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**Chapter 9: Rheological and Mechanical Properties of Polymers**

AE 447: Polymer Technology, Dr. Cattaleeya Pattamaprom

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**Fundamentals of Rheology Seminar**

Adapted from the workshop on Fundamentals of Rheology Seminar by Dr. Abel Gaspar-Rosas TA Instruments – Waters, Inc. April, 2004 Thailand

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**Outline Rheology – Introduction**

Elasticity, Viscosity and Viscoelasticity Rheology tests Steady-state flow Stress Relaxation Creep Recovery Oscillatory Tests Dynamic Frequency sweep Dynamic Temperature ramp Molecular Properties vs. Rheology of Polymers Mechanical Testing

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**Definition of Rheology**

Rheology is the science of deformation and flow of matter. รีโอโลยี (Rheology): ศาสตร์ของรีโอโลยี เป็นศาสตร์ที่กล่าวถึงสมบัติการไหลหรือการเสียรูป รีโอโลยีของพอลิเมอร์ส่วนใหญ่มักกล่าวถึงค่าโมดูลัส และค่าความหนืด

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**Simple Shear Deformation and Shear Flow**

Strain, g = x(t) y0 x(t) F A t = V V y0 1 d x(t) y0 d t y0 Strain Rate, g = . q y A Viscosity, h = t g . z x t g g . = Dg Dt Shear Modulus, G =

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**Range of Rheological Material Behavior**

Rheology: The study of deformation and flow of matter. Range of material behavior Solid Like Liquid Like Ideal Solid Ideal Fluid Classical Extremes Viscous liquid Elastic solid Viscoelastic

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**Classical Extremes: Elastic Solid**

1678: Robert Hooks develops his “True Theory of Elasticity” “The power of any spring is in the same proportion with the tension thereof.” Hooke’s Law: t = G or (Stress = G x Strain) where G is the RIGIDITY MODULUS . G is constant

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**Classical Extremes: Viscosity**

1687: Isaac Newton addresses liquids and steady simple shearing flow in his “Principia” “The resistance which arises from the lack of slipperiness of the parts of the liquid, other things being equal, is proportional to the velocity with which the parts of the liquid are separated from one another.” Newton’s Law: t = is constant whereis the Coefficient of Viscosity

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**Response for Classical Extremes**

Spring Dashpot Purely Viscous Response Newtonian Liquid t = Purely Elastic Response Hookean Solid t = G For the classical extremes cases, what only matters are the values of stress, strain and strain rate. The response is independent of the loading. use mechanical models as analogies for ideal solid and liquid responses

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**Coatings, Food, Cosmetics, Polymers**

Overview about the different rheological material behavior ilustrated by the use of… viscoelastic fluid / solid Ideal viscous Oil, Solvent Newton‘s law Ideal elastic Glass, Steel Hooke‘s law Apparent Yield Point, Thixotropy, … Coatings, Food, Cosmetics, Polymers Physica Messtechnik Gmbh Vor dem lauch 6, Stuttgart, Germany Internet: .de

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**Viscoelasticity Defined**

Range of Material Behavior Solid Like Liquid Like Ideal Solid Most Materials Ideal Fluid Purely Elastic Viscoelastic Purely Viscous Viscoelasticity: Having both viscous and elastic properties

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**Realistic polymer material **

- Materials are in between Elastic solid and Newtonian liquid called “viscoelastic” - Some energy stored, some dissipated. - Some time independent, some time dependent. A typical viscoelastic effect is tackiness (with long strings) which is not occuring with purely viscous liquid (example: water), or purely elastic solid (example: stone) Physica Messtechnik Gmbh Vor dem lauch 6, Stuttgart, Germany Internet: .de

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**Viscoelastic Fluids Polymers**

Viscoelastic behavior, illustrated by use of a dashpot and a spring Polymers Macromolecules are deformed and disentangled low viscosity at rest Macromolecules are entangled and have spherical shapes high viscosity under shear Physica Messtechnik Gmbh Vor dem lauch 6, Stuttgart, Germany Internet: .de

