1 Introduction

The role of liquid bridge has been recognized to be important for many systems and phenomena such as the adhesion of particles to solid surfaces,the consolidation of granules and porous media,and the dispersion of pigments and wetting of powders^[1]. An efficient and accurate algorithm which expresses the effect of the liquid bridge in wet particulate systems is therefore required. Two main theoretical methods are available for obtaining the capillary interactions,either on the basis of the total energy of two solid surfaces interacting through the liquid and the ambient vapor or by the direct calculation of the force as a result of the differential gas pressure across the liquid bridge^[2]. In most reported investigations,the geometry of the liquid bridge is described by the Young-Laplace equation. Within such framework,the capillary interactions are related to the inter-particle separation distance through the capillary suction,the liquid bridge curvature,the ratio of particle radii,and the surface tension force of liquid. Since the initial work on the capillary force in moist particles is studied by Haines^[3-4] and Fisher^[5-6],numerous works on the modeling of particle interaction in the presence of capillary bridges have been reported. Most studies reported so far have focused on how to determine the shape of the capillary bridges and the capillary-bridge force with the approximate approach or the numerical method in the case of the sphere/plate geometry or the sphere/sphere geometry with the uniform or nonuniform size^[7-13],in which the liquid volume is not a constrained condition. Due to the dependency on the degree of saturation,partially saturated granular assemblies exhibit different types of behaviors. It is necessary to understand the capillary interactions between particles with the specific liquid content. However,fewer studies have been reported on the capillary force corresponding to the fixed volume of liquid bridge. On the basis of the total energy principle,the capillary force of a fixed volume liquid bridge was studied by Israelachvili^[14] and Rabinovich et al.^[15] using the Derjaguin approximation for two different surface energies.

Based on the Young-Laplace model,the exact solution of capillary interactions of a fixed volume liquid bridge between two separate nonuniform spheres is investigated in this paper. In Section 2,the governing equations of the liquid bridge with a fixed volume are established,and the capillary force is calculated with the Gorge method. In Section 3,the effects of the inter-particle distance,the contact angle,and the particle size on the capillary interactions are discussed through numerical studies. Finally,conclusions are given in Section 4.

2 Mathematical model

Let us consider an axisymmetric fluid capillary bridge formed between two solid particles with the liquid volume $v$. The bridge between two particles of different sizes takes a complex shape as illustrated in Fig. 1. A rectangular Cartesian coordinate system is chosen,with the point $O$ as the origin,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 $x$. $r_1 $ and $r_2 $ are the particle radii $(r_1 \le r_2) $,$\theta _1 $ and $\theta _2 $ are the liquid-solid contact angles,$d$ is the distance between two unequal-sized spheres,and $\beta _1 $ and $\beta_2$ are the filling angles. The coordinate $x$ of the three-phase contact line,i.e.,the line defining the solid-liquid-gas interface,is denoted by $x=0$ and $x=x_{\rm c}$ for two particles,respectively.

The pressure difference $\Delta p=p_{\rm g} -p_{\rm l}$ (also called the capillary suction or the matrix suction) through the liquid-gas interface is related to the curvature of the liquid bridge and the surface tension of the liquid $\sigma $ with the Young-Laplace equation^[7],

(1)

The capillary force $f_{\rm cap}$ due to the liquid bridge is composed of several contributions,of which the most important ones are the surface tension $\sigma $ and the meniscus capillary suction $\Delta p$. There are three methods to calculate the liquid bridge force,i.e.,the Gorge method,the boundary method,and the boundary pressure method^[16]. For a small amount of liquid held between two particles,a neck will be formed when the liquid bridge stretches^[11]. The capillary force at the neck is calculated with the Gorge method,

(2)

where $y_0 $ is the neck radius of the liquid bridge,as shown in Fig. 2.

The volume of the liquid bridge is given by the integral form,

(3)

where $y_{\rm s} (x)$ is the dotted line,as shown in Fig. 3. According to the geometrical configuration in Fig. 3,the following relationship can be given:

It is important to determine whether the liquid bridge will break with increasing the separation distance between two particles. Lian et al.^[17] proposed the following relationship between the theoretical rupture distance $d_{\rm rupture}$,the volume of liquid bridge $v_{\rm l} $,and the gas-liquid contact angle $\theta $:

(4)

For the sake of convenience,the dimensionless variables are defined as follows:

(5)

where $r_{\rm d} $ is a reference radius.

Substituting Eq. (5) into Eq. (1) ,the dimensionless Young-Laplace can be given as

(6)

with the boundary conditions

(7a)

(7b)

The boundary conditions (7a) and (7b) determine the inclinations of the liquid-gas interface at the contact points,where it terminates on two solid surfaces,respectively.

The governing equation (6) is the two-point non-linear boundary value problem (BVP) with the free boundary (7b). The analytical solution has not been discovered. Therefore,the equations have to be solved numerically. Many techniques for solving BVPs have been suggested. The shooting method is an efficient technique for solving BVPs^[18-19]. With the shooting method,the BVP is reduced to the solution to an initial value problem (IVP) by assuming that initial values would have been given if the ordinary differential equation is an IVP. The boundary value obtained is then compared with the actual boundary value. Using trial and error or some scientific approach,one can try to get as close to the boundary value as possible. When the boundary value calculated is matched with the real boundary value,the exact solution that meets the precision requirement can be obtained.

