Friction Laws at Nanoscale Resumida

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sliding under a normal load and we determine the dependence of fric-tion force on the applied load, contact area, and on the presence ofadhesion at the interface. We focus on amorphous carbon tips anddiamond samples (Fig. 1a), both terminated with hydrogen, because

of the availability of experimental data20 that can be used for compar-ison. We use realistic force fields26, spherical tips that are allowed todeform together with the sample and that have curvature radii Rof upto 30 nm (see Supplementary Fig. S1), and the simulations are per-formed at the temperature of 300 K. The force fields consist of short-range chemical interactions and long-range van der Waals interactions.

Defining contactareais one of the major challenges forunderstandingfriction in nanoscale contacts. Because fundamentally contact is formedbyatomsinteractingacrosstheinterface(Fig.1b),wedefinearealcontact

areatobe Areal~NatAat, where Nat is the numberof atoms of the samplewithin therange ofchemicalinteractions fromthe tipatoms andAat istheaveragesurfaceareaperatom.Incontinuummechanics Aaspisdefinedbythe edge of the contact zone. The definition of the contact edge becomesambiguous when the atomistic nature of the interface dominates itsphysical behaviour27,28. Here, we defineAaspto be the area enclosed bya convex hull around atoms in contact, as shown in Fig. 1c.

We first perform molecular dynamics simulations of normal load-ing and friction in the absence of van der Waals forces. Because boththe tip and the sample are passivated with hydrogen atoms, adhesiondue to short-range chemical forces is negligible. In this case a non-adhesive single asperity (Hertz) model is expected to apply, that is,

Aasp~p 3R=4E 2=3L2=3, where E~ 1{n21

=E1z 1{n

22

=E2

{1

is the effective modulus of the contact, E1and E2are the tip and thesample Youngs moduli, and n1 and n2 are the tip and the samplePoissons ratios, respectively. With the additional assumption ofFf~tAasp, the Hertz model predicts thatFf!L

2=3. Our calculationsshow that Aasp!L

0:7 and Ff!L (Fig. 2a), which yields Ff=tAasp.Additionally, even though Aasp obtained in simulations showsapproximately the same 2/3 power-law dependence on Las in theHertz theory, the effective modulusE*values calculated from fittingthe Hertz model to the simulation data are 61% smaller than thevalue calculated directly from the definition of E* (see Supple-mentary Methods). Our results confirm the conclusions of otherauthors that single-asperity theories break down at the nanoscale1,5.To account for the fact that Ffis not linear withAasp, Wenning andMuser25 suggested that t is not a constant, but varies with contactpressure. Other authors proposed an empirical model in which

mechanics of a nanoscale non-adhesive contact is controlled by load,that is,Ff~mLand the contact area is undefined and unnecessary

5,29.We argue that the break-down of single-asperity theories of friction

is due to the fact that at these length scales the real contact areaArealisdifferent from Aasp (see Fig. 1c) andthatfriction lawsshould be definedin terms ofAreal. Our simulations show that Areal!L(Fig. 2b) andFf~tArealwith constant t (Fig. 2c), which is consistent with the rela-tionFf!L(Fig. 2a). As shown in Table 1, friction force is now pro-portional to contact area at alllength scales aslong as thecontact area iscorrectly defined at each length scale. Also, realizing that it isAreal(or,more fundamentally,Nat) that controls friction at the nanoscale, wecan now understand whyFf!L. Such linear dependence is character-istic of rough contacts (Table 1) and in our case is a result of atomicroughness, as shown by the fact that Areal=Aasp. The above results

demonstrate that a nanoscale contact, which had been previouslyviewed as a single entity, consistsof yet smaller contacts of atomic size.Macroscale roughness theories can be applied to describe the beha-viour of nanoscale contacts (see Supplementary Methods).

