an empirical model for modified bituminous binder master curves

1 23

Materials and Structures ISSN 1359-5997 Mater StructDOI 10.1617/s11527-012-9988-x

An empirical model for modifiedbituminous binder master curves

Seyed Mohammad Asgharzadeh, NaderTabatabaee, Koorosh Naderi & ManfredPartl

1 23

Your article is protected by copyright and all

rights are held exclusively by RILEM. This e-

offprint is for personal use only and shall not

be self-archived in electronic repositories.

If you wish to self-archive your work, please

use the accepted author’s version for posting

to your own website or your institution’s

repository. You may further deposit the

accepted author’s version on a funder’s

repository at a funder’s request, provided it is

not made publicly available until 12 months

after publication.

ORIGINAL ARTICLE

An empirical model for modified bituminous binder mastercurves

Seyed Mohammad Asgharzadeh •

Nader Tabatabaee • Koorosh Naderi •

Manfred Partl

Received: 17 July 2012 / Accepted: 19 November 2012

� RILEM 2012

Abstract Modeling the mechanical behavior of

asphalt binders and mixtures has been the subject of

intensified research in recent decades. Master curves

of the norm of the complex modulus |G*| in the linear

viscoelastic domain are frequently used for modeling,

while phase angle master curves are less frequently

considered for this purpose. Therefore in this research,

an empirical model is introduced for phase angle

master curves of modified and neat bituminous

binders. The model is based on a general form of a

double-logistic (DL) mathematical function. The |G*|

master curve was then modeled using a mutual

relationship between the phase angle and |G*|. Master

curves of three neat and seven modified binders were

generated and used to validate the DL model. The

results showed that the model is capable of properly

predicting the plateau region of phase angle master

curves. In particular, the asymptotic behavior of the

master curves at high frequencies can be modeled

correctly. The model also describes irregularities in

the high temperature range of the phase angle master

curve. In general, model outputs such as the phase

angle value at the plateau, glassy modulus, rheological

index and crossover frequency correctly predict the

behavior of the neat and modified binders.

Keywords Master curve � Modified binders �Phase angle � Modeling � Double logistic � Plateau

1 Introduction

The master curves of neat and modified binders have

been widely generated and interpreted over the years

to characterize the rheological properties of bitumi-

nous binders. In a master curve of dynamic mechan-

ical data, any viscoelastic function, such as the

complex shear modulus |G*| or the phase angle, are

plotted against frequency on a log–log (or semi-log)

scale [4]. Using the master curve of dynamic mechan-

ical data, the three-dimensional characterization of

modulus, time and temperature can be reduced to a

two-dimensional format (modulus, reduced fre-

quency) using the time–temperature superposition

principle [17].

Quantifying the linear viscoelastic rheological

properties of bituminous binders can be performed

by finding predictive models or equations for the

master curves of any rheological parameter in terms of

reduced frequency. Modeling the master curves is

useful because testing is generally laborious, time

consuming, expensive and requires skilled operators

S. M. Asgharzadeh (&) � N. Tabatabaee

Sharif University of Technology, Tehran, Iran

e-mail: [email protected]

K. Naderi

Amirkabir University of Technology, Tehran, Iran

M. Partl

EMPA Swiss Federal Laboratories for Materials Science

and Technology, Duebendorf, Switzerland

Materials and Structures

DOI 10.1617/s11527-012-9988-x

Author's personal copy

[27]. Also, modeling rheological master curves usually

provides a fundamental understanding of the materials

since it allows an estimation of mechanical properties

for a wide range of temperature and loading times or

frequencies that may be experienced in the field but is

not practical to directly simulate in the laboratory [28].

Mathematical models allow such calculations to be

made directly and quickly in a repeatable fashion.

Furthermore, prediction of the stress–strain response of

the binder to model pavement response or predict

pavement performance can only be done realistically

with the use of reasonably accurate mathematical

models of a binder’s viscoelastic behavior [4].

Modeling the mechanical behavior of asphalt

binders and mixtures has been studied since the

1950s. Master curves of binder stiffness modulus were

first introduced by Van der Poel [25]; however, that

modeling was not done mathematically. Improve-

ments in dynamic testing techniques led to the

construction of master curves for dynamic parameters

such as the norm of complex modulus |G*| or the loss

and storage moduli [10, 11, 15]. Modeling and the use

of phase angle master curves have received less

attention, since their construction, interpretation and

application are not as straight forward.

Unlike |G*| master curves, the phase angle d master

curves of polymer modified binders (PMB) substan-

tially differ from those of neat binders. The formation

of entanglements in the bitumen-polymer network in

elastomer modified binders and the presence of a rigid,

tough network in plastomer modified binders lead to

the formation of a plateau zone in phase angle master

curves [1, 2]. The mathematical models available for

neat binders are not capable of describing this

behavior and there is a need to correctly model the

plateau zone.

The mechanical model of the generalized Maxwell

model [6, 12] and the fractional model [23] can predict

the plateau zone and other irregularities in phase angle

master curves, but both models require a compara-

tively large number of parameters than mathematical

models. In these models, the required number of

parameters varies in an arbitrary manner and is

determined for each case by selecting one of the

approaches available for this purpose. The variable

number of parameters and the difficulty of determin-

ing them can affect the uniqueness of the results.

The 2S2P1D model [20] with seven parameters

(including static and glassy modulus) can partially

predict irregularities such as a plateau zone in the PMB

and its corresponding asphalt mixture. One shortcom-

ing of this model is that, for some binders, a third

parabolic parameter must be added to get a good fit at

high temperature ranges. This may increase the

complexity of the model and its parameter estimation

method [20]. A visual inspection of the master curves

may also be needed for parameter estimation, making

the results user-dependent.

Mathematical equations like the Christensen–

Anderson (CA) model [9] developed in the Strategic

Highway Research Program (SHRP) [4] and Chris-

tensen–Anderson–Marasteanu (CAM) model (Maras-

teanu and Anderson [16] cannot predict plateaus or

irregularities in phase angle master curves because

they essentially assume an S-shaped curve. The

equations in the modified CAM model presented by

Zeng et al. [28], however, can be used to fit a simple

form of phase angle master curves for a PMB at low

frequencies. The sigmoidal (or standard logistic)

model presented in the mechanistic empirical pave-

ment design guide (MEPDG) [19] and the generalized

logistic sigmoidal model presented by Rowe et al. [21,

22]) also do not present explicit models for phase

angle diagrams. Yusoff et al. [26, 27] have presented a

thorough discussion of available models for asphalt

master curves and shifting techniques.

Most available models for the mechanical behavior

of bitumen are based on |G*| master curves. The |G*|

master curves follow a similar S-shaped trend for

different binders. However, phase angle master curves

assume various shapes and trends for different mod-

ified binders that can be used to interpret binder

characteristics. Given the lack of an appropriate

mathematical model for the phase angle master curves

of PMBs, it is the objective of this research to focus on

the development of a mathematical model for phase

angle master curves using parameters that have

physical significance. It is hypothesized that a math-

ematical model based on the general shape of a

double-logistic function can be developed to describe

the phase angle master curves of neat and modified

bitumen in its general form. The model will be capable

of predicting the plateau region and other irregularities

in PMB phase angle master curves that cannot be

thoroughly modeled by other available models. This

model can then be used to develop a |G*| model for

that binder as well. In this study, several types of neat

and modified binders are used to validate the newly



introduced model, through its ability to fit various

patterns of phase angle master curves.

2 Modeling approach

The complex modulus G* = |G*|eid comprises an

absolute value |G*| and a phase lag d (phase angle).

The phase angle master curves of most unmodified

binders follow a simple S-shaped diagram in the range

of 0�–90� (Fig. 1a) when tested over a wide range of

temperatures and plotted on a semi-logarithmic scale.

The main scheme for the phase angle master curves of

a large number of PMBs is depicted in Fig. 1b. This

figure represents a group of double logistics (DL)

curves. No algebraic equation has been presented for

asphaltic materials that directly follow such a scheme.

The DL function is described as:

fDL ¼ sgn x� dð Þ � e�x�d

sð Þ2

� �� 1

� �ð1aÞ

where the first term represents a signum function

defined as

sgn x� dð Þ ¼�1 if x\d0 if x ¼ dþ1 if x [ d

8<

:

9=

;ð1bÞ

and the second term is a simplified bell-shaped

Gaussian function [13] with two constants (d and s).

The signum operator converts the bell-shaped Gauss-

ian function to a DL form as shown in Fig. 1c by

flipping the bell-shaped function at its peak point. The

peak point will then form the plateau region in the DL

function.

A double logistic diagram consists of two identical

logistic-shaped curves joined together at point x = d.

It should be noted that, the algebraic equation of the

DL function is entirely different from the logistic

sigmoid function [19, 21, 22] commonly used in

asphalt technology.

In order to use the mathematical DL function for

phase angle master curve data from a modified binder,

appropriate constants must be inserted into Eq. (1a) to

change it to the new function in Eq. (2)

d ¼ dP þ dS � sgn(fred � fPÞ � e�log fred�log fP

Sð Þ2� �

� 1

� �;

ð2Þ

where the phase angle d (in degrees) has been modeled

using four parameters; dP, fP, dS, and S as a function of

the reduced frequency fred. Equation (2) defines a

symmetrical function over point fred = fP (the fre-

quency at the plateau) where the value of the phase

angle is dP (plateau phase angle) as shown in Fig. 1c.

The phase angle master curve rises or falls by the

amount of dS toward two horizontal asymptotes where

the reduced frequency decreases or increases toward

infinity. These values equal d = dP - dS on the right

side and d = dP ? dS on the left side (Fig. 1c). S also

shows how rapidly the rise and fall occurs on both

sides.

Equation (2) only presents a symmetrical form of

the phase angle master curve where the plateau zone

always lies in the middle of the range and the rise and

fall are equal on both sides; however, the general form

of a phase angle master curves is not symmetrical. In

order to define a more general equation, different

coefficients for the left and right side of the plateau

should be applied. However it is not possible to apply

different coefficients at the left and right side of the

plateau when using the signum function, since the

signum function yields equal values at both sides.

