At first, it has to be mentioned that the mineral components contain at least two types of surface groups. The first type includes permanently charged functional groups created by ionic substitution within the crystal structure, namely, by isomorphic substitution of, e.g., Al^{3+} for Si^{4+}, creating a permanent negative charge on the mineral surface, which is compensated externally by cations. These sites are denoted as layer-sites (with symbol =X^{-}). The second type, so called edge-sites, is formed on the edges of the surface structure (with symbol ≡SOH). They have a pH-dependent charge which arises due to the “adsorption” of H+ ions (the protonation proceeds approx. at $\mathrm{p}\mathrm{H}<7:\equiv \mathrm{S}\mathrm{O}{\mathrm{H}}^{0}\to \equiv \mathrm{S}\mathrm{O}{\mathrm{H}}_{{2}^{+}}$) or “desorption” of H^{+} ions (the deprotonation proceeds approx. at $\mathrm{p}\mathrm{H}>7:\equiv \mathrm{S}\mathrm{O}{\mathrm{H}}^{0}\to \equiv \mathrm{S}{\mathrm{O}}^{-}$). Again, the charge is compensated by anions in a case of ≡SO_{2}^{+}, or by cations in a case of ≡SO^{-}. In principle, these surface anionic groups can enter into the coordination sphere of the adsorbed metal ions. The mechanism of this reaction is so called surface complexation in consequence of which the sorption of many species proceeds. At least, there are three basic surface complexation models at hand [2,13,15,17], namely, non-electrostatic chemical equilibrium model (CEM), constant capacitance model (CCM), and double layer diffusion model (DLM).

In principle, a titration curve of mineralogical component type of montmorillonite, kaolinite, magnetite, etc., can be described by means of two protonation, (1) and (2), and one ion exchange, (3), reactions. As it was mentioned above, the first two reactions proceed on the edge-sites and the third one on the layer-sites:

$$\equiv \text{S}{\text{O}}^{-}+{\text{H}}^{+}\leftrightarrow \equiv \text{S}\text{O}{\text{H}}^{0}$$(1)$$\equiv \text{S}\text{O}{\text{H}}^{0}+{\text{H}}^{\mathit{+}}\leftrightarrow \equiv \text{S}\text{O}{{\text{H}}_{2}}^{\mathit{+}}$$(2)$$\equiv \text{X}\text{N}\text{a}+{\text{H}}^{+}\leftrightarrow \equiv \text{X}\text{H}+\text{N}{\text{a}}^{+}$$(3)For these reactions, the equations of equilibrium constants (*K*_{1}, K_{2} or K_{3}) can be written, e. g., using CEM + IExM (chemical equilibrium model, i.e., non-electrostatic model, and ion exchange model), as follows:

$${K}_{1}=\left[\text{S}\text{O}{\text{H}}^{0}\right]/\left(\left[\text{S}{\text{O}}^{-}\left]\cdot \right[{\text{H}}^{+}\right]\right)$$(4)$${K}_{2}=[{{\text{SOH}}_{2}}^{+}]/([{\text{SOH}}^{0}]\cdot [{\text{H}}^{+}]$$(5)$${K}_{3}=\left(\left[\text{X}\text{H}\right]\cdot \left[\text{N}{\text{a}}^{+}\right]\right)/\left(\left[\text{X}\text{N}\text{a}\right]\cdot \left[{\text{H}}^{+}\right]\right)$$(6)Three balance equations, (7) – (9), are in need of the complete description of this system:

$$\sum \mathrm{S}\mathrm{O}\mathrm{H}=\left[\mathrm{S}\mathrm{O}{{\mathrm{H}}_{2}}^{+}\left]+\right[\mathrm{S}\mathrm{O}{\mathrm{H}}^{0}\right]+\left[\mathrm{S}{\mathrm{O}}^{-}\right]$$(7)$$\sum \mathrm{X}=\left[\mathrm{X}\mathrm{N}\mathrm{a}\right]+\left[\mathrm{X}\mathrm{H}\right]\approx \left[{\mathrm{X}}^{-}\right]+\left[\mathrm{X}\mathrm{H}\right]$$(8)$$\begin{array}{l}\sum \mathrm{N}\mathrm{a}=m\cdot \left[\mathrm{X}\mathrm{N}{\mathrm{a}}_{0}\right]+{V}_{0}\cdot \left[\mathrm{N}{\mathrm{a}}_{0}\right]+\\ +{v}_{0H}.{c}_{0H}=m\cdot \left[\mathrm{X}\mathrm{N}\mathrm{a}\right]+{V}_{\mathrm{\Sigma}}\cdot \left[\mathrm{N}{\mathrm{a}}^{+}\right]\end{array}$$(9)Where: *V*_{Σ} (=V_{0} + v_{H} + v_{OH}) is the total volume and *V*_{0} is the starting volume of the aqueous phase [dm^{3}], v_{OH} and v_{H} are the consumptions [dm^{3}] of sodium hydroxide, NaOH, and hydrochloric acid, HCl, respectively, in the course of titration; *m* is the mass of solid phase [kg], and c_{OH} is the concentration of NaOH solution [mol dm^{-3}] used in the titration procedure; [XNa_{0}] and [Na_{0}] are the starting concentrations of sodium in solid [mol kg^{-1}] and in liquid [mol dm^{-3}] phases, respectively.

