Rheology
Antonios Polymer Zavaliangos
Department of Materials Engineering Drexel University
Viscosity
Characterizes resistance to flow Central property for polymers processing since most work is done in the melt. Polymers are COMPLEX FLUIDS! As a result viscosity is not a single number!..
Definition
Shear viscosity
Fh = = & A v
A (length x width) v.t h F
Elongational (Trounton) viscosity
FL E = = Av &
F, v
A (height x width)
Units: Pa.sec (Poise=0.1 Pa.sec)
Viscosity in fiber spinning
Shear flow
Elongational flow
Molecular Origin
The continuous rupture and re-formation of secondary bonds as molecules slide past one-another Continuous supply of irreversible power required to maintain flow This power is converted to heat (viscous heating) Entaglements play a major role in polymers
H H H H H | | | | | -C-C-C-C-C| | | | | H H H H H
Entaglement length
Reptation
How do chains slide past one another? Entaglements place topological constrains on the polymer chain that can be thought as a fictitious tube within which the chain is constrained to move Chain motion is thought to occur in a reptation mode where collaborative motions of sections of the chain (biased by the application of external shear stress) results in macroscopic motion.
Viscosity of typical liquids
Water Oil Honey Molten glass Molten nylon Polymers 0.1 Pa sec 1 Pa sec 5-10 Pa sec 10-20 Pa sec > 100 Pa sec From 1 to 10 Pa sec
5
Viscosity vs Shear rate
In simple liquids viscosity is constant but in polymers it may depend on shear rate
= & = K& n = K& n 1
Zero shear viscosity
Newtonian Plateau (viscosity constant) Linear viscous
Power law Shear thinning
Viscosity vs Shear Rate
In simple liquids viscosity is constant but in polymers it may depend on shear rate
= & = K& n = K& n 1
Newtonian Plateau (viscosity constant) Linear viscous Power law Shear thinning
The role of molecular orientation in viscosity vs shear rate relation
log
Power Law Linear in log-log plot Significant alignment results in a decrease in viscosity Steady state alignment level is achieved. More alignment for fast flow
ORJ
0
n
1
Newtonian Plateau (viscosity constant) Linear viscous
III I II
No structure develops Any alignment is destroyed by random thermal motion Slow flow High strain rate plateau Fully aligned molecules Flow is so fast that thermal motion is not fast enough to destroy it. (difficult to measure)
log &
Power law
Typically n < 1
'LODWHQW
/LQHDU 1HZWRQLDQ
3VHXGRSODVWLF
(shear thinning - pseudoplastic)
Doubling the shear rate requires less than double the power (think why) Although the viscosity decreases the required forces (pressure/stress) are higher for higher shear rate.
Viscosity Models
Power Law (Ostwald; de Waale)
= K& n 1 K = consistency index, n = power law index
Useful but
at zero shear rate -> infinite viscosity at infinite shear rate -> zero viscosity
Bird-Carreau-Yasuda
n 1 0 = (1 + k& ) 0
Has a zero and infinity viscosity and a power law region
Viscosity Models
Menges-Wortberg-Michaeli
K1aT = (1 + K 2 aT & )K3
Other models do exist The question is do they do a better job that those above and do they require more experiments to fit them...
Bingham Fluid
Occasionally (usually when large volume of additive is present (e.g. talk filled PP) plastics exhibit a yield stress. In other words when the shear stress is lower than the yield stress there is no flow.
= 0 + Y & if < Y if > Y
This has major consequences for the flow patterns of the material and must be quantified.
Elongational viscosity
Very difficult to measure For incompressible material, by analogy to elasticity it should be 3x the shear viscosity. This is only true in the Newtonian plateau
Temperature effect on Viscosity
Arrhenius
Very strong (e.g. from 120C to 135C viscosity of PE is reduced by 50%). Higher temperature decreases viscosity and as a result forces are reduced or productivity increases (upper limit degradation of plastic or wear of equipment).
E = Aexp R
Exponential
= a exp( )
Carreau-WLF
K1aT = (1 + K 2 aT & )K3 log aT = a1 a2T T a3
Effect of Mw on Viscosity
cM w M w < M c 0 = 3 cM w.4 M w > M c
e.g., Mc=4000 for PE and Mc=40000 for PS. Why? At low Mw interchain friction controls viscosity while at high Mw entanglements play an important role. The critical value Mc (entanglement molecular weight) defines the boundaries of dominance of the two mechanisms and it depends on the type of the monomer, the chain flexibility and mobility (i.e. implicitly temperature), and represents the minimum chain length required to generate entanglements. The Rouse model (1953) predicts the behavior at low Mw while the reptation model (1971-today) approximates the high Mw behavior (it predicts an exponent of 3).
HIGH MW (GREAT PROPERTIES) PRESENT PARTICULAR CHALLENGES IN PROCESSING!...
Effect of pressure on Viscosity
= 0 exp( (P P0 ))
much less in PE than in PS. Why? Example (PS): 30 bar - 1.03 0 100 bar - 1.105 0 300 bar - 1.35 0 1000 bar - 2.72 0 3000 bar - 20 0 Can be neglected in extrusion maybe important in injection molding
Viscosity of Curing Thermosets
As thermosets cure, their viscosity increases dramatically. The gel point represents a near infinite viscosity limit. A typical model for the viscosity is:
E c g = 0 exp R c g c
C1 +C2 c
Increasing temperature lowers viscosity BUT accelerates crosslinking. Therefore forming of thermosets is limited and must be properly optimized Viscosity characterization of thermosets requires not only rheometry but the evolution of the degree of cure c.
Viscoelastic Effects in Polymer Flow
Die swelling and associated defects
Viscoelastic Effects in Polymer Flow
Melt elasticity.
Viscoelastic Effects in Polymer Flow
Weissenberg effect
Viscoelastic Effects in Polymer Flow
Viscoelastic effects
Think of a coil in its maximum entropy, random coil, which is subjected to a stress field and deforms. When the stress is released the molecule will relax back to its most random state, however this relaxation will take time. This is called Relaxation Time and is associated with the molecular structure, molecular weight and temperature since the entropy of the coil is related to temperature. The application of stress and release of stress are also associated with a characteristic time which could be called the experimental time constant. For a flow process the experimental time is 1/strain rate. The Deborah number , De, has been defined as the ratio between the relaxation time and the experimental time. Large De means the material will behave material will behave like a solid in the operation (this is bad for extrusion). Small De means the material will behave like a liquid (good for processing) Intermediate De means the material will behave as a visco-elastic displaying both solid (elastic) and liquid (viscous) features.
Viscoelastic effects
Criminale-Ericksen-Filbey (CEF) Equations
2 N1 = xx yy = 1& xy 2 N 2 = yy zz = 2& xy
Difficult to measure also strain rate, temperature and pressure dependent.
What level of complexity should I use?
Always use the simplest equations that "work". If the Newtonian fluid equations are good enough, then use them. If you can live with a small modification of Newtonian fluids such as a power-law fluid, then do it. But, be aware of the effects. Remember that complex models need a lot of experiments for parameter identification.