Phase Angle Determination and
Interrelationships within Bituminous
Materials
Geoffrey M. Rowe, Abatech Inc.
45th Petersen Asphalt Research Conference
University of Wyoming
Laramie, Wyoming, July 14-16
Phase angle
d log G *
δ (ω ) = 90 ×
d log ω
G* or E*
1
Objectives
„
To examine the development of phase
angle relationships from stiffness data
Use
„
„
Phase angle is difficult to measure
Data exists which contains no phase
angle information
2
History
„
Various phase angle relationships
exist for binder and mix materials
Binder
„
„
„
Several methods exist
Dobson (BP 1972)
Dickerson and Witt
„
„
„
Developed relationship between G*, frequency
and phase angle
Based on a log-log equation from analysis of
“electric network”
Christensen-Anderson
„
Proposed equation that relates phase to crossover
frequency and rheological index
3
Mixes
„
Bonnaure et al. (AAPT 1977)
„
„
„
⎡ log10 S b − log10 5 ×106
⎤
× 0.974Vb−0.172 ⎥
9
⎣ log10 S b − log10 2 × 10
⎦
φm = 16.36 × Vb0.352 exp ⎢
SHRP A-404 report
„
„
„
Phase linked to binder stiffness
and Vb
Limited for Sb > 5MPa
Various models
Other similar relationships
Hirsch (AAPT 2003)
„
„
φm = −21(log Pc ) 2 − 55 log Pc
Christensen, Pellinen and
Bonaquist
Empirical relationship related to
Hirsch parameters
0.58
⎛
VFA × 3Gb* ⎞
⎜⎜ 20 +
⎟
VMA ⎟⎠
⎝
Pc =
0.58
⎛ VFA × 3Gb* ⎞
⎟⎟
650 + ⎜⎜
VMA
⎝
⎠
Measurement of phase angle
„
„
Measurement of phase
angle has been difficult
Generally G*
measurement is more
reliable than the phase
angle
Tests by Superpave center - 2003
4
Starting point
„
„
Poor historical data measurement
Various equations to evaluate for
mix and binder
Approach
„
„
Consider from visco-elastic view point
and develop understanding from
different materials
Start with binder – then develop for
other materials including mixtures
5
Dickerson and Witt (1974)
„
„
„
Proposed that the phase angle is
related to logG* vs. logω slope
Relationship had some residual δ at
high stiffness
Residual δ could be a result of
measurement artifact rather than
reality
Christensen Anderson (1991-92)
„
„
„
„
„
2 AAPT papers in 1991 and 1992
Important to recognize that instrumentation
and software now 20-years more advanced!
Used log-log suggestion from Dickerson and
Witt
No residual δ at high stiffness, δ = 0 at Eglassy
δ=90 at viscous asymptote
6
Binder
„
CA equation
„
It can be shown that the
log-log relationship is
related to the phase
angle
Dickson and Witt (1974) and used in the development of the CA model.
Binder – logG* vs. logω
„
Equation to note:
⎡ ⎛ ω ⎞β ⎤
δ (ω ) = 90⎢1 + ⎜ ⎟ ⎥
⎢⎣ ⎝ λ ⎠ ⎥⎦
−1
= 90 ×
d log G *
d log ω
We will apply this concept to binders, mastics, mixtures,
polymers etc. to evaluate how robust this technique is.
Note: Center part of relationship is CA model and will only
work for binders/materials conforming to CA relationship.
