Investigation of Molecular Mechanisms of Polyvinylidene Fluoride under the Effects of Temperature, Electric Poling, and Mechanical Stretching Using Molecular Dynamics Simulations

Abstract

This study uses molecular dynamics (MD) simulations to investigate the molecular mechanisms of polyvinylidene fluoride (PVDF) influenced by temperature, electric poling, and mechanical stretching. The β-phase, with all-trans ⟨T⟩ planar zigzag conformation, is known to have the best potential of energy harvesting, while α-phase, with alternating trans ⟨T⟩ and gauche ⟨G⟩ linkages, is more stable in terms of potential energy. By applying an electric field and uniaxial deformation to an amorphous PVDF system, we study the transformation from α- to β-phase and corresponding molecular mechanisms by tracking the molecular chain conformation using the trans percentages (P_T). After complete relaxation of molecular chains, the chain conformations and P_T values indicate a typical distribution pattern of α-phase. Next, we observe that the dipole moment of the system increases significantly with the presence of a strong electric field without immediately affecting the chain conformations. The increment of dipole moment is due to the aligning of side atoms within the chains and the increment becomes more significant with elevated temperature. In contrast, chain conformations change significantly under mechanical stretching. Specifically, before yielding, the total dipole moments are still governed by local orientations of atoms. Later, the chain segments begin to straighten in the large deformation stage, which leads to the increment of the total dipole moment. Our results also show that there exists an optimal temperature window for maximum ⟨G⟩ to ⟨T⟩ transformation rate. Moreover, we look into the synergistic effect of electric poling and mechanical stretching and explain molecular-level mechanisms for this effect. This study contributes to the fundamental understanding of the underlying molecular mechanisms for the piezoelectric PVDF system under different processing conditions.

1. Introduction

Investigating and developing energy harvesting material, which can transform various energy sources such as mechanical loads, vibrations, and human motion into usable electrical energy is a promising trend to facilitate energy usage and reduce the need for battery replacement or recharging. Piezoelectric materials have attracted immense interests in recent years for energy harvesting applications. Polyvinylidene fluoride (PVDF), a typical thermoplastic semicrystalline polymer, can achieve a high piezoelectric coefficient and has been utilized in electrical, filtration, and biomedical applications ^{[1, 2]}. The molecular formula of PVDF, containing 59.4 wt% fluorine (F) and 3 wt% hydrogen (H) atoms, is depicted in Fig. 1(a). Every dipole moment is perpendicular to the backbone in each unit cell due to the presence of large F atoms and high electronegativity ^[2]. There are mainly three crystalline polymorphs for PVDF based on their chain conformations: α-,β-, and γ-phases ^[1], as exhibited in Fig. 1(b). A mixture of α-,β-, and γ-phases is generally observed in bulk PVDF, which usually demonstrates semicrystalline features formed from melts or solutions ^[2]. Among these polymorphs, α-phase is the most common and thermal stable state with aligned polymer chains anti-parallel to each other, in a conformation style of alternating trans-gauche-trans-gauche’ (TGTG’) linkages. It can be formed by cooling from the melt at a moderate or high rate to below 160°C ^[3]. Moreover, at ambient temperature, the directions of these individual dipoles are randomly distributed, resulting in a net-zero dipole moment of the whole system. β-phase processes the highest dipolar moment per unit cell due to its all-trans planar zigzag conformation (TTTT). As a result, PVDF with predominant β-phases has the best piezoelectric, ferroelectric, and pyroelectric properties. γ-phase has a conformation style of TTTTGTTTG’, achieving certain piezo- and pyro-electric properties, while its dipole moment values are several orders lower than those of the β-phase ^{[4, 5]}.

Figure 1.

Schematic diagrams of (a) skeletal formula and (b) representative α-phase (TGTG’), β-phase (TTTT), and γ-phase (TTTTGTTTG’) of PVDF. (c) The schematic diagrams of dipole orientation variations during mechanical stretching and electric poling. Dipole moments are represented by simplified ball-and-stick models.

