A Mesoscopic Particle-based Method for Active-Nematics

Tyler Shendruk

Rudolf Peierls Centre forTheoretical PhysicsOxford University

Physics and Biology of ActiveSystems

23 June, 2015

Motivation

Microscopic Models

• Active colloids• Simulations of motile

microbes

Conclusion:

• Coarse-graining solvents successful in passive soft matter◦ We would like a “toolbox” of similar simulation techniques for active fluids

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www.tnshendruk.com

Motivation

Microscopic Models

• Active colloids• Simulations of motile

microbes

Continuum Models

• Active-nematichydrodynamics

Conclusion:

• Coarse-graining solvents successful in passive soft matter◦ We would like a “toolbox” of similar simulation techniques for active fluids

2 of 26

www.tnshendruk.com

Motivation

Microscopic Models

• Active colloids• Simulations of motile

microbes

Mescoscale Models

• Coarse-grain manymicroscopic entitieswithout calculatingmolecular interactions

Continuum Models

• Active-nematichydrodynamics

Conclusion:

• Coarse-graining solvents successful in passive soft matter◦ We would like a “toolbox” of similar simulation techniques for active fluids

2 of 26

www.tnshendruk.com

Motivation

Microscopic Models

• Active colloids• Simulations of motile

microbes

Mescoscale Models

• Coarse-grain manymicroscopic entitieswithout calculatingmolecular interactions

Continuum Models

• Active-nematichydrodynamics

Conclusion:

• Coarse-graining solvents successful in passive soft matter◦ We would like a “toolbox” of similar simulation techniques for active fluids

2 of 26

www.tnshendruk.com

Colloids in Active Fluids: Anomalous Microrheology andNegative Drag by Foffano, et. al.

Active-NematicHydrodynamic Equations to be Simulated

• Continuity equation Dtρ+ ρ∂iui = 0

• Momentum equation ρDtui = ∂j[Πviscij + Π

nemij + Π

actij

]

◦ viscous stress Πviscij◦ nematic elastic stress Πnemij◦ active stress Πactij = −ζactQij

• force dipole density• ζact positive for extensile and negative for contractile

• Orientation equation DtQij − Sij = ΓHij

◦ generalized nonlinear advection Sij◦ molecular field Hij

• Our goal:

◦ build a meso-scale algorithm to reproduce these active-nematic hydrodynamicson long time and length scales

• Review: recreate Navier-Stokes Equations by mesoscopic MPCD

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www.tnshendruk.com

Active-NematicHydrodynamic Equations to be Simulated

• Continuity equation Dtρ+ ρ∂iui = 0• Momentum equation ρDtui = ∂j

[Πviscij + Π

nemij + Π

actij

]

◦ viscous stress Πviscij◦ nematic elastic stress Πnemij◦ active stress Πactij = −ζactQij

• force dipole density• ζact positive for extensile and negative for contractile

• Orientation equation DtQij − Sij = ΓHij

◦ generalized nonlinear advection Sij◦ molecular field Hij

• Our goal:

◦ build a meso-scale algorithm to reproduce these active-nematic hydrodynamicson long time and length scales

• Review: recreate Navier-Stokes Equations by mesoscopic MPCD

3 of 26

www.tnshendruk.com

Active-NematicHydrodynamic Equations to be Simulated

• Continuity equation Dtρ+ ρ∂iui = 0• Momentum equation ρDtui = ∂j

[Πviscij + Π

nemij + Π

actij

]◦ viscous stress Πviscij◦ nematic elastic stress Πnemij◦ active stress Πactij = −ζactQij• force dipole density• ζact positive for extensile and negative for contractile

• Orientation equation DtQij − Sij = ΓHij

◦ generalized nonlinear advection Sij◦ molecular field Hij

• Our goal:

◦ build a meso-scale algorithm to reproduce these active-nematic hydrodynamicson long time and length scales

