One of the first industrial uses of surfactants was to improve selectivity in ore flotation. In the case of barely soluble oxidised minerals (oxides, silicates...) long-chain ionic surfactants are preferably chosen. An accurate choice must be based on the affinity of the surfactant ions towards the mineral to be recovered. The normal adsorbate–adsorbent bond (generally electrostatic in nature [1,7]) must be strong, so that the mineral is hydrophobised for the lowest surfactant concentrations. Furthermore, to ensure an efficient adhesion of the ore particles to air bubbles, the solid should be covered, for instance, by a surfactant monolayer prior to saturation of the air/aqueous solution interface by surfactant molecules. This can be achieved at very low surfactant concentrations, much lower than the CMC (C/CMC<0.001; Fig. 1). In such systems, the surfactant cannot precipitate due to the low solubility of the solid phase. Different theories have been proposed to describe such systems.

2.1.1 The hemi-micelle hypothesis

2.1.2 Two-dimensional condensation on heterogeneous surfaces

2.1.2.1 Use of ionic surfactants as molecular probes to study surface heterogeneity

Surfactants can be advantageously used for determining the aspect ratio of clay particles [14] or the surface energetic distribution function f_i of the adsorbent. The model adopted was the ‘patches’ model [8,10,12,52], which assumes that adsorption sites having the same energy are grouped on a heterogeneous surface into patches. Each patch is large enough to use the thermodynamics of large dimension phases and the state of an adsorption system in which two interacting molecules adsorbed on neighbour patches can be neglected. The adsorption systems can then be considered as a collection of independent systems. On each homogeneous domain i of the surface, monolayer formation corresponds to a first order transition and a domain i is either empty or full (0–1 approximation of the 2D-condensation [10,12]). Each homogeneous domain is characterised by the value of its area S_i and the value of the normal molecular potential energy −ϕ_a,0,i. The surface coverage θ can then be expressed in a simplified way (equation (1)):

$$\theta \approx \sum _i{S}_i/\mathrm{S}$$

(1)

The position of a step characterising two-dimensional condensation on a domain i, $\Delta {\mu }^{*}$, the undersaturation is obtained from the reduced form of the Cases and Mutaftschiev equation [8,10,12] (equation (2)):

$$\Delta {\mu }^{*}=\mathrm{B}-{\varphi }_{\mathrm{a},0,\mathrm{i}}$$

(2)

with

$$\Delta {\mu }^{*}=\mathrm{k}T\ln C^{*}/\mathrm{CMC}\mathrm{if}\mathrm{T}>{\mathrm{T}}_{\mathrm{Krafft}}$$

$$\mathrm{or}\Delta {\mu }^{*}=\mathrm{k}T\ln C^{*}/{\mathrm{C}}_{\mathrm{s}}\mathrm{if}\mathrm{T}<{\mathrm{T}}_{\mathrm{Krafft}}$$

C_s is the saturation concentration of the crystal–hydrate phase, $C^{*}$, the equilibrium concentration corresponding to the 2D-condensation on the domain i and B a constant for a homologous series of surfactant, depending of the Krafft point [12]. The various energetic domains are filled according to their decreasing energies (equation (2)) and the spatial extension of one lamellar aggregate on the surface is limited by the size of the energetic domain on which surfactant ions condense. The size of patches is not constant. It depends on the area S_i: the more energetic is the considered domain, the smaller is S_i. The heights of the different steps corresponding to the filling of the different homogeneous domains are higher as equilibrium concentrations increase (Fig. 1, curve I, filling of domains 1 to 4). These hypotheses were recently corroborated by Mielczarski et al. [42,43], who used in situ and ex situ Fourier Transform Infrared, NMR and X-ray photoelectron spectroscopy to examine the nature and structure of adsorption layers on apatite contacted with sodium oleate in the range of monolayer formation. The chemisorbed oleate ions prefer to assemble into well-organised close-packed patches with hydrocarbon chains directed towards the solution. Patches size increases with coverage. Structures with the highest degree of ordering and packing were observed at surface coverage values around 0.7 and 1.0. These organised structures of chemisorbed oleate are stabilised by water molecules present in the well-organised structure of the adsorbed layer. In these conditions, the shape of the isotherm is independent of lateral bonds on energetically heterogeneous surfaces. It depends only on a double distribution: on the x-axis, the ϕ_a,0,i distribution and, on the y-axis, the S_i distribution, corresponding to the different homogeneous domains. Therefore, on heterogeneous surface, the experimental isotherms have no precise and predictable shape (Fig. 2) as the shape mainly depends on the distribution of energetic domains on the surface. From these considerations, it appears clearly that 2D-condensation allows the direct determination of the surface energetic distribution function f_i using the experimental derivative isotherm (equation (3)):

