- Email: [email protected]
40 vinylidene fluoride and trifluoroethylene ferroelectric copolymer
Physica A 314 (2002) 714 – 721
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Correlation between molecular dynamics and thermal hysteresis in the 60=40 vinylidene #uoride and tri#uoroethylene ferroelectric copolymer E. L'opez Cabarcosa; ∗ , A.F. Bra˜nab , B. Frickc , F. Batalland a Departamento
de Qu mica-F sica Farmac eutica, Facultad de Farmacia, Universidad Complutense de Madrid, 28040 Madrid, Spain b Instituto de Sistemas Optoelectr onicos y Microtecnolog a, Universidad Polit ecnica de Madrid, Ciudad Universitaria, 28040 Madrid, Spain c Institut Laue-Langevin, 38042 Grenoble Cedex, France d Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain
Abstract Thermal hysteresis associated with ferroelectric to paraelectric phase transition is investigated in a ferroelectric copolymer of vinylidene #uoride and tri#uoroethylene, using X-ray di2raction and incoherent neutron scattering. The lattice spacing derived from the di2ractograms, shows hysteresis at the ferro–paraelectric phase transition. The analysis of the elastic incoherent neutron spectra indicates that besides the structural component, the thermal hysteresis phenomena appears in the dynamics of the copolymer. Furthermore, the area of the hysteresis loop increases with Q until it reaches the Q value corresponding to the Braggs peak. At a given temperature, inside the ferroelectric hysteresis loop, the incoherent quasielastic component di2ers in intensity depending c 2002 Elsevier Science B.V. on recording the spectrum at the heating or at the cooling scan. All rights reserved. PACS: 77.84.Jd; 05.70.Fh; 61.25.Hq Keywords: Hysteresis; Phase transition; Ferroelectric polymer; Elastic neutron scattering; Quasielastic neutron scattering
∗
Corresponding author. E-mail address: [email protected] (E.L. Cabarcos).
c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 1 0 4 3 - 9
E.L. Cabarcos et al. / Physica A 314 (2002) 714 – 721
715
1. Introduction In a macromolecular system, like a polymer, thermal hysteresis appears during a heating–cooling cycle, accompanying the Arst-order phase transitions. In this work, we report on an experimental study of the thermal hysteresis loop associated with the ferroelectric–paraelectric phase transition in a 200 m random copolymer Alm of vinylidene #uoride (VF2 ) and tri#uoroethylene (F3 E) with VF2 molar fraction 60%. This is a semicrystalline polymer (crystallinity ∼0:7), consisting of -((-CH2 -CF2 -)x − (-CHF-CF2 -)1−x )n sequences, which has been studied over the past years [1– 6]. The copolymer exhibits a Curie temperature Tc and upon heating the pseudohexagonal ferroelectric crystals transform to a centrosymmetric paraelectric phase. Recently, a theory of the ferroelectric phase transition, speciAcally developed for these materials, predicts [4] that the area of the thermal hysteresis loop would change not only with the heating rate but also by the application of tensile stress and hydrostatic pressure. 2. Experimental The structural studies were made using wide angle X-ray di2raction (WAXD)at the double-focussing camera for synchrotron radiation on the polymer beamline at HASYLAB, Hamburg. For neutron di2raction experiments we have used the two-axis di2ractometer D1B, at the Institute Laue Langevin (ILL) in Grenoble. The study of the molecular dynamics was performed using elastic and quasielastic incoherent neutron scattering spectrometers IN10 and IN16. A more detailed description of the experimental setup and sample preparation can be found in a previous publication [3]. In these copolymers, provided that the Bragg peaks are avoided, the scattering intensity is almost entirely incoherent and Sinc (Q; ! ≈ 0) is a good measure of the incoherent elastic scattering function for the H atoms of the polymeric chain. To measure Sinc (Q; ! ≈ 0) we have used the “Axed elastic window method” in which the monochromator and analyzer are set to the same energy and only those neutrons that change their energy by an amount smaller than the energy resolution of the instrument are detected. 3. Results and discussion 3.1. Structural hysteresis The structural changes at the ferroelectric transition were followed by means of WAXD. To avoid changes in crystallinity, the sample was heated at a constant rate ◦ of 10 C min−1 above the Curie temperature and then cooled down to room temperature at the same rate. Fig. 1 illustrates a series of WAXD patterns in the range ◦ ◦ O following the above temperature variation. The study of 17 ¡ 2 ¡ 21 ( = 1:5 A) the lattice spacing allows to follow the structural hysteresis at the ferro–paraelectric transition as is shown in Fig. 2. Below the Curie temperature the lattice spacing of the
716
E.L. Cabarcos et al. / Physica A 314 (2002) 714 – 721
Fig. 1. Three-dimensional plot of the WAXD patterns (recorded every 20 s) for the random copolymer of vinylidene #uoride and tri#uoroethylene (60=40%) as a function of temperature. The copolymer was heated ◦ ◦ ◦ at a rate of 10 C min−1 from 25 C up to 127 C, then it was kept at his temperature for ∼10 min and ◦ Anally it was cooled down to 25 C.
