Entropic effect on the rate of dislocation nucleation Seunghwa Ryua,1, Keonwook Kangb, and Wei Caib aDepartment of Physics, and bDepartment of Mechanical Engineering, Stanford University, Stanford, CA 94305 Edited* by William D. Nix, Stanford University, Stanford, CA, and approved January 27, 2011 (received for review November 15, 2010) Dislocation nucleation is essential to our understanding of plastic computing activation free energies (21–23) has been successfully deformation, ductility, and mechanical strength of crystalline ma- applied to dislocation nucleation. terials. Molecular dynamics simulation has played an important We successfully applied the umbrella sampling (21) method role in uncovering the fundamental mechanisms of dislocation nu- to compute the activation free energy for homogeneous and het- cleation, but its limited timescale remains a significant challenge erogeneous dislocation nucleation in copper. Based on this input, for studying nucleation at experimentally relevant conditions. the nucleation rate is predicted using the Becker–Döring theory Here we show that dislocation nucleation rates can be accurately (24). Comparison with direct MD simulations at high stress con- predicted over a wide range of conditions by determining the firms the accuracy of this approach. Both FcðT;γÞ and GcðT;σÞ activation free energy from umbrella sampling. Our data reveal show significant reduction with increasing T, corresponding to S γ 0 092 9k very large activation entropies, which contribute a multiplicative large activation entropies. For example, cð ¼ . Þ¼ B S σ 2 48k factor of many orders of magnitude to the nucleation rate. The and cð ¼ GPaÞ¼ B are observed in homogeneous nuclea- activation entropy at constant strain is caused by thermal expan- tion. We found that ScðγÞ is caused by the anharmonic effect sion, with negligible contribution from the vibrational entropy. The of thermal expansion, with negligible contribution from the vibra- activation entropy at constant stress is significantly larger than tional entropy. The large difference in the two activation entro- that at constant strain, as a result of thermal softening. The large pies, ΔSc ≡ ScðσÞ − ScðγÞ, is caused by thermal softening, which activation entropies are caused by anharmonic effects, showing is another anharmonic effect. Similar behaviors are expected the limitations of the harmonic approximation widely used for to occur in other nucleation processes in solids. rate estimation in solids. Similar behaviors are expected to occur in For simplicity, we begin with the case of homogeneous dislo- other nucleation processes in solids. cation nucleation in the bulk. Even though dislocations often nucleate heterogeneously at surfaces or internal interfaces, ucleation plays an important role in a wide range of physical, homogeneous nucleation is believed to occur in nanoindentation Nchemical, and biological processes (1–6). In the last two (11) and in a model of brittle–ductile transition (25). It also decades, the nucleation of dislocations in crystalline solids has provides an upper bound to the ideal strength of the crystal. attracted significant attention, not only for the reliability of Our model system is a copper single crystal described by the microelectronic devices (7), but also as a responsible mechanism embedded atom method (EAM) potential (26). As shown in for incipient plasticity in nanomaterials (8–10) and nanoindenta- Fig. 1A, the simulation cell is subjected to a pure shear stress tion (11–13). However, predicting the nucleation rate as a func- along ½112. The dislocation to be nucleated lies on the (111) tion of temperature and stress from fundamental physics is plane and has the Burgers vector of a Shockley partial (27), extremely difficult. Because the critical nucleus can be as small bp ¼½112∕6. Fig. 1B shows the shear stress strain relationship as a few lattice spacings, the applicability of continuum theory of the perfect crystal at different temperatures (before disloca- (14) becomes questionable. At the same time, the timescale of tion nucleation). molecular dynamics (MD) simulations is about ten orders of In this work, we predict the nucleation rate based on the – magnitude smaller than the experimental timescale. Hence MD Becker Döring (BD) theory, which expresses the nucleation rate simulations of dislocation nucleation are limited to conditions at per nucleation site as which the nucleation rate is extremely high (15, 16). F T;γ One way to predict the dislocation nucleation rate under com- IBD f þΓ − cð Þ ; [2] ¼ c exp k T mon experimental loading rates (17) is to combine the transition B state theory (TST) (5, 18) and the nudged elastic band (NEB) þ method (19). TST predicts that the nucleation rate per nucleation where f c is the molecular attachment rate, and Γ is the Zeldovich site in a crystal subjected to constant