Capillary Forces Between Two Solid Spheres Linked by a Concave Liquid Bridge:Regions
of Existence and Forces Mapping
David Megias-Alguacil and Ludwig J.Gauckler
Non-Metallic Inorganic Materials,Dept.of Materials,ETH-Zu
¨rich,Zu ¨rich 8093,Switzerland DOI 10.1002/aic.11726
Published online March 23,2009in Wiley InterScience (www.).
This article focus on the capillary interactions arising when two spherical par-ticles are connected by a concave liquid bridge.This scenario is found in many situa-tions where particles are partially wetted by a liquid,like liquid films stabilized with nanoparticles.We analyze different parameters governing the liquid bridge:interpar-ticle paration,wetting angle and liquid volume.The results are compiled in a liquid volume-wetting angle diagram in which the regions of existence (stability)or inexis-tence (instability)of the bridge are outlined and the possible maximum and minimal particle distances for whic
h the liquid bridge may be found.Calculations of the capil-lary forces discriminate tho conditions for which such force is repulsive or attrac-tive.The results are plotted in form of maps that allow an easy understanding of the stability of a liquid bridge and the conditions at which it may be produced for the two
particle model.V
C 2009American Institute of Chemical Engineers AIChE J,55:1103–1109,2009Keywords:liquid bridge,capillary force,wetting force,Laplace,nanoparticle,mapping
Introduction
The inrtion of fine particles in between a liquid film modifies the film geometry,which further forms a liquid bridge with a curved meniscus shape.The contact between the three phas,solid,liquid,and gas,induces the ont of forces between the particles,which will depend on physi-cal-chemical aspects like the wettability of the particles,the geometry of the meniscus as well as the particles size and paration between them.
This scenario has gained an incread attention in the past years becau it may be found in many
practical situations,which reveals itlf as very important to understand the behavior of a wide variety of systems,liquid pha sintering,1liquid foams and emulsions,2and many others.Capillary forces between pairs of particles due to a liquid bridge have been investigated by different 1,3–14
The aim of the prent work is to contribute to this topic by going beyond the mere calculus of such capillary forces,exploring the conditions for which a concave liquid bridge may exist or not.For such a task,we will provide an origi-nal expression for the minimum interparticle distance,not yet reported in literature to our best knowledge,which to-gether with an existing expression for the maximum distance reachable by the particles joined by the bridge,will allow us to build tho regions of possible existence of the bridge,in terms of novel relative liquid volume and wetting angle dia-grams.Afterwards,once identified the corresponding exis-tence regions,the interparticle capillary force will be calcu-lated where appropriate,and according to the results,we will build new liquid volume-wetting angles maps where dis-playing tho regions of attraction/repulsion force character.
Description of the System and Calculus of
Meniscus Geometries
We consider a linear string of aligned solid monosized round particles.The particles are partially wetted by the
Correspondence concerning this article should be addresd to D.hz.ch
V
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liquid,which forms a liquid bridge between the spheres in the so-called pendular state (where the liquid pha is dis-continuous).This is the scenario which may be found in a bubble covered with small solid particles,provided the size of the bubble much bigger than the particle size.Considering that the particles are very small,the effect of gravity is neg-ligible,and no other buoyancy force will be considered.In such a ca,the liquid bridge has a constant pressure 15and the meniscus poss the shape of a surface of revolution 16,17with the same mean curvature everywhere.10The meniscus sh
ape is defined as an arc of one o the Delaunay’s surfa-ces,18which are generated by the rotation around the basis of the Delaunay’s roulettes.15Also,other axisymmetric pro-files of uniform mean curvature,like the nodoid,catenoid,or unduloid,have been considered.10In the prent work,for sake of simplicity,we will assume a meniscus profile described by an arc of circumference,as sketched in Fig-ure 1,under the assumption that the error brought about by this approximation is small in most cas.1,10,16,19
We will consider air as the gaous pha and that the sta-ble liquid bridge features a concave shaped meniscus with wetting angles smaller than 90 ,since hydrophobic solid par-ticles behaves as antifoamers.20
As displayed in Figure 1,R is the solid particle’s radius,x a and y a are the abscissa and coordinate of the contact point between the solid and liquid profile,respectively,a is the half-filling angle,h is the wetting angle,q and L are the principal radius of the liquid meniscus,measured orthogo-nally.H is the surface-to-surface distance between the solid particles,and d is the wetted portion of each hemisphere.The reference system is chon such its origin is the middle point between the spheres and who x -axis lies along the straight line which joins the particles’centers,as shown in Figure 1.
