Introduction

Scandium is the lightest transition metal, with chemical properties similar to those of yttrium and rare-earth elements. Even though it may be seem to be a rare element, its crustal abundance exceeds that of cobalt or molybdenum, and is much greater than that of Au or platinum-group elements. Being a compatible element (Gao et al. 1998), it is taken up by rock-forming minerals and rarely forms accumulations of economic interest.

Because of its price ($580,000/kg of scandium metal in 2022) and unavailability, scandium is not used widely. Scandium iodide is used in high-intensity lighting in mercury vapor lamps (Halka and Nordstrom 2019). Mixtures of scandium carbide and Ti carbide are very hard ceramic materials, with hardness exceeded only by that of diamond. Scandium radionuclides are used in molecular imaging (positron emission tomography imaging) in nuclear medicine (Roesch 2012; Huclier-Markai et al. 2018). A significant area of use is alloying with Al or with Al-Mg. Even relatively small amounts of Sc strongly modify the mechanical and thermal properties of these alloys, making them desirable for the aerospace industry (Röyset 2007; Zakharov 2003). Addition of Sc to Zr–Mo alloys increases their corrosion resistance and improves their performance in nuclear reactors. Another use for scandium is for material is solid oxide fuel cells, in perovskite host with complex composition and structure (Gu et al. 2008; Chen et al. 2016; Zakaria and Kamarudin 2020).

Extending of the uses of scandium depends critically on the development of technologies for cheaper extraction. Röyset (2007) concluded that the production, at least at that time, was controlled by the stockpiles from the former Soviet Union. One option would be to recover scandium from red mud that is the processing waste from \(\hbox {Al}_2\hbox {O}_3\) extraction from bauxite (Pasechnik et al. 2021; Ding et al. 2023). Pasechnik et al. (2021) reported extraction of Sc by crystallization of \(\hbox {NH}_4\hbox {Sc}(\hbox {SO}_4)_2\) and \((\hbox {NH}_4)_3\hbox {Sc}(\hbox {SO}_4)_3\) in sulfuric acid media. Other similar phases, for example \(\hbox {K}_3\hbox {Sc}(\hbox {SO}_4)_3\), can be used for the purposes of extraction (Volkov et al. 1997). Lateritic profiles are, in terms of mineralogy and geochemistry, similar to the red muds, and have also similar speciation of scandium (Chassé et al. 2017).

In this work, we are exploring the possibility of enhancing this technology by determination of the thermodynamic properties of the solid solution FeOOH–ScOOH with goethite structure. Goethite is a very common mineral that precipitates from Fe-rich mining waste. It can preferentially take up or reject elements in its crystal structure, thus partitioning these elements into the solid or into the liquid phase. This partitioning increases the elemental concentration in a certain reservoir and can boost the efficiency of the extracting technologies. For this reason, we synthesized the intermediate solid-solution members of the FeOOH–ScOOH solid solution with goethite structure [after Levard et al. (2018)]. Thermodynamic properties of mixing were determined for this solid solution by powder X-ray diffraction and acid-solution calorimetry. These data can aid in understanding of Sc behavior in mining waste.

Methods and materials

The synthesis of the end and intermediate members of the solid solution followed the protocols of Schwertmann and Cornell (2000) and Levard et al. (2018). Initially, 25 mL of Fe(III)–Sc(III) solution was prepared from the corresponding nitrates, with a total metal molarity of 1 M. The molar Fe/(Sc+Fe) ratios were 1.0, 0.9, 0.8, 0.5, 0.3, 0.2, and 0.0. Afterwards, 45 mL of 5 M KOH were rapidly added to the solution, inducing precipitation. The suspensions were aged in polypropylene bottles at 70 \(^\circ \hbox {C}\) for 10 days. The suspensions were then filtered, washed with copious amounts of deionized water, and the samples were left to dry at room temperature.

The elemental compositions of the samples (Fe, Sc) were determined by inductively-coupled plasma optical emission spectrometry (ICP-OES) using an ARCOS SOP (SPECTRO, Germany) (at the Mining Academy in Freiberg, Germany). For the measurements, 50 mg of each sample were dissolved at 60 \(^\circ \hbox {C}\) in 4 mL 38 wt.% hydrochloric acid, HCl, (VWR, France), diluted with 4 mL deionized water and stabilized with 0.5 mL 65 wt.% nitric acid, \(\hbox {HNO}_3\), (Merck, Germany).

