Multidimensional Characterization and Separation of Ultrafine Particles: Insights and Advances by Means of Froth Flotation

Abstract

Particle systems and their efficient and precise separation are becoming increasingly complex. Therefore, instead of focusing on a single separation feature, a multidimensional approach is needed where more than one particle property is considered. This, however, requires the precise characterization of the particle system, which is especially challenging for fine particles with sizes below 10 µm. This paper discusses the benefits and limitations of different characterization techniques, including optical contour analysis, inverse gas chromatography, flow cytometry, and SEM-based image analysis. The separation of ultrafine particles was investigated for a binary system using froth flotation, where a novel developed flotation apparatus is used. A special focus was placed on the multidimensional evaluation of the separation according to the particle properties of size, shape, and wettability, which was addressed via multivariate Tromp and entropy functions. The results emphasize the intricacy of the flotation process and the complex interaction of the individual particle properties and process parameters. The investigations contribute to the understanding of the characterization of particulate properties as well as the separation behavior of ultrafine particles via froth flotation, especially in the case of a multidimensional approach.

1. Introduction

Separation processes usually focus on a single particle feature, where particles are separated based on differences in that specific particle property only, e.g., classification separates particles according to particle size, gravity separation exploits particle density, and flotation separates particles with different wettabilities. However, in the past, discussions have led to the agreement that separation processes indeed have a dominating separation feature but are governed by more than one property. For example, in the case of flotation, wettability is the main separation property, as hydrophobic particles attach to gas bubbles and are recovered via a froth phase, while hydrophilic particles remain in suspension; other particle properties, such as size, shape, or liberation, also affect the process significantly [1,2,3,4].

The particle size, in particular, influences separation, as particles that are either too fine or too coarse lack efficient separation. Very fine, valuable particles have inefficient particle–bubble collisions and slow flotation kinetics, while fine, unwanted gangue material is recovered unselectively via entrainment and can cause a slime coating to appear on coarser, valuable particles [5,6,7,8,9,10]. Coarse particles have efficient particle–bubble collisions; however, due to the instability of the resulting particle–bubble aggregates, they tend to detach from the bubble [6,7]. Therefore, flotation is most efficient for particles with intermediate size ranges (20 µm–200 µm), while particle systems with a high amount of very fine and very coarse particles result in low recoveries and poor selectivities.

The particle property of shape adds more complexity to the flotation process, and investigations on this topic have reported diverse results. First, one has to divide the flotation process into two sub-regions: the suspension or pulp and the froth zone. Second, one has to consider the recovery mechanism, i.e., true flotation, where particles are actually attached to air bubbles and are recovered in a froth phase, or entrainment, where particles are drawn along into the concentrate within the froth lamella. Studies using flotation devices, where the froth zone is more or less not considered, e.g., mechanical agitator-type flotation cells or micro-flotation, report that particles that are predominantly irregularly shaped should rupture the liquid film between the bubble and the particle more easily compared to those that are spherical. Therefore, higher attachment probabilities and shorter attachment times are obtained, resulting in higher recoveries [2,11,12,13,14,15]. Sygusch et al. [16] used a combination of mechanical agitator-type froth flotation and column flotation, hence including the suspension and the froth zone, and reported higher recoveries for spherical particles with moderate hydrophobicity. However, they also reported that the recovery of irregularly shaped particles increased with a decreasing column length, i.e., a decreasing impact of the froth zone with increasing particle hydrophobicity was observed. Kursun et al. [17], on the other hand, reported higher recoveries for particles with higher elongation and flatness using a column flotation set-up.

While the abovementioned studies all assumed particles to be recovered via true flotation, investigations on the effect of particle shape on entrainment had similarly diverse results. Little et al. [18] and Kupka et al. [19] showed that for their studied systems, entrainment increased with particle roundness, whereas Wiese et al. [20] and Sygusch et al. [16] reported higher entrainment for elongated and irregularly shaped particles. However, Little et al. and Kupka et al. investigated the effect at a platinum concentrator and a scheelite ore beneficiation plant, respectively, whereas Wiese et al. and Sygusch et al. used very specific pure particle systems and lab flotation devices, which makes a comparison of the different studies rather challenging.

Even though the targeted, valuable particles that are meant to be recovered by true flotation need to exhibit some sort of hydrophobicity in order to attach to the hydrophobic gas bubbles, studies showed that there is actually an optimum level of hydrophobicity and that an increase in hydrophobicity does not necessarily always result in an increase in recovery. If only the micro-processes in the suspension zone are considered, then the probability of a particle attaching to an air bubble indeed increases with increasing hydrophobicity. However, the particle properties significantly influence the froth characteristics, and if these strongly hydrophobic particles reach the froth zone, they destabilize the froth by inducing bubble coalescence, which results in froth collapse and, consequently, reduced froth recoveries. Thus, particles with medium hydrophobicity are preferred for efficient flotation, i.e., water contact angles from 30° to 80°, where the suspension as well as the froth zone is taken into account [21,22,23].

The presented studies have already demonstrated the complexity of the flotation process and the diverse results obtained for the effect of certain particle properties. Added to this is the complex interplay of the individual particle properties. For example, the aforementioned optimum level of hydrophobicity or the optimum contact angle varies significantly with the particle shape, as sharp-edged particles rupture the liquid film much quicker and can thus induce bubble coalescence at lower contact angles than spherical particles. However, this also seems to vary with particle size, as finer particles tend to have higher optimum contact angles than coarser ones [16,24,25]. Another example is the entrainment of unwanted hydrophilic gangue material, which is influenced by the particle size, as the entrainment probability increases significantly with decreasing particle size, but it is also affected by the mass density of the particles, as this affects their settling velocity, where smaller values enhance entrainment [26,27].

