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A preliminary computer simulation study of grain boundary segregation
Scripta METALLURGICA
Vol. 14, pp. 127-128, 1980 Printed in the U.S.A.
Pergamon Press Ltd. All rights reserved.
A PRELIMINARY COMPUTERSIMULATION STUDYOF GRAIN BOUNDARYSEGREGATION E.S. Machlin* Columbia University, New York, NY I0027 and A. Levi IBM Research Laboratory, Yorktown Heights, NY I0598
[Received October 30, 1979) A Mie type interatomic potential, modified and f i t to the cohesive energies, l a t t i c e parameters and electronegativities of the pure components, was applied to the interactions between like type atoms as well as between unlike atoms according to the procedure described by Machlin (1). A {310} t i l t type grain boundary((OOl) t i l t axis with Z = 5) was constructed in a fcc system and relaxed according to an algorithm similar to, but not identical with, that used by Smith et al. (2). The structure of the boundary obtained closely approximated that described by Smith et al. (2) and exhibited the trigonal prism form of the 'Bernal' polyhedra found by the l a t t e r authors at the grain boundary. Despite the low excess energy and the zero average volume dilatation associated with this grain boundary, the presence of localized alternate sign planar dilatations in the grain boundary region is apparent in Fig. l which shows two successive planes along the OOl axis normal to the figure. The sign of this planar dilatation alternates along the direction of the t i l t axis as is apparent from the fact that between the horizontal lines shown in the figure four [130] rows f i t in the "plus" plane while five rows f i t in the asterisked plane. Thus, this {310} type t i l t boundary contains both shorter and longer bond lengths than the equilibrium bond length. +
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FIG. l Successive planes along the direction of the t i l t axis [OOl] for a (310) boundary in a f.c.c, system. The horizontal lines bound the region of varying density. Outside these bounds the density is constant. *This research was conducted while the f i r s t author was a Visiting Scientist at the IBM Research Laboratory on sabbatical leave from Columbia University.
127 0036-9748/80/010127-02502.00/0 Copyright (c) 1980 Pergamon Press Ltd.
128
GRAIN BOUNDARY
SEGREGATION
Vol.
14, No.
i
A {910} t i l t type boundary in the fcc cubic system was also constructed and as shown in Fig. 2 is characterized by an array of edge dislocations. In this case, as well, there exist bond lengths that deviate from the equilibrium value; but both shorter and longer bonds are found in or emanating from the same plane normal to the t i l t axis. The calculated excess energy of the {910} type boundary exceeds that of the {310} type boundary for the case of aluminum by a factor of 1.4 in f a i r agreement with that evaluated by Smith et al, (2) of 1.76. The l a t t e r authors used a pseudopotential t h a t provides a more accurate representation for the interatomic potential in aluminum than does the Mie type potential we used. However, the l a t t e r is more generally applicable to alloy systems of a l l types than is the former. Also, in our case, our l a t t i c e sums of interatomic potentials consisted in summing up to a total of 18 neighbors about any atom, both in f i t t i n g the potential parameters and in using them,rather than to a fixed cutoff radius. ÷ + : ~ ÷ x t ~ + +:÷+: + z t ~ + . . ~÷z+x . . , . x ÷ ~ + ~ _++.~.+z+z + ~ ÷ z.+ x +'!"+z.++ z~..,,. ÷
+++++:t+:+++++x+.~:~$~++÷x++++.++~$+++t~.+x++~++ +++++++-+÷I÷,++-++-.+÷t+m÷+++z+z÷x÷++-+=,÷ A (910) type I-+x±I~+I÷I÷I÷~+I+I~+I+.~+I÷l÷I÷+~
I
~÷I÷.~C!.÷I÷I÷..X~I÷X÷I÷..~÷I÷I÷..X~I÷I
FIG. 2
boundary ([OOl] t i l t in a fcc system.
