From Classical to Molecular TRIBOLOGY

Transcription

1 R. Overney NME 498 From Classical to Molecular TRIBOLOGY Motivation In this unit we will discuss tribology (friction and lubrication) down to the molecular scale. But prior to this discussion, we will need to establish some understanding of adhesion and contact mechanics. As adhesion depends on interactions, we will focus first on them. As technology moves more towards miniaturization in novel product developments, it is imperative to integrate interfacial interactions and molecular strain abilities into design strategies. Interfacial forces are on the order of 0-6 to 0-0 N, strong enough, for instance, to freeze gears in micro-electrical mechanical systems (MEMS), to affect the stability of colloidal system, or to wipe out magnetically stored data information in hard drives. Of particular interest are short-range interactions, as compiled in Table below, but also long range interactions such as electromagnetic interactions, and phase interactions that originate from cooperative and structuring phenomena, such entropically cooled molecular fluid cages in close vicinity of polar interfaces (e.g., structuring of water ) or the formation of third interfaces via capillaries, which possess tensile strength that can dominate forces caused by short-range interactions by orders of magnitude. Table : Short Range Interaction Forces Nature of Bond Type of Force Energy (kcal/mol) Distance Ionic bond Coulombic force 80 (NaCl) 40 (LiF).8 Å.0 Å Covalent bond Electrostatic force (wave function overlap) 70 (Diamond) 8 (SiC) N/A Metallic bond free valency electron sea interaction (sometimes also partially covalent (e.g., Fe and W) 6 (Na) 96 (Fe) 0 (W) 4. Å.9 Å. Å Hydrogen Bond a strong type of directional dipole-dipole interaction 7 (HF) Van der Waals (i) dipole-dipole force (ii) dipole-induced dipole force (iii) dispersion forces (charge fluctuation).4 (CH 4 ) significant in the range of a few Å to hundreds of Å Van der Waals Interactions Short range interactions that underlie most processes, such as wetting, mixing, delamination, adhesion, friction, phase separation, and diffusion, in short most

2 o phenomena that matter in life, are typically separated into hydrogen bond interactions, which of great importance for life sciences, and Van der Waals interactions (VdW). Both interaction types are based on dipole-dipole interactions. While hydrogen interaction is a very strong type of directional dipole-dipole interaction, on the order of 0 kcal/mol, Van der Waals interactions are up to an order of magnitude weaker, and composed of three subclasses, i.e.,: o Dipole-dipole forces: Molecules having permanent dipoles will interact by dipoledipole interaction. Dipole-induced dipole forces: The field of a permanent dipole induces a dipole in a non-polar atom or molecule. o Dispersion forces: Due to charge fluctuations of the atoms there is an instantaneous displacement of the center of positive charge against the center of the negative charge. Thus, at a certain moment, a dipole exists and induces a dipole in another atom. Therefore non-polar atoms (e.g. neon) or molecules attract each other. If one considers molecular interactions (i.e., zero order or point particles), all Van der Waals interactions are described in terms of the molecular distance, r, by a /r 6 potential. If one considers the Pauli exclusion principle (i.e., repulsive forces of the electron clouds), the semi-empirical form of the molecular VdW interaction potential, is the Lennard-Jones (LJ) potential, where the repulsive component is empirically chosen in terms of /r. The LJ potential LJ is typically provided in the following two equivalent forms: where C C rep vdw C C vdw rep LJ ( r ) 4 6 r r r 6 ; C 4C vdw C vdw and C rep are characteristic constants. C = C vdw is called the VdW interaction parameter. The empirical constant represents the characteristic energy of interaction between the molecules (the maximum energy of attraction between a pair of molecules)., a characteristic diameter of the molecule (also called the collision diameter), is the distance between two atoms (or molecules) for (r) = 0. The LJ potential is depicted in Figure. rep r The integral form of interaction forces between surfaces of macroscopic bodies through a third medium (e.g., vacuum and vapor) are () r() Figure : Lennard Jones (6-) potential (empirical Van der Waals Potential between two atoms or nonpolar molecules). r

3 Body Interaction Point Interaction R.Overney From Classical to Molecular Tribology NME 498 called surfaces forces. To apply the VdW formalism to macroscopic bodies, one has to integrate the point interaction form presented above. Consequently, the dipole-dipole interaction strength C but also the exponent of the distance dependence becomes geometry dependent. For instance, while for point-point particles the exponent is -6, it is - and - for macroscopic sphere-sphere and sphere-plane interactions, respectively. Thus, while, VdW point particle interactions are very short ranged (~/r 6 ), macroscopic VdW interactions are long ranged (e.g., sphere-sphere: ~/D, where D represents the shortest distance between the two macroscopic objects). Table provides a list of geometry dependent non-retarded VdW interaction strengths and exponents. Table : Van der Waals interaction Potential Geometry of Interaction Interaction Potential (W) Two Atoms Atom-Surface C 6 r C 6D Sphere-Sphere A RR D ( R R ) 6 Plane-Sphere AR 6D Two Cylinders Two Crossed Cylinders Plane-Plane AL D R R ( R R ) A R R 6D A D Two Parallel Chain Molecules CL 5 8 r A π Cρ ρ, with the molecular VdW interaction parameter C, and the number density of the molecules in both solids ρ i i =, is the Hamaker Constant. The interaction parameter of the non-zero dimensional objects, known as the Hamaker constant A π Cρ ρ, where C is the molecular VdW parameter and ρ i i =,) are the molecular number densities of the two interacting objects is approximated via the Lifshitz theory as, h e n n n n A kt 4 8 n n n n n n n n

4 with the dielectric constants,, and refractive indices, n of the media involved (the index is utilized for the media in between), and the absorption frequency e (e.g., for H O: e = x 0 5 Hz). Table provides non-retarded Hamaker constants based on the Lifshitz theory. Table : Non-retarded Hamaker constants for two interacting media across a vacuum (air) (Source: intermolecular & Surface Forces, J. Israelachvili, Academic Press) Dielectric constant Refractive Index Absorption frequency a Hamaker Constant n A medium/air/medium Medium (0 5 s - ) (0-0 ) Acetone Benzene Calcium Flouride Carbon tetrachloride Cyclohexane Ethanol Fused quartz Hydrocarbon (crystal) Iron oxide (Fe O 4 ).97.0 est Liquid He Metals (Au. Ag, Cu) Mica n-pentane n-octane n-dodecane n-tetradecane n-hexadecane Polystyrene Polyvinyl chloride PTFE Water a UV absorption frequencies obtained from Cauchy plots mainly from Hough and White (980) and H. Christenson (98, thesis). It is important to note that the Hamaker constant can be positive or negative depending on the medium in between. In air and vacuum however the interaction is typically attractive, i.e., the Hamaker constant is positive as shown in Table. Adhesion and Surface Energy The energy of adhesion (or just adhesion), W ", i.e., the energy per unit area necessary to separate two bodies ( and ) in contact, defines the interfacial energy as: W '' ; where i (i=,) represent the two surface energies. Assuming two planar surfaces in contact, the Van der Waals interaction energy per unit area is 4

