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Solid solution strengthening in BCC metals I locking of dislocations by atomic sized defects
Scripta
METALLURGICA
Vol. Printed
4, pp. in t h e
69-72, United
1970 States
Pergamon
Press,
Inc
SOLID SOLUTION STRENGTHENING IN BCC METALS I LOCKING OF DISLOCATIONS BY ATOMIC SIZED DEFECTS
E. S. P. Das and M. H. Richman Division of Engineering, Brown University, Providence, R. I.
(Received
July
19,
1969;
Revised
November
19,
1969)
Introduction Recently there have been some discussions on the strength of bcc metals and the contribution of interstitial atom-dislocation bindln~ to the overall strength (1-5).
While there
is some a~reement on the strengthening effect of such interactions, there is no such general ag-~eement on the temperature influence on flow stress (5-8).
The purpose of this paper is
to comment on some phenomenological aspects (usually ignored in literature, which should be considered when analyzin~ the temperature effect) of dislocation binding by solute atoms. The force of solute atom bindin~ on a Volterra tvDe dislocation is calculated from the slope of the curve of interaction energy vs. distance (~,9).
Because of the uncertainties
at the dislocation core, the computation of interaction energy is accurate only at large distances.
Any quantitative estimate of the maximum binding force obtained by applyin ~ the
classical analysis too near the core will be an overestimate (8).
As it is difficult to
obtain an accurate value based on atomic-force calculations (10), there have been some attempts to set a maximum limit and to estimate reasonable values based on experimental data (5,6,8). It may, however, be fruitful to examine the energy-distance curve (the potential well) from a slightly different viewpoint which leads directly to a force-distance curve (force curve) reasonable for the whole re~ion.
It will then be possible to calculate the effect
of interstitial atoms on the strength. In the literature, the elastic interaction enerFv (neglecting the smaller contributions made by modulus, electrical or chemical interactions (4,11-15) is taken to vary as the inverse of the distance from the dislocation.
This hyperbolic law for a Volterra-type
interaction leads to two physlcall v unrealistic characteristics at
r = 0 :
binding energy is infinite and (b) that the force is non zero and infinite.
(a) that the Both are
unrealistic because the interaction ener~v is finite and cannot be greater than the free energy of the dislocation and of the defect (14), and the force must be zero at (Fig. i). suggested:
r = 0
To include these two desirable characteristics, the following alternatives are (a)
Select an arbltrarv smooth shape for the potential well, incorporatin~ the
69
70
SOLID
SOLUTION
STRENGTHENING
IN B C C M E T A L S
I
Vol.
above two physical requirements in the core region (0 < m < mo) ; (b)
4,
No.
i
Require in addition
that this curve coincide with more accurate analysis valid at large distances (r ° < r < -). It should, however, be remembered that though this idea is vet7 attractive as it leads to a physically realistic finite maximum energy and a zero value of force at
r = 0, the final
accurate analysis in the come region is nossible only from the consideration of atomic forces, a computation ver~ difficult to achieve at present.
At the present time, the
potential well may be specified in reruns of the free pamameters
U°
and
too, whose accurate
values are to be determined latem. Several shapes with required characterlstics are possible fop the potential well in the core region.
D. Wilsdorf (15) su~.~ested a Darabolic-hvperbolic well (P-H).
This and
ot~em possible shapes must be examined further, because they lead to different force relations and hence predict considerably different strengthenin~ effects and temperature influence.
Table I presents the results for a core radius = r . It also lists F o o A (the cross-hatched area in the sketch), the other two o important parameters in determinln~ the effect of binding. The differences in F indicate o proportional dlfferences in strengthening. But small differences in A ° predict vastly (maximum binding fomce) and
different tempez.ature dependencies as it appears in the exponent of I. that
It is therefore necessary that Ao, instead of
preferred.
Uo,
F°
or
A
9 = A exp (-f(Uo,Ao)T)--
be known as precisely as possible.
