As discussed in Section 4.2, our goal in characterizing rough surfaces is to determine the size and spatial distribution of contact spots due to asperity–asperity contact. Conventional methods in tribology (Bhus- han, 1990) use an asperity density, η, as a measure of number of spots per unit area and a mean diameter,– d, as a measure of the average spot size. Although η counts the number, it does not involve the size distribution which may be useful for scale-dependent tribological phenomena. In addition, the asperity density under elastic contact theory is found by its statistical relationship to the rms height and the curvature as (Bhushan, 1984a,b)
(4.4.1)
(4.4.2)
where F is the normal load, E is the elastic modulus, and Aa is the apparent area. As has been proved earlier, the rms parameters are not unique for a surface. Therefore, any estimate of the asperity density using these parameters is also not unique.
4.4.1 Observations of Size Distribution for Fractal Surfaces
The multiscale nature of surface roughness suggests that there will be spots of different sizes. As can be observed in Figure 4.3, which is a simulation of a real surface contact, the spots come in all sizes and shapes. As a general trend it is evident that there is a greater number of smaller spots than larger ones.
Consider first a fractal surface of profile dimension D and surface dimension D + 1. In the study of geomorphology of Earth, Mandelbrot (1975) found that the cumulative size distribution of islands on Earth’s surface follows the power law N(a) ≈ a–D/2, where N is the total number of islands with area larger than a and D is the fractal dimension of the coastline of the islands. The coastline is a self-similar curve for which the dimension D is related to the surface dimension Ds by the relation D = Ds – 1. If the surface of, for example, a machined metal or a thin-film magnetic disk is enlarged enough, the hills and valleys would appear similar to that found on Earth’s surface. The contact spots formed by a horizontal cut of such a surface by a contact plane would be analogous to the islands formed by cutting the Earth’s surface by that of the ocean. Therefore, it can be expected that the contact spots would also follow a power law relation of the form:
(4.4.3)
where the distribution is normalized by the area of the largest contact spot aL. The size distribution of contact, n(a), is defined such that the number of contact spots between a and a + da is equal to n(a)da.
For a fractal surface, the distribution n(a) can be obtained from N(a) of Equation 4.4.3 as
(4.4.4)
η σ
σ σ σ
≈ ′′( )
′′
F EAa
d≈
′′
σ σ
N a a
a
L
( )= D2
n a dN da
D a
a a
L
( )= − =2 D2
If any instrument is used to count the number of spots and measure their diameters, the values would obviously be finite and nonzero, respectively. Therefore, conventional parameters such as the contact spot density η and the mean diameter–d can be used to quantify the size distribution of contact spots.
However, note that in the distribution of Equation 4.4.3 the number of the largest spot is unity, whereas the number of spots of area a → 0would tend to infinity. There is obviously a discrepancy between the two methods. One must note that since any measuring instrument is limited by its resolution, not all spots on the contact interface can be detected. In other words, both η and the mean spot diameter are instrument-dependent parameters as has been analytically shown(Majumdar and Bhushan, 1990). The fractal model does not involve these parameters and allows the smallest spot area to tend to zero with their number tending to infinity.
4.4.2 Derivation of Size Distribution for Any Surface
The power law behavior of Equation 4.4.4 is not only an experimental observation, but can be proved rigorously as follows. Consider a rough surface to contact a hard flat plane such that the surface mean and the plane is separated by a distance, d. Then the real area of contact, Ar, can be written in two ways:
(4.4.5)
The first integral is the summation over all the contact spots where aL is the area of the largest contact spot. The second integral is over the probability distribution g(z) of the surface heights as described in Section 4.3.1, where Aa is the apparent surface area. The size distribution n(a) can, hence, be found as
(4.4.6)
where the chain rule is used for the right-hand side.
Now dAr/dz is related to g(z) as
(4.4.7)
The factor dz/da must be derived from the deformation of a single asperity. Consider an asperity of curvature k(a) in contact with a flat hard plane such that the contact area is equal to a and the deformation is equal to δ. Since δ increases in the negative z-direction, dz/da = -dδ/da.
4.4.2.1 Size Distribution under Elastic Deformation
If the contact is under elastic deformation, then according to Hertzian theory(Timoshenko and Goodier, 1970; Johnson, 1985) the asperity deformation δ is related to the area a of contact and curvature k as
(4.4.8) such that dz/da is equal to
(4.4.9) Ar n a a da A g z dz
a
a d
=∫0L ( ) = ∫∞ ( )
n a a dA
da a dA
dz dz da
r r
( )=1 =1
dA
dzr A g d
z d a
=
= − ( )
δ = ( )
π ak a
dz da
k a a
k a dk
= − ( ) da
π +
( )
1
Combining this with dAr/dz, the size distribution ne(a) can be written as
(4.4.10)
This is an important result, because the only assumption that has been made in deriving this result is that the asperity has an ellipsoidal shape of an effective curvature k(a). It is a general relation for any surface — fractal or nonfractal. If the surface deformations are predominantly elastic, then the largest contact spot area, aL, can be found by using Equation 4.4.10 in the equality of Equation 4.4.5 to get
(4.4.11)
where Ω(d) is the cumulative probability distribution defined as
(4.4.12) To use the size distribution, ne(a), of Equation 4.4.10 in any tribological analysis, it is necessary to find the probability distribution, g(z), and the curvature, k(a), as a function of spot area a, and the separation d between the two surfaces. The probability distribution, g(z), can be assumed to be a Gaussian or perhaps some nonsymmetric function. The curvature, k(a), can be obtained by the generalized roughness char- acterization technique discussed in Section 4.3.4. The separation, d, can be found by studying the contact mechanics which depends on the applied normal load on the surface, the size distribution, ne(a), the surface roughness, and the material properties such as hardness and elastic modulus. This will be dealt with in Section 4.5.
