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Tetsuro Tsuji (Kyoto University, Japan)

 The 33rd International Symposium on Rarefied Gas Dynamics (hosted by the German Aerospace Center (DLR))

July 15 - 19 (2024) @ Tagungs- und Veranstaltungshaus Alte Mensa, Göttingen, Germany �(presentation on July 16 13:30-14:10, Micro- and Nanoscale Flows I)

T. Tsuji, S. Mei & S. Taguchi, “Thermo-osmotic slip flows around a thermophoretic microparticle characterized by optical trapping of tracers”, Phys. Rev. Appl. 20, 054061 (2023) Featured in Physics @ APS

Thermally-induced Microflows and

Microparticle Migration

initial state

heating @ 0 s

thermo-osmotic slip flow

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Table of contents

thermal flows

  • Introduction

  • Characterization of thermal flows

  • Kinetic model of thermophoresis

  • Application of thermal flows (& thermophoresis)

speaker: Tetsuro Tsuji (Kyoto University)

title: Thermally-induced Microflows and Microparticle Migration

Phys. Rev. Appl. (2023)

  • what are thermally-induced microflows & thermophoresis?
  • difficulties and challenges?

  • thermal flows around a microparticle in liquid
  • use of the optical trapping of tracers

  • linear Boltzmann equation for a Brownian particle
  • asymptotic analysis and MD simulation

  • thermophoresis in on-chip microfluidic devices
  • microparticle filtration and separation

Phys. Rev. Appl. (2018)

×15

50 µm

mean

flow

Physica A (2018)

3 of 69

Introduction

What is thermally-induced microflows ( = thermal flows)?

  • incompressible NS system is decoupled from temperature fields
  • fluid flows cannot be driven by heating without external force
  • thermal convection (buoyancy-force-driven) → suppressed in small scale

advection

decoupled

flow

NS system

heat

energy eq.

downsizing…

huge resistance

low efficiency

1

  • surface effect ∝ L2
  • bulk effect ∝ L3

characteristic length

surface effect

is important

small scale

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Introduction

What is thermally-induced microflows ( = thermal flows)?

  • surface effect ∝ L2
  • bulk effect ∝ L3

characteristic length

surface effect

is important

characteristic length

mean free path

Kn =

L

= O(1)

Knudsen number (Kn) is a good indicator of

the significance of surface properties

case of gases

gas-surface interaction

rarefied gas

(or molecular gas)

1

large Kn → collision with surfaces > collision with gas molecules

small scale

cf. Boltzmann equation

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Introduction

Sone, Phys. Fluids (1991) http://hdl.handle.net/2433/120983

heated plate

heated plate

ambient pressure

LOW pressure

thermal convection

thermal creep

vacuum

heater

What is thermally-induced microflows ( = thermal flows)?

低温

高温

no-slip b.c. for ordinary viscous fluids

flow over a boundary with

a temperature gradient

thermal creep for rarefied gases

Maxwell (1879)

thermal-stress slip flow

nonlinear-thermal-stress flow

thermal edge flow

other types of

thermal flows

Sone (2007)

2

evidence of

thermal creep

windmill-type flow indicator

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Introduction

vacuum

What is thermally-induced microflows ( = thermal flows)?

低温

高温

no-slip b.c. for ordinary viscous fluids

flow over a boundary with

a temperature gradient

thermal creep for rarefied gases

Maxwell (1879)

thermal-stress slip flow

nonlinear-thermal-stress flow

thermal edge flow

other types of

thermal flows

Sone (2007)

2

applications ?

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Introduction

Sugimoto & Sone (2005)

Taguchi & Tsuji (2022)

  • Knudsen pump

(thermally-driven pump without mechanically moving parts)

X. Wang, et al. Knudsen Pumps: a Review. Microsystems Nanoeng. (2020)

application (ex. 1)

What is thermally-induced microflows ( = thermal flows)?

