Using Weibull Statistics to Analyze Ion Beam Enhanced Adhesion as Measured by the pull Test

Abstract

The adhesion of iron films to single crystal Al2O3 substrates was investigated using a pull test. Chromium (300 keV) or nickel (340 keV) ions were implanted to a fluence of 1 × 1015 ions-cm-2 after film deposition. The adhesion test results were widely scattered due to a random distribution of interfacial flaw sizes controlling the failure nucleation. Because Weibull statistics were developed to describe the failure probability due to a population of flaw-initiated cracks, the Weibull distribution was chosen to analyze the data. Modifications in the adhesion strength due to the ion implantation were reflected in the failure distributions. It was found that the chromium ions improved the adhesion of the Fe/Al2O3 system while the implantation of nickel did not.

Statistical Paradigm for Organic Optoelectronic Devices: Normal Force Testing for Adhesion of Organic Photovoltaics and Organic Light-Emitting Diodes

“…Weibull analysis, particularly with log-likelihood estimators, is best suited to extract differences that are statistically significant between the data from relatively small data sets . Following the method of Pawel et al, we determined if the 15 and 5 min annealing samples were part of the same distribution or came from separate distributions, using a likelihood ratio test (see details in section SD). Again, the p value suggests that within 95% confidence, the different annealing time samples belong to different distributions, which can be distinguished.…”

“…All statistical analyses (ANOVA, Bootstrapping, Wilcoxon rank, and Weibull) were performed using the appropriate packages in the statistical computing language R . The Weibull parameters, scale factor and Weibull modulus, were extracted from a maximization of the log-likelihood method, taking the linear fitting values of a double logarithm of the cumulative Weibull distribution function. ,,, The Weibull parameters are used to extract the average expected failure stress using the Γ function. , The 10 and 50% failure stress quantile denotes the stress value for which 10 or 50% of the population has failed, which is also found from the log-likelihood equation. , p values reported in the text refer to the level of marginal significance within a statistical hypothesis test representing the probability of the occurrence of a given event. For all statistical calculations in this article, the null hypothesis assumed was that there was no difference between the measured breaking strength values.…”

“…All statistical analyses (ANOVA, Bootstrapping, Wilcoxon rank, and Weibull) were performed using the appropriate packages in the statistical computing language R. 43 The Weibull parameters, scale factor and Weibull modulus, were extracted from a maximization of the log-likelihood method, taking the linear fitting values of a double logarithm of the cumulative Weibull distribution function. 22,24,26,44 The Weibull parameters are used to extract the average expected failure stress using the Γ function. 15,25 The 10 and 50% failure stress quantile denotes the stress value for which 10 or 50% of the population has failed, which is also found from the log-likelihood equation.…”

“…15,25 The 10 and 50% failure stress quantile denotes the stress value for which 10 or 50% of the population has failed, which is also found from the log-likelihood equation. 24,26 p values reported in the text refer to the level of marginal significance within a statistical hypothesis test representing the probability of the occurrence of a given event. For all statistical calculations in this article, the null hypothesis assumed was that there was no difference between the measured breaking strength values.…”

“…This so called weakest-link-in-the-chain concept leads to significant variability in the measured breaking strength values. However, this type of failure is well-known in mechanical testing, and the probability model for failure in a normal force pull test has been shown to follow Weibull statistics. ,− …”