L

Ya

b

a

ARRA

KLFND

1

iausaawtb

vofie(&ha&ouL

1d

Particuology 9 (2011) 374– 380

Contents lists available at ScienceDirect

Particuology

j our nal ho me p ag e: www.elsev ier .com/ locate /par t ic

iquid bridge force between two unequal-sized spheres or a sphere and a plane

ouchuan Chena, Yongzhi Zhaoa,∗, Hongli Gaob, Jinyang Zhenga

Institute of Chemical Process Equipment, Department of Chemical and Biochemical Engineering, Zhejiang University, Hangzhou 310027, ChinaMaoming University, Maoming, Guangdong 525000, China

r t i c l e i n f o

rticle history:eceived 8 May 2010eceived in revised form 4 November 2010ccepted 30 November 2010

a b s t r a c t

Liquid bridge force acting between wet particles is an important property in particle characterization. Thispaper deals with liquid bridge force between either two unequal-sized spherical particles or a sphere anda flat plate under conditions where gravitational effect arising from bridge distortion is negligible. In order

eywords:iquid bridge forceilling angleonzero contact angleiscrete element method

to calculate the force of the liquid bridge efficiently and accurately, expressions of liquid configurationand liquid bridge force were derived by building a mechanical model, which assumes the liquid bridgeto be circular in shape between either two unequal-sized spheres or a sphere and a plane. To assessthe accuracy of the numerical results of the calculated liquid bridge forces, they were compared to thepublished experimental data.

© 2011 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

zl&ctcptp

2

2

owsoc

. Introduction

With the development of many industrial processes involv-ng wet particulate systems such as particle size enlargement,gglomeration of mineral particulates and separation of wet gran-lar materials, understanding of the flow property of wet powderystems becomes technologically significant. Liquid bridge forcecting in wet particulate systems is an important property to char-cterize their flow property. An efficient and accurate algorithmhich expresses the effect of the liquid bridge force in wet par-

iculate systems is therefore required to simulate the mechanicalehavior of separating wet granular materials.

Many mechanical models have been proposed to express theolume and the force of the liquid bridge in terms of the shapef the liquid bridge profile, the solid–liquid contact angle and thelling angles (Mehrotra & Sastry, 1980). And many investigatorsstimated the force of liquid bridge by the shape of liquid profileLian, Thornton, & Adams, 1993; Melrose & Wallick, 1967; Heady

Cahn, 1970; Orr, Scriven, & Rivas, 1975), though such analysis isighly complicated for numerical calculation. Simpler approachesssume the meniscus of the liquid profile to be circular arcs (Pietsch

Rumpf, 1967), or even consider the ideal geometry of two spheres

f equal sizes and complete wetting of solids by the bridging liq-id (Melrose, 1966; Lian et al., 1993; Pierrat & Caram, 1997; Urso,awrence, & Adams, 1999). Some models were even limited to

∗ Corresponding author. Tel.: +86 571 87953370; fax: +86 571 87953393.E-mail address: yzzhao@zju.edu.cn (Y. Zhao).

tppaau

674-2001/$ – see front matter © 2011 Chinese Society of Particuology and Institute of Process Eoi:10.1016/j.partic.2010.11.006

ero contact angle while deriving the force and the volume of theiquid bridge (Kruyer, 1958; Cross & Picknett, 1963; Erle, Dyson,

Morrow, 1971; Rabinovich, Esayanur, & Moudgil, 2005), or toalculating the force and the volume of the liquid bridge by charac-erizing the liquid bridge between two unequal-sized spheres andonsidering non-zero contact angles (Mehrotra & Sastry, 1980). Theresent work proposes a mechanical model to analyze the force andhe volume of the liquid bridge, by considering the liquid bridgerofile to be circular in shape between two unequal-sized spheres.