To achieve optimal energy harvesting, the randomly orientated dipoles in natural PVDF need to be aligned, that is, to increase the fraction of β-phase. Mechanical stretching and electric poling have been used as two effective approaches ^[6–9]. The schematics of dipole orientation variation during mechanical stretching and electric poling are shown in Fig. 1(c). For the mechanical stretching method, the α- to β-phase transformation is highly dependent on the stretching rate and temperature ^[2]. Many studies showed that the β-phase is more inclined to form at elevated temperatures (70–100 °C) and a large stretch ratio (3–5) ^[10–12]. Specifically, Sajkiewicz et al. ^[12] carried out experimental studies on drawn PVDF and showed that the transition would happen in the temperature range of 50 °C and 145 °C, and the content of the β-phase fraction would maximize at 87 °C. Another study also suggested that highly oriented β-phase PVDF was achieved at 87 °C under uniaxially or biaxially stretching experiments ^[11]. They controlled the drawing rate of 50 mm min⁻¹ and stretched the sample to a ratio of 6.5. A maximum piezoelectric coefficient d₃₃ of approximately 33 pC N⁻¹ can be reached at this stretch ratio (λ). In addition, Hess et al. ^[13] showed that the annealing of the thin films at 90 °C would promote the transformation from the γ-phase to the β-phase, while much higher temperatures will reduce the transition efficiency. Recent studies have found that β-phase melting occurs in the range 165–172 °C, α-phase crystals in the range 172–175 °C, and γ-phase melting between 175–180 °C ^{[2, 14]}. Matsushige et al. ^[15] carried out stretching experiments to study the crystal transformation of PVDF. The transformation was found at the necking point at a relatively low temperature. However, the sample deformed uniformly without necking and did not experience the crystal transformation above 140 °C. For the electric poling method, the field should be applied as high as 100 kV mm⁻¹ under an elevated temperature of 80–165 °C ^[16]. Moreover, the polar β-phase of PVDF can also be induced by melt crystallization at high-pressure ^[17] or very high cooling rates ^{[3, 18]}, the application of a shearing history^[19], copolymerization^{[2, 4]}, nanoconfinement ^[1], surface charge ^[20], nucleation ^[21], polymer blending ^[22], electrospinning ^[23], etc.

These aforementioned experimental studies propose that mechanical stretching (after a specific strain) and electric poling (high enough voltage) are typical choices to increase the fraction of β-phases in PVDF. Moreover, the α-phase will more likely transform into the β-phase at an elevated temperature. Despite many experimental studies of PVDF, the structure-property relationships at the fundamental molecular level remain poorly understood. The experimental approach provides limited insights into the dynamic change in local atomic arrangement during mechanical stretching or with an electric field. On the other hand, computational simulations can offer a realistic representation of the structures at different size scales and provide detailed insights into deformation processes and mechanisms. Molecular dynamics (MD) simulations are particularly well-positioned to look into detailed dynamics processes of deformation and failure as well as to predict the properties of systems with specific nanostructures. Chen et al. ^[1] investigated the microstructure and mechanical properties of PVDF and PVDF/carbon nanotube with and without a polarization process. They found that the polarization process would help to enhance the percentage of ⟨T⟩ dihedral linkages. MD results also showed that the melting temperature (T_m) of β-phase PVDF is higher than that of α-phase PVDF, and T_m depends on the polymer chain length ^[24]. Bohlén et al. showed the effect of the orientation of single-wall carbon nanotube embedded in the PVDF on mechanical properties of the nanocomposites ^[25]. Moreover, the detailed relaxation processes of electrospun PVDF nanofibers were investigated utilizing atomistic MD simulations and analytical theory ^[26]. Despite these studies, the detailed deformation processes at the molecular level and their correlation with the change of dipole moments during mechanical stretching and electric poling are still not fully known.

The present work studied the structural, mechanical, and piezoelectric properties of PVDF using MD simulations. Our results provide insights into the structure-property relationships of PVDF and the underlying molecular-level deformation mechanisms under mechanical stretching and electric poling. The temperature factor will also be systematically studied. The paper is organized as follows. The “Models and Methodology” section describes the material modeling process and simulation methods. The “Results and Discussion” section shows that the model used in our simulations can capture the key mechanical and physical properties, and we can employ MD simulations to study the effects of temperature, electric field, and mechanical stretching on the structural, mechanical, and piezoelectric properties of PVDF. Moreover, we examine the conformation transformation rate under different temperatures. Finally, we summarize the results and conclusions in the “Conclusions” section.

2. Models and Methodology

The PVDF model system is constructed using the Materials Studio software ^[27]. The skeletal formula of PVDF can be seen in Fig. 1(a). Our previous work shows that short-chain polymer systems exhibit fast chain pull-out and void formation under stretching deformation, which leads to relatively small failure strain ^[28]. Since we focus on chain conformation transformations under large elongations, we choose a longer chain system with 400-monomer-per-chain in this study. Also, the selected chain length balances between the system size and computation efficiency. With similar considerations, the model consists of 100 chains, which lead to 240200 atoms overall. All the simulation work is conducted in the Large-scale Atomic Molecular Massively Parallel Simulator (LAMMPS) MD package ^[29]. The simulation box is set to be periodic in all three dimensions. The Polymer Consistent Force Field (PCFF) ^[30–32] is adopted to model the PVDF. It can capture the thermomechanical and phase transition behaviors of the polymer ^[33]. The total energy is contributed from the following terms:

$$\mathrm{E}={\mathrm{E}}_{\mathrm{B}}+{\mathrm{E}}_{\mathrm{A}}+{\mathrm{E}}_{\mathrm{D}}+{\mathrm{E}}_{\mathrm{I}}+{\mathrm{E}}_{\mathrm{n}\mathrm{b}}$$ 1)

where the sum of bond (E_B), angle (E_A), dihedral (E_D), and improper (E_I) terms denote the bonded interactions, while non-bonded (E_nb) is used to represent the non-bonded interactions. We use PCFF ^[30–32] style parameters and a 9–6 Lennard-Jones potential ^[34] (shown in Eq. 2 below) for the bonded and non-bonded interactions, respectively.