The modelling of the titration system, characterized by reactions (1) – (3), is described in detail elsewhere [13]. In short, the goal of the modelling is to construct the relations (equations) for the calculation of the surface charge [mol kg^{-1}], *Q*_{cal} (= *Q*_{ES} + *Q*_{LS}), consisting of the charge of the edge-sites, *Q*_{ES} (= [SOH_{2}^{+}] - [SO^{-}]), and of the charge of the layer-sites, *Q*_{LS} (= - [X^{-}]). Then, it is necessary to be derived the function (*Q*_{cal})_{i} = (*Q*_{ES})_{i} + (*Q*_{LS})_{i} = f([H^{+}]_{i}) (see Eqs (10) and (11)), applicable for the fitting of experimental data evaluated as (*Q*_{exp})_{i} = f([H^{+}]_{i}), *i* = 1,2,3,…, np, where *np* is the number of experimental points of the given titration curve (see Eq. (12)), and for the determination of the values of *K*_{1}, *K*_{2}, *K*_{3}, ΣSOH and ΣX.

$$\begin{array}{l}({Q}_{\mathrm{E}\mathrm{S}}{)}_{i}=\sum \left\{\mathrm{S}\mathrm{O}\mathrm{H}\phantom{\rule{thinmathspace}{0ex}}\cdot \phantom{\rule{thinmathspace}{0ex}}\left({K}_{1}\phantom{\rule{thinmathspace}{0ex}}\cdot \phantom{\rule{thinmathspace}{0ex}}{K}_{2}\phantom{\rule{thinmathspace}{0ex}}\cdot \phantom{\rule{thinmathspace}{0ex}}{\left[{\mathrm{H}}^{+}\right]}_{i}^{2}+1\right)\right\}/\left(1+{K}_{1}\cdot \left[{\mathrm{H}}^{+}\right]+\right.\\ \left.+{K}_{1}\cdot {K}_{2}\cdot {\left[{\mathrm{H}}^{+}\right]}_{i}^{2}\right)\phantom{\rule{1em}{0ex}}\left[\text{mol}\phantom{\rule{thinmathspace}{0ex}}\text{k}{\text{g}}^{-1}\right]\end{array}$$(10)$$({Q}_{\mathrm{L}\mathrm{S}}{)}_{i}=\sum \left(\mathrm{X}\cdot \left[\mathrm{N}{\mathrm{a}}^{+}\right]\right)/\left(\left[\mathrm{N}{\mathrm{a}}_{i}+{K}_{3}\cdot \right[{\mathrm{H}}^{+}{]}_{i}\right)\phantom{\rule{1em}{0ex}}\left[\text{mol}\phantom{\rule{thinmathspace}{0ex}}\text{k}{\text{g}}^{-1}\right]$$(11)$${\left({Q}_{\mathrm{exp}}\right)}_{i}=\left({V}_{\mathrm{\Sigma}}/m\right)\cdot \left(\left[{C}_{\mathrm{a}}\left]-\right[{C}_{\mathrm{b}}\left]+\right[\mathrm{O}{\mathrm{H}}^{-}\left]-\right[{\mathrm{H}}^{+}\right]\right)\left[\mathrm{m}\mathrm{o}\mathrm{l}\phantom{\rule{thinmathspace}{0ex}}\mathrm{k}{\mathrm{g}}^{1}\right]$$(12)Where *C*_{a} = (*v*_{H} · *c*_{H})/*V*_{Σ} and *C*_{b} = (*v*_{OH} · *c*_{OH})/ *V*_{Σ}, *c*_{H} [mol dm^{-3}] is the concentration of HCl solution used in the titration procedure.

