Università di TorinoDottorato in Scienza ed Alta Tecnologia

Indirizzo Scienze della Terra

Ab initio modelling: Mechanical and Thermodynamic

properties of Calcium Carbonates Polymorphs

Supervisor: Dott. Mauro Prencipe

PhD Student: Crina Georgeta Ungureanu

How the ab initio quantum mechanical calculation method can be applied on thestudy of equilibrium and HP – HT phases?

Quantum calculation have been carried out at pressures between:

0 and 5 GPa (Calcite),

0 to 30 GPa (Aragonite)

and temperatures between 298 and 700 K.

Our Aim

Calcium carbonates :

(trigonal) calcite, (orthorhombic) aragonite, vateriteand (monoclinic) calcite II systems.

This allowed an exclusive investigation of the Mechanical and Thermodynamic Properties of Calcium Carbonate Polymorphs, which represent the underlying

reference for many studies in different fields as :

Mineralogy, Geophysics and Petrography.

Materials Properties from Quantum Calculations

Structural properties:

• Geometries of molecules• Crystal structures (packing)• Density• Defect structures• Crystal morphology• Surface structures• Adsorption• Interface structures

Thermodynamic Properties

• U, Cp, Cv, S• Binding energies• Pressure induced phase transitions• Temperature induced phase transitions• Phase diagrams

Mechanical Properties

• Compressibility• Elastic moduli• Thermal expansion coefficients• Vibrational properties

Why Quantum Mechanical Calculation?

Ab initio → “from first principles” : no experimental data are used and computations are based on quantum mechanics.

Hartree – Fock Self Consistent field SCF method: Born Oppenheimer Approximation

∑∑∑∑∑ −−

−+

−++=

in ni

i

nn nnij ji

ji

n

n

i i

i

rReZ

rre

RR

eZZm

pMp

H2

'

2222

'21

21

22

Nuclear kinetic energy

Potential energy of e-e interaction

Potential nuclei nuclei interaction

Potential energy of electrons nuclei interaction

Electronic kinetic energy

Hohenberg-Kohn theorems:

Hψ = Eψ→ ))r((EE ρ= ))r((E ρ minima ( ) ( ) 2

1

N

ii

ρ ψ=

= ∑r r

( ) ( ) ( ) ( ) ( )KSS ext H xcE T E E Eρ ρ ρ ρ ρ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= + + +⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦r r r r r

Each issue is returned to EXC.

Kohn-Sham implementation:2

2 ( ) ( ) ( )2 eff i i iV r r E r

m⎧ ⎫− ∇ + Ψ = Ψ⎨ ⎬⎩ ⎭

h

classical electronic Coulomb potential

ion Coulomb potential

( ) ( ) ( ) ( )eff ion H XCV r V r V r V r= + +

exchange-correlation potential of electron gas (LDA,GGA)

DFT

The calculation of the vibrational frequencies at Г point for all phases of calcium carbonatestudied, is performed within the harmonic approximation:

vibrational frequencies at central Г point(each volume , pressure):

diagonalizing the mass-weighted Hessian W

special importance of Г point: IR and RAMAN spectra refer to this point.

Frequency harmonic calculations at Γ point: FREQCALC

V (x): Potential energy surface of a systemas a function of coordinates of the nuclei.

( )0

2

21

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂∂

=ji

ij uuxVH

ji

ijij MM

HW =mass-weighted Hessian matrixW: Mi, Mj: masses of atoms i and j.

ui,j: displacement coordinate with respect to equilibrium

1

1),(−

=kT

hjj j

eTn νν Bose-Einstein

statistics

Thermal vs Elastic Properties of a SolidV

VV

j

j

jj ∂

∂−=

∂∂

−=ν

νν

γlnln

Grüneisen parameters

static pressure: PST (V)

zero point pressure: PZP (V)

thermal pressure: PTH (V, T).

From vibrational Frequencies to

Thermo Elastic and Thermodynamic Properties

∑∑−

++⎟⎠⎞

⎜⎝⎛

∂∂

=j

yj

jj

jjT

STje

yVkTy

VkT

VE

P12

γγ

Equation of State (Anderson 1995)

EoS works in Quasi – Harmonic Approximation

Bulk modulus, thermal expansion, specific heat and entropywere obtained in the limit of the quasi-harmonic approximation (Anderson, 1995).

