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Functional Materials Letters Vol. 2, No. 2 (2009) 73–78 © World Scientific Publishing Company
THE USE OF SHAPE-MEMORY ALLOYS FOR MECHANICAL REFRIGERATION LLUÍS MAÑOSA∗ , ANTONI PLANES and EDUARD VIVES Departament d’Estructura i Constituents de la Materia, Facultat de Física, Diagonal 647, 08028 Barcelona, Catalonia, Spain ERELL BONNOT† Departament de Física i Enginyeria Nuclear, ETSEIB, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Catalonia, Spain RICARDO ROMERO‡ IFIMAT, Universidad del Centro de la Provincia de Buenos Aires and CICPBA, Pinto 399, 7000 Tandil, Argentina Received 2 February 2009
This letter reports on stress–strain experiments on a Cu–Zn–Al single crystal performed using a purpose-built tensile device which enables the load applied to the specimen to be controlled while elongation is continuously monitored. From the measured isothermal tensile curves, the stress-induced entropy changes are obtained at different temperatures. These data quantify the elastocaloric effect associated with the martensitic transition in shape-memory alloys. The large temperature changes estimated for this effect, suggest the possibility of using shape-memory alloys as mechanical refrigerators. Keywords: Elastocaloric effect; martensitic transition; shape-memory alloy; magnetocaloric effect.
exhibiting other effects such as the electrocaloric effect5 and the barocaloric effect6 also appear as good candidates for clean refrigeration. Caloric effects are inherent to any thermodynamic system. They are expected to be particularly large in the vicinity of first-order phase transitions where tiny variations of external control parameters lead to significant changes in the extensive thermodynamic quantities. For instance, in magnetic materials, the giant magnetocaloric effect is a consequence of a first-order magnetostructural transition, which encompasses a large entropy change, related to the latent heat of the transition.7 The martensitic transition in shape-memory alloys meets these requirements and therefore large elastocaloric effects take place in these materials. The similarity between the elastocaloric effect in shape-memory alloys and the widely reported giant magnetocaloric effect in materials undergoing magnetostructural transitions will be presented. In a thermodynamic system described by generalized forces {Yi } and displacements {xi }, a differential change in entropy is expressed as n ∂xi C dS = dT + · dYi , (1) T ∂ T {Y j =1,...,n }
Shape-memory alloys are an important class of functional materials which are being used for sensor and actuator applications. Their functionality is linked to the large reversible deformations (up to 10%) they can experience under appropriate changes of temperature and stress which arise from the martensitic transition undergone by these alloys. In the present letter we report on a different functional application, namely the possibility of these alloys being used as mechanical refrigerators. This property is based on the caloric response under the application of an external stress (the elastocaloric effect), which is expected to be large in the vicinity of the martensitic transition.1 The need for environmental-friendly cooling methods has prompted an intense search for materials exhibiting large caloric effects at and around room temperature, which could be an alternative to the classic compressor-based technologies involving gases that are harmful to the environment. Among several caloric effects, the magnetocaloric effect has received increasing interest,2–4 although materials ∗
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i=1
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74 L. Mañosa et al.
where generalized Maxwell relations
∂S ∂Yi
T ,Y j =i
=
∂xi ∂T
(2) Y j =1,...,n
have been used and C is the heat capacity at a constant value of the fields. Each pair of conjugated variables have the same tensorial order. A caloric effect associated with a differential change of a field Yi is quantified by S(0 → Yi ) =
Yi
0
∂xi ∂T
{Y j =1,...,n }
· dYi ,
(3)
when the field is applied isothermally, and T (0 → Yi ) = − 0
Yi
T C
∂xi ∂T
{Y j =1,...,n }
· dYi ,
and T irr (0 → Y ) ≥ T (0 → Y ).
