Advanced Testing and Characterization of Bituminous Materials –
Loizos, Partl, Scarpas & Al-Qadi (eds)
© 2009 Taylor & Francis Group, London, ISBN 978-0-415-55854-9
Phase angle determination and interrelationships
within bituminous materials
G. Rowe
Abatech, Blooming Glen, Pennsylvania, USA
ABSTRACT: The phase angle of bituminous materials is often required when in-sufficient
data exists to define this with a high degree of accuracy. In addition, the measurement of
phase angle has a higher variability than that associated with the complex modulus (G* or
E*). Often historical data consists only as complex modulus versus frequency with no phase
angle information. To enable use of these data sets in current analysis procedures requires
the use of the phase angle information that has to be obtained from mathematical and/or
predictive methods. Phase angle can be calculated from the retardation and relation spectra.
This approach is contrasted to the method developed from the relationship found between
the log-log gradient of complex modulus versus frequency and phase angle. Data is presented
for a range of materials that includes non-bituminous products. It is concluded it would be
reasonable to analyze complex modulus vs. frequency and estimate the phase angle from
numerical procedures.
1
INTRODUCTION
1.1 Background
The phase angle of bituminous material is a parameter that has been growing in importance in specifications in the USA since the early 1990’s following the Strategic Highway
Research Program (SHRP). Since that time a phase angle measurement has been a feature
of the binder specification described in the AASHTO specification M320. In addition, some
states (for example Georgia) specify a minimum phase angle as an additional requirement. In
asphalt mixtures the phase angle is routinely measured and reported using the latest testing
methods as being considered as part of the Mechanistic-Empirical Pavement Design Guide
(ARA, Inc., 2004). In addition, a large volume of data exists in the industry for complex
dynamic modulus, either in extension/compression (E*) or shear (G*) for which no phase
angle is reported. To obtain phase angle information from these data sets using simple methods would advantageous and increase the utility of the data. Currently, methods exist where
Prony series or relaxation/retardation spectra can be fitted to the data with the phase angle
then calculated from the resulting spectra fits. However, this fitting process is relatively complex requiring specialized software and can be numerically confounded depending upon the
amount and quality of data available.
Dickersen and Witt (1974) presented a relationship that linked the phase angle of bitumen
to a relationship between G* and frequency. Christensen and Anderson (1992) further developed this relationship idea by presenting a relationship between phase angle with frequency
and binder parameters, as follows:
δ (ω ) =
90
⎡ ⎛ ω ⎞(1og 2 / R ) ⎤
⎢1+⎜ ⎟
⎥
⎢ ⎝ ω0 ⎠
⎥
⎣
⎦
43
(1)
where δ(ω) is the phase angle at a frequency (expressed as radians/second, ω), ω0 is the crossover frequency and R is the rheological index. This equation for phase angle is based upon the
parameters determined from the Christensen-Anderson (CA) stiffness equation which works
well for unmodified binders.
For asphalt mixtures several researchers have developed relationships between phase angle
and modulus. Bonnaure et al. (1977) developed a relationship that was limited to binder stiffness (Sb) values greater than 5 MPa and less than 2 GPa (when Sb is greater than 2 GPa the
mixture phase angle (φm) is taken to be zero). The relationship used the volume of binder (Vb)
in the prediction and is as follows:
⎡ log10 Sb − log10 5 × 106
φm = 16.36 × Vb0.352 exp ⎢
× 0.974 Vb−0.
9
⎣ log10 Sb − log10 2 × 10
(2)
During the SHRP project Tayebali et al. (1994) developed a relationship linking the phase
angle to the mix stiffness as follows:
φ 0 = 260.096 − 17.172 Ln(S0 )
(3)
where φ0 is the mixture phase angle and S0 is the mixture stiffness. This relationship was
developed from a study of fatigue properties. The subscript to the parameters denotes that
the initial condition is used. Christensen et al. (2003) published a relationship developed from
the Hirsch model. This relationship links the phase angle to binder properties and mixture
volumetrics, as follows:
φ 0 = − 21(log Pc )2 − 55 log Pc
(4)
where
0.58
⎛
VFA × 3Gb* ⎞
⎜⎜ 20 +
⎟
VMA ⎟⎠
⎝
Pc =
⎛ VFA × 3Gb* ⎞
650 + ⎜⎜
⎟⎟
⎝ VMA ⎠
(5)
with VFA being the voids filled with asphalt, Gb* being the complex shear modulus of the
binder and VMA being the percent voids in the mineral aggregate. All of the described relationships for bituminous mixtures are empirical in nature with derived constants from regression analysis of materials.
