Abstract
Poly(vinylidene fluoride) (PVDF) and its copolymers are promising candidates for energy-harvesting devices because of their flexibility, environmental friendliness, lightweight, and high halogen and acid resistance. However, the relatively low piezoelectricity limits their applications. The piezoelectricity of PVDF and its copolymers highly depends on the polar β-phase, while the non-polar α-phase is the most common one. As a result, the β-phase formation and α- to β-phase transformation have attracted much attention in recent years. Stretching is a widely used method to induce the α- to β-phase transformation for the improvement of piezoelectricity. In this work, the influences of the parameters during stretching on phase evolution and piezoelectricity are discussed and summarized. Besides, nontraditional stretching methods are also introduced and discussed. This work will provide important information for preparing high-performance piezoelectric polymer films.
Nomenclature
- PVDF
-
poly(vinylidene fluoride)
- XRD
-
X-ray diffraction
- v
-
drawing speed
- T s
-
stretching temperature
- R
-
elongation ratio
- F(β)
-
relative fraction of the β-phase
- X c
-
crystallinity degree
- TCD
-
tip-to-collector distance
1 Introduction
As potential energy-harvesting materials, poly(vinylidene fluoride) (PVDF) and its copolymers have great advantages such as excellent processing ability, corrosion resistance to acids and alkali, flexibility and thermal stability. PVDF possesses at least five polymorphs, among which the β-phase is the most important one due to its high piezoelectricity. In the initially prepared PVDF films, the non-polar α-phase usually dominates, which does not contribute to piezoelectricity. In order to obtain high-piezoelectric PVDF films containing a high content of the β-phase, a number of methods including stretching, nanofiller addition, poling, and heat treatment have been developed [1,2,3,4,5].
The addition of fillers is widely used to improve the piezoelectricity of PVDFs. In 2020, Chamakh et al. added ZnO star-like fillers (d ≈ 2.5 μm) into PVDF and investigated its effects on the structural and physical properties. The β-phase fraction increases from 44.1 to 66.4% after the addition of 10 wt% ZnO microparticles [6]. Compared to micro-fillers, nanofillers have unique advantages due to their large specific area. In recent years, various nanofillers (carbon nanotubes, vapor-grown carbon fibers, graphene, graphene oxide [GO], metal nanowires, ionic liquid, etc.) have been utilized as reinforcing agents. The enhancement mechanism includes nucleate agents and the interaction between the molecular chains and the nanofillers [4,7,8]. The effects of poling include α- to β-phase transformation and the realignment of dipole moments. In the prepared films without poling, the dipole moments are randomly distributed; thus, the total piezoelectricity is extremely weak or does not exist because of the mutual offset of the dipole moments. During the poling process, the applied high electric field can force the dipole moments to align along the electric field direction, thus inducing piezoelectricity [9–11]. Heat treatment can affect the crystallization process, and thus, proper heat treatments can also result in high β-phase [12,13].
Compared with the above-mentioned methods, stretching has obvious advantages in the α- to β-phase transformation. Phase transformation needs external energy to overcome the energy barrier between the phases. In the poling process, although there is stress applied to the molecular chains, it is an order lower than that of stretching, and thus not enough external energy can be provided. Limited by the breakdown strength, the poling electric field cannot be higher than the ultimate value (usually lower than 150 MV/m); therefore, stretching is a more efficient method to promote crystal phase transformation [14,15]. It should be noted that the piezoelectricity not only depends on the fraction of the electric active β-phase, but also the alignment of the dipole moments is also very important. Therefore, the crystallographic plane orientations should be carried out by high electric poling or other methods, which is not discussed in detail due to the length limit. In some reports, the enhancement of β-phase is considered equal to the enhancement of piezoelectricity, while the following processes are not mentioned.
Traditional mechanical stretching has the advantages of low cost, simple, and practicable. Different from traditional mechanical stretching, the simultaneous poling and stretching process has also been developed. In addition, the stretching process also exists in electrospinning, which has characteristics of simplicity, low cost, and beneficial to small diameters. But the promotion of electrical instabilities within a charged fluid jet and a long jet path in solvent evaporation readily result in nanofibers and ultra-fine fibers. To solve this problem, researchers have developed a technology similar to electrospinning, namely melt electro-writing (MEW), which is distinct from electrospinning as the electrical instability is suppressed due to precisely placed fibers, which can keep a fluid column against breaking at low flow rates. However, it might require high pressure to extrude the material when the filler content is high, which are not achievable by pneumatically driven MEW printers, or lead to nozzle clogging [16,17]. In this work, the influences of stretching parameters on the phase transformation of PVDF and its copolymers are discussed in detail and summarized. Our work will provide a solid foundation for the development of perfect piezoelectric polymers and composites.
2 Influences of the parameters of stretching on the α- to β-phase transformation
2.1 The α-to β-phase transformation mechanism during stretching
During the stretching process, the transformation process from α-crystal to β-crystal of PVDF by mechanical stretching is revealed and is summarized schematically in Figure 1. The α spherulite of PVDF with folded chains can be readily obtained under conventional processing conditions (Figure 1a). When mechanical drawing is applied on the polymer, the transition starts from the middle of the spherulite, where the molecular chains are first extended along the drawing direction (Figure 1b). With the deformation of the α-phase, the conformation is formed in the emerged β-phase gradually (Figure 1c). More and more extended chains formed with the evolution of the deformation of PVDF; therefore, the α-phase transforms into the β-phase under large deformation (Figure 1d). The above is the recrystallization process inside PVDF during stretching [18]. The recrystallization process is also affected by the temperature, as discussed in Section 2.2.1.
![Figure 1
Schematic drawing of the transformation process from α-crystal to β-crystal of PVDF by mechanical stretching [18].](/document/doi/10.1515/phys-2022-0255/asset/graphic/j_phys-2022-0255_fig_001.jpg)
Schematic drawing of the transformation process from α-crystal to β-crystal of PVDF by mechanical stretching [18].
Compared to non-stretched films, the stretched films usually contain a much higher content of β-phase F(β) [19]. The α- to β-phase transformation mechanism during stretching can be described as follows: (1) The applied stress along the stretching direction forces the spherical crystal into a microfiber-like crystal and (2) the fiber-like structure is beneficial for the formation of the TT conformation (β-phase) than the TGTG′ conformation (α-phase). However, the effect of stretching on the crystallinity degree varies with specific processes. Both increment [20,21] and decrement [22,23] have been reported. The increment in the crystallinity degree after stretching may be attributed to the alignment of molecular chains along the stretching direction, and thus, the microcrystalline zone is formed, resulting in the formation of the crystal structure. In contrast, stretching may also destroy the current crystal structure and lead to the formation of a semi-crystal zone, which reduces the crystallinity [22–25]. In the stretched films, the molecular chains mainly align along the stretching direction. Therefore, it is good to work at the 31 mode after subsequential poling.
The parameters of mechanical stretching include the stretching temperature (T s), elongation ratio (R), and stretching speed (ν). Besides, the addition of nanofillers can also affect phase evolution, as discussed in Section 2.2.
