Extremal solutions of $\varphi-$Caputo fractional evolution equations involving integral kernels

1.   Introduction

Poly(vinylidene fluoride) (PVDF) and its copolymers are smart polymers and well known as ferroelectric materials. They are potential for the energy storage capacitors ^[1,2,3,4]. These polymers have a high dielectric constant, high dielectric breakdown strength and high electrical energy density (>20 J/cm³) ^[5]. To reduce the cooperative ferroelectricity in the PVDF crystals, various amounts of comonomers such as chlorotrifluoroethylene (CTFE), hexafluoropropylene (HFP) and trifluoroethylene (TrFE) are added at the beginning of the polymerization and randomly incorporated in PVDF ^[4,5,6,7,8]. It has been reported that both poly(vinylidene fluoride-co-chlorotrifluoroethylene) (P(VDF-CTFE)) and poly(vinylidene fluoride-co-hexafluoropropylene) (P(VDF-HFP)) provide a high energy density of ~25 J/cm³ and break down field > 600 MV/m ^[5,6], which achieved by reducing a crystalline size of the α-phase ^[9]. P(VDF-CTFE) shows a dramatic increase of the energy density with the electric field without saturation until reaching the breakdown field, which is different from a neat PVDF ^[10]. P(VDF-CTFE) random copolymers may have semicrystalline or amorphous structure dependent on CTFE contents. P(VDF-CTFE) containing a small amount of CTFE displays a hexagonal or monoclinic crystalline structure and becomes amorphous when CTFE is higher than 30% ^[11]. According to the beneficial property of P(VDF-CTFE), the main objective of this work is to investigate the effect of CTFE contents on dynamic relaxations of P(VDF-CTFE) copolymers as a function of temperature and frequency. This investigation would help to understand crystal structure and property relationships of these smart polymers. In addition, other than chemical, physical and ferroelectric properties of fluoropolymers, the antibacterial property has been previously found in polymers containing fluorine such as poly(ethylene terephthalate) coated with 2, 3, 5, 6-tetrafluoro-p-phenylenedimethanol and polytetrafluoroethylene ^[12,13]. This is due to fluorine atoms in fluoropolymers hindering the bacterial adhesion. However, there are a few investigations of antibacterial activity of commercial fluoropolymers have been reported. Therefore, in this research, a self–antibacterial property of P(VDF-CTFE) films, containing both fluorine and chlorine atoms was preliminarily studied, compared to neat PVDF. This preliminary result would further help to elucidate about the novel antibacterial polymers in nature.

In this current paper, the crystal structure and thermal property of PVDF and P(VDF-CTFE) were examined using Fourier transform infrared spectroscopy (FTIR), wide-angle X-ray diffraction (WAXD), small-angle X-ray scattering (SAXS) and differential scanning calorimetry (DSC). The dynamic relaxations were investigated using dynamic mechanical analysis (DMA) and broadband dielectric relaxation spectroscopy (DRS). These findings set the stage for understanding the crystal structure, properties, and dynamic behaviors of antibacterial PVDF containing CTFE.

2.   Materials and methods

2.1.   Materials

Solef® 1008 PVDF and Solef® 31008, 31508 and 32008 P(VDF-CTFE) copolymers were obtained and prepared in blown films. The blown films were produced by Brabender single screw extruder with extrusion and die temperature of 250 ℃. P(VDF-CTFE) 31008, 31508 and 32008 blown films containing 10, 15, and 20% of CTFE are referred to as PCTFE10, PCTFE15 and PCTFE20, respectively.

2.2.   Antibacterial activity test

To verify the effect of CTFE on antibacterial property of PVDF and P(VDF-CTFE) films, the antibacterial activity test was performed. Briefly Escherichia coli (E.coli) and Staphylococcus aureus (S. aureus) bacteria were cultured in Nutrient agar. Then 3–5 colonies were suspended into Mueller Hinton Broth (MHB) and incubated at 37 ℃ for 3 h. Each culture was diluted with phosphate buffer solution to 1.5 × 10⁸ colony forming unit per milliliter (CFU/mL). An amount of 100 µL of diluted suspension was transferred to 3 mL of MHB. Each sterile disc (6 mm in diameter) was transferred to the prepared E. coli and S. aureus suspension. After incubation at 37 ℃ for 24 h, the bacterial suspension was serially diluted with phosphate buffer solution. The bacteria in the diluted suspension were counted by a spread plate method. All tests were analyzed in triplicate. The effectiveness of the antibacterial activity (EAA) was calculated according to Eq 1 ^[14]:

where N₀ and N_S refer to the number of CFU/mL for blank (MHB) and the sample respectively.