低温

高温

no-slip b.c. for ordinary viscous fluids

flow over a boundary with

a temperature gradient

thermal creep for rarefied gases

Maxwell (1879)

one-way flow

thermal-stress slip flow

nonlinear-thermal-stress flow

thermal edge flow

other types of

thermal flows

Sone (2007)

thermal-edge flow

one-way flow

2

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Introduction

flow over a boundary with

a temperature gradient

What is thermally-induced microflows ( = thermal flows)?

no-slip b.c. for ordinary viscous fluids

thermal creep for rarefied gases

Maxwell (1879)

thermal-stress slip flow

nonlinear-thermal-stress flow

thermal edge flow

other types of

thermal flows

Sone (2007)

S. Takata & Y. Sone (1995)

  • thermophoresis

force

≈ Kn

force

force toward

cold side

application (ex. 2)

2

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Introduction

flow over a boundary with

a temperature gradient

What is thermally-induced microflows ( = thermal flows)?

no-slip b.c. for ordinary viscous fluids

thermal creep for rarefied gases

Maxwell (1879)

thermal-stress slip flow

nonlinear-thermal-stress flow

thermal edge flow

other types of

thermal flows

Sone (2007)

S. Takata & Y. Sone (1995)

  • thermophoresis

force

≈ Kn

force

force toward

cold side

application (ex. 2)

focus of the talk

deeper understanding of thermophoresis

me hoping to

understand

thermophoresis

2

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Introduction

What is thermally-induced microflows ( = thermal flows)?

低温

高温

flow over a boundary with

a temperature gradient

no-slip b.c. for ordinary viscous fluids

thermal creep for rarefied gases

Maxwell (1879)

thermal-stress slip flow

nonlinear-thermal-stress flow

thermal edge flow

other types of

thermal flows

Sone (2007)

fluid-dynamic limit

Kn = ≪ 1

L

small

generalized slip flow theory (Sone, 2007)

fluid-dynamic-type system (e.g. Stokes set) + slip b.c.

How is thermally-induced microflows analyzed?

flow velocity

unit tangent

slip coef.

temperature

2

/ 4

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Motivation

~ difficulties & challenges ~

me hoping to

understand

thermophoresis

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Motivation

低温

高温

case of gas

  • theory
  • experiment (?)

flow velocity

unit tangent

slip coef.

temperature

generalized slip flow theory

~ difficulties & challenges ~

& various computational tools

3

e.g., DSMC

thermal flows

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Motivation

低温

高温

Arkilic, et al., (2001) Colin, et al., (2004) Graur, et al., (2009) Yamaguchi, et al., (2011) Silva, et al., (2018)

Perrier, et al., (2019) Zhang et al., (2023), etc. .

slip coef. measurement

(NO flow profile)

case of gas

  • theory
  • experiment

mass-flow rate measurements

flow velocity

unit tangent

slip coef.

temperature

generalized slip flow theory

~ difficulties & challenges ~

3

Rojas Cardenas, et al., (2011, 2013), Yamaguchi, et al., (2014, 2016)

thermal slip

shear slip

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Motivation

低温

高温

Fratantonio et al., (2018, 2020)

laser-induced fluorescence

case of gas

  • theory
  • experiment

mass-flow rate measurements

molecular tagging velocimetry

flow velocity

unit tangent

slip coef.

temperature

generalized slip flow theory

~ difficulties & challenges ~

3

flow profile measurement

(NOT for thermal flow)

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Motivation

低温

高温

case of gas

  • theory
  • experiment

flow velocity

unit tangent

slip coef.

temperature

generalized slip flow theory

mass-flow rate measurements

molecular tagging velocimetry

~ difficulties & challenges ~

me hoping to

understand

thermophoresis

challenges

me hoping to

understand

thermophoresis

me hoping to

understand

thermophoresis

3

So far, NO observation of thermal flow profile is made…,

force

and so do thermal flows around thermophoretic particle!!!

naïve question: what is the situation for liquids?

