Research Papers

A Theory of Flexoelectric Membranes and Effective Properties of Heterogeneous Membranes

[+] Author and Article Information
P. Mohammadi

Department of Mechanical Engineering,
University of Houston,
Houston, TX 77204

L. P. Liu

Department of Mathematics,
Department of Mechanical
Aerospace Engineering,
Rutgers University,
Newark, NJ 08854

P. Sharma

Department of Mechanical Engineering,
Department of Physics,
University of Houston,
Houston, TX 77204
e-mail: psharma@uh.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received January 5, 2013; final manuscript received February 14, 2013; published online August 22, 2013. Editor: Yonggang Huang.

J. Appl. Mech 81(1), 011007 (Aug 22, 2013) (11 pages) Paper No: JAM-13-1008; doi: 10.1115/1.4023978 History: Received January 05, 2013; Revised February 14, 2013

Recent developments in flexoelectricity, especially in nanostructures, have lead to several interesting notions such as piezoelectric materials without using piezoelectric materials and enhanced energy harvesting at the nanoscale, among others. In the biological context also, membrane flexoelectricity has been hypothesized to play an important role, e.g., biological mechanotransduction and hearing mechanisms, among others. In this paper, we consider a heterogeneous flexoelectric membrane and derive the homogenized or renormalized flexoelectric, dielectric, and elastic response, therefore, relating the corresponding effective electromechanical properties to its microstructural details. Our work allows design of a microstructure to tailor flexoelectric response, and an illustrative example is given for biological membranes.