$${\mathrm{E}}_{\mathrm{n}\mathrm{b}}=\mathrm{L}\mathrm{J}\left(\mathrm{r}\right)=\mathrm{\epsilon }\left[2{\left(\frac{\mathrm{\sigma }}{\mathrm{r}}\right)}^9-{3\left(\frac{\mathrm{\sigma }}{\mathrm{r}}\right)}^6\right](\mathrm{r}<{\mathrm{r}}_{\mathrm{c}})$$ (2)

where σ is the distance at which E_nb reaches zero, and ε is the depth of the potential well. For the interaction parameters between different kinds of atoms, both σ and ε are calculated based on a commonly used Lorentz-Berthelot mixing rule ^{[28, 35]} :

$${\mathrm{\sigma }}_{\mathrm{A}\mathrm{B}}=({\mathrm{\sigma }}_{\mathrm{A}\mathrm{A}}+{\mathrm{\sigma }}_{\mathrm{B}\mathrm{B}})/2$$ (3)

$${\mathrm{\epsilon }}_{\mathrm{A}\mathrm{B}}=\sqrt{{\mathrm{\epsilon }}_{\mathrm{A}\mathrm{A}}{\mathrm{\epsilon }}_{\mathrm{B}\mathrm{B}}}$$ (4)

where subscripts A and B represent two different types of atoms.

Under the corresponding PCFF potential parameters, we first equilibrate a PVDF unit cell in the NVT ensemble to measure its dipole value at 1 K. Extremely low temperature is used to minimize fluctuations of measured dipole value. A value of 2.25 D is obtained, close to the widely accepted experimental value 2.1 D ^[36]. It’s worth noting that due to the random distribution of atomic and chain orientations, the net dipole value may be close to zero in α-phase PVDF, which is more common in natural PVDF homopolymers ^{[7, 37]}.

To obtain an equilibrated state of the PVDF systems before loading, we subject the systems to an initial energy minimization process. Then, the systems are equilibrated in the NPT ensemble, during which a Nosé-Hoover thermostat method is adopted to control the temperature. The time step is set to be 1 fs. The complete equilibration process is the following: 1) set an initial equilibrium at 500 K for 50 ps so that the system can be fully relaxed in the molten state, and the influence of the initial conformation can be minimized; 2) cool down the system at a cooling rate of 0.5 K/ps from 500 K to 300 K with a temperature interval of 25 K. At each interval, we equilibrate the system using the NPT ensemble with the external pressure in each direction maintained at 0 atm for 10 ps; 3) extra equilibration at four desired temperature 300 K, 350 K, 400 K, and 450 K for 10 ps to ensure the systems has stabilized totally at the equilibrated state. The external pressure is controlled at 0 atm in all three directions during the equilibration. After these steps, fully equilibrated and stress-free systems at desired temperatures are obtained. The dimensions of the equilibrated system are approximately 14nm × 14nm × 14nm.

We first characterize the fundamental physical and thermo-mechanical properties of PVDF system from MD simulations and validate our models. Specifically, we characterize glass transition temperature (T_g) and compare it with experimental results. Dynamical mechanical analysis and Differential Scanning Calorimetry have been widely used to characterize T_g in experiments. The method of tracking the density or specific volume change versus temperature has been commonly used in MD simulations ^{[33, 38–43]}. In addition, previous studies have also characterized T_g through tracking the mean square displacement profiles of the system ^{[33, 43]}. We adopt the specific volume method to characterize T_g in this study. Specifically, we cool down the system at a cooling rate of 0.5 K/ps from 500 K to 100 K with a temperature interval of 25 K using the NPT ensemble at 0 atm. At each 25 K, the system is relaxed for 50 ps to ensure an equilibrium state and alleviate the effect of the cooling rate. The average specific volume is collected during each interval. The change of slope in specific volume against temperature marks the computational T_g. T_m, on the other hand, is an easily measured physical property through experiments but poses challenges on MD simulation to accurately characterize it. Previous studies have characterized T_m by finding the point at which the orientational order parameter of the backbone network completely collapses during heating up process ^{[33, 44, 45]}. In our study, we pick the upper limit value of T_m reported in previous experimental studies ^{[2, 14]}. We also measure Young’s modulus of PVDF by applying a constant engineering strain rate to the system and tracking the stresses in the deformation direction using the viral theorem, consistent with our previous studies ^{[28, 39]}.

Then, electric poling and mechanical stretching are carried out along the x-direction to investigate the chain conformation transformation mechanisms. The intensity of the electric field is controlled at 1 V/nm, similar to previous studies ^[1]. A constant engineering strain rate of 10 ns⁻¹ is chosen to increase computational efficiency in large stretching elongation cases. NPT ensemble is used during these simulations. The temperature effect (300 K - 450 K) on the variation of dipole moments is also considered.

