1 Rajesh Singh Department of Applied Mathematics and Theoretical Physics Splitting and self-propelling active droplets

Splitting active droplets Self-propelling active droplets

3 This work has been done with Michael E. Cates I.Splitting active droplets RS and Cates, Phys. Rev. Lett. 123(14), 148005, 201 9

micrometer size - non-equilibrium processes on the surface create exterior fl ow and may lead to self-propulsion feeding or fuel => break time-reversal symmetry locally Active particles: special colloids 4 Microorganisms Autophoretic particles Ramaswamy Annu. Rev. Condens. Matter Phys. 2010, JSTAT 2017; Cates arXiv:1904.01330 Active matter: active particles in a fl uid

micrometer size - non-equilibrium processes on the surface create exterior fl ow and may lead to self-propulsion feeding or fuel => break time-reversal symmetry locally Active particles: special colloids 4 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 fl uid

micrometer size - non-equilibrium processes on the surface create exterior fl ow and may lead to self-propulsion feeding or fuel => break time-reversal symmetry locally Active particles: special colloids 4 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 fl uid

Incomplete active phase separations 5 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 5 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 5 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 fi eld number of active particles is conserved - use a conserved scalar fi eld

Incomplete active phase separations 5 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 fi eld number of active particles is conserved - use a conserved scalar fi eld start with a theory of phase separation in binary mixtures add minimal terms which break TRS recipe to build an active fi eld theory

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

6 How do binary mixtures (such as oil-water) phase separate from a homogenous mixed phase? Theory of coarsening kinetics in binary fl uid mixtures Consider a scalar fi eld 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)

6 How do binary mixtures (such as oil-water) phase separate from a homogenous mixed phase? Theory of coarsening kinetics in binary fl uid mixtures Consider a scalar fi eld 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 fi eld is conserved: ˙ = r · J Bray, Adv. in Phys. 43, 357 (1994)

6 How do binary mixtures (such as oil-water) phase separate from a homogenous mixed phase? Theory of coarsening kinetics in binary fl uid mixtures Consider a scalar fi eld 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 fl ow di ff usive part from a chemical potential µ The current has three parts: 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 J = rµ + v + p 2D⇤, The scalar fi eld is conserved: ˙ = r · J Bray, Adv. in Phys. 43, 357 (1994)

φ4 fi eld 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 fl ow the fl uid fl ow acts to minimise the deformation of the droplet 7 Hohenberg & Halperin 1977; Bray, Adv in Phys 1994; Kendon et al, JFM 2001; Cates and Tjhung, JFM 2018

φ4 fi eld 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 fl ow the fl uid fl ow acts to minimise the deformation of the droplet 7 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 fi eld 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 fl ow the fl uid fl ow acts to minimise the deformation of the droplet 7 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 fi eld 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 fl ow the fl uid fl ow acts to minimise the deformation of the droplet 7 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 fi eld 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 fl ow the fl uid fl ow acts to minimise the deformation of the droplet 7 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 fi eld theory of passive phase separation ˙ = r · J 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 no fl ow the fl uid fl ow acts to minimise the deformation of the droplet 7 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 fi eld 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 fl ow the fl uid fl ow acts to minimise the deformation of the droplet 7 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?

8 Model H: Ostwald ripening Chemical potential raised at curved interface μs 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

8 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 fi nite system μ∞ = μs Chemical potential raised at curved interface μs 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

8 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 fi nite system μ∞ = μs Chemical potential raised at curved interface μs 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

8 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 fi nite system μ∞ = μs Chemical potential raised at curved interface μs 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 di ff usion 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

9 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⇤,

9 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 fl uid fl ow even in absence of external forces extensile/contractile dipolar fl ow 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 fi eld theory of active phase separation RS et al. PRL 2019, PRR 2020; Cates and Tjhung JFM 2018; Tiribocchi et al. PRL 2015 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 ˙ = r · J Active model H 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 J = rµ + v + p 2D⇤, 10

φ4 fi eld 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 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 J = rµ + v + p 2D⇤, 10

φ4 fi eld 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⇤, 10

Incomplete phase separation ˜ κ < 0 Complete phase separation ˜ κ > 0 Active model H: nucleation and growth 11 -1 1 RS and Cates PRL (2019)

Incomplete phase separation ˜ κ < 0 Complete phase separation ˜ κ > 0 Active model H: nucleation and growth 11 -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 ⟹ γv < 0 RS and Cates PRL (2019) The spontaneous stretching motion results in SSI => splitting the droplet 12

Active model H: self-shearing instability (SSI) A spontaneous stretching motion of the interface A contractile active stress ˜ κ < 0 ⟹ γv < 0 SSI RS and Cates PRL (2019) The spontaneous stretching motion results in SSI => splitting the droplet 12

Active model H: self-shearing instability (SSI) A spontaneous stretching motion of the interface A contractile active stress ˜ κ < 0 ⟹ γv < 0 SSI Ostwald ripening RS and Cates PRL (2019) The spontaneous stretching motion results in SSI => splitting the droplet 12

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 ⟹ γv < 0 SSI Ostwald ripening RS and Cates PRL (2019) The spontaneous stretching motion results in SSI => splitting the droplet 12

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/ 13

Length scale determined by activity SSI Ostwald ripening RS and Cates PRL (2019) From the mechanical tension and fl uid viscosity , we can construct just one quantity with the dimensions of velocity γv 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/ 13

