Scripta Materialia 50 (2004) 181–186 www.actamat-journals.com Kinetics of martensitic transitions in shape-memory alloys ~osa Antoni Planes *, Francisco-Jose Perez-Reche, Eduard Vives, Lluıs Man Departament d’Estructura i Constituents de la Materia, Facultat de Fısica, Universitat de Barcelona, Diagonal, 647, E-08028 Barcelona, Catalonia, Spain Accepted 4 September 2003 Abstract The role of thermal fluctuations in driving the martensitic transition in shape-memory alloys is studied. Experiments show that both athermal and thermally activated kinetics can be obtained in different alloys. We establish that this depends upon the relative importance of the characteristic times associated with the driving field and nucleation.
2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Athermal; Thermally activated; Nucleation; Martensite; Shape-memory 1. Introduction When an external parameter is varied during a firstorder phase transition, the system evolves from a given state (typically metastable) towards a more stable state which corresponds to a minimum of a thermodynamic potential. Such a transition involves overcoming an energy barrier separating the minima corresponding to each phase. Thermal fluctuations are expected to be relevant for the system for it to overcome the energy barrier. Within this scenario––which is the most accepted conventional description of first-order phase transitions––the value of the external parameter (temperature, stress, magnetic field. . .) at which the transition occurs is not always the same upon repeating the experiment, but rather the values at the transition are spread out within a certain range. Such a spread reflects the stochasticity of thermal fluctuations. The phase transitions conforming to the above description are referred to as thermally activated. Furthermore, because these transitions can occur at constant values of all external parameters, they are often called isothermal. In isothermal transitions the amount of resulting phase depends on the value of the external parameter and also depends explicitly on time. The transition exhibits hysteresis and it is rate dependent. Examples of these kinds of transitions are found among the most common first* Corresponding author. Tel.: +34-93-402-1181; fax: +34-93-4021174. E-mail address: toni@ecm.ub.es (A. Planes). order phase transitions as, for instance, the freezing of water on cooling. In nature there are systems which undergo first-order phase transitions that, at least from an experimental point of view, do not conform to the description given above. In these systems, thermal fluctuations do not seem to play any role in the kinetics of the transition. They can only take place while the external parameter is changing, the transition appears to be instantaneous on practical time scales, and the amount of transformed material does not explicitly depend on time and is only a function of the value of the external parameter. These transitions are referred to as athermal [1]. Upon repeating the experiment, they always occur at the same value of the external parameter. Although these transitions also exhibit hysteresis, they do not appear to be rate dependent (at least for the rates available in typical experiments). Actually, the path followed by the system is influenced by the existence of disorder (dislocations, grain boundaries, composition, atomic configuration, etc.) which controls the actual distribution of energy barriers. As the system evolves, it passes through a sequence of metastable states and the kinetics is characterized by jumps (avalanches) from one metastable state to another which occur during very short times [2]. Typical examples of athermal transitions are the magnetization reversal in ferromagnetic systems [3] and structural phase transitions in many solids [4]. A common feature of all these systems is the existence of long-range interactions which arise from compatibility constraints [5]. 1359-6462/$ - see front matter
2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2003.09.021 182 A. Planes et al. / Scripta Materialia 50 (2004) 181–186 The present paper focuses on the study of the martensitic transition in shape-memory alloys. The martensitic transition in these alloys has traditionally been regarded as athermal. However, recent experiments have questioned the athermal character of these transitions, and much debate has arisen. Kakeshita et al. [6] performed electrical resistance measurements under isothermal conditions on Cu–Al–Ni alloys. They found that after a certain incubation time at a temperature above the transition temperature (determined from continuous cooling) a sudden change in the resistance was observed, which was associated with the occurrence of the martensitic transition. After these results, Aspelmeyer et