• Review: recreate Navier-Stokes Equations by mesoscopic MPCD

3 of 26

www.tnshendruk.com

Active-NematicHydrodynamic Equations to be Simulated

• Continuity equation Dtρ+ ρ∂iui = 0• Momentum equation ρDtui = ∂j

[Πviscij + Π

nemij + Π

actij

]◦ viscous stress Πviscij◦ nematic elastic stress Πnemij◦ active stress Πactij = −ζactQij• force dipole density• ζact positive for extensile and negative for contractile

• Orientation equation DtQij − Sij = ΓHij◦ generalized nonlinear advection Sij◦ molecular field Hij

• Our goal:

◦ build a meso-scale algorithm to reproduce these active-nematic hydrodynamicson long time and length scales

• Review: recreate Navier-Stokes Equations by mesoscopic MPCD

3 of 26

www.tnshendruk.com

Active-NematicHydrodynamic Equations to be Simulated

• Continuity equation Dtρ+ ρ∂iui = 0• Momentum equation ρDtui = ∂j

[Πviscij + Π

nemij + Π

actij

]◦ viscous stress Πviscij◦ nematic elastic stress Πnemij◦ active stress Πactij = −ζactQij• force dipole density• ζact positive for extensile and negative for contractile

• Orientation equation DtQij − Sij = ΓHij◦ generalized nonlinear advection Sij◦ molecular field Hij

• Our goal:◦ build a meso-scale algorithm to reproduce these active-nematic hydrodynamics

on long time and length scales

• Review: recreate Navier-Stokes Equations by mesoscopic MPCD

3 of 26

www.tnshendruk.com

Active-NematicHydrodynamic Equations to be Simulated

• Continuity equation Dtρ+ ρ∂iui = 0• Momentum equation ρDtui = ∂j

[Πviscij + Π

nemij + Π

actij

]◦ viscous stress Πviscij◦ nematic elastic stress Πnemij◦ active stress Πactij = −ζactQij• force dipole density• ζact positive for extensile and negative for contractile

• Orientation equation DtQij − Sij = ΓHij◦ generalized nonlinear advection Sij◦ molecular field Hij

• Our goal:◦ build a meso-scale algorithm to reproduce these active-nematic hydrodynamics

on long time and length scales

• Review: recreate Navier-Stokes Equations by mesoscopic MPCD

3 of 26

www.tnshendruk.com

Technique: Multi-Particle Collision Dynamics

Traditional MPCD

• Particle-based algorithm formoderate Péclet numbers.

Each point-like particle i has

1. position xi

2. velocity vi

3. mass mi

Each cell has

1. population Nc

2. centre of mass velocity vcm

3. (moment of inertia Ic)

Advantages

• Finite temperature• No explicit pair-wise interactions

◦ Point-like particles• Highly parallelizable

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Technique: Multi-Particle Collision Dynamics

Streaming

xi (t+ δt) = xi (t) + vi (t) δt

Collision (Andersen MPCD)

vi (t+ δt) = vCM (t) + Ξ

where Ξ is a collision operation

Ξ = ξi︸︷︷︸

noise

−〈ξj

〉Nc︸ ︷︷ ︸

av. cell noise

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† Modelling the separation of macromolecules: A review ofcurrent computer simulation methods, Slater et. al. ,Electrophoresis, 30, 2009.

Technique: Multi-Particle Collision Dynamics

Streaming

xi (t+ δt) = xi (t) + vi (t) δt

Collision (Andersen MPCD)

vi (t+ δt) = vCM (t) + Ξ

where Ξ is a collision operation

Ξ = ξi︸︷︷︸

noise

−〈ξj

〉Nc︸ ︷︷ ︸

av. cell noise

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† Modelling the separation of macromolecules: A review ofcurrent computer simulation methods, Slater et. al. ,Electrophoresis, 30, 2009.

Technique: Multi-Particle Collision Dynamics

Streaming

xi (t+ δt) = xi (t) + vi (t) δt

Collision (Andersen MPCD)

vi (t+ δt) = vCM (t) + Ξ

where Ξ is a collision operation

Ξ = ξi︸︷︷︸

noise

−〈ξj

〉Nc︸ ︷︷ ︸

av. cell noise

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† Modelling the separation of macromolecules: A review ofcurrent computer simulation methods, Slater et. al. ,Electrophoresis, 30, 2009.

