A computational study of the surface structure and reactivity of calcium fluoride

Abstract

Electronic structure calculations based on the density functional theory (DFT) are employed to investigate the electronic structure of fluorite (CaF₂) and the mode and energies of adsorption of water at the main {111} cleavage plane. Electron density plots show the crystal to be strongly ionic with negligible ionic relaxation of the unhydrated surface. We find associative adsorption of water at the surface with hydration energies between 41 and 53 kJ mol⁻¹, depending on coverage. We next employ atomistic simulation techniques to investigate the competitive adsorption of water and methanoic acid at the planar and stepped {111}, {011} and {310} surfaces. The hydration energies and geometries of adsorbed water molecules on the planar {111} surface agree well with those found by the DFT calculations, validating the interatomic potential parameters. Methanoic acid adsorbs in completely different configurations on the three surfaces, but always by one or both oxygen atoms to one or more surface calcium atoms. Molecular Dynamics simulations at 300 K show that the effect of temperature is to increase the difference in adsorption energy between methanoic acid and water at the planar {111} surface. The methanoic acid remains bound to the surface whereas the water molecules prefer to form a droplet of water between the two surface planes. We show in a series of calculations of the co-adsorption of water and methanoic acid that the presence of solvent makes a significant contribution to the final adsorption energies and that the explicit inclusion of solvent in the calculations is necessary to correctly predict relative reactivities of different surface sites, a finding which is important in the modelling of mineral separation processes such as flotation.

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1 Introduction

Fluorite (CaF₂) is a widely distributed mineral and it often occurs in combination with a range of other ores, notably lead and tin ores, and together with calcite and apatites.¹ It is the most important source material for hydrofluoric acid² and as such needs to be efficiently separated from any co-existing minerals, usually by the technique of froth flotation, e.g.refs. 3 and 4. In addition to numerous investigations of the fluorite structure itself, both experimental and computational, e.g., refs. 5–7 its crystal growth and dissolution have been widely studied, e.g., refs. 8–10, as well as its possible application as a substrate for the epitaxial growth of thin films.¹¹ The effect of hydration on the CaF₂ surface structure is of considerable interest in all of these applications, while the adsorption of organic surfactant molecules at the surfaces is important in mineral processing techniques, such as flotation, where the selectivity of the surfactants plays a major rôle in the design of a successful separation process.^3,4

In this paper, we describe our computational investigations of the electronic structure of the fluorite crystal and the surface reactivity towards water and methanoic acid. We use methanoic acid as a model of carboxylic acid surfactants such as oleic acid, and study its adsorption at the major {111}, {011} and {310} surfaces of calcium fluoride, including a series of stepped surface sites. The approach we have chosen to adopt is to use electronic structure calculations based on the density functional theory (DFT) to study the main {111} cleavage plane of fluorite in order to obtain, firstly, details of the geometry and electronic structure of the dry and hydrated surfaces, and secondly, reliable estimates of the hydration energies to compare with when using classical interatomic potential techniques to model larger scale systems. Comparison of the geometries and energies of adsorbed water molecules at the calcium fluoride surface, obtained using both quantum mechanical and classical techniques, will give an indication as to whether the potential model used in the atomistic simulations is capable of reliably modelling fluorite–water interactions. If there is good agreement between the results of the different methods, we can then be confident of applying atomistic simulations to larger systems, including both adsorbing species, which are currently beyond the capability of electronic structure calculations.

2 Methodology

In the electronic structure calculations, the total energy and structure of the simulation system, comprising slabs of solid material separated by a vacuum gap, which together are repeated periodically in three dimensions, was determined using the Vienna Ab Initio Simulation Program (VASP).^12–15 The basic concepts of density functional theory (DFT) and the principles of applying DFT to pseudopotential plane-wave calculations has been extensively reviewed elsewhere.^16–18 Furthermore, this methodology is well established and has been successfully applied to the study of adsorbed atoms and molecules on the surface of ionic materials.^19–22 The VASP program employs ultra-soft pseudo-potentials,^23,24 which allows a smaller basis set for a given accuracy. Within the pseudo-potential approach only the valence electrons are treated explicitly and the pseudo-potential represents the effective interaction of the valence electrons with the atomic cores. In our calculations the core consisted of orbitals up to and including the 1s orbital for fluorine and oxygen and the 3p orbital for Ca (H has no core). The valence orbitals are represented by a plane-wave basis set, in which the energy of the plane-waves is less than a given cutoff (E_cut).