. Computational procedure

.1. The case of two unequal-sized spherical particles

Fig. 1 shows a geometrical representation of the cross-sectionf the liquid bridge between two unequal-sized spherical particlesith their centers located at O1 and O2. Lines O1P1 and O2P2 corre-

pond to the radii r1 and r2 of the two particles. The configurationf the liquid bridge and two spheres is generated by revolving theross-sections around the axis O1O2.

In Fig. 1, � is the nonzero contact angle and O3 is the center of

he circular arc of the liquid bridge interface; P1 and P2 are contactoints at the liquid bridge interface for the two particles; A is theoint of intersection of the extensions O1P1 and O2P2; P3 and C1re points of intersection of O3C1 with the liquid bridge profile P1P2nd the axis O1O2, respectively. And d is the distance between twonequal-sized spheres.

ngineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

Y. Chen et al. / Particuolog

Fs

2

f

O

a

P

T

P

E

(

o

�

2

tci

�

w

˛

ˇ

�

2

ttsa

t1

V

pa

x

y

w

a

a

b

V

w

x

x

V

o

V

ig. 1. Geometrical representation of the liquid bridge between two unequal-sizedpheres.

.1.1. Relationship between filling anglesIn Fig. 1, �1 and �2 are filling angles, related to each other as

ollows:

1Asin �1 = O2A sin �2. (1)

It can be shown that

sin �1

sin �2= O2A

O1A= r2 + P2A

r1 + P1A, (2)

nd

1A cos �1+P2A cos �2 = r1(1 − cos �1)+d + r2(1 − cos �2). (3)

hus,

1A = P2A = r1(1 − cos �1) + d + r2(1 − cos �2)cos �1 + cos �2

. (4)

qs. (4) and (2) can be further simplified to

d + 2r1) tan(

�1

2

)= (d + 2r2) tan

(�2

2

), (5)

r

2 = 2 arctan[

d + 2r1

d + 2r2tan

(�1

2

)]. (6)

.1.2. Principal radii of curvatureWhen the liquid bridge profile is considered as an arc of a circle,

he first principal radius of curvature �1 is equal to the radius of theircular arc P1P3P2 and the second principal radius of curvature �2s equal to the length of line C1P3, and it can be shown that

1 = P1O3 = P1A

(sin ˇ

sin ˛

), (7)

here P1A is given by Eq. (4) and ̨ and ̌ are given by

=[� − (�1 + �2 + 2�)

]2

, (8)

= [� − (�1 + �2)]2

. (9)

Combining Eqs. (4), (7), (8) and (9) leads to

�1 = r1(1 − cos �1) + d + r2(1 − cos �2) cos[(�1 + �2)/2

][ ]

cos �1 + cos �2 cos (�1 + �2 + 2�)/2

= r1(1 − cos �1) + d + r2(1 − cos �2)cos(�1 + �) + cos(�2 + �)

.

(10) V

y 9 (2011) 374– 380 375

Also, from Fig. 1 one obtains

2 = r1 sin �1 − �1[1 − sin(�1 + �)

]. (11)

.1.3. Volume of the liquid bridgeThe volume of the liquid bridge V can be obtained by evaluating

he volume V1 generated by revolution of the arc P1P3P2 aroundhe axis O1O2 and then subtracting the volume V2 of the sphericalegments, generated by the revolution of arcs P1B1 and P2B2 alsoround the axis O1O2, from the volume V1.

The volume of the liquid bridge is first normalized with respecto the sum of volumes of the two spheres as (Mehrotra & Sastry,980; Butt & Kappl, 2009).

