INTRODUCTION
In principle, the complete modulus versus frequency (time of loading) behaviour
of any polymer including bitumens at any temperature can be measured (Shaw
and McKnight, 2005). Data can be shifted relative to the reduced frequency,
so that the various curves can be aligned to form a single curve, called a master
curve. The master curve represents the bitumen or asphalt mixture behaviour
at a given temperature for a large range of frequencies. As discussed by Herh
et al. (1999), the Linear ViscoElastic (LVE) behaviour of unmodified
bitumen and asphalt mixture is bounded by two main transitions. Generally, at
very high frequencies/low temperatures, the elastic modulus (G_{o})
approaches a limiting value called the glassy state modulus. At low frequencies/high
temperatures, material is behaving as the Newtonian fluid.
The principle used to relate the equivalency between frequency and temperature
to construct the master curve is known as Time Temperature Superposition Principle
(TTSP). The master curves can be constructed using a reference temperature (T_{o})
to which all data are shifted and in many cases, the T_{o} value can
arbitrarily be chosen from one of the test temperature. Moreover, the amount
of shifting required at each temperature to form the master curve is called
the shift factor (a_{T}). According to Chailleux
et al. (2006), the construction of master curves only makes sense
where there should be no macromolecular structural rearrangements with temperature
like phase transformations occurs and secondly, the tests are conducted in the
linear viscoelastic region. A lot of attempts have been made by various researchers
in developing the rheological models particularly for unmodified bitumens. However,
none of the comparative studies have been made to compare all the complex modulus
(G*) predictive equations on aged and unaged unmodified and polymer modified
bitumens (PMBs) simultaneously in order to attain a better view on those equations.
In general there are two methods used for representing the Linear Viscoelastic
(LVE) rheological behaviour of bitumen namely as the mathematical and mechanical
models. In the mathematical (phenomenological or constitutive) approach, one
is simply adjusts any mathematical formulation whatsoever to the experimental
main curve, with the quality adjustment being the sole criterion of choice of
formulation. Meanwhile in the mechanical (or analogical) approach, use is made
of the fact the behaviour of linear viscoelastic material can be represented
by a combination of spring and dashpot mechanical models, resulting in a particular
mathematical equations. Historically, the invention of LVE rheological models
is dated circa 1954 when Van der Poel developed the nonlinear multivariable
model called the Van der Poel’s nomograph (VanderPoel,
1954). Since then enormous researchers showed their interest to develop
the models particularly for bituminous binders and asphalt mixtures. The model
can easily be constructed with the aid of Solver function in Excel spreadsheet.
Solver is a powerful function to performing the nonlinear least square regression
to do the fitting task in MS Excel (Pellinen et al.,
2002). In general, a good rheological model should be able to describe as
completely as possible the linear viscoelastic functions of the studied materials.
Therefore, this study was conducted to investigate the predictability of several predictive equations or models to characterize the linear viscoelastic behaviour of aged and unaged unmodified and polymer modified bitumens. Several models namely as the modified Sigmoidal, Generalized Logistic Sigmoidal, Christensen and Anderson (CA), Christensen Anderson and Marasteanu (CAM) and 2S2P1D (2 springs, 2 parabolic elements and 1 dashpot) have been used to investigate the advantages and drawbacks when the models applied to those samples. The correlation between measured and predicted values where then assessed using the goodness of fitting statistics.
RHEOLOGICAL MODELS
The Modified Sigmoidal Model
Mathematically, the G* equation of Sigmoidal model is shown as the following:
where, log (ω) is log reduced frequency, ä is lower asymptote, α
is the difference between the values upper and lower asymptote, β and γ
define the shape of between the asymptotes. However, as the limiting modulus
(G_{o}) is equal to 1GPa, the above equation can be transformed as (Garcia
and Thompson, 2007):
where, Max is the G_{o} at 1 GPa (Bonaquist and
Christensen, 2005). The other parameters are as previously defined.
The Generalized Logistic Sigmoidal Model
Rowe et al. (2008) introduced the generalization
