Chapter 4. Interparticle Forces: A Basis for the Interpretation of Soil Physical Behavior
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J. P. QUIRK
I. INTRODUCTION
Micromorphologists (see, e.g., Brewer, 1988) have made substantial efforts
toward defining soil structure and have generated numerous terms to describe soil
characteristics as observed by the microscope. Although such research has enlarged our knowledge of soils, particularly in relation to soil classification, the
real challenge in defining soil physical behavior lies at a level of discrimination
beyond that of the light microscope.
Soil structure has been defined by Baver (1940) as “the arrangement of soil
particles” which determines the arrangement and size of soil pores. Soil particles
with an equivalent spherical diameter of < 2 pm are described as clay and are
referred to as the “active” fraction of a soil; the principal component of this
fraction is generally the layer lattice silicates which have been extensively described by clay mineralogists (Brindley, 1980). As a result there is a substantial
body of information about the crystalline swelling of smectites and vermiculites
in various homoionic forms. The clay fraction of a soil frequently contains
minerals such as illite, which may be very fine grained, and kaolinite; both of
these minerals have basal spacings which are fixed at all water contents. In the
absence of crystalline swelling, the increase in volume on wetting these materials, or a clay soil containing them, takes place as a result of interparticle
(crystal) interaction. Because of the irregular nature of the clay crystals themselves, especially the distribution of crystal thicknesses and terraced surfaces, the
crystal interaction resulting in swelling cannot be thought of in exactly the same
way as the interaction between the regular 10 A thick aluminosilicate layers
which is the basis of intracrystalline swelling of smectites and vermiculites and
which can be observed by Xray diffraction.
Furthermore, the crystals and lamellae do not exist as separate entities but are
assembled into compound particles which have been described as clay domains
(Aylmore and Quirk, 1960) and as quasicrystals of smectites which also assemble to form clay domains (Quirk and Aylmore, 1971). The swelling or water
uptake by a clay or soil involves the complex interaction between clay domains,
between crystals (or quasicrystals) within a domain, and between the lamellae
within quasicrystals. These interactions are occasioned by the operation of a
number of interparticle forces, some of which are attractive and oppose swelling,
and others which are repulsive and facilitate swelling. The manner in which these
forces vary with the distance of separation of contiguous surfaces is an important
consideration in arriving at a fundamental basis for the swelling of soils and clays
due to the interplay of these forces.
For soils with a texture between clay loam and heavy clay, particle interaction
is expressed as a measurable increase in volume. However, such particle interactions also occur within the interstices of lighter textured soils. This swelling is
INTERPARTICLE FORCES
123
internally accommodated but continues to influence soil physical properties such
as permeability.
Another feature of the “active” clay particles is that their plateshaped character results in the particles (crystals and quasicrystals) within a soil taking up
positions of near parallel alignment. The pore sizes in such an assemblage of
particles will reflect the thickness of the crystals or quasicrystals and there is,
inevitably, a considerable overlap of particle surfaces to give close distances of
approach.
In addressing the operation of interparticle forces, it is necessary to know the
interparticle distances. This has been achieved by the measurement of pore size
distributions using low temperature nitrogen desorption isotherms and other
means (Aylmore and Quirk, 1967; Sills et al., 1973; Murray et al., 1985).
Natural aggregates, even from soils which swell appreciably when fully wet,
have a mass ratio of solid to liquid of 2 or greater reflecting the fact that a soil
should properly be regarded as a condensed system quite distinct from a suspension for which the so1id:liquid ratio may be 0.1 (a 10% suspension) or less.
Bradfield’s (1936) statement that “granulation is flocculation plus” needs to be
modified since the particle arrangement and distances of surface separation in a
floccule are vastly different from those of condensed clay particles within a clay
or soil. For regions of particle surface overlap within a Ca clay domain, the
surfaces are within what colloid scientists refer to as the primary potential minimum, that is the association is face to face and the separation is small. Particles
in a primary minimum are adhesive and are not readily separated. In contrast the
dispersionflocculation transition is the result of a secondary minimum, involves
cardhouse type structures, and is readily reversible. The balance between the
forces when Ca clay particles are within the potential minimum must be delicate
since 15% of Na ions or even less on the exchange sites can dramatically alter the
physical behavior of a soil.
