Int. J. Miner. Process. 56 Ž1999. 1–30
Surface forces and measuring techniques
J.C. Froberg
¨
a
a,b,1
, O.J. Rojas
c,2
, P.M. Claesson
a,)
Laboratory for Chemical Surface Science, Department of Chemistry, Physical Chemistry, Royal Institute of
Technology, S-100 44 Stockholm, Sweden
b
Institute for Surface Chemistry, Box 5607, S-114 86 Stockholm, Sweden
c
Escuela de Ingenieria Quimica, UniÕersidad de Los Andes, Merida 5101, Venezuela
Received 22 March 1998; received in revised form 3 September 1998; accepted 27 November 1998
Keywords: surface forces; double-layer forces; hydrophobic interactions; flotation; mica; surface force
apparatus
1. Introduction
In this chapter we are concerned with interactions between particles, including air
bubbles. Such interactions occurring across aqueous solutions are of special interest
since they determine phenomena in mineral processing, including froth flotation ŽLeja,
1982; Schulze, 1984.. The most important types of colloidal forces will be introduced
and discussed together with short descriptions of the most commonly used techniques
for studying such interactions. This field is large and subject to extensive research and
the account given here is therefore necessarily very concise. The interested reader will
find more extended treatments in some of the books and research articles referred to in
the text. Some results obtained from surface force measurements will also be provided in
order to illustrate what can be learned from such studies.
In flotation we are dealing with complex systems comprising, in addition to the
minerals to be separated, the gangue or non-mineral containing material, other suspended solids, electrolytes, and dissolved and adsorbed polymers. The object of the
)
Corresponding author. Institute for Surface Chemistry, Box 5607, S-114 86 Stockholm, Sweden. Fax:
q46-8-20-89-98; E-mail: [email protected]
1
Fax: q46-8-20-89-98; E-mail: [email protected].
2
Fax: q58-74-402957; E-mail: [email protected].
0301-7516r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.
PII: S 0 3 0 1 - 7 5 1 6 Ž 9 8 . 0 0 0 4 0 - 4
2
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¨
flotation process, to separate unwanted material from the desired product, is usually
facilitated by introduction of additives like surfactants and polymers to the already
complex system. These components interact with one another in a variety of ways. This
leads to formation of various molecular aggregates Žmicelles, polymer–surfactant complexes, etc.. and to adsorption at air–liquid and solid–liquid interfaces ŽFig. 1., which
affect the interactions between the constituents in the flotation system.
The study of interaction forces is vital in order to comprehend, control and optimise
the process. However, as will become apparent in the following, interpretations of
measured interactions are a difficult matter even in relatively simple model systems. It is
advisable to keep this in mind when applying results obtained on model systems to a
real situation where even what might seem to be a minor difference could have a large
effect on the interactions and thus the efficiency of the flotation process.
The interactions that are important for the flotation process are often divided into
three regimes ŽDerjaguin and Duhkin, 1960; Schulze, 1984.: The hydrodynamic regime
which is concerned with the motion of particles, aggregates of particles and air bubbles.
This is important for the actual separation of the mineral from the undesired material.
The second regime is that in which diffusiophoretic forces operate due to lateral
movement of the particle in the electric field emanating from the rising bubble. The third
and final regime is the subject of this chapter, the long-range interactions of molecular
origin. These interactions are important when the distance between the particles is less
than a few hundred nanometers. They determine whether the particles aggregate or not,
and whether the particles attach to the air bubbles constituting the transporting medium.
Another important area deals explicitly with the attachment between the mineral and the
carrier medium. This process is to a large extent governed by the wetting properties of
the solids, which in turn are related to the short-range interactions. This is dealt with in
other chapters in the present volume.
Fig. 1. Illustration of various components present in mineral furnishes.
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2. Force measuring techniques
During the last three decades several different techniques have been developed for
measuring interactions between macroscopic solid surfaces, air–water interfaces, and
between a colloidal particle and a macroscopic surface. Some of these techniques are
briefly described below. The interested reader is referred to the original literature or any
of the recent review articles concerned with force measuring techniques ŽButt et al.,
1995; Israelachvili, 1995; Claesson et al., 1996a,b..
2.1. Interferometric surface forces apparatus
The idea of using optical interferometry to determine the distance between surfaces in
force measuring devices can be traced back to Derjaguin and co-workers ŽDerjaguin et
al., 1954, 1956.. However, the early measurements were not very precise and did not
allow measurements to be carried out at shorter separations than about 100 nm. More
accurate measurements of force–distance curves in air were later obtained in England
ŽTabor and Winterton, 1969; Israelachvili and Tabor, 1972.. The technique was later
improved much further, and measurements in liquid media were made possible ŽIsraelachvili and Adams, 1978.. Modified versions of the interferometric surface force
apparatus, SFA, have been developed in several laboratories ŽKlein, 1983; Parker et al.,
1989; Israelachvili and McGuiggan, 1990.. The basic principles of this instrument are
shown in Fig. 2. The preferred substrate in the SFA is muscovite mica owing to the ease
with which one may obtain molecularly smooth and well defined surfaces of this
material. The substrate surfaces are silvered on the backside and then glued onto curved
silica discs, with a cylindrical surface of radius R f 2 cm, using an epoxy resin. Two
such discs are then mounted in a crossed cylinder geometry which facilitates easy
alignment inside the apparatus chamber.
The surface separation is determined using interferometry by introducing white light
into the optical cavity formed by the silvered back sides of the two mica surfaces. The
emanating light is directed into a spectrometer where a pattern of fringes of equal
chromatic order ŽFECO, Israelachvili, 1973. is observed which can be analysed to give a
distance resolution of 0.1 nm. During the experiment the surface separation is changed
by means of a motor driven positioning rod Ž1 nm sensitivity. or by applying a voltage
to the piezo-electric crystal, on which the upper surface is mounted, which causes a
proportional expansion or contraction in steps as small as 0.1 nm. For every change in
distance, known from the calibration of the distance controlling device measured at large
separation where no force is present, the actual displacement is obtained interferometrically. The magnitude of the force resulting in a deviation, D x, between the expected and
the actual change in separation can be calculated by applying Hooke’s law using the
force constant, k, of the spring which holds the lower surface.
F s kD x
Ž 1.
In order to allow comparison between different experiments and with theoretical
predictions a more suitable quantity is the free energy of interaction per unit area, G,
4
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Fig. 2. Schematic illustration of the main components of the interferometric surface force apparatus is provided
in ŽA.. The stainless steel measuring chamber contains the two interacting surfaces. In a typical experiment,
one mica surface is glued to a silica disc that is attached to a piezoelectric crystal Žtopmost part.. The other
surface, also glued to a silica disc, is mounted on a double cantilever force measuring spring. The surfaces are
oriented in a crossed cylinder configuration wsee ŽB.x. White light enters through the window in the bottom of
the chamber. It had multiple reflections between the silver layers and a standing wave pattern, fringes of equal
chromatic order ŽFECO., is generated wsee ŽC.x. The standing waves exit through the top window and the
wavelength and fringe shape are analysed in a spectrometer.
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between flat surfaces which according to the Derjaguin approximation ŽDerjaguin, 1934.
is related to the force via
F
R
s 2p G
Ž 2.
This relation holds as long as the distance between the surfaces is much smaller than the
radii of the interacting bodies, and while the radii do not become a function of the
separation, i.e., while there is no surface deformation.
