Ferroelectret nanogenerator with large transverse piezoelectric activity

Author’s Accepted Manuscript Ferroelectret nanogenerator with large transverse piezoelectric activity Xiaoqing Zhang, Perceval Pondrom, Gerhard M. Sessler, Xingchen Ma www.elsevier.com/locate/nanoenergy

PII: DOI: Reference:

S2211-2855(18)30331-8 https://doi.org/10.1016/j.nanoen.2018.05.016 NANOEN2724

To appear in: Nano Energy Received date: 4 April 2018 Revised date: 1 May 2018 Accepted date: 6 May 2018 Cite this article as: Xiaoqing Zhang, Perceval Pondrom, Gerhard M. Sessler and Xingchen Ma, Ferroelectret nanogenerator with large transverse piezoelectric activity, Nano Energy, https://doi.org/10.1016/j.nanoen.2018.05.016 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Ferroelectret nanogenerator with large transverse piezoelectric activity Xiaoqing Zhang1, 2, Perceval Pondrom1, Gerhard M. Sessler1*, and Xingchen Ma2 1

. Institute for Telecommunications Technology, TU Darmstadt, Merckstrasse 25, 64283 Darmstadt, Germany

2

. Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Tongji University, Siping Road 1239, 200092 Shanghai, China *E-mail: g. [email protected]

Abstract Energy harvesting from vibrations enables power delivery to low-energy electronics for autonomous devices, wearables, as well as wireless and remote sensing. Here we present a piezoelectric energy harvester consisting of specially designed ferroelectrets based on fluorocarbon polymers that have very large low-frequency transverse piezoelectric coefficients g31 = 3.0 Vm/N. This is much larger than corresponding values for lead zirconate titanate (PZT) or for polyvinylidene fluoride (PVDF) which are presently standard materials for harvester devices. We use these ferroelectrets in miniature energy harvesters that can be classified as Ferroelectret Nanogenerators (FENG’s). Their performance is characterized by a power output of approximately 50 µW for an acceleration of 9.81 m/s2 and a seismic mass of 0.09 g. This compares favorably with the best PZT or PVDF systems, but is accomplished with a soft material that is small, lightweight, and flexible. This permits uses of such harvesters for powering miniature devices, such as small mobile sensing equipment, nano- and microelectronic circuits, body-worn electronics, or medical implants.

Keywords: ferroelectret nanogenerator; FENG; transverse piezoelectric activity; parallel tunnel structure; fluorinated ethylene propylene; vibrational energy

1

1. Introduction Ferroelectrets, also referred to as piezoelectrets, are cellular materials that are internally charged such that they exhibit large piezoelectricity [1]. As has been discussed, the required charge distribution in the cellular structure has to be such that the upper surfaces of all voids are charged to one polarity while the bottom surfaces carry the other polarity charge. This may be achieved by corona or contact charging. Such ferroelectrets have been explored in many studies [1]. Almost all of the early samples were made of polypropylene (PP) [2], which is available in the form of cellular films with nano- to micrometric voids. One drawback of PP ferroelectrets is the poor thermal stability of the charge in this material. This limits its use to temperatures below about 50° C. Efforts to find ferroelectret materials or structures with thermally better charge stability have led to numerous studies. A particularly promising way to solve the problem of thermal stability is the use of laminated samples, consisting of films of polytetrafluoroethylene (PTFE) and/or fluorinated ethylene propylene (FEP), fused together at elevated temperatures with air voids between the layers. Such ferroelectrets have been successfully made some time ago [3-5]. More recent variations are the so-called tubular channel [6-9] - or parallel tunnel samples [10, 11]. All these films are thermally more stable, with working temperatures of 90°C or higher [6, 12, 13]. Another shortcoming of the PP ferroelectrets in many applications is the fact that they essentially only show a strong longitudinal piezoelectric activity, expressed by large d33 (or g33) coefficients, while their transverse d31 or g31 coefficients are very small [1]. This results from the good compressibility (small Young’s modulus Y) of the voided material in the thickness or 3-direction while it is much stiffer in the in-plane or 1- and 2-directions. Since the piezoelectric coefficients are inversely proportional to Y [14, 15], their strong dependence on direction is understandable. In contrast, as was discovered very recently [10, 11], the parallel-tunnel samples also exhibit a strong transverse piezoelectric effect, expressed by very large d31 and g31 coefficients in addition to the large d33 and g33 values. One of the exciting new applications of ferroelectrets is their use in energy harvesting, in particular in the conversion of vibrational energy into electrical energy. While energy harvesting with electrets consisting of fluorocarbon, polyvinylidenefluoride (PVDF) or Nylon polymers as the active material has been pursued for some time [16-22], the use of ferroelectrets in such devices is a much more recent achievement. Starting in 2012, first papers dealing with energy harvesting utilizing the longitudinal piezoelectric d33 effect in ferroelectrets were published [23-28]. Recently, numerous papers on electret or ferroelectret energy harvesters have appeared [29-36]. In this context, the terms 2

Ferroelectret Nanogenerator (FENG) [30, 31] and Ferroelectretic Nanogenerator (FTNG) [35, 36] have been coined for such devices. The term “Nanogenerator” implies that these devices are useful for powering nano- and microelectronic circuits, e. g. sensors and their electronics, with power levels that were originally in the nano-Watt range but have now developed into the µW region. We adopt the term FENG also for our new harvesters, first described in 2016, based on the d31 effect [11]. These devices offer a much larger energy output than the older d33 harvesters. Other types of nanogenerators that have been described in the literature are piezoelectric, triboelectric, and pyroelectric transducers [37-39]. In the following, we first describe the preparation and properties of parallel-tunnel samples that yield the very large transverse piezoelectric d31 and g31 activity. We also explain, for the first time, the significance of characterizing these films with the g31 (and not the d31) coefficient. We then introduce, and give a detailed description of, a significantly improved FENG based on the transverse coefficients. Its performance is theoretically analyzed before experimental results on resonance frequency, harvested power, and other properties are presented. All results on piezoelectric coefficients and generated power are discussed and comparisons with prior results are made utilizing suitable figures of merit.

