Stress-induced long-range ordering snoek atmosphere

Scripta METALLURGICA

Vol. 22, pp. 773-777, 1988 Printed in the U.S.A.

STRESS-Ih~UCED

I.ONG-RANGE ORDERING

Pergamon Press plc All rights reserved

SNOEF ATMOSPHERE

Fang-Xing Jiang and Ton~-Yi Zhang Dept. of Physics Beijing University of Iron & Steel Technology Beijing 100083, China (Received November 23, 1987) (Revised April i, 1988)

Introduction The Cottrell atmosphere(1) and the Snoek atmosphere(2,3) are basic physical concepts of metallurgy, and they explain many facts in the area of material science. The name of the Snoek atmosphere originated from the fact that the physical mechanism of the dynamic ordering Snoek atmosphere is the same as Snoek relaxation(stress-induced ordering) proposed by Snoek(4). However, the concept of the Snoek atmosphere and the concept of the Snoek relaxation involve different subjects. The Snoek atmosphere involves the interaction between a moving screw dislocation and the interstitial atmosphere of the tetragonal distortion around this dislocation, but the gnoek relaxation involves the interaction between the external stress and the interstitial atom of the tetragonal distortion. We discuss the interaction of three entities, the external stress, the screw dislocation and the interstitials, and advance the new concept of the stress-lnduced long-ranGe orderin~ Snoek atmosphere to outline this kind of interaction. We put forward this problem because the character of hydrogen in ~-Fe is different from that of other interstitials and it is difficult to explain by the old theory. Recent results show that hydrogen will soften ultra-high purity iron when the test temperature exceeds a critical value(5,6): it can promote the multiplication and motion of screw dislocations so that it induces >lastic deformation when it enters ~-Fe. 9hese effects have been observed by means of the in situ high-voltage electron microscope(7,8,9). In particular, I~ong's work indicates that hydrogen is trapped in the elastic stress field around the dislocation instead of at the dislocation core(10). The hydrogen-cold-work peak in ~-Fe has a different character from that of other interstitials(11,12). The new thoery based on the new concept proposed by our paper will give a satisfactory explanation to these behaviors(13,14). The Ouasi-Static

Stress-Induced

Tong-Range

Ordering Snoek Atmosphere

It is well known that the elastic interaction energy between the screw dislocation and the interstitial atom of the tetragonal strain field is that(2,3) u i = Acos(@-(i-1)2~/3)/r,

r >b

i=1,2,3

(I)

where the dislocation line and the Burgers vector are both in the C111) direction, the constant A = ~ b G a 3 ( ~ 1 1 - E 2 2 ) / 3 , i=1,2,3 stands for the largest strain component ~11 in the fI00),[010~ and [001} directions, respectively, b=~-~a/2 the Burgers vector and ~ the shear modulus, as shown in Fig.(1). The equilibrium concentrations of the interstitials around the screw dislocation under the Fermi-Dirac distribution can be represented as follows: ci = CoeXp(-ui/kT)/(1+CoeXp(-ui/kT)), The radius of the interstitial

atomsphere

Co,~I,

i=1,2,3

formed around the screw dislocation

773 0036-9748/88 $3.00 + .00 Copyright (c) 1988 Pergamon Press plc

(2) is

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ORDERED SNOEK ATMOSPHERE

defined

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as

r o = A/kT (3) Since the [ 1 1 1 ~ - a x i s is of a trigonal symmetry, the equivalant positions of the lowest energy are a t @ = ~ , 5~/3 and 9/3 radian for ~11 in the CIO0], (010] and {0011 directions, and u i and c i are also of a trlgonal symmetry and the direction of the maximum value ci max is the same as that of the minimum value ui min. T ' " "" m _n other words, Ci,ma~ amd Ui.min are both zn the (211], (121] and {I12] azrections on the (0~I~, (101) and'(~10) planes, respectively, because the dislocation line is in the {111~ direction, as shown in Fi~.(1) and Fig.(2). The interaction energy per unit volume between the interstitials and the screw dislocations is that ~u = 2 ~iuici/a 3 Integrating ~ u over teraction energy U

the volume

(4 )

around

the dis]ocation,

we obtain

the total in-

U = ~'v~udV

(5)

In general, F=-vU since the origin of the coordinate system (xoy) has been chosen on the screw dislocation. _Trstead of F=-vU, the force of the interstitials acting on the screw dislocation is that

(6)

=vr

Since c i and u i are both of tri~onal symmetry, the com!:o~ents F i of ~ produced by the interstitial atmosphere ci at the 1,2 and 3 interstitial positions, respectively, are also of the trigonal symmetry, and its directions are the same as that of Ci.max and Ui,min, as shown in Fig.(3). In other words, the resultant force F of Fi acting on the screw dislocation is zero. Let us discuss the behavior of the interstitial atmosphere when an external stress is applied under the quasi-static condition of the static screw dislocation. When a tensile stress ~ is applied along the ~O01] direction, the elastic interaction energy between the external stress and the strain field of the interstitial atom with ~11 in the (100],[010] and [001] directions is that, respec tively, E I = E 2 = -a3~22~

