Splitting and self-propelling active droplets

Presented in the DAMTP Statistical Physics and Soft Matter Seminar.

Transcript

1. 1 Rajesh Singh Department of Applied Mathematics and Theoretical Physics Splitting and self-propelling active droplets
2. Splitting active droplets Self-propelling active droplets
3. 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
4. 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
5. 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
6. 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
7. 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
8. 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?
9. 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
10. 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
11. 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)
12. 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 <latexit sha1_base64="eq5CDmOoRMQIIXGe5NNNGnPlUps=">AAACJnicbVDLSsNAFJ3UV62vqEs3wSJUxJKIoptCrRuXFewDmhAm00k7dDIJMxOhhHyNG3/FjYuKiDs/xUkbirYeGDiccy93zvEiSoQ0zS+tsLK6tr5R3Cxtbe/s7un7B20RxhzhFgppyLseFJgShluSSIq7Eccw8CjueKO7zO88YS5IyB7lOMJOAAeM+ARBqSRXr9nRkFTsAMqh5yc8Pa3ZPocosfkwdJPb9HxGGmk6l87mUsnVy2bVnMJYJlZOyiBH09Undj9EcYCZRBQK0bPMSDoJ5JIgitOSHQscQTSCA9xTlMEACyeZxkyNE6X0DT/k6jFpTNXfGwkMhBgHnprM8ohFLxP/83qx9G+chLAolpih2SE/poYMjawzo084RpKOFYGIE/VXAw2hqkmqZrMSrMXIy6R9UbWuqubDZbneyOsogiNwDCrAAtegDu5BE7QAAs/gFUzAu/aivWkf2udstKDlO4fgD7TvH/6VprQ=</latexit> (r) = ⇢A ⇢B ⇢A + ⇢B = 1 = +1 <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> B-phase A-phase Bray, Adv. in Phys. 43, 357 (1994)
13. 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 <latexit sha1_base64="eq5CDmOoRMQIIXGe5NNNGnPlUps=">AAACJnicbVDLSsNAFJ3UV62vqEs3wSJUxJKIoptCrRuXFewDmhAm00k7dDIJMxOhhHyNG3/FjYuKiDs/xUkbirYeGDiccy93zvEiSoQ0zS+tsLK6tr5R3Cxtbe/s7un7B20RxhzhFgppyLseFJgShluSSIq7Eccw8CjueKO7zO88YS5IyB7lOMJOAAeM+ARBqSRXr9nRkFTsAMqh5yc8Pa3ZPocosfkwdJPb9HxGGmk6l87mUsnVy2bVnMJYJlZOyiBH09Undj9EcYCZRBQK0bPMSDoJ5JIgitOSHQscQTSCA9xTlMEACyeZxkyNE6X0DT/k6jFpTNXfGwkMhBgHnprM8ohFLxP/83qx9G+chLAolpih2SE/poYMjawzo084RpKOFYGIE/VXAw2hqkmqZrMSrMXIy6R9UbWuqubDZbneyOsogiNwDCrAAtegDu5BE7QAAs/gFUzAu/aivWkf2udstKDlO4fgD7TvH/6VprQ=</latexit> (r) = ⇢A ⇢B ⇢A + ⇢B = 1 = +1 <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> B-phase A-phase The scalar fi eld is conserved: ˙ = r · J Bray, Adv. in Phys. 43, 357 (1994)
14. 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 <latexit sha1_base64="eq5CDmOoRMQIIXGe5NNNGnPlUps=">AAACJnicbVDLSsNAFJ3UV62vqEs3wSJUxJKIoptCrRuXFewDmhAm00k7dDIJMxOhhHyNG3/FjYuKiDs/xUkbirYeGDiccy93zvEiSoQ0zS+tsLK6tr5R3Cxtbe/s7un7B20RxhzhFgppyLseFJgShluSSIq7Eccw8CjueKO7zO88YS5IyB7lOMJOAAeM+ARBqSRXr9nRkFTsAMqh5yc8Pa3ZPocosfkwdJPb9HxGGmk6l87mUsnVy2bVnMJYJlZOyiBH09Undj9EcYCZRBQK0bPMSDoJ5JIgitOSHQscQTSCA9xTlMEACyeZxkyNE6X0DT/k6jFpTNXfGwkMhBgHnprM8ohFLxP/83qx9G+chLAolpih2SE/poYMjawzo084RpKOFYGIE/VXAw2hqkmqZrMSrMXIy6R9UbWuqubDZbneyOsogiNwDCrAAtegDu5BE7QAAs/gFUzAu/aivWkf2udstKDlO4fgD7TvH/6VprQ=</latexit> (r) = ⇢A ⇢B ⇢A + ⇢B = 1 = +1 <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> 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: <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, The scalar fi eld is conserved: ˙ = r · J Bray, Adv. in Phys. 43, 357 (1994)
15. φ4 fi eld theory of passive phase separation ˙ = r · J Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">AAACQ3icbVDLSsNAFJ3UV62vqks3waIIaklE0Y0g6kLEhYJ9YBPKzWTaDk4mcWZSKCH/5sYfcOcPuHGhiFvBSc2iPi4MczjnHu69x4sYlcqynozC2PjE5FRxujQzOze/UF5cqsswFpjUcMhC0fRAEkY5qSmqGGlGgkDgMdLwbk8yvdEnQtKQX6tBRNwAupx2KAalqXb5xvFC5stBoL/kPF0/3B4lHA4eg9QJ4s1Rup86UY9uOvJOqGTnNP1hudDDfUi3Su1yxapawzL/AjsHFZTXZbv86PghjgPCFWYgZcu2IuUmIBTFjKQlJ5YkAnwLXdLSkENApJsMM0jNNc34ZicU+nFlDtlRRwKBzFbUnQGonvytZeR/WitWnQM3oTyKFeH4e1AnZqYKzSxQ06eCYMUGGgAWVO9q4h4IwErHnoVg/z75L6jvVO29qnW1Wzk6zuMoohW0ijaQjfbRETpDl6iGMLpHz+gVvRkPxovxbnx8txaM3LOMfpTx+QWT1rOh</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> 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
16. φ4 fi eld theory of passive phase separation ˙ = r · J Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> 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 <latexit sha1_base64="z4POE0j1bbNVErdJw+wNS33VXIk=">AAACE3icbVDLSsNAFJ3UV62vqEs3g0UQFyURRTdCURCXFewDmlAm00k7dGYSZiZCCfkHN/6KGxeKuHXjzr9x0kbQ1gMXDufcy733BDGjSjvOl1VaWFxaXimvVtbWNza37O2dlooSiUkTRyySnQApwqggTU01I51YEsQDRtrB6Cr32/dEKhqJOz2Oic/RQNCQYqSN1LOPPJ7AC+iFEuHU6xOmkceRHmLE0uss+5HiIc16dtWpORPAeeIWpAoKNHr2p9ePcMKJ0JghpbquE2s/RVJTzEhW8RJFYoRHaEC6hgrEifLTyU8ZPDBKH4aRNCU0nKi/J1LElRrzwHTm96pZLxf/87qJDs/9lIo40UTg6aIwYVBHMA8I9qkkWLOxIQhLam6FeIhMPNrEWDEhuLMvz5PWcc09rTm3J9X6ZRFHGeyBfXAIXHAG6uAGNEATYPAAnsALeLUerWfrzXqftpasYmYX/IH18Q3itJ7T</latexit> µ = F <latexit sha1_base64="XfMZX1CcDqf5PTk2053o0Wr8ijY=">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</latexit> F[ ] = Z ✓ a 2 2 + b 4 4 +  2 (r )2 ◆ dr free energy functional:
17. φ4 fi eld theory of passive phase separation ˙ = r · J Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> 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 <latexit sha1_base64="q4OHB2Zsd98XeZwdyv0le/bQqrI=">AAAB7nicbVBNSwMxEJ2tX7V+VT16CRahXsquKHosevFYwX5Au5Rsmm1Ds0lIskJZ+iO8eFDEq7/Hm//GtN2Dtj4YeLw3w8y8SHFmrO9/e4W19Y3NreJ2aWd3b/+gfHjUMjLVhDaJ5FJ3ImwoZ4I2LbOcdpSmOIk4bUfju5nffqLaMCke7UTRMMFDwWJGsHVSO6721Iid98sVv+bPgVZJkJMK5Gj0y1+9gSRpQoUlHBvTDXxlwwxrywin01IvNVRhMsZD2nVU4ISaMJufO0VnThmgWGpXwqK5+nsiw4kxkyRynQm2I7PszcT/vG5q45swY0KllgqyWBSnHFmJZr+jAdOUWD5xBBPN3K2IjLDGxLqESi6EYPnlVdK6qAVXNf/hslK/zeMowgmcQhUCuIY63EMDmkBgDM/wCm+e8l68d+9j0Vrw8plj+APv8wed0o8Y</latexit> f( ) Hohenberg & Halperin 1977; Bray, Adv in Phys 1994; Kendon et al, JFM 2001; Cates and Tjhung, JFM 2018 <latexit sha1_base64="z4POE0j1bbNVErdJw+wNS33VXIk=">AAACE3icbVDLSsNAFJ3UV62vqEs3g0UQFyURRTdCURCXFewDmlAm00k7dGYSZiZCCfkHN/6KGxeKuHXjzr9x0kbQ1gMXDufcy733BDGjSjvOl1VaWFxaXimvVtbWNza37O2dlooSiUkTRyySnQApwqggTU01I51YEsQDRtrB6Cr32/dEKhqJOz2Oic/RQNCQYqSN1LOPPJ7AC+iFEuHU6xOmkceRHmLE0uss+5HiIc16dtWpORPAeeIWpAoKNHr2p9ePcMKJ0JghpbquE2s/RVJTzEhW8RJFYoRHaEC6hgrEifLTyU8ZPDBKH4aRNCU0nKi/J1LElRrzwHTm96pZLxf/87qJDs/9lIo40UTg6aIwYVBHMA8I9qkkWLOxIQhLam6FeIhMPNrEWDEhuLMvz5PWcc09rTm3J9X6ZRFHGeyBfXAIXHAG6uAGNEATYPAAnsALeLUerWfrzXqftpasYmYX/IH18Q3itJ7T</latexit> µ = F <latexit sha1_base64="XfMZX1CcDqf5PTk2053o0Wr8ijY=">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</latexit> F[ ] = Z ✓ a 2 2 + b 4 4 +  2 (r )2 ◆ dr free energy functional:
