Nanometer Deformation Caused by the Laplace Pressure and the

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Journal of Colloid and Interface Science211,114–121 (1999)Article ID jcis.1998.5978, available online at http://www.idealibrary.com on

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Possibility of Its Effect on Surface Tension Measurements

Bo Zhang1 and Akira Nakajima

Faculty of Science and Engineering, Saga University, 1 Honjo-machi, Saga-shi, Saga 840-8502, Japan

Received June 23, 1998; accepted November 9, 1998

2-methylbutane in atmospheres of their own vapors at a

y ticans

lanterrinrfade-anrcethofocw

anf thd

t olec-u voidt foundt bulkt filmw er itw ters ssurer micsw to aK ther oper-a ctricd mayp ter.C dopt-i rente i (2).Tt n ther suredp de-c ed byF

ssaryb y ofs mo-l tand-i apero olids g them Thea of them pla-n

aturea rfacew dJ solids ntm is-

The deformation of solid surfaces caused by the Laplace pres-ure is analyzed ad its effect on the meniscus force measurements discussed by comparing it with the published experimentalesults of the meniscus force on a surface force apparatus. It isound theoretically that the deformation reaches the order ofeveral nanometers and reduces greatly the meniscus force whenhe meniscus curvature radius is small. It is concluded that theeviation of the measured meniscus force with the surface forcepparatus from equation F 5 4pgR cos u for a nanometer me-iscus is due neither to the hydrogen bonding interaction (Fishernd Israelachvili, Colloids and Surf. 3, 303 (1981)) nor to the shortange solid–solid interaction (Christenson, J. Colloid Interface Sci.21, 170 (1988)). The main reason lies in the deformation of theolid surface due to the Laplace pressure. The nanometer defor-ation of solid surfaces caused by the Laplace pressure is of great

mportance in measuring the meniscus force. © 1999 Academic Press

Key Words: meniscus force; nano liquid film; surface force ap-aratus; Laplace pressure; elastic deformation.

1. INTRODUCTION

It is of great interest both in academia and in industrnow how the concept of the surface tension of liquid, whs based on bulk thermodynamics, can be applied to a n

eter scale and at what limit scale the concept becomealid.The first challenge to the problem was made by McFar

nd Tabor (1). They measured the meniscus force of watmospheres with different relative humidities by measu

he pull-off force between a glass ball and a glass plane sun contact. According to the theory the pull-off force is inendent of the relative humidity. However, McFarlaneabor (1) found in their experiments that the pull-off foecreased suddenly when the relative humidity was less0% and the decrease was dependent on the roughnesslass surfaces. They concluded that the decrease wouldhen the height of the surface roughness was comparable

he thickness of the adsorbed liquid film. Later, Fishersraelachvili (2) made a more precise measurement oeniscus force for water, benzene, cyclohexane,n-hexane, an

1 To whom correspondence should be addressed.

114021-9797/99 $30.00opyright © 1999 by Academic Pressll rights of reproduction in any form reserved.

oho-in-

eingce

d

anthe

curithde

ive vapor pressure in the range of 0–99%. They used mlarly smooth mica in contact surfaces instead of glass to a

he effect of surface roughness on measurement. Theyhat for organic liquids the surface tension theory based onhermodynamics was applicable even when the adsorbedas only a few molecules in thickness. However, for watas quite different. Their experimental results for wahowed that the meniscus force due to the Laplace preeduced to 90% of that expected from bulk thermodynahen the relative vapor pressures is 0.9, correspondingelvin meniscus radius of about 5 nm. They explained

esults in terms of the assumption that the long-range cotive nature of the hydrogen bonding interaction and eleouble layer forces in water film between solid surfaceslay a role in reducing the effective surface tension of wahristenson (3) modified the surface force apparatus by a

ng a double cantilever spring. They obtained very diffexperimental results from those of Fisher and Israelachvilhey found that for organics such as cyclohexane andn-hexane

