Application of Quantum Chemistry in the Study of Flotation Reagents

Abstract

Flotation reagents are significant for modifying the interfacial characteristics of mineral grains to achieve the effective separation of minerals. Since the 1960s, when quantum chemistry was first introduced into the study of flotation reagents, many achievements have been made, although some controversial topics remain. The application of quantum chemistry in the research of flotation reagents for the separation of various minerals in the past decade is herein comprehensively and systematically reviewed. The main directions and gaps of current research are pointed out, the theoretical basis for the design and development of novel flotation reagents is summarized, and more importantly, the potential for the targeting design and development of efficient, selective, and environmentally friendly flotation reagent molecules by means of quantum chemistry is explored.

1. Introduction

1.1. Classification of Flotation Reagents

Froth flotation is a mineral beneficiation method that utilizes the differences in the surface properties of the target minerals and gangue minerals [1]. When target minerals are collected as concentrates, leaving gangue minerals at the bottom of the flotation unit as tailings, this process is known as direct flotation. For reverse flotation, however, the process is reversed, with gangue minerals being collected and valuable minerals being discharged [2].

Flotation reagents are chemicals used in the mineral flotation process that can modify mineral surface properties and improve or reduce the hydrophobicity of minerals, and thereby, the pulp properties and froth stability are more conducive to the separation of minerals [3,4,5,6,7]. They are usually divided into the activator, collector, depressant, frother, pH regulators based on their functions [8]. In the flotation process, the most discussed flotation reagents are collectors and depressants. Collectors are dosed to improve the surface hydrophobicity of the target mineral for a better attachment to air bubbles, and in contrast, depressants are introduced to enhance its surface hydrophilicity [9]. The flotation reagents used in direct flotation and reverse flotation usually share the same names, but act on different objects. In direct flotation, collectors act on target minerals, whereas depressants act on gangue minerals. However, in reverse flotation, the objects are the opposite. Figure 1 illustrates the function and classification of the main flotation reagents. More specifically, information about flotation reagents for common minerals is open and readily available [7,9,10,11]; therefore, they are summarized in Table S1 in the Supplementary Materials.

1.2. Early Exploration of the Structure–Property Relations of Flotation Reagents

A sufficient exploratory study was conducted on the relations between the structure and performance of flotation reagents. In the 1930s, Targgart et al. [12] put forward the “solubility product hypothesis” of the interaction between flotation reagents and minerals, which was considered to be the principle of the earliest selection of the former. Since then, researchers utilized complex dissociation and stability constants to predict the properties of flotation reagents and to select them for the efficient enrichment of ores [13]. In the 1970s, Wang put forward the electronegativity calculation method [14], the critical micelle concentration calculation method [15], and the hydrophile–lipophile balance calculation method [16] for the flotation performance of reagents, and put forward the isotonic volume calculation method for the performance of frothers [17]; meanwhile, the relation between the molecular geometry and selectivity of flotation reagents was also proposed [18]. Based on these calculations, a series of computational criteria were proposed to characterize the performance of flotation reagents, which allowed for their molecular and structural parametrization, and laid a theoretical foundation for the molecular design of flotation reagents [19]. Subsequently, Wang [20] explored the law of the collection effect of collectors, called the “atom enantiomerism law”, which states that, when the target mineral and the polar group in the collector molecule have enantiomorphic atoms (or atomic groups), the collector is efficient. Later on, the general theory of van der Waals forces was used to derive the energy equation for the interaction between flotation reagents and mineral surfaces, and an energy criterion for the selection of flotation reagents was then proposed by Israelachvili et al. [21]. Zhou et al. [22] used the concepts of absolute hardness η, absolute electronegativity x, and chemical potential μ as the theoretical basis, and deduced the acid–base reaction electron transfer number ΔN as a new criterion for the structure–reactivity relation of flotation reagents to predict their performance, which allowed them to obtain better results.

1.3. Development of Quantum Chemistry in Mineral Flotation

Quantum chemistry is an important method in the study of molecular structure, which can provide comprehensive information about it. There are many specific quantum chemical approximation methods, such as the Huckel molecular orbital (HMO) method, the extended Huckel molecular orbital (EHMO) method, the complete neglect of differential overlap (CNDO) method, the intermediate neglect of differential overlap (INDO) method, and the neglect of diatomic differential overlap (NDDO) method. Among them, the best known and most widely used and successful π-electron theory in organic chemistry was the HMO method established by Huckel in 1930 [23]. This method summarized the main chemical phenomena and anticipated the behaviors of molecules in a qualitative or semi-quantitative manner, which made the HMO method a very useful tool for organic chemists. And until about the 1960s, almost all of the work on the calculation of organic molecules with quantum chemistry was derived from the HMO method.

In the 1960s, quantum chemical methods were introduced into the field of flotation reagents to study the structural properties and interaction mechanism of flotation reagents, and certain research results have been realized after 60 years of development. Wang [24] used the HMO method to investigate the structure and properties of flotation reagents when interacting with a mineral surface. Chen [25] designed an HMO program suitable for mineral processing based on quantum chemistry, and used this program to discuss the influences of different substituents and different molecular states on the performance of flotation reagents. In the meantime, Chen et al. [26,27] used the HMO program to explore the relation between the molecular structure of organic flotation reagents and their performance based on the electronegativity of groups and the hard and soft acids and bases (HSAB) theory, and they put forward a formula for calculating the interaction energies of the functional groups of the reagents with mineral surfaces in order to quantitatively study the effects of the polarity of the mineral surface, the polarity of the reagent molecules, and the medium water molecules on the interaction energy of flotation reagents. Wang and Yin [28] subsequently proposed using the acid–base potential scale and the energy level functions for the frontier orbitals of reagent molecules as a criterion to understand the structure performance of flotation reagents. The introduction of this potential scale in the research field of flotation reagents expanded the application scope of the HSAB theory.

The CNDO, INDO, and NDDO methods mentioned above are approximated from the Hartree–Fock method and are known as semi-empirical calculation methods. The advantages of semi-empirical methods are fast computation, small disk space and computer memory required for computation, and thus, an increase in the ability to perform a large system of computations [29]. However, their disadvantages are also obvious. The underestimation of the total energy, the inapplicability for molecules containing unparameterized elements, and erroneous results for molecular systems that are different from those used for parameterization severely limit their development [29]. Therefore, these methods are seldom used in the present, and they have been replaced by the density functional theory (DFT) [30]. The Hartree–Fock method and DFT are categorized as ab initio methods.

With the further development and improvement of the quantum chemical theory, the first-principles studies based on DFT have been widely applied in the calculation and simulation of solid surfaces in the past decade, especially with the development of high-speed computer hardware. The latter makes a large number of calculations and more accurate solutions possible. As a result, the calculation of the structure and properties of mineral surfaces has been much closer to the experimental results, prompting many researchers to adopt the first-principles methodology to uncover the crystal structure, electronic properties, and surface adsorption of minerals (e.g., [31,32,33,34,35,36,37,38,39]).

The applications of DFT include not only static energy calculations but also molecular dynamics simulations. DFT-based molecular dynamics is called ab initio molecular dynamics (AIMD). AIMD and force field-based classical molecular dynamics (CMD) are two of the most discussed simulation methods to describe the thermal and atomic motions in the field of mineral flotation (e.g., [40,41,42,43]). The differences between AIMD and CMD can be categorized into two aspects. First, AIMD is based on the quantum Schrödinger equation, while CMD relies on Newton’s equations of motion. Second, AIMD is based on real physical potentials, while CMD relies on semi-empirical effective potentials that approximate quantum effects [44]. Computational accuracy, computational costs, and application systems are the three primary criteria to be evaluated when selecting molecular dynamics methods. AIMD is more accurate, but the higher computational cost limits its application to small systems containing 500 atoms with a total simulation time of 100 ps, whereas for CMD, it is less accurate, but the relatively low computational cost allows it to be applied to large systems containing millions of atoms with a total simulation time on the order of microseconds [45,46].

1.4. Experimental Methods Alongside Theoretical Calculations

To date, an abundant number of experimental methods have been employed alongside quantum chemical calculations to reveal the mineral–reagent interfacial characteristics and interaction mechanisms, as well as to evaluate the floatability and selectivity of flotation reagents. Among them, the Fourier-transform infrared spectroscopy (FTIR), time-of-flight secondary ion mass spectrometry (Tof-SIMS), and X-ray photoelectron spectroscopy (XPS) methods are mostly used in combination with DFT calculations to determine the chemical compositions of mineral surfaces. Table S2 in the Supplementary Materials illustrates the main experimental methods alongside theoretical calculations and their applications in detail [47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62].

1.5. Classification of Mineral–Reagent Interactions

With the combined application of quantum chemical calculations with a variety of advanced experimental methods, a more comprehensive and in-depth understanding of the atomic interactions between flotation reagents and mineral surfaces has been realized. Based on the strengths of the interactions, they are broadly categorized into two types: chemisorption and physisorption. Additionally, hydrogen bonding, van der Waals (vdW) adsorption, and electrostatic interaction can also be grouped into physisorption. The two most significant criteria used to classify the types of atomic interactions are the interaction energy (ΔE_ads) and interaction distance. The interaction energy can be calculated as follows [63,64]:

ΔE_ads = E_{surface+reagent} − (E_surface + E_reagent)

(1)

where ΔE_ads denotes the adsorption energy, E_{surface+reagent} is the total energy of the mineral surface with the flotation reagent, E_surface represents the energy of the mineral surface, and E_reagent is the energy of the flotation reagent.

The ΔE_ads values for weak, medium, and strong chemisorptions are 40–60, 60–100, and >100 kJ/mol, respectively, and the ΔE_ads values for the van der Waals (vdW) adsorption, hydrogen bonding, and physisorption are 0–5, 20–40, and <40 kJ/mol, respectively [63,64]. The more negative the value of ΔE_ads, the more favorable the adsorption of reagents to mineral surfaces [63,64]. The interaction distance of chemisorption is 0.7–1.3 times the sum of the atomic radii of the bonding atoms, and that of physisorption is greater than 1.3 times the sum of the atomic radii of the interacting atoms. The interaction distance for hydrogen bonding is 1.5~3.5 Å. The shorter the interaction distance, the stronger the electron cloud overlap, and thus, the stronger the bond covalency [63,64].

Notably, there is no absolute criterion for the classification of atomic interactions, and it is difficult to make a conclusion from a single criterion; only a combination of each criterion, such as the interaction energy, interaction distance, DOS, Mulliken/Mulliken bonding population analysis, electronegativity, charge transfer, the energy of the highest occupied molecular orbital (E_HOMO), the energy difference between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) (ΔE_HOMO-LUMO) [65], and so on, can be used to make the most probable judgment [63,64]. The premise of all of these is to ensure the rationality of the adsorption model. Hence, constructing a reliable and reasonable model for mineral–reagent interaction is a prerequisite for quantum chemistry calculations.

The purpose of this review is to comprehensively and systematically describe the application of quantum chemistry in the research of flotation reagents for the flotation separation of minerals and metals (including non-ferrous metals; ferrous metals; rare metals; silicate minerals; rare earth minerals; precious metal minerals; Sn-, W-, Ca-, Mo-, As-, and P-bearing minerals; and other minerals) in the past decade in order to emphasize the key directions of the current research on flotation reagents, to point out the main gaps in the current research on flotation reagents, and more importantly, to summarize the theoretical basis for the design and development of novel flotation reagents. We believe that the current advances would ultimately allow for the exploration of the possibility of molecular design and the development of green, efficient, and selective flotation reagents with targeted effects by means of quantum chemistry.

2. Non-Ferrous Metals

In general, non-ferrous metals are metals that do not contain Fe., i.e., all pure metals are non-ferrous. Among them, quantum chemical studies of flotation reagents have been performed for Cu-, Pb-, and Zn-bearing minerals.

2.1. Non-Ferrous Oxide Minerals

2.1.1. Copper Oxide Minerals

Common copper oxide minerals include malachite, Cu₂(CO₃)(OH)₂; azurite, Cu₃(CO₃)₂(OH)₂; chrysocolla, (Cu_2−xAl_x)H_2−xSi₂O₅(OH)₄·nH₂O; and cuprite, Cu₂O. Most quantum chemical studies of copper oxide minerals have been focused on malachite. Yang et al. [66] investigated the reactivity of aliphatic oxime derivatives including C₇H₁₅CX=NOH (X = H, CH₃, NH₂ or OH, including octanaldoxime, n-octanohydroxamic acid, n-hydroxyoctanimidamide, and methyl n-heptyl ketoxime) as copper flotation collectors towards malachite using the DFT method. The structure–reactivity relations established in this study provided an understanding of the structural demand for aliphatic oximes to recover copper oxide minerals at the atomic scale. Using the same methodology, Yang et al. [67] also studied the chemical reactivity of azolethione derivatives’ (including 1,3,4-oxadiazole-2-thione, 4-amino-5-heptyl-1,2,4-triazole-3-thione, 5-heptyl-1,2,4-triazole-3-thione, 5-heptyl-1,3,4-thiadiazole-2-thione, and 6-heptyl-1,2,4,5-tetrazine-3-thione) collectors on malachite surfaces, which provided an atomic-level understanding of the structure–property relations of azolethione derivatives as chelating reagents for copper mineral flotation. Lu et al. [68] designed a series of amide collectors (including Nhydroxy-N-benzyl butyramide (NHNBB), N-hydroxy-N-benzyl acetamide, N-hydroxy-N-phenyl acetamide and N-hydroxy-Nphenyl butyramide) and used them to achieve the efficient separation of malachite from calcite and quartz in the absence of a frother and activator. By combining quantum chemical calculations, solution chemical analyses, X-ray photoelectron spectroscopy (XPS), and Fourier-transform infrared spectroscopy (FTIR), the results suggested that, in addition to electrostatic attraction, NHNBB could interact with malachite through chemisorption to generate a five-membered ring complex on a malachite surface, thus exhibiting an efficient collection power for malachite.

Furthermore, Chen et al. [69] reported the promotional effect of ammonium sulphate ((NH₄)₂SO₄) as a modifier in malachite sulfidization flotation with butyl xanthate (BX) as a collector. The DFT calculation results indicated that the adsorption energy of HS⁻ produced by the ionization of sodium sulfide (Na₂S, a sulfidizing reagent) in water was reduced by the addition of (NH₄)₂SO₄, which demonstrated that (NH₄)₂SO₄ could improve the adsorption stability of HS⁻ on a malachite surface. However, except for malachite, quantum chemistry studies of flotation reagents for azurite, chrysocolla, and cuprite have been rarely reported.

