Particle Size Effect on the Hydrophobicity and the Natural Floatability of Molybdenite

PARTICLE SIZE EFFECT ON THE HYDROFOBICITY AND THE NATURAL

FLOATABILITY OF MOLIBDENITE

AGUSTÍN FRANCISCO CORREA

Universidad de Concepción1986

(2006 English Edition)

PARTICLE SIZE EFFECT ON THE HYDROFOBICITY AND THE NATURAL FLOATABILITY OF MOLIBDENITE

A Thesis Presented to the Faculty of

the Department of Chemical EngineeringGraduate School

Universidad de Concepción

In Partial Fulfilment of the Requirements for the Degree Master of Science in Engineering and Metallurgy

By Agustín Correa

Concepción, Chile, September 1986

Thesis directed by Professor Dr. Sergio Castro Flores Head of the Program: Dr Igor Wilkomirsky

The original Spanish version of this thesis was revised by the Committee members:

Dr. Sergio MontesDr. Christian Hecker

2006 Revision and Translation by Agustín Correa and Alfredo Correa Tedesco

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Table of ContentsTable of Contents...............................................................................................................3Introduction........................................................................................................................5

Motivation........................................................................................................................5Objectives of the work.....................................................................................................5

Chapter 1 Bibliographic revision...............................................................................71.1 Surface characterisation.......................................................................................7

1.1.1 Surface tension.............................................................................................71.1.2 Adsorption and Surface Free Energy...........................................................81.1.3 Energy and molecular interacting forces and their relationship with the surface free energy.......................................................................................................9

1.2 Contact Angle....................................................................................................111.2.1 Analysis of the terms that compose Young’s equation.............................13

1.3 Adhesion Work of a Liquid...............................................................................141.4 Adhesion Work..................................................................................................151.5 Solid-liquid spreading coefficient......................................................................151.6 Characterisation of a liquid and its interrelation with a liquid through the contact angle measurements..........................................................................................151.7 Theoretical Studies of Models that link and with the surface energies determination.................................................................................................................16

1.7.1 Zisman Model............................................................................................161.7.2 Girifalco-Good’s Model............................................................................181.7.3 Fowkes’ Model.........................................................................................211.7.4 Wu’s Model...............................................................................................241.7.5 Dann’s model.............................................................................................251.7.6 Wetting diagrams.......................................................................................271.7.7 Relationship between the different models................................................29

1.8 Contact angle and adsorption.............................................................................301.9 Determination of by other methods..............................................................321.10 Floatability and natural hydrophobicity.............................................................331.11 Hydrophobicity and Structure............................................................................331.12 Molybdenite: its structure and relationship with hydrophobicity......................341.13 Hydrophobicity and Surface energy of Molybdenite........................................36

Chapter 2 Experimental Part...................................................................................392.1 Materials............................................................................................................392.2 Contact angle measurements on a cylindrical cake...........................................422.3 Contact angle measurements on a massive crystal............................................422.4 Hornby-Leja’s method.......................................................................................432.5 Quantified wetting method................................................................................432.6 Hallimond’s tube Flotation................................................................................452.7 Conventional cell flotation of the sample in presence of quartz.......................462.8 Flotation of a collective of molybdenum samples.............................................47

Chapter 3 Results.......................................................................................................493.1 Contact angle measurements.............................................................................49

3.1.1 Zisman’s model.........................................................................................49

3.1.2 Girifalco & Good’s Model.........................................................................513.1.3 Wetting model (Lucassen-Rynders)..........................................................523.1.4 Fowkes’ model...........................................................................................543.1.5 Wu’s model................................................................................................553.1.6 Dann’s Model............................................................................................563.1.7 Neumann-Good’s model............................................................................573.1.8 Comparison between the values predicted by different models................60

3.2 Techniques where water-methanol solutions were used...................................623.2.1 Hornsby- Leja method...............................................................................623.2.2 Quantified sinking method........................................................................633.2.3 Hallimond tube flotation tests....................................................................653.2.4 Common cells flotation tests......................................................................67

Chapter 4 Discussion.................................................................................................694.1 Zisman’s model application...............................................................................69

4.1.1 Thermodynamic Analysis of the Zisman’s slope......................................694.1.2 Zisman’s slope analysis based on Girifalco-Good’s Model......................754.1.3 Zisman’s slope analyses for measurements with alcohol-water solutions 77

4.2 Good’s Model Application................................................................................784.3 Wetting Diagram (Lucassen-Reynders) Applications.......................................784.4 Fowkes’ Model Application..............................................................................784.5 Wu’s Model Application...................................................................................784.6 Dann’s model Application.................................................................................804.7 Correspondent states equation...........................................................................814.8 Hornsby- Leja’s method....................................................................................814.9 Quantified sinking method................................................................................814.10 Hallimond flotation tube....................................................................................824.11 Conventional flotation cell................................................................................85

Chapter 5 System integral analysis..........................................................................87Chapter 6 Conclusions...............................................................................................91Appendix...........................................................................................................................93

Contact angle measurements.........................................................................................93Addenda and Notes..........................................................................................................97Bibliography...................................................................................................................101Nomenclature...................................................................................................................105

Introduction

MotivationThe mineral molybdenite species has been studied by several authors because its

special properties [2][3][4]. This species and others such as graphite, talc, stibnite are known as highly hydrophobic solids and they exhibit natural floatability [5].

In spite of their high hydrophobicity, a high rate of losses in the flotation processes has been noticed especially in molybdenite at the finest particle sizes.

In order to explain these facts the following statements has been proposed:

1. Energy surface differences amongst the granulometric classes that cause the hydrophobic properties of the finest particles to be partially lost.

2. The finest particles have adverse hydrodynamic conditions in conventional flotation reactors.

Several authors have devoted their efforts to relate the floatability to the hydrophobic properties of molybdenite [7][8][9][10]. Especial attention has been paid on characterising these kinds of properties [11][12][13].

Statements based on the physical chemistry of surfaces were developed and applied for low- surface- energy solids [14][15][16][17].

In spite of the success in the physical chemistry characterisation of the natural hydrophobicity of MoS2, there are no clear evidences in the literature on the true influence of the particle size on the aforementioned properties. On the contrary, an increase in edges/faces ratio is presumed to happen because of the comminution processes; besides, other factors related to the floatability of the finest, such as flotation kinetics and the particle size/bubble, are known to be a stabilising factor in particle-bubble adhesion process.

As previously said, it is necessary to complete the thermodynamics focusing and evaluating quantitatively the particle size effect on the molybdenite surface free energy and the components of its interaction forces with its surroundings. This approach will allow us to reorient the problem and subsequently to evaluate the kinetic effects on a stricter basis.

Objectives of the workThe purpose of the present work is to investigate the influence of the particle size on

the natural hydrophobicity of Molybdenite. We use the contact angle as a measurement technique with liquids of well-known surface free energy and several models are applied to get information about the surface free energy of the solid as a function of its particle size. The critical tension of wetting in methanol-water solutions and the flotation behaviour of fine particles are assessed, in the light of the knowledge of the variation of the polar components of the surface free energy of MoS2; which is the responsible property of the loss of hydrophobicity and floatability of fine particles.

Chapter 1Bibliographic revision

1.1 Surface characterisation

1.1.1 Surface tensionOne of the most suitable properties to characterise a surface is its surface energy or

surface tension. This is caused by the external layers in the boundary of the bulk known as surface zone or interface. The molecules of this boundary layer, whose limits are not well defined, are subject to attractive forces exerted by the adjacent similar molecules. These forces have a resultant force addressed to the bulk of the phase in the normal direction to the surface. These unbalanced forces cause the surface molecules get additional energy by interacting with its surroundings.

The surface tension may considered either as a distributed force per length unit or as energy per surface unit. It is better to give an energetic definition of this property by associating it to the work W that it is done when the area A of the surface changes. For a positive infinitesimal change it will be:

(1-1)

If the first thermodynamic law is taken into account:

(1-2)

Where is the absorbed heat by the system, is the work done by the system and E is the internal energy of the system.

is composed by:

(1-3)

Where is the expansion work and are works not associated to the pressure and volume changes (P = pressure; V = volume).

Applying the Second Law of thermodynamics:

(1-4)

Where is the reversible heat absorbed by the system, T is the absolute temperature and S the entropy.

For a reversible process and according to (1-2), we obtain:

(1-5)

By introducing the free enthalpy definition,

(1-6)

And its differential form:

(1-7)

Also, for a reversible evolution, introducing (1-5) in (1-7)

(1-8)

For an evolution at T and P constant we have

(1-9)

Then we define:

(1-10)

This equation thermodynamically defines as a measure of the free energy per area and it guides us to also define an internal energy per area , an entropy per area , and an enthalpy per area :

(1-11)

and using another Gibbs’ equation

(1-12)

we get:

(1-13)

In these cases we may assume .

Equation (1-13) shows us that the variation of with temperature is a measure of the entropy effects associated to the surface and must not be disregarded to infer [18].

Any possibility of a change in systems where surface energy are involved may be evaluated by these equations either as a free energy variation or as a reversible work done by the system at constant P and T conditions and in electric works absence; so we have:

(1-14)

1.1.2 Adsorption and Surface Free EnergyIn a biphasic system ( and phases) at equilibrium, bounded by an surface with

two components, in each phase we have:

(1-15)

where is the number of moles of substance i in phase and is the chemical potential of component i in phase .

When we consider a liquid phase l in equilibrium with its vapour v; i.e. a solution with components 1 and 2, we get

(1-16)

the Gibbs-Duhem equation is complied in these phases, so

(1-17)

Then we have in (1-16) that

(1-18)

clearing up:

(1-19)

Where the differential coefficients or the chemical equilibrium are defined as the component i superficial excess :

(1-20)

Then we have

(1-21)

Choosing the boundary between both phases so that one of the be null (commonly this presumption is applied to the solvent molecules) [19] we get that

(1-22)

This development gives us a very important relationship between the variation of the surface tension of solutions formed by a solute and its adsorption on the solution/vapour interface [19].

1.1.3 Energy and molecular interacting forces and their relationship with the surface free energy

The origin of the surface tension or the surface free energy may be established on studying the intermolecular forces.

These may be classified [20] in:

1. Attractive forces

a. Primary bonds (chemical)

i. Atomic (homopolars)

ii. Ionic (heteropolars)

iii. Strongly polar (hydrogen bonds)

b. Metallic bonds

c. Secondary bonds

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i. London’s dispersive forces (no-polar/no-polar)

ii. Debye’s forces (induced polar/polar)

iii. Keesom’s forces (polar/polar)

2. Born’s repulsion forces

The chemical and metallic bonds are very strong, with an important free energy of formation and they are associated to the formation of a solid, and in especial cases either in a liquid formation such as water (hydrogen bonds), or mercury (metallic bonds).

The secondary attraction forces between molecules or atoms are known as van der Waals’ forces.

Calculating the surface free energy by associating it to particles bonds and multiplying this by the bonds number per area has been successful only in few cases.

On the contrary, the Lennard-Jones’ electrical potential is more suitable. This potential (E) has the following expression

(1-23)

and its force field (F) in the radial direction:

(1-24)

Whence is the distance between two molecules centres which interact, is a constant associated to Born’s repulsion forces and C is a constant related to the van der Waals forces. These forces cause the surface tension of a system and each one may be analysed in a different form as the usual classification says.

The Keesom’s forces are related to the interaction between the dipolar moments of the molecules. These forces orient the molecules in preferential directions. Their energy is described as:

(1-25)

Where are the dipolar moments, k is the Boltzmann’s constant and is the distance between the forces centres of two molecules of the same kind.

Debye’s forces appear when one or two molecules have a permanent dipole that produces an electric field in its environment that at its time induces other dipoles. Its expression for two dissimilar molecules is

(1-26)

Where is the molecule polarisability.

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The London “dispersion” forces are present in all type of substances; they are additives. They are caused by the fluctuations of the electrons distribution into the atoms. Besides, these fluctuant dipoles induce new dipoles in their adjoining atoms. So, the energy of the “dispersion” forces caused by the interaction of two atoms is calculated this way:

(1-27)

Where are the main frequencies of the electronic fluctuations and is the Planck constant.

The relative importance of the London “dispersive” forces related to the Keesom’s and Debye’s, specially in low energy materials, made Fowkes define the interaction term of his interfacial energy model only as function of these “dispersive” forces, either for solid-liquid or liquid-liquid systems.

1.2 Contact AngleThere are few possibilities when a liquid is brought in contact with a surface; either a

partial spreading or a complete wetting.

The partial wetting phenomena is characterised by the presence of a drop, a spherical vault on top of the solid surface, in case the gravitational forces are not significant [20].

The best measurement of this situation is the contact angle. It is measured in a three-phase boundary defined by the solid surface, and the liquid-vapour interface.

Young was the first to scientifically explain the contact angle phenomena by applying a mechanical model without taking into account the gravitational forces (Figure 1-1).

Considering the tensions as forces per length unit, Young proposed a simple equilibrium of forces by projecting them on the solid-liquid surface plane in the three-phase contact boundary:

(1-28)

Equilibrium conditions requires the resultant of forces to be null in all the points of the plane. So,

(1-29)

This equation is usually known as the first Young’s equation and may be applied only when .

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This approach has been the object of a lot of criticism for not considering the equilibrium force on the perpendicular axis to the solid/liquid plane. This may be avoided by ascertaining that (1.29) describes the phenomena when the component of on the perpendicular axis is absorbed without deformation of the solid surface.

Modified equations of the Young’s relationship have been proposed by considering another three-phase tension forces, but, they have not been well received by the surface chemistry specialists [21].

Young’s equation is of paramount importance when surface tensions are defined as a surface free energy.

A lot of efforts has been made in order to demonstrate Young’s equation by applying thermo dynamical concepts instead of mechanical equilibrium.*

These requirements have been satisfied by Good’s (1952) and Goodrich’s (1969) demonstrations by applying the principle of virtual-work.

The is a valid thermodynamical quantity if it is measured in equilibrium conditions because it is associated to free energies and is usually written as .

If we associate with equilibrium conditions, its measurements must be carried out carefully, especially when hysteresis, rugosity and heterogeneity phenomena may happen [22].

The differences in the values between the receding contact angle and the advance contact angle measurements are caused by the hysteresis phenomena.

They are named this way because when a drop of a liquid is forced to move backward or downward on a solid surface an advance or recede angles are formed. Generally the advance angle is associated to the equilibrium conditions.

The simplest way to analyse the rugosity phenomenon is given by the Wenzel’s equation [21][22]:

(1-30)

Where is a rugosity parameter defined as the ratio between the “actual” area (considering the rugosity) and the apparent area (an ideal plain surface.)

* These two statements do not contradict each other: macroscopic mechanical forces are ultimately derivatives of a suitable free energy (Revisor Note)

: interfacial tension solid/liquid: interfacial tension

liquid/vapour: interfacial tension solid/vapour

: contact angleFigure 1-1: Schematic representation of the contact angle

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On the heterogeneity issue, it is generally accepted that the experimental contact angle is a weighted resultant of the of the different surface places considered, as Cassie developed [22]:

(1-31)

where j represents each kind of superficial place and fraction of the surface of type j.

