Modulus - TA Instruments

Modulus - TA Instruments
9/26/2016




      Fundamentals of Polymer Rheology


                       Sarah Cotts
                     TA Instruments
                  Rubber Testing Seminar
                  CUICAR, Greenville SC



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Rheology: An Introduction




                     Rheology: The study of
                stress-deformation relationships




                              
  
              = Viscosity           
                                             = Modulus


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Modulus - TA Instruments
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Shear Flow in Parallel Plates

                                                #
                     Stress (σ
                             σ)         "=     $%
                                                     !M
                                               
                       Strain (γγ)      =         !θ
                                               
               Strain rate ( )         =       !Ω




                                       r = plate radius
                                       h = distance between plates
                                       M = torque (µN.m)
                                       θ = Angular motor deflection (radians)
                                       Ω= Motor angular velocity (rad/s)


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Open Boundary Rotational Rheometer

            ARES G2

                               Measured
                                 Torque     Transducer
                                (Stress)




                             Sample


                                Applied     Direct Drive
                                Strain or       Motor
                                Rotation
         Controlled Strain


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Closed Die Cavity Rheometer

                                                          RPA Elite
          Measured
            Torque     Transducer
           (Stress)




           Applied     Direct Drive
           Strain or       Motor
           Rotation

                                                       Controlled Strain



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Viscoelasticity


• Viscoelasticity: Having both Viscous and Elastic properties.

• Elastic Behavior: Stress is dependent on Strain
     Hooke’s Law of Elasticity: Modulus = Stress/Strain
• Viscous Behavior: Stress is dependent on Strain Rate
     Newton’s Law of Viscosity: Viscosity = Stress/ Strain Rate

• Viscoelastic materials cannot be fully understood using only one of
  these relationships. Oscillation measurements are able to measure
  stress as a function of both strain and strain rate.




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Hooke’s Law of Elasticity


    • For an Elastic Solid, Stress and
      Strain have a constant
      proportionality σ = E*ε

    • If the material follows Hooke’s
      Law, the deformation will be
      reversible when the stress is
      removed

   The modulus of a Hookean solid will not show any
    time dependence- the stress depends on the
    strain, but not the strain rate



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Newton’s Law of Viscosity

                         • For a Viscous Liquid, Stress is proportional to
                           Strain Rate dε/dt by a coefficient of Viscosity η
                              σ = η* dε/dt
                         • The deformation of a liquid is non-reversible




                                            η
                                   Stress




                                            Strain Rate


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Viscoelastic Behavior
                       σ = E*ε + η*dε/dt
   Kelvin-Voigt Model (Creep)     Maxwell Model (Stress Relaxation)




                                                         L1
                                                                 L2


    Viscoelastic Materials: Force
 depends on both Deformation and
 Rate of Deformation and vice versa.
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Time-Dependent Viscoelastic Behavior




       T is short [< 1s]        T is long [24 hours]

                 Deborah Number [De] = τ/T

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Pitch Drop Experiment

                                                       Long deformation time: pitch behaves
                                                       like a highly viscous liquid
                                                                 9th drop fell July 2013
                                                       Short deformation time: pitch behaves
                                                       like a solid




     Started in 1927 by Thomas Parnell in Queensland, Australia
     http://www.theatlantic.com/technology/archive/2013/07/the-3-most-exciting-words-in-science-right-now-the-pitch-dropped/277919/

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Time-Dependent Viscoelastic Behavior




      Silly Putties have different characteristic relaxation times
      Dynamic (oscillatory) testing can measure time-dependent
       viscoelastic properties more accurately and efficiently by varying
       frequency (deformation time)



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Frequency Sweep- Time Dependent Viscoelastic Properties




                               Frequency of modulus
                                crossover correlates
                                with Relaxation Time




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Storage and Loss of a Viscoelastic Material




                            RUBBER BALL             LOSS
                                          X          (G”)
                              TENNIS
                               BALL
                                               X

                                                   STORAGE
                                                     (G’)




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Oscillation Testing

• The motor applies an oscillating            Deformation
  deformation to the sample at a set
  Amplitude and Frequency.
     Amplitude: Degree of Arc, or % Strain
     Frequency: Hz (cycles per second) or
      CPM (cycles per minute)                        Torque, S*


• The torque transducer measures the
  response of the sample.
     Amplitude: Torque (S*), or Stress
                                                Phase Angle δ
• The phase lag between the deformation
  and the torque is used to determine the
  Elastic and Viscous response.




