Energetics of Sn²⁺ isomorphic substitution into hydroxylapatite: first-principles predictions

Abstract

The energetics of Sn²⁺ substitution into the Ca²⁺ sublattice of hydroxylapatite (HA), Ca₁₀(PO₄)₆(OH)₂, has been investigated within the framework of density functional theory. Calculations reveal that Sn²⁺ incorporation via coupled substitutions at Ca(II) sites is energetically favourable up to a composition of Sn₆Ca₄(PO₄)₆(OH)₂, and further substitutions at Ca(I) sites proceed once full occupancy of Ca(II) sites by Sn²⁺ is achieved. Compositions of Sn_xCa_10−x(PO₄)₆(OH)₂ (x = 4–9) are predominant, with an optimal stoichiometry of Sn₈Ca₂(PO₄)₆(OH)₂, and Sn-substituted HA follows approximately Vegard's law across the entire composition range.

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1 Introduction

Hydroxylapatite [Ca₁₀(PO₄)₆(OH)₂], fluorapatite [Ca₁₀(PO₄)₆(F)₂], and chlorapatite [Ca₁₀(PO₄)₆(Cl)₂] end-members of the apatite group are among the most extensively studied minerals of the apatite supergroup, with generic crystal-chemical formula ^IXM(I)^VII₄M(II)₆(^IVTO₄)₆X₂, where M and T represent species-forming cations (M = Ca²⁺, Pb²⁺, Ba²⁺, Sr²⁺, Mn²⁺, Na⁺, Ce³⁺, La³⁺, Y³⁺, Bi³⁺; T = P⁵⁺, As⁵⁺, V⁵⁺, Si⁴⁺, S⁶⁺, B³⁺), X denotes column monovalent anions (X = OH⁻, F⁻, Cl⁻), and left superscripts indicate ideal coordination numbers.^1,2 Archetypal apatite minerals belong to the hexagonal-dipyramidal class and crystallize in the P6₃/m space group – although lower monoclinic symmetry (space group P2₁/b) can result from orientational ordering of X⁻ anions within the [00z] apatic channels.^3,4 In P6₃/m-structured apatites, ^IXM(I) sites (Wyckoff positions 4f) generally accommodate smaller cation, while ^VIIM(II) sites (Wyckoff positions 6h) are occupied by larger cations.

In particular, hydroxylapatite (HA) is found in igneous and metamorphic environments and in biogenic deposits,^5,6 and is notorious for its facile ion-exchange properties. Owing to its chemical similarity to inorganic bone matrix material, excellent biocompatibility, and favourable surface properties, HA is one of the major components – along with β-tricalcium phosphate, β-Ca₃(PO₄)₂ – of biphasic calcium phosphate (BCP) bioactive ceramics used for bone grafting, implant and prosthesis coatings, and reconstructive surgery.^6–10

The mechanisms and energetics of ion uptake and retention in pristine and doped HA have been subjects of active research in recent years, due to the crucial roles in mediating biological processes played by trace ions found in bones (e.g. Sr²⁺, Zn²⁺, Mn²⁺, F⁻, SiO₄⁴⁻ and CO₃²⁻),^6,11–16 and to potential HA and doped-HA applications in catalysis^17,18 and immobilization of toxic heavy metals and radionuclides in contaminated soil and ground water.^12,19–23