5 A WD (see Table ) D which was obtained by pairwise summation of energies between all the atoms of medium with medium. The summation of atom interactions within the same medium have been neglected, which yields additional energy terms, i.e., A W const. D o consisting of a bulk cohesive energy term (assumed to be constant), and an energy term related to unsaturated "bonds" at the two surfaces in contact (i.e., D = D o ). Notice that contact cannot be defined as D = 0 due to molecular repulsive forces. D o is called the "cutoff distance", which was found astonishingly to be well represented by a material independent constant value of 0.65 nm. The total energy of two planar surfaces at a distance D D o apart is (neglecting the bulk cohesive energy) A A D o W W W. D D D D o o In contact (i.e., D=D o ) W = 0. In the case of isolated surfaces, i.e., D =, A W. Do Thus, in order to words, to separate two planar surfaces, one has to overcome the energy difference W W D W D o A D o Table 4: Surface energies based on Lifshitz theory and experimental values.(source: intermolecular & Surface Forces, J. Israelachvili,, Academic Press) Surface Energy, (mj/m ) which corresponds to the adhesive energy per unit area of W '' =. Hence, the interfacial energy can be expressed as function of the Hamaker constant and the cutoff distance, as or, as A 4Do A Material A Lifshiz Theory A/4 D o Experimental* (0-0 ) {D o =0.65nm} (0 o C) Liquid helium (at 4-.6K) Water Acetone Benzene CCl H o Formamide Methanol.6 8 Ethanol Glycerol Glycol n- Pentane n -Hexadecane n -Octane n -Dodecane Cyclohexane PTFE Polystyrene 6.6. Polyvinyl chloride

6 as can be shown by substituting for cutoff distance 0.65 nm and introducing an average lattice spacing. See Table 4 for surface energy calculations based on the cutoff distance. Capillary Forces via Water Vapor Condensation In the discussion above we have considered a continuous medium in-between the two surfaces to deduce the surface forces. Thereby, we have assumed that this third medium fills up the vacuum space entirely, i.e., does not introduce interfaces. We have to drop this assumption, however, should the third medium form a finite condensed phase within the interaction zone of the two bodies. Any condensed phase within the interaction zone will exhibit interfaces towards the vapor, and thus, if deformed (e.g., stretched) contribute to the acting forces. These new forces, called capillary forces, are on the order of 0-7 N for single asperity contacts with radii of curvatures below 00 nm. Capillary forces are meniscus forces due to condensation. It is well known that microcontacts act as nuclei of condensation. In air, water vapor plays the dominant role. If the radius of curvature of the micro-contact is below a certain critical radius, a meniscus will be formed. This critical radius is defined approximately by the size of the Kelvin radius r K = l/(l/r l + /r ) where r l and r are the radii of curvature of the meniscus. The Kelvin radius is connected with the partial pressure p s (saturation vapor pressure) by LV rk, p RT log ps where L is the surface tension, R the gas constant, T the temperature, V the mol volume and p/p s the relative vapor pressure (relative humidity for water). The surface tension L of water is 0.074N/m (T=0 C) leading to a critical Van der Waals distance of water of L V/RT = 5.4 Å. Consequentially, we obtain for p/p s =0.9 a Kelvin radius of 00 Å. At small vapor pressures, the Kelvin radius gets comparable to the dimensions of the molecules, and thus, the Kelvin equation breaks down. The meniscus forces between two objects of spherical and planar geometry can be approximated, for D «R, as: R D 4R LV 4R L cos F, D / d D / d where SL is the is the interfacial energy between the solid and the liquid (assuming the same solid material), which can be expressed by the product of the liquid surface tension L and the contact angle, as SL = L cos (Young Equation). Furthermore, R is the radius of the sphere, d the length of PQ, see Figure, D the distance between the sphere and the plate, and the meniscus contact angle. The maximum capillary force, found at at D = 0 (contact), is F R d max 4 R cos. 6

7 While this expression estimates the capillary forces of relatively large spheres fairly accurately, the capillary forces of highly wetted nanoscale spheres requires a geometrical factor K, ( cos) K, 4 cos where is the filling angle, which yields a capillary force of F cap 4 R water ( cos ) cos 4 cos. Figure : Capillary meniscus between two objects of spherical and planar geometry Figure : The contact angle measurement. The contact angle is the measure of hydrophobicity (wettability). Left: hydrophilic surface ( < 90 o ). Right: hydrophobic surface ( > 90 o ). The capillary formation is however not only limited by the breakdown of the Kelvin equation on the molecular scale, but a critical humidity level decisive for spreading (collective transport), and thus, capillary neck formation. Scanning force microscopy (SFM) measurements could identify two distinct force regimes, via force displacement analysis involving hydrophilic counter-surfaces in a humid environment, Fig. 4. M. He, A. Blum, D. E. Aston, C. Buenviaje, and R. M. Overney, Journal of chemical physics 4 (), 55 (00) 7

8 Figure 4: (Left) Generic sketch of the functional relationship between the pull-off force and the relative humidity (RH). Regimes I, II and III represent the van der Waals regime, mixed van der Waals capillary regime, and capillary regime decreased by repulsive forces, respectively. (Right) Pull-off force vs. RH measured between a hydrophilic silicon oxide SFM tip and a ultra-smooth silicon oxide wafer. In the low humidity regime at around 5-40 % relative humidity (RH) no capillary through water condensation is formed, and thus the interaction force, F stw, measured is of Van der Waals origin (i.e., involves the two surfaces (s and t) and the medium in between (v = water vapor). Considering the adhesion forces per unit area, W stv, we derived above for planar geometry and employing the Derjaguin approximation (see below), we can express the interaction force (adhesion force) between a sphere-plane arrangement for low RH (i.e., < 5 % RH), as Fstw RW stv where R is the sphere radius of Fig.. If we introduce the interfacial energy A stv W stw and stv 4D o the pull-off force at low humidity is F total F stv RA. 50 4D o 8 R A [N] for RH < 5 %. This relationship would provide either a good estimate for the Hamaker constant A or the relative radius of curvature (in this case the AFM tip radius R). In the high humidity regime (> 40% RH), the adhesive pull force consists of VdW term F stw (w = liquid water) and the capillary force F cap. The total pull of force can be expressed as A F cos total R 4 water for RH > 40%. 4D o while is the contact angle with water. 8

