Active Flows using Coarse-Grained Numerical Methods

NBIA, 11th April, 2024

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. The Effect of Orientation on Active Baths
  6. Applications of AN-MPCD
  7. 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. The Effect of Orientation on Active Baths
  6. Applications of AN-MPCD
  7. 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:
$\vec{v}_i(t+\delta t) = \vec{v}_{CM}(t) + \vec\Xi_i(\vec{r}_\mathrm{CM}, t)$
$\vec{v}_i(t+\delta t) = \vec{v}_{CM}(t) + \textcolor{crimson}{\vec\Xi_i(\vec{r}_\mathrm{CM}, t)}$
  • Malevanets A, Kapral R, J. Chem. Phys., 1999

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
  • Noguchi H, Kikuchi N, Gompper G, Euro. Phys. Lett., 2007

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. The Effect of Orientation on Active Baths
  6. Applications of AN-MPCD
  7. 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 $ $\textcolor{saphire}{ = \alpha_C^\mathrm{P}}$
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

  • Thampi S, Golestanian R, Yeomans J, Euro. Phys. Lett., 2014

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

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. The Effect of Orientation on Active Baths
  6. Applications of AN-MPCD
  7. 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 $

  • Particle-Carried (Original): $\alpha_C^\mathrm{P} = \alpha \rho_C$
  • Cell-Carried: $\alpha_C^\mathrm{C} = \alpha$

  • $\mathcal{S}_C(\rho_C; \sigma_p, \sigma_w) = \frac{1}{2} \left( 1 - \tanh \left( \frac{\rho_C - \av{\rho_C} \left( 1 + \sigma_p \right)}{\av{\rho_C} \sigma_w} \right) \right)$
  • Modulated Cell-Carried: $\alpha_C^\mathrm{MC} (\rho_C) = \alpha_C^\mathrm{C} \mathcal{S}_C(\rho_C)$
  • Modulated Particle-Carried: $\alpha_C^\mathrm{MP} (\rho_C) = \alpha_C^\mathrm{P} \mathcal{S}_C(\rho_C)$
  • Kozhukhov T, Shendruk T, Sci. Adv., 2022
  • Kozhukhov T, Loewe B, Shendruk T, in review (arXiv:2401.17777)
Particle Carried
Cell Carried
Mod. Part. ($\sigma_p=0.4, \sigma_w=0.5$)
Mod. Cell ($\sigma_p=0.4, \sigma_w=0.5$)
$\alpha = 0.1$
  • Kozhukhov T, Loewe B, Shendruk T, in review (arXiv:2401.17777)
Particle Carried
Cell Carried
Mod. Part. ($\sigma_p=0.4, \sigma_w=0.5$)
Mod. Cell ($\sigma_p=0.4, \sigma_w=0.5$)
$\alpha = 0.1$
  • Kozhukhov T, Loewe B, Shendruk T, in review (arXiv:2401.17777)

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 review (arXiv:2401.17777)

Modulation results in improved density distributions

  • Kozhukhov T, Loewe B, Shendruk T, in review (arXiv:2401.17777)

Modulations reduce density-induced drift

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


$\alpha=0.1$
  • Kozhukhov T, Loewe B, Shendruk T, in review (arXiv:2401.17777)

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 review (arXiv:2401.17777)

Summary on modulated AN-MPCD

  • Sigmoidal modulation of cellular activity reduces density fluctuations
  • Density fluctuations are exchanged for active force fluctuations
  • Active turbulence still scales per theoretical expectations
  • Modulated variants have altered turbulence regimes
  • Density induced drift effects on solutes are minimised

  • Kozhukhov T, Loewe B, Shendruk T, in review (arXiv:2401.17777)

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. The Effect of Orientation on Active Baths
  6. Further Applications of AN-MPCD
  7. Conclusion

Colloids in oriented active baths have yet to be explored

  • Colloids in active baths have been studied
  • Effect of orientation has not
  • Colloids are known to have companion defects
  • What interplay is there between defects and colloidal motility?

Wu X, Libchaber A, Phys. Rev. Lett., 2000

Topology results in colloidal companion defects

Activity number: $A = R_C / \ell_\alpha = 0.27$
Activity number: $A = R_C / \ell_\alpha = 1.18$
Activity number: $A = R_C / \ell_\alpha = 7.34$

Sigmoidal Av.
  • Kozhukhov T, Loewe B, Thijssen K, Shendruk T, in prep.

Colloids have a short ballistic regime before diffusing

\[ \begin{align*} \langle \Delta r^2(t) \rangle &= 2dDt + 2 v_0^2 \tau_r t \\ &\quad- 2 v_0^2 \tau_r^2 \left( 1 - e^{-t/\tau_r} \right) \end{align*} \]
$\langle \Delta r^2(t) \rangle = v_0^2t^2 + 2dDt; \quad t << \tau_r$
$\langle \Delta r^2(t) \rangle = 2dD_\mathrm{eff}t; \quad t >> \tau_r$
  • Kozhukhov T, Loewe B, Thijssen K, Shendruk T, in prep.
  • Zottl A, Stark H, J. Phys.: Condens. Matter, 2016

Interplay of anchoring and activity results in critically enhanced motility


$A^* \simeq 2$
  • Kozhukhov T, Loewe B, Thijssen K, Shendruk T, in prep.

No coherent flow in colloidal reference frame past critical activity


$A = 0.27$

$A = 1.18$

$A = 2.45$

$A = 3.54$

$A = 7.34$
  • Kozhukhov T, Loewe B, Thijssen K, Shendruk T, in prep.

Defects cause symmetry breaking in the colloid

$A = 0.27$ $A = 1.18$ $A = 2.45$ $A = 3.54$ $A = 7.34$
  • Kozhukhov T, Loewe B, Thijssen K, Shendruk T, in prep.

Activity controls the location of defects near the colloid

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

Defects have well-defined distribution peaks below critical activity


+1/2 defects

-1/2 defects
  • Kozhukhov T, Loewe B, Thijssen K, Shendruk T, in prep.

Activity causes companion defects to move closer to the colloid


Passive theory: $\langle r \rangle - R_\mathrm{C} \sim R_\mathrm{C}$
  • Kozhukhov T, Loewe B, Thijssen K, Shendruk T, in prep.

Number of nearby defects control colloid motion


$\vec{r}_d$: Vector to nearest defect
$\vec{u}$: Colloid velocity
  • Kozhukhov T, Loewe B, Thijssen K, Shendruk T, in prep.

Summary

  • Oriented active baths result in non-monotonic diffusion
  • Diffusion is closely linked to surrounding topological defects
  • One nearby defect results in more directed motion
  • Kozhukhov T, Loewe B, Thijssen K, 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. The Effect of Orientation on Active Baths
  6. Applications of AN-MPCD
  7. 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., 2024

Applications of mesoscale AN-MPCD


Zahra Valei

Sigmoidal Sum
Valei Z, Baziei O, Loewe B, Shendruk T, in prep., 2024

Applications of mesoscale AN-MPCD


Ryan Keogh

Active Av.

Keogh R, Kozhukhov T, et al., Phys. Rev. Lett. (accepted), 2024

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

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
  • Colloids have non-monotonic diffusion in oriented active baths
  • AN-MPCD provides a strong framework to simulate out of equilibrium mesoscale systems

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 $\sigma_p=0.4, \sigma_w=0.5$