# Unravelling the cleavage-rate relationship from both the experimental and theoretical standpoint: The instance of fluorite dissolution

David Dell'Angelo, Ana Jurković, Tin Klačić, Yann Foucaud, Michael Badawi, Adlane Sayede, Tajana Begović

► To cite this version:

David Dell'Angelo, Ana Jurković, Tin Klačić, Yann Foucaud, Michael Badawi, et al. Unravelling the cleavage-rate relationship from both the experimental and theoretical standpoint: The instance of fluorite dissolution. Journal of Colloid and Interface Science, 2025, 684, pp.844-855. 10.1016/j.jcis.2024.12.242. hal-04998854

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Regular Article

# Unravelling the cleavage-rate relationship from both the experimental and theoretical standpoint: The instance of fluorite dissolution

David Dell'Angelo<sup>a, 1*</sup>, Ana Jurković<sup>b, 1</sup>, Tin Klačić<sup>b, 1*</sup>, Yann Foucaud<sup>c, 1</sup>, Michael Badawi<sup>a, 1</sup>, Adlane Sayede<sup>d, 1</sup>, Tajana Begović<sup>b, 1</sup>

<figure><img src="image_3.png" /></figure>

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## GRAPHICAL ABSTRACT

<figure><img src="image_4.png" /></figure>

## ARTICLE INFO

**Keywords:**

Fluorite dissolution kinetics

Atomic force microscopy (AFM)

Surface wettability

Free energy barrier

Interfacial defect

## ABSTRACT

The phenomenon of solid dissolution into a solution constitutes a fundamental aspect in both natural and industrial contexts. Nevertheless, its intricate nature at the microscale poses a significant challenge for precise quantitative characterization at a foundational level. In this work, the influence across three specific cleavage planes, namely (100), (111), and (110) on the dissolution kinetics of fluorite in aqueous environments was examined from both experimental and theoretical standpoints. For the very first time, the surface potential of fluorite planes during dissolution was measured by means of a fluorite single-crystal electrode. Experimental results indicate that the dissolution of fluorite leads to a marked increase in surface roughness as well as an augmentation in the surface area of all analyzed surfaces. The most significant alteration in roughness is observed on the (111) plane, whereas the most substantial increase in surface area occurs on the (110) plane. In comparison to the (100) crystallographic plane, which demonstrates the slowest dissolution kinetics, the (111) and (110) planes display dissolution at a comparatively expedited rate. Theoretical simulations corroborate this trend, concurrently facilitating an effective examination of the system's free-energy landscape to analyze the dynamics and rates associated with the attachment and detachment of ions to the fluorite surface. Notably, the presence of interfacial defects has the potential to influence the free energy landscape, thereby altering the transition of ions

<footnote><sup>a</sup> Université de Lorraine, CNRS, Laboratoire Lorrain de Chimie Moléculaire, Metz, F-57000, Lorraine, France</footnote>

<footnote><sup>b</sup> University of Zagreb, Faculty of Science, Department of Chemistry, Zagreb, HR-10000, Croatia</footnote>

<footnote><sup>c</sup> Université de Lorraine, CNRS, GeoRessources, Nancy, F-54000, Lorraine, France</footnote>

<footnote><sup>d</sup> UCCS, CNRS, Université d'Artois, Faculté des Sciences Jean Perrin, Lens, 62307, Hauts-de-France, France</footnote>

<footnote>* Corresponding authors.
E-mail addresses: david.dell-angelo@univ-lorraine.fr (D. Dell'Angelo), tklacic@chem.pmf.hr (T. Klačić).</footnote>

<footnote n="1">¹ Current affiliation: Centre for Advanced Laser Techniques, Institute of Physics, Bijenicka 46, 10000 Zagreb, Croatia.</footnote>
into the bulk solution. Ultimately, the interplay of correlations and discrepancies between experimental findings and theoretical predictions is critically examined.

# 1. Introduction

Fluorite (CaF₂) is a calcium fluoride polymorph and the most important source of fluoride ions (F⁻) for industrial purposes. [1–3] Yet, high concentrations of fluoride in wastewater can significantly impact the environment and public health. [4] Consequently, there is an urgent necessity to innovate precipitation techniques that effectively treat fluoride ions from wastewater with high fluoride content while simultaneously enabling the recovery of these ions as valuable resources. [5–7] Through the application of these methodologies, fluoride wastewater may be converted into valuable resources such as fluorite, thus not only alleviating ecological repercussions but also yielding economic advantages through the recovery of beneficial materials. The extracted fluorite may subsequently be utilized for the production of many organic and inorganic chemicals, primarily hydrofluoric acid and aluminum. [8] By virtue of its properties, it is used for the production of optical elements in laser systems, protective layers on other materials, among others. [9] Fluorite can be liberated from non-valuable gangue ores by means of beneficiation process called flotation. [10–12] During this process, the hydrophobic ore attaches to the air bubbles injected into the flotation cell and migrates towards the surface.

To enhance the efficiency of both precipitation and liberation processes, comprehension of the microscopic interfacial characteristics at the surface and the kinetics of fluorite dissolution is imperative, wherein the latter mechanism primarily manifests in two distinct phases. Initially, ions are liberated from the crystal surface as a consequence of surface reactions. This initial phase encompasses the hydration of the surface and the diffusion of the reactive chemical species across the surface, followed by their adsorption, subsequent reactions at the surface, and the eventual desorption of the reaction products alongside their hydration. In this scenario, the surface charge is determined by the interplay of fluoride and calcium desorption. [13] In the subsequent phase, the hydrated dissociated ions migrate through the electrical interface layer (EIL) [14] toward the bulk solution. Generally, the rate-limiting step is represented by the second phase, which consequently dictates the overall rate of crystal dissolution. [15] Nonetheless, the dissolution rate of fluorite is profoundly influenced by the characteristics of the exposed surface, [16] with the impact of surface structure on this dissolution process presenting significant challenges in current research endeavors. [17–19]

From a crystallographic perspective, the fluorite-type crystal lattice is characterized by fluoride ions occupying the eight tetrahedral interstitial sites formed by calcium ions (Ca²⁺), the latter located at the lattice points of the face centre cubic (FCC) unit cell. The prevailing literature predominantly focuses on the thermodynamically stable (111) cleavage plane. [20–22] Within this structure, the repetition of the F-Ca-F pattern perpendicular to the surface results in the annulment of the dipole moment in the direction perpendicular to the (111) plane, yielding facile modulation of the crystal between F-F layers. Consequently, dissolution is expedited in the (110) plane where Ca²⁺ and F⁻ reside in the same layer, while the structural similarities between the (100) and (111) planes lead to a reduced dissolution rate due to the screening effect caused by ions of opposing charges from the adjacent layer, which hinders both dissociation and solvation mechanisms. [16] Although fluorite is characterized by its limited solubility in aqueous environments, [23,24] its dissolution process results in an enhancement of surface roughness attributable to modifications in crystal morphology. In particular, (100) and (111) planes demonstrate the formation of triangular pits and steps throughout the dissolution process, with a predominance of the former over the latter. [21,16] The increase of roughness and the exposure of various crystallographic planes significantly

impact additional surface properties, including but not limited to solid surface wettability, surface energy, and surface potential. [25,26,17]

Furthermore, fluorite dissolution kinetics is affected by the pH of the solution. [27] For instance, very recently it was demonstrated that high pH and the presence of competing ions like HCO₃⁻ and OH⁻ can promote fluoride desorption, further increasing its concentration in groundwater. [28] Although several research groups have observed that an increase in the concentration of H⁺ ions promotes ion dissolution, [29,21,30,22] the role of H⁺ ions in the dissolution process of fluorite is still not completely clear. Recently, one of the authors (Klačić) has proposed a dissolution mechanism involving H⁺ ions and explain the effect of pH on the dissolution rate. [22] According to this mechanism, H⁺ ions bind to the F⁻ ion located on the surface and form HF, which then diffuses through the solution and eventually dissociates as H⁺ and F⁻ ions. This results in 0.33 order of the reaction with respect to H⁺ ions, which is in good accordance with the empirically obtained value of 0.37 as well as the value reported by Cama et al. [21] The fact that H⁺ ions do not appear in the overall reaction of dissolution, indicates that the dissolution mechanism is complex and that oxonium ions could play a catalytic role.

