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Influence of surface roughness on dispersion forces

V.B. Svetovoy, G. Palasantzas

PII: S0001-8686(14)00281-4DOI: doi: 10.1016/j.cis.2014.11.001Reference: CIS 1491

To appear in: Advances in Colloid and Interface Science

Please cite this article as: Svetovoy VB, Palasantzas G, Influence of surface rough-ness on dispersion forces, Advances in Colloid and Interface Science (2014), doi:10.1016/j.cis.2014.11.001

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ACCEPTED MANUSCRIPTInfluence of surface roughness on dispersion forces

V. B. Svetovoy∗

MESA+ Institute for Nanotechnology, University of Twente, PO 217, 7500 AE Enschede, The Netherlands

G. PalasantzasZernike Institute for Advanced Materials University of Groningen, 9747 AG Groningen, The Netherlands

(Dated: November 4, 2014)

Surface roughness occurs in a wide variety of processes where it is both difficult to avoid andcontrol. When two bodies are separated by a small distance the roughness starts to play an im-portant role in the interaction between the bodies, their adhesion, and friction. Control of thisshort-distance interaction is crucial for micro and nanoelectromechanical devices, microfluidics, andfor micro and nanotechnology. An important short-distance interaction is the dispersion forces,which are omnipresent due to their quantum origin. These forces between flat bodies can be de-scribed by the Lifshitz theory that takes into account the actual optical properties of interactingmaterials. However, this theory cannot describe rough bodies. The problem is complicated by thenonadditivity of the dispersion forces. Evaluation of the roughness effect becomes extremely diffi-cult when roughness is comparable with the distance between bodies. In this paper we review thecurrent state of the problem. Introduction for non-experts to physical origin of the dispersion forcesis given in the paper. Critical experiments demonstrating the nonadditivity of the forces and stronginfluence of roughness on the interaction between bodies are reviewed. We also describe existingtheoretical approaches to the problem. Recent advances in understanding the role of high asperitieson the forces at distances close to contact are emphasized. Finally, some opinions about currentlyunsolved problems are also presented.

Keywords: dispersion forces, roughness, nonadditivity, contact, adhesion

Contents

I. Introduction 1

II. Dispersion forces 2A. Early representations 3

1. Van der Waals force 32. Casimir force 3

B. Lifshitz theory 31. Physical ground 42. Fluctuation-dissipation theorem 4

C. Lifshitz formula 51. Real frequency representation 52. Imaginary frequency representation 53. Van der Waals and Casimir limits 64. Convenient representation 6

D. Proximity force approximation 7

III. Critical experiments 7A. Forces between corrugated bodies 8B. Stiction problem 9C. Forces measured with Surface Force

Apparatus 12D. Forces measured with Atomic Force

Microscope 13E. What we learned from the experiments 15

IV. Theoretical description of roughness 15

∗Corresponding author: v.svetovoy@utwente.nl

A. Description of a rough surface 15

1. Characterization 15

2. Distance upon contact and heightdistribution 16

B. Perturbative correction to the force 17

C. Beyond perturbation theory 18

1. Roughness statistics of Au films 19

2. Roughness contribution 19

3. Comparison with the experiments 20

D. What we learned from the theory 21

V. Conclusions 22

Acknowledgments 23

References 23

I. INTRODUCTION

Dispersion forces originate from quantum and thermalfluctuations of electric currents inside of interacting me-dia and in the gap separating them [1–4]. Assuming flatsurfaces, the forces increase as d−α when the distance dbetween bodies decreases. The exponent α is in between3 and 4 depending on the distance. Conditionally thedispersion forces become dominant when the bodies areseparated by the distances smaller than 100 nm. Theyplay an important role in nanoscience and nanotechnol-ogy including micro and nanoelectromechanical devices[5–8]. The role of these forces is also essential in colloid

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and interface science [9–15]. In the latter case the interac-tion happens in a liquid medium, where in addition theelectrostatic forces are involved. The dispersion forcesare closely related to adhesion between bodies under dryconditions [16, 17]. They define the adhesion energy asthe force acting via the gap separating the bodies uponcontact.

Historically different names are used for the forces,which we call here dispersion forces. At distances smallerthan a few nanometers these forces are termed van derWaals forces [1]. At larger distances the same forces arecalled the retarded van der Waals forces or Casimir forces[2, 19]. All these forces have the same physical originrelated to fluctuating currents. To stress this point thegeneral name Casimir-Lifshitz force is in use. Evgeny Lif-shitz [3] was the first who recognized the common originof the van der Waals and Casimir forces. He deduced theso-called Lifshitz formula [18], which is able to predictthe the force between two parallel plates separated bythe distance d using as input parameters the dielectricfunctions of interacting materials. Therefore the bod-ies within the Lifshitz theory are teated macroscopically.The minimal size in the Lifshitz theory is the size wherethe dielectric function is well defined (much larger thaninteratomic distances). The Lifshitz formula interpolatesbetween the van der Waals force at small distances d < 5nm and the pure Casimir force at d > 1 µm. Betweenparallel plates the first decreases as d−3 while the sec-ond one decreases as d−4 when the distance increases. Inthis paper to name all the forces having the same originwe use the general term dispersion forces proposed byLondon for molecules.

The dispersion forces are nonadditive. The force be-tween two molecules depends on the position of a thirdmolecule located nearby. A consequence is that the forcebetween bodies of finite size cannot be calculated aspairwise summation of forces acting between separatemolecules. This is an important point because it com-plicates the calculation of the force in many practicalsituations. Nonadditivity is often cited as a very specificproperty of the dispersion forces. However, the electro-static force can also be nonadditive. The force betweenmetallic sphere and plate cannot be calculated as the sumof forces between infinitesimal capacitors. The reason isthat the charges redistribute in response to the field. Asimilar effect happens for the dispersion forces where po-larization changes with the field.

One limitation of the Lifshitz formula is that the forceis predicted explicitly only between parallel plates. It isnot a principal restriction but rather a computational onerelated to the nonadditivity. Only recently a closed-formexpression for the sphere-plate interaction was presented[20, 21]. Even for numerical calculations the problemwas not an easy task but significant progress was madein the last decade and the force was evaluated for a num-ber of geometrical configurations [22]. The same restric-tion exists for the Derjaguin, Landau, Verwey Overbeek(DLVO) theory of colloidal stability [9, 10] with an addi-

tional complication that includes electrostatic forces.Nonadditivity makes the problem even more difficult

if one would like to calculate the force between randomlyrough bodies. All solids in nature or in laboratories arerough. The roughness can be characterized by two mainparameters that are the root-mean-square (rms) rough-ness w and the lateral correlation length ξ (typical featuresize). While w is much smaller than the distance be-tween bodies d the roughness correction to the force canbe calculated using the perturbation theory. The corre-sponding methods have been developed in relation to theprecise measurements of the Casimir forces at distanceslarger than 100 nm [23–25]. However, when the forcewas measured at distances smaller than 100 nm, strongdeviations from the predictions based on the perturba-tion theory were found [26] even for relatively small rmsroughness w ≪ d. As was explained later [27] this effectis due to significant deviations of the roughness statisticsfrom the normal distribution for some materials.The ultimate problem related to surface roughness is

the evaluation of the force when the rms roughness iscomparable with the distance between the bodies. Atthis moment the problem is not solved but it is clearthat high peaks of the roughness profile play a principalrole. For this reason the problem is closely related to thecontact between two bodies. In principle the force canbe calculated numerically, but practically it can be doneonly for a restricted area. Up to now the roughness effectwas not tackled numerically. The important role of highpeaks, which are rare statistical events, combined withnumerical calculations and experimental data could givea strong push to understand the influence of roughnesson dispersion forces.The purpose of this paper is not only to review existing

experimental and theoretical approaches to the roughnessproblem, which are still in their infancy, but also to givean introduction for non-experts to the methods used tocalculate the dispersion forces in different practical sit-uations. The paper is organized as follows. In ChapterII the physical origin of the dispersion forces is describedand some helpful relations are presented. A review of ex-periments important for understanding of the roughnesseffect is presented in Chapter III. Methods to describerough surfaces and existing theoretical models for theforce taking into account roughness effect are describedin Chapter IV. Finally some conclusions and our visionof the problems that have to be solved are collected inthe last chapter.

II. DISPERSION FORCES

In this chapter we describe the main ideas and resultsof Lifshitz theory of dispersion forces. We try to presenta clear physical picture and give the necessary equationsin the form convenient for practical applications of thetheory. This section can be considered as an introduc-tion to the Lifshitz theory. Quite often it is not realized

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that so different physical phenomena as well-known vander Waals forces acting between macroscopic bodies sepa-rated by a few nanometers gap and rather exotic Casimirforces related to the boundary conditions on perfectly re-flecting mirrors separated by much larger distances areactually intimately related to each other. In terms of theLifshitz theory both originate from fluctuating currents(or polarizations) in macroscopic bodies. The currents inone of the bodies give rise to the electromagnetic field in-teracting with the currents in the other body. If this dis-tance is small, one can neglect the retardation effect andthe resulting interaction reproduces the van der Waalsforces. At larger distances the retardation becomes im-portant and one can reproduce the Casimir forces.

A. Early representations

1. Van der Waals force

Van der Waals [28] introduced weak intermolecularforces to explain deviations from the ideal gas laws butthe physical nature of these deviations stayed unclear. In1930 London [1] showed how one can understand the longdistance behavior of the forces using quantum mechan-ics. It was shown that the attractive forces appear fromelectrostatic interaction of temporary dipoles in atomsor molecules. If R is the distance between atoms, theinteraction energy between them decays according to thepower law: Vint ∼ R−6. The forces associated with thispotential are called dispersion or induced London dipole-dipole forces.

Appearance of attractive forces between neutral atomsnaturally leads to similar forces between any two macro-scopic bodies separated by a distance d. The force be-tween two macroscopic spheres was calculated for the firsttime by Hamaker [29] using pairwise summation of thevan der Waals-London potential between atoms. Iron-ically this famous result happened to be incorrect. Asbecame clear later the dispersion forces are nonadditiveand the pairwise summation gives incorrect results ingeneral. Nevertheless, at some conditions more precisecalculations based on the Lifshitz theory showed that theHamaker approach can be accepted as a convenient firstapproximation [30]. As a particular case, Hamaker foundthe van der Waals force (per unit area) between two par-allel macroscopic slabs that had been derived earlier byde Boer [31] and this result is absolutely precise. As afunction of distance d this force is

FvdW =AH

6πd3, (2.1)

where AH is the so called Hamaker constant [29] account-ing for materials properties of the slabs. Precise mean-ing of this constant and the range of distances where theformula (2.1) is applicable were clarified only after thedevelopment of the Lifshitz theory [3, 4, 18].

2. Casimir force

In 1948 Casimir [2] showed that two uncharged ideallyreflecting mirrors parallel to each other and separated bya gap d will be mutually attracted. The force per unitarea was found to be

FC =π2hc

240d4. (2.2)

It is the famous Casimir formula. An exceptional featureof this formula is that the force depends only on the sep-aration and fundamental constants; no electron chargeenters the formula.The attraction is attributed to quantum fluctuations

of vacuum electromagnetic fields, which are different be-tween the mirrors and outside of them due to presenceof ideally reflecting boundaries. Inside the cavity formedby the mirrors, only the modes that have nodes on themirrors are allowed. Outside of the cavity all the modescan exist. In this situation the radiation pressure fromoutside will be larger than that from inside of the cav-ity. As the result the mirrors will attract each other.According to quantum mechanics each mode of the fieldwith frequency ω has a zero-point energy hω/2. Then theCasimir energy is the change in the zero-point energy ofvacuum due to presence of the boundaries.Zero-point energy interpretation became so popular in

the physics community that the other side of the problemis remembered mostly by experts in the field. Namely,in the same year 1948, Casimir and Polder published apaper [19], where they took into account the retarda-tion effect in the interaction between two induced dipoles.It was found that when the distance between atoms in-creases the interaction energy falls off with distance as∼ 1/R7. It is faster than for the non-retarded van derWaals interaction (∼ 1/R6). Casimir already realized [2]that the retarded interaction of dipoles has to give theforce between macroscopic bodies, which behaves withthe distance precisely as (2.2). However, the final clari-fication of this point was made by Lifshitz [3] who pre-sented the van der Waals (2.1) and Casimir (2.2) forcesas the limit cases of one and the same force at small andlarge distances, respectively.