18. φ4 fi eld theory of passive phase separation ˙ = r · J Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> 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 <latexit sha1_base64="q4OHB2Zsd98XeZwdyv0le/bQqrI=">AAAB7nicbVBNSwMxEJ2tX7V+VT16CRahXsquKHosevFYwX5Au5Rsmm1Ds0lIskJZ+iO8eFDEq7/Hm//GtN2Dtj4YeLw3w8y8SHFmrO9/e4W19Y3NreJ2aWd3b/+gfHjUMjLVhDaJ5FJ3ImwoZ4I2LbOcdpSmOIk4bUfju5nffqLaMCke7UTRMMFDwWJGsHVSO6721Iid98sVv+bPgVZJkJMK5Gj0y1+9gSRpQoUlHBvTDXxlwwxrywin01IvNVRhMsZD2nVU4ISaMJufO0VnThmgWGpXwqK5+nsiw4kxkyRynQm2I7PszcT/vG5q45swY0KllgqyWBSnHFmJZr+jAdOUWD5xBBPN3K2IjLDGxLqESi6EYPnlVdK6qAVXNf/hslK/zeMowgmcQhUCuIY63EMDmkBgDM/wCm+e8l68d+9j0Vrw8plj+APv8wed0o8Y</latexit> f( ) Hohenberg & Halperin 1977; Bray, Adv in Phys 1994; Kendon et al, JFM 2001; Cates and Tjhung, JFM 2018 <latexit sha1_base64="z4POE0j1bbNVErdJw+wNS33VXIk=">AAACE3icbVDLSsNAFJ3UV62vqEs3g0UQFyURRTdCURCXFewDmlAm00k7dGYSZiZCCfkHN/6KGxeKuHXjzr9x0kbQ1gMXDufcy733BDGjSjvOl1VaWFxaXimvVtbWNza37O2dlooSiUkTRyySnQApwqggTU01I51YEsQDRtrB6Cr32/dEKhqJOz2Oic/RQNCQYqSN1LOPPJ7AC+iFEuHU6xOmkceRHmLE0uss+5HiIc16dtWpORPAeeIWpAoKNHr2p9ePcMKJ0JghpbquE2s/RVJTzEhW8RJFYoRHaEC6hgrEifLTyU8ZPDBKH4aRNCU0nKi/J1LElRrzwHTm96pZLxf/87qJDs/9lIo40UTg6aIwYVBHMA8I9qkkWLOxIQhLam6FeIhMPNrEWDEhuLMvz5PWcc09rTm3J9X6ZRFHGeyBfXAIXHAG6uAGNEATYPAAnsALeLUerWfrzXqftpasYmYX/IH18Q3itJ7T</latexit> µ = F <latexit sha1_base64="XfMZX1CcDqf5PTk2053o0Wr8ijY=">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</latexit> F[ ] = Z ✓ a 2 2 + b 4 4 +  2 (r )2 ◆ dr free energy functional: -1 1
19. φ4 fi eld theory of passive phase separation ˙ = r · J Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> 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 <latexit sha1_base64="q4OHB2Zsd98XeZwdyv0le/bQqrI=">AAAB7nicbVBNSwMxEJ2tX7V+VT16CRahXsquKHosevFYwX5Au5Rsmm1Ds0lIskJZ+iO8eFDEq7/Hm//GtN2Dtj4YeLw3w8y8SHFmrO9/e4W19Y3NreJ2aWd3b/+gfHjUMjLVhDaJ5FJ3ImwoZ4I2LbOcdpSmOIk4bUfju5nffqLaMCke7UTRMMFDwWJGsHVSO6721Iid98sVv+bPgVZJkJMK5Gj0y1+9gSRpQoUlHBvTDXxlwwxrywin01IvNVRhMsZD2nVU4ISaMJufO0VnThmgWGpXwqK5+nsiw4kxkyRynQm2I7PszcT/vG5q45swY0KllgqyWBSnHFmJZr+jAdOUWD5xBBPN3K2IjLDGxLqESi6EYPnlVdK6qAVXNf/hslK/zeMowgmcQhUCuIY63EMDmkBgDM/wCm+e8l68d+9j0Vrw8plj+APv8wed0o8Y</latexit> f( ) cost to create an interface <latexit sha1_base64="GxJ7cDOVOO9yyHn3LIfDMaLR0hw=">AAACEXicbVDLSsNAFJ34rPVVdelmsAjdWJOqWBdC0Y3LCvYBTRpuptN26EwSZyZCCfkFN/6KGxeKuHXnzr8xfSy09cCFwzn3cu89XsiZ0qb5bSwsLi2vrGbWsusbm1vbuZ3dugoiSWiNBDyQTQ8U5cynNc00p81QUhAepw1vcD3yGw9UKhb4d3oYUkdAz2ddRkCnkpsr2D0QAtzYTC5tdS91fFS2BxCGgKEdnyTHF147LiVJ1s3lzaI5Bp4n1pTk0RRVN/dldwISCeprwkGplmWG2olBakY4TbJ2pGgIZAA92kqpD4IqJx5/lODDVOngbiDT8jUeq78nYhBKDYWXdgrQfTXrjcT/vFaku2UnZn4YaeqTyaJuxLEO8Cge3GGSEs2HKQEiWXorJn2QQHQa4igEa/bleVIvFa2zonl7mq9cTePIoH10gArIQueogm5QFdUQQY/oGb2iN+PJeDHejY9J64IxndlDf2B8/gBgh5y0</latexit> 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 <latexit sha1_base64="z4POE0j1bbNVErdJw+wNS33VXIk=">AAACE3icbVDLSsNAFJ3UV62vqEs3g0UQFyURRTdCURCXFewDmlAm00k7dGYSZiZCCfkHN/6KGxeKuHXjzr9x0kbQ1gMXDufcy733BDGjSjvOl1VaWFxaXimvVtbWNza37O2dlooSiUkTRyySnQApwqggTU01I51YEsQDRtrB6Cr32/dEKhqJOz2Oic/RQNCQYqSN1LOPPJ7AC+iFEuHU6xOmkceRHmLE0uss+5HiIc16dtWpORPAeeIWpAoKNHr2p9ePcMKJ0JghpbquE2s/RVJTzEhW8RJFYoRHaEC6hgrEifLTyU8ZPDBKH4aRNCU0nKi/J1LElRrzwHTm96pZLxf/87qJDs/9lIo40UTg6aIwYVBHMA8I9qkkWLOxIQhLam6FeIhMPNrEWDEhuLMvz5PWcc09rTm3J9X6ZRFHGeyBfXAIXHAG6uAGNEATYPAAnsALeLUerWfrzXqftpasYmYX/IH18Q3itJ7T</latexit> µ = F <latexit sha1_base64="XfMZX1CcDqf5PTk2053o0Wr8ijY=">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</latexit> F[ ] = Z ✓ a 2 2 + b 4 4 +  2 (r )2 ◆ dr free energy functional: -1 1
20. φ4 fi eld theory of passive phase separation ˙ = r · J Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> 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 <latexit sha1_base64="q4OHB2Zsd98XeZwdyv0le/bQqrI=">AAAB7nicbVBNSwMxEJ2tX7V+VT16CRahXsquKHosevFYwX5Au5Rsmm1Ds0lIskJZ+iO8eFDEq7/Hm//GtN2Dtj4YeLw3w8y8SHFmrO9/e4W19Y3NreJ2aWd3b/+gfHjUMjLVhDaJ5FJ3ImwoZ4I2LbOcdpSmOIk4bUfju5nffqLaMCke7UTRMMFDwWJGsHVSO6721Iid98sVv+bPgVZJkJMK5Gj0y1+9gSRpQoUlHBvTDXxlwwxrywin01IvNVRhMsZD2nVU4ISaMJufO0VnThmgWGpXwqK5+nsiw4kxkyRynQm2I7PszcT/vG5q45swY0KllgqyWBSnHFmJZr+jAdOUWD5xBBPN3K2IjLDGxLqESi6EYPnlVdK6qAVXNf/hslK/zeMowgmcQhUCuIY63EMDmkBgDM/wCm+e8l68d+9j0Vrw8plj+APv8wed0o8Y</latexit> f( ) cost to create an interface <latexit sha1_base64="GxJ7cDOVOO9yyHn3LIfDMaLR0hw=">AAACEXicbVDLSsNAFJ34rPVVdelmsAjdWJOqWBdC0Y3LCvYBTRpuptN26EwSZyZCCfkFN/6KGxeKuHXnzr8xfSy09cCFwzn3cu89XsiZ0qb5bSwsLi2vrGbWsusbm1vbuZ3dugoiSWiNBDyQTQ8U5cynNc00p81QUhAepw1vcD3yGw9UKhb4d3oYUkdAz2ddRkCnkpsr2D0QAtzYTC5tdS91fFS2BxCGgKEdnyTHF147LiVJ1s3lzaI5Bp4n1pTk0RRVN/dldwISCeprwkGplmWG2olBakY4TbJ2pGgIZAA92kqpD4IqJx5/lODDVOngbiDT8jUeq78nYhBKDYWXdgrQfTXrjcT/vFaku2UnZn4YaeqTyaJuxLEO8Cge3GGSEs2HKQEiWXorJn2QQHQa4igEa/bleVIvFa2zonl7mq9cTePIoH10gArIQueogm5QFdUQQY/oGb2iN+PJeDHejY9J64IxndlDf2B8/gBgh5y0</latexit> 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 <latexit sha1_base64="z4POE0j1bbNVErdJw+wNS33VXIk=">AAACE3icbVDLSsNAFJ3UV62vqEs3g0UQFyURRTdCURCXFewDmlAm00k7dGYSZiZCCfkHN/6KGxeKuHXjzr9x0kbQ1gMXDufcy733BDGjSjvOl1VaWFxaXimvVtbWNza37O2dlooSiUkTRyySnQApwqggTU01I51YEsQDRtrB6Cr32/dEKhqJOz2Oic/RQNCQYqSN1LOPPJ7AC+iFEuHU6xOmkceRHmLE0uss+5HiIc16dtWpORPAeeIWpAoKNHr2p9ePcMKJ0JghpbquE2s/RVJTzEhW8RJFYoRHaEC6hgrEifLTyU8ZPDBKH4aRNCU0nKi/J1LElRrzwHTm96pZLxf/87qJDs/9lIo40UTg6aIwYVBHMA8I9qkkWLOxIQhLam6FeIhMPNrEWDEhuLMvz5PWcc09rTm3J9X6ZRFHGeyBfXAIXHAG6uAGNEATYPAAnsALeLUerWfrzXqftpasYmYX/IH18Q3itJ7T</latexit> µ = F <latexit sha1_base64="XfMZX1CcDqf5PTk2053o0Wr8ijY=">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</latexit> F[ ] = Z ✓ a 2 2 + b 4 4 +  2 (r )2 ◆ dr free energy functional: -1 1 Widely separated droplets of varying sizes
21. φ4 fi eld theory of passive phase separation ˙ = r · J Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> 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 <latexit sha1_base64="q4OHB2Zsd98XeZwdyv0le/bQqrI=">AAAB7nicbVBNSwMxEJ2tX7V+VT16CRahXsquKHosevFYwX5Au5Rsmm1Ds0lIskJZ+iO8eFDEq7/Hm//GtN2Dtj4YeLw3w8y8SHFmrO9/e4W19Y3NreJ2aWd3b/+gfHjUMjLVhDaJ5FJ3ImwoZ4I2LbOcdpSmOIk4bUfju5nffqLaMCke7UTRMMFDwWJGsHVSO6721Iid98sVv+bPgVZJkJMK5Gj0y1+9gSRpQoUlHBvTDXxlwwxrywin01IvNVRhMsZD2nVU4ISaMJufO0VnThmgWGpXwqK5+nsiw4kxkyRynQm2I7PszcT/vG5q45swY0KllgqyWBSnHFmJZr+jAdOUWD5xBBPN3K2IjLDGxLqESi6EYPnlVdK6qAVXNf/hslK/zeMowgmcQhUCuIY63EMDmkBgDM/wCm+e8l68d+9j0Vrw8plj+APv8wed0o8Y</latexit> f( ) cost to create an interface <latexit sha1_base64="GxJ7cDOVOO9yyHn3LIfDMaLR0hw=">AAACEXicbVDLSsNAFJ34rPVVdelmsAjdWJOqWBdC0Y3LCvYBTRpuptN26EwSZyZCCfkFN/6KGxeKuHXnzr8xfSy09cCFwzn3cu89XsiZ0qb5bSwsLi2vrGbWsusbm1vbuZ3dugoiSWiNBDyQTQ8U5cynNc00p81QUhAepw1vcD3yGw9UKhb4d3oYUkdAz2ddRkCnkpsr2D0QAtzYTC5tdS91fFS2BxCGgKEdnyTHF147LiVJ1s3lzaI5Bp4n1pTk0RRVN/dldwISCeprwkGplmWG2olBakY4TbJ2pGgIZAA92kqpD4IqJx5/lODDVOngbiDT8jUeq78nYhBKDYWXdgrQfTXrjcT/vFaku2UnZn4YaeqTyaJuxLEO8Cge3GGSEs2HKQEiWXorJn2QQHQa4igEa/bleVIvFa2zonl7mq9cTePIoH10gArIQueogm5QFdUQQY/oGb2iN+PJeDHejY9J64IxndlDf2B8/gBgh5y0</latexit> 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 <latexit sha1_base64="z4POE0j1bbNVErdJw+wNS33VXIk=">AAACE3icbVDLSsNAFJ3UV62vqEs3g0UQFyURRTdCURCXFewDmlAm00k7dGYSZiZCCfkHN/6KGxeKuHXjzr9x0kbQ1gMXDufcy733BDGjSjvOl1VaWFxaXimvVtbWNza37O2dlooSiUkTRyySnQApwqggTU01I51YEsQDRtrB6Cr32/dEKhqJOz2Oic/RQNCQYqSN1LOPPJ7AC+iFEuHU6xOmkceRHmLE0uss+5HiIc16dtWpORPAeeIWpAoKNHr2p9ePcMKJ0JghpbquE2s/RVJTzEhW8RJFYoRHaEC6hgrEifLTyU8ZPDBKH4aRNCU0nKi/J1LElRrzwHTm96pZLxf/87qJDs/9lIo40UTg6aIwYVBHMA8I9qkkWLOxIQhLam6FeIhMPNrEWDEhuLMvz5PWcc09rTm3J9X6ZRFHGeyBfXAIXHAG6uAGNEATYPAAnsALeLUerWfrzXqftpasYmYX/IH18Q3itJ7T</latexit> µ = F <latexit sha1_base64="XfMZX1CcDqf5PTk2053o0Wr8ijY=">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</latexit> 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?
