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Anke LindnerPMMH-ESPCI, Paris, France
Yanan Liu, Olivia du Roure, Brato Chakrabarti, David Saintillan, John Lagrone, Ricardo Cortez, Lisa Fauci
JNNFM seminar - Mai 2020
Morphological transitions of flexible filaments in viscous flows
Flexible fibers and fiber suspensions
Wexler AL, et al, JFM, (2013)
Alvarado Nature Physics 2017
Credit: Dartmouth College
Attia et al (2009)
1m
1mm
Lost circulation problems in oil wells
Schlumberger
Biological objects Flow sensors, valves Industrial applications
Interactions between flexible fibers and flows present in numerous situationsUnderstanding of freely transported fibers and suspension properties still uncomplete.
Aim of the present study
I. Understand microscopic particle dynamics
10mm
II. Understand macroscopic suspension properties
1mm
text
Our experimental model systemFlexible Brownian filaments
Actin filament
𝐿 ∼ 4 − 40 𝜇𝑚𝑑 ∼ 8𝑛𝑚
𝜖 = 𝑑/𝐿 ≪ 1
10 microns
Forces at play and dimensionless numbers
Elasto-viscous number
ҧ𝜇 =viscous force
elastic force
Brownian effect
ℓ𝑝𝐿=
elastic force
Brownian forces
𝜖 = 𝑑/𝐿 ≪ 1Viscous drag force
∼ 𝜇 ሶ𝛾𝐿2
𝐿 ∼ 10𝜇𝑚Brownian forces
∼ 𝑘𝐵𝑇/𝐿
Elastic restoring force∼ 𝐵/𝐿2
Filament-fluidinteractions𝑅𝑒 ≪ 1
Stokes equation
Slenderness𝒄(𝝐)
Actin filaments – wide spread biopolymer
Actin monomersBrownian motion
Spontaneous nucleus
(−) (+)
Pointed end (−)
Barbed end (+)
Actin monomer
(−) (+)
Alexa 488 Phalloidin
Well controlled assembly in the laboratory – stabilized and fluorescently labelled
Persistence length of actin filaments
𝐿 = 17𝜇𝑚 𝐿 = 15𝜇𝑚 𝐿 = 14𝜇𝑚
𝜇 = 1𝑚𝑃𝑎 ⋅ 𝑠 𝜇 = 28𝑚𝑃𝑎 ⋅ 𝑠𝜇 = 7𝑚𝑃𝑎 ⋅ 𝑠
123
Slide
Cover slip
63X objective
~2𝜇𝑚
Bending rigidity:
𝑩 = ℓ𝒑 × 𝒌𝑩𝐓
∼ 𝟕 × 𝟏𝟎−𝟐𝟔𝐍 ⋅ 𝒎𝟐
ℓ𝑝 = 17 ± 1 𝜇𝑚
Persistence length
Filaments in shear flow
Combination of tumbling and periodic deformation
Forgacs &Mason, 1959, Harasim et al, 2013
Becker & Shelley, 2001Also Tornberg & Shelley 2004, Nguyen & Fauci, 2014
Complete picture of the dynamics still missing.
See also Du Roure, AL,
Shelley et al, ARFM, 2019
ExperimentsTheory
Experimental set-up
63X Objective
H=500μm
Z=W
W=150μm
Observation plane
x
y
O
z
Motorized stage
Flow
xO
y
Δy
∆𝑦 ≤ 5𝜇𝑚
𝑢(𝑦) ≈ ሶ𝛾𝑦
• Elasto-viscous numberഥ𝜇=8pgμL4/B ~103 − 107
• Slenderness 𝑐 ∼ 14 ± 2
• Brownian effectℓ𝑝
𝐿~0.5 − 6
•
Experimental set-upExperimental observations
Rodlike tumbling
S bending snake turn
C shape buckling and tumbling
U bending snake turn
𝑳 = 𝟒. 𝟗𝝁𝒎ሶ𝜸 = 𝟐. 𝟕𝒔−𝟏
𝑳 = 𝟓. 𝟓𝝁𝒎ሶ𝜸 = 𝟑. 𝟖𝒔−𝟏
𝑳 = 𝟔. 𝟐𝝁𝒎ሶ𝜸 = 𝟏. 𝟑𝒔−𝟏
𝑳 = 𝟐𝟔. 𝟔𝝁𝒎ሶ𝜸 = 𝟏. 𝟖𝒔−𝟏
𝑳 = 𝟑𝟑. 𝟓𝝁𝒎ሶ𝜸 = 𝟏. 𝟓𝒔−𝟏
𝑥𝑂
Flow
63X objective
Experimental observations
𝑡
Increase elasto-viscous numer ҧ𝜇 ∼ ሶ𝛾𝐿4
Tumbling C bucklingU snake
TurnS snake
turnComplex dynamics
Comparison with numerical simulations
Simulations: Chakrabarti B. & Saintillan D.ҧ𝜇 = 2.0𝐸6 𝑐 = 14.6 ℓ𝑝/𝐿 = 0.83
• Hydrodynamics:
Non-local slender body theory
• Flexibility:
Euler–Bernoulli beam elasticity
• Brownian force distribution
Fluctuation-dissipation theorem
𝒓𝑡 − ҧ𝜇𝑼0 𝒓 = −(Λ(𝑐) + K) 𝒓𝑠𝑠𝑠𝑠 − (𝜎𝒓𝑠)𝑠+ 𝐿/𝑙𝑝𝜁
Viscous drag forces
Elastic restoring forces
Brownian forces
Filament dynamics and morphologies
