On the heterogeneous nucleation of martensite

MATERIALS SCIENCE & ENGINEERING ELSEVIER

Materials Science and Engineering B32 ( 1995 ) 153-158

B

On the heterogeneous nucleation of martensite H.Y. YuS.C. Sanday, B.B. Rath Naval Research Laboratory, Washington, DC 20375-5000, USA

Abstract Martensitic nucleation near inhomogeneities is modeled using linear elasticity. The coherent strain energy due to the formation of a martensite embryo decreases when the inhomogeneity is elastically stiffer than the matrix and vice versa. A maximum reduction of 20% in strain energy is calculated for the case when the embryo is formed near a free surface. The results are consistent with the experimental observations of preferential nucleation of martensite at a free surface. A possible explanation for the nature of the "preexisting martensite embryo" in the Kaufman and Cohen model of a nucleation site is also proposed: the dislocation loop in the parent phase is itself the site for the embryo such that it will transform into martensite during transformation. The calculated critical characteristics of this embryo are in good agreement with the model of Chen and Chiao and their experimental results.

Keywords: Martensitic transformation; Inclusion near interfaces; Nucleation

1. Introduction

Studies of martensitic transformation have been rekindled in recent years. This is due to applications involving the shape memory effect and alloys with high damping characteristics. Martensitic transformations in ceramics are also being discovered and this field is becoming increasingly popular because of the realization of transformation toughening of ceramics. Martensitic transformations are also being found in solid gases such as oxygen and helium. The classical A15 superconducting compounds such as V3Si and Nb3Sn are reported to undergo martensitic transformations just above the superconducting transition temperature. Finally, the importance of hydrogen in metals and the geometrical similarity of the formation of certain metallic hydrides with classical martensitic transformations have prompted a more detailed focus on the characteristics and crystallography of martensitic transformations. Except when the transformation driving force is sufficiently large, as in the case of F e - C o precipitates formed in a Cu matrix reported recently [1] where the nucleation is homogeneous, a straightforward calcula-

1On-site scientist at the Naval Research Laboratory from GeoCenters Inc., Fort Washington, MD, USA. 0921-5107/95/$9.50 © 1995 - Elsevier Science S.A. All rights reserved SSD1 0921-5107(95)03005-0

tion of the energy involved in martensitic nucleation leads to the conclusion that heterogeneous nucleation must be the typical case and therefore special nucleating sites must be considered. This led Kaufman and Cohen [2] to introduce the idea that frozen-in martensite embryos exist. This model remained popular for some years, but a detailed search for such embryos led to a negative conclusion [3]. It has been proposed by Cech and Turnbull [4] that lattice defects are responsible for the nucleation of martensite. It has also been shown experimentally [5-7] that the nucleation of martensite does take place at the location of stress concentrations such as dislocations, stacking faults and grain boundaries. Several theoretical models have been developed for heterogeneous nucleation of martensite. Cohen and Olson [8] have proposed that dislocation arrays with specific structures suitable for the nucleation of martensite are the nucleation sites. Olson and Cohen [9,10] also suggested that an embryo could be formed on a close-packed plane by interacting with a short wall of evenly spaced pre-existing lattice dislocations. The strain energy of a defect interacting with the strain field of the nucleus that is attached to the defect has been computed for dislocation loops [6,11] and for a straight dislocation line [12]. Some recent developments concerning martensitic transformation and heterogeneous martensitic nucleation have been reviewed in detail by Wayman [13] and Cohen [14].

154

H.Y. Yu et al.

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Materials Science and Engineering B32 (1995) 153-158

The dependence of transformed particle fraction on particle size found by means of small particle experiments indicates that martensite nucleates at a free surface or near a free surface for Fe-22Ni-0.49C powder [15]. Magee [15] also proposed that the absence of surface nucleation in the case of Fe-24.2Ni-3.6Mn powder may be due to a slight oxide skin. It has also been shown [16] that for Fe-24Ni-3Mn alloy with a clean surface the sample has a greater concentration of martensite near the surface than in the interior and none of the samples plated with an adherent layer of nickel 0.001 inch thick has any preferential concentration of martensite near the surface. The preferential surface nucleation is thought to be associated with the free surface itself, e.g. the lack of three-dimensional constraint on the transformation shape change at the surface. The increased martensitic transformation temperature in iron-base alloyed thin films is also attributed to this free-surface phenomenon [17]. However, no theoretical model has been provided to explain these phenomena. In this study the effect of inhomogeneities on the strain energy of martensitic transformation will be presented. A possible mechanism of heterogeneous martensitic nucleation is also proposed.