2.2. Non-interferometric techniques
Many different types of non-interferometric surface force techniques have been
developed during the years Žsee e.g., Derjaguin et al., 1978; Peschel et al., 1982; Ducker
et al., 1991; Parker, 1994.. Perhaps the most popular of these techniques is the Atomic
Force Microscope ŽAFM. which apart from being used for imaging also can be applied
to force measurements between a colloidal sized particle and a surface ŽDucker et al.,
1992; Butt et al., 1995; Larson et al., 1997. or between an air bubble and a surface
ŽButt, 1994; Ducker et al., 1994; Fielden et al., 1996.. By using a colloidal probe,
obtained by gluing a spherical particle Žof radius 5–50 mm. onto the cantilever tip, a
known geometry is obtained. A technique in some respects similar to the AFM-colloidal
probe technique is the so-called MASIF, which uses two spherical surfaces, most often
glass Ž R f 1–2 mm., to measure the interaction forces. Both these techniques depend on
electronic determination of the surface separation and spring deflection, rather than
interferometry. The zero for the surface separation is set at the hard wall encountered in
Table 1
Comparison of the three surface force instruments
Mode of operation
Requirement of substrate
Typical time for one force curve
Typical number of data points
Typical radius
Typical force sensitivity
Typical sensitivity in force
normalised by radius
Measures absolute distance
and layer thickness
Measures local radius and
surface deformation
Measures refractive index and
adsorbed amount
SFA
MASIF
AFM colloidal probe
step-wise or continuous
approach
thin sheets of uniform
thickness, molecularly
smooth, transparent,
preferably hard
10–40 min
50–100
1–2 cm
10y7 N
5–10 mNrm
continuous approach
continuous approach
molecularly smooth
preferably hard
well defined geometry
colloidal particle
preferably hard
well defined geometry
0.1–2 min
1000–10 000
0.1–0.2 cm
2=10y8 N
10–20 mNrm
1–2 s
a few hundred
5–50 mm
1=10y10 N
2–20 mNrm
not without use of
interferometry
not without use of
interferometry
not without use of
interferometry
NO
YES
YES
YES
NO
NO
6
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each force run when the surfaces have come into contact, assuming that such a hard wall
is reached ŽSchillen
´ et al., 1997.. This procedure is different from the situation in the
SFA where the zero separation is set at the beginning of each experiment when the two
surfaces are brought into a strong adhesive contact in air. The contact is considered to be
unchanged during the course of the experiment and any change in smallest separation
can be attributed to an adsorbed layer. This constitutes a difference between the
techniques, thus conclusions about layer thicknesses cannot readily be inferred from
AFM-colloidal probe and MASIF experiments. There are on the other hand a wider
variety of surfaces which can be employed, since there is no requirement that the
surfaces are transparent. Furthermore, one can take advantage of the fact that measurements can be performed much faster using these techniques. Some essential features of
the SFA, MASIF and AFM-colloidal probe techniques are summarised in Table 1.
2.3. Thin film balance
Various types of thin film balance techniques can be used for measuring dynamic and
static interactions between two air–liquid interfaces, or between an air–liquid interface
and a solid surface ŽScheludko, 1967; Exerowa and Scheludko, 1971; Bergeron et al.,
1993.. A schematic drawing of the porous frit version of this instrument is shown in Fig.
3. It consists of a gas tight measuring cell. The solution under investigation is placed in
Fig. 3. A schematic illustration of the main components of the thin film balance. A macroscopic foam film is
formed in a hole drilled in a porous glass frit. The surfactant solution is contained in the frit, in the glass
capillary and at the bottom of the closed cell. The film thickness is determined using interferometry. The
reflected light is viewed by a video camera and the intensity of a selected wavelength is measured with a
photomultiplier tube ŽPMT.. The pressure in the measuring cell is varied by means of a syringe pump and
measured by a pressure transducer.
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the bottom of the cell and it is also contained in a porous glass frit with a capillary tube
fused to one of its sides. The foam film is formed in a small hole drilled in the frit. In
the flat portion of the film the disjoining pressure Ž P . equals the capillary pressure,
which in turn equals the difference in the pressure in the gas phase Ž Pg . and in the liquid
phase Ž Pl .
P s Pg y Pl s Pg y Pr q
2g cos u
y D r gh
Ž 3.
r
where Pr is the reference pressure above the capillary Žoften atmospheric pressure., g
the surface tension of the liquid film, u the contact angle between the liquid and the
capillary wall, D r the difference in density between the solution and the surrounding
air, h is the liquid rise in the capillary and g the gravitational constant. By varying the
gas pressure in the measuring cell the disjoining pressure can be changed, and this, of
course, changes the equilibrium film thickness. In this way the repulsive forces can be
determined, commonly referred to as the disjoining pressure isotherm in thin film
balance literature.
3. Interaction energy–distance curves
The total interaction energy between a pair of particles has in general contributions
from many different types of interactions, both repulsive and attractive. These will be
discussed in Section 4 but let us first focus our attention on the total energy–distance
curve and see how its appearance determines the state of the colloidal system. Some
examples of energy–distance curves are provided in Fig. 4. Depending on the balance
Fig. 4. Typical energy vs. distance profiles. Ža. Total interaction energy as the sum of van der Waals and
electrostatic interactions. The inset shows the effect of increased electrolyte concentration andror lower
surface potential. Žb. Notation for different characteristics of the interaction energy profile, also shows the
existence of a secondary minimum present in some systems.
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between the repulsive and attractive components in the particular system, one of several
situations is possible. When the interaction is repulsive at all separations the particles
will always repel each other and the colloidal system will remain in the dispersed state,
which is also the thermodynamically stable state.
On the other hand when the interaction is purely attractive, except at very short
separations, the two particles come close together into an energy minimum and an
aggregate is formed. The colloidal system is then unstable and will rapidly coagulate
Žalso called irreversible flocculation.. In most cases the situation is more complicated
and the total energy of interaction may be repulsive or attractive depending on the
interparticle separation. For instance, the outermost and innermost part of the energy–
distance curve may be attractive whereas it may be repulsive at intermediate separations
ŽFig. 4..
The maximum in the energy distance curve is referred to as the energy barrier, and
when it is small, i.e., comparable to the Brownian energy due to the thermal motion
k B T, the particles may approach each other close enough to reach the primary Žor inner.
energy minimum. The depth of this minimum is often large compared to k B T and the
colloidal system thus becomes coagulated. On the other hand, when the energy barrier is
large compared to k B T the particles cannot reach the primary minimum but they prefer
to be in the secondary Žor outer. minimum. The colloidal system becomes reversibly
flocculated when the depth of the minimum is comparable to k B T. The floc structure
can in this case easily be broken, for example, by stirring or shearing. If the secondary
minimum is absent or small compared to k B T the system remains in a dispersed
metastable state. One says that the colloidal system is kinetically stabilised since the
large energy barrier slows down the coagulation rate.
A colloidal system may thus be unstable, thermodynamically stable or kinetically
stabilised depending on the interactions. These are in turn determined by the properties
of the particles; such as surface charge and dielectric properties, and the properties of the
medium; pH, ionic strength, and concentration of additives such as surfactants, polymers, and polyelectrolytes. It is for the purpose of controlling the interactions in the
flotation process that most additives are used.
4. The interaction energy contributions
A better understanding of how the interactions can be controlled can be obtained by
considering the most important contributions to the total energy–distance curve, the
molecular origin of these contributions, and how the strength of the different interaction
contributions can be regulated by additives. This section will start by considering the
always present van der Waals interaction, and the electrostatic double-layer interaction
present between charged surfaces immersed in electrolyte solutions. These contributions
are considered in the DLVO theory, which is the most widely used framework for
understanding colloidal stability in the absence of adsorbed polymers. The name stems
from the four scientist who developed the theory in the 1940s: Derjaguin and Landau in
Russia and independently Verwey and Overbeek in the Netherlands Žsee Faraday Disc.
