1 Topics to be Covered Polymer Melt Rheology Introduction to Viscoelastic Behavior Time-Temperature Equivalence Chapter 11 in CD (Polymer Science and Engineering)
2 Polymer Melt Rheology
3 δ τ xy Newton s Law z θ v 0 The most convenient way to describe deformation under shear is in terms of the angle θ through which the material is deformed; tanθ = δ z = γ xy Then if you shear a fluid at a constant rate γ xy δ = z = v 0 z dx dz γ xy = dv dz = d dt y Just as stress is proportional to strain in Hooke s law for a solid, the shear stress is proportional to the rate of τ xy γ strain for a fluid xy τ xy = η γ xy
4 Newtonian and Non - Newtonian Fluids Shear Stress.. τ xy = η a (γ) γ Slope = η a Shear Thinning Newtonian Fluid (η = slope) Shear Thickening Strain Rate When the chips are down, what do you use?
5 Variation of Melt Viscosity with Strain Rate Log η a (Pa) Zero Shear Rate Viscosity Log γ (sec -1 ) SHEAR RATES ENCOUNTERED IN PROCESSING Compression Molding Calendering Extrusion Injection Molding Spin Drawing Strain Rate (sec -1 )
6 How Do Chains Move - Reptation D ~ 1/M 2 η 0 ~ M 3
7 Variation of Melt Viscosity with Molecular Weight Log η m + constant Poly(di-methylsiloxane) Poly(iso-butylene) Poly(ethylene) Poly(butadiene) Poly(tetra-methyl p-silphenyl siloxane) Poly(methyl methacrylate) Poly(ethylene glycol) Poly(vinyl acetate) Poly(styrene) η m = K L (DP) 1.0 η m = K H (DP) 3.4 Redrawn from the data of G. C. Berry and T. G. Fox, Adv. Polym. Sci., 5, 261 (1968) Log M + constant 5
8 Entanglements Short chains don t entangle but long ones do - think of the difference between a nice linguini and spaghettios, the little round things you can get out of a tin (we have some value judgements concerning the relative merits of these two forms of pasta, but on the advice of our lawyers we shall refrain from comment).
9 Entanglements Viscosity - a measure of the frictional forces acting on a molecule η m = K L (DP) 1.0 Small molecules - the viscosity varies directly with size At a critical chain length chains start to become tangled up with one another, however Then η m = K H (DP) 3.4
10 Entanglements and the Elastic Properties of Polymer Melts d 0 d d/d C 180 C Depending upon the rate at which chains disentangle relative to the rate at which they stretch out, there is an elastic component to the behavior of polymer melts. There are various consequences as a result of this C Strain Rate at Wall (sec -1 ) Jet Swelling
11 Melt Fracture Reproduced with permission from J. J. Benbow, R. N. Browne and E. R. Howells, Coll. Intern. Rheol., Paris, June-July 1960.
12 Viscoelasticity If we stretch a crystalline solid, The energy is stored in the Chemical bonds If we apply a shear stress to A fluid,energy is dissipated In flow VISCOELASTIC Ideally elastic behaviour Ideally viscous behaviour
13 Viscoelasticity Homer knew that the first thing To do on getting your chariot out in the morning was to put the wheels back in. (Telemachus,in The Odyssey, would tip his chariot against a wall) Robin hood knew never to leave his bow strung
14 Experimental Observations Creep Stress Relaxation Dynamic Mechanical Analysis
15 Creep and Stress Relaxation 5lb 5lb 5lb 5lb Creep - deformation under a constant load as a function of time TIME Stress Relaxation - constant deformation experiment 5lb 4lb 3lb 2lb TIME
