Strain‐Induced Toroidal Polar States in Wrinkled Ferroelectric Polymer by Phase‐Field Simulations

Wrinkled ferroelectric polymer films with out‐of‐plane deformation exhibit exotic strain fields that differ from those of conventional epitaxial film systems, offering a new path toward manipulating electrical and mechanical behaviors in ferroelectrics. However, the direct observation of the domain structure evolution in ferroelectric polymer films during continuous deformation is challenging with current experimental approaches. In this study, the strain‐induced evolution of toroidal domain structures in organic poly(vinylidene fluoride‐ran‐trifluoroethylene) (P(VDF‐TrFE)) films with concentric toroidal wrinkle patterns is comprehensively investigated using phase‐field simulations. The results reveal morphology‐ and thickness‐dependent toroidal polar topology in wrinkled P(VDF‐TrFE) films. External strain loading can modulate this ferroelectric domain, which exhibits robust coupling behavior with a mixed strain field of periodic tensile and compressive strains. It is demonstrated that toroidal wrinkled ferroelectric films provide an effective and unique flexible platform for continuously controllable strain engineering in ferroelectrics and a theoretical framework with potential applications in flexible electronic devices.

Organic ferroelectrics feature superior mechanical flexibility, cost-effectiveness, and high stability, making them ideal for flexible devices like sensors and energy harvesters. [25,26]The flexible organic ferroelectric poly(vinylidene fluorideran-trifluoroethylene) (P(VDF-TrFE), molar ratio 70/30) exhibits a unique toroidal arrangement of electric dipoles, [27] which can be thermally manipulated to achieve pyrotoroidic transition accompanying the Curie transition. [23]The wrinkled P(VDF-TrFE) film that exhibits the unique toroidal polar topology has an internal strain state different from that of conventional epitaxial films due to its distinct morphology.Therefore, a comprehensive investigation into the strain-induced evolution of polar topology is essential to comprehend its underlying physical mechanism.Nevertheless, direct observation of the domain structure evolution in mesoscale ferroelectric thin films during continuous deformation is challenging with current experimental approaches.30] In this study, we investigate the strain-induced evolution of domain structures in wrinkled P(VDF-TrFE) films by visualizing and analyzing the distribution of ferroelectric domains under various tensile strains through phase-field simulation.Unlike conventional epitaxial films, the wrinkled P(VDF-TrFE) films possess a distinct periodic strain distribution within, determined by their self-organized morphology.Our phase-field simulations demonstrate that the toroidal wave structure generates complex strain and domain distributions that can be controlled by external tensile loading.The morphology-induced self-organized strain field determines the toroidal polar topology with antiparallel concentric ferroelectric alignments that are morphology-dependent and thickness-dependent.Additionally, the strain-induced evolution of domain structures in wrinkled thin films provides a convenient and effective approach for continuous strain engineering of ferroelectrics.The morphology of ferroelectric wrinkled films could be developed as a degree of freedom for strain manipulation, enabling the tuning of domain structures and physical properties of flexible ferroelectric systems.

