Capillary forces between surfaces with nanoscale -

Advances in Colloid and Interface ScienceŽ .96 2002 213�230

Capillary forces between surfaces withnanoscale roughness

Yakov I. Rabinovich, Joshua J. Adler,Madhavan S. Esayanur, Ali Ata1, Rajiv K. Singh,

Brij M. Moudgil�

Department of Materials Science and Engineering and Engineering Research Center for ParticleScience and Technology, 205 Particle Science and Technology Bldg., Uni�ersity of Florida,

P.O. Box 116135, Gaines�ille, FL 32611-16135, USA

Abstract

Ž .The flow and adhesion behavior of fine powders approx. less than 10 �m is significantlyaffected by the magnitude of attractive interparticle forces. Hence, the relative humidity andmagnitude of capillary forces are critical parameters in the processing of these materials. Inthis investigation, approximate theoretical formulae are developed to predict the magnitudeand onset of capillary adhesion between a smooth adhering particle and a surface withroughness on the nanometer scale. Experimental adhesion values between a variety ofsurfaces are measured via atomic force microscopy and are found to validate theoreticalpredictions. � 2002 Elsevier Science B.V. All rights reserved.

Keywords: Adhesion; Surface force; Capillary; Nanoscale; Roughness; Atomic force microscope; Rela-tive humidity

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2142. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

2.1. Adhesion to surfaces with nanoscale roughness . . . . . . . . . . . . . . . . . . . . 215

� Corresponding author. Tel.: �1-352-846-1194; fax: �1-352-846-1196.Ž .E-mail address: [email protected] B.M. Moudgil .

1Currently affiliated with the Department of Materials Science and Engineering, Gebze Institute ofTechnology, Turkey.

0001-8686�02�$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.Ž .PII: S 0 0 0 1 - 8 6 8 6 0 1 0 0 0 8 2 - 3

( )Y.I. Rabino�ich et al. � Ad�ances in Colloid and Interface Science 96 2002 213�230214

2.2. Capillary adhesion to surfaces with nanoscale roughness . . . . . . . . . . . . . . 2172.3. A critical relative humidity for capillary forces . . . . . . . . . . . . . . . . . . . . . 221

3. Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2223.1. Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2223.2. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

4. Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2255. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

1. Introduction

In industrial operations ranging from the transport and handling of fine powdersto the removal of abrasive particles after a polishing operation, the magnitude offorces between two surfaces is critical. There are a number of approaches to

� �fundamentally predict these interactions based on van der Waals adhesion 1 orŽ .through surface energy based approaches such as Johnson�Kendall�Roberts JKR

� � Ž . � �2 or Derjaguin�Muller�Toporov DMT 3 adhesion theory. With real surfaces,however, there are a number of difficulties associated with applying these models.One principal concern is the effect nanoscale roughness has on the real area ofcontact and, subsequently, the magnitude of the force of adhesion. In a previous

� �investigation 4,5 , it has been shown that deviations from ideality, as small as 1 nmRMS roughness, may significantly reduce the force of adhesion. A second concern,especially in industrial operations, is the increase in the adhesion force associatedwith the spontaneous formation of a meniscus in the capillary between twoadhered particles or a particle adhered to a surface. This increase may significantlyimpact processing operations.

Concerning capillary forces, there are two critical issues that warrant discussion.Both the magnitude of the capillary force of adhesion and the environmentalconditions, such as relative humidity, that allow a meniscus to form and enhancedadhesion to be observed are important to predict. A number of authors haveinvestigated the effect of relative humidity on condensation in capillaries and the

� �validity of Kelvin’s equation at various conditions 6�10 . An early detailed discus-sion of these phenomena related to particulate adhesion was presented by Coelho

� �and Harnby 11,12 . Utilizing Kelvin’s equation to describe the lesser radius ofcurvature of the meniscus, the Laplace equation to predict the pressure inside themeniscus, and adding the adhesion component produced by the surface tension, adescription of capillary adhesion was developed. Furthermore, the authors believedthat the onset of capillary forces should occur when the pressure inside the liquidannulus is equal to the vapor pressure, as the liquid would otherwise boil.Subsequently, criteria for meniscus formation in terms of the vapor pressure andthe BET constant were developed. Early experiments in general confirmed the

( )Y.I. Rabino�ich et al. � Ad�ances in Colloid and Interface Science 96 2002 213�230 215

basic form of these equations but showed that these criteria significantly overesti-� �mated the critical relative humidity 13 .

� � � �Recently, Marmur 14 and de Lazzer et al. 15 have extended these basictheories for other geometries, environmental conditions and separation distanceswhile more accurately describing the meniscus and its effect of adhesion. DeLazzer et al. incorporate surface tension in their model and discuss capillaryadhesion for a variety of probe geometries including spherical, conical andparabolic. The ratio of the surface tension component to the internal pressurecomponent is also explored and the conditions where they are comparable weredelineated. In both papers, however, surfaces are considered to be ideally smooth.

