×

A theory of flexoelectricity with surface effect for elastic dielectrics. (English) Zbl 1244.78006

Summary: The flexoelectric effect is very strong for nanosized dielectrics. Moreover, on the nanoscale, surface effects and the electrostatic force cannot be ignored. In this paper, an electric enthalpy variational principle for nanosized dielectrics is proposed concerning with the flexoelectric effect, the surface effects and the electrostatic force. Here, the surface effects contain the effects of both surface stress and surface polarization. From this variational principle, the governing equations and the generalized electromechanical Young-Laplace equations are derived and can account for the effects of flexoelectricity, surface and the electrostatic force. Moreover, based on this variational principle, both the generalized bulk and surface electrostatic stresses can be obtained and are composed of two parts: the Maxwell stress corresponding to the polarization and strain and the remainder relating to the polarization gradient and the strain gradient. The theory developed in this paper provides the underlying framework for the analyses and computational solutions of electromechanical problems in nanodielectrics.

MSC:

78A55 Technical applications of optics and electromagnetic theory
78A30 Electro- and magnetostatics
74F15 Electromagnetic effects in solid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aris, R., Vectors, Tensors, and Basic Equations of Fluid Mechanics (1962), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0123.41502
[2] Askar, A.; Lee, P. C.Y.; Cakmak, A. S., A lattice dynamics approach to the theory of elastic dielectrics with polarization gradient, Phys. Rev. B, 1, 3525-3537 (1970)
[3] Brandino, G. P.; Cicero, G., Polarization properties of \((1 \overline{1} 0 0)\) and \((1 1 \overline{2} 0)\) SiC surfaces from first principles, Phys. Rev. B, 76, 085322 (2007)
[4] Bursian, E. V.; Trunov, N. N., Nonlocal piezoelectric effect, Fiz. Tverd. Tela, 16, 1187-1190 (1974)
[5] Camacho, A. S.; Nossa, J. F., Geometric dependence of the dielectric properties of quantum dots arrays, Microelectron. J., 40, 835-837 (2009)
[6] Catalan, G.; Sinnamon, L. J.; Gregg, J. M., The effect of flexoelectricity on the dielectric properties of inhomogeneously strained ferroelectric thin films, J. Phys. Condens. Matter, 16, 2253-2264 (2004)
[7] Chen, H.; Hu, G. K.; Huang, Z. P., Effective moduli for micropolar composite with interface effect, Int. J. Solids Struct., 44, 8106-8118 (2007) · Zbl 1167.74534
[8] Cross, L. E., Flexoelectric effects: charge separation in insulating solids subjected to elastic strain gradients, J. Mater. Sci., 41, 53-63 (2006)
[9] Dequesnes, M.; Rotkin, S. V.; Aluru, N. R., Parameterization of continuum theories for single wall carbon nanotube switches by molecular dynamics simulations, J. Comput. Electron., 1, 313-316 (2002)
[10] Dequesnes, M.; Rotkin, S. V.; Aluru, N. R., Calculation of pull-in voltages for carbon nanotube based nanoelectromechanical switches, Nanotechnology, 13, 120-131 (2002)
[11] Dequesnes, M.; Tang, Z.; Aluru, N. R., Static and dynamic analysis of carbon nanotube-based switches, ASME J. Eng. Mater. Tech., 126, 230-237 (2004)
[12] Eliseev, E. A.; Morozovska, A. N.; Glinehuk, M. D.; Blinc, R., Spontaneous flexoelectric/flexomagnetic effect in nanoferroics, Phys. Rev. B, 79, 165433 (2009)
[13] Eringen, A. C., Balance laws of micromorphic continua revisited, Int. J. Eng. Sci., 30, 805-810 (1992) · Zbl 0754.73079
[14] Fechner, M.; Ostanin, S.; Mertig, I., Effect of the surface polarization in polar perovskites studied from first principles, Phys. Rev. B, 77, 094112 (2008)
[15] Fousek, J.; Cross, L. E.; Litvin, D. B., Possible piezoelectric composites based on flexoelectric effect, Mater. Lett., 39, 289-291 (1999)
[16] Gao, R. P.; Pan, Z. W.; Wang, Z. L., Work function at the tips of multiwalled carbon nanotubes, Appl. Phys. Lett., 78, 1757 (2001)
[17] George, M.; Nair, S. S.; Malini, K. A.; Joy, P. A.; Anantharaman, M. R., Finite size effects on the electrical properties of sol-gel synthesized \(CoFe_2 O_4\) powders: deviation from Maxell-Wagner theory and evidence of surface polarization effects, J. Phys. D: Appl. Phys., 40, 1593-1602 (2007)
[18] Gurtin, M. E.; Murdoch, A. I., A continuum theory of elastic material surfaces, Arch. Ration. Mech. Anal., 57, 291-323 (1975) · Zbl 0326.73001