strain γ can be written as factor (Materials and Methods). The BD theory and TSTonly dif- fers in the frequency prefactor. Whereas TST neglects multiple Fc T;γ recrossing over the saddle point by a single transition trajectory ITST ν − ð Þ ; [1] ¼ 0 exp k T (5), the recrossing is accounted for in the BD theory through the B Zeldovich factor. F T k First, we establish the validity of the BD theory for dislocation where c is the activation free energy, is temperature, and B is Boltzmann’s constant. The frequency prefactor is ν0 ¼ k T∕h, nucleation by comparing it against direct MD simulations at a B σ 2 16 γ 0 135 T 300 where h is Planck’s constant. Note that FcðT;γÞ¼EcðγÞ − TScðγÞ, relatively high stress ¼ . GPa ( ¼ . )at ¼ K, IMD 2 5 × 108 −1 Materials and Methods where Ec and Sc are the activation energy and activation entropy, which predicts ¼ . s ( ). respectively. Here we assume the dependence of Ec and Sc on T The key input to the BD theory is the activation Helmholtz free F T;γ is weak, which is later confirmed numerically for T ≤ 400 K. For energy cð Þ, which is computed by umbrella sampling. The a crystal subjected to constant stress σ, FcðT;γÞ in Eq. 1 should be G T;σ H σ − replaced by the activation Gibbs free energy cð Þ¼ cð Þ Author contributions: S.R. and W.C. designed research; S.R., K.K., and W.C. performed TScðσÞ, where Hc is the activation enthalpy. Because the NEB research; S.R. and K.K. contributed new reagents/analytic tools; S.R. and W.C. analyzed method only computes the activation energy, the contribution data; and S.R., K.K., and W.C. wrote the paper. of Sc is often ignored in rate estimates in solids. Recently, an The authors declare no conflict of interest. approximation of ScðσÞ¼HcðσÞ∕Tm is used (17), where Tm is *This Direct Submission article had a prearranged editor. the surface disordering temperature. This approximation was 1To whom correspondence should be addressed. E-mail: [email protected]. questioned by subsequent MD simulations (20). The magnitude This article contains supporting information online at www.pnas.org/lookup/suppl/ of Sc remains unknown because none of the existing methods for doi:10.1073/pnas.1017171108/-/DCSupplemental. 5174–5178 ∣ PNAS ∣ March 29, 2011 ∣ vol. 108 ∣ no. 13 www.pnas.org/cgi/doi/10.1073/pnas.1017171108 Downloaded by guest on September 26, 2021 3 A B T = 0 K plots Fc as a function of T at γ ¼ 0.092.ForT ≤ 400 K, the data S 9k 100 K closely follow a straight line, whose slope gives c ¼ B. This 200 K 300 K activation entropy contributes a significant multiplicative factor, 2 4 400 K Sc∕k ≈ 10 500 K expð BÞ , to the absolute nucleation rate, and cannot be 600 K ignored. (GPa) xy What causes this rapid drop of activation free energy with σ 1 temperature? Thermal expansion and vibrational entropy are two candidate mechanisms. To examine the effect of thermal expansion, we performed a zero temperature MEP search at 0 γ 0.092, but with other strain components fixed at the equili- 0 0.1 0.2 ¼ γ brated values at T ¼ 300 K. This approach is similar to the quasi- xy harmonic approximation (QHA) (30, 31) often used in free energy calculations in solids, except that, unlike QHA, the Fig. 1. (A) Schematics of the simulation cell. The spheres represent vibrational entropy is completely excluded here. The resulting atoms enclosed by the critical nucleus of a Shockley partial dislocation E~ 2 04 loop. (B) Shear stress–strain curves of the Cu perfect crystal (before disloca- activation energy, c ¼ . eV, is indistinguishable from the tion nucleation) at different temperatures. activation free energy Fc ¼ 2.05 Æ 0.01 eV at T ¼ 300 K com- puted from umbrella sampling. Because atoms do not vibrate umbrella sampling is performed in Monte Carlo simulations in the MEP search, this result shows that the dominant mechan- using a bias potential as a function of the order parameter n, ism for the large ScðγÞ is thermal expansion, whereas the contri- which is chosen as the number of atoms inside the dislocation bution from vibrational entropy is negligible. As temperature loop (Materials and Methods). increases, thermal expansion pushes neighboring atoms further Fig. 2A shows the free energy function FðnÞ obtained from um- apart and weakens their mutual interaction. This expansion brella sampling for the specified ðT;γÞ condition. The maximum makes crystallographic planes easier to shear and significantly of FðnÞ gives the activation free energy Fc ¼ 0.53 Æ 0.01 eV and reduces the free energy barrier for dislocation nucleation. In the critical nucleus size nc ¼ 36. The Zeldovich factor (28), the widely used harmonic approximation of TST, the activation η 1∕2 2 entropy is often attributed to the vibrational degrees of freedom Γ ¼ 0.055, is obtained from Γ ≡ ð2πk TÞ , where η ¼ −∂ FðnÞ∕ Q Q 2 B ν S ∕k N νm ∕ N−1 νa νm νa ∂n as 0 expð c BÞ¼ð i¼1 i Þ ð i¼1 i Þ, where i and i are jn¼nc .