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06

Contact of Single Asperities with Varying Adhesion: Comparing

Continuum Mechanics to Atomistic Simulations

Binquan Luan and Mark O. Robbins

Department of Physics and Astronomy, The Johns Hopkins University,

3400 N. Charles Street, Baltimore, Maryland 21218

(Dated: March 22, 2006)

Abstract

Atomistic simulations are used to test the equations of continuum contact mechanics in nanome-

ter scale contacts. Nominally spherical tips, made by bending crystals or cutting crystalline or

amorphous solids, are pressed into a flat, elastic substrate. The normal displacement, contact

radius, stress distribution, friction and lateral stiffness are examined as a function of load and

adhesion. The atomic scale roughness present on any tip made of discrete atoms is shown to have

profound effects on the results. Contact areas, local stresses, and the work of adhesion change

by factors of two to four, and the friction and lateral stiffness vary by orders of magnitude. The

microscopic factors responsible for these changes are discussed. The results are also used to test

methods for analyzing experimental data with continuum theory to determine information, such

as contact area, that can not be measured directly in nanometer scale contacts. Even when the

data appear to be fit by continuum theory, extracted quantities can differ substantially from their

true values.

PACS numbers: 81.40.Pq 68.35.Np 62.20.Dc 68.37.Ps

1

I. INTRODUCTION

There has been rapidly growing interest in the behavior of materials at nanometer scales

[1]. One motivation is to construct ever smaller machines [2], and a second is to improve

material properties by controlling their structure at nanometer scales [3]. For example,

decreasing crystallite size may increase yield strength by suppressing dislocation plasticity,

and material properties may be altered near free interfaces or grain boundaries. To make

progress, this research area requires experimental tools for characterizing nanoscale prop-

erties. Theoretical models are also needed both to interpret experiments and to allow new

ideas to be evaluated.

One common approach for measuring local properties is to press tips with characteristic

radii of 10 to 1000 nm into surfaces using an atomic force microscope (AFM) or nanoindenter

[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Mechanical properties are then extracted from the

measured forces and displacements using classic results from continuum mechanics [16]. A

potential problem with this approach is that continuum theories make two key assumptions

that must fail as the size of contacting regions approaches atomic dimensions. One is to

replace the atomic structure in the bulk of the solid bodies by a continuous medium with

internal stresses determined by a continuously varying strain field. The second is to model

interfaces by continuous, differentiable surface heights with interactions depending only on

the surface separation. Most authors go further and approximate the contacting bodies by

smooth spheres.

In a recent paper [17], we analyzed the limits of continuum mechanics in describing

nanometer scale contacts between non-adhesive surfaces with curvature typical of experi-

mental probes. As in studies of other geometries [18, 19, 20], we found that behavior in the

bulk could be described by continuum mechanics down to lengths as small as two or three

atomic diameters. However, the atomic structure of surfaces had profound consequences for

much larger contacts. In particular, atomic-scale changes in the configuration of atoms on

nominally cylindrical or spherical surfaces produced factor of two changes in the width of

the contacting region and the stress needed to produce plastic yield, and order of magnitude

changes in friction and stiffness.

In this paper we briefly revisit non-adhesive contacts with an emphasis on the role of

surface roughness. We then extend our atomistic studies to the more common case of

2

adhesive interactions. One important result is that the work of adhesion is very sensitive

to small changes in the positions of surface atoms. Changes in other quantities generally

mirror those for non-adhesive tips, and small differences in the magnitude of these effects

can be understood from geometrical considerations. The results are used to test continuum-

based methods of analyzing AFM measurements of friction and stiffness [4, 5, 6, 7, 8, 9,

10, 11]. We show that the models may appear to provide a reasonable description when

limited information about the true contact structure is available. When the full range

of information accessible to simulations is examined, one finds that the contact area and

pressure distributions may be very different than inferred from the models.

Section II reviews continuum results for contact without and with adhesion, and briefly

describes the effect of surface roughness. The methods used in our atomistic simulations

and the geometries of the tips are described in Sec. III. Section IV presents results for

purely repulsive interactions and Sec. V describes trends with the strength of adhesion. A

summary and conclusions are presented in Sec. VI.

II. CONTINUUM CONTACT MECHANICS

As noted above, contact mechanics calculations assume that the contacting solids are

described by continuum elasticity so that the discrete atomic structure can be ignored. In

most cases the two solids are also assumed to be isotropic with Young’s moduli E1 and E2

and Poisson ratios ν1 and ν2. Then the results depend only on an effective modulus E∗

satisfying:

1/E∗ ≡ (1− ν21)/E1 + (1− ν2

2)/E2. (1)

Three-dimensional crystalline solids are not isotropic, but the theories can still be applied

with an effective E∗ that depends on orientation and is determined numerically [16].

Continuum theories also neglect the atomic structure of the surface. In most cases the

surfaces are assumed to be spherical, with radii R1 and R2. For elastic, frictionless solids the

contact of two spheres is equivalent to contact between a sphere of radius R = (R−11 +R−1

2 )−1

and a flat solid [16]. From Eq. (1), one may then map contact between any two spherical

surfaces onto contact between a rigid sphere of radius R and a flat elastic solid of modulus

E∗. This is the case considered in our simulations, and previous results indicate this mapping

remains approximately correct at atomic scales [17].

3

Non-adhesive contact is described by Hertz theory [16], which assumes solids interact

with an infinitely sharp and purely repulsive “hard-wall” interaction. The surfaces contact

in a circular region of radius a that increases with the normal force or load N pushing the

surfaces together as [16]:

a =

(

3NR

4E∗

)1/3

. (2)

The normal pressure p within the contact has a simple quadratic dependence on the radial

distance from the center r:

p(r) =2aE∗

πR

√

1− r2

a2, (3)

and the surfaces separate slowly outside the contact. The normal displacement of the tip δ

is related to a by:

δH = a2/R = (3NR

4E∗)2/3 , (4)

where the subscript H indicates the Hertz prediction and δH = 0 corresponds to the first

contact between tip and substrate.

Adhesion can be treated most simply in the opposite limits of very short-range interac-

tions considered by Johnson, Kendall and Roberts (JKR) [21] and of infinite range interac-

tions considered by Derjaguin, Muller and Toporov (DMT) [22]. The strength of adhesion

is measured by the work of adhesion per unit area w. In DMT theory the attractive forces

just produce an extra contribution to the normal force, so that N is replaced by N +2πwR

in Eqs. (2) and (4). JKR theory treats the edge of the contact as a crack tip and calculates

the stress by adding the crack and Hertz solutions. The normal force in Eq. (2) is then

replaced by N + 3πwR + [6πwRN + (3πwR)2]1/2

and the equation for δ is modified (Sec.

VB). The two approaches lead to very different functional relations between a and N . For

example, the contact radius goes to zero at pulloff for DMT theory, but remains finite for

JKR. They also predict different values of the pulloff force, Nc, where the surfaces separate.

The normalized pulloff force, Nc/πwR, is -3/2 in JKR theory and -2 for DMT. Finally, the

surfaces separate outside the contact with infinite slope in JKR theory, and gradually in

DMT theory.

The Maugis-Dugdale (M-D) model [23] provides a simple interpolation between the JKR

and DMT limits. The surfaces are assumed to have a hard-wall contact interaction that

prevents any interpenetration, plus a constant attractive force per unit area, σ0, that extends

over a finite distance h0. The work of adhesion is just the integral of the attractive force,

4

implying σ0h0 = w. The M-D model produces coupled equations for the contact pressure

that can be solved to yield a relation between the load, normal displacement, and area. As

discussed further in Section V, the edge of the contact is broadened by the finite interaction

range, making it useful to define three characteristic radii that converge to the JKR value

for a in the limit of short-range interactions.

Maugis introduced a transition parameter [23]

λ ≡(

9Rw2

2πE∗2h30

)1/3

, (5)

that measures the ratio of the normal displacement at pulloff from JKR theory to the

interaction range h0. Tabor [24] had previously defined a similar parameter, µ, that is about

16% smaller than λ for typical interaction potentials [25]. Johnson and Greenwood [25]

have provided an adhesion map characterizing the range of λ over which different models

are valid. For λ > 5 the interaction range is short and JKR theory is accurate, while DMT

is accurate for λ < 0.1. For most materials, both h0 and the ratio w/E∗ are of order 1 nm.

The JKR limit is only reached by increasing R to macroscopic dimensions of micrometers

or larger. JKR theory has been tested in experiments with centimeter scale radii using the

surface force apparatus (SFA) [26] and hemispherical elastomers [27, 28]. Scanning probe

microscope tips typically have R between 10 and 100 nm, and the value of λ ∼ 0.1 to 1 lies

between JKR and DMT limits [4]. The same is true in our simulations, where λ for adhesive

tips varies between 0.1 and 0.75. For this reason we will compare our results to M-D theory

below. We also found it useful to use a simple interpolation scheme suggested by Schwarz

[29]. Both he and Carpick et al. [30] have proposed formulae for the contact radius that

interpolate smoothly between DMT and JKR. These approaches have been attractive in

analyzing experimental data because of their simple analytic forms.

No direct measurement of contact area has been possible in nanometer scale single as-

perity contacts. Instead, the contact area has been determined by measurements of contact

stiffness [4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 25], conductance [9], or friction [4, 5, 6, 7, 9, 10, 11].

The validity of these approaches is not clear [17], and will be tested below. The stiffness

against normal displacements of the surfaces can be determined from the derivative of N

with respect to δ in M-D theory. The tangential stiffness k is normally calculated by assum-

ing friction prevents sliding at the interface, even though all theories described above assume

zero friction in calculating the contact area. With this assumption k = 8G∗a, where G∗ is

5

the effective bulk shear modulus. Relating the friction to contact area requires assumptions

about the friction law. Many authors have assumed that the friction is proportional to area

[4, 5, 6, 7, 8, 9, 10, 11], but simulations [17, 31, 32] and experiments in larger contacts

[33, 34] show that this need not be the case.

The effect of surface roughness on contact has been considered within the continuum

framework [16]. In general, results must be obtained numerically. One key parameter is the

ratio of the root mean squared (rms) roughness of the surface, ∆, to the normal displacement

δ. When ∆/δ < 0.05, results for nonadhesive contacts lie within a few percent of Hertz

theory [16]. As ∆/δ increases, the contact area broadens and the pressure in the central

region decreases. Adhesion is more sensitive to roughness [35]. The analysis is complicated

by the fact that ∆ usually depends on the range of lengths over which it is measured. The

natural upper bound corresponds to the contact diameter and increases with load, while the

lower bound at atomic scales is unclear. The role of roughness is discussed further in Secs.

IV and V.

III. SIMULATION METHODS

We consider contact between a rigid spherical tip and an elastic substrate with effective

modulus E∗. As noted above, continuum theory predicts that this problem is equivalent

to contact between two elastic bodies, and we found this equivalence was fairly accurate

in previous studies of non-adhesive contact [17]. To ensure that any deviations from the

continuum theories described above are associated only with atomic structure, the substrate

is perfectly elastic. Continuum theories make no assumptions about the nature of the atomic

structure and interactions within the solids. Thus any geometry and interaction potentials

can be used to explore the type of deviations from continuum theory that may be produced

by atomic structure. We use a flat crystalline substrate to minimize surface roughness, and

use tips with the minimum roughness consistent with atomic structure. The interactions

are simple pair potentials that are widely used in studies that explore generic behavior [36].

They illustrate the type of deviations from continuum theory that may be expected, but

the magnitude of deviations for real materials will depend on their exact geometry and

interactions.

Atoms are placed on sites of a face-centered cubic (fcc) crystal with a (001) surface. We

6

define a characteristic length σ so that the volume per atom is σ3 and the nearest-neighbor

spacing is 21/6 σ. Nearest-neighbors are coupled by ideal Hookean springs with spring

constant κ. Periodic boundary conditions are applied along the surface of the substrate

with period L in each direction. The substrate has a finite depth D and the bottom is held

fixed. For the results shown below, L = 190.5 σ and D = 189.3 σ. The continuum theories

assume a semi-infinite substrate, and we considered smaller and larger L and D to evaluate

their effect on calculated quantities. Finite-size corrections for the contact radius and lateral

stiffness are negligible for a/D < 0.1 [16], which covers the relevant range of a/R < 0.2.

Corrections to the normal displacement are large enough to affect plotted values. We found

that the leading analytic corrections [37, 38, 39] were sufficient to fit our results at large

loads, as discussed in Sec. VB. Note that previous simulations of AFM contact have used

much shallower substrates (D ∼ 10 σ) [40, 41, 42, 43, 44, 45]. This places them in a very

different limit than continuum theories, although they provide interesting insight into local

atomic rearrangements.

Three examples of atomic realizations of spherical tips are shown in Fig. 1. All are

identical from the continuum perspective, deviating from a perfect sphere by at most σ.

The smoothest one is a slab of f.c.c. crystal bent into a sphere. The amorphous and stepped

tips were obtained by cutting spheres from a bulk glass or crystal, and are probably more

typical of AFM tips [4, 46]. Results for crystalline tips are very sensitive to the ratio η

between their nearest-neighbor spacing and that of the substrate, as well as their crystalline

alignment [31, 32]. We will contrast results for an aligned commensurate tip with η = 1 to

those for an incommensurate tip where η = 0.94437. To mimic the perfectly smooth surfaces

assumed in continuum theory, we also show results for a high density tip with η = 0.05.

In all cases R = 100 σ ∼ 30 nm, which is a typical value for AFM tips. Results for larger

radius show bigger absolute deviations from continuum predictions, but smaller fractional

deviations [17].

Atoms on the tip interact with the top layer of substrate atoms via a truncated Lennard-

Jones (LJ) potential [36]

VLJ = −4ǫi

[

(σ

r

)6

−(σ

r

)12]

− Vcut, r < rcut (6)

where ǫi characterizes the adhesive binding energy, the potential vanishes for r > rcut, and

the constant Vcut is subtracted so that the potential is continuous at rcut. Purely repulsive

7

interactions are created by truncating the LJ potential at its minimum rcut = 21/6 σ. Studies

of adhesion use rcut = 1.5 σ or rcut = 2.2 σ to explore the effect of the range of the potential.

In order to compare the effective strength of adhesive interactions and the cohesive energy

of the solid substrate, we introduce a unit of energy ǫ defined so that the spring constant

between substrate atoms κ = 50 ǫ/σ2. If the solid atoms interacted with a truncated LJ

potential with ǫ and rcut = 1.5 σ, they would have the same equilibrium lattice constant and

nearly the same spring constant, κ = 57 ǫ/σ2, at low temperatures and small deformations.

Thus ǫi/ǫ is approximately equal to the ratio of the interfacial binding energy to the cohesive

binding energy in the substrate.

The elastic properties of the substrate are not isotropic. We measure an effective modulus

E∗ = 55.0 ǫ/σ3 for our geometry using Hertzian contact of a high density tip. This is between

the values calculated from the Young’s moduli in different directions. The sound velocity

is also anisotropic. We find longitudinal sound velocities of 8.5 and 9.5 σ/tLJ and shear

velocities of 5.2 and 5.7 σ/tLJ along the (001) and (111) directions, respectively. Here tLJ is

the natural characteristic time unit, tLJ =√

mσ2/ǫ, where m is the mass of each substrate

atom. The effective shear modulus for lateral tip displacements is G∗ = 18.3 ǫ/σ3.

The simulations were run with the Large-scale Atomic/Molecular Massively Parallel Sim-

ulator (LAMMPS) code [47, 48]. The equations of motion were integrated using the velocity-

Verlet algorithm with time step 0.005 tLJ [36]. Temperature, T , only enters continuum

theory through its effect on constitutive properties, and it is convenient to run simulations

at low temperatures to minimize fluctuations. A Langevin thermostat was applied to solid

atoms to maintain T=0.0001 ǫ/kB, where kB is Boltzmann’s constant. This is about four

orders of magnitude below the melting temperature of a Lennard-Jones solid. The damping

rate was 0.1 t−1LJ , and damping was only applied perpendicular to the sliding direction in

friction measurements.

In simulations, a tip was initially placed about rcut above the substrate. It was then

brought into contact by applying a very small load, with the lateral position kept fixed. The

load was changed in discrete steps and the system was allowed to equilibrate for 350 tLJ

at each load before making measurements. This interval is about 20 times longer than

the time for sound to propagate across the substrate, and allowed full stress equilibration.

Results from loading and unloading cycles showed no noticeable hysteresis. To obtain results

near the pulloff force, we moved the tip away from the substrate at a very slow velocity

8

v = 0.0003 σ/tLJ and averaged results over small ranges of displacement. This approach

was consistent with constant load measurements and allowed us to reach the region that is

unstable at constant load.

To compare to continuum predictions we calculated the stresses exerted on the substrate

by the tip. The force from the tip on each substrate atom was divided by the area per atom

to get the local stresses. These were then decomposed into a normal stress or pressure p on

the substrate, and a tangential or shear stress τsur. The continuum theories described in Sec.

II assume that the projection of the force normal to the undeformed substrate equals the

projection normal to the locally deformed substrate. This is valid in the assumed limits of

a/R << 1 and τsur = 0. It is also valid for most of our simulations (within < 2%), but not for

the case of bent commensurate tips where τsur becomes significant. Normal and tangential

stresses for bent commensurate tips were obtained using the local surface orientation of the

nominally spherical tip. Correcting for the orientation changed the normal stress by less

than 5% of the peak value, and the shear stress by less than 20%.

Friction forces are known to vary with many parameters [49]. Of particular concern is the

dependence on extrinsic quantities such as the stiffness of the system that imposes lateral

motion. Results at constant normal load are often very different than those at fixed height,

motion perpendicular to the nominal sliding direction can raise or lower friction, and the

kinetic friction can be almost completely eliminated in very stiff systems [50, 51]. A full

re-examination of these effects is beyond the scope of this paper. Our sole goal is to measure

the friction in a consistent manner that allows us to contrast the load dependent friction for

different tip geometries and minimizes artifacts from system compliance.

In friction simulations, the tip is sheared at a constant low velocity v′ = 0.01 σ/tLJ

along the (100) direction with a constant normal load. This is typical of AFM experiments

where the low normal stiffness of the cantilever leads to a nearly constant normal load, and

the high lateral stiffness limits lateral motion in the direction perpendicular to the sliding

direction. The measured friction force varies periodically with time as the tip moves by

a lattice constant of the substrate. The time-averaged or kinetic friction during sliding is

very sensitive to both lateral stiffness and load [51]. We focus instead on the peak force,

which is less sensitive. In the limit of low sliding velocities this would correspond to the

static friction. For bent and stepped commensurate tips there is a single strong friction

peak. For incommensurate and amorphous tips, there may be multiple peaks of different size

9

corresponding to escape from different metastable energy minima [32, 50]. The static friction

was determined from the highest of these friction peaks, since lateral motion would stop at

any lower force. With a single peak per period, the time between peaks is ∼ σ/v′ = 100 tLJ .

This is several times the sound propagation time, and the measured force should be close

to the static friction. For incommensurate tips the time between peaks was an order of

magnitude smaller and dynamic effects may be more significant. However, they are not

expected to affect the load dependence significantly, and are much too small to affect the

dramatic difference between incommensurate and other tips.