B. Lifshitz theory

The most detailed theory of dispersion forces betweenmacroscopic bodies was developed by Lifshitz in 1955. Itwas based on the Rytov’s theory of fluctuating electro-magnetic fields [32]. This theory is applicable, in princi-ple, to any bodies, independent on their molecular con-stitution. It includes automatically retardation effectsand the only restriction on the distance is that it mustbe much larger than the interatomic distance. An impor-tant feature of the theory is that it expresses the forcevia macroscopic dielectric functions of the bodies. Anapproach close to the original Lifshitz consideration is

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described briefly in [33]. Generalization of the theory tothe case of liquid gap separating the bodies was devel-oped by Dzyaloshinskii, Lifshitz, and Pitaevskii in 1961[4] and became part of the textbook [18].

In general, the Lifshitz theory can be applied to bodiesof arbitrary shape. However, in a closed form the expres-sion for the dispersion forces can be given only for parallelplates separated by a gap. This is the most important re-striction of the Lifshitz formula, which was recently over-come by the development of numerical procedure [34].

1. Physical ground

It is not possible to calculate the dispersion forces be-tween macroscopic bodies starting from microscopic in-teraction between separate atoms. Pairwise summationwould be valid only for rarefied bodies such as gases. Incondensed bodies the atoms in the neighborhood causean essential change in the properties of the electronicshells, and the presence of a medium between the in-teracting atoms affects the electromagnetic field throughwhich the interaction is established [4]. Nonadditivity isoften cited as a very specific property of the dispersionforces. However, the electrostatic interaction also demon-strates nonadditive properties when charges in bodies areable to redistribute in response to the applied field.

Instead of a ”microscopic” description of the problemone can use a macroscopic point of view, in which inter-acting bodies are treated as continuous media. This ispossible when the bodies are separated by the distanceslarge in comparison with interatomic distances.

The fundamental idea of the theory is that the inter-action between bodies is established through fluctuatingelectromagnetic fields. Such fields are always present in-side and extend beyond material boundaries. A well-known example is the thermal radiation but it has tobe stressed that electromagnetic fluctuations persist evenat zero temperature as zero-point quantum fluctuations.The source of these fluctuations is the electric polariza-tion P(ω, r) or equivalently the electric current densityJ(ω, r) = −iωP(ω, r), where we understand the physicalvalues as Fourier transformed in time. The origin of fluc-tuating currents is easier to understand for metals. Thedensity of quasi-free electrons in a metal fluctuates as thedensity of particles in any gas. If at a point r the densityis smaller than the average value, there will be a currenttending to increase the density at this point.

2. Fluctuation-dissipation theorem

According to the fluctuation dissipation theorem(FDT) [32, 33, 35] in thermal equilibrium the correlationsof these fluctuating currents are related to the dissipationin the medium:

⟨

Jα(ω, r)J∗β(ω

′, r′)⟩

= ωε′′(ω)

(

hω

2+

hω

ehω/kT − 1

)

×

δ(ω − ω′)δ(r− r′)δαβ , (2.3)

where α, β = x, y, z enumerate vector components, andthe dissipation is proportional to the imaginary part ofthe dielectric function ε′′(ω) = Imε(ω). Due to FDT the

very existence of the dispersion forces is closely related to

the dissipation in the materials of interacting bodies. InEq. (2.3) the contributions from the zero-point and ther-mal fluctuations are explicitly separated (the first andsecond term in the brackets). Some additional informa-tion related to the application of FDT for calculating thedispersion forces including nonequilibrium situations canbe found in [36–38].Fluctuating currents are the sources of electromagnetic

field. This field is described by the Maxwell equations.Solutions of these equations can be expressed via theGreen functions. For example, the components of theelectric field Eα(ω, r) are

Eα(ω, r) =i

ω

∫

dr′Gαβ(ω, r, r′)Jβ(ω, r

′), (2.4)

where Gαβ are the components of the Green tensor. It isclear from (2.4) that the Green functions play role of theresponse function in the linear-response theory [39].Using the relation (2.4) together with the FDT (2.3)

and making use of the general properties of the Greentensor we can find the correlation functions for the com-ponents of the electric field expressed via the Green func-tions:

⟨

Eα(ω, r)E∗β(ω

′, r′)⟩

=

2πh coth

(

hω

2kT

)

ImGαβ(ω, r, r′)δ(ω − ω′). (2.5)

The correlation functions of the magnetic field can beeasily found by applying the Maxwell equations [18].The Green tensor is the solution of the equation, which

follows directly from the Maxwell equations:

[

∂α∂β − δαβ

(

∇2 +ω2

c2ε(ω, r)

)]

Gαγ(ω, r, r′) =

4πω2

c2δαβδ(r− r′). (2.6)

Here ε(ω, r) is the nonhomogeneous dielectric functiondescribing the interacting bodies. For example, if twobodies are interacting via a vacuum gap then ε(ω, r) is 1within the gap, it is ε1(ω) in the place filled with the body1 and is ε2(ω) in the space occupied by the body 2. Equa-tion (2.6) has to be solved with proper boundary condi-tions characterizing the field components at the interfaces[18]. In the sense of boundary conditions Gαβ(ω, r, r

′)behaves as Eα(ω, r).The force acting on the body surface is the normal

component of the Maxwell stress tensor. The averagedstress tensor can be expressed via the correlation func-tions (2.5) or finally via the Green functions of the sys-tem (2.6) at r = r′ where r approach the surface from the

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z

x,y

d

ε1

ε2

ε0

FIG. 1: The simplest configuration consisting of two semis-paces separated by a gap, for which the Green function canbe found explicitly

side of the gap. In case of the gap filled with a liquid itis possible to do due to equilibrium between absorptionand emission [4]. This point was recently stressed againby Pitaevskii [40].

C. Lifshitz formula

The explicit form of the Green functions can be eas-ily found for two parallel plates interacting via the longwavelength fluctuations. The simplest configuration istwo semispaces made from different materials with thedielectric functions ε1(ω) and ε2(ω), respectively, sepa-rated by a small gap d filled with the material describedby the dielectric function ε0(ω). This configuration isshown schematically in Fig. 1.

1. Real frequency representation

For parallel plates the force (pressure) acting on thebody 2 (right plate) is calculated via the Green functionstaken at z = z′ → d/2. Solution of the equation (2.6)is very similar to the optical problem in the cavity withadditional complication due to the tensor structure. Thefinal result for the force (per unit area) can be written as

F (T, d) =h

2π2

∞∫

0

dω coth

(

hω

2kT

)

Re

∞∫

0

dq q k0 g(q, ω),

(2.7)where the wave vector in the gap is K = (q, k0) with q

being the x and y components of the vector and k0 =(ε0ω

2/c2 − q2)1/2 being the z-component. The functiong(q, ω) is

g(q, ω) =∑

ν=s,p

rν1rν2e

2ik0d

1− rν1rν2e

2ik0d. (2.8)

Here rν1,2 are the reflection coefficients of the inner sur-faces of the plates (index 1 or 2) for two different polar-izations: ν = s or transverse electric (TE) polarization,and ν = p or transverse magnetic (TM) polarization.

The factor g(q, ω) describes multiple reflections from theinner surfaces of body 1 and 2. The frequency dependentfactor coth (hω/2kT ) originates from the FDT.The reflection coefficients rν1,2 entering in the Lifshitz

formula are nothing else but the Fresnel reflection coef-ficients. In the simple configuration in Fig. 1 these co-efficients are expressed via the z-component of the wavevectors in the gap k0 and in the ith body ki as

rsi =k0 − kik0 + ki

, rpi =εik0 − ε0kiεik0 + ε0ki

, (2.9)

where

k0 =

√

ε0(ω)ω2

c2− q2, ki =

√

εi(ω)ω2

c2− q2. (2.10)

Equation (2.7) is the Lifshitz formula in the real fre-quency representation [33].In many practical applications this representation is

not very convenient because the integrand is a fast os-cillating function due the factor eik0d. However, thereare situations when the real frequency representation isthe only possible or plays an important role in high-lighting underlying physics. It happens, for example,in the non-equilibrium situations, when the plates havedifferent temperatures [38]. In addition, this paper isdirected to the distance range d <∼ 100 nm, where thethermal fluctuations do not play significant role at Taround the room temperature. This is because the dis-tance d is much smaller than the thermal wavelengthλT = hc/kT = 7.6 µm at T = 300 ◦K. If we take intoaccount only quantum fluctuations, then in Eq. (2.7) onehas to change coth(hω/kT ) → 1.

2. Imaginary frequency representation

The problem of fast oscillations is usually avoided bythe contour rotation in the frequency complex plane.This is possible due to analyticity of the integrand. Withthis rotation the force can be expressed as an integral overthe imaginary frequency ω = iζ as:

F (d) =h

2π2

∞∫

0

dζ

∞∫

0

dq q|k0|g(q, iζ), (2.11)

where the low temperature limit T → 0 is assumed asexplained above.To get the reflection coefficients at imaginary frequen-

cies one has to make analytical continuation of (2.9) and(2.10) to the imaginary axis. For example, k0 at ω = iζwill be

k0 = i

√

ε0(iζ)ζ2

c2+ q2. (2.12)

The general sign of the square root is defined by thecondition that any wave decays inside of a material or

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Imk0 > 0. Therefore, the reflection coefficients in thiscase will depend on the dielectric functions at the imagi-nary frequencies ε(iζ). This functions cannot be directlymeasured but can be expressed via the observable func-tion ε′′(ω) with the help of the Kramers-Kronig relation[33]:

ε(iζ) = 1 +2

π

∞∫

0

dωωε′′(ω)

ω2 + ζ2. (2.13)

The general property of ε(iζ) is that this function is real,positive, and monotonously decreases when ζ increases.

The relation (2.13) shows that the dispersion forcesare completely determined by ε′′(ω), which is responsi-ble for the dissipation in the material. Therefore, the

very existence of the forces is intimately related to the

dissipation. This relation is indeed a consequence ofthe fluctuation-dissipation theorem. The same equation(2.13) also shows that the force is sensitive to ε′′(ω) inthe integral sense.

3. Van der Waals and Casimir limits

Let us demonstrate that the Lifshitz formula (2.11)reproduces the van der Waals and Casimir regimes givenby Eqs. (2.1) and (2.2) as limiting cases. The distance dseparating the plates defines a characteristic imaginaryfrequency

ζch = c/2d. (2.14)

It has to be stressed that the real frequency ωch = c/2dnot always plays a similar role due to fast oscillations of

the integrand in (2.7).Let ω0i be a typical frequency of the resonances in the

absorption spectra of the i-th body or in the gap (i = 0).If the distance d is so small that one can neglect theretardation d ≪ c/ω0i for every i = 0, 1, 2, then the vander Waals regime is realized because |ki| ≈ |k0|. In thisregime for the reflection coefficients one has

rsi ≈ 0, rpi ≈ εi(iζ)− ε0(iζ)

εi(iζ) + ε0(iζ). (2.15)

It is interesting to note that the s-polarization does notcontribute. It happens because in the quasi-static ap-proximation, which is equivalent to the non-retardedcase, the s-polarization is reduced to pure magnetic fieldthat freely penetrates non-magnetic materials.

