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Framework to Decompose and Predict Stress Response of Complex Fluids in
Transient, Large Amplitude Flows
Toni BechtelDepartment of Chemical Engineering
Carnegie Mellon UniversityPhD Thesis Proposal
October 30, 2015
Previous Work: Micro-Mechanics of Colloidal Dispersions
2Khair & Bechtel 2015
ProbeBath
Mathematical techniques provided a foundation for current work
Combined macro- and micro-scale forcing leads to:Cross-streamline migration in shear-planeEnhanced orthogonal settling
Published in Journal of Rheology in February 2015
What are Complex Fluids?
3
PaintBlood
Toothpaste Ice Cream
Small Amplitude Oscillatory Shear (SAOS)
4Zhao et. al. 2015 Bird et. al. DPL vol 1. 1977
Viscous/loss modulus
Elastic/storage modulus
Shear rateOscillation FrequencyRelaxation time
Strain rate
time
Shear Stress
Wi = 0.6
Weissenberg #Deborah #
Linear Relaxation modulus
Inverse Fourier Transform Linear memory integral expansion
SAOS
5Large Amplitude Oscillatory Shear (LAOS)
Zhao et. al. 2015
Shear rateOscillation FrequencyRelaxation time
Strain rate
Weissenberg #Deborah #
LAOS
SAOS
time
Shear Stress
Wi = 0.6
Wi = 3
Wi = 6
Wi = 30
Wi = 48
1. Unknown how to use LAOS to predict stress in other flows
2. Current decomposition frameworks are only valid for moderate deformations
Current LAOS Shortcomings
Outline
6
1. Framework to predict weakly nonlinear stress response of complex fluids
Current Work: Model system of Brownian spheroids
Aim 1: Manuscript of current work
Hypothesis: Framework is general and applicable to other systems
2. Decomposition of LAOS that is valid for arbitrary rates of deformation
Aim 1: Generate a library of stress signatures for model system
Aim 2: Develop a systematic decomposition framework from stress signatures
Outline
7
1. Framework to predict weakly nonlinear stress response of complex fluids
Current Work: Model system of Brownian spheroids
2. Decomposition of LAOS that is valid for arbitrary rates of deformation
Aim 1: Generate a library of stress signatures for model system
Aim 1: Manuscript of current work
Hypothesis: Framework is general and applicable to other systems
Aim 2: Develop a systematic decomposition framework from stress signatures
Unknown How to Use LAOS to Predict Stress in Other Flows
8
?Nonlinear, transient stress response
Zhao et. al. 2015
time
Shear Stress
Wi = 0.6
time
Shear Stress
Wi = 6
LAOS
SAOS
Linear Relaxation modulus Linear memory integral expansion
Rheology Can Be Very Different in Different Flows
9Depuit & Squires 2012
Effective viscosity
Wi
Thickening
Thinning
Barnes Intro. to Rheology 1989
A
D
B
C
AUniaxial Extension
DBiaxial Extension
BPlanar Extension Planar Shear
C
Nonlinear Memory Integral Expansion
10
Linear memory integral expansion
SAOS:
Co-rotational memory integral
expansion
Co-rotational rate of strain tensor
1-D Inverse Fourier Transform
time
Bird et. al. DPL vol 1. 1977
How the material relaxesHistory of deformation
This is applicable to a variety of complex fluids!
How Do We Determine the Nonlinear Modulus?
11
Co-rotational memory integral
expansion
Need stress response at two independent frequencies
SAOS:
Nonlinear relaxation modulus
Stress response at single frequency
https://en.wikipedia.org/wiki/Inner_earhttps://en.wikipedia.org/wiki/Flute
Output:
Rogederer 2008 Helmholtz 1957
Input:
Combination tones
Linear:
Nonlinear:
General Framework
12
(1). Calculate the weakly nonlinear stress response
(2). Evaluate co-rotational memory integral expansion
(3). Compare (1) and (2) linear and nonlinear moduli
(4). Predict stress response in a transient flow
RodNewtonian fluid
Model system:
Dilute Suspension of Brownian Rods
13
Rod
Newtonian fluid
Bretherton Constant
Aspect Ratio
Volume fraction
Tobacco Mosaic Virus
Examples:
Jeffery 1922
Micro-Mechanics of Brownian Spheroids
14
Rotary Diffusion Coefficient
Orientational Unit Vector
Shear-rate
TimeFrequencies scaled
by Dr
Normalization Condition
Initial Condition
Orientation distribution function Likelihood of particle oriented
in a differential region about (,)
Kim & Karilla 2005. Leal & Hinch 1971.