16 Creep(%) Redrawn from the data of W. N. Findley, Modern Plastics, 19, 71 (August 1942) 2695psi 2505 psi 2305 psi 2008 psi Creep and Recovery 30 Strain (c) VISCOELASTIC RESPONSE Creep psi Stress applied Stress removed Recovery } deformation Permanent Time psi 1018 psi Time (hours) 8000
17 Strain vs. Time Plots - Creep (a) PURELY ELASTIC RESPONSE Strain (b) PURELY VISCOUS RESPONSE Strain Stress applied Stress removed Time Shear Stress applied Shear Stress removed Permanent Deformation Time Strain (c) VISCOELASTIC RESPONSE Creep Stress applied Stress removed Recovery } Permanent deformation Time
18 Stress Relaxation 10 Redrawn from the data of J.R. McLoughlin and A.V. Tobolsky. J. Colloid Sci., 7, 555 (1952) C 60 0 C The data are not usually reported as a stress/time plot, but as a modulus/time plot. This time dependent modulus, called the relaxation modulus, is simply the time dependent stress divided by the (constant) strain E(t ) = σ (t ) ε 0 Log E(t), (dynes/cm 2 ) C 92 0 C C 80 0 C C Stress relaxation of PMMA C C C C Time (hours)
19 Amorphous Polymers - Range of Viscoelastic Behaviour V s Rubbery Semi- Solid Liquid Like Hard Glass Leather Soft Like Glass T T g
20 Viscoelastic Properties 0f Amorphous Polymers Log E (Pa) 10 Glassy Region Rubbery Plateau Low Molecular Weight Temperature Cross-linked Elastomers Melt Measured over Some arbitrary Time period - say 10 secs
21 Viscoelastic Properties 0f Amorphous Polymers Stretch sample an arbitrary amount, measure the stress required to maintain this strain. Then E(t) = σ(t)/ε Log E(t) (dynes/cm 2 ) Low Molecular Weight Glassy Region Glass Transition Rubbery Plateau High Molecular Weight Log Time (Sec)
22 Time Temperature Equivalence Log E(t) (dynes/cm 2 ) Low Molecular Weight Glassy Region Glass Transition Rubbery Plateau High Molecular Weight Log E (Pa) Log Time (Sec) 6 Rubbery Plateau Glassy Region Low Molecular Weight Temperature Cross-linked Elastomers Melt
23 Relaxation in Polymers First consider a hypothetical isolated chain in space,then imagine stretching this chain instantaneously so that there is a new end - to - end distance.the distribution of bond angles (trans, gauche, etc) changes to accommodate the conformations that are allowed by the new constraints on the ends. Because it takes time for bond rotations to occur, particularly when we also add in the viscous forces due to neighbors, we say the chain RELAXES to the new state and the relaxation is described by a characteristic time τ. How quickly can I do these things?
24 Amorphous Polymers - the Four Regions of Viscoelastic Behavior Log E (Pa) Low Molecular Weight Polymer Melt Temperature GLASSY STATE - conformational changes severely inhibited. Tg REGION - cooperative motions of segments now occur,but the motions are sluggish ( a maximum in tan δ curves are observed in DMA experiments) RUBBERY PLATEAU - τ t becomes shorter,but still longer than the time scale for disentanglement TERMINAL FLOW - the time scale for disentanglement becomes shorter and the melt becomes more fluid like in its behavior
25 Semicrystalline Polymers Motion in the amorphous domains constrained by crystallites α Motions above Tg are often more complex,often involving coupled processes in the crystalline and amorphous domains γ β LDPE LPE Less easy to generalize - polymers often have to be considered individually - see DMA data opposite Redrawn from the data of H. A. Flocke, Kolloid Z. Z. Polym., 180, 188 (1962).