In the case of GC-approach, the acid-base experimental titration data are fitted with (*Q*_{cal})_{i}, in our case by the Newton-Raphson multidimensional non-linear regression method, and the quantity *WSOS/DF* (weighted sum of squares of differences divided by degrees of freedom) is used as a criterion of the goodness-of-fit (the agreement of calculated and experimental data) – if 0.1<WSOS/DF<20, then there is a acceptable agreement between the experimental and calculated data [14]. Its calculation is based on the *Χ*^{2} – test, calculated according to Eq. (13), and then the *WSOS/DF* is obtained by means of Eq. (14):

$${\chi}^{2}=\sum \left\{{\left(SSx\right)}_{i}/\left({\left.{s}_{q}\right)}_{i}^{2}\right)\right\},\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{i}=1,2,3,\dots ..,{n}_{p}$$(13)$$WSOS/DF=\left({\chi}^{2}/{n}_{d}\right)\phantom{\rule{1em}{0ex}}{n}_{d}={n}_{p}-n$$(14)Where, (SSχ)_{i} is the *i*-th square of the deviation of i-the experimental value from the corresponding calculated value, (*s*_{q})_{i} is the estimate of standard deviation (uncertainty) of the *i*-th experimental point, *n*_{p} is the number of experimental points, n_{d} is the number of degrees of freedom, and *n* is the number of model parameters sought during the regression procedure, i.e., *n* = 5 (*K*_{1}, *K*_{2}, *K*_{3},ΣSOH and ΣX) or, if only edge-sites are present, *n* = 3 (*K*_{1}, *K*_{2}, ΣSOH).

If it deals with the mineral assemblage consisting of *J* mineral components, each component takes part individually in the reactions (1) – (3), as it is supposed in a case of CA approach modelling. Of course, the above mentioned model parameters of the *j*-th mineral component (*K*_{1j}, *K*_{2j}, *K*_{3j}, ΣSOH_{j} and ΣX_{j}), the values of which are inserted into the Eqs (10) and (11), have to be known, including the percentage by weight of the *j*-th mineral component (*mP*_{j}, *j* = 1,2,3, ..., *J*) in an assemblage studied. Then, the calculation algorithm of the corresponding CA-code proceeds according to Eq. (15) as follows:

–

At first, for *i* = 1, the surface charges of individual mineral components, (*Q*_{ES})_{ij} (Eq. (10)), and (*Q*_{LS})_{ij} (Eq. (11)), for *j* = 1,2,..., J, are calculated for the given value of concentration [H^{+}]_{i} or – log(pH)_{i}, and, the value of ((*Q*_{cal})_{total})_{1} is obtained.

–

This is repeated in an iteration loop for *i* = 2, 3, ..., *n*_{p}.

–

In the end, altogether there are *n*_{p} values of ((*Q*_{cal})_{total})_{i} = f[H^{+}]_{i}, that is, *n*_{p} points modelling the titration curve of the given mineral assemblage.

$${\left({\left({Q}_{ca1}\right)}_{total}\right)}_{i}={\sum}_{i}{\sum}_{j}\left\{\left(m{P}_{j}/100\right)\cdot \left[{({Q}_{\mathrm{E}\mathrm{S}})}_{ij}+{\left({Q}_{\mathrm{L}\mathrm{S}}\right)}_{ij}\right]\right\}\phantom{\rule{thinmathspace}{0ex}}i=1,2,3,\dots .,{n}_{p}\phantom{\rule{thinmathspace}{0ex}};\phantom{\rule{thinmathspace}{0ex}}j=1,2,3,\dots ,J$$(15)As the goal of the study is the comparison of the calculated (by GC- or CA-code) and experimental titration curves, i.e. of the calculated and experimental surface charges, in such a case, the experimental values [H^{+}]_{i} have to be inserted into Eqs (10) and (11) or in Eq. (15). Also in this case, the quantity *WSOS/DF* can be used as a criterion of the agreement of the modelled and the experimental titration curve.

As it was mentioned above, the CA-approach results, ((*Q*_{cal})_{total})_{i}, can be used as input data into the GC-approach code, and the obtained overall values of *K*_{1}, *K*_{2}, *K*_{3}, ΣSOH and ΣX can be compared with the values of the same parameters resulting from the evaluation of experimental data, (*Q*_{exp})_{i} (by GC-approach code). If the acceptable agreement exists, this implies that the parameters of mineral assemblage can be obtained on the basis of the knowledge of *mP*_{j} and *K*_{1j}, *K*_{2j}, *K*_{3j}, ΣSOH. and ΣX_{j}, values, without experimental determination of the corresponding titration curve.

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