Theories without time-reversal symmetry in soft matter

Online link to view animation on slide 12: https://github.com/rajeshrinet/pystokes-misc/blob/master/gallery/droplets/ssi.gif

Online link to animation on slide 30:
https://github.com/rajeshrinet/pystokes-misc/blob/master/gallery/optofluidic_crystals.gif

Seminar in the Department of Physics, IIT Madras

Transcript

1. Rajesh Singh Department of Applied Mathematics and Theoretical Physics Theories without time-reversal symmetry in soft matter
2. Plan of the talk Experimental observation of incomplete phase separation in active matter Theory of binary fluid mixtures (how do oil-water mixtures phase separate?) Theory of active phase separation (microphase separation) Generalised Stokes laws of colloidal particles with slip (active particles) Phoresis and Stokesian hydrodynamic of active particles How to freeze colloids by heating them? Field-theoretic Particle-based 3 Outlook: field-theoretic and particle-based theories of active matter
3. This work has been done with Michael E. Cates I. Incomplete phase separation in scalar active matter
4. micrometer size - non-equilibrium processes on the surface create exterior flow and may lead to self-propulsion feeding or fuel => break time-reversal symmetry locally Active particles: special colloids 5 Microorganisms Autophoretic particles Ramaswamy Annu. Rev. Condens. Matter Phys. 2010, JSTAT 2017; Cates arXiv:1904.01330 Active matter: active particles in a fluid
5. micrometer size - non-equilibrium processes on the surface create exterior flow and may lead to self-propulsion feeding or fuel => break time-reversal symmetry locally Active particles: special colloids 5 Particle-level dynamics of active particles, unlike Brownian colloids, has no TRS => no inherent Free energy or Boltzmann distribution nonequilibrium steady-state A B C Active particles equilibrium steady-state zero current, TRS Passive particles A B C net current, no TRS Microorganisms Autophoretic particles Ramaswamy Annu. Rev. Condens. Matter Phys. 2010, JSTAT 2017; Cates arXiv:1904.01330 Active matter: active particles in a fluid
6. micrometer size - non-equilibrium processes on the surface create exterior flow and may lead to self-propulsion feeding or fuel => break time-reversal symmetry locally Active particles: special colloids 5 Particle-level dynamics of active particles, unlike Brownian colloids, has no TRS => no inherent Free energy or Boltzmann distribution nonequilibrium steady-state A B C Active particles equilibrium steady-state zero current, TRS Passive particles A B C net current, no TRS Microorganisms Autophoretic particles How to study such nonequilibrium systems in absence of time- reversal symmetry for the particle-level dynamics? Ramaswamy Annu. Rev. Condens. Matter Phys. 2010, JSTAT 2017; Cates arXiv:1904.01330 Active matter: active particles in a fluid
7. Incomplete active phase separations 6 Buttinoni et al PRL 2013 SiO2 beads and a sputtering a thin layer of graphite onto one hemisphere in H2O2 Theurkauff et al PRL 2012 gold-platinum Janus particles in H2O2 Palacci et al Science 2013 A bimaterial colloid: polymer sphere with a hematite cube (dark) in H2O2
8. Incomplete active phase separations 6 Buttinoni et al PRL 2013 SiO2 beads and a sputtering a thin layer of graphite onto one hemisphere in H2O2 Theurkauff et al PRL 2012 gold-platinum Janus particles in H2O2 Palacci et al Science 2013 A bimaterial colloid: polymer sphere with a hematite cube (dark) in H2O2 How to build a theory of such phase separations?
9. Incomplete active phase separations 6 Buttinoni et al PRL 2013 SiO2 beads and a sputtering a thin layer of graphite onto one hemisphere in H2O2 Theurkauff et al PRL 2012 gold-platinum Janus particles in H2O2 Palacci et al Science 2013 A bimaterial colloid: polymer sphere with a hematite cube (dark) in H2O2 How to build a theory of such phase separations? symmetries and conservation laws experiments are for spherical particles - use a scalar field number of active particles is conserved - use a conserved scalar field
10. Incomplete active phase separations 6 Buttinoni et al PRL 2013 SiO2 beads and a sputtering a thin layer of graphite onto one hemisphere in H2O2 Theurkauff et al PRL 2012 gold-platinum Janus particles in H2O2 Palacci et al Science 2013 A bimaterial colloid: polymer sphere with a hematite cube (dark) in H2O2 How to build a theory of such phase separations? symmetries and conservation laws experiments are for spherical particles - use a scalar field number of active particles is conserved - use a conserved scalar field start with a theory of phase separation in binary mixtures add minimal terms which break TRS recipe to build an active field theory
11. 7 How do binary mixtures (such as oil-water) phase separate from a homogenous mixed phase? Theory of coarsening kinetics in binary fluid mixtures Bray, Adv. in Phys. 43, 357 (1994)
12. 7 How do binary mixtures (such as oil-water) phase separate from a homogenous mixed phase? Theory of coarsening kinetics in binary fluid mixtures Consider a scalar field describing the local composition in a symmetric binary mixture <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. 7 How do binary mixtures (such as oil-water) phase separate from a homogenous mixed phase? Theory of coarsening kinetics in binary fluid mixtures Consider a scalar field describing the local composition in a symmetric binary mixture <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 field is conserved: ˙ = r · J Bray, Adv. in Phys. 43, 357 (1994)