Liu, AL et al. PNAS 2018
≈ 250 − 300
≈ 1500 − 2000
First transition: buckling instability
Linear stability analysis:
Becker and Shelley, 2001
AL and Shelley, Elastic fibers in flows, 2015
ҧ𝜇/𝑐 = 304.6 Experimentally observed in straining flows
Kantsler et al, (2012)
Wandersman, AL, et al, Soft Matter, 2010
Quennouz, AL, Shelley et al., JFM, 2015Young and Shelley, PRL, 2007
Numerical simulations
Becker and Shelley, 2001
First observation in shear flow
Buckling J shape
• Force Balance• Torque Balance• Energy conservation
J shape only exists above ҧ𝜇/𝑐 ≈ 1700
𝒙
𝒚
𝜙𝑉𝑠𝑛𝑎𝑘𝑒
𝐶
𝑅
𝑂
Liu, AL et al. PNAS 2018
Second transition: U-turn
Microscopic properties and morphologies
Elastic energy stored Conformation
0: isotropic
1: anisotropic
Dynamics
❑ The elastic energy is a measure of the integrated filament curvature and the conformation dynamics are obtained from the gyration tensor.
❑ Microscopic properties are expected to dictated macroscopic behavior.
Liu, AL et al. PNAS 2018
Filaments in straining flows
Kantsler et al,PRL, (2012)
Stagnation point
Long residence time in homogeneous straining flowHyperbolic flow to provide homogeneous strain rate
Optimized microfluidic flow geometry
Flow focusing
𝑊𝑢
𝑊𝑐
𝑙𝑐
𝑄 𝑥
𝑦
𝑄
Zografos K, et al., Biomicrofluidics, 2016
with M. Oliveira, K. Zografos, J. Fidalgo, U Strathclyde Liu, Zografos, AL et al. in preparation
Optimized microfluidic flow geometry
𝑊𝑢
𝑊𝑐
𝑙𝑐
𝑄 𝑥
𝑦
𝑄
Velocity profile Strain rate profile
Motorized stage to track filaments
𝑥𝑂
Flow
63X objective
• Keep filaments in frame• Limit image blur
Image blur≤ ±0.5𝜇𝑚Keep filament in frame
Δ𝑡 = 50𝑚𝑠
Liu, Zografos, AL et al. in preparation
Experimental observations
Comparison with simulations
With Brownian fluctuations
Chakrabarti, AL, et al, Nature Physics, 2020
Comparison with simulations
With Brownian fluctuations
Without Brownian fluctuations (simulations)
Simulations J. Lagrone, L. Fauci, R. Cortez, Tulane University
Linear stability analysis
Helicoidal shape can be explained by linear stability analysis: odd and even, in and out of plane modes grow simultaneously -> helicoidal shape is obtained.
Weakly non-linear stability analysis
In and out of plane modes couple in weakly non-linear analysis -> helicoidal shape is always obtained.
Chakrabarti, AL, et al, Nature Physics, 2020
Helix shape – time evolution
Radius and length evolution Compression rate
What about the final radius?
Helix shape – radius
Radius as a function of length Radius as a function of the elasto-viscous length
Two length scales in the problem:Filament length and elasto-viscous length
Formation of helicoidal shapes
Chelakkot et al., 2012
Mercader et al., 2010
Pieuchot et al., 2015
hole
53𝑠 53.5𝑠
Under compression In shear
Very general phenomenon,only requires strong enough compression, mostly overlooked so far.
Nguyen & Fauci, 2014Allende et al, 2018
Suspension rheology – flexible fiber suspensions
Tornberg and Shelley, 2004 Becker and Shelley, 2001
Kirchenbuechler et al., 2014See also Huber et al, 2014
No experimental results in the dilute regime. No simulations of flexible Brownian fibers in 3D.
