Strain and grain size of TiO₂ nanoparticles from TEM, Raman spectroscopy and XRD: The revisiting of the Williamson-Hall plot method

The Williamson-Hall (W-H) equation, which has been used to obtain relative crystallite sizes and strains between samples since 1962, is revisited. A modified W-H equation is derived which takes into account the Scherrer equation, first published in 1918, (which traditionally gives more absolute crystallite size prediction) and strain prediction from Raman spectra. It is found that W-H crystallite sizes are on average 2.11 ± 0.01 times smaller than the sizes from Scherrer equation. Furthermore the strain from the W-H plots when compared to strain obtained from Raman spectral red-shifts yield factors whose values depend on the phases in the materials – whether anatase, rutile or brookite. Two main phases are identified in the annealing temperatures (350 °C–700 °C) chosen herein – anatase and brookite. A transition temperature of 550 °C has been found for nano-TiO₂ to irreversibly transform from brookite to anatase by plotting the Raman peak shifts against the annealing temperatures. The W-H underestimation on the strain in the brookite phase gives W-H/Raman factor of 3.10 ± 0.05 whereas for the anatase phase, one gets 2.46 ± 0.03. The new β_tot ²cos²θ-sinθ plot and when fitted with a polynomial yield less strain but much better matching with experimental TEM crystallite sizes and the agglomerates than both the traditional Williamson-Hall and the Scherrer methods. There is greater improvement in the model when linearized – that is the β_totcos²θ-sinθ plot rather than the β_tot ²cos²θ-sinθ plot.