In this work, we aim at examining the dissolution rate of the fluorite surface across three specific cleavage planes, namely (100), (111), and (110), in order to enhance comprehension of the dissolution mechanism concerning particular fluorite crystal orientations. To achieve this objective, experiments primarily focused on the kinetics of dissolution over a period of 84 days to investigate the liberation of calcium and fluoride ions into the aqueous solution. For the very first time, the surface potential of fluorite during dissolution was measured by means of a fluorite single-crystal electrode. To get better insight about the process upon the fluorite surface during dissolution, the influence of calcium and fluoride ions on surface potential was measured. Changes on the fluorite planes during dissolution were also monitored by the optical digital microscopy (ODM), the atomic force microscopy (AFM), and by measuring the contact angle before and after dissolution. Based on the experimental findings, the impact of the formation of an electrical interfacial layer, the accumulation of ions, and the diffusion of ions from this interfacial layer on dissolution kinetics and equilibrium was examined. Concurrent computational simulations were employed to elucidate those nanoscale phenomena that were beyond the reach of the experimental methodologies. These simulations are founded on a quantum mechanical framework of interatomic forces, thereby granting us the ability to explore the atomistic complexities of chemical processes. Moreover, this approach facilitates the identification of reaction mechanisms and critical factors that affect reaction kinetics. Numerous methodologies have been put forth to calculate free-energy profiles of reactions, utilizing either Monte Carlo or molecular dynamics. A range of free-energy techniques have been proposed, such as umbrella sampling, [31] thermodynamic integration, [32–34] metadynamics, [35] adiabatic molecular dynamics, [36] transition path sampling, [37] and transition interface sampling. [38] This study utilizes periodic Density Functional Theory (DFT) simulations to gain key insights into the dissolution mechanism, including the interplay between dissolution and the orientation of a single formula unit situated within the slab, as well as the influence of cleavage on the hydration of interfacial ions. Utilizing this analytical framework, the free energy landscapes along with the dissolution rates pertaining to the fluorite surface across the three distinct cleavage planes were meticulously computed.

The experimental and computational methodologies are presented in the forthcoming section, alongside the specifics employed to ascertain the properties. Section 3 provides the results and the associated analysis and discussion. Concluding remarks are delineated in Section 4.
<caption>**Table 1**</caption>

The roughness average ($R_a$) and the root mean square (RMS) roughness ($R_q$) values of the (111), (110), and (100) single crystal surface planes of fluorite are reported both before and after the dissolution process (The reader may refer to figures S2-S3 for a pictorial representation). The surface areas preceding dissolution (A) and subsequent to dissolution (S) are listed, alongside the Wenzel factor, denoted as *r*.

<table><thead><tr><th colspan="4">Before dissolution</th><th colspan="4">After dissolution</th></tr><tr><th>Plane</th><th><i>R</i><sub>a</sub> / nm</th><th><i>R</i><sub>q</sub> / nm</th><th><i>A</i> / µm<sup>2</sup></th><th><i>R</i><sub>a</sub> / nm</th><th><i>R</i><sub>q</sub> / nm</th><th><i>S</i> / µm<sup>2</sup></th><th><i>r</i></th></tr></thead><tbody><tr><td>(100)</td><td>0.28 ± 0.01</td><td>0.45 ± 0.03</td><td>100</td><td>12.25 ± 3.83</td><td>26.4 ± 6.3</td><td>102 ± 0.63</td><td>1.02 ± 0.01</td></tr><tr><td>(110)</td><td>0.17 ± 0.02</td><td>0.24 ± 0.03</td><td>100</td><td>36.5 ± 5.3</td><td>53.0 ± 10.3</td><td>120 ± 0.86</td><td>1.20 ± 0.01</td></tr><tr><td>(111)</td><td>0.21 ± 0.02</td><td>0.41 ± 0.14</td><td>100</td><td>56.7 ± 19.6</td><td>84.4 ± 28.8</td><td>106 ± 1.98</td><td>1.06 ± 0.02</td></tr></tbody></table>

## 2. Material and methods

**Material.** Fluorite single crystals (one-side polished, 10 mm × 10 mm × 0.5 mm) of (100), (110) and (111) crystallographic planes were purchased from SurfaceNet. Stock solutions of sodium fluoride (NaF) of BioUltra from Sigma-Aldrich and calcium chloride dihydrate (CaCl₂ × 2 H₂O) of BioXtra from Sigma Aldrich were utilized for fluoride ion-selective electrode (F-ISE; Orion 9409BN, Thermo Scientific) and calcium ion-selective electrode (Ca-ISE; 6.0510.100, Metrohm) calibration, in addition to standard buffers (pH = 4.01; pH = 7.00; pH = 9.21; and pH = 10.01 at 298 K; Hamilton) utilized for the calibration of the combined pH electrode (6.0234.100, Metrohm) prior to each measurement. The disodium salt of ethylenediaminetetraacetic acid (EDTA, Fluka) was employed for the standardization of CaCl₂ solutions. For the purpose of modulating the pH levels of the aqueous solutions, hydrochloric acid (HCl) obtained from Sigma Aldrich and sodium hydroxide (NaOH) sourced from Merck were utilized. The water employed in all experimental procedures was meticulously prepared utilizing a Millipore Milli-Q Advantage A10 water purification system, ensuring a conductivity measurement that did not exceed 0.055 µS cm⁻¹.

**ODM and AFM measurements.** Fluorite surfaces corresponding to orientations (100), (110), and (111) were meticulously examined using optical digital microscope (Bruker) and MultiMode 8 (Bruker) atomic force microscope both prior to and following dissolution in order to assess the impact of dissolution on surface topography and roughness. Prior to the scanning procedure, the surface of each individual crystal underwent a cleaning process entailing rinsing with deionized water, followed by drying with ultrapure argon (Messer). The ODM and AFM measurements were conducted under ambient laboratory conditions. ODM images were acquired utilizing an optical digital microscope integrated to AFM and equipped with a 10 × objective (Nikon) and a CCD colour video camera (Sony), employing NanoScope 9.7 software. AFM images were acquired in tapping mode utilizing NCHV-A silicon probes sourced from Bruker, characterized by a nominal spring constant of 40 N/m, a nominal resonant frequency of 320 kHz, and a nominal tip radius of 8 nm. A surface area of 10 µm² for each crystallographic plane was scanned at five distinct locations. The resultant images were subjected to analysis using NanoScope Analysis 2.0 software. Topography served as a qualitative metric for elucidating morphological alterations on the surfaces of all three evaluated fluorite planes. The roughness average, denoted as $R_a$, and the root mean square roughness, referred to as $R_q$, were employed as quantitative metrics to quantify the deviations in height from an ideal planar configuration. The recorded values represent the mean of five discrete surface areas, presented in Table 1 alongside the surface area measurements obtained before, denoted as $A$, and after dissolution, indicated as $S$, in addition to the computed Wenzel roughness coefficient, $r$, which is defined as the ratio $S/A$.

