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Wear 274– 275 (2012) 281– 285

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n the film thickness dependence of shear strengths in sliding, boundary-layerriction

ichael Garveya, Michael Weinertb, Wilfred T. Tysoea,∗

Department of Chemistry and Biochemistry and Laboratory for Surface Studies, University of Wisconsin-Milwaukee, 3210 N Cramer Street, Milwaukee, WI 53211, USADepartment of Physics and Laboratory for Surface Studies, University of Wisconsin-Milwaukee, 2200 E. Kenwood Blvd., Milwaukee, WI 53211, USA

r t i c l e i n f o

rticle history:eceived 12 May 2011eceived in revised form4 September 2011ccepted 21 September 2011vailable online 29 September 2011

a b s t r a c t

The density functional theory calculated pressure-dependent shear strength S of a four-layer slab of KClon a Fe(1 0 0) substrate is compared to previous calculations for a bilayer slab to gauge the effect of filmthickness on the shear properties of the film. It is found that the shear strength varies with pressureas S = S0 + ˛P, where P is the contact pressure. The resulting calculated values for the four-layer slab areS0〈1 0〉 = 62 ± 15 and S0〈1 1〉 = 65 ± 11 MPa while ˛〈1 0〉 and ˛〈1 1〉 are 0.06 ± 0.01. The values are very closeto those calculated for the bilayer slab of S0〈1 0〉 = 64 ± 9 and S0〈1 1〉 =69 ± 8 MPa and ˛〈1 0〉 and ˛〈1 1〉 of

eywords:ensity functional theoryhear strengthressure dependencehickness dependenceronotassium chloride

0.05 ± 0.01, and in reasonable agreement with the experiment values. These results suggest that thethickness of the film does not have a profound effect on the shear properties.

© 2011 Elsevier B.V. All rights reserved.

. Introduction

We have shown previously that the pressure-dependent shearroperties of a thin, model boundary layer film consisting of KCln iron can be calculated using first-principles density functionalheory (DFT) [1]. First, in order to calculate the shear strength inhe absence of an external load, three layers of KCl were placed on

seven layer thick iron slab [1] and the sliding potential was cal-ulated using DFT by laterally displacing the outer layer of the KCllm and then allowing all of the other atoms in the slab to relaxo their minimum energies. It was found that shear occurred at theCl/Fe interface [1]. Thus, since the KCl slab moved rigidly withespect to the iron substrate, the height of the sliding potentialor KCl on iron could be obtained without performing a full slid-ng calculation by simply calculating the energy of the slab locatedn various high-symmetry points on the surface, thereby resultingn a significant saving in computation time [1]. This simplificationccurred because the shear force required to initiate sliding wasuch lower than the shear modulus of the halide film, with the

esult that the film was only minimally distorted during sliding,esulting in a shear plane located at the iron–KCl interface [1].

∗ Corresponding author. Tel.: +1 414 229 5222.E-mail address: wtt@uwm.edu (W.T. Tysoe).

043-1648/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.wear.2011.09.008

It has been shown experimentally in several cases that the shearstrength varies linearly with contact pressure P and can be writtenas S = S0 + ˛P, where S0 is the shear strength at zero pressure and

is a proportionality constant [2–13]. Similar results have beenfound for the shear of bulk materials [14]. In particular, it wasfound experimentally, from friction measurements of a thin film ofKCl on metal surfaces, that S0 = 65 ± 5 MPa and = 0.14 ± 0.02 [15].The experimental measurements were carried out for KCl depositedonto various metal substrates in ultrahigh vacuum where the back-ground pressure was ∼1 × 10−10 Torr. This ensured that the systemwas free of contaminants and that the frictional data were mea-sured for a clean, continuous KCl film on an atomically clean metalsubstrate [15]. The metal foil substrates were all cleaned in ultra-high vacuum and their cleanliness measured in situ using Augerspectroscopy ensuring that the initial substrates were also cleanprior to KCl deposition. While the metal foil substrates were initiallylikely to be rather disordered, following the cleaning and vacuumannealing process, they are likely to become more ordered [16]. KClis not used commercially as a boundary film, but it was chosen as amodel since it is sufficiently simple that it is amenable to analysisby high-level, first-principles quantum calculations.

