Rajesh Singh Department of Applied Mathematics and Theoretical Physics Theories without time-reversal symmetry in soft matter

Plan of the talk Experimental observation of incomplete phase separation in active matter Theory of binary fluid mixtures (how do oil-water mixtures phase separate?) Theory of active phase separation (microphase separation) Generalised Stokes laws of colloidal particles with slip (active particles) Phoresis and Stokesian hydrodynamic of active particles How to freeze colloids by heating them? Field-theoretic Particle-based 3 Outlook: field-theoretic and particle-based theories of active matter

This work has been done with Michael E. Cates I. Incomplete phase separation in scalar active matter

micrometer size - non-equilibrium processes on the surface create exterior flow and may lead to self-propulsion feeding or fuel => break time-reversal symmetry locally Active particles: special colloids 5 Microorganisms Autophoretic particles Ramaswamy Annu. Rev. Condens. Matter Phys. 2010, JSTAT 2017; Cates arXiv:1904.01330 Active matter: active particles in a fluid

micrometer size - non-equilibrium processes on the surface create exterior flow and may lead to self-propulsion feeding or fuel => break time-reversal symmetry locally Active particles: special colloids 5 Particle-level dynamics of active particles, unlike Brownian colloids, has no TRS => no inherent Free energy or Boltzmann distribution nonequilibrium steady-state A B C Active particles equilibrium steady-state zero current, TRS Passive particles A B C net current, no TRS Microorganisms Autophoretic particles Ramaswamy Annu. Rev. Condens. Matter Phys. 2010, JSTAT 2017; Cates arXiv:1904.01330 Active matter: active particles in a fluid

micrometer size - non-equilibrium processes on the surface create exterior flow and may lead to self-propulsion feeding or fuel => break time-reversal symmetry locally Active particles: special colloids 5 Particle-level dynamics of active particles, unlike Brownian colloids, has no TRS => no inherent Free energy or Boltzmann distribution nonequilibrium steady-state A B C Active particles equilibrium steady-state zero current, TRS Passive particles A B C net current, no TRS Microorganisms Autophoretic particles How to study such nonequilibrium systems in absence of time- reversal symmetry for the particle-level dynamics? Ramaswamy Annu. Rev. Condens. Matter Phys. 2010, JSTAT 2017; Cates arXiv:1904.01330 Active matter: active particles in a fluid

Incomplete active phase separations 6 Buttinoni et al PRL 2013 SiO2 beads and a sputtering a thin layer of graphite onto one hemisphere in H2O2 Theurkauff et al PRL 2012 gold-platinum Janus particles in H2O2 Palacci et al Science 2013 A bimaterial colloid: polymer sphere with a hematite cube (dark) in H2O2

Incomplete active phase separations 6 Buttinoni et al PRL 2013 SiO2 beads and a sputtering a thin layer of graphite onto one hemisphere in H2O2 Theurkauff et al PRL 2012 gold-platinum Janus particles in H2O2 Palacci et al Science 2013 A bimaterial colloid: polymer sphere with a hematite cube (dark) in H2O2 How to build a theory of such phase separations?

Incomplete active phase separations 6 Buttinoni et al PRL 2013 SiO2 beads and a sputtering a thin layer of graphite onto one hemisphere in H2O2 Theurkauff et al PRL 2012 gold-platinum Janus particles in H2O2 Palacci et al Science 2013 A bimaterial colloid: polymer sphere with a hematite cube (dark) in H2O2 How to build a theory of such phase separations? symmetries and conservation laws experiments are for spherical particles - use a scalar field number of active particles is conserved - use a conserved scalar field

Incomplete active phase separations 6 Buttinoni et al PRL 2013 SiO2 beads and a sputtering a thin layer of graphite onto one hemisphere in H2O2 Theurkauff et al PRL 2012 gold-platinum Janus particles in H2O2 Palacci et al Science 2013 A bimaterial colloid: polymer sphere with a hematite cube (dark) in H2O2 How to build a theory of such phase separations? symmetries and conservation laws experiments are for spherical particles - use a scalar field number of active particles is conserved - use a conserved scalar field start with a theory of phase separation in binary mixtures add minimal terms which break TRS recipe to build an active field theory

7 How do binary mixtures (such as oil-water) phase separate from a homogenous mixed phase? Theory of coarsening kinetics in binary fluid mixtures Bray, Adv. in Phys. 43, 357 (1994)

7 How do binary mixtures (such as oil-water) phase separate from a homogenous mixed phase? Theory of coarsening kinetics in binary fluid mixtures Consider a scalar field describing the local composition in a symmetric binary mixture AAACJnicbVDLSsNAFJ3UV62vqEs3wSJUxJKIoptCrRuXFewDmhAm00k7dDIJMxOhhHyNG3/FjYuKiDs/xUkbirYeGDiccy93zvEiSoQ0zS+tsLK6tr5R3Cxtbe/s7un7B20RxhzhFgppyLseFJgShluSSIq7Eccw8CjueKO7zO88YS5IyB7lOMJOAAeM+ARBqSRXr9nRkFTsAMqh5yc8Pa3ZPocosfkwdJPb9HxGGmk6l87mUsnVy2bVnMJYJlZOyiBH09Undj9EcYCZRBQK0bPMSDoJ5JIgitOSHQscQTSCA9xTlMEACyeZxkyNE6X0DT/k6jFpTNXfGwkMhBgHnprM8ohFLxP/83qx9G+chLAolpih2SE/poYMjawzo084RpKOFYGIE/VXAw2hqkmqZrMSrMXIy6R9UbWuqubDZbneyOsogiNwDCrAAtegDu5BE7QAAs/gFUzAu/aivWkf2udstKDlO4fgD7TvH/6VprQ= (r) = ⇢A ⇢B ⇢A + ⇢B = 1 = +1 AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS B-phase A-phase Bray, Adv. in Phys. 43, 357 (1994)

7 How do binary mixtures (such as oil-water) phase separate from a homogenous mixed phase? Theory of coarsening kinetics in binary fluid mixtures Consider a scalar field describing the local composition in a symmetric binary mixture AAACJnicbVDLSsNAFJ3UV62vqEs3wSJUxJKIoptCrRuXFewDmhAm00k7dDIJMxOhhHyNG3/FjYuKiDs/xUkbirYeGDiccy93zvEiSoQ0zS+tsLK6tr5R3Cxtbe/s7un7B20RxhzhFgppyLseFJgShluSSIq7Eccw8CjueKO7zO88YS5IyB7lOMJOAAeM+ARBqSRXr9nRkFTsAMqh5yc8Pa3ZPocosfkwdJPb9HxGGmk6l87mUsnVy2bVnMJYJlZOyiBH09Undj9EcYCZRBQK0bPMSDoJ5JIgitOSHQscQTSCA9xTlMEACyeZxkyNE6X0DT/k6jFpTNXfGwkMhBgHnprM8ohFLxP/83qx9G+chLAolpih2SE/poYMjawzo084RpKOFYGIE/VXAw2hqkmqZrMSrMXIy6R9UbWuqubDZbneyOsogiNwDCrAAtegDu5BE7QAAs/gFUzAu/aivWkf2udstKDlO4fgD7TvH/6VprQ= (r) = ⇢A ⇢B ⇢A + ⇢B = 1 = +1 AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS B-phase A-phase The scalar field is conserved: ˙ = r · J Bray, Adv. in Phys. 43, 357 (1994)

7 How do binary mixtures (such as oil-water) phase separate from a homogenous mixed phase? Theory of coarsening kinetics in binary fluid mixtures Consider a scalar field describing the local composition in a symmetric binary mixture AAACJnicbVDLSsNAFJ3UV62vqEs3wSJUxJKIoptCrRuXFewDmhAm00k7dDIJMxOhhHyNG3/FjYuKiDs/xUkbirYeGDiccy93zvEiSoQ0zS+tsLK6tr5R3Cxtbe/s7un7B20RxhzhFgppyLseFJgShluSSIq7Eccw8CjueKO7zO88YS5IyB7lOMJOAAeM+ARBqSRXr9nRkFTsAMqh5yc8Pa3ZPocosfkwdJPb9HxGGmk6l87mUsnVy2bVnMJYJlZOyiBH09Undj9EcYCZRBQK0bPMSDoJ5JIgitOSHQscQTSCA9xTlMEACyeZxkyNE6X0DT/k6jFpTNXfGwkMhBgHnprM8ohFLxP/83qx9G+chLAolpih2SE/poYMjawzo084RpKOFYGIE/VXAw2hqkmqZrMSrMXIy6R9UbWuqubDZbneyOsogiNwDCrAAtegDu5BE7QAAs/gFUzAu/aivWkf2udstKDlO4fgD7TvH/6VprQ= (r) = ⇢A ⇢B ⇢A + ⇢B = 1 = +1 AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS B-phase A-phase Gaussian white noise mechanical part by solving for flow diffusive part from a chemical potential µ The current has three parts: 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 J = rµ + v + p 2D⇤, The scalar field is conserved: ˙ = r · J Bray, Adv. in Phys. 43, 357 (1994)

φ4 field theory of passive phase separation ˙ = r · J Model H 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 J = rµ + v + p 2D⇤, Stokes equation 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 rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I no flow the fluid flow acts to minimise the deformation of the droplet 8 Hohenberg & Halperin 1977; Bray, Adv in Phys 1994; Kendon et al, JFM 2001; Cates and Tjhung, JFM 2018

φ4 field theory of passive phase separation ˙ = r · J Model H AAACQ3icbVDLSsNAFJ3UV62vqks3waIIaklE0Y0g6kLEhYJ9YBPKzWTaDk4mcWZSKCH/5sYfcOcPuHGhiFvBSc2iPi4MczjnHu69x4sYlcqynozC2PjE5FRxujQzOze/UF5cqsswFpjUcMhC0fRAEkY5qSmqGGlGgkDgMdLwbk8yvdEnQtKQX6tBRNwAupx2KAalqXb5xvFC5stBoL/kPF0/3B4lHA4eg9QJ4s1Rup86UY9uOvJOqGTnNP1hudDDfUi3Su1yxapawzL/AjsHFZTXZbv86PghjgPCFWYgZcu2IuUmIBTFjKQlJ5YkAnwLXdLSkENApJsMM0jNNc34ZicU+nFlDtlRRwKBzFbUnQGonvytZeR/WitWnQM3oTyKFeH4e1AnZqYKzSxQ06eCYMUGGgAWVO9q4h4IwErHnoVg/z75L6jvVO29qnW1Wzk6zuMoohW0ijaQjfbRETpDl6iGMLpHz+gVvRkPxovxbnx8txaM3LOMfpTx+QWT1rOh J = rµ + v + p 2D⇤, Stokes equation 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 rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I no flow the fluid flow acts to minimise the deformation of the droplet 8 Hohenberg & Halperin 1977; Bray, Adv in Phys 1994; Kendon et al, JFM 2001; Cates and Tjhung, JFM 2018 AAACE3icbVDLSsNAFJ3UV62vqEs3g0UQFyURRTdCURCXFewDmlAm00k7dGYSZiZCCfkHN/6KGxeKuHXjzr9x0kbQ1gMXDufcy733BDGjSjvOl1VaWFxaXimvVtbWNza37O2dlooSiUkTRyySnQApwqggTU01I51YEsQDRtrB6Cr32/dEKhqJOz2Oic/RQNCQYqSN1LOPPJ7AC+iFEuHU6xOmkceRHmLE0uss+5HiIc16dtWpORPAeeIWpAoKNHr2p9ePcMKJ0JghpbquE2s/RVJTzEhW8RJFYoRHaEC6hgrEifLTyU8ZPDBKH4aRNCU0nKi/J1LElRrzwHTm96pZLxf/87qJDs/9lIo40UTg6aIwYVBHMA8I9qkkWLOxIQhLam6FeIhMPNrEWDEhuLMvz5PWcc09rTm3J9X6ZRFHGeyBfXAIXHAG6uAGNEATYPAAnsALeLUerWfrzXqftpasYmYX/IH18Q3itJ7T µ = F 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 F[ ] = Z ✓ a 2 2 + b 4 4 +  2 (r )2 ◆ dr free energy functional:

φ4 field theory of passive phase separation ˙ = r · J Model H 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 J = rµ + v + p 2D⇤, Stokes equation 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 rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I no flow the fluid flow acts to minimise the deformation of the droplet 8 AAAB7nicbVBNSwMxEJ2tX7V+VT16CRahXsquKHosevFYwX5Au5Rsmm1Ds0lIskJZ+iO8eFDEq7/Hm//GtN2Dtj4YeLw3w8y8SHFmrO9/e4W19Y3NreJ2aWd3b/+gfHjUMjLVhDaJ5FJ3ImwoZ4I2LbOcdpSmOIk4bUfju5nffqLaMCke7UTRMMFDwWJGsHVSO6721Iid98sVv+bPgVZJkJMK5Gj0y1+9gSRpQoUlHBvTDXxlwwxrywin01IvNVRhMsZD2nVU4ISaMJufO0VnThmgWGpXwqK5+nsiw4kxkyRynQm2I7PszcT/vG5q45swY0KllgqyWBSnHFmJZr+jAdOUWD5xBBPN3K2IjLDGxLqESi6EYPnlVdK6qAVXNf/hslK/zeMowgmcQhUCuIY63EMDmkBgDM/wCm+e8l68d+9j0Vrw8plj+APv8wed0o8Y f( ) Hohenberg & Halperin 1977; Bray, Adv in Phys 1994; Kendon et al, JFM 2001; Cates and Tjhung, JFM 2018 AAACE3icbVDLSsNAFJ3UV62vqEs3g0UQFyURRTdCURCXFewDmlAm00k7dGYSZiZCCfkHN/6KGxeKuHXjzr9x0kbQ1gMXDufcy733BDGjSjvOl1VaWFxaXimvVtbWNza37O2dlooSiUkTRyySnQApwqggTU01I51YEsQDRtrB6Cr32/dEKhqJOz2Oic/RQNCQYqSN1LOPPJ7AC+iFEuHU6xOmkceRHmLE0uss+5HiIc16dtWpORPAeeIWpAoKNHr2p9ePcMKJ0JghpbquE2s/RVJTzEhW8RJFYoRHaEC6hgrEifLTyU8ZPDBKH4aRNCU0nKi/J1LElRrzwHTm96pZLxf/87qJDs/9lIo40UTg6aIwYVBHMA8I9qkkWLOxIQhLam6FeIhMPNrEWDEhuLMvz5PWcc09rTm3J9X6ZRFHGeyBfXAIXHAG6uAGNEATYPAAnsALeLUerWfrzXqftpasYmYX/IH18Q3itJ7T µ = F 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 F[ ] = Z ✓ a 2 2 + b 4 4 +  2 (r )2 ◆ dr free energy functional:

φ4 field theory of passive phase separation ˙ = r · J Model H 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 J = rµ + v + p 2D⇤, Stokes equation 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 rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I no flow the fluid flow acts to minimise the deformation of the droplet 8 AAAB7nicbVBNSwMxEJ2tX7V+VT16CRahXsquKHosevFYwX5Au5Rsmm1Ds0lIskJZ+iO8eFDEq7/Hm//GtN2Dtj4YeLw3w8y8SHFmrO9/e4W19Y3NreJ2aWd3b/+gfHjUMjLVhDaJ5FJ3ImwoZ4I2LbOcdpSmOIk4bUfju5nffqLaMCke7UTRMMFDwWJGsHVSO6721Iid98sVv+bPgVZJkJMK5Gj0y1+9gSRpQoUlHBvTDXxlwwxrywin01IvNVRhMsZD2nVU4ISaMJufO0VnThmgWGpXwqK5+nsiw4kxkyRynQm2I7PszcT/vG5q45swY0KllgqyWBSnHFmJZr+jAdOUWD5xBBPN3K2IjLDGxLqESi6EYPnlVdK6qAVXNf/hslK/zeMowgmcQhUCuIY63EMDmkBgDM/wCm+e8l68d+9j0Vrw8plj+APv8wed0o8Y f( ) Hohenberg & Halperin 1977; Bray, Adv in Phys 1994; Kendon et al, JFM 2001; Cates and Tjhung, JFM 2018 AAACE3icbVDLSsNAFJ3UV62vqEs3g0UQFyURRTdCURCXFewDmlAm00k7dGYSZiZCCfkHN/6KGxeKuHXjzr9x0kbQ1gMXDufcy733BDGjSjvOl1VaWFxaXimvVtbWNza37O2dlooSiUkTRyySnQApwqggTU01I51YEsQDRtrB6Cr32/dEKhqJOz2Oic/RQNCQYqSN1LOPPJ7AC+iFEuHU6xOmkceRHmLE0uss+5HiIc16dtWpORPAeeIWpAoKNHr2p9ePcMKJ0JghpbquE2s/RVJTzEhW8RJFYoRHaEC6hgrEifLTyU8ZPDBKH4aRNCU0nKi/J1LElRrzwHTm96pZLxf/87qJDs/9lIo40UTg6aIwYVBHMA8I9qkkWLOxIQhLam6FeIhMPNrEWDEhuLMvz5PWcc09rTm3J9X6ZRFHGeyBfXAIXHAG6uAGNEATYPAAnsALeLUerWfrzXqftpasYmYX/IH18Q3itJ7T µ = F 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 F[ ] = Z ✓ a 2 2 + b 4 4 +  2 (r )2 ◆ dr free energy functional: -1 1

φ4 field theory of passive phase separation ˙ = r · J Model H 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 J = rµ + v + p 2D⇤, Stokes equation 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 rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I no flow the fluid flow acts to minimise the deformation of the droplet 8 AAAB7nicbVBNSwMxEJ2tX7V+VT16CRahXsquKHosevFYwX5Au5Rsmm1Ds0lIskJZ+iO8eFDEq7/Hm//GtN2Dtj4YeLw3w8y8SHFmrO9/e4W19Y3NreJ2aWd3b/+gfHjUMjLVhDaJ5FJ3ImwoZ4I2LbOcdpSmOIk4bUfju5nffqLaMCke7UTRMMFDwWJGsHVSO6721Iid98sVv+bPgVZJkJMK5Gj0y1+9gSRpQoUlHBvTDXxlwwxrywin01IvNVRhMsZD2nVU4ISaMJufO0VnThmgWGpXwqK5+nsiw4kxkyRynQm2I7PszcT/vG5q45swY0KllgqyWBSnHFmJZr+jAdOUWD5xBBPN3K2IjLDGxLqESi6EYPnlVdK6qAVXNf/hslK/zeMowgmcQhUCuIY63EMDmkBgDM/wCm+e8l68d+9j0Vrw8plj+APv8wed0o8Y f( ) cost to create an interface AAACEXicbVDLSsNAFJ34rPVVdelmsAjdWJOqWBdC0Y3LCvYBTRpuptN26EwSZyZCCfkFN/6KGxeKuHXnzr8xfSy09cCFwzn3cu89XsiZ0qb5bSwsLi2vrGbWsusbm1vbuZ3dugoiSWiNBDyQTQ8U5cynNc00p81QUhAepw1vcD3yGw9UKhb4d3oYUkdAz2ddRkCnkpsr2D0QAtzYTC5tdS91fFS2BxCGgKEdnyTHF147LiVJ1s3lzaI5Bp4n1pTk0RRVN/dldwISCeprwkGplmWG2olBakY4TbJ2pGgIZAA92kqpD4IqJx5/lODDVOngbiDT8jUeq78nYhBKDYWXdgrQfTXrjcT/vFaku2UnZn4YaeqTyaJuxLEO8Cge3GGSEs2HKQEiWXorJn2QQHQa4igEa/bleVIvFa2zonl7mq9cTePIoH10gArIQueogm5QFdUQQY/oGb2iN+PJeDHejY9J64IxndlDf2B8/gBgh5y0 0 = p 8a3/9b2 interfacial tension Hohenberg & Halperin 1977; Bray, Adv in Phys 1994; Kendon et al, JFM 2001; Cates and Tjhung, JFM 2018 AAACE3icbVDLSsNAFJ3UV62vqEs3g0UQFyURRTdCURCXFewDmlAm00k7dGYSZiZCCfkHN/6KGxeKuHXjzr9x0kbQ1gMXDufcy733BDGjSjvOl1VaWFxaXimvVtbWNza37O2dlooSiUkTRyySnQApwqggTU01I51YEsQDRtrB6Cr32/dEKhqJOz2Oic/RQNCQYqSN1LOPPJ7AC+iFEuHU6xOmkceRHmLE0uss+5HiIc16dtWpORPAeeIWpAoKNHr2p9ePcMKJ0JghpbquE2s/RVJTzEhW8RJFYoRHaEC6hgrEifLTyU8ZPDBKH4aRNCU0nKi/J1LElRrzwHTm96pZLxf/87qJDs/9lIo40UTg6aIwYVBHMA8I9qkkWLOxIQhLam6FeIhMPNrEWDEhuLMvz5PWcc09rTm3J9X6ZRFHGeyBfXAIXHAG6uAGNEATYPAAnsALeLUerWfrzXqftpasYmYX/IH18Q3itJ7T µ = F 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 F[ ] = Z ✓ a 2 2 + b 4 4 +  2 (r )2 ◆ dr free energy functional: -1 1

