","correspondingAuthor":false,"prefix":"","firstName":"Edgar","middleName":"","lastName":"Dachs","suffix":""}],"badges":[],"createdAt":"2024-02-12 06:51:33","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-3950395/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3950395/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":51034517,"identity":"40a3523d-c52d-45df-98b5-2834a07aa643","added_by":"auto","created_at":"2024-02-13 04:16:04","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":47968,"visible":true,"origin":"","legend":"\u003cp\u003eOxygen packing fraction (OPF) plotted against bulk modulus for different minerals. Bulk modulus and volume data were taken from Holland and Powell (2011). The volume of the oxygen anion was calculated from the data of Shannon (1976).\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-3950395/v1/99398c81245664b900891467.png"},{"id":51034496,"identity":"73409f5c-7918-42ec-bb74-b5afc21e0f10","added_by":"auto","created_at":"2024-02-13 04:16:03","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":83973,"visible":true,"origin":"","legend":"\u003cp\u003eDFT calculated interaction parameters (\u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e) of different substitutions in different minerals as a function of DFT calculated oxygen packing fraction (OPF). The Mg-Al substitution is represented by a solid blue line and blue squares. The Si-Al substitutions are marked by a green solid line and green diamonds, the Si-Al substitutions with isolated substitution polyhedra by a broken green line and light green diamonds. The Mg-Ti\u003csup\u003e4+\u003c/sup\u003e substitutions are marked by a red line and red circles, the Mg-Ca substitutions by a black line and black stars, and the Mg-Fe substitutions by a magenta line and magenta triangles. Open diamond represents the octahedral Si-Al substitution in MgSiO\u003csub\u003e3\u003c/sub\u003e perovskite and is not included in the fit. Inset of the figure shows the whole Mg-Ti\u003csup\u003e4+\u003c/sup\u003e substitution in relation to the other substitutions.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-3950395/v1/549bb5f687eb5477588a4efa.png"},{"id":51034498,"identity":"0ef2fdbe-73d5-49e6-959c-839e3afb69af","added_by":"auto","created_at":"2024-02-13 04:16:03","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":81968,"visible":true,"origin":"","legend":"\u003cp\u003eInteraction parameters (\u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e) of different substitutions and different minerals as a function of the DFT calculated normalised volume difference (V\u003csub\u003e2\u003c/sub\u003e-V\u003csub\u003e1\u003c/sub\u003e)/V\u003csub\u003e2\u003c/sub\u003e. The Mg-Al substitution is represented by a solid blue line and blue squares. The Si-Al substitution is marked by a green solid line and green diamonds. The island and double island silicates are characterised by a (V\u003csub\u003e2\u003c/sub\u003e-V\u003csub\u003e1\u003c/sub\u003e)/V\u003csub\u003e2\u003c/sub\u003e of 0.17 – 0.19 and a \u003cem\u003eW\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e of ca. 170 kJ/mol marked by light green diamonds and are not included in the fit (solid green line). The Mg-Ti\u003csup\u003e4+\u003c/sup\u003e substitutions are marked by a red line and red circles, the Mg-Ca substitutions by a black line and black stars, and the Mg-Fe substitutions by magenta triangles. Inset of the figure shows the whole Mg-Ti\u003csup\u003e4+\u003c/sup\u003e substitution in relation to the other substitutions.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-3950395/v1/0caa1daa3219fa9b5a2b9075.png"},{"id":51034494,"identity":"50edc243-d16d-49e4-8172-305ecc6de66d","added_by":"auto","created_at":"2024-02-13 04:16:03","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":65425,"visible":true,"origin":"","legend":"\u003cp\u003eNormalised volume difference ((V\u003csub\u003e2\u003c/sub\u003e-V\u003csub\u003e1\u003c/sub\u003e)/V\u003csub\u003e2\u003c/sub\u003e) plotted against the oxygen packing fraction (OPF). The Mg-Al, Mg-Ti\u003csup\u003e4+\u003c/sup\u003e and the Si-Al substitutions are represented by blue squares, red circles, and green diamonds, respectively.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-3950395/v1/2e2a16477453692987086b37.png"},{"id":51034674,"identity":"678d22a9-a8cf-48af-87a9-23f090887c2e","added_by":"auto","created_at":"2024-02-13 04:24:06","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":435058,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3950395/v1/7b018de4-b4c7-4f5d-9ea0-c7c887bec0f5.pdf"},{"id":51034497,"identity":"ed5e4821-3dc6-4380-9b44-b54d36b8d9ef","added_by":"auto","created_at":"2024-02-13 04:16:03","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":209446,"visible":true,"origin":"","legend":"","description":"","filename":"TablesSupplementaryMaterial.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3950395/v1/17e1b8a97400aae019e26916.