Capillary bonding

Vincent Richefeu and Farhang Radjai

1 Capillary cohesion

We present here the capillary cohesion resulting from a liquid bridge between two particles. The bridge between two particles of different sizes takes a complex shape as illustrated in figure 1. R1 and R2 are the particle radii, ζ1 and ζ2 are the filling angles (corresponding to the wetted surface of the particles), θ is the wetting angle, and δn is the distance between particles. The axis x coincides with the axis of rotation of the liquid bridge, and the coordinate y describes the profile of the meridian of the bridge as a function of x. The radius of the bridge is denoted y0. The coordinates x of the three-phase contact line (also called triple line), i.e. the line defining the solid-liquid-gas interface, are denoted by xc1 and xc2 respectively for particles 1 and 2.

In the following, we assume that

The pressure difference Δp = pgaz -pliquide (also called capillary suction) through the liquid-gas interface is related to the curvature of the liquid bridge and the surface tension of the liquid σ by the Young-Laplace equation (Hotta et al. [1974], Soulié et al. [2006]):

       [                             ]
            y′′(x)             1
Δp  = σ -----′2---3∕2 - y(x)∘1-+-y′2(x)
        (1+ y  (x ))

The volume of the bridge is given by:

V = π∫xc2y2(x)dx  -  1 πR3 (1 - cosζ )2(2 + cosζ )
      xc1             3    1       1         1
                     1    3         2
                  -  3 πR 2(1 - cosζ2) (2 + cosζ2)

The interparticle distance δn can be expressed in the coordinates xc1 and xc2:

D = R2 (1 - cosζ2)+ xc2 + R1 (1 - cosζ1)- xc1


Figure 1: Geometry of a liquid bridge between two particles of different sizes.

The capillary force can be calculated at the gorge (Hotta et al. [1974], Lian et al. [1993], Mikami et al. [1998]):

F = 2 πy0 σ + πy20 Δp

The relationship between the capillary force and the shape of the liquid bridge is described by a system of nonlinear equations (1) to (4). However, in discrete numerical simulations an explicit expression of fn is needed as a function of interparticle distance δn and volume of liquid bridge V for any set of parameters R1, R2, σ and θ.

2 Implementation

Numerical resolutions of the system of equations have been proposed by several authors (Soulie et al. [2006], Soulié et al. [2006], Scholtes et al. [2009b,a]). The major drawback of these approaches is that they do not provide a physically interpretable form of the capillary law. Moreover, they are not very efficient in terms of computation time. Richefeu  it et al. (Richefeu et al. [2006], Richefeu et al.) proposed a simple expression for the capillary force (figure 2b):

    (     √-----
    {  - κ√R1R2- -dn∕λ  for  dn < 0
fn = ( - κ R1R2e       for  0 ≤ dn ≤ drupture
       0               for  dn > drupture

where κ = 2πσ cosθ (Willett et al. [2000], Bocquet et al. [2002], Herminghaus [2005]), and λ = λ(V,R1,R2) expresses a length scale that governs the exponential decay of the capillary force as a function of the gap (Richefeu et al.). This explicit expression gives an excellent approximation of the capillary force obtained by the integration of the Young-Laplace equation; see figure 2b.

(a) PIC (b) PIC

Figure 2: (a) Representation of the interaction law in the case of a capillary bridge for a given volume of liquid bridge. (b) Capillary force between two particles as a function of δn. The calculation is performed for three values of the volume of the bridge and the reduced radius r (solid line) and compared to the prediction of the Young-Laplace equation (dotted line).

To complete this expression, a failure criterion of the liquid bridge is required. This failure is governed by energy considerations. It corresponds to a minimal liquid-gas surface interface (Pepin et al. [2000]). Using a numerical solution of the Young-Laplace equation, Erle et al. (Erle et al. [1971]) and De Bisschop and Rigole (Bisschop and Rigole [1982]) have proposed two very close solutions for the failure distance. They proposed an empirical criterion based on the filling angle and the radius of the gorge to evaluate the distance of separation. From similar considerations, Lian et al. (Lian et al. [1993]) proposed the following relationship between the debonding distance Drupt, the volume of liquid bridge V and the wetting angle θ:

d      = (1+ θ)V 1∕3
 rupture      2

This failure criterion has been used in various numerical studies (Richefeu et al. [2006], Soulie et al. [2006]). Notice that the debonding distance is not the same as the distance at which the liquid bridge reforms. This hysteresis phenomenon has been discussed in the literature (Pepin et al. [2001], Soulié [2005]) and can be taken into account. A very common solution is to neglect the volume of the adsorbed liquid to the particle surface so that rejoining occurs at contact.

The distribution of liquid bridges in a granular medium is poorly studied (Fournier et al. [2005], Kohonen et al. [2004]). It plays an important role in the force transmission (Richefeu et al. [2009], Radjai and Richefeu [2009]) and in hydro-texturing phenomena (Rondet et al. [2009b,a]). The study of the distribution of the liquid is all the more difficult that the samples are polydisperse and that the configuration is three-dimensional. In the case of polydisperse media, the volume of all liquid bridges cannot be the same. A common way to allocate the water to the grains is to attribute to each liquid bridge a volume of liquid corresponding to a fraction of the total volume of liquid. This distribution cannot be consistent with the thermodynamic equilibrium between gas, solid and liquid phases since it implies a constant pressure in the liquid phase (the gas phase is percolating). For this reason, it is necessary to take into account the capillary suction in the formulation of local laws of cohesion.

Finally, it is sometimes necessary to take into account the viscosity of the liquid. The normal component fnd of the viscous force can be obtained from (Adams and Perchard [1985]):

fd = 6πμR ⋆δ˙R-⋆
 n         n δn

where μ is the dynamic viscosity of the liquid and R is the reduced radius. For the tangential component ftd, it is possible to use the following approximation (Goldman et al. [1967]) which is valid for small values of δn and for small filling angles ζ1 and ζ2:

fd = ( 8-ln R-⋆+ 0.9588) ⋅6πR⋆˙δ .
 t    15   δn                t


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