2. Preparation and properties of parallel tunnel ferroelectrets with large g31 As pointed out above, parallel tunnel films are a relatively new variety of ferroelectret films. They are laminated structures consisting of FEP films fused together at elevated temperatures. Since their preparation has been described before [10, 11], we will only briefly discuss this topic and refer instead to the literature while concentrating in this section on new evaluations of the properties of such samples. The preparation process is schematically shown in Fig. 1. Briefly a sandwich, consisting of a soft pad with two 12.5 µm FEP films on its two surfaces, is clamped between two templates (Fig. 1A and 1B) with grooved structure. Upon application of hot pressing (Fig. 1C), the FEP films are patterned. After removal of the soft pad (Fig. 1D), fusion bonding of the two patterned FEP films is achieved by heating the structure to 320 °C (Fig. 1E). The laminated parallel-tunnel ferroelectret thus formed is detached from the templates (Fig. 1F) whereupon a sample as shown in the SEM images is obtained. The different cross sectional shapes shown are typical and depend on parameters like detailed shape of the templates, structure of the soft pad, pressure during hot pressing etc. 3

Fig.1: Schematic of preparation of laminated parallel tunnel FEP films. A is the template and materials used in this procedure. B is the composition of a stack used for the preparation. C, D, and E are the steps of hot pressing at an elevated temperature of 100 oC, removal of the soft pad and fusion bonding above the melting point of FEP, respectively. F is a schematic view of the completed parallel tunnel film and shows also two SEM images of fabricated parallel tunnel FEP film. The x, y, and z-coordinates are also shown in the Figure, corresponding to the crystallographic 1, 2, and 3-directions, respectively. The samples are electroded by an evaporated Al-nanolayer on their outer surfaces and then poled by contact charging with a triangular voltage in open air, as illustrated in Fig. 2. The desirable charge distribution, shown in Fig. 2A, is generated by micro discharges inside the air tunnels. It consists of charges of opposite polarities on the upper and lower surfaces of the air voids, respectively. The polarization voltage in the present case has triangular shape with a period of 20 ms, as seen in Fig. 2B. It generates hysteresis loops as shown in Fig. 2C [40]. The voltage is applied for about 10 s and thus goes through many cycles. For maximum polarization, it is terminated electronically at a zero crossing. The process is thus similar to one of the poling methods applied for PVDF [40]. The induction charge density on the 4

external electrodes, corresponding approximately to the permanent charge density on the inner surfaces [41], as a function of peak voltage was studied for three samples and the results are depicted in Fig. 2D. This figure shows that with a peak voltage of 2 kV, a charge density of about 0.4 mC/m2 is obtained [11]. This charge density is typical for space-charge ferroelectrets [39] but is small compared to that of polar ferroelectric polymers like PVDF [35]. These results also indicate that a polarization in the sample really exists. This, of course, is also obvious from the appearance of a piezoelectric activity that will be discussed below.

Fig.2: Contact charging of parallel tunnel FEP films to make them piezoelectric, i. e., to be ferroelectrets. A: Schematic of the contact charging procedure and charge distribution in the parallel tunnel FEP film. B: Triangular voltage applied to the FEP sample. C: Measured hysteresis loops of an FEP sample for applied voltages of different magnitude. D: Measured permanent charge density on the external electrodes as a function of the peak voltage. Mechanical properties of the samples in the length or x-direction of the film (see Fig. 1) were determined with the setups depicted in Figs. 3A(1) and 3A(2). The first setup (“Mode I” in Fig. 3A(1)) applies a static force to the sample directly in the longitudinal direction (x-direction in Fig. 1F) and thus generates an extension in this direction and measures the corresponding strain S1. Typical values are shown in Fig.3B. Here, the stress T1 has been calculated from the applied force and the maximal thickness tmax of the sample. The plot yields the Young’s modulus Y1 = T1/S1. Two values of Y1 are shown in Fig. 3B : Normally, the 5

elastic modulus of polymers is determined at a strain of less than 5% (left box). In order to show the nonlinearity of the mechanical properties of the films at large strains, which may occur in vibration energy harvesting, a second value determined at large strain is also shown in the figure (right box). These two values of Y1 are much smaller than those for the solid FEP, which amounts to 0.48 GPa. As shall be discussed below, the small Y1-value is the reason for the large d31 and g31 coefficients of such samples. One may also determine Y1 from the setup labelled “Mode II” in Fig. 3A(2). In this case, the force is applied in the center of the sample in its z-direction. Since the sample is clamped on its two ends, a force F applied will result in a force F/2sin α in each of the two parts of the sample, as seen in Fig. 3A(2). Since the angle α is kept small (typically < 10°), this yields a large enhancement of the force which is depicted in Fig. 3C. If the force actually applied to the sample in the longitudinal direction (F/sin) is related to the strain in the same direction, Young’s modulus can also be obtained in Mode II. The strain in the longitudinal direction as a function of the actually applied force (F) in modes I and II is shown in Fig. 3D. It is seen that, for the same force F employed externally, a larger strain is obtained in mode II. The benefit of the force enhancement realized in the longitudinal direction of the sample in Mode II will be discussed in the energy harvesters described later, where this boost will result in larger output power (see below).

