Session 11: Complex Modulus of Viscoelastic Polymers

Transcription

1 Session 11: Complex Modulus of Viscoelastic Polymers Jennifer Hay Factory Application Engineer Nano-Scale Sciences Division Agilent Technologies To view previous sessions: Page 1

2 Outline Constitutive equation for polymers Material model Macro-scale methods Instrumented indentation method Exemplary results on polyethylene Summary Introduction to next session Page 2

3 Constitutive equation for polymers In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities that is specific to a material or substance, and approximates the response of that material to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in structural analysis, the connection between applied stresses or forces to strains or deformations. - Ask.com ( s*=e*e* Page 3

4 A model of viscoelasticity (Kelvin-Voigt) s(t) e(t) t t Page 4

5 Response to a sinusoidal stimulation What is the relationship between stress and strain? Page 5

6 Invoke constitutive equation E* = s*/e* (s* and e* are phasors) imaginary s* = s 0 e iwt d wt s* e* e* = e 0 e i(wt-d) = e 0 e iwt /e id E* = s* /e* = (s 0 /e 0 )e id E* = (s 0 /e 0 )[cosd + isind] real storage modulus loss modulus loss factor E* = E + ie, where E = (s 0 /e 0 )cosd E = (s 0 /e 0 )sind tand = E /E Page 6

7 Or, invoke constitutive equation G* = t*/g* (t* and g* are phasors) imaginary t* = t 0 e iwt d wt t* g* g* = g 0 e i(wt-d) = g 0 e iwt /e id G* = t* /g* = (t 0 /g 0 )e id G* = (t 0 /g 0 )[cosd + isind] real shear modulus shear loss modulus loss factor G* = G + ig, where G = (t 0 /g 0 )cosd G = (t 0 /g 0 )sind tand = G /G = E /E Page 7

8 Elastic moduli Storage: E /G = (s 0 /e 0 )cosd /(t 0 /g 0 )cosd E /G = E /G = 2(1+n) 3 E G Loss: E /G = (s 0 /e 0 )sind /(t 0 /g 0 )sind E E /G = E /G = 2(1+n) 3 In effect: It does not matter that we are dealing with viscoelastic components; the relationship between normal and shear modulus is the same as for homogeneous elastic materials. G Page 8

9 General objective: E and E (or G and G ) E E For materials which are well described by the Kelvin-Voigt model, we want to know the components of the complex modulus. Page 9

10 Macro-scale methods Dynamic Mechanical Analysis (DMA): imposes normal stresses on a cylindrical sample and returns E* Rheology: imposes oscillating shear stress and returns G* Schematic of rheometric test geometry Page 10

11 Exemplary macro-scale DMA for Nylon 6 0.2% RH/min 1Hz, 50C 0.03% strain TA instruments: TA Instruments Q800 Dynamic Mechanical Analyzer and DMA-RH Accessory Relative humidity (%) Page 11

12 Exemplary macro-scale rheology: gelled PVC Complex shear modulus of PVC in Bi(2- ethyhexyl) phthalate solvent (plasticizer) TA instruments: ARES Rheometer with force transducer and cone/plate fixtures. Page 12

13 Our objective: E and E by indentation Indenter Polymer Use a flat-ended cylindrical punch Apply an oscillating force Measure resulting displacement oscillation Calculate E, E, and tan d Page 13

14 Highly plasticized PVC (3M EAR C ) E-A-R has developed complete lines of high-performance damping materials to meet the widely varying physical and performance requirements of a broad range of applications. They are engineered to provide maximum performance with minimum added weight and thickness. D=353 Page 14

15 Indentation results agree with DMA DMA by TA Instruments Q800; nanoindentation by Agilent G200 NanoIndenter with CSM option and 100um flat punch. Herbert, E.G., Oliver, W.C., and Pharr, G.M., "Nanoindentation and the Dynamic Characterization of Viscoelastic Solids," Journal of Physics D-Applied Physics 41(7), See also: Page 15

16 Flat-punch indentation Surface contact is obvious Contact area is known and constant We are only interested in viscoelastic properties Note: Cylinder diameter should be selected so that: 1000 N/m < E D < 60000N/m. Two tips, D=20mm and D=100mm are a good start. Page 16

17 100-micron-diameter flat punch Page 17

18 Contact mechanics D s w S Elastic component (Sneddon): E r = S/2a E n 1n i r E E E = E r (1-n 2 ) = S(1-n 2 )/2a E = S(1-n 2 )/2a Viscous component (analogy): E = D s w(1-n 2 )/2a i Page 18

19 Contact mechanics Elastic component (Sneddon): D s w S E = S(1-n 2 )/2a Viscous component (analogy): E = D s w(1-n 2 )/2a Page 19

20 Contact mechanics Elastic component (Sneddon): E = S(1-n 2 )/2a D s w S Viscous component (analogy): E = D s w(1-n 2 )/2a Loubet, J.L., Oliver, W.C., and Lucas, B.N., "Measurement of the Loss Tangent of Low- Density Polyethylene with a Nanoindentation Technique," Journal of Materials Research 15(5), , Dr. Lucas Ph.D. Dissertation: Page 20

21 Modeling the dynamic behavior of the head alone K D = K i = D i m = m i z(t) = z 0 ei(wt-) F(t) = F 0 e iwt For a review of the simple-harmonic-oscillator analysis, see A brief detour in session 7. Page 21

22 Compliance, m/n Dynamic calibration of the head alone XP head; indenter in central position. Best fit values are K i = 92.02N/m, m i = 11.6g, D i = 2.66N-s/m measured best fit Frequency, Hz Page 22