Compared to other metal cations such as Sr²⁺ or Zn²⁺ which are widely used in to promote bone formation or retard bone absorption, limited information is available on the energetics of Sn²⁺ incorporation into HA. An early study by Klement and Haselbeck²⁴ reported the synthesis of Sn₈Ca₂(PO₄)₆Cl₂ and Sn₅Ca₅(PO₄)₆F₂ apatites, followed by the preparation of Sn₉Ca(PO₄)₆(OH,Cl,F)₂ apatite by McConnell and Foreman.²⁵ Insertion of Sn²⁺ in dentin (consisting of ∼45 wt% HA) was also recently investigated as a pre-treatment for preventing dental erosive wear, since there is empirical evidence that Sn²⁺ increases bond strength.²⁶ In addition, Sn(II)-doped HA has been proposed as a potential material for removal and immobilization of ⁹⁹Tc contaminant (t_1/2 = 2.13 × 10⁵ years, β⁻ = 294 keV; produced from the nuclear fuel cycle with 6.1 and 5.9% fission yields from the fission of ²³⁵U and ²³⁹Pu) in wastewater or nuclear waste streams/tanks.^23,27,28 Indeed, early experimental ion-exchange studies demonstrated the role of Sn(II) as a reductant of soluble pertechnetate anion (Tc^VIIO₄⁻) in the presence of phosphate or diphosphonate ligands.^29,30 Despite the moderate sorption or retention of ^99/99mTc in pristine HA,^22,31 tin(II)-doped HA was shown to efficiently reduce Tc(VII) and subsequently immobilize Tc(IV), without further reoxidation to volatile Tc₂O₇.^23,27,28

In this work, the energetics of Sn²⁺ substitution for Ca²⁺ in HA has been investigated within the framework of density functional theory (DFT), by predicting the evolution of the excess energy of Sn-doped HA as a function of the composition. This study aims at predicting the impact of Sn(II) substitutions in HA on the structure of bioactive ceramics and candidate materials based on Sn-doped HA for removal and immobilization of radionuclides. Combined Sn²⁺ substitutions at Ca(I) and Ca(II) sites have been systematically assessed and the most energetically favourable compositions of Sn_xCa_10−x(PO₄)₆(OH)₂ (x = 1–10) have been determined. Details of the computational methods utilized in this study are given in the next section, followed by a complete analysis and discussion of our results. A summary of our findings and conclusions is presented in the last section of the manuscript.

2 Computational methods

Total-energy calculations were carried out at 0 K using DFT, as implemented in the Vienna ab initio simulation package (VASP).³² The exchange–correlation energy was calculated using the generalized gradient approximation (GGA), with the parameterization of Perdew, Burke and Ernzerhof (PBE).³³ Standard functionals, such as the PBE or PW91 functionals, were found in previous studies to accurately describe the structural parameters and properties of metal–oxygen systems characterized experimentally.^34–40 Previous first-principles approaches, using for example GTO/B3LYP, have also been successful in describing HA.^41,42

The projector augmented wave (PAW) method was utilized to describe the interaction between valence electrons and ionic cores.^43,44 In the Kohn–Sham (KS) equations, the Ca(3p⁶,4s²), P(3s²,3p³), O(2s²,2p⁴), and Sn(5s²,5p²) electrons were treated explicitly as valence electrons and the remaining core electrons together with the nuclei were represented by PAW pseudopotentials. The KS equation was solved using the blocked Davidson⁴⁵ iterative matrix diagonalization scheme, followed by the residual vector minimization method. The plane-wave cutoff energy for the electronic wavefunctions was set to 500 eV, ensuring the total energy of the system to be converged to within 1 meV per atom. Electronic relaxation was performed with the conjugate gradient method accelerated using the Methfessel–Paxton Fermi-level smearing⁴⁶ with a Gaussian width of 0.1 eV.

The hexagonal structure of HA (space group P6₃/m, IT no. 176; Z = 2) refined by Hughes et al.⁴⁷ using single-crystal X-ray diffraction (XRD) was used as an initial guess in structural optimization calculations. Ionic relaxation calculations of Sn_xCa_10−x(PO₄)₆(OH)₂ (x = 0–10) periodic unit cells were carried out using the quasi-Newton algorithm and Hellmann–Feynman forces acting on atoms were calculated with a convergence tolerance set to 0.01 eV Å⁻¹. Structural optimization was performed without symmetry constraints. The Brillouin zone was sampled using the Monkhorst–Pack k-point scheme⁴⁸ with a 3 × 3 × 3 k-point mesh for all simulated structures.

