Morse potential-based model for contacting composite rough surfaces:Application to self-assembled monolayer junctions

Jonatan A. Sierra-Suarez,1,a) Shubhaditya Majumdar,2,a) Alan J. H. McGaughey,2,3

Jonathan A. Malen,2,3 and C. Fred Higgs III1,2,b)

1Department of Electrical and Computer Engineering, Carnegie Mellon University, 5000 Forbes Avenue,Pittsburgh, Pennsylvania 15213, USA2Department of Mechanical Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh,Pennsylvania 15213, USA3Department of Materials Science and Engineering, Carnegie Mellon University, 5000 Forbes Avenue,Pittsburgh, Pennsylvania 15213, USA

(Received 15 December 2015; accepted 15 March 2016; published online 13 April 2016)

This work formulates a rough surface contact model that accounts for adhesion through a Morsepotential and plasticity through the Kogut-Etsion finite element-based approximation. Compared tothe commonly used Lennard-Jones (LJ) potential, the Morse potential provides a more accurateand generalized description for modeling covalent materials and surface interactions. An extensionof this contact model to describe composite layered surfaces is presented and implemented to studya self-assembled monolayer (SAM) grown on a gold substrate placed in contact with a second goldsubstrate. Based on a comparison with prior experimental measurements of the thermal conduct-ance of this SAM junction [Majumdar et al., Nano Lett. 15, 2985–2991 (2015)], the more generalMorse potential-based contact model provides a better prediction of the percentage contact areathan an equivalent LJ potential-based model. VC 2016 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4945759]

I. INTRODUCTION

The ability to accurately model the interaction of twobodies in contact is of great importance in the field of tribol-ogy, which is the study of interacting surfaces and their asso-ciated friction, lubrication, and/or wear behavior. Inactuality, even surfaces that appear very smooth are rela-tively rough, since they are microscopically comprised ofprotuberances called asperities. Consequently, surfaces thatappear to be in significant contact are actually in contactover an area known as the real area of contact. Situationswhere partial contact of surfaces occurs include micro/nano-scale systems, relatively hard surfaces where interfacialasperities undergo minimal deformation, and mixed lubrica-tion, where the asperities partially support the loads atliquid-mediated interfaces.1–5 In all of these cases, the realarea of contact must be resolved, usually as a function of thecontact load, mechanical properties, surface topography, andadhesive and repulsive surface forces.

The seminal work on the contact of real surfaces is theGreenwood-Williamson (GW) model.6 It was developed tomodel the contact mechanics between two real surfaces bytreating asperities as individual Hertzian hemispherical con-tacts of equal radii and varying heights.7 The GW model canbe used to predict the elastic contact stresses on the asperitiesfrom the two contacting bodies. While this model continuesto be the foundation of most statistical rough surface models,it makes two key assumptions of neglecting both plasticity

and adhesion, which can cause errors while evaluating con-tact of smooth surfaces under large loads.

Asperities on real surfaces usually enter the plastic re-gime even when light loads are applied.8 Adhesion, whichgenerally plays a smaller role in conventional macro-scaleapplications, becomes a stronger contributor to the resultingcontact stress and surface deformation the smoother thesurfaces are.9 While some surface contact models haveadded plasticity10,11 or adhesion12,13 separately to the GWmodeling framework, Chang, Etsion, and Bogy (CEB) devel-oped a series of models to introduce both adhesion and thefull spectrum of surface deformation regimes—elastic,elastic-plastic, and purely plastic.9,14,15 They introducedplasticity effects through the concept of a critical interfer-ence, which is the inter-penetration length (or maximum de-formation) of an asperity into a surface at which plasticityfirst begins.15 Volume conservation and a uniform appliedpressure were then assumed at each of the asperities in orderto account for plasticity. They incorporated adhesion withthe elastic-plastic deformation using their deformed asperityprofiles in conjunction with the Derjaguin-Muller-Toporov(DMT) model.9,16

The two primary models for adhesion are the Johnson-Kendall-Roberts (JKR) model17 and the DMT model.16 TheJKR model allows the adhesive force to affect the surfaceprofile (i.e., shape) of a hemispherical contact, and the forcesoutside of the contact area are neglected. The DMT model,on the other hand, does not allow the adhesive force to affectthe profile of a hemispherical contact. Its profile follows aHertzian profile instead and all adhesive forces, even thoseoutside the contact area, are included. Tabor first postulatedthat these two models were limiting cases of the same

a)J. A. Sierra-Suarez and S. Majumdar contributed equally to this work.b)Author to whom correspondence should be addressed. Electronic mail:

higgs@andrew.cmu.edu. Telephone: (412) 268-2486. Fax: (412) 268-3348

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general adhesive theory.18 In other words, the DMT model iswell-suited for hard materials with a small asperity radiussuch as metals, while the JKR model is better suited for softmaterials with a large asperity radius.19 The adhesive forcedepends on the chosen surface interaction potentialemployed in the adhesive model, the most common beingthe Lennard Jones (LJ) potential.20 Muller et al. developed acomplete model accounting for Tabor’s findings using the LJpotential.21 The Maugis-Dugdale model was the first to de-velop a closed form solution for the JKR-DMT transition.19

A comprehensive chart for determining which adhesionmodel to use for elastic contact was presented by Johnsonand Greenwood.22

The CEB models, due to their generality and simplicity,have been widely adopted and extended by otherauthors.23,24 The assumptions used, however, do not capturethe correct asperity behavior in the elastic-plastic regime.Kogut and Etsion (KE) presented a thorough comparison ofthe CEB models to finite element analysis (FEA) of an asper-ity under deformation and found significant deviation.25

They went on to present a new set of models to capturerough surface contact in the elastic-plastic regime, alsoincluding the effects of adhesion.8,25–27 Their adhesionmodel incorporated the DMT model in conjunction with theLJ potential. While the LJ potential is most often used as afirst approximation for a given material system due to itssimplicity, for many materials it can lead to poor predictionsof physical properties.28

The Morse potential is more general than the LJ poten-tial as it has three free parameters as compared to two freeparameters in the LJ potential.29 It is suitable for manysystems, including molecules and metals.30,31 It has beenessential for representing complex material interactions,such as adhesion of thin films on metal substrates [e.g.,self-assembled monolayers (SAMs) on gold32], and has beenparameterized for both covalent and van der Waals interac-tions.33–38 Herein, we develop an adhesion-based contactmodel for interfaces by incorporating the Morse potentialinto the KE (and as a consequence the DMT) modelingframework. The resulting contact model can be employedfor a wide variety of surface materials, from hard-soft inter-faces to organic-inorganic heterojunctions.