Technique: Multi-Particle Collision Dynamics

vcm

Streaming

xi (t+ δt) = xi (t) + vi (t) δt

Collision (Andersen MPCD)

vi (t+ δt) = vCM (t) + Ξ

where Ξ is a collision operation

Ξ = ξi︸︷︷︸

noise

−〈ξj

〉Nc︸ ︷︷ ︸

av. cell noise

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† Modelling the separation of macromolecules: A review ofcurrent computer simulation methods, Slater et. al. ,Electrophoresis, 30, 2009.

Technique: Multi-Particle Collision Dynamics

random

vcm

Streaming

xi (t+ δt) = xi (t) + vi (t) δt

Collision (Andersen MPCD)

vi (t+ δt) = vCM (t) + Ξ

where Ξ is a collision operation

Ξ = ξi︸︷︷︸

noise

−〈ξj

〉Nc︸ ︷︷ ︸

av. cell noise

4 of 26

www.tnshendruk.com

† Modelling the separation of macromolecules: A review ofcurrent computer simulation methods, Slater et. al. ,Electrophoresis, 30, 2009.

Technique: Multi-Particle Collision Dynamics

Streaming

xi (t+ δt) = xi (t) + vi (t) δt

Collision (Andersen MPCD)

vi (t+ δt) = vCM (t) + Ξ

where Ξ is a collision operation

Ξ = ξi︸︷︷︸

noise

−〈ξj

〉Nc︸ ︷︷ ︸

av. cell noise

4 of 26

www.tnshendruk.com

† Modelling the separation of macromolecules: A review ofcurrent computer simulation methods, Slater et. al. ,Electrophoresis, 30, 2009.

Equations to Simulate

So far. . .

• Continuity equation Dtρ+ ρ∂iui = 0◦ Streaming and collision conserve number of particles

• Momentum equation ρDtui = ∂j[Πviscij +Π

nemij + Π

actij

]◦ Momentum stochastically exchanged but conserved

• Orientation equation DtQij − Sij = ΓHij

• i.e. Navier-Stokes Equations on long time and length scales

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Equations to Simulate

So far. . .

• Continuity equation Dtρ+ ρ∂iui = 0◦ Streaming and collision conserve number of particles

• Momentum equation ρDtui = ∂j[Πviscij +Π

nemij + Π

actij

]◦ Momentum stochastically exchanged but conserved

• Orientation equation DtQij − Sij = ΓHij

• i.e. Navier-Stokes Equations on long time and length scales

5 of 26

www.tnshendruk.com

Equations to Simulate

So far. . .

• Continuity equation Dtρ+ ρ∂iui = 0◦ Streaming and collision conserve number of particles

• Momentum equation ρDtui = ∂j[Πviscij +Π

nemij + Π

actij

]◦ Momentum stochastically exchanged but conserved

• Orientation equation DtQij − Sij = ΓHij

• i.e. Navier-Stokes Equations on long time and length scales

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Example: MPCD Flow Profiles

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Example: MPCD Flow Profiles

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Example: MPCD Flow Profiles

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Example: MPCD Flow Profiles

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Intermission I

Questions on the operation of the traditionalMPCD algorithm?

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Nematic Multi-Particle Collision Dynamics

Traditional MPCD

Each point-like particle i has

1. position xi

2. velocity vi

3. mass mi

Each cell has

1. population Nc

2. centre of mass velocity vcm

3. (moment of inertia Ic)

Nematic MPCD

Each must additionally have

1. orientation ui

2. molecular potential U

3. rotational coefficient γR

4. tumbling parameter λ

Each cell must have

1. order parameter Qc

1.1 scalar order parameter Sc1.2 director nc

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Nematic Multi-Particle Collision Dynamics

Traditional MPCD

Each point-like particle i has

1. position xi

2. velocity vi

3. mass mi

Each cell has

1. population Nc

2. centre of mass velocity vcm

3. (moment of inertia Ic)

Nematic MPCD

Each must additionally have

1. orientation ui

2. molecular potential U

3. rotational coefficient γR

4. tumbling parameter λ

Each cell must have

1. order parameter Qc

1.1 scalar order parameter Sc1.2 director nc

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Nematic Multi-Particle Collision Dynamics