1.2 Log-log relationships
In Dickerson and Witt’s paper the relationship developed made use of the slope (or derivative) of the log-log relationship between the G* and frequency to estimate the phase angle.
Christensen and Anderson (1992) further developed this idea and presented a relationship
between the phase angle and stiffness which can be shown to be the derivative of CA stiffness
equation, as follows:
⎡ ⎛ λ ⎞β ⎤
G * (ω ) = G0 ⎢1 + ⎜ ⎟ ⎥
⎢⎣ ⎝ ω ⎠ ⎥⎦
−1 / β
(6)
then
⎡
⎛ ω⎞
δ (ω ) = 90 ⎢ 1 + ⎜ ⎟
⎝ λ⎠
⎢⎣
β
⎤
⎥
⎥⎦
44
−1
= 90 ×
d log G *
d log ω
(7)
where G*(ω) and δ(ω) are the complex shear modulus and phase angle at a frequency ω. The
parameters λ and β are used to define the shape of the master curve in the same manner as
the crossover frequency and rheological index in equation 1. The implication of this relationship is that it implies two approaches for obtaining an estimation of the phase angle being
1) use of model parameters directly with equation 1 or 5, or 2) using an alternate numerical
procedure to obtain the slope of the log G* versus log frequency relationship. Conceptually,
this relationship should not be limited to one case being unmodified binders but rather it
should be universal in application.
The use of log-log relationships are investigated for a number of materials using numerical
differentiation. In addition, the use of model differential schemes are applied to asphalt mixtures and compared to differentials of other fitting functions such as high-order polynomials.
2
MATERIALS
2.1 Material types and analysis methods
Analysis is applied to a standard asphalt binder; a SBS modified resin binder; polystyrene;
a roofing product (high content of SBS with modified filled binder), and various hot mix
asphalt samples. Several analysis methods are applied to the data sets, as follows:
– A polynomial (order 3 or 4) fit has been applied to many of the master curves and the
differentiated with respect to frequency in an attempt to obtain a good estimation of the
slope of master curves.
– An approximation of the slope at a given point is estimated from consideration of data
points either sides.
– A fit of the discrete spectra as determined using the method defined by Baumgaertel and
Winter (1989).
– By the differential of an equation considered to describe the shape of the master curve.
The equations used to describe the shape of the master curve the CA method as defined
in equation 6 and 7. The standard logistic (Verhulst, 1838) initially adopted by the Asphalt
Institute (1982) and subsequently further developed for use in the AASHTO MechanisticEmpirical Pavement Design Guide (MEPDG) (ARE, Inc., 2004) and the generalized logistic
(Richards, 1959), expressed in a form to be used with the E* definition of hot mix asphalt,
have the functional forms and differentials used to compute the phase angles as follows:
Standard logistic – log E * = δ +
Standard logistic – δ (ω ) = 90 ×
α
(8)
β + γ (log ω )]
]
d log E *
e[
= − 90αγ
2
d log ω
⎡1 + e[ β +γ (log ω )] ⎤
⎣
⎦
β +γ (log ω )
Generalized logistic – log E * = δ +
Generalized logistic – δ (ω ) = 90 ×
1 + e[
α
1/ λ
⎡1 + λ e[ β +γ (log ω )] ⎤
⎣
⎦
d log E *
e [ β + γ (log ω ) ]
= − 90αγ
(1+ 1/ λ )
d log ω
⎡ 1 + λ e [ β + γ (log ω ) ] ⎤
⎣
⎦
(9)
(10)
(11)
2.2 Analysis of various materials
A sample was taken from the data base developed during the SHRP program. With this
material testing has been conducted using the torsion bar (TB) and dynamic shear rheometer
45
(DSR) tests. The master curve for this material using data collected between –23.5 and
63.2°C. The resin modified binder is a clear resin modified by SBS with testing conducted
between –5 and 60°C. Both these data sets are shifted to a reference temperature of 50°C as
illustrated in Figure 1. The conventional (straight run asphalt) binder conforms to the CA
model whereas the SBS resin binder clearly does not conform to the CA model. The analysis
of these binders with various methods is illustrated in Figure 2 which shows fits of the various methods. The conventional binder has a good fit with all methods but some increased
scatter is apparent in the data at the low G*/high δ end of the range when the approx slope
method is used due to inherent variability in the data sets. With the resin modified binder, two
methods (standard logistic and 4th order polynomial) result in a poor estimation at the low
frequency/high temperature end of the data sets where the phase angle turns upwards. The
slope estimated from the discrete spectra fit which results in over 20 fitting parameters for the
curves produces a good estimation of the phase angle in both cases.