2.2 Influences of stretching parameters on the phase transformation
2.2.1 Stretching temperature (T s)
Temperature is an extremely important factor affecting phase transformation. In some reports, the samples are kept at a certain temperature for a short time before and/or after stretching for preheating or eliminating internal stress [26]. Mun et al. stretched the PVDF hollow fiber membranes by 1.5 times for 60 s. Before stretching, the films were kept in distilled water at 60°C for 60 s. After stretching, the annealing process at 50–120°C was applied. It was found that the relative fraction of the β-phase F(β) reached 68.9% and the crystallinity degree was 44.4% when the stretching temperature was 60°C. However, the tensile strength was the lowest [27]. Temperature above 100°C may reduce phase transformation efficiency. When the temperature is above 120°C, the transformation will not occur even when the elongation ratio is higher than 4. The crystallinity degree may increase significantly, and the crystal–amorphous interphase region may be reduced because of the increase of the thickness of the crystalline layer induced by the stretching process [28–30]. Kim et al. stretched the PVDF hollow fiber membranes at a temperature from 30 to 90°C. It was found that 60°C is the right temperature, at which the relative fraction of the β-phase F(β) may reach 70%. Lower or higher temperatures lead to lower F(β) [31]. Sencadas et al. investigated the α- to β-phase transformation during the stretching process at different temperatures (80–140°C). They studied the stress–strain relationship for the samples stretched at 80°C, as shown in Figure 2(a). With the increase of R, the samples undergo three stages subsequently: yielding, necking, and strain hardening. The maximum stress existed at the post-yield softening and a plastic plateau before rupture (R ≈ 8). With stretching, the crystal structure and the phase transformation proceed. For larger deformation, the microscopic deformation mechanism is characterized by the transition from the spherulitic structure to the microfibrillar structure accompanied by the destruction of the lamellar morphology. As mentioned above, PVDF with folded-chain α-spherical crystals can be easily obtained under conventional processing conditions. During stretching, necking and thickness reduction occur in the films. Foldable α-spherical crystal is extended along the stretching direction and gradually spreads toward the unstretched area, i.e., the fragmentation of the crystalline area inside the PVDF matrix. As the deformation occurs, the area of the extended chains expands laterally from the middle of the spherulite. Accompanied with it, the α spherical structure transforms into a microfiber structure, and small blocks of lamellae are torn away from the original lamellae to form a fibrillar structure, which is corresponding to the β-phase. This mechanism induces all-trans planar zigzag conformation into the crystals. Overall, the stretching process is the destruction and reorganization of the internal crystalline region, as shown in Figure 2(b)–(e). For all the samples, F(β) increases with R and reaches a plateau near 5. In contrast, F(β) decreases with temperature, as shown in Figure 2(f)–(g). The temperature increment may decrease the viscosity, leading to the matrix being more ductile [32].
![Figure 2
(a) Stress (σ)–strain (ε) curve for an original α-PVDF sample uniaxially stretched at 80°C. Scanning electron microscopy (SEM) micrographs obtained from samples stretched at 80°C: (b) R = 1, (c) R = 3, (d) R = 4, and (e) R = 5. F(β) as a function of stretching ratio R (f) and stretching temperature T
s (g) [32].](/document/doi/10.1515/phys-2022-0255/asset/graphic/j_phys-2022-0255_fig_002.jpg)
(a) Stress (σ)–strain (ε) curve for an original α-PVDF sample uniaxially stretched at 80°C. Scanning electron microscopy (SEM) micrographs obtained from samples stretched at 80°C: (b) R = 1, (c) R = 3, (d) R = 4, and (e) R = 5. F(β) as a function of stretching ratio R (f) and stretching temperature T s (g) [32].
In order to investigate the mutual influences of T s and R, Ting et al. investigated the influences of biaxial stretching on the piezoelectricity of PVDF thin films by varying the stretching ratio and temperature. From SEM images, the uniformity in the biaxial direction is confirmed. However, the high stretching ratio (R = 4 × 4) can induce porosity and cracks, though it shows better surface chain orientation and smoothness. To avoid crystal defects, the elongation ratio is suggested to be set in between R = 3 × 3 and R = 4 × 4. The β-phase fraction F(β), crystallinity degree and piezoelectric coefficients (d 31 and d 33) of the samples decrease with the temperature for all samples (R = 2 × 2, R = 3 × 3, R = 4 × 4, R = 5 × 5) when the temperature is above 80°C [33]. Li et al. studied the transformation from α- to β-crystal of the PVDF membrane that is carried out under the conditions of different stretching temperatures, stretching rates, and tensile elongation. The stretched films were characterized by Fourier transform infrared (FTIR) spectroscopy and X-ray diffraction (XRD) to analyze quantitatively the transformation from α- to β-crystal. The results indicate that low temperature is conducive to the formation of the β-crystal and the stretching temperature around 100°C is suitable for the transformation under stretching. With the increase of the tensile elongation, the relative crystallinity would also increase and achieve a maximum value when R is above 3. However, the stretching rate has less effect on the transformation from α- to β-crystal [34]. Debili et al. analyzed the stretched films under different temperatures (75, 80, 90, 100, and 110°C) with R = 4 by FTIR and XRD techniques. It was observed that a relatively low temperature (80 and 75°C) induces higher β-phase, while at higher temperatures, the transformation is not sufficient. It is because the semi-nanocrystalline biphasic (α + β) system is formed in the samples stretched at T s > 80°C. In contrast, the single nanocrystalline β-phase is formed when T s ≤ 75°C [34]. The addition of nanofillers also affects the crystal structure in the prepared films. Lee investigated the influences of temperature on the crystal structure and thermal properties of PVDF-HFP/CNF (0.2 wt%) by uniaxially stretching the films (R = 4.5) at 50, 75 and 100°C, respectively. The SEM images confirm the orientation of the CNF and the PVDF-HFP matrix. In the SEM images, the aggregation of CNFs in the matrix is observed in the undrawn samples. When the samples are drawn at 50°C, the PVDF molecular chains align along the stretching direction partially, and it is improved in the samples drawn at 75°C. The samples drawn at 100°C show partial melting of the matrix and wrapping in the CNFs, as shown in Figure 3(a)–(d) [21].
![Figure 3
SEM images of CNF/PVDF-HFP composite films at various stretching temperatures: (a) undrawn, (b) drawn at 50°C, (c) drawn at 75°C, and (d) drawn at 100°C [21].](/document/doi/10.1515/phys-2022-0255/asset/graphic/j_phys-2022-0255_fig_003.jpg)
SEM images of CNF/PVDF-HFP composite films at various stretching temperatures: (a) undrawn, (b) drawn at 50°C, (c) drawn at 75°C, and (d) drawn at 100°C [21].
Guo et al. investigated the stretching-induced phase evolution of PVDF/Gr composites by X-ray scattering. The results show that the α- to β-crystal transformation is based on the orientation evolution of molecular chains and it would not occur when R is less than the threshold value of 1.5. It is concluded that the α- to β-phase transformation goes through an indirect approach via the intermediate state of stretching-induced fragmentation and recrystallization process, rather than a direct evolution, as shown in Figure 4. When T s is 30°C, the adequate concentrated stress on the crystal phase results in the β-phase formation from the initial recrystallization. While in stretched films at 160°C, the α- to β- phase transformation is hindered, which is attributed to the worse compatibility of PVDF and Gr and the high chain mobility at a higher temperature [35]. It is different from most reports in which stretching is carried out from 60 to 100°C, so the effect of temperature is highly dependent on the specific process.
![Figure 4
Schematic diagram of the PVDF/Gr structure evolution induced by tensile deformation [35].](/document/doi/10.1515/phys-2022-0255/asset/graphic/j_phys-2022-0255_fig_004.jpg)
Schematic diagram of the PVDF/Gr structure evolution induced by tensile deformation [35].
