determination of continuous relaxation spectrum based on

© 2017 The Korean Society of Rheology and Springer 115

Korea-Australia Rheology Journal, 29(2), 115-127 (May 2017)DOI: 10.1007/s13367-017-0013-3

www.springer.com/13367

pISSN 1226-119X eISSN 2093-7660

Determination of continuous relaxation spectrum based on the Fuoss-Kirkwood relation

and logarithmic orthogonal power-series approximation

Sang Hun Lee1, Jung-Eun Bae

2 and Kwang Soo Cho

1,*1Department of Polymer Science and Engineering, School of Applied Chemical Engineering,

Kyungpook National University, Daegu 41566, Republic of Korea2Research Center, LG Household & Health Care, Daejeon 34114, Republic of Korea

(Received January 13, 2017; final revision received March 2, 2017; accepted March 27, 2017)

In this study, we suggest a new algorithm for inferring continuous spectrum from dynamic moduli data. Thealgorithm is based on the Fuoss-Kirkwood relation (Fuoss and Kirkwood, 1941) and logarithmic power-series approximation. The Fuoss-Kirkwood relation denotes the existence of the uniqueness of continuousspectrum. If we know the exact equation of dynamic moduli, then continuous spectrum can be inferreduniquely. We used the Chebyshev polynomials of the first kind to approximate dynamic moduli data in dou-ble-logarithmic scale. After the approximation, a spectrum equation can be derived by use of the complexdecomposition method and the Fuoss-Kirkwood relation. We tested our algorithm to both simulated andexperimental data of dynamic moduli and compared our result with those obtained from other methods suchas the fixed-point iteration (Cho and Park, 2013) and cubic Hermite spline (Stadler and Bailly, 2009).

Keywords: continuous relaxation spectrum, linear viscoelasticity, Chebyshev polynomials, Fuoss-Kirkwood

relation, power-series approximation

1. Introduction

In the theory of linear viscoelasticity, relaxation spec-

trum is one of the most important viscoelastic functions. It

is because most of other viscoelastic functions can be cal-

culated from relaxation spectrum and it helps us to under-

stand relaxation mechanisms of materials.

One of such efforts is to find the molecular weight scal-

ing for rheological parameters. Bae and Cho (2015) showed

that quadratic mixing rule is suitable to predict relaxation

modulus for polymer blends. They used continuous relax-

ation spectra to calculate relaxation moduli of mono-

disperse polymers and their blends. Furthermore, they

showed that continuous relaxation spectrum can be used

for the interconversion of linear viscoelastic data which

are measured by the optical measurements of Brownian

particles in the viscoelastic medium (Mason and Weitz,

1995).

However, relaxation spectrum is a hypothetical quantity

which cannot be measured directly. It should be calculated

from other measurable viscoelastic functions. Further-

more, it is a representative ill-posed problem in rheology.

It is because errors in experimental data are apt to be mag-

nified in determined relaxation spectrum (Honerkamp,

1989). It implies that relaxation spectrum has better res-

olution to errors than those of other measurable viscoelas-

tic functions. In the viewpoint of developing molecular

models, this property can be applied to verify their accu-

racy as an acid test.

There are two types of relaxation spectra: discrete and

continuous relaxation spectra (Ferry, 1980). Discrete relax-

ation spectrum is made up of finite relaxation times and

the corresponding relaxation strengths (Baumgärtel and

Winter, 1989). Continuous relaxation spectrum is the

extended version of discrete relaxation spectrum. It con-

tains the infinitesimal contributions of infinite relaxation

times.

There has been a doubt about the uniqueness of relax-

ation spectrum because of the ill-posedness. Malkin and

Masalova (2001) reported that discrete relaxation spec-

trum cannot be determined uniquely. It is because discrete

relaxation spectrum can be differed by any number of

relaxation times.

However, the physically correct meaning of relaxation

spectrum is a continuous function from vanishing times to

a certain terminal relaxation time (Stadler and Bailly,

2009). It denotes that there should be existing only one

continuous relaxation spectrum from the perspective of

physics. This was proven by Fuoss and Kirkwood (1941).

They derived the mathematical relation (the FK relation)

between continuous relaxation spectrum and loss modulus

using inverse Fourier transform and convolution theorem.

There have been a number of noteworthy algorithms

developed by researchers. Baumgärtel and Winter (1989)

developed an algorithm called “IRIS”. IRIS calculates the

“parsimonious” discrete relaxation spectrum based on

nonlinear regression. Cho (2010) suggested an algorithm

determining discrete relaxation spectrum based on the

wavelet transform. It provides a way of determining the*Corresponding author; E-mail: [email protected]

Sang Hun Lee, Jung-Eun Bae and Kwang Soo Cho

116 Korea-Australia Rheology J., 29(2), 2017

number of relaxation times as well as the arrangement of

the relaxation times.

Although it can be a convenient way to determine con-

tinuous relaxation spectrum using the FK relation, it must

be considered both the radius of convergence and branch

cut problem. It is because the FK relation is made up of

complex variable and it is hard to know the exact equation

of loss modulus (Davies and Anderssen, 1997). Instead,

most of the researchers focused on the development of

algorithms using the least-square method of continuous

relaxation spectrum itself.

Honerkamp and Weese (1993) developed an algorithm

based on Tikhonov regularization with nonlinear regres-

sion (NLRG). Although regularization method allows to

select the best fit of given experimental data and prevents

over-fitting the data, NLRG smoothens edges of deter-

mined continuous relaxation spectrum unnecessarily. Roths

et al. (2000) suggested a revised algorithm of Honerkamp

and Weese (1993) with preserving edges of continuous

relaxation spectrum. However, determined continuous

relaxation spectrum shows discontinuous slope which may

be assumed an artifact in double-logarithmic scale. Stadler

and Bailly (2009) developed an iterative algorithm based

on the cubic Hermite spline with regularization method

(CHS). Although CHS shows the remarkable result with

experimental data of PBd-430k (Stadler and van Ruym-

beke 2010), it requires several hours to maximize the pre-

cision of determined continuous relaxation spectrum. Cho

and Park (2013) developed a remarkable iterative algo-

rithm with fast computation time with a precision called

fixed-point iteration (FPI). McDougall et al. (2014) exam-

ined algorithms of CHS, NLRG, and FPI on the same data.

