# The Longitudinal and Transverse Piezoelectric Effects of the Ferroelectric Polymer P(VDF-TrFE)

Christine Revenant, Simon Toinet, Eleanor Lawrence Bright, Mohammed Benwadih

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Christine Revenant, Simon Toinet, Eleanor Lawrence Bright, Mohammed Benwadih. The Longitudinal and Transverse Piezoelectric Effects of the Ferroelectric Polymer P(VDF-TrFE). Macromolecular Materials and Engineering, 2025, 310 (5), 10.1002/mame.202400420. [https://doi.org/10.1002/mame.202400420](https://doi.org/10.1002/mame.202400420)

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# The Longitudinal and Transverse Piezoelectric Effects of the Ferroelectric Polymer P(VDF-TrFE)

Christine Revenant,* Simon Toinet, Eleanor Lawrence Bright, and Mohammed Benwadih

Polyvinylidene fluoride-trifluoroethylene (P(VDF-TrFE)) exhibits outstanding electromechanical properties and is a material of choice for non-volatile memories, energy-harvesting systems, multifunctional actuators, and sensors. However, this polymer remains one of the least understood materials among ferroelectric materials due to its semi-crystalline structure with chains folding in elongated grains through the lamellae and amorphous regions. Here, operando high-resolution X-ray diffraction is exploited to unravel the P(VDF-TrFE) structural evolution upon the first electric field application, called poling. This X-ray technique allows revealing drastic changes in the lamellae with the ordered amorphous (OA) emergence between crystalline nanodomains. The discovery of this OA emergence is critical because it significantly affects the structure, as well as the mechanical, ferroelectric, and piezoelectric properties of the polymer. After poling, even small nanostructural changes lead to measurable effects yielding the piezoelectric coefficients. The longitudinal and transverse piezoelectric effects can be explained by the complex P(VDF-TrFE) structure. The elongated grains consist of lamellae separated by isotropic amorphous (IA) regions. This study not only sheds light on the fundamental mechanisms occurring in P(VDF-TrFE), but also offers guidance for new ferroelectric organic materials for flexible, biocompatible sensor and energy applications.

## 1. Introduction

In organic ferroelectrics, like polyvinylidene fluoride-trifluoroethylene (P(VDF-TrFE)), ferroelectric properties

C. Revenant
Univ. Grenoble Alpes
CEA, IRIG-MEM, NRX
Grenoble 38000, France
E-mail: christine.revenant@cea.fr

S. Toinet, M. Benwadih
Univ. Grenoble Alpes
CEA, LITEN DTNM
Grenoble 38000, France
E. Lawrence Bright
European Synchrotron Radiation Facility
Grenoble 38000, France

<figure><img src="image_2.png" /></figure>

ID The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/mame.202400420

© 2025 The Author(s). Macromolecular Materials and Engineering published by Wiley-VCH GmbH. This is an open access article under the terms of the [Creative Commons Attribution License](https://creativecommons.org/licenses/by/4.0/), which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

DOI: 10.1002/mame.202400420

originate from crystalline phases and amorphous regions. The ordered amorphous (OA) domains link the crystalline lamellae to the isotropic amorphous (IA) phase.[1,2] On the contrary, for ceramics, the ferroelectric properties arise from atomic and domain wall displacements.[3-8] Organic ferroelectrics have become competitive alternatives for flexible energy-harvesting systems, multifunctional actuators,[9,10] and sensors.[8,11,12] Indeed, the P(VDF-TrFE) copolymer is chemically stable, easy to process, and biocompatible. The individual electric dipole in P(VDF-TrFE) originates from the electronegativity difference between the H and F atoms bonded to the polymer chain.[13] During poling, the paraelectric-ferroelectric transition occurs and the dipoles become parallel to the applied electric field.[14] Indeed, ferroelectricity includes piezoelectricity, which is the mutual conversion between electrical charge and mechanical strain in a material. The piezoelectric coefficient, which quantifies such conversion, is large for ferroelectric materials. Among the crystalline phases of P(VDF-TrFE), the β phase is ferroelectric, hence piezoelectric and pyroelectric,[15,16] and has an orthorhombic crystal structure.[17] As example for 75 mol % VDF, the lattice parameters are *a* = 8.86 Å, *b* = 4.62 Å, and *c* = 2.55 Å.[17] When the electrical polarization along the *b*-axis is parallel to the electric field, the P(VDF-TrFE) contracts, i.e., has a negative longitudinal piezoelectric effect along this axis. In contrast, the P(VDF-TrFE) expands along the *a*-axis, i.e., has a positive transverse piezoelectric effect along this axis. A derivative of the P(VDF-TrFE) copolymer is the P(VDF-TrFE-CTFE) terpolymer,[18] which is a relaxor ferroelectric, characterized by smaller crystalline domains and consequently a larger proportion of crystalline-amorphous interfaces. There is a debate whether, at the low temperature ferroelectric state, the macroscopic piezoelectric coefficient in P(VDF-TrFE) is identical to the nanoscopic one in crystalline lamellae or if an additional contribution from the OA regions occurs,[19,20] questioning the role of such regions. Despite intensive research, the origin and role of the OA domains in P(VDF-TrFE) are still unclear and challenging to elucidate due to the complex semi-crystalline structure. It consists of chains folding in elongated grains through the crystalline lamellae and amorphous regions.[19] Moreover, the differences between the longitudinal and transverse piezoelectric effects have so far not been explained on a fundamental point of view. The piezoelectric
properties of the PVDF-based devices are poorly understood, especially near the coercive field. Therefore, advanced structural investigations are necessary to correlate the structural evolution with the ferro- and piezo-electric properties as a function of the applied electric field.

Here, we investigate P(VDF-TrFE) under DC electric field using operando high spatial resolution X-ray diffraction (HRXRD) measurements. In order to distinguish crystalline and OA contributions, HRXRD was measured in a wide range in order to get information about the 3D space. High quality HRXRD was obtained with a challenging transmission synchrotron experiment using a submicrometer X-ray beam in order to get rid of the substrate signal. During poling, the copolymer becomes quasi-amorphous along the polymer chain and at boundaries of the crystalline domains. The OA domains stemming from crystalline domains allow accommodating considerable strain, as well as transitioning from the paraelectric to the ferroelectric state. The longitudinal and transverse structural changes have key implications for the ferro-, piezo-electric and mechanical properties.

## 2. Results

### 2.1. Ferroelectric and Piezoelectric Properties

The curve P(E), i.e., polarization P as a function of the applied electric field E becomes a ferroelectric hysteresis loop near 75 V µm⁻¹ for P(VDF-TrFE) (Figure 1a). At high electric field, the remnant polarization P_r is 7 µC cm⁻², similar to that found in literature and the coercive field E_c is 45 V µm⁻¹, slightly smaller than that found in literature.[19–21]

During poling (Figure 1b), the relative permittivity has a maximum near 75 V µm⁻¹, indicating that the copolymer is more polarizable. A correlation exists between the permittivity maximum and the emergence of the hysteresis loop. After poling, the permittivity maximum shifts to a much lower value of 40 V µm⁻¹. Indeed, a strong link exists between material permittivity and polarization, as permittivity quantifies the amount of energy that can be stored in the form of polarization. At the coercive field, i.e., zero polarization, permittivity is maximum.[10] When polarization occurs, permittivity decreases, since less energy can be stored in the form of polarization.

The electromechanical behavior of PVDF-based devices is quantified by the macroscopic longitudinal d_33 and transverse d_31 piezoelectric coefficients. Their measurements are detailed in Figure S1 (Supporting Information).[22] A d_33 coefficient of -31.9 pC N⁻¹ is obtained and is within the range from -22.3 to -63.5 pC N⁻¹ found in the literature.[20,23,24] This dispersity can be explained by the manufacturing method, the TrFE content, and the frequency.[23,25] Moreover, the d_33 coefficient of PVDF increases with polarization.[26] Besides, the obtained d_31 values vary from 8.95 to 11.5 pC N⁻¹, in agreement with the average value of 11 pC N⁻¹ found in the literature.[27] It has been observed that the d_31 coefficient of a P(VDF-TrFE) capacitor evolves linearly with the electric field, in accordance with the literature.[28] Once the film is sufficiently polarized in order to get the dipole alignment, the piezoelectric coefficients, i.e., strains do not depend on the film thickness.

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>Figure 1. a) Polarization as a function of the applied electric field. The indicated electric field values used for polarization are in V µm⁻¹. b) Relative permittivity as a function of the polarization field during the first and second polarization. Between both, the sample was polarized at -150 V µm⁻¹.</figcaption></figure>

### 2.2. Operando Transmission HRXRD

#### 2.2.1. HRXRD Graphs as a Function of the Electric Field

To provide direct evidence for the OA formation, operando transmission HRXRD (Figure 2a) is carried out on P(VDF-TrFE) capacitors during poling (details in Figure S2, Supporting Information). The HRXRD intensity is plotted versus the momentum transfer q, which is:

$$q = \frac{4\pi}{\lambda} \sin(\theta) \qquad (1)$$

where λ is the X-ray wavelength and θ is the Bragg angle. The HRXRD graphs are plotted as a function of the electric field E used to polarize the device. In order to become ferroelectric, the P(VDF-TrFE) copolymer has to be poled. The evolution of the structure is studied during poling, i.e., as a function of E.

The P(VDF-TrFE) chain along the c-axis can be studied from the wide (001)/(111)/(201)β peak (Figure 2a). As P(VDF-TrFE) copolymer is poled with high electric field E, the (001)β peak disappears and the (111)/(201)β peak decreases significantly. The HRXRD graph of P(VDF-TrFE-CTFE) is shown, as the HRXRD graph of P(VDF-TrFE) becomes similar to the P(VDF-TrFE-CTFE) one in the q range from 2 to 4 Å⁻¹ (Figure 2b). This shows that the P(VDF-TrFE) structure becomes similar to the P(VDF-
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Operando HRXRD measurements a) for P(VDF-TrFE) and b) for P(VDF-TrFE-CTFE). The electric field step is 10 V µm⁻¹. Inset: Integrated intensity of (200)β and (110)β from *q* = 0.8 to 2 Å⁻¹. c) Lattice parameter *c* and domain size *D*<sub>c</sub> of P(VDF-TrFE).</figcaption></figure>
TrFE-CTFE) one in the [001], [111], and [201] directions. Besides, an average crystallite size D is:

$$D = \frac{2\pi}{\Delta q} \qquad (2)$$

where Δq is the Full Width at Half Maximum (FWHM) of the peak. The FWHM of the wide (111)/(201)β copolymer peak yields a very low domain size of 1.5 nm, similar to that of the terpolymer and in agreement with the literature.[29] Upon significant electric field application, chain rotation does not proceed with a long chain, but proceeds only with nanometer-sized crystalline segments, thereby lowering the transformation barriers. The evolution of the P(VDF-TrFE) structure toward an amorphous one enables an easy rotation during polarization at high electric field E similar to what happens for P(VDF-TrFE-CTFE). This is in agreement with the conclusion of Yang et al. to reduce the domain sizes in order to achieve easy rotation of PVDF chains.[30]