Considering,for sake of simplicity,just the upper right quadrant,the liquid and solid profiles are described by the following equations:
y L ðx Þ¼ðq þL ÞÀffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q 2Àx 2p (1)
for the liquid meniscus,and:
y S ðx Þ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR 2Àx ÀH
2
ÀR
2
s (2)for the solid profile.
The principal radii,q and L ,may be described in terms of both reprentative angles,a and h ,or on the other hand,by means of x a .Indeed,geometrical considerations follow in the next relationships.T
he half-filling angle:
a ¼arccos
H =2þR Àx a
R
(3)The radii are q ¼
x a R
H =2þR Àx a ðÞcos h Àffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R 2Àx a ÀH =2ÀR ðÞ2q sin h
(4)
and
L ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR 2Àx a ÀH =2ÀR ðÞ2
q þx a
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R 2Àx a ÀH =2ÀR ðÞ2q cos h þH =2þR Àx a ðÞsin h ÀR
H =2þR Àx a ðÞcos h ÀffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR 2Àx a ÀH =2ÀR ðÞ2q sin h (5)
Let us point out here that the principal radius q is related to the concave–convex character of the liquid meniscus,meanwhile the radius L gives an indication of the meniscus thickness.The liquid volume of the bridge,V ,may be determined for a given distance between particles,H ,by definite integra-tion of both the solid and liquid profiles:
V ¼2p
Z x a
y L ðx Þ½ 2dx À2p
Z x a
H =2
y S ðx Þ½ 2dx
(6)
Introducing Eqs.1–2into Eq.6,solving the integrals gives:V 2p ¼q þL ðÞ2þq 2
h i x a Àx 3a
3Àq þL ðÞx a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq 2Àx 2a q þq 2
arcsin x a q
ÀÀ
x a ÀH =2ðÞ2
3
3R Àx a ÀH =2ðÞ½ ð7ÞSubstituting q and L by Eqs.4and 5,respectively,the volume of the liquid brid
ge is therefore expresd just as a function of the parameter x a and the wetting angle,h ,at each interparticle paration H between the two equal spheres of radius R .Different authors have calculated the liquid bridge vol-ume,most of them in terms of the half-filling angle,a .Par-ticularly,Pietsch and Rumpf 6gave an expression for V in terms of a ,which is fully equivalent to our Eq.7,consider-ing the relationship between x a and a given by Eq.3.Other authors have obtained simpler expressions than Eq.7making assumptions to simplify the calculation and being able of obtaining an explicit expression of V as a function of x a ,but unfortunately,they result in underestimated values 5or overestimated ones,21–23respect to tho obtained with the Eq.
7.
Figure 1.Sketch of the liquid bridge geometry,where
displayed the parameters involved in the calculations.
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It is not possible to derive an explicit expression for x a from Eq.7,for a given value of the liquid volume at a cer-tain paration H and wetting angle h .Thus,x a must be cal-culated numerically.Once this value is obtained,the values of the other geometrical parameters,q and L ,ari immedi-ately by means of Eqs.4and 5.
On what it follows,we will refer to the liquid volume of the bridge throughout the relative volume,V rel ,of the liquid respect to the volume of the sphere,then,V rel ¼3V/(4p R 3).Figure 2displays some liquid bridges profiles for different situations,just considering the upper portion of the spheres and bridges,for sake of clarity.Obrve from Figure 2a,that increasing the distance between the particles,for a given liq-uid volume and wetting angle,the meniscus reduces its ra-dius L ,as expected;the other principal radius,q ,decreas for short distances H ,for later increa as the paration between the particles gets larger,then showing a minimum at larger H for increasing liquid volumes.The overall effect of parating the particles while keeping constant the liquid volume is therefore an elongation of the bridge,which becomes longer and thinner and less concave.