Copyright © 2014 by ASME
Topics: Membranes
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Wang, X., Song, J., Zhang, F., He, C., Zheng, H., and Wang, Z. K., 2010, “Electricity Generation Based on One-Dimensional Group-III Nitride Nanomaterials,” Adv. Mater., 22, pp. 2155–2158. [CrossRef] [PubMed]
Madden, J. D. W., Vandesteeg, N. A., Anquetil, P. A., Madden, P. G. A., Takshi, A., Pytel, R. Z., Lafontaine, S. R., Wieringa, P. A., and Hunter, I. W., 2004, “Artificial Muscle Technology: Physical Principles and Naval Prospects,” IEEE Ocean. Eng. Soc., 29(3), pp. 706–728. [CrossRef]
Gautschi, G., 2002, Piezoelectric Sensorics: Force, Strain, Pressure, Acceleration and Acoustic Emission Sensors, Springer, Berlin.
Nye, J. F., 1985, Physical Properties of Crystals: Their Representation by Tensors and Matrices, reprint ed., Oxford University Press, New York.
Mashkevich, V. S., and Tolpygo, K. B., 1957, “Electrical, Optical and Elastic Properties of Diamond Type Crystals,” Sov. Phys. JETP, 5(3), p. 435–439.
Bursian, E. V., and Trunov, N. N., 1984, “Nonlocal Piezoelectric Effect,” Sov. Physics Solid State, 16(4), pp. 760–762.
Tagantsev, A. K., 1986, “Piezoelectricity and Flexoelectricity in Crystalline Dielectrics,” Phys. Rev. B, 34, pp. 5883–5889. [CrossRef]
Tagantsev, A. K., 1991, “Electric Polarization in Crystals and Its Response to Thermal and Elastic Perturbations,” Phase Trans., 35, pp. 119–203. [CrossRef]
Meyer, R. B., “Piezoelectric Effects in Liquid Crystals,” Phys. Rev. Lett., 22, 918–921 (1969). [CrossRef]
Schmidt, D., Schadt, M., and Helfrich, W., 1972, “Liquid-Crystalline Curvature Elasticity,” Naturforsch, Z, A 27A, p. 277.
Indenbom, V. L., Loginov, V. B., and Osipov, M. A., 1981, Flexoelectric Effect and structure of Crystals. Kristallografiya28, 1157.
Cross, L. E., 2006, “Flexoelectric Effects: Charge Separation in Insulating Solids Subjected to Elastic Strain Gradients,” J. Mater. Sci., 41, pp. 53–63. [CrossRef]
Sharma, N. D., Maranganti, R., and Sharma, P., 2007, “On the Possibility of Piezoelectric Nanocomposites Without Using Piezoelectric Materials,” J. Mech. Phys. Solid., 55, pp. 2328–2350. [CrossRef]
Tagantsev, A. K., Meunier, V., and Sharma, P., 2009, “Novel Electromechanical Phenomena at the Nanoscale: Phenomenological Theory and Atomistic Modeling,” MRS Bulletin, 34(9), 643–647. [CrossRef]
Ma, W., and Cross, L. E., 2001, “Large Flexoelectric Polarization in Ceramic Lead Magnesium Niobate,” Appl. Phys. Lett., 79(19), pp. 4420–4422. [CrossRef]
Ma, W., and Cross, L. E., 2002, “Flexoelectric Polarization in Barium Strontium Titanate in the Paraelectric State,” Appl. Phys. Lett., 81(19), pp. 3440–3442. [CrossRef]
Ma, W., and Cross, L. E., 2003, “Strain-Gradient Induced Electric Polarization in Lead Zirconate Titanate Ceramics,” Appl. Phys. Lett., 82(19), pp. 3923–3925. [CrossRef]
Ma, W., and Cross, L. E., 2006, “Flexoelectricity of Barium Titanate,” Appl. Phys. Lett., 88, p. 232902. [CrossRef]
Catalan, G., Sinnamon, L. J., and Gregg, J. M., 2004, The Effect of Flexoelectricity on the Dielectric Properties of Inhomogeneously Strained Ferroelectric Thin Films,” J. Phys. Condens. Matt., 16(13), pp. 2253–2264. [CrossRef]
Zubko, P., Catalan, G., Buckley, A., Welche, P. R. L., and Scott, J. F., 2007, “Strain–Gradient Induced Polarization in SrTiO3,” Phys. Rev. Lett., 99, p. 167601. [CrossRef] [PubMed]
Fu, J. Y., Zhu, W., Li, N., Cross, L. E., 2006, “Experimental Studies of the Converse Flexoelectric Effect Induced by Inhomogeneous Electric Field in a Barium Strontium Titanate Composition,” J. Appl. Phys., 100, p. 024112. [CrossRef]