al. [7] investigated the Ni–Al alloy system by means of a reflectivity technique. They observed changes in the reflected intensity of a laser beam caused by the appearance of a surface relief at a constant temperature. Very recently, however, Otsuka et al. [8] have pointed to the possibility that the apparently isothermal character found in these alloys could be due to the occurrence of a certain diffusion process. In order to check for this possibility, they investigated the Ti–Ni alloy system which is well known for the absence of significant difussion processes. They performed isothermal electrical resistance experiments similar to those conducted by Kakeshita et al. [6], and did not find any indication of isothermal martensite growth. They concluded that the martensitic transition was not time dependent if diffusion processes were suppressed. It is the aim of the present paper to contribute to the debate of whether the martensitic transition in shapememory alloys is athermal or isothermal by presenting accurate experiments on several Cu-based shape-memory alloys, and giving a simple model which show us up to what extent a transition can be classified as athermal. The model furnishes a unified description of the experimental results showing athermal as well as isothermal characteristics. 2. Experimental results 2.1. Experimental procedures The usual experimental procedure to assess the athermal or isothermal character of martensitic transitions has been the following: the sample is held at a constant temperature while some physical property (electrical resistance, reflectivity, etc.) is continuously monitored. The holding temperature is located slightly above the temperature at which the transition has been detected to start on a continuous cooling, which is assumed to be the socalled martensitic start temperature Ms . Such a procedure has the inconvenience that the temperature where the transition starts during continuous cooling may depend on the cooling rate. Furthermore, as will be shown below, the values may differ significantly from one cycle to another even if they are performed at the same rate. Therefore, the concept of Ms may be ill-defined in some cases and, as a consequence, there is an intrinsic uncertainty in determining how far the holding temperature is from Ms . We have recently proposed [9] the two following different procedures which enable these problems to be circumvented. 2.1.1. Stepwise cooling This procedure resembles the usual process of isothermal holding, but in this case the sample is cooled down in a stepwise manner. Each step consists of an isothermal plateau lasting a time Dt, followed by cooling down to the next plateau at a given rate. It is convenient to test different values for Dt and for the temperature difference between consecutive plateaux (DT ). This procedure is advantageous in that it does not require an a priori determination of Ms because the stepwise cooling is continued until the sample transforms (either during the isothermal plateau or during the short cooling ramp), and hence, for large enough Dt it determines the transition temperature unambiguously with a maximum error of DT . It is worth remarking that by making DT small, it is possible to achieve a very fine tuning for the detection of the isothermal growth of martensite. 2.1.2. Scaling A procedure complementary to that described above consists of cooling the sample through the whole transition, while continuously monitoring some physical parameter. This cooling has to be repeated at different well-controlled cooling rates. Under continuous cooling, athermal and isothermal systems will behave differently as explained below. Let W be a physical quantity which is supposed to depend on the amount of transformed material. If thermal fluctuations are irrelevant, W will depend on temperature but it will not explicitly depend on time WðT Þ. By contrast, for a thermally activated process, the amount of transformed material will also depend on time WðT ; tÞ. The time derivative for the athermal case reads: dW dW dT ¼ ; dt dT dt ð1Þ while for the isothermal case: dW oW dT oW ¼ þ : dt oT dt ot ð2Þ Consider now that dW=dt is measured at different cooling rates and that the ratio ½ðdW=dtÞ=ðdT =dtÞ ¼ _ =T_
