Many-body microhydrodynamics of colloidal particles with active boundary layers

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Many-body microhydrodynamics of colloidal particles with active boundary layers Rajesh Singh, Somdeb Ghose and R. Adhikari The Institute of Mathematical Sciences Chennai, India

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The Institute of Mathematical Sciences Chennai www.imsc.res.in

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3 Microhydrodynamics of a single swimmer Non-equilibrium steady states of active particles in a harmonic potential Introduction Boundary integral equation Outline

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Introduction

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5 Active matter Active matter describes systems driven out of equilibrium by constant consumption of energy, locally, to produce motion.

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5 Active matter Active matter describes systems driven out of equilibrium by constant consumption of energy, locally, to produce motion. • Schools of ﬁsh • Flocks of birds

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5 Active matter Active matter describes systems driven out of equilibrium by constant consumption of energy, locally, to produce motion. • Schools of ﬁsh • Flocks of birds J. Am. Chem. Soc., 2004, 126 (41), 13424–13431 • Catalytic motors

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6 Size Speed ⇠ cm ⇠ m ⇠ µm ⇠ 100µm Microhydrodynamics Scales in the system Features at this scale: • absence of inertia: counterintuitive • vanishing net-external force or torque • instantaneity of interactions • long range interactions

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7 Microhydrodynamics: constraints M ˙ V = I · ndS + Fe I ˙ ⌦ = I r ⇥ · ndS + Te

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7 Microhydrodynamics: constraints M ˙ V = I · ndS + Fe I ˙ ⌦ = I r ⇥ · ndS + Te absence of inertia

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8 Microhydrodynamics: constraints M ˙ V = I · ndS + Fe I ˙ ⌦ = I r ⇥ · ndS + Te Force-free torque-free particles in absence of inertia

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8 • Trivial solution: No motion in absence of external forces and torques • Non-trivial solutions describe the motion of an active particle. I · n dS = 0 I r ⇥ · n dS = 0 Microhydrodynamics: constraints M ˙ V = I · ndS + Fe I ˙ ⌦ = I r ⇥ · ndS + Te Force-free torque-free particles in absence of inertia

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Microhydrodynamics: momentum conservation • Stokes equation = pI + ⌘(rv + rvT ) r · = rp + ⌘r2v = 0 r · v = 0

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Microhydrodynamics: momentum conservation • Boundary condition v = vS, r 2 S n · = f, r 2 S Dirichlet Neumann • Stokes equation = pI + ⌘(rv + rvT ) r · = rp + ⌘r2v = 0 r · v = 0

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10 Boundary layers J. L. Anderson, Ann. Rev. Flu Mech 21(1989) Electrophoretic motion

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10 Boundary layers vS = ✏⇣ 4⇡⌘ ES Interfacial double layer of surface charge and diffused counter-ions determine the vS slip = electrophoretic J. L. Anderson, Ann. Rev. Flu Mech 21(1989) Electrophoretic motion

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11 Boundary layers Ciliary motion J. R. Blake JFM 46 (1971) vS(r) = V + ⌦ ⇥ r + vc(r) slip = ciliary

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11 Boundary layers Ciliary motion J. R. Blake JFM 46 (1971) vS(r) = V + ⌦ ⇥ r + vc(r) slip = ciliary • as a spherical body with an active boundary layer • this leads to a slip velocity on the surface Ideal active particle

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12 vS(r) = V + ⌦ ⇥ r + va(r) boundary velocity = rigid motion + active slip Ideal active particle

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Boundary integral equation for single particle

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14 Boundary integral equation vi(r) = Z ⇥ Gij(r, r0)fj(r0) Kjik(r, r0)nkvS j (r0) ⇤ dS

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14 Boundary integral equation vi(r) = Z ⇥ Gij(r, r0)fj(r0) Kjik(r, r0)nkvS j (r0) ⇤ dS 8⇡⌘ Gij(r) = ij r + rirj r3 8⇡ Kijk(r) = 6 rirjrk r5

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14 reduces a 3-dimensional problem of the solution of Stokes equation to a 2-dimensional integral equation on the boundary of the particles. massive simpliﬁcation for computational purposes! Boundary integral equation vi(r) = Z ⇥ Gij(r, r0)fj(r0) Kjik(r, r0)nkvS j (r0) ⇤ dS 8⇡⌘ Gij(r) = ij r + rirj r3 8⇡ Kijk(r) = 6 rirjrk r5

