Review on Electromechanical Coupling Properties of Biomaterials

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Review

A Review on Electromechanical Coupling Properties of Biomaterials Inseok Chae, Chang Kyu Jeong, Zoubeida Ounaies, and Seong H. Kim ACS Appl. Bio Mater., Just Accepted Manuscript • DOI: 10.1021/acsabm.8b00309 • Publication Date (Web): 07 Sep 2018 Downloaded from http://pubs.acs.org on September 9, 2018

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ACS Applied Bio Materials

A Review on Electromechanical Coupling Properties of Biomaterials Inseok Chae1, Chang Kyu Jeong2, Zoubeida Ounaies3 and Seong H. Kim1,4* 1

Department of Chemical Engineering and Materials Research Institute, Pennsylvania State University, University Park, Pennsylvania 16802, United States

2

Division of Advanced Materials Engineering, Chonbuk National University, Jeonju, Jeonbuk 54896, Republic of Korea

3

Department of Mechanical and Nuclear Engineering and Materials Research Institute, Pennsylvania State University, University Park, Pennsylvania 16802, United States

4

Department of Materials Science and Engineering, Pennsylvania State University, University Park, Pennsylvania 16802, United States

*Corresponding author e-mail: [email protected] Keywords: electromechanical coupling properties, piezoelectricity, polar ordering, cellulose, fibrous protein

Abstract:

Electromechanical coupling properties of biological materials, especially cellulose from

plant cell walls and proteins from animals, are of great interest for applications in biocompatible sensors and actuators, and ecofriendly energy harvesters. Based on their anisotropic nanostructures, cellulose and fibrous proteins such as collagen, silk, keratin, etc. are expected to be piezoelectric; however, this property does not necessarily translate to cellulose- or protein-containing bulk materials. In fact, the values of piezoelectric coefficients reported for cellulose and proteins in the literature vary over several orders of magnitude, which raises the question of whether these are truly intrinsic piezoelectric properties of biological materials or whether they are obscured with other electromechanical coupling processes 1 ACS Paragon Plus Environment

ACS Applied Bio Materials 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

such as electrostriction, flexoelectricity, electrochemical transport, or electrostatic deformation. This critical question about intrinsic and extrinsic electromechanical coupling mechanisms is reviewed in this article. The origin of piezoelectricity of cellulose and collagen (the most widely studied protein for piezoelectricity) is discussed based on their molecular structures. Key requirements to construct macroscopic piezoelectric bio-composites are addressed next in terms of packing orders or arrangements of polar domains in composites. Based on this structural argument, truly piezoelectric responses of macroscopic materials fabricated with or containing cellulose and collagen are found to be extremely difficult to observe or quantify; most values reported in the literature as piezoelectric coefficients of such materials appear to originate from other electromechanical coupling mechanisms. Clarifying these mechanisms is important to properly design electromechanical devices using bio-based materials.

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1. Introduction Electromechanical coupling is a phenomenon whereby an electrical charge is produced when mechanical force is applied to a material or a mechanical strain is developed upon application of electrical bias; probably, the most well-known example of this coupling is piezoelectricity.1-5 Piezoelectricity was discovered as early as 1880 by the Curie brothers,6-7 but it was not until the 1950s that piezoelectricity of biomaterials was formally studied and reported in the literature. Fukada and Bazhenov experimentally measured an induced electric polarization upon applying mechanical stress on simple cuts of wood and bone;3, 8-14 cellulose in woods and collagen in bones were thought to contain individual polar domains responsible for the observed piezoelectricity.3, 8-14 Although the reported piezoelectric coefficients were very small, their studies have spurred further in-depth investigations of piezoelectricity in other biomaterials such as silk, wool, tendon, virus, muscle, etc.3, 15-23 If these materials can be engineered to produce bio-based composite materials with macroscopic piezoelectric responses, such materials could have distinct advantages over piezoelectric ceramics and synthetic polymers because of their excellent biocompatibility, abundance in nature, mechanical flexibility and low density.24-29 This field of research has seen a remarkable resurgence, most notably due to promising applications of biomaterials in medical devices, wearable sensors, flexible actuators and energy harvesting devices.17, 24-33 There is a large number of studies reporting piezoelectric properties of biomaterials consisting of or containing cellulose and various types of proteins, most notably, fibrous proteins.17-43 The piezoelectric coefficients reported in the literature vary over a wide range, covering several orders of magnitude (see Figure 1 in Section 1.1). This variability may imply a large room for improvement in materials performance; however, it could also posit a question about the source of the large discrepancy in the reported values. Piezoelectricity is an intrinsic electromechanical property which arises from the noncentrosymmetry in the crystalline structures.34-36 Therefore, a particular piezoelectric material should have a specific set of piezoelectric coefficient values at a given molecular structure and measurement conditions. The large variance in the reported values for biomaterials in the literature makes it difficult to 3 ACS Paragon Plus Environment