of the Sigmoid model, called the Generalized Logistic Sigmoidal model to predict
the G* of bitumen and can be shown as the following:
where, the parameters are as previously defined above. The λ parameter allows the curve to take a nonsymmetrical shape. Like the modified Sigmoidal model, the G_{o} can be taken as 1GPa, therefore the following equation can be used:
The Christensen Anderson Model (CA)
The G* equation of the Christensen and Anderson (CA) model can be presented
as (Christensen and Anderson, 1992):
where, ω_{c} is the crossover frequency and R is a rheological index. The other parameters are as defined previously.
The Christensen Anderson and Marasteanu Model (CAM)
The Christensen Anderson and Marasteanu (CAM) model was developed to improve
the descriptions of unmodified and modified bitumens. The G* equation can
be shown as follows (Marasteanu and Anderson, 1999):
The introduction of w parameter addresses the issue of how fast or how slow the G* data converge into the two asymptotes (the 45^{o} asymptote and the G_{o} asymptote) as the frequency goes to zero or infinity.
The 2S2P1D Model
The G* of the 2S2P1D model can be shown in the following mathematical
expression (Olard and Benedetto, 2003):
where, i is complex number (i^{2} = 1), ω is frequency, k and
h are exponents with 0<k<h<1,δ is constant, G_{oo} is
the static modulus, G_{o} is the glassy modulus, β is constant
and defined by η = (G_{oo }– G_{o}) β τ,
η is the Newtonian viscosity and τ is characteristic time, function
of temperature. The 2S2P1D model’s representation can be shown in Fig.
1.
Statistical Analysis
Three different statistical analysis have been used in this study, based
on Wu et al. (2008):
The Discrepancy Ratio (R_{i})
where, G*_{p} and G*_{m} are the predicted and measured shifting factors, respectively. The subscript i denotes the data set number. For a perfect fit, R_{i} = 1.
The Mean Normalized Error (MNE)
where, N is the total number of set of data and for a perfect fit, MNE = 0.
The Average Geometric Deviation (AGD)
where for a perfect fit, AGD = 1.
EXPERIMENTAL DESIGN
In this study, three different sources of bitumens i.e., from Middle East,
Russian and Venezuelan were used and regarded as unmodified bitumen 70/100 penetration
grade. These bitumens were blended together with ethylenevinylacetate (EVA)
and styrenebutadienestyrene (SBS), respectively producing the polymer modified
bitumens (PMBs) at three polymer contents, 3, 5 and 7% (by mass). All these
materials underwent the dynamic shear rheometer (DSR) tests in order to obtain
and evaluate the changes in their linear viscoelastic rheological properties
(Airey, 2002). For the master curve construction, the
T_{o} was arbitrarily taken at 10°C and the shifting was done manually,
without assuming any functional form of the shift factor equation. The interested
readers should consult Airey’s research for the detail discussions on rheological
characteristics of aged and unaged unmodified bitumens and PMBs.
RESULTS AND DISCUSSION
Modeling the Modified Sigmoidal Model
Three unknown parameters namely as β, δ and γ were estimated
using the numerical optimization of the test data. Moreover, data at high frequency
is no longer needed since the limiting of maximum modulus (G_{o}) of
1 GPa is specifically used. In this study, the initial guesses values are used,
i.e., β = 1, δ = 1 and γ = 1. The Solver function in MS Excel
then replaces the initial guesses with optimized values, as shown in Table
1 for the unaged and aged unmodified bitumens. It is observed that the δ
is in negative values, showing that the moduli at low frequencies/high temperatures
are small.
Meanwhile the β values, which controls the horizontal position of the turning point, decreases from unaged to aged samples regardless the source of bitumens used. This could attribute to the reason where the oxidation process occurred during ageing increases asphaltenes (polar molecules) content, therefore the samples become harder. In addition, the γ values were found to be more constant for all the samples studied. This phenomenon shows that the ageing process did not influence the steepness of the modified Sigmoidal model’s slope too much. The Russian unmodified bitumen seems to have the highest parameter values. It does because it has very high sulphur content and also its asphaltene content is low and saturates are in different ratio.
The modified Sigmoidal model parameters for the unaged and aged EVA and SBS, PMBs can be shown in Table 2, however, only the Russian unmodified PMBs are shown here for brevity.
Table 1:  Modified
Sigmoidal model’s parameters for unmodified bitumens 