**Contact angle.** The contact angle between deionized water and each fluorite crystallographic plane was quantified under ambient conditions utilizing an optical tensiometer, specifically the Attension Theta Lite (Biolin Scientific), employing the sessile drop methodology. Prior to the

measurements, a calibration of the instrument was performed using a tungsten carbide calibration ball provided with the apparatus. The calibration ball, characterized by a radius of 4 mm ± 1 µm, along with its magnetic holder, was positioned on the sample stage, and the image focus was adjusted until the apex of the intensity gradient, which corresponds to the perimeter of the ball, was maximized and rendered as distinct as feasible. The contrast was optimized to ensure the highest quality of the resultant image. The calibration process and subsequent data analysis were conducted utilizing the software OneAttension (version 4.1.3). A total of five measurements were executed on each fluorite crystallographic plane, namely (100), (110), and (111), both prior to and following dissolution, employing a 5 µL droplet of freshly deionized water. The values reported in Table 2 represent the average values of the contact angles measured on the right and left sides, along with the corresponding standard error.

**Dissolution kinetics.** The dissolution kinetics of three crystallographic planes of fluorite (100), (110) and (111) as a function of time was studied by measuring the concentration of H⁺, Ca²⁺ and F⁻ ions in the solution using an ISE. Kinetic measurements were conducted with F-ISE, Ca-ISE and combined pH electrode. Fluorite single crystals were fixed in poly(methyl methacrylate) (PMMA) holder in a way that only the investigated surface plane was exposed to the aqueous electrolyte solution. The surface area of each crystal plane exposed to the measuring solution was 1 cm². An example of PMMA holder with fixed fluorite crystal is shown in Figure S1. Prior and after the dissolution experiments, the surface of each crystal was rinsed with deionised water to remove any contaminants. Holders with fluorite single crystal were placed in glass bottles filled with 50 mL sodium chloride solution ($c = 1 \times 10^{-3}$ mol dm⁻³), which served as an ionic strength adjuster. Initial pH values of the solution were set at pH = 5.0 with HCl ($c = 0.1$ mol dm⁻³). The measuring systems were thermostated at 298 K during measurements. The electrode potentials of Ca-ISE, F-ISE and pH electrode with respect to the reference Ag/AgCl electrode were measured with Metrohm 913 pH-meter during 84 days of fluorite dissolution. Experiments were conducted without agitation of the solution, thereby allowing dissolution to be governed solely by the diffusion of ions from the interfacial layer to the bulk solution. Time dependence of Ca²⁺ and F⁻ concentrations of the solution during 84 days of dissolution were calculated from measured electrode potentials. Before each measurement, measuring solutions were exposed to UV light for 1 hour to prevent bacteria growth.

**Surface potential measurements during dissolution.** The inner surface potential ($Ψ_0$) of the fluorite crystallographic plane was measured by means of a Single Crystal Electrode (SCrE) during 24 hours of dissolution. The fluorite single crystal (surface area 1 cm²) was fixed on a Plexiglas holder in a way that only the examined surface was exposed to the aqueous electrolyte solution. The measuring system consisted of SCrE and a reference Ag/AgCl electrode immersed in 25 mL sodium chloride solution ($c = 1 \times 10^{-3}$ mol dm⁻³) of pH = 5.0 adjusted with HCl ($c = 0.1$ mol dm⁻³). The CaF₂-SCrE potential with respect to the reference Ag/AgCl electrode was measured with a Metrohm 913 pH-meter. The measurements were conducted without stirring to mitigate
the enhancement of ion diffusion from the interfacial layer, which could potentially influence the dissolution mechanism. The dependence of the inner surface potential of fluorite crystallographic plane on calcium and fluoride ions in sodium chloride solution ($I_c = 1 \times 10^{-3}$ mol dm$^{-3}$) at constant pH = 5.0 was also measured. In light of the impact of ionic strength on $Ψ_0$, a correction to the recorded electrode potential was undertaken, specifically, the observed $Ψ_0$ values at each ionic strength were diminished by the magnitude of the inner surface potential determined through titration with a NaCl solution at a pH of 5.0.

**Theoretical methodology.** Free energies and periodic ab initio DFT calculations have been conducted utilizing the Vienna Ab initio Simulation Program (VASP) [39–42] in accordance with the procedure delineated in our previous studies. [43] A vacuum of 10 Å was systematically maintained between the interface and the upper limit of the cell. The latter was filled with water molecules using the Packmol software. [44] Further, to prevent any drift of the fluorite slab during the simulations, the atoms corresponding to the first layer of the cell were frozen to their equilibrium position. The semi-local Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional in the generalized gradient approximation (GGA) [45] was employed and the projector augmented wave (PAW) [46,47] was used to describe the electron-ion interactions. The Kohn-Sham equations [48] were self-consistently [49] solved until a threshold energy difference of $10^{-7}$ eV between two consecutive cycles was reached. A plane-wave cut-off of 400 eV was employed after convergence. The structural relaxation was performed until the forces were within 0.01 eV/Å. Depending on the slab and surface cleavage employed, sampling of the Brillouin zone was performed using a converged $\times\times\times$ Monkhorst-Pack k-point mesh. A plane-wave cut-off of 400 eV was employed after convergence. Born–Oppenheimer molecular dynamics simulations were executed in the canonical NVT ensemble, with temperature regulation achieved through a Nosé–Hoover thermostat. [50,51] The equations of motion were numerically integrated using a time step of 1 fs and performed with an energy cut-off of 400 eV, a threshold energy difference between two cycles of $10^{-5}$ eV and a thermalization period of 10 ps. The mass of the hydrogen atoms was adjusted to that of a tritium isotope to reduce thermal decoupling between high- and low-frequency modes.

The time-dependent transmission coefficient was obtained utilizing the shake algorithm [52] as incorporated within VASP, by conducting a simulation with a duration of 5 ns, during which the system was maintained at the transition state, with forces calculated but not applied to the calcium and fluorine atoms under investigation that are part of the slab. The final 2 ns of this simulation were dedicated to collecting the positions and velocities of the atoms in the system, with sampling occurring every 5 ps. Following this, 400 additional simulations were performed in both forward and reverse trajectories based on the configurations obtained. Random velocities, characterized by magnitude and direction adjusted for mass, were assigned to the atoms involved in the transition state, conforming to a normal distribution with a standard deviation of $\pm 20\%$ around the mean velocity. Ultimately, the transmission coefficients were determined by averaging the cumulative results from a total of 800 simulations, which were conducted for reactions instigated by the dissociation of Ca/F from the surface, while also confirming the consistency of the calculated transmission coefficient for both forward and reverse reactions.

**Evaluation of the properties.** The impact of dissolution on the wettability characteristics of the (111), (110), and (100) fluorite crystallographic planes was investigated through the assessment of the water contact angle prior to dissolution ($\theta_Y$). Following this evaluation, the Wenzel roughness factor was derived from AFM measurements, allowing for the determination of the relative contact angle subsequent to dissolution ($\theta_W$), in accordance with equation

$$ \cos\theta_W = r\cos\theta_Y \qquad (1) $$

where r represents the roughness coefficient ascertained via AFM and presented in Table 1. In Eq. (1) it was assumed that the measured contact angle of fluorite before dissolution is equal to Young’s contact angle. [53] ODM and AFM results (Figs. 1 and 2) justify that approximation because the surface of the fluorite plane before dissolution is atomically smooth and homogeneous.