9 In the relationships above, we utilized the Derjaguin approximation, * F R W curved which relates the planar adhesive energy per unit area, at distant D, to the interaction force of any two shaped solid objective in close vicinity D, with the combined effective surface curvature R * R R where R*» D. The requirement of R*» D is appropriate for SFM in contact. Capillary forces are very strong forces in comparison to VdW forces, responsible for many processes in nature. In applications such as MEMS they can be detrimental and have to be avoided. planar Contact Mechanics Contact plays an important role in mechanical systems in which there are moving parts. Typically one distinguishes between (i) fully and merely elastic contact, which leads to the Hertzian contact mechanics, (ii) adhesive and elastic contact, which is addressed in the Johnson, Kendall and Roberts (JKR) theory besides others, (iii) plastic contact, which entails abrasive wear, pealing, and crack propagation. Here, we will briefly provide an overview of the Hertzian and JKR theory of contact mechanics. Hertz analyzed the stresses at the contact of two elastic solids, and thereby assumed small strains within the elastic limit. The contact radius a is considered significantly smaller than the radius of curvature R, and the two contacting surfaces, as depicted in Fig. 5, and assumed to be non-conformal. Furthermore, creep at the interface is neglected, i.e., a frictionless contact assumed. Based on these assumptions, the contact radius a, can be expressed as LR a * 4E where L. R* and E* are the applied load (normal force), the effective (combined) contact radius, and the effective (combined) elastic modulus, respectively. * Fig. 5. Contact of two elastic spheres. 9

10 0 Other parameters from the Hertzian contact theory are listed here: Hertz area of contact: * * 4 E LR a A Mutual approach: * a a R a o with 0 L o a a Hertz pressure: * * max 6 R E L p a L p m with the combined Young s modulus and radius of curvature of the two materials ( and ), i.e., E E E * and * R R R. is the Poisson ratio ( 0.5 for polymers). In the case of a sphere-plane contact, R and therefore R * is equivalent to R, the radius of the sphere. This is typically the assumed contact geometry in scanning force microscopy. In the Hertz model adhesive interactions are neglected, as depicted in Figure 6 at zero loads where the contact radius vanishes. Fig. 6. Contact radius as function of the applied for Hertz and JKR contacts.

11 An improvement over the Hertzian theory was provided by Johnson et al. (around 970) with the JKR (Johnson, Kendall, Roberts) theory by considering attractive VdW interactions within the contact zone. The theory yields a contact radius of R * a 6 L R R L R * 4E, which is equivalent to the Hertzian contact expression for = 0. Due to adhesive surface interaction, contacts can be formed during the unloading cycle also in the negative loading (pulling) regime, as shown in Figure 6. Such as the Hertzian theory, the JKR solution is restricted to elastic contacts with sphere-sphere or sphere-plane geometries. The significant differences of the two contact models are illustrated in Figure 6, with a negative loading regime between L = 0 and the instability load, i.e., the adhesion force JKR L L * adh R. An even more involved elastic contact theory (the DMT theory) also considers Van der Waals interactions outside the elastic contact regime, which give rise to an additional load. A schematic representation of the contact models provided below. The theory simplifies to Bradley's Van der Waals model if the two surfaces are separated and significantly appart. In Bradley's model any elastic material deformations due to the effect of attractive interaction forces are neglected. Bradley's non-contact model and the JKR contact model are very special limits explained by the Tabor coefficient. Hertz: JKR: DMT Fig. 7: Contact Mechanical Models: fully elastic model, fully elastic model considering adhesion in the contact zone, fully elastic, adhesive andvan der Waals model.

12 Tribology Contact mechanics, adhesion forces and the properties of the surrounding media play an important role for one of the oldest challenges known to man, tribology, Fig. 8. The science of Tribology (Greek tribos: rubbing) concentrates on Contact Mechanics of Moving Interfaces that generally involve energy dissipation. It encompasses the science fields of Adhesion, Friction, Lubrication and Wear, Fig. 9. Fig. 8: (Top) ~ 7000 B.C. Northern Norway Skier in rock carving. First known use of sleds. (Bottom) 880 B.C. Egypt Transport of Eqyptian colossus on the tomb of Tehuti-Hetep. El-Bersheh. Oldest known depiction of cognitive lubrication. Leonardo da Vinci Fig. 9: (Left) Leonardo da Vinci. (Right) The fields of tribology.

13 Leonardo da Vinci (45-59)) can be named as the father of modern tribology by studying friction, wear, bearing materials, plain bearings, lubrication systems, gears, screw-jacks, and rolling-element bearings. 50 years before Amontons' Laws of Friction were introduced, he had already recorded them in his manuscripts. Hidden or lost for centuries, Leonardo da Vinci's manuscripts were read in Spain a quarter of a millennium later. To the pioneers in tribology one counts besides Leonardo da Vinci also Guillaume Amontons (66-705), John Theophilius Desanguliers (68-744), Leonard Euler (707-78), and Charles-Augustin Coulomb (76-806). These pioneers developed the first laws of tribology which still apply to many engineering problems today, although, on the nanoscale and with increasing molecular complexity involving organic systems, they loose their applicability. Some of the early findings are summarized in the following three laws:. The force of friction is directly proportional to the applied load. (Amontons st Law). The force of friction is independent of the apparent area of contact. (Amontons nd Law). Kinetic friction is independent of the sliding velocity. (Coulomb's Law) These three laws were attributed to dry friction only, as it has been well known since ancient times that lubrication modifies the tribological properties significantly, Fig. 8. However, it took quite a long time until lubrication was studied pragmatically and lubricants were not just listed such as a "cooking formula". It was Nikolai Pavlovich Petrov and Osborne Reynolds around 880, who recognized the hydrodynamic nature of lubrication, and introduced a theory of fluid-film lubrication. Still today, Reynolds' steady state equation of fluid film lubrication v F D is valid for hydrodynamic lubrication of thick films (> m) where the frictional (drag) force, F, is proportional to both the sliding velocity, v, and the bulk fluid viscosity, and inversely proportional to the film lubricant thickness, D. The hydrodynamic theory breaks down below a critical thickness threshold that is expressed in the Stribeck-Curve, Fig. 0. Figure 0: Stribeck Curve (schematic) relates the fluid lubricant thickness,, and the friction coefficient to the Gumbel Number N G = P - ; i.e., the product of the liquid bulk viscosity,, the sliding speed (or more precisely the shaft frequency),, and the inverse of the normal pressure, P. BL: Boundary Lubrication, ML: Mixed Lubrication, EHL: Elastohydrodynamic Lubrication, HL: Hydrodynamic Lubrication (described by Reynolds).