Based on the work discussed above for KCl on an iron substrate,

it was subsequently possible to obtain a theoretical value of thepressure-dependent shear strength of a composite iron–KCl–ironsandwich by calculating the energy of the system as a function ofapplied pressure for various locations of the KCl slab with respect

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82 M. Garvey et al. / Wea

o the iron substrate, rather than having to carry out a more com-lex calculation in which the film was sheared while simultaneous

mposing a normal load [17]. The calculation was carried out forwo layers of KCl sandwiched between iron. The pressure depen-ence of the shear strength was calculated by compressing the slabrom its equilibrium thickness and by plotting the energy of the sys-em as a function of the thickness of the iron–KCl–iron compositelm. The energy was found to vary parabolically with film thicknessnd so was found to deform elastically over the range of pressuressed in the experiment [15]. The normal pressure applied to thelm was calculated from the first derivative of the plot of energyersus normal displacement. It was found that the resulting pres-ure dependence of the shear strength contained both linear (P1)nd quadratic (P2) terms. The first pressure-dependent term aroserom different equilibrium thicknesses of the film located at dif-erent positions on the substrate. This required additional work toe expended against the applied normal load, leading to a sheartrength that depended linearly on pressure, in accord with theonclusions of previous work [2–8]. The second term arose fromhe slightly different compliances of the film located on differ-nt sites and resulted in a quadratic pressure dependence in thehear strength. It was found that this effect was much smallerhan the one that resulted in the linear pressure dependence, thusroving a fundamental theoretical understanding of the experi-entally observed linear dependence. The DFT calculations for a

Cl bilayer sandwich gave S0〈1 0〉 = 64 ± 9 and S0〈1 1〉 = 69 ± 8 MPa,oth in excellent agreement with experiment [17]. However, thisalculation also predicted that both ˛〈1 0〉 and ˛〈1 1〉 had valuesf 0.05 ± 0.01 [17], somewhat lower than the experimentally mea-ured value of 0.14 ± 0.02 [16]. As noted above, for computationalconomy, the DFT calculations were carried out using a KCl bilayer.ne possible explanation for the difference between the experi-ental and theoretical values of may be that it depends on the

hickness of the KCl film, and this issue is explored in the followingections.

The model for determining the pressure-dependent sheartrength exploited the harmonic variation in energy with slab thick-ess suggested by DFT calculations [2]. Extending this model toxplicitly analyze the effect of film thickness on the pressure-ependent shear, the thickness of the film t at some contactressure P is written as P = Y(t − t0)/(t0), where Y is the Young’sodulus of the material that forms the boundary film and t0 is the

lm thickness at zero pressure. The values of Y and t0 are obtainedirectly from the energy change as a function of slab thickness cal-ulated by DFT as shown previously [17]. The shear dependencerises from differences in t(�t) at different locations of the film withespect to the substrate, so that the additional extra work carriedut during sliding is P�t. If a minimum and maximum in the slid-ng potential occur at points a and b, respectively, then the changen the film thickness during sliding is tb − ta. The above equationan be simply rewritten to give t = t0 + (Pt0)/(Y). If the correspond-ng parameters at locations a and b are Ya and ta

0, and Yb and tb0 ,

espectively, then:

t = (tb − ta) = (tb0 − ta

0) + P

(tb0

Yb− ta

0Ya

)(1)

The values of the film thickness and Young’s modulus will varyepending on the location of the slab with respect to the substrate.