22. 8 Model H: Ostwald ripening Chemical potential raised at curved interface μs <latexit sha1_base64="ZAGPOvulsxZljt/eAzG3uVV8924=">AAACC3icbVDLSsNAFJ3UV62vqEs3Q4tQNyURRTdC0Y3LWuwDmhAm00k7dGYSZiZCCd278VfcuFDErT/gzr9x2mahrQcuHM65l3vvCRNGlXacb6uwsrq2vlHcLG1t7+zu2fsHbRWnEpMWjlksuyFShFFBWppqRrqJJIiHjHTC0c3U7zwQqWgs7vU4IT5HA0EjipE2UmCXPZ4Gqto8gVfQiyTCmTdAnKNJ5iVDGoSwOQnsilNzZoDLxM1JBeRoBPaX149xyonQmCGleq6TaD9DUlPMyKTkpYokCI/QgPQMFYgT5WezXybw2Ch9GMXSlNBwpv6eyBBXasxD08mRHqpFbyr+5/VSHV36GRVJqonA80VRyqCO4TQY2KeSYM3GhiAsqbkV4iEyiWgTX8mE4C6+vEzapzX3vObcnVXq13kcRXAEyqAKXHAB6uAWNEALYPAInsEreLOerBfr3fqYtxasfOYQ/IH1+QPRLJpK</latexit> µs(R) = bR <latexit sha1_base64="ypZaUtqe6AylfSkUkDkDZ9KWAX8=">AAACGXicbVDLSgMxFM34rPU16tJNsAgVocyIohuh6MZlLfYBnTLcSTNtaDIzJBmhDv0NN/6KGxeKuNSVf2P6WGjrgcDhnHO5uSdIOFPacb6thcWl5ZXV3Fp+fWNza9ve2a2rOJWE1kjMY9kMQFHOIlrTTHPaTCQFEXDaCPrXI79xT6VicXSnBwltC+hGLGQEtJF82/FE6qti9QhfYi+UQDKvC0KA7wwzL+kxP8DVIT7Go5j3QDX4dsEpOWPgeeJOSQFNUfHtT68Tk1TQSBMOSrVcJ9HtDKRmhNNh3ksVTYD0oUtbhkYgqGpn48uG+NAoHRzG0rxI47H6eyIDodRABCYpQPfUrDcS//NaqQ4v2hmLklTTiEwWhSnHOsajmnCHSUo0HxgCRDLzV0x6YPrRpsy8KcGdPXme1E9K7lnJuT0tlK+mdeTQPjpAReSic1RGN6iCaoigR/SMXtGb9WS9WO/WxyS6YE1n9tAfWF8/rKiffQ==</latexit> µs(R) = 0 bR + µ⇣ Bray, Adv. in Phys. 43, 357 (1994) Widely separated droplets of varying sizes
23. 8 Model H: Ostwald ripening <latexit sha1_base64="b/J8HK825HwlekHO/p+S7cNtiYM=">AAACGHicbZBNS8MwGMfT+TbnW9Wjl+AQ5mGzFUUvg6EX8TTFvcBaSpqlMyxNS5IKo+xjePGrePGgiNfd/DamWw+6+YfAL//neUievx8zKpVlfRuFpeWV1bXiemljc2t7x9zda8soEZi0cMQi0fWRJIxy0lJUMdKNBUGhz0jHH15n9c4TEZJG/EGNYuKGaMBpQDFS2vLMk1tYuT+uw6rDkc8QdMIkM2A9Iy91KA/UaFzNLjLzPbNs1ayp4CLYOZRBrqZnTpx+hJOQcIUZkrJnW7FyUyQUxYyMS04iSYzwEA1ITyNHIZFuOl1sDI+004dBJPThCk7d3xMpCqUchb7uDJF6lPO1zPyv1ktUcOmmlMeJIhzPHgoSBlUEs5RgnwqCFRtpQFhQ/VeIH5FAWOksSzoEe37lRWif1uzzmnV3Vm5c5XEUwQE4BBVggwvQADegCVoAg2fwCt7Bh/FivBmfxtestWDkM/vgj4zJD87YnS8=</latexit> 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 <latexit sha1_base64="ZAGPOvulsxZljt/eAzG3uVV8924=">AAACC3icbVDLSsNAFJ3UV62vqEs3Q4tQNyURRTdC0Y3LWuwDmhAm00k7dGYSZiZCCd278VfcuFDErT/gzr9x2mahrQcuHM65l3vvCRNGlXacb6uwsrq2vlHcLG1t7+zu2fsHbRWnEpMWjlksuyFShFFBWppqRrqJJIiHjHTC0c3U7zwQqWgs7vU4IT5HA0EjipE2UmCXPZ4Gqto8gVfQiyTCmTdAnKNJ5iVDGoSwOQnsilNzZoDLxM1JBeRoBPaX149xyonQmCGleq6TaD9DUlPMyKTkpYokCI/QgPQMFYgT5WezXybw2Ch9GMXSlNBwpv6eyBBXasxD08mRHqpFbyr+5/VSHV36GRVJqonA80VRyqCO4TQY2KeSYM3GhiAsqbkV4iEyiWgTX8mE4C6+vEzapzX3vObcnVXq13kcRXAEyqAKXHAB6uAWNEALYPAInsEreLOerBfr3fqYtxasfOYQ/IH1+QPRLJpK</latexit> µs(R) = bR <latexit sha1_base64="ypZaUtqe6AylfSkUkDkDZ9KWAX8=">AAACGXicbVDLSgMxFM34rPU16tJNsAgVocyIohuh6MZlLfYBnTLcSTNtaDIzJBmhDv0NN/6KGxeKuNSVf2P6WGjrgcDhnHO5uSdIOFPacb6thcWl5ZXV3Fp+fWNza9ve2a2rOJWE1kjMY9kMQFHOIlrTTHPaTCQFEXDaCPrXI79xT6VicXSnBwltC+hGLGQEtJF82/FE6qti9QhfYi+UQDKvC0KA7wwzL+kxP8DVIT7Go5j3QDX4dsEpOWPgeeJOSQFNUfHtT68Tk1TQSBMOSrVcJ9HtDKRmhNNh3ksVTYD0oUtbhkYgqGpn48uG+NAoHRzG0rxI47H6eyIDodRABCYpQPfUrDcS//NaqQ4v2hmLklTTiEwWhSnHOsajmnCHSUo0HxgCRDLzV0x6YPrRpsy8KcGdPXme1E9K7lnJuT0tlK+mdeTQPjpAReSic1RGN6iCaoigR/SMXtGb9WS9WO/WxyS6YE1n9tAfWF8/rKiffQ==</latexit> µs(R) = 0 bR + µ⇣ Bray, Adv. in Phys. 43, 357 (1994) Widely separated droplets of varying sizes
24. 8 Model H: Ostwald ripening <latexit sha1_base64="b/J8HK825HwlekHO/p+S7cNtiYM=">AAACGHicbZBNS8MwGMfT+TbnW9Wjl+AQ5mGzFUUvg6EX8TTFvcBaSpqlMyxNS5IKo+xjePGrePGgiNfd/DamWw+6+YfAL//neUievx8zKpVlfRuFpeWV1bXiemljc2t7x9zda8soEZi0cMQi0fWRJIxy0lJUMdKNBUGhz0jHH15n9c4TEZJG/EGNYuKGaMBpQDFS2vLMk1tYuT+uw6rDkc8QdMIkM2A9Iy91KA/UaFzNLjLzPbNs1ayp4CLYOZRBrqZnTpx+hJOQcIUZkrJnW7FyUyQUxYyMS04iSYzwEA1ITyNHIZFuOl1sDI+004dBJPThCk7d3xMpCqUchb7uDJF6lPO1zPyv1ktUcOmmlMeJIhzPHgoSBlUEs5RgnwqCFRtpQFhQ/VeIH5FAWOksSzoEe37lRWif1uzzmnV3Vm5c5XEUwQE4BBVggwvQADegCVoAg2fwCt7Bh/FivBmfxtestWDkM/vgj4zJD87YnS8=</latexit> 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 <latexit sha1_base64="ZAGPOvulsxZljt/eAzG3uVV8924=">AAACC3icbVDLSsNAFJ3UV62vqEs3Q4tQNyURRTdC0Y3LWuwDmhAm00k7dGYSZiZCCd278VfcuFDErT/gzr9x2mahrQcuHM65l3vvCRNGlXacb6uwsrq2vlHcLG1t7+zu2fsHbRWnEpMWjlksuyFShFFBWppqRrqJJIiHjHTC0c3U7zwQqWgs7vU4IT5HA0EjipE2UmCXPZ4Gqto8gVfQiyTCmTdAnKNJ5iVDGoSwOQnsilNzZoDLxM1JBeRoBPaX149xyonQmCGleq6TaD9DUlPMyKTkpYokCI/QgPQMFYgT5WezXybw2Ch9GMXSlNBwpv6eyBBXasxD08mRHqpFbyr+5/VSHV36GRVJqonA80VRyqCO4TQY2KeSYM3GhiAsqbkV4iEyiWgTX8mE4C6+vEzapzX3vObcnVXq13kcRXAEyqAKXHAB6uAWNEALYPAInsEreLOerBfr3fqYtxasfOYQ/IH1+QPRLJpK</latexit> µs(R) = bR <latexit sha1_base64="ypZaUtqe6AylfSkUkDkDZ9KWAX8=">AAACGXicbVDLSgMxFM34rPU16tJNsAgVocyIohuh6MZlLfYBnTLcSTNtaDIzJBmhDv0NN/6KGxeKuNSVf2P6WGjrgcDhnHO5uSdIOFPacb6thcWl5ZXV3Fp+fWNza9ve2a2rOJWE1kjMY9kMQFHOIlrTTHPaTCQFEXDaCPrXI79xT6VicXSnBwltC+hGLGQEtJF82/FE6qti9QhfYi+UQDKvC0KA7wwzL+kxP8DVIT7Go5j3QDX4dsEpOWPgeeJOSQFNUfHtT68Tk1TQSBMOSrVcJ9HtDKRmhNNh3ksVTYD0oUtbhkYgqGpn48uG+NAoHRzG0rxI47H6eyIDodRABCYpQPfUrDcS//NaqQ4v2hmLklTTiEwWhSnHOsajmnCHSUo0HxgCRDLzV0x6YPrRpsy8KcGdPXme1E9K7lnJuT0tlK+mdeTQPjpAReSic1RGN6iCaoigR/SMXtGb9WS9WO/WxyS6YE1n9tAfWF8/rKiffQ==</latexit> µs(R) = 0 bR + µ⇣ <latexit sha1_base64="LRaCBugJGvpb+fI3gKGqA12rQiQ=">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</latexit> ˙ 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) <latexit sha1_base64="ypZaUtqe6AylfSkUkDkDZ9KWAX8=">AAACGXicbVDLSgMxFM34rPU16tJNsAgVocyIohuh6MZlLfYBnTLcSTNtaDIzJBmhDv0NN/6KGxeKuNSVf2P6WGjrgcDhnHO5uSdIOFPacb6thcWl5ZXV3Fp+fWNza9ve2a2rOJWE1kjMY9kMQFHOIlrTTHPaTCQFEXDaCPrXI79xT6VicXSnBwltC+hGLGQEtJF82/FE6qti9QhfYi+UQDKvC0KA7wwzL+kxP8DVIT7Go5j3QDX4dsEpOWPgeeJOSQFNUfHtT68Tk1TQSBMOSrVcJ9HtDKRmhNNh3ksVTYD0oUtbhkYgqGpn48uG+NAoHRzG0rxI47H6eyIDodRABCYpQPfUrDcS//NaqQ4v2hmLklTTiEwWhSnHOsajmnCHSUo0HxgCRDLzV0x6YPrRpsy8KcGdPXme1E9K7lnJuT0tlK+mdeTQPjpAReSic1RGN6iCaoigR/SMXtGb9WS9WO/WxyS6YE1n9tAfWF8/rKiffQ==</latexit> µs(R) = 0 bR + µ⇣ Bray, Adv. in Phys. 43, 357 (1994) Widely separated droplets of varying sizes
25. 8 Model H: Ostwald ripening <latexit sha1_base64="b/J8HK825HwlekHO/p+S7cNtiYM=">AAACGHicbZBNS8MwGMfT+TbnW9Wjl+AQ5mGzFUUvg6EX8TTFvcBaSpqlMyxNS5IKo+xjePGrePGgiNfd/DamWw+6+YfAL//neUievx8zKpVlfRuFpeWV1bXiemljc2t7x9zda8soEZi0cMQi0fWRJIxy0lJUMdKNBUGhz0jHH15n9c4TEZJG/EGNYuKGaMBpQDFS2vLMk1tYuT+uw6rDkc8QdMIkM2A9Iy91KA/UaFzNLjLzPbNs1ayp4CLYOZRBrqZnTpx+hJOQcIUZkrJnW7FyUyQUxYyMS04iSYzwEA1ITyNHIZFuOl1sDI+004dBJPThCk7d3xMpCqUchb7uDJF6lPO1zPyv1ktUcOmmlMeJIhzPHgoSBlUEs5RgnwqCFRtpQFhQ/VeIH5FAWOksSzoEe37lRWif1uzzmnV3Vm5c5XEUwQE4BBVggwvQADegCVoAg2fwCt7Bh/FivBmfxtestWDkM/vgj4zJD87YnS8=</latexit> 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 <latexit sha1_base64="ZAGPOvulsxZljt/eAzG3uVV8924=">AAACC3icbVDLSsNAFJ3UV62vqEs3Q4tQNyURRTdC0Y3LWuwDmhAm00k7dGYSZiZCCd278VfcuFDErT/gzr9x2mahrQcuHM65l3vvCRNGlXacb6uwsrq2vlHcLG1t7+zu2fsHbRWnEpMWjlksuyFShFFBWppqRrqJJIiHjHTC0c3U7zwQqWgs7vU4IT5HA0EjipE2UmCXPZ4Gqto8gVfQiyTCmTdAnKNJ5iVDGoSwOQnsilNzZoDLxM1JBeRoBPaX149xyonQmCGleq6TaD9DUlPMyKTkpYokCI/QgPQMFYgT5WezXybw2Ch9GMXSlNBwpv6eyBBXasxD08mRHqpFbyr+5/VSHV36GRVJqonA80VRyqCO4TQY2KeSYM3GhiAsqbkV4iEyiWgTX8mE4C6+vEzapzX3vObcnVXq13kcRXAEyqAKXHAB6uAWNEALYPAInsEreLOerBfr3fqYtxasfOYQ/IH1+QPRLJpK</latexit> µs(R) = bR <latexit sha1_base64="ypZaUtqe6AylfSkUkDkDZ9KWAX8=">AAACGXicbVDLSgMxFM34rPU16tJNsAgVocyIohuh6MZlLfYBnTLcSTNtaDIzJBmhDv0NN/6KGxeKuNSVf2P6WGjrgcDhnHO5uSdIOFPacb6thcWl5ZXV3Fp+fWNza9ve2a2rOJWE1kjMY9kMQFHOIlrTTHPaTCQFEXDaCPrXI79xT6VicXSnBwltC+hGLGQEtJF82/FE6qti9QhfYi+UQDKvC0KA7wwzL+kxP8DVIT7Go5j3QDX4dsEpOWPgeeJOSQFNUfHtT68Tk1TQSBMOSrVcJ9HtDKRmhNNh3ksVTYD0oUtbhkYgqGpn48uG+NAoHRzG0rxI47H6eyIDodRABCYpQPfUrDcS//NaqQ4v2hmLklTTiEwWhSnHOsajmnCHSUo0HxgCRDLzV0x6YPrRpsy8KcGdPXme1E9K7lnJuT0tlK+mdeTQPjpAReSic1RGN6iCaoigR/SMXtGb9WS9WO/WxyS6YE1n9tAfWF8/rKiffQ==</latexit> µs(R) = 0 bR + µ⇣ <latexit sha1_base64="LRaCBugJGvpb+fI3gKGqA12rQiQ=">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</latexit> ˙ 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) <latexit sha1_base64="ypZaUtqe6AylfSkUkDkDZ9KWAX8=">AAACGXicbVDLSgMxFM34rPU16tJNsAgVocyIohuh6MZlLfYBnTLcSTNtaDIzJBmhDv0NN/6KGxeKuNSVf2P6WGjrgcDhnHO5uSdIOFPacb6thcWl5ZXV3Fp+fWNza9ve2a2rOJWE1kjMY9kMQFHOIlrTTHPaTCQFEXDaCPrXI79xT6VicXSnBwltC+hGLGQEtJF82/FE6qti9QhfYi+UQDKvC0KA7wwzL+kxP8DVIT7Go5j3QDX4dsEpOWPgeeJOSQFNUfHtT68Tk1TQSBMOSrVcJ9HtDKRmhNNh3ksVTYD0oUtbhkYgqGpn48uG+NAoHRzG0rxI47H6eyIDodRABCYpQPfUrDcS//NaqQ4v2hmLklTTiEwWhSnHOsajmnCHSUo0HxgCRDLzV0x6YPrRpsy8KcGdPXme1E9K7lnJuT0tlK+mdeTQPjpAReSic1RGN6iCaoigR/SMXtGb9WS9WO/WxyS6YE1n9tAfWF8/rKiffQ==</latexit> 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 <latexit sha1_base64="2btw/Rj7G+MrzrPMRvSLgaSzP9c=">AAAB83icbVBNSwMxEM3Wr1q/qh69BIvgqeyKoseiF49VrC10l5JNZ9vQbBKSrFCW/g0vHhTx6p/x5r8xbfegrQ8GHu/NMDMvVpwZ6/vfXmlldW19o7xZ2dre2d2r7h88GplpCi0qudSdmBjgTEDLMsuhozSQNObQjkc3U7/9BNowKR7sWEGUkoFgCaPEOim87+UhKMO4FJNetebX/RnwMgkKUkMFmr3qV9iXNEtBWMqJMd3AVzbKibaMcphUwsyAInREBtB1VJAUTJTPbp7gE6f0cSK1K2HxTP09kZPUmHEau86U2KFZ9Kbif143s8lVlDOhMguCzhclGcdW4mkAuM80UMvHjhCqmbsV0yHRhFoXU8WFECy+vEwez+rBRd2/O681ros4yugIHaNTFKBL1EC3qIlaiCKFntErevMy78V79z7mrSWvmDlEf+B9/gB1LZH1</latexit> R✏ <latexit sha1_base64="F+eCqzv3OsZFbVxCRluB+ci6TEc=">AAACQnicbVDLThsxFPVQ2kJa2rQs2YyIkOiCaAZRlU0lCl2wBEQCVSYdeZw7iRXbM9h3KkWWv62bfgE7PqAbFiDElgXOY8HrSraOzkO+PlkpuMEougjmXs2/fvN2YbH27v3Sh4/1T5/bpqg0gxYrRKFPM2pAcAUt5CjgtNRAZSbgJBvujfWTP6ANL9QxjkroStpXPOeMoqfS+q+kV6A9ct+TXFNm27/tD/cz9RdLLZw556VEQI7rCZSGi0JtTI2bSZ9KSVMbuUnG2WFqd92xt2veH+CXWlpvRM1oMuFzEM9Ag8zmIK2f+11YJUEhE9SYThyV2LVUI2cCXC2pDJSUDWkfOh4qKsF07aQCF655phfmhfZHYThhHyYslcaMZOadkuLAPNXG5Etap8J8u2u5KisExaYP5ZUIsQjHfYY9roGhGHlAmeZ+15ANqK8IfevjEuKnX34O2pvN+GszOtxq7OzO6lggK2SVrJOYfCM7ZJ8ckBZh5C/5T67IdfAvuAxugtupdS6YZZbJownu7gEofrLc</latexit> ˙ R = V ADAceq R ✓ ✏ 2 0V A kBTR ◆ <latexit sha1_base64="QJU3BBIBPOAbE/StBaxbbv8Fauw=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRaheii7ouix6MVjFfsB7VqyabaNzSZLkhXK0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmBTFn2rjut5NbWl5ZXcuvFzY2t7Z3irt7DS0TRWidSC5VK8CaciZo3TDDaStWFEcBp81geD3xm09UaSbFvRnF1I9wX7CQEWys1Lh7OCmb426x5FbcKdAi8TJSggy1bvGr05MkiagwhGOt254bGz/FyjDC6bjQSTSNMRniPm1bKnBEtZ9Orx2jI6v0UCiVLWHQVP09keJI61EU2M4Im4Ge9ybif147MeGlnzIRJ4YKMlsUJhwZiSavox5TlBg+sgQTxeytiAywwsTYgAo2BG/+5UXSOK145xX39qxUvcriyMMBHEIZPLiAKtxADepA4BGe4RXeHOm8OO/Ox6w152Qz+/AHzucPY7COWQ==</latexit> R⇤(t) Bray, Adv. in Phys. 43, 357 (1994) Widely separated droplets of varying sizes
26. 9 Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> 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 <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤,
27. 9 Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> 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 <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤,
28. 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 <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I ˙ = r · J Active model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, 10
29. 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 <latexit sha1_base64="AMPxuO9iYBYmiqIs71LmxRklFT4=">AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow=</latexit> <latexit sha1_base64="AMPxuO9iYBYmiqIs71LmxRklFT4=">AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow=</latexit> <latexit sha1_base64="AMPxuO9iYBYmiqIs71LmxRklFT4=">AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow=</latexit> <latexit sha1_base64="AMPxuO9iYBYmiqIs71LmxRklFT4=">AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow=</latexit> mechanical activity parameter positive (extensile microswimmers) negative (contractile microswimmers) minimal term breaking TRS in mechanical sector ˜ κ Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I ˙ = r · J Active model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, 10
30. 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 <latexit sha1_base64="AMPxuO9iYBYmiqIs71LmxRklFT4=">AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow=</latexit> <latexit sha1_base64="AMPxuO9iYBYmiqIs71LmxRklFT4=">AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow=</latexit> <latexit sha1_base64="AMPxuO9iYBYmiqIs71LmxRklFT4=">AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow=</latexit> <latexit sha1_base64="AMPxuO9iYBYmiqIs71LmxRklFT4=">AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow=</latexit> 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 <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="XJx+pKEirN30bSLrBUI49M+QrtA=">AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK</latexit> v = ˜  0/ Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I ˙ = r · J Active model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">AAACQ3icbVDLSsNAFJ3UV62vqks3waIIaklE0Y0g6kLEhYJ9YBPKzWTaDk4mcWZSKCH/5sYfcOcPuHGhiFvBSc2iPi4MczjnHu69x4sYlcqynozC2PjE5FRxujQzOze/UF5cqsswFpjUcMhC0fRAEkY5qSmqGGlGgkDgMdLwbk8yvdEnQtKQX6tBRNwAupx2KAalqXb5xvFC5stBoL/kPF0/3B4lHA4eg9QJ4s1Rup86UY9uOvJOqGTnNP1hudDDfUi3Su1yxapawzL/AjsHFZTXZbv86PghjgPCFWYgZcu2IuUmIBTFjKQlJ5YkAnwLXdLSkENApJsMM0jNNc34ZicU+nFlDtlRRwKBzFbUnQGonvytZeR/WitWnQM3oTyKFeH4e1AnZqYKzSxQ06eCYMUGGgAWVO9q4h4IwErHnoVg/z75L6jvVO29qnW1Wzk6zuMoohW0ijaQjfbRETpDl6iGMLpHz+gVvRkPxovxbnx8txaM3LOMfpTx+QWT1rOh</latexit> J = rµ + v + p 2D⇤, 10
31. Incomplete phase separation ˜ κ < 0 Complete phase separation ˜ κ > 0 Active model H: nucleation and growth 11 -1 1 RS and Cates PRL (2019)
32. Incomplete phase separation ˜ κ < 0 Complete phase separation ˜ κ > 0 Active model H: nucleation and growth 11 -1 1 RS and Cates PRL (2019)
33. 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
34. 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
35. 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
36. 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
37. Length scale determined by activity SSI Ostwald ripening RS and Cates PRL (2019) ˙ = r · J Active Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I <latexit sha1_base64="XJx+pKEirN30bSLrBUI49M+QrtA=">AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK</latexit> v = ˜  0/ 13
38. 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 <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I <latexit sha1_base64="XJx+pKEirN30bSLrBUI49M+QrtA=">AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK</latexit> v = ˜  0/ 13
39. 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 <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I <latexit sha1_base64="XJx+pKEirN30bSLrBUI49M+QrtA=">AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK</latexit> v = ˜  0/ 13
40. 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 <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I <latexit sha1_base64="XJx+pKEirN30bSLrBUI49M+QrtA=">AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK</latexit> v = ˜  0/ 13
41. 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 <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I <latexit sha1_base64="XJx+pKEirN30bSLrBUI49M+QrtA=">AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK</latexit> v = ˜  0/ 13
42. 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. <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> J = Mrµ + J⇣ + p 2DM⇤, J⇣ = ⇣(r2 )r <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤,
43. 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. <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">AAACf3icbVHbahRBEO0ZL4nrJas++jJkEYWEYSYI6sNCUBQfI7hJYHuz1PTUZJv0XNJdIy6d/g0/zDf/xQd7JhOIiQVFH07Vqa5L1ihpKEl+B+Gdu/fub2w+GD189PjJ1vjps0NTt1rgTNSq1scZGFSywhlJUnjcaIQyU3iUnX3s4kffURtZV99o3eCihNNKFlIAeWo5/snLdur9xH5yO/3LlVfn4Hb5eQv5EJryQoOwPEdFwEuglQBlPzt3RTUreaUg/EF9Y1Zj7hOuF50O4IJntcrNuvSP5RVkClxX4+LE7jkXj5bjSRInvUW3QTqACRvsYDn+xfNatCVWJBQYM0+ThhYWNEmh0I14a7ABcQanOPewghLNwvZtuuilZ/KoqLX3iqKeva6wUJquV5/ZjW5uxjryf7F5S8W7hZVV0xJW4vKjolUR1VF3jCiXGgWptQcgtPS9RmIFftPkT9YtIb058m1wuBenSZx+fTPZ/zCsY5O9YNvsNUvZW7bPvrADNmOC/Qm2g51gNwzCV2EcJpepYTBonrN/LHz/F9i7xpA=</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> J = Mrµ + J⇣ + p 2DM⇤, J⇣ = ⇣(r2 )r <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, <latexit sha1_base64="iU+JbNDhU90L9SncVWcfxeqtL+Y=">AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT</latexit> J⇣ = rµ⇣ + r ⇥ A
44. 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. <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> J = Mrµ + J⇣ + p 2DM⇤, J⇣ = ⇣(r2 )r <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, e ff ective nonlocal chemical potential <latexit sha1_base64="iU+JbNDhU90L9SncVWcfxeqtL+Y=">AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT</latexit> J⇣ = rµ⇣ + r ⇥ A
45. 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. <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> J = Mrµ + J⇣ + p 2DM⇤, J⇣ = ⇣(r2 )r <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">AAACmXicbVFta9swEJa9ty57y1rYl34xCxsd3YIdBuuHFbpXSlmhY0tbiNJwluVGVJZd6VyWCf2n/ZZ927+Z7AS2Nj0Q9/Dcne65u7SSwmAc/wnCGzdv3b6zcrdz7/6Dh4+6j1cPTVlrxoeslKU+TsFwKRQfokDJjyvNoUglP0rPPjTxowuujSjVd5xVfFzAqRK5YICemnR/0QJwmuZ2zz3ffrVP01JmZlZ4Z6mCVIKjRb35L2li6U+O4DapOddoBx/33aWaL751Bu4lPa8hsxT5D2xFWs0zZ6Plj7ZbtzFvdmIHjlZT8eI6HZ53rjPp9uJ+3Fq0DJIF6JGFHUy6v2lWsrrgCpkEY0ZJXOHYgkbBJHcdWhteATuDUz7yUEHBzdi2ol30zDNZlJfaP4VRy/5fYaEwjUyf2UxmrsYa8rrYqMZ8a2yFqmrkis0b5bWMsIyaM0WZ0JyhnHkATAuvNWJT0MDQH7NZQnJ15GVwOOgncT/5+rq3836xjhWyTp6SDZKQN2SH7JIDMiQseBK8DT4Fn8P18F24G+7NU8NgUbNGLln47S+iRtBQ</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">AAACmXicbVFta9swEJa9ty57y1rYl34xCxsd3YIdBuuHFbpXSlmhY0tbiNJwluVGVJZd6VyWCf2n/ZZ927+Z7AS2Nj0Q9/Dcne65u7SSwmAc/wnCGzdv3b6zcrdz7/6Dh4+6j1cPTVlrxoeslKU+TsFwKRQfokDJjyvNoUglP0rPPjTxowuujSjVd5xVfFzAqRK5YICemnR/0QJwmuZ2zz3ffrVP01JmZlZ4Z6mCVIKjRb35L2li6U+O4DapOddoBx/33aWaL751Bu4lPa8hsxT5D2xFWs0zZ6Plj7ZbtzFvdmIHjlZT8eI6HZ53rjPp9uJ+3Fq0DJIF6JGFHUy6v2lWsrrgCpkEY0ZJXOHYgkbBJHcdWhteATuDUz7yUEHBzdi2ol30zDNZlJfaP4VRy/5fYaEwjUyf2UxmrsYa8rrYqMZ8a2yFqmrkis0b5bWMsIyaM0WZ0JyhnHkATAuvNWJT0MDQH7NZQnJ15GVwOOgncT/5+rq3836xjhWyTp6SDZKQN2SH7JIDMiQseBK8DT4Fn8P18F24G+7NU8NgUbNGLln47S+iRtBQ</latexit> <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, e ff ective nonlocal chemical potential <latexit sha1_base64="XbSTBBHNMy7G1Rbi0kv+i7vyfAM=">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</latexit> µs(R) = 0 bR + µ + µ⇣ Chemical potential raised at curved interface: μs <latexit sha1_base64="iU+JbNDhU90L9SncVWcfxeqtL+Y=">AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT</latexit> J⇣ = rµ⇣ + r ⇥ A