∗ = V

4/3�(r31 + r3

2). (12)

Then a rectangular Cartesian coordinate system is chosen, withoint C1 as the origin, the line O1O2 as the axis x, and the line O3C1s the axis y. The algebraic equation of the arc P1P3P2 is

2 + (y − a)2 = b2. (13)

Eq. (13) can be simplified to

2 = a2 + b2 − x2 + 2a√

(b2 − x2), (14)

here

= O3C1 = �1 sin(�2 + �) + r2 sin �2, (15)

nd

= O3P3 = �1. (16)

The volume V1 is then given by

1 =∫ x2

−x1

�y2 dx, (17)

here

1 = C1C2 = �1 cos(�1 + �), (18)

2 = C1C3 = �1 cos(

�2 + �)

. (19)

Then V1 can be obtained as

1 = �{

(a2 + �21)�1

[cos(�1 + �) + cos(�2 + �)

]

−13

�31

[cos3(�1 + �) + cos3(�2 + �)

]− a�2

1

[sin(�1 + �) cos(�1 + �) + sin(�2 + �) cos(�2 + �)

]+ a�2

1(�1 + �2 + 2� − �)}

. (20)

Further, the volume V2 of the spherical segments can bebtained as

2 = �

3

[(2 − 3 cos �1 + cos3 �1)r3

1 + (2 − 3 cos �2 + cos3 �2)r32

].

(21)

The volume V of the liquid bridge can thus be given by

= V1 − V2. (22)

3 uology 9 (2011) 374– 380

2

tppa(t

f

w

(I

�

f

o

f

2

biwa(

d

Fp

Fig. 3. Relationship between filling angles �1 and �2 for different normalized geo-metrical coefficients n.

76 Y. Chen et al. / Partic

.1.4. Liquid bridge forceIn the absence of gravitational effects arising from bridge distor-

ion, the liquid bridge force fl, acting between two unequal-sizedarticles, is given by the sum of two components: (a) the axial com-onent of the surface tension force and (b) the hydrostatic pressurecting on the axially projected area of the liquid bridge on spheresLian et al., 1993). The surface tension force f1 can thus be showno be

1 = 2��2�, (23)

here � is the liquid surface tension.The hydrostatic pressure �p is given by Laplace’s equation

Sirghi, Nakugiri, Sugisaki, & Sugimura, 2000; Pepin, Rossetti,veson, & Simons, 2000) as

p = �(

1�1

− 1�2

). (24)

Force f2, resulting from the hydrostatic pressure, is

2 = ��22�p = ��2

2�(

1�1

− 1�2

). (25)

Then the expression of the force fl of the liquid bridge can bebtained as

l = f1 + f2 = 2��2�+��22�

(1

�1− 1

�2

)= ���2

(�1 + �2

�1

). (26)

.1.5. Critical rupture distanceLiquid bridge force, which depends on the separation distance

etween two spheres, will be stable up to some critical value, thats, the rupture distance, or the maximum separation distance dc at

hich the liquid bridge breaks, determined by the contact angle �nd the volume V of the liquid bridge. If � ≤ 2�/9, d can be shown

c

Lian et al., 1993) to be

c = (1 + 0.5�)V1/3. (27)

ig. 2. Geometrical representation of the liquid bridge between a sphere and alane.

Fig. 4. Liquid bridge force fl as a function of the separation distance d between twoidentical spheres for different liquid bridge volumes V, where r1 = r2 = 2.381 mm..

Fig. 5. Liquid bridge force fl as a function of the separation distance d between twounequal-sized spheres for different liquid bridge volumes V, where r1 = 2.381 mmand r2 = 1.588 mm.

Y. Chen et al. / Particuology 9 (2011) 374– 380 377

Fua

2

ps

Fig. 7. Liquid bridge force fl as a function of the separation distance d between as

ig. 6. Liquid bridge force fl as a function of the separation distance d between twonequal-sized spheres for different liquid bridge volumes V, where r1 = 2.381 mmnd r2 = 1.191 mm.

.2. The case of a spherical particle and a plane

Fig. 2 represents the cross-section of a liquid bridge between alane and a spherical particle with its center at O1. Line O1P2 corre-ponds to the radius r of the particle. The configuration of the liquid

bsaP

Fig. 8. Dependence of the normalized liquid bridge force F* on th

phere and a plane for different liquid bridge volumes V, where r = 2.381 mm.

ridge, the plane and the sphere is generated by revolving the cross-ection around axis O1C2. In Fig. 2, � is the nonzero contact angle

nd O2 is the center of the circular arc of the liquid bridge interface;1 and P2 are contact points at the liquid bridge interface of the par-

e normalized separation distance d* for different V* and R.

