Modelling the rheological properties of bituminous binders using the 2S2P1D Model

Construction and Building Materials 38 (2013) 395–406

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Modelling the rheological properties of bituminous binders using the 2S2P1D Model Nur Izzi Md. Yusoff a,⇑, Damien Mounier b, Ginoux Marc-Stéphane c, Mohd. Rosli Hainin d, Gordon D. Airey e, Hervé Di Benedetto f a

Dept. of Civil & Structural Engineering, Universiti Kebangsaan Malaysia, 43650 Bangi, Selangor, Malaysia Civil Aviation Technical Centre (STAC), CS 30012, 31 Avenue du Maréchal Leclerc, 94385 Bonneuil-Sur-Marne Cedex, France c Laboratoire Régional des Ponts et Chaussées d’Aix en Provence (LRPC), CETE Méditerranée, Pôle d’activités Les Milles, Avenue Albert Einstein, CS 70499, 13593 Aix-en-Provence Cedex 3, France d Faculty of Civil Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Johor, Malaysia e Nottingham Transportation Engineering Centre (NTEC), Dept. of Civil Engineering, University of Nottingham, NG7 2RD Nottingham, United Kingdom f Ecole Nationale des Travaux Publics de l’Etat (ENTPE), Departement Génie Civil et Bâtiment, Rue Maurice Audin, 69518 Vaulx-en-Velin, France b

h i g h l i g h t s " We describe the rheological properties of binders using the 2S2P1D Model. " The model can be used over a wide range of temperatures and frequencies. " Most of the samples show good agreement between the measured and descriptive data.

a r t i c l e

i n f o

Article history: Received 3 May 2012 Received in revised form 2 August 2012 Accepted 14 August 2012 Available online 29 September 2012 Keywords: Modelling Linear viscoelastic region Unmodified bitumen Bitumen–filler mastic Polymer-modified bitumen

a b s t r a c t This study evaluates the suitability of the 2S2P1D Model to describe the rheological properties of a large database of bituminous binders held by the Nottingham Transportation Engineering Centre. This database includes more than 60 different combinations of unmodified bitumen, polymer-modified bitumen and bitumen–filler mastics, unaged and aged samples. The 2S2P1D Model is found to be able to satisfactorily describe the linear viscoelastic rheological (LVE) properties of binders over a wide range of temperatures and frequencies. A comparison between measured and descriptive complex modulus and phase angle data was done using a graphical method and goodness-of-fit statistics. Except for the unaged polymer-modified bitumens, all tested samples show good agreement between the measured and descriptive data. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The behaviour of unmodified bitumens is traditionally evaluated using empirically based tests as part of penetration or viscosity-graded specifications. The primary purpose of these specifications is to grade the unmodified bitumens according to their consistency; therefore, they do not address specific distress modes or ensure good long-term field performance. There is a need for fundamental binder testing as unmodified bitumen tests have proved inadequate to describe the viscoelastic properties needed for a complete rheological evaluation of bitumens. One such fundamental test is the dynamic shear rheometer (DSR), introduced during the Strategic Highway Research Program (SHRP) campaign in the early 1990s [1]. The DSR is regarded as a useful tool to deter⇑ Corresponding author. Tel.: +603 8921 6750. E-mail address: [email protected] (N.I. Md. Yusoff). 0950-0618/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2012.08.038

mine the elastic, viscoelastic and viscous properties of bitumen often using the testing configuration shown in Fig. 1. However, this instrument also has its limitations, as it is unable to reach extreme temperature and/or frequency windows. Alternatively, the introduction of a modelling method is very useful for describing the linear viscoelastic (LVE) rheological properties of materials such as bitumens with windows that cannot be reached by experimental works. The models are used to describe complex modulus (|G|) and phase angle (d) master curves. The technique of determining the master curve is based on the time– temperature superposition principle (TTSP). A standard reference temperature (Tref) is normally selected in the range of the tested temperatures. The amount of shifting required at each temperature to form the master curve is termed the shift factor, aT [2]. Through the use of the master curve and shift factor relationships, it is possible to interpolate stiffness at an expanded range of frequencies and temperatures compared with those at which the data was col-

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N.I. Md. Yusoff et al. / Construction and Building Materials 38 (2013) 395–406

Spindle

Water level Water chamber cover

Bitumen

DSR base plate

DSR body

Fig. 1. Schematic of dynamic shear rheometer testing configuration [2].

lected. If functional forms are fitted to the shape of the master curve plot and to the shift factor relationship this interpolation becomes rapid and easy to apply in computer software. In addition, if a functional form with some thermodynamic basis is used then the resulting equations can be employed to extrapolate the data beyond the observed range of temperatures and frequencies [3]. The use of models to describe these curves can be classified into three groups; nomographs, mathematical and mechanical models [4]. One of such is the 2S2P1D Model. This model is found to be unique in that it can be used to describe the rheological properties of most binders and asphalt mixtures. It was originally calibrated at the Ecole Nationale des Travaux Publics de l’Etat (ENTPE) laboratory using equipment known as an annular shear rheometer (ASR) [5–7]. However, validity of the 2S2P1D Model has yet to be assessed by other types of rheometer. This study therefore evaluates the suitability of the model to fit or describe the rheological properties of a large bituminous binder database held by the Nottingham Transportation Engineering Centre (NTEC). This database includes more than 60 different combinations of unmodified bitumen, polymer-modified bitumen and bitumen–filler mastics, both unaged and aged samples, collected by means of the DSR instrument. Correlations between the measured and descriptive complex modulus and phase angle data were assessed using a graphical method and goodness-of-fit statistics.

Fig. 2. The 2S2P1D Model [7].