46. 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. <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> J = Mrµ + J⇣ + p 2DM⇤, J⇣ = ⇣(r2 )r <latexit 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sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, e ff ective nonlocal chemical potential <latexit sha1_base64="XbSTBBHNMy7G1Rbi0kv+i7vyfAM=">AAACLHicbVBbS8MwGE29O29VH30JDkERRiuKvgjiXnycw11gHeVrlm7BpC1JKszSH+SLf0UQHxziq7/DdBvi7UDgcM75knwnSDhT2nFG1szs3PzC4tJyaWV1bX3D3txqqjiVhDZIzGPZDkBRziLa0Exz2k4kBRFw2gpuq4XfuqNSsTi60cOEdgX0IxYyAtpIvl31ROqr/foBPsdeKIFkXh+EAN/JMy8ZMD/A9Rwf4iKWedxc3IMcfwn3VEPu22Wn4oyB/xJ3SspoippvP3u9mKSCRppwUKrjOonuZiA1I5zmJS9VNAFyC33aMTQCQVU3Gy+b4z2j9HAYS3Mijcfq94kMhFJDEZikAD1Qv71C/M/rpDo862YsSlJNIzJ5KEw51jEumsM9JinRfGgIEMnMXzEZgKlMm35LpgT398p/SfOo4p5UnOvj8sXltI4ltIN20T5y0Sm6QFeohhqIoAf0hF7RyHq0Xqw3630SnbGmM9voB6yPT0orpxA=</latexit> µs(R) = 0 bR + µ + µ⇣ Chemical potential raised at curved interface: μs <latexit sha1_base64="iU+JbNDhU90L9SncVWcfxeqtL+Y=">AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT</latexit> J⇣ = rµ⇣ + r ⇥ A
47. 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 <latexit sha1_base64="KmlrZqHgOXPGYRlr1GxkuG4hVyQ=">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</latexit> = (⇣S0 2 S1) ⇣ 2 ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit 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sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> J = Mrµ + J⇣ + p 2DM⇤, J⇣ = ⇣(r2 )r <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit 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sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">AAACmXicbVFta9swEJa9ty57y1rYl34xCxsd3YIdBuuHFbpXSlmhY0tbiNJwluVGVJZd6VyWCf2n/ZZ927+Z7AS2Nj0Q9/Dcne65u7SSwmAc/wnCGzdv3b6zcrdz7/6Dh4+6j1cPTVlrxoeslKU+TsFwKRQfokDJjyvNoUglP0rPPjTxowuujSjVd5xVfFzAqRK5YICemnR/0QJwmuZ2zz3ffrVP01JmZlZ4Z6mCVIKjRb35L2li6U+O4DapOddoBx/33aWaL751Bu4lPa8hsxT5D2xFWs0zZ6Plj7ZbtzFvdmIHjlZT8eI6HZ53rjPp9uJ+3Fq0DJIF6JGFHUy6v2lWsrrgCpkEY0ZJXOHYgkbBJHcdWhteATuDUz7yUEHBzdi2ol30zDNZlJfaP4VRy/5fYaEwjUyf2UxmrsYa8rrYqMZ8a2yFqmrkis0b5bWMsIyaM0WZ0JyhnHkATAuvNWJT0MDQH7NZQnJ15GVwOOgncT/5+rq3836xjhWyTp6SDZKQN2SH7JIDMiQseBK8DT4Fn8P18F24G+7NU8NgUbNGLln47S+iRtBQ</latexit> <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, e ff ective nonlocal chemical potential <latexit sha1_base64="XbSTBBHNMy7G1Rbi0kv+i7vyfAM=">AAACLHicbVBbS8MwGE29O29VH30JDkERRiuKvgjiXnycw11gHeVrlm7BpC1JKszSH+SLf0UQHxziq7/DdBvi7UDgcM75knwnSDhT2nFG1szs3PzC4tJyaWV1bX3D3txqqjiVhDZIzGPZDkBRziLa0Exz2k4kBRFw2gpuq4XfuqNSsTi60cOEdgX0IxYyAtpIvl31ROqr/foBPsdeKIFkXh+EAN/JMy8ZMD/A9Rwf4iKWedxc3IMcfwn3VEPu22Wn4oyB/xJ3SspoippvP3u9mKSCRppwUKrjOonuZiA1I5zmJS9VNAFyC33aMTQCQVU3Gy+b4z2j9HAYS3Mijcfq94kMhFJDEZikAD1Qv71C/M/rpDo862YsSlJNIzJ5KEw51jEumsM9JinRfGgIEMnMXzEZgKlMm35LpgT398p/SfOo4p5UnOvj8sXltI4ltIN20T5y0Sm6QFeohhqIoAf0hF7RyHq0Xqw3630SnbGmM9voB6yPT0orpxA=</latexit> µs(R) = 0 bR + µ + µ⇣ Chemical potential raised at curved interface: μs <latexit sha1_base64="iU+JbNDhU90L9SncVWcfxeqtL+Y=">AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT</latexit> J⇣ = rµ⇣ + r ⇥ A
48. 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 <latexit sha1_base64="KmlrZqHgOXPGYRlr1GxkuG4hVyQ=">AAACU3icbVHPa9RAFJ6kVetqda1HL4OLUA9dkqLoRSh68dii2xZ2lvAyedkddyYZZl7ENeR/FMGD/4gXD3ayXcG2Phj4+L7385vcauUpSX5G8db2rdt3du4O7t3fffBw+Gjv1NeNkziRta7deQ4etapwQoo0nluHYHKNZ/nyXa+ffUbnVV19pJXFmYF5pUolgQKVDT8Jwi+07tM6LLpWzMEYyIRdKP6Gi9KBbMUSrIV98RUJuDBACwm6/dBlCT/gh0KHccVVIX0eOvXpB3/lrsuGo2ScrIPfBOkGjNgmjrPhd1HUsjFYkdTg/TRNLM1acKSkxm4gGo8W5BLmOA2wAoN+1q5v6fizwBS8rF14FfE1+29FC8b7lclDZr+3v6715P+0aUPl61mrKtsQVvJyUNloTjXvDeaFcihJrwIA6VTYlcsFBBspfMMgmJBeP/kmOD0cpy/HycmL0dHbjR077Al7yvZZyl6xI/aeHbMJk+wb+8X+RCz6Ef2O43j7MjWONjWP2ZWIdy8AVfa0tQ==</latexit> = (⇣S0 2 S1) ⇣ 2 ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> J = Mrµ + J⇣ + p 2DM⇤, J⇣ = ⇣(r2 )r <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">AAACmXicbVFta9swEJa9ty57y1rYl34xCxsd3YIdBuuHFbpXSlmhY0tbiNJwluVGVJZd6VyWCf2n/ZZ927+Z7AS2Nj0Q9/Dcne65u7SSwmAc/wnCGzdv3b6zcrdz7/6Dh4+6j1cPTVlrxoeslKU+TsFwKRQfokDJjyvNoUglP0rPPjTxowuujSjVd5xVfFzAqRK5YICemnR/0QJwmuZ2zz3ffrVP01JmZlZ4Z6mCVIKjRb35L2li6U+O4DapOddoBx/33aWaL751Bu4lPa8hsxT5D2xFWs0zZ6Plj7ZbtzFvdmIHjlZT8eI6HZ53rjPp9uJ+3Fq0DJIF6JGFHUy6v2lWsrrgCpkEY0ZJXOHYgkbBJHcdWhteATuDUz7yUEHBzdi2ol30zDNZlJfaP4VRy/5fYaEwjUyf2UxmrsYa8rrYqMZ8a2yFqmrkis0b5bWMsIyaM0WZ0JyhnHkATAuvNWJT0MDQH7NZQnJ15GVwOOgncT/5+rq3836xjhWyTp6SDZKQN2SH7JIDMiQseBK8DT4Fn8P18F24G+7NU8NgUbNGLln47S+iRtBQ</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, e ff ective nonlocal chemical potential <latexit sha1_base64="XbSTBBHNMy7G1Rbi0kv+i7vyfAM=">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</latexit> µs(R) = 0 bR + µ + µ⇣ Chemical potential raised at curved interface: μs <latexit sha1_base64="iU+JbNDhU90L9SncVWcfxeqtL+Y=">AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT</latexit> J⇣ = rµ⇣ + r ⇥ A Forward Ostwald ripening γϕ > 0 Reverse Ostwald ripening γϕ < 0
49. 15 Steady-states in the plane of e ff ective tensions ˙ = r · J Active Model H <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, <latexit sha1_base64="hHrH1Qh7oykcidiyawuqJauU8m4=">AAACC3icbZC7TsMwFIadcivlVmBkiVohMZUEgWCsYGEsEr1IbRQ57mlr1U4i+6Siirqz8CosDCDEyguw8Ta4bQZoOZKlT/9/jn38B7HgGh3n28qtrK6tb+Q3C1vbO7t7xf2Dho4SxaDOIhGpVkA1CB5CHTkKaMUKqAwENIPhzdRvjkBpHoX3OI7Bk7Qf8h5nFI3kF0sdhAec3ZMq6E7STp9KSf3RaQbOxC+WnYozK3sZ3AzKJKuaX/zqdCOWSAiRCap123Vi9FKqkDMBk0In0RBTNqR9aBsMqQTtpbMdJvaxUbp2L1LmhGjP1N8TKZVaj2VgOiXFgV70puJ/XjvB3pWX8jBOEEI2f6iXCBsjexqM3eUKGIqxAcoUN7vabEAVZWjiK5gQ3MUvL0PjrOJeVJy783L1OosjT45IiZwQl1ySKrklNVInjDySZ/JK3qwn68V6tz7mrTkrmzkkf8r6/AEt45u/</latexit> v/ 0 <latexit sha1_base64="IOOGtSA4k3m0H3LcmZZVWyB+xtw=">AAACEHicbVC7TsMwFHV4lvIKMLJEVAimkiAQjBUsjEWiD6mpKse9ba3aSWTfIKoon8DCr7AwgBArIxt/g9tmgJYjWTo+597r6xPEgmt03W9rYXFpeWW1sFZc39jc2rZ3dus6ShSDGotEpJoB1SB4CDXkKKAZK6AyENAIhtdjv3EPSvMovMNRDG1J+yHvcUbRSB37yEd4wMmcVEE3S/0+lZJ2Uj8e8Owkv7lZxy65ZXcCZ554OSmRHNWO/eV3I5ZICJEJqnXLc2Nsp1QhZwKyop9oiCkb0j60DA2pBN1OJ4tkzqFRuk4vUuaE6EzU3x0plVqPZGAqJcWBnvXG4n9eK8HeZTvlYZwghGz6UC8RDkbOOB2nyxUwFCNDKFPc7OqwAVWUocmwaELwZr88T+qnZe+87N6elSpXeRwFsk8OyDHxyAWpkBtSJTXCyCN5Jq/kzXqyXqx362NaumDlPXvkD6zPH2t6nhA=</latexit> / 0 <latexit sha1_base64="IOOGtSA4k3m0H3LcmZZVWyB+xtw=">AAACEHicbVC7TsMwFHV4lvIKMLJEVAimkiAQjBUsjEWiD6mpKse9ba3aSWTfIKoon8DCr7AwgBArIxt/g9tmgJYjWTo+597r6xPEgmt03W9rYXFpeWW1sFZc39jc2rZ3dus6ShSDGotEpJoB1SB4CDXkKKAZK6AyENAIhtdjv3EPSvMovMNRDG1J+yHvcUbRSB37yEd4wMmcVEE3S/0+lZJ2Uj8e8Owkv7lZxy65ZXcCZ554OSmRHNWO/eV3I5ZICJEJqnXLc2Nsp1QhZwKyop9oiCkb0j60DA2pBN1OJ4tkzqFRuk4vUuaE6EzU3x0plVqPZGAqJcWBnvXG4n9eK8HeZTvlYZwghGz6UC8RDkbOOB2nyxUwFCNDKFPc7OqwAVWUocmwaELwZr88T+qnZe+87N6elSpXeRwFsk8OyDHxyAWpkBtSJTXCyCN5Jq/kzXqyXqx362NaumDlPXvkD6zPH2t6nhA=</latexit> / 0 RS and Cates PRL (2019)
50. 15 Steady-states in the plane of e ff ective tensions ˙ = r · J Active Model H <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, di ff usive sector mechanical sector <latexit sha1_base64="XJx+pKEirN30bSLrBUI49M+QrtA=">AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK</latexit> v = ˜  0/ <latexit sha1_base64="KmlrZqHgOXPGYRlr1GxkuG4hVyQ=">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</latexit> = (⇣S0 2 S1) ⇣ 2 models H (always +ve) <latexit sha1_base64="GxJ7cDOVOO9yyHn3LIfDMaLR0hw=">AAACEXicbVDLSsNAFJ34rPVVdelmsAjdWJOqWBdC0Y3LCvYBTRpuptN26EwSZyZCCfkFN/6KGxeKuHXnzr8xfSy09cCFwzn3cu89XsiZ0qb5bSwsLi2vrGbWsusbm1vbuZ3dugoiSWiNBDyQTQ8U5cynNc00p81QUhAepw1vcD3yGw9UKhb4d3oYUkdAz2ddRkCnkpsr2D0QAtzYTC5tdS91fFS2BxCGgKEdnyTHF147LiVJ1s3lzaI5Bp4n1pTk0RRVN/dldwISCeprwkGplmWG2olBakY4TbJ2pGgIZAA92kqpD4IqJx5/lODDVOngbiDT8jUeq78nYhBKDYWXdgrQfTXrjcT/vFaku2UnZn4YaeqTyaJuxLEO8Cge3GGSEs2HKQEiWXorJn2QQHQa4igEa/bleVIvFa2zonl7mq9cTePIoH10gArIQueogm5QFdUQQY/oGb2iN+PJeDHejY9J64IxndlDf2B8/gBgh5y0</latexit> 0 = p 8a3/9b2 <latexit sha1_base64="hHrH1Qh7oykcidiyawuqJauU8m4=">AAACC3icbZC7TsMwFIadcivlVmBkiVohMZUEgWCsYGEsEr1IbRQ57mlr1U4i+6Siirqz8CosDCDEyguw8Ta4bQZoOZKlT/9/jn38B7HgGh3n28qtrK6tb+Q3C1vbO7t7xf2Dho4SxaDOIhGpVkA1CB5CHTkKaMUKqAwENIPhzdRvjkBpHoX3OI7Bk7Qf8h5nFI3kF0sdhAec3ZMq6E7STp9KSf3RaQbOxC+WnYozK3sZ3AzKJKuaX/zqdCOWSAiRCap123Vi9FKqkDMBk0In0RBTNqR9aBsMqQTtpbMdJvaxUbp2L1LmhGjP1N8TKZVaj2VgOiXFgV70puJ/XjvB3pWX8jBOEEI2f6iXCBsjexqM3eUKGIqxAcoUN7vabEAVZWjiK5gQ3MUvL0PjrOJeVJy783L1OosjT45IiZwQl1ySKrklNVInjDySZ/JK3qwn68V6tz7mrTkrmzkkf8r6/AEt45u/</latexit> v/ 