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Lattice properties in the vicinity of the martensitic transformation in TiNi
Solid State Communications,
Vol. 9, pp. 331—334, 1971.
Pergamon Press.
Printed in Great Britain
LATTICE PROPERTIES IN THE VICINITY OF THE MARTENSITIC TRANSFORMATION IN TiNi N.G. Pace and G.A. Saunders Department of Applied Physics and Electronics, University of Durham, South Road, Durham (Received 2 November 1970 by R. Loudon)
A pronounced peak in the thermal expansion coefficient and thus in the Grüneisen parameter has been found at the temperature corresponding to the martensitic transformation in TiNi. This finding, together with results of ultrasound wave propagation studies, attests to anomalous lattice-dynamical behaviour in the vicinity of the phase transition. Ultrasonic wave attenuation and velocity data are interpreted on the basis of Akhieser.type, phonon viscosity damping. Results obtained are consistent with the hypothesis that a soft phonon mode plays a dominant role in the lattice properties of TiNi near the transition. —
—
AFTER plastic deformation at room temperature, TiNi samples revert to their original undeformed shape when heated above the transition temperature i corresponding to a reversible martensitic transformation. This phenomenon is termed the shape memory effect. In a previous paper an ultrasonic study of the elastic and anelastic properties of TiNi was presented. Considerable differences found between the elastic moduli of the high temperature B2 (CsCI) structure and the low temperature phase comprised of two slightly different but distinct base-centered monoclinic martensites2 were shown to accrue essentially from changes in the free carrier density; the binding energy difference between the two phases was attributed largely to the change in Fermi energy. However, while this concept accounts for the energy difference, it provides no information on the mechanism by which the transformation proceeds. Further studies of lattice properties in the vicinity of T,, now lead to some insight into the microscopic nature of the transition.
A very pronounced peak centred around 7’~ has been found in the linear thermal expansion coefficient (a) (Fig. 1) obtained by a dilatometer technique on arc cast samples. Simultaneous measurements of ultrasonic attenuation and thermal expansion have shown that the maximum in both these properties occurs at the same temperature. The usual displacement of the transition during cooling from that during heating is well demonstrated by the results; all data employed here are those obtained during continuous, slow (0.05°Cmm) cooling, the correct way to drive the martensitic transformation. The peak observed in the thermal expansion is reflected in the Gr~ineisenparameter 7 (= 3a$/C,,) (inset in Fig. 1); the bulk modulus ~ used is that reported previously.’ The specific heat at constant pressure C,, exhibits at the transition a maximum ~ attributed entirely to electron density changes; ~ the measured background specific heat (0.29 x lO~ergs deg’ cm3) is close to that (0.277 x lO~ergs deg~cm3) obtainable from the Dulong and Petit law, and, therefore, has been
331
332
MARTENSITIC TRANSFORMATION IN TiNi
Vol. 9, No.5
152 jI.71 174
I
I.7c
_____________
“
146
_
11. 10
140
I
40
— ‘I
‘S ‘C
V
lWno 30
50
70
90
110
201
20
50
FIG. 1. The linear thermal expansion coefficient of TiNi in the vicinity of the transition. The expansion maximum during heating occurs at a higher temperature than it does during a cooling cycle. of The shows parameter the temperature dependence theinset Grüneisen calculated using 5~= 3a~/C,,. taken here to be the total lattice specific heat, The marked peak in the Gnlneisen parameter strongly evidences a direct connection between the transition mechanism and the lattice vibration spectrum. Somewhat similar behaviour of the Grüneisen parameter has been observed in the perovskites SrTiO 6 The ferroand KTaO3. electric transition 3in~ crystals of this type is
40
SO
~
Tempcrature °C
FIG. 2. The temperature dependence in the vicin-
ity of the transition of the attenuation ye!2) and of longiocity vL (converted to modulus PVL tudinal 15MHz ultrasonic waves propagated through TiNi.
known to be associated with the lowering of the frequency of transverse optic (TO) phonon modes and a consequent incipient lattice instability as the temperature approaches 7., In these circumstances the Grtineisen parameter rises since it becomes approximately inversely proportional to the square of the soft optic mode frequency. ~
ultrasonic wave velocities.”5’6’8 The attenuation and velocity of 15MHz longitudinal waves propagating in TiNi are shown in Fig. 2; the large attenuation peak at T~ is superimposed upon a background attenuation due to Rayleigh scattering by the grains and an apparent attenuation due to diffraction and transducer coupling losses.’ The ultrasonic effects found in SrTiO 3 can be explained quanitatively in terms of an Akhieser-type process with a strongly temperature dependent Grüneisen parameter. ~ Here the hypothesis is tested that the ultrasound wave propagation behaviour near T~in TiNi can also be explained by increased damping due to Akhieser-type interactions with the lattice modes.
As the transition is approached in both TiNi and in the displacive type ferroelectrics there is a large increase in ultrasound absorption accompanied by pronounced decreases in the
In the Akhieser process strains produced by the propagating ultrasonic wave modulate the thermal lattice mode frequencies and thus the equilibrium phonon populations; a finite relaxation time T is required to regain the equilibrium configuration;
~
Vol.9, No.5
MARTENSITIC TRANSFORMATION IN TiNi
the actual populations lag in phase behind the driving strain wave and there is a net energy dissipation during each cycle. Woodruff and of this problem; the approach followed here is based on the extension of their work by Barrett6 for KTaO 3. All the low frequency phonon modes are lumped together into a single ‘mode’ 9 have detailed characterised by aprovided specific aheat C atreatment Gr~1neisen constant Ehrenreichy 1,, and a relaxation time ~0, Similarly, a single ‘mode’, denoted by subscript b, is used to describe attenuation ultrasound all the acoustic due tophonon Akhieser-type 6 modes. The interaction with the soft mode is C 2r (1) 0T5~w 2pv 03(1+w21-2) where 5~ = (Yb r is an effective3 relaxfor ation time, p is the density (6.39 g cm TiNi) and v 0 (= 5.3 x 1O”cm sec’) is the sound velocity when y0 equals Yb, that is in the absence of soft mode damping. The change in velocity Lw (= v v0) resulting from the anharmonic 6 contribution of the soft mode to the velocity is C 2 1 = (2) 0TY 2pv 0 1 + W~T 2 —
—
—
.
The experimental data of ultrasound wave attenuation and velocity shown in Fig. 2 have been used td estimate the soft mode specific heat c0 and the effective relaxation time using equations (1) and (2). The change in attenuation a due to the transition can be readily found by subtracting off the background attenuation; but the velocity contribution is more uncertain: the approach used has been to estimate the deviation of the ultrasound velocity from the straight line entrapolation of the velocity measured from 196°C to 0°C up towards T~.Although the calculated values of C0 and T (Fig. 3) cannot be considered as exact due to this uncertainty in Lw, their order of magnitude and general behaviour are certainly consistent with a phonon mode softening in the vicinity of the transition. For the model used, T can be written as (r0 + 1C0/Cb I Tb); close to the transition C0 is much less than C,, and T is dominated by the relaxation time T0 due to ultrasound interaction with the soft modes. —
‘
333
‘
251
“2
10
t
I
.1
40
~
40
I
\
, SC
70
SO
Tsupsi.t~s
FIG. 3. The temperature tive relaxation time (full variation curve) ‘r of andthe theeffecsoft mode specific heat Ca (dashed curve) obtained by fitting 15MHz ultrasonic attenuation and velocity data to equations (1), (2) and (3) and using the Grüneisen parameter 5~shown in Fig. 2. Apart from the Akhieser-type attenuation, another damping mechanism which could play a significant role is the thermoelastic loss; this results from the temperature changes which take place in the strained regions due to passage of a longitudinal ultrasonic wave; heat flows from the hotter compressed regions to the cooler extended regions. The thermoelastic loss at T~is only about 0.O2dbcm’ at 15MHz; this is a relatively large value for thermoelastic loss but still makes a negligible contribution to the measured attenuation (4dbcm~) at the transition temperature. Furthermore, the shear wave attenuation also goes through a large peak, and shear waves cannot suffer thermoelastic loss. To conclude, the measured lattice properties of TIN! the ultrasonic wave attenuation and velocity, the thermal expansion and Griineisen parameter attest to a marked alteration in lattice behaviour in the vicinity of the martensitic phase transition. Increased Akhieser-type damping by thermal vibrations can account for the ultrasonic effects. And the large value achieved by the Grtineisen parameter is a characteristic expected —
—
334
MARTENSITIC TRANSFORMATION IN TiNi
in a material in which a low lying soft phonon mode develops. However, the hypothesis that this transition does proceed through a soft phonon mode mechanism is contentious. Ultrasonic,
Vol.9, No.5
thermal and lattice transport measurements cannot establish the model conclusively; a neutron diffraction analysis of the phonon dispersion curves would provide a further test.
REFERENCES 1.
PACE N.G. and SAUNDERS G.A., Phil. Mag. 22, 73 (1970).
2.
MARCINKOWSKI M.J., SASTRI A.S. and KOSKIMAKI D., Phil. Mag. 18, 945 (1968).
3.
BERMAN H.A., WEST E.D. and ROZNER A.G., J. appi.. Phys. 38, 4473 (1967).
4. WANG F.E., DeSAVAGE B.F. and BUEHLER W.J., J. appi. Phys. 39, 2166 (1968). 5. 6.
REHWALD W., Solid State Commun. 8, 607 (1970). BARRETT,H.H., Phys. Rev. 178, 743 (1969).
7.
COCHRAN W., Adv. Phys. 9, 387 (1960); 10, 401 (1961).
8.
COWLEY R.A., Phys. Rev. 134, A981 (1964); Phil. Mag. 11, 673 (1965).
9.
WOODRUFF T.O. and EHRENREICH H., Phys. Rev. 123, 1553 (1961).
Un grand pic dans le coefficient d’expansion thermique et ainsi dans le parametre de Grüneisen a été observe a Ia temperature qui correspond ê la transformation martensite dans le TiN!. Ce resultat, et les autres resultats sur la propagation des ondes ultrasons montrent un comportement anorma! de Ia dynamique du rCseau au voisinage de in transition de phase. L’attenuation des ondes ultrasons et les mesures de la vitesse ont etC interpretCes ê partir de Ia théorie de Akheiser (phonon viscosity damping). Les resultats sont en accord avec l’hypothCse qu’une mode de phonons (soft mode) joue un role important dans les propriCtCs du rCseau de TiNi au voisinage de la transformation.
Vol. 9, pp. 331—334, 1971.
Pergamon Press.
Printed in Great Britain
LATTICE PROPERTIES IN THE VICINITY OF THE MARTENSITIC TRANSFORMATION IN TiNi N.G. Pace and G.A. Saunders Department of Applied Physics and Electronics, University of Durham, South Road, Durham (Received 2 November 1970 by R. Loudon)
A pronounced peak in the thermal expansion coefficient and thus in the Grüneisen parameter has been found at the temperature corresponding to the martensitic transformation in TiNi. This finding, together with results of ultrasound wave propagation studies, attests to anomalous lattice-dynamical behaviour in the vicinity of the phase transition. Ultrasonic wave attenuation and velocity data are interpreted on the basis of Akhieser.type, phonon viscosity damping. Results obtained are consistent with the hypothesis that a soft phonon mode plays a dominant role in the lattice properties of TiNi near the transition. —
—
AFTER plastic deformation at room temperature, TiNi samples revert to their original undeformed shape when heated above the transition temperature i corresponding to a reversible martensitic transformation. This phenomenon is termed the shape memory effect. In a previous paper an ultrasonic study of the elastic and anelastic properties of TiNi was presented. Considerable differences found between the elastic moduli of the high temperature B2 (CsCI) structure and the low temperature phase comprised of two slightly different but distinct base-centered monoclinic martensites2 were shown to accrue essentially from changes in the free carrier density; the binding energy difference between the two phases was attributed largely to the change in Fermi energy. However, while this concept accounts for the energy difference, it provides no information on the mechanism by which the transformation proceeds. Further studies of lattice properties in the vicinity of T,, now lead to some insight into the microscopic nature of the transition.
A very pronounced peak centred around 7’~ has been found in the linear thermal expansion coefficient (a) (Fig. 1) obtained by a dilatometer technique on arc cast samples. Simultaneous measurements of ultrasonic attenuation and thermal expansion have shown that the maximum in both these properties occurs at the same temperature. The usual displacement of the transition during cooling from that during heating is well demonstrated by the results; all data employed here are those obtained during continuous, slow (0.05°Cmm) cooling, the correct way to drive the martensitic transformation. The peak observed in the thermal expansion is reflected in the Gr~ineisenparameter 7 (= 3a$/C,,) (inset in Fig. 1); the bulk modulus ~ used is that reported previously.’ The specific heat at constant pressure C,, exhibits at the transition a maximum ~ attributed entirely to electron density changes; ~ the measured background specific heat (0.29 x lO~ergs deg’ cm3) is close to that (0.277 x lO~ergs deg~cm3) obtainable from the Dulong and Petit law, and, therefore, has been
331
332
MARTENSITIC TRANSFORMATION IN TiNi
Vol. 9, No.5
152 jI.71 174
I
I.7c
_____________
“
146
_
11. 10
140
I
40
— ‘I
‘S ‘C
V
lWno 30
50
70
90
110
201
20
50
FIG. 1. The linear thermal expansion coefficient of TiNi in the vicinity of the transition. The expansion maximum during heating occurs at a higher temperature than it does during a cooling cycle. of The shows parameter the temperature dependence theinset Grüneisen calculated using 5~= 3a~/C,,. taken here to be the total lattice specific heat, The marked peak in the Gnlneisen parameter strongly evidences a direct connection between the transition mechanism and the lattice vibration spectrum. Somewhat similar behaviour of the Grüneisen parameter has been observed in the perovskites SrTiO 6 The ferroand KTaO3. electric transition 3in~ crystals of this type is
40
SO
~
Tempcrature °C
FIG. 2. The temperature dependence in the vicin-
ity of the transition of the attenuation ye!2) and of longiocity vL (converted to modulus PVL tudinal 15MHz ultrasonic waves propagated through TiNi.
known to be associated with the lowering of the frequency of transverse optic (TO) phonon modes and a consequent incipient lattice instability as the temperature approaches 7., In these circumstances the Grtineisen parameter rises since it becomes approximately inversely proportional to the square of the soft optic mode frequency. ~
ultrasonic wave velocities.”5’6’8 The attenuation and velocity of 15MHz longitudinal waves propagating in TiNi are shown in Fig. 2; the large attenuation peak at T~ is superimposed upon a background attenuation due to Rayleigh scattering by the grains and an apparent attenuation due to diffraction and transducer coupling losses.’ The ultrasonic effects found in SrTiO 3 can be explained quanitatively in terms of an Akhieser-type process with a strongly temperature dependent Grüneisen parameter. ~ Here the hypothesis is tested that the ultrasound wave propagation behaviour near T~in TiNi can also be explained by increased damping due to Akhieser-type interactions with the lattice modes.
As the transition is approached in both TiNi and in the displacive type ferroelectrics there is a large increase in ultrasound absorption accompanied by pronounced decreases in the
In the Akhieser process strains produced by the propagating ultrasonic wave modulate the thermal lattice mode frequencies and thus the equilibrium phonon populations; a finite relaxation time T is required to regain the equilibrium configuration;
~
Vol.9, No.5
MARTENSITIC TRANSFORMATION IN TiNi
the actual populations lag in phase behind the driving strain wave and there is a net energy dissipation during each cycle. Woodruff and of this problem; the approach followed here is based on the extension of their work by Barrett6 for KTaO 3. All the low frequency phonon modes are lumped together into a single ‘mode’ 9 have detailed characterised by aprovided specific aheat C atreatment Gr~1neisen constant Ehrenreichy 1,, and a relaxation time ~0, Similarly, a single ‘mode’, denoted by subscript b, is used to describe attenuation ultrasound all the acoustic due tophonon Akhieser-type 6 modes. The interaction with the soft mode is C 2r (1) 0T5~w 2pv 03(1+w21-2) where 5~ = (Yb r is an effective3 relaxfor ation time, p is the density (6.39 g cm TiNi) and v 0 (= 5.3 x 1O”cm sec’) is the sound velocity when y0 equals Yb, that is in the absence of soft mode damping. The change in velocity Lw (= v v0) resulting from the anharmonic 6 contribution of the soft mode to the velocity is C 2 1 = (2) 0TY 2pv 0 1 + W~T 2 —
—
—
.
