week ending PRL 105, 015502 (2010) PHYSICAL REVIEW LETTERS 2 JULY 2010
Plasticity and Dislocation Dynamics in a Phase Field Crystal Model
Pak Yuen Chan, Georgios Tsekenis, Jonathan Dantzig, Karin A. Dahmen, and Nigel Goldenfeld Department of Physics, University of Illinois at Urbana-Champaign, Loomis Laboratory of Physics, 1110 West Green Street, Urbana, Illinois, 61801-3080, USA (Received 26 April 2010; published 28 June 2010) The critical dynamics of dislocation avalanches in plastic flow is examined using a phase field crystal model. In the model, dislocations are naturally created, without any ad hoc creation rules, by applying a shearing force to the perfectly periodic ground state. These dislocations diffuse, interact and annihilate with one another, forming avalanche events. By data collapsing the event energy probability density function for different shearing rates, a connection to interface depinning dynamics is confirmed. The relevant critical exponents agree with mean field theory predictions.
DOI: 10.1103/PhysRevLett.105.015502 PACS numbers: 81.16.Rf, 05.10.Cc, 61.72.Cc
Materials yield and deform plastically under large ex- cally vary the strain rate in simulations, and moreover we ternal stress. While the yield surface and the plastic flow relate the critical point underlying plastic flow at finite are well described by various continuum theories [1–3], strain rates to the scaling of magnetic domain wall depin- what happens microscopically during a plastic deformation ning [6,16,17]. We find remarkable agreement between is still not fully understood. On atomic scales, external simulations and analytical mean field theory predictions stress is carried by localized crystal defects, such as dis- of exponents and, in addition, are able to show that the locations and disclinations. They are created under stress strain-rate data exhibit collapse. Our results strongly sup- and interact with each other. Although the properties of port the critical point picture of plasticity and suggest new individual defects and their interactions are well known experiments. We study dislocation avalanches during plas- [4,5], their collective behavior under external stress is tic flows by using the phase field crystal (PFC) model complicated. It gives rise to scale-invariant, power-law- [18,19]. This approach is well-suited to this problem, distributed phenomena [6–12], in strong resemblance to because it can be performed at finite temperature, for large the scaling behavior near a critical point. These phenomena systems, and over long time periods. The PFC model include dislocation slip avalanches of a broad range of describes the dynamics of the local crystalline density field sizes and acoustic emission with a power-law power and has been shown to give an excellent account of numer- spectrum. ous materials properties including polycrystalline solidifi- Recently, evidence has accumulated that these scaling cation, vacancy diffusion, grain growth, grain boundary phenomena reflect an underlying nonequilibrium critical energetics, epitaxial growth, fracture [19], grain coarsen- point [6,7], i.e., a point where the nonequilibrium steady ing [20], elasticity [21], dislocation annihilation, glides, state fluctuations are governed by a diverging correlation and climb [22], as well as vacancy dynamics [23]. The length [13], as appears to be the case in magnetic materials model can be derived from density-functional theory and [14] and even turbulence [15]. Such a critical point would extended to the case of binary alloys [24]. In this paper, we exist at the boundary between two distinct regimes, one of augment the model to treat external shearing forces by which would be a glassy, activated regime, and the other adding an advective term to the dynamics near the bound- would be a genuine plastic flow regime. Up to now, experi- ary. By adjusting the shearing force and measuring the mental and computational data have focused on the glassy resulting avalanche statistics, a data collapse is obtained regime: For example, Weiss et al. have measured the that is consistent with proximity to a domain wall depin- acoustic emission signal from creep deformation experi- ning point at finite temperature [6,17]. ments on single crystal ice and found that the event size The model.