2.1.2. Lead Oxide Minerals

There are many lead oxide minerals, including cerussite, PbCO₃; anglesite, PbSO₄; wulfenite, PbMoO₄; vanadinite, Pb[Cl(VO₄)₃; pyromorphite, Pb₅(PO₄)₃Cl; mimetite, Pb₅(AsO₄)₃Cl; plumbojarosite, PbFe₆[(OH)₆(SO₄)₂; and so forth. In particular, quantum chemical studies on the sulfidization mechanisms of cerussite are of paramount importance. Since lead oxide minerals are more prone to have higher solubility and stronger surface hydration than lead sulfide minerals, conventional sulfhydryl-based reagents are less effective in their flotation. A sulfidization treatment is able to enhance the hydrophobicity of oxide minerals, thus enabling them to be better recovered by the collector. Therefore, the sulfidization–xanthate method is employed as one of the most effective methods for cerussite flotation in industry, with sodium hydrosulfide (NaSH)/sodium sulfide (Na₂S) being added as the sulfidizing reagents [70,71,72].

It is worth noting that the existing literature on DFT studies of cerussite primarily concentrates on HS⁻ adsorption. For instance, Feng et al. [73], employing XPS and DFT, investigated the sulfidization mechanism of cerussite by adsorbing a main hydrolyzate of sulfidizing reagents, a single HS⁻, on a cerussite (110) surface (see Figure 2a). Nevertheless, the experimental results confirmed that a PbS film formed on the cerussite surface, leading to changes in the coordination structure of the Pb atoms on the surface [74,75,76,77,78]. Therefore, to better reflect the actual surface structure of sulfidized cerussite, Tang et al. [79] employed the density functional based tight binding (DFTB+) method to study the microscopic mechanism of cerussite flotation using the sulfidization–xanthate method. As shown in Figure 2b, a sulfidization model of a PbS film covering the PbCO₃ (001) surface was established, resulting in a change in the ligand type of the surface Pb atoms from the O ligand to the S ligand. The results demonstrated that the adsorption energies of a single water molecule or multiple water molecules adsorbed on a sulfidized cerussite (sul-PbCO₃) surface become less negative, whereas the formation energy of BX and water co-adsorbed on a sul-PbCO₃ surface become more negative. As predicted using the HSAB theory, after sulfidization, Pb²⁺ changed its behavior from a hard acid in PbCO₃ to a soft acid in PbS, facilitating its interaction with BX (soft base), but weakening its interaction with water (hard base). For the periodic system calculations, the selection of appropriate crystal planes is a prerequisite for the subsequent calculations of reagent adsorption and molecular dynamics. A detailed protocol for selecting crystal planes can be found in Section S3 of the Supplementary Materials.

2.1.3. Zinc Oxide Minerals

The main zinc oxide minerals are smithsonite, ZnCO₃; hemimorphite, Zn₄(Si₂O₇)(OH)₂·H₂O; zincite, ZnO; willemite, Zn₂SiO₄; and hydrozincite, 3Zn(OH)₂·2ZnCO₃. Similar to cerussite, quantum chemical studies on smithsonite have mainly focused on the discussion of its sulfidization mechanism. For example, Chen et al. [80] developed a model of a smithsonite (101) surface using the DFTB+ method and simulated the adsorption of a monolayer of S²⁻, HS⁻, and OH⁻ ions on the surface to investigate the influences of the sulfidization effect and hydration effect on the adsorption on a smithsonite surface. At low concentrations of sodium sulfide, HS⁻ ions formed a Zn−SH−SH structure on a smithsonite surface, which prevented dodecylamine (DDA) from interacting with the Zn atoms on the smithsonite surface. This was consistent with the flotation practice of smithsonite, which required large quantities of sodium sulfide. However, at high Na₂S concentrations, a ZnS film formed on the smithsonite surface, which facilitated the adsorption of DDA. DFT calculations were also conducted by Zhao et al. [81] to investigate the adsorption mechanism of HS⁻ on a smithsonite (101) surface. The results manifested that HS⁻ ions could spontaneously interact with the top-, bottom-, and bridge-site Zn atoms on a smithsonite surface, resulting in the formation of a stable Zn−S structure on the surface. Meanwhile, at the atomic level, a slight oxidation of smithsonite was also found to have occurred during the sulfidization of smithsonite.

Liu et al. [82] combined theoretical and experimental methods to discuss the influences of six different carboxyl collectors (including linoleic acid (CH₃(CH₂)₄(CH)₂CH₂(CH)₂(CH)₇COOH), oleic acid (CH₃(CH₂)₇(CH)₂(CH₂)₇COOH), lauric acid (CH₃(CH₂)₁₀COOH), stearic acid (CH₃(CH₂)₁₆COOH), palmitic acid (CH₃(CH₂)₁₄COOH), and naphthenic acid (C₆H₁₁COOH)) on smithsonite flotation. The DFT calculations indicated that unsaturated carboxyl collectors exhibit large electrophilicity, whereas benzenoid and saturated carboxyl collectors possess small electrophilicity. Only linoleic acid and oleic acid were considered effective collectors towards smithsonite in the presence of water molecules. It was also found that the adsorption of water molecules weakened the hybridization of the 3s and 3p orbitals and the electrophilicity of the surface Zn atoms, which was detrimental to the adsorption of carboxyl collectors with small electrophilicity.

For hemimorphite (Zn₄Si₂O₇(OH)₂⋅H₂O), Zhao et al. [83] employed the DFT method and the bond valence model to evaluate the correlation between complex stability constants and the bonding strength of the polar groups of flotation collectors (including N-dodecanoylglycine, N-lauroylsarcosine, N-laurylaminoacetic acid, and lauric acid). Through the results, it was believed that the bonding strengths of the polar groups controlled the collecting power for hemimorphite flotation. Jia et al. [84] combined a series of experimental methodologies and DFT calculations to study the adsorption mechanism of a green biosurfactant sodium N-lauroylsarcosinate collector on hemimorphite and quartz surfaces. The DFT results showed that the two O atoms of the carboxyl group in sodium N-lauroylsarcosinate were bonded with Zn and H atoms and adsorbed on a hemimorphite surface in a bidentate bonding structure, indicating that sodium N-lauroylsarcosinate has excellent selectivity for hemimorphite and has a great potential for industrial application in hemimorphite–quartz flotation separation. Zuo et al. [85], employing the DFT method, studied the effects of Na₂S sulfidization and sodium sulfosalicylate (C₇H₅NaO₆S⋅2H₂O) activation on hemimorphite flotation. The calculation results confirmed that the Zn atoms on the hemimorphite (110) surface were separated from the surface O atoms, which resulted in the formation of many vacancies, thereby greatly reducing the steric hindrance effect during sulfidization. Zn(SSA)₂²⁺, formed via the complexation of C₇H₅NaO₆S⋅2H₂O with Zn²⁺, dissolved from the hemimorphite surface and participated in sulfidization reactions, i.e., Zn₄Si₂O₇(OH)₂⋅H₂O→Zn²⁺→Zn(SSA)₂²⁺→ZnS, making the sulfidization process simple and efficient.

2.1.4. Other Non-Ferrous Oxide Minerals

In addition to Cu, Pb, and Zn oxide minerals, the quantum chemistry of flotation for Mg- and Na-containing oxide minerals has been studied in detail. The main magnesium resources utilized globally are magnesite, dolomite, brucite, carnallite, and olivine. The quantum chemical studies of the interactions of flotation reagents with mineral surfaces have been focused on magnesite (MgCO₃) and dolomite (CaMg(CO₃)₂). The research was conducted in two main categories: collectors and depressants. Through DFT calculations, Zhong et al. [86] and Tang et al. [87] investigated the adsorption mechanisms of α-chloro-oleate acid and cetyl phosphate, respectively, as collectors on a magnesite surface. Li et al. [88] compared the effects of dodecylamine and n-octanol as collectors on the removal of impurities in the reverse flotation of magnesite by combining experimental studies and computational simulations. It was shown that the dissolved Ca²⁺ ions in dolomite weaken the selectivity of anion flotation, which is one of the reasons for the difficulty in separating dolomite and magnesite. Whereafter, Liu et al. [89] further employed sodium fatty alcohol polyoxyethylene ether sulfonate (AESNa) as an ion-tolerance collector to understand the reason why AESNa reduces or eliminates the adverse effect of Ca²⁺ ions. The results demonstrated that the EO groups of AESNa were complexed with Ca²⁺ ions via electrostatic interactions, making it difficult for Ca²⁺ ions to access the polar head groups of the collectors, hence leaving magnesite flotation unaffected. For depressants, Sun et al. [90] first introduced ethylenediamine tetra (methylene phosphonic acid) sodium (EDTMPS) as a chelation depressant to improve the separation efficiency of magnesite from quartz with DDA used as a collector in this system. The DFT calculations revealed that the pre-adsorption of EDTMPS weakened the DDA-magnesite interaction; this observation was confirmed by the expansion of the distance between the magnesite surface and the DDA N atoms (from approximately 2.64 Å to 8.00 Å) in the presence of EDTMPS.

Magnesite and dolomite are Mg-bearing minerals with similar crystal structures, which makes their separation a challenge in production. Reverse flotation reagents for magnesite and dolomite are important subjects of quantum chemical studies directed at dolomite. Sun et al. [91] adopted dodecane-1,2-diyl bis (dihydrogen phosphate) (DBDP), and Zhao et al. [92] introduced dimethylaminopropyl lauramide (DPLA) as dolomite collectors in magnesite–dolomite reverse flotation. The DFT simulations, micro-flotation tests, contact angle and zeta potential measurements, and FTIR and atomic force microscopy (AFM) analyses verified their potential for industrial applications. Yao et al. [93] enhanced the flotation separation of dolomite and magnesite using DDA as a collector and using sodium dihydrogen phosphate (SDP) as an activator for dolomite. The DFT calculations suggested that the strength of spontaneous adsorption between dolomite and SDP was much greater than that between magnesite and SDP, indicating that SDP posed a selective effect on the flotation separation of magnesite and dolomite.

Moreover, phosphate flotation plants also prefer to separate apatite and dolomite via reverse flotation, with sulfuric acid (H₂SO₄) being commonly used as a specific apatite depressor. Nevertheless, this inorganic acid is unable to prevent the flotation of dolomite with fatty acid collectors, such as oleate anions. The depression mechanism of H₂SO₄ on apatite flotation is well understood, whereas that on dolomite is unclear. Cao et al. [94] studied the effect of SO₄²⁻ anions on the adsorption of oleate anions on a dolomite (104) surface and compared the adsorption behavior of SO₄²⁻ anions onto both perfect and CO₃-defect dolomite surfaces using DFT. The results indicated that only the SO₄²⁻ anions are adsorbed on the CO₃-defect surface, where they bound to Ca atoms. The rest of the Mg and Ca atoms at the defect sites will further interact with oleate anions to produce new Mg−O and Ca−O ionic bonds. SO₄²⁻ anions and oleate might coexist on a dolomite surface. This phenomenon helped to illustrate the flotation process of dolomite treated with H₂SO₄.

Feldspar-group minerals (KAlSi₃O₈, NaAlSi₃O₈, and CaAl₂Si₂O₈) are both aluminosilicate ores and major sources of Na. Oxalic acid, as a pH modifier, is a commonly used organic substance in feldspar flotation, and it has three different configurations (C₂O₄²⁻, HC₂O₄⁻, and H₂C₂O₄) depending on the pH value of the aqueous solutions. Xue et al. [95] employed DFT calculations, classical molecular dynamic (CMD) simulations, and frequency calculations to understand the adsorption mechanism of oxalic acid at the water–feldspar interface. The results demonstrated that the adsorption of H₂C₂O₄ on a feldspar surface was a physical outer-sphere adsorption with the formation of hydrogen bonds, while the adsorption of C₂O₄²⁻ and HC₂O₄⁻ belonged to inner-sphere adsorption, which preferred to coordinate with Al active sites on a feldspar surface rather than Si active sites. The order of adsorbed substances on feldspar surface was C₂O₄²⁻ > HC₂O₄⁻ > H₂O > H₂C₂O₄. At a lower pH of 0.00, water instead of H₂C₂O₄ would attack the Al active sites to promote Al dissolution. At higher pH values of 2.76 and 6.00, HC₂O₄⁻ and C₂O₄²⁻, but not water, would attack the Al active sites to facilitate Al dissolution; however, at acidic conditions, H⁺ would attack Si active sites instead of Al active sites to promote Si dissolution.

2.2. Non-Ferrous Sulfide Minerals

For non-ferrous sulfide minerals, Liu et al. [96] used DFT calculations to investigate the structure–activity relations between four chelating collectors (including diisobutyl monothiophosphinate (DIBMTPI), diisobutyl monothiophosphate (DIBMTPA), diisobutyl dithiophosphinate (DIBDTPI), and diisobutyl dithiophosphate (DIBDTPA)) and the Cu-, Au-, Ag-, and Pb-bearing sulfide minerals and provided a potential method for the molecular design of new reagents for improving metal recovery. Moreover, Cui et al. [97] introduced the principle of coordination chemistry and the DFT method to study the interactions of methyl xanthate as a collector with galena and sphalerite under weak acidic and neutral conditions, which contributed to a systematic understanding of the interaction mechanism of xanthate collectors on the surfaces of non-ferrous sulfide minerals and provided a theoretical basis for future studies.

2.2.1. Copper Sulfide Minerals

Zhao et al. [98] synthesized two ether thionocarbamates, O-(2-butoxy-1-methylethoxy) isopropyl-N-ethoxycarbonyl thionocarbamate (BMIPECTC), and O-butoxy isopropyl-N-ethoxycarbonyl thionocarbamate (BIPECTC) and studied their collection powers using adsorption measurements, ultraviolet spectra (UV) and FTIR analysis, flotation tests, and DFT calculations. Quantum chemistry calculations revealed that both BIPECTC and BMIPECTC exhibited greater collection power towards copper minerals in terms of binding model simulation with Cu ions, molecular hydrophobicity, and frontier molecular orbital analysis compared to O-isobutyl-N-ethoxycarbonyl thionocarbamate (IBECTC) and O-isopropyl-N-ethyl thionocarbamate (IPETC). Li et al. [99] understood the mechanism of kerosene influence on chalcopyrite floatability in a Mg²⁺-bearing solution (which simulated seawater as flotation medium) by means of DFT calculations and extended the Derjaguin–Landau–Verwey–Overbeek (EDLVO) theory. The results indicated that kerosene was selectively adsorbed on the Mg(OH)₂ surface and formed agglomerates, thus hindering the adsorption of Mg(OH)₂ precipitates on the chalcopyrite surface. When additional kerosene was dosed, hydrophobic agglomerates were also formed due to the adsorption of kerosene on chalcopyrite, which further improved the floatability of chalcopyrite. Mkhonto et al. [100] employed DFT to investigate the adsorption energies, electronic properties, and bonding behavior related to the reactivity of five collectors, including O-butyl- O-butyl-N-butoxycarbonyl-thiocarbamate (BBCTC), N-ethoxycarbonyl-thiocarbamate (BECTC), O-isobutyl-N-butoxycarbonyl-thiocarbamate (IBBCTC), O-isobutyl-N-isobutoxycarbonyl-thiocarbamate (IBIBCTC), and O-isobutyl-N-ethoxycarbonyl-thiocarbamate (IBECTC), with a chalcopyrite (112) surface. The results showed that BECTC and BBCTC are better collectors for the selective flotation of chalcopyrite, indicating that collectors possessing a straight hydrocarbon chain might be preferable to those possessing a branched hydrocarbon chain. Moreover, He et al. [101] proposed a model combining quantum chemistry and machine learning to facilitate the screening of solidophilic reagents for chalcopyrite flotation. A 47-molecule flotation set was built at the level of B3LYP/def2-TZVP under solvation effects to acquire information about their affinity to potential active sites on a chalcopyrite surface (e.g., Fe(II), Cu(I), Cu(II) sites).

