Introduction

The dynamics of molecular motions in systems which do not crystallize, even on slow cooling, but freeze into a glassy state is as yet not fully understood though various attempts have been made [1–6]. Glass-forming liquids and polymers are considered to be complex molecular systems and the structural α-relaxation can not be described by Debye relaxation function with a single relaxation time. In order to get an unambiguous quantitative description of the molecular dynamics in these complex systems, variable temperature and multifrequency study should be performed. It is obvious that there is no single spectroscopy or other technique that can cover the whole dynamical window, which ranges from thousandths of hertz up to thousand of gigahertz on the frequency scale; therefore, several methods have to be used and combined to study the dynamic phenomena. Among the methods to be used there are the quasielastic neutron scattering [6], the mechanical and dielectric relaxation [7–20], the rheo-dielectric method [21, 22], the nuclear magnetic resonance (NMR) [9, 23–27] and others. The relaxation phenomena observed by various techniques are expected to be different projection of the same structural relaxation mechanism. Furthermore, it should be possible to analyze the dynamical data, obtained by different methods, using the same model of motion [6, 28].

The subject of our paper is the poly(vinylidene fluoride) (PVDF) polymer. The PVDF and its copolymers with trifluoroethylene are important for applications in electromechanical and pyroelectric devices [29], in biomedical engineering [30] and recently also in high-density non-volatile memory cells of low writing voltage [31, 32].

PVDF is a semicrystalline polymer presenting pronounced polymorphic crystalline forms. It can crystallize into five possible conformations, three of which are most frequently occurring [33]. Almost planar zig-zag all-trans (TTTT) chain conformation of PVDF, packed in an orthorhombic unit cell exhibits ferroelectric properties—polymorph I (β). The CF2 dipoles are parallel to the orthorhombic b-axis and the long-range interactions are so strong that the Curie temperature is located close below the melting point. Non-polar polymorph II (α) consists of trans–gauche (TGTG′) conformation of chains packed antiparallel in a monoclinic unit cell, whereas in the crystal form III (γ) of PVDF the chains with TTTGTTTG′ conformation are comprised in an orthorhombic polar unit cell. The conditions under which a specific conformation can be obtained depend strongly on the processing, thermal or mechanical treatments that the polymer undergoes. The ferroelectric–paraelectric phase transition is induced by cooperative modification in the chain conformation from all-trans to TGTG′ and TTTGTTTG′.

Despite many experimental and theoretical studies the glass transition still appears an unresolved problem in condensed matter physics. To describe non-Debye character of the α-relaxation at temperatures above the glass transition, the main-chain segmental motion has been accounted for, by employing Havriliak–Negami (HN) empirical equation, where both a distribution of correlation times in the system as well as the correlation of motion [34, 35] is assumed. By using HN formula, both the dielectric and the NMR experiments can be satisfactorily interpreted. The characteristic relaxation time τ(T), obtained from different methods, exhibits non-Arrhenius temperature behaviour, which has later been parameterized by Vogel–Tamman–Fulcher law [36–38] or equivalent Williams–Landel–Ferry equation [7].

The main goal of this work was to examine the molecular dynamics of the PVDF polymer by means of NMR, dielectric and mechanical relaxation. The characterization yields experimental data covering the correlation time widow of ten decades and therefore allows to avoid erroneous dynamic information.

Experimental section

Samples

PVDF powder from Nitrogenous Concern, Tarnów, Poland was used in NMR experiments, whereas the dielectric and mechanical spectroscopy studies were performed with radially oriented PVDF film. The film samples, ~100 μm thick and 11 mm in diameter, were obtained from PVDF powder by hot-pressing at 450 K and 600 MPa for 30 min. After hot-pressing the samples were cooled down to room temperature during ~30 min under the pressure and relaxed at room temperature without pressure for 3–5 days.

To characterize studied materials Differential Scanning Calotymetry (DSC), Wide Angle X-ray Scattering (WAXS) and NIR Raman spectroscopy were performed. DSC measurements were collected over a temperature range of 320–440 K on a Netzsch DSC 200 calorimeter with a scanning rate of 4 and 2 K per minute on heating and cooling, respectively. The sample, in the form of a powder, was placed in aluminium pans with pin hold lids. In the PVDF polymer thermogram (Fig. 1), single peak at 432.8 K related to the melting process is observed. Melting exotherm is shifted to lower temperatures at cooling process and located at 410 K.