Momentum Collision Operation

vi (t+ δt) = vCM (t) + Ξ

Ξ = ξi︸︷︷︸

noise: MB-dist

−〈ξj

〉Nc︸ ︷︷ ︸

av. cell noise

+

(I−1c· δLi

)× r′i︸ ︷︷ ︸

correction to conserve ang mom

0 1 2 3 4 5

Speed, |v|0.0

0.1

0.2

0.3

0.4

0.5

0.6

Pro

bab

ility

,f v

el

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Nematic Multi-Particle Collision Dynamics

Momentum Collision Operation

vi (t+ δt) = vCM (t) + Ξ

Ξ = ξi︸︷︷︸

noise: MB-dist

−〈ξj

〉Nc︸ ︷︷ ︸

av. cell noise

+(I−1c· δLi

)× r′i︸ ︷︷ ︸

correction to conserve ang mom

0 1 2 3 4 5

Speed, |v|0.0

0.1

0.2

0.3

0.4

0.5

0.6

Pro

bab

ility

,f v

el

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Nematic Multi-Particle Collision Dynamics

Momentum Collision Operation

vi (t+ δt) = vCM (t) + Ξ

Ξ = ξi︸︷︷︸

noise: MB-dist

−〈ξj

〉Nc︸ ︷︷ ︸

av. cell noise

+(I−1c· δLi

)× r′i︸ ︷︷ ︸

correction to conserve ang mom

0 1 2 3 4 5

Speed, |v|0.0

0.1

0.2

0.3

0.4

0.5

0.6

Pro

bab

ility

,f v

el

−1.0 −0.5 0.0 0.5 1.0

Orientation, ~u · ~n0.0

0.5

1.0

1.5

2.0

2.5

3.0

Pro

bab

ility

,f o

ri

Orientation Collision Operation

ui (t+ δt) = ηi(βU, S, n) ,

where ηi

is a random orientation drawn from the

equilibrium Maier-Saupe probability distribution

fori = exp

(βUS

d− 1

[d(η · n

)2− 1])∝ exp

(dβUS

d− 1η2n

)

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Nematic Multi-Particle Collision Dynamics

Figure: Nematic S ≈ 1 ; U � kBT

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Nematic Multi-Particle Collision Dynamics

Figure: Nematic S ≈ 1 ; U � kBT

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Nematic Multi-Particle Collision Dynamics

Figure: Nematic S ≈ 1 ; U � kBT

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Isotropic-Nematic Transition

Figure: Isotropic phase (U � kBT )

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Isotropic-Nematic Transition

Figure: Nematic phase (note: topological defects)

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Isotropic-Nematic Transition

0 5 10 15 20 25

Interaction Constant, βU

0.0

0.2

0.4

0.6

0.8

1.0

Glo

bal

Sca

lar

Ord

erP

aram

eter

,S 2D Sc

2D S3D Sc

3D S

MS

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Frank Elastic Coefficient

0 10 20 30 40 50

Interaction Constant, U

0

1000

2000

3000

4000

5000

6000

7000

8000

Fra

nk

Coe

ffici

ent,Ki

(a)

(b)

(c)Fit — ρ = 15

Fit — ρ = 20

Fit — ρ = 25

Splay

Twist

Bend

Isotropic Elasticity

Ki = `2ρUS2 [1 + Ci] /6 ∼ U/a

• i = {splay, twist, bend}• Ci ≈ 0 (rod length small

compared to interaction length)

• ` ≈ a• isotropic elasticity and

one-constant approximationapplies.

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Equations to Simulate

So far. . .

• Continuity equation Dtρ+ ρ∂iui = 0• Momentum equation ρDtui = ∂j

[Πviscij +Π

nemij + Π

actij

]• Orientation equation DtQij−Sij = ΓHij

• Nematohydrodynamics requires coupling momentum and orientation

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Equations to Simulate

So far. . .

