Oscillatory shear rheology of polymers

Transcription

1 Oscillatory shear rheology of polymers Dino Ferri Politecnico Alessandria di Milano, 14/6/22 9 th June 215

2 Outline - elastic solids and viscous liquids (revision) - viscoelastic materials (revision) - the Maxwell model in time domain (revision) - Small Amplitude Oscillatory Shear (SAOS) - linear viscoelastic functions: G ( ), G ( ) e tan ( ) - the Maxwell model in time domain - some examples involving thermoplastic polymers and elastomers - the time-temperature superposition principle (TTS) - complex notation and Cox-Merz rule - Medium and Large Amplitude Oscillatory Shear (MAOS and LAOS)

3 stress strain elastic body viscous liquid strain stress G G constitutive equations d dt Robert Hooke (1678) Isaac Newton (1687) t t 1 time creep and creep recovery t t 1 time t t 1 time stores elastic energy! t t 1 time dissipates energy!

4 stress strain viscoelastic body James Clerk Maxwell (1868) t t 1 time t t 1 time Stores and wastes at the same time elastic energy of deformation! Deborah number: De mat exp Markus Reiner ( ) even the mountains flowed before the Lord (Prophetess Deborah, Judges, 5:5)

5 G(t) (Pa) Maxwell model stress relaxation a constant strain is imposed and the evolution of the stress is followed G 1 stress relaxation.1.1 modulus: G(t) σ(t) γ 1E-3 G(t) G e (t / ) time constant: 1E high De time (s) low De G

6 Rheological experiments in oscillatory regime we need some basics in: rheology rheometry oscillatory shear

7 Small Amplitude Oscillatory Shear by applying a sinusoidal strain or equivalently a cosinusoidal strain rate how does the shear stress evolves with time? elastic solid strain stress ( t) sin( ( t) cos( viscous liquid strain stress t) t) G time T/4 time ( t) G sin( t) ( t) cos( t)

8 Viscoelastic body in oscillatory regime if strain stress ( t) sin( t) ( t) sin( t ) time ( t) sin( t) cos cos( t) sen storage modulus G' ( ) cos G"( ) loss sen modulus ( t) G' ( )sin( t) G"( )cos( t)

9 Physical meaning of the oscillatory viscoelastic functions G is proportional to the elastic energy stored during a cycle: / 2 dt 2 2 G' G proportional to the energy dissipated during a cycle: 2 /ω σγdt πγ 2 G" The loss tangent is an indicator bearing information on which contribution prevails, in the rheological response: elastic (tan <1) or viscous (tan 1) tan G" G'

10 A workflow to obtain oscillatory viscoelastic functions the experimentalist: material geometry sample preparation choice setup of the method the instrument: impose oscillatory measure data viscoelastic stress or strain the response elaboration functions

11 We need to do rheometry to study rheology!

12 The crucial step: cross correlation algorithm Suppose the rheometer imposes a sinusoidal strain (deflection angle): (t) sen(ωt φ) The output signal will be a torque M(t) out of phase (phase angle difference : M sen(ωt M(t) φ) noise After filtering the signal to avoid noise it will be digitalized (248 points/cycle): M sen(ωt M(t) φ)

13 The crucial step: cross correlation algorithm The rheometer calculates the cross correlation coefficient between the output function and the reference one evaluating the following integral: 2N / M sen( t ) sen( t)dt M cos N And the cross correlation coefficient between the output function and that 9 out of phase with respect to the reference one evaluating the following integral: 2N / M sen( t ) cos( t)dt M sen N From the ratio between the values of the two integrals we obtain tan and the value of M

14 moduli (Pa) at low G ~ G ~ Maxwell model in oscillatory regime 2 2 G G' G" G tan Modello di Maxwell ( = 1 s) viscous 1 elastic G' G" tan cr = (rad/s) tan low De high De

15 Moduli (Pa) Isochronal temperature sweep ( will change) Polystyrene M w =17. G' G" 1 6 Elastic 1 5 Elastic Viscous Viscous T ( C) segmental relaxation (T g ) high De low De reptation

16 High Impact Polystyrene TEM micrograph for a HIPS copolymer

17 Moduli (Pa) Isochronal temperature sweep: a tool for comparisons and quantitative analysis 1 1 High Impact Polystyrene G' (5% PB) G' (5% PB) G" (5% PB) G" (5% PB) T ( C)

18 Thermoplastic elastomers: structure thermoplastic elastomeric thermoplastic thermoplastic diblock copolymers elastomeric triblock copolymers thermoplastic thermoplastic elastomeric thermoplastic elastomeric thermoplastic thermoplastic thermoplastic trichain copolymers thermoplastic tetrachain copolymers 18

