Characterization of Mechanical and Electromechanical Properties of Aluminum-Coated Poled Orthotropic PVDF Film

Abstract

Poled PVDF film is a piezoelectric polymer currently utilized in sensing and actuation applications. We investigate the stress–strain behavior of the material as a function of the angle to the stretch direction. These properties were measured using mechanical testing and full-field strain imaging and compared with off-axis analytical formulations. Orthotropic material models are proposed for the elastic strain and charge relationships coupled with Hill’s orthotropic yield function to capture the directional dependence of yield strength in the poled PVDF under high strains. Additionally, the in-plane piezoelectric strain coefficients d₃₁, d₃₂, and d₃₆ were measured to aid in the design of PVDF metamaterials.

1. Introduction

Polyvinylidene fluoride (PVDF) is a commercially available polymer in various forms, including sheet form. While the melting temperature is 170 °C, its glass transition temperature is around −40 °C, which makes the material very flexible in room temperature conditions. In homopolymer form, the primary chains in the polymer have alternating structures of CH₂ and CF₂ groups. During polymerization, defect groups form in the polymer so that the monomer units are reversed in some regions, influencing the crystalline structure and relating to the piezoelectric properties observed in these materials [1]. Creating a piezoelectric material (poling) from a PVDF film consists of stretching the film at an elevated temperature and subjecting it to an electric field. This process leaves the film with orthotropic mechanical and electro-mechanical properties. A thorough understanding of the in-plane properties of this material relative to the electric field is vital for the design of the metamaterial cells. Limited research has studied the orthotropic properties of the PVDF film.

PVDF is a versatile polymer with applications in metamaterials for sensors, actuators, and energy harvesting devices [2,3]. PVDF-based materials are used in electromagnetic interference shielding, wearable biosensors, and acoustic sensors [4,5]. The polymer’s flexibility and biocompatibility enable its use in smart scaffolds and biomedical applications [6]. Research on polyvinylidene fluoride (PVDF) has focused on characterizing its complex electromechanical properties and developing constitutive models. Studies have measured the elastic, dielectric, and piezoelectric constants of poled PVDF films, revealing their frequency dependence and orthorhombic symmetry [7,8]. Nonlinear elastic behavior has been observed and modeled using the Mooney–Rivlin and neo-Hookean approaches [9]. Time-dependent viscoelastic properties have been incorporated into constitutive models [10,11,12]. Molecular modeling has provided insights into PVDF’s piezoelectric effects and switching phenomena [13,14]. Experimental studies have demonstrated nonlinear and time-dependent electromechanical behavior, including pre-stress dependence of electromechanical coefficients [15]. Several groups have characterized various properties of poled PVDF films. Seminara [16] and Vinogradov [17] have developed experimental setups and procedures for the electromechanical and mechanical characterization of these films.

Jones [1] has reviewed the degradation and performance of PVDF under various stress environments for possible application to space mirror concepts. The environmental effects in low-orbit space environments were investigated, and the significant resiliency of the material in these environments was shown. Lee [18] conducted nanoscale characterization using atomic force microscopy of PVDF under stress to measure nanoscale properties and show that the behavior was scale-independent under stress. Wang [19] reviewed the effects of various nanofillers on the piezoelectric response and energy-harvesting properties of PVDF composite films showing that nanofillers can be added without a reduction in the flexibility of the material. Fortunato [20] and Dodds [21] have investigated the piezoelectric properties of PVDF composite films with enhanced piezoelectric response, and PVDF-TrFE thin films enhanced with ZnO nanoparticles, respectively. Fortunato increased the piezoelectric coefficient of PVDF, avoiding the poling process, by inducing an increased β-phase fraction in the PVDF film by adding suitable quantities of nanofillers. Dodds et al., on the other hand, showed that piezoelectric ZnO nanoparticles can enhance the piezoelectric behavior without impacting the flexibility of the PVDF films.

PVDF films have also been extensively used in medical fields, such as smart auxetic stents, biomedical scaffolds, and health monitoring devices. Islam et al. [22] developed a smart auxetic stent, based on a perforated PVDF membrane that features ultrasonic powering, blood flow sensing, and integrated wireless electronics for accessible and continuous post-EVAR (endovascular aneurysm repair) surveillance. Recently, Pan et al. [23] proposed a piezoelectric smart stent for electricity generation driven by blood pressure fluctuation. It was fabricated by fused deposition modeling 3D-printing with a built-in electric field, where the stents are made from composite wires of potassium sodium niobate (KNN) particles and polyvinylidene fluoride-co-hexafluoropropylene (PVDF-HFP) matrix. A comprehensive review of PVDF biomedical scaffolds for tissue engineering is provided in Mokhtari et al. [24] and Ribeiro et al. [25]. In particular, PVDF polymer-based piezoelectric scaffolds that mimic tissue’s electrical microenvironment have been fabricated for bone tissue regeneration [26] and neural tissue engineering [27]. Moreover, in recent years, numerous PVDF-based piezoelectric generators have been developed for healthcare monitoring, such as human sound recording [28], arterial pulse monitoring [29], muscle behavior monitoring [30], and intracochlear sound pressure measurement [31].