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Motivation

低温

高温

case of gas

  • theory
  • experiment

flow velocity

unit tangent

slip coef.

temperature

generalized slip flow theory

mass-flow rate measurements

molecular tagging velocimetry

~ difficulties & challenges ~

me hoping to

understand

thermophoresis

challenges

me hoping to

understand

thermophoresis

me hoping to

understand

thermophoresis

3

Kavokine, Netz, Bocquet,

Fluids at the Nanoscale: From Continuum to Subcontinuum Transport

Ann. Rev. Fluid. Mech. (2021)

NS system seems to be valid much smaller scale than gas,

but still there is a threshold.

naïve question: what is the situation for liquids?

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Motivation

低温

高温

case of liquid

case of gas

  • theory
  • experiment

flow velocity

unit tangent

slip coef.

temperature

generalized slip flow theory

mass-flow rate measurements

molecular tagging velocimetry

~ difficulties & challenges ~

various parameters

4

treatment of

fluid and surface

are different

collision law

(molecular beam experiment)

Kinefuchi, et al., (2017)

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Motivation

case of liquid

  • theory

case of gas

  • theory
  • experiment

flow velocity

unit tangent

slip coef.

temperature

generalized slip flow theory

mass-flow rate measurements

molecular tagging velocimetry

~ difficulties & challenges ~

various parameters

thermal slip coef. K

excess enthalpy in B.L.

Churaev, Derjaguin, Muller

Surface Forces (1987)

same

(formally)

local excess enthalpy

slip coef.

distance from the boundary surface

difficult to measure…

cf.

4

collision law

(molecular beam experiment)

Kinefuchi, et al., (2017)

viscosity

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Motivation

低温

高温

case of liquid

  • theory

case of gas

  • theory
  • experiment

flow velocity

unit tangent

slip coef.

temperature

generalized slip flow theory

mass-flow rate measurements

molecular tagging velocimetry

~ difficulties & challenges ~

various parameters

thermal slip coef. K

excess enthalpy in B.L.

Churaev, Derjaguin, Muller

Surface Forces (1987)

same

(formally)

Continuum theory, Fayolle, Bickel, Würger (2008)

effect of surface charge

effect of hydrophilicity

MD simulation, Fu, Merabia, Joly (2017)

4

collision law

(molecular beam experiment)

Kinefuchi, et al., (2017)

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Motivation

低温

高温

case of liquid

  • theory

case of gas

  • theory
  • experiment

flow velocity

unit tangent

slip coef.

temperature

generalized slip flow theory

mass-flow rate measurements

molecular tagging velocimetry

~ difficulties & challenges ~

various parameters

thermal slip coef. K

excess enthalpy in B.L.

Churaev, Derjaguin, Muller

Surface Forces (1987)

Continuum theory, Fayolle, Bickel, Würger (2008)

effect of surface charge

effect of hydrophilicity

MD simulation, Fu, Merabia, Joly (2017)

same

(formally)

4

collision law

(molecular beam experiment)

Kinefuchi, et al., (2017)

  • experiment

observation of slip flow in thermophoresis

no information on thermal flow profile

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Motivation

低温

高温

case of liquid

  • theory

case of gas

  • theory
  • experiment

flow velocity

unit tangent

slip coef.

temperature

generalized slip flow theory

mass-flow rate measurements

molecular tagging velocimetry

~ difficulties & challenges ~

various parameters

thermal slip coef. K

excess enthalpy in B.L.

Churaev, Derjaguin, Muller

Surface Forces (1987)

Continuum theory, Fayolle, Bickel, Würger (2008)

effect of surface charge

effect of hydrophilicity

MD simulation, Fu, Merabia, Joly (2017)

same

(formally)

4

collision law

(molecular beam experiment)

Kinefuchi, et al., (2017)

  • experiment

observation of slip flow in thermophoresis

thermal flow profile

Fränzl & Cichos (2022)

heated by laser

thermal creep (or thermo-osmosis) on flat surface

Bregulla, et al. (2016)

Fränzl & Cichos (2022)

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Motivation

低温

高温

case of liquid

  • theory

~ difficulties & challenges ~

various parameters

thermal slip coef. K

excess enthalpy in B.L.

