Adsorption Behavior and Wettability of Rhodochrosite Surface: Effect of C18 Fatty Acid Unsaturation

Mineral surface wettability and its regulation by the adsorption of collectors have an important influence on the flotation performance. The adsorption behavior of C18 fatty acid with different unsaturation and its effect on rhodochrosite wettability was investigated with surface tension, contact angle, and atomic force microscopy (AFM) measurements. The results indicated that rhodochrosite hydrophobicity increased with the increasing concentration of fatty acid, along with the maximum contact angle (θmax) between hemimicelle concentration (HMC) and critical micelle concentration (CMC). Oleic acid (OA), linoleic acid (LA), and α-linolenic acid (ALA) had a higher θmax than stearic acid (SA), but the value decreased with the increase of C=C bond number. Besides, preferential adsorption of unsaturated fatty acids on the liquid-air interface can be attributed to the molecule’s steric hindrance resulting from C=C double bond, and the θ kept almost invariant with a higher value of ΓLG than ΓSL until HMC. The oriented monolayer and bilayer structure of fatty acids formed gradually on rhodochrosite surface with increasing concentration. However, the θmax may not necessarily correspond to the beginning of bilayer formation. Cylindrical monolayer and bilayer micelles of SA molecules were observed on rhodochrosite surface at HMC and CMC, respectively. While bilayer structures of unsaturated fatty acids formed before complete coverage of monolayer on rhodochrosite surface because of surface heterogeneity. This work provided a good understanding on the adsorption mechanism of fatty acid on rhodochrosite for flotation.


Introduction
Wettability plays a significant role in numerous industrial processes, such as flotation, oil recovery, coating, detergency, and cosmetics industry [1][2][3]. The wettability of a solid surface could be significantly affected by the adsorption of surfactants owing to the change of interface properties in solid-liquid-air systems, resulting in the extensive use in mineral flotation [4,5]. The mineral surface property, especially the wettability, is the key to flotation. The flotation efficiency mostly depends on using surfactants, especially the collector. Therefore, minerals surface wettability and its regulation by the adsorption of the collector are the core issues in the flotation process [6,7].
Rhodochrosite is the foremost valuable mineral in manganese resources. However, low-grade manganese ore accounts for over 90% of manganese resources in China, which need to be beneficiated [8]. Flotation is regarded as a widely used and effective method for rhodochrosite beneficiation, showing various advantages over other conventional techniques [9][10][11]. There are many kinds of rhodochrosite flotation collectors, including anionic collector [12][13][14], cationic collector [15,16] and nonpolar collector [17]. Among these collectors, the fatty acid has been commonly used due to its low cost and extensive source [18]. The hydrocarbon chain lengths, unsaturation degree, carbon-chain isomerism, and frothing property of fatty acid could influence its collection performance. The solubility, cross-sectional area, and molecular curvature of fatty acid increase with the increase of unsaturation, which significantly affects the effective concentration of fatty acid in aqueous solution and its adsorption activity and capacity on the mineral surface [19]. Therefore, it is significant to explore the adsorption behavior of unsaturated fatty acid and its effect on the wetting behavior of rhodochrosite.
Over the years, many studies about the adsorption mechanism of fatty acid on calcite, fluorite, apatite, and other minerals in flotation have been carried out by researchers, with atomic force microscopy (AFM), infrared spectroscopy, and first-principles calculations methods [20][21][22][23][24]. Ye [20] indicated that the hydrophobicity of collophane can be enhanced by concomitant monolayer and bilayer micelles of oleate above HMC, and effect of adsorbed structure on wettability of collophane was confirmed by MDS. Xie [21] investigated adsorption of fatty acid on fluorapatite (001) surface based on DFT. The results showed that as the carbon chain grew within certain limits, the adsorption became stronger, and fatty acid formed stable chemical adsorption and hydrogen bond adsorption configurations at Ca1 (surf) site of Ca termination. Young [23] suggested that chemisorption of oleate on calcite surface occurs at low oleate concentrations, and the adsorption densities decrease with increasing temperature based on the in-situ FT-NIR/IRS. Moreover, the effect of fatty acid unsaturation degree on mineral hydrophobicity and flotation performances have been investigated. Iwasaki [25] and Purcell [26] et al. suggested that higher recoveries of rutile or hematite can be obtained with higher adsorption acted by oleic acid (OA) than that by linoleic acid (LA) and α-linolenic acid (ALA) at pH greater than 8. It was found that fatty acid with a higher unsaturation degree has less surface activity because of higher solubility. Liu et al. [27] investigated that the effect of iodine value of sodium fatty acids on collophane flotation, which suggested that stronger hydrophobicity of collophane attributes to a higher iodine value, and flotation recovery positively correlates with iodine value. In addition, Wang et al. [28] calculated that the frontier molecular orbital energies difference of LA was lower than that of OA, resulting in lower chemical activity and adsorption stability of OA on diaspore.
However, the mechanism research on fatty acid and rhodochrosite mainly focuses on the effect of structural modification on collecting capacity and solution chemistry for fatty acid. Zhou et al. [29,30] suggested that linoleate hydroxamic acid (LHA) showed superior selectivity in the flotation of rhodochrosite, along with the chemisorption of a five-membered chelate ring of LHA on rhodochrosite surface. The adsorption of LHA onto rhodochrosite leads to the formation of hydrophobic agglomerates of rhodochrosite particles (Mn-LHA compounds or chelates) which enhance the floatability of rhodochrosite. Chen et al. [31] concluded that surface activity and collecting capability of the fatty acid amide collectors increased with the increasing C=C double bond numbers in the long chain. Dai et al. [32] also suggested that hydroxamic acid collectors were adsorbed on the rhodochrosite surface physically and chemically. Besides, the formation of manganese oleate precipitation and non-electrostatic forces between rhodochrosite and sodium oleate were the keys for flotation, which was explained by Qin et al. [33]. Bu et al. [34] indicated that rhodochrosite interacted with oleic acid and sodium dodecyl benzene sulfonate, resulting in a more negative zeta potential and the co-adsorption of the collector and synergist. Moreover, chemical adsorption of the hydroxamic acid on rhodochrosite led to negative shifts for the zeta potential of rhodochrosite, which concluded by Zhao et al. [35]. In summary, research on the adsorption effect of fatty acid on rhodochrosite is limited, especially the effect of the unsaturation degree of fatty acid on the wettability of rhodochrosite. Four fatty acids containing 18 carbon atoms and a different number of C=C double bonds (0~3) were used in this work. By surface tension measurements, contact angle measurements, and AFM measurements, the adsorption properties of fatty acid at solution-air and rhodochrosite-solution interfaces with rhodochrosite wettability have been investigated, which provides a good understanding of the adsorption mechanism of fatty acids on rhodochrosite for flotation applications.

