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 Ày= v0e for 0 <y<Ly=2 dislocation. This measure is similar to the acoustic emis- vðyÞ¼ ð À Þ (3) À Ly y = v0e for Ly=2 <y<Ly sion signal in Weiss et al.’s single crystal ice experiments. As dislocations are generated and interact with each other is the shearing profile and v0 is the magnitude of the in the domain, in addition to the fast avalanching dynam- shearing, is the penetration depth, and control the ics, quasistatic structures, such as grain boundaries, can range and time scale, respectively, of elastic interactions form. These slow dynamics should not be measured be- (phonon excitations)R propagating through the medium cause they are really not part of the avalanches. This leads ð Þ d [21], F f x~ d x is the total free energy, and is the to the distinction between fast-moving and slowly moving thermal noise satisfying the fluctuation-dissipation theo- dislocations introduced by Miguel et al. [9]. In essence, h ð Þ ð 0 0Þi ¼ À r2 ð À 0Þ ð À 0Þ rem x;~ t x~ ;t x~ x~ t t . Here is they introduced a cutoff in dislocation speed and measure the noise amplitude. It is directly proportional to the tem- only dislocations with speed higher than the cutoff. In that perature kBT. The value of v0 controls the magnitude of the way, they tried to retain only the avalanche activities in the shearing force; the penetration depth controls how deep acoustic emission signals. the shearing force extends into the material. In all simula- We employed a different method to tackle this problem. tions we set Ly, so the actual value of does not Instead of simulating a very large system, with all sorts of affect our simulation results. The boundary conditions are dislocation activities, we simulated a moderate size of periodic in x and fixed at y ¼ 0;L. The PFC model we used system with approximately 10 000 atoms. For this system allows propagating sound modes [21] when ¼ 0:9;as size, dislocation avalanches come and go; i.e., not many long as is nonzero and Oð1Þ, we do not expect that our dislocations are left in the system after every avalanche. As scaling results will be sensitive to its precise value. Values a result, no grain boundaries, or slow dynamics, are present of that are Oð10Þ would correspond to very overdamped and we obtain clean avalanche data. It is fair to mention dynamics and could model viscoelastic behavior [21] that that this method severely limits the system size and thus the is outside the scope of our work. resulting avalanche sizes. The system size we chose con- One of the advantages of using the PFC model is that we tains approximately 100 dislocations in the largest ava- do not have to impose any ad hoc assumptions about the lanche events. The trade-off, which we exploit, is the creation and annihilation of dislocations. Recall that in cleanness of the avalanche signal and the speed of the dislocation dynamics simulations, dislocations are treated resulting simulations. Different methods, such as those as elementary particles and usually only the far field inter- we mentioned above, would have to be employed if larger action between dislocations is captured. When dislocations avalanche sizes are desired. get too close to each other (a few atomic spacings), the We count the number of nearest neighbors of each atom, highly nonlinear interaction between them is not captured ni, by using the Delaunay triangulation method in compu- and, more importantly, the annihilation of dislocations is tational geometry [26,27]. Because we have ni ¼ 6 for not accounted for. The standard practice is then to impose every atom in a perfectly triangular crystal, and because some annihilation rules—dislocations of opposite topo- there are no vacancies introduced in the version of the PFC logical charges annihilate when they get too close to each model simulated here (vacancies can be introduced into the other [9].