Calculation of the strength of Snoek relaxation in dilute ternary B.C.C. alloys

~

Acta metall, mater. Vol. 43, No. 2, pp. 705-714, 1995

Pergamon

Copyright ~C, 1995 ElsevierScienceLtd Printed in Great Britain. All rights reserved 0956-7151/95 $9.50 + 0.00

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CALCULATION OF THE STRENGTH OF SNOEK RELAXATION IN DILUTE T E R N A R Y B.C.C. ALLOYS H. NUMAKURA, G. YOTSUIt and M. KOIWA Department of Metal Science and Technology, Kyoto University, Kyoto 606-01, Japan (Received 15 April 1994)

Abstract--The distortion around a pair of an interstitial solute atom and a substitutional solute atom in the b.c.c, lattice has been calculated by the method of molecular statics. The model crystal consists of a finite number of atoms forming a b.¢.c, crystallite and an elastic continuum surrounding the crystallite. The atoms are assumed to interact through pairwise interatomic potentials. The atomic displacements in the vicinity of an interstitial-substitutional solute pair and the elastic displacements at large distances have been calculated for the first to the fifth neighbour configurations. To describe the elastic distortion of the outer region, two elastic displacement fields of isotropic and cylindrical symmetry are employed and are utilized to estimate the volume expansion and the uniaxial distortion due to the pair. Internal friction profiles expected for dilute ternary alloys are calculated by using the results of the simulation. The observed effect of manganese on the Snoek relaxation of carbon in ~ iron has been discussed in terms of the trapping of carbon to manganese and an associated decrease of the uniaxial distortion.

1. INTRODUCTION Interstitial solute atoms in b.c.c, metals exhibit a mechanical relaxation effect, known as the Snoek effect. The presence of substitutional solutes has a strong influence on the Snoek effect [1], particularly if the substitutional solute element is the one which interacts attractively with the interstitial element. In iron, for example, the addition of vandium, chromium or manganese gives rise to a new, extra peak in the profile of the Snoek relaxation of nitrogen [2, 3]. For the snoek relaxation of carbon in ~ iron, on the other hand, the addition of such alloying elements causes no extra peak but the peak height, or the relaxation strength, is markedly reduced [4-6]. The strength of Snoek relaxation is proportional to the interstitial solute concentration and the anisotropy of the strain field around the interstitial solute atom. For dilute F e - M n - C alloys, the solubility of carbon is not affected by the addition of manganese [7]. Therefore, the reduction of the relaxation strength observed in this alloy system should be attributed to a decrease of the strain anisotropy around a carbon atom. A number of theoretical calculations of the lattice distortion around an interstitial carbon in ~ iron have been made by the method of atomistic simulation [8-11] and by the lattice statics [12-14]. The magnitudes of the anisotropy of the strain field evaluated by these methods agree fairly well with the experimental value determined from the measurement of the relaxation strength in single crystals [1]. In the present tPresent address: OMRON Corp., Nagaokakyo, Kyoto 617, Japan.

work, we have calculated the lattice distortion around an interstitial substitutional solute pair in the b.c.c. lattice by extending the method of atomic simulation for an isolated carbon atom developed by Johnson et al. [8]. We have applied an elastic displacement field of cylindrical symmetry and estimated the anisotropy of the distortion around an interstitial substitutional solute pair. When the magnitudes of the lattice distortion and the potential energies of the interstitial atom are given, profiles of Snoek relaxation in dilute ternary alloys can be calculated [15, 16]. Such profiles can directly be compared with experiments and are useful for understanding the interaction between the interstitial and the substitutional solutes. We calculate internal friction curves by using the numerical parameters obtained by the simulation, and discuss the effects observed in F e - M n - C alloys. 2. METHOD OF CALCULATION 2. I. M o d e l

A semidiscrete lattice model and molecular statics technique have been employed, which are essentially the same as those of the earlier studies [8, 17]. The model b.c.c, crystal consists of two regions: the inner, discrete region (region I) and the outer, elastic region (region II). Region I includes 531 lattice points; the atomic site at the centre is taken as the coordinate origin, and the rest are divided into 25 shells according to the distance from the origin. Here we designate the shells by (1, 1, 1), (2, 0, 0) . . . . . by taking ao/2 as the unit distance, where a0 is the lattice constant. The boundary between region I and region II is-beyond 705

706

NUMAKURA et al.: THE SNOEK RELAXATION IN TERNARY B.C.C. ALLOYS

the shell (7, 3, I), which is at thc distance of 7.68 ao/2. The atoms in region II are treated as if they were embedded in an elastic continuum. The elastic displacement u in region II satisfies the equilibrium equation for isotropic elasticity 2(1 - v) grad div u - (1 - 2v) rot rot u = 0

(1)

where v is Poisson's ratio. The general solution of this equation is the spherical harmonics function. A m o n g the infinite number of solutions, we choose the following two functions of isotropic and cylindrical symmetry

1 u~ = C~ = e,,

(2)

t'"