26 Topics to be Covered Simple models of Viscoelastic Behavior Time-Temperature Superposition Principle Chapter 11 in CD (Polymer Science and Engineering)
27 Mechanical and Theoretical Models of Viscoelastic Behavior GOAL - relate G(t) and J(t) to relaxation behavior. We will only consider LINEAR MODELS. (i.e. if we double G(t) [or σ(t)], then γ(t) [or ε(t)] also increases by a factor of 2 (small loads and strains)). TENSILE EXPERIMENT Stress Relaxation Creep E(t ) = σ(t) ε 0 D(t) = ε(t) σ 0 Linear Time Independent Behavior E = 1 D Time Dependent Behavior E(t ) 1 D(t) SHEAR EXPERIMENT G(t) = τ(t) γ 0 J(t) = γ (t) τ 0 G = 1 J G(t) 1 J(t)
28 Simple Models of the Viscoelastic Behavior of Amorphous Polymers Keep in mind that simple creep and recovery data for viscoelastic materials looks something like this Strain (c) VISCOELASTIC RESPONSE Creep Stress applied Stress removed Recovery } Permanent deformation Time
29 Simple Models of the Viscoelastic Behavior of Amorphous Polymers While stress relaxation data look something like this; Log E(t) (dynes/cm 2 ) Low Molecular Weight Glassy Region Glass Transition Rubbery Plateau High Molecular Weight Log Time (Sec)
30 Simple Models of the Viscoelastic Behavior of Amorphous Polymers Extension l 0 Hooke's law σ = Eε l σ I represent linear elastic behavior σ = Eε
31 Simple Models of the Viscoelastic Behavior of Amorphous Polymers Viscous flow v 0 y v Newtonian fluid. τ = ηγ xy σ = η dε dt
32 Strain v s. Time for Simple Models PURELY ELASTIC RESPONSE Strain σ = Eε PURELY VISCOUS RESPONSE σ = η dε dt Strain Stress applied Stress removed Time Permanent Deformation Shear Stress applied Shear Stress removed Time
33 Maxwell Model Maxwell was interested in creep and stress relaxation and developed a differential equation to describe these properties Maxwell started with Hooke s law σ = Eε _dσ dt Then allowed σ to vary with time = Ε _dε dt Writing for a Newtonian fluid σ = η _dε dt Then assuming that the rate of strain is simply a sum of these two contributions _dε dt = _σ η + _1 _dσ Ε dt
34 MAXWELL MODEL - Creep and Recovery Strain Creep and recovery _dε dt = _σ η 0 t Time Strain Recall that real viscoelastic behaviour looks something like this A picture representation of Maxwell s equation 0 t Time
35 Maxwell Model -Stress Relaxation _dε dt = _dε dt _σ η = + 0 _1 _dσ Ε dt In a stress relaxation experiment Hence Where dσ _ σ = _σ η dt σ = σ 0 exp[-t/τ ] t τ t = _η Ε Relaxation time Log (E r /E 0 ) Log (t/τ )
36 Maxwell Model -Stress Relaxation Log E(t) (Pa) The Maxwell model gives curves like this, OR like this Real data looks like this MAXWELL MODEL E(t ) = σ (t) ε 0 = σ 0 ε 0 exp t τ t Time
37 Voigt Model Maxwell model essentially assumes a uniform distribution Of stress.now assume uniform distribution of strain - VOIGT MODEL Picture representation Equation σ(t) = Eε(t) + η dε(t) dt (Strain in both elements of the model is the same and the total stress is the sum of the two contributions) σ 1 σ 2 σ = σ 1 + σ 2
38 Voigt Model - Creep and Stress Relaxation Strain σ 1 σ 2 σ = σ 1 + σ 2 Gives a retarded elastic response but does not allow for ideal stress relaxation,in that the model cannot be instantaneously deformed to a given strain. But in CREEP σ = constant, σ 0 σ(t) = σ 0 = Eε(t) + η dε(t) ε(t) dt + τ t Strain = _ σ η 0 dε(t) dt RETARDED ELASTIC RESPONSE σ ε(t) = _ [1- exp (-t/τ )] Ε 0 τ t - retardation time (η/e) t t 1 t 2 Time