2. Design and Experimental Methods

Measuring the piezoelectric constants of PVDF is crucial because these constants quantify the material’s ability to convert mechanical stress into electrical charge and vice versa. Verifying these constants is essential for applying these models to sensors, actuators, or energy-harvesting devices. In poled PVDF, the alignment of molecular dipoles enhances its piezoelectric response. Still, its mechanical properties, such as stiffness and plasticity, strongly influence this behavior, especially in details with high-stress concentrations such as hinges. Thus, the piezoelectric performance of PVDF is often directionally dependent due to its anisotropic nature. A good understanding of mechanical properties (e.g., Young’s modulus, Poisson’s ratio) and piezoelectric constants allows for precise prediction of how the material will deform under mechanical loads and how much electrical charge it will generate. This relationship is essential for optimizing PVDF’s performance in applications that require specific mechanical and electrical coupling behaviors. The stiffness parameters and piezoelectric strain constants were measured using mechanical testing and non-contact 3D digital image correlation. The key parameter, d₃₁, was measured using three different test methods.

Stiffness parameters: The tensile testing experiments used 50 µm and 75 µm thick metalized PVDF films (PolyK Technologies, Philipsburg, PA, USA). The PVDF specimens were magnetron sputter coated with aluminum to a sheet resistance of 1 Ω/□, corresponding to a roughly 100 nm thick metal film. The specimens were tested in tension according to the ASTM D638 [32] standard for tensile properties of polymer composites. The experiments were performed in a displacement control mode at a rate of 0.5 cm/min. A UV laser cutter machined the dog-bone specimens from flat sheets. The optical strain measurements were performed using a 3D DIC system (Q-400; Dantec Dynamics GmbH, Ulm, Germany, and Skovlunde, Denmark) using 8.1 MP resolution cameras and 35 mm lenses. Before testing, the dog-bone specimens were coated with a flat white paint to conceal the electrodes’ shiny surface and reduce reflections. A pattern was created using black paint applied through a stencil. The cameras were calibrated using a glass calibration plate from the vendor.

Piezoelectric strain coefficients using digital image correlation (DIC): Due to the recent developments in poled PVDF films, there is no standardized testing structure for collecting strain and deflection data on the piezoelectric material. Two methods were employed using the DIC system. A Semiprobe PSM-1000 microscope (Winooski, VT, USA) set to 20× magnification was used to view the material with an AMSCOPE MU300 microscope camera (Irvine, CA, USA) affixed with a 0.5× lens to optimize the field of view. AMSCOPE’s software (Version 4.8) allowed for 1536 × 2048 resolution images to be captured. A 50 µm thick film with metallization (electrodes) on top and bottom was fabricated into a test piece with a 100 × 20 mm active area (area in between electrodes). The long axis of the piece was oriented in the machine direction and was secured along the short end to a non-moving, flat substrate (see Figure 1). The remainder of the test piece was free to expand and contract. The specimen was coated with a flat white paint to reduce reflections, and an irregular pattern was created using a faint layer of air-brushed black paint.

One test method directly measured strain in the 5 × 5.5 mm DIC viewing area and was used to find d₃₁, d₃₂, and d₃₆ coefficients. The other method only measured d₃₁ by tracking the total displacement from the secured edge. Voltages were applied to the electrodes to create expansion and contraction. Then, a secondary voltage was applied between the bottom electrode and the substrate, separated by a thin dielectric. This secured the film to the substrate by electro-adhesion, thus creating a repeatable test by eliminating out-of-plane motion. DIC pictures were taken at voltages ranging from ± 1200 V, corresponding to electric fields ranging over ±24 MV/m.

