Active Flows using Coarse-Grained Numerical Methods

ICMCS, 25th September, 2023

Timofey Kozhukhov (t.kozhukhov@sms.ed.ac.uk), Tyler N. Shendruk (t.shendruk@ed.ac.uk)

View these slides online at www.Kozhukhov.co.uk/

$\definecolor{saphire}{RGB}{0, 50, 95}$ $\definecolor{crimson}{RGB}{193, 0, 67}$ $\definecolor{capri}{RGB}{0, 196, 223}$ $\definecolor{amber}{RGB}{244, 170, 0}$ $\definecolor{plum}{RGB}{129, 2, 98}$ $\definecolor{cerulean}{RGB}{0, 145, 181}$ $\definecolor{ruby}{RGB}{212, 0, 114}$ $\definecolor{cardinal}{RGB}{172, 0, 64}$ $\definecolor{ccinnamon}{RGB}{205, 90, 19}$ $\definecolor{climegreen}{RGB}{41, 188, 41}$ $\definecolor{cgold}{RGB}{141, 116, 74}$ $\definecolor{ctaupe}{RGB}{110, 80, 72}$ $\definecolor{cteal}{RGB}{69, 126, 129}$ $\definecolor{cforestgreen}{RGB}{0, 70, 49}$ $\definecolor{cmahogany}{RGB}{106, 51, 40}$ $\definecolor{csilver}{RGB}{194, 211, 223}$ $\definecolor{coldrose}{RGB}{184, 133, 141}$ $\definecolor{curry}{RGB}{156, 154, 0}$ $\definecolor{cobalt}{RGB}{0, 80, 114}$ $\definecolor{rubydarker}{RGB}{197, 0, 99}$ $\definecolor{purple}{RGB}{56, 6, 92}$ $\definecolor{cardinaldarker}{RGB}{97, 0, 36}$ $\definecolor{ceruleandarker}{RGB}{0, 113, 140}$ $\definecolor{amberlighter}{RGB}{240, 191, 79}$ $\definecolor{amberbrighter}{RGB}{245, 242, 88}$ $\definecolor{onyx}{RGB}{15, 15, 15}$

Table of contents

  1. Introduction & Motivation
  2. Mesoscopic simulation techniques &
    Multi-Particle Collision Dynamics (MPCD)
  3. Active Nematic (AN-) MPCD
  4. Modulated AN-MPCD
  5. Applications of AN-MPCD
  6. Conclusion

Active matter ranges across scales

  • Individual "active" units
  • Local energy injection from environment
  • Energy used to locomote

Youtube: Wonders in the sky (2015)
Rivas D, et al., Soft Matter, 2020

Microtubule bundle systems are active nematics

Rivas D, et al., Soft Matter, 2020

Active Nematic

Microtubule bundle systems are active nematics

Rivas D, et al., Soft Matter, 2020

+1/2

-1/2

Mesoscale active matter has seen rapid recent progress

Thijssen K, Khaladj D, et al., PNAS, 2021
Drechsler M, et al., Mol Biol Cell, 2020
Ray S, et al., Phys. Rev. Lett., 2023

Table of contents

  1. Introduction & Motivation
  2. Mesoscopic simulation techniques &
    Multi-Particle Collision Dynamics (MPCD)
  3. Active Nematic (AN-) MPCD
  4. Modulated AN-MPCD
  5. Applications of AN-MPCD
  6. Conclusion

Active matter simulation methods exist across scales

Microscopic

Vliegenthart G, et al., Sci. Adv., 2020

Macroscopic

Sokolov A, et al., Phys. Rev. X, 2019

Mesoscale methods for this subset of active systems are limited

Our requirements:

  • Includes hydrodynamic interactions
  • Suitable for complex geometries & boundaries
  • Moderate Peclet number
  • Suitable for wide variety of active systems

MD

MD

DPD

DPD

MPCD

MPCD

LB

LB

Stokesian Dyn.

Stokesian Dyn.

Langevin Dyn.

Langevin Dyn.

What is Multi-Particle Collision Dynamics (MPCD)?

