m m m https://github.com/rajeshrinet/pystokes Hydrodynamic and phoretic interactions of active particles in Python no-shear interface crystallization at a no-slip wall Convective rolls of active particles in a harmonic trap experiment & theory electrohydrodynamic flow in a rectangular geometry arrested clusters at a wall in-silico experiments with active matter emergent optofluidic potential bound states of two active particles

m m m https://github.com/rajeshrinet/pystokes Hydrodynamic and phoretic interactions of active particles in Python no-shear interface crystallization at a no-slip wall Convective rolls of active particles in a harmonic trap experiment & theory electrohydrodynamic flow in a rectangular geometry arrested clusters at a wall in-silico experiments with active matter emergent optofluidic potential bound states of two active particles Ronojoy Adhikari Department of Applied Mathematics and Theoretical Physics (DAMTP) University of Cambridge with Rajesh Singh and Mike Cates

Active particles (for this talk) Gollub lab, PRL 2010; Goldstein lab, PRL 2009 Thutupalli et al NJP 2011 Palacci et al Science 2013; Ebbens and Howse SM 2010 microorganisms autophoretic colloids osmophoretic droplets

Active particles (for this talk) Gollub lab, PRL 2010; Goldstein lab, PRL 2009 Thutupalli et al NJP 2011 Palacci et al Science 2013; Ebbens and Howse SM 2010 non-equilibrium processes on the surface drive exterior fluid flow, even when the particle is stationary the fluid stress may react back and cause self-propulsion in absence of external forces and torques fluid flow mediates long-range hydrodynamic interactions (HI) universal mechanisms due to the scale-separation in the slip layer microorganisms autophoretic colloids osmophoretic droplets

Ideal active particle - "spherical cow" A sphere with slip boundary condition boundary velocity = rigid body motion + active slip Such boundary conditions were considered by Derjaguin, Lighthill, Blake ....

Main theoretical questions ‣ What are the forces and torques acting on the particles due to slip ? ‣ How are these modified by the presence of boundaries ? ‣ What is the rigid body motion of the particles under these forces ? ‣ How do we take into account, simultaneously, the many-body character 
 of the hydrodynamic and phoretic interactions between particles ? ‣ How do we promote the dynamical equations to include fluctuations ?
 The problem is classical, motion is governed by Newton's equations. We then need to know:

Expansion of the slip in a complete spherical basis boundary velocity = rigid body motion + active slip Expanding the slip in the basis of tensorial spherical harmonics Y(l)

Expansion of the slip in a complete spherical basis boundary velocity = rigid body motion + active slip Expanding the slip in the basis of tensorial spherical harmonics Y(l) Y(l) are dimensionless, symmetric, irreducible Cartesian tensors of rank l that form a complete, orthogonal basis on the sphere

Expansion of the slip in a complete spherical basis boundary velocity = rigid body motion + active slip Expanding the slip in the basis of tensorial spherical harmonics Y(l) Y(l) are dimensionless, symmetric, irreducible Cartesian tensors of rank l that form a complete, orthogonal basis on the sphere

Boundary integral representation (central idea) ‣ The Stokes equation admits a boundary integral representation, where
 the flow in the bulk can be expressed as an integral of the traction and
 the velocity over the boundaries of the flow. [Lorentz, Odqvist ...]. ‣ This integral can be expressed as a power series in gradients of the 
 Green's functions of Stokes flow (similar to a multipole expansion, but
 exact at the boundaries, not approximate!). (Singh, Ghose,RA, JStat 2015) ‣ Single expression for exterior flow, given a Green's function. No need to solve the Stokes equation for each particular case. ‣ Each irreducible mode of the slip contributes an irreducible mode to the
 exterior fluid flow. SO(3) invariant way to classify possible active flows.

in an unbounded domain of fluid flow flow from the l-th mode has three independent terms: (a) symmetric irreducible gradients of G (b) its curl and, (c) its Laplacian Ghose + RA PRL 2014

in an unbounded domain of fluid flow flow from the l-th mode has three independent terms: (a) symmetric irreducible gradients of G (b) its curl and, (c) its Laplacian Ghose + RA PRL 2014

in an unbounded domain of fluid flow flow from the l-th mode has three independent terms: (a) symmetric irreducible gradients of G (b) its curl and, (c) its Laplacian symmetric antisymmetric trace Ghose + RA PRL 2014

l Symmetric Antisymmetric Trace 1 2 3 in an unbounded domain of fluid flow flow from the l-th mode has three independent terms: (a) symmetric irreducible gradients of G (b) its curl and, (c) its Laplacian symmetric antisymmetric trace Ghose + RA PRL 2014

