First-principles Calculations on Polymorphs of Dicalcium Silicate

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C: Physical Processes in Nanomaterials and Nanostructures

First-Principles Calculations on Polymorphs of Dicalcium Silicate - Belite, a Main Component of Portland Cement Pawel Rejmak, Jorge S. Dolado, Miguel A. G. Aranda, and Andres Ayuela J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b10045 • Publication Date (Web): 15 Feb 2019 Downloaded from http://pubs.acs.org on February 15, 2019

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The Journal of Physical Chemistry

First-principles Calculations on Polymorphs of Dicalcium Silicate Belite, a Main Component of Portland Cement PAWEL REJMAK1, J.S. DOLADO2, MIGUEL A. G. ARANDA3, ANDRES AYUELA2,* 1

Instytut Fizyki PAN, Lotnikow 32/46, 02668, Warszawa, Poland Centro de Física de Materiales (CFM ) CSIC-UPV/EHU- Material Physics Center, Donostia International Physics Center DIPC, Paseo Manuel de Lardizabal n 5, 20018, San Sebastián, Spain 3 ALBA Synchrotron, Carrer de la Llum 2-26, E-08290, Cerdanyola del Vallès, Barcelona, Spain 2

Corresponding author: [email protected]

ABSTRACT We investigate systematically the polymorphism of stoichiometric dicalcium orthosilicates C2S using ab initio studies and interatomic potential functions. Apart from the structural data, here we present the analysis of novel data on phonon dynamics in belite. The latter is related to the number and type of phase transitions in C2S, including the peculiar thermal hysteresis in the route between three low temperature phases and inverse thermal expansion of the lowest temperature phase. The study on the polymorphism of stoichiometric C2S is the necessary step to understand behavior of doped belite based materials.

___________________________________________________________________________________ INTRODUCTION Dicalcium orthosilicate, Ca2SiO4 (C2S), known also as belite, is one of the most widely manufactured materials on the world, as it constitutes about 20% of mass of ordinary Portland cement clinker

1,2

.

There is a growing interest in cements with a large belite content because materials can be produced with low energy consumption and low CO2 footprint 3. Apart from its practical use for construction, C2S is a natural mineral, which is currently attracting a large number of applications, such as a bioceramic component dopants

8-15

4-6

, a CO2 storage material 7, and a matrix for optically active f and d electron

. The characteristic of C2S is that in the temperature range of 500-1500 °C, it is found as

crystallized in five polymorphs shown in Figure 1, an unusually high number for a compound of orthosilicate family A2BX4

1,2,16

. Although belite polymorphs are brought as metastable to room temperature by impurities,

they are also present for the pure orthosilicate as shown using XRD at high temperatures

1,17,18

. Experimentally,

the structures follow the sequence of phases transitions γ -> α'L_>α'H -> α when increasing temperature. On cooling the order changes at low temperatures because a metastable phase known as β inserts between α'L and γ

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structure specially for doped samples. Another peculiarity of C2S is that the most stable γ phase at low temperatures has its lattice parameter expanded with respect to the high temperature phases. (Note that it is seen as a white powder in the production of cement clinkers, known as dusting.) The understanding of polymorphs in C2S by changing temperature provides basic and applied information for many fields. We have thus to investigate the phases and transitions that happen in pure dicalcium silicates.

The understanding of phase transitions in C2S is crucial for cement processing because being slightly doped the high temperature phases are more reactive toward the water and the increase of their fraction facilitates cement hydration

1,2,19

. The  phase content in the pure C2S phases increases with doping,

which provides a downshift in all transition temperatures along the polymorph sequence. By increasing the  phase at room temperature, the C2S hydrophilic reactivity rises, which is one of the most crucial issues in developing more environmental friendly cements 3. The structures of  and  phases are well determined experimentally

20-23

, whereas higher temperature phases are less certain, because usually

they are reported for heavily doped C2S

22-24

. Certainly, for different applications of C2S, e. g. like a

matrix for inorganic phosphorus, the reverse issue may arise, namely how to stabilize the least hydrophilic γ phase upon doping with rare earth or transition metals. Although much is known experimentally on the structural data of the polymorphs in these silicates especially by X-ray diffraction, less is known on their energetics and phonon spectra. Molecular mechanics and firstprinciple calculations related to phonons should therefore be valuable for studying the phase transitions between those polymorphs.

Atomistic simulations using first principles methods have already proven to be of great help in studying complex phases in cement materials 4,25-36. Up to date theoretical studies on C2S usually focused on the β phase, either pure

27,29,35-37

or doped

14,15,27,29,37

study dynamics of several C2S phases

38

, or employed semiempirical interatomic potentials to

. The electronic properties of γ, β, α'L phases with their

possible relation to reactivity with water was presented quite recently

33

. We have then to investigate

the structures, which are frequently difficult to be unambiguously characterized, and involved in C2S polymorphism. In this work, we focus on bulk properties of pure dicalcium silicate, which is the first step before one proceeds to more realistic, yet more complicated issues, such as doping of C2S or looking for the effects of water.