The \(\hbox {H}_2\hbox {O}\) content of the samples was measured by Karl-Fischer-Titration (KFT) (at the Ruhr Universität Bochum, Germany). In this analysis, the sample was heated to 700 \(^\circ \hbox {C}\) and the released volatiles (in this case, steam) were transferred into the solution that was later titrated for its \(\hbox {H}_2\hbox {O}\) concentration. The sample itself did not come in contact with this solution and, therefore, the solution was not contaminated by Fe which could have interfered with the titration. Further details on the method and the instrument used can be found in Johannes and Schreyer (1981).

Phases were identified by powder X-ray diffraction (PXRD) using a D8 ADVANCE (Bruker, Germany) X-ray diffractometer in a Bragg-Brentano \(\theta -\theta\) parallel beam geometry with Cu \(\hbox {K}\alpha\) radiation (\(\lambda = 1.54056\) Å) at \(t = 22.0\pm 0.5\) \(^\circ \hbox {C}\). Diffraction patterns were recorded with a 1D LYNXEYE (Bruker, Germany) detector in the \(2\theta\) range of 5 to \(90^\circ\) with a step size of \(0.020355^\circ\) and a dwell time of 1 s per step using single crystalline silicon and zero-background polymer sample holders. The ceramic X-ray tube was operated at 40 kV and 40 mA. The primary optics are equipped with a Ni monochromator. Lattice parameters were determined by full-profile fits using GSAS (Larson and Dreele 1994).

Acid-solution calorimetry was performed with a commercial Setaram C-80 calorimeter, modified for the purposes of this study. This calorimeter is a Calvet-type twin calorimeter with a sample and reference well. The body of the calorimeter is maintained at a constant pre-selected temperature. After initiation of the reaction, voltage is induced in the thermopiles around the wells due to heat flow between the two wells. The voltage is recorded over time, integrated, and converted to heat with a calibration factor determined in a separate set of experiments.

For our experiments, we exchanged the manufacturer’s assembly, made of sealed Hastelloy cells, by a self-designed assembly (so-called setup). The Hastelloy cells were found to react so strongly with acids that no measurements were possible. The new setup was made completely from \(\hbox {SiO}_2\) glass. A mantle glass piece extended from the calorimeter and housed glass crucible where the calorimetric solvent (water or \(5 \,\hbox {mol}\cdot \hbox {dm}^{-3}\) HCl) was held. The glass crucibles and their contents were positioned in the active zone of the calorimeter, within the thermopile. The setup was open to the ambient atmosphere and the sample was introduced through a glass dropping tube. Our experiments were performed at \(T = 343.15\,\hbox {K}\) (70 \(^\circ \hbox {C}\)). At this temperature, the acid was found not to fume, although evaporation and re-condensation occurred. In a preliminary study, we found that this did not compromise the accuracy of the experiments. A possibility of mixing the solvent by bubbling with air was built into the setup but not utilized in this study.

The samples, in the form of pressed pellets, were dropped from room temperature into the calorimetric solvent at \(T = 343.15 \,\hbox {K}\). The measured heat effect therefore includes the heat content (enthalpy change) between room temperature and \(T = 343.15 \,\hbox {K}\) and the enthalpy of dissolution in the calorimetric solvent. These are the ’drop-solution’ experiments commonly performed in high-temperature calorimetry. The enthalpies calculated from the drop-solution enthalpies refer to the starting (i.e., ambient) temperature, not to \(T = 343.15 \,\hbox {K}\).

The calorimeter was equipped with the possibility of electrical calibration from the manufacturer in the working temperature range (25–200 \(^\circ \hbox {C}\)). The calibration was verified by measurements of enthalpies of dissolution of KCl in pure water at the \(\textit{T} = 298.15 \,\hbox {K}\). The expected heat effect was calculated from Parker (1965); later critically evaluated by Kilday (1980).

For the measurement on the FeOOH-ScOOH solid solution, the glass crucibles were filled with the acid, the setup inserted into the calorimeter and allowed to stabilize for at least 6 h. The samples were prepared in the form of pellets with masses of 5–6 mg. After the stabilization period, the pellets were dropped into the acid through the dropping tube. Complete dissolution occurred after 50–90 min, for most of the samples after 65 min. The time necessary for the dissolution depends, among other factors, on how tightly was the pellet pressed. With sufficient practice, the manual pressing of the pellets was reproducible enough so that the dissolution times did not vary much.

The calorimetric data were integrated by the software provided by the manufacturer (Setaram) and converted to the heat effect with the calibration factor. The results are reported as the mean and two standard deviations of the mean.