� �Fisher and Israelachvili 13 investigated adhesion between smooth mica surfacesin the presence of water and cyclohexane vapors and directly measured the relativecontribution of solid�solid interactions and the capillary effect on the total

� �adhesion force. Rabinovich et al. 16 further explored this area through directmeasurement of the force of adhesion between fused quartz filaments in varioussemi-miscible liquids. Recent investigation has also detailed the important effect ofsoluble contaminants on surfaces in altering the measured forces and critical

� �relative humidity 17 . The effect of surface hydrophobicity and specific surfacegroups on the onset of capillary force has also been subject to direct measurement

� �recently 18,19 . However, the change in roughness possibly associated with thesesurface modifications was not considered.

� �Sirghi et al. 20 further discuss the Marmur and de Lazzer models with theincorporation of local curvature and suggest an analytical solution for capillary andsurface tension adhesion forces using the approximation of a spherical particlegeometry and a symmetrical water meniscus at thermodynamic equilibrium. Theo-retical results are compared with forces measured between a silicon nitride atomic

Ž .force microscopy AFM tip and a platinum covered quartz sample with roughnesson the order of 20 nm. This order of roughness is approximately the samemagnitude as the AFM probe radius and so is not directly applicable to the case ofa particle with a rough surface.

� �Quon et al. 21 were principally concerned with the differences in surfacecharacteristics on the capillary adhesion of surfaces. In their investigation, adhe-sion forces were measured between gold-coated mica crossed cylinders which hadbeen modified by alcohol or methyl self-assembled monolayers. A significant effectof the approximately 2-nm RMS roughness was observed but no theoreticalformulae were developed.

2. Theory

2.1. Adhesion to surfaces with nanoscale roughness

Before developing a predictive model of capillary forces between surfaces, abrief review of adhesion in atmospheres where no liquid annulus is expected to


form is necessary. As mentioned earlier, there are a number of methods tocalculate the force of adhesion between ideal surfaces. In a surface energy basedapproach, the attractive force between a smooth adhering particle and a smoothflat plate of the same material in a dry atmosphere, F , may be expressed in thedry

� �DMT 3 limit of smaller harder particles as

Ž .F � 4��R 1dry

� �or in the JKR 2 limit of larger softer particles as

Ž .F � 3��R 2dry

where � is the surface energy and R is the radius of the particle adhering to a flatsubstrate.

While there has been much discussion in the literature as to which of thesemodels best describes the mechanics of a particle interacting with a surface and,models that describe intermediate behavior based on material properties have

� �been proposed 22,23 , the major barrier to their application lies in the estimationof the surface energy. Surface energy values are often derived from a number ofdifferent sources and may not always incorporate the true fundamental mecha-nisms of adhesion present in a system. These differences occur primarily becausethe separation distance or intimacy of contact between the two surfaces is notdefined or always consistent between experiments. As a result, deviations fromideality either due to surface roughness or plastic deformation may often beincorporated in the surface energy parameter.

An approach where the fundamental interaction parameters are better known isvan der Waals attraction. Although not accounting for polar forces it may beassumed that the bulk of the surface energy term is the result of van der Waalsattraction. The Hamaker constants, A, for a wide variety of materials are known� �24,25 and, hence, the force of adhesion, F , between an adhering sphere and flatdry

� �surface may be calculated as 1,26

ARŽ .F � 3dry 26H

where R is the radius of the adhering particle and H is the separation distancebetween the particle and the substrate, which to predict the force of adhesionbetween ideal surfaces is usually taken to be 0.3�0.4 nm. Because the mechanismsand distance dependent behavior of this force are relatively well defined, moreelaborate models of surface interaction may be developed.

� �In a recent series of publications 4,5 , it was suggested that the force of adhesionbetween a smooth sphere and surface with nanoscale roughness could be expressedthrough


AR 1 1Ž .F � � 4dry 2 2 26H 1 � 58.14R � RMS�� Ž .1 � 1.817RMS�H0 0

where A is the Hamaker constant, R is the radius of the adhering particle, H is0the minimum separation distance between the adhering particle and asperityŽ .0.3�0.4 nm , RMS is the root mean square roughness of the flat surface, and � isthe average peak to peak distance between asperities. In this approach the firstterm in brackets represents the contact interaction of the particle with an asperityand the second term accounts for the non-contact interaction of the particle withan average surface plane. This approach models nanoscale roughness not asnanometer sized spherical asperities with their centers located at an averagesurface plane but rather as the caps of larger asperities with their centers locatedfar below the surface. As a result of modeling the surface in this manner, both theheight and breadth of the asperities are accounted for and significantly closerpredictions of experimental adhesion values are achieved. In the nanoscale regime,it should be noted that the interaction of the particle with the asperity dominatesthe interaction and it is possible to replace the first term in the interaction with a

Ž .surface energy based approach model as expressed by Eq. 5 ,

3��R�2 ARŽ .F � � 5dry 22Ž .� � 58.14R � RMS Ž .6 H � 1.817RMS0

Ž .where variables are equivalent to those in Eq. 4 except the surface energyparameter, �.

