Incorporation of Rare Earths and Yttrium in Calcite: A Critical Re-evaluation

The reported partition coefficients of REE with calcite are reviewed and critically discussed. In some of the reported experimental sets, REE concentrations are found to be supersaturated with respect to individual REE2(CO3)3 but never to REE(OH)3. Although the solutions are unsaturated with respect to individual REY carbonates, REY including Y are incorporated in calcite surfaces, where they are overgrown by calcite. Charge balances may be obtained by building {REY–Na-(CO3)2}n or by exchange of 2Ca2+ against REY3+ + blank space calcite lattice. These surface compounds may either be homogeneously distributed or clustered. Both the size and frequency of clusters increase with [REY]/[Ca] or [ΣREY3+]/[Ca2+] in solution. If these surface precipitates are removed into solutions saturated with respect to ΣREE2(CO3)3, they start growing in the aqueous phase. In this case, the apparent DREY and kREY values decrease with increasing REY concentrations in solution. In previous studies, only the individual distribution coefficients are reported not considering that the entirety of REY determines their behavior in partitioning. Given enough time, these surface clusters equilibrate with the aqueous phase before being overgrown by calcite. In the double logarithmic plots of {REY}/{Ca} versus [REY]/[Ca] or [REY3+]/[Ca2+], two relationships evolve characterizing the REY distribution in marine calcite and experimental calcites grown in Mg2+-free solutions. The double logarithmic plots of partition coefficients of REYi3+ in calcite grown from seawater show a slope exceeding unity, whereas those from fluids without Mg2+ depict slopes less than unity being both in contrast to the Henderson–Kracek rule.