The (001)β peak (Figure 2a) is well defined only up to an applied electric field of 50 V µm⁻¹. In X-ray diffraction (XRD), a lattice spacing d is obtained by:

$$d = \frac{2\pi}{q} \qquad (3)$$

where q is the corresponding peak maximum. The maximum of the (001)β peak is at q = 2.524 Å⁻¹, which yields a lattice parameter c of 2.489 Å, as shown in Figure 2c. The FWHM of the (001)β peak yields the size Dc of the crystalline domains along the c direction (≈9–9.5 nm) as displayed in Figure 2c. Above 50 V µm⁻¹, the lattice parameter c is not defined anymore, due to the rotational disordering of the molecular chains.

The main HRXRD peak corresponds to the (110)/(200) diffraction of the β phase. Without any electric field, this peak is very high, as it is the sum of the (110) and (200) β peaks. The individual electric dipole in P(VDF-TrFE) originates from the electronegativity difference between the H and F atoms bonded to the C-C chain. More precisely, the electronegativity of H is 2.20, whereas F is the most electronegative element with an electronegativity of 3.98. The b-axis is parallel to the CH₂-CF₂ and CHF-CF₂ electric dipoles and the c-axis is parallel to the C-C main chain. As the applied electric field increases, the electric dipole tries to become parallel to the applied electric field. This can occur by a rotation of the C-C chain. As the electric field is applied vertically between the electrodes, the electric dipoles, i.e., the b-axis, tend to orient vertically. Therefore, the normal to the (200) planes, i.e., the a-axis, lies in the horizontal plane. The d-spacing between the (200) planes is parallel to the a-axis and evolves according to the d₃₁ piezoelectric coefficient. When an electric field is applied, the d-spacing between the (200) planes increases as d₃₁ is positive. In reciprocal space, the (200) peak shifts toward smaller q positions. The integration profile of the (110) and (200) peaks is displayed in Figure 2a,b for the copolymer and terpolymer. The intensity is maximum for P(VDF-TrFE) at E = 30 V µm⁻¹, indicating better crystallized β phase. Above E = 40 V µm⁻¹, the intensity slightly decreases, similarly to that of the terpolymer, indicating less crystallinity and more amorphous domains.

### 2.2.2. The (200)/(110) Peaks as a Function of the Electric Field

As we strive to characterize crystalline and OA domains in different directions, it is of the uttermost importance to fit precisely the experimental HRXRD profile with adequate contributions (Figure 3).

The (200)/(110) peak is fitted by Lorentzian profiles, which are well appropriate for fitting wide-angle X-ray scattering curves.[31] Up to 30 V µm⁻¹, the peak is fitted solely by crystalline contributions of (200) and (110) planes, whereas above 40 V µm⁻¹, OA contributions of (200) and (110) planes have to be added, unambiguously indicating the OA emergence. Figure 4 displays the intensity of the different contributions. The sum of (200) and (110) peak intensity corresponds to the crystalline contribution, which decreases above 40 V µm⁻¹. More precisely, the intensity of the (110) peak decreases, whereas that of the (200) peak slightly increases (Figure 4a). The sum of (200) and (110) OA peak intensity corresponds to the OA contribution and significantly increases. More precisely, the OA (110) peak intensity is approximately constant, whereas the OA (200) peak intensity significantly increases (Figure 4b). Therefore, the crystallinity decrease in the b direction indicates that part of the crystalline lattices become OA in this direction and that the OA domains have ordered (200) planes. Finally, the total intensity (crystalline and OA) slightly decreases above 40 V µm⁻¹ (Figure 4c), in accordance with the previous observation in Figure 2a. Part of the crystalline contribution becomes OA domains, which diffract the X-rays less, explaining the global decrease of the HRXRD intensity.

The lattice parameter a is deduced from the q position of the (200)β peak maximum and varies from ~8.89 to 9.50 Å (Figure 5a) as E varies from 0 to 100 V µm⁻¹. The corresponding strain Δa/a₀, where a₀ is the value at E = 0, is positive (tension), reaches a high value of 5% at only 40 V µm⁻¹, and saturates to 7% at 100 V µm⁻¹ (Figure S3, Supporting Information). The partial transformation of the crystalline contribution into an OA one allows accommodating important tensile strain. Globally, the domain size Dₐ varies from 3.5 to 5.0 nm as E changes from 0 to 100 V µm⁻¹ (Figure 5b). This value is in agreement with the lamella thickness of 5–8 nm found in the literature.[19]

The d-spacing between the (110) planes is determined from the q position of the (110)β peak maximum and is linked to the lattice parameters a and b by the equation:

$$\frac{1}{d^2} = \frac{h^2}{a^2} + \frac{k^2}{b^2} \qquad (4)$$

The lattice parameter b is determined by:

$$b = \frac{k}{\sqrt{\frac{1}{d^2} - \frac{h^2}{a^2}}} \qquad (5)$$

For the (110) peak, h = k = 1 and we get:

$$b = \frac{1}{\sqrt{\frac{1}{d^2} - \frac{1}{a^2}}} \qquad (6)$$

Therefore, the d-spacing between the (110) planes and the lattice parameter a allow determining the lattice parameter b which varies from ≈4.95 to 4.93 Å as E goes from 0 to 100 V µm⁻¹
<figure><img src="image_6.png" /><figcaption>Figure 3. a) HRXRD data fit at 0 V µm⁻¹, b) at 50 V µm⁻¹, c) at 100 V µm⁻¹, d) after poling at 0 V µm⁻¹, i.e., at remnant polarization.</figcaption></figure>