They showed that three algorithms give nearly identical

continuous relaxation spectra. This supports the unique-

ness of determined continuous relaxation spectra from

these algorithms indirectly.

On the other hand, some algorithms are based on the

least-square approximation of dynamic moduli. Cho (2013)

suggested an algorithm based on the power series approx-

imation of dynamic moduli data. The method converts the

coefficients of dynamic moduli to those of continuous

relaxation spectrum. As mentioned above, Bae and Cho

(2015) developed an iterative algorithm based on cubic B-

spline with Levenverg-Marquardt method (BLM). BLM

can be applicable not only to dynamic moduli but also to

the Laplace transform of relaxation modulus by changing

its kernel function.

It has been appeared recently application of the FK rela-

tion to determine continuous relaxation spectrum. Anders-

sen et al. (2015) suggested a derivative-based algorithm

using the FK relation and Gureyev iteration. They derived

ordinary differential equation from the FK relation. How-

ever, the algorithm contains multiple convolution proce-

dures, it demands a number of computation time to calculate

thousands of numerical integrations.

In this study, we suggest a new algorithm for determin-

ing continuous relaxation spectrum from dynamic moduli.

Similar with the algorithm of Anderssen et al. (2015), this

method is based on the FK relation and series approxi-

mation. However, this method is not an iterative method.

If the coefficients of dynamic moduli are calculated by

power series approximation, then the continuous relax-

ation spectrum is readily determined. It is only affected by

the precision of curve fitting result to the accuracy of

determined continuous relaxation spectrum. In other words,

the better fit of dynamic moduli, the more precise contin-

uous relaxation spectrum will be determined. This algo-

rithm was tested for simulated data and experimental data

in order to verify the capability of this algorithm.

2. Theoretical Background

2.1. Relaxation spectrumStress relaxation is measuring the stress response of

materials under the certain unit step exertion of strain.

There have been efforts to explain stress relaxation mech-

anisms of materials quantitatively by the models of spring

and dashpot (Ferry, 1980). Generalized Maxwell model is

the most representative one of such models. It consists of

a parallel connection of a finite number of Maxwell mod-

els with different relaxation times and its corresponding

relaxation strengths. Relaxation modulus of the model can

be expressed as the following equation with M relaxation

times.

(1)

where λk and hk are the relaxation time and strength of kth

Maxwell component, respectively. Eq. (1) implies that a

group of relaxation times and corresponding strengths has

a distribution of discrete form. This distribution is called

discrete relaxation spectrum.

If the number of Maxwell elements in Eq. (1) is

increased without limit, then the distribution will be a con-

tinuous function with infinitesimal contributions of each

relaxation time. Therefore, Eq. (1) can be written in the

form of integral equation as follows:

(2)

where H(λ) is continuous relaxation spectrum.

Complex modulus is defined as

(3)

where denotes the Laplace transform of relaxation

modulus, i is the imaginary unit such that , and ω is

the angular frequency.

Dynamic moduli are the real and imaginary parts of

G t( ) = k 1=

M

∑ hk expt

λk

-----–⎝ ⎠⎛ ⎞

G t( ) = ∞–

∞

∫ H λ( )expt

λ---–⎝ ⎠

⎛ ⎞d logλ

G*ω( ) = sG̃ s( )

s=iω = iωG̃ iω( )

G̃ s( )1–

Determination of continuous relaxation spectrum based on the Fuoss-Kirkwood relation and logarithmic orthogonal.....

Korea-Australia Rheology J., 29(2), 2017 117

complex modulus such that

(4)

where and are storage and loss moduli,

respectively. Storage and loss moduli can be expressed in

terms of continuous relaxation spectrum with the help of

Eqs. (2)-(4) as follows:

;

. (5)

Eqs. (2) and (5) denote that H(λ) allows us to intercon-

vert between dynamic moduli and relaxation modulus.

However, it is difficult to measure relaxation modulus

precisely due to the hardware limitation of exerting unit

step function of strain to the specimen. It is impossible to

reproduce initial part of the unit step strain by machine.

Most of researchers prefer to measure dynamic moduli

from oscillatory experiment.

2.2. The Fuoss-Kirkwood relationFuoss and Kirkwood (1941) derived the valuable rela-

tion between continuous relaxation spectrum and loss

modulus. Davies and Anderssen (1997) generalized defi-

nition of the FK relation to storage modulus as follows:

(6)

where Re[…] and Im[…] are equal to the real and imag-

inary parts of complex function, respectively.

Eq. (6) can be derived from Eq. (5) with the convolution

theorem of the Fourier transform. Eq. (6) implies the exis-

tence of the uniqueness of continuous relaxation spectrum.

It is because Fourier transform pairs are uniquely defined

for any continuous function (Boas, 2006).

However, Davies et al. (2016) suggested that the correct

expression of the FK relation has to be the following

equation.

(7)

where ε is an arbitrary real number and 0 < ε << 1.

Note that Eq. (7) includes limit process in contrast to Eq.

(6). Simple replacement of ω by iω gives sometimes use-

less results. As an example, consider loss modulus of the

Maxwell model.

. (8)

Applying Eq. (6) to Eq. (8) gives

. (9)

However, applying Eq. (7) to (8) gives

. (10)

Note that

; . (11)

Then Eq. (10) becomes

. (12)

By the definition of the Dirac delta function, λω is 1.