$$f_i=\mathrm{d}\theta /\mathrm{d}\Delta \mu$$

(3)

The function f_i can be defined as the ratio of site fraction per energy unit.

If the condensation of a second layer is possible, it can be noticed on the adsorption isotherm, by the appearance of a vertical step, at an undersaturation value equal to $\Delta {\mu }_{\mathrm{II}}^{*}$ (Fig. 1, curve II). This step corresponds to the adsorption of surfactant molecules on a surface energetically homogeneous due to the presence of the first layer. Surface phenomena do not influence bilayer formation, as the tail–tail bonds between surfactant molecules are the driving force for such behaviour.

Upon monolayer filling, the surface becomes more and more hydrophobic. In the process presented in Fig. 1 (curves I and II), the surface is hydrophilic for surface coverage values close to 0 and 2, and totally hydrophobic for a surface coverage around 1.

To conclude, it is important to note that micellisation in the bulk solution (C<CMC) and formation of lamellar aggregates at the solid–aqueous solution interface (if the normal adsorbate–adsorbent bonds are strong) are both aggregative processes. However, micellisation generates constant size micelles, whereas the formation of lamellar patches on a surface corresponds to aggregates of various sizes, depending on surface heterogeneity. If the surface of the minerals is very heterogeneous (more than ten domains), the size of the different homogeneous domains is small and if the normal molecular potential energies are continuously distributed, then the isotherm appears as a continuous curve, which is the most frequent case. An example is shown in Fig. 2. Curve I presents the adsorption isotherm of n-dodecyl ammonium chloride on synthetic calcite [26]. It is possible to distinguish two steps in the domain of monolayer formation characterising two homogeneous domains of large extension. The sample was made up of well-developed growth rhomboedra. Their specific area of $0.4{\mathrm{m}}^2{\mathrm{g}}^{-1}$ fits well with the average dimension of the particles (10 μm). The finely dry ground calcite was made up of shapeless sponge-like aggregates without any cleavage plane mixed with a large number of very fine particles. It is a very heterogeneous material and the corresponding isotherm exhibits a quasi-parabolic shape up to an equilibrium concentration of $3.5·{10}^{-4}\mathrm{mol}{\mathrm{l}}^{-1}$, i.e., an isotherm without discontinuity characterising a highly heterogeneous surface (Fig. 2, curve II). The vertical parts of isotherms correspond to the three-dimensional condensation of n-dodecylammonium carbonate.

