Characteristics of Synthesized LDHs

The M(II)/Al ratios of the washed samples were 3.24–3.69 and agreed well with the ratios for the initial solutions (Table 1), most of the differences being <20%. The greater than stoichiometric M(II)/Al ratios may have been caused by some Al lixiviation during excessive washing. Obtaining materials with the desired compositions and exact stoichiometric carbonate contents was difficult, so reactant solutions containing 10% excess carbonate were used to make synthesis of LDHs with the desired compositions easier, albeit with somewhat greater than desired CO₃ ²⁻ contents as revealed by the TG analysis. The composition of the solid samples determined by chemical and TG analyses, as detailed in the Supplementary Material (Figs. S1 and S2), are shown in Table 2. Slight differences only were observed in the TGA curves between Mg-Al-LDH and Zn-Al-LDH endmembers, indicating that the thermal stabilities and temperatures required for the removal of interlayer water, dehydroxylation of the cationic layers, and removal of the interlayer carbonate anion were similar regardless of cation substitution. No alien anions such as NO₃ ⁻ were detected by the FTIR analysis (Figs. S3 and S4).

Table 2 Chemical formulae of the synthesized LDH materials determined from the results of the chemical and TGA analyses and normalized to the Al content, and the BET surface areas of the materials indicating nano-crystalline characteristics of the materials

The BET surface areas between 20 and 90 m² g^-1 (Table 2) indicated nano-particulate characteristics of the material. The TEM images of the synthesized LDHs in fact showed nanoparticles of hexagonal platy shape with diameters of ~100 nm (Fig. S5a,b). The FTIR analyses revealed characteristic vibrations of Mg/Al-hydroxycarbonate phases. The FTIR spectra showed broad but intense hydroxyl stretching bands between 3400 and 3500 cm⁻¹, and weak interlayer water molecule bending bands near 1620–1640 cm⁻¹. The rather sharp intense band at 1360–1385 cm⁻¹ was caused by ν₃ anti-symmetric stretching of interlayer carbonate, shifted from its position in free CO₃ ²⁻ because of strong hydrogen bonding with hydroxyl groups and H₂O molecules in the interlayer space. The bands at <1000 cm⁻¹ were ascribed to both ν₂ and ν₄ modes of carbonate and to M–OH modes, respectively. An intense and sharp band near 450 cm⁻¹ was found in each sample and was assigned to polymerized [AlO₆]³⁻ groups, and single groups with an Al–OH bond length of 0.161 nm typical for LDHs (SM).

The XRD patterns of all the synthesized LDHs contained features common in layered phases (e.g. narrow, symmetrical, and strong reflections at low °2θ values and weaker, less symmetrical reflections at higher °2θ values), as shown in Fig. 1. The hexagonal layer structure with sharp symmetrical peaks for the (003), (006), and (009) planes and broad but still symmetrical peaks for the (015) and (018) planes are characteristic of hydrotalcites (Hofmeister & Von Platen, Reference Hofmeister and Von Platen1992). A well crystalline phase resembling an LDH compound was the only solid phase prepared for the dissolution experiments in the washed samples, but the unwashed samples revealed some other diffractions during the co-precipitation syntheses. The XRD patterns of the unwashed samples contained a diffraction near 30°2θ assigned to a NaNO₃ phase;  no nitrate diffraction was present after the washing treatment, however. A broad diffraction band was also present at 48°2θ, indicating a gibbsite (Al(OH)₃) impurity, which disappeared after 240 days of the dissolution experiment (Fig. 2a).

Fig. 1 XRD patterns for hydrotalcite solid solutions synthesized using various Zn molar ratios

Fig. 2 Powder X-ray diffractograms for the hydrotalcite solid-solutions with mole fraction of Zn = X, where a depicts the patterns for unwashed and washed original co-precipitates and samples collected from the equilibrium dissolution experiment after 240 days, and b the peaks characteristic for the LDH phases of various X _Zn contents used in the Vegard’s law analysis

Crystallographic lattice parameters (a, c) were calculated for all LDH compounds synthesized with various Zn cation contents. The degree of crystallinity of each LDH sample was determined from the basal (003) and (006) reflection intensities and sharpness (Fig. 2b). The broadening of the XRD peaks at 39 and 46°2θ was due to the statistical distribution of the layer stacking sequences of the 2H and 2R polytypes. The layer stacking faults do not affect significantly the distance between the brucite-like layers in the c axis direction, however, because LDH compounds commonly show no long-range ordering, i.e. the respective stacking sequences of the hexagonally or trigonally ordered endmembers are commonly randomly distributed in LDHs.