3 uology

tt

2

P

P

tci

�

w

˛

a

ˇ

s

�

�

2

t

V

taP

V

78 Y. Chen et al. / Partic

icle and the plane; P3 and C1 are points of intersection of O2C1 withhe liquid bridge profile P1P2 and the axis O1C2, respectively.

.2.1. Principal radii of curvatureIn Fig. 2, � is the filling angle, and it can be shown that

1A + P2A cos � = d + r(1 − cos �), (28)

1A = P2A = d + r(1 − cos �)1 + cos �

. (29)

When the liquid bridge profile is considered as an arc of a circle,he first principal radius of curvature �1 is equal to the radius of theircular arc P1P2P3, and the second principal radius of curvature �2s equal to the length of line C1P3, and it can be shown that

1 = P1O2 = P1A

(sin ˇ

sin ˛

), (30)

here P1A is given by Eq. (29) and ̨ and ̌ are given by

=[� − (� + 2�)

]2

, (31)

nd

= � − �

2. (32)

Fig. 9. Dependence of the normalized liquid bridge force F* on th

9 (2011) 374– 380

By combining Eqs. (29), (31) and (32) through (30), it can behown that

1 = d + r(1 − cos �)1 + cos �

cos(�/2)

cos[(� + 2�)/2

] = d+r(1 − cos �)cos � + cos(� + �)

. (33)

Also, from Fig. 2, one obtains

2 = r sin � − �1[1 − sin(� + �)

]. (34)

.2.2. Volume of the liquid bridgeThe volume of liquid bridge V is first normalized with respect to

he volume of the sphere as (Mehrotra & Sastry, 1980).

∗ = V

4/3�r3. (35)

Then, as in Section 2.1.3, a rectangular Cartesian coordinate sys-em is chosen, with point C1 as the origin, line O1C2 as the x axis,nd line O2C1 as the y axis. The volume V1, by revolving the arc1P2P3, can thus be given as

1 = �{

(a2 + �21)�1

[cos(� + �) + cos �

]

−1�3

[cos3(� + �) + cos3 �

]

3 1

− a�21

[sin(� + �) cos(� + �) + sin � cos �

]+a�2

1(� + 2� − �)}

, (36)

e normalized separation distance d* for different R and V*.

uolog

w

a

a

V

Vc

2

eap

2

bwdc

3

3s

u

n

bm�esb

3

Y. Chen et al. / Partic

here

= �1 sin(� + �) + r sin �. (37)

Further, the volume V2 of the spherical segment can be obtaineds

2 = �

3(2 − 3 cos � + cos3 �)r3. (38)

Then, by combining Eqs. (36)–(38) through (22), the volume of the liquid bridge between the plane and the sphere can bealculated.

.2.3. Liquid bridge forceThe method for calculating the force fl is the same as that

mployed in Section 2.1.4. By combining Eqs. (23), (24), (25), (33)nd (34) through (26), the force fl of the liquid bridge between thelane and the sphere can thus be calculated.

.2.4. Critical rupture distanceCritical rupture distance dc is the maximal separation distance

etween a spherical particle and a plane, at which the liquid bridgeill break. The contact angle � and the volume V of the liquid bridgeetermine the critical rupture distance dc. If � ≤ 2�/9, dc can bealculated by Eq. (27).

usad

Fig. 10. Dependence of the normalized liquid bridge force F* on t

y 9 (2011) 374– 380 379

. Results and discussion

.1. Relationship of filling angles between two unequal-sizedpherical particles

Let us define the normalized geometrical coefficient n of twonequal-sized spherical particles as

= d/2 + r2

d/2 + r1. (39)

The relationship between the filling angles �1 and �2 is giveny Eq. (6). In Fig. 3, these relationships are shown for different geo-etrical coefficients. The curves in Fig. 3 reveal that the filling angle

2 of the smaller sphere is larger than �1, except for the case of twoqual-sized spherical particles. This conclusion is identical to thetudy of Mehrotra and Sastry (1980), but their result was limitedy only considering the distance d to be zero.

