Chapter 5 Thermodynamics of Phase Transitions
5.3 Thermodynamics of Martensitic and Bainite Transitions
Fig. 5.3 Schematic representation of G functions of the parent (γA) and marten-sitic (αM) phases (a), ΔG function between the above two phases (b) and ΔG function where ΔG(γA → αM, Ms) = ΔG(αM → γA, As) (c). (Reproduced from Ref. [10] with permission of Elsevier)
alloys, as in Fe-C system, hysteresis between Ms and As can be as large as hundreds of degrees.
When the entropy difference between the two phases is constant, and ΔG(γA→ αM, Ms) = ΔG(αM→ γA, As) [Fig. 5.3(c)], the following equation holds for non-thermoelastic transitions,
T0= (Ms+ As)/2. (5.9)
When there are no magnetic contributions to G, CP of a solid pure ele-ment or an alloy presents a smooth trend, and is approximately equal in the considered temperature range where martensitic transitions usually oc-cur. This implies that the ΔS(γA → αM) can be assumed to be constant.
On the contrary, when magnetic transitions occur, CP shows the so-called λ shape, i.e. a sharp peak at Curie or N´eel temperature. As a consequence, the difference in CP between the parent and martensitic phases can be large, ΔG(γA → αM, T ) is not a linear function of T and the driving forces ΔG(γA → αM, Ms) = ΔG(αM → γA, As). Eq. (5.9) is invalid. This occurs indeed very often. For instance, magnetism is present in Fe-base alloys. Even when the aforementioned hypotheses are not fulfilled, however, Equation (5.9) is commonly used as a reasonable approximation for obtaining T0 in non-thermoelastic martensitic alloys.
In thermoelastic alloys, hysteresis between Ms and As is limited, and thus bracketing is not required to estimate T0. Unfortunately, it has been
168 Chapter 5 Thermodynamics of Phase Transitions
found for some β-brass that As < T0, and thus the determination of T0
requires special care. The origin of this behavior is related to the accumulation of strain energy in the direct reaction (on cooling), which is sufficient to
“prematurely” start the reverse transition on heating. However, it has been recognized that for several thermoelastic transitions, it can be assumed that the elastic contribution becomes negligible at Ms and Af, i.e. for the first plate of martensite to form during the direct transition, and for the last plate to disappear during the inverse transition. Thus, T0 should lie at half way between Msand Af and Eq. (5.9) should be modified to
T0≈ (Ms+ Af)/2. (5.10)
Figure 5.4 reports G(T ) functions of equilibrium phases. For the purpose of illustration, a generic Fe-X system is considered and G curves of γA and αM as a function of composition at different T are shown. For an alloy com-position x0 at a temperature T1 being higher than T0 (Fig. 5.4(a)), γA is stable, and its G(T1) is given by point A. When T is decreased down to T0
(Fig. 5.4(b)), the G values of bothγA andαMare equal (point B). However, according to the common tangent rule, the equilibrium state of the system is a mixture of γA and αM, whose G is given by point B. At any lower T3
(Fig. 5.4(c)), γA (point C) has the possibility to transform intoαMwithout composition change (point C) or into the more stable mixture of αM− γA
(point C), whose compositions are given by the tangent points C and C. Although a thermodynamic driving force exists for a transition ofγA
into the stable mixture ofαM−γA(line CC), this reaction requires diffusion of the components and can be hindered by rapid cooling at low T . Thus, the martensitic reaction, being diffusionless, can occur even with a lower driving force (CC).
The previous thermodynamic scheme is simplified in some regards: (1) In most cases, bothγAandαM are metastable when the martensitic transition occurs. This implies that in the schemes of Figs. 5.3 and 5.4, curves for one or more other phases which have lower G(T ) values should be included, i.e. the
“true” thermodynamic equilibrium is different from that including only γA
and αM. However, this “true” equilibrium can be avoided by quenching and the schemes in Figs. 5.3 and 5.4 are therefore useful to the purpose of describ-ing the martensitic transition. (2) As previously described, the martensitic transition and its reverse occur at different T on cooling and heating, with respect to T0. This is due to the effect of other energetic terms, such as elastic and plastic energy or irreversible frictional energy, which hinder the formation of the martensite and its reverse.