In the application of this model to real systems, a relationship between theexperimentally accessible RMS roughness and the true radius of the asperities, R1must be developed. If the peak-to-peak distance between asperities, �, is known,this may be calculated as

2 Ž .R � � �58.14RMS 61

Note also that the separation distance between the average surface plane andbottom of the adhering particle, H, in this model is

Ž .H � 1.817RMS 7

Very small deviations from an ideally smooth surface can account for largedeviations in adhesion. For example, roughness of only 1 or 2 nm RMS issignificant enough to reduce theoretical adhesion by an order of magnitude or

� �more 4,5 . It is believed that similar effects may be observed in capillary adhesionprocesses.

2.2. Capillary adhesion to surfaces with nanoscale roughness

The geometry of a liquid annulus between an ideally smooth solid sphere andflat substrate is shown in Fig. 1. For simplicity the system may be described by a


Cartesian coordinate system since rotational symmetry is assumed about theŽ .ordinate axis. The capillary adhesion detachment force F between the particleC

and substrate has two principal components. The first force component, F , arisesPfrom the pressure differential, � P, across the meniscus. The second, F , arisesSfrom the vertical component of the surface tension acting along the meniscus. Thesum of these forces may be written as

2 Ž . Ž .F � F � F � � x � P � 2�� xsin � � 8C P S L 1

� �where the pressure difference, � P, in agreement with Laplace’s Equation 27 , isequal to

1 1Ž .� P � � � 9L ž /r x

and � is the surface tension of liquid, R is the radius of the adhering particle, r isLradius of the meniscus, x is horizontal distance of intersection between themeniscus and the adhering particle, � is the contact angle of water of the particle,1and is the angle between the ordinate axis and a radius to the intersection of theparticle with the liquid meniscus. The angle may also be found by the arctangentof the slope of the particle surface at point x,

d yŽ . � arctan 10ž /d x

� �The equilibrium radius of the meniscus, r, is given by Kelvin’s equation 27

�V� �V�L L Ž .r � � 11Ž .N kT ln P�P kT lna S

where V is the molar volume of the liquid, � is the surface tension of the liquid,LN is Avogadro’s number, kT is the product of Boltzmann’s constant and absoluteatemperature, P and P are the vapor pressure and saturated vapor pressure and PSis relative humidity.

Fig. 1. Schematic illustration of the liquid annulus formed in a capillary created by the adhesion of anideally smooth spherical particle to an ideally smooth flat substrate.


Fig. 2. Schematic illustration of the meniscus formed in a capillary created by the adhesion of anideally smooth particle to an asperity on a flat substrate for conditions in which the relative humidity is

Ž . Ž .high a and for which the relative humidity is low b .

The analytical formulae for the calculation of capillary force based on Eqs.Ž . Ž . � � � �8 � 11 were suggested by Marmur 14 and de Lazzer et al. 15 . However, bothworks only describe the force as function of the horizontal distance, x, where themeniscus intersects the adhering particle. Unfortunately, this value is not knownexperimentally. Additionally, to describe the dependence of the capillary force onroughness, the dependence on separation distance between the particle andsubstrate must be developed. Fig. 2 shows pictorially the geometry of a smoothspherical particle adhered to a rough flat surface in the presence of capillarycondensation.

For both high and low humidity ranges Fig. 2a,b, respectively, schematicallydescribe the liquid annulus formed by capillary condensation for high and low

Ž .humidity conditions. For high relative humidity Fig. 2a , to describe the verticaldistance between the surface of the particle at point x and average surface plane,y, the following geometric relationship may be written

Ž . Ž .y � R � H � Rcos � rcos � � � rcos� 121 2

where R is the radius of the adhering particle, H is the separation distancebetween the bottom of the adhering particle and the average surface plane, isthe angle between the ordinate axis and a radius to the intersection of the particlewith the liquid meniscus, r is the lesser radius of the meniscus, and � and � are1 2the contact angles of the particle and flat substrate, respectively.

ŽIf it is suggested that the volume of the liquid annulus is small i.e. x, y and. Ž .r � R , Eq. 12 results in a quadratic equation, which may be solved for to yield

2 2 Ž .rsin� r sin � 2 H � 2 r cos� � cos�1 1 1 2 Ž . � � � � 13( 2R RR

Ž . Ž . Ž . Ž .Eqs. 8 , 9 , 11 and 13 allow calculation of capillary force. An especiallyŽ .simplified solution is obtained for a small contact angle � and � � 1 rad where1 2

Ž .from Eq. 13


Ž .2 r cos� � cos� � 2 H1 22 Ž . � 14R

Ž . Ž .Additionally from Eqs. 8 and 9 it follows that

�� x 2 �� R22L L Ž .F � � 15P r r

2 Ž . Ž .Substituting from Eq. 14 in Eq. 15 the simplified equation for capillaryforce may be derived as

HŽ .F � 4�� Rcos� 1 � 16P L 2 rcos�

where

cos� � cos�1 2 Ž .cos� � 172

Ž .It should be noted that Eq. 16 is applicable only if H � 2 rcos�. In other words,that the liquid meniscus spans the distance from the particle to the average surface

Ž .plane Fig. 2a .The separation distance between the adhering particle and the average surface

plane, H, for spherical asperities with their centers located beneath the averageŽ .surface plane has been shown previously, Eq. 7 , to be approximated by 1.817RMS.