For surface calculations, where two energies are compared, it is important that the total energies are well converged. The degree of convergence depends on a number of factors, two of which are the plane-wave cutoff and the density of k-point sampling within the Brillouin zone. We have by means of a series of test calculations on bulk CaF₂, where these parameters were varied systematically, determined values for E_cut (500 eV) and the size of the Monkhorst–Pack²⁵ k-point mesh (3 × 3 × 3) so that the total energy is converged to within 0.05 eV.

The electronic structure calculations were performed within the generalized-gradient approximation (GGA), using the exchange-correlation potential developed by Perdew and Wang,²⁶ which approach has been shown to give reliable results for the energetics of adsorbates, e.g., water on CaO,²⁷ TiO₂ and SnO₂.²⁸

The larger scale systems were modelled using the less computationally expensive atomistic simulation techniques, based on the Born model of solids,²⁹ where simple parameterised analytical forms are used to describe the forces between atoms. In this work we employ the METADISE code³⁰ to investigate the surface systems by energy minimisation, which is achieved by adjusting the atoms in the system until the net forces on each atom are zero. Energy minimisation simulations will yield adsorption energies, which have previously been shown to give good agreement with experimental surface sampling techniques such as temperature programmed desorption, e.g., ref. 31, as well as lowest energy configurations of the adsorbate/solid interface.

In addition, we employed Molecular Dynamics (MD) simulations to derive the potential parameters for the water–methanoic acid interactions and also to investigate whether the inclusion of temperature in the calculations would affect the adsorption behaviour and/or energies. The MD code used was DL_POLY³² where the integration algorithms are based around the Verlet leap-frog scheme.³³ We used the Nosé–Hoover algorithm for the thermostat,^34,35 as this algorithm generates trajectories in both NVT and NPT ensembles, thus keeping our simulations consistent. The Nosé–Hoover parameters were set at 0.5 ps for both the thermostat and barostat relaxation times. The surface simulations were run for at least 500 ps each (approximately 2.5 × 10⁶ timesteps) as an NVT ensemble, i.e., a constant number of particles, constant volume and a constant temperature of 300 K.

We used a combination of three potential models for a description of the interactions of the various atoms in the systems, namely by Catlow et al., for the calcium fluoride crystal;³⁶ the cvff forcefield for methanoic acid;³⁷ and the water potential model by de Leeuw and Parker.³⁸ The parameters for the interactions between water and methanoic acid with the fluoride surfaces were derived following the approach by Schröder et al.,³⁹ while the water–methanoic acid parameters were fitted to the experimental solvation energy of methanoic acid.⁴⁰ The full potential model is given in Table 1.

Table 1 Potential parameters used in this work (short range cutoff 20 Å)