The major non-chemical contributions are: (1) An elastic stored energy Eel(γA→ αM), i.e. the energy necessary to accommodate the residual strain after the martensite transition, and the subsequent slip or twinning at the habit plane. Eelis reversibly accumulated during the direct reaction and re-leased in the reverse transition. In non-thermoelastic transitions, irreversible plastic deformation also occurs. (2) An irreversible frictional energy or
fric-5.3 Thermodynamics of Martensitic and Bainite Transitions 169
Fig. 5.4 G(T ) functions for parent and martensitic phases in a generic Fe-X system as a function of composition at three different temperatures: (a) T1 > T0; (b) T2= T0; (c) T3< T0. (Reproduced from Ref. [10] with permission of Elsevier)
tional work Efr, which is related to the motion of the interface and the cre-ation of defects during the transition. (3) An interfacial energy contribution γssA, i.e. the energy released due to the creation of the interface betweenγA
andαM. In non-thermoelastic transitions this term is rather small in compari-son with the total driving force, the interface being coherent or semicoherent, and is often neglected.
Both Eeland Efrare of the order of 50 – 100 J·mol−1in thermoelastic tran-sitions, whereas higher values appear in non-thermoelastic ones. The driving force for the martensitic transition can thus be expressed as Eel+ Efr+ γssA.
At Ms combining this equation and Eq. (5.7), the following energy balance
170 Chapter 5 Thermodynamics of Phase Transitions
is established,
ΔG(γA→ αM) = Eel+ Efr+ γssA. (5.11) In light of the aforementioned non-chemical contributions to the energy bal-ance, the different behaviors of thermoelastic and non-thermoelastic marten-sites can be rationalized, and experimental observations can be explained. It has been observed experimentally that the growth and shrinkage of thermoe-lastic martensitic phases take place in a well defined sequential order, with the first plate forming during cooling being the last one to disappear during heating. A necessary condition for this behavior is the absence of plastic ac-commodation of the transitional shape and volume changes. Eel(γA → αM) accumulated in the direct reaction is reversibly recovered during the reverse transition, and therefore helps the reversion of the martensite. In the early stages of the reverse reaction, it can be large enough to promote the transi-tion, even without the help of ΔG(γA→ αM). As a consequence, As may lie below the T0. Moreover, the hysteresis in thermoelastic martensites is limited because of the absence of nucleation and the lower value of elastic energy in comparison with plastic accommodation.
Salzbrenner and Cohen have carried out a complete series of experiments, in order to clarify the role of different contributions to these transitions.
Their main results are: (1) If a single crystal is forced to transform with a single interface, no Eel is accumulated, and the transition proceeds up to completion with the same T at the interface. The thermal hysteresis observed is totally due to Efr, which can be assumed constant since the dimensions of the interface do not change during the transition. Since Ms = Mf, and As = Af, Eq. (5.9) is valid for determination of T0. (2) Multiple interfaces, single crystal samples transform with increasing storage of Eel. Since the transitional shape and volume are constrained, a progressive depression of the transition curves to lower T in the thermal hysteresis cycle. If nucleation takes place at a free corner, the specimen can be considered unconstrained at Ms and Af and Eq. (5.10) can be used. (3) If polycrystalline specimens are considered, Eel is operative even at Ms and Af. Consequently, while T0
is not changed, these temperatures are shifted to lower values and Eqs. (5.9) and (5.10) are invalid.
In the non-thermoelastic case, Eelis not a controlling factor of the reac-tion, and both the direct and reverse transitions require separate nucleation to start. Now the critical driving force to nucleate martensite from the parent phase is the same as that in the reverse transition, i.e. ΔG(γA→ αM, Ms) = ΔG(αM→ γA, As). With the further assumption that CP values of the both phases are similar, Eq. (5.9) is generally considered valid and widely applied.