0 <latexit sha1_base64="IOOGtSA4k3m0H3LcmZZVWyB+xtw=">AAACEHicbVC7TsMwFHV4lvIKMLJEVAimkiAQjBUsjEWiD6mpKse9ba3aSWTfIKoon8DCr7AwgBArIxt/g9tmgJYjWTo+597r6xPEgmt03W9rYXFpeWW1sFZc39jc2rZ3dus6ShSDGotEpJoB1SB4CDXkKKAZK6AyENAIhtdjv3EPSvMovMNRDG1J+yHvcUbRSB37yEd4wMmcVEE3S/0+lZJ2Uj8e8Owkv7lZxy65ZXcCZ554OSmRHNWO/eV3I5ZICJEJqnXLc2Nsp1QhZwKyop9oiCkb0j60DA2pBN1OJ4tkzqFRuk4vUuaE6EzU3x0plVqPZGAqJcWBnvXG4n9eK8HeZTvlYZwghGz6UC8RDkbOOB2nyxUwFCNDKFPc7OqwAVWUocmwaELwZr88T+qnZe+87N6elSpXeRwFsk8OyDHxyAWpkBtSJTXCyCN5Jq/kzXqyXqx362NaumDlPXvkD6zPH2t6nhA=</latexit> / 0 <latexit sha1_base64="IOOGtSA4k3m0H3LcmZZVWyB+xtw=">AAACEHicbVC7TsMwFHV4lvIKMLJEVAimkiAQjBUsjEWiD6mpKse9ba3aSWTfIKoon8DCr7AwgBArIxt/g9tmgJYjWTo+597r6xPEgmt03W9rYXFpeWW1sFZc39jc2rZ3dus6ShSDGotEpJoB1SB4CDXkKKAZK6AyENAIhtdjv3EPSvMovMNRDG1J+yHvcUbRSB37yEd4wMmcVEE3S/0+lZJ2Uj8e8Owkv7lZxy65ZXcCZ554OSmRHNWO/eV3I5ZICJEJqnXLc2Nsp1QhZwKyop9oiCkb0j60DA2pBN1OJ4tkzqFRuk4vUuaE6EzU3x0plVqPZGAqJcWBnvXG4n9eK8HeZTvlYZwghGz6UC8RDkbOOB2nyxUwFCNDKFPc7OqwAVWUocmwaELwZr88T+qnZe+87N6elSpXeRwFsk8OyDHxyAWpkBtSJTXCyCN5Jq/kzXqyXqx362NaumDlPXvkD6zPH2t6nhA=</latexit> / 0 RS and Cates PRL (2019)
51. 15 Steady-states in the plane of e ff ective tensions ˙ = r · J Active Model H <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, di ff usive sector mechanical sector <latexit sha1_base64="XJx+pKEirN30bSLrBUI49M+QrtA=">AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK</latexit> v = ˜  0/ <latexit sha1_base64="KmlrZqHgOXPGYRlr1GxkuG4hVyQ=">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</latexit> = (⇣S0 2 S1) ⇣ 2 models H (always +ve) <latexit sha1_base64="GxJ7cDOVOO9yyHn3LIfDMaLR0hw=">AAACEXicbVDLSsNAFJ34rPVVdelmsAjdWJOqWBdC0Y3LCvYBTRpuptN26EwSZyZCCfkFN/6KGxeKuHXnzr8xfSy09cCFwzn3cu89XsiZ0qb5bSwsLi2vrGbWsusbm1vbuZ3dugoiSWiNBDyQTQ8U5cynNc00p81QUhAepw1vcD3yGw9UKhb4d3oYUkdAz2ddRkCnkpsr2D0QAtzYTC5tdS91fFS2BxCGgKEdnyTHF147LiVJ1s3lzaI5Bp4n1pTk0RRVN/dldwISCeprwkGplmWG2olBakY4TbJ2pGgIZAA92kqpD4IqJx5/lODDVOngbiDT8jUeq78nYhBKDYWXdgrQfTXrjcT/vFaku2UnZn4YaeqTyaJuxLEO8Cge3GGSEs2HKQEiWXorJn2QQHQa4igEa/bleVIvFa2zonl7mq9cTePIoH10gArIQueogm5QFdUQQY/oGb2iN+PJeDHejY9J64IxndlDf2B8/gBgh5y0</latexit> 0 = p 8a3/9b2 <latexit sha1_base64="hHrH1Qh7oykcidiyawuqJauU8m4=">AAACC3icbZC7TsMwFIadcivlVmBkiVohMZUEgWCsYGEsEr1IbRQ57mlr1U4i+6Siirqz8CosDCDEyguw8Ta4bQZoOZKlT/9/jn38B7HgGh3n28qtrK6tb+Q3C1vbO7t7xf2Dho4SxaDOIhGpVkA1CB5CHTkKaMUKqAwENIPhzdRvjkBpHoX3OI7Bk7Qf8h5nFI3kF0sdhAec3ZMq6E7STp9KSf3RaQbOxC+WnYozK3sZ3AzKJKuaX/zqdCOWSAiRCap123Vi9FKqkDMBk0In0RBTNqR9aBsMqQTtpbMdJvaxUbp2L1LmhGjP1N8TKZVaj2VgOiXFgV70puJ/XjvB3pWX8jBOEEI2f6iXCBsjexqM3eUKGIqxAcoUN7vabEAVZWjiK5gQ3MUvL0PjrOJeVJy783L1OosjT45IiZwQl1ySKrklNVInjDySZ/JK3qwn68V6tz7mrTkrmzkkf8r6/AEt45u/</latexit> v/ 0 <latexit sha1_base64="IOOGtSA4k3m0H3LcmZZVWyB+xtw=">AAACEHicbVC7TsMwFHV4lvIKMLJEVAimkiAQjBUsjEWiD6mpKse9ba3aSWTfIKoon8DCr7AwgBArIxt/g9tmgJYjWTo+597r6xPEgmt03W9rYXFpeWW1sFZc39jc2rZ3dus6ShSDGotEpJoB1SB4CDXkKKAZK6AyENAIhtdjv3EPSvMovMNRDG1J+yHvcUbRSB37yEd4wMmcVEE3S/0+lZJ2Uj8e8Owkv7lZxy65ZXcCZ554OSmRHNWO/eV3I5ZICJEJqnXLc2Nsp1QhZwKyop9oiCkb0j60DA2pBN1OJ4tkzqFRuk4vUuaE6EzU3x0plVqPZGAqJcWBnvXG4n9eK8HeZTvlYZwghGz6UC8RDkbOOB2nyxUwFCNDKFPc7OqwAVWUocmwaELwZr88T+qnZe+87N6elSpXeRwFsk8OyDHxyAWpkBtSJTXCyCN5Jq/kzXqyXqx362NaumDlPXvkD6zPH2t6nhA=</latexit> / 0 <latexit sha1_base64="IOOGtSA4k3m0H3LcmZZVWyB+xtw=">AAACEHicbVC7TsMwFHV4lvIKMLJEVAimkiAQjBUsjEWiD6mpKse9ba3aSWTfIKoon8DCr7AwgBArIxt/g9tmgJYjWTo+597r6xPEgmt03W9rYXFpeWW1sFZc39jc2rZ3dus6ShSDGotEpJoB1SB4CDXkKKAZK6AyENAIhtdjv3EPSvMovMNRDG1J+yHvcUbRSB37yEd4wMmcVEE3S/0+lZJ2Uj8e8Owkv7lZxy65ZXcCZ554OSmRHNWO/eV3I5ZICJEJqnXLc2Nsp1QhZwKyop9oiCkb0j60DA2pBN1OJ4tkzqFRuk4vUuaE6EzU3x0plVqPZGAqJcWBnvXG4n9eK8HeZTvlYZwghGz6UC8RDkbOOB2nyxUwFCNDKFPc7OqwAVWUocmwaELwZr88T+qnZe+87N6elSpXeRwFsk8OyDHxyAWpkBtSJTXCyCN5Jq/kzXqyXqx362NaumDlPXvkD6zPH2t6nhA=</latexit> / 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)
52. 16 II. Self-propelling active droplets RS, Tjhung and Cates, Phys. Rev. Research 2(3), 032024(R), 2020 ELSEN TJHUNG MICHAEL E. CATES
53. 17 Poincloux et al PNAS 2011; Hawkins et al Biophys J 2011 Swimming of cells without liquid-crystalline order Cortex
54. 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
55. 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
56. 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
57. 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
58. 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
59. 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 <latexit sha1_base64="U8OTpmJfQMfNZlgcf5DucIC0jXE=">AAAB/nicbVBNS8NAEN3Ur1q/quLJS7AInkoiih6LXjxWsLXQhLLZTNqlmw92J2IJAf+KFw+KePV3ePPfuE1z0NYHA4/3ZnZ2npcIrtCyvo3K0vLK6lp1vbaxubW9U9/d66o4lQw6LBax7HlUgeARdJCjgF4igYaegHtvfD317x9AKh5HdzhJwA3pMOIBZxS1NKgfOAiPWLyTSfDzzElGPB/UG1bTKmAuErskDVKiPah/OX7M0hAiZIIq1betBN2MSuRMQF5zUgUJZWM6hL6mEQ1BuVmxNjePteKbQSx1RWgW6u+JjIZKTUJPd4YUR2rem4r/ef0Ug0s341GSIkRstihIhYmxOc3C9LkEhmKiCWWS67+abEQlZagTq+kQ7PmTF0n3tGmfN63bs0brqoyjSg7JETkhNrkgLXJD2qRDGMnIM3klb8aT8WK8Gx+z1opRzuyTPzA+fwCYGpaJ</latexit>
60. 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 <latexit sha1_base64="U8OTpmJfQMfNZlgcf5DucIC0jXE=">AAAB/nicbVBNS8NAEN3Ur1q/quLJS7AInkoiih6LXjxWsLXQhLLZTNqlmw92J2IJAf+KFw+KePV3ePPfuE1z0NYHA4/3ZnZ2npcIrtCyvo3K0vLK6lp1vbaxubW9U9/d66o4lQw6LBax7HlUgeARdJCjgF4igYaegHtvfD317x9AKh5HdzhJwA3pMOIBZxS1NKgfOAiPWLyTSfDzzElGPB/UG1bTKmAuErskDVKiPah/OX7M0hAiZIIq1betBN2MSuRMQF5zUgUJZWM6hL6mEQ1BuVmxNjePteKbQSx1RWgW6u+JjIZKTUJPd4YUR2rem4r/ef0Ug0s341GSIkRstihIhYmxOc3C9LkEhmKiCWWS67+abEQlZagTq+kQ7PmTF0n3tGmfN63bs0brqoyjSg7JETkhNrkgLXJD2qRDGMnIM3klb8aT8WK8Gx+z1opRzuyTPzA+fwCYGpaJ</latexit> <latexit sha1_base64="DpKpAApI5rASTjlNhiFvmIbPB1E=">AAAB/3icbVDLSsNAFJ3UV62vqODGTbAIrkoiii6LblxWsA9oQ5lMb9qhkwczN2KJWfgrblwo4tbfcOffOE2z0NYDFw7n3Dt37vFiwRXa9rdRWlpeWV0rr1c2Nre2d8zdvZaKEsmgySIRyY5HFQgeQhM5CujEEmjgCWh74+up374HqXgU3uEkBjegw5D7nFHUUt886CE8YP5O6okEsrQXK571zapds3NYi8QpSJUUaPTNr94gYkkAITJBleo6doxuSiVyJiCr9BIFMWVjOoSupiENQLlpvjezjrUysPxI6grRytXfEykNlJoEnu4MKI7UvDcV//O6CfqXbsrDOEEI2WyRnwgLI2sahjXgEhiKiSaUSa7/arERlZShjqyiQ3DmT14krdOac16zb8+q9asijjI5JEfkhDjkgtTJDWmQJmHkkTyTV/JmPBkvxrvxMWstGcXMPvkD4/MHfCqXCw==</latexit> model H
61. 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 <latexit sha1_base64="U8OTpmJfQMfNZlgcf5DucIC0jXE=">AAAB/nicbVBNS8NAEN3Ur1q/quLJS7AInkoiih6LXjxWsLXQhLLZTNqlmw92J2IJAf+KFw+KePV3ePPfuE1z0NYHA4/3ZnZ2npcIrtCyvo3K0vLK6lp1vbaxubW9U9/d66o4lQw6LBax7HlUgeARdJCjgF4igYaegHtvfD317x9AKh5HdzhJwA3pMOIBZxS1NKgfOAiPWLyTSfDzzElGPB/UG1bTKmAuErskDVKiPah/OX7M0hAiZIIq1betBN2MSuRMQF5zUgUJZWM6hL6mEQ1BuVmxNjePteKbQSx1RWgW6u+JjIZKTUJPd4YUR2rem4r/ef0Ug0s341GSIkRstihIhYmxOc3C9LkEhmKiCWWS67+abEQlZagTq+kQ7PmTF0n3tGmfN63bs0brqoyjSg7JETkhNrkgLXJD2qRDGMnIM3klb8aT8WK8Gx+z1opRzuyTPzA+fwCYGpaJ</latexit> <latexit sha1_base64="DpKpAApI5rASTjlNhiFvmIbPB1E=">AAAB/3icbVDLSsNAFJ3UV62vqODGTbAIrkoiii6LblxWsA9oQ5lMb9qhkwczN2KJWfgrblwo4tbfcOffOE2z0NYDFw7n3Dt37vFiwRXa9rdRWlpeWV0rr1c2Nre2d8zdvZaKEsmgySIRyY5HFQgeQhM5CujEEmjgCWh74+up374HqXgU3uEkBjegw5D7nFHUUt886CE8YP5O6okEsrQXK571zapds3NYi8QpSJUUaPTNr94gYkkAITJBleo6doxuSiVyJiCr9BIFMWVjOoSupiENQLlpvjezjrUysPxI6grRytXfEykNlJoEnu4MKI7UvDcV//O6CfqXbsrDOEEI2WyRnwgLI2sahjXgEhiKiSaUSa7/arERlZShjqyiQ3DmT14krdOac16zb8+q9asijjI5JEfkhDjkgtTJDWmQJmHkkTyTV/JmPBkvxrvxMWstGcXMPvkD4/MHfCqXCw==</latexit> model H ⇢ = <latexit sha1_base64="qok9lw7LPnF9frpAN49XCubf42Y=">AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA==</latexit> <latexit sha1_base64="qok9lw7LPnF9frpAN49XCubf42Y=">AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA==</latexit> <latexit sha1_base64="qok9lw7LPnF9frpAN49XCubf42Y=">AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA==</latexit> <latexit sha1_base64="qok9lw7LPnF9frpAN49XCubf42Y=">AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA==</latexit> Cortex