Particle $i$ has:
  1. Position $\vec x_i$
  2. Velocity $\vec v_i$
  3. Mass $m_i$


Cell $C$ has:
  1. Population $\rho_C$
  2. CoM velocity $\vec v_{CM}$
  3. Moment of intertia $I_C$


Streaming: $\vec{x}_i(t+\delta t) = \vec{x}_i(t) + \vec{v}_i(t)\delta t$

Collision:
$\vec{v}_i(t+\delta t) = \vec{v}_{CM}(t) + \vec\Xi_i(\vec{r}_\mathrm{CM}, t)$
$\vec{v}_i(t+\delta t) = \vec{v}_{CM}(t) + \textcolor{crimson}{\vec\Xi_i(\vec{r}_\mathrm{CM}, t)}$

The collision operator controls the material properties

A collision operator $\textcolor{crimson}{\vec \Xi_C}$ acting on cell $C$ must:
  • Include some stochasticity
  • Respect conservation laws (linear momentum, etc)
  • Exchange particle properties (velocities, etc)
Angular momentum conserving Andersen collision operator:
$ \color{crimson}{\Xi_i^{A}} $ $ = \color{cerulean}\underline{\color{black}\vec\xi_i(t) } $ $ - \color{crimson}\underline{\color{black}\langle \vec\xi\rangle_{N_C} } $ $ + \color{amber}\underline{\color{black}{(I_C^{-1}\cdot \delta L_C ) \times \vec{r}_i}} $
where $\delta L_C$ is the angular momentum change

François de Tournemire

Modifying collision operators simulates new materials

Particle $i$ has:
  1. Position $\vec x_i$
  2. Velocity $\vec v_i$
  3. Mass $m_i$
  4. Orientation $\vec u_i$


Cell $C$ has:
  1. Population $\rho_C$
  2. CoM velocity $\vec v_{CM}$
  3. Moment of inertia $I_C$
  4. Local director $\vec n_C$

Velocity collision: $\vec{v}_i(t+\delta t) = \vec{v}_{CM}(t) + \textcolor{crimson}{\vec\Xi_i(\vec{r}_\mathrm{CM}, t)}$
Orientation collision: $\vec{u}_i(t+\delta t) = \vec{n}_C(t) + \textcolor{cerulean}{\vec{\eta}_i(t)}$

$\textcolor{cerulean}{\vec{\eta}_i}$ orientation drawn from a Maier-Saupe distribution \[ \begin{align*} \textcolor{crimson}{\Xi_i^{N}} &= \textcolor{crimson}{\Xi_i^{A}} + (I_C^{-1}\cdot \delta \mathcal{L}_C) \times \vec{r}_i \end{align*} \]
  • Shendruk T, Yeomans J, Soft Matter 2015

Simple changes to collision operators give nematic behaviour

  • Shendruk T, Yeomans J, Soft Matter 2015

Table of contents

  1. Introduction & Motivation
  2. Mesoscopic simulation techniques &
    Multi-Particle Collision Dynamics (MPCD)
  3. Active Nematic (AN-) MPCD
  4. Modulated AN-MPCD
  5. Applications of AN-MPCD
  6. Conclusion

Local force dipoles are necessary for an active-nematic

Rivas D, et al., Soft Matter, 2020

A planar collision rule induces local force dipoles

$\Xi_i^\mathrm{Active} = \Xi_i^{N}$ ${ + \alpha_C \delta t \left( \frac{\kappa_i}{m_i} - \langle \frac{\kappa_j}{m_j} \rangle_C \right) \vec{n}_C } ; \quad \alpha_C = \sum_i^{\rho_C}\alpha_i $ $\textcolor{crimson}{ = \alpha_C^\mathrm{S}}$
Cellular activity $\alpha_C$ - Local force magnitude
  • Kozhukhov T, Shendruk T, Sci. Adv., 2022

