Spherical and Rounded Cone Nano Indenters

SPHERICAL AND ROUNDED CONE NANO INDENTERS

Bernard Mesa Micro Star Technologies

In the present field of nano indentation, spherical tipped indenters made of diamond or sapphire are desirable in numerous applications. A truly spherical tipped cone, as in Fig. 1, is difficult to fabricate at nanometer scale. In practice, a rounded cone may have a geometry similar to Fig. 2. The tip is spherical at the apex but has a transition section which is neither part of the sphere nor the cone. If only a minimal indentation depth is sufficient, such a rounded cone provides acceptable spherical indentations. When deeper indentations are needed, a more precise definition of the area function is required.

Figure 1. Spherical tipped cone profile. Figure 2. Rounded tipped cone profile.

The analysis in the following pages offers a means to calculate the area function of rounded tip indenters with a single equation that is valid for both perfectly spherical and rounded cones.

First, the area function equations for the sphere, the cone and the spherical tipped cone are provided. Then the rounded cone equation and its application are described. The calculated area function values at regular indenting intervals are given in a spread sheet table.

Appendix A shows the equation derivation and Appendix B provides actual examples of rounded cone indenters analysis.

MST manufactures diamond and sapphire cone nano indenters with rounded tips at micrometer and nanometer dimensions. A TEM calibrated with a traceable standard is used to image and measure most of its nano indenters.

The graphic and calculated analysis of rounded conical indenters described here is available on request for purchased indenters. When ordering rounded cone indenters please supply the expected depth of indentation, in addition to the desired tip radius and cone angle.

1 Micro Star Technologies Inc. www.microstartech.com THEORETICAL SPHERE AND CONE AREA FUNCTIONS

Figure 2. Spherical tip cone

A cone indenter with a perfect spherical tip is shown on Fig. 2. The nomenclature used is as follows.

R Sphere radius h Indentation depth r Radius of projected circle at indentation depth α Cone half angle T Transition between cone and sphere C Sphere center P Indenter apex O Cone theoretical apex a Distance from P to O

An indenter area function f(h) allows the calculation of the projected area A of the circle of radius r at indentation depth h. Equation (2) is valid for all conical indenters which are assumed to have a circular symmetry.

r = f(h) (1) A = π r2 (2)

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Figure 3. Spherical Indenter Figure 4. Cone indenter

Simple spherical indenter equations,

r2 = R2 – (R‐h)2 (3)

r2 = 2Rh – h2 (4)

A = π (2Rh – h2) (5)

Simple conical indenter equations,

r = h tan α (6)

A = π h2 tan 2 α (7)

Figure 5. Spherical tip cone

hT Indentation depth at the transition T between sphere and cone

rT Radius of projected circle at transition depth 3 Micro Star Technologies Inc. www.microstartech.com

Equations for the spherical section, when h ≤ hT:

r2 = 2Rh – h2 (4)

A = π (2Rh – h2) (5)

At the transition, when h = hT :

Sin α = (R – hT ) / R (8)

hT = R (1 – Sin α) (9)

rT = R Cos α (10)

Equations for the conical section, when h ≥ hT:

Tan α = r / (a + h) (11)

r = Tan α (a + h) (12)

A = π [Tan α (a + h)]2 (13)

At the transition, when h = hT :

rT = Tan α ( a + hT ) (12)

Sin α = R / (R + a) (14)

a = R ( 1 / Sin α – 1 ) (15)

hT = R ( 1 – Sin α ) (9)

rT = Tan α [R ( 1 / Sin α – 1 ) + ( 1 – Sin α )] (16)

rT = R Tan α ( 1 / Sin α – Sin α ) (17)

2 rT = R ( 1 / Cos α – Sin α / Cos α ) (18)

2 rT = R [ 1– (1 – Cos α)] / Cos α (19)

rT = R Cos α (20)

Which is the same result for rT from the sphere:

rT = R Cos α (10)

4 Micro Star Technologies Inc. www.microstartech.com ACTUAL ROUNDED CONE NANO INDENTERS

Actual diamond nano indenters that approach a perfect spherical tip can only be made with considerable extra time and effort. There are two main reasons. One is the anisotropy of diamond which offers different abrasion rates at different crystal directions. This hampers circular symmetry.

