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INTERACTIONS OF CONSTRAINED FLOW AND

SIZE SCALE ON MECHANICAL BEHAVIOR

W.W. Gerberich, W.M. Mook, M.J. Cordill, and D. Hallman

Chemical Engineering and Materials Science Department

University of Minnesota

421 Washington Ave SE

Minneapolis, MN 55455

wgerb@umn.edu

Summary

Coupled effects between constrained flow, increased strength as a function of decreased sample size, and resulting high stresses affect both modulus and fracture toughness. For submicron size crystalline spheres [1,2], boxes [3], and cubes [4], we have recently shown that dislocation by dislocation events can be followed using a combination of AFM/nanoindentation. This has led to at least three proposed strengthening mechanisms for hardening of small constrained volumes under compression[4]. With the increased stresses, this can produce increased moduli of elasticity in confined volumes small in three dimensions. With increased constrained plasticity this produces increased strength in volumes small in three, two or one dimensions.

Hardening mechanisms for the gigapascal strengths observed in small volumes are addressed. These are explored for Si and Ti nanospheres as well as Ni, Co, and Ni₈₀Fe₂₀ (permalloy) thin films in the 10-40 nm particle radius or thin film thickness[4,5] regime. Because of trapped dislocations between the upper and lower loading surfaces being typically diamond or sapphire, strengths approaching theoretical are often reached with deformation of only tens of nanometers or less. Of course, this is highly dependent upon the length scale of the confining structure. For oxide-covered nanospheres, the length scale can be taken as the volume to contact surface area as given elsewhere[5] by (2/3)r³/a² where r is the sphere radius and a is the contact radius at the upper and lower platen. Alternatively, from contact geometry this length scale is (2/3)r²/d. For deposited thin films, it was previously shown that a measured volume of plastic deformation to contact surface area was empirically given by ah²/a where h is the film thickness and a is the contact radius[6]. These two measures of length scale show that the flow strength of the nanospheres and the hardness of the thin films scale with d^1/2/r and d^1/4/h, respectively. Here, d is the displacement the sphere has been squeezed or the penetration depth of the indenter into the film. It is significant that both of these show strength to be inversely proportional to the smallest dimension of the volume. Additionally, it is interesting that the more constrained nanosphere has strength increasing as d^1/2 compared to film hardness, small in only one dimension, increasing as d^1/4.

With this same length scale approach we have shown that the fracture toughness of a delaminating thin film conforms to R-curve behavior with G_R representing the resistance equivalent to the strain energy release rate at fracture. This has been given by[6]

(1)

where s_ys/E is the strength to modulus ratio, Db/b_o is the incremental growth to initial crack size and a is a constant on the order of 10. In the present paper we use the same approach by equating the local deformation length scale to the fracture process zone length scale. This gives the fracture resistance for the sphere to be

. (2)

Here t_ox is the oxide film thickness on the sphere, and d is the vertical displacement the sphere has been squeezed to the point of fracture. Note that the triggering event is assumed to be a crack nucleated in the less robust oxide film around the sphere such that the defect size is t_ox. In both cases for Eqs. (1) and (2), the leading term is the strain energy density times the smallest dimension of the constrained volume.

We also discuss relationships such as Eq. (2) in terms of the overall size dependence if s_ys obeys a Hall-Petch type relation giving strength proportional to d^1/2/r for the nanosphere. It is emphasized that this is an evolutionary length scale which decreases with increasing displacement, i.e., (2/3)r²/d. Incorporating the length scale dependence of yield strength actually gives G_g’ proportional to d^1/2/r so that both strength and fracture resistance scale with d^1/2/r for very small volumes. Due to the first term in the Hall-Petch relation dominating at very large volumes for both thicker films and larger spheres, this trend would be predicted to reverse. That is, at larger volumes one should find an increase in fracture resistance with scale according to Eqs. (1) and (2) as s_ys becomes more nearly constant.

Most of these relationships are still in their formative stages but have some corroboration with respect to Cu and Au film delamination studies. Regarding fracture experiments on nanospheres, we have only recently fractured a silicon particle in situ in the transmission electron microscope. However, this corroborates some previous indirect analysis using atomic force microscopy based nanoindentation to determine the fracture toughness of silicon nanospheres with radii in the range of 20 to 110 nm. In the full paper we will derive the above relationships and discuss increased strength and fracture resistance of constrained volumes under compression or pressure as having ramifications to friction, wear, and microelectromechanical systems.