17 -4v (3 cos 2 0 -

u2 = C2 ~ ] _ ~

1)e~ - cos 0 sin 0e,

] (3)

where C~ and C 2 are the "strengths" of the fields. The displacement field in region I! is expressed by the superposition of u~ and u:. The distortion around a defect is characterized by the 2 tensor, which is defined as the strain per unit concentration of the defect. The components of the 2 tensor are given by 2,~ ~_= &,~/~cp, where the index p denotes one of the possible equivalent orientations of the defect and cp is the concentration of the defect in orientation p [1]. For a tetragonal defect in a cubic crystal, the 2 tensor has the form

{10 0)

(2~i)=

0

22

0

0

0

)-2

(4)

=

12rtC~ 1 + v

1

v=SEEv'J+EEv'+Ea.c,,+Eb,,c t8) t

/

i

k

n

n

where V o is the potential energy between atoms i and j, the summation index i runs over all atoms in region I, j over atoms in region I that interact with atom i, k over atoms in region II that interact with atom i. The third term is the work done against the forces required to maintain the crystal in mechanical equilibrium, and the fourth term is the elastic energy, where n = 1 for the isotropic field and n = 2 for the cylindrical field. Since the functions uj and u 2 are orthogonal, no cross terms appear in the elastic energy. The coefficients a,, are dependent on the interatomic potential, while b, are given analytically by elasticity theory as follows. 32rc 2 bI = ~/~'

where the first axis is taken to be parallel to the principal axis of the defect. The volume expansion due to a defect, which is generally equal to the trace of the 2 tensor, can be estimated from the strength of the isotropic field u~ by the following formula [18] AV

By using equations (5) and (7), the volume expansion and the shape factor can be determined in terms of CI and C2. In the simulation study of a carbon interstitial in b.c.c, iron by Johnson et al. [8], the field uj was employed as the elastic field for the outer region, but the field u2 was not. They determined the magnitude of C2 by fitting equation (3) to the relaxed positions of the host atoms in the inner region around a carbon interstitial, and estimated the shape factor. In the present calculation, both u~ and u2 are incorporated to determine the volume expansion and the shape factor simultaneously from the elastic distortion of region lI. The energy of the system, U, is defined as the sum of the pairwise interaction energies and the elastic energies

.

(5)

64~z b2 - 135N~

(%)

ll+22v+5v2 (1 - 2v) 2

/~

(9b)

where N is the number of lattice points in region I and # the shear modulus. The force exerted on atom i by atom j is given by V i~r!:

The cylindrical field u2 is useful to evaluate the magnitude of unixial distortion. A tetragonal defect in a cubic crystal gives rise to a relaxation of the compliance S ' [1] 2 c~ 6S' = ~ kT

15~I -

-

).212

47~C2 1 - v

21-22-

f~

1-2v'

where r " = r : - r ' is the vector from atom i to atom j, and the total force on atom i is

F = ~] F ~/+ ~ F '~.

(6)

where c is the defect concentration, ~ the atomic volume of the host crystal, k is the Boltzmann constant and T is the temperature. The relaxation strength observed in experiments is proportional to $S'. The strain anisotropy, or the shape factor, 2 1 - 2 2 , is related to C 2 as [19]

(7)

(10)

F:/= ~f?r! r q

.!

(11)

k

The derivatives of the energy with respect to the elastic variables C,, define the generalized forces, which act across the boundary between region I and region II ~U

aC, =

~ - - Fik ~r ~ ~ T ~ - a.-

2b.C,,.

(12)

NUMAKURA et al.:

THE SNOEK RELAXATION IN TERNARY B.C.C. ALLOYS

For the perfect, undistorted crystal, these forces should vanish, i.e.

0.4

> 0.2

F,,= _ ~ F

ik8~, c~r~ - a,, = 0

707

(13)

(a) \\t Fe-Fe Fe-Mn \~

>,,

0.0 {-

where r0k is the position vector of an atom k in the perfect crystal. As described in the next subsection, the coefficients aj and a 2 can be calculated from these equations. Stable configurations are searched for by the static relaxation technique, by following the procedure of Johnson e t al. [8]. First, an interstitial and/or substitutional solute atom is introduced into the crystal (the positions of the Solute atoms are described later). Initial displacements are given to atoms according to the elastic fields u~ and u2 with various values of C~ and Cz. Smaller or larger values than expected were used to see the effect of initial conditions, but such initial conditions turned out to give essentially the same results. Next, the forces exerted on each atom are calculated, and the atoms are displaced in the direction of the forces so as to decrease the forces between atoms. Then the values of C,, are adjusted so as to reduce the generalized forces F". These two steps are repeated until the relative changes of C, s become less than 10 4 2.2.

Potentials

For the interatomic interactions in region I, empirical pairwise potentials have been adopted for all the four combinations of atom species, namely, the host-host, host-interstitial, host-substitutional and substitutional interstitial atoms. The four potentials are shown in Fig. 1; the curves are designated as Fe-Fe, Fe-C, F e - M n and Mn-C, respectively. All the potential functions are cubic polynomials of the form V(r) = -A

1 (r

-

-

A2) 3 -[- A 3 r -- A 4 .