2. Heterogeneous nucleation near inhomogeneities When the size of the inhomogeneity is much larger than the martensite embryo as shown in Fig. l(a), the problem can be modeled as an embryo formed at point

embryo N~ (~., Ix)

matrix (k, It)

(0, 0, d) near the interface of a bimaterial as shown in Fig. l(b). The Lam6 constants of the parent phase are 2 and At and the Lam6 constants of the inhomogeneity are 2' and At'. The two solids are perfectly bonded together. The elastic constants of the martensite and the parent phase (matrix) are assumed to be the same. The martensite embryo is an ellipsoidal inclusion with semiaxes a, b and c. The habit plane of the martensite embryo is parallel to the interface x3 = 0. Let the stressfree transformation strain of the martensite embryo consist of a uniform dilatation A, an expansion normal to the habit plane and a simple shear s on the habit plane. Thus the stress-free transformation strain components are T

T

A

e11= e22

T

A

e33=~+ .5

3'

~,

T

e13

Eto t = Esurf + Echem + Estrain

embryo (~,, It)

matrix (~., It I1 X 2

xl

(1)

(2)

where Esurf corresponds to the coherent interfacial energy between the matrix and the martensite, Ech.... refers to the chemical driving force of martensitic transformation and Estrain is the coherent strain energy due to the misfit between the two lattices. The elastic strain energy change is made up of two terms E oo+

Estrain =

Ein t

(3)

in which E~ represents the self-strain energy of the embryo when formed within the constraints of the surrounding homogeneous parent phase of infinite extent and Ein t represents the elastic interaction energy between the inhomogeneity and the Bain strain (transformation strain) of the embryo. Following the treatment given by the authors for the inclusion problem in bimaterials [18,19], the constrained strain inside the embryo f2 is C

X3

2

and all other components are zero. The total energy change associated with the transformation is

o0

•

T

e~j = (S~jkt + S~jkt) ekt

(a)

S

(4)

where S0~I is the Eshelby tensor [20] for a homogeneous inclusion in an isotropic infinite solid and S~kt is the coupling tensor due to the presence of the inhomogeneity (details are given in the Appendix). The strain energy change due to the formation of the embryo is given by (3) with

I

inhomogeneity (~', It')

Eoo = _1

f

o0 ao T T T T [( )~Sm,~tbq + 2AtSqkt)ekt --J, emm6ij-2Atei/]eij dff2

Q

(5)

(b) Fig. 1. (a) A martensite embryo near an inhomogeneity. (b) Model for the calculation of the effect of an inhomogeneity on martensitic nucleation.

* ~ ,,~* \ T T + • A t 3 i / k d e k / e O. dff~ Eint = __1 f.J (~,Srnmkl6ij

(6)

H. Y. Yu et al.

/

where 6,). is the Kronecker d. Substituting (1) into (5) and (6), one has Eoo-

Z.3kk_(l+v)(.--.')fl 2 1 ( 6a2/ 4Jr( 1 -- v) r ~ 5 4( 1 -- v) - ~ 5 } (a--2v)(.--.')fl

(q~ '9.,__ ,~..4~a 2 (Z2mkk--3"~)

18

~2 ----[~.Z~nrn33

2

nt-2.Z3333 - ( 2 + 2.)r]

Ein t =

_~2

mmkk

18

+(1-

--.S

*

~zX

Z1313

6

2 (~'ZmmB3 + 2"Z*333)

,

['~Zrnmkk"[-2/lgZ33kk

("-"', + + 6-G d3

Z*mml3"l-Zl*3mm

Z'k, =

f~

r

9(1- v)

*

(11)

fQ S ij*kzdQ

8.

1 fl = . + (3 - 4 v ) .

(9)

Eqs. (8) and (11) show that the elastic interaction energy Eint is negative when the matrix is stiffer than the inhomogeneity, i.e.. > . ', and vice versa. They also show that the magnitude of Eint decreases with increasing d. Let us consider first the case when v= 1/3: (10) becomes E= = 4/, A2 + 4 . r 9 5

9

A~ "}- 4 . $2 15

128 ~ s2 (10)

2 1 r 4d 3

~2 "l- 8._~

25(4 )

4.(1+ V)A~

The volume integrals of the coupling tensors in (8) are obtained by substituting (A2) into (9). They are . (l+v)(1-2v)(.-.')fl E"mk~-zr(1-- V)

. ' + ( 3 - 4v').