Activity magnitude changes algorithm regime


$\alpha < \alpha_\text{eq} = 10^{-3}$

$10^{-3} = \alpha_\text{eq} < \alpha < \alpha_\text{turb} = 2.5\times 10^{-2}$

$\alpha > \alpha_\text{turb} = 2.5\times 10^{-2}$
  • Kozhukhov T, Shendruk T, Sci. Adv., 2022

Active-Nematic (AN-) MPCD Reproduces Active Turbulence

Rivas D, et al., Soft Matter, 2020
  • Kozhukhov T, Shendruk T, Sci. Adv., 2022

AN-MPCD turbulence begins from high bend walls

AN-MPCD succesfully reproduces active turbulence


$\ell_d = \rho_d^{-1/2}$; Theory: $\ell_d \propto \alpha^{-1/2}$

Theory: $v \propto \alpha^{1/2}$
For $\alpha > \alpha_\text{turb} = 2.5\times 10^{-2}$
  • Kozhukhov T, Shendruk T, Sci. Adv., 2022

Active particle model results in density fluctuations

  • Kozhukhov T, Shendruk T, Sci. Adv., 2022

Activity accumulates particles and widens distributions

Suggests an upper limit for effective turbulence regime of $\alpha_\dagger$
  • Kozhukhov T, Shendruk T, Sci. Adv., 2022

Increased activity results in giant number fluctuations

Central Limit Theorem: $\sigma_{N_C}\propto \langle N_C \rangle^{1/2}$

$\sigma_{N_C}=A\langle N_C \rangle^\nu$
  • Kozhukhov T, Shendruk T, Sci. Adv., 2022

Summary on AN-MPCD

  • A simple planar collision rule applies a force dipole
  • Built-in thermostat sets a minimum activity for turbulence
  • Within the turbulence regime, active turbulence scales per theoretical expectations
    • $\alpha > \alpha_\text{turb} = 2.5\times 10^{-2}$
  • However has large density fluctuations, common to active particle models
    • This is problematic for simulation of solutes

Table of contents

  1. Introduction & Motivation
  2. Mesoscopic simulation techniques &
    Multi-Particle Collision Dynamics (MPCD)
  3. Active Nematic (AN-) MPCD
  4. Modulated AN-MPCD
  5. Applications of AN-MPCD
  6. Conclusion

Strategy: Modulate local activity

$\Xi_i^{Active} = \Xi_i^{N}+ {\alpha_C} \delta t \left( \frac{\kappa_i}{m_i} - \langle \frac{\kappa_j}{m_j} \rangle_C \right) \vec{n}_C $

  • Active Sum (Original): $\alpha_C^\mathrm{S} = \alpha \rho_C$
  • Active Average: $\alpha_C^\mathrm{A} = \alpha$

  • $\mathcal{S}_C(\rho_C; \sigma_p, \sigma_w) = \frac{1}{2} \left( 1 - \tanh \left( \frac{\rho_C - \av{\rho_C} \left( 1 + \sigma_p \right)}{\av{\rho_C} \sigma_w} \right) \right)$
  • Sigmoidal Av.: $\alpha_C^\mathrm{S-A}(\rho_C) = \alpha_C^\mathrm{A} \mathcal{S}_C(\rho_C)$
  • Sigmoidal Sum: $\alpha_C^\mathrm{S-S}(\rho_C) = \alpha_C^\mathrm{S} \mathcal{S}_C(\rho_C)$
  • Kozhukhov T, Shendruk T, Sci. Adv., 2022
  • Kozhukhov T, Loewe B, Shendruk T, in prep.
Active Sum
Active Av.
Sigmoid Sum ($\sigma_p=0.4, \sigma_w=0.5$)
Sigmoid Av. ($\sigma_p=0.4, \sigma_w=0.5$)
$\alpha = 0.1$
  • Kozhukhov T, Loewe B, Shendruk T, in prep.
Active Sum
Active Av.
Sigmoid Sum ($\sigma_p=0.4, \sigma_w=0.5$)
Sigmoid Av. ($\sigma_p=0.4, \sigma_w=0.5$)
$\alpha = 0.1$
  • Kozhukhov T, Loewe B, Shendruk T, in prep.