The modification of the wetting angle,Figure 2b,sup-pod that both liquid volume and paration are fixed,is followed mainly by a strong decreasing of the meniscus cur-vature,being flattened as the wetting angle,h ,increas.As in the previous ca,the contact point between the liquid and the solid reduces its coordinate x a and y a ,as well as the half-filling angle,a ,diminishes.
Finally,the effect of the liquid volume is shown in Figure 2c,where the interparticle distance and the wetting angle have been fixed.As expected,the increasing of the amount of liquid follows in increasing thicker meniscus (larger L )as well as an evident reduction of the meniscus curvature (larg
er q ).If the amount of liquid is incread sufficiently,it may cover completely the spheres,and beyond this point,the spheres will be fully immerd in the liquid;this situa-tion clearly does not correspond to a liquid bridge anymore.In other words,the bridge may no longer exist under this circumstance.
According to this discussion,it may be claimed that the liquid bridge has limits,that is,the existence of a liquid bridge for a certain value of liquid volume and wetting angle does not imply that this bridge may be found for all possible distances between the particles.The limits will be outlined in the next ctions.
Regions of existence of a concave liquid bridge
When plotting the liquid volume calculated by means of Eq.7as a ,of the half-filling angle,a ,Figure 3(for the cas h ¼20 and h ¼40 at veral dimensionless distances H/R ),it is obrved that the liquid volume increas for all the interparticle distances as a enlarges,as expected.This trend stops at a certain value of the half-fill-ing ,at a ¼70 when h ¼20 ,and a ¼50 when h ¼40 ,indicating the limiting filling of the space between both solid particles.Indeed,it is clear that the concave liquid bridge cannot hold any amount of liquid for a certain wet-ting angle and paration between the particles.Another sit-uation aris in ca of an excess of liquid should be fol-
lowed by an overfilling of the gap between particles,and the system should tend to transform into a typical bulk suspen-sion.Even another possibility is that the shape of the liquid meniscus changes towards a convex profile,a situation which we will not be allowed in our further considerations.To delimitate the maximum amount of liquid,V max ,which may be hold by a bridge compatible with the prervation of the wetting angle,we consider a limit half-filling angle,a limit ,that can be simply obtained by imposing the critical condition q !1.This corresponds to a liquid bridge with a cylindrical shape,border ca between the concave and con-vex meniscus.Geometrically,Figure 1,it is straight forward to obtain that the limit half-filling angle is:
a limit ¼p
2Àh (8)
For the particular ca h ¼20 ,a limit ¼70 ,which agrees with the finding in Figure
3.
Figure 2.Meniscus and spheres profiles drawn at dif-ferent conditions:(a)h 520 ,V rel 50.1;(b)V rel 50.1,H/R 50.7;(c)h 520 ,H/R 5
0.1.
Figure 3.Relative liquid volume,V rel ,as a function of
the half-filling angle,a ,for veral dimension-less interparticle distances and two wetting angles,h 52
0 and h 540 .
AIChE Journal
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The maximum liquid volume,V max,can then be deter-mined considering such a cylindrical bridge of radius y a and length Hþ2d,ended with two spherical caps of height d. Thus,taking into account the corresponding volumes of the geometrical bodies,we have:
V max¼2p x a y2aÀ2
p d23RÀd
ðÞ(9)
Considering that x a,y a,and d are(Figure1):
x a¼H
2
þRÀR cos a
y a¼R sin a
d¼RÀR cos a 9
>>=
>>;(10)
and that a is in this ca a limit¼p/2Àh,Eq.8,the maximum volume of liquid that the bridge may sustain at each surface-to-surface distance,H,and wetting angle,h,is:
V max;rel¼1
2
3cos2h
H
2R
þ1Àsin h
ðÞ
À1Àsin h
ðÞ22þsin h
ðÞ
!
