Kinetic Arrest of R-B19′ Transformation in Iron-Doped Ti–Ni Shape Memory Alloy

Abstract

We have studied the R-B19′ martensitic transformation behavior of Ti–(50−x)Ni–xFe (at%) alloys. The R-B19′ transformation start temperature (M_s) decreases from 234 K to 102 K as x increases from 2.0 to 3.7 at%. This transformation is suppressed in the alloy with x = 4.0 (4.0Fe alloy), although the equilibrium temperature is estimated to be 104 K. The estimated free energy difference at M_s (driving force) for the R-B19′ transformation increases as M_s decreases from 45 J/mol (M_s = 234 K) to 48 J/mol (M_s = 102 K). If we extrapolate this relation linearly, M_s of the 4.0Fe alloy is expected to be ∼50 K. Nevertheless, the R-B19′ transformation is completely suppressed in the 4.0Fe alloy. Presumably, insufficient thermal activation energy is the main reason for the suppression of the R-B19′ transformation in the 4.0Fe alloy.

Fig. 7 Temperature dependence of free energy difference between the R-phase and the B19′ phase of Ti–(50−x)Ni–xFe alloys. The M_s(B19′) temperature of each alloy is indicated by solid circles.

1. Introduction

Martensitic transformation (MT) appears in many alloys. It has been used to control microstructure of steels and to realize shape memory effects in Ti–Ni alloys.^1,2) The transformation-start temperature (M_s) depends on composition, and its dependence is almost linear in a wide composition range in many alloys. However, the linear relation does not extend to M_s = 0. Usually, the M_s temperature shows a lowest value below which thermally induced martensitic transformation is impossible. The lowest value of M_s depends significantly on alloy system. For example, the lowest M_s of the fcc-bcc transformation in binary Fe–Ni system is ∼50 K (Fe–30.0Ni (at%)),³⁾ that of the B2-B19′ transformation in Ti–Ni binary system is ∼150 K (Ti–51.2Ni (at%)),⁴⁾ that of the B2-R transformation in Ti–Ni–Fe system is ∼220 K (Ti–44.3Ni–5.7Fe (at%)),^5–9) and that of B2-4H transformation in Ti–Pd–Fe system is ∼150 K (Ti–32Pd–18Fe (at%)).^10–12) Presumably, existence of the lowest limit of M_s is related to the kinetics of martensitic transformations.

Alloys whose M_s temperature is close to the lowest limit of the system (and whose M_s has just disappeared) are of interest to study because the isothermal nature of the martensitic transformation clearly appears in such alloys. In fact, TTT diagrams of martensitic transformations are obtained in such alloys; an example is a Ni₄₅Co₅Mn_36.5In_13.5 alloy whose M_s is close to the lowest limit of this system;¹³⁾ another example is a Ti–51.3Ni alloy which does not have M_s while Ti-51.2 alloy has M_s.^4,14) In these alloys, time dependence of martensitic transformation was well explained by assuming a thermal activation process of martensitic transformation. It will help us understand kinetics of martensitic transformation to study the transformation behavior of alloys whose M_s is near the lowest limit.

Ti–(50−x)Ni–xFe system is a representative alloy system in which a successive B2-R-B19′ transformation is observed.^2,5) Many fundamental studies related to the B2-R transformation has been made by using this system. Precursor phenomena of the B2-R transformation such as phonon softening¹⁵⁾ and the appearance of nanoscale domain like microstructure prior to the B2-R transformation was revealed in this system.^6,16–20) In this system, the B2-R transformation temperature decreases linearly as iron content x increases up to 5.7 at%, but the thermally-induced B2-R transformation is suppressed if x is 6 at% and higher.⁶⁾ In this alloy, however, we can detect isothermal nature of the B2-R transformation⁹⁾ although the isothermal B2-R transformation was denied by Kustov et al.²¹⁾ The low temperature state of this alloy is recently referred to a strain glass state^22–24) although some researchers disagree with this termination. Compared to the B2-R transformation, the number of studies on R-B19′ transformation in Ti–(50−x)Ni–xFe system is small. Hwang et al. examined martensitic transformation behavior of a Ti–47Ni–3Fe alloy and showed that M_s is 173 K.^16,17) Choi et al. reported M_s of a Ti–48.0Ni–2.0Fe alloy to be 234 K.⁵⁾ If we assume that M_s decreases linearly as x in this system, the M_s of Ti–46.0Ni–4.0Fe is expected to be 113 K. However, the experimental result by Choi et al. showed that this alloy does not show the R-B19′ transformation down to 2 K.⁵⁾ This means that the R-B19′ transformation of the Ti–(50−x)Ni–xFe system disappears suddenly at a certain composition of 3.0 < x < 4.0. This behavior resembles the sudden disappearance of B2-B19′ transformation of Ti–yNi alloy with y ∼ 51.3 at%.^2,4,14)