2.2.2 Elongation ratio (R)
The elongation ratio is another decisive factor in the stretching process. Generally, F(β) increases with the elongation ratio due to the formation of the fiber-like structure of β-phase molecular chains by the applied stress [36]. It may also cause problems such as structural defects and decrease of crystallinity [23,37]. During the stretching process, the molecular chains are forced to be elongated, and thus, the β-phase is preferred (a(α) = 0.496 nm; a(β) = 0.858 nm). Zhou et al. prepared PVDF/PMMA blend films by biaxial stretching and investigated the effects of the PMMA content and elongation ratio on the crystallinity degree and the β-phase fraction. The results show the presence of the β-phase in the initial crystallized PVDF/PMMA blend films, while F(β) decreases in the pre-heating process. During biaxial stretching, F(β) shows an increasing trend and reaches the maximum value of 93% in PVDF/PMMA (30 wt%) with an elongation ratio R = 2 × 2. It is remarkable that both the reduction of the PMMA content and the increase of the stretching ratio improve the crystallinity degree. The interaction between –C═O groups (PMMA) and –CH2 groups (PVDF) can also induce β-phase formation. Compared to uniaxial stretching, the stress distribution is more uniform, leading to the uniform distribution of the β-phase fraction [38]. Chang et al. investigated the phase transformation of PVDF by a micro-testing machine. It was found that the phase transformation increases with the elongation ratio until 4 and then stays at a plateau. Therefore, for PVDF films, the elongation ratio R = 4 is enough for the α- to β- phase transformation [39]. Ji et al. prepared PVDF hollow fiber membranes by a green melt-spinning and stretching process without an organic solvent and low-molecular-weight diluent. The effect of elongation ratio was investigated by SEM, as shown in Figure 5. It is obvious that with the increase of the elongation ratio, the micro-structure of PVDF becomes fiber-like, which is preferred to form the β-phase [40]. Ozkazanc et al. investigated the effects of the elongation ratio on structures stretched at a constant speed (0.005 mm/s) at 155°C (a temperature much higher than that usually reported). It was found that with the increase of the elongation ratio, the specific volume in the amorphous phase decreases [41,42]. The crystallinity degree may increase with the elongation ratio [43,44]. Wang et al. fabricated high-piezoelectric PVDF films (reverse piezoelectric coefficient: −37 pC/N) by mechanical stretching of the initial crystallized films coated on stretchable poly(vinyl alcohol) substrates with a speed of 80 μm/s at 110°C. The AFM images show that the stretching process results in parallel strip-like structures along the stretching direction, indicating the formation of the extended β-phase. R = 3 is sufficient for phase transformation, while higher R (5) may cause problems such as higher roughness and crystal defects [44]. For biaxial stretching samples, the increase of F(β) has also been observed in biaxial stretching, and a maximum value of 84% is reached at R = 4 × 4 [33]. The phase transformation during the stretching process begins from the middle of α-spherulite and then extends to the neighboring spherule with the increase of the elongation ratio. F(β) increases with the elongation ratio until 3 and then stays at the plateau, indicating R = 3 is enough for sufficient α- to β-phase transformation [34]. Jain et al. investigated F(β) and X c under different stretching conditions by XRD, differential scanning calorimetry (DSC), and FTIR techniques. It was found that both the crystallinity degree X c and F(β) increase with the stretching temperature until 80°C and then decrease [45].
![Figure 5
(a) Schematic description of the preparation process of the melt-spinning hollow fiber membranes. (b) The prepared nascent hollow fibers. The SEM images of the prepared hollow fiber membranes with different elongation ratios: (c) R = 1, outer surface; (d) R = 2, outer surface; (e) R = 3, outer surface; (f) R = 1, inner surface; (g) R = 2, inner surface; and (h) R = 3, inner surface [40].](/document/doi/10.1515/phys-2022-0255/asset/graphic/j_phys-2022-0255_fig_005.jpg)
(a) Schematic description of the preparation process of the melt-spinning hollow fiber membranes. (b) The prepared nascent hollow fibers. The SEM images of the prepared hollow fiber membranes with different elongation ratios: (c) R = 1, outer surface; (d) R = 2, outer surface; (e) R = 3, outer surface; (f) R = 1, inner surface; (g) R = 2, inner surface; and (h) R = 3, inner surface [40].
In summary, for sufficient stretching, R should be at least 3, and in most cases, it exceeds 4. Combined with a moderate temperature, the effects of stretching can be improved. As is well-known, the crystal structure is highly related to the temperature. During stretching, the high temperature can increase the mobility of molecular chains, which makes the stretching more easy and prevents the fracture. However, the ultra-high temperature may result in a decrease in crystallinity degree and β-phase fraction [44,45].
2.2.3 Speed (v)
Speed is also a key factor affecting phase formation. In the current research, the conclusions about the drawn rate on the phase evolution in the reports vary greatly due to differences in experimental techniques and environments. Magniez et al. developed high-β-phase PVDF fibers by a melt-spinning method through unidirectional stretching at 120°C with different elongation ratios (1.25, 1.5, and 1.75). It was found that the β-phase fraction increases with the elongation ratio regardless of the flow rate. When the elongation ratio remains unchanged, the β-phase fraction decreases with the flow rate [46]. In contrast, for hard elastic films prepared by casting or hot pressing methods, when they are subjected to uniaxial stretching, a low stretching rate may lead to insufficient initial layer separation, while an excessively high stretching rate may lead to the destruction of the crystalline region inside the PVDF. However, there is currently significant controversy on the impact of the stretching rate on the internal phase transition and crystallinity of the film. Lei et al. investigated the crystalline morphology and microstructure during the stretching of PVDF hard elastic films under room temperature at different strain rates (0.003–0.034 s−1). It was found that the β-phase content decreases with the strain rate when it increases from 0.003 s−1 (39.8%) to 0.017 s−1 (9.8%) but increases with the strain rate to 0.034 s−1 (26.3%). In other words, when the strain rate is below 0.003 s−1, it is not sufficient to separate the initial lamellae. When the strain rate is >0.003 s−1, the lamellae separation occurs, leading to the lower β-phase content. For a higher faster rate (such as 0.034 s−1), the stress can induce sufficient fragmentation and reorganization of thinner crystals, leading to higher β-phase content [47]. In other reports, the higher the strain rate, the higher the phase transformation, which is completely different from other published reports [48,49]. Li et al. studied α- to β-phase transformation under different conditions (T s, R, and ν). In their experiments, the speed (0.06, 3, 6, 12, and 60 mm/min) has little impact on the transformation, which is completely different from other reports [18]. Regarding crystallinity, it was reported that speed can have a significant effect on it. Lee and Cho stretched PVDF/CNF films (T s = 125°C, R = 4) at various speeds and measured their crystallinity. It was found that with the increase of the temperature, the crystallinity degree decreases gradually (100% min−1: 50.5%; 5,000% min−1: 40.2%) [50]. Tang et al. also reported similar results [51].
In summary, the impact of the stretching rate on the crystal phase transition and crystallinity of thin films is controversial in current reports. As PVDF is a thermoplastic material, the ductility and elastic modulus of the film are usually temperature dependent. At room temperature, the film has low ductility and low internal molecular activity, and the phase transition of the film will start only when the tensile rate reaches a certain threshold. When the stretching temperature is higher, the mobility of molecular chains in PVDF is higher and the stretching rate has little effect on the phase transformation; however, if the stretching rate is very high, the crystallinity degree will decrease significantly.