5.0
d (Å)
4.9 4.8 4.7 4.6 4.5 20
40
60
80
100
120
T (°C)
Fig. 2. Variation of the lattice spacing, d, as a function of temperature at the ferro–paraelectric transition during a heating–cooling cycle.
◦
ferroelectric phase remains constant up to 40 C, then gradually increases and shows a stepwise behavior at the phase transition. The fact that on cooling, the Curie temperature is shifted towards lower temperatures leads to a corresponding shift of the ◦ paraelectric phase lower boundary (∼50 C). To characterize the Q region a2ected by structural hysteresis, we have recorded two neutron di2raction patterns at the phase transition temperature, one during the heating scan and the other when the sample is cooled down. As can be seen in the O −1 and di2ractograms shown in Fig. 3 the background remains constant for Q ¡ 1:2 A
E.L. Cabarcos et al. / Physica A 314 (2002) 714 – 721
717
0.05 Tu = 65 °C Td = 65 °C
S(Q)
0.04
0.03
0.02 0.5
1.0
1.5
2.0
2.5
-1
Q(A ) ◦
Fig. 3. Two neutron di2raction patterns recorded at 65 C during a heating scan and at the same temperature during a cooling scan.
does not show structural hysteresis in this region. The structural changes associated with the transformation of the unit cell during the phase transition are conAned to the O −1 . However, for Q higher than 1:55 A O −1 a small structural interval 1:2 6 Q 6 1:55 A hysteresis (around 1%) can be observed. 3.2. Hysteresis in the dynamics at the ferroelectric phase transition The elastic part of the incoherent neutron scattering, Sinc (Q; ! ≈ 0), was measured as a function of temperature during a heating and cooling cycle with backscattering spectrometers IN10 and IN16. As is shown in Fig. 4, the phase transition appears as a clear step, at the Curie temperature, in the Sinc (Q; ! ≈ 0) versus T plot indicating that some relaxation motions have been activated. As the temperature further increases, the linear decreasing behavior is recovered and on cooling a hysteresis loop develops. What is striking is that the area of the loop depends on Q. Fig. 5 shows that the area of the hysteresis loop increases with Q until it reaches the Q value corresponding to the Bragg peak and then levels o2 for higher Q. The existence of structural hysteresis at the Bragg peak during the Arst-order phase transition is a well-known phenomena. O −1 where What is new is the hysteresis of Sinc (Q; ! ≈ 0) in the Q region below 1:2 A no structural hysteresis is observed. We have also performed quasielastic measurements to investigate the relationship between molecular dynamics and the hysteresis loop. In the ferroelectric phase there is no quasielastic scattering but for T ¿ Tc a quasielastic broadening which increases with temperature is observed [3]. To study the dynamics at the hysteresis loop, we recorded two quasielastic spectra at the same temperature, but one spectrum was taken during the heating scan Tu and the other when the sample is cooled down Td . The ◦ quasielastic spectra were recorded at two selected temperatures: at 63 C in the middle ◦ of the ferroelectric hysteresis loop and at 79 C when the hysteresis loop just closed. The IQNS spectra, together with the spectrum at 2 K which provides the resolution ◦ of the spectrometer, are shown in Fig. 6. It can be seen that at 79 C the quasielastic
718
E.L. Cabarcos et al. / Physica A 314 (2002) 714 – 721 0.008 0.006 Q=0.40A-1 0.004 0.008
0.006 Q=0.60A-1 0.004 0.03 0.02
Q=0.86A-1
0.01 0.03 0.00
Sinc (Q,0)
0.02 Q=1.15A-1
0.01 0.00 0.04
Q=1.41A-1
0.02 0.00 0.01
Q=1.62A-1
0.00 0.03 0.02 0.01
Q=1.79A-1
0.00 0.02 0.01
Q=1.93A-1
0.00 20
40
60
80
100
120
T(°C)
Fig. 4. Plot of Sinc (Q; ! ≈ 0)=Sinc (Q; ! ≈ 0)2 K as a function of temperature for the 60=40 copolymer showing the hysteresis loop as a function of Q at the ferroelectric to paraelectric phase transition.
