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Predicting the dislocation nucleation rate as a function of temperature and stress

Published online by Cambridge University Press:  05 October 2011

Seunghwa Ryu ,
Keonwook Kang  and
Wei Cai
Seunghwa Ryu
Affiliation:
Department of Physics, Stanford University, Stanford, California 94305
Keonwook Kang
Affiliation:
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Maxico 87545
Wei Cai*
Affiliation:
Department of Mechanical Engineering, Stanford University, Stanford, California 94305
*
a)Address all correspondence to this author. e-mail: caiwei@stanford.edu
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Abstract

Predicting the dislocation nucleation rate as a function of temperature and stress is crucial for understanding the plastic deformation of nanoscale crystalline materials. However, the limited time scale of molecular dynamics simulations makes it very difficult to predict the dislocation nucleation rate at experimentally relevant conditions. We recently develop an approach to predict the dislocation nucleation rate based on the Becker–Döring theory of nucleation and umbrella sampling simulations. The results reveal very large activation entropies, which originated from the anharmonic effects, that can alter the nucleation rate by many orders of magnitude. Here we discuss the thermodynamics and algorithms underlying these calculations in greater detail. In particular, we prove that the activation Helmholtz free energy equals the activation Gibbs free energy in the thermodynamic limit and explain the large difference in the activation entropies in the constant stress and constant strain ensembles. We also discuss the origin of the large activation entropies for dislocation nucleation, along with previous theoretical estimates of the activation entropy.

Keywords

dislocationsductilitysimulation
Type
Invited Feature Paper
Information
Journal of Materials Research , Volume 26 , Issue 18 , 28 September 2011 , pp. 2335 - 2354
Copyright
Copyright © Materials Research Society 2011