18 Ž1954. for an account of the development.. We will furthermore briefly describe
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non-DLVO interactions such as solvation, hydrophobic, and steric interactions. There
are several books which treat the subject extensively ŽIsraelachvili, 1991; Evans and
Wennerstrom,
¨ 1994. and the reader is encouraged to consult these for a more extensive
discussion.
4.1. Õan der Waals interactions
Van der Waals interactions exist between molecules and between particles. The most
important molecular contributions are due to interactions between permanent rotating
dipoles ŽKeesom interactions., between a permanent rotating dipole and an induced
dipole ŽDebye interactions., and between two induced dipoles ŽLondon or dispersion
interactions.. The dispersion contribution originates from the motion of electrons around
the nuclei, and it therefore exists also between non-polar molecules and is always
present. Except for highly polar molecules it can be shown that the dispersion contribution accounts for nearly all the van der Waals interaction. The interaction energy is
inversely proportional to the separation between the molecules to the power of six.
The van der Waals interactions between two macroscopic particles can, as a first
approximation, be calculated by summing the interactions between all molecular pairs
between the two bodies. In practise this is done by an integration ŽHamaker, 1937. and
the result is that the distance dependence of the van der Waals interaction between
macroscopic surfaces is much weaker than that between molecules, i.e., the interaction is
more long-ranged.
The expression for two spherical particles of radius R 1 and R 2 separated in Õacuum
by a distance D is ŽHamaker, 1937.
VA s y
A
b
12 a 2 q ab q a
b
q
a 2 q ab q a q b
q 2ln
ž
a 2 q ab q a
a 2 q ab q a q b
/
Ž 4.
where a s Dr2 R 2 , b s R 1rR 2 , and A is the material dependent Hamaker constant.
Only the first term in Eq. Ž1. needs to be considered when the radii of the spheres are
much larger than the surface separation. When this is the case and when the spheres
have the same radii, R, the expression above simplifies to
VA s y
AR
12 D
Ž 5.
The Hamaker method is conceptually easy but unfortunately not very accurate. It
neglects many-body interactions, entropic contributions and retardation. It is also not
very easy to accurately account for the presence of the medium separating the two
interacting particles.
All these effects are considered in an alternative treatment due to Lifshitz et al.
ŽDzyaloshinskii et al., 1961.. Here the interacting particles and the intervening medium
are treated as continuous phases and the interaction is viewed as originating from
interference between fluctuating electromagnetic fields extending beyond the surface of
the particles. The expressions for the interaction energy between bodies of a specific
shape have the same functional form as in the Hamaker treatment, but the equivalent to
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the Hamaker constant is a function of the frequency dependent bulk dielectric properties
of the continuous phases. Rather accurate approximate equations to the Lifshitz expression are often used for calculating the Hamaker constant. The material constants needed
in order to perform the calculations are the static dielectric constants, ´ i , and refractive
indices, n i , of the different phases i. Thus the non-retarded Hamaker constant for two
macroscopic bodies 1 and 2 interacting over a medium 3, is approximately given by
ŽIsraelachvili, 1991.
3
´1 y´3
´2 y´3
A s k BT
4
´1 q´3
´2 q´3
ž
/ž
/
3hne Ž n12 y n 23 .Ž n 22 y n 23 .
q
(
(
8'2 n12 q n 23 n22 q n23
ž (n q n q (n q n /
2
1
2
3
2
2
2
3
Ž 6.
where ne is the main electronic absorption frequency in the UV region, here assumed to
be the same for all three media, k B the Boltzmann constant, h Planck’s constant, and T
the absolute temperature.
Implicit in the discussion so far has been that the Hamaker constant will not be
affected by the separation between the particles, i.e., the distance over which the electric
field has to travel to affect the other body. However, this is not correct when the
separation is so large that the time needed for the electric field emanating from one body
to travel to the other body and being reflected back again becomes comparable to the
period of the fluctuations that generate the field. Thus the phase correlation between the
fields emanating from the two bodies will be lost and the interaction will be weakened.
This phenomenon, called retardation, is important at separations larger than about 5 nm,
and can be taken care of within the framework of the Lifshitz theory, e.g., as described
in the book by Mahanty and Ninham Ž1976..
It can easily be seen from Eqs. Ž5. and Ž6. that the van der Waals interaction is
always attractive between identical entities, and it is always attractive in vacuum even
when the two interacting bodies are not the same. However, when the material
properties of the bodies are different the interaction may become repulsive in a medium.
This is for instance the case for an air bubble interacting with a mineral particle or a
hydrophobised mineral particle across water. Fig. 5 shows the non-retarded van der
Waals interaction for a few different combinations of phases interacting across water.
The figure displays that the van der Waals force can be both attractive and repulsive,
with the latter being the case whenever the dielectric properties of the intervening
medium falls between those of the interacting phases.
The van der Waals interaction is altered by the presence of an adsorbed film
whenever its dielectric properties are different to those of the particles. The non-retarded
van der Waals force between spheres of material 1 and 1X , each coated with a thin layer
of material 2 and 2X of thickness L and LX , respectively, interacting across medium 3 can
be evaluated using the expression ŽNinham and Parsegian, 1970.
R A 232X
A132
A1X 32
A1X 21
A eff
W Ž D. sy
y
y
sy
X q
X
12
D
12 D
Ž D q L. Ž D q L . Ž D q L q L .
Ž 7.
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Fig. 5. Calculated van der Waals forces between two spheres for different combinations of silica, water, air,
and hydrocarbon Žoctane.. The different curves correspond to the following combinations and Hamaker
constants: Ž — . silica–water–silica As 0.83=10y2 0 J, Ž — — . hydrocarbon–water–hydrocarbon As 0.41=
10y2 0 J, Ž – – . air–water–air As 3.7=10y2 0 J, Ž- -. air–water–hydrocarbon Asy0.20=10y2 0 J, and
Ž — - — . air–water–silica Asy0.8=10y2 0 J. Note how a negative Hamaker constant results in a repulsive
van der Waals force.
where A m n o is the Hamaker constant for bodies of material m and o interacting in
medium n, which can be calculated according to Eq. Ž6..
The effective Hamaker constant will consequently be dependent on the separation D:
At large separations it will be dominated by the contribution from the material in the
particles, while the influence of the properties of the adsorbed layer will become
apparent as the separation between the surfaces decreases. The effect of an adsorbed
layer is indicated in Fig. 6a where the dependence of the effective Hamaker constant is
shown as a function of the normalised surface separation. The case of a quartz surface
interacting with an air bubble in water when bearing identical adsorbed hydrocarbon
films, calculated according to Eq. Ž7., is shown by the solid curve, and the case of two
quartz surfaces with adsorbed hydrocarbon films in water is shown by the dashed curve.
For both, the Hamaker constant at small separations will be dominated by the adsorbed
layers while at larger separations it is determined by the properties of the underlying
media. In Fig. 6b, the resulting van der Waals forces between two spherical particles are
given as a function of surface separation for the two systems, the change from a
repulsion to an attraction at 1 nm for the asymmetric system Žsolid curve. reflects the
change in sign of the Hamaker constant whereas the force for the symmetric film
Ždashed curve. remains attractive at all separations.