—The PFC model is given by the free energy distribution follows a power law over 4 decades [7,8]. density [18,19] Miguel et al.’s dislocation dynamics simulations in two dimensions showed event size distributions following a r 4 f ¼ ðr2 þ 1Þ2 þ 2 þ ; (1) power law, with a rate-dependent cutoff, over approxi- 2 2 4 mately 2 decades [9]. Zaiser has reported a data collapse of the event size distribution with different external stresses where r is the undercooling and ðx;~ tÞ is the local density. [6]. The dynamics associated with this free energy is conser- The purpose of this Letter is to approach plastic defor- vative, relaxational and diffusive, and systematically de- mation from the flow side of the nonequilibrium critical rivable from density-functional theory [25]. In order to point, manifested in the strain-rate dependence of the study the plastic response of the PFC model under shear, acoustic emission. Importantly, we are able to systemati- we add a shearing term to the dynamical equation:
0031-9007=10=105(1)=015502(4) 015502-1 Ó 2010 The American Physical Society week ending PRL 105, 015502 (2010) PHYSICAL REVIEW LETTERS 2 JULY 2010
@2 @ F @ form avalanches. To quantify the avalanche activity, we þð Þ ¼ð Þ2r2 þ vðyÞ þ ; (2) 2 @t @x calculate the total speed of all dislocations in the domain: @t P ð Þ V~ðtÞ¼ Ndis t ju~ j, where N ðtÞ is the number of disloca- where i¼1 i dis tions in the system at time t and u~i is the velocity of the ith y= v0e for 0 200 50 tionally to the shear velocity and is primarily used to define 150 the avalanche size, rather than to remove slowly moving 25 N(t) dislocations from the analysis. The signal is then parti- 100 N(t) tioned into individual avalanche events. The probability 0 0 500 1000 1500 50 t distribution of the event energy Z tend 0 ¼ 2ð Þ 0 2500 5000 7500 E V t dt; (5) 200 tbegin 150 where tbegin and tend are the starting and ending time of the event, respectively, can be measured. For cutoff values 100 ¼ N(t) small compared to the signal (i.e., Vcut 15, correspond- 50 ing to the activity of 3–4 dislocations due to thermal creep), the result is insensitive to the cutoff. For each shearing rate, 0 10000 12500 15000 17500 at least 10 different realizations are run to obtain a statis- t tically meaningful result. This results in about 8000 ava- lanche events for each shearing rate. Figure 3 shows the FIG. 1. The number of dislocations in a sheared PFC crystal. Intermittent events with sizes differing in orders of magnitude event size distribution for different shearing rates. We find 2 that the distribution follows a power law for small event are observed. Parameters are ð Þ ¼ 255, ¼ 0:9, v0 ¼ 1:581, ¼ ¼ ¼ ¼ sizes and cuts off at larger sizes, with the cutoff size 0 0:3, 1:5, 40:0, and r 0:5. depending on the shearing rate. The data are somewhat noisier towards the end of large event size because large defect atoms sitting next to a dislocation, we can use the events are rare. latter for convenience. Figure 1 shows the typical time Scaling behavior of the avalanches.—Analogous to the ð Þ dependence of N t from a simulation with parameters scaling behavior in models of crackling noise [14], we dx ¼ 3 =8, dt ¼ 0:025, L ¼ L ¼ 512, ð Þ2 ¼ 255, ¼ x y propose that there is a nonequilibrium critical point v0 ¼ : v ¼ : ¼ : ¼ : ¼ : 0 9, 0 1 581, 0 0 3, 1 5, 40 0, and vc in the system, and we expect, as E !1, the data around r ¼ 0:5. NðtÞ changes as dislocations are being created the critical point to collapse in the form and annihilated. There are intermittent events of creation of ð Þ ð Þ dislocations, with the number of dislocations involved P E; v E f Ev ; (6) ranging from a few to 80. Figure 2 shows the acoustic where PðE; v Þ is the probability distribution of event en- ð Þ emission signal V t in the same simulation. Similar to ergy E and v 1 v0=vc is the reduced shearing rate, ð Þ N t , the signal ranges from 0 to 400, with intermittent with v0 being the shearing rate and vc being the critical