(14)

The parameters Ai and the cut-off distances are listed in Table 1. The potential for the host atoms is the iron potential developed by Johnson [17], which is fitted to the lattice constant and the elastic constants of ~ iron. This is a cubic spline function truncated after the second nearest neighbour distance. The host interstitial solute potential is the Fe-C potential of Johnson e t al. [8]. This potential is matched to the migration energy of carbon, 0.86 eV, zero activation volume and the binding energy between carbon and a vacancy, 0.41 eV. For the interaction between the

IJJ -0.2 0.4

J..5 2.0 Distance / (ao/2)

.o

o. 4 (b) >¢

02

>,

1t

Mn-C\

o.o

/Fe_ C

\ \

e"

w

\

-o.2 -0.4

A \

0.5

I 0 ~..5 Distance / (ao/2)

Fe-Fe Fe C Fe Mn Mn-C A M 43,2

T

2.0

Fig. 1. Interatomic potentials. (a) Fe-Fe (solid curve) and Fe Mn (dashed curve). (b) Fe~C (solid curve) and Mn~C (dashed curve). host substitutional atoms and the substitutionalinterstitial atoms, the F ~ M n and M n - C potentials proposed by Gouzou e t al. [20] are adopted; these potentials are cubic polynomials similar to the Fe-Fe and F e C potentials. The Fe Mn potential has a minimum at a separation 2% larger than that of the Fe Fe potential, thereby reproducing the expansion of the iron lattice by the addition of manganese. The Mn~C potential is derived from the Fe-C potential by modifying the distance scale so as to reduce the equilibrium distance by 7%. The depths of the Fe Mn and Mn-C potentials are kept to be equal to the depths of the corresponding Fe-Fe and F e ~ potentials. With the iron potential adopted for the host atoms, the coefficients a, in equation (8) should be zero because no extra hydrostatic pressure is required to maintain the crystal in equilibrium. However, as found by Johnson [17], this is not so because the surface of region I is not ideally spherical. The values of aNcan be evaluated from equation (13) by calculating the bond sums explicitly, and are found to be: aj = 0.056 e V / ( a 0 ) 3 and a2 = - 2 . 1 x 10 -8 e V / ( a o ) 3. The value ofa~ is larger than that of Johnson by 40%;

Table 1. Potential parameters A~ and cut-off distances r, Potential

.5

A~/eV(ao/2 ) ~

A2/ao/2

A3/eV(ao/2 ) i

Aa/e V

rc/ao/2

6.421488 1.869241 3.260593 9.8400 2.5215 5.1621

2.166371 2.178901 2.144338 1.5636 2.1560 1.5139

3.866806 0.6833555 0.6676555 1.2670 0.64307 1.0217

7.436448 1.581570 1.547967 2.1560 1.5115 1.7219

1.678322 2.097902 2.405594 1.7671 2.4476 1.7708

Reference [17] [8] [20] [20]

708

NUMAKURA et al.:

THE SNOEK RELAXATION IN TERNARY B.C.C. ALLOYS

Table 2. Displacements of host atoms around an isolated interstitial solute atom. Units: ao/2 This work

Johnson et al. [8]

Shell No.

