A time gradient difference algorithm of autocorrelation curve for improving particle size inversion resolution in dynamic light scattering

Abstract

The application discloses a time gradient difference algorithm for improving particle size inversion resolution in dynamic light scattering, and the particle size distribution is inverted through a self-correlation curve in DLS; when an original dynamic light scattering self-correlation function data vector X is used to construct an original self-correlation function matrix A, a first-order difference operation is processed by applying a time gradient difference operation, a new time gradient difference vector X' is generated, and a new time gradient difference matrix is generated; the reconstructed time gradient difference vector X' and the time gradient difference matrix are applied to a DLS inversion algorithm to calculate a high-resolution particle size distribution. The application has the following advantages: the ill-conditioned nature of the DLS inversion problem is reduced, the original decay curve is converted into a curve with obvious peak characteristics, and the peak position difference corresponding to different particle sizes is significant, so that it is easier to distinguish components with close particle sizes in the inversion calculation, the bimodal and multimodal particle size resolution is effectively improved, and the minimum bimodal particle size ratio can reach 1.5 times.

Description

Technical Field

[0001] This invention relates to the field of testing technology, specifically a particle size measurement method. Background Technology

[0002] Dynamic light scattering (DLS) is a widely used technique for measuring the size distribution of nanoparticles and polymers in liquids. Its basic principle is to measure the light intensity fluctuations caused by the Brownian motion of the particles and analyze these fluctuations using an autocorrelation function, thereby retrieving the hydrodynamic diameter distribution of the particles. Due to its ease of operation, lack of sample labeling requirements, and high measurement speed, DLS has become an indispensable characterization method in materials science, biomedicine, environmental science, and other fields.

[0003] However, inverting particle size distribution through autocorrelation curves in DLS is a typical "ill-conditioned problem," meaning that even small measurement errors can lead to huge deviations in particle size distribution results. This is especially true when the sample contains multiple particle sizes that are very close, where the resolution is often insufficient. Effective measurement is even more difficult for multi-peaked samples. Currently, the industry typically requires a particle size ratio greater than 4 times for bimodal particle size resolution. To achieve a particle size resolution of more than 2 times, multi-angle measurements and other methods are required, which complicates the hardware structure and results in larger errors and lower accuracy.

[0004] Existing technical solutions for particle size distribution calculation in DLS mainly involve directly inverting the autocorrelation curve; another method is to obtain the average particle size using the accumulation method, but this does not provide detailed information about the particle size distribution; in terms of hardware structure, a weighted combination of autocorrelation curves from multiple angles can be used to improve resolution. However, these solutions have the following drawbacks: the original DLS autocorrelation function matrix is ​​usually highly correlated, with a large condition number, resulting in low resolution of the inversion results. It cannot resolve multi-peak distributions, i.e., it cannot distinguish between multiple particle size components in the sample, especially when particle sizes are similar. The accumulation method can only provide the average particle size, not the particle size distribution information. Multi-angle measurement methods have complex hardware structures and are difficult to implement. Furthermore, they require separate measurements at different angles before weighted fitting, resulting in inconsistent measurement times and signal error amplification due to data linking at multiple angles, thus reducing accuracy. Summary of the Invention

[0005] This invention is proposed to address the problems of low resolution and strong ill-conditioning in DLS inversion algorithms. Addressing the issue that the high similarity between the original autocorrelation function curves makes it difficult to distinguish features of different particle sizes, resulting in insufficient resolution, this invention reconstructs the original DLS autocorrelation function matrix. This makes the matrix vector eigenvalues ​​more obvious, reduces vector correlation, decreases the condition number of the matrix, and reduces the ill-conditioning of the DLS inversion problem, thereby improving the resolution of DLS particle size distribution inversion.

[0006] The specific technical solution is as follows:

[0007] A time gradient difference algorithm based on autocorrelation curves for improving particle size inversion resolution in dynamic light scattering is proposed. To address the low resolution and ill-conditioned nature of DLS inversion algorithms, when constructing the original autocorrelation function matrix A by inverting the particle size distribution using autocorrelation curves in DLS, a first-order difference operation is performed on the original dynamic light scattering autocorrelation function data vector X to generate a new time gradient difference vector X', thus generating a new time gradient difference matrix. ; Compare the reconstructed temporal gradient difference vector X' with the temporal gradient difference matrix It is applied to the DLS inversion algorithm to calculate the high-resolution particle size distribution.

[0008] The time gradient difference operation includes performing a first-order difference operation on the autocorrelation function curve: obtaining the original dynamic light scattering autocorrelation function data vector X, processing it using the time gradient difference operation, and generating a new time gradient difference vector X'. The time gradient difference operation includes performing a first-order difference operation on the autocorrelation function curve, for example, calculating the difference between the autocorrelation function values ​​at adjacent time delay points, X'. i =X i -X i+1 Calculate the difference in autocorrelation function values ​​at adjacent time delay points, for any column in matrix A (representing particle size). (autocorrelation function curve) After first-order difference processing, a new column is generated. ,in This differential operation aims to highlight the rate of change of the curve on the time delay axis, transforming the original decay curve into a curve with obvious peak characteristics.

[0009] The DLS inversion algorithm refers to:

[0010] Theoretical model column vector construction formula

[0011] The theoretical curve for the nth stage of the single exponential decay model:

[0012]

[0013] Among them, each particle size classification d ℎ,𝑖 diffusion coefficient D i Determined by the Stokes-Einstein equations:

[0014]

[0015]

[0016] In the formula:

[0017] D i Diffusion coefficient (m) 2 / s);

[0018] 𝑘 𝐵 Boltzmann constant (1.38 × 10⁻²³ J / K);

[0019] K: Absolute temperature (K);

[0020] 𝜂: viscosity of the medium (Pa·s);

[0021] 𝑞: Scattering vector;

[0022] n: solvent refractive index;

[0023] τ: Time delay parameter;

[0024] θ: Scattering angle.

[0025] The column vectors of matrix A are composed of theoretical curves for different particle size classifications:

[0026]

[0027] Parameter description:

[0028] 𝑁: Particle size fraction number;

[0029] 𝑀: Number of delay channels;

[0030] Δ𝜏: Sampling time interval.

[0031] The difference interval in first-order difference processing can be arbitrary. For example, adjacent time delay points can be calculated at intervals of 1, 2, or 3.

[0032] Beneficial effects of the technical solution of this invention

[0033] After processing using the time gradient difference method, the condition number of the autocorrelation function matrix can be reduced from 10. 17 The order of magnitude decreased to 10 15 The order of magnitude reduction reduces the ill-conditioned nature of DLS inversion problems. The original decay curve is transformed into a curve with obvious peak characteristics, and the peak positions corresponding to different particle sizes are significantly different, making it easier to distinguish components with similar particle sizes in inversion calculations. This effectively improves the resolution of bimodal and multimodal particle sizes, with the minimum bimodal particle size ratio reaching 1.5 times. Attached Figure Description

[0034] Figure 1 The image is the original autocorrelation matrix. Figure 1The horizontal axis represents the time delay correlation channels, with the unit being the number of channels; the vertical axis represents the normalized light intensity autocorrelation, which is dimensionless.

[0035] Figure 2 To reconstruct the matrix image using the temporal gradient difference method, Figure 2 The horizontal axis represents the correlation channels of the gradient difference over time delay, in units of the number of channels; the vertical axis represents the autocorrelation gradient difference before normalization, without units.

[0036] Figure 3 The residuals of the original autocorrelation curves at 50 nm and 100 nm;

[0037] Figure 4 This represents the residuals of the 50 nm and 100 nm curves after processing using the time gradient difference method. Detailed Implementation

[0038] An autocorrelation curve temporal gradient difference algorithm for improving particle size inversion resolution in dynamic light scattering is proposed. To address the issues of low resolution and strong ill-conditionedness in the DLS inversion algorithm, when inverting the particle size distribution through the autocorrelation curve in DLS and constructing the original autocorrelation function matrix A, a first-order difference operation is performed on the original dynamic light scattering autocorrelation function data vector X, generating a new temporal gradient difference vector X' and a new temporal gradient difference matrix A'. The reconstructed temporal gradient difference vector X' and the temporal gradient difference matrix A' are then applied to the DLS inversion algorithm to calculate a high-resolution particle size distribution.

[0039] The time gradient difference operation includes performing a first-order difference operation on the autocorrelation function curve: obtaining the original dynamic light scattering autocorrelation function data vector X, processing it using the time gradient difference operation, and generating a new time gradient difference vector X'. The time gradient difference operation includes performing a first-order operation on the autocorrelation function curve, for example, calculating the difference between the autocorrelation function values ​​at adjacent time delay points, X'. i =X i -X i+1 Calculate the difference in autocorrelation function values ​​at adjacent time delay points, for any column in matrix A (representing particle size). (autocorrelation function curve) After first-order difference processing, a new column is generated. ,in This differential operation aims to highlight the rate of change of the curve on the time delay axis, transforming the original decay curve into a curve with obvious peak characteristics.

[0040] The DLS inversion algorithm refers to:

[0041] Theoretical model column vector construction formula

[0042] The theoretical curve for the nth stage of the single exponential decay model:

[0043]

[0044] Among them, each particle size classification d ℎ,𝑖 diffusion coefficient D i Determined by the Stokes-Einstein equations:

[0045]

[0046]

[0047] In the formula:

[0048] D i Diffusion coefficient (m) 2 / s);

[0049] 𝑘 𝐵 Boltzmann constant (1.38 × 10⁻²³ J / K);

[0050] K: Absolute temperature (K);

[0051] 𝜂: viscosity of the medium (Pa·s);

[0052] 𝑞: Scattering vector;

[0053] n: solvent refractive index;

[0054] τ: Time delay parameter;

[0055] θ: Scattering angle.