The second reason is the very small dimensions required. At micro and nano meter scales the processes are not precise and repeatable enough to directly produce the desired geometries. These can only be approached by repeating the process in many small steps followed by measurements (usually with an electron microscope) until the required dimensions and tolerances are achieved.

Figs. 6, 7 and 8 show transmission electron microscope (TEM) images of three indenter examples. On the left is the plain TEM image. On the right some graphics have been superimposed. The larger circle indicates the sphere that would fit tangent to the cone sides. An spherical surface in this position would make the ideal spherical indenter.

The smaller circle is a closer approximation to the curve at the indenter tip. If the indentation depths are small in relation to the circle (less than 20% of the small circle radius), the indenter is acceptable as spherical. At deeper indentations the small circle radius would not be a good basis for accurate measurements.

Figure 6. TEM image of indenter VR13211

Figure 7. TEM image of indenter VR13212 5 Micro Star Technologies Inc. www.microstartech.com

Figure 8. TEM image of indenter VR13240

An investigation has been done on the non spherical geometry indenters to determine their area function general equation. There are two equations that provide the projected area as a function of the indentation depth. Equation (21) is applicable to the rounded section of the indenter and equation (12) to the conical section. Appendix A describes in detail the derivation of equation (21).

Radius of the projected circle at an indentation depth h, when h ≤ hT:

2 K 2 r = 2(RP + (RT ‐ RP) Kh / hT)h ‐ h (21)

Radius of the projected circle at an indentation depth h, when h ≤ hT:

r = Tan α (a + h) (12)

In both cases,

A = π r2 (2)

Fig. 9 shows the TEM image of indenter VR13211 with the measurement parameters required by equation (21). The two lines TO and T’O are the cone sides meeting at O. T is the transition where the tip’s curve starts. At point T a perpendicular line extended to the indenters center is the large circle radius or RT. The small circle radius RP is determined at a point where h is 2.5% of hT as explained on the Appendix. Following is the nomenclature for equation (21) and Fig. 9 not defined on page 2.

RP Apex circle radius

RT Transition point circle radius

hT Indentation depth at transition

rT Projected radius at transition depth K Adjustable h coefficient and exponent.

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Figure 9. TEM image with measuring parameters.

MST provides, on request, the analysis of a particular rounded cone indenter. For this purpose, the indenter’s TEM image is measured on a CAD program set to the microscope scale at which the image was taken. Fig. 10 shows the graphic analysis of indenter VR 13211 as an example.

Table 1 is the spread sheet where the parameters have been entered. Equations (21 ) and (12) are used to calculate a series of values for r and A at equally spaced h intervals. Notice that rT (at h = hT = 2.200) is calculated independently with equations (21) and (12). The results differ slightly because the 3 significant decimal precision may round the values in some of the calculations.

The “K factor” is a number used to adjust equation (21). K values fall between 1.00 and 0.70. The value of K is adjusted empirically to minimize the difference between rT calculated and rT measured. On

Table 1 rT calculated with equation (21) is 2.047, rT measured is 2.049 using K = 0.890.

In the appendix several different indenters are measured point by point and compared to the calculated values, showing the validity of equation (21). In the case of a perfect spherical indenter RT = RP = R and, equation (21) becomes equation (4),

2 K 2 K 2 2 r = 2(RP + (RT – RP)Kh / hT)h – h = 2(R + (R – R)Kh / hT)h – h = 2Rh – h (4)

7 Micro Star Technologies Inc. www.microstartech.com

Figure 10. Rounded cone graphic analysis.