Then Eq. (12) can be written as

. (13)

Eq. (13) confirms that the relaxation spectrum of Max-

well model with modulus of unity is the Dirac delta func-

tion of λ = 1/ω. It satisfies the physical meaning of

relaxation spectrum. As mentioned in section 2.1, contin-

uous spectrum consists of infinitesimal contributions of

infinite relaxation times. However, simple substitution of

Eq. (6) often happens to give the same result of Eq. (7) for

some functions such as the extended Cole-Cole model

suggested by Marin and Graessley (1977). In spite of that,

we will employ Eq. (7) in developing an algorithm accord-

ing to the correct expression of the FK relation.

2.3. Ill-posedness of inferring relaxation spectrumIt is well-known that inferring relaxation spectrum from

dynamic moduli data is an ill-posed problem (Honerkamp

1989). It is because errors in experimental data are apt to

be magnified in determined relaxation spectrum. Cho

(2016) explained this problem with mathematics using the

FK relation. They showed how the error effect appears by

adding a sinusoidal perturbation to both exact functions of

relaxation spectrum and modulus. We shall show the ill-

posedness briefly in a graphic manner.

Consider that continuous relaxation spectrum is given

by

, (14)

(15)

where H0(λ) is exact spectrum and HE(λ) represents the

perturbed spectrum. In order to make a difference between

H0(λ) and HE(λ), we added 30% of statistical random

errors to H0(λ) which is represented as δH in Eq. (14).

G*ω( ) = G′ ω( ) + iG″ ω( )

G′ ω( ) G″ ω( )

G′ ω( ) = ∞–

∞

∫λ2ω

2

1 λ2ω

2+

-------------------H λ( )d logλ

G″ ω( ) = ∞–

∞

∫λω

1 λ2ω

2+

-------------------H λ( )d logλ

H1

ω-----

⎝ ⎠⎛ ⎞ =

2

π---Re G″ iω( )[ ] =

2

π---Im G′ iω( )[ ]

H1

ω----

⎝ ⎠⎛ ⎞ =

ε 0→lim

2

π---Re G″ iω ε+( )[ ] =

ε 0→lim

2

π---Im G′ iω ε+( )[ ]

G″ ω( ) = λω

1 λ2ω

2+

-------------------

Re G″ iω( )[ ] = 0

ε 0→lim

2

π---Re G″ iω ε+( )[ ] =

ε 0→lim

2

π---Re

iλω λε+

1 iλω λε+( )2+----------------------------------

= ε 0→lim

2

π---

λε

λε( )2 λω 1–( )2+--------------------------------------

⎩ ⎭⎨ ⎬⎧ ⎫ λε( )2 λω( )2 1+ +

λε( )2 λω 1+( )2+---------------------------------------

⎩ ⎭⎨ ⎬⎧ ⎫

δ x( ) = 1

π---

ε 0→lim

ε

ε2

x2

+--------------

ε 0→lim

λε( )2 λω( )2 1+ +

λε( )2 λω 1+( )2+--------------------------------------- =

λω( )2 1+

λω 1+( )2---------------------

ε 0→lim

2

π---Re G″ iω ε+( )[ ] =

2 λω( )2 2+

λω 1+( )2-------------------------δ λω 1–( )

ε 0→lim

2

π---Re G″ iω ε+( )[ ] = δ

1

ω---- λ–⎝ ⎠

⎛ ⎞

HE λ( ) = H0 λ( ) + δH

H0 λ( ) = 0.1

log10λ( )2 0.01+[ ]2

-------------------------------------------



This is shown in Fig. 1a.

Figure 1b confirms that calculated dynamic moduli from

each of continuous relaxation spectra are almost same. It

is because when we calculate dynamic moduli from HE(λ)

by the numerical integration of Eq. (5), errors in HE(λ) are

being suppressed (Atkinson and Han, 2001). It is assumed

that relaxation spectrum has a higher resolution to errors

than those of dynamic moduli.

3. Development of Algorithm

3.1. Main algorithmIn order to employ Eq. (7) for inferring continuous relax-

ation spectrum, we need an analytical equation to express

dynamic moduli. In general, dynamic moduli vary in log-

arithmic scale as frequency varies in logarithmic scale. It

is plausible to approximate dynamic moduli data in dou-

ble-logarithmic scale by polynomials. Because dynamic

moduli have shapes of simple function in double-logarith-

mic scale. Furthermore, the Weierstrass theorem reads that

any continuous function defined on a closed interval can

be uniformly approximated by polynomials (Atkinson and

Han, 2001).

Since dynamic moduli are assumed as continuous ones,

it is suitable that dynamic moduli data of a finite range of

frequency can be expressed by

(16)

where coefficients {gk} are real numbers, N is polynomial

order, and {Tk(ξ)} is the Chebyshev polynomials of the

first kind. ξ is the Chebyshev domain which is defined by

with (17)

where ν is logω, νmin and νmax are the minimum and max-

imum values of the frequency range, respectively.

Note that ξ is always in . In order to fit

dynamic moduli data by Eq. (16), frequency domain of

logarithmic scale should be changed to ξ by using Eq.

(17). Furthermore, the recursive formula allows us to cal-

culate higher order of {Tk(ξ)} by the definition as follows

(Mason and Handscomb, 2003):

.

(18)

The Chebyshev polynomials of the first kind are one of

orthogonal polynomials. Although curve fitting of the data

by simple polynomials has a unique global minimum of

the sum of square errors, N > 15 usually gives rise to a

numerical problem. If frequency range is tens of decades,

such numerical problem should be solved by using orthog-

onal polynomials. It is because orthogonal polynomials

are more stable for regression than simple polynomials

irrespective of N.