Piezoelectric strain coefficient, d₃₁, using a direct mechanical approach: The third method to measure the piezoelectric coupling coefficient d₃₁ used a mechanical testing system (UStretch; CellScale, Waterloo, ON, Canada). The test piece was a 50 µm thick PVDF film. The active region was 100 × 40 mm, with the long axis oriented in the machine direction. Non-metalized ends extended beyond the active region to provide a clamping location that does not interfere with the piezoelectric strain motion (see Figure 2). The mechanical tests were run under a constant tension force. A small force (1.0 N) was chosen to minimize stretching (ϵ ~0.01%) of the sample but high enough for the CellScale system to maintain a stable force. Voltage is applied to the electrodes at small protrusions in the middle of the test piece, minimizing the mechanical effect on the test film. Potentials ranged over ±2100 V, corresponding to electric fields as high as ±42 MV/m. At each high voltage (HV) input, the sample was cyclically loaded, starting at 0 V and then going to +HV, back to 0 V, then −HV, back to 0 V, and repeating. Cycling in this manner generated the most piezoelectric strain and would potentially show any hysteresis effect.

3. Theoretical Analysis of Mechanical and Electrical Behavior

Poled polyvinylidene fluoride (PVDF) exhibits orthotropy due to its anisotropic structure from the alignment of molecular chains during the poling process. This alignment causes PVDF to have direction-dependent mechanical properties, such as differing stiffness and strength along its principal axes. To accurately capture the material behavior of poled PVDF, particularly in both the elastic and plastic regimes, models must account for this orthotropy. Without considering the directional dependence, predictions of the stress–strain response and deformation under various loads would be inaccurate, leading to errors in design and performance assessments of components made from poled PVDF. Thus, to model the experimental values obtained, we propose to use the classical strain-charge relationship accounting for elastic orthotropy coupled with an orthotropic plasticity model based on Hill’s orthotropic yield function.

The off-axis stiffness can be related to the stiffness in the material directions using the following relationship based on the transformation of the compliance matrix [33]:

$${\displaystyle \frac{1}{E_x}}={\displaystyle \frac{1}{E_1}}{cos}^4\theta +\left[{\displaystyle \frac{2v_{12}}{E_1}}+{\displaystyle \frac{1}{G_{12}}}\right]{sin}^2\theta {cos}^2\theta +{\displaystyle \frac{1}{E_2}}{sin}^4\theta$$

(1)

In this equation, $E_1$ is the machine direction modulus of elasticity, $E_2$ is the transverse modulus, $v_{12}$ is Poisson’s ratio, and $G_{12}$ is the shear modulus. Next, the strain-charge relationship for an electric field perpendicular to the film, ${\mathbb{E}}_3$ is proposed as follows to account for both the elastic stresses and the strains induced due to the piezoelectric effect:

$$\left\{\begin{array}{c}{\mathsf{ϵ}}_1\\ {\mathsf{ϵ}}_2\\ {\mathsf{\gamma }}_{12}\end{array}\right\}=\left[\begin{array}{ccc}{\displaystyle \frac{1}{{\mathrm{E}}_1}}& {\displaystyle \frac{v_{21}}{{\mathrm{E}}_1}}& 0\\ {\displaystyle \frac{v_{12}}{{\mathrm{E}}_1}}& {\displaystyle \frac{1}{{\mathrm{E}}_2}}& 0\\ 0& 0& {\displaystyle \frac{1}{{\mathrm{G}}_{12}}}\end{array}\right]\left\{\begin{array}{c}{\mathsf{\sigma }}_1\\ {\mathsf{\sigma }}_2\\ {\mathsf{\tau }}_{12}\end{array}\right\}+\left[\begin{array}{c}{d}_{31}\\ d_{32}\\ d_{36}\end{array}\right]{\mathbb{E}}_3=\left[\mathrm{S}\right]\left\{\begin{array}{c}{\mathsf{\sigma }}_1\\ {\mathsf{\sigma }}_2\\ {\mathsf{\tau }}_{12}\end{array}\right\}+\left[\begin{array}{c}{d}_{31}\\ d_{32}\\ d_{36}\end{array}\right]{\mathbb{E}}_3$$