3 Results and discussion

3.1 Crystal structure of HA

The crystal unit cell of HA optimized with DFT at the GGA/PBE level of theory is shown in Fig. 1 and the corresponding unit-cell parameters are summarized in Table 1, along with results from crystallographic XRD measurements and previous DFT calculations. The calculated cell dimensions of a = b = 9.542 Å, c = 6.897 Å (c/a = 0.723, α = β = 90.0°, γ = 120.0°) are in good agreement with the values of a = b = 9.432 Å, c = 6.881 Å (c/a = 0.729, α = β = 90.0°, γ = 120.0°) measured by Deer et al.,⁴⁹ and also consistent with the values of Ellis and co-workers¹⁹ (a = b = 9.560 Å, c = 6.863 Å; c/a = 0.718, α = β = 90.0°, γ = 120.0°) and de Leeuw⁵⁰ (a = b = 9.563 Å, c = 6.832 Å; c/a = 0.714, α = β = 90.0°, γ = 120.0°) computed with DFT/GGA.

Fig. 1 Crystal unit cell of hydroxylapatite, Ca₁₀(PO₄)₆(OH)₂ (space group P6₃, IT no. 173; Z = 2), relaxed with DFT at the GGA/PBE level of theory. Top: View along the [001] direction of the apatic channels (c-axis); bottom: side view of the unit cell. Color legend: Ca(I), purple; Ca(II), light blue; P, green; O, red; H, white. Ca(Ia) and Ca(Ib) are labelled distinctively.

Table 1 Calculated and measured structural unit-cell parameters of bulk hydroxylapatite, Ca₁₀(PO₄)₆(OH)₂

	a (Å)	c (Å)	α (°)	γ (°)
a Ellis et al., 2006; ref. 19.b deLeeuw, 2001; ref. 50.c Deer et al., 1992; ref. 49.d Hughes et al., 1989; ref. 47.e McConnell et al., 1966; ref. 25.
This work	9.542	6.897	90	120
DFT(GGA)^a	9.560	6.863	90	120
DFT(GGA)^b	9.563	6.832	90	120
Expt.^c	9.432	6.881	90	120
Expt.^d	9.4166	6.8745	90	120
Expt.^e	9.416(2)	6.883(2)	90	120

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As discussed in details by deLeeuw,⁵⁰ OH groups are preferentially ordered into ⋯O–H⋯O–H⋯ chains along the [001] direction of the apatic channels (c-axis). As a result, the mirror-plane symmetry through the Ca(II) triangle plane of the HA unit cell is broken and the space group symmetry is effectively reduced from P6₃/m (IT no. 176) to P6₃ (IT no. 173). While all six Ca(II) forming apatic channels remain equivalent in the P6₃ space group, the four Ca(I) split into pairs of Ca(Ia) and Ca(Ib) (cf. Fig. 1). The computed [measured]⁴⁷ Ca(II)–O distances are 2.35, 2.36, 2.37, 2.41, 2.48, 2.52 and 2.82 Å [2.343(×2), 2.353, 2.385, 2.509(×2) and 2.710 Å]. The calculated distances between Ca(Ia) and its nine first, second and third oxygen atom neighbours are 2.44(×3), 2.47(×3) and 2.80(×3) Å and the predicted Ca(Ib)–O distances are 2.41(×3), 2.44(×3) and 2.88(×3) Å, i.e., in agreement with the measured distances of 2.403(×3), 2.452(×3) and 2.802(×3) Å.⁴⁷ The calculated P–O bond lengths of 1.55(×3) and 1.56 Å are in line with the experimental estimates of 1.530(×2), 1.534, 1.537 Å.⁴⁷

3.2 Sn substitution into HA

Typical ion-exchange in HA involves, first, adsorption of divalent metal cations onto the HA surface, followed by cationic substitution into the Ca²⁺ sublattice of HA.^51,52 The reaction corresponding to Sn²⁺/Ca²⁺ substitution into HA can be expressed as:

Ca₁₀(PO₄)₆(OH)₂ + xSn²⁺ → Ca_10−xSn_x(PO₄)₆(OH)₂ + xCa²⁺, (1)

where x = 1–10.