For each normal mode, the Grüneisen’s parameter γj was determined through theanalytic first derivative at each volume (pressure) of the 2nd order polynomial resultingfrom the fitting of the numerical νj (V) curves.Static pressures PST were determined by numerical derivative, with respect to V, of the4th order polynomial interpolating the EST(V) curve; zero points (PZP) and thermal (PTH)pressures were obtained by direct application of equations as follow:

PT TV

VTV ⎟

⎠⎞

⎜⎝⎛∂∂

=1),(α

TT TP

KTV ⎟

⎠⎞

⎜⎝⎛∂∂

=1),(α

Num. Deriv BM3 EOSThermal Expansion

TTTvp KVTCC 2α+=∑−

Ν=j

X

Xj

V j

j

eeX

kC2

2

0 )1(

Heat Capacity

∑ ⎥⎦

⎤⎢⎣

⎡−−= − )1ln(),(),( /

0kThj

jjje

kTh

TnkNTVS ννν

Entropy

Applications of ab initio quantum mechanical calculation CRYSTAL code

Thermo- elastic, Equation of State and Thermodynamic Properties of CaCO3 polymorphs

Why Calcium carbonates?

Calcium Carbonate is relatively well - defined system and, assuch can be used as a test of the whole algorithm which hasbeen refined in the present work.

Calcium carbonate crystallizes in nature as:

Calcite R (3+i)c

Aragonite Pmcn

Vaterite Pbnm.

cR−

3

Calcite ( CaCO3 )cR−

3

The calcite uniaxial (-) structure has beendetermined by Bragg in 1914 and hasbeen described traditionally using theNaCl structure as a starting point.

Hexagonal frame: a = 4.989 and c = 17.061 Z = 6 CaCO3

Rhombohedral frame: a = 6.375, α = 46°05’ Z = 2 CaCO3.

CalciteMost calcites are relatively pure. Commonimpurities are:

magnesium, ferrous iron and manganese.

Aragonite Phase relation of Calcium carbonates

The calcite structure is characterized by alternate layers of Ca atoms and CO3 groups along the z axis, and between successive layers the CO3 groups have opposite orientations.

Aragonite

Aragonite occurs as the inorganic constituent of many invertebrate skeletons andsediments derived from them. It also occurs as a primary phase in high – pressuremetamorphic rocks.

The aragonite structure: layers of calcium atoms parallel to (001) along the z axis and separated by CO3 layers. The C atoms do not stack on top of each other along the z axis and are displaced relative to each other in the y direction.

Monoclinic system: Calcite II

A, aragonite; I-V, polymorphic forms of calcite;+ H, J + P, I+W, K+S+M, dashed lines,metastable equilibrium.

This high-pressure (15 kbar ) form ofCaCO3 is monoclinic, space group P21/ca= 6.334 ± 0.002,b= 4.948 ± 0.005,c= 8.033 ± 0.025,β= 107° 54' and Z = 4.

Other Calcium Carbonate Polymorphs

Orthorombic system: Vaterite

Orthorhombic structure with space group Pbnm after Meyer (1959).

Hexagonal structure with space group P63/mmc after Kamhi (1963).

Vaterite (μ - Calcite)

Vaterite is thermodynamically the least stableCaCO3 polymorph, and it rapidly transformsinto calcite or aragonite in aqueous solution(Fricke et al. 2006).

The vibrational spectrum of CaCO3 Aragonite. An ab initio study with the CRYSTAL code

BSC BSD Exp

ΔE 0.054a 5.02311 5.00955 4.96183(1)

b 8.03418 8.02478 7.96914(2)

c 5.85903 5.86119 5.74285(2)

V 236.451 235.621 227.081(1)

Cay 0.41515 0.41527 0.41502(2)

Caz 0.76009 0.75985 0.75985(4)

Cy 0.76258 0.76241 0.7691(1)

Cz -0.08152 -0.08134 -0.0823(1)

O1y 0.92273 0.92163 0.92238(8)

O1z -0.09018 -0.08999 -0.09453(8)

O2x 0.47306 0.47212 0.47499(7)

O2y 0.68097 0.6813 0.68013(5)

O2z -0.08468 -0.08423 -0.08725(7)

dC-O1 1.2877 1.2787 1.2235

dC-O2 1.2983 1.2892 1.3228

dCa-O 2.4306 2.5539 2.5196

Geometrical parameters of aragonite;calculated values obtained with thedifferent basis sets (BSC-D) at theB3LYP level are compared withexperimental results.

a, b and c lattice parameters (Å),V is the volume of the cell in Å3.Cai, Ci, O1i and O2i fractionalcoordinates of the irreducible atoms inthe primitive cell,di−j indicate the shortest interatomic

distances (Å).

