Bhushan B. Handbook of Micro/Nano Tribology, Second Edition

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(14.2a)

and

(14.2b)

The fractal analysis allows the characterization of surface roughness by two parameters

and

, which

are instrument independent and unique for each surface.

(ranging from 1 to 2 for surface proﬁle)

primarily relates to relative power of the frequency contents and

to the amplitude of all frequencies.

is the lateral resolution of the measuring instrument,

is the size of the increment (distance), and

is the frequency of the roughness. Note that if

(

) or

(

) are plotted as a function of

, respectively,

on a log–log plot, then the power law behavior would result in a straight line. The slope of line is related

and the location of the spectrum along the power axis is related to

Figure 14.6 presents the structure function of a thin-ﬁlm rigid disk measured using AFM, noncontact

optical proﬁler (NOP), and stylus proﬁler (SP). A horizontal shift in the structure functions from one

scan to another arises from the change in the lateral resolution.

and

values for various scan lengths

are listed in Table 14.1. We note that fractal dimension of the various scans is fairly constant (1.26 to

1.33); however, C increases/decreases monotonically with

for the AFM data. The error in estimation

FIGURE 14.4

Scale dependence of standard deviation of

surface heights for a glass–ceramic disk, measured using

AFM, SP, and NOP.

FIGURE 14.5

Qualitative description of statistical self-afﬁnity for a surface proﬁle.

()

−

()

−

()

52 2

−

()

π−

()

[]

Γ sin

is believed to be responsible for variation in

. These data show that the disk surface follows a fractal

structure for three decades of length scales.

Majumdar and Bhushan (1991) and Bhushan and Majumdar (1992) developed a fractal theory of

contact between two rough surfaces. This model has been used to predict whether contacts experience

elastic or plastic deformation and to predict the statistical distribution of contact points. For a review of

contact models, see Bhushan (1996b, 1998c).

Based on the fractal model of elastic–plastic contact, whether contacts go through elastic or plastic

deformation is determined by a critical area which is a function of

, hardness, and modulus of

elasticity of the mating surfaces. If the contact spot is smaller than the critical area, it goes through the

plastic deformations and large spots go through elastic deformations. The critical contact area for

inception of plastic deformation for a thin-ﬁlm disk was reported by Majumdar and Bhushan (1991) to

be about 10

–27

, so small that all contact spots can be assumed to be elastic at moderate loads.

The question remains as to how large spots become elastic when they must have initially been plastic

spots. The possible explanation is shown in Figure 14.7. As two surfaces touch, the nanoasperities

(detected by AFM-type of instruments) ﬁrst coming into contact have smaller radii of curvature and are

therefore plastically deformed instantly, and the contact area increases. When load is increased, nanoas-

perities in the contact merge, and the load is supported by elastic deformation of the large-scale asperities

or microasperities (detected by optical proﬁler type of instruments) (Bhushan and Blackman, 1991).

FIGURE 14.6

Structure functions for the roughness data measured at various scan sizes using AFM (scan sizes: 1

1 µm, 10

10 µm, 50

50 µm, and 100

100 µm), NOP (scan size: 250

250 µm), and SP (scan length: 4000 µm),

for a magnetic thin-ﬁlm rigid disk. (From Ganti, S. and Bhushan, B., 1995,

Wear

180, 17–34. With permission.)

TABLE 14.1

Surface Roughness Parameters for a

Polished Thin-Film Rigid Disk

Scan size (µm x µm)

(nm)

1 (AFM) 0.7 1.33 9.8

-4

10 (AFM) 2.1 1.31 7.6

-3

50 (AFM) 4.8 1.26 1.7

-2

100 (AFM) 5.6 1.30 1.4

-2

250 (NOP) 2.4 1.32 2.7

-4

4000 (NOP) 3.7 1.29 7.9

-5

AFM = atomic force microscope; NOP = noncon-

tact optical proﬁler.

Majumdar and Bhushan (1991) and Bhushan and Majumdar (1992) have reported relationships for

cumulative size distribution of the contact spots, portions of the real area of contact in elastic and plastic

deformation modes, and the load–area relationships.

14.4 Friction and Adhesion

14.4.1 Nanoscale Friction

Ruan and Bhushan (1994b) measured friction on the nanoscale using FFM. They reported that atomic-

scale friction of a freshly cleaved, highly oriented pyrolytic graphite (HOPG) exhibited the same period-

icity as that of corresponding topography (also see Mate et al., 1987), Figure 14.8. However, the peaks in

friction and those in corresponding topography proﬁles were displaced relative to each other, Figure 14.9.