Fig. 1
figure 1

DSC curve of the PVDF powder measured on heating and cooling the sample

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Wide angle X-ray scattering measurements of PVDF powder and films were performed with a PANalytical X’Pert PRO X-ray powder diffraction system and Cu Kα radiation. The film was measured in a transmission mode with films placed between two Kapton foils. The measurements were done with Cu Kα focusing X-ray mirror system (Cu Kβ radiation was suppressed to 0.5% of its original level). The diffraction pattern was registered using the PIXcel detector. The data were collected for 2θ range from 10° to 50° with a Δθ step equal to 0.01° during 16 h and worked out at the use of special computer program: X’Pert HighScor Plus version 2.0 (2004). Figure 2 shows an example of the X-ray pattern of hot-pressed PVDF film (the effect of Kapton films was extracted).

Fig. 2
figure 2

Wide-angle X-ray diffraction profile of hot-pressed PVDF film

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We found that the degree of crystallinity of hot-pressed film is 51% and that of the powder is 56%. The amorphous phase of PVDF can be considered as consisting of statistical distribution of various rotational isomeric states.

To characterize local structure of the polymer at the molecular scale both in the crystalline and the amorphous phase NIR Raman spectroscopy was used. The spectra were recorded at room temperature using Bruker IFS 66FRA 106 spectrometer (λinc = 1064 nm, p = 50 mW, Δspectr = 4 cm−1). Figure 3 shows Raman spectra in the fingerprint wavenumber range for PVDF sample in the form of powder and the hot-pressed film. To characterize the phase content we used the assignment of Raman bands of PVDF published earlier [39] with the same notation: νa and νs asymmetric and symmetric stretching vibrations, respectively, δ—bending vibration, r—rocking vibration. At high wavenumbers one observe two overlapping bands at 880 cm−1 originating from stretching vibrations νs(CF2) + ν(CC) in the phase I and phase III and at 874 cm−1 related to a combination of stretching and bending vibrations νs(CC) + δ(CCC) in the phase II. The intensity of the band, characteristic of the phases containing three and more T sequences, is lower than that of the TG conformation for both samples. A strong band at 839 cm−1, due to r(CH2)+νa(CF2), is also present for the phase I and III whereas the r(CH2) vibrations in the polar phase III are evident only as a shoulder. Very strong band at 797 cm−1 is characteristic of the νs(CF2) vibrations in TGTG′ conformation. It appears that Raman spectra of PVDF in the form of powder and hot-pressed film do not differ considerably. The both samples contain all three polymorphous modifications and from the intensity ratio of the band at 797 cm−1 and the band at 839 cm−1 one can state that the hot-pressed sample is a little bit richer in the TG conformation than the PVDF powder.

Fig. 3
figure 3

Raman spectra in the fingerprint wavenumber range for PVDF sample in the form of powder (solid line) and hot-pressed film (thick line)

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Dielectric measurements

Dielectric response of PVDF films was measured in the frequency range 100 Hz–1 MHz using computer controlled HP-4284A LCR Meter. The samples with gold-sputtered electrodes were placed in an Oxford Instruments Cryostat CF 1240, whereas the temperature was changed from 150 to 450 K at a rate of 1 K/min. The real and imaginary parts of the dielectric permittivity (ε* = ε′ − iε″) were obtained as a function of frequency and temperature.

Mechanical measurements

The dynamic mechanical behaviour of PVDF rectangular hot-pressed films (3–5 mm × 5 mm) was studied with NETZSCH DMA–242 using the tensile mode in frequency range 1–25 Hz. The dynamic force was 1 N and the proportion factor amounted to 1.2. The room temperature fatigue was assessed to be ~5% after 4 h. The experiments were performed in the temperature range between 150 and 470 K at a heating rate of 1 K/min and temperature variation of real and imaginary components of the complex Young’s modulus (E* = E′ − iE″) was obtained.

Nuclear magnetic resonance

For the NMR experiment, polymer powder was stored in glass tubes, evacuated at room temperature at ~4 × 10−6 hPa for 20 h to remove oxygen, and sealed under vacuum. The proton spin–lattice relaxation (T 1) measurements were carried out on a SXP 4/100 Bruker pulsed NMR spectrometer at a Larmor frequency of 90 MHz between 200 and 440 K. The temperature of the sample was controlled by means of a continuous nitrogen gas-flow system. An accuracy of the measured temperature was 1.0 K. At least 30-min time was allowed to stabilize the temperature of the samples. Spin–lattice relaxation time was determined by the inversion recovery method using a 180°–τ–90° pulse sequence. The recovery of the magnetization was found to be exponential within experimental error at all temperatures.

