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Stability of grain-boundary motion in the presence of impurities
Acta Materialia 54 (2006) 1595–1596 www.actamat-journals.com
Stability of grain-boundary motion in the presence of impurities Alexander L. Korzhenevskii a, Richard Bausch b, Rudi Schmitz
c,*
a
b
Institute for Problems of Mechanical Engineering, RAS, Bol’shoi prosp. V.O., 61, St. Petersburg, 199178, Russia Institut fu¨r Theoretische Physik IV, Heinrich-Heine-Universita¨t Du¨sseldorf, Universita¨tsstrasse 1, D-40225 Du¨sseldorf, Germany c Institut fu¨r Theoretische Physik C, RWTH Aachen, Templergraben 55, D-52056 Aachen, Germany Received 14 July 2005; received in revised form 16 November 2005; accepted 21 November 2005 Available online 18 January 2006
Abstract The impurity-drag effect in grain-boundary motion has been examined by Cahn who considered the one-dimensional uniform motion of a single planar grain boundary through a dilute atmosphere of impurity atoms. Later Roy and Bauer proposed a two-dimensional model which, allowing deformations of the grain boundary and lateral diffusion of the impurity atoms, gives rise to a morphological instability of the grain boundary. In the present note we point out a serious deficiency of this approach and, after correction, find a significantly different stability behavior. 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Grain-boundary migration; Impurity drag
A convincing explanation for the main properties of the impurity-drag effect in grain boundary motion has been presented in an early paper by Cahn [1]. He considered the one-dimensional uniform motion of a single planar grain boundary in normal direction through an ideal gas of impurity atoms. These were allowed to diffuse parallel to the direction of motion, but in addition are attracted or repelled by the grain boundary. In a later contribution Roy and Bauer [2] proposed a two-dimensional model which allows deformations of the grain-boundary shape as well as lateral diffusion of the impurity atoms. This model gives rise to an inherent instability of the boundary shape in some regime of the grainboundary velocity. The low-velocity threshold of this regime is found to be zero for the limiting case of infinite lateral size of the grain boundary. As shown in the present note, the Roy–Bauer model can be based on a simple dimensional extension of the Cahn model. This kind of approach reveals an improper assumption on the drag force in the Roy–Bauer model which sig*
Corresponding author. Fax: +49 241 953264. E-mail address: [email protected] (R. Schmitz).
nificantly affects the stability properties of the system. In fact, their phenomenological expression for the drag force is compatible with the extended Cahn model only, if this force vanishes identically. Evaluation of the more microscopic result for the impurity-induced friction force implies absence of any instability in the low-velocity regime. Guided by the Roy–Bauer procedure, we introduce Cartesian coordinates x, z parallel and normal to the initially straight grain-boundary line which is driven to move in the z-direction. Deformations of the line at time t will be described by z = Z(x, t) and are caused by coupling to the impurity density C(x, z, t). In terms of the basic quantities Z(x, t) and C(x, z, t) the generalized Cahn model reads Z ot Z ¼ Cro2x Z þ C dzðoz U ÞC þ F ; ð1Þ ot C ¼ Dr ½r þ ðrU ÞC. Here, 1/C and r mean the mobility and surface tension of the grain boundary, U[z Z(x, t)] is the interaction potential with the impurities, and F an external driving force. The density of impurities obeys a Fokker–Planck equation, which for simplicity assumes isotropic diffusion with a coefficient D and with $ ” (ox, oz).
1359-6454/$30.00 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2005.11.023
1596
A.L. Korzhenevskii et al. / Acta Materialia 54 (2006) 1595–1596
For a planar grain boundary, moving with constant velocity V in normal direction, the latter is chosen as the z-axis. Eqs. (1) then essentially reduce to the Cahn model V ¼ GA ðV Þ þ F ; V oz C A ¼ Doz ½oz þ U 0 C A
ð2Þ
where CA(z) means the stationary impurity profile in the co-moving frame, and Z GA ðV Þ C dz U 0 ðz VtÞC A ðzÞ ð3Þ the corresponding drag force. In order to check the stability of the planar boundary shape and of the stationary profile CA(z), we linearize equations (1) in the increments h(x, t) ” Z(x, t) Vt and 0 c(x, z, t) ” C(x, z, t) CA(z) + h(x, t)C A (z). This leads to the set equations Z 2 ðot Crox Þh ¼ C dzU 0 c; ðot Do2x Þc ¼ V oz c þ Doz ½oz þ U 0 c þ C 0A ot h þ V ðC A C 0 Þo2x h; ð4Þ which again refer to the frame, co-moving with constant velocity V. The Roy–Bauer approach tacitly assumes absence of the operators ot and ox in the drag force. From our point of view this amounts to replacing c in the friction integral in (4) by the solution cA of the truncated equation 0 ¼ V oz cA þ Doz ½oz þ U 0 cA .