is plotted as a function of T . It is clear from Eqs. W (1) and (2) that for the athermal case all data points will collapse on a single curve (scaling), while no such collapse or scaling will be observed in the isothermal case. 183 A. Planes et al. / Scripta Materialia 50 (2004) 181–186 2.2. Results Ms (K) TH 255 250 TL 245 3 σ (K) Among all the experimental techniques commonly used to study martensitic transitions, it is worth choosing one being very sensitive to the transformation of a small amount of material. This is the case of the acoustic emission (AE) technique which we have used for the present study. This technique is probably the best suited to monitor the early stages of the transformation. Full details of the experimental system are given elsewhere [9,10]. We have investigated several Cu-based and Ni-based single crystals. In this paper we will present results corresponding to Cu–Al–Ni and Cu–Zn–Al crystals, which transform towards 2H and 18R martensitic structures, respectively. The samples were annealed at 850 C and quenched into iced-water. After this heat treatment, all samples were annealed for several days at room temperature. By doing this we were sure that all diffusion processes had finished and that the vacancy concentration had reached its equilibrium value [11–13]. In addition, the samples were subjected to a large number of cycles (more than 50) before the actual experiments started, so that the state of the sample does not significantly evolve from one cycle to the next during the experiment [14]. An example of the results obtained is shown in Fig. 1. For Cu–Zn–Al, we have not found AE under isothermal conditions (Fig. 1(a)) in any of the experiments, including long-temperature plateaux (Dt > 40 h) separated by step intervals of DT ¼ 0:2 K. By contrast, for Cu–Al– Ni, no isothermal AE is observed for plateaux with DT > 0:8 K, but for DT < 0:8 K, AE was systematically observed at constant temperature (Fig. 1(b)). From these results it could be inferred that thermal fluctuations appear to be irrelevant in Cu–Zn–Al, while they 260 2 1 0 0 1 2 3 4 5 6 dT/dt (K/min) Fig. 2. hMs i and r as a function of dT =dtð¼ T_ Þ. The continuous lines are fits of the proposed model. The temperatures TH and TL obtained from the fit are indicated. are operative in the nucleation of Cu–Al–Ni. The results from continuous cooling are consistent with these statements; good scaling is observed for Cu–Zn–Al (Fig. 1(c)) while there is no scaling at all for Cu–Al–Ni (Fig. 1(d)). Furthermore, for Cu–Al–Ni the transition does not seem reproducible from one cycle to another (performed at the same rate), but rather, the values obtained for Ms spread over a temperature range and exhibit a stochastic character. We have thus measured this quantity for a large number of cycles which enables us to perform a statistical analysis of the results by computing the average value hMs i and its standard deviation r ¼ ðhMs2 i  hMs i2 Þ1=2 . The results obtained are shown in Fig. 2 (solid symbols) as a function of the cooling rate. We would like to point out that any possible fake dependence due to ageing phenomena was taken into account [9]. The mean value of the transition temperature seems to slightly decrease on increasing the cooling rate. More evident is the increase in the standard deviation of the values as the cooling rate increases. 3. Model Fig. 1. AE during stepwise cooling. (a) Cu–Zn–Al, (b) Cu–Al–Ni. Scaling corresponding to measurements at different T_ . (c) Cu–Zn–Al, (d) Cu–Al–Ni. Notice that in the scaled representation the scatter increases for smaller T_ since the relative error in jdN =dT j is proportional to T_ 1 . The measured values of the transition temperature are spread over a certain range due to thermal fluctuations. A suitable theory to explain the behaviour of hMs i and r is based on the analysis of the mean first-passage time [15] between the parent phase and the martensitic phase. Here we briefly summarise this theoretical approach. The probability P ðtÞ for the system to remain in the metastable phase after time t is given by: dP ðtÞ=dt ¼ kðtÞP ðtÞ, where kðtÞ is the transition probability per unit time. As the transition seems to be athermal for certain systems, while it is not in others, our main assumption in reproducing the experimental observations is that thermal fluctuations are only active within two characteristic limiting temperatures: when T > TH , k ¼ 0 184 A. Planes et al. / Scripta Materialia 50 (2004) 181–186 so that no transition is possible, and when T 6 TL , k ¼ 1 so that the transition occurs with absolute certainty. This assumption leads to the following expressions for the average temperature transition hMs i and its fluctuations r2 : hMs i ¼ TH  T_ hti; ð3Þ 2 r2 ¼ T_ 2 ht2 i þ TH2  2T_ TH hti  hMs i : ð4Þ where hti and ht2 i can be obtained once the transition probability is known. Actually, the transition probability is related to the energy barrier between parent and product phases DF. A reasonable assumption is