In order to optimize existing separation technologies, a proper understanding of the interplay and influence of particle properties on the separation process is crucial. The multidimensional characterization of particle systems is a significant part of the process, as the success of the separation can only be evaluated if the addressed particle properties can be assessed accordingly. In the case of flotation, the first challenge would be the correct determination of the wettability or the wetting behavior of a particle. The common expression used for this is the contact angle, i.e., the angle that is formed by a liquid (usually water) on a solid surface. Contact angles below 90° indicate hydrophilicity, and those above 90° indicate hydrophobicity. The measurement of contact angles requires a flat, smooth, and homogeneous surface, which already points out the drawbacks when it comes to particles, as they have rather heterogeneous surfaces that are hardly planar [28]. Furthermore, the actual particle behavior at interfaces is often not solely determined by the contact angle but also by other particle properties, such as shape and size [29].

Furthermore, a lot of different techniques are available that provide information on a single specific particle property. However, when it comes to a multidimensional analysis, the determination of complex property distributions is still a major challenge, and there are only a few possibilities to obtain particle discrete information on more than one property at the same time [30]. One example would be image analysis, as simultaneous information on size and shape can be extracted for individual particles. If this is combined with energy-dispersive X-ray spectroscopy, further information on particle properties, such as composition or density, is obtained [31]. Several studies demonstrate the benefits of this SEM-based mineral liberation analysis for the evaluation of different separation processes [32,33,34]. However, all these studies are carried out on systems with coarse particle sizes, but challenges arise when it comes to the characterization of fine particulates, as the resolution is limited. This particularly creates problems when assessing shape factors. With regard to the multidimensional characterization of particles, flow cytometry is another method that provides simultaneous information on the size and shape of each individual particle, especially for ultrafine particles. It is a common technique in the field of biology to measure the physical and chemical characteristics of biological cells. Particles are individually analyzed as they pass through a laser beam, and the forward, as well as sideward scattering, is detected, from which information on size and shape is obtained, respectively [35]. As a result, a particle discrete dataset is obtained, which, in general, makes it a promising tool for the characterization of particles, i.e., other than non-biological material. However, in contrast to the classical results of size and shape from laser diffraction or image analysis, flow cytometry only offers indirect information on these properties. Therefore, future developments in the assessment of crucial particle properties as well as multidimensional particle characterization are needed, especially for ultrafine particles.

Once this multidimensional particle discrete information is available, it can be used for an enhanced evaluation of the separation process. Schach et al. [33] demonstrated the benefits of this multidimensional evaluation for the case of a density separation process of a cassiterite-bearing skarn ore. By using kernel density estimates, they were able to calculate multidimensional partition curves showing the combined influence of particle size and density on the separation. Buchwald et al. [36] recently presented a general methodology for the description of multidimensional separation processes, also using kernel density estimation. Wilhelm et al. [37] and Sygusch et al. [38], on the other hand, computed multidimensional Tromp functions based on copulas via a parametric modeling approach to evaluate the separation behavior of ultrafine particles by flotation according to particle size and shape. Regardless of the methodology by which multidimensional partition curves are obtained, they can further be used to calculate statistical entropy, thus providing information on the efficiency and the uncertainty of the separation, as presented by Schach et al. [33,39]. All these multivariate approaches allow for a more comprehensive understanding of the separation process and will emerge as crucial methodologies for evaluating separation processes in the future.

This paper reports on the achievements of the project MultiDimFlot, which is part of the priority program SPP 2045, funded by the German Research Foundation (DFG). The project focuses on the development of innovative separation methods, allowing the multidimensional characterization as well as the multidimensional separation of particles below 10 µm, specifically in the context of froth flotation. In the course of the project, a novel separation apparatus is developed that combines the advantages of the high particle-bubble collision rate in a mechanical flotation cell and the fractionating effect of a deep froth based on current knowledge of the flotation separation of ultrafine particles. By using this innovative approach, a new process engineering concept for the separation of ultrafine particles is developed on the basis of several particle characteristics, such as size, shape, and wettability. A binary system—comprising magnetite as the non-floatable fraction and glass particles with differing shapes, either spheres or fragments, as the floatable fraction, where the wettability is adjusted via an esterification reaction—is used. In this way, the influence of the particle shape, size, and wettability can be investigated, once on the entrainment behavior of ultrafine particles when using purely hydrophilic particle systems and on true flotation of ultrafines when hydrophobized glass particles are tested. Furthermore, an intensive characterization of the particle system is presented, with a focus on determining the complex particle property distributions. The benefits and drawbacks of the applied methods are discussed. Finally, a multidimensional approach for characterization, via flow cytometry and SEM-based image analysis, and evaluation, by means of multivariate Tromp and Entropy functions, is presented. Much of what has been achieved throughout the MultiDimFlot project has already been reported to a certain extent in peer-reviewed papers. This paper serves as a summary and offers further aspects beyond the already published work. The presented study demonstrates the characterization and separation behavior of ultrafine particles in the case of froth flotation. However, the results should not be limited to this process, as the acquired knowledge is also transferable to other engineering fields, such as chemical engineering and particle technology.

2. Materials and Methods

2.1. Materials

The ultrafine particle system used as feed for the flotation tests is presented in Figure 1. Within this academic system, differently shaped glass particles, spheres, or fragments represent the floatable fraction, whereas magnetite represents the non-floatable fraction. Both glass particle fractions consist of soda-lime glass and were purchased from VELOX, Germany, as SG7010 and SG3000, respectively. Purchased glass spheres had particle sizes below 10 µm. A size fraction of -10 µm of glass fragments was obtained by milling of coarser glass spheres (SG3000), followed by aero classification. Ultrafine magnetite was purchased from Kremer Pigmente, Germany, and its purity was confirmed via X-ray diffraction. While the glass particles have a density of 2500 kg/m³, magnetite has a density of 5200 kg/m³, resulting in stationary settling velocities of 8.27 × 10⁻⁶ m/s and 2.31 × 10⁻⁵ m/s, respectively (calculation based on spherical particles for the Stokes regime). The particle size distribution, obtained by laser diffraction, is presented in

Table 1. Percentiles of the particle size distribution for glass spheres, glass fragments, and magnetite obtained by laser diffraction.