Materials
Natural rhodochrosite was obtained from Guangxi province, China.
The mineral composition was measured by X-ray diffraction analysis (XRD) (X'Pert PRO, Panalytical Company, Almelo, The Netherlands) (see Figure 1), and the diffraction peak of natural rhodochrosite shows a good agreement with the standard card (PDF#44-1472). The rhodochrosite was cut into a lump of nearly 1 cm in length, width, and thickness, respectively. Then the lump was polished using grit silicon carbide papers of 500, 1000, and 2000 mesh, and polished using a grinding machine (YMP-2B, from Shanghai Metallurgical Equipment Company Ltd., Shanghai, China). Grit silicon carbide papers of 500, 1000, and 2000 mesh were used as the abrasive materials, silk cloth and alumina powders of 1.0, 0.3, and 0.05 mm were used as the polishing materials, both were obtained from Lab Testing Technology (Shanghai) Co., Ltd., Shanghai, China [20]. The ultrasonication with water, ethanol, and ultrapure water was used to wash the lump successively for the contact angle and the AFM measurements.  [35]. In summary, research on the adsorption effect of fatty acid on rhodochrosite is limited, especially the effect of the unsaturation degree of fatty acid on the wettability of rhodochrosite. Four fatty acids containing 18 carbon atoms and a different number of C=C double bonds (0~3) were used in this work. By surface tension measurements, contact angle measurements, and AFM measurements, the adsorption properties of fatty acid at solution-air and rhodochrosite-solution interfaces with rhodochrosite wettability have been investigated, which provides a good understanding of the adsorption mechanism of fatty acids on rhodochrosite for flotation applications.

Materials
Natural rhodochrosite was obtained from Guangxi province, China. The mineral composition was measured by X-ray diffraction analysis (XRD) (X'Pert PRO, Panalytical Company, Almelo, The Netherlands) (see Figure 1), and the diffraction peak of natural rhodochrosite shows a good agreement with the standard card (PDF#44-1472). The rhodochrosite was cut into a lump of nearly 1 cm in length, width, and thickness, respectively. Then the lump was polished using grit silicon carbide papers of 500, 1000, and 2000 mesh, and polished using a grinding machine (YMP-2B, from Shanghai Metallurgical Equipment Company Ltd., Shanghai, China). Grit silicon carbide papers of 500, 1000, and 2000 mesh were used as the abrasive materials, silk cloth and alumina powders of 1.0, 0.3, and 0.05 mm were used as the polishing materials, both were obtained from Lab Testing Technology (Shanghai) Co., Ltd., Shanghai, China [20]. The ultrasonication with water, ethanol, and ultrapure water was used to wash the lump successively for the contact angle and the AFM measurements. The C18 fatty acids including stearic acid (SA), OA and LA used in this work were analytically pure and purchased from Shanghai Aladdin Bio-Chem Technology Co. Ltd. (Shanghai, China). In contrast, ALA was 85% pure and purchased from Shanghai Jizhi Bio-Chem Technology Co. Ltd. (Shanghai, China). The molecular structure and mainly physical properties of the above fatty acids are shown in Figure 2 and Table 1, respectively. Deionized water with the resistivity of 18.25 MΩ·cm was used to prepare the solutions of known concentration. The C 18 fatty acids including stearic acid (SA), OA and LA used in this work were analytically pure and purchased from Shanghai Aladdin Bio-Chem Technology Co. Ltd. (Shanghai, China). In contrast, ALA was 85% pure and purchased from Shanghai Jizhi Bio-Chem Technology Co. Ltd. (Shanghai, China). The molecular structure and mainly physical properties of the above fatty acids are shown in Figure 2 and Table 1, respectively. Deionized water with the resistivity of 18.25 MΩ·cm was used to prepare the solutions of known concentration.
a Sigma-Aldrich catalog. b It is difficult to obtain the values of length and limiting area for ALA due to its inherent molecule characteristics.

Measurements of Surface Tension
The Wilhelmy plate method with a surface tensiometer (JK99B, Powereach, Shanghai, China) under atmospheric pressure was used for surface tension measurement of fatty acid aqueous solutions. The glassware and platinum plate were cleaned and washed by ethyl alcohol and deionized water in sequence, and the platinum plate was treated with alcohol flame to remove any organic contamination. Before sample measurement, the surface tension of deionized water at 25 ℃ was tested to examine the cleanliness of glassware and plate and correct the tensiometer. Approximately 25 mL fatty acid solution was used for each test. Each sample was tested at least three times in succession.

Measurements of Contact Angle
The dynamic contact angle measuring instrument (HARKE-SPCAX3, Beijing Hake test instrument factory, Beijing, China) was used for contact angle (θ) measurements for fatty acid solutions on rhodochrosite with the sessile drop method. The rhodochrosite lump was immersed in a 50 mL fatty acid solution for 20 min. Next, the sample was taken out and dried at atmospheric temperature. The droplet volume used for contact angle test is 2 microliters. In order to avoid the gravity influence of the droplet, the θ was measured 10 s after the droplet falls on the rhodochrosite surface. The "baseline circle" in the instrument's software was used. Each θ was measured three times at different surface areas, and the average value was the final result [41]. After each test, the rhodochrosite lump was successively washed with ethyl alcohol and deionized water. Finally, the lump was heated at 40 ℃ for 30 min for the next test.