Fig. 3: Elastic properties of parallel tunnel FEP samples. A(1): Mode I of force application to the samples. A(2): Mode II of force application to the samples. The right-hand part shows the relation between the force F applied in the vertical direction (corresponding to F/2 in each of the two parts of the sample) and the force F/2sin α in the two parts of the sample, showing 6

the force amplification by a factor 1/sin α. B: Measured stress strain curve obtained in Mode I for a parallel tunnel FEP sample, with two typical values of Y1 evaluated from the slopes of the curve. C: Calculated factor 1/sin α of enhanced force applied in the longitudinal direction of the sample in Mode II. D: Measured strain as a function of applied force in modes I and II.

Fig. 4: Dielectric response of the samples. A: Setup for measuring dielectric resonance spectra. B: Complex capacitance spectrum for a circular shaped parallel tunnel FEP sample with a diameter of 20 mm. C: Complex capacitance spectrum for a rectangular shaped parallel tunnel FEP sample with dimensions of 30 mm in the longitudinal direction and 10 mm width. D: Complex capacitance spectrum for a rectangular shaped parallel tunnel FEP sample with dimensions of 30 mm and 10 mm in the longitudinal direction. Dielectric responses of the parallel tunnel samples in the frequency range of 1 to 100 kHz are presented in Fig. 4. The measurements were taken with a Precision Impedance Analyzer Agilent 4294A for three samples having different dimensions. In all cases, the length extension and the thickness extension modes are visible and marked by LE and TE, respectively. Since the ferroelectrets are mechanically free, the two modes occur at a frequency where the length and the thickness are half a wavelength of the acoustic waves 7

propagating in the respective directions [42]. For all samples, the LE mode has a resonance at a frequency f0 in the low kHz range (at 1.8 to 6.5 kHz) which indicates that the sound velocity v1 = λ×f0 for this mode is of the order of 100 m/s. The TE mode at a frequency of approximately 50 to 60 kHz has, considering the sample thickness of about 300 µm, a velocity of only about 30 m/s. The difference of the velocities for the LE and TE modes reflects the mechanical anisotropy of the films. Such an anisotropy was also found before for cellular PP[42], although the velocities in this material are higher due to its larger stiffness of about 1 GPa, as opposed to the small value of 1.1 MPa (see Fig. 3B) for the present samples. The latter value is also in accord with the result obtained from the simple relation Y1 = v2, when the experimental figures for the density of the samples  = 100 kg/m3 and for the velocity v = 100 m/s (see above) are substituted. From Fig. 4 one can also see that the real parts of the capacitances are about 50 to 90 pF, while the imaginary parts are less than 1 pF in the measured frequency range. Thus, the imaginary parts are almost two orders up to three orders of magnitude lower than the real parts. Therefore, the losses are relatively small. This is partly due to the extremely low conductivity of the FEP.

3. Piezoelectric d and g coefficients 3.1 The coefficients d31 and g31 for parallel-tunnel samples Knowledge of the piezoelectric coefficients of the parallel-tunnel samples is of importance to assess the suitability of these samples for various applications. In general, the longitudinal and the transverse effects are of interest, expressed by the corresponding coefficients d33, g33 and d31, g31, respectively, as introduced in Sect. 1. Other coefficients have also been specified [43], but are of no interest in the present study. The piezoelectric d- and g-coefficients are defined as [43]: d31 = (D3/T1)E = (S1/E3)T,

(1a)

g31 = -(E3/T1)D = (S1/D3)T,

(1b)

where D3, T1, S1, and E3 are the incremental quantities of electric displacement, mechanical stress, mechanical strain, and electric field strength, respectively, referring to the piezoelectric activity of the material. The indices indicate the directions defined in Fig. 1F and in its caption. The quantities following the parentheses have to be kept constant when 8

performing the operation in the parenthesis. As usual, in these equations the initial conditions of D3, T1, S1, and E3 are not considered, i. e. we will ignore pre-existing values of these quantities and take into account only the small incremental quantities referring to the piezoelectric activity. Let’s also assume that the sample dimensions in the 1-, 2-, and 3-directions are L, w, and t, respectively, where L and w are constant for a given sample and t is dependent on location on the film, because of the wavy geometry of the parallel-tunnel samples. By writing the quantities D3, T1, S1, and E3 in terms of electrode charge Q, force F1, length extensionl, and voltage V3, one has D3 = Q/Lw, T1 = F1/wt, S1 = l/L, E3 = V3/t. With Eqs. (1a) and (1b), this yields (2a) (2b) The problem with the definition of d31 in Eq. (2a) is that this quantity depends on the thickness t of the film and thus is dependent on location on the film since the film has a wavy geometry. The expressions for d31 thus change along the 1-axis periodically with the sample thickness. This is different for g31 in Eq. (2b) which is independent of t. This means that it is advantageous to use g31 instead of d31 for the description of the transverse piezoelectric activity of the parallel-tunnel samples and for their use in energy harvesters (see also last paragraph in Sect. 5). Thus only measurements of g31 will be discussed in the following. The importance of using the g- instead of the d-coefficient for samples of non-constant thickness has, to the knowledge of the authors, not been realized before. 3.2 Measurement of g31 of parallel-tunnel samples and experimental results Measurements of g31 are possible with the expressions of Eq. (2b). By using the first expression of this equation, the experiments are performed as follows: One applies a force F1 to the sample and measures the resulting voltage V3 in open circuit (D3 = 0). One now 9

obtains g31 from the first part of Eq. (2b). For applications of ferroelectrets, the samples may be terminated by other impedances. The coefficient g31 may also be determined from the second part of Eq. (2b). In this case a charge Q is put on the large surface (or on the electrode) of the sample and then the resulting displacement l is measured in a mechanically free-running system (T1 = 0). This requires an interferometric measurement and is therefore more difficult to perform than the other experiment to determine g31.

Fig.5: Schematic of measuring setups for determining piezoelectric g31 and d33 coefficients in parallel tunnel FEP films and experimental results. A: Setup for measuring static g31 10

coefficients. B: Output voltage in open circuit as a function of the duration of the applied force. C: Setup for measuring dynamic g31 coefficients. D: Results on the piezoelectric g31 coefficient as a function of frequency. E: Setup for measuring dynamic d33 coefficients in parallel tunnel FEP films. F: Piezoelectric d33 coefficient as a function of frequency. Experimental setups for static and dynamic measurements of g31 according to the first part of Eq. (2b) are shown in Figs. 5A and 5C, respectively. In the static experiments, a static force is applied in the length direction of the sample and the open circuit voltage caused by the transverse piezoelectric effect is measured with an electrometer (Keithley 6514). A typical result is depicted in Fig. 5B. Due to the viscoelastic properties of the film there is some creep in this signal. Evaluation after 10 s yields g31 = 3.0 Vm/N which is a rather high value for this coefficient. Results of the dynamic measurement of another sample, performed with a vibration exciter as illustrated in Fig. 5C, are shown in Fig. 5 D. Values of 0.8 to 0.5 Vm/N in the frequency range from 5 Hz to 100 Hz, which is of interest for energy harvesting, were obtained. The drop of the g31 coefficient with increasing frequency is again due to the viscoelastic properties of the film, i.e. to the increase of the stiffness with frequency. For well poled PVDF, smaller values for g31 of about 0.2 Vm/N are obtained in the literature [44]. Depending on the application, knowledge of the g31 coefficient in different frequency ranges is important, reaching from the static to the dynamic, high-frequency domains. Some applications, such as pressure sensors, require knowledge of the former coefficients, while other applications, for example energy harvesters or sonic and ultrasonic transducers, necessitate information about these coefficients at higher frequencies. Lastly, the measurement of d33, which was described previously for similar fluorocarbon ferroelectrets [13], is depicted in Fig. 5E and results are shown in part F of the figure. The values, obtained from d33 = Q/F, where Q is the charge from, and F is the force applied to, the sample are rather high since only the elevated (and soft) parts of the tunnels, which can be compressed more readily, are contributing [13]. The above values of the piezoelectric coefficients for the parallel tunnel samples are compared in Table I with those of other common piezoelectric materials, namely PZT-5H [45] and PVDF [44]. Also listed in the Table are d33 and g33 values for ferroelectrets from other studies, as indicated. In addition, a Figure of Merit (FoM), defined as d33×g33 or d31×g31 [19] [46], was calculated and is also shown in the Table. The data in the Table will be further discussed in Sect. 7.

11

Table I: Piezoelectric coefficients of some materials. Literature values are in the audio and/or ultrasonic ranges [41, 42, 44, 45]. Values from this work are either static values or pertain to frequencies up to 100 Hz.

Fo d33 Material

g33

[pC/ [Vm N]

/N]

M d33 g33 [TP

Fo d31

g31

[pC/ [Vm N]

/N]

-1

PVDF

640

-33

0.02 13.4 1 0.33

PP ferroelectr

140

13*

et

4 -10. 89 182 0

[TP -1

a ] PZT-5H

M d31 Ref g31 . a ]

-283

23

2

-0.00 93 0.21 6 0.2

2.6

5.0

[45 ] [44 ]

0.

[42

4

]

Fluoropoly mer ferroelectr

350

30*

105 00

et with 12

-

-

-

[41 ]

two-dimen sional void structure -

-

-

Fluoropoly mer parallel-tu nnel

130 0

122*

-

ferroelectr et

(10 - 35 Hz)

32*

3.0

96

* (static)

thi

0.5 -

s

0.8

wo

(10 -

rk

100 Hz) /𝜀0 𝜀𝑟 , taking relative

* Calculated g33from permittivity of 1.2

𝜀0 𝜀𝑟 , taking relative

** Calculated d31from permittivity of 1.2

13

4. FENG utilizing the transverse piezoelectric activity The design of the present FENG is schematically shown in Fig. 6A, some implementations are illustrated in Fig. 6B, and the schematic of the working principle of the FENG is indicated in Fig. 6C. The channels in the ferroelectrets in part C (and also in part A) of the figure extend in the y- (or 1-) direction and are therefore perpendicular to the plane of the figure. Thus Fig 6C shows a cross section of the channels. In the center of Fig. 6C, the sinusoidal deflection of the ferroelectret sample (dashed line) and the current through RL (solid line) over one cycle are shown as a function of time. The current is generated by the periodic stretching of the channels perpendicular to their length: Due to this stretching, the moments of the dipoles formed by the charges inside the channels are periodically changed. This causes changes of the induction charges on the outside electrodes, as seen in the figure, which results in the current I through RL. The device has the shape of a cuboid, as seen in Part B of the figure. This harvester is similar to a recently used device [11], but differs from this in the following important aspects: a. The present harvester is miniaturized. Its volume is reduced by a factor of 40 and its active area (ferroelectret area) by a factor of 12. The dimensions are as low as 3×5×8 mm3. b. The seismic mass has been lowered significantly, from 2 g down to as low as 0.09 g. c. The angle α between the ferroelectret and the horizontal (see Fig. 6A) has also been reduced to values of a few degrees (see below). For characterizing the harvester, it is typically mounted on a shaker table such that it can be excited to vibrations. An accelerometer, placed between the shaker and the harvester, is used to measure the input acceleration of the device.