2.2.4 Nanofiller addition
The addition of nanofillers is a widely used method to prepare PVDF-based composites with a high β-phase fraction [52–55]. The nanofillers mainly work in the initial crystallization stage as nucleate agents, which is beneficial for the direct formation of the β-phase. As a result, the relative β-phase fraction F(β) is quite high in the prepared films containing a high content of nanofillers [20]. Thus, the α- to β-phase transformation is not so obvious in the subsequent stretching process. While for those samples containing lower nanofillers (<threshold value), there is an improvement, which is attributed to the stress concentration (the stretching force tends to be centralized in the nanofiller/PVDF interface). In contrast, when a neat PVDF film is stretched, the stretching force only works on the molecular chains [22]. Mishra et al. investigated the effects of mechanical stretching on the β-phase formation in PVDF/rGO and PVDF/GO composites. The maximum F(β) reaches about 85% (PVDF/GO(0.5 wt%), R = 2, F(β) = 86%; PVDF/rGO(1 wt%), R = 2, F(β) = 84), accompanied with dramatically improved dielectric properties. Compared to non-stretched films, it is more than a 10% increase. This behavior is attributed to the strong interaction between PVDF and GO or rGO, i.e., the maximum stress is applied during the stretching process. The dielectric constant (highly related to the piezoelectricity) also increases with the addition of nanofillers [23]. He et al. prepared PVDF/organosilicate (OS) composites containing 2 and 4 wt% OS and investigated the effects of added nanofillers. Based on previously published reports, it was concluded that the improvement effect of OS is induced by the electrical interfacial interactions between the positively charged OS surface and the partially negative –CF2 bonds of the PVDF molecular chains, as shown in Figure 6 [20]. Yu and Li stretched PVDF/Ni composites at 120°C and investigated the effects of the stretching ratio on the dielectric properties. The results indicated that the addition of Ni particles can strengthen the interaction between Ni particles and the polymer matrix, and thus, the samples containing more Ni nanofillers exhibit a higher rupture ratio and stretching-induced percolation threshold [56]. The stretching process realigns the molecular chains and the added nanofillers attribute to the space-charge accumulation at the two-phase boundary during the stretching process [56]. Jahan et al. explored the polar phase transformation and the crystallinity degree of PVDF/CaCO3/clay composites affected by the CaCO3 fraction and elongation ratio. It was confirmed that the addition of fillers could improve the formation of the β-phase. For stretched (T: 90oC, R: 4–5) samples containing high filler contents (>2 wt% clay), the β-phase predominates regardless of R, and the α-phase even disappears. It means that the added fillers play a significant role in the β-phase formation during the stretching process because of its nucleate effect, which can hinder polymer mobility during melt crystallization, and therefore, more extended chain β-crystals are more easily formed. The added fillers may lead to the decrease of the crystallinity degree, which plays a negative role in piezoelectricity improvement [43].
![Figure 6
Schematic illustration of the proposed interfacial interaction between PVDF and OS for the improvement of β-phase content in PVDF [20].](/document/doi/10.1515/phys-2022-0255/asset/graphic/j_phys-2022-0255_fig_006.jpg)
Schematic illustration of the proposed interfacial interaction between PVDF and OS for the improvement of β-phase content in PVDF [20].
Molecular weight (M w) is important for PVDF’s properties. Mrlík et al. systematically studied the effects of main parameters such as molecular weight, initial sample thickness, stretching and poling. It was found that the highest degree of β-phase (90%) was realized with a moderate molecular weight (M w = 275 kDa) and the highest axial elongation (500%). In the low elongation ratio (<500%) stretching, the piezoelectric charge coefficient d 33 is highly dependent on the M w, while with higher elongation ratio, the M w does not affect it obviously [57]. As mentioned above, the stretching parameters and the nanofiller addition both have significant influences on the phase transformation and piezoelectricity. Furthermore, they also have co-effects between them. For example, in the stretching of composites, the addition of nanofillers may result in crystal defects (sometimes even visible) more easily than pure PVDF films, so the elongation ratio should be lower. While the phase transformation mechanism in the stretching still remains not so clear, it calls for more efforts from the related researchers.
2.3 Nontraditional stretching
In recent years, besides traditional mechanical stretching, several novel methods have been developed to improve the stretching effects [58,59]. Zhang et al. proposed a process combination of simultaneous biaxial stretching and solid-state shear milling (S3M). Compared to traditional uniaxial stretching, both the relative fraction of the β-phase and the total content of β-phase increase as shown in Figure 7 [24].
![Figure 7
(a) Scheme of conformation conversion of the α-phase to the β-phase in PVDF. (b) The relationship between F(β) in PVDF powders and pan-milling cycles. (c) The content of β-phase in milled and unmilled BOPVDF films [24].](/document/doi/10.1515/phys-2022-0255/asset/graphic/j_phys-2022-0255_fig_007.jpg)
(a) Scheme of conformation conversion of the α-phase to the β-phase in PVDF. (b) The relationship between F(β) in PVDF powders and pan-milling cycles. (c) The content of β-phase in milled and unmilled BOPVDF films [24].
Compared to uniaxial stretched films, the biaxial stretched films exhibit more balanced properties. Lu et al. applied simultaneous biaxial stretching on PVDF films and investigated its effects on the crystal structure, dielectric properties, and mechanical properties. It attributes to the applied stress during stretching, which forces the alignment of the PVDF molecular chains into the crystals and transforms to the β-phase conformation. As a result, both F(β) and X c increase (unstretched samples: F(β) = 39%, X c = 43%; stretched samples (R = 3.5 × 3.5): F(β) = 46%, X c = 55%). Compared to uniaxially stretched samples, F(β) is much higher, which may be caused by the higher stretching temperature and stretching rate [60]. Some researchers suggest that the effect of biaxial stretching is reflected in two aspects: (1) the parallel orientation of PVDF crystals reduces the dielectric loss from the αc relaxation in α crystal and (2) the biaxial stretching constrains the amorphous phase in PVDF-HFP and thus the migration loss from impurity ions is reduced [61]. Yang et al. explored the semi-crystalline structure–dielectric property relationship and the electrical conduction in a biaxially oriented PVDF film under high electric fields and high temperatures. In the work, the film is stretched first along the axial (or extrusion) direction, followed by subsequent stretching using tenter clips along the transverse direction. They found that biaxially oriented PVDF films exhibited a DC breakdown strength as high as biaxially oriented polypropylene films [62]. Kang et al. developed a CO2 laser-heated drawing process to prepare PVDF fibers, as shown in Figure 8. In the experiments, the effects of drawing stress and elongation ratio ((a) 110 MPa, R = 4.5; (b) 220 MPa, R = 5.5) were compared. The structure evolution was observed by in situ measurement using synchrotron X-ray radiation. Under a low drawing stress, a moderate neck-deformation occurs, resulting in the mixture of α- and β-phases, in which the β-phase forms within 1 ms after necking, earlier than the α-phase. In contrast, under high drawing stress, a steep neck-deformation occurs, resulting in only the β-phase. Small-angle X-ray scattering images showed that meridional streaks appear at low drawing stress, while a four-pointed pattern appears at high drawing stress [63].
![Figure 8
Schematic diagram of the apparatus for CO2 laser heated-drawing process [63].](/document/doi/10.1515/phys-2022-0255/asset/graphic/j_phys-2022-0255_fig_008.jpg)
Schematic diagram of the apparatus for CO2 laser heated-drawing process [63].