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**Newtonian and Non-Newtonian Behavior of Fluids**

Region = f() 1.000E5 1.000E5 Newtonian Region Independent of 10000 Zero-shear viscosity (o) 1000 t (Pa) h (Pa.s) 100.0 Shear thinning Pseudo-plastic t 10.00 10000 1.000 1.000E-5 1.000E-4 1.000E-3 0.1000 1.000 shear rate (1/s)

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**Elastic Modulus vs. Temp.**

brittle leathery rubbery viscous 4 3 2 1 Area 1: Viscous state: Area 2: Rubbery state Area 3: Leathery state Area 4: Glassy state = stress = F/A = strain = dL/L E = elastic modulus

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**Linear and Non-Linear Stress-Strain Behavior of Solids**

1000 100.0 Linear Region G is constant Non-Linear Region G = f() 100.0 G G' (Pa) osc. stress (Pa) 10.00 s 1.000 1.0000 10.000 100.00 1000.0 % strain

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**Viscosity and Viscoelasticity**

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**Viscosity is . . . . Viscosity: Definition**

synonymous with internal friction. resistance to flow. Variables that Affect Viscosity –for polymer Shear Rate - Molecular Weight Time of Shearing Temperature Pressure Surface Tension

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**Characteristic Diagrams for Newtonian Fluids**

Ideal Yield Stress (Bingham Yield) Pa.s t, Pa s or t, Pa s

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**Viscosity: Temperature dependence**

If we plot on a log scale you can se that 80°C results a two decade change in viscosity! Clearly temperature control is VERY important

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**Shear Thickening (Dilatent)**

Summary of Types of Flow Bingham (Newtonian w/yield stress) Bingham Plastic (shear-thinning w/yield stress) Shear Stress, t Shear Thinning (Pseudoplastic) Newtonian t y Shear Thickening (Dilatent) These rheograms depict non-Newtonian fluids Shear thinning [pseudoplastic] and shear thickening [dilatant] can be classed as power law fluids stress = rate ^n where n is not = 1. The names in parentheses are no longer in common usage and may be misleading. Other fluids exhibit an apparent yield stress, in that they have a non-zero y intercept Yield stress implies an infinite viscosity below that stress, which is physically unlikely; even polycrstalline metals flow over centuries Shear Rate, g

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Viscosity vs. Shear Rate Result: Rheological material behaviour Shear stress & viscosity as a function of the shear rate Newtonian Ideal viscous (e.g.: oil, solvent) Shear thinning (e.g.: typical food, cosmetics, coatings, polymers, …) Shear Thickening Shear thickening (e.g.: PVC plastisol & paper coatings under high shear cond.)

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**Newtonian and Non-Newtonian Behavior of Fluids**

Region = f() 1.000E5 1.000E5 Newtonian Region Independent of 10000 1000 t (Pa) h (Pa.s) 100.0 t 10.00 10000 1.000 1.000E-5 1.000E-4 1.000E-3 0.1000 1.000 shear rate (1/s)

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**Idealized Flow Curve - Polymers**

Power Law Region First Newtonian Plateau h0 = Zero Shear Viscosity h0 = K x MW3.4 Extend Range with Time- Temperature Superposition (TTS) & Cox-Merz log h Measure in Flow Mode on AR1000/AR500 Extend Range with Oscillation & Cox-Merz Second Newtonian Plateau Molecular Structure Compression Molding Extrusion Blow and Injection Molding 1.00E-5 1.00E-4 1.00E-3 0.0100 0.100 1.00 10.00 100.00 1.00E4 1.00E5 shear rate (1/s)

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**Comparison of Newtonian & Shear Thinning Samples**