In order to derive the real and imaginary parts of Eq.

(16), we should define the complex logarithm of iω + ε as

follows:

. (19)

Applying the limit process of ε gives

. (20)

Equation (20) confirms that the substitution of logω by

log(iω + ε) with the limit process of ε is equal to the sub-

stitution of ν by ν + iπ/2. Then, the complex version of

Chebyshev domain can be expressed as

(21)

where

. (22)

f ν( ) exp k 0=

N

∑ gkTk ξ( )⎩ ⎭⎨ ⎬⎧ ⎫

≈

ξ = 2ν νmax– νmin–

νmax νmin–---------------------------------- νmin ν νmax≤ ≤

1– ξ 1≤ ≤

T0 ξ( ) = 1

T1 ξ( ) = ξ

�

Tk 1+ ξ( ) = 2ξTk ξ( ) − Tk 1– ξ( )

log iω ε+( ) = log ω2

ε2

+( ) + i arctanω

ε----

ε 0→lim log iω ε+( )[ ] = ν + i

π

2---

ξν ν→ +iπ /2

= 2 ν +iπ /2( ) νmax– νmin–

νmax νmin–------------------------------------------------------

= ξ + iφ

φ = π

νmax νmin–-----------------------

Fig. 1. (Color online) (a) Original spectrum, H0(λ) and perturbed

spectrum, HE(λ). (b) Calculated dynamic moduli from both con-

tinuous relaxation spectra of H0(λ) and HE(λ). Symbols and lines

denote ones which are related to H0(λ) and HE(λ), respectively.



It can be noticed that is equal to

from Eq. (21). If we substitute ν and ξ of Eq. (16) respec-

tively by and , then Eq. (16) becomes

. (23)

It is difficult to decompose Eq. (23) directly into the real

and imaginary parts. Because we should decompose

of each polynomial order. However, it can be

solved if we define the complex version of Eq. (18). Here,

we shall introduce some useful notations for the real and

imaginary parts of .

;

(24)

where one and two primes denote the real and imaginary

parts of , respectively. Substitution of ξ by

to Eq. (18) gives

.

(25)

Since {gk} are real numbers, the use of Euler’s formula

gives

, (26)

. (27)

Therefore, continuous relaxation spectrum from storage

modulus is

. (28)

On the other hand, continuous relaxation spectrum from

loss modulus is

. (29)

Note that the sign of H(−ν) is affected by the trigono-

metric function of the imaginary part. If the sign of trig-

onometric function is negative, then it is needed to add n

to the imaginary part in order to avoid negative signs of

H(−ν) (Zorn, 1999).

3.2. Verification of main algorithmWe shall use simulated dynamic moduli data to verify

the capability of the main algorithm. Model spectrum is

needed to find out whether the algorithm can determine

the exact spectrum. It is because we do not know the exact

continuous relaxation spectrum of experimental data.

Consider the following model spectrum which was sug-

gested by Baumgärtel and Winter (1992):

. (30)

This spectrum is known to be effective in describing

polydisperse polymer melts (Cho, 2013). We choose the

values of the parameters as H1 = 6.276 × 104 Pa, H2 =

1.27 × 105 Pa, λc = 2.481 s, λmax = 2.564 × 104 s, n1 = 0.25,

and n2 = −0.5. We shall call Eq. (30) as BSW spectrum.

We also choose the double-logarithmic distribution

model as follows:

(31)

where H3 = 1.4737 Pa, H4 = 10−3 H3, λ1 = 1.8328 × 10−2 s,

λ2 = 106λ1, and ω1 = ω2 = 1.5092 rad s–1. It has two peaks

of different heights and a local minimum valley between

them. Eq. (31) was subjected as a model spectrum by Cho

and Park (2013) because this model spectrum gives

severer test to algorithms than the model spectrum of Eq.

ξ ξ iφ+→ ν ν→ +iπ /2

ν +iπ /2 ξ + iφ

f ν iπ

2---+⎝ ⎠

⎛ ⎞ = exp k 0=

N

∑ gkTk ξ + iφ( )⎩ ⎭⎨ ⎬⎧ ⎫

Tk ξ + iφ( ){ }

Tk ξ + iφ( ){ }

Tk′ ξ( ){ } Re Tk ξ + iφ( ){ }[ ]≡ Tk″ ξ( ){ } Im Tk ξ + iφ( ){ }[ ]≡( )

Tk ξ + iφ( ){ }ξ + iφ

T0′ ξ( ) = 1; T0″ ξ( ) = 0

T1″ ξ( ) = ξ; T1″ ξ( ) = φ

Tk 1+ ′ ξ( ) = 2ξTk′ ξ( ) − 2φTk″ ξ( ) Tk 1– ′ ξ( )– ;