A polystyrene material has a complex stiffness and phase angle relationship. The master
curve of this material using isotherms collected between 132 and 269°C is presented in
Figure 3 along with that of a high polymer content roofing material (Rowe and Baumgardner, 2007). For the polystyrene two methods of estimating the log-log slope were used;
1) differential of the Prony series determined from a discrete spectra analysis, and 2) and
approximation of the slope. No functional form models were attempted with this material
since it was obvious these would not apply to the shape of the master curve. For the roofing
1.0E+09
G*
1.0E+09
90
Phase Angle
80
1.0E+08
G*
90
Phase Angle
80
1.0E+08
70
70
1.0E+07
1.0E+05
40
30
60
1.0E+06
50
40
1.0E+05
30
1.0E+04
Phase Angle, deg.
50
G*, Pa
G*, Pa
1.0E+06
Phase Angle, deg.
1.0E+07
60
1.0E+04
20
20
Conventional Binder
1.0E+03
10
1.0E+02
SBS Modified Resin
1.0E+03
10
0
1.0E+00
1.0E+02
1.0E+04
1.0E+06
1.0E+08
1.0E+10
1.0E+02
1.0E+12
0
1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07
o
o
Reduced Frequency, rads/sec Tref = 50 C
Reduced Frequency, rads/sec Tref = 50 C
Figure 1. Master curves for conventional and SBS resin binders, Tref = 50°C.
90
90
Poly, n = 4
Poly Fit, n = 3
CA Model
DS Fit
Slope
80
80
Calculated δ, degrees
Calculated δ, degrees
Approx slope
DS Fit
70
70
60
50
Data items at end of
isotherms where
more noisy
40
30
Standard logistic
60
50
40
30
20
20
10
10
Conventional Binder
SBS Modified Resin
0
0
0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
50
60
70
80
90
Measured δ, degrees
Measured δ, degrees
Figure 2. Measured versus calculated phase angles for conventional (straight run) and SBS resin
modified binders.
46
G*
1.0E+08
90
Phase Angle
1.0E+07
G*
80
80
Roofing Product
Polystrene
70
1.0E+05
60
1.0E+04
50
1.0E+03
40
1.0E+02
30
1.0E+01
20
1.0E+00
1.0E-07
1.0E-05
1.0E-03
1.0E-01
1.0E+01
1.0E+07
70
60
G*, Pa
1.0E+06
Phase Angle, deg.
G*, Pa
90
Phase Angle
1.0E+06
50
40
1.0E+05
Phase Angle, deg.
1.0E+08
30
20
10
1.0E+03
1.0E+04
10
1.0E-04
1.0E-02
o
1.0E+00
1.0E+02
1.0E+04
1.0E+06
1.0E+08
Reduced Frequency, rads/sec Tref = 30oC
Reduced Frequency, rads/sec T ref =132 C
Figure 3. Master curves for polystyrene and roofing product.