Results and Discussion
Certain external stimuli such as mechanical force [31,32] and thermal perturbation [33,34] can induce a buckling-instability mode in thin film systems, resulting in out-of-plane (OOP) deformation.Recent advancements in thin film fabrication have led to the discovery of peculiar instability patterns in flexible ferroelectric thin films, [35][36][37][38] including the regular concentric surface wrinkles ob-served in ferroelectric P(VDF-TrFE) polymer. [27]The concentric "target" structure at the center of the wrinkles features a circular convex deformation, with continuous undulations spreading outwards in a "wave" shape.When subjected to tensile strain, a periodic alternating arrangement between tensile and compressive strain is observed in the strain distribution within the film (Figure 1a).Moreover, the strain distribution at different vertical positions but at the same horizontal position follows the same regularity, as shown in the comparison of the strain distributions at valleys (such as i-position) and peaks (such as iiposition) in the cross-section of Figure 1a.The modulation of the morphology of wrinkles through adjustments to the thickness of the film system, material parameters, or external conditions has been theoretically and experimentally demonstrated to influence the strain state within wrinkles and the distribution of domain structures in ferroelectric films. [20,35,37,38]  exhibits three ferroelectric phase patterns, marked by different symbols and colors, and the background strain distribution map shows the strain ratio between the outermost valley i-position and peak ii-position in Figure 1a.At small wavelengths, the wrinkle wavelength and amplitude are approximately equal, leading to a large strain ratio between the valley and peak.The top surface of the wrinkled P(VDF-TrFE) film contains labyrinthine OOP cdomains (Figure 1c).With an increase in wavelength and a decrease in amplitude, the second phase pattern on the top surface also appears as a labyrinthine OOP domain, while the electric dipoles of the P(VDF-TrFE) layer rotate continuously to form a dipole wave in the shape of a sine function highlighted by the solid green line.Additionally, flux-closure domains are also present between the periodic sinusoidal arrays highlighted by red circular arrows, which contribute to reducing depolarization fields in the overall system. [2,39,40]As both wavelength and amplitude increase gradually, the third phase pattern on the top surface transforms into a periodic toroidal topological domain, containing an internal dipole wave consisting of head-to-tail connected electric dipoles.The peaks and valleys on the top surface correspond to the alternation of the toroidal topological domains along the radius direction.Notably, the toroidal topological state arises when the strain magnitude between the valleys and peaks of the entire P(VDF-TrFE) wrinkle approaches comparability.
The demonstration of the internal polarization distribution and its mechanical manipulation through the development of a third toroidal topological domain structure is vital for the advancement of intelligent nanodevices.In Figure 2, the polarization and strain distributions are depicted for the P(VDF-TrFE) toroidal wrinkled structure subjected to an applied tensile strain of 7.5%.The polarization inside the P(VDF-TrFE) film rotates continuously, forming an in-plane (IP) stable antiparallel concentric arrangement, as demonstrated in Figure 2a, where the different colors represent the rotation angle of the polarization vector.The simulation results are consistent with experimental observations, [27] which have revealed the formation of 180°domain walls between antiparallel arrangements of domain structures.As the toroidal wrinkle constitutes a continuous undulating concentric waveform, the peaks and valleys of the wrinkle structure correspond to periodically distributed compressive and tensile strains (Figure 2b), which ultimately affect the interaction among the elastic, electrostatic, and gradient energies of the system.This interaction results in a continuous rotation of the IP polarization, forming the domain structure depicted in Figure 2a. Figure 2c,d illustrate the distribution of polarization and strain components along the radial direction r and perpendicular to the radial direction  on the red line AB in the wrinkle structure, respectively (the r-axis and -axis come from the polar coordinate system introduced to the circular model).Owing to the continuous undulation of the wrinkle morphology, the strain components  r and   on red line AB also vary with periodic undulation.The maximum magnitude of the strain period consisting of tensile and compressive strains increases gradually outward with the r-axis due to the relaxation behavior of the boundary strain.Therefore, the strain component magnitudes increase gradually in the outward direction of the r-axis, and the polarization components P r and P  follow the strain variations.The ferroelectric domain evolution corresponding to these observations indicates that the polarization has a robust coupling behavior with the periodic strain distribution.
