flexoelectricity

{{Short description|Polarization due to a strain gradient}}

{{ref improve|date=May 2012}}

Flexoelectricity is a property of a dielectric material where there is coupling between electrical polarization and a strain gradient. This phenomenon is closely related to piezoelectricity, but while piezoelectricity refers to polarization due to uniform strain, flexoelectricity specifically involves polarization due to strain that varies from point to point in the material. This nonuniform strain breaks centrosymmetry, meaning that unlike in piezoelectricity, flexoelectric effects occur in both centrosymmetric and asymmetric crystal structures.{{Cite journal |author=Zubko |first1=Pavlo |last2=Catalan |first2=Gustau |last3=Tagantsev |first3=Alexander K. |date=2013 |title=Flexoelectric Effect in Solids |journal=Annual Review of Materials Research |volume=43 |pages=387–421 |bibcode=2013AnRMS..43..387Z |doi=10.1146/annurev-matsci-071312-121634 |hdl-access=free |hdl=10261/99362}} This property is not the same as Ferroelasticity. It plays a critical role in explaining many interesting electromechanical behaviors in hard crystalline materials and core mechanoelectric transduction phenomena in soft biomaterials.{{Cite journal |author=Nguyen |first1=Thanh D. |last2=Mao |first2=Sheng |last3=Yeh |first3=Yao-Wen |last4=Purohit |first4=Prashant K. |last5=McAlpine |first5=Michael C. |date=2013 |title=Nanoscale Flexoelectricity |journal=Adv. Mater. |volume=25 |issue=7 |pages=946–974 |doi=10.1002/adma.201203852 |pmid=23293034|bibcode=2013AdM....25..946N }} Additionally, it is a size-dependent effect that becomes more significant in nanoscale systems, such as crack tips.{{Cite journal|title=Direct observation of huge flexoelectric polarization around crack tips|journal = Nano Lett.|volume = 20|issue = 1|pages = 88–94|author= Hongguang Wang, Xijie Jiang, Yi Wang, Robert W. Stark, Peter A. van Aken, Jochen Mannhart, Hans Boschker|date=2020|doi = 10.1021/acs.nanolett.9b03176| pmid=31851827 | bibcode=2020NanoL..20...88W }}

In common usage, flexoelectricity is the generation of polarization due to a strain gradient; inverse flexoelectricity is when polarization, often due to an applied electric field, generates a strain gradient. Converse flexoelectricity is where a polarization gradient induces strain in a material.{{cite journal| vauthors = Abdollahi A, Domingo N, Arias I, Catalan G | date = 2019| title = Converse flexoelectricity yields large piezoresponse force microscopy signals in non-piezoelectric materials.| journal = Nature Communications| volume = 10| issue = 1| pages = 1266| doi = 10.1038/s41467-019-09266-y| pmid = 30894544| pmc = 6427004| bibcode = 2019NatCo..10.1266A| doi-access= free}}

The electric polarization $P\_i$ due to mechanical strain of $\backslash epsilon\_\{ij\}$ in a dielectric is given by:

:$P\_i\; =\; e\_\{ijk\}\backslash epsilon\_\{jk\}\; +\; \backslash mu\_\{ijkl\}\backslash frac\{\backslash partial\backslash epsilon\_\{jk\}\}\{\backslash partial\; x\_l\}$

where the first term corresponds to the direct piezoelectric effect and the second term corresponds to the flexoelectric polarization induced by the strain gradient.

Here, the flexoelectric coefficient, $\backslash mu\_\{ijkl\}$, is a fourth-rank polar tensor and $e\_\{ijk\}$ is the coefficient corresponding to the direct piezoelectric effect.