Nonlinear regression least-squares method for determining relaxation time spectra for processes with first-order kinetics

Abstract

A simple method is shown for calculating the relaxation time spectrum which controls the rate at which a process following simple first-order kinetics takes place. The method involves unfolding a Fredholm equation of the first kind using least-squares and then using a modified nonlinear regression rather than a linear least-squares technique, thereby avoiding the highly oscillatory solutions which tend to occur with the latter with reduced mesh spacing or an increased number of bins. The validity and accuracy of the method for analyzing experimental data to reproduce various known input spectra are assessed and found to be excellent for data with no experimental error. For data with simulated experimental error with standard deviations up to σ=0.05 the method provides acceptable approximate solutions even though no exact solution is expected. Increasing the magnitude of the experimental error for a single lognormal input spectrum appears to have an increasing but nonsystematic effect upon the uncertainty of the approximate solution. Effects due to increasing the number of bins in the interval over which the spectrum is calculated are assessed and shown not to appreciably change the results, even for up to 60 bins. The methods is shown to be applicable to a wide variety of input spectra including single and double lognormal and box distributions. Importantly, in each of the cases studied the approximate solution appeared both to be unique and to converge toward the known input spectrum. Based upon this validation it is concluded that this method has applicability to a wide range of problems in which simple exponential decay is occurring with a spectrum of time constants. It may also be useful for problems with different kernals than that for first order kinetics; however, careful validation will be required.

An atomically quantized hierarchy of shear transformation zones in a metallic glass

“…Because anelastic relaxation in a metallic glass involves several processes, the temporal evolution of the strain is typically fitted with a linear combination of exponentially decaying terms, exp Àt=s ð Þ, with different time constants, s. Alternatively, a single stretched exponent has been used, exp(À(t/s) b ), 26 implicitly making an assumption about the shape of the relaxation spectrum. Instead, we employ a direct spectrum analysis method 23 by fitting the following functions to the anelastic strain as a function of time:…”

“…Direct spectrum analysis 23 was performed by fitting the relaxation curves, using the Primal-Dual Interior Point Filter Line Search Algorithm. The software package AMPL (Ref.…”

Identification of magnetic minerals related to hydrocarbon authigenesis in venezuelan oil fields using an alternative decomposition of isothermal remanence curves

“…It is noteworthy that, for OMFRC values higher than 2 × 10 −16 spin/g, the error of the EPR signal was lower than 10%, whereas for OMFRC values lower than 2 × 10 −16 spin/g, the instrumental error reached 20%. Cost (1983) introduced the Direct Signal Analysis (DSA) to calculate the relaxation time spectrum that controls the rate at which a process, following first-order kinetics, takes place. Through this method, the relaxation time spectrum was obtained by approximating the experimental curve with a sum of elementary curves.…”

Microscopic description of flow defects and relaxation in metallic glasses

“…[17]. The spectrum of time constants for exponential anelastic relaxation, f (τ ), was determined by nonlinear least-squares fit [17,21]. Significant effort was expended to verify the independence of the analysis results of initial guesses, choice of interval width or number of points, and the method was validated with simulated data that included noise.…”