Open-source, RSE-compliant Python library

Open-source, RSE-compliant Python library

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? Folklore theorem: not important for active matter

Drescher et al PNAS 2011 ? Folklore theorem: not important for active matter

Drescher et al PNAS 2011 Herminghaus et al Soft Matter ? Folklore theorem: not important for active matter

S. Thutupalli, D. Geyer, R. Singh, RA, and H. A. Stone, PNAS 2018 Linear combinations often necessary: 2s + 3t + 4t mode

Experimental flow Theoretical flow S. Thutupalli, D. Geyer, R. Singh, RA, and H. A. Stone, PNAS 2018 Linear combinations often necessary: 2s + 3t + 4t mode

friction tensors for slip slip coefficients Newtonian dynamics: generalized Stokes laws The forces depend on the positions of colloids through the friction tensors obtained in terms of a Green’s function of Stoke’s equation Body Brownian Hydrodynamic We use the above in Newton’s laws to obtain the rigid body motion R. Singh and RA, PRL 2016

Rigid body motion of active colloids RS and Adhikari, EJCM 2017, JPC 2018

Rigid body motion of active colloids Invert for rigid body motion RS and Adhikari, EJCM 2017, JPC 2018

Rigid body motion of active colloids Propulsion tensors relate modes of slip to rigid body motion White noises Mobility matrices connectors for forces and torques Invert for rigid body motion RS and Adhikari, EJCM 2017, JPC 2018

Rigid body motion of active colloids Propulsion tensors relate modes of slip to rigid body motion White noises Mobility matrices connectors for forces and torques Invert for rigid body motion RS and Adhikari, EJCM 2017, JPC 2018

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active particles which swim into a wall, induced a monopole, when stalled by it. This leads to an attractive flow causing crystallization RS and Adhikari PRL 2016.

Faster time-scales at a no-shear surface when compared to a no-slip wall (Thutupalli et al PNAS 2018). viscosity ratio

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RS and Adhikari PRL 2016, Squires JFM 2001

RS and Adhikari PRL 2016, Squires JFM 2001

RS and Adhikari PRL 2016, Squires JFM 2001

What about phoretic interactions?

Known: chemical surface flux at particle boundaries Singh, RA , and Cates JCP 2019 Desired: rigid body motion of particles. Structurally, this is similar to the Taylor-Melchor theory of electrohydrodynamic flows: two
 governing partial differential equations in the volume are coupled only at boundaries.

Generalized Stokes law Many-body phoretic and hydrodynamic interactions RS, Adhikari, and Cates JCP 2019 Use integral representation of both Stokes and Laplace equations, spectral 
 expansion, and Galerkin discretization, to compute both the elastance and friction tensors. Linearity is the key! Many-body slip law

Generalized Stokes law Many-body phoretic and hydrodynamic interactions RS, Adhikari, and Cates JCP 2019 Use integral representation of both Stokes and Laplace equations, spectral 
 expansion, and Galerkin discretization, to compute both the elastance and friction tensors. Linearity is the key! Many-body slip law

Generalized Stokes law Chemical surface flux Many-body slip Exterior fluid flow Chemical interactions Hydrodynamic interactions Many-body phoretic and hydrodynamic interactions RS, Adhikari, and Cates JCP 2019 Use integral representation of both Stokes and Laplace equations, spectral 
 expansion, and Galerkin discretization, to compute both the elastance and friction tensors. Linearity is the key! Many-body slip law

Freezing by heating!

Colloidal tethered to an interface - free to move in the plane Caciagli, Singh, Joshi, RA and Eiser: PRL 2020

Optically trap one of the colloids and study the optofluidic interactions

Puzzle: what causes motion into the hot region ?

