We apply the Finsler geometry (FG) modeling technique to study the electric field-induced strain in ferroelectric polymers. Polyvinylidene difluoride (PVDF) has a negative longitudinal piezoelectric coefficient, which is unusual in ferroelectrics, and therefore the shape changes in this material are hard to predict. We find that the results of Monte Carlo simulations for the FG model are in good agreement with experimental strain-electric field curves of PVDF-based polymers in both longitudinal and transverse directions. This implies that FG modeling is suitable for reproducing the reverse piezoelectric effect in PVDF

Bernard, C.

Cavaille, J. -Y.

Chenal, J. -M.

Diguet, G.

Egorov, V. I.

Jug, G.

Koibuchi, H.

Lame, O.

Maksimova, O. G.

Sebald, G.

Takagi, T.

English

Egorov V. I.

Maksimova O. G.

Koibuchi H.

Jug G.

Bernard C.

Chenal J. -M.

Lame O.

Diguet G.

Sebald G.

Cavaille J. -Y.

Takagi T.

Archivio istituzionale della ricerca - Università dell'Insubria

Journal of Physics: Conference SeriesPAPER • OPEN ACCESSFinsler geometry modeling of reverse piezoelectriceffect in PVDFTo cite this article: V I Egorov et al 2019 J. Phys.: Conf. Ser. 1391 012014 View the article online for updates and enhancements.You may also likeEffect of sodium niobate (NaNbO3)nanorods on -phase enhancement inPolyvinylidene fluoride (PVDF) polymerAbhishek Anand and M C Bhatnagar-Increased dielectric performance of PVDF-based composites by electrochemicalexfoliated graphite additivesYuanzhen Du, Dawei Cao, Yuan Liu et al.-Weak integrability of Hamiltonians on thetwo torus and rigidityJosé Barbosa Gomes and Rafael ORuggiero-Recent citationsOut-of-plane dynamic instability ofnonlocal shear deformable nanoplatesmade of polyvinylidene fluoride materialssubjected to electromechanical forcesMostafa Pirmoradian et al-Electromechanical properties offerroelectric polymers: Finsler geometrymodeling and a Monte Carlo studyV. Egorov et al-This content was downloaded from IP address 193.206.168.223 on 16/11/2021 at 11:06Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing Ltd8th International Conference on Mathematical Modeling in Physical ScienceJournal of Physics: Conference Series 1391 (2019) 012014IOP Publishingdoi:10.1088/1742-6596/1391/1/0120141Finsler geometry modeling of reverse piezoelectriceffect in PVDFV I Egorov1,∗, O G Maksimova1, H Koibuchi2, G Jug3, C Bernard4,5,J-M Chenal6, O Lame6, G Diguet4, G Sebald4, J-Y Cavaillé4,T Takagi4,71 Cherepovets State University, Cherepovets, Russian Federation2 National Institute of Technology (KOSEN), Sendai College, Natori, Japan3 DiSAT, Insubria University, Como, Italy4 ELyTMaX UMI 3757, CNRS-Université de Lyon-Tohoku University International JointUnit, Tohoku University, Sendai, Japan5 Frontier Research Institute for Interdisciplinary Sciences (FRIS), Tohoku University, Sendai,Japan6 Materials Engineering and Science (MATEIS), CNRS, INSA Lyon UMR 5510, Université deLyon, Villeurbanne Cedex, France7 Institute of Fluid Science (IFS), Tohoku University, Sendai, JapanE-mail: ∗rvladegorov@rambler.ruAbstract. We apply the Finsler geometry (FG) modeling technique to study the electricfield-induced strain in ferroelectric polymers. Polyvinylidene difluoride (PVDF) has a negativelongitudinal piezoelectric coefficient, which is unusual in ferroelectrics, and therefore the shapechanges in this material are hard to predict. We find that the results of Monte Carlo simulationsfor the FG model are in good agreement with experimental strain-electric field curves of PVDF-based polymers in both longitudinal and transverse directions. This implies that FG modelingis suitable for reproducing the reverse piezoelectric effect in PVDF.1. IntroductionThe reverse piezoelectric effect in ferroelectrics is widely used for creating actuators to convertthe electrical energy into the mechanical energy, or vise versa. One of the most important goalsin material science is to develop flexible elastic materials with piezoelectric properties. For thispurpose, polyvinylidene difluoride (PVDF)-based polymers need to have increased mechanicalstrength, shock resistance, and flexibility compared with low-weight molecular ferroelectrics.However, PVDF has an unusual shrinking behavior along the direction of the external field, ofwhich the mechanism is not yet completely understood [1]. Furthermore, the shape-change ofthe material caused by the electric field is typically very complex and difficult to