Measurements of Surface Tension
The Wilhelmy plate method with a surface tensiometer (JK99B, Powereach, Shanghai, China) under atmospheric pressure was used for surface tension measurement of fatty acid aqueous solutions. The glassware and platinum plate were cleaned and washed by ethyl alcohol and deionized water in sequence, and the platinum plate was treated with alcohol flame to remove any organic contamination. Before sample measurement, the surface tension of deionized water at 25°C was tested to examine the cleanliness of glassware and plate and correct the tensiometer. Approximately 25 mL fatty acid solution was used for each test. Each sample was tested at least three times in succession.

Measurements of Contact Angle
The dynamic contact angle measuring instrument (HARKE-SPCAX3, Beijing Hake test instrument factory, Beijing, China) was used for contact angle (θ) measurements for fatty acid solutions on rhodochrosite with the sessile drop method. The rhodochrosite lump was immersed in a 50 mL fatty acid solution for 20 min. Next, the sample was taken out and dried at atmospheric temperature. The droplet volume used for contact angle test is 2 microliters. In order to avoid the gravity influence of the droplet, the θ was measured 10 s after the droplet falls on the rhodochrosite surface. The "baseline circle" in the instrument's software was used. Each θ was measured three times at different surface areas, and the average value was the final result [41]. After each test, the rhodochrosite lump was successively washed with ethyl alcohol and deionized water. Finally, the lump was heated at 40°C for 30 min for the next test.

AFM Analysis
The adsorption morphology and thickness of fatty acid on rhodochrosite surface was visualized by the AFM, a Dimension Icon (Bruker, Karlsruhe, Germany). The ScanAsyst-Air tips (Bruker, Karlsruhe, Germany) with peak force tapping mode at 1 Hz scan rate and 512 lines per sample was used for collecting the images. First, the surface of polished rhodochrosite was visualized to record an AFM image for comparison. Then the rhodochrosite lump was immersed in a 50 mL fatty acid solution for 20 min. Next, the sample was taken out and dried at atmospheric temperature. Finally, AFM images of the surface adsorbed fatty acid were recorded. The AFM images were analyzed by Nanoscope Analysis 1.40 software (Bruker, Karlsruhe, Germany) with a first order smoothing firstly, and then the roughness module was used to analyze the entire image roughness. Finally, the corresponding height profiles were obtained with the Section module [42].

Self-Assembly of Fatty Acid in Aqueous Solution
The self-assembly of fatty acids can form micelles in aqueous solutions, depending on their critical micelle concentration (CMC) [43]. The relationship between equilibrium surface tension (γ LG ) and concentration of four fatty acids are presented in Figure 3. As shown in Figure 3, with the increase in the fatty acid concentration of the solutions, the γ LG of all fatty acids decrease until they are stable. The values of the premicellar concentration [44], the CMC, and surface tension at the CMC (γ CMC ) of fatty acid can also be obtained from Figure 3, as shown in Table 2. The premicellar concentration and CMC of OA, LA, and ALA are 1 × 10 −5 mol/L and 1 × 10 −3 mol/L, respectively, which are consistent with other researches [41,45]. The premicellar concentration and CMC of SA are 1 × 10 −6 mol/L and 1 × 10 −4 mol/L, respectively, which are lower than those of OA, LA, and ALA. SA is a saturated fatty acid, and its melting point is higher than unsaturated acids with C=C double bond, resulting in stronger hydrophobicity and lower value of CMC of SA. For unsaturated fatty acids, OA has the smallest value of γ CMC , while ALA has the largest γ CMC , which may be due to that the greater number of C=C bond in ALA, the less hydrophobic of the carbon chain, resulting in higher γ LG when micelles are formed, indicating that the unsaturation degree also affects the self-assembly of fatty acids. This conclusion is supported by other literature [46]. It is considered that small micelles may be formed in the range of fatty acid concentrations ranging from the premicellar concentration to CMC, while spherical micelles may be formed at the CMC. Schematic plots of the correspondent micelles are shown in Figure 4.
Minerals 2020, 10, x; doi: FOR PEER REVIEW www.mdpi.com/journal/minerals solution for 20 min. Next, the sample was taken out and dried at atmospheric temperature. Finally, AFM images of the surface adsorbed fatty acid were recorded. The AFM images were analyzed by Nanoscope Analysis 1.40 software (Bruker, Karlsruhe, Germany) with a first order smoothing firstly, and then the roughness module was used to analyze the entire image roughness. Finally, the corresponding height profiles were obtained with the Section module [42].

Self-assembly of Fatty Acid in Aqueous Solution
The self-assembly of fatty acids can form micelles in aqueous solutions, depending on their critical micelle concentration (CMC) [43]. The relationship between equilibrium surface tension (γLG) and concentration of four fatty acids are presented in Figure 3. As shown in Figure 3, with the increase in the fatty acid concentration of the solutions, the γLG of all fatty acids decrease until they are stable. The values of the premicellar concentration [44], the CMC, and surface tension at the CMC (γCMC) of fatty acid can also be obtained from Figure 3, as shown in Table 2. The premicellar concentration and CMC of OA, LA, and ALA are 1 × 10 −5 mol/L and 1 × 10 −3 mol/L, respectively, which are consistent with other researches [41,45]. The premicellar concentration and CMC of SA are 1 × 10 −6 mol/L and 1 × 10 −4 mol/L, respectively, which are lower than those of OA, LA, and ALA. SA is a saturated fatty acid, and its melting point is higher than unsaturated acids with C=C double bond, resulting in stronger hydrophobicity and lower value of CMC of SA. For unsaturated fatty acids, OA has the smallest value of γCMC, while ALA has the largest γCMC, which may be due to that the greater number of C=C bond in ALA, the less hydrophobic of the carbon chain, resulting in higher γLG when micelles are formed, indicating that the unsaturation degree also affects the self-assembly of fatty acids. This conclusion is supported by other literature [46]. It is considered that small micelles may be formed in the range of fatty acid concentrations ranging from the premicellar concentration to CMC, while spherical micelles may be formed at the CMC. Schematic plots of the correspondent micelles are shown in Figure 4.