The dissolution rate was calculated following the Noyes-Whitney-Nernst-Brunner (NWNB) model [54,55]:

$$ \frac{dc_t}{dt} = \frac{AD}{Vd} (c_s - c_t) \qquad (2) $$

where $c_s$ and $c_t$ are, respectively, the difference in concentration of ions in the saturated solution and concentration of ions in the bulk of the solution. V is the volume of the bulk solution, A is the surface area of the mineral single crystal exposed to the solution, D is the diffusion coefficient of ions, and d is the thickness of the concentration gradient layer. By integrating Eq. (2) and assuming that the equilibrium concentration of $Ca^{2+}$ and $F^-$ ions in saturated solution is different for specific fluorite crystal planes, the dissolution rate $k = \frac{AD}{Vd}$ is evaluated according to the expression:

$$ c_t = c_s (1 - \exp(kt)) \qquad (3) $$

where $c_t$ is the concentration of $F^-$ or $Ca^{2+}$ ions in the bulk of solution at time t, $c_s$ is the equilibrium concentration of $F^-$ and $Ca^{2+}$ ions in saturated fluorite solution for dissolution of specific fluorite plane, i.e. (100), (110) or (111).

Surface energy values were calculated according to the expression:

$$ E_s = \frac{E_{\text{slab}} - n_{\text{slab}} \times \frac{E_{\text{bulk}}}{n_{\text{bulk}}}}{2A} \qquad (4) $$

where $E_{\text{slab}}$ and $E_{\text{bulk}}$ correspond to slab and bulk energies, respectively; A denotes the surface area of the slab, while n represents the number of $CaF_2$ formula units.

Coefficient transmission $k^{\ddagger}(t)$ was calculated according to the expression:

$$ k^{\ddagger}(t) = \frac{\langle \dot{q}(0)\theta [q(t) - q^{\ddagger}] \rangle_{q(0)=q^{\ddagger}}}{\langle \dot{q}(0)\theta [\dot{q}(0)] \rangle_{q(0)=q^{\ddagger}}} \qquad (5) $$

where $\dot{q}(0)$ represents the initial velocity of the reaction (positive for those progressing towards the product state); $\theta [q(t) - q^{\ddagger}]$ signifies the Heaviside function, equalling 1 for reactions originating from the transition state ($q^{\ddagger}$) and being in the product state at time t (otherwise 0), and $\theta [\dot{q}(0)]$ represents the Heaviside function for the initial velocity (i.e., 1 if the reaction begins with positive velocity, else 0). $\langle \dots \rangle_{q(0)=q^{\ddagger}}$ denotes the ensemble average over reactions starting from the transition state. Finally, following the established Eyring formulation, the overall rate constant equals the transition state (TST) rate constant multiplied by the transmission coefficient: [56–58]

$$ k = k^{\text{TST}} \times k^{\ddagger}(t) \quad 0 \le k^{\ddagger} \le 1 \qquad (6) $$

# 3. Results and discussion

## 3.1. Experimental results

The interaction between diminutive solid particulates and macroscopic flat surfaces with aqueous electrolyte solutions induces the accumulation of ions at the interface, culminating in the establishment of an electrical interfacial layer. A thorough examination of precisely defined monocrystalline surfaces, which serve as the fundamental constituents of a singular particle, facilitates an enhanced comprehension of surface reactions and the reciprocal interactions of neighbouring crystal planes regarding the average surface characteristics of particles. Herein, we show the outcomes obtained by ODM, AFM and contact angle measurements, as well as fluorite dissolution kinetics and surface potential.
<figure><img src="image_5.png" /><figcaption>**Fig. 1.** ODM Images of 500 µm × 400 µm surface area of the fluorite crystallographic planes designated as (100), (110), and (111), both prior to (a), (b), (c) and subsequent to dissolution experiments (d), (e), and (f) after 84 days.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 2.** AFM images of the fluorite crystallographic planes designated as (100), (110), and (111), both prior to (a), (b), (c) and subsequent to dissolution experiments (d), (e), and (f) after 84 days.</figcaption></figure>

**Topography and roughness of fluorite surface.** Optical digital microscope (ODM) and atomic force microscope (AFM) were used to visualize the topography of fluorite surfaces on a larger and smaller scale, respectively. Fig. 1 shows representative ODM images, and Fig. 2 AFM images of (100), (110), and (111) fluorite surfaces before and after the performed dissolution experiments. As can be seen from panels 1(a)-(c) and panels 2(a)-(c), the initial surface morphology of all three examined fluorite planes is similar. It exhibits flat and featureless form, both on micro- and nano-scale. Following the dissolution, the morphological features of fluorite surfaces underwent substantial alterations, and every resulting surface exhibits unique topographical attributes. Panel 1(d) shows an ODM image of the (100) fluorite plane after dissolution. Four-sided pyramids of microscopic dimensions can be easily seen on the surface of that single crystal. The observed pyramids are heterogeneous in size and randomly distributed on the fluorite surface. Given the scan size limitations of the instrument, it was not possible to image these artifacts in detail with AFM. Therefore, solely pyramid-free areas on the surface of the (100) fluorite plane were examined by AFM. It was found that these areas have similar surface topography as the initial fluorite surface, with only a few defects visible on the surface (Fig. 1(d)). On the contrary, the dissolution of the (110) fluorite plane produced more pronounced topographical changes on the surface. In this case, a high number of directed protrusions of irregular shape were visible both on ODM (Fig. 1(e)) and AFM (Fig. 2(e)) images. Comparable surface structures have already been mentioned in the literature, [17] and it was noticed that they are formed on (110) fluorite surface from irregular quadrangular pyramids that are elongated in one direction as the dissolution process proceeds. Among all three examined fluorite planes, the most notable change in surface morphology was observed in the case

<caption>**Table 2**</caption>

The static contact angle was quantitatively assessed a the interface between deionized water (V = 5 µl) and the (111), (110), and (100) crystallographic planes, both prior to dissolution ($\theta_Y$) and subsequent to dissolution ($\theta_{exp}$). Furthermore, the listed Wenzel contact angles ($\theta_W$) were computed under conditions of ambient temperature and relative humidity ranging from 25% to 45%.

