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Majumdar, A. et. al. “Characterization and Modeling of Surface...” Handbook of Micro/Nanotribology. Ed. Bharat Bhushan Boca Raton: CRC Press LLC, 1999 © 1999 by CRC Press LLC 4 Characterization and Modeling of Surface Roughness and Contact Mechanics Arun Majumdar and Bharat Bhushan 4.1 4.2 Introduction Why Is Surface Roughness Important? How Rough Is Rough? • How Does Surface Roughness Influence Tribology? 4.3 Surface Roughness Characterization Probability Height Distribution • rms Values and Scale Dependence • Fractal Techniques • Generalized Technique for Fractal and Nonfractal Surfaces 4.4 Size Distribution of Contact Spots Observations of Size Distribution for Fractal Surfaces • Derivation of Size Distribution for Any Surface 4.5 Contact Mechanics of Rough Surfaces Greenwood–Williamson Model • Majumdar–Bhushan Model • Generalized Model for Fractal and Nonfractal Surfaces • Cantor Set Contact Models 4.6 Summary and Future Directions References Appendix 4.1 Appendix 4.2 Appendix 4.3 Abstract Almost all surfaces found in nature are observed to be rough at the microscopic scale. Contact between two rough surfaces occurs at discrete contact spots. During sliding of two such surfaces, interfacial forces that are responsible for friction and wear are generated at these contact spots. Comprehensive theories of friction and wear can be developed if the size and the spatial distributions of the contact spots are known. The size of contact spots ranges from nanometers to micrometers, making tribology a multiscale phenomena. This chapter develops the framework to include interfacial effects over a © 1999 by CRC Press LLC whole range of length scales, thus forming a link between nanometer-scale phenomena and macroscopically observable friction and wear. The key is in the size and spatial distributions, which depend not only on the roughness but also on the contact mechanics of surfaces. This chapter reexamines the intrinsic nature of surface roughness as well as reviews and develops techniques to characterize roughness in a way that is suitable to model contact mechanics. Some general relations for the size distributions of contact spots are developed that can form the foundations for theories of friction and wear. 4.1 Introduction Friction and wear between two solid surfaces sliding against each other are encountered in several dayto-day activities. Sometimes they are used to our advantage such as the brakes in our cars or the sole of our shoes where higher friction is helpful. In other instances such as the sliding of the piston against the cylinder in our car engine, lower friction and wear are desirable. In such cases, lubricants are often used. Friction is usually quantified by a coefficient µ, which is defined as µ= Ff Fn (4.1.1) where Ff is the frictional force and Fn is the normal compressive force between the two sliding bodies. The basic problem in all studies on friction is to determine the coefficient µ. With regard to wear, it is · necessary to determine the volume rate of wear, V, and to establish the conditions when catastrophic failure may occur due to wear. Despite the common experiences of friction and wear and the knowledge of its existence for thousands of years, their origins and behavior are still not well understood. Although the effects of friction and wear can sometimes be explained post-mortem, it is normally very difficult to predict the value of µ and the wear characteristics of the two surfaces. One could therefore characterize tribology as a field that is perhaps in its early stages of scientific development, where phenomena can be observed but can rarely be predicted with reasonable accuracy. The reason for this lies in the extreme complexity of the surface phenomena involved in tribology. Three types of surface characteristics contribute to this complexity: (1) surface geometric structure; (2) the nature of surface forces; and (3) material properties of the surface itself. The lack of predictability in tribology lies in the convolution of effects of surface chemistry, mechanical deformations, material properties, and complex geometric structure. It is very difficult to say which of these is more dominant than the others. Physicists and chemists normally focus on the surface physics and chemistry aspects of the problem, whereas engineers study the mechanics and structural aspects. In a real situation of two macroscopic surfaces sliding in ambient conditions, it is very difficult to separate or isolate the different effects and then study their importance. They can indeed be isolated under controlled conditions such as in ultrahigh vacuum, but how those results relate to real situations is not clearly understood. It is, therefore, of no surprise that the tribology literature is replete with different theories of friction and wear that are applicable at different length scales — macroscopic to atomic scales. It is also of no surprise that a single unifying theory of tribology has not yet been developed. In this chapter we will examine only one aspect of tribology