We first derive expressions for the adhesive pressureand interaction energy per unit area between two macro-scopic bodies where atomic interactions are described usingthe Morse potential in Secs. II A–II C. We then incorporatethese expressions into the KE model and calculate the totaladhesive pressure between the rough bodies in Sec. II D. InSec. II E, we present a method to calculate adhesive pres-sures where one or both bodies in contact are composed oflayers of different materials. In Sec. II F, we describe the an-alytical model used to characterize the roughness parametersof a body. Lastly, we predict the percentage contact area(i.e., the ratio of the real area of contact to total surface areawhich appears to be in contact) between two rough goldsurfaces where one is coated with a SAM in Sec. III. Thesepredictions are validated using interface thermal conduct-ance measurements performed by the authors.39

II. THEORY

A. Derivation of surface pressure using the Morsepotential

The Morse (subscript M) potential between two pointparticles (atoms), separated by a distance r, is

EM;pp ¼ De½e#2aðr#r0Þ # 2e#aðr#r0Þ&; (1)

where the “pp” subscript represents a particle-particle inter-action. Here, De is the depth of the potential well, a describesthe inverse of the width of the well, and r0 is the position ofthe minimum of the well. The 2Dee#aðr#r0Þ term representsLondon dispersion (i.e., van der Waals) interactions, whichare attractive, and the Dee#2aðr#r0Þ term represents exchangerepulsion. Repulsive energy is defined as positive and attrac-tive energy as negative. The difference in behavior betweenthe Morse potential and the LJ potential, given byELJ;pp ¼ 4!½ðr=rÞ12 # ðr=rÞ6&, where ! is the depth of thewell and r is interatomic distance at which ELJ;pp ¼ 0, isshown in Fig. 1(a). The parameters used in plotting theMorse potential describe the interaction between a thiolgroup (-SH group) and a gold atom.32 The LJ potential isconstructed to have the same position and depth of the wellas the Morse potential, but ultimately exhibits a differentenergy landscape at other positions due to its mathematicalformulation. All parameters for the potentials used in Fig.1(a) are listed in Table I.

FIG. 1. (a) Comparison of Morse and LJ potentials for the same energy welldepth and equilibrium separation for a thiol-gold interaction.32 (b)Schematic diagrams representing interactions between two point particles,one point particle and a flat, semi-infinite substrate, and two flat, semi-infinite substrates.

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To study the asperities that describe rough surfaces, thispoint-point potential must be extended to describe interac-tions between macroscopic bodies. Let us first consider theattractive part of the pair potential given byEa

M;pp ¼ #2Dee#aðr#r0Þ. We assume additivity of these inter-actions, such that the net interaction between a single atomand a monatomic substrate is the sum of its interactions withall the atoms in the substrate.43 If we consider a substrate(semi-infinite solid) with a flat surface having a volumetricdensity of atoms q1 at a distance D from an atom (the pointparticle), the attractive interaction energy will be

EaM;ps Dð Þ ¼ # 4pq1De

a3aDþ 2ð Þe#a D#r0ð Þ; (2)

where the subscript “ps” denotes a particle-substrate interac-tion. A schematic diagram of this geometry is shown in Fig.1(b). Following a similar procedure, the attractive interactionenergy per unit area between two substrates, whose surfacesare separated by a distance D, is

Ea00

M;ss Dð Þ ¼ # 4pq1q2De

a4aDþ 3ð Þe#a D#r0ð Þ: (3)

Here, q2 is the volumetric density of atoms in the secondsubstrate. If the second structure is instead a single layer of

atoms (i.e., a surface, such as in a two-dimensional material)with an areal atomic density r2, the attractive interactionenergy per unit area is

Ea00

M;surf#s Dð Þ ¼ # 4pq1r2De

a3aDþ 2ð Þe#a D#r0ð Þ; (4)

where the subscript “surf ” denotes surface. Derivations ofEqs. (2)–(4) are provided in Sec. S1 of the supplementarymaterial.44 The corresponding attractive forces per unitarea are

Fa00

M;ss Dð Þ ¼ #dEa00

M;ss

dD¼ Pa

M;ss

¼ # 4pq1q2De

a3aDþ 2ð Þe#a D#r0ð Þ; (5)

Fa00

M;surf#s Dð Þ ¼ PaM;surf#s ¼ #

4pq1r2De

a2aDþ 1ð Þe#a D#r0ð Þ;

(6)

where PaM;ss and Pa

M;surf#s are the attractive pressures.Until now, we have only considered the attractive inter-

action. To realize a potential with an equilibrium separation,we need to add the repulsive interaction. Following the pro-cedure of Eqs. (2)–(6), we obtain the general form of the re-pulsive interaction for either the substrate-substrate orsurface-substrate configurations to be Pr

M ¼ A1ðaDþ A2Þe#2aðD#r0Þ, and the subsequent total adhesive pressure isPM ¼ Pr

M þ PaM. To determine the unknown coefficients A1

and A2, we first enforce the physical limit that when the dis-tance between the surfaces of the two bodies equals theirequilibrium separation re, zero pressure (i.e., zero force) isfelt by them, i.e., PMðreÞ ¼ 0. We note that re does not equalr0. This assumption maintains a continuity of the physicalpicture of atomic-scale interactions when moving from theatomic to the macro-scale and was employed in the deriva-tion of the DMT model using the LJ potential.19,43,45 We canthen solve for A1 in terms of A2 for the substrate-substrateand surface-substrate cases as

A1;ss ¼4pq1q2De

a3

are þ 2

are þ A2;ss

! "ea re#r0ð Þ; (7)

A1;surf#s ¼4pq1r2De

a2

are þ 1

are þ A2;surf#s

! "ea re#r0ð Þ: (8)

The physical constraint of PMðD!1Þ ¼ 0 is naturally sat-isfied by the form of the Morse potential. There are no morephysical constraints that can be imposed to specify the co-efficient A2.

We generalize the expression for PM by grouping pa-rameters into the coefficients B1, B2, and B3 (defined inTable II), leading to

PM Dð Þ ¼ 4pq1De

a2

! "B1;PM

( B2;PMea re#r0ð Þe#2a D#r0ð Þ # B3;PM

e#a D#r0ð Þh i

;

(9)

where the corresponding interaction energy per unit area is

TABLE I. Atomic interaction and surface topography parameters.

