On the temperature and density dependence of dislocation drag from phonon wind

TL;DR

This work addresses dislocation drag from phonon wind at extreme strain rates by computing second- and third-order elastic constants for Cu and Al from first-principles and feeding these into a phonon-wind drag model. Using VASP for $SOEC$/$TOEC$ extraction and PyDislocDyn for $B(\vartheta,v)$, the study reveals that drag increases with temperature and density and that high-velocity drag is dominated by SOECs, with TOEC effects largely negligible in this regime. A simple analytic form for $B(\vartheta,\sigma)$ is validated, enabling efficient high-rate predictions while reducing the computational burden of TOEC calculations. The results inform dislocation-dynamics modeling and shed light on the non-linear $Y$-$G$ scaling observed in some metals, though pressure effects on this scaling remain insufficient to fully explain the behavior, indicating avenues for further investigation in other materials like Ta and Pb.

Abstract

At extreme strain rates, where fast moving dislocations govern plastic deformation, anharmonic phonon scattering imparts a drag force on the dislocations. In this paper, we present calculations of the dislocation drag coefficients of aluminum and copper as functions of temperature and density. We discuss the sensitivity of the drag coefficients to changes in the third-order elastic constants with temperature and density.

Figures (8)

• Figure 1: Energy (left) and pressure (right) as a function of simulation time step (in fs) calculated with VASP (Vienna Ab initio Simulation Package) for copper at 300K and with lattice constant $a=3.647$Å for an undeformed fcc lattice containing 256 atoms. Note that the experimental value for the lattice constant at ambient pressure is $a=3.6146$Å, the difference being due to the assumed potential used for the simulation.
• Figure 2: Energy as a function of deformation $y$ for deformation types \ref{['eq:c44sym']} (left) and \ref{['eq:cprime']} (right), and including the fitting functions for SOEC and TOEC. Since \ref{['eq:c44sym']} is symmetric in $y$ up to third order (i.e. it does not depend on $y^3$, only positive points are required, and we can only get one SOEC, $C_{44}$ in this case. Eq. \ref{['eq:cprime']} is very asymmetric, allowing us to extract both a SOEC (proportional to $a$) and a TOEC (proportional to $b$).
• Figure 3: Energy (left) as a function of deformation of the rescaling type \ref{['eq:rescale']}, and pressure (right) as a function of density. The plots include the fitting functions for SOEC, TOEC (left), and an "equation of state" polynomial (right) whose derivative at ambient density ($\rho=8.70$g/cm$^3$ in the simulation) yields the bulk modulus with slightly better accuracy than from the energy plot on the left. The latter, however, is required to calculate the TOEC $(C_{111} + 6C_{112} + 2C_{123})$ which is related to coefficient $b$ in the figure legend.
• Figure 4: Energy as a function of deformation $y$ for deformation type \ref{['eq:TOEC1']} for copper at room temperature and ambient pressure, and including the fitting functions for SOEC and TOEC. Note (left plot) that even though the data were corrected by $P\mathrm{d}V$, the lowest energy value is still displaced from zero deformation. Only after including the (volume-dependent) vibrational Helmholtz free energy contribution to $E_\textrm{tot}$ (discussed in the text) do we get a curve whose minimum is located at zero deformation which enables us to compute the TOECs, albeit with significantly greater uncertainty because these constants are very sensitive to those corrections.
• Figure 5: In the top row we show $B(\vartheta,v)$ at ambient conditions for Al and Cu as calculated using the material data determined with VASP as given in Table \ref{['tab:results-300K']}. In the bottom row we show $B(\vartheta,v)$ at high temperature ($T=900$K for Cu, $T=800$K for Al) and the same density as in the top row using the data of Tables \ref{['tab:results-Al-800K_P']} and \ref{['tab:results-900K_P']}. Since the color scale in the bottom row is changed by $T/300$ compared to the top row, we clearly see that the drag coefficient is slightly enhanced at elevated temperatures relative to linear scaling. In each of the four plots, we also show $B(\vartheta)$ at $\beta_{\textrm{t}}=v/c_{\textrm{t}}=0.01$ in the respective lower panels, where velocity $\beta_{\textrm{t}}$ was normalized by the polycrystalline average transverse sound speed $c_{\textrm{t}}$, calculated by averaging over the SOEC using Kröners method Kroener:1958; see also Blaschke:2017Poly.