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Oscillation Testing
 In a purely Elastic material,                          In a purely Viscous material, Stress
 Stress is in phase with the Strain                     is out of phase with the Strain
  Phase angle= 0°                                       Phase angle= 90°




                             Viscoelastic material
                            0˚ < Phase angle< 90°




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     Oscillation Testing- Lissajous-Bowditch Plots


     Elastic Material                   Viscoelastic material                   Viscous Material




                                                                                      Stress
                                                  Stress
            Stress




Strain                                 Strain                          Strain




                     Raw oscillation data can also be plotted as stress vs. strain.
                       An elliptical shape represents a viscoelastic response.




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     Oscillation Testing

   The Phase Angle can be used to separate the stress signal into the elastic and viscous
   components
        G’: Storage Modulus
              Measure of elasticity, or the ability to store energy

              G’ = (Stress/Strain)*cos(δ)

        G”: Loss Modulus
              Measure of viscosity, or the ability to lose energy

              G” = (Stress/Strain)*sin(δ)

        Tan Delta
              Measure of dampening properties
                                                                                                  Loss Modulus




              Tan (δ) = G”/G’

        G*: Complex Modulus
              Measure of resistance to deformation

              G*= Stress/Strain

        η*: Complex Viscosity                               δ
              Measure of resistance to flow                     Storage Modulus
              η* = Stress/Strain Rate




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PDMS Frequency Sweep Comparison Overlay




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Frequency Sweeps on Solids
                              Happy & Unhappy Balls
           108                                                                    100




                    Modulus is the same
                    tan δ is very different
            107
                                                                                             t
                                                                                             a
                                                                                             n


                                                                 Unhappy ball
                                                                                             _
   )                                                                                         d
       ]                                                                                     e
       ²                                                                                 [   lt
       m                                                                                 ]    a
       c
       /                                                                          10-1       (
   ( n
   ' y
     d
   E [

                                                                                             )

            106

                                                        Happy ball


           105                                                                    10-2
             10-1                 100                      101                  102
                                         Freq [rad/s]

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          Applications of Polymer Rheology




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Polymer Characteristics

 Apart from chemical nature, tacticity and microstructure, polymers have
 3 major characteristics which affect processability

     1. Average molecular weight
     2. Molecular weight distribution
     3. Branching (Type and architecture)




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Long Chain Branching: Improving Processability




  • Long Chain Branching (LCB) provides Shear Thinning, Strain
    Hardening and Melt Strength
  • Very effective only one per 10000C needed.
  • Difficult to fully characterize length of branches, number per chain
    and distribution.

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Common Characterization Techniques


Chromatography (SEC)                                       NMR




                  FTIR




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Influence of Molecular Weight on G’ and G”


                       The G’ and G” curves are shifted to lower frequency
                         (longer time) with increasing molecular weight.
                                                                                                   6
                                                                                              10

                                                                                                   5
                                                                                              10
                                                                      Modulus G', G'' [Pa]




                                                                                                   4
                                                                                              10
                                                                                                                                                                          SBR M w [g/mol]
                                                                                                   3                                                                          G' 130 000
                                                                                              10                                                                              G'' 130 000
                                                                                                                                                                              G' 430 000
                                                                                                   2                                                                          G'' 430 000
                                                                                              10                                                                              G' 230 000
                                                                                                                                                                              G'' 230 000
                                                                                                   1
                                                                                              10
                                                                                                            -4             -3            -2        -1        0        1         2         3            4        5
                                                                                                   10               10             10         10        10       10        10        10           10       10
                                                                                                                                               Freq ω [rad/s]
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Influence of MW on Viscosity

                       The zero shear viscosity increases with increasing molecular weight.
                       TTS is applied to obtain the extended frequency range.
                            7
                       10
                                                                                                                                                        SBR Mw [g/mol]
                                                                                                                                                            130 000
                       10
                            6
                                                                                                                                                            230 000                                        The high frequency
                                                                                                                                                            320 000                                        behavior
 Viscosity η* [Pa s]




                                                                                                                                                            430 000                                        (slope -1) is
                            5
                       10                                                                                                                                                                                  independent of the
                                                                                                                                                                                                           molecular weight
                                      Zero Shear Viscosityηo [Pa s]




                            4
                       10                                                                         Zero Shear
                                                                                                  Viscosity


                                                                              6
                                                                      10


                            3
                       10                                                                                        Slope 3.08 +/- 0.39

