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Structure and functional properties of ferroelectric polymers
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ELSEVIER
Advances in Colloid and Interface Science 71-72 (1997) 183-208
ADVANCES IN COLLOID AND INTERFACE SCIENCE
Structure and functional properties of ferroelectric polymers Takeo Furukawa* Department of Chemistry,Facultyof Science, Science Universityof Tokyo, 1-3 Kagurazaka, Shinjuku, Tokyo 162,Japan
Abstract
Ferroelectric polymers represented by vinylidene fluoride copolymers with trifluoroethylene (VDF/TrFE) have been investigated with special interest on the relationship between their dynamical properties and structures. Copolymers containing 50-80 mol% VDF exhibit fast polarization reversal due to the rotation of chain molecules about their axes. Annealing of an extruded copolymer yields a highly crystalline film consisting of regularly stacked large lamellae. The switching transient of these extruded copolymers is consistent with the switching mechanism that progresses through a nucleation-growth process with a considerable waiting time (time between the generation of a nuclei and the onset of its growth) and an accelerated growth velocity. TrFE-rich copolymers exhibit a two-step switching transient, suggesting that polarization reversal occurs via a nonpolar state associated with the copolymer's antiferroelectric-like nature. Piezoelectric and pyroelectric properties were examined for their high frequency and short time characteristics. The time evolution of the charge response induced by laser pulse irradiation consists of intrinsic pyroelectricity and the coupling of thermal expansion and piezoelectric resonance. The intrinsic pyroelectric response for well-annealed VDF-rich copolymers is in the sub-nanosecond range, which promises their use as fast thermal sensors. © 1997 Elsevier Science B.V.
Keywords: Ferroelectric polymer; Vinylidene fluoride/trifluoroethylene copolymer; Polarization reversal; Piezoelectricity; Pyroelectricity; Higher order structure
*Tel.: +81 3 3260 4271; fax: +81 3 3235 2214; e-mail: tfurukaw.ch.kagu.sut.ac.jp 0001-8686/97/$32.00 © 1997 Elsever Science B.V. All rights reserved.
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1. Introduction
Ferroelectric polymers have generated much interest in the last 15 years because of their potential as functional materials for energy transduction and information recording. Polyvinylidene fluoride (PVDF) was the first ferroelectrie polymer for which a D - E (electric displacement and electric field, respectively) hysteresis loop and a fast-switching phenomenon were demonstrated. Incorporation of trifluoroethylene (TrFE) into PVDF produces a random copolymer that exhibits a Curie point, which is undetectable in pure PVDF. Early investigations into the ferroelectric properties of V D F / T r F E copolymers were summarized in my review article [1]. In the early 90s, odd nylons [2] (e.g. nylon7, nylonll) were shown to undergo ferroelectric polarization reversal in melt-quenched and uniaxially-drawn samples. Some aromatic polyamides [3], polyureas, and polyurethanes [4] have also been shown to be ferroelectric. These polymers exhibit D - E hysteresis behavior, but the Curie point is yet to be observed. Solid-state polymers are usually obtained as thin films in which the crystalline and non-crystalline regions coexist. Most experiments on ferroelectric polymers are done on such semicrystalline film samples. Because ferroelectric properties originate from the crystalline region, experimental data are strongly influenced by the higher order structures, such as the degree of crystallinity, orientation, and crystal size, in the samples. For quantitative analysis, therefore, we need samples with well-defined higher order structure. In this article, we examine V D F / T r F E copolymers with special emphasis on their dynamical characteristics. For VDF-rich copolymers, we prepared samples (using various thermal and mechanical treatments) with different higher order structures, and measured their switching characteristics in detail. Concerning TrFE-rich copolymers, we discovered a two-step switching phenomenon that is characteristic of an antiferroeleetric material. Finally, we describe the dynamical aspect of the piezoelectric and pyroelectric properties for VDF-rich copolymers.
2. Structure and basic properties of V D F / T r F E copolymers 2.1. Crystal structure
PVDF consists of a repeat unit (--CH2CF2--)that exhibits a dipole moment /zv = 7 x 10 -30 Cm (2.1 D) associated with positively charged H-atoms and negatively charged F-atoms (Fig. 1). Because such dipoles are rigidly attached to the main chain, their orientation is subject to the conformation and packing of molecules. If the molecule adopts an all-tram conformation and a parallel packing, the dipoles are aligned in one direction, perpendicular to the chain axis. The resulting crystal, called /3, possesses a large spontaneous polarization P, that is responsible for the ferroelectrieity of PVDF. Summing up/z v over the unit volume, we find
T. Furukawa / Adu. Colloid Interface ScL 71-72 (1997) 183-208
185
HH
(A) FF
(B) c =256pm
(c) b =491 pm:
" t !
(~
' Ps= 130mC/m2
: | !
Fig. 1. Molecular (A), chain (B), and crystal(C) structures of PVDF.
Ps = 2l~v/abc = 130 mCm -2
(1)
where a, b, and c are the lattice constants whose values are 858 pm, 491 pm, and 256 pm, respectively. It is known that PVDF has several crystalline polymorphs. Melt-crystallization yields a nonpolar a-phase consisting of TGTG' molecules packed in an antiparaUel manner. The ferroelectric /3-phase is obtained by uniaxial-drawing of a-PVDF. The introduction of a small amount of TrFE into PVDF causes direct crystallization into the /3-phase from the melt. Because the F-atom is slightly larger than the H-atom, in the copolymer the trans bond is favored over the gauche bond due to sterical hindrance occurring in the latter. Of special interest are the copolymers containing 50-80 tool% VDF because they exhibit a Curie point that is absent in pure PVDF. They also show a marked increase in crystallinity as a result of annealing at a temperature between the Curie point and the melting point. We can, therefore, prepare copolymer samples dhat have a variety of higher order structures.
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
186
2.2. Curie transition
Fig. 2 shows the temperature variation of the permittivity e' of a 75/25 mol% copolymer of VDF and TrFE. The rapid increase in e' near 12°C (dielectric anomaly) upon heating is attributed to the onset of cooperative dipolar motions, leading to a ferroelectric-to-paraelectric transition. Melting at higher temperature causes e' to decrease due to loss of cooperativity. Upon cooling, e' first increases and then decreases due to reverse transitions, i.e. crystallization and a paraelectricto-ferroelectric transition. Transition temperatures upon heating are higher than those upon cooling (130°C and 80°C, respectively), thus meaning that these transitions are first-order. Fig. 3 shows the phase diagram for a V D F / T r F E copolymer where the melting point Tm the crystallization point Tcr, and the Curie point T~ upon heating are plotted against composition. These temperatures were determined by using a differential scanning calorimeter (DSC) at a heating and cooling rate of 10°C/rain. The melting point is a minimum at a composition of 80/20 tool%. As the VDF content decreases, Tm increases gradually to 200°C at PTrFE. It also increases with increasing VDF content above 80 mol%. The Tm of a-PVDF is 175°C. The Tcr depends upon composition in a similar manner and is located about 25°C below
rm. The T~ of an equimolar copolymer is located near 65°C (Fig. 3). As the VDF content increases, T~ upon heating increases to coincide with Tm at 80 tool%. This
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Temperature ('C)
Fig. 2. Dielectrictemperaturespectra for a VDF(75)/TrFE(25)copolymer.:
T. Furukawa /Ado. Colloid Interface ScL 71-72 (1997) 183-208
200(
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Molten Phase
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ID
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Paraelectric Phase
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VDF content (mol%)
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Fig. 3. Phase diagram for a V D F / T r F E copolymer (O: melting point; 0 : crystallization point; zx : Curie point on heating; • : Curie point on cooling, [] : ferroelectric-to-antiferroelectric transition point).
implies that melting begins prior to completion of the ferroelectric-to-paraelectric transition. On the other hand, the Curie point disappears in the DSC thermogram when TrFE becomes a major component (> 50 mol%). However, a dielectric anomaly still exists around 50-60°C (triangles with dot) which smoothly connects with Tc of VDF-rich copolymers. Therefore, the region between the circles and triangles in Fig. 3 corresponds to the paraeleetrie or paraeleetric-like phase. VDF-rieh copolymers are in a ferroelectric phase below the Curie point (Fig. 3). On the other hand, TrFE-rieh eopolymers exhibit a double hysteresis loop in the temperature range surrounded by the dashed lines in Fig. 3. The enclosed region could be attributed to an antiferroelectric or an antiferroeleetrie-like phase as will be discussed in section 4.2. 2.3. Polarization reoersal
In the ferroelectric phase, VDF-rich copolymers incorporated with PVDF exhibit
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T. Furukawa /Adv. ColloM Interface ScL 71-72 (1997) 183-208
a typical D - E hysteresis loop as demonstrated in Fig. 4 for a VDF(65)/TrFE(35) copolymer at 20°C with an eleetric field E applied in a sinusoidal form at 1 Hz. In the low field regime ( < 30 MV/m), the D - E relation is nearly linear. As E is increased, hysteresis starts to appear. In the high field regime (> 60 MV/m), we obtain a square hysteresis loop that is independent of the amplitude of E. The intersect of the loop with the abscissa defines the coercive field E c which is ca. 50 MV/m. The intersect with the ordinate gives the remanent polarization Pr ca. 80 m C / m 2, which depends upon not only composition but also higher order structures of the sample. Fig. 5 shows the compositional variation of Pr in which the data plotted are the largest experimental values obtained so far. Also plotted (dashed line) is the value of the spontaneous polarization calculated assuming a rigid dipole. Pr reaches a maximum at a VDF content of 80 tool% and decreases gradually as VDF decreases to 50 tool%. This gradual decrease is attributed to a decrease in the average dipole moment because the dipole moment of a TrFE unit is one-half that of a VDF unit. As the VDF content falls below 50 mol%, Pr decreases rapidly due to a loss of ferroelectricity. The further decrease in P~ in the 80-100 mol% range is attributable to a decrease in crystallinity. The observed P~ of PVDF is ca. 65 m C / m z, which is one-half that predicted by using spontaneous polarization (dashed line), ca. 130 m C / m 2.