2.3.   Sample characterization

2.3.1.   Crystal structure and thermal property

Attenuated total reflectance-Fourier transform Infrared (ATR-FTIR) spectra were acquired using a Thermo Scientific Nicolet 6700 spectrometer, signal averaging 200 scans at a resolution of 2 cm⁻¹.

Wide-angle X-ray diffraction (WAXD) patterns were collected on a Rigaku DMAX/rapid micro diffractometer in transmission mode using a copper point focused source (λ = 0.154 nm) at 50 kV and 40 mA.

Degree of crystallinity and thermal characteristics of PVDF and P(VDF-CTFE) films were performed with a Seiko differential scanning calorimetry (DSC) in the temperature range from −90 to 200 ℃, at a heating rate of 10 ℃/min.

Small-angle X-ray Scattering (SAXS) profiles were collected on a Rigaku SAXS instrument with a CuK_α radiation source (λ = 0.154 nm) and a two-dimensional multi-wire detector with the sample-to-detector distance of 1.5 m. The diameter of the incident X-ray beam at the sample position is approximately 0.8 mm. The diameter of the sensitive area of the detector is 120 mm. Absolute scattering intensities were calculated by comparing the spectra of samples to that of a precalibrated polyethylene (S-2907) secondary standard. Data were azimuthally averaged to yield a one-dimensional profile of the absolute intensity I(q) versus scattering vector q (q = (4π/λ)sinθ, where 2θ is the scattering angle). Lorentz-corrected SAXS intensity (Iq²) versus q will be plotted to gain the details of crystalline microstructures. The one-dimensional correlation function K(r) versus the correlation distance r will calculated based on Eq 2 ^[15]:

The lamellar thickness, amorphous layer thickness and the linear degree of crystallinity in the lamellar stack were determined from the one-dimensional correlation function base on the first intercept of K(r) with the abscissa B and the long period L_p (L_p = B/x_cx_a, where x_c, x_a are the crystalline and amorphous volume fraction). The L_p obtained from the first maximum of the one-dimensional correlation function.

2.3.2.   Dynamic properties

Dynamic mechanical properties were determined in the tension mode using a TA-Q800 dynamic mechanical analysis (DMA), at frequency of 1 Hz and a heating rate of 5 ^oC/min. The storage (E') and loss moduli (E") were determined as a function of the temperatures range from −70 to 160 ℃. The rectangular films of 17 × 6.5 mm and 0.03–0.09 mm thick were used.

Broadband dielectric relaxation spectroscopy (DRS) measurements were performed on a Novocontrol GmbH Concept 40 broadband dielectric spectrometer in the frequency domain range from 0.01 Hz–1 MHz in the temperature range from −100 to 160 ℃. Temperature stability was controlled within ±0.2 ℃. Sample films were sandwiched between brass electrodes of 10 mm in diameter.

