Rheology and Rheometry
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Rheology and Rheometry Paula Moldenaers Department of Chemical Engineering Katholieke Universiteit Leuven W. De Croylaan 46, B-3001 Leuven Tel. 32(0)16 322675 Fax 32(0)16 322991 http://www.cit.kuleuven.ac.be/cit. ρειν = flow Rheology = The Science of Deformation and Flow Why do we need it? - measure fluid properties - understand structure-flow property relations - modelling flow behaviour - simulate flow behaviour of melts under processing conditions Processing conditions shear rate [s-1] Sedimentation 10-6 –10-4 Leveling 10-3 Extrusion 100-102 Chewing 101 -102 Mixing 101-103 Spraying, brushing 103 -104 Rubbing 104 -105 Injection molding 102 -105 coating flows 105 -106 How about Newtonian behaviour? 1. Constant viscosity 2. No time effects 3. No normal stresses in shear flow 4. η(ext)/η(shear) = 3 Symbols: ε° ⇒ D σ⇒T s ⇒σ Newton’s law: T = -pI +η 2D Contents 1. Rheological phenomena 2. Constitutive equations 2.1. Generalized Newtonian fluids 2.2. Linear visco-elasticity 2.3. Non-linear viscoelasticity 3. Rheometry 4. Parameters affecting rheology 1. Rheological Phenomena: Do real melts behave according to Newton’s law? Deviation 1: Mayonnaise: resistance (viscosity) decreases with increasing shear rate: shear thinning Starch solution: resistance increases with increasing shear rate: shear thickening This is a non-linearity: η(γ&) Do real melts behave according to Newton’s law? Deviation 2: example silly putty t < τ t > τ The response of the material depends on the time scale: * Short times: elastic like behaviour * Long times: liquid-like behaviour VISCO-ELASTIC BEHAVIOUR G(t) or G(t,γ) Linear non-linear Do real melts behave according to Newton’s law? Deviation 3: Weissenberg effect Die Swell Normal stresses Do real melts behave according to Newton’s law? deviation 4: Ductless Syphon η(ext) > > η(shear) In summary: Newtonian behaviour real behaviour 1. Constant viscosity 1. Variable viscosity 2. No time effects 2. Time effects 3. No normal stresses in shear flow 3. Normal stresses 4. η(ext)/η(shear) = 3 4. Large η(ext) 3 dim: T = -pI + η (2D) ? Simple shear: σ = η dγ/dt Contents 1. Rheological phenomena 2. Constitutive equations 2.1. Generalized Newtonian fluids 2.2. Linear visco-elasticity 2.3. Non-linear viscoelasticity 3. Rheometry 4. Parameters affecting rheology 2. Constitutive equations 2.1 Generalized Newtonian fluids Examples of non-Newtonian behaviour: Macosko, 1992 2.1 Generalized Newtonian fluids Non-Newtonian behaviour is typical for polymeric solutions and molten polymers e.g. ABS polymer melt (Cox and Macosko) 2.1 Generalized Newtonian fluids T = −pI + f1 (II2D ) ⋅2D The viscosity is now replaced by a function of the second invariant of 2D for a shear flow this becomes: ση=⋅γ2 γ xy 1( &&) and different forms for this function have been proposed 2 2 With: II2D = 1/ 2 (tr 2D – tr (2D) ) tr = trace = sum of the diagonal elements Model 