Thus, the signum function should be replaced by two

Heaviside step functions as defined in Eq. (3a):

HðuÞ ¼ 0 if u\0

1 if u� 0

� �: ð3aÞ

The replacement is

sgn(x� dÞ ¼ Hðx� dÞ � Hðd � xÞ ð3bÞ

or in the terms of a phase angle master curve, it is

dS � sgn(fred� fPÞ ¼ dS �Hðfred� fPÞ � dS �HðfP� fredÞð3cÞ

which with different coefficients on the left and right

sides of fP, it becomes

dS � sgn(fred� fPÞ! dR �Hðfred� fPÞ�dL �HðfP� fredÞ;ð3dÞ

where dR stands for the rise (or fall) on the right side

and dL for the left side of plateau dP. It is then

suggested to preset the value of dR to dP to ensure that

the phase angle always approaches zero at high

frequencies. Similarly to dS, S can vary in the left

and right sides of the plateau (SL and SR) to accom-

modate non-symmetry in the shape of the master



curve. These can also change from denominator to

numerator for simplicity without affecting the con-

cept. All these features can be incorporated into Eq. (2)

to get a final form for the DL model that estimates the

phase angle of unmodified and modified binders as

d ¼ dP � dP �Hðf red � f PÞ � 1� e� SR�log

fredfP

� � 20

@

1

A

þ dL �Hðf P � f redÞ � 1� e� SL�log

fPfred

� � 20

@

1

A;

ð4aÞ0\dP� 90; ð4bÞ0\dP þ dL� 90; ð4cÞ0\SL; SR; ð4dÞ

where d (in degrees) is modeled as a function of fred

using the previously defined parameters dP, dL, fP, SL

and SR. The Heaviside step functions are written in

bold to make the equation more clear. For fred values

larger than fP, the first Heaviside step function is

activated to reduce the phase angle, whereas for fred

values less than fP, the second Heaviside step function

comes into play. Figure 2 shows the scheme of this

model. Parameters dP, fP and SR are referred to as

‘‘base parameters’’ since they generate the general

form of the function at the right side of the plateau.

Equation (4b) keeps the value of dP below 90� to

prevent the model from incorrectly predicting the

phase angle. The parameter dP can be used to

distinguish between different PMBs because it refers

to the phase angle plateau value of the master curves.

The parameter fP determines the horizontal displace-

ment of the master curve over the frequency axis. Its

value depends on the selected reference temperature

for the master curve. Parameter SR indicates how

sharply the master curve approaches zero at high

frequencies. Smaller SR values cause the master curve

to approach the phase angle of zero in a smoother way,

as shown in Fig. 2. This may happen for a PMB with

lower temperature susceptibility at medium and low

temperature ranges.

Parameters dL and SL accommodate the changes in

the shape of the phase angle master curve at the left

side of the plateau compared to the right side. The

horizontal asymptote of the master curve at low

frequencies can be determined by d = dP ? dL. If dL

is positive, the expression yields the maximum value

of the phase angle that should be less than 90� to

produce the correct phase angle values. Equation (4c)

in the model defines this condition. It should be noted

that the model does not necessarily lead to an

asymptotic value of 90� for the phase angle at low

Fig. 1 a S-shaped diagram (common in unmodified binders phase angle master curves), b plateau zone (common in PMB phase angle

master curves), c converting bell-shape function to DL form



frequencies. This is desirable when modeling modified

binders because some PMBs may not show entirely

viscous behavior at high temperatures under testing

conditions. Negative values of dL also make the maxi-

mum value of the model equal to dP at the peak, which is

less than 90� based on Eq. (4b). A larger absolute value of

dL (positive or negative) yield greater rise or fall in the

master curve at the left side of the plateau. Parameter SL

also determines how fast the phase angle approaches its

asymptotic value at low frequencies.

For unmodified binders, the phase angle usually

approaches 90� at high temperatures (low frequencies)

without an intermediate plateau region. The whole

range of phase angle values from 0� to 90� can be

modeled using the base parameters of Eq. (4a). This

happens when dP approaches 90�. In this case, dL and

SL are not needed and it is recommended to preset dL to

zero and SL to SR.

The model in Eq. (4a) estimates the phase angle

master curve of unmodified and modified binders.

Furthermore, the same parameters in Eq. (4) are used

to generate a model for a |G*| master curve. To do this,

the Kramers–Kronig equation as modified by Booij

and Thoone [7] was used as follows:

d xð Þ ffi 90� d log G� xð Þð Þd log x

ð5aÞ

or after integration as

log G� xð Þð Þ ffi 1

90

Zd xð Þd log xþ C ð5bÞ

Equation (5a) relates the phase angle and |G*|. This

has been investigated by Dickinson and Witt [10] and

validated for a wide variety of materials by Ferry [12].

Chailleux et al. [8] also used the above relationship to

produce a new method for master curve shifting.