These three aspects on matter: rugosity, heterogeneity and hysteresis shift the surface from the ideal conditions. They have been studied in an independent way however they are closely related [22].

An experimental method of contact angle measurement that incorporates mechanical vibrations it is very interesting from the thermodynamic point of view, the contact angle will take a unique equilibrium value because this is the minimum energy state [23].

1.2.1 Analysis of the terms that compose Young’s equationAs the angle is a resultant of the surface energy of the phases which are brought in

contact; it is necessary to analyse the origin of the terms in the Young’s equation and to know what information we could extract from them.

: is the surface energy of the liquid phase in presence of its saturated vapour.

: is the surface energy of the solid phase in presence of the saturated vapour of the liquid phase. It is the consequence of the adsorption of the vapour phase onto the solid surface.

The relationship between this solid surface energy and (vacuum condition) is:

(1-32)

where is the equilibrium pressure that may be calculated by the equation of the Gibbs’ isotherm adsorption [14]:

(1-33)

where R is the universal constant of gases; T is the absolute temperature; is the Gibbs’ adsorption in excess; P is the vapour pressure and P* is a very low pressure tending to zero.

Generally, low energy systems get very low values.

: is the solid-liquid interfacial energy. Together with and , it is difficult to evaluate directly.

If we want to characterise the solid surface from an energetic point of view, we notice that Young’s equation is limited in its application.

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We may express Young’s equation as:

(1-34)

and define:

(1-35)

where is the adhesion force or adhesion tension. We notice that this adhesion force may be calculated from (1-34) but the difficult is to clear up from it both and

and then infer .

This task was faced up both by the theoretical models that we are going to show later [24][25], and by the empiric models (such as Zisman’s); when calculating in particular conditions.

We must clarify that sometimes, when certain surfaces are studied, the contact angle measurement is carried out in a system made up of two non-miscible liquids and a solid; i.e. mica, water and several organic liquids [26]. In this case the Young’s equation must be written as:

(1-36)

where W and O subscripts indicate water and organic phases respectively.

Now we are going to introduce definitions based on surface free energy concepts developed before

1.3 Adhesion Work of a LiquidAdhesion work of a liquid WC is the free energy associated to the breakage of a liquid

column and the surfaces creation where new liquid and vapour faces are going to bring in contact (Figure 1-2):

(1-37)

Figure 1-2: Adhesion work; its schematic representation

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1.4 Adhesion WorkAdhesion work Wadh is the free energy associated to the interface A/B separation to

create A and B interfaces. We have for a solid A and liquid B system (Figure 1-3):

(1-38)

On introducing the equilibrium pressure we have

(1-39)

When partial wetting occurs it is more useful to introduce the Young’s equation in (1-39), so we have:

(1-40)

which is the Young-Dupré equation.

1.5 Solid-liquid spreading coefficientThe solid-liquid spreading coefficient K is the difference between the adhesion work

and cohesion work of the liquid:

(1-41)

If , the solid liquid interaction will be strong enough to produce the total wetting of the solid by the liquid. On the contrary, if , the necessary work to overcome the attraction of the molecules of the liquid is not balanced by the attraction between solid and liquid molecules. Whatsoever, the total wetting occurs when .

Replacing in (1.41) the Wadh and WC we have:

(1-42)

1.6 Characterisation of a liquid and its interrelation with a liquid through the contact angle measurements

The Young’s equation, that Zisman [14] qualified as “deceptively simple”, has the inconvenient that it does not distinguish the solid surface energies from the solid-liquid interfacial energies and it is even worse for the adsorptions conditions because these are included in the aforementioned term defined as adhesion tension.

initial finalFigure 1-3: Adhesion work schematic representation

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Different proposed statements about the adhesion tension are going to be observed when we will analyse the models which have been applied in the study of the surface through the contact angle measurements. The Zisman’s empirical model relates the

(adhesion tension) to a particular condition, so well-based assumptions may be

made; i.e. is either in a minimum or is null; is almost equal to , is in a minimum, et cetera.

The deductive model [24][25][27] approaches the problem by proposing a constituent equation for as a function of and , so in this way we have a better approximation to the true value when we combine this equation with Young’s.

1.7 Theoretical Studies of Models that link and with the surface energies determination

1.7.1 Zisman ModelThis model is supported by a searching team. The works in the 1940-1950 decade were

carried out on a theoretical base which linked the contact angle to the solute activity in a solution and to the adhesion tension; i.e. the butanol and butyl-amine activities in the presence of graphite and stibnite related to angle contact measurements [28]. This kind of work permitted to infer hypothesis about the adsorption in the system.

Paramount interest has been reported on determining the adhesion tension in especial situations as those of critical wetting.

This is calculated as:

(1-43)

On studying the low energy surfaces as polystyrene, Zisman related the measurements of contact angles to the liquid superficial tension in contact with the solid surface through a simple linear equation:

(1-44)

where is the value that assumes when and is the slope of the straight line it has got. As we see, this function is valid in the same range as the Young’s equation is .

This way, Zisman introduced a concentrated parameter which allows us to establish a relative scale of surface energies. This shows us that when the liquid tension is less than

, the complete wetting of the solids happens.

The choice of this particular point is based on the usual concepts about the spreading between two similar liquids.

Under these circumstances, we presume that the interfacial tension is null or reaches a minimum and then Young’s equation may be written as:

(1-45)

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where supra-index (C) indicates critic conditions. At the extreme case, would be true for a low energy surface solid ( ) when .

Some authors [29][30] support these last concepts. However, in other cases when the surface energy of solids has been measured by fusion heat techniques serious discrepancies have been found (Gardon 1963).

So, it remained the questions if was a constant of the surface solid, and besides if it was a measure of the solid surface energy independent of the sort of the liquids used in its determination.

Experimentally, Zisman utilised a hydrogen-bond liquid series and he obtained a for polystyrene. This had to be calculated by extrapolation because

surface dissolution caused problems. On the other hand, Zisman got for the same solid utilising non-hydrogen bond liquids.

Later, Good [23] demonstrated that, using a more elaborated model, such discrepancy was caused by Zisman’s model, which he qualified as limited.

A still in force question is the discussion about the critical wetting conditions obtained using liquid mixtures or alcohol water solutions.

ASTM has edited standards in order to carry out this kind of measures utilising formamide-2-ethoxi-ethanol mixtures and water-ethanol solutions [27]. This technique is more flexible because this way it is possible to control the surface liquid tension by changing the concentrations of either the mixtures or the alcohol-water solutions. It was applied on the study of polyethylene, poly-tetrafluoride-ethylene [31], and sulphide [13] [32] surfaces.

This method is criticised because it is not clear whether the preferential adsorption phenomena modifies the surface conditions, so it becomes more or less energetic. Zisman suggests that adsorption happens after the contact; furthermore, it is a consequence and not the cause of wetting conditions [14].

Other authors [23] looked down on this method, basing their opinions on the isotherm preferential adsorption described by the Gibbs’ equation. This phenomena would have a marked influence on the value and on , either which would be different from .

By the way, many researches have been carried out on systems composed by aqueous-alcoholic solutions on low energy surface solids. They have confirmed the adsorption onto polar sites; furthermore a partially covered surface by hydrocarbon chains is obtained even in contact with aqueous-alcoholic solutions of alcohols of low number of carbon such as methanol or ethanol [33][34].

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Lower values of have been obtained when aqueous-alcoholic solutions were utilised

instead of pure liquids; i.e. graphite; with pure liquids a has been obtained

[35], but with butanol and butyl-amine aqueous solutions values of and

have been respectively reported [28].

On the other side, the value of the dispersive component of the graphite surface energy has been reported as [36], furthermore, the calculated either from solutions or liquid mixtures is not suitable to be considered a direct measure of the solid surface energy.

The parameter was only considered as the slope of (1-44), but now it is known as Zisman’s slope and it has got a paramount importance both from the theoretical point of view [23] and in the characterisation of certain situations [11][35].

1.7.2 Girifalco-Good’s ModelThis model [25] was the first based on aspects related to the molecular structure of the

surface which have been brought in contact.

We have explained how limited the Young’s equation is in order to determine the solid surface energy and the (solid-vapour tension) too; and how this may be solved in case were defined satisfactorily as a function of and .

The construction of this model begins with describing a system where molecules of kind ‘a’ and ‘b’ are present. The attraction constants between similar molecules are and , and between different ones is . According to Hildebrand [25], regular solutions comply:

(1-46)

By analogy, Girifalco and Good [25] applied the same relationship between both adhesion and cohesion free energies between two phases; that is between strictly binary systems. In this way:

Figure 1-4: A typical Zisman’s plot

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(1-47)

where is the adhesion free energy between phases ‘a’ and ‘b’;

y are the cohesion free energy of phases ‘a’ and ‘b’

respectively, and is a constant of the ‘ab’ system.

Clearing up we get:

(1-48)

Applying it to the liquid-solid interface:

(1-49)

If we combine it with the Young’s equation by admitting

(1-50)

so:

(1-51)

when is negligible we have

(1-52)

and clearing up again

(1-53)

The importance of this model is that knowing the contact angle between a solid and a liquid, the liquid surface tension and the parameter (that is a constant of the system), it is possible to calculate the solid surface energy.

The main problem will be now how calculate . According of its authors may be calculated using the equation:

(1-54)

where, and are the molecular radii corresponding to the distances between the forces centres of the molecules; and are the molar volumes.

This deduction was elaborated for phases whose kinds of forces are alike, and then the parameter is presumed to be one.

When the kinds of forces are not similar, the follow equation must be applied [37]:

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(1-55)

where C are the constants coefficients of the of the Lennard-Jones potential, defined in (1.24), and the summations are:

(1-56)

The most general expression in order to calculate is:

(1-57)

where are the polarisation constants, the dipolar moments, the ionisation energy, the Boltzmann’s constant and T the absolute temperature.

If one of the both phases is a non-polar type, i.e. the solid; then and we get:

(1-58)

According to this equation depends weakly on the non-polar solid properties.

If both and are null we obtain a system where only the London dispersion forces are working; so:

(1-59)

Model applicationIt is evident this model has a lot of troubles when working on a non well-known

system. Nevertheless some correlations based on this model have been useful in order to get consistent results.

At first, we remember the resolution of the ambiguous that Zisman got when he was studying a polymer using this correlation:

(1-60)

This was proposed by Good on replacing by a linear combination of this kind:

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(1-61)

If we combine the last one with (1-52), it results in the semi empirical equation (1-60).

On the other side, under critical conditions, for a low energy surface solid; ; the mentioned relationship may be used and we get:

(1-62)

In this way if we achieve to evaluate under critical conditions, knowing the

supposed , we are going to obtain .

1.7.3 Fowkes’ ModelFowkes developed a model based on the characteristics of the intermolecular forces

caused by the different sort of intermolecular bonds. Either the metallic or the hydrogen bonds are both of specific nature. However, London’s dispersion forces are always present amongst neighbouring molecules or atoms, even if these are not all of the same substances or elements.

London forces are caused by the fluctuant electronic dipoles interaction with induced dipoles of adjacent molecules. They depend upon the electric properties of volume elements, they are always independent from the temperature and always attractive.

In this way, the surface tension of a condensed phase may be expressed as [24]:

(1-63)

where the supra-indices indicate “d” as London dispersive forces; “h” hydrogen bond; “m” metallic bond; H hydrogen bond; “ ” ; electronic interaction; “i” ionic interaction.

In the mercury (Hg) case:

(1-64)

for the water (W):

(1-65)

and for normal aliphatic and saturated hydrocarbons (HC):

(1-66)

When two phases are brought in contact, the required work to carry a molecule to the solid-liquid interface , from the bulk of each phase, diminishes in a quantity related to the necessary energy to carry the same molecule either to the solid-vapour (S/V) or to liquid vapour (L/V) interface.

Then for a molecule from the solid phase it is:

(1-67)

for a molecule from the liquid phase it is:

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(1-68)

The addition of both works and gives the energy necessary in order to

form the interface S/L:

(1-69)

Fowkes assumes that both works are the same and that both are functions of the dispersion components because he considers that they are the only capable of interacting through the phases boundaries.

So:

(1-70)

Then it remains

(1-71)

The utilisation of the geometric mean is justified by Fowkes in several articles basing on London and Hildebrand concepts [24]. They claimed that the interaction of those phases is defined by the equation:

(1-72)

where N is a constant of each phase; is the polarisation constant; I, the ionisation constant and r, the intermolecular distance.

As Fowkes deduces that:

(1-73)

and we get:

(1-74)

In the same way Girifalco-Good’s model did; now we have an equation that allows us to evaluate as a function of the liquid and solid phases properties.

22

If we replace by the Fowkes’ expression (1-71) in the equation that defines de adhesion work, we get:

(1-75)

Besides, if we combine (1-71) with Young’s equation (1-29), we get:

(1-76)

or in another way:

(1-77)

If it is possible to neglect , we get:

(1-78)

and,

(1-79)

This way, it is possible to evaluate performing only one determination of the contact angle using a liquid whose tensions components are well known.

On the other side, a correlation between vs. may be fitted according to the contact angles measures performed with different well-known-property liquids and so evaluate as a parameter of the corresponding equation (1-78).

The complete wetting condition in Fowkes’ model, neglecting , is given by:

(1-80)

Then for the critical conditions (c) it gets:

Figure 1-5: Typical Fowkes’ diagram

23

(1-81)

or

(1-82)

where is the wetting critical tension

So, only under very special conditions may be assumed as the solid surface energy.

On the other side, we observe that, in order for a solid to be completely wetted by a liquid, it is not enough to have a lower surface tension than the solid surface energy, but instead the aforementioned conditions must be fulfilled.

If the liquid has only dispersive components and may be neglected, we get

(1-83)

and if the solid interacts only by dispersion forces:

(1-84)

By extension, these concepts may be applied to the study of the wetting of solids, which only interact by dispersion forces, utilising water-alcohol solutions mixtures. In these cases we may assume, in a first approximation, that the dispersion forces are like to the water ones; which says that . This way, for a solid with a

according to (1-80), it gives:

(1-85)

This is only true if we assume that there is not preferential adsorption

1.7.4 Wu’s ModelThis model tries to be more general than the Fowkes’. Beginning with the same

Fowkes’ criterion:

(1-86)

where is the summatory of the components which are not “dispersive”. Wu included polar components of the surface tension and named them polar components of the surface tension.

He considered both the polar and dispersive components related to the interaction of the phases that have been brought into contact.

Doing the same analysis as Fowkes did:

(1-87)

Where and they are defined as:

24

(1-88)

replacing in (1-87):

(1-89)

The geometric mean criterion (Fowkes) has been changed by the harmonic mean.