Particle Size in µm	Spheres	Fragments	Magnetite
x10	0.62	0.81	0.73
x50	2.68	3.21	2.53
x90	5.06	7.41	6.72

.

The wettability of the glass particles was modified via an esterification reaction using n-alcohols. The resulting wettability can be controlled by the choice of the alcohol, depending on its alkyl chain length, as shown by Sygusch et al. [40], where longer chain lengths result in higher levels of hydrophobicity/lower levels of wettability. For this study, the glass particles are esterified using 1-hexanol (C6, Carl Roth ≥ 98%, used as received) and 1-decanol (C10, Carl Roth ≥ 99%, used as received). Hence, six particle systems are available for testing: glass spheres and glass fragments, both as pristine, unesterified, hydrophilic glass particles (C0), functionalized glass particles with a moderate hydrophobicity (C6), and functionalized glass particles with a strong hydrophobicity (C10). For all feed systems, ultrafine magnetite is used as received and not altered in any way.

2.2. Contact Angle Measurements

Static and dynamic contact angle measurements were carried out on glass slides with the same glass chemistry, i.e., soda-lime glass. Glass slides underwent the same esterification reaction as the glass particles, i.e., using 1-hexanol and 1-decanol. The contact angles were determined via the sessile drop technique at the air–water interface using the OCA 50 from DataPhysics, Germany. Higher values for contact angles correspond to lower wettabilities and higher hydrophobicities. Static contact angles were measured by placing a drop of deionized water on the glass slide surface. For dynamic contact angle measurements, the water drop is not detached from the syringe, but its volume is increased (advancing contact angle) and decreased (receding contact angle).

2.3. Inverse Gas Chromatography

Glass spheres and fragments in their three different wettability states, as well as magnetite, were analyzed via inverse gas chromatography using the iGC-SEA Surface Energy Analyzer from Surface Measurement Systems, UK. The total surface energy of a solid consists of a dispersive part ${\gamma }_s^d$, representing the non-polar interactions, and a polar part ${\gamma }_s^{sp},$ representing the polar interactions. The latter can further be divided into Lewis acid ${\gamma }_s^{+}$ and Lewis base ${\gamma }_s^{-}$ type of interactions [41]. All IGC measurements are performed with a helium carrier gas flow rate of 10 cm³/min at 0% humidity and 30 °C. The sample was kept in place inside the column using silanized glass wool. Prior to the measurement, the sample was conditioned for 2 h at 120°C. The dead time, i.e., the time it takes to pass through the column without interacting with the sample, was determined using methane, assuming very low interaction with the sample. The dispersive component was determined using a series of n-alkanes (n-hexane to n-decane, all HPLC-grade, Carl Roth), and the evaluation was performed via the Schultz method [42]. The polar component was determined using dichloromethane as a Lewis acid (≥99.8%, Merck) and ethyl acetate as a Lewis base (99.8%, Merck), and the Della Volpe scale [43] was used for calculation according to van Oss, Chaudhury, and Good [41]. All chromatograms were evaluated using the Peak center of mass of the retention curves and the actual fractional surface coverage n/nm of the probe molecule, whereby n is the number of moles adsorbed divided by the number of moles required to cover a theoretical monolayer nm. This can be estimated using the BET value of the sample. All BET values were determined using the Gemini VII 2390 T from Micromeritics, USA, with nitrogen as the probe gas. For better visibility, only the results for the lowest surface coverage of 0.1% are presented within this study.

Furthermore, from the obtained values of dispersive as well as polar surface energy components, the free energy of interaction ΔG_pwb was calculated for a particle (p) interacting with a bubble (b) in water (w). The more negative ΔG_pwb, the more thermodynamically preferable an interaction between the considered phases. The values of water and a superhydrophobic bubble used for the calculations are presented in

Table 2. Specific surface free energy components of water and a hydrophobic bubble at 20 °C used for the calculation of the free energy of interaction ΔG_pwb [43,44].

Fluid	T in K	γd in mJ/m2	γ− in mJ/m2	γ+ in mJ/m2
Water	293	21.8	65.0	10.0
Bubble	-	0.0	0.0	0.0

.

2.4. Flow Cytometry

Flow cytometry analysis is performed with the CyFlow space from PARTEC, now Sysmex, Germany. The forward (FSC) and the sideward scattering (SSC) are measured using logarithmic gains of 200 and 250 and lower and upper limits of 10.0 and 999.9, respectively. For all measurements, the trigger is set for the SSC, and the measurement speed is 1 µL/s. The main light source has a wavelength of 488 nm. A total of 80 mg of particles is suspended in 100 mL milliQ water and dispersed at 11,000 rpm for 1 min using an Ultra-Turrax (dispersion tool S25N-25F) from IKA, Germany. For the measurement, 500µL of this suspension is diluted with 2.5 mL of milliQ water and then analyzed via flow cytometry. Data acquisition stops after detecting 1,000,000 particles.

2.5. Flotation-Based Separation

All flotation experiments are carried out using the MultiDimFlot separation apparatus, which was introduced by Sygusch et al. [16]. It is displayed in Figure 2 and consists of a mechanical agitator-type froth flotation using a bottom-driven Magotteaux machine (12 cm × 12 cm), which is combined with a flotation column (100 cm length, 5 cm inner diameter).