In this work we fill this gap by presenting the results of both interatomic potential functions (IPF) and density functional theory (DFT) level calculations on all five C2S polymorphs. Apart from the ACS Paragon Plus Environment

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structural and electronic data, here we present the analysis of novel data on phonon dynamics in C2S. The latter is related to the number and type of phase transitions in C2S, including the peculiar thermal hysteresis in the route between three low temperature phases and inverse thermal expansion of the lowest temperature phase. Furthermore, the electronic calculations are clearly useful to look at water reactivity given by sites prone to nucleophilic attack as induced by water protons. Our study in the stability versus reactivity of dicalcium silicates could be crucial to clarify the high-temperature structures and to understand the phase transitions in calcium silicates in general, where the specific results play an essential role in the design of ecologically new cements, later by doping. In fact, these efforts of the systematic ab initio study on the polymorphism of stoichiometric C2S also position as the first necessary step to understand the behavior of doped C2S based materials.

MODELS AND METHODS

Selection of input structural models. The initial structures of C2S phases were taken from the following experimental data: γ and β from Ref. 22, α'L and α from Ref. 23, and α'H from Ref. 24. Because of dopants, data on C2S are not always reliable so we must firstly study the models using pure phases. Note that in all cases, we have considered the pure C2S phases, while experiments reported often data for doped C2S. When the experimental structure was resolved with fractional occupations for crystallographic sites, we imposed integer occupation preserving the space group experimentally reported for the initial structure. More specific issues regarding the models of high-temperature phases α'L and α are clarified in the Results and Discussion section below.

Computational details. The atoms and unit cell parameters in the structural models were fully relaxed with both interatomic potentials (IP) and density functional theory (DFT) employing Broyden– Fletcher–Goldfarb–Shanno algorithm 39. IPF calculations were performed using the GULP code 40 with the robust core-shell parametrization developed by Catlow and coworkers

41

. The gradient norm

threshold was set to 10-3 eV/Å. DFT simulations were done with the Quantum Espresso package 42. A generalized gradient exchange – correlation functional was applied

43

. The interactions between core

and valence electrons were treated by norm conserving pseudopotentials

44

. Plane wave basis set was

converged using an energy cutoff of 80 Ry. Brillouin zone was sampled using 4×4×4 k-points Monkhorst-Pack grid 45. Atomic positions and unit cell were optimized using until the differences for total energy and atomic gradient norm dropped below 10-6 and 10-4, respectively, in atomic units.



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IPF calculations are not computationally expensive, thus it is possible to perform structural optimizations, and compute easily quantities that depend on the energy second derivative, such as elastic properties and phonons. Additionally, the GULP code enables structural optimizations either without symmetry constraints, or with symmetries within a certain space group. DFT calculations are much more accurate, but they demand much more computationally; they were then used to relax better the structures previously optimized at the IPF level, and calculate the phonon dispersions along selected directions. The assignment of space groups of optimized structures was achieved with FINDSYM code 46. Structure processing and graphics were prepared with VESTA software 47.

RESULTS

Lattice Parameters and Groups Symmetries. The lattice parameters of relaxed structures are collected in Table 1. The lattice parameters predicted by IP calculations are slightly overestimating the experimental data, while the values calculated at DFT-PBE level are typically in very good agreement. Generally, the best agreement between the computed and experimental results is found for the low temperature phases, and the discrepancies arise for high temperature polymorphs. The γ phase presents a match almost perfect in the lattice parameters between experiment and DFT calculations, agreement which is not surprising since it is the phase at the lowest temperature and the easiest one to be obtained in synthesizing pure C2S form. Furthermore, the calculated values for the monoclinic β phase are matching very well with the experimental ones. In intermediate temperatures, the GULP optimization of the α'L phase within experimental symmetry constrains relaxes to a structure which preserves the space group Pna21 and it can be seen as one third of the input unit cell. Note that the experimental unit cell

23

is three times larger than the unit cell of the γ phase (i. e. 84 atoms vs. 28 atoms). We then

decided to use this smaller cell as the input in DFT calculations and found that the calculated lattice parameters for the small unit cell are in excellent agreement with the experimental ones, when one takes into account the factor three to scale the lattice constant b. In the same temperature region, the obtained results for the phase α'H seems more problematic: this is the only case, where the space group of the calculated structure differs from the experimental one. However, the calculated space group Cmcm contains the experimental one Pnma as a subgroup. It should be stressed that the refinement of the α'H structure was performed on a strongly doped sample with the stoichiometry Ca1.85Na01.5(SiO4)0.85(BO3)0.15

24

, which can thus explain why our pure C2S model somewhat differs.