,

E 3 = -e 3 ~11 ~

(7)

Then the interactlon energy u i between the interstitials stress field of the screw dislocation, is given as u i = u i + El,

and the resultant

i=1,2,3

(8)

Clear_y, ~ , E 2 and E~ are not of trl~onal symmetry around the screw dlslocation. and neither are ul, u~ ard u~. A c c o r d z n ~ to the deflnltzon of a quasl-statlc condition, the concentration of interstitials around the dislocation will achieve a new equilibrium state by diffusion, since the dislocation is static and the interaction energy is changed from ui to u~. mhe equilibrium concentrations c~ of the new atmosphere are also not,of tri~onai symmetry, and c~ in the region from ~ 2 ~ to {~2~] is larger than c I and c 2 in the other regions. The radius of the new atmosphere is approximately equal to r o for Ei<~ kT. In other words, a new kind of interstitial atmosphere is formed around the screw dislocation and is called the stress-induced long-range ordering Snoek atmosphere and the ordering scale is about equal to r o. The stress-induced long-range ordering Snoek atmosphere is different either from the Snoek atmosphere proposed by Schoeck and Seeger(2,3), or from the stress-induced short-range ordering in Snoek relaxation(4). Eq.(4) becomes •

•

|

•

~u

=

2

~..

uic~/a3

.

.

.

.

.

.

(9)

i

In the same way,the components F~ of the new atmosphere acting on the screw dislocation are also not of the trigonal symmetry and F~ in the ~ 1 1 ~ ] direction is larger than F~ and F~ in other directions so that the stress-induced long-range

•

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ORDERED SNOEK ATMOSPHERE

775

ordering Snoek atmosphere produces an additional force Fa~ to act on the screw • 0 " " I I I f I "r dmsl catlon, i.e. Fad:~F=F3-F 2 or Fad=F3-F I and F a d ~ C 3 - C 2. In other words, the additional force is produced by that part of interstitial atoms increased as a result of the nonsymmetry of the interstitial atmosphere or the long-range ordering atmosphere by the external force. When the tensile stress ~ is applied along the ~ 001] direction, the resolved shear stress on the various slip systems is T=~cosAcos@ where ~ and @ are the angles between the stress direction and the normal line of slip plane or slip direction, respectively, and the active slip system is ( 1 1 1 ] ( ~ 2 ) , and the force Fex acting on the unit length of screw dislocation is that ~ex = (~'~)x~=(-~+~)0~b/3 --

(8)

i.e. the magnitude of the force is ~ @ b / 3 , and the direction in [~I0]. It is clear in Fig.(2 ) that the additional force of stress-induced long-range ordering Snoek atmosphere acting on the screw dislocation is at right angles with the largest resolved shear stress Fex , in other words, the additional force can help the external force to move the dislocation but does not resist it. So the character of additional force is different from that of the dragging force of the Snoek atmosphere. Of course, there will be negligible changes in most of the saturated, nearcore portion of the atmosphere. The region adjoining the saturated region is that giving the greatest contribution to the force. The Dynamic Stress-Induced Lon~-Range Ordering Snoek Atmosphere It is difficult to seriously discuss the dynamic distribution of the interstitials around the moving screw dislocation when an external stress is applied. So we simplify the dynamic condition and consider under what condition the dynamic stress-induced long-range ordering Snoek atmosphere will be formed. We assume that the screw dislocation is displaced by 0-x along the (11~) direction when the tensile stress is applied and the quasi-static long-range ordering Snoek atmosphere is formed. Then the potentials u~ of the i~terstitials are also displaced by ~x and the interstitials must be drived by the gradient of the potential of diffuse. The diffusion velocity VI of the interstitial is related to the mobility D/kT by the Einstein mobility relation as follows VI = _ D y e =

_ Dv~iu

I

(11)

where ~ is the chemical potential of the interstitial. tude of VI, Eq.(11) can be simplified into

To estimate the magni-

or VI:III:

u21u

i.e. VI -

Da 3 k'~ ( ~11- ~22 )

(12)

where 1 is the diffusion distance to form the long-range ordering atmosphere. The maximum value of diffusion distance is from the [~11] and [I~I] directions to the [112] direction, i.e. I m a x ~ 2 F r o / 3 . Subst%tuting the diffusivity of interstitial, D=Doexp(-Hm/kT), and ima x into Eq.(12), we can obtain 9D° O~ e xp (-Hm/kT) VI - 2 ~ b G

( 13 )

The dislocation velocity V D is related to the strain rate 6 by the expression: ~DbVD where

~D

is the total length of active dislocations per unit volume.

