Interfacial Forces in Dispersion Scienceand Technology

George KaptayBAY-NANO Research Institute and University of Miskolc, Miskolc, Hungary

Interfacial forces determine many phenomena in dispersion science and technology. Eight types ofinterfacial forces are classified in this article. A general equation for all of them is derived here,with particular equation for each of them (being valid for simplified geometries, such as spheres,cylinders, etc.). As a new element, an interfacial anti-stretching force is introduced in this article,being equivalent to the definition of the interfacial energy in terms of tension as understood byYoung. The differences and similarities between the interfacial gradient force and the interfacialspreading force (the Marangoni force) are shown. The well-known case of the liquid bridgeinduced interfacial force is supplemented by its less known version of a gaseous bridge inducedinterfacial force.

Keywords Capillary forces, disjoining pressure, infiltration, interfacial forces, meniscus

1. INTRODUCTION

The behavior of different dispersed phases (solid parti-cles, liquid droplets, gaseous bubbles) in composites, foamsand emulsions at or near interfaces is mostly determined byinterfacial forces. The basics of the subject of this articlewere founded in the works of Young[1] and Laplace[2] morethan 200 years ago. The majority of forces considered herehave been described in previous reviews.[3–6] However, ageneral approach like this has not been found elsewhereby the author (see a shorter previous version[7]).

By definition, interfacial forces are those which becomezero when all interfacial energies (or all interfacial areas) inthe system are zero. Interfacial forces have been classifiedin[8] according to the number of phases involved and tothe direction of the force relative to the major interfacein the system. This system is slightly corrected in this article(Table 1). The goal of the present article is to derive equa-tions connecting all eight forces with interfacial energies(tensions) and geometrical parameters of the system.

In this article, the terms surface tension, interfacial ten-sion, surface energy, interfacial energy, etc. will all be calledas ‘‘interfacial energy’’ and denoted as rUW, with a unit of

J=m2. Double subscripts are used for phases surroundinginterfaces: v¼ vapor, g¼ gas, l (or L)¼ liquid, s¼ solid,more generally: a, b, c, d or U, and W. The interfacialenergy in this article will be defined in a thermodynamicway as an excess interfacial Gibbs energy of the surface=interface per unit area:[9,10]

rUW � DGUW

xUW; ½1�

where DGUW is the change of the Gibbs energy accompany-ing the transfer of one mole of material from both of thebulk phases U and W to the common interface U=W(J=mol), xUW is the molar interfacial area (m2=mol) ofthe U=W interface.

2. GENERAL EQUATION FOR INTERFACIAL FORCES

Now, let us consider a general scheme of deriving themagnitude and direction of an interfacial force acting ona phase (see Figure 1). Let us select direction x and phasea to be studied. Both selections are made in an arbitraryway. If the present procedure is repeated in three perpen-dicular directions (x, y, z), the resulting force componentsFa,x, Fa,y, and Fa,z can be summed as vectors and in thisway the actual magnitude and direction of the interfacialforce acting on the selected phase a is obtained. The sameprocedure can be applied to derive the interfacial force act-ing on other phases in the system. Now, let us write thetotal interfacial energy of the system as function of x, thatis, while phase a moves along direction x in Figure 1:

GrðxÞ ¼XU;W

AUWðxÞ � rUWðxÞ ½2�

Received 2 October 2010; accepted 28 October 2010.The author is grateful to Professor S. E. Friberg for his valu-

able comments. This work was financed by the OTKA-NKTHproject CK 80255 (Hungary).

This work was partly financed by the TAMOP-4.2.1.B-10/2/KONV-2010-0001 project with support by the European Unionand the European Social Fund.

Address correspondence to George Kaptay, BAY-NANOResearch Institute and University of Miskolc, Egyetemvaros,E=7, 606, Miskolc 3515, Hungary. E-mail: kaptay@hotmail.com

Journal of Dispersion Science and Technology, 33:130–140, 2012

Copyright # Taylor & Francis Group, LLC

ISSN: 0193-2691 print=1532-2351 online

DOI: 10.1080/01932691.2010.548232

130

whereGr is the total interfacial energy of the system (J),AUW

is the interfacial area of the U=W interface (m2), rUW isdefined by Equation (1). According to Newton and Gibbs,the force acting on phase a along direction x is defined as:

Fa;x � � dGrðxÞdx

½3�

The negative sign in Equation (3) indicates that thetransfer of phase a along path x (Figure 1) is spontaneous,that is, takes place towards more negative values of thetotal interfacial Gibbs energy. In other words, Equation(3) describes the inner driving force of the system, arisingas the system is in a macroscopic non-equilibrium state,while local equilibrium is supposed at a molecular levelalong all interfaces. Thus, the driving force described byEquation (3) drives the system towards its macroscopicequilibrium state. In Equations (2)–(3) it was supposed thatthe transfer of phase a along path x (Figure 1) is not con-nected with any change of the bulk Gibbs energies of anyof the phases. It is also supposed that the transfer of phasea along path x takes place infinitely slowly, that is, throughequilibrium steps. Now, let us substitute Equation (2) intoEquation (3) to obtain the final, general equation for theinterfacial force:

Fa;x ¼ �XU;W

AUWðxÞ�drUWðxÞ

dx�XU;W

rUWðxÞ�dAUWðxÞ

dx½4�

TABLE 1Classification of interfacial forces, according to the number of involved phases and according to the

orientation of the force in respect to the major interface in the system

Orientationnumber ofphases Parallel to the main interface Perpendicular to the main interface

Independent of the maininterface

2 Interfacial anti-stretching force Curvature induced interfacial force Interfacial gradient force

3 Interfacial spreading force Interfacial capillary force Interfacial adhesion force

4 Interfacial meniscus force Fluid bridge induced interfacialforce ?

FIG. 1. To the definition of interfacial forces: Phase a moves within

phase b along vector (path) x in a 4-phase (a, b, c, d) system.

INTERFACIAL FORCES 131

From Equation (4) one can conclude that an interfacialforce acting on phase a along path x will appear, if any ofthe interfacial energies rUW or any of the interfacial areasAUW of the system changes when phase a moves along pathx. If none of them change during this process, the interfacialforce is zero. Calculating Fa,x by Equation (4), both positiveand negative values can be obtained. If the calculated valueis positive, the direction of the force coincides with the direc-tion of the arbitrary chosen direction x, and vice versa.

3. THE INTERFACIAL ANTI-STRETCHING FORCE

Let us consider a liquid film of width w stretched on aperfectly wetted frame along direction x (see Figure 2).[4]

To stretch the frame with a constant and infinitely slowrate, a force is needed to compensate the effect of an inter-facial force. This force is called here the ‘‘interfacialanti-stretching force’’ (Fa�str

lv ). For the derivation let usneglect the thickness (d) of the liquid film and the diameterof the frame. Thus, the problem becomes a two-phaseproblem (liquid and vapor). Also, surface tension rlv istaken as a constant, and so only the second term of Equa-tion (4) applies, without summation. The total interfacialarea of the liquid film from Figure 2: Alv(x)¼P � x, whereP is the perimeter (m) around the liquid film, perpendicularto direction x. Substituting this equation into Equation (4),the following equation is obtained:

Fa�strlv ¼ �P � rlv ½5�

The interfacial anti-stretching force has not been con-sidered as a special class of interfacial forces.[3–7] Instead,the experiment in Figure 2 is used to define the surface ten-sion of liquids.[4] Indeed, surface tension (N=m) has anidentical numerical value with that of the interfacialanti-stretching force (N) if w¼ 0.5m and P¼ 1m. In thisarticle surface tension is defined on thermodynamic

grounds by Equation (1) and, thus, Equation (5) is freeto define the interfacial anti-stretching force. This forcecan be used to derive the equilibrium shapes of phases(ie., the Young equation[1]) and equilibrium size of bubblesdetaching from orifices.[11,12]