Ž .Hence, for a small annulus volume and small contact angle hydrophilic surface ,the capillary force between sphere and flat rough samples can be calculated

Ž . Ž . Ž .through Eqs. 7 , 11 and 16 . Note that in the development of this simplifiedmodel of capillary adhesion, the effect of wetting films and the transition from a

� �film to liquid annulus as described in detail by Derjaguin and Churaev 28,29 is notconsidered.

Ž .For the low humidity case Fig. 2b , when the equilibrium radius of the meniscusis small, as described by the Kelvin equation, the liquid annulus is actually formedbetween two spheres, the particle and the asperity. For this situation, an effectiveradius of the interacting particle and asperity, R , should be considered, aseff

� �described by Derjaguin et al. 26 as

2 RR1 Ž .R � 18eff R � R1

where R is the radius of the adhering particle and R is radius of the asperity. In1the case when R is in the range of micrometers and R is on the nanometer scale,1the effective radius is quite small. As a result, very low capillary adhesion ispredicted, for the low relative humidity regime. Significantly greater capillaryadhesion force would result once the relative humidity became significant enoughto produce an annulus that covers the asperity completely and spanned the


distance to the average surface plane. This situation leads to the concept of acritical relative humidity needed to first observe capillary forces on the backgroundof the dry adhesion forces described earlier.

2.3. A critical relati�e humidity for capillary forces

For an ideally smooth sphere adhered to a smooth flat substrate at zeroŽ . Ž .separation distance H � 0 , the classic formula follows from Eq. 16

Ž .F � 4�� Rcos� 19P L

This relationship has no dependence on the relative humidity and, consequently,should have a fixed value even at 0% RH. Experimentally, this clearly is not the

� �case 13,16 . Therefore, there must be a point at which capillary forces first start toappear and below which they are absent. There are at least two approaches to this

� �problem. One of the first was suggested by Coelho and Harnby 11,12 and is basedon the assumption of thermodynamic equilibrium between the liquid meniscus,bulk liquid in the annulus and vapor. Equilibrium also necessitates that the liquidin the annulus is under greater pressure than that of the vapor. Otherwise, theliquid in the meniscus would boil. The critical values of relative humidity, obtainedin within this framework, are on the order of 70�99% RH which are significantly

� �higher than experimentally observed values 13 . This approach has also beenchallenged by the common observation that nitrogen vapor can fill nanometer sized

� Ž .�pores, where, in agreement with Laplace’s Equation Eq. 9 , the negative pressureinside the pore can achieve several hundreds of atmospheres. A possible reason forthis discrepancy is that the liquid in the pores may be supersaturated due to thelack of defects needed to initiate boiling.

A second approach to the problem of a critical humidity was suggested by� �Rabinovich et al. 16 . It was proposed that there is certain annulus volume or

dimension, below which, the adsorbed molecules cannot be considered a macro-scopic phase and, consequently, no surface tension or capillary force would exist ata humidity less than a critical value. The authors after analysis of literature datadetailing force measurements in semi-miscible water�octamethyl cyclotetrasilox-

� �ane mixtures between ideally smooth mica 30,31 , and their own experimentalresults for butanol�water mixtures between silica surfaces, found that the criticalradius of the meniscus corresponded to approximately 1 nm.

Both of these approaches were suggested for ideally smooth solids. As followsŽ .from Eq. 16 , there may be additional criteria for the observation of capillary

adhesion between surfaces with nanoscale roughness. The simplest possibility isthat the critical relative humidity corresponds to the humidity at which the

� Ž .�capillary force described by Eq. 16 is equal to zero. However, a more importantcharacteristic is the point at which the capillary force exceeds the adhesionbetween the ‘dry’ surfaces. At zero or low humidity, only a thin mono-molecular

Ž .film exists on the surface which cannot form a meniscus. In this situation Eqs. 4


Ž .and 5 , developed for dry surfaces, are appropriate. Above the critical humidity,the capillary forces are expected to act in addition to the dry adhesive forces.

Of course this last point brings up an interesting consideration because theregion of the surfaces in the liquid annulus are no longer acting through air butrather through liquid. This is important because the magnitude of van der Waalsforces for solids through water, for instance, are generally significantly less than inair. For example, the Hamaker constant of silica interacting through water is

�20 � �0.8 � 10 J 25 , which is eight times less than the Hamaker constant of silica�20 � �interacting through air, 6.5 � 10 J 25 . In many cases the contribution of van

der Waals force through the liquid annulus should be neglected.To account for this situation, the integration procedure suggested by Derjaguin

� �26 whereby the interaction energies of rings of increasing radii on the flat surfacewith their corresponding areas on the adhered particle are summed to provide thenet force of attraction, may be applied. Assuming that the net attraction within theannulus is small, the van der Waals attraction of the remaining dry surfaces, F , isdryequivalent to a sphere of equal radius but at a distance above the average surfaceplane equal to the height of the meniscus.