Using Fourier expansion of the interaction potential, they calculated interatomic forces between the FFM

tip and the graphite surface. They have shown that variations in atomic-scale lateral force and the observed

displacement can be explained by the variations in intrinsic interatomic forces in the normal and lateral

directions.

14.4.2 Microscale Friction and Adhesion

Friction and adhesion of magnetic head sliders, magnetic media, virgin, treated and coated Si(111) wafers,

and graphite on a microscale have been measured by Kaneko et al. (1988, 1991a), Miyamoto et al. (1990,

1991a,c), Mate (1993a,b), Bhushan et al. (1994a–c,e, 1995a–g, 1997c, 1998), Ruan and Bhushan

(1994a–c), Koinkar and Bhushan (1996a,b, 1997a,b), and Sundararajan and Bhushan (1998).

Koinkar and Bhushan (1996a,b) and Poon and Bhushan (1995a,b) reported that rms roughness and

friction force increase with an increase in scan size at a given scanning velocity and normal force.

Therefore, it is important that while reporting friction force values, scan sizes and scanning velocity

should be mentioned. Bhushan and Sundararajan (1998) reported that friction and adhesion forces are

a function of tip radius and relative humidity (also see Koinkar and Bhushan, 1996b). Therefore, relative

FIGURE 14.7

Schematic of local asperity deformation during contact of a rough surface, upper proﬁle measured

by an optical proﬁler and lower proﬁle measured by AFM; typical dimensions are shown for a polished thin-ﬁlm

rigid disk against a ﬂat slider surface. (From Bhushan, B. and Blackman, G.S., 1991,

ASME J. Tribol.

113, 452–458.

With permission.)

FIGURE 14.8

Gray-scale plots of (a) surface topography and (b) friction force maps of a 1

70×

1 nm area of a freshly cleaved

HOPG showing the atomic-scale variation of topography and friction. Higher points are shown by lighter color. (From Ruan, J.

and Bhushan, B., 1994,

J. Appl. Phys.

76, 5022–5035. With permission.)

humidity should be controlled during the experiments. Care also should be taken to ensure that tip

radius does not change during the experiments.

14.4.2.1 Head Slider Materials

–TiC is a commonly used slider material. In order to study the friction characteristics of this two

phase material, friction of Al

–TiC (70/30 wt%) surface was measured. Figure 14.10 shows the surface

roughness and friction force proﬁles (Koinkar and Bhushan, 1996a). TiC grains have a Knoop hardness

of about 2800 kg/mm

, which is higher than that of Al

grains of about 2100 kg/mm

. Therefore, TiC

grains do not polish as much and result in a slightly higher elevation (about 2 to 3 nm higher than that

of Al

grains). Based on friction force measurements, TiC grains exhibit higher friction force than

grains. The coefﬁcients of friction of TiC and Al

grains are 0.034 and 0.026, respectively, and

the coefﬁcient of friction of Al

–TiC composite is 0.03. Local variation in friction force also arises from

the scratches present on the Al

–TiC surface. A good correspondence between surface slope (also shown

in Figure 14.10) and friction force at scratch locations is observed. (Reasons for this correlation will be

discussed later.) Thus, local friction values of two-phase materials can be measured. Ruan and Bhushan

(1994c) reported that local variation in the coefﬁcient of friction of cleaved HOPG was signiﬁcant, which

arises from structural changes occurring during the cleaving process. The cleaved HOPG surface is largely

atomically smooth but exhibits line-shaped regions in which the coefﬁcient of friction is more than an

order of magnitude larger. Meyer et al. (1992) and Overney et al. (1992) also used FFM to measure

structural variations of a composite surface. They measured friction distribution of mixed monolayer

ﬁlms produced by dipping into a solution of hydrocarbon and ﬂuorocarbon molecules. The resulting

ﬁlm consists of discrete islands of hydrocarbon in a sea of ﬂuorocarbon. They reported that FFM can be

used to image and identify compositional domains with a resolution of ~0.5 nm. These measurements

suggest that friction measurements can be used for structural mapping of the surfaces. FFM measurements

can also be used to map chemical variations, as indicated by the use of the FFM with a modiﬁed FFM

tip to map the spatial arrangement of chemical functional groups in mixed monolayer ﬁlms (Frisbie

et al., 1994). Here, sample regions that had stronger interactions with the functionalized FFM tip exhibited

larger friction.