4. THE CURVATURE INDUCED INTERFACIAL FORCE

The force, arising due to the curvature of an interface iscalled the ‘‘curvature induced interfacial force’’ (Fcur

lv ), beingzero for a flat interface. This force acts perpendicular to theinterface. If this force is divided by the surface area it is con-verted into the ‘‘curvature induced interfacial pressure’’(pcurlv ). Let us consider a small spherical bubble of radius xin a liquid (Figure 3). As the system contains two phases(liquid and gas) and the surface tension of the liquid is con-sidered as constant, only one term remains in Equation (4).The interfacial area equals: Alv(x)¼ 4 � p � x2. Substitutingthis equation into Equation (4), the following equation isobtained: Fcur

lv ¼ �8 � p � x � rlv. Dividing this equation bythe equation for the surface area of the bubble, the equationfor the curvature induced interfacial pressure is obtained:

pcurlv ¼ � 2 � rlvx

½6�

As follows from the negative sign of Equation (6), pcurlv

acts to shrink the smaller phase (the bubble in Figure 3).Equation (6) is a particular case of a more general Laplaceequation[2]:

pcurvlv

�� �� ¼ 1

R1þ 1

R2

� �� rlv ½7�

FIG. 3. To the derivation of the curvature induced interfacial force

(Fcurlv ). Fcur

lv acts to shrink the smaller phase, perpendicular to the curved

liquid=gas interface.

FIG. 2. A liquid film stretched using a frame (to the derivation of the

interfacial anti-stretching force, Fa�strlv ). Fa�str

lv acts along the lv interface,

against stretching the liquid film.

132 G. KAPTAY

where R1 and R2 are the two principal radii of the interfacein the given point.[4] The equivalency of Equations (6)–(7)is shown here to demonstrate that the Laplace equationis one of the particular cases of the general Equation (4).

5. THE INTERFACIAL GRADIENT FORCE

When a particle (droplet, bubble, etc.) is placed in amatrix phase with an interfacial energy gradient, it willbe driven by the so-called ‘‘interfacial gradient force’’(Fgrad

lv ) towards places of lower interfacial energy.[13–22]

This gradient is created by the gradient of any physicalquantity on which it depends, such as composition,temperature, and electric potential:

drUWdx

¼ drUWdT

� dTdx

þXc

drUWdxc

� dxcdx

þ drUWdE

� dEdx

½8a�

where T is absolute temperature (K), xc is the mole fractionof the c-th component in the matrix phase, and E is theelectric potential (V) at the metal=ionic interface if suchis present.

For simplicity let consider a vapor phase of a constantshape and area of Alv, dispersed in a liquid, moving alongvector x. As there are only two phases (liquid and vapor),only one term of Equation (4) remains to determine theequation for the interfacial gradient force:

Fgradlv ¼ �k � Alv �

drlvðxÞdx

½8b�

where k is the numerical factor, taking into account theinfluence of the moving phase on the temperature, concen-tration and=or electric potential gradients inside the matrixphase.[15,19] As follows from Equation (8b), the direction ofthis force will coincide with the direction of the decrease ofthe interfacial energy. As surface tension of liquids usuallydecreases with increased temperature and with the increaseof concentration of a surface active component c, bubblesin liquids are usually driven by the interfacial gradientforce towards regions with a higher temperature and=or ahigher concentration of surface active components.

Similarly to Fgradlv , other forces can be defined along

other types of interfaces, such as Fgradsl . For this case Equa-

tion (8b) is still valid if the subscripts lv are replaced by sub-scripts sl, respectively. However, for interfaces with at leastone solid phase involved, the interfacial gradient forcemight cancel by the well-known ‘‘no-slip’’ hydrodynamicboundary condition at the solid interface, being valid formacroscopic fluid particles. Around microscopic or nano-sized particles surface diffusion along the particle=fluidinterface might provide the necessary mechanism to ensurethat the interfacial gradient force is active even at the solid=fluid interfaces. This is supported by comparison with some

experimental results.[23–26] Finally, it should be mentionedthat migration of phases due to the ‘‘interfacial gradientforce’’ is sometimes called ‘‘thermocapillary con-vection,’’[18] ‘‘Marangoni migration,’’[16] or‘‘Marangoni-induced motion’’[17] in the literature. Thisquestion is discussed below.

6. THE INTERFACIAL SPREADING FORCE

As was documented since the nineteenth century, an oilfilm rapidly spreads over water.[27] The force spreadingliquids over surfaces of another, immiscible liquid surfacesis called the ‘‘interfacial spreading force’’ (F

sprlv ). As shown

in Figure 4a, two immiscible liquids (L, l) and the vapor (v)phase participate in creating the interfacial spreading force.The driving force for spreading is the difference in interfa-cial energies: Drspr� rLv� rlv� rLl. As follows from theYoung equation,[1] the condition of perfect wettability ofthe lower phase L by the upper phase l in v is Drspr� 1.The same is the condition of the positive driving force ofspreading.