Ion	Charges (e)		Core-shell interaction/eV Å^−2
	Core	Shell
F	+1.380	−2.380	101.200000
Oxygen of carbonate group (O)	+0.587	−1.632	507.400000
Oxygen of water (Ow)	+1.250	−2.050	209.449602
Ca	+2.000
Carbon of carbonate group (C)	+1.135
Hydrogen of water (Hw)	+0.400
Doubly-bonded oxygen of methanoic acid (OD)	−0.380
Hydroxy oxygen of methanoic acid (OH)	−0.380
Carbon of methanoic acid (CD)	+0.310
Hydroxy hydrogen of methanoic acid (HO)	+0.350
Hydrogen attached to carbon of methanoic acid (HC)	+0.100
	Buckingham potential
Ion pair	A/eV	ρ/Å	C/eV Å^6
Ca–O	1550.0	0.29700	0.0
Ca–F	1272.8	0.2997	0.0
Ca–Ow	1186.6	0.29700	0.0
Hw–O	396.3	0.23000	0.0
Hw–Ow	396.3	0.25000	10.0
O–O	16372.0	0.21300	3.47
F–F	99731833.99084	0.12013	17.02423
O–Ow	12533.6	0.21300	12.09
Ca–OH	563.64	0.29700	0.0
F–Ow	79785220.99	0.12013	26.78752
F–Hw	715.339	0.2500	10.00
Ca–OD	563.64	0.29700	0.0
OH–O	37898119	0.12013	11.309
OD–O	37898119	0.12013	11.309
OH–F	37898119	0.12013	25.1
OD–F	37898119	0.12013	25.1
Ow–OH	4797.6	0.213	30.2
Ow–OD	4797.6	0.213	30.2
Ow–HO	396.3	0.25	0.0
Ow–HC	396.3	0.25	0.0
Ow–CD	895	0.26	0.0
	Lennard–Jones potential
	A/eV Å^12		B/eV Å^6
Ow–Ow	39344.98		42.15
HC–O	2915.25		4.222
HO–O	2915.25		4.222
CD–O	3315.91		19.846
OD–Hw	1908.1		5.55
OH–Hw	1908.1		5.55
HC–F	2915.25		9.3784
HO–F	2915.25		9.3784
CD–F	3315.91		44.012
	Morse potential
	D/eV	α/Å^−1	r 0/Å
C–O	4.710000	3.80000	1.18000
Hw–Ow	6.203713	2.22003	0.92376
CD–HC	4.66	1.77	1.10
OH–HO	4.08	2.28	0.96
CD–OH	4.29	2.00	1.37
CD–OD	6.22	2.06	1.23
	Three-body potential
	k/eV rad^−2		Θ Õ
Ocore–C–Ocore	1.69000		120.000000
H–Owshell–H	4.19978		108.693195
OH–HO–CD	4.29		112.000000
CD–HC–OH	4.72		110.000000
CD–OD–HC	4.72		120.000000
CD–OD–OH	12.45		123.000000
	Four-body potential
	k/eV rad^−2		Θ Õ
C–Ocore–Ocore–Ocore	0.11290		180.0
	Intermolecular Coulombic interaction (%)
Hw–Ow	50
Hw–Hw	50

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3. Results and discussion

Calcium fluoride has the cubic fluorite crystal structure with space group Fm3m and a = b = c = 5.4323 Å, where each calcium ion is surrounded by eight fluoride ions, which are in turn coordinated to four calcium ions in a tetrahedral arrangement, shown in Fig. 1. The calcium ions are arranged on a cubic face-centred lattice, and if we divide the unit cell into 8 smaller cubes, we find the fluoride ions in the centres of these cubes.⁴¹ The cleavage plane is the {111} surface, which consists of planes of calcium ions in a hexagonal array with a layer of fluoride ions both above and below.⁹ The {111} surface is thus terminated with fluorine atoms and just below the surface are seven-coordinate calcium ions. We first employed DFT methods to investigate the dehydrated {111} surface, calculating lattice parameters of a = b = c = 5.4051 Å, in excellent agreement with experiment. Fig. 2 shows the relaxed {111} surface, including the electron density distribution around the calcium and fluoride ions and interatomic distances, from which it is clear that the ionic relaxation of the surface is small, a dilation of the topmost F–Ca spacing of 0.01 Å, followed by a contraction of the second interlayer distance by 0.02 Å. The contour plots of the electron density show that the crystal is strongly ionic with electron density firmly centred on the anions. The distortion of the electron density round the surface ions is minimal, which leads to the negligible ionic relaxation of the surface layer.

Fig. 1 Bulk structure of CaF₂ showing cubic face-centred calcium lattice with the fluoride ions in the centres of each of eight smaller cubes making up the cubic unit cell (Ca = black, F = pale grey).

Fig. 2 Side view of the relaxed CaF₂ {111} surface showing electron density contour plots and interatomic distances (Ca = dark grey, F = pale grey, contour levels are from 0.05 to 0.35 e Å⁻³ at 0.05 e Å⁻³ intervals, bond lengths in Å).

3.1 Hydrated {111} surface

We next investigated the adsorption of water at the {111} surface to evaluate the energies of adsorption and the relaxed hydrated surface structure. We studied both the adsorption of a full monolayer (i.e. one water molecule per surface calcium ion) and a 50% partial coverage. We used a range of different starting configurations of associatively adsorbed water molecules on the surface to ensure that the final converged configuration would be a global, rather than a local, minimum energy configuration. The hydration energy per water molecule for the partial coverage of 50% was calculated to be −53.4 kJ mol⁻¹ compared to the sum of the energies for the dry surface and an isolated gaseous water molecule, which decreased to −41.4 kJ mol⁻¹ for full monolayer coverage. These calculated hydration energies of approximately 41–53 kJ mol⁻¹ suggest that the water molecules are physisorbed rather than chemisorbed onto the surface. At 50% partial coverage (Fig. 3), the water molecules adsorb almost flatly onto the surface and are much more closely coordinated to the surface than at full monolayer coverage; the calcium–oxygen distance increases from 2.37 Å to 2.62 Å and the fluorine–hydrogen distance from 1.52 Å to 1.67 Å when the coverage is increased. This result suggests that the lattice spacing of fluorine is not large enough to accommodate a full layer of water molecules in an optimum position.