The lattice parameters were calculated (Fig. 3), in which a is the mean cation–cation distance in the brucite-like layer with a = 2d ₁₁₀. The cell parameter c was determined from the equation c/3 = ½{d ₍₀₀₃₎+2d ₍₀₀₆₎}. The unit-cell volume V was calculated from the cell parameters using the equation V = a ² c sin (120), which was simplified to (√3/2) a ² c. Assuming that the brucite-like layer was 4.8 Å wide (Drezdzon, Reference Drezdzon1988), the interlayer space was 2.8 Å, in accordance with the locations of the carbonate anions with their molecular planes parallel to the brucite-like layers. Although of nearly the same ionic radius in octahedral coordination (Shannon, Reference Shannon1976), the inclusion of Zn into the Mg-bearing hydrotalcite slightly increased the cell parameter and shifted the peak position to lower °2θ values, therefore increasing the d spacing. However, Curtius et al. (Reference Curtius, Kaiser, Rozov, Neumann, Dardenne and Bosbach2013) reported no elongation of the octahedra in the c direction by Rietveld analysis of XRD patterns in spite of increasing ionic radii of Ni < Mg and Co < Fe(II) of the LDH endmembers, which showed a lack of dependency on the ionic radius. Nonetheless, a clear linear correlation was found between the cell constants and cation composition ratios, as required by Vegard’s law, even for the c direction (Fig. 3a and b), and thus also for the cell volume (Fig. 3c). The formation of the desired solid solutions without a mixing gap over the whole range of Mg-Zn-bearing hydrotalcite formulations was thus confirmed by the XRD analysis.

Fig. 3 Values for the unit cell a a-axis, b c-axis, and c volumes determined by powder X-ray diffractometry for the LDH samples with various mole fractions of Zn used for the Vegard’s law analysis. The dotted lines indicate linear correlations

Aqueous Solubility Data and SSAS Equilibrium Experiments

As shown in Fig. 4a, apparent equilibrium was reached on the fifth day of the co-precipitation experiments. The Mg concentration was 1.2 μmol/L on that day and for the rest of the experiment. Somewhat lower equilibrium concentrations were reached after day 150 of the dissolution experiments (Fig. 4b). The approximately 3-fold different concentrations at equilibrium in the mother liquor were probably caused by some phase impurities, particularly gibbsite, as indicated by the XRD analysis. In principle, buffering of the Al is thus possible in order to avoid the tricky dissolved Al concentration measurements (Johnson & Glasser, Reference Johnson and Glasser2003), but this approach was not preferred here because of the lack of a gibbsite impurity detected after 240 days in the dissolution experiments. A modified solubility constant for the pure Mg-LDH endmember with the ideal stoichiometry was calculated using the dissolution reaction:

(1) $\left({\text{Mg}}_3,\text{Al},{\left(\text{OH}\right)}_8\right){\left({\text{CO}}_3\right)}_{0.5}·{2H}_{2}O={3\text{MgOH}}^{+}+\text{Al}{{\left(\text{OH}\right)}_4}^{-}+{\text{OH}}^{-}+0.{{5\text{CO}}_3}^{2-}$

Fig. 4 Measured total dissolved concentrations of Al and Mg for a the co-precipitation synthesis solution during the first 7 days of ageing at pH 10.0±0.5 (the dotted line indicates the Al concentration in the presence of gibbsite), and b the dissolution experiment after 240 days (pH 10.2±0.5)