Let us take all interfacial energies and thus Drspr as con-stants. Then, Equation (4) will have 3 terms. The corre-sponding interfacial areas are written from Figure 4a (thethickness of the liquid film l is neglected): Alv¼AlL¼w � x,ALv¼w � (a� x). Substituting these into Equation (4) andtaking into account Drspr� rLv� rlv� rLl, the following

FIG. 4. To the derivation of the interfacial spreading force. (a) A

liquid l, spreading on the surface of another liquid L, in direction x (the

width perpendicular to the article is w). (b) A liquid l with a constant T-

gradient causing a surface tension gradient, leading to the surface liquid

flow due to the interfacial spreading force.

INTERFACIAL FORCES 133

equation is obtained:

Fsprlv ¼ w � Drspr ½9a�

The interfacial spreading force makes liquid l to spreadover the surface of liquid L if Drspr� 1. The interfacialspreading force was first described by Marangoni[27] and,thus, it is called the Marangoni force. A similar phenom-enon can be observed in a two-phase (l=v) system as well,with an interfacial energy gradient along the l=v surface(see[28–30]). In Figure 4b, a liquid with a horizontal T-gradi-ent is shown along direction x by heating the left-hand sideand cooling the right-hand side of a container with liquid l.This T-gradient causes a surface tension gradient alongdirection x. Similarly to the case shown in Figure 4a, aninterfacial spreading force appears in Figure 4b along thesurface in direction x, which causes a similar surface flowin Figures 4a and 4b. The equation for the interfacialspreading force can be obtained by multiplying and divid-ing Equation (9a) by the length of the container, a. Physi-cally, the ratio Drspr=a is the specific driving force for liquidflow in case of Figure 4a. For Figure 4b, the specific drivingforce is the surface tension gradient along direction x:drlv(x)=dx. Substituting the formal equality Drspr=a¼ drlv(x)=x and also Alv¼w � a into Equation (9a) multi-plied by a=a, the following equation is obtained for the caseof Figure 4b:

Fsprlv ¼ Alv �

drlvðxÞdx

½9b�

Equation (9b) is also called the ‘‘Marangoni force.’’Now it is time to compare Equation (9b) with Equation(8b). One can see that the two forces are indeed equal invalues (if k¼ 1 in Equation (8b)), but are opposite in direc-tions: F

sprlv ¼ �Fgrad

lv . Thus, only one of these forces can becalled a ‘‘Marangoni force.’’ Historically it should beEquation (9b), the interfacial spreading force. The interfa-cial gradient force can be called a ‘‘Marangoni forceinduced interfacial gradient force’’ (see Figure 5 and[17]).

7. THE INTERFACIAL CAPILLARY FORCE

Let us consider a three-phase system, with one of thephases (a) situated at the interface of the two other phasesb and c (see Figure 6). An interfacial force will be actingalong the three-phase line in this system, perpendicular tothe b=c interface. As historically this situation was firstdescribed for solid capillaries at liquid=vapor interfaces,[2]

this force will be called here the ‘‘interfacial capillary force’’(F

capa=bc). As three phases have three interfaces and the inter-

facial energies are taken as constant values, Equation (4)for this case will have 3 terms due to Aab(x), Aac(x) andAbc(x). Let us consider phase a being rigid, with constant

interfacial area Aa. Then, interfacial areas Aab(x) andAac(x) are interconnected: Aac(x)¼Aa�Aab(x). As followsfrom Figure 6, phase a covers a DAbc(x) part of the initialb=c interface (A0

bc) and thus: AbcðxÞ ¼ A0bc � DAbcðxÞ. Sub-

stituting these equations into Equation (4), the most gen-eral equation for the interfacial capillary force is obtainedas:

Fcapa=bc ¼ rbc �

dDAbcðxÞdx

þ rac � rab� �

� dAabðxÞdx

½10a�

Now let us apply Equation (10a) for the case of a¼ s,b¼ l, c¼ v, that is, for the solid particle at a liquid=vaporinterface, by taking into account the Young equation:

Fcaps=lv ¼ rlv �

dDAlvðxÞdx

þ dAslðxÞdx

� cosH� �

½10b�

FIG. 6. To the derivation of the interfacial capillary force. The inter-

facial capillary force acts on phase a situated at the interface of a large, flat

interface b=c, perpendicular to this interface b=c (the meniscus is neglected

here as the role of gravity is neglected).

