Master curve properties interrelationships with mix and binder data

Dr. Geoffrey M. RoweAbatech

Presented at the

RAP ETG MeetingRAP ETG MeetingPhoenix, AZ

October 28-29, 2008

Objective

To discuss basic properties of the master curves for binder and mixesTo demonstrate how all properties for MEPDG input can be derived from the mix master curve

Why?

MEPDG based on viscosity aging profilesViscosity is estimated from G* and phase angle

It is possible to derive all this information from the mixture mIt is possible to derive all this information from the mixture master curveaster curve

New concepts

Relationship between slope of log E* (or G*) vs. ω and δGeneralized versus standard logistic functionsKaelble shift factor relationshipNeed for more frequenciesAdditional utility of data from E* master curve

How we have developed ideas

Phase angle data can be deduced from stiffness vs. frequency relationship – we don’t need to measureImprovements to help with determinations

Testing – add a few more frequenciesMaster curve functional – add a parameter to describe non-symmetrical shape of master curveShifting – use Kaelble modification to WLF

Mix vs. binder master curve

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

1.0E+05

1.0E-03 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

Frequency, Hz

Mix

and

Bin

der S

tiffn

ess,

MPa

Binder

Mix

Can use functional forms to describe mix and binder master curves.Alternate – can fit models to describe shape characteristic –for example PMB.

Increase in stiffness due to aggregate volumetrics

Mix to binder properties

Things we needHirsch modelRelationship between phase and G*

Others have shown that we can use Hirsch model to assess quality of RAP dispersion in HMA blends

Mix E* to binder G* - Hirsch

Works well for large range of mixture stiffness valuesPrevious slide is consistent with this information

Binder G* to binder δ

Christensen-Anderson proposed relationship linking G* to δRelationship is based on underlying relationship

ωωδ

log*log90)(

dGd

×=

Log-log relationshipFundamental relationshipApplies to a wide variety of materialsWe have looked at with polymers, asphalt, mixes, etc. etc.CA

SHRP A369SHRP A369

Binder

CA equationIt 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.

Polystyrene

Very good fit with measured vs. calculated

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70 80 90

Measured δ, degrees

Cal

cula

ted

δ, d

egre

es

Approx slopeDS Fit

Polystyrene

Estimated phase angle fits real data very well from the log-log slope information

0

1

2

3

4

5

6

7

8

9

-8 -6 -4 -2 0 2 4

Log Reduced Frequency, rads/sec (Tref = 132C)

G*,

Pa

0

10

20

30

40

50

60

70

80

90

δ, d

egre

es

Log G* Measured phase Est. phase

Phase angleShown – 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” (or E equivalents)Technique can help with analysis

[ ]

[ ]2)(log

)(log

190

log*log90)(

ωγβ

ωγβ

αγω

ωδ+

+

+−=×=

ee

dEd

[ ]

[ ] )/11()(log

)(log

190

log*log90)( λωγβ

ωγβ

λαγ

ωωδ ++

+

+−=×=

ee

dEd

Standard

Generalized

RAP binder propertiesRAP binder properties1. Obtain mixture E*2. Use E* data to back-calculate binder G* using

Hirsch model3. Estimate log-log slope of G* vs. ω plot

a. Method 1 – fit CA modelb. Method 2 – obtain approximate slope from alternate

numerical method4. Use G* and d with 2.2.13 and 2.2.14 to estimate

binder viscositya. Other methods could be used to estimate viscosity

from G* data from analysis of frequency sweep data5. Apply aging and MEPDG parameters to

relationships obtained

Hirsch

( )

1

*3000,200,4100

11

000,10*3

100100,200,4*

−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

××+

−−+⎥

⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ ×+⎟

⎠⎞

⎜⎝⎛ −=

bbm GVFA

VMAVMA

PcVMAVFAGVMAPcE

58.0

58.0

*3650

*320

⎟⎠⎞

⎜⎝⎛ ×

+

⎟⎠⎞

⎜⎝⎛ ×

+=

VMAGVFA

VMAGVFA

Pcb

b

Hirsch model relates volumetrics and stiffness of binder to stiffness of mixtureCan calculate for single points or isotherms

Issues and problems

Various items in current scheme are problematic

Binder is used to dictate shift parametersSymmetric sigmoid

We would like to use better shifting techniques

Consequence – need more data points in isotherms to get better shiftingModification to shift factor relationship

Data quality

More recent testing on master curves for mixes enables more data points to be collected and with better data quality further assessment of models can be consideredNumber of test points/isotherm in present MEPDG scheme is limited resulting in numerical problems in some shifting schemesNeed in many cases to assume model as part of shift development

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E-01 1.0E+00 1.0E+01 1.0E+02

Frequency, Hz

E' o

r E",

MPa

E' E"

Objective of better modelsLeads to better calculations

Spectra calculations and interconversionsBetter definition of low stiffness and high stiffness properties are critical if considering pavement performanceWork looking at obtaining binder properties from mix dataPhase angle interrelationshipsConsiderable evidence that we should be using a non-symmetrical sigmoid function

