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

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approach at a constant loading rate, the indenter is ﬁrst loaded and unloaded typically three times in

succession at a constant loading rate or displacement rate with each of the unloadings terminated at

about 10 to 20% of the peak loading or displacement, respectively, to assure that contact is maintained

between the specimen and the indenter. In a typical indentation experiment, it is usual to have two hold

segments, the ﬁrst one at the end of unloading to 10 to 20% after multiple loading/unloading cycles and

the second one at the peak load just before ﬁnal unloading. The reason for performing multiple loadings

and unloadings is to examine the reversibility of the deformation (hysteresis) and thereby making sure

that the unloading data used for calculation of the modulus of elasticity are mostly elastic. In some

materials, there may be a signiﬁcant amount of creep during the ﬁrst unloading; thus, displacement

recovered may not be entirely elastic, and because of this, the use of ﬁrst unloading curves in the analysis

of elastic properties can sometimes lead to inaccuracies. One way to minimize nonelastic effects is to

include peak load hold periods in the loading sequence to allow time-dependent plastic effects to

diminish. In addition, after multiple loadings, the load is held constant for a period of typically 100 s at

10 to 20% of the peak value while the displacement is carefully monitored to establish the rate of

displacement produced by thermal expansion in the system. To account for thermal drift, the rate of

displacement is measured during the last 80 s of the hold period, and the displacement data are corrected

by assuming that this drift rate is constant throughout the test. Following this hold period, the specimen

is loaded for a ﬁnal time, with another 100 s hold period inserted at peak load to allow any ﬁnal time-

dependent plastic effects to diminish, and the specimen is fully unloaded. The ﬁnal unloading curve is

used for calculations of modulus of elasticity.

For an indentation experiment, the sample is placed on the mounting block. An appropriate region

is selected by observing through the optical microscope. In a typical experiment, the tip of the indenter

is moved toward the surface of the sample by gradually increasing the load on the indenter shaft. With

a constant loading rate of typically 1 µN/s, the tip of the indenter travels downward at a velocity (approach

rate) of about 10 nm/s. When the tip contacts the surface, its velocity drops below 1 nm/s, and the

FIGURE 10.13 Schematics of the Nanoindenter II, (a) showing the major components — the indenter head, an

optical microscope, and an X–Y–Z motorized precision table, (b) showing the details of indenter head and controls

(microscope, which is directly behind the indenter, and the massive U bar are not shown for clarity), (c) showing

the three-plate capacitor and indenter column that form the displacement-sensing system (not to scale). For the sake

of clarity, only the left halves of the 50-mm-diameter, circular outer plates of the capacitor are shown. (Courtesy of

Nano Instruments, Knoxville, TN.)

computer records the displacement of the tip. This is the point at which the indentation experiment

begins. The load is incremented in order to maintain a constant loading rate of about 10% of peak load/s

or to maintain a constant displacement rate of 10% of peak displacement/s. The drift rate of the

nanoindenter is about 0.01 nm/s; therefore, a loading duration of 10 s minimizes the measurement errors.

Loading is followed by unloading and multiple loading/unloading and hold cycles.

Now we describe various steps in some detail based on Anonymous (1991). The ﬁrst step is always

the “Approach segment” in which the tip makes contact with the sample surface. The purpose of the

approach segment is to determine accurately the “zero” of the indenter tip, that is, the values of the load

FIGURE 10.14 (a) Schematic, (b) photograph of the shape of a Berkovich indenter, and (c) indent impression.

and displacement at the point where the tip just touches the sample surface. Based on Anonymous (1991),

these values are obtained in the following manner. The computer moves the sample from the microscope

to a point below the indenter such that the position selected for the initial indent is offset from the

indenter by a user-selected distance and angle (the default values are 50 µm and 180°). With the center

plate of the capacitor (to which the indenter is attached) on its bottom stop, the table moves upward at

a relatively high rate of speed until the indenter contacts the surface and makes the initial surface-ﬁnding

indent. When contact occurs, the indenter is pushed upward, tripping the Z-motor interrupts and

stopping the Z-motor of the table. The table is then moved downward at slow speed for 15 s before being

moved in the X, Y plane until the point on the sample halfway between the initial surface-ﬁnding indent

and the location of the ﬁrst indent is under the indenter. The table is now raised once more, but at slow

speed until contact with the specimen is made once more. This second contact gives the best estimate

of the surface elevation that can be obtained by moving the table alone.

