Dislocation drag from phonon wind in an isotropic crystal at large velocities

ABSTRACT

The anharmonic interaction and scattering of phonons by a moving dislocation, the photon wind, imparts a drag force $vB(v,T,\rho )$ on the dislocation. In early studies, the drag coefficient B was computed and experimentally determined only for dislocation velocities v much less than transverse sound speed, $c_T$. In this paper, we derive analytic expressions for the velocity dependence of B up to $c_T$ in terms of the third-order continuum elastic constants of an isotropic crystal, in the continuum Debye approximation. In so doing we point out that the most general form of the third order elastic potential for such a crystal and the dislocation-phonon interaction requires two additional elastic constants involving asymmetric local rotational strains, which have been neglected previously. We compute the velocity dependence of the transverse phonon wind contribution to B in the range 1–90% $c_T$ for Al, Cu, Fe, and Nb in the isotropic Debye approximation. The drag coefficient for transverse phonons scattering from screw dislocations is finite as $v\to c_T$, whereas B is divergent for transverse phonons scattering from edge dislocations in the same limit. This divergence indicates the breakdown of the Debye approximation and sensitivity of the drag coefficient at very high velocities to the microscopic crystalline lattice cutoff. We compare our results to experimental results wherever possible and identify ways to validate and further improve the theory of dislocation drag at high velocities with realistic phonon dispersion relations, inclusion of lattice cutoff effects, MD simulation data, and more accurate experimental measurements.

KEYWORDS:

• Dislocations in crystals
• drag coefficient
• phonon wind

Acknowledgments

We thank Darby J. Luscher and Benjamin A. Szajewski for enlightening discussions and the anonymous referees for their valuable comments. In particular, the authors are grateful for the support of the Advanced Simulation and Computing, Physics and Engineering Models Program.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Daniel N. Blaschke http://orcid.org/0000-0001-5138-1462

Emil Mottola http://orcid.org/0000-0003-1067-1388

Notes

1 We note that drag is initially dominated by thermal effects at very low dislocation velocities, and that phonon wind becomes important only at velocities of the order of 1% transverse sound speed and higher.

2 Many authors estimate the velocity dependence of the drag coefficient by means of ‘relativistic’ factors $\propto 1/(1-v^2/c^2{)}^m$ with different exponents m and a limiting (sound) speed c based on purely empirical arguments which lack a first-principles theoretical framework, see [16–18, 63, 64] and references therein.

3 Note that here we have removed the cutoff for the dislocation core and expanded for large temperature. Furthermore, in order to compare the two expressions we note that the wave vector cutoff denoted in [39] by $k_{\mathrm{D}}$ is related to the unit cell volume, $k_{\mathrm{D}}\propto {V_{\mathrm{c}}}^{-1/3}$.

Additional information

Funding

This work was performed under the auspices of the U.S. Department of Energy under contract 89233218CNA000001; Los Alamos National Laboratory.