(5)
which leads to the following inequalities which are to be satisfied by the isothermal entropy change and the adiabatic
(7)
For a first-order transition with a given associated hysteresis, the dissipated energy can be estimated as E diss = (Y − Y0 )x, where Y0 is the equilibrium transition field at temperature T , and Y is the actual transition field at this temperature. x is the discontinuity in the generalized displacement. Consideration of dissipative effects leads to a modified Clausius–Clapeyron equation which (under the assumption of constant x) reads dY S 1 d E diss =− + . dT x x d T
(4)
when the field is applied adiabatically. In a typical experiment, the control parameter is one of the components of the force tensor, say Y , while the measured quantity is the displacement component conjugated to this field, x. In the particular case of the elastocaloric effect, the field is a uniaxial tensile stress σ for which the corresponding generalized displacement is the strain ε (or relative elongation along the direction of the applied force). In the case of the magnetocaloric effect, the magnetization M (along the field) and magnetic field H are the generalized displacement and force, respectively. It is worth noticing that in the vicinity of a phase transition, ∂ x/∂ T is expected to be large, since x experiences large changes arising from discontinuities in the displacements at first-order phase transitions. This will lead to giant caloric effects (see Eqs. (3) and (4)). Actually, due to such a discontinuity, the use of Maxwell relations in computing the entropy change at first-order phase transitions gave rise to some controversy. It has been shown,4,8 however, that the above expressions are in general valid, and they lead to the Clausius–Clapeyron equation in the particular case of an ideal first-order phase transition for which Eq. (3) renders the discontinuity in the entropy at the transition, associated with the latent heat. So far, the thermodynamic analysis has assumed equilibrium processes. When considering the effect of irreversibility, the entropy contains a reversible (d S) and irreversible (δSi ≥ 0) contributions, so that the heat exchanged is expressed as δq = d S − δSi , T
temperature change, induced by a change of the field from 0 to Y : Y δq (6) S(0 → Y ) ≥ 0 T
(8)
In general, E diss is weakly dependent on temperature, and thus the last term in the previous equation is small, which shows that the Clausius–Clapeyron equation is still a good approximation. Moreover, in shape-memory alloys, the martensitic transformation takes place close to equilibrium conditions, and dissipative effects are small.9 Therefore, Eqs. (3) and (4) are expected to be valid in evaluating caloric effects associated with this transition. In the above description, we have considered the generalized displacements (and fields) to be independent of one another. Coupling between different degrees of freedom is reflected by an explicit dependence of one coordinate on the remaining ones. In this case, the expressions given in this section are still valid, but it must be taken into account that the entropy changes obtained with the above equations will contain contributions from all degrees of freedom, which are modified when the field is changed from 0 to Y . This is particularly relevant in the case of giant magnetocaloric materials for which there is strong coupling between structure and magnetism and the large magnetic field-induced entropy change has a significant contribution from the entropy change associated with the structural change.10–12 The elastocaloric effect for a solid subjected to an applied uniaxial stress σ with an associated uniaxial strain ε is given by σ ∂ε dσ (9) S(0 → σ ) = ∂T σ 0 and
σ
T (0 → σ ) = − 0
T C
∂ε ∂T
dσ.
(10)
σ
The martensitic transition in shape-memory alloys is first order, and is characterized by discontinuities in the strain ε
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The Use of Shape-Memory Alloys for Mechanical Refrigeration 75
and entropy St (associated with the latent heat of the transition). We will restrict our analysis to non-magnetic shapememory alloys, for which the entropy change has a vibrational origin,13,14 and the strain is not coupled to any other degree of freedom. In the vicinity of the transition, the strain can be expressed as ε(T, σ ) = ε0 + εF [(Tt (σ ) − T )/T ],
(11)
where ε0 is the strain outside the transition region, F is a shape-function and T is a measure of the temperature range over which the transition spreads. Tt (σ ) is the equilibrium transition temperature for a given applied stress σ . In strict equilibrium, T → 0 so that F approaches the Heaviside step function. Using expression (9) and assuming that ε0 and ε are constant, in this equilibrium case the elastocaloric effect in the vicinity of the transition is given by ε − for T ∈ [Tt (0), Tt (σ )] α , (12) S(0 → σ ) = 0 for T ∈ / [Tt (0), Tt (σ )] where α ≡ d Tt /dσ is assumed to be constant. Taking into account the Clausius–Clapeyron equation, α = −ε/St , where St is the transition entropy change, thus, as expected, S(0 → σ ) = St , and T = Tσ −Tt (0) = ασ . Notice that these results confirm the validity of using Maxwell relations in computing the elastocaloric effect, even in the case of an ideal first-order phase transition (involving a discontinuity in the entropy). A