Enthalpy of dilution was calculated for the calorimetric solvent, \(5\, \hbox {mol}\cdot \hbox {dm}^{-3}\) HCl. According to Lemire et al. (2001), the molality of this acid is 5.5735 m HCl. At \(T = 343.15 \,\hbox {K}\), the enthalpy of dilution is \(-0.724\,\hbox {kJ}\cdot \hbox {mol}^{-1}\), calculated from the data presented by Nuys (1943).

Results and discussion

Characterization

All samples were fine-grained powders. The Fe-rich samples had the typical orange-brown (ochreous) color, the two samples with the lowest Fe content were yellowish, and the pure Sc end member was white. Powder X-ray diffraction showed that the samples were phase-pure. All samples could be indexed in the space group Pbnm, with the crystal-structure model of goethite. The same setting (Pbnm) was also used in the study of the FeOOH-ScOOH solid solution by Levard et al. (2018). The XRD peaks shifted as a function of the Fe/(Fe+Sc) ratio (Fig. 1), documenting changes in the lattice parameters. The refined lattice parameters are listed in the Table 1.

Table 1 Lattice parameters of the studied FeOOH–ScOOH samples with goethite structure
Full size table

Metal concentrations, measured by ICP-OES, were converted to the metal fractions with their sum normalized to 1. The Fe/(Fe+Sc) ratios were found to be near the initial ratios in the parent solutions, in agreement with the results of Levard et al. (2018). The \(\hbox {H}_2\hbox {O}\) content, determined by Karl–Fischer titration, was used to calculate excess water content for the formula \(\hbox {Fe}_x\hbox {Sc}_{1-x}\hbox {OOH}\cdot n\hbox {H}_2\hbox {O}\). The values of n are presented for all studied samples in Table 2. They increase with the scandium fraction in the solid product.

Fig. 1
figure 1

Powder X-ray diffraction patterns of the synthesized solid-solution samples in this work. The peaks are labeled with their hkl Bragg indices in the Pbnm setting

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Table 2 Results of the chemical analyses of the studied samples with the concentrations of metals (determined by ICP-OES) and \(\hbox {H}_2\hbox {O}\) (determined by Karl–Fischer titration, KFT)
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Volumes of mixing

The refined lattice parameters document clearly that the studied solid solution is non-ideal in terms of mixing volumes. This conclusion contradicts the assertion of Levard et al. (2018) that the solid solution is ideal. They used, however, arguments based on correlation coefficients of the measured volumes, not the volumes of mixing. The mixing parameters show complex behavior and have to be fitted by Redlich-Kister polynomials in the form

$$\begin{aligned} \Delta _{mix} \phi = x (1-x) \sum _{n=0}^{\theta } L_n{(1-2x)}^n \end{aligned}$$
(1)

where \(\phi\) can be a mixing variable related to the lattice parameters a, b, or c, unit-cell volume V, or enthalpy H. The data and the fits are shown in Fig. 2. For the lattice parameters and unit-cell volume, the fits serve as a visualization and simplification of the data sets, not as a basis for a theoretical model (unlike in the case of the enthalpies of mixing).

Fig. 2
figure 2

Variations of excess lattice parameters for the studied solid solution. The excess lattice parameters \(\Delta _{mix} \phi\) for the composition \(\hbox {Fe}_x\hbox {Sc}_{1-x}\hbox {OOH}\cdot n\hbox {H}_2\hbox {O}\) are calculated as \(\phi _x - (x \phi _{FeOOH} + (1-x) \phi _{ScOOH})\). The horizontal line represents ideal mixing, with \(\Delta _{mix} \phi = 0\). The fits were calculated with equation (1) but are meant only as a guide to the eye. Data from Levard et al. (2018) are shown for comparison

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The trivalent, octahedrally coordinated ions \(\hbox {Fe}^{3+}\) and \(\hbox {Sc}^{3+}\) have different ionic radii. For \(\hbox {Fe}^{3+}\), it is 0.645 Å(in the high-spin configuration), for the larger \(\hbox {Sc}^{3+}\), it is 0.745 Å. The mismatch in the ionic radii is \(\approx\)15 %.

The crystal structure of goethite consists of double chains of octahedra (Fig. 3). Within the chains, the octahedra share edges and these chains extend in the direction c. The double chains condense into a framework by corner-sharing between adjacent chains (Fig. 3). The direction a is perpendicular to the stacking of the sheets defined by the double chains, whereas the direction b lies within these sheets.