14

Fig.6 A: Schematic of experimental setup consisting of FENG and shaker. The ferroelectret film is fixed at both ends on the supporting structure, statically deflected slightly by a seismic mass, and dynamically deflected by this mass in response to the acceleration provided by the electrodynamic shaker. B: Photographs of the FENG. C: Schematic of working principle of FENG. The deflections of the ferroelectret are caused by the acceleration of the seismic mass and result in the output current I (see text).

5. Theory of the g31 energy harvester The stiffness k of a ferroelectret strip film in the direction of its length may be calculated from its Young’s modulus Y, cross sectional area A = t w and length L as k

YA Y t w .  L L

(3)

Young’s modulus Y can be expressed for a small deflection as

Y

F/A FL  , l /L ltw

(4) 15

where F is the force applied to the strip in the direction of its length (1-direction) and l is the dynamic deflection. Since Y depends on t it cannot be defined unambiguously for a ferroelectret strip whose thickness is not constant over the surface. That is why the product Y t, which is independent of t (see Eq. (4)) and thus clearly defined, will be occasionally used instead of Y. As has been pointed out in Sect. 3.1, a clear definition of the piezoelectric charge constant d31 is also not possible since it likewise depends on t (see Eq. (2a)). The piezoelectric voltage constant g31, given by Eq. (2b), however, doesn’t depend on t and thus can be used instead of d31. In a first step, it is necessary to introduce the resonance frequency   and damping ratio

  . The resonance frequency  for α = 90°, when all forces are applied in the direction of the length of the ferroelectret strip, is obtained from its stiffness k and the seismic mass ms as

 

k . ms

(5)

The damping ratio   for α = 90° is calculated from ms, k, and from the damping coefficient c of the ferroelectret film in the direction of its length as

 

c 2 ms k

.

(6)

It will be shown that the resonance frequency ω0 and the damping ratio ζ of the FENG for any angle α can be expressed as a function of respectively   and   , and α. Just as for a FENG with d33 effect, the power generated by a harvester with g31 effect can be derived from its equation of motion. However, since the forces are applied along the two ferroelectret strips which are generally not parallel to each other (s. Fig. 6), it is helpful to write this equation as a vector relation [47]       ms (z  2     (l1  l 2 )   2 (l1  l 2 ))  ms a ,

(7)

  where l1  l1, z zˆ  l1, x xˆ and l 2  l 2, z zˆ  l 2, x xˆ are the respective deflections of both ferroelectret strip films. 16

The z axis is defined as the direction of acceleration and the x axis is the perpendicular axis (s. Fig. 6). For the projection on the z axis, since l1, z  l 2, z  l sin  and l  z sin  , it follows from Eq. (7): ms (z  4    sin 2  z  2 2 sin 2  z )  ms a .

(8)

Due to the symmetry of the construction, the projection of l1 and l2 on the x axis are of opposite sign. Thus, it can be easily shown that the sum of the contributions along the x axis is equal to 0. In the case of a harmonic motion, the ratio z/a can be read from Eq. (8) as

z 1  2 . a   4 j      sin 2   2  2 sin 2 

(9)

Expressing the resonance frequency ω0 of the harvester as function of   and sin α as

0  2  sin 

(10)

and the damping ratio ζ

  2   sin  ,

(11)

z/a follows from Eq. (9)

z  a

1 /( 2  2 sin  ) 2

      j 2 2   sin  1    2   sin   2   sin  



1 /  02 2

   1     2 j  0  0 

.

(12)

Then, using the relation between generated voltage and applied force with the direct piezoelectric g31 effect, as well as Eq. (3) and (4) for the relation between F, l, and the mechanical properties of the film strip, the voltage Voc generated in open circuit by the FENG for the film strip extension l reads Voc  g 31

g Y lt w g Y lt F .  2 31  2 31 w w L L

(13)

With l  z sin  , the voltage generated in open circuit in response to the input acceleration a can be expressed from Eq. (12) and (13), and replacing Y according to Eq. (3) and (4), as 17

Voc  a

2

g 31 Y t sin  L 02

 1    0

2

    2 j  0 



g 31 ms w sin   1    0

2

    2 j  0 

.

(14)

Considering that a ferroelectret film can be represented by an ideal voltage source connected in series with its internal capacitance C, the RMS value of the voltage VR,rms across a finite load resistance Rl may be derived from Eq. (14) using Kirchhoff’s law as

VR ,rms a

 Voc,rms

Rl C 

Rl C  1  Rl C  

2



     1      0 

  

2

2

ms g 31 w sin 

   4  2      0 

  

2

  2  1  Rl C    



.

(15)



The power PR generated in a load resistance Rl is then calculated using the relation PR  VR2 / Rl as

 g C ms   Rl  31  PR w sin    .  2 2 2 a2        2 2      1       4      1  Rl C     0    0    2



(16)



This expression corresponds to that of a FENG based on the d33 effect if d33 is replaced with

g 31C [48]. This coefficient might be described as an “equivalent d33 constant” of the w sin  ferroelectret configuration used in the present FENG in analogy to ferroelectret harvesters with d33 effect. The power Popt generated in the optimal load resistance Ropt = 1/C ω0 is obtained by replacing Rl with the value of Ropt in Eq. (16). For the acceleration a, Popt is equal to Popt a2



2 2 g 31 C ms2 0 g 31 C ms2   .  8 w 2  2 sin 2  8 2 w 2  2 sin 3 

(17)

Per Eq. (17), Popt is proportional to 1/sin³α [11]. dependence of

This dependence is explained partly by the

g 31C on α and partly by the proportionality of ζ to sin α (see Eq. (11)). w sin  18

Thus, to optimize the generated power, the energy harvester should be constructed with an angle α as small as possible. Since a small angle increases the “equivalent d33 constant” and decreases the damping ratio, it results in a large increase of the generated power, especially in the region of the resonance frequency. However, for very small angles α, the increased mechanical tension on the ferroelectret film (see Fig. 3C) is likely to alter its mechanical properties, thus limiting the applicability of the model. It should also be noted that, due to the dependence of PR and Popt in Eqs. (16) and (17) on g31, and not on d31, these P-values are defined independently of the variations of the film strip’s thickness t along the length of the strip. Using g31, instead of d31, it is possible to compare the performance of the FENG presented in this paper with that of energy harvesters of similar design and based on other piezoelectric polymers such as PVDF. Thus, the performance of parallel-tunnel-FEP-ferroelectrets in FENG’s may be better assessed (see also end of Sect. 3.1).