<table><thead><tr><th rowspan="2">Plane</th><th colspan="1">Before</th><th colspan="2">After</th></tr><tr><th>$\theta_Y$ /°</th><th>$\theta_{exp}$ /°</th><th>$\theta_W$ /°</th></tr></thead><tbody><tr><td>(100)</td><td>83.0 ± 0.4</td><td>81.7 ± 1.3</td><td>82.9 ± 0.4</td></tr><tr><td>(110)</td><td>74.9 ± 2.6</td><td>87.1 ± 0.8</td><td>73.6 ± 1.7</td></tr><tr><td>(111)</td><td>75.2 ± 0.5</td><td>72.2 ± 0.5</td><td>74.4 ± 0.6</td></tr></tbody></table>

of (111) surface. Fig. 1(f) shows an ODM image of the (111) fluorite surface subsequent to dissolution experiments. The main characteristics of (111) fluorite plane after dissolution are triangular etch pits with trigonal pyramidal morphology which are mainly coalesced. A detailed inspection of etch pits by AFM (Fig. 2(f)) revealed that they are more or less equilateral with a side length of several micrometers and pit depth around 500 nm. The formation of these types of triangular features in the case of fluorite (111) plane dissolution has already been reported in the literature. [59,21,16] The described changes in surface morphology of three different fluorite planes are also accompanied by changes in surface roughness. Table 1 shows the values of several roughness parameters for the three investigated fluorite surfaces before and after dissolution. Prior to dissolution experiments, all three surfaces have an extremely low surface roughness ($R_q$ < 0.5 nm), which can be attributed to their smooth surface morphology. Dissolution of fluorite caused a significant increase in surface roughness and an increase in the surface area of all tested surfaces. The most pronounced change in roughness parameters ($R_a$ and $R_q$) was obtained in the case of the (111) surface, while the increase in surface area was highest for the (110) surface. This unexpected result can be explained by considering the surface features of those two planes. While the (110) surface of fluorite is covered by a high number of ordered protrusions, the (111) surface is characterized by deep triangular etch pits. Protrusions significantly increase surface area, but they do not affect the surface roughness excessively due to their shallowness. On the contrary, deep and relatively large triangular etch pits do not contribute much to the surface area, but they increase surface roughness.

**Wettability of fluorite surface.** Fluorite demonstrates anisotropic wettability, and the disparities in the recorded contact angles for the (100), (110), and (111) planes prior to dissolution can be attributed to the varying interactions of the distinct planes with water, which arise from their differing surface structures. [26,60,25] The impact of dissolution on the wettability characteristics of the (111), (110), and (100) fluorite crystal cleavages was investigated through the evaluation of the water contact angle prior to dissolution ($\theta_Y$) and subsequent to dissolution ($\theta_{exp}$). The quantified values are presented in Table 2. Before dissolution, The (110) and (111) fluorite facets display comparable contact angles, approximately 75°, while a slightly elevated angle of about 83° is recorded for the (100) plane. The (111) fluorite cleavage exhibits moderate hydrophilicity both before and after the dissolution process, with a marginal reduction in the contact angle observed post-dissolution. This finding suggests a slight enhancement in hydrophilicity of the (111) fluorite surface, consistent with Wenzel's theory, which offers a thermodynamic rationale for the improved surface wettability associated with increased roughness. A similar increase in hydrophilicity was noted for the (100) fluorite plane, whereas a decrease in hydrophilicity was observed for the (110) plane. The results clearly indicate that all three surfaces remain within the classification of hydrophilic surfaces, both prior to and following the dissolution process.
<figure><img src="image_7.png" /><figcaption>**Fig. 3.** The concentrations of F⁻ (left) and Ca²⁺ (right) ions were monitored to elucidate the dissolution phenomena occurring at the (111), (110), and (100) crystallographic planes of fluorite within a 1 mmol dm⁻³ NaCl solution at pH = 5.0 and a temperature of T = 298 K. The surface area of the exposed fluorite crystal is quantified as 1 cm².</figcaption></figure>

<table><caption>**Table 3**

Dissolution parameters in aqueous solution at pH = 5.0, ionic strength $I_c = 1 \times 10^{-3}$ mol dm⁻³, and T = 298 K.</caption><thead><tr><th rowspan="2">Plane</th><th colspan="2">From Fig. 3</th><th colspan="2">Using Eq. (3)</th></tr><tr><th>F<sup>-</sup></th><th>Ca<sup>2+</sup></th><th>F<sup>-</sup></th><th>Ca<sup>2+</sup></th></tr></thead><tbody><tr><td>(100)</td><td>140</td><td>80</td><td>435</td><td>220</td></tr><tr><td>(110)</td><td>240</td><td>130</td><td>255</td><td>145</td></tr><tr><td>(111)</td><td>220</td><td>130</td><td>255</td><td>145</td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>**Fig. 4.** Molar concentration ratios of F⁻ and Ca²⁺ ions in the measuring solution over a span of 84 days, specifically for the dissolution of (111), (110), and (100) fluorite planes at pH = 5.0, with $I_c = 10^{-3}$ mol dm⁻³ and T = 298 K.</figcaption></figure>

**Dissolution kinetics.** The findings pertaining to the three cleavage planes are elucidated in Fig. 3 whereas the final concentrations of fluoride and calcium ions are delineated in Table 3. The experimental data presented suggest that after 84 days of fluorite dissolution, the latter remains incomplete, with the measured concentrations of dissolved ions falling short of the equilibrium concentrations. Incomplete dissolution can be attributed to the lack of stirring - indeed, prior studies have indicated that agitation significantly influences the dissolution rate. [61] Additionally, the surface area of the scrutinized crystal planes exposed to the aqueous solution during measurement was relatively modest (1 cm²), which further impeded the dissolution process. The rates of dissolution among the three scrutinized fluorite crystal faces exhibit variability. The (111) and (110) surfaces dissolve at a more rapid pace, whereas the (100) surface displays the slowest rate. Notably, the maximum concentrations of dissolved calcium and fluoride ions (Table 3) attained differ among the three examined fluorite planes. This discrepancy may stem from the distinct solubility characteristics of these crystal faces as well as from the EIL formation, which could exert an influence on the rate of mineral dissolution. Fig. 4 shows the ratios of molar concentrations between fluoride and calcium ions in the bulk solution during the dissolution of (111), (110), and (100) fluorite planes,

<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Variations in the inner surface potential of the (111), (110), and (100) crystallographic planes over a duration of 24 hours of dissolution, conducted without agitation, in a 1 mmol dm⁻³ NaCl solution at pH = 5.0 and T = 298 K.</figcaption></figure>

derived from the measured data presented in Fig. 3. Although this ratio is expected to be equal to 2, as per the overall equation governing the chemical reaction, a non-stoichiometric ratio of molar concentrations of F⁻ and Ca²⁺ ions was observed. The reduced concentration of fluoride ions in the bulk aqueous solution indicates an elevated concentration of F⁻ ions within the interfacial layer, resulting in a diminished rate of F⁻ ion release from this interfacial region. In contrast to the release of Ca²⁺ ions, the liberation of F⁻ ions into the bulk solution exhibits pH dependency, occurring at a more rapid rate in acidic conditions due to the production of hydrofluoric acid (HF). Under the specified experimental conditions, namely pH = 5, the formation of HF should be negligible within the state of equilibrium (less than 1.4%) [62].

**Surface potential during dissolution.** Fig. 5 delineates the shift in the inner surface potential of the (111), (110), and (100) fluorite crystallographic planes over a 24-hour dissolution period. This shift represents a direct consequence of the fluorite dissolution phenomenon, which involves the release of Ca²⁺ and F⁻ ions into the aqueous phase, in conjunction with the simultaneous processes occurring within the interfacial layer. During the initial phase of dissolution, characterized by a relatively low concentration of constituent ions, the dissolution rate experiences an increase. As the dissolution progresses, the surface potential exhibits a trend towards decreased positivity, potentially signifying an enhanced liberation of Ca²⁺ ions. Given that the most pronounced shift pertained to the (111) plane, in Fig. 6 we systematically assessed the open circuit potential as a function of Ca²⁺ and F⁻ concentrations throughout the dissolution of this specific crystallographic plane. It is noteworthy that during these experimental trials, the concentration of ions within the bulk solution, and thereby the ionic strength, demonstrated an upward trajectory. The corrected sur-
<figure><img src="image_10.png" /><figcaption>**Fig. 6.** Impact of the concentrations of calcium and fluoride ions on the inner surface potential of the (111) fluorite plane within a 1 mol dm⁻³ NaCl solution at pH = 5.0 and T = 298 K.</figcaption></figure>