The distinct difference in strain states on the top and bottom surfaces of P(VDF-TrFE) film under external tensile strain arises from the periodic peak and valley undulation morphology of its toroidal wrinkle structure, as demonstrated by the strain distribution map of the x-z cross-section in Figure 1a.The resultant tensile and compressive strains on the two surfaces cause the phase P(VDF-TrFE) polymer chains to rotate around the OOP zaxis, thereby deflecting the electric dipoles perpendicular to the polymer chains around the z-axis.Consequently, an unusual polarization arrangement occurs along the thickness direction of the wrinkled P(VDF-TrFE) film due to the exotic strain state.The ferroelectric polarization distributions at the top, middle, and bottom interfaces of the 10 nm thick toroidal wrinkled P(VDF-TrFE) film are displayed in Figure 3a.For further discussion, the polarization along the r-axis of the circular film is denoted type 1, whereas the polarization with a large declination from the r-axis is denoted type 2. As illustrated in Figure 3b, the electric dipoles in the compressive strain region belong primarily to type 1, whereas the electric dipoles in the tensile strain region belong primarily to type 2. Notably, the middle layer of the film has a small gap between the tensile and compressive strains, leading to the predominance of type 1 ferroelectric polarization.The domain structures at interfaces at equal distances from the middle layer of the film are the exact opposite.For example, opposite polarization types are induced at the same horizontal position of the top and bottom surfaces due to opposite strains.
To gain microscopic insight into this distinctive IP chiral topological domain, Figure 3c shows the trend of the electric toroidal moments at various interface depths.One can clearly see that the z-component of the toroidal moment G z is far more extensive than the x/y-component G x/y , indicating the existence of IP chiral domains. [41]In addition, the toroidal moment G z exhibits a decreasing trend followed by an increasing trend as the z-axis coordinate gradually increases.It indicates that the more periodic alignments of types 1 and 2 polarizations, the larger the toroidal moment.Consequently, the opposite polarization alignments at the same horizontal position but at different vertical positions can be exploited for storing diverse types of logical data, thereby expanding the prospective applications of ferroelectric materials.
Furthermore, we elucidated the strain-induced evolution of the IP topological domain structure in toroidal wrinkled P(VDF-TrFE) films.As illustrated in Figure 4a,b, the size of the antiparallel chiral domain structure in wrinkled P(VDF-TrFE) films gradually decreases with increasing external tensile strain.The magnitude of the polarization component P r and strain component  r along the r-axis also increases with increasing external tensile strain, and the polarization component varies with the strain component in a corresponding undulating mode (Figure 4c).The results indicate that as the external tensile strain increases, the number of periodic alignments of types 1 and 2 increases gradually, accompanied by an increase in the magnitude of the polarization component (Figure 4c; Figure S1, Supporting Information).This phenomenon ultimately leads to a gradual increase in the z-component of the toroidal moment G z with increasing strain, as illustrated in Figure 4d.Notably, the toroidal moment of the IP chiral domains on the top surface increases rapidly when the external strain exceeds 4%.As the chirality of the toroidal moment can be used as an order parameter in ferroelectric, the possibility of controlling it using external strain fields could open up exciting opportunities for memory devices, nano-motors, nano-sensors, and more.
Our findings suggest that wrinkled P(VDF-TrFE) films exhibit a robust and consistent electromechanical coupling behavior between periodic mechanical strain and polarization.By utilizing the continuous variation of the strain field throughout mechanical manipulation, the ferroelectric wrinkled model can be applied to continuous strain engineering in ferroelectrics.Likewise, the potential mechanism of continuous mechanical modulation by buckling-induced wrinkled film structure investigations could be extended to other film systems to enhance their high performances.