4.2. The electrostatic double-layer interaction
It has been shown that the electrostatic interactions between particles and bubbles
greatly affects the flotation of minerals. The flotation performance is, for example,
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Fig. 6. Ža. The effective Hamaker constant calculated using Eq. Ž7., as a function of normalized surface
separation. The dashed curve is for two quartz surfaces coated with a hydrocarbon layer of thickness T s1
nm. At short separations the effective Hamaker constant is equal to that between the adsorbed hydrocarbon
layers Ž As 0.5=10y2 0 J., whereas at large separations it approaches that between uncoated quartz surfaces
ŽA s 0.83=10y2 0 J.. The solid curve is for the asymmetric system of a quartz particle interacting with an air
bubble when both carry an adsorbed layer of hydrocarbon ŽT s1 nm.. Again at small separations the adsorbed
layers dominate the Hamaker constant, whereas at longer ranges it approaches the value for a quartz particle
interacting with air, which in this case leads to a change in sign. Žb. Calculated van der Waals forces for the
two systems in Ža.. It is worth noting that the change in sign of the Hamaker constant for the asymmetric
system Žsolid line. leads to a situation where the force is repulsive at long range while turning attractive at
small separations, approximately 1 nm. The purely repulsive van der Waals force is for silica interacting with
air across water without any adsorbed layers.
closely correlated to the range of the double-layer force ŽPaulson and Pugh, 1996.. This
interaction, therefore, deserves an extensive study if one aims to control and optimise
the mineral flotation process.
A solidraqueous solution interface is most often charged due to one of several
mechanisms; acid–base equilibria as in the case of silanol groups on silica, desorption of
lattice ions as in the case of clays and minerals such as mica, or adsorption of ionic
surfactants and polyelectrolytes. The surface charge is exactly balanced by the net
charge in the solution outside the surface. In the region immediately outside the surface,
say below 1 nm from the surface, the ions may, in addition to electrostatic forces,
experience other interactions with the surface. These ions are said to be adsorbed and
they build up the so called Stern layer. Outside the Stern layer, in the diffuse layer, the
ions are only affected by electrostatic forces and in this region it is straightforward, in a
mean field approach such as the Poisson Boltzmann ŽPB. model, to calculate the ion
concentration away from the surface. The extension of the diffuse layer will depend on
the surface charge density, ion concentration and valency. A central quantity in this
respect is the Debye length which describes the rate with which the mean potential
decays away from the surface. The Debye-length decreases with increasing ionic
strength but it is independent of the surface charge density and surface potential.
J.C. Froberg
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An electrostatic double-layer force arises when two surfaces come close enough for
their diffuse layers to start to overlap. Before discussing how this force arises it is
valuable to consider the ion distribution in the diffuse layer outside a flat surface as
calculated in the non-linear PB-approximation. The classical text on this subject is the
book by Verwey and Overbeek Ž1948., ‘Theory of the Stability of Lyophobic Colloids’,
more recent treatments are given in the books by Israelachvili Ž1991., and Evans and
Wennerstrom
¨ Ž1994..
The concentration Žnumber density. of ions of type i in the diffuse layer a distance x
from the surface, r i Ž x ., will at equilibrium be related to the bulk concentration r i,` and
the mean potential difference between the bulk solution and layer x, C Ž x ., according to
the Boltzmann relation
r i Ž x . s r i ,` eyz i eC Ž x .r k B T
Ž 8.
where e is the elementary charge and z i the valency of the ion. The potential is related
to the charge density at x via the Poisson equation
Ýz i e ri Ž x . s y´´ 0 Žd 2Crd x 2 .
Ž 9.
i
where ´ 0 is the permittivity of vacuum and ´ is the dielectric constant of the medium.
Combining these two relations one obtains the Poisson–Boltzmann equation
d 2Crd x 2 s yÝ Ž z i e r i ,` r´´ 0 . eyz i eC Ž x .r k B T
Ž 10 .
i
This non-linear second order differential equation in C can be solved analytically. The
first integration gives a relation between the surface potential, C 0 , and the surface
charge density, s 0 ,
1r2
s 0 s 2 k B T´´ 0Ýr i ,` Ž eyz i eC 0 r k B T y 1 .
Ž 11 .
i
A second integration gives an expression for the variation of the mean potential with
the distance from the surface and takes the form
tan h
ž
zeC Ž x .
4 k BT
/
s tan h
zeC 0
ž /
4 k BT
ey k x
Ž 12 .
for a symmetrical electrolyte. For small potentials Ž- 25 mV. Eq. Ž12. reduces to the
Debye–Huckel
relation
¨
C Ž x . s C 0 eyk x
Ž 13 .
where the quantity 1rk is the Debye Žscreening. length mentioned above. It is generally
given for an electrolyte with bulk number density r i,` of species i by
1r2
1rk s
´´ 0 k B T
Ý Ž z i e . 2 ri ,`
i
0
Ž 14 .
J.C. Froberg
et al.r Int. J. Miner. Process. 56 (1999) 1–30
¨
14
Clearly, the Debye-length and the electrostatic potential outside the surface both
decrease rapidly with increasing ionic strength.
To calculate the double layer force per unit area, D P, between two flat surfaces we
start by considering the pressure on a flat surface immersed in an electrolyte solution
which for a surface with surface charge s 0 and with an ion density of r i Ž0. at the
surface, can be written as
P s k BT
ž
Ýr i Ž 0 . y
i
s 02
2 ´´ 0 k B T
/
Ž 15 .
This expression states that a force can only act on a surface by way of transfer of
momentum, the first term, or through the normal component of the Maxwell stress
tensor, second term. The contact value theorem, Eq. Ž15., ŽIsraelachvili, 1991. is exact
in a continuum approach. However, the true values of s 0 and r Ž0. are generally not
known with sufficient precision for an accurate calculation of the small difference
between the two large terms in Eq. Ž15. constituting the origin to the pressure. Within
the Poisson–Boltzmann approximation one can derive an expression for the pressure on
a flat surface by inserting Eq. Ž8.
P s k BT
ž
Ýri ,` eyz i eC 0 r k B T y
i
s 02
2 ´´ 0 k B T
/
Ž 16 .
In order to obtain the net pressure on the flat surface one has to subtract the osmotic
pressure of the bulk solution to obtain
D P s k BT
ž
Ýri ,` eyz i eC 0 r k B T y
i
s 02
2 ´´ 0 k B T
/
y k B TÝr i ,`
Ž 17 .
i
For an isolated surface one finds by applying Eq. Ž11. that the net pressure is zero,
showing the internal consistency of the PB-equation.
One can, in the Poisson–Boltzmann treatment, generalise the contact value theorem,
Eq. Ž15., to any plane between two interacting surfaces. The net normal pressure
between two flat surfaces separated a distance D can in this case be written
Ýri ,` Ž eyz i eC D Ž x . rk B T y 1. y
D P Ž D . s P Ž D . y P Ž `. s k B T
i
sD2 Ž x .
2 ´´ 0 k B T
Ž 18 .
where x takes values between 0 and D, and CD Ž x . and sD Ž x ., respectively are the
surface potential and surface charge density at x when the surfaces are a distance D
apart. The net normal pressure is of course equal at all points between the surfaces when
mechanical equilibrium is established, but will vary with the surface separation. An
indefinite integration of Eq. Ž10. yields
ž
dCD Ž x .
dx
2
/
s
2 k BT
´´ 0
Ýri ,` Ž eyz eC
i
D Ž x . rk B T
qCŽ D. .