FIG. 5. The comparison of the interfacial gradient force and the inter-

facial spreading force in a system containing both a free liquid surface and

a bubble in the liquid. The interfacial spreading force driving the liquid

layer around the bubble towards the lower T region can be considered

as the driving force for the interfacial gradient force, driving the bubble

towards the higher T region.

134 G. KAPTAY

Equation (10b) was used to derive equations forpenetration=infiltration of liquids into porous solids of dif-ferent morphologies: solids with capillaries of differentshapes,[31–33] solids made of closely packed equalspheres,[34,35] solids made of cylindrical fibers,[36–38] solidswith irregular porous structure,[39–42] being the extensionof the Carman equation[43,44–52]). Now, let us apply Equa-tion (10b) to the classical problem of capillary rise. If phasea in Figure 6 is a fixed, solid capillary of inner radius R,partly immersed into the liquid phase l, the interfacialcapillary force will act along the s-l-v three-phase line. Asthe capillary is fixed, this force will move the liquid up ordown in the capillary. The two geometrical parametersAsl(x) and DAlv(x) are described as function of x (with itsvector directed upwards) by the equations: Asl(x)¼2 �R � p � x, DAlv(x)¼ const. Substituting these equationsinto Equation (10b), the interfacial capillary force raisingthe liquid up within the capillary is obtained as:

Fcaps=lv ¼ 2 � R � p � rlv � cosH ½10c�

This equation is the Young-Laplace equation for capil-lary rise.[2] It is shown here to demonstrate that theYoung-Laplace equation is also the consequence of thegeneral Equation (4).

Now, let us consider Figure 6 with phase a being aspherical solid particle of radius R, situated at a liquid=vapor interface, immersed into the liquid at a depth of x(<2R) (see[53–55]). Let us neglect the effect of gravity andthe effect of curved menisci. As the particle is not fixed,the interfacial capillary force will move it up or down, per-pendicular to a large liquid=gas interface. From thegeometry of a sphere and Figure 6: DAlv(x)¼ 2 �R � p � x�p � x2, Asl(x)¼ 2 �R � p � x. Substituting these equations intoEquation (10b), the interfacial capillary force, acting on asolid sphere at the l=v interface is obtained as:

Fcaps=lv ¼ 2 � R � p � rlv � 1þ cosH� x

R

� ½10d�

For a small particle with gravity neglected, the particlewill gain its equilibrium when Fcap

s=lv ¼ 0. Substituting thiscondition into Equation (10d), the equilibrium depth ofimmersion xeq,H of the solid particle with radius R at aliquid=vapor interface is obtained as:

xeq;H ¼ R � ð1þ cosHÞ ½10e�

As follows from Equation (10e), a small solid sphericalparticle will be fully (x� 2R) immersed into a liquid onlyif H¼ 0o. Expressing (1þ cosH) from Equation (10e) andsubstituting it back into Equation (10d), the interfacialcapillary force is obtained in a more simple form:

Fcaps=lv ¼ 2 � p � rlv � xeq;H � x

� �½10f �

As follows from Equation (10f), if x 6¼ xeq,H the inter-facial capillary force will always push the particle back toits equilibrium position. Thus, the interfacial capillaryforce will behave like a spring. This special propertyof the interfacial capillary force makes it suitable toensure stabilization of foams and emulsions by solidparticles).[56–80]

8. THE INTERFACIAL MENISCUS FORCE

As known, the meniscus around particles is usuallycurved.[81] First, let us derive the conditions when the men-iscus remains flat around a floating spherical particle. Ifonly the gravity and the buoyancy forces act on a solidspherical particle of radius R at a flat liquid=gas interface,the condition of mechanical equilibrium of the particle isexpressed as:

q� ¼x2eq;g � 3 � R� xeq;g

� �4 � R3

½11a�

where xeq,g is the equilibrium depth of immersion of theparticle due to gravity only, q� is the dimensionless densitydefined as: q� � ðqs � qvÞ=ðql � qvÞ. On the other hand, theequilibrium depth of immersion of the particle due to theinterfacial capillary force (xeq,H) is written by Equation(10e). There is a special combination of densities and con-tact angle when xeq, g¼ xeq, H. This condition can be foundby substituting xeq,H for xeq,g from Equation (10e) intoEquation (11a):

q�flat ¼ð1þ cosHÞ2 � 2� cosHð Þ

4½11b�

where q�flat is the special value of q� with a property that at

q� ¼ q�flat the meniscus around the spherical particle is flat.