14 Classical Unlubricated Tribology Friction is the compilation of forces that resist sliding motion. It typically entails a start-up effect, which is known as static friction and also referred to as stiction. Stiction is a huge problem in small mechanical devices, such as MEMS. It originates from aging effects during idles times of non-moving surfaces in contact, and, entails capillaries, phase mixing, starvation of lubricants, adhesive and even cohesive bindings. Static friction is uncorrelated to kinetic (dynamic) friction, which is the steady friction force once the adjacent surfaces are in relative motion to each other at constant velocity. These two friction regimes are depicted below in Fig.. Static friction, F stat, if measurable, always exceeds kinetic friction, F kin. F stat (x)= k c x v = 0 F pull F F stat L F kin v = const F pull F kin F pull x o x - F pull L x Figure : Friction force loop: A cuboid in contact with a surface is pulled at constant force F pull, first in one direction and then back again, which results in a friction force loop. At the turning point x, the velocity and pull of force is zero for a certain time. The depicted friction forces are composed of the static friction F stat at the start up and kinetic friction F kin. F stat is a function of the idle time at which F pull = 0. The quality of the contact before the pull off force is applied in any direction is expressed by the contact stiffness k c. Here we assumed the idle times to be the same at x o and x with equivalent elastic startup interfacial deformations that possess an abrupt yield point, at which the two surfaces separate and sliding commences. We assumed the classical simplified viewpoint that the sliding motion is steady at velocity v, which is reflected in a constant kinetic friction force F kin. L expresses the normal load (or weight) of the cuboid. As Figure illustrates, we have to come up with quite a few assumptions to addresses the deformation of the interface prior to sliding giving rise to the static friction, and the velocity relationship of the kinetic aspect of friction. Clearly, we have a problem in coming up with a force balance, reflected by the disconnect of the force vectors. Friction forces are notorious non-conservative forces, which means that do not possess a potential, i.e., cannot be expressed as dv/dx = -F, where V is the potential. After Amontons (or Leonardo da Vinci s) introduction of the friction coefficient, F kin, L the first mathematical approach to tribology was undertaken by Leonard Euler (707-78) with a geometrical resistance theory of "dry" friction the Interlocking Asperity Theory. Euler's theory already provides two terms for friction, i.e., static and dynamic (kineitc) friction. The static friction coefficient is provided by the tangent of the asperity 4

15 angle, i.e., =tan, while the dynamic friction coefficient is reduced by the kinetic term s/gt cos (see derivation below from Overney Thesis in 99). As unlubricated sliding motion was known to man since Ancient times to be inefficient, motion involving wheels were of more practical interest. Energy dissipation involving rolling depend again on the aging processes during the idle time, the interacting forces during rolling, and the material intrinsic response to rolling. 5

16 Charles-Augustin Coulomb (76-806) proposed that the frictional resistance of a rolling wheel or cylinder is proportional to the load P, and inversely proportional to the radius of the wheel. Coulomb's description of rolling friction entirely neglected the material compliance. It was Arsène Dupuit in 840 who argued that the material behind a rolling cylinder would not fully recover after deformation. Thus, a simple torque balance (see below) between the interfacial rolling resistance and the material resisting torque leads to an inverse square root dependence of friction in R. (The rolling resistance is comprised of the product between friction force, F, and cylinder radius, R. The resisting torque is considering the applied load, P, with its lever arm, in which the asymmetric compliance of the material is reflected.) Dupuit's inverse square root relationship of rolling friction with the radius of an elastic cylinder was experimentally confirmed by Tabor in 955, Fig.. Fig. : Dupuit's Torque Balance: Determination of the rolling friction resistance. In the twentieth century the theories of dry friction and lubricated friction were further developed. Solid-like behavior of lubricants in the ultrathin film regime (< m) led to theory of Boundary Lubrication, which was proposed by W.B. Hardy (99). The adhesion concept of friction for dry friction, already proposed by Desanguliers, was applied with great success by Bowden and Tabor to metal-metal interfaces. Adhesion is a term relating to the force required to separate two bodies in contact with each other as discussed above. Desanguliers (74) proposed adhesion as an element in the friction process, a hypothesis which appeared to contradict experiments because of the independence of friction on the contact area (Amontons nd Law). 6

17 Therefore the tribologists rejected Desanguliers' proposal and devoted their attention to a more geometrical hypothesis of friction, the interlocking theory of mechanical asperities by Coulomb and Euler. The contradiction between the adhesive issue and Amontons nd Law cleared up with the introduction of the concept of the real area of contact introduced by Bowden and Tabor. The real area of contact is made up of a large number of small regions of contact, in the literature called asperities or junctions of contact, where atomto-atom contact takes place. The figure below depicts the situation for (a) a general contact with multiple contact zones with contact area a i, and a total real contact area of A r =a i, and a projected (apparent) area A a, and (b) and idealized single asperity contact with contact radius a. Fig. : Real elastic contact. As discussed above, the contact radius can be determined with the Hertz theory (or the JKR theory) assuming that a is much smaller than the radius of curvature of the asperity, R. The theory yields: a LR * ; E ; a «* E 4 E E R where E i and v i (i=,) represent the Young's modulus and Poisson ratio, respectively, of the bodies in contact. L is the load applied. The kinetic friction coefficient is by definition the ratio of the friction force and the applied load, i.e., = F/L. The friction force can be expressed as the product between an interfacial stress,, that has to be overcome in order to slide, and the actual contact area A r. The load can be described with the mean pressure of contact p m, multiplied with A r (for the idealized spherical shape contact, p m =L/a ). Bowden and Tabor assumed that in order for the bodies to slide relative to each other (a) the asperities are plastically deformed; i.e., the mean pressure corresponds to about three times the yield pressure, Y, of the material, i.e., p m,crit.8 Y, and (b) the interfacial stress component corresponds to the shear strength of the soft material crit. 7

18 Consequently the friction coefficient can be expressed as the ratio between the shear strength of the softer material and about three times the yield pressure; i.e., kin F crit Ar crit crit. L p A p. 8Y m,crit Notice that Bowden and Tabor's inelastic adhesive theory provides a constant kinetic friction coefficient as proposed by Amontons. Also with Bowden and Tabor friction was more effectively expresses in terms stresses, i.e., L o pm ; pm. Ar in which the total shear stress,, during a sliding process is comprised of the intrinsic material shear strength, o, plus the compression stress p m, while the friction force F is expressed in terms of the total stress, as F = A r. Thereby, represents the friction coefficient. It is important to note that with this linear separation in the stresses, the friction coefficient is a material insensitive property. It is reduced to a process parameter, and thus, limited in its use for molecular engineering towards materials with low energy dissipation. It also does not provide fundamental insight onto energy dissipation on a molecular level. r m,crit Tribology involving Lubrication In the classical theories of tribology by da Vinci, Amonton, and Coulomb, not much attention was given to the dependence of kinetic friction on the sliding velocity. This clearly changed in the 9th century during the first industrial revolution, at which time lubricants became increasingly important, for instance, in ball and journal bearings. It was Petrov (88), Tower (88), and Reynolds (886) who established that the liquid viscous shear properties determine the frictional kinetics. Reynolds (886) combined the pressure-gradient determined Poisseuille flow with the bearing surface induced Couette flow assuming, based on Petrov's law (Petrov (88)), a no-slip condition at the interface between lubricant and solid. This led to the widely used linear relationship between friction and velocity. Reynolds' hydrodynamic theory of lubrication can be applied to steady state sliding at constant relative velocity and to transient decay sliding (sliding is stopped from an initial velocity v and a corresponding shear stress o ), which leads to the classical Debye exponential relaxation behavior, i.e.; D v o exp t ; o. A D D is the lubricant thickness, A the area of the slider, and the viscosity of the fluid. This classical exponential relaxation behavior, obtained in a thermodynamically well-mixed three dimensional medium, is distorted (e.g., reveals stick-slip phenomena, Fig. 4) when the liquid film thickness is reduced to molecular dimensions. Reynolds hydrodynamic description of lubrication was found to work well for thick lubricant films but to break down for thinner films. One manifestation is that for films on 8