.2. Numerical result of the liquid bridge force

As has been stated, the total liquid bridge force between two

nequal-sized spherical particles can be calculated by solvingimultaneous Eqs. (6), (10), (11), (20), (21) and (26), while contactngle �, the radii r1 and r2 of two different particles, the distance

and the volume of the liquid bridge V are known. Also, force

he normfflized separation distance d* for different � and R.

3 uology

bi

bWuoTsiTwe

eftta

bmt

3

np

F

t

d

fn(s

dcFdcdmog

bivtb

4

tdoihttsortd

lm

A

oF

R

B

C

E

H

K

L

M

M

M

O

P

P

P

R

S

80 Y. Chen et al. / Partic

etween a spherical particle and a plane can be calculated by solv-ng simultaneous Eqs. (33), (34), (36), (38) and (26).

To assess the accuracy of the numerical result of the liquidridge force, comparisons were made with experimental values ofillett, Adams, Johnson, and Seville (2000) who measured the liq-

id bridge force between either a pair of synthetic sapphire spheresr a sphere and a flat sapphire disk by a sensitive microbalance.he larger of the spheres had a radius of 2.381 mm and the smallerpheres with radii of 2.381 mm, 1.588 mm and 1.91 mm. The exper-mental fluid was a DC200 poly(dimethylsiloxane) silicone fluid.he measured viscosity, surface tension, and density of this fluidere 110 mPa s, 20.6 mN/m, and 960 kg/m3, respectively (Willett

t al., 2000).Figs. 4–7 compare respectively the numerical results with

xperimental values taken from Willett’s paper of the liquid bridgeorce as a function of the separation distance for different combina-ions of spheres and also a sphere and a flat plate. In these figures,he curves represent the numerical values calculated by solving thebove simultaneous equations (for � = 0◦).

From comparisons in Figs. 4–7, the numerical values calculatedy above simultaneous equations (for � = 0◦) have excellent agree-ent with the experimental data, testifying to the high accuracy of

he improved mechanical model proposed in the current study.

.3. Normalized liquid bridge force

In order to better analyze computation results, we define theormalized liquid bridge force F* of two unequal-sized sphericalarticles as

∗ = fl��(r1 + r2)

. (40)

And the separation distance d can be normalized with respecto the critical rupture distance dc as

∗ = d

dc. (41)

Figs. 8–10 present the normalized liquid bridge force F* as aunction of the normalized separation distance d* for differentormalized volume of the liquid bridge V*, sphere radius ratio RR = r2/r1, where r1 and r2, are respectively radii of large and smallpheres) and contact angle �.

Fig. 8 shows that the normalized liquid bridge force F* alwaysecreases with d* for the specified R and �. From this figure itan also be seen that with diminishing V*, the descent rate of* increases for given values of � and R. Fig. 9 indicates that F*

ecreases with d* for the specified V* and�. From this figure itan also be seen that with diminishing R, the descent rate of F*

ecreases, for given values of V* and �. Fig. 10 shows that the nor-alized liquid bridge force F* decreases with d* for specified values

f V* = 0.01 and �. With diminishing �, F* increases slightly, for theiven values of V*, R and d*.

These figures indicate that the normalized liquid bridge forceetween two unequal-sized spherical particles depends signif-

cantly on the normalized separation distance, the normalizedolume of the liquid bridge, the sphere radius ratio and the con-act angle, and it increases as the contact angle or the differenceetween particle sizes decreases.