In the present paper, therefore, we study the R-B19′ transformation behavior of Ti–(50−x)Ni–xFe alloys with x between 2.0 and 4.0. First, we derive composition dependence of the equilibrium temperature T₀ of the R-B19′ transformation through the measurements of electrical resistivity. Next, we examine the influence of cooling ratio and the influence of external stress on the transformation behavior. Moreover, we evaluate driving force of the R-B19′ transformation through the evaluation of free energy difference between the R-phase and B19′-phase, which is calculated from specific heat of these phases. By considering all these results, we discuss the reason of the suppression of thermally induced R-B19′ transformation in the Ti–46.0Ni–4.0Fe alloy.

2. Experimental Procedure

Ingots of Ti–(50−x)Ni–xFe (x = 3.3, 3.5, 3.7, 4.0 in at%) alloys were prepared by an arc melting method. The single crystalline rod of Ti–46.0Ni–4.0Fe alloy was grown by the floating zone method (FZ method) at a crystal growth rate is 5.0 mm/h under a high purity argon gas flow. In this study, each alloy is referred to by its iron content. For example, the Ti–46Ni–4.0Fe alloy is referred to as the 4.0Fe alloy. The specimens for electrical resistivity measurements, compressive tests and XRD measurements were cut from these ingots and single crystal and then heat treated at 1273 K for 1 h followed by quenching to ice water. Then the specimens were electropolished in an electrolyte composed of 95 vol% acetic acid and 5 vol% perchloric acid. Electrical resistivity was measured by a four-probe method with a cooling and heating rate 1 K/min. The measurements of thermal expansion under compressive stress were made by attaching a strain gauge to the surface. The stress was applied in the [110] direction of the B2-phase and the calibration of the strain gauge was made by using a quartz plate.

3. Results

Figure 1 shows temperature dependence of electrical resistivity of the 2.0Fe, 3.3Fe, 3.5Fe, 3.7Fe and 4.0Fe alloys measured in the cooling process and the subsequent heating process. The curve of the 2.0Fe alloy is reproduced from previous study.⁵⁾ The characteristic feature of the resistivity curve is essentially the same for the 2.0Fe, 3.3Fe, 3.5Fe and 3.7Fe alloys. In the cooling process (blue curves) of these alloys the resistivity starts to increase sharply in association with the B2-R transformation as indicated by M_s(R). It then starts to decrease sharply in association with the R-B19′ transformation as indicated by M_s(B19′). In the heating process (red dotted curves) the resistivity starts to deviate from the linear relation due to initiation of the B19′-R transformation, then it merges the cooling curve due to its termination A_f(B19′). The resistivity curve of the 4.0Fe alloy is completely different from that of other alloys. It does not show sharp decreases, and the hysteresis between the cooling and heating processes is small. This implies that the thermally induced R-B19′ transformation is suppressed in this alloy.

Fig. 1

Temperature dependence of electrical resistivity (a)–(e). The value is normalized by the value at 293 K. (f) is the difference in resistivity between the cooling process and heating process at M_s(B19′) temperature.