ð5Þ
Thus, one observes that cA(z) and CA(z) obey identical differential equations but differ in the obvious boundary conditions cA(±1) = 0 and CA(±1) = C0. Integration from 1 to z, therefore, leads to the identities D½c0A þ U 0 cA ¼ VcA ; D½C 0A þ U 0 C A ¼ V ðC A C 0 Þ
ð6Þ
which will be used in the first and second equations (4), respectively. If, after these manipulations, cA is changed back to c in the first equation (4), and if the second equation is integrated over z from 1 to +1, then, in terms of the excess concentrations Z qðx; tÞ dz cðx; z; tÞ; Z ð7Þ P dz½C A ðzÞ C 0 one finds ðot Cro2x Þh ¼ V ðC=DÞq; ðot Do2x Þq ¼ VP o2x h.
gested phenomenologically in Ref. [2] for the viscous drag force and for the convective impurity flux along the grain boundary. In the light of our derivation, the result for the drag force should, however, vanish after approximating c by cA. Whereas, namely, from Eqs. (6) one finds the impurity profile CA(z) of the Cahn theory, the result for cA(z) is zero due its trivial boundary conditions. A more physical argument is that the absence of operators ot, ox in the drag force means that in this term the increment h may be considered as a rigid translation, giving rise to the replacement c ! C A ðz þ hÞ C A ðzÞ hC 0A ðzÞ which is zero to linear order in h. Within our treatment the correct way to handle the drag force is to insert the full solution c(x, z, t) of the second equation (4) into the friction integral of the first equation. This requires specification of the potential U, for which, in a first run, we have adopted the triangular form, chosen also by Cahn. For this case, we find in the limit V ! 0 ½ot Cro2x h ¼ G0A ð0Þot h;
ð9Þ
where GA(V) is the drag force (3) of the Cahn theory. Instead of going through the rather involved derivation of this result, we point out that it combines with the lowvelocity behavior of the Cahn expression GA ðV Þ VG0A ð0Þ to yield the plausible behavior G0A ð0Þ½V þ ot h. The drag force, appearing in the result (9), effectively renormalizes the mobility 1/C of the grain boundary but does not affect its stability. A similar argument applies to a renormalization of the surface tension r which will arise, if low-velocity corrections are taken into account. At higher velocities the competition between surface tension and impurity fluxes, pointed out by Roy and Bauer, might have an effect similar to the Mullins–Sekerka instability [3]. A different mechanism derives from the nonmonotonous velocity dependence of the driving force, found by Cahn in the high-velocity regime of equations (2). This can lead to unstable behavior, as generally discussed in Ref. [4], and verified for a concrete model system in Ref. [5]. An exhaustive exploration of these questions is beyond the scope of the present paper and will be discussed elsewhere. Acknowledgements A.L.K. wants to thank the University of Du¨sseldorf for its warm hospitality. This work has been supported by the Deutsche Forschungsgemeinschaft under BA 944/2-1 and by the RFBR under N03-02-04009. References
ð8Þ
Apart from notation, Eqs. (8) are just of the form of the Roy–Bauer model. In fact, the coupling terms on the right-hand sides of (8) agree with the expressions, sug-
[1] [2] [3] [4] [5]
Cahn JW. Acta Metall 1962;10:789. Roy A, Bauer CL. Acta Metall 1975;23:957. Mullins WW, Sekerka RF. J Appl Phys 1963;34:323. 1964;35:444. Fried E. J Elast 1993;31:163. Korzhenevskii AL, Bausch R, Schmitz R. Europhys Lett 2002;59: 533.