Fig. 3 Plan view of the minimum energy structure of the CaF₂ {111} surface with 50% coverage of associatively adsorbed water molecules, showing almost flat adsorption of the water molecules (fluorite shown as framework, water space-filled; Ca = black, F = pale grey, O = black, H = white).

The DFT calculations did not show any dissociation of the water molecules to form a hydroxylated surface at either coverage, so in order to be certain that there was no lower energy configuration with dissociatively adsorbed water molecules, we also simulated a fully hydroxylated surface as a starting configuration. A hydroxyl group was placed above each surface calcium ion and a proton above each surface fluoride ion. However, the dissociatively adsorbed water molecules reassembled to form molecular water. Fig. 4 shows the sequence of reformation of the water molecules on the {111} surface, from initial configuration to a midway snapshot, where tilting of the hydroxyl group towards the proton and lengthening of the H–F bond (from 1.2 Å to 1.7 Å) is observed, to the final configuration with associatively adsorbed water molecules. The distance between the hydroxyl oxygen atom and the proton decreases from an initial hydrogen-bond distance of 2.22 Å to a normal O–H bond of 1.01 Å. The calculated hydration energy of −41.3 kJ mol⁻¹ is identical to the associative starting configuration (monolayer coverage), giving us confidence that the lowest energy configuration had been found. The easy reformation of undissociated water molecules indicates that there is no significant energy barrier to this process.

Fig. 4 Reassembly of dissociatively adsorbed water molecules on the CaF₂ {111} surface: (a) side view showing initial configuration of hydroxylated surface; (b) snapshot during minimisation process showing tilting of hydroxyl group and lengthening of F–H bond; (c) side view of the final configuration showing associatively adsorbed water molecules (crystal shown as framework, water space-filled; Ca = black, F = pale grey, O = black, H = white).

The preference for associatively rather than dissociatively adsorbed water on the main CaF₂ {111} surface agrees qualitatively with previous atomistic and electronic structure calculations of water adsorption at the vacuum interface of MgO, another ionic crystal of cubic space group Fm3m, which showed that dissociative adsorption is energetically unfavourable on the perfect {100} cleavage plane, e.g., refs. 19,31,42,43, and only occurs at defects and low-coordinated surface sites^44–46 or at the liquid water interface where H₃O⁺ species are taken into account.⁴³ Electronic structure calculations of the main TiO₂ cleavage plane, the {110} surface, show hydroxylation of the surface at half coverage.^20,28 However, Lindan et al. found that at full coverage a mixture of associatively and dissociatively adsorbed water molecules is observed, where the water molecule is adsorbed almost flat onto the surface to maximise hydrogen-bonding to oxygen atoms of both the hydroxyl group and the mineral surface.²⁰ This configuration of the associatively adsorbed water molecules on the TiO₂ {110} surface is thus like that on the CaF₂ {111} surface, where almost flat adsorption of the water molecules and a network of hydrogen-bonding is preferred over dissociative adsorption.

Our calculations indicate that the binding of the water molecule's oxygen atom to a surface calcium atom is the main interaction. This finding suggests that increasing the coordination of the surface cation to the bulk value, from seven- to eight-coordinate for the {111} surface, is the driving force behind the adsorption. From these calculations we would therefore suggest that, as with MgO, dissociative adsorption of water takes place at defects and low-coordinated surface sites rather than the higher-coordinated cations of the perfect {111} cleavage plane.

3.2 Hydration of planar and stepped surfaces

In order to study larger-scale systems, we employed atomistic simulation techniques to model two more fluorite surfaces in addition to the {111} surface, namely the {011} and {310} surfaces, as well as two stepped {111} surfaces, as a more realistic model for experimental surfaces. We investigated adsorption of both water and methanoic acid on the surfaces to evaluate their relative structures and adsorption energies.