Ž 19 .
i
where C is a integration constant. Now, by combining Eqs. Ž18. and Ž19. with
dCD Ž x .
sD s y´´ 0
dx
Ž 20 .
J.C. Froberg
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¨
15
we arrive at a simple expression for the pressure in the normal direction for flat surfaces
separated a distance D
D P Ž D . s yk B TÝr i ,` Ž C Ž D . q 1 .
Ž 21 .
i
C is thus a function of the surface separation D, taking the value of y1 for an isolated
surface to give a net pressure of zero. For values of C - y1 the net pressure will be
repulsive while for values larger than y1 it will be attractive. The different types of
possible surface charging mechanisms; constant potential, constant charge, or regulating
surfaces, will give rise to different boundary conditions for the integration of Eq. Ž19.
necessary to find the surface separation corresponding to a given combination of the s 0
Žor c 0 . values for the two surfaces and the C value. The result is an expression for the
surface separation in terms of incomplete elliptic integrals of the first kind. Within this
treatment it is possible to arrive at an expression for the free energy of interaction per
unit area as
D
X
D
X
X
X
H` D P Ž D . d D s k TÝr H` Ž C Ž D . q 1. d D
DG s y
B
i ,`
Ž 22 .
i
When the value of C as a function of the surface separation has been determined the
free energy of interaction can be evaluated, this time in terms of incomplete elliptic
integrals of the second kind. It is then straightforward to obtain the free energy of
interaction for spheres as
Vspheres Ž D .
pR
D
X
X
H` DG Ž D . d D
sy
Ž 23 .
The calculations show that when two surfaces of equal surface potential approach each
other, the double layer interaction will be purely repulsive at all separations, independent
of whether the interaction takes place under constant potential or constant charge.
Devereux and de Bruyn Ž1963. show in their treatment of interaction between dissimilar
surfaces at constant potential, that when the signs of the surface potentials are equal the
surfaces will repel each other at large separations while at shorter distances the
interaction turn attractive. When the potential of the two surfaces have opposite sign the
interaction will be purely attractive. The condition of constant potential for unequal
surfaces does, however, lead to the unrealistic conclusion that the magnitude of the
surface charge density of one of the surfaces approaches infinitely large positive values,
while that of the other approaches infinitely large negative values.
The interaction between dissimilar surfaces at constant charge will, on the other hand,
be purely repulsive for surfaces with the same sign as shown by Bell and Peterson
Ž1972.. However, when the surface charges have opposite signs on the two surfaces, the
interaction turns out to be attractive at long range while turning repulsive at shorter
range. The exception to this is when the two surfaces have equal but opposite surface
charge which leads to a purely attractive force. Fig. 7 shows calculated interaction forces
for two spherical surfaces, when the apparent potentials of the isolated surfaces are 50
and y30 mV both for the case of constant charge and constant potential. The interaction
J.C. Froberg
et al.r Int. J. Miner. Process. 56 (1999) 1–30
¨
16
Fig. 7. Calculated double-layer forces for unequal spheres, with potentials 50 and y30 mV at large
separations, interacting at constant charge Ždashed line. and constant potential Žsolid line. in a 1-mM 1:1
electrolyte.
is at long range attractive for both boundary conditions while it becomes repulsive at
short range for the case of constant charge.
Finally, when the surfaces show a regulation of the surface charge density characterised by the dissociation constant of the chargeable surface groups, a rather complex
variation of the double layer force with surface separation emerges. An account of the
interaction between dissimilar, regulating surfaces was presented by Chan et al. Ž1976.
using both an approximative graphical method giving the main features of the interaction curve, as well as an exact Poisson–Boltzmann treatment. They showed that the sign
of the double layer force may change as often as three times when two such surfaces
approach each other.
To calculate the double-layer force between flat surfaces exactly in the non-linear
PB-approximation is thus possible but rather complicated. For spheres which are large
compared to the range of the force one can use the Derjaguin approximation, Eq. Ž2., to
obtain accurate expressions for the double-layer interaction. It is, however, sometimes
useful to find simple analytical expressions for the double-layer force. The simplest one
is obtained for small surface potentials in the weak overlap approximation, which is
valid at large separations. For the pressure and energy of interaction between two
identical flat surfaces one obtains
P Ž D . s 2 ´´ 0 k 2C 02 eyk D
Ž 24 .
V Ž D . s 2 ´´ 0 kC 02 eyk D
Ž 25 .
and
Using the Derjaguin approximation, one obtains for two spheres of the same radii
F Ž D.
pR
s 2 ´´ 0 kC 02 ey k D
Ž 26 .
J.C. Froberg
et al.r Int. J. Miner. Process. 56 (1999) 1–30
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17
and
V Ž D.
s 2 ´´ 0C 02 ey k D
Ž 27 .
pR
Hence, for this case the double-layer force at large separations decays exponentially
with surface separation and the decay-length is equal to the Debye-length. It turns out
that this conclusion is valid also when the non-linear PB-approximation is solved
exactly. Thus, the range of the double-layer force decreases rapidly when the ionic
strength is increased.
Several assumptions are inherent in the rather primitive model of the Poisson–Boltzmann treatment. The mean-field description of the double-layer breaks down at small
enough separations where several effects will become important: The finite size of the
ions will lead to an increased repulsion resulting from an excluded volume effect. Ion
correlations are neglected, which has been shown to have a substantial effect on the
double layer interaction, even turning what in the PB-model is a repulsive interaction
into an attraction, which, for example, is the case for highly charged surfaces in
solutions of divalent ions. Furthermore, the description of the surfaces as having a
uniform charge will also lead to errors, since at small separation the discreteness of the
surface charge will further invalidate the mean-field treatment ŽGuldbrand et al., 1984;
Kjellander and Marcelja, 1984; Attard et al., 1988; Kjellander, 1996..
4.3. DLVO theory
The DLVO theory considers the total interaction to be due to additive contributions
from the van der Waals interaction and the double layer interaction. As stated above, the
van der Waals interaction is attractive whereas the double-layer interaction is repulsive
between identical surfaces. The van der Waals contribution predominates at small
interparticle distances and the double-layer contribution gives rise to an energy barrier at
larger separations as illustrated in Fig. 4a. The insert to Fig. 4a illustrates that increasing
the ionic strength or lowering the surface potential results in a lower energy maximum.
Fig. 4b shows the situation for a system with a strong van der Waals attraction and a low
double-layer repulsion which results in the existence of a secondary minimum at
relatively large interparticle distances. Clearly, we can see that by just considering the
DLVO-forces one may obtain all the types of energy–distance curves that are required
for having kinetically stable, weakly flocculated or coagulated systems. Further, by
increasing the ionic strength one may go from a kinetically stable colloidal dispersion to
one which is coagulated.
Some calculated DLVO force curves across an aqueous 1 mM 1:1 electrolyte are
shown in Fig. 8a and b. The forces calculated in Fig. 8a are for the case of two solid
surfaces where the van der Waals interaction is characterised by a non-retarded Hamaker
constant of 0.8 = 10y2 0 ŽJ.. The double-layer force is calculated using constant charge
boundary conditions. The upper curve is for the case where both surfaces have a surface
potential Žat large separations. of 50 mV, and here we obtain the classical behaviour
with a repulsion at large separations and an attraction at short distances. The lower curve
is for the case where the surface potentials are unequal, 50 and y30 mV, respectively.
