Introduction

Poly(vinylidene fluoride) (PVDF) is not only a high-performance engineering plastic with high mechanical strength and chemical resistance but is also a multifunctional polymer with piezoelectric, pyroelectric and ferroelectric properties.1 These properties originate from the dipole moment of the CH2CF2 units, which contain a positive H and a negative F. One of the characteristic features of PVDF is the existence of at least four crystalline polymorphs, which are referred to as Forms I, II, III and IV or β, α, γ and δ phases, respectively, and are illustrated in Figures 1 and 2. Form I consists of all trans molecules packed in a parallel manner. Form II consists of trans-gauche-trans-minus gauche () molecules packed in an antiparallel manner. An intermediate conformation of favors parallel packing to generate Form III. There is a parallel version of Form II that is called Form IV. Regarding the molecular packing of these forms, ‘parallel’ and ‘antiparallel’ are related to the dipole orientation in the direction perpendicular to the chain axis. Accordingly, Forms I, III and IV are polar crystals and exhibit spontaneous polarization Ps.

Figure 1
figure 1

The three conformations of PVDF chains observed in different polymorphs (a) all trans, (b) and (c) .

Full size image
Figure 2
figure 2

Crystal structures of PVDF. Black rectangle in each crystal is the crystallographic unit cell. The c-axis is parallel to the chain direction for all polymorphs. The b-axis is vertical direction for Form I and horizontal direction for other polymorphs. (a) Form I is parallel all trans. (b) Form II is antiparallel and up-down . (c) Form III is parallel and up-up . (d) Form IV is parallel and up-down .

Full size image

Form I has the largest Ps of three polar polymorphs of PVDF because all molecular dipoles are aligned in one direction. The spontaneous polarization can be switched by the action of a high electric field, so Form I PVDF is ferroelectric. Both Forms III and IV are polar because of their parallel packing. However, the properties of these polymorphs are not well understood because the samples are usually mixtures of various polymorphs as well as noncrystalline regions.

Spontaneous polarization is the property that reflects ferroelectric cooperative interactions, and can be experimentally determined by D–E hysteresis measurements.2, 3 The amount of switched polarization is called the remnant polarization Pr, which will be close to Ps if a single-crystal sample is measured. However, this is not the case for PVDF. Like most crystalline polymers, the coexistence of noncrystalline regions with crystalline domains is inevitable. Conventional Form I PVDF samples prepared by melt crystallization and cold drawing exhibit a crystallinity of about 50%. The reported values of Pr are scattered in the 50–80 mC m−2 range. Special treatments such as high-pressure crystallization and ultra-drawing can cause considerable increases in crystallinity and Pr. The largest value of Pr reported for PVDF is ca. 100 mC m−2.4

Theoretical prediction of Ps has been an important goal for years. The most primitive prediction can be made simply by summation of monomer dipole moments over a unit volume. Using data for the dipole moments of CF and CH bonds reported in the SI Chemical Data handbook, one obtains an isolated monomer dipole μ=2.1 D and Ps=130 mC m−2. This prediction is based on an assumption of rigid dipoles. In general, a dipole in a crystal receives an additional electric field called a local field that is the sum of the electric field generated by surrounding dipoles, which increases the effective dipole moment because of its polarizability. Assumption of the Lorentz field,5, 6 yields an increase of Ps to 220 mC m−2. The Lorentz field is based on an assumption of point diploes that are located on cubic or higher symmetry lattice points. Extended calculations of field sums for dipoles with a given separation of positive and negative charges and that are located on Form I PVDF lattice points,7, 8 eventually lead to a low local field and a Ps close to that of the rigid-dipole assumption. Recent calculations based on Hartree–Fock formalism,9, 10 molecular dynamics11, 12 and the density functional theory,13, 14 have shown that Ps in Form I PVDF is enhanced to 170–180 mC m−2, primarily because of interchain interactions.

We previously conducted quantum chemical calculations of Form I PVDF using a variety of functionals and basis sets.15 We found that the Perdew–Burke–Ernzerhof zero-parameter hybrid functionals (PBE0/cc-pVTZ)16, 17, 18 can reproduce the lattice parameters given by X-ray diffraction19 reasonably well. In this paper, we extend our calculations to Forms II, III and IV of PVDF and show that PBE0/cc-pVTZ can reproduce their crystal structures well. We extend calculations to simulate the valence X-ray photoelectron spectra (vXPS) and vibrational frequencies of these polymorphs. Finally, we evaluate the spontaneous polarization of polar Forms I, III and IV; similar enhancement is calculated from rigid-dipole criteria in all polar polymorphs.