Morse potential parameters (S-Au)32,39

De (energy well depth) 0.38 eV

a (measure of energy well width) 1.67 ( 10#10 m#1

r0 (position of energy

well minimum)

2.65 ( 10#10 m

LJ potential parameters

! (energy well depth) 0.38 eV (S-Au), 0.0029 eV(C-Au),40 0.0017 eV (Au-Au)40

r (position where energy is zero) 2.360 ( 10#10 m (S-Au)

3.424 ( 10#10 m (C-Au)40

2.934 ( 10#10 m (Au-Au)40

Surface potential parameters

Dc (work of adhesion) 6.60 ( 10#1 J/m2 (Morse—S-Au)

1.35 ( 10#2 J/m2 (LJ—C-Au)

6.98 ( 10#3 J/m2 (LJ—Au-Au)

q1;2 (volumetric number density) 5.89 ( 1028 m#3 (Au)

3.85 ( 1028 m#2 (C)

r2 (areal number density) 4.62 ( 1018 m#2 (S)

re (equilibrium separation) 1.90 ( 10#10 m (Morse—S-Au)

3.17 ( 10#10 m (LJ—C-Au)

2.60 ( 10#10 m (LJ—Au-Au)

Material properties (Au 1 SAM) for effective rough surface

E (elastic modulus)41 4.712 ( 1010 N/m2

" (Poisson ration)41 0.44

H (hardness)42 210 MPa

Asperity properties

rh [root mean-squared(RMS) roughness]

6.30 ( 10#10 m

g (areal asperity density) 2.96 ( 1015 m#2

R (radius of asperity) 6.98 ( 10#8 m

t1 (thickness of S layer) 1.50 ( 10#10 m

t2 (thickness of C layer) 13.5 ( 10#10 m

t3 (thickness of SAM) 15 ( 10#10 m

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E00M Dð Þ ¼ 4pq1De

a3

! "B1;E00M

( B2;E00Mea re#r0ð Þe#2a D#r0ð Þ # B3;E00M

e#a D#r0ð Þh i

:

(10)

For hemispherical asperities, a common form forexpressing the energy and pressure equations is through thework of adhesion Dc, which represents the magnitude of E00Mat its minimum value (i.e., at D ¼ re) and is

DcM ¼ jE00M D ¼ reð Þj

¼ 4pq1De

a3

! "B1;DcM

e#a re#r0ð Þ B2;DcM# B3;DcMð Þ

####

####: (11)

We note that evaluation of Eqs. (9)–(11) requires specifica-tion of re and A2. All other parameters are based on theMorse potential parameters and the crystal structures of thebodies under consideration and thus, known a priori.

We evaluate re and A2 using single-point energy calcula-tions. We first place a single layer of atoms above a substrateof another kind of atoms. A schematic diagram of this struc-ture is shown in Fig. 2(a). The substrate is an fcc solid hav-ing a (111) surface with a lattice constant of 4.08 A (i.e., thatof gold). The surface is a close-packed structure having anearest neighbor distance of 4.997 A (i.e., that of a thiol-based SAM).39,40,46 Our calculations correspond to the casewhere the atoms of the single-layer surface are directlyabove the three-fold hollow sites of the surface of the sub-strate. Only the interaction potential (Morse or LJ) betweenthe layer and the substrate is considered while calculatingthe energy of the system at different separations (details pro-vided in Sec. S2).

The variation of the interaction energy between the sur-face and the substrate (non-dimensionalized by DcLJ) as afunction of the separation distance between them is plottedin Fig. 2(b) for both Morse (blue circles) and LJ (redsquares) potentials, corresponding to a thiol-gold interactionwhose parameters are listed in Table I. The magnitude of theinteraction energy at its minimum value is Dc, and the corre-sponding separation distance is re. We can thus find the ratio

DcM=DcLJ from this calculation, as shown in Fig. 2(b). Wetake this approach because exact analytical expressions forDc exist for an LJ potential and are19

DcLJ;ss ¼pq1q2 4Der6

$ %

16r2e

;

DcLJ;surf#s ¼pq1r2 4Der6

$ %

9r3e

:

(12)

Once DcM is calculated using the results of the single-point energy calculations and Eq. (12), we can analyticallydetermine A2 from Eq. (11). We note, however, that these cal-culations are specific to our system, with A2 equal to #4.9138and DcM equal to 0.66 J/m2. To allow the implementation ofthis model to other surface-substrate systems, we provide gen-eralized expressions describing the variation of re=rNN andDcM=DcLJ as a function of the parameters arNN and r0=rNN

(rNN being the nearest neighbor distance between atoms in thesubstrate) in Appendix A [Eqs. (A1)–(A4)].

The expressions for adhesive pressure and energy perunit area based on the Morse potential, represented by Eqs.(9)–(11), can now be used with the DMT and KE models toestimate the total adhesive force for hemispherical asperitiescontacting a flat substrate.

B. Single asperity deformation

The DMT model provides a method to calculate the ad-hesive force for a deformed hemisphere contacting a flat,undeformable surface. The method to calculate the deformedhemispherical profile is summarized in this section.16 Thehemisphere is assumed to obey Hertzian theory and todeform elastically. The height Z of any point on the surfaceof the hemisphere at a radial distance x from the asperitycenter, following the DMT convention, is

Z a;x;Rð Þ¼1

pRa x2#a2ð Þ1=2# 2a2#x2ð Þtan#1

ffiffiffiffiffiffiffiffiffiffiffiffix2

a2#1

r !" #

;

(13)

where R is the radius of the hemisphere, and a is the radiusof the contact region. A schematic diagram of this setup isshown in Fig. 3(a).

The expression for Z can also be written in terms of theinterference x, which is the centerline deflection of the as-perity, i.e., the maximum deformation of the compressed as-perity.47 It can be seen that Z ¼ 0 when x ¼ a. Eq. (13) was,however, developed for macro-scale deformations. Whenmolecular potentials are taken into account, it is physicallyimpossible for two atoms, or in this case two bodies, to havea separation distance of zero. Thus, Eq. (13) is adjusted suchthat the minimum value of Z equals the equilibrium separa-tion between the surfaces re (derived in Sec. II A) and is fur-ther simplified by substituting the Hertzian relation for thecontact radius

a ¼ ðxRÞ1=2; (14)

thus arriving at

TABLE II. Coefficients in the expressions for pressure [Eq. (9)], energy perunit area [Eq. (10)], and work of adhesion [Eq. (11)] derived for the interac-tion between two substrates (ss) and a surface with a substrate (surf-s)

described by a Morse potential.