J
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50 E o
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-100
-50
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50
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E (MY/m) Fig. 4. D - E hysteresis loops for a VDF(65)/TrFE(35) copolymer at 20°C.
189
T. Furukawa / Ado. Colloid Interface ScL 71-72 (1997) 183-208
140 o
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calculated
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..
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observed
20 0
n
PTrFE
I 20
I 40
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VDF content (mol%)
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1O0 PVDF
Fig. 5. Composition dependence of observed and predicted the remanent polarization Pr for VDF/TrFE copolymers.
3. Switching characteristics of VDF-rich copolymers
3.1. Ferroelectric switching measurement Ferroelectric polarization reversal is commonly examined by means of a D - E hysteresis measurement in which a sinusoidal electricfield E is applied to a sample and the resultingelectricdisplacement D is then measured. A n alternativemethod is a switching measurement, which uses a step-wise E and measures D as a function of time t. Fig. 6 shows a block diagram of the apparatus developed for the fcrroclectric switching measurement. A high step-voltage up to 3.5 kV is applied to a sample through a switch,which is a silicone-controUed rectifier(SCR). The rise time of the applied voltage is ca 1 ~s. For fast (nanosecond) measurements at lower voltages (< 500 V), the S C R is replaced by a mercury rclay.Thc response from the sample is detected by a capacitor or a resistor in terms of charge or current, respectively, and is stored in transient memory. Representative results are shown in Fig. 7 where log D is plotted against log t. Here, D R represents the result obtained by applying E in the direction opposite the existing polarization. Any sharp change in D indicates polarization reversal
190
T. Furukawa /Adv. Colloid Interface Sci. 71-72 (1997) 183-208
MICRO L COMPUTER SYSTEM I"
"l
t.oG I
TIMER I
TRANS' NTI MEMORY 1'
I'/
SWITCH HIGH VOLTAGE SOURCE
_L_ r
Fig. 6. Experimental set-up for ferroelectric switching measurement.
occurring as a result of dipole reorientation. If E is applied in the direction of Pr the resulting electric displacement D F exhibits only a gradual increase because most dipoles are already aligned parallel to the field direction. Such directionality confirms that the material of interest is ferroelectric. Both D R and D F contain contributions from the dielectric response and the dc conduction. By subtracting D F from D R, we obtain the response that arises mostly from ferroelectric polarization reversal. Fig. 8 shows the switching characteristics of a VDF(78)/TrFE(22) sample prepared by rapid cooling from the melt to room temperature (melt-quenched) followed by annealing at 145°C for 1 h. The increment of D corresponds to the amount of reversed polarization equal to double the remanent polarization (i.e. 2P r) which is independent of the strength of E. The position of the O D / O l o g t peak defines the switching time r~, which decreases as E is increased. Fig. 9 shows a plot of log z~ versus the reciprocal of E. The straight line indicates an exponential relationship between ~-~ and E: r s = ~'~o e x p ( E a / E )
(2)
where E a is the activation field and r,0 is the switching time at an infinite field, which were determined to be 0.8 G V / m and 10 ns, respectively. 3. 2. Effect o f annealing on structure and switching characteristics
To examine structure-property relationship with respect to ferroelectric polarization reversal, we prepared two kinds of samples; melt-quenched and extruded (extruded from the melt). In the melt-quenched sample, there is no preferred orientation of molecules, whereas in the extruded sample, molecules are well-ori-
T. Furukawa / Adv. Colloid Interface ScL 71-72 (1997) 183-208
191
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log t (s) Fig. 7. Time evolutionof the electric displacement D inducedby a stepped electricfield E applied in the directionopposite(D R) to the remanent polarization P~ and in the same direction(Dr).
ented. Both samples were subjected to annealing at elevated temperatures to increase crystallinity. Fig. 10 shows the effect of annealing on the switching characteristics of an extruded VDF(78)/TrFE(22) copolymer. The as-extruded sample exhibits essentially no polarization reversal. As the sample is annealed at a temperature just above the Curie point (128°C), polarization reversal starts to appear. Annealing at higher temperatures causes further increase in the amount of reversed polarization as well as sharpening of the a D / a l o g t peak. The field dependence of zs is basically the same as that seen in an unoriented sample (Fig. 9). We examined the structure of these two samples (melt-quenched and extruded) before and after annealing by using a scanning electron microscope (SEM). Figs. l l A - 1 1 C show SEM images of the fractured surface parallel to the extrusion direction (vertical) and to the film thickness (horizontal). The as-extruded sample (llA) exhibits vertical striation associated with oriented chain molecules. Annealing at 128°C gives rise to horizontal stripes (liB), suggesting a formation of crystalline lamellae. As the annealing temperature is increased to 145°C, extremely large lamellae 100 nm thick and 1/zm wide are stacked in a parallel manner along the extrusion direction (llC). The SEM image of the fractured surface perpendicular to the extrusion direction (liD) provides a better view of these stacked lamellae. Comparing the results from the switching measurements (Fig. 10) and the SEM measurements (Fig. 11), we find that annealing causes a rapid increase in crys-
192
T. Furukawa /Adv. Colloid Interface Sci. 71-72 (1997) 183-208
0.15
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0.15
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log t (s) Fig. 8. Switching characteristics of melt-quenched and annealed VDF(78)/TrFE(22) copolymers at 20°C.
tallinity, leading to a marked increase in the amount of reversed polarization. The extremely sharp OD/Ologt peak of a well annealed sample would be due to very large lamellae. The melt-quenched samples have similar structural changes due to annealing, i.e. a marked increase in crystallinity and in the growth of large lamellae, as
T, Furukawa / Adv. Colloid Interface Sci. 71-72 (1997) 183-208
193
0v
- 2 --
o
-6-I
5
10
, I,
l
I
15
20
25
lIE
(m/GV)
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Fig. 9. Switching time ~'s versus the reciprocal of the applied electric field E for a VDF(78)/TrFE(22) copolymer.
revealed by SEM measurements (analogous to Fig. C but wihout preferred orientation). Switching characteristics also show a significant increase in Pr as well as a sharpening of the OD/Ologt peak as shown in Fig. 10 for a well-annealed sample. Fig. 12 shows a comparison of the switching transients of well-annealed samples of melt-quenched and extruded copolymers. The amount of reversed polarization of the melt-quenched sample is 20% larger than that of the extruded. From a structural viewpoint, polarization reversal of V D F / T r F E copolymers occurs as a result of the rotation of chain molecules about their axes. Therefore, molecules oriented in the thickness direction (i.e. E direction) do not contribute to polarization reversal. Quantitatively, molecules oriented with angle 0 with respect to the thickness direction contribute to Pr as a coefficient sin 0. Assuming that molecules are randomly oriented in a melt-quenched sample, the expected Pr is ~r/4 that of a completely oriented sample, for samples with the same crystallinity. Thus, the difference in Pr for the extruded and melt-quenched samples is due to molecular orientation, which confirms that chain rotation is responsible for polarization reversal in V D F / T r F E copolymers. Another important difference between the two samples is the shape of the switching transient. The extruded sample undergoes a very sharp change in D, whereas the melt-quenched sample exhibits a rather gradual change (Fig. 12). The time evolution of the switching transient reflects how polarization reversal pro-
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
194
0.25 0.20
annealed at 145"C
0.15 E 0 c~ 0.10
137"C
0.05
as-extruded
v
128"C
0.00 -4
-3
-2
-1
0
log t (s)
0.8
annealed at 145°C
0.6 o *" 0.4 v
137"C
0
0.2 0.0 -z
128"C ' -3
'~
~ -2
as-extruded -1
0
log t (s) Fig. 10. Switching characteristics of extruded and annealed VDF(78)/TrFE(22) copolymers at 20°C.
gresses after the application of the step-wise electric field. The switching process in crystalline lamellae would be affected by their orientation with respect to this applied field. Because the extruded copolymer consists of well-aligned lamellae, the observed switching transient may represent that of each lamella. On the other
T. Furukawa /Ado. Colloid Interface ScL 71-72 (1997) 183-208
:
4J
~
,,
195
i~
(,
., i (A)
(B)
(C)
(D)
5oonm
Fig. ll. SEM images of the fractured surface of a VDF(78)/TrFE(22) film (A) as-extruded, (B) annealed at 128°C, and (C, D) annealed at 145°C.
hand, the switching transient of the melt-quenched sample is subject to an 'orientation average', which results in a much broader spectrum of the transient. In the following, we analyze the switching transient of extruded copolymers.