Surface roughness and coefﬁcient of friction of various head slider materials were measured by Koinkar

and Bhushan (1996a). For typical values, see Table 14.2. Macroscale friction values for all samples are

higher than microscale friction values; the reasons are presented in the following subsection.

FIGURE 14.9 Schematic of surface topography and fric-

tion force maps shown in Figure 14.8. The oblate triangles

and circles correspond to maxima of topography and fric-

tion force, respectively. There is a spatial shift between the

two. (From Ruan, J. and Bhushan, B., 1994, J. Appl. Phys.

76, 5022–5035. With permission.)

Miyamoto et al. (1990) measured adhesive force of four tips made of tungsten, Al

–TiC, Si

, and

SiC tips in contact with unlubricated, polished SiO

-coated thin-ﬁlm rigid disk and a disk lubricated

with 2-nm functional lubricant (with hydroxyl end groups, Z-Dol). Nominal radii for all tips were about

5 µm. Adhesive force data are presented in Table 14.3. Mean adhesive forces of the tungsten, Al

–TiC,

, and SiC tips on a disk medium coated with the functional lubricant are about 10% of those for

an unlubricated disk surface. The adhesive force of the ceramic tips is lower than that for the tungsten

tip. The adhesive forces of the SiC tip show very low values, even for an unlubricated disk. A good

correlation was found between adhesive forces measured by the AFM and the coefﬁcient of macroscale

static friction. They also reported that adhesive force increased almost linearly with an increase in the

tip radius. (Also see Sugawara et al., 1993; Bhushan et al., 1998).

14.4.2.2 Magnetic Media

Bhushan and co-workers measured friction properties of magnetic media including polished and textured

thin-ﬁlm rigid disks, MP, BaFe and ME tapes, and PET tape substrate. For typical values of coefﬁcients

of friction of polished and textured, thin-ﬁlm rigid disks and MP, BaFe and ME tapes, PET tape substrate,

see Table 14.4. In the case of magnetic disks, similar coefﬁcients of friction are observed for both lubricated

and unlubricated disks, indicating that most of the lubricant (although partially thermally bonded) is

squeezed out from between the rubbing surfaces at high interface pressures, consistent with liquids being

poor boundary lubricant (Bowden and Tabor, 1950). Coefﬁcient of friction values on a microscale are

much lower than those on the macroscale. When measured for the small contact areas and very low loads

used in microscale studies, indentation hardness and modulus of elasticity are higher than at the mac-

roscale (data to be presented later). This reduces the real area of contact and the degree of wear. In

addition, the small apparent areas of contact reduces the number of particles trapped at the interface,

and thus minimizes the “plowing” contribution to the friction force (Bhushan et al., 1995d,f).

Miyamoto et al. (1991b) reported the coefﬁcient of friction of an unlubricated disk with amorphous

carbon and SiO

overcoats against the diamond tip to be 0.24 and 0.36, respectively. The coefﬁcients of

friction of disks lubricated with 2-nm-thick perﬂuoropolyether lubricant ﬁlms were 0.08 for functional

lubricant (with hydroxyl end groups, Z-Dol) on SiO

overcoat, 0.10 for functional lubricant on carbon

overcoat, and 0.19 for nonpolar lubricant (Krytox 157FS L) on carbon overcoat. They found that the

coefﬁcient of friction of a 4-nm-thick lubricant ﬁlm was about twice that of a 2-nm-thick ﬁlm. Mate

(1993a) measured the coefﬁcient of friction of unlubricated polished and textured disks and with a

lubricant ﬁlm with ester end groups (Demnum SP) against a tungsten tip with a tip radius of 100 nm.

The coefﬁcients of friction of unlubricated polished disks and with 1.5-nm-thick lubricant ﬁlm were

0.5 and 0.4, respectively, and of unlubricated textured disks and with 2.5-nm-thick lubricant ﬁlm were

0.5 and 0.2, respectively.