Introduction
Rare earth elements (REE) and yttrium (Y) behave similarly and are henceforth referred to as REY (Bau and Dulski 1995). When Y is absent in cited experiments, the term REE is used. This suite of REY is ubiquitous in nature and is particularly incorporated in omnipresent calcite. The heterogeneous, onion-shell-like (Eq. 1) and homogeneous partitioning (Eq. 2) was first applied to describe the partition of Ra 2+ with barium sulfate (Doener and Hoskins 1925) and chromate (Henderson and Kracek 1927), respectively.  (Table 1). There are mainly four different procedures by which REE partition coefficients with calcite are determined: 1. REE in diagenetic marine carbonates and reefal microbialites are related to REE in seawater (Fig. 1a). Parekh et al. (1977) extrapolated the coprecipitated fraction of REE in plots of REE in limestone samples versus their REE in associated acid insoluble residue. The derived apparent D REE values decrease from 1388 (La) to 464 (Lu). Scherer and Seitz (1980) analyzed calcitic cements from coral reefs of the Bahamas with about 15 mol% MgCO 3 . After oxidation of the organic matter by H 2 O 2 for 24 h, the acid insoluble residue was determined. With the help of these residues, the individual, total REE abundance in carbonates was corrected and related to individual [REE i ] in modern seawater yielding D REE values between 120 and 530 for La and Sm, respectively. The trend of average D REE values of 296 ± 30 for Heron microbialites, Great Barrier Reef, Australia (Webb and Kamber 2000) and 212-256 (except Ce with 327) in carboniferous to quaternary limestones from Japan (Toyama and Terakado 2019) resemble those reported by Scherer and Seitz (1980). After ultrasonically removing of Fe-Mn coatings in a reducing bath, Palmer (1985) determined REE abundance in calcitic foraminiferal skeletons from the Atlantic. The removed coatings contained about 1000 times more REE than the remaining calcite skeletons with D REE values of 125 (La) and 73 (Yb). 2. Steady-state condition in suspension of 5 g of calcite seeds in 2 L of phosphate-cleaned seawater was achieved by bubbling a CO 2 /N 2 mixture with pCO 2 of 0.0031, additional stirring by glass propeller and constant addition of calcite-saturated solution containing the REE spike Mucci 1993, 1995). The determination of the quantity of overgrowth by chemical means was not easy because of recovering all calcite sticking to the walls of the reactor. Their reported logarithmic D REE values decrease from 3.6 to 1.9 for La and Yb, respectively (Fig. 1b). Tanaka and Kawabe (2006) precipitated calcite on seed crystals suspended in Ca 2+ -Na + solution by bubbling N 2 containing 1% of CO 2 through the solution and operating a magnetic stirrer chip in order to yield (1) k = log REY in ∕ REY fin ∕ log Ca in ∕ Ca fin 1 3 a homogeneous overgrowth on calcite seeds. Their initial REY concentrations ranged from 10 ppb in general and 20 ppb for Pr, Sm, Tb and Tm. The calcite overgrowth was derived from charge and mass balances. The developed log(D REE ) values cover a range between 2.5 and 3.6 with a broad maximum of the intermediate REE (Fig. 1b). The uncertainty of D REE values between the extreme runs is about a factor of 3.7. Voigt et al. (2017) reported log(D REE ) values of runs with either La or Yb and with both elements together. Using the constant addition technique, the experiments were performed under 1 atm CO 2 , pH 6-6.4 and variable amounts of seeds. Log(D REE ) decreases with both  (Parekh et al. 1977), Mg-calcite cements of reefs (Scherer and Seitz 1980), and averages of biogenic carbonates of various coral species (Sholkovitz and Shen 1995;Agaki et al. 2004;Wyndam et al. 2004), all related to REE 3+ in modern seawater increasing log(REE) and log(Ω − 1) (Ω = calcite saturation index) in solution indicating an inhibiting effect of REE on the growth of calcite. 3. An evaporation technique (no stirring) under constant addition of mother solution and spike was applied by Toyama and Terakado (2014). In a pre-phase, calcite seeds were grown on movable silica glass plates. In these experiments, the initial volume of the calcite-saturated solution with 15.6 g NaCl/l and REE spike was maintained by refilling the reactor. The overgrowth on calcite seeds was determined by electronical weighing of the plates. The calcite growth was exclusively diffusion-controlled, i.e., [Ca] and [REE] decreased around the growing calcite crystals. Although the initial [REE] in runs A and B differed by a factor of 10, the final [REE] differed only by factors 2-3 (Fig. 2). Experiments A1 + A2 and B1 + B2 represent results after 122 and 37 h of calcite growth, respectively. Log(D REE ) of light REE (LREE) and heavy REE and Y (HREY) decrease in runs A-1 and (A-2) from 25-10 and 65-25, and in runs (B-1) and (B-2) from 70-40 and 95-65, respectively. 4. CO 2 exsolution from a bicarbonate solution was applied by Terakado and Masuda (1988). According to the experimental procedure, Eq. (1) is used to quantify the coefficient k REE (Fig. 1b). The initial solution contained about 350 mg/l Ca. No seeds were applied. In different runs, they used a prepared spike solution of REE 3+ covering a range of f = 0.2 to 7, where f is the multiplication factor of their standard spike composition that was added to the Ca(HCO 3 ) 2 solution. Although the individual REE concentrations in the initial solution vary between 40 and 0.9 ppb for Nd and Lu, respectively, the derived apparent k REE values differ between each f run but are almost similar within each f run for all REE (Fig. 3a). Their k REE values increase from f = 0.2 to f = 1 and then decrease with the further increase in REE abundance.
In the above reviewed work, the values of D REE or k REE depend on the state of calcite saturation (Voigt et al. 2017), growth rate of calcite (Zhong and Mucci 1995;Toyama and Terakado 2014;Voigt et al. 2017), salinity (Webb and Kamber 2000), concentration of REY (Terakado and Masuda 1988;Toyama and Terakado 2014), chemical complexation   (Tanaka et al. 2004;Voigt et al. 2017), temperature, CO 2 partial pressure, pH and performance of experiments.
Irrespective of the performance of REE partitioning during calcite precipitation, there are at least three important aspects affecting REY partitioning in calcite: • High REY concentrations induce precipitation of separate phases such as REY carbonates and hydroxides due to which REY abundance in solutions and consequently D REE values decrease; • Ion exchange of REY 3+ against Ca 2+ form variously composed surface compounds with different kinds of charge balance; • Chemical complexation in solutions.
The aim of this contribution is to search for a common process which could explain the wide spread of experimental partition coefficients D REE . Are REY really homogeneously distributed in calcite or do they agglomerate to flatspread or linear surface clusters which are then overgrown by calcite? The size of such compounds may show a strong dependence of partition coefficients on the sum of REY in solution because all REE and Y behave similarly.
An outlook on REY partitioning between alcite and seawater is given because one aspect of some published studies was to gain a deeper insight into the development of REY in seawater in deep time (Shields and Webb 2004;Tanaka et al. 2004;Tanaka and Kawabe 2006;Voigt et al. 2017;Toyama and Terakado 2019). Because calcite is omnipresent in the earth's crust and preferentially precipitated from seawater, the partitioning of REY between calcite and its solution has also become a very important tool in hydrochemistry, where limestone aquifers play an important role (Johannesson et al. 1997;Möller et al. 2003;Siebert et al. 2014). Zhong and Mucci (1995) reported that REE concentrations immediately decreased after addition of the REE spike to the calcite suspension followed by a slow process (Fig. 4a). After this initial fast decrease in La and Yb concentration in solution (what they called: adsorption step), the further decrease of Yb is much less than of La with time. This difference in behavior may be due to differences in dehydration enthalpies which are less for the bigger La 3+ than for the smaller Yb 3+ favoring adsorption of La 3+ but retarding the adsorption of Yb 3+ . About 25 and 40% of added Yb and La, respectively, is "adsorbed" causing high surface concentrations of REE. Except at thermodynamic equilibrium, the zeta potential of calcite is always negative (Moulin and Roques 2003) which promotes the adsorption REY 3+ onto calcite surfaces.