Length scale determined by activity SSI Ostwald ripening RS and Cates PRL (2019) From the mechanical tension and fl uid viscosity , we can construct just one quantity with the dimensions of velocity γv 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 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/ 13

Length scale determined by activity SSI Ostwald ripening RS and Cates PRL (2019) From the mechanical tension and fl uid viscosity , we can construct just one quantity with the dimensions of velocity γv 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 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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/ 13

Length scale determined by activity SSI Ostwald ripening RS and Cates PRL (2019) From the mechanical tension and fl uid viscosity , we can construct just one quantity with the dimensions of velocity γv 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 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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/ 13

14 Activity in di ff usive 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= 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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 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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⇤,

14 Activity in di ff usive 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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 J = rµ + J⇣ + p 2D⇤, AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT J⇣ = rµ⇣ + r ⇥ A

14 Activity in di ff usive 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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 J = rµ + J⇣ + p 2D⇤, e ff ective nonlocal chemical potential AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT J⇣ = rµ⇣ + r ⇥ A

14 Activity in di ff usive 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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 J = rµ + J⇣ + p 2D⇤, e ff ective nonlocal chemical potential 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 µs(R) = 0 bR + µ + µ⇣ Chemical potential raised at curved interface: μs AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT J⇣ = rµ⇣ + r ⇥ A

14 Activity in di ff usive sector Cates and Tjhung JFM 2018; Tjhung et al PRX 2018 The active contributions can reverse the sign of chemical fl ux and thus the sign of e ff ective 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⇤, e ff ective nonlocal chemical potential 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 µs(R) = 0 bR + µ + µ⇣ Chemical potential raised at curved interface: μs AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT J⇣ = rµ⇣ + r ⇥ A

14 Activity in di ff usive sector Cates and Tjhung JFM 2018; Tjhung et al PRX 2018 The active contributions can reverse the sign of chemical fl ux and thus the sign of e ff ective tension => reverse Ostwald ripening e ff ective tension in di ff usive 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⇤, e ff ective nonlocal chemical potential 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 µs(R) = 0 bR + µ + µ⇣ Chemical potential raised at curved interface: μs AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT J⇣ = rµ⇣ + r ⇥ A

14 Activity in di ff usive sector Cates and Tjhung JFM 2018; Tjhung et al PRX 2018 The active contributions can reverse the sign of chemical fl ux and thus the sign of e ff ective tension => reverse Ostwald ripening e ff ective tension in di ff usive 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⇤, e ff ective nonlocal chemical potential 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 µs(R) = 0 bR + µ + µ⇣ Chemical potential raised at curved interface: μs AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT J⇣ = rµ⇣ + r ⇥ A Forward Ostwald ripening γϕ > 0 Reverse Ostwald ripening γϕ < 0

15 Steady-states in the plane of e ff ective 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)

15 Steady-states in the plane of e ff ective tensions ˙ = r · J Active Model H 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 J = rµ + J⇣ + p 2D⇤, di ff usive 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 RS and Cates PRL (2019)

15 Steady-states in the plane of e ff ective tensions ˙ = r · J Active Model H 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 J = rµ + J⇣ + p 2D⇤, di ff usive 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 fi eld theories is a generic consequence of an e ff ective interfacial tension - mechanical (causing fl ow) or di ff usive (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)

16 II. Self-propelling active droplets RS, Tjhung and Cates, Phys. Rev. Research 2(3), 032024(R), 2020 ELSEN TJHUNG MICHAEL E. CATES

17 Poincloux et al PNAS 2011; Hawkins et al Biophys J 2011 Swimming of cells without liquid-crystalline order Cortex

17 Poincloux et al PNAS 2011; Hawkins et al Biophys J 2011 It has been experimentally found that the self- propulsion of a cell in a bulk 3D fl uid, far from any surfaces, is possible if the cell cortex can vary in thickness Swimming of cells without liquid-crystalline order Cortex

17 Poincloux et al PNAS 2011; Hawkins et al Biophys J 2011 It has been experimentally found that the self- propulsion of a cell in a bulk 3D fl uid, far from any surfaces, is possible if the cell cortex can vary in thickness A varying thickness of the cellular cortex is possible due to a higher concentration of myosin motors at the back of the cell Swimming of cells without liquid-crystalline order Cortex

17 Poincloux et al PNAS 2011; Hawkins et al Biophys J 2011 It has been experimentally found that the self- propulsion of a cell in a bulk 3D fl uid, far from any surfaces, is possible if the cell cortex can vary in thickness A varying thickness of the cellular cortex is possible due to a higher concentration of myosin motors at the back of the cell This results in the cortex becoming more more contractile at the rear of the cell Swimming of cells without liquid-crystalline order Cortex

17 Poincloux et al PNAS 2011; Hawkins et al Biophys J 2011 It has been experimentally found that the self- propulsion of a cell in a bulk 3D fl uid, far from any surfaces, is possible if the cell cortex can vary in thickness A varying thickness of the cellular cortex is possible due to a higher concentration of myosin motors at the back of the cell This results in the cortex becoming more more contractile at the rear of the cell A phase- fi eld model without any long-range vectorial/tensorial order was lacking Swimming of cells without liquid-crystalline order Cortex

Poincloux et al PNAS 2011 cell cortex can vary in thickness: more conc. of myosin motors at the back => more contractile Cortex RS, and Cates Phys Rev Lett 2019; RS, Tjhung and Cates, Phys Rev Research 2020 18 Active droplet: Active model H + Model H