G ¼

ixs a

ð3Þ

where the parameters are as previously defined. The 2S2P1D Model consists of seven parameters and the |G| is shown in the following expression:

G ðxÞ ¼ G0 þ

Gg  G0 1 þ aðixsÞ

k

þ ðixsÞh þ ðixbsÞ1

ð4Þ

where k and h are exponents with 0 < k < h < 1, a is constant, G0 is the static modulus when x ? 0, Gg is the glassy modulus when x ? 1. Meanwhile, b is constant and defined by

g ¼ ðGg  G0 Þbs

ð5Þ

where g is Newtonian viscosity and s is the characteristic time, a function of temperature. s evolution can be approximated by a shift factor law such as the William, Landel and Ferry (WLF) and Arrhenius equations in the range of temperatures observed in the laboratory [6]:

2. The 2S2P1D Model

s ¼ aT ðTÞ  so

The 2S2P1D Model, an abbreviation of the combinations of two springs, two parabolic creep elements and one dashpot, is a model to describe the rheological properties of binders and asphalt mixtures [4–8]. This model, as shown in Fig. 2, is based on the generalisation of the Huet–Sayegh Model. According to Olard and Di Benedetto [5], the parabolic element is an analogical model with a parabolic creep function:

The 2S2P1D equation can be separated into parts and rewritten in the following form [9]:

 h t JðtÞ ¼ a

AðxÞ ¼ aðxsÞk  cos

ðixsÞh G ¼ aCðh þ 1Þ

ð2Þ

where J(t) is the creep function, h is the exponent such that 0 < h < 1 (h = 0 (elastic) and h = 1 (viscous)), a is the dimensionless constant (1/k), k is relaxation time, C is gamma function,1 t is the loading time, s is the characteristic time (which value pffiffiffiffiffiffiffi varies only with tem2 perature), i is the complex number ði ¼ 1Þ and x is frequency. For the linear dashpot, the equation can be written as: Gamma function is defined as CðnÞ ¼

R1 0

tn1 et dt [5].

ð7Þ

    kp hp þ ðxsÞh  cos 2 2

and

and, subsequently the complex modulus equation is shown as: 

Gg  Go 1 þ A þ iB

where

ð1Þ

s

1

G ðxÞ ¼

ð6Þ

BðxÞ ¼ ðxbsÞ1  aðxsÞk  sin



   kp hp  ðxsÞh  sin 2 2

The imaginary and real parts of |G| can be separated as follows:

Gg  G0 1 þ A þ iB ðGg  G0 Þ  ð1 þ AðxÞ  iBðxÞÞ ð8Þ ¼ G0 þ ð1 þ AðxÞÞ2 þ B2 ðxÞ " # " # ðGg  G0 Þ  ð1 þ AðxÞÞ ðGg  G0 Þ  ðBðxÞÞ ¼ G0 þ þi ð9Þ 2 ð1 þ AðxÞÞ þ B2 ðxÞ ð1 þ AðxÞÞ2 þ B2 ðxÞ

G ðxÞ ¼

N.I. Md. Yusoff et al. / Construction and Building Materials 38 (2013) 395–406

397

Table 1 The 2S2P1D parameters functions. Parameter

Function(s)

h k

Controlled the slope at low values of G00 in the Cole–Cole diagram Controlled the slope at high values of G00 in the Cole–Cole diagram Controlled the slope at the low temperatures/high frequencies in the |G| master curve and the height of the pinnacle point of the Cole–Cole diagram Controlled the slope at the high temperatures/low frequencies of the |G| master curve where the higher the value of b, the higher the values of g and |G|

d

b

Fig. 3a. Graphical representation of the model’s parameters in term of Cole–Cole diagram.

and a yielded the same values for all tested samples [7]. Earlier, Pellinen et al. used the 2S2P1D Model and found good agreement between the measured and descriptive data of asphalt mixtures [8]. In this study, the complex moduli and phase angles (from experiments) are randomly or manually shifted to construct master curves. This is due to the high degree of freedom of this approach [11], although the lack of a functional form of the shift factors versus temperature relationship makes it impossible to describe any shifting outside the range of the experimental data. Most of the shift factor equations are basically empirical and might not be strictly obeyed by all materials [12]. The model’s parameters for all samples are changed using a trial-and-error method until good complex modulus and phase angle master curves, Cole–Cole diagram and Black diagram are obtained. The model’s parameters are fitted manually where the initial values of those parameters were selected based on the work done by Di Benedetto et al. The trial-and-error process is done using an Excel spread sheet until the most suitable parameters are obtained. The reference temperature, Tref is arbitrarily chosen at a temperature of 10 °C for the sake of comparison with Di Benedetto et al.’s work. 3. Goodness-of-fit

Fig. 3b. Graphical representation of the model’s parameters in term of complex modulus master curve.

   0  G0  ð1 þ AðxÞÞ G  ðBðxÞÞ þi ¼ G0 þ DEN DEN

ð10Þ

where G0 = Gg  G0 and DEN = (1 + A(x))2 + B2(x). It has to be emphasised that this model only needs seven parameters to entirely determine the rheological properties of materials. However, G0 for bitumens is commonly very close to zero and therefore the parameters can be reduced to six. G0, Gg, k, h, a, and b are illustrated graphically in Fig. 3a and b respectively. The Cole–Cole diagram2 (Fig. 3a) is not shown in Section 5, as rheological tests are not conducted at very low temperatures (less than 5 °C). Table 1 explains the functions of k, h, a and b in greater detail. Meanwhile the phase angle, d, is shown as

d ¼ tan1

 00  G G0

ð11Þ

where G0 and G00 are storage and loss moduli. Di Benedetto et al. conducted dynamic tests on nine binders and four asphalt mixtures (with one mixture design). They found that the model fitted the experimental data reasonably well, even though anomalies were still seen particularly for the phase angles between 50° and 70° [5,6]. The model has also been used to describe the rheological properties of bitumen–filler mastics and parameters such as k, h 2

The Cole–Cole diagram is a graph of the loss modulus G00 as a function of the storage modulus G0 [10].

Five different statistical methods were used to indicate the goodness-of-fit between the measured and descriptive data and can be shown as follows [13–15]. 3.1. Standard error ratio (Se/Sy) The standard error of estimation, Se and standard error of deviation, Sy can be defined as follows:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P b Þ2 ðY  Y Se ¼ ðn  qÞ

ð12Þ

and

sP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðY  YÞ2 Sy ¼ ðn  1Þ

ð13Þ

where n is sample size, q is the number of independent variables in b is descriptive |G| and Y is the mean the model, Y is tested |G|, Y value of tested |G|. A small value of Se/Sy indicates a good correlation between the measured and descriptive data [13]. 3.2. Coefficient of determination (R2)

R2 ¼ 1 

 2 ðn  qÞ Se  ðn  1Þ Sy

ð14Þ

where the parameters are as defined in Eqs. (12) and (13). For a perfect fit, R2 = 1. The criteria for the Se/Sy and R2 goodness-of-fit statistics are shown in Table 2.

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Table 2 Criteria of the goodness of fit statistics [13].