lj

/2

l~

A&

Ax,

21x3

Ax t

Ax 2

Ax,

1 2 3 4 5 6 7 8 9 10 11 12 13 14

0 1 2 1 0 2 3 2 3 2 4 1 3 4

0 1 0 I 0 2 I 0 l 2 0 I 3 2

1 0 I 2 3 1 0 3 2 3 1 4 0 1

0 -0.043 0.015 0.024 0 0.011 -0.013 0.006 - 0.001 0.010 -0.006 0.005 0.006 0.006

0 -0.043 0 0.024 0 0.011 -0.009 0 0.001 0.010 0 0.005 -0.006 0.004

0.225 0 (I.018 0.047 0.048 -0.004 0 0.020 0.007 0.017 0.002 0.019 0 -0.001

0 -0.042 -0.014 0.025 0 -0.010 -0.012 0.007 0 0.010 -0.005 0.006 -0.005 -0.004

0 -0.042 0 0.025 0 0.010 0.008 0 0.001 0.010 0 0.006 0.005 0.003

0.225 0 0.017 0.045 0.046 -0.004 0 0.018 0.005 0.015 0.002 0.016 0 -0.001

the cause of this discrepancy is not clear, but it may come from a difference in computational precision because the calculation involves a large number of summation of small quantities. The coefficient a2 is practically zero because the cylindrical field u2 is orthogonal to hydrostatic (isotropic) displacements. The numerical values of b,,s are calculated by inserting N = 5 3 1 , v = 0 , 2 5 and I~ = 1.40eV/(a0) 3 into equations (ga) and (9b): b I = 0.0696 eV/(a 0)e and b 2 = 0.0106 eV/(a0) 6. The values of Poisson's ratio v and the shear modulus /~ are obtained by Voigt average of the anisotropic elastic constants of the model iron crystal given in Johnson's paper [17]. 3. RESULTS 3. I. Lrolated interstitial solute To check if the use of the cylindrical field u 2 gives a reasonable estimate of the uniaxial distortion, an isolated interstitial solute atom in an octahedral site is studied first. An interstitial solute atom is inserted at the position (0, 0, l). Note, however, that the origin for the elastic fields u~ and m is set at the central host-atom position, (0, 0, 0). The displacements of the host atoms after relaxation are shown in Table 2, together with those reported by Johnson et al. Here the host atoms are designated by the coordinates (l~, 12, 13) measured from the interstitial atom. The host atom position in the perfect crystal is thus r = l l e ~ + 1 2 e 2 + 1 3 e 3 , where el = (a0/2, 0, 0), e 2 = (0, ao/2, 0), and e3 = (0, 0, al/2 ).

Table 3. Displacements o f host a t o m s around a substitutional solute atom. Units: a~,,'2 Shell No.

lj

l2

/3

Ax I

1 2 3 4 5 6 7 8 9 10 11

1 2 2 3 2 4 3 4 4 3 5

1 0 2 1 2 0 3 2 2 3 I

1 0 0 I 2 0 1 0 2 3 I

0.0097 0.0009 0.0024 0.0011 0.0037 0.0001 0.0012 0.0007 0.0009 0.0015 0.0(103

~x 2

Ax 3

0.0097 0 0.0024 0.0006 0.0037 0 0.0012 0.0005 0.0006 0,0015 0.0001

0.0097 0 0 0.0006 0.0037 0 0.0004 0 0.0006 0.0015 0.0001

The displacements obtained in this study are slightly, but systematically, larger than those calculated by Johnson et al. F r o m the present simulation, the coefficients CI and C 2 are found to be 0.0151 f~ and 0.033 ~, respectively. These give the following values for the volume expansion and the shape factor: AV = 0.34 ~ and 21 - 22 = 0.62. As summarized later in Table 5, these values are in reasonable agreement with the experimental values for carbon in ~ iron viz. A V = ( 0 . 6 8 - 0 . 9 ) f ~ [21] and 121- 221= 0.78 ~ 0.87 [22, 23], and the agreement is slightly better than the previous results of Johnson et al. They found that imposing the elastic field u t did not affect significantly the atomic configurations near the interstitial solute atom. Here it appears that the application of u2 is not very important for the atomic displacements either, but distinguishable differences are found in the elastic parameters. 3.2. Isolated substitutional solute Since no description is given by Gouzou et al. as to atomic displacements or elastic distortion calculated with the F e - M n potential, we have simulated an isolated substitutional solute atom by using this potential. A substitutional solute atom is placed at the position (0, 0, 0) by replacing the host atom. In this calculation, only the field uj is used for the elastic distortion. The displacements of the host atoms around the substitutional solute atom are shown in Table 3 for some close neighbour shells. One can see that the displacements are propagating in (111) directions. The strength of the isotropic elastic displacement field is found to be C~ = 0.0060 O. This leads to AV = 0.13 fL which is larger than the experimental value of 0.06 f~ for manganese in c~ iron [21]. 3.3, Interstitial~ubstitutional

solute atom pairs

Interstitial-substitutional solute atom pairs of the first to the fifth neighbour configurations have been simulated. The geometrical configurations of the pairs are shown in Fig. 2(a-f). As the initial condition, the interstitial solute atom is placed at (0, 0, 1), except for the alternative configuration of the second neighbour pair, and the substitutional solute atom at

N U M A K U R A el al.:

THE SNOEK R E L A X A T I O N IN T E R N A R Y B.C.C. ALLOYS

(a)

(b)

709

(c)

(f)

(e)

(d)

Fig. 2. Geometrical configurations of the interstitial substitutional solute atom pairs. (a) The first neighbour pair, (b) and (c) the second neighbour pair, (d) the third neighbour pair, (e) the fourth neighbour pair, (f) the fifth neighbour pair. 'i' and 's' denote respectively the interstitial and the substitutional atom. The double circle indicates the central site (0, 0, 0).

