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

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(10.28)

where

FIGURE 10.56 Optical images of worn DLC, ion

beam–sputtered Al

, and RF-sputtered SiC ﬁlms

deposited on Al

–TiC substrates and the bare sub-

strate, slid against a single-crystal sapphire ball

(diameter = 3 mm) for 2 h at a 10 mN normal load,

3.5 mm stroke length, 0.1 Hz frequency, 0.7 mm s

–1

average linear speed, and under ambient temperature

of 22 ± 1°C and humidity of 45 ± 5% RH. (From

Gupta, B.K. and Bhushan, B., 1995, Wear 190,

110–122. With permission.)

gk=

()

FIGURE 10.57 Pole tip recession as a function of sliding distance as measured

with an atomic force microscope for (a) uncoated and ion beam carbon-coated,

and (b) uncoated and cathodic arc carbon-coated Al

–TiC heads run against

MP tapes.

FIGURE 10.58 (a) Schematic diagram of the deﬂection measurement

of a bent beam using a nanoindenter and (b) load–deﬂection curve for

a warped composite-beam substrate.

where a is the radius of the membrane, t is its thickness, and the function g(k) depends on the membrane

and its geometry. If the geometry and elastic constants of the membrane are known, the tension can be

accurately evaluated. This technique can only be used to study tensile residual stresses since compressive

stresses buckle the membrane when the substrate is removed.

10.6.4 Microwear Measurements Using Modiﬁed Nanoindentation

Wu and Lee (1994) modiﬁed the nanoindenter/nanoscratch technique for microwear studies. A piezo-

electric pusher with a servo control was employed to implement a reciprocating horizontal motion at

the indenter tip for implementing a microwear test. Through a lock-in detection scheme, they measured

friction force.

10.7 Other Applications of Nanoindentation Techniques

Nanoindentation techniques have been used for measurement of time-dependent viscoelastic/plastic

properties (creep/relaxation), nanofracture toughness, and nanofatigue.

10.7.1 Time-Dependent Viscoelastic/Plastic Properties

Most materials including ceramics and even diamond are found to creep at temperatures well below half

their melting points, even at room temperature. Indentation creep and indentation load relaxation (ILR)

tests are used for measurement of the time-dependent ﬂow of materials. These offer an advantage of

being able to probe the deformation properties of a thin ﬁlm as a function of indentation depth and

location.

FIGURE 10.59 Indentation creep data for Si at different temperatures. (From Li, W.B. et al., 1991, Acta Metall.

Mater. 39, 3099–3110. With permission.)

−

()

12 1 ν

In the indentation creep test, the hardness indenter maintains its load over a period of time under

well-controlled conditions, and changes in indentation size are monitored (Westbrook, 1957; Mulhearn

and Tabor, 1960/61; Atkins et al., 1966; Walker, 1973; Hooper and Brookes, 1984; Li et al., 1991). The

analysis of creep is more complex than that of creep data obtained using a conventional technique because

of the shape of the tip, indentation stress acting on the sample decreases with time as the contact area

increases. Chu and Li (1977, 1979, 1980) developed an impression creep test on a macroscale which used

a circular tip with a ﬂat end. Log (hardness) decreases (linearly for most metals) with log (time at load).

Nanoindentors are also used for indentation creep studies (Li et al., 1991; Lucas and Oliver, 1992; Raman

and Berriche, 1990, 1992). Figure 10.59 shows the plot of log (indentation hardness) as a function of log

(time) for silicon at different temperatures. Indentation creep is inﬂuenced by a large number of variables

such as the plastic deformation properties of the material, diffusion constants, normal load of indenter,

duration of the indentation, and the test temperature. Li et al. (1991) reported that for temperatures

between 27°C and melting, the mechanism of dislocation glide plasticity dominates the indentation creep

process.

