Which is more informative between creep and relaxation ...

© 2017 The Korean Society of Rheology and Springer 79

Korea-Australia Rheology Journal, 29(2), 79-86 (May 2017)DOI: 10.1007/s13367-017-0010-6

www.springer.com/13367

pISSN 1226-119X eISSN 2093-7660

Which is more informative between creep and relaxation experiments?

Kwang Soo Cho*

Department of Polymer Science and Engineering, School of Applied Chemical Engineering, Kyungpook National University, Daegu 41566, Republic of Korea

(Received September 23, 2016; final revision received January 17, 2017; accepted January 18, 2017)

We present mathematical analysis which compares linear viscoelastic measurements such as creep andrelaxation. The analysis is focused on which one is more informative. Since the intervals of relaxation time(or retardation time) of most polymeric materials are much wider than the interval of observation time ofconventional rheological measurement, it is not possible to extract perfectly the whole information of mate-rial from any rheological measurement. The mathematical analysis is to manifest which experimentalmethod can obtain more information within the same interval of observation. Although the analysis ofJäckle and Richert (2008) means that the width of retardation is wider than that of relaxation, the resultsof the analysis hold for only viscoelastic solid because their analysis is based on dielectric relaxation. Ouranalysis shows that creep experiment is more informative than relaxation experiment as for viscoelasticfluid.

Keywords: relaxation modulus, creep compliance, relaxation spectrum, retardation spectrum, Laplace trans-

form

1. Introduction

Viscoelastic measurement is one of characterization

methods for structure of polymeric materials (Du et al.,

2015; Kim et al., 2016; Seo et al., 2015). However, most

users of viscoelastic measurement do not concern how

appropriate the measurement methods adopted are for

their objectives. Three representative methods for linear

viscoelastic measurement are creep, relaxation, and dynamic

tests. Linear viscoelastic theory addresses that the results

of the three measurements are equivalent and their mate-

rial functions can be convertible to others in principle

(Cho, 2016). Although the relations between the material

functions (or viscoelastic functions) are mathematically

exact, conversion of experimental data is not easy because

of finite range of measurement and errors. Hence, spe-

cially designed algorithms are necessary (Evans et al.,

2009; He et al., 2004; Kim et al., 2015; Kwon et al., 2016;

Plazek et al., 1979; Tassieri et al., 2016; Valko and Abate,

2004). These algorithms are classified to use of empirical

equation (Plazek et al., 1979); use of relaxation or retar-

dation spectra (He et al., 2004); direct conversion by use

of numerical Fourier or Laplace transforms (Evans et al.,

2009; Kim et al., 2015; Kwon et al., 2016; Tassieri et al.,

2016; Valko and Abate, 2004). When viscoelastic spec-

trum is involved, reliable algorithms of spectrum should

be selected (Bae and Cho, 2015; Baumgaertel and Winter,

1989; Cho, 2013; Cho and Park, 2013; Honerkamp and

Weese, 1993; Stadler and Bailly, 2009).

Compared with numerical analysis on data conversions,

it is hard to find a literature which concerns the compar-

ison between measurement methods from the viewpoint of

information. The few literatures are Anderssen et al. (2008)

and Jäckle and Richert (2008). Using a discrete model for

viscoelastic solid, Anderssen et al. (2008) proved that the

conversion from relaxation modulus to creep compliance

is more stable than the reverse conversion. The work of

Jäckle and Richert (2008) is remarkable because the paper

shows that relaxation test is faster than creep test. Since

their paper considers dielectric relaxation, their result is

valid for only viscoelastic solid. On the other hand, as for

viscoelastic fluid, relaxation time could be longer than

retardation time depending on relaxation time spectrum.

We show that the inequality of Jäckle and Richert (2008)

does not hold for viscoelastic fluid. The main purpose of

this paper is to show that creep test is more informative

than relaxation test at least for viscoelastic fluids.