Nakamura et al. produced α-PVDF films by extrusion with R up to 9 at 160°C (10°C lower than the melting temperature T m) to obtain β-PVDF films. It was found that nearly all α-phase transformed into the oriented β-phase with increasing R, even when the drawing occurs near T m. The results indicate that the oriented structure and high F(β) can be attributed to the arrangement of the crystal/amorphous series along the drawing direction [64]. Chen et al. prepared high-performance piezoelectric polymer films by simple melt stretching of slightly crosslinked PVDF solution for sensor applications. Upon controlled temperature and stretching ratio, strong chain orientation or extension proceeds prior to crystallization, leading to the generation of an ultrahigh content of polar β-phase (∼96%) directly from the melt. The β-phase adopts an oriented nanofibril morphology (with a diameter of 21.6 nm), while the nanofibrils are further densely and parallelly aligned to fill the film space. Attributed to such a nanoscale phase morphology, the melt-stretched PVDF film shows high optical transmittance (∼90%), good flexibility, and superior mechanical properties. A flexible piezoelectric device constructed on the β-phase nanofibrils-filled PVDF film is demonstrated to have excellent electrical output in sensing dynamic pressure (0–250 kPa) and temperature variation (3.3 mV/°C) with a broad response range. Figure 9 shows a schematic illustration on the fabrication of PVDF films by a strategy of melt stretching at high temperature, which adopts a three-step process [65].
![Figure 9
Fabrication of PVDF films by melt stretching. (a) Schematic illustration of vacuum-assisted hot compression to prepare pristine PVDF sheets. (b) Preparation of slightly crosslinked PVDF by γ-ray irradiation. (c) Schematic drawing of home-made two-drum extensional rheometer and the corresponding melt stretching process. The digital photograph of the prepared PVDF film (160°C, 40×) is also shown. (d) Thermal program [64].](/document/doi/10.1515/phys-2022-0255/asset/graphic/j_phys-2022-0255_fig_009.jpg)
Fabrication of PVDF films by melt stretching. (a) Schematic illustration of vacuum-assisted hot compression to prepare pristine PVDF sheets. (b) Preparation of slightly crosslinked PVDF by γ-ray irradiation. (c) Schematic drawing of home-made two-drum extensional rheometer and the corresponding melt stretching process. The digital photograph of the prepared PVDF film (160°C, 40×) is also shown. (d) Thermal program [64].
2.3.1 Simultaneous stretching and poling
In most reports, the stretching and poling processes are independent of each other. In the stretching process, phase transformation finishes completely, and therefore, the effects of poling on phase transformation are not so obvious [35,66]. If stretching and poling are applied simultaneously, better piezoelectricity may be generated. Mahadeva et al. applied sequential stretching and corona poling to fabricate piezoelectric PVDF thin films and investigated the effects of poling time and grid voltage on phase transformation and piezoelectricity. It was found that the poling time and grid voltage have no substantial influences on phase transformation, indicating that phase transformation mainly occurs during the stretching process. However, poling has decisive effects on piezoelectricity due to its function on dipole moment alignment [67]. Kumar and Perlman used the simultaneous stretching and corona poling method to fabricate PVDF-TrFE (PVDF/TrFE = 75:25 or 50:50 mol%) copolymer films. The maximum piezoelectric constant d 31 of 75:25 films reaches 48 pC/N under optimal conditions (R = 4.5, T = 60°C), which is about 1.7 times of those fabricated by separated poling after stretching. The enhancement is attributed to the ferroelectric domain rotation and step rotation of molecular dipoles at 60°C in the field direction. In addition, the optimal poling temperature for PVDF (80°C) is higher than that for PVDF-TrFE (60°C) [68]. Both simultaneous stretching and poling have been utilized in 3D printing techniques in recent years. Lee and Tarbutton utilized an innovative method electric poling-assisted additive manufacturing (EPAM) to print PVDF filament rods under a strong electric field in which poling and stretching are applied simultaneously. Figure 10 shows a schematic of the preparation process. A high electric field is applied to the molten polymer column between the nozzle tip and the printing bed (poling) during extrusion (stretching), which leads to molecular chain alignment and phase transformation to the β-phase [69]. Huan et al. developed a simultaneously stretched and static electric poled (SSSEP) process to enhance the piezoelectricity of PVDF-HFP films. The applied apparatus is shown in Figure 11. In this work, the elongation ratio is kept at 4.5, equal to the tooth ratio of Gear 1 and Gear 2, and by varying the poling temperature and the poling electric field, different samples were prepared. The maximum piezoelectric coefficient d 33 reaches 24 pC/N under optimal conditions (T p = 65°C, E max = 160 MV/m). From the results, it can also be found that the saturation of the poling electric field is about 150 MV/m [70]. Because of its complexity, the application of the simultaneous stretching and poling method is limited.
![Figure 10
The principle in the proposed printing process of piezoelectric PVDF devices: (a) experimental results of electric poling and the mechanical stretching process in the EPAM process and (b) extrusion results of the filament type of the PVDF polymer [69].](/document/doi/10.1515/phys-2022-0255/asset/graphic/j_phys-2022-0255_fig_010.jpg)
The principle in the proposed printing process of piezoelectric PVDF devices: (a) experimental results of electric poling and the mechanical stretching process in the EPAM process and (b) extrusion results of the filament type of the PVDF polymer [69].
![Figure 11
Apparatus of SSSEP for preparing piezoelectric PVDF-HFP films [70].](/document/doi/10.1515/phys-2022-0255/asset/graphic/j_phys-2022-0255_fig_011.jpg)
Apparatus of SSSEP for preparing piezoelectric PVDF-HFP films [70].