and that is the effect of surface geometric structure. It will be shown that friction and wear depend on surface phenomena that occurs over several length scales, starting from atomic scales and extending to macroscopic “human” scales, where objects, motion, and forces can be studied by the human senses. Atomic-scale studies focus on the nature of surface forces and the displacements that atoms undergo during contact and sliding of two surfaces. This book has several chapters devoted to this important new field of nanotribology. One must, however, remember that friction is still a macroscopically observable phenomena. Hence, there must be a link or a bridge between the atomic-scale phenomena and the macroscopically observable motion and measurable forces. This chapter takes a close look at surface geometric structure, or surface roughness, and attempts to formulate a methodology to form this link between all the length scales. © 1999 by CRC Press LLC FIGURE 4.1 Appearance of surface roughness under repeated magnification up to the atomic scales, where atomic steps are observed. First, the influence of surface roughness in tribology is established. Next, the complexities of the surface microstructure are discussed, and then techniques to quantify the complex structures are developed. The final discussion will demonstrate how to combine the knowledge of surface microstructure with that of surface forces and properties to develop comprehensive models of tribology. The reader will find that the theories and models are not all fully developed and much research remains to be done to understand the effects of surface roughness, in particular, and tribology, in general. This, of course, means that there is tremendous opportunity for contributions to understand and predict tribological phenomena. 4.2 Why Is Surface Roughness Important? Solid surfaces can be formed by any of the following methods: (1) fracture of solids; (2) machining such as grinding or polishing; (3) thin-film deposition; and (4) solidification of liquids. It is found that most solid surfaces formed by these methods are not smooth. Perfectly flat surfaces that are smooth even on the atomic scale can be obtained only under very carefully controlled conditions and are very rare in nature. Therefore, it is of most practical importance to study the characteristics of naturally occurring or processed surfaces which are inevitably rough. However, the first question that the reader may ask is how rough is rough and when does one call something smooth? 4.2.1 How Rough Is Rough? Smoothness and roughness are very qualitative and subjective terminologies. A polished metal surface may appear very smooth to the touch of a finger, but an optical microscope can reveal hills and valleys and appear rough. The finger is essentially a sensor that measures surface roughness at a lateral length scale of about 1 cm (typical diameter of a finger) and a vertical scale of about 100 µm (typical resolution of the finger). A good optical microscope is also a roughness sensor that can observe lateral length scales of the order of about 1 µm and can distinguish vertical length scales of about 0.1 µm. If the polished metal surface has a vertical span of roughness (hills and valleys) of about 1 µm, then the person who uses the finger would call it a smooth surface and the person using the microscope would call it rough. This leads to what one may call a “roughness dilemma.” When someone asks whether a surface is rough or smooth, the answer is — it depends!! It basically depends on the length scale of the roughness measurement. This problem of scale-dependent roughness is very intrinsic to solid surfaces. If one uses a sequence of high-resolution microscopes to zoom in continuously on a region of a solid surface, the results are quite dramatic. For most solid surfaces it is observed that under repeated magnification, more and more roughness keeps appearing until the atomic scales are reached where roughness occurs in the form of atomic steps (Williams and Bartlet, 1991). This basic nature of solid surfaces is shown graphically for a surface profile in Figure 4.1. Therefore, although a surface may appear very smooth to the touch of a finger, it is rough over all lateral scales starting from, say, around 10–4 m (0.1 mm) to about 10–9 m (1 nm). In addition, the roughness often appears random and disordered, and does not seem to follow © 1999 by CRC Press LLC FIGURE 4.2 (a) Schematic diagram of two surfaces in static contact against each other. Note that the contact takes place at only a few discrete contact spots. (b) When the surfaces start to slide against each other, interfacial forces act on the contact spots. any particular structural pattern (Thomas, 1982). The randomness and the multiple roughness scales both contribute to the complexity of the surface geometric structure. It is this complexity that is partly responsible for some of the problems in studying friction and wear. The multiscale structure of surface roughness arises due to the fundamentals of physics and thermodynamics of surface formation, which will not be discussed in this chapter. What will be discussed is the following. Given the complex multiscale roughness structure of a surface, (1) how does it influence tribology; (2) how does one quantify or characterize the structure; and (3) how does one use these characteristics to understand or study tribology? 