(2)

where S is the compliance, $d_{31}, d_{32}, d_{36}$ are piezoelectric strain coefficients and ϵ and σ are the inplane strain and stress components, respectively. The strain in the 1′–2′ direction, as the material is rotated, can then be described as follows, where

$$\left\{\begin{array}{c}{{\mathsf{ϵ}}_1}^{\prime }\\ {{\mathsf{ϵ}}_2}^{\prime }\\ {{\mathsf{\gamma }}_{12}}^{\prime }\end{array}\right\}=\left[\begin{array}{ccc}{m}^2& n^2& mn\\ n^2& m^2& -mn\\ -2mn& 2mn& m^2{-n}^2\end{array}\right]\left[S\right]{\left[\begin{array}{ccc}{m}^2& n^2& 2mn\\ n^2& m^2& -2mn\\ -mn& mn& m^2{-n}^2\end{array}\right]}^{-1}\left\{\begin{array}{c}{\mathsf{\sigma }}_1\\ {\mathsf{\sigma }}_2\\ {\mathsf{\tau }}_{12}\end{array}\right\}+\left[\begin{array}{ccc}{m}^2& n^2& mn\\ n^2& m^2& -mn\\ -2mn& 2mn& m^2{-n}^2\end{array}\right]\left[\begin{array}{c}{d}_{31}\\ d_{32}\\ d_{36}\end{array}\right]{\mathbb{E}}_3$$

(3)

with $m=\cos \theta$, $n=\sin \theta$, where $\theta$ is measured from the reference (1–2) to the (1′–2′) directions. The anisotropic plasticity is simulated with Hill’s orthotropic yield function. Hill’s orthotropic yield function is widely used to simulate anisotropic plasticity in materials with directionally dependent properties. This yield criterion extends the classical von Mises isotropic yield function by introducing distinct parameters for different principal material directions. This allows the model to capture variations in yield strength and plastic deformation behavior across these directions. By incorporating the orthotropic yield function, we can capture the responses to stresses along their various axes. For the PVDF material studied, the yield function $Q_p$ is given by

$$Q_p=F{\left({\sigma }_{22}-{\sigma }_{33}\right)}^2+G{\left({\sigma }_{33}-{\sigma }_{11}\right)}^2+H{({\sigma }_{11}-{\sigma }_{22})}^2+2L{{\sigma }_{23}}^2+2M{{\sigma }_{31}}^2+2N{{\sigma }_{12}}^2-1$$

(4)

In the equation, Hill’s function involves multiple parameters, F, G, H, L, M, and N, that correspond to the materials state of anisotropy, allowing for a more accurate prediction of plastic deformation. In terms of volume, the model assumes incompressibility during plastic deformation, meaning the material’s volume remains constant.

Finite element simulation: An axisymmetric dog-bone geometry with dimensions shown in Figure 3 is created along the y-direction and modeled using the Solid Mechanics module in the COMSOL Multiphysics finite element software (Version 6.2, COMSOL, Boston, MA, USA). For capturing the plasticity, Hill’s orthotropic yield function is considered, for which the Hill’s parameters are chosen as (50, 100, 50, 25, 50, 25) MPa, along with 1.07 GPa isotropic tangent modulus and 3.82 GPa for the initial modulus. The values correspond to the orthotropic properties relative to the machine direction, with the highest values corresponding to the machine direction. The lower end of the dog-bone specimen is designated as a fixed constraint boundary whereas the opposite end receives a prescribed displacement along the positive y-direction. To study the mechanical stress–strain response for the variation of the machine direction orientation relative to the applied displacement only, a rotated coordinate system (Z-X-Z sequence) is introduced to the linear elastic material.

Including the geometric nonlinearity and physics-controlled extra fine mesh (Figure 3), a stationary analysis is carried out to compute the stress–strain response at the center point (0, 0, 0) of the axisymmetric test specimen by linearly increasing the prescribed displacement from 0 to 4 mm.

4. Results and Discussion

Mechanical behavior: The stress versus strain response for the four specimens studied is shown in Figure 4. The results show a large variability in the response based on the stretching direction relative to the applied loading. Strains over 18% were observed when the stretching direction was 0°, 15°, or 45° from the machine direction, with a significant drop in the ultimate strength for the 45° and 90° specimens. The 15° and 45° specimens showed a higher strain to failure and were comparable to the 90° and 0° specimens, which may be attributed to the axial-shear coupling response. A significant reduction in failure strain and ultimate stress is also observed when the machine direction is at 90° to the applied load. A closer look at the stress–strain response below 1% strain shows a linear elastic behavior and is shown in Figure 4b and also illustrates the orthotropic behavior of the material. The elastic modulus ranged from 3.82 GPa in the machine direction to 1.64 GPa in the transverse direction. A comparison of the stiffness predicted from Equation (1) and the slope from the experimental results is shown in Figure 5 showing a relatively good approximation of the initial stiffness using the equation provided. This equation can be used to understand the relative stiffness of the material when loaded away from the stretching direction, shows the relative sensitivity of the elastic stiffness to the loading orientation, and can be used in the linear regime of behavior.