2. Mathematical Analysis on Time Scales

2.1. Material functions and characteristic timesRelaxation modulus can be expressed by the following

equation

(1)

where and are, respectively, moduli at and

at t = 0 and φ(t) is a dimensionless function of time which

satisfies the following properties

; ; . (2)

If the modulus at infinity, is zero then we shall call

G t( ) = G∞ + Go G∞–( )φ t( )

G∞ Go t = ∞

φ 0( ) = 1 φ ∞( ) = 0dφ

dt------ 0≤

G∞*Corresponding author; E-mail: [email protected]

Kwang Soo Cho

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

the material fluid and otherwise, solid. Of course, it is

clear that .

Relaxation modulus can be expressed in terms of relax-

ation spectrum H(λ) as follows

. (3)

Then it is clear that

(4)

and

(5)

where

. (6)

Since can be regarded as the probability density

function of relaxation time λ, we define the nth moment of

relaxation time as follows:

. (7)

Application of Laplace transform to Eq. (5) gives a con-

venient way to calculate the moments of relaxation time.

The Laplace transform of φ(t) is given by

. (8)

Then the followings are immediate consequences from

Eq. (8):

,

,

, (9)

�

.

If we denote Jo and J∞ as the compliances at t = 0 and

t = ∞, respectively, then the creep compliance of visco-

elastic solid is given by

(10)

where ψ(t) is a dimensionless function which has the same

properties of φ(t), Eq. (2).

Introduction of retardation spectrum L(τ), we know the

followings

(11)

and

(12)

where . Then we can define the nth moment

of retardation time in the same way of Eqs. (7) and

(9).

If G∞

= 0, then creep compliance cannot be bounded by

J∞ < ∞. The creep compliance of fluid is given by

. (13)

Here, J∞ is no longer the compliance at t = ∞. Although it

is clear that Jo = J∞ − Jd at t = 0, we need to mention the

meaning of the three parameters J∞, Jd, and ηo.

Linear viscoelastic theory addresses that the zero-shear

viscosity ηo and steady state compliance Je can be calcu-

lated from relaxation modulus

; . (14)

Using the relation between relaxation modulus and

creep compliance, we can identify that ηo of Eq. (13) is

the zero-shear viscosity and Je = J∞. We can also recognize

the meaning of Jd from Jo = J∞ − Jd.

2.2. Viscoelastic solidWe shall use the following relation between the Laplace

transforms of creep compliance and relaxation modulus in

order to derive relation between and :

. (15)

As for solid, because of GoJo = G∞J∞

= 1, Eq. (15) gives

the following relation:

. (16)

Taking limit of on both sides of Eq. (16), we

obtain

. (17)

Equation (17) is the viscoelastic version of the work of

Jäckle and Richert (2008). Since it is usual that Go >> G∞,

mean retardation time is much longer than mean relax-

ation time. As for polymer, we can consider Go and G∞ as

the moduli of glassy and rubbery states, respectively. Then

it is usual that . This means that we have to

observe creep test thousand times longer than relaxation

test in order to get equivalent information of material. This

analysis seems to tell us that creep test could be useless in

an unlucky case.

One may consider that variance is more important than

mean. The relation between variations of retardation and

Go G∞> 0≥

G t( ) = G∞ + ∞–

∞

∫ H λ( )e t/λ–d logλ

Gd Go G∞–≡ = ∞–

∞

∫ H λ( )d logλ

φ t( ) = ∞–

∞

∫ H λ( )e t/λ–d logλ

H λ( )H λ( )Gd

------------≡

H λ( )

λn⟨ ⟩

∞–

∞

∫≡ λn

H λ( )d logλ = 0

∞

∫ λn 1–

H λ( )dλ

φ s( ) = 0

∞

∫ φ t( )e st–dt =

∞–

∞

∫H λ( )

s λ1–

+--------------d logλ

φ 0( ) = λ⟨ ⟩

φ1( )

0( ) = λ2⟨ ⟩–

φ2( )

0( ) = 2 λ3⟨ ⟩

φn( )

0( ) = 1–( )nn! λn 1+⟨ ⟩

J t( ) = J∞ J∞ Jo–( )ψ t( )–

Jd J∞ Jo– = ∞–

∞

∫ L τ( )d logτ≡

ψ t( ) = ∞–

∞

∫ L τ( )e t/λ–d logλ

L τ( ) L τ( )/Jd≡τn⟨ ⟩

J t( ) = J∞ + t

ηo

----- Jd– ψ t( )