By integrating the phase angle function of Eq. (4a)

in terms of fred instead of x, a direct function for the

|G*| master curve is obtained:

log G�j j ¼ log G�0 � dP

90�H f red ¼ f p

� �

log fredð Þ � p1=2

2 � SR

erf SR logfred

fP

� ��

� logðfPÞ �dP þ dL

dP

�þ dL

90�Hðf p � f redÞ

�log fredð Þ þ p1=2

2 � SL

erf SL logfP

fred

� ��

þ dP

90log fred;

ð6Þ

where erf(x) (error function) is defined as

erfðxÞ ¼ 2

Zx

0

e�t2

dt

0

@

1

A=p1=2 ð7Þ

The parameters in Eq. (6) are the same as those in

Eq. (4a) and the conditions in Eq. (4b) and (4c) hold.

Also, as described for Eq. (4a), only one Heaviside

step function is called in depending on the value of fred.

The parameter G0* is an integration constant that

accommodates the required vertical shift for the |G*|

curve.

It is possible to calculate rheological parameters for

the model to evaluate the binder characteristics. These

parameters include the glassy modulus |Gg*|, crossover

frequency and rheological index (R) as defined by

Anderson et al. [3]. The glassy modulus indicates the

hardness of the binder at extremely low temperatures.

The lower the glassy modulus, the softer the binder,

and an improved cracking behavior of the binder at

low temperatures is expected. R is directly propor-

tional to the width of the relaxation spectrum. When

R increases, the master curve flattens and the binder

becomes less temperature susceptible [3, 9]. The

frequency at which R is calculated is called the

crossover frequency (fC) and can be regarded as a

hardness parameter [9]. It frequently correlates with

the gel point [18], but its value changes with the

reference temperature. A further description of these

parameters and the formulation needed to calculate

them from the DL model is explained in Appendix A.

The DL model presented above can also be used to

model asphalt mixture master curves. However, the

Fig. 2 DL model parameters for two PMBs phase angle master

curves



data used for model evaluation in this research is

limited to asphalt binders. An explanation of how to

apply this model to asphalt mixture master curves is

presented in Appendix B.

3 Model validation

Combining Eqs. (4) and (6) yields a new model for the

master curves of the phase angle and |G*| for neat and

modified binders. Note that, despite the complex

appearance of the equations, fitting these functions to

master curve data simply yields the parameters of the

model. Three neat and seven modified binders are used

to generate master curves using the DL model in this

research. The neat binders consist of two Pen.85/100

bitumen samples from different sources (Neat-1 and

Neat-3) and an aged Pen.40/50 bitumen (Neat-2). The

modified binders comprise one commercial (Styrelf

13-80) and six laboratory-made unaged PMBs. These

PMBs were produced using the Neat-1 binder with

different types and amounts of modifiers in the lab.

These modifiers and their concentrations, by the

weight of the neat bitumen, are polyphosphoric acid

(PPA) at 0.5 and 2.5 %, crumb rubber (CR) at 9 and

18 %, styrene–butadiene–styrene (SBS) at 7 %, and

ethylene vinyl acetate (EVA) at 6 %. Table 1 shows

the properties of these modified binders. The detailed

description of the preparation of the PMBs was

presented elsewhere [24].

The data required to generate master curves was

gathered using a dynamic shear rheometer (DSR)

MCR301 Anton Paar at several temperatures from

-30 to 88 �C, depending on the binders. For the EVA

modified binder, high temperature testing was lim-

ited to 64 �C because the melting of different

crystalline structures [1] in the EVA polymer causes

discontinuities in the master curve, making the

time–temperature superposition principle (TTSP)

inapplicable [12]. Frequency sweep data in the

frequency range of 0.1–10 Hz in a strain-controlled

mode was used. Strain amplitudes of 0.1 % or lower

was used at low temperatures to assure linear

viscoelastic response.

Shifting the isotherms to generate the master curves

was done for G0 and G00 using US200/32 V2.3

commercial software. The reference temperature of

40 �C was chosen. Then, the nonlinear least square

method was used to estimate the model parameters

using Microsoft Excel 2010 solver tools. Model fitting

can be applied to any viscoelastic parameter, including

|G*|, phase angle d, G0 or G00 or a combination of them.

By fitting the d master curve, dP, dL, fP, SL and SR can

be estimated and the best fit of |G*| then used to

determine |G0*,| as done by other researchers for other

models [23]. However, the approach used in this

research is the simultaneous fitting of the models over

G0 and G00 [5, 14] where the target is to minimize the

summation of the square of relative errors (SSRE) as

shown in Eq. (8):

Table 1 Physical properties of binders

Continuous PG

ASTM D7643

Pen. @ 25 �C

ASTM D5

TR&B

ASTM D36

PIa Stiffness @ -30 �C,

60 s, unaged

ASTM D6648

m-value @ -30 �C,

60 s, unaged

ASTM D6648

Binder (�C) (dmm) (�C) – (MPa) –

Styrelf 13–80 – 65 58 1.31 – –

’SBS7 75.8–22.3 54 82 4.74 516 0.186

EVA6 74.9–20.5 46 71 2.78 746 0.190

PPA2.5 91.9–23.8 40 79 3.61 729 0.214

CR18 75.0–31.6 60 56 0.66 451 0.209

CR9 66.8–27.7 66 52 -0.14 603 0.208

Neat-1 (85/100) 60.7–26.2 86 48 -0.35 781 0.196

Neat-2 (40/50) – – – – – –

Neat-3 (85/100) – – – – – –

PPA0.5 67.3–26.3 61 52 -0.23 688 0.249

a PI = 1952–500 9 Log(Pen.)-20 9 TR&B/50 9 Log(Pen.)-TR&B-120



SSREG0 þ SSREG00ð Þ ¼Xn

i¼1

1� G0model fið ÞG0measured fið Þ

� �2

þ 1� G00model fið ÞG00measured fið Þ

� �2 !