It is suKgested o too, be the basis on which a particulam curve should be
The next step would be to calculate quantitativelv the amount of hardeninK
produced in a matmix, which cannot be done unless the mean separation distance between two defects on the dislocation line is known (there is sufficient reason to believe that this need not equal that obtained from a complete random distribution (16,17). Table I also lists
F ° for the case of carbon in iron, where the tetra~onal defects
lock both screw and edge dislocations with about the same maximum binding energy (12-16,18).
Discussions As mentioned earlier, the well must (i) coincide with accurate elastic analysis valid outside the core;
(2) have a finite maximum at
r = 0
and
(3) have zero slope at
r = 0.
Under the method proposed, the well is specified by means of two analytical functions, one valid for the core
0 < r < r°
and another outside,
r° < r < ®
The two functions have
the same ordinates and first derivatives at
r = r . It is not, however, clear if the o second derivatives should also be e~ual at this distance. Justification for any one model over the other must come from reasons other than
physical measurements of external stress alone.
We shall look at the values of
Uo,
F0
and A for a final selection--all the three must have reasonable values. H-well is o unsuitable as its U is ~. H-L has finite U but is still unsuitable as F ~ 0. o o o This leaves H-P, H-C, H-Cu wells with a finite U and a zero F . Let us considem rheim o o fomce curves. H-P and H-C wells have a discontinuity in the slope at m = too, while H-Cu leads to a gradual maximum in the force, a Damabolic f o m c e curve, similam to the one
Vol.
4, No.
1
SOLID
SOLUTION
STRENGTHENING
considered by Fleischer in his rapid hardening theory. well is more attractive. eliminating any model? A
o
to
IN BCC M E T A L S
I
71
From this it may appear that H-Cu
But should we use the discontinuity in the slope as a basis for At this point the answer is not clear.
However, we can look into
for a possible answer. H-P and H-C wells give a value for A (A is proportional o o the work done by activated length of dislocation) equal to about 2/3rds that of H-Cu
well.
From Eq. (i), it is evident that these models predict vastly different temperature
dependencies of flow stress, and at least in principle we should be able to select one shape for the well in the core by a careful study of temperature dependence.
Work is in
progress to include the non-random nature of the spatial distribution of defects also in the quantitative estimate of temperature dependency of flow stress and will be published at a later date. In conclusion, a continuous smooth shape with a finite maxima is suggested to be more reasonable for the potential well than the truncated hyperbolic well. desirable values of
U°
and
F°
are analyzed.
Values of
A°
Three wells with
must be used for a further
selection.
References i.
E. Das and M. H. Richman, Brown University Research Report, Division of Engineering, GKI312/I, July 1969.
2.
N . F . Mort, W. Shockly et al. (ed.), Imperfections in nearly perfect crystals, Wiley Sons, p. 178 (1952).
3.
J. Friedel, a) Dislocations, Pergamon, N. Y. (1964); b) Electron microscopy and strength of crystals, G."Thomas et al., Ed., Interscience, N. Y., p. 505 (1963).
q.
R . L . Fleischer, a) Acta Met., Vol. i0, p. 835 (1962); b) J. Appl. Phys., Vol. 33, p. 350~ (1962).
5.
R . L . Fleischer, Acta Met., Vol. 15, p. 113 (1967).
6.
R . L . Fleischer, a) Scripta Met., Vol. 2, p. 113 (1968); b) Scripta Met., Vol. 2, p. 573 (1968).
7.
R.J. Arsenault, Scripta Met., Vol. 2, p. 99 (1968).
8.
J . W . Christian, a) Scripta Met., Vol. 2, p. 569 (1968); b) Scripta Met., Vol. 2, p. 677 (1968).
9.
A. Kumar, Acta Met., Vol. 16, p. 333 (1968).
i0.
J . P . Hirth and Morris Cohen, a) Scripta Met., Vol. 3, Vol. 3, p. 311 (1968).
p. 107 (1968); b) Scripta Met.,
ii.
A . H . Cottrell and B. A. Bilby, Proc. Phys. Soc. A, Vol. 62, p. ~9 (1949).
12.
A . W . Cochardt, et al., Acta Met., Vol. 3, p. 533 (1955).
13.
G. Schoek and A. Seeger, Acta Met., Vol. 7, p. 469 (1959).
14.
N.F.
15.