We first modelled the unhydrated surfaces to calculate their stabilities, after which we hydrated the surfaces to evaluate any changes in stability. The surface stabilities are measured by the surface energy, which is calculated as follows:

where U_s is the energy of the ions in the surface simulation cell and U_b is the energy of an equal number of bulk ions, while A is the area of the surface. A low, positive value for the surface energy indicates a stable surface, which will be important in the morphology of the mineral. The surface energies of the dry and hydrated surfaces are collected in Table 2, where the surface energy of the hydrated surface is calculated with respect to bulk water. The {111} surface is clearly the most stable of the three surfaces considered, both in dry and aqueous conditions, in agreement with the fact that this is experimentally the perfect cleavage plane of calcium fluoride.⁴¹ The dry {310} surface is very unstable, but its stability is increased substantially when the surface is hydrated, making it now more stable than the {011} surface.

Table 2 Surface energies of the calcium fluoride surfaces

Surface energies/J m^−2
Surface	Unhydrated	Hydrated
{111}	0.52	0.40
{011}	0.82	0.90
{310}	1.56	0.67

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We first considered adsorption of water on the planar {111} surface, where comparison to the equivalent DFT calculations gives an indication of the accuracy of the potential model. The water molecules adsorb flat onto the surface at a Ca–O distance of 2.47 Å and H–F distances of 2.13–2.18 Å. As suggested by the DFT calculations, the Ca–Ca interatomic spacing of 3.85 Å is too small for a water molecule to adsorb on each calcium ion, and hence only 50% of the available adsorption sites are covered by water molecules. We again calculated the adsorption energies with respect to isolated gaseous water molecules to enable direct comparison with experimental techniques such as temperature programmed desorption (TPD). We calculated a hydration energy of −61.8 kJ mol⁻¹ at a partial coverage of 50%, which is in acceptable agreement with the DFT result of −53.4 kJ mol⁻¹. The discrepancy in the hydration energies is due to the fact that the atomistic simulations predict a completely flat mode of adsorption for the water molecules, with close coordination of both hydrogens to surface fluoride ions, while the water molecules in the DFT calculations adsorb slightly tilted, with one hydrogen atom pointing away from the surface, giving very different H–F distances for the two hydrogens (1.52 Å and 2.85 Å). When a full monolayer is adsorbed, the average adsorption energy drops to 38.5 kJ mol⁻¹, compared to 41.4 kJ mol⁻¹ for the DFT calculations. Due to lack of space on the surface for a full monolayer, the water molecules do not adsorb flat onto the surface and the lesser binding between surface fluoride ions and hydrogen atoms leads to an even better agreement for the adsorption energies between the two computational techniques. On the {011} and {310} surfaces, the Ca–Ca spacings are large enough easily to accommodate a full monolayer of water. On the {011} surface, the water molecules adsorb in an upright fashion, without significant H–F interactions. However, the increased stability of the hydrated {310} surface is due to flat adsorption of the water molecules, similar to the {111} surface, and an extensive network of hydrogen-bonding between both hydrogens and surface fluoride ions.

As ‘real’ surfaces are never completely free from defects, we have also included stepped surface sites in our calculations. We considered two steps on the {111} surface that differ in the orientation of the F₂ groups, which either lean backwards at an obtuse angle of 135° with respect to the underlying plane or forwards at an acute angle of 45° (Fig. 5). From the adsorption energies in Table 3, we see that hydration of the acute step edge is less favourable than the planar surface, due to the restricted space available for the adsorbing water molecule under the step. However, the more open adsorption site at the obtuse step edge, combined with the step ions' lower coordination number, makes hydration of this step more exothermic than the planar surface.

Fig. 5 (a) Acute and (b) obtuse steps on the CaF₂ {111} surface (Ca = black, F = pale grey).

Table 3 Adsorption energies of water and methanoic acid at the surfaces

Adsorption energies/kJ mol^−1
Surface	Water	Methanoic acid	Methanoic acid in water
Planar {111}	−38.5	−56.3	−96.7
Acute {111}	−29.1	−90.8	−34.1
Obtuse {111}	−50.8	−79.5	−21.2
{011}	−33.4	−102.4	—
{310}	−250.7	−110.9	—

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3.3 Adsorption of methanoic acid