he measured pull-off force increased with the decrease ielative vapor pressure, and for water, although the meaull-off force decreased as the relative vapor pressurereased, the decrease was much smaller than that obtainisher and Israelachvili (2).It appears that more experimental evidences are nece

efore making any decisive conclusions about the validiturface tension theory based on bulk thermodynamics toecular scale liquids. On the other hand, precisely undersng the experimental results may be more important. This pffers an analysis about the effect of deformation of a surface caused by the Laplace pressure on measurineniscus force, which has been neglected until now.nalysis provides some insight about the measurementseniscus force and will be helpful with regards to the exation of experimental results of the meniscus force.There have been many studies published in the liter

bout the non-Hertz contact between a ball and a plane suith surface adhesion. Johnsonet al. (4) proposed a so-calleKR model considering the adhesion force between twourface. Derjaguinet al. (5) proposed a somewhat differeodel from JKR, which is called the DMT model. The d

cussion of the difference between the JKR and the DMTm eri edf er-a llarf turm was tingc heot nu( anW

thm ot fora thebm ent

aced wih surm thm

2

eet anp pha ticr rigb .w ttea thep isct

A

wpa

N rma-t sf

w ss,w rma-t termso

w ces.S ee )

E sid-e

ssurei peri-m s ass validb highL iscus

re anda

115NANOMETER DEFORMATION

odels was made by Tabor (6). The JKR model was genzed by Pollocket al. (7). The DMT model was also developurther by Mulleret al. (8, 9). Fogden and White (10) genlized the JKR model by introducing condensation capi

orce. Maugis (11) gave a general theory by using fracechanics with a Dugdale model of adhesion forces. It

hown that the JKR and the DMT theories are the limiases of the general theory. When applying the general to the case of a liquid meniscus, Maugis and Gauthier-Ma12) found similar results to those obtained by Fogden

hite (10).The deformation of solid surface has a great effect oneasurement of the pull-off force. Christenson (3) pointed

hat the pull-off force measurements by means of surfacepparatus do not give any information on the validity ofulk surface tension for a very high (r , 4 nm) curvatureeniscus, at least not in the absence of a complete treatm

he surface deformation problem.In this paper, we will focus our attention on the surf

eformation caused by the Laplace pressure only, shoow the deformation may affect the meniscus force meaent, which is a critical problem for the measurement ofeniscus force by using such a surface force apparatus.

2. THEORY

.1. Meniscus Force Theory without Deformation

The contact geometry between two cylindricals or betwwo spheres can be equivalent to that between a spherelane surface. Therefore, assuming a contact between a snd a plane surface will not lose any generality of the analyesults. Furthermore, plane surface is assumed to be aody. And then the deformation can be illustrated as Fighere the Laplace pressure is uniformly distributed in a werea of radiusa. The meniscus force is then given byroduct of the Laplace pressure by the area of the men

hat is

F 5 pA. [1]

ccording to the Young–Laplace equation, we have

p 5 gS 1

r 11

1

r 2D , [2]

hereg is the surface tension of the liquid andr1, r2 are tworincipal curvature radii of the meniscus. Noting thatr2 @ r1

nd denotingr1 as r , we obtain

p 5g

r. [3]

al-

yes

ryeld

eutce

t of

nge-e

nd aerealid

1,d

us,

ow let us at first consider the case that the surface defoion is neglected. In such cases the areaA of the meniscus iound to be

A 5 pa2 5 2pRh [4]

hen h ! R, where h is the nominal meniscus thicknehich is equal to the real meniscus thickness is nondefo

ion cases. The meniscus radius can also be expressed inf the meniscus thicknessh and is given by

r 5h

2 cosu, [5]

hereu is the contact angle of the liquid on the solid surfaubstituting expressions forp andA into Eq. [1] and noting thxpression of Eq. [5] for the meniscus radiusr , we find that (1

F 5 4pRg cosu. [6]

quation [6] is the meniscus force expression without conring the deformation of solid surfaces.When the meniscus radius is large and the Laplace pre

s low, Eq. [6] has been verified by many researchers exentally. However, when the meniscus radius become

mall as several nanometers, Eq. [6] may become inecause the solid surface may be deformed due to theaplace pressure and then the solid surface within the men