(Figure 5a). Hence, for 80 mol % VDF, the experimental lattice parameters are a = 8.89 Å, b = 4.95 Å, and c = 2.49 Å before poling. These values are similar to those found in the literature.[17] The strain Δb/b₀, where b₀ is the value at E = 0, is negative (compression) and saturates to -0.5% (Figure S3, Supporting Information). By contrast to the large tensile strain in the a direction, over one order of magnitude smaller compressive strain is created in the b direction and the diffracted intensity of the OA contribution is larger for the (200) peak than for the (110) one. This confirms that the OA contribution is a response to large strain accumulation. The domain sizes D₁₁₀ (Figure S4a, Supporting Information) and Dₐ allow determining the domain size Dₛ along b. It is first constant (≈10–11 nm), then decreases near 30 V µm⁻¹, before saturating to 6 nm (Figure 5b). Globally, the domain size Dₛ evolves from 10.7 to 6.1 nm, i.e., a decrease of 40% as E changes from 0 to 100 V µm⁻¹. The large Dₛ decrease cannot be explained by the very small decrease of the lattice parameter b, but is due to smaller (200) planes in order to accommodate strain. At low electric field, the domains have an elliptic section with a ratio Dₛ/Dₐ ~3, the major (resp. minor) axis is along b (resp. a), indicating that the crystalline domains are elongated along the b-axis (Figure S4b, Supporting Information). In contrast, at high electric field, the domains tend toward a circular section with Dₛ/Dₐ ~1.3. The surface of the elliptic section decreases by 17% from 0 to 100 V µm⁻¹ (Figure S4c, Supporting Information). Above 50 V µm⁻¹, the crystalline domains have a much smaller size in the c direction than in the a or b direction,

indicating that the crystalline nanodomains become platelets during poling. The decrease of the domain size and the increase in the crystalline-amorphous interfacial area yield a higher relative permittivity, in agreement with previous work.[32]

Once the lattice parameters a and b have been determined, the nanoscopic piezoelectric coefficients d₃₁ and d₃₃ are obtained from the strain in the a and b direction, respectively, between the end of poling (100 V µm⁻¹) and after poling (0 V µm⁻¹) at the remnant polarization Pᵣ (Figure S5, Supporting Information). More precisely, the strain S₃₁ is:

$$S_{31} = (a - a_r)/a_r \qquad (7)$$

where aᵣ is the value at Pᵣ. This strain allows obtaining the nanoscopic piezoelectric strain d₃₁ by the relation:

$$S_{31} = d_{31} \times E \qquad (8)$$

One gets d₃₁ ~ 9.1 pm V⁻¹ ± 1 pm V⁻¹, which is in the range of the macroscopic one. Similarly, the strain S₃₃ calculated between 100 V µm⁻¹ and 0 V µm⁻¹ is:

$$S_{33} = (b - b_r)/b_r \qquad (9)$$

where bᵣ is the value at Pᵣ. This strain allows obtaining the nanoscopic piezoelectric strain d₃₃ by the relation:

$$S_{33} = d_{33} \times E \qquad (10)$$
<figure><img src="image_7.png" /><figcaption>**Figure 4.** a) Intensity of the fitted peaks for the crystalline (200), (110) peaks and the sum of both. b) Intensity of the fitted peaks for the OA (200), OA (110) peaks and the sum of both. c) Intensity of the fitted peaks for the crystalline contribution, OA contribution, and the sum of both. The indication R stands for the remnant polarization.</figcaption></figure>

One obtains $d_{33} \sim -29.1 \text{ pm V}^{-1} \pm 2 \text{ pm V}^{-1}$, which is in good agreement with the macroscopic one. The macroscopic strain is similar to that of the crystalline nanodomains, in agreement with a work at low temperature ferroelectric state.[19] Interestingly, the $d_{33}$ coefficient is negative and high in absolute value, whereas the $d_{31}$ coefficient is positive and small, indicating different piezoelectric behaviors due to tension and compression.

The average parameter $a_{OA}$, deduced from the maximum of the broad (200) OA peak, is approximately constant ~10.3 Å as a function of E (Figure 5c) and is slightly larger than the lattice parameter a. The position of the (110) OA peak and the parameter $a_{OA}$ allow determining the parameter $b_{OA}$ that is also constant ≈5.0 Å (Figure 5c) and slightly larger than the lattice parameter b. Therefore, the OA domains are less dense than the crystalline ones. The width of the (200) OA peak yields the size $D_{OAa}$ of the OA domains in the a direction. Similarly, the width of the (110) OA peak yields the size $D_{OA110}$ of the OA domains in the [110] direction. The domain sizes $D_{OA110}$ and $D_{OAa}$ allow determining the domain size $D_{OAb}$ in the b direction. The $D_{OAb}$ (resp. $D_{OAb}$)

is approximately constant ≈4 nm (resp. 7 nm) as a function of E (Figure 5d) and is slightly smaller than $D_a$ (resp. slightly larger than $D_b$) at high electric field. The ratio $D_{OAb}/D_{OAA}$ ~1.7 is approximately constant (Figure S4b, Supporting Information).

