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

Transcription

1 Morse potential-based model for contacting composite rough surfaces: Application to self-assembled monolayer junctions Jonatan A. Sierra-Suarez 1,, Shubhaditya Majumdar 2,, Alan J.H. McGaughey 2,3, Jonathan A. Malen 2,3 & C. Fred Higgs III 1,2,* 1 Department of Electrical and Computer Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh 15213, USA 2 Department of Mechanical Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh 15213, USA 3 Department of Materials Science and Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh 15213, USA These authors had equal contributions Corresponding author *(C. Fred Higgs III) higgs@andrew.cmu.edu, Ph: (412) , Fax: (412)

2 ABSTRACT This work formulates a rough surface contact model that accounts for adhesion through a Morse potential and plasticity through the Kogut-Etsion finite element-based approximation. Compared to the commonly-used Lennard-Jones (LJ) potential, the Morse potential provides a more accurate and generalized description for modeling covalent materials and surface interactions. An extension of this contact model to describe composite layered surfaces is presented and implemented to study a selfassembled monolayer (SAM) grown on a gold substrate placed in contact with a second gold substrate. Based on a comparison with prior experimental measurements of the thermal conductance of this SAM junction [Nano Letters, 15, (2015)], the more general Morse potential-based contact model provides a better prediction of the percentage contact area than an equivalent LJ potential-based model. I. INTRODUCTION The ability to accurately model the interaction of two bodies in contact is of great importance in the field of tribology, which is the study of interacting surfaces and their associated friction, lubrication, and/or wear behavior. In actuality, even surfaces that appear very smooth are relatively rough since they are microscopically comprised of protuberances called asperities. Consequently, surfaces that appear to be in significant contact are actually in contact over an area known as the real area of contact. Situations where partial contact of surfaces occurs include micro/nano-scale systems, relatively hard surfaces where interfacial asperities undergo minimal deformation, and mixed lubrication, where the asperities partially support the loads at liquid-mediated interfaces. 1 5 In all of these cases, the real area of contact must be resolved, usually as a function of the contact load, mechanical properties, surface topography, and adhesive and repulsive surface forces. The seminal work on the contact of real surfaces is the Greenwood-Williamson (GW) model. 6 It was developed to model the contact mechanics between two real surfaces by treating asperities as individual Hertzian hemispherical contacts of equal radii and varying heights. 7 The GW model can be used to predict the elastic contact stresses on the asperities from the two contacting bodies. While this model continues to 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 contact of smooth surfaces under large loads. Asperities on real surfaces usually enter the plastic regime even when light loads are applied. 8 Adhesion, which generally plays a smaller role in conventional macro-scale applications, becomes a stronger contributor to the resulting contact stress and surface deformation the smoother the surfaces are. 9 While some surface contact models have added plasticity 10,11 or adhesion 12,13 separately to the GW modeling framework, Chang, Etsion, and Bogy (CEB) developed a series of models to introduce both adhesion and the full spectrum of surface deformation regimes elastic, elastic-plastic, and purely plastic. 9,14,15 They introduced plasticity effects through the concept of a critical interference, which is the inter-penetration length (or maximum deformation) of an asperity into a surface at which plasticity first begins. 15 Volume conservation and a uniform applied pressure were then assumed at each of the asperities in order to account for plasticity. They incorporated adhesion with the elastic-plastic deformation using their deformed asperity profiles in conjunction with the Derjaguin-Muller-Toporov (DMT) model. 9,16 The two primary models for adhesion are the Johnson-Kendall-Roberts (JKR) model 17 and the DMT model. 16 The JKR model allows the adhesive force to affect the surface profile (i.e., shape) of a 2

3 hemispherical contact and the forces outside of the contact area are neglected. The DMT model, on the other hand, does not allow the adhesive force to affect the profile of a hemispherical contact. Its profile follows a Hertzian profile instead and all adhesive forces, even those outside the contact area, are included. Tabor first postulated that these two models were limiting cases of the same general adhesive theory. 18 In other words, the DMT model is well-suited for hard materials with a small asperity radius such as metals, while the JKR model is better suited for soft materials with a large asperity radius. 19 The adhesive force depends on the chosen surface interaction potential employed in the adhesive model, the most common being the Lennard Jones (LJ) potential. 20 Muller et al. developed a complete model accounting for Tabor s findings using the LJ potential. 21 The Maugis-Dugdale model was the first to develop a closed form solution for the JKR-DMT transition. 19 A comprehensive chart for determining which adhesion model to use for elastic contact was presented by Johnson and Greenwood. 22 The CEB models, due to their generality and simplicity, have been widely-adopted and extended by other authors. 23,24 The assumptions used, however, do not capture the correct asperity behavior in the elastic-plastic regime. Kogut and Etsion (KE) presented a thorough comparison of the CEB models to finite element analysis (FEA) of an asperity under deformation and found significant deviation. 25 They went on to present a new set of models to capture rough surface contact in the elastic-plastic regime, also including the effects of adhesion. 8,25 27 Their adhesion model incorporated the DMT model in conjunction with the LJ potential. While the LJ potential is most often used as a first approximation for a given material system due to its simplicity, for many materials it can lead to poor predictions of physical properties. 28 The Morse potential is more general than the LJ potential as it has three free parameters as compared to two free parameters in the LJ potential. 29 It is suitable for many systems, including molecules and metals. 30,31 It has been essential for representing complex material interactions, such as adhesion of thin films on metal substrates [e.g., self-assembled monolayers (SAMs) on gold 32 ] and has been parameterized for both covalent and van der Waals interactions Herein, we develop an adhesion-based contact model for interfaces by incorporating the Morse potential into the KE (and as a consequence the DMT) modeling framework. The resulting contact model can be employed for a wide variety of surface materials, from hard-soft interfaces to organic-inorganic heterojunctions. We first derive expressions for the adhesive pressure and interaction energy per unit area between two macroscopic bodies where atomic interactions are described using the Morse potential in Secs. II A, II B and II C. We then incorporate these expressions into the KE model and calculate the total adhesive pressure between the rough bodies in Sec. II-D. In Sec. II-E, we present a method to calculate adhesive pressures where one or both bodies in contact are composed of layers of different materials. In Sec. II F, we describe the analytical model used to characterize the roughness parameters of a body. Lastly, we predict the percentage contact area (i.e., the ratio of the real area of contact to total surface area which appears to be in contact) between two rough gold surfaces where one is coated with a SAM in Sec. III. These predictions are validated using interface thermal conductance measurements performed by the authors. 39 II. THEORY A. Derivation of surface pressure using the Morse potential The Morse (subscript M) potential between two point particles (atoms), separated by a distance r, is 3

4 E M,pp = D e [e 2α(r r 0) 2e α(r r 0) ], (1) where the pp subscript represents a particle-particle interaction. Here, D e is the depth of the potential well, α describes the inverse of the width of the well, and r 0 is the position of the minimum of the well. The 2D e e α(r r 0) term represents London dispersion (i.e., van der Waals) interactions, which are attractive, and the D e e 2α(r r 0) term represents exchange repulsion. Repulsive energy is defined as positive and attractive energy as negative. The difference in behavior between the Morse potential and the LJ potential, given by E LJ,pp = 4ε[(σ r) 12 (σ r) 6 ], where ε is the depth of the well and σ is interatomic distance at which E LJ,pp = 0, is shown in Fig. 1(a). The parameters used in plotting the Morse potential describe the interaction between a thiol group (-SH group) and a gold atom. 32 The LJ potential is constructed to have the same position and depth of the well as the Morse potential, but ultimately exhibits a different energy landscape at other positions due to its mathematical formulation. 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 well depth 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. TABLE I. Atomic interaction and surface topography parameters. Morse Potential Parameters (S-Au) 32,39 D e (Energy well depth) α (Measure of energy well width) r 0 (Position of energy well minimum) 0.38 ev 1.67x10-10 m x10-10 m 4