define and quantify the intrinsic piezoelectric properties of such materials; as a result, it is difficult to properly design suitable piezoelectric devices using biomaterials. Piezoelectricity is not the only electromechanical coupling phenomenon; there exist other phenomena which intrinsically convert mechanical force to electrical energy, and vice versa, such as electrostriction and flexoelectricity.35-40 Moreover, extrinsic factors such as electrochemical ion migration and electrostatic effects can also contribute to electromechanical signals measured.41-45 Consequently, the electrical response under applied mechanical force (or vice versa) could be a combination of piezoelectric and other electromechanical effects. It is difficult to quantify the piezoelectric response without ruling out the contributions from other effects, particularly when the piezoelectric effect may be small or the origin of piezoelectric response is unclear. This paper provides a critical review on piezoelectric properties of cellulose- and protein-based biomaterials. First, the piezoelectric coefficients reported in the literature for cellulose- and protein-based biomaterials are compiled in Sections 1. The large discrepancy among the reported values raises important questions regarding proper understanding of electromechanical coupling mechanisms and improved quantification of electromechanical properties of materials. For this reason, Section 2 covers intrinsic and extrinsic electromechanical coupling mechanisms by using thermodynamic principles and molecular structure arguments. Section 3 discusses molecular origin and mechanism of piezoelectricity in cellulose and collagen as well as the relationship between piezoelectricity and orientation of polar domains. Then, Section 4 revisits electromechanical coupling behaviors reported for specific materials produced in four different ways – raw materials, chemically treated materials, mechanically stretched and aligned composites, and composites produced under electric fields. Lastly, perspectives on remaining challenges and suggestions for future studies of piezoelectric biomaterials are discussed in Section 5. 1.1. The large discrepancy among the reported values of piezoelectricity in biomaterials In Fig. 1, the reported values of piezoelectric coefficients for cellulose- and protein-based materials are compared alongside with those of the most-extensively characterized piezoelectric ceramic


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(lead zirconate titanate, PZT) and polymer (polyvinylidene fluoride, PVDF).46-50 The literature values for PZT and PVDF converge within a small range whereas the values found for cellulose- and protein-based materials spread over several orders of magnitude; in the case of d31, the values for cellulose and protein are distributed over 6 to 7 orders of magnitude (Fig. 1b), which begs the question: what is the true value of the intrinsic piezoelectricity of these biomaterials? In the papers reporting really large piezoelectric coefficients, it has been claimed that specific preparation methods induced the polar ordering of biomaterials.51-56 However, this claim was made based on the magnitude of the measured coefficient, i.e., no independent characterization data supporting the polar ordering of piezoelectric domains was provided. Without such evidence, it is difficult to confirm whether the preparation methods used in those studies indeed increased the piezoelectricity of biomaterials or not. The details of reported piezoelectric properties of cellulose and protein in the literatures are explained in the following section.

Figure 1. (a) Longitudinal, (b) transverse and (c) shear piezoelectric coefficients of cellulose- and protein-based biomaterials reported in literature.3, 8-10, 12, 15-19, 21-23, 28, 31-33, 51-64 The piezoelectric coefficients of PZT and PVDF were included for the comparison.46-50

1.2. Piezoelectricity of cellulose- and protein-based biomaterials in the literature



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As the main component of the cell walls in plants, cellulose is the most abundant crystalline biopolymer on earth.65 For that reason, it is of great interest to harvest the unique properties originating from the crystalline nature of cellulose; piezoelectricity is one such property that cellulose can potentially provide. Early reports on piezoelectricity of cellulose focused on measurements of electromechanical response of simple cuts of woods in the 1950s and 1960s.12-13, 66 Fukada speculated that the polar ordering of hydroxyl groups in crystalline cellulose domains could produce a piezoelectric response;8,