Combining this with the Young’s equation and neglecting we have:

(1-90)

If we place the critical wetting condition, it simplifies to:

(1-91)

It is interesting to observe that if and , it implies, whatsoever , that . So, one of the wetting conditions is reached when a solid is brought in contact

with a liquid with dispersive and polar components alike.

The validation of the model calls for knowing and , so it would be necessary to perform two measurements with different liquids; producing, in this way, two equations that permit us to solve the problem.

An alternative mathematical method will be shown in the discussion of the present work in order to manage this model.

1.7.5 Dann’s model Both, Wu’s and Dann’s [27] models hypothesis are similar. We may consider Dann’s

model as an expanded expression of the Fowkes’ model; utilising besides, the interaction of the polar components the same way Fowkes claimed for dispersive forces.

The expression that foretells is:

(1-92)

The author claims that the treatment based on the geometric means has had better results than those which utilised harmonic means.

If the Young’s equation is introduced in (1-92) and neglecting , it results that:

(1-93)

25

The method consists in considering and as parameters to be optimised in a linear correlation and using the performed measurements in the presence of different liquids as fitting points:

(1-94)

where indexes each measured point ; ; .

This method has been also used in order to study high energy solids surfaces.

It is well known that it is very difficult to find liquids which do not wet completely these sort of surfaces (mica, oxides, et cetera).

Contact angle measurements on this sort of surfaces are performed in front of a water-saturated n-hydrocarbon system.

The Young’s expression for this new system is:

(1-95)

Where is the solid-saturated hydrocarbon interfacial tension; is the water-saturated n-hydrocarbon interfacial tension.

It is correct to assume that this kind of hydrocarbons interacts only by London’s dispersion forces; so applying Fowkes’ equation:

(1-96)

and

(1-97)

For Dann’s equation is applied:

(1-98)

When these expressions are introduced into the modified Young’s equation (1.95), it gets:

(1-99)

In this equation and are measured and has been obtained from

Fowkes’ model. However, the and terms remain independent from the hydrocarbon utilised.

Then, measurements may be performed with two different saturated hydrocarbons getting two independent equations with two unknown factors; so, the system of equations is solved and and are calculated

Dann took the credit for claiming for a geometric mean expression of the polar interaction. The solid water interaction term has been expressed as by other authors, so, they are not compromised with an expression as function of the polar components of both solid and water.

26

This happens because the London’s dispersive components are rather known [16], on the other hand, this does not happen with the so named “polar” components that embrace metallic bonds, hydrogen bonds and neither with Keesom’s and Debye’s bonds too.

1.7.6 Wetting diagramsThe wetting diagram [38] is a different way of display the measurement data. In

this diagram we represent the adhesion tensions as ordinates axis vs. the surface tension of the liquid utilised in the measure as abscissa axis.

The adhesion tension was defined as:

and it is calculated as

The region between the straight lines and has got a physical meaning. The first straight line is the geometrical place when and the latter;

is the geometrical place when the wetting critical conditions happens.

The data processing in this diagram, is carried out performing a linear correlation between vs. :

(1-100)

where m is the slope of this straight line and Cte is the intersection point of this straight line with y-axis.

This methodology has been followed by several researchers [21], [32] on studying the minerals wettability with water-alcohol solutions or tensoactive agents.

Otherwise, other authors have utilised second order correlations between and [39].

Generally, a wetting diagram will offer us the following information:

1. Critical wetting conditions by the intersection of the modelled straight line (or another function) with the identity function.

2. The differential wetting between two solid species. Generally, the slope of the straight line m is very well defined. The crossed zone in Figure 1-6 will show us the complete wetting of solid #1 under the same conditions at which solid #2 is partially wetted (Hornsby-Leja, 1980).

27

3. To establish a relative quantification of the superficial excess of the tensoactive agents between liquid-vapour, solid-vapour and solid- liquid interfaces.

If we analyse the derivate of the studied function, either in a point or as a whole in case a linear model has been performed [21] we have:

(1-101)

According to the Gibbs’ isotherm (1-22), if we replace the differentiation in both interfaces and if we consider that there exists an equilibrium state either, we obtain that:

(1-102)

or, in another way:

(1-103)

If the slope m is negative, as it happens usually, then , in a quantity that depends upon of m. If the absolute value of m is large, it may be considered , and we could claim for a null adsorption in the solid-vapour interface.

Finch and Smith have reported positive slopes for high surface energy solids, so this means that its tensoactive adsorption tension is predominant in the solid-

vapour interface over the solid-liquid adsorption.

This situation was reported on a determination on a molybdenite species using methanol-water solutions [32].

Figure 1-6: Wetting diagram

28

1.7.7 Relationship between the different models The models that we have presented must have point of coincidences between them,

because they have been generally applied successfully to the study of low energy solids surface, and even in high energy solids surface as micas and metallic surfaces.

The Zisman’s model treatment offers its versatility as advantage; its method is plain.

It must be noticed that when the aforementioned methods have been applied the calculated coincided with the liquids surface tensions of low interfacial energy.

Among these resemblances, we will mention the one between Zisman’s and Good’s models. The latter based his demonstration on Taylor series expansion of his expression model. He made this development in and establishing as independent of and he got:

(1-104)

which will be convergent when .

Having into account , we get:

(1-105)

If the expansion is truncated at the second term and we compare it with Zisman’s equation (1-29) we have that:

(1-106)

Both models, Fowkes’ and Good’s are very alike because they have a very similar starting point. If we compare their equations, (1-52) and (1-78), and we equalise them, we get:

(1-107)

and if we are in the presence of a solid and a liquid which only interact by dispersive forces then [37].

The Fowkes’ model has been the most frequently used because it obtains a paramount piece of information, which is the dispersion component of the solid surface energy. On the contrary, to obtain Girifalco-Good’s parameter, a tedious calculation is required in its model original model version.

However on account of the Fowkes’ model has got only one parameter, this model is not flexible in order to interpret situations as Parekh and Applan describe in the study on coals wetting [11]. This case, the coal species were differentiated between them by the

29

different contact angle they form with the very same liquids but they converged to a same critical wetting condition.

When the high energy solids problem is raised, it is necessary to consider the polar components in the interfacial interaction. Wettability of quartz in water is a typical case. If we apply Fowkes’ criterion , quartz will not be completely wetted by

water. If we replace the energy surfaces by their experimental values; and

, its geometric mean gives 42 mJm-2, less than the experimental value

(according to literature data reported for silica and water [24]).

On account of this, some authors [17] define the work adhesion not only with dispersion components (d), as Fowkes did, but they also include ionic (i) and metallic (m) components that in a first approximation may be considered as independent:

(1-108)

So, we observe that the condition for complete wetting, , will be fulfilled on increasing appreciably the adhesion work by means of its polar component rise.

These anomalies happen when Fowkes’ model is applied, and they were the base for constructing new models [27], where the dispersive forces are present and the remainder are embraced into the polar term (1-90)(1-93).

It must be noticed that the treatment of the surface energy based on the determination of dispersion and polar components are not accepted by all the authors, i.e. Good [23]. He precisely argues, amongst other things, that the signs of dispersion and induction energy in a solid will not be independent of the solid-liquid system. Besides, he claims to pay attention to an orientation term whose characteristics would be closely related to the surface entropy of the system interface.

1.8 Contact angle and adsorption.The deductive models of contact angle have restrictions in theirs use when we are in

presence of adsorption phenomena even in solid-pure liquid systems

In these situations, the proposal of a state equation that links the adsorption with the surface free energies ( ) is of paramount significance [39].

Authors have calculated in low energy surface solids using these concepts [39].

The construction and justification of this state equation is as follows: First, the Gibbs’ equation is set up for each of the interfaces which form the contact angle,

(1-109)

where subscript refers to each interface, E is the internal energy per surface unit, is the vapour adsorption at each interface and µ is the adsorbent chemical potential in each face.

Differentiating and introducing the first principle of thermodynamics:

30

(1-110)

and, expanding it for each interface, we get:

(1-111)

(1-112)

(1-113)

This differential equation system permits one solution as function of and :

(1-114)

and its solution may be written down as:

(1-115)

These expressions are considered by Good as an equation of corresponding states.

By replacing these equations in (1.34), we get:

(1-116)

This equation permits us to conclude that, in a wetting diagram, the adhesion tension (that is calculated as ) will have a good correlation as function of , only when is constant.

This will happen in solid-liquid systems when the adsorption is null or constant.

On the contrary, if the correlation is not good we may presume that adsorption is happening and it is different for each liquid in contact with the solid.

This thermodynamic point of view allows us to set up constitutive equations which express as function of and , even in an empiric form.

According to this theory, in order to calculate , a methodology has been developed applying the Girifalco - Good’s model that we are going to explain now.

Once the aforementioned correlation was satisfied, is calculated:

1. Guessing an arbitrary , may be close to .

2. Using this value, we calculate, in each measure (i), as function of and

:

(1-117)

3. With this value; ; the Good’s parameter is calculated for each measurement:

(1-118)

31

4. vs. are correlated through a straight line function:

with (1-119)

Steps are repeated from step 1, for different until

(1-120)

is achieved; since when .

This methodology is not only suitable to estimate but also to estimate and in each contact angle measurement; both of them will give us an idea of the degree of interaction between different liquids with the surface solid we are studying.

1.9 Determination of by other methodsOther methods; that have been applied to fine-size-hydrophobic solids, have been

developed in order to determine critical wetting conditions related to the solid surface free energy.

1. Hornsby-Leja’s method [12]

It consists in bringing up in contact the solid species on study with methanol-water solutions of different surface tension. Furthermore; we could define wetting and no-wetting zones up to when we establish a wetting with the precision permitted by the error of the technique.

We must clarify that this method was suggested as a possible technique in order to separate hydrophobic materials i.e. PTFE and coals.

2. Sinking time method [10]

This method measures the time a particle lasts in sinking in different surface tension. So, the ultimate surface tension indicates that the sinking time is infinite and it is calculated by extrapolation. This value may be linked to the immersion free energy variation (air-solid-liquid system):

(1-121)

and applying Young’s equation:

(1-122)

The immersion condition is given by . If we accept the approximation the critical wetting tension is going to be associated at . This should

be modified if gravitational forces and particle shapes must be taken into account in the immersion process [10].

3. Gamma flotation [13]This method may be carried out using a tube-Hallimond flotation with different water-

alcohol solutions whose composition determines the superficial tension of the liquid. The data are represented as a recovery vs. solution surface tension function.

32

It is observed than the recovery keeps on nearly constant as the surface tension solution decreases with respect to water’s. However, at a given point, the recovery drops dramatically, almost linearly. Their authors associate with the value at which the extrapolated aforementioned fall intersects the abscissa axis.

1.10 Floatability and natural hydrophobicityThe minerals flotation is based on reducing the surface free energy of the species to be

floated, using certain kind of chemical reagents. Furthermore, the hydrophobicity of the species rises and it acquires the property of adhering spontaneously to air bubbles. Hence, the interest to study the minerals species that are naturally hydrophobic comes up .

Natural flotability and hydrophobicity has been considered as synonyms and both had been related to solids with low surface free energy.

The “hydrophobic-solid” term indicates that such a solid does not have affinity with water.

This terminology has been ambiguous to describe the liophobic behaviour of a surface.

The aforementioned solid introduced in a water-gas system adheres to the gaseous phase, forming a partial-solid/gas interface as a result of the interfacial tensions balance. The particle-bubble adhesion may be quantified through the contact angle measure and its value will depend upon the hydrophobic grade of the particle we are studying on.

1.11Hydrophobicity and StructureRelating natural hydrophobic solids to their crystallographic structure has been a

constant source of concern to scientific searchers and therefore to outline generalisations on the subject [5].

The basic hypothesis is that the behaviour of these species is linked to the available free forces they have in order to interplay with its environment and these for their part with interatomic forces of the crystalline lattice.

It has been found that a crystal or mineral particle shows hydrophobicity when it cleavage breaks preferably residual bonds (van der Waals). This way, the particle will interact with its environment mainly through London’s dispersion forces [5].

However, on especial occasions a violent breakage as it happens in minerals comminution, may break another sort of stronger bonds, such as the covalent ones, generating more energetic sites. These sites are capable to interact with its environment in a more energetic way i.e. oxidation surface reactions, adsorption et cetera. This mechanism increases the solid surface energy and whatsoever spoils its hydrophobic properties.

The energetic character of a bond is associated to the equilibrium distance between the atomic centres of interaction.

Furthermore, the analysis of the crystalline structure of these solids results a suitable method to foretell their hydrophobic properties.

33

The laminated crystalline structure is a common property of the inorganic hydrophobic solids (graphite, talc, mercury oxide, boric acid, molybdenite, et cetera).

The graphite crystalline lattice is a typical case, having a laminar structure in hexagonal arrangement, a 1.42 Å interatomic distance and its links between layers with 3.4 Å bonds.

Boric acid is a special case, despite having an ionic behaviour, it is naturally hydrophobic because it forms perfect hexagonal layers through hydrogen bonds; however; such sheets interplay through van der Waals bonds.

The exception is given by micas and chlorites, despite having a layered structure, they present substitutions in the by aluminium, which cause unbalanced charges building up a poly-functional structure. So, polar groups appear on the crystal surface that render it hydrophilic.

1.12 Molybdenite: its structure and relationship with hydrophobicity

Molybdenum sulphide crystallises in two different habits. The hexagonal habit is the most common with and axes; the other is the rhombohedric habit with and . These frameworks were discovered and described, in natural crystals by Dickinson and Pauling [2].

The hexagonal habit is characterised by two-layer MoS2, whose molybdenum atoms have prismatic coordination, trigonal with 6 atoms of sulphur. Each cell is composed by two molecules of MoS2.

The shortest distance between Mo and S atoms is 2.41±0.6 Å. The faces of a imaginary prism of Mo-S6 are 3.15 ± 0.02 Å wide. Whereas, the distance between adjacent layers is 3.49 Å long.

The rhombohedric structure happens in some deposits [41], it has a similar arrangement of Molybdenum atom layers and the very same type of trigonal coordination.

Other possible frameworks has been claimed for the molybdenite; they are: a) rhombohedric, with 3 molecules per cell unity (3R), b) two hexagonal with two molecules per unity cell (2H); and c) trigonal with two molecules per cell unity. However, 2H1 and 3R (Figure 1-9) are the only ones type of structures which have been detected.

34

These details of structure cause the anisotropic phenomena to be present in Molybdenite, which is a typical property of hydrophobic species. For example; the microhardness of the (001) plane is 320 MPa and the (100) plane’s is 9000 MPa. Furstenau has called faces to the planes determined by sheets (001) and edges the plane (100) perpendicular to the aforementioned.

According to Gaudin’s theories, in a molybdenite crystal, we may foretell that the breakages of van der Waals’ bonds produce cleavages in the face plane; besides; this sort of bonds has got its centres on the S-atoms of different layers. This plane (001) may be characterised as hydrophobic.

On the other side, breakages of the molybdenum-sulphur chemical bonds will cause cleavages through the edge planes; so these planes are the most energetic.