In addition to the research on flotation reagents for chalcopyrite itself, the reagents used to separate chalcopyrite from pyromorphite, pyrite, and galena have attracted a great deal of attention. Molybdenite and chalcopyrite have similar floatability under the action of collectors; hence, it is difficult to selectively separate them. Therefore, depressants are essential to achieve the necessary Cu-Mo separation. Conventional inorganic depressants, such as lime, cyanide, and sulfides, inevitably cause low selective depression for the flotation separation of various sulfide minerals. Therefore, organic depressants have become one of the research focuses in recent years because of their biodegradability, excellent selectivity, high flexibility, potential modifications, and rich sources. The adsorption mechanisms of multiple depressants such as L-cysteine [102], rhodanine-3-acetic acid [103], disodium carboxymethyl trithiocarbonate [104], thioglycolic acid [105], 3-amino-5-mercapto-1,2,4-triazole [106], 2-((5-mercapto-1,3,4-thiadiazol-2-yl)thio)acetic acid [107], and 5-amino-1,3,4-thiadiazole-2-thiol [108] on a chalcopyrite surface were studied using DFT supplemented with micro-flotation and bench-scale flotation experiments and various advanced characterization technologies, including ultraviolet-visible (UV-vis) spectroscopy, FTIR, contact angle and XPS analysis, time-of-flight secondary ion mass spectrometry (Tof-SIMS) measurements, adsorption capacity, and zeta potential measurements, which helped to offer a theoretical basis of the molecular design of organic depressants with respect to chalcopyrite.

Pyrite (FeS₂) often associates with chalcopyrite, which is a common gangue mineral in copper flotation. Therefore, depressants are often required to restrict the flotation of pyrite to acquire qualified concentrates during chalcopyrite–pyrite flotation separation. In this case, Wu et al. [109] analyzed the sodium butyl xanthate (SBX) adsorption on Cu- and Fe-deficient chalcopyrite surfaces using DFT calculations and examined the galvanic effect on flotation behaviors using mixed mineral flotation tests. The DFT calculations revealed that the Cu-/Fe-deficient surface could not facilitate SBX adsorption. The galvanic interaction of the chalcopyrite–pyrite couple increased pyrite recovery by means of copper activation but decreased chalcopyrite recovery due to a weaker SBX adsorption, making their flotation separation more difficult. Mkhonto et al. [110] adopted DFT combined with micro-flotation tests, electronic property analysis, and XPS to study the interaction mechanisms of three thiocarbamate collectors, Sallyl-N-diethyl-dithiocarbamate (ADEDTC), O-isopropyl-N-diethyl-thionocarbamate (IPDETC), and IPETC, on pyrite (100) and reconstructed chalcopyrite (112) surfaces. The results demonstrated that the active sites on the chalcopyrite surface were Cu atoms rather than Fe atoms. Among the three collectors, the adsorption of ADEDTC was the strongest, and its adsorption on chalcopyrite was stronger than that on the pyrite surface. Zhang et al. [111] systematically investigated the chalcopyrite–pyrite separation mechanism at high alkaline conditions using adsorption studies, DFT calculations, flotation tests, FTIR analysis, and zeta potential measurements. The flotation tests and various measurements indicated that the flotation separation of chalcopyrite and pyrite could be realized at highly alkaline conditions. The DFT calculations further confirmed that the adsorption of SBX on chalcopyrite Cu sites was stronger than on pyrite Fe sites. Since the hydroxyl ion had a stronger affinity towards pyrite, it could adsorb efficiently on the pyrite surface rather than on the chalcopyrite surface.

Complex and refractory Cu-Pb polymetallic sulfide minerals such as galena and chalcopyrite have always been difficult to separate because of their similar surface wettability. Hence, in galena–chalcopyrite flotation separation, depressants should be added to enlarge the difference in their floatability, which is regarded as the most common method in their separation. Liu et al. [112] systematically investigated the selectivity of three mercapto acids (mercaptoacetic acid, 3-mercaptoisobutyric acid, and 3-mercaptopropionic acid) for the separation of galena and chalcopyrite using flotation experiments combined with first-principles calculations. Both the experimental and calculation results demonstrated that mercapto acids have a higher affinity towards chalcopyrite, and, among them, the selectivity of 3-mercaptopropionic and 3-mercaptoisobutyric acids are better than that of mercaptoacetic acid, which makes them the most selective depressants in the improved flotation separation of chalcopyrite and galena. Zhang et al. [113] synthesized a novel depressant, dithiocarbamated poly (acrylamide-allyamine) (DTC-PAA), and employed it as a depressant for galena in chalcopyrite–galena flotation separation. The DFT calculation results showed that when using O-isopropyl-N-ethyl thiocarbamate (IPETC) as the collector, DTC-PAA was chemisorbed onto galena surface Pb sites through the dithiocarbamate groups in DTC-PAA, consequently achieving the effective flotation separation of galena from chalcopyrite. These studies have enabled the development of reagent molecules with high selectivity by quantum chemical means, exemplifying the potential for the effective separation of Cu-Pb polymetallic sulfide minerals.

Quantum chemical studies on flotation reagents for covellite (CuS) have also been reported. Porento and Hirva [114] performed ab initio calculations to study the interaction of three different sulfhydryl surfactants, 1,1,1-butanetrithiol (BTT), diethyl dithiocarbamate, and ethyl xanthate, with a covellite (001) surface. The results suggested that the mineral–reagent interaction of BTT was the strongest, making it a potential collector for the covellite flotation. Ma et al. [115], using single-mineral and mixed-mineral flotation tests, in conjunction with adsorption measurements, an FTIR analysis, and DFT calculations, studied the collection mechanism of ethyl isobutyl xanthogenic acetate (EIBXAC) in the flotation of secondary copper sulfide minerals (for instance, covellite and digenite). The results confirmed that EIBXAC could be chemisorbed on the covellite (001) and digenite (001) surfaces, showing the possibility of using EIBXAC to collect secondary copper sulfide minerals. Botero et al. [116] studied the interaction mechanisms of potassium amyl xanthate (PAX) and O-isopropyl-N-ethyl thionocarbamate (IPETC) as collectors on a covellite surface. The DFT calculations predicted that PAX bonded to the surface Cu atoms via the C–S and C=S groups, whereas IPETC bonded to the surface Cu atoms only through the C=S group.

Nevertheless, quantum chemical studies of two other common copper sulfide minerals, chalcocite (Cu₂S) and bornite (Cu₃FeS₃), have been reported only in terms of surface properties and electronic structures [117,118], and they do not report on the interaction mechanisms involving reagent adsorption. As a consequence, their study may become one of the main directions for future research.

2.2.2. Lead Sulfide Minerals

Galena, PbS, is the most abundant lead mineral in nature, often associated with sphalerite, chalcopyrite, and pyrite, and thus, the flotation separation reagents for it are the invariable subject of research for many scholars. Ma et al. [119], Jia et al. [120], and Jia et al. [121] synthesized novel collectors, including S-benzoyl-N,N-diethyldithiocarbamate (BEDTC), β-oxo thioamide surfactant 3-(ethylamino)-N-phenyl-3-thioxopropanamide (EAPhTXPA), and trimethylacetyl thiobenzamide (TTBA), respectively, and investigated their interaction mechanisms in the flotation separation of galena and sphalerite via flotation tests, adsorption measurements, FTIR and XPS analyses, and DFT calculations. The results showed that all of these collectors have stronger collection powers than traditional collectors and have greater selectivity towards galena against sphalerite. The BEDTC acted as a bidentate ligand, bonding with galena Pb atoms through the carbonyl O and thiol S atoms to form two different adsorption geometries, one with two distinct Pb atoms to form a bullet-shaped complex, and the other with the same surface Pb atom to form a six-membered ring complex [119]. Whereas EAPhTXPA interactes with PbS by forming Pb–O, Pb–N, and Pb–S bonds [120], TTBA interacts with PbS via the formation of Pb–O and Pb–S bonds [121]. In addition, Zhang et al. [122] developed a reagent scheme consisting of aerofloat collectors as well as Zn²⁺ and SO₃²⁻ depressants for the flotation separation of galena from sphalerite-rich sulfide minerals. Ab initio molecular dynamics (AIMD) simulations and static calculations were used to investigate the mechanism of the employed reagent scheme on the flotation separation of galena and sphalerite at the atomic level. The results suggested that Zn²⁺ and SO₃²⁻ have synergistic effects on depressing sphalerite, whereas among them, aerofloat collectors exhibit greater selectivity towards galena.

Based on the DFT calculation, Zhang et al. [123] designed and synthesized polymaleamide-propyl dithiocarbamate (PMA-PDTC) as a novel depressant and investigated its depression effect in separating galena from chalcopyrite. Wei et al. [124] employed three 2-mercaptobenzimidazole derivatives, including 1-benze-2-mercapto-benzimidazole (BMBI), 1-ethyl-2-mercapto-benzimidazole (EMBI), and 1-propyl-2-mercapto-benzimidazole (PMBI), as chelating collectors to understand their collection mechanisms in the flotation separation of galena from pyrite via lab-scale flotation tests and DFT simulations. The results indicated that the floatability of these collectors follows the increasing order of EMBI < PMBI < BMBI. Later on, Chen et al. [125] adopted computational simulations and the microcalorimetry method to study the adsorption of xanthate, dithiocarbamate, and dithiophosphate on pyrite and galena surfaces. The results revealed that pyrite Fe atoms are more active than galena Pb atoms, and the reagents coordinated mainly to the surfaces through interactions between their S atoms and surface Pb/Fe atoms. The adsorption of xanthate on the pyrite surface was stronger than that on the galena surface, while those of dithiocarbamate and dithiophosphate were the opposite, and they showed good selectivity in the separation of pyrite and galena. Furthermore, Dong et al. [126] used a novel collector S-benzyl-N-ethoxycarbonyl thiocarbamate (BET) to study its interaction mechanism for selectively separating galena from a polymetallic sulfide ore using flotation experiments, adsorption tests, and an FTIR spectra analysis in conjunction with DFT. The results manifested that BET is chemisorbed on the galena surface via the formation of normal covalent bonds between carbonyl S atoms and surface Pb atoms and back donation covalent bonds between carbonyl O atoms and surface Pb atoms. BET showed strong collection power and good selectivity for galena and was therefore considered to have a wide range of industrial application prospects.

Quantum chemical studies of flotation reagents for jamesonite (Pb₄FeSb₆S₁₄) have also been reported. Cui et al. [127] adopted the mixed collectors of sodium diethyldithiocarbamate (DDTC) and sodium diisobutyl dithiophosphinate (3418A) to enhance the flotation of jamesonite, and the adsorption mechanism was studied by means of flotation experiments; FTIR, scanning electron microscopy, and energy-dispersive X-ray spectroscopy (SEM-EDS) analyses; and dispersion-corrected density functional theory (DFT-D) calculations. In this study, the dispersion correction was added to make the calculation results more consistent with the experimental values, as the dispersion interaction caused by the incorrect long-range behavior of the correlation potential was corrected [127]. The results showed that the DDTC/3418A mixture with a 2:1 molar ratio exhibits excellent selectivity and significantly enhances the flotation performance of natural refractory Pb-Sb-Zn minerals. Such mixture could perform synergistic chemisorption on a jamesonite surface, with 3418A rapidly overcoming the binding interactions of the surrounding molecules from the micelles and then forming effective adsorption on the jamesonite surface, while DDTC reinforces the hydrophobic layer through the formation of H^…N hydrogen bonds. Additionally, Li et al. [128], using flotation tests supplemented with DFT calculations, compared the depression performance of ten kinds of organic depressants for the flotation of marmatite, jamesonite, and pyrite. The results showed that jamesonite could be well depressed via pyrogallic acid and 4-amino-hydroxybenzene, indicating that the presence of a benzene ring in the molecule could enhance the depression performance. This study also proposed that the calculations of frontier orbitals could well explain the interactions between sulfide minerals and organic depressants, and the hydrophilicity of organic depressants should be taken into account when applying frontier orbitals to investigate their depression effects on sulfide minerals.

2.2.3. Zinc Sulfide Minerals

Sphalerite and wurtzite are two polymorphs of ZnS and represent the most widespread zinc sulfide minerals in nature. Of these two, the quantum chemistry of sphalerite flotation reagents has been studied since the beginning of the 21st century, with research topics being mainly focused on non-activated collection, activated collection, and the development and mechanistic study of collectors and depressants. Regarding the non-activated collection of sphalerite, Liu et al. [129] tested three thiophenol collectors, including 2-fluoro thiophenol, 2-hydroxy thiophenol, and 2-amino thiophenol, for the flotation of marmatite (Fe-bearing sphalerite) without the addition of an activator, copper sulfate (CuSO₄). Both the flotation tests and quantum chemical calculations confirmed that 2-amino thiophenol has the strongest collection power among the three reagents.

For the activated collection, Porento et al. [130] employed ab initio cluster model calculations, and Liu et al. [131,132,133] adopted DFT calculations to investigate the effect of Cu atoms on the adsorption of ethyl xanthate (EX) on sphalerite (111) and (110) surfaces, respectively. These studies revealed that Cu adsorbed on sphalerite S atoms [133] and Cu substituted with sphalerite Zn atoms [132] could lead to the activation of sphalerite. The S 3p orbitals of EX and the Cu 3d orbitals of Cu-activated sphalerite overlapped exactly to the maximum extent near the Fermi level (E_F), implying stable chemisorption. Moreover, Sarvaramini et al. [134] combined flotation tests and DFT simulations to study the interactions of the collector DIBDTPI with un-activated and Pb-activated sphalerite. Unlike copper activation, it was impossible to substitute Zn cations in the lattice with Pb at the surface due to the larger van der Waals radius of Pb and the pronounced lattice deformations of the sphalerite structure. Dissolved collectors are attached to the Pb-activated sphalerite surface via adsorbed Pb cations or Pb(OH)₂. The adsorbed Pb cations, in return, are able to attach DIBDTPI through forming two bidentate covalent bonds between the collector S atoms and Pb cations. The interactions of surface Pb(OH)₂ with the collector is realized by the formation of covalent bonds between the S head of DIBDTPI and the Pb cations of Pb(OH)₂. Aside from these, Long et al. [131] performed a DFT study to investigate the interaction mechanism of collector EX with non-activated/Cu-activated sphalerite (110) surfaces in the absence and presence of water molecules. The calculation showed that the adsorption of water molecules drastically changes the properties of the sphalerite surface, leading to a decrease in the reactivity of surface Zn atoms with xanthate, but the presence of water has a rare effect on the properties of the Cu-activated sphalerite surface.