Fig. 3
figure 3

Polyhedral model of the crystal structure of goethite, projected slightly off 001. The double octahedral chains run almost perpendicular to the projection plane

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Only the variations of the lattice parameter c can be fitted by a one-parameter model, using Eq. (1). In that case, the chains respond to the Fe–Sc substitution by regular expansion dictated by the difference of the ionic radii of \(\hbox {Fe}^{3+}\) and \(\hbox {Sc}^{3+}\). Slight positive deviation from the ideality in the elongation direction of the chains (direction c) is likely caused by the strain owing to placing ions with different radii randomly along the chain direction. The response in the a and b directions is more complex and must be fitted with equation (1) with three adjustable parameters. When the smaller \(\hbox {Fe}^{3+}\) is substituted into ScOOH with the larger \(\hbox {Sc}^{3+}\) ion, the lattice parameter a reacts by essentially ideal decrease up to \(\hbox {X}_{goe}\approx 0.4\) (Fig. 2a). In the Fe-rich region, however, the lattice parameter a deviates significantly and positively from ideality. This behavior can be explained by the strain induced by placing the larger \(\hbox {Sc}^{3+}\) into a structure that contains predominantly the smaller \(\hbox {Fe}^{3+}\) ion. In other words, the stacking distance of the sheets expands in the a direction much more than expected for an ideal solution that obeys the Vegard’s law. The b direction within the sheets, a small negative deviation from ideality is observed (Fig. 2b). The precise mechanism of how the structure accommodates the strain remains unknown because it would require refinement of atomic positions and evaluation of octahedral deformation and tilting of the chains.

The interplay of variations of the lattice parameters results in peculiar changes of the unit-cell volume (Fig. 2d). In the Sc-rich region, the unit-cell volume deviates negatively from ideality, but in the Fe-rich region, it deviates positively.

Enthalpies of mixing in the FeOOH–ScOOH solid solution

Using our calorimetric data (Table 3), the deviation of the enthalpies of mixing from the ideal behavior can be evaluated. There is, however, another variable that complicates the thermodynamic properties of this solid solution. It is the hydration state of the samples, expressed in the number of \(\hbox {H}_2\hbox {O}\) molecules in the formula \(\hbox {Fe}_x\hbox {Sc}_{1-x}\)OOH\(\cdot n\hbox {H}_2\hbox {O}\). This variable n can be eliminated with the assumption that the excess \(\hbox {H}_2\hbox {O}\) is physisorbed and behaves as bulk liquid water. The thermochemical cycle for the corresponding calculation is outlined in Table 4. The assumption mentioned above can be expressed as \(\Delta H_2 = 0\,\hbox {kJ}\cdot \hbox {mol}^{-1}\). The enthalpy of dilution (\(\Delta H_3\), \( {-}0.724\,\hbox {kJ}\cdot \hbox {mol}^{-1}\)) was calculated as explained above. The resulting calculated drop-solution enthalpies for \(\hbox {Fe}_x\hbox {Sc}_{1-x}\)OOH phases with no excess \(\hbox {H}_2\hbox {O}\) are listed in Table 3. Because the amount of excess \(\hbox {H}_2\hbox {O}\) is not large and the dilution enthalpy is relatively small, the differences between \(\Delta H_1\) (enthalpies of dissolution with excess \(\hbox {H}_2\hbox {O}\)) and \(\Delta H_4\) (enthalpies of dissolution without excess \(\hbox {H}_2\hbox {O}\)) are small.

Table 3 Experimentally measured drop-solution enthalpies of the studied samples in \(5 \,\hbox {mol}\cdot \hbox {dm}^{-3}\) HCl at \(T = 343.15 \,\hbox {K}\) (values \(\Delta H_1\))
Full size table
Table 4 Thermochemical cycle to subtract the excess \(\hbox {H}_2\hbox {O}\) and to calculate the enthalpies of mixing in the FeOOH-ScOOH solid solution
Full size table

Using the \(\Delta H_4\) values, the enthalpies of mixing can be calculated according to reaction (5) in Table 4. The enthalpies of mixing (\(\Delta H_5\)) are listed in Table 3. All the values are small and positive but they deviate from \(0\, \hbox {kJ}\cdot \hbox {mol}^{-1}\), even when considering the uncertainties (all given at the 95 % confidence level). Within the given uncertainties, the calculated values can be fitted with a simple, one-parameter mixing model according to equation (1). The mixing parameter W for this solid solution is \(15.2\pm 1.0 \,\hbox {kJ}\cdot \hbox {mol}^{-1}\) for the equation \(\Delta _{mix} H = W x (1 -x)\). For this case of the one-parameter, so-called regular solution, x can represent either \(\hbox {X}_{{Fe}}\) or \(\hbox {X}_{{Sc}}\) because the function is symmetric around \(\hbox {X}_{{Fe}} = \hbox {X}_{{Sc}} = 0.5\). The data and the fit are shown together in Fig. 4.