Poincloux et al PNAS 2011 cell cortex can vary in thickness: more conc. of myosin motors at the back => more contractile Cortex RS, and Cates Phys Rev Lett 2019; RS, Tjhung and Cates, Phys Rev Research 2020 18 Active droplet: Active model H + Model H active model H AAAB/nicbVBNS8NAEN3Ur1q/quLJS7AInkoiih6LXjxWsLXQhLLZTNqlmw92J2IJAf+KFw+KePV3ePPfuE1z0NYHA4/3ZnZ2npcIrtCyvo3K0vLK6lp1vbaxubW9U9/d66o4lQw6LBax7HlUgeARdJCjgF4igYaegHtvfD317x9AKh5HdzhJwA3pMOIBZxS1NKgfOAiPWLyTSfDzzElGPB/UG1bTKmAuErskDVKiPah/OX7M0hAiZIIq1betBN2MSuRMQF5zUgUJZWM6hL6mEQ1BuVmxNjePteKbQSx1RWgW6u+JjIZKTUJPd4YUR2rem4r/ef0Ug0s341GSIkRstihIhYmxOc3C9LkEhmKiCWWS67+abEQlZagTq+kQ7PmTF0n3tGmfN63bs0brqoyjSg7JETkhNrkgLXJD2qRDGMnIM3klb8aT8WK8Gx+z1opRzuyTPzA+fwCYGpaJ

Poincloux et al PNAS 2011 cell cortex can vary in thickness: more conc. of myosin motors at the back => more contractile Cortex RS, and Cates Phys Rev Lett 2019; RS, Tjhung and Cates, Phys Rev Research 2020 18 Active droplet: Active model H + Model H active model H AAAB/nicbVBNS8NAEN3Ur1q/quLJS7AInkoiih6LXjxWsLXQhLLZTNqlmw92J2IJAf+KFw+KePV3ePPfuE1z0NYHA4/3ZnZ2npcIrtCyvo3K0vLK6lp1vbaxubW9U9/d66o4lQw6LBax7HlUgeARdJCjgF4igYaegHtvfD317x9AKh5HdzhJwA3pMOIBZxS1NKgfOAiPWLyTSfDzzElGPB/UG1bTKmAuErskDVKiPah/OX7M0hAiZIIq1betBN2MSuRMQF5zUgUJZWM6hL6mEQ1BuVmxNjePteKbQSx1RWgW6u+JjIZKTUJPd4YUR2rem4r/ef0Ug0s341GSIkRstihIhYmxOc3C9LkEhmKiCWWS67+abEQlZagTq+kQ7PmTF0n3tGmfN63bs0brqoyjSg7JETkhNrkgLXJD2qRDGMnIM3klb8aT8WK8Gx+z1opRzuyTPzA+fwCYGpaJ AAAB/3icbVDLSsNAFJ3UV62vqODGTbAIrkoiii6LblxWsA9oQ5lMb9qhkwczN2KJWfgrblwo4tbfcOffOE2z0NYDFw7n3Dt37vFiwRXa9rdRWlpeWV0rr1c2Nre2d8zdvZaKEsmgySIRyY5HFQgeQhM5CujEEmjgCWh74+up374HqXgU3uEkBjegw5D7nFHUUt886CE8YP5O6okEsrQXK571zapds3NYi8QpSJUUaPTNr94gYkkAITJBleo6doxuSiVyJiCr9BIFMWVjOoSupiENQLlpvjezjrUysPxI6grRytXfEykNlJoEnu4MKI7UvDcV//O6CfqXbsrDOEEI2WyRnwgLI2sahjXgEhiKiSaUSa7/arERlZShjqyiQ3DmT14krdOac16zb8+q9asijjI5JEfkhDjkgtTJDWmQJmHkkTyTV/JmPBkvxrvxMWstGcXMPvkD4/MHfCqXCw== model H

Poincloux et al PNAS 2011 cell cortex can vary in thickness: more conc. of myosin motors at the back => more contractile Cortex RS, and Cates Phys Rev Lett 2019; RS, Tjhung and Cates, Phys Rev Research 2020 18 Active droplet: Active model H + Model H active model H AAAB/nicbVBNS8NAEN3Ur1q/quLJS7AInkoiih6LXjxWsLXQhLLZTNqlmw92J2IJAf+KFw+KePV3ePPfuE1z0NYHA4/3ZnZ2npcIrtCyvo3K0vLK6lp1vbaxubW9U9/d66o4lQw6LBax7HlUgeARdJCjgF4igYaegHtvfD317x9AKh5HdzhJwA3pMOIBZxS1NKgfOAiPWLyTSfDzzElGPB/UG1bTKmAuErskDVKiPah/OX7M0hAiZIIq1betBN2MSuRMQF5zUgUJZWM6hL6mEQ1BuVmxNjePteKbQSx1RWgW6u+JjIZKTUJPd4YUR2rem4r/ef0Ug0s341GSIkRstihIhYmxOc3C9LkEhmKiCWWS67+abEQlZagTq+kQ7PmTF0n3tGmfN63bs0brqoyjSg7JETkhNrkgLXJD2qRDGMnIM3klb8aT8WK8Gx+z1opRzuyTPzA+fwCYGpaJ AAAB/3icbVDLSsNAFJ3UV62vqODGTbAIrkoiii6LblxWsA9oQ5lMb9qhkwczN2KJWfgrblwo4tbfcOffOE2z0NYDFw7n3Dt37vFiwRXa9rdRWlpeWV0rr1c2Nre2d8zdvZaKEsmgySIRyY5HFQgeQhM5CujEEmjgCWh74+up374HqXgU3uEkBjegw5D7nFHUUt886CE8YP5O6okEsrQXK571zapds3NYi8QpSJUUaPTNr94gYkkAITJBleo6doxuSiVyJiCr9BIFMWVjOoSupiENQLlpvjezjrUysPxI6grRytXfEykNlJoEnu4MKI7UvDcV//O6CfqXbsrDOEEI2WyRnwgLI2sahjXgEhiKiSaUSa7/arERlZShjqyiQ3DmT14krdOac16zb8+q9asijjI5JEfkhDjkgtTJDWmQJmHkkTyTV/JmPBkvxrvxMWstGcXMPvkD4/MHfCqXCw== model H ⇢ = AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA== AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA== AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA== AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA== Cortex