B1 B2 B3

PMðDÞ PM;ssq2

aaDþ A2;ssð Þ are þ 2ð Þ

are þ A2;ssð ÞaDþ 2

PM;surf#s r2aDþ A2;surf#sð Þ are þ 1ð Þ

are þ A2;surf#sð ÞaDþ 1

E00MðDÞ E00M;ss

q2

aare þ 2ð Þð2aDþ 2A2;ss þ 1Þ

4 are þ A2;ssð ÞaDþ 3

E00M;surf#s r2are þ 1ð Þð2aDþ 2A2;surf#s þ 1Þ

4 are þ A2;surf#sð ÞaDþ 2

DcM DcM;ssq2

aðare þ 2Þð2are þ 2A2;ss þ 1Þ

4ðare þ A2;ssÞare þ 3

DcM;surf#s r2ðare þ 1Þð2are þ 2A2;surf#s þ 1Þ

4ðare þ A2;surf#sÞare þ 2

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Z x; !xð Þ ¼xp

!x2 # 1ð Þ1=2 # 2# !x2ð Þ tan#1ffiffiffiffiffiffiffiffiffiffiffiffiffi!x2 # 1p$ %h i

þ re;

(15)

where !x ¼ x=a. Eq. (15) is valid for elastic deformations, butit has been shown that the elastic deformation assumption isnot adequate for most scenarios since plastic deformationoccurs.23 In order to account for both elastic and plastic de-formation, we follow the approach taken by Kogut andEtsion,26 where Z is further non-dimensionalized withrespect to the critical interference xc, which is the interfer-ence at which plasticity begins (details in Appendix B).Thus, the separation Z in the elastic regime, from Eq. (15),can be expressed as

Z x; !xð Þxc

¼ 1

pxxc

f !xð Þ þ re

xcfor 0 ) x=xc ) 1; (16)

where f ð!xÞ ¼ ð!x2 # 1Þ1=2 # ð2# !x2Þ tan#1ðffiffiffiffiffiffiffiffiffiffiffiffiffi!x2 # 1p

Þ: Thisnon-dimensionalization allows for the study of the Hertzianprofile independent of material properties. For the deforma-tion regimes where x=xC * 1 (elastic-plastic regimes), we

use the FEA derived dimensionless separations found byKogut and Etsion25,26

Z x; !xð Þxc

¼ 0:951

pxxc

! "1:153

f !xð Þ þ re

xcfor 1 ) x=xc ) 6;

(17)

and

Z x; !xð Þxc

¼ 0:457

pxxc

! "1:578

f !xð Þ þ re

xcfor 6 ) x=xC ) 110:

(18)

Kogut and Etsion validated Eqs. (17) and (18) for a range ofvalues of the plasticity index w, which Greenwood andWilliamson showed to be directly related to the critical inter-ference as w / x#0:5

c .6 We specifically note that the inter-ference x of the contacting asperities must fall within therange of 0 to 110 xC in order for the FEA data used to beaccurate. The various deformation regimes used in this studyare listed in Table III.

C. Morse potential-based adhesion model based onasperity deformation

We now derive expressions for the adhesive pressurebetween a single hemispherical asperity and a substrate inter-acting through a Morse potential. The asperity may be a solidstructure (having a volumetric density of atoms q1) or hollowand composed of a single layer of atoms on its surface (hav-ing an areal density of atoms r2). Derjaguin et al. found theadhesive force between atoms in a slice along the surface ofthe hemisphere (i.e., a ring with radius x and thickness dx)and the flat surface to be16

dFadc ¼ 2pxPMðZÞdx: (19)

It is important to note that the pressure PMðZÞ contains theeffect of all the atoms behind the exposed surface (if any).Thus, if we integrate Eq. (19) for all x, we get the total adhe-sive force between a deformed hemispherical asperity and asubstrate to be

Fadc ¼ 2p

ð1

a

xPMðZÞdx: (20)

Strictly speaking, the upper limit of the integral shouldbe the hemispherical radius R; but it can be approximated asinfinity without significant error if PM;ssðZÞ! 0 at separa-tions much smaller than R, as is the case here.16 Now, fromEq. (20), we define Fad

c ða ¼ 0Þ ¼ Fad0 ¼ 2pRDc as the point

contact adhesive force. We use Fad0 to normalize the total

contact adhesive force in the same manner as Kogut andEtsion by changing the integrating variable to !x ¼ x=a andthen dividing throughout by Fad

0 .26 The variables x; a; r0,and re are also non-dimensionalized using xc. The normal-ized force of adhesion for an asperity in contact (in any de-formation regime), with respect to the point contact adhesive

FIG. 2. (a) Schematic diagram of the structure for single-point energy calcu-lations between a two-dimensional surface of atom type 1 (yellow) placed atdifferent heights above a substrate of atom type 2 (pink). (b) Interactionenergy, non-dimensionalized by DcLJ, using both Morse (red squares) andLJ (blue circles) potentials for a thiol-gold interaction (parameters in TableI) plotted as a function of separation distance D between the surface and thesubstrate. The magnitude of the interaction energy at its minimum value(where D¼ re) is equal to Dc.

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force, is found from Eq. (20) using the expression for PM

from Eq. (9). A general form of this expression is

Fadc xð ÞFad

0

¼ G1;Fad

ð1

1

G2;Fadc

ea re#r0ð Þe#2 axcð Þ Zxc

–r0xcð Þ

h

#G3;Fadc

e# axcð Þ Zxc

–r0xcð Þi!xd!x; (21)

where the coefficients G1, G2, and G3 for the substrate-substrate and surface-substrate cases are listed in Table IV.For any deformation regime, the corresponding separation Z[Eqs. (16)–(18)] can be used in Eq. (21).

In the case of a hemispherical asperity not in contactwith the surface, the adhesive force from the DMT model is

Fadnc ¼ 2pR

ð1

Da

PMðhÞdh; (22)

where h is the separation between a slice of thickness dhwithin the asperity and the flat surface, and Da is the

minimum separation between the asperity and flat surface, asshown in Fig. 3(b). Using the expression for PM from Eq. (9)and normalizing with Fad

0 , we derive the non-contact adhe-sive force to be

Fadnc Dað ÞFad

0

¼ G1;Fadnc

G2;Fadnc

ea re#r0ð Þe#2 axcð Þ Daxc

–r0xcð Þ

h

#G3;Fadnc

e# axcð Þ Daxc

–r0xcð Þi; (23)

where the coefficients G1, G2, and G3 for the substrate-substrate and surface-substrate cases are listed in Table IV.

D. Extension to two-body contact problem

Until now, we have derived expressions for the adhesiveinteraction of a single asperity with a substrate. To model theinteraction between two real bodies, we analyze the interac-tion of a number of asperities (representing the rough surfaceof one body) with a substrate. The two-body contact problemcan be visualized as shown in Fig. 4. A balance between anyexternally applied force, the adhesive force, and the contactforce must exist for the bodies to be in equilibrium. In thissection, we present the equations necessary to calculate eachof these forces and the percentage contact area for the two-body system. In order to simplify the problem, the twobodies with rough surfaces are transformed into an equiva-lent system of one rough body (Body I) in contact with asmooth, rigid body (Body II). The geometrical properties ofBody I are equal to the sum of the geometrical properties ofthe original two rough bodies. The elastic modulus of Body I

FIG. 3. (a) Schematic diagram of ahemispherical asperity in contact witha flat surface and undergoing deforma-tion. The flat surface is assumed to berigid. (b) Schematic diagram of a hem-ispherical asperity at a height Da abovea flat surface.

TABLE III. Range of contact regimes.