Wettability of Rhodochrosite
The dependence of θ of rhodochrosite on the total concentration of four fatty acids in aqueous solutions is shown in Figure 5. As shown in Figure 5, the θ keeps almost constant over the low range of fatty acid concentration. While the θ and surface hydrophobicity of rhodochrosite increases with

Wettability of Rhodochrosite
The dependence of θ of rhodochrosite on the total concentration of four fatty acids in aqueous solutions is shown in Figure 5. As shown in Figure 5, the θ keeps almost constant over the low range of fatty acid concentration. While the θ and surface hydrophobicity of rhodochrosite increases with the increasing fatty acid concentration. When the concentration of SA is 5 × 10 −6 mol/L, and the concentration of OA, LA and ALA are 5 × 10 −5 mol/L, the maximum contact angle (θ max ) is 67.50 • , 76.20 • , 75.87 • , and 72.60 • , respectively. It suggests that the rhodochrosite may be the most hydrophobic between hemimicelle concentration (HMC) and CMC for fatty acids. However, there is a more marked decrease in θ for all fatty acids with a further increase in concentrations. When it exceeds CMC, the θ decreases rapidly, and rhodochrosite becomes hydrophilic with the rise in concentration, indicating that the mineral surface becomes hydrophilic again with increasing fatty acid concentration.
Minerals 2020, 10, x; doi: FOR PEER REVIEW www.mdpi.com/journal/minerals 76.20°, 75.87°, and 72.60°, respectively. It suggests that the rhodochrosite may be the most hydrophobic between hemimicelle concentration (HMC) and CMC for fatty acids. However, there is a more marked decrease in θ for all fatty acids with a further increase in concentrations. When it exceeds CMC, the θ decreases rapidly, and rhodochrosite becomes hydrophilic with the rise in concentration, indicating that the mineral surface becomes hydrophilic again with increasing fatty acid concentration.
The θmax for three unsaturated acids are significantly larger than that for SA. The reason may be that the greater the unsaturation degree, the lower the melting point (Table 1), and the easier it is to dissolve and disperse in water. Figure 5 also shows that for OA, LA, and ALA, the θmax decrease with increase in C=C bond number of fatty acids, the reason may be that the more C=C double bond, the larger the unsaturation degree, the molecular distance and limit area of unsaturated fatty acid (data in Table 1: OA < LA < ALA). Therefore, the adsorption of unsaturated fatty acid on the surface of rhodochrosite is more affected by the steric hindrance caused by its curved structure [47]. Moreover, there is also an increase in chain disorder within the adsorbed layer as the unsaturation increases, hence increased effective surface space required, resulting in lower packing density of unsaturated alkyl chains. This is also supported by Wood et al. [48]. At the same concentration, the number of fatty acid molecules adsorbed on the rhodochrosite surface are OA > LA > ALA, and the adsorption density of fatty acid on rhodochrosite is proportional to its molecule number, resulting in the weaker hydrophobicity of rhodochrosite with an increase in C=C bond number of fatty acids. The θ max for three unsaturated acids are significantly larger than that for SA. The reason may be that the greater the unsaturation degree, the lower the melting point (Table 1), and the easier it is to dissolve and disperse in water. Figure 5 also shows that for OA, LA, and ALA, the θ max decrease with increase in C=C bond number of fatty acids, the reason may be that the more C=C double bond, the larger the unsaturation degree, the molecular distance and limit area of unsaturated fatty acid (data in Table 1: OA < LA < ALA). Therefore, the adsorption of unsaturated fatty acid on the surface of rhodochrosite is more affected by the steric hindrance caused by its curved structure [47]. Moreover, there is also an increase in chain disorder within the adsorbed layer as the unsaturation increases, hence increased effective surface space required, resulting in lower packing density of unsaturated alkyl chains. This is also supported by Wood et al. [48]. At the same concentration, the number of fatty acid molecules adsorbed on the rhodochrosite surface are OA > LA > ALA, and the adsorption density of fatty acid on rhodochrosite is proportional to its molecule number, resulting in the weaker hydrophobicity of rhodochrosite with an increase in C=C bond number of fatty acids.

Dependence of Wettability and Surface Tension
According to literature, the critical surface tension of wetting can be estimated by extrapolation of cosθ versus surface tension γ LG plots to cosθ = 1 [49,50]. As shown in Figure 6, it appears that there is no linear correlation between cosθ and γ LG . However, the cosθ is correlated with γ LG within a certain range of γ LG . .98 mN/m, respectively. However, the slope of cosθ-γLG curves quickly increase with a marked increase in cosθ which eventually approaching 1 with a further decrease in γLG, indicating that as the bulk concentration continues to rise, the surface of rhodochrosite changes from hydrophobic to hydrophilic, and finally strongly hydrophilic. At this time, the γLG of SA, OA, LA, and ALA correspond to the minimum values of 33, 24, 29 and 30, respectively.

Relationship between the Adhesion and Surface Tension
According to literature, the relationship between adhesion tension(γLG·cosθ) and γLG for both hydrophilic and hydrophobic solids is possible to be established [51][52][53], as shown in Equation (1): where a and b are constants, depending on the solid surface property. Besides, Combining the Young's equation and Gibbs equation, Lucassen-Reynders proposed a simple method to analyze the relationship between adsorption and equilibrium wetting, as shown in Equation (2) [54]: where ΓSG, ΓSL and ΓLG represent the excess surface concentrations of surfactants at the solid-air, solid-liquid, and liquid-air interfaces, respectively. Assuming that ΓSG ≈ 0, it is possible to obtain the ΓSL/ΓLG value via calculating the slope of the γLG·cosθ-γLG curve. Therefore, combining Equations (1) and (2), Equation (3) can be obtained.

