Fourier transform distribution function of relaxation times; application and limitations

2015 ◽  
Vol 154 ◽  
pp. 35-46 ◽  
Author(s):  
Bernard A. Boukamp
Keyword(s):  
Distribution Function ◽  
Fourier Transform ◽  
Relaxation Times

Temperature Dependence of Dynamic Properties of Elastomers. Relaxation Distributions

1952 ◽  
Vol 25 (4) ◽  
pp. 720-729 ◽  
Author(s):  
John D. Ferry ◽  
Edwin R. Fitzgerald ◽  
Lester D. Grandine ◽  
Malcolm L. Williams
Keyword(s):  
Distribution Function ◽  
Temperature Dependence ◽  
Dynamic Properties ◽  
Relaxation Times ◽  
Dynamic Mechanical Properties ◽  
Relaxation Processes ◽  
The Real ◽  
Reduced Frequency ◽  
Dynamic Rigidity ◽  
Composite Curve

Abstract By the use of reduced variables, the temperature dependence and frequency dependence of dynamic mechanical properties of rubberlike materials can be interrelated without any arbitrary assumptions about the functional form of either The definitions of the reduced variables are based on some simple assumptions regarding the nature of relaxation processes. The real part of the reduced dynamic rigidity, plotted against the reduced frequency, gives a single composite curve for data over wide ranges of frequency and temperature; this is true also for the imaginary part of the rigidity or the dynamic viscosity. The real and imaginary parts of the rigidity, although independent measurements, are interrelated through the distribution function of relaxation times, and this relation provides a check on experimental results. First and second approximation methods of calculating the distribution function from dynamic data are given. The use of the distribution function to predict various types of time-dependent mechanical behavior is illustrated.

NOTE ON THE VLASOV EQUATION FOR PLASMAS

1963 ◽  
Vol 41 (12) ◽  
pp. 1960-1966 ◽  
Author(s):  
Ta-You Wu ◽  
M. K. Sundaresan
Keyword(s):  
Distribution Function ◽  
Fourier Transform ◽  
Vlasov Equation ◽  
Hermite Polynomials ◽  
Infinite Matrix ◽  
Mean Square ◽  
Initial Value ◽  
Positive Ions ◽  
The Mean ◽  
The Fourier Transform

The linearized Vlasov equation is solved as an initial value problem by expanding (the Fourier components of) the distribution function in a series of Hermite polynomials in the momentum, with coefficients which are functions of time. The spectrum of frequencies is given by the eigenvalues of an infinite matrix. All the frequencies ω are real, extending from small values of order ω2 = k2(u22), where (u22) is the mean square velocity of the positive ions (of mass M), to [Formula: see text], where ω1, (u12) are the plasma frequency and mean square velocity of the electrons (of mass m). The classic work of Landau solves the Vlasov equation for (the Fourier transform of) the potential for which he obtains the "damping", whereas Van Kampen and the present writers solve the equation for (the Fourier transform of) the distribution function itself. While the present work gives results equivalent to those of Van Kampen, the method is simpler and in fact elementary.

scholarly journals Optimized Process Parameters for a Reproducible Distribution of Relaxation Times Analysis of Electrochemical Systems

Batteries ◽  
2019 ◽  
Vol 5 (2) ◽  
pp. 43 ◽  
Author(s):  
Markus Hahn ◽  
Stefan Schindler ◽  
Lisa-Charlotte Triebs ◽  
Michael A. Danzer
Keyword(s):  
Distribution Function ◽  
Electrochemical Impedance ◽  
Regularization Parameter ◽  
Relaxation Times ◽  
Measurement Data ◽  
Complex Impedance ◽  
Lithium Ion ◽  
Parameter Study ◽  
Model Free ◽  
The Impact

The distribution of relaxation times (DRT) analysis offers a model-free approach for a detailed investigation of electrochemical impedance spectra. Typically, the calculation of the distribution function is an ill-posed problem requiring regularization methods which are strongly parameter-dependent. Before statements on measurement data can be made, a process parameter study is crucial for analyzing the impact of the individual parameters on the distribution function. The optimal regularization parameter is determined together with the number of discrete time constants. Furthermore, the regularization term is investigated with respect to its mathematical background. It is revealed that the algorithm and its handling of constraints and the optimization function significantly determine the result of the DRT calculation. With optimized parameters, detailed information on the investigated system can be obtained. As an example of a complex impedance spectrum, a commercial Nickel–Manganese–Cobalt–Oxide (NMC) lithium-ion pouch cell is investigated. The DRT allows the investigation of the SOC dependency of the charge transfer reactions, solid electrolyte interphase (SEI) and the solid state diffusion of both anode and cathode. For the quantification of the single polarization contributions, a peak analysis algorithm based on Gaussian distribution curves is presented and applied.

Magnetische Relaxation in Glycerin

1969 ◽  
Vol 24 (1) ◽  
pp. 143-153 ◽  
Author(s):  
F . Noack ◽  
G . Preissing
Keyword(s):  
Distribution Function ◽  
Magnetic Relaxation ◽  
Relaxation Times ◽  
Nuclear Magnetic Relaxation ◽  
Frequency Range ◽  
Glycerol Water ◽  
Correlation Times ◽  
Pure Glycerol ◽  
New Type ◽  
Nuclear Magnetic

AbstractProton nuclear magnetic relaxation times in pure glycerol and glycerol water mixtures have been measured in the frequency range from 450 kHz . . . 120 MHz from -20 °C .. . 70 °C. The results cannot be interpreted in terms of the well-known distributions of dielectric correlation times (Log-Gaussian, Cole-Davidson etc.), as has been proposed recently. Instead a new type of distribution function, called "diffusion-distribution", is introduced.

A Selective Determination of the Proton Nuclear Relaxation Times of the Alkaloid Vindoline: An Extension via Fourier Transform Measurements

1974 ◽  
Vol 52 (5) ◽  
pp. 829-832 ◽  
Author(s):  
L. D. Hall ◽  
Caroline M. Preston
Keyword(s):  
Fourier Transform ◽  
Relaxation Times ◽  
Lattice Relaxation ◽  
Spin Lattice Relaxation ◽  
Nuclear Relaxation ◽  
Transform Method ◽  
Fourier Transform Method ◽  
Selective Determination ◽  
Spin Lattice

A Fourier Transform method has been used to measure the spin–lattice relaxation times of essentially all the protons of the alkaloid, vindoline. It is shown that even for a molecule of this size substantial and potentially useful differences exist in the experimental relaxation times which reflect the degree of crowding of each proton by other protons.

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