19 the order of ten molecular diameters, the stress in the film does not allow the tension to return to zero. It was also found that the motion in the steady state sliding regime was disrupted, exhibiting a stick-slip-like slider motion (Israelachvili et al. (990)). The term Boundary Lubrication is used to describe a lubricant that is reduced in thickness to molecular dimension and effectively reduces friction between two opposing solid surfaces. Hardy et al. (9) recognized that molecular properties, such as molecular weight and molecular arrangement, are governing the frictional force. This confined concept of lubrication, often visualized by two highly ordered opposing films with shear taking place somewhere in between the two layers, contains many of the rate dependent manifestations of frictional sliding; e.g., stick-slip, ultra-low friction, transitions from high to low friction, phase transitions, dissipation due to dislocations (e.g., gauche and cis-transformations), and memory effects. stick-slip in ultrathin lubricants classical lubricant stop sliding Fig.4: Solid behaviour of a fluid film. If the thickness of the lubricant layer is greater than a critical value of about ten molecular diameters, then the shear strength remains constant during sliding. Once the sliding process stops, the film will reduce tension exponentially. However, the film shows solid-like behaviour if the film thickness is below that threshold. During the sliding process the shear strength shows a stick-slip behavior and after sliding has stopped, the film remains strained. It was Israelachvili et al. (990) who, based on surface forces apparatus (SFA) experiments and computer simulations, provided a molecular picture of the stick-slip behavior caused by the lubricant material. The major achievement of this work was to draw our attention to the molecular structure of the lubricant, which is often different from the bulk, and unstable during the sliding motion. Israelachvili interpreted the sticklip behavior observed for nanometer thick simple liquids, such as hexadecane, as in-plane structuring within the liquid caused by compression forces and "freezing-melting" transitions due to shear. The simple concept of a freezing-melting transition is based on a common perception of the two distinctive parts of a stick-slip occurrence: the solid (Hookian)-like sticking part and the liquid (Newtonian)-like slipping part. But a deformation of a solid can be both, coordinated or uncoordinated, and thus can exhibit both solid-like and liquid-like behavior. For instance, most of the plastic yielding processes are uncoordinated. On the other hand, slipping within a solid, along a crystal plane in a thermally activated strainrelease process for instance, is a highly coordinated molecular process (Blunier et al. (99)). 9

20 Similar arguments can be made for a liquid. For example, stick-slip behaviors were observed in more complex fluidic systems by Reiter et al. (994), who compared a molecularly "wet" lubricant film with a "dry" self-assembled monolayer lubricant. They concluded that sliding in liquid films is the result of slippage along an interface. In other words, the degree of molecular cooperation determined the frictional resistance. The concept of local-versus-cooperative yield to shear is briefly illustrated here with a frictional-load study of a molecularly entangled polymer melt obtained in a SFM study of Buenviaje et al. (999). Each of the curves presented in Figure 5(a) represents a polymer film of polyethylene co-propylene of distinctly different degree of entanglement. Films of thickness above 0 nm exhibit the strongest entanglement strength. Films of 0 nm thickness or thinner are fully disentangled. The reason for the film thickness-dependent entanglement strength is given by the substrate distance-dependent shear strength during the spin coating process of the thin films. For entangled films SFM friction studies exhibit a critical applied load (identified by P t, and the thickness t) that separates two friction regimes: One identified by a high friction coefficient and the other by a low friction coefficient. At loads below P t the friction coefficients are high, indicating plastic yielding during sliding. In these plastic regimes of sliding, molecular cooperation is low, leading to high local shear stresses compensated by local yielding of the material. Above the critical load, the friction coefficient drops, independent of the film thickness, to a low value of.0, corresponding to the value obtained from the fully disentangled film. Note that the polymer molecules in the 0-nm-thick film experience high substrate tangential stresses during the spin coating process. Hence the disentangled polymer molecules can be considered to be aligned preferentially along the substrate surface as sketched in Fig. 5(b). This leads to a decrease of the structural entropy the closer the material is to the solid substrate surface. Considering the matching friction coefficient of 0. above P t for thicker films, we can assume that any entangled film above a critical load exhibits a similar molecular collective response toward shear as the 0 nm film during spin coating. The critical Fig. 5. (a) SFM friction measurements at a speed m/s: Cooperative molecular response of polyethylene co-propylene to frictional shear forces as a function of the applied load. P t (t corresponds to the thickness of the polymer film) represents the critical activation load at which collective sliding is energetically more favorable than local plastic yielding. Adapted from Buenviaje et al. (999). (b) Sketch of the degree of disentanglement in the vicinity to the solid substrate surface. 0

21 load and its related pressure represent a barrier that has to be overcome before a collective phenomenon is activated. With the discussion of shear in entangled polymer systems we have introduced structural entropy as one of the key players that affect frictional resistance in lubricants. We found that the structural entropy was affected by the load of the slider, which introduces an activation barrier in the form of a critical pressure. The terminology used here resembles the one of the Eyring theory of molecular liquid transport (Glasstone et al. (94)). Eyring discussed a pure liquid at rest in terms of a thermal activation model. The individual liquid molecules experience a "cage-like" barrier that hinders molecular free motion, because of the close packing in liquids. To escape from the cage an activation barrier needs to be surmounted. In Eyring's model, two processes are considered in order to overcome the potential barrier: (i) shear stresses and (ii) thermal fluctuations. The potential barrier in the thermal activation model is depicted in Figure 6 indicating the barrier modification by the applied pressure force P, and shear stress. Briscoe et al. (98) picked up on this idea to interpret the frictional behavior observed on molecularly smooth monolayer systems. Starting from the overall barrier height E = Q+P (Fig. 6) that is repeatedly overcome during a discontinuous sliding motion, using a Boltzmann distribution to determine the average time for single molecular barrier-hopping, and assuming a regular series of barriers and a high stress limit (/kt > ), the following shear strength versus velocity v relationship was derived P (Briscoe et al. (98)): k BT v ln Q P. vo The barrier height, E, is composed of the process activation energy Q, the compression energy P where P is the pressure acting on the volume of the junction, and the shear energy, where is the shear strength acting on the stress activation volume. T represents the absolute temperature. The stress activation volume can be conceived as a process coherence volume and interpreted as the size of the moving segment in the unit shear process, whether it is a part of a molecule or a dislocation line. The most critical parameter in equation (), v o, is a characteristic velocity related to the frequency of the process and to a jump distance. From Eyring s equation iso-relationships can be directly deduced as comprised in Table 5. Thus Eyring's model predicts a linear relationship of friction (the product of the shear strength and the active process area) in pressure and temperature and a logarithmic relationship in velocity. The model has been verified in lubrication experiments of solid (soap-like) lubricants by Briscoe and liquid lubricants by He et al. (00) within three logarithmic decades of velocities, and recently also by Knorr et al (00) who has extended the model. While Briscoe et al. (98) employed a SFA that confines and Q Fig. 6: Potential barrier in a lubricant based on Eyring's thermodynamic "cage-model". The normal pressure P and the shear stress are modifying the barrier height Q. Modified from Briscoe et al. (98).