23 contact Modeling the dynamic behavior of the system K f KS c D cs K i D i m i +m s m i z(t) = z 0 e i(wt-) F(t) = F 0 e iwt Page 23

24 contact Modeling the dynamic behavior of the system K f KS c D cs K i D i m i +m s m i z(t) = z 0 e i(wt-) F(t) = F 0 e iwt Page 24

25 contact Modeling the dynamic behavior of the system K f K D KS c D cs K i D i m i +m s m i F(t) = F 0 e iwt z(t) = z 0 e i(wt-) m z(t) = z 0 ei(wt-) F(t) = F 0 e iwt K = [1/K f + 1/S] -1 +K i D = D i +D s m = m i +m s 0 Page 25

26 Stiffness of the contact D m K F(t) = F 0 e iwt z(t) = z 0 ei(wt-) D m K F(t) = F 0 e iwt z(t) = z 0 ei(wt-) K = [1/K f + 1/S] -1 +K i D = D i +D s m = m i +m s 0 Page 26 w cos z F m K mw z F K S K i f cos 1 1 f i K m K z F S 1 cos w

27 Damping of the contact 0 Dw F sin z0 0 w sin D i D s F z 0 K D D s w F z 0 0 sin D i w m z(t) = z 0 ei(wt-) F(t) = F 0 e iwt K = [1/K f + 1/S] -1 +K i D = D i +D s m = m i +m s 0 Page 27

28 Components of the complex modulus Elastic : E = S(1-n 2 )/2a, where Viscous: E = D s w (1-n 2 )/2a where S F z cos 2 K f K i mw 1 1 D s w F z 0 0 sin D i w Ratio: tand = E /E = D s w/s Page 28

29 Hardware requirements Agilent G200 NanoIndenter with XP or DCM head Continuous-stiffness measurement option (CSM) Flat-ended cylindrical punch Page 29

30 Hardware requirements Agilent G200 NanoIndenter with XP or DCM head Continuous-stiffness measurement option (CSM) Flat-ended cylindrical punch Software requirements NanoSuite 5.10A, build 148 or later Test methods: G-Series XP CSM Flat Punch Complex Modulus OR G-Series DCM CSM Flat Punch Complex Modulus Page 30

31 Test method and documentation Method is automatically installed with the CSM option. Page 31

32 Basic procedure 1. The face of the indenter is pressed into full contact with the test material. 2. The indenter is vibrated at a discrete number of frequencies and the response is measured at each. Page 32

33 User-Controlled Inputs Required: Editable post-test: Page 33

34 Testing polyethylene Very low density polyethylene (VLDPE) Low-density polyethylene (LDPE) Linear-low-density polyethylene (LLDPE) High-density polyethylene (HDPE) Hay, J. and Herbert, E., "Measuring the Complex Modulus of Polymers by Instrumented Indentation Testing," Experimental Techniques April, Agilent Application note: Page 34

35 Results for HDPE, Cycle view Page 35 Measuring Substrate- Independent Modulus July 13, 2010

36 New polymer test method Page 36 October 5, 2010

37 Storage Modulus / MPa Storage modulus for all four PE samples HDPE (27C) LLDPE (27C) LDPE (27C) Frequency/Hz VLDPE (27C) Page 37

38 Storage Modulus / MPa Comparison with literature HDPE results: G.M. Odegard, T.S. Gates, and H.M. Herring, Characterization of viscoelastic properties of polymeric materials through nanoindentation, Experimental Mechanics 45(2): (2005) LDPE results: J. Capodagli and R. Lakes, Isothermal viscoelastic properties of PMMA and LDPE over 11 decades of frequency and time: a test of timetemperature superposition, Rheol. Acta. 47: (2008). HDPE by DMA (23C) HDPE (27C) LLDPE (27C) LDPE (27C) LDPE by DMA (23C) Frequency/Hz VLDPE (27C) Page 38

39 Loss Factor Loss factor for all four PE samples HDPE (27C) LDPE (27C) LLDPE (27C) VLDPE (27C) Frequency/Hz Page 39

40 Loss Factor Comparison with literature HDPE (27C) LDPE (27C) LLDPE (27C) VLDPE (27C) LDPE by DMA (23C) Frequency/Hz Page 40

41 Summary We use the constitutive equation, viscoelasticity in polymers s*=e*e*, to comprehend The Kelvin-Voigt material model comprises a Hookean spring and a Newtonian damper in parallel; with this model, all deformation is completely elastic, only viscously so. Macro-scale methods for measuring complex modulus are dynamic mechanical analysis (DMA) and rheometry. Dynamic indentation can be used to measure local complex modulus; results on bulk materials match DMA. Page 41

42 Session 12: Complex Shear Modulus of Gelatin and Soft Biological Tissue Wednesday, November 13, 2013, 11:00 (New York) Abstract Abstract: Biological tissue and simulants thereof can be characterized by instrumented indentation using test methods for polymers. However, if the experiment is not well designed, the measured response can easily be on the order of experimental uncertainty. Rigorous uncertainty analysis provides firm guidance for experimental choices. Specifically, it provides guidance on the diameter of the punch which should be used for a particular material. Or, if the punch diameter cannot be changed, uncertainty analysis tells us the most compliant material which can be tested with that punch. The analysis and test method are demonstrated for edible gelatin and bovine muscle tissue To register: Page 42

43 Suggested reading for next session Agilent application note: Complex Shear Modulus of Commercial Gelatin by Instrumented Indentation ( Agilent application note: In Vitro Complex Shear Modulus of Bovine Muscle Tissue ( Page 43