The total lateral stiffness of the system, k, corresponds to the derivative of F with lateral

tip displacement evaluated at a potential energy minimum. Since the tip is rigid, k is

determined by displacements in the substrate and at the interface. The interfacial stiffness

ki and substrate stiffness ksub add in series because stress is transmitted through interfacial

interactions to the substrate. Thus the total stiffness is [17, 51]:

k−1 = k−1sub + k−1

i . (7)

If the tip were not rigid, it would also contribute a term to the right-hand-side of Eq. (7).

We evaluate k from the derivative of F during sliding, making the assumption that the

results are in the quasisatic limit. For bent and stepped commensurate tips there is a single

potential energy minima, and for amorphous tips one minimum dominated the periodic

force. For incommensurate tips, there are many closely spaced minima and we evaluate k

from the derivative in the minimum preceding the largest friction peak. Due to the small

magnitude of forces and short time intervals, the relative errors in these values are as big

as 50%. To estimate the lateral stiffness in the substrate, ksub, we fix the relative positions

of those substrate atoms that lie inside the contact, and move them laterally at a slow

velocity. The total force between these atoms and the rest of the substrate is measured

and its derivative with respect to distance gives the lateral stiffness in the substrate. In

principal, there might also be some relative displacement between atoms in the contact that

is not captured by this approach, but the results for the substrate stiffness are consistent

with continuum predictions.

Values of the adhesion energy per unit area w were obtained for flat, rigid surfaces of

the same nominal geometry as the tip. For bent crystal tips (Fig. 1), the tip was just

flattened back into a crystal. For stepped tips, we used an unstepped crystal with the same

10

spacing and interactions. For amorphous tips, an amorphous solid was cleaved with a flat

surface rather than a sphere. The resulting surfaces were then brought into contact with

the substrate and allowed to equilibrate at zero load. At the low temperatures used here,

the adhesion energy is just the potential energy difference between contacting and separated

configurations.

IV. NONADHESIVE CONTACTS

A. Pressure distribution

Figure 2 contrasts the distribution of normal pressure p under five tips: (a) dense, (b)

bent commensurate, (c) bent incommensurate, (d) amorphous and (e) stepped. In each case,

R = 100 σ and the dimensionless load is N/(R2E∗) = 0.0018. Hertz theory predicts the

same pressure distribution (solid lines) for all tips. Points show the actual local pressure on

each substrate atom as a function of radial distance r from the center of the spherical tip,

and circles in (c) and (d) show the average over bins of width σ. Clearly, small deviations in

atomic structure lead to large changes in the mean pressure and the magnitude of pressure

fluctuations. We find that these deviations become larger as N is decreased, and the contact

radius drops closer to the atomic size.

One possible source of deviations from Hertz theory is friction, but we find the mean

tangential forces are small in most cases. The exception is the bent commensurate tip (Fig.

2(b)), where the tangential stress rises with r and is comparable to the normal stress near

the edge of the contact. This result is not surprising given the high friction measured for

commensurate tips below, and reflects the strong tendency for atoms in the substrate to

remain locked in epitaxial registry with atoms in the tip. However, the deviation from Hertz

theory is in the opposite direction from that expected from friction. Since this contact was

made by gradually increasing the load, friction should decrease the contact size rather than

broadening it.

Another possible origin of the deviations from Herts theory is surface roughness. From

continuum theory (Sec. II), this is characterized by the ratio of rms surface roughness ∆ to

normal displacement δ. The normal displacement for all tips is about the same, δ ≈ 1.5 σ,

but ∆ is difficult to define. The reason is that there is no unique definition of the surface

11

height for a given set of atomic positions. For example, one might conclude that ∆ = 0 for

the substrate, since all atoms lie on the same plane. However, if a tip atom were moved

over the surface with a small load, its height would increase as it moved over substrate

atoms and be lowest at sites centered between four substrate atoms [50]. For the parameters

used here, the total height change is about 0.33 σ. Similar deviations from a sphere are

obtained for the bent commensurate and incommensurate tips. The height change decreases

as the ratio of the nearest-neighbor spacing to the Lennard-Jones diameter for interfacial

interactions decreases, and is only 0.0007 σ for the dense tip. Amorphous and stepped

tips have additional roughness associated with variations in the position of atomic centers

relative to a sphere. The total variation is about σ, or about three times the height change

as an atom moves over the substrate. A reasonable estimate is that ∆/δ < 0.1 for the bent

commensurate and incommensurate tips, ∆/δ < 10−3 for the dense tip, and ∆/δ ∼ 0.3 for

the amorphous and stepped tips. However, the ambiguity in ∆ is one of the difficulties in

applying continuum theory in nanoscale contacts.

The closely spaced atoms on the dense tip approximate a continuous sphere, and the

resulting pressure distribution is very close to Hertz theory (Fig. 2(a)). Results for the

bent commensurate tip are slightly farther from Hertz theory. The deviations can not be

attributed to roughness, because fluctuations at a given r are small, and the pressure in the

central region is not decreased. The main change is to smear the predicted sharp pressure

drop at the edge of the contact. This can be attributed to the finite range of the repulsive

potential between surfaces.

We can estimate the effective interaction range by the change in height of an atom,

dh = 0.04 σ, as p/E∗ decreases from 0.1 to 0. The effective range is much smaller for the

dense tip because ∼ 400 times as many atoms contribute to the repulsive potential. In Hertz

theory [16], the separation between surfaces only increases with distance (r − a) from the

edge of the contact as (8/3π)(r − a)3/2(2a)1/2/R. Equating this to dh gives r − a ≈ 1 σ

for the bent commensurate tip, which is very close to the range over which the edge of the

contact is shifted from the Hertz prediction. Note that this analysis predicts that the shift

in the edge of the contact will grow as√R, and simulations with larger R confirm this.

However, the fractional change in a decreases as 1/√R. The larger values of pressure at

low r result from the greater stiffness of the repulsive potential as p increases. All of the

above effects could be included in continuum theory by changing the form of the repulsive

12

potential [52].

For bent incommensurate and amorphous tips (Fig. 2 (c) and (d)), the variations in pres-

sure at a given r are as large as the mean (circles) [53]. While all atoms on the commensurate

tip can simultaneously fit between substrate atoms, minimizing fluctuations in force, atoms

on the incommensurate tip sample all lateral positions and experience a range of forces at

a given height. The mean pressure for the incommensurate tip remains fairly close to the

commensurate results, but there is slightly more smearing at large r due to the variations

in relative height of substrate and tip atoms. The mean pressure on the amorphous tip

shows the depression at small r and increase at large r that are expected for rough tips in

continuum theory [16]. The magnitude of the central drop is about 18%, which is consistent

with ∆/δ ∼ 0.2 in continuum theory (Fig. 13.12 of Ref. [16]). The lack of a noticeable drop

for incommensurate tips implies that the effective ∆/δ < .03. The implication is that the

incoherent height changes on amorphous tips contribute to the effective roughness in con-

tinuum theory, while the atomic corrugation on bent tips does not. The effective roughness

in both cases is about 0.1 σ smaller than our estimates for ∆ above.

Results for stepped tips show the largest deviations from Hertz theory, and they are

qualitatively different than those produced by random roughness. The terraced geometry of

this tip (Fig. 1) is closest to that of a flat punch. In continuum theory, the pressure on a

flat punch is smallest in the center, and diverges as the inverse square root of the distance

from the edge. The simulation results show qualitatively similar behavior. The main effect

of atomic structure is to cut off the singularity at a distance corresponding to an atomic

separation. Similar effects are observed in simulations of other geometries [18, 19, 20]. Note

that the terraces are only flat because the sphere was cut from a crystal that was aligned

with the substrate. We also examined tips cut from a crystal that was slightly tilted away

from the (001) direction [46]. This produces inclined terraces that contact first along one

edge. The resulting pressure distribution is very different, and closest to the continuum

solution for contact by an asymmetric wedge.

Figure 2 has an important general implication about the probability P (p) of finding a

local pressure p at a point in the contact. For smoothly curved surfaces, continuum theory

predicts that the derivative of the pressure diverges at the edge of the contact [16]. Thus

P (p) → 0 as p → 0 [54]. The finite resolution at atomic scales always smears out the

change in p, leading to a non-zero value of P (0). Indeed, the approximately constant value

13

of dp/dr near the contact edge in Fig. 2 leads to a local maximum in P at p = 0. Similar

behavior is observed for randomly rough atomic contacts [55] and in continuum calculations

for piecewise planar surfaces [56, 57].

Plastic deformation is usually assumed to occur when the deviatoric shear stress τs exceeds

the yield stress of the material. In Hertz theory, τs reaches a maximum value at a depth of

about 0.5a. The pressure variations at the surface shown in Fig. 2 lead to changes in both

the magnitude and position of the peak shear stress [17]. Factors of two or more are typical

for amorphous and stepped tips. Thus tip geometry may have a significant impact on the

onset of yield. Of course atomistic effects also influence the yield stress at nanometer scales,

and a full evaluation of this effect is left to future work. Saint-Venant’s principal implies

that the pressure distribution should become independent of tip geometry at depths greater

than about 3a, but the shear stress at these depths is substantially smaller than peak values

and yield is unlikely to occur there.

B. Variations with load

Figure 3 shows the load dependence of (a) normal displacement, (b) radius, (c) friction

and (d) lateral stiffness for the same tips as Fig. 2. Each quantity is raised to a power

that is chosen so that Hertz theory predicts the indicated straight line. A small finite-depth

correction (∼ 2%) is applied to the Hertz prediction for δ (Eq. (8)).

As also found for cylindrical contacts [17], the normal displacement shows the smallest

deviation from Hertz theory because it represents a mean response of many substrate atoms.

Results for all bent crystals are nearly indistinguishable from the straight line. Results

for the stepped surface are lower at small loads. Since the entire tip bottom contacts

simultaneously, it takes a larger load to push the tip into the substrate. The amorphous

results are shifted upwards by a fairly constant distance of about 0.2 σ. We define the origin

of δ as the tip height where the first substrate atom exerts a repulsive force on the tip. This

is strongly dependent on the height of the lowest tip atom, while subsequent deformation is

controlled by the mean tip surface. Agreement with Hertz is greatly improved by shifting

the amorphous curve by this small height difference. Note that the zero of δ is difficult to

determine experimentally and is usually taken as a fit parameter. If this is done, even results

for the amorphous system yield values of R and E∗ that are within 10% of their true values.

14

Thus measurements of δ, interpreted with continuum theory for spheres, can provide useful

values of elastic properties at nanometer scales.

As expected from the observed pressure distributions (Fig. 2), the contact radius is

generally larger than the Hertz prediction. The shift is smallest for the dense tip because it

approximates a continuous surface and the high density leads to a repulsive potential that

rises more than a hundred times more rapidly than for other tips. Results for bent crystal

and amorphous tips are shifted upwards by fairly load-independent offsets of ∼ 1 − 3 σ,

leading to large fractional errors at low loads (up to 100%). The stepped crystal shows

qualitatively different behavior, with a rising in discrete steps as sequential terraces come

into contact. Note that the size of the first terrace is not unique, but depends on the registry

between the bounding sphere and crystalline lattice [46]. Larger deviations may be observed

when the first step has very few atoms. Such tips may be more likely to be chosen for AFM

studies because they tend to give sharper images.

In order to predict the friction between surfaces, one must make assumptions about how

F depends on area and load. The straight line in Fig. 3(c) corresponds to a friction force

that is proportional to load. Static friction values for bent and stepped commensurate

surfaces are consistent with this line and a coefficient of friction µ ≡ F/N = 0.63. Analytic

[32] and simple numerical [58] models show that this is a general feature of mated surfaces

where each tip atom has the same lateral position relative to substrate atoms. The friction

on amorphous and incommensurate surfaces is always lower and scales more closely with

the contact area, as indicated by broken line fits to F ∝ N2/3 and discussed further in

Sec. VB [59]. Many authors have made this assumption in fitting experimental data, but

it is not obvious why it should hold. The friction per unit area between flat amorphous

surfaces decreases as the square root of the area, but rises linearly with the normal pressure

[32]. Wenning and Muser have noted that these two factors may combine for spherical

contacts to give a net friction that rises linearly with area [31]. However, their argument

would predict that the frictional force in a cylinder-on-flat geometry would not scale with

area, and our previous simulations found that it did [17]. Continuum theory predicts that

the lateral stiffness k = 8G∗a, and should follow the straight line in Fig. 3(d). Measured

values of the total stiffness (open symbols) are always substantially lower. This is because

continuum theory assumes that there is no displacement at the interface, only within the

substrate. In reality, the frictional force is always associated with atomic scale displacements

15

of interfacial atoms relative to a local energy minimum [50, 58]. The derivative of force with

displacement corresponds to an interfacial stiffness ki that adds in series with the substrate

contribution (Eq. (7)) [17, 51]. Our numerical results show that ki can reduce k by more

than an order of magnitude, particularly for tips where F is small. We also evaluated the

substrate stiffness ksub by forcing all contacting substrate atoms to move together. These

results (filled symbols) lie much closer to the Hertz prediction. Only the stepped tip shows

large discrepancies, and these are correlated with the large deviation between the measured

and predicted contact radii.

V. ADHESIVE CONTACTS

A. Pressure distribution

Figure 4 compares the calculated pressure distribution in adhesive contacts with the

Maugis-Dugdale prediction (lines). A bent commensurate tip was used to minimize devia-

tions from continuum predictions for a sphere. Results for two different rcut are presented to

indicate difficulties in fitting longer-range interactions to M-D theory. The work of adhesion

was calculated for unbent surfaces (Sec. III) with ǫi/ǫ = 0.5, yielding w = 1.05 ǫ/σ2 and

1.65 ǫ/σ2 for rcut = 1.5 σ and 2.2 σ, respectively. This leaves only one fitting parameter in

M-D theory. For the dashed lines, the width of the attractive interaction h0 = w/σ0 was

chosen to coincide with the range of the atomic potential. The dotted line shows a fit with

h0 = 0.8 σ for rcut = 2.2 σ, which gives better values for the pulloff force, but poorer radii

(Sec. VB).

In M-D theory, it is common to identify two radii, a and c, with the inner and outer

edges of the plateau in the pressure, respectively [23, 25]. For r < a the surfaces are in

hard-sphere contact, and for a < r < c they are separated and feel the constant attraction.

The continuously varying interactions between atoms in our simulations lead to qualitatively

different behavior. There is no sharp transition where the surfaces begin to separate, and

the attraction shows a smooth rise to a peak, followed by a decay. To facilitate comparison

to continuum theories, we introduce the three characteristic radii indicated by arrows for

each rcut. The innermost, ra, corresponds to the point where the interaction changes from

repulsive to attractive and can be calculated in M-D theory. The outermost, rc, corresponds

16

to c in M-D theory – the point where interactions end. The middle, rb, corresponds to the

peak attraction. It was also studied by Greenwood [52] for contact of a featureless sphere

that interacted with the flat substrate through Lennard-Jones interactions. He found rb lay

close to the JKR prediction for contact radius at large loads. All three radii converge in the

limit of repulsive interactions or the JKR limit of an infinitely narrow interaction range.

At small radii the atomistic results for p lie above M-D theory, and they drop below near

r/σ = 10. These deviations can be attributed to the increasing stiffness of the repulsion

between tip and substrate atoms with increasing pressure. Just as in the non-adhesive

case, the stiffer interactions in the center of the tip lead to bigger pressures for a given

interpenetration. The change in pressure with separation produces less smearing at the

edge of the repulsive region of the contact than for the non-adhesive case (Fig. 2(b)), and

the values of ra are very close to M-D theory. This can be understood from the fact that

surfaces separate much more rapidly in JKR theory (∝ (r − a)1/2) than in Hertz theory

(∝ (r − a)3/2) (Sec. IV). Thus the same height change in the repulsive region corresponds

to a much smaller change in radius. Of course the finite range of the attractive tail of the

potential leads to a broad region of adhesive forces out to rc. The continuous variation

of p in the attractive tail is only crudely approximated by M-D theory. The difficulty in

determining the optimum choice for h0 increases with the range of interactions, as discussed

further below.

Figure 5 shows the effect of tip geometry on pressure distribution. We found that the

work of adhesion was very sensitive to tip geometry. To compensate for this effect, we varied

ǫi (Table I) to yield the same w for the tips in Fig. 5. Then all tips should have the same

pressure distribution in continuum theory. The M-D predictions for p are not shown because

even the bent commensurate tips produce significantly different results (Fig. 4). Instead,

we compare other tips to the bent commensurate tip.

As for non-adhesive tips, local fluctuations in pressure are small for commensurate tips

(Fig. 5(a) and (d)) and comparable to the mean pressure for incommensurate or amorphous

tips (Fig. 5(b) and (c)). Note however, that the fluctuations become smaller in the adhesive

regime (large r). This is because the potential varies more slowly with height, so fluctuations

in the separation between atoms have less effect on p. One consequence is that the outer

edge of the contact is nearly the same for commensurate and incommensurate tips. The radii

for the amorphous tip are significantly larger than bent tips, presumably because of a much

17

greater effective roughness. As for nonadhesive tips, the mean pressure on incommensurate

tips is close to the commensurate results. Adhesion reduces the roughness-induced drop

in pressure in the central region of the amorphous tip. For the stepped tip, the contact

radius is dominated by the size of the terraces, but adhesive tails are visible outside the

edge of each terrace. Only rc is easily defined for the stepped tips. This increases in a

nearly stepwise manner, and its load dependence is not shown below. For the amorphous

and incommensurate tips, radii are determined from the mean pressure at a given radius

(open circles). Errorbars are less than 0.5 σ.

B. Variations of radius and displacement with load

Figure 6(b) compares the measured contact radii for the tips of Fig. 5 to M-D theory as

load is varied. The simulation results for rc (open symbols) decrease with decreasing load

as ra (closed symbols) decreases to zero. All interactions in the contact are then adhesive.

As rc continues to drop, the area contributing to adhesion drops, and the load rises back

toward zero. This regime is not considered in the original M-D theory. The extension to

ra = 0 by Johnson and Greenwood (JG) [25] is shown by a dashed line in the figure. Along

this line the stress in the contact is constant, giving N = −σ0 πr2c .

As for non-adhesive tips, the contacts tend to be larger than continuum theory predicts.

However, the shift for bent tips is smaller than in the non-adhesive case, and the commen-

surate and incommensurate results are closer, as expected from Fig. 5. Larger deviations

are observed for the amorphous tip, with radii typically 2 or 3 σ larger than predicted. The

deviation becomes most pronounced at negative loads, where the amorphous tip remains in

contact well below the predicted pulloff force.

Figure 6(a) compares the value of rb to the JKR prediction for contact radius. As found

by Greenwood [52], the numerical results are close to JKR at large loads, but deviate at

negative loads because M-D and JKR predict different pulloff forces. Since JKR assumes a

singular adhesive stress at the radius of the contact, it seems natural that its predictions lie

closest to the position of the peak tensile stress.

Figure 7 shows the calculated radii for bent and amorphous tips with the same interaction

energy ǫi/ǫ = 0.5. The small changes in tip geometry lead to a roughly four fold variation

in both w and Nc (Table I). The largest w is obtained for commensurate tips because all

18

atoms can optimize their binding coherently. Atoms on incommensurate tips sample all

positions relative to the substrate, and can not simultaneously optimize the binding energy.