φ4 field theory of passive phase separation ˙ = r · J Model H 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 J = rµ + v + p 2D⇤, Stokes equation 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 rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I no flow the fluid flow acts to minimise the deformation of the droplet 8 AAAB7nicbVBNSwMxEJ2tX7V+VT16CRahXsquKHosevFYwX5Au5Rsmm1Ds0lIskJZ+iO8eFDEq7/Hm//GtN2Dtj4YeLw3w8y8SHFmrO9/e4W19Y3NreJ2aWd3b/+gfHjUMjLVhDaJ5FJ3ImwoZ4I2LbOcdpSmOIk4bUfju5nffqLaMCke7UTRMMFDwWJGsHVSO6721Iid98sVv+bPgVZJkJMK5Gj0y1+9gSRpQoUlHBvTDXxlwwxrywin01IvNVRhMsZD2nVU4ISaMJufO0VnThmgWGpXwqK5+nsiw4kxkyRynQm2I7PszcT/vG5q45swY0KllgqyWBSnHFmJZr+jAdOUWD5xBBPN3K2IjLDGxLqESi6EYPnlVdK6qAVXNf/hslK/zeMowgmcQhUCuIY63EMDmkBgDM/wCm+e8l68d+9j0Vrw8plj+APv8wed0o8Y f( ) cost to create an interface AAACEXicbVDLSsNAFJ34rPVVdelmsAjdWJOqWBdC0Y3LCvYBTRpuptN26EwSZyZCCfkFN/6KGxeKuHXnzr8xfSy09cCFwzn3cu89XsiZ0qb5bSwsLi2vrGbWsusbm1vbuZ3dugoiSWiNBDyQTQ8U5cynNc00p81QUhAepw1vcD3yGw9UKhb4d3oYUkdAz2ddRkCnkpsr2D0QAtzYTC5tdS91fFS2BxCGgKEdnyTHF147LiVJ1s3lzaI5Bp4n1pTk0RRVN/dldwISCeprwkGplmWG2olBakY4TbJ2pGgIZAA92kqpD4IqJx5/lODDVOngbiDT8jUeq78nYhBKDYWXdgrQfTXrjcT/vFaku2UnZn4YaeqTyaJuxLEO8Cge3GGSEs2HKQEiWXorJn2QQHQa4igEa/bleVIvFa2zonl7mq9cTePIoH10gArIQueogm5QFdUQQY/oGb2iN+PJeDHejY9J64IxndlDf2B8/gBgh5y0 0 = p 8a3/9b2 interfacial tension Hohenberg & Halperin 1977; Bray, Adv in Phys 1994; Kendon et al, JFM 2001; Cates and Tjhung, JFM 2018 AAACE3icbVDLSsNAFJ3UV62vqEs3g0UQFyURRTdCURCXFewDmlAm00k7dGYSZiZCCfkHN/6KGxeKuHXjzr9x0kbQ1gMXDufcy733BDGjSjvOl1VaWFxaXimvVtbWNza37O2dlooSiUkTRyySnQApwqggTU01I51YEsQDRtrB6Cr32/dEKhqJOz2Oic/RQNCQYqSN1LOPPJ7AC+iFEuHU6xOmkceRHmLE0uss+5HiIc16dtWpORPAeeIWpAoKNHr2p9ePcMKJ0JghpbquE2s/RVJTzEhW8RJFYoRHaEC6hgrEifLTyU8ZPDBKH4aRNCU0nKi/J1LElRrzwHTm96pZLxf/87qJDs/9lIo40UTg6aIwYVBHMA8I9qkkWLOxIQhLam6FeIhMPNrEWDEhuLMvz5PWcc09rTm3J9X6ZRFHGeyBfXAIXHAG6uAGNEATYPAAnsALeLUerWfrzXqftpasYmYX/IH18Q3itJ7T µ = F 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 F[ ] = Z ✓ a 2 2 + b 4 4 +  2 (r )2 ◆ dr free energy functional: -1 1 Widely separated droplets of varying sizes

φ4 field theory of passive phase separation ˙ = r · J Model H 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 J = rµ + v + p 2D⇤, Stokes equation 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 rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I no flow the fluid flow acts to minimise the deformation of the droplet 8 AAAB7nicbVBNSwMxEJ2tX7V+VT16CRahXsquKHosevFYwX5Au5Rsmm1Ds0lIskJZ+iO8eFDEq7/Hm//GtN2Dtj4YeLw3w8y8SHFmrO9/e4W19Y3NreJ2aWd3b/+gfHjUMjLVhDaJ5FJ3ImwoZ4I2LbOcdpSmOIk4bUfju5nffqLaMCke7UTRMMFDwWJGsHVSO6721Iid98sVv+bPgVZJkJMK5Gj0y1+9gSRpQoUlHBvTDXxlwwxrywin01IvNVRhMsZD2nVU4ISaMJufO0VnThmgWGpXwqK5+nsiw4kxkyRynQm2I7PszcT/vG5q45swY0KllgqyWBSnHFmJZr+jAdOUWD5xBBPN3K2IjLDGxLqESi6EYPnlVdK6qAVXNf/hslK/zeMowgmcQhUCuIY63EMDmkBgDM/wCm+e8l68d+9j0Vrw8plj+APv8wed0o8Y f( ) cost to create an interface AAACEXicbVDLSsNAFJ34rPVVdelmsAjdWJOqWBdC0Y3LCvYBTRpuptN26EwSZyZCCfkFN/6KGxeKuHXnzr8xfSy09cCFwzn3cu89XsiZ0qb5bSwsLi2vrGbWsusbm1vbuZ3dugoiSWiNBDyQTQ8U5cynNc00p81QUhAepw1vcD3yGw9UKhb4d3oYUkdAz2ddRkCnkpsr2D0QAtzYTC5tdS91fFS2BxCGgKEdnyTHF147LiVJ1s3lzaI5Bp4n1pTk0RRVN/dldwISCeprwkGplmWG2olBakY4TbJ2pGgIZAA92kqpD4IqJx5/lODDVOngbiDT8jUeq78nYhBKDYWXdgrQfTXrjcT/vFaku2UnZn4YaeqTyaJuxLEO8Cge3GGSEs2HKQEiWXorJn2QQHQa4igEa/bleVIvFa2zonl7mq9cTePIoH10gArIQueogm5QFdUQQY/oGb2iN+PJeDHejY9J64IxndlDf2B8/gBgh5y0 0 = p 8a3/9b2 interfacial tension Hohenberg & Halperin 1977; Bray, Adv in Phys 1994; Kendon et al, JFM 2001; Cates and Tjhung, JFM 2018 AAACE3icbVDLSsNAFJ3UV62vqEs3g0UQFyURRTdCURCXFewDmlAm00k7dGYSZiZCCfkHN/6KGxeKuHXjzr9x0kbQ1gMXDufcy733BDGjSjvOl1VaWFxaXimvVtbWNza37O2dlooSiUkTRyySnQApwqggTU01I51YEsQDRtrB6Cr32/dEKhqJOz2Oic/RQNCQYqSN1LOPPJ7AC+iFEuHU6xOmkceRHmLE0uss+5HiIc16dtWpORPAeeIWpAoKNHr2p9ePcMKJ0JghpbquE2s/RVJTzEhW8RJFYoRHaEC6hgrEifLTyU8ZPDBKH4aRNCU0nKi/J1LElRrzwHTm96pZLxf/87qJDs/9lIo40UTg6aIwYVBHMA8I9qkkWLOxIQhLam6FeIhMPNrEWDEhuLMvz5PWcc09rTm3J9X6ZRFHGeyBfXAIXHAG6uAGNEATYPAAnsALeLUerWfrzXqftpasYmYX/IH18Q3itJ7T µ = F 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 F[ ] = Z ✓ a 2 2 + b 4 4 +  2 (r )2 ◆ dr free energy functional: -1 1 Widely separated droplets of varying sizes What happens next?

9 Model H: Ostwald ripening Chemical potential raised at curved interface μ s AAACC3icbVDLSsNAFJ3UV62vqEs3Q4tQNyURRTdC0Y3LWuwDmhAm00k7dGYSZiZCCd278VfcuFDErT/gzr9x2mahrQcuHM65l3vvCRNGlXacb6uwsrq2vlHcLG1t7+zu2fsHbRWnEpMWjlksuyFShFFBWppqRrqJJIiHjHTC0c3U7zwQqWgs7vU4IT5HA0EjipE2UmCXPZ4Gqto8gVfQiyTCmTdAnKNJ5iVDGoSwOQnsilNzZoDLxM1JBeRoBPaX149xyonQmCGleq6TaD9DUlPMyKTkpYokCI/QgPQMFYgT5WezXybw2Ch9GMXSlNBwpv6eyBBXasxD08mRHqpFbyr+5/VSHV36GRVJqonA80VRyqCO4TQY2KeSYM3GhiAsqbkV4iEyiWgTX8mE4C6+vEzapzX3vObcnVXq13kcRXAEyqAKXHAB6uAWNEALYPAInsEreLOerBfr3fqYtxasfOYQ/IH1+QPRLJpK µs(R) = bR AAACGXicbVDLSgMxFM34rPU16tJNsAgVocyIohuh6MZlLfYBnTLcSTNtaDIzJBmhDv0NN/6KGxeKuNSVf2P6WGjrgcDhnHO5uSdIOFPacb6thcWl5ZXV3Fp+fWNza9ve2a2rOJWE1kjMY9kMQFHOIlrTTHPaTCQFEXDaCPrXI79xT6VicXSnBwltC+hGLGQEtJF82/FE6qti9QhfYi+UQDKvC0KA7wwzL+kxP8DVIT7Go5j3QDX4dsEpOWPgeeJOSQFNUfHtT68Tk1TQSBMOSrVcJ9HtDKRmhNNh3ksVTYD0oUtbhkYgqGpn48uG+NAoHRzG0rxI47H6eyIDodRABCYpQPfUrDcS//NaqQ4v2hmLklTTiEwWhSnHOsajmnCHSUo0HxgCRDLzV0x6YPrRpsy8KcGdPXme1E9K7lnJuT0tlK+mdeTQPjpAReSic1RGN6iCaoigR/SMXtGb9WS9WO/WxyS6YE1n9tAfWF8/rKiffQ== µs(R) = 0 bR + µ⇣ Bray, Adv. in Phys. 43, 357 (1994) Widely separated droplets of varying sizes

9 Model H: Ostwald ripening AAACGHicbZBNS8MwGMfT+TbnW9Wjl+AQ5mGzFUUvg6EX8TTFvcBaSpqlMyxNS5IKo+xjePGrePGgiNfd/DamWw+6+YfAL//neUievx8zKpVlfRuFpeWV1bXiemljc2t7x9zda8soEZi0cMQi0fWRJIxy0lJUMdKNBUGhz0jHH15n9c4TEZJG/EGNYuKGaMBpQDFS2vLMk1tYuT+uw6rDkc8QdMIkM2A9Iy91KA/UaFzNLjLzPbNs1ayp4CLYOZRBrqZnTpx+hJOQcIUZkrJnW7FyUyQUxYyMS04iSYzwEA1ITyNHIZFuOl1sDI+004dBJPThCk7d3xMpCqUchb7uDJF6lPO1zPyv1ktUcOmmlMeJIhzPHgoSBlUEs5RgnwqCFRtpQFhQ/VeIH5FAWOksSzoEe37lRWif1uzzmnV3Vm5c5XEUwQE4BBVggwvQADegCVoAg2fwCt7Bh/FivBmfxtestWDkM/vgj4zJD87YnS8= J(R) = rµ(R) = µ1 µs(R) The current is then A single droplet coexists with vapour at in a finite system μ ∞ = μ s Chemical potential raised at curved interface μ s AAACC3icbVDLSsNAFJ3UV62vqEs3Q4tQNyURRTdC0Y3LWuwDmhAm00k7dGYSZiZCCd278VfcuFDErT/gzr9x2mahrQcuHM65l3vvCRNGlXacb6uwsrq2vlHcLG1t7+zu2fsHbRWnEpMWjlksuyFShFFBWppqRrqJJIiHjHTC0c3U7zwQqWgs7vU4IT5HA0EjipE2UmCXPZ4Gqto8gVfQiyTCmTdAnKNJ5iVDGoSwOQnsilNzZoDLxM1JBeRoBPaX149xyonQmCGleq6TaD9DUlPMyKTkpYokCI/QgPQMFYgT5WezXybw2Ch9GMXSlNBwpv6eyBBXasxD08mRHqpFbyr+5/VSHV36GRVJqonA80VRyqCO4TQY2KeSYM3GhiAsqbkV4iEyiWgTX8mE4C6+vEzapzX3vObcnVXq13kcRXAEyqAKXHAB6uAWNEALYPAInsEreLOerBfr3fqYtxasfOYQ/IH1+QPRLJpK µs(R) = bR AAACGXicbVDLSgMxFM34rPU16tJNsAgVocyIohuh6MZlLfYBnTLcSTNtaDIzJBmhDv0NN/6KGxeKuNSVf2P6WGjrgcDhnHO5uSdIOFPacb6thcWl5ZXV3Fp+fWNza9ve2a2rOJWE1kjMY9kMQFHOIlrTTHPaTCQFEXDaCPrXI79xT6VicXSnBwltC+hGLGQEtJF82/FE6qti9QhfYi+UQDKvC0KA7wwzL+kxP8DVIT7Go5j3QDX4dsEpOWPgeeJOSQFNUfHtT68Tk1TQSBMOSrVcJ9HtDKRmhNNh3ksVTYD0oUtbhkYgqGpn48uG+NAoHRzG0rxI47H6eyIDodRABCYpQPfUrDcS//NaqQ4v2hmLklTTiEwWhSnHOsajmnCHSUo0HxgCRDLzV0x6YPrRpsy8KcGdPXme1E9K7lnJuT0tlK+mdeTQPjpAReSic1RGN6iCaoigR/SMXtGb9WS9WO/WxyS6YE1n9tAfWF8/rKiffQ== µs(R) = 0 bR + µ⇣ Bray, Adv. in Phys. 43, 357 (1994) Widely separated droplets of varying sizes

9 Model H: Ostwald ripening AAACGHicbZBNS8MwGMfT+TbnW9Wjl+AQ5mGzFUUvg6EX8TTFvcBaSpqlMyxNS5IKo+xjePGrePGgiNfd/DamWw+6+YfAL//neUievx8zKpVlfRuFpeWV1bXiemljc2t7x9zda8soEZi0cMQi0fWRJIxy0lJUMdKNBUGhz0jHH15n9c4TEZJG/EGNYuKGaMBpQDFS2vLMk1tYuT+uw6rDkc8QdMIkM2A9Iy91KA/UaFzNLjLzPbNs1ayp4CLYOZRBrqZnTpx+hJOQcIUZkrJnW7FyUyQUxYyMS04iSYzwEA1ITyNHIZFuOl1sDI+004dBJPThCk7d3xMpCqUchb7uDJF6lPO1zPyv1ktUcOmmlMeJIhzPHgoSBlUEs5RgnwqCFRtpQFhQ/VeIH5FAWOksSzoEe37lRWif1uzzmnV3Vm5c5XEUwQE4BBVggwvQADegCVoAg2fwCt7Bh/FivBmfxtestWDkM/vgj4zJD87YnS8= J(R) = rµ(R) = µ1 µs(R) The current is then A single droplet coexists with vapour at in a finite system μ ∞ = μ s Chemical potential raised at curved interface μ s AAACC3icbVDLSsNAFJ3UV62vqEs3Q4tQNyURRTdC0Y3LWuwDmhAm00k7dGYSZiZCCd278VfcuFDErT/gzr9x2mahrQcuHM65l3vvCRNGlXacb6uwsrq2vlHcLG1t7+zu2fsHbRWnEpMWjlksuyFShFFBWppqRrqJJIiHjHTC0c3U7zwQqWgs7vU4IT5HA0EjipE2UmCXPZ4Gqto8gVfQiyTCmTdAnKNJ5iVDGoSwOQnsilNzZoDLxM1JBeRoBPaX149xyonQmCGleq6TaD9DUlPMyKTkpYokCI/QgPQMFYgT5WezXybw2Ch9GMXSlNBwpv6eyBBXasxD08mRHqpFbyr+5/VSHV36GRVJqonA80VRyqCO4TQY2KeSYM3GhiAsqbkV4iEyiWgTX8mE4C6+vEzapzX3vObcnVXq13kcRXAEyqAKXHAB6uAWNEALYPAInsEreLOerBfr3fqYtxasfOYQ/IH1+QPRLJpK µs(R) = bR AAACGXicbVDLSgMxFM34rPU16tJNsAgVocyIohuh6MZlLfYBnTLcSTNtaDIzJBmhDv0NN/6KGxeKuNSVf2P6WGjrgcDhnHO5uSdIOFPacb6thcWl5ZXV3Fp+fWNza9ve2a2rOJWE1kjMY9kMQFHOIlrTTHPaTCQFEXDaCPrXI79xT6VicXSnBwltC+hGLGQEtJF82/FE6qti9QhfYi+UQDKvC0KA7wwzL+kxP8DVIT7Go5j3QDX4dsEpOWPgeeJOSQFNUfHtT68Tk1TQSBMOSrVcJ9HtDKRmhNNh3ksVTYD0oUtbhkYgqGpn48uG+NAoHRzG0rxI47H6eyIDodRABCYpQPfUrDcS//NaqQ4v2hmLklTTiEwWhSnHOsajmnCHSUo0HxgCRDLzV0x6YPrRpsy8KcGdPXme1E9K7lnJuT0tlK+mdeTQPjpAReSic1RGN6iCaoigR/SMXtGb9WS9WO/WxyS6YE1n9tAfWF8/rKiffQ== µs(R) = 0 bR + µ⇣ 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 ˙ R = J(R) 2 b / R  1 R⇤(t) 1 R For a system with many droplets, is set by mean radius μ ∞ R*(t) AAACGXicbVDLSgMxFM34rPU16tJNsAgVocyIohuh6MZlLfYBnTLcSTNtaDIzJBmhDv0NN/6KGxeKuNSVf2P6WGjrgcDhnHO5uSdIOFPacb6thcWl5ZXV3Fp+fWNza9ve2a2rOJWE1kjMY9kMQFHOIlrTTHPaTCQFEXDaCPrXI79xT6VicXSnBwltC+hGLGQEtJF82/FE6qti9QhfYi+UQDKvC0KA7wwzL+kxP8DVIT7Go5j3QDX4dsEpOWPgeeJOSQFNUfHtT68Tk1TQSBMOSrVcJ9HtDKRmhNNh3ksVTYD0oUtbhkYgqGpn48uG+NAoHRzG0rxI47H6eyIDodRABCYpQPfUrDcS//NaqQ4v2hmLklTTiEwWhSnHOsajmnCHSUo0HxgCRDLzV0x6YPrRpsy8KcGdPXme1E9K7lnJuT0tlK+mdeTQPjpAReSic1RGN6iCaoigR/SMXtGb9WS9WO/WxyS6YE1n9tAfWF8/rKiffQ== µs(R) = 0 bR + µ⇣ Bray, Adv. in Phys. 43, 357 (1994) Widely separated droplets of varying sizes