Tk 1+ ″ ξ( ) = 2ξTk″ ξ( ) + 2φTk′ ξ( ) Tk 1– ″ ξ( )–

Re f ν iπ

2---+⎝ ⎠

⎛ ⎞ = exp k 0=

N

∑ gkTk′ ξ( )⎩ ⎭⎨ ⎬⎧ ⎫

cos k 0=

N

∑ gkTk″ ξ( )⎩ ⎭⎨ ⎬⎧ ⎫

Im f ν iπ

2---+⎝ ⎠

⎛ ⎞ = exp k 0=

N

∑ gkTk′ ξ( )⎩ ⎭⎨ ⎬⎧ ⎫

sin k 0=

N

∑ gkTk″ ξ( )⎩ ⎭⎨ ⎬⎧ ⎫

H ν–( ) = 2

π---exp

k 0=

N

∑ gkTk′ ξ( )⎩ ⎭⎨ ⎬⎧ ⎫

sin k 0=

N

∑ gkTk″ ξ( )⎩ ⎭⎨ ⎬⎧ ⎫

H ν–( ) = 2

π---exp

k 0=

N

∑ gkTk′ ξ( )⎩ ⎭⎨ ⎬⎧ ⎫

cos k 0=

N

∑ gkTk″ ξ( )⎩ ⎭⎨ ⎬⎧ ⎫

H λ( ) = H1

λ

λc

------⎝ ⎠⎛ ⎞

n1

+ H2

λ

λc

------⎝ ⎠⎛ ⎞

n2

⎩ ⎭⎨ ⎬⎧ ⎫

expλ

λmax

-----------⎝ ⎠⎛ ⎞–

⎩ ⎭⎨ ⎬⎧ ⎫

H λ( ) = H3 exp1

ω1

------– log10

λ

λ1

------⎝ ⎠⎛ ⎞

2

⎩ ⎭⎨ ⎬⎧ ⎫

+ H4 exp1

ω2

------– log10

λ

λ2

------⎝ ⎠⎛ ⎞

2

⎩ ⎭⎨ ⎬⎧ ⎫

Fig. 2. (Color online) Regression results of dynamic moduli

data: Dynamic moduli data were calculated from (a) BSW and

(b) DP model spectra by the numerical integration. Symbols are

dynamic moduli data. Lines are regression curves.



(30). We shall call Eq. (31) as double peak spectrum (DP).

With the help of Eq. (5), dynamic moduli can be calcu-

lated from the numerical integration from both BSW and

DP spectra.

Figure 2 shows results of regression for dynamic moduli

data which is generated from both model spectra. It can be

noticed that the results of regression by Eq. (16) fit well

with their original data. Note that results of regression in

different N were omitted for both model spectra. It is

because they become nearly same when N > 13.

Figure 3 shows determined spectra from loss modulus

data of both model spectra. We chose N = 6 for BSW and

N = 13 for DP spectra. Two vertical dash-dot lines denote

the effective range of relaxation time, which is determined

mathematically from the given range of the data. It was

proven by Davies and Anderssen (1997). According to

their research, the effective range of the determined relax-

ation time should have the interval such as

(32)

where ωmin and ωmax are the minimum and maximum fre-

quencies of the modulus data, respectively. Eq. (32)

implies that the effective range of relaxation time is nar-

rower than the range of frequency of the modulus data.

We shall indicate the Davies-Anderssen interval [Eq. (32)]

as vertical lines in the figures of calculated spectrum.

Figure 3a shows that the determined spectrum is similar

to the original BSW spectrum. On the other hand, Fig. 3b

shows the severe fluctuation of the determined spectrum

for DP spectrum. It can be explained by the Runge’s phe-

nomenon which occurs in polynomial interpolation. It

seems that the fluctuation is occurring due to the use of

high order polynomials more than the data requires. Even

though the Chebyshev polynomials are more stable than

simple polynomials for regression, it does not guarantee

that the Chebyshev polynomials are independent on

Runge’s phenomenon especially for higher N. Further-

more, the loss modulus data have a sudden change of

slopes in a narrow range of frequency, which is analogous

to the local minimum valley of the spectrum. This can be

the reason that N should be larger than required for the

plausible result of regression.

In order to solve this problem, we should use lower

order polynomials. However, higher order polynomials are

needed to fit the experimental data which have a broad

range of frequency. Consequently, we should consider

some modifications of the main algorithm to satisfy these

requirements.

3.3. Point-wise calculationIn order to satisfy the requirements as mentioned in the

previous section, it can be a solution to use low order

polynomials in a partitioned range of frequency. However,

there is not an absolute benchmark to solve an ambiguity

of dividing experimental data into several partitions.

Hence, we shall introduce a modified version of the main

algorithm, which does not need to consider the ambiguity.

We shall call this algorithm as the point-wise calculation.

Consider the following definition of each partition for

experimental data as

with (33)

where {νp} are frequency points of the whole range of the

data, Δν is half of the width of each Ip, and Ndata is the

number of frequency points in the whole range of fre-

quency. We shall call the number of collected data points

in each Ip as Nsamp.

Eq. (33) implies that every frequency point of the data

becomes the center of each Ip. In other words, the number

of divided partitions is same with Ndata. Then, the dynamic

moduli data in each Ip can be expressed by

(34)

where denote the coefficients of the pth partition.

Therefore, calculated spectrum at νp from storage mod-

ulus is

eπ /2–

ωmin( ) 1–------------------ λ

eπ /2

ωmax( ) 1–-------------------≤ ≤

Ip = ν νp|νp– Δν– ν νp≤ ≤ Δν+{ } p = 1, 2, ..., Ndata

f ν νp–( ) = exp k 0=

N

∑ gk

p( )Tk ξ ξp–( )

⎩ ⎭⎨ ⎬⎧ ⎫

gk

p( ){ }

Fig. 3. (Color online) Comparison of original and determined

relaxation spectra from loss modulus data by the main algo-

rithm: (a) BSW and (b) DP model spectra. Symbols are deter-

mined spectra. Lines are original spectra. Dash-dot vertical lines

indicate the effective range of determined relaxation spectra.



.

(35)

On the other hand, calculated spectrum at νp from loss

modulus is

.

(36)

Note that every Ip has the same N and Nsamp.

Figure 4 illustrates the algorithm of the point-wise cal-

culation in a graphic manner. It is expected that the point-

wise calculation allows us to approximate dynamic mod-

uli data with lower order polynomials without loss of the

precision. As mentioned in section 1, the accuracy of

determined continuous relaxation spectrum is only

affected by the precision of regression for dynamic moduli

data.

4. Results and Discussion

4.1. Comparison of determined relaxation spectrafrom storage and loss moduli

According to Eq. (7), continuous relaxation spectrum

can be calculated either from storage and loss moduli.