Introducing a new variable x = 2d|k0| it is easy to findfrom Eq. (2.11) with the help of Eq. (2.15)

F (d) =h

16π2d3

∞∫

0

dζ

∞∫

0

dxx2

(ε1+ε0)(ε2+ε0)(ε1−ε0)(ε2−ε0)

ex − 1, d ≪ c

ω0i,

(2.16)where εi are functions of ζ. The double integral heredepends only on the dielectric functions but not on the

distance. Comparing (2.16) with the van der Waals force(2.1) one finds that the Hamaker constant is defined as

AH =3h

8π

∞∫

0

dζ

∞∫

0

dxx2

(ε1+ε0)(ε2+ε0)(ε1−ε0)(ε2−ε0)

ex − 1. (2.17)

This is a precise definition of the Hamaker constant ex-pressed via the dielectric properties of involved materials.In the opposite limit when the distance is so large that

d ≫ c/ω0i (but still d ≪ λT to neglect the thermal ef-fects) we can introduce new variables x = 2d|k0| andp = xζch/ζ in (2.11). In these terms the argument ofthe dielectric functions εi(iζ) will be ζ = ζchx/p. Bydefinition p ≥ 1 and natural value of x is x ∼ 1 due tothe exponential dependence. It means that ζ <∼ ζch andbecause ζch is small we can take the static values of thedielectric functions εi(iζ) ≈ ε0i. With this simplificationwe can write Eq. (2.11) as

F (d) =hc

32π2d4

∞∫

1

dp

p2

∞∫

0

dxx3g(x, p) |εi=ε0i, ζch ≪ ω0i

(2.18)Again, the double integral here is a number that dependson permittivities of the materials. In this case both po-larizations contribute and the force decreases faster withthe distance increase due to the retardation effect. Incomparison with the van der Waals force in Eq. (2.18)there is an extra factor ζch responsible for the retarda-tion. If we choose in (2.18) the materials to be idealmetals, ε0i → ∞ (i = 1, 2), and the gap to be vac-uum, ε00 = 1, then both the reflection coefficients arersi = rpi = 1 and we reproduce the Casimir result (2.2).Equations (2.16) and (2.18) present the limiting cases

and these equations can be used only to obtain roughestimates in the range of separation typical for MEMSand NEMS. For example, for gold Eq. (2.16) can beapplied with a reasonable precision for d <∼ 5 nm [41]and Eq. (2.18) is working well for d ∼ 1 µm [42] (forlarger d the thermal correction becomes important). Inthe range of separation 1 nm < d < 1 µm it is preferableto use full Eq. (2.11).

4. Convenient representation

Concluding the discussion of the Lifshitz formula let uspresent a representation convenient for numerical evalu-ation of the force. Introducing in Eq. (2.11) new vari-ables x and t according to the relations x = 2|k0|d andtx = ζ/ζch the force can be presented in the form

F (d) =hc

32π2d4

∑

ν=s,p

1∫

0

dt

∞∫

0

dxx3 rν1rν2e

−x

1− rν1rν2e

−x. (2.19)

The Fresnel reflection coefficients are

rsi =1−

√

1 + t2 (εi − ε0)

1 +√

1 + t2 (εi − ε0),

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rpi =εi − ε0

√

1 + t2 (εi − ε0)

εi + ε0√

1 + t2 (εi − ε0). (2.20)

Here all the dielectric functions are functions of theimaginary frequency expressed via t and x as εi(iζ) =εi(iζchtx). The convenience of this representation is re-lated to simplicity of the integration range that is a stripeof unit width and effectively finite length.

The reflection coefficients (2.20) are presented for thickslabs made of material i. If the body i consists of dif-ferent layers, then the reflection coefficients have to begeneralized. For this case the coefficients can be foundin Refs. [43, 44].

D. Proximity force approximation

As was already mentioned, the force between two bod-ies can be presented in an explicit form only for paral-lel plates. While the stress tensor approach is valid foran arbitrary geometry of the bodies, the Green functioncannot be found in a closed form. Recently numericalmethods were developed [34] to evaluate the stress ten-sor for arbitrary geometries, but here a simple analyticalapproach is presented.

Under some conditions one can find approximate re-sults for a non-planar geometry. This approximation,called the proximity force approximation (PFA), was ap-plied for the first time for calculation of the dispersionforces by Derjaguin [45]. It was discussed in detail for ap-plications in nuclear physics [46], and it was derived forbodies of arbitrary shapes in colloidal science [47]. Theapproximation can be applied when the smallest sepa-ration between bodies d is much smaller than a typicalcurvature R of the bodies [48], d ≪ R. If this conditionis fulfilled, locally one can change the curved surfacesby flat patches, calculate the force between the opposedpatches as between parallel plates, and take the sum overall patches. The scheme of the procedure is presented inFig. 2.

The size of patches must be much larger than d. Ifthis size is still small in comparison with R, then onecan change the sum over the patches by the integral.The precision of the PFA is of the order of d/R and themain problem of this approximation is that the precisionis poorly controlled. The PFA can be also applied toother nonadditive forces. For example, one can apply itto the electrostatic force, for which the answers for someproblems are known precisely [49].

Let us consider as an illustrative example the electro-static force between a cylinder parallel to a plate. Theforce (per unit area) between two plates in vacuum is

F plel (d) =

V 20

8πd2, (2.21)

where V0 is the potential difference between the plates.The local distance between the cylinder and the plateis d(x) = R + d − (R2 − x2)1/2, where R is the radius

dA

d

FIG. 2: Scheme of application of the proximity force approx-imation.

of the cylinder and x is the horizontal coordinate in theplane of Fig. 2. The local distance can be approximatedas d(x) ≈ d+ x2/2R. The area of rectangular patches isdA = Ldx, where L is the length of the cylinder. Thenthe force between the cylinder and the plate within PFAwill be

F c,plel (d) ≈ V 2

0

8π

∞∫

−∞

Ldx

(d+ x2/2R)2 =

V 20 L

√R

8√2d3/2

, (2.22)

where we changed the integration limits to infinity withinthe precision d/R. The precise electrostatic result is [50]:

F c,plel (d) =

LV 20

∆ln−2

(

d+R−∆

d+R+∆

)

, (2.23)

where ∆ = [(d + R)2 − R2]1/2. Expanding (2.23) in aseries of d/R one finds that the relative correction to PFAresult (2.22) is 9/8(d/R). The latter is in agreement withthe expectations.The PFA was invented to calculate the dispersion

forces in the configurations of sphere-plate and crossedcylinders that were used to measure the forces. In bothcases the van der Waals and Casimir forces calculatedfrom the plate-plate interaction (2.1) and (2.2) are

FvdW =AHR

6d2, FC(d) =

π3R

360· hcd3

. (2.24)

For the sphere-plate configuration R is the radius of thesphere and for the crossed cylinders R =

√R1R2, where

R1 and R2 are the radii of cylinders.The PFA integration can be applied to the Lifshitz for-

mula (2.11) to find the force between sphere and plate.In this formula the dependence on the distance d is con-tained in the exponent e−d|k0|. This distance has to bechanged by the local distance d(x, y) and the formula hasto be integrated over the (infinite) area in x − y plane.After explicit integration one finds:

F sp(d) = − hR

2π

∑

ν=s,p

∞∫

0

dζ

∞∫

0

dqq ln

(

rν1rν2e

−2|k0|d

1− rν1rν2e

−2|k0|d

)

(2.25)This equation is usually considered as the Lifshitz for-mula for the sphere-plate configuration at zero tempera-ture.

III. CRITICAL EXPERIMENTS

The importance of roughness on dispersion forces hasbeen fully recognized relatively recently. This recognition

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was ignited by a few critical experiments performed indifferent fields where these forces play important role. Inthis chapter we describe these experiments and reviewrelated papers.

A. Forces between corrugated bodies

In this section we consider the dispersion interactionwith intentionally corrugated bodies. It differs from ran-dom surface roughness but firmly establishes nonadditiv-ity of dispersion forces that is used further for randomroughness.

The Casimir force is strongly geometry dependent[22, 51, 52] and it was already stressed that the dis-persion forces in general are not additive. One of thewell-investigated case is the interaction between a flatplate and a corrugated plate. The force between flat andsinusoidally corrugated plates made of ideal metal wascalculated for the first time in Ref. [52] taking into ac-count non-additivity effects (beyond the proximity forceapproximation). It was found that for corrugation wave-length λ much larger than the distance between bodiesλ ≫ d, the pairwise summation works well but in theopposite limit, λ ≪ d, the deviation from PFA is signifi-cant. The case of rectangular corrugation was consideredin [53]. Strong deviation from PFA was found over a widerange of λ, even though the surface was composed onlyof flat segments.

Strong deviation from PFA for rectangular corruga-tion one can understand qualitatively in the followingway. The Casimir force is associated with confined elec-tromagnetic modes with wavelength comparable to theseparation between the interacting objects. When λ ≪ dthese modes fail to penetrate into the trenches, renderingthe Casimir force on the corrugated surface equal to a flatone. On the other hand, for sufficiently deep trenches theadditivity predicts the effective area of interaction thatis equal to the fraction of solid surface.

Besides many theoretical arguments the nonadditiv-ity was tested only recently by direct experiments. Forthese experiments corrugated plates provide a convenientplatform. The fist experiment where the normal Casimirforce between aluminium-coated plate with small sinu-soidal corrugation and a large sphere was performed withthe use of the atomic force microscope technique [54].The force was measured in the range of separations be-tween bodies from 0.1 to 0.9 µm. While this measure-ment showed deviations from PFA, the interpretation ofthe deviation is still controversial [52, 55].

The first conclusive experiment has been done by Chanet al. in 2008 [56] (see also [57]). In this experiment oneof the interacting objects is a silicon plate with nanoscalehigh aspect ratio rectangular corrugations. The other ob-ject was a gold-coated glass ball with a radius of 50 µm.Two identical balls are stacked and glued onto a microme-chanical torsional oscillator [58, 96] that is often used tomeasure the Casimir force. The oscillator consisted of

FIG. 3: Detail of the experiment [56]. (a) Cross section ofrectangular trenches in silicon with a period of 400 nm anddepth of 0.98µm. (b) Top view of the trenches. (c) Scheme ofthe experimental setup including the torsional oscillator, goldcovered spheres, and silicon trench array. (d) Measurementscheme with electrical connections. (Courtesy H. B. Chan)

a 3.5 µm thick, 500 µm square silicon plate suspendedby two torsional rods. Some details of the experimentalconfiguration are shown in Fig. 3.Three plates were used for measurements: one with a

corrugation period of 400 nm, one with a period of 1µm,and one with a flat surface. The distance between the balland the plate was varied with a piezoelectric actuator.The resonance frequency of the torsional oscillator waschanged in response to the attraction between the balland the plate. This frequency change is proportional tothe gradient of the force between bodies.It was found that for the flat sample the measured force

is in a reasonable agreement with the force predicted the-oretically. The latter was calculated using the Lifshitzformula for sphere-plate (2.25) with the tabulated opti-cal data [60] for used materials (Au and Si). Note thatroughness of gold layer on the sphere can be neglectedbecause it was small in comparison with the distance be-tween bodies. For the sample with a corrugation periodof λ = 400 nm the measured gradient of the force F ′(d)is shown in Fig.4a. The observed value is compared withthe force gradient predicted by PFA that is pF ′

flat(d),where p = 0.5 is the fraction of solid surface on the sam-ple and Fflat(d) is the force for the flat sample. One cansee that the measured force gradient is larger than thatpredicted assuming the additivity. Figure 4b shows theparameter

ρ(d) =F ′(d)

pF ′flat(d)

. (3.1)

Deviation of this parameter from 1 characterizes the ef-

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fect of nonadditivity. One can see that this deviationis more than 20%, however, it is less than the value pre-dicted theoretically [53] for perfect reflectors. It was con-cluded [56] that both effects nonadditivity and finite con-ductivity of the materials play role.