[0056] The column vectors of matrix A are composed of theoretical curves for different particle size classifications:

[0057]

[0058] Parameter description:

[0059] 𝑁: Particle size fraction number;

[0060] 𝑀: Number of delay channels;

[0061] Δ𝜏: Sampling time interval.

[0062] The difference interval in first-order difference processing can be arbitrary. For example, adjacent time delay points can be calculated at intervals of 1, 2, or 3.

[0063] This invention transforms the original autocorrelation function curve into its rate of change or difference form on the time delay axis. This transformation converts the originally smoothly decaying curve into a curve with peak characteristics, enhancing the differences between characteristics of different particle sizes. The time gradient difference method includes performing first-order or higher-order difference operations on the autocorrelation function curve. The first-order difference operation includes calculating the difference between the values ​​of the autocorrelation function at adjacent time delay points. The peak position of the peak characteristic curve varies with particle size. The condition number of the reconstructed time gradient difference matrix is ​​reduced compared to the condition number of the original autocorrelation function matrix.

[0064] refer to Figure 1 and Figure 2 A comparison of the original autocorrelation function matrix and the reconstructed matrix using the time gradient difference method. The original autocorrelation matrix simply shows the decay of particle size over time, without any significant characteristic changes. Reconstructing the matrix using the autocorrelation time gradient difference method reveals a significant enhancement in the characteristic changes between particle sizes, transforming the previous time decay into a peak shift. The matrix condition number increases from 10. 17 dropped to 10 15 .

[0065] refer to Figure 3 and Figure 4 The residual between the original autocorrelation curves for the two particle sizes was 1.26, while the residual between the two particle size curves using the time gradient difference method was 2.83, indicating a 2.25-fold improvement in the correlation coefficient. This indirectly suggests a 2.25-fold improvement in resolution. For example, previously, the method could distinguish between bimodal particles with a particle size ratio of 1:3; now, it can distinguish between bimodal particles with a ratio of 1:1.33. Therefore, it can be proven that the time gradient difference method based on autocorrelation curves is an effective way to improve resolution.

Claims

1. A time gradient difference algorithm for autocorrelation curves to improve particle size inversion resolution in dynamic light scattering, characterized in that: To address the issues of low resolution and strong ill-conditioning in DLS inversion algorithms, when inverting particle size distribution through autocorrelation curves in DLS and constructing the original autocorrelation function matrix A, a first-order difference operation is performed on the original dynamic light scattering autocorrelation function data vector X using temporal gradient difference, generating a new temporal gradient difference vector X' and a new temporal gradient difference matrix A'. The reconstructed temporal gradient difference vector X' is then compared with the temporal gradient difference matrix A'. It is applied to the DLS inversion algorithm to calculate high-resolution particle size distribution; The time gradient difference operation includes performing a first-order difference operation on the autocorrelation function curve: obtaining the original dynamic light scattering autocorrelation function data vector X, processing it using the time gradient difference operation, and generating a new time gradient difference vector X'. The time gradient difference operation includes performing a first-order difference operation on the autocorrelation function curve to calculate the difference between the autocorrelation function values ​​at adjacent time delay points, X'. i =X i -X i+1 ; Calculate the difference in autocorrelation function values ​​at adjacent time delay points, for any column in matrix A. After first-order difference processing, a new column is generated. ,in ; This differential operation aims to highlight the rate of change of the curve on the time delay axis, transforming the original decay curve into a curve with obvious peak characteristics; The DLS inversion algorithm refers to: Theoretical model column vector construction formula The theoretical curve for the i-th stage of the single exponential decay model: Among them, each particle size classification d h,i diffusion coefficient D i Determined by the Stokes-Einstein equations: In the formula: D i : Diffusion coefficient; k B Boltzmann constant; T: Absolute temperature; η: viscosity of the medium; q: Scattering vector; n: solvent refractive index; τ: Time delay parameter; θ: Scattering angle.

2. The time gradient difference algorithm for autocorrelation curves used to improve particle size inversion resolution in dynamic light scattering according to claim 1, characterized in that: The column vectors of matrix A are composed of theoretical curves for different particle size classifications: Parameter description: N: Particle size fraction number; M: Number of delay channels; Δτ: Sampling time interval.

3. The time gradient difference algorithm for autocorrelation curves used to improve particle size inversion resolution in dynamic light scattering according to claim 1, characterized in that: The difference interval in first-order difference processing can be arbitrary.

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