Theoretically, both spectra should be the same. However,

it is usual that the both spectra are very similar but dif-

ferent (Cho, 2013).

In order to represent a difference of determined spectra

and its original model spectrum, we defined the sum of

square errors of determined spectrum as follows:

(37)

where HN(λ) and HExact(λ) denote the determined spectrum

by the point-wise calculation and its original model spec-

trum, respectively.

Figure 5 shows results of determined spectra from stor-

age and loss moduli by the point-wise calculation for both

model spectra. Figure 5a confirms that both determined

spectra from storage and loss moduli are nearly identical

to the original BSW spectrum. However, Fig. 5b shows

the deviation between determined spectra from storage

and loss moduli for DP spectrum. This also can be noticed

by the of each determined spectrum in Table 1. has

smaller value in the determined spectrum from loss mod-

ulus than that of determined spectrum from storage mod-

ulus. Although both determined spectra from storage and

loss moduli are more plausible than the result of Fig. 3b,

it is clear that the determined spectrum from loss modulus

is more acceptable than that from storage modulus.

This can be explained by the difference of shapes

between storage and loss moduli data. The shape of loss

modulus is more similar to the shape of original spectrum

H νp–( ) = 2

π---exp

k 0=

N

∑ gk

p( )Tk′ ξ ξp–( )

⎩ ⎭⎨ ⎬⎧ ⎫

sin k 0=

N

∑ gk

p( )Tk″ ξ ξp–( )

⎩ ⎭⎨ ⎬⎧ ⎫

H νp–( ) = 2

π---exp

k 0=

N

∑ gk

p( )Tk′ ξ ξp–( )

⎩ ⎭⎨ ⎬⎧ ⎫

cos k 0=

N

∑ gk

p( )Tk″ ξ ξp–( )

⎩ ⎭⎨ ⎬⎧ ⎫

χH

2 =

1

Ndata

---------- k 0=

Ndata

∑ 1HN λ( )

HExact λ( )--------------------–⎝ ⎠

⎛ ⎞2

χH

2χH

2

Fig. 4. (Color online) Schematic illustration of the point-wise

calculation which consists of (i) determination of coefficients

from the regression for the experimental data of the pth partition;

(ii) calculation of the spectrum at the pth frequency point using

Eqs. (35) or (36); (iii) repetition of the same procedure over the

whole frequency points.Fig. 5. (Color online) Comparison of determined relaxation

spectra from storage and loss moduli by the point-wise calcu-

lation: (a) BSW and (b) DP model spectra. The dash-dot vertical

lines are Eq. (32).



than that of storage modulus. Even though we calculated

continuous relaxation spectrum by the use of lower order

polynomials, this difference cannot be filled completely.

In addition, it is worthy to consider that relaxation spec-

trum has higher resolution to the errors than those of

dynamic moduli. Small regression errors are magnified in

determined spectrum. Therefore, we will use only loss

modulus data for continued sections.

4.2. Effects of calculation conditionsThere are two parameters in the algorithm of the point-

wise calculation, which are the order of polynomials, N

and the number of collected data points in the partition,

Nsamp. In order to check the effect of parameters of the

point-wise calculation, we assigned different sets of N and

Nsamp.

Figure 6 shows results of determined spectra in different

sets of N for both model spectra. We fixed Nsamp = 8 for

BSW model and Nsamp = 15 for DP model spectra. How-

ever, Fig. 7 shows results of determined spectra in differ-

ent sets of Nsamp for both model spectra. In contrast to Fig.

6, we fixed N = 8 and N = 14 for BSW and DP model

spectra, respectively.

It seems that determined spectra are not different

between results with different sets of N and Nsamp for BSW

model spectrum in both Fig. 6a and Fig. 7a. However, Fig.

6b and Fig. 7b show the same tendency that local mini-

mum valley of determined spectrum is more plausible

Table 1. of determined relaxation spectra from storage and

loss moduli data by the point-wise calculation.

BSW Model DP Model

From From

7.5869 × 10−4 1.8685 × 101

1.0271 × 10−4 8.1319 × 10−3

χH

2

χH

2χH

2

G′ ω( ) G′ ω( )G″ ω( ) G″ ω( )

Fig. 6. (Color online) Comparison of determined relaxation

spectra in different sets of N and original spectra: (a) BSW and

(b) DP model spectra. Nsamp was fixed at 8 and 15 for each model

spectrum, respectively. The dash-dot vertical lines are Eq. (32).

Fig. 7. (Color online) Comparison of determined relaxation

spectra in different sets of Nsamp and original spectra: (a) BSW

and (b) DP model spectra. N was fixed at 7th and 14th for each

model spectrum, respectively. The dash-dot vertical lines are Eq.

(32).

Table 2. of determined relaxation spectra in differentsets of N and Nsamp by the point-wise calculation.

BSW Model

N Nsamp Nsamp N

2

8

2.4846 × 10−2 14

7

8.1674 × 10−5

5 1.2034 × 10−4 11 6.8829 × 10−5

7 5.4804 × 10−5 8 5.4804 × 10−5

DP Model

N Nsamp Nsamp N

9

15

1.8475 × 10−2 23

14

2.6892 × 100

11 8.1319 × 10−3 19 1.9507 × 10−2

14 1.3008 × 10−3 15 1.3008 × 10−3

χH

2

χH

2χH

2

χH

2χH

2



with the set of N + 1 = Nsamp than the other sets for DP

model spectrum. Table 2 shows of determined spectra

in each set of N and Nsamp. of N + 1 = Nsamp is the most

smallest value in the Table 2. This implies that the

N + 1 = Nsamp condition is the best condition for describing

the loss modulus data of DP model spectrum. It is because

dynamic moduli data do not contain errors.