The “plus” referred to by Bradfield encompasses a range of substances which
are capable of stabilizing a soil aggregate against the disruptive forces of slaking
due to immersion in water or exposure of the soil surface to intense rain. These
stabilizing substances are sometimes referred to as cements and are considered to
include the clay particles themselves, iron and aluminium oxides, or their precursors, silica, calcium carbonate, and parts of the organic fraction of soils. The
profound effect of organic matter in influencing the water stability of aggregates
remains largely unexplained. However, the variation of stability for a particular
soil type depending on management and period under an arable phase is well
documented (Bradfield, 1936; Greenland et al., 1962). The site of action of these
stabilizing agents within the porous matrix of a soil has received very little
attention. Quirk (1978) has defined soil stabilization by means of additives as
“the strategy of placing the most appropriate material at the most efficacious
place within the soil structure or pore space so that the desired strength may be
J. P. QUIRK
124
achieved, most economically for agricultural or engineering purposes.” Such an
aspiration may be fulfilled with an increased knowledge of soils as porous materials, particularly the micropores, and with advances in polymer science in
relation to organic and inorganic polymers. The establishment of pillared complexes (Frenkel and Shainberg, 1980; Pinnavaia, 1983) at strategic points within
a soil may be one such approach; such research also has the potential to extend
our knowledge of the basis for the stability of natural soil aggregates.
11, INTERPARTICLE FORCES
Swelling is one way in which the reaction between clay and water is manifest.
The forces which give rise to a greater separation of two interacting clay particles
(swelling) are restrained by the suction in the soil water, by Londonvan der
Waals forces, and by ionion correlation forces (Kjellander er al., 1988). The
forces which facilitate swelling are osmotic repulsive forces which arise from the
interaction of diffuse double layers between contiguous particles and the hydration of the exchangeable cations and their perturbation of the normal hydrogenbonded structure of water in the vicinity of the clay/solution interface (water
structural forces).
A. SUCTIONIN A POROUS
MATERIAL:
CAPILLARY
CONDENSATION
The pressure deficiency, P,, across a curved meniscus is given by the YoungLaplace equation,
P,=
?(t+ A)
cos8
where y is the surface tension of the liquid, r , and r2 are the two principal radii of
curvature, and 8 is the contact angle which for readily wet surfaces is zero so that
cos 8 = I . Within a porous material, such as soil, P, is expressed as a suction
which is transmitted throughout the soil water.
For a cylindrical capillary for which r l = r2 the pressure deficiency is 2 y / r ,
and in a slitshaped pore for which r2 % r , , the pressure deficiency is given by
yJr,.
The condensation of liquid into pores from the vapor phase (capillary condensation) is described by the Kelvin equation which gives the relative vapor pressure (PIP”) in equilibrium with a meniscus in a cylindrical capillary of radius r ,
125
INTERPARTICLE FORCES
= exp
(  rRT)
2vy
(2)
Po
where p is the vapor pressure above the meniscus, po is is.e saturated vapor
pressure of the liquid at temperature T (K), V is the molar volume of the liquid, y
its surface tension, and R the gas constant. Equation (2) can be rewritten,
and since the height of rise, h, in a capillary above a free water surface is given
by the relation p gh = 2y/r, the lowering of the vapor pressure can be expressed
in terms of h ,
where M is the molecular weight of the liquid and g is the gravitational constant.
For a slitshaped pore r is replaced by d, the slit width, in Eq. (3). In Table I the
values of pIpo and hydrostatic suction corrggonding to particular sizes of slitshaped pore have been calculated using Eqs. (3) and (4). Even for small pores it
is assumed that the bulk properkdof water, density and surface tension, are
applicable.
Table I shows the magnitude of the pressure deficiency (suction) residing in the
water in a porous material for a range of circumstances. This suction is transmitted to the pore walls and acts to draw the walls of pores closer together. The
suction is thus rightly considered to act in concert with other attractive forces to
resist swelling. These considerations provide the basis for soil watersuction
relationships (the moisture characteristic). For a clay soil, as the suction is
increased the interacting plates are brought close together and shrinkage takes
place.