REFERENCES

1. Gerberich, W.W., Cordill, M.J., Mook, W.M., Moody, N.R., Perrey, C.R., Carter, C.B., Mukherjee, R., and Girshick, S.L., Acta Mater., vol. 53, 2215-2229, 2005.

2. Gerberich, W.W., Jungk, J.M., Cordill, M.J., Mook, W.M., Boyce, B., Friedmann, T., Moody, N.R. and Yang, D., Intern. J. Fracture, 2005 (accepted).

3. Mook, W.M., Jungk, J.M., Cordill, M.J., Moody, N.R., Sun, Y., Xia, Y., and Gerberich, W.W., Z. Metallkd., vol. 95, 416-424, 2004.

4. Cordill, M.J., Chamber, M.M., Hallman, D., Lund, M., Perrey, C.R., Carter, C.B., Kortshagen, U. and Gerberich, W.W., “Plasticity Responses in Ultra-Small Confined Cubes and Films,” 2005 (in preparation).

5. Gerberich. W.W., Mook. W.M., Perrey. C.R., Carter. C.B., Baskes. M.I., Mukherjee. R., Gidwani. A., Heberlein. J., McMurry. P.H. and Girshick. S.L., J. Mech. Phys. Solids, vol. 51, 979-992, 2003.

6. Gerberich. W.W., Jungk, J.M., Li, M., Volinsky, A.A., Hoehn, J. W. and Yoder, K., Intern. J. Fracture vol. 119/120, 287-405, 2003.

William W. Gerberich, Professor

Summary of professional activities March 2003 through April 2004

Teaching:

Fall 2003: MatS 4221 -- Materials Design and Performance (Undergraduate Lecture/Lab)

Spring 2004; MatS 8004 -- Mechanical Properties (Graduate Lecture)

University/Department Service:

- Institute of Technology, Donaldson's Seminar Series Committee

- Leader, Nanomechanical Test Facility, IPRIME

Professional Service:

- Editorial Board of Review, Metallurgical and Materials Transactions

- Advisory Board, Key Engineering Materials

- Scientific Advisory Board, J. Mechanics and Materials

- Editorial Committee, Japan Institute of Metals, Materials Transaction

- Editorial Advisory Board, J. Adhesion Science and Technology

- Co-Editor, Volume 8, Elsevier Encyclopedia of Comprehensive Structural Integrity, (2003)

- Co-Editor, Special Issue on Fracture Nanomechanics, Intern. J. Fracture, vol. 119/120 (2003)

- Associate Editor, J. Strength, Fracture and Complexity

Invited Presentations:

"A Length Scale for Thin Film Deformation," June 19, ASME 2003 Mechanics and Materials Summer Meeting, Scottsdale, AZ

"Nanomechanics of Ultra-Small Volumes," June 20, ASME 2003 Mech. & Matls., keynote address, Scottsdale, AZ

"The Nanomechanical Response of Nanoparticles and Their Films," July 18, IGERT Workshop on Nanoparticle Science and Engineering

"Scaling Rules from Nanopaticles," July 28, Summer Workshop on ALD Films, Albuquerque

Scaling Rules from Nanoparticles, August 28, Departmental Colloquium, Cornell University, Ithaca, NY

"The Bottom-Up Approach for Designing Nanoparticle Structures," September 17, 2002 International Conf. On Nanotechnology and PM2, Providence

"Scaling Rules from Nanoparticles," September 19, Engineering Departmental Seminar, Brown University, Providence, RI

"Deconvolution of Mechanical Behavior at the Nanoscale," The Adhesion Length Scale Problem," October 13, SES 2003 40^th Annual Society of Engineering Science Meeting, Ann Arbor, MI

"Scaling Rules from Nanoparticles and Films," November 20, ASME International Mechanical Engineering Congress, Washington, D.C.