The N,N-dimethyldi-thiocarbamate (DMDC), the lowest homologue of dialkyldithiocarbamate salts, has been found to be effective in chalcopyrite flotation and to significantly depress Cu-activated marmatite (the Fe-rich variety of sphalerite), with excellent selectivity in Cu-Zn sulfide minerals. However, the separation mechanisms of Cu-Zn sulfide minerals are unclear. Therefore, Qin et al. [135] adopted UV-vis spectroscopy, FTIR, and a first-principles study to investigate the effects of the sodium salt of DMDC with or without BX on the flotation of chalcopyrite, marmatite, and sphalerite. It was found that the presence of Fe could hinder the activation of sphalerite, and thus, marmatite was more likely to be depressed than sphalerite. DMDC could enhance the recovery of chalcopyrite at pH 7.5, but it is detrimental to the recovery of Cu-activated sphalerite/mamatite in the presence of BX. Meanwhile, the Cu-activated marmatite was depressed more obviously. The adsorption of DMDC on mineral surfaces occurred through the interaction of S 3p orbitals with 3p, the 3d orbitals of Cu atoms, and 3d orbitals of Zn atoms. The electrons transferred from the Cu and Zn atoms to S atoms, respectively.

Zhu et al. [136] exploited a green depressant 1-hydroxyethylidene-1,1-diphosphonic acid (HEDP) for sphalerite and investigated its interaction mechanism in the flotation separation of sphalerite and galena via zeta potential tests, FTIR, XPS, ToF-SIMS, and DFT calculations. The results demonstrated that when using sodium isobutyl xanthate (SIBX) as a collector, HEDP selectively depresses sphalerite rather than galena, which achieves effective flotation separation results. This observation was attributed to the selective adsorption of HEDP on the sphalerite surface, but the pre-adsorbed HEDP on the galena surface could be easily substituted via SIBX. HEDP was attached to the sphalerite surface by coordinating Zn atoms through two distinct phosphonic acid groups, generating a six-membered chelating ring.

Previous research works have not examined the depression of Ca via the Fe content in the Fe-bearing sphalerite, nor has convincing microscopic evidence of Ca adsorption been obtained to support the depression of Ca-containing ions on the flotation of Fe-bearing sphalerite. Thus, Zhang et al. [137] investigated the effects of the Fe content and the presence of Ca on the flotation depression of marmatite under high-alkalinity environments. The following conclusions could be obtained through the results of various experiments and calculations. The increased adsorption of hydroxides (especially iron hydroxide) and Ca significantly hinders the adsorption of Cu and xanthate and increases the hydrophilicity of a high-iron sphalerite surface, which ultimately leads to an increase in the depression degree of high-iron sphalerite with the increasing Fe content.

2.2.4. Other Non-Ferrous Sulfide Minerals

Cao et al. [138] investigated the adsorption mechanism of Pb²⁺ as an activator on a stibnite (Sb₂S₃) (010) surface using DFT calculations, inductively coupled plasma mass spectrometry (ICP-MS) experiments, and micro-flotation tests. The calculation results indicated that BX could be adsorbed on both the Sb sites on the non-activated surface and the Pb sites on the Pb-activated surface, and that BX adsorption on the Pb-activated surface is more stable.

Apart from stibnite, quantum chemical studies of flotation reagents acting on other non-ferrous sulfide minerals containing Ni, Co, Sb, Hg, Cd, Bi, Al, Mg, Na, Sr, Ba, etc., for instance, pentlandite, (Ni,Fe)₉S₈; violarite, Ni₂FeS₄; millerite, NiS; linnaeite, Co₃S₄; carrollite, CuCo₂S₃; cobaltite, CoAsS; cinnabar, HgS; corderoite, Hg₃S₂(Cl,Br)₂; etc., are rarely reported.

3. Ferrous Metals

“Ferrous metals” is the industrial term for Fe, Cr, and Mn, including the alloys of these three metals. Iron is widely distributed in nature, and it is the earliest discovered and most abundant metal, of which the most important and most industrially utilized mineral sources are magnetite, hematite, and ilmenite. Previous quantum chemical studies of flotation reagents for Fe-bearing minerals concentrated mostly on these three minerals, followed by Mn-bearing minerals. However, at present, no quantum chemical studies on flotation reagents for Cr-bearing minerals have been reported.

3.1. Iron Oxide Minerals

For hematite, Fe₂O₃, its separation from quartz via reverse flotation has been the primary focus of research. Reverse flotation is the most commonly used industrial beneficiation method for hematite. In reverse cationic flotation, carboxymethyl cellulose (CMC), dextrin, and modified starch are among the most efficient hematite depressants in practice, while quartz is first activated by metal ions (for instance, Ca²⁺ ions) and then collected by sodium oleate (OL) and DDA, which has been widely applied commercially. By using AFM in conjunction with DFT, Li et al. [139] studied the depression of hematite in the oleate–starch–hematite reverse flotation system. The results showed that oleate and starch alone could be adsorbed onto the hematite (001) surface through the formation of covalent bonds. Nevertheless, in the oleate–starch–hematite system, the presence of starch hindered the adsorption of oleate on the hematite surface. In addition, the depression mechanism of causticized cassava starch (CCS) in the reverse flotation system of DDA-CCS-hematite was also studied by Zhang et al. [140]. The DFT calculations demonstrated that starch has a stronger depression effect on a hydrated hematite surface than quartz, whereas the collector RNH₃⁺ (one of the main components of DDA) has a better selective adsorption to the hydrated surface of starch-modified quartz, leading to the selective separation of hematite and quartz. Moreover, by employing flotation tests, zeta potential measurements, XPS, and CMD simulations, Wang et al. [141] used carboxymethyl chitosan (CMCS) as a depressant to investigate its interaction mechanism in the reverse flotation of quartz from hematite using collector DDA. Similar to starch, the results suggested that CMCS can prevent the adsorption of DDA on a hematite surface, while it rarely affects the interaction of DDA with a quartz surface, and thus, CMCS has a better selective depression ability than starch. Therefore, CMCS can replace starch as one of the effective depressants in the hematite–quartz reverse flotation system in practice. Liu et al. [142,143,144] screened and firstly introduced N, N-bis (2-hydroxyethyl)-N-methyl dodecyl ammonium chloride (BHMDC), N,N-dimethyl-N′-(2-hydroxyethyl)-N′-dodecyl-1,3-propanediamine (DMPDA), and N, N-Dimethyl-N′-dodecyl-1,3-propanediamine (DPDA) as collectors in the reverse flotation of hematite based on DFT calculations, and employed flotation experiments combined with various surface characterization methods, including zeta potential measurements, XPS, and FTIR, to study the interaction mechanisms of these collectors on quartz and hematite surfaces. The results indicated that all three collectors have higher selectivity than the conventional collector DDA. BHMDC and DMPDA are mainly adsorbed on mineral surfaces through hydrogen bonding and electrostatic interaction. The introduction of hydroxyl groups can improve the surface activity of the collectors and selective adsorption on the target mineral surfaces. The excellent selectivity of DPDA, on the other hand, can be attributed to a greater number of active sites and a larger polar group size.

Studies on ilmenite (FeTiO₃) collectors have also been widely reported. Zhang et al. [145] systematically assessed the adsorption mechanisms of the commonly used collectors with O-containing functional groups on the ilmenite (104) surface via DFT calculations. The adsorption energy results manifested that, for ilmenite beneficiation, the O-containing pnictogen compounds are the best among all of the O-containing collectors, and the collection performance decreases along the row of arsenic acid, phosphorous acid, and phosphoric acid ester. A partial density of states (PDOS) analysis further confirmed that the hybridization of ilmenite Ti 3d orbitals and collector O 2p orbitals contributes to the adsorption of collectors on an ilmenite surface, in which the O 2p electrons occupy the empty Ti 3d orbitals to complete the coordination shell of surface Ti atoms. Furthermore, the adsorption mechanisms of the collectors 2-ethyl-2-hexenoic hydroxamic acid (EHHA) [146], α-hydroxyoctyl phosphonic acid (HPA) [147], and benzohydroxamic acid (BHA) [148] on the ilmenite surface were evaluated via DFT simulation supplemented with flotation tests and multiple surface characterization tools. The experimental results confirmed the superior affinity of the three collectors for ilmenite, in which EHHA may be chemisorbed on the ilmenite surface in the form of 5-membered chelates, HPA may be chemisorbed on the ilmenite surface through the formation of HPA²⁻ species, and the strong selectivity of BHA is attributed to the abundance of adsorption sites and the solid adsorption of five-membered rings. These works provide new ideas for the screening of solidophilic functional groups for the reagent design of ilmenite beneficiation and help to increase the understanding of the adsorption mechanisms of O-containing collectors on oxide mineral surfaces.

Ren et al. [149] studied the adsorption mechanism of salicylhydroxamic acid (SA) as a novel collector on a coulmbite surface using DFT calculations, flotation tests, and zeta potential determination. The results indicated that the dianion of SA exhibits a higher atomic charge value, HOMO energy, and greater dipole moment, and therefore, it has a stronger collecting power for coulmbite. In addition, Rath et al. [150] used DFT calculations to compare the interaction mechanism between oleate, as a collector, and magnetite (111), hematite (110), and goethite (010) surfaces. The results suggested that magnetite forms the most stable complexes with oleate, followed by hematite, and the least stable complexes are formed by goethite; meanwhile, this trend was verified by the contact angle measurements and flotation studies of hematite, magnetite, and goethite with OL at different pH and collector concentrations. However, quantum chemical studies of flotation reagents for other iron oxide minerals, including maghemite (γ-Fe₂O₃), limonite (a mixture of iron oxides and hydroxides, mostly goethite), and siderite (FeCO₃), have not been reported.

3.2. Iron Sulfide Minerals

Pyrite, marcasite, and pyrrhotite are the three most common iron sulfide minerals, and they commonly coexist with other metal sulfides, such as copper, lead, and zinc sulfide minerals. The collection and depression of iron sulfides is essential for the recovery of other valuable metal sulfides. Hence, many DFT studies have been undertaken to investigate the collection and depression mechanisms in the flotation separation of pyrite, marcasite, and pyrrhotite from their associated minerals.

Xanthates are the most important and commonly used collectors in pyrite flotation. Therefore, it is necessary to understand the flotation chemistry of pyrite with xanthates during flotation. Han et al. [151] combined experimental and computational studies to evaluate the flotation chemistry between pyrite and isomeric xanthates (i.e., BX and iso-BX). The quantum chemical calculations elucidated that the iso-BX presents higher reactivity than that of the corresponding BX based on the frontier molecular orbital theory of chemical reactivity. Yang et al. [152], using the DFT method, studied the quantum chemical properties of xanthate derivatives, ROC=SS⁻ (R = amyl, ethyl, or isobutyl), and their interactions with Fe(OH)₂⁺, Fe(OH)²⁺, Fe³⁺, and pyrite cluster under solvation effects. Based on the results of the electronic chemical potential and frontier orbital energies, the flotation performance of the alkyl xanthates was predicted to be in a decreasing order from amyl xanthate (AMX), through iso-BX to EX. According to these studies, the experimental results of the xanthate collectors agree with the theoretical prediction, suggesting that the quantum chemical calculation is one of the effective tools for the rational design and selection of flotation reagents.

Kumar et al. [153] compared the adsorption of 2-mercaptobenzothiazole (MBT) on chalcopyrite and pyrite surfaces via DFT, explaining the selectivity of MBT towards pyrite in flotation. Mkhonto et al. [154] designed and synthesized a novel collector di-sodium 2,6-dithio-4-butyl-amino-1,3,5-triazine (SDTBAT) and investigated its adsorption mechanism on a pyrite surface using experimental methods combined with the computational DFT with dispersion correction and U-parameter (DFT-D3+U), demonstrating that SDTBAT has the potential to be an alternative collector to xanthates due to its high collection power in the separation of sulfide minerals.

NaOH, cyanide, and lime (CaO) are the three most common depressants in the flotation separation of sulfide minerals. The flotation of polymetallic sulfides is generally carried out at a pH of about 12, and at pH values below 12.5, the main components of CaO dissolved in water are the calcium hydroxyl ions, [Ca(OH)]⁺. Zhao et al. [155,156] employed DFT to investigate the depression mechanisms of cyanide and [Ca(OH)]⁺ with the pyrite (100), marcasite (010), and pyrrhotite (001) surfaces. The calculation results indicated that the depression effect of both cyanide and [Ca(OH)]⁺ is in the increasing order of pyrite < pyrrhotite < marcasite. As shown in Figure 3a–c, after CN⁻ adsorption, the C atom interacts with one Fe atom on the pyrite surface; for marcasite, the C atom interacts with one S atom, while the N atom interacts with one Fe atom on the surface; and for pyrrhotite, only the N atom interacts with one Fe atom on the surface. The different adsorption configurations resulted in different charge transfers and adsorption energies, causing different flotation behaviors of these three iron sulfide minerals. After [Ca(OH)]⁺ adsorption, for marcasite and pyrite, the O atom interacts with one Fe atom, the Ca atom interacts with two surface S atoms, and there exists a Ca–Fe anti-bonding on the pyrite surface. For pyrrhotite, the Ca atom is attached to three S atoms on the pyrrhotite surface (see Figure 3d–f). Li et al. [157] also reported that the adsorption of [Ca(OH)]⁺ on the pyrite (100) surface is greater than the adsorption of (OH)⁻. After adsorption, partial surface S atoms are covered by the Ca atoms of [Ca(OH)]⁺, which is unfavorable for Cu activation, and thus, the Cu activation of pyrite after depression via lime is tougher than after depression via sodium hydroxide.

Hydrogen peroxide (H₂O₂) is another efficient pyrite depressant. Cao et al. [158] studied the interaction between H₂O₂ and hydrated pyrite (100) surfaces using DFT calculations. The results showed that the H₂O₂ molecule is prone to react with the pyrite surface to generate an H₂O molecule and one S=O bond.

3.3. Manganese Oxide Minerals

Minerals with a high Mn content are not common, and there are seven most typical types of industrially mined Mn-bearing minerals: pyrolusite, MnO₂; “psilomelane” (a mixture of manganese oxides and hydroxides; mMnO·MnO₂·nH₂O); manganite, MnO (OH)₂; hausmannite, Mn₃O₄; braunite, Mn₂O₃; rhodochrosite, MnCO₃; and alabandite, MnS. The first six of them belong to the oxide minerals, while the last one is a sulfide. There are a few quantum chemical studies on flotation reagents for Mn oxide minerals, including rhodochrosite, but none were reported for Mn sulfide minerals. Zhao et al. [159] first introduced a novel collector, tert-butyl benzohydroxamic acid (TBHA), into the flotation of rhodochrosite, and compared the interaction mechanisms of TBHA and a commonly used collector BHA on the mineral surface using DFT calculations as well as micro-flotation tests, zeta potential measurements, and XPS analysis. Both the experimental and theoretical investigation results suggested that TBHA has a stronger collecting power to rhodochrosite than BHA, i.e., the substitution of tert-butyl groups on benzene rings greatly enhances the affinity of the hydroxamic acid to rhodochrosite. Consequently, TBHA is considered to be an ideal candidate collector for the flotation of rhodochrosite or other oxide minerals.