Perazzo, PNAS, 2017
Suspension rheology – 2D predictions (non-Brownian)
First normal stress difference
Shear viscosity
Starts to grow at buckling threshold
Buckling induces shear thinning
Sharp onset of non-Newtonian properties at buckling threshold
Suspension rheology – 3D Brownian
Chakrabarti, AL et al, in preparation
Normal stress difference
❑ Brownian rotational noise induces shear thinning as well as normal stress differences already for rigid rods
❑ Buckling threshold blurred due to Brownian fluctuations as well as 3D orientation distributions
Brownian rigid fibers from: EJ Hinch and LG Leal., JFM, 1972.
Viscosity
Suspension rheology – Gyration tensor
What about experiments?
Average orientation Gyration tensor contribution
❑ With increasing Pe number fibers become more and more aligned with flow direction
❑ Initiation of U-turns lead to a sharper decrease, corresponding to folded aligned conformations
❑ Change in scaling observed slightly earlier for viscosity and normal stress difference indicates importance of buckling events on rheology.
Experimental measurements – monodisperse suspensions
(+)Spectrinactin seed
A drop of filamentsuspension
Polylysine charged slide:attracting filaments to
the surface
Length distribution
0 5 10 15 200
200
400
600
800
1000
1200
0 10 20 30 400
50
100
150
0 5 10 15 200
200
400
600
800
Short Middle Long
𝜑 = 7.5%𝜑 = 0.4% 𝜑 = 23.9%
Experimental measurements - rheometer
𝑤1 𝑤2
Suspension𝜇1
Reference fluid𝜇2
𝑤1
𝑤2=𝜇1𝜇2
𝑄 𝑄
W = 600𝜇𝑚
W
0 500 1000 1500 2000
200
400
600
800
1000
1200
Intensity profile
Error function
Wall
Interface
Y channel has proven to be very sensitive to small viscosity differences
Gachelin, AL et al, PRL, 2013
Viscosity measurements – very preliminary
10-2
10-1
100
101
102
1.00
1.05
1.10
1.15
1.20
10-2
10-1
100
101
102
1.00
2.00
3.00
4.00
5.00
6.00
Flow rate increasing
Actin filament suspension
Reference fluid
WallsMiddle
ത𝐿 = 4.3𝜇𝑚 𝜑 = 0.4%Entangled suspension
Summary – fiber morphologies
Fiber morphologies in shear and compressive flows flow
Comprehensive understanding
Suspension properties
10-2
10-1
100
101
102
1.00
1.05
1.10
1.15
1.20
Preliminary results
To be continued!
Olivia du Roure (ESPCI)
David Saintillan (UCSD, USA)
Ricardo Cortez (Tulane, USA)
Lisa Fauci (Tulane, USA)
Monica Oliveira (Strathclyde)
PhD students and Post-Docs
Yanan Liu
Brato Chakrabarti (UCSD)
Joana Fidalgo (Strathclyde)
John Lagrone (Tulane)
Colleagues
Biochemistry:Antoine Jégou & Guillaume RometLemonne (IJM)
Further reading
Review papers on fluid structure interactions and flexible fibers
❑ O. du Roure, A. Lindner, E. Nazockdat and M. Shelley, “Dynamics of flexible fibers in viscous flows and fluids” Annu. Rev. Fluid Mech. 2019. 51:539–72
❑ Fluid-Structure Interactions in Low-Reynolds-Number Flows, editors Camille Duprat and Howard Stone, RSC Soft Matter Series
Papers on which the talk was based
❑ B. Chakrabarti, Y. Liu, J. LaGrone, R. Cortez, L. Fauci, O. du Roure, D. Saintillan, A. Lindner, Flexible filaments buckle into helicoidal shapes in strong compressional flows, Nature Physics, 2020
❑ Y. Liu, B. Chakrabarti, D. Saintillan, A. Lindner, O. du Roure , “Tumbling, buckling, snaking: Morphological transitions of elastic filaments in shear flow”, PNAS, 2018
❑ B. Chakrabarti, Y. Liu, O. du Roure, A. Lindner, and D. Saintillan, Signatures of elastoviscous buckling in the dilute rheology of stiff polymers, in preparation
❑ Y. Liu, K. Zografos, J. Fidalgo, C. Duchene, C. Quintard, T. Darnige, V. Filipe, S. Huille, O. du Roure, M. Oliveira, A. Lindner, Optimized hyperbolic microchannels for the mechanical characterization of bio-particles, in preparation
❑ N. Quennouz, M. Shelley, O. du Roure, A. Lindner, Transport and buckling dynamics of an elastic fiber in a viscous cellular flow, J. Fluid Mech. (2015)