Fu, J. Y., Zhu, W., Li, N., Smith, N. B., and Cross, E. L., 2007, “Gradient Scaling Phenomenon in Microsize Flexoelectric Piezoelectric Composites,” Appl. Phys. Lett., 91, p. 182910. [CrossRef]
Eliseev, E. A., Morozovska, A. N., Glinchuk, M. D., and Blinc, R., 2009, “Spontaneous Flexoelectric/Flexomagnetic Effect in Nanoferroics,” Phys. Rev. B, 79, p. 165433. [CrossRef]
Eliseev, E. A., Glinchuk, M. D., Khist, V., Skorokhod, V. V., Blinc, R., and Morozovska, A. N., 2011, “Linear Magnetoelectric Coupling and Ferroelectricity Induced by the Flexomagnetic Effect in Ferroics,” Phys. Rev. B, 84(17), p. 174112. [CrossRef]
Maranganti, R., and Sharma, P., 2009, “Atomistic Determination of Flexoelectric Properties of Crystalline Dielectrics,” Phys. Rev. B, 80, p. 054109. [CrossRef]
Majdoub, M. S., Sharma, P., and Cagin, T., 2008, “Enhanced Size-Dependent Piezoelectricity and Elasticity in Nanostructures Due to the Flexoelectric Effect,” Phys. Rev. B, 77, p. 125424. [CrossRef]
Majdoub, M. S., Sharma, P., and Cagin, T., 2009, “Dramatic Enhancement in Energy Harvesting for a Narrow Range of Dimensions in Piezoelectric Nanostructures,” Phys. Rev. B, 78, p. 121407(R). [CrossRef]
Sharma, N. D., Landis, C. M., and Sharma, P., 2010, “Piezoelectric Thin-Film Superlattices Without Using Piezoelectric Materials,” J. Appl. Phys., 108, p. 024304. [CrossRef]
Gharbi, M., Sun, Z. H., White, K., El-Borgi, S., and Sharma, P., 2011, “Flexoelectric Properties of Ferroelectrics and the Nanoindentation Size-Effect,” Int. J. Solid. Struct., 48, p. 249. [CrossRef]
Kalinin, S. V., and Meunier, V., 2008, “Electronic Flexoelectricity in Low-Dimensional Systems,” Phys. Rev. B, 77(3), p. 033403. [CrossRef]
Dumitrica, T., Landis, C. M., and Yakobson, B. I., 2002, “Curvature Induced Polarization in Carbon Nanoshells,” Chem. Phys. Lett., 360(1–2), pp. 182–188. [CrossRef]
Baskaran, S., Thiruvannamalai, S., Heo, H., Lee, H. J., Francis, S. M., Ramachandran, N., and Fu, J. Y.2010, “Converse Piezoelectric Responses in Nonpiezoelectric Materials Implemented via Asymmetric Configurations of Electrodes,” J. Appl. Phys., 108, p. 064116. [CrossRef]
Chandratre, S., and Sharma, P., 2012, “Coaxing Graphene to be Piezolectric,” Appl. Phys. Lett., 100, p. 023114. [CrossRef]
Naumov, I., Bratkovsky, A. M., and Ranjan, V., 2009, “Unusual Flexoelectric Effect in Two-Dimensional Noncentrosymmetric sp2-Bonded Crystals,” Phys. Rev. Lett., 102(21), p. 217601. [CrossRef] [PubMed]
Petrov, A. G., Spassova, M., and Fendler, J. H., 1996, “Flexoelectricity and Photoflexoelectricity in Model and Biomembranes,” Thin Solid Films, 284, p. 845. [CrossRef]
Petrov, A. G., 1998, “Mechanosensitivity of Cell Membranes, Role of Liquid Crystalline Lipid Matrix Liquid Crystals,” Chem. Struct., 3319, p, 306.
Kuczynski, W., and Hoffmann, J., 2005, “Determination of Piezoelectric and Flexoelectric Polarization in Ferroelectric Liquid Crystals,” Phys. Rev. E, 72(4), p. 041701. [CrossRef]
Spector, A., Deo, N., Grosh, K., Ratnanather, J., and Raphael, R., 2005, “Electromechanical Models of the Outer Hair Cell Composite Membrane,” J. Memb. Biol., 209(2–3), pp. 135–152. [CrossRef]
Harden, J., Chambers, M., Verduzco, R., Luchette, P., Gleeson, J. T., Sprunt, S., and Jákli, A., 2010, “Giant Flexoelectricity in Bent-Core Nematic Liquid Crystal Elastomers,” Appl. Phys. Lett., 96(10), p. 102907. [CrossRef]
Jewell, S. A., 2011, “Living Systems and Liquid Crystals,” Liq. Cryst., 38(11–12), pp. 1699–1714. [CrossRef]
Petrov, A. G., “Flexoelectric Model for Active Transport,” Physical and Chemical Bases of Biological Information Transfer, Plenum Press, New York, pp. 111–125.
Petrov, A. G., 2002, “Flexoelectricity of Model and Living Membranes,” Biochim. Biophys. Acta, 1561, pp. 1–25. [CrossRef] [PubMed]