FIGURE 10.15 Different segments of a typical constant-loading indentation experiment.

TABLE 10.2 Examples of Two Typical Indentation Experiments

Segments

Rate Maximum Value

(a) Constant Loading Experiment

Approach 10 nm/s —

Loading at constant loading rate 12 mN/s 120 mN

Unloading at constant unloading rate 12 mN/s 10–20% of 120 mN

Multiple loading/unloading cycles

(typically three times)

•

Hold for 100 s Data rate 1/s 100 points

Loading at constant loading rate 12 mN/s 120 mN

Hold for 100 s Data rate 1/s 100 points

Unloading at constant unloading rate 12 mN/s 0 mN

(b) Constant Displacement Experiment

Approach 10 nm/s —

Loading at constant displacement rate 20 nm/s 200 nm

Unloading at constant displacement rate 20 nm/s 10–20% of 200 nm

(typically three times)

•

Hold for 100 s 1 point/s 100 points

Loading at constant displacement rate 10 nm/s 200 nm

Hold for 100 s 1 point/s 100 points

Unloading at constant displacement rate 10 nm/s 0 nm

In routine experiments the number of segments does not exceed more than

12, but if needed this number could be high.

After this second surface-ﬁnding indent is made, the indenter is left in contact with the surface under

a very small load with the displacement-sensing capacitor near the center of its travel. At this point the

system is allowed to monitor changes in indenter displacement under constant load, and when the drift

rate becomes smaller than the user-prescribed maximum (usually 0.05 nm/s), the displacement of the

indenter is recorded, establishing an initial estimate of the elevation of the sample surface.

The indenter is then raised to near the top of its travel using the coil/magnet assembly (the elevation

of the table remains ﬁxed for the rest of the experiment), and the table is moved so that the chosen location

for the ﬁrst indent of the speciﬁed shape is under the indenter. The indenter is now lowered toward the

surface at a rate of several hundred nanometers per second until the “surface search distance” is reached.

The surface search distance is a user-speciﬁed distance (usual 1000 to 2000 nm) above the estimated

elevation of the sample surface. At this point, the rate of approach to the surface is decreased to approx-

imately 10 nm/s, and the load–displacement values that are constantly recorded are used to calculate the

stiffness of the system as reﬂected initially in the stiffness of the very ﬂexible leaf springs that support the

indenter shaft. When the indenter ﬁnally reaches the surface, a large increase in stiffness is sensed, and

when the stiffness increases by a factor of 4, the approach phase of the indentation process is complete.

The computer now discards all but the last 50 sets of load–displacement data taken during the

approach. A plot of load vs. displacement for these data reﬂects the point of contact of the indenter with

the sample surface in terms of a very sharp change in slope of the load–displacement plot (see

Figure 10.16). For an approach rate of 10 nm/s and factor of four increase in stiffness, experience has

shown that surface contact is made at the 13th or 14th data point from the end of the 50-data-point set.

The zero points for both load and displacement are then taken as the averages of the loads and displace-

ments of 12th and 13th data sets from the end of the approach data. For many materials this procedure

identiﬁes the sample surface to within 0.1 to 0.2 nm. However, for very soft materials such as many

polymers or for other approach rates and stiffness-factor increases, the user may ﬁnd it advisable to plot

the approach segment data and, if necessary, change the algorithm used to deﬁne the precise point of

contact with the sample surface.

Once surface contact is established, the other segments of the indentation process are carried out as

prescribed in the programmed indentation experiment. The ﬁnal segment always involves load removal.

When the voltage on the indenter coil passes the displacement voltage at which the surface was detected

in the approach portion of the cycle, the current through the coil is ﬁxed while the raw data are recorded

on the hard disk, and plotted on the computer monitor. The indenter is then raised well away from the

surface in preparation for moving the sample to the position of the next indent. For subsequent indents

in a given series of indents, the initial estimate of surface position used is that found in making the

previous indent.

For each indentation step, load voltages, displacement (penetration depth or indentation depth)

voltages, and real time are recorded in separate ﬁles. These raw voltage data are converted to load vs.

displacement data by using load and displacement calibration constants. From the displacement data,

the contact depth is calculated for calculations of the hardness. The slope of the unloading curve is used

to calculate the modulus of elasticity.