In a typical ILR test, the indenter is ﬁrst pushed into the sample at a ﬁxed displacement rate until a

predetermined load or displacement is achieved and the position of the indenter is then ﬁxed. The

material below the indenter is elastically supported and will continue to deform in a nonelastic manner,

thereby tending to push the indenter farther into the sample. Load relaxation is achieved by conversion

of elastic strain in the sample into inelastic strain in the sample. During the test, the load and position

of the indenter and the specimen are continuously monitored. Normally the indenter motion is held

constant and the changes in the load are monitored as a function of time. It is possible to obtain the

plastic indentation rate from the indentation load and total depth information during the relaxation run

(Hart and Solomon, 1973; Chu and Li, 1980; Nastasi et al., 1993). The resulting load relaxation data are

reported in the form of log (indentation pressure) as a function of log (plastic indentation strain rate)

(Hannula et al., 1985; LaFontaine et al., 1990a,b; Wu, 1991).

The indentation pressure is calculated by dividing the load by the projected area of the indenter. Once

the plastic indentation depth is known as a function of time, the projected area is determined experi-

mentally as described earlier. The plastic indentation strain rate — [(1/h)(dh/dt), where h is the current

FIGURE 10.60 Indentation load relaxation data of a (111) single-crystal aluminum sample at 25°C; l is the load,

is the plastic indentation depth at the beginning of the relaxation, and

x is the plastic indentation rate. (From

LaFontaine, W.R. et al., 1990, J. Mater. Res. 5, 2100–2106. With permission.)

indentation depth] — is calculated in the manner similar to that for bulk relaxation data. First, the plastic

indentation depth is calculated which is equal to the total depth minus its elastic depth based on load

vs. time data. The plastic indentation depth vs. time data is divided into segments. Over each segment,

the plastic indentation depth is assumed to vary linearly with time. At the midpoint of each segment the

slope of the plastic indentation depth curve is determined to obtain the plastic indentation rate. The

average plastic indentation depth is used along with the corresponding average load over the same time

interval to determine the indentation pressure (LaFontaine et al., 1990b). Figure 10.60 shows typical log

(indentation pressure) vs. log (normalized plastic indentation rate normalized with the depth at the

beginning of the relaxation experiment) ﬂow curves for a (111) single-crystal aluminum.

Another technique to measure strain rate sensitivity of submicron ﬁlms was developed by Mayo and

Nix (1988a,b). They developed two procedures. In the ﬁrst procedure, known as the constant rate of

loading test, individual indentations are performed at a prescribed loading rate that is varied from one

indentation to another by about a factor of 2. The values of the indentation pressure and strain rates

from tests performed at different loading rates were compared at a common indentation depth. In a

second related technique known as the loading rate change test, the loading rate is held constant at a

speciﬁed depth, but is suddenly changed to a new value and the subsequent changes in pressure and

strain rate are monitored.

Mayo et al. (1990, 1992) also developed yet another procedure for determining the strain rate sensitivity

of nanophase materials in order to conduct tests at the high loading rates required for very hard materials.

In this procedure, as soon as the indenter contacts the sample surface, the indenter loading rate is

instantaneously increased to a high value and this value is maintained until the indenter reaches a

prescribed displacement. The load is held constant at this point, and the load is monitored as a function

of time. The initial fast descent rate of the indenter produces a substantial amount of creep during a

constant-hold period. A range of descent rates are realized ranging from high values at the beginning of

the hold period to smaller values as the material stops deforming. These correspond to a range of strain

rates, and several stress–strain pairs can be obtained (Raman and Berriche, 1990, 1992).

