Rheology of commercial polyolefins: Relating the viscoelastic behaviour to microstructure 1
2
I. Stratigakis , C. Tsenoglou and A.D. Gotsis
1
1
Department of Mineral Resources Engineering, Technical University of Crete, 73100 Chania (
[email protected] ;
[email protected]) 2 School of Chemical Engineering, National Technical University of Athens, Greece (
[email protected])
Molecular architecture of commercial polyethylenes
T.U.C
Estimation of Bn from elongational rheology data Strain sensitivity parameter, β (0 ≤ β ≤ 2)
• Branched vs. Linear macromolecular chains – Commercial polyethylenes are polydisperse – LDPE has long chain branches on the main chain and on its branches – HDPE is considered to have no branches (or very few)
• Inverse measure of network connectivity strength ( ) √ − m+0.5 0.6 β = βu 2 − e • βu depends on molecular structure • βu ≥ 1 : linear polymers
Rheological behaviour of commercial polyethylene melts
• m = −0.5: uniaxial; m = 1: equibiaxial extension • βu depends on the branching number, Bn √ (−α 3 Bn)
βu ≈ 2e
• Scope – Consistent predictions about the rheological behaviour of polyethylenes in the melt state during processing – Processing flows affect material’s physical and mechanical properties.
– Bn, average number of long chain branches per molecule – α , fitting parameter; not a universal constant but depends on the MWD breadth and the details of branching
The eXtended Pom-Pom model
• Constitutive modelling – Successful modeling is achieved using a class of differential equations that are based on Boltzmann’s superposition principle of fading memory and the idea of the separation of time and strain effects – Emphasis is put on elongational flows – Constitutive equations are provided both in integral and differential form
The temporary network model
• Constitutive equation of the UCM/Lodge type ∇
σ i +λ (σ )−1σi = 2GiD ∇ σ i,
upper-convected time derivative of the extra-stress tensor
• Non-linear parameter fitting Table 1: XPP parameters for fitting the LDPE Riblene melt, T = 150oC, Mw=160.3 (kg/mol ), PI = 12, Ea=50.4 (kJ/mol )
λ (σ )−1, relaxation time tensor with an extra function
i 1 2 3 4 5 6
D, rate of deformation tensor
Lodge’s rubber-like liquid with a damping function
σ=
∫t
µ (t − t ′)h(I1, I2)C−1(t ′,t)dt ′
C−1(t,t), deformation tensor (affine deformation)
µ (t − t) =
qi 1 1 2 2 6 6
XPP model λ0,si (sec) αi 0.0002 0.1 0.0025 0.1 0.03 0.05 0.53 0.05 7.3 0.016 74.1 0.016
Table 2: XPP parameters for fitting the HDPE Eraclene melt, T = 170oC, Mw=196.6 (kg/mol ), PI = 16, Ea=32.5 (kJ/mol )
−∞
N Gi − (t − t ′ )/τi , ∑i=1 τi e
Maxwell parameters λ0,bi (sec) Gi (Pa) 0.001 137370 0.01 59813 0.1 30000 1 12869 10 3000 100 645
memory function
h(I1, I2), damping function accounts for non-linear behaviour
i 1 2 3 4 5 6
Damping function; common for shear and elongational flows
Maxwell parameters λ0,bi (sec) Gi (Pa) 0.001 290000 0.01 150000 0.1 57721 1 20178 10 5683 100 1900
qi 1 1 1 1 1 2
XPP model λ0,si (sec) αi 0.0002 0.1 0.002 0.1 0.025 0.1 0.33 0.1 3.6 0.1 41.7 0.05
• Wagner-1 single exponential for shear flows: ( √ ) h(I2) = exp −n I2 − 3 = exp(−nγ )
qi, number of arms on either side of the pom-pom
• Tsenoglou’s power-law for elongational flows:
λ0,bi, backbone orientation relaxation times (from DMA) λ0,si, stretch relaxation times (physically constrained λ0,bi−1 < λ0,si ≤ λ0,bi)
h(λ ) = λ −β γ , magnitude of shear deformation
• The fit is good
λ ≡ exp(ε ), principal stretch ratio; ε : Hencky strain
• Fitting of the non-linear parameters is done manually; αi = 0.1/qi
The amount of vertical displacement of the curves of stepstrain measurements at different strains gives the form of h(λ ) and the value of the parameters n and β
• The scalar parameter, ai, controls the cut-off viscosity at the onset of steady-state
Conclusions • The values of the non-linear fitting parameters n and β can be the same. Therefore, a single damping function can be used for simple shear and uniaxial extensional flows. • The strain sensitivity parameter evaluated from shear measurements may be used for the prediction of the extensional viscosity growth • The non-linear parameters, qi of the XPP model may be used as a measure of branching in the melt. However, their values are inferred from extensional rheology measurements only • The stretching relaxation times, λ0,si of the XPP model may also be correlated with the corresponding relaxation times found from single step-strain measurements
Branched polypropylenes fitted with different values of β
LDPE at different strain rates
References
fitted with β = 0.25
1. M.H. Wagner and J. Meissner, Makromol. Chem., 181, 1533-1550, 1980 2. A.D. Gotsis, B.L.F Zeevenhoven, and C. Tsenoglou, J. of Rheol., 48, 895-914, 2004 3. C. Tsenoglou, E. Voyiatzis and A.D. Gotsis, J.Non-Newt.Fl.Mech., 138, 33-43, 2006 4. R.P. Lagendijk, A.H. Hogt, A. Buijtenhuijs and A.D. Gotsis, Polymer, 42, 10035-10043, 2001 5. A.D. Gotsis, B.L.F. Zeevenhoven and A.H. Hogt, Polym. Eng. Sci., 44, 973-982, 2004 6. W.M.H. Verbeeten, G.W.M. Peters and F.P.T. Baaijens, J. of Rheol., 45, 823-843, 2001
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