The larger height fluctuations on amorphous tips lead to even greater reductions in w.

In the simulations, the pulloff force, Nc, corresponds to the most negative load where the

surfaces remain in contact and the ri can be defined. Its normalized value, Nc/πwR, is equal

to -3/2 in JKR theory, -2 in DMT theory, and lies between these values for M-D theory.

Table I shows normalized results for various tips. As expected from the good agreement in

Figs. 6 and 7, results for bent tips lie between JKR and DMT values and can be fit with M-D

theory. The values for stepped and amorphous tips lie outside the bounds of M-D theory.

This is an important result because pulloff forces are frequently used to measure the work

of adhesion. Based on continuum theory, one would expect that the uncertainty in such

measurements is less than the difference between JKR and DMT predictions. Our results

show factor of two deviations for stepped tips, which may be common in scanning probe

experiments. Other simulations showed that the stepped tip values are strongly dependent

on the size of the first terrace, as well as any tilt of the terraces or incommensurability.

It might seem surprising that the stepped tip has a smaller pulloff force than the bent

tip, because the entire first terrace can adhere without any elastic energy penalty. However

this effect is overcome by the greater contact size for bent tips: The radius of the first

terrace, rt ∼ 6 σ, is smaller than the values of rb and rc at pulloff for bent tips. As the

adhesion is decreased, the predicted contact size at pulloff will drop below rt and the stepped

tip may then have a larger pulloff force than predicted. This limit can also be reached by

increasing the width of the first terrace. For a tip that lies entirely within a sphere of radius

R, r2t < R2 − (R− d)2 ≈ 2dR where d is the layer spacing in the crystal. For our geometry

this corresponds to rt < 12 σ, which is about twice the value for our tip. As noted above,

terraces with even smaller radii may be preferentially selected for imaging studies and will

lead to lower |Nc|.As rcut increases, it becomes harder for the M-D model to simultaneously fit both the

radii and the pulloff force. Figure 8 shows simulation data for a bent commensurate tip

with rcut = 2.2 σ. Using the value of h0 = 1.2 σ (Fig. 4) reproduces all radii fairly

well at large loads, but gives a substantially different pulloff force, −906 ǫ/σ instead of

−872 ǫ/σ. Decreasing h0 to 0.8 σ fits the pulloff force, and improves the fit to ra at low N .

However, the predicted values for ra at large N are slightly too high and the values for rc

19

are shifted far below (∼ 2 − 3 σ) simulation data. This failure is not surprising given the

crude approximation for adhesive interactions in M-D theory. As might be expected, the

best values for the pulloff force are obtained by fitting the region near the peak in the force,

rather than the weak tail (Fig. 4).

It should be noted that for bent crystalline and amorphous tips, all of our results for r

can be fit accurately to M-D theory if E∗, w, and R are taken as adjustable parameters. The

typical magnitude of adjustments is 10 to 30%, which is comparable to typical experimental

uncertainties for these quantities in nanoscale experiments. Indeed one of the common goals

of these experiments is to determine any scale dependence in continuum parameters. For

this reason it would be difficult to test continuum theory in scanning probe experiments

even if r could be measured directly.

Experiments can more easily access the variation of normal displacement with load. This

requires subtracting the height change due to the normal compliance of the machine control-

ling the tip, which is difficult for standard AFM cantilevers, but possible in stiffer devices

[12, 13, 14]. The absolute zero of δ is not fixed, but must be fitted to theory. Figure 9

shows two measures of the normal displacement in our simulations. One, δtip, corresponds

to the experimentally accessible tip displacement. We associate the zero of δtip with the

point where the first substrate atom exerts a repulsive force. The second, δsur, is the actual

depression of the lowest substrate atom on the surface relative to the undeformed surface.

The two differ because of the interfacial compliance normal to the surface, which is assumed

to vanish in continuum theory.

The simulation results for δ are more sensitive to the finite sample depth D and lateral

periodicity L than other quantities. To account for this, the predictions of M-D and JKR

theory are shifted by the leading analytic corrections in a/D [37, 38, 39]:

δ = δH(1− ba

D) + δadhesion(1− d

a

D) , (8)

where δH is the Hertz prediction (Eq. (4)), b and d are fit parameters, and δadhesion =

−√

2wπa/E∗ for JKR theory and −(2σ0/E∗)√c2 − a2 for M-D theory. We obtained b = 0.8

from simulations with dense, non-adhesive tips (Fig. 3) and d = 0.3 from simulations with

dense, adhesive tips. Results for dense tips are then indistinguishable from the fit lines in

Figs. 3(a) and 9. Values of b and d from numerical studies of continuum equations are of

the same order [37, 38, 39], but to our knowledge these calculations have not considered our

20

geometry where L ≈ D.

With the finite-size corrections, continuum results for δ lie very close to the simulation

data for bent tips. Agreement is best for δsur because neither it nor continuum theory

include the interfacial compliance. The choice of zero for δtip is not precisely consistent with

M-D theory. The first repulsive interaction would correspond to the first non-zero ra, which

occurs at a slightly negative δ in M-D theory. For the parameters used here, the shift in δ

is about 0.24 σ, which is about twice the discrepancy between δtip and M-D theory at large

loads. This implies that the interfacial compliance produces a comparable correction.

The effects of interfacial compliance are biggest for the most negative values of δtip.

Here δtip decreases monotonically as N increases to zero. In contrast, δsur is flat and then

increases back towards zero. In this regime, ra = 0 and the only interaction comes from the

attractive tail of the potential. The net adhesive force gradually decreases (N increases)

as the magnitude of the separation between tip and substrate, ∆δ ≡ δtip − δsur, increases

(inset). Most of this change occurs in δtip, while the displacement of the surface relaxes

back to zero as the attractive force on it goes to zero. The JG extension to M-D theory

is expressed in terms of δtip and provides a good description of its change with N (dashed

line), even with the assumption of a constant attractive σ0.

For amorphous tips, both values of δ are substantially above M-D theory. The shifts

are bigger than in the non-adhesive case, about 0.35 σ vs. 0.25 σ. This is correlated with

the larger than predicted pulloff force. Based on the increase in |Nc|, the effective work

of adhesion appears to be larger by about 30% than that measured for a flat surface. The

stronger adhesive contribution to the total normal force leads to a larger value of δ. One may

understand these changes in terms of the effect of surface roughness. Atomic-scale roughness

on a nominally flat surface prevents many atoms from optimizing their binding energy. As δ

and the contact area shrink, the long wavelength height fluctuations become irrelevant and

no longer prevent the few atoms remaining in the contact from adhering. Thus while the

large load values of δ can be fit to continuum predictions with the measured w and a simple

shift in the origin of δ, the small N values correspond to a larger work of adhesion. The

magnitude of the increase (∼ 30%) is modest given that the incommensurate tip has about

twice as large a w as the amorphous tip for the same interaction energy ǫi.

The data for stepped tips are qualitatively different than the others. As for the non-

adhesive case, δ is lower than the continuum prediction at large loads, because the flat tip

21

is harder to push into the substrate. The deviation increases rapidly at negative loads, with

a sharp drop near N = 0 where the contact shrinks to the first terrace. As noted above,

the radius of the first terrace is smaller than the radius at pulloff predicted by continuum

theory, and the pulloff force is less than half the predicted value.

C. Friction and lateral stiffness

Scanning probe microscopes can most easily detect friction forces and the lateral com-

pliance of the tip/substrate system [4, 5]. Figure 10 shows F as a function of load for five

different adhesive tips. As for non-adhesive tips, tip geometry has a much larger effect on

F than other quantities, and values for bent commensurate and incommensurate tips differ

by two orders of magnitude.

Since the friction was measured at constant load (Sec. III), only values in the stable

regime, dδtip/dN > 0 could be obtained. Even in this regime, we found that the tip tended

to detach at N > Nc. This was particularly pronounced for commensurate tips. Indeed the

bent and stepped commensurate tips detached after the first peak in the friction force for

all the negative loads shown in Fig. 10(a). At loads closer to the pulloff force, detachment

occurred at even smaller lateral displacements.

As noted above, bent commensurate tips have the strongest adhesion energy because all

atoms can simultaneously optimize their binding. For the same reason, the adhesion energy

changes rapidly as atoms are displaced laterally away from the optimum position, allowing

pulloff above the expected Nc. The extent of the change depends on the sliding direction

and registry [40, 49, 50]. We consider sliding in the (100) direction (Sec. III), where atoms

move from points centered between substrate atoms towards points directly over substrate

atoms. This greatly reduces the binding energy, leading to detachment at less than half the

pulloff force. Note that changes in binding energy with lateral displacement lead directly

to a lateral friction force [50] and the bent and stepped commensurate tips also have the

highest friction. We suspect that tips with higher friction may generally have a tendency to

detach farther above Nc during sliding than other tips.

At the macro scale, friction is usually assumed to be directly proportional to load. All

the tips have substantial friction at zero load, due to adhesion. The friction force also varies

non-linearly with N , showing discrete jumps for the stepped tip, and curvature for the other

22

tips, particularly near Nc. The curvature at small N is reminiscent of the dependence of

radius on load. Several authors have fit AFM friction data assuming that F scales with

contact area, and using a continuum model to determine the area as a function of load

[4, 5, 6, 7, 8, 9, 10, 11]. The dotted lines in Fig. 10 show attempts to fit F using the area

from JKR theory with w adjusted to fit the pulloff force and the proportionality constant

chosen to fit the friction at high loads. None of the data is well fit by this approach.

The solid lines show fits to an expression suggested by Schwarz [29] that allows one to

simply interpolate between DMT and JKR limits as a parameter t1 is increased from 0 to

1. Reasonable fits are possible for non-stepped tips when this extra degree of freedom is

allowed. Values of the friction per unit area are within 25% of those determined below from

the true area (Table II), but the values of w and t1 do not correspond to any real microscopic

parameters. For example, for the three non-stepped tips where a direct measurement gives

w = 0.46 ǫ/σ2, the fits give w = 0.41 ǫ/σ2, 0.55 ǫ/σ2 and 0.94 ǫ/σ2 for commensurate,

amorphous and incommensurate tips, respectively. The value of t1 varies from 0.1 to 0.5

to 1.5, respectively. Note that the value of t1 = 1.5 for incommensurate tips is outside the

physical range. In this case, and some experiments [6], the friction rises more slowly than

the JKR prediction, while M-D theory and simpler interpolation schemes [29, 30] always

give a steeper rise. Such data seem inconsistent with F scaling with area.

Our simulations allow us to test the relationship between F and area without any fit

parameters. However, it is not obvious which radius should be used to determine area.

Figure 11 shows friction plotted against r2a, r2b and r2c . The stepped tip is not shown, since

only rc is easily defined and it increases in one discontinuous jump over the range studied.

For all other tips the friction is remarkably linear when plotted against any choice of radius

squared. In contrast, plots of N vs. r2 show significant curvature.

For F to be proportional to area, the curves should also pass through the origin. This

condition is most closely met by r2a, except for the incommensurate case. The idea that

friction should be proportional to the area where atoms are pushed into repulsion seems

natural. The other radii include attractive regions where the surfaces may separate far

enough that the variation of force with lateral displacement is greatly reduced or may even

change phase. However, in some cases the extrapolated friction remains finite as ra goes to

zero and in others it appears to vanish at finite ra.

Also shown in Fig. 11 are results for non-adhesive tips (open circles). Values for in-

23

commensurate and amorphous tips were doubled to make the linearity of the data more

apparent. Only the bent commensurate data shows significant curvature, primarily at small

N . As noted in Sec. IV, friction is proportional to load for this tip, and the curvature is

consistent with this and a2 ∼ N2/3. No curvature is visible when adhesion is added, but the

fact that the linear fit extrapolates to F = 0 at ra/σ ∼ 4 suggests that the linearity might

break down if data could be obtained at lower loads.

Results for adhesive and non-adhesive amorphous tips can be fit by lines through the

origin (not shown), within numerical uncertainty. The same applies for non-adhesive bent

incommensurate tips, but adding adhesion shifts the intercept to a positive force at ra = 0.

It would be interesting to determine whether this extrapolation is valid. Friction can be

observed with purely adhesive interactions, since the only requirement is that the magnitude

of the energy varies with lateral displacement [50]. However, it may also be that the friction

on incommensurate tips curves rapidly to zero as ra → 0. Indeed one might expect that for

very small contacts the tip should behave more like a commensurate tip, leading to a more

rapid change of F with area or load. The only way to access smaller ra is to control the tip

height instead of the normal load. This is known to affect the measured friction force [49],

and most experiments are not stiff enough to access this regime.

Figure 12 shows the lateral stiffness as a function of the three characteristic radii. Except

for the stepped tip (not shown), k rises linearly with each of the radii. As for non-adhesive

tips, the slope is much smaller than the value 8G∗ predicted by continuum theory (solid lines)

because of the interfacial compliance (Eq. (7)). The intercept is also generally different from

the origin, although it comes closest to the origin for ra. As for friction, it seems that the

repulsive regions produce the dominant contribution to the stiffness.

Results for non-adhesive tips are also included in Fig. 12. They also are linear over the

whole range, and the fits reach k = 0 at a finite radius ak. This lack of any stiffness between

contacting surfaces seems surprising. Note however that linear fits to Fig. 3(b) also would

suggest that the radius approached a non-zero value, a0, in the limit of zero load. Moreover,

the values of a0 and ak follow the same trends with tip geometry and have similar sizes.

The finite values of a0 and ak can be understood from the finite range of the repulsive part

of the interaction. As long as atoms are separated by less than rcut there is a finite interaction

and the atoms are considered inside a. However, the force falls rapidly with separation, and

atoms near this outer limit contribute little to the friction and stiffness. If δh is the distance

24

the separation must decrease to get a significant interaction, then a0 = (2Rδh)1/2 at the

point where the first significant force is felt. Taking the estimate of δh = 0.04 σ from Sec.

IV, then a0 ∼ 3 σ, which is comparable to the observed values of a0 and ak. The shift is

smaller for the adhesive case because there are still strong interactions when ra goes to zero.

The larger shifts for the amorphous tip may reflect roughness, since the first point to contact

need not be at the origin. We conclude that the linear fits in Fig. 12 go to zero at finite

radius, because ra overestimates the size of the region that makes significant contributions to

forces, particularly for non-adhesive tips. Note that the plots for friction with non-adhesive

tips (Fig. 11) are also consistent with an offset, but that the offset appears much smaller

when plotted as radius squared.

The slope of the curves in Fig. 11 can be used to define a differential friction force

per unit area or yield stress τf ≡ ∂F/∂πr2a. It is interesting to compare the magnitude of

these values (Table II) to the bulk yield stress of the substrate τy. Assuming Lennard-Jones

interactions, the ideal yield stress of an fcc crystal in the same shearing direction is 4 to 10

ǫσ3, depending on whether the normal load or volume is held fixed. The commensurate tip

is closest to a continuation of the sample, and the force on all atoms adds coherently. As

a result τf is of the same order as τy, even for the non-adhesive tip. Values for adhesive

amorphous and incommensurate tips are about one and two orders of magnitude smaller

than τy, respectively. This reflects the fact that the tip atoms can not optimize their registry

with the substrate. Removing adhesive interactions reduces τy by an additional factor of

about four in both cases.

In continuum theory, k = 8G∗r, and the slope of fits in Fig. 12 could then be used

to determine the effective shear modulus G∗. However, as noted above, the interfacial

compliance leads to much lower stiffnesses. To illustrate the magnitude of the change we

quote values of G′ ≡ (1/8)∂k/∂r in Table II. All values are below the true shear modulus

G∗ = 18.3 ǫ/σ3 obtained from the substrate compliance alone (Fig. 3). As always, results

for bent commensurate tips come closest to continuum theory with G′/G∗ ∼ 0.7. Values

for adhesive amorphous and incommensurate tips are depressed by factors of 3 and 20

respectively, and removing adhesion suppresses the value for amorphous tips by another

factor of four.

Carpick et al. noted that if friction scales with area and k with radius, then the ratio

F/k2 should be constant [5, 6, 7, 60]. Defining the frictional force per unit area as τf and

25

using the expression for k from continuum theory, one finds 64G∗2F/πk2 = τf . In principal,

this allows continuum predictions to be checked and τf to be determined without direct

measurement of contact size. Figure 13 shows:

τ efff ≡ 64(G∗)2F/πk2 (9)

as a function of N for different tip geometries and interactions. Except for the non-adhesive

stepped tip, the value of F/k2 is fairly constant at large loads, within our numerical accuracy.

Some of the curves rise at small loads because the radius at which F reaches zero in Fig.

11 tends to be smaller than that where k reaches zero in Fig. 12. These small radii are

where continuum theory would be expected to be least accurate. Note that the deviations

are larger for the non-adhesive tips, perhaps because the data extends to smaller radii.

The data for stepped tips are of particular interest because the contact radius jumps

in one discrete step from the radius of the first terrace to the radius of the second. The

friction and stiffness also show discontinuous jumps. Nonetheless, the ratio F/k2 varies

rather smoothly and even has numerical values close to those for other tips. The most

noticeable difference is that the data for the nonadhesive stepped tip rises linearly with

load, while all other tips tend to a constant at high load. These results clearly demonstrate

that success at fitting derived quantities like F and k need not imply that the true contact

area is following continuum theory.

The curves for τ efff in Fig. 13 are all much higher than values of the frictional stress τf

obtained directly from the friction and area (Table II). Even the trends with tip structure

are different. The directly measured frictional stress decreases from bent commensurate to

amorphous to bent incommensurate, while τ efff is largest for the amorphous and smallest

for the bent commensurate tip. These deviations from the continuum relation are directly

related to the interfacial compliance ki. The continuum expression for the lateral stiffness

neglects ki and gives too small a radius at each load. This in turn over-estimates the frictional

stress by up to two orders of magnitude. Similar effects are likely to occur in experimental

data.

Experimental plots of F/k2 have been obtained for silicon-nitride tips on mica and sodium

chloride [6, 7, 60] and on carbon fibers [8]. Data for carbon fibers and mica in air showed a

rapid rise with decreasing N at low loads [8, 60]. For mica the increase is almost an order of

magnitude, which is comparable to our results for non-adhesive bent incommensurate tips.

26

This correspondence may seem surprising given that the experiments measured adhesion in

air. However, the adhesive force was mainly from long-range capillary forces that operate

outside the area of contact. Following DMT theory, they can be treated as a simple additive

load that does not affect the contact geometry. In contrast, data for mica in vacuum is

well fit by JKR theory, implying a strong adhesion within the contact [6, 7]. The measured

value of F/k2 is nearly constant for this system, just as in our results for most adhesive

tips. Results for carbon fibers in vacuum [8] show a linear rise like that seen for nonadhesive

stepped tips.

