Polymer Melt Rheology. Introduction to Viscoelastic Behavior. TimeTemperature Equivalence


 Merryl McBride
 1 years ago
 Views:
Transcription
1 Topics to be Covered Polymer Melt Rheology Introduction to Viscoelastic Behavior TimeTemperature 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(dimethylsiloxane) Poly(isobutylene) Poly(ethylene) Poly(butadiene) Poly(tetramethyl psilphenyl 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, JuneJuly 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 Crosslinked 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 Crosslinked 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 TimeTemperature 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 Crosslinked 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
Chapter 8 Maxwell relations and measurable properties
Chapter 8 Maxwell relations and measurable properties 8.1 Maxwell relations Other thermodynamic potentials emerging from Legendre transforms allow us to switch independent variables and give rise to alternate
More informationDesign information. Delrin. Module III. acetal resin ONLY BY DUPONT
Delrin ONLY BY DUPONT Module III acetal resin Design information Registered trademarks of E.I. du Pont de Nemours and Company The miracles of science is a trademark of E.I. du Pont de Nemours and Company
More informationSpecific volume of polymers. Influence of the thermomechanical history
Specific volume of polymers Influence of the thermomechanical history CIPDATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN Beek, Maurice H.E. van der Specific volume of polymers : influence of the thermomechanical
More informationLearning Objectives for AP Physics
Learning Objectives for AP Physics These course objectives are intended to elaborate on the content outline for Physics B and Physics C found in the AP Physics Course Description. In addition to the five
More informationUnderstanding the FiniteDifference TimeDomain Method. John B. Schneider
Understanding the FiniteDifference TimeDomain Method John B. Schneider June, 015 ii Contents 1 Numeric Artifacts 7 1.1 Introduction...................................... 7 1. Finite Precision....................................
More information2 ONE DIMENSIONAL MOTION
2 ONE DIMENSIONAL MOTION Chapter 2 OneDimensional Motion Objectives After studying this chapter you should be able to derive and use formulae involving constant acceleration; be able to understand the
More informationFigure 2.1: Center of mass of four points.
Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would
More information4.1 Modelling with Differential Equations
Chapter 4 Differential Equations The rate equations with which we began our study of calculus are called differential equations when we identify the rates of change that appear within them as derivatives
More informationSYLLABUS. Cambridge International AS and A Level Physics. Cambridge Advanced
SYLLABUS Cambridge International AS and A Level Physics 9702 For examination in June and November 2015 Cambridge Advanced Changes to syllabus for 2015 This syllabus has been updated. Significant changes
More informationHigh Strain R. 33. High Strain Rate and Impact Experiments. Part D 33
1 High Strain R 33. High Strain Rate and Impact Experiments Part D 33 K. T. Ramesh Experimental techniques for highstrainrate measurements and for the study of impactrelated problems are described. An
More informationPQM Supplementary Notes: Spin, topology, SU(2) SO(3) etc
PQM Supplementary Notes: Spin, topology, SU(2) SO(3) etc (away from the syllabus, but interesting material, based on notes by Dr J.M. Evans) 1 Rotations and Noncontractible Loops Rotations in R 3 can
More informationHighDimensional Image Warping
Chapter 4 HighDimensional Image Warping John Ashburner & Karl J. Friston The Wellcome Dept. of Imaging Neuroscience, 12 Queen Square, London WC1N 3BG, UK. Contents 4.1 Introduction.................................
More informationIrreversibility and Heat Generation in the Computing Process
R. Landauer Irreversibility and Heat Generation in the Computing Process Abstract: It is argued that computing machines inevitably involve devices which perform logical functions that do not have a singlevalued
More informationfor some scalar field φ(x, t). The field φ(x, t) is called the potential, or velocity potential, for u.