ð8Þ

where G0model and G00model are the real and imaginary

parts of |G*|model, respectively. These components are

calculated using the DL model at frequency fi. G0mea-

sured and G00measured are the laboratory measured coun-

terparts of the above components and n is the number of

data points. Equation (8) provides elative errors instead

of absolute errors. This is preferred because G0 and G00

values vary over several orders of magnitude. The use

of logarithmic errors for the above parameters is

another option for modeling. The selection of method

is a user preference. Further evaluation and refinement

of these methods is a subject for future research.

4 Results and discussion

The results of applying the DL model to three neat and

seven modified binders are shown in Table 2. The six

parameters of the model, as well as Gg*, rheological

index (R) and fc, are the main outcomes of the model.

Table 2 also presents the SSRE values as calculated

from Eq. (8), the Pearson coefficient of R2 for |G*| and

d, as well as the number of data points used for master

curve generation. The model generates high R2 values

for |G*| and the phase angle for all binders. This shows

the general goodness of fit for the model with the small

amount of scatter in the experimental results. Using R2

to compare a model is not recommended under these

conditions because all R2 values were very high and

close to one another. A visual comparison of the model

is recommended in this case. Figure 3 shows d and

|G*| master curves for the binders as well as the fit of

the DL model. It is clear in the figure that the phase

angle master curves of different modified binders

cover comparatively diverse patterns and features,

which makes the set of the selected binders an

appropriate choice for validating the model.

Styrelf 13–80 is the first binder shown in Fig. 3. It

corresponds exactly to the concept for which the DL

model has been generated. Its phase angle master

curve consists of an intermediate plateau region at

about 70�, a rise toward 90� at low frequencies, and a

decline toward zero at high frequencies. The DL

model precisely fits the master curves as shown in

Fig. 3. Table 2 also confirms the phase angle plateau

at about 70�. Furthermore, it can be seen in the figure

that the |G*| master curve shows a nearly constant

slope in the same frequency range (about

0.01–100 Hz) where the phase angle plateau occurs.

These observations confirm the fundamental assump-

tions used in the modeling (Eq. 5).

Two PMBs modified with 7 % SBS (SBS7) and

6 % EVA (EVA6) follow similar phase angle master

curve patterns as shown in Fig. 3. They both show a

peak at intermediate frequencies followed by a decline

toward lower frequencies (high temperatures). The

model captures the general pattern of the master

curves for both binders. Table 2 shows that both

models result in negative values for dL which means

that these two binders do not exhibit further viscous

behavior at higher temperatures. In Fig. 3, the |G*|

master curve shows a smaller slope at low frequencies

followed by an inflexion point that corresponds to the

peak point of its phase angle master curve. The model

also captures this behavior.

The binder modified with 2.5 % PPA (PPA2.5)

shows a phase angle master curve with low frequency

dependency as shown in Fig. 3. It is difficult to

visually determine an exact region at which the

plateau occurs. The model reveals a plateau phase

angle value of about 47�. Visual inspection confirms

this selection. A high PI value of 3.6 for PPA2.5

(Table 1) confirms the low frequency dependency, or

temperature susceptibility, of the binder. The rheo-

logical index, R calculated using the DL model, shows

the highest value (3.1) for PPA2.5 in Table 2,

indicating low temperature susceptibility as well [3,

9]. Parameter dP for PPA2.5 has the lowest value

among the binders. This shows that the phase angle

increased only slightly from zero to the point where

the plateau occurs, which, again, indicates the low

temperature susceptibility of the binder. The low

crossover frequency of this binder indicates a hard and

consistent binder with a low penetration value

(40 dmm). This is expected from a binder modified

with a high dose of PPA. The parameters of the model

correctly predict the above-mentioned behavior.