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15 r2 = 0 = S, r 2 S r · n = q, r 2 S Laplace equation Boundary conditions Electrostatic analogy (r) = Z  G(r r0) (r0) @G @n S(r0) d S

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15 r2 = 0 = S, r 2 S r · n = q, r 2 S Laplace equation Boundary conditions Electrostatic analogy (r) = Z  G(r r0) (r0) @G @n S(r0) d S vi(r) = Z ⇥ Gij(r, r0)fj(r0) Kjik(r, r0)nkvS j (r0) ⇤ dS

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15 Stokes equation Boundary conditions rp + ⌘r2v = 0 v = vS, r 2 S n · = f, r 2 S r2 = 0 = S, r 2 S r · n = q, r 2 S Laplace equation Boundary conditions Electrostatic analogy (r) = Z  G(r r0) (r0) @G @n S(r0) d S vi(r) = Z ⇥ Gij(r, r0)fj(r0) Kjik(r, r0)nkvS j (r0) ⇤ dS

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16 Electrostatic analogy (r) = Z  G(r r0) (r0) @G @n S(r0) d S vi(r) = Z ⇥ Gij(r, r0)fj(r0) Kjik(r, r0)nkvS j (r0) ⇤ dS

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16 ﬂow ~ potential traction ~ charge We can understand the microhydrodynamic phenomena guided by electrostatic analogies. The electrostatic equation is for scalar potential while the Stokes equation is for vector ﬁelds. Expect lots of indices in the formulation! Electrostatic analogy (r) = Z  G(r r0) (r0) @G @n S(r0) d S vi(r) = Z ⇥ Gij(r, r0)fj(r0) Kjik(r, r0)nkvS j (r0) ⇤ dS

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17 Coordinate system ⇢ R r ⇢

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18 • For discretisation of the integral equation, including both single and double layer, we expand the boundary ﬁelds in tensorial spherical harmonics Y(l). Tensorial spherical harmonics Y(l)

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18 • For discretisation of the integral equation, including both single and double layer, we expand the boundary ﬁelds in tensorial spherical harmonics Y(l). Tensorial spherical harmonics Y(l) Y(l)(ˆ ⇢) = ( 1)l⇢l+1r(l) 1 ⇢ • Y(l) are symmetric traceless in all indices • Orthogonal basis function on sphere surface

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19 Galerkin expansion of the boundary ﬁelds in Y(l) vi(r) = Z ⇥ Gij(r, r0)fj(r0) Kjik(r, r0)nkvS j (r0) ⇤ dS

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19 Galerkin expansion of the boundary ﬁelds in Y(l) f(Rn + ⇢n ) = X 2l + 1 4⇡a2 F(l+1) n Y(l)( ˆ ⇢n ) v(Rn + ⇢n ) = X 1 l!(2l 1)!! V(l+1) n Y(l)( ˆ ⇢n ) Notation: implies a l-fold contraction between a tensor of rank l and a higher rank tensor vi(r) = Z ⇥ Gij(r, r0)fj(r0) Kjik(r, r0)nkvS j (r0) ⇤ dS

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19 Galerkin expansion of the boundary ﬁelds in Y(l) f(Rn + ⇢n ) = X 2l + 1 4⇡a2 F(l+1) n Y(l)( ˆ ⇢n ) v(Rn + ⇢n ) = X 1 l!(2l 1)!! V(l+1) n Y(l)( ˆ ⇢n ) Notation: implies a l-fold contraction between a tensor of rank l and a higher rank tensor v(r) = P h G(l+1)(r, R) F(l+1) K(l+1)(r, R) V(l+1) i vi(r) = Z ⇥ Gij(r, r0)fj(r0) Kjik(r, r0)nkvS j (r0) ⇤ dS

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G(l+1)(r, R) = al ✓ 1 + a2 4l + 6 r2 ◆ r(l)G(r, R) 20 Flow expression as derivatives of Green’s function v(r) = P h G(l+1)(r, R) F(l+1) K(l+1)(r, R) V(l+1) i

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G(l+1)(r, R) = al ✓ 1 + a2 4l + 6 r2 ◆ r(l)G(r, R) 20 Flow expression as derivatives of Green’s function K(l+1)(r, R) = 4⇡al+1 (l 1)!(2l + 1)!! ✓ 1 + a2 4l + 6 r2 ◆ r(l 1)K(r, R) v(r) = P h G(l+1)(r, R) F(l+1) K(l+1)(r, R) V(l+1) i