## 2.3. Structural Model

Figure 6a illustrates the sizes and orientation of the crystalline domains before poling. As $D_b$ decreases during poling, the OA domains probably protrude out of the crystalline domains, decreasing the surface of the (200) planes and lowering the tensile strain between the (200) planes (Figure 6b).

Before poling, a crystalline domain is comprised of $N_{b0}$ unit cells:

$$N_{b0} = \frac{D_{b0}}{b_0} \qquad (11)$$

where the subscript 0 denotes the values at zero electric field. At the end of poling at 100 V µm⁻¹, a crystalline domain is comprised of $N_{b100}$ unit cells in the b direction:

$$N_{b100} = \frac{D_{b100}}{b_{100}} \qquad (12)$$

where the subscript 100 denotes the values at E = 100 V µm⁻¹. Similarly, the OA contribution is comprised of $N_{OAb100}$ unit cells:

$$N_{OAb100} = \frac{D_{OAb100}}{b_{OAb100}} \qquad (13)$$

As the sum of $N_{b100}$ and $N_{OAb100}$ should be equal to $N_{b0}$, we have:

$$\frac{D_{b0}}{b_0} = \frac{D_{b100}}{b_{100}} + \frac{D_{OAb100}}{b_{OAb100}} \qquad (14)$$

Finally, the OA contribution has an average size $D_{OAb100}$:

$$D_{OAb100} = \left( \frac{D_{b0}}{b_0} - \frac{D_{b100}}{b_{100}} \right) \times b_{OAb100} \qquad (15)$$

One obtains $D_{OAb100} = 4.4$ nm, whereas HRXRD puts in evidence larger OA domains with a 6.9 nm size. Therefore, OA contribution may also originate from the IA phase, which may be polarized at high electric field on a 2.5 nm length (Figure 6c). Indeed, a structural change in amorphous domains may occur due to polarization resulting in an improvement of the crystallinity degree.[33]

## 2.4. Domain Dipoles

Once the structure of the crystalline and OA domains has been determined, the study can concentrate on the domain dipole, which is the sum of the individual dipoles in a domain. Upon positive electric field application, the domain dipoles align along the upward direction, as depicted in Figure 7a. Parallel (resp. perpendicular) to the surface, the electrostatic repulsion
<figure><img src="image_8.png" /><figcaption>Figure 5. a) Lattice parameters in the crystalline domains. b) Crystalline domain sizes. c) Average parameters in the OA domains. d) OA domain sizes. The indication R stands for the remnant polarization.</figcaption></figure>

(resp. attraction) among the positively aligned domains leads to an elongation (resp. contraction). The observed increase of $d_{31}$ (resp. $d_{33}$ in absolute value) with $E$ can be explained by a better orientation of the dipoles parallel to $E$ leading to an increase of the repulsion parallel to the surface and of the contraction perpendicular to the surface.

In addition, IA regions exist between the lamellae formed by the crystalline and OA domains. The average size $D_{IAa}$ of these regions in the $a$ direction can be estimated using the interaction forces between dipoles. First, two dipoles with parallel moments and placed at a distance $D$ have an interaction force $F_{31}$ (Figure S6a, Supporting Information):

$$F_{31} = \frac{3p_1p_2}{4\pi\epsilon D_{31}^4} \qquad (16)$$

with $p_1$ and $p_2$ the dipole moments and $\epsilon$ the permittivity (details in SI). Moreover, two dipole moments along the dipole axis have an interaction force $F_{33}$ (Figure S6b, Supporting Information):

$$F_{33} = -\frac{6p_1p_2}{4\pi\epsilon D_{33}^4} \qquad (17)$$

For the same distance $D$, $F_{33}$ is twice larger than $F_{31}$, resulting in a larger electromechanical effect. This explains partially why the $d_{33}$ coefficient is larger in absolute value than the $d_{31}$ one. As the ratio $d_{33}/d_{31}$ is larger than 2, the distance between dipole centers should be larger in the $a$ direction than in the $b$ one. The average size $D_{IAa}$ is estimated by using the ratio $d_{33}/d_{31}$ of 3.2 at the end of poling and the ratio of the dipolar forces between both directions. Perpendicularly to the surface, the distance $D_{33}$ between the dipole centers is:

$$D_{33} = (D_b + D_{OAb}) / 2 \qquad (18)$$

Parallel to the surface, the distance $D_{31}$ between the dipole centers is:

$$D_{31} = D_{IAa} + (D_a + D_{OAb}) / 2 \qquad (19)$$

By assuming that:

$$\frac{F_{33}}{F_{31}} \sim \frac{d_{33}}{d_{31}} \qquad (20)$$
<figure><img src="image_9.png" /><figcaption>**Figure 6.** Crystalline and OA domains a) before poling and b) at the end of poling. The crystalline, OA, and IA domains are in blue, with blue lines, and in white, respectively. The chain axis along *c* is represented with orange lines in the (*ac*) plane and by orange points in the (*ab*) plane. c) Details of crystalline and OA domains.</figcaption></figure>
<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 7.** a) Upon large electric field in the upward direction, the dipoles are oriented in this direction and the actuator curvature is positive. b) Upon reverse electric field smaller than $E_c$, only the dipoles of the OA domains are downward. If identical domains are aligned parallel to the surface, no electrostatic attraction occurs. If the crystalline and OA domains of adjacent columns are shifted, electrostatic attraction occurs and the curvature is negative. c) Above $E_c$, the dipoles of the crystalline lamellae are also downward and the curvature is positive. The crystalline, OA, and IA domains are in blue, with blue lines, and in white, respectively. The P(VDF-TrFE) layer, the electrodes, and the substrate are represented in blue, grey, and yellow, respectively. d) SEM image of the P(VDF-TrFE) film surface showing elongated grains. e) Cross-section of an elongated grain of the SEM image: lamellae consisting of crystalline and OA nanodomains. The lamellae are separated by amorphous (IA) regions.</figcaption></figure>