10000 Standard oil A 1000 100.0 viscosity (Pa.s) 10.00 Xanthan/Gellan Fructose Soln. 1.000 0.1000 This slide illustrates the importance of shear thinning in a fluid ["Orbitz"]. The two standard oils are at the extremes of viscosity. The fluid can exist at both these extremes depending on the shear rate. Thus at rest the fluid is stiffer than plasticine, but at moderate shear rates it is as fluid as milk This kind of behavior allows for easy dispensing and handling, but robust structure and thus stability in its container Standard oil B 1.000E-3 1.000E-6 1.000E-4 1.000 10.00 100.0 1000 shear rate (1/s)

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**Non-Newtonian, Time Dependent Fluids**

Thixotropy A decrease in apparent viscosity with time under constant shear rate or shear stress, followed by a gradual recovery, when the stress or shear rate is removed. Rheopexy An increase in apparent viscosity with time under constant shear rate or shear stress, followed by a gradual recovery when the stress or shear rate is removed. Also called Anti-thixotropy or negative thixotropy. Reference:Barnes, H.A., Hutton, J.F., and Walters, K., An Introduction to Rheology, Elsevier Science B.V., ISBN

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**Non-Newtonian, Time Dependent Fluids**

Rheopectic Viscosity Shear Rate = Constant Thixotropic time

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**Time-Dependent Viscoelastic Behavior: **

Solid and Liquid Properties of "Silly Putty" T is short [< 1s] T is long [24 hours] Deborah Number [De] = /

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**Storage and Loss of Viscoelastic Material**

SUPER BALL LOSS TENNIS BALL X STORAGE

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Rheology Tests

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**Some Types of Rheometers**

Typical Rheometer: Low shear rate/Oscillatory Capillary Rheometer: High shear rate

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**Rheology:Choosing Tests and Conditions**

Geometries: Typical Rheometer Max. 3° Max. 3° Delta < 1.2 Delta < 1.2 0.25mm < gap < 2mm Ref.: Physica

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**1. Stepped Ramp or Steady-Shear Flow**

A series of logarithmic stress steps allowed to reach steady state, each one giving a single viscosity data point: Shear Rate Time h = s / g/ (d dt) Shear Thinning Region Viscosity This shows how and equilibrium flow curve is constructed from many stress steps. Each step is allowed to reach equilibrium, before a viscosity calculation is made. The inset box shows a step reaching steady state [shear rate approaching a constant]and from the shear rate asymptote and the known stress the viscosity is derived Lower stress steps are performed as creep tests, see later sections Shear Rate, 1/s

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**HDPE LLDPE LDPE Typical steady shear flow experiment**

Polyethylene 150 C HDPE LDPE LLDPE 100000 viscosity (Pa.s) 10000 PE can be divided into 3 categories based on long and short chain branching: HDPE has no branching and is much more visous, being processed at higher temps., LDPE is highly branched with long and short chain branches, resulting in a lower overall viscosity. In between in LLDPE with some short chain branching. The graph shows a family of 9 resins with varying degrees of branching etc. 1000 1.000E-4 0.1000 1.000 10.00 1.00E-3 shear rate (1/s)

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**Idealized Full Flow Curve - Polymers**

Power Law Region First Newtonian Plateau h0 = Zero Shear Viscosity h0 = K x MWc 3.4 Extend Range with Time- Temperature Superposition (TTS) & Cox-Merz log h Measure in Flow Mode on AR1000/AR500 Extend Range with Oscillation & Cox-Merz Second Newtonian Plateau Molecular Structure Compression Molding Extrusion Blow and Injection Molding 1.00E-5 1.00E-4 1.00E-3 0.0100 0.100 1.00 10.00 100.00 1.00E4 1.00E5 shear rate (1/s)

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**At very high shear rate Melt fracture**

Higher Shear Rate

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**2. Stress Relaxation Experiment**

Strain is applied to sample instantaneously (in principle) and held constant with time. Stress is monitored as a function of time (t). Strain time

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**Response of Classical Extremes**