ARŽ . Ž . Ž .F H � F H � y � 20dry dry 0 2Ž .6 H � 2 rcos�0

Where H is shortest distance between the two surfaces, y is the maximum0height of the miniscus, A is Hamaker’s constant, R is the radius of the adheringparticle, and � is the contact angle. After a significant relative humidity has beenachieved, the total force of adhesion between the two surfaces should be calculatedas

Ž .F � F � F 21C dry

Ž . Ž .Where F and F are calculated with Eqs. 16 and 20 , respectively. Note thatC dryŽ .the radius of the meniscus and, therefore, the height in Eq. 20 increase in

magnitude quickly with increasing relative humidity. Hence, it follows that at highrelative humidity the contribution of van der Waals attractive forces becomes smalland adhesion is principally dominated by the capillary force.

3. Experimental

3.1. Materials

Adhesion force as a function of relative humidity, was measured between 20 and40 �m glass microspheres from Duke Scientific Corp. and a variety of substrates.The spheres were found to have an RMS roughness of less than 0.2 nm. Silicasubstrates included an oxidized, to approximately 180 nm, silicon wafer of 0.2 nm

Ž .RMS roughness provided by Dr Arwin Linkoping University, Sweden ; a plasma¨Ž .enhanced chemical vapor deposited PE-CVD 2-�m thick coating of 0.3 nm RMS


ŽFig. 3. Atomic force microscopy images of 2 � 2 �m areas with a total vertical scale of 20 nm 10. Ž . Ž .nm�division of the rougher silica samples utilized in this investigation. a is the PE-CVD silica and b

is the PE-CVD silica etched in ammonia�peroxide solution. Images were taken using contact modeAFM.

roughness silica on silicon supplied by Motorola Company; and a sample of thePE-CVD silica boiled in a 1�1 mixture of hydrogen peroxide and ammonia for 6 hproducing an RMS roughness of 0.7 nm. Images of non-etched and etchedPE-CVD silica surfaces are shown in Fig. 3a,b. Measurements were also preformedbetween the spheres and a sapphire substrate supplied by MTI Corp. of 0.3 nm


RMS roughness, a sputtered titanium surface deposited on silicon of 1.4 nm RMSroughness, and a sputtered silver surface of 3.0 nm RMS roughness both acquiredfrom Motorola Corporation.

All silica, glass and alumina surfaces were cleaned by rinsing in acetone andmethanol followed by repeated rinsing and boiling in acidified deionized water forat least 8 h. Metal surfaces were not boiled. Cleaning was preformed immediatelyprior to experimentation. Also before each experiment, particles were glued to

Ž .tapping mode TESP rectangular atomic force microscopy AFM cantilevers,supplied by Digital Instruments Inc. using a low melting temperature resin, Epon R1004f from Shell Chemicals Company.

3.2. Methods

Surface force was measured on a Digital Instruments Nanoscope IIIa according� �to the methods described by Ducker et al. 32 . The sphere, attached to a cantilever

of known spring constant, was positioned close to the flat substrate. Then, as thesubstrate was moved towards and away from the particle by a piezoelectric tube,the deflection of the cantilever was monitored by a laser that reflects from the topof the cantilever onto a position sensitive photodiode. In this manner, the forcebetween the two surfaces as a function of separation distance was obtained. Theforce�distance profiles and values presented are normalized by dividing themeasured force by the radius of the sphere. In other words, the data are presentedin terms of energy per unit area of flat surfaces. This enables determination of theforce for different geometries and sizes of particles as long as the range of theforces is much less than the particle’s radius of curvature. Surface roughness wasmeasured using contact mode AFM and images are presented without filtering butwith plane fitting.

To control the atmosphere, an environmental chamber was constructed thatŽ .encased the entire AFM without microscope . The chamber was fitted with a

recirculating fan and the AFM was placed on a vibration isolation pad within thechamber. Relative humidity was monitored by two Fisher Scientific High AccuracyThermo-Hygrometers located at the top and bottom of the chamber. In a typicalexperiment, the atmosphere within the chamber was saturated with water in dishesand then allowed to dry over time. To achieve low relative humidity, desiccant wasopened or pure nitrogen gas was used to flush the chamber. Each reportedadhesion data point is the average of at least 20 measurements taken at differentlocations on the surface.

The adhesion force was found to increase as a function of applied load and thenbecome constant above a critical loading value. This suggests that a certain amountof pressure is necessary to bring surfaces into intimate contact and allow time

� �enough for a capillary to form 33 . In this system, an applied normalized loadgreater than approximately 2 N�m was found to produce relatively constant resultsand all adhesion data are reported after applying a load slightly greater than thisvalue.