An interesting corollary of the simple model developed here is that the solid solution studied should not be continuous. The positive enthalpies of mixing can be offset by the contribution of the configurational entropy to the Gibbs free energy in the form \(G^{ideal} = RT[x \ln x + (1-x) \ln (1-x) ]\). This function reaches its minimum at \(x = 0.5\) of \(-1.72\,\hbox {kJ}\cdot \hbox {mol}^{-1}\) at \(T = 298.15\,\hbox {K}\). For the regular model presented here, \(H^{ex}(x = 0.5) = +3.8\,\hbox {kJ}\cdot \hbox {mol}^{-1}\). Thus, there should be a small miscibility gap; some of the intermediate compositions reported in this and other studies (Levard et al. 2018) are probably metastable and persist because of the kinetic barrier to exsolution.

Fig. 4
figure 4

Enthalpies of mixing (\(\Delta _{mix} H\), equal to \(\Delta H_5\) in Tables 3 and 4) as a function of the molar Fe/(Fe+Sc) ratio in the studied solid solution. The horizontal line shows the ideal behavior (\(\Delta _{mix} H = 0\)). The fit, shown by the solid curve, results from equation (1) with a single fit parameter \(W = 15.2\pm 1.0 \,\hbox {kJ}\cdot \hbox {mol}^{-1}\)

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Enthalpies of mixing in similar systems

To our best knowledge, the only other solid-solution series with goethite (\(\alpha\)-FeOOH) as an end member that was investigated in terms in thermodynamics is \(\alpha\)-FeOOH–AlOOH. The \(\alpha\)-AlOOH phase with the goethite-type structure is the mineral diaspore. Using calorimetry, Majzlan and Navrotsky (2003) showed that the solid solution is non-ideal, with a large parameter \(W = 79 \pm 14\,\hbox {kJ}\cdot \hbox {mol}^{-1}\). Such parameter W suggests that the equilibrium solubility of Al in \(\alpha\)-FeOOH is negligible. Indeed, the solid solution is not continuous, but the calculated equilibrium \(x_{AlOOH}\) is in stark conflict with common natural occurrence or laboratory syntheses of Al-goethite well exceeding 10 mol.% AlOOH (e.g., Lewis and Schwertmann (1979)). The discrepancy was rationalized by a thermodynamic model where \(\alpha\)-FeOOH is being mixed with a hypothetical AlOOH phase (not diaspore) whose solubility is much higher than that of diaspore (Königsberger 2013). Bazilevskaya et al. (2011) argued that Al preferentially clusters when inserted into the structure of goethite and the solid solution is inhomogeneous on microscopic level. The solid solution \(\alpha\)-FeOOH–MnOOH (goethite-groutite) is also not continuous. Just as in the previous case, \(\hbox {Mn}^{3+}\) seems to form clusters in the goethite matrix (Scheinost et al. 2001). In this case, however, the reason therefore is the electronic configuration of \(\hbox {Mn}^{3+}\) and the associated Jahn-Teller distortion of the coordination octahedra. A recent study (He et al. 2023) proposed that the substitution of \(\hbox {Fe}^{3+}\) into the NiOOH structure is possible but up to 25 % and Fe enters this structure in its low-spin state. If also applicable to other systems, the switch between spin states of the \(\hbox {Fe}^{3+}\) would constitute yet another complication that would need to be reflected in the thermodynamic models.

The difference between the mixing parameters W between \(\alpha\)-FeOOH–AlOOH and \(\alpha\)-FeOOH–ScOOH solid solutions is difficult to rationalize. They are similar in terms of the mismatch of cation radii (17 % for \(\hbox {Fe}^{3+}\)-\(\hbox {Al}^{3+}\), 15 % for \(\hbox {Fe}^{3+}\)-\(\hbox {Sc}^{3+}\)). They are also similar in terms of the mismatch in the molar volume of the entire structure. The difference in the molar volumes was defined by Davies and Navrotsky (1983) as

$$\begin{aligned} \Delta V = \frac{V_2 - V_1}{V_{{\overline{12}}}} \end{aligned}$$
(2)

where \(V_1\) and \(V_2\) are the molar volumes of the two end members (\(V_2\) being larger than \(V_1\)) and \(V_{{\overline{12}}}\) is the average volume of the two end members. For the FeOOH–AlOOH solid solution, \(\Delta V = 16.0\) %, for the FeOOH-ScOOH solid solution, \(\Delta V = 14.5\) %. It appears, though, the \(\hbox {Al}^{3+}\) is too small for the goethite-type structure, at the lower limit of what this structure type can tolerate. Diaspore does not precipitate from natural aqueous solutions at surface temperatures, and is not the stable phase in the system \(\hbox {Al}_2\hbox {O}_3\)-\(\hbox {H}_2\hbox {O}\).