• Continuity equation Dtρ+ ρ∂iui = 0• Momentum equation ρDtui = ∂j

[Πviscij +Π

nemij + Π

actij

]• Orientation equation DtQij−Sij = ΓHij

• Nematohydrodynamics requires coupling momentum and orientation

11 of 26

www.tnshendruk.com

Equations to Simulate

So far. . .

• Continuity equation Dtρ+ ρ∂iui = 0• Momentum equation ρDtui = ∂j

[Πviscij +Π

nemij + Π

actij

]• Orientation equation DtQij−Sij = ΓHij

• Nematohydrodynamics requires coupling momentum and orientation

11 of 26

www.tnshendruk.com

Equations to Simulate

So far. . .

• Continuity equation Dtρ+ ρ∂iui = 0• Momentum equation ρDtui = ∂j

[Πviscij +Π

nemij + Π

actij

]• Orientation equation DtQij−Sij = ΓHij

• Nematohydrodynamics requires coupling momentum and orientation

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Shear Alignment:Velocity→Orientation Couplings

Flow-Induced Rotation

• Discretised Jeffery’s equation for a slender rodδuHI,iδt = χHI

[ui · ω + λ

(ui ·D − uiuiui : D

)]• bare tumbling parameter λ• heuristic shear coupling coefficient χHI

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Tumbling & Shear Alignment

0 5 10 15 20

position, y

−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

velo

city

,u

( y)

• Lees-Edwards BCs• If λ < 1 then tumbling

behaviour

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Tumbling & Shear Alignment

−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Tumbling Parameter, λ

0

10000

20000

30000

40000

50000

Per

iod

,P

2π/(γ̇χHI

√1− λ2

)

ξHI = 1.0

ξHI = 0.5

ξHI = 0.1

γR = 0.0001

γR = 0.001

γR = 0.01

0 5 10 15 20

position, y

−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

velo

city

,u

( y)

• Lees-Edwards BCs• If λ < 1 then tumbling

behaviour

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Tumbling & Shear Alignment

−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Tumbling Parameter, λ

0

5000

10000

15000

20000

25000

Per

iod

,Pχ

HI

2π/(γ̇√

1− λ2)

ξHI = 1.0

ξHI = 0.5

ξHI = 0.1

γR = 0.0001

γR = 0.001

γR = 0.01

0 5 10 15 20

position, y

−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

velo

city

,u

( y)

• Lees-Edwards BCs• If λ < 1 then tumbling

behaviour

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Tumbling & Shear Alignment

0.0 0.2 0.4 0.6 0.8 1.0

Tumbling Parameter, λ′ = λ (15S + 48S4 + 42) /105S0

5000

10000

15000

20000

25000

Per

iod

,Pχ

HI

2π/(γ̇√

1− λ2)

ξHI = 1.0

ξHI = 0.5

ξHI = 0.1

γR = 0.0001

γR = 0.001

γR = 0.01

0 5 10 15 20

position, y

−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

velo

city

,u

( y)

• Lees-Edwards BCs• If λ < 1 then tumbling

behaviour

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Tumbling & Shear Alignment

Expect:• If λ > 1 then shear

aligning

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Tumbling & Shear Alignment

Expect:• If λ > 1 then shear

aligning

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Tumbling & Shear Alignment

x̂

ŷ

Expect:• If λ > 1 then shear

aligning1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

Tumbling Parameter, λ

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Les

lieA

ngl

e,θ L

tan θL = [(λ− 1) / (λ+ 1)]1/2ξHI = 1.0

ξHI = 0.5

ξHI = 0.1

γR = 0.0001

γR = 0.001

γR = 0.01

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Tumbling & Shear Alignment

x̂

ŷ

Expect:• If λ > 1 then shear

aligning1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Tumbling Parameter, λ′ = λ (15S + 48S4 + 42) /105S0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Les

lieA

ngl

e,θ L

tan θL = [(λ− 1) / (λ+ 1)]1/2ξHI = 1.0

ξHI = 0.5

ξHI = 0.1

γR = 0.0001

γR = 0.001

γR = 0.01

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Equations to Simulate

So far. . .

