Journal of Colloid and Interface Science245,215–218 (2002)doi:10.1006/jcis.2001.7985, available online at http://www.idealibrary.com on

NOTE

Direct Measurement of the Laplace Pressure in a Very Thin Liquid Film

Meniscus force plays an important part in magnetic storage de-vices, scanning probe microscopy, and micromachines, because themeniscus force increases steeply as the clearance between solid sur-faces decreases down to submicrometers. Recently increased exper-imental and analytical studies on the subject have been publishedfrom the point of view of mechanical engineering. Meniscus forceis generated by the Laplace pressure within the thin liquid film.Investigating the Laplace pressure is a direct way to clarify thegeneration mechanism of meniscus force. However, no appropriatemeasuring method of the Laplace pressure is available for a verythin liquid film confined by solid surfaces. This paper proposes asimple method of direct measurement of the Laplace pressure forthe cases. The principle of the measurement method is described.Measurement is conducted using the measuring device. Experimen-tal results on the Laplace pressure show good agreement with thetheoretical predictions. C© 2002 Elsevier Science

Key Words: nanotechnology; surface force; meniscus force; theLaplace pressure.

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1. INTRODUCTION

Surface tension or surface free energy receives ever-increasing interesengineers, researchers, and professors in fields such as magnetic storage sscanning probe microscopy, micromachines, and traditional physical chemThe surface tension of condensation of water vapor and/or very thin lilubricant film on the hard disk surface causes stiction (1), which often spthe magnetic storage data. Overcoming stiction is still a formidable task inmagnetic disk drive industry (2). When soft materials such as bacteria flaare observed by atomic force microscopy (AFM), the water meniscus forceto an excessive adhesion force, which in turn results in specimen damagefriction force microscopy (FFM) it is also found that meniscus force has a geffect on the observing results (4). The meniscus force is considered onemajor sources of load in micromachines (5).

Many measuring methods of surface tension of liquid have been develoAll these methods are summarized comprehensively in a book by AdamsoThe capillary rise method is generally considered to be the most accurathe all methods, being improved by many investigators (6). The other typmethods are the maximum bubble pressure method, the drop weight methoring method, Wilhelmy slide method, the pendent drop method, and the sedrop or bubble method (6).

All these methods measure surface tension of liquid in macro scale. Howthe surface roughness of a magnetic disk in the data zone is less than 2RMS, and that of magnetic slider is in the same order (7). An overlay of very(about 2 nm) of liquid lubricant is used over the magnetic disk surface to mainlow friction, low wear, and corrosion resistance (8). In such a contact geomthe liquid film is confined between two solid surfaces in a clearnance ofthan 10 nm. It is also true in the cases of micromachines and scanning

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microscopes such as AFM and FFM. There are increasing needs for meassurface tension of liquid in nanoclearnance in industry.

The contact geometry between a ball and a plane surface (9) or ccylindrical surfaces (10, 11) has been used to measure the meniscus forcemicroclearance. The geometry of both the ball-on-plane method and thecylindrical surfaces is shown in Fig. 1, in which a meniscus is formed arothe contact area by capillary condensation or by dropping a liquid. The Lappressurep within the meniscus is given by the Young–Laplace equation (6)

p = γ

r, [1]

whereγ is the surface tension andr the meniscus radius. The meniscus forF acting on the ball is then obtained by the product of the Laplace pressuthe meniscus area and is finally expressed for rigid surfaces by the equati

F = 4πRγ cosθ, [2]

whereθ is the contact angle of the liquid on the solid surface. It is known, frEq. [2], that meniscus forceF is proportional to the surface tensionγ and isindependent of meniscus radiusr . Orret al.(12) gave a more general and preciexpression for Eq. [2], which is applicable to the cases that liquid has diffecontact angles on the ball and the plane surfaces. Fisheret al. (10) found thatEq. [2] can apply to a meniscus radius as small as 0.5 nm for cyclohexanebenzene and about 5 nm for water. However, later Christenson (11) showdifferent experimental result from that of (10) by using a modified surface foapparatus (SFA) which had a double cantilever instead of a single cantilIt is said that the measurements do not give any immediate information ovalidity of the bulk surface tension for very high curvature systems, at leasin the absence of a complete treatment of the surface deformation prob(11). Theoretical analysis by Zhang and Nakajima (13) also shows that, wthe meniscus radius is less than several nanometers, the Laplace pressubecome so high that the solid surfaces will be deformed, the measured puforce will reduced greatly as the curvature radius of meniscus decreasesEq. [2] will not hold anymore. In addition to surface deformation, other posseffects are discussed in detail (14) based on experimental results. Thereforemust be taken when the above method is used and the clearance is in senanometer order.