9 Model H: Ostwald ripening AAACGHicbZBNS8MwGMfT+TbnW9Wjl+AQ5mGzFUUvg6EX8TTFvcBaSpqlMyxNS5IKo+xjePGrePGgiNfd/DamWw+6+YfAL//neUievx8zKpVlfRuFpeWV1bXiemljc2t7x9zda8soEZi0cMQi0fWRJIxy0lJUMdKNBUGhz0jHH15n9c4TEZJG/EGNYuKGaMBpQDFS2vLMk1tYuT+uw6rDkc8QdMIkM2A9Iy91KA/UaFzNLjLzPbNs1ayp4CLYOZRBrqZnTpx+hJOQcIUZkrJnW7FyUyQUxYyMS04iSYzwEA1ITyNHIZFuOl1sDI+004dBJPThCk7d3xMpCqUchb7uDJF6lPO1zPyv1ktUcOmmlMeJIhzPHgoSBlUEs5RgnwqCFRtpQFhQ/VeIH5FAWOksSzoEe37lRWif1uzzmnV3Vm5c5XEUwQE4BBVggwvQADegCVoAg2fwCt7Bh/FivBmfxtestWDkM/vgj4zJD87YnS8= J(R) = rµ(R) = µ1 µs(R) The current is then A single droplet coexists with vapour at in a finite system μ ∞ = μ s Chemical potential raised at curved interface μ s AAACC3icbVDLSsNAFJ3UV62vqEs3Q4tQNyURRTdC0Y3LWuwDmhAm00k7dGYSZiZCCd278VfcuFDErT/gzr9x2mahrQcuHM65l3vvCRNGlXacb6uwsrq2vlHcLG1t7+zu2fsHbRWnEpMWjlksuyFShFFBWppqRrqJJIiHjHTC0c3U7zwQqWgs7vU4IT5HA0EjipE2UmCXPZ4Gqto8gVfQiyTCmTdAnKNJ5iVDGoSwOQnsilNzZoDLxM1JBeRoBPaX149xyonQmCGleq6TaD9DUlPMyKTkpYokCI/QgPQMFYgT5WezXybw2Ch9GMXSlNBwpv6eyBBXasxD08mRHqpFbyr+5/VSHV36GRVJqonA80VRyqCO4TQY2KeSYM3GhiAsqbkV4iEyiWgTX8mE4C6+vEzapzX3vObcnVXq13kcRXAEyqAKXHAB6uAWNEALYPAInsEreLOerBfr3fqYtxasfOYQ/IH1+QPRLJpK µs(R) = bR AAACGXicbVDLSgMxFM34rPU16tJNsAgVocyIohuh6MZlLfYBnTLcSTNtaDIzJBmhDv0NN/6KGxeKuNSVf2P6WGjrgcDhnHO5uSdIOFPacb6thcWl5ZXV3Fp+fWNza9ve2a2rOJWE1kjMY9kMQFHOIlrTTHPaTCQFEXDaCPrXI79xT6VicXSnBwltC+hGLGQEtJF82/FE6qti9QhfYi+UQDKvC0KA7wwzL+kxP8DVIT7Go5j3QDX4dsEpOWPgeeJOSQFNUfHtT68Tk1TQSBMOSrVcJ9HtDKRmhNNh3ksVTYD0oUtbhkYgqGpn48uG+NAoHRzG0rxI47H6eyIDodRABCYpQPfUrDcS//NaqQ4v2hmLklTTiEwWhSnHOsajmnCHSUo0HxgCRDLzV0x6YPrRpsy8KcGdPXme1E9K7lnJuT0tlK+mdeTQPjpAReSic1RGN6iCaoigR/SMXtGb9WS9WO/WxyS6YE1n9tAfWF8/rKiffQ== µs(R) = 0 bR + µ⇣ 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 ˙ R = J(R) 2 b / R  1 R⇤(t) 1 R For a system with many droplets, is set by mean radius μ ∞ R*(t) AAACGXicbVDLSgMxFM34rPU16tJNsAgVocyIohuh6MZlLfYBnTLcSTNtaDIzJBmhDv0NN/6KGxeKuNSVf2P6WGjrgcDhnHO5uSdIOFPacb6thcWl5ZXV3Fp+fWNza9ve2a2rOJWE1kjMY9kMQFHOIlrTTHPaTCQFEXDaCPrXI79xT6VicXSnBwltC+hGLGQEtJF82/FE6qti9QhfYi+UQDKvC0KA7wwzL+kxP8DVIT7Go5j3QDX4dsEpOWPgeeJOSQFNUfHtT68Tk1TQSBMOSrVcJ9HtDKRmhNNh3ksVTYD0oUtbhkYgqGpn48uG+NAoHRzG0rxI47H6eyIDodRABCYpQPfUrDcS//NaqQ4v2hmLklTTiEwWhSnHOsajmnCHSUo0HxgCRDLzV0x6YPrRpsy8KcGdPXme1E9K7lnJuT0tlK+mdeTQPjpAReSic1RGN6iCaoigR/SMXtGb9WS9WO/WxyS6YE1n9tAfWF8/rKiffQ== µs(R) = 0 bR + µ⇣ smaller drops shrink and bigger drops grow material is transported from small drops to large ones by diffusion AAAB83icbVBNSwMxEM3Wr1q/qh69BIvgqeyKoseiF49VrC10l5JNZ9vQbBKSrFCW/g0vHhTx6p/x5r8xbfegrQ8GHu/NMDMvVpwZ6/vfXmlldW19o7xZ2dre2d2r7h88GplpCi0qudSdmBjgTEDLMsuhozSQNObQjkc3U7/9BNowKR7sWEGUkoFgCaPEOim87+UhKMO4FJNetebX/RnwMgkKUkMFmr3qV9iXNEtBWMqJMd3AVzbKibaMcphUwsyAInREBtB1VJAUTJTPbp7gE6f0cSK1K2HxTP09kZPUmHEau86U2KFZ9Kbif143s8lVlDOhMguCzhclGcdW4mkAuM80UMvHjhCqmbsV0yHRhFoXU8WFECy+vEwez+rBRd2/O681ros4yugIHaNTFKBL1EC3qIlaiCKFntErevMy78V79z7mrSWvmDlEf+B9/gB1LZH1 R✏ 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 ˙ R = V ADAceq R ✓ ✏ 2 0V A kBTR ◆ AAAB7XicbVBNSwMxEJ2tX7V+VT16CRaheii7ouix6MVjFfsB7VqyabaNzSZLkhXK0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmBTFn2rjut5NbWl5ZXcuvFzY2t7Z3irt7DS0TRWidSC5VK8CaciZo3TDDaStWFEcBp81geD3xm09UaSbFvRnF1I9wX7CQEWys1Lh7OCmb426x5FbcKdAi8TJSggy1bvGr05MkiagwhGOt254bGz/FyjDC6bjQSTSNMRniPm1bKnBEtZ9Orx2jI6v0UCiVLWHQVP09keJI61EU2M4Im4Ge9ybif147MeGlnzIRJ4YKMlsUJhwZiSavox5TlBg+sgQTxeytiAywwsTYgAo2BG/+5UXSOK145xX39qxUvcriyMMBHEIZPLiAKtxADepA4BGe4RXeHOm8OO/Ox6w152Qz+/AHzucPY7COWQ== R⇤(t) Bray, Adv. in Phys. 43, 357 (1994) Widely separated droplets of varying sizes

10 Stokes equation 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 rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I Bray, Adv. in Phys. 1994; Kendon et al, JFM 2001; Cates and Tjhung, JFM 2018 -1 1 Model H: coarsening kinetics Nucleation Ostwald ripening Coalescence Bulk phase separation ˙ = r · J Model H 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 J = rµ + v + p 2D⇤,

10 Stokes equation 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 rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I Bray, Adv. in Phys. 1994; Kendon et al, JFM 2001; Cates and Tjhung, JFM 2018 -1 1 Model H: coarsening kinetics Nucleation Ostwald ripening Coalescence Bulk phase separation Active particles create dipolar fluid flow even in absence of external forces extensile/contractile dipolar flow along the swimming axis Ramaswamy, Annu. Rev. Condens. Matter Phys. 2010 ˙ = r · J Model H 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 J = rµ + v + p 2D⇤,

φ4 field theory of active phase separation RS et al. PRL 2019, PRR 2020; Cates and Tjhung JFM 2018; Tiribocchi et al. PRL 2015 Stokes equation 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 rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I ˙ = r · J Active model H 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 J = rµ + v + p 2D⇤, 11

φ4 field theory of active phase separation RS et al. PRL 2019, PRR 2020; Cates and Tjhung JFM 2018; Tiribocchi et al. PRL 2015 ⌃A = (˜  )S AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow= AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow= AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow= AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow= mechanical activity parameter positive (extensile microswimmers) negative (contractile microswimmers) minimal term breaking TRS in mechanical sector ˜ κ Stokes equation 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 rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I ˙ = r · J Active model H AAACQ3icbVDLSsNAFJ3UV62vqks3waIIaklE0Y0g6kLEhYJ9YBPKzWTaDk4mcWZSKCH/5sYfcOcPuHGhiFvBSc2iPi4MczjnHu69x4sYlcqynozC2PjE5FRxujQzOze/UF5cqsswFpjUcMhC0fRAEkY5qSmqGGlGgkDgMdLwbk8yvdEnQtKQX6tBRNwAupx2KAalqXb5xvFC5stBoL/kPF0/3B4lHA4eg9QJ4s1Rup86UY9uOvJOqGTnNP1hudDDfUi3Su1yxapawzL/AjsHFZTXZbv86PghjgPCFWYgZcu2IuUmIBTFjKQlJ5YkAnwLXdLSkENApJsMM0jNNc34ZicU+nFlDtlRRwKBzFbUnQGonvytZeR/WitWnQM3oTyKFeH4e1AnZqYKzSxQ06eCYMUGGgAWVO9q4h4IwErHnoVg/z75L6jvVO29qnW1Wzk6zuMoohW0ijaQjfbRETpDl6iGMLpHz+gVvRkPxovxbnx8txaM3LOMfpTx+QWT1rOh J = rµ + v + p 2D⇤, 11

φ4 field theory of active phase separation RS et al. PRL 2019, PRR 2020; Cates and Tjhung JFM 2018; Tiribocchi et al. PRL 2015 ⌃A = (˜  )S AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow= AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow= AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow= AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow= mechanical activity parameter positive (extensile microswimmers) negative (contractile microswimmers) minimal term breaking TRS in mechanical sector ˜ κ Contractile along the normal to the interface ˜ κ < 0 Extensile along the normal to the interface ˜ κ > 0 ⌃A ⇠ (r )(r ) ⌃A ⇠ (r )(r ) = 1 = +1 AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK v = ˜  0/ Stokes equation 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 rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I ˙ = r · J Active model H 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 J = rµ + v + p 2D⇤, 11

Incomplete phase separation ˜ κ < 0 Complete phase separation ˜ κ > 0 Active model H: nucleation and growth 12 -1 1 RS and Cates PRL (2019)

Incomplete phase separation ˜ κ < 0 Complete phase separation ˜ κ > 0 Active model H: nucleation and growth 12 -1 1 RS and Cates PRL (2019)

Active model H: self-shearing instability (SSI) A spontaneous stretching motion of the interface A contractile active stress ˜ κ < 0 ⟹ γ v < 0 RS and Cates PRL (2019) The spontaneous stretching motion results in SSI => splitting the droplet 13

Active model H: self-shearing instability (SSI) A spontaneous stretching motion of the interface A contractile active stress ˜ κ < 0 ⟹ γ v < 0 SSI RS and Cates PRL (2019) The spontaneous stretching motion results in SSI => splitting the droplet 13

Active model H: self-shearing instability (SSI) A spontaneous stretching motion of the interface A contractile active stress ˜ κ < 0 ⟹ γ v < 0 SSI Ostwald ripening RS and Cates PRL (2019) The spontaneous stretching motion results in SSI => splitting the droplet 13

Active model H: self-shearing instability (SSI) A self-shearing instability splits larger droplets by stretching them. Ostwald ripening: smaller drops shrink and bigger drops grow. The result is a dynamic steady-state maintained by SSI. Self-shearing instability Small drops disappear Big drops grow in size Steady-state dynamics A spontaneous stretching motion of the interface A contractile active stress ˜ κ < 0 ⟹ γ v < 0 SSI Ostwald ripening RS and Cates PRL (2019) The spontaneous stretching motion results in SSI => splitting the droplet 13

Length scale determined by activity SSI Ostwald ripening RS and Cates PRL (2019) ˙ = r · J Active Model H 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 J = rµ + v + p 2D⇤, Stokes equation 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 rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK v = ˜  0/ 14

Length scale determined by activity SSI Ostwald ripening RS and Cates PRL (2019) From the mechanical tension and fluid viscosity , we can construct just one quantity with the dimensions of velocity γ v η Vv = v/⌘ ˙ = r · J Active Model H 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 J = rµ + v + p 2D⇤, Stokes equation 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 rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK v = ˜  0/ 14

Length scale determined by activity SSI Ostwald ripening RS and Cates PRL (2019) From the mechanical tension and fluid viscosity , we can construct just one quantity with the dimensions of velocity γ v η Vv = v/⌘ Ostwald process gives another rate V ( ¯ R) = ˙ ¯ R / M / 2 B ¯ R2 ˙ = r · J Active Model H 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 J = rµ + v + p 2D⇤, Stokes equation 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 rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK v = ˜  0/ 14

Length scale determined by activity SSI Ostwald ripening RS and Cates PRL (2019) From the mechanical tension and fluid viscosity , we can construct just one quantity with the dimensions of velocity γ v η Vv = v/⌘ Ostwald process gives another rate V ( ¯ R) = ˙ ¯ R / M / 2 B ¯ R2 ¯ R / ✓ v 2 B ⌘M ◆ 1/2 ⇠ v 1/2 Balancing the two rates, we get scaling for droplet size ˙ = r · J Active Model H 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 J = rµ + v + p 2D⇤, Stokes equation AAADlHiclVJNbxMxEHV2+SjhKwGJC5cVEVIqSLRbgcqBSilVJbigopK2UpxGs14nseL96Ho2InL9h/g53Pg3OMkekjQ5MJLl5/dmnscjh5kUCn3/b8Vx791/8HDvUfXxk6fPntfqLy5UWuSMd1kq0/wqBMWlSHgXBUp+leUc4lDyy3ByMtcvpzxXIk1+4izj/RhGiRgKBmipQb3yu0XDVEZqFttN0wRCCSZ7RznC8nCtD8xqytQcbSuhLEpxjT8XoxjMtT41FPkvXPSqcx4Z/X/1x8ZQusPZtjKBLINV+dy8pzcFRJu37jA/ajUpChlxvbQypeX+uqc1pesM5TeFmDa3vSUbi/2dQoviMAemA6Mjc7sj63Zz7N9MdVBr+G1/Ed5dEJSgQco4G9T+0ChlRcwTZBKU6gV+hn0NOQomuanSQvEM2ARGvGdhAjFXfb0YmfHeWibyhmluV4Legl2t0BCreW82MwYcq01tTm7TegUOP/W1SLICecKWFw0L6WHqzX+oF4mcM5QzC4DlwvbqsTHYiaH9x/MhBJtPvgsuDtrBx7b/40Oj86Ucxx55Td6QJgnIIemQr+SMdAlz6s6h03GO3VfuZ/fEPV2mOpWy5iVZC/f7P6QyNjI= rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK v = ˜  0/ 14

Length scale determined by activity SSI Ostwald ripening RS and Cates PRL (2019) From the mechanical tension and fluid viscosity , we can construct just one quantity with the dimensions of velocity γ v η Vv = v/⌘ Ostwald process gives another rate V ( ¯ R) = ˙ ¯ R / M / 2 B ¯ R2 ¯ R / ✓ v 2 B ⌘M ◆ 1/2 ⇠ v 1/2 Balancing the two rates, we get scaling for droplet size ˙ = r · J Active Model H AAACQ3icbVDLSsNAFJ3UV62vqks3waIIaklE0Y0g6kLEhYJ9YBPKzWTaDk4mcWZSKCH/5sYfcOcPuHGhiFvBSc2iPi4MczjnHu69x4sYlcqynozC2PjE5FRxujQzOze/UF5cqsswFpjUcMhC0fRAEkY5qSmqGGlGgkDgMdLwbk8yvdEnQtKQX6tBRNwAupx2KAalqXb5xvFC5stBoL/kPF0/3B4lHA4eg9QJ4s1Rup86UY9uOvJOqGTnNP1hudDDfUi3Su1yxapawzL/AjsHFZTXZbv86PghjgPCFWYgZcu2IuUmIBTFjKQlJ5YkAnwLXdLSkENApJsMM0jNNc34ZicU+nFlDtlRRwKBzFbUnQGonvytZeR/WitWnQM3oTyKFeH4e1AnZqYKzSxQ06eCYMUGGgAWVO9q4h4IwErHnoVg/z75L6jvVO29qnW1Wzk6zuMoohW0ijaQjfbRETpDl6iGMLpHz+gVvRkPxovxbnx8txaM3LOMfpTx+QWT1rOh J = rµ + v + p 2D⇤, Stokes equation 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 rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK v = ˜  0/ 14

15 Activity in diffusive sector Cates and Tjhung JFM 2018; Tjhung et al PRX 2018 ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. 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 J = Mrµ + J⇣ + p 2DM⇤, J⇣ = ⇣(r2 )r 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 AAACmXicbVFta9swEJa9ty57y1rYl34xCxsd3YIdBuuHFbpXSlmhY0tbiNJwluVGVJZd6VyWCf2n/ZZ927+Z7AS2Nj0Q9/Dcne65u7SSwmAc/wnCGzdv3b6zcrdz7/6Dh4+6j1cPTVlrxoeslKU+TsFwKRQfokDJjyvNoUglP0rPPjTxowuujSjVd5xVfFzAqRK5YICemnR/0QJwmuZ2zz3ffrVP01JmZlZ4Z6mCVIKjRb35L2li6U+O4DapOddoBx/33aWaL751Bu4lPa8hsxT5D2xFWs0zZ6Plj7ZbtzFvdmIHjlZT8eI6HZ53rjPp9uJ+3Fq0DJIF6JGFHUy6v2lWsrrgCpkEY0ZJXOHYgkbBJHcdWhteATuDUz7yUEHBzdi2ol30zDNZlJfaP4VRy/5fYaEwjUyf2UxmrsYa8rrYqMZ8a2yFqmrkis0b5bWMsIyaM0WZ0JyhnHkATAuvNWJT0MDQH7NZQnJ15GVwOOgncT/5+rq3836xjhWyTp6SDZKQN2SH7JIDMiQseBK8DT4Fn8P18F24G+7NU8NgUbNGLln47S+iRtBQ 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 J = rµ + J⇣ + p 2D⇤,