Water Oil t=0.00s t=1.05s t=1.10s t=1.15s t=1.20s t=0.00s t=0.45s t=0.55s t=0.75s t=0.85s Similar in mechanism to like-charge attraction (Squires and Brenner PRL 2000)

Water Oil t=0.00s t=1.05s t=1.10s t=1.15s t=1.20s t=0.00s t=0.45s t=0.55s t=0.75s t=0.85s Thermophoresis: quadrupolar flow Similar in mechanism to like-charge attraction (Squires and Brenner PRL 2000)

Water Oil t=0.00s t=1.05s t=1.10s t=1.15s t=1.20s t=0.00s t=0.45s t=0.55s t=0.75s t=0.85s Thermophoresis: quadrupolar flow Monopolar flow once the colloid is stalled Similar in mechanism to like-charge attraction (Squires and Brenner PRL 2000)

Water Oil Flow-induced attraction 0 1 t=0.00s t=1.05s t=1.10s t=1.15s t=1.20s t=0.00s t=0.45s t=0.55s t=0.75s t=0.85s Thermophoresis: quadrupolar flow Monopolar flow once the colloid is stalled Similar in mechanism to like-charge attraction (Squires and Brenner PRL 2000)

Water Oil Flow-induced attraction 0 1 t=0.00s t=1.05s t=1.10s t=1.15s t=1.20s t=0.00s t=0.45s t=0.55s t=0.75s t=0.85s Thermophoresis: quadrupolar flow Monopolar flow once the colloid is stalled Similar in mechanism to like-charge attraction (Squires and Brenner PRL 2000)

Water Oil Flow-induced attraction 0 1 t=0.00s t=1.05s t=1.10s t=1.15s t=1.20s t=0.00s t=0.45s t=0.55s t=0.75s t=0.85s Thermophoresis: quadrupolar flow Monopolar flow once the colloid is stalled Similar in mechanism to like-charge attraction (Squires and Brenner PRL 2000)

Variants of complete crystallisation at boundaries Petroff et al PRL 2015 (T. Majus)

Variants of complete crystallisation at boundaries Petroff et al PRL 2015 (T. Majus)

Variants of complete crystallisation at boundaries Petroff et al PRL 2015 (T. Majus)

Arrested crystallisation due to competing interactions Light on Light o Palacci et al Science 2013: chemical reactions in presence of light make colloids move (active) Crystallisation at a wall in presence of light Crystals melt as light is turned off

Arrested crystallisation due to competing interactions Theurkauff et al PRL 2012, Buttinoni et al PRL 2013: Dynamic clusters of autophoretic particles Cluster size increases with speed of isolated colloids Light on Light o Palacci et al Science 2013: chemical reactions in presence of light make colloids move (active) Crystallisation at a wall in presence of light Crystals melt as light is turned off

Arrested crystallisation due to competing interactions Petroff et al PRL 2015: Crystallization has also been reported for microorganisms , T. Majus, at a plane wall. But there is no arrest of clustering here. The activity here is biological and these colloids interact by hydrodynamic interactions alone Theurkauff et al PRL 2012, Buttinoni et al PRL 2013: Dynamic clusters of autophoretic particles Cluster size increases with speed of isolated colloids Light on Light o Palacci et al Science 2013: chemical reactions in presence of light make colloids move (active) Crystallisation at a wall in presence of light Crystals melt as light is turned off

RS, Adhikari, and Cates JCP 2019

RS, Adhikari, and Cates JCP 2019

Formalism to study the hydrodynamic and phoretic interactions between colloids that does not need to resolve explicit fluid or phoretic degrees of freedom. Solution obtained in terms of irreducible modes of slip and phoretic flux. Boundary conditions in the flow modify the active forces and torques, and thus, determine the collective behaviour. Summary

‣ Grid-free method to compute hydrodynamic and phoretic interactions between spheres with slip boundary conditions ‣ Identify mechanisms using interactions computed from each irreducible mode ‣ Deepen theoretical understanding and suggest fresh experiments. ‣ Tool to play around with! Formalism to study the hydrodynamic and phoretic interactions between colloids that does not need to resolve explicit fluid or phoretic degrees of freedom. Solution obtained in terms of irreducible modes of slip and phoretic flux. Boundary conditions in the flow modify the active forces and torques, and thus, determine the collective behaviour. Summary

‣ Grid-free method to compute hydrodynamic and phoretic interactions between spheres with slip boundary conditions ‣ Identify mechanisms using interactions computed from each irreducible mode ‣ Deepen theoretical understanding and suggest fresh experiments. ‣ Tool to play around with! Formalism to study the hydrodynamic and phoretic interactions between colloids that does not need to resolve explicit fluid or phoretic degrees of freedom. Solution obtained in terms of irreducible modes of slip and phoretic flux. Boundary conditions in the flow modify the active forces and torques, and thus, determine the collective behaviour. Summary Thank You !