4. Silicate Minerals

Silicate minerals are the most diverse and abundant minerals in the Earth’s crust, of which more than 1600 species are known, accounting for nearly 27% of all known minerals. The quantum chemical studies on the flotation of silicate minerals mainly include quartz, kaolinite, montmorillonite, talc, chlorite-group minerals, mica-group minerals, zircon, and serpentine.

4.1. Quartz

For the reverse flotation separation of quartz from magnetite, Huang et al. [160] studied the adsorption mechanism of a cationic collector, ethane-1,2-bis(dimethyl-dodecyl-ammonium bromide) (EBAB), on quartz and magnetite surfaces by means of DFT calculations and various experiments. The results showed that EBAB has higher collection power than the traditional surfactant dodecylammonium chloride (DAC) and has superior selectivity for quartz against magnetite. The interaction of EBAB with quartz and magnetite is mainly realized via electrostatic attraction. The unique properties of EBAB, such as the high positive grouping Mulliken charge of –CH₂N⁺(CH₃)₂(CH₂)₂(CH₃)₂N⁺CH₂– and the strong air–water interface reactivity, make it a superior collector for the reverse flotation desilication from Fe-bearing minerals. Liu et al. [161] also reported that the protonation species of the collectors DDA, dodecyl-propyl ether amine, and fatty amine ethoxylate (AC1201) can be easily adsorbed on a quartz surface through electrostatic attraction, and among them, AC1201 is found to be the most effective collector.

Reverse flotation desilication with the OL anionic collector and Ca²⁺ activator is considered to be the common beneficiation method for the purification of hematite. The introduction of sodium humate (HM), which selectively depresses hematite, is also critical. Zhang et al. [162] systematically investigated the reverse flotation separation mechanism of hematite and quartz using flotation experiments, zeta potential measurements, XPS, and DFT calculations, with full consideration of the effect of surface hydroxylation. The results elucidated that the Ca²⁺ activator is highly selective to quartz, and the Ca²⁺ ions chemisorbed on the quartz surface provides targeted reaction sites for the subsequent OL adsorption, whereas the depressant HM is poorly selective to quartz.

Although the activation of metal ions on the quartz surface has been demonstrated [163,164,165,166,167], the microscopic mechanism of quartz activation is unknown, and the activating components in metal ion-activated quartz have been controversial. Luo and Chen [168] investigated the activation mechanisms of Ca²⁺, Fe³⁺, Cu²⁺, and Pb²⁺ on a quartz surface for the first time by using DFT. The hydroxylation of metal ions was revealed to be a prerequisite for their adsorption on a quartz surface. During the adsorption process, the bonds between the metal ions and the –OH of the metal–hydroxyl complexes are broken, and the released –OH subsequently combines with the surface H atoms to form free water molecules, leaving the exposed metal ions attached to the surface O atoms.

The collection and depression of quartz in the presence of activator Ca²⁺ ions have also been reported. Gong et al. [169] used DFT calculations in conjunction with various experimental techniques to study the adsorption mechanism of collector OL on a quartz surface in the presence of Ca²⁺ ions. The results demonstrated that Ca²⁺ enhances the recovery and adsorption density of OL on quartz, and the functional species of activated quartz was considered to be Ca(OH)⁺. The adsorption of Ca(OH)⁺ on a quartz (101) surface decreases the space resistance and electrostatic repulsion and therefore further enhances the adsorption of an oleate anion. Furthermore, when quartz is separated from Ca-bearing minerals, such as apatite, calcite, fluorite, and scheelite, quartz is usually difficult to depress because a quartz surface is activated by Ca²⁺ ions dissolved from the Ca-bearing minerals. Hence, Wang et al. [170] adopted citric acid (CA) as a depressant and OL as a collector for Ca²⁺-activated quartz, and analyzed the depression mechanism of CA through a series of experiments, surface characterization, and calculation means. The results suggested that the depression of CA on Ca²⁺-activated quartz can be attributed to two aspects: first, CA^3- (the main CA species in CA aqueous solution at pH 10–12) can desorb Ca²⁺ ions adsorbed on the quartz surface, thus reducing the number of active sites on the quartz surface for subsequent OL adsorption; second, the pre-adsorption of CA on Ca²⁺ ions hinders OL adsorption on the quartz surface.

Mao et al. [171] evaluated the effect of adding grinding aids, triisopropanolamine (TIPA) and triethanolamine (TEA), during the grinding process on the flotation behaviors of quartz in the system while using DDA as a collector via FTIR spectroscopy, XPS analysis, quantum chemistry, and solution chemistry calculations. The results showed that the combination of TIPA/TEA grinding aids can have a synergistic effect with DDA, and its introduction improves the adsorption of DDA, thus increasing the recovery of quartz. The mixed cationic collectors for quartz flotation have been investigated by Monte et al. [172] using AFM, solution chemistry, and quantum chemical calculations. The results indicated that the combined use of etherdiamine (D) and ethermonoamine (M) in the weight ratio of 3D:1M at pH 10.5 has the best recovery and selectivity. The synergistic effect of D and M on the quartz flotation may be attributed to the reorganization of the collectors.

4.2. Kaolinite

By implementing quantum mechanical calculations, Sun et al. [173] and Hu et al. [174] reported the different flotation behaviors of collector DDA adsorbs on kaolinite (001) and (00$\overline{1}$) surfaces. The results manifested that the (001) surface easily adsorbs cationic collectors and exhibits hydrophobicity, while the (00$\overline{1}$) surface readily interacts with reagents with high electronegativity groups (–O–, –N–, and F–) and exhibits hydrophilicity; thereby, the interaction of DDA with the (001) surface is stronger. DDA adsorption on the (001) surface as well as the self-aggregation between the (00$\overline{1}$) surface and the edge planes makes kaolinite aggregates hydrophobic with good floatability in acidic solutions. Nevertheless, in alkaline solutions, the hydrophilic (00$\overline{1}$) surface is exposed in the presence of DDA, and hence, flotation is not realized. Liu et al. [175] used dynamics simulations and quantum chemistry calculations to evaluate the floatability of three tertiary amines, N,N-dimethyl-dodecyl amine (DRN), N,N-diethyl-dodecyl amine (DEN), and N,N-dipropyl-dodecyl amine (DPN), on the kaolinite (001) surface, and the results indicated that the collection power is in a decreasing order from DEN, through DPN to DRN. The influence of non-polar and polar groups on the adsorption of three carboxyl hydroxamic acids, 2-carboxyl-6-methylcyclohexane carboxamic acid (CMCA), 3-bis(hydroxycarbamoyl) undecanoic acid (BHUA), and 2-bis(hydroxycarbamoyl) octyl maleate (BHOM), on the kaolinite (001) surface was investigated by Wang et al. [176] using quantum chemical calculations and the molecular dynamic method. The results showed that the collecting capacity of three reagents is BHOM > BHUA > CMCA, and the interaction between the O and N atoms of the BHUA and H atoms of kaolinite is realized through the formation of hydrogen bonds. Shen et al. [177] studied the substituent effect of three cationic collectors (DDA, dodecyl diethanolamine, and dodecyl dihydroxyethyl methyl ammonium chloride (BHMAC)) with different head group structures for kaolinite flotation using the DFT approach. The calculated results deduced that the increase in the number of substituents in the dodecylamine head group can significantly increase its Connolly surface, head group charge and solvent-accessible surface, which is conductive to decreasing the collector consumption. Therefore, BHMAC with a large head group charge can be stably adsorbed on a kaolinite surface.

4.3. Other Silicate Minerals

By employing DFT, sedimentation, contact angle, and adsorption measurements, Peng et al. [178] investigated the effects of the head group type and alkyl chain length on the adsorption of alkylamine cationic collectors on the montmorillonite (001) surface. The research results indicated that the adsorption energies of C_n alkylamine cations (where n is the number of carbon atoms) increases with the decreasing substitution degree of –CH₃ groups to H_n atoms in the head group. Moreover, the adsorption energies increase with the n value increasing from 12 to 16, but it changes slightly when n is beyond 16.

The coverage of target minerals via montmorillonite and kaolinite is an important factor contributing to poor flotation selectivity. Organic depressants with small molecular weights can significantly improve this situation. Thus, Luo et al. [179] studied the adsorption of nine selected organic carboxylate depressants with small molecules (including acetic acid, oxalate acid, lactic acid, succinic acid, citric acid, tartaric acid, salicylic acid, phthalic acid, and gallic acid) on the montmorillonite Na-(001) and kaolinite (001) surfaces using DFT calculations. The adsorption energy and energy difference of frontier orbitals indicated that hydrogen phthalate ions (i.e., the anion of phthalic acid) had the strongest interaction with both types of surfaces. Meanwhile, the results of the bond populations and atomic Mulliken charges showed that organic carboxylate can stably adsorb on the kaolinite (001) surface through strong hydrogen bonding and electrostatic force, while the strong electrostatic interaction between the interlayer Na atoms of montmorillonite and acetate acid ions causes Na atoms to break away from the montmorillonite Na-(001) surface.

Extensive studies have also been conducted on ions and depressants (CMC like), as well as mineral particles [180,181,182], but few studies have employed the quantum chemical method [183,184]. CMC is a high-molecular-weight polysaccharide that is widely used to depress talc. It is known that the addition of metal cations can reinforce the depression of CMC on the talc flotation. Therefore, Luo et al. [183,184] reported the adsorption mechanism of Ca and Al ions on the talc (001) basal surface using DFT calculations. The results demonstrated that [Ca(OH)(H₂O)₃]⁺ and [Ca(H₂O)₆]²⁺ are effective hydrated components for the adsorption of surface Ca²⁺ ions. Their adsorption on the talc surface is due to the fact that the H 1s orbitals of the H₂O ligands hybridize with the surface O 2s and O 2p orbitals to form hydrogen bonds. In contrast, the adsorption mechanism of Al on the talc surface is due to the hybridization of the H 1s orbitals of the H₂O ligand in Al(OH)₃(H₂O) (as the preferred hydrate structure) with the surface O 2p orbitals to form hydrogen bonds.

Chlorite is a kind of common gangue mineral associated with valuable minerals, such as barite, cassiterite, fluorite, scheelite, wolframite, etc., and thus, the depression of chlorite is an unavoidable problem in the flotation separation process. Li et al. [185] evaluated the depression mechanism of tetrasodium glutamate diacetate (TGD) in the direct flotation separation of chlorite and specularite (a variety of hematite) on the surface of chlorite using surface charge measurements, FTIR, XPS, and DFT calculations. The results manifested that TGD favorably chemisorbs on the chlorite surface through the formation of O–Mg bonds, suggesting that TGD can be used as a potential chlorite depressant in the flotation of Fe-bearing silicate minerals.

The recent study by Wang et al. [186] introduced metal-ion-modified starch in the flotation of fine minerals, and they investigated the depression mechanism of Pb-starch on chlorite flotation using zeta potential measurements, FTIR, XPS, and DFT calculations [187]. The results showed that the glucose molecule (C₆H₁₂O₆, as a single starch molecule) forms a glucose-Pb²⁺ complex with Pb²⁺, which is chemisorbed on the silica-terminated surface by forming three covalent bonds between Pb²⁺ and O atoms. Pb(II)-benzohydroxamic acid (Pb-BHA) is adopted as a highly selective collector in the oxide mineral flotation, which belongs to another metal–organic complex system. He et al. [188], using AFM, Raman spectroscopy, diffuse reflectance infrared Fourier transform spectroscopy (DRIFTS), and first-principles calculations, comprehensively studied the adsorption mechanism of Pb-BHA on a typical oxide mineral mica (001) surface. The results revealed that the mono-coordinated complex Pb(BHA)⁺ has a better affinity for the mica surface than the di-coordinated complex Pb(BHA)₂ and produces a higher adsorption energy, indicating that Pb(BHA)⁺ is the more effective species for mica flotation than Pb(BHA)₂.

Gao et al. [189] employed the ethyl-functional group (EFG) database to compute reaction-free energies of solidophilic groups with possible metal hydroxide ions as the primary indicator for screening zircon flotation collectors. Both the practical and theoretical results confirmed that the performance of the screened zircon flotation reagents (n-octyl phosphoric acid (nOPA) and octyl hydroxamic acid (OHA)) is much better than that of octanoic acid, indicating that employing quantum chemistry to assist the screening and molecular design of flotation reagents is feasible and meaningful.

In the flotation separation of serpentine and sulfide minerals, due to the preferential solubility of hydroxyl ions, serpentine is positively charged under weakly alkaline conditions, and it is thereby easy to cover the sulfide minerals via electrostatic attraction. To address this, Li et al. [190] adopted sodium tripolyphosphate (STPP) as a novel depressant for the flotation separation of serpentine from pyrite and investigated its interaction mechanism on a serpentine (001) surface using inductively coupled plasmaoptical emission spectroscopy (ICP-OES), XPS, the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory, and DFT. The results demonstrated that the O anions in STPP can chelate with the Mg cations exposed on the serpentine surface with the generation of P–O–Mg bonds, resulting in the adsorption of STPP on serpentine. The accelerated dissolution of Mg²⁺ as well as the adsorption of negatively charged STPP on the serpentine surface eventually results in a negative surface charge of serpentine. Hence, the interaction between serpentine and pyrite changes from electrostatic attraction to electrostatic repulsion, which results in the destruction of the hetero-coagulation between them, thereby achieving their selective flotation separation.

Quantum chemical studies of flotation reagents for other silicate minerals, such as sillimanite, asbestos, olivine, epidote-group minerals, tourmaline-supergroup minerals, pyroxenes, amphiboles, and zeolites, have not been reported.

5. Flotation of Tungsten and Tin

The world’s tungsten resource reserves are relatively rich, and more than fifty W-bearing mineral species are known, but only wolframite and scheelite have economic value in mining. More than a hundred Sn mineral species are known, but the Sn-bearing minerals of industrial significance are limited and include cassiterite, SnO₂; stannite, Cu₂FeSnS₄; franckeite, Pb₅Sb₂Sn₃S₁₄; teallite, PbSnS₂; and cylindrite, Pb₃Sb₂Sn₄S₁₄. Currently, quantum chemical studies of W and Sn mineral flotation reagents have been carried out extensively, with the most attention focused on the flotation of wolframite, scheelite, and cassiterite.

5.1. Wolframite

As for one of the main W ore minerals, the development of highly efficient collectors for wolframite flotation is of great significance for the comprehensive utilization of tungsten resources. Lu et al. [191] synthesized a hydroxamic acid collector N-substituted phenyl octyl hydroxamic acid (NPOHA) in wolframite flotation and studied the interaction mechanism with NPOHA and wolframite using DFT calculations in conjunction with flotation tests and various advanced surface characterization technologies. The results indicated that NPOHA has stronger electron-donating ability, chemical reactivity, and hydrophobicity than the traditional collectors, OHA and BHA, and exhibits excellent collection ability for wolframite via chemisorption on the surface through the formation of C–O–Fe and N–O–Fe bonds.