we obtain:

$$2 \frac{D_{31}^4}{D_{33}^4} = \frac{d_{33}}{d_{31}} \qquad (21)$$

The values of $d_{33}/d_{31}$, $D_a$, $D_b$, $D_{OAa}$, and $D_{OAb}$ at 100 V µm⁻¹ yield $D_{IAa} \sim 2.8$ nm (Figure 6b), which is in the range of values

found in the literature. More precisely, Sharma et al. found a 6 nm distance between crystalline domains of P(VDF-TrFE) by piezoresponse force microscopy (PFM) and explain that this large value for domain walls in ferroelectrics may be due to the finite size of the probing tip and to the presence of amorphous phase.[34] Hafner et al. estimated the distance between lamellae to ≈10 nm by atomic force microscopy (AFM).[19]
We give now one example of a technological device whose behavior can be explained by our fundamental findings. Let us consider an actuator made of a P(VDF-TrFE) capacitor on a flexible substrate.[10] When an electric field is in the same direction as the dipoles (Figure 7a), the elongation parallel to the surface film forces the substrate to bend downward and a positive curvature of the actuator occurs.[10] On the contrary, when an electric field smaller than $E_c$ is in opposite direction to the initial polarization, the dipoles in the OA domains align first in the down direction, due to their high polarizability (Figure 7b). Therefore, an attraction parallel to the surface occurs between the crystalline and OA domains if the crystalline and OA domains are alternate (Figure 7b). The contraction parallel to the surface forces the substrate to bend upward, leading to a negative curvature occurring at the permittivity maximum.[10] Even though the sum of the dipoles is zero at $E_c$, the dipoles still interact with each other. It is observed that the curvature is larger for the negative one than for the positive one at equal electric field value.[10] The improvement in the negative curvature is attributable to the enhanced force due to closer dipoles. Above $E_c$, the curvature becomes positive again, as all the dipoles align in the same direction as the applied electric field (Figure 7c).[10] Therefore, the coercive field $E_c$ corresponds also to the threshold between the rotation of the dipoles in the OA domains and that in the crystalline lamellae. As the sum of the dipoles is zero at $E_c$, it means that the dipoles of the OA domains are equal, in absolute value, to those of the crystalline lamellae. The OA domains, which are less dense than the crystalline ones, have a higher polarizability, i.e., a higher permittivity. This confirms that the observed permittivity maximum is correlated to the high polarizability of the OA domains and puts in evidence the major role of the OA domains in the dielectric and ferroelectric properties of P(VDF-TrFE). In addition, a scanning electron microscope (SEM) image was performed to characterize the P(VDF-TrFE) surface (Figure 7d). Elongated grains, sometimes called fibrils, with a length of several micrometers and a width in the nanometer range are visible. The grains consist of lamellae separated by amorphous regions.[14,19,20] Our results show that the lamellae are not fully crystalline, but consist of crystalline and OA domains (Figure 7e).

As the macroscopic piezoelectric coefficients correspond to the nanoscopic ones of the crystalline nanodomains, it means that no strain of the OA domains intervenes in the piezoelectric coefficient, due to the columnar structure of the crystalline and OA domains. Indeed, the crystalline nanodomains imposes the contraction/elongation of the layers for the $d_{33}$ coefficient and of the columns for the $d_{31}$ coefficient.

## 2.5. Mechanical and Electric Properties

During poling, the drastic structural changes may modify the mechanical properties. Indeed, Young's modulus reported in the literature varies significantly for P(VDF-TrFE). Mechanical measurements yield a mean value of 3.3 GPa on a raw (non-poled) P(VDF-TrFE) film,[9] whereas a value of 2.4 GPa is used for poled P(VDF-TrFE) in an analytical model, which fits well the actual curvature.[9] A Young's modulus of 1.7 GPa (resp. 0.3) for the crystalline (resp. amorphous) domain was measured in P(VDF-TrFE) lamellae at low temperature ferroelectric state,[19] showing

that Young's modulus Y depends on the crystalline/amorphous characteristics. Along the b axis, the dipoles interact through electrostatic forces $F_{33} \propto \frac{1}{\epsilon_r D^4}$, which are also:

$$F_{33}/A = \sigma_{33} = Y \times S_{33} \qquad (22)$$

with $\sigma_{33}$ the stress (force/area) and $S_{33}$ the strain. For small variations, one obtains:

$$\frac{\Delta F_{33}}{F_{33}} = \frac{\Delta Y}{Y} + \frac{\Delta S_{33}}{S_{33}} \qquad (23)$$

$$\frac{\Delta Y}{Y} = -\frac{\Delta \epsilon_r}{\epsilon_r} - 4 \times \frac{\Delta D_{33}}{D_{33}} - \frac{\Delta S_{33}}{S_{33}} \qquad (24)$$

The expression $\Delta Y/Y$ yields $\Delta Y$ as a function of the applied electric field, taking the value of Y before poling, i.e., 3.3 up to 30 V $\mu m^{-1}$ and the $\epsilon_r$ values reported in Figure 1b during the first polarization. Then, Y globally decreases up to 1.9 GPa at $E = 100 V \mu m^{-1}$ (Figure 8a). This value is between that used for the analytical model and that of the crystalline domains.[9,19] The OA contribution, which appears above 40 V $\mu m^{-1}$, softens P(VDF-TrFE), decreasing the average Young's modulus.