14. 7 How do binary mixtures (such as oil-water) phase separate from a homogenous mixed phase? Theory of coarsening kinetics in binary fluid mixtures Consider a scalar field describing the local composition in a symmetric binary mixture <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 flow diffusive 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 field is conserved: ˙ = r · J Bray, Adv. in Phys. 43, 357 (1994)
15. φ4 field 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 flow the fluid flow acts to minimise the deformation of the droplet 8 Hohenberg & Halperin 1977; Bray, Adv in Phys 1994; Kendon et al, JFM 2001; Cates and Tjhung, JFM 2018
16. φ4 field 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 flow the fluid flow acts to minimise the deformation of the droplet 8 Hohenberg & Halperin 1977; Bray, Adv in Phys 1994; Kendon et al, JFM 2001; Cates and Tjhung, JFM 2018 <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 field 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 flow the fluid flow acts to minimise the deformation of the droplet 8 <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 field 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=">AAADlHiclVJNbxMxEHV2+SjhKwGJC5cVEVIqSLRbgcqBSilVJbigopK2UpxGs14nseL96Ho2InL9h/g53Pg3OMkekjQ5MJLl5/dmnscjh5kUCn3/b8Vx791/8HDvUfXxk6fPntfqLy5UWuSMd1kq0/wqBMWlSHgXBUp+leUc4lDyy3ByMtcvpzxXIk1+4izj/RhGiRgKBmipQb3yu0XDVEZqFttN0wRCCSZ7RznC8nCtD8xqytQcbSuhLEpxjT8XoxjMtT41FPkvXPSqcx4Z/X/1x8ZQusPZtjKBLINV+dy8pzcFRJu37jA/ajUpChlxvbQypeX+uqc1pesM5TeFmDa3vSUbi/2dQoviMAemA6Mjc7sj63Zz7N9MdVBr+G1/Ed5dEJSgQco4G9T+0ChlRcwTZBKU6gV+hn0NOQomuanSQvEM2ARGvGdhAjFXfb0YmfHeWibyhmluV4Legl2t0BCreW82MwYcq01tTm7TegUOP/W1SLICecKWFw0L6WHqzX+oF4mcM5QzC4DlwvbqsTHYiaH9x/MhBJtPvgsuDtrBx7b/40Oj86Ucxx55Td6QJgnIIemQr+SMdAlz6s6h03GO3VfuZ/fEPV2mOpWy5iVZC/f7P6QyNjI=</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I no flow the fluid flow acts to minimise the deformation of the droplet 8 <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 field 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 flow the fluid flow acts to minimise the deformation of the droplet 8 <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 field 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 flow the fluid flow acts to minimise the deformation of the droplet 8 <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 field 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 flow the fluid flow acts to minimise the deformation of the droplet 8 <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. 9 Model H: Ostwald ripening Chemical potential raised at curved interface μ s <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. 9 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 finite system μ ∞ = μ s Chemical potential raised at curved interface μ s <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. 9 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 finite system μ ∞ = μ s Chemical potential raised at curved interface μ s <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. 9 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 finite system μ ∞ = μ s Chemical potential raised at curved interface μ s <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 diffusion <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=">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</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. 10 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. 10 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 fluid flow even in absence of external forces extensile/contractile dipolar flow along the swimming axis Ramaswamy, Annu. Rev. Condens. Matter Phys. 2010 ˙ = r · J Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤,
28. φ4 field theory of active phase separation RS et al. PRL 2019, PRR 2020; Cates and Tjhung JFM 2018; Tiribocchi et al. PRL 2015 Stokes equation <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⇤, 11
29. φ4 field theory of active phase separation RS et al. PRL 2019, PRR 2020; Cates and Tjhung JFM 2018; Tiribocchi et al. PRL 2015 ⌃A = (˜  )S <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⇤, 11
30. φ4 field theory of active phase separation RS et al. PRL 2019, PRR 2020; Cates and Tjhung JFM 2018; Tiribocchi et al. PRL 2015 ⌃A = (˜  )S <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=">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</latexit> J = rµ + v + p 2D⇤, 11
31. Incomplete phase separation ˜ κ < 0 Complete phase separation ˜ κ > 0 Active model H: nucleation and growth 12 -1 1 RS and Cates PRL (2019)
32. Incomplete phase separation ˜ κ < 0 Complete phase separation ˜ κ > 0 Active model H: nucleation and growth 12 -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 ⟹ γ v < 0 RS and Cates PRL (2019) The spontaneous stretching motion results in SSI => splitting the droplet 13
34. Active model H: self-shearing instability (SSI) A spontaneous stretching motion of the interface A contractile active stress ˜ κ < 0 ⟹ γ v < 0 SSI RS and Cates PRL (2019) The spontaneous stretching motion results in SSI => splitting the droplet 13
35. Active model H: self-shearing instability (SSI) A spontaneous stretching motion of the interface A contractile active stress ˜ κ < 0 ⟹ γ v < 0 SSI Ostwald ripening RS and Cates PRL (2019) The spontaneous stretching motion results in SSI => splitting the droplet 13
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 ⟹ γ v < 0 SSI Ostwald ripening RS and Cates PRL (2019) The spontaneous stretching motion results in SSI => splitting the droplet 13
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/ 14
38. Length scale determined by activity SSI Ostwald ripening RS and Cates PRL (2019) From the mechanical tension and fluid viscosity , we can construct just one quantity with the dimensions of velocity γ v η Vv = v/⌘ ˙ = r · J Active Model H <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/ 14