19 19 Some common thermoplastic elastomers PS PI SI PS PB SB Styrene Isoprene Styrene-Butadiene PS PI PS SIS PS PB PS SBS Styrene-Isoprene-Styrene Styrene-Butadiene-Styrene PS EB PS SEBS Styrene-Ethylene-co-Butylene-Styrene

20 automotive Thermoplastic elastomers: applications personal care sports and toys 2

21 Moduli (Pa) Isochronal temperature sweep: a tool to detect morphological transitions 1 6 SIS copolymer (styrene-isoprene-styrene) PS PI PS 1 5 T OOT T ODT.4 G' G".2 tan T ( C) 1..8 Loss tangent

22 Isochronal temperature sweep : T g, ODT and crosslonking of a tetrachain SBS copolymer (S=38%, B=62%) tg PS.75 T g, PS OOT PB.5 T g, PB PS PS.25 ODT+retic. PS T( C)

23 Isochronal temperature sweep : T g and ODT of linear SEBS copolymers Increasing M w tg 1..8 M w =62K M w =8K M w =213K OOT.6 T g of EB T g of PS.4.2 crankshaft of -(CH 2 ) n - units PS EB PS T( C)

24 Acrylonitrile-butadiene-styrene copolymers TEM micrograph for an ABS copolymer

25 Isothermal and isochronal strain sweep ( is allowed to change): linear viscoelastic region of SAN and ABS copolymer melts 1. G'/G' ( =) T=22 C =1 rad/s SAN ( % rubber) ABS (SAN + 25 % rubber) strain(%) The linear viscoelastic region for melts is wide and its amplitude depends on structural parameters!

26 G' (GPa) Isothermal and isochronal strain sweep ( is allowed to change): the linear viscoelastic region of HIPS in the solid state.2.18 =1 rad/s =1 rad/s HIPS T=95 C Strain (%)

27 G' (Pa) Isothermal frequency sweep ( is allowed to change) T Polystyrene M w =17. T=16 C T=17 C T=18 C T=19 C T=2 C T=21 C T=22 C T=23 C T=24 C T=25 C T=26 C (rad/s) Low De high De

28 G' (Pa) Time-temperature superposition principle (TTS) If the material exhibits several relaxation times at T : i i ( T ) i T i 1,... n which are all multiplied at a new T by the same shift factor a T : (T) (T ) a i 1,...n 1 6 Polystyrene The curves perfectly overlap when shifted along the frequency axis to form a MASTER CURVE M w =17. T=16 C T=17 C T=18 C T=19 C T=2 C T=21 C T=22 C T=23 C T=24 C T=25 C T=26 C a T (rad/s)

29 Shift factor as a function of temperature experimental data best fit 1 2 a T Polystyrene M w = Williams-Landel-Ferry (WLF) equation T ( C) Loga T C C 2, 1, (T (T T T ) )

30 Log G' (Pa) To describe real cases more than one relaxation time has to be taken into account! 5 4 Polystyrene M w = slope=2 experimental data single relaxation time 5 relaxation times a T (rad/s)

31 Storage modulus (Pa) But nowadays we are able to predict LVE of linear polymers! M w =23 kda M w =12 kda M w =18 kda 2 C Polystyrene angular frequency (rad/s)

32 tan TTS sometimes breaks down: T g of polystyrene T=13 C T=15 C T=11 C T=115 C T=12 C Polystyrene M w = Tan (rad/s) D. Ferri, L. Castellani, Macromolecules, 34, (21) is not affected by geometry uncertainties: it is suited to highlight fine details

33 Frequency sweep; a tool to estimate the molecular weight between entanglements G' G RT M e segmental relaxation G terminal relaxation

34 shear moduli (Pa) Master curves to determine the molecular weight between entanglements of polystyrene 1 5 T red =2 C.28 MPa M e =1835 g/mol G' G tan 1 tan monodisperse polystyrene M w = a T (rad/s) D. Ferri, F. Greco, Macromolecules, 39, (26).1

35 shear moduli (Pa) Master curves to determine the molecular weight between entanglements of polyisoprene 1 6 T red =6 C 1 M e =635 g/mol MPa monodisperse polyisoprene M w =148 G' G" tan a T (rad/s) D. Ferri, F. Greco, Macromolecules, 39, (26) 1 tan

36 Moduli (Pa) Low frequency moduli for LLDPE/LDPE blends 2 15 T=19 C =.1 rad/s G" 1 5 log (G")= log (G") i i G' log (G')= LDPE % i log (G') i The positive deviation is a signature of immiscibility!