simulate.Finsler geometry (FG) technique has been successfully applied for modeling the deformationof materials with an anisotropy of mechanical properties like rubbers and soft biological materials[2, 3]. Also, the shape transformation of liquid crystal elastomers under external electric fieldshas been studied by this technique [4]. In the present work, we further extend the FG modelfor describing the reverse piezoelectric effect in ferroelectric polymers. In the FG model, Finslermetric is used instead of Euclidean metric, which is always used in simulations of surface8th International Conference on Mathematical Modeling in Physical ScienceJournal of Physics: Conference Series 1391 (2019) 012014IOP Publishingdoi:10.1088/1742-6596/1391/1/0120142models for membranes. The Finsler metric is defined by using an internal degree of freedom σ,which represents the dipole moment in the case of PVDF. This approach precisely or effectivelyimplements the role of the positional and directional degrees of freedom in polymers.2. The modelMore detailed description of 3D FG model is reported in [5]. For the simulation of a field-induced deformation, we modify the Hamiltonian in [5] by taking into account an additionaldipole interaction with the external electric field. In this section, the 3D FG approach is brieflyoutlined. The continuous Hamiltonian S1 corresponding to the Gaussian bond potential is givenby:S1 =∫√ggab∂~r∂xa∂~r∂xbd3x, (1)where ~r is a position vector of a three-dimensional body with coordinates xa(a = 1, 2, 3). Thesymbol gab is the inverse of metric tensor gab and g is its determinant.To define the discrete Hamiltonian, we use a thin cylindrical body (Fig.1(a)). This body isdiscretized by tetrahedrons using the Voronoi tessellation. We introduce a variable σi, whichcorresponds to the direction of the dipole moment at the vertex i. The discrete metric tensor hasthe form: gab =v−212 0 00 v−213 00 0 v−214 . This form is obtained from Euclidean metric by replacingits diagonal elements with v−2ij , where vij is the Finsler length given by the projection σi · ~tij ofσi on link ~tij of the tetrahedron such thatvij =√1− |σi · ~tij |2 + v0, (2)where ~tij is an unit tangential vector of the bond ij (Fig.1(b))), and v0 = 0.001 is a cutoff.Almost all ferroelectric polymers have a dipole moment, which is oriented perpendicular to thecarbon skeleton (the structure of β-PVDF is illustrated in Fig.1(c)).(a) (b)(c)Figure 1. (a) The three-dimensional thin cylinder discretized by tetrahedrons, (b) a tetrahedronon which the discrete Hamiltonian is defined, and (c) the molecular structure of β-PVDF.Replacing the integration in Eq.(1) with a sum over tetrahedrons and including the symmetricterms obtained by index replacement with the factor 1/4, we have a discrete version of S1 [5].With some extra terms, the discrete Hamiltonian for the FG model of the ferroelectric polymeris given by:8th International Conference on Mathematical Modeling in Physical ScienceJournal of Physics: Conference Series 1391 (2019) 012014IOP Publishingdoi:10.1088/1742-6596/1391/1/0120143S = γS1 + κS2 − bS3 − cS4 + U3D + Uvol, S1 =∑ij Γijl2ij , Γij =(1/N̄)∑tet γij(tet),S2 =∑i(1− cos(φi − π/3)), S3 =∑i σi · ~E, S4 =∑i(σi · ~E)2,U3D ={∞ (Vtet ≤ 0)0 (Vtet > 0), Uvol ={0 |V − V0| ≤ ∆V )∞ (otherwise),(3)where lij denotes a length of the bond ij. The coefficients Γij(Γij =Γji) are given by the sum oftetrahedrons sharing the bond ij, and γij(tet) for the tetrahedron of vertices 1234 are determinedvia components of the metric tensor such that [5]:γ12 =14(v12v13v14+ v21v23v24), γ13 =14(v13v12v14+ v31v32v34), γ14 =14(v14v12v13+ v41v42v43),γ23 =14(v23v21v24+ v32v31v34), γ24 =14(v24v21v23+ v42v41v43), γ34 =14(v34v31v32+ v43v41v42).(4)The second term κS2 in Eq.(3) plays a role of deformation strength against bending andshear deformation, both of which have an influence on the material shape. φi in S2 is aninternal angle of the triangle. The third term bS3 is an energy of the dipole interaction with theexternal electric field, which causes the piezoelectric effect. cS4 is the potential describing thequadratic electrostrictive effect. U3D is a constraint for the volume of the tetrahedron for notbeing negative. To prevent the volume from changing, the last term Uvol is introduced, where∆V is a mean value of the volume of one tetrahedron and V0 is an initial volume of the wholecylinder which is determined by the simulation without Uvol in the absence of