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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