10.2.3.5 Acoustic Emission Measurements during Indentation

AE measurement is a very sensitive technique to monitor cracking of the surfaces and subsurfaces. The

nucleation and growth of cracks result in a sudden release of energy within a solid; then some of the

energy is dissipated in the form of elastic waves. These waves are generated by sudden changes in stress

and in displacement that accompany the deformation. If the release of energy is sufﬁciently large and

rapid, then elastic waves in the ultrasonic frequency regime (AE) will be generated and these can be

detected using PZTs via expansion and compression of the PZT crystals (Yeack-Scranton, 1986; Scruby,

1987; Bhushan, 1996).

Weihs et al. (1992) used an AE sensor to detect cracking during indentation tests using the nanoin-

denter. The energy dissipated during crack growth can be estimated by the rise time of the AE signal.

They mounted a commercial transducer with W-impregnated epoxy backing for damping underneath

the sample as shown in Figure 10.17. The transducer converts the AE signal into voltage that is ampliﬁed

by oscilloscopes and used for continuous display of the AE signal. Any correlation between the AE signal

and the load–displacement curves can be observed. (Also see Wu et al., 1990b and Wu, 1991.)

10.2.3.6 Nanoscratch and Tangential Force Measurements

Several micro- and nanoscratch testers are commercially available, such as the Taber shear/scratch tester

model 502 with a no. 139-58 diamond cutting tool (manufactured by Teledyne Taber, North Tonawanda,

NY) for thick ﬁlms; Revetest automatic scratch tester (manufactured by Centre Suisse d’ Electronique et

de Microtechnique S.A, CH-2007, Neuchatel, Switzerland) for thin ﬁlms (Perry, 1981, 1983; Steinmann

et al., 1987; Sekler et al., 1988), and nanoindenter for ultrathin ﬁlms (Wu et al., 1989, 1990b; Wu, 1990,

1991; Anonymous, 1991; Bhushan et al., 1995, 1996, 1997; Bhushan and Gupta, 1995; Gupta and Bhushan,

1995a,b; Patton and Bhushan, 1996; Bhushan and Li, 1997; Li and Bhushan, 1998b,c).

FIGURE 10.16 A plot of load vs. displacement (expressed in volts) for a typical approach segment of an indentation

obtained using the lowest load range and with an approach rate of about 10 nm/s. A factor of four increase in stiffness was

used as the criterion for terminating this segment. The very sharp “knee” at the right end of the plot indicates surface

contact. Data near the knee of the curve are shown on an expanded scale in the inset. Note that the displacement voltage

changes by only 0.1 mV (corresponding to a displacement of about 1.3 nm) between the surface contact and the end of

the approach segment. Thus, it is obvious that the position of the surface (i.e., displacement voltage corresponding to the

surface contact) can be speciﬁed to about ±0.1 nm. (Courtesy of Nano Instruments, Knoxville, TN.)

We describe the nanoscratch and tangential force option which allows making of the scratches of

various lengths at programmable loads. Tangential (friction) forces can also be measured simultaneously

(Anonymous, 1991). The additional hardware for the tangential force option includes a set of proximity

(capacitance) probes for measurement of lateral displacement or force in the two lateral directions along

x and y, and a special “scratch collar” which mounts around the indenter shaft with hardness indenter,

Figure 10.18. The scratch collar consists of an aluminum block, mounted around the indenter shaft, with

four prongs descending from its base. Two of these prongs hold the proximity probes and the set screws

set them in place, while the other two prongs hold position screws (and corresponding set screws). The

position screws serve a dual purpose; they are used to limit the physical deﬂection of the indenter shaft,

and they are used to lock the indenter shaft in place during tip change operations. A scratch block is

mounted on the end of the indenter shaft, in line with the proximity probes and the positioning screws.

Finally, the scratch tip itself is mounted on the end of the indenter shaft, covering the scratch tip. The

scratch tip is attached to the scratch block with two Allen head screws. The scratch tip can be a Berkovich

indenter or a conventional conical diamond tip with a tip radius of about 1 to 5 µm and an included

angle of 60 to 90° (typically 1 µm of tip radius with 60°

of included angle, Wu et al., 1990a,b; Wu, 1991).

A larger included angle of 90° may be desirable for a more durable tip. The tip radius should not be very

small as it will get blunt readily.