ROUNDED CONE AREA FUNCTION

SERIAL NUMBER: VR13211 APEX RAD . RP : 0.487 CONE ANGLE 2α: 62.3

DATE: 5/26/2008 TRANSITION RAD. RT : 2.391 MEASURED rt: 2.049

INITIALS: BM TRASITION DEPTH hT : 2.200 APEX DIST. a: 1.195 FACTOR K : 0.894 ROUNDED SECTION CONICAL SECTION 2 K 2 2 2 r = 2(Rp + (Rt ‐Rp)Kh /hT)h ‐ h A = π r r = Tan α (a + h) A = π r INDENTATION DEPTH CALCULATED RADIUS CALCULATED AREA INDENTATION DEPTH CALCULATED RADIUS CALCULATED AREA h µ r µ A µ2 h µ r µ A µ2 0.100 0.327 0.336628 2.200 2.052 13.228805 0.200 0.478 0.716947 2.300 2.112 14.019593 0.300 0.600 1.132313 2.400 2.173 14.833337 0.400 0.709 1.578472 2.500 2.233 15.670034 0.500 0.808 2.052567 2.600 2.294 16.529687 0.600 0.901 2.552455 2.700 2.354 17.412294 0.700 0.990 3.076430 2.800 2.415 18.317855 0.800 1.074 3.623077 2.900 2.475 19.246372 0.900 1.155 4.191191 3.000 2.536 20.197843 1.000 1.233 4.779722 3.100 2.596 21.172269 1.100 1.310 5.387744 3.200 2.656 22.169649 1.200 1.384 6.014429 1.300 1.456 6.659027 1.400 1.527 7.320855 1.500 1.596 7.999286 1.600 1.664 8.693741 1.700 1.730 9.403681 1.800 1.796 10.128605 1.900 1.860 10.868042 2.000 1.923 11.621551 2.100 1.986 12.388713 2.200 2.047 13.169135

Table 1. Rounded cone projected area calculation. 8 Micro Star Technologies Inc. www.microstartech.com APPENDIX A

ROUNDED CONE AREA FUNCTION EQUATION DERIVATION

Consider the rounded cone indenter shown on Fig. A1. The rounded section curve starts at the transition point T. A circle of radius RT is drawn tangent to the cone at this point with the vertical distance to the apex P, hT . At a smaller distance from P, h3, another circle is drawn with radius R3. Similarly several more circles are drawn at h2, h1 and hP. The smallest circle conforms to the tip such that its radius RP is also valid at P when h = 0.

Figure A1. Circles tangential to rounded cone. A perfectly spherical projection radius r is given by equation (4), r2 = 2Rh – h2 (4) This equation is not directly applicable to a rounded cone like in Fig. A1 because R is not a constant. It is apparent that the value of the radii Rn changes with the value of h. As the distance h gets larger the radii of the tangent circles also get larger. So R must be a function of h, R = f(h) (A1) From the rounded cone geometry the following corresponding values are found,

R = RP when h = 0 (A2)

R = RT when h = hT (A3) A possible equation for R(h) could be,

R(h) = RP + Mh (A4)

RT = RP + MhT (A5)

M = (RT ‐ RP) / hT (A6)

R(h) = RP + (RT – RP)h / hT (A7)

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And substituting in equation (4),

2 2 r = 2(RP + (RT – RP)h / hT)h – h (A8) To test this equation, a careful measurement is made of the r values at equally spaced intervals of h on indenter VR13211 TEM image, as illustrated on Fig. A2 . For clarity, not all values are shown. All the measured values are inserted in Table A1.

Figure A2. r versus h measurements on indenter VR13211.

The calculated values of r and A on Table A1 are derived with equation (A8). Fig. A3 shows a plot comparison of the measured and calculated values of A. The divergence indicates that an equation to define R(h) for a rounded cone is not exactly linear as equation (A7). A modification was tried by adding a coefficient and exponent to h on equation (A9). Both were tested separately but it was found that their optimal values were similar. The same value, designated K, was chosen for exponent and coefficient,

K R(h) = RP + (RT – RP)Kh / hT (A9)

2 K 2 r = 2(RP + (RT – RP)Kh / hT)h ‐ h (21)

Table A2 uses equation (21) to calculate r and A from the measured values. Fig. 4A shows the plot. K was adjusted to the value 0.894 as shown. To find the adjusted optimal value for a particular rounded cone only the measured value of rT is needed. Therefore only the values shown on Fig. 10 are needed to generate the Table 1, on page 8.