n the case of KCl on iron, it was found that the energetically mosttable – and also the geometrically lowest – site was that in whichhe potassium and chloride ions were located directly above an iron

tom in the substrate. This is contrary to what might be expectedased on a hard-sphere model, and arose because of chemical bond-

ng between the iron atoms and the halide ions [1]. If the bondingetween the film and the substrate is rather local, the thickness of

Fig. 1. Depiction of the structure used for the density functional theory calculationsshowing four layers of KCl sandwiched between iron.

the film might not be expected to strongly influence t0 and Y. How-ever, the second term explicitly depends on the thickness of thefilm. Calculations for the bilayer slab revealed that the second termwas much smaller than the first, thereby resulting in an overall lin-ear variation of shear strength with pressure [17]. However, theexistence of this term, and the possibility of longer-range bondinginteractions warrant an examination of this effect.

2. Theoretical methods

The KCl/Fe(1 0 0) system was previously modeled using a bilayerof KCl that was placed between Fe(1 0 0) slabs, which are seveniron layers thick [17] and the results are compared in the follow-ing to those for four layers of KCl placed between Fe(1 0 0) slabs.The calculations are performed for a (2 × 2) iron substrate cell withKCl placed epitaxially on the substrate and the calculations arecarried out using cyclic boundary conditions to mimic an infinitesurface. This orientation is chosen since X-ray diffraction mea-surements of thicker KCl films have shown them to have a (1 0 0)orientation [18,19]. The calculated bulk lattice spacing for KCl of0.627 nm agrees well with the experimental value of 0.629 nm. Theexperimental lattice spacing for iron is 0.287 nm, so that the latticemismatch between KCl and iron is ∼11%. It is assumed in the calcu-lation that the KCl film is strained to so that it is epitaxial with theiron substrate and the structure of the iron plus KCl system is opti-mized to yield the total energy. The structure of the iron–KCl–ironsandwich with four KCl layers used for the calculation is shown inFig. 1. Thus, the KCl film is strained to accommodate to the underly-ing iron lattice. Such pseudomorphic growth of strained films hasbeen observed in other systems where the energy gain by form-ing an epitaxial film is larger than the energy required to strainthe lattice. As the films become thicker, the strain energy increaseseventually resulting in a loss of epitaxy [20]. It has been shown pre-viously that shear of the KCl/metal slab occurs at the film–metalinterface since the shear modulus of the KCl film is larger than theshear strength of the interface [1]. Thus, such a configuration that

uses the smallest (2 × 2) unit cell will lead to the maximum energychange due to shear between the film and substrate. The use oflarger substrate unit cells that reduce the strain of the adsorbedfilm will result in lower heights of the sliding potentials and thus

r 274– 275 (2012) 281– 285 283

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Table 1Harmonic potential fitting parameters to the data displayed in Fig. 3.

E0 (×10−14 J) d0 (Å) Y (Pa)

Atop 4.811837846 13.263 4.8635

M. Garvey et al. / Wea

ower calculated shear strengths. Thus, a comparison of the calcu-ated value of shear strength, in particular the value of S0 (the sheartrength at zero applied pressure) with the experimental value willrovide information on the nature of the interface. The role of dif-erent asperity contact geometries on sliding has been simulatedsing molecular dynamics methods [21].

The effect of an applied normal load is modeled by varying theistance d between the outer layers of the iron slab (Fig. 1); that is,y changing the distance between the topmost layer of the upperlab and the bottom layer of the lower slab. These outermost ironayers are chosen to be sufficiently far away from the sliding inter-ace that they can be uniformly displaced to mimic an externallypplied load. In addition, to maintain cyclic boundary conditionsormal to the contacting surfaces, the slab for the DFT calculationontains two, symmetrically equivalent Fe–KCl interfaces.