Davies and Navrotsky (1983) proposed that \(\Delta V\), not the mismatch in ionic radii, is of greater importance for the assessment of the ideality of a solid solution. It is so because large, more complex structures, where the substituting ions fill only small fraction of the total volume, are more capable of accommodating strain created by the substitution. Indeed, the \(\hbox {Fe}^{3+}\)-\(\hbox {In}^{3+}\) substitution in rhomboclase [nominally (\(\hbox {H}_3\hbox {O}\))\(\hbox {Fe}_2\)(\(\hbox {SO}_4\))\(\cdot 2\hbox {H}_2\hbox {O}\)] is accompanied only by small positive enthalpy of mixing, with \(W = 4.26 \pm 0.32\,\hbox {kJ}\cdot \hbox {mol}^{-1}\) (Bärthel and Majzlan 2024). In this case, \(\Delta V = 4.7\) %, even though the mismatch in the ionic radii is more than 19 %.

The FeOOH-GaOOH solid solution with goethite structure is also known to be continuous (Krehula et al. 2013). Santos et al. (2011), on the other hand, proposed that only a small fraction of the \(\hbox {Ga}^{3+}\) ions substitute for the \(\hbox {Fe}^{3+}\) ions. The rest, according to their results, should occupy empty octahedral sites, grain boundaries, and other defects. The \(\Delta V\) for this solid solution is only 4 % and the mismatch in ionic radii is equally only 4 %. Thermodynamic properties of this solid solution are not known. It could be predicted, though, that the solid solution is nearly ideal, because of the small volume mismatch.

Thermodynamics of ScOOH

The answer to the question of partitioning of Sc during precipitation of goethite requires several pieces of information. One of them, the thermodynamics of the FeOOH-ScOOH solid solution, was sought and found in this work. The other one are the solubility products (\(\log K_{sp}\)) of the end members. This value is well known for goethite (Lemire et al. 2013). Using the critically evaluated data from that source, the \(\log K_{sp}\) value for the dissolution reaction \(\alpha\)-\(\hbox {FeOOH} + 3\hbox {H}^+ \rightarrow \hbox {Fe}^{3+} + 2\hbox {H}_2\hbox {O}\) is +0.17.

For ScOOH, Nikolaychuk (2016) lists \(\Delta _f G^o\) of \(-1007.2\,\hbox {kJ}\cdot \hbox {mol}^{-1}\), referring to the compilation of Baes and Mesmer (1976). He also compiled values of \(\Delta _f G^o\)(\(\hbox {Sc}^{3+}\),aq). Three of four cluster very tightly around the average of \(-586.5\,\hbox {kJ}\cdot \hbox {mol}^{-1}\), the fourth one (from a Russian compilation Thermicheskie konstanty veschiestv) is slightly off, at \(-583.89\,\hbox {kJ}\cdot \hbox {mol}^{-1}\). Since the values from the compilation Thermicheskie konstanty veschiestv seem to be consistently outliers for other phases and aqueous species, we adopt here \(-586.5\,\hbox {kJ}\cdot \hbox {mol}^{-1}\) for \(\Delta _f G^o\)(\(\hbox {Sc}^{3+}\),aq). One of the \(\Delta _f G^o\) values stems from a critical review of Travers et al. (1976) who provided references to older studies. It seems that the review of Travers et al. (1976) was also largely incorporated into the tables of Wagman et al. (1982).

Using these data, we obtain \(\log K_{sp}\) for the dissolution reaction \(\alpha\)-\(\hbox {ScOOH} + 3\hbox {H}^+ \rightarrow \hbox {Sc}^{3+}\) + 2\(\hbox {H}_2\hbox {O}\) of +9.38. Without further data, it is difficult to judge, even roughly, the accuracy of this datum. We can compare the thermodynamic properties of the Me(OH)\(_3\) solids to verify if there is a similar difference in the \(\log K_{sp}\) values for the Fe and Sc phases.