Stress Relaxation Experiment Deformation Strain time Response of Classical Extremes Elastic Viscous Hookean Solid Newtonian Fluid Stress stress for t>0 is 0 stress for t>0 is constant Stress time time

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**Stress Relaxation Experiment (cont’d)**

Response of Material Visco elastic Stress decreases with time starting at some high value and decreasing to zero. Stress time For small deformations (strains within the linear region) the ratio of stress to strain is a function of time only. This function is a material property known as the STRESS RELAXATION MODULUS, G(t) G(t) = s(t)/

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**3. Creep Recovery Experiment**

Stress is applied to sample instantaneously, t1, and held constant for a specific period of time. The strain is monitored as a function of time (t). The stress is reduced to zero, t2, and the strain is monitored as a function of timet. Stress t1 t2 time

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**Response of Classical Extremes**

Creep Recovery Experiment Deformation Stress t1 time t2 Response of Classical Extremes Elastic Viscous Stain rate for t>t1 is constant Strain for t>t1 increase with time Strain rate for t >t2 is 0 Stain for t>t1 is constant Strain for t >t2 is 0 Strain Strain time t1 time t2 t1 t2

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**Creep Recovery Experiment: Response of Viscoelastic Material**

Recovery = 0 (after steady state) / Recoverable Strain Strain t t 1 2 time Strain rate decreases with time in the creep zone, until finally reaching a steady state. In the recovery zone, the viscoelastic fluid recoils, eventually reaching a equilibrium at some small total strain relative to the strain at unloading. Reference: Mark, J., et.al., Physical Properties of Polymers ,American Chemical Society, 1984, p. 102.

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**4.Oscillatory Testing of Polymers (Dynamic Mechanical Tests)**

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**Dynamic Mechanical Testing**

Deformation An oscillatory (sinusoidal) deformation (stress or strain) is applied to a sample. The material response (strain or stress) is measured. The phase angle , or phase shift, between the deformation and response is measured. Response Phase angle d

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**Dynamic Mechanical Testing Viscoelastic Material Response**

Phase angle 0° < d < 90° Stress Strain

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**DMA Viscoelastic Parameters: The Complex, Elastic, & Viscous Stress**

The stress in a dynamic experiment is referred to as the complex stress t* Phase angle d Strain, g t* = t' + it" Complex Stress, t*

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**DMA Viscoelastic Parameters**

The Complex Modulus: Measure of materials overall resistance to deformation. G* = Stress*/Strain G* = G′ + iG″ The Elastic (Storage) Modulus: Measure of elasticity of material. The ability of the material to store energy. G' = (stress*/strain)cos The Viscous (loss) Modulus: The ability of the material to dissipate energy. Energy lost as heat. G" = (stress*/strain)sin Tan Delta: Measure of material damping - such as vibration or sound damping. Tan = G"/G'

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**Storage and Loss of Viscoelastic Material**

SUPER BALL LOSS TENNIS BALL X STORAGE

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**4.1 Frequency Sweep Deformation Time Stress or Strain**

The material response to increasing frequency (rate of deformation) is monitored at a constant amplitude (stress or strain) and temperature. Stress or Strain Time USES Viscosity Information - Zero Shear h, shear thinning Elasticity (reversible deformation) in materials MW & MWD differences Polymer Melts and Polymer solutions. Finding Yield in gelled dispersions High and Low Rate (short and long time) modulus properties. Extend time or frequency range with TTS

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**Frequency Sweep: Material Response**

Transition Region Rubbery Plateau Region Terminal Region Glassy Region log G'and G" 1 2 Storage Modulus (E' or G') Loss Modulus (E" or G") log Frequency (rad/s or Hz)

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**Typical Oscillatory Data**

G” G’ Curve of a frequency sweep used to talk about oscillation modes; talk about other ways to run the experiment like time sweeps, temp sweeps; ability to do in both controlled strain and stress mode and advantage of each the waveform and why that is important(and unique to us)

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**Time-Dependent Viscoelastic Behavior: **