ΔE (in m Hartree) BSC- BSD (- 3766.163915688 Hartree)→BSD

better than BSC.

mmm orthorhombic cell of aragonite:Z = 4 CaCO3N = 20 atoms;

60 vibrational modes can be classified according to the Irreducible Representations of the mmm point group as follows:

Γtot = 9Ag + 6Au + 6B1g + 9B1u + 6B2g + 6B2u + 9B3g + 6B3u.

24 external vibrations (E): 68,2 – 293,4 cm-1

33 internal vibrational modes (I): 692,9 – 1599 cm-1

Where: 9A1g + 6B1g + 6B2g + 9B3g Raman active8B1u + 5B2u + 5B3u IR active

6A1u inactive1B1u+1B2u+1B3u acoustic.

The Raman and IR vibrational modes calculated using BSD basis set and the B3LYPHamiltonian and were compared with experimental data: the RMS = 4.1 cm-1.Two exceptions are noted:• lowest frequencies < 300 cm-1

• highest >1080 cm-1, for which calculation overestimates the experimental valuesas for Raman and IR frequencies (LO-TO) between 9.9 and 12.9 cm-1.

Ab-initio quantum-mechanical calculation on aragonite at high pressure. Elastic and Thermodynamic properties and comparison with experimental data

Structure and vibrational frequencies (at the Г point) of aragonite have been calculated from first principles, by using the hybrid Hartree-Fock/DFT B3LYP Hamiltonian, at different unit cell volumes, in the 185-242 Å3 range.

Starting with the equilibrium geometry of theorthorhombic aragonite primitive cell, newgeometries have been obtained for reducedvolumes of the unit cell which correspond to staticpressures (1st term of EoS formula) ranging fromPst = 0 GPa at V = 236.44 Å3 (equilibriumgeometry) to Pst = 28.62 GPa at V = 185.46 Å3.

0 400 800 1200 16000

5

10

15

g j

n (cm -1)

gj

M ean g=1.637

Calculated Grüneisen’s parameter values (γj),at equilibrium volume, as a function of frequency (νj).

γj calculated γj experimental

νj(cm-1) γj νj(cm-1) γj

155.3 1.30 155 1.20

180.5 2.35 180 2.60

211.2 1.48 209 1.20

696.1 0.15 702 0.14

707.8 0.21 710 0.20

1092.1 0.16 1084 0.16

0 500 1000 1500

0.01

0.1

1

Ther

mal

pre

ssur

e(G

Pa)

300K 400K 500K 600K 700K

ν(cm-1)

Calculated thermal pressure contribution of phonons, as a function of the normal modes frequency

210 215 220 2250

2

4

6

8

Pres

sure

(GPa

)

Volume(A3)

EXP CALC

Calculated (scaled volume) isothermal curve at 298 K: comparison with experimental values (Martinez et al., 1996).

Excellent agreement between the calculated total pressure curve at 298 K, as a function of the scaled volume, and the experimental one (Martinez et al. 1996).

300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

80

KBM3

KNumDeriv

EXPBul

k M

odul

us (G

Pa)

Temperature (K)

P= 0 GPa

Calculated BM3 and Num_Deriv bulkmodulus as a function of T (298÷973 K)

at P = 0 GPa. Comparison with experimental data.

0 1 2 3 4 50

20

40

60

80

100

Bul

k M

odul

us(G

Pa)

Pressure(GPa)

KBM3

KNumDerix

EXP

T=298K

Calculated BM3 and NumDeriv bulk modulus as a function of P (0÷5 GPa) at

T = 298 K . Comparison with experimental results.

400 600 800 10000

2x10-5

4x10-5

6x10-5

8x10-5

1x10-4

a(K

-1)

Temperature (K)

aBM3 aNumDerivaEXP

Calculated BM3 and NumDeriv thermal expansion values as a function of

T (273÷973 K) at P=0 GPa. Comparison with experimental data.

T (K) BM3 (K-1) NumDeriv (K-1) EXP (K-1)

298 6.06E-05 5.88E-05 6.36E-05

373 6.40E-05 6.33E-05 6.76E-05

473 6.97E-05 6.99E-05 7.01E-05

573 7.52E-05 7.94E-05 7.14E-05

673 8.06E-05

773 8.59E-05

873 9.11E-05

973 9.60E-05

Better agreement between Thermal expansion BM3 and experimental values.

0 50 100 150 200 250 300

0

20

40

60

80

100

Entr

ophy

(J*m

ole-1

*K-1

)

Temperature (K)

SCALC

SEXP

Calculated Entropy values as a function of T (20-300K) at P = 0 GPa.