3.   Results and discussion

3.1.   Antibacterial activity test

The verified antibacterial property of PVDF and P(VDF-CTFE) is provided in Table 1. The antibacterial efficiency of MHB (blank), PVDF and P(VDF-CTFE) copolymers against E. coli and S. aureus was reported as the effectiveness of the antibacterial activity (% EAA) and the average number of survival bacteria colonies (CFU/mL). PVDF and all P(VDF-CTFE) displayed the antibacterial efficiency against both E. coli and S. aureus and the effectiveness increases with increasing CTFE contents. P(VDF-CTFE) samples designated the better effectiveness against S. aureus than E. coli, compared to neat PVDF. PCTFE20 yielded the antibacterial effectiveness up to 73.01 and 70.21% for S. aureus and E. coli, respectively. In addition, the findings presented that the EAA of PVDF is higher than 50% for both S. aureus and E. coli. This indicates that PVDF itself has rather a good antibacterial property. The finding also shows that introducing chlorine atoms on polymer chains helps improving the antimicrobial activity. This might be due to chlorine atoms of CTFE enhancing antifouling performances by precluding the protein accumulating on polymer surfaces. The mechanisms how to kill bacteria is still unclear. However, from our preliminary experiments it is more likely that polymers containing fluorine and chlorine atoms act as anti-biofilms and then reject the adhesion of bacteria ^[12,13,16]. This preliminary investigation indicated that polymers containing both chlorine and fluorine atoms are more efficiency to prevent the adhesion of bacteria on the surface ^[17] than that having only fluorine atoms.

3.2.   FTIR characterization

FTIR spectra was used to confirm the state of crystalline phases of PVDF and P(VDF-CTFE). Figure 1 displays the comparative FTIR spectra of PVDF and P(VDF-CTFE) in the region from 550 to 1500 cm⁻¹. All films exhibit strong absorption bands at ~614 and 763 cm⁻¹, which have been assigned to CF₂ bending in the α-phase PVDF. The peaks at ~795 and 854 cm⁻¹ correspond to CH₂ in-plane bending. The absorption bands at ~975, 1148, and 1210 cm⁻¹ is attributed respectively to the CH₂ out-of-plane bending, CF₂ symmetric stretching, and CF₂ asymmetric stretching. These bands have been found to be the characteristic of the vibration absorption of the TGTGʹ conformation, anti-parallel orientation of C–F bonds (nonpolar α-phase) ^{[11,18,19,20,21]}. P(VDF-CTFE) samples exhibit a small peak at 840 cm⁻¹, which correspond to the vibrational frequencies of an all-trans conformation, two neighboring chains parallel to each other called polar β-phase ^[22].

3.3.   WAXD

Figure 2 displays WAXD pattern of PVDF compared to P(VDF-CTFE) films at 2θ region from 10º to 50º. Owing to the previous investigation ^[23], the prominent peaks were assigned to the peaks of the α-crystalline phase. Neat PVDF shows the prominent peaks of α (100), α (020), α (110), α (120), α (130) and α (200) ^[19]. Compared to neat PVDF, P(VDF-CTFE) exhibits three prominent peaks of α (100), α (110) and α (200). The crystal plane of (020), (120) and (130) disappeared for P(VDF-CTFE) due to the CTFE molecules disrupt the molecular chain packing of neat PVDF, which confirms by the degree of crystallinity shown in Table 2. Unlike FTIR results, no β-crystalline phase was observed for P(VDF-CTFE) copolymers. This is probably due to a tiny β-crystalline phase was formed.

Diffraction peaks were deconvoluted with PeakFit® software by fitting with a Gaussian curve, using a linear baseline. The degree of crystallinity (X_cw) was obtained from the fraction of the area under the crystalline peaks to the total area of the amorphous halo and crystalline ^[24]. As results given in Table 2, the X_cw decreases with increasing CTFE contents, which is in good agreement with that observed from DSC technique, illustrating how the bulky CTFE disrupts crystal formation.

