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Chapter 4. Interparticle Forces: A Basis for the Interpretation of Soil Physical Behavior

Chapter 4. Interparticle Forces: A Basis for the Interpretation of Soil Physical Behavior

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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 X-ray 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



internally accommodated but continues to influence soil physical properties such

as permeability.

Another feature of the “active” clay particles is that their plate-shaped 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


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

dispersion-flocculation transition is the result of a secondary minimum, involves

card-house 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



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.


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 London-van der

Waals forces, and by ion-ion 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).





The pressure deficiency, P,, across a curved meniscus is given by the YoungLaplace equation,


?(t+ A)


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 slit-shaped pore for which r2 %- r , , the pressure deficiency is given by


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 ,



= exp

( - rRT)




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 slit-shaped 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


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 water-suction

relationships (the moisture characteristic). For a clay soil, as the suction is

increased the interacting plates are brought close together and shrinkage takes


Table I

Relative Vapor Pressure and Suction of Water at which Slit-Shaped Pores

of Various Widths Begin to Empty at 298°K

Slit width (nm)

Relative vapor pressure ( p l p , )

Hydrostatic suction (MPa)







I .38









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 sec-I

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 pair-wise 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 pair-wise 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 semi-infinite, thick, flat, parallel plates is given



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

10-20 J, respectively. The principal contribution to this later value is in the

electronic frequency (3 X 10’5 sec- I), however, the Keesom (dipole-dipole),

Debye (dipole-induced dipole), and other effects contribute less than 10% be-



cause the frequencies involved are in the microwave region (10" sec-I). 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



121~ 0




+ 2t)2





+ t)*


and the pressure by


+ 2t)3


+ 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 mica-water-mica.

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 mlm-2 (erg cm-2). 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 mlm-2 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

Mica-Water-Mica in Relation to Distance of Separation ( D ) and Plate Thickness (Oa

Surface separation (nm)
















Energy ( d m - * )



Pressure (MPa)

t = lnm







The Hamaker constant for eqs. (7) and (8) is 2.2












“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


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 I-nm 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 slit-shaped 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.






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 London-van 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.



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 double-layer 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 Debye-Huckel 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 (K-I) 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)


( 2 cash YG - 2 cash U ) ’ Q

= K


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:”



[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









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 just-mentioned

theory. The equation of Langmuir (1938) can then be applied,

P, = 2 RTc ( C O SU~ - 1)


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


[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



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 10-4 to 10-1 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 potential-regulating 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)]












- 19







- 12



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 Ca-illite (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,. Calcium-kaolinite 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 Ca-Wyoming montmorillonite around

- 10 mV, and for Ca-Muloorina illite the potentials ranged from -6 to - 17 mV


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 double-layer 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 Na-Wyoming montmorillonite the whole surface of 750 m2g-1 is

involved in the swelling process whereas the interactions for Ca-montmorillonite

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









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