Comparison with experimental data.0 50 100 150 200 250 3000

20

40

60

80

100

Hea

t Cap

acity

(J*m

ole-1

*K-1

)

Temperature (K)

CpBM3

CpNumDeriv

CpEXP

Cv

Calculated Heat capacity at constant volume (Cv) and two at constant pressure

Cp BM3 and CpNumDeriv as a function of T (20÷300 K) at P=0 GPa.

Comparison with experimental values.

KVTCC VP2α+=

Using the α, KT and CV values, it was possible to calculate the following Grüneisen ratio

= 1.898

at P = 0 GPa and T = 298 K, which appears to be in good agreement

(over-estimation of 3%) with respect to the experimental room temperature value of 1.843.

VT CVKαγ =

The Gruneisen ratio is a very important thermodynamic parameter which helps to quantify the

relationship between the thermal and elastic properties of a solid.

The pressure (or volume) and temperature variation of γj contributesignificantly to the evaluation of EoS parameters (ex: Pthermal) and thus on

exact (P,T) point computation of many thermodynamic functions.

Using BM3 method, which uses a wider sampling of (P, T) space, we obtain more stable and reliable (if compared with experimental ones) values of the

thermodynamic functions. (*)

Ungureanu C.G., Prencipe M., Cossio R. “Ab initio quantum-mechanical calculation of CaCO3 aragonite at high pressure: thermodynamic properties and comparison with experimental data” Eur. J. Min. 2010, 22, 693–701.

Thermodynamic properties of Calcite at high pressure:an ab initio quantum - mechanical calculation

Optimized geometries, staticenergies and zone-centrevibrational frequencies (νj, j =1,3n-3; n = 10 atoms in the unitcell) were determined for a set of10 different cell volumes (V)ranging from 356.0839 to 385.7618A3) at B3LYP and WC1LYP level.

0 200 400 600 800 1000 1200 1400 1600

0.00

0.05

0.10

0.15

0.20

0.25

Ther

mal

Pre

ssur

e(G

Pa)

ν(cm-1)

300K 400K 500K 600K 700K

Thermal pressure contribution Pth of phonons, calculated as a function of

the normal modes frequency, at five different temperatures.

Raman and IR vibrationalfrequencies of CaCO3 calcite (0 ÷5 GPa pressure range) has beencalculated at Г point with theperiodic ab initio CRYSTAL09code using full geometryoptimization at constant volume,high quality localized basis set andWC1LYP hybrid Hamiltonianfunction.

0 1 2 3 4 50

20

40

60

80

100

K (G

Pa)

Pressure(GPa)

KCALC

KEXPWC1LYP

Calculated bulk modulus K at WC1LYP level as a function of P (0÷5 GPa) at T = 298 K. Comparison

with experimental values (Salje & Viswanathan, 1976).

300 400 500 600 7000

1x10-5

2x10-5

3x10-5

α(K

-1)

Temperature(K)

αCALC

αEXP

WC1LYP

Calculated thermal expansion α at WC1LYP level as a function of T (298 -671 K) at P = 0 GPa: comparison with

experimental values (Salje & Viswanathan, 1976).

Ungureanu C.G., Prencipe M., Cossio R. “Mechanical, Elastic andThermodynamic properties of CaCO3 calcite at high pressure: an ab initioquantum-mechanical calculation”, Eur. J. Min. , submitted

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

Diff.RMS=0.31 J/mole*K

S(J/

mol

e*K

)

Temperature(K)

SCALC

SEXPWC1LYP

Calculated Entropy values S as afunction of T (20-1000K) at P=0 GPa.Comparison with experimental (20-

300K; 350 ÷ 750K; 298 and 1000 K).

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

Cp(

J/m

oleK

)

Temperature(K)

WC1LYP CpCALC

CpEXP

Diff.RMS=1.07 J/moleK

Calculated Heat capacity at constant pressure (CP) at WC1LYP level as a function of T (20 ÷1000 K) at P = 0 GPa. Comparison with experimental (20 - 300K; 350 ÷ 750 K. 298 and 1000 K).

Although we note small difference between thermodynamic properties calculated at B3LYP and at WC1LYP level, a general

better agreement between WC1LYP calculated values and experimental ones

can be reported.

*S at T = 298K, was calculated taking intoaccount the phonon dispersion.