the a p p r o p r i a t e p o s i t i o n s as s h o w n in each figure. In the alternative c o n f i g u r a t i o n o f the s e c o n d n e i g h b o u r pair, Fig. 2(c), the interstitial a t o m is placed at the p o s i t i o n (1, 1,0) a n d the s u b s t i t u t i o n a l a t o m at (0, 0, 0). T h e central site, (0, 0, 0), indicated by the d o u b l e circle, is a l w a y s t a k e n as the c o o r d i n a t e origin for the elastic d i s p l a c e m e n t fields. A l t h o u g h the seco n d , third a n d f o u r t h n e i g h b o u r p a i r s do n o t p o s s e s s the uniaxial s y m m e t r y a l o n g the third c o o r d i n a t e axis, all the c a l c u l a t i o n s h a v e been d o n e by a p p l y i n g u~ a n d u 2 in the s a m e w a y as the c a l c u l a t i o n o f an isolated interstitial a t o m . In the fifth n e i g h b o u r c o n f i g u r a t i o n , the distance b e t w e e n the interstitial

a n d s u b s t i t u t i o n a l a t o m s is o u t o f the i n t e r a c t i o n r a n g e o f the M n C p o t e n t i a l f u n c t i o n , so t h a t the t w o solute a t o m s do n o t interact directly. T a b l e 4 s h o w s the d i s p l a c e m e n t s o f a t o m s a r o u n d the interstitial solute a t o m at the first n e i g h b o u r site o f the s u b s t i t u t i o n a l solute a t o m . I n this table, a t o m s are d e n o t e d by the shell n u m b e r s with respect to the interstitial a t o m , w h o s e p o s i t i o n is n o w t a k e n as (0, 0, 0); each shell m is divided into two, m + a n d m - , a c c o r d i n g to the sign o f the third c o o r d i n a t e . T h e s u b s t i t u t i o n a l a t o m is d e n o t e d by " 1 - " . While the first a n d the s e c o n d c o m p o n e n t s o f the displacem e n t s are nearly e q u a l to t h o s e a r o u n d an isolated

Table 4. Displacements of host atoms around the first neighbour pair of the interstitiM-substitutional solute atoms [of. Fig. 2(a)]. Units: a0/2. The interstitial atom position is (0, 0, -0.069) after relaxation Shell No.

l,

12

1+ 1 2 3+ 3 4+ 4-5+ 5-6+ 6 7 8+ 8-9+ 9-10+ 1011 + 11 12 + 1213 14+ 14--

0 0 1 2 2 1 1 0 0 2 2 3 2 2 3 3 2 2 4 4 I I 3 4 4

0 0 1 0 0 1 I 0 0 2 2 1 0 0 1 1 2 2 0 0 1 1 3 2 2

l~

-

-

1 1 0 1 1 2 2 3 3 1 1 0 3 3 2 2 3 3 1 1 4 4 0 1 1

Axj

0 0 0.034 --0.008 0.011 0.016 0.025 0 0 0.008 0.009 -0.010 0.005 0.006 0 -0.001 0.007 0.010 -0.004 --0.005 0.004 0.005 0.005 -0.004 --0.005

A.x2

0 0 0.034 0 0 0.016 0.025 0 0 0.008 -0.009 0.007 0 0 0.001 0.001 0.007 0.010 0 0 0.004 0.005 0.005 0.003 0.003

Ax 3

0.246 0.096 0.079 0.085 0.056 0.104 0.026 0.106 0.030 0.068 0.074 0.070 0.085 0.052 0.075 0.064 0.082 0.054 0.072 0.068 0.084 0.053 0.070 0.069 0.070

Ax3- 0.069 0.177 0.165 0.010 0.014 0.013 0.035 - 0.043 0.317 -0.039 0.001 0.005 0.001 0.016 -0.017 0.006 0.005 0.013 -0.015 0.003 -0.001 0.015 -0.016 0.001 0 0.001

710

NUMAKURA et al.:

THE SNOEK RELAXATION IN TERNARY B.C.C. ALLOYS

Table 5. Displacement fields and binding energies of interstitial-substitutional solute atom pairs Coetticient/10-2 f~ Strain parameters Defect Isolated interstitial

CI Experiment C in • Fe Simulation¶ 1.24 This study 1.51 Isolated Experiment substitutional Mn in ~ Fe This study 0.60 i s pair 1st neighbour 1.35 2rid neighbour 1.89 3rd neighbour 2.07 4th neighbour 2.09 5th neighbour 2.07 tRef. [21]. :~Ref. [22]. §Ref. [23]. 'lRef. [8].

interstitial atom (Table 2), the third component, Ax3, is very different. The distance between the two atoms at the first neighbour positions, i.e. above and below the interstitial atom, is 1 . 2 4 6 - ( - 1.096) = 2.342, which is shorter than the corresponding value for an isolated interstitial atom, 1.225 x 2 = 2.450, by 4.4%. This comes from the small inter-atomic distance between the substitutional and the interstitial atoms, which is due directly to the nature of the Mn C potential. Apparently, Ax 3 is significantly large even for the shells which are quite distant from the interstitial atom. However, this happens simply because the interstitial atom is shifted towards the substitutional atom from the original position by about 0.069a0/2, in the negative direction. In the last column of the table are listed the numerical values of the quantity Ax3 - 0.069 ao/2. After this subtraction, the atomic displacements turn out to have the same trend as those in Table 2 for the neighbour shells farther than the fifth shell. However, the strengths of the displacement fields in the outer region are found to be smaller than those for an isolated interstitial atom, as described below. The elastic parameters of all the defects calculated are summarized in Table 5, together with some relevant experimental data. The two configurations of the second neighbour pair have been found to give essentially the same results. For the interstitialsubstitutional solute pairs, the total volume expansion is equal to the sum of those associated with the individual defects for the third, fourth and fifth neighbour pairs, but is notably smaller for the first neighbour pair. The shape factor is also small for the first neighbour pair; it is 0,8 times the value for an isolated interstitial atom. An interstitial atom in the first neighbour site contributes to the relaxation of only 64% in magnitude, compared with an isolated interstitial atom. The close pair configurations are associated with small distortion and the total energy of the system containing the close pairs has been found to be low. One can define the binding energy of an interstitial-