FIG. 1. Contact geometry between a sphere and a plane surface befofter deformation due to the Laplace pressure.

is not a sphere again, which is expected to introduce errors intoE

eer rav meo ane t tp ssum nexs thed

2

fora suf led isca sols atuo e isu reT thee tep ofat

w

a

w esw

ag eaf

The maximum deflection occurs at the center of the meniscusa

w et t de-p efor-m hapew me-n . [4],w is as b-s g thed

2

kingi e tofi them ig. 1a anda de-fe

Sg

w

i of as de-fl Re-p usr fiest

116 ZHANG AND NAKAJIMA

q. [4] and then Eq. [6].The deviation of experimental results from Eq. [6] has b

eported when measuring the meniscus force in undersatuapor pressure, which produces a meniscus radius in nanorder (2, 3). The deviation is especially obvious for water. Mxplanations of the deviation have been proposed, buossibility that the deformation caused by the Laplace preay attribute to the deviation is completely ignored. In the

ection, this possibility will become evident in terms ofeformation analysis based on the theory of elasticity.

.2. Deformation due to the Laplace Pressure

In measuring the meniscus force by means of surfacepparatus (SFA), the pull-off force between the contact

aces of two cylinders crossing each other at a right angetected by a cantilever spring. Since the size of the menrea is very small compared with the curvature radius ofurfaces, it is reasonable to neglect the effect of the curvf solid surfaces on the deformation. The Laplace pressurniformly distributed tensile pressure within the meniscus aherefore the problem is simplified to a problem of findinglastic deflection of solid surface due to a uniformly distriburessure over a circle area of radiusa on a plane boundaryn infinite body. According to elasticity, the deflectionw out of

he load area is given (13) as

54pr

pE*@E

0

p/ 2

Î1 2 ~a/r !2sin2qdq

2 S1 2a2

r 2D E0

p/ 2 dq

Î1 2 ~a/r !2sin2qG [7]

nd within the load area as

w 54pa

pE* E0

p/ 2

Î1 2 ~r /a!2sin2cdc, [8]

hereE* is the composite Young modulus of solid surfachich is defined by

1

E*5

1 2 n12

E11

1 2 n22

E2,

nd p the Laplace pressure. Substitutingr 5 a into Eq. [8]ives the deflection at the boundary of the meniscus ar

ollows:

wa 54pa

pE*. [9]

ntedter

yheret

cer-isusidrea

a.

d

,

as

rea and is given by

wmax 52pa

E*[10]

hich is p/2 times the deflectionwa at the boundary. Sinche deflection due to the Laplace pressure is differenending on the position, the solid surface shape after dation is changed. This change in the solid surface sill of course invalidate the relationships between theiscus area and the meniscus thickness defined by Eqhich is derived from the assumption that solid surfacephere of radiusR, and therefore Eq. [6]. In the next suection, the expression of the meniscus force considerineformation will be derived.

.3. Modified Meniscus Force Expression

In order to obtain the meniscus force expression tanto account the deformation of solid surfaces, we havnd the relationship between the meniscus area andeniscus thickness after deformation. Let us refer to Fgain, which shows the meniscus geometry beforefter the deflection. It is known from Fig. 1 that the

ormed meniscus thicknessh9 is given by the followingquation:

h9 5 h 1 wmax 2 wa. [11]

ubstituting Eqs. [9] and [10] forwa and wmax into Eq. [11]ives

h9 5 h 12~p 2 2! pa

pE*5 h 1 bpa, [12]

here

b 52~p 2 2!

pE*[13]

s a parameter depending only on the material constantolid surface andh is the meniscus thickness before theection, which is called the nominal meniscus thickness.lacing h in Eq. [5] with h9 and substituting the meniscadiusr into Eq. [3], we find that the Laplace pressure satishe following equation:

p 52g cosu

h95

2g cosu

h 1 bpa. [14]

Because the Laplace pressurep is positive (tensile pressure),t

T

S

w

E n thm sidi res ion[ nt ot urfc f thc

q.[z isct rcet useb ta f ts scuf

ftet

T sbt ,t

thl mn

E n ofs counto them olids iscusf [6]t nde nifi-c , butt [17]g e andt ] isr

w ity,w

I nt

a icalrtm0 sk thec hite

n.