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eThe packing fraction of the oxygen sublattice: Its impact on the heat of mixing\u003c/p\u003e","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eTo describe the heat of mixing (\u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eH\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e) more accurate, a decomposition of the macroscopic interaction energies into interaction energies between elements on different crystallographic sites has been proposed, termed the micro-f approach (Powell et al. \u003cspan class=\"CitationRef\"\u003e2014\u003c/span\u003e). It was used by these authors to extend these micro-f parameters from minerals where good data exist to minerals where no or little data exist without considering the physical differences between them. Such an extension is, however, questionable. The different polyhedron sizes of the substituted and substituent cations generate local strain heterogeneities and hence strain energy in the solid solution giving rise to the heat of mixing/disordering (e.g., Boffa Ballaran et al. 1998, Tarantino et al. \u003cspan class=\"CitationRef\"\u003e2003\u003c/span\u003e, Carpenter \u003cspan class=\"CitationRef\"\u003e2003\u003c/span\u003e). Strain was found to play an important role in generating heat of mixing in many studies (e.g., Davies and Navrotsky \u003cspan class=\"CitationRef\"\u003e1983\u003c/span\u003e, Christian 1975, Geiger \u003cspan class=\"CitationRef\"\u003e2001\u003c/span\u003e, Urusov \u003cspan class=\"CitationRef\"\u003e2001\u003c/span\u003e). Furthermore, it is to be expected that the generated strain energy depends on the stiffness of the mineral (e.g., Urusov \u003cspan class=\"CitationRef\"\u003e2001\u003c/span\u003e, Tarantino et al. \u003cspan class=\"CitationRef\"\u003e2003\u003c/span\u003e). Interaction energy differences between minerals might thus be correlated to differences in mineral elasticity. In biotite, for example, the energetic effect of the Tschermak\u0026acute;s substitution is lower than half of that in pyroxene (Dachs and Benisek \u003cspan class=\"CitationRef\"\u003e2019\u003c/span\u003e, Benisek et al. \u003cspan class=\"CitationRef\"\u003e2007\u003c/span\u003e), although the difference in end-member volumes is similar in both minerals (Dachs and Benisek \u003cspan class=\"CitationRef\"\u003e2019\u003c/span\u003e, Etzel et al. \u003cspan class=\"CitationRef\"\u003e2007\u003c/span\u003e).\u003c/p\u003e\n\u003cp\u003eHowever, a systematic investigation of\u0026nbsp;\u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eH\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e versus mineral stiffness is difficult because cation ordering has also a significant influence on the interaction energies, especially when coupled substitutions are considered, where the degree of local charge balances complicates the situation (Vinograd et al. \u003cspan class=\"CitationRef\"\u003e2007\u003c/span\u003e, Benisek and Dachs \u003cspan class=\"CitationRef\"\u003e2020\u003c/span\u003e). The decomposition of the macroscopic interaction energies into interaction energies between elements on a single crystallographic site (microscopic interaction energies), however, allows a comparison excluding the effect of cation ordering and local charge balances. We present here results from density functional calculations on microscopic interaction energies for a number of petrological relevant substitutions and their dependence on mineral elasticity.\u003c/p\u003e"},{"header":"2. Experimental methods","content":"\u003cp\u003e\u003cstrong\u003eCalculations using the density functional theory (DFT)\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eQuantum-mechanical calculations were based on the DFT plane wave pseudopotential approach implemented in the CASTEP code (Clark et al. 2005) included in the Materials Studio software from Biovia\u0026reg;. The calculations used the local density approximation (LDA) for the exchange-correlation functional (Ceperley and Alder \u003cspan class=\"CitationRef\"\u003e1980\u003c/span\u003e) and norm-conserving pseudopotentials to describe the core-valence interactions. Studying the Mg-Fe substitution, the LDA\u0026thinsp;+\u0026thinsp;U (2.5 eV) method and a ultrasoft potential was used. For the k-point sampling of the investigated unit cells, a Monkhorst-Pack grid (spacing of 0.04 \u0026Aring;\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e) was used (Monkhorst and Pack \u003cspan class=\"CitationRef\"\u003e1976\u003c/span\u003e) and convergence was tested by performing calculations using a denser k-point grid. The structural relaxation was calculated by applying the BFGS algorithm (Pfrommer et al. \u003cspan class=\"CitationRef\"\u003e1997\u003c/span\u003e), where the convergence threshold for the force on an atom was 0.01 eV \u0026Aring;\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e. The excess internal energy of mixing (\u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eE\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e) was calculated using the single defect method, which is a particularly effective way to calculate it using DFT methods (e.g., Sluiter and Kawazoe \u003cspan class=\"CitationRef\"\u003e2002\u003c/span\u003e). The obtained value represents \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eE\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e of the disordered state (Vinograd and Sluiter \u003cspan class=\"CitationRef\"\u003e2006\u003c/span\u003e, Vinograd and Winkler 2010).\u003c/p\u003e\n\u003cp\u003eThe difference between \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eE\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e and \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eH\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e is given by the volume term \u003cem\u003eP\u003c/em\u003e \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e, where \u003cem\u003eP\u003c/em\u003e is the pressure and \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e is the excess volume of mixing, which is typically less than 1 J/bar/mol. At a pressure of 0 Pa, where the results of this study are compiled, \u003cem\u003eP\u003c/em\u003e \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e is zero and \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eE\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e = \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eH\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e.