The selective adsorption mechanisms of collectors used to separate wolframite from fluorite and calcite via flotation have also been reported. Sun et al. [192] adopted 2-(benzylthio)-acetohydroxamic acid (BTHA) and Shuai et al. [193,194] used 3-dodecylamine propyl amidoxime (DPA) and N-(2-aminoethyl)-3-aminopropyltrisiloxane (AATS) as collectors to successfully separate wolframite from fluorite and calcite, respectively. A study of adsorption mechanisms manifested that BTHA adsorbed on the wolframite surface can produce chemisorption and hydrogen bonding, and its adsorption on the surface is realized by the formation of an Fe–S bond and five-membered hydroxycarbamate –(O,O)–Fe ring through the hydroxycarbamate and thioether groups in BTHA. On the other hand, DPA and AATS are selectively adsorbed on the wolframite surface through the electrostatic interaction of positively charged –C(NOH)N⁺H₃ and –N⁺H₃ groups, respectively.

Huang et al. [195] investigated the flotation mechanism of Pb(II)-activated wolframite with BHA as a collector. The DFT calculation results indicated that the Pb(II) activation on BHA adsorption is indirect. The activation mechanism of Pb(II) in wolframite flotation under neutral conditions might be due to the traction effect of Pb(II) ions on water molecules around Mn sites on the wolframite surface that decreases the hydration, thereby making it easier for them to be adsorbed by BHA.

By comparing the activation ability of the Pb²⁺ ion with Ca²⁺, Mn²⁺, and Fe²⁺ ions, Zhao et al. [196] studied the activation mechanism of lead nitrate on both wolframite and scheelite with BHA as a collector. As shown in Figure 4, the coordination models of BHA with Ca(OH)⁺, Mn(OH)⁺, Fe(OH)⁺, and Pb(OH)⁺ were simulated, with their binding energies being calculated using the DFT method. The results showed that BHA exhibits the strongest binding ability to Pb(OH)⁺, suggesting that BHA can preferentially coordinate with Pb²⁺ ions compared with Ca²⁺, Mn²⁺, and Fe²⁺ ions. The effective activation was realized by the adsorption of Pb on mineral surfaces and the subsequent complexation of Pb²⁺ ions with the collector.

5.2. Scheelite

Apart from wolframite, scheelite is another critical source of tungsten. Quantum chemical research on the scheelite flotation reagents mainly focuses on the development of novel collectors, among which scheelite collectors for scheelite–calcite flotation separation is a hot research topic.

Zhao et al. [197] introduced cyclohexyl hydroxamic acid (CHA), Lyu et al. [198] adopted propyl 3,4,5-trihydroxybenzoate (PG), Wei et al. [199] introduced p-methyl-benzohydroxamic acid (MBHA) and p-bromine-benzohydroxamic acid (BBHA), and Ni et al. [200] employed hexane-1,6-didodecyldimethylammonium bromide (HDDA) as new collectors for scheelite flotation, and they studied their collection mechanisms towards scheelite using experimental and computational methods. The results showed that the binary ions of CHA exhibit higher atomic charge values, HOMO energies, larger dipole moments, and binding energies with Ca²⁺ than the conventional collector BHA, and thus have a higher capability to collect scheelite than BHA. Although both MBHA and BBHA are substitutes for aryl hydroxamic acids, the bonding groups in the molecule of MBHA are better electron donors than those of BHA and BBHA, which contributes to the better adsorption effect and collection performance of MBHA on scheelite. The study of the adsorption mechanisms confirmed that PG is chemisorbed on the scheelite surface through the formation of five-membered rings by means of its ortho oxygens, whereas HDDA is adsorbed on the scheelite surface through the electrostatic interaction between the negatively charged scheelite surface and the positively charged HDDA molecules.

Yin et al. [201] evaluated the structure–activity relation and mechanisms of collectors including carboxylic acid and BHA and depressants including sodium hexametaphosphate (SHAP), sodium silicate, and oxalic acid in scheelite flotation through flotation tests, quantum chemical calculation, and flotation solution computational chemistry. The results suggested that BHA is adsorbed on the scheelite surface through the formation of –O– and =O metal bonds in ionic and molecular forms; however, carboxylic acid is adsorbed on the scheelite surface in the ionic form. The recovery of scheelite decreases sequentially from SHAP and sodium silicate to oxalic acid when using OL as a collector, indicating that the group’s electronegativity is the dominant factor to determine the depression performance in the scheelite flotation, i.e., the larger the electronegativity, the more pronounced the depression effect on the scheelite flotation.

For the efficient separation of scheelite from calcite, Huang et al. [202] introduced DPA; Deng et al. [203] used N-(6-(hydroxyamino)-6-oxohexyl) benzamide (NHOB), N-(6-(hydroxyamino)-6-oxohexyl) octanamide (NHOO), N-(6-(hydroxyamino)-6-oxohexyl) decanamide (NHOD), and N-(4-(hydroxyamino)-4-oxobutyl) octanamide (NOBO); Liu et al. [204] synthesized N-tetradecyl-isopropanolamine (NTIA); and Huang et al. [205] brought AATS as collectors and evaluated the adsorption mechanisms on a scheelite surface using DFT calculations combined with flotation tests and other advanced characterization methods. The results showed that NHOD may be chemisorbed on scheelite through the formation of NHOD-W surface complexes in addition to the electrostatic attraction with Ca cations on the scheelite surface. After the NTIA treatment, the scheelite surface is preferentially modified, and the hydrophobicity is enhanced. Meanwhile, the NTIA molecules can be selectively adsorbed on the scheelite surface under the effect of hydrogen bonding of the hydrophilic N–H groups of NTIA and the strong electrostatic field. Nevertheless, similar to the interaction mechanism of AATS on the wolframite surface, DPA and AATS are adsorbed selectively on the scheelite surface through the electrostatic interaction of the positively charged –C(NOH)N⁺H₃ and –N⁺H₃ groups with the negatively charged scheelite surface, respectively.

5.3. Cassiterite

The non-polar carbon chain length of the surfactant has been reported to have a significant effect on the collection ability of cassiterite (SnO₂). By using a theoretical approach in conjunction with experimental methods, Lu et al. [206] designed and synthesized a series of N-phenyl hydroxamic acids (NPHA) with different carbon chain lengths, including N-phenyl acetohydroxamic (NPHA-2), N-phenyl butyrohydroxamic acid (NPHA-4), N-phenyl hexanohydroxamic acid (NPHA-6), and N-phenyl octanohydroxamic acid (NPHA-8), and they first introduced them as collectors for the selective flotation of cassiterite from quartz and feldspar. Jin et al. [207] evaluated the effects of five alkyl hydroxamates, including hexyl hydroxamate (C6), octyl hydroxamate (C8), decyl hydroxamate (C10), dodecyl hydroxamate (C12), and tetra-decyl hydroxamate (C14), on the flotation behavior of cassiterite. Both pieces of research revealed that increasing the carbon chain length can improve the collection ability of surfactants on cassiterite. The study demonstrated that NPHA-8 exhibits the strongest collection power towards cassiterite and can be chemisorbed on cassiterite surfaces by forming C–O–Sn and N–O–Sn bonds, accompanied by proposing two interaction geometries, as shown in Figure 5. The first interaction scheme involves the same Sn atom of the cassiterite surface (Figure 5a) that leads to a five-membered chelating ring, whereas the second involves two different Sn atoms (Figure 5b), resulting in the formation of an irregular complex. As for hydroxamates, those with longer carbon chains can be more easily absorbed on the cassiterite surface by chelating six-membered rings, and after surface hydration, hydroxamates adsorb on the mineral surfaces through a hydrogen bonding interaction.

Moreover, it was proven that some of the sulfhydryl minerophilic groups used for the flotation of sulfide minerals can be developed as preferable hydrophobic groups for the flotation of oxide minerals, e.g., cassiterite. For instance, Qi et al. [208], taking carbon disulfide, octylamine, methyl acrylate, and hydroxylamine as raw materials, synthesized a special surfactant, S-[(3-hydroxyamino)-propoxy]-N-octyl dithiocarbamate (DTCHA), and uncovered the effects of the dithiocarbamate ester group in the hydroxamic acid flotation of cassiterite using AFM, XPS, FTIR, and DFT calculations. The self-assembled DTCHA aggregates are clearly observed on the surface of cassiterite, which suggests that the dithiocarbamate ester group, with its electrostatic interaction and electron-donating ability, might be another reaction center of DTCHA. However, this study only identified the bonding interactions between the dithiocarbamate of DTCHA with Sn during the reaction of DTCHA with Sn(IV) ions in aqueous solutions, whereas that with Sn(IV) atoms on the surface of cassiterite could not be confirmed.

Many scholars have also developed a number of novel cassiterite collectors, for example, styryl phosphonate mono-iso-octyl ester (SPE108) [47], BTHA [209], styrene phosphonic acid (SPA) [210], and 2-carboxyethylphenylphosphinic acid (CEPPA) [211], and used experimental methods in conjunction with theoretical methods to study their effects on cassiterite flotation, revealing their adsorption mechanisms on the cassiterite surface. Furthermore, quantum chemical studies on the adsorption mechanism of surfactants on a Pb-activated cassiterite surface have also been reported. Quartz, a primary gangue mineral associated with cassiterite, is quite difficult to depress once activated by metal ions. Tian et al. [212] used cupferron as a collector in the selective flotation of cassiterite from quartz and investigated its interaction mechanism on mineral surfaces. The results indicated that, with the presence of a low dosage of Pb²⁺ ions, the cupferron exhibits a greater collection power and selectivity towards cassiterite compared to quartz. Sn⁴⁺ ions and the pre-adsorbed Pb²⁺ ions on the cassiterite surface offer dominant reactive sites for cupferron adsorption, and as a consequence, the effective flotation of cassiterite from quartz is achieved. Furthermore, He et al. [213] studied the interfacial microstructure and adsorption mechanism of a collector BHA on the Pb-activated cassiterite (110) surface. The results indicated that BHA chemisorbs on the cassiterite surface through intermolecular interactions (i.e., hydrogen bonding).

In the flotation separation of cassiterite and fluorite, Cu²⁺ ions have been found to have a strong selective depression effect on fluorite. Therefore, Gong et al. [214,215], using experimental and DFT methods, studied the effect and mechanism of Cu²⁺ on the flotation separation of cassiterite from fluorite with styrene phosphonic acid (SPA) and 2-carboxyethylphenylphosphinic acid (CEPPA) as collectors at pH = 4. It could be inferred from the results that Cu²⁺ consumes SPA and CEPPA in the pulp, leaving little free SPA and CEPPA to be adsorbed on the fluorite surface, whereas the adsorption of SPA and CEPPA on cassiterite was unaffected. This makes Cu²⁺ an effective depressant for the flotation separation of cassiterite from fluorite.

6. Calcium-Bearing Minerals

6.1. Fluorite

Fluorite is a strategic Ca-bearing mineral, and the study of its flotation separation from co-existing minerals such as barite, calcite, scheelite, and cassiterite has been a hot research topic in recent years. Duan et al. [216] adopted octylamino-bis-(propanohydroxamic) acid (OPHA) as a novel fluorite collector and evaluated its adsorption mechanism on mineral surfaces during the separation of fluorite from barite. The research found that, compared with barite, OPHA⁺ and OPHA²⁻ (the main species of OPHA in neutral and basic environments) are more strongly adsorbed on fluorite surfaces, with the preferential adsorption mechanism involving the replacement of H₂O molecules.

The effects of novel collectors, octylamino-bis-(butanohydroxamic acid) (OBHA) [217] and saponified tricarboxylic acid (TA) [218], on the flotation separation of fluorite from calcite were reported using micro-flotation experiments and a series of surface characterization technologies in conjunction with DFT calculations. The results indicated that the hydroxamic acid group of OBHA contributes to its adsorption on fluorite, and TA can be chemisorbed on the fluorite surface by generating a TA-Ca complex through a bidentate coordination.

Meanwhile, the effects of novel collectors and depressants on the flotation separation of fluorite from scheelite were also reported. Miao et al. [219] investigated the adsorption mechanism of N-decanoylsarcosine sodium (SDAA) on the fluorite and scheelite surfaces through flotation tests, adsorption capacity detections, zeta potential tests, FTIR analysis, crystal chemistry, and DFT calculations. The results showed that because of the electrostatic effect, the negatively charged SDAA is more inclined to adsorb on the positively charged fluorite surface. The SDAA has greater chemical interaction and more electron transfer with Ca atoms on the fluorite surface with the formation of a Ca-SDAA complex. Zhang et al. [220], using micro-flotation tests, infrared spectroscopy (IR), zeta potentials, XPS, and DFT, studied the depression mechanism of sodium polyacrylate (PA-Na) for scheelite and fluorite flotation. The results indicated that PA-Na depresses fluorite better than scheelite, and PA-Na can be chemisorbed on both scheelite and fluorite surfaces, with a stronger adsorption occurring on fluorite. The significant difference in the adsorption behaviors of Na on these two surfaces suggests that SDAA/PA-Na is an effective collector/depressant in the fluorite–scheelite flotation separation. He et al. [221] evaluated the roles of Pb²⁺ in the flotation separation of fluorite and scheelite using in situ AFM force curves and first-principles calculations in conjunction with other experimental methods. The results deduced that BHA adsorption on a fluorite surface is ascribed to the strong electrostatic attraction; however, it is hard for BHA to bind to the negatively charged and hydroxylated Ca sites on the scheelite surface; the formed BHA-Pb complex exhibits attraction/repulsion on the scheelite/fluorite surfaces, which indicates that Pb²⁺ is capable of facilitating the flotation of scheelite but depresses the flotation of fluorite.

As for the flotation separation of fluorite and cassiterite, the depression towards fluorite has been broadly studied. For instance, Wang et al. [222,223] employed 2-phosphonobutane-1,2,4-tricarboxylic acid (PBTCA) and the disodium salt of adenosine 5′-triphosphate (Na₂ATP) as eco-friendly depressants for fluorite to achieve the efficient flotation separation of cassiterite from fluorite. According to the DFT calculation results, Wang et al. [222] proposed a model, called the “mono-polar group” interaction, that includes PBTCA being chemisorbed on the fluorite surface through the interaction between the Ca sites in fluorite and the phosphate groups in PBTCA (see Figure 6a). Nevertheless, Wang et al. [223] also found that Na₂ATP adsorbs on the fluorite surface in the form of “horizontal adsorption”, with the adsorption occurring at multiple Ca sites on the surface (see Figure 6b), and the selective adsorption of Na₂ATP on fluorite can be ascribed to the chemisorption between the Ca²⁺ ions on the surface and the phosphate groups of Na₂ATP. In addition, the adsorption of PBTCA and Na₂ATP on the fluorite surface prevents the adsorption of the collector NaOL, suggesting that PBTCA and Na₂ATP are highly efficient depressants for the separation of fluorite and cassiterite via direct flotation.