Table 2:  Modified
Sigmoidal model’s parameters for polymer modified bitumens 

Like the unaged and aged unmodified bitumens, the ä parameter obtained
for the aged and unaged PMBs are small, showing that the elastic modulus at
low frequencies closely to zero. It was also observed that the present of polymer
modification for unaged samples decrease the β values. In general the discussions
for all modified samples are similar to the unmodified bitumens except for several
samples. For instance, the difference observed between the Russian RTFOT 5 and
7% EVA PMBs were quite significant where the parameter values seem to be dispersed
compare to the other samples. As discussed by Chen et
al. (2002) the critical network might form with the modification around
5% and leads to the partial breakdown of a polymer network in the samples. Moreover,
the presence of waves in certain G* curves cannot be predicted by the modified
Sigmoidal model, suggesting that this model is not able to predict the linear
viscoelastic behaviour of highly modified bitumen.
The Generalized Logistic Sigmoidal Model
As mentioned earlier, the difference between the Generalized Logistic Sigmoidal
and the modified Sigmoidal models is the introduction of ë which allows
the curve to take a nonsymmetric shape (Rowe et al.,
2008). The initial values for the model are used, i.e., δ = 0, β
= 1, γ = 1, λ = 0 and these four unknown fitting parameters are still
estimated using numerical optimization of the test data. Table
3 shows the Generalized Logistic Sigmoidal model parameters obtained for
the unaged and aged unmodified bitumens. Like the modified Sigmoidal function,
the δ which represents the minimum G* values is observed to be in negative
values. The β coefficient values are slightly decreased due to the samples
become harder. Meanwhile, the λ values increase for the unaged and aged
samples. Analysis of the materials used in this study showed that the presence
of ë did not play a significant role in since its value is approaching
a unity and this parameter might be useful for modeling the asphalt mixture.
Meanwhile, Table 4 shows the Generalized Logistic Sigmoidal
model parameters obtained for the EVA PMBs. The small values of δ were
observed at low frequencies/high temperatures. The inconsistency is detected
since this model is not capable of predicting the presence of the waves curve
especially at the intermediate to high temperatures in the mixture. This observation
could also relate to the inconsistency of γ and λ. However, as the
temperature increases, the curve reverts back to a unique slope associated with
the Newtonian asymptote found for unmodified bitumens. The β are observed
to be decreased from the unaged to the aged samples due to hardening process.
Table 3:  The
Generalized Logistic Sigmoidal model’s parameters for unmodified
bitumens 

Table 4:  The
Generalized Logistic Sigmoidal model’s parameters for polymer modified
bitumens 

The parameters of the model for the unaged and aged SBS PMBs can also be found
in Table 4. The δ was also found to be in negative values
as observed for the other samples. Meanwhile, the β,γ and λ were
inconsistent for the SBS PMBs. This could be explained by the presence of elastomeric
behaviour of SBS. The effect of ageing on the polymer dominant regions of behaviour
for the SBS PMBs relate to a shifting of the rheological properties towards
greater viscous response (Airey, 2002).
The Christensen and Anderson Model (CA)
In this model, the G_{o} was taken as 1GPa to avoid the underestimation
of the value during the optimization process. Table 5 shows
the CA model parameters for unaged and aged unmodified bitumens. The w_{o}
and R were observed to be increase and decrease, respectively from the unaged
to aged samples. This condition could attribute to the hardness parameters with
the presence of higher asphaltenes content in aged samples. Moreover, it was
observed the width of relaxation spectrum becomes smaller as the sample aged.
This model, in general, predicts precisely the complex modulus at temperature
of 10 to 70°C.
Meanwhile, Table 6 shows the CA parameters for the unaged
and aged EVA PMBs. Generally, the w_{o} coefficient values are increased
from unaged to aged samples and on the other hand, the width of relaxation spectrum
becomes smaller. This behaviour give supports to some authors who relate the
R to the binder asphaltene content, finding that it grows as this polar molecules
is elated (Silva et al., 2004). However, like
the previous models, this model was also not able to predict the presence of
special importance elements such as the semicrystalline structure in EVA PMBs
and elastomeric SBS PMBs.
Table 5:  The
CA model’s parameters for unmodified bitumens 