axis)
We have systematically tested the effect so far of only the f i r s t two of the following four possible variables available for our modified Mie potential: atomic diameter (D), cohesive energy (E), electronegativity (X), and quadrupole p o l a r i z a b i l i t y (P). The following results were obtained. For the case of exchange of grain boundary sited atoms with l a t t i c e sited atoms* i t was found that: I. for Esolute = Esolvent, Xsolute = Xsolvent, Psolute = Psolvent = O, solute atoms with D < Dsolvent tend to segregate to sites having average values of nearest-neighbor bond lengths smaller than that for l a t t i c e equilibrium and solute atoms with D > Dsolvent tend to segregate to sites having average values of nearest-neighbor bond lengths larger than that for l a t t i c e equilibrium; however, for certain values of Dsolute/Dsolvent specific to particular sites, local shuffles are i n i t i a t e d upon relaxation after exchange leading to violation of these rules: 2. for Dsolute = Dsolvent, Xsolute = Xsolvent, Psolute = Psolvent = 0 and Esolute < Esolvent, solute atoms tend to segregate to a l l sites having strained bonds regardless of the sign of the strain or the magnitude of the average strain: 3. for Dsolute = Dsolvent, Xsolute -- Xsolvent, Psolute = Psolvent = 0 and Esolute > Esolvent, solute atoms are repelled from sites having associated strained bonds. No violations of the l a t t e r two rules were found. For the case of segregation of a solute atom at the largest i n t e r s t i t i a l Site at a grain boundary not involving exchange with a grain boundary sited atom i t was found that there is an optimum valu---e of Dsolute < Dsolvent at which segregation of the solute atom is strongly favored. Further+, this optimum value of Dsolute exceeds the diameter of the largest i n t e r s t i t i a l space in the fcc crystal of the solvent host ( i . e . there are i n t e r s t i t i a l spaces in the 910 and 310 grain boundaries that are larger than those in the host crystal l a t t i c e . ) We g r a t e f u l l y acknowledge the contributions of C. Bennett, S. Kirkpatric~i and P. Chaudhari to the development of the relaxation computer program which was adapted for use in the present study. REFERENCES I. 2.
E.S. Machlin, Acta Met. 22, 95 (1974). D.A. Smith, V. Vitek and~.C. Pond, Acta Met. 25, 475 (1977).
*A block of the fcc host metal was constructed in the computer and the exchange process consisted in exchanging a solute atom in the center of the fcc block with a solvent atom at a grain boundary in the array containing the l a t t e r . Decrease of the sum of the energies of both arrays upon such exchange is the c r i t e r i o n for the tendency to segregate at the grain boundary and the value of this increment is the energy of segregation ( i . e . negative value corresponds to segregation tendency).
Vol. 14, pp. 127-128, 1980 Printed in the U.S.A.
Pergamon Press Ltd. All rights reserved.
A PRELIMINARY COMPUTERSIMULATION STUDYOF GRAIN BOUNDARYSEGREGATION E.S. Machlin* Columbia University, New York, NY I0027 and A. Levi IBM Research Laboratory, Yorktown Heights, NY I0598
[Received October 30, 1979) A Mie type interatomic potential, modified and f i t to the cohesive energies, l a t t i c e parameters and electronegativities of the pure components, was applied to the interactions between like type atoms as well as between unlike atoms according to the procedure described by Machlin (1). A {310} t i l t type grain boundary((OOl) t i l t axis with Z = 5) was constructed in a fcc system and relaxed according to an algorithm similar to, but not identical with, that used by Smith et al. (2). The structure of the boundary obtained closely approximated that described by Smith et al. (2) and exhibited the trigonal prism form of the 'Bernal' polyhedra found by the l a t t e r authors at the grain boundary. Despite the low excess energy and the zero average volume dilatation associated with this grain boundary, the presence of localized alternate sign planar dilatations in the grain boundary region is apparent in Fig. l which shows two successive planes along the OOl axis normal to the figure. The sign of this planar dilatation alternates along the direction of the t i l t axis as is apparent from the fact that between the horizontal lines shown in the figure four [130] rows f i t in the "plus" plane while five rows f i t in the asterisked plane. Thus, this {310} type t i l t boundary contains both shorter and longer bond lengths than the equilibrium bond length. +
Y~
z
1{
z
z
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Z
+
+
+
+
+
+
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FIG. l Successive planes along the direction of the t i l t axis [OOl] for a (310) boundary in a f.c.c, system. The horizontal lines bound the region of varying density. Outside these bounds the density is constant. *This research was conducted while the f i r s t author was a Visiting Scientist at the IBM Research Laboratory on sabbatical leave from Columbia University.
127 0036-9748/80/010127-02502.00/0 Copyright (c) 1980 Pergamon Press Ltd.
128
GRAIN BOUNDARY
SEGREGATION
Vol.
