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)