3. Results and Discussion

3.1. Model validation

To validate our model of PVDF, we compare the physical and thermo-mechanical properties to the results in other studies, as listed in Table 1. The density of PVDF at 300 K (room temperature) measured from our simulations is 1.59 g/cm³, slightly lower than the values reported in the literature. We attribute this to the fact that our system is fully amorphous without pre-existing crystallites. The specific volume (inverse to the density) as a function of temperature is shown in Fig. 2, and the change of slope marks the computational T_g, which is 254 ± 5 K. Despite possible uncertainties in the calculation process, the bulk value of T_g agrees with experimental measurements. The characterized T_g and T_m informed from literature (which is selected as the upper value from previous characterizations ^{[2, 14]}, i.e., 453 K) provide a reasonable temperature range to investigate the molecular chain conformation transformation mechanisms. In the following sections, we select four typical temperatures, i.e., 300 K, 350 K, 400 K, and 450 K, to study the effect of temperature and explore chain conformations and underlying mechanisms. In addition, Young’s modulus of our model agrees with a previous study ^[6] in general and the slight discrepancy is probably due to the longer chains used in our study.

Table 1.

Comparison of density, T_g, and Young’s modulus of PVDF with other studies. The density and Young’s modulus listed are measured at 300 K.

	Density (g/cm3)	Tg (K)	Young’s modulus (GPa)
Literature	1.66–1.78 [1]	253 [7]	5.3 [6]
This work	1.59	254±5	5.6±1.4

Figure 2.

Specific volume as a function of temperature from 100 K to 500 K, where the slope change marks the computational T_g.

The capability of our model in capturing these physical and thermo-mechanical properties shows the accuracy of our model and associated parameters. In the following sections, we use the model to investigate molecular mechanisms of PVDF under the effects of temperature, electric poling, and mechanical stretching.

3.2. Effect of Temperature

The fundamental differences between α-,β- and γ- phases are the distributions of PVDF backbone dihedral angles. For example, the α-phase has a TGTG’ style chain conformation, resulting in two similar height peaks corresponding to two different dihedral angles in the probability distribution of dihedrals. The other conformations are similar, but they have different peak values. To quantitatively analyze the chain conformations, two superimposed Gaussian distribution functions are adopted here to depict the probability distributions of dihedrals within the backbone in the same way as Chen et al. ^[6]. The equation is shown as follows ^[46]:

$$\mathrm{f}(\mathrm{\varphi })={\mathrm{A}}_{\mathrm{G}}\mathrm{e}\mathrm{x}\mathrm{p}(-{\left(\mathrm{\varphi }-{\mathrm{\mu }}_{\mathrm{G}}\right)}^2/(2{\mathrm{\sigma }}_{\mathrm{G}}^2))+{\mathrm{A}}_{\mathrm{T}}\mathrm{e}\mathrm{x}\mathrm{p}(-{\left(\mathrm{\varphi }-{\mathrm{\mu }}_{\mathrm{T}}\right)}^2/(2{\mathrm{\sigma }}_{\mathrm{T}}^2))$$ (5)

where f(ϕ) describes the occurrence probability of dihedral angle ϕ. A_G, A_T, μ_G, μ_T, ${\mathrm{\sigma }}_{\mathrm{G}}^2$ and ${\mathrm{\sigma }}_{\mathrm{T}}^2$ are the parameters of normal distributions of Gauche ⟨G⟩ and Trans ⟨T⟩ dihedral angles.

To explore the effect of temperature on the conformational transformation mechanisms of PVDF chains, we study dihedral distributions between 300 K to 450 K. Fig. 3 displays the density against time curves during the 10 ps equilibrium process mentioned in the ‘Models and Methodology’. The density results indicate that the systems have reached equilibrium at the investigated temperatures.

Figure 3.

Density of the system during the 10 ps equilibrium process before polarizing or stretching. Constant densities with minimal fluctuations can be observed at investigated temperatures, indicating that the systems have reached equilibrium.

Fig. 4 shows the statistical results using Eq. 5 to fit the raw data. Two major peaks at ∼65° and ∼167° can be observed in all cases, corresponding to the ⟨G⟩ and ⟨T⟩ dihedral angles in PVDF, respectively. It is a typical distribution style of dihedral angles for a thermal-stable α-phase structure. As the temperature increases, both peaks show a decreasing trend, while at the same rate. This suggests that the chain conformations become more random. We also observe that the angles corresponding to the two peaks become slightly closer upon increasing the temperature.

Figure 4.

Probability distribution of dihedral angles for equilibrated conformations at different temperatures.

We note that for an ideal Gauche-trans angle distribution of polymer chains, the peak angles appear at 60° and 180°. The slight deviation of peak angles observed from our simulations can be explained by two reasons. First, the fitting process leads to certain deviations from the peak angles in the raw data in Fig. 4. The fitted first peak is higher, i.e., closer to 180°, and the fitted second peak is lower, i.e., closer to 60°. Second, the long-chain amorphous system in our case does not consist of entirely ideal α-phases segments, even when the system is fully equilibrated. The high ambient temperature and corresponding high entropy play an essential role in this phenomenon. These two reasons explain the deviated peak angles from 60° and 180°. A similar phenomenon has been observed in the work of Chen et al. ^[6]. In experiments, a mixture of α-,β-, and γ-phases is generally observed in bulk PVDF ^[2].

