PHYSICAL REVIEW B VOLUME 46, NUMBER 12 15 SEPTEMBER 1992-II

van der Waals forces in atomic force microscopy operating in liquids: A spherical-tip model

N. GarciaDepartamento de Fisica de la Materia Condensada, Uni Uersidad A utonoma de Madrid, Madrid 28049, Spain

and FISINTEC, Ruperto Chapi 19, Alcobendas, Madrid 28100, Spain

Vu Thien BinhDepartement de Physique des Materiaux, Uni Uersite Claude Bernard Lyon 1, 69622 Villeurbanne, France

(Received 21 January 1992}

In this paper we discuss the van der Waals forces in atomic force microscopy when operating inliquids. We show that these forces are almost cancelled out for spherical tips immersed in liquids andseem to be in agreement with recent in situ electrochemistry results. Experiments as well as an analogywith Archimedes s principle are proposed. Furthermore, we present experiments showing such spheri-cal tips of hundred nanometers which can be fabricated in a reproducible manner and are stable.

Goodman and Garcia' presented a theory for van derWaals (vdW) forces acting on a spherical body due to thepresence of a dielectric surface and its efFects on cantilev-er displacements used in atomic force microscopy(AFM). Previous experiments ' have indicated thatAFM's operate more stably in water than in air or in avacuum. Experimental confirmation of this behavior wasobtained with AFM studies in electrochemical solutionsin situ in which single atoms were observed with goodregularity and resolution, for many diferent electrochem-ical solutions. This means that AFM's generally operatemore stably in liquids. In this work we explain that thisstabilization phenomenon is caused by the compensationof the long-range vdW forces, which leaves only the verylocal "high resolution" repulsive forces on the tip. Wepresent, as a guide and also as a pedagogical point, ananalogy between the vdW forces in liquids andArchimedes's principle. We propose an experiment witha spherical tip, since spherical tips can be fabricated in areproducible manner. *

vdW forces are due to the induced dipole interactionbetween two neutral bodies and the polarizability of theirmedia. Goodman and Garcia' represented a theory forthe force, f,p, on a sphere of radius r and dielectric con-stant e„placed in vacuum at a distance z from a flat sam-ple surface with a dielectric constant c., :

Mossotti formula:

SPc., —1 c, —1

+1 ' ' cs+2(3)

including the case of metals for which the value of thedielectric constant is infinity. The formulas are valid forthe interaction between the tip and a plane sample whichcan be either a solid, a liquid, or a gas sample surface.

Now consider the case of a spherical tip completely im-mersed in a liquid and above a solid sample as in Fig. 1.The vdW forces acting on the tip are due to the presenceof the liquid medium and the solid sample. By symmetry,the resulting vdW force on the spherical tip due to thevolume of liquid contained in the region defined byD =2 (z+r) is zero. Then the remaining vdW forcesacting on the sphere are (i) the attractive force f, withthe solid sample given by formulas (1)—(3) and substitutedthe solid parameters P„s„andB, for P, , s,p, and B,p,and (ii) the force fI, due to the liquid above the volumedelimited by the distance D (Fig. 1). This latter force isattractive for the liquid but it is repulsive with respect tothe sample, i.e., the tip moves away from the sample un-der a force given also by formulas (1)—(3), but substitutedthe liquid parameters P&, sI, and Bt for P,p, s,p, and B,p.Therefore, for the case z &&r, the net force on the sphere1S

r 1, = —B,sP sp 2 (1+5)2 F&otg =f. ft =

z' (4)

the minus sign indicates that the force is attractive,5=z/2r, 8 p is the Hamaker constant' of the samplegiven by

Bsp = 4+I spl t (2)

where K=1.41 eV is a universal constant determinedfrom experimental He surface scattering data byHoinkes. " P, and P, are related to the sample and tippolarizabilities, respectively. They can be calculatedfrom their dielectric constants by using the Clausius-

From Eq. (4) we see that the magnitude and directionof F„„&is proportional to the difference of the P's of thesolid and the liquid. Two consequences can then be de-duced:

(i) For a given metallic sample (c,~ oo, so P, is practi-cally equal to 1) in a liquid environment of water (sl -80,we have PI-0.975) the resulting vdW force is then re-duced to only 2.5% of its value if the tip is working in avacuum or air environment. Indeed in this latter case, asthe dielectric constant of vacuum or air is —1, the corre-

46 7946 1992 The American Physical Society

46 BRIEF REPORTS

sponding P value is practically zero. This may explainwhy in electrochemistry experiments done in the pres-ence of a liquid, the AFM cantilever could be stabi-lized and achieved high resolution: the resulting long-range vd%' force, E„„&,is reduced to practically zero,then only the short-range repulsive interactions'remain. There is always a very substantial reduction ofthe force for any fluid, even if the dielectric constant is

small, as can be deduced from Eq. (4).(ii) This reduction leads to more stability of the micro-

scope because the snap to contact due to the large vd%'forces in the liquid are drastically reduced. Moreover,the force acting now on the tip is only the local repulsiveone, without the fluctuations that could be introduced bythe vdW forces.