The energy of the slab was evaluated as a function of dsing spin-polarized DFT calculations, which were performed byn implementation of the all-electron, full-potential linearizedugmented-plane wave (FLAPW) method, flair [22–24] using theeneralized gradient approximation (GGA) functional of Perdewt al. [25] for exchange-correlation and self-consistent convergenceriteria of better than 10−5 eV. The geometry was optimized byllowing all atoms to relax until the forces on all unrestricted atomsere less than 10−2 eV/Å. The resulting plots of energy versus d aresed to calculate the pressure-dependent shear strength and thetrategy for accomplishing this is discussed below.

It was found previously that, for the KCl film on iron [1], theeight of the sliding potential could be calculated from the energyf the slab located at high-symmetry points on the surface; theinimum and maximum energies at these positions correspond to

urning points in the sliding potentials since the KCl layer was onlyinimally distorted during sliding. This occurs since shear occurs

t the KCl–substrate interface as discussed above [1]. Accordingly,he calculations in this work were performed for a four-layer KCllm with the Cl− and K+ ions in the KCl layer located (i) above irontoms in the substrate (designated atop), (ii) above bridge sites inhe Fe(1 0 0) substrate (designated bridge) and (iii) above four-foldollow sites in the substrate (designated hollow).

. Results and discussion

The shear of a model, four-layer KCl film sandwiched betweenron slabs is compared with previous results for a two-layer slabo gauge the effect of film thickness of the shear properties of thelm [17]. It was found previously for the two-layer slab that thealue of S0 calculated from the height of the sliding potential wasn good agreement with experiment, while the predicted value of ˛

as substantially lower than the experimentally measured value.n order to carry out DFT calculations most efficiently using periodicoundary conditions, it is assumed that the KCl film is commensu-ate (that is, is epitaxial) with the underlying iron and is modeledy a KCl slab on a (2 × 2) Fe(1 0 0) substrate. The resulting sheartrength is taken to be directly proportional to the contact area.n the case of incommensurate interfaces, it has been shown thathis assumption is not always valid [26,27]. The validity of the

odel and its underlying assumptions are tested by comparing theheoretical predictions with experimental measurements of sheartrength collected in ultrahigh vacuum for thin, well-characterizedCl films. Agreement between the calculated and measured val-es of shear strength will suggest that the assumptions made inhe model are valid, at least for this particular system. Thus, the

greement between theory and experiment for the two-layer slabmplies that the assumptions discussed above are valid for the

odel KCl boundary layer since it correctly reproduces the experi-ental value of S0. The discrepancy between the experimental and

Bridge 4.811837567 13.337 4.7658Hollow 4.811837435 13.374 4.7196

calculated values of suggest that some modifications to the modelare required that affect the value of ˛, while maintaining the keyingredients so that the value of S0 remains unchanged. The follow-ing explores whether changing the film thickness can accomplishthis since the experiments were performed by evaporating KCl ontometals substrates and by measuring the friction coefficient for filmsthat just covered the substrate. However, because of the immobil-ity of the halide film on the metal substrate, a second layer formsprior to completely saturating the first layer [19].

The potential energy curves as a function of d/2 (to take accountof the presence of two interfaces in the model) for a four-layer KClfilm placed between Fe(1 0 0) slabs are shown in Fig. 2 for the potas-sium and chloride ions located at (a) atop, (b) bridge and (c) hollowpositions. These are fit to a parabolic potential [17] and the fits areshown as solid lines through the data points and the resulting fit-ting parameters are summarized in Table 1. The resulting fits arereplotted together on the same scale in Fig. 3 for direct compar-ison. This reveals that the potential energy curves for the halidefilms located at different positions with respect to the surface aredifferent, which gives rise to the pressure-dependent sliding poten-tial as the atoms slide over each other. This indicates that, as wasfound for the bilayer KCl film, the film is most stable when thepotassium and chloride ions are located above iron atoms in thesubstrate. In the case of the bilayer films, this was found to be pre-dominantly due to some bonding between the Cl− ions in the filmand the iron atoms in the substrate [1]. This bonding interaction wasstronger at the atop sites than when the halide ions were locatedover bridge or four-fold sites. The different energies at the minimaof the potentials yield pressure-independent sliding potentials, andthus values of S0. In addition, the value of d/2 at the minima of eachof the potentials is different giving rise to different thicknesses ofthe film at different locations of the KCl slab with respect to theiron substrate. Additional work must be expended against the nor-mal force to overcome this energy. Finally, the shapes of the curvesare slightly different, resulting in slightly different Young’s mod-uli of the film at different sites, corresponding to the second termin Eq. (1). These effects give rise to a pressure-dependent energydifference while sliding from site a to b as:

�E = (Eb − Ea) + P(tb0 − ta

0) + P2

(tb0

Yb− ta

0Ya

)(2)

and thus predicts both a linear and a quadratic pressure depen-dence in the sliding potential.

The data in Fig. 3 indicate, as noted above for the bilayer KClfilm, that the atop location of the potassium and chloride ions isthe most stable. Thus, the heights of the sliding potentials are cal-culated for sliding via the bridge site (from an atop to a bridge toan atop site, along the 〈1 0〉 direction), or via the hollow site (froman atop to a hollow to an atop site, along 〈1 1〉) and was done pre-viously [17]. The pressure-dependent height of the potential as afunction of normal pressure P is plotted in Fig. 4, up to 700 MPacorresponding to the hardness of iron [28]. The linear terms (�) forsliding along the 〈1 0〉 and 〈1 1〉 directions are shown and the cor-rections to those values arising from the quadratic terms in Eq. (2)

are also plotted (�). As was found for the bilayer slab, the quadraticpart of the sliding potential is much smaller than the linear part.Thus, the majority of the external work is expended in overcom-ing the atomic-scale roughness of the surface, and the differences in

284 M. Garvey et al. / Wear 274– 275 (2012) 281– 285

13.513.012.5

-4.811837

-4.811836

-4.811835

Bridge

Tota

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Hollow

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rgy

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(d/2)/Å

14.013.513.012.512.0-4.811838

-4.811837

-4.811836

-4.811835 Atop

Tota

l Ene

rgy

per I

nter

face

/10-1

4 J

(d/2)/Å

a

b

c

Fig. 2. Plot of the energy of a four-layer KCl slab sandwiched between iron as afunction of d/2, where d is the distance between the outermost edges of the slab.R(

ca

tdttts

Fig. 3. Plots of the fits to the results of the calculations shown in Fig. 1 for thepotassium and chloride ions on the KCl located on (a) atop, (b) bridge and (c) hollowsites on the iron substrate. (Fitting parameters are given in Table 1.)

6004002000

0.0

0.5

1.0

1.5

Δ E

/10-2

1 J

P/MPa

<10> E(P)<10> E(P 2)<11> E(P)<11> E(P 2)

Fig. 4. Plots of the linear change in the height of the sliding potential along the〈1 0〉 and 〈1 1〉 directions (�) and the effect on these potentials caused by adding thequadratic pressure-dependent term (�).

7006005004003002001000

60

70

80

90

100

110

<11>α = 0.057

S0 = 64.6 MPa

S/M

Pa

P/MPa

<10><11>

<10>α = 0.054

S0 = 61.9 MPa

esults are shown for potassium and chloride ions on the KCl located on (a) atop,b) bridge, and (c) hollow sites on the iron substrate. Lines are fits to the data.

ompliances at different slab locations, although they do play a role,re rather minor.

The resulting sliding potentials (Eq. (2)) are used to calculatehe shear strength as a function of pressure [2] and the results areisplayed in Fig. 5. Because the second term in Eq. (2) is small,

his results in a linear variation of shear strength with pressure,hereby enabling the values of S0 and to be calculated fromhe intercept and slopes of the curves, respectively. Note that, inpite of the quite large differences in the heights of the sliding

Fig. 5. Calculated plots of shear strength S versus contact pressure P for sliding alongthe 〈1 1〉 (�) and 〈1 0〉 (�) directions.