We follow Chen’s work ^[1] in which the trans percentage P_T can be evaluated by the following equation:

$${\mathrm{P}}_{\mathrm{T}}={\mathrm{A}}_{\mathrm{T}}/({\mathrm{A}}_{\mathrm{G}}+{\mathrm{A}}_{\mathrm{T}})\times 100$$ (6)

where P_T indicates the percentage of ⟨T⟩ dihedrals in all PVDF chains, A_G and A_T are prefactors of normal distributions of the ⟨G⟩ and ⟨T⟩ dihedrals fitted using Eq. 5. Chen’s work ^[1] indicated that higher P_T is related to a higher fraction of β-phase. However, this conclusion ignores the existence of γ-phase in PVDF, as shown in Fig. 1(b). In our study, we use P_T as one of several indicators for chain conformations.

Since this phenomenon does not necessarily indicate that the average dihedral angles will become larger, we also calculate the average dihedral angles listed in Table 2, which quantitatively describes our findings. ⟨G⟩ peak angle, A_G, ⟨T⟩ peak angle, and A_T are listed in Table 2 under different temperatures, as well as P_T(%) calculated using Eq. 6 and the averaged angle. As shown in Fig. 3, these equilibrated systems are thermally stable at investigated temperatures, P_T value is very close to 50%, indicating the initial arrangement consists of mainly α-phase. The P_T and the averaged angle values are similar (or a slight-increasing trend) upon the increase of temperatures, while A_G and A_T both decrease. It suggests that the temperature alone does not necessarily induce conformation transformation from ⟨G⟩ to ⟨T⟩ but does result in more randomly configured chains.

Table 2.

The fitted parameters of the probability distribution of dihedral angles at different temperatures.

T(K)	⟨G⟩ angle (°)	AG	⟨T⟩ angle (°)	AT	PT(%)	averaged angle (°)
300	63.5	0.0130	167.5	0.0136	51.1	107.4
350	64.2	0.0120	166.9	0.0128	51.6	107.7
400	64.8	0.0112	166.0	0.0121	51.9	107.9
450	65.1	0.0104	165.4	0.0115	52.5	107.8

3.3. Effect of Electric Poling

We then study the effect of electric poling. The electric field is applied along the x-direction. Fig. 5 shows the change of dihedral distributions at four different temperatures when applying the electric field. The distributions are collected every 10 ps. The peak corresponding to the ⟨G⟩ dihedral linkages decreases while the ⟨T⟩ one increases. These results show the direct effect of electric poling on dihedral distribution. Comparing the cases from different temperatures, we find that electric poling becomes more effective on chain conformations upon increasing temperature, consistent with the experiment results ^[16].

Figure 5.

Distribution of dihedral angles under (a) 300 K, (b) 350 K, (c) 400 K, and (d) 450 K during electric poling.

Fig. 6 shows averaged angles, P_T, and dipole variations of PVDF under 300 K, 350 K, 400 K, and 450 K during electric poling. The averaged angles and P_T values increase with time, suggesting that electric poling can induce chain conformation transformation. However, with the current electric field, the chain conformation changes are less significant than those under mechanical stretching, which will be discussed later. Due to simulation limitations on spatiotemporal scales, the whole process of chain conformation transformation may not be fully captured herein. We expect that the chain conformation changes under the electric field extend to a much longer time scale, which far exceeds the capability of MD simulations. The chain conformation changes should also depend on the applied electric voltage. With these limitations in mind, our main objective in this study is to reveal the underlying mechanisms of dipole moment changes during electric poling at a relatively small time scale.

Figure 6.

(a) The averaged angles, (b) P_T, and (c) dipole variation of PVDF under four different temperatures during electric poling.

The effect of temperature can be observed in Fig. 6. It indicates that elevated temperature promotes the segmental movement ability in the electric poling, consistent with previous studies ^[16]. This phenomenon suggests that though the temperature factor cannot induce polarization alone, it can promote the increase of dipole by enhancing the effects of other factors, e.g., electric poling. Fig. 6 shows that dipole values can increase effectively because of the applied electric field. Our simulation results also indicate that the significant dipole moment increment under electric poling is mainly due to the local alignment of H and F atoms within the chains. The reason for the weakened influence (or slowing down of the growth rate of dipole moment) at 450 K is the tradeoff between the local alignment and the randomness induced by high temperature.