Fig. 4. Typical approaching and retracting force curves for the glass sphere interacting with theoxidized silicon wafer sample at 54% RH. Arrows indicate the direction of travel either approachingŽ . Ž .leftward pointing arrow or retracting rightward pointing arrow .

4. Results and discussion

A characteristic force�distance curve for the smoothest silica sample at 54% RHis shown in Fig. 4. Upon approach of the two surfaces, as separation distancesbecome small, long range van der Waals attractive forces are observed to pull thesphere towards the surface starting at approximately 8 nm separation distance.When the gradient of the attractive forces becomes greater than the springconstant, the mechanical equilibrium of the cantilever is overcome and the spherejumps into contact with the flat surface. After the point of contact, pressure is stillapplied to the surfaces up to a known load and then the surfaces are pulled apartfrom one another. Due to the magnitude of the adhesion forces, the particleattached to the tip of the cantilever does not separate from the substrate as thecantilever starts to be pulled away. However, when the deflection force of thecantilever equals the force of adhesion, the sphere releases from the surface, andjumps to its equilibrium position. From the product of this distance and the springconstant of the cantilever, the force of adhesion may be obtained. This method isoften necessary because, due to the limited range of the photodiode, evidenced bythe flat portion of the retraction curve, the minimum in the retraction curve cannotalways be directly observed. However, for cases in which both the minimum andcritical separation distance may be observed, the agreement between the two isgood.

The dependence of adhesion force as a function of relative humidity for thesilica samples of different roughness is given in Fig. 5. Of interest first is thesignificant decrease in the magnitude of the adhesion force with increase inroughness in the low relative humidity regime. It can be observed that an increasefrom approximately 0.2 to 0.7 nm in surface RMS roughness decreases theobserved dry adhesion by over an order of magnitude. Note also that the adhesionbetween the smoothest samples results in a surface energy of 60 mJ�m2 for JKR


Fig. 5. Force of adhesion as a function of relative humidity for silica surfaces of increasing roughness.Ž .Circles represent the oxidized silicon wafer 0.2 nm measured RMS roughness , squares represent the

Ž .PE-CVD substrate 0.3 nm measured RMS roughness , and triangles represent the etched PE-CVDŽ .surface 0.7 nm measured RMS roughness . Solid lines are theoretical predictions for both the dry

� Ž .� � Ž .�adhesion horizontal lines from Eq. 5 and capillary adhesion regimes Eq. 16 based on theinformation in Table 1.

� �adhesion mechanics which is in the range reported in the literature 34�36 . Thisgeneral behavior of a regime where, as a function of relative humidity, there is noincrease in the magnitude of the force of adhesion was found for all the silicasamples. However, as expected for each sample, there is a critical relative humidityat which the force of adhesion begins to increase. Additionally, for each substrate,the transition does not result in a jump to a single adhesion value predicted by Eq.Ž .19 for ideally smooth surfaces. Instead adhesion forces increases as relativehumidity increases.

As may be observed from the experimental data, there is also a clear trend in theonset of the capillary forces with increasing surface roughness. At low surfaceroughness the transition occurs at approximately 25% RH, which is significantlylower than the 70�99% RH predicted by the thermodynamic equilibrium model

� �suggested by Coelho and Harnby 11,12 . As a result, it may be suggested thatliquid inside the annulus is metastable rather than in thermodynamic equilibrium.This relative humidity corresponds to a radius of approximately 0.4 nm whichcoincides with the critical radius derived from the experiments of Fisher and

� �Israelachvili 13 for water and cyclohexane vapors between mica surfaces. Thisvalue is comparable but smaller than the critical radius, 1 nm, found by Rabinovich

� �et al. 16 for nearly saturated solutions of water in butanol and by Christension� �and coworkers 30,31 for water in octamethyl cyclotetrasiloxane. It may be possible

Ž .that the geometry of the surfaces, their radii, the medium vapor or solution , andcondensate composition also has an impact on this critical radius.

�Also presented in Fig. 5 are the theoretical predictions of the dry horizontalŽ .� � Ž .�lines from Eq. 5 and capillary forces from Eq. 16 for the silica surfaces. It

appears that the relatively simple theoretical approach to the capillary adhesionbetween surfaces with nanoscale roughness adequately predicts the experimental


Table 1Characteristics of the substrates used in this investigation and the results of analysis by the proposedmodels

Flat substrate Measured RMS Average peak- Measured Fit to capillaryroughness to-peak dry adhesion adhesion RMSŽ .nm distance force roughness

Ž . Ž . Ž .nm mN�m nm

Oxidized silicon 0.2 490 380 0.22PE-CVD silica 0.3 650 340 0.34Etched PE-

CVD silica 0.7 130 23 1.20Sapphire 0.3 250 150 0.38Titanium 1.4 215 31 1.20Silver 3.0 610 316 0.58

data. For all samples, a surface energy of 60 mJ�m2, a contact angle of 20�, and asurface tension of 72 mN�m was applied. The surface parameters and fitted valuesof surface roughness are presented in Table 1. Average peak-to-peak distances