The structures proposed for the α phase at high temperatures were trigonal with the space group P3m1 48,17,23

, or hexagonal with the P63mc group (Eysel 1970). Both structures differ in the orientation of the ACS Paragon Plus Environment

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SiO4 tetrahedra. The hexagonal phase has all the tetrahedra pointing in the same direction, parallel to the c axis, while the tetrahedra of the trigonal structure flip alternatively between positive and negative in the +c and -c orientation. We then performed calculations on the trigonal and hexagonal cells, and respectively we found that the converged cells are reduced or expanded in comparison to the experimental unit cells. As mentioned before, the peculiar feature of C2S is the fact that the γ phase at low temperatures has the lowest density. A significant increase in the density occurs at intermediate temperatures for the α'L and β phases, a trend which is well reproduced by our calculations, particularly for the β phase. At high temperatures the hexagonal form for the α polymorph with respect to the γ phase is predicted to be slightly less dense, while our calculated structure having the trigonal group is denser. Although the trigonal result seems to be puzzling, it can be directly compared to the experiments that show that even the highest temperature structure remains denser than the γ phase.

Atomic Distances and Coordination Numbers. Table 2 presents the averaged interatomic distances and the coordination numbers for some selected pairs of elements. We again found excellent agreement between the experimental and calculated data for the γ phase at a low temperature, and unusual differences for the stable phases at high temperatures. However, one should remember that the peaks of diffraction spectra in X-ray experiments serve to identify the crystalline translational symmetry, while the atomic positions within the unit cells are often refined by imposing some additional assumptions that constrain the interatomic distances. Therefore, it cannot be excluded that certain differences are rather due to the approximations applied in the refinement of experimental data rather than to our theoretical approach. The average distance between Si–O bonds in all phases is nearly constant (about 1.65 Å), although there is a weak shortening of these bonds (of the order of 0.01 Å) when going from the low temperature phases to the high temperature ones. More prominent changes can be found in the Ca–O bonds. We have considered that the O atoms are coordinated to the Ca atoms if the distances are below 3 Å. In the γ phase, both non-equivalent crystallographic sites with Ca atoms (Ca1 and Ca2) are coordinated to six O atoms, a number which corresponds with the most stable coordination for the Ca-O bond according to the Pauling's rules

49

.

Experimental data for α' species reported higher coordination numbers (CN), namely 7-8. In contrast, we found lower CNs for the Ca1 sites, namely 5 in the case of the α'L phase and 4 in the case of α'H. The CNs of the Ca2 sites are 7 and 6 for the α'L and α'H phases, respectively. For the β phase, both experimental and DFT data agree well, predicting CNs of 7 in the Ca1 sites and 8 for the Ca2 ones. Finally, the Ca–O coordination in the phase α depends on the assigned lattice symmetry which can be ACS Paragon Plus Environment


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either trigonal or hexagonal. Note that for the trigonal structure, the Ca2 site is split into two sites, and this structure has actually 3 types of Ca sites in total. The Ca1 sites in the trigonal α structure are predicted to be coordinated with 10 O atoms, while the Ca2a and Ca2b sites remain as six-fold coordinated. The hexagonal α structure has the Ca atoms bound to 7 (Ca1) or 6 (Ca2) O atoms. The average distance of Ca-O bonds is shortest in the orthorhombic phases (about 2.4 Å), it is somewhat larger than for the β phase (over 2.5 Å), and it achieves the highest value for the α-phase (up to nearly 2.8 Å). Apart from Ca–O distances we checked the Ca–Si distances and the overall coordination of Ca to (SiO4) tetrahedra, as given in the last column of Table 2. The changes in Ca–(SiO4) coordinations are less rapid throughout the phases than the change in Ca–O bonding. Regarding the coordination patterns of the Ca1 and Ca2 ions, respectively, we have (i) corrdination by 4 and 5 tetrahedra in distances about 3.3 and 3.5 Å, (ii) coordination by 5 tetrahedra with Ca–Si distance about 3.4 Å in α'L polymorph for each type of Ca site, and (iii) coordination by 5 and 6 tetrahedra with average Ca–Si spacing of 3.4 and 3.7 Å. For bonds Ca-Si, the Ca-Si distances calculated and the coordination between Ca atoms with (SiO4) tetrahedral are generally in better agreement with experiment than those for the Ca–O bonds. Particularly, for high-temperature phases, even small rotations of the units (SiO4) about its center of mass can cause significant changes within the pattern of Ca–O bonds. We also look at Ca–Ca distances as strengthening of the Ca–Ca repulsion in denser C2S phases was deemed critical to increasing hydraulic reactivity (Jost 1977). Considering the Ca–Ca distances below 4Å, we found that the average Ca-Ca coordination for both types of Ca sites is equal to 7 for the γ, α'H, β and hexagonal α phases, and to 6 for the α'L phase. The highest coordination Ca-Ca, being 8 or 9, was found in the trigonal polymorph α. The average Ca-Ca distances in the α' and β polymorphs are about 0.1-0.25 Å shorter than the γ phase values. In general, increasing the Ca coordination is expected to increase the repulsion between Ca ions so they will be more prone to dissolution in water. Consequently, the γ phase with larger Ca-Ca distances would be the phase with lower tendecy to dissolution in water. However, the dissolution of belite based just on the intercationic distances is a first approach because further aspects, such as the electronic behavior and the kinetics aspects as an inert hydrated layer, have to be taken into account 50.

