Viscoelasticity. Basic Notions & Examples. Formalism for Linear Viscoelasticity. Simple Models & Mechanical Analogies. Non-linear behavior

Transcription

1 Viscoelasticity Basic Notions & Examples Formalism for Linear Viscoelasticity Simple Models & Mechanical Analogies Non-linear behavior

2 Viscoelastic Behavior Generic Viscoelasticity: exhibition of both viscous and elastic properties, depending on the time scale over which an external stress is applied Specific Effects: Dilatancy: viscosity increases with the rate of shear ( shear-thickening ) Pseudo-plasticity: viscosity decreases with the rate of shear ( shear-thinning ) Thixotropy: viscosity decreases with duration of stress Rheopecticity: viscosity increases with duration of stress

3 Bouncing Drops

4 Viscoelastic Flows Climbing a Rotating Rod: Example of a Weissenberg instability

5 Viscoelastic Flows Die Swell: Another Weissenberg instability

6 Viscoelastic Flows Tubeless Siphon: Due to extensional elastic stress

7 Viscoelastic Flow Movies

8 Momentum Balance Integrate Newton s Second Law over a volume element Z V m d v dt dv = Z S m ˆndS+ Gives a generalized equation of motion for a complex fluid: vi For a simple viscous fluid, ij = P ij + + v j, x we obtain the Navier-Stokes Eq: j x i Z V m v t + ( v ) v = + f fdv v t + ( v ) v = P + 2 v + f

9 Creeping Flow Limit Dissipative terms dominate over inertial terms: + f =0 ij (t) =F[ ij (t 0 t), ij (t 0 t)] ij = v i x j + v j x i ij = u i x j + u j x i generalized Stokes flow for a complex fluid c.f.: Stokes flow for a simple viscous fluid: r 2 v + f = rp

10 Rheological Measurements Apply a controlled strain; measure the stress response - Relaxation after step strain - Response to steady strain rate - Dynamic response to oscillatory strain Apply a controlled stress; measure the strain response - Creep after step stress - Dynamic response to oscillatory stress generalized Stokes flow for a complex fluid

11 Simple Shear Consider unidirectional shear deformation between parallel plates u Sheared Material. v Sheared Material Solid-Like Liquid-Like Scalar parameters:,,

12 Linear Solid and Liquid u Sheared Material. v Sheared Material = G = Hookian Solid Newtonian Liquid Typical VE material shares features of each

13 Rheometer

14 Cone-Plate Rheometry Uniform strain rate:

15 Stress Relaxation Consider applying a small fixed step strain γ 0 and measuring the resulting stress (linear response) For all but perfect elastic solids, stress will decrease with time: Define the linear relaxation shear modulus: G(t) = (t)/ 0 Equilibrium shear modulus (residual elasticity):

16 Creep Under a weak, constant shear stress σ 0, most viscoelastic materials slowly deform (creep): Defines the linear creep compliance: J(t) = (t)/ 0

17 Prototype VE Solid and Liquid G 0 G 0 η η = G = Maxwell VE Liquid Voigt VE Solid

18 Maxwell Fluid Strain is additive: = E + v G 0 Stress is uniform: η Gives: E = v = = /G 0

19 Response to a Step Shear Strain: Maxwell Fluid Step Strain : Stress Relaxation γ 0 is the step shear strain

20 General Stress Relaxation Apply a series of infinitessimal fixed step strains δγ i at time t i The resulting stress σ(t) is a linear superposition of responses: (t) = X i G(t t i ) i = X i G(t t i ) i t i Go to the continuum limit of infinitessimal step durations:

21 General Linear Viscoelastic Fluid Can do the same for a series of step stresses: Constraint:

22 Response to a Step Shear Stress: Maxwell Fluid Step Stress : Creep: γ (t) is the shear strain

23 Voigt Fluid Stress is additive: Strain is uniform: = E + v E = v = G 0 η Gives: G 0 (t)+ (t) = (t)

24 Response to a Step Shear Stress: Voigt Solid Step Stress : Creep:

25 Steady Shear For a constant applied rate of shear strain: Defines the viscosity: Maxwell model: Z 1 = G 0 exp( 0 t/ ) dt = G 0