The solubility product was not formulated using the activities of the free cations as usual. It was defined by hydroxo-complexes MgOH⁺ and Al(OH)₄ ⁻, which removed the extreme surplus of OH⁻. The Al(OH)₄ ^- species concentration was, however, almost the same as the total Al concentration at pH > 10. The MgOH⁺ species concentrations were calculated using the geochemical speciation modeling code Visual Minteq 3.1 with the built-in Davies equation for the ion activity product calculations from the default thermodynamic database (Gustafsson, Reference Gustafsson2020). The Visual Minteq simulation results indicated also that all aqueous solutions of the dissolution experiments analysed at day 240 were undersaturated with respect to any secondary mineral phases, including magnesite (MgCO₃) and gibbsite. Equation (1) allowed the logarithmic solubility product to be formulated:

(2) $log{K}_{\mathrm{SP}},{}_{Mg-LDH}=3log\left({\text{MgOH}}^{+}\right)+log\left(\text{Al},{\left(\text{OH}\right)}_4^{-}\right)+log\left({\text{OH}}^{-}\right)+0.5log\left({\text{CO}}_3^{2-}\right)$

where the curved brackets indicate the activity of the respective ion. For the long-term steady state reached in the dissolution experiments after day 240, the dissolution of the LDHs gave almost the same Mg/Al ratios for the solutes and solids, indicating that almost congruent dissolution had occurred. For the pure Mg₃-Al-LDH endmember in this dissolution experiment (Fig. 4b), a logK _sp,_Mg-LDH of −32.2±0.2 could be calculated from the ion activity product (Eq. 2). Even though it was for an uncommon solubility product formulation, this value can be translated using the Visual Minteq code into another uncommon solubility product formulation to compare with the solubility of manasseite found in a previous study (Allada et al., Reference Allada, Peltier, Navrotsky, Casey, Johnson, Berbeco and Sparks2006):

(3) ${\text{Mg}}_{0.74}{\text{Al}}_{0.26}{\left(\text{OH}\right)}_2{\left({\text{CO}}_3\right)}_{0.13}·0.39{\text{H}}_{2}O+{2H}^{+}=0.74{\text{Mg}}^{2+}+0.26{\text{Al}}^{3+}+0.13{\text{CO}}_3^{2-}$

The logarithmic solubility product for this dissolution reaction was (Allada et al., Reference Allada, Peltier, Navrotsky, Casey, Johnson, Berbeco and Sparks2006):

(4) $log{K}^{\ast }{}_{\text{SP}},{}_{\text{Mg}-\text{LDH}}=0.74log\left({\text{Mg}}^{2+}\right)+0.26log\left({\text{Al}}^{3+}\right)+0.13log\left({{\text{CO}}_3}^{2-}\right)+2\text{pH}=9.82.$

Using the same formulation, the present study gave a logK* _SP,_Mg-LDH of 9.45, which corresponded to the uncertainty for the value found by Allada et al. (Reference Allada, Peltier, Navrotsky, Casey, Johnson, Berbeco and Sparks2006). Rozov et al. (Reference Rozov, Berner, Kulik and Diamond2011) calculated a traditional thermodynamic solubility product using the stoichiometric free ion activity product equation:

(5) $log{K}^{*}{}_{\text{SP}},{}_{\text{Mg}-\text{LDH}}=3log\left({\text{Mg}}^{2+}\right)+log\left({\text{Al}}^{3+}\right)+0.5log\left({\text{CO}}_3^{2-}\right)+8log\left({\text{OH}}^{-}\right)=-68.9$

Using this formulation, the present dissolution experiment gave a logK* _SP,_Mg-LDH of −73.4, which was somewhat lower than the value found by Rozov et al. (Reference Rozov, Berner, Kulik and Diamond2011). However, Eq. 5 is quite sensitive to pH, meaning that even a minor uncertainty in the pH drastically increased uncertainty in the calculated logK* _SP,_Mg-LDH. For example, an uncertainty of 0.5 pH units (e.g. between 10.0 and 10.5) would explain the 4 units difference. Corroborating the results against previously published data confirmed the present data as promising to be used for further modeling.

