# A Monolithic Micro-Tensile Tester for Investigating Silicon Dioxide Polymorph Micromechanics, Fabricated and Operated Using a Femtosecond Laser

^{*}

## Abstract

**:**

## 1. Introduction

_{2}, fractures according to the weakest-link model [27,28]. Surface flaws act as stress concentrators where nucleation of cracks can take place. Its breaking strength is, therefore, dictated by the presence of surface flaws rather than by the intrinsic strength of the Si–O bond (which has a particularly high strength, estimated in the order of 21 GPa [29]). At the nano-scale, silica glass is found to exhibit unconventional behavior such as “pseudoductility” [30]. Molecular dynamics simulations suggest that a fracture in silica glass proceeds through the growth and coalescence of nanoscale cavities [31,32]. Custers [33] in 1949 and more recently, Celarie et al. [34] and Bellouard [35] experimentally verified the existence of nanoductile mode and plastic flow. Despite these works, the mechanical behavior of SiO

_{2}at microscales remains largely unexplored, due to the inherent experimental difficulties associated with it.

## 2. Microtensile Tester Design and Working Principle

#### 2.1. Working Principle

**Figure 1.**(

**a**) Monolithic micro-tensile tester seen through an optical microscope. The overall dimensions of the system are 15 mm × 15 mm. Part A is the loading cell and part B is the displacement amplification sensor. The test beam (#3 in part A) is also shown magnified in the inset. (

**b**) Graphical representation of the micro-tensile tester. Note that the dimensions of the device used here are not the real ones, but were chosen for a clearer understanding of the device’s operation. (

**c**) Photoelasticity image of the test beam. (

**d**) Magnified view of the system’s displaced lever beam.

**Figure 2.**Illustration of the loading cell’s working principle. The two loading bars (red areas) expand during femtosecond laser exposure resulting in the loading of the middle (blue) test beam.

#### 2.2. Test Beam Dimensioning

_{LB}, A

_{TB}be the cross-sections of the laser-exposed beams (“LB”) and the test beam (“TB”), respectively, and likewise, σ

_{LB}and σ

_{TB}the corresponding stresses in both beams. The stress in the test beam is simply expressed as:

_{LB}is the strain induced by the volume variation induced to the laser-affected beams, E

_{TB}is the elastic modulus of fused silica, E

_{LB}is the elastic modulus of the composite material formed in the fused silica matrix following laser-exposure, and n is the number of stressors induced in the system.

#### 2.3. Loading Cell

_{2}as the stiffness of the laser-affected volume and k

_{1}as the stiffness of the non-affected sidebars.

_{1}and k

_{2}and is given by the following equation:

_{P}respectively (Figure 3d). For ε

_{applied}= 1 μm, W

_{P}= 10 μm, we find that the corresponding shear stress transferred from pristine material to the laser affected zone is σ

_{z}= 24 MPa. This value is rather low and justifies the assumption considered in the first place.

**Figure 3.**(

**a**) 2D visualization of the loading process during the writing of the stressors. The mechanical energy that develops due to the localized material expansion, results in the principal loading of the sidebars. (

**b**) Simplified spring model of the loading process. (

**c**) Parameters used to describe a single stressor. (

**d**) Close-up view of the stress development due to shear stress taking place at the interface of the laser-affected and laser-unaffected zones. According to the shear-lag model developed in [52,53,54]: the displacement of the pristine material u

_{0}(z) is known; the unknown displacement of the laser-affected zones is denoted by u(z). The corresponding shear strain in the laser-affected zone is [u(z) − u

_{0}(z)]/W

_{P}.

#### 2.4. Lever Amplification Mechanism

_{1}in Figure 4a) anchored to a fixed body. Two bars (bars 1 and 2 in Figure 4b) fixed at their one end (from the kinematics point of view each bar is equivalent with two pivots) form a hinge, which is used to interface the linear motion of the actuator with the angular motion of the pivot. The loading cell (illustrated as a dashpot in Figure 4a) forming the input of the mechanism is attached at one end while the output of the mechanism is at the other end of the lever (part B in Figure 1a and magnified in Figure 1d). The second stage of the amplification acts exactly like the first one, scaling up the input linear motion even more. The amplification level for small angles θ of the end-effector bar (Figure 4a) is simply the ratio of the output displacement of the mechanism and the input displacement induced in the system.