ð11Þ
where V max has been made dimensionless by dividing by the sphere’s volume.On view of Eq.11,lar
ger parations enable higher amounts of liquid,same trend than decreasing the wetting angle at a certain distance,H.
If considering the situation from the point of view of the distances instead of the volumes,one can realize that the paration H which appears in Eq.11corresponds to the minimum distance at which the particles can stay for a cer-tain amount of liquid in a stable situation.Indeed,when a certain relative volume of liquid in respect to the solid vol-ume and wetting angle are impod,the particles must locate,at least,at distances H!H min.Thus,we can re-write Eq.11to obtain an explicit expression for H min:
H min R ¼22V max;rel
þ1Àsin h
ðÞ22þsin h
ðÞ
3cos h
À1Àsin h
ðÞ"#
(12)
Notice from Figure3that larger distances between the particles are able to hold more liquid,in agreement with Eq.12.
If by any reason,the particles are clor than this paration H min,the particles should move apart from each other until the condition H!H min is satisfied,if the condition of having a concave meniscus is demanded.In ca that the particles are not allowed to displace ,spatial restrictions due to their number and/or distribution,there exists an excess of liq-uid which should induce a change of meniscus geometry,this becoming convex,or in ca that the liquid amount is highly excessive,the system should evolve towards a suspension of solid particles fully immerd in the liquid and the liquid bridge configuration is destroyed.
Figure4displays a map diagram with some of tho rela-tive minimum distances,H min/R,calculated using Eq.12.There,the solid lines show the pairs V relÀh which delimit the values of liquid volume and wetting angle where a liquid bridge may be found at distances higher(restrictive condi-tion)than tho indicated by the corresponding line.As an example:a liquid bridge with V rel¼1.5and h¼30 can
only be found at distances larger than2times the particle ra-dius R(H/R[2).The values of the portion limited by the curve H/R[0and both axis correspond to the situation in which the bridge can be found when the particles are in contact.
In contrast,the bridge cannot be elongated indefinitely, and therefore,it exists a maximum distance between par-ticles,beyond that the liquid bridge breaks,H break.There have been efforts about the determination of this maximum ,considering numerical evaluations of the solu-tions of the Young–Laplace equation8or the distance at which the half-filling angle,a,is a minimum.12The rupture of the liquidfilm occurs when the distance between the par-ticles is large enough.24Lian et al.25propod the following expression for moderate wetting angles:
H break
R
¼1þ
h
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4
3
p V rel
3
r
(13)
This Eq.13was found to agree with some experimental and theoretical work17,26,27bad on the well-known Rayleigh instabilities suffered by a stretched cylindrical filament,and therefore it will be ud in this work.
We draw the Eq.13in Figure4(dotted lines),and obrve that the distances at which the liquid bridges break increa when increasing the liquid volume,V rel,and the wetting angle,h.Assuming the Raylei
gh instabilities as the main reason for the bridge rupture,it follows that the effect of such instabilities is more pronounced when thefilament becomes thinner,which is the ca of a liquid bridge
of Figure4.Map,h2V rel,showing the combination of dimensionless minimum(solid lines),H min/R,
and maximum(dotted lines)distances,H max/
R,between which the liquid bridges may be
found.
The solid symbol line reprents the frontier between
regions of existence(Region I)or inexistence(Region II)of
the bridge.Shadowed area:e text.