The purpose of this study is to design and make an apparatus that cansure the Laplace pressure and meniscus force of liquid film confined withnanometer clearance between solid surfaces. The principle of the methodscribed at first, and then the method is used to measure both the Laplace prand the meniscus force. Experimental results are discussed by comparingthe predictions from the Young–Laplace equation.

2. METHOD OF MEASURING THE LAPLACE PRESSURE

2.1. Principle of the Method

Figure 2 shows the schematics of the method of measuring the Laplacesure. Two glass plates are put in contact with each other. However, it is geneknown that the real contact occurs only in some limited local area, and texists spacing in most of the overlapped area because of the geometrica

5 0021-9797/02 $35.00C© 2002 Elsevier Science

All rights reserved.

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FIG. 1. Schematics of meniscus force measuring apparatus of a ballplane or two cross-cylindrical surfaces.

of the two glass plates. The spacing changes from place to place. Liquid fiformed in the spacing by adding a little amount of liquid or by capillary codensation of vapor. The spreading of liquid film in the spacing can be obseeven with the naked eye, and the spread area of the film can be easily meaalthough the boundary is generally not regular. The liquid film attracts theglass plates together because of the Laplace pressure. The contact loadby the Laplace pressure is dependent on the magnitude of the pressurthe spread area of liquid film. In this study, we propose a method of measuthe contact load which is equal to the meniscus force by means of measurinfriction force between the two plates, instead of measuring the pull-off forcexisting methods do. To do so, a load cell is placed on the left-end side in FiThe upper plate is pushed against the load cell at its end when the lower plmoved by a screw feed (right side in Fig. 2), and the friction force is then deteby the load cell. The friction coefficient between the two plates is calibratedfore test. A dead load is put on the upper plate, the friction force is measand then the friction coefficient is obtained. The area of liquid film is measuby using tracing paper with meshes of 1 mm2 on its surface, and the clearancethe liquid film boundary is measured by the interference fringes formed onglass plate surface. It is possible to measure spacing as small as 20 nm winaked eye, and if spectroscopic ellipsometers are used a subnanometer scan be measured (15).

The glass plates used in this study are of borosilicate glass of 1 mmthickness, 26 mm in width, and 76 mm in length. The refractive index is 1.5the Abbe’s number is 57, and the flatness is 0.01 mm. A strain gauge loadwas used.

2.2. Friction Coefficient

Friction coefficient of the contact surfaces must be measured under lubriccondition with the testing liquid, because contact area is enclosed by a li

FIG. 2. Schematics of the shear Laplace pressure meter.

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TABLE 1Friction Coefficient in Air, Water, and Acetone

Variation percentage MediumFriction coefficient range (µmax-µmin)/µ (%) in spacing

0.42–0.43 2 Air0.39–0.51 29 Water0.39–0.44 12 Acetone

film in measuring the Laplace pressure, and the friction coefficient at conarea is dependent not only on solid surfaces in contact but also on the lubricconditions. According to the definition of friction coefficient we have

F = µ(W +mg), [3]

whereF is the friction force measured by load cell,W the normal load appliedby means of dead weight, andm is the mass of the upper plate. Comparwith the dead weight the weight of the upper glass plate is much smaller anegligible. Therefore, Eq. [3] can be simplified into

F = µW. [4]

It is worth noting that, when measuring the friction coefficient, the glplates should be immersed in testing liquid to minimize the effect of additioload caused by the Laplace pressure on friction force. Friction coefficiethen estimated by using the method of least squares from the applied loameasured friction force. The results are shown in Table 1. Friction coefficifor the cases in air, water, and acetone are 0.42, 0.41, and 0.41, respectiveall the three cases the correlation coefficient is larger than 0.99, showing areliability of the results by the method of least squares. Liquid films of waand acetone give almost the same friction coefficient.

2.3. Measurement of Spacing

The spacing between two glass plate surfaces is measured by using theference fringe. Clear interference fringes are found when two glass plateput together. The relationship between the interference fringes and spaccalibrated by means of Newton’s ring between a steel ball of 16 mm in diamand a glass plate. Newton’s ring is observed under an optical microscopethe radius of the Newton’s rings is measured. The calibration result is giveFig. 3, which shows both Newton’s ring and spacing as functions of the distfrom central point of contact. Spacing between the ball and the plate surfaccalculated by the following equation:

H = R−√

R2 − x2, [5]

whereR is the radius of the ball andx the distance from the central point of thcontact. The radiusR of the ball used in this study is 8 mm.

FIG. 3. The Newton’s ring and spacing.

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It is found that the radius of the most inside Newton’s ring is 0.019 mm, whgives a spacing of 23 nm according to Eq. [5]. The secondary and third Nerings appeared at radii of 0.067 and 0.094 mm, respectively. The spacithese radii are 280 and 552 nm, respectively. This suggests that the diffein spacing between two neighboring Newton’s ring is 272 nm.