(12)

and when d = a, (8) and (11 ) give

15(1- v) ~24 9(1- v)

+ . ( 7 - 5v) 2 s 30(1 - v)

1

(8)

and dQ = dx 1 dx 2 dx 3. In this study the shape of the embryo in the earliest stage of nucleation is assumed to be spherical as proposed in the Olson-Cohen model [9]. A spherical nucleus leads to a closed-form solution for the strain energy in the presence of inhomogeneities, since the mean value of the harmonic function of a sphere is equal to its value at the center. For a spherical embryo (a = b = c), (7) reads

E=_2.(l+ v)&2q

m

where

where r is the volume of the embryo, S ijkz dQ,

*

"~rnml3 -- "~13rnm= 0

)

~Sr.~* 2 1~2"mm13q- 2 . (Z3"313-k y 1"333)]

Zij~, =

10d 4]

,

. A s ((32 + 2.) ~

12a2+9a4/

32at(l-v)d 3 .(1+v)-5dS-

* __ * i Y~1333 -- Y~3313 --

~-

(

("-'"')fir2

+ ( 3 2 + 2")~mm33] *

3

1 1

• ~'~1313--

2

(l+v)-2(3-v)Sd2

(7)

+ (3)]. + 2")(Zmm33 -- 2r)] ( 3 ; t + 2 . ) a 2 Y*

3a2/

3a4 / + 10d4 } + [(2- 7 v ) ( . - . ' ) f l

[~l,Zmmkk 4- 2 . Z 3 3 k k

( 2 Z 1 3 1 3 -- Z ' ) -

1 (

Z3333 8 e r ( 1 - v ) ( " - " ' ) f l

2

2

155

Materials Science and Engineering B32 (1995) 153-158

(13)

f o r . > . ' and

Eint 2 ( ~ A 2 ) + 2 4 7 ( ~ ) r

15 27 ( 4 . ) + 128 ~ s2

960

21 ( ~ ) ~2 + 1 ~ A~ (14)

156

H.Y. Yu et al.

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Materials Science and Engineering B32 (1995) 153-158

for j~ ~/~'. For pure shear transformation strain, i.e. A = ~ = 0 , a reduction of about 19.5% in the strain energy is obtained when the nucleation takes place near a free surface. Taking typical values of A + ~ = 0.05 and s = 0.18 for steels [21], the reduction in energy is 22.6% when A = 0 and 22.4% when ~ = 0. These numbers show that regardless of the transformation strain composition the strain energy will be reduced about 20% when the embryo is formed near a free surface or void, while the strain energy will increase by about 20% when the embryo is nucleated near a hard inclusion. This is consistent with the experimental observations on the lack of preferential concentration of martensite near the surface when the surface is coated with a harder layer. For more detailed calculations the embryo can be assumed to be in the shape of an oblate spheroid with a = b ">c. By substituting (A1) into (9) and (7), one has

X3

~X 2 Xl dislocation loop

embryo

Fig. 2. A martensite embryo formed by the transformation of a dislocation loop.

E~o _ 2/2(l+V) A2_~ 7r/~ c ~ 2 + 3~(1+ ( i - - v )v) aCA~ r 9( 1 - v) 4( 1 - v) a where 7rkt(2- v) c 2 + s 8 ( 1 - v) a

(15)

(b~ + --s,h2]l/2 p=

where v is the Poisson ratio of the matrix and the embryo and (15) is the same as the expression obtained by Christian [21]. The interaction energy Eint can then be obtained numerically by using (8), (9) and (A2).

(17)

8

is the cut-off distance such that two dislocation elements do not interact when they are closer than this distance. After the transformation the strain energy of the martensite disc, using an ellipsoidal thin disc as an approximation, is given by (10) for zero dilatation as

3. Nature of the pre-existing embryo The idealized shape of a plate of martensite is lenticular, i.e. a minimum strain energy configuration, a situation much like that involved in the formation of a mechanical twin. The model proposed here is shown in Fig. 2, where a circular dislocation loop with radius a and Burgers vector with edge component be and shear component b~ is assumed to exist in a parent phase of infinite extent. Unlike the model given by Chen and Chiao [6] where the martensite is nucleated around the loop, it is assumed that the loop itself, i.e. the distorted parent phase, will transform into a martensite embryo with Bain transformation strain e~3=~ and elY3= e~l = s/2 in the shape of a thin disc with radius a and thickness c. Before the transformation the strain energy of the dislocation loop is [22]

Eembryo 7rkt C [2~2 + (2 -r 8( 1 - V) a

V)Sz]

(18)

Let us assume that the strain energy of the embryo is provided entirely by the strain energy of the dislocation loop, i.e. Eloop = Eembryo,and use the approximations r~ = :Tra2be,

rs = 7raZ bs

(19)

which may be deduced from the relationship given by Eshelby [23] for the equivalence of inclusions and dislocation loops. The critical dimension a* of the embryo or the dislocation loop can be estimated from (16)-(19) as