\u003c/p\u003e\n\u003cp\u003eUnit cells with element substitutions involving different charges, i.e., the Mg-Al substitution, were generated by charging the unit cell itself. The energy calculations of charged unit cells can be corrected (Makov and Payne \u003cspan class=\"CitationRef\"\u003e1995\u003c/span\u003e), which was not feasible during this study because this correction is not implemented in the Materials Studio\u0026acute;s CASTEP code. However, as shown below, the results were verified by transforming the microscopic interaction energies into macroscopic ones, where no charged unit cells exist, and compared with independently derived DFT-calculated macroscopic interaction parameters.\u003c/p\u003e\n\u003cp\u003eGenerally, the symmetry constraints of a unit cell that incorporates a single defect are destroyed generating a cell with \u003cem\u003eP1\u003c/em\u003e symmetry. This local structural situation cannot be observed by X-ray diffraction because this technique averages over hundreds of unit cells and does not yield the correct crystal system of a given single unit cell. However, in a real crystal, a given unit cell cannot change its symmetry independently from their neighbouring cells. The unit cell with a defect will probably have a symmetry that is lowered compared to the end-member unit cell, but it will be higher than \u003cem\u003eP1\u003c/em\u003e symmetry. These considerations were used to generate the crystal structure of the unit cell that contains a single defect.\u003c/p\u003e\n\u003cp\u003eThe investigation focused on the octahedral Mg-Al substitution. The studied minerals are partly characterised by different octahedral sites. If the preferred site for Al is not known, the interaction energies were calculated for each octahedral site. In the case that one site is characterised by a significant lower interaction energy, ordering on this site was assumed (e.g., Al on M1 in olivine). On the other hand, if the energies are similar for different sites, they were combined into one single crystallographic site (e.g., Al on M1\u0026thinsp;+\u0026thinsp;M2 in wadsleyite). Table\u0026nbsp;1 presents detailed information on the end-member choice and in consequence on the crystallographic site where mixing was assumed to take place (given in the supplementary material).\u003c/p\u003e"},{"header":"3. Results","content":"\u003cp\u003eTo express the elastic properties of a mineral, the bulk modulus or compressibility is the mostly used parameter, however, for rare minerals it is not well known. During the search for another parameter, the question raised, which structural units determine the stiffness of a mineral. For minerals containing an oxygen sublattice, its packing fraction is likely to be a good proxy for the elastic properties since it dominates the occupancy of the unit cell volume. The oxygen packing fraction (OPF) is a parameter that has the great advantage that it is easily available for poor investigated minerals because OPF can be defined as\u003c/p\u003e\n\u003cp\u003e\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; OPF = n\u003csub\u003eO\u003c/sub\u003e \u003cem\u003eV\u003c/em\u003e\u003csub\u003eO\u003c/sub\u003e/(\u003cem\u003eV\u003c/em\u003e\u003csub\u003eUC\u003c/sub\u003e), \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;1)\u003c/p\u003e\n\u003cp\u003ewhere n\u003csub\u003eO\u003c/sub\u003e is the number of oxygens in the unit cell, V\u003csub\u003eO\u003c/sub\u003e is the volume of the oxygen anion calculated from the data of Shannon (1976), and V\u003csub\u003eUC\u003c/sub\u003e is unit cell volume. The correlation between OPF and bulk modulus for different minerals is shown in Fig. 1 demonstrating that OPF increases from 51 to 71 % during a bulk-modulus increase from 300 to 2500 kbar. The data are from the compilation of Holland and Powell (2011) and range from tridymite with low bulk modulus over all mineral groups that contain oxygen to stishovite with a high bulk modulus (including 95 minerals).\u003c/p\u003e\n\u003cp\u003e\u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eH\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e is defined as the difference between the enthalpy of a solid solution (\u003cem\u003eH\u003c/em\u003e\u003csub\u003eAB\u003c/sub\u003e) at a given composition and the enthalpy of its mechanical mixture (\u003cem\u003eX\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e \u003cem\u003eH\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e + \u003cem\u003eX\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e \u003cem\u003eH\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e), i.e.,\u003c/p\u003e\n\u003cp\u003e\u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eH\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e = \u003cem\u003eH\u003c/em\u003e\u003csub\u003eAB\u003c/sub\u003e \u0026ndash; (\u003cem\u003eX\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e \u003cem\u003eH\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e + \u003cem\u003eX\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e \u003cem\u003eH\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e), \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;2)\u003c/p\u003e\n\u003cp\u003ewhere \u003cem\u003eX\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e, \u003cem\u003eX\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e and \u003cem\u003eH\u003c/em\u003e\u003csub\u003eA,\u0026nbsp;\u003c/sub\u003e\u003cem\u003eH\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e represent the mole fractions and the enthalpy of the A and B component, respectively. \u0026nbsp;\u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eH\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e of different substitutions in different minerals was investigated by DFT using the single defect method (Sluiter and Kawazoe 2002). The data were then described by a symmetric Margules model,\u003c/p\u003e\n\u003cp\u003e\u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eH\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e = \u003cem\u003eX\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e \u003cem\u003eX\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e, \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; 3)\u003c/p\u003e\n\u003cp\u003ewhere \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e is the interaction energy between elements on a single crystallographic site (micro \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e). This parameter is listed in Tab. 1 (given in the supplementary material) and plotted against OPF of the Mg-end members in Fig. 2.\u003c/p\u003e\n\u003cp\u003eThe Mg-Al substitution was investigated in 16 minerals (ordered by increasing \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e): melilite, chlorite, biotite, cordierite, amphibole, talc, pseudobrookite, pyroxene, sulphate (4 x hydrated), olivine, wadsleyite, sulphate (1 x hydrated), ilmenite, Mg-wolframite, anhydrous sulphate, and ringwoodite that has the spinel structure. As can be seen from Fig. 2, a linear positive correlation of \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e versus OPF was found (fit parameters are given in the supplementary material). The Mg-Al substitution in biotite, as an example, has an energetic effect that is less than 1/2 of that in pyroxene, and less than 1/6 of that in spinel. This behaviour is expected to be a consequence of the different oxygen packing fractions, which are 53.5 % in biotite, 59.1 % in pyroxene, and 65 % in spinel. In addition to the Mg-Al substitution, the Si-Al substitution has been investigated in 8 minerals. Here, two trends were observed. One, where the tetrahedra are connected with each other (chain and sheet silicates) investigated in 4 minerals (ordered by increasing \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e): biotite, amphibole, pyroxene, perovskite. The second trend is from island and double island silicates defined by 4 minerals, i.e., melilite, olivine, wadsleyite, ringwoodite. This group is characterised by a much flatter positive correlation. MgSiO\u003csub\u003e3\u003c/sub\u003e perovskite, with \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eSiAl\u003c/sub\u003e/OPF = 577.4/77.1, formed an outlier and was not included in the fit. The interaction energies of the Mg-Ca substitution were investigated in 7 minerals (ordered by increasing OPF): MgO-CaO solid solution, amphibole, olivine, pyroxene, carbonate, garnet, and perovskite). This substitution is characterised by a flat \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e/OPF correlation, whereas the Mg-Ti\u003csup\u003e4+\u003c/sup\u003e substitution shows a steep slope that is defined by 4 minerals (ordered by increasing \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e): biotite, pyroxene, olivine, ringwoodite. Finally, 7 minerals were studied for the Mg-Fe substitution finding ideal to slightly negative\u0026nbsp;D\u003cem\u003eH\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e behaviour (ordered by increasing OPF): biotite, brucite, pyroxene, olivine, spinel, garnet, and perovskite.\u003c/p\u003e\n\u003cp\u003eUsing the interaction energies of the separate Mg-Al and Si-Al substitutions (micro \u003cem\u003ew\u003c/em\u003e\u0026acute;s, i.e., \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eMgAl\u003c/sub\u003e and \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eSiAl\u003c/sub\u003e) in combination with the underlying cross-site interaction (\u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003ecross\u003c/sub\u003e), the macroscopic interaction energy for the Tschermak\u0026acute;s substitution (\u003cem\u003eW\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eTS\u003c/sub\u003e) can be calculated (e.g., Powell et al., 2014):\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eW\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eTS\u003c/sub\u003e = \u0026frac14; (4 \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eMgAl\u003c/sub\u003e + \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eSiAl\u003c/sub\u003e \u0026ndash; 2 \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003ecross\u003c/sub\u003e),\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;\u0026nbsp;4)\u003c/p\u003e\n\u003cp\u003ewhere the cross-site interaction is given by the energies of the reciprocal reaction:\u003c/p\u003e\n\u003cp\u003e\u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003ecross\u003c/sub\u003e = \u003cem\u003eH\u003c/em\u003e(Mg\u003csup\u003eM\u003c/sup\u003eSi\u003csup\u003eT\u003c/sup\u003e) + \u003cem\u003eH\u003c/em\u003e(Al\u003csup\u003eM\u003c/sup\u003eAl\u003csup\u003eT\u003c/sup\u003e) \u0026ndash; \u003cem\u003eH\u003c/em\u003e(Mg\u003csup\u003eM\u003c/sup\u003eAl\u003csup\u003eT\u003c/sup\u003e) \u0026ndash; \u003cem\u003eH\u003c/em\u003e(Al\u003csup\u003eM\u003c/sup\u003eSi\u003csup\u003eT\u003c/sup\u003e). \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; 5)\u003c/p\u003e\n\u003cp\u003eSuch calculations were performed for the Mg-Al biotite in order to verify the DFT results of charged unit cells. Using eq (4), \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eMgAl\u003c/sub\u003e = 82.5 kJ/mol, \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eSiAl\u003c/sub\u003e = 95.6 kJ/mol, and \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003ecross\u003c/sub\u003e of 175.1 kJ/mol, the macroscopic interaction energy of \u003cem\u003eW\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eTS\u003c/sub\u003e = 18.8 kJ/mol is obtained. It represents the interaction energy with ordering of Al on M1 and T1 sites, but without strict local-charge balance (without short-range ordering). To compare it with DFT values obtained without using charged unit cells, a Mg-biotite (KMg\u003csub\u003e3\u003c/sub\u003e[(OH)\u003csub\u003e2\u003c/sub\u003eAlSi\u003csub\u003e3\u003c/sub\u003eO\u003csub\u003e10\u003c/sub\u003e], phlogopite), a Tschermak substituted biotite (KMg\u003csub\u003e2\u003c/sub\u003eAl[(OH)\u003csub\u003e2\u003c/sub\u003eAl\u003csub\u003e2\u003c/sub\u003eSi\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e10\u003c/sub\u003e], eastonite) and intermediate solid-solution biotites were investigated. Using 10 different super cells for the intermediate biotites (50:50 composition with randomly distributed Al on M1 and T1 with sizes up to 16 formula units), a \u003cem\u003eW\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eTS\u003c/sub\u003e of 19.0 +/- 3.0 kJ/mol was obtained, which is in good agreement with the \u003cem\u003eW\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eTS\u003c/sub\u003e calculated via micro \u003cem\u003ew\u003c/em\u003e\u0026acute;s (18.8 kJ/mol). This value is slightly lower than an experimentally derived value of 25.4 kJ/mol (Dachs and Benisek 2019) because of different ordering schemes. The same calculations were performed for the diopside \u0026ndash; CaTs solid solution, (\u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eMgAl\u003c/sub\u003e = 213.6 kJ/mol, \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eSiAl\u003c/sub\u003e = 285.6 kJ/mol, and a \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003ecross\u003c/sub\u003e = 422.0 kJ/mol) yielding a \u003cem\u003eW\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eTS\u003c/sub\u003e of 74.0 kJ/mol. The uncharged cells delivered 74.8 +/- 3.7 kJ/mol, also in good agreement with \u003cem\u003eW\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eTS\u003c/sub\u003e obtained via micro \u003cem\u003ew\u003c/em\u003e\u0026acute;s. This value is larger than the experimentally derived value of ~ 24 kJ/mol (Benisek et al. 2007), if a symmetric fit is applied to their data. The difference comes mainly from the fact that the experimental data were derived using a partly disordered CaTs end-member (Benisek and Dachs 2020).\u0026nbsp;\u003c/p\u003e"},{"header":"4. Discussion","content":"\u003cp\u003eDavies and Navrotsky (\u003cspan class=\"CitationRef\"\u003e1983\u003c/span\u003e) correlated the interaction parameter (\u003cem\u003eW\u003c/em\u003e) with a normalised volume difference (\u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csub\u003enorm\u003c/sub\u003e) as a quantity to include the strain, i.e.,\u003c/p\u003e\n\u003cp\u003e\u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csub\u003enorm\u003c/sub\u003e = (V\u003csub\u003e2\u003c/sub\u003e \u0026ndash; V\u003csub\u003e1\u003c/sub\u003e) / V\u003csub\u003e2\u003c/sub\u003e, 6)\u003c/p\u003e\n\u003cp\u003ewhere V\u003csub\u003e2\u003c/sub\u003e and V\u003csub\u003e1\u003c/sub\u003e are the end-member volumes. The investigated minerals were also studied in this respect by the DFT methods, and the corresponding results are shown in Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e and compiled in Table 1 (given in the supplementary material), demonstrating good correlations, if the different substitutions are separated.\u003c/p\u003e\n\u003cp\u003eThe \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e/\u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csub\u003enorm\u003c/sub\u003e and \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e/OPF correlations yielded similar results, i.e., the Mg-Ti\u003csup\u003e4+\u003c/sup\u003e has a steep and the Mg-Ca substitution has a very flat slope. They have also similar R-squared values (see supplementary material). However, the Si-Al substitution in MgSiO\u003csub\u003e3\u003c/sub\u003e perovskite is no longer an outlier, as is the case for the \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e/OPF correlation. From these points of view, there would be no need to postulate a relationship between \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e and OPF or between \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e and another parameter describing the mineral elasticity. However, \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csub\u003enorm\u003c/sub\u003e may also incorporate information on the packing density of the oxygen sublattice. To test this idea, \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csub\u003enorm\u003c/sub\u003e was plotted against the oxygen packing fraction (Fig. \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e) demonstrating a good \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csub\u003enorm\u003c/sub\u003e/OPF correlation.