(iii) The resulting vdW force, E«,», could be either at-tractive or repulsive with respect to the sample, depend-ing on the respective values of e, and et (see Fig. l). Forexample, in the case of mica this force will be repulsive inthe presence of many liquids.

The remarkable point of Eq. (4) is that it is analogousto the Archimedes's principle which states that if a bodyis immersed in a liquid it experiences an upward forceequal to the weight of the liquid which will fill the space

es —1

e, +l.

Ps= Pi

(a)

gs —1

c, +1

Ps& Pi

D

ErQO

6

Ps&0((c)

FIG. 1. Schematic drawings of the different parameters andvd% forces acting on a spherical tip immersed in a liquid and atthe proximity of a solid sample. (a) The resulting vdW force,Ft t ] is zero when f, = fr for P, =Pl (s, =s, ); (1}F„„,is attrac-tive when f, & fl, i.e., for P, &PI, (c) F„„,is repulsive whenf, &fi, i.e., for P, & P, .

FIG. 2. Formation of spherical tips by surface diffusion witha small angle conical tip. (2) Three-dimensiona1 representationof the profile of a 0' cone angle tip, at -80% of its solid dropdetachment time, calculated by using surface diffusion law; (b)experimental W "spherica1 tip" geometry; (c) experimental Cu"spherical tip" geometry.

7948 BRIEF REPORTS 46

occupied by the body. To make the analogy with theArchimedes's equation we replace the density of the bodyand liquid by the p's, the gravity by (—,'K) and the volumeof the body by (P, r /z ). In other words, in Archimedes'sprinciple it is the difference in densities which accountsfor the actual weight of the body, while with the vdWforces it is the difference in the polarizabilities betweenthe solid sample and the liquid medium which accountfor the net force acting on the tip. Note that the analogyholds only if the whole spherical body is completely im-mersed in the liquid. If this is not the case, for example,the spherical tip is half immersed, the net vdW force in-creases because the liquid also attracts the tip towardsthe sample. The resulting force is then very complicatedto calculate.

This theory is exact for spherical bodies and thereforeshould be used with "spherical tips. " The experimentalproblem is to obtain, in a reproducible manner, a tip end-ing with a spherical geometry. The evolution of the tipshape, when heated in vacuum, is basically governed bysurface diffusion law. The surface atoms, at high temper-atures, move from regions of high curvature towards re-gions of low curvature in order to minimize the total en-

ergy of the system. For a conical tip with an anglegreater than 6', the surface diffusion will blunt the tip,but for small angle tips ( (6') a constriction will developjust under the apex region. The diameter of this constric-tion will decrease with the heating time, ending in the de-tachment of a solid drop. This has been studied theoreti-cally and confirmed experimentally. ' We show, Fig.2(a), the morphology in three dimensions, just before thedetachment of the solid drop, of a 0' angle tip (-80% ofthe ovulation time) calculated by using only the surfacediffusion law. Experimentally, by stopping the heatingbefore the detachment of the solid drop, we can then ob-tain a stable tip presenting a nearly spherical ball at itsend. Figures 2(b) and 2(c) are scanning electron micros-

copy images of W and Cu tips, respectively, obtained byheating under vacuum and presenting this geometry.The dimension of the solid drop will depend on the initialdimension of the tip, but it is usually around a few hun-dreds of nanometers. Differences exist between the ex-perimental profile and the calculated ones; they should beexplained by the experimental conditions (heating, vacu-um). It is important, however, to note that the solid dropformation is governed by the surface diffusion law(Herring's law), ' the fabrication of such a tip is thenreproducible and available for any material. It could beargued that the model described here is for a spherical tipwhich is not the geometry of the tips presented due to theneck which connects the ending ball and the shank.However, the mass of the neck is negligible compared tothe ball and the neck is farther away from the samplethan the ball, thus for any possible configuration betweenthe tip and the surface. The neck, therefore, has negligi-ble contribution to the vdW forces and our tips are quitegood experimental approaches of a "spherical tip." Webelieve that it is worthwhile to use this tip geometry inAFM since the interactions can be well estimated andconsidered theoretically.

In conclusion, by operating AFM in a liquid, the re-sulting vdW force acting on a spherical body is drastical-ly reduced compared to its value when operating in vacu-um or in air. The equations reported here applied to aspherical tip. Such a tip is prepared in a reproduciblemanner, just by heating in vacuum a conical tip with asmall angle.

Discussions with Frank O. Goodman are appreciated,as well as the contribution of V. Semet for the three-dimensional tip profile simulations. The Department de

Physique des Materiaux is "Unite Associe au Centre Na-tional de la Recherche Scientifique. "

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