6. Experimental results for the g31 FENG Several measurements have been performed with harvesters designed according to Fig. 6. First, the dependence of the resonance frequency on sin α was determined. Since the measurement of α for small α’s is difficult, the angle was, specifically for these measurements, determined in a slightly modified setup shown in Fig. 7A. The modification consists in a movable sidewall of the fixture. Compared to a stretched ferroelectret (α = 0), one obtains a ferroelectret with an angle α by sliding the movable part by an amount L to the left. Under the assumption that the seismic mass just keeps the sample taut but does not elongate it, the angle α is given by √





 √



(18)

The approximation term holds for small α’s and is inaccurate by less than 3.4 percent for sin α smaller than 0.5. The value of sin α resulting from the displacement L is plotted in Fig. 7B. Experimental results for the resonance frequency f0 ( 0 as function of sin α are 0/ plotted in Fig. 7C. The experimental data in the figure may be compared with the expected dependence as obtained from Eqs. (3), (5), and (10):

ω0= √

sin α.

(19) 19

This calculated dependence is also shown in the figure, using Yt = 60 N/m, w = 10 mm, L = 30 mm, and ms = 0.16 g. According to Eq. (19), the measured resonance frequency is expected to increase proportionally to sin α. The actually observed rise of the resonance frequency is not quite as steep as predicted. However, the agreement of the experimental with the calculated data is satisfactory, considering the uncertainties with respect to determining  with the above-described method.

20

Fig.7 A: Schematic of the setup used in measurements of the dependence of resonance frequency on angle α in energy harvesters. B: Calculated values of sinα as a function of the displacement L of the movable part in the setup. C: Measured dependence of resonance frequency on sin α. The angles α were determined by the displacement of the movable part of the fixture with Eq. (18). The normalized power is defined as 2

g Pn  P  , a

(20)

where P is the power generated for the acceleration a and g is the gravity of earth. The power Pn generated by two g31 FENG’s with different-size ferroelectrets and different seismic masses as function of frequency is plotted in Figs. 8 A and B. These figures show the customary resonance peaks with relatively large quality-values of  25, corresponding to damping ratios of 0.02. The harvested power is 57 µW and 109 µW for the seismic masses of 0.09 g and 0.3 g, respectively. As will be discussed below, these are very high power outputs, considering the small seismic masses. The resonance frequencies are in the preferred range below 100 Hz [48], where ambient vibrations are often present [49].

21

Fig. 8: A and B: Measured normalized power (referred to an acceleration of 1 g = 9.81 cm/s2), generated by two g31 energy harvesters having different parameters. The samples were excited by a shaker in the frequency range of 30 to 150 Hz. C: Setup of energy harvester to power LED. A photograph is shown on the right. To demonstrate the harvested power, the FENG was used together with a rectifier bridge to power a blue light-emitting diode (LED), as shown in Fig. 8 C, left. A photograph of the setup is seen in the right-side part of the figure. An acceleration of the harvester setup of 0.3 g is sufficient to power the LED.

7. Summary and Discussion. In this paper the development of ferroelectrets with a parallel tunnel structure, having strong transverse piezoelectric activity and thus a large g31 coefficient, and their use in an efficient energy harvester is discussed. The ferroelectret samples were made of FEP films fused together such that a regular arrangement of tunnels is created between the films. 22

After charging, the samples show electro-activity characterized by a piezoelectric voltage constant g31 that assumes static values up to 3.0 Vm/N and dynamic values of 0.5 to 0.8 Vm/N at frequencies up to 100 Hz, which is the important frequency range for energy harvesting. These values are compared with those for other piezoelectric substances in Table I. The high values of the longitudinal voltage coefficient g33 of the ferroelectrets, have been realized recently [50] [51]. The reason for these high values is easily recognized from the relationship g33 = d33/ε as being due the small permittivity of only slightly higher than 1 of this class of piezoelectric materials. The high transverse g31 coefficient of the parallel-tunnel samples, however, is reported here for the first time, although there were already some indications from the energy harvesting data published in 2016 [11]. Due to the large values of the piezoelectric d and g coefficients, the parallel-tunnel samples also yield high Figures of Merit (FoM) d33×g33 and d31×g31 (see Table I). Particularly the high transverse coefficient may be of use in other applications too, such as transmit-receive systems with sonic and ultrasonic transducers [46] and pressure sensors. The performance of the new FENG’s is illustrated in Fig. 8. For seismic masses of 0.09 and 0.3 g, normalized peak power outputs of 57 and 109 µW, respectively, were achieved. These values may be used to determine the angle α, which is too small to be measured with the setup in Fig. 7C. If one calculates α from the first part of Eq. (17), substituting the experimental values a = 9.81 m/s2, g31 = 0.7 Vm/N, C = 7.0 or 8.8 pF, ms = 0.09 or 0.3 g, ω0 = 58 or 57 Hz, w = 5 or 10 mm, Ϛ = 0.015 (taken from the quality values of typical resonance curves) for the two peak powers, one obtains α-values of about 1 to 2°. Such values are consistent with a visual inspection of the harvesters. Next, we will look at the progress that has been made with our FENG’s in recent years. This is illustrated in Table II where various ferroelectret harvesters are compared among themselves. Here, not only the normalization with respect to acceleration, defined in Eq. (29), is carried out but also a normalization with respect to seismic mass. From Eq. (17), considering the dependence of



on ms, one finds that the power output is

proportional to m3/2. This is the mass dependence that has been found in several studies (see, e. g., Ref. [51], Fig. 6 and also other references). An additional mass dependence through the factor Ϛ2 in Eq. (17) was not found in these studies. This question is still under investigation and we will use the m3/2 dependence for now. The “twice normalized” power in the Table is used to examine the progress that has been achieved with these harvesters.