spectra exactly superimposed whereas that is not the case for the spectra recorded at ◦ 63 C. At the ferroelectric–paraelectric transition the IQNS spectrum recorded during ◦ the heating (Tu =63 C) shows higher intensity that the one registered during the cooling ◦ (Td = 63 C) even though the width of both curves is the same. We performed the analysis of the spectra using an incoherent quasielastic function given by a Debye relaxation process in a restricted volume and considering that only a fraction n(T ) of all the protons contributes to the motion [3,7,8]. Then Sinc (Q; !) is
E.L. Cabarcos et al. / Physica A 314 (2002) 714 – 721
719
60/40
30
area (K)
v = 1K/m
25
20
0.0
0.5
1.0
1.5
2.0
-1
Q (Å ) Fig. 5. Area of the hysteresis loop as a function of the scattering vector Q.
6000
2500 Q = 1.94 Å
Sinc(Q,ω)
4000
Q = 1.94 Å
Tu = 63 °C
-1
2000
T u = 79 °C
Td = 63 °C
T d = 79 °C
2K
1500
2K
3000 1000 2000 500
1000 0 -15
Sinc (Q,ω)
5000
-1
0 -10
-5
0
5
10
15
E (µeV)
(a)
(b)
-10
-5
0
5
10
15
E (µeV)
Fig. 6. Sinc (Q; !) versus energy transfer recorded, Arst during the heating scan and later during the cooling scan, at the temperatures indicated in the Agure.
given by Sinc (Q; !) = e−u
2
Q2
{n(T )F0 (Q)(!)
+ n(T )[1 − F0 (Q)]L(!; ) + [1 − n(T )](!)} ;
(1)
2
where u is the mean-square amplitude of the proton vibration, F0 (Q) is the Fourier transform of the volume where the proton moves and L(!; ) is a lorentzian function of half-width half-maximum (Q). The elastic incoherent structure factor can then be described by the function FEISF (n; Q) = 1 − n(T )[1 − F0 (Q)] :
(2)
By Atting the measured spectra to Eq. (1) we have obtained (Q) and F0 (Q). The values of (Q) as a function of Q are shown in Fig. 7(a2) and (b2) and it can be seen that (Q) of the spectra Tu superimposed with (Q) of the spectra Td at the two
720
E.L. Cabarcos et al. / Physica A 314 (2002) 714 – 721 1.0
0.4 0.2
1
(a2)
(a1)
0 3
1.0 Tu = 79 °C Td = 79 °C
FEISF
0.8
2
0.6 0.4 0.2
0.0
Γ (µeV)
2
0.6
Γ (µeV)
FEISF
3
Tu = 63 °C Td = 63 °C
0.8
1
(b2)
(b1) 0.5
1.0
1.5
-1
2.0
0.5
1.0
1.5
0 2.0
Q (Å-1)
Q(A )
Fig. 7. (a) FEISF and (Q) as a function of Q in the temperature interval of the phase transition. (b) FEISF and (Q) as a function of Q outside the temperature interval of the phase transition.