In the PFA picture, the total interaction is a sum oftwo contributions: interaction with the top surface ofthe corrugated plate (fraction p) and interaction with thebottom surface (fraction 1−p). In the experiment [56] thelatter contribution is negligible because the trenches aredeep (∼ 1µm) and interaction with the bottom surface issmall. In Ref. [61] the depth of trenches was comparablewith the distance between bodies and the contributionof the bottom surface played role. The measured forcewas compared with advanced calculations based on thescattering theory [34, 62, 63] that includes both geometryeffects and the optical properties of the material. In thisexperiment a reasonable agreement with the theory wasfound. Therefore, nonadditivity of dispersion forces isnow proven experimentally and is in a good agreementwith theoretical expectations.

Recently detailed investigation of interaction betweenmetallic gratings with the sizes on the level of 100nm anda sphere was performed [64]. Corrugations in a metalliclayer can behave differently from that in Si because col-lective surface modes such as surface plasmons can beexcited in metallic gratings. It is known that the sur-face plasmons affect the force in a non-trivia manner [65].Very significant deviations from the PFA predictions wereobserved, however, the interpretation of these deviationsis still unclear.

B. Stiction problem

Because of the very small sizes involved in micro andnanoelectromechanical systems (MEMS or NEMS), sur-face forces are dominant, and they can generate a se-vere problem resulting in malfunction some of the MEMSdevices or making fabrication impossible. The prob-lem is spontaneous stiction between separate elements ofMEMS devices. It is an important limitation in bringingMEMS to the broader market. The problem was identi-fied already in the 1990’s [16, 66, 67]. MEMS structuresare typically made by forming a layer of material on topof a sacrificial layer above another material with the fol-lowing wet etching of the sacrificial layer. For example,it is desirable to fabricate transducers with suspendedstructures that have a minimum gap distance and largearea. Drying after rinsing is the final fabrication stepthat can collapse such microstructures resulting in per-manent adherence. Schematically the situation is shownin Fig.5. Strong capillary forces pull the surfaces togetherbut when the liquid is dried out the surfaces can stuckpermanently due to presence of the dispersion forces. Ithappens if a restoring elastic force cannot overcome at-traction induced by the dispersion forces.

This strongly undesirable effect makes impossible fab-

FIG. 4: Results of the experiment [56]. (a) Measured gradientof the Casimir force (dots). The solid curve is the theoreti-cal prediction assuming additivity but taking into accountfinite conductivity of the material. (b) Parameter ρ definedby Eq. (3.1). Experimental data for λ = 400 nm are shownby solid squares and for λ = 1 µm are shown by open circles.The theoretical predictions are shown by the dashed and solidlines for the first and second case, respectively. (Courtesy H.B. Chan).

rication of many MEMS devices or restricts their func-tioning. The latter happens if two surfaces approach tooclose to each other during the device operation. Thereason can be related to surface forces (capillary, electro-static) or inertial forces (shock, rapid air flow). Due tojump to contact the surfaces stuck permanently and thedevice cannot operate anymore. Figure 6 shows failuresof realistic MEMS devices. It was realized that coatingmaterials, surface roughness, and environmental agingcan influence the autoadhesion.

The problem of adhesion between macromachined sur-faces was tackled experimentally using microfabricatedcantilevers hanging above a substrate [17, 69–71]. Atheory and experimental method for measuring surfaceenergy of cantilever beams has been proposed by Mas-trangelo and Hsu [72, 73]. This theory describes the roleof capillary forces in bringing beams into contact with

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1

2

3

FIG. 5: Schematic view of a membrane suspended above asubstrate. 1 is the substrate, 2 is liquid, and 3 is the mem-brane. Drying meniscus pulls the surfaces together. When theliquid is vaporized the membrane can stuck to the substrate ifthe restoring elastic force is smaller than the dispersion forcesacting on the membrane.

FIG. 6: Permanent stiction in MEMS devices [68]. The arrowsshow adhered elements. (a) Stiction of soft microcantileversto the substrate. (b) Microstructured elements in a microma-chined accelerometer after impact loading. (Courtesy Y. P.Zhao).

the substrate and determines critical beam lengths forbeam collapse. Adhesion of the dried cantilever beamis predicted by considering the elastic energy in the de-formed beam, which is attempting to pull the beam up offthe substrate, and the surface energy that is promotingcontinued adhesion.

The theory was refined further by de Boer andMichalske [69] who proposed and realized also a prac-

FIG. 7: Adhered cantilever [17]. (a) Schematic image of thecantilever adhered to the substrate. The crack length is s, theadhered length is t. Voltage can be be applied to the actuationpad to produce an electrostatic load. (b)-(e) Interferogramsof cantilever beams at a fixed actuation voltage but differentsurface roughnesses. (Courtesy M. P. de Boer).

tical method to measure the adhesion energy. They fab-ricated an array of cantilever beams made of polysilicon.The beams with a width of 20µm were separated from Sisubstrate by a gap of 1.8 µm and their length was variedfrom 10 to 500 µm. One set of samples was dried in airafter rinsing and the other set was treated with a molecu-lar coating of octadecyltrichlorosilane. The samples fromthe first set were hydrophilic and from the second set werehydrophobic.

Short beams were free but starting from some length(172 and 225µm for hydrophilic and hydrophobic, respec-tively) the beams were adhered. The adhered cantilevershad so-called S-shape shown in Fig. 7a. According tothe theory [69, 72] from the shape of the beams one candeduce the adhesion energy. An important parameterhere is the crack length s that is unattached length ofthe beam. The attached length t can be calculated ast = L− s, where L is the length of the beam. The shapeof adhered cantilevers were observed with the help of aninterferometer. Inteferograms similar to those in Fig.7b-ewere recorded and analyzed. It was shown that the shapeof S-shaped cantilevers is in good agreement with expec-tations and can be used for precise determination of theadhesion energy. Although the surface roughness in thisexperiment was not changed, it was stressed that it canplay a significant role in the apparent adhesion energy.

Controlled actuation of cantilevers was used [70, 71]instead of spontaneous adhesion to get more detailed in-formation on the surface forces acting in the adhesionarea (landing pad). For this purpose an actuation pad

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under each beam was fabricated [74]. The cantilever andlanding pad were at ground potential, while a large po-tential can be applied to the actuation pad (see Fig. 7a).Detailed theoretical description and measurement of thebeam shape under combined action of the electrostaticand adhesive forces was made in Ref. [71].

Chemical and capillary forces result in the strongestadhesion between surfaces [75] but these forces can beeliminated using coating with a low-surface-energy hy-drophobic molecular monolayer [76, 77]. However, thedispersion forces cannot be eliminated and they posea fundamental limit to the adhesion between microma-chined surfaces. Influence of surface roughness on theadhesion energy was investigated in a special experimentwith the actuated cantilevers [17]. The main purpose ofthis experiment was to clarify role of the van der Wallsforces in adhesion of micromachined surfaces.

Polysilicon cantilevers 1500 µm long and 30 µm wideseparated from the substrate by a gap of 1.9µm were usedin this experiment. To make the cantilevers freestand-ing and also to eliminate capillary forces at the inter-face, a hydrophobic monolayer coating of perfluorodecyl-trichlorosilane (FDTS, CF3(CF2)7(CH2)2SiCl3) was ap-plied with a water contact angle of 110◦ in a solvent-basedprocess [76]. Roughness of the lower layer of polysilicon,which defines the landing pad, was accomplished by ther-mal oxidation in dry O2 at 900 ◦C for increasing times.The rms surface roughness increases with the oxidationtime. This occurs because the polysilicon grains are ran-domly oriented, and dry oxidation proceeds at differentrates on the various orientations of silicon.

To perform the adhesion test freestanding cantileverswere brought into contact with the substrate by apply-ing a voltage to the actuation pad. Using phase-steppinginterferometry, the full deflection curve of the cantileverswas determined to nanometer-scale accuracy. The result-ing interferograms are shown in Fig. 7b-e for an appliedload of Vpad = 50 V. They qualitatively indicate a de-crease in adhesion for an increase in surface roughness.The adhesion energy was extracted from the beam shapeunder action of electrostatic and adhesive forces as de-scribed in [71].

The result is presented in Fig. 8 as a function of theaverage surface separation dave that was calculated us-ing actual surface topography (see [17] for details). Forrelatively smooth surfaces it is natural to assume thatthe adhesion energy per unit area Γ is defined by the dis-persion forces between parallel plates separated by thedistance dave [74]. This distance (later we call it dis-tance upon contact) is defined by one highest asperitywhere the bodies comes to a contact. The energy due tothe van der Waals force in the contact area is negligible incomparison with that in the non-contact area. This limitcase is shown by the blue curve. The other limit caseis realized for relatively rough surfaces when the disper-sion force over the non-contact area is negligible. In thiscase the adhesion energy is given by the van der Walls

FIG. 8: Adhesion energy as a function of averaged distancebetween rough bodies [17]. The blue curve shows the limitof smooth surfaces separated by the distance dave. The blackline gives the adhesion energy for the limit of single asperitycontact. The triangles show the energy calculated from theAFM images and the squares give the energy determined fromthe observed shapes of cantilevers. (Courtesy M. P. de Boer).

interaction in the area of contact. It is estimated as [13]

Γ =1

A

(

AHR

6d0

)

, (3.2)

where A is the nominal area of contact, R is the radiusof contacting asperity, and d0 = 0.2 nm is the cutoff sep-aration. This limit is shown in Fig. 8 by the black line.

Influence of surface roughness on Γ was estimatedusing the actual surface topographies of the landing-pad and structural polysilicon layers. The topographieswere measured by tapping-mode atomic force microscopy(AFM). For each pixel in the AFM images the local dis-tance dloc was determined. The surface energy was cal-culated as the sum of dispersion energies over all pixelsopposing each other. These calculations are shown bythe green triangles in Fig. 8. At last the experimental re-sults for Γ determined from the shape of cantilevers arepresented by brown squares.

From Fig. 8 one can conclude that the energy deter-mined experimentally and that estimated from the AFMimages are in a reasonable agreement with each other.However the limits of smooth surfaces and single asper-ity contact lie well below the experimental data. It meansthat the surface roughness gives a considerable contribu-tion to the adhesion energy and simple theoretical modelsstrongly underestimate Γ. The situation does not changefor a relative humidity up to 80%, therefore, capillarycondensation does not play a role.

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C. Forces measured with Surface Force Apparatus

Dispersion interaction across liquid medium is often ac-companied by the electrostatic interaction. Both of theseinteractions are nonadditive and for this reason the inclu-sion of roughness into consideration is a difficult problem.Despite the fact that most of natural surfaces are rough,in very few experiments the influence of surface rough-ness on the dispersion forces was reported.

Two main methods to measure the dispersion forcesin liquid environment are in use. One uses a surfaceforce apparatus (SFA) where the force is measured be-tween two macroscopically curved surfaces over a rela-tively large area of nominal contact. This method is ableto measure absolute normal distances with angstromeresolution. The second method uses AFM to measurethe forces. In this case the interaction area is muchsmaller and determination of the absolute separation be-tween bodies is a difficult problem. The second methodwe consider in the next section.

With the help of SFA the effect of surface roughness onthe contact mechanics was studied for various polymericsurfaces with application of external load [78]. The forcewas measured in the crossed-cylinder geometry with acurvature radius of R ≈ 2 cm. Experiments were con-ducted in atmosphere of dry nitrogen or with lubricantoil between surfaces. Randomly rough surfaces were pre-pared by spin coating of polymer films onto mica. Rough-ness was controlled by varying the temperature of thepolymer solution, the solvent composition, the rate ofevaporation, and the relative humidity when depositedonto a substrate. The rms roughness was varied from 3to 220 nm.