4.2.2 How Does Surface Roughness Influence Tribology? Consider two multiscale rough surfaces (belonging to two solid bodies), as shown in Figure 4.2a, in contact with each other without sliding and under a static compressive force of Fn . Since the surfaces are not smooth, contact will occur only at discrete points which sustain the total compressive force. Figure 4.3 shows a typical contact interface which is formed of contact spots of different sizes that are spatially distributed randomly over the interface. The spatial randomness comes from the random nature of surface roughness, whereas the different sizes of spots occur due to the multiple scales of roughness. For a given load, the size of spots depends on the surface roughness and the mechanical properties of the contacting bodies. If this load is increased, the following would happen. The existing spots will increase in size, new spots will appear, and two or more spots may coalesce to form a larger spot. This is depicted by computer simulations of real surfaces in Figure 4.3. The surface in this case has isotropic statistical properties; that is, it does not have any texture or bias in any particular direction. It is evident that even for an isotropic surface the shapes of the contact spots are not isotropic and can be quite irregular and complex. In addition, when the load is increased, there are no set rules that the contact spots follow. Thus, the static problem itself is quite difficult to analyze. But one must, nevertheless, attempt to do so since these contact spots play a critical role in friction as explained below. Consider the two surfaces to slide against each other. To do so, one must overcome a resistive tangential or frictional force Ff . It is clear that this frictional force must arise from the force interactions between the two surfaces that act only at the contact spots, as shown in Figure 4.2b. Since the normal load-bearing © 1999 by CRC Press LLC FIGURE 4.3 Qualitative illustration of behavior of contact spots (dark patches) on a contact interface under different loads. (a) At very light loads only few spots support the load; (b) at moderate loads the contact spots increase in size and number; (c) at high loads the contact spots merge to form larger spots and the number further increases. capacity depends on the contact spot size, it is reasonable to assume that the tangential force is also size dependent. Therefore, to predict the total frictional force, it is very important to determine the size distribution, n(a), of the contact spots such that the number of spots between area a and a + da is equal to n(a)da. In addition to the interactions at each spot, there could be tangential force interactions between two or more contact spots. This is because the contact spots cannot operate independently of each other since they are connected by the solid bodies that can sustain some elastic or plastic deformation. So one can imagine the contact spots to be connected by springs whose spring constant depends on the elasticity/plasticity of the contacting materials. Because the number of contact spots is very large, the mesh of contact spots and springs thus forms a very complicated dynamic system. The deformations of the springs are usually localized around the contact spot and so the proximity of two spots influences their dynamic interactions. Therefore, it is also important to determine the spatial distribution, ∆(ai ,aj), of contact spots where ∆ is equal to the average closest distance between a contact spot of area ai and a spot of area aj . In other words, the frictional force, Ff , is a cumulative effect that arises due to force interactions at each spot and also dynamic force interactions between two or more spots. This can be written in a mathematical form as Ff = ∫ τ(a) n(a) a da + {dynamic interaction terms} aL (4.2.1) 0 where τ(a) is the shear stress on a contact spot of area a, aL is the area of the largest contact spot, and · n(a) is the size distribution of contact spots. Similarly, the total volume rate V of wear that is removed from the surface can be written as V˙ = ∫ v˙ (a) n(a) da aL 0 © 1999 by CRC Press LLC (4.2.2) where ν· (a) is the volume rate of wear at the microcontact of area a. It can be seen that as long as the size distribution n(a) is known, tribological phenomena can be studied at the scale of the contact spots. Let us concentrate on the first term on the right-hand side of Equation 4.2.1. This term adds up the tangential force on each contact spot starting from areas that tend to zero to the upper limit aL, which is the area of the largest contact spot. Recent studies have shown that the shear stress τ(a) is not a constant and can be size dependent. In other words, the frictional phenomena at the nanometer scale can