10

10

Criteria

R2

Se/Sy

Excellent Good Fair Poor Very poor

P0.90 0.70–0.89 0.40–0.69 0.20–0.39 60.19

60.35 0.36–0.55 0.56–0.75 0.76–0.89 P0.90

Table 3 The model parameters for unaged unmodified bitumens. Pen grade

G0 (Pa)

Gg (Pa)

k

h

a

s

b

10/20 35/50 40/60 70/100 160/220

0 0 0 0 0

1.00E+09 1.00E+09 1.00E+09 1.00E+09 1.00E+09

0.22 0.22 0.22 0.22 0.22

0.64 0.64 0.64 0.64 0.64

4.00 4.00 4.00 4.00 4.00

6.00E03 1.50E03 3.00E04 6.00E05 3.50E05

90 90 90 90 45

Complex Modulus, |G*| (Pa)

8

10

6

10

4

10

10/20 10/20 (2S2P1D) 40/60 40/60 (2S2P1D) 160/220 160/220 (2S2P1D)

2

10

0

10

-10

-5

10

5

0

10

10

10

10

10

Reduced Frequency (Hz) 90

Table 4 Comparisons between NTEC and Di Benedetto et al.’s model parameters. PG

G0 (Pa)

Gg (Pa)

k

h

a

s

b

1 2 1 2 3

10/20

0 0 0 0 0

1.00E+09 2.00E+09 1.00E+09 2.00E+09 9.00E+08

0.22 0.19 0.22 0.18 0.21

0.64 0.52 0.64 0.55 0.55

4.00 2.20 4.00 2.00 2.30

6.00E03 2.50E03 1.00E04 1.90E03 8.00E05

90 800 200 320 400

40/60 50/70 50/70

1 – NTEC, 2 – Olard et al. [6], 3 – Delaporte et al. [7].

3.3. The discrepancy ratio (ri)

Phase Angle (Degrees)

a

80

Sourcea



ri ¼

jG jp jG jm

60 50 40

10/20 10/20 (2S2P1D) 40/60 40/60 (2SP1D) 160/220 160/220 (2S2P1D)

30 20 10

ð15Þ

0

where |G|p and |G|m are the descriptive and measured complex modulus respectively. The subscript i denotes the data set number. For a perfect fit, ri = 1.

-10

! ei ; R

ð17Þ

Ri ¼

jG jp P jG jm

jG jm =jG jp

for jG jp j < jG jm

10

70 60 50 40 30 20

i¼1

for

10

10

Reduced Frequency (Hz)

Phase Angle (Degrees)

ð16Þ

3.5. The Average Geometric Deviation (AGD)

jG jp =jG jm

5

10

80

where N is the total number of sets of data, and for a perfect fit, MNE = 0.

(

0

10

90

    N   100 X jG jm  jG jp  MNE ¼    N i¼1  jG jp

N Y

-5

10

3.4. The Mean Normalised Error (MNE)

AGD ¼

70

10

) ð18Þ

where AGD = 1 for a perfect fit. The implementation of statistical analysis in this study was done with the aid of the MATLAB (The MatWorks Inc., Natick, Massachusetts) program. 4. Experimental programs 4.1. Materials Various types of materials from conducted experiments and the Nottingham Transportation Engineering Centre (NTEC) DSR database were used in this study. These include unaged and aged unmodified bitumens, polymer-modified bitumens

0 0 10

10/20 10/20 (2S2P1D) 40/60 40/60 (2S2P1D) 160/220 160/220 (2S2P1D) 2

10

4

10

6

10

8

10

10

10

Complex Modulus, |G*| (Pa) Fig. 4. (a) Complex modulus, (b) phase angle and (c) Black diagrams of unaged unmodified bitumens (Tref = 10 °C).

(PMBs) and bitumen–filler mastics. 70/100 penetration grade bitumen was mixed with a plastomeric ethylene-vinyl acetate (EVA) polymer and an elastomeric styrene butadiene styrene (SBS) polymer to produce various polymer-modified bitumens. Three polymer contents (by mass) at 3%, 5% and 7% were used to create several combinations of the EVA– and SBS–polymer-modified bitumens [16,17]. The unaged bitumen–filler mastics were taken from previous research conducted by Wu [18,19] and Liao [20,21]. The first group consists of three different fillers namely gritstone, limestone and cement, mixed with bitumen at 40% (by

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N.I. Md. Yusoff et al. / Construction and Building Materials 38 (2013) 395–406 Table 5 The model parameters for bitumen filler mastics (40% by volume). Materiala

G0 (Pa)

Gg (Pa)

k

h

a

s

b

Gritstone Limestone Cement

6000 10 120

4.00E+09 2.00E+09 2.00E+09

0.21 0.21 0.21

0.55 0.55 0.55

4.00 2.30 2.30

1.10E04 4.00E04 3.00E04

250 250 250

Fillers were mixed with a 100/150 penetration grade bitumen.

Table 6 The model parameters for bitumen–filler mastics (by weight).

a

Fillera

(%)

G0 (Pa)

Gg (Pa)

k

h

a

s

b

Gritstone

35 65

0 8

1.00E+09 1.00E+09

0.21 0.21

0.59 0.59

2.30 2.30

7.00E04 5.20E03

100 100

Limestone

35 65

0 5

1.20E+09 1.50E+09

0.21 0.21

0.59 0.59

2.30 2.30

7.00E04 2.00E03

150 150

Cement

35 65

0 5

1.30E+09 1.40E+09

0.21 0.21

0.55 0.55

2.30 2.30

1.50E04 1.50E03

250 250

Complex Modulus, |G*| (Pa)

a

10

10

8

10

6

10

4

10

cement cement (2S2P1D) limestone limestone (2S2P1D) gritstone gritstone (2S2P1D)

2

10

0

10 -10 10

-5

0

10

5

10

90

cement cement (2S2P1D) limestone limestone (2S2P1D) gritstone gritstone (2S2P1D)

Mode of loading: controlled-strain. Temperatures: 10–75 °C (with the interval of 5 °C). Frequencies: 0.01–10 Hz. Spindle geometries: 8 mm (diameter) and 2 mm gap (10–35 °C) and 25 mm (diameter) and 1 mm gap (25–75 °C).  Strain amplitude: within the LVE response, dependent on |G| of each material used.

   

For the construction of the |G| and d master curves, Tref is arbitrarily taken at 10 °C. The curve is shifted manually, without assuming any shift factor forms.