ηo = 0

∞

∫ G t( )dt Je = 1

ηo

2-----

0

∞

∫ tG t( )dt

φ s( ) ψ s( )

sG s( )sJ s( ) = 1

Go

G∞

-------1

ψ s( )----------- =

1

φ s( )---------- +

Go G∞–

G∞

------------------s

s 0→

s 0→lim

ψ s( )

φ s( )----------- =

τ⟨ ⟩λ⟨ ⟩

-------- = Go

G∞

-------

Go/G∞ 103≈


Korea-Australia Rheology J., 29(2), 2017 81

relaxation times can be obtained by differentiation of Eq.

(16):

. (18)

Taking limit, Eq. (18) gives the ration between variances

of relaxation and retardation times:

. (19)

Equation (19) implies that the width of retardation times

is wider than that of relaxation times. This means that a

finite interval of observation time cannot include the

whole width of retardation times while the same interval

can include most effects of relaxation times.

In summary, the analysis of Jäckle and Richert (2008)

seems to support the superiority of relaxation test. How-

ever, we shall show that the conclusion is not valid for vis-

coelastic fluid.

2.3. Viscoelastic fluidIt is questionable whether the retardation time of fluid is

really longer than the relaxation time. Since viscoelastic

fluid means , relaxation modulus is given by

and creep compliance is given by Eq. (13).

Use of Eq. (15) yields the following relation between

and :

. (20)

From the definition of Laplace transform of ψ(t), we

know that exists. However, both denomi-

nator and numerator of the right hand side of Eq. (20) go

to zero as s goes to zero. To find a relation between

and , we need to use L’Hopital’s rule:

(21)

.

Here, we used differentiation two times.

To derive the ratio , we need to use Eq. (9) and

the following approximations:

; ; . (22)

Note that is usual. The last identity of Eq. (22)

is the consequence from Eq. (14) with . Exploiting

Eq. (22), we obtain

. (23)

Although the positivity of the denominator is obvious,

the positivity of the numerator can be proven by the defi-

nition of the moments:

(24)

.

However, Eq. (24) does not mean neither

nor . The inequality depends on relaxation time

spectrum. We have an example:

. (25)

Here, the exponent n is considered as a positive real. For

this spectrum, we have

. (26)

Application of Eq. (26) gives

. (27)

The right hand side is smaller than unity if ν > νo and

larger than unity if ν < νo. The value of νo is about 0.17.

Thus, the inequality of Jäckle and Richert (2008) does not

hold for viscoelastic fluid. The power-law spectrum of Eq.

(25) is called BSW spectrum (Baumgaertel et al., 1990)

which agrees well with viscoelasticity of monodisperse

polymer melts in flow regime. Examples of the exponent

are 0.22 for polystyrene and 0.42 for polybutadiene.

It is questionable whether can be an obstacle

in obtaining material information from creep test. In the

next section, we shall show examples that creep test could

be more advantageous than relaxation test even though

under the assumption that both static tests are

conducted on the same interval of observation time.

3. Test by a Toy Model

3.1. Toy ModelHere, we shall show that even if , creep test

could be more informative than relaxation test. Since

long-time observation of relaxation test is very limited by

the sensitivity of torque sensor, it is known that creep test

is better than relaxation test. Both static tests suffer from

Go

G∞

-------–ψ

1( )s( )

ψ s( )[ ]2

------------------ = φ

1( )s( )

φ s( )[ ]2

-----------------– + Go

G∞

------- − 1

Vτ

Vλ

----- = τ2⟨ ⟩ τ⟨ ⟩2–

λ2⟨ ⟩ λ⟨ ⟩2–

------------------------ = Go

G∞

------- >> 1

G∞ = 0

G t( ) = Goφ t( )φ s( )

ψ s( )

ψ s( ) = J∞ηos 1+( )Goφ s( ) ηo–

JdηoGos2φ s( )