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J.C. Froberg
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¨
Fig. 8. Calculated DLVO-forces Žsolid lines. for interacting spheres in 1-mM 1:1 electrolyte, dashed lines
represent the electrostatic double layer force only Žconstant charge.. Ža. Two solid surfaces with a non-retarded
Hamaker constant of 0.8=10y2 0 J, upper curves are for equal potentials Ž50 mV. while the lower are for
unequal spheres Ž50 and y30 mV.. Žb. One solid surface interacting with an air bubble resulting in a
non-retarted Hamaker constant y0.8=10y2 0 J and therefore repulsive van der Waals forces, upper and lower
curves correspond to potentials as given in Ža..
In this case the total interaction is purely attractive, even though the double-layer
contribution turns repulsive at short separations Ždashed line.. Some experimental data
for the case of two mica surfaces interacting across aqueous KBr solutions are shown in
Fig. 9. In 0.1-mM KBr, the long-range force is dominated by a double-layer force,
Fig. 9. Experimental data for mica interacting across a 0.1-mM Kbr solution ŽI. and a 10-mM KBr Ž'.. The
solid lines are best fit for the DLVO theory, showing that an extra hydration repulsion is present at high ionic
strengths.
J.C. Froberg
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¨
19
whereas the short range force is attractive due to the van der Waals interaction. This
behaviour is as expected from DLVO theory. In the 10 mM KBr solution the long-range
force is still dominated by a double layer force. However, in this case no attraction is
seen at short separations, which can be explained by the appearance of a short-range
repulsive hydration force not considered in the DLVO theory Žsee below.. The data in
Fig. 8b are calculated for the case of a repulsive van der Waals force Žwith a Hamaker
constant equal to y0.8 = 10y2 0 ŽJ.., as for the case of an air bubble interacting with
quartz across water. The upper curve, which is purely repulsive, is for the case of both
potentials being 50 mV. When the potentials instead are unequal, 50 and y30 mV, the
long-range force is attractive whereas the short range force is repulsive. It is seen that
the repulsive van der Waals interaction prevents the air-bubble from attaching to the
quartz particle. To make the quartz particle hydrophobic does not change the fact that
the van der Waals force is repulsive, even though the Hamaker constant for hydrocarbon–water–air is considerably less negative, y0.2 = 10y2 0 ŽJ.. However, if also the air
bubble carries an adsorbed surfactant layer, the van der Waals force at short separations
becomes attractive as shown in Fig. 6b. The attachment of air bubbles to hydrophobised
mineral particles are often alternatively explained in terms of an additional attraction due
to hydrophobic interactions.
4.4. Hydrophobic interactions
For the solid minerals to float, their surfaces must be made hydrophobic, that is,
wetted only partially by water. In a series of papers Yoon and co-workers have
advocated that the additional attractive force needed to explain why air bubbles do
attach to hydrophobised mineral particles is a long-range hydrophobic interaction
ŽRabinovich et al., 1993a,b; Yotsumoto and Yoon, 1993a,b; Rabinovich and Yoon,
1994; Yoon and Ravishankar, 1994; Yoon and Mao, 1996; Yoon and Ravishankar,
1996a,b; Ravishankar and Yoon, 1997; Yoon et al., 1997.. At this point it is worthwhile
to consider what evidence there are for such a long-range attraction between hydrophobic surfaces and how it may be related to the hydrophobic interaction between molecules
that drives the self-assembly of surfactants in aqueous and other hydrogen bonding
solutions ŽRamadan et al., 1983; Evans and Ninham, 1986; Bergenstahl
˚ and Stenius,
1987; Beesley et al., 1988; Warnheim
and Jonsson,
1988; Jonstromer
et al., 1990;
¨
¨
¨
Warnheim
et al., 1990.. Self-assembly of surfactants has been much discussed, for
¨
instance by Tanford in his famous book about the hydrophobic effect ŽTanford, 1980..
One conclusion was that the hydrophobic effect was short-range and proportional to the
size of the non-polar molecule. The comparison of the temperature dependence of the
changes in free energy, entropy and enthalpy of micelle formation in water and
hydrazine ŽRamadan et al., 1983. has provided a good understanding for the association
process of surfactants.
One can, artificially but pedagogically, divide the free energy change for the
association process into two main contributions; one that is due to transfer of non-polar
groups from the polar solvent and their confinement in the micelle Ž D Htr and D Str ..
Here both the enthalpy contribution and the entropy contribution are negative and the
resulting free energy Ž DGtr s D Htr y TD Str . is also negative. This is the dominant
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¨
contribution to the solvophobic effect in non-structured hydrogen bonding liquids such
as hydrazine and water at high temperatures Žwell above 1008C. ŽEvans and Ninham,
1986.. Thus, in this case the process is enthalpy driven. In water at temperatures well
below 1008C there is also a second contribution due to changes in the water structure
itself which accompany the removal of non-polar substances from the aqueous environment. Here both the enthalpic Ž D Hstr . and entropic Ž D Sstr . contributions are large and
positive. In this case one may write, even though the two contributions are, of course,
not independent of each other, DG s D Htr q D Hstr y TD Str y TD Sstr , and the total
enthalpy and entropy change is dominated by the structural contribution. Hence, one can
say that at low temperature the micellisation process in water is entropy driven and that
it is due to the release of water clathrates around the hydrocarbon chains. However, the
most remarkable finding is that despite the fact that both D Hstr and D Sstr vary strongly
with temperature, the free energy change accompanying micelle formation is rather
temperature insensitive. Hence, DGstr s D Hstr y TD Sstr is close to zero at all temperatures and for this reason it has been suggested that it is misleading to view the
hydrophobic effect as being due to structural changes in the solvent ŽEvans and Ninham,
1986..
It is thus clear that there is a short-range attractive hydrophobic interaction between
non-polar molecules in aqueous solutions. The question then naturally arises if there is a
similar attraction between non-polar surfaces in water, and in that case which distance
dependence it has. The first measurements by Israelachvili and Pashley Ž1982. indicated
that an additional attraction in excess of the van der Waals force indeed exists between
hydrophobic surfaces, and they reported an exponential distance dependence of this
interaction with a decay constant of 1 nm. Later more strongly hydrophobic surfaces
were studied and a decay length of the force of 1.4 nm was reported by Pashley et al.
Ž1985.. Later experiments showed evidence for much more long-ranged forces that were
better fitted to a two step exponential function
0
0
V hph s Vhph1
eyD r l1 q V hph2
eyD r l 2
Ž 28 .
with one decay constant of 1–3 nm dominating for separations up to 10 nm and one of
around 5–15 nm dominating at larger separations. Hence, it is clear that there do exist
an additional long-range attraction between hydrophobic surfaces, but it is not clear if it
should be regarded as a hydrophobic attraction akin to the hydrophobic effect between
molecules. From the high interfacial tension between hydrocarbon and water it is evident
that the short-range interaction between hydrocarbon surfaces immersed in aqueous
solutions should be strongly attractive. However, there is no theory that convincingly
explain why the effect should extend out to 100 nm.