Relationship between the Adhesion and Surface Tension
According to literature, the relationship between adhesion tension (γ LG ·cosθ) and γ LG for both hydrophilic and hydrophobic solids is possible to be established [51][52][53], as shown in Equation (1): where a and b are constants, depending on the solid surface property. Besides, Combining the Young's equation and Gibbs equation, Lucassen-Reynders proposed a simple method to analyze the relationship between adsorption and equilibrium wetting, as shown in Equation (2) [54]: where Γ SG , Γ SL and Γ LG represent the excess surface concentrations of surfactants at the solid-air, solid-liquid, and liquid-air interfaces, respectively. Assuming that Γ SG ≈ 0, it is possible to obtain the Γ SL /Γ LG value via calculating the slope of the γ LG ·cosθ-γ LG curve. Therefore, combining Equations (1) and (2), Equation (3) can be obtained.
From Equations (1) and (2), the critical surface tension of solid wetting would be determined, and surfactant adsorption on the liquid-air and solid-liquid interfaces would be analyzed. As weak interaction (such as Lifshitz-van der Waals interactions) presented in the surfactant and solid, the slope of the γ LG ·cosθ-γ LG curve is less than 1, while the slope is greater than 1 as strong interaction presented (such as electrostatic interaction) [55,56].
The variation of γ LG ·cosθ with γ LG of fatty acid solutions was presented in Figure 7. As shown in Figure 7, there is no direct linear trend between γ LG ·cosθ and γ LG on the rhodochrosite-fatty acid solutions-air interface. However, within a certain concentration range, the γ LG ·cosθ changes linearly with the γ LG . For SA, with reduction in γ LG , the value of a varies from 0.57 to 1.97, the γ LG ·cosθ decreases and suddenly increases around γ CMC . In the initial adsorption period, the ratio of Γ SL to Γ LG (a) is less than 1, that is, Γ SL < Γ LG , indicating that the adsorption density on the solid-liquid interface is lower than that on the air-liquid interface. As γ LG continues to decrease, the a value increases to 1.24 (~1), which shows that the Lifshitz-van der Waals interactions between fatty acid molecules and rhodochrosite is the primary force of adsorption. While the electrostatic force interaction mainly drives the adsorption, Γ SL > Γ LG and the slope value >1 [55]. In the case of OA, LA, and ALA, the variations of γ LG ·cosθ and γ LG are similar. The γ LG ·cosθ decrease with the decreasing γ LG and then increase rapidly at γ CMC with no stable region in the γ LG ·cosθ-γ LG curves. From the values of the slope, Van der Waals force is the primary interaction in the initial adsorption process. However, the steric hindrance effect and chain disorder produced by the molecular structure with C=C double bond makes it more difficult for unsaturated fatty acids to adsorb on rhodochrosite [47], resulting in the ratio of Γ SL to Γ LG slightly smaller than 1. While the electrostatic interaction comes into operation, Γ SL is much larger than Γ LG , which causes a marked increase in the value of a.
Minerals 2020, 10, x; doi: FOR PEER REVIEW www.mdpi.com/journal/minerals and surfactant adsorption on the liquid-air and solid-liquid interfaces would be analyzed. As weak interaction (such as Lifshitz-van der Waals interactions) presented in the surfactant and solid, the slope of the γLG·cosθ-γLG curve is less than 1, while the slope is greater than 1 as strong interaction presented (such as electrostatic interaction) [55,56]. The variation of γLG·cosθ with γLG of fatty acid solutions was presented in Figure 7. As shown in Figure 7, there is no direct linear trend between γLG·cosθ and γLG on the rhodochrosite-fatty acid solutions-air interface. However, within a certain concentration range, the γLG·cosθ changes linearly with the γLG. For SA, with reduction in γLG, the value of varies from 0.57 to 1.97, the γLG·cosθ decreases and suddenly increases around γCMC. In the initial adsorption period, the ratio of ΓSL to ΓLG (a) is less than 1, that is, ΓSL < ΓLG, indicating that the adsorption density on the solid-liquid interface is lower than that on the air-liquid interface. As γLG continues to decrease, the value increases to 1.24 (~1), which shows that the Lifshitz-van der Waals interactions between fatty acid molecules and rhodochrosite is the primary force of adsorption. While the electrostatic force interaction mainly drives the adsorption, ΓSL > ΓLG and the slope value >1 [55]. In the case of OA, LA, and ALA, the variations of γLG·cosθ and γLG are similar. The γLG·cosθ decrease with the decreasing γLG and then increase rapidly at γCMC with no stable region in the γLG·cos θ-γLG curves. From the values of the slope, Van der Waals force is the primary interaction in the initial adsorption process. However, the steric hindrance effect and chain disorder produced by the molecular structure with C=C double bond makes it more difficult for unsaturated fatty acids to adsorb on rhodochrosite [47], resulting in the ratio of ΓSL to ΓLG slightly smaller than 1. While the electrostatic interaction comes into operation, ΓSL is much larger than ΓLG, which causes a marked increase in the value of .

Work of Adhesion
Because the solid-liquid interfacial tension can be expressed by the following equation [57]: where WA is the adhesion work of the liquid to the solid surface. According to Young's equation, the θ depends on the solid-liquid interfacial tension(γSL), solid-air interfacial tension (γSG), and liquid-air interfacial tension (γLG), as shown in Equation (5):

Work of Adhesion
Because the solid-liquid interfacial tension can be expressed by the following equation [57]: where W A is the adhesion work of the liquid to the solid surface. According to Young's equation, the θ depends on the solid-liquid interfacial tension(γ SL ), solid-air interfacial tension (γ SG ), and liquid-air interfacial tension (γ LG ), as shown in Equation (5): Equation (6) can be obtained by introducing Equation (4) into Young's Equation (5): Minerals 2020, 10, 905 9 of 17 The W A of fatty acid solutions to the rhodochrosite surface was calculated, as shown in Figure 8. It is suggested that the W A for the aqueous solution of all fatty acids decreases first and then with increasing concentration. Interestingly, the minima values of the W A of all fatty acids are also observed at the concentration around the HMC.
Equation (6) can be obtained by introducing Equation (4) into Young's Equation (5): The WA of fatty acid solutions to the rhodochrosite surface was calculated, as shown in Figure 8. It is suggested that the WA for the aqueous solution of all fatty acids decreases first and then with increasing concentration. Interestingly, the minima values of the WA of all fatty acids are also observed at the concentration around the HMC.