Modulation results in energy input fluctuations

  • Active Sum: $\alpha_C^\mathrm{S} = \alpha \rho_C$
  • Active Average: $\alpha_C^\mathrm{A} = \alpha$
  • Sigmoidal Sum: $\alpha_C^\mathrm{S-S}(\rho_C) = \alpha_C^\mathrm{S} \mathcal{S}_C(\rho_C)$
  • Sigmoidal Av.: $\alpha_C^\mathrm{S-A}(\rho_C) = \alpha_C^\mathrm{A} \mathcal{S}_C(\rho_C)$
  • Kozhukhov T, Loewe B, Shendruk T, in prep.

Modulated AN-MPCD reproduces active turbulence


$\ell_d = \rho_d^{-1/2}$; Theory: $\ell_d \propto \alpha^{-1/2}$

Theory: $v \propto \alpha^{1/2}$
  • Kozhukhov T, Loewe B, Shendruk T, in prep.

Modulation results in improved density distributions

  • Kozhukhov T, Loewe B, Shendruk T, in prep.

Modulations reduce density band width

  • Kozhukhov T, Loewe B, Shendruk T, in prep.

Modulations reduce density-induced drift

Fick's law: Density-induced drift $\propto \left| \nabla \rho \right|$


$\alpha=0.1$

$\chi_\mathrm{NGM}=\frac{d}{d+2} \frac{\Delta r^4}{|\Delta r^2|^2}-1$
  • Kozhukhov T, Loewe B, Shendruk T, in prep.

Giant number fluctuation analysis reveal algorithm regimes


$\sigma_{N_C}=A\langle N_C \rangle^\nu$
$A=\sigma_{N_C}|_{\langle N_C \rangle=1}$
  • Kozhukhov T, Loewe B, Shendruk T, in prep.

Table of contents

  1. Introduction & Motivation
  2. Mesoscopic simulation techniques &
    Multi-Particle Collision Dynamics (MPCD)
  3. Active Nematic (AN-) MPCD
  4. Modulated AN-MPCD
  5. Applications of AN-MPCD
  6. Conclusion

Applications of mesoscale AN-MPCD


Louise Head

Active Av.
D.P.Rivas, L.C.Head, D.H.Reich, T.N.Shendruk, R.L.Leheny, in prep., 2023

Applications of mesoscale AN-MPCD


Zahra Valei

Sigmoidal Sum

Applications of mesoscale AN-MPCD


Ryan Keogh

Active Av.

Keogh R, Kozhukhov T,et al., Submitted 2023

Applications of mesoscale AN-MPCD


Kira Koch

Mark Curtis-Rose
Sigmoidal Av.

Applications of mesoscale AN-MPCD


Benjamín Loewe

Humberto Híjar
Loewe B, Shendruk T, NJP 2022

Active Sum
Related: Marcias-Duran E, et al., Soft Matter, 2023

Applications of mesoscale AN-MPCD


Benjamín Loewe
Positive overcharge
Negative overcharge
Active Av. Loewe B, Shendruk T, in prep.

Summary

  • A force dipole reproduces active turbulence for sufficient activity
    • $\alpha > \alpha_\text{turb} \simeq 2.5\times 10^{-2}$
  • Density dependent activity modulation reduces density fluctuations
    • Sigmoidal Av. for general use
    • Sigmoidal Sum for minimising low density regions
  • AN-MPCD provides a strong framework to simulate out of equilibrium mesoscale systems

t.shendruk@ed.ac.uk, t.kozhukhov@sms.ed.ac.uk

View these slides online at www.Kozhukhov.co.uk/


Acknowledgements

Tyler N. Shendruk
Benjamín Loewe
Ryan Keogh
Jack Paget
Louise Head
Humberto Híjar
Zahra Valei
François de Tournemire
Oleksandr Baziei
Kira Koch
Mark Curtis-Rose
Kristian Thijssen

Appendix Slides

Above is for Sig-Av. Sig-Sum very similar.
Chosen params are $\sigma_p=0.4, \sigma_w=0.5$