7
Standard asphalt
Standard asphalt
8
Standard asphalt
„
Four methods used to obtain log-log gradient
„
„
„
„
1
2
3
4
–
–
–
–
slope – with polynomial – 3rd order
CA equation
DS fit
slope approx
3
2
log G*
1
log ωr
G’=
Standard asphalt
90
„
All three methods
produce a good fit
of the data sets
and determination
of the phase lag
r2 >0.99 in all
instances
Poly Fit, n = 3
CA Model
DS Fit
80
70
Calculated δ, degrees
„
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
Measured δ, degrees
9
Standard asphalt
90
„
All three methods
produce a good fit
of the data sets
and determination
of the phase lag
r2 >0.99 in all
instances
Poly Fit, n = 3
CA Model
DS Fit
Slope
80
70
Calculated δ , degrees
„
60
50
Data items at end of
isotherms where
more noisy
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
Measured δ, degrees
SBS modified resin
10
SBS modified resin
SBS modified resin
„
Two methods used
„
„
„
„
„
1
2
3
4
-
3
slope – Poly=4
DS fit
Approx slope
Standard logistic
2
log G*
1
log ωr
CA equation would not be expected to fit
STANDARD LOGISTIC - ALTERNATE FORMAT
log( E*) = D +
A
1 + e − B (logω − M )
d log( E*)
e − B (ω − M )
= AB +
d log ω
1 + e − B (log ω − M )
[
]
2
11
SBS modified resin
90
„
„
Poly fit is poor fit at
one end – maybe if
increase order or
use some other
function would be
better
Standard logistic
also has same issue
Log-log from
approximate slope
gives same numbers
as DS and measured
Poly, n=4
80
Approx slope
DS Fit
70
Calculated δ , degrees
„
Standard logistic
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
Measured δ, degrees
SBS modified resin
„
„
The logistic
function is not
capable of
predicting the turn
up at low
frequencies
The polynomial fit
gets this wrong
Function form may
consist of two CA
models added! –
more work in this
area
90.0
80.0
P h ase an gle, d eg rees
„
70.0
60.0
50.0
40.0
DS Fit
30.0
Measured
20.0
Poly, n=4
Appox Slope
10.0
0.0
-4.00
Standard logistic
-2.00
0.00
2.00
4.00
6.00
8.00
Log Reduced Frequency, rads/sec (Tref = 50C)
12
Polystyrene
Polystyrene
13
Polystyrene
„
Two methods used
„
„
„
1 - DS fit
2 - Approx slope
Equations (simple function forms) would
not be expected to fit
Polystyrene
Very good fit
with measured
vs. calculated
Approx slope
80
DS Fit
70
Calculated λ, degrees
„
90
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
Measured λ, degrees
14
Polystyrene
9
Measured phase
Est. phase
8
80
7
70
6
60
5
50
4
40
3
30
2
20
1
10
0
δ , degrees
Estimated phase
angle fits real data
very well from the
log-log slope
information
G*, Pa
„
90
Log G*
0
-8
-6
-4
-2
0
2
4
Log Reduced Frequency, rads/sec (Tref = 132C)
Roofing product
„
Roofing material
„
„
„
„
„
8.75 % Radial SBS Polymer
61.25 % Vacuum Distilled
Asphalt
30 % Calcium Carbonate
Filler
Master curve considered in
range -24 to 75oC – this
range gives a good fit in
linear visco-elastic region
After 75oC structure in
material starts to change
and material is not behaving
in a thermo-rheologically
simple manner
15
Roofing product
„
90
Simple slope
method shows
validity
Better fitting
functions give
better prediction
log-log slope
80
Standard logistic
Predicted δ , degrees
„
70
Generalized (Gompertz) logistic
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
Measured δ, degrees
Roofing product
Generalized
logistic clearly
provides a better
prediction of
phase response
compared to
standard logistic
60
50
δ , degrees
„
40
30
20
Measured
log-log slope
10
Standard Logistic
Generalized (Gompertz) logistic
0
-12
-10
-8
-6
-4
-2
0
Log Reduced Frequency, rad/sec (Tref=-24C)
16
Adhesive product
„
Master curve for
a material used
for fixing road
markers
Adhesive product
The better fit
model produces a
result close to that
obtained from the
slope
approximation
90
log-log slope
80
Standard logistic
Predicted δ , degrees
„
70
Generalized (Gompertz) logistic
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
Measured δ, degrees
17
Adhesive product
Generalized logistic
– in this case the
limiting Gompertz
condition produces
the best fit
45
Measured
40
log-log slope
Standard Logistic
35
δ , degrees
„
Generalized (Gompertz) logistic
30
25
20
15
10
5
0
-10
-8
-6
-4
-2
0
2
4
Log Reduced Frequency, rad/sec (Tref=16C)
Thin surfacing on PCC
„
Material mixed
with aggregate
and used as a
thin surfacing
material on
concrete bridge
decks
18
Thin surfacing on PCC
„
log-log
relationship
produces phase
angle
Better fitting
relationship
produces better
fit
90
log-log slope
80
Standard logistic
Predicted δ , degrees
„
70
Generalized (Gompertz) logistic
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
Measured δ, degrees
Thin surfacing on PCC
Standard
logistic
underestimates
the peak phase
angle response
50
Measured
log-log slope
45
Standard Logistic
Generalized (Gompertz) logistic
40
35
δ , degrees
„
30
25
20
15
10
5
0
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
Log Reduced Frequency, rad/sec (Tref=-16C)
19
What we have
„
Relationship holds well for large number of
different materials investigated
„
„
When fits are poor, suspect
„
„
„
Applies to polymers, mastics, mixtures, binders,
etc, etc.