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G(l+1)(r, R) = al ✓ 1 + a2 4l + 6 r2 ◆ r(l)G(r, R) 20 Flow expression as derivatives of Green’s function • F(l+1) and V(l+1) are symmetric irreducible in their last l indices • This leads to systematic classiﬁcation of the ﬂow terms K(l+1)(r, R) = 4⇡al+1 (l 1)!(2l + 1)!! ✓ 1 + a2 4l + 6 r2 ◆ r(l 1)K(r, R) v(r) = P h G(l+1)(r, R) F(l+1) K(l+1)(r, R) V(l+1) i

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Irreducible part of the traction • Using angular momentum algebra, each F(l+1) and V(l+1) can be written as a sum of three irreducible tensors of rank (l − 1), l and (l + 1). • Each F(I) and V(l) can then be written as in terms of F(lσ) and V(lσ), which are individually symmetric irreducible tensors of rank l − σ. • Irreducible parts of the traction for l=0, 1 and 2 F(1) i = Fe i Monopole (3) aF(2) ij = Sij + 1 2 ✏ijkTk Irreducible dipole (5) Anti-symmetric dipole (3)

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22 Irreducible part of the velocity V(1) = V Va 1 a V(2) = s + ✏ · (⌦ ⌦a) 1 a2 V3 = 2 3 (2) ✏ · + 3 5 (2) d Corresponding irreducible parts of the velocity

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22 Irreducible part of the velocity V(1) = V Va 1 a V(2) = s + ✏ · (⌦ ⌦a) 1 a2 V3 = 2 3 (2) ✏ · + 3 5 (2) d Corresponding irreducible parts of the velocity Va = hvai where the active translational velocity of the particle is and the active rotational velocity is ⌦a = 3 2a hˆ ⇢ ⇥ vai Here < - > indicates average over boundary.

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23 Fluid ﬂow at any order l has only three independent terms: irreducible gradient of the Green’s function, its curl and its Laplacian

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23 Fluid ﬂow at any order l has only three independent terms: irreducible gradient of the Green’s function, its curl and its Laplacian Fl = ✓ 1 + a2 4l + 6 r2 ◆ v(l ) = 0 = 1 = 2 l = 0 F0G · Q(10) — — l = 1 F1rG Q(20) 1 2 r ⇥ G · Q(21) — l = 2 F2rrG Q(30) 2 3 r r ⇥ G Q(31) 2 5 r2G Q(32) l = 3 F3rrrG Q(40) 3 4 rr · r ⇥ G Q(41) 36 35 rr2G Q(42) . . . . . . . . . . . . l Fl (l+1)r(l)G Q(l0) l l+1 (l) r(l) ⇥ G Q(l1) l(l+1) 2(2l+1) (l)r(l 2)r2G Q(l2)

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l Symmetric Antisymmetric Degenerate 0 1 2 3 Monopole 1/r

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l Symmetric Antisymmetric Degenerate 0 1 2 3 Symmetric irreducible dipole 1/r^2

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l Symmetric Antisymmetric Degenerate 0 1 2 3 Antisymmetric dipole 1/r^2

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l Symmetric Antisymmetric Degenerate 0 1 2 3 Symmetric irreducible quadrupole 1/r^3

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l Symmetric Antisymmetric Degenerate 0 1 2 3 Antisymmetric quadrupole 1/r^3

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l Symmetric Antisymmetric Degenerate 0 1 2 3 Degenerate quadrupole 1/r^3

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l Symmetric Antisymmetric Degenerate 0 1 2 3 Antisymmetric octupole 1/r^3

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31 One-body solution To determine velocity coefficients in terms of traction coefficients, expand the fluid velocity in the boundary integral equation and integrate on the surface of the particle

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31 One-body solution To determine velocity coefficients in terms of traction coefficients, expand the fluid velocity in the boundary integral equation and integrate on the surface of the particle Rigid body motion of active particles including the active contributions, generalising Stokes law for one particle. V = Va + Fe 6⇡⌘a l=0 Va = hvai ⌦ = ⌦a + Te 8⇡⌘a3 l=1, antisymmetric ⌦a = 3 2a hˆ ⇢ ⇥ vai

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32 Many-body solution The net flow is a superposition of flow due to each particle, because of linearity of the Stokes equation v = N X m=1 vm(r, Rm) The presence of n-th particle changes the boundary condition on m-th particle. And hence the traction and velocity boundary condition has to be satisfied simultaneously. This means a large systems of linear equations which must be solved simultaneously to obtain the irreducible coefficients and hence the RBM of the many active particles.