3.3. Anal, sis of switching transient It is generally accepted that polarization reversal in a ferroelectric is based on a nucleation-growth mechanism [5]. When an electric field is applied, polarization reversal starts with the generation of the nuclei of reversed domains, followed by nuclei growth that is seen as domain-wall motions. For single crystal ferroelectrics such as BaTiO 3, such domain-wall motions can be observed by using an optical microscope. This mechanism is consistent with the observed coercive field being much lower than phenomenologieal prediction. Assuming random nucleation, the number of generated nuclei at time t is N = N 0 ( 1 - e -R')
(3)
196
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
0.25 0.20
E
extruded
0.15
c~ 0.10 0.05
0.00
I
f
I
I
I
-4
-3
-2
-1
0
log t (s) Fig. 12. Comparison of the shape of switching transient for melt-quenched and as-extruded VDF(78)/TrFE(22) copolymers.
where R is the nucleation probability and N O is the total number of available nucleation sites. If the generated nuclei grow m-dimensionally at a velocity o, the total volume of all reversed domains V (extended volume) is given by -d d N
where F is a shape factor. As growth proceeds further, ingestion of nucleation sites occurs. Growing domains then coalesce. Taking ingestion and coalescence into consideration, the actual reversed polarization P given by Avrami [5] is P = 2Psi1 - e -g ]
(5)
where Ps is the spontaneous polarization. Fig. 13 shows a schematic of the nucleation-growth process for the ease of two-dimensional growth. Substituting Eqs. (3) and (4) into Eq. (5) and then integrating, we obtain an analytical expression for P, which can then be simplified as P = 2P~[1 - e - ( t / r s ) ' ]
(6)
Here, the exponent n is related to the growth dimension m. When the growth velocity v is small (t :~ l / R ) , polarization reversal occurs after most nucleation sites are activated. For this case, n = m results [5]. On the other hand, if v is very large (t < l / R ) , polarization reversal is completed as soon as a small amount of nuclei are generated, thus yielding n = m + 1 [5].
197
T. Furukawa / Adv. Colloid Interface Sci. 71-72 (1997) 183-208
ole
@ 0 (A)
(B)
(C)
Generated Nuclei Sites
Growth of Sites
Coalescence of Sites
Fig. 13. Schematic of the nucleation-growth mechanism for ferroelectric polarization reversal.
The value of n determines the shape of the switching transient curve in terms of the sharpness of the aD/alogt peak, as represented in Fig. 14 for observed and predicted (using Eq. [6]) peaks for the melt-quenched and extruded copolymers. The corresponding n value of the extruded copolymer is larger than 5, which requires an unrealistic growth dimension of 4 to 5. The observed transient also shows departure from the predicted transient when t < ~-s/10. To explain such a large n value and to reproduce the switching transient over the entire range of t, we introduce three assumptions into our analysis: (a) v is not constant but has an acceleration a; (b) generated nuclei must wait a time Tw before they start to grow; and (c) generated nuclei have a certain initial volume before growth. Incorporating these assumptions, the extended volume is expressed as
6 • Extruded n =6 o Melt-Quenched ~ / ~ 4
i
Calculated 3
~
'
~
0
-0.6
-0.4
-0.2
0.0
0.2
0.4
Iogt/~:s Fig. 14. Comparison of observed aD/alogt peak with that predicted from an exponential function.
T. Furukawa /Adv. Colloid Interface Sci. 71-72 (1997) 183-208
198
O,
I
, -3= -5
. i,,/" -4
IIIIII
n=5.3
,; a,v, ro,~. -3
I -2
-
log t (s) Fig. 15. Observed and fitted switching transients for extruded and well-annealed V D F / T r F E copolymers. ! dM~
V= J~[tJ"~" J F(½a(t-t w - s ) z + v ( t - t w - s ) +r2)mds o~ dt]s
(7)
Fig. 15 shows the results of fitting the observed transient for the extruded and annealed copolymer with this expression. We find that the observed transient is well reproduced. The a and 7", account for the sharp rise in D with time t. The radius of the nuclei r 0 contributes to the initial gradual increase in D. In practice, the even-order nonlinear permittivity also contributes to the initial gradual increase in D as we previously reported [6]. We have succeeded in reproducing the observed very sharp switching transient of extruded and annealed samples of copolymers by introducing a variety of factors, such as a and Tw. Because such samples consist of regularly-stacked large lamellae, we believe that the observed sharp switching transient represents the events occurring in respective crystalline lamellae. Much broader switching transients in unoriented copolymers can be interpreted by taking into consideration the averaging effect arising from orientation distribution.
4. Switching characteristics of TrFE-rich copolymers
4.1. Double hysteresis and two-step switching charactens"acs As stated earlier, the amount of reversed polarization rapidly decreases as the VDF content is decreased to below 50 mol%. Although there remains a dielectric
T. Furukawa /Ado. Colloid Interface Sci. 71-72 (1997) 183-208
199
anomaly, the ferroelectric-to-paraelectrie transition in such samples is undetectable by DSC measurements. In this section, we look at the loss of ferroelectricity in TrFE-rich copolymers. Fig. 16 shows the D - E relation for uniaxially-drawn and annealed V D F / T r F E copolymers with compositions 65/35, 47/53, and 37/63 mol%. The square hysteresis loop that is characteristic of VDF-rich copolymers is transformed into a double hysteresis loop as TrFE becomes a major component (> 50 mol%). Double hysteresis suggests that polarization reversal progresses through a nonpolar state. The VDF(47)/TrFE(53) copolymer experiences the nonpolar state just after the applied field is reversed, whereas the 37/63 copolymer drops to the nonpolar state at zero field. Fig. 17 shows D - E hysteresis measurements at various temperatures for the VDF(47)/TrFE(53) copolymer. As the temperature is decreased to -40°C, double hysteresis is lost and transforms into a regular single loop. This implies that the polar phase is more stable than the nonpolar phase below -40°C. The ferroelectric switching measurements for the VDF(47)/TrFE(53) copolymer in Fig. 18 show a two-step switching transient, thus confirming that polarization reversal progresses through a nonpolar state in the time domain. The dependence of the time constant on field strength is stronger for the polarization change from
100
S
~'~'~65/35
50
E O E
v
0
-50 -
-100 -100
I -50
0
I 50
100
E (MY/m) Fig. 16. D - E hysteresis loops for VDF(65)/TrFE(35), 47/53, and 37/63 copolymers at room temperature.
200
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
60 40 20
_E o E
0
v
-20 -40
-
-60
-
c0OC 7. "'-....40oC
25°C
-150
I
I
-100
-50
0
I
I
50
100
150
E (MY/m) Fig. 17. D - E hysteresis loops for a VDF(47)/TrFE(53) copolymer at 25, 0, - 10 and -40°C.
negative to zero than that from zero to positive. These two time constants are unified when the E reaches 90 M V / m . Fig. 19 shows plots of these two time constants versus the 1/17, obtained at various temperatures. As the temperature is decreased to -40°C, we obtain a single switching process in accord with the regular square hysteresis loop at the same temperature.
4.2. Possibility of anti-ferroelectricity There are two possibilities that give rise to a double hysteresis loop or a two-step switching phenomenon. One is that the ferroelectric undergoes a first-order transition. Near the Curie point, both the ferroeleetric and paraeleetric phases are at an energy minimum for such a ferroelcctric. Just above the Curie point, the external field induces a paraelectric-to-ferroelectric transition, resulting in a double hysteresis loop. The other possibility is that the material in concern is the antfferroelectric consisting of two sublattices that have spontaneous polarizations opposite in direction. The application of an external field causes polarization reversal of one of the sublattices, thus inducing a transition to the ferroelectric phase. Here we have shown that TrFE-rich copolymers exhibit a double hysteresis loop as well as a two-step switching phenomenon. One question that arises is whether the copolymer is an antiferroelectric or a first-order ferroelcctric. It is known that the ferroelectric transition of VDF-rich copolymers is first-order as revealed by
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
201
+i iit °070; f 0"06F
0.00 -5
-4
-3
-2
-1
0
-1
0
Iogt(s)
0.10 !
E
0 •08
O E
0.06
o0 ) 0.04 ¢~
8O 7
~
60
50
0.02 0.00 -5
-4
-3
-2 Iogt(s)
Fig. 18. Switching characteristics of a VDF(47)/TrFE(53) copolymer at 25°C.
thermal hysteresis of the transition behavior. However, no clear indication of double hysteresis has been reported so far. As the VDF content is decreased to equimolar composition, thermal hysteresis is lost and the transition becomes second-order. As the transition of TrFE-rich copolymers is second order, we expect that the two-step switching arises from their antiferroelectric or antiferroelectric-like nature. Fig. 20 shows a schematic of the structural change during polarization reversal for TrFE-rich copolymers of VDF. Structure A describes the paraelectric phase in which molecules adopt a random conformation, resulting in disordered
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
202
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0-1 v
69
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25 °C
30
(m/GV)
Fig. 19. Switching time r s against I/E for the two-step switching transient of a VDF(47)/TrFE(53) copolymer at various temperatures.
dipole orientation. Structure C expresses the antiferroelectric phase consisting of all-tram molecules packed in an ant/parallel manner. An alternative structure, B, assumes a random packing of all-tram molecules. To answer which structural change is occurring, we must determine the crystal structure of TrFE-rich copolymers at temperatures where the two-step switching is observed.