39. Length scale determined by activity SSI Ostwald ripening RS and Cates PRL (2019) From the mechanical tension and fluid viscosity , we can construct just one quantity with the dimensions of velocity γ v η Vv = v/⌘ Ostwald process gives another rate V ( ¯ R) = ˙ ¯ R / M / 2 B ¯ R2 ˙ = r · J Active Model H <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/ 14
40. Length scale determined by activity SSI Ostwald ripening RS and Cates PRL (2019) From the mechanical tension and fluid viscosity , we can construct just one quantity with the dimensions of velocity γ v η Vv = v/⌘ Ostwald process gives another rate V ( ¯ R) = ˙ ¯ R / M / 2 B ¯ R2 ¯ R / ✓ v 2 B ⌘M ◆ 1/2 ⇠ v 1/2 Balancing the two rates, we get scaling for droplet size ˙ = r · J Active Model H <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/ 14
41. Length scale determined by activity SSI Ostwald ripening RS and Cates PRL (2019) From the mechanical tension and fluid viscosity , we can construct just one quantity with the dimensions of velocity γ v η Vv = v/⌘ Ostwald process gives another rate V ( ¯ R) = ˙ ¯ R / M / 2 B ¯ R2 ¯ R / ✓ v 2 B ⌘M ◆ 1/2 ⇠ v 1/2 Balancing the two rates, we get scaling for droplet size ˙ = r · J Active Model H <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/ 14
42. 15 Activity in diffusive sector Cates and Tjhung JFM 2018; Tjhung et al PRX 2018 ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. <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> <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> 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=">AAACSXicbVBNTxRBEO1ZVHD9YMUjl4kbjQm6mSEYuZAQ4WCMB0xcINnerDU9NdChp2fsriEsnf57XLx58z948aAxnuxZ9oBgJZ1+efX16mW1kpaS5FvUWbh1+87i0t3uvfsPHi73Hq3s26oxAoeiUpU5zMCikhqHJEnhYW0QykzhQXay0+YPTtFYWemPNK1xXMKRloUUQIGa9D7xrFK5nZbhc+/8s62XjmvIFHheNmuc8IxmS5zB3DteAh1nRSicOH6OBN6vcfvZkFvf9Vcn8fdBQg7+RXfS6yeDZBbxTZDOQZ/NY2/S+8rzSjQlahIKrB2lSU1jB4akUOi7vLFYgziBIxwFqKFEO3YzkT5+Gpg8LioTnqZ4xl7tcFDaVmKobE+x13Mt+b/cqKFic+ykrhtCLS4XFY2KqYpbW+NcGhSkpgGAMDJojcUxGBAUzG9NSK+ffBPsrw/SV4Pkw0Z/+83cjiW2yp6w5yxlr9k2e8v22JAJdsG+s5/sV/Ql+hH9jv5clnaiec9j9k90Fv4Cc2m1jw==</latexit> J = rµ + J⇣ + p 2D⇤,
43. 15 Activity in diffusive sector Cates and Tjhung JFM 2018; Tjhung et al PRX 2018 ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. <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 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. 15 Activity in diffusive sector Cates and Tjhung JFM 2018; Tjhung et al PRX 2018 ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. <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⇤, effective 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. 15 Activity in diffusive sector Cates and Tjhung JFM 2018; Tjhung et al PRX 2018 ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. <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⇤, effective 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: μ s <latexit sha1_base64="iU+JbNDhU90L9SncVWcfxeqtL+Y=">AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT</latexit> J⇣ = rµ⇣ + r ⇥ A
46. 15 Activity in diffusive sector Cates and Tjhung JFM 2018; Tjhung et al PRX 2018 The active contributions can reverse the sign of chemical flux and thus the sign of effective tension => reverse Ostwald ripening ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. <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="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, effective 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: μ s <latexit sha1_base64="iU+JbNDhU90L9SncVWcfxeqtL+Y=">AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT</latexit> J⇣ = rµ⇣ + r ⇥ A
47. 15 Activity in diffusive sector Cates and Tjhung JFM 2018; Tjhung et al PRX 2018 The active contributions can reverse the sign of chemical flux and thus the sign of effective tension => reverse Ostwald ripening effective tension in diffusive sector <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 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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 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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⇤, effective 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: μ s <latexit sha1_base64="iU+JbNDhU90L9SncVWcfxeqtL+Y=">AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT</latexit> J⇣ = rµ⇣ + r ⇥ A
48. 15 Activity in diffusive sector Cates and Tjhung JFM 2018; Tjhung et al PRX 2018 The active contributions can reverse the sign of chemical flux and thus the sign of effective tension => reverse Ostwald ripening effective tension in diffusive sector <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 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⇤, effective 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: μ s <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 γ ϕ > 0 Reverse Ostwald ripening γ ϕ < 0
49. 16 Steady-states in the plane of effective 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. 16 Steady-states in the plane of effective tensions ˙ = r · J Active Model H <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, diffusive 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. 16 Steady-states in the plane of effective tensions ˙ = r · J Active Model H <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, diffusive 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 field theories is a generic consequence of an effective interfacial tension - mechanical (causing flow) or diffusive (causing Ostwald ripening) - becoming negative. This is achieved by adding non-local terms, which break TRS, to equilibrium model H RS and Cates PRL (2019)
52. 17 II. Phoresis and Stokesian hydrodynamics of active particles Michael E. Cates Ronojoy Adhikari