26 Oscillatory Shear Apply a simple harmonic shear strain: Shear stress response is simple harmonic with a phase shift: In-phase response is elastic: (δ=0) Out-of-phase response is viscous: (δ=π/2) General response is a sum of of these two:

27 Complex Modulus Complex modulus contains the storage and loss components: Given by the Fourier transform of the relaxation modulus G(t): G ( )=i Z 1 0 G(t)exp( i t)dt

28 Applied Oscillatory Stress gives: J ( )=i Z 1 0 J(t)exp( i t)dt Dynamic Creep Compliance

29 Oscillatory Maxwell Fluid Can show: G 0 (!) = G 0 2! 2 (1 + 2! 2 ) G 00 (!) = G 0! (1 + 2! 2 ) Cole-Cole Plot

30 Flow of Complex Fluids Complex fluids exhibit non-newtonian behavior in steady state flow conditions (non-linear response): Newtonian Thinning Thickening Thickening/Thinning behavior is due to microstructural changes Numerous phenomenological constitutive relations e.g. power-law fluids: = ( ) = apple n

31 Shear Thinning Fluids Strong deformation leads (eventually) to micro-structural rearrangements that lower resistance to further deformation Typical flow-induced changes: - break-up of clusters - flow alignment of domains - disentanglement of polymers Reversible vs permanent thinning

32 Shear Thinning Fluids Dense colloidal suspensions: Reversible thinning behavior for high enough concentrations

33 Shear Thinning Fluids Concentrated polymer fluids: PDMS Almost always reversible thinning behavior

34 Shear Thickening Fluids Strong deformation leads (eventually) to micro-structural rearrangements that increases resistance to further deformation Typical flow-induced changes: - transient cluster formation (jamming) - order-to-disorder transitions - formation and resistance of topological entanglements Reversible vs permanent thickening

35 Shear Thickening-Example 1 Polymer-Colloid Shake Gels Strong deformation leads to transient extension of polymers Extended polymers form bridges between colloids Transient network is formed Example: Laponite-PEO mixtures: J. Zebrowski, et al., Coll. Surf Sci. A,( 2006)

36 Shake Gel Phase Diagram Fluid Gel Thicken Fluid

37 Shake Gel Relaxation Φ=1.25 wt % Relaxation time is a decreasing function of PEO concentration!

38 Shear Thickening-Example 2 Concentrated starch suspensions Strong deformation leads to transient jamming of starch grains Transient stress pillars are formed that resist deformation Suspension flows if not strongly perturbed

39 Shear Thickening Instability Driven concentrated starch suspensions:

40 Transient Solid-Like Behavior Walking on corn starch solutions: Rapid steps: solid response Slow steps: fluid response

41 Chicago Experiments Impact experiments: SR Waitukaitis & HM Jaeger, Nature 487, (2012)

42 Chicago Experiments Displacement field near rod: Interpretation: formation of transient, localized solid support pillar SR Waitukaitis & HM Jaeger, Nature 487, (2012)

43 Non-Newtonian Pipe Flow Pressure-driven flow of a power-law fluid in a cylinder (radius R and length L; long axis in the z direction) Creeping flow limit without body forces: 1 r d dr (r rz) = dp dz = Constitutive equation: rz(r) =apple (r) n = apple Flow solutions: 1/n PR R v z (r) = 2L 1+1/n r =0 P L =) rz(r) = dvz dr apple 1 n r R 1+1/n Pr 2L

44 Yield Stress Fluids Fluids that are apparently solid-like below a critical stress σ c The solid-like state is usually a very slowly creeping fluid Models: Bingham Fluid: = c + p for > c Hershel-Buckley Fluid: (Newtonian at high shear rates) = c + n (non-newtonian at high shear rates) for > c 0 <n<1

45 Yield Stress Fluids Model Bingham fluid: Bentonite clay particles in water: 10 wt % (no apparent flow)

46 Bingham Fluid in Pipe Flow Low stress region near center is below σ c and thus not sheared (plug flow) All shear occurs near the higher stress regions near the wall Flow solution is a combination of the two regimes: