**6.5.1. Partitioning of divalent cations**

Partitioning of divalent cations is defined by the chemical equilibrium expressing the divalent cation (Me2+) exchange between apatite and aqueous solutions [135]:

$$\text{Me}^{2+}\left(\text{aq}\right) + \text{Ca}-\text{apate}\left(\text{ss}\right) \rightarrow \text{Ca}^{2+}\left(\text{aq}\right) + \text{Me}-\text{apate}\left(\text{ss}\right) \tag{31}$$

where (*aq*) and (*ss*) refer to the aqueous solution and to the solid solution, respectively. The equilibrium constant of **Eq. 31** can be written as

$$K\_D \left(\frac{Me^{2+}}{Ca}\right) = \frac{\frac{\mathcal{X}\_{Me-\text{-}apaste}}{\mathcal{X}\_{Ca-\text{-}apaste}}}{\frac{\mathcal{M}\_{Me^{2+}}}{\mathcal{M}\_{Ca^{2+}}}} = \frac{K\left(T\right)\_{Ca-\text{-}apaste}}{K\left(T\right)\_{Mn-\text{-}apaste}} \frac{\mathcal{X}\_{Ca-\text{-}apaste}}{\mathcal{X}\_{Me-\text{-}apaste}} \frac{\mathcal{Y}\_{Me^{2+}}}{\mathcal{Y}\_{Ca^{2+}}} \tag{32}$$

where *x* is the molarfraction in apatite solid solution, *m* is the molality in water, *λ* is the activity coefficient of the component in the solid solution, *K*(*T*) is the solubility product of the endmember at temperature *T* and *γ* is the ion activity in the aqueous solution, the ratio of which in wateris assumed to be equal to one. The activity coefficients in regular solid-solution model are described by Margules parameters14 [136] and can be approximated by the elastic energy due to the deformation of the host crystal lattice around the substituted cation [135],[137]:

$$W\_{G\_{\vec{\psi}}} = 4\pi N\_A E \left[ \frac{r\_i}{2} \left( r\_j - r\_i \right)^2 + \frac{1}{3} \left( r\_j - r\_i \right)^3 \right] \tag{33}$$

where *N*<sup>A</sup> is the Avogadro number, *E* is the Young's modulus of the crystal, *r*<sup>i</sup> is the ionic radius of cation normally occupying the site in the *i*-compound (Ca in apatite) and *r*<sup>j</sup> is the ionic radius of the substituted cation in compound *j*. The elasticity of hydroxyapatite gives *E* = 114 ± 2 GPa.

At low concentrations (XMe−apatite << 1) like those of trace elements in biogenic apatites, **Eq. 32** is reduced to the relationships [135]:

$$K\_D \left(\frac{Me^{2+}}{Ca}\right) = \frac{K\left(T\right)\_{Ca-quad}}{K\left(T\right)\_{Me-quad}} \exp\left(\frac{-W\_{G\_{McCa}}}{RT}\right) = \exp\left(-\frac{G\_{ideal} + W\_{McCa}}{RT}\right) \tag{34}$$

where the term *exp* (-ΔGideal/RT) is the free enthalpy change of the **reaction 31**, equivalent to the ratio of end-member solubility products. Unlike carbonates, the data of solubility prod‐ ucts and thermodynamics for end-member apatites are scarce. When no data are available for the solubility and enthalpy of formation of the end-members, it is assumed that the elastic energy term dominates over the partitioning, i.e. ΔGideal ≪ WGMeCa. Promising ways for obtaining the enthalpies of formation and the substitution energies are the first-principle calculations [138] and the atomistic modeling [135],[139].

For heterovalent substitutions, the equilibrium reaction becomes complex since complemen‐ tary substitutions are necessary to maintain the charge balance in the crystal. Typically, the substitution of trivalent elements of important rare-earth series requires the compensation by Na+ for Ca2+ at an adjacent site or even more complex substitution scheme involving carbo‐ nate groups or fluorine. In that case, most thermodynamic data required for the calculation of the equilibrium constant are not available. Among the series of elements with the same charge and substitution scheme, the pattern of equilibrium constants, or of distribution coefficients, can be approximated by combining **Eqs. 33** and **34** [135]:

$$\text{In } \quad \gamma\_1 = \chi\_2^2 \mathbb{E}\left[A\_2 + 2\chi\_1(A\_1 - A\_2)\right] \text{ and } \ln \quad \gamma\_2 = \chi\_1^2 \left[A\_1 + 2\chi\_2(A\_2 - A\_1)\right].$$

