Deformation Mechanisms of Nanoporous Oxide Glasses: Indentations and Finite Element Simulation

2.1 MaterialsThe glass AlPO4 (AP glass) was synthesized following the sol-gel route reported previously[27]. To tailor the pore size of the AP glass, varying amounts of distilled water were used to dissolve the precursors. The molar ratios of water, aluminum lactate, and phosphoric acid in the sol were adjusted to 423:1:1, 471:1:1, and 509:1:1, respectively.The glass 95.24SiO2-4.76Al2O3 (SA glass) was prepared via the following procedure[26]: 0.19g aluminum lactate was dissolved in 10 mL distilled water, and then mixed with the desired amount of TEOS dissolved in 8 mL isopropanol 1 mol/L HCl solution was added in the mixture to adjust the pH to a value of 3.0. After stirring overnight, a clear solution was obtained, then aged and dried at 100oC to form transparent bulk xerogel. The dry xerogels were converted to transparent glass after sintering at 600oC for several hours. To modify the pore size of SA glass, compositions with varying aluminum content (98SiO₂-1Al₂O₃ ,and 96SiO95.24SiO₂-4.76Al₂O₃, and 95SiO₂-5Al₂O₃) were prepared by altering the mass of aluminum lactate in the recipe. For a pure porous SiO2 glass, instead of aluminum lactate, citrate in varied amounts (0.14mL, 1mL, 2mL and 3mL) were dissolved in 4 mL distilled water, respectively, and subsequently mixed with 2.7 mL TEOS (6 mmol) dissolved in 10 mL ethanol. The aging and sintering processes were carried out under the same conditions as those used for the SA glass. 2.2 CharacterizationsThe amorphous structure of glasses was identified by X-ray powder diffraction (XRD) using filtered Cu-Ka radiation in Bragg–Brentano geometry on a Bruker D8 Advance (Germany) diffractometer (diffraction angle 2q ranges from 10° to 90°). The thermodynamic features of glasses were obtained from a Differential Scanning Calorimeter (DSC): NETZSCH STA449F3 instrument (NETZSCH Germany). The heating rate was 10 K/min. The transmittance spectra of the porous glasses were collected from 250 nm to 800 nm with the step size of 1 nm using a UV-Vis spectrometer (Lambda 950, PerkinElmer).The porous structures of glasses were characterized by the N2 absorption–desorption isotherms measured at 77 K on a volumetric adsorption analyzer Autosorb iQ Station (Quantachrome, FL, USA). Prior to analysis, samples weighing about 0.25 g were outgassed at 110 °C for at least 24 h under vacuum until a residual pressure of <0.6 mmHg was reached. The surface area was determined based on the Brunauer–Emmett–Teller (BET) equation[28]. The total pore volume, Vpore, was obtained from the amount adsorbed at a P/P0 ratio of about 0.99. The pore size distributions were obtained using the Barrett–Joyner–Halenda (BJH) method assuming a cylindrical pore model[29].The porous structures of the glass specimens were observed using the Scanning Electron Microscope (SEM) JSM-IT800 (JEOL, Japan), and the element mapping analysis was performed on the scanned area using the Energy Dispersive Spectrometer (EDS) equipped on the SEM instrument. The pieces of porous glasses were also ground into fine powder and further inspected using the Transmissive Electron Microscope (TEM) JEM-F200 (JEOL, Japan).The instrumented indentations were performed using Bruker Triboindenter TI-980 (Minneapolis, MN, USA) equipped with a Berkovich indenter. Over 10 quasi-static indentations with the maximum depth of 3 μm were performed in the displacement-control mode. The applied penetration rate is 300 nm/s. Assume the glasses are isotropic, and their elastic moduli are readily calculated from the relation [30]:  , where the elastic modulus and the Poisson’s ratio for diamond indenter [31] are Ei = 1141GPa, νi = 0.07, and assume the Poisson’s ratio for the silica νs = 0.19. The hardness and elastic moduli were measured according to Oliver-Pharr method[32]. The creep indentation tests with 60 s holding time at the peak load were carried out. The creep indentation tests were performed after thermal equilibrium for 20 minutes. During each indentation, the thermal drift was monitored at the level of < 0.1 nm/s. The applied constant loading rate in the creep indentations is 1 mN/s. The indentation rate jump tests were also conducted for the strain rate sensitivity m of the glass samples, and the penetration rate shifted between 300 nm/s and 30 nm/s for 6 times during the loading process. At least 6 indents with the maximum penetration depth of 4 μm were carried out in the displacement-control mode. Microhardness tests were also performed on a Vickers hardness tester (Q10A+, QNESS in Austria) for a comparison with the instrumented indentations by the Berkovich probe.2.3 Finite Element ModellingAs the difference between the load-depth curves obtained from the 2D axisymmetric and the 3D Berkovich model is negligible[33], so the less expensive 2D axisymmetric model in ABAQUS/Standard[34] was adopted to simulate the indentation process for simplicity. In the simulation, a conical indenter with half apex angle of 70.3o, which is equivalent to the Berkovich indenter, was applied. It is justified to treated indenter as rigid body, because the Young's modulus of the diamond indenter is almost 100 times higher than the porous glasses. A cylinder with the diameter of 600 mm and the height of 300 mm was created as glass sample to be indented, and it was meshed with 132365 CAX4-type quadrilateral elements, as displayed in Fig.6a. A finer mesh was designed in the contact area (plastic zone) for an accurate representation of the stress distribution under the indenter tip, while a coarser mesh was used further away from the contact zone to reduce the computational load.The following boundary conditions were set as follow: the nodes along the axis of revolution are free to move only along the axis, whereas all the degrees of freedom for the nodes at the bottom of the sample were set to zero. The indenter-sample interaction was modeled with a ‘surface-to-surface’ contact discretization. The sample surface was taken as the slave surface, which cannot penetrate the master surface thus the diamond indenter in the FEM simulation. Contact perpendicular to the indenter, the sample is more compliant and needs to be meshed with finer elements.The nanoindentation test was simulated by using two subsequent load steps, one for the loading part and the other for the unloading part. During the loading step, the indenter tip moves down along the axis of symmetry until the maximum depth (3 mm for porous glasses, and other specific values reported in literature for other materials) was reached. During the unloading step, the indenter tip retracted to its original position.Significant densification (up to 20%) has been observed in the indentation of fused silica[35], and even stronger densifying processes were expected to occur in the nanoporous glasses. Therefore, a constitutive model[35] with a double linear stress-strain curve was proposed to account for the densification of glass under load, as displayed in Fig.6b. The glasses start to densify when the applied stress reaches the yield strength sy, which correlates to the hardness through a constraint factor C. A higher flow stress sr in the porous glass due to the densification was given by Larsson’s solution[36]. When the plastic strain reaches 30%, the flow stress sr is in the following equation:       (1)where the hardness H and elastic moduli E of porous glasses have been measured previously. Took the values of the measured E and calculated sy and sr as input data, the FE simulation of indentation can be conducted.To be noted, considering the effects of densification and hydrostatic compression, a more precise constitutive model[37] has been introduced previously and described the stress/strain field of fused silica using a yield surface in the Drucker-Prager-Cap elliptical shape[38,39]. However, due to the lack of some crucial parameters e.g. densification, critical hydrostatic pressure of the porous glasses at current stage, the advanced FE simulation of the deformation in porous glasses can be adopted when relevant data become available in future.