Poincloux et al PNAS 2011 cell cortex can vary in thickness: more conc. of myosin motors at the back => more contractile Cortex RS, and Cates Phys Rev Lett 2019; RS, Tjhung and Cates, Phys Rev Research 2020 18 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 F = Z ✓ a 2 2 + b 4 4 +  2 (r )2 + d 2 2 + ◆ dr Couples the two fi elds by the free energy and active stress Active droplet: Active model H + Model H active model H AAAB/nicbVBNS8NAEN3Ur1q/quLJS7AInkoiih6LXjxWsLXQhLLZTNqlmw92J2IJAf+KFw+KePV3ePPfuE1z0NYHA4/3ZnZ2npcIrtCyvo3K0vLK6lp1vbaxubW9U9/d66o4lQw6LBax7HlUgeARdJCjgF4igYaegHtvfD317x9AKh5HdzhJwA3pMOIBZxS1NKgfOAiPWLyTSfDzzElGPB/UG1bTKmAuErskDVKiPah/OX7M0hAiZIIq1betBN2MSuRMQF5zUgUJZWM6hL6mEQ1BuVmxNjePteKbQSx1RWgW6u+JjIZKTUJPd4YUR2rem4r/ef0Ug0s341GSIkRstihIhYmxOc3C9LkEhmKiCWWS67+abEQlZagTq+kQ7PmTF0n3tGmfN63bs0brqoyjSg7JETkhNrkgLXJD2qRDGMnIM3klb8aT8WK8Gx+z1opRzuyTPzA+fwCYGpaJ AAAB/3icbVDLSsNAFJ3UV62vqODGTbAIrkoiii6LblxWsA9oQ5lMb9qhkwczN2KJWfgrblwo4tbfcOffOE2z0NYDFw7n3Dt37vFiwRXa9rdRWlpeWV0rr1c2Nre2d8zdvZaKEsmgySIRyY5HFQgeQhM5CujEEmjgCWh74+up374HqXgU3uEkBjegw5D7nFHUUt886CE8YP5O6okEsrQXK571zapds3NYi8QpSJUUaPTNr94gYkkAITJBleo6doxuSiVyJiCr9BIFMWVjOoSupiENQLlpvjezjrUysPxI6grRytXfEykNlJoEnu4MKI7UvDcV//O6CfqXbsrDOEEI2WyRnwgLI2sahjXgEhiKiSaUSa7/arERlZShjqyiQ3DmT14krdOac16zb8+q9asijjI5JEfkhDjkgtTJDWmQJmHkkTyTV/JmPBkvxrvxMWstGcXMPvkD4/MHfCqXCw== model H ⇢ = AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA== AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA== AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA== AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA== Cortex

Poincloux et al PNAS 2011 cell cortex can vary in thickness: more conc. of myosin motors at the back => more contractile Cortex RS, and Cates Phys Rev Lett 2019; RS, Tjhung and Cates, Phys Rev Research 2020 18 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 F = Z ✓ a 2 2 + b 4 4 +  2 (r )2 + d 2 2 + ◆ dr Couples the two fi elds by the free energy and active stress Active droplet: Active model H + Model H active model H AAAB/nicbVBNS8NAEN3Ur1q/quLJS7AInkoiih6LXjxWsLXQhLLZTNqlmw92J2IJAf+KFw+KePV3ePPfuE1z0NYHA4/3ZnZ2npcIrtCyvo3K0vLK6lp1vbaxubW9U9/d66o4lQw6LBax7HlUgeARdJCjgF4igYaegHtvfD317x9AKh5HdzhJwA3pMOIBZxS1NKgfOAiPWLyTSfDzzElGPB/UG1bTKmAuErskDVKiPah/OX7M0hAiZIIq1betBN2MSuRMQF5zUgUJZWM6hL6mEQ1BuVmxNjePteKbQSx1RWgW6u+JjIZKTUJPd4YUR2rem4r/ef0Ug0s341GSIkRstihIhYmxOc3C9LkEhmKiCWWS67+abEQlZagTq+kQ7PmTF0n3tGmfN63bs0brqoyjSg7JETkhNrkgLXJD2qRDGMnIM3klb8aT8WK8Gx+z1opRzuyTPzA+fwCYGpaJ AAAB/3icbVDLSsNAFJ3UV62vqODGTbAIrkoiii6LblxWsA9oQ5lMb9qhkwczN2KJWfgrblwo4tbfcOffOE2z0NYDFw7n3Dt37vFiwRXa9rdRWlpeWV0rr1c2Nre2d8zdvZaKEsmgySIRyY5HFQgeQhM5CujEEmjgCWh74+up374HqXgU3uEkBjegw5D7nFHUUt886CE8YP5O6okEsrQXK571zapds3NYi8QpSJUUaPTNr94gYkkAITJBleo6doxuSiVyJiCr9BIFMWVjOoSupiENQLlpvjezjrUysPxI6grRytXfEykNlJoEnu4MKI7UvDcV//O6CfqXbsrDOEEI2WyRnwgLI2sahjXgEhiKiSaUSa7/arERlZShjqyiQ3DmT14krdOac16zb8+q9asijjI5JEfkhDjkgtTJDWmQJmHkkTyTV/JmPBkvxrvxMWstGcXMPvkD4/MHfCqXCw== model H ⇢ = AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA== AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA== AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA== AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA== Cortex AAACUXicdVFNbxMxEJ0sBUoKNMCRi0WEVC7RLmoFF6QClx6LIG2lbIhmvZOsVa/XsmcR0Wr/Igc48T964dAKb5oD/WAky0/vzbPHz5nVynMc/+5Fdzbu3ru/+aC/9fDR4+3Bk6dHvqqdpLGsdOVOMvSklaExK9Z0Yh1hmWk6zk4/dvrxN3JeVeYLLy1NS1wYNVcSOVCzQZFmlc79sgxbk35WixLbr8379l3K9J1X5zeO8rZJUdsCRWq9asXOFZfBTGOb2kK9+q/Qnw2G8ShelbgJkjUYwroOZ4OfaV7JuiTDUqP3kyS2PG3QsZKa2n5ae7IoT3FBkwANluSnzWriVrwMTC7mlQvLsFix/zoaLH03ZegskQt/XevI27RJzfO300YZWzMZeXnRvNaCK9HFK3LlSLJeBoDSqTCrkAU6lBw+oQshuf7km+Do9SjZG8Wfdof7H9ZxbMJzeAE7kMAb2IcDOIQxSPgBZ3AOF71fvT8RRNFla9Rbe57BlYq2/gIefrbT ⌃A = ↵ (r )(r )