In the following, the endpoints of pristine hydroxylapatite and fully-substituted Sn-hydroxylapatite are referred to as CaHA and SnHA, respectively, and the normalized composition of Sn-substituted hydroxylapatite is simply written as Sn_xCa_1−xHA (x = 0.1–1). In addition, square-bracket notations [p + q] indicate occupancies p and q on adjacent Ca(II) triangles, as shown in Fig. 2 depicting Ca(II)/Sn(II) cationic arrangements in apatic channels for normalized compositions of Sn-substituted hydroxylapatite, Sn_xCa_1−xHA (x = 0.2–0.4). Sn²⁺ substitutions at Ca(Ia) and Ca(Ib) sites are designated as Sn(Ia) and Sn(Ib), respectively.

Fig. 2 Ca(II)/Sn(II) cationic arrangements in apatic channels for normalized compositions of Sn-substituted hydroxylapatite, Sn_xCa_1−xHA (x = 0.2–0.4). Square-bracket notations [p + q] indicate occupancies p and q on adjacent Ca(II) triangles. Color legend: Ca, light blue; Sn, light purple.

In order to determine the most energetically favourable compositions of the solid solution Sn_xCa_1−xHA (x = 0.1–0.9) relative to the CaHA and SnHA reference end-members, the excess energy was calculated as:

E_x = E(Sn_xCa_1−xHA) − xE(SnHA) − (1 − x)E(CaHA), (2)

where negative values of E_x correspond to energetically favourable configurations of the Sn_xCa_1−xHA solid solution. The resulting E_x calculated for various configurations of one to nine Sn²⁺/Ca²⁺ substitutions at Ca(I) and Ca(II) sites within the HA unit cell are displayed in Fig. 3, with their corresponding numerical values reported in Table 2. As shown in Fig. 3, substitutions occur preferentially at Ca(II) sites up to a composition of Sn_0.6Ca_0.4HA and further substitutions at Ca(I) sites proceed only once full occupancy of Ca(II) sites by Sn²⁺ is achieved (solid black line). This finding is similar to previous Rietveld analyses of XRD data on Pb²⁺- or (Pb²⁺,Sr²⁺)-substituted HA synthesized in aqueous environment.^19,53,54 Let us note that the lowest excess-energy configurations [1 + 0] and [2 + 1]-mer for Sn_0.1Ca_0.9HA and Sn_0.3Ca_0.7HA, respectively, are not energetically favourable (E_x > 0), while Sn²⁺ incorporation via coupled substitutions at Ca(II) sites for the [1 + 1]-trans configuration of Sn_0.2Ca_0.8HA is essential energy neutral. Compositions of Sn_xCa_1−xHA (x = 0.4–0.9) are predicted to be predominant (E_x < 0), with an optimal stoichiometry of Sn_0.8Ca_0.2HA, and optimal configurations of [3 + 1] (x = 0.4), [3 + 2] (x = 0.5), [3 + 3] (x = 0.6), [3 + 3] + 1 Sn(Ia) (x = 0.7), [3 + 3] + 1 Sn(Ia) + 1 Sn(Ib) (trans) (x = 0.8) and [3 + 3] + 2 Sn(Ia) + 1 Sn(Ib) (x = 0.9).