1: Power Law = (n−1)/ 2 2 parameters T = −pI + k ⋅ II2D ⋅2D 100 n n−1 σxy = kγ& η = kγ& n=0.4, k=1 n=0.8, k=1 10 n=1.4, k=0.01 1 300 η 0.1 250 0.01 200 0.001 150 σ 0.001 0.01 0.1 1 10 100 1000 100 shear rate (s-1) 50 n<1 : shear-thinning 0 0 200 400 600 800 1000 1200 n>1: shear-thickening shear rate (s−1) Model 2: Ellis model η 1 = 3 parameters η ()1−n 0 1+⋅k γ& n−1 1000 ⎡ II ⎤ η ( σ ) =+1 ⎢ ⎥ log η η ⎢ k ⎥ 0 0 ⎣ ⎦ 100 n-1 η 10 1 0.1 0.001 0.01 0.1 1 10 100 1000 shear rate (s-1) Model 3: CROSS model ηη− ∞ 1 = 4 parameters ηη− ()1−n 0 ∞ 1+⋅k γ& 1000 log η0 3D 100 n-1 ηη− 1 η 10 ∞ = 2 ()12−n / ηη− ∞ 0 1+⋅(kII2D ) 1 log η∞ 0.1 1e-3 1e-2 1e-1 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 shear rate (s-1) Special case: plastic Behaviour Yield stress σ < σy → γ& = 0 , σ = G ⋅ γ No flow, only deformation σσ>→y γ& ≠0 What have we gained in generalized Newtonian? Newtonian behaviour real behaviour 1. Constant viscosity 1. Variable viscosity 2. No time effects 2. Time effects 3. No normal stresses in shear flow 3. Normal stresses 4. η(ext)/η(shear) = 3 4. Large η(ext) 3 dim: T = -pI + η (2D) T = -pI + η(II2D) (2D) Simple shear: σ = η dγ/dt Contents 1. Rheological phenomena 2. Constitutive equations 2.1. Generalized Newtonian fluids 2.2. Linear visco-elasticity 2.3. Non-linear viscoelasticity 3. Rheometry 4. Parameters affecting rheology 2. Constitutive equations 2.2 Linear visco-elasticity Reference: 2 extremes Hooke’s Law Newton’s Law (solid mechanics) (fluid mechanics) σ = G.γ σ= η .dγ/dt G = modulus (Pa) η = viscosity (Pa.s) Material property material property Time effects (linear visco-elastic phenomena): Example 1: creep Apply constant stress σ Compliance γ()t Jt()= σ0 Time effects (linear visco-elastic phenomena): Example 2: stress relaxation upon step strain Apply constant strain Modulus G(t) = σ(t)/γ Example for molten LDPE [Laun, Rheol. Acta, 17,1 (1978)] How to descibe this behaviour? Example: differential models Phenomenological models Hookean spring σ1 = G0γ1 Newtonian dashpot σ = η ⋅ γ Maxwell Kelvin-Voigt 2 0 & 2 Maxwell model: γ = γ12+ γ σσ==12σ γγ&&=+12γ& σσ& 1 2 γ& =+ G0 η0 ⎛ η0 ⎞ σ + ⎜ ⎟ση& = 0γ& ⎝G0 ⎠ στ+=σ& η0γ& τ = relaxation time Example: Stress relaxation upon step strain for a Maxwell element 12 G0 10 γ =≥γ 0 t 0 8 τ ) dσ t 6 στ+=0 G( dt 4 tG==0;σγ00 2 0 σ()t 0.01 0.1 1 10 ==Gt( ) G0 ⋅exp(−t/ τ) γ ω [rad/s] Generalized Maxwell model to describe molten polymers: σTOT = ∑ σi i στii+=σ&&iηiγ Gt( ) =−∑G0i exp( t/τi ) i Relaxation functions: For a simple Maxwell model: single exponential (single relaxation time τ): Gt()= G0 exp(−t/ τ) For a generalized Maxwell fluid (discrete number of relaxation times τI: Gt( ) =−∑Giiexp( t/ τ ) We can replace the discrete relaxation times by a continuous spectrum: ∞ ⎛ −t ⎞ Gt()= ∫ F(τ)exp⎜ ⎟dτ 0 ⎝ τ ⎠ Or based on a logaritmic time scale : the relaxation spectrum is defined by: ∞ ⎛ −td⎞ τ Gt()= ∫ H(τ)exp⎜ ⎟ 0 ⎝ τ ⎠ τ Time effects (linear visco-elastic phenomena): Example 