Table 6:  The
CA model’s parameters for polymer modified bitumens 

Table 7:  The
CAM model’s parameters for unmodified bitumens 

The CAM Model
The CAM model was proposed to improve the CA model and by comparing these
two models, it can be noted that the Log 2/R is equivalent to the v of the CAM
model (Silva et al., 2004). In this study, the
glassy modulus was taken as 1 GPa for all unaged and aged samples. The CAM model
coefficients for the unaged and aged unmodified bitumens can be shown as in
Table 7.
It was observed that the v and R values decrease and increase, respectively,
from unaged to aged unmodified bitumens. As frequency approach zero, the PAV
aged samples reaches the 45° asymptote faster than unaged bitumens. The
w_{c} values are decreased from unaged to aged bitumens, suggesting
that samples become stiffer due to the ageing process. Meanwhile, Table
8 shows the CAM model parameters for the EVA and SBS PMBs, aged and unaged,
respectively. The v values for all samples except the PAV Middle East EVA PMBs
show constant value with the modification of 3 to 7%. Moreover, the R values
increase as the asphaltenes content increases. In contrast, the w is decrease
from 3 to 7%, showing the 3% of modification reaches the 45° asymptote faster
than the 7% with the lower value.
Table 8:  The
CAM model’s parameters for polymer modified bitumens 

Table 9:  The
2S2P1D model’s parameters for unmodified bitumens 

The 2S2P1D Model
Table 9 shows the 2S2P1D model parameters for all bitumens
used in this study. The G_{oo} is equal to zero and only six constants
of the model are needed to be determined. It was observed that the G_{oo},
G_{o}, k and h are consistent for all studied materials.
Moreover, it was found that the β linked to the Newtonian viscosity η of the model has a large influence in this domain of behaviour. The influence of the binder ageing affects the β, where this parameter keeps increased from unaged to the aged. As the materials become harder, the δ values increased. The 2S2P1D model parameters for EVA and SBS PMBs are shown in Table 10. The δ was found to be more consistent for the SBS PMBs compared to the EVA PMBs. This is probably due to the fact that the presence of crystalline structure at different temperatures increases the complexity of the bitumens. Moreover, the DSR compliance errors may also occur for PMBs samples. As the percentage of the modifier is elevated, the β increases. Like the other samples, for the aged samples the G_{oo}, G_{o}, k and h parameters can be taken similar for all studied samples.
Table 10:  The
2S2P1D model’s parameters for polymer modified bitumens 

Statistical Analysis
To evaluate the performance of the predictive equation, the correlation
of the measured and predicted values was assessed using the R_{i}, MNE
and AGD. The R_{i} is used to observe the predicted data’s dispersion
from the equality line, with one as the perfect value. In this study, the interval
of 2% from the equality line was used until R_{i} reaches interval of
10%. Table 11 shows the goodness of fit statistics for the
unaged and aged unmodified bitumens. It was observed that all of the models
studied show a good correlation when the R_{i} = 0.95 1.05. This finding
indicates that all of the LVE rheological models used in this study can predict
the aged and unaged unmodified bitumens precisely. However, the modified Sigmoidal
model shows the most outstanding correlation, followed by the Generalized Logistic
Sigmoidal, CAM and CA models. On the other hand, the 2S2P1D model seems to have
the worst correlation. This could attribute to the reason where during the modeling
work, the compliance error from the DSR machine was taking into account, therefore
the predicted G* values were slightly higher compare to the experimental data
at high frequencies. Moreover, each of this model parameter has a direct relation
with the construction of G* master curve, the Black diagram and also the ColeCole
diagram. Conversely, the other models used only rely on the construction of
G* master curve.
The modified Sigmoidal, Generalized Logistic Sigmoidal and CAM models show
the best correlation with 10% of R_{i} for the aged unmodified bitumens.
However, it also showed that the CA and 2S2P1D models data tabulated closely
to the equality line. The Generalized Logistic Sigmoidal model shows the best
correlation, showing the presence of ë plays a significant role for the
aged samples. Meanwhile the modified Sigmoidal and CAM models also predict the
measured data really well. Interestingly, the 2S2P1D model shows better correlation
compare to the CA model. This finding was in good agreement with the previous
study where the CA model was not able to predict the linear viscoelastic properties
of bitumens particularly at higher and lower temperatures (Christensen
and Anderson, 1992).
It was observed that for the unaged PMBs (Table 12), the
Generalized Logistic Sigmoidal and modified Sigmoidal models had comparably
good results, followed by the CAM, CA and 2S2P1D models. However, the modified
Sigmoidal model predicts the behaviour of PMBs in better way with the lowest
AGD and highest MNE values, respectively. This indicates that the addition of
ë in generalized logistic sigmoidal did not play a significant role in
predicting the linear viscoelastic behaviour of PMBs.
Table 11:  Goodness
of fit statistical analysis for the unmodified bitumens 