This reasoning was useful in order to explain not only the molybdenite anisotropic behaviour of its electric and thermal conductivities [42], but also its hydrophobic behaviour as a function of particle size [6].

According to Chander and Fuerstenau, generally, and particularly for molybdenite, the smaller its particle size is, the larger its possibility of producing cleavages through edge planes are; furthermore a loss of flotability and natural hydrophobicity is produced.

Figure 1-7: MoS2 crystalline structure.

35

This happens when the surfaces become richer in strong bonds of marked polarity, furthermore, they become richer also from the energetic point of view.

This is of paramount importance for its concentration-flotation process, because these phenomena explain in a simply way the low recoveries observed in molybdenite-fine particles during the aforementioned process.

However, Trahar and Warren [9] do not agree with these statements. They claim that largest particles may become richer energetically on being subjected to tensions during the comminution processes without dissipating energies through breakages.

1.13 Hydrophobicity and Surface energy of Molybdenite Contact angle measures, ‘pzc’ determinations on different cleavages planes and

Hallimond’s tube flotation tests (all considered to be pH functions) have been carried out in order to confirm the aforementioned hypothesis [6].

This approach has allowed some authors to obtain interesting conclusions related to chemical superficial reactions that have an influence on molybdenite hydrophobic/ hydrophilic behaviour [8][6][43].

However, there is a lack in scientific literature related to the straight links between the surface free energy and the particle size. This may be achieved through a suitable interpretation of the contact angle measurements, because this is an energetic sign of its composing faces.

A suitable implementation of wetting thermodynamic models may reach an adequate estimation of the surface free energy and its dispersion and polar components. It is very convenient in this sort of treatments to carry out direct or indirect measures of the contact angle of the solids in contact with different pure liquids.

The parameters that the evaluation of this models yield; particularly; surface energy of different granulometric classes, will be an evidence for the edges-faces sites theory and its links with the particle size.

Figure 1-8: Fragment of a MoS2 layer. S atoms are represented by hollow circles. Doble circles lay on the (110) plane.

36

Solids of low surface energy, hydrophobicity and natural flotability are terms that have been utilised as synonyms with a few exceptions as Warren and Trahar [9].

Either the surface energy ( ) and its components or the parameters of the

different angle contact models, (Zisman); (Girifalco-Good), will allow us to set up and inherent hydrophobicity ranking in a more precise way.

The natural flotability will be a consequence of hydrophobicity, however we do not have to forget that the hydrodynamic aspects that are present in the flotation process.

Flotation process can be defined as a probabilistic event

(1-123)

where is the flotation probability, is the particle-bubble collision probability, is the adhesion probability and is the probability of setting up a bubble-solid

system.

We observe that the flotation probability is determined by two terms, and (defined chiefly by the hydrodynamics of the system) and a third one (related to an equilibrium system and associated to thermodynamic properties such as solid surface energy or the contact angle of the system.)

In this way, we aim to evaluate the different meaning between floatability and hydrophobicity such as Trahar [9] did on treating the flotation behaviour of fine particles.

Though, it is not right to link straight and strongly the contact angle with the flotation behaviour (natural or induced), however; it is impossible not to take into account the

(a) (b)

Figure 1-9: Views of different MoS2 crystals: (a) Simple polytype in ‘ab’ plane. (b) Simple polytypes along the ‘c’ axis. Figures (a) and (b) can be considered as top views and side

views of variuos simple polytypes of MoS2, if the plaquettes were in their base planes on an horizontal plane. Black circles represent Molibdenum atoms and open circles represent

Sulfur atoms. In (a) the solid circle inside the open circle represents Mo atoms and S located on top of others.

(a) (b)

(c) (d) (e)

Figure 1-10: Mo1−xS2 crystalization model. Triangles indicate fragments of Mo1−xS2 layers. (a) Random arrangement of MoS2 layers (b) Piling order of layers with random orientation with respect to ‘c’ axis and 001-plane. (c) Hexagonal MoS2 with imperfect piling ordering

(d) hexagonal MoS2 in antiparallel orientation (e) Rombohedric MoS, all layers are in parallel orientations.

37

extreme conditions that the equilibrium thermodynamic defines, such as the complete wetting conditions that prevent genuine flotation of a mineral species.

38

Chapter 2Experimental Part

2.1 Materials

Molybdenite mineral: A sample from a molybdenite concentrate of Chuquicamata Division was used. This

was sized in six granulometric classes using a Cyclosizer equipment.

Only one of them was larger than 44 µm. Every sample was washed and purified. They were cleaned thoroughly with diethyl ether to eliminate the flotation agents remaining, and then lixiviated in 1:1 H2SO4 solution. After this, they were thoroughly washed with distilled water until the complete removing of sulphate into de wasted water. At once, the samples were treated overnight with a 40 g/L-CNNa solution in order to eliminate the surface impurities such as Cu, Fe and Re. These might be present either as substitutions in the crystal lattice of molybdenite or as sulphide remains disseminated into the Molybdenite. The excess of cyanide was removed through a profuse washing with distilled water, subsequently, particles were dried up at room temperature.

The chemical analyses are shown in Table 2.1.

Table 2-1: Chemical Analyses. Content of Mo and Cu, Fe and Re impurities in the molybdenite samples after being purified.

Cyclosizer class Size[m]

Re[%]

Fe[%]

Cu[%]

Mo[%]

+C1 >44 .0047 .622 .594 49.7 C1/C2 44-33 .018 .487 .343 50.57 C2/C3 33-23 .019 .507 .281 52.96 C3/C4 23-16 .017 .0468 .299 54.95 C4/C5 16-12 - - - - C5 <12 .29 .543 .293 54.96

Liquids:Glycerine, formamide, di-I-methane, tetra- Br- ethane and methanol (p.a.) were used

(Table 2.2). The water was bi-distilled in a quartz-equipment.

MIBC (methyl-iso butyl carbinol), technical grade commonly used in flotation.

Table 2-2: Properties of the used liquids (20o C, after Fowkes).

Liquid [kgm3]

L

[mNm1]

[mNm1] [mN-½m½]

[mNm1]

Water 998 72.1 21.8 .7 6.5 .2 50.2Glycerine 1261 63.4 37 4 9.6 .5 26.4Formamide 1133 58.4 39.5 7 10.8 .95 18.9Di I methane 3325 50.8 48.5 9 13.7 1.4 2.3Tetra Br ethane 2960 49.7 49.7 14.2 -

: Density: Liquid surface tension.

: Dispersive component of the surface tension

: Fowkes’ ordinate

: polar component of the surface tension

Equipment: Cyclosizer classifier Paul Weber–Marsch U Apparetebau Stuggart hydraulic press Cathetometer Tripod with an adjustable platform and indicator of horizontally Transparent Chamber Dosage injection Spray Microscope with goniometer Hallimond flotation tube with a 40 cm- height and 3 cm diameter column. Magnetic stirrer Cylindrical containers with pressurised nitrogen. Agitair L-500 flotation cell. 20 mL test tubes Modified test tubes with an underflow discharge 25 mL tubes

Synthetic molybdenite: MoS2 Merck (p.a.)

Molybdenite massive crystal:A high purity crystal was cut in a prismatic shape so the polished “face” plane was a

1.5 cm side square and the polished “edge” plane was a 1.5 cm x 0.8 cm rectangle. The polishing was carried out utilizing baize and an alumina power; this polishing was repeated after each one of the contact angle measurements, previously washed by methanol.

Methyled Silica: Aerosil reagent (p.a.) purity

High purity quartz:A sample of quartz was ground at <74 µm, treated with HCl to eliminate Fe, and then

profusely washed with distilled water until the complete chloride removal.

Table 2-3: Performed measurements carried out by the Kossen and Heerjes’ method.(a) Sample +C1 (> 44 m)

Liquid (1Ev) h ×103 cos Fadh Wadh

Water .824 4.24 .127 9.17 62.8 Glycerine .799 2.4 .430 27.2 90.6 Formamide .818 2.08 .541 31.6 90.0 Di I methane .831 0.12 .959 48.7 99.5 Tetra Br ethane .800 nil 1.0 49.7 99.4

(b) Sample C2/C1 (44-33 m)Liquid (1Ev) h ×103 cos Fadh Wadh

Water .821 4.40 .160 13.4 58.6 Glycerine .821 2.96 .222 14.1 77.5 Formamide .821 1.84 .605 35.3 93.7 Di I methane .813 0.04 .986 50.1 101 Tetra Br ethane .820 nil 1.0 49.7 99.4

(c) Sample C3/C2 (33-23 m)Liquid (1Ev) h ×103 cos Fadh Wadh


(d) Sample C4/C3 (23-16 m)Liquid (1Ev) h ×103 cos Fadh Wadh

Water .881 4.20 .064 4.63 67.4 Glycerine .861 2.40 .450 28.5 91.9 Formamide .799 1.92 .578 33.6 91.8 Di I methane .844 nil 1.0 50.8 102 Tetra Br ethane .817 nil 1.0 49.7 99.4

(e) Sample C5/C4 (16-12 m)Liquid (1Ev) h ×103 cos Fadh Wadh Water .849 3.45 .270 19.4 91.4 Glyceriene .851 2.40 .506 32.2 95.6 Formamide .859 1.95 .586 34.2 92.6 Di I methane .858 0.10 .982 49.9 100.7 Tetra Br ethane .860 nil 1.0 49.7 99.4

(f) Sample C5 (< 12 m)Liquid (1Ev) h ×103 cos Fadh Wadh


(g) Sample p.a.Liquid (1Ev) h ×103 cos Fadh Wadh

Water .851 4.25 .116 8.35 63.7 Glycerine .781 2.55 .370 23.5 86.9 Formamide .820 1.90 .589 34.4 92.8 Di I methane .782 nil 1.0 50.8 102 Tetra Br ethane .812 nil 1.0 49.7 99.4

2.2 Contact angle measurements on a cylindrical cakeThe Kossen and Heertjes’s method [1] was applied. It consists in performing the

measure of the equilibrium height of a lying drop on a compressed cake (Table 2-3.)

41

The molybdenite cylindrical cakes have approximately a 1.5 cm diameter and a variable height. They were obtained by applying a 7-ton force.

The measure equipment was a small closed transparent chamber that contained the tablet inside with a device that assures it to be laid down horizontally. The measurement was carried out with a cathetometer. A drop of liquid was deposited over the surface of the cake using a syringe, and then the interior atmosphere is saturated with a vaporiser. These steps are repeated until the drop reached to a constant height. The authors of this method suggest doing the final measurement within an hour, in our case we decided for a 3-hour operation to assure the equilibrium conditions. Especial care was taken with the illumination in order to minimise the problems of reflection and background which difficult the observation during the test.

The formula (A-184) from the appendix was applied in the calculations

(2-124)

Defining

(2-125)

where is the liquid density, h is the equilibrium height of the drop, g the acceleration of natural gravity, EV is the factor of packaging of the cylinder cake. In these tablets the packaging factor is simply calculated by:

(2-126)

where M is the pellet mass and is the mineral density and is the cake volume.

M was obtained by weighing, of molybdenite was considered as 4801 kg m-3. The geometric measures of the cake were carried out with high precision instruments in the Metrology Division Department of the Metallurgical School of Engineering (Universidad de Concepción.)

2.3 Contact angle measurements on a massive crystalThe direct method of the laying drop was applied utilising a microscope with a

goniometer. A series of measures were performed in different zones of the crystal before reaching to an ultimate value. The measurements were performed in the aforementioned closed chamber.

42

Table 2-4: Contact angles measurements on a molybdenite crystal(a) Edge plane

Liquid cos Fadh Wadh

Water 662

.407.03

29.3 101

Glicerine 503

.643.04

40.8 104

Formamide 422

.743.02

43.4 102

Di I methane 242

.914.03

46.4 97.2

Tetra Br ethane 152

.966.01

48.0 97.7

(b) Faces plane

Liquid cos Fadh Wadh

Water 846

.105.1 7.56 79.6

Glicerine 633

.454.05

28.8 92.2

Formamide 555

.574.06

44.5 91.9

Di I methane 171

.956.04

48.6 99.4

Tetra Br ethane 0 1.0 49.7 99.4 : contact angle in sxagesimal degrees Fadh y Wadh: Idem before

2.4 Hornby-Leja’s methodSeveral methanol-bi-distilled solutions were prepared. 10 mL of each solutions were

poured in a 25 mL test tube. A little sample of mineral was brought in contact with a water-methanol solution into the test tube. These tubes were tapped with Parafilm and shaken during 30 seconds. After a few hours we observed if the complete wetting of the sample was carried out.

This way the intervals of wetting of the surface tension variable (methanol concentration) were determined.

This interval of wetting was diminished with a series of runs until getting the finest possible determination.

The p.a. Molybdenite and Methylated silica were tested besides the granulometric class samples.

2.5 Quantified wetting methodThis is a new method proposed by the authors of this work. The procedure to be

followed is similar to the Hornsby-Leja’s method. The test tubes utilised in our method were modified with an underflow discharge in order to remove the sunken material and so quantify the process.

43

The sunk fraction is denominates sink and the floated float.

The concentration of methanol (%) in the solution vs. float (%); and the surface tension the solutions vs. float (%) were plotted for every tested sample.

Table 2-5: Measurments by quantified sink method(a) Fraction +C1 ( > 44 m)Float CMet LV [mNm1]6.4 5 63.9

13.0 10 58.0 11.6 14 54.5 1.9 20 50.0 2.0 25 46.9 1.2 30 44.0

(b) Fraction C2/C1 (4433 m)Float CMet LV [mNm1]6.4 5 63.9 13.0 10 58.0 11.6 14 54.5 1.9 20 50.0 2.0 25 46.9 1.2 30 44.0

(c) Fraction C3/C2 (3323 m)Float CMet LV [mNm1]13.9 5 63.9 19.9 10 58.0 10.0 14 54.5 2.6 20 50.0 1.9 25 46.9 0.7 30 44.0

(d) Fraction C4/C3 (2316 m)Float CMet LV [mNm1]13.8 5 63.9 10.7 10 58.0 10.8 14 54.5 9.31 20 50.0 3.1 25 46.9 1.4 30 44.0

(d) Fraction C4/C3 (2316 m)Float CMet LV [mNm1]13.8 5 63.9 10.7 10 58.0 10.8 14 54.5 9.31 20 50.0 3.1 25 46.9 1.4 30 44.0 (e) Fraction C5 ( < 12 m)Float CMet LV [mNm1]18.7 0 72.0 25.7 2 68.0 24.5 5 63.9 8.0 10 58.0 10.7 14 54.5 5.7 20 50

(f) MoS2 (p.a)Float CMet LV [mNm1] 16.1 5 63.9 11.3 10 58.0 2.0 14 54.5 1.2 20 50.0 0.2 25 46.9 0.0 30 44.0

CMet: percentage volume in aqueous solution LV: surface tension of the solution

44

2.6 Hallimond’s tube FlotationFlotation tests were performed in Hallimond’s tube adapted for fine particles (Figure 2-

11). See Table 2-6.