\u003c/p\u003e\n\u003cp\u003e\u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csub\u003enorm\u003c/sub\u003e, as used in the study of Davies and Navrotsky (\u003cspan class=\"CitationRef\"\u003e1983\u003c/span\u003e) to parameterise the strain, contains thus also information about the packing density of the oxygen sublattice. For minerals with the same formula but different volumes (olivine versus ringwoodite), the \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csub\u003enorm\u003c/sub\u003e/OPF correlation is easy to explain, because the volume of the ringwoodite is small compared to that of olivine increasing \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csub\u003enorm\u003c/sub\u003e from olivine to ringwoodite (from 0.258 to 0.361) as it is the case with their oxygen packing fractions (from 59.1 to 65.0%). However, for minerals with different formulae, the \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csub\u003enorm\u003c/sub\u003e/OPF correlation is not intuitively understandable. \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csub\u003enorm\u003c/sub\u003e decreases generally with an increase of the number of atoms in the formula unit. It may be that the increase of the number of different elements and different polyhedra in the structure increases the probability that the mineral becomes less dense. However, the data of melilite should also be mentioned in this context. Melilite\u0026acute;s formula has neither a large nor a small number of elements. Nevertheless, the Mg-Al substitution in melilite is characterised by a very low strain (\u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csub\u003enorm\u003c/sub\u003e = 0.042), which is certainly a consequence of its open packed structure (OPF\u0026thinsp;=\u0026thinsp;49.9%). The oxygen packing fraction may, therefore, be the dominant factor for determining \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csub\u003enorm\u003c/sub\u003e.\u003c/p\u003e\n\u003cp\u003eThe coordination number of the substituted cation does not always have the same value within a certain substitution and is quite heterogenous for the Mg-Ca substitution. The Ca site in different minerals increases from 6 to 12 with a simultaneous increase of OPF. These circumstances may result in a flattening of the \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eMgCa\u003c/sub\u003e/\u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csub\u003enorm\u003c/sub\u003e and \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eMgCa\u003c/sub\u003e/OPF correlations. In MgSiO\u003csub\u003e3\u003c/sub\u003e perovskite, as another example, Si(Al) has not tetrahedral but octahedral coordination. This might be a reason for producing the perovskite outlier in the \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e/OPF correlation. On the other hand, melilite has a tetrahedral Mg(Al)-site and plots on the Mg-Al line, where all other minerals have Mg(Al) in octahedral coordination. Melilite\u0026acute;s \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eMgAl\u003c/sub\u003e is slightly negative, which is a surprising observation, because cation substitutions are thought to produce zero (substitutions without local strain heterogeneities) or positive \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eH\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e in the case that local strain heterogeneities exist (Tarantino et al., \u003cspan class=\"CitationRef\"\u003e2003\u003c/span\u003e). However, the generated strain for the Mg-Al substitution in melilite is small (\u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csub\u003enorm\u003c/sub\u003e = 0.042).\u003c/p\u003e"},{"header":"5. Conclusions","content":"\u003cp\u003eIt was shown in this study that the interaction energies of some substitutions strongly depend on the oxygen packing fraction, i.e., the Mg-Al and Mg-Ti exchanges and that of the Si-Al in chain and sheet silicates. This dependence can be understood as follows: The denser the oxygen packing, the stronger the impact of the local strain heterogeneities \u0026ndash; caused by a substituted cation \u0026ndash; will be on the whole unit cell. The strain heterogeneities of the Si-Al substitutions in island silicates may be absorbed by the less rigid neighbouring polyhedra producing the flat \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e/OPF correlation. On the other hand, the \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e\u003csub\u003eMgCa\u003c/sub\u003e/OPF correlation may be superimposed by the increasing coordination numbers with increasing OPF.\u003c/p\u003e\n\u003cp\u003eThere is also a \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e/\u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eV\u003c/em\u003e\u003csub\u003enorm\u003c/sub\u003e correlation, which is, however, not easily applicable to substitutions involving cations with different charges because the volume of the charged end members is only accessible by DFT methods.