2.1.2.2 The use of the θ vs Δμ plot to characterise the state of the adsorbed layer

It was difficult in the past to study in situ the state of the adsorbed layer by direct means of investigation. In contrast, the organisation and structure of either bulk micellar and lyotropic phases or of the hydrated-crystal state were well documented [6,39]. Then, Cases and Mutaftschiev [10] decided to develop an equilibrium equation allowing to compare the state of the adsorbed layer with that of a reference well-known phase. The Frumkin–Fowler isotherm that characterises two-dimensional condensation can be simplified using the following assumptions: (i) a complete adsorption layer (θ=1) on an homogeneous domain has the same structure as the top lattice plane of the surface of a crystal of the same substance (T<T_Krafft), hydrated-crystal state (the top bilayer [10]) or as the top lattice plane of the surface of an aggregate of the lamellar lyotropic phase (T>T_Krafft, liquid crystal state); (ii) the maximum lateral binding energy per molecule ω is then the same in the adsorbed layer and in the reference phase; (iii) in a first approximation, the difference in entropic terms of kinetics origin between a molecule adsorbed on a foreign substrate and a molecule being part of the top layer of the reference phase of the same substance can be neglected. In other words, surface phenomena do not influence the structure and properties of the adsorbed layer as compared to those of the reference phase. Comparison of the state of the adsorbed layer with that of a reference phase can therefore be accomplished by subtracting the two equilibrium equations, between adsorbed layers on an homogeneous domain i and dilute solutions and between reference phase and bulk solution at saturation (equation (4)):

$$kT\ln C_{\mathrm{e},\mathrm{n}}/{\mathrm{C}}_{\mathrm{o},\mathrm{n}}=\Delta \mu =({\varphi }_{0,0}-{\varphi }_{\mathrm{a},0})+(\omega /2)(1-2\theta )+\mathrm{k}T\ln (\theta /1-\theta )$$

(4)

where C_o,n is the equilibrium concentration of the reference phase for ionic surfactants with n carbons in the hydrocarbon chain and ϕ_0,0 is half the normal work of desorption of a molecule in the half-crystal position of the reference phase [10].

The equilibrium concentration of the reference phase corresponds to: (i) the saturation concentration of the hydrated crystal phase for a surfactant with n carbon atoms in the hydrocarbon chain C_s,n if T<T_Krafft, or (ii) the saturation concentration of the lamellar lyotropic phase C_0,L,n if T>T_Krafft. This latter value is not known. As the conformation of the alkyl chains for lamellar reference phase is the same as that of micelles [40], the following equation can be written for a homologous series of surfactants (equation (5)):

$$C_{0,\mathrm{L},\mathrm{n}}/{\mathrm{CMC}}_n=\mathrm{A}=\mathrm{const}$$

(5)

On an energetically heterogeneous surface, each domain i can be characterised by the position of the step, and the corresponding undersaturation $\Delta {\mu }_i^{*}$ is obtained by posing θ=1/2 in equation (4), because of the evident symmetry of Frumkin–Fowler isotherms with respect to θ=1/2. This leads to equation (2).

Two important inferences can be made: (i) the θ vs Δμ plot can be used to unambiguously determine the surface energetic distribution function f_i; (ii) in this plot, if the hypothesis concerning the similarity in organisation and structure of the reference phase and the adsorbed layer are fulfilled, then the isotherms obtained for an homologous series of surfactant on a solid have to superimpose. Such overlapping is often observed [49] (Figs. 3 and 4).

The separation of minerals such as apatite, barite, fluorite, magnesite, by flotation from other salt-type minerals such as calcite is extremely complex owing to the close similarity between their physicochemical properties. Up to recently, such problems remained largely unsolved. Some of the most crucial problems were:

• – the real influence of the nature of the normal bond;
• – solubility and its consequences.

The real influence of the nature of the normal bond. These minerals are concentrated by flotation with anionic collectors such as long-chained fatty acid or soaps. It was usually assumed that [1,27,44]: (i) surfactants included in the first adsorbed layer, are chemisorbed at the mineral surface; (ii) in the flotation of these minerals, conditioning is important because time is required for collector–mineral interactions to occur.

Solubility and its consequences. Salt-type minerals are characterised by solubilities that are higher than those of most oxides and silicate minerals. For fluorite-sodium oleate system, calcium cations can precipitate the anionic surfactants, provided the solubility of the calcium salt is exceeded. What are the consequences of this phenomenon?

In this last case, thermodynamic models have to agree with three-dimensional condensation of great dimension phases to take into account the fact that surfactants precipitate in the bulk solution and that, as surfactants can precipitate on the surface (3D-condensation on substrate), the amount of surfactant ‘abstracted’ or ‘retained’ on the surface may go far beyond monolayer capacity.