5 LJ Potential Parameters 0.38 ev (S-Au) ε (Energy well depth) ev (C-Au) ev (Au-Au) 40 σ (Position where energy is zero) m (S-Au) m (C-Au) m (Au-Au) 40 Surface Potential Parameters J/m 2 (Morse S-Au) Δγ (Work of adhesion) J/m 2 (LJ C-Au) J/m 2 (LJ Au-Au) ρ 1,2 (Volumetric number density) σ 2 (Areal number density) m -3 (Au) m -2 (C) m -2 (S) m (Morse S-Au) r e (Equilibrium separation) m (LJ C-Au) m (LJ Au-Au) Material Properties (Au and SAM) E (Elastic modulus) N/m 2 ν (Poisson ration) H (Hardness) MPa Asperity Properties σ h [Root mean-squared (RMS) roughness] m η (Areal asperity density) m -2 R (Radius of asperity) t 1 (Thickness of S layer) t 2 (Thickness of C layer) t 3 (Thickness of SAM) m m m m 5

6 To study the asperities that describe rough surfaces, this point-point potential must be extended to describe interactions between macroscopic bodies. Let us first consider the attractive part of the pair a potential given by E M,pp = 2D e e α(r r 0). We assume additivity of these interactions, such that the net interaction between a single atom and a monatomic substrate is the sum of its interactions with all the atoms in the substrate. 43 If we consider a substrate (semi-infinite solid) with a flat surface having a volumetric density of atoms ρ 1 at a distance D from an atom (the point particle), the attractive interaction energy will be a E M,ps (D) = 4πρ 1D e α 3 (αd + 2)e α(d r0), (2) where the subscript ps denotes a particle-substrate interaction. A schematic diagram of this geometry is shown in Fig. 1(b). Following a similar procedure, the attractive interaction energy per unit area between two substrates, whose surfaces are separated by a distance D, is a" E M,ss (D) = 4πρ 1ρ 2 D e α 4 (αd + 3)e α(d r0). (3) Here, ρ 2 is the volumetric density of atoms in the second substrate. 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 σ 2, the attractive interaction energy per unit area is a" E M,surf s (D) = 4πρ 1σ 2 D e α 3 (αd + 2)e α(d r0), (4) where the subscript surf denotes surface. Derivations of Eqs. (2)-(4) are provided in Sec. S1 of the supplementary material. 44 The corresponding attractive forces per unit area are a" F M,ss (D) = de a" M,ss dd a = P M,ss = 4πρ 1ρ 2 D e α 3 (αd + 2)e α(d r 0), (5) a" F M,surf s (D) a = P M,surf s = 4πρ 1σ 2 D e (αd + 1)e α(d r0). (6) α 2 a a where P M,ss and P M,surf s are the attractive pressures. Until now, we have only considered the attractive interaction. To realize a potential with an equilibrium separation, we need to add the repulsive interaction. Following the procedure of Eqs. (2)-(6), we obtain the general form of the repulsive interaction for either the substrate-substrate or surfacesubstrate configurations to be P M r = A 1 (αd + A 2 )e 2α(D r 0) and the subsequent total adhesive pressure is P M = P M r + P M a. To determine the unknown coefficients A 1 and A 2, we first enforce the physical limit that when the distance between the surfaces of the two bodies equals their equilibrium separation r e, zero pressure (i.e., zero force) is felt by them, i.e. P M (r e ) = 0. We note that r e does not equal r 0. This assumption maintains a continuity of the physical picture of atomic-scale interactions when moving from the atomic to the macro-scale and was employed in the derivation of the DMT model using the LJ 6

7 potential. 19,43,45 We can then solve for A 1 in terms of A 2 for the substrate-substrate and surface-substrate cases as A 1,ss = 4πρ 1ρ 2 D e α 3 ( αr e + 2 αr e + A 2,ss ) e α(r e r 0 ), (7) A 1,surf s = 4πρ 1σ 2 D e αr e + 1 α 2 ( ) e α(r e r 0 ). αr e + A 2,surf s (8) The physical constraint of P M (D ) = 0 is naturally satisfied by the form of the Morse potential. There are no more physical constraints that can be imposed on Eqs. (9)-(11) to specify the coefficient A 2. TABLE II. Coefficients in the expressions for pressure [Eq. (9)], energy per unit area [Eq. (10)], and work of adhesion [Eq. (11)] derived for the interaction between two substrates (ss), and a surface with a substrate (surf-s) described by a Morse potential. B 1 B 2 B 3 P M (D) P M,ss ρ 2 α P M,surf s σ 2 (αd + A 2,ss )(αr e + 2) (αr e + A 2,ss ) (αd + A 2,surf s )(αr e + 1) (αr e + A 2,surf s ) αd + 2 αd + 1 E M " (D) " E M,ss ρ 2 α " E M,surf s σ 2 (αr e + 2)(2αD + 2A 2,ss + 1) 4(αr e + A 2,ss ) (αr e + 1)(2αD + 2A 2,surf s + 1) 4(αr e + A 2,surf s ) αd + 3 αd + 2 Δγ M,ss ρ 2 α (αr e + 2)(2αr e + 2A 2,ss + 1) 4(αr e + A 2,ss ) αr e + 3 Δγ M Δγ M,surf s σ 2 (αr e + 1)(2αr e + 2A 2,surf s + 1) 4(αr e + A 2,surf s ) αr e + 2 We generalize the expression for P M by grouping parameters into the coefficients B 1, B 2, and B 3 (defined in Table II), leading to P M (D) = ( 4πρ 1D e α 2 ) B 1,PM [B 2,PM e α(r e r 0 ) e 2α(D r 0) B 3,PM e α(d r 0) ], where the corresponding interaction energy per unit area is (9) 7

8 E M " (D) = ( 4πρ 1D e α 3 ) B 1,EM " [B 2,EM " e α(r e r 0 ) e 2α(D r 0) B 3,EM " e α(d r 0) ]. (10) For hemispherical asperities, a common form for expressing the energy and pressure equations is through the work of adhesion Δγ, which represents the magnitude of E M " at its minimum value (i.e., at D = r e ) and is Δγ M = E M " (D = r e ) = ( 4πρ 1D e α 3 ) B 1,ΔγM e α(r e r 0 ) (B 2,ΔγM B 3,ΔγM ). We note that evaluation of Eqs. (9)-(11) requires specification of r e and A 2. All other parameters are based on the Morse potential parameters and the crystal structures of the bodies under consideration and thus, known a priori. We evaluate r e and A 2 using single-point energy calculations. We first place a single layer of atoms above a substrate of another kind of atoms. A schematic diagram of this structure is shown in Fig. 2(a). The substrate is an fcc solid having a (111) surface with a lattice constant of 4.08 Å (i.e., that of gold). The surface is a close-packed structure having a nearest neighbor distance of Å (i.e., that of a thiol-based SAM). 39,40,46 Our calculations correspond to the case where the atoms of the single-layer surface are directly above the three-fold hollow sites of the surface of the substrate. Only the interaction potential (Morse or LJ) between the layer and the substrate is considered while calculating the energy of the system at different separations (details provided in Sec. S2). The variation of the interaction energy between the surface and the substrate (nondimensionalized by Δγ LJ ) as a function of the separation distance between them is plotted in Fig. 2(b) for both Morse (blue circles) and LJ (red squares) potentials, corresponding to a thiol-gold interaction whose parameters are listed in Table I. The magnitude of the interaction energy at its minimum value is Δγ and the corresponding separation distance is r e. We can thus find the ratio Δγ M /Δγ LJ from this calculation, as shown in Fig. 2(b). We take this approach because exact analytical expressions for Δγ exist for an LJ potential and are 19 (11) Δγ LJ,ss = πρ 1ρ 2 (4D e σ 6 ) 16r e 2 Δγ LJ,surf s = πρ 1σ 2 (4D e σ 6 ) 9r e 3. (12) 8