The tests were carried out with the molybdenite granulometric fractions and p.a. Molybdenite.

All the different samples were treated in independent tests and using different water-methanol solutions.

The gas was nitrogen; it was supplied from a pressurised balloon; it entered to a general line that worked as a reservoir. Its flow was controlled and measured with a flow-meter. The gases were washed before entering the Hallimond’s tube with KOH, pyrogalol and water. The tests were performed with an additional open column with water in order to avoid a gas overpressure at the very beginning of all the tests. This column presented a resistance to the flow-gas equal to the resistance produced by the Hallimond’s tube.

A. Collector tubeB. Rubber plugC. Frosted unionD. Magnetic barE. Porous glass stabe F. Magnetic stirrerG. Water levelH. Water trampI. RotameterJ. Valve

Figure 2-11: Hallimond’s tube, modified for fine particles.

45

The experiment was a semi batch process; the operation time was fixed in three minutes. The amount of solid sample was of 2 grams and the gas flow was also fixed constant to 60 cm3/minute. During the operation time the pulp was homogenised by a magnetic stirrer bar.

Table 2-6: Flotation test in Hallimond tube. Recovery percentage obtained for each especies vs. Methanol concentration CMet and it surface tension L

% RecoveryCMet L Test 1 Test 2 Test 3 Test 4 Test 5 Test 6

% (v/v) [mNm1] >44m 44–33m 33–23m 23–16m <12m MoS2 (p.a.) 0 72 62 55 60 56 44 85 2 68 - - - - - 70.7 5 63.9 - - - - - 29.3 8 60 42.1 54 62 56 24 22 10 58 58 66 56 - - - 12 56.2 47 47 55 55 7.8 23 14 54.5 47 - 55 42 60.2 - 16 53 32 52 30 40 32 21 18 51.5 36 57 - 52 26 - 21 49.4 13.4 35 35 41 13 20 25 46.9 31 21 39 25 10 - 40 39 32 38 29 31 28 26 100 23 - 13 13 5 19 -

2.7 Conventional cell flotation of the sample in presence of quartz

Semi–batch tests were performed on a composite sample containing 11 g of each granulometric fraction and 554 g of quartz in order to simulate the presence of a quartz gangue. The solids concentration in the pulp was 16 %.

The only reagent utilised was MIBC (methyl-isobutil-carbinol), as a frother: three droplets. The cell was a conventional Agitair L-500 one.

The samples of the concentrate were removed from the cell at scheduled times (flotation kinetics). The recoveries are calculated through molybdenum analyses.

The flotation nitrogen flow was kept constant.

46

Table 2-7: Flotation test in conventional cell for each fraction in the presence of quartz

% RecoveryTime Test 1 Test 2 Test 3 Test 4 Test 5 Test 6

minutes >44 m 4433 m 3323 m 2216 m < 12 m MoS2 (p.a) 1 - 31.5 57.7 51.0 23.7 18 2 79.1 60.8 67.9 58.6 57.1 53.9 3 83.1 - - - - - 4 - 77.6 81.9 65.3 71.5 62.4 5 84.4 - - - - - 6 - 90.8 84.9 67.2 74.1 63.3 7 85.3 - - - - - 8 - 91.1 84.9 69.3 75.3 67.2

2.8 Flotation of a collective of molybdenum samplesThe same method as described before was followed.

The sample of the concentrates was separated by Cyclosizer in granulometric classes of the same size as those which composed the feed. This allowed us to calculate the recoveries of each fraction through a granulometric analysis.

Table 2-8: Flotation test in a conventional cell of a collective of granulometric samples.

% No flotadoTime Total Test 1 Test 2 Test 3 Test 4 Test 5 Test 6

minutes > 44 m 4433 m 3323 m 2316 m < 12 m MoS2 (p.a.) 1 87.2 86.7 87.2 87.6 86.9 87.2 87.3 2 76.3 75.0 77.2 76.9 75.9 75.6 76.6 4 51.3 53.9 55.1 53.2 50.1 50.4 45.1 8 27.1 31.6 29.6 28.3 25.6 26.6 23.2 16 9.3 11.2 8.92 9.25 9.37 10.3 8.0

47

Chapter 3Results

3.1 Contact angle measurementsThe results are shown in Table 2-3; they were processed according to the previously

described models.

3.1.1 Zisman’s modelThe parameters of this model were obtained using the minimum least square technique

and are shown in Table 3-9. This model fits well according to the correlation coefficients an its plotting (Figure 3-12).

The obtained critical wetting tensions range around a mean value of 49.5 mJ/m2. There is no significant variation between the ’s obtained considering the analytical precision of the experimental data.

The largest absolute values of the Zisman’s slope belong to largest size fractions (<16 µm) and p.a. Molybdenite.

Figure 3-12: Wettability of different types of molybdenite, according to Zisman’s model (A: water, G: glycerine, F: formamide, I: di-I-methane y B: tetra-Br-ethane).

There is a marked difference between slopes values of edge and face surfaces; face surface slope is larger.

Table 3-9: Results after modelling cos vs. LV for each species acording to Zisman’s model [cos = 1 + (c)]

Species c

[mNm1] ×102 Regression

coefficient > 44m 49.9

3.6 4.92 .3 .99

4433m 50.2 2.6

5.40 .3 .995

3323m 50.0 4.5

4.94 .4 .98

2316m 50.3 3.2

4.76 .3 .99

1612m 49.0 5.5

3.36 .4 .97

< 12m 47.9 6.7

3.20 .4 .95

Figure 3-13: Wettability of different types of molibdenites, accordind to Good’s model (A: water, G: glycerine, F: formamide, I: di-I-methane y B: tetra-Br-etane).

MoS2 (p.a.) 50.8 1.5

5.22 .15 .99

Edge 47.9 1.8

2.42 .087

.99

Face 49.3 2.8

4.00 .22 .99

3.1.2 Girifalco & Good’s ModelThis model fits well too, according to the correlation coefficients and its plots (Figure

3-13). In this case, ranges narrowly around a mean value of 50.0 mJ/m2.

The p parameter is larger for the largest sizes (>16 µm) and for MoS2 p.a.

There is also a marked difference between p parameter values of edge and face surfaces; face surface slope is larger.

Table 3-10: Results after modeling cos vs. LV relationship for each species according to Girifalco-Good semiempiric model [ ].

Species c

[mNm1]p Regression

coefficient > 44m 50.2 44.9 .98 4433m 50.5 49.4 .99 3323m 50.3 44.8 .97 2316m 50.5 43.4 .98 1612m 49.7 31.1 .99 < 12m 48.9 29.9 .98 MoS2 (p.a.) 51.3 49.0 .99 Edge 48.7 22.1 .993Face 49.8 36.8 .997

51

3.1.3 Wetting model (Lucassen-Rynders)The correlation coefficient is acceptable for every fitting but little less than the value

we got in previous models (Figure 3-14).

The obtained are very similar for each of the species on study and ranges around 50 mJ/m2 as a mean value. The m slope is quite different for MoS2 p.a. and the largest sizes from the finest diameters. This difference appears too between the m slope of the edge and face planes. Face’s are larger (Table 3-11)

Figure 3-14: Wettability of different types of molibdenites, accordind to Lucassen-Rynders’ model (A: water, G: glicerine, F: formamide, I: di-I-metane y B: tetra-Br-etane).

52

Table 3-11: Results after modelling cos vs. LV relationship for each species according to the Lucassen-Reynders semiempirical model [Fadh = mLV + Cte].

Species c

[mNm1]m Regression

coefficient> 44m 50.5 2.5 .96 4433m 50.6 2.8 .98 3323m 50.7 2.5 .95 2316m 50.8 2.6 .97 1612m 49.7 1.4 .98 < 12m 48.9 1.3 .97 MoS2 (p.a.) 50.8 2.6 .97 Edge 49.1 .77 .96 Face 50.0 1.85 .993

53

3.1.4 Fowkes’ model

This model is suitable for obtaining . The parameters (Table 3-12) are calculated on the basis of the least square technique, they show uncertainty; such as the obtained as an average in each one of the measures.

The value of is larger for the finest sizes (< 16 µm) compared with the largest sizes

and MoS2 p.a. The “edge” surface gets a larger than the “face” surface. On watching Figure 3.4 it could be appreciated that the correlation in “edge” surfaces is not good, although the experimental points are on a straight line this function does not match the point (0,-1) as the Fowkes’ model demands.

Figure 3-15: Wettability of different types of molybdenite, according to Fowkes’ model (A: water, G: glicerine, F: formamide, I: di-I-metane y B: tetra-Br-etane).

54

Table 3-12: Results after modelling wetting conditions for each species according to Fowkes’ model [

].

Species sd*

[mNm1]s

d** [mNm1]

> 44m 50.8 1.6

49.8 4.2

44–33m 48.9 3.6

47.9 6.5

33–23m 51.1 3.7

51.0 6.7

23–16m 52.6 1.2

53.0 2.7

16–12m 55.5 5.7

62.6 1.9

< 12m 54 7 61.6 2.1

MoS2 (p.a.) 51.5 1.4

50.3 3.1

Edge 63.7 30

63 30

Face 54.9 3.4

58.6 10

* Calculated as aparameter of equation: .

** Calculated as an average of each determination

3.1.5 Wu’s model

The and foretold by this model are shown in Table 3.5. The method used to calculate these parameters will be described in Chapter 4.

It may be observed that is very similar for the different models we are applying. However, the smaller the particle is, the larger its polar component is. It could be observed also a larger polar component for the “edge” surface than the polar component of the “face” surface.

All the values this model obtains have a high range of uncertainty.

Table 3-13: Resultados del tratamiento de mojabilidad de acuerdo a la ecuación de Wu.

Species sp

[mNm1]s

d [mNm1]

s [mNm1]

(sp/s) × 1

02 > 44m .75 .3 48.4

1.2 49.4 .4 1.5

4433m 1.8 48.2 50.0 3.6 3323m 2.4

3.1 45.8 2.2

48 .2 5.0

55

2316m 1.8 .6 48.5 2.3

50.3 2.0

3.6

1612m 5 4.3 46.6 1.8

51.6 2.6

9.7

< 12m 5.3 4.8

44.0 1.5

48.7 2.3

11.0

MoS2 (p.a.) .9 .3 49.0 2.8

49.9 2.5

1.8

Edge 9.7 3.8

41.7 .7 51.4 3 18.8

Face 3.4 2.2

45.8 1.3

49.2 .95

6.9

3.1.6 Dann’s Model Dann and Wu’s model are aimed at calculate the polar and dispersion components of

the solid surface energy . The resolutions method was analysed and explained in Equation (1-94).

By observing the results (Table 3-14), it can be noticed that the largest particles and MoS2 p.a. have got a larger dispersion component than the finest sizes. On the contrary, the 12-16 µm and <12 µm fractions have got the largest polar components. The comparison between both “edge” and “face” shows a similar situation to the mineral species ones; having the “edge” surfaces a larger polar component.

Table 3-14: Results for treatment of wettability according to Dann’s equation

Species sp

[mNm1]s

d [mNm1]

s [mNm1]

(sp/

s)×102 > 44m - 51.4 51.4 - 44–33m - 51.2 52.7 - 33–23m 3.4 × 105 51.2 51.2 0 23–16m .025 51.8 51.8 .05 16–12m 2.37 46.4 48.8 4.9 < 12m 2.58 44.6 47.2 5.5 MoS2 (p.a.) - 52.8 52.8 - Edge 6.21 43.8 50.0 12.4 Face 0.76 48.1 48.8 1.6

56

3.1.7 Neumann-Good’s modelA procedure based on the state equation described in Section 1.8 was applied.

The calculated values of and for each of liquid-molybdenite species pair were written down in Table 3-15.

As it might be expected, the larger the liquid surface tension is, the larger the liquid-solid tension becomes; and increases as we get closer to critical-wetting conditions. The calculated values of (solid-vapour tension), (interfacial solid-water tension) and the solid-water interaction parameter are shown in Table 3-16.

The calculated for each MoS2 species are similar.

On the contrary, the calculated for the largest particles are almost nearly twice of the < 16 µm values’. The same happens when we compare “edge” and “face” surfaces

values. The “face” values double the “edge’s” values.

Inversely; the interaction parameter of the finest particles is larger than those of the largest’s. The “edge’s” surface has got also a larger interaction parameter than the “face’s”.

The opposite trends of and is obvious if we keep in mind the constitutive Equation (1-119).

Figure 3-16: (interfacial tension solid/water) as a function of the particle size. According to Neumann-Good’s model.

EDGES

FACES

MoS2 (p.a.)

57

Both parameters and are plotted against the size particle (Figure 3-16 and Figure 3-17).

Figure 3-17: (solid/water interaction parameter) as a function of particle size according to Neumann-Good model.

EDGES

FACES

MoS2 (p.a.)

58

Table 3-15: Results after applying the equation of state (Neumann-Good)(a) Species > 44 m

Liquid SL

[mNm1] SL calculated

Water 60.4 .501 Glicerine 24.0 .808 Formamide 19.6 .843 Di I methane 2.5 .980 Tetra Br ethane* 1.5* .988

59

(b) Species 4433 mLiquid SL

SL calculated


(c) Species 3323 mLiquid SL

SL calculated


(d) Species 2316 mLiquid SL

SL calculated

Water 55.9 .555 Glicerine 22.5 .821 Formamide 17.4 .861 Di I methane* 0.2* .998 Tetra Br ethane* 1.3* .99

(e) Species 1612 mLiquid SL

SL calculated

Water 31.1 .756 Glicerine 18.3 .856 Formamide 16.3 .872 Di I methane .6 .995 Tetra Br ethane* .8* .994

(f) Species < 12 mLiquid SL

SL calculated

Water 29.6 .763 Glicerine 21.0 .832 Formamide 17.8 .857 Di I methane 3.4 .972 Tetra Br ethane* .8* .993

(g) Species MoS2 p.a.Liquid SL

SL calculated

Water 60.4 .520 Glicerine 28.5 .773 Formamide 17.6 .860 Di I methane* 1.2* .990 Tetra Br ethane* 2.3* .982

(h) Species bordeLiquid SL

SL calculated

Water 20.3 .837 Glicerine 8.8 .930 Formamide 6.2 .950 Di I methane 3.2 .974 Tetra Br ethane 1.6 .987

(i) Species caraLiquid SL

SL calculado

Water 43 .657 Glicerine 21.8 .826 Formamide 17.1 .864 Di I methane 2.0 .984 Tetra Br ethane* .9* .993

* wetting complete conditions

Table 3-16: SV, SW, SW determined by the equation of state (Neumann-Good) for each fraction

Species SV [mNm1]

SW [mNm1]

SW

[mNm1] × 103

60

> 44m 51.2 60.4 .516 8.000 4433m 52.1 65.5 .476 8.000 3323m 51.2 59.4 .524 8.000 2316m 51.5 55.5 .555 7.960 1612m 50.5 31.1 .756 7.839 < 12m 50.5 29.6 .763 7.970 MoS2 (p.a.) 52.0 60.4 .520 7.937 Edge 49.6 20.3 .837 8.000 Face 50.6 43.0 .657 7.970

SV: interfacial tension solid-vaporSW: interfacial tension solid-waterSW: Good’s parameter of solid-water interaction: relation parameters SL = SL + 1

3.1.8 Comparison between the values predicted by different modelsIt is illustrative to compare the experimental vs. the predicted values by each

model for different kinds of MoS2; these are shown in Table 3.9. The parameters obtained by different models are introduced in their equations and so getting the fitting value that we must compare with the experimental value. It is difficult to define which model is the best; but; it is possible to asses that the Fowkes’ model does not fit well neither the finest particles behaviour (<16 µm) nor the measures done on the MoS2 massive crystal; especially the “edge” surface.