15 Activity in diffusive sector Cates and Tjhung JFM 2018; Tjhung et al PRX 2018 ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. AAACf3icbVHbahRBEO0ZL4nrJas++jJkEYWEYSYI6sNCUBQfI7hJYHuz1PTUZJv0XNJdIy6d/g0/zDf/xQd7JhOIiQVFH07Vqa5L1ihpKEl+B+Gdu/fub2w+GD189PjJ1vjps0NTt1rgTNSq1scZGFSywhlJUnjcaIQyU3iUnX3s4kffURtZV99o3eCihNNKFlIAeWo5/snLdur9xH5yO/3LlVfn4Hb5eQv5EJryQoOwPEdFwEuglQBlPzt3RTUreaUg/EF9Y1Zj7hOuF50O4IJntcrNuvSP5RVkClxX4+LE7jkXj5bjSRInvUW3QTqACRvsYDn+xfNatCVWJBQYM0+ThhYWNEmh0I14a7ABcQanOPewghLNwvZtuuilZ/KoqLX3iqKeva6wUJquV5/ZjW5uxjryf7F5S8W7hZVV0xJW4vKjolUR1VF3jCiXGgWptQcgtPS9RmIFftPkT9YtIb058m1wuBenSZx+fTPZ/zCsY5O9YNvsNUvZW7bPvrADNmOC/Qm2g51gNwzCV2EcJpepYTBonrN/LHz/F9i7xpA= 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 J = Mrµ + J⇣ + p 2DM⇤, J⇣ = ⇣(r2 )r AAACmXicbVFta9swEJa9ty57y1rYl34xCxsd3YIdBuuHFbpXSlmhY0tbiNJwluVGVJZd6VyWCf2n/ZZ927+Z7AS2Nj0Q9/Dcne65u7SSwmAc/wnCGzdv3b6zcrdz7/6Dh4+6j1cPTVlrxoeslKU+TsFwKRQfokDJjyvNoUglP0rPPjTxowuujSjVd5xVfFzAqRK5YICemnR/0QJwmuZ2zz3ffrVP01JmZlZ4Z6mCVIKjRb35L2li6U+O4DapOddoBx/33aWaL751Bu4lPa8hsxT5D2xFWs0zZ6Plj7ZbtzFvdmIHjlZT8eI6HZ53rjPp9uJ+3Fq0DJIF6JGFHUy6v2lWsrrgCpkEY0ZJXOHYgkbBJHcdWhteATuDUz7yUEHBzdi2ol30zDNZlJfaP4VRy/5fYaEwjUyf2UxmrsYa8rrYqMZ8a2yFqmrkis0b5bWMsIyaM0WZ0JyhnHkATAuvNWJT0MDQH7NZQnJ15GVwOOgncT/5+rq3836xjhWyTp6SDZKQN2SH7JIDMiQseBK8DT4Fn8P18F24G+7NU8NgUbNGLln47S+iRtBQ 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 J = rµ + J⇣ + p 2D⇤, AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT J⇣ = rµ⇣ + r ⇥ A

15 Activity in diffusive sector Cates and Tjhung JFM 2018; Tjhung et al PRX 2018 ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. 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 AAACf3icbVHbahRBEO0ZL4nrJas++jJkEYWEYSYI6sNCUBQfI7hJYHuz1PTUZJv0XNJdIy6d/g0/zDf/xQd7JhOIiQVFH07Vqa5L1ihpKEl+B+Gdu/fub2w+GD189PjJ1vjps0NTt1rgTNSq1scZGFSywhlJUnjcaIQyU3iUnX3s4kffURtZV99o3eCihNNKFlIAeWo5/snLdur9xH5yO/3LlVfn4Hb5eQv5EJryQoOwPEdFwEuglQBlPzt3RTUreaUg/EF9Y1Zj7hOuF50O4IJntcrNuvSP5RVkClxX4+LE7jkXj5bjSRInvUW3QTqACRvsYDn+xfNatCVWJBQYM0+ThhYWNEmh0I14a7ABcQanOPewghLNwvZtuuilZ/KoqLX3iqKeva6wUJquV5/ZjW5uxjryf7F5S8W7hZVV0xJW4vKjolUR1VF3jCiXGgWptQcgtPS9RmIFftPkT9YtIb058m1wuBenSZx+fTPZ/zCsY5O9YNvsNUvZW7bPvrADNmOC/Qm2g51gNwzCV2EcJpepYTBonrN/LHz/F9i7xpA= J = Mrµ + J⇣ + p 2DM⇤, J⇣ = ⇣(r2 )r 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 AAACmXicbVFta9swEJa9ty57y1rYl34xCxsd3YIdBuuHFbpXSlmhY0tbiNJwluVGVJZd6VyWCf2n/ZZ927+Z7AS2Nj0Q9/Dcne65u7SSwmAc/wnCGzdv3b6zcrdz7/6Dh4+6j1cPTVlrxoeslKU+TsFwKRQfokDJjyvNoUglP0rPPjTxowuujSjVd5xVfFzAqRK5YICemnR/0QJwmuZ2zz3ffrVP01JmZlZ4Z6mCVIKjRb35L2li6U+O4DapOddoBx/33aWaL751Bu4lPa8hsxT5D2xFWs0zZ6Plj7ZbtzFvdmIHjlZT8eI6HZ53rjPp9uJ+3Fq0DJIF6JGFHUy6v2lWsrrgCpkEY0ZJXOHYgkbBJHcdWhteATuDUz7yUEHBzdi2ol30zDNZlJfaP4VRy/5fYaEwjUyf2UxmrsYa8rrYqMZ8a2yFqmrkis0b5bWMsIyaM0WZ0JyhnHkATAuvNWJT0MDQH7NZQnJ15GVwOOgncT/5+rq3836xjhWyTp6SDZKQN2SH7JIDMiQseBK8DT4Fn8P18F24G+7NU8NgUbNGLln47S+iRtBQ AAACmXicbVFta9swEJa9ty57y1rYl34xCxsd3YIdBuuHFbpXSlmhY0tbiNJwluVGVJZd6VyWCf2n/ZZ927+Z7AS2Nj0Q9/Dcne65u7SSwmAc/wnCGzdv3b6zcrdz7/6Dh4+6j1cPTVlrxoeslKU+TsFwKRQfokDJjyvNoUglP0rPPjTxowuujSjVd5xVfFzAqRK5YICemnR/0QJwmuZ2zz3ffrVP01JmZlZ4Z6mCVIKjRb35L2li6U+O4DapOddoBx/33aWaL751Bu4lPa8hsxT5D2xFWs0zZ6Plj7ZbtzFvdmIHjlZT8eI6HZ53rjPp9uJ+3Fq0DJIF6JGFHUy6v2lWsrrgCpkEY0ZJXOHYgkbBJHcdWhteATuDUz7yUEHBzdi2ol30zDNZlJfaP4VRy/5fYaEwjUyf2UxmrsYa8rrYqMZ8a2yFqmrkis0b5bWMsIyaM0WZ0JyhnHkATAuvNWJT0MDQH7NZQnJ15GVwOOgncT/5+rq3836xjhWyTp6SDZKQN2SH7JIDMiQseBK8DT4Fn8P18F24G+7NU8NgUbNGLln47S+iRtBQ 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 J = rµ + J⇣ + p 2D⇤, effective nonlocal chemical potential AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT J⇣ = rµ⇣ + r ⇥ A

15 Activity in diffusive sector Cates and Tjhung JFM 2018; Tjhung et al PRX 2018 ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. AAACf3icbVHbahRBEO0ZL4nrJas++jJkEYWEYSYI6sNCUBQfI7hJYHuz1PTUZJv0XNJdIy6d/g0/zDf/xQd7JhOIiQVFH07Vqa5L1ihpKEl+B+Gdu/fub2w+GD189PjJ1vjps0NTt1rgTNSq1scZGFSywhlJUnjcaIQyU3iUnX3s4kffURtZV99o3eCihNNKFlIAeWo5/snLdur9xH5yO/3LlVfn4Hb5eQv5EJryQoOwPEdFwEuglQBlPzt3RTUreaUg/EF9Y1Zj7hOuF50O4IJntcrNuvSP5RVkClxX4+LE7jkXj5bjSRInvUW3QTqACRvsYDn+xfNatCVWJBQYM0+ThhYWNEmh0I14a7ABcQanOPewghLNwvZtuuilZ/KoqLX3iqKeva6wUJquV5/ZjW5uxjryf7F5S8W7hZVV0xJW4vKjolUR1VF3jCiXGgWptQcgtPS9RmIFftPkT9YtIb058m1wuBenSZx+fTPZ/zCsY5O9YNvsNUvZW7bPvrADNmOC/Qm2g51gNwzCV2EcJpepYTBonrN/LHz/F9i7xpA= 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 J = Mrµ + J⇣ + p 2DM⇤, J⇣ = ⇣(r2 )r 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 AAACSXicbVBNTxRBEO1ZVHD9YMUjl4kbjQm6mSEYuZAQ4WCMB0xcINnerDU9NdChp2fsriEsnf57XLx58z948aAxnuxZ9oBgJZ1+efX16mW1kpaS5FvUWbh1+87i0t3uvfsPHi73Hq3s26oxAoeiUpU5zMCikhqHJEnhYW0QykzhQXay0+YPTtFYWemPNK1xXMKRloUUQIGa9D7xrFK5nZbhc+/8s62XjmvIFHheNmuc8IxmS5zB3DteAh1nRSicOH6OBN6vcfvZkFvf9Vcn8fdBQg7+RXfS6yeDZBbxTZDOQZ/NY2/S+8rzSjQlahIKrB2lSU1jB4akUOi7vLFYgziBIxwFqKFEO3YzkT5+Gpg8LioTnqZ4xl7tcFDaVmKobE+x13Mt+b/cqKFic+ykrhtCLS4XFY2KqYpbW+NcGhSkpgGAMDJojcUxGBAUzG9NSK+ffBPsrw/SV4Pkw0Z/+83cjiW2yp6w5yxlr9k2e8v22JAJdsG+s5/sV/Ql+hH9jv5clnaiec9j9k90Fv4Cc2m1jw== J = rµ + J⇣ + p 2D⇤, effective nonlocal chemical potential 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 µs(R) = 0 bR + µ + µ⇣ Chemical potential raised at curved interface: μ s AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT J⇣ = rµ⇣ + r ⇥ A

15 Activity in diffusive sector Cates and Tjhung JFM 2018; Tjhung et al PRX 2018 The active contributions can reverse the sign of chemical flux and thus the sign of effective tension => reverse Ostwald ripening ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. 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 J = Mrµ + J⇣ + p 2DM⇤, J⇣ = ⇣(r2 )r 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 J = rµ + J⇣ + p 2D⇤, effective nonlocal chemical potential 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 µs(R) = 0 bR + µ + µ⇣ Chemical potential raised at curved interface: μ s AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT J⇣ = rµ⇣ + r ⇥ A

15 Activity in diffusive sector Cates and Tjhung JFM 2018; Tjhung et al PRX 2018 The active contributions can reverse the sign of chemical flux and thus the sign of effective tension => reverse Ostwald ripening effective tension in diffusive sector 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 = (⇣S0 2 S1) ⇣ 2 ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. 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 J = Mrµ + J⇣ + p 2DM⇤, J⇣ = ⇣(r2 )r AAACmXicbVFta9swEJa9ty57y1rYl34xCxsd3YIdBuuHFbpXSlmhY0tbiNJwluVGVJZd6VyWCf2n/ZZ927+Z7AS2Nj0Q9/Dcne65u7SSwmAc/wnCGzdv3b6zcrdz7/6Dh4+6j1cPTVlrxoeslKU+TsFwKRQfokDJjyvNoUglP0rPPjTxowuujSjVd5xVfFzAqRK5YICemnR/0QJwmuZ2zz3ffrVP01JmZlZ4Z6mCVIKjRb35L2li6U+O4DapOddoBx/33aWaL751Bu4lPa8hsxT5D2xFWs0zZ6Plj7ZbtzFvdmIHjlZT8eI6HZ53rjPp9uJ+3Fq0DJIF6JGFHUy6v2lWsrrgCpkEY0ZJXOHYgkbBJHcdWhteATuDUz7yUEHBzdi2ol30zDNZlJfaP4VRy/5fYaEwjUyf2UxmrsYa8rrYqMZ8a2yFqmrkis0b5bWMsIyaM0WZ0JyhnHkATAuvNWJT0MDQH7NZQnJ15GVwOOgncT/5+rq3836xjhWyTp6SDZKQN2SH7JIDMiQseBK8DT4Fn8P18F24G+7NU8NgUbNGLln47S+iRtBQ 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 J = rµ + J⇣ + p 2D⇤, effective nonlocal chemical potential 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 µs(R) = 0 bR + µ + µ⇣ Chemical potential raised at curved interface: μ s AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT J⇣ = rµ⇣ + r ⇥ A

15 Activity in diffusive sector Cates and Tjhung JFM 2018; Tjhung et al PRX 2018 The active contributions can reverse the sign of chemical flux and thus the sign of effective tension => reverse Ostwald ripening effective tension in diffusive sector 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 = (⇣S0 2 S1) ⇣ 2 ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. 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 J = Mrµ + J⇣ + p 2DM⇤, J⇣ = ⇣(r2 )r 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 J = rµ + J⇣ + p 2D⇤, effective nonlocal chemical potential 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 µs(R) = 0 bR + µ + µ⇣ Chemical potential raised at curved interface: μ s AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT J⇣ = rµ⇣ + r ⇥ A Forward Ostwald ripening γ ϕ > 0 Reverse Ostwald ripening γ ϕ < 0

16 Steady-states in the plane of effective tensions ˙ = r · J Active Model H 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 J = rµ + J⇣ + p 2D⇤, AAACC3icbZC7TsMwFIadcivlVmBkiVohMZUEgWCsYGEsEr1IbRQ57mlr1U4i+6Siirqz8CosDCDEyguw8Ta4bQZoOZKlT/9/jn38B7HgGh3n28qtrK6tb+Q3C1vbO7t7xf2Dho4SxaDOIhGpVkA1CB5CHTkKaMUKqAwENIPhzdRvjkBpHoX3OI7Bk7Qf8h5nFI3kF0sdhAec3ZMq6E7STp9KSf3RaQbOxC+WnYozK3sZ3AzKJKuaX/zqdCOWSAiRCap123Vi9FKqkDMBk0In0RBTNqR9aBsMqQTtpbMdJvaxUbp2L1LmhGjP1N8TKZVaj2VgOiXFgV70puJ/XjvB3pWX8jBOEEI2f6iXCBsjexqM3eUKGIqxAcoUN7vabEAVZWjiK5gQ3MUvL0PjrOJeVJy783L1OosjT45IiZwQl1ySKrklNVInjDySZ/JK3qwn68V6tz7mrTkrmzkkf8r6/AEt45u/ v/ 0 AAACEHicbVC7TsMwFHV4lvIKMLJEVAimkiAQjBUsjEWiD6mpKse9ba3aSWTfIKoon8DCr7AwgBArIxt/g9tmgJYjWTo+597r6xPEgmt03W9rYXFpeWW1sFZc39jc2rZ3dus6ShSDGotEpJoB1SB4CDXkKKAZK6AyENAIhtdjv3EPSvMovMNRDG1J+yHvcUbRSB37yEd4wMmcVEE3S/0+lZJ2Uj8e8Owkv7lZxy65ZXcCZ554OSmRHNWO/eV3I5ZICJEJqnXLc2Nsp1QhZwKyop9oiCkb0j60DA2pBN1OJ4tkzqFRuk4vUuaE6EzU3x0plVqPZGAqJcWBnvXG4n9eK8HeZTvlYZwghGz6UC8RDkbOOB2nyxUwFCNDKFPc7OqwAVWUocmwaELwZr88T+qnZe+87N6elSpXeRwFsk8OyDHxyAWpkBtSJTXCyCN5Jq/kzXqyXqx362NaumDlPXvkD6zPH2t6nhA= / 0 AAACEHicbVC7TsMwFHV4lvIKMLJEVAimkiAQjBUsjEWiD6mpKse9ba3aSWTfIKoon8DCr7AwgBArIxt/g9tmgJYjWTo+597r6xPEgmt03W9rYXFpeWW1sFZc39jc2rZ3dus6ShSDGotEpJoB1SB4CDXkKKAZK6AyENAIhtdjv3EPSvMovMNRDG1J+yHvcUbRSB37yEd4wMmcVEE3S/0+lZJ2Uj8e8Owkv7lZxy65ZXcCZ554OSmRHNWO/eV3I5ZICJEJqnXLc2Nsp1QhZwKyop9oiCkb0j60DA2pBN1OJ4tkzqFRuk4vUuaE6EzU3x0plVqPZGAqJcWBnvXG4n9eK8HeZTvlYZwghGz6UC8RDkbOOB2nyxUwFCNDKFPc7OqwAVWUocmwaELwZr88T+qnZe+87N6elSpXeRwFsk8OyDHxyAWpkBtSJTXCyCN5Jq/kzXqyXqx362NaumDlPXvkD6zPH2t6nhA= / 0 RS and Cates PRL (2019)

16 Steady-states in the plane of effective tensions ˙ = r · J Active Model H 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 J = rµ + J⇣ + p 2D⇤, diffusive sector mechanical sector AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK v = ˜  0/ AAACU3icbVHPa9RAFJ6kVetqda1HL4OLUA9dkqLoRSh68dii2xZ2lvAyedkddyYZZl7ENeR/FMGD/4gXD3ayXcG2Phj4+L7385vcauUpSX5G8db2rdt3du4O7t3fffBw+Gjv1NeNkziRta7deQ4etapwQoo0nluHYHKNZ/nyXa+ffUbnVV19pJXFmYF5pUolgQKVDT8Jwi+07tM6LLpWzMEYyIRdKP6Gi9KBbMUSrIV98RUJuDBACwm6/dBlCT/gh0KHccVVIX0eOvXpB3/lrsuGo2ScrIPfBOkGjNgmjrPhd1HUsjFYkdTg/TRNLM1acKSkxm4gGo8W5BLmOA2wAoN+1q5v6fizwBS8rF14FfE1+29FC8b7lclDZr+3v6715P+0aUPl61mrKtsQVvJyUNloTjXvDeaFcihJrwIA6VTYlcsFBBspfMMgmJBeP/kmOD0cpy/HycmL0dHbjR077Al7yvZZyl6xI/aeHbMJk+wb+8X+RCz6Ef2O43j7MjWONjWP2ZWIdy8AVfa0tQ== = (⇣S0 2 S1) ⇣ 2 models H (always +ve) AAACEXicbVDLSsNAFJ34rPVVdelmsAjdWJOqWBdC0Y3LCvYBTRpuptN26EwSZyZCCfkFN/6KGxeKuHXnzr8xfSy09cCFwzn3cu89XsiZ0qb5bSwsLi2vrGbWsusbm1vbuZ3dugoiSWiNBDyQTQ8U5cynNc00p81QUhAepw1vcD3yGw9UKhb4d3oYUkdAz2ddRkCnkpsr2D0QAtzYTC5tdS91fFS2BxCGgKEdnyTHF147LiVJ1s3lzaI5Bp4n1pTk0RRVN/dldwISCeprwkGplmWG2olBakY4TbJ2pGgIZAA92kqpD4IqJx5/lODDVOngbiDT8jUeq78nYhBKDYWXdgrQfTXrjcT/vFaku2UnZn4YaeqTyaJuxLEO8Cge3GGSEs2HKQEiWXorJn2QQHQa4igEa/bleVIvFa2zonl7mq9cTePIoH10gArIQueogm5QFdUQQY/oGb2iN+PJeDHejY9J64IxndlDf2B8/gBgh5y0 0 = p 8a3/9b2 AAACC3icbZC7TsMwFIadcivlVmBkiVohMZUEgWCsYGEsEr1IbRQ57mlr1U4i+6Siirqz8CosDCDEyguw8Ta4bQZoOZKlT/9/jn38B7HgGh3n28qtrK6tb+Q3C1vbO7t7xf2Dho4SxaDOIhGpVkA1CB5CHTkKaMUKqAwENIPhzdRvjkBpHoX3OI7Bk7Qf8h5nFI3kF0sdhAec3ZMq6E7STp9KSf3RaQbOxC+WnYozK3sZ3AzKJKuaX/zqdCOWSAiRCap123Vi9FKqkDMBk0In0RBTNqR9aBsMqQTtpbMdJvaxUbp2L1LmhGjP1N8TKZVaj2VgOiXFgV70puJ/XjvB3pWX8jBOEEI2f6iXCBsjexqM3eUKGIqxAcoUN7vabEAVZWjiK5gQ3MUvL0PjrOJeVJy783L1OosjT45IiZwQl1ySKrklNVInjDySZ/JK3qwn68V6tz7mrTkrmzkkf8r6/AEt45u/ v/ 0 AAACEHicbVC7TsMwFHV4lvIKMLJEVAimkiAQjBUsjEWiD6mpKse9ba3aSWTfIKoon8DCr7AwgBArIxt/g9tmgJYjWTo+597r6xPEgmt03W9rYXFpeWW1sFZc39jc2rZ3dus6ShSDGotEpJoB1SB4CDXkKKAZK6AyENAIhtdjv3EPSvMovMNRDG1J+yHvcUbRSB37yEd4wMmcVEE3S/0+lZJ2Uj8e8Owkv7lZxy65ZXcCZ554OSmRHNWO/eV3I5ZICJEJqnXLc2Nsp1QhZwKyop9oiCkb0j60DA2pBN1OJ4tkzqFRuk4vUuaE6EzU3x0plVqPZGAqJcWBnvXG4n9eK8HeZTvlYZwghGz6UC8RDkbOOB2nyxUwFCNDKFPc7OqwAVWUocmwaELwZr88T+qnZe+87N6elSpXeRwFsk8OyDHxyAWpkBtSJTXCyCN5Jq/kzXqyXqx362NaumDlPXvkD6zPH2t6nhA= / 0 AAACEHicbVC7TsMwFHV4lvIKMLJEVAimkiAQjBUsjEWiD6mpKse9ba3aSWTfIKoon8DCr7AwgBArIxt/g9tmgJYjWTo+597r6xPEgmt03W9rYXFpeWW1sFZc39jc2rZ3dus6ShSDGotEpJoB1SB4CDXkKKAZK6AyENAIhtdjv3EPSvMovMNRDG1J+yHvcUbRSB37yEd4wMmcVEE3S/0+lZJ2Uj8e8Owkv7lZxy65ZXcCZ554OSmRHNWO/eV3I5ZICJEJqnXLc2Nsp1QhZwKyop9oiCkb0j60DA2pBN1OJ4tkzqFRuk4vUuaE6EzU3x0plVqPZGAqJcWBnvXG4n9eK8HeZTvlYZwghGz6UC8RDkbOOB2nyxUwFCNDKFPc7OqwAVWUocmwaELwZr88T+qnZe+87N6elSpXeRwFsk8OyDHxyAWpkBtSJTXCyCN5Jq/kzXqyXqx362NaumDlPXvkD6zPH2t6nhA= / 0 RS and Cates PRL (2019)