There is another similar transition in steel, namely, bainite transition. In a far-reaching paper, Zener (1946) attempted to give a rational thermody-namic description of the phase transitions in steel. He assumed that bainite growth is diffusionless, any carbon supersaturation in bainitic ferrite is re-lieved subsequent to growth, by partitioning into the residual austenite. He
5.3 Thermodynamics of Martensitic and Bainite Transitions 171
believed that unlike martensite, there is no Eel associated with the growth of bainite. Thus bainite should form at a temperature just below the cor-responding equilibrium temperature T0b, where γA and ferrite (αF) of the same composition have an identical G value. This is schematically shown in Fig. 5.5.
Fig. 5.5 Schematic illustration of the origin of the T0b curve on the phase dia-gram. The T0curve incorporates a strain energy term for the ferrite, illustrated on the diagram by raising the Gibbs free energy curve for ferrite by an appropriate quantity.
Hultgren at the time proposed a model for the role of substitutional al-loying elements in steel; at high T where diffusion rates are reasonable, these elements can redistribute during transition in a way consistent with equilib-rium. The transition was then said to occur under “orthoequilibrium” con-ditions. This contrasts with “paraequilibrium” in which the substitutional alloying elements are unable to partition, although C, which is a fast diffus-ing interstitial element, redistributes between the phases until its chemical potential is uniform throughout.
Another important solid transition in steel is the eutectoid transition with the product phase peralite, which consists of cementite and ferrite. The mechanism of pearlite transition was believed to be initiated by the nucleation of cementite. This led to the contrasting suggestion that bainite is initiated by the nucleation of ferrite. Hultgren put these ideas together and proposed that upper bainite begins with the nucleation and growth of ferrite with a paraequilibrium C concentration, causing the residual austenite to become enriched in C. This bainitic ferrite, unlike the ferrite associated with pearlite, was considered to have a rational Kurdjumov-Sachs or Nishiyama-Wasserman orientation relationship withγAin which it grows. This was used to illustrate
172 Chapter 5 Thermodynamics of Phase Transitions
the observed difference in ferrite morphologies in bainite and pearlite. Bainitic ferrite was always found to consist of individual plates or sheaves whereas the ferrite in pearlite apparently formed alternating plates of a regularly spaced two-phase lamellar aggregate. The enrichment of austenite with C should eventually cause the paraequilibrium precipitation of cementite from austenite in a region adjacent to the bainitic ferrite. At this time, pearlitic cementite was thought to bear a rational orientation relation to the austenite grain into which the pearlite colony grows whereas bainitic cementite should be randomly orientated to the austenite in which it precipitated. This process of ferrite and subsequent cementite precipitation then repeated, giving rise to the sheaf of bainite. Thus, the upper bainite is similar to pearlite but growing under paraequilibrium conditions and different in the orientation relations with austenite. Later, Hillert pointed out an important distinction between pearlite and upper bainite; in the former case, the ferrite and cementite phases grow cooperatively, whereas in the latter case, the plates of bainitic ferrite form first with the precipitation of cementite being a subsequent reaction.
A bainitic microstructure is far from equilibrium. The free energy change accompanying the formation of αF in a Fe-0.1C wt% alloy at 813 K is
−580 J·mol−1, whereas that for the formation of an equilibrium mixture of allotriomorphic ferrite and austenite at the same T is −1050 J·mol−1. Consequently, the excess energy of αF is about 470 J·mol−1 relative to al-lotriomorphic ferrite, equivalent to about 0.04 in units of RTm. This is about an order of magnitude larger than the stored energy of a severely deformed pure metal. It is small, however, when compared with highly metastable materials such as rapidly-quenched liquids which solidify as supersaturated solutions, or multilayered structures having a high density of interface (Table 5.1). Thus, bainitic steel can be welded whereas all the other materials listed with higher stored energy would not survive the welding process.