12

his

original hypothesis was quite remarkable since the complete crystal structure of cellulose was not known at that time.8, 12-13, 67-69 Inspired by these early reports, a large number of research has been carried out to measure the piezoelectric properties of cellulose and develop cellulose-based engineering materials that take advantage of such properties. Table 1 compiles and compares the piezoelectric coefficients of cellulose-based biomaterials reported in the literature along with the source of cellulose and preparation methods. Note that there are two different units (pC/N and pm/V) listed for the coefficients, but they are thermodynamically equivalent (see Section 2.1). Initially, Fukada reported that only the shear piezoelectric effect, quantified by the coefficient d14 (= -d25), could be measured for cellulose in woody tissues while longitudinal and transverse effects could not be observed.11, 13 He noted that the piezoelectric response of a material requires polar arrangement of domains throughout the entire material under test.34, 70 Crystalline cellulose microfibrils in cell walls of woody tissues are aligned uniaxially and most likely in an anti-parallel fashion on average; at least, there is no experimental evidence that cellulose microfibrils in plant cell walls are assembled in the parallel direction so that individual dipoles of hydroxyl groups of cellulose add up to a macroscopic dipole. 71-73

11, 13,

Based on this conjecture, Fukada claimed that the longitudinal and transverse piezoelectric

polarizations of the uniaxially aligned cellulose crystallites are zero owing to this expected anti-parallel arrangement.11,

13

Interestingly, non-zero transverse piezoelectric effects, quantified using transverse

coefficients of d31 and d32, were reported later, and further motivated the need for an in-depth and critical discussion of cellulose in particular and biomaterials in general.28, 32, 51-52, 57-58


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Various engineering methods have been proposed and tested to prepare cellulose-based piezoelectric biomaterials; those methods include mechanical stretching, electric field-driven orientation, pressurized filtration, hybrid mixture in synthetic polymers, etc.28, 32, 51-52 A seminal paper on piezoelectric properties of cellulose-based biomaterials was published by J. Kim and colleagues in 2006 introducing electroactive paper (EAPap) made of cotton pulp.20,

53-56, 74-75

This paper revived interests in

electromechanical devices based on cellulosic materials. In a follow-up paper from the same group, d31 value as high as 1424.5 pm/V was measured,54 which is the highest value reported for a cellulose-based material and even higher than that of most piezoelectric ceramics.33, 46-50, 53-56 Table 1. Reported values of piezoelectricity in cellulose-based biomaterials in the literature. The coefficients with pC/N and pm/V represent the direct and converse piezoelectric effect, respectively. The data are categorized with piezoelectric tensors: longitudinal (d33), transverse (d31) and shear (d14 and d15) piezoelectric effects. Coefficient Longitudinal (d33)

Transverse (d31)

Source Birch wood Wood microfibers

Ramie Japanese cypress Long leaf pine Carpinus tschonoskii Acer mono Pine wood Electro-active paper (EAPap)

Preparation Pressurized filtration and drying Pure cellulose Hybrid (50 wt%) with BaTiO3 Embedded in polydimethylsiloxane (PDMS) AC electric field applied Simple cut Simple cut Simple cut Simple cut Simple cut Wet-drawing ratio of 1 1.5 1.8 2.0 Mechanical stretching DC electric field applied 0 V/mm 10 V/mm 20 V/mm 40 V/mm Corona poling at 100 V 200 V 300 V


Piezoelectric coefficient 5.7 ± 1.2 pC/N 0.4 pC/N 5 pC/N

Ref. 51 28

11.5 ± 2.3 pC/N 210 pm/V -0.0015 pC/N 0.002 pC/N 0.002 pC/N 0.001 pC/N 278 pm/V

32 52 57

3.4 ± 2.3 pC/N 5.9 ± 0.8 pC/N 10.1 ± 0.5 pC/N 16.5 ± 1.9 pC/N 30.6 pC/N

58 56

33 55

2.3 ± 0.1 pC/N 3.1 ± 0.18 pC/N 5.4 ± 0.36 pC/N 10.6 ± 0.77 pC/N 53 2.3 ± 0.9 pC/N 3.5 ± 0.4 pC/N 5.4 ± 0.3 pC/N


Shear (d14 = -d25)

400 V Electro spinning Spin-cast Simple cut Simple cut Simple cut Simple cut Untreated simple cut (cellulose I) Treated with NaOH (cellulose II) Treated with NH3 (cellulose III) Simple cut Simple cut Simple cut Simple cut

Japanese cypress Long leaf pine Carpinus tschonoskii Acer mono Hinoki wood

Maple Spruce Ramie bundle Rayon bundle

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16.7 ± 0.2 pC/N 1424.5 pm/V 454.5 pm/V 0.1 pC/N 0.05 pC/N 0.12 pC/N 0.05 pC/N 0.03 pC/N 0.1 pC/N 0.12 pC/N -0.086 pC/N, -0.08 pm/V -0.17 pC/N, -0.076 pm/V -0.27 pC/N, -0.21 pm/V -0.026 pC/N, -0.026 pm/V

54 57

8

12

Similar to cellulose, wide ranges of piezoelectric properties for protein-based biomaterials can be found in the literature and they are listed in Table 2. In earlier reports, very small piezoelectric coefficients (sub-pC/N and pm/V level) were reported from simple cuts of animal bones, tendons and skins at a mm- or cm-scale.9-11,

67, 76

One of the major components responsible for the piezoelectric

response of these biological materials was identified as collagen,10, 22, 66, 77-78 which is the main structural protein in various connective tissues in animal bodies; this drew significant attention to designing and developing collagen-based piezoelectric biomaterials. Other proteins such as silk, keratin, elastin, myosin, actin and lysozyme were also reported to have piezoelectric properties.3, 10, 18, 23, 31, 61-63 Through carefullydesigned engineering approaches, protein-based materials with much higher coefficient values have been demonstrated.15-16, 18, 60 Table 2. Selected values of piezoelectricity in protein-based biomaterials from the literatures. Coefficients with pC/N and pm/V represent the direct and converse piezoelectric effect, respectively. Coefficient Longitudinal (d33)

Source Silk Wool Horse femur Horse achilles tendon Human bone M13 phage virus Spider silk

Preparation Simple cut

Simple cut Liquid-crystalline films Embedded inside nano-wells Embedded in PDMS


Piezoelectric coefficient 0.023 pm/V 0.0033 pm/V 0.0033 pm/V 0.066 pm/V 7-9 pC/N 7.8 pm/V, 6.1 pm/V 0.36 pm/V

Ref. 10

59 15 16 31

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Fish swim bladder Lysozyme

Transverse (d31)

Shear (d14 = -d25)

Embedded in PDMS Aggregate films of monocle lysozyme tetragonal lysozyme Purified elastin fiber Simple cut Simple cut Simple cut Simple cut Diagonal cut Vertical cut Horizontal cut

1.07 ± 0.27 pC/N 3.13 ± 2.07 pC/N 0.75 pm/V 0.02 pm/V 0.01 pm/V 0.0033 pm/V 0.013 pm/V 2250 pC/N 600 pC/N 200 pC/N

Human bone

Simple cut

0.12 pC/N, 0.12 pm/V

9

Silk Wool Horse femur Horse achilles tendon

Simple cut Simple cut Simple cut Simple cut Mechanical stretching by zone drawing (heating) methanol-wet drawing water-wet drawing Simple cut Simple cut Simple cut Elongated (30%) cut

-1.09 pm/V -0.066 pm/V -0.21 pm/V -1.9 pm/V

10

Porcine thoracic aorta Silk Wool Horse femur Horse achilles tendon Human cornea

Silk Skin Wool Horn Psoas muscles of a rabbit

Shear (d15)

22 pC/N

17 61

62 10

60

18 -0.7 pC/N -0.4 pC/N -0.05 pC/N 0.2 pC/N 0.1 pC/N 1.8 pC/N -0.396 pC/N

3

64 23

Myosin Actin

Extracted from psoas muscle

0.01 pC/N 0.05 pC/N

Bovine horn keratin

Simple cut measured at 150°C

2.5 pC/N

63

Animal cortical bone

Simple cut

0.1-0.3 pm/V

21

Collagen fibril

Collagen fibril on gold substrate

1 pm/V

22

Bovine achilles tendon Cortical bone

Collagen nanofibril Simple cut

2 pm/V 0.3 pm/V

19

Reviewing the piezoelectric coefficient values in the literature for cellulose- and protein-based biomaterials, the critical question naturally arises − why are there such large variations spanning over several orders of magnitude in the reported values? It should be noted that there are several mechanisms or processes that are not piezoelectric but could result in electromechanical coupling effects. These nonpiezoelectric effects need to be ruled out to confirm the piezoelectricity of cellulose- and protein-based materials. This confirmation is important for two reasons: 1) properly identifying the mechanism responsible for electromechanical coupling in biomaterials fills a fundamental knowledge gap, and 2) the 9 ACS Paragon Plus Environment


nature of the coupling mechanism affects the development of devices based on these materials and their engineering applications. For that purpose, Section 2 reviews various electromechanical coupling processes including piezoelectricity, to provide background for the subsequent sections.

2. Principles of electromechanical coupling processes For proper electromechanical applications of biomaterials, it is important to understand the origins and characteristics of various electromechanical coupling phenomena. Fig. 2 shows a list of electromechanical coupling processes classified as intrinsic and extrinsic materials responses. The intrinsic processes are electromechanical coupling responses that are related to the crystal symmetry and molecular structure of the material; they include piezoelectricity, electrostriction and flexoelectricity. The extrinsic processes usually originate from external factors not directly related to the molecular structure, such as electrochemical migrations of injected charges or mobile ions, or elastic deformation due to externally applied or produced electrostatic forces. In some cases, extrinsic effects can be misconstrued as inherent responses if sample preparation or experimental measurements are not carefully designed. In addition, non-piezoelectric intrinsic responses, such as electrostriction for example, could be misinterpreted as the result of piezoelectric responses, if not careful.41, 79-80

Figure 2. Diagram of electromechanical coupling phenomena with the intrinsic and extrinsic processes.


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2.1. Intrinsic processes Piezoelectricity, electrostriction and flexoelectricity couple mechanical quantities (strain and stress) with electrical quantities (polarization, electric field and dielectric displacement or flux) in different ways. Fig. 3 shows how each coupling mechanism results in a characteristic electromechanical response. The direct (dir) and converse (con) coefficients of each coupling are included with mechanical variables [strain () and stress ()] and electrical variables [polarization (), electric field ( ) and dielectric susceptibility ()]. The unit of each coupling is included in the square brackets. Piezoelectricity is the linear coupling between strain and electric field, whereas electrostriction is the non-linear (quadratic) coupling between strain and electric field. Flexoelectricity relates an induced polarization to strain gradient. The coefficients of these intrinsic electromechanical couplings are derived from the energy functions in thermodynamics.40,

81-82

The concept and derivation of coefficients of these

electromechanical couplings are explained in this section.



Figure 3. The relationships between mechanical variables and electrical variables in the piezoelectric, electrostrictive and flexoelectric couplings. Note that in the flexoelectric coupling, ′ and ′ represent the strain gradient and electric field gradient, respectively.

2.1.1. Piezoelectricity Piezoelectricity is the linear electromechanical coupling between mechanical and electrical domains. Fig. 4 graphically explains the basic mechanism of direct and converse piezoelectric effects. The direct piezoelectric effect refers to an electric polarization induced by an applied mechanical stress. The converse piezoelectric effect relates a physical strain induced under an electric field. Piezoelectricity requires lack of inversion symmetry, in other words, noncentrosymmetric arrangement of dipole moments within a material. Individual dipole components can form a region of local alignment, which is called a ‘domain’.83 Such domains are typically microscopic in size. When the direction of dipoles across domains is random, the material has no (or very weak) overall polarization because the polarity of each domain is averaged out by its neighbors.34 In this case, the net piezoelectricity is zero (or negligible, at best), even though each microscopic domain has a non-zero dipole. The piezoelectric response of a material will be maximum when all domains in the material are aligned in the same direction.34 In Fig. 4, the red arrows represent dipole directions of noncentrosymmetric domains. When they are preferentially aligned along one direction, the material containing such domains can exhibit piezoelectricity. Ferroelectricity is a subset of piezoelectricity whereby the resultant non-zero spontaneous electric polarization can be reversed by applying an electric field larger than the coercive field.83-85


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Figure 4. Schematic illustration of direct (left) and converse (right) piezoelectric effects. The red arrows represent electric dipoles and blue arrows represent the applied mechanical stress.

The piezoelectric coefficients can be derived from the Gibbs free energy. The Gibbs free energy of a solid material (G) has variables with intensive variables (stress and electric field) and its exact differential is,49, 81 dG = - S · dT - · d - · d

(1)

where T, S, , , and are the temperature, entropy, strain, stress, polarization and electric field, respectively. The second derivatives of the Gibbs free energy give the direct ( ) and converse ( )

piezoelectric coefficients. According to the Maxwell relation, the reversal of the order of differentiation should provide the same result:

=

-(

∂ G ∂ ∂

)=(

∂ ∂

)= -(

∂ G ∂ ∂

)=(

∂ ∂

) = !"

(2)

This is the example of piezoelectric coefficients derived from the Gibbs free energy. Using the same method of derivation from other state functions, other piezoelectric coefficients can be obtained. For example, derivation from the Helmholtz free energy provides the piezoelectric coefficients that relate applied strain (or polarization) to resulting electric field (or stress), i.e., (

∂#$ ∂%&'

) and (

∂(&' ∂)$

). Determining

which coefficient or energy function to use in analysis depends on which experimental variables are being controlled in measurements.