Churaev, Derjaguin, Muller

Surface Forces (1987)

Continuum theory, Fayolle, Bickel, Würger (2008)

effect of surface charge

effect of hydrophilicity

MD simulation, Fu, Merabia, Joly (2017)

same

(formally)

4

  • experiment

observation of slip flow in thermophoresis

thermal flow profile

Fränzl & Cichos (2022)

heated by laser

thermal creep (or thermo-osmosis) on flat surface

Bregulla, et al. (2016)

Fränzl & Cichos (2022)

me hoping to

understand

thermophoresis

challenges

force

but still NO thermal flows profile around thermophoretic particle!!!

Thermal flow profile can be investigated in liquid.

This situation is better than the case of gas …,

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Motivation

~ difficulties & challenges ~

me hoping to

understand

thermophoresis

Let’s observe thermal flow around thermophoretic particle and check if thermal flow is indeed the origin of thermophoresis

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Table of contents

  • Introduction

  • Characterization of thermal flows

  • Kinetic model of thermophoresis

  • Application of thermal flows (& thermophoresis)

speaker: Tetsuro Tsuji (Kyoto University)

title: Thermally-induced Microflows and Microparticle Migration

Phys. Rev. Appl. (2023)

  • what are thermally-induced microflows & thermophoresis?
  • difficulties and challenges?

  • thermal flows around a microparticle in liquid
  • use of the optical trapping of tracers

  • linear Boltzmann equation for a Brownian particle
  • asymptotic analysis and MD simulation

  • thermophoresis in on-chip microfluidic devices
  • microparticle filtration and separation

Phys. Rev. Appl. (2018)

×15

50 µm

mean

flow

Physica A (2018)

25 of 69

Introduction

Thermophoresis → Motion along temperature gradient

ions, biomolecules, nano- & microparticles

  • Thermophoretic velocity V

thermophoretic mobility

  • fluid/particle dependent
  • DT > 0 → toward cold
  • DT < 0 → toward hot

T: fluid temperature

100 μm

hot

cold

cold

hot

Tsuji, et al. Micro & Nano Lett., (2017)

in Water

in NaOH

Piazza, Soft Matter (2008)

Wurger, Rep. Prog. Phys. (2010)

Mechanism…?

1

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Introduction

thermo-osmotic

slip flow

counteraction

particle motion

(thermophoresis)

Thermophoresis of colloids

gas Takata, et al (1995), Sone (2007) etc.

liquid Ruckenstein (1981), Fayolle, et al (2008) etc.

theory

slip flow

Piazza, Soft Matter (2008)

Wurger, Rep. Prog. Phys. (2010)

Molecular gas dynamics

Macroscopic, Ionic, Electric force

Mechanism…?

Thermophoresis → Motion along temperature gradient

ions, biomolecules, nano- & microparticles

COLD

HOT

flow velocity

unit tangential vector

slip coef.

temperature

SLIP BOUNDARY CONDITION

1

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Introduction

thermo-osmotic

slip flow

counteraction

particle motion

(thermophoresis)

Thermophoresis of colloids

gas Takata, et al (1995), Sone (2007) etc.

liquid Ruckenstein (1981), Fayolle, et al (2008) etc.

theory

Piazza, Soft Matter (2008)

Wurger, Rep. Prog. Phys. (2010)

Mechanism…?

Thermophoresis → Motion along temperature gradient

ions, biomolecules, nano- & microparticles

Weinert & Braun, PRL (2008)

exp

Evaluation of thermo-osmotic slip flows around thermophoretic particles

slip flow

COLD

HOT

challenge

1

Fränzl & Cichos,

Nature Commun. (2022)

difficulty

  • few tracers in region-of-interest…
  • target particle must be < 10 µm

to suppress thermal convection

e.g., near surface

(thermophoresis & Brownian motion of tracers)

Optical trapping

of TRACERS� (flow indicators)

This study

laser

tracer

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Phys. Rev. Appl. (2023)

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CONCEPT OF EXPERIMENT

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Trap on a circular path

  • radially constrained
  • free to move in circumferential direction

Trap on a circular path

  • radially constrained
  • free to move in circumferential direction

CONCEPT OF EXPERIMENT

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Experiment

target

optical trap on

circular path

dia.