53. 18 Nonequilibrium processes - in a thin layer on the surface drive exterior fluid flow - diverse mechanisms. The resulting fluid stress may react back and self-propel Feeding or fuel => break time-reversal symmetry locally nonequilibrium steady-state A B C Active particles equilibrium steady-state zero current, TRS Passive particles A B C net current, no TRS Active particles: special colloids Microorganisms Autophoretic particles Ramaswamy Annu. Rev. Condens. Matter Phys. 2010, JSTAT 2017; Cates arXiv:1904.01330 Particle-level dynamics of active particles, unlike Brownian colloids, has no time-reversal symmetry => no inherent Free energy or Boltzmann distribution Active matter: active particles in a fluid How to study active matter systems in absence of time- reversal symmetry for particle-level dynamics?
54. Stokes law Micrometer size => neglect inertia Newton’s equation becomes FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic 19
55. Stokes law Micrometer size => neglect inertia Newton’s equation becomes FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic Stokes law for a no-slip sphere <latexit sha1_base64="MObKdaVqbh7UWWwNBggtIoJRMLc=">AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0Y0nE10YoCtJlBfuAJobJdNIOnUzCzEQooZ/gxl9x40IRty7d+TdO2gjaeuDC4Zx7ufceP2ZUKsv6Mgpz8wuLS8Xl0srq2vqGubnVlFEiMGngiEWi7SNJGOWkoahipB0LgkKfkZY/uMr81j0Rkkb8Vg1j4oaox2lAMVJa8sx9J0Sq7wfw2qN3NXgBD0+dmDpEIejDH6/pUc8sWxVrDDhL7JyUQY66Z3463QgnIeEKMyRlx7Zi5aZIKIoZGZWcRJIY4QHqkY6mHIVEuun4oRHc00oXBpHQxRUcq78nUhRKOQx93ZmdKKe9TPzP6yQqOHdTyuNEEY4ni4KEQRXBLB3YpYJgxYaaICyovhXiPhIIK51hSYdgT788S5pHFfukYt0cl6uXeRxFsAN2wQGwwRmoghqogwbA4AE8gRfwajwaz8ab8T5pLRj5zDb4A+PjG886mzA=</latexit> FH i = 6⇡⌘bV i <latexit sha1_base64="5Laazrb3aq86EUiJDLep8vA/DVk=">AAACFXicbVDLSsNAFJ3UV62vqEs3g0UQ1JKIr41QFMRlBfuAJobJdNIOnUzCzEQooT/hxl9x40IRt4I7/8ZJG0RbDwycOede7r3HjxmVyrK+jMLM7Nz8QnGxtLS8srpmrm80ZJQITOo4YpFo+UgSRjmpK6oYacWCoNBnpOn3LzO/eU+EpBG/VYOYuCHqchpQjJSWPHMfHpw4MXWIQtCHTohUzw9gw6Nw7+d35dG7GjyHlmeWrYo1Apwmdk7KIEfNMz+dToSTkHCFGZKybVuxclMkFMWMDEtOIkmMcB91SVtTjkIi3XR01RDuaKUDg0joxxUcqb87UhRKOQh9XZktKie9TPzPaycqOHNTyuNEEY7Hg4KEQRXBLCLYoYJgxQaaICyo3hXiHhIIKx1kSYdgT548TRqHFfu4Yt0clasXeRxFsAW2wS6wwSmogmtQA3WAwQN4Ai/g1Xg0no03431cWjDynk3wB8bHN+sPnCU=</latexit> 6⇡⌘bVi + FP i = 0 19
56. Stokes law Micrometer size => neglect inertia Newton’s equation becomes FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic Stokes law for a no-slip sphere <latexit sha1_base64="MObKdaVqbh7UWWwNBggtIoJRMLc=">AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0Y0nE10YoCtJlBfuAJobJdNIOnUzCzEQooZ/gxl9x40IRty7d+TdO2gjaeuDC4Zx7ufceP2ZUKsv6Mgpz8wuLS8Xl0srq2vqGubnVlFEiMGngiEWi7SNJGOWkoahipB0LgkKfkZY/uMr81j0Rkkb8Vg1j4oaox2lAMVJa8sx9J0Sq7wfw2qN3NXgBD0+dmDpEIejDH6/pUc8sWxVrDDhL7JyUQY66Z3463QgnIeEKMyRlx7Zi5aZIKIoZGZWcRJIY4QHqkY6mHIVEuun4oRHc00oXBpHQxRUcq78nUhRKOQx93ZmdKKe9TPzP6yQqOHdTyuNEEY4ni4KEQRXBLB3YpYJgxYaaICyovhXiPhIIK51hSYdgT788S5pHFfukYt0cl6uXeRxFsAN2wQGwwRmoghqogwbA4AE8gRfwajwaz8ab8T5pLRj5zDb4A+PjG886mzA=</latexit> FH i = 6⇡⌘bV i <latexit sha1_base64="5Laazrb3aq86EUiJDLep8vA/DVk=">AAACFXicbVDLSsNAFJ3UV62vqEs3g0UQ1JKIr41QFMRlBfuAJobJdNIOnUzCzEQooT/hxl9x40IRt4I7/8ZJG0RbDwycOede7r3HjxmVyrK+jMLM7Nz8QnGxtLS8srpmrm80ZJQITOo4YpFo+UgSRjmpK6oYacWCoNBnpOn3LzO/eU+EpBG/VYOYuCHqchpQjJSWPHMfHpw4MXWIQtCHTohUzw9gw6Nw7+d35dG7GjyHlmeWrYo1Apwmdk7KIEfNMz+dToSTkHCFGZKybVuxclMkFMWMDEtOIkmMcB91SVtTjkIi3XR01RDuaKUDg0joxxUcqb87UhRKOQh9XZktKie9TPzPaycqOHNTyuNEEY7Hg4KEQRXBLCLYoYJgxQaaICyo3hXiHhIIKx1kSYdgT548TRqHFfu4Yt0clasXeRxFsAW2wS6wwSmogmtQA3WAwQN4Ai/g1Xg0no03431cWjDynk3wB8bHN+sPnCU=</latexit> 6⇡⌘bVi + FP i = 0 19 The fluid flow boundary condition on the surface of a no-slip sphere, with velocity and angular velocity , is given as: V i Ω i v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) is the radius vector of the -th particle. ρi i
57. Stokes law Micrometer size => neglect inertia Newton’s equation becomes FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic Stokes law for a no-slip sphere <latexit sha1_base64="MObKdaVqbh7UWWwNBggtIoJRMLc=">AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0Y0nE10YoCtJlBfuAJobJdNIOnUzCzEQooZ/gxl9x40IRty7d+TdO2gjaeuDC4Zx7ufceP2ZUKsv6Mgpz8wuLS8Xl0srq2vqGubnVlFEiMGngiEWi7SNJGOWkoahipB0LgkKfkZY/uMr81j0Rkkb8Vg1j4oaox2lAMVJa8sx9J0Sq7wfw2qN3NXgBD0+dmDpEIejDH6/pUc8sWxVrDDhL7JyUQY66Z3463QgnIeEKMyRlx7Zi5aZIKIoZGZWcRJIY4QHqkY6mHIVEuun4oRHc00oXBpHQxRUcq78nUhRKOQx93ZmdKKe9TPzP6yQqOHdTyuNEEY4ni4KEQRXBLB3YpYJgxYaaICyovhXiPhIIK51hSYdgT788S5pHFfukYt0cl6uXeRxFsAN2wQGwwRmoghqogwbA4AE8gRfwajwaz8ab8T5pLRj5zDb4A+PjG886mzA=</latexit> FH i = 6⇡⌘bV i <latexit sha1_base64="5Laazrb3aq86EUiJDLep8vA/DVk=">AAACFXicbVDLSsNAFJ3UV62vqEs3g0UQ1JKIr41QFMRlBfuAJobJdNIOnUzCzEQooT/hxl9x40IRt4I7/8ZJG0RbDwycOede7r3HjxmVyrK+jMLM7Nz8QnGxtLS8srpmrm80ZJQITOo4YpFo+UgSRjmpK6oYacWCoNBnpOn3LzO/eU+EpBG/VYOYuCHqchpQjJSWPHMfHpw4MXWIQtCHTohUzw9gw6Nw7+d35dG7GjyHlmeWrYo1Apwmdk7KIEfNMz+dToSTkHCFGZKybVuxclMkFMWMDEtOIkmMcB91SVtTjkIi3XR01RDuaKUDg0joxxUcqb87UhRKOQh9XZktKie9TPzPaycqOHNTyuNEEY7Hg4KEQRXBLCLYoYJgxQaaICyo3hXiHhIIKx1kSYdgT548TRqHFfu4Yt0clasXeRxFsAW2wS6wwSmogmtQA3WAwQN4Ai/g1Xg0no03431cWjDynk3wB8bHN+sPnCU=</latexit> 6⇡⌘bVi + FP i = 0 Model an active particles as a sphere with slip boundary condition Boundary velocity = rigid body motion + active slip v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) 19 The fluid flow boundary condition on the surface of a no-slip sphere, with velocity and angular velocity , is given as: V i Ω i v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) is the radius vector of the -th particle. ρi i