9 FIG. 2. (a) Schematic diagram of the structure for single-point energy calculations between a two-dimensional surface of atom type 1 (yellow) placed at different heights above a substrate of atom type 2 (pink). (b) Interaction energy, non-dimensionalized by Δγ LJ, using both Morse (red squares) and LJ (blue circles) potentials for a thiolgold interaction (parameters in Table I) plotted as a function of separation distance D between the surface and substrate. The magnitude of the interaction energy at its minimum value (where D = r e ) is equal to Δγ. 9

10 FIG. 3. (a) Schematic diagram of a hemispherical asperity in contact with a flat surface and undergoing deformation. The flat surface is assumed to be rigid. (b) Schematic diagram of a hemispherical asperity at a height D a above a flat surface. Once Δγ M is calculated using the results of the single-point energy calculations and Eq. (12), we can analytically determine A 2 from Eq. (11). We note, however, that these calculations are specific to our system, with A 2 equal to and Δγ M equal to 0.66 J/m 2. To allow the implementation of this model to other surface-substrate systems, we provide generalized expressions describing the variation of r e / and Δγ M /Δγ LJ as a function of the parameters α and r 0 / ( being the nearest neighbor distance between atoms in the substrate) in Appendix A [Eqs. (A1)-(A4)]. The expressions for adhesive pressure and energy per unit area based on the Morse potential, represented by Eqs. (9)-(11), can now be used with the DMT and KE models to estimate the total adhesive force for spherical asperities contacting a flat substrate. B. Single asperity deformation The DMT model provides a method to calculate the adhesive force for a deformed hemisphere contacting a flat, undeformable surface. The method to calculate the deformed sphere profile is summarized in this section. 16 The sphere is assumed to obey Hertzian theory and to deform elastically. The height Z of any point on the surface of the hemisphere at a radial distance x from the asperity center, following the DMT convention, is Z(a, x, R) = 1 πr [a(x2 a 2 ) 1/2 (2a 2 x 2 )tan 1 ( x2 a 2 1)], (13) where R is the radius of the hemisphere and a is the radius of the contact region. A schematic diagram of this setup is shown in Fig. 3(a). 10

11 The expression for Z can also be written in terms of the interference ω, which is the centerline deflection of the asperity, i.e., the maximum deformation of the compressed asperity. 47 It can be seen that Z = 0 when x = a. Eq. (13) was, however, developed for macro-scale deformations. When molecular potentials are taken into account, it is physically impossible for two atoms, or in this case two bodies, to have a separation distance of zero. Thus, Eq. (13) is adjusted such that the minimum value of Z equals the equilibrium separation between the surfaces r e (derived in Sec. II A) and is further simplified by substituting the Hertzian relation for the contact radius thus arriving at a = (ωr) 1/2, (14) Z(ω, x ) = ω π [(x 2 1) 1/2 (2 x 2 )tan 1 ( x 2 1)] + r e, (15) where x = x/a. Eq. (15) is valid for elastic deformations, but it has been shown that the elastic deformation assumption is not adequate for most scenarios since plastic deformation occurs. 23 In order to account for both elastic and plastic deformation, we follow the approach taken by Kogut and Etsion, 26 where Z is further non-dimensionalized with respect to the critical interference ω c, which is the interference at which plasticity begins (details in Appendix B). Thus, the separation Z in the elastic regime, from Eq. (15), can be expressed as Z(ω, x ) ω c = 1 ω f(x ) + r e π ω c ω c for 0 ω ω c 1, (16) where f(x ) = (x 2 1) 1/2 (2 x 2 )tan 1 ( x 2 1). This non-dimensionalization allows for the study of the Hertzian profile independent of material properties. For the deformation regimes where ω ω C 1 (elastic-plastic regimes), we use the FEA derived dimensionless separations found by Kogut and Etsion 25,26 Z(ω, x ) = ω c π ( ω ) f(x ) + r e ω c ω c for 1 ω ω c 6, (17) and TABLE III. Range of contact regimes. Z(ω, x ) = ω c π ( ω ) f(x ) + r e ω c ω c Regime for 6 ω ω C 110. Dimensionless Separation (18) Non-Contact ω/ω c 0 Elastic 0 ω/ω c 1 Elastic-Plastic I 1 ω/ω c 6 Elastic-Plastic II 6 ω/ω c 110 Plastic ω/ω c

12 Kogut and Etsion validated Eqs. (17) and (18) for a range of values of the plasticity index ψ, which Greenwood and Williamson showed to be directly related to the critical interference as ψ ω c We specifically note that the interference ω of the contacting asperities must fall within the range of 0 to 110ω C in order for the FEA data used to be accurate. The various deformation regimes used in this study are listed in Table III. C. Morse potential-based adhesion model based on asperity deformation We now derive expressions for the adhesive pressure between a single hemispherical asperity and a substrate interacting through a Morse potential. The asperity may be a solid structure (having a volumetric density of atoms ρ 1 ) or hollow and composed of a single layer of atoms on its surface (having an areal density of atoms σ 2 ). Derjaguin et al. found the adhesive force between atoms in a slice along the surface of the hemisphere (i.e. a ring with radius x and thickness dx) and the flat surface to be 16 df c ad = 2πxP M (Z)dx. (19) It is important to note that the pressure P M (Z) contains the effect of all the atoms behind the exposed surface (if any). Thus, if we integrate Eq. (19) for all x, we get the total adhesive force between a deformed spherical asperity and a substrate to be F ad c = 2π xp M (Z)dx. a Strictly speaking, the upper limit of the integral should be the hemispherical radius R, but it can be approximated as infinity without significant error if P M,ss (Z) 0 at separations much smaller than R, as is the case here. 16 Now, from Eq. (20), we define F c ad (a = 0) = F 0 ad = 2πRΔγ as the point contact adhesive force. We use F 0 ad to normalize the total contact adhesive force in the same manner as Kogut and Etsion by changing the integrating variable to x = x/a and then dividing throughout by F 0 ad. 26 The variables ω, α, r 0, and r e are also non-dimensionalized using ω c. The normalized force of adhesion for an asperity in contact (in any deformation regime), with respect to the point contact adhesive force, is found from Eq. (20) using the expression for P M from Eq. (9). A general form of this expression is (20) F ad c (ω) ad = G F 1,F ad [G 2,Fc ad e α(r e r 0 ) e 2(αω Z c)( ωc r 0 ωc ) G 3,Fc ade (αω c)( Z ωc r 0 ωc ) ] x dx, 0 1 (21) where the coefficients G 1, G 2, and G 3 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 spherical asperity not in contact with the surface, the adhesive force from the DMT model is ad = 2πR P M (h)dh, D a F nc (22) 12