5. Piezoelectric and pyroelectric properties As far as application is concerned, ferroelectric polymers have received much attention as soft transducer materials because of the polymers" piezoelectric and pyroelectric properties [7]. This section describes our recent work on the dynamical aspect of these properties, particularly in the high frequency or the short time ranges. The piezoelectric constant dq and the pyroelectric constants Pi are defined as
203
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
+P
-p
Nonpolar P=O Fig. 20. Structural changes during the two-step switching transient for TrFE-rich copolymers.
dq =
-~J/e
Pi
t OT /e
(i = 1 - 3 , j
= 1-6)
(7)
where D is the electrical displacement, E is the electric field, X is the stress, and T is temperature. The uniaxially-drawn and poled V D F / T r F E film has a C=v symmetry. Therefore, the matrix form of d u and Pi are 0
dij =
0 d31
0 0 d32
0 0 d33
0 d24 0
dis 0 0
0 0 0
(8)
(9) Here, 1 refers to the draw direction, 3 to the poling direction, and 2 is orthogonal to both 1 and 3. For undrawn samples, d31 = d32 and d15 = d24. The values of
204
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
nonzero components are not a material constant but depend upon the remanent polarization Pr imparted by poling. For a VDF(65)/TrFE(78) copolymer, we obtain a good linear relationship with respect to d31/s~l and P3 as demonstrated in Fig. 21. Pr depends upon the higher order structure of the sample and its maximum value is limited to the spontaneous polarization of the crystalline regions. 5.1. Piezoelectric resonance The piezoelectric constant at high frequencies is measured by using a resonance method. Fig. 22 shows the dielectric spectra for a V D F / T r F E copolymer over a broad frequency range at room temperature. The spectra show two kinds of resonance [8] superimposed on a broad dielectric relaxation. The resonance near 400 kHz is assigned to the length extension (LE) mode vibration and that around 40 MHz and its odd order harmonics are attributed to the thickness extension (TE) mode vibration. Here the sample used for the measurement was 20 /zm thick, 10 mm long, and 5 mm thick. Sample deformations associated with LE mode and TE mode resonances are depicted in the figure. In the former the length extension is acompanied by thickness and width contractions, whereas the thickness extension occurs without length and width contractions in the latter.
120
60
100
-
50
80
-
40
-
30
-
20
-
10
3 v
20
0
7~
I
0
20
40
60
80
100
Pr (me/m2) Fig. 21. Dependence of the piezoelectric constant dal/sll and the pyroelectric constant P3 on the remanent polarization Pr for a VDF(65)/TrFE(35) copolymer.
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
lo[
205
x
8~ M o d e
61-
TE Mode
~. . . . . . . . . . . . . . . . . . X
4
2 0 4
5
6
7
8
9
Iogf (Hz)
5 4
%
TE Mode
3-
LE Mode
2
::~:~
0
"
4
5
i
~. . . . . . . . .
t
I
6
7
!
.,=
"
8
9
Iogf (Hz) Fig. 22. Frequency spectra of the complex permittivity E* = e - is" containing piezoelectric resonance for a V D F / T r F E copolymer.
The dielectric spectra observed can be described by [9]
E33 ffi 8~3
k321 tan a ) 1 + 1-k~-------~ a tan b
(10)
206
T. Furukawa /Adv. Colloid Interface Sci. 71-72 (1997) 183-208
with °' a = 2v
v
(11)
ps~
and
o,t b = 2v
/ cg v
(12)
P
where ~o is the angular frequency, p is the density, l is length, t is thickness, s is the elastic compliance, c is the elastic stiffness constant, and k is the electromechanical coupling coefficient defined by
ag, k32~-
xE ,g33511
k~3=
x o 833C33
(13)
Here e x is the free permittivity and e x is the clamped permittivity. Using Eq. (10), we can reproduce the observed spectra by adjusting the constants as best-fit parameters. In this case, we obtained k 3 = 0.27 and k31 = 0.09, which lead to estimates of the piezoelectric constants d31 and e33 by the use of the elastic and dielectric constants appearing in Eq. (13). We are also able to determine the frequency spectra of the clamped permittivity e x (dashed curves in Fig. 22), which is completely independent of piezoelectric resonance. The broad relaxation observed has been attributed to dipolar fluctuations in the crystalline regions [10]. 5.2. Pyroelectric d y n a m i c s
The dynamical characteristics of the pyroelectricity can be measured by the use of laser pulse heating [11]. Fig. 23 shows the time evolution of the charge response from a freely suspended VDF(75)/TrFE(25) sample after the irradiation of a 7 ns YAG laser pulse. The time spectra consist of four processes: (a) initial fast rise ( < 10 ns); (b) TE-mode damped oscillation (10 ns-1 ~s); (c) LE-mode damped oscillation (1 ~s-10 ms); and (d) final decrease due to thermal diffusion outward from the sample (10 ms-10 s). The TE mode and LE mode oscillations in the charge response occur as a result of rapid thermal expansion coupled with piezoelectric effects. Comparing Figs. 22 and 23 reveals that two types of mechanical vibration contribute to dielectric and pyroelectric spectra through piezoelectric coupling. The former was measured on the frequency domain whereas the latter on the time domain. Therefore, Fourier transform of one of the spectra is necessary to make comparison on the same (frequency or time) domain. Detailed analysis is now in progress. We expect to obtain further information about the dynamics of not only dielectric and pyroelectric properties but also piezoelectric, thermal and mechanical properties of V D F / T r F E copolymers.
207
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
40 LE Mode
':':~::
300Q..
20-
(3 10-
0 -'0
-8
-6
-4
-2
0
Iogt (s) Fig. 23. Time spectra of the charge response for a VDF(75)/TrFE(25) copolymer induced by laser pulse irradiation.
Regarding the fast response, by removing the TE-mode component, we obtained the time spectra describing fast pyroelectric dynamics. The spectra consist of two components. The first is an initial rapid rise attributed to the intrinsic pyroelectricity that appears prior to thermal expansion in parallel with an integrated laser 30 25
20 (3 10 5 0 0
I 50
I 1O0
I 150
I 200
I 250
t (ns) Fig. 24. Fast pyroelectric response for a VDF/TrFE copolymer.
300
208
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
pulse. This property holds promises in using V D F / T r F E copolymers as very fast pyroelectric detectors. The second component is the gradual rise (time constant of 100 ns) associated with dipolar fluctuations in accord with the dielectric relaxation in Fig. 22.
6. Concluding remarks Structure-property relationship has been examined for V D F / T r F E copolymers over the entire range of composition. Annealing is a very efficient technique for controlling higher order structures such as the degree of crystallinity and the crystalline size of this copolymer. It was found that the crystalline size does affect the ferroeleetric switching characteristics of VDF-rich copolymers. The meltquenched and annealed sample consisting of regularly packed large crystalline lamellae exhibited unusually sharp switching transient for which quantitative interpretation was given. The uniaxially-drawn and annealed TrFE-rich copolymers exhibited two-step switching phenomena implying their antiferroelectric or antiferroeleetric-like nature. Further investigation using such well-defined samples would lead to microscopic understanding for the V D F / T r F E eopolymers. Concerning functional properties were presented dielectric frequency spectra and pyroeleetrie time spectra in that piezoelectric contribution associated with mechanical resonance were commonly observed. The results provided rich information about thermo-eleetro-meehanical dynamics of V D F / T r F E eopolymers as well as key data for fast sensor applications.
References [1] T. Furukawa,Phase Transitions, 18 (1989) 143. [2] J.W. Lee, Y. Takase, B.A. Newmanand J.I. Scheinbeim,J. Polym.Sci. Polym.Phys.,29 (1991) 273. [3] Y. Murata, K. Tsunashima, N. Koizumi,K. Ogami, F. Hosokawaand K. Yokoyama,Jpn. J. Appl. Phys., 32 (1993) L849. [4] S. Tasaka, T. Shouk and N. Inagaki,Jpn. J. Appl. Phys.,31 (1992) L1086. [5] T. Furukawa,M. Date, M. Ohuchi and A. Chiba, J. Appl. Phys.,56 (1984) 148. [6] T. Furukawa,H. Kodama,O. Uchinokura and Y. Takahashi, Ferroelectrics, 135 (1992) 401. [7] T. Furukawa,IEEE Trans. Electr. Insul., 24, (1989) 375. [8] K. Kogaand H. Ohigashi, J. Appl. Phys.,59 (1986) 2142. [9] T. lkeda, Fundamentalsof piezoelectric materials (in Japanese), Ohms (1990). [10] T. Furukawa,Y. Tajitsu, X. Zhang and G.E. Johnson, Ferroelectrics, 171 (1995) 33. [11] R.G. Kepler and R.A. Anderson,J. Appl. Phys.,49, (1978) 4918.