6.2. Other Calcium-Bearing Minerals

The flotation separation of calcite from co-existing Ca-bearing minerals, e.g., scheelite, magnesite, and fluorite, is a challenging problem due to their similar physicochemical characteristics. For calcite, as for another significant Ca-bearing mineral apart from fluorite, flotation reagent quantum chemistry studies have also been reported. For instance, Zhang et al. [224] employed PA-Na as a calcite depressant and investigated its depression mechanism during the selective flotation of scheelite from calcite using micro-flotation experiments, zeta potentials, XPS, and DFT calculations. The results demonstrated that PA-Na is able to chemisorb on both calcite (104) and scheelite (111) surfaces, but the absolute value of the adsorption energy of PA-Na on the calcite surface is larger than that on the scheelite surface, revealing that the use of PA-Na can depress calcite more easily in the flotation separation of calcite from scheelite. The STPP is another calcite depressant; its effects on the improvement of the flotation separation efficiency of magnesite from calcite was investigated by Wang et al. [225] using infrared spectroscopy, XPS testing, potentiodynamic potential measurements, and quantum chemistry calculations. The DFT calculation showed that STPP can be strongly bound to the Ca sites on the calcite surface but weakly bound to the Mg sites on the magnesite surface. Therefore, STPP is believed to be an effective depressant during the flotation separation of magnesite from calcite. Jin et al. [226] assessed the effect of the combination of Al³⁺ ions and glucose on the flotation separation of calcite and fluorite using micro-flotation experiments, SEM, XPS, solution chemistry, and DFT calculations. The results indicated that the introduction of Al³⁺ ions leads to the formation of Al(OH)₃ and Al₂O₃ clusters on the calcite surface; the glucose and Al₂O₃ glucose generated on calcite surface improves the adsorption through the Al–O bonds, which increases the reactive sites on the surface, thereby increasing the selective depression of glucose on calcite.

In addition to fluorite and calcite, the quantum chemistry studies of reagents used for the flotation of other Ca-bearing minerals, such as gypsum (CaSO₄·2H₂O) and anorthite (CaAl₂Si₂O₈), have not drawn much attention.

7. Rare Earth Minerals

Rare earth elements (REEs) are widely used in catalytic materials, communications, biomedical materials, magnetic materials, and many other advanced technological fields. Due to the different metallogenic regularity, a wide variety of rare earth minerals have been discovered with complex compositions, which led to the development and utilization of REEs to be a very thorny problem [227]. Flotation is a conventional method used to separate rare earth minerals (REMs) from gangue minerals. However, REM flotation still faces constraints, especially when REMs and their associated gangue minerals share a similar surface chemistry.

7.1. Bastnaesite

Bastnaesite-(Ce), (Ce,La)F(CO₃), as one of the few REMs with mining value, is a semi-soluble salt mineral. It contains the Ce subgroup or the lighter REE and is often associated with barite, calcite, and quartz. Currently, the development of novel collectors for bastnaesite-(Ce) flotation and the quantum chemical studies of its collection mechanism is the focus of attention.

OHA [228] and di(2-ethylhexyl) phosphate (DEHPA), dibutyl phosphate (DBP), and tributyl phosphate (TBP) [229] were adopted as collectors, and their adsorption mechanisms on a bastnaesite-(Ce) surface were studied using a combination of computational and experimental technologies. Although the mechanism for hydroxamate chemisorption has been believed to be related to bidentate chelation in previous studies [230,231,232,233], the research results of Wanhala et al. [228] showed that monodentate conformation also plays a significant role in collector adsorption on a bastnaesite-(Ce) surface. At a low surface loading, the monodentate chemisorption is dominate, whereas at higher concentrations, the bidentate chemisorption is more prominent. The flotation is more effective at a low hydroxamate dosage because of the formation of micelles and the adsorption of gangue are avoided. During the flotation separation of bastnaesite-(Ce) and calcite, hydroxamate exhibits poor selectivity at pH 9. A ligand that forms a bidentate complex at low surface coverages might induce greater selectivity for bastnaesite-(Ce). Moreover, the research results obtained by Fan et al. [229] demonstrated that the chemical activity of the three phosphate collectors is in the decreasing order of DEHPA ≥ DBP >> TBP. The interaction of DEHPA with bastnaesite-(Ce) is realized through the formation of the Ce–O–P bonds between the O atoms in the P(=O)–O– group of DEHPA and the surface Ce(III) atoms, and two 2-ethylhexyl groups of DEHPA are oriented outwardly for the attachment to bubbles, leading to the flotation enrichment of bastnaesite-(Ce).

In order to improve the recovery of REE, it is essential to develop collectors with strong collection and selectivity capabilities. Yao et al. [234] brought phenylpropyl hydroxamic acid (PHA) as a novel collector for the flotation separation of bastnaesite-(Ce) from calcite. The study indicated that the PHA is chemisorbed on the bastnaesite-(Ce) surface through the formation of a Ce–O bond with Ce, and it was deduced that the hydroxamic acid forms a five-membered chelated hydroxamic group with Ce on the surface, thereby improving the hydrophobicity of bastnaesite-(Ce). Not only has the effect of OBHA in the flotation separation of fluorite from calcite [217] been reported, but its influence on the flotation separation of bastnaesite-(Ce) from barite and calcite has also been studied by Duan et al. [235]. Zeta potential measurements, FTIR, XPS analysis, and DFT calculations were performed to investigate the selective adsorption mechanism of OBHA. The results suggested that OBHA is chemisorbed on the bastnaesite-(Ce) surface, while the interaction between OBHA and barite/calcite is weak. Referring to the concept of “lattice matching” between flotation reagents and mineral surfaces [236,237,238], the OBHA matches well with the bastnaesite-(Ce) surface, but mismatches with barite and calcite surfaces, resulting in the superior selectivity of OBHA for bastnaesite-(Ce).

Cao et al. [239] explored the adsorption mechanisms of the Al³⁺ and Fe³⁺ ions as surface modifiers on the bastnaesite-(Ce) surface using DFT calculations, XPS analysis, adsorption isotherm study, and adsorption kinetic investigations. The results suggested that the Al³⁺ and Fe³⁺ ions are adsorbed on the bastnaesite-(Ce) surface through the interaction between the Al/Fe hydroxide species and the O atoms of one of the two main functional groups on bastnaesite-(Ce) (i.e., the ≡CeOH⁰ groups), as shown in Figure 7. The adsorption data and kinetic of Al³⁺ and Fe³⁺ ions are fitted relatively well to the Freundlich equations and the pseudo-second-order model, respectively. The Freundlich constants demonstrate a favorable process for the adsorption of Al³⁺ and Fe³⁺ ions, and at least two O atoms of the surface ≡CeOH⁰ groups are complexed with every adsorbed metal hydroxide species.

7.2. Other Rare Earth Minerals

In addition to bastnaesite-(Ce) and monazite-(Ce), (Ce,La,Y,Th)(PO₄) is the other most important naturally occurring REE-bearing mineral. Sarvaramini et al. [240] evaluated the interaction mechanisms between cerium hydroxides (i.e., the major REE metal in minerals) and hydroxamic acid collectors during the flotation of bastnaesite-(Ce) and monazite-(Ce) using DFT simulations combined with experiments. The results indicated that the considerable floatability present under moderately alkaline (pH = 7–9) conditions is related to the adsorption of hydroxamic acids on the mineral surfaces. Based on the DFT simulations, at a pH of 7–9, the interactions between the solvated Ce(OH)₂⁺ and Ce(OH)²⁺ and heptyl-hydroxamic acid (HHA) anions leads to the formation of [Ce(OH)₂(HHA)_x(H₂O)_y]^1−x (x(y=)=1(5),2(1),3(0)) and [Ce(OH)(HHA)_x(H₂O)_y]^2−x (x(y=)=1(6),2(3),3(1)) complexes, respectively. It was found that the HHA anion interacts directly through the formation of two covalent bonds between the two polar head O atoms and Ce in the hydroxide complex. Nevertheless, the formation of such covalent bonds breaks a few electrostatic or covalent bonds between the Ce and water molecules that exist in the first hydration shell of the REM cations. Hence, it is reasonable to speculate that the complexes formed in the electric double layer of the semi-soluble minerals and the complexes that came from the other REEs belonging to bastnaesite-(Ce) and monazite-(Ce) all contribute to the interactions between REM and hydroxamic acid collectors.

8. Other Minerals

8.1. Lithium-Bearing Minerals

Lithium is one of the most important rare metals, and it is broadly distributed in nature and widely used in various fields. In nature, most Li-bearing minerals have been found to be silicates and phosphates, while others are rare, of which spodumene, lepidolite, amblygonite, and petalite are the main mineral sources of Li. Quantum chemical studies of flotation reagents for spodumene and lepidolite have been broadly investigated so far, especially for spodumene collectors and activators.

By combining experiments with DFT calculations, the flotation behaviors and adsorption mechanisms of collectors including oleic acid [241,242,243], arsenic acid (BAA) [243], and a mixture of OL and DDA [244] on a spodumene surface were uncovered. The results showed that the oleic acid anion is adsorbed on the spodumene surface through the formation of a multicomponent ring [242]. The carboxyl groups of the ionic–molecular complexes of oleic acid interact with the Al sites on the spodumene surface via chemisorption [241]. Compared with oleic acid, the electronegativity of the solidophilic O atoms of BAA is stronger, with the strength of the C–O bond being greater than that of the As–O bond. Hence, when BAA is adsorbed on the mineral surface, the bonding strength of solidophilic O atoms to the adsorption sites is higher [243]. Moreover, a certain amount of DDA is able to significantly enhance the adsorption of NaOL on the spodumene (001) surface, and its interaction mechanism involves the bonding of DDA ions to oleate and the subsequent adsorption on the surface Al sites. The increased adsorption of the complex (DDA ions/oleate) on the spodumene surface increases the surface coverage and consequently improves the ordering of the collector [244].

However, the adsorption strength of the oleate anions on the hydrated mineral surface is greatly reduced due to the interaction between the surface metal sites and water molecules. Therefore, in the OL system, it is very difficult for spodumene to float in the absence of external activating ions [242]. Based on this, the activation mechanisms of spodumene in different systems with Ca²⁺ [245,246,247,248], Fe³⁺ [248], and Pb²⁺ [249] ions as activators were reported. The results revealed that the main hydrolyzed species of Ca²⁺ ions are [Ca(OH)]⁺ and Ca(OH)₂, with the latter possessing a larger steric hindrance to the adsorption of collectors on the Al sites, leading to the collector being adsorbed on the Ca sites of the spodumene surface in the form of bidentate adsorption. Therefore, Ca(OH)₂ could not effectively facilitate the flotation of spodumene. After the [Ca(OH)]⁺ treatment, the O atoms of OL are bonded to the surface Ca and Al atoms, which greatly increases the adsorption strength of the collector. Therefore, [Ca(OH)]⁺ is considered to be the only effective active component in the flotation process [245,246,247,248]. For Fe activation, the bond population value of Ca–O indicated that the Ca–O bond consists of partially covalent and partially ionic components. On the contrary, the Fe–O bond presents a stronger covalent nature through the hybridization of the 3d orbitals of Fe atoms and the 2p orbitals of O atoms. After the adsorption of Fe³⁺, the density of states (DOS) near the E_F of the spodumene surface is much stronger than that after the adsorption of Ca²⁺, and thus, Fe³⁺ plays a more significant role than Ca²⁺ [248]. When Pb²⁺ is introduced, the adsorption mode of the benzene ring perpendicular to the spodumene surface prefers the bidentate mode between the Pb²⁺ ions of the pre-adsorbed solvated Pb(OH)⁺ complex and two O^2– ions of BHA, resulting in a higher adsorption energy (−270.00 kJ/mol). Based on the coordination chemistry analysis, the differences in the adsorption energies and adsorption modes of BHA on the non-activated and Pb²⁺-activated spodumene surfaces can be understood from the differences in the coordination numbers and bond dissociation energies [249]. In general, using a combination of experimental and computational methods, the activation of spodumene flotation via these metal ions has been confirmed.

Lepidolite is another crucial mineral resource of lithium, for which a few quantum chemical studies of flotation reagents are available. For example, Huang et al. [250] synthesized amidoxime as a collector for the efficient collection of lepidolite and compared it with a commonly used collector, DDA, to understand the collecting performances and adsorption mechanism of amidoxime using experimental and computational methods. The results suggested that amidoxime is adsorbed on the lepidolite surface via electrostatic attraction through the −CH₂NH(CH₂)₂C(NOH)N⁺H₃ group as the adsorption site. Compared with DDA, amidoxime is more easily adsorbed on the lepidolite surface and it is more effective in improving the hydrophobicity of the mineral.

8.2. Molybdenum-Bearing Minerals

So far, among the known eighty-eight molybdenum-bearing mineral species, molybdenite, MoS₂, has the most extensive existence, the most abundant content, and the largest mining value. As for wulfenite, Pb(MoO₄); molybdite, MoO₃; jordisite, MoS₂; ilsemannite, Mo₃O₈·nH₂O; etc., although their distributions are also very wide, their quantities are not large, and they are often exploited as trace minerals associated with molybdenite.

At present, the quantum chemical studies of Mo-bearing minerals have been focused on the collection of molybdenite. Wu et al. [251,252] introduced potassium cetyl phosphate (PCP) and N-(N-butyl) thiophosphoric triamide (NBPT) as collectors and used DFT calculations in combination with many surface characterization methods to uncover their adsorption mechanisms on a molybdenite surface. The results demonstrated that NBPT strongly interacts with molybdenite edges by hybridizing the S 3p orbitals of NBPT with the Mo 4d orbitals of molybdenite to form strong covalence bonds, whereas PCP chemisorbs on the edges via the hybridization of the O 2p orbitals of the P−O radical in phosphate groups with the Mo 4d orbitals of molybdenite, forming two high ionicity bonds, and thereby, both NBPT and NBPT could greatly facilitate the flotation of molybdenite.

The Ca²⁺ ions can depress the adsorption of collectors on the molybdenite surface, leading to the poor flotation performance of the mineral in a high-Ca aqueous solution. Therefore, to reduce the deleterious effect of Ca²⁺ ions on the flotation of molybdenite, Wan et al. [253] used hydrocarbon oils (HOs) as a collector combined with aromatic hydrocarbon (AH) as a synergistic collector and employed flotation tests, contact angle measurements, zeta potential, XPS, DFT calculations, and CMD simulations to investigate the adsorption mechanism of AH on a molybdenite surface and its synergistic mechanism with HO. The results manifested that AH is physically adsorbed on the molybdenite (100) surface through the formation of cation-π bonds with the Mo atoms of molybdenite, which enhances the hydrophobicity of the mineral surface and facilitates the adsorption of HO on the surface, and as a result, the unfavorable effect of Ca²⁺ ions on flotation is weakened. With the Ca²⁺ and [Ca(OH)]⁺ ions being the most common ionic forms of the Ca in a solution, Sun et al. [254] evaluated their adsorption characteristics and mechanisms on the molybdenite (001) and (100) surfaces via experiments and DFT simulations. The results indicated that both Ca²⁺ and [Ca(OH)]⁺ ions prefer to adsorb on the edges of the molybdenite (100) surface. For the (100) surface, the Ca²⁺ ions prefer to chemisorb on the Mo-top site on the molybdenite S-(100) surface by forming Ca−S bonds, whereas the [Ca(OH)]⁺ ions prefer to adsorb on the top of the S atom at the Mo-(100) surface through the formation of the strong covalent Mo−O and S−Ca bonds.