13 where h is the separation between a slice of thickness dh within the asperity and the flat surface, and D a is the minimum separation between the asperity and flat surface, as shown in Fig. 3(b). Using the expression for P M from Eq. (9) and normalizing with F 0 ad, we derive the non-contact adhesive force to be F ad nc (D a ) ad = G F 1,Fnc ad [G 2,Fnc ad e α(r e r 0 ) e 2(αω c)( D a ωc r 0 ωc ) G 3,Fnc ad e (αω c)( D a ωc r 0 ωc ) ], 0 (23) where the coefficients G 1, G 2, and G 3 for the substrate-substrate and surface-substrate cases are listed in Table IV. TABLE IV. Coefficients in the expressions for contact [Eq. (21)] and non-contact [Eq. (23)] adhesive pressures derived for the interaction between two substrates (ss) and a surface with substrate (surf-s) described by a Morse potential. The coefficients B 2 and B 3 are listed in Table II. G 1 G 2 G 3 F c ad (ω) F 0 ad F ad c,ss (ω) (αω) e α(re r0) Z (αr e + 2) [(αω c ) ( ) + A ω 2,ss ] c ad F 0 B 2,ΔγM,ss B 3,ΔγM,ss (αr e + A 2,ss ) ad F c,surf s (ω) ad F 0 (αω) e α(r e r 0 ) (αr e + 1) [(αω c ) ( ) + A ω 2,surf s ] c B 2,ΔγM,surf s B 3,ΔγM,surf s (αr e + A 2,surf s ) Z (αω c ) ( Z ω c ) + 2 (αω c ) ( Z ω c ) + 1 F ad nc (D) ad F 0 F ad nc,ss (D) ad F 0 ad F nc,surf s (D) ad F 0 1 B 2,ΔγM,ss B 3,ΔγM,ss (αr e + 2) [2(αω c ) ( D ω c ) + 2A 2,ss + 1] 4(αr e + A 2,ss ) 1 B 2,ΔγM,surf s B 3,ΔγM,surf s (αr e + 1) [2(αω c ) ( D ω c ) + 2A 2,surf s + 1] 4(αr e + A 2,surf s ) (αω c ) ( D ω c ) + 3 (αω c ) ( D ω c )

14 FIG. 4. Conversion of two real surfaces to a statistical representation. The real surfaces are mapped to a flat surface in contact with a rough surface comprised of a density of spherical asperities of radii R, which follow a statistical distribution [Eq. (25)], allowing for the height of each asperity to vary about the mean line. D. Extension to two-body contact problem Until now, we have derived expressions for the adhesive interaction of a single asperity with a substrate. To model the interaction between two real bodies, we analyze the interaction of a number of asperities (representing the rough surface of one body) with a substrate. The two-body contact problem can be visualized as shown in Fig. 4. A balance between any externally applied force, the adhesive force, and the contact force must exist for the bodies to be in equilibrium. In this section we present the equations necessary to calculate each of these forces and the percentage contact area for the two-body system. In order to simplify the problem, the two bodies with rough surfaces are transformed into an equivalent system of one rough body (Body I) in contact with a smooth, rigid body (Body II). The geometrical properties of Body I are equal to the sum of the geometrical properties of the original two rough bodies. The elastic modulus of Body I is calculated using Eqn. (B3). The surface of Body I is represented in a statistical manner as a distribution of hemispherical asperities of uniform radii R with heights varying based on a specified probability distribution φ(z). The total non-dimensional adhesive force F ad between the asperities and the surface is calculated as the sum of all the adhesive forces (i.e. contact and non-contact) weighted by the distribution of asperity heights for all of the deformation regimes. Thus, using Eqs. (21) and (23), we arrive at d F ad = Fad A n H = 2πηRΔγ ( F nc ad (d z + r e ) φ (z )dz H d +110ω c F 0 ad F ad c (z d ) + ad φ (z )dz ), (24) d F 0 where η is the areal density of asperities. Here, any variable with the superscript *, except for F ad, is dimensionless and is normalized using the standard deviation of surface heights, σ h. F ad is normalized by the nominal contact area A n and the hardness H. d represents the non-dimensional separation between the flat surface and the mean-line of asperity heights. The distribution φ has each of its variables normalized by σ h. For a normal distribution of asperity heights, 25 φ (z ) = σ h 2πσ s e 1 2 (σ h σs )2 (z ) 2. (25) Alternatively, a deterministic approach could be used to calculate the percentage contact area and the mean separation between our two surfaces In addition to the complexity of numerically solving the elastic and plastic constitutive equations for deformation, such an approach would require an iterative optimization method in order to calculate the balance between the adhesive and contact stresses. While such a deterministic approach may shed additional insight, we believe that a statistical model better represents our system due to its isotropic nature and Angstrom-scale RMS roughness of the surface (Table I). The measurements of Majumdar et al. 39 were also statistically averaged representations of the thermal conductance values. The potential difference between statistical and deterministic models is thus left for future study. 14

15 E. Composite asperity model for modeling layered structures We now extend our formulation to model adhesion between two surfaces where one of them has a thin film grown on it. The thin film is treated as an incompressible coating that follows the deformation behavior of the underlying substrate, such as in the case of an organic SAM grown on a metal or dielectric substrate. 55 Modeling the surface topography of such a system requires the asperities to be composed of multiple materials in a layered configuration, thus creating a composite asperity. Assuming substrate effects dominate, all materials of the composite asperity are assumed to have the same elastic modulus, Poisson ratio, and hardness, while having independent Morse (or LJ) parameters, layer thicknesses, and distances from the flat surface. The contribution of each layer in the composite asperity to the adhesive pressure is incorporated by adjusting its distance from the flat surface. For example, considering only elastic deformations, each slice within the asperity will now have a height Z from the surface [derived from Eq. (16)] given by Z(ω, r ) ω c = 1 ω f(r ) + r e + t π ω c ω c ω c for 0 ω ω c 1, (26) where t is the minimum separation of a hemisphere composed of one kind of atoms within the composite asperity and the surface of the asperity. Similar adjustments are made for the elastic-plastic deformation regimes [Eqs. (C1) and (C2)]. The remaining calculations are the same as given by Eqs. (19)-(25). The contribution of each material layer to the adhesive pressure is modeled by subtracting the adhesive pressure of a smaller hemisphere from the adhesive pressure of the larger hemisphere, thus creating the hemispherical layer. F. Characterization of an experimental surface A rough surface can be geometrically modeled as being composed of multiple spherical asperities arranged with an areal density η. The asperities have a mean radius R with a standard deviation of asperity heights σ s, which is mathematically correlated to the standard deviation of surface heights σ h. 25 How these quantities can be obtaining from atomic force microscopy (AFM) measurements has been studied in detail by Bush et al., 56 Gibson, 57 and McCool. 58 The quantities η, R, and σ s were originally derived by Nayak, 59 who built upon the work of Longuet-Higgins. 60 Nayak found that for random and isotropic surfaces having a Gaussian distribution of surface heights, a surface could be completely characterized by its spectral moments corresponding to height, slope, and curvature. Kotwal and Bhushan 61 studied non-gaussian surfaces with kurtosis and skew but such an analysis is extremely complex and beyond the scope of this study. More recent work has focused on the fractal-like behavior of surfaces and addressed variations with 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 not exhibit any trend in the roughness parameters between different AFM scans (see Sec. S3). Instead, random variations in σ h, R, and η are found between scans. Poon and Bhushan 65 also found that AFM surfaces do not follow a trend in roughness properties with scan size. We speculate that the dominant cause of our variations is related to the surface manufacturing process Furthermore, given the Angstrom-scale roughness of our surfaces, we do not believe that it is meaningful to consider them from a fractal perspective. We thus follow McCool s approach to characterize the surface topography. 58 From metrology data (i.e., AFM images) of a given surface, the parameters η, R, σ h, and σ s can be calculated from 15

