A Photoacoustic Method to Measure the Young’s Modulus of Plant Tissues

Abstract

Background

The PA method combines optical absorption with acoustic detection of laser-generated ultrasound signals to enable high-resolution and high-speed imaging and determination of the mechanical properties of materials. A measurement of a single point takes only a few seconds; thus, the PA method is high throughput and allows for extracting spatially varying mechanical properties of materials, which is critical in characterizing heterogeneous materials such as biological tissues. As the PA method is non-contact, it precludes damaging the sample surfaces during the measurements.

Objective

This study explores the ability of a non-contact and high throughput photoacoustic (PA) method to extract the elastic moduli of bioenergy sorghum tissues, i.e., rind, pith, and vascular bundle, in the axial direction.

Methods

A pulsed laser generated a collimated circular beam, which was expanded from 3 mm to 15 mm by a pair of convex lenses. To increase the light absorption, red ink was applied to the sample surface. A focused laser beam from a vibrometer was also delivered at the same location to measure the local surface displacement in the vertical direction. The built-in camera of the vibrometer was used as a monitor.

Results

The elastic modulus of the bioenergy sorghum rind was significantly larger than the moduli of the pith and fiber bundles, thus indicating that rind tissues were much stiffer. The statistical results show statistically significant differences among the elastic moduli of the different tissues.

Conclusions

These measurements agree well with studies that have implemented other characterization techniques, thus attesting to the utility of the PA technique in characterizing sorghum and other plants going forward.

Appendices

Appendix A

This Appendix presents the derivation of equation (1) [90, 93], which was used to determine the elastic modulus of the tissues from PA signals. The PA method is based on solving a wave propagation equation of motion $\frac{{d\tau_{z} }}{dr} + F_{z} = \rho \frac{{\partial^{2} u_{z} }}{{\partial t^{2} }}$ in a linear viscoelastic material under shear deformations with the following linear viscoelastic model $\tau_{z} = G\gamma_{z} + \mu \frac{{d\gamma_{z} }}{dt}$, where $\tau_{z}$ is the shear stress and the shear strain is given as $\gamma_{z} = \frac{{du_{z} }}{dr}$ and G and μ are the shear elastic modulus and viscosity, respectively . The wave propagation equation of motion is then written by $G\frac{{d^{2} u_{z} }}{{dr^{2} }} + \mu \frac{d}{dt}\left( {\frac{{d^{2} u_{z} }}{{dr^{2} }}} \right) + F_{z} = \rho \frac{{\partial^{2} u_{z} }}{{\partial t^{2} }}$ and with an algebraic manipulation we have:

$$\frac{{\partial^{2} u_{z} }}{{\partial t^{2} }} - \left( {C_{T}^{2} + \eta \frac{\partial }{\partial t}} \right)\Delta_{ \bot } u_{z} = F_{z}$$

(A.1)

where $u_{z}$ is the displacement, z is the incident direction of the laser, $C_{T}$ is the shear wave velocity, $\Delta_{ \bot }$ is the 2D Laplacian operator, $\eta = \frac{\mu }{\rho }$ is the kinematic viscosity which depends on the viscosity μ and the density ρ, and $F_{z}$ is the equivalent excitation laser force. In the laser-illuminated area with a radius near 50μm, the plant tissue is assumed to be elastically isotropic [83]. The relation between the wave velocity and the shear modulus G is written as:

$$C_{T} = \sqrt {\frac{G}{\rho }}$$

(A.2)

The Young’s modulus and shear modulus are related by Poisson’s ratio $\upsilon$:

$$G = \frac{E}{2(1 + \upsilon )}$$

(A.3)

The plant tissue is generally assumed to be incompressible with a Point ratio $\upsilon$ near 0.5. The shear modulus can be further estimated as:

$$G \approx \frac{E}{3}$$

(A.4)

Substituting equation (A.4) into equation (A.2); the shear velocity $C_{T}$ can be estimated as:

$$C_{T} \approx \sqrt {\frac{E}{3\rho }}$$

(A.5)

For a pulsed Gaussian laser beam, the equivalent excitation force can be given by:

$$F_{Z} = \alpha \Gamma I_{0} f(t)\exp \left( { - \alpha z - \frac{{r^{2} }}{{R^{2} }}} \right)$$

(A.6)

where $\alpha$ is the light absorption coefficient, $\Gamma$ is the Gruneisen parameter, $I_{0}$ is the initial laser intensity, $f(t)$ is the delta function which is used to represent the laser pulse, r is the radial coordinate, and R is the waist radius of the Gaussian laser beam. Because the laser pulse duration can usually be short with a magnitude order of ns, the heat conduction has been neglected. The Substituting equations (A.5), (A.6) into equation (A.1), the PA-generated shear wave equation can be rewritten as:

$$\frac{{\partial^{2} u_{z} }}{{\partial t^{2} }} - \left( {C_{T}^{2} + \eta \frac{\partial }{\partial t}} \right)\Delta_{ \bot } u_{z} = \alpha \Gamma I_{0} f(t)\exp \left( { - \alpha z - \frac{{r^{2} }}{{R^{2} }}} \right)$$

(A.7)

To analytically solve the vertical displacement at the center of the laser beam, a Hankel transform is conducted on both sides of equation (A.7) which was discussed in in [90, 93]. By considering a Gaussian laser and delta function pulse, the displacement at the center of the laser beam is given by:

$$u_{z} = \frac{{\alpha \Gamma I_{0} R\delta }}{{\rho C_{T} }}\frac{{\frac{{\sqrt {E/3\rho } }}{R}t}}{{1 + \frac{4\mu }{{\rho R^{2} }}t + \left( {\frac{{\sqrt {E/3\rho } }}{R}t} \right)^{2} }}$$

(A.8)

where $\delta$ is the laser pulse width. By differentiating $u_{z}$ on t, the displacement $u_{z}$ reaches its peak at $t_{\max }$:

$$t_{\max } = \frac{R}{{\sqrt {E/3\rho } }}$$

(A.9)

The estimation of Young’s modulus from the PA method can be affected by the size of the laser beam [90, 93]. To account for the size effect, a parameter K, which is ratio of the focal size to the diffraction length, needs to be first calibrated:

$$E = \frac{{3KR^{2} \rho }}{{t_{\max }^{2} }}$$

(A.10)

Appendix B

This Appendix presents the statistical results for density and elastic moduli comparison of the tissues.

Table 1B Two-way ANOVA table of tissue densities. The tissue factor contains three levels: rind, pith, and bundle, while the internode factor contains 7 levels: IN3-9

Table 2B Multiple comparisons of tissue densities

Table 3B Two-way ANOVA table of tissue elastic moduli. The tissue factor contains three levels: rind, pith, and bundle, while the internode factor contains 7 levels: IN3-9. Square root transformation was applied to the modulus

Table 4B Multiple comparisons of tissue elastic moduli