FIGURE 10.17 (a) Schematic of nanoindenter with an AE transducer, (b) schematic of commercial transducer with

W-impregnated epoxy backing for damping. (From Weihs, T.P. et al., 1992, in Thin Films: Stresses and Mechanical

Properties III, Symp. Proc., W.E. Nix, et al., eds., Vol. 239, pp. 361–370, Materials Research Society, Pittsburgh. With

permission.)

During scratching a load is applied up to a speciﬁed indentation load or up to a speciﬁed indentation

depth, and the lateral motion of the sample is measured. In addition, of course, load and indentation

depth are monitored. Scratches can be made either at the constant load or at ramp-up load. Measurement

of lateral force allows the calculations of the coefﬁcient of friction during scratching. The resolution of

the capacitance proves to measure tangential load is about 50 µN; therefore, a minimum load of about

0.5 mN can be measured or a minimum normal load of about 5 mN should be used for a sample with

coefﬁcient of friction of about 0.1. Microscopy of the scratch produced at ramp-up load allows the

measurement of critical load required to break up of the ﬁlm (if any) and scratch width and general

observations of scratch morphology.

Additional parameters that are used to control the scratch are scratch length (µm), draw acceleration

(µm/s

), and draw velocity (µm/s). The latter parameters control the speed with which the scratch is

performed. The default values of 10 µm/s

and 10 µm/s provide safe rates for performing the scratch.

Draw velocity is limited by the maximum rate of data acquisition (during a scratch the maximum rate

is approximately 2/s) and the length of the desired scratch. Thus, a scratch with a desired 20 points over

1 mm must have a draw velocity no greater than 100 µm/s.

The lateral deﬂection calibration is performed with a calibrated cantilever beam (of known stiffness)

mounted on the specimen tray. Lateral force applied to the indenter shaft with the beam can be used to

determine response of the indenter shaft in micrometers per volt of probe or newton per volt (from

known lateral stiffness of the indenter shaft). Calibration for the cross talk of the probe (resulting from

the probe surface not being parallel to the axis of motion in the vertical direction) also needs to be

performed. For this calibration, the scratch tip is moved up and down and any output of the proximity

probe in volts of vertical motion per volt of proximity probe is measured.

10.3 Analysis of Indentation Data

An indentation curve is the relationship between load W and displacement (or indentation depth or

penetration depth) h, which is continuously monitored and recorded during indentation. Stress–strain

curves, typical indentation curves, the deformed surfaces after tip removal, and residual impressions of

indentation for ideal elastic, rigid–perfectly plastic and elastic–perfectly plastic and real elastic–plastic

solids are shown in Figure 10.19. For an elastic solid, the sample deforms elastically according to Young’s

modulus, and the deformation is recovered during unloading. As a result, there is no impression of the

indentation after unloading. For a rigid–perfectly plastic solid, no deformation occurs until yield stress

is reached, when plastic ﬂow takes place. There is no recovery during unloading and the impression

FIGURE 10.18 Schematic of the tangential force option hardware (not to scale and the front and rear prongs not

shown). (Courtesy of Nano Instruments, Knoxville, TN.)

remains unchanged. In the case of elastic–plastic solid, it deforms elastically according to Young’s modulus

and then it deforms plastically. The elastic deformation is recovered during unloading. In the case of an

elastic–perfectly plastic solid, there is no work hardening.

All engineering surfaces follow real elastic–plastic deformation behavior with work hardening

(Johnson, 1985). The deformation pattern of a real elastic–plastic sample during and after indentation

is shown schematically in Figure 10.20. In this ﬁgure we have deﬁned the contact depth (h

) as the depth

of indenter in contact with the sample under load. The depth measured during the indentation (h)

includes the depression of the sample around the indentation in addition to the contact depth. The

depression of the sample around the indentation (h

= h – h

) is caused by elastic displacements and must

be subtracted from the data to obtain the actual depth of indentation or actual hardness. At peak load,

the load and displacement are W

max

and h

max

, respectively, and the radius of the contact circle is a. Upon

unloading, the elastic displacements in the contact region are recovered and, when the indenter is fully

withdrawn, the ﬁnal depth of the residual hardness impression is h

Schematic of a load–displacement curve is shown in Figure 10.21. Based on the work of Sneddon

(1965) to predict the deﬂection of the surface at the contact perimeter for a conical indenter and a

paraboloid of revolution, Oliver and Pharr (1992) developed an expression for h

at the maximum load

(required for hardness calculation) from h

max

FIGURE 10.19 Schematics of stress–strain curves, typical indentation curves, deformed surfaces after tip removal,

and residual impressions of indentation, for ideal elastic, rigid–perfectly plastic, elastic–perfectly plastic (ideal), and

real elastic–plastic solids.