Structural Dependence of Fatty Acid on the Wettability of Rhodochrosite Surface
The dependencies of γLG, θ, γLG·cosθ, and WA on concentration and the results have been replotted in Figure 9 for four fatty acids. The variations in adhesional data as a function of increasing concentration can be divided into four stages. The probable diagrammatic drawing of the adsorbed fatty acid on the air-liquid interface and rhodochrosite-liquid interface in the four stages is presented in Figure 10.
In the case of SA, as shown in Figure 9a, in stage 1, stable adsorption films are difficult to form on both liquid-air and rhodochrosite-liquid interfaces because of very low concentration. Therefore, the γLG, θ, γLG·cosθ, and WA all change little until 1 × 10 −7 mol/L. In stage 2, the γLG, γLG·cosθ, and WA gradually decrease with increasing concentration, while the θ become larger and larger. The stable unsaturated adsorption film begins to form with SA molecules, resulting in enhanced hydrophobicity of rhodochrosite. At the early step of stage 2, under the action of Lifshitz-van der Waals forces, SA molecules adsorb with the hydrophilic group pointing to the rhodochrosite and the hydrophobic group to the solution. On the basis of results in Figure 7, ΓSL ≈ ΓLG, leading to a slow increase in θ. Then, the anionic head group of SA begins to interact with the rhodochrosite surface via electrostatic forces, molecules preferentially adsorb at rhodochrosite-liquid interface and ΓSL ≈ 2ΓLG. As a result, the θ increases sharply attributed to the relatively greater extent of decrease in γLG·cosθ than γLG. Besides, there are the θmax and minimal γLG·cosθ as the concentration increasing to 5 × 10 −5 mol/L, around the HMC, suggesting that a saturated monolayer adsorption film has formed at the rhodochrosite-liquid interface. In stage 3, SA molecules kept adsorbing at the liquid-air interface, the γLG decreases monofonically, and the θ decreases significantly. Then, the micelles will form with the addition of SA molecules and adsorb onto monolayer film at the rhodochrosite surface again through carbon-chain association, which leads to enhanced hydrophilicity of rhodochrosite. In stage 4, as the SA concentration reaches 1 × 10 −4 mol/L (CMC), the γLG decreases to a minimum and remains stable, a saturated adsorption film forms with SA molecules at the liquid-air interface. However, the θ continues to decrease, and γLG·cosθ and WA increase with a further increase in concentration. Hence the rhodochrosite is more hydrophilic attributed to the formation of the bilayer structure at a solidliquid interface.

Structural Dependence of Fatty Acid on the Wettability of Rhodochrosite Surface
The dependencies of γ LG , θ, γ LG ·cosθ, and W A on concentration and the results have been replotted in Figure 9 for four fatty acids. The variations in adhesional data as a function of increasing concentration can be divided into four stages. The probable diagrammatic drawing of the adsorbed fatty acid on the air-liquid interface and rhodochrosite-liquid interface in the four stages is presented in Figure 10. The variation of γLG, θ, γLG·cosθ, and WA with concentration for OA, LA, and ALA are similar. As shown in Figure 9 b-d, in stage 2, the θ increases slowly with the decreases in γLG and γLG·cosθ for OA, while the θ keeps constant with the reductions in γLG and γLG·cosθ for LA and ALA. The reason may be that the unsaturated acid molecules mainly adsorb on the solid-liquid interface by van der Waals force with lower concentration. Meanwhile, in contrast to SA, the presence of C=C double bond makes unsaturated acids molecules more polar, reactive, and bent [58], which may increase the nonlinear tendency of molecules, and reduce the adsorption space and amount of unsaturated fatty acid molecules on the surface of rhodochrosite [59]. On the other hand, the values of ΓSL are smaller than ΓLG in stage 2 for OA, LA, and ALA from the results in Figure 7, suggesting the dominant adsorption of acids on liquid-air interface. On the basis of Young's equation (Equation (5)), when θ < 90°, the reduction of γLG will result in the decrease of θ, while the decline in the γLG·cosθ will lead to the increase of θ. Therefore, the θ remains almost stable at stage 2 attributed to the competition between γLG and γLG·cosθ [4]. In stage 3, the adsorption density of OA, LA, and ALA in the rhodochrositeliquid interface is higher than that in the air-liquid interface (OA: ΓSL ≈ 5ΓLG, LA: ΓSL ≈ 7ΓLG, ALA: ΓSL ≈ 9ΓLG). At this time, the driving force for adsorption should be a stronger interaction (electrostatic attraction), which leads to an increase in γSL, a significant increase in the θ with enhanced hydrophobicity of rhodochrosite. When the concentration arriving at 5 × 10 −5 mol/L, a saturated film forms at the liquid-air and rhodochrosite-liquid interface, resulting in the θmax and strongest hydrophobicity of rhodochrosite. Therefore, as the concentration increases, the excessive acid molecules will be adsorbed on the saturated film, immediately forming a double-layer structure at the rhodochrosite-liquid interface (stage 4).