face potential data, which were modified to accommodate the effects of ionic strength illustrated in Fig. 6, unambiguously indicate that the inner surface potential of fluorite is contingent upon the concentration of calcium ions. Upon elevating the concentration of calcium ions from 2 mmol dm⁻³ to 10 mmol dm⁻³ at a pH of 5, the inner surface potential of the fluorite plane undergoes an increase of 15 mV. This modification signifies a markedly greater enhancement in surface potential in comparison to the analogous addition of hydrogen ions. [22] Prior investigations [29,63,64] have similarly asserted that the impact of calcium ions on the dissolution process is more substantial than that of H⁺ ions. Therefore, it can be deduced that calcium ions function as the potential-determining species at the (111) fluorite/aqueous electrolyte solution interface. Fig. 6 further indicates that the inner surface potential remains unaffected by the concentration of fluoride ions. A similar conclusion was reached by Miller et al. [29,63,65] based on electrophoretic measurements of fluorite particles. This observation suggests that fluoride ions do not adsorb directly onto the fluorite surface; rather, they reside within the diffuse layer, a finding corroborated by an initial molecular dynamics. [30] This phenomenon may be attributed to the significant hydration of fluoride ions [63] and the highly organized structure of interfacial water molecules.

## 3.2. Theoretical results

In order to enhance the efficiency of the flotation process, a comprehensive understanding of the intricate microscopic interfacial characteristics at the surface and the dynamics of the dissolution of fluorite is deemed imperative. Indeed, the initial hydration plays a pivotal role in the overall dissolution process and can only be understood by theoretical simulations. Hence, in order to assess the influence of water on the dissolution process, theoretical simulations were conducted with a systematic approach. Primarily, cohesive energies pertaining to a single formula unit were analyzed for the three surfaces, accompanied by the calculations of the surface tensions; subsequently, simulations were performed in a vacuum setting, followed by simulations in water to compare the outcomes of both approaches. From these simulations, free energy profiles were derived to extrapolate the dissolution rate constant values. The concluding section juxtaposes the theoretical findings with the experimental data.

**Cohesive energies at the interface.** Figure S5 illustrates the depiction used throughout this theoretical section of a singular formula unit, which may exhibit either a linear or bent arrangement within the slab. Hereafter, *fu* refers to one formula unit, and whenever the latter is represented in red within the slab, this signifies that the relative cohesive energy (*E*<sub>coh</sub>) is computed for the specific position occupied by the *fu*. Figure S6 delineates the model for the three cleavage planes, namely (110), (100), and (111). In order to facilitate a fair comparison amongst the slabs, approximately equivalent thicknesses were established for

each of the three cleavage configurations. To assist the reader in replicating the findings presented, Figure S7 illustrates the initial coordinates employed in the calculation of the surface energies. In accordance with prior research, [25,66] simulations indicate that the surface energy values, determined using Eq. (4), are ranked as follows: (111) (0.336 J/m²) < (110) (0.572 J/m²) < (100) (0.772 J/m²). Therefore, with respect to specific surface energy, the (111) plane demonstrates the lowest surface energy, rendering it more prone to cleavage. Nonetheless, other aspects may play a critical role in ascertaining the *E*<sub>coh</sub> for one *fu* and subsequently the energy required to detach the *fu* under investigation from the surface. This concept is herein referred as the dissolution cost associated with a singular *fu*. For instance, the *fu* may extend to encompass the two outermost layers. Figure S6(b) presents a few instances demonstrating how this extension, in conjunction with its spatial arrangement and configuration within the slab, may affect *E*<sub>coh</sub>. Moreover, the atomic environment surrounding the *fu* may impact the relative dissolution cost. For instance, let us consider the scenario where we aim to remove one *fu* from the slab's interfacial layer in the presence of a defect (i.e., one *fu* is absent). Figure S8 depicts an analysis conducted on the neighbouring *fu* environments associated with the three distinct cleavage planes. We contrast situations wherein the second *fu* elected for removal is either in direct proximity to the defect created by the removal of the first *fu* (Nearest (N)) or one *fu* distant (Close (C)). The most pronounced C-N energy disparity is observed within the (110) cleavage plane, underscoring that the dissolution cost is relatively more sensitive to the existence of defects within the (110) surface. The latter aspect calls into question the distinction regarding dissolution costs associated with the simultaneous versus the continuous removal of *fus* from the surface. Figure S9 elucidates the most notable outcomes within this context, specifically achieved for the (100) cleavage plane (the other cleavage planes not illustrated demonstrate a consistent trend). The profiles illustrated reflect the energetic disparities arising from the extraction of between 1 to 3 *fus*. The red graph within the figure represents the simultaneous removal of *fus*, whereas the blue profile pertains to the sequential removal methodology (i.e., one after another, as illustrated in the visual depiction within the dashed blue box). Above all, as expected, the least costly removal of a *fu* in terms of energy expenditure pertains to a *fu* with the two fluorine atoms situated within the first (interfacial) layer (with the energy differential quantified at 0.45 eV). The energetic gain realized by the successive removal of three *fus* approximates 2.5 eV, thereby illustrating the influence of surface relaxation throughout the dissolution process.

**Role of the surfactant water molecules during the dissolution process.** The comparison between simulations conducted in a water environment and those in a vacuum sheds light on the solvation effect by water. Figure S10 displays three snapshots from the gradual growth simulation of a fluoride ion originating from the (111) cleavage plane. Under vacuum conditions, the fluoride ion carries both F⁻ and Ca²⁺ atoms as it moves, whereas in water - consistently with our experimental findings - the fluoride ion undergoes hydration and does not transport additional atoms into the solution. Specifically, F⁻ ion traces the path between two neighbouring Ca atoms from the slab (the reader may refer to the sequence illustrated in Fig. 7). The dissolution of a singular Ca²⁺ atom from the (111) interfacial layer is depicted in Figure S11. Analogous to F⁻, in vacuum conditions Ca²⁺ carries other ions in the bulk solution. Unlike the dissolution of fluoride, Ca²⁺ in a water environment is not entirely hydrated and carries additional F⁻ and Ca²⁺ ions from the slab. The overall dissolution of calcium appears to be stoichiometric (i.e., CaF₂ units are liberated in the bulk solution), with no isolated F⁻ ions released into the bulk solution. Fig. 8 presents the progressive dissolution simulations conducted on the (110) interfacial plane for both fluoride and calcium ions. In water, the fluoride is readily released in the bulk solution given its initial, exposed position within the (110) plane. Dealing with the dissolution of the Ca²⁺ ion, the latter transports two additional F⁻ ions, ultimately culminating in the release of one *fu* into
<figure><img src="image_11.png" /><figcaption>**Fig. 7.** Dissolution of an individual F⁻ ion from the (100) cleavage plane. The selected F ion (a), which is accentuated in red colour (b), delineates a trajectory between two adjacent calcium atoms within the slab (c-d), culminating at the apex of the ridge (d). Subsequently, F⁻ is liberated into the bulk solution due to the hydration effect facilitated by water molecules. (For interpretation of the colours in the figure(s), the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 8.** A few instances of the simulations generated for the dissolution of one F⁻ ion (left) and one Ca²⁺ ion (right) from the (110) cleavage plane, with the extents delineated by the x and y axes, while the z axis denotes the axial distance from the cleavage plane.</figcaption></figure>

the bulk solution. Finally, Figure S12 contrasts the dissolution processes of F⁻ and Ca²⁺ in water with respect to the (100) plane, which epitomizes the most constrained scenario throughout the dissolution process (see section below). Throughout the simulation, it is observable that the F⁻ ion adheres to the trajectory of a singular interfacial calcium atom prior to its transition to the bulk solution. The Ca²⁺ ion manifests less impediment during the dissolution process, promptly undergoing hydration while carrying along an F⁻ ion that is subsequently released into the bulk solution. Notably, during the dissolution process, an additional fluorine atom belonging to the slab interface may temporarily establish a connection with the dissolved Ca²⁺ ion (Figure S13), extending itself towards the bulk solution. Upon the failure of this connection, the fluorine atom returns to its original position within the outermost layer of the slab.