Stability of grain-boundary motion in the presence of impurities Alexander L. Korzhenevskii a, Richard Bausch b, Rudi Schmitz
c,*
a
b
Institute for Problems of Mechanical Engineering, RAS, Bol’shoi prosp. V.O., 61, St. Petersburg, 199178, Russia Institut fu¨r Theoretische Physik IV, Heinrich-Heine-Universita¨t Du¨sseldorf, Universita¨tsstrasse 1, D-40225 Du¨sseldorf, Germany c Institut fu¨r Theoretische Physik C, RWTH Aachen, Templergraben 55, D-52056 Aachen, Germany Received 14 July 2005; received in revised form 16 November 2005; accepted 21 November 2005 Available online 18 January 2006
Abstract The impurity-drag effect in grain-boundary motion has been examined by Cahn who considered the one-dimensional uniform motion of a single planar grain boundary through a dilute atmosphere of impurity atoms. Later Roy and Bauer proposed a two-dimensional model which, allowing deformations of the grain boundary and lateral diffusion of the impurity atoms, gives rise to a morphological instability of the grain boundary. In the present note we point out a serious deficiency of this approach and, after correction, find a significantly different stability behavior. 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Grain-boundary migration; Impurity drag
A convincing explanation for the main properties of the impurity-drag effect in grain boundary motion has been presented in an early paper by Cahn [1]. He considered the one-dimensional uniform motion of a single planar grain boundary in normal direction through an ideal gas of impurity atoms. These were allowed to diffuse parallel to the direction of motion, but in addition are attracted or repelled by the grain boundary. In a later contribution Roy and Bauer [2] proposed a two-dimensional model which allows deformations of the grain-boundary shape as well as lateral diffusion of the impurity atoms. This model gives rise to an inherent instability of the boundary shape in some regime of the grainboundary velocity. The low-velocity threshold of this regime is found to be zero for the limiting case of infinite lateral size of the grain boundary. As shown in the present note, the Roy–Bauer model can be based on a simple dimensional extension of the Cahn model. This kind of approach reveals an improper assumption on the drag force in the Roy–Bauer model which sig*
Corresponding author. Fax: +49 241 953264. E-mail address: [email protected] (R. Schmitz).
nificantly affects the stability properties of the system. In fact, their phenomenological expression for the drag force is compatible with the extended Cahn model only, if this force vanishes identically. Evaluation of the more microscopic result for the impurity-induced friction force implies absence of any instability in the low-velocity regime. Guided by the Roy–Bauer procedure, we introduce Cartesian coordinates x, z parallel and normal to the initially straight grain-boundary line which is driven to move in the z-direction. Deformations of the line at time t will be described by z = Z(x, t) and are caused by coupling to the impurity density C(x, z, t). In terms of the basic quantities Z(x, t) and C(x, z, t) the generalized Cahn model reads Z ot Z ¼ Cro2x Z þ C dzðoz U ÞC þ F ; ð1Þ ot C ¼ Dr ½r þ ðrU ÞC. Here, 1/C and r mean the mobility and surface tension of the grain boundary, U[z Z(x, t)] is the interaction potential with the impurities, and F an external driving force. The density of impurities obeys a Fokker–Planck equation, which for simplicity assumes isotropic diffusion with a coefficient D and with $ ” (ox, oz).
1359-6454/$30.00 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2005.11.023
1596
A.L. Korzhenevskii et al. / Acta Materialia 54 (2006) 1595–1596
For a planar grain boundary, moving with constant velocity V in normal direction, the latter is chosen as the z-axis. Eqs. (1) then essentially reduce to the Cahn model V ¼ GA ðV Þ þ F ; V oz C A ¼ Doz ½oz þ U 0 C A
ð2Þ
where CA(z) means the stationary impurity profile in the co-moving frame, and Z GA ðV Þ C dz U 0 ðz VtÞC A ðzÞ ð3Þ the corresponding drag force. In order to check the stability of the planar boundary shape and of the stationary profile CA(z), we linearize equations (1) in the increments h(x, t) ” Z(x, t) Vt and 0 c(x, z, t) ” C(x, z, t) CA(z) + h(x, t)C A (z). This leads to the set equations Z 2 ðot Crox Þh ¼ C dzU 0 c; ðot Do2x Þc ¼ V oz c þ Doz ½oz þ U 0 c þ C 0A ot h þ V ðC A C 0 Þo2x h; ð4Þ which again refer to the frame, co-moving with constant velocity V. The Roy–Bauer approach tacitly assumes absence of the operators ot and ox in the drag force. From our point of view this amounts to replacing c in the friction integral in (4) by the solution cA of the truncated equation 0 ¼ V oz cA þ Doz ½oz þ U 0 cA .