"Crack-tip Plasticity Effects on the Interfacial Failure of Thin Gold Films," December 2, Fall MRS, Boston

"Mechanical Behavior of Amorphous ALD Alumina Films," December 4, Fall MRS, Boston

"Road Maps for Reliable Nanostructures," January 27, NNI Interagency Workshop in Instrumentation and Metrology for Nanotechnology

"Mechanical Behavior of Films, Nanospheres and Nanobumps," Spring MRS, San Francisco

"Property Determination of ALD Films Via Nanoindenation, March 2004, TMS Annual Meeting, Charlotte, NC

"Mechanical Performance of ALD Films and Nanolaminates," April 2004, ICMCTF Meeting, San Diego

"Constitutive Modeling from the Bottom Up for Mechanical Behavior," April 13, 2004, Spring MRS, San Francisco

"Tailoring Nanolaminate ALD Films for Dynamic Toughness," April 14, 2004, Spring MRS, San Francisco

Publications:

Approximately 12 refereed, coauthored papers, see attached list.

1. A.A. Volinsky, J.B. Vella and W.W. Gergerich, "Fracture Toughness, Adhesion and Mechanical Properties of Low-k Dielectric Thin Films Measured by Nanoindentation, " Thin Solid Films (2003) in press.

2. W.W. Gerberich, W.M. Mook, C.R. Perrey, C.B. Carter, M.I. Baskes, R. Mukherjee, A. Gidwani, J. Heberlein, P.H. McMurry and S.L. Girshick, “Superhard Silicon Nanospheres,” J. Mech. Phys. Solids, 51/6 (2003) p. 979-992.

3. W.W. Gerberich, J.M. Jungk, M. Li, A.A. Volinsky, J.W. Hoehn and K. Yoder, "Length Scales for the Fracture of Nanostructures," Intern. J. Fracture (2003) 119/120 pp. 387-405.

4. H. Kutomi, A. Daugela, W.W. Gerberich, H. Fujii and T.J. Wyrobek, "Nanoscale Friction Reduction and Fatigue Monitoring due to Ultrasonic Excitation," Tribology Int. 36(4-6) (2003) pp. 255-259.

5. A.A. Volinsky, N.R. Moody, and W.W. Gerberich, "Fiducial Mark and CTOA Estimates of Thin Film Adhesion," Intern. J. Fracture 119/120 (2003) pp. 431-439.

6. W.W. Gerberich, J.M. Jungk, and W.M. Mook, "Crack-Dislocation Interactions," in Encyclopedia of Comprehensive Structural Integrity, Vol. 8 , Elsevier (2003) pp. 357-382.

7. A.A. Volinsky, D.F. Bahr, M.D. Kriese, N.R. Moody and W.W> Gerberich, "Nanoindentation Methods in Interfacial Fracture Testing," in Encyclopedia of Comprehensive Structural Integrity, Vol. 8, Chapter 13, Elsevier (2003).

8. W.M. Mook, J.M. Jungk, M.J. Cordill, N.R. Moody, Y. Sun, Y. Xia and W.W. Gerberich, "Geometry and Surface State Effects on the Mechanical Response of Au Nanostructures," Zeits Metallkunde, 95(2004) pp. 416-424,

9. J. Jungk, W. Mook, M. Cordill, D. Bahr, N. Moody, J. Hoehn, M. Chambers, and W. Gerberich, "A Length Scale Based Hardening Model for Ultra Small Volumes," J. Mater. Research, accepted (2004).

10. W.W. Gerberich and B. Mook, "De Petites Boules Tres Dures," Pour La Science (2003) 9 p. 2.

11. W. Gerberich, W. Mook, M. Cordill, C.B. Carter, C. Perrey, J. Heberlein, and S. Girshick, "Reverse Plasticity in Single Crystal Silicon Nanospheres," submitted, Intern. J. Plasticity (2004).

12. Y.S. Garif, W.W. Gerberich, and A.V. Pocius, "Thermally Activated Parameters of Self-Adhesion in Acrylic Pressure Sensitive-like Networks," J. Adhesion, accepted (2004).

13. T.J. Hermel, S.F. Hahn, K.A. Chaffin, W.W. Gerberich, et al., "Role of Molecular Architecture in Mechanical Failure of Glassy/Semicrystalline Block Copolymers: ECE vs CECEC lamellae."Macromolecules 36(16) (2003) 6280.

14. T.J. Hermel, S.F. Hahn, K.A. Chaffin, W.W. Gerberich, et al., "Role of Molecular Architecture in Mechanical Failure of Glassy/Semicrystalline Block Copolymers: CEC vs CECEC lamellae."Macromolecules 36(16) (2003) 2190-2193.

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