22 pressurizes the film over several square microns, He et al. used a SFM system in which the contact is on the order of the lubricant molecular dimension. Table 5: Eyring Model Equations for Shear-induced Contact o p () o ' T (6) o '' ln v (9) o kt ln v v o Q (4) (5) k v ln v o o ' Q p (7) (8) Q p ktln v o'' kt o (0) () He et al. determined the degree of interfacial structuring and its effect on lubrication of n-hexadecane and octamethylcyclotetra-siloxane (OMCTS). For spherically shaped OMCTS molecules, only an interfacial "monolayer" was found; in contrast, a -nm-thick entropically cooled layer was detected for n-hexadecane in the boundary regime to an ultra-smooth silicon wafer. SFM measurements of the two lubricants (with similar chemical affinity to silicon) identified the molecular shape of n-hexadecane responsible for augmented interfacial structuring. Consequently, interfacial liquid structuring was found to reduce lubricated friction, Fig. 7(a,b). Again as reasoned above, these results can be discussed in terms of a collective phenomenon, i.e., in terms of increased molecular coordination in n-hexadecane versus OMCTS. Figure 7 (a): Logarithmic F F (v)-plots. F F (v) = F o +ln(v[m/s]): "dry" contact (8% relative humidity) with F o =6.4 nn and =0.9 nn, OMCTS lubricated with F o =. nn and =.4 nn, and n-hexadecane (n-c 6 H 4 ) lubricated with F o =7. nn and =.5 nn. The measurements were obtained with rectangular SFM cantilevers ( N/m) at 00 nn load and C, both feedback-controlled. Figure 7 (b): Stress activation length, /A. (A area of contact) for OMCTS, n-hexadecane and dry contact. The inset provides a linear relationship between friction and temperature at a velocity of m/s and a normal load of 00 nn. Adapted from He et al. (00).

23 Molecular Motion, Energetics and Time Temperature Superposition For small and unbranched molecules in the liquid phase, such as simple alkanes, it is possible that the confined materials undergo a pressure-induced phase reconstruction, which leads to material properties that deviate significantly from the bulk, as discussed above with the stick-slip behavior, Fig. 4. Larger and more complex (branched) molecules in the liquid or melt phase are less likely to exhibit pressure-induced phase reconstruction due to internal constraints and poor mixing within the contact area. This was shown by Drummond et al. (000) in surface forces apparatus (SFA) shear experiments. They found that the linear friction-velocity dependence does not apply for branched hydrocarbon lubricants. Also Drummond discussed "molecular lubrication" in terms of a logarithmic friction-velocity relationship, which is in accordance with the above-discussed thermal activation model, the solid lubricant SFA study by Briscoe et al. (98), and the liquid lubricant SFM study by He et al. (00). Motion of molecules in condensed matter (solids and liquids) involve kinetic activated processes. Activated processes generally entail, based on Boltzmann statistics, exponential expressions. Hence, it was not surprising when Doolittle came up with the following empirical description of the property that describes fluid motion, i.e., the viscosity. Based on his experimental observations, he expressed the viscosity as * v ln exp const. v f This expression relates the viscosity to the ratio of the smallest free volume per molecule v *, at which diffusional motion is possible, and the averaged free volume per molecule v f (T) at a given temperature T. It is interesting to note that the limiting lower temperature at which v f (T) converges to v * also defines the glass transition temperature T g. By considering the viscosity at two different temperatures, T and T, both above T g, we can write based on Doolittle s equation * * T v v exp. T v f,t v f,t. To express v f in terms of T, we assume the free volume to expand linearly with increasing temperature, i.e., v v v T. f,t f,t m T With this linear approximation for T > T, we assumed v m to be constant. Thereby, is equivalent to an expansion coefficient. We combine the two equations and obtain after some rearrangements * v T T v T f,t ln, * T v T T v f,t which indicates that the mobility ratio depends on the temperature shift T -T. This leads to the definition of the shift factor a T, i.e.,

24 T log 0 log0 at. T ref where T ref is an arbitrary chosen reference temperature. For polymers, T ref is typically T g +50 o C and based on the well-known Williams, Landel, Ferry (WLF) equation, the shift factor can be empirically expressed as log C T T g 0 at. C T Tg with C and C 0.6 for many polymers. Experiments reveal, as illustrated in Fig. 8 that by changing the temperature, the shear responses, i.e., the elastic G in-phase response and the viscous G out-of-phase response are simply shifted along the time axis. The same could have been accomplished by shifting the time (or frequency) window of observation. This phenomenon is known as the time-temperature equivalence, and is the basis for many theories (e.g., the Rouse theory) and leads to the equivalence of the two ratios of viscosity and relaxation time, i.e., T T a T. T T that can now be expressed in both cases by a single factor, the shift factor a T. log G G T G T log G G T G T log a T log a T log t log t Fig. 8. Time-Temperature Superposition expresses a mere lateral shift of the physical property values (e.g., the components of the complex shear modulus G* G +ig ) regarding the time (t) axis, when the temperature is switched from T to T. The shift of the isotherm E T to overlap the reference isotherm T, at, is known as the shift factor. As periodic stress experiments are typically performed over a constant time frame (respectively frequency window) for all temperatures, G and G response isotherms have to be shifted accordingly into an equivalent time-temperature window. Thereby, the choice of the reference window (i.e., the reference temperature T ref ) is arbitrary. This process of shifting data, as shown in Fig. 5, is known as time-temperature superposition principle, and leads to a continuous data composition curve, also referred to as master curve. In the example provided in Fig. 9, all data was collected within the time interval from 0 to 0 4 seconds. For the shifts, -8 o C was chosen as the reference temperature. Data from other isotherms (Fig. 9 - left) were shifted until a smooth master curve was 4