39 Summary What do the strain/time plots look like?
40 Problems with Simple Models I can t t do creep And I can t t do stress relaxation The maxwell model cannot account for a retarded elastic response The voigt model does not describe stress relaxation Both models are characterized by single relaxation times - a spectrum of relaxation times would provide a better description NEXT - CONSIDER THE FIRST TWO PROBLEMS THEN -THE PROBLEM OF A SPECTRUM OF RELAXATION TIMES
41 Four - Parameter Model Elastic + viscous flow + retarded elastic E M Eg CREEP ε σ_ σ 0 t σ = + η + _ [1- exp (-t/τ )] M Ε 0 M Ε 0 M t E V η V Strain Retarded or Anelastic Response Stress applied Elastic Response Stress removed } Permanent deformation Time η M
42 Distributions of Relaxation and Retardation Times We have mentioned that although the Maxwell and Voigt models are seriously flawed, the equations have the right form. What we mean by that is shown opposite, where equations describing the Maxwell model for stress relaxation and the Voigt model for creep are compared to equations that account for a continuous range of relaxation times. E(t ) = Stress Relaxation E(t ) = 0 E(τ t )exp( t/τ t )dτ t D(t) = D( τ t )1 [ exp( t/ τ t )] d 0 E 0 exp ( t / τ t ) Creep D(t) = D 0 [ 1 exp( t/ )] τ t τ t These equations can be obtained by assuming that relaxation occurs at a rate that is linearly proportional to the distance from equilibrium and use of the Boltzmann superposition principle. We will show how the same equations are obtained from models.
43 The Maxwell - Wiechert Model _dε dt = _σ η = _σ η = _σ η + _1 _dσ Ε dt + _1 _dσ Ε dt + _1 _dσ Ε dt E 1 E 2 E 3 Consider stress relaxation 0 _dε dt = 0 σ = σ exp[-t/τ ] 1 0 t1 σ = σ exp[-t/τ ] 2 0 t2 σ = σ exp[-t/τ ] 3 t3 η 1 η 2 η 3
44 Distributions of Relaxation and Retardation Times Stress relaxation modulus E(t) = σ(t)/ε 0 σ(t) = σ + σ + σ E(t) = σ exp (-t/τ 01 ) + σ exp (-t/τ 02 ) + σ exp (-t/τ 03) ε 01 0 Or, in general ε 02 0 E(t) = Σ E n exp (-t/τ tn ) where E = n ε 03 0 σ ε 0n 0 Similarly, for creep compliance combine voigt elements to obtain D(t) =ΣD n [1- exp (-t/τ tn )]
45 Log E(t) (dynes/cm 2 ) Distributions of Relaxation and Retardation Times Example - the Maxwell - Wiechert Model With n = Glassy Region Glass Transition Rubbery Plateau Melt Log Time (Sec) Log E(t) (Pa) Log time (min) E(t) = Σ E n exp (-t/τ tn ) n = 2 4
46 Time - Temperature Superposition Principle Log E(t) (dynes/cm 2 ) Recall that we have seen that there is a time - temperature equivalence in behavior Log E (Pa) 10 9 Glassy Region 6 Glass 10 Transition 5 9 Rubbery 4 8 Plateau Low Molecular High 5 Weight Molecular Weight Log Time (Sec) 8 7 Rubbery Plateau Glassy Region Low Molecular Weight Temperature Cross-linked Elastomers Melt This can be expressed formally in terms of a superposition principle
47 Time Temperature Superposition Principle - Creep T 3 > T 2 > T 1 ε(t) T σ0 G' T 3 T 2 T 1 log t Master Curve at T 0 0 C ε(t) T σ0 log a T 0 T 0 [log t - log a T ] T
48 Stress Relaxation Modulus dynes/cm 2 Time Temperature Superposition Principle - Stress Relaxation Stress Relaxation Data C C C C C C C C -0.0 C +25 C +50 C Log Shift Factor Master Curve at 25 0 C SHIFT FACTOR vs TEMPERATURE Temperature 0 C Time (hours)
49 Relaxation Processes above T g - the WLF Equation From empirical observation Log a = -C (T - T ) 1 s For Tg > T < ~(Tg C) T C + (T - T ) 2 s Originally thought that C 1 and C 2 were universal constants, = and 51.6, respectively,when T s = Tg. Now known that these vary from polymer to polymer. Homework problem - show how the WLF equation can be obtained from the relationship of viscosity to free volume as expressed in the Doolittle equation
50 Semi - Crystalline Polymers NON - LINEAR RESPONSE TO STRESS.SIMPLE MODELS AND THE TIME - TEMPERATURE SUPERPOSITION PRINCIPLE DO NOT APPLY Temperature Tm Tg Amorphous melt Rigid crystalline domains Rubbery amorphous domains Rigid crystalline domains Glassy amorphous domains