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M. Garvey et al. / Wea

otentials (Fig. 4), the pressure-dependent shear strength curvesre quite similar leading to rather isotropic shear for the Fe–KClnterface. This occurs because the differences in heights of theotentials are compensated by the different lattice periodicitieslong the 〈1 0〉 and 〈1 1〉 directions. The calculated values for theour-layer KCl slab are S0〈1 0〉 = 62 ± 15 and S0〈1 1〉 = 65 ± 11 MPahile ˛〈1 0〉 and ˛〈1 1〉 = 0.06 ± 0.01. The corresponding values for

he bilayer slab are S0〈1 0〉 = 64 ± 9, S0〈1 1〉 = 69 ± 8 MPa, and ˛〈1 0〉nd ˛〈1 1〉 equal 0.05 ± 0.01 [2]. Within the error limits, the val-es for the bilayer and four-layer KCl slabs are identical andoth are in reasonable agreement with the experimental valuesf S0 = 65 ± 5 MPa and = 0.14 ± 0.02, although again the calcu-ated values of are somewhat lower than the experimentalalue.

The shear properties along the 〈1 0〉 and 〈1 1〉 directions are iden-ical within experimental error suggesting that the calculated shearroperties of the KCl film are isotropic. Since the experiments werearried out using annealed metal foil substrates [16], in principle,he shear properties should be averaged over all substrates orien-ations for the theoretical data to be compared with experiment.ince, in the case of the KCl film, the shear is isotropic, such aver-ging is not necessary in this case, although this is likely not to beenerally true for other films.

The calculated values of S0 and indicate that the thickness ofhe alkali halide (in this case, KCl) slab has a negligible effect on thehear properties of the film. Eq. (2) suggests that there should be anxplicit dependence on films thickness in the quadratic term but theesults in Fig. 3 indicate that this term is much smaller than the lin-ar term, so that its overall contribution to the pressure-dependenthear strength (at least for a model KCl film) is negligible. The resultslso suggest that both the pressure independent part of the poten-ial (that yields S0) and the linear pressure-dependent term (thatields ˛) are not strongly affected by changing the thickness ofhe film implying that the interaction between the slab and theubstrate is dominated by local effects.

Thus, while the calculated value of is not in good agreementith experiment, suggesting that other sources for this discrep-

ncy should be sought, the values of S0 do agree with experiment.s discussed above, the calculations assume that the KCl film is

n registry with the iron substrate [2]. This enables calculationso be carried out using periodic boundary conditions as requiredor the DFT calculations and the assumption to be made that theateral friction force scales with contact area. It could be arguedhat the discrepancy between the calculated and measured valuesf arises because the interface is not commensurate. This, how-ver, would result in lower values of S0, whereas the calculatednd experimental values are in good agreement, suggesting thathe commensurate model is valid.

. Conclusions

First-principles DFT calculations are used to determine the shearroperties of four-layer KCl films sandwiched between slabs of

ron by considering the potential energy curves calculated for aCl film located at high-symmetry sites with respect to the ironubstrate. These calculations are compared to similar ones forilayer slabs in order to gauge the effects of film thickness on thehear properties. The resulting calculated values for the four-layerlab are S0〈1 0〉 = 62 ± 15 and S0〈1 1〉 = 65 ± 11 MPa, and ˛〈1 0〉 and

〈1 1〉 = 0.06 ± 0.01. These values are in excellent agreement withrevious results for the bilayer slab and in reasonable agreementith experiment. These results suggest that the slab thickness doesot have a profound effect on the tribological properties of the film,

[

– 275 (2012) 281– 285 285

and that the local bonding between the KCl slab and the iron sub-strate plays an important role. The results further imply that theassumptions of the model, namely that the film is in registry withthe substrate is true for the tribological interface.

Acknowledgments

This work was supported by the National Science Foundationunder grant number CHE-0654276 and the Office of Naval Research.

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