◦
temperatures. On the other hand, at 63 C the variation of the FEISF as a function of Q shows appreciable di2erences as it is illustrated in Fig. 7(a1). Our results strongly suggest that the di2erences observed in the amplitude of Sinc (Q; !) (and therefore in the FEISF ) can only arise from n(T ) and u2 . Hysteresis in n(T ) means that at a given temperature the number of protons in motion at Tu is di2erent from the number involved at Td . At a given temperature, the fraction of each phase varies depending if we are heating or cooling. Clearly, this structural contribution to the hysteresis is taken into account through n(T ) because the number of protons involved in the dynamics is greater in the paraelectric than in the ferroelectric phase. The possible existence of metastable states can also give rise to hysteretic e2ects which will appear as additional contributions to n(T ). The hysteresis in u2 could be attributed to anharmonic e2ects of the vibrational modes at the phase transition. In polymers, the anharmonicity parameter is very sensitive to the excess of free volume and in this copolymer the lattice spacing shows a dramatic increase at the ferroelectric transition (see Fig. 2) which supports the idea of the anharmonic origin of the hysteresis in u2 . 4. Conclusions The copolymer of vinylidene #uoride and tri#uoroethylene 60=40% molar, shows thermal hysteresis in the structure and in the dynamics at the ferroelectric phase transition. The area of the dynamical hysteresis loop increases as a function of Q reaching a maximum at the Braggs peak. The analysis of the spectra using an incoherent scattering function given by a Debye relaxation process in a restricted volume permits to
E.L. Cabarcos et al. / Physica A 314 (2002) 714 – 721
721
interpret the di2erences observed in Sinc (Q; !) at the phase transition, and to separate a contribution due to the anharmonicity of the amplitude of the proton vibration from a more general structural contribution. References [1] T. Furukawa, G.E. Johnson, H.E. Bair, Y. Tajitsu, A. Chiba, E. Fukada, Ferroelectrics 32 (1981) 61. [2] A.J. Lovinger, Science 220 (1983) 1115. [3] E. L'opez Cabarcos, F. Batallan, B. Frick, T.A. Ezquerra, F.J. Balta Calleja, Phys. Rev. B 50 (1994) 13 214. [4] S. Ikeda, H. Suda, Phys. Rev. E 56 (1997) 3231. [5] A.V. Bune, V.M. Fridkin, S. Ducharme, L.M. Blinov, S.P. Palto, A.V. Sorokin, S.G. Yudin, A. Zaklin, Nature 391 (1998) 874. [6] Q.M. Zhang, V. Bharti, X. Zhao, Science 280 (1998) 2101. [7] M. Bee, Quasielastic Neutron Scattering. Principles and Applications, Hilger, Bristol, UK, 1988. [8] T. Springer, Quasielastic Neutron Scattering for the Investigation of Di2usive Motions in Solids and Liquids, Springer, Berlin, 1972.
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Correlation between molecular dynamics and thermal hysteresis in the 60=40 vinylidene #uoride and tri#uoroethylene ferroelectric copolymer E. L'opez Cabarcosa; ∗ , A.F. Bra˜nab , B. Frickc , F. Batalland a Departamento
de Qu mica-F sica Farmac eutica, Facultad de Farmacia, Universidad Complutense de Madrid, 28040 Madrid, Spain b Instituto de Sistemas Optoelectr onicos y Microtecnolog a, Universidad Polit ecnica de Madrid, Ciudad Universitaria, 28040 Madrid, Spain c Institut Laue-Langevin, 38042 Grenoble Cedex, France d Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain
Abstract Thermal hysteresis associated with ferroelectric to paraelectric phase transition is investigated in a ferroelectric copolymer of vinylidene #uoride and tri#uoroethylene, using X-ray di2raction and incoherent neutron scattering. The lattice spacing derived from the di2ractograms, shows hysteresis at the ferro–paraelectric phase transition. The analysis of the elastic incoherent neutron spectra indicates that besides the structural component, the thermal hysteresis phenomena appears in the dynamics of the copolymer. Furthermore, the area of the hysteresis loop increases with Q until it reaches the Q value corresponding to the Braggs peak. At a given temperature, inside the ferroelectric hysteresis loop, the incoherent quasielastic component di2ers in intensity depending c 2002 Elsevier Science B.V. on recording the spectrum at the heating or at the cooling scan. All rights reserved. PACS: 77.84.Jd; 05.70.Fh; 61.25.Hq Keywords: Hysteresis; Phase transition; Ferroelectric polymer; Elastic neutron scattering; Quasielastic neutron scattering
∗
Corresponding author. E-mail address: [email protected] (E.L. Cabarcos).