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

ψ s 0→( ) = τ⟨ ⟩

ψ 0( )φ 0( )

s 0→lim ψ s( ) =

s 0→lim

J∞Goηoφ s( ) + Go J∞ηos 1+( )φ1( )

s( )

JdGoηo 2sφ s( ) s2φ

1( )s( )+[ ]

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

= s 0→lim

2J∞Goηoφ1( )

s( ) + Go J∞ηos 1+( )φ2( )

s( )

JdGoηo 2φ s( ) 4sφ1( )

s( )+s2φ

2( )s( )+[ ]

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

= J∞Goφ

1( )0( )

Jdηo

--------------------------- + Goφ

2( )0( )

2Jdηo2

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

τ⟨ ⟩/ λ⟨ ⟩

J∞ηo = λ2⟨ ⟩λ⟨ ⟩

---------- Jdηo≈ ηo = Go λ⟨ ⟩ J∞Go = λ2⟨ ⟩

λ⟨ ⟩2----------

Jo << J∞

Je = J∞

τ⟨ ⟩λ⟨ ⟩

--------λ⟨ ⟩ λ

3⟨ ⟩ λ2⟨ ⟩

2

–

λ⟨ ⟩2 λ2⟨ ⟩

----------------------------------≈

λ⟨ ⟩ λ3⟨ ⟩− λ

2⟨ ⟩2

= 1

2---

∞–

∞

∫ ∞–

∞

∫ xy3

x3y 2x

2y2

–+( )H x( )H y( )d log x d logy

= 1

2---

∞–

∞

∫ ∞–

∞

∫ x y–( )2xyH x( )H y( )d log x d log y

= 1

2---

0

∞

∫ 0

∞

∫ x y–( )2H x( )H y( )dx dy > 0

τ⟨ ⟩/ λ⟨ ⟩ > 1

τ⟨ ⟩/ λ⟨ ⟩ 1≤

H λ( ) = ν λ/λmax( )ν 0 λ λmax≤ ≤

0 otherwise⎩⎨⎧

λn⟨ ⟩ = ν

0

∞

∫ λn 1– λ

λmax

-----------⎝ ⎠⎛ ⎞

ν

dλ = νλmax

n

0

1

∫ tn 1– ν+

dt = νλmax

n

n ν+--------------

τ⟨ ⟩λ⟨ ⟩

--------1 ν+( )

ν 3 ν+( ) 2 ν+( )----------------------------------≈

τ⟨ ⟩/ λ⟨ ⟩ > 1

τ⟨ ⟩/ λ⟨ ⟩ > 1

τ⟨ ⟩/ λ⟨ ⟩ > 1

Kwang Soo Cho


uncertainty at short time regime because of the imperfec-

tion of input step function. To emphasize the effect of the

distribution of retardation and relaxation times, we shall

assume no measuring error and no effect of the imperfec-

tion of input function. Although creep test suffers from

inertia effect more significantly, we ignore inertia effect

for the simplicity of analysis because it can be removed by

the method developed by Kim et al. (2015). At first, both

static tests are also assumed to be conducted in the same

time interval such that

. (28)

This time interval of observation is plausible because

most experiments are conducted in such intervals.

As a toy model for fluid, we select the following relax-

ation modulus

(29)

with G1 = 1 Pa; G2 = 103 Pa; λ1 = 1 s; λ2 = 10−5 s. For the

model, creep compliance is given by

. (30)

Equation (15) allows us to express the material param-

eters of creep compliance in terms of those of relaxation

modulus:

; ;

; . (31)

In this case, we know that

; . (32)

Then the ratio of characteristic times is given by

. (33)

Figure 1 shows the relaxation modulus and creep com-

pliance of the model. The dotted lines represent the whole

behavior while the symbols represent the hypothetical

measurements. The shape of relaxation modulus gives us

an insight that short-time mode represents glassy behavior

and long-time mode fluid behavior. Although the hypo-

thetical data are generated from the toy model, assume

that we do not know the model. We shall recover the

model from the hypothetical data of creep test and show

that relaxation data cannot recover it.