                                                                              5
                                                                      10

                                                                                  1000 00
                            2
                       10                                                                              M olecilar weight M w [Daltons]




                                 -4                                                          -3            -2             -1             0         1         2        3          4            5
                            10                                   10                               10               10              10         10        10       10         10       10
                                                                                                                 Frequency ω aT [rad/s]

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      Materials Evaluated: Polyolefin Elastomers


                POE                   Co-Monomer          Density (g/cm3)         Long Chain Branching

               EB-1                       Butene                0.901                      High

               EB-2                       Butene                0.870                     Medium

               EB-3                       Butene                0.870                      High

             EO-1                         Octene                0.870                     Medium

                                • The level of Long Chain Branching (LCB) is difficult to
                                  directly measure in POEs
                                      Amount and length of short chain branching interferes
                                       with NMR
                                • Characterize differences in LCB using rheology


L. J. Effler,et.al., DuPont Dow Elastomers, Freeport, TX, Presented
at Society of Plastics Engineers 2005 ANTEC Boston, MA                                            TAINSTRUMENTS.COM




      Conventional Rheology of Polyolefin Elastomers

                         1,000,000

                                                                                       EB-1: High LCB
                                                                                       EB-2: Med LCB
                                                                                       EB-3: High LCB
                                                                                       EO-1: Med LCB
                  (poise)




                            100,000
      (b /α )η (Poise)
       Viscosity
           T αT η




                             10,000




                              1,000
                                   0.01            0.1           1           10            100            1000
                                                         Angular Frequency
                                                                  α ω (rad/s) (rad/sec)
L. J. Effler,et.al., DuPont Dow Elastomers, Freeport, TX, Presented
at Society of Plastics Engineers 2005 ANTEC Boston, MA                                            TAINSTRUMENTS.COM




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           Conventional Rheology of Polyolefin Elastomers
                             10


                                                                                                 EB-1: High LCB
                                                                                                 EB-2: Med LCB
                                                                                                 EB-3: High LCB
                                                                                                 EO-1: Med LCB
                 Tan Delta




                              1




                             0.1
                                0.01                      0.1         1               10             100             1000

                                                                        α ω (rad/s) (rad/sec)
                                                                Angular Frequency
L. J. Effler,et.al., DuPont Dow Elastomers, Freeport, TX, Presented
at Society of Plastics Engineers 2005 ANTEC Boston, MA                                                     TAINSTRUMENTS.COM




                 Extensional Viscosity of Polyolefin Elastomers
                 3000000
                                       ε& = 0.1 s -1   190 °C


                 2500000



                 2000000
                                                                                                EB-1: High LCB
                                                                                                EB-2: Med LCB
   η e (Poise)




                                                                                                EB-3: High LCB
                 1500000                                                                        EO-1: Med LCB


                 1000000



                   500000



                               0
                                   0             2         4    6    8       10      12    14       16     18      20

                                                                          Time (s)

 L. J. Effler,et.al., DuPont Dow Elastomers, Freeport, TX, Presented
 at Society of Plastics Engineers 2005 ANTEC Boston, MA                                                    TAINSTRUMENTS.COM




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Linear Viscoelasticity

                      In the Linear Viscoelastic Region, the Stress
                      signal is a sine wave, and meaningful
                      rheology measurements can be made.




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Linear Viscoelasticity

               At higher strains, stress
             becomes non-sinusoidal.
          Modulus is an approximation.




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Linear Viscoelasticity




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Non-Linear Viscoelasticity- Higher Harmonics

SAOS                γ(t) =       γ0 sin(ωt)

                    τ(t) =       τ0 sin(ωt+δ)



LAOS
                    γ(t) =       γ0 sin(ωt)


  τ(t) = τ1 sin(ω1t+ϕ1) + τ3 sin(3ω1t+ϕ3) + τ5 sin(5ω1t+ϕ5) + ...
          ∞
      =   Σ τ sin(nω +ϕ )
          n=1
                n            1    n

          odd
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  Non-Linear Viscoelasticity- Higher Harmonics

                                                         Sample Response




                                                                                                                      =

                                              +                                                          +
          Fundamental                                            3rd Harmonic                                                    5th Harmonic
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  Large Amplitude Oscillatory Shear (LAOS)



                                                          1200                      10000

          8000                                            1000
                                                                                     8000
                                                          800
          6000                                                                       6000
                                                          600
          4000                                                                       4000
                                                          400
          2000                                            200                        2000
 τ [Pa]




                                                                           τ [Pa]
                                                                   γ [%]