16 η = < (d2 ξ d 2 x) 2 >/< (dξ dx) 2 > 6π 3, (27) R = 3 π (28) 8 < (d 2 ξ d 2 x) 2 > 1/2, σ h =< ξ 2 > 1/2, (29) and σ s = σ h ( (ηrσ h ) 2 ) 1/2, (30) where ξ is the measured height (above the mean-line) and the operator <> denotes a spatial arithmetic average. The derivatives are calculated along a set of straight lines parallel to the horizontal axis of an AFM scan. The derivatives are numerically 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 to highlight the difference in adhesive behavior predicted by a Morse potential and an equivalent LJ potential [Fig. 1(a)]. The non-dimensional adhesive (F ad = F ad /A n H) force between a flat gold substrate and a rough thiol surface (i.e., a single layer of thiol groups) is plotted in Fig. 5(a) as a function of d using Eq. (24) (Morse potential-based, solid blue line) and using the KE model 25 (LJ potential-based, solid red line) with appropriate values of Δγ (calculated separately for the Morse and LJ cases), r 0, and σ. Details of the formulation involving surface-substrate adhesion based on the LJ potential are described in Appendix D and were derived as an extension to the KE model. The roughness parameters η, R, σ h, and σ s are obtained from AFM measurements of a gold surface, as outlined in Sec. II F, and are listed in Table I for one particular AFM measurement from a set of separate measurements (details in Sec. S3). It must be noted that the roughness parameters listed in Table I are for the new, effective rough surface for which the AFM measurements of the surface heights were doubled before using Eqs. (27)-(30). Expressions for the total non-dimensional contact pressure P cs and the percentage contact area A between deforming asperities 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 under contact is achieved when P ad = F ad /A n = P cs, (i.e., a balance between the adhesive and contact pressures). This separation is found by identifying the point of intersection between the adhesive and contact pressure curves, as shown in Fig. 5(a). If an external load is applied to the top surface, then it is added to P ad. This equilibrium separation is then used to find the percentage contact area A, whose behavior as a function of d [using Eq. (E2)] is 16

17 shown in the inset of Fig. 5(a). The Morse potential-based adhesion model predicts 1.74 times the percentage contact area than the LJ based model, therein emphasizing the importance of the specialized framework based on the Morse potential. B. Application to thin films: Self-assembled monolayer on gold Thin films of SAMs provide a convenient and simple system to tailor interfacial properties of surfaces. 46 They have gained widespread use in nanoscience with applications in surface functionalization, 66,67 electrochemistry, 68,69 electronics, 70,71 and thermoelectrics A thiol-based SAM on a gold substrate is the most widely-studied configuration with detailed studies performed on its preparation, 46 structure, 75,76 and transport properties. 39,77 79 Many experiments for probing the transport properties of SAMs require the formation of a SAM junction. 39,77 81 Here, a SAM grown on one substrate has another substrate (usually a metal) brought in contact with it either through high-energy deposition techniques such as sputtering or evaporation 82 or through transfer printing. 83 The latter technique is widely used, as it does not damage the underlying SAM layer during the formation of the junction This technique has recently been used to create SAM junctions comprised of metal-dielectric 77,78 and metal-metal substrates 39 to study the junction thermal conductance. Experimental measurements of the junction thermal conductance, 39,78 however, do not agree with predictions from molecular dynamics simulations. 39,86,87 We hypothesize that a major source of this discrepancy is incomplete contact between the two surfaces one being the SAM grown on a metal/dielectric and the other being the bare transfer-printed metal. 17

18 a.! 10 0 Morseadhesion F* ad 10 1 A* (%)! 80! 60! 40! LJadhesion 20! 10 2 b.! 0! -2! -1! 0! 1! 2! d*" Contact reaction d* d*" S! R t 1! R t 3! C! Au! R t 1 t 2! c.! 10 4 R! S Au(Morse) Presure(MPa) Contact presure C Au(LJ) Au Au(LJ) Meanseparation, d(nm) x

19 FIG. 5. (a) Comparison of the non-dimensional adhesive pressure calculated using the Morse potential-based (solid blue line) and LJ potential-based (solid red line, similar to the KE model 25 ) contact models, and the nondimensional contact stress, plotted as a function of non-dimensional mean separation (d*) between a 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 adhesion curves and the contact reaction pressure curve give the equilibrium separation between the surfaces and are used to calculate the percentage contact area. (b) Spherical 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 carbon atoms (thickness of t 2 t 1 ) and an inner region of gold atoms (radius of R t 3 ). There is an additional single thiol layer at the inner gold surface. (c) Adhesive pressure 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. Using our Morse potential-based contact model in conjunction with the composite asperity model, we can predict the percentage contact area between two rough metal substrates where one has a SAM grown on its surface. We choose a system composed of a 1,10-decanedithiol SAM grown on a gold substrate brought in contact with another gold substrate, which is identical to the system recently studied by Majumdar et al. 39 The asperities are composed of three materials thiol (denoted by S), carbon (denoted by C), and gold (denoted by Au: we also neglect the contribution of hydrogen atoms) as shown in Fig. 5(b). The planar substrate is gold. The thiol layers are single-atom thick and are treated as surfaces. The thickness of the carbon layer is assumed to be 12 Å, given that the SAM layer has an average thickness of 16 Å, 39 which is commensurate to that measured in other experiments. 46,75 The asperities defining the carbon shell are assumed to be shifted 1.5 Å (t 1 ) and 13.5 Å (t 2 ) from the asperity surface. The inner thiol layer is shifted by 15 Å (t 3 ), which is also where the Au section of the asperity is assumed to begin. The thiol-gold interaction is modeled using a Morse potential while the carbon-gold and gold-gold interactions are modeled using LJ potentials (and the KE model for their adhesion). We calculate a plasticity index of 12±3 for our gold samples [using Eq. (B4)], which is consistent with those reported in literature for sputtered gold. 41 The adhesive pressure between the flat gold substrate and each of the three components in the asperities and the contact pressure are plotted in Fig. 5(c) as a function of the mean separation between the two surfaces. The contribution from the two different thiol layers is added together into a single thiol adhesion curve. The material properties used to plot the curves in Fig. 5(c) are listed in Table I. The total adhesive pressure is the sum of the individual contributions from each material within the asperity and is dominated (>99%) by the thiol-gold adhesion involving the thiol layer closest to the asperity surface. Materials deeper in the asperity and thus further away from the flat substrate exhibit a lower adhesion. The equilibrium separation between the surfaces is found from the intersection of the total adhesive pressure and contact pressure, which is then used to find the percentage contact area using Eq. (E2). Using surface topography data obtained from our AFM scans of the gold substrate used to create the SAM junction in Ref. 39, we predict a percentage contact area A M of 56 ± 25% 88 between the two surfaces. The percentage contact area is strongly correlated to the RMS surface roughness, as shown in the sensitivity analysis (details in Sec. S4). The uncertainty value of 25% is obtained by averaging the uncertainty for each AFM scan, details of which are provided in the Supporting Information of Ref. 39. To demonstrate the importance of using the Morse potential-based contact model within the composite asperity model, we also predict the percentage contact area by using an equivalent LJ 19