10 Micro Star Technologies Inc. www.microstartech.com ROUNDED CONE AREA FUNCTION ‐ TEST

SERIAL NUMBER: VR13211 APEX RAD . RP : 0.487

DATE: 5/26/2008 TRANSITION RAD. RT : 2.391

INITIALS: BM TRASITION DEPTH hT : 2.200

ROUNDED SECTION 2 2 2 r = 2(Rt + (Rt ‐Rp)h/hT)h ‐ h A = π r INDENTATION DEPTH MEASURED RADIUS CALCULATED RADIUS MEASURED AREA CALCULATED AREA

h µ rm µ r µ Am µ2 A µ2 0.100 0.299 0.324 0.280862 0.328953 0.200 0.447 0.473 0.627718 0.703831 0.300 0.580 0.598 1.056832 1.124633 0.400 0.694 0.712 1.513104 1.591359 0.500 0.794 0.818 1.980573 2.104010 0.600 0.886 0.921 2.466138 2.662585 0.700 0.972 1.020 2.968126 3.267085 0.800 1.053 1.117 3.483426 3.917509 0.900 1.132 1.212 4.025712 4.613857 1.000 1.210 1.306 4.599606 5.356130 1.100 1.287 1.398 5.203637 6.144327 1.200 1.363 1.490 5.836353 6.978448 1.300 1.437 1.582 6.487291 7.858494 1.400 1.510 1.672 7.163145 8.784464 1.500 1.581 1.762 7.852602 9.756359 1.600 1.651 1.852 8.563356 10.774178 1.700 1.720 1.941 9.294088 11.837921 1.800 1.786 2.030 10.021040 12.947589 1.900 1.853 2.119 10.787001 14.103181 2.000 1.918 2.207 11.557052 15.304697 2.100 1.983 2.295 12.353650 16.552138 2.200 2.049 2.383 13.189666 17.845503 Table A1. Measured and calculated values of r and A using equation (A8), without K

Figure A3. Plot of measured and calculated values of A, without K

11 Micro Star Technologies Inc. www.microstartech.com ROUNDED CONE AREA FUNCTION ‐ TEST

SERIAL NUMBER: VR13211 APEX RAD . RP : 0.487

DATE: 5/26/2008 TRANSITION RAD. RT : 2.391

INITIALS: BM TRASITION DEPTH hT : 2.200 FACTOR K : 0.894 ROUNDED SECTION 2 K 2 2 r = 2(Rt + (Rt ‐Rp)Kh /hT)h ‐ h A = π r INDENTATION DEPTH MEASURED RADIUS CALCULATED RADIUS MEASURED AREA CALCULATED AREA

h µ rm µ r µ Am µ2 A µ2 0.100 0.299 0.327 0.280862 0.336628 0.200 0.447 0.478 0.627718 0.716947 0.300 0.580 0.600 1.056832 1.132313 0.400 0.694 0.709 1.513104 1.578472 0.500 0.794 0.808 1.980573 2.052567 0.600 0.886 0.901 2.466138 2.552455 0.700 0.972 0.990 2.968126 3.076430 0.800 1.053 1.074 3.483426 3.623077 0.900 1.132 1.155 4.025712 4.191191 1.000 1.210 1.233 4.599606 4.779722 1.100 1.287 1.310 5.203637 5.387744 1.200 1.363 1.384 5.836353 6.014429 1.300 1.437 1.456 6.487291 6.659027 1.400 1.510 1.527 7.163145 7.320855 1.500 1.581 1.596 7.852602 7.999286 1.600 1.651 1.664 8.563356 8.693741 1.700 1.720 1.730 9.294088 9.403681 1.800 1.786 1.796 10.021040 10.128605 1.900 1.853 1.860 10.787001 10.868042 2.000 1.918 1.923 11.557052 11.621551 2.100 1.983 1.986 12.353650 12.388713 2.200 2.049 2.047 13.189666 13.169135 Table A2. Measured and calculated values of r and A using equation (21), with K = 0.894