61

Table 3-17: Comparison of obtained data vs. Predicted for each species and different models(a) Fraction > 44 m

Liquid cos cos calculated Experimental Zisman Good DM Fowkes Dann Wu

Water .127 .087 .046 .045 0.73 .070 .124 Glicerine .430 .337 .301 .288 .368 .376 .369 Formamide .541 .582 .538 .527 .540 .543 .539 Di I methane .959 .956 .962 .979 .953 .966 .952 Tetra Br ethane 1.000 1.000* 1.000* 1.06 1.024* * .973

(b) Fraction 4433 mLiquid Experimental Zisman Good DM Fowkes Dann Wu Water .160 .177 .130 .129 0.91 .058 .070 Glicerine .222 .287 .253 .233 .343 .393 .427 Formamide .605 .557 .512 .492 .510 .563 .600 Di I methane .986 .968 .979 .985 .916 .990 .983 Tetra Br ethane 1.000 1.000* 1.000* 1.000* .986 * .969

(c) Fraction 3323 mLiquid Experimental Zisman Good DM Fowkes Dann Wu Water .114 .087 .037 .035 0.071 .070 .052 Glicerine .319 .338 .310 .299 .372 .374 .430 Formamide .711 .585 .545 .539 .544 .541 .599 Di I methane .914 .960 .969 .993 .959 .963 .947 Tetra Br ethane 1.000* 1.000* 1.04* 1.070* 1.030* * .918

(d) Fraction 2316 mLiquid Experimental Zisman Good DM Fowkes Dann Wu Water .064 .033 +.008 .060 0.057 .035 .068 Glicerine .450 .376 .343 .285 .392 .407 .430 Formamide .578 .614 .572 .532 .567 .573 .604 Di I methane 1.000 .976 .982 1.0* .987 .983 .989 Tetra Br ethane 1.000 1.000* 1.05* 1.08* 1.06* * .976

(e) Fraction 1612 mLiquid Experimental Zisman Good DM Fowkes Dann Wu Water .270 .227 .254 .257 .032 .186 .078 Glicerine .506 .516 .494 .481 .430 .556 .566 Formamide .586 .684 .658 .642 .609 .695 .735 Di I methane .982 .940 .952 .948 1.04* .960 .995 Tetra Br ethane 1.000 .976 1.000* 1.000* 1.11* .932 .936

(f) Fraction < 12 mLiquid Experimental Zisman Good DM Fowkes Dann Wu Water .290 +.229 .248 .262 .044 .182 .076 Glicerine .466 .504 .479 .474 .411 .541 .547 Formamide .560 .664 .637 .626 .587 .677 .709 Di I methane .927 .907 .919 .914 1.013* .927 .943 Tetra Br ethane 1.000 .942 .965 .963 1.09* .895 .878

(g) Sample p.a.Liquid Experimental Zisman Good DM Fowkes Dann Wu Water .116 .107 .067 .06 .067 .058 .113 Glicerine .370 .342 .313 .285 .378 .394 .385 Formamide .589 .603 .571 .532 .550 .563 .557 Di I methane 1.000 1.0* 1.03* 1.00 .966* .992 .970 Tetra Br ethane 1.000 1.06 1.1* 1.08* 1.03* * .986

(h) EdgesLiquid Experimental Zisman Good DM Fowkes Dann Wu Water .407 .417 .438 .437 .038 .407 .247 Glicerine .643 .625 .609 .600 .532 .761 .684 Formamide .743 .746 .725 .718 .723 .893 .828 Di I methane .914 .930 .934 .941 1.187* * .912 Tetra Br ethane .966 .956 .970 .979 1.27* * .825

(i) FacesLiquid Experimental Zisman Good DM Fowkes Dann Wu Water .105 .092 .122 .129 .037 .071 .003 Glicerine .454 .436 .407 .398 .423 .472 .481 Formamide .574 .636 .601 .590 .600 .623 .650 Di I methane .956 .940 .948 .955 1.03* .954 .963 Tetra Br ethane 1.000 .984 1.005* 1.02* 1.1* .968 .918

* wetting critical condition or total predicted by the modelsDM: Wetting model (Lucassen-Reynders).

62

3.2 Techniques where water-methanol solutions were used

3.2.1 Hornsby- Leja methodThe obtained by applying this method are shown in Table 3.10. They are all similar

for the different sorts of Molybdenite except for the largest particle

fraction (> 44 µm), with a and the finest particle fraction (<12 µm) with

a .

Additionally, the Methylated silica is shown in order to observe the behaviour of an almost ideal hydrophobic species.

Table 3-18: Critical methanol in water solution concentrations of wetting determined by the Hornsby-Leja’s for different species

Species Cmet [%]

c [mNm1]

> 44m 2122 49 4433m 1314 55 3323m 1314 552316m 1112 56.5< 12m 89 59.5MoS2 (p.a.) 1314 55Methilade Silica 4849 36

Cmet: percentage volume of methanol in water solutionc: solution surface tension (critical)

63

3.2.2 Quantified sinking methodThe values obtained for each species are shown in Table 3-19. Their plot may be

appreciated in Figure 3-18 (% Float vs. solution interfacial tension) and in Figure 3-19 (% Float vs. methanol (v/v) )

From a concentration methanol value, the float percentage begins to fall in a more or less dramatic way. This point is named or Cmax (Table 3-19 and Table 3-20) and it has got a value between 55 and 59 mNm-1 (14 and 9 % of methanol solution) for >16 µm size fractions, and it ranges between 63 and 66 mNm-1 (5 and 3 % methanol solution) for <12 µm and MoS2 p.a.

(a) (b)

(c) (d)

(e) (f)Figure 3-18: % Float vs. , quantified sink model for the different fractions.

64

The curve falls down to a point where the percent float becomes a constant. This point is named or Cbreak . It ranges in the 48.5 mMm-1 to 50 mMm-1 range (25 and 20 % methanol), for almost all of the granulometric fractions, but it is slightly different for MoS2 p.a.: 55 mNm-1 (14 % methanol solution.)

(a) (b)

(c) (d)

(e) (f)Figure 3-19: % Float vs. % methanol, quantified sink model for the different fractions.

65

Table 3-19: Surface tension associated to characteristic points in the curve of float % vs. LV of the methanol-water solution, according the quantified sink model

Species break [mNm1]

max

[mNm1] J

> 44m 50 57 14.5 4433m 50 57 13.7 3323m 50 59 11.5 2316m 48.5 55 13.2 < 12m 50 66 5.9 MoS2 (p.a.) 55 63 15.9

J: homogeneity index

Table 3-20: Methanol concentration in aqueous solution associated to characteristic points in the float % vs. methanol concentration according the quantified sink model

Species Cbreak

[% methanol v/v] Cmax

[% methanol v/v] > 44m 20 11 4433m 20 10 3323m 20 9 2316m 25 14 < 12m 10 3.5 MoS2 (p.a.) 14 5

3.2.3 Hallimond tube flotation testsThe results we have obtained by this technique are shown in Table 2-6 and they are

plotted for each fraction and MoS2 p.a. in Figure 3-20(f) (Recovery vs. methanol %). It may be observed two characteristic point: or Cb (break point) and or Cf, which are defined in Table 3-21.

The point where recovery falls dramatically is similar in the largest size fractions and it ranges between 53 and 58 mNm-1 (10 and 16 % methanol).

Both the finest fraction, < 12 µm, and MoS2 p.a. show a marked sensitivity to the addition of methanol, that results in a dramatic fall of the recovery.

On the contrary, a point defined as “break up” ranges over a 50 mNm-1 value in the largest sizes but the 22-33 µm fraction this point is not well defined.

The < 12 µm fraction yields erratic answers in its low flotability zone, probably due to the mechanical drag phenomena.

On the other hand, MoS2 p.a. shows a higher flotability than the other species in distilled water.

66

Table 3-21: Hallimond flotation. Values obtained from recovery vs. surface tension (methanol concentration) of the aqueous solution curves

Species break up Cbreak up f Cf [mNm1] [% methane v/v] [mNm1] [% methane v/v]

> 44m 50 20 58 10 4433m 47 25 5313.7 16 3323m - - 55 12 2316m 45 30 57 11 < 12m 55 12 72 0 MoS2 (p.a.) 60 8 72 0

(a) (b)

(c) (d)

(e) (f)Figure 3-20: % recuperation vs. , Hallimond flotation for the different fractions.

67

break up, Cbreak up: Surface tension and methanol concentration at which the flotation undergoes a sudden increasef, Cf : Surface tension and methanol concentration at which the flotation undergoes a sudden increase

3.2.4 Common cells flotation testsThe performed tests for each granulometric fraction in the presence of quartz are shown

in Table 2-7.

Values of and K were obtained (Table 3.14) when these test were fitted according to a first order kinetic model.

The values of the largest-size fractions were the highest. Nevertheless the K values do not show a noticeable trend.

There were not noticeable differences among the different size fractions when a flotation test using only a frother (MIBC) as flotation reagent was performed on a sample composed with a mixture of all the fraction sizes on study (Table 2-8). The tested sample was composed by a mixture of all the size fractions on study.

68

Table 3-22: Flotation in conventional cell in the presence of quartz. Kinetic parameters for each species for the expression R = R(1eKt).

Fraction R K[m1]

> 44m .85 1.154433m .94 .50 3323m .85 .68 2316m .70 .88 1612m .78 .53 < 12m .69 .54

(a) (b)

(c) (d)

(e) (f)Figure 3-21: % Float vs. % methanol v/v, Hallimod flotation for the different fractions.

69

Chapter 4Discussion

The relationship between contact angle and surface free energy in the molybdenite case is of basic-science and technological interest. If a suitable link between particle size and polar and dispersion components of its surface free energy is established, it is possible to obtain a significant advance in order to correlate hydrophobicity with natural flotability; so answering to the hypothesis that postulates the most hydrophilic behaviour to the finest size particles.

The approach of this discussion will include the analyses of the obtained values of the superficial tension and the parameters of the different wetting thermodynamic models applied on studying molybdenite surfaces.

The correlation of these backgrounds; with Hallimond’s flotation-tube tests and regulating solution surface tension; will allow us a better comprehension of the concepts that have caused the present study: the relationship between hydrophobicity and flotability.

4.1 Zisman’s model applicationThe results presented in the section before permit us to learn that is a low sensitivity

parameter in order to express the hydrophobicity differences between all the kind of MoS2 studied. In effect, the values are around 49.5 mJm-2 without exhibiting an especial trend. However, it was noticed that the Zisman’s slope showed a marked trend with the particle size. Similar results in carbons were reported by Parekh and Applan [11]. So, it may be concluded that analysing this slope is more useful than doing it with the ; in order to characterise the relative hydrophobicity of a set of samples.

4.1.1 Thermodynamic Analysis of the Zisman’s slopeFirstly, the Young’s equation will be written in using energetic terms:

(4-127)

where is the adhesion work and is the cohesion work of the liquid.

There is an advantage in writing down this equation this way because it is suitable to be analysed on the basis of the contribution of the different kind of forces that took part in the adhesion work.

If we define as:

(4-128)

on introducing (4.1) it gets:

(4-129)

Now, if the Laskowski and Kitchener concepts [17] are accepted as we did in (1.108) it is possible to write in a general form:

(4-130)

where is the dispersive component of the adhesion work, is the polar component.

Working on (4.4) it gets:

(4-131)

Now, the slope is defined as a function of the polar and dispersive forces. Then:

(4-132)

where is the London forces contribution and the contribution of the polar one to the Zisman’s slope .

Now we get:

(4-133)

(4-134)

Some conditions have to be satisfied in order to develop a consistent explanation about the causes of hydrophobic behaviours.

When the solid has got a low energetic surface and negative. By the way, when polar interactions are coming up at the liquid–solid system the absolute value of drops; obviously; only a describes the situation.

When is satisfied under any condition, slope will be modified at least in qualitative terms. This approach is suitable in order to analyse the effect of the hydrophobicity on the slope. The soundness of these suppositions will demand to introduce and mathematical expressions.

It is advisable to use handy expressions as function of the solid free energy because the relationship between and the hydrophobicity grade are of paramount importance.

Dann’s model gives us two suitable expressions:

(4-135)

(4-136)

On appearing polar interactions in the system, the wetting behaviour will not depend only upon the solid but on the global liquid-solid system.

It is necessary to introduce correlations that links independently and with . The following empiric relationships have been found out over 50 mNm-1 for the particular liquids set used in this work (Figure 4.1):

(4-137)

(4-138)

where and are positive values from the absolute values of the slopes shown in 4.1 Figure. C1 and C2 are intersection points on extrapolating the straight lines fitted with X-axis ( ). The particular values are = 1.2, C1 = 91.7, = 2.2 and C2 = 49.96.

By deriving the (4-133) expression and applying the (4-137) correlation it gets:

Figure 4-22: Dispersion component Ld and polar L

p of the liquids vs. surface tension of the liquid L (A: water, G: glycerine, F: formamide, I: di I methane

and B: tetra Br ethane)

73

(4-139)

Where

(4-140)

and taking into account (1-37):

(4-141)

according to (4-137); :

(4-142)

on replacing into (4-139)

(4-143)

(4-144)

and finally

(4-145)

The same procedure may be applied on ; joining (4-134) with (4-136) it gets:

(4-146)

Where

(4-147)

according to (4-138),

(4-148)

it comes out

74

(4-149)

Finally, taking (4-136) and (4-138) into account

(4-150)

Looking through the and (4.19) and (4.24); valid for ; it is

noticed that they are proportional to and respectively. If we also give the constants its numerical value is negative while is positive.

Now, some especial cases will be analysed:

1. A solid –liquid system where both faces interact only by London’s forces.It will be applied to low energy surface solid and liquid . So,

and ; therefore and .

For this especial case (4-139) is:

(4-151)

Then

(4-152)

that may be written down as:

(4-153)

It may be noticed that varies with the liquid superficial tension, so the slope will not be constant and a straight line is not possible.

However; if we applied (4-152) to the wetting critical condition, where both the adhesion work (Wa) and the cohesion work (Wc) are equals, we arrive at the Good’s relationship which was demonstrated in the field of low energy solids and liquids.

