Chen-Yang Liu, Roland Keunings, Christian Bailly
UCL, Louvain la Neuve, Belgium
Dynamics of complex fluids: 10 years on, Cambridge, October 2-5 2006
OldNew
IdeasQuestions
…about viscosity, plateau modulus and Rouse chains
Objectives - outline
• Some old and recent results suggest there are still significant inconsistencies/questions about the LVE predictions of tube models
• Three examples :
– Why is Z-dependence of the plateau modulus is less than predicted ?
– Is the 3.4 power law fully understood after all ?
– Is Rouse really Rouse ?
Plateau modulus and zero shear viscosity :
questions about constraint release and fluctuations
1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01 0.1 1 101E-3
0.01
0.1
1
10
τd τR
τeG', G"/Ge
ωτe
G' G"
Ze = 100
Gexpl determination
Gapp = G '(wmin G '' )
limZÆ •
(Gapp ) = GN0
Ferry (1980)
t (Gmin" ) ª t e.t R w(Gmin
" )
Minimum G’ method
Gexp l
Low polydispersity model polymers (anionic polymerization)-Polybutadiene-Polyisoprene-Polystyrene
Systems analysed
Raju VR; Menezes EV; Marin G; Graessley WW; Fetters LJ. Macromolecules 1981 1668 Struglinski MJ; Graessley WW. Macromolecules 1985 2630 Colby RH; Fetters LJ; Graessley WW. Macromolecules 1987 2226 Rubinstein M; Colby RH. J. Chem. Phys. 1988 5291 Baumgaertel M; Derosa ME; Machado J; Masse M; Winter HH. Rheol. Acta 1992 75 Wang SF; Wang SQ; Halasa A; Hsu WL. Macromolecules 2003 5355
Getro JT; Graessley WW. Macromolecules 1984 2767 Santangelo PG; Roland CM. Macromolecules 1998 3715 Watanabe et al. Macromolecules 2004 1937; and 2000 499Abdel-Goad M; Pyckhout-Hintzen W; Kahle S; Allgaier J; Richter D; Fetters LJ. Macromolecules 2004 8135
Onogi S; Masuda T; Kitagawa K. Macromolecules 1970 109 Graessley WW; Roovers J Macromolecules 1979 959 Schausberger A; Schindlauer G; Janeschitz-Kriegl H. Rheol. Acta 1985 220 Lomellini P. Polymer 1992 1255
Liu, He, Keunings, Bailly
Polymer (2006)
2.1 Dependence of GN0 on ZeDependence of Gexpl on Z
Red
uced
Gex
p
LM Theory : Exact CLF treatment + CR
Likhtman and McLeish Macromolecules (2002)
MW dependence of plateau modulus less than predicted by advanced tube models:
Liu et al. Macromolecules (2006)
Dependence of Gexpl on Z
10 100 10000.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
G'min G'' integral 3.56 G''max G'min 3.56 G''max
Z
Normalized
G0 app
G'min of L-M model K-N model
100k 1M20k Mw (g/mol)
Nor
mal
ized
Gep
xtl
2.2 Comparison experimental data with predictions
Excellent accuracy for the terminal relaxation time
Significant stress deviations for low Mw samples
Relaxation time-modulus contradiction
Experimental data vs. predictions of LM theory
Inconsistency for the value of
G0F ; G0
NF 1- m.Z - 1/2ÈÎ
˘˚
t 0F ; t 0
NF 1- m.Z - 1/2ÈÎ
˘˚2
Non-permanententanglements
Fluctuations
3.4
0 MW3.4
D MW-2.3
Experimental scaling:
0
MW
Zero shear viscosity
CLF of Probe chainsUnaffected (?)
Tube Motion suppressed
Separate contributions of tube motion from CLF
Idea goes back to Ferry and coworkers (1974-81)
Put a small amount of short chains in a very high MW matrix
Probe rheology
CLF of Probe chainsunaffected
Tube Motion suppressed
Separate contributions of tube motion from CLF
Key question : is there a MW dependence of the retardation factor ?
RF =
τ d of probe in Maτrixτ d of probe Self−elτ
Probe rheology
CLF of Probe chainsunaffected
Tube Motion suppressed
Separate contributions of tube motion from CLF
If yes, there should be a contribution of tube motions to the non reptation scaling of viscosity !