Results and discussion

Dielectric relaxation

The vacuum dipole moment of vinylidene fluoride unit μ(VDF) = 7.07 × 10−30 Cm originates from the distribution of positively charged protons and negatively charged fluorine ions and lies in the plane of C–C bond (see Fig. 4). The dipoles are attached to the main chain allowing a study of the dielectric response related to segmental motions, local modes and reorientations leading from all-trans conformation in the crystalline state to TGTG′ and TTTGTTTG′ conformations.

Fig. 4
figure 4

Schematic drawing of the spatial dipole moment arrangement of vinylidene fluoride (VDF) unit

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Normalized dielectric relaxation function can be expressed as:

$$ \frac{{\varepsilon^{*} (\omega ) - \varepsilon_{\infty } }}{{\varepsilon_{\text{s}} - \varepsilon_{\infty } }} = {\Upphi^{*}} (\omega ), $$
(1)

where ε ∞ and ε s denote the non-relaxed and completely relaxed permittivity value, respectively.

The relaxation function Φ*(ω) of polymeric systems can be satisfactorily described by the Havriliak–Negami (HN) empirical equation [34, 35], which assumes a distribution of correlation times in the system and also the correlation of the motion, and is defined by:

$$ {\Upphi^{*}} (\omega ) = \frac{1}{{[1 + ({\text{i}}\omega \tau_{\text{HN}} )^{\alpha } ]^{\beta } }} . $$
(2)

For the imaginary part of the permittivity it can be rewritten as:

$$ \frac{{\varepsilon^{\prime \prime } (\omega )}}{{\varepsilon_{\text{s}} - \varepsilon_{\infty } }} = \sin \left[ {\beta \arctan \left( {\frac{{\sin \left( {\frac{\alpha \pi }{2}} \right)}}{{\left( {\omega \tau_{\text{HN}} } \right)^{ - \alpha } + \cos \left( {\frac{\alpha \pi }{2}} \right)}}} \right)} \right] \times \left[ {1 + 2\left( {\omega \tau_{\text{HN}} } \right)^{\alpha } \cos \left( {\frac{\alpha \pi }{2}} \right) + \left( {\omega \tau_{\text{HN}} } \right)^{2\alpha } } \right]^{ - \beta /2} , $$
(3)

where α (0 < α ≤ 1) and β (0 < αβ ≤ 1) are two parameters characterizing the symmetric and asymmetric broadening of the dielectric band, respectively and τ HN is a characteristic time of the relaxation process. It should be noticed that the Eq. 1 leads to a simple Debye law for α = β = 1 and to the Cole–Cole [40] for β = 1 and Davidson–Cole [41] function for α = 1.

Dielectric response in the low-temperature range, shown in Fig. 5a, is characteristic of segmental motion, i.e., freezing of dipolar motions in the amorphous phase of the semicrystalline polymer. The response is similar to that reported earlier for PVDF [10, 15]. The dielectric dispersion shows cusp-like temperature dependence, whereas the dielectric absorption increases with temperature and the maxima shift towards higher temperature with increasing frequency. The results obtained for normalized imaginary part of the permittivity ε″(ω)/ε″max at different temperatures are shown in Fig. 5b. It is clearly visible only at 260 K the loss curve is well defined in the whole measuring frequency window. Therefore, the fitting procedure of Eqs. 1–3 was performed in the following way: first we have determined α and β parameters at T = 260 K and than we have adopted the obtained values for the other temperatures. The solid lines in Fig. 5b show the best least-squares fit to HN equation with α = 0.56 and β = 0.36. The agreement between experimental and calculated curves is satisfactory. The temperature behaviour of the dielectric relaxation time τ HN determined from the above fitting procedure will be discussed below and combined with that of the mechanical and NMR relaxation times, characteristic of the same process.

Fig. 5
figure 5

a Temperature dependence of real and imaginary part of the dielectric permittivity of radially oriented PVDF at various frequencies. b Normalized imaginary part of the dielectric permittivity at 240 (filled square), 250 (star) 260 (open square), 270 (inverted triangle), 280 (open circle) and 290 K (diamond). The solid lines represent the HN fit with α = 0.56 and β = 0.36

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Mechanical relaxation

Figure 6 shows mechanical response of PVDF in the temperature range of the segmental motions. Similar temperature variation of piezoelectric and electrostrictive response as well as mechanical storage and loss moduli were reported in PVDF earlier [14, 19, 42, 43] but here we measured both mechanical and dielectric response using the same hot-pressed PVDF films with well-defined degree of crystallinity and polymorphs content.