3 IRROTATIONAL FLOWS, aka POTENTIAL FLOWS Irrotational flows are also known as potential flows because the velocity field can be taken to be the gradient of a 3.1 Velocity potential. That is, an irrotational
More information6 WORK and ENERGY. 6.0 Introduction. 6.1 Work and kinetic energy. Objectives
6 WORK and ENERGY Chapter 6 Work and Energy Objectives After studying this chapter you should be able to calculate work done by a force; be able to calculate kinetic energy; be able to calculate power;
More informationClassical Mechanics. Joel A. Shapiro
Classical Mechanics Joel A. Shapiro April 21, 2003 Copyright C 1994, 1997 by Joel A. Shapiro All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted
More informationW = F d. rule for calculating work (simplest version) The work done by a force can be calculated as. W = F d,
In example c, the tractor s force on the trailer accelerates it, increasing its kinetic energy. If frictional forces on the trailer are negligible, then the increase in the trailer s kinetic energy can
More informationPrinciples of DirectOperated Regulators
Principles of DirectOperated Regulators Introduction Pressure regulators have become very familiar items over the years, and nearly everyone has grown accustomed to seeing them in factories, public buildings,
More informationOPRE 6201 : 2. Simplex Method
OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2
More informationNMR Pulse Spectrometer PS 15. experimental manual
NMR Pulse Spectrometer PS 15 experimental manual NMR Pulse Spectrometer PS 15 Experimental Manual for MS Windows For: MS Windows software Winner Format: MS Word 2002 File: PS15 Experimental Manual 1.5.1.doc
More informationChapter 18. Power Laws and RichGetRicher Phenomena. 18.1 Popularity as a Network Phenomenon
From the book Networks, Crowds, and Markets: Reasoning about a Highly Connected World. By David Easley and Jon Kleinberg. Cambridge University Press, 2010. Complete preprint online at http://www.cs.cornell.edu/home/kleinber/networksbook/
More informationMASTER'S THESIS. Graphical User Interface for Scania Truck And Road Simulation. Johan Thorén. Civilingenjörsprogrammet
2000:103 MASTER'S THESIS Graphical User Interface for Scania Truck And Road Simulation Johan Thorén Civilingenjörsprogrammet Institutionen för Maskinteknik Avdelningen för Fysik 2000:103 ISSN: 14021617
More informationSensing and Control. A Process Control Primer
Sensing and Control A Process Control Primer Copyright, Notices, and Trademarks Printed in U.S.A. Copyright 2000 by Honeywell Revision 1 July 2000 While this information is presented in good faith and
More informationThe Turbulence Problem
Leonardo da Vinci s illustration of the swirling flow of turbulence. (The Royal Collection 2004, Her Majesty Queen Elizabeth II) The Turbulence Problem An Experimentalist s Perspective Robert Ecke Turbulent
More informationFeedback Control of a Nonholonomic Carlike Robot
Feedback Control of a Nonholonomic Carlike Robot A. De Luca G. Oriolo C. Samson This is the fourth chapter of the book: Robot Motion Planning and Control JeanPaul Laumond (Editor) Laboratoire d Analye
More informationFEM Simulation of Nonisothermal Viscoelastic Fluids
FEM Simulation of Nonisothermal Viscoelastic Fluids Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften Der Fakultät für Mathematik der Technischen Universität Dortmund vorgelegt
More informationHow to Use Expert Advice
NICOLÒ CESABIANCHI Università di Milano, Milan, Italy YOAV FREUND AT&T Labs, Florham Park, New Jersey DAVID HAUSSLER AND DAVID P. HELMBOLD University of California, Santa Cruz, Santa Cruz, California
More informationThe critical current of superconductors: an historical review
LOW TEMPERATURE PHYSICS VOLUME 27, NUMBER 9 10 SEPTEMBER OCTOBER 2001 The critical current of superconductors: an historical review D. DewHughes* Oxford University, Department of Engineering Science,
More informationThere is no such thing as heat energy
There is no such thing as heat energy We have used heat only for the energy transferred between the objects at different temperatures, and thermal energy to describe the energy content of the objects.
More informationMpemba effect from a viewpoint of an experimental physical chemist. by Nikola Bregović
Mpemba effect from a viewpoint of an experimental physical chemist by Nikola Bregović Introduction Checking my email a few weeks ago, I found a message from my close friend with the title This seams like
More information