Modeling of Hysteretic Behavior in Ferroelectric Polymers Micka€ el Lallart, Ga€ el Sebald, Jean-Fabien Capsal, Benjamin Ducharne, Daniel Guyomar Universit e De Lyon, INSA-Lyon, LGEF EA 682, F-69621, France Correspondence to: M. Lallart (E - mail: mickael.lallart@insa-lyon.fr) Received 27 July 2015; accepted 29 September 2015; published online 28 October 2015 DOI: 10.1002/polb.23939 ABSTRACT: Controlling the polarization state of ferroelectric mate- rials, and more particularly piezoelectric polymers, is critical to ensure good operation of actuators or sensors using such energy conversion mechanisms. More specifically, the modeling and pre- diction of the hysteretic behavior of such materials is a critical aspect for the fabrication of robust and accurate devices. The pur- pose of this article is to present a model based on mathematical functions describing hysteretic behavior as a sum of elementary polarizations arising from combined avalanche and saturation physical effects. Predicted responses show good agreement with experimental measurements, and extension of the model for taking into account electric field-induced crystallization during operations is presented. Finally, the proposed model is simple to implement and does no require heavy computational and memory requirements, as it relies on pure mathematical func- tions and only requires unidimensional distribution of elementary polarizations. V C 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2016, 54, 499–508 KEYWORDS: crystallization; ferroelectricity; modeling INTRODUCTION Ferroelectric materials have been used as trans- ducers for actuation and sensing for several decades, 1,2 through the use of the electromechanical coupling they offer. Originally, ferroelectric ceramics have been of significant interest for applications like sonars, inkjet transducers and so on, then followed by ferroelectric crystals that exhibit giant electromechanical effects. Nowadays however, particu- lar trends and demands in terms of applications in flexible transducers, for instance artificial muscles, flexible sensors for Structural Health Monitoring or haptic devices, and ferro- electric transistors and memories, 3–13 have drawn significant attention on electroactive polymers (EAP). Ferroelectric poly- mers are a particular class of EAPs, that feature long-term remnant polarization, 14 allowing them to be used as piezo- electric transducers, for example. More specifically, polyvinyli- dene difluoride (PVDF) materials have been widely used for this purpose. However, polarization of this material requires stretching to allow the appearance of a polar b-phase that exhibits ferroelectric behavior. Fluorinated PVDF, namely poly(vinylidenefluoride-co-trifluoroethylene) (P(VDF-TrFE)) copolymer, allows polarization without stretching, making its processing easier while featuring similar remnant polarization that b-PVDF. 15–19 Other copolymers, such as poly(vinylidene fluoride-hexafluoropropylene) (P(VDF-HFP)) or poly(vinyli- dene fluoride - chlorotrifluoroethylene) (P(VDF-CTFE)) feature high energy density (that makes them particularly suitable for ferroelectric capacitors and transistors), but exhibit lower remnant polarization, limiting their application as sensors or actuators. 20,21 In all cases, however, being able to accurately predict the behav- ior of such transducers is of prior importance to ensure good sensing and actuation abilities. Such predictions should also be performed using as little computation as possible, to be able to quickly control the transducer using embedded controllers and/or limit the consumption in energy-constrained systems. In addition to ab initio studies that are not implementable due to their very high computational requirements, 22 many macro- scopic models for assessing the hysteretic behavior of ferroelec- tric devices have been proposed. 23 The simplest model consists of shifting an anhysteretic curve by the coercive field, the sign of the shift being given by the rate of change in the polarization. However, such an approach works well only for a particular electric field magnitude, and cannot relate first polarization or minor loops, for example. One of the first full models, initially developed for ferromagnetic elements, lies in the Preisach approach, 24 that consists in a two-dimensional distribution of elementary loops (called “hysterons”), whose polarization direc- tion switch depends on the electric field evolution direction. Although quite accurate and widely used, the Preisach model has no analytical form, hence requiring high computational requirements in addition to high memory capacity due to the two-dimensional aspect of the model. Enhancement of the Preisach model, such as the work presented by Tsang et al. in Ref. 15, consisted in introducing analytical models in combination with Preisach approach in order to lighten the computational effort. Another mathematical formulation of the Preisach model has been done by Krasnosel’skii, leading to the so-called Krasnosel’skii-Pokrovskii (KP) model. 25 However, V C 2015 Wiley Periodicals, Inc. WWW.MATERIALSVIEWS.COM JOURNAL OF POLYMER SCIENCE, PART B: POLYMER PHYSICS 2016, 54, 499–508 499 JOURNAL OF POLYMER SCIENCE WWW.POLYMERPHYSICS.ORG FULL PAPER