**Figure 4.**(

**a**) Kinematics of the displacement amplification mechanism. The circles represent one-degree-of-freedom rotation pivot joints. The bars represent rigid links. The dash lines indicate the kinematics being operated. The end-effector is body B. (

**b**) Scanning electron microscope view of part of the displacement amplification sensor fabricated using femtosecond laser machining and chemical etching

#### 2.5. Mechanical Guidance

**Figure 5.**(

**a**) Mechanical guidance principle of the loading cell. (

**b**) The characteristic force-deformation curve of the guidance which consists of four parallel leaf-springs is shown here. The parameters (w = 100 μm, l = 5000 μm and t = 230 μm) chosen for this graph correspond to the parameters of the fabricated device. Here, the guidance is used in its quasi-linear regime.

## 3. Experimental Results

#### 3.1. Manufacturing

#### 3.2. Stressors Model Validation and Optimization

**Figure 6.**(

**a**) A finite element analysis of the deformed system is illustrated. The same mechanism used for the amplification of the strain in the tensile tester is also used stand-alone (attached only to bulk fused silica-bar 1) for the quantitative evaluation of the stressors model. The red boxes in bar 1 represent the machined stressors. (

**b**) Using white light interferometry for measuring the profile of the deformed device, a significant volume variation was found and attributed to out-of-plane motion.

_{porous}is the effective elastic modulus of the porous material with porosity p, E

_{0}is the elastic modulus of solid material, p

_{c}is the porosity at which the effective elastic modulus becomes zero and f is the parameter dependent on the pore geometry [63]. From the images by the work of Canning et al. [60], the pores in the laser-affected zones are of complex shapes and interconnected. Based on these images, we estimate the porosity of the zones to range between 0.4 and 0.5. The value of the characteristic exponent f is for almost all the investigated materials in the range of 1.10–1.70 [64]. Porous materials with high concentration in surface heterogeneities and cracks have a low value of this parameter. Here, as an example we will use a conservative value (f = 1.20). Finally, as noted by Wagh et al. [65] fittings of experimental data for different materials often give p

_{c}= 1 and this is the value we adopt for this example.

**Figure 7.**Analytical model for different values of elastic modulus for the laser-affected zones and experimental data of the induced volume variation in the device. The experimental data were corrected with the measured factor of out-of-plane motion displacement of the device.

**Table 1.**Estimation of the laser-affected zones elastic modulus of the using the Phani-Nuyogi empirical relationship [63].

Elastic Modulus of Laser-Affected Zones for Different Porosity Parameters | E_{porous} (GPa) |
---|---|

p = 0.40 | 41.1 |

p = 0.45 | 37.8 |

p = 0.50 | 34.0 |

**Figure 8.**(

**a**) Method for measuring laser-induced volume variation in fused silica using cantilever deflection. The laser exposure takes place only near the anchoring point of the cantilever and only in its upper-half thickness and forms a bimorph composite structure that induces local bending of the cantilever. The deflection measured at the tip of the cantilever is effectively amplified by the length of the cantilever. The two different configurations result to different absolute values of induced volume variations, due to the spherical aberration effect. (

**b**) A sketch map of focusing across a plane surface from air into the sample. O

_{1}is the crossing point of the light axis across the interface, F

_{0}is the geometrical focus in air, F

_{1}is the focus under the paraxial approximation, f

_{d}= |O

_{1}F

_{1}| = n|O

_{1}F

_{0}| is the focusing depth, F

_{3}is the focus of the marginal rays, and Δ = |F

_{1}F

_{3}| is the foci range (focal displacement).

**Table 2.**Volume variation measured using the microcantilevers bending method. Three cantilevers were measured.

For Given Exposure Conditions (NA), Pulse Energy + Writing Speed | Tip Deflection (μm) | Mean Value of the Volume Variation % | ||
---|---|---|---|---|

Loading case 1 | 42.0 | 41.5 | 42.0 | 0.0011 |

Loading case 2 | 53.0 | 52.5 | 53.0 | 0.0009 |

_{d}is defined as the focusing depth. This leads to lower intensity distribution in the laser-affected zone and finally a lower state of induced volume variation.

