Vesicles and Red Blood Cells
in Microcapillary FlowsGerhard Gompper and Hiroshi Noguchi∗
Institut für Festkörperforschung, and Institute for Advanced Simulations,Forschungszentrum Jülich, Germany∗

Institute for Solid State Physics, University of Tokyo, Japan

Soft Matter Hydrodynamics
Cells and vesicles in flow:• Red blood cells in microvessels:Diseases such as diabetes reduce deformability of red blood cells!

Soft Matter Hydrodynamics
Example: Flow behavior of malaria-infected red blood cells in microchannels
Just after infection:

Late stage:

Diameter: 8 µm

6 µm

J.P. Shelby et al., Proc. Natl. Acad. Sci. 100 (2003)

4 µm

2 µm

Mesoscale Flow Simulations
Complex fluids: length- and time-scale gap between• atomistic scale of solvent• mesoscopic scale of dispersed particles (colloids, polymers, membranes)−→ Mesoscale Simulation Techniques
Basic idea:

Examples:• drastically simplify dynamics• Lattice Boltzmann Method (LBM)

on molecular scale• respect conservation laws formass, momentum, energy• DissipativeParticle

Dynamics(DPD)• Multi-Particle-Collision Dynamics(MPC)

Alternative approach: Hydrodynamic interactions via Oseen tensor

Mesoscale Hydrodynamics Simulations
Multi-Particle-Collision Dynamics (MPC)• coarse grained fluid• point particles• off-lattice method• collisions inside “cells”• thermal fluctuations

A. Malevanets and R. Kapral, J. Chem. Phys. 110 (1999)
A. Malevanets and R. Kapral, J. Chem. Phys. 112 (2000)

Multi-Particle Collision Dynamics (MPC)
Flow dynamics: Two step process
Streaming• ballistic motionri(t + h) = ri(t) + vi(t)hMulti-Particle Collision Dynamics (MPC)
Flow dynamics: Two step process
Streaming

Collisioni• ballistic motionri(t + h) = ri(t) + vi(t)h• mean velocity per cellPni
1v̄i(t) = ni j∈Ci vj (t)• rotation of relative velocity by angle αvi′ = v̄i + D(α)(vi − v̄i)

Mesoscale Flow Simulations: MPC• Lattice of collision cells: breakdown of Galilean invariance• Restore Galilean invariance exactly: random shifts of cell lattice

T. Ihle and D.M. Kroll, Phys. Rev. E 63 (2001)

Mesoscale Flow Simulations: MPC• Lattice of collision cells: breakdown of Galilean invariance• Restore Galilean invariance exactly: random shifts of cell lattice

T. Ihle and D.M. Kroll, Phys. Rev. E 63 (2001)

Low-Reynolds-Number Hydrodynamics• Reynolds number Re = vmaxL/ν ∼ inertia forces / friction forces
For soft matter systems with characteristic length scales of µm:
Re ≃ 10−3• Schmidt number Sc = ν/D ∼ momentum transp. / mass transp.
Gases: Sc ≃ 1, liquids: Sc ≃ 103
1000

α=130,ρ=30α=130, ρ=5α=90, ρ=5α=45, ρ=5α=15, ρ=5Sc

100
1010.10.010.1h110M. Ripoll, K. Mussawisade, R.G. Winkler and G. Gompper, Europhys. Lett. 68 (2004)

Other MPC Methods• Anderson thermostat (MPC-AT-a):
Choose new relative velocities from Maxwell-Boltzmann distribution• Angular-momentum conservation (MPC-AT+a):
Modify collision rule to conserve angular momentum• Importance of angular-momentum conservation:
Rotating fluid drop with different viscosity in circular Couette flowΩ

R1dAngularvelocity

yR2vθ/Ω0a10

5+a-aprofile:

2m1/m0=50x05r/a10H. Noguchi, N. Kikuchi, G. Gompper, Europhys. Lett. 78 (2007); I.O. Götze, H. Noguchi, G. Gompper, Phys. Rev. E 76 (2007)