Ž .derived from Eq. 5 correspond adequately with those taken from the images ofthe surfaces and the measured adhesion values. Fitted values of the roughness,based on adhesion forces measured in the capillary regime, are also similar tomeasured values. The measurement of roughness on this scale is subject tomeasurement technique and conditions. Although roughness trends and generalmagnitude are easily discerned, there may still be error associated with backgroundnoise, imaging conditions, and uniformity of the substrates. Additionally, in realitythere is a distribution of distances between asperities. To more accurately correlateexperimental and theoretical predictions, these surface characteristics may need tobe more accurately described. However, given the level of sophistication employed,the models are adequate.

The adhesion as a function of relative humidity for a number of other materialsinvestigated are presented in Fig. 6. The surface characteristics and fitting parame-ters are also detailed in Table 1. The shape of the curves are very similar to that ofsilica. As roughness increases, the dry force of adhesion decreases significantly, as

Ž .predicted by Eq. 5 . The relative humidity at which capillary forces begin todominate the interaction also increases with increasing roughness, in agreement

Ž .with Eq. 16 . The absolute values of the forces also correspond well to those in thesilica system indicating the importance of surface roughness in controlling adhe-sion over other surface properties such as the Hamaker constant or polar forces.The model of capillary forces between surfaces with nanoscale roughness devel-oped in this investigation is also in relative agreement with the experimental results

� �of Quon et al. 21 in which the adhesion of mica coated with rough gold layers,functionalized by surfactants, was measured.

The one material that did not behave as predicted in either the dry adhesion orcapillary adhesion regimes was silver. Although the RMS roughness as measured


ŽFig. 6. Force of adhesion as a function of relative humidity for substrates of sapphire circles:. Ž .measured RMS roughness of 0.3 , titanium squares: measured RMS roughness of 1.4 , and silver

Ž .triangles: measured RMS roughness 3.0 nm . Solid lines are theoretical predictions for both dry� Ž .� � Ž .�adhesion horizontal lines from Eq. 5 and capillary adhesion Eq. 16 regimes based on the

information in Table 1.

by AFM was 3.0 nm, the capillary force model most closely fit to an RMSroughness of 0.53 nm. Similarly, in the dry adhesion region, it is seen that themagnitude of the adhesion force is significantly greater than expected for a surfacewith a 3.0 nm RMS roughness. Although the surface profile did not show anyunusual characteristics, this contradiction may be understood if the higher plastic-ity of silver is accounted for. It is possible that asperities of this size combined withthe relatively low yield strength of silver resulted in asperities under the adheringparticle being plastically deformed. The deformation would result in a more

� �intimate contact between the surfaces and increase the dry adhesion forces 37 .The surfaces would also be allowed to come closer together such that the radius ofthe meniscus needed to observe capillary forces is lower.

It should be noted that the minimum critical relative humidity needed to observecapillary forces corresponds to approximately 25% for smooth surfaces. This valuemay either be explained by the minimum size of macroscopic liquid phase or by theeffect of surface roughness in the range of 0.2 nm RMS. In this regime, it may nolonger be appropriate to describe surfaces in terms of macroscopic parameters.Instead, atomic scale considerations may need to be developed for specific inter-faces. In the case of greater roughness, it is more clearly observed that the criticalhumidity is controlled by the surface roughness rather than the minimum possibleradius of the liquid meniscus.

The adhesion forces presented in this investigation show sizable deviation fromideally smooth surfaces. This is significant considering that surface roughness wasonly varied by a few nanometers. These examples are meant to illustrate theabsolutely critical nature that even minuscule surface roughness plays a major rolein controlling the adhesion of surfaces. Many industrial powders and surfacescontain asperities that are significantly larger than explored here, but these datasuggest that it may be the smallest scale of roughness that principally controls the


interaction of surfaces whether they are in the dry or humid atmospheric regime.The formulas presented here are greatly simplified, but even these basic ap-proaches can elucidate the possible role of nanoscale roughness if the geometry ofthe surface is known.

5. Summary

In this investigation, a theoretical framework is developed to predict the magni-tude and onset of capillary and dry adhesion forces in the presence of nanoscaleroughness. This basic framework was validated with direct measurement of surfaceforces using atomic force microscopy. It is found that there is a significant decreasein dry adhesion force for a variety of surfaces with only an increase of surfaceroughness of a nanometer or two. Similarly, capillary adhesion was also substan-tially lower in the presence of nanoscale roughness. In addition to this lowermagnitude, the critical relative humidity, where capillary forces are first observed,increases as roughness on the nanoscale increases. Although other scales ofroughness may be present on real surfaces, it is suggested that it is the smallestscale of roughness that primarily controls the adhesion of surfaces.