As the temperature for precipitated phases is always below the Krafft point, the reference phase to take into account is the topmost layer of the hydrated crystal phase. Therefore, to study these systems, two domains have to be taken into account:

• – the domain where the 2D-condensed phases are stable, i.e., Δμ<0 or C_e,n<C_s,n; equation (2) can be written in a simplified form (equation (2′))

  $$\Delta {\mu }^{*}={\varphi }_{0,0}-{\varphi }_{\mathrm{a},0,\mathrm{i}}$$

  (2′)

  in this domain ϕ_0,0−ϕ_a,0,i<0 or ϕ_0,0<ϕ_a,0,i;
• – the domain of 3D-condensation, i.e., Δμ=0 or C_e,n=C_s,n and ϕ_0,0=ϕ_a,0.

This result is of fundamental importance for understanding the formation of two- or three-dimensional condensed layers on a foreign substrate. It means that a condensed two-dimensional phase on a solid, i.e., an adsorbed condensed layer, is possible only if the work necessary to break a normal adsorbate–adsorbent bond is higher than half the separation work needed to break a molecule in the half crystal position of the reference phase [10,12]. In the other cases (Δμ=0 or >0), the formation of a three-dimensional condensed phase on the surface or in the bulk is more favourable. This result is independent of the nature of the normal adsorbate–adsorbent bond.

In addition, special care must be taken for correctly interpreting adsorption experiments:

• – in the range of 3D-condensation, surface coverage loses its physical meaning and terms such as ‘statistical surface coverage’ have to be used; the distribution on the surface of the amount abstracted is not known;
• – 3D-condensation on the surface and its characteristic infinite step (Fig. 2) are not always detected on experimental isotherms, because they depend on saturation concentration, dissolution kinetics of the mineral, volume of solution and surface area of the adsorbent; typically, these two last factors are simultaneously accounted for by changing the ratio solid weight to solution volume; the advent of the maximum often observed [51] on the ‘adsorption isotherms’ is reduced if the solid/liquid ratio is increased and, at high solid–liquid concentrations, the isotherm levels off at the statistical bilayer without any step characterising surface precipitation.

In conclusion, in all cases two-dimensional condensation takes place before three-dimensional condensation.

In order to illustrate the above-mentioned conclusions, it is necessary to present the results obtained by Mielczarski et al. [44,45] on the molecular effects in monolayer formation of oleate on fluorite and on the dynamics of fluorite–oleate interactions:

• – quantitative evaluation of self-assembled monolayers (composition, orientation, adsorbed amount) of oleate on cleaved (111) and polished fluorite surfaces by means of infrared external reflection spectroscopy [44] reveals two types of interaction of oleate groups with calcium surface atoms: (i) on cleaved fluorite surface, a bidentate-like bonding was revealed, with an orientation angle of 80° between the asymmetric stretching vibration of the COO groups and the normal to the surface (position almost parallel to the interface). This bidentate is characterised by an absorbance band at 1536 cm⁻¹. Such a situation is predominant for cleaved fluorite (up to 75% of adsorbed molecules); (ii) the second calcium oleate surface species is the unidentate-like form, with an absorbance band at 1575 cm⁻¹; in that case, the average orientation angle is equal to 56°; the amount of this surface species is related to the number of crystallographic defects present on the natural cleavage plane of fluorite, since their amount increases with polishing;
• – the perturbation found naturally or occurring during polishing or sample preparation of the perfect cleavage plane results in the formation of both bidentate-like and unidentate carboxylate surface complex, as well as the intermediate forms;
• – these first results suggest that in the case of perfect conditions for a well-organised monolayer, only one type of carboxylate group (the bidentate-like form) with the band at 1536 cm⁻¹ should be observed. The importance of this band characterises the order of the lamellar aggregates present on the surface below monolayer capacity;
• – the nature, structure and kinetics of formation of mono and multilayers have also been examined by internal reflection spectroscopy [45].