Poincloux et al PNAS 2011 cell cortex can vary in thickness: more conc. of myosin motors at the back => more contractile Cortex RS, and Cates Phys Rev Lett 2019; RS, Tjhung and Cates, Phys Rev Research 2020 18 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 F = Z ✓ a 2 2 + b 4 4 +  2 (r )2 + d 2 2 + ◆ dr Couples the two fi elds by the free energy and active stress Active droplet: Active model H + Model H active model H AAAB/nicbVBNS8NAEN3Ur1q/quLJS7AInkoiih6LXjxWsLXQhLLZTNqlmw92J2IJAf+KFw+KePV3ePPfuE1z0NYHA4/3ZnZ2npcIrtCyvo3K0vLK6lp1vbaxubW9U9/d66o4lQw6LBax7HlUgeARdJCjgF4igYaegHtvfD317x9AKh5HdzhJwA3pMOIBZxS1NKgfOAiPWLyTSfDzzElGPB/UG1bTKmAuErskDVKiPah/OX7M0hAiZIIq1betBN2MSuRMQF5zUgUJZWM6hL6mEQ1BuVmxNjePteKbQSx1RWgW6u+JjIZKTUJPd4YUR2rem4r/ef0Ug0s341GSIkRstihIhYmxOc3C9LkEhmKiCWWS67+abEQlZagTq+kQ7PmTF0n3tGmfN63bs0brqoyjSg7JETkhNrkgLXJD2qRDGMnIM3klb8aT8WK8Gx+z1opRzuyTPzA+fwCYGpaJ AAAB/3icbVDLSsNAFJ3UV62vqODGTbAIrkoiii6LblxWsA9oQ5lMb9qhkwczN2KJWfgrblwo4tbfcOffOE2z0NYDFw7n3Dt37vFiwRXa9rdRWlpeWV0rr1c2Nre2d8zdvZaKEsmgySIRyY5HFQgeQhM5CujEEmjgCWh74+up374HqXgU3uEkBjegw5D7nFHUUt886CE8YP5O6okEsrQXK571zapds3NYi8QpSJUUaPTNr94gYkkAITJBleo6doxuSiVyJiCr9BIFMWVjOoSupiENQLlpvjezjrUysPxI6grRytXfEykNlJoEnu4MKI7UvDcV//O6CfqXbsrDOEEI2WyRnwgLI2sahjXgEhiKiSaUSa7/arERlZShjqyiQ3DmT14krdOac16zb8+q9asijjI5JEfkhDjkgtTJDWmQJmHkkTyTV/JmPBkvxrvxMWstGcXMPvkD4/MHfCqXCw== model H ⇢ = AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA== AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA== AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA== AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA== Cortex Active stress couples the two models at the interface ΣA Fluid fl ow depends on the sign of the active stress ΣA 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 ⌃A = ↵ (r )(r )

RS, Tjhung and Cates, Phys Rev Research 2020 19 Active droplet: propulsion mechanism

symmetry breaking (rotational) by initial perturbation or noise RS, Tjhung and Cates, Phys Rev Research 2020 19 Active droplet: propulsion mechanism

symmetry breaking (rotational) by initial perturbation or noise RS, Tjhung and Cates, Phys Rev Research 2020 19 ‣ Spontaneous symmetry breaking => self-propulsion ‣ Displaced ψ-droplet is locked in the front of φ-droplet ‣ This steady-state is maintained by the active fl ow due to the excess contractile stress at the back. Active droplet: propulsion mechanism

symmetry breaking (rotational) by initial perturbation or noise RS, Tjhung and Cates, Phys Rev Research 2020 19 ‣ Spontaneous symmetry breaking => self-propulsion ‣ Displaced ψ-droplet is locked in the front of φ-droplet ‣ This steady-state is maintained by the active fl ow due to the excess contractile stress at the back. Active droplet: propulsion mechanism surface tension gradient between front and rear of the droplet 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 ⌃A = ↵ (r )(r )

