INTERPARTICLE FORCES

Long-range particle interactions associated with elec-trical double layers and van der Waals forces are

dis-INTERPARTICLE FORCES 175 Figure 7.1 Interparticle forces at the particle level: (a) skeletal forces by external loading, (b) particle level forces, and (c) contact level forces (after Santamarina, 2003).

cussed in Chapter 6. These interactions control the flocculation–deflocculation behavior of clay particles in suspension, and they are important in swelling soils that contain expanding lattice clay minerals. In denser soil masses, other forces of interaction become impor-tant as well since they may influence the intergranular stresses and control the strength at interparticle con-tacts, which in turn controls resistance to compression and strength. In a soil mass at equilibrium, there must be a balance among all interparticle forces, the pres-sure in the water, and the applied boundary stresses.

Interparticle Repulsive Forces

Electrostatic Forces Very high repulsion, the Born repulsion, develops at contact points between particles.

It results from the overlap between electron clouds, and it is sufficiently great to prevent the interpenetra-tion of matter.

At separation distances beyond the region of direct physical interference between adsorbed ions and hy-dration water molecules, double-layer interactions pro-vide the major source of interparticle repulsion. The theory of these forces is given in Chapter 6. As noted there, this repulsion is very sensitive to cation valence, electrolyte concentration, and the dielectric properties of the pore fluid.

Surface and Ion Hydration The hydration energy of particle surfaces and interlayer cations causes large repulsive forces at small separation distances between unit layers (clear distance between surfaces up to about 2 nm). The net energy required to remove the last few

layers of water when clay plates are pressed together may be 0.05 to 0.1 J / m². The corresponding pressure required to squeeze out one molecular layer of water may be as much as 400 MPa (4000 atm) (van Olphen, 1977).

Thus, pressure alone is not likely to be sufficient to squeeze out all the water between parallel particle sur-faces in naturally occurring clays. Heat and / or high vacuum are needed to remove all the water from a fine-grained soil. This does not mean, however, that all the water may not be squeezed from between interparticle contacts. In the case of interacting particle corners, edges, and faces of interacting asperities, the contact stress may be several thousand atmospheres because the interparticle contact area is only a very small pro-portion (_⬍⬍1%) of the total soil cross-sectional area in most cases. The exact nature of an interparticle con-tact remains largely a matter for speculation; however, there is evidence (Chapter 12) that it is effectively solid to solid.

Hydration repulsions decay rapidly with separation distance, varying inversely as the square of the dis-tance.

Interparticle Attractive Forces

Electrostatic Attractions When particle edges and surfaces are oppositely charged, there is attraction due to interactions between double layers of opposite sign.

Fine soil particles are often observed to adhere when dry. Electrostatic attraction between surfaces at differ-ent potdiffer-entials has been suggested as a cause. When the

gap between parallel particle surfaces separated by dis-tance d at potentials V₁ and V₂ is conductive, there is an attractive force per unit area, or tensile strength, given by (Ingles, 1962)

⫺6 2

4.4_⫻ 10 (V_1⫺ V )_{2 2}

F_{⫽ 2} N / m (7.1)

d

where F is the tensile strength, d is in micrometers, and V₁ and V₂are in millivolts. This force is indepen-dent of particle size and becomes significant (greater than 7 kN / m² or 1 psi) for separation distances less than 2.5 nm.

Electromagnetic Attractions Electromagnetic at-tractions caused by frequency-dependent dipole inter-actions (van der Waals forces) are described in Section 6.12. Anandarajah and Chen (1997) proposed a method to quantify the van der Waals force between particles specifically for fine-grained soils with various geomet-ric parameters such as particle length, thickness, ori-entation, and spacing.

Primary Valence Bonding Chemical interactions between particles and between the particles and adja-cent liquid phase can only develop at short range. Co-valent and ionic bonds occur at spacings less than 0.3 nm. Cementation involves chemical bonding and can be considered as a short-range attraction.

Whether primary valence bonds, or possibly hydro-gen bonds, can develop at interparticle contacts with-out the presence of cementing agents is largely a matter of speculation. Very high contact stresses be-tween particles could squeeze out adsorbed water and cations and cause mineral surfaces to come close to-gether, perhaps providing opportunity for cold weld-ing. The activation energy for soil deformation is high, in the range characteristic for rupture of chemical bonds, and strength behavior appears in reasonable conformity with the adhesion theory of friction (Chap-ter 11). Thus, in(Chap-teratomic bonding between particles seems possible. On the other hand, the absence of co-hesion in overconsolidated silts and sands argues against such pressure-induced bonding.

Cementation Cementation may develop naturally from precipitation of calcite, silica, alumina, iron ox-ides, and possibly other inorganic or organic com-pounds. The addition of stabilizers such as cement and lime to a soil also leads to interparticle cementation. If two particles are not cemented, the interparticle force cannot become tensile; they loose contact. However, if a particle contact is cemented, it is possible for some interparticle forces to become negative due to the ten-sile resistance (or strength) of the cemented bonds.

There is also an increase in resistance to tangential force at particle contacts. However, when the bond breaks, the shear capacity at a contact reduces to that of the uncemented contacts.

An analysis of the strength of cemented bonds should consider three cases: (i) failure in the cement, (ii) failure in the particle and (iii) failure at the ce-ment–particle interface. The following equation can be derived (Ingles, 1962) for the tensile strength T per unit area of soil cross section:

1 n

 ⫽^T Pk冉 冊^{1 ⫹e} _冘ⁿ _A (7.2)

i 1

where P is the bond strength per contact zone, k is the mean coordination number of a grain, e is the void ratio, n is the number of grains in an ideal breakage plane at right angles to the direction of T, and Ai is the total surface area of the ith grain.