4.4.2.1.1 Gaussian Height Distribution
If the surface heights have a Gaussian distribution, then the size distribution reduces to
(4.4.13)
where σ is the standard deviation of the surface heights. The largest contact spot area, aL, can be found from Equation 4.4.12 to be
(4.4.14)
where erfc( ) is the complementary error function. Thus one would get a relation between aL and the nondimensional height d/σ.
4.4.2.1.2 Fractal Surface
If the surface is a fractal with a profile dimension of D, then we saw earlier that the radius of curvature, R, behaved as R = aD/2/GD–1 such that k(a) = GD–1/aD/2. Therefore the size distribution, ne(a), varies with spot size as
n a A k a g d a
a k a
dk
e da
( )= a ( ) ( )
π +
( )
1
k a a
k a dk
da da d
g d
aL
( ) + ( )
=π ( )
∫0 1 Ω( )
Ωd g z dz
( )=∫d∞ ( )
n a A k a a
a k a
dk da
e
e
a d
( )= ( )
π +
( )
π
−
1 2
2 2σ2
σ
k a a
k a dk
da da d
e
a
d
L ( ) + ( ) = π π ⋅ ( )
∫ 1 2 −
2
0 22 2
σ σ
σ
erfc
(4.4.15)
It is evident that the variation of ne(a) ≈ a–(1+D/2) is exactly what was observed by Mandelbrot(1975) for the distribution of islands on Earth’s surface and given in Equation 4.4.4. This is a mathematical derivation of the experimentally observed distribution.
The largest island size, aL, can be found for a Gaussian fractal surface by using the relation k(a) = GD–1/aD/2 in Equation 4.4.14 to get the relation
(4.4.16)
Using the fact that , we find that
(4.4.17)
4.4.2.2 Size Distribution under Plastic Deformation
When the contact is under plastic deformation, then using volume conservation the area of contact is found to follow (Chang et al., 1987):
(4.4.18)
where δc(a) is the critical deformation such that when δ > δc, the deformation is plastic and when δ <
δc the deformation is elastic. It is clear that when δ = δc, the relation for elastic deformation in Equation 4.4.8 is retrieved. In this case, dz/da is equal to
(4.4.19)
More will be discussed on the critical deformation δc in the section on contact mechanics whereas now it will be stated that it depends on the contact area a and curvature as δc = φ/k(a), where φ depends on the material properties. By using this, the size distribution under plastic deformation is
(4.4.20)
For a Gaussian height distribution, one can follow the analysis for elastic deformation to obtain the relation for the largest island. For a fractal surface with profile dimension, D, such that k(a) = GD–1/aD/2, the size distribution can be shown to follow the relation:
n a A G g d a
D
e a
D
( )= D ( )
π
−
( )−
( )+
1 2 1
2 2
a
G
d
L e
D
D d
2 2
1 2
2
2
2 2
( )−
( )− −
= π π σ ( σ )
σ
erfc
σ =G( ) ( )D−1 L2−D 2Ψ( )D
a L
D d
L e d
D
= π π
( )
−
( )− 2
2
2 2
2
2
2 2
Ψ
erfc σ
σ
δ δ
= ( )+ ( )
π
c a ak a
2 2
dz da
d da
k a a
k a dk da
= − c + ( )
π + ( )
1
2 δ 1
n a A k a g d
a k
dk da
a k
dk
p da
( )= a ( ) ( )
π − π + +
φ
2
1 2 1
3
(4.4.21)
4.4.2.3 Size Distribution under Elastic-Plastic Deformations
In the next section it will be shown that when surface contact involves a combination of elastic and plastic deformations there exists a critical contact spot area, ac, such that all spots smaller than the critical area, a < ac, deform plastically and all spots larger than ac, a > ac, deform elastically. Under these circumstances, the largest contact spot area, aL, must be found by the equation:
(4.4.22)
where ne(a) and np(a) are given by Equations 4.4.10 and 4.4.20, respectively. If the surface is assumed to have a Gaussian surface height distribution, Equation 4.4.22 reduces to
(4.4.23)
The relation between the largest contact spot, aL, and the nondimensional separation d/σ for a magnetic tape surface in Figure 4.28. It is assumed here that φ = 0.072 and ac = 0.35 àm2 as will be shown in the next section. The relation between curvature k(a) as a function of a is obtained from the generalized roughness characterization technique discussed in Section 4.3.4.