D . K . Wilsdorf.
Fiore and C. L. Bauer, Pro~I-ess in Materials Science, Vol. 13, p. 85 (1968). Private Com~mnicatlon.
72
SOLID SOLUTION STRENGTHENING IN BCC METALS I
Vol. 4, No. I
References (Cont'd) 16.
P. M. Kelly, Scripta Met., Vol. 3, p. 149 (196g).
17.
M. H. Richman and M. Cohen.
18.
O. Schoek, Scripta Met., Vol. 3, p. 239 (1969).
Unpublished.
\
',,,11-- V o l t e r r a
model
k ~J u l-
o b_
Di s t a n c e FIG.1 F o r c e - D i ~ t a n c e C u r v e
= fora
Dislocation-Defect
Interaction
Table I Ener~-Distance Curves and Maximum Force Description of I well
Equation
Core 0
<
r
<
V'
,l
r
o
o
<
U
r
<
(H)
2
0---0o l-
F~xlO 5 dynes
Area% A O
~
U
o
U--- V- ro
Hyperbolic
Hyp.-Parabolic (H-P)
F max
Outside
r
0
3.3
o
U
2 Uo
0.66
2.15
0.33U
U 0.65 _~O r o
2.12
0.35U
U U=-0.5 _~o r r o
U 0.5 o r
1.65
0.5 U
U 3 r + ( ? ) 2 ) U = - O . 5 _.oo r o o o
U 0.5 o r
1.65
0.5 U
U=- T ~-- ro
r
o
0.86r
Hyp.-Cosine (H-C)
U=-U cos o
Hyp.-binear (H-L)
u=_Uo(Z_ ~E_))2r
HVD. -Cubic (H-Cu)
U=+Uo(I_ ~ )
I~o
o
D
0
U U=-0.65
r
o
O
o
o
o
Binding force exerted on a dislocatio~ by carbon interstitials ina-iron based on a U = 0.5 eV and r = b = 2.48 A. O
O
%Area shown hatched in Fig. 3. ACKNOWLEDGEMENT The above work was supported by the National Science Foundation, Grant GKI312.
o
o
METALLURGICA
Vol. Printed
4, pp. in t h e
69-72, United
1970 States
Pergamon
Press,
Inc
SOLID SOLUTION STRENGTHENING IN BCC METALS I LOCKING OF DISLOCATIONS BY ATOMIC SIZED DEFECTS
E. S. P. Das and M. H. Richman Division of Engineering, Brown University, Providence, R. I.
(Received
July
19,
1969;
Revised
November
19,
1969)
Introduction Recently there have been some discussions on the strength of bcc metals and the contribution of interstitial atom-dislocation bindln~ to the overall strength (1-5).
While there
is some a~reement on the strengthening effect of such interactions, there is no such general ag-~eement on the temperature influence on flow stress (5-8).
The purpose of this paper is
to comment on some phenomenological aspects (usually ignored in literature, which should be considered when analyzin~ the temperature effect) of dislocation binding by solute atoms. The force of solute atom bindin~ on a Volterra tvDe dislocation is calculated from the slope of the curve of interaction energy vs. distance (~,9).
Because of the uncertainties
at the dislocation core, the computation of interaction energy is accurate only at large distances.
Any quantitative estimate of the maximum binding force obtained by applyin ~ the
classical analysis too near the core will be an overestimate (8).
As it is difficult to
obtain an accurate value based on atomic-force calculations (10), there have been some attempts to set a maximum limit and to estimate reasonable values based on experimental data (5,6,8). It may, however, be fruitful to examine the energy-distance curve (the potential well) from a slightly different viewpoint which leads directly to a force-distance curve (force curve) reasonable for the whole re~ion.
It will then be possible to calculate the effect
of interstitial atoms on the strength. In the literature, the elastic interaction enerFv (neglecting the smaller contributions made by modulus, electrical or chemical interactions (4,11-15) is taken to vary as the inverse of the distance from the dislocation.
This hyperbolic law for a Volterra-type
interaction leads to two physlcall v unrealistic characteristics at
r = 0 :
binding energy is infinite and (b) that the force is non zero and infinite.