All flotation tests are conducted in batch mode using a rotational speed of 600 rpm and an airflow rate of 0.9 l/min, resulting in a superficial gas velocity of 0.76 cm/s. The feed consists of glass particles and magnetite in a weight ratio of 1:9, respectively, with a pulp density of 4.8% (w/w). Poly(ethylene glycol) (PEG, Carl Roth with a molecular weight of 10,000 g/mol) is used as a flotation frother. As the glass particles are available in three different wettability states (via esterification prior to flotation), no other reagents like collectors or modifiers are used for flotation. Thus, no chemical conditioning is needed. Using an Ultra Turrax (dispersion tool S25N-25F) from IKA, Germany, the particles are dispersed for 1 min at 11,000 min⁻¹ in a 10⁻² M KCl aqueous solution with a PEG concentration of 10⁻⁵ M. The resulting dispersion has a pH of 9. During flotation, the froth is scraped off every 10 s, with concentrates being taken after 1, 2, 4, 6, and 8 min. After dewatering and drying, the flotation tests are evaluated by gravimetric analysis as well as X-ray fluorescence (with the Bruker S1 TITAN handheld device) of all concentrates and tailings for mass balancing and for obtaining the chemical composition, respectively. Individual trials are further evaluated via laser diffraction (HELOS, Sympatec) to determine the particle size and mineral liberation analysis (MLA) to obtain the composition of particle systems and particle discrete information on size and shape.

2.6. Mineral Liberation Analysis

Representative sample splits were mixed with graphite powder (as dispersing inert spacers with low atomic weight) and embedded in epoxy resin. The dried resin was sliced, rotated, and remounted in order to reduce effects of settling [45]. MLA measurements were carried out using a ThermoFisher (formerly FEI) Quanta 650F MLA system equipped with two Bruker Quantax X-Flash 5030 energy-dispersive X-ray spectroscopy (EDS) detectors. BSE images and EDX analyses were seamlessly integrated using FEI’s MLA suite, version 3.1.4, to obtain false-color images. In the BSE images, the gray scale’s lower limit was established at the epoxy resin (<20), while the upper limit was set to copper (245–255). Mapping with one EDX measurement point per defined grain resulted in a comprehensive characterization of each mineral phase and its distribution across the entire sample, where the mapping was conducted using Extended BSE liberation analysis (XBSE mode) [31,46]. The measurement was conducted using an acceleration voltage of 15 kV, a probe current of 10 nA, a horizontal field width of 250 µm, and a frame resolution of 1000 pixels. With this setting, a spatial resolution of 0.25 µm per pixel is achieved. Further details regarding the measurement procedure can be found in Bachmann et al. [47].

False-color MLA images are used to compute bivariate entropy functions based on the workflow that is outlined by Wilhelm et al. [37], which holds information on the uncertainty of the separation. Entropy values close to zero indicate high probabilities that particles are recovered in either the concentrate or the tailings (separation functions of zero and one), while values close to one represent high levels of uncertainty (separation function of 0.5). Particles that consist of less than four pixels are excluded from the analysis.

3. Results and Discussion

3.1. Characterization of Ultrafine Particles

3.1.1. Characterizing (de)Wetting(Ability)

The separation process of flotation is based on differences in the wettability of the particles to be separated. Therefore, said wettability is characterized for the particles used in this study, i.e., glass spheres and glass fragments, both in different wettability states, and magnetite. A more detailed characterization of the glass particles is presented in Sygusch et al. [40], where not only spheres and fragments are investigated but also glass fibers with different glass chemistry, with a special focus on the particle shape and the degree of hydrophobization. Some of those results are presented here, again with great interest in the particle shape and wettability state of the glass particles, but this time in comparison to magnetite and with a special focus on flotation separation.

As the determination of contact angles requires flat substrates, measurements are conducted on glass slides rather than on the actual glass particles. For a proper comparison, glass slides with the same chemical composition as the glass particles are used, and the esterification is performed identically. Figure 3 displays the results of the static contact angles as well as the dynamic ones, i.e., advancing and receding, for pristine glass slides (C0) and those esterified with 1-hexanol (C6) and 1-decanol (C10) as their cumulative frequency distributions. The corresponding average values are given in

Table 3. Average values and corresponding 95% confidence intervals of static, receding, and advancing contact angles (CA) measured via optical contour analysis on glass slides with wettability states: hydrophilic C0, moderately hydrophobic C6, and strongly hydrophobic C10.

Wettability Level	Static CA in °	Receding CA in °	Advancing CA in °
C0	38.2 ± 2.4	10.6 ± 0.4	47.9 ± 0.8
C6	83.7 ± 1.2	65.6 ± 0.1	86.2 ± 0.1
C10	103.9 ± 1.0	94.8 ± 0.2	108.2 ± 0.1

. An increase is reported for all contact angles as the number of carbon atoms increases, i.e., from C0 to C6 to C10, indicating that the glass slides’ wettability is decreasing and their hydrophobicity is increasing. This can be attributed to the change of functional groups on the glass surface, as hydrophilic hydroxyl groups from the pristine glass are exchanged by hydrophobic alkyl chains from the alcohol used for esterification, i.e., either hexane or decane if 1-hexanol or 1-decanol is used, respectively. While the results for the advancing contact angles are close to those obtained for the static ones, the receding contact angles show smaller numbers for all cases. This difference between the advancing and receding contact angle is referred to as hysteresis and provides information on the heterogeneity of the surface. While the hysteresis value is rather large for C0 glass slides (37°), the difference between the advancing and receding contact angle is reduced for the C6 (21°) and C10 (13°) wettability states, indicating that longer alkyl chains not only result in a more hydrophobic but also more homogeneous glass surface.

The results of the contact angle measurement demonstrate that the glass was successfully esterified and defined wettability states can be adjusted. In order to obtain information on the particle wettability, all three materials are analyzed via inverse gas chromatography to determine their surface energy. In this way, the influence of the particle shape (spheres vs. fragments) is investigated, and the wettability of magnetite is also accessible.

Table 4. Disperse surface energies γ^d of glass spheres, glass fragments, and magnetite for surface coverages n/nm of 0.1% of probe molecules. The error bars are obtained via error propagation considering the involved regressions.