90
90
Approx slope
80
log-log slope
80
DS Fit
Standard logistic
Predicted δ , degrees
Calculated δ, degrees
70
60
50
40
30
20
Polystyrene
10
70
Generalized (Gompertz) logistic
60
50
40
30
20
Roofing Product
10
0
0
0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
50
60
70
80
90
Measured δ, degrees
Measured δ, degrees
Figure 4. Measured versus calculated phase angles for polystyrene and roofing product.
product two functional forms were applied, the standard logistic and the generalized logistic
functional forms. The generalized logistic analysis resulted in the limiting Gompertz (1825)
case of the Richards curve. The results from the phase angle estimation are presented in
Figure 4 which demonstrates that both methods provide very good estimations of the phase
angle. The discrete spectra fit and the approximate slope methods provide the best estimations. With the roofing material the two logistic models are evaluated. From the results of
the analysis it is clear that the use of the generalized logistic provides a better fit that the
standard logistic since it more accurately captures the shape of the master curve.
The key aspect of obtaining a good fit of the measured phase angle data is that the curve
adopted to describe the master curve (or part of the master curve being considered) must be
a good representation of the shape. The models that more accurately define the shape of the
master curve are the generalized logistic function or discrete spectra fits.
2.3 Hot mix asphalt
The analysis has been applied to several HMA material data sets. For example the Center of
Advanced Infrastructure Testing (CAIT) produced data on a series of mixtures representing
modified and unmodified materials. The fit obtained from analysis of the standard logistic
function is given in Figure 5 which shows a reasonable correspondence with the measured
data.
Use of this same method has been applied to data taken from frequency sweep tests conducted in shear developed as part of the SHRP test program. Data from this test is presented
47
Mixtures are made with
PG grades are in accordance with AASHTO
M320 specifications except the AR-HMA which
is a crumb rubber (20%)
modified PG64-22. Mix
details are given in Bennert et al, 2004.
Figure 5. Phase angle calculated using standard logistic curve (equation 9) vs. measured for four HMA
mixtures containing conventional and modified binders (E* and δ data obtained from Bennert et al.,
2004).
40
35
δ , degrees
30
25
20
15
10
Measured
5
Approx. Log-Log
0
2.0
2.5
3.0
3.5
4.0
4.5
Log G*, MPa
Figure 6. Mixture complex modulus and phase angle measured in shear with the calculated phase
from the approximation of the logG* vs. logω slope.
in Figure 6 on a plot of G* versus δ, commonly referenced as a Black Space Plot. The
differences observed in predicted from measured results occurs at the lower end of the stiffness range as the complex modulus G* reduces below 1,000 MPa. It should be noted as the
stiffness reduces the accuracy of measurements is often questionable and certainly the problem of fitting data at the extremes of the data ranges is often problematic.
In a study conducted with the Asphalt Institute (Hakimzadeh-Khoee, 2009) it was observed
that at the highest test temperature some densification of the specimens was taking place.
The results obtained from the E* and δ test for one of the specimens is illustrated in Figures 7
and 8. Figure 8 also shows how the predictive relationship from the differential of the standard logistic equation compares over a much wider range of stiffnesses. It was concluded from
48
4.50
60
E*
Measured phase
4.00
50
3.50
40
3.00
30
2.50
20
Data from 40°C
2.00
δ , degrees
Log E* (MPa)
Calculated phase
10
1.50
0
1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04
Reduced frequency, Hz (T ref = 20C)
Figure 7. Mixture complex modulus and phase angle measured in shear with the calculated phase
from the approximation of the logG* vs. logω slope.
60
Mix 1 measured
Fitted, standard logistic
50
δ, degrees
40
30
20
10
0
–2
–1
0
1
2
3
4
5
Log E*
Figure 8. Mixture complex modulus and phase angle measured in shear with the calculated phase
(standard logistic, equation 9) from the approximation of the logG* vs. log ω slope.
this data that the secondary compaction of the material during the testing was affecting the
phase angle result which can be clearly observed by inspecting the calculated versus measured
phase angle relationships.
The concept of using differential of the complex modulus vs. frequency relationship
appears to be sound for mixtures and it has functionality in that it shows where mixture data
has poor phase angle measurement.
3
DISCUSSION
The discrete spectra analysis (Baumgaertel and Winter, 1989) is more commonly applied
to visco-elastic data when both modulus and phase angle are available and the elastic and
49
G”
ge
g1
g2
g3
g4
g5
η1
η2
η3
η4
η5
g4*
g4”
G*
G’
g3*
g3”
g2*
ge
g1*
g5”
g5*
g1”
g2”
g5’
g4’
g3’
g2’
g1’
Figure 9. Representation of G* made up from relaxation spectra components.