(a) that the Both are
unrealistic because the interaction ener~v is finite and cannot be greater than the free energy of the dislocation and of the defect (14), and the force must be zero at (Fig. i). suggested:
r = 0
To include these two desirable characteristics, the following alternatives are (a)
Select an arbltrarv smooth shape for the potential well, incorporatin~ the
69
70
SOLID
SOLUTION
STRENGTHENING
IN B C C M E T A L S
I
Vol.
above two physical requirements in the core region (0 < m < mo) ; (b)
4,
No.
i
Require in addition
that this curve coincide with more accurate analysis valid at large distances (r ° < r < -). It should, however, be remembered that though this idea is vet7 attractive as it leads to a physically realistic finite maximum energy and a zero value of force at
r = 0, the final
accurate analysis in the come region is nossible only from the consideration of atomic forces, a computation ver~ difficult to achieve at present.
At the present time, the
potential well may be specified in reruns of the free pamameters
U°
and
too, whose accurate
values are to be determined latem. Several shapes with required characterlstics are possible fop the potential well in the core region.
D. Wilsdorf (15) su~.~ested a Darabolic-hvperbolic well (P-H).
This and
ot~em possible shapes must be examined further, because they lead to different force relations and hence predict considerably different strengthenin~ effects and temperature influence.
Table I presents the results for a core radius = r . It also lists F o o A (the cross-hatched area in the sketch), the other two o important parameters in determinln~ the effect of binding. The differences in F indicate o proportional dlfferences in strengthening. But small differences in A ° predict vastly (maximum binding fomce) and
different tempez.ature dependencies as it appears in the exponent of I. that
It is therefore necessary that Ao, instead of
preferred.
Uo,
F°
or
A
9 = A exp (-f(Uo,Ao)T)--
be known as precisely as possible.
It is suKgested o too, be the basis on which a particulam curve should be
The next step would be to calculate quantitativelv the amount of hardeninK
produced in a matmix, which cannot be done unless the mean separation distance between two defects on the dislocation line is known (there is sufficient reason to believe that this need not equal that obtained from a complete random distribution (16,17). Table I also lists
F ° for the case of carbon in iron, where the tetra~onal defects
lock both screw and edge dislocations with about the same maximum binding energy (12-16,18).
Discussions As mentioned earlier, the well must (i) coincide with accurate elastic analysis valid outside the core;
(2) have a finite maximum at
r = 0
and
(3) have zero slope at
r = 0.
Under the method proposed, the well is specified by means of two analytical functions, one valid for the core
0 < r < r°
and another outside,
r° < r < ®
The two functions have
the same ordinates and first derivatives at
r = r . It is not, however, clear if the o second derivatives should also be e~ual at this distance. Justification for any one model over the other must come from reasons other than
physical measurements of external stress alone.
We shall look at the values of
Uo,
F0
and A for a final selection--all the three must have reasonable values. H-well is o unsuitable as its U is ~. H-L has finite U but is still unsuitable as F ~ 0. o o o This leaves H-P, H-C, H-Cu wells with a finite U and a zero F . Let us considem rheim o o fomce curves. H-P and H-C wells have a discontinuity in the slope at m = too, while H-Cu leads to a gradual maximum in the force, a Damabolic f o m c e curve, similam to the one
Vol.
4, No.
1
SOLID
SOLUTION
STRENGTHENING
considered by Fleischer in his rapid hardening theory. well is more attractive. eliminating any model? A
o
to
IN BCC M E T A L S
I
71
From this it may appear that H-Cu
But should we use the discontinuity in the slope as a basis for At this point the answer is not clear.
However, we can look into
for a possible answer. H-P and H-C wells give a value for A (A is proportional o o the work done by activated length of dislocation) equal to about 2/3rds that of H-Cu
well.
From Eq. (i), it is evident that these models predict vastly different temperature
dependencies of flow stress, and at least in principle we should be able to select one shape for the well in the core by a careful study of temperature dependence.