Solid and Liquid Properties of "Silly Putty" T is short [< 1s] T is long [24 hours] Deborah Number [De] = /

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**Dynamic Moduli of a Polymer Melt vs. Frequency**

1.00E7 PDMS at 20°C 100000 100000 10000 10000 h* 100000 G' (Pa) 1000 1000 G'' (Pa) h* (Pa.s) 100.0 100.0 10000 G" 10.00 10.00 1000 1.000 1.000 G' 100.0 0.1000 0.1000 1.000E-4 1.000E-3 0.1000 1.000 10.00 100.0 1000 ang. frequency (rad/sec)

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**HDPE LLDPE LDPE Typical steady shear flow experiment**

Polyethylene 150 C HDPE LDPE LLDPE 100000 viscosity (Pa.s) 10000 PE can be divided into 3 categories based on long and short chain branching: HDPE has no branching and is much more visous, being processed at higher temps., LDPE is highly branched with long and short chain branches, resulting in a lower overall viscosity. In between in LLDPE with some short chain branching. The graph shows a family of 9 resins with varying degrees of branching etc. 1000 1.000E-4 0.1000 1.000 10.00 1.00E-3 shear rate (1/s)

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**Cox-Merz Rule . h (g) = h*(w) Polydimethylsiloxane**

Creep or Steady shear Flow Dynamic Frequency Sweep . h (g) = h*(w) Cox-Merz data for PDMS software allows the plotting of oscillation data [rad/s] and flow data as shear rate "shear rates" up to max freq in rad/s can thus be achieved without artifact 250 CSL, 628 AR, 1000 WB This technique allows medium to high shear measurements to be made without edge instability Is an empirical test - it either works or it doesn't Dynamic data gives high shear rates unattainable in flow

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**4.2 Dynamic Temperature Ramp: Material Response**

Glassy Region Transition Region Rubbery Plateau Region Terminal Region log E' (G') and E" (G") Loss Modulus (E" or G") Storage Modulus (E' or G') Temperature

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**Time – Temp Superposition:**

Time – Temp Superposition: can be used for shifting “moduluse” in creep test (E(t)), relaxation test (G(t)), dynamic oscillatory test ( G’, G”) t 1/f “ Time scale (or frequency) of stress applies and temperature has a similar influent on mechanical properties.! “ short time low temp. ( high f ) = /G long time high temp. ( low f ) works because t and T affect the relaxation time of polymer in same way < Relaxation time determines mechanical properties

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**Relationship between Molecular Properties and Rheology of Polymers**

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**Effect of Molecular Properties on Rheology**

Molecular Structure MW & MWD Chain Branching and Cross-linking Interaction of Fillers with Matrix Polymer Single or Multi-Phase Structure Viscoelastic Properties As a function of:: Strain Rate(frequency) Strain Amplitude Temperature Processability & Product Performance

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**Molecular Structure - Effect of Molecular Weight**

Glassy Region Rubbery Plateau Region Transition Region MW has practically no effect on the modulus below Tg log E' (G') High MW Med. MW Low MW Temperature

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**Effect of Molecular Weight on Viscosity and Elasticity**

MWc = critical MW MWc’ 3.4 log o MWc log Je o log MW log MW Ref. Graessley, Physical Properties of Polymers, ACS, c 1984.

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**Sensitive to molecular weight, MW **

Zero Shear Viscosity o Sensitive to molecular weight, MW For low MW (no entanglements) o is proportional to MW For MW > Critical MW, o is proportional to MW3.4.

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**Effect of Molecular Weight**

h0 =K x MW3.4 h0 Increasing MW log h log g

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**Effect of Molecular Weight Distribution on h0**

A Polymer with a broad MWD exhibits non-Newtonian flow at a lower rate of shear than a polymer with the same h0 but has a narrow MWD Narrow MWD Log Viscosity (Pa.s) Broad MWD Log Shear Rate (1/s)

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x x y y

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Compressive strength Tensile strength

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Physical Testing

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