c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 1 0 4 3 - 9
E.L. Cabarcos et al. / Physica A 314 (2002) 714 – 721
715
1. Introduction In a macromolecular system, like a polymer, thermal hysteresis appears during a heating–cooling cycle, accompanying the Arst-order phase transitions. In this work, we report on an experimental study of the thermal hysteresis loop associated with the ferroelectric–paraelectric phase transition in a 200 m random copolymer Alm of vinylidene #uoride (VF2 ) and tri#uoroethylene (F3 E) with VF2 molar fraction 60%. This is a semicrystalline polymer (crystallinity ∼0:7), consisting of -((-CH2 -CF2 -)x − (-CHF-CF2 -)1−x )n sequences, which has been studied over the past years [1– 6]. The copolymer exhibits a Curie temperature Tc and upon heating the pseudohexagonal ferroelectric crystals transform to a centrosymmetric paraelectric phase. Recently, a theory of the ferroelectric phase transition, speciAcally developed for these materials, predicts [4] that the area of the thermal hysteresis loop would change not only with the heating rate but also by the application of tensile stress and hydrostatic pressure. 2. Experimental The structural studies were made using wide angle X-ray di2raction (WAXD)at the double-focussing camera for synchrotron radiation on the polymer beamline at HASYLAB, Hamburg. For neutron di2raction experiments we have used the two-axis di2ractometer D1B, at the Institute Laue Langevin (ILL) in Grenoble. The study of the molecular dynamics was performed using elastic and quasielastic incoherent neutron scattering spectrometers IN10 and IN16. A more detailed description of the experimental setup and sample preparation can be found in a previous publication [3]. In these copolymers, provided that the Bragg peaks are avoided, the scattering intensity is almost entirely incoherent and Sinc (Q; ! ≈ 0) is a good measure of the incoherent elastic scattering function for the H atoms of the polymeric chain. To measure Sinc (Q; ! ≈ 0) we have used the “Axed elastic window method” in which the monochromator and analyzer are set to the same energy and only those neutrons that change their energy by an amount smaller than the energy resolution of the instrument are detected. 3. Results and discussion 3.1. Structural hysteresis The structural changes at the ferroelectric transition were followed by means of WAXD. To avoid changes in crystallinity, the sample was heated at a constant rate ◦ of 10 C min−1 above the Curie temperature and then cooled down to room temperature at the same rate. Fig. 1 illustrates a series of WAXD patterns in the range ◦ ◦ O following the above temperature variation. The study of 17 ¡ 2 ¡ 21 ( = 1:5 A) the lattice spacing allows to follow the structural hysteresis at the ferro–paraelectric transition as is shown in Fig. 2. Below the Curie temperature the lattice spacing of the
716
E.L. Cabarcos et al. / Physica A 314 (2002) 714 – 721
Fig. 1. Three-dimensional plot of the WAXD patterns (recorded every 20 s) for the random copolymer of vinylidene #uoride and tri#uoroethylene (60=40%) as a function of temperature. The copolymer was heated ◦ ◦ ◦ at a rate of 10 C min−1 from 25 C up to 127 C, then it was kept at his temperature for ∼10 min and ◦ Anally it was cooled down to 25 C.
5.0
d (Å)
4.9 4.8 4.7 4.6 4.5 20
40
60
80
100
120
T (°C)
Fig. 2. Variation of the lattice spacing, d, as a function of temperature at the ferro–paraelectric transition during a heating–cooling cycle.