C2

AV/~ 0.684).9t

)'t -22 0.78~-0.87§

3.20 3.30

0.28 0.34 0.06t

0.60 0.62

2.65 3.33 3.30 3.30 3.43

0.13 0.31 0.43 0.47 0.47 0.47

0.50 0.63 0.62 0.62 0.65

B/eV

0.20 0.05 0.01 -0.03 (0)

substitutional pair as the difference between the total energy of a reference system that contain the solute atoms far apart and the energy of a system containing the pair. Here we take as a reference system the fifth neighbour configuration, in which the interstitial and the substitutional atoms do not interact directly. The binding energies thus obtained are listed in the last column of Table 5; it is 0.20 eV for the first neighbour pair and 0.05 eV for the second neighbour pair. 4. DISCUSSION 4.1. Estimation o f A V and I)q -- 221 For the purpose of estimating the elastic distortion around a defect or a defect complex, the superposition of the isotropic field u~ and the cylindrical field u2 has been applied to the outer region of the model crystal. The volume expansion and the shape factor for an isolated interstitial solute atom obtained by this method are in accordance with the previous results of Johnson et al., as well as the experimental data for carbon in c~ iron. This method appears to give reasonable results also for the interstitial substitutional pairs. In the calculations the coordinate origin of the elastic field was not located at the interstitial atom position. However, even the two configurations for the second neighbour pair, Fig. 2(b) and (c), have given essentially the same results. This suggests that the exact matching of the defect axes and the coordinate axes of the uniaxial field is not very important. Although the results are consistent with the characteristics of the interatomic potentials, the potentials are all empirical ones, and the host-substitutional and substitutional-interstitial potentials do not have sound physical basis. It is indeed desirable to develop a model which better describes atomic interactions in the ternary alloy of interest. If an appropriate model is available, one can apply the present method to estimate the elastic distortion around a complex of solute atoms.

NUMAKURA et al.: 4.2. Effect o [ substitutional relaxation profile

THE SNOEK RELAXATION IN TERNARY B.C.C. ALLOYS

solute

on

the

Shock

As mentioned in the Introduction, the presence of substitutional solutes which attract interstitial solutes influences the profile of Snoek relaxation. For example, reduction of the relaxation strength is one of the characteristic effects of manganese on the Snoek relaxation of carbon in e iron. In a dilute ternary alloy where the interaction between the substitutional and interstitial solute atoms exhibits the feature found by the present simulation viz. the uniaxial distortion is smaller for the trapped interstirials, the integrated strength of the Snoek relaxation would be decreased by the addition of the substitutional solute. If the migration energy of an interstitial solute atom, the binding energy to a substitutional atom, and the magnitude of the tetragonal distortion are known for the interstitial sites around a substitutional solute atom, one can calculate the spectrum of the Snoek relaxation [15, 16]. Here we show some results of such calculations and discuss the observed effect in F e - M n - C alloys. This theory [15] considers the migration of interstitial solute atoms over octahedral sites in the presence of substitutional impurity atoms under stress, on the assumption that jumps of the interstitial atoms occur according to a potential energy, as illustrated schematically in Fig. 3. It is assumed that the concentration of the substitutional and the interstitial solutes are low, namely < 1 at.%, and that the substitutional atoms are distributed randomly and are immobile. We further assume that (1) the influence of the substitutional solute atom extends up to the second neighbour octahedral interstitial position [16], and (2) the interstitial atoms are distributed over the three types of interstitial sites viz. the first neighbour sites, the second neighbour sites and farther sites,

E12 3 I I I I

I I I I

I I I I

Fig. 3. Potential energy diagram for interstitial solute atoms migrating over octahedral sites in the neighbourhood of a substitutional atom (solid circle). Numbers 1, 2 and 3 denote respectively the first, second and third neighbour interstitial sites. E~2 and E~ are the activation energies of the jumps between the sites 1--,2 and 2 4 l .