Fig. 3 Excess energy, E_x, as a function of the normalized composition of Sn-substituted hydroxylapatite Sn_xCa_1−xHA (x = 0.1–1) unit-cell calculated at the GGA/PBE level of theory. Legend: combined Sn substitutions at Ca(I) and Ca(II) sites, green triangles; Sn substitution at Ca(I) sites only, blue diamonds; Sn substitution at Ca(II) sites only, red circles; pristine Ca-hydroxylapatite (CaHA), pink square; fully-substituted Sn-hydroxylapatite (SnHA), black square. The most energetically favourable configurations at fixed compositions are connected by a solid black line. Square-bracket notations [p + q] correspond to neighbour configurations, as shown in Fig. 2.

Table 2 Excess energy (E_x) and lattice parameters of Sn_xCa_1−xHA (x = 0–1) unit-cell structures optimized at the GGA/PBE level of theory. Square-bracket notations [p + q] indicate Sn occupancies p and q on adjacent Ca(II) triangles and Sn substitutions at Ca(Ia) and Ca(Ib) sites are referred to as Sn(Ia) and Sn(Ib), respectively

Composition	Sn(I)	Sn(II)	Configuration	Ex (eV)	a (Å)	b (Å)	c (Å)	V (Å^3)
0%	0	0	CaHA	0.0	9.542	9.542	6.897	543.9
10%	0	1	[1 + 0]	0.108	9.619	9.637	6.916	553.9
	1	0	[0 + 0], 1 Sn(Ia)	0.244	9.571	9.571	6.947	551.2
	1	0	[0 + 0], 1 Sn(Ib)	0.277	9.567	9.567	6.960	551.7
20%	0	2	[1 + 1]-trans	0.003	9.714	9.771	6.896	563.5
	0	2	[1 + 1]-cis	0.132	9.729	9.694	6.922	563.3
	0	2	[2 + 0]	0.217	9.667	9.668	6.949	562.1
	1	1	[1 + 0], 1 Sn(Ia)	0.308	9.650	9.657	6.967	560.8
	1	1	[1 + 0], 1 Sn(Ib)	0.361	9.638	9.657	6.964	560.3
	2	0	2 Sn(Ia)	0.378	9.576	9.576	7.021	557.5
	2	0	1 Sn(Ia), 1 Sn(Ib) (trans)	0.415	9.587	9.587	7.015	558.3
	2	0	2 Sn(Ib)	0.477	9.592	9.592	7.005	558.1
	2	0	1 Sn(Ia), 1 Sn(Ib) (cis)	0.538	9.600	9.600	6.993	558.2
30%	0	3	[2 + 1]-mer	0.061	9.704	9.755	6.953	571.9
	0	3	[2 + 1]-fac	0.155	9.749	9.710	6.984	571.8
	1	2	[1 + 1]-trans, 1 Sn(Ia)	0.246	9.764	9.826	6.940	572.6
	0	3	[3 + 0]	0.345	9.690	9.701	7.008	569.9
	1	2	[1 + 1]-cis, 1 Sn(Ia)	0.367	9.758	9.714	6.979	570.9
	1	2	[1 + 1]-cis, 1 Sn(Ib)	0.410	9.760	9.721	6.976	571.2
40%	0	4	[3 + 1]	−0.189	9.883	9.740	7.022	583.4
	0	4	[2 + 2]-cis	0.032	9.819	9.785	7.009	582.6
	0	4	[2 + 2]-trans	0.082	9.797	9.783	7.002	579.9
	1	3	[2 + 1]-mer, 1 Sn(Ia)	0.279	9.757	9.790	7.002	581.6
	1	3	[2 + 1]-mer, 1 Sn(Ib)	0.284	9.746	9.794	6.998	581.7
	2	2	[1 + 1]-trans, 1 Sn(Ia), 1 Sn(Ib) (trans)	0.313	9.800	9.791	7.063	580.6
	1	3	[2 + 1]-fac, 1 Sn(Ia)	0.333	9.799	9.749	7.039	580.7
	1	3	[2 + 1]-fac, 1 Sn(Ib)	0.387	9.794	9.745	7.032	580.4
	2	2	[1 + 1]-trans, 2 Sn(Ia)	0.400	9.780	9.750	7.058	578.1
50%	0	5	[3 + 2]	−0.320	9.772	9.794	7.110	588.5
	1	4	[2 + 2]-trans, 1 Sn(Ia)	0.120	9.811	9.797	7.106	592.1
	1	4	[2 + 2]-cis, 1 Sn(Ia)	0.157	9.861	9.810	7.078	592.7
60%	0	6	[3 + 3]	−0.603	9.741	9.740	7.231	594.2
	1	5	[3 + 2], 1 Sn(Ia)	−0.334	9.762	9.857	7.239	604.5
	1	5	[3 + 2], 1 Sn(Ib)	−0.282	9.789	9.808	7.209	600.1
	2	4	[2 + 2]-trans, 1 Sn(Ia), 1 Sn(Ib) (trans)	0.028	9.825	9.712	7.230	603.7
	2	4	[2 + 2]-trans, 2 Sn(Ia)	0.071	9.799	9.753	7.222	604.6
	2	4	[2 + 2]-cis, 1 Sn(Ia), 1 Sn(Ib) (trans)	0.154	9.868	9.775	7.199	604.2
	2	4	[2 + 2]-cis, 2 Sn(Ia)	0.242	9.842	9.784	7.189	602.4
70%	1	6	[3 + 3], 1 Sn(Ia)	−0.600	9.769	9.769	7.336	606.4
	1	6	[3 + 3], 1 Sn(Ib)	−0.540	9.767	9.765	7.339	606.4
80%	2	6	[3 + 3], 1 Sn(Ia), 1 Sn(Ib) (trans)	−0.695	9.773	9.775	7.512	621.5
	2	6	[3 + 3], 2 Sn(Ia)	−0.663	9.774	9.777	7.498	620.6
	2	6	[3 + 3], 1 Sn(Ia), 1 Sn(Ib) (cis)	−0.343	9.810	9.809	7.403	617.0
90%	3	6	[3 + 3], 2 Sn(Ia), 1 Sn(Ib)	−0.610	9.839	9.850	7.620	639.6
	3	6	[3 + 3], 1 Sn(Ia), 2 Sn(Ib)	−0.350	9.816	9.815	7.552	630.1
100%	4	6	SnHA	0.0	9.867	9.868	7.646	644.8