Active droplet: simulations 20 RS, Tjhung and Cates, Phys Rev Research 2020

Active droplet: simulations 20 RS, Tjhung and Cates, Phys Rev Research 2020

Active droplet: simulations 20 RS, Tjhung and Cates, Phys Rev Research 2020

Active droplet: theory 21 vr(r = R) = ˜ vr(r = R) = 0, v✓(r = R) = ˜ v✓(r = R), ( ˜) · ˆ r = (r · ˆ r) ˆ r r , at r = R. 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 no- fl ux continuity of tangential surface slip velocity Cauchy stress is related to the surface tension RS, Tjhung and Cates, Phys Rev Research 2020

In the co-moving frame of the droplet, the fl ow speed at in fi nity equals the swim speed U Active droplet: theory 21 vr(r = R) = ˜ vr(r = R) = 0, v✓(r = R) = ˜ v✓(r = R), ( ˜) · ˆ r = (r · ˆ r) ˆ r r , at r = R. 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 no- fl ux continuity of tangential surface slip velocity Cauchy stress is related to the surface tension RS, Tjhung and Cates, Phys Rev Research 2020

In the co-moving frame of the droplet, the fl ow speed at in fi nity equals the swim speed U Active droplet: theory 21 vr(r = R) = ˜ vr(r = R) = 0, v✓(r = R) = ˜ v✓(r = R), ( ˜) · ˆ r = (r · ˆ r) ˆ r r , at r = R. 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 no- fl ux continuity of tangential surface slip velocity Cauchy stress is related to the surface tension RS, Tjhung and Cates, Phys Rev Research 2020 vx = @# @y , vy = @# @x , vr = 1 r @# @✓ , v✓ = @# @r . 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 #(r, ✓) = f(r) sin ✓ AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== stream function r4# = 0 AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==

In the co-moving frame of the droplet, the fl ow speed at in fi nity equals the swim speed U Active droplet: theory 21 vr(r = R) = ˜ vr(r = R) = 0, v✓(r = R) = ˜ v✓(r = R), ( ˜) · ˆ r = (r · ˆ r) ˆ r r , at r = R. 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 no- fl ux continuity of tangential surface slip velocity Cauchy stress is related to the surface tension Interior Exterior RS, Tjhung and Cates, Phys Rev Research 2020 vx = @# @y , vy = @# @x , vr = 1 r @# @✓ , v✓ = @# @r . 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 AAACvnicjVFdS8MwFE3r9/ya+uhLcAg+6GhFUJDB0BcfFdwU1jFus3SLSz9M0rFS+icFH/w3plvVuQnzQsjJuffkJue6EWdSWdaHYS4tr6yurW+UNre2d3bLe/tNGcaC0AYJeSieXZCUs4A2FFOcPkeCgu9y+uQObvP805AKycLgUSURbfvQC5jHCChNdcrvw046ymqOJ4CkTgRCMeDOUO99qiD7pnCSnTqvMXSxFiRZ7WyxYjSlEF8t7EzjhVpncv7RF8R/2oqsWuqUK1bVGgeeB3YBKqiI+075zemGJPZpoAgHKVu2Fal2ml9JOM1KTixpBGQAPdrSMACfynY6tj/Dx5rpYi8UegUKj9lpRQq+lInv6kofVF/O5nLyr1wrVt5VO2VBFCsakEkjL+ZYhTifJe4yQYniiQZABNNvxaQP2iClJ56bYM9+eR40z6u2VbUfLir1m8KOdXSIjtAJstElqqM7dI8aiBjXBhgvxsCsm57pm+Gk1DQKzQH6FeboE0Je4SI= 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 #(r, ✓) = f(r) sin ✓ AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== stream function r4# = 0 AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==

In the co-moving frame of the droplet, the fl ow speed at in fi nity equals the swim speed U Active droplet: theory 21 vr(r = R) = ˜ vr(r = R) = 0, v✓(r = R) = ˜ v✓(r = R), ( ˜) · ˆ r = (r · ˆ r) ˆ r r , at r = R. 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 no- fl ux continuity of tangential surface slip velocity Cauchy stress is related to the surface tension fl ow should remain fi nite => condition at in fi nity => no- fl ux condition => C = D = 0 B = U A = − UR2 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 fext = A r + Br + Cr ln r + Dr3 Interior Exterior RS, Tjhung and Cates, Phys Rev Research 2020 vx = @# @y , vy = @# @x , vr = 1 r @# @✓ , v✓ = @# @r . AAACvnicjVFdS8MwFE3r9/ya+uhLcAg+6GhFUJDB0BcfFdwU1jFus3SLSz9M0rFS+icFH/w3plvVuQnzQsjJuffkJue6EWdSWdaHYS4tr6yurW+UNre2d3bLe/tNGcaC0AYJeSieXZCUs4A2FFOcPkeCgu9y+uQObvP805AKycLgUSURbfvQC5jHCChNdcrvw046ymqOJ4CkTgRCMeDOUO99qiD7pnCSnTqvMXSxFiRZ7WyxYjSlEF8t7EzjhVpncv7RF8R/2oqsWuqUK1bVGgeeB3YBKqiI+075zemGJPZpoAgHKVu2Fal2ml9JOM1KTixpBGQAPdrSMACfynY6tj/Dx5rpYi8UegUKj9lpRQq+lInv6kofVF/O5nLyr1wrVt5VO2VBFCsakEkjL+ZYhTifJe4yQYniiQZABNNvxaQP2iClJ56bYM9+eR40z6u2VbUfLir1m8KOdXSIjtAJstElqqM7dI8aiBjXBhgvxsCsm57pm+Gk1DQKzQH6FeboE0Je4SI= 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 #(r, ✓) = f(r) sin ✓ AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== stream function r4# = 0 AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==