3.2. Laplace transform from experimental dataKim et al. (2015) and Kwon et al. (2016) showed that

if we get Laplace transforms of creep compliance or relax-

ation modulus then other linear viscoelastic functions can

be calculated. Fulchiron et al. (1993) also showed that

Laplace transform of relaxation modulus can be converted

to relaxation spectrum by use of the Padé-Laplace method.

Shimhambhatla and Leonov (1993) also applied the Padé-

Laplace method to calculation of discrete relaxation mod-

ulus. Cho (2016) developed a method to calculate the

Laplace transform of relaxation modulus from storage

modulus data and a method to convert the Laplace trans-

form to continuous relaxation spectrum. Hence, it can be

said that if we know the Laplace transform then we can

calculate any other viscoelastic functions.

As shown in Fig. 1, the hypothetical creep data include

the two time intervals where creep compliance behaves

. It is easily understood that at very short time

regime

. (34)

In very long time regime, we know that the term of zero

shear viscosity in Eq. (13) gives rise to the terminal

behavior such that

. (35)

Ignoring Jo, it can be said that the hypothetical creep

data contains almost information of the viscoelastic fluid.

On the other hand, the hypothetical relaxation data look

like a single mode Maxwell model.

103–s tmin≈ t tmax≤ ≤ 10s≈

G t( ) = G1et/λ

1–

+ G2et/λ

2–

J t( ) = J∞ + t

ηo

----- − J∞ Jo–( )e t/τ–

ηo = G1λ1 + G2λ2 Jo = 1

G1 G2+----------------- =

1

Go

------

J∞ = G1λ1

2 + G2λ2

2

ηo

2------------------------------ τ =

Goλ1λ2

ηo

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

λ⟨ ⟩ = ηo

Go

------ τ⟨ ⟩ = Goλ1λ2

ηo

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

τ⟨ ⟩λ⟨ ⟩

-------- = Go

ηo

------⎝ ⎠⎛ ⎞

2

λ1λ2 10≈

J t( ) t∝

1 ψ t( )– t∝

J t( ) t∝

Fig. 1. (Color online) Relaxation modulus (Eq. (30)) and creep

compliance (Eq. (31)) of the toy model. The symbols represent

the hypothetical data and the dotted lines represent the whole

behavior of the model.



Laplace transforms of creep compliance and relaxation

modulus can be decomposed to three parts as follows

(36)

and

. (37)

It is easily understood that the middle terms of Eqs. (36)

and (37) can be approximated by numerical integral of

data. Since sampling time tn is given uniformly in loga-

rithmic scale, the numerical integration should be done by

the following equation:

(38a)

and

(38b)

where Gn and J

n are, respectively, values of relaxation

modulus and creep compliance measured at t = tn. It is

clear that Eq. (38) is better than the integration scheme in

linear scale of time.

As for the initial and terminal parts of , one may

use the following approximations:

(39)

and

(40)

where Gex and λex can be determined by the regression for

the long time data by use of

. (41)

As for the initial and terminal parts of , we can use

the following approximations

(42)

and

(43)

Of course, Jex and ηex can be determined by the regres-

sion for the long-time region of creep data.

Figure 2 compares the Laplace transforms obtained

from Eqs. (38a), (39), and (40) with the Laplace transform

obtained from Eqs. (38b), (42), and (43). The

solid line is the exact one obtained from Eq. (29):

. (44)

Figure 2 indicates that creep experiment is more infor-

mative than relaxation experiment.

4. Test by a More Realistic Model

Since the toy model in the previous section has no retar-

dation time distribution, it is not appropriate for the direct

comparison of the widths of retardation and relaxation

spectra. Furthermore, the toy model is an extreme simpli-

fication of linear viscoelasticity of polymer fluid. In this

section, we shall analyze linear viscoelasticity of polymer

melts with very narrow molecular weight distribution

(polydispersity index is less than 1.1).

Marin and Graessley (1977) applied modified Cole-Cole

model to polymer melts of very narrow molecular weight

distribution. The model fits experimental data very well.

The model in terms of Laplace transform of creep com-

pliance is given by

. (45)

This model gives retardation spectrum analytically as

follows:

(46)

where . Relaxation spectrum can also be obtain-

ed analytically. However, the equation is very complicate.

