When measuring the length and diameter of rod-shaped nanoparticles by using depolarized dynamic light scattering (DDLS), it is often necessary to combine the inversion algorithms for calculation. Commonly used inversion algorithms include exponential fitting and Tikhonov regularization algorithms. The exponential fitting method will produce fitting errors due to improper setting of the initial value, while in the Tikhonov regularization algorithm, when inverting rod-shaped particles, the error of the horizontally depolarized signal will significantly amplify the diameter inversion bias. In order to improve the accuracy of particle inversion, scholars apply neural networks to the field of spherical particle inversion, which effectively improves the accuracy of inversion, and some scholars apply neural networks to non-spherical particles, but not to the length and size of rod-shaped particles. In order to improve the accuracy and repeatability of the inversion, we propose a generalized regression neural network (GRNN) combining Tikhonov and parameter optimization, which can effectively carry out the inversion of rod-shaped particle size.

The DDLS measurements are performed on the rod-shaped particle samples to obtain the scattering signals of vertical polarization and horizontal depolarization, from which the light intensity autocorrelation function is obtained, and the attenuation linewidth distribution is obtained by the Tikhonov regularized inversion of the autocorrelation function. The light intensity autocorrelation function and the attenuation linewidth distribution are sorted out into the dataset required for the Tikhonov-GRNN model, and the dataset is divided into the data set is divided into training samples and validation samples. After the training is completed, the data in the validation samples are inverted and analyzed: the predicted attenuation linewidth distributions in the vertical polarization and horizontal depolarization directions are obtained by the vertical vertical (VV)-GRNN and vertical horizontal (VH)-GRNN models, the horizontal diffusion coefficients and the rotational diffusion coefficients of the motion of the rod-like particles are calculated respectively, and the lengths and diameters of the rod-like particles are solved by the Tirado-Garcia de la Torre (TG) model.

Before the experiment, 300 particles from experimental samples are observed by transmission electron microscopy first. Results show that actual sizes of the rod particles used in the experiment coincided are mostly consistent with the nominal size (Figs. 4 and 5, Table 1), so the use of nominal lengths and diameters has no effect on experimental results. In this paper, experiments are carried out on two different sizes of rod-shaped nanoparticles. The experiments use the DDLS method to measure the samples, and the data set required for the Tikhonov-GRNN model is obtained. After the training is completed, the autocorrelation function of light intensity is input for the prediction, and the attenuation linewidth distributions of VV and VH are obtained, which led to inverse results of lengths and diameters of rod-shaped particles (Table 4). Experimental results show that the inversion accuracy of the Tikhonov-GRNN model is better than inversion results of the traditional algorithm (Table 5), which is closer to the real size. After repeating experiments many times, repeatabilities of two particle lengths and diameters are 4.5%, 1.6% and 9.8%, 8.9%, respectively, which are better than 5.9%, 16.3% and 3.2%, 17.5% of the traditional Tikhonov regularization algorithm. All these can prove that the stability of the Tikhonov-GRNN model is stronger.

When using DDLS method to measure the size of rod-shaped particles, the results of current inversion algorithms are not stable due to the complexity of the mathematical model of non-spherical particles. Therefore, based on the nonlinear mapping and generalization ability of GRNN, a GRNN combining Tikhonov’s regularization algorithm and parameter optimization is proposed, which can invert the length and diameter of rod-shaped particles. The method uses DDLS to measure the light intensity autocorrelation function in the VV polarization and VH depolarization directions of the rod particles, respectively, and obtains the attenuation linewidth by Tikhonov regularization inversion, and constructs a training dataset by combining the attenuation linewidth with the corresponding light intensity autocorrelation function. The trained VV-GRNN and VH-GRNN predict the attenuation linewidth to obtain the translational diffusion coefficients and rotational diffusion coefficients of the rod particles, and then calculate the size of the rod particles. Two kinds of gold nanorod particles are measured, and the experimental results show that the Tikhonov-GRNN algorithm can realize the inversion of the length and diameter of the rod particles, and compared with the Tikhonov regularization algorithm, the accuracy of the Tikhonov-GRNN inversion algorithm is improved, while repeatabilities of the length and diameter are lower than 4.5% and 9.8%, respectively.