Regime Dimensionless separation

Non-contact x=xc ) 0

Elastic 0 ) x=xc ) 1

Elastic-plastic I 1 ) x=xc ) 6

Elastic-plastic II 6 ) x=xc ) 110

Plastic x=xc * 110

TABLE IV. Coefficients in the expressions for contact [Eq. (21)] and non-contact [Eq. (23)] adhesive pressures derived for the interaction between two sub-strates (ss) and a surface with a substrate (surf-s) described by a Morse potential. The coefficients B2 and B3 are listed in Table II.

G1 G2 G3

Fadc ðxÞFad

0

Fadc;ssðxÞFad

0

ðaxÞ ea re#r0ð Þ

B2;DcM;ss# B3;DcM;ss

ðare þ 2Þ axcð Þ Zxc

( )þ A2;ss

h i

ðare þ A2;ssÞaxcð Þ

Z

xc

! "þ 2

Fadc;surf#sðxÞ

Fad0

ðaxÞ ea re#r0ð Þ

B2;DcM;surf#s# B3;DcM;surf#s

ðare þ 1Þ axcð Þ Zxc

( )þ A2;surf#s

h i

ðare þ A2;surf#sÞaxcð Þ

Z

xc

! "þ 1

FadncðDÞFad

0

Fadnc;ssðDÞFad

0

1

B2;DcM;ss# B3;DcM;ss

ðare þ 2Þ 2 axcð Þ Dxc

( )þ 2A2;ss þ 1

h i

4ðare þ A2;ssÞaxcð Þ

D

xc

! "þ 3

Fadnc;surf#sðDÞ

Fad0

1

B2;DcM;surf#s# B3;DcM;surf#s

ðare þ 1Þ 2 axcð Þ Dxc

( )þ 2A2;surf#s þ 1

h i

4ðare þ A2;surf#sÞaxcð Þ

D

xc

! "þ 2

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is calculated using Eq. (B3). The surface of Body I is repre-sented in a statistical manner as a distribution of hemispheri-cal asperities of uniform radii R with heights varying basedon a specified probability distribution /ðzÞ. The total non-dimensional adhesive force Fad+ between the asperities andthe surface is calculated as the sum of all the adhesive forces(i.e., contact and non-contact) weighted by the distributionof asperity heights for all of the deformation regimes. Thus,using Eqs. (21) and (23), we arrive at

Fad+ ¼ Fad

AnH¼ 2pgRDc

H

ðd+

#1

Fadnc d+ # z+ þ r+eð Þ

Fad0

/+ z+ð Þdz+

0

B@

þðd+þ110x+c

d+

Fadc z+ # d+ð Þ

Fad0

/+ z+ð Þdz+

1

CA; (24)

where g is the areal density of asperities. Here, any variablewith the superscript *, except for Fad, is dimensionless and isnormalized using the standard deviation of surface heights,rh. Fad is normalized by the nominal contact area An and thehardness H. d+ represents the non-dimensional separationbetween the flat surface and the mean-line of asperityheights. The distribution / has each of its variables normal-ized by rh. For a normal distribution of asperity heights,25

/+ z+ð Þ ¼ rhffiffiffiffiffiffi2pp

rs

e#12

rhrsð Þ

2z+ð Þ2 : (25)

Alternatively, a deterministic approach could be used tocalculate the percentage contact area and the mean separa-tion between our two surfaces.47–54 In addition to the com-plexity of numerically solving the elastic and plasticconstitutive equations for deformation, such an approachwould require an iterative optimization method in order tocalculate the balance between the adhesive and contactstresses. While such a deterministic approach may shedadditional insight, we believe that a statistical model betterrepresents our system due to its isotropic nature andAngstrom-scale RMS roughness of the surface (Table I). The

measurements of Majumdar et al.39 were also statisticallyaveraged representations of the thermal conductance values.The potential difference between statistical and deterministicmodels is thus left for future study.

E. Composite asperity model for modeling layeredstructures

We now extend our formulation to model adhesionbetween two surfaces where one of them has a thin filmgrown on it. The thin film is treated as an incompressiblecoating that follows the deformation behavior of the underly-ing substrate, such as in the case of an organic SAM grownon a metal or dielectric substrate.55 Modeling the surface to-pography of such a system requires the asperities to be com-posed of multiple materials in a layered configuration, thuscreating a composite asperity. Assuming substrate effectsdominate, all materials of the composite asperity areassumed to have the same elastic modulus, Poisson ratio,and hardness, while having independent Morse (or LJ) pa-rameters, layer thicknesses, and distances from the flatsurface.

The contribution of each layer in the composite asperityto the adhesive pressure is incorporated by adjusting its dis-tance from the flat surface. For example, considering onlyelastic deformations, each slice within the asperity will nowhave a height Z from the surface [derived from Eq. (16)]given by

Z x; !rð Þxc

¼ 1

pxxc

f !rð Þ þ re

xcþ t

xcfor 0 ) x=xc ) 1; (26)

where t is the minimum separation of a hemisphere com-posed of one kind of atoms within the composite asperityand the surface of the asperity. Similar adjustments are madefor the elastic-plastic deformation regimes [Eqs. (C1) and(C2)]. The remaining calculations are the same as given byEqs. (19)–(25). The contribution of each material layer to theadhesive pressure is modeled by subtracting the adhesivepressure of a smaller hemisphere from the adhesive pressureof the larger hemisphere, thus creating the hemisphericallayer.

F. Characterization of an experimental surface

A rough surface can be geometrically modeled as beingcomposed of multiple hemispherical asperities arranged withan areal density g. The asperities have a mean radius R witha standard deviation of asperity heights rs, which is mathe-matically correlated to the standard deviation of surfaceheights rh.25 How these quantities can be obtaining fromatomic force microscopy (AFM) measurements has beenstudied in detail by Bush et al.,56 Gibson,57 and McCool.58

The quantities g, R, and rs were originally derived byNayak,59 who built upon the work of Longuet-Higgins.60

Nayak found that for random and isotropic surfaces having aGaussian distribution of surface heights, a surface could becompletely characterized by its spectral moments corre-sponding to height, slope, and curvature. Kotwal andBhushan61 studied non-Gaussian surfaces with kurtosis and

FIG. 4. Conversion of two real surfaces to a statistical representation. Thereal surfaces are mapped to a flat surface in contact with a rough surfacecomprised of hemispherical asperities of radii R, which follow a statisticaldistribution [Eq. (25)], allowing for the height of each asperity to vary aboutthe mean line.

