Modeling TiO2׳s refractive index function from bulk to nanoparticles

- TiO2׳s refractive index function (RIF) is modeled from bulk to nanoparticles.
- Results are solved from extinction spectra by generalized oscillator based model.
- For both rutile and anatase, the estimated RIFs reveal two distinct regions.
- Starting from ca. 200 nm, the amplitudes of the RIF decreases on crystal diameter.
- Starting from particle diameter ca. 3 nm, blue shift starts to increase the band gap.

In recent decades, the use of nanomaterials has become very common. Different nanomaterials are being used in over 1600 consumer products. Nanomaterials have been defined as having at least one dimension in the range of 1−100 nm. Such materials often have unique properties. Despite some warnings of applying bulk optical constants for nano size materials, stated already in 1980s, bulk constants are still commonly used in the light scattering measurements of nano size particles. Titanium dioxide is one of the materials that is manufactured and used as an engineered nanomaterial in increasing quantities. Due to the aforementioned facts, it is quite crucial for successful research and production of nanoparticles to find out the dependence of the refractive index function (RIF) of the material on its crystal size. We have earlier performed several ab initio computations for obtaining the dependence of the RIF of TiO2 on the crystal or on the cluster size, for particles of size up to ca. 2 nm. Extending the calculations to greater sizes has turned out to be infeasible due to the unbearable increase in computational time. However, in this study we show how the crystal-size-dependent-RIF (CS-RIF), for both rutile and anatase can be modeled from measured extinction or turbidity spectra of samples with varying crystal and particle sizes. For computing the turbidity spectrum, we constructed a model including primary crystals whose distributions were parameterized by mean and standard deviation, and also including aggregates consisting of mean sized primary particles, parameterized just by mean aggregate size. Mainly because of the long computing times Mie calculation was used in the computation of extinction spectra. However, in practical process applications, the obtained RIF will be used together with the T-matrix method. We constructed the RIFs used in the model using generalized oscillator model (GOM) as expanded to crystal size dependence. The unknown parameters of the model were solved using nonlinear least squares estimation. When the crystal size becomes smaller than the bulk size the shape of the estimated CS-RIFs reveal two distinct regions for both rutile and anatase. In the first region, starting apparently already from ca. 200 nm, the height of both the real part and the imaginary part of CS-RIF decreases on crystal diameter. However, the band gap remains constant. In the second region, starting when the crystal diameter is decreased to ca. 3 nm, a blue shift starts to increase the band gap. The band gap dependence on crystal size is quite consistent with the existing experimental values. Consequently, it is of great importance to use CS-RIF in light scattering measurements for nanoparticle size determination. Neglecting this, the smaller particles in the size distribution will have too small values, already for sub-micrometer particles, naturally distorting also the mean value. To our knowledge, this is the first time ever that a CS-RIF from bulk to 1 nm size is determined for any material.