• Continuity equation Dtρ+ ρ∂iui = 0• Momentum equation ρDtui = ∂j

[Πviscij +Π

nemij + Π

actij

]• Orientation equation DtQij − Sij = ΓHij

• Backflow next

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Equations to Simulate

So far. . .

• Continuity equation Dtρ+ ρ∂iui = 0• Momentum equation ρDtui = ∂j

[Πviscij +Π

nemij + Π

actij

]• Orientation equation DtQij − Sij = ΓHij

• Backflow next

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Backflow:Orientation→Velocity Coupling

Torque Balance

Γnet,i = ΓHI,i + Γcol,i + Γext,i = 0

ΓHI,i = −Γcol,i − Γext,i

Hydrodynamic Drag Passes Angular Momentum To Fluid

The MPCD collision operator modified to

Ξi,c = ξi−〈ξj

〉Nc

+(I−1c· [δLc + δLc]

)× r′i

where δLc =∑Nci δLi = −

∑Nci ΓHI,iδt

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Equations to Simulate

So far. . .

• Continuity equation Dtρ+ ρ∂iui = 0• Momentum equation ρDtui = ∂j

[Πviscij + Π

nemij +Π

actij

]• Orientation equation DtQij − Sij = ΓHij

• nematic-MPCD for fluctuating nematohydrodynamics

◦ Shendruk and Yeomans, Soft Matter, 2015, 11, 5101.

• Remaining: Activity

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Equations to Simulate

So far. . .

• Continuity equation Dtρ+ ρ∂iui = 0• Momentum equation ρDtui = ∂j

[Πviscij + Π

nemij +Π

actij

]• Orientation equation DtQij − Sij = ΓHij

• nematic-MPCD for fluctuating nematohydrodynamics◦ Shendruk and Yeomans, Soft Matter, 2015, 11, 5101.

• Remaining: Activity

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Equations to Simulate

So far. . .

• Continuity equation Dtρ+ ρ∂iui = 0• Momentum equation ρDtui = ∂j

[Πviscij + Π

nemij +Π

actij

]• Orientation equation DtQij − Sij = ΓHij

• nematic-MPCD for fluctuating nematohydrodynamics◦ Shendruk and Yeomans, Soft Matter, 2015, 11, 5101.

• Remaining: Activity

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Intermission II

Questions on the operation of thenematic-MPCD algorithm?

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Proposal: Active MPCD

Collision Operator

Passive Fluids

• MPCD collisions are artificial operationsthat stochastically exchange velocities whileconserving energy and momentum

Active Fluids

• Active materials convert chemical potentialenergy to spontaneously drive dynamics

• MPCD collision operations can be conceivedthat lead to various classes of active matter

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Dipolar Active-Nematic MPCD

Dipole-Force Andersen Collision

• MPCD cells are subject to dipolar force

Ξi,c = ξi︸︷︷︸

noise

−〈ξj

〉︸ ︷︷ ︸

av. noise

+(I−1c·[δLc + δLc

])× r′i︸ ︷︷ ︸

ang. mom.

+ τ [αact − |vi (t)|](κinc

)︸ ︷︷ ︸activity

−〈τ [αact − |vj (t)|]

(κjnc

)〉Nc︸ ︷︷ ︸

residuals

where

• τ relaxation coefficient◦ Extensile 0 < τ ≤ 1◦ Contractile −1 ≤ τ < 0

• αact out-of-equilibrium speed• κi = ±1 based on plane:

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Dipolar Active-Nematic MPCD

Dipole-Force Andersen Collision

• MPCD cells are subject to dipolar force

Ξi,c = ξi︸︷︷︸

noise

−〈ξj

〉︸ ︷︷ ︸

av. noise

+(I−1c·[δLc + δLc

])× r′i︸ ︷︷ ︸

ang. mom.

+ τ [αact − |vi (t)|](κinc

)︸ ︷︷ ︸activity

−〈τ [αact − |vj (t)|]

(κjnc

)〉Nc︸ ︷︷ ︸

residuals

where

• τ relaxation coefficient◦ Extensile 0 < τ ≤ 1◦ Contractile −1 ≤ τ < 0

• αact out-of-equilibrium speed• κi = ±1 based on plane:

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Dipolar Active-Nematic MPCD

Dipole-Force Andersen Collision

• MPCD cells are subject to dipolar force

Ξi,c = ξi︸︷︷︸

noise

−〈ξj

〉︸ ︷︷ ︸

av. noise

+(I−1c·[δLc + δLc

])× r′i︸ ︷︷ ︸

ang. mom.