The relaxation spectrum can be calculated by use of the

Fuoss-Kirkwood relation (Fuoss and Kirkwood, 1941).

maxmin

min max

num

0

( ) ( ) ( )

( ) ( ) d ( ) d ( ) d

TI

tt

st st st

t t

G s G s G s

G s G t e t G t e t G t e t

∞

− − −

≡ ≡ ≡

= + +∫ ∫ ∫

�

� �

�

��

maxmin

min max

num

0

( ) ( ) ( )

( ) ( ) d ( ) d ( ) d

I T

tt

st st st

t t

J s J s J s

J s J t e t J t e t J t e t

∞

− − −

≡ ≡ ≡

= + +∫ ∫ ∫

�

� �

�

��

( )1

11

num 1 1

1

1( ) log

2n n

Nst st n

n n n n

nn

tG s t G e t G e

t

+

−

− − +

+ +

=

= +∑�

( )1

11

num 1 1

1

1( ) log

2n n

Nst st n

n n n n

nn

tJ s t J e t J e

t

+

−

− − +

+ +

=

= +∑�

G s( )

min

min1

1

0

( ) d (1 )t

t sst

I

GG s G e t e

s

−−

≈ = −∫�

ex

max

1

ex max ex

ex 1

ex

exp ( )( ) dt st

T

t

G t sG s G e e t

s

λλ

λ

−

∞

− −

−

⎡ ⎤− +⎣ ⎦≈ =+

∫�

ex

ex

log ( ) logt

G t Gλ

≈ −

J s( )

min minmin

1 1 min

2

0 min min

1( ) d

t s t st

st

I

J t J t se eJ s e t

t t s

− −

−

− −

≈ =∫�

max

max

max

ex ex

ex ex

1( ) d

t s

st

T

t

t t s eJ s J e t J

s sη η

−∞

−⎛ ⎞ ⎛ ⎞+

≈ + = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∫�

sG s( )

sJ s( )[ ]1–

1 2

1 2

1 2

( )1 1

s ssG s G G

s s

λ λ

λ λ= +

+ +

�

1 2

1 2

o

o 1 2

1( )

1 ( ) 1 ( )

J JsJ s J

s s sα α

η τ τ= + + +

+ +

�

2

21

1 sin( )

1 2 cos

k k k

kk k

JL

ζ πατ

π ζ πα ζ=

=

+ +

∑

ζk = τ/τk( )αk

Fig. 2. (Color online) Laplace transform : Solid line rep-

resents the exact one; circle and triangle represent the calculated

ones from the hypothetical data of relaxation modulus and creep

compliance, respectively.

sG s( )

Kwang Soo Cho


.

(47)

Derivation of Eq. (48) and necessary definitions are

shown in Appendix.

We determine the parameters by use of the experimental

data measured by Stadler and Ruymbecke (2010). As

shown in Fig. 3, the model describes the experimental data

very accurately. The curve fitting was done by use of

Laplace transform of creep compliance which can

be calculated from loss modulus data (Cho, 2016). Details

of curve fitting are found in Cho (2016). The values of the

parameters are

(48)

Since we know the values of the parameters, we can cal-

culate retardation and relaxation spectra by use of Eqs.

(46) and (47), respectively. Figure 4 shows the normalized

spectra: = H(λ)/Go and = L(t)/Jd. Note that here,

Go = Jo−1 and Jd = J1 + J2. As shown in Fig. 4, both relax-

ation and retardation spectra consist of two peaks. Retar-

dation spectrum looks wider than relaxation spectrum

because the two peaks of relaxation spectrum are sharper

than those of retardation spectrum. This result seems to

agree with the analysis of Jäckle and Richert (2008), even

though the spectra are those of viscoelastic fluid.

5. Discussion

We have shown that mean retardation time of viscoelas-

tic fluid might be smaller than the mean relaxation time

depending on relaxation time spectrum, which is different

from the analysis of Jäckle and Richert (2008). It is clear

that their analysis holds for viscoelastic solid because it

was proved mathematically.