Petrov, A. G., 2006, “Electricity and Mechanics of Biomembrane Systems: Flexoelectricity in Living Membranes,” Anal. Chim. Acta, 568(1–2), pp. 70–83. [CrossRef] [PubMed]
Petrov, A. G., 2007, “Flexoelectricity and Mechanotransduction,” Current Topics in Membranes, Vol. 58: Mechanosensitive Channels, O. P.Hamil, ed., Elsevier/Academic Press, Galveston, TX, pp. 121–150.
Raphael, R. M., Popel, A. S., and Brownell, W. E., 2000, “A Membrane Bending Model of Outer Hair Cell Electromotility,” Biophys. J., 78(6), pp. 2844–2862. [CrossRef] [PubMed]
Brownell, W. E., Spector, A. A., Raphael, R. M., and Popel, A. S., 2001, “Micro- and Nanomechanics of the Cochlear Outer Hair Cell,” Ann. Rev. Biomed. Eng., 3, pp. 169–194. [CrossRef]
Brenemann, K. D., and Rabbitt, R. D., 2009, “Piezo- and Flexoelectric Membrane Materials Underlie Fast Biological Motors in the Ear,” Mater. Res. Soc. Symp. Proc., 1186E, pp. 1186–JJ06-04. [PubMed]
Steigmann, D. J., 2009, “Analysis of Nonlinear Electrostatic Membranes,” J. Elast., 97(1), pp. 97–101. [CrossRef]
Evans, L. C., 1998, Partial Differential Equations, American Mathematical Society, Providence, RI.
Toupin, R. A., 1956, “The Elastic Dielectric,” J. Rational Mech. Anal., 5, pp. 849–914.
Petrov, A. G., 1999, The Lyotropic State of Matter: Molecular Physics and Living Matter Physics, Gordon and Breach Science Publishers, Amsterdam.
Eshelby, J. D., 1957, “The Determination of the Elastic Field of an Ellipsoidal Inclusion and Related Problems,” Proc. R. Soc. London, Ser. A, 241, pp. 376–396. [CrossRef]
Eshelby, J. D., 1961, “Elastic Inclusions and Inhomogeneities,” Progress in Solid Mechanics, II, I. N.Sneddon and R.Hill, eds., North Holland, Amsterdam, pp. 89–140.
Li, S., 2000, “The Micromechanics Theory of Classical Plates: A Congruous Estimate of Overall Elastic Stiffness,” Int. J. Solid. Struct., 37(40), pp. 5599–5628. [CrossRef]
Liu, L. P., James, R. D., and Leo, P. H., 2006, “Magnetostrictive Composites in the Dilute Limit,” J. Mech. Phys. Solids, 54(5), pp. 951–974. [CrossRef]
Liu, L. P., 2013, “Polynomial Eigenstress Inducing Polynomial Strain of the Same Degree in an Ellipsoidal Inclusion and Its Applications,” Math. Mech. Solid, 18(2), pp. 168–180. [CrossRef]
Liu, L. P., James, R. D., and Leo, P. H., 2008, “New Extremal Inclusions and Their Applications to Two–Phase Composites,” Arch. Rational Mech. Anal. (accepted).
Liu, L. P., James, R. D., and Leo, P. H., 2007, “Periodic Inclusion—Matrix Microstructures With Constant Field Inclusions,” Met. Mat. Trans. A, 38, pp. 781–787. [CrossRef]
Ashcroft, N. W., and Mermin, N. D., 1976, Solid State Physics, Brooks/Cole, Cengage Learning.
Vigdergauz, S. B., 1986, “Effective Elastic Parameters of a Plate With a Regular System of Equal-Strength Holes,” Inzhenernyi Zhurnal: Mekhanika Tverdogo Tela: MIT, 21, pp. 165–169.


Grahic Jump Location
Fig. 1

Mechanism of flexoelectricity in 2D crystalline membranes such as graphene (adapted from Dumitrica et al. [31])

Grahic Jump Location
Fig. 2

A representative volume element of two-phase heterogeneous membrane: (a) a simple laminate; (b) inclusions embedded in a continuous matrix

Grahic Jump Location
Fig. 3

Effective flexoelectric of protein inclusions in lipid bilayer: (a) the effective constant γe = fe/ae as a function of aspect ratio of protein ellipsoid. The volume fraction of protein is assumed to be 0.1, and (b) the effective constant γe = fe/ae as a function of volume fraction of inclusion. The solid curve is predicted by Eq. (4.38); the dashed curve is calculated by Eq. (5.2) assuming ax/ay = 1.



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