2

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Control experiment

Tracer

thermophoresis

20 μm

laser heating

no trapping laser

no target particle

laser heating

20 μm

Thermophoresis

of target particle�(d = 7 µm)

no trapping laser

with target particle

(freely movable)

with trapping laser

no target particle

  • Control experiment confirms that target & tracers are both directed toward colder region

target particle

hot

cold

heating start

t = 0 s

3

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Main result

target particle

(target particle is fixed)

Hypothesis

If the slip flow exists, the tracers are directed to hotter side

4

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Main result

R = 4.4 μm

R = 5.4 μm

R = 6.5 μm

  • Tracers move toward HOT
  • Small trap radius R (=near sphere surface)FASTER

hot

cold

start heating

t = 0 s

target particle

Hypothesis

If the slip flow exists, the tracers are directed to hotter side

(target particle is fixed)

4

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Main result

R = 4.4 µm

R = 5.4 µm

R = 6.5 µm

HOT

intensity distribution

t = 25 s

θ/π

target particle

Hypothesis

If the slip flow exists, the tracers are directed to hotter side

(target particle is fixed)

  • Tracers move toward HOT
  • Small trap radius R (=near sphere surface)FASTER

COLD

COLD

symbol: experiment

curve: model fitting

4

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Slip coefficient

R = 4.4 µm

R = 5.4 µm

R = 6.5 µm

intensity distribution

t = 25 s

θ/π

  • Tracers move toward HOT
  • Small trap radius R (=near sphere surface)FASTER

Curve fitting model

  • Axisymmetric state
  • Heat conduction eq. with linear background temperature gradient
  • Stokes Eq. & slip B.C.
  • Drift-diffusion Eq. in θ direction

HOT

COLD

COLD

symbol: experiment

curve: model fitting

estimation of slip coef.

from experimental data

slip B.C.

slip coef.

5

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Slip coefficient

R = 4.4 µm

R = 5.4 µm

R = 6.5 µm

intensity distribution

t = 25 s

θ/π

  • Tracers move toward HOT
  • Small trap radius R (=near sphere surface)FASTER

Curve fitting model

  • Axisymmetric state
  • Heat conduction eq. with linear background temperature gradient
  • Stokes Eq. & slip B.C.
  • Drift-diffusion Eq. in θ direction

HOT

COLD

COLD

symbol: experiment

curve: model fitting

5

/ 7

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Slip coefficient

R = 4.4 µm

R = 5.4 µm

R = 6.5 µm

intensity distribution

t = 25 s

θ/π

  • Tracers move toward HOT
  • Small trap radius R (=near sphere surface)FASTER

Curve fitting model

  • Axisymmetric state
  • Heat conduction eq. with linear background temperature gradient
  • Stokes Eq. & slip B.C.
  • Drift-diffusion Eq. in θ direction

HOT

COLD

COLD

symbol: experiment

curve: model fitting

slip B.C.

5

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Slip coefficient

R = 4.4 µm

R = 5.4 µm

R = 6.5 µm

intensity distribution

t = 25 s

θ/π

  • Tracers move toward HOT
  • Small trap radius R (=near sphere surface)FASTER

Curve fitting model

  • Axisymmetric state
  • Heat conduction eq. with linear background temperature gradient
  • Stokes Eq. & slip B.C.
  • Drift-diffusion Eq. in θ direction

HOT

COLD

COLD

symbol: experiment

curve: model fitting

tracer density

diffusion coef.

drift

thermophoretic velocity

flow velocity

5

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Phys. Rev. Appl. (2023)

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Slip coefficient

intensity distribution

t = 25 s

θ/π

Curve fitting model

  • Axisymmetric state
  • Heat conduction eq. with linear background temperature gradient
  • Stokes Eq. & slip B.C.
  • Drift-diffusion Eq. in θ direction

HOT

COLD

COLD

tracer density

diffusion coef.

drift

thermophoretic velocity

flow velocity

flow speed − thermophoretic speed

diffusion

=

  1. Obtain γ by fitting to exp data
  2. Compute slip coefficient

related to slip coef.