The experimental data of ultrasound wave attenuation and velocity shown in Fig. 2 have been used td estimate the soft mode specific heat c0 and the effective relaxation time using equations (1) and (2). The change in attenuation a due to the transition can be readily found by subtracting off the background attenuation; but the velocity contribution is more uncertain: the approach used has been to estimate the deviation of the ultrasound velocity from the straight line entrapolation of the velocity measured from 196°C to 0°C up towards T~.Although the calculated values of C0 and T (Fig. 3) cannot be considered as exact due to this uncertainty in Lw, their order of magnitude and general behaviour are certainly consistent with a phonon mode softening in the vicinity of the transition. For the model used, T can be written as (r0 + 1C0/Cb I Tb); close to the transition C0 is much less than C,, and T is dominated by the relaxation time T0 due to ultrasound interaction with the soft modes. —
‘
333
‘
251
“2
10
t
I
.1
40
~
40
I
\
, SC
70
SO
Tsupsi.t~s
FIG. 3. The temperature tive relaxation time (full variation curve) ‘r of andthe theeffecsoft mode specific heat Ca (dashed curve) obtained by fitting 15MHz ultrasonic attenuation and velocity data to equations (1), (2) and (3) and using the Grüneisen parameter 5~shown in Fig. 2. Apart from the Akhieser-type attenuation, another damping mechanism which could play a significant role is the thermoelastic loss; this results from the temperature changes which take place in the strained regions due to passage of a longitudinal ultrasonic wave; heat flows from the hotter compressed regions to the cooler extended regions. The thermoelastic loss at T~is only about 0.O2dbcm’ at 15MHz; this is a relatively large value for thermoelastic loss but still makes a negligible contribution to the measured attenuation (4dbcm~) at the transition temperature. Furthermore, the shear wave attenuation also goes through a large peak, and shear waves cannot suffer thermoelastic loss. To conclude, the measured lattice properties of TIN! the ultrasonic wave attenuation and velocity, the thermal expansion and Griineisen parameter attest to a marked alteration in lattice behaviour in the vicinity of the martensitic phase transition. Increased Akhieser-type damping by thermal vibrations can account for the ultrasonic effects. And the large value achieved by the Grtineisen parameter is a characteristic expected —
—
334
MARTENSITIC TRANSFORMATION IN TiNi
in a material in which a low lying soft phonon mode develops. However, the hypothesis that this transition does proceed through a soft phonon mode mechanism is contentious. Ultrasonic,
Vol.9, No.5
thermal and lattice transport measurements cannot establish the model conclusively; a neutron diffraction analysis of the phonon dispersion curves would provide a further test.
REFERENCES 1.
PACE N.G. and SAUNDERS G.A., Phil. Mag. 22, 73 (1970).
2.
MARCINKOWSKI M.J., SASTRI A.S. and KOSKIMAKI D., Phil. Mag. 18, 945 (1968).
3.
BERMAN H.A., WEST E.D. and ROZNER A.G., J. appi.. Phys. 38, 4473 (1967).
4. WANG F.E., DeSAVAGE B.F. and BUEHLER W.J., J. appi. Phys. 39, 2166 (1968). 5. 6.
REHWALD W., Solid State Commun. 8, 607 (1970). BARRETT,H.H., Phys. Rev. 178, 743 (1969).
7.
COCHRAN W., Adv. Phys. 9, 387 (1960); 10, 401 (1961).
8.
COWLEY R.A., Phys. Rev. 134, A981 (1964); Phil. Mag. 11, 673 (1965).
9.
WOODRUFF T.O. and EHRENREICH H., Phys. Rev. 123, 1553 (1961).
Un grand pic dans le coefficient d’expansion thermique et ainsi dans le parametre de Grüneisen a été observe a Ia temperature qui correspond ê la transformation martensite dans le TiN!. Ce resultat, et les autres resultats sur la propagation des ondes ultrasons montrent un comportement anorma! de Ia dynamique du rCseau au voisinage de in transition de phase. L’attenuation des ondes ultrasons et les mesures de la vitesse ont etC interpretCes ê partir de Ia théorie de Akheiser (phonon viscosity damping). Les resultats sont en accord avec l’hypothCse qu’une mode de phonons (soft mode) joue un role important dans les propriCtCs du rCseau de TiNi au voisinage de la transformation.