Energetic Stabilities. The energetic stabilities of the different C2S species are shown in Table 3. The lowest energy corresponds to the γ phase in IPF and DFT calculations. IP calculations predict that the β phase is less stable than α'H, while DFT finds that the β polymorph has slightly lower energy than the α'H one. IPF calculations have the α’L phase slightly above to the most stable γ phase. The α’L phase ACS Paragon Plus Environment

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becomes nearly degenerated but below the β phase in DFT calculations. However, the α’L -β difference keeps the energetic order making the α’L phase more stable. In fact, this finding indicates that the β phase can be considered as metastable. The α phases at high-temperature, particularly the hexagonal structure, have significantly higher energies than the orthorhombic polymorphs and the  phase. We have thus found that the energetic sequence of phases generally matches the experimental order with temperature. We can justify that the α'H phase transforms into the β phase that has higher energy in slow cooling because dopants are incorporated into the phase, an effect that can be induced also on pressure or strain, as shown in Figure 2.

Elastic properties. Bulk moduli were calculated in GULP using the full matrix of elastic constants, according to the Voigt, Reuss and Hill definitions

51

, as included in Table 4. The Voigt and Reuss

definitions of bulk moduli correspond to the limits of the uniform strain and stress within the sample, respectively. The Hill value is defined as the arithmetic average of previous two. Additionally, we fitted the plot of energy versus volume to the third order Murnaghan equation of state 52 given by (1)

V 1 V (1− B') 1 + ( ) − } E= E 0 +B V 0 { B ' V 0 B' (B'− 1) V 0 B '− 1

where the minimum energy E0 is modified by the bulk modulus B and its pressure derivative B’ , as shown in Fig. 2. The values of bulk moduli calculated in several ways for a given phase are very similar. However, the β polymorph has the Voigt bulk modulus almost 30 GPa higher than the Reuss value. Apart from the hexagonal α polymorph, which is predicted to be much softer than other phases, the elastic properties of different polymorphs are rather similar. The predicted values of average Young moduli and Poisson's ratios are close to the reported experimental values of about 130-140 GPa

53

and

0.3 54, respectively. The calculated bulk moduli for the β and γ phases are smaller than those reported experimentally

55

. These finding can be explained because the IPF results give too much expanded

structures, hence the lower bulk moduli.

Phonons. Phonon dispersion curves were calculated for various directions using the GULP code, and, for the Γ – X directions, which we found particularly important for our considerations as in the discussion given below, also at the DFT level. There is generally a good agreement between both levels of theory. The notable feature of the Γ – X dispersion curve for the γ polymorph, is the crossing between acoustic modes and the lowest optical modes. Also the two lowest acoustic modes are nearly degenerate. For the α'L phase, one can see in Fig 3 the highest acoustic modes touching the lowest



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optical modes for long wavevectors (q~π/3) and softening mildly at longer qs. In the β phase, we found that the highest acoustic modes are strongly coupled to optical modes from about the middle of the Γ – X direction to the edge of Brillouin zone. At high qs, the softening of other acoustic modes becomes clearly visible, which is larger for the expanded structures of IP calculations in comparison to DFT data with more compressed lattice parameters. For the α'H phase, the acoustic modes in IPF calculations become imaginary in the middle of the Γ – X direction for q~0.33, while in DFT predicts additionally that some optical frequencies are already imaginary for the Γ point. The softening in the acoustic branches at q~0.33 is indicative of modulation in these phase structures before or during the phase transition, while the imaginary optical modes would indicate the instability of phase transitions along such vibrations. Note that there are also quite serious differences between the α'H structures optimized at the IPF and DFT levels. Finally, for trigonal α phase our calculations found that the lowest frequencies of optical bands are also imaginary. The phonon branches for these two phases can be found looking at Fig S1 in Supplementary Information. Certain rough estimations of free energies at IPF level in harmonic and quasi-harmonic approximations are studied. In order to look at the phase sequence and transition temperatures, we calculated the free energy in harmonic and quasi-harmonic approach. The vibrational contribution to free energy Fvib at a given temperature T was calculated according to the formula (2)

1  Fvib =   hiq + kTln(1 − exp(− hiq / kT )) , 2  q i 

where h and k are the Planck’s and Boltzman’s constants, and the phonon frequencies qi are summed over the wavelengths q and the branches i. The elastic contribution to free energy Felas due to the thermal expansion with temperature T of each phase can be expressed as (3)

F elas=

1 B V 0 α2 T 2 , 2

where B is the bulk modulus,

α is the thermal expansion coefficient and

V 0 is the equilibrium

volume at 0 K. The thermal expansion coefficient in the quasi-harmonic limit is calculated by the following formula (4)

α=

γ DG⋅ C v B⋅ V 0 ,

where C v is the constant volume heat capacity (calculated with GULP from vibrational partition function), and (5)