2.1.2. Electrostriction Electrostriction is present in all dielectric materials, where deformation results under an applied electric field.40, 86 Fig. 5 illustrates the electrostriction process. Unlike piezoelectricity, electrostriction 13 ACS Paragon Plus Environment


does not require alignment of permanent dipoles across all domains; in theory, all polarizable materials can exhibit electrostriction behavior.87 Also, in contrast to piezoelectricity (linear electromechanical coupling), electrostriction has a nonlinear (quadratic) relationship between mechanical strain and electric field.87 The electrostrictive effect is generally small in most dielectric materials; however, in some polar materials, electrostriction can be quite large, for example, high electrostrictive strains were reported for relaxor ferroelectric ceramics with the perovskite crystalline structure.88-91 Also, exceptionally high electrostrictive response was reported in electron-irradiated poly(vinylidene fluoride-trifluoroethylene) copolymer and in relaxor ferroelectric poly(vinylidene fluoride-trifluoroethylene-chlorofluoroethylene) terpolymer.92-95

Figure 5. Schematic illustration of electrostriction. The red arrows represent electric dipoles.

Same as piezoelectricity, electrostrictive coefficients can be derived from the energy functions. From the Gibbs free energy, the direct electrostrictive effect shows a quadratic relationship between the induced strain and applied electric field. The converse electrostrictive effect is the linear relationship betwen dielectric susceptibility ( * ) and applied stress ( ). Taking the third derivative of the Gibbs free


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) and converse (+ * ) electrostrictive coefficients are obtained and energy in equation 1, the direct (+ *

shown to be equal via the Maxell relationship as follows:40

, - =

-(

∂. G ∂ ∂ ∂/

)=(

∂0 ∂ ∂-

)= - (

∂. G ∂ ∂/ ∂

)=(

∂1∂

!" ) = ,-

(3)

2.1.3. Flexoelectricity Flexoelectricity originates from the polarization induced by a strain gradient.38 Fig. 6 schematically explains the principle of flexoelectricity. When a dielectric material is deformed in a nonuniform manner such as through bending for example, a strain or stress gradient is generated in the material; this gradient can break the inversion symmetry by resulting in non-uniform displacement of atoms.96-98 Thus, same as electrostriction, flexoelectricity can also appear in all dielectric materials including centrosymmetric crystalline structure. In contrast to piezoelectricity, flexoelectricity is a sizedependent electromechanical coupling phenomenon. It is usually much smaller than piezoelectricity in large-scale bulk materials because the strain gradient is negligible in a large object, but it becomes significant at smaller scales such as nano-size dimensions.39 Due to this size effect, piezoelectricity can sometimes be overshadowed by flexoelectricity in small-scale material measurements. Flexoelectricity has been measured in many biological samples such as a bilayer lipid membrane, hair cell, etc.37

Figure 6. Schematic illustration of flexoelectricity. The red arrows represent electric dipoles.



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Derivation of the flexoelectric coefficients requires a spatial coordinate axis, because flexoelectricity includes spatial gradient terms. In order to derive the most commonly used direct flexoelectric coefficient which couples the polarization and strain gradient, electric Gibbs energy (G2) with the state variables of strain and electric field needs to be used.81-82 The differential of electric Gibbs energy can be expressed as: dG2 = - S · dT + · d - · d

(4)

In order to derive the flexoelectric coefficients, the volume density of electric Gibbs energy (φ) with the spatial coordinate axis (2* ) needs to be employed,82 G2 = φ · 23 24 25

(5)

Omitting the - S · dT term for simplicity, the exact differential of φ can be derived, dφ [ (2* ), (2* )] = (2* )· d (2* ) - (2* ) · d (2* )

(6)

The partial differential of dφ with respect to 2* is d

∂ φ [ (2* ), (2* )]= d9 : ∂ 78

= ′ · d (2* ) + (2* )· d′ - ′ · d (2* ) - (2* ) · d′

(7)

The derivation of the volume density of the electric Gibbs energy with the gradient of electric field and strain provides,

=

-(

∂; < ∂#:$

)

and

= (

∂; < ∂%:&'

)

(8)

Finally, the direct coefficient (μ * ) and converse coefficient (μ * ) are related to each other via the

Maxell relationship:


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μ - = - (

∂ ;

Review on Electromechanical Coupling Properties of Biomaterials

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