Entropy change in the B2 → B19′ martensitic transformation in TiNi alloy

Thermochimica Acta 602 (2015) 30–35

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Entropy change in the B2 ! B190 martensitic transformation in TiNi alloy Resnina Natalia * , Belyaev Sergey Saint-Petersburg State University, Universitetsky pr. 28, Saint-Petersburg 198504, Russia

A R T I C L E I N F O

A B S T R A C T

Article history: Received 26 August 2014 Received in revised form 29 December 2014 Accepted 5 January 2015 Available online 17 January 2015

Different methods for calculating the entropy change in the thermoelastic martensitic transformation were analysed. A new way of estimating the transformation entropy was proposed based on the simple relationship between the heat released during the forward martensitic transformation and the finish temperature of the forward phase transition. The value of the B2 ! B190 transformation entropy was calculated for Ti50Ni50 alloy using the different methods. It was shown that the values of DSA!M calculated based on the “entropy argument”, “Clausius–Clapeyron relation” and through the calorimetric data (proposed in the present work) were close to each other and gave values which were approximately equal to 0.105 J/(g K). As the method proposed in the present work is simpler than others, it might successfully be used to estimate the transformation entropy and the contributions of the latent heat, elastic and dissipative energies. ã 2015 Elsevier B.V. All rights reserved.

Keywords: DSC Martensitic transformation Entropy change Thermodynamics TiNi alloy

1. Introduction The shape memory effects found in many alloys are caused by the thermal elastic martensitic transformation [1,2], which was discovered by Kurdyumov and Khandros in 1949 [3]. Initially, this type of solid–solid transformation was initiated in the alloys by temperature or stress changes [4]. Later, it was found that the martensitic transformation may be induced by hydrostatic pressure [5], ultrasound [6], neutron irradiation [7] or magnetic fields [8]. In spite of the method for initiation of thermal elastic martensitic transformation in the alloy, the parameters of this phase transition are determined from the thermodynamic balance between the change in free Gibbs energy (DGch), elastic energy (Eel) and dissipative energy (Edis), sometimes called “frictional work” [3,9]:

DGch ¼ Eel þ Edis :

(1)

In [9], it was shown that the elastic energy Eel stored in the alloy during forward martensitic transformation was responsible for the temperature ranges of the forward and reverse martensitic transitions. Dissipative energy (frictional work) Edis was a cause of the transformation hysteresis. Hence, the temperatures of the

* Corresponding author. Tel.: +7 9119949636; fax: +7 8124287079. E-mail addresses: [email protected] (R. Natalia), [email protected] (B. Sergey). http://dx.doi.org/10.1016/j.tca.2015.01.004 0040-6031/ ã 2015 Elsevier B.V. All rights reserved.

martensitic transformation and the energy contributions in the thermodynamic balance are connected to each other, and one of them may be used for estimation of the others [14–17]. According to the determination DG ¼ DH  T  DS (DH = enthalpy, DS = entropy changes in the transformation), one may give the Eq. (1) for forward austenite ! martensite transformation as follows:

DGA!M ¼ DHA!M  T  DSA!M ¼ Eel þ Edis : ch

(2)

This equation should be written at three different temperatures: T0 (thermodynamic equilibrium temperature), Ms and Mf (the start and finish temperatures of austenite ! martensite transition). It is necessary to take into consideration that Eel = Edis = 0 at T0 temperature, and Eel = 0 at Ms temperature [9– 16]. Thus, one may obtain the following Eqs. (3)–(5):

DHA!M  T 0  DSA!M ¼ 0atT 0 temperature

(3)

DHA!M  Ms  DSA!M ¼ EdisatMstemperature

(4)

DHA!M  Mf  DSA!M ¼ Eel þ EdisatMftemperature

(5)

The equation for dissipative energy (frictional work) may be easily obtained from Eqs. (3) and (4):

R. Natalia, B. Sergey / Thermochimica Acta 602 (2015) 30–35

Edis ¼ ðT 0  Ms Þ  DS

A!M

:

(6)