Slip coef. K

-0.50 μm2/s K

-0.45 μm2/s K

-0.44 μm2/s K

Experiments with different laser power

(78 mW)

(47 mW)

(30 mW)

5

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Slip coefficient

intensity distribution

t = 25 s

θ/π

Curve fitting model

  • Axisymmetric state
  • Heat conduction eq. with linear background temperature gradient
  • Stokes Eq. & slip B.C.
  • Drift-diffusion Eq. in θ direction

HOT

COLD

COLD

tracer density

diffusion coef.

drift

thermophoretic velocity

flow velocity

Slip coef. K

-0.50 μm2/s K

-0.45 μm2/s K

-0.44 μm2/s K

Experiments with different laser power

(78 mW)

(47 mW)

(30 mW)

-0.58 μm2/s K

K obtained from

thermophoresis

good agreement

control experiment

5

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Effect of surface modification

SO FAR

carboxylate-modified

amine-modified

Effect of

surface modification?

ζ = - 46.0 mV

ζ = - 23.7 mV

target particle

Hypothesis

If the slip flow exists, the tracers are directed to hotter side

6

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Effect of surface modification

θ/π

hot

ζ = - 46.0 mV

ζ = - 23.7 mV

Weak zeta potential weak flow

Qualitatively consistent

hot

hot

cold

cold

heating

t = 0 s

7 µm

  • Charged particle

Fayolle, Bickel, Wurger, PRE (2008)

thin EDL (electric double layer) limit, weak surface charge (Debye-Hückel approx.)

surface potential

Slip velocity

viscosity

electric permittivity

thermal conductivity

heating

t = 0 s

6

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Short summary

  • thermally-induced microflows along the surface of microparticles drive thermophoresis in liquids
  • similar experiments for gases (i.e. thermal flow profile measurement, thermal flow around microparticles) are open problems & future work

7

T. Tsuji, S. Mei & S. Taguchi, �“Thermo-osmotic slip flows around a thermophoretic microparticle characterized by optical trapping of tracers”, �Phys. Rev. Appl. 20, 054061 (2023) �Featured in Physics @ APS

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Phys. Rev. Appl. (2023)

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Table of contents

  • Introduction

  • Characterization of thermal flows

  • Kinetic model of thermophoresis

  • Application of thermal flows (& thermophoresis)

speaker: Tetsuro Tsuji (Kyoto University)

title: Thermally-induced Microflows and Microparticle Migration

Phys. Rev. Appl. (2023)

  • what are thermally-induced microflows & thermophoresis?
  • difficulties and challenges?

  • thermal flows around a microparticle in liquid
  • use of the optical trapping of tracers

  • linear Boltzmann equation for a Brownian particle
  • asymptotic analysis and MD simulation

  • thermophoresis in on-chip microfluidic devices
  • microparticle filtration and separation

Phys. Rev. Appl. (2018)

×15

50 µm

mean

flow

Physica A (2018)

45 of 69

Effect of mass in thermophoresis

mechanism based on thermal flows neglects the effect of mass

force

  • Navier-Stokes system �+ slip b.c.
  • Boltzmann system

Epstein (1929), Brock (1962),

Takata & Sone (1995), Sone (2007) etc.

  • light polymer tends to show smaller DT
  • change the sign (negative thermophoresis)

Stadelmaier & Kohler (2009)

  • thermophoresis of polymers in alkanes
  • molecular weight Mw dependence

DT(Mw) DT() (normalized)

molecular weight

different mechanism other than thermal flow?