**A fu nearby the interfacial slab.** In the preceding section, it was observed that the dissolution process may involve the release of ions as well as of *fus*. Consequently, at this juncture, the inquiry may concern the rapidity by which a *fu* present in the bulk solution interacts with the interfacial layer. We present the scenario pertaining to the (100) cleavage, with the (111) and (110) orientations exhibiting analogous behaviour. Simulations reveal that two key factors may promote the interaction of a *fu* with the slab. Primarily, the presence of a Ca²⁺ ion within the solution can strengthen the attraction between CaF₂ and the slab, as depicted in figure S14, where a bridge is established by the fluoride atom originating from the (100) surface. Additionally, the presence of interfacial defects (Figure S14) may also contribute to this interaction. In such situations, the adsorption mechanism involves the initial inter-

<caption>**Table 4**</caption>

In this table, a quantitative comparison is presented between the experimental data and theoretical simulations carried out within this study regarding the dissolution constant values. The latter was derived by multiplying the coefficient of transition by $k^2$ (the height of the barrier, as illustrated in Fig. 9), in accordance with Eq. (6).

<table><thead><tr><th rowspan="2">Plane</th><th colspan="2">k/s<sup>-1</sup><sub>(Exp.)</sub></th><th colspan="2">k/s<sup>-1</sup><sub>(Theory)</sub></th></tr><tr><th>F<sup>-</sup></th><th>Ca<sup>2+</sup></th><th>F<sup>-</sup> (k<sup>2</sup>(t))</th><th>Ca<sup>2+</sup> (k<sup>2</sup>(t))</th></tr></thead><tbody><tr><td>(100)</td><td>0.004</td><td>0.005</td><td>0.001 (0.48)</td><td>0.002 (0.51)</td></tr><tr><td>(110)</td><td>0.043</td><td>0.044</td><td>0.015 (0.57)</td><td>0.018 (0.60)</td></tr><tr><td>(111)</td><td>0.027</td><td>0.033</td><td>0.008 (0.54)</td><td>0.023 (0.56)</td></tr></tbody></table>

action of the *fu* with water molecules in its surroundings, followed by the protonation of F⁻ atoms within the *fu*. Thereafter, a water molecule facilitates the formation of a bridge between the slab and the *fu*, with the final step being the enhancement of adsorption due to interaction with a fluorine atom from the slab.

**Free energy profiles.** The disparity in energy levels between the valleys and peaks depicted in the profiles of the subsequent figures elucidates the free energy requisite for activation during the dissolution reaction, thereby revealing a considerable degree of variation across distinct reactions. This heterogeneity serves as an indication of the intricate nature of the dissolution reaction, which involves traversing through intermediate states that are both energetically and structurally distinct prior to the completion of the process. Within each profile, the particular configuration corresponding to the transition states at the saddle point is presented, while the relative values for dissolution constants are enumerated in Table 4. Overall, it was observed that for both F⁻ and Ca²⁺ ions, the maximum energy barrier is encountered during the dissolution from the (100) cleavage plane. Following this, the barriers diminish for the (111) and (110) planes in the context of the F⁻ ion, whereas for the Ca²⁺ ion, the order is inverted with barriers decreasing for the (110) and (111) planes. The results derived from simulations for the (100) cleavage plane are graphically depicted in Fig. 9. The findings obtained from simulations concerning the (100) cleavage plane are visually represented in Fig. 9, wherein the system encounters an energy barrier of 0.151 eV. A top-down perspective (Fig. 9(a)) illustrates that the F⁻ atom is positioned between two Ca²⁺ ions, leading to increased resistance during its removal, as evidenced in the side view (Fig. 9(b)). Subsequent rearrangement of bonds results in the formation of an intermediate state, overcoming additional barriers that may arise from interactions with water molecules (Fig. 9(c)), until the ion ultimately detaches into the bulk solution, thereby completing the dissolution process. In the scenario of Ca²⁺ dissolution from the (100) cleavage plane, the process is characterized by increased friction with adjacent atoms attributable to the larger dimensions of the Ca²⁺ ion. This leads to an initially lower barrier of 0.03 eV at the onset of dissolution, succeeded by a more substantial barrier of 0.142 eV, with the apex coinciding with the Ca²⁺ ion attaining the same height (i.e., the z value) as the neighbouring F⁻ atoms (Fig. 9(f)). As expected, this barrier is wider compared to the F⁻ ions, partly due to interactions with water molecules in the vicinity. In particular, these interactions may encompass intricate processes over dissolution that entail solvation, dissociation as well as alterations of the slab structure. Solvation engenders a hydration shell that enhances the stability of the ions within the aqueous environment, with a coordination number that may fluctuate in accordance with the distance from the vacated slab surface. The dissolution of F⁻ ions from the (110) cleavage plane, Figure S15, exhibits an intriguing landscape. The top view emphasizes that the F⁻ atom is bound to two Ca atoms in the interfacial layer, but not inserted as in the case of the (100) plane. Consequently, the barrier is smaller at 0.094 eV compared to the dissolution from the (100) plane. In particular, the valley profile stems from the attraction of F⁻ to one of the adjacent Ca atoms (Figure S15(b)). Further along the dissolution process, the F⁻ ion may be attracted to another Ca²⁺ ion, potentially
<figure><img src="image_13.png" /><figcaption>**Fig. 9.** Free energy profile associated with the dissolution of fluoride ions ($F^−$) (left) and calcium ions ($Ca^{2+}$) (right) from the (100) surface.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 10.** Interfacial free energy profiles following the dissolution of both $F^−$ and $Ca^{2+}$ ions from (100) surface. Water has been omitted for clarity. Black arrows indicate where the defects are located: 1 fu (b) and, respectively, 1 (d) and 2 (e) fluoride ions.</figcaption></figure>

leading to the appearance of a second peak in the free energy profile (Figure S15(d)) before completion of the dissolution. Additionally, the interaction of the $F^−$ ion with surrounding water molecules can be observed in Figure S15(e). Dealing with $Ca(110)$, Figure S16 illustrates a quite similar profile, characterized by a marginally reduced barrier height of 0.090 eV in comparison to the dissolution process of $F^−$ ions from the same plane. It is noteworthy that in this particular scenario, the dissolution of $Ca^{2+}$ ions unfolds in conjunction with interactions involving both $Ca^{2+}$ and $F^−$ atoms as depicted in Figure S17(c)-(d). The depicted top view in Figure S17(a) serves as a reminiscent of the initial configuration concerning the dissolution mechanism of $F^−$ ions from the (111) surface. Notably, the fluoride ion $F^−$ is nestled amidst three neighbouring $Ca^{2+}$ atoms, thereby affording a relatively more spacious environment compared to the dissolution process from the (100) surface. Consequently, an observable reduction in barrier height of 0.110 eV is discernible in contrast to the dissolution from the (100) surface. Furthermore, it is discernible that the $F^−$ ion exhibits the propensity to dissolve following interaction with a singular surrounding $Ca^{2+}$ ion as depicted in Figure S17(c) prior to completing the dissolution process. Finally, the dissolution of $Ca^{2+}$ ions from the (111) surface (Figure S18) is characterized by a barrier height of 0.082 eV; this is followed by a subsequent dissolution process within the bulk solution accompanied by the dissolution of other $CaF_2$ units.