\u003c/p\u003e\n\u003cp\u003eThe calculated \u003cem\u003ew\u003c/em\u003e\u003csup\u003eH\u003c/sup\u003e parameters represent the interaction energies between elements on a single crystallographic site. However, the degree of local charge balances and the degree of cation ordering influence the macroscopic interaction energies as demonstrated by DFT calculations (Dachs and Benisek \u003cspan class=\"CitationRef\"\u003e2019\u003c/span\u003e, Benisek and Dachs \u003cspan class=\"CitationRef\"\u003e2020\u003c/span\u003e). Other theoretical studies have also shown that \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eH\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e increases with temperature because of such effects, as demonstrated for garnets, diopside-jadeite solid solution, carbonites, and halides (e.g., Vinograd and Sluiter \u003cspan class=\"CitationRef\"\u003e2006\u003c/span\u003e, Vinograd et al. \u003cspan class=\"CitationRef\"\u003e2007\u003c/span\u003e, Vinograd et al., \u003cspan class=\"CitationRef\"\u003e2009\u003c/span\u003e, Vinograd and Winkler 2010). If one end-member is involved in the disordering processes as is the case with Tschermak\u0026acute;s substituted pyroxene and melilite, \u003cspan style=\"text-align: start;color: rgb(0, 0, 0);background-color: rgb(255, 255, 255);font-size: 11px;\"\u003e\u0026Delta;\u003c/span\u003e\u003cem\u003eH\u003c/em\u003e\u003csup\u003emix\u003c/sup\u003e can decrease with temperature (Sack and Ghiorso \u003cspan class=\"CitationRef\"\u003e2017\u003c/span\u003e, Sack \u003cspan class=\"CitationRef\"\u003e2021\u003c/span\u003e).\u003c/p\u003e\n\u003cp\u003eThe use of OPF-dependent interaction energies for substitutions that are characterised by large strains and the formulation of appropriate models for cation ordering should further contribute to the reliability of future mixing models.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis work was supported by grants from the Austrian Science Fund (FWF), project number P33904, which is gratefully acknowledged. We thank E. Forsthofer for maintaining the Materials Studio software at the Salzburg University.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor Contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eED initiated the study. AB had the idea, worked out the conception and design, did the calculations and drafted the manuscript. Both authors contributed to the analysis of data, read, and approved the final manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflict of interest\u003c/strong\u003e\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAll available data are given in the text and supplementary material.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eBenisek A, Etzel K, Cemic L (2007) Thermodynamic mixing behaviour of synthetic Ca-Tschermak-diopside pyroxene solid solutions: II. Heat of mixing and activity-composition relationships. Phys Chem Minerals 34:747-755.\u003c/li\u003e\n \u003cli\u003eBenisek A, Dachs E (2020) Excess enthalpy of mixing of mineral solid solutions derived from density-functional calculations. Phys Chem Minerals 47: 15.\u003c/li\u003e\n \u003cli\u003eBoff Ballaran T, Carpenter MA, Domeneghetti MC, Salje EKH, Tazzoli V (1998) Structural mechanism of solid solution and cation ordering in augite-jadeite pyroxenes II: a microscopic perspective. Am Mineral 83: 419-433.\u003c/li\u003e\n \u003cli\u003eCarpenter MA (2003) Microscopic strain, macroscopic strain and the thermodynamic of phase transitions in minerals. In: Gramaccioli CM (ed) European notes in mineralogy, energy modelling in minerals, vol 4. European Mineralogy Union, pp 311-346.\u003c/li\u003e\n \u003cli\u003eCeperley DM, Alder BJ (1980) Ground state of the electron gas by a stochastic method.\u003cem\u003e\u0026nbsp;Phys Rev Lett\u003c/em\u003e \u003cstrong\u003e45:\u0026nbsp;\u003c/strong\u003e566-569.\u003c/li\u003e\n \u003cli\u003eClark SJ, Segall MD, Pickard CJ, Hasnip PJ, Probert MIJ, Refson K, Payne MC (2005) First principles methods using CASTEP. Z Kristallogr 220: 567-570.\u003c/li\u003e\n \u003cli\u003eDachs E, Benisek A (2019) A new activity model for Mg-Al biotites determined through an integrated approach. Contrib Mineral Petrol 174: 76.\u003c/li\u003e\n \u003cli\u003eDavies PK, Navrotsky A (1983) Quantitative correlations of deviations from ideality in binary and pseudobinary solid solutions. J Solid State Chem 46: 1-22.\u003c/li\u003e\n \u003cli\u003eEtzel K, Benisek A, Dachs E, Cemic L (2007) Thermodynamic mixing behavior of synthetic Ca-Tschermak-diopside pyroxene solid solutions: I. Volume and heat capacity of mixing. Phys Chem Minerals 34: 733-746.\u003c/li\u003e\n \u003cli\u003eGeiger CA (2001) Thermodynamic mixing properties of binary oxide and silicate solid solutions determined by direct measurements: The role of strain. EMU Notes Mineral 3, Budapest, E\u0026ouml;tv\u0026ouml;s Univ Press, 71-100.\u003c/li\u003e\n \u003cli\u003eHolland TJB, Powell R (2011) An improved and extended internally consistent thermodynamic dataset for phases of petrological interest, involving a new equation of state for solids. J metamorphic Geol 29: 333-383.\u003c/li\u003e\n \u003cli\u003eMakov G, Payne MC (1995) Periodic boundary conditions in ab initio calculations. Phys Rev B 51: 4014-4022.\u003c/li\u003e\n \u003cli\u003eMonkhorst HJ, Pack JD (1976) On special points for Brillouin zone integrations. Phys Rev B 13: 5188.\u003c/li\u003e\n \u003cli\u003ePfrommer BG, Cote M, Louie SG, Cohen ML (1997) Relaxation of crystals with the quasi-Newton method. J Comput Phys 131: 233-240.\u003c/li\u003e\n \u003cli\u003ePowell R, White RW, Green ECR, Holland TJB, Diener JFA (2014) On parameterizing thermodynamic descriptions of minerals for petrological calculations. Journal of Metamorphic Geology 32: 245-260.