117NANOMETER DEFORMATION

he solution of Eq. [14] for the Laplace pressurep is

p 5Îh2 1 8bag cosu 2 h

2ba. [15]

herefore the meniscus force is expressed by

F 5 pa2p 5pa@Îh2 1 8bag cosu 2 h#

2b. [16]

ince

a2 5 2Rh

e have

F 5pÎ2Rh @Îh2 1 8bÎ2Rhg cosu 2 h#

2b. [17]

quation [17] is the expression of the relationship betweeeniscus force and the nominal meniscus thickness con

ng the effect of the deformation caused by the Laplace pure. Equation [17] is much different from Eq. [6]. Equat17] indicates that the meniscus force is not only dependehe surface tension, the contact angle, and the sphere survature radius, but is also dependent on elasticity oontact surface and the nominal meniscus thickness.Although it is not obvious that Eq. [17] will be equal to E

6] when Eq. [17] is applied to a rigid solid surface whoseb isero, it is quite easy to show the results. For the same menhicknessh, Eq. [17] always gives a smaller meniscus fohan Eq. [6]. This is due to the fact that the deflection cay the Laplace pressure at the meniscus center isp/2 times that the boundary, which decreases the curvature radius oolid surface. Therefore, according to Eq. [6] the meniorce is reduced.

The variationd in the meniscus thickness before and ahe deflection is given by

d 5Îh2 1 8bÎ2Rhg cosu 2 h

2. [18]

he variation depends not only on the material preopertieoth solid surfaceb and liquid surface tensiong, but also on

he nominal meniscus thicknessh. Whenh approaches infinityhe differences will approach zero.

For elastic sphere surface, Fogden and White (10) andater Maugis and Gauthier-Manuel (12) showed that theiscus force is given by

F 5 3pRg cosu. [19]

eer-s-

nacee

us

d

hes

r

of

ene-

quation [19] shows that the effect of the elastic deflectioolid surfaces on the meniscus force can be taken into acnly by changing the constant 4 in Eq. [6] into 3, and againeniscus force is independent of the elasticity of the s

urfaces. Equations [6] and [19] suggest that the menorce is not continuous and there will be a jump from Eq.o Eq. [19]. Considering the continuity between rigid alastic bodies, we may reasonably try to find physical sigance behind the discontinuity between Eqs. [6] and [19]here are not any explanations in the literature. Equationives a continuous relationship between the meniscus forc

he elasticity of solid surfaces. For comparison, Eq. [17ewritten in a similar form to Eq. [6] as

F 5 4kpRg cosu, [20]

here k is the correcting factor considering the elastichich is expressed by

k 52

1 1 Î1 1 8Î2R

h Sbg

h Dcosu

. [21]

t is obvious that the correcting factork is dependent only ohe dimensionless parameter

x 5 Î2R

h Sbg

h Dcosu [22]

nd decreases asx increases. Figure 2 shows the numeresult of the relationship between the correcting factork andhe dimensionless parameterx. As a typical example,R 5 30m, h 5 10 nm,g 5 70 mN/m, and 1/b 5 50 GPa, andu 5are put into Eq. [22] andx 5 0.343 is obtained, which give5 0.68. Comparing Eqs. [19] and [20], we know that

orrecting factor in the model proposed by Fogden and W

FIG. 2. The correcting factork considering the effect of elastic deflectio

( odu hicn , wa t oF perm ncs blee ae 0)