70 60 50 40 30 20 10 0 -10 10

-5

0

10

5

10

10

10

10

Reduced Frequency (Hz) 90 80

Phase Angle (Degrees)

The rheological properties of various materials were determined using dynamic mechanical methods consisting of temperature and frequency sweeps in an oscillatory-type testing mode performed within the LVE region. The oscillatory tests were conducted on a Bohlin Gemini dynamic shear rheometer (DSR) using two parallel plate testing geometries consisting of 8 mm diameter plates with a 2 mm testing gap and 25 mm diameter plates with a 1 mm testing gap. In this study, the samples were prepared using a hot pour method and a silicone mould method. In the hot pour method, the gap between the upper and lower plates was set to a desired height of 50 lm plus the required testing gap, either at the proposed testing temperature or at the mid-point of an expected testing temperature range. Once the gap has been set, a sufficient quantity of hot bitumen (typically between 100 and 150 °C) was poured onto the lower plate of the DSR to ensure a slight excess of material appropriate to the chosen testing geometry. The upper plate of the DSR was then gradually lowered to the required nominal testing gap plus 50 lm. The bitumen that has been squeezed out between the plates was then trimmed flush to the edge of the plates using a hot spatula or blade. After trimming, the gap was closed by a further 50 lm to achieve the required testing gap as well as a slight bulge around the circumference of the testing geometry (periphery of the test specimen) [22]. Meanwhile, for the silicone mould method, the hot bitumen is poured into either an 8 mm or 25 mm diameter silicone mould of height approximately 1.5 times that of the recommended testing gap for the two geometries, namely 3 mm and 1.5 mm for the 8 mm and 25 mm geometries. The testing gap is set at a height of 50 lm plus 1 mm or 2 mm. Once the bitumen has cooled, either by means of short-term refrigeration or by natural cooling, the bitumen disc (typically at ambient temperatures) is removed from the mould and centred on the lower plate of the DSR. The upper plate is then lowered to the required gap plus 50 lm, the excess bitumen is trimmed with a hot spatula and the gap further closed to its final testing height [22]. Amplitude sweep tests are first conducted to determine the LVE region of the bituminous binders based on the point where |G| has decreased to 95% of its initial value [23]. After obtaining the limiting strain, frequency sweep tests are carried out under the following test conditions:

Phase Angle (Degrees)

80

4.2. Dynamic mechanical analysis

10

Reduced Frequency (Hz)

Fillers were mixed with a 50 penetration grade bitumen.

weight) [18,19]. The second group, originally from the work of Liao [20,21], also consists of gritstone, limestone and cements mixed with bitumens. Two filler contents (by mass) at 35% and 65% were used to create six combinations of bitumen– filler mastics. Two bitumens, 50 and 100/150 penetration grade, were used as the base bitumens [18–21].

10

10

70 60 50 40 cement

30

cement (2S2P1D) limestone

20

limestone (2S2P1D) gritstone

10

gritstone (2S2P1D)

0 0 10

10

2

10

4

10

6

10

8

10

10

Complex Modulus, |G*| (Pa) Fig. 5. (a) Complex modulus, (b) phase angle and (c) Black diagrams of unaged bitumen–filler mastics (Tref = 10 °C).

4.3. Ageing procedures Short- and long-term laboratory ageing of the unmodified bitumens and polymer-modified bitumens were performed using the Rolling Thin Film Oven Test (RTFOT, ASTM D 2872) and the pressure ageing vessel (PAV, AASHTO PP1) respectively. The standard ageing procedures of 163 °C and 75 min for the RTFOT and 100 °C, 2.1 MPa and 20 h for the PAV were used [17]. Meanwhile, the Thin Film

400

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Table 7 The model parameters for bitumen-filler mastics obtained by Delaporte et al. [24] (40% by volume). Material

G0 (Pa)

Gg (Pa)

k

h

a

s

b

B5070 B5070LS40 B5070LSS40 B5070S40

0 150 200 500

9.00E+08 6.00E+09 7.00E+09 9.00E+09

0.21 0.21 0.21 0.21

0.55 0.55 0.55 0.55

2.30 2.30 2.30 2.30

1.00E04 6.00E05 6.00E05 5.00E04

400 400 1200 30,000

Oven Test (TFOT, ASTM D1754) was used to age the bitumen–filler mastics at 1, 3, 5, 10 and 20 h of ageing times [20,21]. All the aged binders then subjected to DMA to evaluate changes in their rheological properties.

5. Results and discussion 5.1. Unaged unmodified bitumens Table 3 shows the 2S2P1D Model parameters for different types of penetration grade, unaged and unmodified bitumens used in this study. The parameters are fitted manually using a trial-and-error method until a good curve is obtained. As aforementined, the same parameters are used to plot the complex modulus and phase angle master curves, the Black diagram and the Cole–Cole diagram. However, the Cole–Cole diagram is not shown here because this study did not involve measurements at very low temperatures. In general, the unaged bitumens tend to show Newtonian behaviour at low frequencies and/or high temperatures. G0 is too small and can be neglected and only six parameters of the model need to be determined. Five of the model’s parameters; Gg, G0, k, h, and a are the same for all samples, regardless of the penetration grade of the bitumens. The b values for 10/20, 35/50, 40/60 and 70/100 penetration grade bitumens are the same due to the fact that they have the same slope at low frequencies and/or high temperatures. As expected the 160/220 penetration grade bitumen with the lowest viscosity value has the smallest b. The presence of a higher asphaltene content, which contributes to the hardness of bitumens, plays a significant role in determining b. It increases as the asphaltenes content increases. Table 4 shows a comparison between the NTEC dataset and the work of Di Benedetto et al. [5–7]. Several values are noticeably different when examining the NTEC dataset and that of Di Benedetto et al. For instance, the Gg values obtained by Di Benedetto et al.

vary from 0.9  109 to 2  109 Pa, whilst the value of 1  109 Pa was used in this study. The k and h values are almost the same, even though a slightly higher value of h is observed. Di Benedetto et al. show higher and lower values of b and d compared to the NTEC dataset. These findings might be attributed to several factors. First, different crude sources used by the respective studies might play a significant role in influencing the viscosity of the unaged and unmodified bitumens. Secondly, the use of different types of rheometers may affect the precision of rheological data. Di Benedetto et al. used the annular shear rheometer (ASR), while the dynamic shear rheometer (DSR) is used in this study. Wu and Airey [19] show that the use of ASR allows continuous measurements of |G| over seven decades (in norm), from about 1  103 to 1  1010 Pa. It is worth mentioning that a can be of thought as an appropriate parameter to assess the occurrence of measurement errors of |G|. A higher value of a indicates that |G| is exposed to measurement errors and vice versa. Finally, examples of the 2S2P1D Model of |G| and d master curves and the Black diagram for some of the tested unaged and unmodified bitumens (10/20, 40/60 and 160/ 220) are presented in Fig. 4. From these figures, it is observed that the 2S2P1D Model satisfactorily fits the experimental results.