The strain rate sensitivity of materials is measured in terms of stress exponent, n, which is deﬁned by

the equation,

Plastic indentation rate = A (indentation pressure)

(10.29)

where A and n are the constants. The stress exponent is found as a slope of a log–log plot of plastic

indentation rate (or strain rate) and indentation pressure. In the ILR test, the continuous change in the

contact area results in continuous changes in both plastic indentation rate and pressure. Thus, data from

a single indentation test, which may span several orders of magnitude in both strain rate and pressure,

are sufﬁcient to determine the stress exponent. Stress exponent can be used to deﬁne the superplasticity

FIGURE 10.61 Schematic of a standard specimen used for measurement of fracture toughness of materials in

tension.

of a material. The variations in stress exponent reﬂect the changes that may take place when the sub-

structure generated at high strain rate approaches equilibrium condition (Mayo and Nix, 1988a).

10.7.2 Nanofracture Toughness

Fracture toughness, K

of a material is a measure of its resistance to the propagation of cracks and the

ratio H/K

is an index of brittleness, where H is the hardness. Resistance to fracture is a strong function

of crack pattern. It is typically measured in a test in which a specimen containing a sharp crack of known

length, c, is subjected to an applied stress s, which is increased during test until the sample fractures

(Lawn, 1993), Figure 10.61. The magnitude of the stresses near the crack tip are determined by the stress

intensity factor K

which, in turn, depends on s, c, and the specimen geometry (A)

(10.30)

The term A provides correction for the thickness-to-width ratio of the material. Units of stress-intensity

factor are MPa . With more intense stress or with deeper cracks, the stress intensity becomes sufﬁcient

for the fracture to progress spontaneously. This threshold stress intensity is a property of the material

and is called the critical stress intensity factor, K

, or the fracture toughness of the material. Ceramics

generally have relatively low fracture toughness, consequently, it is an important property to be considered

for the selection of ceramics for industrial applications.

Indentation fracture toughness is a simple technique for determination of fracture toughness (Palm-

quist, 1957; Lawn and Wilshaw, 1975; Evans and Charles, 1976; Lawn and Evans, 1977; Lawn and Marshall,

1979; Lawn et al., 1980; Antis et al., 1981; Chantikul et al., 1981; Chiang et al., 1981, 1982; Henshall and

Brookes, 1985; Cheng et al., 1990; Cook and Pharr, 1990; Choi and Salem, 1993; de Boer et al., 1993;

Lawn, 1993; Pharr et al., 1993; Bhushan et al., 1996). The indentation cracking method is especially useful

for measurement of fracture toughness of thin ﬁlms or small volumes. This method is quite different

from conventional methods in that no special specimen geometry is required. Rather, the method relies

on the fact that when indented with a sharp indenter, most brittle materials form radial cracks and the

lengths of the surface traces of the radial cracks (for deﬁnition of crack length, see Figure 10.62) have

been found to correlate reasonably well with fracture toughness. By using simple empirical equations,

fracture toughness can then be determined from simple measurement of crack length.

In microindentation, cracks at relatively high indentation loads of several hundred grams are on the

order of 100 µm in length and can be measured optically. However, to measure toughness of very thin

ﬁlms or small volumes, much smaller indentations are required. However, a problem exists in extending

the method to nanoindentation regime in that there are well-deﬁned loads, called cracking thresholds,

below which indentation cracking does not occur in most brittle materials (Lankford, 1981). For a Vickers

indenter, cracking thresholds in most ceramics are about 25 g. Pharr et al. (1993), Li et al. (1997), and

Li and Bhushan (1998a) have found that the Berkovich indenter (a three-sided pyramid) with the same

depth-to-area ratio as a Vickers indenter (a four-sided pyramid) has a cracking of the thresholds very

similar to that of the Vickers indenter. They showed that cracking thresholds can be substantially reduced

FIGURE 10.62 Schematic of Vickers

indentation with radial cracks.

KA c

=πσ

by using sharp indenters, i.e., indenters with smaller included tip angles, such as a three-sided indenter

with the geometry of the corner of a cube. Studies using a three-sided indenter with the geometry of a

corner of a cube have revealed that cracking thresholds can be reduced to loads as small as 0.5 g, for

which indentations and crack lengths in most materials are submicron in dimension.