γ DG is the so-called Debye-Grueneisen parameter. This latter parameter can be approximated as 1 γ DG= ⋅ B '− 0.9 ,56 2 ACS Paragon Plus Environment

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where B ' is the pressure derivative of the bulk modulus B , which was obtained from the Murnaghan equation of state, as indicated above. The free energy versus temperature in the harmonic and quasi-harmonic approximations for the phases at low-temperature is schematically shown in Fig. 4. The curves for the γ and α' phases lie close to each other. The curves of free energy for the α'L and γ phases intersect about 800 K in the quasi-harmonic approach. This transition temperature is in good agreement with reported experiments. The β curve intersects the γ line at high temperatures, indicating the meta-stability of the β polymorph. This has also been suggested in experimental works. Regarding the α phases, the free energy curves are always much higher than the related plots of the orthorhombic and β phases. Note that the free energy curve for hexagonal structure is about 3 eV above the corresponding plot for the trigonal one, as it has been commented above.

DISCUSSION (1) All four C2S polymorphs are clearly identified in theoretical calculations: γ, α'L, β and trigonal α. For the α'L structure, we proposed a modified structure which unit cell is 1/3 of the reported experimental one

23

. Of course, it is possible that due to the modulation certain superstructures are

experimentally obtained. For instance, regarding the α'H phase, we optimized a new structure given in recent experiments 24. We were able to locate a local minimum at the IPF level, but at the more reliable DFT level we found that it is an unstable stationary point, as indicated by the presence of imaginary phonon modes (see below). The difference between our calculations and experiments may be just because they differ in chemical composition: our models are dealing with stoichiometric C2S, while the experimental data report it doped (with sodium borate). Therefore the issue, whether the α'H and α'L phases are really distinct or simply the α'L polymorph at high-temperature is the modulated superstructure of the α'H, cannot be unambiguously addressed here. In the case of the high-temperature α phase we suggest that it fits better to the trigonal structure rather than the hexagonal one. Note that the latter one is predicted at DFT level to be less stable by about 2.5 eV than the trigonal structures. This difference is so large that even the high-temperature contributions to free energy would likely do not change the order of stability between these phases. Nevertheless, the conclusive answer for the issues of the high-temperature α'H and α phases could be obtained by ab initio molecular dynamic simulations involving large supercells, beyond the actual scope of this work. A partial answer is given below looking at estimations of the free energies for these phases. Different element substitution(s) have been studied in C2S which stabilize high temperature polymorphs at room temperature. There is not space here to describe in detail all studies on this matter. ACS Paragon Plus Environment


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We highlight that the behavior of isovalent doping (for instance Ca2+ substitution by Mg2+) 57,58 is quite different from aliovalent doping (for instance Si4+ by Al3+)

59

. Doping in C2S can be even more

challenging to characterize as several element substitution can take place simultaneously, (for instance Na+ and B3+ incorporation) 24. Few comments must be added in order to bring these results into contact with experiments. Firstly, in DFT the energy ordering of the phases is nearly correct. The α'H phase is somewhat higher than the β phase, which lies slightly above the α'L. These three phases have a comparable energy, when calculated at zero, thus one can guess that other effects such as thermal contributions to free energy can change stability order. It is right that the phase ordering looks much worse with interatomic potentials, but this is approximated method and may not reproduce well the energy ordering, that is why we mainly comment on the DFT results. Secondly, regarding the hexagonal α phase, the structure seems to be very complicated to be obtained even in experiments. This hexagonal phase is so high in energy that it is not very likely that even thermal contributions could make it make it stable at high T (at least for stoichiometric C2S). We here propose that the trigonal alpha phase is more plausible. A conclusive answer could be given by ab initio molecular dynamics at high T or further X-ray diffraction experiments at a high temperature, a story which is beyond the actual scope of this work. (2) The slight shortening of Si – O bonds with the increase of temperature is not sufficient to explain changes in C2S structures and its properties. The distances and coordination numbers of Ca – Si bonds vary less between the different phases than the corresponding quantities of the pairs Ca – O and Ca – Ca. Clearly the differences in the Ca – O coordination between phases are more significant. The most stable γ phase has octahedral coordination for all Ca ions, which is the preferable coordination for this cation in a simple ionic model of solids 49. We found that more reactive species have at least one type of Ca ions with this optimal coordination number being broken. Generally, the high-temperature phases for the Ca-O bonds have higher CNs with longer average distances than the γ phase. It was assumed that higher hydraulic reactivity of C2S polymorphs correlates first with higher Ca coordination

29

, suggestion that is generally in line with our results. We can add to this rule that the

hydraulic activity of the C2S polymorph is increased when there are deviations from the most favorable octahedral value, both in the case of over- and under- coordination, as illustrated by the Ca1 sites in α' species. Another issue possibly responsible for the destabilization of denser C2S phases on water is second in the shortening of Ca – Ca distances, which results in larger repulsion ACS Paragon Plus Environment