7ll

according to thermal equilibrium. Adjustable parameters in a potential energy diagram are therefore the activation energies EIE, E21 and E23, where Em, denotes the activation energy of the jump from the ruth neighbour site to the nth neighbour site. The binding energies for the first and the second nearest neighbour sites are given respectively by B1 = EIE - E21 + E23 - E and B2 = E23 - E. In Figs 4 and 5 are shown potential energy diagrams (a) and calculated internal friction vs temperature curves for different concentrations of the substitutional solute. The curves are those expected for dilute, polycrystalline iron alloys containing carbon in solution measured in torsion with a frequency of 1 Hz. The numerical parameters used in the calculations are as follows: shear modulus # concentration of interstitial solute Co shape factor for the first neighbour sites shape factor for farther sites binding energy of the first neighbour pair B~ binding energy of the second neighbour pair B2 attempt frequency Zo ~ migration energy E

= 86 GPa, = 0.04 at.%, = 0.62, = 0.78, = 0.20 eV, = 0.05 eV, =4.3 × 1 0 1 a s = 0.86 eV.

-j,

The last two are those for the thermally activated jump of the interstitial solute atom unaffected by the substitutional solute atom. In the diagram of Fig. 4, the potential energies of the first and second neighbour sites are lowered by B~ and B2 with the levels of the saddle point energies kept unchanged; the activation energies are thus simply the sum of the original value and the binding energy: E l 2 = E + B l , E z l = E23 = E + Be. The contributions to the internal friction of the interstitial atoms in the first and second neighbour sites appear at higher temperatures than that of the others. The resultant internal friction profile is similar to the experimental profile observed for Fe-V-N alloys [2]. Figure 5 shows the case where variations of saddle point energies are introduced. In the present work we have not calculated saddle point energies, but Gouzou et al. [20] calculated the energy map for a carbon atom in the vicinity of a manganese atom by using the same interatomic potentials; they found that the activation energy of the jump of a carbon atom from the first to the second neighbour site is lower than the activation energy in the absence of a mangenese atom by about 0.1 eV. On the basis of this result, the activation energy E12 is lowered from the original value of 0.86 eV to 0.76 eV in the example shown in Fig. 5. Then the contribution of the interstitial atoms in the first neighbour site appears at lower temperatures than the temperature of the original Snoek peak. It is important to note that, for a given binding energy, an extra peak, which is the contribution of

712

N U M A K U R A et al.:

THE SNOEK RELAXATION IN T E R N A R Y B.C.C. ALLOYS

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T/K Fig. 5. (a) An energy diagram of an interstitial solute atom where the potential energy and the saddle point energy are changed for both the first and second neighbour sites. EIz=0.76eV, Ezt=0.61eV, Ez3=0.91eV (B 1=0,20eV B2 = 0.05 eV). (b), (c) and (d) Calculated internal friction profiles for the substitutional solute concentrations of 0.04, 0.2 and 1 at.%; the curves have the same meanings as in Fig. 4.

NUMAKURA

et al.:

THE SNOEK

RELAXATION

" t r a p p e d " interstitial atoms, may appear at either lower or higher temperatures, depending on the saddle point energies. In the experimental profile of Fe Mn C alloys, however, the Snoek peak shows only a slight shift to lower temperatures and broadening, but no extra peak appears [7, 24]. These features indicate that the contributions of trapped carbon atoms appear at the same temperature range as the original Snoek peak. Assuming that the attempt frequencies of trapped and untrapped carbon atoms are approximately equal, it is thus suggested that the activation energies of trapped carbon atoms are not very different from the activation energy of untrapped ones. Figure 6 shows another potential energy diagram and a set of internal friction profiles calculated from the diagram. In this example, the binding energies for the first and the second neighbour sites and the shape factor of the first neighbour site are arbitrarily chosen as 0.14 eV, 0.02 eV and 0.54 so as to reproduce the features of the experimental profile of F e - M n - C alloys; other parameters are the same as those of the previous examples. There is only one peak, but it consists of the contributions of the interstitial atoms in the first, second and farther sites. The peak becomes smaller and broader and is shifted to a lower temperature, as the concentration of the substitutional solute is increased. Several experimantal investigations have been made on the interaction between carbon and manganese in e iron. In the early studies by Wert [4] and Graham and Wuttig [5], it was concluded that manganese had no effect on the internal friction or magnetic after-effect of carbon. However, these issues have been revisited recently and it has turned out that both the internal friction and magnetic after-effect exhibit characteristic influences of manganese [7, 24]: decrease of the peak height, increase of the peak width, and a slight shift of the peak toward lower temperatures. Apart from these mechanical/'magnetic relaxation studies, Abe et al. investigated the aging behaviour of the electrical resistivity of low-carbon steels; they reported that the binding energy of a Mn C pair was 0.46 eV [25], but corrected later to 0.26 eV [26]. If the binding energy was so large, the terminal solubility of carbon would be increased significantly by the addition of manganese through trapping. Assuming the first neighbour site of a substitutional solute atom be the trap site, the concentration of the trapped interstitial solute is given by

t 1 - 2cs + 2cs exp ~ where co is the total concentration of the interstitial solute and c~ the concentration of the substitutional solute. The fraction of the trapped interstitials, C~rap/CO, is equal to, for example, 0.29 for c, = 0.01,