20 potential to describe the thiol-gold adhesion (as described in Sec. III A). Keeping all other parameters the same (as listed in Table I), we predict a percentage contact area A LJ of 31 ± 17%. We assess the accuracy of the predicted percentage contact areas from the Morse and LJ models by using them to compare experimental measurements of junction thermal conductance G exp performed by Majumdar et al., 39 to area-corrected predictions G junc M/LJ from MD simulations. For an Au- (1,10-decanedithiol)-Au junction, G junc exp was obtained using frequency domain thermoreflectance (FDTR) and represents the thermal conductance of the entire SAM junction. 39 Assuming that the SAM itself has a negligible thermal resistance compared to the interfaces, G junc exp can be approximated as two interface thermal conductances in series (one at each gold substrate). Based on the fabrication technique described in Ref. 39, the interface on which the SAM is grown is assumed to be perfect. Its interface thermal conductance is represented by G int MD, the MD predicted thermal conductance of a perfect thiolgold interface and is 226 ± 18 MW/m 2 -K. 39 The other interface is comprised of a rough gold substrate int transfer printed onto the SAM surface. Its interface thermal conductance G M/LJ is estimated using the Morse- or LJ-based contact models. The final prediction of G junc M/LJ is thus given by junc, 1 junc = 1 int + 1 int. (31) G M/LJ G MD G M/LJ We follow the methodology described by Seong et al. 41 and Prasher and Phelan 89 to obtain the int area-corrected prediction of interface thermal conductance G M/LJ using 3/2 ( A M/LJ ) = ( int ) η φ (z )dz d πa 2 int R G +, MD 2k H a R [ ] G M/LJ where a a is the mean radius of the contact region of asperities given by [πη φ (z )dz ] 0.5, and k H is the harmonic mean of the thermal conductivities of the A M/LJ d contacting bodies, which here is given by 2k SAM k Au /(k SAM + k Au ), where k SAM (4 ± 1 W/m-K) and k Au (185 ± 10 W/m-K) are the thermal conductivities of the SAM and the gold substrate. 39 The first term inside the square brackets on the right-hand side of Eq. (32) represents the thermal conductance for the regime where the mean free path (MFP) of the heat carriers (in this case, atomic vibrations or phonons) is comparable to the size (represented by a R ) of the contact regions. The second term represents the regime where the MFPs are much smaller than a a. For the case where the MFPs are much smaller than a R, the interface thermal conductance is the Maxwell constriction conductance as described by Prasher and Phelan. 89 int Using Eq. (32), we find G M to be 112 ± 45 MW/m 2 int -K and G LJ to be 61 ± 26 MW/m 2 -K, based on averaging over all AFM measurements (details of the uncertainty calculation are provided in Sec. S5). We now use Eq. (31), finding G junc M to be 73 ± 20 MW/m 2 -K and G junc LJ to be 47 ± 17 MW/m 2 -K. From the FDTR measurements, the experimental value of the junction thermal conductance G junc exp is 65 ± 7 MW/m 2 -K. 39 We independently estimate the degree of agreement between the experiment and Morse or LJ predictions using an overlapping coefficient. 90 Assuming that each (32) 20

21 prediction can be represented by a normal distribution, the overlapping coefficient is equal to the area of the overlapping region between the two distributions (details are provided in Sec. S6). When comparing with G junc exp, we find the overlapping coefficient to be 0.50 with the Morse-based prediction and 0.38 with the LJ-based prediction. 91 The Morse potential is a physically-accurate description of the thiol-gold bond present in our junctions (as shown in Ref. 32 and used by us in Ref. 39). Thus, an adhesion model derived using the Morse potential is a better representation of our system. It should also provide the more accurate prediction of percentage contact area compared to using an equivalent LJ potential. Our finding of better agreement between the Morse prediction and the experimental measurements of the SAM junction thermal conductance support these arguments. This study can also potentially explain the discrepancy between electronic conductance measurements of SAMs made by different research groups Measurements using nano-transfer printed metal films, AFM tips, STM tips, and sputtered metal films as electrodes are also sensitive to the areal contact between the molecular layer and the surface of the metal film. Some of these works measured the surface roughness of their films and even attributed the observed variations of measured electronic conductance to it, 92,94 96 especially for interfaces having strong, covalent bonds (e.g., the thiol-gold interface studied in this work). None, however, use rigorous mathematical analysis as we have done here to quantify the areal contact at the metal-molecular layer interface. IV. SUMMARY We formulated a rough surface contact model that accounts for plasticity through the FEAderived model of Kogut and Etsion, and adhesion through the use of the DMT model in conjunction with the Morse potential. Our Morse potential-based contact model is especially useful for covalently bonded materials whose interactions cannot be accurately described using an LJ potential. A composite asperity model was derived to extend this study to a layered substrate and applied to a system comprised of a thiol-based SAM on a gold substrate in contact with another rough gold substrate, as shown in Fig. 5(b). Generalized expressions for calculating the work of adhesion and equilibrium separations between a surface and a substrate interacting through a Morse potential were also presented, which only required knowledge of the Morse potential parameters and the substrate lattice constant. The percentage contact area between these rough bodies was found to be 56 ± 25% when using the Morse-based adhesion model and 31 ± 17% for the LJ-based model. Using the Morse prediction of percentage contact area provided better agreement between experimentally measured and MD predicted values of the thermal conductance of a thiol-based SAM junction than the LJ prediction. The Morse potential-based contact model will be an important tool for researchers to quantify how charge and energy transport through covalent interfaces are influenced by incomplete areal contact. ACKNOWLEDGMENTS We acknowledge support from the National Science Foundation (NSF) CAREER Award (Award No. ENG ), the NSF Award (Award No. CCF ), and the American Chemical Society (ACS) PRF DNI Award (Award No. PRF51423DN10). 21

22 APPENDIX APPENDIX A. Generalized expressions to calculate r e and A 2 Calculating r e : We find a numerical fit for the variation of the ratio r e / obtained from the single-point energy calculations for a range of values of α and r 0 / to be r e = C 1 (α ) C 2 + C 3 for 3 α 14, (A1) where C 1 = e 11.88( r 0 ) 13.32e 1.078( r 0 ) C 2 = ( r 0 ) ( r 0 ) ( r ) ( r 3 0 ) ( r 0 ) C 3 = ( r 0 ) } for 0.79 r (A2) For our case, = l c / 2, where l c is the lattice constant. Eq. (A1) has a maximum error of 8% with respect to the energy calculations for our chosen ranges of α and r 0 /, which are sufficient to describe both covalent and van der Waals bonding characteristics. Calculating A 2 : We find a numerical fit for the variation of the ratio Δγ M /Δγ LJ obtained from the single-point energy calculations for a range of values of α and r 0 / to be Δγ M Δγ LJ = E 1 (α ) E 2 + E 3 for 3 α 14, (A3) where E 1 = e 2.172( r 0 ) e 1.796( r 0 ) E 2 = ( r 0 ) E 3 = ( r 0 ) ( r 0 ) ( r 0 ) } for 0.79 r (A4) Eq. (A3) has a maximum error of 12% with respect to the energy calculations for our chosen ranges of α and r 0 /. Details regarding these fits are provided in Sec. S2. APPENDIX B. Expression for critical interference and plasticity index The critical interference is ω c = ( πkh 2E ) 2 R, (B1) 22

23 where K is the hardness coefficient, 15 K = ν, (B2) H is the Vickers hardness of the softer material, ν is the Poisson ratio of the softer material, E is the Hertz elastic modulus, 2 1 E = 1 ν ν 2, E 1 E 2 2 (B3) E 1 and E 2 are the Young s moduli for the sphere and surface, and ν 1 and ν 2 are their Poisson ratios. The plasticity index Ψ is calculated as follows Ψ = 2E πkh (σ s R )1/2. (B4) APPENDIX C. Plastic deformation regimes for composite asperities For the deformation regimes where ω ω C 1 (elastic-plastic regime), we adjust the FEA derived dimensionless separations found by Kogut and Etsion to be 25,26 Z(ω, x ) = ω c π ( ω ) f(x ) + r e + t ω c ω c ω c for 1 ω ω c 6, (C1) and Z(ω, x ) = ω c π ( ω ) f(x ) + r e + t ω c ω c ω c for 6 ω ω C 110. (C2) APPENDIX D. Adhesive pressure and energy between a surface and a substrate for an LJ 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 (surface) and a substrate interacting through an LJ potential are P LJ,surf s (D) = πρ 1σ 2 (4εσ 6 ) 4 [( r 10 e 2r e D ) ( r 4 e D ) ], (D1) " (D) = πρ 1σ 2 (4εσ 6 ) E LJ,surf s 6r e 3 [ 1 3 (r e D ) 9 ( r e D )3 ], (D2) 23