16 Steady-states in the plane of effective tensions ˙ = r · J Active Model H 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 J = rµ + J⇣ + p 2D⇤, diffusive sector mechanical sector AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK v = ˜  0/ 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 = (⇣S0 2 S1) ⇣ 2 models H (always +ve) AAACEXicbVDLSsNAFJ34rPVVdelmsAjdWJOqWBdC0Y3LCvYBTRpuptN26EwSZyZCCfkFN/6KGxeKuHXnzr8xfSy09cCFwzn3cu89XsiZ0qb5bSwsLi2vrGbWsusbm1vbuZ3dugoiSWiNBDyQTQ8U5cynNc00p81QUhAepw1vcD3yGw9UKhb4d3oYUkdAz2ddRkCnkpsr2D0QAtzYTC5tdS91fFS2BxCGgKEdnyTHF147LiVJ1s3lzaI5Bp4n1pTk0RRVN/dldwISCeprwkGplmWG2olBakY4TbJ2pGgIZAA92kqpD4IqJx5/lODDVOngbiDT8jUeq78nYhBKDYWXdgrQfTXrjcT/vFaku2UnZn4YaeqTyaJuxLEO8Cge3GGSEs2HKQEiWXorJn2QQHQa4igEa/bleVIvFa2zonl7mq9cTePIoH10gArIQueogm5QFdUQQY/oGb2iN+PJeDHejY9J64IxndlDf2B8/gBgh5y0 0 = p 8a3/9b2 AAACC3icbZC7TsMwFIadcivlVmBkiVohMZUEgWCsYGEsEr1IbRQ57mlr1U4i+6Siirqz8CosDCDEyguw8Ta4bQZoOZKlT/9/jn38B7HgGh3n28qtrK6tb+Q3C1vbO7t7xf2Dho4SxaDOIhGpVkA1CB5CHTkKaMUKqAwENIPhzdRvjkBpHoX3OI7Bk7Qf8h5nFI3kF0sdhAec3ZMq6E7STp9KSf3RaQbOxC+WnYozK3sZ3AzKJKuaX/zqdCOWSAiRCap123Vi9FKqkDMBk0In0RBTNqR9aBsMqQTtpbMdJvaxUbp2L1LmhGjP1N8TKZVaj2VgOiXFgV70puJ/XjvB3pWX8jBOEEI2f6iXCBsjexqM3eUKGIqxAcoUN7vabEAVZWjiK5gQ3MUvL0PjrOJeVJy783L1OosjT45IiZwQl1ySKrklNVInjDySZ/JK3qwn68V6tz7mrTkrmzkkf8r6/AEt45u/ v/ 0 AAACEHicbVC7TsMwFHV4lvIKMLJEVAimkiAQjBUsjEWiD6mpKse9ba3aSWTfIKoon8DCr7AwgBArIxt/g9tmgJYjWTo+597r6xPEgmt03W9rYXFpeWW1sFZc39jc2rZ3dus6ShSDGotEpJoB1SB4CDXkKKAZK6AyENAIhtdjv3EPSvMovMNRDG1J+yHvcUbRSB37yEd4wMmcVEE3S/0+lZJ2Uj8e8Owkv7lZxy65ZXcCZ554OSmRHNWO/eV3I5ZICJEJqnXLc2Nsp1QhZwKyop9oiCkb0j60DA2pBN1OJ4tkzqFRuk4vUuaE6EzU3x0plVqPZGAqJcWBnvXG4n9eK8HeZTvlYZwghGz6UC8RDkbOOB2nyxUwFCNDKFPc7OqwAVWUocmwaELwZr88T+qnZe+87N6elSpXeRwFsk8OyDHxyAWpkBtSJTXCyCN5Jq/kzXqyXqx362NaumDlPXvkD6zPH2t6nhA= / 0 AAACEHicbVC7TsMwFHV4lvIKMLJEVAimkiAQjBUsjEWiD6mpKse9ba3aSWTfIKoon8DCr7AwgBArIxt/g9tmgJYjWTo+597r6xPEgmt03W9rYXFpeWW1sFZc39jc2rZ3dus6ShSDGotEpJoB1SB4CDXkKKAZK6AyENAIhtdjv3EPSvMovMNRDG1J+yHvcUbRSB37yEd4wMmcVEE3S/0+lZJ2Uj8e8Owkv7lZxy65ZXcCZ554OSmRHNWO/eV3I5ZICJEJqnXLc2Nsp1QhZwKyop9oiCkb0j60DA2pBN1OJ4tkzqFRuk4vUuaE6EzU3x0plVqPZGAqJcWBnvXG4n9eK8HeZTvlYZwghGz6UC8RDkbOOB2nyxUwFCNDKFPc7OqwAVWUocmwaELwZr88T+qnZe+87N6elSpXeRwFsk8OyDHxyAWpkBtSJTXCyCN5Jq/kzXqyXqx362NaumDlPXvkD6zPH2t6nhA= / 0 To summarise, incomplete phase separation in active scalar field theories is a generic consequence of an effective interfacial tension - mechanical (causing flow) or diffusive (causing Ostwald ripening) - becoming negative. This is achieved by adding non-local terms, which break TRS, to equilibrium model H RS and Cates PRL (2019)

17 II. Phoresis and Stokesian hydrodynamics of active particles Michael E. Cates Ronojoy Adhikari

18 Nonequilibrium processes - in a thin layer on the surface drive exterior fluid flow - diverse mechanisms. The resulting fluid stress may react back and self-propel Feeding or fuel => break time-reversal symmetry locally nonequilibrium steady-state A B C Active particles equilibrium steady-state zero current, TRS Passive particles A B C net current, no TRS Active particles: special colloids Microorganisms Autophoretic particles Ramaswamy Annu. Rev. Condens. Matter Phys. 2010, JSTAT 2017; Cates arXiv:1904.01330 Particle-level dynamics of active particles, unlike Brownian colloids, has no time-reversal symmetry => no inherent Free energy or Boltzmann distribution Active matter: active particles in a fluid How to study active matter systems in absence of time- reversal symmetry for particle-level dynamics?

Stokes law Micrometer size => neglect inertia Newton’s equation becomes FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic 19

Stokes law Micrometer size => neglect inertia Newton’s equation becomes FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic Stokes law for a no-slip sphere AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0Y0nE10YoCtJlBfuAJobJdNIOnUzCzEQooZ/gxl9x40IRty7d+TdO2gjaeuDC4Zx7ufceP2ZUKsv6Mgpz8wuLS8Xl0srq2vqGubnVlFEiMGngiEWi7SNJGOWkoahipB0LgkKfkZY/uMr81j0Rkkb8Vg1j4oaox2lAMVJa8sx9J0Sq7wfw2qN3NXgBD0+dmDpEIejDH6/pUc8sWxVrDDhL7JyUQY66Z3463QgnIeEKMyRlx7Zi5aZIKIoZGZWcRJIY4QHqkY6mHIVEuun4oRHc00oXBpHQxRUcq78nUhRKOQx93ZmdKKe9TPzP6yQqOHdTyuNEEY4ni4KEQRXBLB3YpYJgxYaaICyovhXiPhIIK51hSYdgT788S5pHFfukYt0cl6uXeRxFsAN2wQGwwRmoghqogwbA4AE8gRfwajwaz8ab8T5pLRj5zDb4A+PjG886mzA= FH i = 6⇡⌘bV i AAACFXicbVDLSsNAFJ3UV62vqEs3g0UQ1JKIr41QFMRlBfuAJobJdNIOnUzCzEQooT/hxl9x40IRt4I7/8ZJG0RbDwycOede7r3HjxmVyrK+jMLM7Nz8QnGxtLS8srpmrm80ZJQITOo4YpFo+UgSRjmpK6oYacWCoNBnpOn3LzO/eU+EpBG/VYOYuCHqchpQjJSWPHMfHpw4MXWIQtCHTohUzw9gw6Nw7+d35dG7GjyHlmeWrYo1Apwmdk7KIEfNMz+dToSTkHCFGZKybVuxclMkFMWMDEtOIkmMcB91SVtTjkIi3XR01RDuaKUDg0joxxUcqb87UhRKOQh9XZktKie9TPzPaycqOHNTyuNEEY7Hg4KEQRXBLCLYoYJgxQaaICyo3hXiHhIIKx1kSYdgT548TRqHFfu4Yt0clasXeRxFsAW2wS6wwSmogmtQA3WAwQN4Ai/g1Xg0no03431cWjDynk3wB8bHN+sPnCU= 6⇡⌘bVi + FP i = 0 19

Stokes law Micrometer size => neglect inertia Newton’s equation becomes FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic Stokes law for a no-slip sphere AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0Y0nE10YoCtJlBfuAJobJdNIOnUzCzEQooZ/gxl9x40IRty7d+TdO2gjaeuDC4Zx7ufceP2ZUKsv6Mgpz8wuLS8Xl0srq2vqGubnVlFEiMGngiEWi7SNJGOWkoahipB0LgkKfkZY/uMr81j0Rkkb8Vg1j4oaox2lAMVJa8sx9J0Sq7wfw2qN3NXgBD0+dmDpEIejDH6/pUc8sWxVrDDhL7JyUQY66Z3463QgnIeEKMyRlx7Zi5aZIKIoZGZWcRJIY4QHqkY6mHIVEuun4oRHc00oXBpHQxRUcq78nUhRKOQx93ZmdKKe9TPzP6yQqOHdTyuNEEY4ni4KEQRXBLB3YpYJgxYaaICyovhXiPhIIK51hSYdgT788S5pHFfukYt0cl6uXeRxFsAN2wQGwwRmoghqogwbA4AE8gRfwajwaz8ab8T5pLRj5zDb4A+PjG886mzA= FH i = 6⇡⌘bV i AAACFXicbVDLSsNAFJ3UV62vqEs3g0UQ1JKIr41QFMRlBfuAJobJdNIOnUzCzEQooT/hxl9x40IRt4I7/8ZJG0RbDwycOede7r3HjxmVyrK+jMLM7Nz8QnGxtLS8srpmrm80ZJQITOo4YpFo+UgSRjmpK6oYacWCoNBnpOn3LzO/eU+EpBG/VYOYuCHqchpQjJSWPHMfHpw4MXWIQtCHTohUzw9gw6Nw7+d35dG7GjyHlmeWrYo1Apwmdk7KIEfNMz+dToSTkHCFGZKybVuxclMkFMWMDEtOIkmMcB91SVtTjkIi3XR01RDuaKUDg0joxxUcqb87UhRKOQh9XZktKie9TPzPaycqOHNTyuNEEY7Hg4KEQRXBLCLYoYJgxQaaICyo3hXiHhIIKx1kSYdgT548TRqHFfu4Yt0clasXeRxFsAW2wS6wwSmogmtQA3WAwQN4Ai/g1Xg0no03431cWjDynk3wB8bHN+sPnCU= 6⇡⌘bVi + FP i = 0 19 The fluid flow boundary condition on the surface of a no-slip sphere, with velocity and angular velocity , is given as: V i Ω i v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) is the radius vector of the -th particle. ρi i

Stokes law Micrometer size => neglect inertia Newton’s equation becomes FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic Stokes law for a no-slip sphere AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0Y0nE10YoCtJlBfuAJobJdNIOnUzCzEQooZ/gxl9x40IRty7d+TdO2gjaeuDC4Zx7ufceP2ZUKsv6Mgpz8wuLS8Xl0srq2vqGubnVlFEiMGngiEWi7SNJGOWkoahipB0LgkKfkZY/uMr81j0Rkkb8Vg1j4oaox2lAMVJa8sx9J0Sq7wfw2qN3NXgBD0+dmDpEIejDH6/pUc8sWxVrDDhL7JyUQY66Z3463QgnIeEKMyRlx7Zi5aZIKIoZGZWcRJIY4QHqkY6mHIVEuun4oRHc00oXBpHQxRUcq78nUhRKOQx93ZmdKKe9TPzP6yQqOHdTyuNEEY4ni4KEQRXBLB3YpYJgxYaaICyovhXiPhIIK51hSYdgT788S5pHFfukYt0cl6uXeRxFsAN2wQGwwRmoghqogwbA4AE8gRfwajwaz8ab8T5pLRj5zDb4A+PjG886mzA= FH i = 6⇡⌘bV i AAACFXicbVDLSsNAFJ3UV62vqEs3g0UQ1JKIr41QFMRlBfuAJobJdNIOnUzCzEQooT/hxl9x40IRt4I7/8ZJG0RbDwycOede7r3HjxmVyrK+jMLM7Nz8QnGxtLS8srpmrm80ZJQITOo4YpFo+UgSRjmpK6oYacWCoNBnpOn3LzO/eU+EpBG/VYOYuCHqchpQjJSWPHMfHpw4MXWIQtCHTohUzw9gw6Nw7+d35dG7GjyHlmeWrYo1Apwmdk7KIEfNMz+dToSTkHCFGZKybVuxclMkFMWMDEtOIkmMcB91SVtTjkIi3XR01RDuaKUDg0joxxUcqb87UhRKOQh9XZktKie9TPzPaycqOHNTyuNEEY7Hg4KEQRXBLCLYoYJgxQaaICyo3hXiHhIIKx1kSYdgT548TRqHFfu4Yt0clasXeRxFsAW2wS6wwSmogmtQA3WAwQN4Ai/g1Xg0no03431cWjDynk3wB8bHN+sPnCU= 6⇡⌘bVi + FP i = 0 Model an active particles as a sphere with slip boundary condition Boundary velocity = rigid body motion + active slip v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) 19 The fluid flow boundary condition on the surface of a no-slip sphere, with velocity and angular velocity , is given as: V i Ω i v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) is the radius vector of the -th particle. ρi i

Stokes law Micrometer size => neglect inertia Newton’s equation becomes FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic Stokes law for a no-slip sphere AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0Y0nE10YoCtJlBfuAJobJdNIOnUzCzEQooZ/gxl9x40IRty7d+TdO2gjaeuDC4Zx7ufceP2ZUKsv6Mgpz8wuLS8Xl0srq2vqGubnVlFEiMGngiEWi7SNJGOWkoahipB0LgkKfkZY/uMr81j0Rkkb8Vg1j4oaox2lAMVJa8sx9J0Sq7wfw2qN3NXgBD0+dmDpEIejDH6/pUc8sWxVrDDhL7JyUQY66Z3463QgnIeEKMyRlx7Zi5aZIKIoZGZWcRJIY4QHqkY6mHIVEuun4oRHc00oXBpHQxRUcq78nUhRKOQx93ZmdKKe9TPzP6yQqOHdTyuNEEY4ni4KEQRXBLB3YpYJgxYaaICyovhXiPhIIK51hSYdgT788S5pHFfukYt0cl6uXeRxFsAN2wQGwwRmoghqogwbA4AE8gRfwajwaz8ab8T5pLRj5zDb4A+PjG886mzA= FH i = 6⇡⌘bV i AAACFXicbVDLSsNAFJ3UV62vqEs3g0UQ1JKIr41QFMRlBfuAJobJdNIOnUzCzEQooT/hxl9x40IRt4I7/8ZJG0RbDwycOede7r3HjxmVyrK+jMLM7Nz8QnGxtLS8srpmrm80ZJQITOo4YpFo+UgSRjmpK6oYacWCoNBnpOn3LzO/eU+EpBG/VYOYuCHqchpQjJSWPHMfHpw4MXWIQtCHTohUzw9gw6Nw7+d35dG7GjyHlmeWrYo1Apwmdk7KIEfNMz+dToSTkHCFGZKybVuxclMkFMWMDEtOIkmMcB91SVtTjkIi3XR01RDuaKUDg0joxxUcqb87UhRKOQh9XZktKie9TPzPaycqOHNTyuNEEY7Hg4KEQRXBLCLYoYJgxQaaICyo3hXiHhIIKx1kSYdgT548TRqHFfu4Yt0clasXeRxFsAW2wS6wwSmogmtQA3WAwQN4Ai/g1Xg0no03431cWjDynk3wB8bHN+sPnCU= 6⇡⌘bVi + FP i = 0 Model an active particles as a sphere with slip boundary condition Boundary velocity = rigid body motion + active slip v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) What is slip? 19 The fluid flow boundary condition on the surface of a no-slip sphere, with velocity and angular velocity , is given as: V i Ω i v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) is the radius vector of the -th particle. ρi i

What is slip? Charged colloid in a non-conducting fluid Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md FP = qE = charge x Electric field 20 E

What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md FP = qE = charge x Electric field 20 E

What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: exterior flow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md FP = qE = charge x Electric field 20 E

What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: exterior flow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md FP = qE = charge x Electric field Charged colloid in a conducting fluid 20 E