When we considered adsorption of methanoic acid at the same surface sites, we found that the methanoic acid molecules adsorb onto the planar surfaces in three distinctly different fashions. On the {011} surface, the lattice spacing is large enough to allow full monolayer coverage with one methanoic acid molecule per surface calcium. The methanoic acid molecule adsorbs with both oxygen ions to two surface calcium ions, bridging between them (Fig. 6). The doubly bonded oxygen ion closely coordinates to the calcium ion at a distance of 2.2 Å while the oxygen atom of the hydroxyl group is at 2.65 Å from the second calcium ion. The hydrogen of the hydroxyl group relaxes into the surface and coordinates to a fluorine atom at 2.4 Å. The lattice spacing on the {310} surface is also large enough to accommodate full monolayer coverage (Fig. 7). The doubly bonded oxygen ion is again bonded to a surface calcium ion at 2.15 Å, while more loosely coordinated to calcium ions further away in the next layer (3.93 Å). Furthermore, the doubly bonded oxygen ions coordinate to hydroxyl hydrogens of other adsorbed methanoic acid molecules (2.4 Å), while the hydrogen bonded to the carbon atoms coordinates to surface fluoride ions (2.35 Å).

Fig. 6 Side view of the {011} surface, showing the crystal as a lattice framework (Ca = black, F = pale grey) and the methanoic acid molecule (space-filled, O = black, C = pale grey, H = white) coordinated by its oxygens to two surface calcium ions.

Fig. 7 Side view of the {310} surface, showing the crystal as a lattice framework (Ca = black, F = pale grey) and the methanoic acid molecule (space-filled, O = black, C = pale grey, H = white) closely coordinated by its doubly bonded oxygen atom to a surface calcium ion and hydrogen-bonding to a surface fluorine.

Due to the much smaller interatomic distance on the {111} surface, only a 50% coverage of methanoic acid can be accommodated, which is a reasonable coverage if we compare it with experimental work by Mielczarski et al., who observed a 30% coverage of oleic acid, which is a carboxylic acid with a long carbon chain instead of the hydrogen of methanoic acid.⁴⁷ The molecules adsorb in a fairly flat configuration onto the surface, bridging between two calcium ions, with both oxygen ions coordinated to a calcium at 2.2 Å for the doubly bonded oxygen ion and at 2.9 Å for the oxygen ion of the hydroxyl group (Fig. 8). The hydrogen atom of the hydroxyl group coordinates to two surface fluoride ions at 2.5 and 2.7 Å. When adsorbed at the step edges, we see that the trend in adsorption energies is reversed from the hydration pattern. More energy is now released upon adsorption at the acute step edge than at the obtuse edge, while both steps are calculated to be more favourable adsorption sites than the terraces of the planar surface. The reason for the higher exothermicity at the acute step edge is the fact that in addition to the same interactions between methanoic acid and the terrace atoms, as shown for adsorption on the planar surface, the doubly bonded oxygen atom also bonds to a low-coordinated calcium ion on the step edge, hence bridging the gap between step and terrace, which was not possible in the adsorption of water. The hydrogens also interact with fluoride ions both on the edge and the terrace, leading to a network of hydrogen-bonding between the surface and adsorbate. It is these multiple interactions that cause the methanoic acid adsorption at the steps to be more exothermic than on the planar surfaces.

Fig. 8 Plan view of the {111} surface with adsorbed methanoic acid molecule, showing the crystal as a lattice framework (Ca = black, F = pale grey) and the methanoic acid molecule (space-filled O = black, C = pale grey, H = white).

3.4 The effect of temperature: Molecular Dynamics simulations of the (111) surface

As energy minimisation techniques do not take into account temperature, in this section we employ Molecular Dynamics simulations (MD) to explicitly investigate the effect of temperature on the interaction of water and methanoic acid at the {111} plane at 300 K. We started the simulations with a monolayer of water adsorbed at the surface. However, during the simulation water molecules desorbed from the surface and diffused through the gap between the two surfaces, forming a droplet (Fig. 9). From the mean square deviation, the diffusion coefficient of the water molecules is calculated to be 1.2 × 10⁻⁹ m² s⁻¹, which is identical to that calculated for water molecules in a box of pure liquid water at 300 K under NPT conditions.³⁸ The hydration energy of −32.7 kJ mol⁻¹ at 300 K is less than the intermolecular interactions between water molecules themselves, calculated to be −43.0 kJ mol⁻¹ at the same temperature in agreement with experiment (−43.4 kJ mol⁻¹).⁴⁸ Hence, at this low water density, we may conclude that on energetic grounds the water molecules on the fluorite {111} plane prefer not to adsorb to the surface but to cluster together. We next repeated the calculation with methanoic acid as the adsorbate rather than water. This time, however, the methanoic acid remained bound to the {111} surface rather than diffuse through the gap. Clustering of the methanoic acid molecules also took place, but unlike the water molecules, surface diffusion of the methanoic acid was followed by the formation of clusters at the surface, in a similar adsorption pattern as was observed in the energy minimisation calculations above, by both oxygen atoms to surface calcium ions (Fig. 10). Similar clustering of methanoic acid was observed experimentally by Iwasawa et al. to occur at terraces on the TiO₂ (110) surface.⁴⁹ The energy of interaction of methanoic acid with the surface is calculated at −91 kJ mol⁻¹, higher than in the energy minimisation simulations. Hence, the effect of temperature, included in calculations through the employment of Molecular Dynamics simulations, is to exacerbate the difference in adsorption energies between water and methanoic acid at the fluorite (111) surface. The MD simulations also showed that although the methanoic acid diffuses along the surface, it remains adsorbed, while competitive interactions between the water molecules themselves outweigh interactions between the solid surface and water molecules, which as a result leave the surface.