New and interesting results can be presented as follows (Fig. 5).

• (i) Fluorite immersed in basic oleate solution shows surface phenomena dependent on the initial oleate concentration.
• (ii) At a diluted concentration of 5·10⁻⁶ M, the adsorbed amount does not exceed 0.25 of a monolayer after 2 or 6 h of adsorption (2D-condensation domain).
• (iii) In dilute solution, for a concentration equal to 9·10⁻⁶ M, a maximum of adsorption of about 4 monolayers is observed after 4 h. Microscopic studies already reveal the presence on the surface of three-dimensional aggregates looking like round and elongated patches. After 19 h of adsorption, the surface species undergo dissolution and the adsorbed amount is about 0.2 statistical surface coverage. At this low coverage, a broad band centred at 1560 cm⁻¹ appears due to a wide distribution of the various conformations of carboxylate surface groups. This seems to agree with the presence of very small patches on the surface, with a high ratio of external, less oriented molecules.
• (iv) At 3.3·10⁻⁵ M, the amount of surface precipitated calcium oleate presents a maximum after 6 h adsorption at about 20 statistical monolayers. After 8 h, the amount of calcium oleate abstracted decreases down to 8 statistical monolayers. Microscopic and AFM studies show that the patches already described occupy only a small part of the surface and that their height changes from 10 to 600 nm, i.e., 5 to 285 monolayers in a rigid conformation. These results clearly show that 3D-condensation (nucleation and growth) is responsible for the formation of calcium oleate surface structures. The dissolution of surface products with an increase in adsorption time can then be assigned to Ostwald ripening, which transfers oleate and calcium ions from the surface to the bulk solution to favour the growth of calcium oleate particles more crystallised than on the surface.
• (v) At a concentration of about 1·10⁻⁴ M sodium oleate, the intensities of the aliphatic stretching vibration bands in the spectra indicate the monolayer coverage after 30 min of adsorption. The carboxylate absorbance region exhibits a doublet at 1574 and 1537 cm⁻¹ similar to those observed at lower initial oleate concentrations as well as an additional sharp band at 1563 cm⁻¹ assigned to sodium oleate [50], which disappears after 1 h. Longer adsorption time results in the disappearance of the band at 1563 cm⁻¹, whereas the total adsorbed amount increases up to about two statistical monolayers, the highest coverage found at this concentration. After 4 h, lower intensities indicate a coverage of about 1.2 statistical monolayer.
• (vi) Surface products formed at a solution concentration of 5·10⁻⁴ M sodium oleate were found to be similar to those formed at 1·10⁻⁴ M after a short adsorption time. The difference is that at higher concentrations, the same adsorption products are observed at short and long adsorption times. After 4 h contact with oleate solution, there is still sodium oleate in the surface product. The total amount of oleate abstracted never exceeds a statistical coverage. The microscopic picture shows small three-dimensional patches, unevenly distributed on the mineral surface. At this high concentration, it was suggested that the presence of mixed calcium-sodium oleate micelles in solution prevents the formation of surface aggregates [45]. The unravelling of such complex phenomena was only possible by using in situ real time experiments. These results were later confirmed using other methods [46]. The use of infrared external reflection spectroscopy coupled with AFM and optical microscopy was also successfully applied to study the nature and structure of surface oleate species abstracted on calcite in spite of the overlapping of a small spectroscopic signal from surface species with a very strong absorption signal from the substrate [41].

As thermodynamically predicted, surfactant ions have at first, as much affinity for surface products than for particles that precipitate in solution. For this reason, these systems are considered as systems with weak normal adsorbate–adsorbent bonds. Some of the above-developed considerations can be applied to short chain surfactants, such as potassium xanthates (which are commonly used as collectors in the selective flotation of sulphide minerals) when the mechanisms presented here are not hidden or complicated by associated phenomena resulting from superficial electrochemical reactions [11,60].