After poling, the P(VDF-TrFE) actuator is typically used with an applied electric field between 0 (remnant) and 100 V $\mu m^{-1}$. In this range, the cumulative lattice displacements change the distance between the dipole centers $D_{33}$ from 6.54 to 6.46 nm and $D_{31}$ from 7.18 to 7.38 nm (Figure 8b). Although the electrostatic forces vary as $D^{-4}$, they have a linear behavior with respect to D within the range of operation (Figure 8c,d). Indeed, the Taylor expansion of F is:

$$F \propto D^{-4} \propto D_{0}^{-4} \left( 1 - 4 \frac{\Delta D}{D_{0}} \right) \qquad (25)$$

with $D = D_{33}$ or $D_{31}$, $D_0 = D$ at 0 V $\mu m^{-1}$ and $\Delta D = D - D_0 \le 0.2$ nm $\ll D_0$. This expression shows that F depends linearly on D.

The complementary role of the OA domains compared to the crystalline ones allows electrostatic interactions and finally ferro- and piezo-electric behavior.

## 3. Conclusion

In conclusion, operando HRXRD experiment was used to probe the structural evolution in PVDF-based polymers upon electric field application. During poling, most of the OA contributions initiate at boundaries of crystalline nanodomains and spread perpendicularly to the surface. Between these OA contributions, IA domains may become OA due to polarization. The OA domains accommodate considerable strain, allow the paraelectric-ferroelectric transition, and decrease Young's modulus. Consisting of crystalline and OA nanodomains, the lamellae are separated by IA regions. This complex structure explains the behavior of PVDF-based devices, especially near the coercive field. Moreover, the nanoscopic piezoelectric coefficients are in agreement with the macroscopic ones. The different longitudinal and transverse piezoelectric behaviors originate from the
**Supporting Information**

Supporting Information is available from the Wiley Online Library or from
the author.

**Acknowledgements**

The authors acknowledge the European Synchrotron Radiation Facil-
ity (ESRF) for provision of synchrotron radiation facilities under pro-
posal MA5418. The data are available at doi.esrf.fr/10.1515/ESRF-ES-
937817538

**Conflict of Interest**

The authors declare no conflict of interest.

**Data Availability Statement**

The data that support the findings of this study are openly avail-
able in [ESRF] at [DOI: 10.1515/ESRF-ES-937817538], reference number
[937817538].

**Keywords**

actuator, amorphous, piezoelectric coefficient, P(VDF-TrFE), Young's modulus

Received: November 12, 2024
Revised: January 16, 2025
Published online: February 6, 2025

<footnote>[1] Y. Huang, G. Rui, Q. Li, E. Allahyarov, R. Li, M. Fukuto, G.-J. Zhong, J.-Z. Xu, Z.-M. Li, P. L. Taylor, L. Zhu, *Nat. Commun.* **2021**, *12*, 675.</footnote>

<footnote>[2] G. Rui, E. Allahyarov, Z. Zhu, Y. Huang, T. Wongwirat, Q. Zou, P. L. Taylor, L. Zhu, *Respons. Mater.* **2024**, 2, e20240002.</footnote>

<footnote>[3] R. Cohen, *Nature* **1992**, *358*, 136.</footnote>

<footnote>[4] W. Gao, Y. Zhu, Y. Wang, G. Yuan, J.-M. Liu, *J. Materiom.* **2020**, *6*, 1.</footnote>

<footnote>[5] S. H. Xiao, W. F. Jiang, K. Luo, J. H. Xia, L. Zhang, *Mater. Chem. Phys.* **2011**, *127*, 420.</footnote>

<footnote>[6] C. Mansour, M. Benwadih, G. A. Chahine, C. Revenant, *AIP Adv.* **2020**, *10*, 065204.</footnote>

<footnote>[7] C. Mansour, M. Benwadih, C. Revenant, *AIP Adv.* **2021**, *11*, 085302.</footnote>

<footnote>[8] C. Revenant, S. Minot, S. Toinet, E. L. Bright, R. Ramos, M. Benwadih, *Sens. Actuator A-Phys.* **2024**, *377*, 115738.</footnote>

<footnote>[9] S. Toinet, M. Benwadih, H. Szambolics, C. Revenant, D. Alincant, M. Bordet, J.-F. Capsal, N. Della-Schiava, M.-Q. Le, P.-J. Cottinet, *Materials* **2024**, *17*, 2135.</footnote>

<footnote>[10] S. Toinet, M. Benwadih, H. Szambolics, S. Minot, C. Revenant, M. Bordet, N. Della Schiava, M.-Q. Le, P.-J. Cottinet, *Macromol. Mater. Eng.* **2024**, *2400132*.</footnote>

<footnote>[11] A. Petritz, E. Karner-Petritz, T. Uemura, P. Schäffner, T. Araki, B. Stadlober, T. Sekitani, *Nat. Commun.* **2021**, *12*, 2399.</footnote>

<footnote>[12] J. Faudou, M. Benwadih, A. Aliane, C. Revenant, D. Grinberg, M.-Q. Le, P.-J. Cottinet, *Adv. Electron. Mater.* **2024**, *2400456*.</footnote>