Table I
Relative Vapor Pressure and Suction of Water at which SlitShaped Pores
of Various Widths Begin to Empty at 298°K
Slit width (nm)
Relative vapor pressure ( p l p , )
Hydrostatic suction (MPa)
1000
100
10
0.999
0.137
0.990
I .38
0.900
14.4
I
0.350
144
126
J. P. QUIRK
B. LONDONVANDER WMS FORCES
In 1930 London applied quantum mechanics to derive the force between two
apolar atoms arising from their mutually induced polarization. The basis of the
attractive force is that the fluctuating dipole of one atom polarizes the other one
and consequently the two atoms attract each other. The frequency of the fluctuations is of the order of the electronic frequency and is taken as 3 X 1015 secI
corresponding to the first ionization potential for the Bohr hydrogen atom. These
forces are also referred to as dispersion forces because of their link to the
dispersion of electromagnetic radiation.
In 1937, Hamaker introduced the idea that, for conglomerates of atoms in two
interacting macroscopic bodies, the London forces are pairwise additive; that is
the interaction of all atoms in both bodies contributes to the energy and force of
interaction (Verwey and Overbeek, 1948; Mahanty and Ninham, 1976; Israelachvili, 1985). The constant, A, which governs such interactions is referred to
as the Hamaker constant and is given by
A = $ h u o Q .2r r 2 q 2
where h is Planck’s constant, a0 is the static polarizability of the atoms, uo is the
characteristic electronic frequency, and q is the number of atoms per unit volume
of interacting bodies. Because of the additivity principle the energy of interaction
decreases much more slowly with distance than that between individual atoms
which decays as the inverse sixth power of distance.
This treatment is referred to as the microscopic basis for the Hamaker constant. The assumption of simple pairwise additivity inherent in Eq. (5) ignores
the influence of neighboring atoms on the interaction between any pair of atoms.
The problem of additivity is completely avoided in Lifshitz’s macroscopic theory
(Mahanty and Ninham, 1976) in which the atomic structure is ignored and the
forces between large bodies, treated as continuous media, are derived in terms of
such bulk properties as the dielectric constants and refractive indices.
The attractive pressure for two semiinfinite, thick, flat, parallel plates is given
by
A
PA = 6nD3
where D is the distance of separation of the plates. The value of A has been
determined experimentally for muscovite mica surfaces separated by air and
water (Israelachvili, 1985); the values reported were 13.5 X 10*0 and 2.2 X
1020 J, respectively. The principal contribution to this later value is in the
electronic frequency (3 X 10’5 sec I), however, the Keesom (dipoledipole),
Debye (dipoleinduced dipole), and other effects contribute less than 10% be
INTERPARTICLE FORCES
127
cause the frequencies involved are in the microwave region (10" secI). This
low frequency contribution is referred to as the static or zero frequency contribution and is affected by temperature (Israelachvili, 1985).
For a plate of thickness, t, the van der Waals interaction energy per unit area is
given by
"+
VAT A
121~ 0
1
2
(D
+ 2t)2
1


(D
+ t)*
(7)
and the pressure by
(D
+ 2t)3
(D
+ t)3
These equations are applicable over distances of separation from 0.2 (the
surface granularity) to 7 nm. Because of the finite time required for the propagation of the electromagnetic radiation the attractive energy is reduced when D
approaches c/u, where c is the velocity of light. The forces are then said to be
retarded and decline more rapidly with increasing D when the separation exceeds
about 7 nm for the system micawatermica.