(10.3a)

where e = 0.72 for the conical indenter, Œ = 0.75 for the paraboloid of revolution, and Œ= 1 for the

ﬂat punch; S

max

is the stiffness (= 1/compliance) equal to the slope of unloading curve (dW/dh) at the

maximum load. Oliver and Pharr assumed that behavior of Berkovich indenter is similar to that of conical

indenter, since cross-sectional areas of both types of indenters varies as the square of the contact depth

and their geometries are singular at the tip. Therefore, for Berkovich indenter, e ~ 0.72. Thus, h

is slightly

larger than plastic indentation depth (h

), which is given by

(10.3b)

We note that Doerner and Nix (1986) had underestimated h

by assuming that h

= h

. Based on the

ﬁnite element analysis of the indentation process, Laursen and Simo (1992) showed that h

cannot be

assumed equal to h

for indenters which do not have ﬂat punch geometry.

FIGURE 10.20 Schematic representation of the indenting process illustrating the depression of the sample around

the indentation and the decrease in indentation depth upon unloading. (From Oliver, W.C. and Pharr, G.M., 1992,

J. Mater. Res. 7, 1564–1583. With permission.)

FIGURE 10.21 Schematic of load–displacement curve.

hh W S

=−

max max max

hh W S

=−

max max max

For a Vickers indenter with ideal pyramidal geometry (ideally sharp tip), projected contact-area-to-

depth relationship is given as (Doerner and Nix, 1986; Bhushan, 1996)

(10.4a)

Since the area-to-depth relationship is equivalent for both typical Berkovich and Vickers pyramids,

Equation 10.4 holds for the Berkovich indenter as well. Although we have derived a slightly different

expression for A(h) presented in Equation 10.2 for the assumed Berkovich indenter geometry, we use

Equation 10.4 for A(h) in this chapter, as this relationship is most commonly used in the analysis of the

indentation hardness data.

The indenter tip is generally rounded so that ideal geometry is not maintained near the tip

(Figure 10.22). To study the effect of tip radius on the elastic–plastic deformation (load vs. displacement

curve), Shih et al. (1991) modeled the blunt-tip geometry by a spherical tip of various radii. They derived

a geometric relationship (assuming no elastic recovery) between the projected contact area of the indenter

to the actual contact depth. Figure 10.23a shows the measured contact area vs. indentation depth data

by Pethica et al. (1993) for nickel and by Doerner and Nix (1986) for annealed a-brass. From this ﬁgure,

it seems that a tip radius of 1 µm ﬁts the data best. If there is elastic recovery, the experimental data are

smaller than what they should be, and then the tip radius would be even larger than 1 µm. Shih et al.

(1991) used the ﬁnite-element method to simulate an indentation test. They showed that load–indenta-

tion depth data obtained using nanoindenter for nickel by Pethica et al. (1983) can be ﬁtted with a

simulated proﬁle for a tip radius of about 1 µm, Figure 10.23b.

As shown in Figure 10.20, the actual indentation depth, h

, produces a larger contact area than would

be expected for an indenter with an ideal shape. For the real indenter used in the measurements, the

nominal shape is characterized by an area function F(h

), which relates projected contact area of the

indenter to the contact depth (Equation 10.4a),

(10.4b)

The functional form must be established experimentally prior to the analysis (to be described later).

10.3.1 Hardness

Berkovich hardness HB (or H

) is deﬁned as the load divided by the projected area of the indentation.

It is the mean pressure that a material will support under load. From the indentation curve, we can

obtain hardness at the maximum load as

(10.5)

where W

max

is the maximum indentation load and A is the projected contact area at the peak load. The

contact area at the peak load is determined by the geometry of the indenter and the corresponding contact

depth h

using Equation 10.3a and 10.4b. A plot of hardness as a function of indentation depth for

polished single-crystal silicon (111), with and without tip shape calibration, is shown in Figure 10.24.

FIGURE 10.22 Schematic of an indenter tip with a

nonideal shape. The contact depth and the effective

depth are also shown.

= 24 5

AFh

()

HB W A=

max