The Margules expressions for activity coefficients are based on the Lewis-Randall standard state (pure substance in the same phase and the same temperature and pressure as the mixture); therefore, they must obey the pure-component limit (lim *xi* →1 *γ<sup>i</sup>* =1). The parameters *A*1 and *A*2 are simply related to the activity coefficients at infinite dilution: lim *γ*<sup>1</sup> <sup>∞</sup> <sup>=</sup> *<sup>A</sup>*<sup>1</sup>

$$\frac{G^{\circ E}}{RT} = \sum \sum \propto\_i \propto\_j \prod\_{i$$

<sup>14</sup> The Margules equation expresses the Gibbs free energy (*G*E) of binary liquid mixture (*x*1 + *x*<sup>2</sup> = 1) in the symmetric form [136]:

*G <sup>E</sup> RT* = *A*1*x*<sup>1</sup> + *A*2*x*2; where *A*1 = *A* + *B* and *A*2 = *A* - *B*. Applying the partial molar derivative produces the expression for the activity coefficients:

and lim *γ*<sup>2</sup> <sup>∞</sup> <sup>=</sup> *<sup>A</sup>*2. For the binary mixture, where *γ*<sup>1</sup> <sup>∞</sup> <sup>=</sup>*γ*<sup>2</sup> ∞⇒ *<sup>A</sup>*<sup>1</sup> <sup>=</sup> *<sup>A</sup>*2, the Margules equation collapses to Porter equation. The multicomponent version of the Margules equation is [136]:

$$K\_D = K\_D^0 \exp\left(-4\pi N\_A E\_{\rm eff} \left[\frac{r\_0}{2} \left(r\_f - r\_0\right)^2 + \frac{1}{3} \left(r\_f - r\_0\right)^3\right] / RT\right) \tag{35}$$

where *E*eff is the effective Young's modulus and *r*<sup>0</sup> is the optimum radius for maximum equilibrium constant *K*<sup>D</sup> 0 , all of which will depend on the charge of the considered series of elements. These parameters can be adjusted to experimental data such as partition coeffi‐ cients between minerals and liquids and lead to parabola-like curves, the position and curvature of which depend on the charge of the element. This approach was applied so far only to rare-earth elements in apatite, where the relative partition coefficients were extrapo‐ lated from magmatic temperatures around 800°C to low temperatures appropriate for fossil diagenesis [135].

## **6.5.2. Complexation of metal cations**

( ) ( ) 2 3 <sup>1</sup> <sup>4</sup>

where *N*<sup>A</sup> is the Avogadro number, *E* is the Young's modulus of the crystal, *r*<sup>i</sup> is the ionic radius of cation normally occupying the site in the *i*-compound (Ca in apatite) and *r*<sup>j</sup> is the ionic radius of the substituted cation in compound *j*. The elasticity of hydroxyapatite gives *E* = 114 ± 2 GPa.

At low concentrations (XMe−apatite << 1) like those of trace elements in biogenic apatites, **Eq. 32**

where the term *exp* (-ΔGideal/RT) is the free enthalpy change of the **reaction 31**, equivalent to the ratio of end-member solubility products. Unlike carbonates, the data of solubility prod‐ ucts and thermodynamics for end-member apatites are scarce. When no data are available for the solubility and enthalpy of formation of the end-members, it is assumed that the elastic energy term dominates over the partitioning, i.e. ΔGideal ≪ WGMeCa. Promising ways for obtaining the enthalpies of formation and the substitution energies are the first-principle

For heterovalent substitutions, the equilibrium reaction becomes complex since complemen‐ tary substitutions are necessary to maintain the charge balance in the crystal. Typically, the substitution of trivalent elements of important rare-earth series requires the compensation by Na+ for Ca2+ at an adjacent site or even more complex substitution scheme involving carbo‐ nate groups or fluorine. In that case, most thermodynamic data required for the calculation of the equilibrium constant are not available. Among the series of elements with the same charge and substitution scheme, the pattern of equilibrium constants, or of distribution coefficients,

<sup>14</sup> The Margules equation expresses the Gibbs free energy (*G*E) of binary liquid mixture (*x*1 + *x*<sup>2</sup> = 1) in the symmetric form

*RT* = *A*1*x*<sup>1</sup> + *A*2*x*2; where *A*1 = *A* + *B* and *A*2 = *A* - *B*. Applying the partial molar derivative produces the expression for

<sup>2</sup> *<sup>A</sup>*<sup>1</sup> <sup>+</sup> <sup>2</sup>*x*<sup>2</sup> (*A*<sup>2</sup> <sup>−</sup> *<sup>A</sup>*1) . The Margules expressions for activity coefficients are based on the Lewis-Randall standard state (pure substance in the same phase and the same temperature and pressure as the mixture); therefore, they must obey the pure-component limit

*γ<sup>i</sup>* =1). The parameters *A*1 and *A*2 are simply related to the activity coefficients at infinite dilution: lim *γ*<sup>1</sup>