1

/ 5

Physica A (2018)

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Effect of mass in thermophoresis

■Adiabatic piston problem Gruber & Piasecki (1999)

  • Fix the piston ➡ No net force
  • Mass of the piston M
  • Mass of gas molecule m

  • Freely movable piston ➡ Motion toward hot side

toward hotter side

2

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Physica A (2018)

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Effect of mass in thermophoresis

■Adiabatic piston problem Gruber & Piasecki (1999)

  • Fix the piston ➡ No net force
  • Mass of the piston M
  • Mass of gas molecule m

  • Freely movable piston ➡ Motion toward hot side

toward hotter side

asymmetry of fluctuation

negative thermophoresis?

2

/ 5

Physica A (2018)

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Effect of mass in thermophoresis

■Adiabatic piston problem Gruber & Piasecki (1999)

  • Mass of the piston M
  • Mass of gas molecule m

asymmetry of fluctuation

negative thermophoresis?

COLD

HOT

HOT

COLD

HOT

HOT

COLD

COLD

analogy

2

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Physica A (2018)

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Linear Boltzmann equation

HOT

HOT

COLD

COLD

  • mass of nanoparticle M
  • mass of gas molecule m

mixture with large mass difference

velocity distribution function of nanoparticles

nanoparticle

collision term between nanoparticles and gas molecules

gain

loss

nanoparticle

velocity of nanoparticles

gain

loss

velocity of gas molecules

direction of relative position

linear

Boltzmann-type

equation

3

velocity distribution function of gas molecules

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Linear Boltzmann equation

HOT

HOT

COLD

COLD

  • mass of nanoparticle M
  • mass of gas molecule m

mixture with large mass difference

given function (gas is not disturbed by the nanoparticle)

linear

Boltzmann-type

equation

velocity distribution function of nanoparticles

nanoparticle

gain

loss

nanoparticle

(non-local collision)

“grazing” collision (due to large mass difference)

velocity of nanoparticles

gain

loss

collision term between nanoparticles and gas molecules

3

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Linear Boltzmann equation

velocity of nanoparticles

given function (gas is not disturbed by the nanoparticle)

linear

Boltzmann-type

equation

velocity distribution function of nanoparticles

gain

loss

(non-local collision)

“grazing” collision (due to large mass difference)

gain

loss

collision term between nanoparticles and gas molecules

mass ratio (small)

(non-dimensional) temperature gradient (small)

asymptotic analysis

3

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Asymptotic analysis

set of linear PDEs for Φi,j

diffusion scaling

4

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Asymptotic analysis

: differential operators

is a Maxwellian

set of linear PDEs for Φi,j

κ

λ

4

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Asymptotic analysis

density ratio

set of linear PDEs for Φi,j

mass of a nanoparticle

mass of gas molecules in the same volume as nanoparticle

4

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Asymptotic analysis

integration 2nd order eqs. by velocity space

Drift-diffusion Eq.

2nd order:

conservation of mass

0

set of linear PDEs for Φi,j

4

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Drift-diffusion equation

number density

drift velocity

(DT >0 → motion toward colder side)

heavy-mass limit

Duhr & Braun, Proc. Natl. Acad. Sci. USA (2006)

thermophoretic mobility

5

mass of gas molecules in

the same volume as nanoparticle

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Drift-diffusion equation

  • toward hot or cold?

5

HOT

HOT

COLD

COLD

?

?

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Drift-diffusion equation

  • toward hot or cold?

toward cold

toward hot

5

unit vector�in x1 direction

center of nanopartice

vector pointing

to surface

temperature gradient

nanoparticle diameter R → ∞

equivalent

toward hotter side

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Drift-diffusion equation

  • toward hot or cold?

toward cold

toward hot

Mw:molecular weight

weight dependence for macromolecules:

Stadelmaier & Kohler, (2008, 2009)

DT(Mw) DT() (normalized)

molecular weight

same functional form

  • similarity with experiments?