Computational Details

All calculations were performed at the PBE0/cc-pVTZ level using the CRYSTAL 09 software.20 The PBE0 hybrid functional is defined as:

where are electron densities with spin up or down, respectively, and are the generalized gradient approximation functional,21 exchange functional and Hartree–Fock exchange energy, respectively.

By considering reasonable calculation results for Form I PVDF crystals,15 we determined the lattice constants and atomic positions of Forms II, III and IV of PVDF by geometry optimization with the Bloch periodic boundary condition using primitive cells. From the band structure and density of states, we simulated vXPS of the PVDF crystal forms. We also examined the normal mode vibrations of these polymorphs with PBE0/cc-pVTZ//PBE0/6-311G** level22 calculations and compared the results with infrared (IR) and Raman data. Finally, we derived Ps for the polymorphs through the localized Wannier function approach,23 using the calculated results for atomic positions and electronic distributions. The Wannier function of the n-th band at the lattice point is expressed as

where V is the volume of the unit cell, is the reciprocal vector, describes the Bloch orbital and the integration is performed over the Brillion zone. The polarization is calculated as

where e is the elementary charge and ZA and are the charge and position of each nucleus in a unit cell, respectively. The Mulliken population was also calculated so as to compare the results with the classical model of polarization.

Results and Discussion

The crystal structure of each polymorph was obtained by optimizing the corresponding initial structure in which chains with conformations of all trans (Form I), (Forms II and IV) or (Form III) were packed in a parallel (Forms I, III and IV) or an antiparallel (Form II) manner. Calculations of all crystal polymorphs were converged and every atomic position and electronic wave function at the ground state were obtained.

Packing energy

The total energy per CH2CF2 unit is −276.9385 Ha for Form I, −276.9396 Ha for Form II, −276.9395 Ha for Form III and −276.9394 Ha for Form IV. These energy values include the formation energy of atoms from nuclei and electrons and that of polymer molecules from these atoms. The relative energy to Form I is −2.9 kJ mol−1 for Form II, −2.7 kJ mol−1 for Form III and −2.5 kJ mol−1 for Form IV. These small differences are considered the reason why PVDF has multiple polymorphs.

Crystal structure

Lattice constants can be determined because the dispersion force is appropriately expressed in this calculation method. Calculated lattice constants are listed and compared with experimental values19, 24 in Table 1. The lattice constant along the chain direction, c, of each crystal form agrees with the corresponding experiment value within the experimental error. In addition, a and b also agree fairly well even though they are directions that are not connected by valence bonds and are generally difficult to determine by calculation. Thus, the calculated structure can be regarded as precise.

Table 1 Lattice constants of PVDF crystals. (Å for a, b and c. ° for β.). Calculated value (experimental value) and the relative deviation are shown in each column
Full size table

In the calculated Form I crystal, the bond length of C-H is 1.090 Å, C-F is 1.368 Å and C-C is 1.528 Å, and the angle of H-C-H is 108.0°, F-C-F is 105.4° and C-C-C is 114.4°. The lattice constant c and the angle of C-C-C is slightly, but significantly, larger than those of polyethylene (2.534 Å and 112°). This may be caused by the repulsion between fluorine atoms that are bonded to different carbon atoms. The C-F bond is longer than its typical value (1.34 Å) and the angle of F-C-F is smaller than that of a tetrahedron (109.5°). The F-F distance of 2.176 Å is less than twice the van der Waals radius of a fluorine of 1.35 Å. This suggests that two fluorine atoms bonded to the same carbon atom have some attractive interaction, which will be discussed further later. It should be noted that the statistical deflection of chains determined by the X-ray structural analysis19 could not be modeled in this calculation because just one monomer unit was repeated with periodic boundary condition.

Band structure and density of states

The band structures of the different polymorphs are shown in Figure 3. At about −35 eV, two split electron bands can be seen that are the 2s electrons of fluorine. This 2 eV gap provides further evidence for a weak attractive interaction between the two fluorine atoms of each CF2 unit; in other words, the fluorine atoms have a weak chemical bond and thereby the C-F bonds are weakened. Two groups of dispersed bands can be seen between −24 eV and −21 eV, which are σ bands, and between −20 eV and −18 eV, which are πb bands, respectively. Because these orbitals involve C-C bonds and are distributed along the c-axis, corresponding bands are dispersed along the c* direction, although they are degenerated along other directions.

Figure 3
figure 3

Band structures of polymorphs of PVDF (a) Form I, (b) Form III, (c) Form IV and (d) Form II. The crystallographic axes in primitive cells and the positions of k-points in reciprocal lattices are taken as indicated. For every form, a3 axis is the fiber axis and the direction is taken as the crystallographic axes form the right-handed system.