From a continuum analysis of the carbon fiber data, the frictional stress was estimated

to be τf ∼ 300 MPa assuming a bulk shear stress of G∗ = 9.5 GPa [8]. Note that Fig. 13

would suggest τf/G∗ ∼ 0.1 to 0.3, while the true values (Table II) are as low as 0.0002. The

data on carbon fibers could be fit with the bulk shear modulus, but data on mica and NaCl

[6, 60] indicated that G∗ was 3 to 6 times smaller than bulk values. Our results show that the

interfacial compliance can easily lead to reductions of this magnitude and a corresponding

increase in τf , and that care must be taken in interpreting experiments with continuum

models.

VI. DISCUSSION AND CONCLUSION

The results described above show that many different effects can lead to deviations be-

tween atomistic behavior and continuum theory and quantify how they depend on tip ge-

ometry for simple interaction potentials (Fig. 1). In general, the smallest deviations are

observed for the idealized model of a dense tip whose atoms form a nearly continuous sphere,

although this tip has nearly zero friction and lateral stiffness. Deviations increase as the

geometry is varied from a bent commensurate to a bent incommensurate to an amorphous

tip, and stepped tips exhibit qualitatively different behavior. Tip geometry has the smallest

effect on the normal displacement and normal stiffness (Fig. 3 and 9) because they reflect

an average response of the entire contact. Friction and lateral stiffness are most affected

(Fig. 3 and 10), because they depend on the detailed lateral interlocking of atoms at the

interface.

One difference between simulations and continuum theory is that the interface has a finite

normal compliance. Any realistic interactions lead to a gradual increase in repulsion with

27

separation rather than an idealized hard-wall interaction. In our simulations the effective

range over which interactions increase is only about 4% of the atomic spacing, yet it impacts

results in several ways. For bent commensurate tips it leads to an increase in pressure in

the center of the contact (Figs. 2 and 4). The pressure at the edge of nonadhesive contacts

drops linearly over about 2 σ, while continuum theory predicts a diverging slope. The width

of this smearing grows as the square root of the tip radius and leads to qualitative changes in

the probability distribution of local pressures [54, 55, 56, 57]. These effects could be studied

in continuum theories with soft-wall interactions. The normal interfacial compliance also

leads to the offset in linear fits of F vs. r2a and k vs. ra (Figs. 11 and 12). Fits to the friction

and local stiffness extrapolate to zero at finite values of ra because atoms at the outer edge

of the repulsive range contribute to ra but interact too weakly to contribute substantially

to F and k. This effect is largest for non-adhesive tips.

Approximating a spherical surface by discrete atoms necessarily introduces some surface

roughness. Even bent crystalline tips have atomic scale corrugations, reflecting the variation

in interaction as tip atoms move from sites nestled between substrate atoms to sites directly

above. Amorphous and stepped tips have longer wavelength roughness associated with their

random or layered structures respectively. This longer wavelength roughness has a greater

effect on the contacts. For non-adhesive interactions, incommensurate and amorphous tips

have a lower central pressure and wider contact radius than predicted for ideal spheres.

These changes are qualitatively consistent with continuum calculations for spheres with

random surface roughness [16]. However the effective magnitude of the rms roughness ∆ is

smaller than expected from the atomic positions. The correlated deviations from a sphere

on stepped tips, lead to qualitative changes in the pressure distribution on the surface (Fig.

2 and 5). However, these changes are also qualitatively consistent with what continuum

mechanics would predict for the true tip geometry, which is closer to a flat punch than a

sphere. We conclude that the usual approximation of characterizing tips by a single spherical

radius is likely to lead to substantial errors in calculated properties. Including the true tip

geometry in continuum calculations would improve their ability to describe nanometer scale

behavior. Unfortunately this is rarely done, and the atomic-scale tip geometry is rarely

measured. Recent studies of larger tips and larger scale roughness are an interesting step in

this direction [61].

Roughness also has a strong influence on the work of adhesion w (Table I). Values of

28

w were determined independently from interactions between nominally flat surfaces. For a

given interaction strength, commensurate surfaces have the highest w, because each atom can

optimize its binding simultaneously. The mismatch of lattice constants in incommensurate

geometries lowers w by a factor of two, and an additional factor of two drop is caused by

the small (∆ ∼ 0.3 σ) height fluctuations on amorphous surfaces. In continuum theory,

these changes in w should produce nearly proportional changes in pulloff force Nc, and tips

with the same w and h0 should have the same Nc. Measured values of Nc differ from these

predictions by up to a factor of two. It is particularly significant that the dimensionless

pulloff force for amorphous and stepped tips lies outside the limits provided by JKR and

DMT theory. Experimentalists often assume that these bounds place tight limits on errors

in inferred values of w.

In the case of amorphous tips the magnitude of Nc is 30% higher than expected. The

higher than expected adhesion in small contacts may reflect a decrease in effective roughness

because long-wavelength height fluctuations are suppressed. Stepped tips show even larger

deviations from continuum theory that are strongly dependent on the size of the first terraces

[46]. Tips selected for imaging are likely to have the smallest terraces and the largest

deviations from continuum theory.

Adding adhesion introduces a substantial width to the edge of the contact, ranging from

the point where interactions first become attractive ra to the outer limits of attractive

interactions rc (Fig. 4). As the range of interactions increases, it becomes increasingly

difficult to fit both these characteristic radii and the pulloff force with the simple M-D

theory (Fig. 8). For short-range interactions, good fits are obtained with the measured w

for bent tips. Data for amorphous tips can only be fit by increasing w, due to the reduction in

effective roughness mentioned above (Fig. 6). For stepped tips the contact radius increases

in discrete jumps as successive terraces contact the surface.

The normal interfacial compliance leads to significant ambiguity in the definition of the

normal displacement as a function of load (Fig. 9). Continuum theory normally includes

only the substrate compliance, while experimental measures of the total tip displacement

δtip include the interfacial compliance. The substrate compliance was isolated by following

the displacement of substrate atoms, δsur and found to agree well with theory for bent tips

in the repulsive regime. Johnson and Greenwood’s extension of M-D theory [25] includes the

interfacial compliance in the attractive tail of the potential. It provides a good description of

29

δtip in the regime where ra = 0. Here δtip−δsur increases to the interaction range h0. Results

for amorphous tips show the greater adhesion noted above. Stepped tips follow continuum

theory at large loads but are qualitatively different at negative loads.

The most profound effects of tip geometry are seen in the lateral stiffness k and friction

F , which vary by one and two orders of magnitude respectively. Continuum theories for k

do not include the lateral interfacial compliance ki. This adds in series with the substrate

compliance ksub (Eq. (7)). Except for commensurate tips, ki << ksub and the interface

dominates the total stiffness [17]. Experiments have also seen a substantial reduction in the

expected lateral stiffness from this effect [51, 60].

The friction on non-adhesive commensurate tips (bent or stepped) increases linearly with

load, as frequently observed for macroscopic objects. In all other cases, F is a nonlinear

function of load. Our ability to directly measure contact radii allowed us to show that F

scales linearly with contact area for incommensurate, amorphous and adhesive bent com-

mensurate tips. These tips also show a linear scaling of k with radius. While these scalings

held for any choice of radius, the linear fits are offset from the origin. It appears that the

effective area contributing to friction and stiffness is often a little smaller than the area of

repulsive interactions corresponding to ra. As noted above, the offset from ra appears to

correspond to the finite range over which repulsive forces rise at the interface.

Experimental data [4, 5, 6, 7, 8, 9, 10, 11] for friction and stiffness have been fit to

continuum theory with the assumptions that F ∝ r2 and k ∝ r, but without the offsets

seen in Figs. 11 and 12. We showed that our data for bent and amorphous tips could be fit

in this way (Fig. 10), but that the fit parameters did not correspond to directly measured

values. This suggests that care should be taken in interpreting data in this manner.

We also examined the ratio F/k2. In continuum theory, this is related to the friction per

unit area τ efff through Eq. (9). Our results for F/k2 (Fig. 13) show the range of behaviors

observed in experiments, with a relatively constant value for adhesive cases, a rapid increase

at low loads in some nonadhesive cases, and a linear rise for non-adhesive stepped tips.

The directly measured values of τf (Table II) are smaller than τ efff by up to two orders

of magnitude, and have qualitatively different trends with tip geometry. The difference is

related to a reduction in the stiffness k due to interfacial compliance. This reduces the

inferred value of bulk shear modulus G′ and increases the calculated contact area at any

given area. We expect that experimental results for F/k2 will produce similar overestimates

30

of the true interfacial shear force.

It remains unclear why F and k should follow the observed dependence on ra. Analytic

arguments for clean, flat surfaces indicate that F is very sensitive to structure, with the forces

on commensurate, incommensurate and disordered surfaces scaling as different powers of

area [32, 50]. Only when glassy layers are introduced between the surfaces, does the friction

scale in a universal manner [32, 62, 63, 64]. Wenning and Muser [31] have argued that the

friction on clean, amorphous tips rises linearly with area because of a cancellation of two

factors, but have not considered k. Naively, one might expect that the length over which

the force rises is a constant fraction of the lattice spacing and that k is proportional to

F . However, the friction traces change with load and do not always drop to zero between

successive peaks. We hope that our results will motivate further analytic studies of this

problem, and simulations with glassy films and more realistic potentials.

While we have only considered single asperity contacts in this paper, it is likely that the

results are relevant more broadly. Many experimental surfaces have random roughness on

all scales that can be described by self-affine fractal scaling. Continuum models of contact

between such surfaces show that the radius of most of contacts is comparable to the lower

length scale cutoff in fractal scaling [56, 57]. This is typically less than a micrometer, sug-

gesting that typical contacts have nanometer scale dimensions where the effects considered

here will be relevant.

Acknowledgments

We thank G. G. Adams, R. W. Carpick, K. L. Johnson, M. H. Muser and I. Srid-

har for useful discussions. This material is based upon work supported by the National

Science Foundation under Grants No. DMR-0454947, CMS-0103408, PHY99-07949 and

CTS-0320907.

31

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33

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34

TABLE I: Relations between interaction strength ǫi, work of adhesion w, Nc and dimensionless

pulloff force Nc/(πwR) for tips with different atomic-scale geometries. The last column gives the

dimensionless pulloff force from M-D theory, NM−Dc , with the measured w and h0 = 0.5 σ. Values

are accurate to the last significant digit.

Tip geometry ǫiǫ w( ǫ

σ2 ) |Nc|( ǫσ )

|Nc|πwR

|NM−Dc |πwR

Commensurate 0.213 0.46 256 1.77 1.74

0.5 1.05 569 1.72 1.64

Incommensurate 0.535 0.46 258 1.79 1.74

0.5 0.45 238 1.68 1.74

Amorphous 1.0 0.46 326 2.26 1.74

0.5 0.23 136 1.88 1.83

Stepped 0.213 0.46 104 0.72 1.74

0.5 1.05 168 0.51 1.64

TABLE II: Frictional stresses τf and apparent moduli G′ evaluated from the derivatives of fits in

Figs. 11 and 12, respectively, as a function of tip geometry and work of adhesion w. Values of G′

are less than the effective shear modulus of the substrate, G∗ = 18.3 ǫ/σ3, and frictional stresses

are much smaller than expected from continuum expressions (Fig. 13). Statistical errorbars are

indicated in parentheses.

Tip geometry w(ǫσ−2) τf (ǫσ−3) G′(ǫσ−3)

Commensurate 0 1.35(7) 13.7(4)

0.46 1.82(4) 12.8(3)

Incommensurate 0 0.0044(7) 0.70(5)

0.46 0.0151(8) 1.0(3)

Amorphous 0 0.056(4) 1.53(5)

0.46 0.24(2) 4.3(6)

35

FIG. 1: Snapshots of atoms near the center regions (diameter 50 σ) of spherical tips with average

radius R=100 σ. From top to bottom, tips are made by bending a crystal, cutting a crystal

or cutting an amorphous solid. Three ratios η of the atomic spacing in bent crystals to that

in the substrate are considered; a dense case η = 0.05, a commensurate case η = 1, and an

incommensurate case η = 0.94437. The step structure of cut crystalline tips is not unique, leading

to variations in their behavior.

36

FIG. 2: (Color online) Local normal pressure vs. radius for five different tip geometries (a) dense

tip, (b) bent commensurate crystal, (c) bent incommensurate crystal, (d) amorphous, and (e)

stepped crystal. All are non-adhesive, and have the same nominal radius R = 100 σ, ǫi/ǫ = 1, and

normalized load N/(R2E∗) = 0.0018. Solid lines show the prediction of Hertz theory, dots show

the pressure on each surface atom, and circles in (c) and (d) show the mean pressure in radial bins

of width σ. Squares in (b) show the component of the tangential force directed radially from the

center of the contact. The azimuthal component is nearly zero, and tangential forces are much

smaller for other tips.

37

FIG. 3: Dimensionless plots of powers of (a) normal displacement δ, (b) contact radius a, (c) static

friction F , and (d) lateral stiffness k vs. the cube root of the normal force N for non-adhesive

tips with R = 100 σ. The powers of δ, a and k are chosen so that continuum theory predicts the

indicated straight solid lines. In (c), the solid line corresponds to F ∝ N , while dashed and dotted

lines are fits to F ∝ N2/3. The continuum predictions for δ and r are followed by a dense tip (stars).

Also shown are results for bent commensurate (pentagons), bent incommensurate (squares), amor-

phous (triangles) and stepped commensurate tips (circles). In (d), the total lateral stiffness (open

symbols) lies well below the continuum prediction because of the interfacial compliance. The stiff-

ness from the substrate alone (filled symbols) scales with radius, as expected from Hertz theory.

Numerical uncertainties are comparable to the symbol size.

38

FIG. 4: (Color online) Local pressure vs. radius for bent commensurate tips with rcut = 1.5 σ (filled

squares) and rcut = 2.2 σ (open squares) and with R = 100 σ, ǫi/ǫ = 0.5 and N/(R2E∗) = 0.0016.

Dashed lines show Maugis-Dugdale fits with the measured work of adhesion and h0 fit to the range

of the interactions. A dotted line shows a fit for rcut = 2.2 σ with h0 = 0.8 σ that improves

agreement with the measured pulloff force (Fig. 8). Characteristic radii ra, rb and rc are defined

by the locations where the radially averaged pressure first becomes zero, is most negative, and

finally vanishes, respectively.

39

FIG. 5: (Color online) Normal pressure vs. radius for four different tip geometries: (a) bent

commensurate crystal, (b) bent incommensurate crystal, (c) amorphous, and (d) stepped crystal.

In all cases the tip radius R = 100 σ, the normalized load N/(R2E∗) = 0.0023 and the surface

energy w = 0.46 ǫ/σ2. Dots show the normal pressure on each surface atom. In (b) and (c), circles

show the mean pressure in radial bins of width σ, and lines show the bent commensurate results

from (a). Horizontal dashed lines are at zero pressure.

40

FIG. 6: Variation of contact radii with external load for bent commensurate (squares), bent

incommensurate (triangles) and amorphous (circles) tips with R = 100 σ. In (a), the radius rb

where the force is most attractive is compared to JKR theory (solid line). In (b), values of ra (filled

symbols) and rc (open symbols) are compared to M-D theory (solid lines) with the JG extension

for rc when ra = 0 (dashed line). All continuum fits use the independently measured surface energy

w = 0.46 ǫ/σ2 and an interaction range h0 = 0.5 σ that is consistent with the potential range.

Numerical uncertainty in radii is comparable to the symbol size.

41

FIG. 7: Contact radii as a function of normal load for (a) bent commensurate, (b) bent incom-

mensurate, and (c) amorphous tips with ǫi = 0.5ǫ. While all have the same interaction strength,

the work of adhesion varies by more than a factor of four. Broken lines show M-D predictions

for ra (triangles) and rc (circles) with the indicated values of w and h0 = 0.5 σ. The value of rb

(squares) is closest to the JKR result (solid line). Deviations from the continuum predictions show

the same trends as in Fig. 6. Numerical uncertainty in radii is comparable to the symbol size.

42

FIG. 8: Fits of M-D theory (broken lines) to rc (squares) and ra (circles), and of JKR theory

(solid line) to rb (triangles) for a bent commensurate tip with rcut = 2.2 σ. For h0 = 1.2 σ (dashed

line) M-D theory fits both rc and ra at large loads, but the magnitude of the pulloff force is too

large. Decreasing h0 to 0.8 σ (dotted lines) gives excellent agreement with the pulloff force, but

not the radii (Fig. 4). Numerical uncertainty in radii is comparable to the symbol size.

43

FIG. 9: Normal displacement as a function of external load measured from (a) the depression of

the lowest substrate atom δsur and (b) the displacement of the tip relative to the height where the

first substrate atom exerts a repulsive force δtip. The tips are the same as in Fig. 6. The JKR

prediction is indicated by dotted lines, the M-D prediction by solid lines, and the JG extension for

ra = 0 with dashed lines. All are corrected for the finite dimensions of the substrate as described

in the text, and uncertainties are comparable to the symbol size. The difference ∆δ ≡ δtip − δsur

reflects the interfacial compliance, and is shown for the bent incommensurate tip in the inset.

44

FIG. 10: (Color online) Static friction F vs. load N for the indicated tip geometries with same

systems as in Fig. 6. Attempts to fit the data by assuming F is proportional to the area predicted

by JKR theory and the interpolation scheme of Ref. [29] are shown by dotted and solid lines,

respectively. Dashed lines are separate linear fits for the stepped tip for the cases where one or two

terraces are in repulsive contact. Numerical uncertainties are comparable to the symbol size.

45

FIG. 11: (Color online) Static friction F plotted against radius squared for the indicated tip

geometries with same parameters as in Fig. 6. Values of r2a (filled circles), r2b (open squares), and

r2c (filled triangles) are shown for adhesive tips. Open circles show F vs. a2 for non-adhesive tips

with data in (b) and (c) multiplied by a factor of two for clarity. Dashed lines show unconstrained

linear fits to each data set. Numerical uncertainties are comparable to the symbol size.

46

FIG. 12: (Color online) Lateral stiffness k plotted against radius for the indicated tip geometries

with same parameters as in Fig. 6. Values of ra (filled circles), rb (open squares), and rc (filled

triangles) are shown for adhesive tips. Open circles show F vs. a for non-adhesive tips. Broken

lines show unconstrained linear fits to each data set, and the solid line indicates the slope predicted

by continuum theory. Numerical uncertainties are comparable to the symbol size, except for the

adhesive data in (b) where they may be up to 50%.

47

FIG. 13: (Color online) Ratio of friction to stiffness squared as a function of load for the indicated

tip geometries (a) without and (b) with adhesion (w = 0.46 ǫ/σ2). Lines are guides to the eye.

Numerical uncertainties are comparable to the symbol size, except for the bent incommensurate

data in (b), where they may be as large as 50%.

48

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06

Contact of Single Asperities with Varying Adhesion: Comparing Continuum

Mechanics to Atomistic Simulations

Binquan Luan and Mark O. RobbinsDepartment of Physics and Astronomy, The Johns Hopkins University,

3400 N. Charles Street, Baltimore, Maryland 21218(Dated: March 22, 2006)

Atomistic simulations are used to test the equations of continuum contact mechanics in nanometerscale contacts. Nominally spherical tips, made by bending crystals or cutting crystalline or amor-phous solids, are pressed into a flat, elastic substrate. The normal displacement, contact radius,stress distribution, friction and lateral stiffness are examined as a function of load and adhesion.The atomic scale roughness present on any tip made of discrete atoms is shown to have profoundeffects on the results. Contact areas, local stresses, and the work of adhesion change by factors oftwo to four, and the friction and lateral stiffness vary by orders of magnitude. The microscopicfactors responsible for these changes are discussed. The results are also used to test methods foranalyzing experimental data with continuum theory to determine information, such as contact area,that can not be measured directly in nanometer scale contacts. Even when the data appear to befit by continuum theory, extracted quantities can differ substantially from their true values.