1106DOI10.1002/aic Published on behalf of the AIChE May2009Vol.55,No.5AIChE Journal
small liquid volume and/or wetting angle,situations in which
the meniscus thickness,identified with the radius L,gets
smaller.We have also checked the behavior of the liquid
bridge for unrestricted distances,and we found that there is
a certain paration beyond that the liquid volume is not lon-
ger prerved,and the height of the contact point between
the liquid and the solid surface,y a,Figure1,is zero.This
distance is larger than H break,Eq.13,and shorter than the
Rayleigh–Plateau limit(H\2p L),this latter result was also
found by Chen et al.28
With the combination of minimum and maximum distan-
ces,Figure4becomes a map which reflects what combina-
tions of liquid amounts,V rel,and wetting angles,h,are able
to produce a liquid bridge between certain interparticle dis-
tances,H min/R\H/R\H max/R.shown in Figure 4.the
lines which delimit the minimum distances(solid lines)cross
tho lines which indicate the break or maximum distances
(dashed lines).The line which connects the points where
H min¼H break(symbols)is the frontier between two regions: one below such a border line,Region I,which corresponds
to possible combinations of liquid volume,V rel,and wetting
angle,h,which follow in a liquid bridge with a concave
shape;and another region,Region II,where the bridge can-
not prerve such a geometrical form.Also notice that a point belonging to Region II should need larger distances H min than the bridge rupture distance,H break,which obvi-ously has not any meaning.
Figure4indicates that when the wetting angle is very high,only liquid bridges poor in liquid content can be formed.On contrary,a reduction of h allows higher amounts of liquid.Interesting to note is the fact that a bridge with a liquid volume more than double that the sphere’s volume is not possible to exist,for any wetting angle.
Figure4offers the regions of existence of liquid bridges. The shadowed area in Figure4is an exemplary ca of the region of existence of liquid bridges:a liquid bridge who liquid content and w
etting angle belongs to such an area may exist only for distances larger than1particle radius and smaller than2R.
The results allow discriminating if for a certain pair V relÀh,a concave liquid bridge may exist,and in such a positive ca,between which ranges of interparticle distances the bridge may be found.
Forces mapping
Once the geometrical parameters which describe the liquid bridge(inside Region I)are calculated for a certain value of particle size and paration,wetting angle and relative vol-ume of liquid respect to the solid sphere volume,the capil-lary forces between a pair of solid particles arising from the interaction between the solid and liquid surfaces may be cal-culated.On what it follows,negative forces indicate attrac-tion whereas a positive force is repulsive.
It is well known,that the capillary force consists of two components.Thefirst is a surface tension term acting at the wetting perimeter,tangent to the meniscus at the interction with the solid surface and directed towards the liquid.The cond component comes from the pressure difference across the curved air-liquid interface,which can be described by the Laplace–Young equation,computed over the axially pro-jected wetted area of each particle.Thus,the total capillary force may be expresd in
dimensionless form as:
F cap
c R
¼À2p sin a sinðaþhÞÀp R sin2a1
L
À1
q
(14)
where the first term corresponds to the wetting force and the cond to the Laplace force,and c is the surface tension of the liquid.
For a certain value of interparticle distance,H,relative volume of liquid respect to the solid,V rel,and wetting angle, h,the dimensionless capillary force,Eq.14,may be obtained at the conditions.Figure
5displays some results when the wetting angle is h¼20 ,at different liquid volumes and dis-tances between the solid particles.Notice the truncation of the plots according to the corresponding H min and H break,for each liquid volume,V rel,discusd earlier.
In general terms,it is obrved that the capillary force shows two characters,repulsive and attractive,arising mainly from the behavior of the Laplace component.As the princi-pal radii q and L have opposite signs this component may feature both characters.Repulsive Laplace forces are only obrved in a small ction of Region I,Figure4,for liquid volumes,V rel,smaller than0.1and wetting angles below 79.2 ,conditions where the liquid meniscus is highly curved. For small distances and small liquid amounts,the Laplacian repulsion is able to dominate over the always attractive wet-ting force.This behavior is quite limited and beyond a cer-tain distance between the solid surfaces,the capillary force becomes attractive.This is also the ca when the liquid volume is larger;then,the total force is attractive in any combination V relÀh where it exists,Figure4.Under this circumstance,the attractive capillary force increas its mag-nitude for increasing volumes of liquid due to an increa in contact area between the liquid and solid.
The maxima(minima in attraction)showed by F cap obeys to the behavior of the Laplace component,which also shows a maximum.The maxima displayed in Figure5is found
at Figure5.Dimensionless capillary force,F cap,scaled by the product c R,Eq.14,as a function of the
dimensionless paration between particles,
for the relative liquid volumes indicated in the
legend,at h520 .
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