•
4
,,
, e '
ELSEVIER
Advances in Colloid and Interface Science 71-72 (1997) 183-208
ADVANCES IN COLLOID AND INTERFACE SCIENCE
Structure and functional properties of ferroelectric polymers Takeo Furukawa* Department of Chemistry,Facultyof Science, Science Universityof Tokyo, 1-3 Kagurazaka, Shinjuku, Tokyo 162,Japan
Abstract
Ferroelectric polymers represented by vinylidene fluoride copolymers with trifluoroethylene (VDF/TrFE) have been investigated with special interest on the relationship between their dynamical properties and structures. Copolymers containing 50-80 mol% VDF exhibit fast polarization reversal due to the rotation of chain molecules about their axes. Annealing of an extruded copolymer yields a highly crystalline film consisting of regularly stacked large lamellae. The switching transient of these extruded copolymers is consistent with the switching mechanism that progresses through a nucleation-growth process with a considerable waiting time (time between the generation of a nuclei and the onset of its growth) and an accelerated growth velocity. TrFE-rich copolymers exhibit a two-step switching transient, suggesting that polarization reversal occurs via a nonpolar state associated with the copolymer's antiferroelectric-like nature. Piezoelectric and pyroelectric properties were examined for their high frequency and short time characteristics. The time evolution of the charge response induced by laser pulse irradiation consists of intrinsic pyroelectricity and the coupling of thermal expansion and piezoelectric resonance. The intrinsic pyroelectric response for well-annealed VDF-rich copolymers is in the sub-nanosecond range, which promises their use as fast thermal sensors. © 1997 Elsevier Science B.V.
Keywords: Ferroelectric polymer; Vinylidene fluoride/trifluoroethylene copolymer; Polarization reversal; Piezoelectricity; Pyroelectricity; Higher order structure
*Tel.: +81 3 3260 4271; fax: +81 3 3235 2214; e-mail: tfurukaw.ch.kagu.sut.ac.jp 0001-8686/97/$32.00 © 1997 Elsever Science B.V. All rights reserved.
PllS0001-8686(97)00017-1
184
7". Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
1. Introduction
Ferroelectric polymers have generated much interest in the last 15 years because of their potential as functional materials for energy transduction and information recording. Polyvinylidene fluoride (PVDF) was the first ferroelectrie polymer for which a D - E (electric displacement and electric field, respectively) hysteresis loop and a fast-switching phenomenon were demonstrated. Incorporation of trifluoroethylene (TrFE) into PVDF produces a random copolymer that exhibits a Curie point, which is undetectable in pure PVDF. Early investigations into the ferroelectric properties of V D F / T r F E copolymers were summarized in my review article [1]. In the early 90s, odd nylons [2] (e.g. nylon7, nylonll) were shown to undergo ferroelectric polarization reversal in melt-quenched and uniaxially-drawn samples. Some aromatic polyamides [3], polyureas, and polyurethanes [4] have also been shown to be ferroelectric. These polymers exhibit D - E hysteresis behavior, but the Curie point is yet to be observed. Solid-state polymers are usually obtained as thin films in which the crystalline and non-crystalline regions coexist. Most experiments on ferroelectric polymers are done on such semicrystalline film samples. Because ferroelectric properties originate from the crystalline region, experimental data are strongly influenced by the higher order structures, such as the degree of crystallinity, orientation, and crystal size, in the samples. For quantitative analysis, therefore, we need samples with well-defined higher order structure. In this article, we examine V D F / T r F E copolymers with special emphasis on their dynamical characteristics. For VDF-rich copolymers, we prepared samples (using various thermal and mechanical treatments) with different higher order structures, and measured their switching characteristics in detail. Concerning TrFE-rich copolymers, we discovered a two-step switching phenomenon that is characteristic of an antiferroeleetric material. Finally, we describe the dynamical aspect of the piezoelectric and pyroelectric properties for VDF-rich copolymers.
2. Structure and basic properties of V D F / T r F E copolymers 2.1. Crystal structure
PVDF consists of a repeat unit (--CH2CF2--)that exhibits a dipole moment /zv = 7 x 10 -30 Cm (2.1 D) associated with positively charged H-atoms and negatively charged F-atoms (Fig. 1). Because such dipoles are rigidly attached to the main chain, their orientation is subject to the conformation and packing of molecules. If the molecule adopts an all-tram conformation and a parallel packing, the dipoles are aligned in one direction, perpendicular to the chain axis. The resulting crystal, called /3, possesses a large spontaneous polarization P, that is responsible for the ferroelectrieity of PVDF. Summing up/z v over the unit volume, we find
T. Furukawa / Adu. Colloid Interface ScL 71-72 (1997) 183-208
185
HH
(A) FF
(B) c =256pm
(c) b =491 pm:
" t !
(~
' Ps= 130mC/m2
: | !
Fig. 1. Molecular (A), chain (B), and crystal(C) structures of PVDF.
Ps = 2l~v/abc = 130 mCm -2
(1)
where a, b, and c are the lattice constants whose values are 858 pm, 491 pm, and 256 pm, respectively. It is known that PVDF has several crystalline polymorphs. Melt-crystallization yields a nonpolar a-phase consisting of TGTG' molecules packed in an antiparaUel manner. The ferroelectric /3-phase is obtained by uniaxial-drawing of a-PVDF. The introduction of a small amount of TrFE into PVDF causes direct crystallization into the /3-phase from the melt. Because the F-atom is slightly larger than the H-atom, in the copolymer the trans bond is favored over the gauche bond due to sterical hindrance occurring in the latter. Of special interest are the copolymers containing 50-80 tool% VDF because they exhibit a Curie point that is absent in pure PVDF. They also show a marked increase in crystallinity as a result of annealing at a temperature between the Curie point and the melting point. We can, therefore, prepare copolymer samples dhat have a variety of higher order structures.
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
186
2.2. Curie transition
Fig. 2 shows the temperature variation of the permittivity e' of a 75/25 mol% copolymer of VDF and TrFE. The rapid increase in e' near 12°C (dielectric anomaly) upon heating is attributed to the onset of cooperative dipolar motions, leading to a ferroelectric-to-paraelectric transition. Melting at higher temperature causes e' to decrease due to loss of cooperativity. Upon cooling, e' first increases and then decreases due to reverse transitions, i.e. crystallization and a paraelectricto-ferroelectric transition. Transition temperatures upon heating are higher than those upon cooling (130°C and 80°C, respectively), thus meaning that these transitions are first-order. Fig. 3 shows the phase diagram for a V D F / T r F E copolymer where the melting point Tm the crystallization point Tcr, and the Curie point T~ upon heating are plotted against composition. These temperatures were determined by using a differential scanning calorimeter (DSC) at a heating and cooling rate of 10°C/rain. The melting point is a minimum at a composition of 80/20 tool%. As the VDF content decreases, Tm increases gradually to 200°C at PTrFE. It also increases with increasing VDF content above 80 mol%. The Tm of a-PVDF is 175°C. The Tcr depends upon composition in a similar manner and is located about 25°C below
rm. The T~ of an equimolar copolymer is located near 65°C (Fig. 3). As the VDF content increases, T~ upon heating increases to coincide with Tm at 80 tool%. This
50
40
_
30 _
Cooling/ -
20 oo
~
o
_.,o-
•
~
10 0
I 40
I 60
I 80
Heating
I 100
I 120
o
•
o I 140
I 160
Temperature ('C)
Fig. 2. Dielectrictemperaturespectra for a VDF(75)/TrFE(25)copolymer.:
T. Furukawa /Ado. Colloid Interface ScL 71-72 (1997) 183-208
200(
187
Molten Phase
(
150
-
100 _
ID
so,
~.
Paraelectric Phase
Tc
" ~
i I
I !
Anti-ferroelectric
0
Phase
/
?
/
Forroe/ectric P h a s e
/
-50 -
J
I 0 PTrFE
20
/D'I 40
I
I
60
80
VDF content (mol%)
1O0 PVDF
Fig. 3. Phase diagram for a V D F / T r F E copolymer (O: melting point; 0 : crystallization point; zx : Curie point on heating; • : Curie point on cooling, [] : ferroelectric-to-antiferroelectric transition point).
implies that melting begins prior to completion of the ferroelectric-to-paraelectric transition. On the other hand, the Curie point disappears in the DSC thermogram when TrFE becomes a major component (> 50 mol%). However, a dielectric anomaly still exists around 50-60°C (triangles with dot) which smoothly connects with Tc of VDF-rich copolymers. Therefore, the region between the circles and triangles in Fig. 3 corresponds to the paraeleetrie or paraeleetric-like phase. VDF-rieh copolymers are in a ferroelectric phase below the Curie point (Fig. 3). On the other hand, TrFE-rieh eopolymers exhibit a double hysteresis loop in the temperature range surrounded by the dashed lines in Fig. 3. The enclosed region could be attributed to an antiferroelectric or an antiferroeleetrie-like phase as will be discussed in section 4.2. 2.3. Polarization reoersal
In the ferroelectric phase, VDF-rich copolymers incorporated with PVDF exhibit
188
T. Furukawa /Adv. ColloM Interface ScL 71-72 (1997) 183-208
a typical D - E hysteresis loop as demonstrated in Fig. 4 for a VDF(65)/TrFE(35) copolymer at 20°C with an eleetric field E applied in a sinusoidal form at 1 Hz. In the low field regime ( < 30 MV/m), the D - E relation is nearly linear. As E is increased, hysteresis starts to appear. In the high field regime (> 60 MV/m), we obtain a square hysteresis loop that is independent of the amplitude of E. The intersect of the loop with the abscissa defines the coercive field E c which is ca. 50 MV/m. The intersect with the ordinate gives the remanent polarization Pr ca. 80 m C / m 2, which depends upon not only composition but also higher order structures of the sample. Fig. 5 shows the compositional variation of Pr in which the data plotted are the largest experimental values obtained so far. Also plotted (dashed line) is the value of the spontaneous polarization calculated assuming a rigid dipole. Pr reaches a maximum at a VDF content of 80 tool% and decreases gradually as VDF decreases to 50 tool%. This gradual decrease is attributed to a decrease in the average dipole moment because the dipole moment of a TrFE unit is one-half that of a VDF unit. As the VDF content falls below 50 mol%, Pr decreases rapidly due to a loss of ferroelectricity. The further decrease in P~ in the 80-100 mol% range is attributable to a decrease in crystallinity. The observed P~ of PVDF is ca. 65 m C / m z, which is one-half that predicted by using spontaneous polarization (dashed line), ca. 130 m C / m 2.
J
100
50 E o
=
-50 -
-100
-
I
I
-100
-50
0
I
I
50
100
E (MY/m) Fig. 4. D - E hysteresis loops for a VDF(65)/TrFE(35) copolymer at 20°C.
189
T. Furukawa / Ado. Colloid Interface ScL 71-72 (1997) 183-208
140 o
120 100 E
calculated
80'
E ~,_ 6 0 40
..
/
observed
20 0
n
PTrFE
I 20
I 40
I 60
VDF content (mol%)
I 80
1O0 PVDF
Fig. 5. Composition dependence of observed and predicted the remanent polarization Pr for VDF/TrFE copolymers.
3. Switching characteristics of VDF-rich copolymers
3.1. Ferroelectric switching measurement Ferroelectric polarization reversal is commonly examined by means of a D - E hysteresis measurement in which a sinusoidal electricfield E is applied to a sample and the resultingelectricdisplacement D is then measured. A n alternativemethod is a switching measurement, which uses a step-wise E and measures D as a function of time t. Fig. 6 shows a block diagram of the apparatus developed for the fcrroclectric switching measurement. A high step-voltage up to 3.5 kV is applied to a sample through a switch,which is a silicone-controUed rectifier(SCR). The rise time of the applied voltage is ca 1 ~s. For fast (nanosecond) measurements at lower voltages (< 500 V), the S C R is replaced by a mercury rclay.Thc response from the sample is detected by a capacitor or a resistor in terms of charge or current, respectively, and is stored in transient memory. Representative results are shown in Fig. 7 where log D is plotted against log t. Here, D R represents the result obtained by applying E in the direction opposite the existing polarization. Any sharp change in D indicates polarization reversal
190
T. Furukawa /Adv. Colloid Interface Sci. 71-72 (1997) 183-208
MICRO L COMPUTER SYSTEM I"
"l
t.oG I
TIMER I
TRANS' NTI MEMORY 1'
I'/
SWITCH HIGH VOLTAGE SOURCE
_L_ r
Fig. 6. Experimental set-up for ferroelectric switching measurement.
occurring as a result of dipole reorientation. If E is applied in the direction of Pr the resulting electric displacement D F exhibits only a gradual increase because most dipoles are already aligned parallel to the field direction. Such directionality confirms that the material of interest is ferroelectric. Both D R and D F contain contributions from the dielectric response and the dc conduction. By subtracting D F from D R, we obtain the response that arises mostly from ferroelectric polarization reversal. Fig. 8 shows the switching characteristics of a VDF(78)/TrFE(22) sample prepared by rapid cooling from the melt to room temperature (melt-quenched) followed by annealing at 145°C for 1 h. The increment of D corresponds to the amount of reversed polarization equal to double the remanent polarization (i.e. 2P r) which is independent of the strength of E. The position of the O D / O l o g t peak defines the switching time r~, which decreases as E is increased. Fig. 9 shows a plot of log z~ versus the reciprocal of E. The straight line indicates an exponential relationship between ~-~ and E: r s = ~'~o e x p ( E a / E )
(2)
where E a is the activation field and r,0 is the switching time at an infinite field, which were determined to be 0.8 G V / m and 10 ns, respectively. 3. 2. Effect o f annealing on structure and switching characteristics
To examine structure-property relationship with respect to ferroelectric polarization reversal, we prepared two kinds of samples; melt-quenched and extruded (extruded from the melt). In the melt-quenched sample, there is no preferred orientation of molecules, whereas in the extruded sample, molecules are well-ori-
T. Furukawa / Adv. Colloid Interface ScL 71-72 (1997) 183-208
191
0.0 -0.5
,l,,~, f EDR/ o
'-pr ~ ' |
''~;
E
-2.0 -2.5 -3.0
-3
I
I
I
I
-2
-1
0
1
2
log t (s) Fig. 7. Time evolutionof the electric displacement D inducedby a stepped electricfield E applied in the directionopposite(D R) to the remanent polarization P~ and in the same direction(Dr).
ented. Both samples were subjected to annealing at elevated temperatures to increase crystallinity. Fig. 10 shows the effect of annealing on the switching characteristics of an extruded VDF(78)/TrFE(22) copolymer. The as-extruded sample exhibits essentially no polarization reversal. As the sample is annealed at a temperature just above the Curie point (128°C), polarization reversal starts to appear. Annealing at higher temperatures causes further increase in the amount of reversed polarization as well as sharpening of the a D / a l o g t peak. The field dependence of zs is basically the same as that seen in an unoriented sample (Fig. 9). We examined the structure of these two samples (melt-quenched and extruded) before and after annealing by using a scanning electron microscope (SEM). Figs. l l A - 1 1 C show SEM images of the fractured surface parallel to the extrusion direction (vertical) and to the film thickness (horizontal). The as-extruded sample (llA) exhibits vertical striation associated with oriented chain molecules. Annealing at 128°C gives rise to horizontal stripes (liB), suggesting a formation of crystalline lamellae. As the annealing temperature is increased to 145°C, extremely large lamellae 100 nm thick and 1/zm wide are stacked in a parallel manner along the extrusion direction (llC). The SEM image of the fractured surface perpendicular to the extrusion direction (liD) provides a better view of these stacked lamellae. Comparing the results from the switching measurements (Fig. 10) and the SEM measurements (Fig. 11), we find that annealing causes a rapid increase in crys-
192
T. Furukawa /Adv. Colloid Interface Sci. 71-72 (1997) 183-208
0.15
E o
0.10
v
0.05
0.0q
-5
-4
-3
-2
-1
0
1
-1
0
1
10g t (S)
0.20 E
o
L
100 90
0.15
0.10
"o
0.05 0.00 1 I
-5
-4
-3
-2
log t (s) Fig. 8. Switching characteristics of melt-quenched and annealed VDF(78)/TrFE(22) copolymers at 20°C.
tallinity, leading to a marked increase in the amount of reversed polarization. The extremely sharp OD/Ologt peak of a well annealed sample would be due to very large lamellae. The melt-quenched samples have similar structural changes due to annealing, i.e. a marked increase in crystallinity and in the growth of large lamellae, as
T, Furukawa / Adv. Colloid Interface Sci. 71-72 (1997) 183-208
193
0v
- 2 --
o
-6-I
5
10
, I,
l
I
15
20
25
lIE
(m/GV)
30
Fig. 9. Switching time ~'s versus the reciprocal of the applied electric field E for a VDF(78)/TrFE(22) copolymer.
revealed by SEM measurements (analogous to Fig. C but wihout preferred orientation). Switching characteristics also show a significant increase in Pr as well as a sharpening of the OD/Ologt peak as shown in Fig. 10 for a well-annealed sample. Fig. 12 shows a comparison of the switching transients of well-annealed samples of melt-quenched and extruded copolymers. The amount of reversed polarization of the melt-quenched sample is 20% larger than that of the extruded. From a structural viewpoint, polarization reversal of V D F / T r F E copolymers occurs as a result of the rotation of chain molecules about their axes. Therefore, molecules oriented in the thickness direction (i.e. E direction) do not contribute to polarization reversal. Quantitatively, molecules oriented with angle 0 with respect to the thickness direction contribute to Pr as a coefficient sin 0. Assuming that molecules are randomly oriented in a melt-quenched sample, the expected Pr is ~r/4 that of a completely oriented sample, for samples with the same crystallinity. Thus, the difference in Pr for the extruded and melt-quenched samples is due to molecular orientation, which confirms that chain rotation is responsible for polarization reversal in V D F / T r F E copolymers. Another important difference between the two samples is the shape of the switching transient. The extruded sample undergoes a very sharp change in D, whereas the melt-quenched sample exhibits a rather gradual change (Fig. 12). The time evolution of the switching transient reflects how polarization reversal pro-
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
194
0.25 0.20
annealed at 145"C
0.15 E 0 c~ 0.10
137"C
0.05
as-extruded
v
128"C
0.00 -4
-3
-2
-1
0
log t (s)
0.8
annealed at 145°C
0.6 o *" 0.4 v
137"C
0
0.2 0.0 -z
128"C ' -3
'~
~ -2
as-extruded -1
0
log t (s) Fig. 10. Switching characteristics of extruded and annealed VDF(78)/TrFE(22) copolymers at 20°C.
gresses after the application of the step-wise electric field. The switching process in crystalline lamellae would be affected by their orientation with respect to this applied field. Because the extruded copolymer consists of well-aligned lamellae, the observed switching transient may represent that of each lamella. On the other
T. Furukawa /Ado. Colloid Interface ScL 71-72 (1997) 183-208
:
4J
~
,,
195
i~
(,
., i (A)
(B)
(C)
(D)
5oonm
Fig. ll. SEM images of the fractured surface of a VDF(78)/TrFE(22) film (A) as-extruded, (B) annealed at 128°C, and (C, D) annealed at 145°C.
hand, the switching transient of the melt-quenched sample is subject to an 'orientation average', which results in a much broader spectrum of the transient. In the following, we analyze the switching transient of extruded copolymers.