5.2. Unaged bitumen–filler mastics Mastics, composed of bitumen and filler, is an intermediate material between bitumen and an asphalt mixture [7]. The presence of mineral filler cannot be neglected, as it brings important effects such as improved strength, plasticity, amount of voids, resistance to water action and resistance to weathering [20]. Three types of fillers; cement, gritsone and limestone, are used in this study [20,21]. The model parameters for each sample are shown in Table 5 (by volume) and Table 6 (by weight). Meanwhile, Fig. 5 shows the plot of |G| and d master curves and the Black diagram of measured and descriptive data of the unaged bitumen–filler mastics (Table 5). In general, the k, h and b values are found to be the same for all samples used in this study. This indicates that the samples have the same slope at low frequencies and/or high temperatures. The presence of mineral filler is clearly observed where the G0 values are higher than those of the unaged and unmodified bitumens. Results show that the G0 values are double for limestone and cement

Table 8 The model parameters for unaged polymer-modified bitumens. Percent

Modifier

Source

G0 (Pa)

Gg (Pa)

k

h

a

s

0

Control

Middle East Russian Venezuelan

0 0 0

1.00E+09 1.00E+09 1.00E+09

0.21 0.21 0.21

0.55 0.55 0.55

2.30 3.50 2.30

5.00E05 5.00E05 1.50E05

300 150 200

3 5 7

EVA

Middle East

0 0 0

1.00E+09 1.00E+09 1.00E+09

0.21 0.21 0.21

0.55 0.55 0.55

3.50 5.00 5.00

7.00E05 7.00E05 1.00E04

900 2500 6000

3 5 7

Russian

0 0 0

1.00E+09 1.00E+09 1.00E+09

0.21 0.21 0.21

0.55 0.55 0.55

3.50 3.00 2.30

5.00E05 6.00E05 6.00E05

600 1000 6000

3 5 7

Venezuelan

0 0 0

1.00E+09 1.00E+09 1.00E+09

0.21 0.21 0.21

0.55 0.55 0.55

2.30 2.30 2.30

2.00E05 2.00E05 2.00E05

1200 7000 10,000

Russian

0 0 0

1.00E+09 1.00E+09 1.00E+09

0.21 0.21 0.21

0.55 0.55 0.55

2.30 2.30 2.30

3.00E05 3.00E05 5.00E06

1500 3000 20,000

VE

0 0 0

1.00E+09 1.00E+09 1.00E+09

0.21 0.21 0.21

0.55 0.55 0.55

2.30 2.30 2.30

2.00E05 2.00E05 2.00E05

700 1300 2500

3 5 7 3 5 7

SBS

b

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N.I. Md. Yusoff et al. / Construction and Building Materials 38 (2013) 395–406 Table 9 The model parameters for the aged unmodified bitumens.

10

10

8

Methoda Source

6

Unaged RTFOT PAV

Middle East 0 0 0

1.00E+09 0.21 0.55 2.30 5.00E05 300 1.00E+09 0.21 0.55 4.00 7.00E05 700 1.00E+09 0.21 0.55 5.00 4.00E04 1000

Unaged RTFOT PAV

Russian

0 0 0

1.00E+09 0.21 0.55 3.50 5.00E05 150 1.00E+09 0.21 0.55 5.00 5.00E05 400 1.00E+09 0.21 0.55 5.00 4.00E04 1500

Unaged RTFOT PAV

Venezuelan 0 0 0

1.00E+09 0.21 0.55 2.30 1.50E05 200 1.00E+09 0.21 0.55 3.50 3.20E05 500 1.00E+09 0.21 0.55 4.50 1.00E04 1500

Complex Modulus, |G*| (Pa)

10

10

4

10

3% EVA 3% EVA (2S2P1D) 5% EVA 5% EVA (2S2P1D) 7% EVA 7% EVA (2S2P1D) 70/100 70/100 (2S2P1D)

2

10

0

10

10

-10

10

-5

0

5

10

10

a

10

10

Reduced Frequency (Hz) 90

3% EVA 3% EVA (2S2P1D) 5% EVA 5% EVA (2S2P1D) 7% EVA 7% EVA (2S2P1D) 70/100 70/100 (2S2P1D)

Phase Angle (Degrees)

80 70 60 50 40 30 20 10 0 -10 10

-5

0

10

5

10

10

10

10

Reduced Frequency (Hz) 90

Phase Angle (Degrees)

80 70 60

30 20 10 0 0 10

k

h

a

s

b

RTFOT – Rolling Thin Film Oven Test, PAV – Pressure Ageing Vessel.

Further investigation was done to fit the 2S2P1D Model using the bitumen–filler mastics at 35% and 65% (by weight). The results are shown in Table 6. At 35%, all samples tend to exhibit Newtonian behaviour at very low frequencies where the G0 values are equal to zero. However, as the percentage increases up to 65%, the presence of mineral filler slowly appears at low frequencies even though these values are apparently small. This indicates that the considerably higher |G| for the 65% bitumen–filler mastics might be caused by the filler skeleton being present in the bitumen–filler system. The 2S2P1D Model is also able to satisfactorily describe the rheological properties of the bitumen–filler mastics (Fig. 5). Delaporte et al. [24] used different types of bitumen–filler mastics to evaluate the influence of different fillers on the 2S2P1D Model parameters. Fillers such as limestone (noted as LS), a mixture of limestone and ultrafine particles (noted as LSS) and a new type of filler composed of ultrafine particles (noted as S) mixed with a 50/70 penetration grade bitumen affect the model parameters, as shown in Table 7. It was observed that a, k and h are the same regardless of the selected filler mastics when a comparison between the NTEC (Table 5) and Delaporte et al. [24] (Table 7) rheological data was made. These three parameters were not dependent on the filler content and they were fixed by the binder. The G0 values are different depending on the type of filler and binder used. Delaporte et al. [24] observed higher Gg compared to the values obtained in Table 5. This result is expected since they used a harder bitumen (50/70 penetration grade) compared to the softer 100/150 penetration grade bitumen used in this study. The b values reported in [24] are also higher compared to the NTEC data. 5.3. Unaged polymer modified bitumens

50 40

G0 (Pa) Gg (Pa)

3% EVA 3% EVA (2S2P1D) 5% EVA 5% EVA (2S2P1D) 7% EVA 7% EVA (2S2P1D) 70/100 70/100 (2S2P1D) 2

10

4

10

6

10

8

10

10

10

Complex Modulus, |G*| (Pa) Fig. 6. (a) Complex modulus, (b) phase angle and (c) Black diagrams of unaged polymer-modified bitumens (Tref = 10 °C).

bitumen–filler mastics, whereas the G0 values for gritstone bitumen–filler mastics are four times higher compared to the unaged and unmodified bitumens. The G0 values for bitumen–filler mastics can no longer be neglected. Delaporte et al. [7] found that the G0 values are higher due to the existence of solid contacts between particles.