Table 5.1 Excess energy of metastable materials adapted from Turnbull
Example Excess energy (RTm)
Highly supersaturated solution 1
Amorphous solid 0.5
Artificial multilayers 0.1
Bainite 0.04
Cold-deformed metal 0.003
The concepts of equilibrium, metastable equilibrium and indeed con-strained equilibrium remain useful in spite of the large excess energy, which can be applied to αF in the interpretation of the transition mechanism and to the design of modern steel.
The atom-probe experiments have established that there is no redistri-bution of substitutional solutes during the bainite transition. These experi-ments cover the finest conceivable scale for chemical analysis. They rule out any mechanism which requires the diffusion of substitutional solutes. This includes the local equilibrium modes of growth. By contrast, all
experimen-5.3 Thermodynamics of Martensitic and Bainite Transitions 173
tal data show that pearlite grows with the diffusion of substitutional solute atoms. Cr, Mo, Si, and Co have been shown to partition at the reaction front.
The extent of partitioning is smaller for Mn and Ni, especially at large un-dercoolings, but there is localised diffusion. These observations are expected because pearlite is the classic example of a reconstructive transition.
Solutes in iron affect the relative stabilities ofγAandαF. This thermody-namic effect is identical to all transitions. We have seen, however, that sub-stitutional solutes do not diffuse at all during displacive transitions whereas they are required to do so during reconstructive transition. It is for this rea-son that the observed effect of solutes, on the rate of transition, is larger for reconstructive than for displacive transition as shown in Fig. 5.6.
Fig. 5.6 Time-temperature-transition (TTT) diagrams showing the greater retard-ing effect that Mn has on a reconstructive transition compared with its influence on a displacive transition.
Bainite forms at somewhat higher T where C can escape from the plate within a fraction of a second. Its original composition cannot thus be mea-sured directly. There are three possibilities. C may partition during growth so that the ferrite may never contain any excess C. The growth may on the other hand be diffusionless with C being trapped by the advancing interface.
Finally, there is intermediate case in which some C may diffuse with the remainder being trapped to leave the ferrite partially supersaturated.
Diffusionless growth requires that transition occurs below T0b, when the free energy ofαF becomes smaller than that ofγA of the same composition.
Growth without diffusion can only occur if C concentration of γA lies to the left of the T0b curve. It is found experimentally that transition to αF
does stop at T0bboundary. The balance of evidence is that the growth ofαF
below Bs involves the successive nucleation and martensitic growth of sub-units, followed in upper αF by the diffusion of C into the surroundingγA. The possibility that a small fraction of C is nevertheless partitioned during growth cannot entirely be ruled out, but there is little doubt on whether αF
174 Chapter 5 Thermodynamics of Phase Transitions
is at first substantially supersaturated with C.
The chemical potential is not uniform in steel when the bainite reaction stops. The reaction remains incomplete in that the fraction of αF is less than expected from a consideration of equilibrium betweenγAandαF. This
“incomplete reaction phenomenon” explains why the degree of transition to αF is zero at Bs (starting transition temperature of bainite) and increases with undercooling below Bsin steel where other reactions do not overlap the formation ofαF.
Although the bainite reaction stops before equilibrium is reached, the remainingγAcontinues to decompose by reconstructive transition, albeit at a greatly reduced rate. Pearlite often forms sluggishly after αF. The delay between the cessation of αF and the start of pearlite varies with the steel composition and transition temperature. In one example, the bainite reaction stopped in a matter of minutes, with pearlite growing from the residual γA
after some 32 h at T = 723 K. In another example, isothermal reaction to lower αF at 751 K was completed within 30 min, but continued heat treatment for 43 days caused the incredibly slow reconstructive transition to two different products. One of these was alloy-pearlite which nucleated at the austenite grain boundaries and developed as a separate reaction. The other involved the irregular, epitaxial and reconstructive growth of ferrite from the existing αF. The extent of ferrite growth in 43 days was comparable to the thickness of the bainite plates, which took just a few seconds to form. The two-stage decomposition ofγA is more difficult to establish for plain carbon steel where the reaction rates are high, with the pearlite reaction occurring a few seconds afterαF.