2. The solid interacts only by London’s forces while the liquid interacts by both dispersion and polar forces.This may be applied on low energy solids when they are brought in contact with

liquids.

The follow conditions are fulfilled: and .

75

The expression obtained was (4-144). If we take into account that de adhesion

work will have only a dispersion component and that for critical

wetting conditions the adhesion work equals the liquid cohesion work, we get:

(4-154)

The slope defined this way is practically a reference state in order to study the loss of hydrophobicity when “ionic” sites appear onto the surfaces.

3. The most complete case; and .This was developed from (4-139) to (4-150).

The expressions for and will be (4-145) and (4-150). The quotients

and are plotted in Figure 4-23 and Figure 4-24 against the tension of the

liquid in order to obtain a relationship independently of the and of the solids on study.

On introducing (4-135) and (4-143) we are giving a thermodynamic meaning to the Zisman’s slope. It quits as a function of and , however; the function drifts away from the linearity Zisman claimed for.

This inconvenient does not allow us to make a strict interpretation of the Zisman’s slope. However, it is quite clear that the occurrence of little polar components on the surface of a hydrophobic material, such as molybdenite is, will modify the slope of a plot.

Figure 4-23: Effect of the surface tension of the liquid LV on the polar component of the slope, (s

p)

76

It will be interesting in the preset work to evaluate (4-154). We call it reference slope value of a hydrophobic solid which only interacts through dispersion forces, in a liquids set with dispersive and polar components.

When the numerical values are introduced in this equation; for the present case it gets:

(4-155)

This values is the greatest of all values obtained in this work for the slopes of different species of molybdenite, whose slope maximum absolute value was

.

4.1.2 Zisman’s slope analysis based on Girifalco-Good’s ModelIt was demonstrated that in a previous section. This was done based on the

Good’s model when its mathematical expressions are developed in Taylor series around the critical wetting conditions.

The independence of from was assumed.

Now, if the same development is carried out until the first order term it gets:

(4-156)

where sub indices or supra indices ‘c’ refer to critical conditions.

Figure 4-24: Effect of the surface tensions of the liquid LV on the dispersion component of the slope (s

d)

77

On equalising with Zisman’s expression (1.29) and knowing that for critical wetting conditions relationship it gets:

(4-157)

We observe that if does not vary with we get .

Now,

(4-158)

When the state equation was applied in the calculation the equation (1.119) was obtained; as linear function of . This allows us to write:

(4-159)

where both factors of the right member are positives.

As the solid has less affinity with liquids, it says, it is more hydrophobic, will

increase.

It gets:

(4-160)

When we are working on solids that do not have the characteristics of the low energy surfaces, the Zisman’s slope will be modified as the hydrophobicity increases.

This relationship is useful in the case we are studying; molybdenite with a . The aforementioned expression yields .

The same way while (4-154) belongs to a solid that interacts only trough dispersion forces with a liquid that has got polar component; corresponds to the Zisman’s slope of a solid in its minimum of hydrophobicity state among different kinds of surfaces of specie with the very same .

A similar situation to the molybdenite was described in a paper about coal wetting [35].

Here it is observed too that the slopes are lesser in absolute value than that we named reference slope ; it says the maximum hydrophobicity state; and slope values close to

for a sub-bituminous specie and for a lignite that are the lest rank species (56.1% and 46.2%, respectively).

78

4.1.3 Zisman’s slope analyses for measurements with alcohol-water solutions

The interpretation of the Zisman’s slope obtained from measures performed using water-alcohol solutions has not been established yet. Nevertheless, several authors give an interpretation based on the absorption phenomena that reports the m slope of the wetting model; as we marked in (1-102).

If we express the equation model as:

(4-161)

and we develop in Taylor’s series from , it gets:

(4-162)

As

(4-163)

it results:

(4-164)

On truncating the develop in the first order term and equalising it with Zisman’s expression; it gets:

(4-165)

There is preferential adsorption in the solid-vapour interface if m > 0, [21]. Then if

(4-166)

Then, if in a Zisman’s diagram for water- alcohol solutions, it is a possible

indication of adsorption.

When implies , that is the Good’s relationship he claims to be valid in the proximity of for low energy solids in front of pure liquids.

4.2 Good’s Model ApplicationThis model that correlates linearly gives in similar results of critical

wetting tension to the other models (Figure 3-13).

The p slope of this model may be associated to more or less grade of hydrophobicity (Figure 6.1b) as the Zisman’s slope does.

79

4.3 Wetting Diagram (Lucassen-Reynders) Applications When experimental data were treated using this model, it might be possible to establish

a suitable correlation between (Figure 3-14), this means a low equilibrium pressure ; therefore this pressure might be considered roughly as a constant in the measure range [39][44].

Again, there is a likeness between the values obtained of different granulometric fractions, MoS2 p.a. and crystal surfaces, besides; there is also a likeness with the values obtained using other models (Table 3-11). Nevertheless, the difference is given by the slope m (Figure 6-28), this is obvious if the relationship between Zisman’s slope and m is taking into account according to (4-165).

A linear model has been chosen but we could have adopted a second order one. This will take the advantage of showing whether or not our system gets closer to the low energy solids behaviour, where for the adhesion tension has got an analytical

maximum. Unfortunately, the liquid pure set is not enough in the zone; so the parabolic model would not have whatsoever, the necessary freedom degrees to describe this situation.

4.4 Fowkes’ Model ApplicationIf the results obtained with this model are compared it is noticed that the less the

particle size is the more the dispersion component is (Table 3-12). This not only

quantifies but is showing the poor versatility this model has. It may be noticed that only one parameter is not enough to describe those conditions whose wetting critical conditions are alike but the water-solid contact angle are different.

Fowkes attributes these deviations to the existence of large values of equilibrium and to all little dirty, rough and heterogeneous composition surfaces.

All seems to show that it is not satisfactory to express the adhesion work only as a function of London’s dispersion components. Laskowski et al. have claimed for incorporating polar components to the adhesion work; see (1-108). Thus, the idea of considering Wu and Dann’s Model has come up.

4.5 Wu’s Model ApplicationThe results show a similar for the different size classes, MoS2 p.a. and crystal

surfaces (Table 3-13).

80

The proportion of tension components signals the difference. So, the quotient is chosen as polarity index; see (A-185).

(a) (b)

(c) (d)

(e) (f)

(g) (h)

(i)Figure 4-25: s

p and sd obtained by means of a graphical/numerical, Wu’s model, for the

each fraction

81

The highest polarity indices are observed in both “16-12 µm” and “<12 µm” and a very low one in the “MoS2 p.a.”.

When we compare the results between the two different of molybdenite crystal surfaces we realise that the edge surface has a higher polarity than the face surface.

The results of working with Wu’s equation are shown in Figure 4-25; they were calculated using a graphical-numerical method.

Each contact angle measure generates a hyperbole in the plane as we have written down here:

(4-167)

where .

In order to carry out the calculations only is useful the hyperbola branch between the asymptotes:

(4-168)

and

(4-169)

in the region of the plane where both, and are positives.

Thus, the solution will be given in the intersection values of the hyperbolas that each measure generates.

It may be observed that the obtained solutions are different for each pair of liquids; this signals that a marked dispersion in the found values is going to be produced. It is preferable to obtain solutions using the intersections of the hyperbola generated by Methylen Iodide because this is the best defined curve and the closest one to the critical wetting conditions.

4.6 Dann’s model ApplicationThe same conclusions we obtained in the Wu’s model Application may be expounded.

The obtained values are shown in Table 3-14 and a noticeable polarity difference may be observed according to the particle size of the granulometric classes and the MoS2 p.a. that trends to behave as the largest size granulometric surfaces. Besides, a different polarity is observed between the edge and face crystal surfaces

When we try to calculate the dispersive and polar components applying these both models: Wu’s and Dann’s we come up against the lack of precision that the dispersion and polar components of the utilised liquids have (Table 2-2).

82

Both, the and values obtained used to have high uncertainties, specially . These uncertainties could be higher than the resolution uncertainty.

4.7 Correspondent states equation This kind or treatment was proposed by Wade and Neumann [39]; it authorises to set

out thermodynamically relationships between and further away beyond the Young’s equation. This may be applied to the study of minerals surfaces as molybdenite.

Previously it might take into account that there is a good correlation between (wetting diagram) for each variety. This condition is necessary in order to

assure that is constant, it says, independent of the liquid that forms a contact angle with the solid.

When was calculated for each granulometric fraction, MoS2 p.a., and massive crystal, their values were almost equal (Table 3-16).

So, the hydrophobicity may be estimated calculating and for the water-surface solid system. Amongst the granulated classes the highest values of indicate less affinity between solid and water (Figure 3-16). On the contrary, the least hydrophobic fractions have the highest (water-solid) parameter, showing this way more affinity for water (Figure 3-17).

4.8 Hornsby- Leja’s methodA certain correlation is observed between and the particle size. The value of the

middle-of-the-range fractions are alike to those reported by Garshva [10] for the immersion critical condition ( ). Coincidence was almost complete for the

methylated Silica case ( ).

The determination was very precise because its surface is very homogenous, it does not happen the same with Molybdenite species.

The behaviour of the MoS2 natural species is owing perhaps to a definite heterogeneity of its superficial properties; even between the same granulometric classes particles.

The Hornsby-Leja’s method actually gives a modified critical wetting condition owing to the air bubble formed during the previous agitation, producing this way a certain foam or froth. This may explain the similarity between our wetting results and those reported by Garshva.

4.9 Quantified sinking methodThis method is similar to the Hornsby Leja’s; but this case both, floated (F) and sunk masses are quantified.

Owing to the heterogeneity of the minerals species on study this new method was designed. It may be established as a critical wetting condition , the situation when the floating masses percent is almost null.

83

This way, the quantification allows perhaps a better definition of the wetting and also determining the dispersion conditions. The Hornsby Leja’s method produces an uncertainty in the value owing to the heterogeneity of natural materials. This has just happened in Molybdenite case. When the wetted and sunk masses are quantified it is possible to define any sort of criterion in order to precise much better the value. Besides, the wetting distribution may help us to understand both the surface sites of dispersion and polar of the same specie.

A break-up point (Fb, ) more or less well defined gives similar values to those obtained by contact angle method, but the synthetic specie. The ; where both the float fraction (Fmax) and the dispersion conditions attain its maximum (Figure 3-18 and Figure 3-19).

The kindness of the curves allows us to introduce a parameter of sensitivity J between the surface tensions and their respective float percentage.

(4-170)

This index might show the grade of homogeneity J of each species related to the wetting properties; and going the higher index to the more homogenous surfaces (Table 3-19).

On applying this index; MoS2 p.a. surface was the most homogeneous and on the other side “<12 µm” fraction surface was the most heterogeneous. This conclusion is in agreement with what we were just expecting because the finest fraction “<12 µm” has been subjected to different comminution mechanisms.

4.10Hallimond flotation tubeThe flotation of molybdenite using water alcohol solutions was carried out in order to

find a link between flotation, wetting conditions and the hydrophobicity established by the contact angle models.

It was observed that flotation with distilled water produces a high recovery on MoS2

p.a. and medium particles fractions but a low recovery for the “<12 µm” fraction.

When the surface tension decreases the recovery of the largest fractions keep on constant, while both “<12 µm” fraction and MoS2 p.a. low their recovery dramatically (Figure 4-25c and Figure 4-25d.) We have defined previously the flotation as a probabilistic phenomenon (1-123) including probabilities of hydrodynamic and thermodynamic origin.

The relative weighting of these factors seems to depend upon the characteristics of the liquid medium; if the flotation is carried out in water the recovery obeys to the hydrophobicity scale observed in the contact angles measures and the process happens mainly under thermodynamic control.

However, when the surface tension diminishes, flotation remains subjected to the dynamic conditions of the system because the suitable thermodynamic conditions of flotation have been spoiled.

84

Therefore, the drop in the recovery of both MoS2 p.a. and “<12 µm” fraction may be explained because they are the less advantaged form the hydrodynamic point of view (Figure 4-26 and Figure 4-27.)

A has been also defined; it is the point when the specie floats by dragging ( ) and a is the point where flotation drops dramatically. These points indicate a largest sizes trend to float and dragging when is in a range between 45 to 50 mNm-1. Both, “<12 µm” and MoS2 species do not produce these parameters well defined because their ill-dynamic-condition to float (Figure 3-21e and Figure 3-21f) . It may be noticed that the point where the dramatic fall in recovery begins ( ) is near to reported by Garshva [10].

Figure 4-26: Hallimond flotation. Recovery % vs. surface tension of the solution LV curves

85

Whereas, the point where the recovery begins to be performed by dragging is similar to wetting. Qualitative explanations have been given on these three latest methods behaviour and mainly in Hallimond’s flotation. Being the flotation away from the equilibrium conditions, it is not convenient to get from its information concentrate thermodynamic parameters related to free energies changes ( , wetting, immersion et cetera.)

86

It is convenient to determine these parameters by other methods (contact angle, et cetera) and then to explain their influence on such phenomena, if this is possible and reasonable from the thermodynamic point of view.

4.11Conventional flotation cellPlain tests were performed in order to establish links with the classical flotation; the

only reagent utilised was an alcoholic frother. When the different fractions were floated independently and removed form a quartz gangue, the ultimate recovery parameter (R) seems to be higher in the larger sizes fractions.

Where as, on floating a collective of these samples all the R parameters obtained are very similar.

Figure 4-27: Hallimond flotation. Recovery % vs. methanol v/v % of the aqueous solution

87

Furthermore, the nearer to the real conditions the systems are, the more difficult is to extract conclusions about the relationship between particle size and both, hydrophobicity and flotability.

88

Chapter 5System integral analysis

It was demonstrated that it is possible to detect the particle size effect associated to the hydrophobicity by using a measure method of contact angle on pressurised pellets. This is the main achievement of the present work, so the Kossen and Hertjes’ method is authorised as a tool to be applied on natural hydrophobicity systems.

Both, mineral molybdenite and synthetic MoS2 exhibit a marked natural hydrophobicity in agreement to literature reports. This is clearly shown in the measures in all, cos measurements, Hornsby Leja’s wetting technique and flotation test using methanol-water solutions.

The crystallographic character of molybdenite determines the presence of both polar and no polar superficial sites, being the non polar which interact only by London’s dispersion forces. This determines a inherently anisotropic surface from the point of view of its hydrophobicity.

Thermodynamically, the surface free energy of molybdenite will be given by the addition of both polar and dispersion contributions.

The surface free energy value for molybdenite is associated to with the definite predominance of dispersion forces over polar ones.

On the contrary, the little fraction of polar components determines a change in hydrophobicity that is associated to the particle size or to the cleavage plane in the crystal case.

So, the analysis of contact angle measures, processed through the different models treatment, indicates a wetting condition similar for all the species on study. Whereas, it is shown a higher polarity in the less size particle fractions and also in the “edge” sites compared to the “face” sites of the massive crystal.

These conclusions strongly bear out Fuerstenau’s hypothesis which claims that the fine particle size fractions hydrophobicity has been spoiled by the occurrence of polar sites.