Fig. 6
figure 6

Real and imaginary part of the complex dynamical modulus at various frequencies

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The frequency range of our dynamic mechanical measurements in PVDF polymer was not broad enough to define the whole mechanical loss modulus curve. As the E″(ω, T) response determines the relaxation frequencies ω and the relaxation time τ HN ~ 1/ω we used the mechanical response in the temperature regime (Fig. 6) to calculate the characteristic time τ HN values from the maximum of the absorption curves E″(T). It should also be noticed that in the glass transition range, the behaviour of the mechanical loss modulus E″ (Fig. 6) is similar to that of dielectric losses ε″ (Fig. 5a), i.e., they increase with increasing frequency.

Nuclear magnetic resonance

Proton spin–lattice relaxation time T 1 versus inverse of temperature for PVDF powder, plotted in a logarithmic scale, is shown in Fig. 7. The data display a single, quite broad and asymmetric minimum of 0.157 s at 329.9 K, related to segmental motion in the amorphous phase. The uncertainty of the measurements is about ± 8% and is represented by the size of the symbols. Above 430 K a sudden decrease in T 1 vs 1000/T was observed (not shown in Fig. 7). This change is attributed to the melting process in polymer and its value is in a good agreement with the corresponding melting temperature (see Fig. 1).

Fig. 7
figure 7

1H spin–lattice relaxation time versus reciprocal temperature in PVDF polymer at the Larmor frequency of 90.00 MHz. Solid line represents the best theoretical fit of the HN function with α = 0.59 and β = 0.35

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A qualitative analysis of the spin–lattice relaxation behaviour was based on the assumption that temperature dependence of T 1, as governed by the dipolar interaction modulated by the motional process, can be written as a linear combination of spectral densities J(ω L) [44]:

$$ \frac{1}{{T_{1} }} = C[J(\omega_{\text{L}} ) + 4J(2\omega_{\text{L}} )] , $$
(4)

where ω L denotes Larmor frequency and C is a constant related to the fraction of the second moment, which corresponds to the dipolar interaction averaged by the motional process under consideration. The spectral density J(ω) can be expressed in terms of motional correlation times by means of equations depending on the theoretical or semiempirical model chosen to describe the molecular dynamics. In our case, the spectral density is based on Havriliak–Negami relaxation function [35] and is defined by [45]:

$$ J(\omega ) = \frac{2}{\omega }\sin \left[ {\beta \arctan \left( {\frac{{(\omega \tau_{\text{HN}} )^{\alpha } \sin \left( {\frac{\alpha \pi }{2}} \right)}}{{1 + (\omega \tau_{\text{HN}} )^{\alpha } \cos \left( {\frac{\alpha \pi }{2}} \right)}}} \right)} \right] \times \left[ {1 + (\omega \tau_{\text{HN}} )^{2\alpha } + 2(\omega \tau_{\text{HN}} )^{\alpha } \cos \left( {\frac{\delta \pi }{2}} \right)} \right]^{ - \beta /2} . $$
(5)

For the systems, which deviate most strongly from Arrhenius behaviour, the temperature dependence of a correlation time τ HN can be analyzed using the Vogel–Tamman–Fulcher (VTF) formula [36–38]:

$$ \tau_{\text{HN}} = \tau_{0} \exp \left( {\frac{B}{{T - T_{\text{VTF}} }}} \right), $$
(6)

where τ 0 is the relaxation time in the limit of high temperature, B is related to the apparent activation energy \( \left( {B = \left[ {E_{\text{a}} \left( {1 - T_{\text{VTF}} /T} \right)^{2} } \right]/R} \right) \), and T VTF is the empirical Vogel–Tamman–Fulcher temperature, which is usually 50 K lower than the glass transition temperature T g, below which the segmental motion of the main chain is completely absent. The dynamic glass transition temperature (T DRSg ) can be calculated [46]:

$$ T_{\text{g}}^{\text{DRS}} = T_{\text{ref}} = \frac{B}{{\ln (1/\tau_{0} )}} + T_{\text{VTF}} $$
(7)

where T ref is defined as the temperature at which the segmental relaxation time is 1 s (τ max = 1 s).

The fragility, F which is a measure of the ability of a material to change its conformation across the glass transition region [47], can be calculated from the VTF fitting parameters using [48]:

$$ F = \frac{{B/T_{\text{ref}} }}{{\ln (10)(1 - T_{\text{VTF}} /T_{\text{ref}} )^{2} }} $$
(8)

Larger fragility has been correlated with stronger intermolecular coupling or larger segmental size [48, 49].

Formalism mathematically equivalent to the VTF equation and also often used to characterize polymeric systems is the Williams–Landel–Ferry (WLF) equation [7]:

$$ \log a_{10} \left[ {\tau_{\text{HN}} (T)} \right] = \frac{{C_{1} (T - T^{*} )}}{{\left[ {T - \left( {T^{*} - C_{2} } \right)} \right]}} $$
(9)

where C 1 and C 2 are constants and T* is a reference temperature.