3.4.   DSC

As seen in Figure 3a, the melting temperature (T_m), a small endothermic transition (T_cd) and glass transition temperature (T_g) are commonly observed for all PVDF and P(VDF-CTFE) samples. Upon cooling, the polymers show sharp exotherms associated with the recrystallization temperature at temperatures lower than T_m (Figure 3b). The inset in Figure 3a displays the glass transition temperature of the samples, which about −37, −36, −26 and −20 ℃, respectively. The transition around 50–60 ℃, denoted as T_cd, has been previously assigned to the conformation disorder (T_cd) transition in the PVDF crystals ^[19], which is clearly observed for P(VDF-CTFE) compared to neat PVDF. Although this peak has been assigned to the Curie transition (T_c), ferroelectric-paraelectric transition (TF-P), or the transformation of polar (β, γ, and δ) phases to nonpolar α-phases ^[15], it is not the case for our investigation. To prove this case, a stretched PVDF film (ST-PVDF), containing almost completely polar β-crystalline phases and an unstretched PVDF film, containing only nonpolar α-crystalline phases (see inset in Figure 3c) were prepared and tested. As seen in Figure 3c, this transition remained the same enthalpy for both stretched and unstretched PVDF films. Therefore, this peak cannot be the transformation of polar to nonpolar phases and the strong transition was observed for all P(VDF-CTFE) copolymers, which contain tiny β-crystalline phases compared to the neat PVDF. In addition, this transition cannot be a Curie temperature, since T_c was previously reported that it relates to mole fractions of VF₂ in PVDF copolymers so T_c of neat PVDF was expected to near 200 ℃ ^[25]. In addition, T_c of P(VDF-CTFE) should be higher than 50 ℃ and suppose to reduce with increasing CTFE contents. Therefore, from our finding, we proposed that this transition (T_cd) is attributed to the conformation disorder in the nonpolar α-crystalline phases.

To confirm the degree of crystallinity obtained from WAXD, the degree of crystallinity was estimated from the enthalpy of T_m peaks (in the temperature range of 100−190 ℃) divided by the ideal melting enthalpy of 100% crystallinity PVDF ( = 104.5 J/g) ^[19,26]. Compared with WAXD result, similar trend was observed; the degree of crystallinity (X_c) decreases as the amount of CTFE increases (see Table 2).

3.5.   SAXS

Figure 4 shows the Lorentz-corrected SAXS intensity (Iq²) vs. q and the one-dimensional correlation function for PVDF and P(VDF-CTFE) samples. As seen in Figure 4, there is no significant difference between P(VDF-CTFE) and PVDF films. The long period (L_p), lamellar thickness (l_c = x_cL_p), amorphous layer thickness (l_a = x_aL_p) and the linear degree of crystallinity in the lamellar stack (X_L = l_c/(l_c + l_a)) were determined, which are about 89, 69, 26 and 0.73 Å, respectively. The findings suggest that the amount of CTFE does not significantly impact on the value of the long period, lamellar thickness, amorphous layer thickness and linear degree of crystallinity of neat PVDF. This verified that for neat PVDF and P(VDF-CTFE) copolymers with different amounts of CTFE 10–20% (PCTFE10, PCTFE15 and PCTFE20), CTFE molecules do not affect the overall shape of copolymers but to an extent alter the internal arrangements of the PVDF chain. As a result, PVDF and P(VDF-CTFE) have very similar SAXS profiles and different WAXD patterns in some 2θ positions.

3.6.   DMA

The plots of storage (E') and loss moduli (E") vs. temperatures for PVDF and P(VDF-CTFE) samples are shown in Figure 5. At temperatures above the T_g (about –37 ℃), two relaxations were clearly observed for PVDF (α₁ and α₂) and P(VDF-CTFE) copolymers (α₁ and α_c).

To clarify the process above T_g, tan δ (tan δ = E"/E') was plotted as a function of temperatures (Figure 6). Relative to E", three relaxations (α₁, α_c and α₂) can be seen for P(VDF-CTFE) polymers. Previously, the α₁ and α₂ have been assigned to the segmental motion of amorphous chains and the local conformational rearrangement of the α-crystalline phase, respectively ^[19]. The α_c transition was only observed for P(VDF-CTFE) copolymers, therefore the α_c transitions are more likely associated to the motions within regions of CTFE ^[27]. This transition will be further elucidated along with DRS studies. Interestingly the loss magnitude of the α_c was much higher when the CTFE content increased, PCTFE20 > PCTFE15 > PCTFE10. This tendency follows a typical motion of amorphous chain segments, which indicates that amorphous chains are free from crystal restraints. PCTFE20 requires the lowest energy to move because it contains the lowest degree of crystallinity.