The phase angle master curve for binders modified

with 9 and 18 % crumb rubber (CR9 and CR18,

respectively) are different from the others. At the left

side of the plateau, which is about 53� for CR18 (59�



Ta

ble

2R

esu

lts

of

fitt

edD

Lm

od

els

for

thre

en

eat

and

sev

enm

od

ified

bin

der

s

Par

amet

ers

Un

its

Bin

der

s

Sty

relf

13

–8

0S

BS

7%

EV

A6

%P

PA

2.5

%C

R1

8%

CR

9%

Nea

t-1

85

/10

0N

eat-

24

0/5

0N

eat-

38

5/1

00

PP

A0

.5%

d Pd

egre

es6

8.4

06

3.9

55

6.5

44

7.6

75

3.0

65

9.1

39

09

09

09

0.0

0

f PH

z1

.E-1

9.E

-31

.E-1

3.E

-14

.E-1

1.E

?0

5.E

-41

.E-5

1.E

-34

.E-5

SR

–0

.14

30

.11

80

.12

00

.11

10

.12

60

.14

20

.11

40

.10

50

.12

20

.10

5

d LD

egre

es3

7.8

-2

2.2

-5

6.5

24

.31

9.4

24

.70

00

0.0

0

SL

–0

.25

70

.98

50

.23

70

.27

31

.55

30

.89

80

.11

40

.10

50

.12

20

.10

5

|G0*|

GP

a1

.7E

-4

2.2

E-

52

.1E

-5

1.7

E-

47

.0E

-5

3.3

E-

52

.7E

-5

2.4

E-

43

.5E

-5

8.2

E-

5

|Gg*|

GP

a0

.74

0.5

60

.94

1.1

40

.47

0.5

40

.83

0.7

70

.99

0.8

7

f cH

z4

.E?

03

1.E

?0

31

.E?

03

5.E

?0

16

.E?

02

6.E

?0

31

.E?

04

1.E

?0

31

.E?

04

3.E

?0

3

R–

1.6

92

.15

2.3

23

.10

2.3

61

.89

1.8

52

.01

1.7

42

.02

SS

RE

G0

?S

SR

EG00

–0

.95

0.5

51

.07

0.9

53

.35

4.0

35

.40

8.8

31

0.2

11

.26

(R2)d

–0

.99

78

0.9

95

50

.97

66

0.9

87

70

.97

58

0.9

91

80

.99

80

0.9

97

40

.99

57

0.9

99

1

(R2)G

*–

0.9

99

30

.99

99

0.9

99

90

.99

98

0.9

98

90

.99

83

0.9

99

90

.99

65

0.9

95

30

.99

95

n–

14

31

43

12

31

58

14

21

43

12

11

43

12

71

31



Fig. 3 Phase angle and |G*|

master curves and fitted DL

model for 10 binders



for CR9), a rise toward 80� (85� for CR9), and then a

decline at temperatures up to 88 �C occurs. This

differs from all other binders in this study although in

general SBS modified binders show a similar pattern.

The DL model follows the phase angle master curve at

high frequencies down to the plateau region at lower

frequency ranges, but not for the last decline at very

high temperatures (extremely low frequencies) as

shown in Fig. 3. This is especially the case for CR18.

However, this is not a major concern for the model

because, at very high temperatures (low frequency

regions in the master curve), complications such as

distortion of sample geometry may occur, making the

high temperature test results unreliable. Furthermore,

these temperatures are well above common high

pavement service temperatures. Therefore, generating

master curves for these temperatures does not have

practical significance and is mostly for the sake of

completeness over the entire temperature range.

CR18, in comparison with CR9, shows a smaller dP

and a higher R. It was expected that CR18 would show

more elastic behavior and lower temperature suscep-

tibility because of its higher modification level. The

higher softening point temperature and higher PI value

(Table 1) also confirm these findings.

If the phase angle plateau value approaches 90 in

the model estimation, the model is applied to a binder

having no intermediate temperature plateau region.

This is especially the case for neat binders, where the

entire range of phase angle master curve can be

modeled using only the base parameters. As shown in

Fig. 3, the three neat binders follow the general form

of a simple logistic function. Hence, the model is able

to fit these binder master curves using a reduced

number of parameters (dP, fP, SR and G0*). This may

result in higher SSRE values for neat binders com-

pared to PMBs, as shown in Table 2. Although using

all six parameters can improve the goodness of fit for

neat binders, the authors believe that it is not

reasonable to model a relatively simple behavior with

a more complicated model, especially when the simple

model goodness is sufficiently high, as shown in

Fig. 3.

The binder modified with 0.5 % PPA (PPA0.5) is a

binder with little modification that does not show

irregularities in the master curve, as shown in Fig. 3.

The generated master curve by the DL model for this

binder is like a neat binder with dP = 90� and dL = 0�,

as presented in Table 2.

Comparison of the glassy modulus in Table 2

demonstrates the capability of the model in predicting

binder low temperature properties. As it was expected,

the |Gg*| for binder Neat-1 reduced from 0.83 to

0.54 GPa, 0.47 and 0.56 GPa for CR9, CR18, and

SBS7, respectively. This shows the softening effects

of crumb rubber and SBS. The results of the bending

beam rheometer (BBR) in Table 1 show similar

stiffness reduction trends for the same binders.

Table 2 shows increased glassy modulus of 0.87 and

1.14 GPa for PPA0.5 and PPA2.5, respectively.

However, the BBR stiffness values in Table 1 do not

show increase for these two binders. The increased

glassy modulus of these binders can be attributed to

the increased slope of the stiffness curve (indicated by

the m value in Table 2). This implies that the stiffness

has not yet reached its asymptotic value at -30 �C.

5 Summary and conclusion

A simple model with six parameters based on the

general shape of a double logistic function has been

introduced to estimate the phase angle and |G*| master

curves of neat and modified binders in their linear

viscoelastic domains. The model was applied to

master curves of selected binders covering tempera-

tures from -30 �C to about 88 �C and more than 15

decades of reduced frequency values, which is a large

domain for validating a model.