Work is in
progress to include the non-random nature of the spatial distribution of defects also in the quantitative estimate of temperature dependency of flow stress and will be published at a later date. In conclusion, a continuous smooth shape with a finite maxima is suggested to be more reasonable for the potential well than the truncated hyperbolic well. desirable values of
U°
and
F°
are analyzed.
Values of
A°
Three wells with
must be used for a further
selection.
References i.
E. Das and M. H. Richman, Brown University Research Report, Division of Engineering, GKI312/I, July 1969.
2.
N . F . Mort, W. Shockly et al. (ed.), Imperfections in nearly perfect crystals, Wiley Sons, p. 178 (1952).
3.
J. Friedel, a) Dislocations, Pergamon, N. Y. (1964); b) Electron microscopy and strength of crystals, G."Thomas et al., Ed., Interscience, N. Y., p. 505 (1963).
q.
R . L . Fleischer, a) Acta Met., Vol. i0, p. 835 (1962); b) J. Appl. Phys., Vol. 33, p. 350~ (1962).
5.
R . L . Fleischer, Acta Met., Vol. 15, p. 113 (1967).
6.
R . L . Fleischer, a) Scripta Met., Vol. 2, p. 113 (1968); b) Scripta Met., Vol. 2, p. 573 (1968).
7.
R.J. Arsenault, Scripta Met., Vol. 2, p. 99 (1968).
8.
J . W . Christian, a) Scripta Met., Vol. 2, p. 569 (1968); b) Scripta Met., Vol. 2, p. 677 (1968).
9.
A. Kumar, Acta Met., Vol. 16, p. 333 (1968).
i0.
J . P . Hirth and Morris Cohen, a) Scripta Met., Vol. 3, Vol. 3, p. 311 (1968).
p. 107 (1968); b) Scripta Met.,
ii.
A . H . Cottrell and B. A. Bilby, Proc. Phys. Soc. A, Vol. 62, p. ~9 (1949).
12.
A . W . Cochardt, et al., Acta Met., Vol. 3, p. 533 (1955).
13.
G. Schoek and A. Seeger, Acta Met., Vol. 7, p. 469 (1959).
14.
N.F.
15.
D . K . Wilsdorf.
Fiore and C. L. Bauer, Pro~I-ess in Materials Science, Vol. 13, p. 85 (1968). Private Com~mnicatlon.
72
SOLID SOLUTION STRENGTHENING IN BCC METALS I
Vol. 4, No. I
References (Cont'd) 16.
P. M. Kelly, Scripta Met., Vol. 3, p. 149 (196g).
17.
M. H. Richman and M. Cohen.
18.
O. Schoek, Scripta Met., Vol. 3, p. 239 (1969).
Unpublished.
\
',,,11-- V o l t e r r a
model
k ~J u l-
o b_
Di s t a n c e FIG.1 F o r c e - D i ~ t a n c e C u r v e
= fora
Dislocation-Defect
Interaction
Table I Ener~-Distance Curves and Maximum Force Description of I well
Equation
Core 0
<
r
<
V'
,l
r
o
o
<
U
r
<
(H)
2
0---0o l-
F~xlO 5 dynes
Area% A O
~
U
o
U--- V- ro
Hyperbolic
Hyp.-Parabolic (H-P)
F max
Outside
r
0
3.3
o
U
2 Uo
0.66
2.15
0.33U
U 0.65 _~O r o
2.12
0.35U
U U=-0.5 _~o r r o
U 0.5 o r
1.65
0.5 U
U 3 r + ( ? ) 2 ) U = - O . 5 _.oo r o o o
U 0.5 o r
1.65
0.5 U
U=- T ~-- ro
r
o
0.86r
Hyp.-Cosine (H-C)
U=-U cos o
Hyp.-binear (H-L)
u=_Uo(Z_ ~E_))2r
HVD. -Cubic (H-Cu)
U=+Uo(I_ ~ )
I~o
o
D
0
U U=-0.65
r
o
O
o
o
o
Binding force exerted on a dislocatio~ by carbon interstitials ina-iron based on a U = 0.5 eV and r = b = 2.48 A. O
O
%Area shown hatched in Fig. 3. ACKNOWLEDGEMENT The above work was supported by the National Science Foundation, Grant GKI312.
o
o