Table 8 shows the model parameters of EVA and SBS-modified bitumens and, for simplification, Gg is taken as 1  109 Pa. The EVA copolymer was blended with three different crude sources whilst the SBS was blended with two crude sources. The result shows that Gg, G0, k and h are similar for all unaged polymer-modified bitumens but varies in the values of a, s and b. a is more consistent for the SBS-modified bitumens than those of the EVAmodified bitumens. This is probably due to the presence of a semi-crystalline structure at different temperatures increasing the complexity of the mixtures. Interested readers should consult the works of Airey for a thorough discussion of the rheological properties of EVA and SBS PMBs in the LVE region [2,16,17]. Like the unaged unmodified bitumens, the testing errors might also occur on |G| data of PMBs. Results show that the 2S2P1D Model is able to simulate the effect of polymer modification as b increases when the percentage of modifier used is increased. The presence of polymer modification increases the mixture’s resistance to flow. The plot of |G| and d master curves and the Black diagram of one unaged polymer modified bitumen (3% EVA) used in this study are shown in Fig. 6. Only 3% EVA-modified bitumen is discussed

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N.I. Md. Yusoff et al. / Construction and Building Materials 38 (2013) 395–406 Table 10 The model parameters for the aged bitumen–filler mastics (40% by volume).

10

Complex Modulus, |G*| (Pa)

10

8

10

k

h

a

s

b

Gritstone

1 3 5 10 20

200 12,000 8000 600 600

1.40E+09 1.60E+09 1.70E+09 1.40E+09 1.00E+09

0.21 0.21 0.21 0.21 0.21

0.55 0.55 0.55 0.55 0.55

2.30 2.30 2.30 2.30 2.30

9.50E04 6.00E04 8.00E04 2.00E03 1.00E02

150 150 150 150 600

Limestone

1 3 5 10 20

8 10 10 10 3000

1.50E+09 1.50E+09 1.50E+09 1.20E+09 1.30E+09

0.21 0.21 0.21 0.21 0.21

0.55 0.55 0.55 0.55 0.55

2.30 2.30 2.30 2.30 2.30

5.50E04 8.00E04 1.00E03 2.00E03 2.00E03

150 150 150 150 800

6

4

10

unaged unaged (2S2P1D) RTFOT RTFOT (2S2P1D) PAV PAV (2S2P1D)

2

10

10 -10 10

as most of the rheological models are also unable to describe the behaviour of highly modified bitumens.

-5

0

10

5

10

10

10

10

5.4. Aged unmodified bitumen

Reduced Frequency (Hz) 90

unaged unaged (2S2P1D) RTFOT RTFOT (2S2P1D) PAV PAV (2S2P1D)

80

Phase Angle (Degrees)

Time (h) G0 (Pa) Gg (Pa)

10

0

70 60 50 40 30 20 10 0 -10 10

-5

0

10

5

10

10

10

5.5. Aged bitumen–filler mastics

90 80 70 60 50 40 30 20 10 0 0 10

unaged unaged (2S2p1D) RTFOT RTFOT (2S2p1D) PAV PAV (2S2P1D) 10

2

10

4

10

6

An important change in behaviour is experimentally observed due to the binder ageing mainly at high temperatures and low frequencies (Table 9). Fig. 7 shows the |G| and d angle master curves and Black diagram of aged unmodified bitumens. The parameters such as Gg, G0, k and h are consistent for all materials. This indicates that these materials are independent of the ageing process. Moreover, b that is linked to Newtonian viscosity g of the model has a large influence in this domain of behaviour. The influence of binder ageing on the model’s parameters mainly affects b. The values of b increase from unaged to aged unmodified bitumens. As the materials become harder, a is increased. Olard et al. [6] had similar findings, where a increased from 2.3 to 3.0 when the curve is plotted using the aged 50/70 unmodified penetration grade bitumen. Therefore, it can be inferred that a also can be used as an ‘‘ageing’’ indicator, where the aged samples show higher a values compared to the unaged samples.

10

Reduced Frequency (Hz)

Phase Angle (Degrees)

Material

10

8

10

10

Complex Modulus, |G*| (Pa) Fig. 7. (a) Complex modulus, (b) phase angle and (c) Black diagrams of aged unmodified bitumens (Tref = 10 °C).

here for brevity, with the understanding that the commentary applies to all studied samples. It is observed that the model is only capable of describing the complex modulus master curves satisfactorily, but not the phase angle master curve. The quality of data is further evaluated using the Black diagram, and it is found that the model has its limitations, as it partially confirms the rheological properties of polymer-modified bitumens. This finding is expected

An attempt has also been made to model the aged bitumen–filler mastics using the 2S2P1D Model (Table 10). Only two filler mastics (gritstone and limestone) are used in this ageing study, as cement bitumen–filler mastics do not produce good data. This model is able to simulate the aged bitumen–filler mastics with the same quality as the aged unmodified bitumens. The gritstone and limestone bitumen–filler mastics show the same values of k, h, a and b for the ageing time from 1 to 10 h. However, as the ageing time is increased up to 20 h, b increases up to 600 (for gritstone) and 800 (for limestone) respectively. Like the unaged bitumen–filler mastics, the values of Gg need to be taken into account due to the existence of solid contacts between particles in the mixtures. The acidic gritstone bitumen–filler mastic shows a slight difference of Gg with times compared to the limestone bitumen–filler mastic. As expected, |G| of the gritstone bitumen–filler mastic increases as the ageing time increases. It is believed that this phenomenon is caused by the adsorption of heavier fractions from bitumen onto the mineral surface of the limestone bitumen–filler mastics [21]. Fig. 8 shows an example of the rheological data (|G| and d) and the Black diagram of aged cement bitumen–filler mastics used in this study. Graphically it is observed that the 2S2P1D Model describes the measurements satisfactorily. 5.6. Aged polymer-modified bitumens The model parameters for all the aged PMBs are shown in Table 11. Meanwhile, Fig. 9 shows the simulations of |G| and d

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N.I. Md. Yusoff et al. / Construction and Building Materials 38 (2013) 395–406 Table 11 The model parameters for the aged polymer-modified bitumens. 8

Complex Modulus, |G*| (Pa)