3.4. Effect of Mechanical Stretching

We discuss the effect of mechanical stretching in this section. During the deformation of the PVDF system, the stress σ is calculated by the following equation ^[29]:

$$\mathrm{\sigma }=\frac{1}{\mathrm{V}}\left(\mathrm{N}{\mathrm{k}}_{\mathrm{B}}\mathrm{T}+\frac{1}{3}\sum _{\mathrm{i}}^{{\mathrm{N}}^{\mathrm{\prime }}}{\mathrm{r}}_{\mathrm{i}}\cdot {\mathrm{f}}_{\mathrm{i}}\right)$$ (7)

where N is the number of atoms in the system, V is the system volume, k_B is the Boltzmann constant, and T is the temperature. The stress vs. strain curves under different temperatures are shown in Fig. 7. Due to the strain-rate-dependent mechanical properties, we note that caution must be taken when directly comparing stresses or strengths measured in MD simulations to experiments. With the decrease of temperature, Young’s modulus becomes higher. A turning of the stress-strain curve, marking the yielding of the systems, occurs at a strain level of around 10–15%. But for the 450 K case, the curve does not have an apparent downward trend, and it shows a plateau instead. After the yielding stage, the chain segments start to orient parallel to the stretching direction ^[47]. We note that a limitation of the force field used in our simulations is that it cannot capture the bond-breaking events during chain scission. Thus, the stress calculation at large deformation regime and the failure strain cannot be predicted accurately. However, our previous studies show that the chain transformation mechanisms under large deformation are largely conserved using non-reactive force fields ^{[28, 39]}.

Figure 7.

Stress vs. strain curves under 300 K, 350 K, 400 K, and 450 K.

Fig. 8 shows the van der Waals, bond, angle, and dihedral energy evolutions during mechanical stretching at different temperatures. We note that the energy variations are very modest under the electric poling, which further supports our claim that the chain conformation change under electric poling is minor. As shown in Fig. 8, van der Waals energy variation is dominant in small strain regime during mechanical stretching. This is consistent with our previous study that van der Waals interaction experiences dramatic changes during mechanical deformation of polymeric systems ^[48]. Besides, we find that bond and angle terms contribute moderately to the energy increments in the whole deformation process, while dihedral energy seems to only increase after a certain strain (10 ~ 12 %). This strain level is coincident with the yielding point in the stress-strain curve. This observation indicates that the dihedral linkages within polymer chains only begin to transform after the yield point rather than from the very beginning. This finding is consistent with previous experimental observations ^[15]. Moreover, the energy increments will become less significant upon increasing temperature. Compared to the low-temperature cases, the dihedral energy contributes little to the total energy evolution at high temperatures, e.g., 450 K.

Figure 8.

The energy increments of van der Waals, bond, angle, and dihedral at 300 K, 350 K, 400 K, and 450 K during mechanical stretching.

Fig. 9 shows the distributions of dihedral angles at different strain levels during mechanical stretching for different temperatures. Compared with the electric field results, mechanical stretching promotes chain conformation change significantly. Comparing the four cases, we find that as the temperature rises, both peaks of the dihedral angle distribution curves go down due to the randomness induced by high temperature. This is similar to the conclusion in Section 3.2. A more significant decreasing trend of the right (higher) peak with increasing temperature can also be seen, consistent with more randomly configured chains.

Figure 9.

Distribution of dihedral angles of PVDF during mechanical stretching under (a) 300 K, (b) 350 K, (c) 400 K, and (d) 450 K.

Fig. 10 (a) and 10(b) show that average angles and the P_T value are increasing with the stretching deformation under different temperatures. The steeper slope is observed after a strain level of around 100%, though the rising temperature tends to decrease the slope. This observation indicates that large strain promotes the ⟨G⟩ to ⟨T⟩ transformation. It is worth noting that the P_T at a large strain level under high temperature is lower than those under low temperature, indicating the random chain distribution resulting from high temperature. Fig. 10(c) shows the dipole evolutions during mechanical stretching at four different temperatures. We note that dipole evolutions show large variations among different trials as the chain conformation changes are subject to significant randomness. However, we confirm that the variations do not influence the general trend of dipole evolutions and their temperature dependence. Compared to Fig. 6(c), the increasing rate of the dipole is more gradual during mechanical stretching. The significant increment of dipole values is coincident with the rapid change of chain conformations at large strain regime. It agrees with the experimental results that the transformation tends to initiate at the necking point of the PVDF specimen ^[15].

Figure 10.

(a) The averaged angles, (b) P_T, and (c) dipole variations of PVDF under four different temperatures during mechanical stretching.

We also observe that the dipole curves representing 350 K and 400 K rise faster than the other two. According to the structure of PVDF (Fig. 1(a)), the presence of fluorine atoms induces a dipole moment perpendicular to the chain in each monomer unit ^[2]. Therefore, when stretched, the backbone of PVDF is straightened, which leads to better alignment of local dipoles, thus resulting in the growth of the total dipole moment. Increasing temperature is also conducive to increasing dipole values during mechanical stretching. However, at 450 K, which is close to T_m, there will be significant thermal movements that decrease the tendency of ordered alignment of chains.

From the results, we deduce that the γ-phase may form during the transformation from α- to β-phase. As aforementioned, γ-phase also contributes to the piezo- and pyro-electric properties of PVDF, though the dipole value of γ-phase is several orders of magnitude lower than that of the β-phase. This explains the delayed increment of dipole moments than the change of dihedral distributions, as shown in Fig. 10. In experiments, it has been observed that the β-phase is more favorably formed in the 343 K to 373 K and at a 300 % to 500 % strain ^{[7, 10, 49]}. This phenomenon shows that the dipole growth resulting from mechanical stretching is mainly caused by the straightened backbone, promoting the chains to transform into β-phase. We note this mechanism is different from that during electric poling.