Figure A4. Plot of measured and calculated values of A, with K = 0.894

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SPHERICAL CONE TEST

To confirm the validity of equation (21), a theoretical spherical cone is drawn on Fig. A5. The dimensions are tested on Table A3. Fig. A6 plots the comparison of measured and calculated values of

A, which are identical. The value of K is irrelevant since (RT ‐ RP) = 0. Table A4 is the complete area function calculation for the spherical cone based on equations (21) and (22).

Figure A5. Spherical cone measurements

13 Micro Star Technologies Inc. www.microstartech.com ROUNDED CONE AREA FUNCTION ‐ TEST

SERIAL NUMBER: SPHRCON APEX RAD . RP : 1.744

DATE: 5/29/2008 TRANSITION RAD. RT : 1.744

INITIALS: BM TRASITION DEPTH hT : 1.007 FACTOR K : 1.000 ROUNDED SECTION 2 K 2 2 r = 2(Rt + (Rt ‐Rp)Kh /hT)h ‐ h A = π r INDENTATION DEPTH MEASURED RADIUS CALCULATED RADIUS MEASURED AREA CALCULATED AREA

h µ rm µ r µ Am µ2 A µ2 0.050 0.415 0.415 0.541061 0.540040 0.100 0.582 0.582 1.064133 1.064372 0.150 0.708 0.708 1.574767 1.572995 0.200 0.811 0.811 2.066291 2.065911 0.250 0.900 0.900 2.544690 2.543119 0.300 0.978 0.978 3.004883 3.004619 0.350 1.048 1.048 3.450424 3.450411 0.400 1.111 1.111 3.877734 3.880495 0.450 1.169 1.169 4.293178 4.294871 0.500 1.222 1.222 4.691290 4.693539 0.550 1.271 1.271 5.075058 5.076500 0.600 1.316 1.316 5.440786 5.443752 0.650 1.358 1.358 5.793612 5.795296 0.700 1.397 1.397 6.131160 6.131132 0.750 1.433 1.433 6.451226 6.451261 0.800 1.466 1.466 6.751773 6.755681 0.850 1.497 1.497 7.040337 7.044393 0.900 1.526 1.526 7.315751 7.317398 0.950 1.553 1.553 7.576921 7.574694 1.000 1.577 1.577 7.812918 7.816283 1.007 1.580 1.581 7.842672 7.848851

Table A3. Measured and calculated values of r and A for perfect spherical cone

Figure A6. Spherical cone plot of measured and calculated values 14 Micro Star Technologies Inc. www.microstartech.com