<footnote>[13] K. Uneda, S. Horike, Y. Koshiba, K. Ishida, *Polymer* **2022**, *249*, 124822.</footnote>

<footnote>[14] N. Meng, X. Ren, X. Zhu, J. Wu, B. Yang, F. Gao, H. Zhang, Y. Liao, E. Bilotti, M. J. Reece, H. Yan, *J. Mater. Chem. C* **2020**, *8*, 16436.</footnote>

<footnote>[15] J. Seo, J. Y. Son, W.-H. Kim, *Mater. Lett.* **2019**, *238*, 294.</footnote>

<footnote>[16] S. Toinet, M. Benwadih, S. Tardif, J. Eymery, C. Revenant, *J. Polym. Res.* **2022**, *29*, 456.</footnote>

<footnote>[17] H. Ohigashi, K. Koga, *Jpn. J. Appl. Phys.* **1982**, *21*, L455.</footnote>

<footnote>[18] Y. Cho, D. Ahn, J. B. Park, S. Pak, S. Lee, B. O. Jun, J. Hong, S. Y. Lee, J. E. Jang, J. Hong, S. M. Morris, J. I. Sohn, S. N. Cha, J. M. Kim, *Adv. Electron. Mater.* **2016**, *2*, 1600225.</footnote>

<footnote>[19] J. Hafner, S. Benaglia, F. Richheimer, M. Teuschel, F. J. Maier, A. Werner, S. Wood, D. Platz, M. Schneider, K. Hradil, F. A. Castro, R. Garcia, U. Schmid, *Nat. Commun.* **2021**, *12*, 152.</footnote>

<footnote>[20] I. Katsouras, K. Asadi, M. Li, T. B. van Driel, K. S. Kjær, D. Zhao, T. Lenz, Y. Gu, P. W. M. Blom, D. Damjanovic, M. M. Nielsen, D. M. de Leeuw, *Nat. Mater.* **2016**, *15*, 78.</footnote>

<footnote>[21] W. Xia, Z. Xu, Q. Zhang, Z. Zhang, Y. Chen, *J. Polym. Sci., Part B:Polym. Phys.* **2012**, *50*, 1271.</footnote>

<footnote>[22] S. J. Rupitsch, in *Piezoelectric Sensors and actuators. Topics in Mining, Metallurgy and Materials Engineering*, Springer, Berlin, Heidelberg **2019**.</footnote>

<footnote>[23] H. Jiang, J. Yang, F. Xu, Q. Wang, W. Liu, Q. Chen, C. Wang, X. Zhang, G. Zhu, *Energy Convers. Manage.* **2020**, *211*, 112771.</footnote>

<footnote>[24] Y. Liu, H. Aziguli, B. Zhang, W. Xu, W. Lu, J. Bernholc, Q. Wang, *Nature* **2018**, *562*, 96.</footnote>

<footnote>[25] J. Yang, Q. Chen, F. Xu, H. Jiang, W. Liu, X. Zhang, Z. Jiang, G. Zhu, *Adv. Electron. Mater.* **2020**, *6*, 2000578.</footnote>

<footnote>[26] R. Tao, J. Shi, M. Rafiee, A. Akbarzadeh, D. Therriault, *Mater. Adv.* **2022**, *3*, 4851.</footnote>

<footnote>[27] P.-H. Ducrot, I. Dufour, C. Ayela, *Sci. Rep.* **2016**, *6*, 19426.</footnote>

<footnote>[28] M. Marandi, J. Tarbutton, *Procedia Manuf.* **2019**, *34*, 666.</footnote>

<footnote>[29] T. Wongwirat, Z. Zhu, G. Rui, R. Li, P. Laoratanakul, H. He, H. Manuspiya, L. Zhu, *Macromolecules* **2020**, *53*, 10942.</footnote>

<footnote>[30] L. Yang, X. Li, E. Allahyarov, P. L. Taylor, Q. M. Zhang, L. Zhu, *Polymer* **2013**, *54*, 1709.</footnote>

<footnote>[31] Y. Mao, Y. Su, B. S. Hsiao, *Eur. Polym. J.* **2016**, *81*, 433.</footnote>

<footnote>[32] S. Nayak, H. T. Ng, A. Pramanick, *Appl. Phys. Lett.* **2020**, *117*, 232903.</footnote>

<footnote>[33] X. Bai, H. Li, X. Chu, M. Liu, H. Li, H. Wang, X. Sun, S. Yan, *J. Polym. Sci.* **2024**, *62*, 1742.</footnote>

<footnote>[34] P. Sharma, T. Reece, D. Wu, V. M. Fridkin, S. Ducharme, A. Gruverman, *J. Phys.: Condens. Matter* **2009**, *21*, 485902.</footnote>

<footnote>[35] C. Carbone, M. Benwadih, G. D'Ambrogio, M.-Q. LE, J.-F. Capsal, P.-J. Cottinet, *Polymers* **2021**, *13*, 2166.</footnote>

<footnote>[36] Y. Li, W. Feng, L. Meng, K. M. Tse, Z. Li, L. Huang, Z. Su, S. Guo, *Mater. Des.* **2021**, *199*, 109415.</footnote>

<footnote>[37] A. Pipertzis, K. Asadi, G. Floudas, *Macromolecules* **2022**, *55*, 2746.</footnote>

<footnote>[38] A. Aliane, M. Benwadih, B. Bouthinon, R. Coppard, F. Domingues-Dos Santos, A. Daami, *Org. Electron.* **2015**, *25*, 92.</footnote>