**Presence of interfacial defects.** We have explored how defects within the outermost layer of the slab can impact the dissolution of $F^−$ and $Ca^{2+}$ ions. For instance, in Fig. 10, we start from the relative highest barriers encountered during dissolution from the (100) plane (see Fig. 9). In the inset of Fig. 10, the absence of one fu is highlighted by a black arrow. Consequently, the relative free energy profile displays a significant decrease in barrier height, allowing the fluoride ion to transition more

easily to the bulk solution due to the defect's presence. Further analysis in insets (d)-(e) reveals scenarios where 1 and 2 $F^−$ ions are absent, respectively. Notably, the absence of one $F^−$ not only reduces the barrier height but also induces a shift in its position. Specifically, when one $F^−$ ion is missing, we observe the persistence of two distinct barriers during dissolution (d); conversely, these barriers merge into a single one when 2 $F^−$ ions are absent.

**Dissolution rates and comparison to the experiments.** Table 4 ultimately presents the computed values of the rate constants according to the Eq. (6); included within the same table are the experimental constant values for the sake of comparison. Additionally, the values of the transmission coefficients are documented for further analysis. These coefficients are intricately linked to the free energy profile, shedding light on the underlying mechanisms at play: a diminutive energetic barrier facilitates a higher frequency of recrossing, subsequently leading to a decrease in the transmission coefficient. Figure S19, elucidates the mean time-varying transmission coefficient values obtained for both $F^−$ and $Ca^{2+}$, alongside the three cleavage planes. Except for $Ca(111)$, our findings show that a lower energy barrier results in a smaller value of $k^‡(t)$. Upon analysis, rate constants encompass a range of two orders of magnitude and show a substantial agreement within the same order of magnitude to the experiments, along with a consistent trend. Nevertheless, several discrepancies between experiments and theoretical simulations exist, such as the peculiar contrasting trend observed between (110) and (111) cleavages in the context of $Ca^{2+}$ dissolution. Albeit intriguing, these discrepancies warrant further investigation to better understand the underlying mechanisms governing the observed phenomena. These variances are also apparent with respect to the non-stoichiometric dissolution characteristics that have been empirically observed across all crystallographic planes; conversely, theoretical sim-
ulations suggest that non-stoichiometry is predominantly observable for the (100) plane, possibly due to the considerable energy barrier which could obstruct stoichiometric dissolution. However, as previously mentioned, the dissolution of a particular *fu* may depend on the selection of the two fluorine atoms within the unit: do they reside within the same outermost, interfacial layer (i.e., bent *fu*, with both F atoms oriented towards the aqueous medium)? Or do they occupy the two outermost layers (i.e., still bent *fu*, but with one F oriented towards the aqueous phase and the other towards the bulk - as depicted in Figure S9)? As earlier demonstrated, the variance between these scenarios could be as considerable as 0.45 eV. Furthermore, experimental investigations reveal that the (110) plane displays both the most pronounced measured enhancement in surface area and the most accelerated recrystallization process, attributed to the rapid surface reactions of ions liberated from the crystal lattice. Theoretical simulations appear to align with these results: they indicate that fluorine becomes hydrated subsequent to dissolution and can be adsorbed more readily on the surface in comparison to calcium; conversely, $Ca^{2+}$ aids in the transport of additional $F^-$ and $Ca^{2+}$ ions throughout the dissolution process. Ultimately, experimental investigations and free energy profiles conducted in the context of defects underscore the necessity of integrating surface orientation and atomic-scale structure, including the nature and quantity of steps, when formulating precise kinetic models for dissolution and crystal growth. By elucidating the mechanisms governing the morphology of fluorite during the dissolution process, this study has the potential to inform the advancement of novel methodologies for the manipulation of mineral surfaces across diverse chemical environments, which may have significant implications for disciplines such as geochemistry, environmental science, and materials engineering.

## 4. Concluding remarks

With the goal of enhancing comprehension of the dissolution mechanism relevant to distinct fluorite crystal orientations, this study illustrated the dissolution rates of the fluorite surface across three specific cleavage planes, namely (100), (111), and (110). Following dissolution, the most significant alteration in roughness was noted on the (111) surface, whereas the most substantial increase in surface area was observed for the (110) surface. Both empirical experiments and theoretical simulations indicated that the (111) and (110) surfaces exhibit a more accelerated dissolution rate, in contrast to the (100) surface, which demonstrated the slowest rate. In particular, the presence of defects may influence and modify the apex of the dissolution barrier. With the exception of certain discrepancies, the theoretical results demonstrate a correlation within the same order of magnitude as the experimental outcomes, revealing a consistent pattern. Clearly, to contrast the experimentally observed enhancement in surface area and roughness, the theoretical analysis requires advanced tool such as first-principles-based potential energy surfaces to examine these formations at the mineral interface. To this end, drawing upon prior research experience, [43,67] we are presently developing a machine learning potential capable of incorporating long-range interactions, which are essential for simulating broader effects at the mineral-water interface.

### CRediT authorship contribution statement

**David Dell'Angelo:** Writing – review & editing, Writing – original draft, Validation, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Ana Jurković:** Writing – original draft, Methodology, Investigation, Data curation.
**Tin Klačić:** Validation, Supervision, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Yann Foucaud:** Writing – review & editing, Validation, Supervision.
**Michael Badawi:** Writing – review & editing, Validation, Resources, Supervision, Funding acquisition.
**Adlane Sayede:** Writing – review & editing.
**Tajana Begović:** Conceptualization, Formal

analysis, Investigation, Methodology, Supervision, Validation, Writing – review & editing.

### Supporting information

The Supporting Information document is available free of charge and encompasses (a) surface data derived from experimental measurements, both pre- and post-dissolution; (b) initial configurations for the three surfaces used for theoretical calculations; and (c) instances pertaining to the dissolution process acquired through theoretical simulations. Further details and additional data are available from the corresponding authors upon reasonable request.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgements

This work has been supported by Croatian Science Foundation under the POLYMIN2 project (IP-2020-02-9571) and European Regional Development Fund (infrastructural project CluK, Grant No. KK.01.1.1.02.0016). DD acknowledges project NEST funded under the National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.3 -Call for tender No. 1561 of 11.10.2022 of Ministero dell'Università edella Ricerca (MUR), funded by the European Union NextGeneration EU. DD, YF and MB acknowledge the Agence Nationale de la Recherche under France 2030 (contract ANR-22-EXSS-0006) for support in the context of the InnovTech project (PEPR Sous-Sol), as well as the industrial chair MULTIMINE funded by ArcelorMittal, Région Grand Est, Metz Métropôle and Université de Lorraine. This work was granted access to the HPC resources of TGCC under the allocations 2023-A0140810433 and 2024-A0160810433 made by GENCI.

## Appendix A. Supplementary material

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.jcis.2024.12.242.

### Data availability

Data will be made available on request.

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