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References

REFERENCES

1.Li, X., Wei, Y., Lu, L., Lu, K., and Gao, H.: Dislocation nucleation governed softening and maximum strength in nano-twinned metals. Nature 464, 877 (2010).Google Scholar
2.Li, J.: The mechanics and physics of defect nucleation. MRS Bull. 32, 151 (2007).Google Scholar
3.Zhu, T., Li, J., Ogata, S., and Yip, S.: Mechanics of ultra-strength materials. MRS Bull. 34, 167 (2009).CrossRefGoogle Scholar
4.Li, J., Van Vliet, K.J., Zhu, T., Yip, S., and Suresh, S.: Atomistic mechanisms governing elastic limit and incipient plasticity in crystals. Nature 418, 307 (2002).Google Scholar
5.Schuh, C.A., Mason, J.K., and Lund, A.C.: Quantitative insight into dislocation nucleation from high-temperature nanoindentation experiments. Nature Mater. 4, 617 (2005).Google Scholar
6.Schall, P., Cohen, I., Weitz, D.A., and Spaepen, F.: Visualizing dislocation nucleation by indenting colloidal crystals. Nature 440, 319 (2006).CrossRefGoogle ScholarPubMed
7.Nix, W.D., Greer, J.R., Feng, G., and Lilleodden, E.T.: Deformation at the nanometer and micrometer length scales: Effects of strain gradients and dislocation starvation. Thin Solid Films 515, 315 (2007).Google Scholar
8.Izumi, S., Ohta, H., Takahashi, C., Suzuki, T., and Saka, H.: Shuffle-set dislocation nucleation in semiconductor silicon device. Philos. Mag. Lett. 90, 707 (2010).CrossRefGoogle Scholar
9.Xu, G., Argon, A.S., and Ortiz, M.: Critical configurations for dislocation nucleation from crack tips. Philos. Mag. A 75, 341 (1997).CrossRefGoogle Scholar
10.Frank, F.: Symposium on the Plastic Deformation of Crystalline Solids; Carnegie Institute of Technology and Office of Naval Research, Pittsburgh, PA, 1950; p. 89.Google Scholar
11.Aubry, S., Kang, K., Ryu, S., and Cai, W.: Energy barrier for homogeneous dislocation nucleation: Comparing atomistic and continuum models. Scripta Mater. 64, 1043 (2011).CrossRefGoogle Scholar
12.Tschopp, M.A., Spearot, D.E., and McDowell, D.L.: Atomistic simulations of homogeneous dislocation nucleation in single crystal copper. Model. Simul. Mater. Sci. Eng. 15, 693 (2007).CrossRefGoogle Scholar
13.Bringa, E.M., Rosolankova, K., Rudd, R.E., Remington, B.A., Wark, J.S., Duchaineau, M., Kalantar, D.H., Hawreliak, J., and Belak, J.: Shock deformation of face-centered-cubic metals on subnanosecond timescales. Nat. Mater. 5, 805 (2006).Google Scholar
14.Zhu, T., Li, J., Samanta, A., Leach, A., and Gall, K.: Temperature and strain-rate dependence of surface dislocation nucleation. Phys. Rev. Lett. 100, 025502 (2008).CrossRefGoogle ScholarPubMed
15.Hanggi, P., Talkner, P., and Borkovec, M.: Reaction-rate theory: Fifty years after Kramers. Rev. Mod. Phys. 62, 251 (1990).Google Scholar
16.Becker, R. and Döring, W.: The kinetic treatment of nuclear formation in supersaturated vapors. Ann. Phys. (Weinheim) 24, 719 (1935).Google Scholar
17.Eyring, H.: The activated complex in chemical reactions. J. Chem. Phys. 3, 107 (1935).Google Scholar
18.Vineyard, G.H.: Frequency factors and isotope effects in solid state rate processes. J. Phys. Chem. Solids 3, 121 (1957).CrossRefGoogle Scholar
19.Jónsson, H., Mills, G., and Jacobsen, K.W.: Nudged elastic band method for finding minimum energy paths of transitions. In Classical and Quantum Dynamics in Condensed Phase Simulations; Berne, B.J., Ciccotti, G., and Coker, D.F., Eds.; World Scientific: New York; 1998; pp. 385–404.CrossRefGoogle Scholar
20.Voter, A.F.: Introduction to the kinetic Monte Carlo Metho; Springer: New York; 2007.Google Scholar
21.Jin, C., Ren, W., and Xiang, Y.: Computing transition rates of thermally activated events in dislocation dynamics. Script. Mater. 62, 206 (2010).CrossRefGoogle Scholar
22.Chandler, D.: Introduction to Modern Statistical Mechanics; Oxford University Press: Oxford, UK; 1987.Google Scholar
23.Ren, W. E. and Vanden-Eijnden, E.: String method for the study of rare events. Phys. Rev. B 66, 052301 (2002).Google Scholar
24.Ren, W. E. and Vanden-Eijnden, E.: Finite temperature string method for the study of rare events. J. Phys. Chem. B 109, 6688 (2005).Google Scholar
25.Ryu, S., Kang, K., and Cai, W.: Entropic effect on the rate of dislocation nucleation. Proc. Natl. Acad. Sci. USA 108, 5174 (2011).Google Scholar
26.Frenkel, D. and Smit, B.: Understanding Molecular Simulation: From Algorithms to Applications, Academic Press: San Diego, CA; 2002.Google Scholar
27.Kocks, U.F., Argon, A.S., and Ashby, M.F.: Thermodynamics and kinetics of slip. Prog. Mater. Sci. 19, 1 (1975).Google Scholar
28.Nye, J.F.: physical Properties of Crystals: Their Representation by Tensors and Matrices; Clarendon Press: New York; 1957.Google Scholar
29.Xiao, H., Bruhns, O.T., and Meyers, A.: Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mech. 124, 89105 (1997).CrossRefGoogle Scholar