When the measurement range includes the glass transition temperature T g, obtained from calorimetry or dilatometry, it appears natural to choose T g as T*. If T g was defined in a consistent way by measurement of thermal or volumetric changes at fixed scan rate Q, usually 1 K/min, then the parameters C 1 and C 2 appeared to have universal values of 17.4 and 51.6 K, respectively [46]. C 1 and C 2 are related to the VTF parameters (if T g is used as a reference temperature) through the relations C 2 = T g − T VTF and C 1 = B/(2.303C 2).

To analyzed the NMR relaxation data Eq. 4 after inserting Eqs. 5 and 6 was used to fit the experimental data on Fig. 7, in the temperature range from 272 to 430 K. The best fit was judged as that with the lowest χ 2 value. The solid line in Fig. 7 represents the best result of this routine with α = 0.59 and β = 0.35. These values are close to those obtained from the dielectric measurements above and the agreement between experimental and calculated curves is acceptable. The values of the characteristic time τ HN (T) in the temperature range covered experimentally can be determined for a given set of α and β parameters, using Eqs. 4–6.

The temperature dependence of characteristic relaxation time τ HN, obtained from dielectric and NMR spectroscopy experiments together with mechanical relaxation data is shown in Fig. 8.

Fig. 8
figure 8

Temperature dependence of the HN relaxation times for the α-process of PVDF obtained by mechanical (triangle), dielectric (star) and NMR (circle) measurements. The solid line corresponds to the Vogel–Tamman–Fulcher fit defined by Eq. 6

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It is visible that the values of τ HN (T) obtained by different relaxation experimental techniques show the same, non-Arrhenius temperature behaviour and are of similar magnitude. Therefore, the temperature behaviour of the relaxation times from dielectric, mechanical and NMR can be parameterized by one Vogel–Tamman–Fulcher law (Eq. 6). The fitted values of τ 0, B and T VTF are listed in Table 1 along with the calculated T ref and F values. As can be seen in the Fig. 8 (solid line), the VTF law fits very well the temperature dependence of the correlation time τ HN of the macroscopic α-relaxation. The small deviation of the correlation times (triangles) obtained from the mechanical relaxation is due to the different method of the calculation τ HN in relation to those acquired from dielectric and nuclear magnetic relaxations.

Table 1 Vogel–Tamman–Fulcher parameters for the α-relaxation process
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As seen from Table 1, the temperature T VTF is 51 K lower than the glass transition temperature T g. In addition the calorimetrical T g [50] and the calculated dynamic glass transition T DRSg values are consistent.

From the equivalent Williams–Landel–Ferry equation (Eq. 9) we obtained C 1 = 12.2 and C 2 = 56 K. Though these values are not very adjacent to the so called “universal” ones (C 1 = 17.5 and C 2 = 52 K, respectively) they are still, however, within the range of numbers commonly available for various polymers [9, 26].

Compatible values of α and β parameters of the Havriliak–Negami equation have been determined from the fitting procedure of the dielectric (α = 0.56, β = 0.36) and NMR (α = 0.59, β = 0.35) data. These values indicate that the segmental main-chain motion are intermediate correlated (α ≈ 0.57) and strongly distributed (αβ ≈ 0.21). The value of C constant (Eq. 4) amounts to 1.86 × 109 s−2, and is of the expected order of magnitude.

Conclusion

We have shown that the combined dielectric, nuclear proton and mechanical relaxation measurements, performed over a wide temperature and frequency range on PVDF polymer, are a powerful tools for the characterization of molecular dynamics and the α-relaxation in such complex systems.

Havriliak–Negami empirical equation was used to analyze the relaxation data and the obtained results show that the dynamics of segmental motion can be well described by the same HN parameters (α ≈ 0.57 and β ≈ 0.35) in a wide time range. This indicates that the correlations between local conformational transitions in the amorphous phase of PVDF are intermediate and characterized by a pronounced distribution of the correlation times. As PVDF is a semicrystalline polymer it is not possible to determine directly “stretching” of the exponential behaviour, therefore we characterized the temperature dependence of the relaxation times. The temperature variation of the characteristic times τ HN, deduced from our analysis, deviates from the Arrhenius behaviour and can be described by Vogel–Tamman–Fulcher, or equivalent Williams–Landel–Ferry equations. The values of τ HN (T) determined by different relaxation experimental techniques consistently fit to Vogel–Tamman–Fulcher law with the same values of τ 0 = 1.23 × 10−12 s, B = 1388 K and T VTF = 182 K in the whole temperature range.