Probe rheology
10% Probe in Matrix
Probe Rheology
▪ G’ ω2 and G’’ ω
▪ G’ and G’’ cross-point close to G’’max
5 10 50 100 500
2.5
5
7.5
10
CR model: RF ~ 2.5
slope = - 0.3Retardation factors
Z
PBD PI PS Ref 45 Ref 46
Retardation Factors as a function of Z
meltSelfprobeofMatrixinprobeofRF
d
d
−=ττ
10 10010-5
10-4
10-3
10-2
10-1
100
3.4
3.1
τd
Z
in Maτrix in Self- elτ
τ d
τ d/Z
3
Probe rheology
CR parameter:
Cv = 1 or 0: with or without CRDoi (1981, 1983)Milner and McLeish (1998)Likhtman and McLeish (2002)
τ d/Z
3
Probe rheology
Probe Rheology vs Tracer Diffusion
Lodge (1999)Wang (2003)
Two entangled environments:in Self-melt or in High Mw Matrix
DM
2
Rouse region :
Longitudinal modes and « is it Rouse ? »
PBD 1.2M Master Curve
10-3 10-1 101 103 105 107104
105
106
G' G'' Slope 0.71 CutRouse
G', G'' (Pa)
ω (rad/s)
τpeak ~ a few multiples of τe
10-1 100 101 102104
105
106
107
Slope = 0.71
G' G''
G', G'' (Pa)
ω (rad/s)
PBD 1.2M –80 oC
G’’ - (A.ω0.71)
10-1 100 101 1020
1x105
2x105
G'' - (A.ω0.71)
G'' (Pa)
ω (rad/s)
10-3 10-1 101 103 105 1070.0
2.0x105
4.0x105
1.15E62.44E5
G'' CutRouse
G'' (Pa)
ω (rad/s)
Relaxation strength ~ 1/4 GN0
PBD 1.2M Master Curve Linear-Log
Longitudinal Modes
Shape of relaxation peak ~ Maxwell
105 106 107 1080
1x105
2x105
3x105
G', G'' (Pa)
ω/ω ax
Maxωell G' Maxωell G'' G' G''
102 104 106 108
0.0
0.2
0.4
0.6
0.8
1.0
1/τe = 106
1/τR = 100
G', G''
ω (rad/s)
G' of longiτudinal odes G'' of longiτudinal odes G' of Maxωell τx = 3 τe G'' of Maxωell τx = 3 τe
Slippage of a polymer chain through entanglement links.
Likhtman-McLeish Macromolecules 2002 Lin Macromolecules 1984
τpeak = 3τe
LM prediction vs Maxwell
Redistribution of monomers along the tube
Conclusions - Questions
▪ There seem to be inconsistencies of tube model predictions for time/stress and CR/CLF balance
▪ Probably some of the inconsistencies come from the non-universality of real chains.
▪ Several possible reasons :
a chain hits entangled constraint before reaching Rouse behavior
local stiffness effects
interchain correlations
▪ Moreover: the assumption that fluctuations are unaffected in bimodal blends can be wrong if fluctuations depend on the environment
Gexp l = 2
pG"(w)d ln
- •
+ •
Ú w
Ferry (1980)
Published methods for Gexpl determination
G” Integral method
-3 -2 -1 0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
G"/G"
max
ω/ωax
in the terminal region
(Kramer-Kronig principle)
Gexp l
G ''max
= 2.303 2p
G"(w / wmax )G ''max
d log w / wmax( )- •
+ •
ÚÈ
ÎÍÍ
˘
˚˙˙= 3.56
Raju et al. Macromolecules (1981)
If the shape is universal, Gapp must be proportional to the maximum of the terminal G” peak
Maximum G” method
Published methods for Gexpl determination
G’’max vs. Z : data vs. predictions
Too strong Z dependence
G” m
ax /
Gex
pl
PBD 99K in 1.2M Matrix
τe▪ Probe in Matrix vs. Probe Self-melt
▪ Probe in Matrix vs. Matrix
10-1 100 101 102 103 104 105 106 107
104
105
G', G'' (Pa)
ω (rad/s)
Probe-99k G' Probe-99k G'' Probe-39k G' Probe-39k G'' Probe-14k G' Probe-14k G''
10-1 100 101 102 103 104 105 106 107
104
105
G', G'' (Pa)
ω (rad/s)
Probe-99k G' Probe-99k G'' Probe-39k G' Probe-39k G'' Probe-13k G' Probe-13k G''
Probe Rheology vs. LM Model without CR
▪ Same horizontal shift factors for ALL: 5.2 106
Vertical shift factor: (1 – fmatrix2) GN
0
▪ Horizontal shift factors: 5.2 106; 4 106; 2 106
Data from: Likhtman and McLeish (2002)
Z = 63, 24, 9; Constraint release parameter cv = 0
Graessley (1980)
Evaluation of the τd
▪ Narrow G’’ peak
▪ Retardation of the τd
Suppression of tube motions
Two Key Results for Probe Chain
100 101 102 103 104 105
104
105
-1/4
-1/2
G', G'' (Pa)
ω (rad/s)
PBD-99K probe Ze-63 LM odel ωiτ Cv = 0
CLF for Well-entangled case
▪ Excellent agreement with model w/o CR
Likhtman and McLeish (2002)Vertical shift: (1 – fmatrix2) GN
0
102 103 104 105 106104
105
PBD-14K Subtraction of Rouse modes 10% PBD-14K in Matrix Subtraction of diluted Matrix
5.1G'' (Pa)
ω (rad/s)