2b:[

[ l / l-2c-7~1/2 32a* n /(be_l._bs)

/ - 1 +(2-v)b~[ln( )

32a* / - 2 ] (be2~b~ffs)X/2j

2

Eloop 2 tea

4(7r(ff-v)a[2b~[ln(~)-I

_7r [2b~+(2-v)b~]

1

2

(20)

Eq. (2) becomes + ( 2 - v)b~ [ln ( ~ ) - 2]1

(16) Etot = ~a*2 CAgchem + 27ra*2 ~11

(21)

H. Y. Yu et al.

/

Materials Science and Engineering B32 (1995) 153-158

for Estrain = 0, where Agchem is the chemical free-energy change per unit volume of martensite formed and is negative at temperatures T< TO (TO being the equilibrium temperature at which the martensite and its parent phase coexist) and ~'11is the interfacial energy per unit area. The critical thickness of the embryo is obtained by letting Eto t = O, which gives 2~1 (22) A&h~m Using the proposed model, the values of a* and c* may be estimated for ZrO 2 particles in order to compare them with the values estimated from the model of Chen and Chiao [6]. Adopting the material parameters ~ = 0 , s=0.154, bs=0.52 nm, Agchem=2.04Xl08 J m-3 and VII= 0.2 J m 2 used by Chen and Chiao, one has from (20) and (22)

Appendix The Eshelby tensor Si/~kt and the tensor S*kt in Eq. (4) are 1

c* = 1.96 nm

(23)

These are comparable with the values of a* = 18.1 nm and c* = 3.3 nm given by Chen and Chiao which were qualitatively supported by their experimental (transmission electron microscopy) results. No definite argument may be given as to which model is more valid. The experimental evidence of the e-martensite embryo nucleating from a single stacking fault [5] could support both models, since the stacking fault could be considered as a loop of Shockley-Frank partial dislocations [22]. The advantage of the proposed model rests mostly on its simplicity.

4. Conclusions The foregoing arguments have been presented to demonstrate that martensite favors nucleation near inhomogeneities with stiffness less than the parent phase, such as voids and free surfaces. This conclusion can also be applied to other solid state transformations that involve strain energy changes. A model of the preexisting martensite embryo in Kaufman and Cohen's theory has also been proposed. In this new model the dislocation loops in the parent phase are themselves the so-called "pre-existing embryos". The strain energy stored in this distorted parent phase (the dislocation loop) provides the strain energy needed for the nucleation of the martensite embryo during transformation. The advantage of this model is its simplicity.

+

This work has been partially sponsored by the Office of Naval Research through the Naval Research Laboratory.

(~ljk(~i,"1- (~],jl(~ik)-- 2 v¢~j6k,]

(11)

and .

S mmkk ~"

(l+v)(1-2v)(tt-,u')fl

II

(ID,33

~(1 - v) _1_

.

t

(1 t-v)(tt-_/j 4~(1 - v) )fl[( 1 _4v)~i,~33 - 2x30,333,, ]

533kk =

,

II

(1+

Sl3k~=

4 ~ ( 1 - v)

333) ~P,31+ 2X3@,..

*

(1--2V)(/AZ-fl')flf"~ll 4er(1 -- 1"P) \z~l

53333

1 {(tt- t t ' ) f l [ - ( 1 8zr(1- v)

atom33 "~-

--

4(2 +

'3333

-70~33-2x30,3,3)n

n + 2x3F,33333 ,, 4V)F,3333

II ,~ 2~11 1 V)X3(I),333 -- ZX31¥,3333]

+ 2[(2 - 7 v)(kt-/t')fl + (1 - 'l,')[At(fl-31)](~l,133} S 1~333 =

11 _ 4 = ( 11 - v)[(/~-/~ t )fl(x3F,33313- O,~3 4x30 11313 2~II

x .

- x3 ' 3313 ) * (1 -

•

(1 -- mY) (/A--/A')3

S mml3 "~-

- 4 - - ~ 1 = -~)

'( fl -

3'

11

II

lI

(2F3313- 3 (I~,13 -- 2X31I),133)

(A2) • (/'g --/'g ; )fl [( II S3313 = ~ ( ~ 7 ~ ~,1 -- 41,,)F,3313

1I

- 2x3F 33133+ 41)OI113

+ 4 ( 1 + 1-')23(I),313 i, -t- ~ 2._n 1 /zfl-/~'fl' (1)1,113 Z~x3qJ'3313] 4~r

S1313

Acknowledgment

6j, + ,t,I.,aj

1 - v) [tp' ij ' +(1 -

S i/kl

c* -

a* = 16.7 nm,

157

~7~(i 7 ~ [FI"313+ 2x3Fl,I13,33 _2(1_v)~I,111_4x3dp,iu3_., 2~1I , , Z;X3qJ,1313] 8,7l: /V-l- .'