3.7.   DRS

The dielectric constant ε' and dielectric loss ε" were collected from DRS measurement. At high temperature with low frequency, the conduction loss created from the motion of the impurity ions obscures some dipole relaxations. The first derivative of ε' (${\varepsilon }_{der}^{"} = -\frac{\pi }{2}\frac{\partial {\varepsilon }^{'}\left(f\right)}{\partial lnf}$) has been used to elucidate the relaxation of the conduction-free dielectric loss, helpful for identifying a dipole relaxation of materials with high conduction losses. In addition, previously the derivative loss (ε"_der) has been shown to partially resolve overlapping relaxations ^[28]. Figure 7 shows the temperature dependence of ε"_der at a frequency of 1 Hz of the neat PVDF and P(VDF-CTFE) copolymers. Four relaxations were clearly observed for PVDF (β, α₁, α₂ and α_MWS) and five relaxations were observed for the P(VDF-CTFE) (β, α₁, α_c, α₂ and α_MWS). Both PVDF and P(VDF-CTFE) copolymers exhibit the β and α₁ relaxations at the same temperature range below and above T_g, respectively. At temperature above T_g, PVDF exhibits the α₂ relaxation and a Maxwell–Wagner–Sillars (MWS) interfacial polarization (α_MWS) between the amorphous and crystalline phases, while the P(VDF-CTFE) displays an extra relaxation, referred to as the α_c relaxation.

To deep investigation, the dielectric relaxation spectroscopy data processing was performed. The frequency dependence of ε" at various temperatures was plotted and fit with the appropriate empirical Havriliak-Negami equation as shown in Eq 3 ^[29]:

The characteristic relaxation time (τ_HN) and dielectric relaxation strength (Δε) were obtained by fitting with Eq 3. The frequency maximum was calculated from Eq 4 ^[30]:

where ω is the angular frequency. m and n are the shape parameters. σ₀ is the dc conductivity (S/cm), ε₀ is the permittivity of free space and the parameter s relates to the type of conduction process.

PVDF and P(VDF-CTFE) copolymers clearly exhibit four strong relaxations, denoted as β, α₁, α₂, and α_MWS as temperatures increase. The β relaxations were observed below T_g. This relaxation appeared within a temperature range from −100 to −50 ℃. Above −50 ℃ this relaxation was ceased and taken over by the α₁ relaxation (see Figure 7). The α₂ relaxations were observed from −10 to 160 ^oC and the α_MWS begins to occur at ~100 ^oC.

The temperature dependence of all relaxations, except α₁ process, follows the Arrhenius law, ƒ_max = ƒ₀exp[–(E_a/RT)] as shown in Figure 8. The α₁ process has the temperature dependence following the Vogel-Fulcher-Tammann (VFT) relation, ƒ_max = ƒ₀exp[–DT₀/(T–T₀)] ^[31].

Below T_g, the β process exhibits an Arrhenius behavior with the activation energy range of 43–57 kJ/mol as given in Table 3, which is previously assigned to the local relaxation, arising from rotational motions of amorphous chains. The second relaxation appears above T_g but lower than T_cd, denoted as the α₁ process, and followed the nonlinear VFT behavior. Therefore, the α₁ relaxation is assigned to the segmental motion of amorphous chains, which agrees well with previous investigations ^[19]. The α₂ relaxation that is slower than the α₁ process exhibits Arrhenius temperature dependence which represents the typical local relaxation behavior with the activation energy range of 73–91 kJ/mol (see Table 3). Since this relaxation was observed at the temperature below T_cd and follows Arrhenius equation, it is more likely related to a local non-cooperative relaxation. Furthermore, as given in Table 4 the Δε (α₂) is twice higher than the Δε (β) but less than the Δε (α₁). According to the evidence from the structure and thermal properties in previous sections and our previous works, this relaxation relates to the local conformational rearrangement of the nonpolar α-crystalline phases of VDF ^[19]. It has been reported that the conformation disorder could change between TGTGʹ and GTG'T conformation ^[32,33]. In addition, this relaxation was clearly observed although CTFE is incorporated, which has no significant difference of the activation energy with increasing CTFE contents. According to FTIR results, PVDF completely contains the nonpolar α-crystalline phase, while P(VDF-CTFE) contains slightly polar β-crystalline phases. This confirms that the α₂ process involves with only the nonpolar α-crystalline phases.