Although we included the determined spectrum with

N + 1 = Nsamp condition for comparison of other deter-

mined spectra with different conditions, the result is not

meaningful for the real application of the point-wise cal-

culation due to experimental errors. It is because the con-

dition of N + 1 = Nsamp cannot exclude the experimental

error in the data by the regression.

4.3. Error effectIn the previous sections, we dealt with simulated data

which are free from any errors. However, real experimen-

tal data have errors which are originated from various rea-

sons. In order to mimic experimental data, we added 5%

of statistical random errors to the simulated dynamic mod-

uli data from the BSW model spectrum. We generated 100

sets of error-contaminated data in order to investigate

error effect on spectrum calculation.

Figure 8 shows the average of the spectra calculated

from the 100 sets of error-contaminated data. The vertical

lines behind individual symbols represent the range of the

spectrum values at the corresponding relaxation times.

The error-contaminated data consist of 36 data points and

Nsamp is fixed by 22. Figure 8a is the result from N = 8 and

Fig. 8b from N = 5. Because of the error in data, lower

order of polynomial seems to provide better result than

higher order. As shown in Fig. 8, the vertical lines of N =

5 is shorter than those of N = 8. The errors in raw data

result in rapid variation around the exact values of data,

which is called waviness. Higher order polynomial is sen-

sitive to describe the rapidly varying data while lower

order polynomial gives slowly varying curve. This

explains why higher order polynomial is inferior to lower

order polynomial in fitting dynamic data with errors

although higher order polynomial is better than lower

order polynomial in fitting dynamic data without errors.

On the other hand, Fig. 9 compares the effect of Nsamp

with fixing the order of polynomial, N = 5. The lengths of

the vertical lines are reduced by the increase of Nsamp. The

effect of Nsamp can be explained by the fact that the more

data gives the better regression result.

Table 3 summarizes the results of both Fig. 8 and Fig. 9

in terms of . Note that is calculated from the aver-

aged spectrum. The evaluation by gives the same with

that by the length of vertical lines. The same effects of N

and Nsamp are found for the error-contaminated data from

the DP spectrum. Because of duplication, we omitted the

result from the DP spectrum.

χH

2

χH

2

χH

2χH

2

χH

2

Fig. 8. (Color online) Comparison of determined BSW spectra

for different sets of N with 5% of statistical random errors: (a) N

= 8th and (b) 5th. Nsamp was fixed at 22. The dash-dot vertical lines

are Eq. (32).

Fig. 9. (Color online) Comparison of determined BSW spectra

for different sets of Nsamp with 5% of statistical random errors for

BSW model spectrum: (a) Nsamp = 14 and (b) 22. N was fixed at

5th. The dash-dot vertical lines are Eq. (32).



We found the optimized N and Nsamp for the 5% error-

contaminated data from both the BSW and DP spectra.

Figure 10 shows the optimized results from the two model

spectra, which are calculated by the point-wise calculation

as before. Figure 10a shows the BSW spectrum can be

inferred overcoming the effect of errors in the raw data.

On the other hand, Fig. 10b shows that the error bars in

spectrum (vertical lines behind the symbols) are signifi-

cantly high at the both ends of the boundary of relaxation

time as well as in the valley. Furthermore, the average

spectrum deviates significantly from the original DP spec-

trum at the same regimes of relaxation time. Hence, Fig.

10 implies that the DP model spectrum is a more severe

test than the BSW model spectrum. Furthermore, 5% error

is much higher than the error levels expected for usual

experimental data (Bae and Cho, 2015). It is interesting

that the BSW spectrum is more similar to inferred spectra

from usual experimental data of polymers than the DP

spectrum (Baumgärtel and Winter, 1992; Stadler and van

Ruymbeke, 2010). Since the DP spectrum is a severe test,

most researchers test their algorithms with the DP spec-

trum (Bae and Cho, 2015; Cho and Park, 2013; Honer-

kamp and Weese, 1993; Stadler and Bailly, 2009).

4.4. Analysis on truncation errorIn the previous section, we investigated the effect of

experimental error by using the simulated data which sta-

tistical errors are added. However, since our algorithm is

based on polynomial approximation, the spectrum calcu-

lation contains inevitably truncation error. Therefore, it is

demanded to analyze the truncation errors in order to clar-

ify the errors in the spectrum calculated from the point-

wise calculation. The error analysis will be made by use

of the remainder theorem.

We can define that the loss modulus data can be expressed

exactly with the infinite series of Eq. (34) as follows:

. (38)

However, since we use finite order of polynomial, the

remainder theorem of Taylor expansion gives

(39)

where

, (40)

; . (41)

PN(ξ) and RN(ξ) denote the truncated series with a finite

order of N and the remainder term of series expansion,

respectively. Therefore, the exact equation of continuous

relaxation spectrum becomes

(42)

where

; , (43)

; . (44)

Note that , , , and are the real functions

of . Application of the addition formula of trigono-

metric function to Eq. (42) gives

(45)

where

. (46)

denotes the determined spectrum by the trun-

cated series. It is convenient for the analysis of the effect

G″ ν νp–( ) = exp k 0=

∞

∑ gkTk ξ ξp–( )⎩ ⎭⎨ ⎬⎧ ⎫

G″ ν νp–( ) = GN″ ν νp–( )exp RN ξ ξp–( ){ }

GN″ ν νp–( ) = exp PN ξ ξp–( ){ }

PN ξ( ) = k 0=

N

∑ gkTk ξ( ) RN ξ( ) = logG″ ν( )GN″ ν( )-----------------

⎩ ⎭⎨ ⎬⎧ ⎫

H νp–( ) = 2

π---exp PN′ RN′+( )cos PN″ RN″+( )

PN′ = Re PN ξ iφ+( )[ ] PN″ = Im PN ξ iφ+( )[ ]

RN′ = Re RN ξ iφ+( )[ ] RN″ = Im RN ξ iφ+( )[ ]

PN′ PN″ RN′ RN″ξ ξp–

H νp–( ) = HN νp–( )exp RN′( ) cosRN″ sin RN″ tan PN″–( )

HN νp–( ) = 2

π---exp PN′( )cos PN″

HN νp–( )

Table 3. of determined relaxation spectra in differentsets of N and Nsamp with 5% of statistical random errors bythe point-wise calculation.

