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Solution hardening of titanium by oxygen
Scripta METALLURGICA
Vol. 3, pp. 917-922, 1969 Printed in the United States
Pergamon Press, Inc.
SOLUTION H.~RDENING OV TITANIUM BY OXYGEN
W. R. Tyson Physics Department Tren~ University Peterborough, Ontario, Canada
(Received October 9, 1969)
The marked solid solution strengthening of titanium by oxygen is well established
(1) but the details of the interaction between dislocations
oxygen interstitials
is not yet fully understood.
and
In an earlier paper (2)
it was shown that elastic interaction of the defect strain fields is insufficient to explain the observed interaction energy of about 1.4 eV., and it was suggested that the strengthening
is due to obstruction of atomic motions
within the core of a gliding dislocation.
Subsequently,
Tang et al. (5)
pointed out that elastic anisotropy should be taken into account, although they concluded that it would have only a small effect in the case of titanium.
It is the purpose of this note to review the calculation of the elastic
interaction energy using anisotropic elasticity and to present some atomistic calculations that shed some qualitative
light on the problem.
Elastic anisotropy will affect both the strain fields ~ sik of the interstitials and the stress fields ~ D of the dislocations. However, the former ik effect is taken into account implicitly when $ s is determined from X-ray ik measurements of lattice parameter changes. Due to the symmetry of the hexagonal lattice and of the octahedral hole, (~0 ~, ~ )
using the conventional
hexagonal unit cell (x,//a, x3//c).
~s
must be of the form ik orientation of Cartesian axes for the Since the change in lattice parameters
c and a must be proportional to the atom fraction x of solute atoms,
~,
and
~3 may be found by extrapolation to x = 1 which corresponds to complete filling of the octahedral
interstices;
using the results of Andersson et al.
(4) we find 6, a .028 and ~ = .087. The volume element appropriate to these strains is the volume per interstitial atom at x = l, i.e. about ~ 2 . The effect of elastic anisotropy on dislocations on the prism plane is to modify the non-zero stresses of screw and edge dislocations (5), but because of the small deviation from isotropy found in titanium the changes are slight. The interaction energy U Ds calculated by the method of Cochardt et al. (6) applied to the c.p.h, lattice (2) using average elastic constants 917
918
SOLUTION HARDENING OF TITANIUM BY OXYGEN
= .598 x i0 ~z d / c m 2, ~ = 0.34
Vol. 3, No. 12
and isotropic elasticity is 0.14 eV for
edge dislocations, which is reduced to 0.12 eV by anisotropic elasticity. The interaction energy with screw dislocations is zero in both cases. Evidently, the elastic interaction energy with isolated interstitial atoms is an order of magnitude lower than the observed activation energy. However,
it is well established (7) that long range order appears at high
concentrations, and some short-range order ~n the form of interstitial pairs may exist at the concentrations of interest in solution hardening studies (below 1 at.%).
The principal strains within a volume element of 2ca 2 con-
taining such a dipole may be inferred from Weissman's results to have a maximum tetragona]ity of (0,0~.05); this would produce an interaction energy with edge dislocations of nearly the same magnitude as that due to isolated interstitial atoms, and an interaction energy with screw dislocations considerably smaller than this.
Even if a large fraction of the oxygen atoms were
present as interstitial pairs, the elastic distortion would appear to be insufficient to explain the observed hardening. Direct calculation of the lattice energy of crystal defects using appropriate interatomic potentials and high-speed digital computers has recently become possible, and this approach has been used to investigate the interaction between a dislocation and an interstitial atom in a planar square lattice.
The interatomic potentials (see Fig. I) were chosen to produce an
.s
"I ~ "
,.s
2r
o
-i.o FIG. I Interatomic potentials between host atoms A and interstitial atoms B. elastically isotropic two-dimensional lattice with a square unit cell of lattice parameter = 1 stable for small and large deformations.
The unit
cell undergoes a 20% expansion on introduction of an interstitial atom which interacts with the host atoms through the A-B potential (Fig. I).
Fig. 2
shows the relaxed atomic configuration for an edge dislocatio~ and interstitial far removed from the dislocation and in two sites below the extra half plane.