2.3.2 Electrospinning
Stretching occurs during the electrospinning process, and therefore, usually no individual stretching is required in the preparation of electrospun samples. Figure 12 shows a typical electrospinning process for the preparation of PVDF-HFP fibers [71]. The stress induced by the injection on the mixture solution has a decisive effect on the alignment of the molecular chains as well as the added nanofillers as demonstrated by the electrospinning-hot press method for the preparation of surface modified graphene (SMG)/PVDF composites in Figure 13 [72]. The key parameters of electrospinning are the solvent fraction, tip-to-collector distance (TCD), flow rate, and voltage setting [73]. Gee et al. applied the electrospinning method to fabricate piezoelectric nanofiber membranes. By varying the parameters, the optimization setting for piezoelectricity was determined (solvent ratio: 60 vol% DMF + 40 vol% acetone; TCD = 16 cm; flow rate = 0.8 mL/h; voltage setting = 14 kV), at which the β-phase fraction reaches 91.0%. Based on the statistical analysis of the mean β-phase fraction and signal-to-noise ratio, it is concluded that the solvent ratio and flow rate are the two factors contributing most to the β-phase fraction [73]. Similar to traditional stretching, the stretching effect during electrospinning is also affected by the addition of nanofillers. Khalifa et al. investigated the synergistic effect of electrospinning and nano-alumina trihydrate (n-ATH) on the crystal structure and piezoelectric properties of PVDF nanofibers. The added n-ATH can enhance the surface of the electrospun droplets, which leads to the decrease in the thickness. When the n-ATH increases to 10%, F(β) reaches the highest value of 70.1% and the highest output voltage of 840 mV, both of which are much higher than those of pure PVDF nanofibers. When the n-ATH content is above 10%, F(β) decreases, which may be attributed to the dilution effect [74]. Lei et al. investigated the dipole arrangement during electrospinning. Compared with mechanical spinning, the electrospun fibers can generate voltage after deformation, while the mechanical-spun fibers do not exhibit piezoelectricity. This result indicates that although both processes can produce high-β-fraction PVDF fibers, the applied voltage during electrospinning can force the dipole moments along the electric field, and thus exhibit piezoelectricity. For mechanical-spun fibers, the dipole moments are randomly distributed, and thus the fibers have no piezoelectricity. In other words, electrospinning includes both stretching and poling functions [75]. Lei et al. developed an electrospinning process to produce PVDF membranes of desired phases by simultaneously controlling the collector temperature and the flow rate during electrospinning. The α-, β- or γ-phase dominant PVDF nanofibrous membranes can be achieved by adjusting the electrospinning parameters and heat treatment during or after electrospinning [76]. The influences of the parameters can be described briefly as follows: (a) the voltage can influence the nanofiber creation and the formation of the β-phase to some extent, and an excessively high voltage may lead to instability of jets and decrease the solvent evaporation rate; (b) TCD influences the diameter of the nanofibers (the lager the TCD, the smaller the diameter); (c) high feed rate can lead to the formation of the β-phase and uniform fibers, while excessively high rates create unstable jets and disruption of electrospinning; and (d) solvent fraction can influence the solution viscosity and the nanofiber diameter, while it may lead to electrospraying and exhibit negative effects on the β-phase formation if the solvent fraction is too high [77–83]. Yang et al. added GO into PVDF to prepare nanofiber mats by electrospinning. By heating the PVDF/GO nanofiber mat at 140°C for 1 h, the PVDF/rGO mat was produced by the reduction of GO to rGO. The β-phase content was improved with the GO (or rGO) content until 2 wt%, and therefore, the power generation capability was increased. Compared to PVDF/GO, the output voltage of PVDF/rGO was higher due to fewer functional groups on the carbon atoms [84]. In addition, it was reported that natural substances were added into PVDF. Lee et al. prepared PVDF/cellulose nanocrystal (CNC) nanocomposite fibers (CNC content: 0, 1, and 3 wt%) by dry-jet wet spinning, followed by mechanical stretching with the drawing ratio R = 8. The fraction of β-phase appears to be slightly higher in CNC containing as-spun nanocomposite fibers, which may be due to the synergistic effect of a low stretching degree and heterogeneous nucleation. When R = 8, all polymer chains are supposed to be highly aligned and thereby the fraction of β-phase dramatically increases. However, the PVDF/CNC (3 wt%) composites had a 10% lower β-fraction and instead contained more γ phase because of the hindrance caused by the defects during the β-phase formation [85]. Issa et al. studied the physicomechanical properties of electrospun PVDF/cellulose nanofibers. It was found that the hydroxyl groups on cellulose would force the molecular chains in PVDF to be in the trans-conformation, during which the cellulose particles could act as nucleation agents. Moreover, the total crystallinity degree and the formation of β-crystals were improved mainly by the molecular chains and crystal orientation in electrospinning, while a little additional contribution from the cellulose dispersed in the electrospun PVDF [86].
![Figure 12
Schematic of the synthesis of PVDF-HFP fibers by the electrospinning process [71].](/document/doi/10.1515/phys-2022-0255/asset/graphic/j_phys-2022-0255_fig_012.jpg)
Schematic of the synthesis of PVDF-HFP fibers by the electrospinning process [71].
![Figure 13
(a) The preparation process of SMG/PVDF composites by an electrospinning-hot pressing method. SEM images of electrospun SMG (16 wt%)/PVDF membrane at (b) low and (c) high magnification [72].](/document/doi/10.1515/phys-2022-0255/asset/graphic/j_phys-2022-0255_fig_013.jpg)
(a) The preparation process of SMG/PVDF composites by an electrospinning-hot pressing method. SEM images of electrospun SMG (16 wt%)/PVDF membrane at (b) low and (c) high magnification [72].
From these studies, it is evident that electrospinning is a suitable method to produce piezoelectric fibers or films of low thickness.
3 Summary
The influences of stretching on phase transformation are investigated and the results are summarized in Table 1. It is worth noting that the parameters of stretching are not independent of each other on the properties of the materials. As a result, during the development of a suitable method for PVDF or its copolymers, a lot of work has to be done before the optimization parameters can be determined. Another point noteworthy noting is that the effect of one parameter reported in one study may differ from that of other reports, or even be contradictory.
A brief summary of stretching effects on the β-phase formation of PVDF and its copolymers
Factors | Positive effect | Negative effect | Remarks |
---|---|---|---|
Temperature | Eliminate the internal stress and improve the ductility [32] | When the temperature is above 100°C, the phase transformation decreases or even disappears | The appropriate T s is usually set from 60 to 100°C |
At higher temperatures, the weaker filler–matrix compatibility and higher chain mobility limit the phase transformation [35] | |||
Elongation rate | Increase the crystallinity (possible) [42,46] | Decrease the crystallinity (possible) [23] | The elongation is set from 3 to 5 |
Speed | Improve the transformation from α- to β-phase (high speed) [47] | Lower phase transformation efficiency [48,49] | |
Decreases the crystallinity degree (high speed) [50,51] | |||
Nanofiller addition | Improve phase transformation due to the stress concentration on the PVDF/nanofiller interface [20,22] | Reduces the crystallinity induced by phase transformation and nanofiller re-orientation [22,23] | |
Extremely lower content nanofillers (<threshold value) may hinder phase transformation [22] | |||
Increase the dielectric constant [22] | |||
The agglomeration of nanofillers at higher concentrations results in chain confinement, thus leading to lower piezoelectricity [23] | |||
Increase the crystallinity degree [20] | |||
Simultaneous stretching and poling | Molecular chain alignment and β-phase formation [69] | Complicate operation | T p > 60°C, R > 4, E max > 150 MV/m |
Molecular dipole moments rotation [68] |
4 Conclusions
Stretching is an important method to improve the α- to β-phase transformation efficiently in PVDF films. In this work, the effects of stretching parameters (temperature, elongation ratio, stretching speed, and nanofiller addition) are discussed. It is worth noting that for most parameters, there is a proper value at which the optimal effects can be achieved. Another aspect that should be pointed out is that the effects are clearly influenced by specific conditions. For example, in some reports, the crystallinity degree increases after stretching, while in others, completely different results are observed. In addition, the nontraditional stretching method is introduced briefly. This work can help researchers in the piezoelectric polymer field to understand the mechanism of phase transformation during stretching and to fabricate high piezoelectric polymer/composite films.
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Funding information: This work was financially supported by the National Natural Science Foundation of China (No. 51703015) and Fundamental Research Funds for the Central Universities (No. 2020CDJQY-A008).
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Conflict of interest: The authors state no conflict of interest.
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Data availability statement: The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.