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33 The formal solution for the rigid body motion of passive particles is written in terms of mobility matrices Many-body solution ⌦n = ⌦a n + X h µRT nm · Fe m + µRR nm · Te m + ⇡(R,l+1) nm · V(l+1) m i Vn = Va + X h µTT nm · Fe m + µTR nm · Te m + ⇡(T,l+1) nm · V(l+1) m i

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34 The formal solution for the rigid body motion of active particles can then be written in terms of mobility and propulsion matrices Many-body solution ⌦n = ⌦a n + X h µRT nm · Fe m + µRR nm · Te m + ⇡(R,l+1) nm · V(l+1) m i Vn = Va + X h µTT nm · Fe m + µTR nm · Te m + ⇡(T,l+1) nm · V(l+1) m i

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⌦n = ⌦a n + X h µRT nm · Fe m + µRR nm · Te m + ⇡(R,l+1) nm · V(l+1) m i 35 The formal solution for the rigid body motion of active particles can then be written in terms of mobility and propulsion matrices Many-body solution Vn = Va + X h µTT nm · Fe m + µTR nm · Te m + ⇡(T,l+1) nm · V(l+1) m i The above clearly shows that particle can both rotate and translate in the absence of external forces and torques Rajesh Singh, Somdeb Ghose and R. Adhikari J. Stat. Mech. (2015)

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36 Summary of the particulate theory Stokes equation

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36 Summary of the particulate theory Stokes equation Integral representation

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36 Summary of the particulate theory Stokes equation Integral representation Galerkin expansion

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36 Summary of the particulate theory Stokes equation Integral representation Galerkin expansion Sum of irreducible parts

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36 Summary of the particulate theory Stokes equation Integral representation Galerkin expansion Sum of irreducible parts Classiﬁcation of ﬂow

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36 Summary of the particulate theory Stokes equation Integral representation Galerkin expansion Sum of irreducible parts Classiﬁcation of ﬂow ⌦n = ⌦a n + X h µRT nm · Fe m + µRR nm · Te m + ⇡(R,l+1) nm · V(l+1) m i Vn = Va + X h µTT nm · Fe m + µTR nm · Te m + ⇡(T,l+1) nm · V(l+1) m i Eliminating unknowns

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36 Summary of the particulate theory Stokes equation Integral representation Galerkin expansion Sum of irreducible parts Classification of flow • Generalisation of Stokes equation for active particles • Rigid body motion in terms of mobility and propulsion matrices • The infinitely many propulsion matrices, as compared to only four mobility matrices, account for the rich and diverse phenomena shown by active dynamical systems ⌦n = ⌦a n + X h µRT nm · Fe m + µRR nm · Te m + ⇡(R,l+1) nm · V(l+1) m i Vn = Va + X h µTT nm · Fe m + µTR nm · Te m + ⇡(T,l+1) nm · V(l+1) m i Eliminating unknowns

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Microhydrodynamic ﬂow around C. Reinhardtii

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Microswimming of C. Reinhardtii Flagella driven by dyenin motors J. S. Guasto, K. A. Johnson, J. P. Gollub, PRL (2010) Microhydrodynamic ﬂow around C. Reinhardtii

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Microswimming of C. Reinhardtii Flagella driven by dyenin motors J. S. Guasto, K. A. Johnson, J. P. Gollub, PRL (2010) Microhydrodynamic ﬂow around C. Reinhardtii

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Microswimming of C. Reinhardtii Flagella driven by dyenin motors J. S. Guasto, K. A. Johnson, J. P. Gollub, PRL (2010) Time-dependent flow ﬁeld Microhydrodynamic ﬂow around C. Reinhardtii

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39 V ( t ) = V0 + V1 exp( iwt ) + V2 exp(2 iwt ) Fit the following experimental quantities • translational motion to fix the degenerate quadrupole. • location of the stagnation point from the center of the swimmer to fix the symmetric irreducible dipole and quadrupole Sum the contributions to the flow from these three modes and compare ... Microhydrodynamic ﬂow around C. Reinhardtii 0 0.25 0.50 0.75 1.00 −0.2 0 0.2 0.4 0.6 0.8 t/T V(t) / Vmax v cm fit

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Experiment J. S. Guasto, K. A. Johnson, J. P. Gollub PRL (2010) Theory S. Ghose, R. Adhikari PRL (2014)