Table I1 gives the van der Waals energy and pressure for varying distances of
plate separation for t = 1 nm, the thickness of an elementary aluminosilicate
layer in a montmorilloniteor vermiculite, and also for the circumstances when t is
much greater than D. At close distances of approach (0.25 nm) the energy of interaction is about 10 mlm2 (erg cm2). This may be compared with 107 d m  2
obtained by Bailey and Kay (1967) for the pristine cleavage of muscovite in
water. McGuiggan and Israelachvili (1988) reported values of 7 to 10 mlm2 for
the adhesion energy between two molecularly smooth muscovite surfaces at
Table II
van der Waals Interaction Energy and Attractive Pressure between Surfaces for the System
MicaWaterMica in Relation to Distance of Separation ( D ) and Plate Thickness (Oa
Surface separation (nm)
0.25
0.5
1.o
2.0
4.0
8.7
9.3
1.9
2.3
0.36
0.6
0.05
0.006
0.04
~~~~~~~~~~~
Energy ( d m  * )
t=lnm
T*D
Pressure (MPa)
t = lnm
t%D
a
74
75
8.7
9.3
The Hamaker constant for eqs. (7) and (8) is 2.2
0.15
0.9
1.2
X
0.08
0.15
J.
0.005
0.018
128
J. P. QUIRK
“contact” in water. Quirk and Pashley (1991a) have discussed the nature of
“contact” and have concluded that the mica surfaces in adhesive “contact” in
aqueous solutions are probably separated by two layers of water; they also
discussed the special role of H 3 0 + in enabling surfaces to be brought into
“contact.
From Table I1 it may be noted that the interaction energy and pressure, for
surfaces separated by water, decrease markedly with increasing distance of separation and that beyond Inm separation the magnitude of these quantities is
relatively small. For distance of separation of 1 nm or less Table I1 shows that the
energy and pressure of interaction are similar for t = 1 nm and t % D. Through
the use of Eq. (8) it can be shown, particularly for small separations, that the
attractive pressures calculated for a plate thickness of 5 nm are not very much
less than when t
D. Using Eq. (8) it can be calculated that the attractive
pressure between two plates 5 nm thick at a distance of 5 nm is 7 kPa whereas
when the same plates are at a distance of 0.5 or 1 nm the pressures are 9.3 and
1.2 MPa. These calculations can be considered in relation to the two separate sets
of slitshaped pores which would exist within a clay matrix; one set for which the
surface separation would be similar to the crystal thickness, for example, 5 nm,
and the other set in regions of crystal overlap for which the surface separation
might be I nm or less. The attractive pressure in the smaller pores, as seen from
these calculations, is several orders of magnitude greater than in the larger pores.
From the information in Tables I and I1 it is possible to conclude that for slitshaped pores greater than 1 nm, the van der Waals forces make only a minor
contribution to the forces resisting swelling in the usual suction regimes in soils.
”
C. OSMOTIC
REPULSIVEFORCES:
DIFFUSE
DOUBLELAYER
THEORY
The DLVO theory,* for the stability of colloidal suspensions, combines
Gouy’s original ideas on the diffuse distribution of counterions at particle surfaces in an aqueous environment with the Londonvan der Waals forces; the
Gouy treatment leads to the osmotic repulsive force for two interacting surfaces
(Langmuir, 1938). The DLVO theory incorporates a Stern layer as modification
of the Gouy double layer to allow for the fact that the counterions, as point
charges, can approach a surface without any limit and this gives rise to impossibly high concentrations. The double layer is divided into two parts, a Stern layer
approximately two water monolayers thick (5.5 A) in which there is a rapid fall
in potential to the value at the Gouy plane; the behavior of ions in the diffuse part
*The DLVO theory is a result of the independent work of Derjaguin and Landau in the USSR and
of Verwey and Overbeek (1948) in the Netherlands during World War 11.
INTERPARTICLE FORCES
129
of the double layer is governed by the Gouy potential. The total surface charge is
balanced by counterions in the Stem layer and by the excess of counterions over
coions in the Gouy layer.