The equilibrium depth of immersion of the particle will bein the interval between xeq,g and xeq,H. When q� > q�flat,then xeq, g> xeq> xeq,H, that is, the gravity force pushesthe particle lower than the interfacial force and vice versa(see Figure 7).

If two floating solid particles appear close to each otherat the same liquid=vapor interface, the menisci aroundthem will overlap, leading to the appearance of the socalled ‘‘interfacial meniscus force’’ (Fmen

ss=lv). This force actsparallel to the macroscopic liquid=vapor interface betweenthe two neighboring particles.[82–84] The ‘‘interfacial menis-cus force’’ was called also as ‘‘lateral capillary force.’’[84]

There are four phases involved in this situation: twosolid particles (s1 and s2), the liquid (l) phase and the vapor(v) phase. As all interfacial energies are taken as constantvalues, Equation (4) will have five terms, due to As1l(x),As1v(x), As2l(x), As2v(x) and Alv(x). The following two obvi-ous relationships exist between them: As1l(x)¼

INTERFACIAL FORCES 135

As1�As1v(x) andAs2l(x)¼As2�As2v(x). Substituting theseequations into Equation (4) and taking into account theYoung equation, the general equation for the interfacialmeniscus force is obtained as:

Fmenss=lv ¼ �rlv �

�dAlvðxÞ

dx� dAs1lðxÞ

dx� cosH1

� dAs2lðxÞdx

� cosH2

� ½12a�

where Hs1 and Hs2 are the contact angles of the liquid onsolid particles s1 and s2, respectively. Equation (12a) canbe solved only if the equations for the shapes for all inter-faces are found by solving the Laplace equation coupledwith the effect of gravity. The analytical solution of thiscomplex problem is not known.[82–84] The followingapproximated equation exists for the interaction of twofloating particles with radii R1 and R2

[83]:

Fmen;floatss=lv ffi � 8 � p

9� R

31 � R3

2 � g2 � ql � qvð Þ2

rlv � xþ R1 þ R2ð Þ � Dq�1 � Dq�2

½12b�

where g¼ 9.81m=s2 (the acceleration due to gravity), Dq�i isdefined as: Dq�i � q�i � q�i;flat. The force is attracting whenits sign is negative and vice versa. This force is inverselyproportional to the distance between the centres ofthe spheres (similarly to forces due to gravity,electro-magnetic fields, etc.). This force is proportional tothe R3

1 � R32 � ql � qvð Þ2�g2=rlv expression showing that the

force is due to the compensation of the curved l=v surfacearea of the meniscus between the particles while theyapproach each other. This is the only interfacial force beinginversely (and not linearly) proportional to the interfacialenergy. This is because a larger interfacial energy allowssmaller curvature of the meniscus at the same gravity forcebeing proportional to R3

1 � R32 � g2. It also follows that for

particles between two immiscible liquids with samedensities (ql� qv¼ 0) the interfacial meniscus force will be

zero, as there will be no curvature around the floating par-ticles due gravity (the same is true in microgravity environ-ment, g¼ 0). The force described by Equation (12b) isproportional to �Dq�1 � Dq�2. It means that the force isattracting between two similar particles with menisci ofthe same sign around them (both curved down or bothcurved up). The force is repulsing when two particles withdissimilar menisci meet, and the force becomes zero if themeniscus around at least one of the particles is flat. Finallyit should be reminded that the approximated Equation(12b) is not valid at rlv¼ 0.

The horizontal extension of a curved meniscus around aspherical particle (wmen) equals

[83] (Figure 7):

wmen ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

rlvql � qvð Þ � g

r:

It is identical to the so-called ‘‘capillary length,’’[4] beingindependent of the radius of the particle. When the dis-tance between the particles is larger than 2wmen, the interfa-cial meniscus force becomes negligible.