37 Isothermal frequency sweep test: PP/carbon nanofibers composites (L ~ 1 m, d~1 nm) G' (Pa) 1 5 T=21 C PP 1.5% 3% 6% 1% (rad/s)

38 G normalized (Pa) Isothermal and isochronal time sweep test: PE rigradation C LLDPE 21 C 215 C 25 C 2 C 195 C time (min) g/cm 3

39 Complex numbers notation z * e i (cos i sen ) if * e i t * e i( t ) we can define a complex modulus : G * ( ) ' G ( ) " ig ( ) G * ' (G ) 2 " (G ) 2 and write: * G * * a complex viscosity can be defined: * ( ) G * ( )

40 Viscosity (Pa s) The Cox-Merz rule W. P. Cox and E. H. Merz, J. Polym. Sci., 28, 619 (1958) Polystyrene M w =17. ( ) ( ) We can use data from an oscillatory experiment to predict non linear properties! 1 2 time domain (cone-plate) time domain (capillary rheometer) master curve in oscilaltory regime (parallel plates) a T (rad/s) or shear rate (s )

41 Some suggestions and problems! Isochronal temperature sweeps: ensure the linearity with respect to the relaxation at study choose dt/dt for a certain to allow thermal pseudo-equilibrium Isothermal frequency sweeps: ensure the linearity with respect to the different values decide if the sweep has to begin at low or high frequencies allow the sample to equilibrate pay attention to flow instabilities: edge fracture pay attention to inertia effects due to the geometry pay attention to inertia effects due to the transducers

42 Large Amplitude Oscillatory Shear (LAOS) Allows changes of and independently! In the linear viscoelastic regime the Boltzmann superposition principle holds: t ( t ) G( t t') ( t') dt' which allows the moduli to be calculated: G' ( ) G( s) sen( s) ds G" ( ) G( s)cos( s) ds In te case of large strain amplitudes ( >>1): ( t ) sen( n t ) n odds n n n n, n (, n ( ) )

43 stress (Pa) stress (Pa) Non linearity is evident only at hig strain values! 8 PS M w =18 KDa strain 5% tempo (s) PS M w =18 KDa -2-4 strain 4% time (s)

44 stress (Pa) Lissajous plot Experimental warnings: fluid inertia, viscous heating, wall slip PS M w =18 KDa 2 % 4 % 5 % 5 % strain (%)

45 LAOS and MAOS of Polystyrene PS M w =18 KDa.1 MAOS I 3 /I or I 5 /I.1 1E-3 LAOS 1E strain (%) It is particularly useful to test the validity of constitutive models!

46 Non linear behavior is the rule in polymer processing industry! But we cannot escape the reality that most real world phenomena are highly non linear. The division of viscoelastic behavior into two categories, linear and non-linear, suggests that non-linear behavior is somehow exceptional, but this point of view does not reflect reality. The late matematician Stanislav Ulam noted that this is like classifying all animals that are not elephants as nonelephants non-linearity Stanislav Ulam linearity

47 References [1] J. M. Dealy, K. F. Wissbrun, Melt Rheology and its Role in Plastics Processing, Van Nostrand Reinhold, New York, 199. [2] J. D. Ferry, Viscoelastic Properties of Polymers, 3rd edition, Wiley, New York, 198. [3] N. G. McCrum, B. E. Read, G. Williams, Anelastic and Dielectric Effects in Polymeric Solids, Wiley, New York, [4] I. M. Ward, Mechanical Properties of Solid Polymers, Wiley, New York, [5] C. W. Macosko, RHEOLOGY. Principles, Measurements and Applications, VCH Publishers, New York [7] G. Marin, Oscillatory Rheometry, in Rheological Measurements, edited by A. A. Collier and D. W. Clegg, Elsevier Appl. Science Publ. LTD, 1998 (chap. 1) [8] A. J. Giacomin and J. M. Dealy, Large-Amplitude Oscillatory Shear, in Techniques in Rheological Measurement, edited by A. A. Collier, Chapman & Hall, 1993 (chap. 4) [9] D. Ferri, Materiali polimerici amorfi a T<T g e semicristallini a T<T m, Polimeri termoplastici nello stato fluido, Giornate didattiche dell AIM su CARATTERIZZAZIONE MECCANICO- DINAMICA DI MATERIALI POLIMERICI, Gargnano, Maggio 25.