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Photon Correlation Spectroscopy for Nano-particle Diameter Measurement with Weighted Nonnegative Least Squares
Abstract
To reduce the effect of noise on inversion result of grain diameter in nano-particle diameter measurement using photon correlation spectroscopy, a nano-particle diameter computing method is proposed based on photon correlation spectroscopy with nonnegative least squares. Photon correlation spectroscopy itself is as the weight to derive discrete model of inversion algorithm and avoid the influence of data fluctuation close to zero. The 90 nm, 190 nm and mixed latex particles are measured by the photon correlation spectroscopy equipment and compared with the traditional nonnegative least squares. The 30 experimental data in 60 seconds indicate that in the inversion of unimodal paticle group, the results of present method is close to traditional nonnegative least squares but variance of multiple repeated measurement is smaller which proves good repeatability of present method; in the inversion of multimodal particles, the results of present method are much closer to true values of diameters, however, the results of nonnegative least squares deviate more from true values. Experimental data of different measurement time show that in a short period of time, variance of present method is smaller and it can obtain more accurate results in a shorter period of time.
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The method of traditional cumulant is a standard technique used to analyze dynamic light-scattering data. However, the different baseline values influence the results measured by using dynamic light-scattering. A method of moment-cumulant, which fits the baseline automatically, is proposed. The correlation function is used as the model function to inverse the particle diameter and distribution factor pi directly. An experiment is done with latex particle of 90 nanometers in laboratory. Compared with the two cumulant models, it is shown that the particle diameter and particle distribution factor pi are unstable in width fit range and lead to deviation with traditional cumulant, while measurement results is good in stability and not influenced by correlation time with the cumulant method of baseline auto-fitting.
Weighted cumulant which combined traditional cumulant with Frisken's cumulant was proposed to reverse nano-particle diameter with dynamic light scattering. The effect of correlation time was reduced when particle diameter was reversed. The 90 nm and 190 nm particles were measured with dynamic light scattering equipment. Traditional cumulant, Frisken's cumulant and weighted cumulant were used to reverse nano-particle diameter. The experimental result shows that the particle diameter error and standard deviation which is reversed with traditional cumulant is increased when the correlation time increases and the correlation faction is around zero; the particle diameter error and standard deviation which is reversed with Frisken's cumulant is big when the correlation time is not big enough and correlation data is less; the weighted cumulant inheritance the advantages of traditional cumulant and Frisken's cumulant, and the disadvantages of traditional cumulant and Frisken's cumulant are overcame. So the reverse result is stability overall situations and have the advantage of lower calculate time.
In order to simulate the dynamic light scattering signal of the bimodal distribution ultrafine particles, through setting up auto-regressive (AR) module of dynamic light scattering random process. The simulation signal of the dynamic light scattering can be acquired by this method. The light scattering signals of 10 nm and 90 nm, 200 nm and 1000 nm with bimodal distribution particles are respectively simulated by the computer. The autocorrelation function of the simulated light intensity is nearly identical to its theory autocorrelation function. The relative error of inverting particle size by double exponential is less than 3.55%. Analyzing influence of parameters such as the module order, sample time, sample frequency, simulation data length on simulation precision, the relation of simulation precision and parameters can be drawn. When the order is lower than the threshold order, the simulation precision is highly affected by the module order. The simulation precision will increase with the increase of the module. When the order is higher than the threshold order, the simulation precision is lowly affected by the module order. Therefore, the threshold order can be used in simulation. Selecting the certain sample time, the simulation date length and simulation percision will increase with the increase of the sample freguency.
The impact of scattering angle on nano-particles measurement is studied by using the method of correlation spectroscopy. The system of nano-particles experiment using the method of correlation spectroscopy is reformed to cany out the measurement of nano-particles at different scattering angles. Particles of 30-nanometer and 200-nanometer were mixed in different proportions, which were measured at scattering angles of 90 degrees and 165 degrees. It shows that accurate measurements can be realized at the two angles for the single particle group. While for the mixed particle group, the experiment results are affected by the mixing ratio of two kinds of particles. Better measurement results were got at the scattering angle of 165 degrees. Therefore the usage of large scattering angle contributes to the inversion of particle size for the hybrid nano-particles size measurement.
A Posteriori Choice Strategies based on Morozov discrepancy principle is adopted in order to choose the Optimum Regularization Parameter in the Inverse Algorithm of the Photon Correlation Spectroscopy particle sizing techniques. Using the method analyzed the simulation experimental data of single peake and multimodel peak particles system are analyzed. For the single peake particles system, the peake value is correct when the noise of the experimental data varies from 0 to 0.05. The peake value is incorrect when the noise of the experimental data is greater than 0.05. The distributing width decreases with the decrease of the original value of Regularization Parameter when the original value of Regularization Parameter varies from 0.00002 to 2. The distributing width decreases with the increase of the convergence error when the convergence error varies from 0.00005 to 50. For the multimodel peak particles system, the peak value incline to the average value when the particles diameter discrepancy is small. The smaller particle size distributing appear as the noise when the diameter discrepancy is large.