As the temperature T0 may be estimated as T0 = (Af + Ms)/2 [9– 16], then the dissipative energy is equal to:
 Af  Ms (60)  DSA!M : Edis ¼ 2 The equation for elastic energy may be found from Eqs. (3),(5) and (6) as follows: T 0  DSA!M  Mf  DSA!M ¼ Eel þ ðT 0  Ms Þ  DSA!M : Eel ¼ ðMs  Mf Þ  DSA!M

(7)

Transformation temperatures may be easily measured from the calorimetric or resistivity curves obtained on cooling and heating of the alloy through the temperature range of the phase transition. However, it is difficult to calculate the change in free Gibbs energy and elastic and dissipative energies because the value of the entropy changes in the transformation (transformation entropy) is impossible to obtain directly from the experimental data. In [10–12], the value of the transformation entropy change was estimated using complex thermodynamics analysis of the calorimetric data. In this analysis, the main assumption (entropy argument) about the null change in the entropy of the universe “specimen–surroundings” during transformation was made to obtain the equations for estimation of transformation entropy. In [11], this analysis was used to estimate DSA!M and the energy contributions for thermally induced martensitic transformation in polycrystalline Fe–24Pt alloy, single-interface and multiple-interfaces thermally induced transition in Cu–27.7Al–2.3 Ni single crystal. In [12], this method was applied to determine the entropy change, enthalpy, and elastic and dissipative energies in the case of the thermally induced transformation under stress or for stressinduced transformation. In these studies, the DSA!M was calculated using the equation:

DSA!M ¼

ZMf

dQ A!M dT T

(8)

Ms

if the difference between the heat capacity measured in the fully austenite or fully martensite state was equal to zero (CpA–CpM = 0), or

DSA!M ¼

ZMf

dQ A!M dT T

(9)

Af

if DCp = Cp A–CpM 6¼ 0. In both equations dQA!M is the increment in heat released on cooling, T is the sample temperature, Ms and Mf

Fig. 1. Dependence of heat flow on temperature obtained on cooling of the Ti50Ni50 alloy through the temperature range of the forward martensitic transformation.

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are the start and finish temperatures of the forward martensitic transformation, and Af is the finish temperature of the reverse transition. At the same time, in [13], it was proposed that the entropy argument postulated in [10–12] was not correct because it was based on the assumption that a variation in transformation entropy was zero during the forward and reverse transformations owing to the absence of heat dissipation. In [13], it was shown that dissipation energy was the work spent against the frictional force of moving interfaces, and was thus revealed as heat. In this case, an increase in entropy occurred. Hence, an assumption about null entropy could not be applied to describe the thermoelastic martensitic transformation. The authors of [13–16] proposed that the transformation entropy should be calculated through a Clausius–Clapeyron-like relation: ! ds DSA!M (10) ¼ dT et where ds was a variation in stress acting in the alloy, dT was a variation in transformation temperature, DSA!M was a transformation entropy and et was the transformation strain measured as a value of the plateau during the stress-induced transformation in a tensile test. Thus, DSA!M may be calculated using the following equation:

DSA!M ¼ 

ds  et : dT

(11)

However, there is a huge difficulty in estimating et during a tensile test because the value of the pseudoelastic plateau depends on many parameters [2,18]. For instance, in [18,19], the et value was measured as the unelastic strain recovered on unloading. At the same time, it is well known that the pseudoelastic unloading is not observed in equiatomic Ti50Ni50 alloy [2,20]. Thus, in this alloy, the et cannot be measured as in [18,19], and another method of determining the et should be found. Therefore, both known methods of determining the transformation entropy change are very complex and need to make many mathematical calculations or carry out many experiments. It is necessary to find a simple way to estimate the DS value using the experimental data that avoids the difficulties described above. It may be done using an Eq. (12) between the heat released during forward martensitic transformation QA!M from one side and the latent heat, elastic energy and dissipation energy [10]: Q A!M ¼ DHA!M þ Eel þ Edis

(12)

Fig. 2. Dependence of the entropy change value calculated according to “entropy argument” on the sample temperature.