ROUNDED CONE AREA FUNCTION

SERIAL NUMBER: SPHRCON APEX RAD . RP : 1.744 CONE ANGLE 2α: 50.0

DATE: 5/29/2008 TRANSITION RAD. RT : 1.744 MEASURED rt: 1.580

INITIALS: BM TRASITION DEPTH hT : 1.007 APEX DIST. a: 2.382 FACTOR K : 1.000 ROUNDED SECTION CONICAL SECTION 2 K 2 2 2 r = 2(Rp + (Rt ‐Rp)Kh /hT)h ‐ h A = π r r = Tan α (a + h) A = π r INDENTATION DEPTH CALCULATED RADIUS CALCULATED AREA INDENTATION DEPTH CALCULATED RADIUS CALCULATED AREA h µ r µ A µ2 h µ r µ A µ2 0.050 0.415 0.540040 1.007 1.580 7.845816 0.100 0.582 1.064372 1.050 1.600 8.046176 0.150 0.708 1.572995 1.100 1.624 8.282329 0.200 0.811 2.065911 1.150 1.647 8.521899 0.250 0.900 2.543119 1.200 1.670 8.764883 0.300 0.978 3.004619 1.250 1.694 9.011283 0.350 1.048 3.450411 1.300 1.717 9.261099 0.400 1.111 3.880495 1.350 1.740 9.514331 0.450 1.169 4.294871 1.400 1.764 9.770978 0.500 1.222 4.693539 1.450 1.787 10.031040 0.550 1.271 5.076500 1.500 1.810 10.294518 0.600 1.316 5.443752 0.650 1.358 5.795296 0.700 1.397 6.131132 0.750 1.433 6.451261 0.800 1.466 6.755681 0.850 1.497 7.044393 0.900 1.526 7.317398 0.950 1.553 7.574694 1.000 1.577 7.816283 1.007 1.581 7.848851

Table A4. Spherical cone complete area function calculation.

15 Micro Star Technologies Inc. www.microstartech.com APPENDIX B AREA FUNCTION EQUATION TESTS

Following is the complete set of data for three indenters analyzed with equation (21) and graphically measured to test the equation’s validity.

ROUNDED CONE INDENTER VR13211 The data is already presented in the previous pages but is repeated here for easier access.

Figure B1. Original TEM image and basic graphics

Figure B2. r versus h measurements.

16 Micro Star Technologies Inc. www.microstartech.com ROUNDED CONE INDENTER VR13211

ROUNDED CONE AREA FUNCTION ‐ TEST

SERIAL NUMBER: VR13211 APEX RAD . RP : 0.487

DATE: 5/26/2008 TRANSITION RAD. RT : 2.391

h µ rm µ r µ Am µ2 A µ2 0.100 0.299 0.327 0.280862 0.336628 0.200 0.447 0.478 0.627718 0.716947 0.300 0.580 0.600 1.056832 1.132313 0.400 0.694 0.709 1.513104 1.578472 0.500 0.794 0.808 1.980573 2.052567 0.600 0.886 0.901 2.466138 2.552455 0.700 0.972 0.990 2.968126 3.076430 0.800 1.053 1.074 3.483426 3.623077 0.900 1.132 1.155 4.025712 4.191191 1.000 1.210 1.233 4.599606 4.779722 1.100 1.287 1.310 5.203637 5.387744 1.200 1.363 1.384 5.836353 6.014429 1.300 1.437 1.456 6.487291 6.659027 1.400 1.510 1.527 7.163145 7.320855 1.500 1.581 1.596 7.852602 7.999286 1.600 1.651 1.664 8.563356 8.693741 1.700 1.720 1.730 9.294088 9.403681 1.800 1.786 1.796 10.021040 10.128605 1.900 1.853 1.860 10.787001 10.868042 2.000 1.918 1.923 11.557052 11.621551 2.100 1.983 1.986 12.353650 12.388713 2.200 2.049 2.047 13.189666 13.169135 Table B1. Measured and calculated values of r and A using equation (21), K = 0.894

17 Micro Star Technologies Inc. www.microstartech.com ROUNDED CONE INDENTER VR13211

Figure B3. Plot of measured and calculated values of A

ROUNDED CONE AREA FUNCTION

SERIAL NUMBER: VR13211 APEX RAD . RP : 0.487 CONE ANGLE 2α: 62.3

DATE: 5/26/2008 TRANSITION RAD. RT : 2.391 MEASURED rt: 2.049

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ROUNDED CONE INDENTER VR13212

Figure B4. Original TEM image and basic graphics

Figure B5. r versus h measurements.

