^{1,a)}

### Abstract

We suggest a new algorithm for relaxation time spectrum, which is based on the power series approximation of dynamic modulus and relaxation time spectrum. Through the regression of dynamic modulus as a polynomial of the logarithm of frequency, the method converts the coefficients of the modulus to those of relaxation time spectrum. The algorithm provides relaxation time spectrum as stable as regularization method.

This work was supported by Mid-career Research Program through NRF Grant funded by the MEST (No. 2009-0083966) and by Kyungpook National University Research Fund, 2012.

I. INTRODUCTION

II. THEORY

A. Series expression of dynamic moduli

B. Relaxation spectrum by Taylor series

C. Analysis on truncated series

III. RESULTS AND DISCUSSION

A. Test for simulated data without partitioning

B. Test for simulated data with partitioning

C. Effect of errors

D. Application to experimental data

IV. CONCLUSIONS

### Key Topics

- Polynomials
- 34.0
- Relaxation times
- 16.0
- Computational complexity
- 2.0
- Elastic moduli
- 2.0
- Integral equations
- 2.0

## Figures

Effect of the order of polynomial for the regression of dynamic moduli. (a) Modulus data (symbols) and moduli obtained from regression (line). Moduli data are generated by the spectrum of Eq. (43) without statistical errors. (b) The calculated spectra for various orders of polynomials. The open symbols in (b) are the spectra calculated from the regression of loss modulus and the closed symbols are the spectra from the regression of storage modulus. The line behind the symbols is Eq. (43) .

Effect of the order of polynomial for the regression of dynamic moduli. (a) Modulus data (symbols) and moduli obtained from regression (line). Moduli data are generated by the spectrum of Eq. (43) without statistical errors. (b) The calculated spectra for various orders of polynomials. The open symbols in (b) are the spectra calculated from the regression of loss modulus and the closed symbols are the spectra from the regression of storage modulus. The line behind the symbols is Eq. (43) .

Test for the regression using data partitioning. (a) Moduli generated from the given spectrum Eq. (45) (line) and regression results. The different symbols denote the regressions for different partitions of data. All regressions were done by without the use of an orthogonal polynomial. (b) Comparison of the given spectrum (line) and spectra calculated from storage modulus data (symbols), which are obtained from partitioned data. (c) Comparison of the given spectrum (line) and spectra calculated from loss modulus.

Test for the regression using data partitioning. (a) Moduli generated from the given spectrum Eq. (45) (line) and regression results. The different symbols denote the regressions for different partitions of data. All regressions were done by without the use of an orthogonal polynomial. (b) Comparison of the given spectrum (line) and spectra calculated from storage modulus data (symbols), which are obtained from partitioned data. (c) Comparison of the given spectrum (line) and spectra calculated from loss modulus.

Spectrum calculated using an orthogonal polynomial and data partitioning. Modulus data used are the same as those in Fig. 2 . Regression for loss modulus is mad by use of the Chebyshev polynomial of the first kind. (a) The 12th order polynomial is used for the regression of loss modulus. (b) The 13th order polynomial is used for the regression of loss modulus.

Spectrum calculated using an orthogonal polynomial and data partitioning. Modulus data used are the same as those in Fig. 2 . Regression for loss modulus is mad by use of the Chebyshev polynomial of the first kind. (a) The 12th order polynomial is used for the regression of loss modulus. (b) The 13th order polynomial is used for the regression of loss modulus.

Effect of regression of loss modulus by use of Eq. (1) for the entire data. The Chebyshev polynomial is used. (a) Loss modulus data of Fig. 2 and the regression result. The order of polynomial is 17. (b) Spectra calculated by (symbols) and the given spectrum (line).

Effect of errors. The errors are generated from the Gaussian distribution with the mean of zero and the standard deviation of unity and added to the modulus data of Fig. 1 . (a) The error level is 5% and the regression order is 8. (b) The error level is 5% and the regression order is 10. (c) The error level is 3% and the regression order is 8. The dotted lines indicate the range of modulus data. The error bar is the standard deviation calculated from 11 spectra, which are calculated from independently noised data. When the error level is fixed, a lower order of regression gives better stability of spectrum, and when the order of regression is fixed, a lower level of error gives a more stable spectrum.

Effect of errors. The errors are generated from the Gaussian distribution with the mean of zero and the standard deviation of unity and added to the modulus data of Fig. 1 . (a) The error level is 5% and the regression order is 8. (b) The error level is 5% and the regression order is 10. (c) The error level is 3% and the regression order is 8. The dotted lines indicate the range of modulus data. The error bar is the standard deviation calculated from 11 spectra, which are calculated from independently noised data. When the error level is fixed, a lower order of regression gives better stability of spectrum, and when the order of regression is fixed, a lower level of error gives a more stable spectrum.

Precision of regression ( ) and spectrum ( ) as functions of the regression order. These measures of precision were calculated from the data in Fig. 1 with or without error. (a) Effect of error on the precision of regression. (b) Effect of error on the precision of spectrum. The insertion is the semilogarithm plot of the main plot. Although decreases monotonically as *N* increases, shows local minimum.

Precision of regression ( ) and spectrum ( ) as functions of the regression order. These measures of precision were calculated from the data in Fig. 1 with or without error. (a) Effect of error on the precision of regression. (b) Effect of error on the precision of spectrum. The insertion is the semilogarithm plot of the main plot. Although decreases monotonically as *N* increases, shows local minimum.

Application to experimental data. The modulus data were measured by ^{ Schausberger et al. (1985) } for nearly monodisperse polystyrene (PS6). (a) Regression of the whole data of loss modulus with the order of 10. (b) Regression of the whole data of loss modulus with the order of 12. (c) Comparison of spectra calculated by and . These regression orders show local minimum of for .

Application to experimental data. The modulus data were measured by ^{ Schausberger et al. (1985) } for nearly monodisperse polystyrene (PS6). (a) Regression of the whole data of loss modulus with the order of 10. (b) Regression of the whole data of loss modulus with the order of 12. (c) Comparison of spectra calculated by and . These regression orders show local minimum of for .

Application of data partitioning for the data in Fig. 7 . The partitioning was done by for the partition I, for the partition II, and for the partition III. The regression orders are 5 for the partitions I and II and 11 for the partition III. (a) Spectra from data partitioning (symbols) and regression spectrum (line). The regression spectrum was obtained using Eq. (47) for the data shown as symbols in (a). (b) Modulus data and calculated moduli from the regression spectrum [line in (a)].

Application of data partitioning for the data in Fig. 7 . The partitioning was done by for the partition I, for the partition II, and for the partition III. The regression orders are 5 for the partitions I and II and 11 for the partition III. (a) Spectra from data partitioning (symbols) and regression spectrum (line). The regression spectrum was obtained using Eq. (47) for the data shown as symbols in (a). (b) Modulus data and calculated moduli from the regression spectrum [line in (a)].

## Tables

Constants for the power series expansion.

Constants for the power series expansion.

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