3: Oscillatory experiments STRAIN: γγ= 0 sin(ωt) γγ= 0 exp(itω) STRAIN RATE γγ& ==00ωcos(ωtt) γsin(ω+90°) γω& = ieγ0 xp(itω) HOOKEAN SOLID σγ==GGγ0 sin(ωt) σγ= Gi0 exp( ωt) NEWTONIAN FLUID ση==γ& ηγ0ωsin(ωt +90°) ση==iiγ0ωexp( ωt) ηiγω VISCOELASTIC MATERIAL σσ=+el σvisc σγ=+Giωηγ σω=+()Giηγ complex modulus: σγ= G∗ ∗ σγ=+Gt0 sin(ωδ) ∗ σγ=+Gt0[]sin(ω).cos(δ) cos(ωt).sin(δ) ∗∗ σδ=⋅(Gtcos( )) γ00sin(ω) +(Gtsin(δ)) ⋅γcos(ω) σω=⋅[]Gt'sin()+G"⋅cos()ωtγ0 σγ=+('GiG") G'' Storage and Loss modulustanδ= G' Dynamic moduli of a Maxwell fluid 10 G0 22 G ωτ 1 τ G'( ) 0 ω = 22 1+ ωτ a] P [ 0.1 G ωτ G"( ) 0 Moduli ω = 22 1+ ωτ 0.01 G' G" 0.001 0.01 0.1 1 10 100 w [rad/s] G’ and G” for a generalized Maxwell fluid: ∞ ∞ N ⎛ −τi s ⎞ GG"(=⋅ωω∫ s)cos(s)ds=ω∫ ⎜ ∑Gie⎟ ⋅ cos(ωs)ds 0 0 ⎝ i =1 ⎠ N ωτ G i = ∑ i 2 i =1 1+ ()ωτi ∞ GG'(=⋅ωω∫ s)sin(s)ds 0 N ()ωτ 2 G i = ∑ i 2 i =1 1+ ()ωτi What have we gained in linear visco-elasticity? Newtonian behaviour real behaviour 1. Constant viscosity 1. Variable viscosity 2. No time effects 2. Time effects 3. No normal stresses in shear flow 3. Normal stresses 4. η(ext)/η(shear) = 3 4. Large η(ext) 3 dim: T = -pI + η (2D) G(t), H(τ),… Simple shear: σ = η dγ/dt fully descibes linear VE Contents 1. Rheological phenomena 2. Constitutive equations 2.1. Generalized Newtonian fluids 2.2. Linear visco-elasticity 2.3. Non-linear viscoelasticity 3. Rheometry 4. Parameters affecting rheology 2. Constitutive equations 2.3. Non-linear visco-elasticity Example 1: Steady state shear flow σ η = xy σ xy γ& N1 = σ xx −σ yy σ xx −σ yy Ψ1 = 2 N2 = σ yy −σ zz γ& σ yy −σ zz Ψ2 = 2 LDPE, Laun et al. PIB in decalin, Keentok et al. γ& Example 2: Non-linear stress relaxation upon step strain LDPE, Laun et al. Example 3: Stress evolution upon inception of shear or elongational flow What have we gained in non-linear visco-elasticity? Newtonian behaviour real behaviour 1. Constant viscosity 1. Variable viscosity 2. No time effects 2. Time effects 3. No normal stresses in shear flow 3. Normal stresses 4. η(ext)/η(shear) = 3 4. Large η(ext) 3 dim: T = -pI + η (2D) no universal model Simple shear: σ = η dγ/dt Contents 1. Rheological phenomena 2. Constitutive equations 2.1. Generalized Newtonian fluids 2.2. Linear visco-elasticity 2.3. Non-linear viscoelasticity 3. Rheometry 4. Parameters affecting rheology 3. Rheometry Why ? 1. Input for Constitutive Equations 2. Quality control 3. Simulate industrial flows A rheometer is an instrument that measures both stress and deformation. Ù indexer Ù viscometer What do we want to measure? - steady state data - small strain (LVE) functions - large strain deformations Introduction: classifications ÎKinematics : shear vs elongation Îhomogeneous vs non-homogeneous vs complex flow fields Î type of straining: + - - small : G’(ω), G”(ω), η (t), η (t), G(t), σy - large : G’(ω,γ), G”(ω,γ), η+(t,γ), η-(t ,γ), G(t ,γ), η(t,ε) γ& γ& - steady : η( ), Ψ1( ) Î shear rheometry : Drag or pressure driven flows.