PACS numbers: 81.40.Pq 68.35.Np 62.20.Dc 68.37.Ps

I. INTRODUCTION

There has been rapidly growing interest in the behav-ior of materials at nanometer scales [1]. One motivationis to construct ever smaller machines [2], and a second isto improve material properties by controlling their struc-ture at nanometer scales [3]. For example, decreasingcrystallite size may increase yield strength by suppress-ing dislocation plasticity, and material properties maybe altered near free interfaces or grain boundaries. Tomake progress, this research area requires experimentaltools for characterizing nanoscale properties. Theoreti-cal models are also needed both to interpret experimentsand to allow new ideas to be evaluated.

One common approach for measuring local propertiesis to press tips with characteristic radii of 10 to 1000 nminto surfaces using an atomic force microscope (AFM) ornanoindenter [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Me-chanical properties are then extracted from the measuredforces and displacements using classic results from con-tinuum mechanics [16]. A potential problem with thisapproach is that continuum theories make two key as-sumptions that must fail as the size of contacting re-gions approaches atomic dimensions. One is to replacethe atomic structure in the bulk of the solid bodies bya continuous medium with internal stresses determinedby a continuously varying strain field. The second isto model interfaces by continuous, differentiable surfaceheights with interactions depending only on the surfaceseparation. Most authors go further and approximatethe contacting bodies by smooth spheres.

In a recent paper [17], we analyzed the limits of con-tinuum mechanics in describing nanometer scale contactsbetween non-adhesive surfaces with curvature typical ofexperimental probes. As in studies of other geometries[18, 19, 20], we found that behavior in the bulk couldbe described by continuum mechanics down to lengths

as small as two or three atomic diameters. However,the atomic structure of surfaces had profound conse-quences for much larger contacts. In particular, atomic-scale changes in the configuration of atoms on nominallycylindrical or spherical surfaces produced factor of twochanges in the width of the contacting region and thestress needed to produce plastic yield, and order of mag-nitude changes in friction and stiffness.

In this paper we briefly revisit non-adhesive contactswith an emphasis on the role of surface roughness. Wethen extend our atomistic studies to the more commoncase of adhesive interactions. One important result isthat the work of adhesion is very sensitive to smallchanges in the positions of surface atoms. Changes inother quantities generally mirror those for non-adhesivetips, and small differences in the magnitude of these ef-fects can be understood from geometrical considerations.The results are used to test continuum-based methodsof analyzing AFM measurements of friction and stiffness[4, 5, 6, 7, 8, 9, 10, 11]. We show that the models mayappear to provide a reasonable description when limitedinformation about the true contact structure is available.When the full range of information accessible to simu-lations is examined, one finds that the contact area andpressure distributions may be very different than inferredfrom the models.

Section II reviews continuum results for contact with-out and with adhesion, and briefly describes the effect ofsurface roughness. The methods used in our atomisticsimulations and the geometries of the tips are describedin Sec. III. Section IV presents results for purely repul-sive interactions and Sec. V describes trends with thestrength of adhesion. A summary and conclusions arepresented in Sec. VI.

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II. CONTINUUM CONTACT MECHANICS

As noted above, contact mechanics calculations assumethat the contacting solids are described by continuumelasticity so that the discrete atomic structure can beignored. In most cases the two solids are also assumed tobe isotropic with Young’s moduli E1 and E2 and Poissonratios ν1 and ν2. Then the results depend only on aneffective modulus E∗ satisfying:

1/E∗ ≡ (1− ν21 )/E1 + (1 − ν22)/E2. (1)

Three-dimensional crystalline solids are not isotropic, butthe theories can still be applied with an effective E∗ thatdepends on orientation and is determined numerically[16].Continuum theories also neglect the atomic structure

of the surface. In most cases the surfaces are assumedto be spherical, with radii R1 and R2. For elastic, fric-tionless solids the contact of two spheres is equivalent tocontact between a sphere of radius R = (R−1

1 + R−12 )−1

and a flat solid [16]. From Eq. (1), one may then mapcontact between any two spherical surfaces onto contactbetween a rigid sphere of radius R and a flat elastic solidof modulus E∗. This is the case considered in our simula-tions, and previous results indicate this mapping remainsapproximately correct at atomic scales [17].Non-adhesive contact is described by Hertz theory [16],

which assumes solids interact with an infinitely sharp andpurely repulsive “hard-wall” interaction. The surfacescontact in a circular region of radius a that increases withthe normal force or load N pushing the surfaces togetheras [16]:

a =

(

3NR

4E∗

)1/3

. (2)

The normal pressure p within the contact has a simplequadratic dependence on the radial distance from thecenter r:

p(r) =2aE∗

πR

√

1− r2

a2, (3)

and the surfaces separate slowly outside the contact. Thenormal displacement of the tip δ is related to a by:

δH = a2/R = (3NR

4E∗)2/3 , (4)

where the subscript H indicates the Hertz prediction andδH = 0 corresponds to the first contact between tip andsubstrate.Adhesion can be treated most simply in the oppo-

site limits of very short-range interactions considered byJohnson, Kendall and Roberts (JKR) [21] and of infi-nite range interactions considered by Derjaguin, Mullerand Toporov (DMT) [22]. The strength of adhesion ismeasured by the work of adhesion per unit area w. In

DMT theory the attractive forces just produce an ex-tra contribution to the normal force, so that N is re-placed by N + 2πwR in Eqs. (2) and (4). JKR theorytreats the edge of the contact as a crack tip and cal-culates the stress by adding the crack and Hertz solu-tions. The normal force in Eq. (2) is then replaced by

N + 3πwR +[

6πwRN + (3πwR)2]1/2

and the equationfor δ is modified (Sec. VB). The two approaches leadto very different functional relations between a and N .For example, the contact radius goes to zero at pullofffor DMT theory, but remains finite for JKR. They alsopredict different values of the pulloff force, Nc, where thesurfaces separate. The normalized pulloff force,Nc/πwR,is -3/2 in JKR theory and -2 for DMT. Finally, the sur-faces separate outside the contact with infinite slope inJKR theory, and gradually in DMT theory.The Maugis-Dugdale (M-D) model [23] provides a sim-

ple interpolation between the JKR and DMT limits. Thesurfaces are assumed to have a hard-wall contact interac-tion that prevents any interpenetration, plus a constantattractive force per unit area, σ0, that extends over afinite distance h0. The work of adhesion is just the in-tegral of the attractive force, implying σ0h0 = w. TheM-D model produces coupled equations for the contactpressure that can be solved to yield a relation betweenthe load, normal displacement, and area. As discussedfurther in Section V, the edge of the contact is broadenedby the finite interaction range, making it useful to definethree characteristic radii that converge to the JKR valuefor a in the limit of short-range interactions.Maugis introduced a transition parameter [23]

λ ≡(

9Rw2

2πE∗2h30

)1/3

, (5)

that measures the ratio of the normal displacement atpulloff from JKR theory to the interaction range h0. Ta-bor [24] had previously defined a similar parameter, µ,that is about 16% smaller than λ for typical interactionpotentials [25]. Johnson and Greenwood [25] have pro-vided an adhesion map characterizing the range of λ overwhich different models are valid. For λ > 5 the inter-action range is short and JKR theory is accurate, whileDMT is accurate for λ < 0.1. For most materials, both h0

and the ratio w/E∗ are of order 1 nm. The JKR limit isonly reached by increasing R to macroscopic dimensionsof micrometers or larger. JKR theory has been tested inexperiments with centimeter scale radii using the surfaceforce apparatus (SFA) [26] and hemispherical elastomers[27, 28]. Scanning probe microscope tips typically have Rbetween 10 and 100 nm, and the value of λ ∼ 0.1 to 1 liesbetween JKR and DMT limits [4]. The same is true inour simulations, where λ for adhesive tips varies between0.1 and 0.75. For this reason we will compare our resultsto M-D theory below. We also found it useful to use asimple interpolation scheme suggested by Schwarz [29].Both he and Carpick et al. [30] have proposed formulaefor the contact radius that interpolate smoothly between

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DMT and JKR. These approaches have been attractivein analyzing experimental data because of their simpleanalytic forms.No direct measurement of contact area has been possi-

ble in nanometer scale single asperity contacts. Instead,the contact area has been determined by measurementsof contact stiffness [4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 25],conductance [9], or friction [4, 5, 6, 7, 9, 10, 11]. Thevalidity of these approaches is not clear [17], and will betested below. The stiffness against normal displacementsof the surfaces can be determined from the derivativeof N with respect to δ in M-D theory. The tangentialstiffness k is normally calculated by assuming frictionprevents sliding at the interface, even though all theoriesdescribed above assume zero friction in calculating thecontact area. With this assumption k = 8G∗a, where G∗

is the effective bulk shear modulus. Relating the frictionto contact area requires assumptions about the frictionlaw. Many authors have assumed that the friction is pro-portional to area [4, 5, 6, 7, 8, 9, 10, 11], but simulations[17, 31, 32] and experiments in larger contacts [33, 34]show that this need not be the case.The effect of surface roughness on contact has been

considered within the continuum framework [16]. In gen-eral, results must be obtained numerically. One keyparameter is the ratio of the root mean squared (rms)roughness of the surface, ∆, to the normal displacementδ. When ∆/δ < 0.05, results for nonadhesive contactslie within a few percent of Hertz theory [16]. As ∆/δincreases, the contact area broadens and the pressure inthe central region decreases. Adhesion is more sensitiveto roughness [35]. The analysis is complicated by thefact that ∆ usually depends on the range of lengths overwhich it is measured. The natural upper bound corre-sponds to the contact diameter and increases with load,while the lower bound at atomic scales is unclear. Therole of roughness is discussed further in Secs. IV and V.

III. SIMULATION METHODS

We consider contact between a rigid spherical tip andan elastic substrate with effective modulus E∗. As notedabove, continuum theory predicts that this problem isequivalent to contact between two elastic bodies, andwe found this equivalence was fairly accurate in previousstudies of non-adhesive contact [17]. To ensure that anydeviations from the continuum theories described aboveare associated only with atomic structure, the substrateis perfectly elastic. Continuum theories make no assump-tions about the nature of the atomic structure and inter-actions within the solids. Thus any geometry and in-teraction potentials can be used to explore the type ofdeviations from continuum theory that may be producedby atomic structure. We use a flat crystalline substrateto minimize surface roughness, and use tips with the min-imum roughness consistent with atomic structure. Theinteractions are simple pair potentials that are widely

used in studies that explore generic behavior [36]. Theyillustrate the type of deviations from continuum theorythat may be expected, but the magnitude of deviationsfor real materials will depend on their exact geometryand interactions.Atoms are placed on sites of a face-centered cubic (fcc)

crystal with a (001) surface. We define a characteris-tic length σ so that the volume per atom is σ3 and thenearest-neighbor spacing is 21/6 σ. Nearest-neighbors arecoupled by ideal Hookean springs with spring constantκ. Periodic boundary conditions are applied along thesurface of the substrate with period L in each direction.The substrate has a finite depth D and the bottom isheld fixed. For the results shown below, L = 190.5 σ andD = 189.3 σ. The continuum theories assume a semi-infinite substrate, and we considered smaller and largerL and D to evaluate their effect on calculated quanti-ties. Finite-size corrections for the contact radius andlateral stiffness are negligible for a/D < 0.1 [16], whichcovers the relevant range of a/R < 0.2. Corrections tothe normal displacement are large enough to affect plot-ted values. We found that the leading analytic correc-tions [37, 38, 39] were sufficient to fit our results at largeloads, as discussed in Sec. VB. Note that previous sim-ulations of AFM contact have used much shallower sub-strates (D ∼ 10 σ) [40, 41, 42, 43, 44, 45]. This placesthem in a very different limit than continuum theories, al-though they provide interesting insight into local atomicrearrangements.Three examples of atomic realizations of spherical tips

are shown in Fig. 1. All are identical from the continuumperspective, deviating from a perfect sphere by at mostσ. The smoothest one is a slab of f.c.c. crystal bentinto a sphere. The amorphous and stepped tips wereobtained by cutting spheres from a bulk glass or crys-tal, and are probably more typical of AFM tips [4, 46].Results for crystalline tips are very sensitive to the ra-tio η between their nearest-neighbor spacing and thatof the substrate, as well as their crystalline alignment[31, 32]. We will contrast results for an aligned commen-surate tip with η = 1 to those for an incommensuratetip where η = 0.94437. To mimic the perfectly smoothsurfaces assumed in continuum theory, we also show re-sults for a high density tip with η = 0.05. In all casesR = 100 σ ∼ 30 nm, which is a typical value for AFMtips. Results for larger radius show bigger absolute devi-ations from continuum predictions, but smaller fractionaldeviations [17].Atoms on the tip interact with the top layer of sub-

strate atoms via a truncated Lennard-Jones (LJ) poten-tial [36]

VLJ = −4ǫi

[

(σ

r

)6

−(σ

r

)12]

− Vcut, r < rcut (6)

where ǫi characterizes the adhesive binding energy, thepotential vanishes for r > rcut, and the constant Vcut

is subtracted so that the potential is continuous at rcut.Purely repulsive interactions are created by truncating

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FIG. 1: Snapshots of atoms near the center regions (diameter50 σ) of spherical tips with average radius R=100 σ. Fromtop to bottom, tips are made by bending a crystal, cutting acrystal or cutting an amorphous solid. Three ratios η of theatomic spacing in bent crystals to that in the substrate areconsidered; a dense case η = 0.05, a commensurate case η = 1,and an incommensurate case η = 0.94437. The step structureof cut crystalline tips is not unique, leading to variations intheir behavior.

the LJ potential at its minimum rcut = 21/6 σ. Studiesof adhesion use rcut = 1.5 σ or rcut = 2.2 σ to explorethe effect of the range of the potential.

In order to compare the effective strength of adhesiveinteractions and the cohesive energy of the solid sub-strate, we introduce a unit of energy ǫ defined so that thespring constant between substrate atoms κ = 50 ǫ/σ2. Ifthe solid atoms interacted with a truncated LJ poten-tial with ǫ and rcut = 1.5 σ, they would have the sameequilibrium lattice constant and nearly the same springconstant, κ = 57 ǫ/σ2, at low temperatures and small de-formations. Thus ǫi/ǫ is approximately equal to the ratioof the interfacial binding energy to the cohesive bindingenergy in the substrate.

The elastic properties of the substrate are not isotropic.We measure an effective modulus E∗ = 55.0 ǫ/σ3 forour geometry using Hertzian contact of a high densitytip. This is between the values calculated from theYoung’s moduli in different directions. The sound ve-locity is also anisotropic. We find longitudinal soundvelocities of 8.5 and 9.5 σ/tLJ and shear velocities of 5.2and 5.7 σ/tLJ along the (001) and (111) directions, re-spectively. Here tLJ is the natural characteristic timeunit, tLJ =

√

mσ2/ǫ, where m is the mass of each sub-strate atom. The effective shear modulus for lateral tipdisplacements is G∗ = 18.3 ǫ/σ3.

The simulations were run with the Large-scaleAtomic/Molecular Massively Parallel Simulator(LAMMPS) code [47, 48]. The equations of mo-tion were integrated using the velocity-Verlet algorithmwith time step 0.005 tLJ [36]. Temperature, T , only en-ters continuum theory through its effect on constitutiveproperties, and it is convenient to run simulations at

low temperatures to minimize fluctuations. A Langevinthermostat was applied to solid atoms to maintainT=0.0001 ǫ/kB, where kB is Boltzmann’s constant.This is about four orders of magnitude below themelting temperature of a Lennard-Jones solid. Thedamping rate was 0.1 t−1

LJ , and damping was onlyapplied perpendicular to the sliding direction in frictionmeasurements.

In simulations, a tip was initially placed about rcutabove the substrate. It was then brought into contact byapplying a very small load, with the lateral position keptfixed. The load was changed in discrete steps and the sys-tem was allowed to equilibrate for 350 tLJ at each loadbefore making measurements. This interval is about 20times longer than the time for sound to propagate acrossthe substrate, and allowed full stress equilibration. Re-sults from loading and unloading cycles showed no no-ticeable hysteresis. To obtain results near the pulloffforce, we moved the tip away from the substrate at avery slow velocity v = 0.0003 σ/tLJ and averaged resultsover small ranges of displacement. This approach wasconsistent with constant load measurements and allowedus to reach the region that is unstable at constant load.

To compare to continuum predictions we calculated thestresses exerted on the substrate by the tip. The forcefrom the tip on each substrate atom was divided by thearea per atom to get the local stresses. These were thendecomposed into a normal stress or pressure p on thesubstrate, and a tangential or shear stress τsur. The con-tinuum theories described in Sec. II assume that the pro-jection of the force normal to the undeformed substrateequals the projection normal to the locally deformed sub-strate. This is valid in the assumed limits of a/R << 1and τsur = 0. It is also valid for most of our simulations(within < 2%), but not for the case of bent commen-surate tips where τsur becomes significant. Normal andtangential stresses for bent commensurate tips were ob-tained using the local surface orientation of the nominallyspherical tip. Correcting for the orientation changed thenormal stress by less than 5% of the peak value, and theshear stress by less than 20%.

Friction forces are known to vary with many parame-ters [49]. Of particular concern is the dependence on ex-trinsic quantities such as the stiffness of the system thatimposes lateral motion. Results at constant normal loadare often very different than those at fixed height, motionperpendicular to the nominal sliding direction can raiseor lower friction, and the kinetic friction can be almostcompletely eliminated in very stiff systems [50, 51]. Afull re-examination of these effects is beyond the scopeof this paper. Our sole goal is to measure the friction ina consistent manner that allows us to contrast the loaddependent friction for different tip geometries and mini-mizes artifacts from system compliance.

In friction simulations, the tip is sheared at a constantlow velocity v′ = 0.01 σ/tLJ along the (100) directionwith a constant normal load. This is typical of AFMexperiments where the low normal stiffness of the can-

5

tilever leads to a nearly constant normal load, and thehigh lateral stiffness limits lateral motion in the directionperpendicular to the sliding direction. The measured fric-tion force varies periodically with time as the tip movesby a lattice constant of the substrate. The time-averagedor kinetic friction during sliding is very sensitive to bothlateral stiffness and load [51]. We focus instead on thepeak force, which is less sensitive. In the limit of low slid-ing velocities this would correspond to the static friction.For bent and stepped commensurate tips there is a sin-gle strong friction peak. For incommensurate and amor-phous tips, there may be multiple peaks of different sizecorresponding to escape from different metastable energyminima [32, 50]. The static friction was determined fromthe highest of these friction peaks, since lateral motionwould stop at any lower force. With a single peak perperiod, the time between peaks is ∼ σ/v′ = 100 tLJ .This is several times the sound propagation time, andthe measured force should be close to the static friction.For incommensurate tips the time between peaks was anorder of magnitude smaller and dynamic effects may bemore significant. However, they are not expected to af-fect the load dependence significantly, and are much toosmall to affect the dramatic difference between incom-mensurate and other tips.The total lateral stiffness of the system, k, corresponds

to the derivative of F with lateral tip displacement eval-uated at a potential energy minimum. Since the tip isrigid, k is determined by displacements in the substrateand at the interface. The interfacial stiffness ki and sub-strate stiffness ksub add in series because stress is trans-mitted through interfacial interactions to the substrate.Thus the total stiffness is [17, 51]:

k−1 = k−1sub + k−1

i . (7)

If the tip were not rigid, it would also contribute a termto the right-hand-side of Eq. (7).We evaluate k from the derivative of F during sliding,

making the assumption that the results are in the qua-sisatic limit. For bent and stepped commensurate tipsthere is a single potential energy minima, and for amor-phous tips one minimum dominated the periodic force.For incommensurate tips, there are many closely spacedminima and we evaluate k from the derivative in the min-imum preceding the largest friction peak. Due to thesmall magnitude of forces and short time intervals, therelative errors in these values are as big as 50%. To esti-mate the lateral stiffness in the substrate, ksub, we fix therelative positions of those substrate atoms that lie insidethe contact, and move them laterally at a slow velocity.The total force between these atoms and the rest of thesubstrate is measured and its derivative with respect todistance gives the lateral stiffness in the substrate. Inprincipal, there might also be some relative displacementbetween atoms in the contact that is not captured by thisapproach, but the results for the substrate stiffness areconsistent with continuum predictions.Values of the adhesion energy per unit area w were

obtained for flat, rigid surfaces of the same nominal ge-ometry as the tip. For bent crystal tips (Fig. 1), the tipwas just flattened back into a crystal. For stepped tips,we used an unstepped crystal with the same spacing andinteractions. For amorphous tips, an amorphous solidwas cleaved with a flat surface rather than a sphere. Theresulting surfaces were then brought into contact withthe substrate and allowed to equilibrate at zero load. Atthe low temperatures used here, the adhesion energy isjust the potential energy difference between contactingand separated configurations.

IV. NONADHESIVE CONTACTS

A. Pressure distribution

Figure 2 contrasts the distribution of normal pressurep under five tips: (a) dense, (b) bent commensurate, (c)bent incommensurate, (d) amorphous and (e) stepped.In each case, R = 100 σ and the dimensionless load isN/(R2E∗) = 0.0018. Hertz theory predicts the samepressure distribution (solid lines) for all tips. Pointsshow the actual local pressure on each substrate atomas a function of radial distance r from the center of thespherical tip, and circles in (c) and (d) show the aver-age over bins of width σ. Clearly, small deviations inatomic structure lead to large changes in the mean pres-sure and the magnitude of pressure fluctuations. We findthat these deviations become larger as N is decreased,and the contact radius drops closer to the atomic size.One possible source of deviations from Hertz theory

is friction, but we find the mean tangential forces aresmall in most cases. The exception is the bent commen-surate tip (Fig. 2(b)), where the tangential stress riseswith r and is comparable to the normal stress near theedge of the contact. This result is not surprising giventhe high friction measured for commensurate tips below,and reflects the strong tendency for atoms in the sub-strate to remain locked in epitaxial registry with atomsin the tip. However, the deviation from Hertz theory is inthe opposite direction from that expected from friction.Since this contact was made by gradually increasing theload, friction should decrease the contact size rather thanbroadening it.Another possible origin of the deviations from Herts

theory is surface roughness. From continuum theory(Sec. II), this is characterized by the ratio of rms sur-face roughness ∆ to normal displacement δ. The normaldisplacement for all tips is about the same, δ ≈ 1.5 σ,but ∆ is difficult to define. The reason is that there is nounique definition of the surface height for a given set ofatomic positions. For example, one might conclude that∆ = 0 for the substrate, since all atoms lie on the sameplane. However, if a tip atom were moved over the surfacewith a small load, its height would increase as it movedover substrate atoms and be lowest at sites centered be-tween four substrate atoms [50]. For the parameters used

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FIG. 2: (Color online) Local normal pressure vs. radius forfive different tip geometries (a) dense tip, (b) bent commensu-rate crystal, (c) bent incommensurate crystal, (d) amorphous,and (e) stepped crystal. All are non-adhesive, and have thesame nominal radius R = 100 σ, ǫi/ǫ = 1, and normalizedload N/(R2E∗) = 0.0018. Solid lines show the prediction ofHertz theory, dots show the pressure on each surface atom,and circles in (c) and (d) show the mean pressure in radialbins of width σ. Squares in (b) show the component of thetangential force directed radially from the center of the con-tact. The azimuthal component is nearly zero, and tangentialforces are much smaller for other tips.

here, the total height change is about 0.33 σ. Similardeviations from a sphere are obtained for the bent com-mensurate and incommensurate tips. The height changedecreases as the ratio of the nearest-neighbor spacing tothe Lennard-Jones diameter for interfacial interactionsdecreases, and is only 0.0007 σ for the dense tip. Amor-phous and stepped tips have additional roughness as-sociated with variations in the position of atomic cen-ters relative to a sphere. The total variation is aboutσ, or about three times the height change as an atommoves over the substrate. A reasonable estimate is that∆/δ < 0.1 for the bent commensurate and incommensu-rate tips, ∆/δ < 10−3 for the dense tip, and ∆/δ ∼ 0.3for the amorphous and stepped tips. However, the ambi-guity in ∆ is one of the difficulties in applying continuumtheory in nanoscale contacts.

The closely spaced atoms on the dense tip approximatea continuous sphere, and the resulting pressure distribu-

tion is very close to Hertz theory (Fig. 2(a)). Results forthe bent commensurate tip are slightly farther from Hertztheory. The deviations can not be attributed to rough-ness, because fluctuations at a given r are small, and thepressure in the central region is not decreased. The mainchange is to smear the predicted sharp pressure drop atthe edge of the contact. This can be attributed to thefinite range of the repulsive potential between surfaces.

We can estimate the effective interaction range by thechange in height of an atom, dh = 0.04 σ, as p/E∗

decreases from 0.1 to 0. The effective range is muchsmaller for the dense tip because ∼ 400 times as manyatoms contribute to the repulsive potential. In Hertz the-ory [16], the separation between surfaces only increaseswith distance (r − a) from the edge of the contact as(8/3π)(r − a)3/2(2a)1/2/R. Equating this to dh givesr− a ≈ 1 σ for the bent commensurate tip, which is veryclose to the range over which the edge of the contact isshifted from the Hertz prediction. Note that this analysispredicts that the shift in the edge of the contact will growas

√R, and simulations with larger R confirm this. How-

ever, the fractional change in a decreases as 1/√R. The

larger values of pressure at low r result from the greaterstiffness of the repulsive potential as p increases. All ofthe above effects could be included in continuum theoryby changing the form of the repulsive potential [52].

For bent incommensurate and amorphous tips (Fig.2 (c) and (d)), the variations in pressure at a given rare as large as the mean (circles) [53]. While all atomson the commensurate tip can simultaneously fit betweensubstrate atoms, minimizing fluctuations in force, atomson the incommensurate tip sample all lateral positionsand experience a range of forces at a given height. Themean pressure for the incommensurate tip remains fairlyclose to the commensurate results, but there is slightlymore smearing at large r due to the variations in relativeheight of substrate and tip atoms. The mean pressureon the amorphous tip shows the depression at small rand increase at large r that are expected for rough tipsin continuum theory [16]. The magnitude of the centraldrop is about 18%, which is consistent with ∆/δ ∼ 0.2 incontinuum theory (Fig. 13.12 of Ref. [16]). The lack ofa noticeable drop for incommensurate tips implies thatthe effective ∆/δ < .03. The implication is that theincoherent height changes on amorphous tips contributeto the effective roughness in continuum theory, while theatomic corrugation on bent tips does not. The effectiveroughness in both cases is about 0.1 σ smaller than ourestimates for ∆ above.

Results for stepped tips show the largest deviationsfrom Hertz theory, and they are qualitatively differentthan those produced by random roughness. The ter-raced geometry of this tip (Fig. 1) is closest to thatof a flat punch. In continuum theory, the pressure ona flat punch is smallest in the center, and diverges asthe inverse square root of the distance from the edge.The simulation results show qualitatively similar behav-ior. The main effect of atomic structure is to cut off

7

the singularity at a distance corresponding to an atomicseparation. Similar effects are observed in simulations ofother geometries [18, 19, 20]. Note that the terraces areonly flat because the sphere was cut from a crystal thatwas aligned with the substrate. We also examined tipscut from a crystal that was slightly tilted away from the(001) direction [46]. This produces inclined terraces thatcontact first along one edge. The resulting pressure dis-tribution is very different, and closest to the continuumsolution for contact by an asymmetric wedge.Figure 2 has an important general implication about

the probability P (p) of finding a local pressure p at apoint in the contact. For smoothly curved surfaces, con-tinuum theory predicts that the derivative of the pressurediverges at the edge of the contact [16]. Thus P (p) → 0as p → 0 [54]. The finite resolution at atomic scales al-ways smears out the change in p, leading to a non-zerovalue of P (0). Indeed, the approximately constant valueof dp/dr near the contact edge in Fig. 2 leads to a localmaximum in P at p = 0. Similar behavior is observed forrandomly rough atomic contacts [55] and in continuumcalculations for piecewise planar surfaces [56, 57].Plastic deformation is usually assumed to occur when

the deviatoric shear stress τs exceeds the yield stress ofthe material. In Hertz theory, τs reaches a maximumvalue at a depth of about 0.5a. The pressure variationsat the surface shown in Fig. 2 lead to changes in boththe magnitude and position of the peak shear stress [17].Factors of two or more are typical for amorphous andstepped tips. Thus tip geometry may have a significantimpact on the onset of yield. Of course atomistic ef-fects also influence the yield stress at nanometer scales,and a full evaluation of this effect is left to future work.Saint-Venant’s principal implies that the pressure distri-bution should become independent of tip geometry atdepths greater than about 3a, but the shear stress atthese depths is substantially smaller than peak valuesand yield is unlikely to occur there.

B. Variations with load

Figure 3 shows the load dependence of (a) normal dis-placement, (b) radius, (c) friction and (d) lateral stiffnessfor the same tips as Fig. 2. Each quantity is raised toa power that is chosen so that Hertz theory predicts theindicated straight line. A small finite-depth correction(∼ 2%) is applied to the Hertz prediction for δ (Eq. (8)).As also found for cylindrical contacts [17], the normal

displacement shows the smallest deviation from Hertztheory because it represents a mean response of manysubstrate atoms. Results for all bent crystals are nearlyindistinguishable from the straight line. Results for thestepped surface are lower at small loads. Since the entiretip bottom contacts simultaneously, it takes a larger loadto push the tip into the substrate. The amorphous resultsare shifted upwards by a fairly constant distance of about0.2 σ. We define the origin of δ as the tip height where

FIG. 3: Dimensionless plots of powers of (a) normal dis-placement δ, (b) contact radius a, (c) static friction F , and(d) lateral stiffness k vs. the cube root of the normal forceN for non-adhesive tips with R = 100 σ. The powers of δ, aand k are chosen so that continuum theory predicts the indi-cated straight solid lines. In (c), the solid line corresponds to

F ∝ N , while dashed and dotted lines are fits to F ∝ N2/3.The continuum predictions for δ and r are followed by a densetip (stars). Also shown are results for bent commensurate(pentagons), bent incommensurate (squares), amorphous (tri-angles) and stepped commensurate tips (circles). In (d), thetotal lateral stiffness (open symbols) lies well below the con-tinuum prediction because of the interfacial compliance. Thestiffness from the substrate alone (filled symbols) scales withradius, as expected from Hertz theory. Numerical uncertain-ties are comparable to the symbol size.

the first substrate atom exerts a repulsive force on thetip. This is strongly dependent on the height of the low-est tip atom, while subsequent deformation is controlledby the mean tip surface. Agreement with Hertz is greatlyimproved by shifting the amorphous curve by this smallheight difference. Note that the zero of δ is difficult todetermine experimentally and is usually taken as a fit pa-rameter. If this is done, even results for the amorphoussystem yield values of R and E∗ that are within 10% oftheir true values. Thus measurements of δ, interpretedwith continuum theory for spheres, can provide usefulvalues of elastic properties at nanometer scales.

As expected from the observed pressure distributions(Fig. 2), the contact radius is generally larger than theHertz prediction. The shift is smallest for the dense tipbecause it approximates a continuous surface and thehigh density leads to a repulsive potential that rises more

8

than a hundred times more rapidly than for other tips.Results for bent crystal and amorphous tips are shiftedupwards by fairly load-independent offsets of ∼ 1 − 3 σ,leading to large fractional errors at low loads (up to100%). The stepped crystal shows qualitatively differ-ent behavior, with a rising in discrete steps as sequentialterraces come into contact. Note that the size of thefirst terrace is not unique, but depends on the registrybetween the bounding sphere and crystalline lattice [46].Larger deviations may be observed when the first stephas very few atoms. Such tips may be more likely to bechosen for AFM studies because they tend to give sharperimages.

In order to predict the friction between surfaces, onemust make assumptions about how F depends on areaand load. The straight line in Fig. 3(c) corresponds toa friction force that is proportional to load. Static fric-tion values for bent and stepped commensurate surfacesare consistent with this line and a coefficient of frictionµ ≡ F/N = 0.63. Analytic [32] and simple numerical [58]models show that this is a general feature of mated sur-faces where each tip atom has the same lateral positionrelative to substrate atoms. The friction on amorphousand incommensurate surfaces is always lower and scalesmore closely with the contact area, as indicated by brokenline fits to F ∝ N2/3 and discussed further in Sec. VB[59]. Many authors have made this assumption in fittingexperimental data, but it is not obvious why it shouldhold. The friction per unit area between flat amorphoussurfaces decreases as the square root of the area, butrises linearly with the normal pressure [32]. Wenningand Muser have noted that these two factors may com-bine for spherical contacts to give a net friction that riseslinearly with area [31]. However, their argument wouldpredict that the frictional force in a cylinder-on-flat ge-ometry would not scale with area, and our previous simu-lations found that it did [17]. Continuum theory predictsthat the lateral stiffness k = 8G∗a, and should follow thestraight line in Fig. 3(d). Measured values of the totalstiffness (open symbols) are always substantially lower.This is because continuum theory assumes that there isno displacement at the interface, only within the sub-strate. In reality, the frictional force is always associatedwith atomic scale displacements of interfacial atoms rel-ative to a local energy minimum [50, 58]. The derivativeof force with displacement corresponds to an interfacialstiffness ki that adds in series with the substrate contribu-tion (Eq. (7)) [17, 51]. Our numerical results show thatki can reduce k by more than an order of magnitude,particularly for tips where F is small. We also evalu-ated the substrate stiffness ksub by forcing all contactingsubstrate atoms to move together. These results (filledsymbols) lie much closer to the Hertz prediction. Onlythe stepped tip shows large discrepancies, and these arecorrelated with the large deviation between the measuredand predicted contact radii.

FIG. 4: (Color online) Local pressure vs. radius for bentcommensurate tips with rcut = 1.5 σ (filled squares) andrcut = 2.2 σ (open squares) and with R = 100 σ, ǫi/ǫ = 0.5and N/(R2E∗) = 0.0016. Dashed lines show Maugis-Dugdalefits with the measured work of adhesion and h0 fit to the rangeof the interactions. A dotted line shows a fit for rcut = 2.2 σwith h0 = 0.8 σ that improves agreement with the measuredpulloff force (Fig. 8). Characteristic radii ra, rb and rc aredefined by the locations where the radially averaged pressurefirst becomes zero, is most negative, and finally vanishes, re-spectively.

V. ADHESIVE CONTACTS

A. Pressure distribution

Figure 4 compares the calculated pressure distributionin adhesive contacts with the Maugis-Dugdale prediction(lines). A bent commensurate tip was used to minimizedeviations from continuum predictions for a sphere. Re-sults for two different rcut are presented to indicate diffi-culties in fitting longer-range interactions to M-D theory.The work of adhesion was calculated for unbent surfaces(Sec. III) with ǫi/ǫ = 0.5, yielding w = 1.05 ǫ/σ2 and1.65 ǫ/σ2 for rcut = 1.5 σ and 2.2 σ, respectively. Thisleaves only one fitting parameter in M-D theory. Forthe dashed lines, the width of the attractive interactionh0 = w/σ0 was chosen to coincide with the range ofthe atomic potential. The dotted line shows a fit withh0 = 0.8 σ for rcut = 2.2 σ, which gives better values forthe pulloff force, but poorer radii (Sec. VB).

In M-D theory, it is common to identify two radii, aand c, with the inner and outer edges of the plateau inthe pressure, respectively [23, 25]. For r < a the surfacesare in hard-sphere contact, and for a < r < c they areseparated and feel the constant attraction. The contin-uously varying interactions between atoms in our sim-

9

ulations lead to qualitatively different behavior. Thereis no sharp transition where the surfaces begin to sepa-rate, and the attraction shows a smooth rise to a peak,followed by a decay. To facilitate comparison to contin-uum theories, we introduce the three characteristic radiiindicated by arrows for each rcut. The innermost, ra, cor-responds to the point where the interaction changes fromrepulsive to attractive and can be calculated in M-D the-ory. The outermost, rc, corresponds to c in M-D theory– the point where interactions end. The middle, rb, cor-responds to the peak attraction. It was also studied byGreenwood [52] for contact of a featureless sphere thatinteracted with the flat substrate through Lennard-Jonesinteractions. He found rb lay close to the JKR predictionfor contact radius at large loads. All three radii convergein the limit of repulsive interactions or the JKR limit ofan infinitely narrow interaction range.

At small radii the atomistic results for p lie above M-D theory, and they drop below near r/σ = 10. Thesedeviations can be attributed to the increasing stiffnessof the repulsion between tip and substrate atoms withincreasing pressure. Just as in the non-adhesive case, thestiffer interactions in the center of the tip lead to biggerpressures for a given interpenetration. The change inpressure with separation produces less smearing at theedge of the repulsive region of the contact than for thenon-adhesive case (Fig. 2(b)), and the values of ra arevery close to M-D theory. This can be understood fromthe fact that surfaces separate much more rapidly in JKRtheory (∝ (r− a)1/2) than in Hertz theory (∝ (r− a)3/2)(Sec. IV). Thus the same height change in the repulsiveregion corresponds to a much smaller change in radius.Of course the finite range of the attractive tail of thepotential leads to a broad region of adhesive forces out torc. The continuous variation of p in the attractive tail isonly crudely approximated by M-D theory. The difficultyin determining the optimum choice for h0 increases withthe range of interactions, as discussed further below.

Figure 5 shows the effect of tip geometry on pressuredistribution. We found that the work of adhesion wasvery sensitive to tip geometry. To compensate for this ef-fect, we varied ǫi (Table I) to yield the same w for the tipsin Fig. 5. Then all tips should have the same pressuredistribution in continuum theory. The M-D predictionsfor p are not shown because even the bent commensuratetips produce significantly different results (Fig. 4). In-stead, we compare other tips to the bent commensuratetip.