Figure 1(f) shows the difference in resistivity between the cooling process and heating process at M_s(B19′) temperature. An example is indicated in Fig. 1(c). The value is normalized by the value of the cooling process. The change in resistivity of the 3.7Fe alloy is obviously small compared with other alloys. This implies that the amount of the product phase (B19′-phase) is small in this alloy. Therefore, we may consider that the B2-B19′ transformation is partly suppressed in the 3.7Fe alloy.

The transformation temperatures obtained by resistivity curves are summarized in Fig. 2. In the figure, the values reported previously for the 3.0Fe alloy^16,17) and the 0.0Fe alloy (B2-B19′ transformation)²⁾ is also shown. Both M_s(B19′) and A_f(B19′) decreases as iron content increases, but the composition dependence is not linear. We evaluated the equilibrium temperature of the R-B19′ transformation as T₀ = (M_s(B19′) + A_f(B19′))/2,²⁵⁾ and the value is also shown in Fig. 2. Composition dependence of T₀ is also not linear. By fitting the composition dependence of T₀ using a third order polynomial, T₀ of the 4.0Fe alloy is estimated to be 104 K. Below this temperature, B19′-phase should be the stable phase of the 4.0Fe alloy. It is likely that the R-B19′ transformation is suppressed kinetically for some reason.

Fig. 2

Temperature dependence of R-B19′ transformation start temperature M_s, reverse transformation finish temperature A_f and the equilibrium temperature T₀. The equilibrium temperature for the B2-R transformation is also shown. The data of x = 3 are reproduced from Refs. 16, 17 and that of x ≤ 2 and x ≥ 4 from Ref. 5.

In the case that kinetics dominates a martensitic transformation, the transformation shows a clear time dependence. Recently, time dependent nature of martensitic transformation has been revealed in many alloys.^{4,13,14,21,26–29)} We checked time dependence of B2-B19′ transformation by changing the cooling and heating rate. Figure 3 shows resistivity curves of the 3.7Fe alloy measured with cooling rates of 0.5 K/min, 2 K/min, 5 K/min and 10 K/min. We notice that M_s(B19′) decreases as the cooling rate increases. In addition, the difference in resistivity between the cooling curve and heating curve decreases as the cooling rate increases. These results imply that the R-B19′ transformation proceeds isothermally, which is consistent with a previous report by Kustov et al.³⁰⁾ Considering the time dependent nature, we cannot deny the occurrence of thermally-induced R-B19′ transformation in the 4.0Fe alloy if the cooling rate is extremely slow.

Fig. 3

Temperature dependence of electrical resistivity of Ti–46.3Ni–3.7Fe (at%) alloy measured at different cooling and heating ratio.

Although thermally-induced R-B19′ transformation in the 4.0Fe alloy is completely suppressed within our experimental time scale, application of external stress may assist this transformation. To clarify this, we measured thermal expansion under compressive stress by using a single crystal of the 4.0Fe alloy, and the result is shown in Fig. 4. In the cooling process under zero stress, the alloy shows a sharp expansion starting from 244 K. This temperature agrees with the B2-R transformation temperature. Under compressive stress of 100 MPa and 200 MPa, the specimen contracts sharply at 244 K in association with the B2-R transformation. There is an obvious hysteresis between cooling and heating curves, and the hysteresis under 200 MPa is larger than that under 100 MPa. One characteristic feature is that the strain recovery is incomplete after the thermal cycle under 200 MPa. That is, residual strain of 0.19% appears even after the removal of the stress. In order to understand the behavior, X-ray diffraction profile after the thermal cycle under 200 MPa was examined. As shown in Fig. 5, we detected reflections of the B19′-phase in addition to those of the B2-phase. This implies that thermally-induced R-B19′ transformation occurs partly in the specimen when thermal cycle is made under 200 MPa. This again implies that R-B19′ transformation is kinetically suppressed.

Fig. 4

Strain-temperate curves measured in the cooling process and subsequent heating process of Ti–46Ni–4Fe single crystal. Compressive stress is applied in the [110] on B2-phase.

Fig. 5

X-ray profile of Ti–46Ni–4.0Fe single crystal measured at 298 K after a thermal cycle under 200 MPa along [110] direction on B2-phase.