Based on fracture mechanics analysis, Lawn et al. (1980) developed a mathematical relationship

between fracture toughness and indentation crack length, given as

(10.31)

where W is the applied load and B is an empirical constant depending upon the geometry of the indenter

(also see Lawn, 1993; Pharr et al., 1993). Antis et al. (1981) conducted a study on a number of brittle

materials chosen to span a wide range of toughnesses. They indented with a Vickers indenter at several

loads and measured crack length optically. They found a value of B = 0.016 to give good correlation

between the toughness values measured from the crack length and the ones obtained using more con-

ventional methods. Mehrotra and Quinto (1985) used a Vickers indenter to measure fracture toughness

of the coatings. Pharr et al. (1993) tested several bulk ceramics listed in Table 10.7 using Vickers, Berkovich,

and cube corner indenters. They found that the fracture toughness equation can be applied for the data

TABLE 10.7 Typical Mechanical Properties

of Materials Tested by Pharr et al. (1993)

Material E(GPa) H(GPa) K

(MPa )

Soda lime glass 70 5.5 0.70

Fused quartz 72 8.9 0.58

(111) Silicon 168 9.3 0.70

(111) Sapphire 403 21.6 2.2

300 16.3 4.0

FIGURE 10.63 Indentations in fused quartz made with the cube corner indenter showing radial cracking at inden-

tation loads of (a) 12 g and (b) 0.45 g. (From Pharr, G.M. et al., 1993, in Mechanical Properties and Deformation

Behavior of Materials Having Ultra-Fine Microstructures, M. Nastasi et al., eds., pp. 449–461, Kluwer Academic,

Dordrecht. With permission.)

























obtained with all three indenters provided a different empirical constant was used for a cube corner

indenter. The constant B for the Vickers and Berkovich indenter was found to be about 0.016 and for

cube corner it was about 0.032. Pharr et al. (1993) further reported that predominant cracks formed

with Vickers or Berkovich indenters are cone cracks and with cube corner indenter, predominant cracks

were radial cracks, Figure 10.63. Bhushan et al. (1996) reported that cracks formed in microcrystalline

ceramic material (glass–ceramic) with Vickers indenter are radial cracks, Figure 10.64. Note that cracks

propagate in a zigzag manner. The interlocked crystal morphology is responsible for propagation in a

zigzag manner. By using Equation 10.31, the fracture toughness for this material is calculated and

presented in Table 10.4.

Chantikul et al. (1981) developed a relationship between fracture toughness and the indentation

fracture strength and the applied load given as,

(10.32)

where s

is the fracture strength after indentation at a given load and c is an empirical constant (0.59).

Advantage of this analysis is that the measurement of crack length is not required. Mecholsky et al. (1992)

used this analysis to calculate fracture toughness of diamond ﬁlms on silicon. They indented the ﬁlms

at various indentation loads of 3 to 9 kg and then fractured in four-point ﬂexure to measure fracture

strength. The data were then used to get fracture toughness. Equation 10.32 was found to hold for the

measurements. They reported a fracture toughness of 6 MPa and 12-µm-thick diamond ﬁlms on

silicon on the order of 2 MPa .

For fracture toughness measurement of ultra-thin ﬁlms ranging from 100 nm to few micrometers,

indentation or four-point ﬂexure techniques cannot be used. Because of shallow indentation depths

required in the indentation technique, it is difﬁcult to measure a radial crack length under even SEM.

Li et al. (1997, 1998a) developed a novel technique based on nanoindentation in which through-thickness

cracking in the coating is detected from a discontinuity observed in the load–displacement curve and

energy released during the cracking is obtained from the curve. Based on the energy released, fracture

mechanics analysis is then used to calculate fracture toughness. A cube corner is preferred because the

through-thickness cracking of hard ﬁlms can be accomplished at lower loads (Li and Bhushan, 1998a).