To formulate a binary SSAS equilibrium model, the solubility products of the other pure endmember assumed to form the solid solution first had to be determined using the formulation:

(6) $log{K}_{\text{SP}},{}_{\text{Zn}-\text{LDH}}=3log\left({\text{Zn}}^{+}\right)+log\left(\text{Al},{\left(\text{OH}\right)}_4^{-}\right)+log\left({\text{OH}}^{-}\right)+0.5log\left({\text{CO}}_3^{2-}\right)$

The solubility constant for the dissolution experiment after day 240 (sample SMZ11 in Table 3) calculated using Eq. 6 was logK _sp,_Zn-LDH = −37.7±0.6 and, therefore, more than five units lower than for the pure Mg endmember.

Table 3 Ion concentrations found after 240 days of the aqueous solubility experiments using the LDH materials of various mole fractions of Zn (X _Zn)

Thermodynamic equilibrium in the binary (Mg+Zn)₃-Al-LDH system was described by the equivalence of the chemical potentials of the solid and aqueous phase components, i.e. by the mass action expressions:

(7) ${\left({\text{Zn}}^{+}\right)}^3\left(\text{Al},{{\left(\text{OH}\right)}_4}^{-}\right)\left({\text{OH}}^{-}\right){\left({{\text{CO}}_3}^{2-}\right)}^{0.5}=K_{\text{SP}},{}_{\text{Zn}-\text{LDH}}{X}_{\text{Zn}-\text{LDH}}{\lambda }_{\text{Zn}-\text{LDH}}$

and:

(8) ${\left({\text{Mg}}^{+}\right)}^3\left(\text{Al},{{\left(\text{OH}\right)}_4}^{-}\right)\left({\text{OH}}^{-}\right){\left({{\text{CO}}_3}^{2-}\right)}^{0.5}=K_{\text{SP}},{}_{\text{Mg}-\text{LDH}}\left(1,-,X_{\text{Zn}-\text{LDH}}\right){\lambda }_{\text{Mg}-\text{LDH}}$

The solid-phase activities were given as the products of the appropriate mole fraction X and activity coefficient λ. Dividing Eq. 7 by Eq. 8 and removing the common ions gave the Berthelot–Nernst distribution law:

(9) $X_{\text{Zn}-\text{LDH}}/\left(1,-,X_{\text{Zn}-\text{LDH}}\right)=D\left({\text{Zn}}^{+}\right)/\left({\text{MgOH}}^{+}\right)$

where the Berthelot–Nernst distribution coefficient D is given by:

(10) $D=K_{\text{SP}},{}_{\text{Mg}-\text{LDH}}{\lambda }_{\text{Mg}-\text{LDH}}/K_{\text{SP}},{}_{\text{Zn}-\text{LDH}}{\lambda }_{\text{Zn}-\text{LDH}}$

Equation 9 is the classic expression for a SSAS system at thermodynamic equilibrium. Another expression for SSAS equilibrium can be obtained by simply adding together Eqs. 7 and 8 to give:

(11) ${\Sigma \Pi }_{\text{eq}}=\left(\left({\text{ZnOH}}^{+}\right),+,\left({\text{MgOH}}^{+}\right)\right)\left(\text{Al},{{\left(\text{OH}\right)}_4}^{-}\right)\left({\text{OH}}^{-}\right){\left({{\text{CO}}_3}^{2-}\right)}^{0.5},$

where the term on the left-hand side is the “total solubility product” variable ΣΠ, which is related via the expression on the right-hand side to give the total solubility product constant ΣΠ_eq at thermodynamic equilibrium with the aqueous solute composition (Glynn & Reardon, Reference Glynn and Reardon1990; Lippmann, Reference Lippmann1980). Unlike the endmember solubility product K _SP, which remains constant regardless of the composition of the aqueous phase (at stoichiometric saturation with a given solid solution composition), ΣΠ_eq varies as a function of the solid solution composition in terms of the fraction X _Zn-LDH of the trace metal in the host LDH phase. The ΣΠ_eq constant can be calculated from the measured solute concentrations using Eq. 11. Alternatively, ΣΠ_eq can be calculated if the activity fractions of the substituting cation in the aqueous phase at thermodynamic equilibrium are known. The solute activity fractions for the (Mg,Zn)₃-Al-LDH system were defined as:

(12) ${\chi }_{\text{Zn},\text{aq}}=\left({\text{Zn}}^{+}\right)/\left(\left({\text{Zn}}^{+}\right),+,\left({\text{MgOH}}^{+}\right)\right)$

and:

(13) ${\chi }_{\text{Mg},\text{aq}}=\left({\text{Mg}}^{+}\right)/\left(\left({\text{ZnOH}}^{+}\right),+,\left({\text{Mg}}^{+}\right)\right)$

Substituting these two equations into Eq. 11 and rearranging the equation to express the activities of the solid-phase components in terms of their activity coefficients and mole fractions gives:

(14) $\Sigma \Pi \text{eq}=\frac{1}{\frac{{\chi }_{\text{Zn},\text{aq}}}{K_{\text{SP},\text{Zn}-\text{LDH}}{\lambda }_{\text{Zn}-\text{LDH}}}+\frac{{\chi }_{\text{Mg},\text{aq}}}{K_{\text{sp},\text{Mg}-\text{LDH}}{\lambda }_{\text{Mg}-\text{LDH}}}}$

In principle, Lippmann ΣΠ–X–χ phase diagrams can be constructed using the total solubility product Eqs. 11 and 14 and the appropriate mole and activity fractions (or solute concentrations) as coordinates (Lippmann, Reference Lippmann1980). Analogous to phase diagrams for binary gas or binary liquid systems or for solid/melt systems, a Lippmann phase diagram is a plot of a solidus curve (the total solubility product defined by Eq. 11) and a solutus curve (the isothermal crystallization curve defined by Eq. 14) with the variable ΣΠ as the ordinate. The solidus and solutus curves both form a characteristic loop that is common in phase diagrams involving the (complete) solubility of two phases in each other. The loop became wider as the difference between the solubility products of the endmembers increased. The logK _SP values for the endmembers were five units different, so a considerable loop opening was expected to indicate non-ideal SSAS equilibrium behavior.

A Lippmann diagram differs from a traditional phase diagram in that it uses aqueous activity products and activity fractions rather than element concentrations and mole fractions to define the compositions of all phases in the system. Two different scales were used for the abscissa, an activity-fraction scale χ_Zn,aq relating to the aqueous-phase composition and a mole-fraction scale X _Zn-LDH defining the solid-phase composition. The activity-fraction scale required a degree of ion pairing and/or complexation with other ions in the aqueous solution to be considered. The effect of speciation is important in alkaline solutions of cementitious solidification/stabilization systems (e.g. Kulik & Kersten, Reference Kulik and Kersten2002) in which cations that form complete or almost complete solid-solution series may not have similar aqueous complexing affinities, as in this case for Mg and Zn. A speciation model of ion complexation in the aqueous solution, such as the Visual Minteq code used here, was therefore required to construct Lippmann diagrams from experimental concentration data. The solidus and solutus curves were then plotted and used to predict the solubility of any solid solution at thermodynamic equilibrium. This was done in a similar way to phase diagrams used for binary solid/melt systems but with solutus instead of liquidus curves to give the solid-phase and aqueous-phase compositions for the series of possible thermodynamic SSAS equilibrium states.

For {ZnOH⁺} << {MgOH⁺} at pH > 10, Eq. 14 can be simplified in logarithmic terms to:

(15) ${log\Sigma \Pi }_{\text{eq}}=-{log\chi }_{\text{Zn},\text{aq}}+{log\lambda }_{\text{Zn}-\text{LDH}}-37.7$