As seen in Figure 7, the α_c process was observed for P(VDF-CTFE) copolymers but absent for neat PVDF, consistent to DMA results. Unfortunately, this process is merging with the α₁ relaxation, and results to the broadening of the α₁ relaxation for PCTFE10 and PCTFE15 but slightly excluded for PCTFE20. However, the α_c process of PCTFE20 merges with the α₁ relaxation when frequency is higher than 100 Hz. From the author's knowledge, this relaxation is first time observed. However, unfortunately we are unable to elucidate more information of this relaxation because this weak relaxation exhibits at the frequency closed to that of the segmental relaxation. Based on DMA results, this relaxation is clearly observed for all P(VDF-CTFE) samples with higher loss modulus magnitudes when CTFE increases. This behavior is more likely reflected to the segmental relaxation ^[34]. Therefore, we proposed that the α_c process is the molecular motion of CTFE chain segments, which is accompanied by VDF segments.

The α_MWS process shows the Arrhenius temperature dependence with the high activation energy (E_a > 100 kJ/mol). Due to a high crystallinity in PVDF and P(VDF-CTFE) copolymers, the moving of free ions from amorphous phases is blocked by the crystalline phases and causes a strong increase in ε' at low frequency and high temperatures. Therefore, the α_MWS process was proposed to a manifestation of MWS interfacial polarization, arising in multiphase systems. The sharp increase of ε" at highest temperatures and low frequency is due to electrode polarization (EP), contributed from the accumulation of mobile ions at an electrode interface ^[19].

The strength parameter (D) was obtained from the VFT behavior of the α₁ relaxation, which represents the topology of potential energy governing the segmental reconfigurations of polymers, reflecting the nature of bond interactions and the configurational entropy available to the system ^[19,30]. Typically, D was used to express the fragility of polymers, higher D corresponding to lower fragility and higher the configurational entropy available to the system. As seen in Table 3, the fragility decreases with increasing CTFE contents because of lower degree of crystallinity (see Table 2) and greater entropy of the system. Comparisons of dielectric relaxation strength (Δε) of β, α₁, and α₂ processes, at selected temperatures, are provided in Table 4. The Δε of β, α₁, and α₂ relaxations increases with rising CTFE content. An increment of Δε is mainly influenced by an increase of the dipole moment of CTFE structure.

4.   Conclusions

Neat PVDF had completely nonpolar α-crystalline phases, while P(VDF-CTFE) copolymers contained slightly polar β-crystalline phases. Three transition temperatures were observed for all samples, which are the glass transition temperature (T_g), the melting temperature (T_m), and the conformation disorder in nonpolar α-crystalline phases (T_cd). There were no significant changes on the long period, lamellar thickness, amorphous layer thickness and linear degree of crystallinity when CTFE was copolymerized with VDF up to 20%. The degrees of crystallinity decreased with increasing CTFE contents. At temperatures above T_g, three relaxations (α₁, α₂ and α_c) were clearly observed by DMA for P(VDF-CTFE) copolymers. Four relaxation processes were clearly observed in DRS spectra for the neat PVDF which are attributed to the local motion of amorphous chains (β), the segmental relaxation of amorphous chains (α₁), the local conformational rearrangement of the TGTGʹ conformation (α₂) and the process arising from Maxwell–Wagner–Sillars interfacial polarization (α_MWS). Other than four relaxations, the extra relaxation was observed for P(VDF-CTFE) copolymers, which was more likely associated to the molecular motion of CTFE chain segments (α_c). P(VDF-CTFE) copolymers are effective on inhibiting viable growth of both S. aureus and E. coli up to 73.01 and 70.21%, respectively. The antibacterial effectiveness increased with increasing CTFE contents for both S. aureus and E. coli. Chlorine atoms existing in fluoropolymers enhanced an antibacterial ability.

Acknowledgments

The author would like to special thank Prof. James Runt, The Penn State University, USA for all supports. Authors thanks the Faculty of Science, Burapha University, Thailand.

Conflict of Interest

The authors declare no conflict of interest.