Key parameters from the master curves such as the

phase angle plateau, glassy modulus, rheological

index, and crossover frequency were extracted from

the model for all binders. Their results were in good

agreement with the expectations from the modified

binders and other tests.

It is shown that the model can properly predict the

plateau region, a common PMB characteristic, at the

intermediate or high temperature ranges of phase

angle master curves. Furthermore, this model appro-

priately estimates the master curves at low tempera-

tures (high frequencies) and extrapolates below this

temperature range because it yields an asymptotic

value of zero at extremely high frequencies. This

model is also applicable at high temperatures for a

large variety of PMBs, especially those with low

variations in that range. This includes all binders in

this research except CR18. The phase angle master

curves for the neat binders and the SBS, EVA, and



PPA modified binders were all well described by the

DL model. A good fit of the |G*| master curve for all

binders at all frequency ranges was also achieved by

the model. The reduced slope of the |G*| master curves

of SBS and EVA modified binders at their low

frequency ranges were also properly predicted.

For neat binders, this model requires only about

half as many parameters as for the modified binders to

provide an acceptable fit, even though the point of

strength of the DL model is its ability to closely model

different modified binders.

This investigation clearly demonstrates that |G*|

master curves alone, for modeling purposes, do not

sufficiently characterize binder properties because

they mask some of the characteristic differences

among the binders, whereas phase angle master curves

clearly reveal them.

Phase angle master curves of modified binders vary

significantly from binder to binder. Therefore, vali-

dating the DL model using a large amount of master

curve data from other laboratories could be beneficial.

The DL function generated in this research identified

parameters that are expected to have physical signif-

icance. The correctness of this assumption was briefly

evaluated in the current research, but it is also

suggested as a topic of further research to calibrate

the model parameters with other tests results.

Appendix A

The glassy modulus of the binders is calculated using

Eq. (6) when simplified as:

Rheological index R, as defined by Anderson et al.

[3], can be calculated using the glassy modulus. This

parameter shows how rapidly the behavior of asphalt

cement changes overtime and thus indicates the time-

dependency of the asphalt binder [3]. R can be

calculated as the difference between the glassy

modulus and the modulus of the binder at a point

where the loss and storage moduli are equal (or the

phase angle is 45�). The crossover frequency fC can be

calculated by setting Eq. (4a) to 45� and extracting the

logarithm of the reduced frequency as a function of the

model parameters:

LogfC ¼ LogfP þH f C � f Pð Þ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�LN 45

dP

� r

SR

�H f P � f Cð Þ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�LN dPþdL�45

dL

� r

SL

ðA:2Þ

The above equation denotes that Log(fC) is located

on either side of the frequency at the plateau (fP) and

can be calculated using either of the Napierian

logarithms.

The complex modulus at crossover frequency Gfc*

can then be calculated by substituting Log(fC) from

Eq. (A.2) into Eq. (6). Finally, R can be calculated as:

R ¼ LogG�gG�fc

ðA:3Þ

Appendix B

The DL model is also capable of modeling asphalt

mixture master curves using the same parameters

log G�g

¼ lim

log fred!1log G�j j ¼ log G�0

þ limlog fred!1

1

90

� dP �H f red � f Pð Þ

logðfredÞ �p1=2

2 � SRerf SR log

fred

fP

� �� log fPð Þ �

dP þ dL

dP

� �þ dP � log fred

!

¼ 1

90�dP � lim

log fred!1log fredð Þ � p1=2

2 � SRlim

log fred!1erf SR log

fred

fP

� �� log fPð Þ �

dP þ dL

dP

� ��

þ limlog fred!1

dP � log fred

�þ log G�0

¼ 1

90dP

p1=2

2 � SR� log fPð Þ � dP

dP þ dL

dP

� �

þ log G�0 ¼ 1

90dP

p1=2

2 � SRþ dP þ dLð Þ � log fPð Þ

� �þ log G�0

ðA:1Þ



presented above. The high frequency asymptotes of

the master curves of asphalt mixtures can be obtained

using the same method as for asphalt binders. Here, the

phase angle approaches zero and |E*| approaches the

glassy modulus. For low frequency ranges (fred B fP),

the phase angle approaches dP ? dL. The phase angle

master curve of an asphalt mixture usually shows an

intermediate peak and declines on both sides at higher

and lower frequencies. Based on Eq. (6), the |G*| low

frequency asymptote will be zero except fordL =

-dP, where it approaches a fixed value of |G0*|-10^

(p0.5dP/180SL). It is generally believed that asphalt

mixtures show the behavior of a viscoelastic solid.

These materials approach an equilibrium modulus of

|Ge*| at very low frequencies [28], which happens only

when dL = -dP. This is not a limitation, but an

advantage, and the number of parameters will reduce

to five. The phase angle for extremely low frequencies

will also be zero for dL = -dP. This is a commonly

used assumption for asphalt mixtures [28].