Wagman et al. (1982) list \(\Delta _f G^o\) for Sc(OH)\(_3\) of \(-1233.3\,\hbox {kJ}\cdot \hbox {mol}^{-1}\), without citing the original work or specifying the crystal structure of the phase (the phase was reported as crystalline). Nikolaychuk (2016) compiled the values of other sources. With the exception of the outlier from Thermicheskie konstanty veschiestv, they cluster tightly around the mean of \(-1233.2\,\hbox {kJ}\cdot \hbox {mol}^{-1}\). The value of \(-1226.0\,\hbox {kJ}\cdot \hbox {mol}^{-1}\), proposed by Travers et al. (1976), is also slightly off. It seems, though, that all these values are variations of the original experimental result published by Oka (1938) in Japanese. For the dissolution reaction \(\text{Sc(OH)}_3 + 3\hbox {H}^+ \rightarrow \hbox {Sc}^{3+} + 3\hbox {H}_2\hbox {O}\), the resulting \(\Delta _r G = {-}64.7\,\hbox {kJ}\cdot \hbox {mol}^{-1}\), corresponding to \(\log K_{sp} = +11.33\). These values can be seen as tentative, however. Feitknecht and Schindler (1963) reported that amorphous \(\hbox {Sc}_2\hbox {O}_3 \cdot n\hbox {H}_2\hbox {O}\) precipitates are converted as a rule to ScOOH, not to Sc(OH)\(_3\), thus questioning as to what was actually investigated in the original work. Further, they wrote that the \(\hbox {Me(OH)}_3\) phases (probably also those including Sc) were obtained as a rule in high-temperature - high-pressure syntheses.

Table 5 A list of solubility products (\(\log K_{sp}\)) for phases with the stoichiometry \(\hbox {Me(OH)}_3\) and their dissolution reactions \(\hbox {Me(OH)}_3 + 3\hbox {H}^+ \rightarrow \hbox {Me}^{3+} + 3\hbox {H}_2\hbox {O}\) at \(T = 298.15 \,\hbox {K}\)
Full size table

A set of \(\log K_{sp}\) values for the phases with the stoichiometry Me(OH)\(_3\) is listed in Table 5. It can be readily observed that the \(\log K_{sp}\) value for Sc(OH)\(_3\) is much higher than that of Fe(OH)\(_3\), even though the \(\log K_{sp}\) values for Fe(OH)\(_3\) refer to X-ray amorphous precipitates. Hence, the difference between \(\log K_{sp}\) values of the corresponding Fe and Sc phases appears to be systematic. The \(\log K_{sp}\) for ScOOH, presented above and based on the compilation of Nikolaychuk (2016), can be considered roughly correct. Further work will be needed to test its accuracy and distinguish between the thermodynamic properties of \(\alpha\)-ScOOH and \(\gamma\)-ScOOH.

Partitioning of Sc between goethite and aqueous solution

The program MBSSAS (Glynn 1991) calculates Lippmann diagrams from the solubility products (\(\log K_{sp}\)) and thermodynamic properties of solid solution. These diagrams visualize the composition of the co-existing solid and liquid phases. The code uses an expression based on Guggenheim equations (Guggenheim 1952)

$$\begin{aligned} G^{ex}= & x_{BA} x_{CA}RT\left( a_0 + a_1(x_{BA}-x_{CA}\right) \nonumber \\ & +a_2\left( x_{BA}-x_{CA})^2 +...\right) \end{aligned}$$
(3)

for a solid solution \(B_{1-x}C_xA\). In our case, B = \(\hbox {Fe}^{3+}\), C = \(\hbox {Sc}^{3+}\), and A = [OOH]\(^{3-}\). R is the universal gas constant, and T is the thermodynamic temperature. In the case of a regular solid solution, all terms except \(a_0\) are set to 0. Assuming that the excess entropies are 0 J\(\cdot \hbox {mol}^{-1}\cdot \hbox {K}^{-1}\) throughout the solid solution, \(G^{ex} = H^{ex}\). Then

$$\begin{aligned} x_{BA} x_{CA}RT a_0 = G^{ex} = H^{ex} = W x (1-x) \end{aligned}$$
(4)

Comparing the left- and right-hand sides of the equation gets \(a_0 = W/RT\). \(a_0\) is therefore a dimensionless parameter, thus, according to Glynn (1991), “emphasizing the empirical nature of the Guggenheim model”. For the FeOOH-ScOOH solid solution, \(a_0 = 6.13\).