Particle System	γdspheres in mJ/m2	γdfragments in mJ/m2	γdmagnetite in mJ/m2
C0	53.9 ± 3.1	44.9 ± 2.3	156.6 ± 40.0
C6	48.7 ± 3.8	47.1 ± 2.1	-
C10	39.5 ± 1.5	36.8 ± 1.5	-

presents the results of the dispersive surface energy of glass spheres and glass fragments, both in their defined wettability states, and magnetite. While the contact angle measurement on flat surfaces cannot distinguish between different shapes, the surface energy results for spheres and fragments show that there is an influence of shape, as the values differ for all investigated hydrophobicity levels. The largest difference is observed for pristine glass particles (C0). The esterification of glass particles reduces said difference. For glass spheres, there is a reduction in surface energy with an increasing alkyl chain length (from C0 to C6 to C10). This is not observed in the same way for the glass fragments, as the C6 particles show a slightly higher surface energy than the pristine ones, even though not with significance. However, the surface energy is reduced again when looking at the C10 fragments, which even exhibit the lowest surface energy of all analyzed particles. Interestingly, the surface energy results exhibit a similar trend to the contact angles measured on glass slides. However, their reduction is not as drastic as could have been expected, based on the increase in static contact angle by around 45° and 65° from C0 to C6 and C10, respectively. Although the same trend is derived, determining the wettability via contact angle measurements alone is not sufficient for proper characterization, as the results have shown that the particle shape influences the surface energy, and thus the wettability, considerably.

The surface energy obtained by inverse gas chromatography for magnetite is rather high compared to that of the glass particles, even to those in their pristine state. The heterogeneity of the magnetite surface results in a large error (deviation); however, even with this deviation in consideration, its surface energy is still considerably larger. Therefore, the difference in surface energy/wettability provides a good basis for the separation by flotation. This is supported by the results for the theoretical energy of interaction between the particles and a hydrophobic air bubble in water, i.e., as is the case of flotation, which is displayed in Figure 4. The more negative ΔG_pwb becomes, the more thermodynamically favorable the interaction of a particle with an air bubble. While all values of ΔG_pwb obtained for glass spheres and fragments are negative, regardless of their wettability state, the value obtained for magnetite is positive. Again, despite the large error bars for magnetite, there is still a sufficiently large difference between ΔG_pwb of magnetite and the glass particles. Based on these results, magnetite is expected not to attach to any air bubbles, while the interaction probability of the glass particles increases with increasing alkyl chain length, i.e., with increasing hydrophobicity.

3.1.2. Characterizing Shape and Size

The particles are characterized by automated mineralogy with regard to their size and shape. An exemplary false-color image is displayed in Figure 5, where glass spheres and magnetite particles are depicted in red and blue, respectively.

Information on particle size and shape is extracted from these images, from which bivariate probability densities of area-equivalent diameter (Equation (1)) and aspect ratio (Equation (2), using the minimum and maximum Feret diameter as Feret_min and Feret_max, respectively, which were obtained as outlined in [38]) are computed. The results are presented in Figure 6.

$$\mathrm{area}-\mathrm{equivalent}\mathrm{diameter}=2\sqrt{{\displaystyle \frac{\mathrm{projected}\mathrm{area}}{\pi }}}$$

(1)

$$\mathrm{aspect}\mathrm{ratio}={\displaystyle \frac{{\mathrm{Feret}}_{\min }}{{\mathrm{Feret}}_{\max }}}$$

(2)

All three fractions have a narrow particle size distribution, with sizes from around 1 µm to 5 µm, but varying aspect ratios. While fragments and magnetite have particles with aspect ratios between roughly 0.4 and 1, the aspect ratio of spheres is more concentrated at values around 0.9. The data points for low aspect ratios, i.e., more elongated particles, for spheres can be a result of segmentation limits. As seen in Figure 5, some spherical particles seem to stick together, thus creating larger, non-spherical particles with lower aspect ratios, i.e., higher elongation. Obtaining descriptors of particle shape for these ultrafine particles based on MLA images is challenging because of the limitations in resolution, as discussed in more detail by Sygusch et al. [38]. Although Figure 5 is captured with the highest possible resolution of 0.25 µm per pixel, it is still not sufficient to accurately compute other particle shape descriptors, such as circularity or roundness, which might be more suitable to describe particles from the aspect of flotation and its micro processes in the pulp and the froth zones. Therefore, the aspect ratio, which provides information on the elongation, is chosen as a representative of particle shape for this study as its extraction from the available image data is less complex and thus more reliable. Furthermore, the distribution of aspect ratio for spheres and fragments is significantly different, thus providing the necessary basis for investigating different shapes with regard to their separation behavior.

Figure 7 displays the results of the flow cytometry measurements for glass spheres, glass fragments, and magnetite, where sideward scattering is presented against forward scattering, i.e., shape vs. size information.

An individual plot is obtained for each particle fraction, which could be described as a fingerprint for that material. The results for the different particle shapes (spheres vs. fragments), in particular, show significant variation. While the scattering for fragments and magnetite is rather diverse, that for spheres shows a clear pattern. It is important to keep in mind that sideward scattering is influenced by particle size as well, which is why it increases along with forward scattering. Although the method has the drawback of only providing indirect values for particle size and shape yet (which would theoretically be possible via mathematical operations and reference particle systems with known properties), but only distribution patterns, it is still a promising method for the multidimensional characterization of particulate materials that should be further developed and elaborated.

3.2. Separation of Ultrafine Particles

The results of the flotation tests are presented in different subsections, starting with the separation of ultrafine particles of the same shape and looking into the influence of wettability. The next section deals with the impact of shape on particles with similar levels of wettability. Finally, a multivariate approach is presented, including the combined influence of particle size, shape, and wettability. The recovery R of material i is defined as the amount of said material recovered into the concentrate in relation to the amount of that material in the feed, calculated according to Equation (3), where c_i,c and c_i,f represent the grade/concentrations of said material and m_c and m_f indicate the mass of the concentrate c and the feed f, respectively. The cumulative recoveries are given for different concentrates after defined flotation times.

$$R_i={\displaystyle \frac{c_{i,c}·m_c}{c_{i,f}·m_f}}$$

(3)

All tests are carried out under comparable conditions, where only the glass fraction of the particle system is changed (see Section 2.2). For particle systems consisting of hydrophilic particle fractions, i.e., hydrophilic spheres or fragments and magnetite, it is assumed that they are recovered solely by entrainment. Based on this, the effect of particle shape on the entrainment of ultrafine particles is investigated. Hydrophilic glass particles are expected to be entrained more easily than hydrophilic magnetite because they have a lower density, and thus a lower settling velocity. As a result, they would be distributed in the suspension more evenly. Therefore, they have a higher chance of reaching the pulp-froth interface and finally being entrained within the froth lamella. The glass particles that are esterified and thus exhibit moderate (C6) and strong hydrophobicity levels (C10) are expected to attach to the hydrophobic gas bubbles and will be recovered as particle–bubble aggregates, i.e., they are recovered by true flotation, though, due to the very fine particle sizes, entrainment is still expected to some extent.