10

Method (%) Modifier Source G0 Gg (Pa) (Pa) RTFOT

3 5 7

PAV

3 5 7

RTFOT

5 7

PAV

5 7

RTFOT

5 7

PAV

5 7

RTFOT

3 5 7

PAV

3 5 7

RTFOT

3 5 7

PAV

3 5 7

6

10

1 Hr 1 Hr (2S2P1D) 3 Hrs 3 Hrs (2S2P1D) 5 Hrs 5 Hrs (2S2P1D) 10 Hrs 10 Hrs (2S2P1D) 20 Hrs 20 Hrs (2S2P1D)

4

10

2

10

-10

-5

10

0

10

5

10

10

10

10

Reduced Frequency (Hz) 90

1 Hr 1 Hr (2S2P1D) 3 Hrs 3 Hrs (2S2P1D) 5 Hrs 5 Hrs (2S2P1D) 10 Hrs 10 Hrs (2S2P1D) 20 Hrs 20 Hrs (2S2P1D)

Phase Angle (Degrees)

80 70 60 50 40 30

EVA

ME

RU

VE

SBS

RU

VE

k

H

a

s

b

0 0 0

1.00E+09 0.21 0.55 4.50 1.00E04 1500 1.00E+09 0.21 0.55 5.00 3.00E04 2000 1.00E+09 0.21 0.55 5.00 2.00E04 10,000

0 0 0

1.00E+09 0.21 0.55 7.00 5.00E04 2500 1.00E+09 0.21 0.55 7.00 2.00E03 3000 1.00E+09 0.21 0.55 7.00 2.00E03 10,000

0 0

1.00E+09 0.21 0.55 5.00 8.00E05 8000 1.00E+09 0.21 0.55 5.00 2.00E04 20,000

0 0

1.00E+09 0.21 0.55 5.00 8.00E04 3000 1.00E+09 0.21 0.55 7.00 9.00E04 25,000

0 0

1.00E+09 0.21 0.55 5.00 1.00E04 8000 1.00E+09 0.21 0.55 3.00 3.00E05 60,000

0 0

1.00E+09 0.21 0.55 7.00 5.00E04 10,000 1.00E+09 0.21 0.55 7.00 4.00E04 60,000

0 0 0

1.00E+09 0.21 0.55 5.00 8.00E05 1.00E+09 0.21 0.55 5.00 9.00E05 1.00E+09 0.21 0.55 5.00 3.00E05

1500 1000 5000

0 0 0

1.00E+09 0.21 0.55 7.00 4.50E04 1.00E+09 0.21 0.55 7.00 2.00E04 1.00E+09 0.21 0.55 5.00 3.00E04

1500 3000 5000

0 0 0

1.00E+09 0.21 0.55 4.00 3.00E05 1.00E+09 0.21 0.55 5.00 1.50E05 1.00E+09 0.21 0.55 5.00 3.00E05

1800 8000 8000

0 0 0

1.00E+09 0.21 0.55 5.00 1.00E04 1.00E+09 0.21 0.55 5.00 3.00E05 1.00E+09 0.21 0.55 5.00 3.00E04

6000 5000 8000

20 10 0 -10 10

-5

0

10

5

10

10

10

10

Reduced Frequency (Hz) 90

Phase Angle (Degrees)

80 70 60 50 40 1 Hr 1 Hr (2S2P1D) 3 Hrs 3 Hrs (2S2P1D) 5 Hrs 5 Hrs (2S2P1D) 10 Hrs 10 Hrs (2S2P1D) 20 Hrs 20 Hrs (2S2P1D)

30 20 10 0 0 10

2

10

4

10

6

10

creases, the polymer-rich phase plays a more dominant role in determining the rheological properties of bitumens. The effect of ageing can be seen as an alteration of the Black diagram waves associated with the different EVA crystalline structures and a general reduction in polymer modification [2]. Olard and Di Benedetto [5] found an identical problem for the PMBs and they called this property the partial time–temperature superposition principle (PTTSP) as shifting procedures give only a unique and continuous |G| master curves. However, the PTTSP might or might not be applicable for certain conditions (or percentage) of polymer modifications and not suitable to be used on highly polymer-modified bitumens. It was observed that the TTSP and PTTSP are unable to describe the branching and discontinuous waves in the master curves that show the presence of highly semi-crystalline EVA copolymer [2]. 6. Statistical analysis 6.1. Graphical representation

8

10

10

10

Complex Modulus, |G*| (Pa) Fig. 8. (a) Complex modulus, (b) phase angle and (c) Black diagrams of aged bitumen–filler mastics (Tref = 10 °C).

master curves and the Black diagram of one of the aged SBS PMBs used in this study. The Gg, G0, k and h values are the same for all samples. The b value which is linked to the Newtonian viscosity g of the 2S2P1D Model has a large influence in this domain of behaviour, where b increases as polymer modification is increased. For some samples, the changes of a value depend on the percentage of modification and the crude source. As the temperature in-

Comparisons between the measured and descriptive |G| and d data of all tested samples are shown in Fig. 10a and b. The measured and descriptive values are equated by matching two values in a normal scale graph. If the matching points are fairly distributed around the equality line, then the model should have a good correlation with the measured data [13]. Graphically, it is observed that the 2S2P1D Model is able to accurately describe |G| values for the unaged and aged unmodified bitumens and bitumen–filler mastics (Fig. 10a). However, the descriptive |G| of the unaged PMBs are slightly scattered from the equality line particularly for highly modified bitumens. Meanwhile, Fig. 10b shows a comparison between measured and descriptive d. It is found that the unaged and aged unmodified bitumens show good correlation

404

N.I. Md. Yusoff et al. / Construction and Building Materials 38 (2013) 395–406 10

10

10

Predicted Complex Modulus, |G*| (Pa)

Complex Modulus, |G*| (Pa)

10

8

10

6

10

RTFOT RTFOT (2S2P1D) PAV PAV (2S2P1D) 3% SBS RTFOT 3% SBS RTFOT (2S2P1D) 3% SBS PAV 3% SBS PAV (2S2P1D)

4

10

2

10

unaged unmodified bitumens aged unmodified bitumens unaged bitumen-filler mastics aged bitumen-filler mastics unaged polymer modified bitumens aged polymer modified bitumens

8

10

6

10

4

10

2

10

a 0

0

10

10 -10 10

-5

0

10

5

10

10

10

0

10

2

10

90

80

80

Predicted Phase Angle (Degrees)

Phase Angle (Degress)

90

70 60 50 40

20 10

RTFOT RTFOT (2S2P1D) PAV PAV (2S2P1D) 3% SBS RTFOT 3% SBS RTFOT (2S2P1D) 3% SBS PAV 3% SBS PAV (2S2P1D)

0 -10 10

-5

0

10

5

10

10

6

8

10

10

10

10

Measured Complex Modulus, |G*| (Pa)

Reduced Frequency (Hz)

30

4

10

10

10

10

Reduced Frequency (Hz)

70 60 50 40 30 20

b

10 0 0

10

20

30

40

50

60

70

80

90

Measured Phase Angle (Degrees) Fig. 10. Graphical comparison between measurements and descriptive of (a) complex modulus and (b) phase angles data.