3

osY a,w rcm s ad f 2a mr

lacp inam ngm mi n dc isct ickn

S umv sc olis

ickn deo teT orm ofm yc soa m

ed byt2

TABLE 1

118 ZHANG AND NAKAJIMA

10) is 34, which is a constant independent of the Young m

lus of solid surfaces, the curvature radius, the meniscus tess, and the surface tension of liquid. At the present timere not able to claim that our theory is correct and thaogden and White (10) is incorrect. Of course, more exental evidence is necessary before we make the final co

ion, but it is difficult, if not impossible, to give a reasonaxplanation about the jump behavior from a rigid body tolastic body for the theory given by Fogden and White (1

3. NUMERICAL RESULTS

.1. Deflection due to the Laplace Pressure

For mechanical properties of the solid surface, the compoung’s modulusE* is chosen to be from 20 to 200 GPhich covers materials from mica (well used in surface foeasurement) to ceramics (well used in magnetic headisks). Analysis is conducted for typical surface tensions ond 75 mN/m. All the data used in the calculation are sumized in Table 1.

The variation in meniscus thickness due to the Lapressure is given in Fig. 3 as a function of the nomeniscus thickness for three different composite Youoduli. The variation increases quickly at first with the no

nal meniscus thickness, reaches a maximum, and thereases slowly. It is easy to show that the nominal menhickness giving the maximum variation in the meniscus thess is expressed by

hm 5 S4bÎRg cosu

3Î2 D 2/3

. [23]

ubstiting Eq. [23] into Eq. [18] shows that the maximariation increases with the product ofbg. Figure 3 also showlearly this trend; that is, the variation increases for soft surfaces and/or for higher liquid of surface tension.As to the magnitude of the variation in the meniscus th

ess, Fig. 3 illustrates that, although the variation is depenn various parameters, it is generally in several nanomeherefore, it is not necessary to consider the variation facro meniscus. However, when the nominal thicknesseniscus is also in several nanometer orders as formed b

ondensation liquid in undersaturated vapor, it is quite reable to expect that the change in the meniscus thickness

Data Used in Calculation

Parameter Value

E* (GPa) 20, 50, 200g (mN/m) 23, 75R (mm) 30u 0

-k-efi-lu-

n.

ite

end3a-

el

’s-e-

us-

d

-ntrs.aa

then-ay

FIG. 3. Increase in the meniscus thickness due to the deflection caushe Laplace pressure for (a) E*5 200 GPa, (b) E*5 50 GPa, and (c) E*50 GPa.

have a significant effect on the meniscus force or the pull-offf

3

cuf iF odu eaa treb d/a surf t ot l mn st ism wht Thr usf them

that enc minm api thL foh s ist

het ero weg sols s, tmt

wv re.Nt

I scuf d trd .

to theL .

119NANOMETER DEFORMATION

orce.

.2. Effect of the Deflection on Meniscus Force

The analytical results of the variation in the menisorce with the nominal meniscus thickness are givenig. 4 for different values of the composite Young’s mlus and surface tensions. The meniscus force decrs the nominal meniscus thickness decreases. Thisecomes stronger for higher surface tension liquid anlower composite Young’s modulus. However, the

ace tension of liquid seems to have much greater effeche dependence of the meniscus force on the nominaiscus thickness than the composite Young’s modulu

he solid surfaces. For water, whose surface tensionN/m, the meniscus force begins to decrease even

he nominal meniscus thickness is as large as 40 nm.esult suggests that we should take care when Eq. [6] isor water, which may give us an overestimation ofeniscus force.Equation [6] given by McFarlane and Tabor (1) shows

he meniscus force remains unchanged even when the mus thickness changes. This is not true unless the noeniscus thickness is very large. The analysis of this p

ndicates that the effect of the deformation caused byaplace pressure on the measurements of the meniscusas to be taken into account when the meniscus thicknes

he order of several nanometers.There are many examples of experimental evidence w

he measured meniscus force deviates from Eq. [6]. In ordbtain a very small meniscus radius the experimentsenerally conducted on condensation films trapped inurfaces in the presence of vapor. Under such conditioneniscus radius is related to the relative vapor pressureq/qs by