Morphology of Adsorbed Fatty Acid on Rhodochrosite Surface
In most research, mica is often used as a substrate because of its flatness [60]. Therefore, to accurately determine the fatty acid adsorption layer height, a very smooth rhodochrosite surface is required. A ground and polished rhodochrosite surface was used for the substrate in our research. The micrograph (Figure 11a) indicates that the surface of rhodochrosite is relatively flat, showing a certain streak, and accompanied by tiny voids. The surface roughness (Ra) of 5.77 nm can be obtained from the two-dimensional AFM image (Figure 11b). The 2D AFM images of the rhodochrosite surface treated by four fatty acids solutions with two different initial concentrations are shown in Figure 12. The homologous height of adsorbed fatty acid on the rhodochrosite surface is shown in Figure 13. The two concentrations used are Premicellar concentration and CMC mentioned above in Table 2. Comparing Figure 12 with Figure 11, it is suggested that spotted, spherical, laminated, and block adsorption structures are evident on the surface of rhodochrosite after treated with fatty acid. From the height distribution diagram, the small and large columned micelles morphologies of adsorbed fatty acid were observed, and their heights were obtained. Moreover, by comparing the measured heights of the adsorption layer with the theoretical molecular length of the fatty acid, estimation for monolayer, bilayer, or multilayer structures of adsorbed fatty acid were carried out [20].
In the case of SA, the theoretically calculated molecular length range from 2.62 to 2.66 nm ( Table  1). When the concentrations were 1 × 10 −6 mol/L and 1 × 10 −4 mol/L, the adsorption layer heights were in the range of 2.24-2.64 nm and 3.25-5.67 nm, respectively. This indicates that the small cylindrical micelles with monolayer formed at HMC, while large columned micelles with monolayer and bilayer formed at CMC (Figure 12a,b, Figure 13a,b). For OA, LA, and ALA, the adsorption layer heights were In the case of SA, as shown in Figure 9a, in stage 1, stable adsorption films are difficult to form on both liquid-air and rhodochrosite-liquid interfaces because of very low concentration. Therefore, the γ LG , θ, γ LG ·cosθ, and W A all change little until 1 × 10 −7 mol/L. In stage 2, the γ LG , γ LG ·cosθ, and W A gradually decrease with increasing concentration, while the θ become larger and larger. The stable unsaturated adsorption film begins to form with SA molecules, resulting in enhanced hydrophobicity of rhodochrosite. At the early step of stage 2, under the action of Lifshitz-van der Waals forces, SA molecules adsorb with the hydrophilic group pointing to the rhodochrosite and the hydrophobic group to the solution. On the basis of results in Figure 7, Γ SL ≈ Γ LG , leading to a slow increase in θ. Then, the anionic head group of SA begins to interact with the rhodochrosite surface via electrostatic forces, molecules preferentially adsorb at rhodochrosite-liquid interface and Γ SL ≈ 2Γ LG . As a result, the θ increases sharply attributed to the relatively greater extent of decrease in γ LG ·cosθ than γ LG . Besides, there are the θ max and minimal γ LG ·cosθ as the concentration increasing to 5 × 10 −5 mol/L, around the HMC, suggesting that a saturated monolayer adsorption film has formed at the rhodochrosite-liquid interface. In stage 3, SA molecules kept adsorbing at the liquid-air interface, the γ LG decreases monofonically, and the θ decreases significantly. Then, the micelles will form with the addition of SA molecules and adsorb onto monolayer film at the rhodochrosite surface again through carbon-chain association, which leads to enhanced hydrophilicity of rhodochrosite. In stage 4, as the SA concentration reaches 1 × 10 −4 mol/L (CMC), the γ LG decreases to a minimum and remains stable, a saturated adsorption film forms with SA molecules at the liquid-air interface. However, the θ continues to decrease, and γ LG ·cosθ and W A increase with a further increase in concentration. Hence the rhodochrosite is more hydrophilic attributed to the formation of the bilayer structure at a solid-liquid interface.
The variation of γ LG , θ, γ LG ·cosθ, and W A with concentration for OA, LA, and ALA are similar. As shown in Figure 9b-d, in stage 2, the θ increases slowly with the decreases in γ LG and γ LG ·cosθ for OA, while the θ keeps constant with the reductions in γ LG and γ LG ·cosθ for LA and ALA. The reason may be that the unsaturated acid molecules mainly adsorb on the solid-liquid interface by van der Waals force with lower concentration. Meanwhile, in contrast to SA, the presence of C=C double bond makes unsaturated acids molecules more polar, reactive, and bent [58], which may increase the non-linear tendency of molecules, and reduce the adsorption space and amount of unsaturated fatty acid molecules on the surface of rhodochrosite [59]. On the other hand, the values of Γ SL are smaller than Γ LG in stage 2 for OA, LA, and ALA from the results in Figure 7, suggesting the dominant adsorption of acids on liquid-air interface. On the basis of Young's equation (Equation (5)), when θ < 90 • , the reduction of γ LG will result in the decrease of θ, while the decline in the γ LG ·cosθ will lead to the increase of θ. Therefore, the θ remains almost stable at stage 2 attributed to the competition between γ LG and γ LG ·cosθ [4]. In stage 3, the adsorption density of OA, LA, and ALA in the rhodochrosite-liquid interface is higher than that in the air-liquid interface (OA: Γ SL ≈ 5Γ LG , LA: Γ SL ≈ 7Γ LG , ALA: Γ SL ≈ 9Γ LG ). At this time, the driving force for adsorption should be a stronger interaction (electrostatic attraction), which leads to an increase in γ SL , a significant increase in the θ with enhanced hydrophobicity of rhodochrosite. When the concentration arriving at 5 × 10 −5 mol/L, a saturated film forms at the liquid-air and rhodochrosite-liquid interface, resulting in the θ max and strongest hydrophobicity of rhodochrosite. Therefore, as the concentration increases, the excessive acid molecules will be adsorbed on the saturated film, immediately forming a double-layer structure at the rhodochrosite-liquid interface (stage 4).

Morphology of Adsorbed Fatty Acid on Rhodochrosite Surface
In most research, mica is often used as a substrate because of its flatness [60]. Therefore, to accurately determine the fatty acid adsorption layer height, a very smooth rhodochrosite surface is required. A ground and polished rhodochrosite surface was used for the substrate in our research. The micrograph (Figure 11a) indicates that the surface of rhodochrosite is relatively flat, showing a certain streak, and accompanied by tiny voids. The surface roughness (Ra) of 5.77 nm can be obtained from the two-dimensional AFM image (Figure 11b).