The origin of them would be associated to the way the fine particles have been generated. So, the produced particles by comminution would have more probability of being broken in their covalent bonds Mo-S then, there must be an increase in polar sites occurrence on their surfaces. Another superficial phenomenon may explain these facts such as the crystalline deformation by collision without being associated necessarily to bonds breakage [9].

On the other hand, MoS2 p.a., a species prepared by chemical synthesis, shows a high hydrophobicity behaviour, similar to crystal low energy planes; this suggests an absence of polar sites.

We learned that the relative hydrophobicity of the different studied samples is better correlated by both the Zismam’s slope and the wetting diagram than the obtained.

Despite the validity of applying some models used in this work may be debatable, it must be stressed that generally the gathered information for molybdenite is coherent and permitted a better characterisation of the samples. Some of the more conflictive suppositions on applying these models are: adsorption absence (e =0), clean, homogenous and smooth surfaces, and finally the forces components of the superficial sites are independent and there is not any interaction amongst them.

The value obtained for molybdenite in this work ( 50 mJm-2) is strongly in disagreement with the dates reported elsewhere in literature. . Some authors have reported a for molybdenite and for synthetic molybdenite. Both were obtained using water-methanol solutions and applying the gamma flotation method [45]. Other authors have reported similar values using the same technique [32].

Hornsby and Leja pointed, when they exposed their work [12] on selective separation of hydrophobic solids: “The measure of the wetting critical tension may depend significantly upon the kind of surfactant used to prepare the aqueous solutions series”.

It is not evident either, that the water-methanol solutions do not adsorb onto the solid surface, not modifying this way the estimation of , whatsoever. On the contrary, methanol adsorption on low energy surfaces has been detected [33] [34].

We have mentioned previously that different graphite value have been got depending whether both alcohol and amine-water solutions [28] or pure liquids are utilised [11].

Measures performed on a molybdenite disk [32], and a molybdenite specie both using methanol-water solutions are precedents that point out the methanol adsorption on molybdenite surface. The first case has a positive slope in the Lucassen-Reynders’ model so indicating a preferential adsorption in the solid-vapour interface in agreement with (1-103). This adsorption will be higher than the solid liquid adsorption one and at least comparable with that of the liquid-vapour [46][47]

There are not precedents that report studies of wetting in molybdenite surfaces using pure liquids but water [6][7].

The present work is a contribution on the subject because the pure liquids in the contact angles measures were taken into account. This has a clear advantage over the use of mixtures because in the later case the interpretations will not be valid whatsoever “until a correct theory of the thermodynamic of wetting for mixtures had not been elaborated” (R. Good, 1977) [23].

Another contribution has been developed by the quantitative sinking method. This method allows assessing that the behaviour of the sinking and wetting is heterogeneous even inside a granulometric fraction.

The relative hydrophobicity scale of the different fractions serves to explain the behaviour of these very fractions, this added to the assessment of the deterioration of the thermodynamic probabilities of adhesion for fine particles, fulfils the probabilistic scope of flotation that this work is claiming for.

Table 5-23: Bibliographic records of wetting determinations carried out on different molybdenite species.

Obtained parameters

Value Technique Variety Ref.

Contact angle in water

80.5

Laying drop Molybdenite crystal [6]60.5

c of wetting[mNm1]

26Flotation and contact angle in water-alcohol solutions

MoS2 (synthetic)[13]29 MoS2

31 Mo1xS2

29.2 Contact angle in water-methanol solution

Molybdenite in cleavage plane [32]

41.7c of sinking

[mNm1]55 Sinking time in aqueous

solutionsMoS2 p.a. [10]

60

91

Chapter 6Conclusions

1. Different granulometric fractions of mineral molybdenite, synthetic molybdenite sulphide both p.a. and both “face” and “edge” crystals, and edge showed the same value of obtained by measurements of contact angles using pure liquids of lesser surface tension than water.

2. The idea of correlating hidrophobicity with particle size by means of a revealed this parameter is not sensitive enough to this purpose.

3. A more elaborated treatment of data, applying well known models, allowed us to determine the components of free energy surface components of Molybdenite. Thus varies in range from 41.7 to 48.5 mJm-2 (Wu’s model); 43.8 to 52.8 mJ/m-2 (Dann’s model) and from 48.9 to 63.7 mJm-2 (Fowkes’ model). On the other hand, varies between 0.75 and 9.7 mJm-2 (Wu’s model) and between 0 and 6.21 mJm-2 (Dann’s model).

4. A complementary analysis of theses models allowed us to conclude that the slope of some of them is a parameter which suitably correlates to the relative hydrophobicity of the samples. A thermodynamic interpretation of Zisman’s model was elaborated, giving so a well defined theoretical sense to the observations and fulfils the use of this slope as a relative hydrophobicity criteria of a set of samples.

5. Most of the experimental techniques to measure (except for the contact angle methods, such as Hornsby-Leja’s method and both microflotation and immersion in methanol-water solutions) failed to precise a well defined value due to the existence of and heterogeneous surface behaviour.

6. A new technique of quantified sinking are put forward in the present work. This allows us to define an anisotropy index of wetting which must be of paramount importance in mineral science.

7. It is a well known fact that fine particles are quite hydrophilic but from the previous arguments in molybdenite case; they keep on being hydrophobic enough to perform natural flotability behaviour. So, we suggest that the loss of thermodynamic probability of flotation would be smaller than the loss of hydrodynamic probability; due to the difficulties of adhesion between fine particles and large bubbles.

(a) (b)

(c) (d)

(e) Figure 6-28: Slopes of the different models as a function of particle size. (a) Ziman’s slope

vs. particle size (b) Good’s model slope vs. particle size (c) Slope of wetting diagram vs. particle size (d) cosθ in water vs. particle size (e) Good’s model vs. particle size.

Appendix

Contact angle measurementsThere are several methods in order to measure contact angles, such as: the vertical

cylinder, capillary methods, the slanting plate and the lying or hanging drop or bubble.

Generally, these methods are of direct measure of the angle or through trigonometric relationships measures of the shapes the bubbles have got [20].

In our case, when the contact angle measures must be performed on fines particles lesser than 50 µm size, the choice of the methods are restricted precisely by this characteristic.

The contact angles measures should be indirect in these conditions; they must be associated to a capillary rise time [48] or geometrical dimensions. This later method must be utilised in our study not only for the material porosity but having into account that a drop in equilibrium on a porous solid rises to such a dimension that the gravitational forces should not be excluded whatsoever when a mechanic or energetic balance is performed.

The chosen method (Kossen-Heeertjes, 1965) [1] has got a theoretical based on all three, the Young, Laplace and Padday’s equations.

In order to obtain a final equation of the aforementioned method some aspects must be analysed:

1. The relationship between and the shape of a drop that has been brought in contact with a smooth surface.

This case, the different energies of the system included the potential one should define .

The drop shape on study will be a kind of cylinder with it lateral surfaces much lesser than the basis’ [49].

Applying an energy balance of virtual displacements when an infinitesimal amount of liquid is added we have:

(A-171)

The potential energy EP is:

(A-172)

Where V is the drop volume, h is the drop height, is the liquid density and g is the gravity acceleration.

The height h keeps on constant in the process, so:

(A-173)

Being dV = hdA. In this conditions dA = dALV = dASL ; where the subscripts define both interfaces; liquid-solid (SL) and liquid-vapour (LV).

Then it gets:

(A-174)

If Young’s equation is guessed:

(A-175)

Where

The (A.5) formula is utilised to measure contact angles on smooth surfaces when the mass drop acquires paramount importance and is difficult to be measured directly [49].

2. The same situation as 1; but in the presence of rough surface.

This case, the energy balance must consider the surface porosity since the measure will be performed on a pressed pellet.

The liquid that is brought in contact with this kind of surfaces will not form a steady drop but the pellet will suck it up. When the liquid saturates the surface a well-defined shaped drop is formed; so; its height can be measured.

The Young’s equation has the same form when is applied to a porous solid but the

and will be:

(A-176)

(A-177)

Where ES is the superficial porosity defined as the wetted surface fraction.

The Young’s equation quiets:

(A-178)

Where is the apparent contact angle of liquid onto a porous surface.

On replacing the first factor of (A.8)

(A-179)

By analogy with (A.5)

(A-180)

And replacing in (A.9)

(A-181)

According to Kossen and Heertjes [1] Es and EV (volumetric porosity) are linked by:

(A-182)

Replacing (1 – Es) in (A.11) gives a quadratic equation:

(A-183)

And resolving yields:

(A-184)

If B h2 < 0.1; (A14) may be simplified as:

(A-185)

It must be noticed the characteristic of the drop height (h); it is the ultimate height the drop reaches after the surface solid is saturated and one of the curvature radii becomes infinite.

Both, this method and the capillary rise are the most applied in liquid-powder cos calculations.

97

Addenda and NotesThese are responses to questions raised by the Committee at the time of the oral

presentation of the work:

Section 1.7, page 21The s is a constant. On the contrary c is not a constant; generally it varies according

to the different series of aqua alcoholic solutions, glycerine mixtures et cetera that are used in its determination.

Section 2.5 The question of whether the comminution enhances or not the loss of hydrophobicity

is precisely the subject of the present work.

Generally, the behaviour of fines particles are based on the Fuerstenau’s postulates summarised by him in a modified figure of the Klassen’s original work.

This has been completed with the “edges & faces” theory that has been built by Gaudin, Miaw and Spedden. This theory was useful in order to give an staight and plain explanation of the loss of fine particles in a flotacion process; they claim a greater hydrophilicity and superficial energy for these sort of particles. This properties have not been confirmed until nowadays or tested experimentally. Then, it was necessary to use laborious methods as Kossen and Heertjes’ in order to be applied without changing the samples sizes.

It must be stressed that Trahar and Warren do not accept the Fuerstenau’s postulates whatsoever; they employed arguments based on theoretical developments of the given explanations and certain observations made on experiments. They impugned the “Kelvin effect” as an explanation of the loss of hydrophobicity because it is only valid for particles sizes under 0.1 mm.

Similar concerns were reported about the comminution processes: The largest particles; do they have more high energies sites than small ones because they have been subject to impacts which did not result in breakage?

Until now Flotability and hydrophobicty have been taken as synonyms especially when these properties are inherent, and the surface has a low eneregy. On the contrary, Trahar and Warren claim for separating both properties. This concept influenced on the execution and conclusions of the present work.

We introduce here a paragraph of these authors about this subject: “However , it may be argued just as convincingly that large particles have more high energy sites than small ones because they have been subject to impacts which did not result in breakage” (W.J Trahar and L.J. Warren, 1976. The flotability of very fine particlesa review. Int. Journal of Min. Processing, 3:103–131, 1976.)

Section Conclusions, page 102This technique aims to be justified in the Discussion chapter. It has been described only

the operative part during the development of the method. Besides, we considered so boastfully to develop it in the same item where we explain methods claimed by first line authors.

This method ought to have a stricter justification. It must be noticed that we have avoided to put the b on an equal footing with the thermodynamic concept of c (wetting critical point) and max with i (immersion critical tension).

values

In the finest fractions, the higher the values of were in the finest fractions the greater their statistical dispersion values were.

This behaviour may be caused by the rift between the conditions the model states. This case, the model establishes that a straight line passes by the (0, -1) coordinates point.

In some ways it is true that we cannot propose differences into from a statistical point of view; but the observed dispersion and the forced change in the slopes could indicate nonetheless that we should turn to ; either another kind of interaction or another models.

valuesThe obtained values come from the average of the different determinations. We may

presume a difference, from a statistical point of view, in the polarity between the edge surfaces and the face ones in a massive crystal (9.7 ± 3.8 mJ/m2 vs. 3.4 ± 2.2 mJ/m2). We may assume there are not any significant differences between the values of the different granulometric fractions from a strict statistical point of view.

This observed statistical dispersion reminds us the Girifaco and Good’s advices when they criticise the models which use the forces decomposition; these forces should not be independent of the system they interact with.

Variations between the obtained in this work and the previous ones.This was analysed in Chapter 5 “System integral analysis”, basing on concepts of

Good, Fowkes and the determinations reported by Ottewill and Murphy. Besides we show a graph elaborated by Dann who worked in this subject and calculated the dispersive components of binary systems as alcohol-water solutions series et cetera.

For example; if we have a solid whose wetting critical tension ( ) is 50 mJ/m2

determined with normal-hydrocarbon-liquid series it will get a 36 mJ/m2 value for c with the ASTM mixtures and 28 mJ/m2 value for c with ethanol-water solutions series (see Figure Addenda-29.)

Figure Addenda-29: Critical surface tension conversion curves: A, pure hydrocarbon liquid series; B, mixed glycol, polyglycol, and ASTM liquid series; C, ethanol/water series.

[Extracted from Journal of Colloid and Interface Science, Vol. 22, No. 2, February 1970]

101

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106

NomenclatureVariable Description A Area B g / 2L

B Associated parameter to the Born’s repulsion forces (Lennard-Jones equation)

C Associted parameter to van der Waal’s forces (Lennard-Jones equation) Cte Constant C1, C2 Constants in (4-137) and (4-138) E Internal energy1 Ev Packaging factorEs Surface porosity of the wetted areaF Forces of a Lennard-Jones potentialFi percent of mass floated in a liquid of superficial tension “i” Fadh Adhesion force or tensionG Free energyH EnthalpyI Ionisation energyJ Homogeneity index in (4-170)K Spreading coefficientMab Attraction constant between ‘a’ and ‘b’ phasesN Constant in (1-72)P Pressure Padh particle-bubble attaching probabilityPCH Collision probabilityPes Stabilisation probabilityPF Flotation probabilityR Ideal Gases Universal constantR Ultimate recovery S EntropyT TemperatureV Molar VolumeW WorkWadh Adhesion workWc Cohesion workg Gravitational accelerationfi Surface fractionh Height of the droph Planck constantk Specific rate constantk Boltzmann’s constantm Lucassen-Rynders’ slopen Number of moles p Girifalco- Good’s model slopeq Heat adsorbed by the system

r Interaction distancer Rugosity factor Zisman’s slope Polarisation constant Parameter in (4-138)0 Relationship parameter between SL y SL in (1-119) Surface tensiond Dispersive (London) Componentsp Polar componentsSV Solid vapour interfacial tensionSL Solid-liquid interfacial tensionLV Liquid vapour interfacial tensionc Critical tension of complete wettingi Critical tension of immersioni Surface excess of adsorptionSV Solid-vapour interface adsorptionLV Liquid vapour interface adsorptionSL Solid liquid interface adsorptionW Difference between two values of variable W Contact angle Apparent contact angle Chemical potential Dipolar moment Parameter in (4-137) 0 Fundamental frequency of electronic vibratione Adsorption equilibrium pressure in the solid Density Interaction parameter (Girifalco-Good’s model) Relationship parameter between SL and SL in (1-119)

Documents

Particle Size Effect on the Hydrophobicity and the Natural Floatability of Molybdenite