Exchange of Mg 2+ and REE 3+ Against Ca 2+ in Calcite Surface
A similar kinetic behavior is reported for Mg 2+ exchange against Ca 2+ doped by radioactive 45 Ca 2+ in rhombohedral faces of calcite (Möller 1973) (Fig. 4b). With increasing Mg 2+ in solution, 45 Ca 2+ -doped Ca 2+ in the calcite surface decreases. With increasing Mg 2+ /Ca 2+ in solution, the Mg 2+ /Ca 2+ in the calcite surface approach distinct ratios of 1 and 3 is indicated by significant changes in slopes (Fig. 4c). These ratios resemble those of the compositions of the minerals dolomite and huntite. The ratio of 1 suggests that Mg 2+ 1 3

Fig. 4
Exchange of Ca 2+ in calcite surface against foreign ions. a Drop of REE concentration when adding the spike to the calcite seed suspension (Zhong and Mucci 1995); b exchange kinetic of Mg 2+ against Ca 2+ in calcite surface; c exchange of Mg 2+ against Ca 2+ in calcite surface as a function of Mg 2+ /Ca 2+ in solution (Möller 1973;Möller and Sastri 1973;Möller and De Lucia 2019) and Ca 2+ are either randomly distributed or are arranged in alternating lines of Ca 2+ and Mg 2+ . Rhombohedral faces of calcite present lines of Ca 2+ and CO 3 2− . Thermodynamical estimates suggest that the arrangement of Mg 2+ and Ca 2+ in separate lines on rhombohedral surfaces of calcite has a minimum in free energy (Möller and Rajagopalan 1976). Using single crystals, it was shown that only about one molecular layer of calcite takes part in this type of ion exchange (Möller and Sastri 1974;Pokrovski and Sholkovitz 2001). The Mg 2+ distribution in the calcite surface layer must have an effect on REY partitioning between calcite and seawater.
No specific studies of REY 3+ exchange against Ca 2+ in calcite surfaces are reported. Here the experience with REE 3+ in groundwater may help out. The REY 3+ /Ca 2+ values in groundwater from limestone aquifers are only about 2‰ of that in the dissolving calcite (Table 5; Fig. 5). Furthermore, [REY 3+ ]/[Ca 2+ ] values of seamount limestones are 10 2 -10 4 times higher than their corresponding ratios in seawater (Tanaka et al. 2003;Miura et al. 2004;Toyama and Terakado 2019) indicating that during recrystallization REY from seawater are incorporated due to exchange against Ca 2+ . This process explains the high DREE values in limestones (Parekh et al. 1977) and disqualifies limestones as reliable reference material to derive DREE values in seawater carbonate systems.