What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: exterior flow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md FP = qE = charge x Electric field the colloid still moves & exterior flow decays as 1/r3 colloidal charge is balanced by a diffused cloud of counter ions AAAB+HicbVDLSsNAFL2pr1ofjbp0M1gEVyURi26EoiAuK9gHtLFMppN26GQSZiZCDf0SNy4UceunuPNvnLRZaOuBgcM593LPHD/mTGnH+bYKK6tr6xvFzdLW9s5u2d7bb6kokYQ2ScQj2fGxopwJ2tRMc9qJJcWhz2nbH19nfvuRSsUica8nMfVCPBQsYARrI/Xtci/EeuQH6OahgS6R07crTtWZAS0TNycVyNHo21+9QUSSkApNOFaq6zqx9lIsNSOcTku9RNEYkzEe0q6hAodUeeks+BQdG2WAgkiaJzSaqb83UhwqNQl9M5nFVIteJv7ndRMdXHgpE3GiqSDzQ0HCkY5Q1gIaMEmJ5hNDMJHMZEVkhCUm2nRVMiW4i19eJq3TqlurOndnlfpVXkcRDuEITsCFc6jDLTSgCQQSeIZXeLOerBfr3fqYjxasfOcA/sD6/AG9MJHU FP = 0 Charged colloid in a conducting fluid 20 E

What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: exterior flow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md FP = qE = charge x Electric field the colloid still moves & exterior flow decays as 1/r3 colloidal charge is balanced by a diffused cloud of counter ions AAAB+HicbVDLSsNAFL2pr1ofjbp0M1gEVyURi26EoiAuK9gHtLFMppN26GQSZiZCDf0SNy4UceunuPNvnLRZaOuBgcM593LPHD/mTGnH+bYKK6tr6xvFzdLW9s5u2d7bb6kokYQ2ScQj2fGxopwJ2tRMc9qJJcWhz2nbH19nfvuRSsUica8nMfVCPBQsYARrI/Xtci/EeuQH6OahgS6R07crTtWZAS0TNycVyNHo21+9QUSSkApNOFaq6zqx9lIsNSOcTku9RNEYkzEe0q6hAodUeeks+BQdG2WAgkiaJzSaqb83UhwqNQl9M5nFVIteJv7ndRMdXHgpE3GiqSDzQ0HCkY5Q1gIaMEmJ5hNDMJHMZEVkhCUm2nRVMiW4i19eJq3TqlurOndnlfpVXkcRDuEITsCFc6jDLTSgCQQSeIZXeLOerBfr3fqYjxasfOcA/sD6/AG9MJHU FP = 0 Charged colloid in a conducting fluid 20 E Anderson, Ann Rev Fluid Mech (1989) E

What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: exterior flow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md FP = qE = charge x Electric field the colloid still moves & exterior flow decays as 1/r3 colloidal charge is balanced by a diffused cloud of counter ions AAAB+HicbVDLSsNAFL2pr1ofjbp0M1gEVyURi26EoiAuK9gHtLFMppN26GQSZiZCDf0SNy4UceunuPNvnLRZaOuBgcM593LPHD/mTGnH+bYKK6tr6xvFzdLW9s5u2d7bb6kokYQ2ScQj2fGxopwJ2tRMc9qJJcWhz2nbH19nfvuRSsUica8nMfVCPBQsYARrI/Xtci/EeuQH6OahgS6R07crTtWZAS0TNycVyNHo21+9QUSSkApNOFaq6zqx9lIsNSOcTku9RNEYkzEe0q6hAodUeeks+BQdG2WAgkiaJzSaqb83UhwqNQl9M5nFVIteJv7ndRMdXHgpE3GiqSDzQ0HCkY5Q1gIaMEmJ5hNDMJHMZEVkhCUm2nRVMiW4i19eJq3TqlurOndnlfpVXkcRDuEITsCFc6jDLTSgCQQSeIZXeLOerBfr3fqYjxasfOcA/sD6/AG9MJHU FP = 0 Charged colloid in a conducting fluid 20 E Anderson, Ann Rev Fluid Mech (1989) E How does this neutral object move?

What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: exterior flow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md FP = qE = charge x Electric field the colloid still moves & exterior flow decays as 1/r3 colloidal charge is balanced by a diffused cloud of counter ions AAAB+HicbVDLSsNAFL2pr1ofjbp0M1gEVyURi26EoiAuK9gHtLFMppN26GQSZiZCDf0SNy4UceunuPNvnLRZaOuBgcM593LPHD/mTGnH+bYKK6tr6xvFzdLW9s5u2d7bb6kokYQ2ScQj2fGxopwJ2tRMc9qJJcWhz2nbH19nfvuRSsUica8nMfVCPBQsYARrI/Xtci/EeuQH6OahgS6R07crTtWZAS0TNycVyNHo21+9QUSSkApNOFaq6zqx9lIsNSOcTku9RNEYkzEe0q6hAodUeeks+BQdG2WAgkiaJzSaqb83UhwqNQl9M5nFVIteJv7ndRMdXHgpE3GiqSDzQ0HCkY5Q1gIaMEmJ5hNDMJHMZEVkhCUm2nRVMiW4i19eJq3TqlurOndnlfpVXkcRDuEITsCFc6jDLTSgCQQSeIZXeLOerBfr3fqYjxasfOcA/sD6/AG9MJHU FP = 0 Charged colloid in a conducting fluid 20 E Anderson, Ann Rev Fluid Mech (1989) E How does this neutral object move? it is not rigid - diffuse clouds of counter-ions move in opposite direction of the charged particle

What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: exterior flow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md FP = qE = charge x Electric field the colloid still moves & exterior flow decays as 1/r3 colloidal charge is balanced by a diffused cloud of counter ions AAAB+HicbVDLSsNAFL2pr1ofjbp0M1gEVyURi26EoiAuK9gHtLFMppN26GQSZiZCDf0SNy4UceunuPNvnLRZaOuBgcM593LPHD/mTGnH+bYKK6tr6xvFzdLW9s5u2d7bb6kokYQ2ScQj2fGxopwJ2tRMc9qJJcWhz2nbH19nfvuRSsUica8nMfVCPBQsYARrI/Xtci/EeuQH6OahgS6R07crTtWZAS0TNycVyNHo21+9QUSSkApNOFaq6zqx9lIsNSOcTku9RNEYkzEe0q6hAodUeeks+BQdG2WAgkiaJzSaqb83UhwqNQl9M5nFVIteJv7ndRMdXHgpE3GiqSDzQ0HCkY5Q1gIaMEmJ5hNDMJHMZEVkhCUm2nRVMiW4i19eJq3TqlurOndnlfpVXkcRDuEITsCFc6jDLTSgCQQSeIZXeLOerBfr3fqYjxasfOcA/sD6/AG9MJHU FP = 0 Charged colloid in a conducting fluid Slip flow AAACI3icbVBNS8MwGE7n15xfVY9egkPw4mhFUQRh6sXjBPcBax1pmm5haVqSdDBK/4sX/4oXD8rw4sH/YtrtoJsvhDw8z/Mm7/t4MaNSWdaXUVpaXlldK69XNja3tnfM3b2WjBKBSRNHLBIdD0nCKCdNRRUjnVgQFHqMtL3hXa63R0RIGvFHNY6JG6I+pwHFSGmqZ145IVIDL4AteA1PoMMQ7zMCHS9ivhyH+kpH2VNauDBi6U2WVRxRmHpm1apZRcFFYM9AFcyq0TMnjh/hJCRcYYak7NpWrNwUCUUxI/rdRJIY4SHqk66GHIVEummxYwaPNOPDIBL6cAUL9ndHikKZD6yd+axyXsvJ/7RuooJLN6U8ThThePpRkDCoIpgHBn0qCFZsrAHCgupZIR4ggbDSsVZ0CPb8yougdVqzz2vWw1m1fjuLowwOwCE4Bja4AHVwDxqgCTB4Bq/gHXwYL8abMTE+p9aSMevZB3/K+P4BCpGklQ== V = hvAi Electrophoretic slip vA = ✏⇣ 4⇡⌘ Es 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 AAACJXicbVDLSgMxFM34rPVVdekmWAQ3lhkp6EKhKoLLCvYBnbZk0jttaCYzJJlCHeZn3PgrblxYRHDlr5g+Ftp6IHA451xy7/EizpS27S9raXlldW09s5Hd3Nre2c3t7VdVGEsKFRryUNY9ooAzARXNNId6JIEEHoea178d+7UBSMVC8aiHETQD0hXMZ5RoI7Vzl4NW4gZE9yjh+DrFV/jU9SWhiQuRYjwU7hNogtOk6EbMNTSdpD0f37VUO5e3C/YEeJE4M5JHM5TbuZHbCWkcgNCUE6Uajh3pZkKkZpRDmnVjBRGhfdKFhqGCBKCayeTKFB8bpYP9UJonNJ6ovycSEig1DDyTHK+o5r2x+J/XiLV/0UyYiGINgk4/8mOOdYjHleEOk0A1HxpCqGRmV0x7xJSkTbFZU4Izf/IiqZ4VHLvgPBTzpZtZHRl0iI7QCXLQOSqhe1RGFUTRM3pF72hkvVhv1of1OY0uWbOZA/QH1vcP54qlgQ== 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 AAACI3icbVBNS8MwGE7n15xfVY9egkPw4mhFUQRh6sXjBPcBax1pmm5haVqSdDBK/4sX/4oXD8rw4sH/YtrtoJsvhDw8z/Mm7/t4MaNSWdaXUVpaXlldK69XNja3tnfM3b2WjBKBSRNHLBIdD0nCKCdNRRUjnVgQFHqMtL3hXa63R0RIGvFHNY6JG6I+pwHFSGmqZ145IVIDL4AteA1PoMMQ7zMCHS9ivhyH+kpH2VNauDBi6U2WVRxRmHpm1apZRcFFYM9AFcyq0TMnjh/hJCRcYYak7NpWrNwUCUUxI/rdRJIY4SHqk66GHIVEummxYwaPNOPDIBL6cAUL9ndHikKZD6yd+axyXsvJ/7RuooJLN6U8ThThePpRkDCoIpgHBn0qCFZsrAHCgupZIR4ggbDSsVZ0CPb8yougdVqzz2vWw1m1fjuLowwOwCE4Bja4AHVwDxqgCTB4Bq/gHXwYL8abMTE+p9aSMevZB3/K+P4BCpGklQ== V = hvAi AAACI3icbVBNS8MwGE7n15xfVY9egkPw4mhFUQRh6sXjBPcBax1pmm5haVqSdDBK/4sX/4oXD8rw4sH/YtrtoJsvhDw8z/Mm7/t4MaNSWdaXUVpaXlldK69XNja3tnfM3b2WjBKBSRNHLBIdD0nCKCdNRRUjnVgQFHqMtL3hXa63R0RIGvFHNY6JG6I+pwHFSGmqZ145IVIDL4AteA1PoMMQ7zMCHS9ivhyH+kpH2VNauDBi6U2WVRxRmHpm1apZRcFFYM9AFcyq0TMnjh/hJCRcYYak7NpWrNwUCUUxI/rdRJIY4SHqk66GHIVEummxYwaPNOPDIBL6cAUL9ndHikKZD6yd+axyXsvJ/7RuooJLN6U8ThThePpRkDCoIpgHBn0qCFZsrAHCgupZIR4ggbDSsVZ0CPb8yougdVqzz2vWw1m1fjuLowwOwCE4Bja4AHVwDxqgCTB4Bq/gHXwYL8abMTE+p9aSMevZB3/K+P4BCpGklQ== V = hvAi 20 E Anderson, Ann Rev Fluid Mech (1989) E How does this neutral object move? it is not rigid - diffuse clouds of counter-ions move in opposite direction of the charged particle

What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: exterior flow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md FP = qE = charge x Electric field the colloid still moves & exterior flow decays as 1/r3 colloidal charge is balanced by a diffused cloud of counter ions AAAB+HicbVDLSsNAFL2pr1ofjbp0M1gEVyURi26EoiAuK9gHtLFMppN26GQSZiZCDf0SNy4UceunuPNvnLRZaOuBgcM593LPHD/mTGnH+bYKK6tr6xvFzdLW9s5u2d7bb6kokYQ2ScQj2fGxopwJ2tRMc9qJJcWhz2nbH19nfvuRSsUica8nMfVCPBQsYARrI/Xtci/EeuQH6OahgS6R07crTtWZAS0TNycVyNHo21+9QUSSkApNOFaq6zqx9lIsNSOcTku9RNEYkzEe0q6hAodUeeks+BQdG2WAgkiaJzSaqb83UhwqNQl9M5nFVIteJv7ndRMdXHgpE3GiqSDzQ0HCkY5Q1gIaMEmJ5hNDMJHMZEVkhCUm2nRVMiW4i19eJq3TqlurOndnlfpVXkcRDuEITsCFc6jDLTSgCQQSeIZXeLOerBfr3fqYjxasfOcA/sD6/AG9MJHU FP = 0 a mechanism to drive exterior flow. The resulting fluid stress may cause self-propulsion Charged colloid in a conducting fluid Slip flow AAACI3icbVBNS8MwGE7n15xfVY9egkPw4mhFUQRh6sXjBPcBax1pmm5haVqSdDBK/4sX/4oXD8rw4sH/YtrtoJsvhDw8z/Mm7/t4MaNSWdaXUVpaXlldK69XNja3tnfM3b2WjBKBSRNHLBIdD0nCKCdNRRUjnVgQFHqMtL3hXa63R0RIGvFHNY6JG6I+pwHFSGmqZ145IVIDL4AteA1PoMMQ7zMCHS9ivhyH+kpH2VNauDBi6U2WVRxRmHpm1apZRcFFYM9AFcyq0TMnjh/hJCRcYYak7NpWrNwUCUUxI/rdRJIY4SHqk66GHIVEummxYwaPNOPDIBL6cAUL9ndHikKZD6yd+axyXsvJ/7RuooJLN6U8ThThePpRkDCoIpgHBn0qCFZsrAHCgupZIR4ggbDSsVZ0CPb8yougdVqzz2vWw1m1fjuLowwOwCE4Bja4AHVwDxqgCTB4Bq/gHXwYL8abMTE+p9aSMevZB3/K+P4BCpGklQ== V = hvAi Electrophoretic slip vA = ✏⇣ 4⇡⌘ Es 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 AAACI3icbVBNS8MwGE7n15xfVY9egkPw4mhFUQRh6sXjBPcBax1pmm5haVqSdDBK/4sX/4oXD8rw4sH/YtrtoJsvhDw8z/Mm7/t4MaNSWdaXUVpaXlldK69XNja3tnfM3b2WjBKBSRNHLBIdD0nCKCdNRRUjnVgQFHqMtL3hXa63R0RIGvFHNY6JG6I+pwHFSGmqZ145IVIDL4AteA1PoMMQ7zMCHS9ivhyH+kpH2VNauDBi6U2WVRxRmHpm1apZRcFFYM9AFcyq0TMnjh/hJCRcYYak7NpWrNwUCUUxI/rdRJIY4SHqk66GHIVEummxYwaPNOPDIBL6cAUL9ndHikKZD6yd+axyXsvJ/7RuooJLN6U8ThThePpRkDCoIpgHBn0qCFZsrAHCgupZIR4ggbDSsVZ0CPb8yougdVqzz2vWw1m1fjuLowwOwCE4Bja4AHVwDxqgCTB4Bq/gHXwYL8abMTE+p9aSMevZB3/K+P4BCpGklQ== V = hvAi AAACI3icbVBNS8MwGE7n15xfVY9egkPw4mhFUQRh6sXjBPcBax1pmm5haVqSdDBK/4sX/4oXD8rw4sH/YtrtoJsvhDw8z/Mm7/t4MaNSWdaXUVpaXlldK69XNja3tnfM3b2WjBKBSRNHLBIdD0nCKCdNRRUjnVgQFHqMtL3hXa63R0RIGvFHNY6JG6I+pwHFSGmqZ145IVIDL4AteA1PoMMQ7zMCHS9ivhyH+kpH2VNauDBi6U2WVRxRmHpm1apZRcFFYM9AFcyq0TMnjh/hJCRcYYak7NpWrNwUCUUxI/rdRJIY4SHqk66GHIVEummxYwaPNOPDIBL6cAUL9ndHikKZD6yd+axyXsvJ/7RuooJLN6U8ThThePpRkDCoIpgHBn0qCFZsrAHCgupZIR4ggbDSsVZ0CPb8yougdVqzz2vWw1m1fjuLowwOwCE4Bja4AHVwDxqgCTB4Bq/gHXwYL8abMTE+p9aSMevZB3/K+P4BCpGklQ== V = hvAi 20 E Anderson, Ann Rev Fluid Mech (1989) E How does this neutral object move? it is not rigid - diffuse clouds of counter-ions move in opposite direction of the charged particle

What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: exterior flow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md FP = qE = charge x Electric field the colloid still moves & exterior flow decays as 1/r3 colloidal charge is balanced by a diffused cloud of counter ions AAAB+HicbVDLSsNAFL2pr1ofjbp0M1gEVyURi26EoiAuK9gHtLFMppN26GQSZiZCDf0SNy4UceunuPNvnLRZaOuBgcM593LPHD/mTGnH+bYKK6tr6xvFzdLW9s5u2d7bb6kokYQ2ScQj2fGxopwJ2tRMc9qJJcWhz2nbH19nfvuRSsUica8nMfVCPBQsYARrI/Xtci/EeuQH6OahgS6R07crTtWZAS0TNycVyNHo21+9QUSSkApNOFaq6zqx9lIsNSOcTku9RNEYkzEe0q6hAodUeeks+BQdG2WAgkiaJzSaqb83UhwqNQl9M5nFVIteJv7ndRMdXHgpE3GiqSDzQ0HCkY5Q1gIaMEmJ5hNDMJHMZEVkhCUm2nRVMiW4i19eJq3TqlurOndnlfpVXkcRDuEITsCFc6jDLTSgCQQSeIZXeLOerBfr3fqYjxasfOcA/sD6/AG9MJHU FP = 0 a mechanism to drive exterior flow. The resulting fluid stress may cause self-propulsion Charged colloid in a conducting fluid Slip flow AAACI3icbVBNS8MwGE7n15xfVY9egkPw4mhFUQRh6sXjBPcBax1pmm5haVqSdDBK/4sX/4oXD8rw4sH/YtrtoJsvhDw8z/Mm7/t4MaNSWdaXUVpaXlldK69XNja3tnfM3b2WjBKBSRNHLBIdD0nCKCdNRRUjnVgQFHqMtL3hXa63R0RIGvFHNY6JG6I+pwHFSGmqZ145IVIDL4AteA1PoMMQ7zMCHS9ivhyH+kpH2VNauDBi6U2WVRxRmHpm1apZRcFFYM9AFcyq0TMnjh/hJCRcYYak7NpWrNwUCUUxI/rdRJIY4SHqk66GHIVEummxYwaPNOPDIBL6cAUL9ndHikKZD6yd+axyXsvJ/7RuooJLN6U8ThThePpRkDCoIpgHBn0qCFZsrAHCgupZIR4ggbDSsVZ0CPb8yougdVqzz2vWw1m1fjuLowwOwCE4Bja4AHVwDxqgCTB4Bq/gHXwYL8abMTE+p9aSMevZB3/K+P4BCpGklQ== V = hvAi Electrophoretic slip vA = ✏⇣ 4⇡⌘ Es 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 AAACJXicbVDLSgMxFM34rPVVdekmWAQ3lhkp6EKhKoLLCvYBnbZk0jttaCYzJJlCHeZn3PgrblxYRHDlr5g+Ftp6IHA451xy7/EizpS27S9raXlldW09s5Hd3Nre2c3t7VdVGEsKFRryUNY9ooAzARXNNId6JIEEHoea178d+7UBSMVC8aiHETQD0hXMZ5RoI7Vzl4NW4gZE9yjh+DrFV/jU9SWhiQuRYjwU7hNogtOk6EbMNTSdpD0f37VUO5e3C/YEeJE4M5JHM5TbuZHbCWkcgNCUE6Uajh3pZkKkZpRDmnVjBRGhfdKFhqGCBKCayeTKFB8bpYP9UJonNJ6ovycSEig1DDyTHK+o5r2x+J/XiLV/0UyYiGINgk4/8mOOdYjHleEOk0A1HxpCqGRmV0x7xJSkTbFZU4Izf/IiqZ4VHLvgPBTzpZtZHRl0iI7QCXLQOSqhe1RGFUTRM3pF72hkvVhv1of1OY0uWbOZA/QH1vcP54qlgQ== 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 AAACI3icbVBNS8MwGE7n15xfVY9egkPw4mhFUQRh6sXjBPcBax1pmm5haVqSdDBK/4sX/4oXD8rw4sH/YtrtoJsvhDw8z/Mm7/t4MaNSWdaXUVpaXlldK69XNja3tnfM3b2WjBKBSRNHLBIdD0nCKCdNRRUjnVgQFHqMtL3hXa63R0RIGvFHNY6JG6I+pwHFSGmqZ145IVIDL4AteA1PoMMQ7zMCHS9ivhyH+kpH2VNauDBi6U2WVRxRmHpm1apZRcFFYM9AFcyq0TMnjh/hJCRcYYak7NpWrNwUCUUxI/rdRJIY4SHqk66GHIVEummxYwaPNOPDIBL6cAUL9ndHikKZD6yd+axyXsvJ/7RuooJLN6U8ThThePpRkDCoIpgHBn0qCFZsrAHCgupZIR4ggbDSsVZ0CPb8yougdVqzz2vWw1m1fjuLowwOwCE4Bja4AHVwDxqgCTB4Bq/gHXwYL8abMTE+p9aSMevZB3/K+P4BCpGklQ== V = hvAi AAACI3icbVBNS8MwGE7n15xfVY9egkPw4mhFUQRh6sXjBPcBax1pmm5haVqSdDBK/4sX/4oXD8rw4sH/YtrtoJsvhDw8z/Mm7/t4MaNSWdaXUVpaXlldK69XNja3tnfM3b2WjBKBSRNHLBIdD0nCKCdNRRUjnVgQFHqMtL3hXa63R0RIGvFHNY6JG6I+pwHFSGmqZ145IVIDL4AteA1PoMMQ7zMCHS9ivhyH+kpH2VNauDBi6U2WVRxRmHpm1apZRcFFYM9AFcyq0TMnjh/hJCRcYYak7NpWrNwUCUUxI/rdRJIY4SHqk66GHIVEummxYwaPNOPDIBL6cAUL9ndHikKZD6yd+axyXsvJ/7RuooJLN6U8ThThePpRkDCoIpgHBn0qCFZsrAHCgupZIR4ggbDSsVZ0CPb8yougdVqzz2vWw1m1fjuLowwOwCE4Bja4AHVwDxqgCTB4Bq/gHXwYL8abMTE+p9aSMevZB3/K+P4BCpGklQ== V = hvAi 20 E Anderson, Ann Rev Fluid Mech (1989) E How does this neutral object move? it is not rigid - diffuse clouds of counter-ions move in opposite direction of the charged particle force-free motion from slip: exterior field: driven particles self-phoresis: active particles