The diffuse doublelayer theory involves two dimensionless parameters. One
concerns the balance between the electrical forces attracting a counterion to the
surface and the diffusion of counterions away from the surface. The balance
between these two opposing tendencies is expressed as the ratio of the electrical
and thermal energies, zeJl,lkT, in which z is the charge on the counterion, e is
the electronic charge, +G is the electrical potential at the origin of the diffuse
layer (Gouy potential), k is the Boltzmann constant, and T is the temperature
(OK).The other dimensionless parameter concerns the product of half the distance of separation of the Gouy planes of the interacting surfaces taken as 2x and
K from the DebyeHuckel theory for strong electrolyte solutions; K has the
dimensions of a reciprocal length and is given by
where ni(o) is the number concentration of ions far from the surface, z is the
dielectric constant, e is the electronic charge, and z is the valency of the ions in
solution. At 25°C the magnitude of the Debye length (KI) in A is 3 . 0 4 / G for a
1: 1 electrolyte, 1.76/< for a 2: 1 electrolyte, and 1.52/& for a 2:2 electrolyte;
c is the molar concentration.
The general equation for the case of interacting or overlapping diffuse double
layers for symmetric electrolytes has been presented by Verwey and Overbeek
( 1948)
dYlh
( 2 cash YG  2 cash U ) ’ Q
= K
(10)
where x is the distance from the midplane, Yc = z e q G / k T is the reduced or
scaled electric potential at the Gouy lane, and U is the reduced potential at the
midplane where dY/& = 0, Y = U ,and x = 0. Integration gives
1 I:”
=
dY
[2 cosh Yc  2 cosh Ull/z
This leads to an elliptic integral of the first kind for which tables are available;
it can therefore be solved numerically to obtain the potential distance curve and,
in particular, the midplane electric potential for different values of Y,, of plate
separation, of concentration (contained in K), and of counterion valency. Kemper
and Quirk ( 1970) have provided a nomogram which illustrates the interrelationship of these variables (see also Bresler et al., 1982).
The starting point for the application of this theory is the relationship
130
J. P. QUIRK
uG=
(7)
2nekT
sinhT
Yc
(12)
in which uGis the surface density of charge at the Gouy p.,ne and n is the
number concentration of ions.
It has been acknowledged for a long time that there was no satisfactory way of
estimating 3rC from the crystallographic charge of a surface on account of the
rapid fall in potential between the surface itself and the Gouy plane. Zeta potential values have not been considered satisfactory, except in a general way, because of the uncertainty of the position of the shear plane.
When appropriate values of the Gouy plane potential are available then the
midplane potential can be arrived at by the application of the justmentioned
theory. The equation of Langmuir (1938) can then be applied,
P, = 2 RTc ( C O SU~  1)
(13)
to obtain the swelling pressure P, due to the excess of ions at the midplane in
relation to the bulk concentration, c; Derjaguin and Churaev (1989) refer to this
swelling pressure as the electrostatistical component of the disjoining pressure.
1. Surface Potentials
To a significant extent the difficulty concerning Yc [Eq. (12)] has been clarified
(Chan et af.,1984) by the reinterpretation of the coion exclusion measurements
of Edwards et al. (1965), for a montmorillonite and illite saturated with alkali
metal ions. They derived an equation, based on double layer theory, which
enabled the Gouy plane potential for a clay to be obtained from the measured
volumes of coion (chloride) exclusion with respect to concentration. For an
interface of area, A, the volume of exclusion is given by
Vex = A
2
[I  exp(ze+,/2kT)]
It may be noted that for Gouy potentials of  150 mV the volume of exclusion
is within 5% of 2 / multiplied
~
by the surface area. This is the circumstance to
which Schofield’s (1947) equation was applied; he proposed negative adsorption
(coion exclusion) as a measure of surface area. However, this approach is not
justified as the Gouy potentials for clays are generally less negative than  100
mV (Table 111).
Equation (14) can be rewritten by noting that the slope of the plot of volume of
exclusion against 2 / that
~ is Vex ~ / 2 has
, the dimensions of a surface area which
is denoted as A, so that
131
INTERPARTICLE FORCES
Thus, when the surface area of a clay is known the surface potential can be
calculated by the application of Eq. (15). For a smectite the area, A, can be
calculated from crystallographic parameters and chemical composition and for an
illite or kaolinite the nitrogen surface area is used. Chan et al. (1984) have
reported the surface potential values shown in Table 111.