Charge balance
Substitution is maximum if the size of the foreign ion is comparable with the substituted Ca 2+ in calcite leading to least lattice distortion. Thus, Na + fits perfectly into Ca 2+ position (Table 2). REY are either slightly larger or smaller than Ca 2+ . Secondly, the charge balance has to be achieved in the substitution process. There are different possibilities: associated

Fig. 5
Relationship between REE in groundwater and limestone aquifer rocks (Möller and Siebert 2016). Note that the given ratios are similar, although limestones are from different geological periods (Avedat Group: Eocene; Judea Group: Cenomanian). For details refer to "Appendix 1" □ blank space in the lattice. In all reviewed experiments of partitioning of ΣREY between calcite and solutions, the concentrations of incorporated Na + exceed by far REY 3+ concentrations (Fig. 6). The excess Na + amounts are suggested to be due to occupation of Na + in crystal defects (White 1975;Busenberg and Plummer 1985;Lakstanov and Stipp 2004). Excess of Na + in the lattice may also be compensated by substitution of CO 3 2− by HCO 3− .  (Cesbron 1989;Kim et al. 2018).

Formation of REY carbonates
In their experiments, Voigt et al. (2017) reported saturation states Ω of calcite and Lahydroxylbastnasite (La(OH)CO 3 ) of 1 to 11 and − 1.5 to + 0.6, respectively. In any case, phases such as REY 2 (CO 3 ) 3 , REY(OH) 3 or REY(OH)CO 3 may play a role in partitioning of REY. Based on the initial concentrations in experimental solutions or seawater (Table 6), the saturation indices of REY carbonates and hydroxides in each experiment and in seawater are estimated (Table 7). SI values of REE hydroxides are all negative. The precipitation of REE carbonates (Fig. 3b, c) is mostly excluded by the authors of all the above-cited studies based on logarithms of solubility products, logK sp , of REE carbonates of Smith and Martell (1976) being about two orders of magnitude higher than the corresponding values of Spahiu and Bruno (1995). The initial SI of REE carbonate under conditions of the  (Table 8), and results are summarized in Table 3.
Often the less abundant REE and the very abundant Y are not determined in biogenic calcite and its diagenetic products. Particularly Y cannot be neglected in ΣREY. {Y} in calcite is approached by using [Y]/[Er] in seawater multiplied by [Er] in calcite. Other absent REY are determined in a similar way. REY partitioning in marine calcite depends on the composition of local seawater which is not always given by the authors. In these cases, REE in seawater reported by Sholkovitz and Schneider (1991) is used and Y is taken from the compilation of Bruland and Lohan (2003).

Discussion
Different from solution without Mg 2+ , calcite from seawater environments is subjected to ion exchange of Mg 2+ against Ca 2+ in their surfaces (Sect. 2.1). In seawater, the Mg 2+ /Ca 2+ values are about 5 and the surface ratio of {Mg 2+ }/{Ca 2+ } is about 1 (Fig. 4c). Thus, calcite surfaces expose only half of their theoretical surface Ca 2+ in seawater. The minimum of free energy in the surface is obtained when the distribution of Ca 2+ and Mg 2+ resembles that in dolomite, i.e., alternating lines of Ca 2+ and Mg 2+ (Möller and Rajagopalan 1976). If such a surface structure is present, REY exchange only takes place in Ca 2+ lines facing the solution. Thus, partitioning of REY between calcite and natural and artificial seawater can hardly be the same as in Mg 2+ -free systems. 1 3