25 Increasing Temperature R.Overney From Classical to Molecular Tribology NME 498 obtained (Fig. 9 right). Each shift of isotherm identifies a shift factor a T (T) at its specific temperature. The reason to aspire a smooth composition curve is based on the fact that natural systems are generally well behaved. They show only disruptive behavior under extreme conditions, such as phase transitions, material yield or rupture and spontaneous reactions. Also around their relaxation frequency, a shift can show incommensurate behavior that is discussed further below in terms of entropic cooperativity. Experiment Analysis: Time Temperature Superposition Experimental Time Frame shift of the -50 o C data -8 o C a T (T= -66 o C) Master Curve (data composition) - 40 o C Fig. 9. (Left) Elasticity E data collected over a time and temperature scale of 0 to 0 4 seconds and -8 to -40 o C, respectively. The corresponding temperatures of the extreme values are indicated in the graph. (Right) The Master Curve (composite line on the very right) is obtained by shifting the different isotherms along the time axis. A representative shift is illustrated with the -66 o C isotherm. The data that has not been shifted out of the original time frame (in this particular case it is the -8 o C isotherm) was chosen arbitrary and defines the reference temperature (or reference isotherm). The superposition principle provides access to the activation energy E a (more precisely the apparent activation energy) via the shift factors, as they can be expressed as E a exp, kbt where k B is the Boltzmann constant and T the absolute temperature. This leads with the former equations for the shift factor to Ea ln at. R T TR Thereby, we have introduced the universal gas constant R and expressed the energy on a molar basis. If we plot now log 0 a T versus the inverse temperature /T, we can determine 5

26 the activation energy E a. This is illustrated in Fig. 0 (see inset) for friction isotherms, which brings us back to tribology. Fig. 0. The shift factors obtained from shifting the friction isotherm data onto a master curve, are compiled in the inset and expressed as function of the inverse temperature. From the slope an activation barrier of 8. kcal/mol could be determined. The material analyzed here is polystyrene. The energy corresponds to the phenyl rotation (also known as transition). Molecular Tribology We return to our friction discussion and take a molecular perspective. Common to the three studies by Briscoe, He, and Drummond is that they operate on a single material phase that is disrupted or relaxed over a very specific lateral length scale. In the Eyring model, the length scale is deduced by assuming a regular series of barriers, separated by a virtual jump distance. The distance is embedded in v o, the characteristic velocity, which is the product of the jump distance and the frequency of the process. Briscoe et al. (98) used the lattice constant of the highly oriented monolayers as the virtual jump distance. It was assumed that the process frequency was related to the vibrational frequency of the molecules (0 s - ), neglecting sliding velocity, temperature, and pressure effects. He et al. (00) assumed a jump distance of 0. nm and considered frequencies between a perfectly structured alkane layer (0 Hz) and the bulk fluid (0-0 5 Hz, estimated from infrared absorption data for typical covalent bonds). With these assumptions He determined total "jump-energies" of J. That indeed material imposed modes can affect the friction dissipation process had been shown by Overney et al. (994) on a molecularly corrugated lipid solid surface, Fig.. This study avoided two levels of difficulties Briscoe et al. (98) and He et al. (00) encountered: (a) large contact areas of SFA studies, and (b) complex rheology with unknown structure parameters as in liquid lubricant studies. It involved contact dimensions on the order of nm, and the crystalline form a bilayer model-lipid-lubricant with in-plane lattice spacings of 0.6 and. nm. The study mainly focused on the effect of the depth of the corrugation potential (barrier height) on the static and dynamic friction 6

27 force. This is illustrated in Figure in the form of stick-slip amplitude plotted as a function of the drag direction (i.e., sliding with respect to the anisotropic row-like film structure). Figure. (Left) SFM molecular stick-slip measurements of a bilayer lipid system (5-(4'-N,N-dihexadecylamino) benzylidene barbituric acid). (a) and (b) are adapted from Overney et al. (994a) and (c) from Overney et al. (994b). (a) High amplitude frictional stick-slip behavior is observed for scans perpendicular to molecular rows as imaged in 5(c). F st, static friction, is assigned to the maximum force occurrence. The average value corresponds to the dynamic friction value, F dyn, determined on largescale micrometer scans. (b) A 0 degrees out of row direction scan leads to decreased frictional stick-slip behavior due to smaller molecular corrugations. (c) nm SFM lateral force image of a highly structured lipid bilayer. Two crystalline domains with a boundary are imaged. The anisotropic row-like structure is responsible for directional dependent friction forces. The molecular corrugation between the rows is larger than the molecular corrugation in between a single row. (Right) Large scale friction contrast related to underlying molecular orientation. Relevant to the discussion about jump distance in lubrication events is Overney's discussion about the sliding speed and its effect on the slip distance. They demonstrated that within sliding speeds of 6 nm/s to 00 nm/s, the jump distance corresponded to the lattice spacing. At higher velocities, however, they could observe jumps over multiples of lattice distances and found the jump length distribution to become increasingly stochastic at higher velocities. They proposed molecular (or atomistic) friction as a white-noise driven system, which obeys a Gaussian fluctuation-dissipation relation. Hence, based on this finding one should consider discussing kinetic friction in terms of a statistical fluctuation model and understand the jump distance as a statistical quantity. Overney s study had been motivated by the Fig. : Atomistic Model of Friction: The Prandtl-Tomlinson Model Prandtl-Tomlinson Model (90), Fig., that considered atomistic friction as an instability motion of a spring systems that is moved in respect of a corrugated surface potential leading to an effective potential with temporarily local minima. 7

28 Ever since the beginning of 990s experiments and theories have continued to revolve around the Prandl-Tomlinson model, ignoring the fact that the modes for dissipation do not have to be rigid and limited to atomically or molecularly corrugated surfaces. The reason for the early success in obtaining a basic insight into liquid lubrication is due to the recognition of underlying fundamental theories such as fluid mechanics, or more general rheology, and thermodynamics. These theories have been applied with significantly less success to dry friction, i.e., solid organic coatings. Since Bowden and Tabor s adhesive-concept of energy dissipation in dry friction, friction has been dealt with from a surface science perspective involving a variety of process descriptions (e.g., wear phenomena and stick-slip processes, and tribochemistry), while material specific relaxation properties moved more into the background. It was first Grosch (950) and then Tabor who considering molecular relaxation properties responsible for energy dissipation in polymer rubbers. This triborheological description of frictional dissipation faced significant resistance by surface bonding-debonding theories pioneered by Schallamach and others. One of the shortcomings in was the lack of access to material intrinsic molecular modes of relaxations during phenomenological sliding motions. Large and rough contact regimes imposed convoluted signals containing aspects of sliding and cutting among others, and trapped the slow relaxing polymer phases in non-equilibrium states. Also, the material within the interfacial sliding region was rather poorly defined. This situation has changed with the inception of scanning force microscopy and the work by Sills, Knorr and Overney, who addressed the tribo-rheological complexity of elastomeric friction with a simplistic methodology, dubbed intrinsic friction analysis (IFA), with which they have correlated characteristic signatures of the dissipation mechanism, e.g. the activation energy, with the tribological process. The characteristic signatures of the dissipation process can be determined by the superposition of friction-rate isotherms and from the critical velocity corresponding to the maximum in the friction force. This is illustrated in Figure with superposed frictionvelocity isotherms obtained with SFM measurements on a polystyrene melt. An activation barrier of 8 kcal/mol (.5 ev) is deduced from the apparent Arrhenius behavior of the thermal a T shift factor in the inset Fig.. The value coincides with the kcal/mol activation energy for the -relaxation process, i.e. the segmental relaxation of the PS backbone. In this case, the activation barrier overcome during the course of frictional sliding corresponds directly with the molecular relaxation within the bulk elastomer. The friction peak in Fig. is recognized as an analogue to a spectroscopic peak in the frequency space. It reflects the competition between material and experimental timescales. At low sliding velocities, the contact stress stored in the soft material is capable of relaxing (through internal friction modes) before an asperity can slip to the next contact site. In this region, the friction force increases logarithmically with velocity, which is consistent to an activated molecular relaxation process in the soft material. Above the critical velocity, the probing tip is driven to the next contact site before the material can respond internally through viscoelastic relaxation. Thus with increasing velocity, the tip experiences fewer and fewer dissipative relaxation events per jump, and consequently, the friction force decreases. In other words, molecular scale dynamics can be deduced from local friction measurements. 8