( 14) Then the

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ORDERED SNOEK ATMOSPHERE

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ratio of V I to V D is

vI Vo

-

9~DDo 2~o~

exp(-Hm/kT)

( 15 )

We define the critical temperature Tc of the dynamic stress-induced long-range ordering Snoek atmosphere induced by the external stress when VI/VD--I, i.e. when T~Tc, VI/VD~I. In other words, the critieal condition is that when the diffusion velocity V I of the interstitials to form the long-range ordering Snoek atmosphere is faster than the valocity V D of the dislocation mobility, the stress-induced long-range Snoek atmosphere will be formed. Discussion Let us emphasize the difference between the Snoek atmosphere and the stressinduced long-range ordering Snoek atmosphere. It is clear that the direction of drag force produced by Snoek atmosphere is opposite to that of the moving dislocation and the drag force resists the motion of dimlocation. However, the stress-induced long-range ordering Snoek atmosphere can produce an additional force to reduce the resistance of the motion of dislocation. According to Eqs. (12) and (13), the diffussion of interstitials derived by the external stress can maintain the long-range ordering Snoek atmosphere around the moving dislocation under the dynamic conditions. On the one hand, the applied stress is resolved as a shear stress on the slip plane to act on the screw dislocation directly, on the other hand, the applied stress makes the interstitials form a long-range ordering Snoek atmosphere to produce an additional force to act on the screw dislocation indirectly, So the external force is the key to understanding the difference between the Snoek atmosphere and the stress-induced long-range ordering Snoek atmosphere. For the Snoek atmosphere, the drag force is expected to appear when the dislocation begins to move, and to be a maximum when the dislocation is at a critical velocity, i.e. ro/~, w h e r e ~ is relaxation time. In contrast to the Snoek atmosphere, the additional force of long-range ordering Snoek atmosphere is expected to be a maximum when the dislocation is a static, and to decrease monotonically when the dislocation begins to move, and the long-range ordering Snoek atmosphere will disappear when VI=V D. In other words, the effect of the interstit~al atmosphere around the screw dislocation transforms gradually from the stress-induced long-range ordering Snoek atmosphere to the Snoek atmosphere when the static dislocation begins to move under the external force. According to Eq.(15), the formation of the stress-induced long-range ordering Snoek atmosphere is also dependent on the diffusivity of interstitials except for the external force. The larger the diffusivity, the easier the formation of long-range ordering Snoek atmosphere under the dynamic condition. Based on this new concept, the diffusivity of hydrogen i n ~ - F e is larger than that of other interstitials(11) so that it is merely possible for hydrogen to form the stressinduced long-range ordering Snoek atmosphere in ~-Fe. Therefore, the different character of hydrogen in ~-Fe from that of other interstitials is expected to be explained and the mechanism of hydrogen-induced softening(13) and the mechanism of S-K relaxation for hydrogen i n ~ - F e ( 1 4 ) are advanced. Re ferences (I) (2) (3) (4) (5)

A.H.Cottrell and A.B.Bilby, Proc. Roy. Soc., A62,-(1949), 49. A.W. Cochardt, G.Schoeck and H.Wiedersich, Acta Met., 3(1955), 533. G.Schoeck and A.Seeger, Acta Met., 7(1959), 469. J.L.Snoek, Physica, 6(1939), 591. H.Kimura, et al., Hydrogen Effect on Metals(Eds. I.M. Bernstein & A.W.Thompson), ASM, (1980), 191. (6) H.Kimura, H.Matsui and S.Moriya, Scripta Met., 11(1977), 473. (7) T. Tabata and H.E.Birnbaum, Scripta Met., 17(1983), 947.

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(8) (9) (10) (11) (12) (13) (14)

ORDERED SNOEK ATMOSPHERE

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T.Tabata and H.K.Birnbaum, Scripta Met., 18(1984), 231. I.M.Robersnn, et al., Scripta Met., 18(1984), 841. G.W.Hong and J.Y.Lee, Acta Met., 32(1984), 1581. J.P.Hirth, Metall. Trans., 11A(1980), 861. E.Sakamoto.and M.Shimata, J.Phys.(Paris), 42(1981), No.t0, Suppl. C5-I09. T.Y.Zhang, W.Y.Chu and Y.M.Xiao, Scientia Sinica(A), XXIX(1986), 1157. F.X.Jiang and T.Y. Zhang, Scripta Met., in press.

[11I] I:111)

Z'

[..~2'] ~ ~ -

1 ~121~

[ ~ ' 1 0 ] ~ [ 0 1 1 ] y' X @

~ig.(2)

Fig.(1)

[,111]

/

D15]

/

=

//

(111_~

F2

3

,

,,'" " t /

Fi6.(3)

Fig.(1). Orientatiomal relationships of interstitial atoms in the neighborhood of a screw dislocation in bcc lattice. Fig.(2). The (111~ zone showing relevant planes. Fig.(3). The components FI, F 2 and F3 of the interstitial atmosphere acting on the screw dislocation in the (I 1 I] direction as shown in the (111) plane.