BSW Model

N Nsamp Nsamp N

822

8.3301 × 10−1 145

1.9126 × 10−2

5 1.5848 × 10−2

22 1.5848 × 10−2

χH

2

χH

2χH

2

Fig. 10. (Color online) Optimized results of determined spectra

with 5% of statistical random errors for both model spectra: (a)

BSW and (b) DP model spectra. The dash-dot vertical lines are

Eq. (32).



of truncation error to define the relative difference

between and as

. (47)

With the help of Eqs. (45) and (46), Eq. (47) can be

rewritten by

. (48)

If is a good approximation then the truncation

error is bounded by a small positive number, say

δ << 1: for all n. Then the following inequality

is the immediate consequence:

; . (49)

Then, Eq. (48) becomes

. (50)

Consider the modulus data as the one contaminated

errors which can be expressed by

(51)

where εv is the relative error at the frequency of ν = logω.

Then Eq. (39) can be modified as follows

. (52)

Note that can be regarded as the experimental data

of loss modulus (or error-contaminated simulated data).

Since , the error term of Eq. (52) can be expressed

by

(53)

where

. (54)

If we extend the meaning of the upper bound, δ as the

following:

. (55)

We can use Eq. (50) as the estimator of the total error.

Then we can estimate δ at each frequency as follows

. (56)

Figure 11 shows the and for the inferred

spectra of Fig. 10 which contain statistical errors as well

as truncation ones. To compare and eas-

ily, they are normalized by their mean values. Although all

the points of and are not superposed,

overall shapes are similar to each other. Figure 11 indi-

cates again that the DP model spectrum is a more severe

test than the BSW model spectrum. It is observed that

lower consistency between and is found

in Fig. 11b than in Fig. 11a. The error analysis confirms

that the accuracy of approximating loss modulus data is

the most important factor for the determination of accurate

spectrum.

4.5. Application to experimental dataIn order to test our algorithm, we used loss modulus data

of poly(1,3-butadiene) 430k (PBd 430 K) which were

measured by Stadler and van Ruymbeke (2010). The data

have a wide frequency range of 14 decades. The Mw of

polybutadiene is about 450,000 g/mol. The data is shown

H νp–( ) HN νp–( )

δH νp–( ) = 1HN νp–( )H νp–( )-------------------–

δH νp–( ) = 11

exp RN′( ) cosRN″ sin RN″ tan PN″–( )------------------------------------------------------------------------------------–

PN ν( )RN ν( )

RN ν( ) <δ

RN′ δ< RN″ δ<

δH νp–( ) 11

eδ

cosδ sinδ tanPN″–( )----------------------------------------------------–≈ ΦH νp–( )≡

Gε″ ν( ) = G″ ν( ) 1 εv+( )

Gε″ ν( ) = GN″ ν( )exp RN ξ( ){ } 1 εv+( )

Gε″

εv <1<

1 εv+( )exp RN ξ( ){ } = exp RN ξ( ){ }

RN ξ( ) RN ξ( ) + log 1 εv+( )≡

RN ξ( ) + log 1 εv+( ) <δ <<1

δ logGN″ ν( )Gε″ ν( )-----------------

⎩ ⎭⎨ ⎬⎧ ⎫

≈

ΦH νp–( ) δH νp–( )




Fig. 11. (Color online) Comparison of δH(−νp) and ΦH(−νp) for

the results of Fig. 10: (a) BWS and (b) DP model spectra. Values

on all vertical axis are normalized by its mean values.

Fig. 12. (Color online) Loss modulus data of PBd 430k (Stadler

and van Ruymbeke 2010). Symbol and line indicate the original

data and the result of regression, respectively.



with the regression result in Fig. 12.

Since we do not know the exact spectrum of real exper-

imental data, we compared our result with those obtained

from other algorithms such as fixed-point iteration method

by Cho and Park (2013) and cubic Hermite spline method

by Stadler and Bailly (2009). We chose Nsamp = 130 and N

= 6 for the calculation. The comparison is shown in Fig.

13. It confirms that our algorithm gives a spectrum nearly

identical to those from the fixed-point iteration and cubic

Hermite spline methods within the effective range. Hence,

our algorithm is at least equivalent to the previous algo-

rithms. Although the two previous algorithms are mainly

based on least-square method, our algorithm is based on

mathematically exact equation, the FK relation. Hence, it

can be said that our method is superior to the previous

algorithms from the mathematical viewpoint.

5. Conclusions

We suggested a new algorithm called the point-wise cal-

culation for inferring continuous relaxation spectrum from

dynamic moduli data. The algorithm of the point-wise cal-

culation is based on the analytical equation of double log-

arithmic orthogonal polynomials and the FK relation.

We confirmed the algorithm of the point-wise calcula-

tion works well with both simulated and experimental

data. The success of the point-wise calculation mainly

originates from the use of Eq. (16) which is flexible in

curve fitting of loss modulus data with the precision.

We analyzed the error of determined spectrum using the

remainder theorem. The error analysis denotes that the

regression accuracy of loss modulus data is the most

important factor for determining continuous relaxation

spectrum in the algorithm of the point-wise calculation.

Acknowledgment

This work (2013R1A1A2055232) was supported by

Mid-Career Researcher Program through NRF grant funded

by the MEST.

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determination of continuous relaxation spectrum based on

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