Normal force-distance curves measured on several dif-ferent randomly rough surfaces all show an extended ex-ponentially repulsive force at close distances. The repul-sion is dominated by asperities and their bulk elastic orplastic properties. At large separations, the forces wereless repulsive due to the attractive short-range van derWaals and/or solvation forces between the initially con-tacting asperities. The adhesion energy versus rough-ness showed no obvious correlation at a fixed preload.Moreover, the observed value of adhesion was very large,Γ ≈ 28 mJ/m2. For comparison, the value of Γ be-tween oxidized polysilicon surfaces [17] was smaller than10 µJ/m2. Large adhesion energy was explained by thesegmental entanglements of the polymers, which is cor-related with the real contact area [79]. The results ofthe discussed paper are consistent with the old state-ment [80] that the real contact area is proportional tothe applied load and with the later works [81, 82] on theeffect of roughness and material stiffness on the adhesionof solids.

Different method to prepare rough surfaces was usedin Ref. [83]. The rough surfaces were polymer repli-cas of substrates of different roughness and composition.The replicas were fabricated following a technique sim-ilar to microcontact printing [84], using a UV-curable

polyurethane. This methods made it possible to keep themechanical properties and surface chemistry of the sub-strates constant, while varying the roughness. The fric-tion between surfaces prepared in this way and smoothbare mica was measured in dry air or in the presence oflubricating oil using the SFA.

The experiments described in [78, 83] are related tothe main point of our interest but do not tell much onthe roughness dependence of the dispersion forces. Thisis because the adhesion investigated in these papers wasnot dominated by the dispersion forces as, for example,in [17] but was defined by much stronger forces relatedto the segmental entanglements of the polymers.

There is lack of unambiguous experimental data oninteraction forces between rough surfaces in electrolytesolutions. The first surface force measurements acrossa liquid medium between a rough metal and a ceramicsurface with a control of the electrochemical potentialwas performed in Ref. [85]. These experiments revealhow increasing level of surface roughness and dissimi-larity between the potentials of the interacting surfacesinfluence the strength and range of electric double layer,dispersion forces, hydration, and steric forces and howthis contributes to deviations from DLVO theory (otherthan roughness deviations are reviewed in [15]). Rough-ness and hydration effects [86] are known to influencecolloidal forces, adhesion, contact mechanics, friction andwear [87, 88].

An electrochemical surface force apparatus (EC-SFA)was used to control interaction forces between dissimi-lar surfaces in situ in an electrolyte [89]. This apparatusis well suited to study the forces between rough elec-trodes. It has combined capabilities to simultaneouslycontrol the electrode surface potential, measure interac-tion forces, and visualize the contact mechanics. Thesystem used for measurements in Ref. [85] is illustratedin Fig. 9. The forces were measured between gold elec-trode with varied surface roughness and a molecularlysmooth self-assembled amino-silane monolayer on micain a 1.0 mM aqueous solution of HNO3. The surface po-tential of the gold electrode was varied in situ. Threesamples with different roughness were used. One samplewas atomically smooth gold film (Au-1) with rms rough-ness w = 3 A prepared by template stripping from micasurface [90]. The second sample (Au-2) was preparedby physical vapor deposition (PVD) and was moderatelyrough (w = 12 A). The third sample (Au-3) was furtherroughened by electrochemical treatment of the depositedAu film (w = 17 A). Although the third sample onlyslightly rougher compared to the second one, it has adistinctly different surface morphology.

Figure 10 shows the force-distance profiles measuredfor positive electrode potentials (∆U > 0), where theelectrode and SAM surfaces are like-charged (the aminegroups in SAM are fully protonated charging the surfacepositively). The results show that interaction forces mea-sured for atomically smooth gold surfaces are fitted verywell by DLVO-theory using constant potential boundary

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FIG. 9: Scheme of the experiment with the electrochemicalSFA [85]. The electrochemical three-electrode cell consists ofthe gold working electrodes (WE), a platinum counter elec-trode (CE), and an Ag|AgCl reference electrode. The WE is aglass sphere or cylinder covered with gold. Interaction of theWE is investigated with the mica sheet covered with amino-silane monolayer. In this setup an external electrochemicalpotential is applied to the gold electrode and force-distancecurves are measured. At the potential Upzc (the potentialof zero charge) the effective surface potential of gold is zero.(Courtesy J. N. Israelachvili).

conditions down to separations of around 8 nm. Below8nm, the Au-1 electrode shows a jump in to about 4nm,where the appearance of short ranged hydration forcesoverpowers the attractive contribution from the disper-sion forces. The solid line describing the fit for Au-1data includes DLVO interactions and an additional termtaking into account repulsions due to hydration forces.This repulsive contribution agrees well with the litera-ture value [91] for specifically adsorbed nitrate ions atthe positively charged SAM surface.

The force profiles and the repulsive hard wall sepa-ration measured for the two rough surfaces are shiftedoutwards by approximately 3 nm and 3.5 nm for Au-2and Au-3, respectively. Moreover, an additional expo-nentially repulsive component is superimposed on theforce profile (see Fig. 10) for rough samples and a pro-nounced ’kink’ in the slope of the repulsive force can beseen. These effects represents an additional steric repul-sion force arising from the elastic deformation of the out-ermost asperities at the rough surfaces. These asperitiesextend out to up to 10− 15 nm.

It was found also that even small differences in theroughness morphology lead to observable changes in theforce profiles. Particularly, the differences in ”jump-in”behavior show that attractive dispersion forces are weak-

FIG. 10: Results of the experiment [85]. (a) Representativeforce-distance profiles measured during approach at externallyapplied positive potentials (∆U = U − Upzc). Differentlyrough gold electrodes are indicated as Au-1,2,3. The inset (b)shows the semi-log plot of the same data shown in (a). Thesolid lines corresponds to theoretical fits (normalized by theradius of curvature, R) using the theoretical model explainedin [85]. (Courtesy J. N. Israelachvili).

ened by the presence of surface roughness and dependon the height distribution and number density of thenano-scale asperities. Qualitatively similar effects wereobserved for opposite charged surfaces (∆U < 0).It has to be noted that in general the surface force ap-

parats are better suited for investigation of surface forcesat very small separations and assume application of sig-nificant load. In comparison with the force measurementswith AFM they investigate larger area and independentlymeasure the distance between bodies with high precision.On the other hand, the AFM method can give more de-tailed information about the force at larger separationwithout application of external load. This method is re-viewed in the next section.

D. Forces measured with Atomic Force Microscope

Precise measurement of the dispersion forces withAFM was performed for the first time by Mohideen andRoy in 1998 [92]. A typical scheme for AFM experimentsis shown in Fig.11. Instead of a sharp tip they attached asphere to the AFM cantilever to increase the magnitudeof the force. For this configuration the effective area ofinteraction is ∼ πRd, where R is the sphere radius. Forpractical reasons it was possible to use the sphere with aradius of 100 µm. The interaction area is much smallerthan that in SFA where R = 2 cm but with a soft can-tilever (spring constant k ≈ 0.02 N/m) it was possible tomeasure quite weak forces ∼ 10 pN. In this experiment

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FIG. 11: Scheme of the AFM experiment. The sphere isattached to the AFM cantilever which position is read outby a laser beam. The plate is mounted on a piezo allowingcontrole of the distance between bodies. Inset shows SEMimage of the sphere with magnified area shown as AFM image.

both the sphere and the plate were covered with 300 nmof aluminum and 20nm of Au/Pd alloy to prevent oxida-tion of Al. The distance between bodies was controlledwith a piezo.

The main problem of the AFM method is determi-nation of the absolute distance between the bodies. Itis reduced to determination of the parameter d0 that isthe absolute separation on contact. This parameter isusually extracted from the electrostatic force measuredat relatively large distances where the dispersion forcesare small. Significant progress in determination of d0was reached during later development [58, 93, 94] butthe errors in the force versus separation distance are stilldominated by the errors in d0.

In the first experiment [92] roughness of the bodieswas quite significant up to 35 nm on each surface. Asthe result the distance upon contact was determined asd0 = 120 ± 5 nm. In this situation roughness has togive a significant contribution to the force. It was esti-mated using the perturbation theory (see Sec. IVB) as50% at closest separations. In the following developmentof the method most of the experiments were performedusing Au [95] instead of Al to cover the dielectric bod-ies. This eliminated the problem of oxide formation andresulted in significant reduction of roughness applyingwell-developed deposition methods. In the most preciseexperiments [95–97] the roughness contribution to theforce was on the level of the experimental errors.

However, the precise experiments were performed atdistances of the order of 100 nm. In this case rms rough-ness in a few nanometers is not important but at smallerseparations it can play very significant role. Special in-vestigation of the roughness effect at smallest possibleseparations was undertaken in Ref. [26] (see some addi-tional details in [98]). Relatively stiff cantilevers havebeen used to reach as small separations as possible be-tween sphere (R = 100 µm) and plate before jump tocontact. The sphere was covered with 100 nm Au in an

FIG. 12: AFM images of deposited Au films on Si substrate.Successive images correspond to the film thickness 100, 200,400, 800, 1600 nm, scan size is 0.5 µm, all but the last on thesame color scale.

electron-beam evaporator. Silicon wafers were coated inthe same way by Au to different thicknesses between 100and 1600nm. All of these films have different rms rough-ness w and different feature size ξ (correlation length) asone can see Fig. 12. The value of w increases with thefilm thickness from 1.5 nm to 10.1 nm while ξ is between22 and 42 .

The force was measured in air in the same way asshown in Fig. 11. The cantilever spring constant k, theresidual contact potential Vc, and the distance upon con-tact d0 all three parameters were determined from theelectrostatic calibration similar to [95]. Much stiffer can-tilevers with k = 0.235N/m have been used. It was foundthat the parameter d0 correlates very well with the rough-ness of investigated surfaces. Nearly linear relation (seeinset in Fig. 13) was found d0 = C(w + wsph), wherewsph = 3.5 nm is the sphere roughness and the constantC ≈ 3.7. It is important to stress that the value of d0 isconsiderably larger than the rms roughness of interact-ing surfaces. At the smallest possible separations (justbefore jump to contact) the error in the force versus sep-aration d is about 10% and it is dominated by the errorin d0 that is ∆d0 ∼ 1 nm.

The results for the force vs distance are presented inFig. 13 for the films of different thicknesses. Thin films(100, 200, and 400 nm) are in reasonable agreement withthe theoretical expectations that take into account somedeviations of dielectric functions of deposited gold fromthe single crystal material and account for the roughnesscorrections using the perturbation theory (see Sec. IVB).For these films the force is well described by the powerlaw [99] F ∼ d−α, where α is between 2 and 3. However,the thick films show very different behavior. There is sig-nificant deviation from expected scaling. The theoreticalcurve (black) including the roughness correction is notable to describe the observed force. It seems like the per-turbation theory is still applicable even for the roughest

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FIG. 13: Casimir force measured for different rough surfaceson a log-log scale for various Au film thicknesses. The theo-retical curves for 100nm and 1600nm films are shown by solidlines. The inset shows the electrostatically calibrated valuesof d0 as a function of the rms roughness.

film 1600 nm. For this film w ≈ 10 nm and the minimaldistance is d = d0 ≈ 50 nm. The relative perturbationcorrection is estimated as 6(w/d0)

2 ≈ 0.24. It is muchsmaller than the observed deviation. Explanation of thisphenomenon is presented in Sec. IVC.