Equation (12b) refers to floating particles. However,particles can be larger than the thickness of the liquid layer.When the particles are supported against gravity by the flatfixed solid surface with some liquid around them, they arecalled ‘‘immersed’’ particles. Immersed particles alsoexperience the effect of interfacial meniscus force, whenthey appear close enough to each other. The approximatedsolution of Equation (12a) for two vertically immersed cyl-indrical particles of radii R1 and R2 is written as[84]:

Fmen;immss=lv ffi �p � rlv �

R1 � R2

xþ R1 þ R2ð Þ � cosH1 � cosH2 ½12c�

Equations (12b) and (12c) are similar in that both areinversely proportional to the distance between the centresof the particles. However, the curvature of the meniscusin the case of immersed particles has nothing to do withgravity. It is rather determined by the contact anglesaround the two particles. The two immersed particles willbe attracting each other, if they are similarly wetted ornon-wetted and vice versa. If the contact angle aroundany of the particles is 90o, the interfacial meniscus forceis zero. This force has been used to interpret theself-arrangement of nanotube arrays.[85]

9. THE FLUID BRIDGE INDUCED INTERFACIALFORCE

Two solid particles connected by a small liquid bridge ina vapor environment are shown in Figure 8a. The ‘‘liquidbridge induced interfacial force’’ (Fl�br

ss=lv ) will attract orrepulse the two particles.[86,87] This force is called also‘‘normal capillary force’’[87] for making a difference fromthe ‘‘lateral capillary force.’’[84] As in this case also two

FIG. 7. The shapes of the liquid=vapor menisci around three different

particles with the meniscus curved down (particle s1), with the flat menis-

cus (particle s2) and with the meniscus curved up (particle s3). The width

of the meniscus is shown by wmen.

136 G. KAPTAY

particles appear at the liquid=vapor interface, the generalEquation (12a) will remain valid. An analytical solutionexists only in the limit of negligible liquid volume and zeroseparation (x) between two spherical particles of radii R1

and R2 wetted by the liquid with contact angles of H1

and H2:[86,87]

limx!0

Fl�brss=lv ¼ �2 � p � R1 � R2

R1 þ R2� rlv � ðcosH1 þ cosH2Þ ½13�

For more realistic cases of x> 0 and higher liquid volumesthe force will have a lower absolute value than that calcu-lated by Equation (13) and will be gradually approachingzero by increasing the volume of the liquid and=or x(see[87]). This force is attracting when its sign is negativeand vice versa. Thus, the liquid bridge induced interfacialforce will attract the two solid particles if the particlesare wetted by the liquid (cosH1þ cosH2> 0, or at equalcontact angles: H1¼H2< 90o) and vice versa.

Two solid particles connected by a gaseous bridge in aliquid environment are shown in Figure 8b. From the simi-larity of Figures 8a and 8b it is clear that the gaseous bridgeinduced interfacial force is opposite of the liquid bridgeinduced interfacial force:

Fg�brss=lv ¼ �Fl�br

ss=lv ½14�

This force is attracting when its sign is negative and viceversa. Thus, the gaseous bridge induced interfacial forcewill attract the two solid particles if the particles are not

wetted by the liquid (cosH1þ cosH2< 0, or at equal contactangles: H1¼H2> 90o) and vice versa.

10. THE INTERFACIAL ADHESION FORCE

Let us consider two phases a and b in a larger phase c,the particles being far from the outer interface of phase c.If the particles are also far from each other, the interfacialforce acting on both of them will be zero. However, if theparticles are close enough to each other (yet not touchingeach other) the molecules=atoms in their interfaces willgenerate some change in the energetic states of each other.In other words, the ac and bc interfacial energies willbecome functions of particle separation x and, therefore,an interfacial force will arise.[5,6,88,89] As for like particlesthis force leads to strong adhesion between them, this forceis called here the ‘‘interfacial adhesion force.’’ The sameforce might also be repulsing between unlike particles, asfollows from the term ‘‘disjoining pressure.’’[5] The interfa-cial adhesion force is responsible for re-distribution of par-ticles during solidification of composites.[90–104]