Membranes
Hydrodynamics of Membranes and Vesicles

Equilibrium Vesicle Shapes
Minimize curvature energy for fixed area A = 4πR02 and reduced volume V ∗ = V /V0, whereV0 = 4πR03/3:

stomatocyte

discoctyte

U. Seifert, K. Berndl, and R. Lipowsky, Phys. Rev. A 44 (1991)

prolateSimulations of Membranes
Modelling of membranes on different length scales:atomisticcoarse-grained

solvent-freetriangulated

Simulations of Membranes
Dynamically triangulated surfaces

Hard-core diameter σ
Tether length L: σ < L < √3 σ--> self-avoidance

Dynamic triangulation:

G. Gompper & D.M. Kroll (2004)

Membrane Hydrodynamics
Interaction between membrane and fluid:• Streaming step:bounce-back scattering of solvent particles on triangles• Collision step:
membrane vertices are included in MPC collisions

implies impenetrable membranewith no-slip boundary conditions.

H. Noguchi and G. Gompper, Phys. Rev. Lett. 93 (2004); Phys. Rev. E 72 (2005)

Membrane Hydrodynamics
Vesicle Dynamics in Shear Flow

Vesicles in Shear Flow
Parameter: shear rate γ̇
Variables:• Reduced volume V ∗• Shape• Membrane viscosity ηmb• Internal viscosity ηinlow viscosity

Behavior in shear flow:• Tank-treading• Tumbling• Swinging (vacillating-breathing, trembling)• Shape transformations

high viscosity

Swinging, Vacillating-breathing, Tremblingγ̇ = 1.8s−1γ̇ = 1.7s−1

Transition fom tumbling toswinging

withincreasing

shear rate γ̇
V. Kantsler and V. Steinberg,
Phys. Rev. Lett. 96 (2006)

inclination angle θ
Theory: C. Misbah, Phys. Rev. Lett. 96 (2006); H. Noguchi and G. Gompper, Phys. Rev. Lett. 98 (2007);
P.M. Vlahovska and R.S. Garcia, Phys. Rev. E 78 (2007); V.V. Lebedev et al., Phys. Rev. Lett. 99 (2007) ...

Swinging of Fluid Vesicles: Theory
Shape dynamics:

Phase diagram:

Mechanism:→ tumbling
H. Noguchi and G. Gompper, Phys. Rev. Lett. 98 (2007)→ tank-treadingVesicles with Viscosity Contrast in Bulk
• Two-dimensional vesicles in shear
flow(a) Tank-treading:• Employ MPC-AT+a (with angular15momentum conservation)• Change viscosity contrast λ by va-(b) Swinging:rying mass of fluid particles
25(c) Tumbling:

This work:
Beaucourt:
KS theory:20θ [◦ ]

15
100550-50246810

λS. Messlinger, B. Schmidt, H. Noguchi and G. Gompper, Phys. Rev. E 80 (2009)

1015

202530 tγ̇

Vesicles with Viscosity Contrast near Wall

1.08
1.04

FL Rp /(kB T )Vesicle in gravitational field near wall:

100



1.00
10rcm
FG0.96
ycm

111111111111111
000000000000000
000000000000000
111111111111111
000000000000000
111111111111111
(a)

λ=1:
λ=2:
λ=3:
λ=4:
λ=7:
λ = 10 :Oseen:−2ycm:(a)10.50.92

2(b)
180

Lift force FL balanced by gravitational

Lift force depends on viscosity

Simulation
Oseen

160
1402/(kB T Rp )FL ycmforce FG

contrast λ = ηin/ηout

5ycm /Rp

120
100
80
60
40
20002

46
λ

S. Messlinger, B. Schmidt, H. Noguchi and G. Gompper, Phys. Rev. E 80 (2009)

810Membrane Hydrodynamics
Vesicle and Cells in Capillary FlowCapillary Flow: Fluid Vesicles• small flow velocities: vesicle axis perpendicular to capillary axis −→ no axial symmetry!
• discocyte-to-prolate transition with increasing flow
H. Noguchi and G. Gompper, Proc. Natl. Acad. Sci. USA 102 (2005)