4. Discussion

In order to understand the reason for the suppression of the R-B19′ transformation in the 4.0Fe alloy, we roughly estimated free energy difference Δg between the R-phase and the B19′-phase in Ti–(50−x)Ni–xFe alloys using experimentally obtained specific heat. Figure 6 shows specific heat of the 4.0Fe alloy (R-phase) and that of the 2.0Fe alloy (B19′-phase) reproduced from a previous paper.⁵⁾ Specific heat of the 4.0Fe alloy is obviously higher than that of the 2.0Fe alloy. Assuming that the difference in specific heat essentially arises from the difference in crystal structure, we estimate molar free energy difference Δg (= g^R − g^B19′) as a function of temperature.

Fig. 6

Temperature dependence of specific heat of Ti–48Ni–2.0Fe (B19′-phase) and Ti–46Ni–4.0Fe (R-phase) alloys reproduced from Ref. 5.

The molar free energy difference, Δg is given as Δg = Δh − TΔs, where Δh is molar enthalpy difference and Δs is molar entropy difference between the R-phase and the B19′-phase. They are given by the specific heat difference ΔC_p (= C_p^R − C_p^B19′) as follows   

\begin{equation*} \Delta h = \Delta h_{0} + \int\limits_{0}^{T}\Delta C_{p}\,dT \end{equation*}

  

\begin{equation*} \Delta s = \Delta s_{0} + \int\limits_{0}^{T}\frac{\Delta C_{p}}{T}\,dT \end{equation*}

We assume that Δs₀ is zero from the third law of the thermodynamics, and Δh₀ is determined so that Δg = 0 at the equilibrium temperature T₀ for each specimen. For the evaluation of Δg, we need the specific heat of both the R-phase and the B19′-phase of an alloy. However, we can obtain either the R-phase or the B19′-phase for an alloy in the temperature range of C_p measurements. If we use C_p^R of the 4.0Fe alloy and C_p^B19′ of the 2.0Fe alloy to estimate ΔC_p, then the calculated value of Δs at the M_s temperature (234 K) of the 2.0Fe alloy becomes −4.0 J/mol K. This value is apparently larger than the Δs evaluated from the latent heat of the 2.0Fe alloy (−2.2 J/mol K).³¹⁾ This deviation is apparently caused by the difference of C_p^R between the 2.0Fe and the 4.0Fe alloys. We need to correct ΔC_p so as to compensate for the difference. As an attempt, we corrected ΔC_p as ΔC_p = 0.6 (C_p^R(4.0Fe) − C_p^B19′(2.0Fe)). By this correction, Δs at 234 K becomes −2.2 J/mol, which agrees with experimental value. If we use the corrected value of ΔC_p, we obtain temperature dependence of Δg of the 2.0Fe, 3.3Fe, 3.5Fe, 3.7Fe and 4.0Fe alloys as shown in Fig. 7.

Fig. 7

Temperature dependence of free energy difference between the R-phase and the B19′ phase of Ti–(50−x)Ni–xFe alloys. The M_s(B19′) temperature of each alloy is indicated by solid circles.

On each Δg curve of Fig. 7, we indicated the transformation start temperature M_s(B19′) by red circles. The value of Δg at M_s is usually called the driving force of martensitic transformation. The driving force increases slightly as M_s decreases. It is 43 J/mol for the 2.0Fe alloy (M_s(B19′) = 234 K) and 48 J/mol for the 3.7Fe alloy (M_s(B19′) = 102 K). The M_s(B19′) temperature dependence of the driving force can be fitted as Δg = 64.8 − 0.155M_s(B19′), which is given by the dotted red line in Fig. 7. We notice that this line intersects with the Δg curve of the 4.0Fe alloy near 50 K and the value of Δg at the intersection is 52 J/mol. This implies that insufficient driving force is not the main reason for the suppression of R-B19′ transformation in the 4.0Fe alloy. We need to consider the contribution of thermal activation process for the R-B19′ transformation.