Fig. 9 Water droplet formation between two {111} surfaces in a Molecular Dynamics simulation at NVT and 300 K.

Fig. 10 Adsorption and clustering of methanoic acid molecules at the {111} surface in a Molecular Dynamics simulation at NVT and 300 K.

Of course, in “real” mineral separation processes, water and the organic flotation reagents co-exist and we have hence extended our calculations to include both water and methanoic acid in the simulations.

3.5 Co-adsorption of water and methanoic acid

The adsorption energies from the energy minimisations for both water and methanoic acid onto the three surfaces are collected in Table 3. The hydration energies for the different calcium fluoride surfaces vary considerably due to the presence (or absence) of hydrogen-bonding to the surface fluoride ions, in addition to the calcium–oxygen interactions. The adsorption energies for methanoic acid onto the mineral surfaces are considerably larger than the hydration energies on the dominant {011} and planar and stepped {111} surfaces due to the capability of the acid molecules to bridge, by their oxygen atoms, between two or more surface calcium atoms and the close hydrogen-bonding to surface fluoride ions. These calculations would therefore suggest that it would be energetically preferential for the methanoic acid molecules to adsorb to these surfaces, displacing the water molecules from the adsorption sites, which was borne out by the MD simulations of water and methanoic acid at the {111} surface at 300 K, where the water desorbed from the surface while the methanoic acid remained adsorbed. However, in order to verify whether this assumption based on separate calculations of the adsorbates is valid, we repeated the calculations of methanoic acid adsorption at the planar and stepped {111} surfaces, but this time including a layer of water in the simulations, hence studying the competitive adsorption of water and methanoic acid directly. The adsorption energies were now calculated with respect to the hydrated surface and a solvated methanoic acid molecule. In order to model the co-adsorption of methanoic acid and water at the fluorite surfaces, we needed to derive interatomic potential parameters for the water–methanoic acid interactions. We fitted these potential parameters to the experimental solvation energy of methanoic acid, in a series of Molecular Dynamics simulations of a methanoic acid molecule in a box of 255 water molecules (Fig. 11). The final parameters thus derived (Table 1) gave a solvation energy for methanoic acid of −41.7 kJ mol⁻¹, compared to the experimental value of −47.4 kJ mol⁻¹.⁴⁰

Fig. 11 Average configuration of methanoic acid molecule in a simulation cell of 255 water molecules during a Molecular Dynamics simulation at NPT and 300 K (the apparently dissociated water molecules are, in fact, water molecules, but shown split up as an artefact of the periodic boundary conditions).