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Taking into account the Eqs. (3), (6), (7) and T0 = (Af + Ms)/2, Eq. (12) may be rewritten as: Q A!M ¼ 

Af þ Ms A  Ms  DSA!M þ ðMs  Mf Þ  DSA!M þ f 2 2

 DSA!M ¼ Mf  DSA!M

(13)

It is seen that the transformation entropy may be simply estimated as a ratio of the heat released during forward martensitic transformation to the finish temperature of the forward martensitic transition:

DSA!M ¼

Q A!M Mf

(14)

Both values of QA!M and Mf may be easily found from simple calorimetric measurements carried out on the cooling and heating of the sample through the temperature range of the martensitic transformation. Thus, the value of the transformation entropy may be estimated through a simple DSC experiment. There is a question about which method gives the true value of the transformation entropy and what is the difference between the values of DSA!M estimated by different methods. Solving this question will allow us to find the right way to estimate the DSA!M value and any terms in the thermodynamic balance Eq. (1) for thermoelastic martensitic transformation. To correctly estimate the transformation entropy, it is necessary to calculate the value of DSA!M for the same TiNi alloy using the different methods described previously and to determine the optimal method for calculation, which was the aim of the present work.

Fig. 3. Stress–strain diagrams obtained on deformation of the Ti50Ni50 samples at different temperatures at which the sample is in austenite phase (a) and dependence of the stress s p on deformation temperature found in the Ti50Ni50 sample (b).

2. Experimental procedure Commercial Ti–50.0 at.% Ni alloy (bought in company MATEKSMA Ltd., Moscow, Russia) wire samples with a diameter of 0.5 mm were preliminarily annealed at a temperature of 773 K for 2 h. After this heat treatment, the structure of the alloy was homogeneous and at high temperatures no any other phases or precipitates except cubic B2 phase were found. On cooling the alloy underwent forward martensitic transformation from the cubic B2 phase to the monoclinic B190 phase and on subsequent heating the B190 phase transformed to B2 phase. Hence the B2 $ B190 thermoelastic martensitic transformation occurred in the studied sample. A sample with an average mass of 5 mg (length: 5 mm) was used for a calorimetric study in a differential scanning calorimeter (Mettler Toledo 822e). The sample was cooled and heated in the temperature range of 413 K–293 K at a cooling/heating rate of 10 K/ min. Using the calorimetric curve, the transformation temperatures (Ms = 337 K, Mf = 328 K, As = 355 K, Af = 369 K) and the transformation heat of the forward transformation (QA!M = 34.4 J/g) were determined. Standard uncertainties u were u(Ms) = 1 K, u (Mf) = 1 K, u(As) = 1 K, u(Af) = 1 K and u(QA!M) = 0.1 J/g. The ds /dT value was estimated through stress–strain diagrams obtained at different temperatures close to the Af temperature. The wire samples (length: 100 mm) were subjected to tension up to 22% at different temperatures in the range of 348 K–433 K (in austenite state). To achieve the temperature of 413–433 K, the sample was heated to these temperatures. To attain temperatures

Fig. 4. Dependence of Ms temperature (a) and recoverable strain (the value of the shape memory effect) (b) on the stress acting on cooling and heating. The Ms and et values were measured on e(T) curves obtained on cooling and heating of the Ti50Ni50 alloy under a stress of 50, 100, 150 and 200 MPa.

R. Natalia, B. Sergey / Thermochimica Acta 602 (2015) 30–35

lower than 413 K, the sample was first heated to 413 K and then cooled to the required temperature. This procedure was carried out to provide the austenite structure of the sample at all deformation temperatures. Stress–strain diagrams were obtained using the testing machine Lloyd 30k Plus equipped with a thermal chamber. The value of the stress was measured by a stress sensor, and the strain was measured using a video-extensometer (base length: 40 mm). The temperature was measured by a thermocouple that touched the sample. The strain rate was 4105 s1. Standard uncertainties u were u(Td) = 1 K, u(e) = 0.0001 and u(s ) = 1 MPa. The dMs/ds value was estimated through e(T) curves obtained on cooling and heating of the sample under stress of 50, 100, 150 and 200 MPa. The wire samples (length: 100 mm) were put into the testing machine Lloyd 30k Plus equipped with a thermal chamber, heated up to 420 K, loaded up to a certain stress and subjected to cooling and heating in the temperature range of 420– 300 K. The cooling/heating rate was 5 K/min. Standard uncertainties u were u(Ms) = 1 K, u(e) = 0.0001 and u(s ) = 1 MPa. 3. Results and discussion 3.1. Calculation of the transformation entropy using Eq. (9) based on the entropy argument To use Eqs. (8) or (9) for calculation of the entropy change during B2 ! B190 transformation, it is necessary to use the dependence of the heat flow on the temperature obtained on cooling with a constant temperature rate (Fig. 1). It is seen that the peak of the heat release is observed on cooling due to a forward martensitic transformation. Moreover, levels of heat flow before and after transformation do not coincide, and hence, the heat capacity of austenite phase CAp is not equal to the heat capacity of martensite phase CMp. In this case, to calculate the entropy changes in the transformation (DSA!M), it is necessary to use Eq. (9) and carry out the following procedure: 1. Divide the values of dH (heat flow) by the cooling rate (b) to