\u003c/li\u003e\n \u003cli\u003eSack RO, Ghiorso MS (2017) Ti\u003csup\u003e3+\u003c/sup\u003e - and Ti\u003csup\u003e4+\u003c/sup\u003e - rich fassaites at the birth of the solar system: Thermodynamics and applications. \u0026nbsp;American Journal of Science 317: 807-845.\u003c/li\u003e\n \u003cli\u003eSack RO (2021) Thermochemistry of melilites I. Towards resolving and inconsistency in nebular condensation calculations. American Journal of Science 321: 424-457.\u003c/li\u003e\n \u003cli\u003eShannon RD (1976) Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. Acta Cryst. A32: 751.\u003c/li\u003e\n \u003cli\u003eSluiter MHF, Kawazoe Y (2002) Prediction of the mixing enthalpy of alloys. Europhys Lett 57: 526-532.\u003c/li\u003e\n \u003cli\u003eTarantino SC, Carpenter MA, Domeneghetti MC (2003) Strain and local heterogeneity in the forsterite-fayalite solid solution. Phys Chem Minerals 30: 495-502.\u003c/li\u003e\n \u003cli\u003eUrusov VS (2001) The phenomenological theory of solid solutions. \u003cem\u003eIn\u003c/em\u003e Geiger CA (ed): Solid solution in silicate and oxide systems. EMU Notes Mineral 3, Budapest, E\u0026ouml;tv\u0026ouml;s Univ Press, 121-153.\u003c/li\u003e\n \u003cli\u003eVinograd VL, Sluiter MHF (2006) Thermodynamics of mixing in pyrope-grossular, Mg\u003csub\u003e3\u003c/sub\u003eAl\u003csub\u003e2\u003c/sub\u003eSi\u003csub\u003e3\u003c/sub\u003eO\u003csub\u003e12\u003c/sub\u003e- Ca\u003csub\u003e3\u003c/sub\u003eAl\u003csub\u003e2\u003c/sub\u003eSi\u003csub\u003e3\u003c/sub\u003eO\u003csub\u003e12\u003c/sub\u003e, solid solution from lattice dynamics calculations and Monte Carlo simulations. Am Mineral 91: 1815-1830.\u003c/li\u003e\n \u003cli\u003eVinograd VL, Gale JD, Winkler B (2007) Thermodynamics of mixing in diopside-jadeite, CaMgSi\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e6\u003c/sub\u003e-NaAlSi\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e6\u003c/sub\u003e, solid solution from static lattice energy calculations. Phys Chem Minerals 34: 713-725.\u003c/li\u003e\n \u003cli\u003eVinograd VL, Sluiter MHF, Winkler B (2009) Subsolidus phase relations in the CaCO\u003csub\u003e3\u003c/sub\u003e-MgCO\u003csub\u003e3\u003c/sub\u003e system predicted from the excess enthalpies of supercell structures with single and double defects. Phys Rev B 79: 104201.\u003c/li\u003e\n \u003cli\u003eVinograd VL, Winkler B (2010) An efficient cluster expansion method for binary solid solutions: application to the halite-sylvite, NaCl-KCl, system. Rev Mineral Geochem 71: 413-436.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[{"identity":"adc311c1-1e1c-47ba-815d-9321e0a1333f","identifier":"10.13039/501100002428","name":"Austrian Science Fund","awardNumber":"P33904","order_by":0}],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"University of Salzburg","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"interaction energy, excess enthalpy of mixing, stiffness, oxygen packing density, density functional theory","lastPublishedDoi":"10.21203/rs.3.rs-3950395/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3950395/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe heat of mixing of some petrological relevant substitutions (i.e., Mg-Al, Si-Al, Mg-Ti, Mg-Ca, and Mg-Fe) was investigated systematically in silicates, titanates, tungstates, carbonates, oxides, hydroxides, and sulphates by density functional theory calculations (e.g., melilite, chlorite, biotite, brucite, cordierite, amphibole, talc, pseudobrookite, pyroxene, olivine, wadsleyite, ilmenite, MgWO\u003csub\u003e4\u003c/sub\u003e, ringwoodite (spinel), perovskite, pyrope-grossular, magnesite-calcite, MgO-CaO, anhydrous and different hydrated MgSO\u003csub\u003e4\u003c/sub\u003e). A specific substitution is characterised by different microscopic interaction energies in different minerals, e.g., the octahedral Mg-Al exchange on a single crystallographic site in pyroxene has a microscopic interaction energy that is more than twice compared to that in biotite. A comparative investigation of the heat of mixing using microscopic interaction energies on a single crystallographic site has the advantage that they are not influenced by cation ordering. They could be successfully correlated with the stiffnesses of the minerals, which in turn were scaled to the oxygen packing fraction, a parameter that is easily available for poorly investigated minerals. With this information, the interaction energies of a certain substitution can be transferred from minerals where they are well-known to mineral groups where they are less- or unknown. Using the cross-site terms and the microscopic interaction energies, the macroscopic interaction energies of the coupled substitution, e.g., Mg+Si = Al+Al, of biotite and pyroxene were calculated, which are, however, affected by cation ordering and different degrees of local charge balance, for which appropriate models are necessary.\u003c/p\u003e","manuscriptTitle":"The packing fraction of the oxygen sublattice: Its impact on the heat of mixing","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-02-13 04:15:57","doi":"10.21203/rs.3.rs-3950395/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"9dfeab46-19f3-438e-b2a0-0f9cdcf50c70","owner":[],"postedDate":"February 13th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":28706040,"name":"Physical Chemistry"}],"tags":[],"updatedAt":"2024-02-13T04:15:57+00:00","versionOfRecord":[],"versionCreatedAt":"2024-02-13 04:15:57","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3950395","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3950395","identity":"rs-3950395","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}