Fokker-Planck Equation(FPE)
Rate of DeformationRate of Brownian Rotation
Vorticity Straining Rigidity
Weakly Nonlinear Stress Response
15
Total stress
Kim & Karrila 2005
Solvent stress
Solvent viscosity
Identity Tensor
Thermodynamic Pressure
Rate of strain tensor
Orientational Average
Stress from particles
Regular perturbation expansion
Calculate the stress
Substitute expansion into FPE to obtain equations for and
Linear
Weakly Nonlinear
Nonlinear
SAOS: O(Wi)MAOS: O(Wi2)
LAOS
Fokker-Planck Equation(FPE)
General Framework
16
(1). Calculate the weakly nonlinear stress response
(2). Evaluate co-rotational memory integral expansion
(3). Compare (1) and (2) linear and nonlinear moduli
(4). Predict stress response in a transient flow
RodNewtonian fluid
Model system:
Linear Stress Response: Micro-Mechanics
17
Liquid-likeSolid-like
Compare to co-rotational memory integral expansion:
SAOS: O(Wi)
MAOS: O(Wi2)
LAOS
1. Be linear in the flow2. Oscillate at input frequencies
SAOS: O(Wi)
MAOS: O(Wi2)
LAOS
18
Weakly nonlinear response
Linear response
1. Be quadratic in the flow
2. Oscillate at frequencies
Nonlinear Stress Response: Micro-Mechanics
Nonlinear Relaxation Modulus
19
Weakly nonlinear response
Linear response
Stress response from micro-mechanics
Co-rotational memory integral
expansion
timettt
Compare the yy-stress component at
Abdel-Khalik et. al 1974
2-D Inverse Fourier Cosine Transform
General Framework
20
(1). Calculate the weakly nonlinear stress response
(2). Evaluate co-rotational memory integral expansion
(3). Compare (1) and (2) linear and nonlinear moduli
(4). Predict stress response in a transient flow
RodNewtonian fluid
Model system:
Start-up and Cessation of Steady Shearr = 1000 rest
steady shearrest
Cone and Plate Rheometer
LAOS
MAOSSAOS
21Strand, Kim & Karrila 1987
Wi = 3
Wi = 1
Wi = 0.5
Used MAOS to predict nonlinear stress response
Numerical SolutionFramework
Outline
22
1. Framework to predict weakly nonlinear stress response of complex fluids
Current Work: Model system of Brownian spheroids
2. Decomposition of LAOS that is valid for arbitrary rates of deformation
Aim 1: Generate a library of stress signatures for model system
Aim 1: Manuscript of current work
Hypothesis: Framework is general and applicable to other systems
Aim 2: Develop a systematic decomposition framework from stress signatures
General Framework
23
(1). Calculate the weakly nonlinear stress response
(2). Evaluate co-rotational memory integral expansion
(3). Compare (1) and (2) linear and nonlinear moduli
(4). Predict stress response in a transient flow
Other Model Systems:
Colloidal dispersions Active Suspensions
Outline
24
1. Framework to predict weakly nonlinear stress response of complex fluids
Current Work: Model system of Brownian spheroids
2. Decomposition of LAOS that is valid for arbitrary rates of deformation
Aim 1: Generate a library of stress signatures for model system
Aim 1: Manuscript of current work
Hypothesis: Framework is general and applicable to other systems
Aim 2: Develop a systematic decomposition framework from stress signatures
Strongly Nonlinear Stress Response
25Zhao et. al. 2015
time
Shear Stress
Wi = 0.6
time
Shear Stress
Wi = 3
Nonlinear
Weakly NonlinearLinear
time
Shear Stress
Wi = 48
26
Fourier Transform (FT) Rheology:
Hyun, et. al. 2011
scaled frequency
scaled Fourier mode
Fourier Transform Rheology
Zhao et. al. 2015
151 Fourier modes
Ewoldt et. al. 2008
time
Shear Stress
Wi = 0.6
Wi = 3
Wi = 6
Wi = 30
Wi = 48
What can we learn about the microstructure in this
strongly nonlinear regime?
Dilute Suspension of Brownian Spheroids
27
Spheroids
Newtonian fluid
Bretherton Constant
Aspect Ratio
Volume fraction
Oblate Nearly Spherical
Prolate
Strongly Nonlinear Stress Decomposition
28
(1). Calculate the strongly nonlinear stress response
Fokker-Planck Equation(FPE)
Numerical solutions
Stress from particles
(2). Repeat over a range of Bretherton constants
Change in orientation
Hypothesis: From these stress signatures we can infer physically meaningful details about the microstructure of complex fluids
Spectrum of Stress Responses
29Khair 2015 (submitted)
time
Shear Stress
Wi = 300B ~ 0
time
Shear Stress
Wi = 10B = 1
Strand, Kim & Karrila 1987
Rapid oscillations
Dominated by rotation (vorticity) Dominated by alignment (strain)
Smooth oscillations
scaled frequency
scaled Fourier mode
Hyun, et. al. 2011Zhao et. al. 2015
Summary and Timeline
30
Manuscript: Predicting
weakly nonlinear
stress
Building toolset for numerical
scheme
Generating stress
response library
Manuscript: Strongly
nonlinear stress decomposition
Thesis
Nov 2015
May 2016
Jan 2017
June 2017
Jan 2018
May 2018
1. Developed a framework to predict weakly nonlinear stress response of complex fluids Model system of Brownian spheroids Hypothesis: framework is general and could be used on a variety of systems
2. Require a framework to decompose stress response for arbitrary deformations Model system of Brownian spheroids Hypothesis: library of stress signatures provides physical insight into microstructure
31
Cone and Plate Rheometer
Shear Stress:
Normal Stress Differences:
Shear Stress:
Normal Stress Differences:
Start-up and Cessation of Steady Shear Flow
32
A
B
A
BNumerical SolutionFramework
Shear Stress
33
Wi = 3
Wi = 1
Wi = 0.5
Numerical SolutionFramework
Start-up and Cessation of Steady Shear Flow1st NSD
Start-up and Cessation of Steady Shear Flow
34
Numerical SolutionFrameworkA
A
2nd NSD
(Planar) Shear Planar Extensional
Uniaxial Extensional Biaxial Extensional
Barnes Introduction to Rheology 1989
Previous Numerical Solution to FPE
36Strand, Kim & Karrila 1987
Impractical for high Wi Current procedure only valid for rods (B=1)
Fokker-Planck Equation(FPE)
Edge Fracturing in Polymer Melts
37
Polystyrene melt (145 kg/mol)
Worm-like Micelles
38Zhao et. al. 2015