30.Bechmann, R., Ballato, A.D., and Lukaszek, T.J.: Higher-order temperature coefficients of the elastic stiffinesses and complinaces of alpha-quartz. In Proceedings of the Institute of Radio Engineers, Vol. 50; 1962; p. 1812.Google Scholar
31.Whalley, E.: Use of volumes of activation for determining reaction mechanisms. In Advances in Physical Organic Chemistry; Gold, V., Ed.; Academic Press: London; 1964; pp. 93–162.Google Scholar
32.Tonnet, M.L. and Whalley, E.: Effect of pressure on the alkaline hydrolysis of ethyl acetate in acetone–water solutions. Parameters of activation at constant volume. Can. J. Chem. 53, 3414 (1975).Google Scholar
33.Overton, W.C. Jr. and Gaffney, J.: Temperature variation of the elastic constants of cubic elements. I. Copper. Phys. Rev. 98, 969 (1955).CrossRefGoogle Scholar
34.Cahn, J.W. and Nabarro, F.R.N.: Thermal activation under shear. Philos. Mag. A 81, 1409 (2001).CrossRefGoogle Scholar
35.Cottrell, A.H.: Thermally activated plastic glide. Philos. Mag. Lett. 82, 65 (2002).CrossRefGoogle Scholar
36.Mishin, Y., Sorensen, M.R., and Voter, A.F.: Calculation of point-defect entropy in metals. Philos. Mag. A 81, 2591 (2001).CrossRefGoogle Scholar
37.Gupta, D., Ed.: Diffusion Processes in Advanced Technological Materials; Springer: New York; 2002; p.140.Google Scholar
38.Kittel, C.: Introduction to Solid State Physics, 8th ed.; Wiley: New York; 2004.Google Scholar
39.Argon, A.S., Andrews, R.D., Godrick, J.A., and Whitney, W.: Plastic deformation bands in glassy polystyrene. J. Appl. Phys. 39, 1899 (1968).Google Scholar
40.DiMelfi, R.J., Nix, W.D., Barnett, D.M., Holbrook, J.H., and Pound, G.M.: An analysis of the entropy of thermally activated dislocation motion based on the theory of thermoelasticity. Phys. Status Solidi B 75, 573 (1976).CrossRefGoogle Scholar
41.DiMelfi, R.J., Nix, W.D., Barnett, D.M., and Pound, G.M.: The equivalence of two methods for computing the activation entropy for dislocation motion. Acta Mater. 28, 231 (1980).CrossRefGoogle Scholar
42.Kemeny, G. and Rosenberg, B.: Compensation law in thermodynamics and thermal death. Nature 243, 400 (1973).Google Scholar
43.Yelon, A., Movagha, M., and Branz, H.M.: Origin and consequences of the compensation (Meyer–Neldel) law. Phys. Rev. B 46, 12243 (1992).CrossRefGoogle ScholarPubMed
44.Born, M.: Thermodynamics of crystals and melting. J. Chem. Phys. 7, 591 (1939).CrossRefGoogle Scholar
45.Brochard, S., Hirel, P., Pizzagalli, L., and Godet, J.: Elastic limit for surface step dislocation nucleation in face-centered cubic metals: Temperature and step height dependence. Acta Mater. 58, 4182 (2010).Google Scholar
46.Khantha, M., Pope, D.P. and Vitek, V.: Dislocation screening and the brittle-to-ductile transition: A Kosterlitz–Thouless type instability. Phys. Rev. Lett. 74, 684 (1994).Google Scholar
47.Mishin, Y., Mehl, M.J., Papaconstantopoulos, D.A., Voter, A.F., and Kress, J.D.: Structural stability and lattice defects in copper: Ab initio, tight-binding, and embedded-atom calculations. Phys. Rev. B 63, 224106 (2001).Google Scholar
48.Hirth, J.P. and Lothe, J.: Theory of Dislocations; Krieger: New York; 1992.Google Scholar
49.Parrinello, M. and Rahman, A.: Crystal structure and pair potentials: A molecular dynamics study. Phys. Rev. Lett. 45, 1196 (1980).Google Scholar
50.Anderson, H.C.: Molecular dynamics at constant pressure and/or temperature. J. Chem. Phys. 72, 2384 (1980).Google Scholar
51.Hoover, W.G.: Canonical dynamics: Equilibrium phase space distribution. Phys. Rev. A 31, 1695 (1985).Google Scholar
52.Zhu, T., Li, J., Van Vliet, K.J., Ogata, S., Yip, S., and Suresh, S.: Predictive modeling of nanoindentation-induced homogeneous dislocation nucleation in copper. J. Mech. Phys. Sol. 52, 691 (2004).CrossRefGoogle Scholar
53.Ngan, A.H.W., Zuo, L., and Wo, P.C.: Size dependence and stochastic nature of yield strength of micron-sized crystals: A case study on Ni3Al. Proc. Royal Soc. A 462, 1661 (2006).Google Scholar
54.Ryu, S. and Cai, W.: Validity of classical nucleation theory for Ising models. Phys. Rev. E 81, 030601(R) (2010).Google Scholar
55.Zhu, T., Li, J., Samanta, A., Kim, H.G., and Suresh, S.: Interfacial plasticity governs strain rate sensitivity and ductility in nanostructured metals. Proc. Natl. Acad. Sci. USA 104, 3031 (2007).Google Scholar
56.Foiles, S.M.: Evaluation of harmonic methods for calculating the free energy of defects in solids. Phys. Rev. B 49, 14930 (1994).Google Scholar
57.de Koning, M., Miranda, C.R., and Antonelli, A.: Atomistic prediction of equilibrium vacancy concentraions in Ni3Al. Phys. Rev. B 66, 104110 (2002).CrossRefGoogle Scholar
58.Ryu, S. and Cai, W.: Comparison of thermal properties predicted by interatomic potential models. Model. Simul. Mater. Sci. Eng. 16, 085005 (2008).Google Scholar
59.Yelon, A., Movaghar, B., and Crandall, R.S.: Multi-exitation entropy: Its role in thermodynamics and kinetics. Rep. Prog. Phys. 69, 1145 (2006).CrossRefGoogle Scholar