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These computational results are overall consistent with the reported synthesis of Sn₈Ca₂(PO₄)₆Cl₂ and Sn₅Ca₅(PO₄)₆F₂ apatites by Klement and Haselbeck,²⁴ as well as with the preparation of Sn₉Ca(PO₄)₆(OH,Cl,F)₂ apatite by McConnell and Foreman.²⁵ Although the fully-substituted SnHA end-member was predicted to crystallize in the P6₃ space group, isomorphic to CaHA, the formation of SnHA is comparatively less likely to occur owing to its higher energy cost.

The computed cell dimensions of Sn₉Ca(PO₄)₆(OH)₂ are a = 9.84 Å, b = 9.85 Å, c = 7.62 Å, i.e. larger than the XRD estimates of a = b = 9.45 Å, c = 6.89 Å determined by McConnell and Foreman²⁵ for Sn₉Ca(PO₄)₆(OH,Cl,F)₂ apatite. Interestingly, as found in the latter experimental study, calculations predict a volume expansion as a result of Sn/Ca substitutions in HA, although McConnell and Foreman pointed out that a simple rigid-ion model suggests instead a slight reduction of the unit cell size, based on the comparative Ahrens ionic radii (A-IR) of Sn²⁺ (0.93 Å) and Ca²⁺ (0.99 Å).⁵⁵ However, several caveats to the aforementioned A-IR rigid-ion model prediction are worth considering. Indeed, no coordination number (CN) or polyhedral distortion were accounted for in this A-IR model, where the radii are independent of the structure type. A subsequent CN-dependent rigid-ion model by Shannon and Prewitt⁵⁶ introduced effective ionic radii (‘IR’) [and crystal radii (CR)], with values of ‘IR’(Ca²⁺) = 1.00–1.35 Å [CR(Ca²⁺) = 1.14–1.49 Å] for CN = 6–12 and ‘IR’(Sn²⁺) = 1.22 Å [CR(Sn²⁺) = 1.36 Å] for CN = 8. A revision of this ‘IR’ model including polyhedral distortion provided refined estimates of ‘IR’(Ca²⁺) = 1.00–1.34 Å [CR(Ca²⁺) = 1.14–1.48 Å] for CN = 6–12 and ‘IR’(Sn²⁺) = 1.22–1.32 Å [CR(Sn²⁺) = 1.36–1.46 Å] for CN = 7–9.⁵⁷