The force in the sphere-plate configuration was mea-sured with AFM also in intervening liquids. Munday andCapasso [100] measured the force between gold coveredbodies in ethanol. In polar liquids electrostatic calibra-tion of the system (determination of k, Vc and d0) is in-appropriate due to strong reduction of the force. Insteadthe hydrodynamic drag force has been used for calibra-tion. In this experiment the force was measured for dis-tances as small as 35 nm and the effect of surface rough-ness clearly manifested itself in the data. It was foundthat a simple theory for roughness correction based onPRA (see Sec. IVB) overestimates the force at small sep-arations. Few information on the roughness morphologywas presented. More detailed analysis of the electrostaticforces has been done in [101] including the screening ef-fect due to dissolved NaI in ethanol. In this experimentthe measured force was found to be in a good agreementwith the theoretical prediction including the roughnesscorrection. Comparison of the dispersion forces betweengold-gold and gold-silica immersed in bromobenzene isdiscussed in [102]. It was observed that the force be-tween gold and silica becomes repulsive as predicted bythe Lifshitz theory. The roughness contributed to theforce was not discussed in detail.

E. What we learned from the experiments

The first conclusion that can be driven from the ex-periments with corrugated bodies is that the dispersionforces are not additive. In general, one cannot calculate

the force using the pairwise summation method. Insteadmuch more complicated approaches developed in the last15 years have to be used. The additivity is applicable ifthe corrugation wavelength is much larger than the dis-tance between bodies. It is not very clear to what extendwe can transfer this conclusions to random roughness.Both AFM experiments in air and experiments with

adhered cantilevers demonstrate that the dispersionforces deviate very significantly from expectations basedon simple theories. Typically the forces at small separa-tions are much larger than expected. The situation forinteraction in liquids is not that clear and needs moreanalysis.Experiments with SFA give very detailed information

about the forces for bodies in contact at significant ex-ternal load. When the separation distance increases theybecome less informative. An important conclusion thatcan be drawn from these experiments is significant influ-ence of roughness distribution on the dispersion forces.

IV. THEORETICAL DESCRIPTION OF

ROUGHNESS

The Lifshitz formula (2.19) does not take into accountinevitable roughness of interacting bodies. When rmsroughness of the bodies is much smaller than the sep-aration distance, then the roughness influence on theforce can be calculated using the perturbation theory (seeSec. IVB). However, when the distance becomes compa-rable with the roughness, the perturbation theory cannotbe applied. The problem of short distance separationsbetween rough bodies is one of the unresolved problems.In this chapter we give introduction into interaction oftwo rough plates or a sphere and a plate and describethe current state of the problem.

A. Description of a rough surface

1. Characterization

Suppose there is a rough plate with a surface profilethat can be described by the function h(x, y), where xand y are the in-plane coordinates. An approximationfor this function provides, for example, an AFM scan ofthe surface. It gives the height hij at the pixel positionxi = ∆ · i and yj = ∆ · j, where i, j = 1, 2, ..., N and ∆ isthe pixel size related to the scan size as L = ∆·N . We candefine the mean plane of the rough plate as the averagedvalue of the function h(x, y): h = A−1

∫

dxdyh(x, y),where A is the area of the plate. This definition assumesthat the plate is infinite. In reality we have to deal witha scan of finite size, for which the mean plane is at

hav =1

N2

∑

i,j

h(xi, yj). (4.1)

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The difference h−hav, although small, is not zero and isa random function of the scan position on the plate [103].This difference becomes larger the smaller the scan sizeis. Keeping in mind this point, which can be important insome situations, we can consider (4.1) as an approximatedefinition of the mean plane position.

An important characteristic of a rough surface is therms roughness w, which is given as

w =1

N2

∑

i,j

[h(xi, yj)− hav]2. (4.2)

It has the meaning of the surface width. More detailedinformation on the rough surface can be extracted fromthe height-difference correlation function defined for aninfinite surface as

g(R) =1

A

∫

dxdy [h(r+R)− h(r)]2, (4.3)

where r = (x, y) and R = r′ − r.A wide variety of surfaces, as for example, deposited

thin films far from equilibrium, exhibit the so called self-affine roughness which is characterized besides the rmsroughness amplitude w by the lateral correlation lengthξ (indicating the average lateral feature size), and theroughness exponent 0 < H < 1 [104–106]. Small values ofH ∼ 0 corresponds to jagged surfaces, while large valuesof H ∼ 1 to a smooth hill valley morphology. For aspecial case of the self-affine rough surfaces g(R) scalesas

g(R) =

{

R2H , R ≪ ξ,2w2, R ≫ ξ.

(4.4)

The parameters w, ξ and H can be determined fromthe measured height-difference correlation function g(R).This function can be extracted approximately from theAFM scans of the surface. An example is shown inFig. 14. The top image is an AFM scan of 100 nm thickAu film. The bottom image shows the correlation func-tion extracted from the image and its approximation withEq.(4.4). From Eq.(4.2) it is found w ≈ 1.5nm and fromthe power-law behavior of the correlation function it isfound ξ ≈ 22.4 nm and H ≈ 0.9. The latter value showsthat deposited gold is described by smooth hills and val-leys.

To find roughness correction to the force one has toknow (see Sec. IVB) the spectral density σ(k) of theheight-height correlation function C(R). The latter isrelated to g(R) as g(R) = 2w2−C(R). An analytic formof the spectral density for a self-affine surface is given by[107]

σ(k) =CHw2ξ2

(1 + k2ξ2)1+H

, C =2

1− (1 + k2cξ2)

−H. (4.5)

Here C is a normalization constant [106, 107] and kc =2π/Lc is the cutoff wavenumber.

FIG. 14: (a) AFM image of a deposited gold film 100nm thick.The image size is 1 × 1 µm2. (b) The correlation functionextracted from the image (red circles). Blue solid line is thelinear fit (on log-log scale) that allows determination of bothparameters ξ and H.

2. Distance upon contact and height distribution

The absolute distance separating two bodies is a pa-rameter of principal importance for determination of thedispersion forces. In fact, when the bodies are broughtinto gentle contact they are still separated by some dis-tance d0, which we call the distance upon contact due tosurface roughness. This parameter has a special signifi-cance for weak adhesion, which is mainly due to van derWaals forces across an extensive noncontact area [17]. Inthe modern precise measurements of the dispersion forces[8, 94–96] d0 is the main source of errors. The distanceupon contact is usually considerably larger than the rmsroughness because it is defined by the highest asperities.It is important to clearly understand the origin of d0, itsdependence on the lateral size L of involved surfaces, andpossible uncertainties in its value [108].

Consider two parallel rough plates facing each other.Each surface can be described by the roughness profilehi(x, y) (i = 1, 2 for the plate 1 or 2) as shown in Fig.15a.The averaged value over a large area is assumed to bezero 〈hi(x, y)〉 = 0. Then the local distance between the

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d0

combined rough

smooth

d0

(a) h1(x,y) h

2(x,y)

(b) h1(x,y)+h

2(x,y)

FIG. 15: Contact of two rough surfaces. (a) Two rough platesin contact. Roughness of each plate, hi(x, y), is counted fromthe mean plane shown by the dashed lines. The distancebetween bodies is the distance between these mean planes.(b) The interaction between two rough plates is equivalentto the interaction between a smooth plate and a rough platewith the roughness given by the combined profile h(x, y). Thedistance upon contact, d0, is the maximal peak within a givenarea.

plates is

d(x, y) = d− h1(x, y)− h2(x, y). (4.6)

This distance depends on the combined rough profileh(x, y) = h1(x, y) + h2(x, y). Indeed, the averaged lo-cal distance has to give 〈d(x, y)〉 = d. We can define thedistance upon contact d0 as the largest distance d = d0,for which d(x, y) becomes zero.

It is well known from contact mechanics [109] that thecontact of two elastic rough plates is equivalent to thecontact of a rough hard plate and an elastic flat platewith an effective Young’s modulus E and a Poisson ratioν. Here we analyze the contact in the limit of zero loadwhen both bodies can be considered as hard. Then wecan consider an equivalent configuration of a flat platefacing a rough plate with the roughness given by thecombined profile h(x, y) as shown in Fig. 15b. In thisconfiguration d0 is defined as the maximal peak of thecombined profile.

Consider a combined image with the size L. It is as-sumed that this size is much larger than the correlationlength ξ presenting typical feature size in the image. Itmeans that the image area can be divided into a largenumber N2 = L2/ξ2 of cells. The height of each cell (as-perity) can be considered as a random variable h. Theprobability to find h smaller than some value z can bepresented in a general form

P (z) = 1− e−φ(z), (4.7)

where the ”phase” φ(z) is a nonnegative and nondecreas-ing function of z. Note that (4.7) is just a convenient wayto represent the data: instead of the cumulative distribu-tions P (z) we are using the ”phase” φ(z), which carriesmore information about the tails of the distribution.

For a given asperity the probability to find its heightabove d0 is 1 − P (d0), then within the area of nominalcontact one asperity will be higher than d0 if

e−φ(d0)(

L2/ξ2)

= 1 or φ(d0) = ln(

L2/ξ2)

. (4.8)

This condition can be considered as an equation for theheight of the height asperity due to a sharp exponentialbehavior of the distribution tail. To solve (4.8) we haveto know the function φ(z), which can be found from theroughness topography.The cumulative distribution P (z) can be extracted

from combined images by counting pixels with the heightbelow z. The probability density function f(z) can be ex-pressed as f(z) = dP/dz. The ”phase” is calculated asφ(z) = − ln(1−P ). The results are presented in Fig.16b.The procedure of solving Eq.(4.8) is shown schematicallyby thin dashed lines, and the solution itself is the red solidcurve in Fig.16a. The blue dots with the error bars showthe values of d0 determined directly from the combinedAFM images taken on different scales L. The errorbarsdemonstrate variation of d0 determined from different lo-cal areas of the size L2 of homogeneous gold films [108].Note that d0 is considerably larger than w for all thescales L ≫ ξ.It has to be mentioned that the normal distribution

fails to describe the data for Au films at large z. Otherknown distributions cannot satisfactory describe the datafor all z. Asymptotically at large |z| the data can bereasonably well fit with the generalized extreme valueGumbel distributions (green lines in Fig. 16b) [110]:

lnφ(z) =

{

−αz, z → −∞βz, z → ∞ (4.9)

It has to be stressed that the extreme value statisticsdescribes quite well the tails of the distribution of allgold films investigated in [108].

B. Perturbative correction to the force

Suppose that the force per unit area between two flatplates is F (d). If the rms roughness w of the combinedprofile h(x, y) is small, w ≪ d, but the correlation lengthis large, ξ ≫ d, we can use the proximity force approxi-mation and calculate the roughness correction additively.Averaging the force for local distances d(x, y) and ex-panding it in power series in w/d one can present theenergy between rough plates as

F(d) = 〈F (d(x, y))〉 ≈ F (d) +F ′′

2

⟨

h2⟩

, (4.10)

where⟨

h2⟩

= w2 = w21 + w2

2 is the square of the com-bined rms roughness and F is the force between tworough plates. Equation (4.10) defines the PFA rough-ness correction δF(d) = F ′′(d)w2/2. This correction wasused in all early studies to estimate the roughness effect.

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2 4 6 8 10 1215

20

25

30

35

40

45

L [µm]

d0 [

nm

]

films: 100 nm above 400 nm

ξ=26 nm

(a)

−20 −10 0 10 20 30 40

10−6

10−4

10−2

100

102

z [nm]

φ(z

)

−10 0 10 20 30 40

0.2

0.4

0.6

0.8

z [nm]

P(z

)

ln(L2/ξ

2)

d0(L)(b)

FIG. 16: (a) Distance upon contact as a function of the lengthscale. Dots with the error bars are the values calculated fromthe AFM images. The solid curve is the theoretical expec-tation according to Eq. (4.8). (b) Statistics of the surfaceroughness. Four 10 × 10 µm2 images were used. The maingraph shows the ”phase” as a function of z. The green solidlines show the best fits at large positive and negative z. Thered dashed lines demonstrate the solution of Eq. (4.8). Theinset shows the cumulative distribution P (z).