In order to initiate thermally-induced martensitic transformation, a part of the specimen must overcome a potential barrier which lies on the path of martensitic transformation. Let Q be the height of the potential barrier, then the probability P to overcome the potential barrier is expected to be given as P = A exp(−Q/k_BT), where A is a constant and k_B is Boltzmann factor. The martensitic transformation is expected to start within an observation time when P exceeds a certain value. This is an interpretation for the isothermal nature of martensitic transformations.³²⁾ The value of P is large when Q/k_BT is small, meaning that M_s temperature will be significantly influenced by the value Q/k_BT.

Considering the fact that driving force (Δg) caused by supercooling is necessary for a martensitic transformation, we may expect that the value of Q decrease as Δg increases. Then Q is expected to decrease as temperature decrease. If the decease of Q by temperature decrease is large enough, Q/k_BT also decreases as temperature decreases, resulting in the increase in the probability P to overcome the potential barrier. However, if the decrease of Q is not large enough, Q/k_BT may increase as temperature decreases. In this case, P is not large enough for the martensitic transformation to initiate within the experimental time scale. We may regard this state to be the kinetically arrested state.

Figure 8(a) schematically illustrates Q and Δg on the transformation path, and Fig. 8(b) illustrates temperature dependence of Q. We consider that the probability P of the martensitic transformation is constant on the dotted line because Q/k_BT is constant on this line. We may assume that transformation occurs within our experimental time scale when Q is below this line. In Fig. 8(b), martensitic transformation in Alloy-A is possible in the temperature range indicated by “D”. Martensitic transformation does not occur in Alloy-A below T_A as well as above M_s within experimental time scale. The state below T_A can be regarded as thermally arrested stage. Martensitic transformation in Alloy-B does not occur at any temperature because Q is always larger than the line “A”.

Fig. 8

A schematic illustration showing the free energy on the transformation path from the R-phase to the B19′ phase (a), and the temperature dependence of the potential barrier Q (b). The probability to overcome the potential barrier is constant on the dotted line “A” where Q/k_BT is constant. Martensitic transformation occurs within our experimental time scale when Q is below this line (region “D” for Alloy-A). Martensitic transformation is suppressed in Alloy-B because Q is above the line “A” for all the temperature range.

In the case of Ti–(50−x)Ni–xFe alloys, the lowest M_s of the B2-B19′ transformation is 102 K (3.7Fe) alloy. This implies that at 102 K of the 3.7Fe alloy, the value of Q/k_BT is small enough for the martensitic transformation. The suppression of the B2-B19′ transformation in the 4.0Fe alloy implies that the value of Q/k_BT does not become low enough at any temperature like Alloy-B in Fig. 8(b). As temperature decreases, the numerator Q may decrease due to the increase of Δg as shown in Fig. 7, but Q/k_BT is not low enough because of the decrease of denominator. The application of external stress is expected to decrease Q through the increase of Δg. The application of 200 MPa (Fig. 4) is estimated to increase Δg by 80 J/mol, which must decrease Q substantially. This explains the reason why R-B19′ transformation occurred under compressive stress. For further discussion, we need to know the function Q(T) or Q(Δg(T)), which is a challenging subject in the future.

5. Conclusions

We have studied composition dependence of the R-B19′ transformation in Ti–(50−x)Ni–xFe alloys. The M_s temperature decreases as x increases up to x = 3.7, but the thermally induced martensitic transformation is suppressed when x = 4.0 (4.0Fe alloy). The R-B19′ transformation of the 4.0Fe alloy occurs when a thermal cycle is made under compressive stress of 200 MPa. The free energy difference at M_s (driving force) for the R-B19′ transformation increases as M_s decreases. The driving force expected for the 4.0Fe alloy is 52 J/mol, which can be obtained by cooling below 50 K. Insufficient thermal activation energy at this temperature will be the main reason for the suppression of B2-B19′ transformation in the 4.0Fe alloy. That is, B2-B19′ transformation is thermally arrested in the 4.0Fe alloy.

Acknowledgment

The present study was supported by Kakenhi from JSPS (No. 19H02460).

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