Partitioning of individual REE
Except Ce, which often displays anomalous behavior because of oxidation to Ce(IV), REE seemingly behave alike in partitioning with respect to calcite (Terakado and Masuda 1988), irrespective of their actual concentrations in the fluid phase (Fig. 3a). Although the concentrations of La and Nd differ from Eu by factors of about 30 in the initial solution of these experiments, their derived k REE values are comparable for all REE in each experiment defined by f. The f-dependent changes of REY partitioning give evidence of different processes. Starting with the highest concentration of REE (f = 7), k REE values increase with decreasing REE concentration (f = 4 and 1) achieving maximum at about f = 1. This increase suggests that decreasing density of adsorbed REE 3+ avoids nucleation of REY carbonates being separated from the surface. Indeed, LREE are supersaturated with respect to REE carbonates (Fig. 3b, c). The lowest k REE values are obtained at f = 0.2. Contrasting these results are those of Toyama and Terakado (2014) in which the initial differences of REE concentrations by a factor of 10 are not reflected in D REE values (Fig. 2) (Fig. 7). The solid line (Eq. 5a) characterizes Mg 2+ -free systems based on Toyama and Terakado's (2014) and Voigt's et al. (2017) results. The average of eight experiments of Tanaka and Kawabe (2006) does not fit Eq. (5a). The dashed line (Eq. 6a) is based on results of Mg 2+ -dominated solutions such as seawater (Zhong and Mucci 1995;Parekh et al. 1977;Scherer and Seitz 1980;Palmer 1985;Toyama and Terakado 2019).
Using the dissolved species REY 3+ and Ca 2+ instead of concentrations of REY and Ca correlations (7a) and (8a) evolve (Fig. 7b). The overall fit of all reported values including the average of Tanaka and Kawabe (2006) is better than in Fig. 7a. Contrasting the total concentration of individual REE i , the dissolved REE i 3+ species yield trend lines that seem to merge at low REY concentrations with the trend line representing partitioning between calcite and seawater (Fig. 7b).

Partition of the Entirety of REY
Although it is known that REY substitute each other in minerals, it is still common practice to consider the individual REE 3+ and not the entirety of all REY as one "species" in partitioning between calcite and solutions. For instance, the individual log(D La ) and log(D Yb ) are slightly higher than those derived from mixtures of both elements (Voigt et al. 2017).
Equations (9a) and (12a) in Fig. 7c (7a) and (8a). There is, however, a significant difference in using either total concentrations or dissolved species. Using the dissolved species, the resulting regression lines seemingly merge at very low concentrations of REY. The different slopes in Fig. 7 give evidence of the influence of Mg 2+ on the incorporation of REY. The pre-factors are significantly smaller in the absence of Mg 2+ than in its presence.

Partition Coefficients as Functions of REY Concentrations in Solution
Dividing Eqs. (5a)-(12a) by the correspondent ratios of either REY and Ca or REY 3+ and Ca 2+ in solution, the REY partition coefficients are obtained as functions of their corresponding ratios [Eqs. (5b)-(12b) in Table 4]. The resultant D REY values as function of the ratios of REY and Ca yield subparallel trends for D REY values in seawater and in Mg 2+ -free solutions (Fig. 8a). A different result is obtained for D REY values, if the tervalent species in solution are considered. D REY in seawater increases with REY, whereas D REY in Mg 2+ -free solution decreases with increasing REY (Fig. 8b).     (Fig. 7b and 11b)]. Equation (7a) is seemingly the best representation in terms of linear regression line, whereas Eq. (11a) could also be fitted by a curve.