Genet et al. [23] noted that in most experimental con-figurations the condition ξ ≫ d is broken and PFA can-not be applied. In Refs. [24, 25] a theory was developed,which is not restricted by the condition ξ ≫ d. Withinthis theory the roughness correction is expressed via thespectral density of the rough surface σ(k) as

δF(d) =

∫

d2k

(2π)2ρ(k, d)σ(k), (4.11)

where ρ(k, d) is the roughness response function derivedin [25]. This function is quite complicated and we donot provide it here refereeing to the original paper. ThePFA result (4.10) is recovered from (4.11) in the limit ofsmall wavenumbers k → 0 when ρ(k, d) → F ′′(d)/2. Theroughness power spectrum is normalized by the condi-tion

∫

d2kσ(k)/(2π)2 = w2. The spectrum itself can beobtained from AFM scans and in the case of self-affinerough surfaces is given by Eq. (4.5).

Let us enumerate the conditions at which the PFA re-

sult given by Eq.(4.10) is valid. (i) The lateral dimensionsof the roughness ξ must be much smaller than the systemsize L, ξ ≪ L. This is usually the case in experiments.(ii) The rms roughness w must be small compared to theseparation distance, w ≪ d. This condition means thatroughness is treated as perturbative effect. (iii) The lat-eral roughness dimensions must be much larger than thevertical dimensions, w ≪ ξ [25]. The last two assump-tions are not always satisfied in the experiment.In the plate-plate configuration the force per unit area

can be calculated as F(d). For the sphere-plate config-uration, which is used in most of the experiments, theforce is calculated with the help of PFA as Fsp(d) =2πR

∫∞

dF(t)dt. In contrast with the roughness correc-

tion the latter relation is justified for d ≪ R, which holdstrue for most of the experimental configurations. We usethe sphere-plate configuration to illustrate the roughnesseffect. The deposited gold films can be considered as self-affine. The perturbative correction to the Casimir forcewas calculated for the sphere-plate geometry [26, 27, 111]using smoothest spheres with the parameters w = 1.8nm, ξ = 22 nm, and H = 0.9 [112]. The plate rough-ness was different in dependence on thickness of the de-posited Au films. We use the optical data for gold filmsmeasured in [113]. It was found that the PFA limit isquickly recovered for increasing correlation length. Devi-ations from PFA prediction for real films were found tobe about 1-5% in the range of distances d = 50−200 nm.Therefore, for real rough surfaces the scattering theory

gives a few percent correction to the force compared tothe PFA. This difference is difficult to measure. How-ever, at small separations both PFA and perturbationtheory fail since the rms roughness becomes comparablein size to the separation distance. It is important to cal-culate the roughness effect when d is comparable withw. At the moment there is no a theoretical instrumentto estimate the effect except a direct numerical analysissimilar to that used in [114] (see also [22] as a review ofthe method). It would be interesting to do a full numer-ical analysis for rough films in close proximity when theperturbation theory does not work. However, even thisapproach seems quite complicated for real rough surfaces.On the other hand, it is experimentally possible to go tosufficiently small distances as was discussed in Ch. III.A class of problems that can be solved using a relativelysimple semi-analytical approach is presented in the nextsection.

C. Beyond perturbation theory

Here we are going to present an approach that allowsus to calculate the roughness correction beyond pertur-bation theory. The approach is based on the experimen-tal fact that the contact of rough bodies happens at thedistance d0, which is considerably larger than the rmsroughness w. The distance upon contact depends on thelateral scale L but for L > 1 µm it is typically d0 > 3w.

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FIG. 17: AFM megascan of the 100 nm film (a) and thesphere (b). The insets show the highlighted areas at highermagnifications.

This fact was established for gold films in many papers[26, 58, 93–95, 97], where the Casimir force was mea-sured, and checked directly by analysis of the AFM im-ages [108]. However, at the moment there are no datafor other materials and restrictions of the method arenot clear yet.

The general idea of the method is very simple [27]. Iftwo bodies are gently touch they are still separated bythe distance d0. If the area of nominal contact includesmany asperities, the real contact happens only in oneor a few highest asperities. The contribution of thesehighest peaks to the force has to be calculated precisely.It is possible to do because the distance between highpeaks is large. Normal asperities with the height ∼ wcan still be calculated using the perturbation theory. Letus consider now the details of the approach.

1. Roughness statistics of Au films

Detailed analysis of the roughness statistics of differentgold films was performed by van Zwol et al. [108]. Thefilms were already described in Sec. IIID. These films

were optically characterized [113] in a wide frequencyrange from 0.03 < ω < 9 eV and were used to mea-sure the roughness contribution to the dispersion forces[26]. The AFM images of the films presented in Fig. 12cannot be used to extract detailed information on thefilm roughness. This is because the scan size L = 0.5µmis too small to draw conclusions on the statistics fromone image. Practically it means that one has to analyzea large number of images to get reliable data but eventhis is difficult because the film is not absolutely homoge-neous: images taken from different local places can poorcorrelate.This practical problem was resolved [108] by collect-

ing so-called megascans for each film. These are highresolution AFM scans with the size up to 40 × 40 µm2

and a lateral resolution of 4096× 4096 pixels. The max-imal area that was possible to scan on the sphere was8×8µm2 (2048×2048 pixels). All images were flattenedwith linear filtering; for the sphere the parabolic filteringwas used to exclude the effect of curvature. The imagesof 100 nm film and the sphere are shown in Fig. 17(a)and (b), respectively. Each image can be zoomed a fewtimes due to high resolution. The detailed statistics iscollected just from one image. In this way we can besure that nonhomogeneity of the film plays a minimalrole.From all the images the cumulative distribution P (z)

was determined. Using this distribution the ”phase” φ(z)has been calculated according to Eq. (4.7) and the prob-ability density function was calculated as f(z) = dP/dz.In Fig. 18 the result is presented for the image of theroughest 1600 nm thick Au film. As was already men-tioned logarithm of the ”phase” at large positive andnegative z is well described by the linear behavior thatis a signature of the extreme value statistics. Strong de-viation of the tails from the normal distribution is wellvisible in the inset. The real film has much more highpeaks than predicted by the normal distribution of theasperity heights. The latter is important to calculate theroughness effect beyond perturbation theory.

2. Roughness contribution

We can imagine a rough surface as a large number ofasperities with the height ∼ w and lateral size ξ, andoccasional high peaks and deep pits. These peaks arehigh in the sense that their height is considerably largerthan w, say > 3w. The situation can be visualized as alawn covered with grass and occasional high trees stand-ing here and there. For this reason the model describedin this subsection sometimes is called ”lawn and threes”model.The interacting surfaces are separated by the distance

d > d0 > 3w. Normal asperities with the height ∼ wcan be treated perturbatively because even at contact(w/d)2 ≪ 1 and the perturbation theory can be applied.For the high peaks the local distance between the bodies

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−40 −20 0 20 40 60 80−6

−4

−2

0

2

z (nm)

log

10φ(z

)

−40 0 40 80−6

−4

−2

z (nm)

log

10f(

z)

w=10. 1 nm

ξ=42 nm

FIG. 18: The phase as a function of z for a 1600 nm goldfilm. The circles are the actual data extracted from the AFMimage via P (z). At large positive and large negative heightslog10 φ(z) is well fitted by linear functions of z as it is shownby the straight lines. The curved line is a polynomial fitat intermediate z. The inset shows the probability densityfunction f(z). It demonstrates a significant deviation fromthe normal distribution.

becomes small and the perturbation theory fails. Be-cause the high peaks are rare the average distance be-tween them l is large. If l ≫ d then we can calculatethe contribution of the high peaks independently on eachother. Interaction of one high peak with a flat surface canbe taken into account precisely using developed numeri-cal or sometimes analytical methods [22].

The number of asperities N with a height d1 > 3wand lateral size ξ on the area L2 is given by the equationsimilar to Eq. (4.8)

N =L2

ξ2e−φ(d1). (4.12)

The average distance between these peaks is

l =L√N

= ξeφ(d1)/2. (4.13)

One can choose the parameter d1 in such a way thatl ≫ d. Its actual value lies in the interval 3w < d1 < d0.If it happens that d1 > d0 than the perturbation theorycan be applied for all asperities and there is no need toseparate high asperities.

Suppose for a moment that the PFA can be appliedto any roughness topography. Then the force betweenrough surfaces can be calculated as

F(d) =

∫ d0

d1

. . .+

∫ −d′

1

−d′

0

. . .+

∫ d1

−d′

1

dzf(z)F (d−z), (4.14)

where F (d) is the force between flat surfaces, f(z) =dP/dz is the probability density function determined

from the images of rough surfaces. The first integral isthe contribution of high peaks, the second one accountsfor the deep pits with the corresponding parameters d′1and d′0, and the last term is the contribution of normalroughness.We know that application of PFA to normal asperities

is wrong but it is possible to apply the perturbation the-ory to the last term that has to be generalized as (onlythe roughness correction)

δFPT (d) =F ′′(d)

2!

∫

d2k

(2π)2ρ(k, d)σ(k). (4.15)

Here σ(k) is the Fourier spectrum of the roughness mea-sured or calculated according Eq. (4.5). The functionρ(k, d) measures the deviation from the PFA and it isthe same function introduced in Eq. (4.10). One can usefor ρ(k, d) the expressions found in [24, 25].The second term in (4.14) can be neglected because the

force for deep pits is small and their number is not large.On the contrary, the first term in (4.14) is important andthe PFA is applicable to this term because the high peaksdo not influence each other. The final result for the forcebetween rough surfaces can be written as

F(d) = F (d) + δFPT (d) + δFPFA(d), (4.16)

where δFPFA is the contribution of the high peaks givenby the relation

δFPFA(d) =

∫ d0

d1

dzf(z)×[

F (d− z)− F (d) + F ′(d)z − F ′′

2!z2]

. (4.17)

There last three terms in the square brackets excludethe contribution that was already included in δFPT (d).Equations (4.16) and (4.17) give us the possibility to cal-culate the effect of roughness beyond the perturbationtheory.

3. Comparison with the experiments

We have seen already in Sec.III B and IIID that experi-ments demonstrate significant deviation of the roughnesscontribution to the force (or adhesion energy) from ex-pectations. The described above model that is going be-yond the perturbation theory have a chance to settle thecontradiction. It is difficult to make predictions for theexperiment [17], where the adhesion energy between oxi-dized polysilicon surfaces was measured. This is becausethe detailed roughness of polysilicon surfaces was not col-lected. On the other hand, all the necessary informationfor the experiment [26] is available. In this experimentvery strong deviation from the expected scaling of theforce with distance was found. This is an optimal situa-tion to check a new model. It was done for the first timeby Broer et al. [27].

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50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

d (nm)

Forc

e (

nN

)

50 60 70 80 90 1000

0.1

0.2

0.3

0.4

d (nm)

Forc

e (

nN

) 1

2

3

FIG. 19: The force between a Au-covered sphere (R = 50µm)and a plate (1600 nm thick Au). The circles (green) are theexperimental data. The vertical and horizontal bars showthe experimental errors for a few points. The solid (blue)line is the result of the model. Naive application of the PFAto the force between rough bodies is shown by the dashed(red) curve. The inset shows different contributions to theforce. 1 (black) is the force F (d) between smooth surfaces.2 (red) is the contribution of the high peaks. 3 (blue) is theperturbation theory correction. The sum of all three curvesgives the solid line in the main panel.

Gold films can be considered as self-affine. The spec-tral density of roughness for such films is given byEq. (4.5) with the parameters w, ξ, and H determineddirectly from the images. The dielectric functions of thefilms needed for calculations were measured in [113]. Thisfunctions were used to evaluate the first term in Eq.(4.16)that is given by the Lifshitz formula (Eq. (2.25) forthe sphere-plate configuration) and the function ρ(k, d)needed for the second term. The last term responsible forthe contribution of the high peaks is given by Eq. (4.17).This is the term that has to be responsible for the strongchange of the force scaling. The probability density f(z)is extracted from the AFM images of the films. In thesimplest approach it is assumed that the force F (d) en-tering under the integral in (4.17) is also given by (2.25).This assumption means that the high asperities are pre-sented as pillars with the cross section ξ × ξ and a flatcap.