The interaction energies
U Ds in the two sites, in units of
Vol. 3, No. 12
SOLUTION HARDENING OF TITANIUM BY OXYGEN
919
O
(a)
(b)
Dislocation
Interstitial
( ( ( (4) Interstitial in Site 2
(c) ~nterstitial in Site I
FIG. 2 Relsxed atomic configurations for separated and interacting dislocation and interstitial the "chemical bond energy" which is the depth of the A-B potential well, were calculsted from
U Os = U D÷s - ( U e + U ~) where U D÷~
is the total
energy of the dislocation-solute configuration (Figs. 2c or 2d), and U ~ S are the energies of the separate dis]ocatio~ and irterst~tial (F~gs. 2a aod 2b).
Finite ]attices of the same size were used in calculating these ener-
gies, and the correction for image effects was neglected. reported in Table l, where
The results are
U D5 is separated into a part due to a chanEe in
the energies of the A-B bonds, and a part due to a change in the lattice energy (i.e. the elastic strain energy).
For comparison, the elastic inter-
action energy (lattice strain energy) calculated by Cochardt et al.'s method is zero for site i and -.39 for site 2.
The interstitial is repelled from
site 1 and attracted to site 2 by approximately the same total energy.
920
SOLUTION HARDENING OF TITANIUM BY OXYGEN
TABLE
Vol. 3, No. 12
1
Interaction Energies (unit:
A-B bond energy)
Chemical Bond
Lattice Strain
Total U Ds
Site i
+ 1.O0
-
.22
+ .78
Site 2
-
-
.67
-
.06
.73
An interstitial site in the planar square lattice has four near neighbours in the unstrained lattice and three near neighbours when slip by an amount b/2 has occurred.
Based on a hard sphere model, the radius of the
hole is thereby decreased by 63%, and this reduction in available space should cause a considerable increase in lattice strain when the dislocation passes over an interstitial atom.
Analogous reasoning was presented earlier
(2) as one possible cause of the hardening in titanium.
However, the present
results show that, while this effect may still be important for screw dislocations, the dilatation below the extra half plane of an edge dislocation drastically modifies the atomic arrangement so that, for the dislocation of Fig. 2a, the space availab]e in site 1 within the core is practically the same as the interstitial space in the perfect lattice.
For the particular
interstitial atom considered here, which is very "hard" and causes a change in lattice spacing of about 20%, the elastic strain energy is reduced when the interstitial moves to site 1 because of the reduction in number of nearest neighbours, thereby reducing the energy required to displace them. For the same reason, the energy of chemical binding is increased by 1.O, which produces an overall repulsion.
In site 2, the hydrostatic tension of
the dislocation reduces the strain energy of the interstitial by an amount close to that calculated from linear elasticity, and since there are still four near neighbours to this site the chemical bond energy is practically unchanged. It is not known how well two-body interatomic potentials would describe the cohesion of titanium, and even less how adequate they would be for oxygen atoms in interstitial solid solution.
A detailed atomistic calculation for
the h.c.p, titanium lattice would thus give results of doubtful quantitative significance.
Nevertheless, the work described above suggests that the
breaking of chemical bonds in the dislocation core may be a significant source of strengthening in interstitial Ti-O alloys.
Calculations by Roos
Vol. 3, No. 12
SOLUTION HARDENING OF TITANIUM BY OXYGEN
(8) on the analogous
921
Zr - 0 system have shown that the cohesive energies of
such alloys can be very large, and hence considerable
energy may be required
to break a Ti - 0 bond. The author is grateful to S. Weissman for a pre-publlcation
copy of his
paper (reference 7), and to Trent University for the provision of computer time. REFERENCES
i.
~. Conrad, Acta Met. 1__4, 1631 (Iq66).
2.
W. R. Tyson, Can. Met. Quart. ~, 501 (1968).
5.
R. Tang, J. Kratochvil and H. Conrad, Scripta Met. ~, 485 (1969).
4.
S. Anders~on, sca~.
B. Collen, V. Kuylenstierna and A. MaRne]i,
Acta Chem.
1~, 164-] (1957).
5.
Y. T. Chou, J. Appl. Phys. 5_~4, 429 (]965).
6.
A. W. Cochardt. (1955).
7.
S. Weissman and A. Shrier, "Strain D~stribution in Oxidized Al~ha Titarium Crystals", Intern~tior~l Conference on Titanium, Lon~on~ Msy 1968.
8.
B. ~oos,
G. S. Schoeck,
and H. Wiedersich,
Ark. ~ys. (Sweden] 25, 563 (]963).