From the good fit of the experimental results to Eq. (14), that is a plot of Vex
against 2 / for
~ a range of concentrations from 104 to 101 M of alkali metal
chloride solutions (Edwards et al., 1965), it would seem that clay surfaces
behave as constant potential surfaces. The authors described this finding as
surprising because the surface charge of clays results from isomorphous substitution in the crystal lattice and thus a constant charge behavior for the double layer
would have been expected. It was concluded that, over the above concentration
range, there must be a potentialregulating mechanism for which there is currently no theoretical treatment. Miller and Low (1990) reported similar results to
those of Edwards et al. (1965) and using Eq. (15) arrived at a similar conclusion.
There must, of course, be some limit to the constant potential behavior, otherwise the Gouy layer charge would exceed the crystallographic charge at high
concentrations. Kemper and Quirk (1972) have reported a considerable decrease
in the negative electrokinetic potentials at concentrations above 0.1 M NaCl for
Na clays.
According to Eq. (12), for the potential to remain constant with increasing
concentration, uGmust increase and hence fewer ions would be accommodated
in the Stem layer. This does not accord with the Langmuir treatment, on which
the Stem theory is based, which requires that more ions should be accommodated
in the Stem layer with increasing concentration. Horikawa et al. (1988), using
electrokinetic measurements, showed that there is some dependency of zeta
potential on concentration. However, the observed behavior resembles that of
constant potential more closely than constant charge.
Table 111
Surface Potential (mV) for Fithian Illite and Wyoming Montmorillonite
Saturated with Alkali Metal Ions [Eq. (15)]
Ion
Mineral
Li
Na
K
Rb
Illite
Montmorillonite
65
90
51
 19
5
69
44

CS
0
 12
J. P. QUIRK
I32
There is only a limited amount of information for the surface potentials of Ca
clays. The coion exclusion results of Edwards et af. (1965) for Caillite (Fithian)
indicate a surface potential of  1 I mV. From osmotic efficiency measurements
on clay plugs, Kemper and Quirk (1 972) obtained values of zeta potentials for Ca
clays at concentrations in the vicinity of 0.1 M CaCl,. Calciumkaolinite and CaWillalooka illite had potentials of about  10 mV; Fithian illite and Wyoming
montmorillonite had potentials, respectively, of 20 and 25 mV. Horikawa et
af. (1988) found zeta potential values for CaWyoming montmorillonite around
 10 mV, and for CaMuloorina illite the potentials ranged from 6 to  17 mV
to
M CaCl,. In general terms it would
over the concentration range of
seem that the surface potentials of Ca clays could be considered to be in the range
of  10 to 20 mV. That is, a reduced electrical potential at the Gouy plane (Yc)
of 0.8 to I .6 which contrasts with values for Na clays of 2.0 to 3.0.
2. Swelling Pressures
In Table 1V a comparison is made of the swelling pressures obtained when the
diffuse doublelayer theory is applied for reduced Gouy electrical potential values of Yc = 1, Yc = 2 , and Yc = 3 with a Gouy plane separation of 20 8, in a 0.1
M solution of a 1: 1 and a 2:2 electrolyte.
In making a comparison of the swelling behavior of Na and Ca clays two
features need to be considered. First the Ca clays have a smaller negative potential than Na clays so that in Table 1V for YG = 1 and a 0. I M solution of a 2:2
electrolyte, the swelling pressure is 0.01 MPa but for Yc = 3 in the presence of a
1: 1 electrolyte and also at 0.1 M the swelling pressure is 0.67 MPa. Second, for a
clay such as NaWyoming montmorillonite the whole surface of 750 m2g1 is
involved in the swelling process whereas the interactions for Camontmorillonite
are between the external surfaces of the quasicrystals after the crystalline swelling to a d(001) value of 19 A is complete at a suction of about 10 MPa (see Fig. 6
in Section V).
Table IV
Calculated Swelling Pressure (Mh)for Na and Ca Clay
Surfaces with a Separation of 20 between the Gouy
Planes and with Varying Reduced Electric Potentials
in 1:1 and 2:2 Electrolyte Solutions (0.1 M )
Reduced surface potentials
Electrolyte type
Y, = I
YG = 2
Y, = 3
I:I
212
0.09
0.33
0.01
0.05
0.67
0.09