obtain the value of dQM = dH/b. 2. Divide the value of dQM(T) by the current temperature. 3. Plot the dQM/T(T) curve and integrate this curve from Af to Mf.

All these mathematical operations have been done using the calorimetric curve given in Fig. 1 and Origin software, which allows one to make an integration using the graph data. Fig. 2 shows dependence of entropy change on temperature and according to [10–12], the value of entropy change in transformation DSA!M is

Fig. 5. Estimation of the heat released during the forward martensitic transformation in the Ti50Ni50 alloy.

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Table 1 Mf temperaturea measured in different ways and the value of transformation entropyb estimated through the calorimetric data using the Eq. (14). All calorimetric data were carried out at a pressure p = 0.1 MPa. Way for Mf measurement

Mf, K

DSA!M, J/(g K)

As intersection of tangent lines As a temperature where the heat release is stopped As a temperature of 90% enthalpy

328 325 331

0.105 0.106 0.104

a Standard uncertainties u are u(Mf) = 1 K, u(p) = 10 kPa. Combined expanded uncertainty Uc is Uc(r) = 0.0004 J/(g K) (0.95 level of confidence). b Combined expanded uncertainty Uc is Uc(DS) = 0.0004 J/(g K) (0.95 level of confidence).

equal to value of entropy change measured at Mf temperature. It is equal to 0.104 J/(g K). This value is larger than the value DS = 0.077 J/(g K) found in [14]. 3.2. Calculation of the transformation entropy through the Clausius– Clapeyron-like equation using Eq. (11) To calculate the change in entropy through the Clausius– Clapeyron equation, the ds /dT value should be found. Fig. 3a shows the stress–strain diagrams obtained on tension at different temperatures. It is seen that an increase in deformation temperature results in an increase in yield stress that corresponds to the formation of stress-induced martensite. The value of the stress s p (stress of the pseudoelastic plateau) was measured as the intersection of the tangent lines on the s (e) curves, and the dependence of s p on the deformation temperature is given in Fig. 3b. An increase in deformation temperature from 348 K to 393 K leads to a linear increase in the pseudoelasticity stress from 205 MPa to 415 MPa. A further increase in deformation temperature results in a decrease in the stress increment, and finally, a change in the deformation temperature from 413 K to 433 K does not influence the value of s p. According to [19], an increase in deformation temperature must lead to a linear increase in pseudoelasticity stress. Such behaviour was found only in the temperature range of 348 K to 393 K. Thus, the ds /dT value was estimated in this temperature range and it was equal to 7.97 MPa/K. To estimate the value of DSA!M according to Eq. (11) it is necessary to find a value of et. In [18,19], this value was determined as the value of strain recovered on pseudoelastic unloading. However, pseudoelastic unloading has not been found in the studied alloys (Fig. 3a); hence a value of et can not be found as described in [18,19]. That is why the value of the transformation strain was determined as a plateau length obtained on loading and it was equal to 8.54%. Using Eq. (11), the value of the transformation entropy was calculated and was equal to DSA!M ¼ 0:681MPa K . To express the value of transformation entropy in J/(g K), it is necessary to divide the value of DS measured in MPa/K by the alloy density (6.5 g/cm3), which gives DSA!M = 0.1047 J/(g K). This value is close to the value calculated using Eq. (9). At the same time, the value of et determined as the length of the plateau observed on loading is not equal to the transformation strain because this value may include both transformation strain (et) induced by the appearance of the martensite phase and plastic strain (epl) that appears due to a plastic accommodation of the stress during martensite transformation. The value of epl does not varied on unloading at the same time, the value of et should disappear. However, as seen from Fig. 3a, the equiatomic TiNi alloy does not demonstrate the pseudoelastic behaviour on unloading, hence there is no possibility of finding the value et exactly. Therefore, it may be concluded that the value of the transformation entropy DSA!M = 0.1047 J/(g K) is not true.