An additional complication arises from the poorly-defined coordination environment of Ca²⁺ in HA, as discussed in previous studies.^58,59 In fact, the coordination environment of Ca(II) can be described as either 4 (∼2.36 Å) + 2 (∼2.51 Å) + 1 (∼2.71 Å) or 5 (∼2.35 Å) + 2 (∼2.51 Å) + 1 (∼2.71 Å), while the coordination environment of Ca(I) is best described as 6 (∼2.43 Å) + 3 (∼2.79 Å).⁵⁹ Owing to the 2 : 3 ratio of Ca(I) : Ca(II) in HA, a simplified, single CN of 8.4 was proposed, assuming a relaxed 2.9 Å cutoff for the Ca–O first-shell coordination – a shorter cutoff of 2.6 Å would result in CN = 6. Since ‘IR’(Ca²⁺) = 1.00–1.18 Å [CR(Ca²⁺) = 1.14–1.32 Å] for CN = 6–9, and ‘IR’(Sn²⁺) = 1.22–1.32 Å [CR(Sn²⁺) = 1.36–1.46 Å] for CN = 7–9, a volume increase of Sn_xCa_1−xHA unit cells concomitant with Sn substitutions for Ca(I)/Ca(II) in HA is therefore predicted by the ‘IR’ model.

The evolution of the optimized unit-cell volume of Sn_xCa_1−xHA (x = 0–1) as a function of the composition is shown in Fig. 4, for all the configurations listed in Table 2. Calculations predict a nearly linear increase of the unit-cell volume of the most energetically favourable configurations. Sn_xCa_1−xHA (x = 0.1–0.6) equilibrium structures resulting from substitutions at Ca(II) sites closely follow Vegard's law for ideal mixture up to x = 0.4, followed by a breakaway in volume increase for x = 0.5 ([3 + 2] configuration) and x = 0.6 ([3 + 3] configuration) from Vegard's predictions. Subsequent Sn(Ia) and Sn(Ib) substitutions tend to gradually bring the Sn_xCa_1−xHA solid solution back to an ideal Vegard's mixing behaviour. It is also worth noting that among the compositions of Sn_xCa_1−xHA (x = 0.4–0.9) predicted to be predominant (E_x < 0), the volume corresponding to the optimal stoichiometry of Sn_0.8Ca_0.2HA ([3 + 3] + 1 Sn(Ia) + 1 Sn(Ib)) (trans) configuration is the closest to Vegard's prediction within this compositional range. In addition, at low Sn content (see x = 0.1–0.2 in Fig. 4), the most energetically unfavourable structures with Sn substitutions at Ca(I) sites only (cf. Table 2 and Fig. 3) exhibit the largest departure from their corresponding Vegard's volumes. These findings suggest a strong interplay between structure and energetics in Sn_xCa_1−xHA solid solutions across the complete compositional range.