Mix and binder stiffness, MPa
1.0E+05
E*, psi (Measured)
Phase (Hirsch)
Phase (Bonnaure et al.)
90.0
1.0E+04
80.0
1.0E+03
70.0
1.0E+02
60.0
1.0E+01
50.0
1.0E+00
40.0
1.0E-01
30.0
1.0E-02
20.0
1.0E-03
10.0
??
1.0E-04
1.0E-03
Phase Angle, degrees
E*, psi (Calc.)
E*, Binder (Hirsch)
Phase (Slope)
Phase (SHRP A-404)
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
0.0
1.0E+07
Reduced frequency, Hz
Figure 10. Mix and binder complex modulus and various estimations of mixture phase angle.
viscous parts of the complex modulus are used. A representation of how visco-elastic
components—expressed as loss and storage modulus in shear (g' and g" respectively) are
considered as spectra components to produce the calculation of phase of total G*, G' and G"
is illustrated in Figure 9. While this figure shows the shear (G*) version of the graph it should
be noted that this also applies equally to the use of the extensional form, E*. A fit of the
discrete spectra can be applied when any visco-elastic parameter is described as a function
of frequency or loading time (e.g. G*, E*, E(t), G(t), S(t) etc.). Thus if only G* or E* data
exists, such as that found in many older references, it is still possible to determine phase angle
information from an analysis of the spectra information or from fitting the master curve the
log-log slope information from the master curve.
Recent discussion between Daniel and Rowe (2008) have suggested the utility of this
approach. It is possible to determine the phase angle information for both binder and mixture, and the complex shear modulus of the binder given just the complex dynamic modulus
50
master of a mixture. An example of an analysis conducted using mixture properties is given
in Figure 10. The mixture stiffness is shown and compared to the Hirsch predictive equation. This data shows close agreement. In addition, the graph shows the mixture phase angle
deduced from the Hirsch model along with that determined from the dlogE*/dlogω relationship. These two curves are in very good agreement and contrast to the poor fits of phase
angle obtained from the method developed by Bonnaure et al. (1977) and the SHRP-A-404
method (Tayebali et al., 1994). In addition, the binder E* curve is also shown. The two lowest
points on this curve were estimations since they were beyond the limit of the Hirsch model
for back-calculation. Regardless of this, using the remainder of the data it is still possible to
fit the binder stiffness curve to a CA model (Christensen and Anderson, 1992) and use this
with equation 7 to obtain the phase angle information. This type of analysis could be applied
to any type of mix where the effective binder properties is required, for example a blend of
virgin mix with recycled asphalt pavement.
Traditionally phase angle of binder and mixtures has presented a complex verification of
properties. Only recently an article in the Society of Rheology commented on the difficulty
that occurs due to a lack of calibration standards for phase angle measurement (Velankar and
Giles, 2007). However, accepting the relationships that exist and using these to verify and check
the adequacy of phase angle measurements will assist in this aspect of lack of calibration.
4
CONCLUSIONS
The analyses of experimental data obtained from testing various materials have been conducted. These materials have been very wide ranging, covering asphalt binders; polymer
modified binders; roofing compounds; polymers (such as polystyrene); hot mix asphalt, and
many other materials.
The use of interrelationships between the stiffness and frequency dependency allows the
use of two methods for obtaining the phase angle information using a fundamental analysis approach. The first of these is to fit discrete spectra to the complex modulus master
curve whereas the second makes use of the log-log relationship between the complex stiffness
modulus and frequency and shows how this relates to phase angle. The spectrum fit involves
sophisticated software in the implementation of analysis methods (Baumgaertel and Winter,
1989). While these have been implemented in a number of software packages the use of
dlogG*/dlogω (or dlogE*/dlogω) provides a simple rapid method for obtaining the phase
angle information.
The experimentation with many materials has confirmed this validity of this approach
which is essentially similar to that originally suggested by Christen and Anderson (1992) in the
analysis of binder with their differential analysis of the CA equation for binder master curves.