The experimental observation that β-phase is more favorably formed in a specific temperature range intrigues us to investigate further the general distribution of the chain conformation transformation rate curve from T_g to T_m. When the temperature increases from 300 K to a higher temperature, the movement capacity of the chain segments increases. The transformation rate also begins to increase gradually. When the temperature increases to a temperature near T_m, the thermal movements of molecules become violent. The local ⟨T⟩ dihedral linkages are unstable, greatly influenced by the thermal motion of atoms. The transformation rate begins to decrease gradually. Therefore, there exists a temperature value, T_max, that can lead to maximum (or fastest) conformation changes, and T_max should be between T_g to T_m.

The standard principle to determine T_max of polymers macroscopically is to measure the thermodynamic or physical property changes. The isothermal cooling method is used here. Specifically, four systems equilibrated in the melt (550 K), are relaxed directly at 300 K, 350 K, 400 K, and 450 K for 150 ps at 0 atm pressure in separate NPT simulations. The total conformation transformation rate can be determined by measuring the reciprocal time t_1/2 obtained from the cooling process. Fig. 11(a) shows the volume-time curves at different temperatures. As time goes on, there is a noticeable trend of volume convergence in all systems, and the concurrent volume values lift with the increase of temperature. Besides, Fig. 11(b) depicts the (v_t − v_∞)/(v₀ − v_∞) vs. time curve during the isothermal cooling process, where v₀, v_t and v_∞ represent the volume at the initial time, time t, and final time respectively. The dotted line represents the volume value of 0.5, and its intersection with the different curves describes the state when the volume of the respective systems is reduced by half. The corresponding time is t_1/2. Results show that the case of “400 K” corresponds to the shortest reciprocal time, followed by 350 K, 300 K, and 450 K. We conclude that T_max lies between 350 K and 400 K.

Figure 11.

(a) Volume-time curve and (b) (v_t − v_∞)/(v₀ − v_∞) vs. time curve during the isothermal cooling process. v₀, v_t, and v_∞ represent the volume at the initial time, time t, and final time, respectively.

Similar conclusions can be found in previous experimental studies. Matsushige et al. ^[9] and Hsu et al. ^[50] show a critical temperature range for the maximum chain conformation transformations. Using the wide-angle-X-ray-scattering technique, Sajkiewicz et al. ^[8] found the optimal temperature is 360 K. Our results are in good agreement with these experimental studies.

3.5. Molecular-Level Mechanisms for Chain Conformation Changes

To better understand the underlying mechanisms responsible for the dipole moment increment during stretching of PVDF, the local P₂ order parameter is used ^[51]. A system with a higher local P₂ value means a higher local ordered structure. In a microscopic aspect, P₂ of the (i)th carbon atom can be computed as ^{[52, 53]}:

$${\mathrm{P}}_2(\mathrm{i})={⟨\left(3{\mathrm{c}\mathrm{o}\mathrm{s}}^2{\mathrm{\theta }}_{\mathrm{i}\mathrm{j}}-1\right)/2⟩}_{\mathrm{j}}$$ (8)

where θ_ij is the angle between the local vectors of adjacent chain segments, i.e., the vector from the (i − 1)th carbon atom to the (i + 1)th carbon atom in the segment of interest, and the vector from the (j − 1)th carbon atom to the (j + 1)th carbon atom in another segment. The average in Eq. 8 uses all of the (j) neighboring carbon atoms of carbon atom (i) that lie within a cutoff distance r_ij ≤ r_p2. For an ideal conformation that chains align along the same direction, all θ_ij equal to 0, which gives rise to P₂(i) = 1; if the two chains are completely perpendicular, θ = 90°, corresponding to P₂(i) = −0.5; and for two randomly distributed chains, P₂(i) approaches 0.

We post-process the simulation trajectories by applying the coordinates of atoms to calculate the local P₂ for every C atom. 1 nm is chosen to be the r_p2 in our simulations. We draw the atomic local P₂ patterns during the progression of mechanical stretching by using the Open Visualization Tool ^[54]. Fig. 12 shows patterns of PVDF at strain values of 0, 100%, 200%, 300%, and 400%. The P₂ maps reflect the extent of local ordered ⟨T⟩ chain segments. Blue indicates lower P₂ levels while red indicates higher values. In the stretching process, the local P₂ value becomes larger, showing the increase in the proportion of ⟨T⟩ dihedral linkages. This observation indicates mechanical stretching leads to a larger fraction of straight segments of the chain, which promotes the possibility of forming local ⟨T⟩ dihedral linkages, i.e., increasing dipole value. In addition, from these maps, an obvious heterogeneity of P₂ distribution is observed, especially in the strain = 400% case. The results suggest many aligning segments but not uniformly distributed at large deformation. It has been shown that the heterogeneous transformation distribution plays a critical role during the transformation of α- to β- phase ^[8]. We further show here that the ⟨G⟩ to ⟨T⟩ transformation occurs in the regions with the local P₂ higher than the average value.