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ROUNDED CONE INDENTER VR13212

ROUNDED CONE AREA FUNCTION ‐ TEST

SERIAL NUMBER: VR13212 APEX RAD . RP : 0.325

DATE: 5/26/2008 TRANSITION RAD. RT : 2.201

INITIALS: BM TRASITION DEPTH hT : 2.600 FACTOR K : 0.945 ROUNDED SECTION 2 K 2 2 r = 2(Rt + (Rt ‐Rp)Kh /hT)h ‐ h A = π r INDENTATION DEPTH MEASURED RADIUS CALCULATED RADIUS MEASURED AREA CALCULATED AREA

h µ rm µ r µ Am µ2 A µ2 0.100 0.218 0.265 0.149301 0.221414 0.200 0.354 0.387 0.393692 0.469973 0.300 0.463 0.486 0.673460 0.741843 0.400 0.558 0.574 0.978179 1.035064 0.500 0.641 0.655 1.290821 1.348293 0.600 0.717 0.731 1.615058 1.680511 0.700 0.788 0.804 1.950753 2.030897 0.800 0.856 0.874 2.301958 2.398764 0.900 0.923 0.941 2.676414 2.783523 1.000 0.989 1.007 3.072858 3.184657 1.100 1.055 1.071 3.496671 3.601707 1.200 1.119 1.133 3.933780 4.034263 1.300 1.181 1.194 4.381771 4.481948 1.400 1.241 1.255 4.838307 4.944421 1.500 1.300 1.314 5.309292 5.421364 1.600 1.357 1.372 5.785083 5.912486 1.700 1.414 1.429 6.281288 6.417513 1.800 1.470 1.486 6.788668 6.936189 1.900 1.526 1.542 7.315751 7.468274 2.000 1.582 1.597 7.862539 8.013542 2.100 1.637 1.652 8.418743 8.571779 2.200 1.693 1.706 9.004587 9.142781 2.300 1.749 1.760 9.610135 9.726355 2.400 1.805 1.813 10.235387 10.322317 2.500 1.861 1.865 10.880344 10.930490 2.600 1.917 1.917 11.545004 11.550708

Table B3. Measured and calculated values of r and A using equation (21), K = 0.945

20 Micro Star Technologies Inc. www.microstartech.com ROUNDED CONE INDENTER VR13212

Figure B6. Plot of measured and calculated values of A

ROUNDED CONE AREA FUNCTION

SERIAL NUMBER: VR13212 APEX RAD . RP : 0.325 CONE ANGLE 2α: 54.6

DATE: 5/26/2008 TRANSITION RAD. RT : 2.201 MEASURED rt: 1.917

INITIALS: BM TRASITION DEPTH hT : 2.600 APEX DIST. a: 1.131 FACTOR K : 0.945 ROUNDED SECTION CONICAL SECTION 2 K 2 2 2 r = 2(Rp + (Rt ‐Rp)Kh /hT)h ‐ h A = π r r = Tan α (a + h) A = π r INDENTATION DEPTH CALCULATED RADIUS CALCULATED AREA INDENTATION DEPTH CALCULATED RADIUS CALCULATED AREA h µ r µ A µ2 h µ r µ A µ2 0.100 0.265 0.221414 2.600 1.926 11.650186 0.200 0.387 0.469973 2.700 1.977 12.283063 0.300 0.486 0.741843 2.800 2.029 12.932678 0.400 0.574 1.035064 2.900 2.081 13.599031 0.500 0.655 1.348293 3.000 2.132 14.282122 0.600 0.731 1.680511 3.100 2.184 14.981952 0.700 0.804 2.030897 3.200 2.235 15.698521 0.800 0.874 2.398764 3.300 2.287 16.431827 0.900 0.941 2.783523 3.400 2.339 17.181872 1.000 1.007 3.184657 3.500 2.390 17.948656 1.100 1.071 3.601707 3.600 2.442 18.732177 1.200 1.133 4.034263 1.300 1.194 4.481948 1.400 1.255 4.944421 1.500 1.314 5.421364 1.600 1.372 5.912486 1.700 1.429 6.417513 1.800 1.486 6.936189 1.900 1.542 7.468274 2.000 1.597 8.013542 2.100 1.652 8.571779 2.200 1.706 9.142781 2.300 1.760 9.726355 2.400 1.813 10.322317 2.500 1.865 10.930490 2.600 1.917 11.550708

Table B4. Rounded cone indenter projected area calculation.