be quite different from that at macroscales (Mate et al., 1987; Israelachvili et al., 1988; McGuiggan et al., 1989; Landman et al., 1990). In addition, the shear stress is strongly influenced by the different types of surface forces (Israelachvili, 1992). Some of the chapters in this book have concentrated on studying the nature of τ(a) when a is at the atomic or nanometer scales. The second term on the right hand side of Equation 4.2.1 represents the dynamic spring–mass interactions between the contact spots. Although this depends on the spatial and size distributions, it is unclear what the functional form would be. However, it is not insignificant since collective phenomena such as onset of sliding and stick-slip depend upon these types of interactions. Recently, there has been some interest in studying this as a percolation or a self-organized critical phenomena (Bak et al., 1988). The onset of sliding friction can be pictured as follows. When an attempt is made to slide one surface against another, the force on a contact spot can be released and distributed among neighboring spots. The forces in at least one of these spots may exceed a critical level creating a cascade or an avalanche. The avalanche may turn out to be limited to a small region or become large enough so that the whole surface starts sliding. During this process, the interface evolves into a self-organized critical system insensitive to the details of the distribution of initial disorder. This type of analysis has been used to provide a physical interpretation of the Guttenberg–Richter relation between earthquake magnitude and its frequency (Sornette and Sornette, 1989; Knopoff, 1990; Carlson et al., 1991). In summary, the basic problem of tribology can be divided as follows: (1) to determine the size, n(a), and spatial distribution of contact spots which depends on the surface roughness, normal load, and mechanical properties; (2) to find the tangential surface forces at each spot; (3) to determine the dynamic interactions between the spots; and, finally, (4) to find the cumulative effect in terms of the frictional force, Ff . 4.3 Surface Roughness Characterization A rough surface can be written as a mathematical function: z = f (x,y), where z is the vertical height and x and y are the coordinates of a point on the two-dimensional plane, as shown in Figure 4.4a. This is typically what can be obtained by a roughness-measuring instrument. The surface is made up of hills and valleys often called surface asperities of different lateral and vertical sizes, and are distributed randomly on the surface as shown in the surface profiles in Figure 4.4b. The randomness suggests that one must adopt statistical methods of roughness characterization. It is also important to note that because of the involvement of so many length scales on a rough surface, the characterization techniques must be independent of any length scale. Otherwise, the characterization technique will be a victim of the “rough or smooth” dilemma as discussed in Section 4.2.1. 4.3.1 Probability Height Distribution One of the characteristics of a rough surface is the probability distribution (Papoulis, 1965) g(z), of the surface heights such that the probability of encountering the surface between height z and z + dz is equal to g(z)dz. Therefore, if a rough surface contacts a hard perfectly flat surface* and it is assumed that the *Although hard flat surfaces are rarely found in nature, we make the assumption because contact between two rough surfaces can be reduced to the contact between an equivalent surface and a hard flat surface (see Section 4.4). © 1999 by CRC Press LLC FIGURE 4.4 (a) Schematic diagram of a rough surface whose surface height is z(x, y) at a coordinate point (x, y). (b) A vertical cut of the surface at a constant y gives surface profile z(x) with a certain probability height distribution. distribution g(z) remains unchanged during the contact process, then the ratio of real area of contact, Ar , to the apparent area, Aa, can be written as Ar = Aa ∫ () ∞ g z dz = σ d ∫ g (z ) dz ∞ d σ (4.3.1) where d is the separation between the flat surface, σ is the standard deviation of the surface heights, and – z = z/σ is the nondimensional surface height. The real area of contact, Ar , is usually about 0.1 to 10% of the apparent area and is the sum of the areas of all the contact spots. Therefore, the probability distribution, g(z), can be used to determine the sum of the contact spot areas but does not provide the crucial information on the size distribution, n(a). In addition, it contains no information concerning the shape of the surface asperities. It is often found that the normal or Gaussian distribution fits the experimentally obtained probability distribution quite well (Thomas, 1982; Bhushan, 1990). In addition, it is simple to use for mathematical calculation. The bell-shaped normal distribution (Papoulis, 1965) which has a variance of unity is given as (   z − zm g z = exp − 2  2π () – 1 )  − ∞ < z < ∞ 2  (4.3.2) where zm is the nondimensional mean height. The mean height and the standard