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skew but such an analysis is extremely complex and beyondthe scope of this study. More recent work has focused on thefractal-like behavior of surfaces and addressed variationswith AFM scan parameters.50,62,63

As opposed to surfaces explored in multi-scale studies,64

the sputtered thin-film surface discussed in Sec. III does notexhibit any trend in the roughness parameters between dif-ferent AFM scans (see Sec. S3). Instead, random variationsin rh; R; and g are found between scans. Poon andBhushan65 also found that AFM surfaces do not follow atrend in roughness properties with scan size. We speculatethat the dominant cause of our variations is related to the sur-face manufacturing process. Furthermore, given theAngstrom-scale roughness of our surfaces, we do not believethat it is meaningful to consider them from a fractal perspec-tive. We thus follow McCool’s approach to characterize thesurface topography.58 From metrology data (i.e., AFMimages) of a given surface, the parameters g, R, rh, and rs

can be calculated from

g ¼h d2n=d2x$ %2i=h dn=dxð Þ2i

6pffiffiffi3p ; (27)

R ¼ 3ffiffiffipp

8h d2n=d2xð Þ2i1=2; (28)

rh ¼ hn2i1=2 ; (29)

and

rs ¼ rh 1# 0:0003717

gRrhð Þ2

!1=2

; (30)

where n is the measured height (above the mean-line), andthe operator hi denotes a spatial arithmetic average. Thederivatives are calculated along a set of straight lines parallelto the horizontal axis of an AFM scan. The derivatives arenumerically calculated using central difference schemes(providing second-order error with respect to grid spacing).The derivatives are only calculated for the interior points,i.e., the edge points are not considered.

III. RESULTS

A. Comparison of Morse and LJ adhesive pressures

We now use the example of a thiol-gold interaction tohighlight the difference in adhesive behavior predicted by aMorse potential and an equivalent LJ potential [Fig. 1(a)].The non-dimensional adhesive (Fad+ ¼ Fad=AnH) forcebetween a flat gold substrate and a rough thiol surface (i.e., asingle layer of thiol groups) is plotted in Fig. 5(a) as a func-tion of d+ using Eq. (24) (Morse potential-based, solid blueline) and using the KE model25 (LJ potential-based, solid redline) with appropriate values of Dc (calculated separately forthe Morse and LJ cases), r0, and r. Details of the formulationinvolving surface-substrate adhesion based on the LJ poten-tial are described in Appendix D and were derived as anextension to the KE model. The roughness parameters g, R,rh, and rs are obtained from AFM measurements of a gold

surface, as outlined in Sec. II F, and are listed in Table I forone particular AFM measurement from a set of separatemeasurements (details in Sec. S3). It must be noted that theroughness parameters listed in Table I are for the new, effec-tive rough surface for which the AFM measurements of thesurface heights were doubled before using Eqs. (27)–(30).Expressions for the total non-dimensional contact pressureP+cs and the percentage contact area A+ between deformingasperities and the substrate are provided in Appendix E [Eqs.(E1) and (E2)]. The contact reaction pressure using Eq. (E1)is also plotted in Fig. 5(a) (dashed line).

The equilibrium separation between the bodies undercontact is achieved when P+ad ¼ Fad+=An ¼ P+cs (i.e., a bal-ance between the adhesive and contact pressures). This sepa-ration is found by identifying the point of intersectionbetween the adhesive and contact pressure curves, as shownin Fig. 5(a). If an external load is applied to the top surface,then it is added to P+ad. This equilibrium separation is thenused to find the percentage contact area A+, whose behavioras a function of d+ [using Eq. (E2)] is shown in the inset ofFig. 5(a). The Morse potential-based adhesion model pre-dicts 1.74 times the percentage contact area than the LJbased model, therein emphasizing the importance of the spe-cialized framework based on the Morse potential.

B. Application to thin films: Self-assembled monolayeron gold

Thin films of SAMs provide a convenient and simplesystem to tailor interfacial properties of surfaces.46 Theyhave gained widespread use in nanoscience with applicationsin surface functionalization,66,67 electrochemistry,68,69 elec-tronics,70,71 and thermoelectrics.72–74 A thiol-based SAM ona gold substrate is the most widely studied configurationwith detailed studies performed on its preparation,46 struc-ture,75,76 and transport properties.39,77–79

Many experiments for probing the transport properties ofSAMs require the formation of a SAM junction.39,77–81 Here,a SAM grown on one substrate has another substrate (usuallya metal) brought in contact with it either through high-energydeposition techniques such as sputtering or evaporation82 orthrough transfer printing.83 The latter technique is widelyused, as it does not damage the underlying SAM layer duringthe formation of the junction.83–85 This technique has recentlybeen used to create SAM junctions comprised of metal-dielec-tric77,78 and metal-metal substrates39 to study the junctionthermal conductance. Experimental measurements of the junc-tion thermal conductance,39,78 however, do not agree with pre-dictions from molecular dynamics simulations.39,86,87 Wehypothesize that a major source of this discrepancy is incom-plete contact between the two surfaces—one being the SAMgrown on a metal/dielectric and the other being the baretransfer-printed metal.

Using our Morse potential-based contact model in con-junction with the composite asperity model, we can predict thepercentage contact area between two rough metal substrateswhere one has a SAM grown on its surface. We choose a sys-tem composed of a 1,10-decanedithiol SAM grown on a goldsubstrate brought in contact with another gold substrate, which

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is identical to the system recently studied by Majumdar et al.39

The asperities are composed of three materials—thiol (denotedby S), carbon (denoted by C), and gold (denoted by Au: wealso neglect the contribution of hydrogen atoms) as shown inFig. 5(b). The planar substrate is gold. The thiol layers aresingle-atom thick and are treated as surfaces. The thickness ofthe carbon layer is assumed to be 12 A, given that the SAMlayer has an average thickness of 16 A,39 which is commensu-rate to that measured in other experiments.46,75 The asperitiesdefining the carbon shell are assumed to be shifted 1.5 A (t1)and 13.5 A (t2) from the asperity surface. The inner thiol layeris shifted by 15 A (t3), which is also where the Au section ofthe asperity is assumed to begin. The thiol-gold interaction ismodeled using a Morse potential, while the carbon-gold and

gold-gold interactions are modeled using LJ potentials (and theKE model for their adhesion). We calculate a plasticity indexof 12 6 3 for our gold samples [using Eq. (B4)], which is con-sistent with those reported in literature for sputtered gold.41

The adhesive pressure between the flat gold substrateand each of the three components in the asperities and thecontact pressure are plotted in Fig. 5(c) as a function of themean separation between the two surfaces. The contributionfrom the two different thiol layers is added together into asingle thiol adhesion curve. The material properties used toplot the curves in Fig. 5(c) are listed in Table I. The total ad-hesive pressure is the sum of the individual contributionsfrom each material within the asperity and is dominated(>99%) by the thiol-gold adhesion involving the thiol layer

FIG. 5. (a) Comparison of the non-dimensional adhesive pressure calculated using the Morse potential-based (solid blue line) and LJ potential-based (solid redline, similar to the KE model25) contact models, and the non-dimensional contact stress, plotted as a function of non-dimensional mean separation (d*) betweena rough thiol surface and a smooth gold substrate. Inset: Percentage contact area A* plotted as a function of d*. The points of intersection between the adhesioncurves and the contact reaction pressure curve give the equilibrium separation between the surfaces and are used to calculate the percentage contact area. (b)Hemispherical asperity composed of a thin film (1,10-decanedithiol SAM) on a gold substrate. There is an outer layer of thiols, followed by a shell of carbonatoms (thickness of t2 – t1) and an inner region of gold atoms (radius of R – t3). There is an additional single thiol layer at the inner gold surface. (c) Adhesivepressure versus mean separation of asperities for different material components of the asperity—thiol (blue line), carbon (red line), and gold layers (green line)with the planar gold substrate. The contact reaction pressure (dashed line) for the asperity is also plotted and is not affected by the structure of the asperity.The adhesive pressure is dominated by the outermost thiol layer and also dictates the final predicted value of the percentage contact area.