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- Parametric simulation of stagnation point flow of motile microorganism hybrid nanofluid across a circular cylinder with sinusoidal radius
- Fractal-fractional advection–diffusion–reaction equations by Ritz approximation approach
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- Flat and bent branes with inner structure in two-field mimetic gravity
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- Weighted survival functional entropy and its properties
- Bioconvection effect in the Carreau nanofluid with Cattaneo–Christov heat flux using stagnation point flow in the entropy generation: Micromachines level study
- Study on the impulse mechanism of optical films formed by laser plasma shock waves
- Analysis of sweeping jet and film composite cooling using the decoupled model
- Research on the influence of trapezoidal magnetization of bonded magnetic ring on cogging torque
- Tripartite entanglement and entanglement transfer in a hybrid cavity magnomechanical system
- Compounded Bell-G class of statistical models with applications to COVID-19 and actuarial data
- Degradation of Vibrio cholerae from drinking water by the underwater capillary discharge
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- Study on the inter-porosity transfer rate and producing degree of matrix in fractured-porous gas reservoirs
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- Controlling the physical field using the shape function technique
- Significance of heat and mass transport in peristaltic flow of Jeffrey material subject to chemical reaction and radiation phenomenon through a tapered channel
- Complex dynamics of a sub-quadratic Lorenz-like system
- Stability control in a helicoidal spin–orbit-coupled open Bose–Bose mixture
- Research on WPD and DBSCAN-L-ISOMAP for circuit fault feature extraction
- Simulation for formation process of atomic orbitals by the finite difference time domain method based on the eight-element Dirac equation
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- Vibration sensitivity minimization of an ultra-stable optical reference cavity based on orthogonal experimental design
- Effect of dysprosium on the radiation-shielding features of SiO2–PbO–B2O3 glasses
- Asymptotic formulations of anti-plane problems in pre-stressed compressible elastic laminates
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- A novel compact highly sensitive non-invasive microwave antenna sensor for blood glucose monitoring
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- Optimal interatomic potentials using modified method of least squares: Optimal form of interatomic potentials
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- Research on absolute ranging technology of resampling phase comparison method based on FMCW
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- Effects of Joule heating and reaction mechanisms on couple stress fluid flow with peristalsis in the presence of a porous material through an inclined channel
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- Application of IoT network for marine wildlife surveillance
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- Joint optimization of two-dimensional warranty period and maintenance strategy considering availability and cost constraints
- Numerical investigation of the flow characteristics involving dissipation and slip effects in a convectively nanofluid within a porous medium
- Spectral uncertainty analysis of grassland and its camouflage materials based on land-based hyperspectral images
- Application of low-altitude wind shear recognition algorithm and laser wind radar in aviation meteorological services
- Investigation of different structures of screw extruders on the flow in direct ink writing SiC slurry based on LBM
- Harmonic current suppression method of virtual DC motor based on fuzzy sliding mode
- Micropolar flow and heat transfer within a permeable channel using the successive linearization method
- Different lump k-soliton solutions to (2+1)-dimensional KdV system using Hirota binary Bell polynomials
- Investigation of nanomaterials in flow of non-Newtonian liquid toward a stretchable surface
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- Special Issue on Advanced Topics on the Modelling and Assessment of Complicated Physical Phenomena - Part III
- Some standard and nonstandard finite difference schemes for a reaction–diffusion–chemotaxis model
- Special Issue on Advanced Energy Materials - Part II
- Rapid productivity prediction method for frac hits affected wells based on gas reservoir numerical simulation and probability method
- Special Issue on Novel Numerical and Analytical Techniques for Fractional Nonlinear Schrodinger Type - Part III
- Adomian decomposition method for solution of fourteenth order boundary value problems
- New soliton solutions of modified (3+1)-D Wazwaz–Benjamin–Bona–Mahony and (2+1)-D cubic Klein–Gordon equations using first integral method
- On traveling wave solutions to Manakov model with variable coefficients
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- Special Issue on Predicting pattern alterations in nature - Part I
- Modeling the monkeypox infection using the Mittag–Leffler kernel
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Articles in the same Issue
- Regular Articles
- Dynamic properties of the attachment oscillator arising in the nanophysics
- Parametric simulation of stagnation point flow of motile microorganism hybrid nanofluid across a circular cylinder with sinusoidal radius
- Fractal-fractional advection–diffusion–reaction equations by Ritz approximation approach
- Behaviour and onset of low-dimensional chaos with a periodically varying loss in single-mode homogeneously broadened laser
- Ammonia gas-sensing behavior of uniform nanostructured PPy film prepared by simple-straightforward in situ chemical vapor oxidation
- Analysis of the working mechanism and detection sensitivity of a flash detector
- Flat and bent branes with inner structure in two-field mimetic gravity
- Heat transfer analysis of the MHD stagnation-point flow of third-grade fluid over a porous sheet with thermal radiation effect: An algorithmic approach
- Weighted survival functional entropy and its properties
- Bioconvection effect in the Carreau nanofluid with Cattaneo–Christov heat flux using stagnation point flow in the entropy generation: Micromachines level study
- Study on the impulse mechanism of optical films formed by laser plasma shock waves
- Analysis of sweeping jet and film composite cooling using the decoupled model
- Research on the influence of trapezoidal magnetization of bonded magnetic ring on cogging torque
- Tripartite entanglement and entanglement transfer in a hybrid cavity magnomechanical system
- Compounded Bell-G class of statistical models with applications to COVID-19 and actuarial data
- Degradation of Vibrio cholerae from drinking water by the underwater capillary discharge
- Multiple Lie symmetry solutions for effects of viscous on magnetohydrodynamic flow and heat transfer in non-Newtonian thin film
- Thermal characterization of heat source (sink) on hybridized (Cu–Ag/EG) nanofluid flow via solid stretchable sheet
- Optimizing condition monitoring of ball bearings: An integrated approach using decision tree and extreme learning machine for effective decision-making
- Study on the inter-porosity transfer rate and producing degree of matrix in fractured-porous gas reservoirs
- Interstellar radiation as a Maxwell field: Improved numerical scheme and application to the spectral energy density
- Numerical study of hybridized Williamson nanofluid flow with TC4 and Nichrome over an extending surface
- Controlling the physical field using the shape function technique
- Significance of heat and mass transport in peristaltic flow of Jeffrey material subject to chemical reaction and radiation phenomenon through a tapered channel
- Complex dynamics of a sub-quadratic Lorenz-like system
- Stability control in a helicoidal spin–orbit-coupled open Bose–Bose mixture
- Research on WPD and DBSCAN-L-ISOMAP for circuit fault feature extraction
- Simulation for formation process of atomic orbitals by the finite difference time domain method based on the eight-element Dirac equation
- A modified power-law model: Properties, estimation, and applications
- Bayesian and non-Bayesian estimation of dynamic cumulative residual Tsallis entropy for moment exponential distribution under progressive censored type II
- Computational analysis and biomechanical study of Oldroyd-B fluid with homogeneous and heterogeneous reactions through a vertical non-uniform channel
- Predictability of machine learning framework in cross-section data
- Chaotic characteristics and mixing performance of pseudoplastic fluids in a stirred tank
- Isomorphic shut form valuation for quantum field theory and biological population models
- Vibration sensitivity minimization of an ultra-stable optical reference cavity based on orthogonal experimental design
- Effect of dysprosium on the radiation-shielding features of SiO2–PbO–B2O3 glasses
- Asymptotic formulations of anti-plane problems in pre-stressed compressible elastic laminates
- A study on soliton, lump solutions to a generalized (3+1)-dimensional Hirota--Satsuma--Ito equation
- Tangential electrostatic field at metal surfaces
- Bioconvective gyrotactic microorganisms in third-grade nanofluid flow over a Riga surface with stratification: An approach to entropy minimization
- Infrared spectroscopy for ageing assessment of insulating oils via dielectric loss factor and interfacial tension
- Influence of cationic surfactants on the growth of gypsum crystals
- Study on instability mechanism of KCl/PHPA drilling waste fluid
- Analytical solutions of the extended Kadomtsev–Petviashvili equation in nonlinear media
- A novel compact highly sensitive non-invasive microwave antenna sensor for blood glucose monitoring
- Inspection of Couette and pressure-driven Poiseuille entropy-optimized dissipated flow in a suction/injection horizontal channel: Analytical solutions
- Conserved vectors and solutions of the two-dimensional potential KP equation
- The reciprocal linear effect, a new optical effect of the Sagnac type
- Optimal interatomic potentials using modified method of least squares: Optimal form of interatomic potentials
- The soliton solutions for stochastic Calogero–Bogoyavlenskii Schiff equation in plasma physics/fluid mechanics
- Research on absolute ranging technology of resampling phase comparison method based on FMCW
- Analysis of Cu and Zn contents in aluminum alloys by femtosecond laser-ablation spark-induced breakdown spectroscopy
- Nonsequential double ionization channels control of CO2 molecules with counter-rotating two-color circularly polarized laser field by laser wavelength
- Fractional-order modeling: Analysis of foam drainage and Fisher's equations
- Thermo-solutal Marangoni convective Darcy-Forchheimer bio-hybrid nanofluid flow over a permeable disk with activation energy: Analysis of interfacial nanolayer thickness
- Investigation on topology-optimized compressor piston by metal additive manufacturing technique: Analytical and numeric computational modeling using finite element analysis in ANSYS
- Breast cancer segmentation using a hybrid AttendSeg architecture combined with a gravitational clustering optimization algorithm using mathematical modelling
- On the localized and periodic solutions to the time-fractional Klein-Gordan equations: Optimal additive function method and new iterative method
- 3D thin-film nanofluid flow with heat transfer on an inclined disc by using HWCM
- Numerical study of static pressure on the sonochemistry characteristics of the gas bubble under acoustic excitation
- Optimal auxiliary function method for analyzing nonlinear system of coupled Schrödinger–KdV equation with Caputo operator
- Analysis of magnetized micropolar fluid subjected to generalized heat-mass transfer theories
- Does the Mott problem extend to Geiger counters?