The data listed in Table 3 show that on the planar {111} surface, the presence of water increases the adsorption energy of methanoic acid, the reason for which becomes clear if we compare the adsorption pattern of the methanoic acid with that of water at the same surface sites. The methanoic acid only replaces one adsorbed water molecule at the planar surface and as the intermolecular interactions between the water molecules themselves (43 kJ mol⁻¹) or with the methanoic acid (40 kJ mol⁻¹) are very similar, the water molecules have no preference for interacting with either the methanoic acid or each other. The regular adsorption pattern of the water on the surface is not disturbed by the presence of the surfactant, but the adsorbate is stabilised by the formation of a network of hydrogen-bonded interactions to neighbouring water molecules. However, at the stepped surface sites the co-adsorption of water lowers the adsorption energies for methanoic acid (Table 3), both processes becoming much less exothermic. Again, the reason is two-fold, based on both the geometry of the surface sites and the relative adsorption energies of the surfactant and the water molecules. Hydration of the stepped sites (29–51 kJ mol⁻¹) is energetically similar to the planar fluorite surface (average ∼38.5 kJ mol⁻¹). Binding of the methanoic acid to the steps is stronger than water (average ∼85 kJ mol⁻¹) and it therefore remains closely bound to the step site even in the presence of water, as shown in Fig. 12 for the obtuse step. However, its presence at the step disturbs the regular pattern of water adsorption, at least in the immediate vicinity of the step, leading to a smaller adsorption energy. Thus, before the addition of water to the system we find similar adsorption energies for the three different surface sites on the {111} surface (56–91 kJ mol⁻¹), but once water has been introduced in the calculations, the adsorption energies show a much bigger variation with adsorption site (21–97 kJ mol⁻¹) and even a reversal of the relative stabilities, indicating that we need to include solvent effects explicitly if we are to predict realistic adsorption behaviour.

Fig. 12 Co-adsorption of water and methanoic acid at the obtuse step on the {111} surface (fluorite as framework, methanoic acid space-filled, water as triangles; Ca = dark grey, F = pale grey, O = black, C = grey, H = white).

4 Conclusions

We have shown in this work that computational techniques are well placed to provide insight at the atomic level into the interactions between substrate and adsorbate molecules. We have presented calculations of the adsorption of water and methanoic acid at calcium fluoride surfaces, using a combination of computational techniques. Accurate density functional theory calculations were used to obtain the electronic structure of the {111} surface together with hydration modes and energies; and atomistic simulation techniques to elucidate the geometry and relative adsorption energies of water and methanoic acid at a range of different surface sites. From our simulations we can draw the following conclusions.

Electron density plots generated by DFT calculations of the {111} surface show calcium fluoride to be a strongly ionic crystal, with no discernible distortion in the surface layer with respect to bulk layers, leading to minimal ionic relaxation of the surface.

Associative adsorption of water is preferred at the {111} surface, without significant energy barrier to reformation of dissociated water molecules into molecular water. The hydration energy is dependent upon coverage and both the decrease in hydration energy and lesser coordination to the surface upon increasing coverage indicates repulsive interactions between the adsorbed water molecules.

Modelling hydration of the {111} surface using atomistic simulation techniques gives similar hydration energies and configurations of the adsorbed water molecules to the DFT calculations, although there may be some overbonding of the H–F hydrogen-bonding in the interatomic potential approach.

Adsorption of methanoic acid up to full monolayer coverage is possible on both {011} and {310} surfaces, but on the {111} surface, due to the smaller calcium–calcium distance, only adsorption up to 50% is preferred. On this surface, the methanoic acid molecules adsorb by their oxygen atoms to two calcium atoms, forming a bridge between them. This mode of adsorption is particularly favourable, which is also seen at the stepped sites on the {111} surface. On both the {011} and the dominant {111} fluorite surfaces the energies of adsorption of methanoic acid compared to water show that adsorption of methanoic acid is energetically more favourable and hence methanoic acid should compete effectively with water for adsorption at these surfaces. Molecular Dynamics simulations of the two adsorbates at the {111} surface show that the effect of temperature is to widen the gap in adsorption energies between methanoic acid and water. The latter adsorbate leaves the surface and forms a water droplet between the {111} planes.

Simulations of methanoic acid adsorption in the presence of water bear out the suggestion that methanoic acid competes effectively with water as an adsorbate, as the methanoic acid remains adsorbed at the {111} terrace and steps forming close interactions with the surrounding water molecules. However, these latter calculations have also shown that interactions between surfactant and water molecules can have a radical effect on adsorption behaviour, and it is therefore not sufficient to calculate the interactions of surfactant molecules with mineral surfaces in isolation, as the presence of solvent in the calculations makes a significant contribution to the final adsorption energies and relative stabilities of the surface sites. The implication of these findings for the search for flotation reagents is that we need to explicitly include solvent in the calculations if we are to successfully predict the affinity of the mineral for particular surfactants.

Acknowledgements

We acknowledge the Engineering and Physical Sciences Research Council, grant no. GR/N65172/01, and the Royal Society, grant no. 22292, for funding and the Minerals and Materials Consortia for the provision of computer time on the Cray T3E.

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