In the co-moving frame of the droplet, the fl ow speed at in fi nity equals the swim speed U Active droplet: theory 21 vr(r = R) = ˜ vr(r = R) = 0, v✓(r = R) = ˜ v✓(r = R), ( ˜) · ˆ r = (r · ˆ r) ˆ r r , at r = R. 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 no- fl ux continuity of tangential surface slip velocity Cauchy stress is related to the surface tension fl ow should remain fi nite => condition at in fi nity => no- fl ux condition => C = D = 0 B = U A = − UR2 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 fext = A r + Br + Cr ln r + Dr3 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 fint = ˜ A r + ˜ Br + ˜ Cr ln r + ˜ Dr3 fl ow should remain fi nite => no- fl ux condition => continuity of slip => ˜ A = ˜ C = 0 ˜ B + ˜ CR2 = 0 ˜ B + 3 ˜ CR2 = − 2U Interior Exterior RS, Tjhung and Cates, Phys Rev Research 2020 vx = @# @y , vy = @# @x , vr = 1 r @# @✓ , v✓ = @# @r . 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 #(r, ✓) = f(r) sin ✓ AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== stream function r4# = 0 AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==

In the co-moving frame of the droplet, the fl ow speed at in fi nity equals the swim speed U Active droplet: theory 21 vr(r = R) = ˜ vr(r = R) = 0, v✓(r = R) = ˜ v✓(r = R), ( ˜) · ˆ r = (r · ˆ r) ˆ r r , at r = R. 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 no- fl ux continuity of tangential surface slip velocity Cauchy stress is related to the surface tension fl ow should remain fi nite => condition at in fi nity => no- fl ux condition => C = D = 0 B = U A = − UR2 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 fext = A r + Br + Cr ln r + Dr3 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 fint = ˜ A r + ˜ Br + ˜ Cr ln r + ˜ Dr3 fl ow should remain fi nite => no- fl ux condition => continuity of slip => ˜ A = ˜ C = 0 ˜ B + ˜ CR2 = 0 ˜ B + 3 ˜ CR2 = − 2U Interior Exterior no Stokes paradox! RS, Tjhung and Cates, Phys Rev Research 2020 vx = @# @y , vy = @# @x , vr = 1 r @# @✓ , v✓ = @# @r . 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 #(r, ✓) = f(r) sin ✓ AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== stream function r4# = 0 AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==

In the co-moving frame of the droplet, the fl ow speed at in fi nity equals the swim speed U Active droplet: theory 21 vr(r = R) = ˜ vr(r = R) = 0, v✓(r = R) = ˜ v✓(r = R), ( ˜) · ˆ r = (r · ˆ r) ˆ r r , at r = R. 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 no- fl ux continuity of tangential surface slip velocity Cauchy stress is related to the surface tension fl ow should remain fi nite => condition at in fi nity => no- fl ux condition => C = D = 0 B = U A = − UR2 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 fext = A r + Br + Cr ln r + Dr3 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 fint = ˜ A r + ˜ Br + ˜ Cr ln r + ˜ Dr3 fl ow should remain fi nite => no- fl ux condition => continuity of slip => ˜ A = ˜ C = 0 ˜ B + ˜ CR2 = 0 ˜ B + 3 ˜ CR2 = − 2U Interior Exterior no Stokes paradox! The self-propulsion speed is then 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 U = 16⌘ = 0↵ 16⌘ RS, Tjhung and Cates, Phys Rev Research 2020 vx = @# @y , vy = @# @x , vr = 1 r @# @✓ , v✓ = @# @r . 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 #(r, ✓) = f(r) sin ✓ AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== stream function r4# = 0 AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==