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Squirmer model

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49 Lighthill (1952) and Blake (1971) considered a sphere with axisymmetric slip velocity vS ⇢ = ˆ ⇢ · vS = 0, vS = ˆ · vS = 0, vS ✓ = ˆ ✓ · vS = B1 sin ✓ + B2 sin ✓ cos ✓, Squirmer model

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49 Lighthill (1952) and Blake (1971) considered a sphere with axisymmetric slip velocity vS ⇢ = ˆ ⇢ · vS = 0, vS = ˆ · vS = 0, vS ✓ = ˆ ✓ · vS = B1 sin ✓ + B2 sin ✓ cos ✓, This can be written more compactly as vS = (ˆ ⇢ˆ ⇢ I)  B1p + ✓ pp I 3 ◆ · ˆ ⇢ Here p is the orientation vector and B1 and B2 are two constants. M. J. Lighthill. Commun. Pure. Appl. Math., (1952). ; J. R. Blake JFM 46 (1971) Squirmer model

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50 The active slip velocity on the surface of the particle is vS = (ˆ ⇢ˆ ⇢ I)  B1p + ✓ pp I 3 ◆ · ˆ ⇢ Squirmer model

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50 The active slip velocity on the surface of the particle is vS = (ˆ ⇢ˆ ⇢ I)  B1p + ✓ pp I 3 ◆ · ˆ ⇢ Writing this in tensorial spherical harmonics we have vS = 2 3 B1p + B1Y(2) · p + ✓ pp I 3 ◆ ·  3 5 B2Y(1) + B2Y(3) Active translational velocity of the squirmer is then Va n = hvSi = 2B1 3 p Squirmer model

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51 Flow ﬁeld around the squirmer Squirmer model

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Active particles in a harmonic trap

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53 Active particles in a harmonic potential The equation of motion of an isolated active particle in a trap where F =-kR is the harmonic force and k is spring constant. ˙ R = F 6⇡⌘a + Va where p is the orientational vector. The self-propulsion of the active particle is Va n = hvSi = 2B1 3 p

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53 The stable solution of this equation is particle pointing radially outward on the surface of a sphere of radius R⇤ = 6⇡⌘avs/k Active particles in a harmonic potential The equation of motion of an isolated active particle in a trap where F =-kR is the harmonic force and k is spring constant. ˙ R = F 6⇡⌘a + Va where p is the orientational vector. The self-propulsion of the active particle is Va n = hvSi = 2B1 3 p

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53 The stable solution of this equation is particle pointing radially outward on the surface of a sphere of radius R⇤ = 6⇡⌘avs/k Active particles in a harmonic potential R W Nash, R Adhikari, J Tailleur and M E Cates PRL (2010); R Singh, S Ghose and R. Adhikari J. Stat. Mech. (2015) The equation of motion of an isolated active particle in a trap where F =-kR is the harmonic force and k is spring constant. ˙ R = F 6⇡⌘a + Va where p is the orientational vector. The self-propulsion of the active particle is Va n = hvSi = 2B1 3 p

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54 Equation of motion in a trap

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54 One body self-propulsion Equation of motion in a trap

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54 Two-body rotation from the Stokeslet vorticity One body self-propulsion Equation of motion in a trap

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55 Ω Ω x y z Flow around a trapped active particle

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55 Ω Ω x y z The leading term in the angular update equation destabilises the stable state of particles stuck at trap surface pointing outwards. Flow around a trapped active particle

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56 Dynamics of two squirmers in a trap

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56 Dynamics of two squirmers in a trap Two particles initialised on diametrically opposite points are stable as the angular velocity is zero. For angles less than 180, they get rotated towards each other and swim into the trap leading to the formation of an orbit.

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57 Convective rolls of squirmers in the trap Stable state in absence of HI

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57 Convective rolls of squirmers in the trap Stable state in absence of HI Destabilization of the initial state

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57 Convective rolls of squirmers in the trap Stable state in absence of HI Destabilization of the initial state Convection by one-body terms in velocity and rolls by leading order monopolar vorticity

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59 • Formalism for studying hydrodynamic interactions between active colloids that does not need explicit fluid degrees of freedom. Result is expressed in terms of familiar mobility matrices and newly introduced propulsion matrices. • As fluid degrees of freedom are not needed, orders of magnitude more particles can be simulated, e. g. 10^4 particles in harmonic trap. Fast summation methods can increase this to 10^6 and beyond. • Formalism can be extended easily to active colloidal flows near a single wall, between parallel walls, and in periodic volumes. • A complete and tractable microscopic model for active matter. Summary

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Thank you for your attention!