he Kelvin equation as

r 5 2gV

RT ln~q/qs!, [24]

hereR is the gas constant (58.31451 J/Kz mol), V the molarolume of the liquid, andT (K) is the absolute temperatuoting the relationship between the meniscus thicknessh9 and

he meniscus radius of Eq. [5], we have

h9 5 22gV cosu

RT ln~q/qs!. [25]

n order to find the direct relationship between the meniorce and the relative vapor pressure, it is necessary to finelationship between the nominal meniscus thicknessh and theeformed meniscus thicknessh9. Substituting Eq. [14] into Eq

sn-sesnd

or-ne-of75enised

tis-alerercein

retoreidhe

she

FIG. 4. Change in the meniscus force as a result of deflection dueaplace pressure for (a) E*5 200 GPa, (b) E*5 50 GPa, and (c) E*5 20 GPa

[12] and noting thata2 5 2Rh, we obtain the followinge

N

a

C ipb ssut tiod r ac

mus atr xar ateC thm largw get cec

thm disc ntr ns( edt mics als a ia uluo hee tiou hee tha

aporp

120 ZHANG AND NAKAJIMA

xpression forh:

Îh 5 Î2Rb2g2cos2u

h92 1 h9 2Î2Rbg cosu

h9. [26]

oting that

F 5 pa2p 5 p~2Rh!2g cosu

h95 4pRg cosu

h

h9

nd substituting Eq. [26], we have

F 54pgR cosu

h9

3 F Î2Rb2g2cos2u

h92 1 h9 2Î2Rbg cosu

h9 G 2

or

F

4pR5

g cosu

h9 F Î2Rb2g2cos2u

h92 1 h9 2Î2Rbg cosu

h9 G 2

.

[27]

ombining Eq. [27] with Eq. [25], we find the relationshetween the meniscus force and the relative vapor pre

aking into account the effect of the solid surface deformaue to the Laplace pressure. Numerical examples for wateyclohexane are given in Fig. 5.Figure 5 indicates that the measured meniscus force is

maller than that predicted by Eq. [6] for water evenelative vapor pressure as high as 0.9, but for cycloheeduction in the meniscus force is much less than womparing Figs. 5a–5c, it is known that the reduction ineniscus force with the relative vapor pressure becomeshen the solid surface is softer. The analytical results sug

hat it is almost impossible to measure the meniscus forondensation films by using Eq. [6], at least for water.

4. DISCUSSION

In this section the analytical results of the reduction ineniscus force with the relative vapor pressure will be

ussed in some detail by comparing with the experimeesults obtained by Fisher and Israelachvili (2) and Christe3) on their surface force apparatus. The solid surfaces usheir experiments are shown schematically in Fig. 6. Theheets of 1–3mm thickness are glued to silica cylindricurfaces with an epoxy resin. Young’s modulus of micbout 34 GPa (14), which gives a composite Young’s modf 17 GPa. The curvature radius of cylindrical surfaces in txperiments ranged from 15–25 mm. Therefore the condised in the analytical results in Fig. 5c are very similar to txperimental conditions. It was found in their experiments

rennd

chaner.eerstof

e-alonina

ss

irnsirtFIG. 5. Change in the meniscus force as a function of relative v

ressure for (a) E*5 200 GPa, (b) E*5 50 GPa, and (c) E*5 20 GPa.

f elav ed one tc ns( fort tef lysi xpn rfad

es ,a ref pew twew thao cuf

tivee n tr ovb d bi enc ish s (3 dg sui thea foo efom ai ina

5. CONCLUSIONS

lyzeda ussedb sur-f chest mea-s . It isc ce bytd thes y liei lacep wills theirl ntalr olids ce inm

niver-s r J. N.I F. E.T siona unda-t

A

111

1 ed.,

1 .

