Full length articleStationary dislocation motion at stresses significantly below the Peierls stress: Example of shuffle screw and dislocations in silicon
Graphical abstract
Introduction
As the intrinsic lattice resistance to a dislocation motion, the Peierls stress is the minimum stress required to start the motion of a static straight dislocation at zero temperature [1]. This is a key controlling parameter in many mesoscale models such as crystal plasticity [2] and dislocation dynamics [3], [4], [5]. Extensive research has been devoted to determining the Peierls stress for different materials using first principle calculations [6], [7], [8], [9], [10], [11], molecular dynamics (MD) [12], [13], [14], [15], [16], [17], [18], as well as the Peierls-Nabarro model [19], [20], [21], [21], [22], [23], [24]. In different constitutive equations, it is often assumed that below the Peierls stress dislocation motion is impossible [2], [3], [4], [5], [13], [25]. Here we found that the Peierls stress should be exceeded for starting the motion of a dislocation, but not to maintain its stationary motion. This concept can be understood through an analysis of the equation of motion of dislocations at the continuum level. The energy balance equation for describing the motion of a dislocation [26] reads aswhere is the applied shear stress acting on the dislocation within the slip plane along the slip direction, is the magnitude of the Burgers vector, is the dislocation velocity, and are the Peierls energy per unit length and the lattice resistance stress, respectively, which are periodic along the dislocation path and dependent on the applied stress and are the kinetic energy and effective mass of a dislocation per unit length, is the dissipation force per unit length induced by the electron/phonon drag and the wave radiation from a fast moving dislocation. For with and Eq. (1) can be re-written as
It is clear from Eq. (2) that, at a zero initial velocity, a dislocation does not move until . That is why the Peierls stress is considered as a counterpart of the dry friction, assuming that the stationary dislocation motion at is impossible.
However, in contrast to the constant dry friction along the glide plane, the lattice resistance stress, is periodic along . It is smaller than almost everywhere, changes the sign, and may add to along the half of the period. Besides, as long as the initial dislocation velocity and kinetic energy are high enough, for the dislocation under a shear stress of a change of the kinetic energy in Eq. (1) may effectively overcome the Peierls stress. These two features allow the following phenomenon: a stationary dislocation motion at . Here the stationary dislocation motion refers to a state in which the dislocation velocity is a time-independent periodic function of with a period of . Here the minimum when a stationary motion becomes possible is defined as the dynamic Peierls stress, . In a pioneering work, through including the radiation of elastic waves into the elasticity theory for dislocation motion, Al’shitz et al. [27] analytically predicted that the stationary motion of a dislocation under a fraction of the Peierls stress is possible as long as its velocity exceeds a critical value. Similarly, in an independent work with a simple atomic scale model, Crowley et al. also analytically showed that a stationary dislocation motion in materials under a shear below the static Peierls stress might happen and the external strain required for such phenomenon would strongly depend on the shape of the interatomic potential [28]. Note that the dynamic Peierls stress [27], [28] was considered as a function of dislocation velocity, which is just a dissipation stress. Here we define the dynamic Peierls stress, as the minimum at which a stationary motion is possible. The existence of and whether it is significantly lower than the traditional Peierls stress, was not discussed in the literature. Later on, in a simple 2D lattice, such a phenomenon was confirmed through computer simulations at the atomic scale. The elastic wave emitted from a moving dislocation was characterized by Koizumi et al. in MD [29]. In contrast, at the continuum level, the dynamics of fast moving dislocations in materials [30] under high strain-rate deformation is usually studied through an extension of the Peierls-Nabarro model. Those models either incorporate the static Peierls stress [25], [31] as one controlling parameter into the mobility law by assuming that a negligible drag has been induced by the elastic wave radiating from the moving dislocations, or simply approximate it as a viscous drag [32], [33].
There obviously exists a need to perform the large-scale atomistic simulations of the motion of a dislocation below the Peierls stress in realistic materials, especially high-Peierls-barrier materials, such as silicon or other materials with covalent or ionic bonds. In particular, how far below the Peierls stress is a stationary dislocation motion possible and what are the conditions for its realization? For a continuum study of the high strain rate plastic deformation below the Peierls stress at the meso- and macro-scales, one needs a set of the constitutive equations for a single dislocation to be justified and calibrated by atomistic simulations or experiments. This will lead to an explicit incorporation of the concept of the dynamic Peierls stress, into the mesoscale computer models and in turn, a significant expansion of their predictive capability.
In this paper, to confirm the existence of and to study the dislocation motion in a stress range of we perform a series of MD simulations of the motion of shuffle screw and also dislocations in silicon at 1 K and 300 K. Taking the dissipation force, calibrated from the MD simulations as an input, the numerical solution of Eq. (1) is also pursued. In this way, an iterative loop of linking the atomistic and continuum-level analysis is developed and closed. Surprisingly, the dynamic Peierls stresses at 1 K is 5–7 times smaller than the static Peierls stress. We also found that the critical initial velocity above which a dislocation can maintain a stationary motion at . After a calibration of all the material parameters in the constitutive equation (1), it well describes the MD simulation results.
Section snippets
Definition of the driving force for the motion of a dislocation
In contrast to the quasi-static processes, the definition of a driving force for the motion of a dislocation under dynamic loading is nontrivial because it is not equal to the averaged shear stress over the sample or external surface. Based on Weertman’s analysis [34], the force per unit dislocation length in the direction of the Burger’s vector can be calculated as follows:Here is the stress distribution along the dislocation Burger’s direction within the slip plane,
Molecular dynamics simulations
Here, silicon (Si) is chosen as a representative material for verifying the possibility of a stationary dislocation motion below the static Peierls stress and the definition of a dynamic Peierls stress. The simulations are started at a low initial temperature of () using a Nos-Hoover thermostat [35] to exclude the effect of the finite temperature on the dislocation motion. The atomic interactions are described using the Stillinger-Weber (SW) potential [36], which correctly describes the
Results and discussion
The mobility of both the initially non-moving dislocation in a static loading scenario and the dislocation carrying a high initial velocity in a dynamic loading scenario has been studied in great details here. The stationary dislocation velocity versus the applied shear stress is shown in Fig. 2. It is clear that both static and dynamic loading generate the same dislocation velocity for larger than the static Peierls stress, which was determined as the applied shear stress for
Conclusions
To summarize, we have proved and quantified the stationary motion of a dislocation in Si at using molecular dynamics simulations and solving the equation of motion for dislocations. The Peierls stress is usually considered as a dry friction or a direct reflection of the yield strength. We introduced a concept of the dynamic Peierls stress below which a stationary dislocation motion is impossible for dislocations with any initial velocities. Starting with a qualitative
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
HC acknowledges supports by Shanghai Sailing Program (20YF1409400) and NSFC of China (52005186). VIL acknowledges supports of NSF (CMMI-1943710 and DMR-1904830), ONR (N00014-19-1-2082), ARO (W911NF-17-1-0225), and Iowa State University (Vance Coffman Faculty Chair Professorship). LX acknowledges the NSF support (CMMI-1930093 and DMR-1807545). VIL and LX acknowledge XSEDE computing resources under MSS170015 and MSS170003. XCZ acknowledges the support from NSFC of China (51725503).
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