3.2.1. Separation Based on Wettability of Particles with the Same Shape

Figure 8 displays the results of the flotation tests for spheres and fragments as Fuerstenau upgrading curves, where the influence of the different wettability states of the glass particles is investigated: hydrophilic C0 (red), moderately hydrophobic C6 (blue), and strongly hydrophobic C10 (green). All flotation tests show a higher selectivity for glass, regardless of shape or state of wettability. As expected, even for the purely hydrophilic systems, the recovery of glass is higher than that of magnetite. This is most probably a result of their lower density and thus lower settling velocity, as already mentioned above, even though it is not excluded that some hydrophilic glass particles reach the concentrates attached to gas bubble surfaces. An increase in hydrophobicity, i.e., a decrease in wettability, is accompanied by an increase in the recovery of glass particles, as the particles are more likely to attach to gas bubbles. As a result, more particle–bubble aggregates are formed that can be recovered into the froth [48,49]. The selectivity (in favor of the glass particles) also increases along with the hydrophobicity of the glass particles for both shapes. This effect can be attributed to the froth characteristics that are influenced by the properties of the particle system, as hydrophobic particles result in a dryer froth, which carries less water and thus also less gangue material. A more detailed analysis and discussion can be found in work by Sygusch et al. [16].

3.2.2. Separation Based on Shape of Particles with Similar Wettability States

The influence of shape on the separation of spheres (black circles) and fragments (pink diamonds) at similar levels of wettability is investigated, and the results are presented as Fuerstenau upgrading curves in Figure 9.

The results for the hydrophilic systems show that the maximum recovery of fragments is higher than that of spheres, which, at the same time, contradicts and supports what has been reported in the literature. Little et al. [18], for example, demonstrated that the entrainment of chromite particles increased with roundness, whereas Wiese et al. [20] reported higher entrainment for particles that are less spherical.

For the system with moderately hydrophobic glass particles (C6), a slightly higher average recovery is obtained for spheres than for fragments. These results are contradictory to what has been reported in the literature, where higher recoveries are obtained for fragments. This is usually attributed to their edgy and /or rough surface that induces the rupture of the liquid film more easily and hence results in higher probabilities of attaching to gas bubbles than spherical particles [2,11,12,13]. However, these studies were conducted in different flotation setups using different experimental parameters as well as other particle systems with much coarser sizes. Since many of these settings have a significant influence on the flotation process, it does not provide a good basis for comparison.

A comparison of systems with strongly hydrophobic glass particles is also not as straightforward because the flotation using fragments-C10 was accompanied by strong bubble coalescence and froth collapse. Although additional flotation tests were carried out using said particle system, concentrates could only be taken for two of those, but with very low mass pull and poor recoveries below 20%. On the other hand, for the flotation tests using spheres-C10, recoveries of almost 100% are achieved, which is most probably a result of the particle–froth interaction. The froth characteristics are not only influenced by the particle wettability, as particles that are strongly hydrophobic can cause bridging effects and thus induce bubble coalescence, as presented by Johansson et al. [21], but also by the particle shape. Sharp-edged particles rupture the liquid film much quicker and at lower contact angles than spherical particles, which is why this effect is only observed for fragments and not for spheres [24,25]. Additionally, Ulusoy et al. [50] demonstrated that particles with higher elongation ratios exhibit higher apparent hydrophobicities compared to spherical particles. This not only adds to the shape effect but indirectly enhances the wettability effect as well. A more detailed discussion can be found in work by Sygusch et al. [16].

3.2.3. Separation Based on Multiple Particle Properties

So far, the separation tests have been analyzed in a more standard way, where only one particle property was considered at a time, either shape or wettability, and the particle system was considered as a bulk. This section will now focus on the multidimensional analysis of the separation, i.e., looking into the effect of particle shape and size at the same time for different wettability scenarios using particle discrete data. One set of flotation tests has been analyzed with MLA, from which bivariate Tromp functions, as well as entropy functions, are computed. The results of the Tromp functions are discussed in Sygusch et al. [38], whereas this paper will focus on the entropy functions.

The entropy function holds information on the uncertainty of the separation of a particle. Particles with entropy values close to zero have a high probability of being recovered in either the concentrates or tailings, corresponding to separation functions of zero and one. Entropy values close to one indicate high levels of uncertainty, and one cannot state whether the particle reports to the concentrate or the tailings, which would correspond to a separation function value of 0.5 [33]. The entropy functions are computed for the individual particle fractions, i.e., separate functions for glass particles and magnetite. In this way, the influence of the particle properties on the particle behavior can be investigated, on the one hand, for the effect of shape, size, and wettability of the glass fraction and, on the other hand, for the entrainment behavior of magnetite.

Figure 10 displays the computed entropy functions for glass spheres (upper row) and glass fragments (lower row), with increasing hydrophobicity from left to right (from C0 to C6 to C10). The most diverse results, with entropy values from 0.3 up to 1, are obtained for the hydrophilic systems. Hence, a clear influence of the particle properties on the entrainment behavior is observed. For hydrophilic spheres, the highest entropy, i.e., the highest uncertainty, corresponds to particles with very high aspect ratios across all sizes and for very fine particles over a range of aspect ratios. The entropy is lowest for particles with sizes between 2 µm and 6 µm and aspect ratios between roughly 0.7 and 0.9. Hydrophilic fragments have the highest uncertainty for particles with high aspect ratios, but the entropy drops significantly to around 0.3 for very fine particles.