Figure 12.

Deformation snapshots of PVDF at a strain range from 0 to 400 %. The atoms are colored by their corresponding local P₂ values. We only apply contour color to the backbone of PVDF.

We also draw the conformational change of two molecular chains during the mechanical stretching process in Fig. 13. Random orientation of backbone is observed in the initial state. When stretched, the two chains begin to align in the stretching direction. In this process, chain segments are straightened gradually, consistent with previous work ^[47]. A highlighted inset shows the local structure composition at strain = 400%, where atoms are colored based on their types. If not restricted, chain segments will get closer and straighter, forming the β-phase. However, this tendency does not necessarily lead to the β-phase. If restricted, the chain segments will form the γ-phase firstly. The β- and γ-phase representative structures are shown in the inset of Fig. 13. This phenomenon is observed for many chains in our simulations. The observed chain conformation change provides a molecular-level explanation for the dipole moment increment during mechanical stretching.

Figure 13.

The conformational change of two PVDF chains during the mechanical stretching process. The atoms are colored by their types in the inset.

3.6. Synergistic Effect of Electric Poling and Mechanical Stretching

The electric field directly impacts dipole moment by aligning side atoms (H and F atoms) in the chains, while the stretching is more effective in inducing the transformation of the backbone. As a result, both electric poling and mechanical stretching contribute to the growth of dipole but with different mechanisms. In many processing conditions, poling and mechanical stretching are applied synergistically to improve the piezoelectric properties of PVDF ^[9].

To examine the potential synergistic effect of poling and mechanical stretching, we first perform a 200% mechanical stretching simulation in the X direction at 400 K. This deformation level has not yet resulted in a significant increase in dipole, as shown in Fig. 10(c). After the stretching process, the electric field is then applied in the Y direction with an intensity of only 0.1 V/nm for as long as 50 ps. The dipole is collected every 0.05 ps. In comparison, the electric poling process is also applied to a model without stretching. A weaker intensity of 0.1 V/nm is used so that it does not dominate the dipole moment increment. The results of the dipole increment under poling with and without stretching are shown in Fig. 14. Due to the lower value of electric field intensity, the overall dipole moment is smaller than the result in Fig. 6(c). The dipole moment increases more rapidly for the pre-stretched PVDF system than the case without stretching. The results demonstrate the synergistic effect between electric poling and mechanical stretching. As the pre-stretching straightens the backbone of PVDF chains, as discussed in Section 3.4 and shown in Fig. 9 and 10, the H and F atoms within the chains can be better aligned under an electric field. Our study thus provides important insights into the molecular mechanisms that lead to the synergistic effect of mechanical stretching and electric poling. We note that the electrospinning technology can induce simultaneous mechanical stretching and electric poling when producing polymer nanofibers ^{[2, 55, 56]}.

Figure 14.

Synergistic effect of electric poling and mechanical stretching on dipole increment compared to the pure poling case under 400 K.

4. Conclusions

This paper investigates the chain conformation transformation of PVDF under the effect of temperature, electric poling, and mechanical stretching using MD simulations. We have systematically studied the underlying mechanisms that contribute to the mechanical and piezoelectric properties of the PVDF system.

We first show that the primary thermal and physical properties predicted by our model agree well with other studies. Two superimposed Gaussian distribution functions can depict the dihedral distributions of the PVDF backbone. We find that at equilibrium states, α-phase segments are dominant. The effective ⟨G⟩ to ⟨T⟩ transformation is difficult to occur under the influence of temperature alone.

Dipole values can increase considerably under a strong electric field. We observe that the change of total dipole moment is much more significant than the effect of P_T change. Our simulations indicate that electric poling increases the dipole moment values by aligning H and F atoms, significantly increasing the dipole moment in the poling direction. It is also observed that increasing the temperature is conducive to increasing the overall dipole moment.

Under mechanical stretching, the transformation from ⟨G⟩ to ⟨T⟩ dihedral linkages can be observed within chains after the yielding point. We deduce that γ-phase segments are formed before forming large-scale β-phase structures, resulting from the confinement effect of the surrounding chain segments. Only after reaching a certain strain level, the chain segments begin to straighten significantly and dipole moments start to rise quickly. The results of local P₂ maps indicate that the heterogeneous deformation plays a critical role in transforming from α- to β-phase, especially at large deformation.

Lastly, we show that the dipole increment of PVDF under electric poling after pre-stretching is much more significant than pure poling without pre-stretching, thus demonstrating the synergistic effect of electric poling and mechanical stretching. We also explain the synergistic effect at the molecular level, which is broadly applicable for the typical processing or fabrication technologies used on piezoelectric polymers, such as electrospinning.

In summary, our study systematically illustrates the molecular mechanisms of PVDF under different stimuli. The computational approach and the insights of deformation mechanisms obtained from this paper can be broadly applied to other piezoelectric materials. Future studies are planned to reveal more detailed structural transformation mechanisms and develop a molecularly informed coarse-grained model of PVDF to enable investigation at a larger spatiotemporal scale.