r aI

121NANOMETER DEFORMATION

or water the measured meniscus force decreased as the rapor pressure decreased although the rate of decreasifferent between Fisher and Israelachvili and Christensxperiments. The difference was due to the change inantilever spring of the surface force apparatus by Christe3). As to the mechanism of the decrease of the meniscushey gave a different explanation based on liquid–solid inace adhesion (2) and solid–solid interaction (3). The anan this paper, however, indicates that the most possible eation of the decrease in the meniscus force is the solid sueformation caused by the Laplace pressure.Besides water, they also measured the meniscus forc

ome organic liquid such as benzene, cyclohexane,n-hexanend 2-methylbutane and found that they were much diffe

rom water in behavior. According to the analysis in this pae suggest that the great difference in surface tension beater and the organic liquid may be the only reasonrganic liquid shows different behavior from water in menis

orce measurement.Although the theoretical prediction gives only a qualita

stimation on the dependence of the meniscus force oelative vapor pressure, the theoretical model may be impry considering solid surface structure more precisely an

ntroducing another surface adhesion force besides the mus force. In the theory we assume that the solid surfaceomogeneous infinite body. However, in their experiment), the solid surface was composed by a mica sheet gluelass. Such a laminar structure of the solid surface will re

n a different deformation behavior from that given innalysis and then a different dependence of the meniscusn the relative vapor pressure. A detail analysis of the dation of the solid surface in the surface force apparatus

ts effect on the measured pull-off force will be givennother paper.

FIG. 6. Schematics of the solid surface used in experiments by Fishesraelachvili (2) and Christenson (3).

tivewas’sheoncer-isla-ce

for

ntr,ent

s

heedyis-a

2,onlt

rcer-nd

The deformation caused by the Laplace pressure is anand its effect on the meniscus force measurements is discy comparing with the published experimental results of

ace force apparatus. It is found that the deformation reahe order of several nanometers and reduces greatly theured meniscus force when the meniscus radius is smalloncluded that the deviation of the measured meniscus forhe surface force apparatus from equationF 2 4pgR cosu isue neither to the hydrogen bonding interaction (2) nor tohort range solid–solid interaction (3). The only reason man the deformation of the solid surface due to the Lapressure. The theory also predicts that the organic liquidhow a much different behavior from water because ofower surface tension. This is in agreement with experimeesults. Therefore, it is clear that the deformation of a surface due to the Laplace pressure is of great importaneniscus force measurements.

ACKNOWLEDGMENTS

The authors express their gratitude to Professor K. Kato at Tohoku Uity for his continued encouragement. Thanks are also due to Professosraelachvili at University of California at Santa Barbara and Professoralke at University of California at San Diego for their helpful discusbout the paper. This work gratefully is supported by the Sumitomo Fo

ion.

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2. Fisher, L. R., and Israelachvili, J. N.,Colloids and Surf.3, 303 (1981).3. Christenson, H. K.,J. Colloid Interface Sci.121,170 (1988).4. Johnson, K. L., Kendal, K., and Roberts, A. D.,Proc. R. Soc. London

324,301 (1971).5. Derjaguin, B. V., Muller, V. M., and Toporov, Y. P.,J. Colloid Interface

Sci.53, 314 (1975).6. Tabor, D.,J. Colloid Interface Sci.58, 2 (1977).7. Pollock, H. M., Maugis, D., and Barquins, M.,Appl. Phys. Lett.33, 798

(1978).8. Muller, V. M., Derjaguin, B. V., and Toporov, Y. P.,Colloids and Surf.7,

251 (1983).9. Muller, V. M., Yushchenko, V. S., and Derjaguin, B. V.,J. Colloid

Interface Sci.77, 91 (1989).0. Fogden, A., and White, L. R.,J. Colloid Interface Sci.138,414 (1990).1. Maugis, D.,J. Colloid Interface Sci.150,243 (1992).2. Maugis, D., and Gauthier-Manuel, B.,J. Adhesion Sci. Technol.8, 1311

(1994).3. Timoshenko, S. P., and Goodier, J. N., “Theory of Elasticity,” 3rd

McGraw-Hill International, New York, 1982.4. Matsuoka, H., and Kato, T.,Trans. ASME, J. Tribology118,832 (1996)

nd