This assumption is justified by the following consider-ation. A few high peaks can give significant contributionif the local distance between the peak and the opposingsurface is very small. The actual condition is d−d0 ≪ ξ.If this condition is true, then the shape of the pillar cap isnot very important. It becomes important if d − d0 ∼ ξbut in this case the contribution of high peaks to theforce is small.

The result of calculations and the experimental datafrom [26] are shown in Fig. 19 for the roughest film

1600 nm, but similar situation is observed for the films800 nm and 1200 nm thick. It is obvious that the modelis able to explain the behavior of the force at small sepa-rations. This is due to the contribution of high peaks asthe inset demonstrates (curve 2). It increases very sharpwith the distance. On the other hand, the contributionof normal asperities is never large as the curve 3 shows.If one would naively apply the additivity to calculate

the effect of roughness in the dispersion forces (we callthe approach ”naive PFA”), the result would correspondto the dashed line in Fig.19. For this naive PFA the forceis given by the expression similar to (4.14)

F(d) =

∫ d0

−d′

0

dzf(z)F (d− z). (4.18)

At small distances this curve coincides with the exactmodel because the contribution of normal asperities issmall but high asperities can be included additively. Atlarger distances there is some deviations between thecurves but it is not large. This deviation is due to the per-turbation theory contribution (4.15), which never domi-nates. On can conclude that the reason for strong devi-ation of the force from normal scaling is found and thisreason is large contribution of high asperities which lo-cally approach the opposite surface very close.Influence of the shape of high peaks on the force was

analyzed in detail [111] in spite of arguments above thatit should not be significant. One has to note that theshape cannot be determined from AFM scans becausethe tip radius is comparable with the curvature radius ofa peak. The following model shapes were considered: arectangular pillar with a flat or spherical cap, an ellip-soid with half-axes d0 and ξ/2, and a cone with heightd0 and bottom diameter ξ. For a flat pillar the forcebetween a peak and a flat surface was calculated withthe Lifshitz formula (2.19). For a pillar with a sphericalcap the force was evaluated using sphere-plate interactionthat does not assume applicability of PFA [115]. For anellipsoid and cone the force was calculated numericallywith a finite-difference time-domain (FDTD) [116] pro-gram called Meep [117]. It was found that the all possibleshapes of peaks are nearly equivalent except of the cone.The conical shape predicts the force to be too small toexplain the experiment. One could expect it because thetip of the cone gives very small contribution to the force(small area). Due to the shape the force variation issmaller than 5% nearby the contact.

D. What we learned from the theory

In general roughness contribution to the dispersionforces is a complicated problem when the distance be-tween interacting bodies become comparable with therms roughness, d ∼ w. The difficulty is related to thenonadditivity of the dispersion forces. Similar problemappears for the electrostatic interaction between roughbodies.

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Naive PFA approach neglects the nonadditivity prob-lem. The force is calculated as the sum of indepen-dent forces between small parallel patches. For randomlyrough surfaces the force together with the roughness con-tribution is given by Eq.(4.18). To use this expression onehas to know very detailed information about the prob-ability density function. The tails of the distributionare especially important. The probability density canbe extracted from AFM images of the interacting sur-faces. The probability to find a high peak on the roughsurface is very important because the high peaks can ap-proach locally close to the opposite surface and give largecontribution to the force. A strong disadvantage of thenaive PFA approach is that the precision of this methodcannot be controlled. It is a good approximation if thecorrelation length is much larger than the distance be-tween bodies, ξ ≫ d. Nevertheless, in many situationthe approximation works well even if this condition isbroken.

At large distances, d ≫ w, the roughness effect canbe treated perturbatively. A method to calculate theroughness correction in this situation was developed [24].It is based on the scattering theory and it can be appliedeven if the condition ξ ≫ d does not fulfilled. One hasto know the roughness power spectrum that can be de-termined from AFM images of the interacting surfaces.In this situation the roughness correction is treated as asmall correction to the Lifshitz formula that describes themain contribution to the force. This correction is givenby Eq. (4.15).

When the surfaces are close to contact the perturba-tive correction does not describe the situation as wasdemonstrated experimentally [17, 26]. However, the ra-tio (w/d)2 stays small even at contact because the min-imal distance between bodies is d0 > 3w and one wouldexpect that the perturbation theory is applicable. Theexplanation of the puzzle was given for gold films wherethe problem is clearly observed [26]. Careful analysis ofthese films [108] revealed significant deviations from thenormal distribution. The number of high peaks for goldfilms was much larger than predicted by the normal dis-tribution. This property gave an idea for the ”lawn andtrees” model [27]. In this model asperities with normalheight ∼ w are treated perturbatively, while contributionof high asperities > 3w can be calculated precisely. Be-cause the distance between high asperities is large theydo not influence each other and they can be treated ad-ditively. This approach successfully explained the exper-imental data. It also gives a hint why the proximity forceapproximation works better than it is expected: at smallseparations high peaks play more important role thannormal asperities, but for these peaks the PFA can beapplied.

V. CONCLUSIONS

The main purpose of this paper is to analyze the influ-ence of natural roughness on the dispersion interaction oftwo closely separated bodies. We did not discuss in de-tail loaded contact when elastic or plastic deformations ofsome asperities takes place (for recent analysis see [118]).A convenient way to investigate influence of roughness onthe dispersion forces is analysis of contactless interactionbetween bodies approaching the contact between highestasperities.The problem considered here is important for a wide

range of applications where one has to tune forces actingin air or in liquid environment between components ofdifferent micro or nanodevices. These devices have ar-eas large enough but gap small enough for these forcesto draw components together and even lock them per-manently. Roughness also influences adhesion, friction,wetting, and stiction.For non-experts in the dispersion forces we gave the

introduction of the forces between flat and gently curvedsurfaces. Special attention is paid on the physical un-derstanding of common origin of such forces as van derWaals, Casimir, and Casimir-Lifshitz. The physical basisof the Lifshitz theory [4] was briefly presented. This the-ory explains the dispersion forces as interaction betweenbodies due to electromagnetic field induced by currentfluctuations in the bodies. We provided a number ofhelpful relations to use in different practical situations.There are many experiments where the effect of rough-

ness was important but only in a few critical experimentsthis effect was investigated in detail. We reviewed theseexperiments in Chapter III. Nonadditivity of the disper-sion forces was proven experimentally using the interac-tion between corrugated plates and a sphere [56]. Thefact of nonadditivity is a serious obstacle in the devel-opment of the theory. After careful investigation of thestiction problem that results in malfunction of many mi-croelectromechanical devices it becomes clear that it ispossible to exclude strong adhesion forces like capillaryof chemical forces, but it is not possible to switch offthe dispersion interaction. Special investigation of adhe-sion due to this interaction revealed the significant roleof roughness [17]. The effect of roughness happened tobe considerably stronger than one would expect for in-teraction between flat surfaces.Influence of roughness was investigated also using the

surface force apparats. This method is better designedfor investigation of strong adhesion. It was demonstratedthat attractive dispersion forces depend on the detailsof the height distribution of the nano-scale asperities[85]. The forces measured with atomic force microscopeor with torsional oscillator also showed the dependenceon roughness. However, in most cases it was done atrather large distances when the roughness effect is small.Only in one experiment influence of roughness was in-vestigated specifically at the smallest possible distancebetween sphere and plate covered with deposited gold.

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In this experiment it was demonstrated that roughnesschanged significantly the expected scaling of the forcewith the distance [26]. Moreover, it seemed that the ob-served effect cannot be explained by any existed theoryuntil recently.

The state of the theory was reviewed is Chapter IV.First, we described the ways to characterize a rough sur-face with emphasis on the minimal distance between bod-ies. This distance d0 is an important parameter for in-teraction at smallest separations. For gold films d0 isconsiderably larger than rms roughness, d0 = (3 − 5)w,depending on the size of the interaction area [108]. Theroughness correction to the force can be expressed explic-itly if the distance between bodies is much larger than therms roughness, d ≫ w. In this case one can apply theperturbation theory using as a small parameter the ratiow/d. As input data one has to know also the spectraldensity of the rough surface. For a smooth roughnessprofile, when the correlation length ξ is large, ξ ≫ d, it ispossible to use additivity and the correction can be eas-ily found. In the opposite situation, which is quite oftenthe case, one has to use a complicated approach basedon the scattering theory. Nevertheless, the explicit resultfor the correction exists [24].

Experiments demonstrate that significant deviationsfrom theoretical expectations are observed for the pa-rameter (w/d)2 ∼ 0.01 where the perturbation theoryhas to work fine. To understand this phenomena care-ful analysis of the roughness statistics was undertaken fordeposited gold films. It was found that the probability tofind a high peak is considerably higher than it is expectedfrom the normal distribution [108]. High peaks and deeppits for gold films are described much better with the ex-treme value statistics. Despite that only a small area iscovered with high peaks, they can be important becausethese peaks locally can approach very close to the oppos-ing surface. Using these observations the so-called ”lawnand trees” model was proposed [27] to calculate the forceclose to contact. In this model normal asperities (∼ w)are treated perturbatively, while the contribution of highasperities is calculated precisely. The latter is possiblebecause the distance between high peaks is large and theycan be taken into account additively. This model repro-duced successfully the experimental data for rough goldfilms. It also explained why the naive approach basedon the proximity force approximation (additivity) oftenworks better than it is expected.

Serious investigation of roughness on the dispersionforces has started relatively recently. Many questions are

still open. Here we would like to enumerate some impor-tant problems that have to be analyzed. Few experimentscharacterize the influence of roughness on the dispersionforces in liquid environment. The surface force apparatusis able to give valuable information but it is better de-signed for very small separations and flat surfaces. AFMexperiments in dielectric liquids and electrolytes couldfill this gap. There was no special attention to roughnesscontributions in the existing AFM experiments in liquids.It was demonstrated that the ”lawn and trees” model

was able to reproduce the experimental data for goldfilms. How general the model is? At this moment it is dif-ficult to answer the question because careful investigationof the roughness of different materials is needed. It has tobe stressed that a few AFM images with the size ∼ 1µmare not sufficient to collect the data on roughness. Dif-ferent materials especially those used in microtechnologyhave to be carefully characterized. It is clear that thereis a class of materials that has roughness similar to thatof gold. It is obvious also that some materials cannot bedescribed in this way, for example, materials with ter-races. Is it possible to explain roughness effect for someintermediate situations? If roughness can be describedby the normal distribution, still real contact happens athigh peaks. Will the model work in this case?

In any case evaluation of the roughness contributionseems not possible in the situation when the distance iscomparable with the rms roughness, w ∼ d. The onlypossible approach here is numerical. This approach hasbeen developed very successfully [22]. It was applied tocalculate the force between many geometrical configura-tions but the roughness effect never has been tackled.These calculations combined with careful characteriza-tion of the surfaces and model predictions could give solidground for the roughness problem.

Acknowledgments

VBS is grateful to L. B. Boinovich for the idea to writethis review. Both authors thanks P. van Zwol and W.Broer for collaboration in experimental and theoreticalaspects of the roughness effect. GP acknowledges sup-port from Zernike Institute for Advanced Materials andMaterial Innovation Institute M2i (project MC3.05242).The authors benefited from exchange of ideas within theESF Research Network CASIMIR.

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· Influence of surface roughness on the dispersion forces is reviewed.

· Nonadditivity of the forces complicates theoretical description.

· Strong deviations from the expected force scaling are observed experimentally.

· Statistics of Au films shows increased number of high asperities.

· Close to contact the high asperities give significant contribution to the force.

Highlights