Coefﬁcient of microscale friction values reported by Miyamoto et al. (1991b), by Mate (1993a) and

by Bhushan et al. (1995g) (to be reported later in this section) are larger than those reported by Bhushan

et al. (1994a–c,e, 1995a–f, 1997c) in Table 14.4. Miyamoto et al. made measurements with a three-sided

pyramidal diamond tip at large loads of 500 nN to tens of micronewtons and Mate et al. made measure-

ments with a soft tungsten tip from 30 to 300 nN, as compared to Bhushan et al.’s measurements made

using the Si

tip at lower loads ranging from 10 to 150 nN. High values reported by Miyamoto et al.

and Mate et al. may arise from plowing contribution at higher normal loads and differences in the friction

properties of different tip materials. Bhushan et al. (1995f) have reported that the coefﬁcient of friction

on microscale is a strong function of normal load. The critical load at which an increase in friction occurs

corresponds to surface hardness. At high loads, the coefﬁcient of friction on microscale increases toward

values comparable with those obtained from macroscale measurements. The increase in the value of

microscale friction at higher loads is associated with interface damage and associated plowing.

In order to elegantly show any correlation between local values of friction and surface roughness,

Bhushan (1995b) measured the surface roughness and friction force of a gold-coated ruling with rect-

angular girds. Figure 14.11 shows the surface roughness proﬁle, the slopes of roughness proﬁle taken

along the sliding direction, and the friction force proﬁle for the ruling. We note that friction force changes

signiﬁcantly at the edges of the grid. Friction force is high locally at the edge of grid with a positive slope

and is low at the edge of the grid with a negative slope. Thus, there is a strong correlation between the

slope of the roughness proﬁles and the corresponding friction force proﬁles.

Bhushan et al. (1994c,e, 1995a–f, 1997c) examined the relationship between local variations in micros-

cale friction force and surface roughness proﬁles for magnetic media. Figures 14.12 and 14.14 show the

surface roughness map, the slopes of roughness proﬁle taken along the sliding direction, and the friction

force map for textured and lubricated disks, and an MP tape, respectively. Bhushan and Ruan (1994a)

noted that there is no resemblance between the friction force maps and the corresponding roughness

maps; e.g., high or low points on the friction force map do not correspond to high or low points on the

roughness map. By comparing the slope of roughness proﬁles taken in the tip sliding direction and

friction force map, we observe a strong correlation between the two. (For a clearer correlation, see gray-

scale plots of surface roughness slope and friction force proﬁles for FFM tip sliding in either directions

in Figures 14.13 and 14.15).

TABLE 14.2 Surface Roughness (σ and P-V distance), Micro- and Macroscale Friction, Microscratching/Wear, and

Nano- and Microhardness Data for Various Samples

Surface Roughness

nm (1 × 1 µm)

Coefﬁcient of friction

Scratch Depth

at 60 µN (nm)

Wear Depth at

60 µN (nm)

Hardness (GPa)

Macroscale

Nano at

2 mN MicroSample σ P-V

Microscale Initial Final

0.97 9.9 0.03 0.18 0.2–0.6 3.2 3.7 24.8 15.0

–TiC 0.80 9.1 0.05 0.24 0.2–0.6 2.8 22.0 23.6 20.2

Polycrystalline 2.4 20.0 0.04 0.27 0.24–0.4 9.6 83.6 9.6 5.6

Mn–Zn ferrite

Single–crystal (110) 1.9 13.7 0.02 0.16 0.18–0.24 9.0 56.0 9.8 5.6

Mn–Zn ferrite

SiC (α-type) 0.91 7.2 0.02 0.29 0.18–0.24 0.4 7.7 26.7 21.8

Peak-to-valley distance.

Obtained using silicon nitride ball with 3 mm diameter in a reciprocating mode at a normal load of 10 mN, reciprocating

amplitude of 7 mm, and average sliding speed of 1 mm/s. Initial coefﬁcient of friction values were obtained at ﬁrst cycle (0.007 m

sliding distance) and ﬁnal values at a sliding distance of 5 m.

TABLE 14.3 Mean Values and Ranges of Adhesive Forces

between Unlubricated and Lubricated SiO

-Coated Disks

and Tips Made of Various Materials

Tip Material

Adhesive Force (µN)

Without Lubricant

Functional Liquid Lubricant

(2.0 nm)

Tungsten 12.1 (2.32–13.9) 1.09 (0.67–1.60)

–TiC 5.17 (3.64–6.92) 0.25 (0.078–0.47)

1.88 (1.00–2.82) 0.07 (0–0.16)

SiC 0.21 (0.13–0.32) 0.030 (0–0.09)

From Miyamoto, T. et al., 1990, ASME J. Tribol. 112, 567–572. With

permission.