Capillary Flow: Red Blood Cells• Spectrin network induces shear elasticity µ of composite membrane• Elastic parameters: κ/kB T = 50, µR02/kB T = 5000

Capillary Flow: Elastic Vesicles
Elastic vesicle:• curvature and shear elasticity (κ = 20 kB T , µ = 110 kB T /R02 )• model for red blood cells

parachute shape

Capillary Flow: Elastic Vesicles
Elastic vesicle:• curvature and shear elasticity• model for red blood cells

Tsukada et al., Microvasc. Res. 61 (2001)

Capillary Flow: Red Blood Cells
Shear elasticity suppresses prolate shapes (large deformations)
Flow velocity at discocyte-to-parachute transition

bending rigidity
Implies for RBCs: vtrans ≃ 0.2mm/s for Rcap = 4.6µm

shear modulus

RBC Clustering & Alignment in Flow
Physiological conditions: Hematocrit (volume fraction of RBCs) H = 0.45
Lower in narrow capillaries HT = 0.1...0.2
Therefore: Hydrodynamic interactions between RBCs very important
Note: No direct attractive interactions considered!

RBC Clustering & Alignment in Flow
Low hematocrit HT :• Single vesicles more deformed → move faster• Effective hydrodynamic attraction stabilizes clusters
J.L. McWhirter, H. Noguchi, G. Gompper, Proc. Natl. Acad. Sci. 106 (2009)

RBC Clustering & Alignment in Flow
Low hematocrit HT :G(z*nb)6

v*0=7.7=10=104

Positional correlation function
a20012345z*nb

0.7

Probability for cluster size ncl

0.6P(ncl)

0.5
0.4

b0.3
0.2
0.1

Clustering tendency increases with

001234ncl

567increasing flow velocity

RBC Clustering & Alignment in Flow
Low hematocrit HT :G(z*nb)6

v*0=7.7=10=104

Positional correlation function
a2
00123

4

5

z*nb

0.7

Probability for cluster size ncl

0.6P(ncl)

0.5
0.4

b0.3
0.2
0.1

Clustering tendency increases with

001234ncl

567increasing flow velocity

RBC Clustering & Alignment in Flow
High hematocrit HT :disordered discocytealigned parachute

zig-zag slipper

J.L. McWhirter, H. Noguchi, G. Gompper, Proc. Natl. Acad. Sci. 106 (2009)

RBC Clustering & Alignment in Flow
High hematocrit HT :disordered discocytealigned parachute

zig-zag
Skalak, Science (1969)

Clustering & Alignment in Flow
Phase diagram:Hematocrit HT = 0.28/L∗ves
Transition to zig-zag phase despite
higher flow resistance than alignedparachute phase!
2468

10flow velocity v0∗12Vesicles in Structured Channels

Vesicle motion through zig-zag shapedchannel (Lx = 100µm):
time-dependent flow

Vesicles in Structured Channels

Vesicle motion through zig-zag shaped
channel (Lx = 100µm):
time-dependent flow

H. Noguchi, G. Gompper, L. Schmid, A. Wixforth, and T. Franke, EPL 89 (2010)

Vesicles in Structured Channels

s1s2

s4
e4

e3
s3e2
e1

∆l /<l >0.2lye10.3

0.1

Large reduced volume V :∗e2• Fast flows: Symmetric shape oscillations

0s1

0.8lxV*0.91s3s4• Slow flows: Orientational oscillations
Smaller reduced volume V :∗s2e3• Symmetric double tail• Asymmetric single taile4e2
20µm

Summary• Mesoscale simulation techniques are powerful tool to bridge thelength- and time-scale gap in complex fluids• Multi-particle-collision dynamics well suited for hydrodynamics ofembedded particles: colloids, polymers, vesicles, RBCs
• Vesicles in shear flow: tank-treading, tumbling, swinging, lift force• Red blood cells in capillary flow: shear elasticity implies parachuteshapes, hydrodynamic clustering and alignment• Vesicles in structured channels: single- and double-tailed shapes

Review: G. Gompper, T. Ihle, D.M. Kroll, R.G. Winkler, Adv. Polym. Sci. 221, 1 (2009)