21 Micro Star Technologies Inc. www.microstartech.com ROUNDED CONE INDENTER VR13240

Figure B7. Original TEM image and basic graphics

Figure B8. r versus h measurements.

22 Micro Star Technologies Inc. www.microstartech.com ROUNDED CONE INDENTER VR13240

ROUNDED CONE AREA FUNCTION ‐ TEST

SERIAL NUMBER: VR13240 APEX RAD . RP : 0.875

DATE: 5/26/2008 TRANSITION RAD. RT : 1.596

INITIALS: BM TRASITION DEPTH hT : 0.850 FACTOR K : 0.765 ROUNDED SECTION 2 K 2 2 r = 2(Rt + (Rt ‐Rp)Kh /hT)h ‐ h A = π r INDENTATION DEPTH MEASURED RADIUS CALCULATED RADIUS MEASURED AREA CALCULATED AREA

h µ rm µ r µ Am µ2 A µ2 0.050 0.283 0.303 0.251607 0.287644 0.100 0.412 0.433 0.533267 0.588405 0.150 0.514 0.534 0.829996 0.897253 0.200 0.604 0.621 1.146103 1.211948 0.250 0.689 0.698 1.491380 1.531054 0.300 0.766 0.768 1.843348 1.853534 0.350 0.833 0.833 2.179917 2.178582 0.400 0.892 0.893 2.499652 2.505544 0.450 0.947 0.950 2.817409 2.833874 0.500 1.000 1.003 3.141593 3.163104 0.550 1.051 1.054 3.470206 3.492828 0.600 1.101 1.103 3.808242 3.822686 0.650 1.149 1.150 4.147534 4.152356 0.700 1.194 1.194 4.478768 4.481550 0.750 1.237 1.237 4.807168 4.810004 0.800 1.278 1.279 5.131113 5.137476 0.850 1.318 1.319 5.457336 5.463746

Table B5. Measured and calculated values of r and A using equation (21), K = 0.765

23 Micro Star Technologies Inc. www.microstartech.com ROUNDED CONE INDENTER VR13240

Figure B9. Plot of measured and calculated values of A

ROUNDED CONE AREA FUNCTION

SERIAL NUMBER: VR13240 APEX RAD . RP : 0.875 CONE ANGLE 2α: 68.6

DATE: 5/28/2008 TRANSITION RAD. RT : 1.596 MEASURED rt: 1.319

INITIALS: BM TRASITION DEPTH hT : 0.850 APEX DIST. a: 1.086 FACTOR K : 0.765 ROUNDED SECTION CONICAL SECTION 2 K 2 2 2 r = 2(Rp + (Rt ‐Rp)Kh /hT)h ‐ h A = π r r = Tan α (a + h) A = π r INDENTATION DEPTH CALCULATED RADIUS CALCULATED AREA INDENTATION DEPTH CALCULATED RADIUS CALCULATED AREA h µ r µ A µ2 h µ r µ A µ2 0.050 0.303 0.287644 0.850 1.321 5.479301 0.100 0.433 0.588405 0.900 1.355 5.765977 0.150 0.534 0.897253 0.950 1.389 6.059963 0.200 0.621 1.211948 1.000 1.423 6.361258 0.250 0.698 1.531054 1.050 1.457 6.669863 0.300 0.768 1.853534 1.100 1.491 6.985777 0.350 0.833 2.178582 1.150 1.525 7.309001 0.400 0.893 2.505544 1.200 1.559 7.639534 0.450 0.950 2.833874 1.250 1.594 7.977377 0.500 1.003 3.163104 1.300 1.628 8.322529 0.550 1.054 3.492828 1.350 1.662 8.674990 0.600 1.103 3.822686 0.650 1.150 4.152356 0.700 1.194 4.481550 0.750 1.237 4.810004 0.800 1.279 5.137476 0.850 1.319 5.463746

Table B6. Rounded cone indenter projected area calculation.

24 Micro Star Technologies Inc. www.microstartech.com