3.3. Anal, sis of switching transient It is generally accepted that polarization reversal in a ferroelectric is based on a nucleation-growth mechanism [5]. When an electric field is applied, polarization reversal starts with the generation of the nuclei of reversed domains, followed by nuclei growth that is seen as domain-wall motions. For single crystal ferroelectrics such as BaTiO 3, such domain-wall motions can be observed by using an optical microscope. This mechanism is consistent with the observed coercive field being much lower than phenomenologieal prediction. Assuming random nucleation, the number of generated nuclei at time t is N = N 0 ( 1 - e -R')
(3)
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0.25 0.20
E
extruded
0.15
c~ 0.10 0.05
0.00
I
f
I
I
I
-4
-3
-2
-1
0
log t (s) Fig. 12. Comparison of the shape of switching transient for melt-quenched and as-extruded VDF(78)/TrFE(22) copolymers.
where R is the nucleation probability and N O is the total number of available nucleation sites. If the generated nuclei grow m-dimensionally at a velocity o, the total volume of all reversed domains V (extended volume) is given by -d d N
where F is a shape factor. As growth proceeds further, ingestion of nucleation sites occurs. Growing domains then coalesce. Taking ingestion and coalescence into consideration, the actual reversed polarization P given by Avrami [5] is P = 2Psi1 - e -g ]
(5)
where Ps is the spontaneous polarization. Fig. 13 shows a schematic of the nucleation-growth process for the ease of two-dimensional growth. Substituting Eqs. (3) and (4) into Eq. (5) and then integrating, we obtain an analytical expression for P, which can then be simplified as P = 2P~[1 - e - ( t / r s ) ' ]
(6)
Here, the exponent n is related to the growth dimension m. When the growth velocity v is small (t :~ l / R ) , polarization reversal occurs after most nucleation sites are activated. For this case, n = m results [5]. On the other hand, if v is very large (t < l / R ) , polarization reversal is completed as soon as a small amount of nuclei are generated, thus yielding n = m + 1 [5].
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T. Furukawa / Adv. Colloid Interface Sci. 71-72 (1997) 183-208
ole
@ 0 (A)
(B)
(C)
Generated Nuclei Sites
Growth of Sites
Coalescence of Sites
Fig. 13. Schematic of the nucleation-growth mechanism for ferroelectric polarization reversal.
The value of n determines the shape of the switching transient curve in terms of the sharpness of the aD/alogt peak, as represented in Fig. 14 for observed and predicted (using Eq. [6]) peaks for the melt-quenched and extruded copolymers. The corresponding n value of the extruded copolymer is larger than 5, which requires an unrealistic growth dimension of 4 to 5. The observed transient also shows departure from the predicted transient when t < ~-s/10. To explain such a large n value and to reproduce the switching transient over the entire range of t, we introduce three assumptions into our analysis: (a) v is not constant but has an acceleration a; (b) generated nuclei must wait a time Tw before they start to grow; and (c) generated nuclei have a certain initial volume before growth. Incorporating these assumptions, the extended volume is expressed as
6 • Extruded n =6 o Melt-Quenched ~ / ~ 4
i
Calculated 3
~
'
~
0
-0.6
-0.4
-0.2
0.0
0.2
0.4
Iogt/~:s Fig. 14. Comparison of observed aD/alogt peak with that predicted from an exponential function.
T. Furukawa /Adv. Colloid Interface Sci. 71-72 (1997) 183-208
198
O,
I
, -3= -5
. i,,/" -4
IIIIII
n=5.3
,; a,v, ro,~. -3
I -2
-
log t (s) Fig. 15. Observed and fitted switching transients for extruded and well-annealed V D F / T r F E copolymers. ! dM~
V= J~[tJ"~" J F(½a(t-t w - s ) z + v ( t - t w - s ) +r2)mds o~ dt]s
(7)
Fig. 15 shows the results of fitting the observed transient for the extruded and annealed copolymer with this expression. We find that the observed transient is well reproduced. The a and 7", account for the sharp rise in D with time t. The radius of the nuclei r 0 contributes to the initial gradual increase in D. In practice, the even-order nonlinear permittivity also contributes to the initial gradual increase in D as we previously reported [6]. We have succeeded in reproducing the observed very sharp switching transient of extruded and annealed samples of copolymers by introducing a variety of factors, such as a and Tw. Because such samples consist of regularly-stacked large lamellae, we believe that the observed sharp switching transient represents the events occurring in respective crystalline lamellae. Much broader switching transients in unoriented copolymers can be interpreted by taking into consideration the averaging effect arising from orientation distribution.
4. Switching characteristics of TrFE-rich copolymers
4.1. Double hysteresis and two-step switching charactens"acs As stated earlier, the amount of reversed polarization rapidly decreases as the VDF content is decreased to below 50 mol%. Although there remains a dielectric
T. Furukawa /Ado. Colloid Interface Sci. 71-72 (1997) 183-208
199
anomaly, the ferroelectric-to-paraelectrie transition in such samples is undetectable by DSC measurements. In this section, we look at the loss of ferroelectricity in TrFE-rich copolymers. Fig. 16 shows the D - E relation for uniaxially-drawn and annealed V D F / T r F E copolymers with compositions 65/35, 47/53, and 37/63 mol%. The square hysteresis loop that is characteristic of VDF-rich copolymers is transformed into a double hysteresis loop as TrFE becomes a major component (> 50 mol%). Double hysteresis suggests that polarization reversal progresses through a nonpolar state. The VDF(47)/TrFE(53) copolymer experiences the nonpolar state just after the applied field is reversed, whereas the 37/63 copolymer drops to the nonpolar state at zero field. Fig. 17 shows D - E hysteresis measurements at various temperatures for the VDF(47)/TrFE(53) copolymer. As the temperature is decreased to -40°C, double hysteresis is lost and transforms into a regular single loop. This implies that the polar phase is more stable than the nonpolar phase below -40°C. The ferroelectric switching measurements for the VDF(47)/TrFE(53) copolymer in Fig. 18 show a two-step switching transient, thus confirming that polarization reversal progresses through a nonpolar state in the time domain. The dependence of the time constant on field strength is stronger for the polarization change from
100
S
~'~'~65/35
50
E O E
v
0
-50 -
-100 -100
I -50
0
I 50
100
E (MY/m) Fig. 16. D - E hysteresis loops for VDF(65)/TrFE(35), 47/53, and 37/63 copolymers at room temperature.
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60 40 20
_E o E
0
v
-20 -40
-
-60
-
c0OC 7. "'-....40oC
25°C
-150
I
I
-100
-50
0
I
I
50
100
150
E (MY/m) Fig. 17. D - E hysteresis loops for a VDF(47)/TrFE(53) copolymer at 25, 0, - 10 and -40°C.
negative to zero than that from zero to positive. These two time constants are unified when the E reaches 90 M V / m . Fig. 19 shows plots of these two time constants versus the 1/17, obtained at various temperatures. As the temperature is decreased to -40°C, we obtain a single switching process in accord with the regular square hysteresis loop at the same temperature.
4.2. Possibility of anti-ferroelectricity There are two possibilities that give rise to a double hysteresis loop or a two-step switching phenomenon. One is that the ferroelectric undergoes a first-order transition. Near the Curie point, both the ferroeleetric and paraeleetric phases are at an energy minimum for such a ferroelcctric. Just above the Curie point, the external field induces a paraelectric-to-ferroelectric transition, resulting in a double hysteresis loop. The other possibility is that the material in concern is the antfferroelectric consisting of two sublattices that have spontaneous polarizations opposite in direction. The application of an external field causes polarization reversal of one of the sublattices, thus inducing a transition to the ferroelectric phase. Here we have shown that TrFE-rich copolymers exhibit a double hysteresis loop as well as a two-step switching phenomenon. One question that arises is whether the copolymer is an antiferroelectric or a first-order ferroelcctric. It is known that the ferroelectric transition of VDF-rich copolymers is first-order as revealed by
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
201
+i iit °070; f 0"06F
0.00 -5
-4
-3
-2
-1
0
-1
0
Iogt(s)
0.10 !
E
0 •08
O E
0.06
o0 ) 0.04 ¢~
8O 7
~
60
50
0.02 0.00 -5
-4
-3
-2 Iogt(s)
Fig. 18. Switching characteristics of a VDF(47)/TrFE(53) copolymer at 25°C.
thermal hysteresis of the transition behavior. However, no clear indication of double hysteresis has been reported so far. As the VDF content is decreased to equimolar composition, thermal hysteresis is lost and the transition becomes second-order. As the transition of TrFE-rich copolymers is second order, we expect that the two-step switching arises from their antiferroelectric or antiferroelectric-like nature. Fig. 20 shows a schematic of the structural change during polarization reversal for TrFE-rich copolymers of VDF. Structure A describes the paraelectric phase in which molecules adopt a random conformation, resulting in disordered
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
202
j
0-1 v
69
"2~ o
"3
"4
o
-5
5
I 10
I 15 1/E
I 20
I 25
i
25 °C
30
(m/GV)
Fig. 19. Switching time r s against I/E for the two-step switching transient of a VDF(47)/TrFE(53) copolymer at various temperatures.
dipole orientation. Structure C expresses the antiferroelectric phase consisting of all-tram molecules packed in an ant/parallel manner. An alternative structure, B, assumes a random packing of all-tram molecules. To answer which structural change is occurring, we must determine the crystal structure of TrFE-rich copolymers at temperatures where the two-step switching is observed.