0 for T > TH k¼ m 1 ð5Þ for T 6 T 0 DFðT Þ H where m0 is a characteristic energy per unit time related to thermal fluctuations. The energy barrier can be evaluated from a freeenergy functional which is adequate to describe the martensitic transitions in shape-memory alloys. It has been shown [16–18] that a good candidate is a Landau free energy with two anharmonically coupled order parameters: a shear homogeneous strain e and the amplitude g of an inhomogeneous shear strain or shuffle which corresponds to an incipiently unstable phonon. These two order parameters are suitable to describe the crystallographic transition mechanism [19]. Symmetry considerations allow the following free-energy expansion: 1 1 1 1 F ¼ m x2 g2 þ bg4 þ cg6 þ C 0 e2  jeg2 ; 2 4 6 2 ð6Þ where C 0 is the shear elastic constant and x, the frequency of the incipiently unstable phonon. The remaining constants are assumed to be positive. Minimization with respect to e leads to an effective free energy, which after suitable rescaling can be expressed in the following simple form: 2 f ¼ ða  1Þg2e þ g2e ðg2e  1Þ ; ð7Þ where ge is the reduced order parameter. The preceding equation contains a single dimensionless control parameter a, which is temperature dependent a¼ 16m x2 c 3ðb  2j2 2 Þ C0 ¼ T  TL : TH  TL ð8Þ Notice that the term ðb  2j2 =C 0 Þ must become negative for the first-order martensitic transition to take place. A typical feature of martensitic materials is the temperature softening for shearing {1 1 0} planes along Æ1 1 0æ directions, made evident by the decrease of C 0 and x2 on reducing temperature. In this model, such a softening shows up by a decrease of a when the temperature is reduced. Within the context of the present study, this free energy has to be considered as a local free energy adequate to account for the onset of the martensitic Fig. 3. Landau free-energy as a function of a. transition at Ms . From this viewpoint, the parameter a should be considered as an effective parameter related to renormalized values of C 0 and x in the spirit of the local soft-mode theory as proposed by Guenin and Clapp [20]. We have plotted the reduced free energy as a function of the reduced order parameter for selected values of a in Fig. 3. For a > 43 there is only a minimum, corresponding to the parent phase; as a decreases, two symmetric minima develop and for a ¼ 1 all three minima have the same energy value and the two phases are equally stable. This corresponds to the situation of local equilibrium between parent and product phases. At this point, if thermal fluctuations are operative, they can drive the system from the parent phase to the product phase. This situation corresponds to the temperature TH . Upon further cooling the softening of C 0 and x lead to a reduction of a so that the minima corresponding to the product phase become deeper and the energy barrier decreases, thus increasing the probability of transformation in the isothermal case (see Eq. (5)). In the athermal limit the existence of the energy barrier prevents the transformation to the product phase, and hence, the transformation can only take place when there is no energy barrier; this corresponds to a ¼ 0, and the minimum corresponding to the parent phase no longer exists. This situation is the limit of stability of the parent phase (temperature TL ) that is, below this temperature the system is sure to be in the product phase. For a 6 1 (T 6 TH ), the probability that the system transforms to the product phase is obtained from Eq. (5) and from the expression for the barrier within the proposed model. It is given by m : ð9Þ 9a þ ð4  aÞ3=2  8 pffiffiffi where m ¼ m0 ð1 þ 3 3Þ=DFðTH Þ is a characteristic frequency for nucleation. The dependence of the transition probability on a is depicted in Fig. 4. The discontinuity at a ¼ 1 reflects the fact that the nucleation of the product phase is not possible for T > TH while its probability is finite at T ¼ TH . kðaÞ ¼ A. Planes et al. / Scripta Materialia 50 (2004) 181–186 185 small. However, if this time becomes comparable to sdr rate-dependent effects will occur, which are not associated with thermal fluctuations. From the three time scales introduced, the following scenarios can be contemplated: Fig. 4. Transition probability k divided by m as a function of the reduced temperature a for the model presented in Section 3. Within the preceding model a transition must be classified as (ideally) athermal if it always occurs at TL , independently of the rate at which temperature is swept. This means that fluctuations are not operative in this limit. In any other case, the transition is, strictly speaking, thermally activated. In this case, the model described above enables one to compute the mean first passage time which, through Eqs. (3) and (4), leads to values of hMs i and r2 as functions of T_ =mðTH  TL Þ. We have used these expressions to perform a simultaneous non-linear minimum-v2 fitting to the experimental data shown in Fig. 2 for Cu–Al–Ni. The values obtained are TH ¼ 256  1 K, TL ¼ 248  1 K and m ’ 5  102 s1 . The fit (shown as a continuous line in Fig. 2) agrees remarkably well with experimental data, and reproduces the increase of r with T_ and also the slightly decreasing behaviour of Ms . 