Main theoretical questions The problem is classical and the motion is governed by Newton’s equations. We then need to know: FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic 21

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 particles under these forces? ‣ How do we take into account, simultaneously, the many-body character of the hydrodynamic and phoretic interactions? The problem is classical and the motion is governed by Newton’s equations. We then need to know: FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic 21

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 particles under these forces? ‣ How do we take into account, simultaneously, the many-body character of the hydrodynamic and phoretic interactions? The problem is classical and the motion is governed by Newton’s equations. We then need to know: , where is the normal component of the fluid stress. This is to be obtained from the Stokes equation FH i = ∫ f dSi f = ̂ ρi ⋅ σ FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic 21

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 particles under these forces? ‣ How do we take into account, simultaneously, the many-body character of the hydrodynamic and phoretic interactions? The problem is classical and the motion is governed by Newton’s equations. We then need to know: , where is the normal component of the fluid stress. This is to be obtained from the Stokes equation FH i = ∫ f dSi f = ̂ ρi ⋅ σ FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic 21 r · v = 0, r · + ⇠ = 0, fluid velocity = pI + ⌘(rv + (rv)T ) r · v = 0, r · + ⇠ = 0, fluid stress Boundary velocity = rigid body motion + active slip active slip boundary condition v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i )

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 particles under these forces? ‣ How do we take into account, simultaneously, the many-body character of the hydrodynamic and phoretic interactions? The problem is classical and the motion is governed by Newton’s equations. We then need to know: Given the slip, we seek to generalise Stokes law for active particles. f , where is the normal component of the fluid stress. This is to be obtained from the Stokes equation FH i = ∫ f dSi f = ̂ ρi ⋅ σ FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic 21 r · v = 0, r · + ⇠ = 0, fluid velocity = pI + ⌘(rv + (rv)T ) r · v = 0, r · + ⇠ = 0, fluid stress Boundary velocity = rigid body motion + active slip active slip boundary condition v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i )

Boundary integral representation of Stokes equations Fluid velocity at any point in the bulk is given in terms of integrals on the surface of the colloids Odqvist 1930, Jackson 1962, Ladyzhenskaia 1969, Pozrikidis 1992 force per unit area (traction) boundary velocity 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 v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi 22

Boundary integral representation of Stokes equations Fluid velocity at any point in the bulk is given in terms of integrals on the surface of the colloids Odqvist 1930, Jackson 1962, Ladyzhenskaia 1969, Pozrikidis 1992 force per unit area (traction) boundary velocity AAAC1nicfVJba9swFJbdXbrslm6PexELgxbaYI+W9WVQuocN9tLRpQ1ExhzLciwqWUaSA8G4Dy1jr/tte9uP2H+YnHiwJmEHhD6d7zvSuSgpBTc2CH55/ta9+w8ebj/qPX7y9Nnz/s6LC6MqTdmIKqH0OAHDBC/YyHIr2LjUDGQi2GVy9aHlL2dMG66Kr3ZeskjCtOAZp2CdK+7/nsU1AVHm0OySRInUzKXbat3svT8gvLCYnPIpnuCPf3UkYXZNvF/fPce82ctcxFK7gcQH+HPHd9dOQcqN2v2VvEgOtiY6V427YBk1++9TbQVReh7zXtwfBMNgYXgdhB0YoM7O4v5PkipaSVZYKsCYSRiUNqpBW04Fa3qkMqwEegVTNnGwAMlMVC/G0uA3zpPiTGm3XCMX3n8japCmzdQpJdjcrHKtcxM3qWx2HNW8KCvLCrp8KKsEtgq3M8Yp14xaMXcAqOYuV0xz0ECt+wltE8LVktfBxdtheDQMvhwOTk67dmyjV+g12kUheodO0Cd0hkaIeufe3Lvxbv2xf+1/878vpb7XxbxEd8z/8QfErunV v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi potential single layer double layer Laplace equation r2 = 0 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 (r) = Z h ˜ G(r, ri)˜(ri) ˜ K↵(ri, r)ˆ ⇢↵ (ri) i dSi 22

Boundary integral representation of Stokes equations Fluid velocity at any point in the bulk is given in terms of integrals on the surface of the colloids Green’s function Stress tensor Odqvist 1930, Jackson 1962, Ladyzhenskaia 1969, Pozrikidis 1992 force per unit area (traction) boundary velocity 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 v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi potential single layer double layer Laplace equation r2 = 0 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 (r) = Z h ˜ G(r, ri)˜(ri) ˜ K↵(ri, r)ˆ ⇢↵ (ri) i dSi 22

Boundary integral representation of Stokes equations Fluid velocity at any point in the bulk is given in terms of integrals on the surface of the colloids Green’s function Stress tensor Odqvist 1930, Jackson 1962, Ladyzhenskaia 1969, Pozrikidis 1992 force per unit area (traction) boundary velocity flow satisfies boundary conditions at non-particle boundaries 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 v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi potential single layer double layer Laplace equation r2 = 0 AAACy3icdVFNa9wwEJWdfqTbr2177MV0KWwgWezQklwKITk0UAop7SaBlVnGsnYtIktGGrfdKD72D/bWa39J5Y2hzWY7IPTmzRv0NJNVUliM419BuHHn7r37mw96Dx89fvK0/+z5qdW1YXzMtNTmPAPLpVB8jAIlP68MhzKT/Cy7OGrrZ1+5sUKrL7ioeFrCXImZYICemvZ/08qKIc20zO2i9Jczzda7HSoU0kMxn1AUMufufbOi2XY386kTTbPVqakV8xKa4VrNTif64HMKsir+I9xeMUULQEdNof/2Lb2vN+K9p/nnNulN+4N4FC8jug2SDgxIFyfT/k+aa1aXXCGTYO0kiStMHRgUTPKmR2vLK2AXMOcTDxWU3KZuuYsmeu2ZPJpp44/CaMn+2+GgtK1ZrywBC7taa8l1tUmNs/3UCVXVyBW7fmhWywh11C42yoXhDOXCA2BGeK8RK8AAQ7/+dgjJ6pdvg9PdUfJ2FH96Mzg47MaxSV6SV2RIErJHDsgxOSFjwoLjQAXfgu/hx9CGl+HVtTQMup4X5EaEP/4A6NznwA== (r) = Z h ˜ G(r, ri)˜(ri) ˜ K↵(ri, r)ˆ ⇢↵ (ri) i dSi 22

Boundary integral representation of Stokes equations Fluid velocity at any point in the bulk is given in terms of integrals on the surface of the colloids Green’s function Stress tensor the Green’s function and the Stress tensor satisfy Stokes equation the integral admits analytical solution by Galerkin discretization for smooth boundaries like spheres problem reduced from the bulk three-dimensional flow to the two-dimensional surfaces of the colloids Odqvist 1930, Jackson 1962, Ladyzhenskaia 1969, Pozrikidis 1992 force per unit area (traction) boundary velocity flow satisfies boundary conditions at non-particle boundaries 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 v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi potential single layer double layer Laplace equation r2 = 0 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 (r) = Z h ˜ G(r, ri)˜(ri) ˜ K↵(ri, r)ˆ ⇢↵ (ri) i dSi 22

Ritz-Galerkin discretization 23 boundary velocity v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) Mazur and Van Saarloos, Physica A 1982; Hess 2015; RS et al. JSTAT 2015, PRL 2016, JOSS 2020 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 v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi

Ritz-Galerkin discretization 23 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 vA i Ri + ⇢i = 1 X l=1 V(l) i · Y(l 1)(ˆ ⇢i ), boundary velocity v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) Mazur and Van Saarloos, Physica A 1982; Hess 2015; RS et al. JSTAT 2015, PRL 2016, JOSS 2020 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 v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi

Ritz-Galerkin discretization 23 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 vA i Ri + ⇢i = 1 X l=1 V(l) i · Y(l 1)(ˆ ⇢i ), boundary velocity v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) Mazur and Van Saarloos, Physica A 1982; Hess 2015; RS et al. JSTAT 2015, PRL 2016, JOSS 2020 Expansion of boundary fields in tensorial spherical harmonics Y(l) - dimensionless, symmetric, irreducible Cartesian tensors of rank l that form a complete, orthogonal basis on the sphere Y(l)(ˆ ⇢) = ( 1)l⇢l+1r(l)⇢ 1 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 Y (0) = 1, Y (1) ↵ = ˆ ⇢↵, Y (2) ↵ = 3ˆ ⇢↵ ˆ ⇢ ↵ 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 v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi

Ritz-Galerkin discretization 23 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 vA i Ri + ⇢i = 1 X l=1 V(l) i · Y(l 1)(ˆ ⇢i ), boundary velocity v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) Mazur and Van Saarloos, Physica A 1982; Hess 2015; RS et al. JSTAT 2015, PRL 2016, JOSS 2020 Expansion of boundary fields in tensorial spherical harmonics Y(l) - dimensionless, symmetric, irreducible Cartesian tensors of rank l that form a complete, orthogonal basis on the sphere Y(l)(ˆ ⇢) = ( 1)l⇢l+1r(l)⇢ 1 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 Y (0) = 1, Y (1) ↵ = ˆ ⇢↵, Y (2) ↵ = 3ˆ ⇢↵ ˆ ⇢ ↵ 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 v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi

Ritz-Galerkin discretization 23 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 vA i Ri + ⇢i = 1 X l=1 V(l) i · Y(l 1)(ˆ ⇢i ), boundary velocity v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) Mazur and Van Saarloos, Physica A 1982; Hess 2015; RS et al. JSTAT 2015, PRL 2016, JOSS 2020 Expansion of boundary fields in tensorial spherical harmonics Y(l) - dimensionless, symmetric, irreducible Cartesian tensors of rank l that form a complete, orthogonal basis on the sphere Y(l)(ˆ ⇢) = ( 1)l⇢l+1r(l)⇢ 1 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 Y (0) = 1, Y (1) ↵ = ˆ ⇢↵, Y (2) ↵ = 3ˆ ⇢↵ ˆ ⇢ ↵ 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 v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi irreducible coefficients AAACD3icbVC5TgMxFPRyhnAFKGksIlBool0EgjKChjJI5JCyIfI6bxMr3kP2W0S02j+g4VdoKECIlpaOv8E5CkgYydJo5j17PF4shUbb/rYWFpeWV1Zza/n1jc2t7cLObl1HieJQ45GMVNNjGqQIoYYCJTRjBSzwJDS8wdXIb9yD0iIKb3EYQztgvVD4gjM0Uqdw5CI84PieVEE3S92AYd/zaf0uLUlXi17AjrOsUyjaZXsMOk+cKSmSKaqdwpfbjXgSQIhcMq1bjh1jO2UKBZeQ5d1EQ8z4gPWgZWjIAtDtdJwjo4dG6VI/UuaESMfq742UBVoPA89MjtLqWW8k/ue1EvQv2qkI4wQh5JOH/ERSjOioHNoVCjjKoSGMK2GyUt5ninE0FeZNCc7sl+dJ/aTsnJXtm9Ni5XJaR47skwNSIg45JxVyTaqkRjh5JM/klbxZT9aL9W59TEYXrOnOHvkD6/MHLHWdWg== V(l )

Ritz-Galerkin discretization 23 Fluid flow due to the irreducible mode lσ known body forces and torques FP TP known slip coefficients 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 v(r) = G(1s) · FP + G(2a) · TP + X l =2s ⇧(l ) V(l ) AAACmnicbVFbb9MwFHbCZaPcCnvgAR4sKqRWQJVMIPZSaQyJi3gZaO2G6hI5jtNac+zIPplUWflR/BXe+Dc4aZC6jSNZ/vSd79zTUgoLUfQnCG/cvHV7Z/dO7+69+w8e9h89nlldGcanTEttzlJquRSKT0GA5Gel4bRIJT9Nzz80/tMLbqzQ6gTWJV8UdKlELhgFTyX9XyTVMrPrwn/uok6cqH86UlBYMSrd+7omqVgOt0XfW9HLbYqYlW7ZRjyaEFsViZOTuEklVA7rGrcp09zNuhJDOaoJyzT8c/xoydfxqL5UjawobPK3gaNXvaQ/iMZRa/g6iDswQJ0dJ/3fJNOsKrgCJqm18zgqYeGoAcEkr3uksryk7Jwu+dxDRQtuF65dbY1feCbDuTb+KcAtux3haGGbVr2yGcRe9TXk/3zzCvKDhROqrIArtimUVxKDxs2dcCYMZyDXHlBmhO8VsxU1lIG/ZrOE+OrI18Fsfxy/HUff3gwOj7p17KKn6Dkaohi9Q4foMzpGU8SCJ8Ek+Bh8Cp+FR+GX8OtGGgZdzB66ZOHJX5pz0PU= vA i Ri + ⇢i = 1 X l=1 V(l) i · Y(l 1)(ˆ ⇢i ), boundary velocity v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) Mazur and Van Saarloos, Physica A 1982; Hess 2015; RS et al. JSTAT 2015, PRL 2016, JOSS 2020 Expansion of boundary fields in tensorial spherical harmonics Y(l) - dimensionless, symmetric, irreducible Cartesian tensors of rank l that form a complete, orthogonal basis on the sphere Y(l)(ˆ ⇢) = ( 1)l⇢l+1r(l)⇢ 1 AAACc3icbZHLSiQxFIZT5Vy059Y64MaFwZ4BG8amygu6aRDduHRgWh26eopTqdNWMHUxOSU0Rb3APJ4738KNe9MXxFYPBP585/9JchIVShryvDvHXXj3/sPHxaXGp89fvn5rLq+cmbzUAnsiV7m+iMCgkhn2SJLCi0IjpJHC8+jqeNw/v0FtZJ79oVGBgxQuMzmUAsiisPn/779q02vXXf9XcF1CzMd7v12HAagigW6QAFWBTvI6rKaofmbcbj/hIEKCurvzRmAOTWxbQYyKYD4bNltex5sUfy38mWixWZ2GzdsgzkWZYkZCgTF93ytoUIEmKRTWjaA0WIC4gkvsW5lBimZQTWZW85+WxHyYa7sy4hP6PFFBaswojawzBUrMy94YvtXrlzQ8GFQyK0rCTEwPGpaKU87HH8BjqVGQGlkBQkt7Vy4S0CDIflPDDsF/+eTX4my74+91vN+7rcOj2TgW2RrbYJvMZ/vskJ2wU9Zjgt07q866w50Hd83dcH9Mra4zy3xnc+VuPQLIbL+H Y (0) = 1, Y (1) ↵ = ˆ ⇢↵, Y (2) ↵ = 3ˆ ⇢↵ ˆ ⇢ ↵ 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 v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi irreducible coefficients AAACD3icbVC5TgMxFPRyhnAFKGksIlBool0EgjKChjJI5JCyIfI6bxMr3kP2W0S02j+g4VdoKECIlpaOv8E5CkgYydJo5j17PF4shUbb/rYWFpeWV1Zza/n1jc2t7cLObl1HieJQ45GMVNNjGqQIoYYCJTRjBSzwJDS8wdXIb9yD0iIKb3EYQztgvVD4gjM0Uqdw5CI84PieVEE3S92AYd/zaf0uLUlXi17AjrOsUyjaZXsMOk+cKSmSKaqdwpfbjXgSQIhcMq1bjh1jO2UKBZeQ5d1EQ8z4gPWgZWjIAtDtdJwjo4dG6VI/UuaESMfq742UBVoPA89MjtLqWW8k/ue1EvQv2qkI4wQh5JOH/ERSjOioHNoVCjjKoSGMK2GyUt5ninE0FeZNCc7sl+dJ/aTsnJXtm9Ni5XJaR47skwNSIg45JxVyTaqkRjh5JM/klbxZT9aL9W59TEYXrOnOHvkD6/MHLHWdWg== V(l )

l Symmetric (ls) Antisymmetric (la) Trace (lt) 1 2 3 Fluid flow due to the irreducible mode lσ RS et al. JSTAT 2015, PRL 2016, JOSS 2020 24

l Symmetric (ls) Antisymmetric (la) Trace (lt) 1 2 3 SO(3) invariant way to classify active flows. The -th mode of the flow decays as in an unbounded geometry. It has three independent terms: symmetric irreducible gradients of a Green’s function G of Stokes equation its curl and, its Laplacian. l v ∝ r−l ∇l−2(∇ × G) ∇(l−1)G ∇l−3(∇2G) Fluid flow due to the irreducible mode lσ RS et al. JSTAT 2015, PRL 2016, JOSS 2020 24

l Symmetric (ls) Antisymmetric (la) Trace (lt) 1 2 3 SO(3) invariant way to classify active flows. The -th mode of the flow decays as in an unbounded geometry. It has three independent terms: symmetric irreducible gradients of a Green’s function G of Stokes equation its curl and, its Laplacian. l v ∝ r−l ∇l−2(∇ × G) ∇(l−1)G ∇l−3(∇2G) Fluid flow due to the irreducible mode lσ RS et al. JSTAT 2015, PRL 2016, JOSS 2020 Charged colloid in a non-conducting fluid 24

l Symmetric (ls) Antisymmetric (la) Trace (lt) 1 2 3 SO(3) invariant way to classify active flows. The -th mode of the flow decays as in an unbounded geometry. It has three independent terms: symmetric irreducible gradients of a Green’s function G of Stokes equation its curl and, its Laplacian. l v ∝ r−l ∇l−2(∇ × G) ∇(l−1)G ∇l−3(∇2G) Fluid flow due to the irreducible mode lσ RS et al. JSTAT 2015, PRL 2016, JOSS 2020 Charged colloid in a non-conducting fluid Charged colloid in a conducting fluid 24