The entropy functions for the corresponding magnetite particles that were mixed with the glass fractions are displayed in Figure 11. These results strongly suggest that there is an influence of the glass particle shape on the entrainment behavior of magnetite when mixed with hydrophilic glass particles. Magnetite mixed with spheres has low entropy values for coarse particles, and the uncertainty increases with decreasing particle size. However, magnetite mixed with fragments shows the opposite trend, as entropy values increase with particle size.

When the hydrophobicity of glass spheres is increased (from C0 to C6 and C10), the entropy values increase significantly across all sizes and aspect ratios. For fragments, the change in entropy depends on the particle properties considered. While the entropy is increased for very fine fragments (<2 µm), it stays more or less constant for coarser fragments and even decreases slightly for fragments with an aspect ratio around 1. If the hydrophobicity of fragments is increased even further, the entropy decreases significantly. However, one has to keep in mind that the fragments-C10 flotation tests suffered from froth collapse, which most definitely will influence the entropy function.

While the entropy functions for magnetite mixed with hydrophilic glass particles vary significantly depending on the glass fraction it is mixed with, there is no such variation observed when mixed with hydrophobic glass particles. Magnetite mixed with C6-esterified particles exhibits entropy functions of around 0.2 across all sizes and aspect ratios, i.e., no influence of shape or size is observed. If mixed with strongly hydrophobic glass particles, the entropy decreases even further to almost zero since almost all magnetite from these systems reports to the tailings, as shown in the Fuerstenau upgrading curves in Figure 8 and Figure 9.

3.3. General Discussion

In the previous sections, it was shown how the particles were analyzed to determine their wettability and wetting ability using different characterization techniques, focusing on the one hand on the influence of particle shape (spheres vs. fragments) and on the other hand on the characterization of the particle system used for flotation.

The wettability analysis of glass particles and identically treated glass slides showed that their surface was successfully modified via esterification. An increase in the length of the alkyl chain of the alcohol used for esterification results in a decrease in particle surface energy and glass slide contact angle. A comparison of these techniques demonstrated how the particle shape influences surface energy, as the energy of fragments is lower compared to that of spheres, which would not be accessible via contact angle measurements only. The difference in surface energy and the effect of shape is also observed during the flotation tests using strongly hydrophobic glass particles, as only the particle system with fragments induces a froth collapse, while glass spheres result in a stable froth.

Using the particle surface energy that is measured with inverse gas chromatography, the energy of interaction between a particle and a bubble in water can be calculated, as would be the case for flotation. While positive values were obtained for magnetite, indicating that the probability of interaction is rather low, negative values were obtained for glass spheres for all three wettability states, which serves as a good basis for separation. However, negative values of ΔG are expected for hydrophobic particles (C6 and C10); there is not much difference to the hydrophilic glass particles. This is despite the fact that the contact angle measurements indicate hydrophilicity for the C0 state, as well as the observation of a distinct behavior during flotation and in phase-transfer tests demonstrated by Sygusch et al. [40]. This demonstrates that the comparison of different analytical techniques is sometimes challenging, as the methods themselves operate very differently. Contact angle measurements are carried out on flat surfaces using liquids, and a relatively large part of the surface is analyzed, offering a relatively fast determination of wettability. The surface energy is obtained from the interaction of probe molecules with dry particulates, making it a very sensitive technique with the ability to determine minor differences on the particle surface. One drawback of the IGC is that information is only obtained on the bulk and not on the individual particles. Obtaining particle discrete data of surface energy for the whole particle system and correlating this to the separation behavior would significantly aid in further understanding the separation of these fine particles, thus contributing to the multidimensional scope of the project.

While this is not yet accessible for surface energy via IGC, flow cytometry offers the possibility of providing particle discrete information on their wettability. Besides forward and sideward scattering, from which information on particle shape and size is obtained, a flow cytometer can also measure fluorescence intensities. For this, particles would need to exhibit some sort of fluorescence, which could be induced by using fluorescent dyes. By using dyes that specifically bind to hydrophobic surfaces, information on their hydrophobicity could be obtained. Particles to which the dye adheres would exhibit fluorescence, while none is detected for those that are hydrophilic, as they have no interaction with the dye [51,52,53]. The first tests were carried out within the scope of this study; however, a suitable procedure for staining has not been identified yet for the particle systems used here. Nevertheless, if this were to be successful, it would significantly contribute to the multidimensional characterization of fine particles, as particle discrete information on size, shape, and wettability would be obtained at the same time with only one measurement.

The characterization of the particle system and the evaluation of the flotation tests were carried out using automated mineralogy, a method that provides quantitative single-particle information on shape and size, as well as qualitative information on their composition. It is a common and well-established analytical technique that is a standard tool for analyzing mineral systems. However, it is designed for coarser particle systems, whereas the characterization of fine particulates, as used in this study, is challenging. Due to the limited resolution, the computation of shape descriptors is not as reliable as it is for coarse particles, and not all shape descriptors can be properly assessed. Another aspect that should be kept in mind when dealing with 2D images is the stereological bias that arises for planar surface sections, as only 2D information is obtained [54,55]. This typically results in underestimated particle sizes, since the particles are cut at a random plane during the sample preparation process. This introduces a certain bias to the computed shape descriptors as their orientation within the epoxy resin framework significantly influences which part of the particle is actually seen on the screened surface. One possibility for visualizing the true 3D structures of particles without stereological error is measurement via X-ray computed tomography (CT). Detailed information on the distributions of specific surface area and particle size obtained from X-ray CT for glass spheres used in this study are presented in work by Ditscherlein et al. [56]. Although the particle properties are better represented by the 3D dataset, its acquisition is very time-consuming, and the overall sample size is much smaller, resulting in fewer particles and smaller datasets. Consequently, it can be challenging to achieve statistically representative quantities using this methodology. Furthermore, only structural information is obtained via CT, but detailed information on the composition of the particles cannot be assessed. Therefore, the standard method for the evaluation of larger sets of samples, as is the case for the separation tests in this study, remains the automated mineralogy, and thus 2D image data, as they combine image data as well as mineralogical information.