According to the relation between the normalized light field autocorrelation function (ACF) and the intensity ACF and the decay rate distribution function, the effect of baseline values on the effective diameter and polydispersity of nano-particles was analyzed. Two different types of baseline methods: Autoslope and extended, were discussed and the results show that the baseline values will change with the baseline methods. Experiments of three samples also testify that the larger the baseline values are, the smaller the effective diameter and the polydispersity are. The reason that the results measured by autoslope ate more stable than those measured by extended was analyzed.
To solve the difficulty of PCS (Photon Correlation Spectroscopy) for online particle sizing in high concentrated suspension, a new back scattering optic was designed and the back-scattering PCS method was proposed through analyzing the influence of the concentration on the PCS. The PCS and BSPCS were systematically tested using small polystyrene latex particles with a wide range of particle concentrations as well as 50 nm, 100 nm and 500 nm diameters in suspension. The results show that the multiple-scattering can be suppressed with back scattering PCS and this method suitable to online particle sizing in high concentrated suspension.
Multiangle dynamic light scattering (MDLS) technique can give better particle-size distribution (PSD) than single-angle dynamic light scattering (SDLS) technique. However, the choice of scattering angles is affected by the measuring particles in MDLS. Unimodal simulation distribution of 100 nm and 500 nm and bimodal simulation distribution of 300 nm and 600 nm are respectively measured at one, three, six and nine scattering angles and are inversed to obtain the PSD. This results show that MDLS can give better PSD with the increase of number of angles. Furthermore, the PSD has a little modification when there are more than one angle for 100 nm particles or more than three angles for 600 nm particles. A dilute bimodal suspension of polystyrene latex standard spheres mixed in a number ratio of 5:1 is measured at one, three, five and ten angles. The results show that one angle only can give one peak and more than three angles can give two peaks. The number ratio is closer to the true value with the increase of number of angles. Though MDLS can give better PSD than SDLS, the improvement of PSD become less obvious with the increase of number of angles. In some cases, the PSD may become worse with the increase of angle numbers because the calibration noise of scattering angle and measurement noise of light-intensity correlation function are added.
For the low accuracy of single-level inversion methods to dynamic light scattering, a novel Multi-level Tikhonov regularization inversion (ML-TIK) method combining the Tikhonov regularization method with cascadic multi-grid technique was developed. Firstly, this method divided the original problem into several sub-inversion problems with different grid spaces by a multi-grid technique. Then, from the coarsest scale to the finest scale, each sub-inversion problem was inverted by single-level Tikhonov regularization (TIK) method. Finally, the Particle Size Distribution (PSD) was successively obtained by solving several sub-inversion problems. This method effectively reduces the ill-condition of the original equations. At noise levels 0, 0.005 and 0.01, the simulation data of 200~650 nm bimodal distribution particles were respectively inverted by the TIK and ML-TIK. The results indicate that the inversion PSD of ML-TIK is more consistent with that of the theoretical one and it has better smoothness. Comparing to TIK, the ML-TIK can reduce the peak value error by 8.19% and relative error by 0.4482. However, when the noise level is 0.005 and 0.01, the PSD of TIK has not obvious bimodal features. Therefore, the ML-TIK has improved the inversion accuracy and noise immunity. Inversion results of 60 and 200 nm experimental data verify above conclusions.
The field of particle size distribution (PSD) characterization and measurement has experienced a renaissance over the past ten years. This revitalization has been driven by advances in electronics, computer technology and sensor technology in conjunction with the market pull for PSD methods embodied in cost effective user friendly instrumentation. The renaissance can be characterized by at least four activities. (1) End user innovation exemplified by techniques such as hydrodynamic chromatography (HDC), capillary hydrodynamic fractionation (CHDF) and field flow fractionation methods (SdFFF, FlFFF, and ThFFF). (2) Revitalization of older instrumental methods such as gravitational and centrifugal sedimentation; (3) Evolution of research grade instrumentation into low cost, routine, user friendly instrumentation exemplified by dynamic light scattering (DLS). (4) The attempt to meet extremely difficult technical challenges such as: (a) providing a single hybrid instrument with high resolution over a very broad dynamic range (4+ decades in size; e.g., Fraunhofer/Mie; photozone sensing/DLS); (b) PSD measurement of concentrated dispersions (acoustophoretic, dielectric measurements, fiber optic DLS (FOQELS)); (c) in-situ process particle size sensors (in-line or at line, e.g., FOQELS); (d) routine measurement of particle shape and structure (e.g., image analysis). Instrumental methods resulting from these activities are discussed in terms of measurement principles and the strengths and weaknesses of these methods for characterizing PSDs. Business and societal driving forces will impact customer perceived instrumentation and knowledge needs for the 21st century and the ability to meet the specific difficult technical challenges in particle size distribution characterization mentioned above. Anticipated progress toward meeting these technical challenges is discussed in conjunction with the associated anticipated advances in required technologies.