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R. Natalia, B. Sergey / Thermochimica Acta 602 (2015) 30–35

Table 2 Comparison of the change in transformation entropy according to Eq. (13). Way for DS estimation

a b c d

estimated by the different methods and heat releasede during forward martensitic transformation calculated

Equation

Source

DSA!M,

QTA!M, J/g

J/(g K) Through the “entropy argument”

DSA!M ¼

ZMf

[10–12]

0.104a

34.1

[13–16]

0.1047b

34.3

[13–16]

0.114c

37.4

[given work]

0.105d

34.4

dQ A!M dT T

Af

Through the Clausius–Clapeyron relation using s (e) curve

ds  et dT

DSA!M ¼  Through the Clausius–Clapeyron relation using e(T) curve

ds  et dT

DSA!M ¼  Through the calorimetric data

DSA!M ¼

a b c d e

Q A!M Mf

Combined expanded uncertainty Uc is Uc(DS) = 0.006 J/(g K) (0.95 level of confidence). Combined expanded uncertainty Uc is Uc(DS) = 0.005 J/(g K) (0.95 level of confidence). Combined expanded uncertainty Uc is Uc(DS) = 0.02 J/(g K) (0.95 level of confidence). Combined expanded uncertainty Uc is Uc(DS) = 0.0004 J/(g K) (0.95 level of confidence). Combined expanded uncertainty Uc is Uc(QT) = 0.2 J/g (0.95 level of confidence).

On the other hand, the ds /dT and et values may be measured using the e(T) curves obtained on cooling and heating under a constant load. In this case, the ds /dT value is determined as an inverse value of the dMs/ds and the et value – as the maximum strain recovered on heating (the value of the shape memory effect). In the present work, the e(T) curves were obtained on cooling and heating of the Ti50Ni50 alloy under a different constant stress that varied from 50 to 200 MPa. The value of the Ms temperature has been determined as the intersection of the tangent lines on e(T) curve, and the dependence of this temperature on the stress acting on cooling is shown in Fig. 4a. It is found that the dependence of Ms(s ) is linear and the dMs/ds coefficient is equal to 0.06 K/MPa, hence the ds /dT value is equal to 16.67 MPa/K. Comparison of the results shown in Figs. 3b and 4a shows that the ds /dT coefficient measured from e(T) curves is higher than the same coefficient measured from s (e) curves. This may be due to the different martensite structure that appears on cooling or under stress, but as this question is out of the present study it should be studied further and published elsewhere. Fig. 4b shows the dependence of the recoverable strain (the value of the shape memory effect) on the stress acting on cooling. An increase in the stress from 0 to 150 MPa results in a rise in the recoverable strain up to 4.46%. Further increase in the stress up to 200 MPa does not influence the recoverable strain and it may be concluded that the value of 4.46% is the maximum recoverable strain found in the studied sample; hence, it is equal to et (transformation strain). Using the values of ds /dT = 16.67 MPa/K and the et = 4.46%, the value of the transformation entropy is found to be equal to 0.114 J/(g K). This value is somewhat higher than the DSA!M value measured using the s (e) curves. Apparently, this is due to some mistake in the determination of the Ms temperature, causing a mistake in the value of the dMs/ds coefficient. 3.3. Calculation of the entropy change directly from the calorimetric data (Eq. (14)) To estimate the change in transformation entropy using Eq. (14), it is necessary to determine the value of the heat released during the