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closest to the asperity surface. Materials deeper in the asper-ity and thus further away from the flat substrate exhibit alower adhesion. The equilibrium separation between thesurfaces is found from the intersection of the total adhesivepressure and contact pressure, which is then used to find thepercentage contact area using Eq. (E2). Using surface topog-raphy data obtained from our AFM scans of the gold sub-strate used to create the SAM junction in Ref. 39, we predicta percentage contact area A+M of 56625% (Ref. 88) betweenthe two surfaces. The percentage contact area is strongly cor-related to the RMS surface roughness, as shown in the sensi-tivity analysis (details in Sec. S4). The uncertainty value of25% is obtained by averaging the uncertainty for each AFMscan, details of which are provided in the supplementary ma-terial of Ref. 39.

To demonstrate the importance of using the Morsepotential-based contact model within the composite asperitymodel, we also predict the percentage contact area by usingan equivalent LJ potential to describe the thiol-gold adhesion(as described in Sec. III A). Keeping all other parameters thesame (as listed in Table I), we predict a percentage contactarea A+LJ of 31617%.

We assess the accuracy of the predicted percentage con-tact areas from the Morse and LJ models by using them tocompare experimental measurements of junction thermal con-ductance Gjunc

exp , performed by Majumdar et al.,39 to area-corrected predictions Gjunc

M=LJ from MD simulations. For an Au-(1,10-decanedithiol)-Au junction, Gjunc

exp was obtained usingfrequency domain thermoreflectance (FDTR) and representsthe thermal conductance of the entire SAM junction.39

Assuming that the SAM itself has a negligible thermal resist-ance compared to the interfaces, Gjunc

exp can be approximatedas two interface thermal conductances in series (one at eachgold substrate). Based on the fabrication technique describedin Ref. 39, the interface on which the SAM is grown isassumed to be perfect. Its interface thermal conductance isrepresented by Gint

MD, the MD predicted thermal conductanceof a perfect thiol-gold interface, and is 226618 MW/m2 K.39

The other interface is comprised of a rough gold substratetransfer printed onto the SAM surface. Its interface thermalconductance Gint

M=LJ is estimated using the Morse- or LJ-basedcontact models. The final prediction of Gjunc

M=LJ is thus given by

1

GjuncM=LJ

¼ 1

GintMD

þ 1

GintM=LJ

: (31)

We follow the methodology described by Seong et al.41

and Prasher and Phelan89 to obtain the area-corrected predic-tion of interface thermal conductance Gint

M=LJ using

1

GintM=LJ

¼ 1

gÐ1

d+ /+ z+ð Þdz+

!1

pa2RGint

MD

þ1#

ffiffiffiffiffiffiffiffiffiffiffiffiA+M=LJ

q! "3=2

2kHaR

2

64

3

75;

(32)

where aR is the mean radius of the contact region of asper-

ities given byffiffiffiffiffiffiffiffiffiffiffiffiA+M=LJ

q½pgÐ1

d+ /+ðz+Þdz+&#0:5, and kH is the

harmonic mean of the thermal conductivities of the

contacting bodies, which here is given by 2kSAMkAu

=ðkSAM þ kAuÞ, where kSAM (461 W/m K) and kAu (185610W/m K) are the thermal conductivities of the SAM and thegold substrate.39 The first term inside the square brackets onthe right-hand side of Eq. (32) represents the thermal con-ductance for the regime where the mean free path (MFP) ofthe heat carriers (in this case, atomic vibrations or phonons)is comparable to the size (represented by aR) of the contactregions. The second term represents the regime where theMFPs are much smaller than aR. For the case where theMFPs are much smaller than aR, the interface thermal con-ductance is the Maxwell constriction conductance asdescribed by Prasher and Phelan.89

Using Eq. (32), we find GintM to be 112645 MW/m2 K

and GintLJ to be 61626 MW/m2 K, based on averaging over

all AFM measurements (details of the uncertainty calcula-tion are provided in Sec. S5). We now use Eq. (31), find-ing Gjunc

M to be 73620 MW/m2 K and GjuncLJ to be 47617

MW/m2 K. From the FDTR measurements, the experimen-tal value of the junction thermal conductance Gjunc

exp is6567 MW/m2 K.39 We independently estimate the degreeof agreement between the experiment and Morse or LJpredictions using an overlapping coefficient.90 Assumingthat each prediction can be represented by a normal distri-bution, the overlapping coefficient is equal to the area ofthe overlapping region between the two distributions(details are provided in Sec. S6). When comparing withGjunc

exp , we find the overlapping coefficient to be 0.50 withthe Morse-based prediction and 0.38 with the LJ-basedprediction.91

The Morse potential is a physically accurate descriptionof the thiol-gold bond present in our junctions (as shown inRef. 32 and used by us in Ref. 39). Thus, an adhesion modelderived using the Morse potential is a better representationof our system. It should also provide the more accurate pre-diction of percentage contact area compared to using anequivalent LJ potential. Our finding of better agreementbetween the Morse prediction and the experimental measure-ments of the SAM junction thermal conductance supportsthese arguments. This study can also potentially explain thediscrepancy between electronic conductance measurementsof SAMs made by different research groups.92–96 The meas-urements using nano-transfer printed metal films, AFM tips,STM tips, and sputtered metal films as electrodes are alsosensitive to the areal contact between the molecular layerand the surface of the metal film. Some of these works meas-ured the surface roughness of their films and even attributedthe observed variations of measured electronic conductanceto it,92,94–96 especially for interfaces having strong, covalentbonds (e.g., the thiol-gold interface studied in this work).None, however, use rigorous mathematical analysis as wehave done here to quantify the areal contact at the metal-molecular layer interface.