21

. The shortening of the

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average distances Ca – Ca with respect to the γ phase is clearly seen in our results for the α'L and β structures. A word of caution must be added when discussing reactivity in structural terms, for instance, water reactivity is reported to have higher scale dynamic effects in interfaces between water and crystal surfaces 50. The general view of C2S phase transitions is shown schematically in Figure 1. The most expanded structures correspond to the orthorhombic γ followed by the trigonal α at high-temperature, which can be seen as the most ordered ones. At low-temperature, the γ phase along the [011] direction appears to be built of parallel negative chains of (SiO4)4- tetrahedra compensated by positive chains with two types of Ca2+ sites. On the other side, the α phase at high-temperature has two types of parallel chains, one consisting only of Ca2+ cations, and the second one, with overall negative charge, alternating the SiO44- tetrahedral with the Ca2+ ions. At intermediate temperatures, the α' and β phases can be regarded as structures where there is partial intercalation in the silicate chains with Ca2+ ions. These phases have associated a misalignment of the silica tetrahedral, with respect to the a direction in the α' phases, and in the case of the β phase a slight shearing in the b direction. It seems that all transitions might be regarded as displacive ones, as previously suggested 16.

(3) Elastic properties and pressure– averaged values for different phases are quite similar, except very soft hexagonal α, which is not likely anyway. Note that the careful inspection of the data presented in Ref. 60 showed, that initial β at IPF level converged to α'H one. Similar issue in IPF calculations was reported elsewhere

61

. These findings suggest that experiments have to be performed having the

polymorphs under pressure. A further discussion about E vs. V can be pursued following results of Fig. 2 . The less stable phases would be stabilized by pressure or compressive strain

62,63

, as shown in our plot. However, when there

is a pressure stabilization of phases, we also have some pressure thresholds as limits to get those changes. Other issues with to be related are the decrease of unit cell volume with temperature, a finding that is behind the irreversibility of the transitions involving the β phase.

(4) Phonons and temperature. Vibrational lattice properties are related to the thermodynamic properties of belite phases. Phonons are calculated for low temperature phases in the range from 0 to 125-150 cm1

and shown in Fig 1. There are two acoustic phonon branches for the γ phase almost degenerated with

low velocities and a third acoustic branch with high velocity. The phonons for the γ phase seem following text book examples reported for standard and stable crystalline phases

64,65

. However, the

phonon spectrum for the β phase departs largely from having those trends found in bulk crystals. The ACS Paragon Plus Environment


three acoustic branches are running separately for any q value.

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The acoustic phonons largely

hybridized with optical phonons for q > 1/3. At the boundary, two acoustic modes are mixing with a low lying acoustic branch, so that is difficult to consider anymore whether this mode just purely acoustic or optical. The acoustic optical mixing can be taken as a source of phonon softening at the zone boundary. A fact that is behind the experimental twining and superstructure, reported difficult to be establish using optical and X-ray techniques for the β phase. Furthermore, the phonon branches of α’L phase show similar trends to those for β phases. The acoustic branches are non-degenerated and strongly hybridized with the optical modes for sizeable q values. But, the behavior of of α’L phonons are not that large at the boundary as for β phonons. We last comment in more detail the phonon modes for the γ and β phases at the border zone at the X point, as shown in Fig. 5. For the γ phase the SiO4 tetrahedra at the unit cell boundary remain along the c direction fixed while other tetrahedra and Ca atoms are rotating in the center, where the shiftings of Ca and O atoms are parallel or nearly orthogonal in the ab plane. This is a mode that can be considered mostly optical after the crossing with acoustic modes at lower q along the ΓX direction in reciprocal space. However, for the β phase all the Ca atoms and SiO4 tetrahedra are shifting with a highly acoustic like component along a defined direction, which indicates a strong coupling between the ab plane and the c direction. This is related to the partial softening of the acoustic modes in the β phase after mixing with optical modes and showing low energies at the X point. The differences between the modes for the γ and β phases at the X point are related to the behavior of these polytopes and how they transform between phases with temperature.

CONCLUSIONS We performed systematic computational studies of the experimentally reported Ca2SiO4 polymorphs, employing both interatomic potential calculations and density-functional theory. The calculated energetic stability of the studied polymorphs follows the sequence γ < α' ≤ β < α, finding that agree well with the order of phase transitions driven by temperature. The polymorphs at an intermediate temperature, the monoclinic-β, and orthorhombic-α' phases, can be considered in between the most stable phases, the orthorhombic γ structure at low-temperature and the α polymorph at hightemperature. We found that the high-temperature α phase prefers the trigonal structure more likely than a hexagonal one. Therefore we suggest that phase transitions in C2S can be mostly regarded as displacive. The γ and αh phases show a coupling between optical and acoustic modes. The aL’ shows a phonon softening so that it becomes modulated. Also the β phase presents this softening; however, it seems to be partial with an energy barrier. There are then two classes of transformations. ACS Paragon Plus Environment