Measurements of forces between different types of hydrophobic surfaces agree on
that there is an additional attraction in excess of the van der Waals interaction. However,
they do not agree on the range of the attractive force. It is of the order of 10 nm for
slightly hydrophobic surfaces ŽHerder, 1990., for plasma polymers ŽParker et al.,
1994b., polymerised Langmuir–Blodgett layers ŽWood and Sharma, 1994. and some
silanated surfaces ŽParker and Claesson, 1994., whereas it is of the order of 100 nm for
non-polymerised Langmuir–Blodgett deposited films ŽClaesson and Christenson, 1988.
and some other silanated surfaces ŽParker et al., 1994a; Rabinovich and Derjaguin,
J.C. Froberg
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¨
21
1988.. It is clear that at least some of the data describing the most long-range attractive
forces can be rationalised in terms of cavity formation between the surfaces ŽParker et
al., 1994a., whereas the interpretation of the data obtained for Langmuir–Blodgett layers
are complicated by the limited stability of such films ŽEriksson et al., 1997.. The
literature on the subject was recently reviewed by Hato Ž1996., who also studied double
chained surfactants with tails terminated by methyl- or hydroxide groups. He found that
the most hydrophobic surface showed the least long-range attraction. However, the
attraction in the shorter distance regime, D - 5–20 nm increased with increasing surface
hydrophobicity. Hence, the more long-range part of the measured force can not be
viewed as a hydrophobic interaction. Further support for this view is that the range of
the attraction for surfaces of equal hydrophobicity Žas judged by contact angle measurements. prepared in different ways are very different. This is illustrated in Fig. 10 where
the interactions between three different kinds of hydrophobic surfaces, all having
advancing contact angles above 908, are shown. The attraction measured between the
three different hydrophobic surfaces vary substantially in range. The most hydrophobic
surfaces, the plasma polymerised layers of hexamethyldisiloxane, show the least longrange attraction. In fact, the range of the attraction is in this case consistent with that of
a van der Waals force ŽParker et al., 1994b.. A more long-range attraction is seen for the
uncharged silanated glass surfaces. In this case the range of the attraction is larger than
expected for a van der Waals force and one has to conclude that another attractive force
is present. Mica coated with a deposited layer of dimethyldioctadecylammonium ions
ŽDDOA. are also uncharged. This surface is more heterogeneous than the others
mentioned above, as judged from the contact angle hysteresis. It is remarkable that the
most long-range attraction is present between these surfaces.
Fig. 10. Force normalised by radius between the three different kinds of hydrophobic surfaces. The
macroscopic contact angles of water on each surface is given within parenthesis. Solid line DDOA coated mica
surfaces ŽQA s948, Q R s608., ŽB. glass silanated with 3,3-dimethylbutyldimethylchlorosilane ŽQA s1008,
Q R s 708., Ž`. glass coated with a plasmapolymer film of hexamethyldisiloxane Ž QA s1098, Q R s988..
22
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¨
Hence, from the examples above it is clear that the contact angle does not correlate
with the range of the attraction. Instead the structure of the hydrophobic surface is very
important for the range of the attractive force. Interestingly, this also influences how
easily air-bubbles nucleate on the surfaces and the lateral mobility of the surface
molecules. Both these phenomena have in some theories been suggested to be important
parameters that determine the range and magnitude of the attractive force between
hydrophobic surfaces ŽParker et al., 1994a; Yaminsky et al., 1996..
The subject of ‘hydrophobic interactions’ between macroscopic surfaces and their
range is still controversial and opposing views and interpretations can be found in the
literature. It would go to far to discuss this controversial subject any further here, but in
the remainder of this chapter we will take a pragmatic view. We will use the
experimental finding that an additional attraction does exist between macroscopic
hydrophobic surfaces, but we will avoid giving this attraction a molecular interpretation.
When one adds an exponential attraction to the DLVO-forces, as in the extended DLVO
theory of Yoon et al., and calculate the forces between a hydrophobised mineral particle
and an air-bubble under the same conditions as in Fig. 8b, one sees that the short-range
interaction now is attractive and the air bubble will be attached to the surface once the
force barrier has been overcome ŽFig. 11..
4.5. Structural or solÕation forces
When the DLVO-forces were discussed it was implicitly assumed that the liquid
between the surfaces was a structureless continuum, characterised by its bulk dielectric
properties. However, at short separations this is not a good approximation and the
molecular nature of the liquid has to be considered. The presence of the surface will
Fig. 11. The solid lines show the expected force between spheres after adding a hydrophobic attraction of the
form Fr R ŽmNrm. sy150 expŽy Dr l ., where l s1.5 nm, to the DLVO-forces in Fig. 8b. The Hamaker
constant for the quartz–water–air bubble system is y0.8=10y2 0 J, the surface potentials where in Ža. equal
on both surfaces, 50 mV, while in Žb. they where 50 and y30 mV. The long dashed curves correspond to
double-layer forces only, whereas the short dashed ones are obtained after adding the van der Waals force.
J.C. Froberg
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23
modify the packing of liquid molecules outside it, and the density of the liquid will vary
with the separation from the surface in an oscillatory fashion approaching the bulk value
a few molecular layers away from the surface. For spherical molecules the periodicity of
the oscillations is close to the diameter of the molecule. When two particles come
sufficiently close to each other their solvent layers start to overlap. This results in a
change in liquid density next to the surface that also varies in an oscillatory fashion with
particle separation. This change in liquid density is the cause of the oscillating structural
force observed between smooth solid surfaces immersed in various liquids ŽChristenson,
1988; Israelachvili, 1991..
The situation in water is, because of its hydrogen bonding properties, more complicated than in non-polar liquids. Measurements of the forces acting between mica
surfaces in electrolyte solutions have shown that once the salt concentration is high
enough, typically above 1–10 mM depending on the electrolyte, a short-range repulsive
force decaying exponentially with a decay constant of about 1 nm dominates the
interaction ŽPashley, 1981a,b; Pashley and Israelachvili, 1984.. One example of such a
force curve is provided in Fig. 9. This force, known as a hydration force, is in the case
of mica presumably due to dehydration of adsorbed cations. It has been given a
theoretical interpretation by Gruen and Marcelja in terms of a surface induced polarisation of the liquid water ŽGruen and Marcelja, 1983.. We note that high precision
measurements have shown that at short separations an oscillatory force profile is
superimposed on the exponentially decaying hydration force ŽIsraelachvili and Pashley,
1983. indicating, not surprisingly, that normal packing constraints as well as specific
surface–water interactions are influencing the short-range forces between hydrophilic
surfaces in aqueous solutions. It has been convincingly demonstrated that hydration
forces may affect the stability of colloidal dispersions ŽHealy et al., 1978. but it has to
our knowledge not yet been shown that hydration forces are important in the flotation
process.
4.6. Interactions due to the presence of polymers
High-molecular-weight polymers, used as depressants and flocculating agents comprises an important component in mineral processing. Organic regulating agents, represented by natural polymers Žor their derivatives., such as modified cellulose, starch, and
tannins are also commonly used ŽLeja, 1982.. The presence of polymers in solution
andror on particle surfaces affects the interparticle forces in a way that depends on
whether they adsorb to the particle surface irreversibly, reversibly or not at all. Other
factors which determine the sign, range and magnitude of polymer induced forces are,
e.g., solventrpolymer interactions, surface coverage and polymer concentration ŽFleer et
al., 1993.. The different types of forces which may be generated by polymers are
discussed briefly below.
4.6.1. Forces between surfaces carrying irreÕersibly adsorbed polymers
Models describing the interaction between particles carrying irreversibly adsorbed
flexible polymers have been developed by de Gennes Ž1979. and by Scheutjens and
Fleer ŽFleer et al., 1993.. These theories are applicable when the polymer adsorp-
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¨
tionrdesorption rate is slow compared with how fast the polymer coated interfaces
approach each other. Under such circumstances the total amount of polymer on the
surfaces is independent of the surface separation and the system is not in true
equilibrium with bulk solution. However, it is often the case that the speed of approach
is sufficiently slow for the irreversibly adsorbed polymers to adopt the most favourable
conformation for each surface separation. Hence, there is an equilibrium within the
layer. This condition is referred to as quasi-equilibrium or restricted equilibrium.