Vaterite (μ - Calcite). Ab initio quantum mechanical calculation of normal vibrational modes at Γ point

(Å) BSD Exp.

a 6.69422 7.15

b 8.50694 8.48

c 4.52341 4.13

V ( Å3) 257.597 250.4142

Cax 0 0

Cay 0 0

Caz 0 0

Cx -0.3693 0.67

Cy 0.25 0.25

Cz 0.25 0.157

O1x -0.45677 0.471

O1y 0.25 0.25

O1z 0.27 0.157

O2x 0.45677 0

O2y 0.25 0.118

O2z 0.27 0.67

dCa-O 2.3841

dC-O1 1.2794

dC-O2 1.2917

The equilibrium geometryof the orthorhombic vaterite Pnma

and vibrational frequencies calculation

(CRYSTAL code, using hybrid Hartree-Fock/DFT B3LYP Hamiltonian and BSD).

The cell parameters (Å) obtained are:

a = 6.69422, b = 8.50694, c = 4.52341α=β = γ = 90° V = 257.597 Å3

and E = -3.766173E+03 Ha, at B3LYP level.

Dielectric properties

The Born effective charge anddielectric tensors were obtained withBSD and B3LYP calculation, using theCPHF method which is available in thenew test version CRYSTAL09.

Components Calculated

xx 2.1615

yy 2.5534

zz 2.388

The orthorhombic cell of vaterite Pnma contains 4 CaCO3 formula units, for a total of N = 20 atoms; its 60 vibrational modes can be classified according to the Irreducible Representations of the mmm point group as follows:

Γtot = 7Ag + 8Au + 5B1g + 9B1u + 5B2g + 9B2u + 7B3g + 7B3u.

Where: 7A1g + 5B1g + 5B2g + 7B3g Raman active9B1u + 9B2u + 7B3u IR active

8A1u inactive1B1u+1B2u+1B3u translations.

The spectrum of vaterite is divided in two frequency regionsseparated by a relatively large gap (320 cm-1)

from 352 cm-1 (B2g) to 672 cm-1 (B1u) "external" modes (E),frequencies larger than 672 cm-1 correspond to the "internal" modes (I).

Calcite II vibrational frequencies

calculation at Γ pointB3LYP WC1LYP Exp.

E (ha) -3.76E+03 -3.77E+03(Merrill &

Bassett 1975)

a 6.46320917 6.37962599 6.334

b 5.03907252 5.00426778 4.984

c 8.19823057 8.12532514 8.033

β (°) 108.026772 107.567684 107.9

ρ(g/cm-

3) 2.615 2.684 2.77

V(A3) 253.898004 247.305581 241.3

Cax 0.25 0.25 0.234(2)

Cay -0.25 -0.25 0.738(2)

Caz 0.25 0.25 0.217(1)

Cx 0.25 0.25 0.260(8)

Cy 0.25 0.25 0.253(9)

Cz -0.5 -0.5 0.504(3)

O1x 0.38 0.38 0.380(7)

O1y 0.12 0.12 0.156(3)

O1z -0.37 -0.37 0.637(3)

O2x 0.12 0.12 0.134(7)

O2y 0.12 0.12 0.088(4)

O2z 0.37 0.37 0.381(3)

O3x 0.25 0.25 0.221(6)

O3y -0.5 -0.5 0.490(4)

O3z -0.5 0.5 0.467(3)

Geometrical parameters of CaCO3;calculated values obtained with twodifferent functionals (B3LYP andWC1LYP) are compared withexperimental results.

CalciteII is monoclinic, P21/ca = 6.334 ± 0.020%, b = 4.948 ±0.015%, c = 8.033 ± 0.025%, andβ = 107.9°.

According to Irreducible Representations of the mmm point group as follows:Γtot = 15Ag + 15Au + 15Bg + 15Bu

Where: 15Ag + 15Bg Raman active15Bu IR active15A1u inactive1B1u+1B2u+1B3u

translation.

•We analysed these polymorphs of calcium carbonates and

• obtained appropriate results for the geometry of crystallographic cell and for the vibrational normal modes,

• using ab initio quantum mechanical calculation

We can conclude that:

CRYSTAL code may be considered one of the best modelization method dedicated to the study of the solid phases (especially 3D crystalline

structures)

not only at the equilibrium, but even more under HP- HT conditions, which plays an important role in

mineralogy, geophysics and sedimentary petrology.

The vibration frequencies obtained with ab initio quantum mechanical calculation, in comparison with

Debye 1911 - the frequencies in a crystal are distributed as though the solid were an isotropic elastic continuum rather than an aggregate of particles, and Einstein

1907 - all vibrational modes have the same frequency models,provide the appropriate underlying in the determination of Mechanical and

Thermodinamical Properties of the crystalline systems as calcium carbonate type.

Conclusions