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In our theoretical studies we found several indicators that link lower stability with chemical reactivity against water for a given structure. Regarding structures, the less stable and more reactive phases have that the coordination of one type of Ca ions deviates significantly from the most favorable octahedral one. Lower stability and higher reactivity of the polymorphs are correlated with the soft structures, as shown in the phonon dispersion curves. The polymorphs with higher hydraulic activity, such as the α'L and β phases, exhibit a softening of acoustic phonon modes in the direction corresponding to the relative displacement between the Ca ions and the SiO4 tetrahedra. The most stable γ structure shows the monotonic dispersion of acoustic modes, however, the crossing between acoustic and optical modes implies a different susceptibility of these solid to phase transitions in high temperatures, as observed experimentally, SUPPORTING

which

can

be

INFORMATION

ACKNOWLEDGMENTS This

behind

Phonon

work

was

the

small

reactivity

curves

for

the

supported

by

Project FIS2016-76617-P

dispersion partially

toward high

water.

temperature.

of the Spanish Ministry of Economy and Competitiveness MINECO, the Basque Government under the ELKARTEK project (SUPER), and the University of the Basque Country (GrantNo. IT-756-13).




Page 14 of 27

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Table 1 Experimental and calculated unit cell parameters: space group, unit cell volume (Vcell), lattice vectors (a, b, c), selected angles (γ) and density (ρ). Space Vcell (Å3) a (Å) b (Å) c (Å) γ (°)1 ρ (g/cm3) group γ Exp.

Pnma

385.8

5.081

6.764

11.224

2.97

GULP

Pnma

404.6

5.168

6.803

11.508

2.83

DFT

Pnma

393.4

5.118

6.792

11.318

2.91

α'L Exp.

Pna21

370.02

5.601

6.9642

9.500

3.09

GULP

Pna21

382.1

5.485

6.924

10.061

2.99

DFT

Pna21

372.2

5.392

6.962

9.914

3.07

α'H Exp.

Pnma

344.7

5.455

6.843

9.235

3.32

GULP

Cmcm

378.7

5.850

6.927

9.346

3.02

DFT

Cmcm

375.7

5.809

7.023

9.208

3.05

β Exp.

P21/c

345.9

5.512

6.758

9.314

94.6

3.31

GULP

P21/c

380.2

5.824

7.013

9.314

93.0

3.01

DFT

P21/c

353.6

5.570

6.801

9.361

94.3

3.24

α Exp.3

P3m1

194.2

5.532

5.532

7.327

2.95

GULP

P3m1

188.3

5.622

5.622

6.881

3.04

DFT

P3m1

186.6

5.620

5.620

6.822

3.07

GULP

P63mc

207.8

5.511

5.511

7.897

2.75

DFT

P63mc

199.4

5.506

5.506

7.593

2.87

Only non-right angle for monoclinic β phase specified. The values of remaining angles are implied by the lattice symmetry (90 or 120º). 2The experimental unit cell content is (Ca2SiO4)12. GULP optimized structure can be represented by the unit cell of 1/3 of experimental one, hosting (Ca2SiO4)4, and this smaller cell was also used as the input for DFT calculations. 3 Hexagonal P63mc constructed from experimental trigonal P3m1 by rotation of one tetrahedron. 1



Page 20 of 27

Table 2 The average interatomic distances RAB (in Å) along with coordination numbers of atom A by atoms B (CNAB). RSi-O RCa1-O RCa2-O RCa1-Ca RCa2-Ca RCa1-Si RCa2-Si CNCa1-O CNCa2-O CNCa1-Ca CNCa2-Ca CNCa1-Si CNCa2-Si γ Exp.1

1.618

2.355 (6)

2.411 (6)

3.689 (8)

3.806 (6)

3.252 (4)

3.490 (5)

GULP

1.641

2.407 (6)

2.453 (6)

3.752 (8)

3.869 (6)

3.296 (4)

3.537 (5)

DFT

1.660

2.368 (6)

2.408 (6)

3.724 (8)

3.834 (6)

3.266 (4)

3.517 (5)

α'L2 2.601 (7.83)3

3.683 (7.5)3

3.679 (5.17)3

Exp.

1.619

GULP

1.633

2.381 (5)

2.553 (7)

3.660 (6)

3.652 (6)

3.430 (5)

3.477 (5)

DFT

1.652

2.348 (5)

2.495 (7)

3.610 (6)

3.669 (6)

3.415 (5)

3.435 (5)

α'H Exp.

1.629

2.645 (8)

2.389 (7)

3.669 (8)

3.589 (8)

3.274 (5)

3.471 (5)

GULP

1.632

2.380 (4)

2.381 (6)

3.681 (6)

3.626 (8)

3.338 (5)

3.684 (6)

DFT

1.653

2.339 (4)

2.365 (6)

3.661 (6)

3.624 (8)

3.335 (5)

3.666 (6)

β Exp.