<sup>∞</sup> <sup>=</sup>*γ*<sup>2</sup>

<sup>∞</sup> <sup>=</sup> *<sup>A</sup>*<sup>1</sup>

∞⇒ *<sup>A</sup>*<sup>1</sup> <sup>=</sup> *<sup>A</sup>*2, the Margules equation collapses to Porter

exp exp *Ca apatite GMeCa ideal MeCa*

æ ö æ ö - æ ö <sup>+</sup> ç ÷ <sup>=</sup> ç ÷ = -ç ÷ è ø è ø è ø (34)

ë û (33)

é ù = -+ - ê ú

2 3 *ij i G A ji ji <sup>r</sup> W NE r r r r*

*Me K T <sup>W</sup> G W <sup>K</sup> Ca K T RT RT*

p

322 Apatites and their Synthetic Analogues - Synthesis, Structure, Properties and Applications

( ) ( )

calculations [138] and the atomistic modeling [135],[139].

can be approximated by combining **Eqs. 33** and **34** [135]:

<sup>2</sup> *<sup>A</sup>*<sup>2</sup> <sup>+</sup> <sup>2</sup>*x*<sup>1</sup> (*A*<sup>1</sup> <sup>−</sup> *<sup>A</sup>*2) and ln *<sup>γ</sup>*<sup>2</sup> <sup>=</sup> *<sup>x</sup>*<sup>1</sup>

(*xi* − *xj* )

<sup>∞</sup> <sup>=</sup> *<sup>A</sup>*2. For the binary mixture, where *γ*<sup>1</sup>

equation. The multicomponent version of the Margules equation is [136]:

.

*Me apatite*


is reduced to the relationships [135]:

*D*

[136]: *G <sup>E</sup>*

(lim *xi* →1

*G <sup>E</sup>*

ln *γ*<sup>1</sup> = *x*<sup>2</sup>

and lim *γ*<sup>2</sup>

the activity coefficients:

*RT* <sup>=</sup>∑ ∑ *xixj Aij* <sup>+</sup> *Bij*

*i*< *j*

2

+

The complexation of metal cations in aqueous fluids involves binding with a broad range of molecules from simple inorganic ones (e.g. carbonates, phosphates and sulfates) to complex organic ones (humic acids, amino acids, proteins, enzymes, etc.). For molecules with several bonding sites and structural flexibility (e.g. multidentate or chelator), the complexation is thermodynamically favored with respect to the complexation with several monodentates having one bonding site; the process is named chelation. Chelators can be adsorbed on mineral surfaces while remaining complexed to metallic cations. The pattern of the partition coeffi‐ cients associated with this process was measured for rare-earth elements complexed with humic acids and manganese oxides. It shows null fractionation along the whole series; the effect of chelation is therefore to screen the trace element in the crystal or ligand field and to suppress the fractionation associated with ionic radius variations and tetrad effects, and most of the anomalies associated with redox of Ce [140]. Similar effects might occur for the adsorption of chelated metals on other mineral surfaces and in particular phosphates. In addition to chelators, the transition metals also form complexes with proteins and enzymes that interact with bones and teeth in living organisms and may influence their incorporation in bioapatite [135].

#### **6.5.3. Diffusion processes**

Solid-state diffusion in crystals is a thermally activated process governed by the enthalpy of formation and of migration of defects and usually well described by the Arrhenius relation [135]:

$$\mathbf{D} = \mathbf{D}\_0 \exp\left(-\frac{\Delta \mathbf{H}\_\mathbf{s}}{\mathbf{R}\mathbf{T}}\right) \tag{36}$$

where *D*<sup>0</sup> is the pre-exponential factor corresponding to the diffusion coefficient at infinite temperature and Δ*H*<sup>a</sup> is the activation enthalpy (or energy) of the diffusion process. The extrapolation of high-temperature diffusion data of trace elements in apatite shows that these processes are inefficient at temperatures below 300°C, which cover the conditions of diage‐ netic alteration up to low-grade metamorphism [135],[141].

The differing initial and boundary conditions imposed in three sets of diffusion experiments:


consequently resulting in different solutions to the diffusion equation. However, in all cases, the process can be described as one-dimensional, concentration-independent diffusion [141].

**Fig. 17.** The summary of data diffusion for cations and anions in apatite (a) [142] and the diffusion of Sm and Nd for various minerals and oxides (a) [141].

A plot of diffusivities of various cations and anions in apatite is shown in **Fig. 17**(**a**). The diffusivities of Mn are similarto those of Sr and about an order of magnitude slowerthan those of Pb. On the other hand, the diffusion of Mn2+ in apatite is about two orders of magnitude faster than the diffusion of (trivalent) REE when coupled substitutions according to **Eqs. 4** and **5** are involved [141],[142]. The diffusion coefficients of Nd and Sm in various minerals and related oxides are plotted in **Fig. 17**(**b**). The diffusion of REE in apatite is relatively fast; when only simple REE exchange is involved, it is among the fastest in rock-forming minerals for which the data exist. Even when the chemical diffusion involving coupled exchange is considered, REE transport in apatite is considerably faster than the REE diffusion in other accessory minerals [141].