Although shorter characteristic time implies that the

measurement can be completed earlier, it does not mean

that the test is more informative. The counter example was

stated in Sec. 2. The learning from the toy model implies

that the time interval representing the effect of retardation

time (or relaxation time) should be located in the time

interval of observation. As shown in Fig. 4, although

retardation spectrum looks like wider than the relaxation

spectrum, the time interval including the two peaks of the

retardation spectrum (Δlogτ) is narrower than that of

relaxation spectrum (Δlogλ). Actually, the contributions

from the outside of Δlogτ (or Δlogλ) are much smaller

than those from the interval of Δlogτ (or Δlogλ). Since

the distance between the two peaks of relaxation spectrum

is wider than that of retardation spectrum, we need longer

observation time interval (in logarithmic scale not in linear

scale) in order to extract the same amount of information

from relaxation experiment compared with creep test.

Furthermore, relaxation test is erroneous when stress is

outside of the sensitivity of torque sensor, which occurs

often in long time regime. On the other hand, creep exper-

iment is relatively free from such long time problem. As

shown in Fig. 4, Δlogτ is located in longer time regime

than Δlogλ. Different from the relaxation spectrum, the

two peaks of the retardation spectrum of Fig. 4 have

nearly same height. Considering the position of effective

time interval and the ratio of peak heights, creep test is

more informative than relaxation test because most visco-

elastic characteristics of fluid are likely to appear in the

usual observation window of creep test.

Baumgaertel and Winter (1989) considered discrete spec-

tra which model linear viscoelastic fluid as follows:

(49)

and

1 1 1 2

Re Re Im

2 1 2 1 2 2 1 2 1 2 1

Re Im Re Re Im

2 ( )[ ( ) ( )]( )

[ ( ) ( )] 4 ( )[ ( ) ( )]

V U VH

U V V U V

λ λ λλ

π λ λ λ λ λ

− − −

− − − − −

−=

− + +

sJ s( )

.7835.0,3605.0

s,10482.2s, 136.2

,Pa 10004.1,Pa 10274.7

,Pa 10162.1s,-Pa 10658.2

21

7

21

19

o

17

2

16

1

6

o

==

×==

×=×=

×=×=

−

−−−−

−−

αα

ττ

η

JJ

J

H t( ) L t( )

1

( ) k

Nt

kk

G t G eλ−

=

= ∑

Fig. 3. (Color online) Dynamic moduli of polybutadiene. The

experimental data were measured by Stadler and Ruymbeke

(2010). The graph is the Fig. 12 in Chapter 5 of Cho (2016).

Details of regression method are found in the book.

Fig. 4. (Color online) Retardation and relaxation spectra of the

modified Cole-Cole model for PBD 430 K data of Fig. 3.



(50)

They derived that

. (51)

Detailed derivation of the inequality is found in Cho

(2016). Although Eq. (51) implies ,

it does not mean

; . (52)

It is because the ratios of mean times and variances

depend on {Gk} and {Jk}. However, Eq. (51) supports the

inequality Δlogτ < Δlogλ as shown in Fig. 4.

6. Conclusions

According to the analysis on dielectric relaxation by

Jäckle and Richert (2008), relaxation test is more advan-

tageous than creep test because both mean and variance of

retardation time are larger than those of relaxation time.

Although their analysis is exact, it is applicable to only

viscoelastic solid because their analysis is for dielectric

relaxation.

We show that as for viscoelastic fluid, mean retardation

time could be smaller than mean relaxation time depend-

ing on relaxation spectrum. We also show that even if

shape of retardation spectrum in double logarithmic scale

looks like wider than that of relaxation spectrum as for

viscoelastic fluid, most viscoelastic characteristics of fluid

are likely to appear in the usual observation window of

creep test rather than in that of relaxation test.

Acknowledgement

This work was supported by the Mid-Career Researcher

Program through the National Research Foundation of

Korea (NRF) funded by the Ministry of Education, Sci-

ence and Technology (2013R1A1A2055232).