23

Table II. Recent progress in FENG’s [11, 28, 50, 52]: Comparison among various of our implementations. The normalized power is referred to an acceleration of 1 g = 9.81 m/s2 and a seismic mass of 1 g. The adjustment is made by taking into consideration that the power output is proportional to a2 and m3/2. We call these power values therefore “twice normalized”.

Type of Harvester Normalized [Reference]

Resonance

Size of

Power

Frequency

Active

(W)

(Hz)

Area 2

(cm ) Electret FEP

[52]

Ferroelectret g33

20

35

4

0.6

400

3

0.16

130

7

70

30

3

2000

58

0.4

IXPP [28] Ferroelectret g33 cross-tunnel FEP [50] Ferroelectret g31 parallel-tunnel FEP [11] Ferroelectret g31 parallel-tunnel FEP, advanced design (ms=0.09 g) [this paper, Fig. 24

8A]

The Table shows that the twice normalized power of the new ferroelectret energy harvesters with transverse activity (g31) is much larger than that of harvesters with longitudinal (g33) effect. The reason for this observation is not only the large g31 coefficient but also the force enhancement discussed in Figs. 3C and 3D. The fact that ferroelectret materials with large g31 coefficient and thus large power output in energy harvesters are available now suggests to further explore intensively the use of these materials in this field. The hope is that even better arrangements using the transverse piezoelectric effect can be found in this very new and still relatively unexplored field. Apart from the comparison of FENG’s among themselves, a comparison of these devices with other types of energy harvesters is of interest. For such an assessment, the FoM d×g discussed in Table 1 above, has been suggested [19, 46]. However, for a more comprehensive characterization, other properties have to be considered in addition to d31 and g31. For example, power output should be “normalized” by such important parameters as applied acceleration, location and bandwidth of the resonance frequency, size of the active part of the harvester, and seismic mass, to name just a few of the most essential properties. Correspondingly, a number of FoM’s have been defined that take into consideration some of these parameters [48, 53, 54]. One such FoM, recently evaluated for “small scale

25

piezoelectric energy harvesters”, is the “Normalized Volumetric Power Density” (NVPD) PN, defined as [48] (21) where P is the harvested power in µW, v the volume of the piezoelectric material in mm3 (“active volume”), f0 the resonant frequency in Hz, and a the acceleration in terms of g (9.81 m/s2). This may be applied to our two FENG’s: Using P = 57 or 109 µW, v = 8 or 16 mm3 (assuming an average sample thickness of 0.2 mm), and f0 =58 or 57 Hz, one obtains for both devices described in this paper. This performance may be compared with that of the 14 “small scale” piezoelectric energy harvesters utilizing the ceramic materials PZT or AlN as active materials in devices based on the longitudinal or transverse piezoelectric effects as described in [48]. These harvesters have active volumes of the piezoelectric material between 0.02 and 464 mm3, as compared to the 8 or 16 mm3 for our harvesters. The PN-values of these 14 harvesters are between 1.9×10-1 and 9×10-5. Only one of them has a larger PN than the harvesters described in the present paper. Thus, we now have a strong and rapidly developing competitor to the ceramic piezoelectric materials in energy harvesting. This is even more significant because the polymer ferroelectrets have definite advantages, such as low cost, environmental suitability, mechanical flexibility, and better impedance matching to the vibrational sources. Considering the fact that the g33 ferroelectret harvesters have only been around since 2012 and the g31 FENG’s since 2016, while harvesters with the conventional piezoelectric materials have been discussed for more than 20 years [53, 55], the development of the new devices is very promising. Since it is reasonable to assume that the rapid development of the ferroelectret materials and the associated harvesters will continue for some time, their competitiveness should even increase. This ought to help the ferroelectrets to find their place in the energy- harvesting field.

Acknowledgments The authors are grateful to Prof. Siegfried Bauer from the University of Linz for stimulating comments on the paper. Financial support from the Deutsche Forschungsgemeinschaft (DFG 26

392020380) and from the Natural Science Foundation of China (NSFC 11374232 and 61761136004) is also gratefully acknowledged.