For mixtures a differential form of the generalized logistic sigmoid as proposed by Rowe et al.,
(2009) provides a reliable means of estimation of phase angle for many filled materials.
REFERENCES
ARE, Inc. 2004. Guide for Mechanistic-Empirical Design of New and Rehabilitated Pavement Structures, Final Report, Part 2. Design Inputs, Chapter 2 Material Characterization. National Cooperative Highway Research Program, Transportation Research Board, National Research Council.
Baumgaertel, M. and Winter, H.H. 1989. Determination of discrete relaxation and retardation time
spectra from dynamic mechanical data. Rheol Acta 28. pp. 511–519.
Bennert, T., Maher, A. and Smith, J. 2004. Evaluation of Crumb Rubber in Hot Mix Asphalt. Center
for Advanced Infrastructure and Transportation (CAIT), Rutgers Asphalt/Pavement Laboratory
(RAPL), Rutgers University, Department of Civil and Environmental Engineering, 623 Bowser
Road, Piscataway, NJ 08854.
Bonnaure, F., Gest, G., Gravois, A. and Uge, P. 1977. A New Method of Predicting the Stiffness of
Asphalt Paving Mixtures. Proceedings, Association of Asphalt Paving Technologists.
Christensen, D.W. and Anderson, D.A. 1992. Interpretation of Dynamic mechanical Test Data for Paving
Grade Asphalt Cements. Journal, Association of Asphalt Paving Technologists, Volume 61, pp. 67–116.
51
Christensen, D.A., Pellinen, T. and Bonaquist, R.F. 2003. Hirsch Model for Estimating the Modulus of
Asphalt Concrete. Journal of Asphalt Technologists, Volume 72. pp. 97–121.
Dainel, J. and Rowe, G.M. 2008. Discussions on Phase Angle Relationships, FHWA Asphalt Fundamental Properties and Advanced Modeling Expert Task Group, minutes on web - www.asphaltmodelsetg.
org/Tampa%20Models%20ETG%20Minutes%20Feb%202008.pdf
Dickinson, E.J. and Witt, H.P. 1974. The Dynamic Shear Modulus of Paving Asphalts as a Function of
Frequency. Transactions of the Society of Rheology 18:4, 591–606.
Gompertz, B. 1825. On the Nature of the Function Expressive of the Law of Human Mortality, and on a
New Mode of Determining the Value of Life Contingencies. Phil. Trans. Roy. Soc. London, Vol. 115,
pp. 513–585.
Hakimzadeh-Khoee, S., Rowe, G.M. and Blankenship, P. 2009. Evaluation of Aspects of E* Test with
using HMA Specimens with Varying Void Contents. Paper submitted to Transportation Research
Board Annual Meeting.
Richards, F.J. 1959. A Flexible Growth Function for Empirical Use. Journal of Experimental Botany,
Vol. 10, No 29, pp. 290–300.
Rowe, G.M. and Baumgardner, G. 2007 Evaluation of the Rheological Properties and Master Curve
Development for Bituminous Binders Used in Roofing. Journal of ASTM International, Vol. 4, No. 9.
Rowe, G.M., Baumgardner, G. and Sharrock, M.J. 2009. Functional forms for master curve analysis of
bituminous materials. Paper submitted to the 7th International Symposium on Advanced Testing and
Characterization of Bituminous Materials Organized by RILEM TC 206 ATB, Rhodes, Greece.
Tayebali, A.A., Deacon, J.A., Coplantz, J.S., Finn, F.N. and Monismith, C.L. 1994. Fatigue Response
of Asphalt-Aggregate Mixes, Part II Extended Test Program. Strategic Highway Research Program,
National Research Council, Washington DC. Report SHRP-A-404.
The Asphalt Institute. 1982. Research and Development of the Asphalt Institute Thickness Design
Manual (MS-1) Ninth Edition. The Asphalt Institute, RR-82-2.
Velanker, S.S. and Giles, D. 2007. How do I know if my phase angles are correct? Rheology Bulletin,
76(2), July, pp. 8–20.
Verhulst, P.F. 1838. Notice sur la loi que la population poursuit dans son accroissement. Correspondance
mathématique et physique 10: pp. 113–121.
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