For a random and isotropic assembly of spheres of diameter d, Eq. (7.2) becomes

 ⫽T d (12Pk_⫹ e) (7.3)

For a random and isotropic assembly of rods of length l and diameter d

 ⫽_T Pk (7.4)

d(l_⫹d / 2)(1 _⫹e)

Bond strength P is evaluated in the following way (Fig.

7.2) for two cemented spheres of radius R. It may be shown that

(R_⫺ cos_)

cosh _⫽ R sin_ (7.5)



so for known_,_can be computed. Then, for cement failure,

P_⫽  ⫻ c 2 (7.6)

where _c is the tensile strength of the cement; for sphere failure,

P_{ ⫽ ⫻ }s (_)2 (7.7) where _{ ⫽}R sin _, and _s is the tensile strength of the sphere, and for failure at the interface

INTERPARTICLE FORCES 177

Figure 7.2 Contact zone failures for cemented spheres.

sin_{ 2}

P_{ⴖ ⫽ ⫻1 ⫻} 2_R (1 _⫺ cos_) (7.8)



where _1 is the tensile strength of the interface bond.

In principle, Eq. (7.6), (7.7), or (7.8) can be used to obtain a value for P in Eq. (7.2) enabling computation of the tensile strength _Tof a cemented soil.

The behavior of cemented soils can depend on the timing of cementation development. Artificially ce-mented soils are often loaded after cementation has developed, whereas cementation develops during or af-ter overburden loading in natural soils. In the former case, the particles and cementation bonding are loaded together and contact forces can become negative de-pending on the tensile resistance of cementation bond-ing. The distribution and magnitude of skeletal forces are therefore influenced by both geometric arrange-ment of particles and the cearrange-mentation bonding at the particle contacts. In the latter case, on the other hand, the contact forces induced by external loading are de-veloped before cementation coats the already loaded particles. In this case, it is possible that cementation creates extra forces at particle contacts. In some

ce-mented natural materials, if the soil is unloaded from high overburden stress, elastic rebound may disrupt ce-mented bonds.

Cementation allows interparticle normal forces to become negative, and, therefore, the distribution and evolution of skeletal forces may be different than in uncemented soils, even though the applied external stresses are the same. Thus, the stiffness and strength properties of a soil are likely to differ according to when and how cementation was developed. How to account for this in terms of effective stress is not yet clear.

Capillary Stresses Because water is attracted to soil particles and because water can develop surface tension, suction develops inside the pore fluid when a saturated soil mass begins to dry. This suction acts like a vacuum and will directly contribute to the effective stress or skeletal forces. The negative pore pressure is usually considered responsible for apparent and tem-porary cohesion in soils, whereas the other attractive forces produce true cohesion.

When the soil continues to dry, air starts to invade into the pores. The air entry pressure is related to the pore size and can be estimate using the following equa-tion, assuming a capillary tube as shown in Fig. 7.3a:

2awcos 

Pˆc_⫽ (7.9)

rp

where P^ˆc is the capillary pressure at air entry, aw is the air–water interfacial tension,is contact angle de-fined in Fig. 7.3, and rp is the tube radius. For pure water and air, aw depends on temperature, for exam-ple, it is 0.0756 N / m at 0_C, 0.0728 N / m at 20_C, and 0.0589 N / m at 100_C. If the capillary pressure Pc

(_⫽ ua ⫺ uw, where ua and uw are the air and water pressures, respectively) is larger than P ,^ˆc then air in-vades the pore.²Since soil has pores with various sizes, the water in the largest pores is displaced first followed by smaller pores. This leads to a macroscopic model of the soil–water characteristic curve (or the capillary pressure–saturation relationship), as discussed in Sec-tion 7.11.

If the water surrounding the soil particles remains continuous [termed the ‘‘funicular’’ regime by Bear (1972)], the interparticle force acting on a particle with radius r can be estimated from

2It is often assumed that u_a⫽ 0 (for gauge pressure) or 1 atm (for absolute pressure). However, this may not be true in cases such as rapid water infiltration when air in the pores cannot escape or the air boundary is completely blocked.

Capillary Tube Representing a Pore

2rp

d_c

θ

(a) (b)

u_a

u_w P_c=ρwgd_c= r_p

^ 2σawcosθ

Figure 7.3 Capillary tube concept for air entry estimation: (a) capillary tube and (b) bundle of capillary tubes to represent soil pores with different sizes.

2r2 aw cos

2 ˆ

Fc_{⫽ }r Pc_⫽ (7.10)

rp

where rp is the size of the pore into which the air has entered. Since the fluid acts like a membrane with neg-ative pressure, this force contributes directly to the skeletal forces like the water pressure as shown in Fig.

7.4a.

As the soil continues to dry, the water phase be-comes disconnected and remains in the form of me-nisci or liquid bridges at the interparticle contacts [termed the ‘‘pendular’’ regime by Bear (1972)]. The curved air–water interface produces a pore water ten-sion, which, in turn, generates interparticle compres-sive forces. The force only acts at particle contacts in contrast to the funicular regime, as shown in Fig. 7.4b.

The interparticle force generally depends on the sep-aration between the two particles, the radius of the liq-uid bridge, interfacial tension, and contact angle (Lian et al., 1993). Once the water phase becomes discontin-uous, evaporation and condensation are the primary mechanisms of water transfer. Hence, the humidity of the gas phase and the temperature affect the water va-por pressure at the surface of water menisci, which in turn influences the air pressure ua.