REE partitioning in calcite and aragonite in seawater
The evaluation of D REY values in marine carbonates has to consider that, independently of actually measured REE species, all REY are present in calcite.
The marine calcite and cleaned limestones cluster at log{La}/{Ca} of about − 5.5 to − 7 when plotted either against REY i or ΣREY i . Limestones and cements of corals are of diagenetic origin, i.e., recrystallization under exchange of seawater. Microbialites show enhanced REY i taken up during their recrystallization (Scherer and Seitz 1980;Wyndam et al. 2004). The foraminifera of Palmer are very high in REY which may be caused by exchange with their highly enriched REY of their coatings.

Surface Processes Influencing Partitioning of REY
The regression lines of the marine calcite and the results of Zhong and Mucci in artificial seawater reveal enhanced data of log{La 3+ }/{Ca 2+ } compared with experimental results in the absence of Mg 2+ (Figs. 7, 8). The slopes of these regression lines exceed unity (Fig. 7) or zero (Fig. 8) Contrasting this result, the slope in Mg 2+ -free systems is less than unity, when using dissolved species (Fig. 8b). Thus, the question arises, why does {REY i }/{Ca} and D REY increase in the presence of Mg 2+ -bearing solution and why does the ratio decrease in the absence of Mg 2+ in Fig. 8b, whereas in both systems D REY increases when related to increasing total concentrations (Fig. 8a)?
In Mg 2+ -free solutions, the first step is adsorption of REY onto the surface of calcite. Moving across the surface, the adsorbed REY may come across already exchanged REY 3+ for Ca 2+ in the surface layer. Due to distortion of the anion layer, its exchange for a neighboring Ca 2+ is favored, hence a linear structure of cluster starts growing as shown for Mg 2+ exchange in calcite surfaces (Möller and Rajagopalan 1976). With the growth of calcite, large amounts of Ca 2+ have to be adsorbed and some of them may replace REY in lattice sites of calcite surface. Thus, the {REY}/{Ca} value in the surface layer that will be overgrown by calcite loses some REY i .
In the presence of Mg 2+ , REY 3+ exchange against Ca 2+ only occurs in remaining Ca 2+ lines, which, together with associated CO 3 2-, dominate about 50% of the calcite surface in 1 3 seawater. The exchange of REY against Mg 2+ is unlikely because of the great difference in ionic ratios (Table 2). Along with growth of calcite, almost all Mg 2+ has to be substituted by Ca 2+ , a process during which also REY 3+ may be incorporated particularly because of their enhanced charge. Compared to a homogeneous distribution of REY on calcite surface in Mg 2+ -free solution, REY are enriched during growth of calcite. Each line or flatspread surface cluster consists of n lines presenting 2 or 2n sites for ongoing ion exchange, respectively (Fig. 9). The tendency to form surface clusters depends on the change of reaction free energy in the surface of calcite. Loss of free energy is higher for HREY 3+ than for LREE 3+ because the former are smaller than Ca 2+ ions (Table 2). Once clusters formed, the enthalpy for removing a REE from a cluster is enhanced compared to removal of an isolated REY 3+ . These clusters seemingly build at REY concentrations in solutions much below the solubility products of REY carbonates. Na + and REY 3+ are probably placed near to each other. These assemblages correspond to REY incorporation in either line or surface clusters of the average composition of {REY-Na-(CO 3 ) 2 } as suggested by Tanaka and Kawabe (2006). Voigt et al. (2017) suggested {REY(OH)(CO 3 ) 2 }. Any homogeneous distribution is only achieved when the concentration of REY 3+ in the calcite surface is extremely low keeping them clearly separated during growth of calcite. Although the saturation index for ΣREY carbonates in seawater is − 7.5 (Table 7), small REY clusters may still form (Fig. 9). The formation of clusters is the source of high enrichment of REY in growing calcite.
The scatter in Fig. 7 may be caused by different processes: (1) The natural material was not clean enough and thus inappropriate to be used for determination of D REY coefficients and/or (2) the biogenic-sourced calcite differs from inorganically precipitated low magnesian calcite by enhanced REY 3+ abundances in this material and this enhancement can be species-dependent. REE + Y in seawater do not form REY 2 (CO 3 ) 3 , but the less soluble phosphates may be precipitated (Spahiu and Bruno 1995) and may be enclosed in growing calcite. For instance, log(K sp ) of LaPO 4 is − 40.01, i.e., much lower than for La carbonate. REY-phosphate clusters may also be present in biogenic carbonates. Microbialites show enhanced REY taken up during their recrystallization (Scherer and Seitz 1980;Wyndam et al. 2004). The foraminifera of Palmer are very high in REY which may be caused by exchange with their highly enriched REY of their coatings.