90

Phase Angle (Degrees)

80 70

6.2. Goodness-of-fit statistics

60 50 40 30 20 10 0 0 10

RTFOT RTFOT (2S2P1D) PAV PAV (2S2P1D) 3% SBS RTFOT 3% SBS RTFOT (2S2P1D) 3% SBS PAV 3% SBS PAV (2S2P1D) 10

2

10

4

10

6

10

8

10

10

Reduced Frequency (Hz) Fig. 9. (a) Complex modulus, (b) phase angle and (c) Black diagrams of aged polymer-modified bitumens (Tref = 10 °C).

between measured and descriptive data. However, the descriptive data of unaged and aged PMBs, and aged bitumen–filler mastics are slightly scattered from the equality line. Di Benedetto et al. also found the same problem, where this model was partially satisfied in fitting |G| and d master curves of PMBs [5,6].

The goodness-of-fit statistics between the measured and descriptive |G| and d data are shown in Tables 12 and 13 respectively. In summary, the Se/Sy for all samples have ‘‘excellent’’ correlations between the measured and descriptive |G| data. Similarly, the Se/Sy for d of unaged and aged unmodified bitumens and unaged bitumen–filler mastics show ‘‘excellent’’ correlations. Conversely, the Se/Sy for aged bitumen–filler mastics and PMBs have ‘‘good’’ correlations, whereas the Se/Sy for unaged PMBs shows ‘‘poor’’ correlation between measured and the descriptive data. All samples show ‘‘excellent’’ correlations of R2 for |G| data. However for d, the R2 has ‘‘good’’ correlations for unaged and aged PMBs. As expected, the unaged PMBs show ‘‘poor’’ correlations. However, it is observed that the use of R2 and Se/Sy is not really suitable for describing mismatches between measured and descriptive data. The discrepancy ratio ri is used to observe the model data’s tabulation from the equality line. A perfect value is equal to 1. When the discrepancy ratio is larger or smaller than 1, it measures how much wider the model interval had to be to cover the observed number of cases [15]. With the interval of 1 ± 0.04 used in this study, the unaged unmodified bitumen data dispersed close to

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N.I. Md. Yusoff et al. / Construction and Building Materials 38 (2013) 395–406 Table 12 Statistical analysis for the complex modulus master curves. Se/Sy

R2

AGD

96.27 82.89

0.171 0.069

0.971 0.995

1.090 1.092

8.733 9.421

76.31 45.55

84.93 57.30

0.145 0.114

0.979 0.987

1.116 1.212

11.318 19.197

51.13 62.75

59.05 71.96

0.102 0.171

0.990 0.971

1.253 1.161

25.631 16.034

Se/Sy

R2

AGD

MNE

Sample

Condition

Discrepancy ratio, ri (%) 0.96–1.04

0.92–1.08

0.88–1.12

0.84–1.16

0.80–1.20

Unmodified bitumen

Unaged Aged

26.43 39.29

48.87 64.88

72.10 75.00

87.78 80.51

Bitumen–filler mastics

Unaged Aged

25.30 12.62

47.82 25.00

65.55 35.27

PMBs

Unaged Aged

13.69 17.37

29.46 35.19

41.43 50.19

MNE

Table 13 Statistical analysis for the phase angle master curve. Sample

Condition

Discrepancy ratio, ri (%) 0.96–1.04

0.92–1.08

0.88–1.12

0.84–1.16

0.80–1.20

Unmodified bitumen

Unaged Aged

88.81 71.28

98.96 90.63

99.91 98.96

100.00 100.00

100.00 100.00

0.107 0.163

0.989 0.974

1.018 1.032

1.787 3.064

Bitumen–filler mastics

Unaged Aged

61.06 40.10

81.79 69.31

91.05 87.50

96.03 92.39

97.48 93.94

0.294 0.405

0.914 0.837

1.063 1.107

5.500 10.395

PMBs

Unaged Aged

36.81 40.57

65.16 67.69

76.47 85.03

83.80 93.60

89.10 96.73

0.808 0.357

0.345 0.873

1.110 1.069

12.893 6.736

the equality line, followed by the aged unmodified bitumens, unaged bitumen–filler mastics, aged filler mastics and aged PMBs. As expected, the unaged PMBs showed the worst correlation between the measured and descriptive |G| data. As discussed earlier, the 2S2P1D Model is unable to satisfactorily fit or describe the rheological properties of highly modified polymers. Like the ri, the MNE and AGD are also used to observe the difference between measured and descriptive data. The unaged unmodified bitumens show the most outstanding correlation for both |G| and d, followed by the aged unmodified bitumens, unaged bitumen–filler mastics, aged PMBs, aged bitumen–filler mastics and unaged PMBs. An observation is made where the ageing process renders the aged PMBs more stable compared to the unaged samples. The ageing process may also possibly affect the breakdown in polymer structures, therefore decreasing the effect of the polymerrich phase network in the mixtures. Based on Tables 12 and 13, it is found that the AGD is not always reliable at detecting the goodness-of-fit, as only small differences between measurements and model are observed. It is therefore recommended that the ri and MNE provide the best means of identifying the goodness-of-fit statistics for model and experiments associated with the rheological data over a wide range of temperatures and frequencies. 7. Conclusions  The 2S2P1D Model is quite simple in its formulation, as a combination of springs, dashpot and parabolic elements. The calibration of this model is not a very difficult task and is good for bituminous binders over a wide range of frequencies and/ or temperatures. The model can be thought of as unique, as its parameters are relatable to the construction of complex modulus and phase angle master curves, the Black diagram and the Cole–Cole diagram. It also takes into account the presence of polymers, mastics and ageing.  This model is able to fit or describe the rheological properties of unmodified bitumens (unaged and aged), bitumen–filler mastics (unaged and aged) and polymer-modified bitumens (aged). However, the model failed to fit the rheological properties of

unaged polymer-modified bitumens particularly with high modifications. The fit is even worse when a comparison is made for the measured and descriptive phase angle (d) data.  For modelling purposes, the Gg values can be taken as 1  109 Pa (in shear) for unmodified and polymer-modified bitumens, both unaged and aged samples. They approach a limiting maximum modulus at very low temperatures.  However, for the unaged and aged bitumen–filler mastics, the Gg values vary. The Gg values depend on the percentage and type of mineral fillers used. This phenomenon occurs due to the existence of physical interaction in the mixture.

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