Appendix: Spectra of the Modified Cole-Cole Model

The original Cole-Cole model was invented for linear

dielectrics (Cole and Cole, 1941). Retardation spectrum of

the original Cole-Cole model is defined as

. (A.1)

Cole and Cole (1941) used the Fuoss-Kirkwood relation

(Fuoss and Kirkwood, 1941) and obtained the following

retardation spectrum:

. (A.2)

From linear viscoelastic theory, it is known that

. (A.3)

Then the modified Cole-Cole model used by Marin and

Graessley (1977) gives the following retardation spectrum

(A.4)

where

. (A.5)

Application of Eq. (A.3) to Eq. (A.1) gives dynamic

compliances as follows

(A.6)

where and . We know that complex

modulus is given by

; .

(A.7)

According to Davies and Anderssen (1997), the Fuoss-

Kirkwood relation can be expressed by

. (A.8)

For simplicity, we shall drop the limit. Application of

Eqs. (A.6) and (A.7) to Eq. (A.8) yields

(A.9)

where real functions of frequency, URe(ω), UIm(ω), VRe(ω),

and VIm(ω) are defined as follows:

.

(A.10)

A tedious calculation gives

where .

1

o

1o

1

1o

( ) (1 )

.

k

k

Nt

kk

Nt

kk

tJ t J J e

tJ J e

τ

τ

η

η

−

−

=

−

−

∞

=

= + + −

= + +

∑

∑

min 1 1 2 2 1 maxN Nλ λ τ λ τ τ λ λ

−

≡ < < < < < < ≡�

min min max maxλ τ τ λ< < <

1τ

λ<

22

221

τ τ

λ λ

−

<

−

CC

ˆ1 ( )d log

1 (i ) 1 i

L

α

τ

τ

τ ω τω

∞

−∞

=

+ +∫

CC

2

CC CC

1 ( ) sinˆ( )1 2( ) cos ( )

L

α

α α

τ τ πα

τ

π τ τ πα τ τ

=

+ +

i( ) i ( ) [ ( )]

sJ J sJ s

ωω ω

=

′ ′′− = �

2

21

1 sin( )

1 2 cos

k k k

kk k

JL

ζ πατ

π ζ πα ζ=

=

+ +

∑

k

k

k

α

τζ

τ

⎛ ⎞≡ ⎜ ⎟⎝ ⎠

2

o 21

2

21

o

1 cos( ) ;

1 2 cos

1 sin( )

1 2 cos

k k

kk

k k k

k k

kk

k k k

zJ J J

z z

zJ J

z z

θω

θ

θω

η ω θ

=

=

+′ = +

+ +

′′ = ++ +

∑

∑

θk = 1

2---παk zk = τkω( )

αk

G′ ω( ) = J′ ω( )

J′ ω( )[ ]2

J″ ω( )[ ]2

+-------------------------------------------- G″ ω( ) =

J″ ω( )

J′ ω( )[ ]2

J″ ω( )[ ]2

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

{ } { }0 0

1 2 2lim Im (i ) lim Re (i )H G Gε ε

ω ε ω ε

ω π π→ →

⎛ ⎞ ′ ′′= + +⎜ ⎟⎝ ⎠

2

Re Re Im

2 2 2 2 2 2

Re Im Re Re Im

1 2 ( )

( ) 4 ( )

V U VH

U V V U Vω π

−⎛ ⎞=⎜ ⎟

− + −⎝ ⎠

Re Im Re Im(i ) ( ) i ( ); (i ) ( ) i ( )J U U J V Vω ω ω ω ω ω′ ′′= + = +

2

Re o 21

2

Im Re21

2

Im 21

o

1 1 1 cos2( ) ,

2 1 1 2 cos

1 sin2( ) ( ),

2 1 2 cos

1 1 1 2cos2( ) cos2

2 1 2 cos 1

k k

kk

k k k k

k k k

kk k k

k k k

kk

k k k k

zU J J

z z z

J zU V

z z

J zV

z z z

θω

θ

θω ω

θ

θω θ

η ω θ

=

=

=

⎛ ⎞+= + +⎜ ⎟

+ + +⎝ ⎠

= − = −+ +

⎛ ⎞−= − + +⎜ ⎟

+ + +⎝ ⎠

∑

∑

∑

(A.11)

zk = τkω( )αk

Kwang Soo Cho


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