Table 12:  Goodness
of fit statistical analysis for the polymer modified bitumens 

Meanwhile, the 2S2P1D model shows the worst correlation where the predicted
data seems dispersed from the equality line. As discussed by Olard and Di Benedetto,
this model was not suitable to be used and from their study, it only conform
behaviour of PMBs at low temperature (Olard and Benedetto,
2003). The Generalized Logistic Sigmoidal shows the best correlation in
term of MNE for the aged PMBs, even though it has comparable values for the
R_{i} at 0.951.05 and AGD with the modified Sigmoidal model. The CA
model also shows good correlation with the values of MNE and AGD. This could
attribute to the reason where the CAM model cannot predict the complexity behaviour
of the aged PMBs. In general, all of the LVE rheological models suffer from
similar drawbacks where they cannot predict the linear viscoelastic behaviour
of unaged and aged PMBs precisely. The presence of semicrystalline EVA and
SBS modified can be linked to the breakdown of the molecular structure of the
copolymer to form a lower molecular weight polymer substructure changes the
rheological properties of materials (Airey, 2002).
CONCLUSIONS
Several conclusions can be drawn from the study as:
• 
It was observed that all the linear viscoelastic (LVE) rheological
models are able to predict the unaged and aged unmodified bitumens satisfactorily.
For modeling purposes, the glassy modulus for unmodified and polymer modified
bitumens (PMBs), aged and unaged samples, approaching the value of 1 GPa.
Moreover, the elastic modulus (G_{oo}) value is really small and
in most cases, this value can be neglected 
• 
However, these predictive models suffer from the drawback
where they did not able to predict the linear viscoelastic behaviour particularly
for the unaged PMBs due to the presence of semicrystalline EVA and SBS
modified, rendering the breakdown in time temperature superposition principle 
• 
For the unaged and aged unmodified bitumens, the modified
Sigmoidal and Generalized Logistic Sigmoidal models were shown to be the
best models, followed by the CAM, CA and 2S2P1D models. In addition, the
Generalized Logistic Sigmoidal and modified Sigmoidal models show the most
outstanding correlations for the unaged and aged PMBs 
•  The
CAM model improves the CA model curve fitting particularly at extreme
zones of complex modulus master curve 
• 
The presence of additional λ parameters in modified generalized
logistic sigmoidal model which cater for the nonsymmetrical shape of the
curve seems did not play a significant role since shape of the complex modulus
master curves were not too different from each other. This model might suitable
to be used for the asphalt mixture and in many cases of bituminous binders,
the modified Sigmoidal model is appropriate 
• 
The 2S2P1D model was statistically observed to be the worst
model compares to the others, but this model still can be thought as a unique
rheological model. The model consists of seven unknown parameters, relates
to the construction of the complex modulus master curve, the Black and ColeCole
diagrams. In contrast, the other models coefficients merely related to the
construction of complex modulus master curves 
ACKNOWLEDGMENTS
The authors would like to thank the Malaysian Ministry of Higher Education (MOHE) and Universiti Kebangsaan Malaysia (UKM) for funding this study. The authors also owe many thanks to the Nottingham Transportation Engineering Centre (NTEC) for providing data used in this paper.