58. Stokes law Micrometer size => neglect inertia Newton’s equation becomes FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic Stokes law for a no-slip sphere <latexit sha1_base64="MObKdaVqbh7UWWwNBggtIoJRMLc=">AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0Y0nE10YoCtJlBfuAJobJdNIOnUzCzEQooZ/gxl9x40IRty7d+TdO2gjaeuDC4Zx7ufceP2ZUKsv6Mgpz8wuLS8Xl0srq2vqGubnVlFEiMGngiEWi7SNJGOWkoahipB0LgkKfkZY/uMr81j0Rkkb8Vg1j4oaox2lAMVJa8sx9J0Sq7wfw2qN3NXgBD0+dmDpEIejDH6/pUc8sWxVrDDhL7JyUQY66Z3463QgnIeEKMyRlx7Zi5aZIKIoZGZWcRJIY4QHqkY6mHIVEuun4oRHc00oXBpHQxRUcq78nUhRKOQx93ZmdKKe9TPzP6yQqOHdTyuNEEY4ni4KEQRXBLB3YpYJgxYaaICyovhXiPhIIK51hSYdgT788S5pHFfukYt0cl6uXeRxFsAN2wQGwwRmoghqogwbA4AE8gRfwajwaz8ab8T5pLRj5zDb4A+PjG886mzA=</latexit> FH i = 6⇡⌘bV i <latexit sha1_base64="5Laazrb3aq86EUiJDLep8vA/DVk=">AAACFXicbVDLSsNAFJ3UV62vqEs3g0UQ1JKIr41QFMRlBfuAJobJdNIOnUzCzEQooT/hxl9x40IRt4I7/8ZJG0RbDwycOede7r3HjxmVyrK+jMLM7Nz8QnGxtLS8srpmrm80ZJQITOo4YpFo+UgSRjmpK6oYacWCoNBnpOn3LzO/eU+EpBG/VYOYuCHqchpQjJSWPHMfHpw4MXWIQtCHTohUzw9gw6Nw7+d35dG7GjyHlmeWrYo1Apwmdk7KIEfNMz+dToSTkHCFGZKybVuxclMkFMWMDEtOIkmMcB91SVtTjkIi3XR01RDuaKUDg0joxxUcqb87UhRKOQh9XZktKie9TPzPaycqOHNTyuNEEY7Hg4KEQRXBLCLYoYJgxQaaICyo3hXiHhIIKx1kSYdgT548TRqHFfu4Yt0clasXeRxFsAW2wS6wwSmogmtQA3WAwQN4Ai/g1Xg0no03431cWjDynk3wB8bHN+sPnCU=</latexit> 6⇡⌘bVi + FP i = 0 Model an active particles as a sphere with slip boundary condition Boundary velocity = rigid body motion + active slip v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) What is slip? 19 The fluid flow boundary condition on the surface of a no-slip sphere, with velocity and angular velocity , is given as: V i Ω i v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) is the radius vector of the -th particle. ρi i
59. What is slip? Charged colloid in a non-conducting fluid Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field <latexit sha1_base64="hMTuaFHHVlhxnHIgBP8YKM1gC0A=">AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md</latexit> FP = qE = charge x Electric field 20 E
60. What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field <latexit sha1_base64="hMTuaFHHVlhxnHIgBP8YKM1gC0A=">AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md</latexit> FP = qE = charge x Electric field 20 E
61. What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: exterior flow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field <latexit sha1_base64="hMTuaFHHVlhxnHIgBP8YKM1gC0A=">AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md</latexit> FP = qE = charge x Electric field 20 E
62. What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: exterior flow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field <latexit sha1_base64="hMTuaFHHVlhxnHIgBP8YKM1gC0A=">AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md</latexit> FP = qE = charge x Electric field Charged colloid in a conducting fluid 20 E
63. What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: exterior flow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field <latexit sha1_base64="hMTuaFHHVlhxnHIgBP8YKM1gC0A=">AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md</latexit> FP = qE = charge x Electric field the colloid still moves & exterior flow decays as 1/r3 colloidal charge is balanced by a diffused cloud of counter ions <latexit sha1_base64="nsMPlhhzHUVilPM8UiU2UfaRRrU=">AAAB+HicbVDLSsNAFL2pr1ofjbp0M1gEVyURi26EoiAuK9gHtLFMppN26GQSZiZCDf0SNy4UceunuPNvnLRZaOuBgcM593LPHD/mTGnH+bYKK6tr6xvFzdLW9s5u2d7bb6kokYQ2ScQj2fGxopwJ2tRMc9qJJcWhz2nbH19nfvuRSsUica8nMfVCPBQsYARrI/Xtci/EeuQH6OahgS6R07crTtWZAS0TNycVyNHo21+9QUSSkApNOFaq6zqx9lIsNSOcTku9RNEYkzEe0q6hAodUeeks+BQdG2WAgkiaJzSaqb83UhwqNQl9M5nFVIteJv7ndRMdXHgpE3GiqSDzQ0HCkY5Q1gIaMEmJ5hNDMJHMZEVkhCUm2nRVMiW4i19eJq3TqlurOndnlfpVXkcRDuEITsCFc6jDLTSgCQQSeIZXeLOerBfr3fqYjxasfOcA/sD6/AG9MJHU</latexit> FP = 0 Charged colloid in a conducting fluid 20 E
64. What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: exterior flow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field <latexit sha1_base64="hMTuaFHHVlhxnHIgBP8YKM1gC0A=">AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md</latexit> FP = qE = charge x Electric field the colloid still moves & exterior flow decays as 1/r3 colloidal charge is balanced by a diffused cloud of counter ions <latexit sha1_base64="nsMPlhhzHUVilPM8UiU2UfaRRrU=">AAAB+HicbVDLSsNAFL2pr1ofjbp0M1gEVyURi26EoiAuK9gHtLFMppN26GQSZiZCDf0SNy4UceunuPNvnLRZaOuBgcM593LPHD/mTGnH+bYKK6tr6xvFzdLW9s5u2d7bb6kokYQ2ScQj2fGxopwJ2tRMc9qJJcWhz2nbH19nfvuRSsUica8nMfVCPBQsYARrI/Xtci/EeuQH6OahgS6R07crTtWZAS0TNycVyNHo21+9QUSSkApNOFaq6zqx9lIsNSOcTku9RNEYkzEe0q6hAodUeeks+BQdG2WAgkiaJzSaqb83UhwqNQl9M5nFVIteJv7ndRMdXHgpE3GiqSDzQ0HCkY5Q1gIaMEmJ5hNDMJHMZEVkhCUm2nRVMiW4i19eJq3TqlurOndnlfpVXkcRDuEITsCFc6jDLTSgCQQSeIZXeLOerBfr3fqYjxasfOcA/sD6/AG9MJHU</latexit> FP = 0 Charged colloid in a conducting fluid 20 E Anderson, Ann Rev Fluid Mech (1989) E
65. What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: exterior flow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field <latexit sha1_base64="hMTuaFHHVlhxnHIgBP8YKM1gC0A=">AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md</latexit> FP = qE = charge x Electric field the colloid still moves & exterior flow decays as 1/r3 colloidal charge is balanced by a diffused cloud of counter ions <latexit sha1_base64="nsMPlhhzHUVilPM8UiU2UfaRRrU=">AAAB+HicbVDLSsNAFL2pr1ofjbp0M1gEVyURi26EoiAuK9gHtLFMppN26GQSZiZCDf0SNy4UceunuPNvnLRZaOuBgcM593LPHD/mTGnH+bYKK6tr6xvFzdLW9s5u2d7bb6kokYQ2ScQj2fGxopwJ2tRMc9qJJcWhz2nbH19nfvuRSsUica8nMfVCPBQsYARrI/Xtci/EeuQH6OahgS6R07crTtWZAS0TNycVyNHo21+9QUSSkApNOFaq6zqx9lIsNSOcTku9RNEYkzEe0q6hAodUeeks+BQdG2WAgkiaJzSaqb83UhwqNQl9M5nFVIteJv7ndRMdXHgpE3GiqSDzQ0HCkY5Q1gIaMEmJ5hNDMJHMZEVkhCUm2nRVMiW4i19eJq3TqlurOndnlfpVXkcRDuEITsCFc6jDLTSgCQQSeIZXeLOerBfr3fqYjxasfOcA/sD6/AG9MJHU</latexit> FP = 0 Charged colloid in a conducting fluid 20 E Anderson, Ann Rev Fluid Mech (1989) E How does this neutral object move?