◦
ferroelectric phase remains constant up to 40 C, then gradually increases and shows a stepwise behavior at the phase transition. The fact that on cooling, the Curie temperature is shifted towards lower temperatures leads to a corresponding shift of the ◦ paraelectric phase lower boundary (∼50 C). To characterize the Q region a2ected by structural hysteresis, we have recorded two neutron di2raction patterns at the phase transition temperature, one during the heating scan and the other when the sample is cooled down. As can be seen in the O −1 and di2ractograms shown in Fig. 3 the background remains constant for Q ¡ 1:2 A
E.L. Cabarcos et al. / Physica A 314 (2002) 714 – 721
717
0.05 Tu = 65 °C Td = 65 °C
S(Q)
0.04
0.03
0.02 0.5
1.0
1.5
2.0
2.5
-1
Q(A ) ◦
Fig. 3. Two neutron di2raction patterns recorded at 65 C during a heating scan and at the same temperature during a cooling scan.
does not show structural hysteresis in this region. The structural changes associated with the transformation of the unit cell during the phase transition are conAned to the O −1 . However, for Q higher than 1:55 A O −1 a small structural interval 1:2 6 Q 6 1:55 A hysteresis (around 1%) can be observed. 3.2. Hysteresis in the dynamics at the ferroelectric phase transition The elastic part of the incoherent neutron scattering, Sinc (Q; ! ≈ 0), was measured as a function of temperature during a heating and cooling cycle with backscattering spectrometers IN10 and IN16. As is shown in Fig. 4, the phase transition appears as a clear step, at the Curie temperature, in the Sinc (Q; ! ≈ 0) versus T plot indicating that some relaxation motions have been activated. As the temperature further increases, the linear decreasing behavior is recovered and on cooling a hysteresis loop develops. What is striking is that the area of the loop depends on Q. Fig. 5 shows that the area of the hysteresis loop increases with Q until it reaches the Q value corresponding to the Bragg peak and then levels o2 for higher Q. The existence of structural hysteresis at the Bragg peak during the Arst-order phase transition is a well-known phenomena. O −1 where What is new is the hysteresis of Sinc (Q; ! ≈ 0) in the Q region below 1:2 A no structural hysteresis is observed. We have also performed quasielastic measurements to investigate the relationship between molecular dynamics and the hysteresis loop. In the ferroelectric phase there is no quasielastic scattering but for T ¿ Tc a quasielastic broadening which increases with temperature is observed [3]. To study the dynamics at the hysteresis loop, we recorded two quasielastic spectra at the same temperature, but one spectrum was taken during the heating scan Tu and the other when the sample is cooled down Td . The ◦ quasielastic spectra were recorded at two selected temperatures: at 63 C in the middle ◦ of the ferroelectric hysteresis loop and at 79 C when the hysteresis loop just closed. The IQNS spectra, together with the spectrum at 2 K which provides the resolution ◦ of the spectrometer, are shown in Fig. 6. It can be seen that at 79 C the quasielastic
718
E.L. Cabarcos et al. / Physica A 314 (2002) 714 – 721 0.008 0.006 Q=0.40A-1 0.004 0.008
0.006 Q=0.60A-1 0.004 0.03 0.02
Q=0.86A-1
0.01 0.03 0.00
Sinc (Q,0)
0.02 Q=1.15A-1
0.01 0.00 0.04
Q=1.41A-1
0.02 0.00 0.01
Q=1.62A-1
0.00 0.03 0.02 0.01
Q=1.79A-1
0.00 0.02 0.01
Q=1.93A-1
0.00 20
40
60
80
100
120
T(°C)
Fig. 4. Plot of Sinc (Q; ! ≈ 0)=Sinc (Q; ! ≈ 0)2 K as a function of temperature for the 60=40 copolymer showing the hysteresis loop as a function of Q at the ferroelectric to paraelectric phase transition.
spectra exactly superimposed whereas that is not the case for the spectra recorded at ◦ 63 C. At the ferroelectric–paraelectric transition the IQNS spectrum recorded during ◦ the heating (Tu =63 C) shows higher intensity that the one registered during the cooling ◦ (Td = 63 C) even though the width of both curves is the same. We performed the analysis of the spectra using an incoherent quasielastic function given by a Debye relaxation process in a restricted volume and considering that only a fraction n(T ) of all the protons contributes to the motion [3,7,8]. Then Sinc (Q; !) is
E.L. Cabarcos et al. / Physica A 314 (2002) 714 – 721
719
60/40
30
area (K)
v = 1K/m
25
20
0.0
0.5
1.0
1.5
2.0
-1
Q (Å ) Fig. 5. Area of the hysteresis loop as a function of the scattering vector Q.