ð5Þ
Thus, one observes that cA(z) and CA(z) obey identical differential equations but differ in the obvious boundary conditions cA(±1) = 0 and CA(±1) = C0. Integration from 1 to z, therefore, leads to the identities D½c0A þ U 0 cA ¼ VcA ; D½C 0A þ U 0 C A ¼ V ðC A C 0 Þ
ð6Þ
which will be used in the first and second equations (4), respectively. If, after these manipulations, cA is changed back to c in the first equation (4), and if the second equation is integrated over z from 1 to +1, then, in terms of the excess concentrations Z qðx; tÞ dz cðx; z; tÞ; Z ð7Þ P dz½C A ðzÞ C 0 one finds ðot Cro2x Þh ¼ V ðC=DÞq; ðot Do2x Þq ¼ VP o2x h.
gested phenomenologically in Ref. [2] for the viscous drag force and for the convective impurity flux along the grain boundary. In the light of our derivation, the result for the drag force should, however, vanish after approximating c by cA. Whereas, namely, from Eqs. (6) one finds the impurity profile CA(z) of the Cahn theory, the result for cA(z) is zero due its trivial boundary conditions. A more physical argument is that the absence of operators ot, ox in the drag force means that in this term the increment h may be considered as a rigid translation, giving rise to the replacement c ! C A ðz þ hÞ C A ðzÞ hC 0A ðzÞ which is zero to linear order in h. Within our treatment the correct way to handle the drag force is to insert the full solution c(x, z, t) of the second equation (4) into the friction integral of the first equation. This requires specification of the potential U, for which, in a first run, we have adopted the triangular form, chosen also by Cahn. For this case, we find in the limit V ! 0 ½ot Cro2x h ¼ G0A ð0Þot h;
ð9Þ
where GA(V) is the drag force (3) of the Cahn theory. Instead of going through the rather involved derivation of this result, we point out that it combines with the lowvelocity behavior of the Cahn expression GA ðV Þ VG0A ð0Þ to yield the plausible behavior G0A ð0Þ½V þ ot h. The drag force, appearing in the result (9), effectively renormalizes the mobility 1/C of the grain boundary but does not affect its stability. A similar argument applies to a renormalization of the surface tension r which will arise, if low-velocity corrections are taken into account. At higher velocities the competition between surface tension and impurity fluxes, pointed out by Roy and Bauer, might have an effect similar to the Mullins–Sekerka instability [3]. A different mechanism derives from the nonmonotonous velocity dependence of the driving force, found by Cahn in the high-velocity regime of equations (2). This can lead to unstable behavior, as generally discussed in Ref. [4], and verified for a concrete model system in Ref. [5]. An exhaustive exploration of these questions is beyond the scope of the present paper and will be discussed elsewhere. Acknowledgements A.L.K. wants to thank the University of Du¨sseldorf for its warm hospitality. This work has been supported by the Deutsche Forschungsgemeinschaft under BA 944/2-1 and by the RFBR under N03-02-04009. References
ð8Þ
Apart from notation, Eqs. (8) are just of the form of the Roy–Bauer model. In fact, the coupling terms on the right-hand sides of (8) agree with the expressions, sug-
[1] [2] [3] [4] [5]
Cahn JW. Acta Metall 1962;10:789. Roy A, Bauer CL. Acta Metall 1975;23:957. Mullins WW, Sekerka RF. J Appl Phys 1963;34:323. 1964;35:444. Fried E. J Elast 1993;31:163. Korzhenevskii AL, Bausch R, Schmitz R. Europhys Lett 2002;59: 533.