In the co-moving frame of the droplet, the fl ow speed at in fi nity equals the swim speed U Active droplet: theory 21 vr(r = R) = ˜ vr(r = R) = 0, v✓(r = R) = ˜ v✓(r = R), ( ˜) · ˆ r = (r · ˆ r) ˆ r r , at r = R. 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 AAADEXicdVLLbtQwFHXCo2V4dApLNhEjYCq1owRVgs1IFWxYFsS0lcaj0Y3jZKzaSbBvRowi/wIbfoUNCxBiy44df4MzDaiPcCXLR+ec+/AjLqUwGIa/Pf/a9Rs3NzZv9W7fuXtvq799/8gUlWZ8wgpZ6JMYDJci5xMUKPlJqTmoWPLj+PRVox8vuTaiyN/hquQzBVkuUsEAHTXf9p4u57W2Qz1+u/NkTFHIhNdL+48bh7uUOgvFBUfo8J0XnJVKnuKQxoVMzEq5raZGZArsXpvTIVmqRbbAHcqSAukC8IJJW+saZqAUdBTPIZZg/5v5t3KHtNdVZ91ml76vIKHIP2ANGFh3tFFv3h+Eo3AdwVUQtWBA2jic93/RpGCV4jkyCcZMo7DEWQ0aBZPc9mhleAnsFDI+dTAHxc2sXr+oDR47JgnSQruVY7Bmz2fUoEwzunMqwIW5rDVklzatMH0xq0VeVshzdtYorWSARdB8jyARmjOUKweAaeFmDdgCNDB0n6i5hOjyka+Co2ejKBxFb/YHBy/b69gkD8kjMiQReU4OyGtySCaEeR+9z95X75v/yf/if/d/nFl9r815QC6E//MP/AMBzg== 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 AAADEXicdVLLbtQwFHXCo2V4dApLNhEjYCq1owRVgs1IFWxYFsS0lcaj0Y3jZKzaSbBvRowi/wIbfoUNCxBiy44df4MzDaiPcCXLR+ec+/AjLqUwGIa/Pf/a9Rs3NzZv9W7fuXtvq799/8gUlWZ8wgpZ6JMYDJci5xMUKPlJqTmoWPLj+PRVox8vuTaiyN/hquQzBVkuUsEAHTXf9p4u57W2Qz1+u/NkTFHIhNdL+48bh7uUOgvFBUfo8J0XnJVKnuKQxoVMzEq5raZGZArsXpvTIVmqRbbAHcqSAukC8IJJW+saZqAUdBTPIZZg/5v5t3KHtNdVZ91ml76vIKHIP2ANGFh3tFFv3h+Eo3AdwVUQtWBA2jic93/RpGCV4jkyCcZMo7DEWQ0aBZPc9mhleAnsFDI+dTAHxc2sXr+oDR47JgnSQruVY7Bmz2fUoEwzunMqwIW5rDVklzatMH0xq0VeVshzdtYorWSARdB8jyARmjOUKweAaeFmDdgCNDB0n6i5hOjyka+Co2ejKBxFb/YHBy/b69gkD8kjMiQReU4OyGtySCaEeR+9z95X75v/yf/if/d/nFl9r815QC6E//MP/AMBzg== no- fl ux continuity of tangential surface slip velocity Cauchy stress is related to the surface tension fl ow should remain fi nite => condition at in fi nity => no- fl ux condition => C = D = 0 B = U A = − UR2 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 fext = A r + Br + Cr ln r + Dr3 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 fint = ˜ A r + ˜ Br + ˜ Cr ln r + ˜ Dr3 fl ow should remain fi nite => no- fl ux condition => continuity of slip => ˜ A = ˜ C = 0 ˜ B + ˜ CR2 = 0 ˜ B + 3 ˜ CR2 = − 2U Interior Exterior no Stokes paradox! The self-propulsion speed is then 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 U = 16⌘ = 0↵ 16⌘ RS, Tjhung and Cates, Phys Rev Research 2020 vx = @# @y , vy = @# @x , vr = 1 r @# @✓ , v✓ = @# @r . 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 #(r, ✓) = f(r) sin ✓ AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ== stream function r4# = 0 AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ== depends on surface gradient and does not depend on radius of particle

22 Active droplet: theory and simulations RS, Tjhung and Cates, Phys Rev Research 2020 (c) (d)

Ramaswamy, Adv. in Phys. 2001; RS, Tjhung and Cates, Phys Rev Research 2020 Many active droplets 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 F[ , ] = Z N X i=1 ⇣a 2 2 i + b 4 4 i + a0 2 2 i + b0 4 4 i i i +  2 (r i)2 + 0 2 (r i)2 + X j6=i 0 i j ⌘ dr.

Ramaswamy, Adv. in Phys. 2001; RS, Tjhung and Cates, Phys Rev Research 2020 Many active droplets 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 F[ , ] = Z N X i=1 ⇣a 2 2 i + b 4 4 i + a0 2 2 i + b0 4 4 i i i +  2 (r i)2 + 0 2 (r i)2 + X j6=i 0 i j ⌘ dr. this term leads to repulsion between the φ-droplets, and thus, precludes any overlap

Ramaswamy, Adv. in Phys. 2001; RS, Tjhung and Cates, Phys Rev Research 2020 Many active droplets 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 F[ , ] = Z N X i=1 ⇣a 2 2 i + b 4 4 i + a0 2 2 i + b0 4 4 i i i +  2 (r i)2 + 0 2 (r i)2 + X j6=i 0 i j ⌘ dr. this term leads to repulsion between the φ-droplets, and thus, precludes any overlap

Ramaswamy, Adv. in Phys. 2001; RS, Tjhung and Cates, Phys Rev Research 2020 Many active droplets 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 F[ , ] = Z N X i=1 ⇣a 2 2 i + b 4 4 i + a0 2 2 i + b0 4 4 i i i +  2 (r i)2 + 0 2 (r i)2 + X j6=i 0 i j ⌘ dr. this term leads to repulsion between the φ-droplets, and thus, precludes any overlap Active-Crowley-like instability: A linear array of active droplets is unstable in top panel (a). This instability is opposite to that in an array of sedimenting particles. There is no instability in (b) due to PBC.

24 incomplete phase separation in active scalar fi eld theories is a generic consequence of an e ff ective interfacial tension - mechanical (causing fl ow) or di ff usive (causing Ostwald ripening) - becoming negative. SSI Ostwald ripening spontaneous symmetry breaking on the surface of active droplets => surface slip fl ow => exterior fl uid fl ow. The resulting fl uid stress causes force-free self-propulsion. This provides a new method to simulate many droplets 1. Splitting active droplets 2. Self-propelling active droplets Summary: splitting and self-propelling droplets