IN TERNARY

B.C.C. A L L O Y S

713

B1 = 0.26 eV and T = 1000 K. The untrapped interstitials are distributed over the octahedral interstitial sites which are not the first neighbour sites of the substitutional solute. F r o m the geometry of the b.c.c. lattice, the fraction of such sites is 1 - 2 q and

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714

NUMAKURA et al.: THE SNOEK RELAXATION IN TERNARY B.C.C. ALLOYS

amounts to 98% of the total number of octahedral sites for c~ = 0.01. The number of interstitial atoms soluble over these sites can reasonably be assumed to be 0.98 times the solubility in the unalloyed matrix. This quantity corresponds to the concentration of the untrapped interstitial solute in the alloy matrix, and constitutes 1 - 0.29 = 0.71 of the total interstitial atoms in solution for the conditions above. Therefore, the terminal solubility of the interstitial solute in the alloy matrix containing 1 at.% substitutional solute is 0.98/0.71 = 1.38 times the terminal solubility in the unalloyed matrix. This means that the terminal solubility of carbon in ~ iron would be increased by 38% at 1000 K by the addition of 1 at.% manganese. However, this does not agree with the experimental results [7]; the termianl solubility of carbon at 1000 K is almost unchanged by the addition of manganese. Therefore, the binding energy must be considerably lower, probably less than 0.2 eV. A binding energy of 0.14 eV would give rise to an increase of the solubility at the same temperature by about 9%. This can be reconciled with the experimental observation, within the accuracy of the experimental data. In summarizing, the effect of manganese on the Snoek relaxation of carbon in c~ iron can be accounted for in terms of local attractive interaction between carbon and manganese atoms with a binding energy < 0 . 2 e V . The decrease of the relaxation strength is attributable to reduction of the uniaxial distortion around a carbon atom by the presence of manganese atoms. 5. CONCLUSIONS The distortion around a substitutional-interstitial solute atom pair is estimated theoretically by the semidiscrete model calculation with the use of the isotropic and cylindrical elastic displacement fields. The calculated magnitudes of the volume expansion and the uniaxial distortion exhibit a trend which can be understood from the nature of the interatomic potentials. If any appropriate physical models for the atomic interactions are available, this method will be

useful for the cvaluation of the elastic distortion due to defect complexes. REFERENCES 1. A. S. Nowick and B. S. Berry, Anelastic Relaxation in Crystalline Solids. Academic Press, New York (1972). 2. L. J. Dijkstra and R. J. Sladek, Trans, metall. Soc. A.LM.E. 197, 69 (1953). 3. I. G. Ritchie and R. Rawlings, Acta metall. 15, 491 (1967). 4. C. Wert, Trans. metall. Soc. A.1.M.E. 194, 602 (1952). 5. R. H. Graham and M. Wuttig, Scripta metall. 3, 9 (1969). 6. K. Ushioda, N~ Yoshinaga, H. Saitoh and O. Akisue, Defect Diffusion Forum 95-98, 375 (1993). 7. H. Saitoh and K. Ushioda, 1SIJ Int. 29, 960 (1989). 8. R. A. Johnson, G. J. Dienes and A. C. Damask, Acta metall. 12, 1215 (1964). 9. D. S. Richter and D. O. Welch, Scripta metall. 12, 831 (1978). 10. B. O. Hall, Metall. Trans. A 11, 1315 (1980). 1l. V. Rosato, Acta metall. 37, 2759 (1989). 12. J. W. Flocken, Phys. Rev. B 4, 1187 (1971). 13. V. K. Tewary, J. Phys. F: Met. Phys, 3, 1515 (1973). 14. A. Sato, Y. Watanabe and T. Mura, J. Phys. Chem. Solids 49, 529 (1988). 15. M. Koiwa, Phil. Mag. 24, 81 (1971). 16. M. Koiwa, Phil. Mag. 24, 107 (1971). 17. R. A. Johnson, Phys. Rev. 134, A1329 (1964). 18. J. D. Eshelby, Solid St. Phys. 3, 79 (1956). 19. H. B. Huntington and R. A. Johnson, Acta metall. 10, 281 (1962). 20. J. Gouzou, J. Wrgria and L. Habraken, C. r. M. metall. Rep. 33, 65 (1972). 21. W. B. Pearson, A Handbook of Lattice Spacings and Structures of Metals and Alloys. Pergamon Press, New York (1958). 22. H. Ino and Y. Inokuti, Acta metall. 20, 157 (1972). 23. J. C. Swartz, J. W. Schilling and A. J. Schwoeble, Acta metalL 16, 1359 (1968). 24. G. Yotsui, Master thesis, Graduate School of Kyoto University (1990) (in Japanese). 25. H. Abe, T. Suzuki and S. Okada, Trans. Japan Inst. Metals 25, 215 (1984). 26. H. Abe, "The interaction of interstitial solute with substitutional solute in the solid solution of iron and its role in the formation of annealing textures in lowcarbon steels". A note for the lecture given at The Central Iron and Steel Research Institute in Beijing (1988).