In order to convert the above BVP with a free end point $X_{\rm c}$ into a common BVP,other unknown variables $W$ and $Z$ are introduced by

(8)

Since $X_{\rm c} $ is an unknown constant,

(9)

Define a new independent variable $t$ and apply a variable substitution as follows:

(10)

Taking the derivative of Eq. (3) with respect to $x$ and substituting Eqs. (8) --(10) into the Young-Laplace equation (6) yield a set of first-order ordinary differential equations dependent on the variable $t$ given as follows:

(11a)

(11b)

(11c)

(11d)

(11e)

For Eq. (11e),

(12)

The corresponding boundary conditions can be rewritten as follows:

(13a)

(13b)

(13c)

(13d)

(13e)

(13f)

in which

Equations (11a)$-$(13f) are a typical two-point BVP dependent on the variable $t$ in the interval [0,1]. The boundary condition (13f) makes sure that the liquid bridge has a fixed volume $V$. It can be found that the liquid bridge profile $Y$ is formulated in terms of the particle separation $D$,the particle diameters $R_1 $ and $R_2 $,the contact angles $\theta _1 $ and $\theta _2 $,and the liquid volume $V$.

First,given values of geometric parameters $R_1 $,$R_2 $,$\theta _1 $,$\theta _2 $,and $D$,with Eqs. (11) $-$(12) ,a set of exact solutions of the liquid bridge profile $Y$ and the capillary suction $H$ that meet precision requirement can be obtained with the shooting method in terms of the capillary suction liquid volume $V$. Then,the capillary force $F_{\rm cap}$ is completely determined through Eq. (2) .

3 Numerical results and discussion

With the reference radius $r_{\rm d} =r_2 $,and $\theta _1 =\theta _2 =\theta $,the inter-particle contact force and the capillary suction are investigated for different parameters.

Figure 4 presents the dimensionless liquid bridge force $F_{\rm cap} $ and the capillary suction $H$ as functions of the dimensionless separation distance $D$ for different dimensionless volumes of the liquid bridge $V$ with the sphere radius ratio $R_1/R_2=1$ and the contact angle $\theta =15^{\circ}$. For a fixed volume $V$,it is found that the liquid bridge force and the capillary suction decrease when the separation distance is increased. Since the liquid volume is constant,as expected from Eqs. (2) ,(3) ,and (6) ,increasing the separation distance leads to the decrease of the neck radius of liquid bridge as well as the variation of the radius of the meridian,as illustrated in Fig. 4. Moreover,we observe that the liquid bridge force and the capillary suction decline faster for a lower value of the liquid volume. For the dimensionless liquid volume $V=0.35$,Fig. 5 shows the variation of geometrical configuration of liquid bridge between two particles with different values of ratio of particle size. The dotted line is the surface of solid sphere between two particles for different values of the inter-particle distance.

For the contact angle $\theta =15^{\circ}$ and the dimensionless separation distance $D=0.01$,the relationship between the dimensionless liquid bridge force $F_{\rm cap} $,the capillary suction $H $,and the ratio of particle size ${R_1 }/{R_2 }$ is depicted in Fig. 6. It can be observed that for the same size ratio,the capillary force and the capillary suction are both larger for a lower value of the liquid volume. For a fixed liquid volume,the capillary force and the capillary suction decrease with the decrease of the ratio of particle size. For the dimensionless liquid volume $V=0.35$,the variation of geometrical configuration of the liquid bridge between two particles with different values of the ratio of particle size is presented in Fig. 7.

In order to investigate the effect of the contact angle on capillary interactions between two particles with the fixed liquid volume,when the given ratio of particle size ${R_1 } / {R_2 }=1$,and the dimensionless inter-particle distance $D=0.01$,the results of the relationship between the dimensionless capillary force,the dimensionless capillary suction,and the contact angle are shown in Fig. 8. The increase of the contact angle results in a reduction of the capillary force and the capillary suction. For the same contact angle,the capillary force and the capillary suction are higher for a lower value of the liquid volume.

Figure 9 gives the variation of geometrical configuration of the liquid bridge with different contact angles when the dimensionless liquid volume $V=0.35$. It indicates that the reduction of the radius of the meridian is due to the increase of the amount of liquid between the two spheres.

4 Conclusions

In this study,for a fixed volume liquid bridge,a numerical model for the calculation of the capillary interactions between two spherical particles is established. It is shown that,when the volume of liquid bridge is known,the exact solution of the capillary suction and the capillary force can be obtained with the shooting method. The effects of several parameters such as the distance between two spheres,the ratio of sphere radii,and the liquid-solid contact angles are also studied. These numerical results indicate that the liquid bridge force between two unequal-sized spherical particles depends significantly on the separation distance,the ratio of sphere radii,and the contact angle,and it increases as the contact angle or the inter-particle distance decreases.

Acknowledgements Thanks to the financial support provided to the first author in the form of a visiting scholarship by the China Scholarship Council.