Conclusion
In this study, we developed a phase-field model to investigate the morphology-dependent phase diagram and domain evolution behavior of a three-dimensional concentric toroidal wrinkled P(VDF-TrFE) film under external tensile strains.Our phase-field simulations demonstrate that a toroidal polar topology state exists only when the strains between the valleys and peaks of the entire P(VDF-TrFE) wrinkle are nearly identical and that this topological state is dependent on the thickness of the film.Moreover, the opposite strain states on the top and bottom surfaces of the film determine the opposite alignment of the ferroelectric polarization along the thickness direction.By applying external tensile strain, chiral domains with an antiparallel concentric polarization arrangement can be modulated.As the external strain increases, the size of the antiparallel domain structure decreases, and the toroidal moment G z increases.The observed behavior of the wrinkled films can be explained by the robust coupling between ferroelectric polarization and the complex strain field induced by external loading.Unlike previous studies on epitaxial films, the strain field in our wrinkled films is a concentric periodic strain field with mixed tensile and compressive strains.These wrinkled ferroelectric films with opposite polarization alignment have potential applications in flexible sensors, bioelectric control, and other fields.

Experimental Section
Phase-Field Simulation: The ferroelectric transition and domain structure in P(VDF-TrFE) films were described by polarization vector P = (P x ,P y ,P z ).The temporal evolution of the polarization field was solved by the time-dependent Ginzburg-Landau (TDGL) equation: where r is the spatial coordinate, t is the evolution time, L is the kinetic coefficient that was related to the domain evolution, and F is the total free energy that includes the contributions from the Landau free energy, the gradient energy, the elastic energy, the electrostatic energy: The Landau free energy density f Land is given by, f Land =  ij P i P j +  ijkl P i P j P k P l + ijklmn P i P j P k P l P m P n , where  ij ,  ijkl ,  ijklmn are the Landau free energy coefficients.The gradient energy density f grad can be described as follows, where G ijkl are the gradient energy coefficients, and P i,j = P i x j .
The elastic energy density f elas can be written as where C ijkl is the elastic stiffness tensor, e ij is the elastic strain,  ij and  0 ij are the total local strain and eigenstrain, respectively.And  0 ij = Q ijkl P k P l , where Q ijkl are the electrostrictive coefficients.
The electrostatic energy density f ele is expressed as, where E i is the electric field component,  0 is the vacuum permittivity, and  ij is the dielectric constant.
In the simulation, the ferroelectric thin film model was chosen with an electrical open-circuit boundary condition, as shown in Figure S2 (Supporting Information).Periodic boundary conditions were selected for the in-plane direction of the film.The investigated film structure features concentric wrinkles with a thickness of 10 nm and a radius of 180 nm.The wavelength of the wrinkles was set at 40 nm, and the amplitude was 3 nm.The temperature was set to 25 °C.The finite element method was used to solve the TDGL equations, and the values of the parameters in the simulation were listed in Table S1 (Supporting Information).The electrical governing equations and open-circuit boundary conditions were as follows: D i ⋅ n = 0 (on the boundary) , (8)   where D i is the electric displacement.The mechanical equilibrium equations and boundary conditions were as follows: ij ⋅ n = 0(on the top surface), u i = 0(on the bottom surface), (10)   where  ij is the stress.

Figure 1 .
Figure 1.Phase diagram of wrinkled P(VDF-TrFE) films under tensile loading varying wavelength and amplitude.a) Schematic illustration of the strain distribution in a wrinkled P(VDF-TrFE) film under tensile loading.The lower box is the strain distribution map of the cross-section.b) Phase diagram of wrinkled P(VDF-TrFE) films as a function of wrinkle wavelength and amplitude at a tensile strain of 7.5%.The colored background contours represent the strain ratio between the outermost valleys and peaks of wrinkles in (a).c) Top-view OOP polarization component P [001] and cross-sectional polarization distributions of the three ferroelectric domain patterns.The color of the polarization on the top surface represents the magnitude of the polarization component P [001] , and each polarization arrow on the cross-sectional plane represents the polarization direction as depicted in the colored disk.
Figure 1b displays the wavelength-amplitude phase diagram of these wrinkles obtained through morphological engineering.The diagram

Figure 2 .
Figure 2. Antiparallel concentric topological domains in the toroidal wrinkled P(VDF-TrFE) film under tensile loading.a) The IP domain structure of P(VDF-TrFE) film at 7.5% external tensile strain.b) The corresponding IP strain distribution map of P(VDF-TrFE) films.c) Distribution of  r and P r along the r-axis on the red line AB in (a,b).d) Distribution of   and P  along the -axis on the red line AB.

Figure 3 .
Figure 3. Ferroelectric polarization and strain distribution at different depths of the interfaces within the wrinkled P(VDF-TrFE) film under an external tensile strain of 7.5%.a) Ferroelectric domain structure at the top, middle, and bottom interfaces of the wrinkled P(VDF-TrFE) film.b) Polarization and strain distributions at the top, middle, and bottom interfaces of the quarter of the wrinkled P(VDF-TrFE) film (type 1 is the ferroelectric polarization along the r-axis of the film, and type 2 is the ferroelectric polarization with a large declination from the r-axis).c) Toroidal moments are a function of the z-axis location (the bottom surface is the location of z = 0).

Figure 4 .
Figure 4. Polarization and strain distribution of wrinkled P(VDF-TrFE) films at various tensile strains.a) Domains and b) strain distribution maps of wrinkled P(VDF-TrFE) films at 5%, 7.5%, and 10% tensile strain, respectively.c) Distribution of the polarization component P r and strain component  r along the r-axis on line AB at different tensile strains.d) The z-component of the toroidal moment G z on the top surface of the wrinkled P(VDF-TrFE) film as a function of the external tensile strain.