62. 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 <latexit sha1_base64="xrbmyzmhPDH11ajAYU1ekas0Yd8=">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</latexit> 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 <latexit sha1_base64="U8OTpmJfQMfNZlgcf5DucIC0jXE=">AAAB/nicbVBNS8NAEN3Ur1q/quLJS7AInkoiih6LXjxWsLXQhLLZTNqlmw92J2IJAf+KFw+KePV3ePPfuE1z0NYHA4/3ZnZ2npcIrtCyvo3K0vLK6lp1vbaxubW9U9/d66o4lQw6LBax7HlUgeARdJCjgF4igYaegHtvfD317x9AKh5HdzhJwA3pMOIBZxS1NKgfOAiPWLyTSfDzzElGPB/UG1bTKmAuErskDVKiPah/OX7M0hAiZIIq1betBN2MSuRMQF5zUgUJZWM6hL6mEQ1BuVmxNjePteKbQSx1RWgW6u+JjIZKTUJPd4YUR2rem4r/ef0Ug0s341GSIkRstihIhYmxOc3C9LkEhmKiCWWS67+abEQlZagTq+kQ7PmTF0n3tGmfN63bs0brqoyjSg7JETkhNrkgLXJD2qRDGMnIM3klb8aT8WK8Gx+z1opRzuyTPzA+fwCYGpaJ</latexit> <latexit sha1_base64="DpKpAApI5rASTjlNhiFvmIbPB1E=">AAAB/3icbVDLSsNAFJ3UV62vqODGTbAIrkoiii6LblxWsA9oQ5lMb9qhkwczN2KJWfgrblwo4tbfcOffOE2z0NYDFw7n3Dt37vFiwRXa9rdRWlpeWV0rr1c2Nre2d8zdvZaKEsmgySIRyY5HFQgeQhM5CujEEmjgCWh74+up374HqXgU3uEkBjegw5D7nFHUUt886CE8YP5O6okEsrQXK571zapds3NYi8QpSJUUaPTNr94gYkkAITJBleo6doxuSiVyJiCr9BIFMWVjOoSupiENQLlpvjezjrUysPxI6grRytXfEykNlJoEnu4MKI7UvDcV//O6CfqXbsrDOEEI2WyRnwgLI2sahjXgEhiKiSaUSa7/arERlZShjqyiQ3DmT14krdOac16zb8+q9asijjI5JEfkhDjkgtTJDWmQJmHkkTyTV/JmPBkvxrvxMWstGcXMPvkD4/MHfCqXCw==</latexit> model H ⇢ = <latexit sha1_base64="qok9lw7LPnF9frpAN49XCubf42Y=">AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA==</latexit> <latexit sha1_base64="qok9lw7LPnF9frpAN49XCubf42Y=">AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA==</latexit> <latexit sha1_base64="qok9lw7LPnF9frpAN49XCubf42Y=">AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA==</latexit> <latexit sha1_base64="qok9lw7LPnF9frpAN49XCubf42Y=">AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA==</latexit> Cortex
63. 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 <latexit sha1_base64="xrbmyzmhPDH11ajAYU1ekas0Yd8=">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</latexit> 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 <latexit sha1_base64="U8OTpmJfQMfNZlgcf5DucIC0jXE=">AAAB/nicbVBNS8NAEN3Ur1q/quLJS7AInkoiih6LXjxWsLXQhLLZTNqlmw92J2IJAf+KFw+KePV3ePPfuE1z0NYHA4/3ZnZ2npcIrtCyvo3K0vLK6lp1vbaxubW9U9/d66o4lQw6LBax7HlUgeARdJCjgF4igYaegHtvfD317x9AKh5HdzhJwA3pMOIBZxS1NKgfOAiPWLyTSfDzzElGPB/UG1bTKmAuErskDVKiPah/OX7M0hAiZIIq1betBN2MSuRMQF5zUgUJZWM6hL6mEQ1BuVmxNjePteKbQSx1RWgW6u+JjIZKTUJPd4YUR2rem4r/ef0Ug0s341GSIkRstihIhYmxOc3C9LkEhmKiCWWS67+abEQlZagTq+kQ7PmTF0n3tGmfN63bs0brqoyjSg7JETkhNrkgLXJD2qRDGMnIM3klb8aT8WK8Gx+z1opRzuyTPzA+fwCYGpaJ</latexit> <latexit sha1_base64="DpKpAApI5rASTjlNhiFvmIbPB1E=">AAAB/3icbVDLSsNAFJ3UV62vqODGTbAIrkoiii6LblxWsA9oQ5lMb9qhkwczN2KJWfgrblwo4tbfcOffOE2z0NYDFw7n3Dt37vFiwRXa9rdRWlpeWV0rr1c2Nre2d8zdvZaKEsmgySIRyY5HFQgeQhM5CujEEmjgCWh74+up374HqXgU3uEkBjegw5D7nFHUUt886CE8YP5O6okEsrQXK571zapds3NYi8QpSJUUaPTNr94gYkkAITJBleo6doxuSiVyJiCr9BIFMWVjOoSupiENQLlpvjezjrUysPxI6grRytXfEykNlJoEnu4MKI7UvDcV//O6CfqXbsrDOEEI2WyRnwgLI2sahjXgEhiKiSaUSa7/arERlZShjqyiQ3DmT14krdOac16zb8+q9asijjI5JEfkhDjkgtTJDWmQJmHkkTyTV/JmPBkvxrvxMWstGcXMPvkD4/MHfCqXCw==</latexit> model H ⇢ = <latexit sha1_base64="qok9lw7LPnF9frpAN49XCubf42Y=">AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA==</latexit> <latexit sha1_base64="qok9lw7LPnF9frpAN49XCubf42Y=">AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA==</latexit> <latexit sha1_base64="qok9lw7LPnF9frpAN49XCubf42Y=">AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA==</latexit> <latexit sha1_base64="qok9lw7LPnF9frpAN49XCubf42Y=">AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA==</latexit> Cortex <latexit sha1_base64="nGXfS5XSNpfoRoMwlS4Q+lOeS/4=">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</latexit> ⌃A = ↵ (r )(r )
64. 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 <latexit sha1_base64="xrbmyzmhPDH11ajAYU1ekas0Yd8=">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</latexit> 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 <latexit sha1_base64="U8OTpmJfQMfNZlgcf5DucIC0jXE=">AAAB/nicbVBNS8NAEN3Ur1q/quLJS7AInkoiih6LXjxWsLXQhLLZTNqlmw92J2IJAf+KFw+KePV3ePPfuE1z0NYHA4/3ZnZ2npcIrtCyvo3K0vLK6lp1vbaxubW9U9/d66o4lQw6LBax7HlUgeARdJCjgF4igYaegHtvfD317x9AKh5HdzhJwA3pMOIBZxS1NKgfOAiPWLyTSfDzzElGPB/UG1bTKmAuErskDVKiPah/OX7M0hAiZIIq1betBN2MSuRMQF5zUgUJZWM6hL6mEQ1BuVmxNjePteKbQSx1RWgW6u+JjIZKTUJPd4YUR2rem4r/ef0Ug0s341GSIkRstihIhYmxOc3C9LkEhmKiCWWS67+abEQlZagTq+kQ7PmTF0n3tGmfN63bs0brqoyjSg7JETkhNrkgLXJD2qRDGMnIM3klb8aT8WK8Gx+z1opRzuyTPzA+fwCYGpaJ</latexit> <latexit sha1_base64="DpKpAApI5rASTjlNhiFvmIbPB1E=">AAAB/3icbVDLSsNAFJ3UV62vqODGTbAIrkoiii6LblxWsA9oQ5lMb9qhkwczN2KJWfgrblwo4tbfcOffOE2z0NYDFw7n3Dt37vFiwRXa9rdRWlpeWV0rr1c2Nre2d8zdvZaKEsmgySIRyY5HFQgeQhM5CujEEmjgCWh74+up374HqXgU3uEkBjegw5D7nFHUUt886CE8YP5O6okEsrQXK571zapds3NYi8QpSJUUaPTNr94gYkkAITJBleo6doxuSiVyJiCr9BIFMWVjOoSupiENQLlpvjezjrUysPxI6grRytXfEykNlJoEnu4MKI7UvDcV//O6CfqXbsrDOEEI2WyRnwgLI2sahjXgEhiKiSaUSa7/arERlZShjqyiQ3DmT14krdOac16zb8+q9asijjI5JEfkhDjkgtTJDWmQJmHkkTyTV/JmPBkvxrvxMWstGcXMPvkD4/MHfCqXCw==</latexit> model H ⇢ = <latexit sha1_base64="qok9lw7LPnF9frpAN49XCubf42Y=">AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA==</latexit> <latexit sha1_base64="qok9lw7LPnF9frpAN49XCubf42Y=">AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA==</latexit> <latexit sha1_base64="qok9lw7LPnF9frpAN49XCubf42Y=">AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA==</latexit> <latexit sha1_base64="qok9lw7LPnF9frpAN49XCubf42Y=">AAACCHicbVDLSsNAFJ34rPVVdenCYBHcWBIRdCMU3bisYB/QlDKZ3LRDJ8kwcyOWkKUbf8WNC0Xc+gnu/Bunj4W2HrhwOOfemXuPLwXX6Djf1sLi0vLKamGtuL6xubVd2tlt6CRVDOosEYlq+VSD4DHUkaOAllRAI19A0x9cj/zmPSjNk/gOhxI6Ee3FPOSMopG6pQMP4QHH72QKgjzzVD+59GSfn3hS87xbKjsVZwx7nrhTUiZT1LqlLy9IWBpBjExQrduuI7GTUYWcCciLXqpBUjagPWgbGtMIdCcbL5DbR0YJ7DBRpmK0x+rviYxGWg8j33RGFPt61huJ/3ntFMOLTsZjmSLEbPJRmAobE3uUih1wBQzF0BDKFDe72qxPFWVosiuaENzZk+dJ47TiOhX39qxcvZrGUSD75JAcE5eckyq5ITVSJ4w8kmfySt6sJ+vFerc+Jq0L1nRmj/yB9fkDFcqaoA==</latexit> Cortex Active stress couples the two models at the interface ΣA Fluid fl ow depends on the sign of the active stress ΣA <latexit sha1_base64="nGXfS5XSNpfoRoMwlS4Q+lOeS/4=">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</latexit> ⌃A = ↵ (r )(r )
65. RS, Tjhung and Cates, Phys Rev Research 2020 19 Active droplet: propulsion mechanism
66. symmetry breaking (rotational) by initial perturbation or noise RS, Tjhung and Cates, Phys Rev Research 2020 19 Active droplet: propulsion mechanism
67. 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
68. 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 <latexit sha1_base64="nGXfS5XSNpfoRoMwlS4Q+lOeS/4=">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</latexit> ⌃A = ↵ (r )(r )
69. Active droplet: simulations 20 RS, Tjhung and Cates, Phys Rev Research 2020
70. Active droplet: simulations 20 RS, Tjhung and Cates, Phys Rev Research 2020
71. Active droplet: simulations 20 RS, Tjhung and Cates, Phys Rev Research 2020
72. 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. <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> 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
73. 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. <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> 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
74. 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. <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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==</latexit> 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 . <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">AAACvnicjVFdS8MwFE3r9/ya+uhLcAg+6GhFUJDB0BcfFdwU1jFus3SLSz9M0rFS+icFH/w3plvVuQnzQsjJuffkJue6EWdSWdaHYS4tr6yurW+UNre2d3bLe/tNGcaC0AYJeSieXZCUs4A2FFOcPkeCgu9y+uQObvP805AKycLgUSURbfvQC5jHCChNdcrvw046ymqOJ4CkTgRCMeDOUO99qiD7pnCSnTqvMXSxFiRZ7WyxYjSlEF8t7EzjhVpncv7RF8R/2oqsWuqUK1bVGgeeB3YBKqiI+075zemGJPZpoAgHKVu2Fal2ml9JOM1KTixpBGQAPdrSMACfynY6tj/Dx5rpYi8UegUKj9lpRQq+lInv6kofVF/O5nLyr1wrVt5VO2VBFCsakEkjL+ZYhTifJe4yQYniiQZABNNvxaQP2iClJ56bYM9+eR40z6u2VbUfLir1m8KOdXSIjtAJstElqqM7dI8aiBjXBhgvxsCsm57pm+Gk1DQKzQH6FeboE0Je4SI=</latexit> <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> #(r, ✓) = f(r) sin ✓ <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> stream function r4# = 0 <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit> <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit> <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit> <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit>
75. 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. <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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==</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> 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 . <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">AAACvnicjVFdS8MwFE3r9/ya+uhLcAg+6GhFUJDB0BcfFdwU1jFus3SLSz9M0rFS+icFH/w3plvVuQnzQsjJuffkJue6EWdSWdaHYS4tr6yurW+UNre2d3bLe/tNGcaC0AYJeSieXZCUs4A2FFOcPkeCgu9y+uQObvP805AKycLgUSURbfvQC5jHCChNdcrvw046ymqOJ4CkTgRCMeDOUO99qiD7pnCSnTqvMXSxFiRZ7WyxYjSlEF8t7EzjhVpncv7RF8R/2oqsWuqUK1bVGgeeB3YBKqiI+075zemGJPZpoAgHKVu2Fal2ml9JOM1KTixpBGQAPdrSMACfynY6tj/Dx5rpYi8UegUKj9lpRQq+lInv6kofVF/O5nLyr1wrVt5VO2VBFCsakEkjL+ZYhTifJe4yQYniiQZABNNvxaQP2iClJ56bYM9+eR40z6u2VbUfLir1m8KOdXSIjtAJstElqqM7dI8aiBjXBhgvxsCsm57pm+Gk1DQKzQH6FeboE0Je4SI=</latexit> <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> #(r, ✓) = f(r) sin ✓ <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> stream function r4# = 0 <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit> <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit> <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit> <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit>