FIGURE 14.10 Gray-scale plots of surface topography (σ = 1.12 nm), slope of the roughness proﬁles taken along

the sliding direction (the horizontal axis) (mean = –0.003, σ = 0.015), and friction force map (mean = 28.5 nN, σ =

4.0 nN; Al

grains: mean = 24.8 mN, σ = 1.85 nN and TiC grains: mean = 32.7 nN, σ = 2.6 nN) for a Al

–TiC

slider for a normal load of 950 nN.

To further verify the relationship between surface roughness slope and friction force values, to eliminate

any effect resulting from nonuniform composition of disk and tape surfaces, Bhushan and Ruan (1994a)

measured a polished natural (IIa) diamond. Repeated measurements were made along one line on the

surface. Highly reproducible data were obtained, Figure 14.16. Again, the variation of friction force

correlates to the variation of the slope of the roughness proﬁles taken along the sliding direction of the

tip. This correlation has been shown to hold for various magnetic disks, magnetic tapes, polyester tape

substrates, silicon, graphite and other materials (Bhushan et al., 1994a,c–e, 1995a–d, 1997a, 1998; Ruan

and Bhushan, 1994b,c).

We now examine the mechanism of microscale friction, which may explain the resemblance between

the slope of surface roughness proﬁles and the corresponding friction force proﬁles (Bhushan and Ruan,

1994a; Ruan and Bhushan, 1994b,c). There are three dominant mechanisms of friction: adhesive, adhesive

and roughness (ratchet), and plowing. As a ﬁrst order, we may assume these to be additive. The adhesive

mechanism alone cannot explain the local variation in friction. Let us consider the ratchet mechanism.

According to Makinson (1948), we consider a small tip sliding over an asperity making an angle θ with

the horizontal plane, Figure 14.17. The normal force (normal to the general surface) applied by the tip

to the sample surface W is constant. Friction force F on the sample varies as a function of the surface

roughness. It would be a constant µ

W for a smooth surface in the presence of “adhesive” friction

mechanism. In the presence of a surface asperity, the local coefﬁcient of friction µ

in the ascending part is

(14.3)

Since µ

tan θ is small on a microscale, Equation 14.3 can be rewritten as

(14.4)

indicating that in the ascending part of the asperity one may simply add the friction force and the asperity

slope to one another. Similarly, on the right-hand side (descending part) of the asperity,

TABLE 14.4 Surface Roughness (σ), Microscale and Macroscale Friction, and Nanohardness Data of Thin-Film

Magnetic Rigid Disk, Magnetic Tape, and Magnetic Tape Substrate (PET) Samples

Sample

σ (nm)

Coefﬁcient of

Microscale Friction

Coefﬁcient of

Macroscale Friction

Nanohardness

(GPa)/

Normal Load

(µN)

NOP

AFM

250 × 250 µm

1 × 1 µm

10 × 10 µm

1 × 1 µm

10 × 10 µm

Mn–Zn

–TiCFerrite

Polished,

unlubricated

disk

2.2 3.3 4.5 0.05 0.06 — 0.26 21/100

Polished,

lubricated disk

2.3 2.3 4.1 0.04 0.05 — 0.19 —

Textured,

lubricated disk

4.6 5.4 8.7 0.04 0.05 — 0.16 —

MP tape 6.0 5.1 12.5 0.08 0.06 0.19 — 0.30/50

Barium-ferrite

tape

12.3 7.0 7.9 0.07 0.03 0.18 — 0.25/25

ME tape 9.3 4.7 5.1 0.05 0.03 0.18 — 0.7 to 4.3/75

PET tape

substrate

33 5.8 7.0 0.05 0.04 0.55 — 0.3/20 and

1.4/20

Scan area; NOP = noncontact optical proﬁler; AFM = atomic force microscope.

Numbers are for polymer and particulate regions, respectively.

µ= =µ+

()

−µ

()

10 0

1FW tan tan .θθ

µµ+

~ tan ,θ

FIGURE 14.11 (a) Surface roughness map, (b) slope of the roughness proﬁles taken in the sample sliding direction

(the horizontal axis), and (c) friction force map for a gold-coated ruling at a normal load of 155 nN. (From Bhushan,

B., 1995, Tribol. Int. 28, 85–95. With permission.)