5. Piezoelectric and pyroelectric properties As far as application is concerned, ferroelectric polymers have received much attention as soft transducer materials because of the polymers" piezoelectric and pyroelectric properties [7]. This section describes our recent work on the dynamical aspect of these properties, particularly in the high frequency or the short time ranges. The piezoelectric constant dq and the pyroelectric constants Pi are defined as
203
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
+P
-p
Nonpolar P=O Fig. 20. Structural changes during the two-step switching transient for TrFE-rich copolymers.
dq =
-~J/e
Pi
t OT /e
(i = 1 - 3 , j
= 1-6)
(7)
where D is the electrical displacement, E is the electric field, X is the stress, and T is temperature. The uniaxially-drawn and poled V D F / T r F E film has a C=v symmetry. Therefore, the matrix form of d u and Pi are 0
dij =
0 d31
0 0 d32
0 0 d33
0 d24 0
dis 0 0
0 0 0
(8)
(9) Here, 1 refers to the draw direction, 3 to the poling direction, and 2 is orthogonal to both 1 and 3. For undrawn samples, d31 = d32 and d15 = d24. The values of
204
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
nonzero components are not a material constant but depend upon the remanent polarization Pr imparted by poling. For a VDF(65)/TrFE(78) copolymer, we obtain a good linear relationship with respect to d31/s~l and P3 as demonstrated in Fig. 21. Pr depends upon the higher order structure of the sample and its maximum value is limited to the spontaneous polarization of the crystalline regions. 5.1. Piezoelectric resonance The piezoelectric constant at high frequencies is measured by using a resonance method. Fig. 22 shows the dielectric spectra for a V D F / T r F E copolymer over a broad frequency range at room temperature. The spectra show two kinds of resonance [8] superimposed on a broad dielectric relaxation. The resonance near 400 kHz is assigned to the length extension (LE) mode vibration and that around 40 MHz and its odd order harmonics are attributed to the thickness extension (TE) mode vibration. Here the sample used for the measurement was 20 /zm thick, 10 mm long, and 5 mm thick. Sample deformations associated with LE mode and TE mode resonances are depicted in the figure. In the former the length extension is acompanied by thickness and width contractions, whereas the thickness extension occurs without length and width contractions in the latter.
120
60
100
-
50
80
-
40
-
30
-
20
-
10
3 v
20
0
7~
I
0
20
40
60
80
100
Pr (me/m2) Fig. 21. Dependence of the piezoelectric constant dal/sll and the pyroelectric constant P3 on the remanent polarization Pr for a VDF(65)/TrFE(35) copolymer.
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
lo[
205
x
8~ M o d e
61-
TE Mode
~. . . . . . . . . . . . . . . . . . X
4
2 0 4
5
6
7
8
9
Iogf (Hz)
5 4
%
TE Mode
3-
LE Mode
2
::~:~
0
"
4
5
i
~. . . . . . . . .
t
I
6
7
!
.,=
"
8
9
Iogf (Hz) Fig. 22. Frequency spectra of the complex permittivity E* = e - is" containing piezoelectric resonance for a V D F / T r F E copolymer.
The dielectric spectra observed can be described by [9]
E33 ffi 8~3
k321 tan a ) 1 + 1-k~-------~ a tan b
(10)
206
T. Furukawa /Adv. Colloid Interface Sci. 71-72 (1997) 183-208
with °' a = 2v
v
(11)
ps~
and
o,t b = 2v
/ cg v
(12)
P
where ~o is the angular frequency, p is the density, l is length, t is thickness, s is the elastic compliance, c is the elastic stiffness constant, and k is the electromechanical coupling coefficient defined by
ag, k32~-
xE ,g33511
k~3=
x o 833C33
(13)
Here e x is the free permittivity and e x is the clamped permittivity. Using Eq. (10), we can reproduce the observed spectra by adjusting the constants as best-fit parameters. In this case, we obtained k 3 = 0.27 and k31 = 0.09, which lead to estimates of the piezoelectric constants d31 and e33 by the use of the elastic and dielectric constants appearing in Eq. (13). We are also able to determine the frequency spectra of the clamped permittivity e x (dashed curves in Fig. 22), which is completely independent of piezoelectric resonance. The broad relaxation observed has been attributed to dipolar fluctuations in the crystalline regions [10]. 5.2. Pyroelectric d y n a m i c s
The dynamical characteristics of the pyroelectricity can be measured by the use of laser pulse heating [11]. Fig. 23 shows the time evolution of the charge response from a freely suspended VDF(75)/TrFE(25) sample after the irradiation of a 7 ns YAG laser pulse. The time spectra consist of four processes: (a) initial fast rise ( < 10 ns); (b) TE-mode damped oscillation (10 ns-1 ~s); (c) LE-mode damped oscillation (1 ~s-10 ms); and (d) final decrease due to thermal diffusion outward from the sample (10 ms-10 s). The TE mode and LE mode oscillations in the charge response occur as a result of rapid thermal expansion coupled with piezoelectric effects. Comparing Figs. 22 and 23 reveals that two types of mechanical vibration contribute to dielectric and pyroelectric spectra through piezoelectric coupling. The former was measured on the frequency domain whereas the latter on the time domain. Therefore, Fourier transform of one of the spectra is necessary to make comparison on the same (frequency or time) domain. Detailed analysis is now in progress. We expect to obtain further information about the dynamics of not only dielectric and pyroelectric properties but also piezoelectric, thermal and mechanical properties of V D F / T r F E copolymers.
207
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
40 LE Mode
':':~::
300Q..
20-
(3 10-
0 -'0
-8
-6
-4
-2
0
Iogt (s) Fig. 23. Time spectra of the charge response for a VDF(75)/TrFE(25) copolymer induced by laser pulse irradiation.
Regarding the fast response, by removing the TE-mode component, we obtained the time spectra describing fast pyroelectric dynamics. The spectra consist of two components. The first is an initial rapid rise attributed to the intrinsic pyroelectricity that appears prior to thermal expansion in parallel with an integrated laser 30 25
20 (3 10 5 0 0
I 50
I 1O0
I 150
I 200
I 250
t (ns) Fig. 24. Fast pyroelectric response for a VDF/TrFE copolymer.
300
208
T. Furukawa /Adv. Colloid Interface ScL 71-72 (1997) 183-208
pulse. This property holds promises in using V D F / T r F E copolymers as very fast pyroelectric detectors. The second component is the gradual rise (time constant of 100 ns) associated with dipolar fluctuations in accord with the dielectric relaxation in Fig. 22.
6. Concluding remarks Structure-property relationship has been examined for V D F / T r F E copolymers over the entire range of composition. Annealing is a very efficient technique for controlling higher order structures such as the degree of crystallinity and the crystalline size of this copolymer. It was found that the crystalline size does affect the ferroeleetric switching characteristics of VDF-rich copolymers. The meltquenched and annealed sample consisting of regularly packed large crystalline lamellae exhibited unusually sharp switching transient for which quantitative interpretation was given. The uniaxially-drawn and annealed TrFE-rich copolymers exhibited two-step switching phenomena implying their antiferroelectric or antiferroeleetric-like nature. Further investigation using such well-defined samples would lead to microscopic understanding for the V D F / T r F E eopolymers. Concerning functional properties were presented dielectric frequency spectra and pyroeleetrie time spectra in that piezoelectric contribution associated with mechanical resonance were commonly observed. The results provided rich information about thermo-eleetro-meehanical dynamics of V D F / T r F E eopolymers as well as key data for fast sensor applications.
References [1] T. Furukawa,Phase Transitions, 18 (1989) 143. [2] J.W. Lee, Y. Takase, B.A. Newmanand J.I. Scheinbeim,J. Polym.Sci. Polym.Phys.,29 (1991) 273. [3] Y. Murata, K. Tsunashima, N. Koizumi,K. Ogami, F. Hosokawaand K. Yokoyama,Jpn. J. Appl. Phys., 32 (1993) L849. [4] S. Tasaka, T. Shouk and N. Inagaki,Jpn. J. Appl. Phys.,31 (1992) L1086. [5] T. Furukawa,M. Date, M. Ohuchi and A. Chiba, J. Appl. Phys.,56 (1984) 148. [6] T. Furukawa,H. Kodama,O. Uchinokura and Y. Takahashi, Ferroelectrics, 135 (1992) 401. [7] T. Furukawa,IEEE Trans. Electr. Insul., 24, (1989) 375. [8] K. Kogaand H. Ohigashi, J. Appl. Phys.,59 (1986) 2142. [9] T. lkeda, Fundamentalsof piezoelectric materials (in Japanese), Ohms (1990). [10] T. Furukawa,Y. Tajitsu, X. Zhang and G.E. Johnson, Ferroelectrics, 171 (1995) 33. [11] R.G. Kepler and R.A. Anderson,J. Appl. Phys.,49, (1978) 4918.