Measuring the Inverse Piezoelectric Effect of Bone

Introduction

The piezoelectric effect occurs when a potential difference is generated across two surfaces of a material in response to an applied force. It was first discovered in 1880 by Jacques and Pierre. Since then, piezoelectric materials (e.g. wood, bone, quartz and other crystals) have been researched extensively for their practical applications. An image of the piezoelectric effect as well as the inverse process is shown below.

Figure 1: Left: The normal piezoelectric effect, where the applied force induces a potential difference. Right: The inverse effect, where an applied potential difference induces a mechanical deformation of the material.

The piezoelectric properties of bone were first demonstrated by Fukada and Yasuda in 1956. They found that collagen, the main structural protein in bone, is naturally piezoelectric. It is known that this piezoelectricity contributes to natural bone growth. Because of this, collagen is being studied for its potential applications in tissue engineering and regeneration. Due to its compatibility with existing tissue within the body, collagen is viewed as a desirable candidate to be a biomaterial.

Inverse Piezoelectric Theory

A piezoelectric material with a voltage V applied across two of its surfaces can be modeled as a capacitor, for which the equation V=EX applies, where E is the electric field and X is the distance between the surfaces. For such a material, the strain S can be found as S=dE, where d is the 6x3 piezoelectric constant matrix. The vector S is broken down into three linear strain components, and three shear strain components, the latter of which is relevant for bone. For bone, the nonzero constant in the matrix is given by

where ∆u is the displacement measured, X is the thickness of the sample, and h is the length of the sample.

In the following equation relating strain to electric field in the inverse piezoelectric effect, S_i for i=1,2,3 is a linear strain in the x,y,z directions, and i=4,5,6 is a shear strain in the plane perpendicular to the x,y,z directions. E_j is the applied electric field in the corresponding direction.

Figure 2: This image shows the shear strain occurring in the bone, with the negative z side of the bone glued to the mount to fix this side in place. The voltage is applied in the x direction, and the collagen fibers run in the z direction. Mirror and measurement arm are relevant for the quadrature interferometer, explained in the experimental setup.

Experimental Setup

We used quadrature interferometry to measure the displacement of the bone, ∆u. An image of our interferometer as well as a schematic are shown below. A λ=633 nm red laser is steered into a CP, which adds a 90°phase shift to the y-component of the beam’s electric field. The beam is then divided into the reference arm and the measurement arm by the NPBS. The reference arm portion encounters the LP, which removes the phase shift introduced by the CP. The measurement arm remains circularly polarized, and an additional phase difference is added to both x and y-components by the bone displacement. Solving for the displacement from this phase difference gives

After reflecting, the beams recombine in the NPBS and are sent to the PBS, which separates the x and y-components of the electric field. The intensity of these fields are then measured by the two photodetectors X and Y, across which the voltage is proportional to the measured intensity. Their signals are sent to an oscilloscope, and plotting the voltages against one another creates a circle, whose center (Vxc ,V_yc) is found by averaging over all V_x and V_y values separately. We then define ∆V_x=V_x -V_xc and ∆V_y=V_y -V_yc , and determine ∆ϕ_m to be

Figure 3: Image of our interferometer setup.

Figure 4: Schematic of the interferometer with parts labeled. Note that the reference and measurement arms are swapped from the Figure 3, which physically changes nothing.

An image of our bone mount at the end of the measurement arm is shown below. A cylindrical holder (in maroon) was 3D printed using Autodesk Inventor, and the negative z side of the bone sample was epoxied into a notch in the holder. The Inventor file is attached at the bottom of this webpage. A mirror was also epoxied to positive y side of the bone as seen in Figure 2 above. A voltage was applied across the bone sample using leads attached to pieces of conductive tape that were placed on the positive and negative x sides. The measurement arm beam from the interferometer reflected off of the most positive z part of the mirror in order to measure the maximum displacement ∆u resulting from the shear strain.

Figure 5: An image of the bone mount. The maroon cylindrical holder was 3D printed, and the bone was inserted into the notch in the mount. The mirror is shown attached to the right side of the bone, and the conductive tape was attached to the top and bottom sides of the bone sample, with leads from the scope and voltage source attached to this.

The bone samples were rectangular prisms with a thickness, X, of 0.25’’ and a square face with width, h, of 1’’. These values were used with the first equation from the theory section to convert from the slope of the graph of V vs Δu to the piezoelectric constant.

Results and Analysis

The analysis of the data was completed in Matlab, and the live script used for this is attached at the end of the page. Additionally, the raw data sets for a test piezoelectric material, as well as all five trials of good collected bone data are attached.

For the test piezo, we got a value of -423 ± 4 pm/V for the piezoelectric constant. A previous group had found values in the range of 400 to 600 pm/V, so we felt confident about our setup. Thus, we began making measurements of the bone, and we were able to get five trials worth of good data with the bone oriented correctly. The results for each trial are summarized in the table below. In each case, the uncertainty on the value is higher than the actual value for the piezoelectric constant. This is because there was a lot of noise in the experiment that we were unable to get rid of. The mount for the bone was 3-D printed quite loosely, and so this was one source of noise. Additionally, the 3-D printed mount stuck out from where it was held by the kinematic mount, and so this lever arm likely contributed to vibrations that affected the results as well.

Table 1: Results from our trials. The data set numbers correspond to the names of the .mat files attached at the bottom of this webpage.

Even with an uncertainty higher than the slope in each case, part of the reason we feel like our results were reasonable nonetheless is because when we swapped the leads across the bone, we measured a resulting slope with the sign flipped, which occurred between datasets 6 and 8/9. The opposite sign for datasets 4 and 5 compared to dataset 6 is just because we used a different bone sample, which was likely oriented in the opposite direction relative to the resulting piezoelectric effect.

We did a weighted average of the five trials, and we got a final result for the piezoelectric constant of 4.1 ± 2.9 pm/V. Previous studies have measured the piezoelectric effect of bone to be anywhere from 0.7 to 8 pm/V, so our value falls in this range.

References

• 1. Fukada, E. and Yasuda, I. On the piezoelectric effect of bone, J. Phys. Soc. Japan,12, 1158-1162 , 1957. https://doi.org/10.1143/JPSJ.12.1158.

  2. Biranche Tandon, Jonny J. Blaker, Sarah H. Cartmell, Piezoelectric materials as stimulatory biomedical materials and scaffolds for bone repair, Acta Biomaterialia, Volume 73, 2018, Pages 1-20, ISSN 1742-7061, https://doi.org/10.1016/j.actbio.2018.04.026.

  3. Jacob, J., More, N., Kalia, K. et al. Piezoelectric smart biomaterials for bone and cartilage tissue engineering. Inflamm Regener 38, 2 (2018). https://doi.org/10.1186/s41232-018-0059-8.

  4. Nano Letters 2004, 4, 7, 1253–1256, Publication Date:June 4, 2004, https://doi.org/10.1021/nl049453i.