Load–displacement curves of indentations made at 30, 100, and 200 mN peak indentation loads

together with the SEM micrographs of indentations on the cathodic arc carbon coating on silicon are

shown in Figure 10.65. Steps are found in all loading curves as shown by arrows in Figure 10.65a. In the

30-mN SEM micrograph, in addition to several radial cracks, ringlike through-thickness cracking is

observed with small lips of material overhanging the edge of indentation. The step at about 23 mN in

FIGURE 10.64 Optical images of Vickers indentation made

on a glass–ceramic substrate at 500 g load.

cfI













()

the loading curves of indentations made at 30 and 100 mN peak indentation loads result from the ringlike

through-thickness cracking. The step at 175 mN in the loading curve of indentation made at 200 mN

peak indentation load is caused by spalling.

No steps were observed in the loading curve of indentation made at 20 mN peak indentation load

which suggests that the coating under the indenter was not separated instantaneously from the bulk

coating via the ringlike through-thickness cracking but occurred over a period of time. At 30 mN peak

indentation load, partial ringlike spalling is observed around the indenter and the other parts of the ﬁlm

bulged upward. This partial ringlike spalling is believed to result in the step in the loading curve. Absence

of long steps in the loading curve for uncoated silicon reported by Li et al. (1997, 1998a) suggest that

the steps in the loading curve on the coating result from the ﬁlm cracking. Based on their work, the

fracture process progresses in three stages: (1) ﬁrst ringlike through-thickness cracks form around the

indenter by high stresses in the contact area, (2) delamination and buckling occur around the contact

area at the ﬁlm/substrate interface by high lateral pressure, (3) second ringlike through-thickness cracks

and spalling are generated by high bending stresses at the edges of the buckled ﬁlm, see Figure 10.66. In

the ﬁrst stage, if the ﬁlm under the indenter is separated from the bulk ﬁlm via the ﬁrst ringlike through-

thickness cracking, a corresponding step will be present in the loading curve. If discontinuous cracks

form and the ﬁlm under the indenter is not separated from the remaining ﬁlm, no step appears in the

loading curve because the ﬁlm still supports the indenter and the indenter cannot suddenly advance into

the material. In the second stage, for the ﬁlms used in the present study, the advances of the indenter

FIGURE 10.65 (a) Load-displacement curves of indentations made at 30, 100, and 200 mN peak indentation loads

using the cube corner indenter and (b) the SEM micrographs of indentations on the cathodic arc carbon ﬁlm on

silicon. Arrows indicate steps during loading portion of the load–displacement curve. (From Li, X. et al., 1997, Acta

Mater. 45, 4453–4461. With permission.)

during the radial cracking, delamination, and buckling are not big enough to form steps in the loading

curve because the ﬁlm around the indenter still supports the indenter, but generate discontinuities which

change the slope of the loading curve with increasing indentation loads. In the third stage, the stress

concentration at the end of the interfacial crack cannot be relaxed by the propagation of the interfacial

crack. With an increase in indentation depth, the height of the bulged ﬁlm increases. When the height

reaches a critical value, the bending stresses caused by the bulged ﬁlm around the indenter will result in

the second ringlike through-thickness crack formation and spalling at the edge of the buckled ﬁlm as

shown in Figure 10.66, which leads to a step in the loading curve. This is a single event and results in

the separation of the part of the ﬁlm around the indenter from the bulk ﬁlm via cracking through ﬁlms.

The step in the loading curve is totally from the ﬁlm cracking and not from the interfacial cracking or

the substrate cracking.

The area under the load–displacement curve is the work performed by the indenter during elastic–plas-

tic deformation of the ﬁlm/substrate system. The strain energy release in the ﬁrst/second ringlike cracking

and spalling can be calculated from the corresponding steps in the loading curve. Figure 10.67 shows a

FIGURE 10.66 Schematic of various stages in nanoindentation fracture for the ﬁlm/substrate system.

FIGURE 10.67 Schematic of a load–displacement curve, showing

a step during the loading cycle and associated energy release.