- Stability analysis, phase plane analysis, and isolated soliton solution to the LGH equation in mathematical physics
- Effects of Joule heating and reaction mechanisms on couple stress fluid flow with peristalsis in the presence of a porous material through an inclined channel
- Bayesian and E-Bayesian estimation based on constant-stress partially accelerated life testing for inverted Topp–Leone distribution
- Dynamical and physical characteristics of soliton solutions to the (2+1)-dimensional Konopelchenko–Dubrovsky system
- Study of fractional variable order COVID-19 environmental transformation model
- Sisko nanofluid flow through exponential stretching sheet with swimming of motile gyrotactic microorganisms: An application to nanoengineering
- Influence of the regularization scheme in the QCD phase diagram in the PNJL model
- Fixed-point theory and numerical analysis of an epidemic model with fractional calculus: Exploring dynamical behavior
- Computational analysis of reconstructing current and sag of three-phase overhead line based on the TMR sensor array
- Investigation of tripled sine-Gordon equation: Localized modes in multi-stacked long Josephson junctions
- High-sensitivity on-chip temperature sensor based on cascaded microring resonators
- Pathological study on uncertain numbers and proposed solutions for discrete fuzzy fractional order calculus
- Bifurcation, chaotic behavior, and traveling wave solution of stochastic coupled Konno–Oono equation with multiplicative noise in the Stratonovich sense
- Thermal radiation and heat generation on three-dimensional Casson fluid motion via porous stretching surface with variable thermal conductivity
- Numerical simulation and analysis of Airy's-type equation
- A homotopy perturbation method with Elzaki transformation for solving the fractional Biswas–Milovic model
- Heat transfer performance of magnetohydrodynamic multiphase nanofluid flow of Cu–Al2O3/H2O over a stretching cylinder
- ΛCDM and the principle of equivalence
- Axisymmetric stagnation-point flow of non-Newtonian nanomaterial and heat transport over a lubricated surface: Hybrid homotopy analysis method simulations
- HAM simulation for bioconvective magnetohydrodynamic flow of Walters-B fluid containing nanoparticles and microorganisms past a stretching sheet with velocity slip and convective conditions
- Coupled heat and mass transfer mathematical study for lubricated non-Newtonian nanomaterial conveying oblique stagnation point flow: A comparison of viscous and viscoelastic nanofluid model
- Power Topp–Leone exponential negative family of distributions with numerical illustrations to engineering and biological data
- Extracting solitary solutions of the nonlinear Kaup–Kupershmidt (KK) equation by analytical method
- A case study on the environmental and economic impact of photovoltaic systems in wastewater treatment plants
- Application of IoT network for marine wildlife surveillance
- Non-similar modeling and numerical simulations of microploar hybrid nanofluid adjacent to isothermal sphere
- Joint optimization of two-dimensional warranty period and maintenance strategy considering availability and cost constraints
- Numerical investigation of the flow characteristics involving dissipation and slip effects in a convectively nanofluid within a porous medium
- Spectral uncertainty analysis of grassland and its camouflage materials based on land-based hyperspectral images
- Application of low-altitude wind shear recognition algorithm and laser wind radar in aviation meteorological services
- Investigation of different structures of screw extruders on the flow in direct ink writing SiC slurry based on LBM
- Harmonic current suppression method of virtual DC motor based on fuzzy sliding mode
- Micropolar flow and heat transfer within a permeable channel using the successive linearization method
- Different lump k-soliton solutions to (2+1)-dimensional KdV system using Hirota binary Bell polynomials
- Investigation of nanomaterials in flow of non-Newtonian liquid toward a stretchable surface
- Weak beat frequency extraction method for photon Doppler signal with low signal-to-noise ratio
- Electrokinetic energy conversion of nanofluids in porous microtubes with Green’s function
- Examining the role of activation energy and convective boundary conditions in nanofluid behavior of Couette-Poiseuille flow
- Review Article
- Effects of stretching on phase transformation of PVDF and its copolymers: A review
- Special Issue on Transport phenomena and thermal analysis in micro/nano-scale structure surfaces - Part IV
- Prediction and monitoring model for farmland environmental system using soil sensor and neural network algorithm
- Special Issue on Advanced Topics on the Modelling and Assessment of Complicated Physical Phenomena - Part III
- Some standard and nonstandard finite difference schemes for a reaction–diffusion–chemotaxis model
- Special Issue on Advanced Energy Materials - Part II
- Rapid productivity prediction method for frac hits affected wells based on gas reservoir numerical simulation and probability method
- Special Issue on Novel Numerical and Analytical Techniques for Fractional Nonlinear Schrodinger Type - Part III
- Adomian decomposition method for solution of fourteenth order boundary value problems
- New soliton solutions of modified (3+1)-D Wazwaz–Benjamin–Bona–Mahony and (2+1)-D cubic Klein–Gordon equations using first integral method
- On traveling wave solutions to Manakov model with variable coefficients
- Rational approximation for solving Fredholm integro-differential equations by new algorithm
- Special Issue on Predicting pattern alterations in nature - Part I
- Modeling the monkeypox infection using the Mittag–Leffler kernel
- Spectral analysis of variable-order multi-terms fractional differential equations
- Special Issue on Nanomaterial utilization and structural optimization - Part I
- Heat treatment and tensile test of 3D-printed parts manufactured at different build orientations