Team:NYU Shanghai/Model
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Introduction
In piezomaterials, deformation results in asymmetric shift of ions or dipoles inducing a change in electric polarization and electric potential is generated. This piezoelectric conversion between mechanical and electric enegry is therefore utilzied in many industries such as actuators and sensor devices.
As we discussed the project, fish-scale derived collagen has these piezoelectric properties. Human bones, muscles, and tendons are also having these effect and lots of studies are done for medical applications [2,3]. We took those previous researches as the examples of microanalysis of collagen’s piezoelectricity. We wanted to look into how stress-generated electrical phenomena can play a role in bone tissue engineering and conduct the analysis of fish-scale collagen in silico. In this analysis, we firstly focus on the piezoelectricity in bones and then move on that in fish scales.
Piezoelectricity in Bone
The piezoelectricity in bone is firstly reported by Yasuda in 1954 [4]. They attributed these properties to the collagen fibers. They also identified that isolated type I collagen fibrils have unipolar axial polarizations and behave as other piezoelectric materials such as quartz. It holds the piezoelectric constant d15 1[pC/N]. The piezoresponse to the force allow microscopy to show unidirectional polarization along the collagen fibril axis and a negligible amount of radial and vertical piezoresponse. When the angle between the bone axis and pressure is 45-50 ̊, a higher constant was observed. Primarily, the effect was defined by Strain-Generated Potentials (SGP) by:
where i indicates the ith direction;dii is the component of piezoelectric stress tensor; L is the thickness; σs is solution conductivity; βii is the component of dielectric permittivity tensor, and t is the time [5]. The relaxation time is reported to be very quick, 0.5-50 μs.
This is the equation assuming the piezoelectric material dry, but the real-life situation is more complicated
and harder to detect the piezoelectricity because of the fluid and ions surrounding the material. Charged
fluid would bind to the solid with an opposite charge, which might cause the spatially non-uniform distribution of
charges. This boundary is called a slip plane and the zeta potential is defined as the electric potential at this plane
relative to the moving fluid. The effect of fluid should be dependent on some factors: the direction, amplitude,
and the velocity of fluid. The new form of SGP (Equation 1) is rewritten by using ν for viscosity, ξ for Zeta
potential, Δp for pressure gradient.
This equation shows that the increased viscosity and conductance of fluid decreases the generated potential. The
discovery of the piezoelectric potentials led to the development of technologies of electrical stimulation for tissue
growth. Particularly, osteogenic electrical stimulation is being discovered by using solenoid coils or Helmholtz
but the relevance of using piezo materials for osteoblasts cell growth is hardly known.
Later on, the class model of piezoelectric potential is updated with two additional factors: 1) non-linear
relationship between stress and polarization, and 2) additional components that determine the stress gradient. The additional terms states the new polarization, Pi, in the relation to strain gradient as:
where ϵijk is the piezoelectric tensor; εjk is the strain tensor; γklij is the electric tensor; ∂ϵkl⁄∂xj is the spatial strain gradient; χij is the dielectric susceptibility; and Ej is the electric field.
However, these tensors are not consistent with obtained experimental results because the sign of the tensors
cannot be determined only by the theory. The ferroelectric property of the bone is further investigated by
the stress-induced reorientation of dipoles. In the updated model, the fixed amplitude of the stress loading rate
causes the dipole rotation, and a high loading rate leads to the maximum displacement of the specimen. Since bone is
fully surrounded by the periosteum, the surface generates potential while mechanical stress is applied close to
that periosteum. The deformation is described by:
where du⁄dt is the rate of displacement to the surface; Px is the x component of polarization vector; a and b are
given constants; and du/dt is the velocity of resorption. When du⁄dt > 0, it is bone resorption; and dU⁄dt < 0 for bone decomposition. The piezoelectric polarization contributes to the stimulus for bone regeneration. To simplify the model, the Elastic modulus of isotropic solid is taken place. The stress tensor (σ) and electric displacement (D) is defined as:
where u is the displacement field; ε is the strain tensor; ϵT is the transpose of the piezoelectric stress tensor (ϵ); E(ψ) is the electric field; β is the dielectric permittivity vector; ρ∗ is the reference density (1.0g/cm3), and I is the identity operator. μ(ρ) and λ(ρ) are Lame’s coefficients of the piezo materials.
Piezoelectricity in Fish Scales
According to the previous research on fish scales [6], piezoelectric coupling coefficients, d, has a much higher value than that of bones, which indicates the fish scale is more efficient in making electric signals by given mechanical stimuli. On the other hand, the stress tensor, ϵ, has smaller numbers, which suggests the smaller deformation gives the bigger piezoelectric potential on the surface.
Future Studies
We have quantitatively shown that piezoelectric effect is bigger in fish scale than known bone collagen. However, we have not yet quantified how chemo-electrical signals trigger remodeling process. The multiscale (Collagen as ECM and Cells) analysis is necessary to develop the applied use of fish scale.
References
1] M. Mohammadkhah, D. Marinkovic, M. Zehn, S. Checa. A review on computer modeling of bone piezoelectricity and its application to bone adaptation and regeneration Bone, 127 (2019), pp. 544-555 https://doi.org/10.1016/j.bone.2019.07.024
[2] B. Miara, E. Rohan, M. Zidi, B. Labat ”Piezomaterials for bone regeneration design̶homogenization approach.” Journal of the Mechanics and Physics of Solids, 53 (11) (2005), pp. 2529-2556
[3] A.H. Rajabi, M. Jaffe, T.L. Arinzeh ”Piezoelectric materials for tissue regeneration: a review Actator Biomaterial.,” 24 (2015), pp. 12-23
[4] I. Yasuda ”On the piezoelectric activity of bone” J Jpn Orthop Surg Soc, 28 (3) (1954), p. 267
[5] N. Petrov ”On the electromechanical interaction in physiologically wet bone” Biomechanics, 2 (1975), pp. 43-52
[6] Ghosh, S. K. Mandal, D. High-performance bio-piezoelectric nanogenerator made with fish scale. Appl. Phys. Lett. 109, 103701 (2016).