Acta Met. 2, 553
Vol. 3, pp. 917-922, 1969 Printed in the United States
Pergamon Press, Inc.
SOLUTION H.~RDENING OV TITANIUM BY OXYGEN
W. R. Tyson Physics Department Tren~ University Peterborough, Ontario, Canada
(Received October 9, 1969)
The marked solid solution strengthening of titanium by oxygen is well established
(1) but the details of the interaction between dislocations
oxygen interstitials
is not yet fully understood.
and
In an earlier paper (2)
it was shown that elastic interaction of the defect strain fields is insufficient to explain the observed interaction energy of about 1.4 eV., and it was suggested that the strengthening
is due to obstruction of atomic motions
within the core of a gliding dislocation.
Subsequently,
Tang et al. (5)
pointed out that elastic anisotropy should be taken into account, although they concluded that it would have only a small effect in the case of titanium.
It is the purpose of this note to review the calculation of the elastic
interaction energy using anisotropic elasticity and to present some atomistic calculations that shed some qualitative
light on the problem.
Elastic anisotropy will affect both the strain fields ~ sik of the interstitials and the stress fields ~ D of the dislocations. However, the former ik effect is taken into account implicitly when $ s is determined from X-ray ik measurements of lattice parameter changes. Due to the symmetry of the hexagonal lattice and of the octahedral hole, (~0 ~, ~ )
using the conventional
hexagonal unit cell (x,//a, x3//c).
~s
must be of the form ik orientation of Cartesian axes for the Since the change in lattice parameters
c and a must be proportional to the atom fraction x of solute atoms,
~,
and
~3 may be found by extrapolation to x = 1 which corresponds to complete filling of the octahedral
interstices;
using the results of Andersson et al.
(4) we find 6, a .028 and ~ = .087. The volume element appropriate to these strains is the volume per interstitial atom at x = l, i.e. about ~ 2 . The effect of elastic anisotropy on dislocations on the prism plane is to modify the non-zero stresses of screw and edge dislocations (5), but because of the small deviation from isotropy found in titanium the changes are slight. The interaction energy U Ds calculated by the method of Cochardt et al. (6) applied to the c.p.h, lattice (2) using average elastic constants 917
918
SOLUTION HARDENING OF TITANIUM BY OXYGEN
= .598 x i0 ~z d / c m 2, ~ = 0.34
Vol. 3, No. 12
and isotropic elasticity is 0.14 eV for
edge dislocations, which is reduced to 0.12 eV by anisotropic elasticity. The interaction energy with screw dislocations is zero in both cases. Evidently, the elastic interaction energy with isolated interstitial atoms is an order of magnitude lower than the observed activation energy. However,
it is well established (7) that long range order appears at high
concentrations, and some short-range order ~n the form of interstitial pairs may exist at the concentrations of interest in solution hardening studies (below 1 at.%).
The principal strains within a volume element of 2ca 2 con-
taining such a dipole may be inferred from Weissman's results to have a maximum tetragona]ity of (0,0~.05); this would produce an interaction energy with edge dislocations of nearly the same magnitude as that due to isolated interstitial atoms, and an interaction energy with screw dislocations considerably smaller than this.
Even if a large fraction of the oxygen atoms were
present as interstitial pairs, the elastic distortion would appear to be insufficient to explain the observed hardening. Direct calculation of the lattice energy of crystal defects using appropriate interatomic potentials and high-speed digital computers has recently become possible, and this approach has been used to investigate the interaction between a dislocation and an interstitial atom in a planar square lattice.
The interatomic potentials (see Fig. I) were chosen to produce an
.s
"I ~ "
,.s
2r
o
-i.o FIG. I Interatomic potentials between host atoms A and interstitial atoms B. elastically isotropic two-dimensional lattice with a square unit cell of lattice parameter = 1 stable for small and large deformations.
The unit
cell undergoes a 20% expansion on introduction of an interstitial atom which interacts with the host atoms through the A-B potential (Fig. I).
Fig. 2
shows the relaxed atomic configuration for an edge dislocatio~ and interstitial far removed from the dislocation and in two sites below the extra half plane.