24 F c ad (ω) F 0 ad = 9(ω/ω 4 c) 2(r e /ω c ) [(r e/ω c ) ( r 10 e/ω c ) ] x dx, Z/ω c Z/ω c 1 (D3) F ad nc (D) ad = 3 F 0 2 [(r e D ) (r e D ) 9 ]. (D4) APPENDIX E. Contact stress and contact area The total non-dimensional contact stress P cs and the percentage contact area A between deforming asperities and the surface are given by 25 P cs = P cs A n H = 2πηRσ d hkω +ω c d +6ω c c ( I I d d +ω c d +110ω c I d +6ω c K I1 ), d +110ω c (E1) and A = A d +ω c d +6ω c = πηrσ A h ω c ( I I n d d +ω c + 2 I 1 ), d +110ω c d +110ω c I d +6ω c (E2) where I α = ( z d α ) φ (z )dz ω c (E3) Each of the four integrals represent the elastic and plastic deformation regimes listed in Table III. 24

25 References 1 R.W. Carpick, D.F. Ogletree, and M. Salmeron, J. Colloid Interface Sci. 211, 395 (1999). 2 C.F. Higgs, S.H. Ng, L. Borucki, I. Yoon, and S. Danyluk, J. Electrochem. Soc. 152, G193 (2005). 3 B.A. Krick, J.R. Vail, B.N.J. Persson, and W.G. Sawyer, Tribol. Lett. 45, 185 (2012). 4 P.M. Johnson and C.M. Stafford, ACS Appl. Mater. Interfaces 2, 2108 (2010). 5 C. Greiner, M. Schäfer, U. Popp, and P. Gumbsch, ACS Appl. Mater. Interfaces 6, 7986 (2014). 6 J.A. Greenwood and J.B.P. Williamson, Proc. R. Soc. London. Ser. A. Math. Phys. Sci. 295, 300 (1966). 7 H. Hertz, J. Für Die Reine Und Angew. Math. 92, 156 (1882). 8 L. Kogut and I. Etsion, J. Appl. Mech. 69, 657 (2002). 9 W.-R. Chang, I. Etsion, and D.B. Bogy, J. Tribol. 110, 50 (1988). 10 H.A. Francis, Wear 45, 221 (1977). 11 H. Ishigaki, I. Kawaguchi, and S. Mizuta, Wear 54, 157 (1979). 12 K.N.G. Fuller and D. Tabor, Proc. R. Soc. A 345, 327 (1975). 13 S.K. Roy Chowdhury and H.M. Pollock, Wear 66, 307 (1981). 14 W.R. Chang, I. Etsion, and D.B. Bogy, J. Tribol. 110, 57 (1988). 15 W.R. Chang, I. Etsion, and D.B. Bogy, J. Tribol. 109, 257 (1987). 16 B.. Derjaguin, V.. Muller, and Y.. Toporov, J. Colloid Interface Sci. 53, 314 (1975). 17 K.L. Johnson, K. Kendall, and A.D. Roberts, Proc. R. Soc. A 324, 301 (1971). 18 D. Tabor, J. Colloid Interface Sci. 58, 2 (1977). 19 D. Maugis, J. Colloid Interface Sci. 150, 243 (1992). 20 J.E. Jones, Proc. R. Soc. A 106, 463 (1924). 21 V.. Muller, V.. Yushchenko, and B.. Derjaguin, J. Colloid Interface Sci. 77, 91 (1980). 22 K.L. Johnson and J.A. Greenwood, J. Colloid Interface Sci. 192, 326 (1997). 23 Z. Liu, A. Neville, and R.L. Reuben, J. Tribol. 124, 27 (2002). 24 A.A. Polycarpou and I. Etsion, J. Tribol. 120, 296 (1998). 25 L. Kogut and I. Etsion, Tribol. Trans. 46, 383 (2003). 26 L. Kogut and I. Etsion, J. Colloid Interface Sci. 261, 372 (2003). 27 L. Kogut and I. Etsion, J. Tribol. 126, 34 (2004). 25

26 28 F. Jensen, Introduction to Computational Chemistry (John Wiley and Sons, 2007). 29 P.M. Morse, Phys. Rev. 34, 57 (1929). 30 L.A. Girifalco and V.G. Weizer, Phys. Rev. 114, 687 (1959). 31 A.K. Rappe, C.J. Casewit, K.S. Colwell, W.A. Goddard III, and W.M. Skiff, J. Am. Chem. Soc. 114, (1992). 32 R. Mahaffy, R. Bhatia, and B.J. Garrison, J. Phys. Chem. B 101, 771 (1997). 33 J. Tersoff, Phys. Rev. Lett. 56, 632 (1986). 34 D. Qian, W.K. Liu, and R.S. Ruoff, J. Phys. Chem. B 105, (2001). 35 Y. Wang, D. Tomnek, and G.F. Bertsch, Phys. Rev. B 44, 6562 (1991). 36 R.N. Costa Filho, G. Alencar, B.-S. Skagerstam, and J.S. Andrade, Europhys. Lett. 101, (2013). 37 T.H. Fang and C.I. Weng, Nanotechnology 11, 148 (2000). 38 A.J. Weymouth, T. Hofmann, and F.J. Giessibl, Science 343, 1120 (2014). 39 S. Majumdar, J.A. Sierra-Suarez, S.N. Schiffres, W.-L. Ong, C.F. Higgs III, A.J.H. McGaughey, and J.A. Malen, Nano Lett. 15, 2985 (2015). 40 T. Luo and J. Lloyd, J. Heat Transfer 132, (2010). 41 M. Seong, P.K. Singh, and S. Sinha, J. Appl. Phys. 113, (2013). 42 G. V Samsonov, in edited by G. V Samsonov (Springer US, Boston, MA, 1968), pp V.M. Muller, B.V. Derjaguin, and Y.P. Toporov, Colloids Surf. 7, 251 (1983). 44 See supplemental material at [URL] for details regarding equations describing macroscopic (pointsubstrate, surface-substrate and substrate-substrate) interactions, single points energy calculations, AFM measurements, sensitivity and uncertainty analyses, and calculation of the overlapping coefficient. 45 V.M. Muller, V.S. Yushchenko, and B. V Derjaguin, J. Colloid Interface Sci. 92, 92 (1983). 46 J.C. Love, L.A. Estroff, J.K. Kriebel, R.G. Nuzzo, and G.M. Whitesides, Chem. Rev. 105, 1103 (2005). 47 K.L. Johnson, Contact Mechanics (Cambridge university press, 1987). 48 M.N. Webster and R.S. Sayles, J. Tribol. 108, 314 (1986). 49 P. Põdra and S. Andersson, Wear 207, 79 (1997). 50 B. Bhushan, Tribol. Lett. 4, 1 (1998). 51 X. Ai and K. Sawamiphakdi, J. Tribol. 121, 639 (1999). 52 I. Polonsky and L. Keer, J. Tribol. 122, 30 (2000). 53 T. Nogi and T. Kato, J. Tribol. 119, 493 (1997). 26