For the plastic region, the stress–strain response using Hill’s orthotropic yield function is shown in Figure 6, showing the ability to capture the plastic region directional dependence of yield strength in the poled PVDF material relative to the loading direction. In the figure, the symbols represent the experimental data, and the line patterns follow the scaled model results. The results show that Hill’s orthotropic yield function can capture the variation in the yield initiation point in the loading relative to the machine direction. The corresponding stress–strain response in the plastic region is shown in Figure 7 which depicts the ability to capture directional dependence of yield strength in the poled PVDF dog-bone model. Further, the Von Mises stress contours for four distinct orientations (0°, 15°, 45°, 90°) at 3% strain show shear-axial strain coupling in the off-axis specimens, which can also be seen from the modeling results. Figure 8 shows the optical micrographs of the failed region of all four specimens. From these experiments and observations of the failure locations, we observe the dependence of the failure mode on the direction of the applied load to the machine direction. Most deformation is observed when the applied load is 15° and 45° from the machine direction. In the specimen loaded with the machine direction at 90° to the applied load, the failure plane started at 45° relative to the loading direction. Still, it quickly transitioned to fracture along the transverse to the films’ stretching direction, as seen in Figure 8d.

Piezoelectric Strain Coefficients d₃₁, d₃₂, d₃₆: Three separate test methods were used on two specimens to measure the piezoelectric strain coefficient d₃₁. Figure 9 shows electric field-induced strain in the machine direction as a function of the electric field applied across the film. The slope of a linear fit gives the value for d₃₁ in pC/N. The pull test (linear fit shown on plot) and DIC displacement methods yielded the same value with different precisions, d₃₁ = 24.4 ± 1.0 pC/N and d₃₁ = 24.4 ± 0.1 pC/N, respectively. Directly measuring strain with the DIC system resulted in an 8% lower value, d₃₁ = 22.4 ± 0.3 pC/N. For reference, the manufacturer reports a value of 28 ± 1.4 pC/N, while another manufacturer (Measurement Specialties, Inc., Hampton, PA, USA) reports a d₃₁ value of 23 pC/N. No measurable hysteresis effect was observed in the data at these electric fields.

As seen in the d₃₁ data, directly measuring the strain coefficient with the DIC is the most challenging. The small strains and a limited field of view make detecting differential movement in the sample difficult. Nevertheless, the DIC system measured the much smaller motion corresponding to the d₃₂ and d₃₆ coefficients. The DIC detected a small strain in the traverse direction (see Figure 10). A value of d₃₂ = 3.3 ± 0.3 pC/N was found, roughly an order of magnitude less than d₃₁. Although the estimated error in the data points is large, the p-value of the slope is less than 0.01%, indicating a statistically significant effect. The shear data associated with d₃₆ is shown in Figure 11. No statistically significant shear strain was detected; thus, d₃₆ = 0 ± 0.3 pC/N.

5. Conclusions

Poled (piezoelectric) PVDF material properties were measured and compared with theory and other sources. The mechanical behavior of the specimens exhibited significant variability in stress–strain responses based on the orientation of the machine direction relative to the applied load. Strains over 18% were observed for specimens stretched at 0°, 15°, and 45°, with the 45° specimen showing a notable reduction in ultimate strength. The 0° and 15° specimens demonstrated comparable strength, with the 15° specimen achieving higher strain to failure. However, when stretched at 90°, a considerable reduction in both failure strain and ultimate stress was observed, with the failure mode transitioning along the transverse direction. The initial stiffness prediction from the provided model showed a good approximation to the experimental data, validating its utility for assessing stiffness sensitivity to loading orientation. Hill’s orthotropic yield function can model the anisotropic yield behavior.

In the piezoelectric strain coefficient measurements, the d₃₁ coefficient was measured using three different methods, yielding similar results, though slightly lower than the manufacturer’s reported value. The DIC displacement method provided the most precise measurements. For the d₃₂ coefficient, the DIC system detected small but significant strains in the transverse direction, while no significant shear strain was observed for d₃₆, indicating it is negligible. These findings highlight the complexities in measuring piezoelectric strain coefficients, particularly for smaller coefficients like d₃₂ and d₃₆, where statistical significance was harder to achieve.

PVDF can endure up to 18% strain before failing. Compare this to a typical ceramic-based piezoelectric material, PZT, which cannot reliably exceed 0.1% strain without fracturing [34]. Although the strain coefficient, d₃₁ = 24.4 pC/N, is 3.5–11 times less than ceramics [35], it is the authors’ opinion that the advantages in flexibility, biocompatibility, and micro-manufacturability of PVDF will enable a wide range of applications inaccessible to ceramic devices.