Acknowledgements

The authors acknowledge the financial support of the Engineering ResearchŽ .Center ERC for Particle Science and Technology at the University of Florida,

Ž .The National Science Foundation NSF grant No. EEC-94-02989, and the Indus-trial Partners of the ERC.

References

� � Ž .1 H.C. Hamaker, Physica 4 1937 1058.� � Ž .2 K.L. Johnson, K. Kendall, A.D. Roberts, Proc. R. Soc. Lond. A: Math. Phys. Sci. 324 1971 301.� � Ž .3 B.V. Derjaguin, V.M. Muller, Y.P. Toporov, J. Colloid Interface Sci. 53 1975 314.� � Ž .4 Y.I. Rabinovich, J.J. Adler, A. Ata, B.M. Moudgil, R.K. Singh, J. Colloid Interface Sci. 232 2000

10.� � Ž .5 Y.I. Rabinovich, J.J. Adler, A. Ata, B.M. Moudgil, R.K. Singh, J. Colloid Interface Sci. 232 2000

17.� � Ž .6 L.M. Skinner, J.R. Sambles, Aerosol Sci. 3 1972 199.� � Ž .7 L.R. Fisher, J.N. Israelachvili, J. Colloid Interface Sci. 80 1981 528.� � Ž .8 L.R. Fisher, R.A. Gamble, J. Middlehurst, Nature 290 1981 575.� � Ž .9 J.C. Melrose, Langmuir 5 1989 293.

� � Ž .10 W. Pietsch, H. Rumpf, Chemie-Ing. Techn. 39 1967 885.� � Ž .11 M.C. Coelho, N. Harnby, Powder Tech. 20 1978 197.� � Ž .12 M.C. Coelho, N. Harnby, Powder Tech. 20 1978 201.� � Ž .13 L.R. Fisher, J.N. Israelachvili, Colloids Surf. A: Physicochem. Eng. Aspects 3 1981 303.� � Ž .14 A. Marmur, Langmuir 9 1993 1922.� � Ž .15 A. de Lazzer, M. Dreyer, H.R. Rath, Langmuir 15 1999 4551.


� � Ž .16 Y.I. Rabinovich, T.G. Movchan, N.V. Churaev, P.G. Ten, Langmuir 7 1991 817.� � Ž .17 M.M. Kohonen, H.K. Christenson, Langmuir 16 2000 7285.� � Ž .18 M. Fuji, K. Machida, T. Takei, T. Watanbe, M. Chikazawa, Langmuir 15 1999 4584.� � Ž .19 X. Xiao, L. Qian, Langmuir 16 2000 8153.� � Ž .20 L. Sirghi, N. Nakagiri, K. Sugisaki, H. Sugimura, O. Takai, Langmuir 16 2000 7796.� � Ž .21 R.A. Quon, A. Ulman, T.K. Vanderlick, Langmuir 16 2000 8912.� � Ž .22 N.A. Burnham, A.J. Kulik, in: B. Bhushan Ed. , Handbook of Micro�Nano Technology, 2nd ed.,

CRC Press, New York, 1999 247.� � Ž .23 K.L. Johnson, J.A. Greenwood, J. Colloid Interface Sci. 192 1997 326.� � Ž .24 B.V. Derjaguin, N.V. Churaev, Y.I. Rabinovich, Adv. Colloid Interface Sci. 28 1988 197.� � Ž .25 Y.I. Rabinovich, N.V. Churaev, Colloid J. USSR 52 1990 256.� � Ž .26 B.V. Derjaguin, Kolloid Z. 69 1934 155.� �27 A.W. Adamson, Physical Chemistry of Surfaces, 2nd ed., Interscience Publishers, New York, 1967.� � Ž .28 B.V. Derjaguin, N.V. Churaev, Wetting Films, Nauka, Moscow 1984 .� �29 B.V. Derjaguin, N.V. Churaev, V.M. Muller, Surface Forces, Plenum Publishing Corp., New York,

1987.� � Ž .30 H.K. Christenson, J. Colloid Interface Sci. 104 1985 234.� � Ž .31 H.K. Christenson, C.E. Blom, J. Chem. Phys. 86 1987 419.� � Ž .32 W.A. Ducker, T.J. Senden, Langmuir 8 1992 1831.� �33 A. Ata, Synthesis of Composite Particles by Dry Coating Techniques, Ph.D. Dissertation, Univer-

Ž .sity of Florida, Gainesville 1998 .� � Ž . Ž .34 R.S. Bradley, Phil. Mag. Ser. 7 13 1932 853.� �35 V.V. Yaminsky, R.K. Yusupov, A.A. Amelina, V.A. Pchelin, E.D. Shchukin, Colloid J. USSR 37

Ž .1975 918.� � Ž .36 V.V. Yaminsky, B.W. Ninham, R.M. Pashley, Langmuir 14 1998 3223.� �37 R.K. Singh, A. Ata, J. Fitz-Gerald, Y.I. Rabinovich, W. Hendrickson, Kona: Powder Part. 15

Ž .1997 121.

Documents

Capillary forces between surfaces with nanoscale -