Proposal v5

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








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


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


General Framework

16





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


expansion

timettt

Compare the yy-stress component at

Abdel-Khalik et. al 1974

2-D Inverse Fourier Cosine Transform

General Framework

20





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








General Framework

23





Other Model Systems:

Colloidal dispersions Active Suspensions

Outline

24








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


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)


Edge Fracturing in Polymer Melts

37

Polystyrene melt (145 kg/mol)

Worm-like Micelles

38Zhao et. al. 2015

Documents

Proposal v5