References

1. Airey GD (2002) Rheological evaluation of ethylene vinyl

acetate polymer modified bitumens. Const Build Mater

16(8):473–487

2. Airey GD (2003) Rheological properties of styrene butadi-

ene styrene polymer modified road bitumens. Fuel

82(14):1709–1719. doi:10.1016/s0016-2361(03)00146-7

3. Anderson DA, Christensen DW, Bahia H (1991) Physical

properties of asphalt cement and the development of per-

formance-related specifications. J Assoc Asphalt Paving

Technol 60:437–475

4. Anderson DA, Christensen DW, Bahia HU, Dongre R,

Sharma M, Antle CE, Button J (1994) Binder character-

ization and evaluation, vol 3: physical characterization,

SHRP-A-369. Strategic Highway Research Program,

National Research Council, Washington DC

5. Baumgaertel M, Winter H (1989) Determination of discrete

relaxation and retardation time spectra from dynamic

mechanical data. Rheol Acta 28(6):511–519

6. Bird RB, Armstrong RC, Hassager O (1987) Dynamics of

polymeric liquids, vol 1, fluid mechanics, 2nd edn. Wiley,

New York

7. Booij H, Thoone G (1982) Generalization of Kramers–

Kronig transforms and some approximations of relations

between viscoelastic quantities. Rheol Acta 21(1):15–24

8. Chailleux E, Ramond G, Such C, de La Roche C (2006)

A mathematical-based master-curve construction method

applied to complex modulus of bituminous materials. Road

Mater Pavement Des 7(1):75–92

9. Christensen D, Anderson DA (1992) Interpretation of

dynamic mechanical test data for paving grade asphalt

cements. J Assoc Asphalt Paving Technol 61:67–116

10. Dickinson EJ, Witt HP (1974) The dynamic shear modulus

of paving asphalts as a function of frequency. Trans Soc

Rheol 18(4):591–606

11. Dobson G, Jongepier RD, Monismith C, Puzinauskas V,

Busching H, Warden WD (1969) The dynamic mechanical

properties of bitumen. J Assoc Asphalt Paving Technol

38:123–139

12. Ferry JD (1980) Viscoelastic properties of polymers, 1st

edn. Wiley, New York

13. Guo H (2011) A simple algorithm for fitting a gaussian

function [DSP tips and tricks]. Signal Process Mag IEEE

28(5):134–137

14. Jensen EA (2002) Determination of discrete relaxation

spectra using simulated annealing. J Non Newtonian Fluid

Mech 107(1–3):1–11

15. Jongepier R, Kuilman B, Schmidt R, Puzinauskas V, Rostler

F (1969) Characteristics of the rheology of bitumens.

J Assoc Asphalt Paving Technol 38:98–122

16. Marasteanu O, Anderson DA (1999) Improved model for

bitumen rheological characterization. In: Eurobitume work-

shop on performance related properties for bitumens binder,

Luxembourg

17. Marateanu M, Anderson D (1996) Time-temperature

dependency of asphalt binders—an improved model. J As-

soc Asphalt Paving Technol 65:408–448

18. Menard KP (1998) Dynamic mechanical analysis. Wiley

Online Library

19. NCHRP (2004) Guide for mechanistic–empirical design of

new and rehabilitated pavement structures. NCHRP Report

01-37A. National Cooperative Highway Research Program,

Transportation Research Board, National Research Council,

Washington DC

20. Olard F, Di Benedetto H (2003) General 2S2P1D model and

relation between the linear viscoelastic behaviours of bitu-

minous binders and mixes. Road Mater Pavement Des

4(2):185–224

21. Rowe GM, Baumgardner G, Sharrock MJ (2008) A gen-

eralized logistic function to describe the master curve

stiffness properties of binder mastics and mixtures. In:

45th Petersen Asphalt Research Conference. Laramie,

Wyoming, July 14–16

22. Rowe GM, Baumgardner G, Sharrock MJ (2009) Functional

forms for master curve analysis of bituminous materials. In:

Loizos A, Partl MN, Scarpas T, Al-Qadi IL (eds) Advanced

testing and characterization of bituminous materials. CRC

Press, Boca Raton, pp 81–91

23. Stastna J, Zanzotto L, Ho K (1994) Fractional complex

modulus manifested in asphalts. Rheol Acta 33(4):344–

354

24. Tabatabaee N, Teymourpour P (2010) Rut resistance

evaluation of mixtures made with modified asphalt

binders. Paper presented at the 11th International Con-

ference on Asphalt Pavements, ISAP, Japan, August

2010

25. Van der Poel C (1954) A general system describing the

visco elastic properties of bitumens and its relation to rou-

tine test data. J Appl Chem 4(5):221–236

26. Yusoff NIM, Chailleux E, Airey GD (2011a) A comparative

study of the influence of shift factor equations on master

curve construction. Int J Pavement Res Technol 4(6):324–

336



http://dx.doi.org/10.1016/s0016-2361(03)00146-7

27. Yusoff NIM, Shaw MT, Airey GD (2011) Modelling the

linear viscoelastic rheological properties of bituminous

binders. Const Build Mater 25:2171–2189

28. Zeng M, Bahia H, Zhai H, Turner P (2001) Rheological

modeling of modified asphalt binders and mixtures. J Assoc

Asphalt Paving Technol 70:403–441



an empirical model for modified bituminous binder master curves

Documents