external electricfield.3. The simulation resultsIn this work, we use a cylinder of size (N,NB, NT , Ntet) = (10346, 69964, 116041, 56422),where N,NB, NT , Ntet denote the total number of vertices, bonds, triangles, and tetrahedrons,respectively. The ratio of the cylinder height and its diameter is equal to 0.125. For thiscylinder, the simulation of electric field-induced deformation is carried out by Monte Carlomethod. The Monte Carlo (MC) updates of locations of vertices and σ are performed usingthe standard Metropolis algorithm. The uniform external electric field is applied along the axisof the cylinder i.e. along the z axis. The snapshots in Figs. 2(a)-(d) show how the shape ofthe cylinder is changed. We can see that the height shrinks and the diameter expands withincreasing electric field.The absolute value of thickness strain |εz| and diameter strain εd vs. the external electric fieldE for different values of the bending stiffness are presented in Figs.2(e),(f). The values of c andb =√c were chosen by data analysis for the best correspondence of the experimental and thesimulation data. We compare our simulation results for longitudinal shrinking with experimentaldata of highly electrostrictive networks of β-PVDF [6]. For the transverse stretching comparison,the strain-field diagram of P(VDF-TrFE-CFE) films is used [7]. To summarize, the simulationresults are in good agreement with experimental data in both longitudinal and transversedirections.From the comparison of the energy cS4 and the physical dielectric energy, we obtaincE2kBT/a3=ε0∆εE2exp [4], where ε0=8.85×10−12[F/m], ∆ε is the dielectric anisotropy, a is thelattice spacing, kBT is given by kBT = 4×10−21[J] at room temperature. If the experimentaldata εz or εd at Eexp in units of [106V/m] are comparable with those simulation data at E,then we conclude that E = 10−6Eexp [4]. In this case, we have a = (ckBTE/ε0∆εEexp)1/3 '(7.7×10−8) (c/∆ε)1/3 [m]. Using the value of ∆ε= 0.25 [8] and the input c= 0.024[MV−2], wefind that a' 3.5×10−8[m] in β-PVDF model, which is clearly larger than the van der Waals8th International Conference on Mathematical Modeling in Physical ScienceJournal of Physics: Conference Series 1391 (2019) 012014IOP Publishingdoi:10.1088/1742-6596/1391/1/0120144(a) (c)0 5 1005101520Eexp[MV/m]  κ=0κ=0.4κ=1.8Exp [6][%]|εz|c=0.024(e)(b) (d)0 50 1000246Eexp [MV/m]  κ=0.4κ=0.5κ=0.7Exp [7]c=0.0003εd[%](f)Figure 2. Snapshots of the cylinder with the external field: (a),(b) Eexp = 0, and (c),(d)Eexp=20 with the bending stiffness value κ=0.4. (e) The thickness-strain |εz| v.s. Eexp and (f)the diameter-strain εd vs. Eexp with several different κ values.distance DVWD ' 10−10[m]. In P(VDF-TrFE-CFE) model with c = 0.0003[MV−2], we haveless value of the lattice spacing a'8.1×10−9[m], which is reasonable because the experimentalsamples of P(VDF-TrFE-CFE) were thinner than ones of β-PVDF.4. ConclusionWe propose a simulation model on the basis of Finsler geometry for describing the reversepiezoelectric effect in PVDF. The simulation results of deformation, induced by the electricfield, are in good agreement with reported experimental data for both longitudinal shrinkingand transverse stretching. Hence, we conclude that this model is suitable to predict the shapetransformation of PVDF-based actuators.AcknowledgmentsThis work is supported in part by the Collaborative Research Project of the Institute of FluidScience, Tohoku University, and JSPS KAKENHI Grant Number JP17K05149.References[1] Katsouras I et al 2016 Nature materials 15.1 78.[2] Koibuchi H et al 2018 J. Phys. Conf. Series 1141 012018.[3] Takano Y and Koibuchi H 2017 Phys. Rev. E 95 042411.[4] Proutorov E, Matsuyama N and Koibuchi H 2018 J. Phys. Condens. Matter 30 405101.[5] Osari K and Koibuchi H 2017 Polymer 114 355.[6] Casalini R and Roland C M 2001 Appl. Phys. Lett. 79 2627.[7] Ren K, Liu S Lin M, Wang Y and Zhang Q M 2008 Sensors and Actuators A: Physical 143(2) 335-342.[8] Osaki S 1995 Polymer J. 28 323.

Finsler geometry modeling of reverse piezoelectric effect in PVDF

Egorov, V

Maksimova, O

Koibuchi, H

Jug, G

Chenal, J-M

Cavaillé, J-Y

Takagi, T

Archive Ouverte en Sciences de l'Information et de la Communication

Cavaillé, J-y

Hal-Diderot

https://irinsubria.uninsubria.it/bitstream/11383/2119521/2/Egorov_2019_J._Phys.__Conf._Ser._1391_012014.pdf

Finsler geometry modeling of reverse piezoelectric effect in PVDF

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