As for non-adhesive tips, local fluctuations in pressureare small for commensurate tips (Fig. 5(a) and (d)) andcomparable to the mean pressure for incommensurate oramorphous tips (Fig. 5(b) and (c)). Note however, thatthe fluctuations become smaller in the adhesive regime(large r). This is because the potential varies more slowlywith height, so fluctuations in the separation betweenatoms have less effect on p. One consequence is that theouter edge of the contact is nearly the same for com-mensurate and incommensurate tips. The radii for the

TABLE I: Relations between interaction strength ǫi, work ofadhesion w, Nc and dimensionless pulloff force Nc/(πwR) fortips with different atomic-scale geometries. The last columngives the dimensionless pulloff force from M-D theory, NM−D

c ,with the measured w and h0 = 0.5 σ. Values are accurate tothe last significant digit.

Tip geometry ǫiǫ

w( ǫσ2 ) |Nc|(

ǫσ) |Nc|

πwR

|NM−D

c|

πwR

Commensurate 0.213 0.46 256 1.77 1.740.5 1.05 569 1.72 1.64

Incommensurate 0.535 0.46 258 1.79 1.740.5 0.45 238 1.68 1.74

Amorphous 1.0 0.46 326 2.26 1.740.5 0.23 136 1.88 1.83

Stepped 0.213 0.46 104 0.72 1.740.5 1.05 168 0.51 1.64

amorphous tip are significantly larger than bent tips, pre-sumably because of a much greater effective roughness.As for nonadhesive tips, the mean pressure on incom-mensurate tips is close to the commensurate results. Ad-hesion reduces the roughness-induced drop in pressure inthe central region of the amorphous tip. For the steppedtip, the contact radius is dominated by the size of theterraces, but adhesive tails are visible outside the edgeof each terrace. Only rc is easily defined for the steppedtips. This increases in a nearly stepwise manner, and itsload dependence is not shown below. For the amorphousand incommensurate tips, radii are determined from themean pressure at a given radius (open circles). Errorbarsare less than 0.5 σ.

B. Variations of radius and displacement with load

Figure 6(b) compares the measured contact radii forthe tips of Fig. 5 to M-D theory as load is varied. Thesimulation results for rc (open symbols) decrease withdecreasing load as ra (closed symbols) decreases to zero.All interactions in the contact are then adhesive. Asrc continues to drop, the area contributing to adhesiondrops, and the load rises back toward zero. This regime isnot considered in the original M-D theory. The extensionto ra = 0 by Johnson and Greenwood (JG) [25] is shownby a dashed line in the figure. Along this line the stressin the contact is constant, giving N = −σ0 πr

2c .

As for non-adhesive tips, the contacts tend to be largerthan continuum theory predicts. However, the shift forbent tips is smaller than in the non-adhesive case, and thecommensurate and incommensurate results are closer, asexpected from Fig. 5. Larger deviations are observed forthe amorphous tip, with radii typically 2 or 3 σ largerthan predicted. The deviation becomes most pronouncedat negative loads, where the amorphous tip remains incontact well below the predicted pulloff force.Figure 6(a) compares the value of rb to the JKR pre-

diction for contact radius. As found by Greenwood [52],the numerical results are close to JKR at large loads, but

10

FIG. 5: (Color online) Normal pressure vs. radius for fourdifferent tip geometries: (a) bent commensurate crystal, (b)bent incommensurate crystal, (c) amorphous, and (d) steppedcrystal. In all cases the tip radius R = 100 σ, the nor-malized load N/(R2E∗) = 0.0023 and the surface energyw = 0.46 ǫ/σ2. Dots show the normal pressure on each sur-face atom. In (b) and (c), circles show the mean pressure inradial bins of width σ, and lines show the bent commensurateresults from (a). Horizontal dashed lines are at zero pressure.

deviate at negative loads because M-D and JKR predictdifferent pulloff forces. Since JKR assumes a singular ad-hesive stress at the radius of the contact, it seems naturalthat its predictions lie closest to the position of the peaktensile stress.

Figure 7 shows the calculated radii for bent and amor-phous tips with the same interaction energy ǫi/ǫ = 0.5.The small changes in tip geometry lead to a roughly fourfold variation in both w and Nc (Table I). The largest wis obtained for commensurate tips because all atoms canoptimize their binding coherently. Atoms on incommen-surate tips sample all positions relative to the substrate,and can not simultaneously optimize the binding energy.The larger height fluctuations on amorphous tips lead toeven greater reductions in w.

In the simulations, the pulloff force, Nc, correspondsto the most negative load where the surfaces remain incontact and the ri can be defined. Its normalized value,Nc/πwR, is equal to -3/2 in JKR theory, -2 in DMT the-ory, and lies between these values for M-D theory. TableI shows normalized results for various tips. As expected

FIG. 6: Variation of contact radii with external load forbent commensurate (squares), bent incommensurate (trian-gles) and amorphous (circles) tips with R = 100 σ. In (a),the radius rb where the force is most attractive is compared toJKR theory (solid line). In (b), values of ra (filled symbols)and rc (open symbols) are compared to M-D theory (solidlines) with the JG extension for rc when ra = 0 (dashed line).All continuum fits use the independently measured surface en-ergy w = 0.46 ǫ/σ2 and an interaction range h0 = 0.5 σ thatis consistent with the potential range. Numerical uncertaintyin radii is comparable to the symbol size.

from the good agreement in Figs. 6 and 7, results forbent tips lie between JKR and DMT values and can befit with M-D theory. The values for stepped and amor-phous tips lie outside the bounds of M-D theory. This isan important result because pulloff forces are frequentlyused to measure the work of adhesion. Based on con-tinuum theory, one would expect that the uncertainty insuch measurements is less than the difference betweenJKR and DMT predictions. Our results show factor oftwo deviations for stepped tips, which may be common inscanning probe experiments. Other simulations showedthat the stepped tip values are strongly dependent on thesize of the first terrace, as well as any tilt of the terracesor incommensurability.It might seem surprising that the stepped tip has a

smaller pulloff force than the bent tip, because the en-tire first terrace can adhere without any elastic energypenalty. However this effect is overcome by the greatercontact size for bent tips: The radius of the first terrace,rt ∼ 6 σ, is smaller than the values of rb and rc at pullofffor bent tips. As the adhesion is decreased, the predictedcontact size at pulloff will drop below rt and the steppedtip may then have a larger pulloff force than predicted.This limit can also be reached by increasing the widthof the first terrace. For a tip that lies entirely within a

11

FIG. 7: Contact radii as a function of normal load for (a)bent commensurate, (b) bent incommensurate, and (c) amor-phous tips with ǫi = 0.5ǫ. While all have the same interactionstrength, the work of adhesion varies by more than a factorof four. Broken lines show M-D predictions for ra (triangles)and rc (circles) with the indicated values of w and h0 = 0.5 σ.The value of rb (squares) is closest to the JKR result (solidline). Deviations from the continuum predictions show thesame trends as in Fig. 6. Numerical uncertainty in radii iscomparable to the symbol size.

sphere of radius R, r2t < R2 − (R − d)2 ≈ 2dR where dis the layer spacing in the crystal. For our geometry thiscorresponds to rt < 12 σ, which is about twice the valuefor our tip. As noted above, terraces with even smallerradii may be preferentially selected for imaging studiesand will lead to lower |Nc|.As rcut increases, it becomes harder for the M-D model

to simultaneously fit both the radii and the pulloff force.Figure 8 shows simulation data for a bent commensuratetip with rcut = 2.2 σ. Using the value of h0 = 1.2 σ (Fig.4) reproduces all radii fairly well at large loads, but givesa substantially different pulloff force, −906 ǫ/σ insteadof −872 ǫ/σ. Decreasing h0 to 0.8 σ fits the pulloff force,and improves the fit to ra at low N . However, the pre-dicted values for ra at large N are slightly too high andthe values for rc are shifted far below (∼ 2− 3 σ) simula-tion data. This failure is not surprising given the crudeapproximation for adhesive interactions in M-D theory.As might be expected, the best values for the pulloff forceare obtained by fitting the region near the peak in theforce, rather than the weak tail (Fig. 4).

It should be noted that for bent crystalline and amor-

FIG. 8: Fits of M-D theory (broken lines) to rc (squares) andra (circles), and of JKR theory (solid line) to rb (triangles) fora bent commensurate tip with rcut = 2.2 σ. For h0 = 1.2 σ(dashed line) M-D theory fits both rc and ra at large loads,but the magnitude of the pulloff force is too large. Decreasingh0 to 0.8 σ (dotted lines) gives excellent agreement with thepulloff force, but not the radii (Fig. 4). Numerical uncertaintyin radii is comparable to the symbol size.

phous tips, all of our results for r can be fit accuratelyto M-D theory if E∗, w, and R are taken as adjustableparameters. The typical magnitude of adjustments is 10to 30%, which is comparable to typical experimental un-certainties for these quantities in nanoscale experiments.Indeed one of the common goals of these experimentsis to determine any scale dependence in continuum pa-rameters. For this reason it would be difficult to testcontinuum theory in scanning probe experiments even ifr could be measured directly.

Experiments can more easily access the variation ofnormal displacement with load. This requires subtract-ing the height change due to the normal compliance ofthe machine controlling the tip, which is difficult forstandard AFM cantilevers, but possible in stiffer devices[12, 13, 14]. The absolute zero of δ is not fixed, but mustbe fitted to theory. Figure 9 shows two measures of thenormal displacement in our simulations. One, δtip, corre-sponds to the experimentally accessible tip displacement.We associate the zero of δtip with the point where the firstsubstrate atom exerts a repulsive force. The second, δsur,is the actual depression of the lowest substrate atom onthe surface relative to the undeformed surface. The twodiffer because of the interfacial compliance normal to thesurface, which is assumed to vanish in continuum theory.

The simulation results for δ are more sensitive to the fi-nite sample depth D and lateral periodicity L than otherquantities. To account for this, the predictions of M-Dand JKR theory are shifted by the leading analytic cor-

12

FIG. 9: Normal displacement as a function of external loadmeasured from (a) the depression of the lowest substrate atomδsur and (b) the displacement of the tip relative to the heightwhere the first substrate atom exerts a repulsive force δtip.The tips are the same as in Fig. 6. The JKR prediction is in-dicated by dotted lines, the M-D prediction by solid lines, andthe JG extension for ra = 0 with dashed lines. All are cor-rected for the finite dimensions of the substrate as describedin the text, and uncertainties are comparable to the symbolsize. The difference ∆δ ≡ δtip − δsur reflects the interfacialcompliance, and is shown for the bent incommensurate tip inthe inset.

rections in a/D [37, 38, 39]:

δ = δH(1 − ba

D) + δadhesion(1− d

a

D) , (8)

where δH is the Hertz prediction (Eq. (4)), b and d are

fit parameters, and δadhesion = −√

2wπa/E∗ for JKR

theory and −(2σ0/E∗)√c2 − a2 for M-D theory. We ob-

tained b = 0.8 from simulations with dense, non-adhesivetips (Fig. 3) and d = 0.3 from simulations with dense,adhesive tips. Results for dense tips are then indistin-guishable from the fit lines in Figs. 3(a) and 9. Valuesof b and d from numerical studies of continuum equa-tions are of the same order [37, 38, 39], but to our knowl-edge these calculations have not considered our geometrywhere L ≈ D.With the finite-size corrections, continuum results for δ

lie very close to the simulation data for bent tips. Agree-ment is best for δsur because neither it nor continuumtheory include the interfacial compliance. The choice ofzero for δtip is not precisely consistent with M-D theory.The first repulsive interaction would correspond to thefirst non-zero ra, which occurs at a slightly negative δ inM-D theory. For the parameters used here, the shift in δis about 0.24 σ, which is about twice the discrepancy be-tween δtip and M-D theory at large loads. This impliesthat the interfacial compliance produces a comparable

correction.The effects of interfacial compliance are biggest for the

most negative values of δtip. Here δtip decreases monoton-ically as N increases to zero. In contrast, δsur is flat andthen increases back towards zero. In this regime, ra = 0and the only interaction comes from the attractive tailof the potential. The net adhesive force gradually de-creases (N increases) as the magnitude of the separationbetween tip and substrate, ∆δ ≡ δtip − δsur, increases(inset). Most of this change occurs in δtip, while thedisplacement of the surface relaxes back to zero as theattractive force on it goes to zero. The JG extension toM-D theory is expressed in terms of δtip and provides agood description of its change with N (dashed line), evenwith the assumption of a constant attractive σ0.For amorphous tips, both values of δ are substantially

above M-D theory. The shifts are bigger than in the non-adhesive case, about 0.35 σ vs. 0.25 σ. This is correlatedwith the larger than predicted pulloff force. Based on theincrease in |Nc|, the effective work of adhesion appearsto be larger by about 30% than that measured for a flatsurface. The stronger adhesive contribution to the totalnormal force leads to a larger value of δ. One may un-derstand these changes in terms of the effect of surfaceroughness. Atomic-scale roughness on a nominally flatsurface prevents many atoms from optimizing their bind-ing energy. As δ and the contact area shrink, the longwavelength height fluctuations become irrelevant and nolonger prevent the few atoms remaining in the contactfrom adhering. Thus while the large load values of δ canbe fit to continuum predictions with the measured w anda simple shift in the origin of δ, the small N values cor-respond to a larger work of adhesion. The magnitude ofthe increase (∼ 30%) is modest given that the incommen-surate tip has about twice as large a w as the amorphoustip for the same interaction energy ǫi.The data for stepped tips are qualitatively different

than the others. As for the non-adhesive case, δ is lowerthan the continuum prediction at large loads, becausethe flat tip is harder to push into the substrate. The de-viation increases rapidly at negative loads, with a sharpdrop near N = 0 where the contact shrinks to the firstterrace. As noted above, the radius of the first terrace issmaller than the radius at pulloff predicted by continuumtheory, and the pulloff force is less than half the predictedvalue.

C. Friction and lateral stiffness

Scanning probe microscopes can most easily de-tect friction forces and the lateral compliance of thetip/substrate system [4, 5]. Figure 10 shows F as afunction of load for five different adhesive tips. As fornon-adhesive tips, tip geometry has a much larger effecton F than other quantities, and values for bent commen-surate and incommensurate tips differ by two orders ofmagnitude.

13

FIG. 10: (Color online) Static friction F vs. load N forthe indicated tip geometries with same systems as in Fig. 6.Attempts to fit the data by assuming F is proportional to thearea predicted by JKR theory and the interpolation schemeof Ref. [29] are shown by dotted and solid lines, respectively.Dashed lines are separate linear fits for the stepped tip forthe cases where one or two terraces are in repulsive contact.Numerical uncertainties are comparable to the symbol size.

Since the friction was measured at constant load (Sec.III), only values in the stable regime, dδtip/dN > 0 couldbe obtained. Even in this regime, we found that the tiptended to detach at N > Nc. This was particularly pro-nounced for commensurate tips. Indeed the bent andstepped commensurate tips detached after the first peakin the friction force for all the negative loads shown inFig. 10(a). At loads closer to the pulloff force, detach-ment occurred at even smaller lateral displacements.As noted above, bent commensurate tips have the

strongest adhesion energy because all atoms can simul-taneously optimize their binding. For the same reason,the adhesion energy changes rapidly as atoms are dis-placed laterally away from the optimum position, al-lowing pulloff above the expected Nc. The extent ofthe change depends on the sliding direction and registry[40, 49, 50]. We consider sliding in the (100) direction(Sec. III), where atoms move from points centered be-tween substrate atoms towards points directly over sub-strate atoms. This greatly reduces the binding energy,leading to detachment at less than half the pulloff force.Note that changes in binding energy with lateral displace-ment lead directly to a lateral friction force [50] and thebent and stepped commensurate tips also have the high-est friction. We suspect that tips with higher frictionmay generally have a tendency to detach farther above

TABLE II: Frictional stresses τf and apparent moduli G′ eval-uated from the derivatives of fits in Figs. 11 and 12, respec-tively, as a function of tip geometry and work of adhesion w.Values of G′ are less than the effective shear modulus of thesubstrate, G∗ = 18.3 ǫ/σ3, and frictional stresses are muchsmaller than expected from continuum expressions (Fig. 13).Statistical errorbars are indicated in parentheses.

Tip geometry w(ǫσ−2) τf (ǫσ−3) G′(ǫσ−3)

Commensurate 0 1.35(7) 13.7(4)0.46 1.82(4) 12.8(3)

Incommensurate 0 0.0044(7) 0.70(5)0.46 0.0151(8) 1.0(3)

Amorphous 0 0.056(4) 1.53(5)0.46 0.24(2) 4.3(6)

Nc during sliding than other tips.

At the macro scale, friction is usually assumed to be di-rectly proportional to load. All the tips have substantialfriction at zero load, due to adhesion. The friction forcealso varies non-linearly with N , showing discrete jumpsfor the stepped tip, and curvature for the other tips, par-ticularly near Nc. The curvature at small N is reminis-cent of the dependence of radius on load. Several authorshave fit AFM friction data assuming that F scales withcontact area, and using a continuum model to determinethe area as a function of load [4, 5, 6, 7, 8, 9, 10, 11]. Thedotted lines in Fig. 10 show attempts to fit F using thearea from JKR theory with w adjusted to fit the pulloffforce and the proportionality constant chosen to fit thefriction at high loads. None of the data is well fit bythis approach. The solid lines show fits to an expressionsuggested by Schwarz [29] that allows one to simply in-terpolate between DMT and JKR limits as a parametert1 is increased from 0 to 1. Reasonable fits are possiblefor non-stepped tips when this extra degree of freedom isallowed. Values of the friction per unit area are within25% of those determined below from the true area (Ta-ble II), but the values of w and t1 do not correspond toany real microscopic parameters. For example, for thethree non-stepped tips where a direct measurement givesw = 0.46 ǫ/σ2, the fits give w = 0.41 ǫ/σ2, 0.55 ǫ/σ2

and 0.94 ǫ/σ2 for commensurate, amorphous and incom-mensurate tips, respectively. The value of t1 varies from0.1 to 0.5 to 1.5, respectively. Note that the value oft1 = 1.5 for incommensurate tips is outside the physi-cal range. In this case, and some experiments [6], thefriction rises more slowly than the JKR prediction, whileM-D theory and simpler interpolation schemes [29, 30]always give a steeper rise. Such data seem inconsistentwith F scaling with area.

Our simulations allow us to test the relationship be-tween F and area without any fit parameters. However,it is not obvious which radius should be used to deter-mine area. Figure 11 shows friction plotted against r2a,r2b and r2c . The stepped tip is not shown, since only rc iseasily defined and it increases in one discontinuous jumpover the range studied. For all other tips the friction

14

FIG. 11: (Color online) Static friction F plotted againstradius squared for the indicated tip geometries with same pa-rameters as in Fig. 6. Values of r2a (filled circles), r2b (opensquares), and r2c (filled triangles) are shown for adhesive tips.Open circles show F vs. a2 for non-adhesive tips with data in(b) and (c) multiplied by a factor of two for clarity. Dashedlines show unconstrained linear fits to each data set. Numer-ical uncertainties are comparable to the symbol size.

is remarkably linear when plotted against any choice ofradius squared. In contrast, plots of N vs. r2 show sig-nificant curvature.