Fig. 4 Evolution of the unit-cell volume of Sn_xCa_1−xHA (x = 0–1) as a function of the normalized composition calculated at the GGA/PBE level of theory. Legend: combined Sn substitutions at Ca(I) and Ca(II) sites, green triangles; Sn substitution at Ca(I) sites only, blue diamonds; Sn substitution at Ca(II) sites only, red circles; pristine Ca-hydroxylapatite (CaHA), pink square; fully-substituted Sn-hydroxylapatite (SnHA), black square. The most energetically favourable configurations at fixed compositions are connected by a solid black line. The dashed black line represents the empirical Vegard's law for an ideal mixture.

The density of states for the most energetically favourable structures from pure HA to fully-substituted Sn-HA were also computed in order to understand the electronic structure underlying the physicochemical properties of these compounds. As shown in Fig. 5, all the structures are semiconductors, with a band gap decreasing from 5.3 eV for pure HA to 3.0 eV for fully-substituted Sn-HA. The top of the valence space is dominated by O(2p) states, with increasing O(2p)–Sn(5s) hybridization with Sn doping. The bottom of the conduction space possesses predominantly Sn(5p) character. Both Ca and P electronic states are not expected to have significant impact on the physicochemical properties of these compounds since their electronic states are much deeper/higher in the valence/conduction space.

Fig. 5 Total and partial density of states of the most energetically favourable configurations at fixed compositions from pure HA (top) to fully-substituted Sn-hydroxylapatite (10 Sn-HA) (bottom) calculated at the GGA/PBE level of theory. The Fermi level is set to zero.

4 Conclusions

The energetics of Sn²⁺ substitution for Ca²⁺ in bulk hydroxylapatite (HA) has been investigated within the framework of density functional theory. Calculations of the excess energy, E_x, of Sn-doped HA as a function of the composition predict that compositions of Sn_xCa_10−x(PO₄)₆(OH)₂ (x = 4–9) are predominant, with an optimal stoichiometry of Sn₈Ca₂(PO₄)₆(OH)₂ (corresponding to a [3 + 3] + 1 Sn(Ia) + 1 Sn(Ib)-trans configuration). At low Sn content, the lowest excess-energy configurations [1 + 0] and [2 + 1]-mer for SnCa₉(PO₄)₆(OH)₂ and Sn₃Ca₇(PO₄)₆(OH)₂, respectively, are not energetically favourable (E_x > 0), while Sn²⁺ incorporation via coupled substitutions at Ca(II) sites for the [1 + 1]-trans configuration of Sn₂Ca₈(PO₄)₆(OH)₂ is essential energy neutral. Calculations also show that Sn²⁺ incorporation at Ca(II) sites is likely to occur up to a composition of Sn₆Ca₄(PO₄)₆(OH)₂, and further substitutions at Ca(I) sites proceed once full occupancy of Ca(II) sites by Sn²⁺ is achieved.

Sn substitution into the Ca²⁺ sublattice of HA results in a steady increase of the unit-cell volume that follows approximately Vegard's law for ideal mixture across the entire compositional range. This volume increase is consistent with predictions from Shannon's rigid ion model accounting for polyhedral distortion and coordination environment. A slight departure from Vegard's volume predictions occurs for x = 0.5 ([3 + 2] configuration) and x = 0.6 ([3 + 3] configuration). Among the predominant compositions (i.e., E_x < 0) of Sn_xCa_1−xHA (x = 0.4–0.9), the volume of the optimal Sn_0.8Ca_0.2HA ([3 + 3] + 1 Sn(Ia) + 1 Sn(Ib))-trans structure is the closest to Vegard's prediction, thus stressing the importance of the structure–energetics relationship in Sn_xCa_1−xHA solid solutions.

Acknowledgements

Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000. We thank Robert Moore and Mark Rigali (Sandia National Laboratories) for stimulating discussions and John Vienna, Jeff Serne, Matt Asmussen and Nik Qafoku (Pacific Northwest National Laboratory) for useful information.

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