pulses of various sizes. In order to measure the avalanche shearing rate. and are two critical exponents. As v ! ¼ event size, we introduce a cutoff to the signal Vcut 15 for 0, PðE; v Þ tends to a power law PðE; v Þ E . Figure 4 ¼ shear velocity v0 0:765. The cutoff is increased propor- shows an attempt to collapse the data, by using the equiva- lent scaling form PðE; v Þ v gðEv Þ, with ¼ 1:5, ¼ 2, and vc ¼ 1:5 and universal scaling function gðxÞ¼ 500 x fðxÞ shown in the collapse. Logarithmic binning is 400 performed and singletons are ignored to obtain PðE; v Þ. 300 We obtain a satisfactory data collapse over 4 decades, with V(t) 200 100 0 0 2500 5000 7500 500 400 300 V(t) 200 100 0 10000 12500 15000 17500 t FIG. 2. The total speed of defect atoms in a sheared PFC crystal. Intermittent events with sizes differing in orders of FIG. 3 (color online). The probability distribution of the event magnitude are observed. Parameters are ð Þ2 ¼ 255, ¼ 0:9, energy during dislocation avalanches, for different values of the v0 ¼ 1:581, 0 ¼ 0:3, ¼ 1:5, ¼ 40:0, and r ¼ 0:5. shearing rates. 015502-3 week ending PRL 105, 015502 (2010) PHYSICAL REVIEW LETTERS 2 JULY 2010 03-25939 ITR (MCC) (G. T. and K. D.) and DOE Subcontract No. 4000076535 (J. D.). [1] A. Cottrell, Dislocations and Plastic Flow in Crystals (Oxford University Press, New York, 1953). [2] A. Mendelson, Plasticity: Theory and Application (Macmillan, New York, 1968). [3] W. Hosford, Mechanical Behavior of Materials (Cambridge University Press, Cambridge, England, 2005). [4] J. Hirth and J. Lothe, Theory of Dislocations (Wiley, New York, 1982). FIG. 4 (color online). Data collapse of the probability distri- [5] F. 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Rev. Lett. 88, 245701 (2002). analogies to the depinning transition of magnetic domain [19] K. R. Elder and M. Grant, Phys. Rev. E 70, 051605 (2004). walls [6,17,31], although one exponent that may deviate in [20] H. M. Singer and I. Singer, Phys. Rev. E 74, 031103 simulations is discussed in Ref. [32]. The exponents found (2006). in our collapse are consistent with the domain wall depin- [21] P. Stefanovic, M. Haataja, and N. Provatas, Phys. Rev. Lett. 96, 225504 (2006). ning picture. As argued previously, mean field theory 73 (MFT) is expected to give exact scaling results in this [22] J. Berry and M. Grant, Phys. Rev. E , 031609 (2006). [23] P. Y. Chan, N. Goldenfeld, and J. Dantzig, Phys. Rev. E 79, case. The MFT values for the exponent is ¼ 1:5 035701(R) (2009). [6,17]. The MFT value for the exponent can be calcu- [24] K. Elder, N. Provatas, J. Berry, P. Stefanovic, and M. ¼ lated from the MFT of Refs. [16,17,31]tobe 2. These Grant, Phys. Rev. B 75, 064107 (2007). MFT values for the exponents and lead to a satisfac- [25] S. Majaniemi and M. Grant, Phys. Rev. B 75, 054301 tory collapse of the numerical avalanche size distributions (2007). at different shear rates. At zero temperature the critical [26] J. Sack and J. Urrutia, Handbook of Computational shear rate is vc ¼ 0. Our simulations, however, are per- Geometry (North-Holland, Amsterdam, 2000). formed at finite temperature T. Temperature-induced dis- [27] J. O’Rourke, Computational Geometry in C (Cambridge location creep causes the critical shear rate to appear to be University Press, Cambridge, England, 1998). v ðTÞ! T ! [28] T. Richeton, P. Dobron, F. Chmelik, J. Weiss, and F. nonzero, with the apparent c 0 as 0. It also 424 causes the scaling collapse in Fig. 4 to be less precise than Louchet, Mater. Sci. Eng. A , 190 (2006). [29] D. Dimiduk, C. Woodward, R. LeSar, and M. Uchic, in zero temperature studies. We hope to report on a theo- Science 312, 1188 (2006). retical investigation of the temperature dependence and a [30] M. Koslowski, Philos. Mag. 87, 1175 (2007). comparison with experiments at finite shear rate in a future [31] S. Zapperi, P. Cizeau, G. Durin, and H. Stanley, Phys. Rev. publication. B 58, 6353 (1998). We thank C. Miguel and J. T. Uhl for helpful conversa- [32] M. Miguel, A. Vespignani, M. Zaiser, and S. Zapperi, tions. We acknowledge support from NSF Grant No. DMR Phys. Rev. Lett. 89, 165501 (2002). 015502-4