14, No.
i
A {910} t i l t type boundary in the fcc cubic system was also constructed and as shown in Fig. 2 is characterized by an array of edge dislocations. In this case, as well, there exist bond lengths that deviate from the equilibrium value; but both shorter and longer bonds are found in or emanating from the same plane normal to the t i l t axis. The calculated excess energy of the {910} type boundary exceeds that of the {310} type boundary for the case of aluminum by a factor of 1.4 in f a i r agreement with that evaluated by Smith et al, (2) of 1.76. The l a t t e r authors used a pseudopotential t h a t provides a more accurate representation for the interatomic potential in aluminum than does the Mie type potential we used. However, the l a t t e r is more generally applicable to alloy systems of a l l types than is the former. Also, in our case, our l a t t i c e sums of interatomic potentials consisted in summing up to a total of 18 neighbors about any atom, both in f i t t i n g the potential parameters and in using them,rather than to a fixed cutoff radius. ÷ + : ~ ÷ x t ~ + +:÷+: + z t ~ + . . ~÷z+x . . , . x ÷ ~ + ~ _++.~.+z+z + ~ ÷ z.+ x +'!"+z.++ z~..,,. ÷
+++++:t+:+++++x+.~:~$~++÷x++++.++~$+++t~.+x++~++ +++++++-+÷I÷,++-++-.+÷t+m÷+++z+z÷x÷++-+=,÷ A (910) type I-+x±I~+I÷I÷I÷~+I+I~+I+.~+I÷l÷I÷+~
I
~÷I÷.~C!.÷I÷I÷..X~I÷X÷I÷..~÷I÷I÷..X~I÷I
FIG. 2
boundary ([OOl] t i l t in a fcc system.
axis)
We have systematically tested the effect so far of only the f i r s t two of the following four possible variables available for our modified Mie potential: atomic diameter (D), cohesive energy (E), electronegativity (X), and quadrupole p o l a r i z a b i l i t y (P). The following results were obtained. For the case of exchange of grain boundary sited atoms with l a t t i c e sited atoms* i t was found that: I. for Esolute = Esolvent, Xsolute = Xsolvent, Psolute = Psolvent = O, solute atoms with D < Dsolvent tend to segregate to sites having average values of nearest-neighbor bond lengths smaller than that for l a t t i c e equilibrium and solute atoms with D > Dsolvent tend to segregate to sites having average values of nearest-neighbor bond lengths larger than that for l a t t i c e equilibrium; however, for certain values of Dsolute/Dsolvent specific to particular sites, local shuffles are i n i t i a t e d upon relaxation after exchange leading to violation of these rules: 2. for Dsolute = Dsolvent, Xsolute = Xsolvent, Psolute = Psolvent = 0 and Esolute < Esolvent, solute atoms tend to segregate to a l l sites having strained bonds regardless of the sign of the strain or the magnitude of the average strain: 3. for Dsolute = Dsolvent, Xsolute -- Xsolvent, Psolute = Psolvent = 0 and Esolute > Esolvent, solute atoms are repelled from sites having associated strained bonds. No violations of the l a t t e r two rules were found. For the case of segregation of a solute atom at the largest i n t e r s t i t i a l Site at a grain boundary not involving exchange with a grain boundary sited atom i t was found that there is an optimum valu---e of Dsolute < Dsolvent at which segregation of the solute atom is strongly favored. Further+, this optimum value of Dsolute exceeds the diameter of the largest i n t e r s t i t i a l space in the fcc crystal of the solvent host ( i . e . there are i n t e r s t i t i a l spaces in the 910 and 310 grain boundaries that are larger than those in the host crystal l a t t i c e . ) We g r a t e f u l l y acknowledge the contributions of C. Bennett, S. Kirkpatric~i and P. Chaudhari to the development of the relaxation computer program which was adapted for use in the present study. REFERENCES I. 2.
E.S. Machlin, Acta Met. 22, 95 (1974). D.A. Smith, V. Vitek and~.C. Pond, Acta Met. 25, 475 (1977).
*A block of the fcc host metal was constructed in the computer and the exchange process consisted in exchanging a solute atom in the center of the fcc block with a solvent atom at a grain boundary in the array containing the l a t t e r . Decrease of the sum of the energies of both arrays upon such exchange is the c r i t e r i o n for the tendency to segregate at the grain boundary and the value of this increment is the energy of segregation ( i . e . negative value corresponds to segregation tendency).