Full size image

Incidentally, the first conduction band of the Form I PVDF crystal has a lower energy than the vacuum level and is shaped like a quadratic function. As a result, this crystal may function as a wide-band-gap n-type semiconductor.25, 26, 27 In contrast, the other forms may not behave as semiconductors because the conduction bands of these crystals are higher than the vacuum level and are also degenerated.

Comparison of density of states and experimental vXPS

To simulate the vXPS of Forms I–IV, we used the energy levels of the valence regions for the density of states curves depicted in Figure 4, and estimated the intensity of vXPS from the relative photoionization cross-section for Al Kα radiation using the Gelius intensity model.28, 29 For the relative atomic photoionization cross-section, we used the theoretical values determined by Yeh that are presented in Table 2.30

Figure 4
figure 4

Density of states (DOS) of the four polymorphs of PVDF.

Full size image
Table 2 Relative photoionization cross-section of each atomic orbital for H, C and F atoms (relative to C 2s)
Full size table

For experimental vXPS of PVDF (shown in Figure 5), we can infer from simulated spectra that the intense peaks in the energy range of 30–40 eV are related to the F2s orbitals of PVDF. Two broad peaks between 15 eV and 25 eV are caused by the main C2s and partial F2s and 2p contributions, respectively. Two intense, broad peaks at lower energy (8–15 eV) are attributed to the F2p contribution to pσ(C2p-F2p) bonding at around 14 eV, and p lone-pair orbitals at around 10 eV. The simulated and experimental spectra show good agreement.

Figure 5
figure 5

Comparison of DOSs and experimental vXPS of the polymorphs of PVDF.

Full size image

Lattice vibration

The lattice vibration in each polymorph was analyzed and the dispersion relations of phonons were obtained. The phonon energies of IR-active modes at the Γ point for Form I are listed and compared with observed IR absorption bands in Table 3. The energies of almost all bands agree well, suggesting that the calculated packing state and conformation of chains are correct. Thus, vibration modes can be precisely assigned. Vibrational analyses for Forms II and III revealed that the simulated IR and Raman spectra agreed well with the experimental ones (details of these comparisons are provided in the Supplementary Information).

Table 3 Vibrational frequencies of Form I crystal of PVDF
Full size table

Spontaneous polarization

Forms I, III and IV of PVDF are polar structures. Ps perpendicular to the chain axis of each crystal form was calculated using the localized Wannier function as an equation (3). The results are listed in Table 4. The results obtained for Form I are almost the same as those reported by Nakhmanson et al.13, 14 even though their calculation method was different. As mentioned in the introduction, classical calculations give Ps of ca. 130 mC m−2 for PVDF. The larger value of the calculated Ps suggests that the polarization is enhanced in the Form I crystal; that is, the spontaneous polarization is stabilized by Coulomb interactions, resulting in ferroelectric cooperation. As reported in the previous paper,15 the calculated Mulliken charge for Form I shows the enhancement of polarization of the C-H bonding. The deficiency of electron at a hydrogen atom is calculated to be 0.289, that is four times larger than the classical estimation of 0.067.33 As the obtained excess of electron at a fluorine atom is 0.185 and classical estimation is 0.219, the enhancement of polarization of the C-F bonding is not so much. Thus, the enhancement of Ps in Form I crystal can be attributed, at least partially, to the enhanced polarization of the C-H bonding. Similar enhancement is found to occur in crystal Forms III and IV.

Table 4 Calculated and experimental spontaneous polarization perpendicular to the chain axis of different polymorphs of PVDF
Full size table

Because samples consisting of only crystal Forms III or IV cannot be obtained for PVDF, Ps for these crystal forms has not been determined experimentally. Thus, this is the first report estimating the values of Ps for Forms III and IV of PVDF. The values of Ps are listed in Table 4. Ps of Form IV is nearly half that of Form I and is consistent with the crystal structure because the chain includes gauche conformation and the dipoles incline by 60° from the direction of total polarization. In contrast, Ps of Form III is much smaller even though the inclination of dipoles is the same as that of Form IV. This may be related to the difference of ferroelectric cooperation in these two polymorphs.

In this study, the structures and electronic states of all polymorphs of PVDF were calculated by the density functional theory using the generalized gradient approximation combined with the Hartree–Fock exchange energy. Crystal structures of all polymorphs were precisely reproduced. Electronic structures and phonon dispersions were calculated and agreed closely with experimental values. The amount of spontaneous polarization in all polar polymorphs was also determined. The spontaneous polarization behavior of these polymorphs strongly demonstrates the importance of precise quantum chemical treatment to examine electric properties, including ferroelectricity. Among various approximation levels used in quantum chemical calculations, the PBE0/cc-VTZ method is shown to be suitable for modeling the electronic behavior in ferroelectric polymer systems. This may be one of the most promising methods to help understand the mechanism behind various properties of functional polymers as well as with designing new ones.