4. Discussion From the results presented above it can be concluded that, as already mentioned, athermal transitions do not occur strictly at finite temperatures. Nevertheless, in systems that evolve by processes of avalanches from one metastable state to another, there are three characteristic time scales involved in the transition whose relative importance controls the kinetics observed in these systems. The typical time scale for thermal fluctuations is given by sfl ¼ m1 and is related to the passage probability per unit time. On the other hand, the rate of change of the external field (temperature in our case) is characterized by the following characteristic time: sdr ¼ ðTH  TL Þ=T_ . Notice that when sfl  sdr , fluctuations become irrelevant, and the transition exhibits an athermal character. For Cu–Al–Ni we have estimated that this occurs for cooling rates above 30 K/min, while for Cu–Zn–Al, rates above 0.1 K/min are sufficient. Finally, once the energy barrier is overcome, the system takes a given characteristic time to decay to the subsequent metastable state (sav ). This time is not given by the proposed model, but it is expected to be very • When sav  sdr  sfl , the effect of thermal fluctuations will be irrelevant, and the transition will exhibit metastability, an athermal character and rate-independent properties. This is, for instance, the case of the studied Cu–Zn–Al alloy, for which TH  TL is very small [9]. • When sav  sdr ’ sfl the effect of thermal fluctuations will be observable and the transition will exhibit metastability and an isothermal character. This is the case, for instance, of the Cu–Al–Ni alloy for the studied range of cooling rates. • When sav  sfl  sdr the transition is expected to follow an equilibrium path. An important feature which remains open is the following; how do the different time scales, and particularly sdr which depends on TH  TL , relate to physical properties of a given system? The different observed behaviour of systems transforming to different martensitic structures suggests that relevant time scales are intimately related to the different accommodation mechanisms of the shape-change necessary to minimize the stored elastic energy during the transformation. Actually, microgliding and microtwinning are the operative mechanisms in Cu-based shape-memory alloys transforming to the 18R and 2H phases respectively [21]. Finally, we would like to point out that the relaxation from one metastable state to another metastable state both in the athermal as well as in the thermally activated cases is an irreversible process with associated energy dissipation. It is therefore interesting to consider briefly the dissipation occurring during the martensitic transformation, which is directly related to the hysteresis of the transition. This problem has been very controversial in the past [22]. In any case, for thermally induced transitions it has been well established that the dominant mechanism of hysteresis is the relaxation of elastic strain energy. With this idea in mind, it has been conjectured that all energy losses can be considered, in thermodynamic language, as mechanical work that the system dissipates without entropy production [23]. This is supported by the fact that elastic energy is released in the form of elastic waves which are detected as AE. It must be stressed that, once these elastic waves are absorbed in the neighbourhood of the system, the dissipative work finally gives rise to entropy production. The preceding interpretation seems to be more adequate for transitions displaying an athermal character which appear to be independent of cooling rate. Regarding the results presented in the present paper, this interpretation 186 A. Planes et al. / Scripta Materialia 50 (2004) 181–186 is therefore more adequate for the transition to the 18R structure (in which thermal fluctuations are in practice not operative) than for the 2H . Whether or not this is the case still remains an open issue. Very precise calorimetric measurements in samples under very wellcontrolled thermomechanical treatments are needed to answer this important question. Acknowledgements This work has received financial support from the CICyT (Spain), project MAT2001-3251, and from the CIRIT (Catalonia), project 2001SGR00066. F.J.P acknowledge DGCyT for a Ph.D. Grant. References [1] Cao W, Krumhansl JA, Gooding RJ. Phys Rev B 1990;41:11319. [2] Vives E, Ortın J, Ma~ nosa Ll, Rafols I, Perez-Magrane R, Planes A. Phys Rev Lett 1994;72:1694. [3] Bertotti G. Hysteresis in magnetism. New York: Academic Press; 1998. [4] Nishiyama Z. Martensitic transformations. New York: Academic Press; 1978. 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