We investigate the effect of van der Waals adhesion on contact beha-viour by adding these forces to the tipsample interactions and byperforming additional molecular dynamics simulations. As shown inFig. 3a, the relation Ff~tAreal still holds, which demonstrates thatfriction is controlled by the short-range (chemical) interactions evenin the presence of dispersive forces. However, unlike the non-adhesivecase,here Ffis a sublinearfunctionofL(Fig. 3b), which is consistent withpredictions of adhesive single-asperitymodels (see Table1). We fit Ff(L)to the MaugisDugdale model using a convenient approximation pro-posed by Carpick, Ogletree, and Salmeron (COS)30 and later physically

justified by Schwarz31. The fits show an excellent agreement with both

a

b

c

Atoms in contact

Surface area per atom

Contact edge

Figure 1| Contact between an amorphous carbon tip and a diamondsample. a, Far view, showing contact geometry. Golden and red atomscorrespond to C and H, respectively. b, Close view. Solid red and goldensticks represent covalent bonds. Translucent pink sticks represent repulsiveinteractions.c, Contact area definitions. Red circles represent sample atoms

within the range of chemical interactions from tip atoms. Contact area peratomAat is representedby grey hexagons.Realcontact areaAreal isthesumofthe areas of hexagons. The contact area Aaspof an asperity is enclosed by theedge (solid line) of the contact zone.

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the MaugisDugdale theory and withexperimental results (see MethodsSummary), which validates our modelling approach. For instance, thefitted value of Tabors32parametermTis 0.19, which falls withinthe rangeof 0.110.22 calculated directly from theory for our system.

However, the fact that adhesive single-asperity theory fits both oursimulation data and experimental data very well should not be taken asevidence that thetheory represents thecorrect physics.We argue that theflexibility of the MaugisDugdale model (that is, three fitting para-meters) masks the models physical deficiencies. For instance, we cancheckwhetherthe elastic restoring force Lel, calculated bysubtracting vander Waals forces LvdW from the total load L, follows Hertzs predictions,

thatis,whether Areal!L2=3el .AscanbeseenfromFig.3c, Areal!Lel,which

means that thecorrect physical picture is that of a rough(multi-asperity)contact rather than of a single-asperity one (see Table 1).

Why then doesFfdepend sublinearly onL? This behaviour is dueto the presence of adhesion forces LvdW, which for a spherical tip incontact with a flat sample do not scale linearly withAreal(triangles inFig. 3c). As a result the total load L~LelzLvdWis not proportional toArealand henceFf~tAreal is a sublinear function ofL. Our modelstherefore predict that as adhesion in the contact is increased, a trans-ition takes place from linear to sublinear dependence ofFfonL. The

opposite effect can be achieved by increasing the roughness of the

10

8

6

Fric

tionforce,

Ff

(nN)

4

2

0

10

8

6

Frictionforce,

Ff

(nN)

4

2

0

300

a

b

c

200

100

0

Normalforce(nN)

0.51

40 0 40 80

Load, L(nN)

100 12020 20 60

0 2 4

Real contact area, Areal(nm2)

6 108 97531

0 2 4


6 108

LelAreal

= 0.21

corresponds to T= 0.19

LvdWAreal

L

Linear fit

Ff

COS fitting

Ff

100

200

Figure 3| Mechanics of adhesive contacts. a, Friction force versus realcontact area.b, Friction force versus load.c, Contributions to the total loadas a function of real contact area: total load L (full squares); van der WaalscontributionLvdW(empty triangles); elastic restoring force Lel~L{LvdW(empty circles).Ffis measured over three lattice periods of the samplessurface. Theerrorbarsin a and b correspond to a standard deviationin thesemeasurements.

6

5

4

3

Fr

ictionforce,

Ff

(nN)

2

1

0

25

20

15

10

Realcontac

tarea,

Areal

(nm

2)

5

0

6

a

b

c

5

4

3

Frictionforce,

Ff

(nN) Linear fit

0 1 2 3


4 65

0 4 8 12

Load/tip radius, L/R(nN/nm)

16 182 6 10 14

0 20 40

Load, L(nN)

60 10080

Ff

Linear fit

R= 30 nm

R= 20 nm

R= 5 nm

R= 10 nm

Ff

2

1

0

Figure 2| Mechanics of non-adhesive contacts. a, Friction force versusload. b, Real contact area versus load. c, Friction force versuscontact area.Ffis measured over five latticeperiods of thesamplessurface. Theerror bars inaand c correspond to a standard deviation in these measurements. Linear

extrapolation toFf?

0 leads to finite values ofArealandL, but it is not clearwhether such extrapolation should be linear. Measurements at very smallloads carry large uncertainty owing to the small number of atoms in contactand to thermal vibrations.

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interface. Behaviour consistent with single-asperity theories can beexpected in the limit ofAreal

Friction Laws at Nanoscale Resumida

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