5

unit vector�in x1 direction

center of nanopartice

vector pointing

to surface

temperature gradient

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Table of contents

  • Introduction

  • Characterization of thermal flows

  • Kinetic model of thermophoresis

  • Application of thermal flows (& thermophoresis)

speaker: Tetsuro Tsuji (Kyoto University)

title: Thermally-induced Microflows and Microparticle Migration

Phys. Rev. Appl. (2023)

  • what are thermally-induced microflows & thermophoresis?
  • difficulties and challenges?

  • thermal flows around a microparticle in liquid
  • use of the optical trapping of tracers

  • linear Boltzmann equation for a Brownian particle
  • asymptotic analysis and MD simulation

  • thermophoresis in on-chip microfluidic devices
  • microparticle filtration and separation

Phys. Rev. Appl. (2018)

×15

50 µm

mean

flow

Physica A (2018)

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Application of thermophoresis

thermophoresis in on-chip microfluidic devices

  • concentration of dilute particles
  • selective transport of the mixture of particles

detection of nano/micro particles (cell, biomolecules, virus, etc.)

Kawaguchi, et al (2012) J. Phys.: Condense. Matter

1

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Application of thermophoresis

thermophoresis in on-chip microfluidic devices

  • concentration of dilute particles
  • selective transport of the mixture of particles

microfluidic device

microchannel

local heating

concept of selective transport by thermophoresis

DT > 0

DT < 0

Kawaguchi, et al (2012) J. Phys.: Condense. Matter

detection of nano/micro particles (cell, biomolecules, virus, etc.)

1

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Microfluidic device

  • PDMS microchannel rectangular cross-section
  • glass substrate with �thin-film Au electrodemicroheater (20 µm width)
  • pressure-difference controlresolution ~ 0.01 Pa

test section

(bottom view)

channel height ≈ 17 μm

2

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Observation of thermophoresis

×15 speed

50µm

  • polystyrene (PS)�(Φ=1µm, COOH modified)
  • tris-HCl solution�(10 mM)
  • no mean flow

electrode

themophobic

PS beads are

repelled from

hot region

Test section

electric current (0.34 W)

3

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Thermophoretic filtration

  • mean flow

×15 speed

50µm

pressure-

driven flow

local concentration of PS beads

thermophoretic velocity

mean flow

100-fold concentration

4

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Particle-flow control

  • PS particle flow control �at bifurcation channel
  • application to micro and �nano particles

electrode

×15 speed

electrode

×15 speed

microparticle

(Φ=1 µm)

nanoparticle

(Φ=100 nm)

Current OFF

Tsuji et al., Micromachines (2019)

5

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Thermophoresis of silica beads

×15 speed

50µm

Thermophilic

Attraction

to hot region

  • silica beads (Φ=1 µm, COOH modified)

×15 speed

50µm

  • silica beads (Φ=1 µm, no surface modification)

Thermophobic

Repulsion�from hot region

6

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Selective filtration

  • PS & silica mixed sol.(Φ=1µm, COOH modified)
  • tris-HCL 10 mM sol.
  • mean flow~4 µm/s

PS & silica

d = 1 μm

locally heated

pressure-driven flow

same experiment with different optical filter

PS & silica

d = 1 μm

invisible

PS

silica

Tsuji, et al., Phys. Rev. Appl. (2018a)

Tsuji, et al., Phys. Rev. Appl. (2018b)

PS

100 nm

laser heating

7

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Table of contents

Thank you for your attention

  • Introduction

  • Characterization of thermal flows

  • Kinetic model of thermophoresis

  • Application of thermal flows (& thermophoresis)

speaker: Tetsuro Tsuji (Kyoto University)

title: Thermally-induced Microflows and Microparticle Migration

Phys. Rev. Appl. (2023)

  • what are thermally-induced microflows & thermophoresis?
  • difficulties and challenges?

  • thermal flows around a microparticle in liquid
  • use of the optical trapping of tracers

  • linear Boltzmann equation for a Brownian particle
  • asymptotic analysis and MD simulation

  • thermophoresis in on-chip microfluidic devices
  • microparticle filtration and separation

Phys. Rev. Appl. (2018)

×15

50 µm

mean

flow

Physica A (2018)