**Figure 9.**(

**a**) Side view of the laser-affected bars that load the tensile tester. Optimized pattern structure of the sequential scanning of stressors for achieving uniform loading conditions. (

**b**), (

**c**) Parameters used to describe a single stressor. (

**d**) Simplified spring model of the loading process.

_{2}as the laser-affected volume, k

_{1}as the stiffness of the laser unaffected sidebars under loading as a spring arranged in parallel to the elongated spring, and k

_{3}as a parallel spring to k

_{1}and k

_{2}(as shown in Figure 3a,b).

#### 3.3. Third-Harmonics Generation (THG) as an In Situ Metrology Tool

**Figure 10.**(

**a**) Illustration of the formation of the third harmonic generation signal (THG) across the edge of a test sample. Regions one and two are close to the side edge of the material. In region one there is a decay of the signal, which is attributed to internal reflections. The spike in region two indicates reflection at the interface and thanks to it we can define the edge of the material. (

**b**) Illustration of the THG signal across the surface of a calibration pattern made of known-size grooves. The red labels are extracted from the THG trace.

#### 3.4. Stress Monitoring through Photoelasticity

_{1}− C

_{2}= C = 3.55 × 10

^{−12}Pa

^{−1}for fused silica at the microscope’s operated wavelength of 546 nm and is related to the piezo-optical coefficients by:

_{11}and π

_{22}are the piezo-optic constants for fused silica and n is the refractive index of the material at 546 nm (the wavelength used by the instrument). T is the thickness of the sample and R is the measured retardance, respectively.

_{f}is the final retardance, R

_{m}the retardance measured in the test beam, k is the number of orders, and λ is the wavelength at which the instrument operates.

#### 3.5. Example of Stress Measurements on a Silica Micro-Beam

^{2}in the center of the test beam (Figure 11).

**Figure 11.**Stress map of the test beam just before its failure. The stress is uniformly distributed along the length and the thickness of the test beam.

**Figure 12.**A typical stress–strain curve is illustrated. The elastic modulus is estimated at 72.8 GPa with a relative error of 2.1 GPa.

**Figure 13.**Tested specimen under the scanning electron microscope (

**right**) and a virtual experiment using finite element analysis simulation software illustrated (

**left**). The circled area has a stress concentration factor of 1.15.

#### 3.6. Error Analysis for the Experimental Technique

_{R}is the uncertainty of the retardance measurement, t is the thickness of the test beam, u

_{t}is the uncertainty of the thickness measurements, and is C = 3.55 × 10

^{−12}Pa

^{−1}as indicated in Section 3.4.

_{L}is the uncertainty of the measured length, Δl is the output displacement of the displacement amplification sensor and u

_{ε}is the uncertainly of the measured displacement, normalized with the amplification factor of the sensor.

## 4. Conclusions and Outlook

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

#### Derivation of the Shear-Lag Model

_{P}in Figure 3d. In the model, the laser affected material undergoes uniform displacement u

_{o}(z) = zε

_{applied}; the displacement field in the pristine region is assumed to be uniform throughout the thickness of the material. The shear stress in the pristine region can be expressed as:

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**MDPI and ACS Style**

Athanasiou, C.-E.; Bellouard, Y.
A Monolithic Micro-Tensile Tester for Investigating Silicon Dioxide Polymorph Micromechanics, Fabricated and Operated Using a Femtosecond Laser. *Micromachines* **2015**, *6*, 1365-1386.
https://doi.org/10.3390/mi6091365

**AMA Style**

Athanasiou C-E, Bellouard Y.
A Monolithic Micro-Tensile Tester for Investigating Silicon Dioxide Polymorph Micromechanics, Fabricated and Operated Using a Femtosecond Laser. *Micromachines*. 2015; 6(9):1365-1386.
https://doi.org/10.3390/mi6091365

**Chicago/Turabian Style**

Athanasiou, Christos-Edward, and Yves Bellouard.
2015. "A Monolithic Micro-Tensile Tester for Investigating Silicon Dioxide Polymorph Micromechanics, Fabricated and Operated Using a Femtosecond Laser" *Micromachines* 6, no. 9: 1365-1386.
https://doi.org/10.3390/mi6091365