The separation of ultrafine particles via froth flotation with a special focus on the influence of particle shape, size, and wettability is studied using a novel self-built separation apparatus that combines mechanical agitator type and column flotation. The results show that an increase in hydrophobicity generally promotes the recovery of the valuable fraction. These results are in line with the literature, as it is assumed that the particle–bubble attachment is more efficient with increasing hydrophobicity [48,49]. However, the results also show that there is a limit to the degree of hydrophobicity. The strongly hydrophobic fragments destabilize the froth to such an extent that froth collapse is induced and flotation is no longer possible. Similar results have been reported by Johansson et al. [21] and Schwarz et al. [57], who determined an optimum contact angle of around 65°, while everything above resulted in reduced recoveries. However, while the particle systems used in their studies were much coarser, the results of this study are the first for ultrafine fractions, showing that the optimum contact angle and the hydrophobicity limit are not only affected by particle shape but also by particle size.

The influence of particle shape on separation is not as straightforward, and the results obtained in this study are different from those presented in the literature. Considering the particle systems, where the recovery is expected to occur by true flotation, which assumes the particle is attached to a gas bubble, spheres-C6 show a slightly higher recovery than fragments-C6. As stated before, one would expect that, due to their rough edges, the attachment of fragments to gas bubbles is more efficient than for spheres, as the liquid film is ruptured more easily, which has also been presented in other flotation studies [2,11,12,13]. Regarding the influence of shape on entrainment, where the particles do not attach to bubbles, but are recovered as part of the froth within the lamella, there are not too many studies found in the literature. However, the results obtained here, on the one hand, support and, on the other hand, contradict some of those results.

Looking at all the different results, the ones from this study and those from the literature, for both investigations, the effect on true flotation or entrainment reveals that a comparison is actually rather challenging. Many studies use different flotation setups (e.g., micro-flotation, mechanical cell, industrial trials, and self-built devices), different experimental conditions (rpm, airflow rate, cell volume, and reagent regime), and different particle systems (in terms of size and composition), thus obtaining only very system-specific results.

Furthermore, the findings demonstrate that it can be quite difficult to have a system that is actually comparable so that the outcome can only be a result of the induced change, for example, a different shape or a different wettability state. Although the composition of glass particles is the same, with the shape being the only difference, it does not only influence the recovery in terms of the attachment mechanisms taking place in the pulp zone but also influences the froth, which in turn has a significant influence on the result. This complex interplay of parameters makes it difficult to actually attribute a certain behavior to a single property. Nevertheless, the datasets obtained from these flotation tests provide a lot of information on the separation behavior and will be used to model the process in the following. In this way, the understanding of the micro-processes that occur during flotation can be enhanced.

Another important aspect of this project is the multidimensional perspective on ultrafine particle separation processes, i.e., not only looking into one particle property at a time but considering at least two—in this case, particle shape, size, wettability, and density. One approach to accomplishing this is the computation of multidimensional Tromp functions from particle discrete data. In this way, a map of the separation probability for each particle with specific particle property vectors is obtained, and the combined influence of said properties is displayed. Bivariate Tromp functions regarding aspect ratio and area-equivalent diameter for a certain test set of this study are presented in work by Sygusch et al. [38]. These Tromp functions are then used to compute bivariate entropy functions, which are displayed in the previous section and provide information on the uncertainty of separation for the individual particles. The results of the Tromp and entropy functions show quite some unexpected variations, depending on the particle system, especially as these significant differences are not observed in or can be explained with the classic flotation results (recovery, mass–water–pull), as discussed in work by Sygusch et al. [38]. However, MLA might not be the most suitable method for the analysis of such ultrafine particle systems (2D information and resolution limit), and the resulting Tromp and entropy functions should be questioned to determine their validity. Nonetheless, the innovative multivariate approach provides essential information on the separation behavior of individual particles and offers new possibilities for evaluating complex separation processes.

4. Conclusions

This work provides an overview of what has been achieved in the project MultiDimFlot, which is part of the priority program MehrDimPart SPP 2045, funded by the German Research Foundation DFG. Some of the presented results have been published before and discussed in more detail in different papers, which are all open access and available. Unfortunately, not all work conducted throughout the project can be presented here in one single paper, which is why the focus is put on the two main investigations, namely, the characterization and the separation of ultrafine particles by froth flotation. The main conclusions of the study are summarized as follows:

• The wettability of glass particles and glass slides can be modified via an esterification reaction using n-alcohols, whereby the length of the alkyl chain controls the resulting level of hydrophobicity.
• The particle wettability is influenced by the particle shape.
• The commonly used contact angle is not sufficient to accurately describe particle wettability.
• The particle-specific characterization of size and shape is challenging for ultrafine particles, as analytical techniques are not well-equipped for these size ranges.
• A novel separation apparatus was introduced, combining the advantages of mechanical and column flotation to enhance the separation of ultrafine particles.
• Different separation behaviors are observed, especially with regard to the particle shape, but also depending on the particle wettability.
• Differently shaped particles seem to exhibit different optimum levels of wettability.
• A multivariate approach was demonstrated for the multidimensional evaluation of the separation process, offering the possibility to show the combined influence of particle shape and size for different wettability scenarios.

This project contributes to a better understanding of fine particle flotation, in particular, the complex effect of particle shape in combination with the particle properties of size and wettability. Beyond that, the results of the project are not only valid for the mineral processing industry in terms of the beneficiation of valuable minerals from primary or secondary ores but will also be useful for other fields, such as chemical engineering or particle technology—for example, in the separation or classification of pigments.