27 54 Y.Y.-Z. Hu, G.C. Barber, and D. Zhu, Tribol. Trans. 42, 443 (1999). 55 F.W. Delrio, C. Jaye, D.A. Fischer, and R.F. Cook, Appl. Phys. Lett. 94, 1 (2009). 56 A.W. Bush, R.D. Gibson, and G.P. Keogh, J. Tribol. 101, 15 (1979). 57 R.D. Gibson, in Rough Surfaces, edited by T.R. Thomas (Longman, New York, 1982), pp J.I. McCool, Wear 107, 37 (1986). 59 P.R. Nayak, J. Tribol. 93, 398 (1971). 60 M.S. Longuet-Higgins, Philos. Trans. R. Soc., A 250, 157 (1957). 61 C.A. Kotwal and B. Bhushan, Tribol. Trans. 39, 890 (1996). 62 C.K. Bora, E.E. Flater, M.D. Street, J.M. Redmond, M.J. Starr, R.W. Carpick, and M.E. Plesha, Tribol. Lett. 19, 37 (2005). 63 S. You and M.P. Wan, Langmuir 29, 9104 (2013). 64 J.A. Greenwood, Proc. R. Soc. A 393, 133 (1984). 65 C.Y. Poon and B. Bhushan, Wear 190, 76 (1995). 66 Y. Xia, X.-M. Zhao, and G.M. Whitesides, Microelectron. Eng. 32, 255 (1996). 67 M. Geissler, H. Schmid, A. Bietsch, B. Michel, and E. Delamarche, Langmuir 18, 2374 (2002). 68 H.O. Finklea, Electroanal. Chem. 19, 109 (1996). 69 D.M. Adams, L. Brus, C.E.D. Chidsey, S. Creager, C. Creutz, C.R. Kagan, P. V. Kamat, M. Lieberman, S. Lindsay, R.A. Marcus, R.M. Metzger, M.E. Michel-Beyerle, J.R. Miller, M.D. Newton, D.R. Rolison, O. Sankey, K.S. Schanze, J. Yardley, and X. Zhu, J. Phys. Chem. B 107, 6668 (2003). 70 A. Nitzan and M.A. Ratner, Science 300, 1384 (2003). 71 D.K. Aswal, S. Lenfant, D. Guerin, J. V. Yakhmi, and D. Vuillaume, Anal. Chim. Acta 568, 84 (2006). 72 P. Reddy, S.Y. Jang, R.A. Segalman, and A. Majumdar, Science 315, 1568 (2007). 73 S.K. Yee, J.A. Malen, A. Majumdar, and R.A. Segalman, Nano Lett. 11, 4089 (2011). 74 J.A. Malen, S.K. Yee, A. Majumdar, and R.A. Segalman, Chem. Phys. Lett. 491, 109 (2010). 75 M.D. Porter, T.B. Bright, D.L. Allara, and C.E.D. Chidsey, J. Am. Chem. Soc. 109, 3559 (1987). 76 C.D. Bain, E.B. Troghton, Y.-T. Tao, J.E.G.M. Whitesides, and R.G. Nuzzo, J. Am. Chem. Soc. 111, 321 (1989). 77 R.Y. Wang, R.A. Segalman, and A. Majumdar, Appl. Phys. Lett. 89, 1 (2006). 78 M.D. Losego, M.E. Grady, N.R. Sottos, D.G. Cahill, and P. V Braun, Nat. Mater. 11, 502 (2012). 79 F. Sun, T. Zhang, M.M. Jobbins, Z. Guo, X. Zhang, Z. Zheng, D. Tang, S. Ptasinska, and T. Luo, Adv. 27

28 Mater. 26, 6093 (2014). 80 P.J. O Brien, S. Shenogin, J. Liu, P.K. Chow, D. Laurencin, P.H. Mutin, M. Yamaguchi, P. Keblinski, and G. Ramanath, Nat. Mater. 12, 118 (2012). 81 T. Meier, F. Menges, P. Nirmalraj, H. Hölscher, H. Riel, and B. Gotsmann, Phys. Rev. Lett. 113, (2014). 82 L.T. Cai, H. Skulason, J.G. Kushmerick, S.K. Pollack, J. Naciri, R. Shashidhar, D.L. Allara, T.E. Mallouk, and T.S. Mayer, J. Phys. Chem. B 108, 2827 (2004). 83 M.A. Meitl, Z.-T. Zhu, V. Kumar, K.J. Lee, X. Feng, Y.Y. Huang, I. Adesida, R.G. Nuzzo, and J.A. Rogers, Nat. Mater. 5, 33 (2006). 84 X.-M. Zhao, Y. Xia, and G.M. Whitesides, J. Mater. Chem. 7, 1069 (1997). 85 J.W. Kim, K.Y. Yang, S.H. Hong, and H. Lee, Appl. Surf. Sci. 254, 5607 (2008). 86 T. Luo and J.R. Lloyd, Int. J. Heat Mass Transfer 53, 1 (2010). 87 L. Hu, L. Zhang, M. Hu, J.-S. Wang, B. Li, and P. Keblinski, Phys. Rev. B 81, 1 (2010). 88 This value differs from the percentage contact area reported by Majumdar et al. (Ref. 39) after further refinement of our model and averaging of more AFM measurements. 89 R.S. Prasher and P.E. Phelan, J. Appl. Phys. 100, (2006). 90 H.F. Inman and E.L. Bradley, Commun. Stat. Theor. M 18, 3851 (1989). 91 junc junc junc In Ref. 39, we used modified versions of Eqs. (31) and (32) to compare G M with GMD, where GMD equals G int MD /2 = 113 ± 9 MW/m 2 junc -K. We found G M to be 102 ± 26 MW/m 2 -K, which differs from the area-corrected junction thermal conductance reported in Ref. 39 due to further refinement of our model and averaging of more AFM measurements. 92 H.B. Akkerman and B. de Boer, J. Phys. Condens. Matter 20, (2008). 93 X.D. Cui, A. Primak, X. Zarate, J. Tomfohr, O.F. Sankey, A.L. Moore, T.A. Moore, D. Gust, G. Harris, and S.M. Lindsay, Science 294, 571 (2001). 94 V.B. Engelkes, J.M. Beebe, and C.D. Frisbie, J. Phys. Chem. B 109, (2005). 95 D. Guerin, C. Merckling, S. Lenfant, X. Wallart, S. Pleutin, and D. Vuillaume, J. Phys. Chem. C 111, 7947 (2007). 96 A. Salomon, D. Cahen, S. Lindsay, J. Tomfohr, V.B. Engelkes, and C.D. Frisbie, Adv. Mater. 15, 1881 (2003). 28

29 a.! a.# Energy (ev) 19 x 10 Morse LJ LJ Energy (ev)! LJ! LJ! Morse 0.2 Morse! 0.4 Morse! r"(å)" 5 5 r (Å) 6! 10 x 10 b.! b.# EM,pp" EM,ps" EM,ss" ε = 0.38 ev! σ = 2.36 Å! De = 0.38 ev! r0 = 2.65 Å! α = 1.47 Å-1! 6

30 a. Atom 1 Move 2-D surface Atom 2 b. 2 Morse LJ E surf-s / ΔγLJ 1 0 γm / γlj 1 re D (Å) 5 6 7

31 a. b. Hemispherical%% Asperity% Z Flat% surface% ω Hemispherical%% Asperity% x R r 0 h x D a Flat% surface% R 2a

32

33 a.! 10 0 Morse adhesion F* ad 10 1 A* (%)! 80! 60! 40! LJ adhesion 20! b.! ! -2! -1! 0! 1! 2! d*" Contact reaction d* d*" S! R t 1! R t 3! C! Au! R t 1 t 2! c.! 10 4 R! S Au (Morse) Pressure (MPa) Contact pressure C Au (LJ) Au Au (LJ) Mean separation, d (nm) x 10 9