In the Laplace pressure measuring, the glass plates are quite flat, and inence fringes are so far apart from one another that they can be detectedeven with the naked eye. However, it is helpful to use a magnifying scopescale in order to measure the distance between interference fringes accuIt is assumed that spacing changes linearly from one Newton ring to the n

3. MEASUREMENT OF THE LAPLACE PRESSURE

3.1. Determination of the Laplace Pressure

According to the Pascal’s principle of hydrostatics, the pressure in the lifilm is constant everywhere, and equal to the Laplace pressure. Therefortotal load applied by the Laplace pressure ispA and is related to the frictionforce by the following equation:

F = (pA+mg)µ, [6]

where p is the Laplace pressure,A the liquid film area,m the mass of uppeplate, andµ the friction coefficient at the solid interface. Therefore the Laplpressure can be obtained by rewriting Eq. [6]:

p = F/µ−mg

A. [7]

In this study, the mass of the upper glass platem is about 6 g,mg is much lessthan F/µ, and Eq. [7] is simplified as follows by ignoring the weight of th

FIG. 4. Laplace pressure as a function of spacing (a) in water and (bacetone.

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FIG. 5. Change of meniscus force with dwell time for water.

upper plate:

p = F

µA. [8]

All the other surface forces, except the Laplace pressure, are neglected bethe van der Waals force between solid surfaces is about two orders less thLaplace force in magnitude even when spacing is as small as 1 nm (16).

3.2. The Laplace Pressure as a Function of Spacing

Measurement of the Laplace pressure was conducted for water and acThe measurement results are shown in Fig. 4 as a function of spacing. Oother hand, the Laplace pressure is related to spacing by the Young–Laequation as follows, if it is assumed that the contact angleθ is zero:

p = 2γ

H, [9]

whereγ is surface tension of liquid andH is spacing. In Eq. [9] it is assumed thameniscus radiusr in Eq. [1] is equal to the half of the spacingH . The calculatingresults of Eq. [9] is also given in Fig. 4 for comparison, usingγ = 75 mN/m forwater andγ = 23 mN/m for acetone. The experimental results agree quite wwith Eq. [9]. The maximum Laplace pressure measured in this study is 10.5for water and 1.5 MPa for acetone. It is worth noting that the Laplace presmeasure in this study is tensile stress, and not compressive stress, althouworld of pressure is used.

3.3. The Laplace Pressure as a Function of Dwell Time

The liquid film between the test plate surfaces is formed by adding liqfrom the outside. The liquid film at the beginning is therefore relatively thiwhich gives a low Laplace pressure. However, the liquid film vaporizes fromboundary, and the liquid film becomes thinner and thinner. Figure 5 showchange in the Laplace pressure with dwell time for water. The Laplace presincreases almost proportionally with dwell time.

4. CONCLUSION

A simple method of measuring the Laplace pressure is proposed, whicalled the shear Laplace pressure meter. The method measures meniscuby means of measuring the friction force and the meniscus area at thetime. Therefore, the Laplace pressure is known as well as the meniscusThe liquid film thickness is measured by using interference fringes. The meament results for both water and acetone was compared with theoretical reof the Young–Laplace equation. Good agreements between the measurresults and theoretical results verify the validity of the method. The met

) inis simple in both principle and structure. Compared with the existing pull-offmethod of measuring meniscus force, the present method has the advantages that

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Received June 6, 2001; accepted September 20, 2001

218 NO

(i) meniscus is not disrupted in measuring, (ii) both meniscus force andLaplace pressure are measured, and (iii) it has strong resistance to vibrati

ACKNOWLEDGMENT

The work is appreciatively supported partly by the Ministry of EducatiScience, Sports, and Culture through a Grant-in-Aid for Science Rese(No. 11650153).

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(1995).6. Adamson, A. W., “Physical Chemistry of Surfaces.” Wiley, New York, 1977. Wang, S., and Komvopoulos, K.,Trans. ASME J. Tribol.119,830 (1997).

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10. Fisher, L. R., and Israelachvili, J. N.,Colloids Surf.3, 303 (1981).11. Christenson, H. K.,J. Colloid Interface Sci.121,170 (1988).12. Orr, F. M., Scriven, L. E., and Rivas, A. P.,J. Fluid Mech. 67, 723

(1975).13. Zhang, B., and Nakajima, A.,J. Colloid Interface Sci.211, 114

(1999).14. Christenson, H. K.,J. Phys. Chem.97,12,034 (1993).15. Li, Y., Trans. ASME J. Tribol.119,681 (1998).16. Matsuoka, H., and Kato, T.,Trans. ASME J. Tribol.118,832 (1996).

Bo Zhang1

Akira Nakajima

Faculty of Science and EngineeringSaga University1 Honjo-machi, Saga-shiSaga 840-8502, Japan

s.” 1 To whom correspondence should be addressed. Fax: +81-952-288587.E-mail: zhang@me.saga-u.ac.jp.