              0                                           0
                                                                                        0

          -2000                                           -200
                                                                                     -2000
                                                          -400
          -4000
                                                                                     -4000
                                                          -600
          -6000
                                                          -800                       -6000
          -8000                                           -1000
                                                                                     -8000
                  PIB, ω =0.1 Hz, γ 0=1000%, T=140°C
                                                           -1200                             PIB, ω=0.1 Hz, γ 0=1000%, T=140°C
              0.0 π       0.5 π     1.0 π      1.5 π   2.0 π                        -10000
                                  t×ω [Pa]                                                   -1200    -600      0      600       1200
                                                                                                              γ [%]

“The potential of large amplitude oscillatory shear to gain an insight into the long-chain branching structure of polymers”
ACS Meeting 2008, Florian J. Stadler, Sunil Dhole, Adrien Leygue, Christian Bailly – Université Catholique de Louvain (UCL)

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Large Amplitude Oscillatory Shear (LAOS)

                                                 300000



                                                 200000



                                                 100000
  Shear stress (Pa)




                                                       0
                      -8   -6   -4          -2             0          2            4            6           8


                                                 -100000

                                                                                Branched
                                                                                Linear
                                                 -200000                        5% Branched in Linear



                                                 -300000
                                                  Strain rate (s-1)

“The potential of large amplitude oscillatory shear to gain an insight into the long-chain branching structure of polymers”
ACS Meeting 2008, Florian J. Stadler, Sunil Dhole, Adrien Leygue, Christian Bailly – Université Catholique de Louvain (UCL)

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Long Chain Branching Index




“The potential of large amplitude oscillatory shear to gain an insight into the long-chain branching structure of polymers”
ACS Meeting 2008, Florian J. Stadler, Sunil Dhole, Adrien Leygue, Christian Bailly – Université Catholique de Louvain (UCL)

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Long Chain Branching

Shear stress (Pa)
 75000
            High visc. PP
            Low visc. PP
 50000      95% Low visc. PP+5% high visc. PP
            50% Low visc. PP+50% high visc. PP

 25000

                                                                    Secondary loops
                                                                     not affected by
     0
                                                                    AMW and MWD


-25000

                                                 •Secondary loops can be associated
                                                 with linear polymer architecture
-50000
                                                 •There has to be a mathematical
                                                 condition to account for loops
-75000
             -10              -5                  0             5              10

                                       Shear rate (s-1)

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Lissajous-Bowditch Plots

                                                                          Ewoldt, R.H; McKinley, G. H. “On
                                                                          secondary loops in LAOS via self-
                                                                          intersection of Lissajous-Bowditch
                                                                          Curves.” Rheologica Acta 49, no 2.
                                                                          February 12th, 2010. 213-219.




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      Rheological characterization of EPDMs


                                           ML(1+4) at 125 C




Burhin, Henri G. Polymer Process Consult                      TAINSTRUMENTS.COM




      Rheological characterization of EPDMs




Burhin, Henri G. Polymer Process Consult                      TAINSTRUMENTS.COM




                                                                                        21
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      LAOS testing of EPDMs




Burhin, Henri G. Polymer Process Consult   TAINSTRUMENTS.COM




      Long Chain Branching Index- EPDMs




Burhin, Henri G. Polymer Process Consult   TAINSTRUMENTS.COM




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Dynamic Mechanical Analysis


•Is DMA Rheology, or Thermal Analysis?
•Oscillation measurements- E’, E”, Tan Delta
•Characterize viscoelastic materials as a function of time,
 temperature, stress and strain.




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Glass Transition of PSA measured by DMA




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Effect of Crosslinking

                                Mc = MW between
                                     crosslinks

                                            120

                                            160


                                            300


                                            1500
                                            9000

                                   30,000



                  Temperature

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Effect of Aging on Elastomer O Rings




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Conclusions


•Closed-Die RPA rheometer can match the viscoelastic
 measurements of the ARES
•Rheological measurements within the linear visocelastic
 region can be used to compare MW for non-branched
 polymers
•Effects of branching are seen in linear, small-amplitude
 rheology measurements, especially at low frequency
•Large Amplitude Oscillatory Shear (LAOS) clearly
 differentiates linear and branched polymers.
•DMA measurements also show differences in viscoelastic
 properties. These can indicate differences in chemistry or
 molecular structure, but also relate to bulk properties.


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                       Thank You

          The World Leader in Thermal Analysis,
             Rheology, and Microcalorimetry




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