1.606

2.528 (7)

2.503 (8)

3.606 (7)

3.548 (7)

3.293 (5)

3.587 (6)

GULP

1.634

2.420 (5)

2.432 (6)

3.637 (5)

3.607 (7)

3.374 (5)

3.598 (5)

DFT

1.650

2.523 (7)

2.525 (8)

3.637 (7)

3.569 (7)

3.315 (5)

3.613 (6)

α4 Exp.

1.559

2.707 (7)

2.194 (6)trig 2.792 (6)trig 2.194 (6)hex

3.682 (6)

3.678 (8)

3.382 (5)

3.682 (6)

GULPtrig

1.622

2.798 (10)

2.228 (6) 2.689 (6)

3.749 (5)

3.458 (8) 3.815 (6)

3.387 (5)

3.834 (6) 3.534 (6)

DFTtrig

1.639

2.784 (10)

2.230 (6) 2.637 (6)

3.770 (9)

3.417 (8) 3.850 (8)

3.398 (5)

3.842 (6) 3.516 (6)

GULPhex

1.623

2.648 (7)

2.569 (6)

3.751 (6)

3.800 (8)

3.297 (4)

3.747 (6)

DFThex

1.631

2.662 (7)

2.495 (6)

3.703 (6)

3.726 (8)

3.428 (5)

3.708 (6)

1

Data given for initial structure used in calculation. If the experimental structure was resolved with fractional occupations of atomic position, the intergal occupation preserving experimental space group was imposed. 2The experimental unit cell content is (Ca2SiO4)12. GULP optimized structure can be represented by the unit cell of 1/3 of experimental one, hosting (Ca2SiO4)4, and this smaller cell was also used as the input for DFT calculations. 3RCa-B distances and CNCa-B averaged for all 8 crystallographic Ca sites in experimental cell. 4Hexagonal P63mc constructed from experimental trigonal P3m1 by rotation of one tetrahedron. The trigonal symmetry split single Ca2 site from hexagonal structure into two types. Data for trigonal and hexagonal species denoted with subscripts.


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Table 3 Relative energetic stabilities of C2S polymorphs (ΔE), calculated per single orthorhombic cell content ([Ca2SiO4]4). ΔE (eV) γ

α'L

α'H

β

αtrigonal

αhexagonal

GULP

0.00

0.19

0.53

1.38

2.81

6.42

DFT

0.00

0.90

1.15

1.07

3.99

6.58

Table 4 Elastic properties of C2S polymorphs obtained with IPF calculations. Bulk modulus (B) according to Voigt, Reuss and Hill definitions, along with the bulk modulus and its pressure derivative (B') obtained from fit to the third order Murnaghan equation of state, shear (G) modulus according to Voigt, Reuss and Hill definitions, Young modulus (E)1 and Poisson ratio (υ).2 B, K and E in GPa, B' and υ dimensionless. γ

α'L

α'H

β

αtrigonal

αhexagonal

BReuss

91.7

102.8

113.0

70.3

99.1

61.3

BVoigt

94.6

104.1

115.0

99.4

99.4

67.5

BHill

93.1

103.5

114.0

84.9

99.3

64.4

BMurnaghan

95.4

104.4

117.3

99.9

100.1

66.5

B'Murnaghan

4.809

4.261

4.530

3.351

4.109

5.974

GReuss

50.1

49.5

49.6

34.2

43.6

13.4

GVoigt

51.5

51.8

56.7

48.0

50.2

15.5

GHill

50.8

50.6

53.1

41.1

46.9

14.5

E

129.0

130.5

137.9

106.2

121.6

40.5

υ

0.269

0.289

0.298

0.292

0.296

0.395

E calculated from formula E = 9G/(3+(G/B)), where G and B are Hill values. Poisson ratio calculated from formula υ = (3-

1

2

2(G/K))/(6+2 (G/K)), where G and B are Hill values.



Page 22 of 27

Figure 1 Overview of C2S polymorph structures (DFT optimized models) along with the experimental ordering and temperatures of phase transitions. The projection is in the direction 100, except the α phase, for which the direction is fixed along 001.


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Figure 2 Energy versus volume for the considered phases of calcium disilicates. The phases are denoted according to the following order: γ, dot-red; α'L , diamond-blue; α'H , square-violet; β, asteriskgreen; α trigonal, up triangle-brown; α hexagonal, down triangle-grey.



Page 24 of 27

Figure 3 Phonon dispersion curves for the three most important phases to be considered along the Γ-X direction in reciprocal space. Left panels indicates results obtained using model potentials; right, using density functional theory. Frequencies are ranging from 0 to 125 cm-1 (150 cm-1 for the γ phase).


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.

Figure 4 Free energy differences with respect to the gamma phase. Note the transition γ-αL about 700 K, and to other phases at even higher temperatures. Colored boxes indicate the region of stability of each C2S phase given inside



Figure 5 Example of phonon modes for a low lying curve at the X point for the γ and β phases.


Page 26 of 27

Page 27 of 27 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41


Phonon mode of β phase

b a

c

at the X point


First-principles Calculations on Polymorphs of Dicalcium Silicate

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