There are three phases involved in this situation: a, b,and c. Vector x is chosen as the separation (distancebetween closest interfaces) of phases a and b. Phases aand b are considered rigid, that is, their interfacial areasare kept constant. Thus Equation (4) will have only twoterms, due to the x-dependent interfacial energies. To sim-plify the geometrical situation, two equal cubic solid parti-cles are considered, turned to each other by their equalparallel sides. In this configuration the following simplifiedequalities are valid: Aac¼Abc�A. Substituting these equa-tions into Equation (4) the interfacial adhesion force iswritten as:

Fadhabc ¼ �A � drac

dxþ drbc

dx

� �½15a�

To make Equation (15a) explicit, one has to express theinterfacial energies as function of x. To find this relation-ship, let us first define the boundary conditions:

i. at infinite separation (x! 1) the interfacial energieshave their standard, x-independent values:rac(1)¼ rac, rbc(1)¼ rbc,

ii. at zero separation (x¼ 0) the two particles touch eachother and so the sum of their interfacial energiesbecomes the interfacial energy of the ab interface:rac(0)þ rbc(0)¼ rab. Formally this can be divided intotwo parts as: rac(0)¼ p � rab and rbc(0)¼ (1� p) � rabwith parameter 0< p< 1 (later parameter p will fallout).

The interaction energies are assumed to change inverselyproportional to square of x. Then, one can write thefollowing approximated equations for the separationdependent interfacial energies, in accordance with the

FIG. 8. To the derivation of the fluid bridge induced interfacial force.

(a) Two solid particles connected by a liquid bridge in a vapor. (b) Two

solid particles connected by a gaseous bridge in a liquid.

INTERFACIAL FORCES 137

above boundary conditions:

racðxÞ ¼ rac þ p � rab � rac� �

� d

d þ x

� �2

½15b�

rbcðxÞ ¼ rbc þ ð1� pÞ � rab � rbc� �

� d

d þ x

� �2

½15c�

where d – is the diameter of a molecule (atom) in phase c.Substituting Equations (15b) and (15c) into Equation(15a), the following final equation is obtained:

Fadhabc ¼ 2 � A � Drabc �

d2

ðd þ xÞ3½15d�

Equation (15d) is valid for two, parallel slabs. For othergeometries see other works.[5,6,88,89] The key parameter inEquation (15) is Drabc. Its definition follows from theabove derivation:

Drabc � rab � rac � rbc ½15e�

Since Equation (15e) was introduced by Derjaguin[88]

and later by Hamaker,[89] two incorrect equations havebeen published to replace it. One of them isDrabc� rab� rac,

[90] the other one was introduced by thisauthor[7,96] Drabc� 2 � rab� rac� rbc. Both of these equa-tions should be disregarded.

�

From Equation (15e) it is obvious that Drabc�Drbac.The interfacial adhesion force is attractive when its valueis negative and vice versa. The following cases are worthto discuss:

1. For two equal phases a¼ b: rab¼ 0 and rac¼ rbc, thatis, Drabc¼�2 � rac< 0. Thus, the interfacial adhesionforce between identical phases is always attractive.

2. When phase c is a vacuum, rab< racþ rbc as there isalways some adhesion energy between a and b missingat an interface with a vacuum. Thus, Drabc< 0 and,thus, the interfacial adhesion force trough a vacuum isalso always attractive.

3. When phase c is a vapor, or especially if it is a liquid orsolid, Drabc and the interfacial adhesion force can haveany sign, that is, it can be attractive or repulsive.

11. CONCLUSIONS

In this article, all known interfacial forces being essentialfor dispersion science and technology have been reviewed.The interfacial forces are classified and are divided intoeight different classes, depending on the number and geo-

metrical arrangement of the phases and on the directionof the given interfacial force (Table 1). Equations to calcu-late the interfacial forces for all these 8 cases are derivedfrom the same general equation (4) being the consequenceof ideas of Newton and Gibbs. All known interfacial forcesand pressures are shown to be particular cases of this gen-eral Equation (4). As a new element, an interfacialanti-stretching force is introduced in this article, beingequivalent to the definition of the interfacial energy interms of tension as understood by Young. Also, the differ-ences and similarities between the interfacial gradient forceand the interfacial spreading force (the Marangoni force)are shown. Further, the well-known case of the liquidbridge induced interfacial force is supplemented by its lessknown version of a gaseous bridge induced interfacialforce. It is expected that further possible interfacial forceswould also follow from the same general equation (4) incomplex situations arising during analyzing scientificand technological challenges in dispersion science andtechnology.

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