6000
2500 Q = 1.94 Å
Sinc(Q,ω)
4000
Q = 1.94 Å
Tu = 63 °C
-1
2000
T u = 79 °C
Td = 63 °C
T d = 79 °C
2K
1500
2K
3000 1000 2000 500
1000 0 -15
Sinc (Q,ω)
5000
-1
0 -10
-5
0
5
10
15
E (µeV)
(a)
(b)
-10
-5
0
5
10
15
E (µeV)
Fig. 6. Sinc (Q; !) versus energy transfer recorded, Arst during the heating scan and later during the cooling scan, at the temperatures indicated in the Agure.
given by Sinc (Q; !) = e−u
2
Q2
{n(T )F0 (Q)(!)
+ n(T )[1 − F0 (Q)]L(!; ) + [1 − n(T )](!)} ;
(1)
2
where u is the mean-square amplitude of the proton vibration, F0 (Q) is the Fourier transform of the volume where the proton moves and L(!; ) is a lorentzian function of half-width half-maximum (Q). The elastic incoherent structure factor can then be described by the function FEISF (n; Q) = 1 − n(T )[1 − F0 (Q)] :
(2)
By Atting the measured spectra to Eq. (1) we have obtained (Q) and F0 (Q). The values of (Q) as a function of Q are shown in Fig. 7(a2) and (b2) and it can be seen that (Q) of the spectra Tu superimposed with (Q) of the spectra Td at the two
720
E.L. Cabarcos et al. / Physica A 314 (2002) 714 – 721 1.0
0.4 0.2
1
(a2)
(a1)
0 3
1.0 Tu = 79 °C Td = 79 °C
FEISF
0.8
2
0.6 0.4 0.2
0.0
Γ (µeV)
2
0.6
Γ (µeV)
FEISF
3
Tu = 63 °C Td = 63 °C
0.8
1
(b2)
(b1) 0.5
1.0
1.5
-1
2.0
0.5
1.0
1.5
0 2.0
Q (Å-1)
Q(A )
Fig. 7. (a) FEISF and (Q) as a function of Q in the temperature interval of the phase transition. (b) FEISF and (Q) as a function of Q outside the temperature interval of the phase transition.
◦
temperatures. On the other hand, at 63 C the variation of the FEISF as a function of Q shows appreciable di2erences as it is illustrated in Fig. 7(a1). Our results strongly suggest that the di2erences observed in the amplitude of Sinc (Q; !) (and therefore in the FEISF ) can only arise from n(T ) and u2 . Hysteresis in n(T ) means that at a given temperature the number of protons in motion at Tu is di2erent from the number involved at Td . At a given temperature, the fraction of each phase varies depending if we are heating or cooling. Clearly, this structural contribution to the hysteresis is taken into account through n(T ) because the number of protons involved in the dynamics is greater in the paraelectric than in the ferroelectric phase. The possible existence of metastable states can also give rise to hysteretic e2ects which will appear as additional contributions to n(T ). The hysteresis in u2 could be attributed to anharmonic e2ects of the vibrational modes at the phase transition. In polymers, the anharmonicity parameter is very sensitive to the excess of free volume and in this copolymer the lattice spacing shows a dramatic increase at the ferroelectric transition (see Fig. 2) which supports the idea of the anharmonic origin of the hysteresis in u2 . 4. Conclusions The copolymer of vinylidene #uoride and tri#uoroethylene 60=40% molar, shows thermal hysteresis in the structure and in the dynamics at the ferroelectric phase transition. The area of the dynamical hysteresis loop increases as a function of Q reaching a maximum at the Braggs peak. The analysis of the spectra using an incoherent scattering function given by a Debye relaxation process in a restricted volume permits to
E.L. Cabarcos et al. / Physica A 314 (2002) 714 – 721
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