In situ nanoindentation was employed to probe the mechanical properties of individual polycrystalline titania (TiO2) microspheres. The force-displacement curves captured by a hybrid scanning electron microscope/scanning probe microscope (SEM/SPM) system were analyzed based on Hertz’s theory of contact mechanics. However, the deformation mechanisms of the nano/microspheres in the nanoindentation tests are not very clear. Finite element simulation was employed to investigate the deformation of spheres at the nanoscale under the pressure of an AFM tip. Then a revised method for the calculation of Young’s modulus of the microspheres was presented based on the deformation mechanisms of the spheres and Hertz’s theory. Meanwhile, a new force-displacement curve was reproduced by finite element simulation with the new calculation, and it was compared with the curve obtained by the nanoindentation experiment. The results of the comparison show that utilization of this revised model produces more accurate results. The calculated results showed that Young’s modulus of a polycrystalline TiO2 microsphere was approximately 30% larger than that of the bulk counterpart.
Nano/microspheres have wide application in supercapacitors, biosensors, drug delivery, and catalysts [
The extremely small dimensions of the nano/microspheres present a formidable challenge for experimental studies of their mechanical properties. Nanoindentation is currently the only viable approach to measure the mechanical properties of nano/microspheres. Armini et al. and Chen characterized the mechanical properties of microspheres for drug delivery and polymer microspheres using the nanoindentation method [
As an important wide-bandgap semiconductor, TiO2 has been formed into nanostructured materials that have extensive applications in many fields, especially in photocatalysis, solar cells, and electrochromic devices [
The nanocrystalline TiO2 microspheres were prepared using the same process described by Crepaldi et al. [
In situ nanoindentation experiments were conducted using a hybrid SEM/SPM system. The hybrid system relied on high-magnification SEM as a visual feedback system to indent the microspheres accurately with a cantilever probe. A more detailed description of the experimental setup previously can be found in the recent literature [
After the nanoindentation tests, FEM was used to investigate the deformation mechanisms of the nano/microspheres and to validate the calculated results. The FEM is shown in Figure
(a) Model of indentation of TiO2 microsphere. (b) Selected stress distribution to illustrate the deformation mechanisms.
The morphology and structure of the TiO2 microspheres are shown in Figure
(a) SEM image of TiO2 microspheres. (b) TEM image of a single TiO2 microsphere. (c) HRTEM image of the microsphere shown in (b). (d) XRD patterns of TiO2 microspheres. (e) Selected-area electron diffraction (SAED) pattern recorded on the microsphere shown in (b).
Figure
To quantitatively measure Young’s modulus of the TiO2 microspheres by Hertz’s model, force-displacement curves should be extracted. However, the force-displacement curves produced by the hybrid SEM/SPM system were the sum of the real penetration depth (
Next, the actual force-displacement curves were fitted with the general Sneddon’s expression (
The reduced elastic modulus for the test sample
Next, the way to extract the single penetration depth (
Schematic indentation of microsphere material by conical indenter. (a) Real penetration depth (
The effective modulus of elasticity
When the indenter tip and the substrate are the same material (the same material is silicon in this study), the reduced elastic modulus and the effective modulus of elasticity have the relationship
After the force-penetration depth curves were obtained, Young’s modulus of the TiO2 microspheres with various radii was calculated via (
(a) Force-displacement curves extracted from nanoindentation experiments and finite element simulations. (b) Young’s modulus as a function of microsphere diameter. The dash line represents Young’s modulus of bulk titania.
Meanwhile, the results extracted from the unrevised force-displacement curve were also calculated, which show that neglecting the compressive deformation of the sphere leads to 10–15% error. FEM was used to verify the calculation results by reproducing the force-displacement curves and comparing the simulated curves with the experimental curves. FE models were created based on physical properties of each component in experiments. Young’s modulus used in simulations was the results calculated before, while the Poisson ratio was estimated from literature [
In this study, we have used a hybrid SEM/SPM system to perform in situ nanoindentation on TiO2 microspheres, and we have investigated the deformation mechanisms for indenting a microsphere by finite element simulations. The penetration depth was extracted by removing the compressive deformation of the microsphere to obtain more accurate results. The calculated Young’s modulus of the TiO2 microspheres is larger than that of their bulk counterpart as a result of size effect. Additionally, the results were verified using FEM by reproducing the force-displacement curves and comparing the simulated curves with the experimental curves. It showed that there was a good agreement between them. This work provides a useful guide for interpretation of the data obtained by indenting nano/microspheres and obtains a relatively accurate result. Further experimental investigation is necessary to verify its applicability to many other nano/microspheres.
The authors declare that there is no conflict of interests regarding the publication of this paper.
Special Projects for Development of National Major Scientific Instruments and Equipments (2012YQ03007508), National Natural Science Fund Project (11374027), and Key Scientific and Technological Project of Shanxi Province (20130321011-04) are gratefully acknowledged.