U

W

9 (2011) 374– 380

. Conclusions

A detailed analysis of the liquid bridge force between eitherwo unequal-sized spherical particles or a sphere and a flat plate isescribed by using an improved mechanical model which is capablef analyzing the force and the volume of the liquid bridge by assum-ng that the liquid bridge profile is circular in shape. Expressionsave been derived from this model in order to calculate accuratelyhe volume of the liquid bridge, the filling angles and the force ofhe liquid bridge. Results of computations by using these expres-ions indicate that the liquid bridge force depends significantlyn the separation distance, the volume of the liquid bridge, theatio of sphere radii and the contact angle. Specifically, the force ofhe liquid bridge increases with decreasing contact angle and withecreasing difference between particle sizes.

Much greater accuracy and efficiency in computation to simu-ate the mechanical behavior of separating wet granular materials

ay be obtained by using the above model.

cknowledgments

This research is financially supported by the Science Foundationf Chinese University and the Zhejiang Provincial Natural Scienceoundation of China (Grant No. Y1100636).

eferences

utt, H. J., & Kappl, M. (2009). Normal capillary forces. Advances in Colloid and Inter-face Science, 146, 48–60.

ross, N. L., & Picknett, R. C. (1963). The liquid layer between a sphere and a planesurface. Transactions of the Farady Society, 59, 846–855.

rle, M. A., Dyson, D. C., & Morrow, N. R. (1971). Liquid bridges between cylinders,in a torus, and between spheres. AIChE Journal, 17, 115–121.

eady, R. B., & Cahn, J. W. (1970). An analysis of the capillary forces in liquid-phasesintering of spherical particles. Metallurgical Transactions, 1, 185–189.

ruyer, S. (1958). The penetration of mercury and capillary condensation in packedspheres. Transactions of The Farady Society, 54, 1758–1767.

ian, G., Thornton, C., & Adams, M. J. (1993). A theoretical study of the liquid bridgeforces between two rigid spherical bodies. Journal of Colloidal and Interface Sci-ence, 161, 138–147.

ehrotra, V. P., & Sastry, K. V. S. (1980). Pendular bond strength between unequal-sized spherical particles. Powder Technology, 25, 203–214.

elrose, J. C. (1966). Model calculations for capillary condensation. AIChE Journal,12, 986–994.

elrose, J. C., & Wallick, G. C. (1967). Exact geometrical parameters for pendularring fluid. The Journal of Physical Chemistry, 71, 3676–3677.

rr, F. M., Scriven, L. E., & Rivas, A. P. (1975). Pendular rings between solids-meniscusproperties and capillary force. Journal of Fluid Mechanics, 67, 723–742.

epin, X., Rossetti, D., Iveson, S. M., & Simons, S. J. R. (2000). Modeling the evolu-tion and rupture of pendular liquid bridges in the presence of large wettinghysteresis. Journal of Colloidal and Interface Science, 232, 289–297.

ierrat, P., & Caram, H. S. (1997). Tensile strength of wet granular materials. PowderTechnology, 91, 83–93.

ietsch, W., & Rumpf, H. (1967). Adhesion capillary pressure liquid volume and angleof contact of a liquid bridge between 2 spheres. Chemie Ingenieur Technik, 39,885–893.

abinovich, Y. I., Esayanur, M. S., & Moudgil, B. M. (2005). Capillary forces betweentwo spheres with a fixed volume liquid bridge: Theory and experiment. Lang-muir, 21, 10992–10997.

irghi, H., Nakugiri, N., Sugisaki, K., Sugimura, H., & Takai, O. (2000). Effect of sampletopography on adhesive force in atomic force spectroscopy measurements inair. Langmuir, 16, 7796–7800.

rso, M. E. D., Lawrence, C. J., & Adams, M. J. (1999). Pendular, funicular, and capillarybridges: Results for two dimensions. Journal of Colloidal and Interface Science,220, 42–56.

illett, C. D., Adams, M. J., Johnson, S. A., & Seville, J. P. K. (2000). Capillary bridgesbetween two spherical bodies. Langmuir, 16, 9396–9405.