IV. SUMMARY

We formulated a rough surface contact model thataccounts for plasticity through the FEA-derived model ofKogut and Etsion, and adhesion through the use of the DMT

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model in conjunction with the Morse potential. Our Morsepotential-based contact model is especially useful for cova-lently bonded materials whose interactions cannot be accu-rately described using an LJ potential. A composite asperitymodel was derived to extend this study to a layered substrateand applied to a system comprised of a thiol-based SAM ona gold substrate in contact with another rough gold substrate,as shown in Fig. 5(b). Generalized expressions for calculat-ing the work of adhesion and equilibrium separationsbetween a surface and a substrate interacting through aMorse potential were also presented, which only requiredknowledge of the Morse potential parameters and the sub-strate lattice constant. The percentage contact area betweenthese rough bodies was found to be 56625% when using theMorse-based adhesion model and 31617% for the LJ-basedmodel. Using the Morse prediction of percentage contactarea provided better agreement between experimentallymeasured and MD predicted values of the thermal conduct-ance of a thiol-based SAM junction than the LJ prediction.The Morse potential-based contact model will be an impor-tant tool for researchers to quantify how charge and energy

transport through covalent interfaces are influenced byincomplete areal contact.

ACKNOWLEDGMENTS

We acknowledge support from the National ScienceFoundation (NSF) CAREER Award (Award No. ENG-1149374), the NSF Award (Award No. CCF-0811770), andthe American Chemical Society (ACS) PRF DNI Award(Award No. PRF51423DN10).

APPENDIX A: GENERALIZED EXPRESSIONS TOCALCULATE re AND A2

Calculating re: We find a numerical fit for the variationof the ratio re=rNN obtained from the single-point energy cal-culations for a range of values of arNN and r0=rNN to be

re

rNN¼ C1 arNNð ÞC2 þ C3 for 3 ) arNN ) 14; (A1)

where

C1 ¼ #1:373( 105e#11:88

r0rNN

$ %# 13:32e

#1:078r0

rNN

$ %

C2 ¼ 1:933r0

rNN

! "5

# 9:226r0

rNN

! "4

þ 16:530r0

rNN

! "3

#15:850r0

rNN

! "2

þ 12:000r0

rNN

! "# 8:558

C3 ¼ 1:138r0

rNN

! "# 0:328

9>>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>>;

for 0:79 ) r0

rNN) 1:96: (A2)

For our case, rNN ¼ lc=ffiffiffi2p

, where lc is the lattice constant. Eq. (A1) has a maximum error of 8% with respect to the energy cal-culations for our chosen ranges of arNN and r0=rNN , which are sufficient to describe both covalent and van der Waals bondingcharacteristics.

Calculating A2: We find a numerical fit for the variation of the ratio DcM=DcLJ obtained from the single-point energy cal-culations for a range of values of arNN and r0=rNN to be

DcM

DcLJ

¼ E1 arNNð ÞE2 þ E3 for 3 ) arNN ) 14; (A3)

where

E1 ¼ 204:10e#2:172

r0rNN

$ %þ 0:0504e

1:796r0

rNN

$ %

E2 ¼ #0:3229r0

rNN

! "2

þ 2:096r0

rNN

! "# 4:49

E3 ¼ 0:3135r0

rNN

! "2

# 1:407r0

rNN

! "þ 1:683

9>>>>>>>>=

>>>>>>>>;

for 0:79 ) r0

rNN) 1:96: (A4)

Eq. (A3) has a maximum error of 12% with respect to the

energy calculations for our chosen ranges of arNN and

r0=rNN . Details regarding these fits are provided in Sec. S2.

APPENDIX B: EXPRESSION FOR CRITICALINTERFERENCE AND PLASTICITY INDEX

The critical interference is

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xc ¼pKH

2E

! "2

R; (B1)

where K is the hardness coefficient,15

K ¼ 0:454þ 0:41"; (B2)

H is the Vickers hardness of the softer material, " is thePoisson ratio of the softer material, E is the Hertz elasticmodulus,

1

E¼ 1# "2

1

E1þ 1# "2

2

E2; (B3)

E1 and E2 are the Young’s moduli for the hemisphere andsurface, and "1 and "2 are their Poisson ratios. The plasticityindex W is calculated as follows:

W ¼ 2E

pKH

rs

R

! "1=2

: (B4)

APPENDIX C: PLASTIC DEFORMATION REGIMES FORCOMPOSITE ASPERITIES

For the deformation regimes where x=xC * 1 (elastic-plastic regime), we adjust the FEA derived dimensionlessseparations found by Kogut and Etsion to be25,26

Z x;!xð Þxc

¼ 0:951

pxxc

! "1:153

f !xð Þþ re

xcþ t

xcfor 1)x=xc) 6;

(C1)

and

Z x;!xð Þxc

¼0:457

pxxc

! "1:578

f !xð Þþ re

xcþ t

xcfor 6)x=xC)110:

(C2)

APPENDIX D: ADHESIVE PRESSURE AND ENERGYBETWEEN A SURFACE AND A SUBSTRATE FOR ANLJ POTENTIAL

The analogous expressions of Eqs. (9)–(10), (21), and(23) for describing adhesive pressure, energy per unit area,and work of adhesion between a single layer of atoms (sur-face) and a substrate interacting through an LJ potential are

PLJ;surf#s Dð Þ ¼ pq1r2 4!r6ð Þ2r4

e

re

D

! "10

# re

D

! "4" #

; (D1)

E00LJ;surf#s Dð Þ ¼ pq1r2 4!r6ð Þ6r3

e

1

3

re

D

! "9

# re

D

! "3" #

; (D2)

Fadc xð ÞFad

0

¼ 9 x=xcð Þ2 re=xcð Þ

ð1

1

re=xc

Z=xc

! "4

# re=xc

Z=xc

! "10" #

!xd!x; (D3)

Fadnc Dð ÞFad

0

¼ 3

2

re

D

! "3

# 1

3

re

D

! "9" #

: (D4)

APPENDIX E: CONTACT STRESS AND CONTACTAREA

The total non-dimensional contact stress P+cs and the per-centage contact area A+ between deforming asperities andthe surface are given by25

P+cs ¼Pcs

AnH¼ 2pgRrhKx+c

3

ðd+þx+c

d+

I1:5 þ 1:03

ðd+þ6x+c

d+þx+c

I1:425

0

B@

þ1:4

ðd+þ110x+c

d+þ6x+c

I1:263 þ 3

K

ð1

d+þ110x+c

I1

1

CA;

(E1)

and

A+ ¼ A

An¼ pgRrhx+c

ðd+þx+c

d+

I1 þ 0:93

ðd+þ6x+c

d+þx+c

I1:136

0

B@

þ0:94

ðd+þ110x+c

d+þ6x+c

I1:146 þ 2

ð1

d+þ110x+c

I1

1

CA;

(E2)

where

Ia ¼ z+ # d+

x+c

! "a

/+ z+ð Þdz+: (E3)

Each of the four integrals represents the elastic and plasticdeformation regimes listed in Table III.

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145306-13 Sierra-Suarez et al. J. Appl. Phys. 119, 145306 (2016)

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