deviation can be found from a roughness measurement z(x,y) as © 1999 by CRC Press LLC zm = σ= 1 Lx Ly 1 Lx Ly ∫ ∫ ( ) Lx 0 Ly z x , y dx dy = 0 ∫ ∫ [( ) ] Lx 0 Ly 0 2 z x , y − z m dx dy = 1 NxNy Nx Ny ∑ ∑ z (x , y ) i i =1 1 NxNy (4.3.3) j j =1 Ny ∑ ∑[ z ( x , y ) − z ] Nx i i =1 j m 2 (4.3.4) j =1 Here, Lx and Ly are the lengths of surface sample, whereas Nx and Ny are the number of points in the x and y lateral directions, respectively. The integral formulation is for theoretical calculations, whereas the summation is used for calculating the values from finite experimental data. Although used extensively, the normal distribution has limitations in its applicability. For example, it has a finite nonzero probability for surface heights that go to infinity, whereas a real surface ends at a finite height, zmax, and has zero probability beyond that. Therefore, the normal distribution near the tail is not an accurate representation of real surfaces. This is an important point since it is usually the tail of the distribution that is significant for calculating the real area of contact. Other distributions, such as the inverted chi-squared (ICS) distribution, fit the experimental data much better near the tail of the distribution (Brown and Scholz, 1985). This is given for zero mean and in terms of nondimensional – height, z, as (ν 2) (z g (z ) = Γ( ν 2) ν 4 max −z )( ) ν 2 −1 ( e z − z max ) ν 2 − ∞ < z < z max (4.3.5) – which has a variance of 2ν and a maximum height zmax = ν ⁄ 2 . The advantage of the ICS distribution is it has a finite maximum height, as does a real surface, and has a controlling parameter ν, which gives a better fit to the topography data. The Gaussian and the ICS distributions are shown in Figure 4.5. Note that as ν increases, the ICS distribution tends toward the normal distribution. Brown and Scholz (1985) FIGURE 4.5 Comparison of the Gaussian and the ICS distributions for zero mean height and nondimensional – surface height z. © 1999 by CRC Press LLC found that the surface heights of a ground-glass surface were not symmetric like the normal distribution but were best fitted by an ICS distribution with ν = 21. 4.3.2 RMS* Values and Scale Dependence A rough surface is often assumed to be a statistically stationary random process (Papoulis, 1965). This means that the measured roughness sample is a true statistical representation of the entire rough surface. Therefore, the probability distribution and the standard deviation of the measured roughness should remain unchanged, except for fluctuations, if the sample size or the location on the surface is altered. The properties derived from the distribution and the standard deviation are therefore unique to the surface, thus justifying the use of such roughness characterization techniques. Because of simplicity in calculation and its physical meaning as a reference height scale for a rough surface, the rms height of the surface is used extensively in tribology. However, it was shown by Sayles and Thomas (1978) that the variance of the height distribution is a function of the sample length and in fact suggested that the variance varied as σ2 ≈ L (4.3.6) where L is the length of the sample. This behavior implies that any length of the surface cannot fully represent the surface in a statistical sense. This proposition was based on the fact that beyond a certain length, L, the surface heights of the same surface were uncorrelated such that the sum of the variances of two regions of lengths L1 and L2 can be added up as ( ) ( ) ( ) σ 2 L1 + L2 = σ 2 L1 + σ 2 L2 (4.3.7) They gathered roughness measurements of a wide range of surfaces to show that the surfaces follow the nonstationary behavior of Equation 4.3.6. However, Berry and Hannay (1978) suggested that the variance can be represented in a more general way as follows: σ 2 ≈ Ln (4.3.8) where n varies between 0 and 2. If the exponent n in Equation 4.3.8 is not equal to zero of a particular surface, then the standard deviation or the rms height, σ, is scale dependent, thus making a rough surface a nonstationary random process. This basically arises from the multiscale structure of surface roughness where the probability distribution of a small region of the surface may be different from that of the larger surface region as depicted in Figure 4.4b. If the larger segment follows the normal distribution, then the magnified region may or may not follow the same distribution. Even if it does follow the normal distribution, the rms σ can still be different. Other statistical parameters that are also used in tribology (Nayak, 1971, 1973) are the rms slope, σ′x , and rms curvature, σ″x , defined as σ ′x = 1 Lx ∫ 0 Lx ( ) 2  ∂z x , y    dx =  ∂x    1 N −1 N −1 ∑ i =i ( ) ( ) z x , y − z x , y  i j  i+1 j    ∆x   2 (4.3.9) *The rms values (of height, slope, or curvature) are related to the corresponding standard deviation, σ, of a surface in the following way: rms2 = σ2 + zm2, where zm is the mean value. In this chapter it will be assumed that zm = 0; that is, the mean is taken as the reference, such that rms = σ. © 1999 by CRC Press LLC