Non-ionic polymers may give rise to an attractive bridging interaction when polymer
segments belonging to the same molecule are bound to two surfaces, whereas polyelectrolytes may generate bridging forces once segments from the same macromolecule are
close to both surfaces ŽDahlgren et al., 1993.. Bridging is important when the surface
coverage is low. This situation may arise when the polymer concentration is low or
when the time for adsorption is short. In such cases it is possible for the polymer to
adopt a larger number of favourable conformations when the surfaces are at intermediate
distances from each other compared to when they are far apart, resulting in an increase
in entropy and this is the molecular mechanism behind the attraction. However, at small
enough separations the number of available polymer conformations decreases again
which results in a repulsion. The range of the attraction is determined by the length of
the polymer tails extending from the surface.
As the surface coverage increases the effect of bridging polymers becomes less
important, whereas force contributions arising from interactions between polymers
adsorbed onto different surfaces become dominant. Such contributions are: Ži. an
osmotic contribution including changes in ideal entropy of mixing and changes in
solvation of the polymer segments, and Žii. an elastic contribution due to a reduction of
the number of possible conformations of the polymer chains at smaller surface separations.
Under restricted equilibrium conditions, the second contribution always becomes
increasingly repulsive as the surface separation is decreased, and it dominates at small
enough separations. The osmotic contribution may be either repulsive or attractive
depending on the solvent quality. It is unfavourable for polymer chains to interpenetrate
in good solvents, whereas such a process is favourable in poor solvents. The solvent
quality, which is a measure of the strength of the interaction between individual
segments compared to those between segment and solvent, is usually expressed by the
x-parameter. The x-parameter is larger than 0.5 in good and less than 0.5 in poor
solvents. Under poor solvency conditions the outer part of the interaction between
polymer coated surfaces is thus expected to be attractive.
The most effective steric stabilisers are often diblock copolymers where one of the
blocks, the anchor block, is strongly adsorbed to the surface whereas the other block, the
buoy, experiences good solvency conditions and thus extends far out into the solution.
One example of such a block copolymer is B 8 E 41 , where the butylene block ŽB. anchors
the polymer to hydrophobic surfaces or to air–water interfaces whereas the ethylene
oxide block ŽE. extends into the solution. The forces measured between two hydrophobised mica surfaces coated with a layer of B 8 E 41 are shown in Fig. 12. Here, we see that
adsorption of the block copolymer completely masks the attractive van der Waals and
hydrophobic forces that act between uncoated hydrophobic surfaces, and instead a
J.C. Froberg
et al.r Int. J. Miner. Process. 56 (1999) 1–30
¨
25
Fig. 12. Force normalised by radius as a function of absolute surface separation between hydrophobised mica
surfaces across a 0.1-mM KBr solution containing 50 ppm B 8 E 41.
purely repulsive steric force is encountered. The functional form of the measured
interaction agrees well with forces calculated using lattice mean field theory ŽSchillen
´ et
al., 1997..
The force Ždisjoining pressure. acting between two air–water interfaces in a single
foam film stabilised by B 8 E 41 were measured employing the thin film balance technique. Just as was the case for hydrophobic solid surfaces coated with this polymer, the
interaction is completely dominated by a steric force ŽFig. 13.. The range of the force is
about the same, 22–23 nm, at all B 8 E 41 concentrations Ž20–300 ppm.. This indicates
that the length of the longest tails is the same at all polymer concentrations. However, as
Fig. 13. Disjoining pressure isotherms for single foam films stabilised by the diblock copolymer B 8 E 41. In
addition to the polymer, the solutions also contained 10 mM KBr. The diblock copolymer concentrations were
20 ppm Žl., 50 ppm Že., 100 ppm Ž'., and 300 ppm Ž^.. The measurements were carried out with the thin
film balance technique.
26
J.C. Froberg
et al.r Int. J. Miner. Process. 56 (1999) 1–30
¨
the polymer concentration is increased the steric interaction at smaller distances becomes larger, indicating that as the adsorbed amount increases it becomes increasingly
unfavourable to compress the adsorbed layers. Another feature that is evident from the
data displayed in Fig. 13 is that one may distinguish three regimes in the disjoining
pressure isotherm. In the outermost regime the film thickness decreases rapidly with
increasing pressure. However, once the disjoining pressure has reached about 2 kPa, the
isotherm becomes considerably steeper. The film thickness at this transition is observed
at larger separations for more concentrated polymer solutions Žfrom 18.5 nm in the 20
ppm solution to 21 nm in the 300 ppm B 8 E 41 solution.. The reason for this change in
character of the interaction is not clear. At very high pressures the slope of the disjoining
pressure isotherm decreases again. We propose that this is due to a reduction in the
adsorbed amount at high pressures. This depletion of the polymer from the interface
eventually leads to rupture of the foam film at high pressures.
4.6.2. Forces between surfaces carrying reÕersibly adsorbed polymers
The theoretical models developed by de Gennes Ž1987. and Scheutjens and Fleer
ŽFleer et al., 1993. both predict that under full equilibrium conditions, i.e., when
polymers are free to desorb as the particles approach each other, purely attractive forces
are generated by homopolymers Žde Gennes, 1982.. The reason for this is that, as some
polymers desorb from the surfaces, a greater fraction of the remaining polymers will
attach to both surfaces and a bridging attraction will pull them towards each other.
Hence, there cannot be any steric stabilisation by adsorbed homopolymers under full
equilibrium conditions, but the reason that homopolymers can be used as steric
stabilisers is that, due to a very slow desorption, they are retained on the surface during
a collision event and the interactions are thus better described by theories for interactions between surfaces carrying irreversibly adsorbed polymers.
4.6.3. Forces due to non-adsorbing polymers
The concentration of polymer segments of non-adsorbing polymers is for entropic
reasons lower close to a surface than in bulk solution. An attractive osmotic force is
generated when the polymer depleted layer extends all the way between the surfaces.
The magnitude of this so-called depletion attraction increases strongly with the polymer
concentration. In concentrated systems the magnitude of the force is sufficient to cause
flocculation ŽFeigin and Napper, 1980; Fleer et al., 1993..
5. Final remarks
It can be said that the interactions between like interfaces are fairly well understood,
and the mathematical treatment allows for fairly easy calculations of the interactions
present. However, systems more apt to flotation most often involves interfaces which are
different, for instance with respect to their surface charge Žor potential., their ability to
deform under load, and whose affinity for the additives present in most applied systems
differ. Another question not dwelled upon here is the fact that presence of divalent and
other ions of higher valency will necessitate further assumptions in the treatment of the
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¨
27
double-layer interactions. Yet another complication is the vast presence of hydrophobic
molecules which imply that hydrophobic interactions should be taken into account, and
today there is no widespread agreement on how to do this. It therefore becomes
necessary to invoke more approximate relations for asymmetric systems which facilitate
calculations for estimates of the interactions present, methods which might have less
predictive power. It is nevertheless essential to obtain an assessment of the magnitude
and range of the interaction forces to be expected in a certain system, in order to be able
to predict its behaviour and make proper adjustments to the formulations used. The
development of new techniques such as the AFM and Thin Film Balance, also makes it
possible to experimentally determine the interaction profile for air-bubble–particle
systems. Increased understanding of the interactions in such systems may provide a new
means to enhance the efficiency of certain processes by removing constituents rather
than adding a few more, which in turn can turn out to be a more efficient way to obtain
an ecologically acceptable and therefore economical process.
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