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30

Xiaoqing Zhang received the Ph.D. degree from the Tongji University, China, in 2001. From 2001 to 2004 she worked as a post doctoral at the Darmstadt University of Technology, Germany. She joined in the Tongji University and started to lead the Group of Electrets and Functional Dielectrics there in 2005. In 2010, she was appointed full professor at the School of Physics Science and Engineering, Tongji University. She is a committee member of the IEEE International Symposium on Electrets and the Chinese National Conference on Electrets (CNCE). Her current research interests are electro-active polymers, flexible transducers and energy harvesters based on ferroelectrets and electrets. Perceval Pondrom was born in Neuilly-sur-Seine, France in 1983. He received the Diplôme d’Ingénieur from Institut Supérieur d’Électronique de Paris, France and the Diplom from Technische Universität Darmstadt, Germany in 2007. He worked as research engineer at the company MED-EL (Innsbruck, Austria) from 2007 to 2010. From 2010 to 2016 he worked on the field of piezoelectret sensor applications at the Institute for Telecommunications Engineering and the research group System Reliability and Machine Acoustics SzM of the Technische Universität Darmstadt, Germany. Since 2016 he has done volunteer work in the Centre of Art, Culture and Spirituality Kalahrdaya in Calcutta, India. Gerhard M. Sessler is a graduate of the University of Goettingen in Germany. He joined Bell Laboratories in Murray Hill, N.J. and was co-inventor of the first polymer electret microphone which became the predominant microphone type. In 1975, he was appointed a professor at the University of Technology in Darmstadt, Germany and was involved in MEMS technology. In 1983, he and his co-workers designed the first MEMS condenser microphones which are now the leading microphone types worldwide. Presently, he is an Emeritus Professor and he and his group are working on the new ferroelectret materials and their applications. Xingchen Ma received the bachelor’s degree in physics from the Suzhou University of Science and Technology, China, in 2015. She is currently pursuing the Ph.D. degree with the School of Physics Science and Engineering, Tongji University. Her research interests include flexible piezoelectret film-transducers and green micro-energy harvesters based on electret films.

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Figure captions Fig.1: Schematic of preparation of laminated parallel tunnel FEP films. A is the template and materials used in this procedure. B is the composition of a stack used for the preparation. C, D, and E are the steps of hot pressing at an elevated temperature of 100 oC, removal of the soft pad and fusion bonding above the melting point of FEP, respectively. F is a schematic view of the completed parallel tunnel film and shows also two SEM images of fabricated parallel tunnel FEP film. The x, y, and z-coordinates are also shown in the Figure, corresponding to the crystallographic 1, 2, and 3-directions, respectively. Fig.2: Contact charging of parallel tunnel FEP films to make them piezoelectric, i. e., to be ferroelectrets. A: Schematic of the contact charging procedure and charge distribution in the parallel tunnel FEP film. B: Triangular voltage applied to the FEP sample. C: Measured hysteresis loops of an FEP sample for applied voltages of different magnitude. D: Measured permanent charge density on the external electrodes as a function of the peak voltage. Fig. 3: Elastic properties of parallel tunnel FEP samples. A(1): Mode I of force application to the samples. A(2): Mode II of force application to the samples. The right-hand part shows the relation between the force F applied in the vertical direction (corresponding to F/2 in each of the two parts of the sample) and the force F/2sin α in the two parts of the sample, showing the force amplification by a factor 1/sin α. B: Measured stress strain curve obtained in Mode I for a parallel tunnel FEP sample, with two typical values of Y1 evaluated from the slopes of the curve. C: Calculated factor 1/sin α of enhanced force applied in the longitudinal direction of the sample in Mode II. D: Measured strain as a function of applied force in modes I and II. Fig. 4: Dielectric response of the samples. A: Setup for measuring dielectric resonance spectra. B: Complex capacitance spectrum for a circular shaped parallel tunnel FEP sample with a diameter of 20 mm. C: Complex capacitance spectrum for a rectangular shaped parallel tunnel FEP sample with dimensions of 30 mm in the longitudinal direction and 10 mm width. D: Complex capacitance spectrum for a rectangular shaped parallel tunnel FEP sample with dimensions of 30 mm and 10 mm in the longitudinal direction. Fig.5: Schematic of measuring setups for determining piezoelectric g31 and d33 coefficients in parallel tunnel FEP films and experimental results. A: Setup for measuring static g31 coefficients. B: Output voltage in open circuit as a function of the duration of the applied force. C: Setup for measuring dynamic g31 coefficients. D: Results on the piezoelectric g31 coefficient as a function of frequency. E: Setup for measuring dynamic d33 coefficients in parallel tunnel FEP films. F: Piezoelectric d33 coefficient as a function of frequency. 32

Fig.6 A: Schematic of experimental setup consisting of FENG and shaker. The ferroelectret film is fixed at both ends on the supporting structure, statically deflected slightly by a seismic mass, and dynamically deflected by this mass in response to the acceleration provided by the electrodynamic shaker. B: Photographs of the FENG. C: Schematic of working principle of FENG. The deflections of the ferroelectret are caused by the acceleration of the seismic mass and result in the output current I (see text). Fig.7 A: Schematic of the setup used in measurements of the dependence of resonance frequency on angle α in energy harvesters. B: Calculated values of sinα as a function of the displacement L of the movable part in the setup. C: Measured dependence of resonance frequency on sin α. The angles α were determined by the displacement of the movable part of the fixture with Eq. (18). Fig. 8: A and B: Measured normalized power (referred to an acceleration of 1 g = 9.81 cm/s2), generated by two g31 energy harvesters having different parameters. The samples were excited by a shaker in the frequency range of 30 to150 Hz. C: Setup of energy harvester to power LED. A photograph is shown on the right.

Tables Table I: Piezoelectric coefficients of some materials. Literature values are in the audio and/or ultrasonic ranges [41, 42, 44, 45]. Values from this work are either static values or pertain to frequencies up to 100 Hz. Table II. Recent progress in FENG’s [11, 28, 50, 52]: Comparison among various of our implementations. The normalized power is referred to an acceleration of 1 g = 9.81 m/s2 and a seismic mass of 1 g. The adjustment is made by taking into consideration that the power output is proportional to a2 and m3/2. We call these power values therefore “twice normalized”.

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Research Highlights 1. Layered fluorocarbon ferroelectrets with soft, reproducible structure are described 2. They exhibit transverse piezoelectricity with large, thermally stable coefficients 3. Using these, a force-amplifying Ferroelectret Nanogenerator (FENG) is designed 4. The FENG converts efficiently vibrational to electric power with large output

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Graphical abstract

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