The retention mechanisms of non-ionic surfactants at the silica surface were carefully described by Klimenko et al. [29] and Levitz and co-workers [31–37]. The adsorption isotherms of alkyl or alkyl phenol polyoxyethylene glycol on hydrophilic silica surface present a ‘Langmuirian shape’ with no clear inflexion point. Reversible adsorption occurs below the CMC ($0.1\mathrm{CMC}<{\mathrm{C}}_{\mathrm{e}}<\mathrm{CMC}$). All the isotherms reach a plateau around the CMC. This plateau decreases when the polar chain length increases. Fluorescence spectroscopy of a built-in chromophore or an extrinsic probe (pyrene) shows that adsorption is an aggregative process. Adsorbed molecules are involved in finite surface aggregates. The average aggregation number between 0.2<θ<0.8 turns out to be in the same range as the size of micelles in solution measured well above the CMC. For higher surface coverage, surface aggregates coalesce to form a bilayer. These results can be understood as follows: these surfactants are anchored to the surface through a direct bond between the POE chains and the solid surface. Normal adsorbate–adsorbent bonds responsible for adsorption are due to hydrogen bonding between surface silanols and oxygen atoms of the ethoxy-groups of the non-ionic surfactant [22] (for this reason, maximum adsorption always occurs at the Zero Point of Charge of the solid [38]). The strength of this bond is similar to that existing between POE chains and water molecules in the bulk solution. This explains why micelles are forming on the surface near the CMC. In contrary to the case described in §2, where the strong bond between the solid and the ionic headgroups of the surfactant directs the hydrophobic tail of the surfactant towards the aqueous solution, non-ionic molecules tend to take a configuration avoiding the direct contact of the hydrophobic tails with the aqueous solution from very low surface coverage values. This can occur through the formation of surface micelles, according to a process described by Levitz and co-workers. In others words, in these systems, ‘the weak normal interaction between surfactant polar groups and solid allows to minimise the hydrophobic interactions through an aggregative process and to optimise the surface aggregate curvature in order to reduce steric repulsion of surfactant polar chains’, as claimed by Levitz [33]. Formation of surface micelles was confirmed by ellipsometry studies [38,58] and other methods [34] including microcalorimetry: upon micelle formation, the differential enthalpies of displacement of water by polyethylenic surfactants on silica and kaolin are slightly endothermic always lower than 10 kJ and close to the values obtained for micellisation enthalpies [19,20].

In these systems, the surface always remains hydrophilic, whatever the apparent surface coverage.

Two different cases can be presented according to the Krafft point.

3.2.1 Adsorption isotherms are carried out at a temperature higher than the Krafft point

3.2.2 Adsorption isotherms are carried out at a temperature lower than the Krafft point

In these kind of systems, it was also shown that counter-ions play an important part in adsorption phenomena. Adsorption isotherms of sodium tetradecyl sulfonate (TS), sodium dodecylsulfonate (DS) and sodium decylbenzene sulfonate (DBS) on kaolinite, determined at a temperature below the Kraff point, are presented in the θ vs Δμ plot (Fig. 6) [47,48]. In that case, the hydrated-crystal state was chosen as a reference phase. It was assumed (§2.1.2.2) that if the organisation of the adsorbed layer is identical to that of the top reticular plane of the reference phase, the isotherms built with a homologous series of surfactants superimpose. The non-superimposition of the tetradecyl sulfonate isotherm with the two others has to be discussed according to two conclusions:

• – one may assume that the adsorbed layer structure is different from that of the reference phase; this should be assigned to some influence of the kaolinite surface; however, the superimposition of the DS and DBS isotherms does not agree with this assumption; therefore, this hypothesis can be rejected;
• – one may assume that the normal bond between solid surface and the adsorbed surfactant ions is different in the TS-kaolinite system compared to that in the DS- and DBS-kaolinite systems.