+ τ [αact − |vi (t)|](κinc

)︸ ︷︷ ︸activity

−〈τ [αact − |vj (t)|]

(κjnc

)〉Nc︸ ︷︷ ︸

residuals

where

• τ relaxation coefficient◦ Extensile 0 < τ ≤ 1◦ Contractile −1 ≤ τ < 0

• αact out-of-equilibrium speed• κi = ±1 based on plane:

xcm

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Dipolar Active-Nematic MPCD

Dipole-Force Andersen Collision

• MPCD cells are subject to dipolar force

Ξi,c = ξi︸︷︷︸

noise

−〈ξj

〉︸ ︷︷ ︸

av. noise

+(I−1c·[δLc + δLc

])× r′i︸ ︷︷ ︸

ang. mom.

+ τ [αact − |vi (t)|](κinc

)︸ ︷︷ ︸activity

−〈τ [αact − |vj (t)|]

(κjnc

)〉Nc︸ ︷︷ ︸

residuals

where

• τ relaxation coefficient◦ Extensile 0 < τ ≤ 1◦ Contractile −1 ≤ τ < 0

• αact out-of-equilibrium speed• κi = ±1 based on plane:

xcm

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Dipolar Active-Nematic MPCD

Dipole-Force Andersen Collision

• MPCD cells are subject to dipolar force

Ξi,c = ξi︸︷︷︸

noise

−〈ξj

〉︸ ︷︷ ︸

av. noise

+(I−1c·[δLc + δLc

])× r′i︸ ︷︷ ︸

ang. mom.

+ τ [αact − |vi (t)|](κinc

)︸ ︷︷ ︸activity

−〈τ [αact − |vj (t)|]

(κjnc

)〉Nc︸ ︷︷ ︸

residuals

where

• τ relaxation coefficient◦ Extensile 0 < τ ≤ 1◦ Contractile −1 ≤ τ < 0

• αact out-of-equilibrium speed• κi = ±1 based on plane:

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Equations to Simulate

So far. . .

• Continuity equation Dtρ+ ρ∂iui = 0• Momentum equation ρDtui = ∂j

[Πviscij + Π

nemij + Π

actij

]• Orientation equation DtQij − Sij = ΓHij

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Wall and Defect Pairs Formation

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Wall and Defect Pairs Formation

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Spontaneous Flows

0 50 100 150 200

x

0

10

20

30

40

50y

0.004

0.008

0.012

0.016

0.020

0.024

0.028

|~u|

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Active Colloidal Liquid Crystals

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Conclusion

Recapitulation

• MPCD traditional mesoscalealgorithm for isotropic solvent

• MPCD for nematic liquid crystals◦ Isotropic-nematic transition◦ Topological defect dynamics◦ Frank elastic coefficients◦ Nematodynamics with backflow◦ Tumbling and aligning

• MPCD for active nematic◦ Modified collision operator◦ Dipolar active fluid

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Defect Annihilation Dynamics

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Defect Annihilation Dynamics

100 101 102 103 104

Time, t

10−5

10−4

10−3

10−2

Top

olog

ical

Def

ect

Den

sity

,ρD

χHI

1.0

0.0

100 101 102 103 10410−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

RD

(a)

Annihilation Rate

• RD = −ρ̇D ∼ t−(ν+1)

• Liu and Muthukumar: ν = 6/7• Varies in time: ν = 0.74± 0.02

for t ∈[10, 103

]but

ν = 0.83± 0.04 overt ∈

[102, 104

]

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MotivationMulti-Particle Collision Dynamics AlgorithmExample: Flows

Nematic-MPCDNMPCD AlgorithmIsotropic-Nematic TransitionNematodynamics

Active-MPCDDipolar-MPCD Algorithm

Preliminary ResultsConclusion