The properties of nanoparticles are related to their structures and sizes, and studying methods for measuring the length and diameter of rod-shaped nanoparticles is of practical significance. Transmission electron microscopy has high resolution and can provide detailed morphological features of rod-shaped nanoparticles. However, electron microscopy can only observe a small number of particles, and the measurement results lack statistical significance. The dynamic light scattering method can quickly characterize the particle size and size distribution of nanoparticles, but since this method assumes that the measured particles are spherical, it cannot accurately measure the size of rod-shaped particles. The depolarization dynamic light scattering method can obtain the length and diameter of rod-shaped nanoparticles by measuring the translational and rotational diffusion coefficients of particles in Brownian motion. It is necessary to fit the translational and rotational attenuation linewidths separately for obtaining the translational and rotational diffusion coefficients of the Brownian motion of rod-shaped nanoparticles. Exponential fitting algorithms are commonly adopted in fitting the attenuation linewidth, but they are greatly affected by the initial value. When the initial value is not suitable, the measurement results will deviate from the true value. To this end, a Tikhonov regularization algorithm is proposed to invert the vertical and horizontal polarization autocorrelation functions obtained from depolarized dynamic light scattering experiments, thereby putting forward a method for acquiring the translational and rotational attenuation linewidths.
The experimental device employs a 532 nm vertically polarized solid-state laser as the light source, and a Glen Thompson lens is placed at a 90° scattering angle position. The lens divides the scattered light into two optical paths of horizontal polarization and vertical polarization. On each path, a single-mode fiber is utilized to receive the scattered light signal, which is then fed into a photomultiplier tube. After receiving the scattered light signal, the normalized autocorrelation function of light intensity is obtained by real-time calculation of a large dynamic range high-speed digital correlator. Additionally, the temperature control system maintains the sample cell temperature at 25 ℃. During the experiment, the experimental device is covered with a shell to prevent interference from stray light and reduce measurement errors. Three different sizes of gold nanorod samples are purchased, and four different concentrations of gold nanorod samples are set for depolarization dynamic light scattering measurements. The autocorrelation functions of vertical and horizontal polarization directions of samples with different concentrations are obtained. The Tikhonov regularization algorithm is adopted to invert the autocorrelation function to obtain the translational and rotational attenuation linewidths. After converting the attenuation linewidth into diffusion coefficient, the Tirado-Garcia de la Torre (TG) model can be leveraged to fit the length and diameter of rod-shaped nanoparticles. Since rod-shaped gold nanoparticles are surrounded by an adsorption layer in the liquid, the adsorption layer increases the size of the rod-shaped gold nanoparticles, making their size slightly larger than the actual size in the liquid. Therefore, we have corrected the three sets of length and diameter data obtained from depolarized dynamic light scattering measurements. The measurement results are compared with those of a transmission electron microscope to verify the feasibility of this method.
After Tikhonov regularization inversion of the horizontal polarization autocorrelation function, a single-peak attenuation linewidth distribution can be obtained, and the mixed attenuation linewidth can be obtained from its peak [Fig. 7(a)]. After performing Tikhonov regularization inversion on the vertical polarization autocorrelation function, a bimodal attenuation linewidth distribution is obtained, and the translational attenuation linewidth can be acquired from its left peak [Fig. 7(b)]. The original concentrations of the three samples are all 0.1 mg/ml, and samples with different concentrations of 0.10, 0.07, 0.05, and 0.03 mg/ml respectively are obtained by diluting them. The experimental data show that the autocorrelation functions of light intensity of samples with different concentrations coincide, with consistent measurement results (Fig. 3).
We propose to employ the Tikhonov regularization algorithm to invert the autocorrelation functions in the horizontal and vertical polarization directions, respectively and thus to obtain the translational and rotational attenuation linewidths. After converting the attenuation linewidths into diffusion coefficients, the length and diameter of rod-shaped nanoparticles can be fitted using the TG model. The experimental results show that after removing the adsorption layer after correction, the length and diameter measurements of three sets of rod-shaped gold nanoparticles obtained using the depolarization dynamic light scattering method based on Tikhonov inversion are within 8% of the measurement results of transmission electron microscopy. This indicates that the corrected measurement results are consistent with the measurement results of transmission electron microscopy. The experimental data demonstrate that the autocorrelation functions of light intensity of samples with different concentrations basically coincide, and the measurement results remain consistent.