Morphology of Adsorbed Fatty Acid on Rhodochrosite Surface
In most research, mica is often used as a substrate because of its flatness [60]. Therefore, to accurately determine the fatty acid adsorption layer height, a very smooth rhodochrosite surface is required. A ground and polished rhodochrosite surface was used for the substrate in our research. The micrograph (Figure 11a) indicates that the surface of rhodochrosite is relatively flat, showing a certain streak, and accompanied by tiny voids. The surface roughness (Ra) of 5.77 nm can be obtained from the two-dimensional AFM image (Figure 11b). The 2D AFM images of the rhodochrosite surface treated by four fatty acids solutions with two different initial concentrations are shown in Figure 12. The homologous height of adsorbed fatty acid on the rhodochrosite surface is shown in Figure 13. The two concentrations used are Premicellar concentration and CMC mentioned above in Table 2. Comparing Figure 12 with Figure 11, it is suggested that spotted, spherical, laminated, and block adsorption structures are evident on the surface of rhodochrosite after treated with fatty acid. From the height distribution diagram, the small and large columned micelles morphologies of adsorbed fatty acid were observed, and their heights were obtained. Moreover, by comparing the measured heights of the adsorption layer with the theoretical molecular length of the fatty acid, estimation for monolayer, bilayer, or multilayer structures of adsorbed fatty acid were carried out [20].
In the case of SA, the theoretically calculated molecular length range from 2.62 to 2.66 nm ( Table  1). When the concentrations were 1 × 10 −6 mol/L and 1 × 10 −4 mol/L, the adsorption layer heights were in the range of 2.24-2.64 nm and 3.25-5.67 nm, respectively. This indicates that the small cylindrical micelles with monolayer formed at HMC, while large columned micelles with monolayer and bilayer formed at CMC (Figure 12a,b, Figure 13a,b). For OA, LA, and ALA, the adsorption layer heights were The 2D AFM images of the rhodochrosite surface treated by four fatty acids solutions with two different initial concentrations are shown in Figure 12. The homologous height of adsorbed fatty acid on the rhodochrosite surface is shown in Figure 13. The two concentrations used are Premicellar concentration and CMC mentioned above in Table 2. Comparing Figure 12 with Figure 11, it is suggested that spotted, spherical, laminated, and block adsorption structures are evident on the surface of rhodochrosite after treated with fatty acid. From the height distribution diagram, the small and large columned micelles morphologies of adsorbed fatty acid were observed, and their heights were obtained. Moreover, by comparing the measured heights of the adsorption layer with the theoretical molecular length of the fatty acid, estimation for monolayer, bilayer, or multilayer structures of adsorbed fatty acid were carried out [20].
Minerals 2020, 10, x; doi: FOR PEER REVIEW www.mdpi.com/journal/minerals to the low compressibility of SA molecules resulting from the attraction between hydrocarbon chains of adjacent SA molecules. Therefore, a tight monolayer can be obtained in the adsorption process [34]. However, unsaturated fatty acids have large molecular spacing (Table 1) and weak intermolecular attraction due to the spatial hindrance caused by C=C double bond in their carbon chains. This will cause the monolayer structure to be loose during the adsorption process, resulting in the interspace on the surface easily overlaid by double-layer structures.
Minerals 2020, 10, x FOR PEER REVIEW 13 of 17  In the case of SA, the theoretically calculated molecular length range from 2.62 to 2.66 nm (Table 1). When the concentrations were 1 × 10 −6 mol/L and 1 × 10 −4 mol/L, the adsorption layer heights were in the range of 2.24-2.64 nm and 3.25-5.67 nm, respectively. This indicates that the small cylindrical micelles with monolayer formed at HMC, while large columned micelles with monolayer and bilayer formed at CMC (Figure 12a,b, Figure 13a,b). For OA, LA, and ALA, the adsorption layer heights were in the range of 2.97-10.30 nm, 2.09-5.52 nm, 2.43-10.18 nm at the concentration of 1 × 10 −5 mol/L, respectively, while in the range of 5.43-18.96 nm, 4.65-7.66 nm, and 5.07-10.44 nm at 1 × 10 −3 mol/L, respectively. It is suggested that the major monolayer formed accompanied by bilayer and multilayer structures at HMC. As the concentration increased to CMC, large columned micelles formed with bilayer and multilayer structures, along with further increase surface coverage (Figure 12c-h, Figure 13c-h). The concomitant monolayer, bilayer, and multilayer structure of unsaturated fatty acid molecules on the same surface were observed, which shows that before the completion of monolayer coverage on rhodochrosite surface, patches of bilayer structures formed, and the hydrophobic association of the hydrocarbon chains takes place. This effect may be attributed to the heterogeneity of mineral surface [61] and the self-assembly of unsaturated fatty acids in aqueous solution [20]. In contrast, the bilayer structure of SA did not appear before the monolayer coating, which may be due to the low compressibility of SA molecules resulting from the attraction between hydrocarbon chains of adjacent SA molecules. Therefore, a tight monolayer can be obtained in the adsorption process [34]. However, unsaturated fatty acids have large molecular spacing (Table 1) and weak intermolecular attraction due to the spatial hindrance caused by C=C double bond in their carbon chains. This will cause the monolayer structure to be loose during the adsorption process, resulting in the interspace on the surface easily overlaid by double-layer structures.

Conclusions
The θ of rhodochrosite increases with the increase in fatty acid concentration, reaching a maximum between HMC and CMC. Fatty acids containing C=C double bond have a higher maximum θ than SA, but the maximum decreases with an increase in the number of C=C double bonds. As the SA concentration increases, a monolayer with oriented structure can be formed on rhodochrosite surface with enhanced hydrophobicity, while the formation of bilayer structure appears at a concentration around its CMC. The steric hindrance caused by the introduction of C=C double bond leads to the preferential adsorption of OA, LA, and ALA on the liquid-air interface. The θ remains almost constant with the increasing concentration until HMC, mainly contributed to the lower value of Γ SL than Γ LG . Electrostatic interaction is the main driving force for the significant enhancement of rhodochrosite hydrophobicity. AFM researches suggest that small cylindrical monolayer and large cylindrical bilayer micelles formed on rhodochrosite surface as SA concentration arriving at its HMC and CMC, respectively. The concomitant monolayer and bilayer micelles adsorbed on rhodochrosite at HMC for OA, LA, and ALA, while the coexisting bilayer and multilayer structures on rhodochrosite at CMC due to the surface heterogeneity. Therefore, there is an appropriate concentration of fatty acid between HMC and CMC for rhodochrosite flotation.