The interaction energies
U Ds in the two sites, in units of
Vol. 3, No. 12
SOLUTION HARDENING OF TITANIUM BY OXYGEN
919
O
(a)
(b)
Dislocation
Interstitial
( ( ( (4) Interstitial in Site 2
(c) ~nterstitial in Site I
FIG. 2 Relsxed atomic configurations for separated and interacting dislocation and interstitial the "chemical bond energy" which is the depth of the A-B potential well, were calculsted from
U Os = U D÷s - ( U e + U ~) where U D÷~
is the total
energy of the dislocation-solute configuration (Figs. 2c or 2d), and U ~ S are the energies of the separate dis]ocatio~ and irterst~tial (F~gs. 2a aod 2b).
Finite ]attices of the same size were used in calculating these ener-
gies, and the correction for image effects was neglected. reported in Table l, where
The results are
U D5 is separated into a part due to a chanEe in
the energies of the A-B bonds, and a part due to a change in the lattice energy (i.e. the elastic strain energy).
For comparison, the elastic inter-
action energy (lattice strain energy) calculated by Cochardt et al.'s method is zero for site i and -.39 for site 2.
The interstitial is repelled from
site 1 and attracted to site 2 by approximately the same total energy.
920
SOLUTION HARDENING OF TITANIUM BY OXYGEN
TABLE
Vol. 3, No. 12
1
Interaction Energies (unit:
A-B bond energy)
Chemical Bond
Lattice Strain
Total U Ds
Site i
+ 1.O0
-
.22
+ .78
Site 2
-
-
.67
-
.06
.73
An interstitial site in the planar square lattice has four near neighbours in the unstrained lattice and three near neighbours when slip by an amount b/2 has occurred.
Based on a hard sphere model, the radius of the
hole is thereby decreased by 63%, and this reduction in available space should cause a considerable increase in lattice strain when the dislocation passes over an interstitial atom.
Analogous reasoning was presented earlier
(2) as one possible cause of the hardening in titanium.
However, the present
results show that, while this effect may still be important for screw dislocations, the dilatation below the extra half plane of an edge dislocation drastically modifies the atomic arrangement so that, for the dislocation of Fig. 2a, the space availab]e in site 1 within the core is practically the same as the interstitial space in the perfect lattice.
For the particular
interstitial atom considered here, which is very "hard" and causes a change in lattice spacing of about 20%, the elastic strain energy is reduced when the interstitial moves to site 1 because of the reduction in number of nearest neighbours, thereby reducing the energy required to displace them. For the same reason, the energy of chemical binding is increased by 1.O, which produces an overall repulsion.
In site 2, the hydrostatic tension of
the dislocation reduces the strain energy of the interstitial by an amount close to that calculated from linear elasticity, and since there are still four near neighbours to this site the chemical bond energy is practically unchanged. It is not known how well two-body interatomic potentials would describe the cohesion of titanium, and even less how adequate they would be for oxygen atoms in interstitial solid solution.
A detailed atomistic calculation for
the h.c.p, titanium lattice would thus give results of doubtful quantitative significance.
Nevertheless, the work described above suggests that the
breaking of chemical bonds in the dislocation core may be a significant source of strengthening in interstitial Ti-O alloys.
Calculations by Roos
Vol. 3, No. 12
SOLUTION HARDENING OF TITANIUM BY OXYGEN
(8) on the analogous
921
Zr - 0 system have shown that the cohesive energies of
such alloys can be very large, and hence considerable
energy may be required
to break a Ti - 0 bond. The author is grateful to S. Weissman for a pre-publlcation
copy of his
paper (reference 7), and to Trent University for the provision of computer time. REFERENCES
i.
~. Conrad, Acta Met. 1__4, 1631 (Iq66).
2.
W. R. Tyson, Can. Met. Quart. ~, 501 (1968).
5.
R. Tang, J. Kratochvil and H. Conrad, Scripta Met. ~, 485 (1969).
4.
S. Anders~on, sca~.
B. Collen, V. Kuylenstierna and A. MaRne]i,
Acta Chem.
1~, 164-] (1957).
5.
Y. T. Chou, J. Appl. Phys. 5_~4, 429 (]965).
6.
A. W. Cochardt. (1955).
7.
S. Weissman and A. Shrier, "Strain D~stribution in Oxidized Al~ha Titarium Crystals", Intern~tior~l Conference on Titanium, Lon~on~ Msy 1968.
8.
B. ~oos,
G. S. Schoeck,
and H. Wiedersich,
Ark. ~ys. (Sweden] 25, 563 (]963).
Acta Met. 2, 553