Sigmoid

Standard logistic

Generalized logistic

log( *) (log )Ee

= ++ +δ

αβ γ1 ω

λωγβλαδ

/1)log(1[*)log( ++

+=e

E [

Why generalized logistic

Allows non-symmetric sigmoid format consistent with asphalt material behaviorBinder CA equation also based on non-symmetric behavior

1

1.5

2

2.5

3

3.5

4

4.5

5

-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

x

y

Generalized logistic example

Lower asymptoteEquilibrium modulus = 98 MPa

Upper asymptote = Equilibrium modulus = 22.3 GPa

Note T= λ

Generalized logistic example

1

1.5

2

2.5

3

3.5

4

4.5

5

-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

x

y

Generalized logisticGeneralized logistic curve (Richard’s) allows use of non-symmetrical slopesIntroduction of additional parameter λ

When λ = 1 equation becomes standard logisticWhen λ tends to 0 – then equation becomes Gompertzλ must be positive for analysis of mixtures since negative values will not have asymptote and produces unsatisfactory inflection in curveMinimum value of inflection occurs at 1/e –or 36.8% of relative height

Minimum inflection

Standard logistic inflection

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-6 -4 -2 0 2 4 6

x

y λ=-0.5λ=0.0 Gompertzλ=0.6λ=1.0 Logisticλ=2.0

Typical range in inflection values

Kaelble shift factors

Kaelble shift factors

Working with materials from MEPDG E* database – observed that shifting works best with Kaelble modification to WLF equation (Arrhenius and WLF – do not work)

||)(

a log2

1

g

gT TTC

TTC−+

−−=

Data from MEPDG database

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E-07 1.0E-05 1.0E-03 1.0E-01 1.0E+01 1.0E+03

Reduced Frequency, T ref = -21.1 oC

E*, M

Pa

TLA Modified - E*

AC-20 - E*

Kaelble shift factors

WLF, Arrhenius, polynomial fits to shift factors are unstable as data is extrapolated to extreme conditionsKaelble provides a sigmoid shift factor relationship

-10

-5

0

5

10

15

20

25

-60 -40 -20 0 20 40 60 80

Temperature oC

log

a T

Data obtained in this range

Pavements temperatures in this range

WLFArrheniusKaelble

Tg = -20oC

-10

-5

0

5

10

15

20

25

-60 -40 -20 0 20 40 60 80

Temperature oC

log

a T

Data obtained in this range

Pavements temperatures in this range

WLFArrheniusKaelble

Tg = -20oC

Kaelble shift factors

-4

-3

-2

-1

0

1

2

3

4

5

-20 -10 0 10 20 30 40 50 60

Temperature C

Log

a T

Gordon and and Shaw shiftsKaelble fit

Typical fit – have analyzed all MPA mixes in MEPDG data base and all give similar fits.

NeedsPhase 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 phaseCan use method to assess data quality –often measurement of phase is poorReduces need to always measure phase –can be easily deducedCan go back to old historical data and obtain phase information

Example RAP – mix to binder

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

1.0E+05

1.0E-03 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

Frequency, Hz

Mix

and

Bin

der S

tiffn

ess,

MPa

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

Phas

e A

ngle

, deg

rees

E*, psi (Calc.) E*, psi (Measured)E*, Binder (Hirsch) Phase (Hirsch)Phase (Slope) Phase (Bonnaure et al.)Phase (SHRP A-404)

Ref: Jo Daniel

??

Example RAP – binder G* & δ

1.0E+04

1.0E+05

1.0E+06

1.0E+07

1.0E+08

1.0E+09

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 1.0E+08

freq (rad/s)

G* (

Pa)

0.0

15.0

30.0

45.0

60.0

75.0

90.0

δ, d

egre

es

G* Back-calculated G* CA FitPhase - from dy/dx CA fit Phase - from approx slope, CA FitPhase - DS on Back-calculated Phase - from approx slope, Back-calculated

Ref: Jo Daniel

Problems with older data

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E+06

1.0E+07

1.0E+08

1.0E+09

1.0E+10

1.0E+11

-8.00 -6.00 -4.00 -2.00 0.00 2.00 4.00 6.00 8.00

Log Reduced Frequency (Hz)

Log

E* (m

ix) o

r G* (

bind

er),

Pa

0

10

20

30

40

50

60

70

80

90

Phas

e A

ngle

(deg

rees

)

Mix E*, Measured Binder G*, measuredE*, Hirsh Pa Gb est, PaCA fit to Gb Hirsch Mix PhaseBinder phase, measured Binder phase, Pred., Hirsch, CA fit, fixed GgBinder Phase, Approx Slope from best fit area Binder Phase, CA Method

ALF5, AC10ALF5, AC10

SummaryE* vs. ω mixture data provides significantly more information than currently assumed

Mixture phase angleBinder G* and δTemperature shift factorsCan determine for individual isotherms if needed

RecommendationsIncrease test frequencies - does not significantly increase preparation/testing timeUse free shifting and generalized logistic sigmoidInvestigate relationships further, Hirsch, CA models etc.