66. What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: exterior flow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field <latexit sha1_base64="hMTuaFHHVlhxnHIgBP8YKM1gC0A=">AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md</latexit> FP = qE = charge x Electric field the colloid still moves & exterior flow decays as 1/r3 colloidal charge is balanced by a diffused cloud of counter ions <latexit sha1_base64="nsMPlhhzHUVilPM8UiU2UfaRRrU=">AAAB+HicbVDLSsNAFL2pr1ofjbp0M1gEVyURi26EoiAuK9gHtLFMppN26GQSZiZCDf0SNy4UceunuPNvnLRZaOuBgcM593LPHD/mTGnH+bYKK6tr6xvFzdLW9s5u2d7bb6kokYQ2ScQj2fGxopwJ2tRMc9qJJcWhz2nbH19nfvuRSsUica8nMfVCPBQsYARrI/Xtci/EeuQH6OahgS6R07crTtWZAS0TNycVyNHo21+9QUSSkApNOFaq6zqx9lIsNSOcTku9RNEYkzEe0q6hAodUeeks+BQdG2WAgkiaJzSaqb83UhwqNQl9M5nFVIteJv7ndRMdXHgpE3GiqSDzQ0HCkY5Q1gIaMEmJ5hNDMJHMZEVkhCUm2nRVMiW4i19eJq3TqlurOndnlfpVXkcRDuEITsCFc6jDLTSgCQQSeIZXeLOerBfr3fqYjxasfOcA/sD6/AG9MJHU</latexit> FP = 0 Charged colloid in a conducting fluid 20 E Anderson, Ann Rev Fluid Mech (1989) E How does this neutral object move? it is not rigid - diffuse clouds of counter-ions move in opposite direction of the charged particle
67. What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: exterior flow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field <latexit sha1_base64="hMTuaFHHVlhxnHIgBP8YKM1gC0A=">AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md</latexit> FP = qE = charge x Electric field the colloid still moves & exterior flow decays as 1/r3 colloidal charge is balanced by a diffused cloud of counter ions <latexit sha1_base64="nsMPlhhzHUVilPM8UiU2UfaRRrU=">AAAB+HicbVDLSsNAFL2pr1ofjbp0M1gEVyURi26EoiAuK9gHtLFMppN26GQSZiZCDf0SNy4UceunuPNvnLRZaOuBgcM593LPHD/mTGnH+bYKK6tr6xvFzdLW9s5u2d7bb6kokYQ2ScQj2fGxopwJ2tRMc9qJJcWhz2nbH19nfvuRSsUica8nMfVCPBQsYARrI/Xtci/EeuQH6OahgS6R07crTtWZAS0TNycVyNHo21+9QUSSkApNOFaq6zqx9lIsNSOcTku9RNEYkzEe0q6hAodUeeks+BQdG2WAgkiaJzSaqb83UhwqNQl9M5nFVIteJv7ndRMdXHgpE3GiqSDzQ0HCkY5Q1gIaMEmJ5hNDMJHMZEVkhCUm2nRVMiW4i19eJq3TqlurOndnlfpVXkcRDuEITsCFc6jDLTSgCQQSeIZXeLOerBfr3fqYjxasfOcA/sD6/AG9MJHU</latexit> FP = 0 Charged colloid in a conducting fluid Slip flow <latexit sha1_base64="rHZKegjpSzYExDVJRbDCLhblYfc=">AAACI3icbVBNS8MwGE7n15xfVY9egkPw4mhFUQRh6sXjBPcBax1pmm5haVqSdDBK/4sX/4oXD8rw4sH/YtrtoJsvhDw8z/Mm7/t4MaNSWdaXUVpaXlldK69XNja3tnfM3b2WjBKBSRNHLBIdD0nCKCdNRRUjnVgQFHqMtL3hXa63R0RIGvFHNY6JG6I+pwHFSGmqZ145IVIDL4AteA1PoMMQ7zMCHS9ivhyH+kpH2VNauDBi6U2WVRxRmHpm1apZRcFFYM9AFcyq0TMnjh/hJCRcYYak7NpWrNwUCUUxI/rdRJIY4SHqk66GHIVEummxYwaPNOPDIBL6cAUL9ndHikKZD6yd+axyXsvJ/7RuooJLN6U8ThThePpRkDCoIpgHBn0qCFZsrAHCgupZIR4ggbDSsVZ0CPb8yougdVqzz2vWw1m1fjuLowwOwCE4Bja4AHVwDxqgCTB4Bq/gHXwYL8abMTE+p9aSMevZB3/K+P4BCpGklQ==</latexit> V = hvAi Electrophoretic slip vA = ✏⇣ 4⇡⌘ Es <latexit sha1_base64="PVIeaH1jWVueUX0FzMolPsiAd4o=">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</latexit> <latexit sha1_base64="PVIeaH1jWVueUX0FzMolPsiAd4o=">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</latexit> <latexit sha1_base64="PVIeaH1jWVueUX0FzMolPsiAd4o=">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</latexit> <latexit sha1_base64="PVIeaH1jWVueUX0FzMolPsiAd4o=">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</latexit> <latexit sha1_base64="rHZKegjpSzYExDVJRbDCLhblYfc=">AAACI3icbVBNS8MwGE7n15xfVY9egkPw4mhFUQRh6sXjBPcBax1pmm5haVqSdDBK/4sX/4oXD8rw4sH/YtrtoJsvhDw8z/Mm7/t4MaNSWdaXUVpaXlldK69XNja3tnfM3b2WjBKBSRNHLBIdD0nCKCdNRRUjnVgQFHqMtL3hXa63R0RIGvFHNY6JG6I+pwHFSGmqZ145IVIDL4AteA1PoMMQ7zMCHS9ivhyH+kpH2VNauDBi6U2WVRxRmHpm1apZRcFFYM9AFcyq0TMnjh/hJCRcYYak7NpWrNwUCUUxI/rdRJIY4SHqk66GHIVEummxYwaPNOPDIBL6cAUL9ndHikKZD6yd+axyXsvJ/7RuooJLN6U8ThThePpRkDCoIpgHBn0qCFZsrAHCgupZIR4ggbDSsVZ0CPb8yougdVqzz2vWw1m1fjuLowwOwCE4Bja4AHVwDxqgCTB4Bq/gHXwYL8abMTE+p9aSMevZB3/K+P4BCpGklQ==</latexit> V = hvAi <latexit sha1_base64="rHZKegjpSzYExDVJRbDCLhblYfc=">AAACI3icbVBNS8MwGE7n15xfVY9egkPw4mhFUQRh6sXjBPcBax1pmm5haVqSdDBK/4sX/4oXD8rw4sH/YtrtoJsvhDw8z/Mm7/t4MaNSWdaXUVpaXlldK69XNja3tnfM3b2WjBKBSRNHLBIdD0nCKCdNRRUjnVgQFHqMtL3hXa63R0RIGvFHNY6JG6I+pwHFSGmqZ145IVIDL4AteA1PoMMQ7zMCHS9ivhyH+kpH2VNauDBi6U2WVRxRmHpm1apZRcFFYM9AFcyq0TMnjh/hJCRcYYak7NpWrNwUCUUxI/rdRJIY4SHqk66GHIVEummxYwaPNOPDIBL6cAUL9ndHikKZD6yd+axyXsvJ/7RuooJLN6U8ThThePpRkDCoIpgHBn0qCFZsrAHCgupZIR4ggbDSsVZ0CPb8yougdVqzz2vWw1m1fjuLowwOwCE4Bja4AHVwDxqgCTB4Bq/gHXwYL8abMTE+p9aSMevZB3/K+P4BCpGklQ==</latexit> V = hvAi 20 E Anderson, Ann Rev Fluid Mech (1989) E How does this neutral object move? it is not rigid - diffuse clouds of counter-ions move in opposite direction of the charged particle
68. What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: exterior flow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field <latexit sha1_base64="hMTuaFHHVlhxnHIgBP8YKM1gC0A=">AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md</latexit> FP = qE = charge x Electric field the colloid still moves & exterior flow decays as 1/r3 colloidal charge is balanced by a diffused cloud of counter ions <latexit sha1_base64="nsMPlhhzHUVilPM8UiU2UfaRRrU=">AAAB+HicbVDLSsNAFL2pr1ofjbp0M1gEVyURi26EoiAuK9gHtLFMppN26GQSZiZCDf0SNy4UceunuPNvnLRZaOuBgcM593LPHD/mTGnH+bYKK6tr6xvFzdLW9s5u2d7bb6kokYQ2ScQj2fGxopwJ2tRMc9qJJcWhz2nbH19nfvuRSsUica8nMfVCPBQsYARrI/Xtci/EeuQH6OahgS6R07crTtWZAS0TNycVyNHo21+9QUSSkApNOFaq6zqx9lIsNSOcTku9RNEYkzEe0q6hAodUeeks+BQdG2WAgkiaJzSaqb83UhwqNQl9M5nFVIteJv7ndRMdXHgpE3GiqSDzQ0HCkY5Q1gIaMEmJ5hNDMJHMZEVkhCUm2nRVMiW4i19eJq3TqlurOndnlfpVXkcRDuEITsCFc6jDLTSgCQQSeIZXeLOerBfr3fqYjxasfOcA/sD6/AG9MJHU</latexit> FP = 0 a mechanism to drive exterior flow. The resulting fluid stress may cause self-propulsion Charged colloid in a conducting fluid Slip flow <latexit sha1_base64="rHZKegjpSzYExDVJRbDCLhblYfc=">AAACI3icbVBNS8MwGE7n15xfVY9egkPw4mhFUQRh6sXjBPcBax1pmm5haVqSdDBK/4sX/4oXD8rw4sH/YtrtoJsvhDw8z/Mm7/t4MaNSWdaXUVpaXlldK69XNja3tnfM3b2WjBKBSRNHLBIdD0nCKCdNRRUjnVgQFHqMtL3hXa63R0RIGvFHNY6JG6I+pwHFSGmqZ145IVIDL4AteA1PoMMQ7zMCHS9ivhyH+kpH2VNauDBi6U2WVRxRmHpm1apZRcFFYM9AFcyq0TMnjh/hJCRcYYak7NpWrNwUCUUxI/rdRJIY4SHqk66GHIVEummxYwaPNOPDIBL6cAUL9ndHikKZD6yd+axyXsvJ/7RuooJLN6U8ThThePpRkDCoIpgHBn0qCFZsrAHCgupZIR4ggbDSsVZ0CPb8yougdVqzz2vWw1m1fjuLowwOwCE4Bja4AHVwDxqgCTB4Bq/gHXwYL8abMTE+p9aSMevZB3/K+P4BCpGklQ==</latexit> V = hvAi Electrophoretic slip vA = ✏⇣ 4⇡⌘ Es <latexit sha1_base64="PVIeaH1jWVueUX0FzMolPsiAd4o=">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</latexit> <latexit sha1_base64="PVIeaH1jWVueUX0FzMolPsiAd4o=">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</latexit> <latexit sha1_base64="PVIeaH1jWVueUX0FzMolPsiAd4o=">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</latexit> <latexit sha1_base64="PVIeaH1jWVueUX0FzMolPsiAd4o=">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</latexit> <latexit sha1_base64="rHZKegjpSzYExDVJRbDCLhblYfc=">AAACI3icbVBNS8MwGE7n15xfVY9egkPw4mhFUQRh6sXjBPcBax1pmm5haVqSdDBK/4sX/4oXD8rw4sH/YtrtoJsvhDw8z/Mm7/t4MaNSWdaXUVpaXlldK69XNja3tnfM3b2WjBKBSRNHLBIdD0nCKCdNRRUjnVgQFHqMtL3hXa63R0RIGvFHNY6JG6I+pwHFSGmqZ145IVIDL4AteA1PoMMQ7zMCHS9ivhyH+kpH2VNauDBi6U2WVRxRmHpm1apZRcFFYM9AFcyq0TMnjh/hJCRcYYak7NpWrNwUCUUxI/rdRJIY4SHqk66GHIVEummxYwaPNOPDIBL6cAUL9ndHikKZD6yd+axyXsvJ/7RuooJLN6U8ThThePpRkDCoIpgHBn0qCFZsrAHCgupZIR4ggbDSsVZ0CPb8yougdVqzz2vWw1m1fjuLowwOwCE4Bja4AHVwDxqgCTB4Bq/gHXwYL8abMTE+p9aSMevZB3/K+P4BCpGklQ==</latexit> V = hvAi <latexit