8.3. Arsenic-Bearing Minerals

For arsenopyrite, which is generally associated with chalcopyrite, its flotation separation from chalcopyrite is considered to be crucial. In Section 2.2.1, the quantum chemical studies for chalcopyrite collectors have been well discussed, which prompts us to focus here on the arsenopyrite depressants. Through the combination of experimental and quantum chemical methods, Sun et al. [255,256] analyzed the depression mechanism of sodium m-nitrobenzoate (m-NBO) on the arsenopyrite surface. The results manifested that the presence of m-NBO and the NH₄⁺ activator reduces the adsorption capacity of the collector BX on the arsenopyrite surface. The –NOO– in m-NBO adsorbs on the Fe site known as the primary activation site on the arsenopyrite surface via covalent bonds, thereby indicating that m-NBO has a strong depression effect on arsenopyrite. Later on, Dong et al. [257] studied the depression mechanism of sodium thioglycallate (STG) towards arsenopyrite in the flotation separation of chalcopyrite and arsenopyrite. The results demonstrated that STG prefers to adsorb on arsenopyrite rather than on chalcopyrite and hinders the adsorption of BX on the arsenopyrite surface, which further improves the wettability of arsenopyrite. The STG is chemically bonded with As and Fe sites on the arsenopyrite surface through its –SH group, and it forms hydrogen bonds with water molecules via the –COO– group at the top of the molecule, eventually generating a bridge-like structure among STG, arsenopyrite, and water molecules. This structure forms a stable hydrophilic film on the arsenopyrite surface. As a result, the floatability of arsenopyrite is severely reduced.

Wang et al. [258] investigated the promotion effect of the Ca²⁺ ions on the arsenopyrite depressant, CMC. The results suggested that, due to the inactivity of the Fe, As, and S atoms of arsenopyrite, the depression of arsenopyrite by CMC alone is not satisfactory, but its depression is significantly enhanced in the presence of Ca²⁺ ions. This is ascribed to the adhesion of hydroxyl calcium ions on the surface, whose electrophilic strength is larger than that of the pristine surface, and CMC is capable of chemisorbing on the arsenopyrite surface by inducing the bridging effect of the calcium hydroxyl ions.

Quantum chemical studies of flotation reagents towards other As-bearing minerals, such as realgar (AsS), orpiment (As₂S₃), domeykite, liroconite, and enargite, are very limited. For instance, Yekeler et al. [259] investigated the interactions between As(III) ions in realgar, orpiment, and arsenopyrite with two of the most popular collector ions, i.e., ethyl dithiocarbonate (C₂H₅OCS₂⁻) and ethyl trithiocarbonate (C₂H₅SCS₂⁻), using DFT at the B3LYP/6-31G** level. By comparing the stabilities of the major flotation products, namely As(III) xanthates and dixanthogens, the following conclusions were drawn: first, the O atom showed a greater electron releasing property than the S atom in xanthates; second, when the O atom was replaced by the S atom in C₂H₅OCS₂⁻, the energies for the formations of As(III) xanthates and dixanthogens increased. Therefore, C₂H₅OCS₂⁻ is preferred in these formations.

8.4. Phosphorus-Bearing Minerals

Phosphorus, produced from phosphate ores, is a significant chemical raw material, but also a crucial mineral raw material that is used for the production of phosphate fertilizers. Currently, the main types of industrially utilized P-bearing minerals are apatite-group minerals that include fluorapatite (Ca₅(PO₄)₃F), chlorapatite (Ca₄(PO₄)₃Cl), and hydroxyapatite (Ca₅(PO₄)₃(OH)), followed by svanbergite (SrAl₃(PO₄)(SO₄)(OH)₆), vivianite (Fe₃(PO₄)₂·8H₂O), and others.

Among them, apatite-group minerals are the primary source of phosphorus, which is essential for the production of fertilizers. Gordeijev et al. [260] uncovered the interactions of models of oleoyl sarcosine (N-oleoyl-N-methyl-amino carboxylic acid) with the Ca²⁺ ion and the surface Ca of apatite using quantum chemical methods. The study results proved that oleoyl sarcosine can form either bidentate or tridentate structures with the Ca²⁺ ions. However, the tridentate complex may not always form on a more extended surface due to the steric hindrance and the unavailability of active Ca sites. The functional group has an influence on the interaction between N-substituted sarcosine and the apatite surface Ca sites. According to the model shown in this study, when interacting with a single-surface Ca site, the interaction strength would mainly depend on the electronic nature of the functional group. Peng et al. [261] synthesized a novel surfactant, N-(2-hydroxy-1, 1-dimethylethyl) dodecylamine (HDMEA), by introducing an additional hydrogen-bonding functional group into the conventional collector molecule DDA and explored the role of HDMEA in the separation of apatite from quartz via reverse flotation through experiments and DFT calculations. The results showed that the adsorption of HDMEA on quartz mainly relies on the electrostatic interactions and hydrogen bonds that involve the –NH– and –OH groups, whereas its collection ability for apatite is hindered by the steric hindrance effect induced by HDMEA. Therefore, HDMEA is considered an efficient reagent for the flotation separation of apatite from quartz.

As for the flotation of fluorapatite, Ca₅(PO₄)₃F, many quantum chemical research studies have been undertaken in order to understand its collection mechanisms. Xie et al. [262] investigated the influences of the carbon chain length, carbon chain isomerism, and number of C=C double bonds on fatty acid adsorption on the fluorapatite (001) surface using DFT calculations. The results revealed that the effect of fatty acids on the surface adsorption of fluorapatite depends mainly on the interaction between the fatty acid O atom and the surface Ca atom. Fatty acid collectors could form stable adsorption configurations at the surface Ca1 site, and as the carbon chain increases within certain limits, the adsorption becomes stronger; a certain amount of C=C double bonds poses little effect on the adsorption strength. The interaction between 4-methylheptanoic acid and the fluorapatite surface is stronger than that between 6-methylheptanoic acid and the fluorapatite surface, indicating that the change in the substitution configuration of fatty acid has a great influence on the adsorption of fatty acid on fluorapatite. The co-adsorption mechanism of kerosene and fatty acid on the fluorapatite (001) surface was also studied by Du et al. [263]. The results show that there is a synergistic effect of the chain–chain interaction between fatty acid molecules. The interpenetrating adsorption of kerosene and fatty acids in an oriented arrangement on the surface of fluorapatite can increase the adsorption strength and density, and ultimately, the flotation of fluorapatite is improved by the synergistic effect between kerosene and fatty acid. In addition, Zou et al. [264] investigated the role of the hydrogen bond induced by ricinoleic acid (RA) molecules in the fluorapatite flotation. The results of the quantum chemical calculations indicated that the formation of hydrogen bonds in calcium ricinoleate (Ca(RA)₂) precipitates at pH 9 leads to the close packing of RA⁻ anions on the fluorapatite surface, which promotes the flotation of fluorapatite.

The effects of metals ions (Mg²⁺ and Al³⁺ ions) on fluorapatite flotation using fatty and hydroxamic acids as collectors were evaluated by Eskanlou et al. [265]. The results indicated that fatty acid establishes a stronger interaction with untreated and Mg²⁺/Al³⁺-treated fluorapatite compared to hydroxamates, and Mg²⁺ ions are more favorable than Al³⁺ ions in fluorapatite flotation using both fatty acid and hydroxamates.

Although hydroxamic acid collectors were widely used to separate different target minerals from gangue minerals, such as scheelite [197,199], cassiterite [206], and monazite [240], the mechanism of applying hydroxamic acid to separate collophane from dolomite remains unclear. Hence, Yu et al. [266] used a combination of experimental and quantum chemical calculations to study the flotation performance of hydroxamic acid, a chelating collector, on collophane and dolomite. The results elucidated that hydroxamic acid molecules could form O–O five-membered rings with Ca ions and then adsorb on a collophane surface via chemical bonding. The adsorption of AH at the collophane/water interface was larger than that on the dolomite surface, which is an essential reason for the effective separation of collophane from dolomite.

8.5. Minerals Containing Precious Metals

Precious metals are non-ferrous metals with a low content in the Earth’s crust, a high melting point, high price, and high specific gravity (10.4–22.4), such as gold, silver, platinum group elements (platinum, iridium, osmium, and palladium), and so on. As the most commonly used sulfide mineral collectors in the mining industry, the adsorption mechanism of sulfhydryl collectors not only on the surface of metal sulfide minerals, but also on the surface of precious metal minerals has received wide attention. Yekeler et al. [267], employing DFT, studied the interaction energies between Ag⁺ ions in acanthite (Ag₂S) and a series of sulfhydryl collector anions, namely diethyl dithiocarbamate [(C₂H₅)₂NCS₂⁻], ethyl dithiocarbamate (C₂H₅NHCS₂⁻), ethyl dithiocarbonate (or xanthate) (C₂H₅OCS₂⁻), ethyl trithiocarbonate (C₂H₅SCS₂⁻), and ethyl dithiophosphate [(C₂H₅O)(OH)PS₂⁻] ions. The results demonstrated that the reactivity of these collectors follows the order of (C₂H₅)₂NCS₂⁻ > C₂H₅NHCS₂⁻ > C₂H₅OCS₂⁻ > C₂H₅SCS₂⁻ > (C₂H₅O)(OH)PS₂⁻. Wei et al. [268] evaluated the collecting performance and interaction mechanism of three sulfhydryl collectors, including ammonium dibutyl dithiophosphate (ADD), AMX, and DIBDTPI, with the Au (100) surface using the DFT and CMD methods. The results indicated that the collection power of three collectors towards gold follows the decreasing order of DIBDTPI > ADD > AMX. The Au-collector interaction on the surface is realized through the bonding of surface Au atoms and collector S atoms. The DOS results further suggested that the delocalization of the S 3p orbital is increased upon the adsorption of collectors, thus enhancing the nephelauxetic effect, which proved that the covalency of the Au−S bond has been facilitated.

Using DFT calculations and CMD simulations, Liu et al. [269] assessed the reactivities and adsorption states of DIBDTPA and DIBMTPA on the surfaces of gold, pyrite, and oxidized pyrite to improve the fundamental understanding of the surface chemistry of monomeric gold in the thiophosphate collector flotation of sulfide minerals. The results revealed that DIBMTPA exhibits effective collection power for gold and excellent selectivity against pyrite. This was attributed to the lower head group charge and lower reactivity of DIBMTPA, and the reactive atoms of =S and −O in DIBMTPA present better electron donation ability than those of =S and −S in DIBDTPA. It was also found that an oxidized pyrite surface contributes to the high selectivity of DIBMTPA in the flotation of gold from pyrite, whereas the selectivity of DIBDTPA in the flotation of gold from pyrite is limited.

9. Conclusions and Prospects

Since quantum chemistry was first applied to the field of flotation reagents, some great achievements have been accomplished over the past six decades. Quantum chemical studies on flotation reagents have been reported for a wide range of minerals, but due to the complexity of crystal structures and the limitations of computing power, there are also a number of minerals, e.g., Ni-, Co-, Sb-, Hg-, Cd-, Bi-, Al-, Mg-, Na-, Sr-, and Ba-bearing non-ferrous sulfide minerals, Cr-bearing minerals, some iron oxide minerals (maghemite, goethite, and siderite), some silicate minerals (sillimanite, asbestos, olivine-group minerals, epidote-group minerals, tourmaline-supergroup minerals, pyroxenes, amphiboles, and zeolites), and Ca-bearing minerals, apart from fluorite and calcite, that remain undiscovered, and therefore, they need to be studied in the future.

Moreover, for some of the minerals exhibiting strong surface hydrophilicity, their simulations often need to account for the effects of hydration. However, a model containing the mineral surface, the hydration layer structure, and the flotation reagents often consists of several hundred atoms. This leads to high computational costs, and it is difficult to perform simulations with the current computing power and computational software. As a result, for these minerals, only the surface properties and electronic structures are known, which makes studies on the mineral–reagent interaction mechanisms one of the main subjects for future research on these minerals.

Abundant novel types of collectors with high efficiency and selectivity were synthesized and discovered, with most of them agreeing with the “atom enantiomerism law”, confirming that O-containing surfactants are more applicable to oxide minerals, while S-containing surfactants are more applicable to sulfide minerals. Meanwhile, the effects of the type and geometry size of the head groups of these reagents on the flotation performance have also been reviewed. In addition, the hydrocarbon chain length, carbon chain isomerism, degree of saturation, and the degree of substitution of the flotation reagents also have great influences on the flotation behaviors of different minerals. When it comes to an unsaturated carbon chain and branched hydrocarbon chain, the results are even controversial.

Although the current studies are abundant but discrete, the majority of them focus only on one or a few flotation reagents and use various experimental methods in conjunction with DFT calculations to determine their interaction mechanisms on mineral surfaces, lacking systematic discussion. Hence, the design and discussion of experiments and calculations should be systematic and purpose-oriented to obtain specific and meaningful results as well as to minimize the existing controversy.

The effects of metal cations on the mineral–reagent interaction are diverse. The addition of metal cations can not only act as an activator to promote the subsequent collector adsorption, but also reinforces the depression of the depressant on mineral flotation. Nevertheless, for some minerals, such as quartz, once activated by metal cations, they are not easily depressed. In recent quantum chemistry studies, it has been found that a prerequisite for the adsorption of metal cations on mineral surfaces is the hydroxylation of metal cations [37,168]. The factors influencing the roles of metal cations are both direct (e.g., type, radius, valence state, and dosage) and indirect (e.g., other associated minerals in the pulp, pH, and other reagents). However, due to the limitations of the current computational model, the influence of other reagents on the roles of metal cations in flotation has hardly been investigated.

This also leads to a direction for future research, namely the assembly of reagent molecules on mineral surfaces, that is, the co-adsorption of multiple reagent molecules on mineral surfaces. In flotation practice, the combined use of multiple reagents is frequent (the mixed reagents could all be polar, or one could be polar and the other could be non-polar), but simulations of the assembly of reagent molecules on mineral surfaces through the means of quantum chemistry are still rare, and this is considered to be one of the two future directions for the application of quantum chemistry to flotation reagents.

The second future direction is to incorporate machine learning through the comprehensive evaluation and screening of a series of performance indexes, such as the indexes from the reagent aspect, including group electronegativity, solubility product, and the stability constants of complexes, and indexes from the mineral interaction aspect, including the interaction energy, interaction distance, adsorption configuration, density of state, Mulliken population, and charge density, in order to achieve the targeting design and development of novel, efficient, and green flotation reagents.