Determination of Arbitrary Moments of Aerosol Size Distributions from Measurements with a Differential Mobility Analyzer

A method to determine arbitrary moments of aerosol size distributions from differential mobility analyzer measurements has been proposed. The proposed method is based on a modification of the algorithm developed by Knutson and Whitby to calculate the moments of electrical mobility distributions. For this modification, the electrical mobility and the charge distribution have been approximately expressed by power functions of the particle diameter. To evaluate the validity of the approximation, we have carried out numerical simulations for typical size distributions. We have found that for typical narrowly distributed aerosols such as polystyrene latex particles and particles that arise in the tandem differential mobility analyzer configuration, the distribution parameters can be accurately determined by this method. For a log-normally distributed aerosol, the accuracy of the distribution parameters determined by this method has been evaluated as a function of the geometric standard deviation. We have also compared the accuracy of the proposed method with other existing methods in the case of the asymmetric Gaussian distribution.     

The effects of timing jitter in sampling systems

Timing jitter generally causes a bias (systematic error) in the amplitude estimates of sampled waveforms. Equations are developed for computing the bias in both the time and frequency domains. Two principle estimators are considered: the sample mean and the so-called Markov estimator used in some equivalent-time sampling systems. Examples are given using both real and simulated data. It is shown that the bias that results from using the sample mean as an estimator can be approximated in the frequency domain by a simple filter function. The Markov estimator is shown to asymptotically converge to the population median. It is therefore an unbiased estimator for monotonic waveforms sampled with jitter distributions having a median of zero     

Measurement of 100 nm and 60 nm Particle Standards by Differential Mobility Analysis

The peak particle size and expanded uncertainties (95 % confidence interval) for two new particle calibration standards are measured as 101.8 nm ± 1.1 nm and 60.39 nm ± 0.63 nm. The particle samples are polystyrene spheres suspended in filtered, deionized water at a mass fraction of about 0.5 %. The size distribution measurements of aerosolized particles are made using a differential mobility analyzer (DMA) system calibrated using SRM^® 1963 (100.7 nm polystyrene spheres). An electrospray aerosol generator was used for generating the 60 nm aerosol to almost eliminate the generation of multiply charged dimers and trimers and to minimize the effect of non-volatile contaminants increasing the particle size. The testing for the homogeneity of the samples and for the presence of multimers using dynamic light scattering is described. The use of the transfer function integral in the calibration of the DMA is shown to reduce the uncertainty in the measurement of the peak particle size compared to the approach based on the peak in the concentration vs. voltage distribution. A modified aerosol/sheath inlet, recirculating sheath flow, a high ratio of sheath flow to the aerosol flow, and accurate pressure, temperature, and voltage measurements have increased the resolution and accuracy of the measurements. A significant consideration in the uncertainty analysis was the correlation between the slip correction of the calibration particle and the measured particle. Including the correlation reduced the expanded uncertainty from approximately 1.8 % of the particle size to about 1.0 %. The effect of non-volatile contaminants in the polystyrene suspensions on the peak particle size and the uncertainty in the size is determined. The full size distributions for both the 60 nm and 100 nm spheres are tabulated and selected mean sizes including the number mean diameter and the dynamic light scattering mean diameter are computed. The use of these particles for calibrating DMAs and for making deposition standards to be used with surface scanning inspection systems is discussed.     

Reliability of conformance tests

Conformance testing is considered from a statistical point of view. An s-confidence interval is found for the reliability that an implementation of a software package complies with specifications of a standard. Determination of whether it complies depends on a conformance test, which is written directly from the standard. Although the conformance test is written directly from the standard it does not test all possible software parameter-settings that invoke the standard. Thus, statistical inference is necessary. A general s-confidence interval for the reliability is given when the specification requires that the implementation passes all the tests in the conformance test suite. The conformance test is made of disjoint homogeneous partitions. The failure probability of the software is based on a weighted linear combination of the partition failure probabilities. An example is included     

Estimation of the stress-threshold for the Weibull inverse powerlaw

Lifetime data on stress rupture of copper joints made with certain lead-free solders suggest that specimens under `stress below a certain threshold' run indefinitely without failure. A commonly used model for this type of data is the Weibull inverse power law that includes a threshold. If the threshold is unknown, this estimation problem presents several difficulties for statistical treatment. The largest problem is: as the threshold approaches the minimum of the data (stresses) the likelihood approaches infinity, thus there is no global maximum. A modified maximum likelihood approach, in the spirit of Cohen is used to resolve this problem. The method is similar to Cohen's, but interesting differences occur for censored data. The results show that modifications of the Cohen method produce estimates of the parameters in the Weibull inverse power threshold law     

Stochastic Modeling of a New Spectrometer

A new spectrometer for classifying aerosol particles according to specific masses is being considered (Ehara et al. 1995). The spectrometer consists of concentric cylinders which rotate. The instrument is designed so that an electric field is established between the cylinders. Thus, aerosol particles injected into the spectrometer are subjected to a centrifugal force and an electric force. Depending on the balance between these two forces, as well as Brownian motion, charged particles either pass through the space between the cylinders or stick to either cylinder wall. Particles which pass through are detected. Given the rotation rate, voltage drop and physical dimensions of the device, we calculate the probability of detection in terms of particle density, diameter and charge. This is the transfer function. In this work, the focus is on situations where Brownian motion is significant. To solve for the transfer function, the trajectory of a particle in the spectrometer is modeled with a stochastic differential equation. Laminar flow is assumed. Further, attention is restricted to spherical particles with uniform density. The equation is solved using both numerical and Monte Carlo methods. The agreement between methods is excellent.