76. 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. <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> 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 <latexit sha1_base64="WXJYAI40qotJO8pIC0/hXjn3k5E=">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</latexit> 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 . <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">AAACvnicjVFdS8MwFE3r9/ya+uhLcAg+6GhFUJDB0BcfFdwU1jFus3SLSz9M0rFS+icFH/w3plvVuQnzQsjJuffkJue6EWdSWdaHYS4tr6yurW+UNre2d3bLe/tNGcaC0AYJeSieXZCUs4A2FFOcPkeCgu9y+uQObvP805AKycLgUSURbfvQC5jHCChNdcrvw046ymqOJ4CkTgRCMeDOUO99qiD7pnCSnTqvMXSxFiRZ7WyxYjSlEF8t7EzjhVpncv7RF8R/2oqsWuqUK1bVGgeeB3YBKqiI+075zemGJPZpoAgHKVu2Fal2ml9JOM1KTixpBGQAPdrSMACfynY6tj/Dx5rpYi8UegUKj9lpRQq+lInv6kofVF/O5nLyr1wrVt5VO2VBFCsakEkjL+ZYhTifJe4yQYniiQZABNNvxaQP2iClJ56bYM9+eR40z6u2VbUfLir1m8KOdXSIjtAJstElqqM7dI8aiBjXBhgvxsCsm57pm+Gk1DQKzQH6FeboE0Je4SI=</latexit> <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> #(r, ✓) = f(r) sin ✓ <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> stream function r4# = 0 <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit> <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit> <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit> <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit>
77. 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. <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> 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 <latexit sha1_base64="WXJYAI40qotJO8pIC0/hXjn3k5E=">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</latexit> fext = A r + Br + Cr ln r + Dr3 <latexit sha1_base64="K5fNFMGNfwV2HMOSYp/mjxvkcdI=">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</latexit> 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 . <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> #(r, ✓) = f(r) sin ✓ <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> stream function r4# = 0 <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit> <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit> <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit> <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit>
78. 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. <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> 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 <latexit sha1_base64="WXJYAI40qotJO8pIC0/hXjn3k5E=">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</latexit> fext = A r + Br + Cr ln r + Dr3 <latexit sha1_base64="K5fNFMGNfwV2HMOSYp/mjxvkcdI=">AAACOHicbVBBaxNBFJ5tbRtjbVM9ehkMgiCEXVuxl0JMevBmCiYtZLdhdvK2HTI7u8y8LQ3L/Kxe/BnepBcPivTaX9DZZBVt/WDge9/3Hm/eF+dSGPT9b97K6qO19Y3G4+aTzadb262dZyOTFZrDkGcy0ycxMyCFgiEKlHCSa2BpLOE4nvUr//gCtBGZ+ozzHKKUnSmRCM7QSZPWp+S0DBEukZZCobUHNEw0404TcgrlB2tLbd/UVc/q37RvdSgV/VMfWn1a7trmpNX2O/4C9CEJatImNQaT1tdwmvEiBYVcMmPGgZ9jVDKNgkuwzbAwkDM+Y2cwdlSxFExULg639JVTpjTJtHsK6UL9e6JkqTHzNHadKcNzc9+rxP954wKT/cglkhcIii8XJYWkmNEqRToVGjjKuSOMa+H+Svk5c8Ghy7oKIbh/8kMyetsJ3nX8o712t1fH0SAvyEvymgTkPemSj2RAhoSTK3JNfpCf3hfvu/fLu1m2rnj1zHPyD7zbO3wvrpA=</latexit> 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 . <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> <latexit 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sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> #(r, ✓) = f(r) sin ✓ <latexit 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sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> stream function r4# = 0 <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit> <latexit 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79. 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. <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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==</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> 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 <latexit sha1_base64="WXJYAI40qotJO8pIC0/hXjn3k5E=">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</latexit> fext = A r + Br + Cr ln r + Dr3 <latexit sha1_base64="K5fNFMGNfwV2HMOSYp/mjxvkcdI=">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</latexit> 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 <latexit sha1_base64="7vNZJLFXtdJgi1A4prgGzjN4jms=">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</latexit> U = 16⌘ = 0↵ 16⌘ RS, Tjhung and Cates, Phys Rev Research 2020 vx = @# @y , vy = @# @x , vr = 1 r @# @✓ , v✓ = @# @r . <latexit 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sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> #(r, ✓) = f(r) sin ✓ <latexit 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sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> stream function r4# = 0 <latexit 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80. 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. <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> <latexit sha1_base64="Uw7PEdVyMGMl/1v+zk77fWah96E=">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</latexit> 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 <latexit sha1_base64="WXJYAI40qotJO8pIC0/hXjn3k5E=">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</latexit> fext = A r + Br + Cr ln r + Dr3 <latexit sha1_base64="K5fNFMGNfwV2HMOSYp/mjxvkcdI=">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</latexit> 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 <latexit sha1_base64="7vNZJLFXtdJgi1A4prgGzjN4jms=">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</latexit> U = 16⌘ = 0↵ 16⌘ RS, Tjhung and Cates, Phys Rev Research 2020 vx = @# @y , vy = @# @x , vr = 1 r @# @✓ , v✓ = @# @r . <latexit 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sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> <latexit sha1_base64="hKNy4FgjBiHSVFiFJmXdkr8j76U=">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</latexit> #(r, ✓) = f(r) sin ✓ <latexit 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sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> <latexit sha1_base64="sN7KLzHZPaKKT7Sd0X19Ye3/d4M=">AAACC3icbVDLSsNAFJ3UV62vqEs3oUVoQUoigm6EohuXFewDmlAm00k7dDIJMzeFErp346+4caGIW3/AnX/jNM1CWw8MnHvOvdy5x485U2Db30ZhbX1jc6u4XdrZ3ds/MA+P2ipKJKEtEvFIdn2sKGeCtoABp91YUhz6nHb88e3c70yoVCwSDzCNqRfioWABIxi01DfL7gRLGFHAVXnmZqR2HVRlzVVMLOq+WbHrdgZrlTg5qaAczb755Q4ikoRUAOFYqZ5jx+Cleg8jnM5KbqJojMkYD2lPU4FDqrw0u2VmnWplYAWR1E+Alam/J1IcKjUNfd0ZYhipZW8u/uf1EgiuvJSJOAEqyGJRkHALImsejDVgkhLgU00wkUz/1SIjLDEBHV9Jh+Asn7xK2ud1x6479xeVxk0eRxGdoDKqIgddoga6Q03UQgQ9omf0it6MJ+PFeDc+Fq0FI585Rn9gfP4AW32amQ==</latexit> stream function r4# = 0 <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit> <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit> <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit> <latexit sha1_base64="BGLmSROIXCFnj0F+OZGyYJxm5Ek=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJVECroRim5cVrAPaGK5mU7aoZNJmJkUSuzCX3HjQhG3/oY7/8ZJ24W2HrhwOOfemXtPkHCmtON8W4WV1bX1jeJmaWt7Z3fP3j9oqjiVhDZIzGPZDkBRzgRtaKY5bSeSQhRw2gqGN7nfGlGpWCzu9TihfgR9wUJGQBupax95AgIOD1l14o1A6gHVcOWUunbZqThT4GXizkkZzVHv2l9eLyZpRIUmHJTquE6i/cy8yAink5KXKpoAGUKfdgwVEFHlZ9P9J/jUKD0cxtKU0Hiq/p7IIFJqHAWmMwI9UIteLv7ndVIdXvoZE0mqqSCzj8KUYx3jPAzcY5ISzceGAJHM7IrJACQQbSLLQ3AXT14mzfOK61Tcu2q5dj2Po4iO0Qk6Qy66QDV0i+qogQh6RM/oFb1ZT9aL9W59zFoL1nzmEP2B9fkDTnWVnQ==</latexit> depends on surface gradient and does not depend on radius of particle
81. 22 Active droplet: theory and simulations RS, Tjhung and Cates, Phys Rev Research 2020 (c) (d)
82. Ramaswamy, Adv. in Phys. 2001; RS, Tjhung and Cates, Phys Rev Research 2020 Many active droplets <latexit sha1_base64="6LcMrs+OYZ1GcthqvydaZD3ihew=">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</latexit> 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.
83. Ramaswamy, Adv. in Phys. 2001; RS, Tjhung and Cates, Phys Rev Research 2020 Many active droplets <latexit sha1_base64="6LcMrs+OYZ1GcthqvydaZD3ihew=">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</latexit> 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
84. Ramaswamy, Adv. in Phys. 2001; RS, Tjhung and Cates, Phys Rev Research 2020 Many active droplets <latexit sha1_base64="6LcMrs+OYZ1GcthqvydaZD3ihew=">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</latexit> 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
85. Ramaswamy, Adv. in Phys. 2001; RS, Tjhung and Cates, Phys Rev Research 2020 Many active droplets <latexit sha1_base64="6LcMrs+OYZ1GcthqvydaZD3ihew=">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</latexit> 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.
86. 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