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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
2 of 26
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
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 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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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
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