Conclusions
A critical re-evaluation of REY partitioning coefficients with respect to calcite reveals that the reported, wide-spread experimental D REY values reflect that two processes have been overlooked.
In the absence of Mg 2+ , the whole calcite surface is accessible for REY 3+ exchange against Ca 2+ . The result of this exchange is reduced under the influence of continuing overgrowth of the next calcite layer. This leads to sub-proportional correlation of REY between the aqueous phase and the bulk of calcite.
The Mg 2+ ion exchange against Ca 2+ in calcite surface layers interfere with the exchange of REY 3+ against Ca 2+ . All rhombohedral faces of calcite show a composition of Mg 2+ /Ca 2+ of about 1 in seawater. Thus, only half of the theoretical numbers of Ca 2+ ions are accessible for REY exchange against Ca 2+ . REY are slightly concentrated in these remaining Ca 2+ lines. The growth of calcite requests that most of the Mg 2+ has to be removed and substituted by Ca 2+ and REY 3+ . REY 3+ may be enriched because of their enhanced charge. These processes together may explain any over-proportional relationship of REY/Ca between the bulk of calcite and solution. Because REY partitioning in limestone is thought to be a key in understanding REY development in seawater, the effect of the presence of Mg 2+ has to be considered.
Two different trend lines evolve: one for Mg 2+ free systems and the other for artificial and natural seawater. Because of subtle exchange processes, the sum of all REY evidence that REY/Ca in solution are not linearly correlated with {ΣREY}/{Ca} in the bulk of calcite as expected by the Henderson and Kracek rule. It seems that REY form clusters in the surface of growing calcite, even if the solubility product of REY carbonates is not reached.

Appendix 2
Compilation of the initial concentrations in experimental solutions is compiled in Table 6. The logarithms of individual activities, log( a REE 3+ ), log( a CO 3 2− ) and log( a OH − ) are estimated using PHREEQC with "lln" database (Table 7). From these data, the individual log(IAP i ) is estimated after Eq. (5) and the results are compiled in Table 7. Using the corresponding log(K sp,i ) (Table 6), the individual saturation indices are calculated after Eq. (6). Results are tabulated in the lower part of Table 7.
Index i refers to the REE 3+ involved in the experiment.
To estimate the saturation indices of the entirety of REE carbonates, the respective log(IAP carb ) and log(K sp , Σ REY ) are estimated after Eqs. (7) and (8). This activity product can hardly be lower than K sp of La but could be higher than K sp of Lu.
Dealing with the entirety of REY is limited by the unawareness of the corresponding solubility product of the whole suite, logK sp,ΣREY , a value that cannot be determined and is therefore approximated by the sum of individual K sp weighted by the corresponding activity fraction of REY 3+ species in solution (Table 7). SI values of the entirety of REY carbonates are approximated in Table 7, in which log(IAP ΣREY ) of REY carbonates are estimated from log(Σ a REY 3+ ) and log( a CO 3 2− ) obtained by PHREEQC and its "llnl" database (Parkhurst and Appelo 2013). Log(IAP ΣREY ) of the entirety of REE carbonates is given by (log( a ΣREY 3+ )× 2 + 3×log( a CO 3 2− ). The derived values of SI ΣREY = log(IAP ΣREY ) − log(K sp,ΣREY ) of the ΣREY carbonate are given in detail in Table 7 and are summarized in Table 3. The entirety of REY carbonates is supersaturated in Terakado and Masuda's and Zhong and Mucci's experiments.