For F to be proportional to area, the curves shouldalso pass through the origin. This condition is mostclosely met by r2a, except for the incommensurate case.The idea that friction should be proportional to the areawhere atoms are pushed into repulsion seems natural.The other radii include attractive regions where the sur-faces may separate far enough that the variation of forcewith lateral displacement is greatly reduced or may evenchange phase. However, in some cases the extrapolatedfriction remains finite as ra goes to zero and in others itappears to vanish at finite ra.

Also shown in Fig. 11 are results for non-adhesivetips (open circles). Values for incommensurate and amor-phous tips were doubled to make the linearity of the datamore apparent. Only the bent commensurate data showssignificant curvature, primarily at small N . As noted inSec. IV, friction is proportional to load for this tip, andthe curvature is consistent with this and a2 ∼ N2/3. Nocurvature is visible when adhesion is added, but the factthat the linear fit extrapolates to F = 0 at ra/σ ∼ 4 sug-gests that the linearity might break down if data could

be obtained at lower loads.

Results for adhesive and non-adhesive amorphous tipscan be fit by lines through the origin (not shown),within numerical uncertainty. The same applies for non-adhesive bent incommensurate tips, but adding adhesionshifts the intercept to a positive force at ra = 0. It wouldbe interesting to determine whether this extrapolation isvalid. Friction can be observed with purely adhesive in-teractions, since the only requirement is that the magni-tude of the energy varies with lateral displacement [50].However, it may also be that the friction on incommen-surate tips curves rapidly to zero as ra → 0. Indeed onemight expect that for very small contacts the tip shouldbehave more like a commensurate tip, leading to a morerapid change of F with area or load. The only way to ac-cess smaller ra is to control the tip height instead of thenormal load. This is known to affect the measured fric-tion force [49], and most experiments are not stiff enoughto access this regime.

Figure 12 shows the lateral stiffness as a function of thethree characteristic radii. Except for the stepped tip (notshown), k rises linearly with each of the radii. As for non-adhesive tips, the slope is much smaller than the value8G∗ predicted by continuum theory (solid lines) becauseof the interfacial compliance (Eq. (7)). The intercept isalso generally different from the origin, although it comesclosest to the origin for ra. As for friction, it seems thatthe repulsive regions produce the dominant contributionto the stiffness.

Results for non-adhesive tips are also included in Fig.12. They also are linear over the whole range, and the fitsreach k = 0 at a finite radius ak. This lack of any stiff-ness between contacting surfaces seems surprising. Notehowever that linear fits to Fig. 3(b) also would suggestthat the radius approached a non-zero value, a0, in thelimit of zero load. Moreover, the values of a0 and ak fol-low the same trends with tip geometry and have similarsizes.

The finite values of a0 and ak can be understood fromthe finite range of the repulsive part of the interaction.As long as atoms are separated by less than rcut thereis a finite interaction and the atoms are considered in-side a. However, the force falls rapidly with separation,and atoms near this outer limit contribute little to thefriction and stiffness. If δh is the distance the separa-tion must decrease to get a significant interaction, thena0 = (2Rδh)1/2 at the point where the first significantforce is felt. Taking the estimate of δh = 0.04 σ fromSec. IV, then a0 ∼ 3 σ, which is comparable to the ob-served values of a0 and ak. The shift is smaller for theadhesive case because there are still strong interactionswhen ra goes to zero. The larger shifts for the amor-phous tip may reflect roughness, since the first point tocontact need not be at the origin. We conclude that thelinear fits in Fig. 12 go to zero at finite radius, becausera overestimates the size of the region that makes signifi-cant contributions to forces, particularly for non-adhesivetips. Note that the plots for friction with non-adhesive

15

FIG. 12: (Color online) Lateral stiffness k plotted againstradius for the indicated tip geometries with same parametersas in Fig. 6. Values of ra (filled circles), rb (open squares),and rc (filled triangles) are shown for adhesive tips. Opencircles show F vs. a for non-adhesive tips. Broken lines showunconstrained linear fits to each data set, and the solid lineindicates the slope predicted by continuum theory. Numericaluncertainties are comparable to the symbol size, except for theadhesive data in (b) where they may be up to 50%.

tips (Fig. 11) are also consistent with an offset, but thatthe offset appears much smaller when plotted as radiussquared.

The slope of the curves in Fig. 11 can be used to definea differential friction force per unit area or yield stressτf ≡ ∂F/∂πr2a. It is interesting to compare the magni-tude of these values (Table II) to the bulk yield stress ofthe substrate τy. Assuming Lennard-Jones interactions,the ideal yield stress of an fcc crystal in the same shear-ing direction is 4 to 10 ǫσ3, depending on whether thenormal load or volume is held fixed. The commensuratetip is closest to a continuation of the sample, and theforce on all atoms adds coherently. As a result τf is ofthe same order as τy, even for the non-adhesive tip. Val-ues for adhesive amorphous and incommensurate tips areabout one and two orders of magnitude smaller than τy,respectively. This reflects the fact that the tip atoms cannot optimize their registry with the substrate. Removingadhesive interactions reduces τy by an additional factorof about four in both cases.

In continuum theory, k = 8G∗r, and the slope of fitsin Fig. 12 could then be used to determine the effectiveshear modulus G∗. However, as noted above, the in-terfacial compliance leads to much lower stiffnesses. Toillustrate the magnitude of the change we quote valuesof G′ ≡ (1/8)∂k/∂r in Table II. All values are below the

true shear modulus G∗ = 18.3 ǫ/σ3 obtained from thesubstrate compliance alone (Fig. 3). As always, resultsfor bent commensurate tips come closest to continuumtheory with G′/G∗ ∼ 0.7. Values for adhesive amorphousand incommensurate tips are depressed by factors of 3and 20 respectively, and removing adhesion suppressesthe value for amorphous tips by another factor of four.Carpick et al. noted that if friction scales with area

and k with radius, then the ratio F/k2 should be con-stant [5, 6, 7, 60]. Defining the frictional force per unitarea as τf and using the expression for k from continuum

theory, one finds 64G∗2F/πk2 = τf . In principal, this al-lows continuum predictions to be checked and τf to bedetermined without direct measurement of contact size.Figure 13 shows:

τefff ≡ 64(G∗)2F/πk2 (9)

as a function of N for different tip geometries and in-teractions. Except for the non-adhesive stepped tip, thevalue of F/k2 is fairly constant at large loads, withinour numerical accuracy. Some of the curves rise at smallloads because the radius at which F reaches zero in Fig.11 tends to be smaller than that where k reaches zero inFig. 12. These small radii are where continuum theorywould be expected to be least accurate. Note that thedeviations are larger for the non-adhesive tips, perhapsbecause the data extends to smaller radii.The data for stepped tips are of particular interest be-

cause the contact radius jumps in one discrete step fromthe radius of the first terrace to the radius of the second.The friction and stiffness also show discontinuous jumps.Nonetheless, the ratio F/k2 varies rather smoothly andeven has numerical values close to those for other tips.The most noticeable difference is that the data for thenonadhesive stepped tip rises linearly with load, whileall other tips tend to a constant at high load. These re-sults clearly demonstrate that success at fitting derivedquantities like F and k need not imply that the true con-tact area is following continuum theory.The curves for τefff in Fig. 13 are all much higher than

values of the frictional stress τf obtained directly fromthe friction and area (Table II). Even the trends withtip structure are different. The directly measured fric-tional stress decreases from bent commensurate to amor-phous to bent incommensurate, while τefff is largest forthe amorphous and smallest for the bent commensuratetip. These deviations from the continuum relation aredirectly related to the interfacial compliance ki. The con-tinuum expression for the lateral stiffness neglects ki andgives too small a radius at each load. This in turn over-estimates the frictional stress by up to two orders of mag-nitude. Similar effects are likely to occur in experimentaldata.Experimental plots of F/k2 have been obtained for

silicon-nitride tips on mica and sodium chloride [6, 7, 60]and on carbon fibers [8]. Data for carbon fibers andmica in air showed a rapid rise with decreasing N atlow loads [8, 60]. For mica the increase is almost an or-

16

FIG. 13: (Color online) Ratio of friction to stiffness squaredas a function of load for the indicated tip geometries (a) with-out and (b) with adhesion (w = 0.46 ǫ/σ2). Lines are guidesto the eye. Numerical uncertainties are comparable to thesymbol size, except for the bent incommensurate data in (b),where they may be as large as 50%.

der of magnitude, which is comparable to our results fornon-adhesive bent incommensurate tips. This correspon-dence may seem surprising given that the experimentsmeasured adhesion in air. However, the adhesive forcewas mainly from long-range capillary forces that operateoutside the area of contact. Following DMT theory, theycan be treated as a simple additive load that does notaffect the contact geometry. In contrast, data for micain vacuum is well fit by JKR theory, implying a strongadhesion within the contact [6, 7]. The measured valueof F/k2 is nearly constant for this system, just as in ourresults for most adhesive tips. Results for carbon fibersin vacuum [8] show a linear rise like that seen for nonad-hesive stepped tips.

From a continuum analysis of the carbon fiber data,the frictional stress was estimated to be τf ∼ 300 MPaassuming a bulk shear stress of G∗ = 9.5 GPa [8]. Notethat Fig. 13 would suggest τf/G

∗ ∼ 0.1 to 0.3, whilethe true values (Table II) are as low as 0.0002. The dataon carbon fibers could be fit with the bulk shear mod-ulus, but data on mica and NaCl [6, 60] indicated thatG∗ was 3 to 6 times smaller than bulk values. Our re-sults show that the interfacial compliance can easily leadto reductions of this magnitude and a corresponding in-crease in τf , and that care must be taken in interpretingexperiments with continuum models.

VI. DISCUSSION AND CONCLUSION

The results described above show that many differenteffects can lead to deviations between atomistic behav-ior and continuum theory and quantify how they dependon tip geometry for simple interaction potentials (Fig.1). In general, the smallest deviations are observed forthe idealized model of a dense tip whose atoms form anearly continuous sphere, although this tip has nearlyzero friction and lateral stiffness. Deviations increase asthe geometry is varied from a bent commensurate to abent incommensurate to an amorphous tip, and steppedtips exhibit qualitatively different behavior. Tip geom-etry has the smallest effect on the normal displacementand normal stiffness (Fig. 3 and 9) because they reflectan average response of the entire contact. Friction andlateral stiffness are most affected (Fig. 3 and 10), be-cause they depend on the detailed lateral interlocking ofatoms at the interface.

One difference between simulations and continuumtheory is that the interface has a finite normal compli-ance. Any realistic interactions lead to a gradual in-crease in repulsion with separation rather than an ide-alized hard-wall interaction. In our simulations the effec-tive range over which interactions increase is only about4% of the atomic spacing, yet it impacts results in severalways. For bent commensurate tips it leads to an increasein pressure in the center of the contact (Figs. 2 and 4).The pressure at the edge of nonadhesive contacts dropslinearly over about 2 σ, while continuum theory predictsa diverging slope. The width of this smearing grows asthe square root of the tip radius and leads to qualitativechanges in the probability distribution of local pressures[54, 55, 56, 57]. These effects could be studied in con-tinuum theories with soft-wall interactions. The normalinterfacial compliance also leads to the offset in linear fitsof F vs. r2a and k vs. ra (Figs. 11 and 12). Fits to thefriction and local stiffness extrapolate to zero at finitevalues of ra because atoms at the outer edge of the re-pulsive range contribute to ra but interact too weakly tocontribute substantially to F and k. This effect is largestfor non-adhesive tips.

Approximating a spherical surface by discrete atomsnecessarily introduces some surface roughness. Evenbent crystalline tips have atomic scale corrugations, re-flecting the variation in interaction as tip atoms movefrom sites nestled between substrate atoms to sites di-rectly above. Amorphous and stepped tips have longerwavelength roughness associated with their random orlayered structures respectively. This longer wavelengthroughness has a greater effect on the contacts. For non-adhesive interactions, incommensurate and amorphoustips have a lower central pressure and wider contact ra-dius than predicted for ideal spheres. These changes arequalitatively consistent with continuum calculations forspheres with random surface roughness [16]. However theeffective magnitude of the rms roughness ∆ is smallerthan expected from the atomic positions. The correlated

17

deviations from a sphere on stepped tips, lead to qual-itative changes in the pressure distribution on the sur-face (Fig. 2 and 5). However, these changes are alsoqualitatively consistent with what continuum mechanicswould predict for the true tip geometry, which is closerto a flat punch than a sphere. We conclude that theusual approximation of characterizing tips by a singlespherical radius is likely to lead to substantial errors incalculated properties. Including the true tip geometryin continuum calculations would improve their ability todescribe nanometer scale behavior. Unfortunately this israrely done, and the atomic-scale tip geometry is rarelymeasured. Recent studies of larger tips and larger scaleroughness are an interesting step in this direction [61].

Roughness also has a strong influence on the workof adhesion w (Table I). Values of w were determinedindependently from interactions between nominally flatsurfaces. For a given interaction strength, commensu-rate surfaces have the highest w, because each atom canoptimize its binding simultaneously. The mismatch oflattice constants in incommensurate geometries lowersw by a factor of two, and an additional factor of twodrop is caused by the small (∆ ∼ 0.3 σ) height fluctua-tions on amorphous surfaces. In continuum theory, thesechanges in w should produce nearly proportional changesin pulloff force Nc, and tips with the same w and h0

should have the same Nc. Measured values of Nc differfrom these predictions by up to a factor of two. It is par-ticularly significant that the dimensionless pulloff forcefor amorphous and stepped tips lies outside the limitsprovided by JKR and DMT theory. Experimentalists of-ten assume that these bounds place tight limits on errorsin inferred values of w.

In the case of amorphous tips the magnitude of Nc is30% higher than expected. The higher than expectedadhesion in small contacts may reflect a decrease in ef-fective roughness because long-wavelength height fluctu-ations are suppressed. Stepped tips show even largerdeviations from continuum theory that are strongly de-pendent on the size of the first terraces [46]. Tips selectedfor imaging are likely to have the smallest terraces andthe largest deviations from continuum theory.

Adding adhesion introduces a substantial width to theedge of the contact, ranging from the point where inter-actions first become attractive ra to the outer limits ofattractive interactions rc (Fig. 4). As the range of inter-actions increases, it becomes increasingly difficult to fitboth these characteristic radii and the pulloff force withthe simple M-D theory (Fig. 8). For short-range inter-actions, good fits are obtained with the measured w forbent tips. Data for amorphous tips can only be fit byincreasing w, due to the reduction in effective roughnessmentioned above (Fig. 6). For stepped tips the contactradius increases in discrete jumps as successive terracescontact the surface.

The normal interfacial compliance leads to significantambiguity in the definition of the normal displacementas a function of load (Fig. 9). Continuum theory nor-

mally includes only the substrate compliance, while ex-perimental measures of the total tip displacement δtipinclude the interfacial compliance. The substrate compli-ance was isolated by following the displacement of sub-strate atoms, δsur and found to agree well with theory forbent tips in the repulsive regime. Johnson and Green-wood’s extension of M-D theory [25] includes the interfa-cial compliance in the attractive tail of the potential. Itprovides a good description of δtip in the regime wherera = 0. Here δtip− δsur increases to the interaction rangeh0. Results for amorphous tips show the greater adhe-sion noted above. Stepped tips follow continuum theoryat large loads but are qualitatively different at negativeloads.

The most profound effects of tip geometry are seenin the lateral stiffness k and friction F , which vary byone and two orders of magnitude respectively. Contin-uum theories for k do not include the lateral interfacialcompliance ki. This adds in series with the substratecompliance ksub (Eq. (7)). Except for commensuratetips, ki << ksub and the interface dominates the totalstiffness [17]. Experiments have also seen a substantialreduction in the expected lateral stiffness from this effect[51, 60].

The friction on non-adhesive commensurate tips (bentor stepped) increases linearly with load, as frequently ob-served for macroscopic objects. In all other cases, F is anonlinear function of load. Our ability to directly mea-sure contact radii allowed us to show that F scales lin-early with contact area for incommensurate, amorphousand adhesive bent commensurate tips. These tips alsoshow a linear scaling of k with radius. While these scal-ings held for any choice of radius, the linear fits are offsetfrom the origin. It appears that the effective area con-tributing to friction and stiffness is often a little smallerthan the area of repulsive interactions corresponding tora. As noted above, the offset from ra appears to corre-spond to the finite range over which repulsive forces riseat the interface.

Experimental data [4, 5, 6, 7, 8, 9, 10, 11] for frictionand stiffness have been fit to continuum theory with theassumptions that F ∝ r2 and k ∝ r, but without theoffsets seen in Figs. 11 and 12. We showed that our datafor bent and amorphous tips could be fit in this way (Fig.10), but that the fit parameters did not correspond todirectly measured values. This suggests that care shouldbe taken in interpreting data in this manner.

We also examined the ratio F/k2. In continuum the-ory, this is related to the friction per unit area τefffthrough Eq. (9). Our results for F/k2 (Fig. 13) show therange of behaviors observed in experiments, with a rela-tively constant value for adhesive cases, a rapid increaseat low loads in some nonadhesive cases, and a linear risefor non-adhesive stepped tips. The directly measuredvalues of τf (Table II) are smaller than τefff by up totwo orders of magnitude, and have qualitatively differenttrends with tip geometry. The difference is related to areduction in the stiffness k due to interfacial compliance.

18

This reduces the inferred value of bulk shear modulus G′

and increases the calculated contact area at any givenarea. We expect that experimental results for F/k2 willproduce similar overestimates of the true interfacial shearforce.It remains unclear why F and k should follow the ob-

served dependence on ra. Analytic arguments for clean,flat surfaces indicate that F is very sensitive to struc-ture, with the forces on commensurate, incommensurateand disordered surfaces scaling as different powers of area[32, 50]. Only when glassy layers are introduced betweenthe surfaces, does the friction scale in a universal man-ner [32, 62, 63, 64]. Wenning and Muser [31] have arguedthat the friction on clean, amorphous tips rises linearlywith area because of a cancellation of two factors, buthave not considered k. Naively, one might expect thatthe length over which the force rises is a constant frac-tion of the lattice spacing and that k is proportional toF . However, the friction traces change with load and donot always drop to zero between successive peaks. Wehope that our results will motivate further analytic stud-ies of this problem, and simulations with glassy films and

more realistic potentials.While we have only considered single asperity contacts

in this paper, it is likely that the results are relevantmore broadly. Many experimental surfaces have randomroughness on all scales that can be described by self-affinefractal scaling. Continuum models of contact betweensuch surfaces show that the radius of most of contactsis comparable to the lower length scale cutoff in fractalscaling [56, 57]. This is typically less than a micrometer,suggesting that typical contacts have nanometer scale di-mensions where the effects considered here will be rele-vant.

Acknowledgments

We thank G. G. Adams, R. W. Carpick, K. L. Johnson,M. H. Muser and I. Sridhar for useful discussions. Thismaterial is based upon work supported by the NationalScience Foundation under Grants No. DMR-0454947,CMS-0103408, PHY99-07949 and CTS-0320907.

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