29 Fig. : (Left) Schematic of IFA shifting. (Right) Friction force-velocity, F(v) isotherms for polystyrene (M = 96.5 kg/mol) above T g = 7 K, superposed using the method of reduced variables with a reference temperature of 40 K. Inset: From the Arrhenius behavior of a T, an average activation energy, E A, of 8 kcal/mol identifies the -relaxation as responsible for frictional dissipation. Due to time-temperature equivalence, the set of horizontal shifts, {a T }, provide the means to determine the apparent activation energy, E a, if the process behaves in an Arrhenius manner, i.e., ln( at ) Ea R T (/ ) P where ln(a T ) is a linear function of the reciprocal temperature, T -, at isobaric conditions (Fig. ). The apparent activation energy is composed of both an enthalpic energy component and a cooperative entropic energy, i.e., E a = H* +TS*. While noncooperative processes are described in terms of the dynamic enthalpy H* alone,] cooperative processes also include the dynamic entropy, S*. In these cases, vertical shifting of the friction force by F F is necessary, as also illustrated in Fig. with two isotherms that differ not only in ln(v) but also in F F. As described in our previous work, frictional vertical shifting, F F, is directly related to the dynamic entropy via, TS * F F ' where is the contact area normalized stress activation volume, i.e., /A. The IFA approach reveals the thermally active modes within the probing regime of the SFM tip, which can be thought of either to represent a device specific contact, such as a MEMS contact, or an asperity (roughness spot) of a larger contact. As materials can possess multiple thermally active modes that are either shifted thermally or locally, the SFM tip can pick either of these modes. This sketched in Fig. 4 with polystyrene at different loading conditions. 9

30 Fig. 4: Different mode coupling for low and high loads (pressures). Polystyrene is known to exhibit at the free surface a higher concentration of phenyls. In conventional friction study, where friction is expressed in terms of the load, the friction forces reveal kinks at critical loads that are dependent on both temperature and sliding velocities. These two regimes (branches I and II) of friction could be analyzed by IFA and revealed branch-specific apparent energies of E a,i = 8. kcal/mol (~ 0.0 ev) and E a,ii =.7 ± kcal/mol (~ 0.94 ev), Fig. 5. By inspection, E a,i corresponds to the activation energy E of the -relaxation in PS and originates from the hindered rotation of phenyl groups around the C-C bond with the backbone. E a,ii exceeds by about 4 kcal/mol the experimentally and theoretically determined activation energy E of 7-8 kcal/mol for the -relaxation in PS. The transition forces F * (v) and F * (T) (see Fig. 5(a)). We deduced from F * (v) T isotherms an activation energy of the phenyl loading capacity of 9.9 ± kcal/mol. By employing a kinetic theory by Starkweather, the above mentioned excess energy of ~4 kcal/mol could be assessed and treated as a cooperative entropic contribution. Thereby, the apparent activation energy E a, underlying the intrinsic friction dissipation process, was considered to be H RT, E a where R is the universal gas constant. As per Starkweather, the enthalpic contribution H was substituted into the following expression kt f e e, h obtained from the theory of absolute reaction rates, where S is the activation entropy, T R is the relaxation temperature, and k B and h denote the Boltzmann constant and the Plank constant, respectively. This led to an expression for the apparent process activation energy of the form, Ea RTR [ ln(kbtr / hfr )] TRS, where f R represents the frequency at which the relaxation peak is observed. For a purely activated dissipation process, i.e., a process without cooperativity, the entropic term T R S vanishes. From the IFA data of the vertical shift (see Fig. 6 below) and peak information as in Fig., the 4 kcal/mol could be attributed to cooperativity within the -relaxation. H RT S R 0

31 b) c) FIG. 5: (a) Isothermal friction-load plots for atactic PS (spun cast onto silicon, MW = 96.5k, Polymer Source, Inc.) subdivided at critical loads L k * (k =,, ) into two friction regimes with friction coefficients I and II. The dashed line follows friction values at critical loads. (b,c) IFA master curves of glassy PS at a load of (b) -4 nn (< L * T = -9.9 nn). Upper Inset: Phenyl group rotation around PS backbone (relaxation). (c) IFA at 40 nn above the critical load (< L * T = 7.9 nn). Upper Inset: Local translational backbone motion (-relaxation). (b,c) Lower Insets: Thermal ln(a T ) vs. (/T) shift factor analyses. Fig. 6: Polysturene at 40 nn (load > L * T = -7.9 nn) over a temperature range including T g. Cooperative effect manifests itself by vertical F shifting necessary for collapsing the data to the master curve. T < T g : -relaxation. Inset: Magnification of F for the -relaxation (T < T g ).

32 Summary We have started our discussion on tribology with contact forces, in particular adhesion forces and capillary forces, and contact mechanics, which considers the deformation aspect of contact. Thereby, we limited the discussion first to elastic contact. Following these introductory fundamental aspects of contact, we addressed dry and lubricated friction in classical phenomenological terms. First steps towards a molecular description of friction dissipation, were taken with the Eyring activation model, which led us to discuss critical length scales. Molecular stick-slip phenomena and logarithmic friction-velocity behavior directed us to consider molecular modes of rotation and translation that couple to the sliding motion and extract energy from it, as illustrated in Figure 7. The ultimate physical quantity of friction dissipation, i.e., heat is thus a consequence of a two step process: (i) entropic alignment of molecular modes, followed by (ii) mode coupling to the vibrational modes. Fig. 7: Entropic alignment of rotational modes (indicated with the path on the right) after coupling with the directional motion of a sliding tip (asperity) initiates frictional dissipation. The entropic alignment indicated here with the equivalent direction of rotation withdraws first energy from the slider, and then decays (relaxes) over time due to coupling with rotional modes, leading to heat generation, i.e. the ultimate form of friction dissipation.