Poor phase angle measurement
Poor model format
Very important to fit correct functional form
model if you plan to use dy/dx to get slope
G* method
„
If we have G* alone - or E* - we can
determine phase angle via a fit of the
discrete relaxation spectrum
20
VE Solid representation
G”
ge
g1
g2
g3
g4
g5
η1
η2
η3
η4
η5
g4*
G*
G’
g3*
g3”
g2*
ge
g1*
g5”
g5*
g1”
g2”
g4”
g5’
g4’
g3’
g2’
g1’
A discrete relaxation spectrum can be fitted to G* since G* is a
vector sum of the G' and G" components.
Example – G* only
„
Material – Roofing product
21
Asphalt mixtures (L1)
„
„
„
Asphalt mixtures –
CAIT NJ
Fit is reasonable –
but since MEPDG
data format the
slope interpolation
is not so good
Ideally would be
better if more
points per decade
Asphalt mixtures (L2)
Data suggests possible
problem with measured
phase angle at highest
temperature
4.50
60
Mix 1 measured
Mix 2
Mix 3
40
60
E*
4.00
Measured phase
50
Calculated phase
30
20
3.50
40
3.00
30
2.50
20
δ , degrees
Log E* (MPa)
Mix 1
50
δ, degrees
„
10
0
-2
-1
0
1
2
3
4
5
Log E*
2.00
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
Lines are calculated from
dy/dx of standard logistic
sigmoid fit
Reduced Frequency, Hz (Tref=20C)
22
Asphalt mixtures (L3)
Highest temperature
isotherm not quite in
agreement with data
from log-log
45
90
80
70
δ, calculated
„
40
35
60
50
40
30
20
δ , degrees
30
10
25
20
0
15
0
BLACK SPACE
10
Measured
5
10
20
30
40
50
60
70
80
90
δ, measured
Approx. Log-Log
0
2.0
2.5
3.0
3.5
4.0
4.5
Log G*, MPa
Asphalt mixes
„
„
„
Generally reasonable fits are obtained
Looking at data sets – often problem
at extremes – e.g. highest or lowest
test temperatures and/or frequencies
Issues still exist with equipment and
testing procedures!
23
Needs
„
„
„
„
Phase angle important in some MEPDG work
– used to derive viscosity – can obtain from
back-calculation of Gb* from mix data and
then use log-log slope or dy/dx of CA model
to obtain phase
Can use method to assess data quality –
often measurement of phase is poor
Reduces need to always measure phase –
can be easily deduced
Can go back to old historical data and obtain
phase information
Example RAP – mix to binder
Mix and Binder Stiffness, MPa
Ref: Jo Daniel
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+05
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
Frequency, Hz
24
Example RAP – binder G* & δ
G* Back-calculated
G* CA Fit
Phase - from dy/dx CA fit
Phase - from approx slope, CA Fit
Phase - DS on Back-calculated
Phase - from approx slope, Back-calculated
1.0E+09
90.0
75.0
1.0E+08
60.0
45.0
δ, degrees
G* (Pa)
1.0E+07
1.0E+06
30.0
1.0E+05
15.0
1.0E+04
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
freq (rad/s)
1.0E+05
1.0E+06
1.0E+07
0.0
1.0E+08
Ref: Jo Daniel
Summary
„
„
„
Shown – for a wide variety of
materials – that –
δ=90(dlogG*/dlogω)
Analysis is consistent with that
produced by discrete spectra analysis
of G* or G’G”
Technique can help with analysis
25