Fig. 5
figure 5

Lippmann diagram calculated for the solid solution \(\alpha\)-FeOOH-ScOOH. The input data are listed in the text. The diagram was calculated with the software MBSSAS (Glynn 1991). The compositions of the starting solutions (black circles) are connected with tie lines to the compositions of the resulting solids (white circles), plotted on the solidus curve. The compositions of the starting solutions are plotted in the upper part of the diagram to indicate that these solutions were initially supersaturated. The precise values of \(\Sigma \Pi\) for these solutions (defined as \(a_A(a_B+a_C)\) for the solid solution \(B_{1-x}C_xA\)) cannot be, however, calculated. For this reason, the upper part of the vertical axis has no numerical values attached to it

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The calculated Lippmann diagram (Fig. 5) shows the composition of the coexisting liquid and solid phases for the solid solution \(\alpha\)-FeOOH–ScOOH and an aqueous solution. For the investigated \(\alpha\)-FeOOH–ScOOH solid solution, \(\hbox {Sc}^{3+}\) is predicted to be strongly partitioned into the liquid phase. Thus, formation of goethite would harbor scandium in the liquid phase and could be helpful for the following extraction step.

Using the data from this and previous studies, distribution coefficient for \(\hbox {Sc}^{3+}\) between aqueous solution and the solid solution can be calculated. The distribution coefficient was defined by Shtukenberg et al. (2006), their equation 18 as

$$\begin{aligned} D_{eq} = \frac{K_{CA}}{K_{BA}} \cdot \frac{f_{CA}}{f_{BA}} \end{aligned}$$
(5)

retaining their notation, for a solid solution \(B_{1-x}C_xA\), where K is the solubility product and f is the activity coefficient of a component in the solid solution. Because of the large difference in the solubility products of FeOOH and ScOOH, the value of \(D_{eq}\) is controlled to a large extent by their ratio and modified only somewhat by the ratio of the activity coefficients. If this solid solution was ideal, \(D_{eq} \approx 10^{-10}\). Calculating the activity coefficients for the regular solid solution model as \(RT \ln f_{CA} = W x^2_{BA}\), the value of distribution coefficient drops to \(D_{eq} \approx 10^{-12}\), confirming the tendency of \(\hbox {Sc}^{3+}\) to remain in the aqueous solution.

The results of the syntheses in this work and in Levard et al. (2018) oppose the calculated distribution coefficient because the molar ratios of Fe and Sc are similar in the aqueous and in the resulting solid phase. The initial composition of the aqueous solutions and the composition of the resulting solids are plotted as black and white circles, respectively, in Fig. 5. It can be seen, with one exception, that the products are Sc-poorer than the starting solution, but the difference is small and not controlled by thermodynamics. Shtukenberg et al. (2006) provided theoretical framework for kinetics of crystallization from strongly supersaturated aqueous solutions that produce solid solutions with similar molar ratio of the substituting components. These models apply also in our case.

The conclusions based on thermodynamics of the solid solution qualitatively agree with the findings of Qin et al. (2021) who determined that \(\hbox {Sc}^{3+}\) associates with goethite predominantly via adsorption. Namely, the thermodynamics predicts that \(\hbox {Sc}^{3+}\) should remain in the aqueous phase where it can be associated with the goethite surfaces by adsorption. Ulrich et al. (2019) found that Sc in lateritic profiles migrates from the primary pyroxenes into goethite and could be possibly taken up in the structure of goethite, in contrast to the results of this study. Chassé et al. (2017) found that Sc associates with iron oxides, particularly goethite, in lateritic ores. Scandium is lost when goethite recrystallizes to hematite (Ulrich et al. 2019). These studies emphasize the importance of adsorption and the complex pathway to crystalline goethite. Goethite is likely never produced directly from the aqueous solution. Instead, it is a product of recrystallization of the X-ray amorphous precursor ferrihydrite that is able, owing to its large surface area and variable local coordination, uptake plethora of ions and molecules.

Selective precipitation of \(\hbox {Fe}^{3+}\) and partitioning of \(\hbox {Sc}^{3+}\) into the aqueous solution was observed during crystallization of jarosite (Dutrizac and Chen 2009). We assume that the underlying thermodynamics is similar as in the case of goethite. The difference lies in the fact that jarosite crystallizes directly from the aqueous media, without a complex precursor. In summary, the results of this work are applicable to recrystallization of iron oxides with Sc load or formation of iron oxides at elevated temperatures. Under surface temperatures, adsorption and hindered exsolution kinetics control scandium speciation.