As discussed below, the term logλ_Zn-LDH is small and can, therefore, be neglected at first approximation. However, because {ZnOH⁺} << {MgOH⁺} at pH > 10, all the χ_Zn,aq values are <0.01 and a reliable solutus cannot easily be plotted on a linear χ_Zn,aq scale, as is common in Lippmann diagrams (Fig. 5a). When plotting the χ_Zn,aq values on a logarithmic scale, however, all the χ_Zn,aq values for the dissolution experiment data were in fact on a perfectly straight line between the endmember solubilities (Fig. 5b), in particular as the minor experimental scattering of the data vanished on the extended log-log scale. Unfortunately, the conode tie concept as common for phase diagrams does not work here. This is again due to the low Zn concentrations in comparison to those of Mg. The large difference between the solubility products of the endmembers involves a strong preferential partitioning of the less soluble endmember toward the solid phase, which explains why Zn-poor solid solutions are in equilibrium with Mg-rich aqueous solutions. Therefore, Zn does not play a role in equations where both cation concentrations are summed (solidus) but only in equations where both cations appear as a quotient (solutus).

Fig. 5 Lippmann diagram for the (Zn + Mg)₃Al-LDH system with total solubility product data for the dissolution experiments after 240 days (dots) using a a linear x-axis scale and b a logarithmic x-axis scale fitted using model solidus and solutus curves including the (dashed line) minimum stoichiometric saturation line

The description of the thermodynamic properties of a non-ideal solid solution and consequently of the solid-phase activity coefficients requires an understanding of the excess free energy of mixing of the solid solution, i.e. of the difference between the free energy of mixing for the actual system and an ideal solid solution, as shown below (Ganguly, Reference Ganguly and Geiger2001):

(16) $G^E=G^M+G^{M,\text{id}}$

The Gibbs free energy of mixing G ^M of a solid solution can be considered as the difference between the actual free energy of the solid solution and the free energy of a compositionally equivalent mechanical mixture of the endmember components. For an ideal binary solid solution, the free energy of mixing of endmember components i and j with the composition X _i,j is given by:

(17) $G^{M,\text{id}}=RT\left(X_i,,ln,X_i,+,X_j,,ln,X_j\right)$

No experimental data are available, however, for the Gibbs free energies of the mixing functions for the composition of the (Mg+Zn)₃-Al-LDH or, therefore, for the solid-phase activity coefficients for the system. Empirical relationships for the composition-dependence of the solid-phase activity coefficients obtained by Gibbs–Duhem integration using the Guggenheim subregular model for the excess free energy of mixing in a binary solid solution has been used successfully to fit solubility data for various binary SSAS systems (Glynn & Reardon, Reference Glynn and Reardon1990), as shown below:

(18) $G^E=X_i{X}_{j}RT\left({\alpha }_0,+,{\alpha }_1,\left(X_i,-,X_j\right)\right)$

The two dimensionless empirical coefficients α₀ and α₁ for the Guggenheim expansion equation allowed the expressions for the solid phase activity coefficients in the binary (Mg+Zn)₃-Al-LDH system shown below to be derived:

(19) ${\text{ln}\lambda }_{\text{Mg}-\text{LDH}}={X_{\text{Zn}-\text{LDH}}}^2\left({\alpha }_0,-,{\alpha }_1,\left(3,-,4,X_{\text{Zn}-\text{LDH}}\right)\right)$

(20) ${\text{ln}\lambda }_{\text{Zn}-\text{LDH}}={X_{\text{Mg}-\text{LDH}}}^2\left({\alpha }_0,-,{\alpha }_1,\left(4,X_{\text{Mg}-\text{LDH}},-,1\right)\right)$

Structural information such as mixing gaps in the binary (Mg+Zn)₃-Al-LDH solid-solution behavior can be used to estimate α₀ and α₁ more accurately than the commonly used ideal solid solution assumption (i.e. G ^M = 0). The analysis results of Vegard’s law presented above, however, did not indicate the presence of a miscibility gap, so a complete solid-solution series in the SSAS system was expected, i.e. the system was expected not to deviate far from the ideal system. This meant that (ignoring any difference between activity and concentration in the aqueous phase, i.e. aqueous activity coefficient γ_i,j = 1 at infinite dilution) the distribution coefficients could be defined as:

(21) $D=\frac{X_{\text{Zn}-\text{LDH}}/X_{\text{Mg}-\text{LDH}}}{\left({\text{Zn}}^{-}\right)/\left({\text{Mg}}^{-}\right)}=\frac{K_{\text{SP},\text{Mg}-\text{LDH}}{\lambda }_{\text{Mg}-\text{LDH}}}{K_{\text{SP},\text{Zn}-\text{LDH}}{\lambda }_{\text{Zn}-\text{LDH}}}$

Substitution of the definitions of the solid-phase activity coefficients λ_Mg-LDH and λ_Zn-LDH into the Guggenheim subregular model gives the expression (Glynn & Reardon, Reference Glynn and Reardon1990):

(22) $D=\frac{K_{\text{SP},\text{Mg}-\text{LDH}}}{K_{\text{SP},\text{Zn}-\text{LDH}}}exp\left(\left(2,X_{\text{Zn}-\text{LDH}},-,1\right),a_0,+,\left(6,\left(X_{\text{Zn}-\text{LDH}}^2,-,X_{\text{Zn}-\text{LDH}}\right),+,1\right),a_1\right)$

Given the two distribution coefficients at a specified solid-solution composition, the MBSSAS code uses expressions equivalent to this equation and a Newton–Raphson procedure to calculate a ₀ and a ₁ (Glynn, Reference Glynn1991). The Lippmann solidus and solutus lines fitted to the present data using the MBSSAS code for a ₀ = 0.5 and a ₁ = −0.5 are shown in Fig. 5. Most of the solutus data were on a steep part of the solutus line at very low Zn concentrations. This part was, therefore, zoomed in using a logarithmic scale for the x-axis as mentioned above (Fig. 5b). The logarithmic scale solubility diagram indicated very well the excellent fit of the Lippmann equation to the experimentally acquired data according to Eq. 15, and may, therefore, be used readily to predict Zn mobility.

Having calculated a ₀ and a ₁ for the binary (Mg+Zn)₃-Al-LDH solid-solution, the MBSSAS calculates (as the last step) the solid activity coefficients at infinite dilution of the respective Zn-LDH and Mg-LDH endmember components using the equations below:

(23) ${\lambda }_{\text{Mg},\text{XZn}-\text{LDH}=1}=exp\left(a_0,+,a_1\right)=1$

(24) ${\lambda }_{\text{Zn},\text{XZn}=0}=exp\left(a_0,-,a_1\right)=2.7$

These “Henry’s law activity coefficients” support the almost ideal solid-solution aqueous-solution equilibrium behavior of the binary (Mg+Zn)₃Al-LDH system. The straight minimum stoichiometric saturation curve relating the stoichiometric-saturation solubilities to the solid-solution composition is also shown in Fig. 5a (Glynn, Reference Glynn1991), and the equation is:

(25) ${\Sigma \Pi }_{\text{mss}}=\frac{K_{\text{SP},\text{Mg}-\text{LDH}}^x{K}_{\text{SP},\text{Zn}-\text{LDH}}^{1-x}{\lambda }_{\text{Mg}}^x{\lambda }_{\text{Zn}}^{1-x}}{X_{\text{Zn}}^{1-x}{X}_{\text{Mg}}^x}$

The “stoichiometric saturation” concept has been introduced for kinetic dissolution experiments, in which relatively insoluble solid solutions have been found to react as physical mixtures of the endmember single-component phases over short reaction times (hours) (Plummer & Busenberg, Reference Plummer and Busenberg1987). Equation 25 gives a series of possible χ_Zn,aq/ΣΠ pairs that an aqueous solution may reach theoretically at stoichiometric saturation with respect to a given solid composition with a fixed X _Zn-LDH. This is commonly called the “equal-G curve” (Gamsjäger, Reference Gamsjäger1985) but it is clearly not relevant for the present data. At longer equilibration times typical of environmental settings, a LDH solid solution reacts more like a multi-component solid and does not maintain a constant composition, as shown in the dissolution experiments. However, the minimum saturation curve may eventually become important in industrial settings, a point described in more detail by Glynn et al. (Reference Glynn, Reardon, Plummer and Busenberg1990).