forward martensitic transformation. In the present work, this has been done using the experimental data presented in Fig. 1 by the STARe software of the Mettler Toledo apparatus. Fig. 5 shows the method of measuring the heat released during the forward martensitic transformation. Taking into account the difference between the heat capacities of the austenite and martensite phases, the baseline is given by the spline connection of the heat flow before and after transformation. The value of the heat released during the forward transition is equal to 34.4 J/g. The value of the Mf temperature may be found in different ways: as the intersection of the tangent lines, as the temperature where the heat release is stopped and, for instance, as the point where the enthalpy attains 90% of the maximum value. These methods give different values of the finish temperature of the forward martensitic transformation and one may assume that this influences significantly the value of DSA!M measured according to Eq. (14). Table 1 shows the values of the Mf temperature measured in the different ways and the calculated values of DSA!M. It is seen that variation in the transformation temperature results in a very small variation in transformation entropy. The lowest Mf temperature was 325 K (measured as the temperature where the heat release was stopped) and the highest temperature was 331 K (measured as the temperature of 90% of the transformation enthalpy); thus, the variation between the maximum and the minimum values of the Mf temperature was 6 K. This gave a variation of the DS from 0.106 J/ (g K) to 0.104 J/(g K) and it was less than 2% from the average value of the entropy equal to 0.105 J/(g K). From Table 1, it is seen that the average value of DSA!M corresponds to the change in transformation entropy estimated at an Mf temperature equal to 328 K. This is the temperature measured as the intersection of the tangent lines and hence, for correct estimation of the DS value, the Mf temperature should be measured by this method; moreover, this way of measuring the transformation temperature is described in the ASTM standard (F2004-00) [21]. Therefore, the value of the change in transformation entropy estimated using Eq. (14) is equal to 0.105 J/(g K) and this is close to those obtained using the entropy argument or through the Clausius–Clapeyron relation using the s (e) curves.

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3.4. Comparison of the transformation entropy values estimated using different methods Table 2 shows the values of transformation entropy estimated by the different methods. The values of DSA!M vary between 0.104 and 0.114 J/(g K). To determine which method of calculation is most reliable, the values of DSA!M were used to estimate the heat released during the forward martensitic transformation using Eq. (13) and taking into account that the Mf was equal to 328 K. The estimated values of QTA!M were compared to the QA!M value found on the calorimetric curve (34.4 J/g). In Table 2, it is seen that the estimated values of QTA!M are close to the experimental value if the transformation entropies are calculated according to Eqs. (9), (11) (using the s (e) curve) and (14). Hence, there is no significant difference between these methods for estimation of the DSA!M value. As the method proposed in the present paper (Eq. (14)) is simpler than that described in [10–12] and [13–16], it may be used to estimate the transformation entropy and contributions of the latent heat, and elastic and dissipative energies to the thermodynamic balance Eq. (1). These terms were calculated for the studied sample using the value of transformation entropy estimated through Eq. (14), transformation temperatures and Eqs. (3), (6’) and (7). These values were equal to DHA!M = 37.06 J/(g K), Eel = 0.945 J/(g K), and Edis = 1.68 J/(g K). 4. Conclusions The obtained results may be summarized as follows: 1. A simple way to measure the transformation entropy through

the values of the parameters measured on the calorimetric curve is proposed. 2. The transformation entropy was estimated using the different methods described in [10–16] and proposed in the present paper for the Ti50Ni50 alloy. The value of DSA!M for B2 ! B190 martensitic transformation was calculated based on the entropy argument, Clausius–Clapeyron relation and from calorimetric data (proposed in the present work). It was shown that the value of the transformation entropy does not depend on the calculation method and it was approximately equal to 0.105 J/(g K). This value of the transformation entropy allows one to calculate the value of the heat release QTA!M close to the experimental value QA!M measured through the calorimetric data. 3. As the method proposed in the present work is simpler than the methods based on the entropy argument [10–12] and the Clausius–Clapeyron relation described in [13–16], it may be used for calculation of the transformation entropy and contributions of the latent heat, and elastic and dissipative energies in the thermodynamic balance equation.

Acknowledgement This work has been carried out through the financial support of Saint-Petersburg State University projects (projects number 0.37.177.2014, 6.37.147.2014).

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