25 Lighthill, CPAM 1952; Blake JFM 1971; RS et al. PRL 2016, JPC 2018 Generalized Stokes laws active slip boundary condition v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) irreducible slip coefficients: AAACD3icbVC5TgMxFPRyhnAFKGksIlBool0EgjKChjJI5JCyIfI6bxMr3kP2W0S02j+g4VdoKECIlpaOv8E5CkgYydJo5j17PF4shUbb/rYWFpeWV1Zza/n1jc2t7cLObl1HieJQ45GMVNNjGqQIoYYCJTRjBSzwJDS8wdXIb9yD0iIKb3EYQztgvVD4gjM0Uqdw5CI84PieVEE3S92AYd/zaf0uLUlXi17AjrOsUyjaZXsMOk+cKSmSKaqdwpfbjXgSQIhcMq1bjh1jO2UKBZeQ5d1EQ8z4gPWgZWjIAtDtdJwjo4dG6VI/UuaESMfq742UBVoPA89MjtLqWW8k/ue1EvQv2qkI4wQh5JOH/ERSjOioHNoVCjjKoSGMK2GyUt5ninE0FeZNCc7sl+dJ/aTsnJXtm9Ni5XJaR47skwNSIg45JxVyTaqkRjh5JM/klbxZT9aL9W59TEYXrOnOHvkD6/MHLHWdWg== V(l )

25 FH i = T T ij ·V i T R ij ·⌦ i 1 X l =1s (T, l ) ij · V(l ) j , TH i = RT ij ·V i RR ij ·⌦ i 1 X l =1s (R, l ) ij · V(l ) j . generalised friction tensors for slip boundary condition Solution of boundary integral equation gives Lighthill, CPAM 1952; Blake JFM 1971; RS et al. PRL 2016, JPC 2018 Generalized Stokes laws active slip boundary condition v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) irreducible slip coefficients: AAACD3icbVC5TgMxFPRyhnAFKGksIlBool0EgjKChjJI5JCyIfI6bxMr3kP2W0S02j+g4VdoKECIlpaOv8E5CkgYydJo5j17PF4shUbb/rYWFpeWV1Zza/n1jc2t7cLObl1HieJQ45GMVNNjGqQIoYYCJTRjBSzwJDS8wdXIb9yD0iIKb3EYQztgvVD4gjM0Uqdw5CI84PieVEE3S92AYd/zaf0uLUlXi17AjrOsUyjaZXsMOk+cKSmSKaqdwpfbjXgSQIhcMq1bjh1jO2UKBZeQ5d1EQ8z4gPWgZWjIAtDtdJwjo4dG6VI/UuaESMfq742UBVoPA89MjtLqWW8k/ue1EvQv2qkI4wQh5JOH/ERSjOioHNoVCjjKoSGMK2GyUt5ninE0FeZNCc7sl+dJ/aTsnJXtm9Ni5XJaR47skwNSIg45JxVyTaqkRjh5JM/klbxZT9aL9W59TEYXrOnOHvkD6/MHLHWdWg== V(l )

25 FH i = T T ij ·V i T R ij ·⌦ i 1 X l =1s (T, l ) ij · V(l ) j , TH i = RT ij ·V i RR ij ·⌦ i 1 X l =1s (R, l ) ij · V(l ) j . generalised friction tensors for slip boundary condition Solution of boundary integral equation gives Lighthill, CPAM 1952; Blake JFM 1971; RS et al. PRL 2016, JPC 2018 Generalized Stokes laws Consistent with the linearity of Stokes flow The generalised friction tensors relate the modes of slip and hydrodynamic forces The expression for the generalised friction tensors is obtained in terms of a Green’s function of the Stokes equation active slip boundary condition v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) irreducible slip coefficients: AAACD3icbVC5TgMxFPRyhnAFKGksIlBool0EgjKChjJI5JCyIfI6bxMr3kP2W0S02j+g4VdoKECIlpaOv8E5CkgYydJo5j17PF4shUbb/rYWFpeWV1Zza/n1jc2t7cLObl1HieJQ45GMVNNjGqQIoYYCJTRjBSzwJDS8wdXIb9yD0iIKb3EYQztgvVD4gjM0Uqdw5CI84PieVEE3S92AYd/zaf0uLUlXi17AjrOsUyjaZXsMOk+cKSmSKaqdwpfbjXgSQIhcMq1bjh1jO2UKBZeQ5d1EQ8z4gPWgZWjIAtDtdJwjo4dG6VI/UuaESMfq742UBVoPA89MjtLqWW8k/ue1EvQv2qkI4wQh5JOH/ERSjOioHNoVCjjKoSGMK2GyUt5ninE0FeZNCc7sl+dJ/aTsnJXtm9Ni5XJaR47skwNSIg45JxVyTaqkRjh5JM/klbxZT9aL9W59TEYXrOnOHvkD6/MHLHWdWg== V(l )

25 FH i = T T ij ·V i T R ij ·⌦ i 1 X l =1s (T, l ) ij · V(l ) j , TH i = RT ij ·V i RR ij ·⌦ i 1 X l =1s (R, l ) ij · V(l ) j . generalised friction tensors for slip boundary condition Solution of boundary integral equation gives Lighthill, CPAM 1952; Blake JFM 1971; RS et al. PRL 2016, JPC 2018 Generalized Stokes laws Consistent with the linearity of Stokes flow The generalised friction tensors relate the modes of slip and hydrodynamic forces The expression for the generalised friction tensors is obtained in terms of a Green’s function of the Stokes equation We use the above in Newton’s laws to obtain the rigid body motion FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic active slip boundary condition v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) irreducible slip coefficients: AAACD3icbVC5TgMxFPRyhnAFKGksIlBool0EgjKChjJI5JCyIfI6bxMr3kP2W0S02j+g4VdoKECIlpaOv8E5CkgYydJo5j17PF4shUbb/rYWFpeWV1Zza/n1jc2t7cLObl1HieJQ45GMVNNjGqQIoYYCJTRjBSzwJDS8wdXIb9yD0iIKb3EYQztgvVD4gjM0Uqdw5CI84PieVEE3S92AYd/zaf0uLUlXi17AjrOsUyjaZXsMOk+cKSmSKaqdwpfbjXgSQIhcMq1bjh1jO2UKBZeQ5d1EQ8z4gPWgZWjIAtDtdJwjo4dG6VI/UuaESMfq742UBVoPA89MjtLqWW8k/ue1EvQv2qkI4wQh5JOH/ERSjOioHNoVCjjKoSGMK2GyUt5ninE0FeZNCc7sl+dJ/aTsnJXtm9Ni5XJaR47skwNSIg45JxVyTaqkRjh5JM/klbxZT9aL9W59TEYXrOnOHvkD6/MHLHWdWg== V(l )

26 Active Brownian motion: theory FH i + FP i + ˆ F i = 0, TH i + Body Brownian Hydrodynamic T T ij ·Vj T R ij · ⌦j + FP i + ˆ Fi 1 X l =1s (T, l ) ij · V(l ) j = 0, RT ij ·Vj RR ij · ⌦j + TP i + ˆ Ti 1 X l =1s (R, l ) ij · V(l ) j = 0. Brownian Body Mazur and Van Saarloos, Physica A 1982; Ladd, JCP 1988; RS et al. JSTAT 2015, PRL 2016, JPC 2018

26 Active Brownian motion: theory FH i + FP i + ˆ F i = 0, TH i + Body Brownian Hydrodynamic invert for rigid body motion T T ij ·Vj T R ij · ⌦j + FP i + ˆ Fi 1 X l =1s (T, l ) ij · V(l ) j = 0, RT ij ·Vj RR ij · ⌦j + TP i + ˆ Ti 1 X l =1s (R, l ) ij · V(l ) j = 0. Brownian Body Mazur and Van Saarloos, Physica A 1982; Ladd, JCP 1988; RS et al. JSTAT 2015, PRL 2016, JPC 2018

26 Active Brownian motion: theory FH i + FP i + ˆ F i = 0, TH i + Body Brownian Hydrodynamic Propulsion tensors new tensors that relate modes of slip to rigid body motion Mobility matrices well-known for passive suspension Vi = µT T ij · FP j + µT R ij · TP j + q 2kBTµT T ij · ⌘T j + q 2kBTµT R ij · ⇣R j + 1 X l =2s ⇡(T, l ) ij · V(l ) j + VA i ⌦i = µRT ij · FP j + µRR ij · TP j | {z } Passive + q 2kBTµRT ij · ⇣T j + q 2kBTµRR ij · ⌘R j | {z } Brownian + 1 X l =2s ⇡(R, l ) ij · V(l ) j + ⌦A i | {z } Active there are infinitely many propulsion tensors in contrast to the four mobility matrices White noises the correlated noise does not depend on propulsion tensors invert for rigid body motion T T ij ·Vj T R ij · ⌦j + FP i + ˆ Fi 1 X l =1s (T, l ) ij · V(l ) j = 0, RT ij ·Vj RR ij · ⌦j + TP i + ˆ Ti 1 X l =1s (R, l ) ij · V(l ) j = 0. Brownian Body Mazur and Van Saarloos, Physica A 1982; Ladd, JCP 1988; RS et al. JSTAT 2015, PRL 2016, JPC 2018

26 Active Brownian motion: theory FH i + FP i + ˆ F i = 0, TH i + Body Brownian Hydrodynamic Propulsion tensors new tensors that relate modes of slip to rigid body motion Mobility matrices well-known for passive suspension Vi = µT T ij · FP j + µT R ij · TP j + q 2kBTµT T ij · ⌘T j + q 2kBTµT R ij · ⇣R j + 1 X l =2s ⇡(T, l ) ij · V(l ) j + VA i ⌦i = µRT ij · FP j + µRR ij · TP j | {z } Passive + q 2kBTµRT ij · ⇣T j + q 2kBTµRR ij · ⌘R j | {z } Brownian + 1 X l =2s ⇡(R, l ) ij · V(l ) j + ⌦A i | {z } Active there are infinitely many propulsion tensors in contrast to the four mobility matrices White noises the correlated noise does not depend on propulsion tensors invert for rigid body motion T T ij ·Vj T R ij · ⌦j + FP i + ˆ Fi 1 X l =1s (T, l ) ij · V(l ) j = 0, RT ij ·Vj RR ij · ⌦j + TP i + ˆ Ti 1 X l =1s (R, l ) ij · V(l ) j = 0. Brownian Body Mazur and Van Saarloos, Physica A 1982; Ladd, JCP 1988; RS et al. JSTAT 2015, PRL 2016, JPC 2018

Known: chemical surface flux at the particle boundaries jA = D⇢ · rc AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA== AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA== AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA== AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA== Desired: rigid body motion of the particles 27 RS et al. JPC 2018, JCP 2019, JOSS 2020 What about phoretic interactions?

Known: chemical surface flux at the particle boundaries jA = D⇢ · rc AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA== AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA== AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA== AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA== Desired: rigid body motion of the particles 27 RS et al. JPC 2018, JCP 2019, JOSS 2020 What about phoretic interactions?

Known: chemical surface flux at the particle boundaries jA = D⇢ · rc AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA== AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA== AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA== AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA== Desired: rigid body motion of the particles 27 Hydrodynamic interactions Exterior Fluid Flow Phoretic interactions Many-Body Slip Chemical Surface Flux RS et al. JPC 2018, JCP 2019, JOSS 2020 What about phoretic interactions?

Known: chemical surface flux at the particle boundaries jA = D⇢ · rc AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA== AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA== AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA== AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA== Desired: rigid body motion of the particles 27 Hydrodynamic interactions Exterior Fluid Flow Phoretic interactions Many-Body Slip Chemical Surface Flux RS et al. JPC 2018, JCP 2019, JOSS 2020 What about phoretic interactions?

RS et al. JCP 2019, JOSS 2020 Active Brownian motion: simulations ˙ Ri = Vi, ˙ pi = ⌦i ⇥ pi . ‣ The positions and orientations of the colloids are updated as ‣ The Steric repulsion is modelled using the truncated Lennard-Jones potential ‣ Problem reduced to the choice of a Green’s function and surface flux (or slip) ‣ The slip is fixed by choosing leading modes which match experimental data ‣ The boundary conditions in the bulk flow is implemented by choosing appropriate Green’s functions. No need to simulate the fluid explicitly, just like in Coulomb's law for evaluating electrostatic interactions 28 28

RS et al. JCP 2019, JOSS 2020 Active Brownian motion: simulations ˙ Ri = Vi, ˙ pi = ⌦i ⇥ pi . ‣ The positions and orientations of the colloids are updated as ‣ The Steric repulsion is modelled using the truncated Lennard-Jones potential ‣ Problem reduced to the choice of a Green’s function and surface flux (or slip) ‣ The slip is fixed by choosing leading modes which match experimental data ‣ The boundary conditions in the bulk flow is implemented by choosing appropriate Green’s functions. No need to simulate the fluid explicitly, just like in Coulomb's law for evaluating electrostatic interactions Experimental flow Theoretical flow slip expansion truncated to l=3. Thutupalli, Geyer, RS, Adhikari, and Stone, PNAS 2018 28 28

RS et al. JCP 2019, JOSS 2020 Active Brownian motion: simulations ˙ Ri = Vi, ˙ pi = ⌦i ⇥ pi . ‣ The positions and orientations of the colloids are updated as ‣ The Steric repulsion is modelled using the truncated Lennard-Jones potential ‣ Problem reduced to the choice of a Green’s function and surface flux (or slip) ‣ The slip is fixed by choosing leading modes which match experimental data ‣ The boundary conditions in the bulk flow is implemented by choosing appropriate Green’s functions. No need to simulate the fluid explicitly, just like in Coulomb's law for evaluating electrostatic interactions GitHub.com/rajeshrinet/PyStokes [20K downloads] Experimental flow Theoretical flow slip expansion truncated to l=3. Thutupalli, Geyer, RS, Adhikari, and Stone, PNAS 2018 28 28

RS et al. JCP 2019, JOSS 2020 Active Brownian motion: simulations ˙ Ri = Vi, ˙ pi = ⌦i ⇥ pi . ‣ The positions and orientations of the colloids are updated as ‣ The Steric repulsion is modelled using the truncated Lennard-Jones potential ‣ Problem reduced to the choice of a Green’s function and surface flux (or slip) ‣ The slip is fixed by choosing leading modes which match experimental data ‣ The boundary conditions in the bulk flow is implemented by choosing appropriate Green’s functions. No need to simulate the fluid explicitly, just like in Coulomb's law for evaluating electrostatic interactions GitHub.com/rajeshrinet/PyStokes [20K downloads] Experimental flow Theoretical flow slip expansion truncated to l=3. Thutupalli, Geyer, RS, Adhikari, and Stone, PNAS 2018 28 PNAS 2018 - with the labs of Dr. Thutupalli (Bangalore) and Prof. Stone (Princeton) J Phy Chem C 2018 - with the lab of Prof. T Pradeep at IIT Madras Sci Rep and Phy Rev E 2017 - with the lab of Prof. Banerjee at IISER Kolkata PRL 2020 - with the lab of Prof. Eiser at Cavendish Laboratory, University of Cambridge PRL 2020: Hamiltonian description of green algae Volvox dance near boundaries due to HI 28

RS et al. JCP 2019, JOSS 2020 Active Brownian motion: simulations ˙ Ri = Vi, ˙ pi = ⌦i ⇥ pi . ‣ The positions and orientations of the colloids are updated as ‣ The Steric repulsion is modelled using the truncated Lennard-Jones potential ‣ Problem reduced to the choice of a Green’s function and surface flux (or slip) ‣ The slip is fixed by choosing leading modes which match experimental data ‣ The boundary conditions in the bulk flow is implemented by choosing appropriate Green’s functions. No need to simulate the fluid explicitly, just like in Coulomb's law for evaluating electrostatic interactions GitHub.com/rajeshrinet/PyStokes [20K downloads] Experimental flow Theoretical flow slip expansion truncated to l=3. Thutupalli, Geyer, RS, Adhikari, and Stone, PNAS 2018 28 PNAS 2018 - with the labs of Dr. Thutupalli (Bangalore) and Prof. Stone (Princeton) J Phy Chem C 2018 - with the lab of Prof. T Pradeep at IIT Madras Sci Rep and Phy Rev E 2017 - with the lab of Prof. Banerjee at IISER Kolkata PRL 2020 - with the lab of Prof. Eiser at Cavendish Laboratory, University of Cambridge PRL 2020: Hamiltonian description of green algae Volvox dance near boundaries due to HI 28

Colloids tethered to an interface - free to move in the plane Caciagli, RS et al. PRL 2020 D Joshi Erika Eiser A Caciagli 29

Colloids tethered to an interface - free to move in the plane Caciagli, RS et al. PRL 2020 D Joshi Erika Eiser A Caciagli 29

30 Optically trap one of the colloids and study the optofluidic interactions Caciagli, RS et al. PRL 2020

30 Optically trap one of the colloids and study the optofluidic interactions Caciagli, RS et al. PRL 2020

30 Optically trap one of the colloids and study the optofluidic interactions Caciagli, RS et al. PRL 2020

31 Water Oil Puzzle: what causes motion into the hot region? Caciagli, RS et al. PRL 2020

31 Water Oil 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 FT = µT µ? rT1 1 Thermophoresis into the interface Puzzle: what causes motion into the hot region? Caciagli, RS et al. PRL 2020

31 Water Oil FP 1 AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= Monopolar flow once the colloid is stalled 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 FT = µT µ? rT1 1 Thermophoresis into the interface Puzzle: what causes motion into the hot region? Caciagli, RS et al. PRL 2020

31 Water Oil FP 1 AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= Monopolar flow once the colloid is stalled 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 FT = µT µ? rT1 1 Thermophoresis into the interface Puzzle: what causes motion into the hot region? Flow-induced attraction FH FH 0 1 Caciagli, RS et al. PRL 2020

31 Water Oil FP 1 AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= Monopolar flow once the colloid is stalled 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 FT = µT µ? rT1 1 Thermophoresis into the interface Puzzle: what causes motion into the hot region? Flow-induced attraction FH FH 0 1 Caciagli, RS et al. PRL 2020

31 Water Oil FP 1 AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= Monopolar flow once the colloid is stalled 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 FT = µT µ? rT1 1 Thermophoresis into the interface Puzzle: what causes motion into the hot region? The optofluidic force can written as the gradient of a potential. FH Flow-induced attraction FH FH 0 1 Caciagli, RS et al. PRL 2020

31 Water Oil FP 1 AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40= Monopolar flow once the colloid is stalled 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 FT = µT µ? rT1 1 Thermophoresis into the interface Puzzle: what causes motion into the hot region? The optofluidic force can written as the gradient of a potential. FH Flow-induced attraction FH FH 0 1 Caciagli, RS et al. PRL 2020 An attractive optofluidic potential centred about the hot region 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 (r) = µT 4⇡⌘µ?µk  1 1 + h r⇤ + 2 1 + h3 r⇤3 @zT1.

Summary: field-theoretic and particle-based theories of active matter Incomplete phase separation in scalar active matter A self-shearing instability (SSI), due to active contractile stress, interrupts growth of droplets by splitting them. The result is a dynamic steady-state maintained by the self- shearing instability and Ostwald ripening. SSI Ostwald ripening Analytical and numerics-friendly formalism to study phoresis and Stokesian hydrodynamics of colloids with surface slip. The non-equilibrium steady state due to heating (freezing by heating) admits an effective equilibrium description 32 Field-theoretic Particle-based theories of active matter Particle-based Phoresis and Stokesian hydrodynamics without resolving fluid degrees of freedom