single crystal of Cu–Zn–Al, obtained by melting metals of 99.999% purity, was grown by the Bridgman method. The composition of the crystal was determined from EDX to be Cu68.13 Zn15.74Al16.13. The martensitic transition temperature under cooling without external stress is Ms = 234 K. A rectangular sample with cylindrical heads was machined from the ingot. The body of the sample has flat faces 35 mm long, 3.95 mm wide and 1.4 mm thick. The crystallographic direction of the tensile axis is close to the [001] direction. The sample was annealed at 1073 K for 30 min, cooled down to room temperature in air and finally aged for 2 h in boiling water. This heat treatment ensures that the sample is in the ordered state, free from internal stresses and the vacancy concentration is close to the equilibrium state. Typically, stress–strain experiments are carried out by means of tensile machines that control the sample displacement while the load is continuously measured. In order to perform mechanical experiments that can be directly compared to the corresponding experiments in other caloric systems (where the field is the control parameter and the generalized displacement is the measured quantity), we have developed a purpose-built experimental set-up which enables external control of the load applied to the sample, while the elongation is
continuously monitored. The system uses special grips which adapt to the heads of the specimen. The upper grip is attached to a load cell hanging from the ceiling and the lower grip holds a container that plays the role of a dead load. The load is increased or decreased at a well-controlled rate by supplying or removing water by means of a pump. A cryofurnace enables the experiments to be conducted at different temperatures. Details of the device can be found in Ref. 15. Typical stress–strain hysteresis loops at three selected temperatures are illustrated in Fig. 1. On increasing the stress, strain linearly increases (elastic behavior of the hightemperature phase) until a certain stress where the martensitic transition starts, giving rise to a large increase of strain (about 8%) at an almost constant stress. The elastic behavior of the martensite is obtained when the stress is further increased. Upon unloading, the strain decreases with a certain hysteresis (about 10 MPa) at the martensitic transition. An increase in the transition stress with increasing temperature is clearly observed. In Fig. 2(a), we show the stress–strain curves corresponding to the loading branches measured at different temperatures. Here the control parameter (σ ) is plotted on the horizontal axis while the measured parameter (ε) is plotted on the vertical axis. Note that with this representation, the curves are similar to the corresponding isothermal field-magnetization curves used to compute the magnetocaloric effect in magnetic materials. From the curves in Fig. 2(a) it is possible to compute the elastocaloric effect by means of Eq. (9). In order to minimize numerical errors, we first numerically compute the area A(T ) below the curve up to a certain stress. Results are shown in Fig. 2(b), where the molar value V = 7.52 cm3 mol−1 has been used to achieve the energy units. The entropy change is then obtained from the temperature derivative of this data set as σ dA d . (13) εdσ = S = dT 0 dT
Fig. 1. Stress–strain hysteresis loops at different temperatures. From top to bottom: T = 310.5 K (green), T = 305.4 K (red) and T = 296.1 K (blue). (color online)
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76 L. Mañosa et al.
(a) Fig. 3. Stress-induced entropy change at selected values of the stress ranging from 105 to 143 MPa. The color code (and symbols) correspond to the stress value indicated in Fig. 2(a) by a vertical line. The continuous lines are fits based on the Eq. 14. (color online)
(b) Fig. 2. (a) Stress–strain curves at selected temperatures. From right to left, T= 310.5, 309.4, 307.5, 306.0, 305.4, 303.1, 302.0, 299.9, 298.8, 297.9, 296.1, 295.3 and 294.6 K. (b) Area below the stress–strain curves. The color code (and symbols) correspond to the maximum stress indicated as a vertical line in Fig. (a). (color online)
The obtained results are plotted in Fig. 3, which shows the stress-induced entropy change S as a function of temperature, for the different levels of applied stress. It is interesting to point out that the maximum stressinduced entropy change remains constant within the errors (independent of temperature and stress). This value corresponds to the whole entropy change of the martensitic transition St . As previously mentioned, the temperature and stress dependence of the strain can be expressed by Eq. (11). In our case, a convenient choice for the shape-function is F (z) = tanh−1 (z), which leads to the following expression for the stress-induced entropy change Tt (σ ) − T St 1 + tanh . (14) S(0 → σ ) = 2 T This expression has been fitted to the stress-induced entropy change data given in Fig. 2(a), and the results for each value of the applied stress are shown as continuous lines. By averaging over the whole set of curves, we obtain the value St = −1.21 ± 0.15 J mol−1 K−1 . It is interesting to compare the value obtained for the whole entropy change (St ) to the value given by the
Fig. 4. Transition stress as a function of temperature. The line corresponds to a linear fit.
Clausius–Clapeyron equation. Figure 4 shows the transition stress as a function of temperature, where σt has been taken as the inflection point of each isothermal stress–strain curve. A linear rise in σt with increasing temperature is observed, with a slope dσt /d T = 2.01 MPa/K. By taking an average value for the discontinuity of strain ε = 0.075 ± 0.005 (see Fig. 5(a)), we obtain St = −1.15 ± 0.05 J/mol K. The good agreement of this value with that obtained from the elastocaloric effect, suggests that irreversible effects are negligible for the martensitic transition under study. To support this assertion, we have computed the area of the hysteresis loops obtained at different temperatures, which quantify the dissipated energy in a f full cycle E diss (to within a good approximation it is twice the energy dissipated in each branch, E diss). Results are shown in Fig. 5(b). It is evident that there is no significant dependence of the dissipated energy on temperature, which leads to a vanishing second term in Eq. (8), thus confirming that the Clausius–Clapeyron equation is valid to obtain the entropy change at the martensitic transition for this class of materials. In order to explore the possibilities of the elastocaloric effect for mechanical refrigeration, it is important to estimate the temperature change associated with the application of a
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The Use of Shape-Memory Alloys for Mechanical Refrigeration 77
(a)
(b)
Fig. 5. (a) Strain discontinuity at different values of temperature and (b) dissipated energy computed from the area of hysteresis loops, as a function of temperature.
given stress, which under the assumption of constant C, can be obtained as T (15) T ≈ − S. C In the temperature range of interest here, for Cu–Zn–Al, C 25 J K−1 mol−1 ,16 so for an adiabatic drop in stress σ (involving the whole transition) of about 30 MPa, the maximum expected temperature change is 15 K. Notice that this value is several orders of magnitude larger than the typical values given by the conventional elastic behavior in solids far from any phase transition. It is worth mentioning that the expected temperature change for mechanical refrigeration is even larger than the values reported for the magnetocaloric materials for magnetic fields up to 2 T (achievable by permanent magnets). Hence, the 15 K value estimated is competitive with the values reported for pure Gd (6 K) and Gd–Si–Ge (7 K).17 Another figure of merit to be considered is the refrigerant capacity, which is defined as2 R= S(T )d T ST, (16)
the strong dependence of transition temperature on stress (see Fig. 4), and also by the lack of a critical point (as occurring in the magnetic case) above which the first-order transition no longer occurs. Actually, the upper bound for the applied stress is imposed by the elastic limit of the cubic phase. In summary by means of stress–strain experiments, it has been shown that shape-memory alloys exhibit a large elastocaloric effect. The estimated temperature changes associated with this effect are larger than those reported for giant magnetocaloric materials for fields achievable by permanent magnets. In addition, the energy losses associated with the first-order character of the transition have been found to be weak. Present results suggest that shape-memory alloys have a potential use for close-to-room temperature mechanical refrigeration. This new functional property adds to the well-known shape-memory properties in this class of alloys, and opens-up the possibility of designing multi-functional devices based on shape-memory alloys.
which in the case of elastocaloric effect is equal to R = −εσ , where σ is the change of stress necessary to change the transition temperature by T (S and ε are assumed to be constants). For the investigated alloy, we obtain R = 2.5 J cm−3 for σ = 30 MPa. The interest in mechanical refrigeration is that σ can be chosen in a broad range of values (see Fig. 3). In contrast, this is not the case for giant magnetocaloric materials for which large entropy changes are only obtained in a relatively narrow temperature interval. The main difference is due to the fact that while tensile experiments can be performed at temperatures well above the transition temperature at zero stress, in magnetic experiments, the field is always applied close to the transition temperature at zero field. Such a different experimental procedure is made possible by
We acknowledge financial support from CICyT (Spain) under project MAT2007-62100 and from DURSI (Generalitat de Catalunya) through project 2005SGR00969.
T
Acknowledgments
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