[19] Hu, S.L., Shen, S., 2010. Variational principles and governing equations in nano-dielectrics with the flexoelectric effect. Sci. China G., in press, doi:10.1007/s11433-010-0098-x.; Hu, S.L., Shen, S., 2010. Variational principles and governing equations in nano-dielectrics with the flexoelectric effect. Sci. China G., in press, doi:10.1007/s11433-010-0098-x.
[20] Kuang, Z. B., Nonlinear Continuum Mechanics (2002), Shanghai Jiaotong University Press: Shanghai Jiaotong University Press Shanghai, (in Chinese)
[21] Kuang, Z. B., Some problems in electrostrictive and magnetostrictive materials, Acta Mech. Solida, 20, 219-227 (2007)
[22] Kuang, Z. B., Some variational principles in elastic dielectric and elastic magnetic materials, Eur. J. Mech. A/Solids, 27, 504-514 (2008) · Zbl 1154.74338
[23] Kuang, Z. B., Some variational principles in electroelastic media under finite deformation, Sci. China G, 51, 1390-1402 (2008) · Zbl 1147.74021
[24] Kuang, Z. B., Internal energy variational principles and governing equations in electroelastic analysis, Int. J. Solids Struct., 46, 902-911 (2009) · Zbl 1215.74025
[25] Landau, L. D.; Lifshitz, E. M., Electrodynamics of Continuum Media (1960), Pergamon Press: Pergamon Press Oxford · Zbl 0122.45002
[26] Landau, L.D., Lifshitz, E.M., 2000. Statistical Physics, Course of Theoretical Physics, 3rd ed., vol. 5. Butterworth-Heinemann, Oxford.; Landau, L.D., Lifshitz, E.M., 2000. Statistical Physics, Course of Theoretical Physics, 3rd ed., vol. 5. Butterworth-Heinemann, Oxford.
[27] Majdoub, M. S.; Sharma, P.; Cagin, T., Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect, Phys. Rev. B, 77, 125424 (2008)
[28] Majdoub, M. S.; Sharma, P.; Cagin, T., Dramatic enhancement in energy harvesting for a narrow range of dimensions in piezoelectric nanostructures, Phys. Rev. B, 78, 121407 (2008)
[29] Maranganti, R.; Sharma, N. D.; Sharma, P., Electromechanical coupling in nonpiezoelectric materials due to nanoscale size effects: Green’s function solutions and embedded inclusions, Phys. Rev. B, 74, 014110 (2006)
[30] Maranganti, R.; Sharma, P., Atomistic determination of flexoelectric properties of crystalline dielectric, Phys. Rev. B, 80, 054109 (2009)
[31] Mindlin, R. D., Micro-structure in linear elasticity, Arch. Ration. Mech. Anal., 16, 51-78 (1964) · Zbl 0119.40302
[32] Mindlin, R. D., Polarization gradient in elastic dielectrics, Int. J. Solids Struct., 4, 637-642 (1968) · Zbl 0159.57001
[33] Mindlin, R. D., Continuum and lattice theories of influence of electromechanical coupling on capacitance of thin dielectric films, Int. J. Solids Struct., 5, 1197-1208 (1969)
[34] Ogden, R. W., Nonlinear Elastic Deformation (1984), Ellis Horwood, Halsted Press: Ellis Horwood, Halsted Press Chichester, New York · Zbl 0541.73044
[35] Sahin, E.; Dost, S., A strain-gradients theory of elastic dielectrics with spatial dispersion, Int. J. Eng. Sci., 26, 1231-1245 (1988)
[36] Sharma, N. D.; Maranganti, R.; Sharma, P., On the possibility of piezoelectric nanocomposites without using piezoelectric materials, Journal of the Mechanics and Physics of Solids, 52, 2328-2350 (2007) · Zbl 1171.74016
[37] Shen, S.; Kuang, Z. B., An active control model of laminated piezothermoelastic plate, Int. J. Solids Struct., 36, 1925-1947 (1999) · Zbl 0942.74051
[38] Slavchov, R.; Ivanov, T.; Radoev, B., Effect of the surface polarizability on electrostatic screening in semiconductors, J. Phys.: Condens. Matter, 18, 5873-5879 (2006)
[39] Stratton, J. A., Electromagnetic Theory (1941), McGraw-Hill: McGraw-Hill New York · Zbl 0022.09303
[40] Tagantsev, A. K., Piezoelectricity and flexoelectricity in crystalline dielectrics, Phys. Rev. B, 34, 5883-5889 (1986)
[41] Toupin, R. A., The elastic dielectric dielectrics, J. Rat. Mech. Anal., 5, 849-914 (1956) · Zbl 0072.23803
[42] Wang, Z. Q.; Zhao, Y. P.; Huang, Z. P., The effects of surface tension on the elastic properties of nano structures, Int. J. Eng. Sci. (2009)
[43] Yang, J. S., An Introduction to the Theory of Piezoelectrocity (2004), Kluwer Academic Publishers: Kluwer Academic Publishers Boston
[44] Zhang, X.; Sharma, P., Size dependency of strain in arbitrary shaped anisotropic embedded quantum dots due to nonlocal dispersive effects, Phys. Rev. B, 72, 195345 (2005)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.