*{12 + 2(tiff--/2'fl')(I){Ill

158

H.Y. Yu et al. [(3-

S1212-

8 o z ( 1 - v)

/

Materials Science and Engineering B32 (1995) 153-158

n n 2~11 1 4v)l-" 1212+ 2X31-' 12123-- AX3t'Pj212] ~

R2 =[(x, -X'l) 2 +

/~ - p'

p

+8

+

-

4(1 -

v)x3(I)lll21

,

1 [/u-p'

+--

References

,

(/.2 -- ~U;)~

2x3rl~2,33

~ 2~I1 --ZX3q) 1213J1

I t~t\lTr]l

U

ll

2~11

S 1312 - 8,7['( 1 - v ) [F'1312 + 2(x3F'13123 - x31"P'1312)

--

4 ])X3 (I)I,I112]

1 {/u-/u'

4zr/,u + ,u'

tU -- ~ut

,ufl + ,u'fl'

ll

IJ~II112

where

~l=!

1 R i dff2',

Fl=f R1 dr2' f2

O"

=!

1

aft',

~ n = f ~ll dx3,

I

+ 2V11112)- 2 ( P f l - P p : w 112]

8Jr I p + /u'

)q '/2

+/,')

[(3 - 4v)Fl1213 +

8~r(1 - v)

+ (x3 + d-x

and dr2' -- dx'~ dx~ dx~.

x [(1 - 2 v ) ( 2 p ' - p)fl - (1 - "L~' ")/.l .p. . .J~J,1212 . n

S 1213 ="

RI = [(Xl _ xrl )2 + (x 2 _ x~)2 + (x 3 - ,~A-- -~3/"' ~2]1/2j

F II= f R 2 d r ' n

(~II = f u~jH dx 3

\

[1] M. Lin, G.B. Olson and M. Cohen, Acta Metall., 41 (1992) 253. [2] L. Kaufman and M. Cohen, Prog. Met. Phys., 7(1958) 165. [3] M.H. Korenko and M. Cohen, Scr. Metall., 8 (1974) 751. [4] R.E. Cech and D. Turnbull, Trans. A1ME, 206 (1956) 124. [5] J.W. Brooks, M.H. Loretto and R.E. Smallman, Acta Metall., 27(1979) 1839. [6] I.-W.Chen and Y.-H. Chiao, Acta Metall., 33 (1985) 1827. [7] T. Saburi and S. Nenno, Proc. Int. Conf. on Martensitic Transformations, 1986, p. 176. [8] M. Cohen and G.B. Olson, Trans. Jpn. Inst. Met., Suppl., 17 (1976)93. [9] G.B. Olson and M. Cohen, Metall. Trans. A, 7(1976) 1897. [ 10] G.B. Olson and M. Cohen, Metall. Trans. A, 7{ 1976) 1905. [11] K.E. Eastefling and A.R. Th61dn, Acta Metall., 24 (1976) 333. I12] M. Suesawa and H.E. Cook, Acta Metall., 28(1980) 423. I13] C.M. Wayman, in H.I. Aaronson, D.E. Laughlin, R.E Sekerka and C.M. Wayman (eds.), Proc. Int. Conf. on Solid Phase Transformations, TMS-AIME, Warrendale, PA, 1981,p. 1119. [14] M. Cohen, Mater. Trans., JIM, 33 (1992) 178. [15] C.L. Magee, Metall. Trans., 2(1971 ) 2419. [ 16] V. Raghavan and M. Cohen, Metall. Trans., 2 ( 1971 ) 2409. [17] H. Warlimont, Trans. A1ME, 221 (1961) 1270. [ 18] H.Y. Yu and S.C. Sanday, Proc. R. Soc. Lond. A, 434 ( 1991 ) 520. [19] H.Y. Yu and S.C. Sanday, Proc. R. Soc. Lond. A, 439(1992) 659. [20] J.D. Eshelby, Proc. R. Soc. Lond. A, 241 (1957) 376. [21] J.W. Christian, Acta Metall., 6 (1958) 377. [22] J.P. Hirth and J. Lothe, Theory of Dislocations, McGrawHill, New York, 1968, p. 144. [23] J.D. Eshelby, in I.N. Sneddon and R. Hill (eds.), Progress in Solid Mechanics, Vol. 2, North-Holland~ Amsterdam, 1961, p. 89.