sha1_base64="rHZKegjpSzYExDVJRbDCLhblYfc=">AAACI3icbVBNS8MwGE7n15xfVY9egkPw4mhFUQRh6sXjBPcBax1pmm5haVqSdDBK/4sX/4oXD8rw4sH/YtrtoJsvhDw8z/Mm7/t4MaNSWdaXUVpaXlldK69XNja3tnfM3b2WjBKBSRNHLBIdD0nCKCdNRRUjnVgQFHqMtL3hXa63R0RIGvFHNY6JG6I+pwHFSGmqZ145IVIDL4AteA1PoMMQ7zMCHS9ivhyH+kpH2VNauDBi6U2WVRxRmHpm1apZRcFFYM9AFcyq0TMnjh/hJCRcYYak7NpWrNwUCUUxI/rdRJIY4SHqk66GHIVEummxYwaPNOPDIBL6cAUL9ndHikKZD6yd+axyXsvJ/7RuooJLN6U8ThThePpRkDCoIpgHBn0qCFZsrAHCgupZIR4ggbDSsVZ0CPb8yougdVqzz2vWw1m1fjuLowwOwCE4Bja4AHVwDxqgCTB4Bq/gHXwYL8abMTE+p9aSMevZB3/K+P4BCpGklQ==</latexit> V = hvAi 20 E Anderson, Ann Rev Fluid Mech (1989) E How does this neutral object move? it is not rigid - diffuse clouds of counter-ions move in opposite direction of the charged particle
69. What is slip? Charged colloid in a non-conducting fluid V = qE 6⇡⌘b size-dependent velocity: exterior flow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric field <latexit sha1_base64="hMTuaFHHVlhxnHIgBP8YKM1gC0A=">AAACI3icbVDLSgNBEJz1GeNr1aOXwSB4CruiKIIQFMVjBKOBJIbZSW8yZPbhTK8kLPkXL/6KFw+KePHgvzgbV/BVMFBd3U1PlRdLodFx3qyJyanpmdnCXHF+YXFp2V5ZvdRRojjUeCQjVfeYBilCqKFACfVYAQs8CVde/zjrX92C0iIKL3AYQytg3VD4gjM0Uts+aAYMe55PT6+r9JDe0K/6xFRNhAGmvMdUF+iAnkjgqASnvgDZGbXtklN2xqB/iZuTEslRbdsvzU7EkwBC5JJp3XCdGFspUyi4hFGxmWiIGe+zLjQMDVkAupWOPY7oplE61I+UeSHSsfp9I2WB1sPAM5OZA/27l4n/9RoJ+vutVIRxghDyz0N+IilGNAuMdoQytuXQEMaVMH+lWSKMo4m1aEJwf1v+Sy63y+5u2TnfKVWO8jgKZJ1skC3ikj1SIWekSmqEkzvyQJ7Is3VvPVov1uvn6ISV76yRH7DePwCtj6Md</latexit> FP = qE = charge x Electric field the colloid still moves & exterior flow decays as 1/r3 colloidal charge is balanced by a diffused cloud of counter ions <latexit sha1_base64="nsMPlhhzHUVilPM8UiU2UfaRRrU=">AAAB+HicbVDLSsNAFL2pr1ofjbp0M1gEVyURi26EoiAuK9gHtLFMppN26GQSZiZCDf0SNy4UceunuPNvnLRZaOuBgcM593LPHD/mTGnH+bYKK6tr6xvFzdLW9s5u2d7bb6kokYQ2ScQj2fGxopwJ2tRMc9qJJcWhz2nbH19nfvuRSsUica8nMfVCPBQsYARrI/Xtci/EeuQH6OahgS6R07crTtWZAS0TNycVyNHo21+9QUSSkApNOFaq6zqx9lIsNSOcTku9RNEYkzEe0q6hAodUeeks+BQdG2WAgkiaJzSaqb83UhwqNQl9M5nFVIteJv7ndRMdXHgpE3GiqSDzQ0HCkY5Q1gIaMEmJ5hNDMJHMZEVkhCUm2nRVMiW4i19eJq3TqlurOndnlfpVXkcRDuEITsCFc6jDLTSgCQQSeIZXeLOerBfr3fqYjxasfOcA/sD6/AG9MJHU</latexit> FP = 0 a mechanism to drive exterior flow. The resulting fluid stress may cause self-propulsion Charged colloid in a conducting fluid Slip flow <latexit sha1_base64="rHZKegjpSzYExDVJRbDCLhblYfc=">AAACI3icbVBNS8MwGE7n15xfVY9egkPw4mhFUQRh6sXjBPcBax1pmm5haVqSdDBK/4sX/4oXD8rw4sH/YtrtoJsvhDw8z/Mm7/t4MaNSWdaXUVpaXlldK69XNja3tnfM3b2WjBKBSRNHLBIdD0nCKCdNRRUjnVgQFHqMtL3hXa63R0RIGvFHNY6JG6I+pwHFSGmqZ145IVIDL4AteA1PoMMQ7zMCHS9ivhyH+kpH2VNauDBi6U2WVRxRmHpm1apZRcFFYM9AFcyq0TMnjh/hJCRcYYak7NpWrNwUCUUxI/rdRJIY4SHqk66GHIVEummxYwaPNOPDIBL6cAUL9ndHikKZD6yd+axyXsvJ/7RuooJLN6U8ThThePpRkDCoIpgHBn0qCFZsrAHCgupZIR4ggbDSsVZ0CPb8yougdVqzz2vWw1m1fjuLowwOwCE4Bja4AHVwDxqgCTB4Bq/gHXwYL8abMTE+p9aSMevZB3/K+P4BCpGklQ==</latexit> V = hvAi Electrophoretic slip vA = ✏⇣ 4⇡⌘ Es <latexit sha1_base64="PVIeaH1jWVueUX0FzMolPsiAd4o=">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</latexit> <latexit sha1_base64="PVIeaH1jWVueUX0FzMolPsiAd4o=">AAACJXicbVDLSgMxFM34rPVVdekmWAQ3lhkp6EKhKoLLCvYBnbZk0jttaCYzJJlCHeZn3PgrblxYRHDlr5g+Ftp6IHA451xy7/EizpS27S9raXlldW09s5Hd3Nre2c3t7VdVGEsKFRryUNY9ooAzARXNNId6JIEEHoea178d+7UBSMVC8aiHETQD0hXMZ5RoI7Vzl4NW4gZE9yjh+DrFV/jU9SWhiQuRYjwU7hNogtOk6EbMNTSdpD0f37VUO5e3C/YEeJE4M5JHM5TbuZHbCWkcgNCUE6Uajh3pZkKkZpRDmnVjBRGhfdKFhqGCBKCayeTKFB8bpYP9UJonNJ6ovycSEig1DDyTHK+o5r2x+J/XiLV/0UyYiGINgk4/8mOOdYjHleEOk0A1HxpCqGRmV0x7xJSkTbFZU4Izf/IiqZ4VHLvgPBTzpZtZHRl0iI7QCXLQOSqhe1RGFUTRM3pF72hkvVhv1of1OY0uWbOZA/QH1vcP54qlgQ==</latexit> <latexit sha1_base64="PVIeaH1jWVueUX0FzMolPsiAd4o=">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</latexit> <latexit sha1_base64="PVIeaH1jWVueUX0FzMolPsiAd4o=">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</latexit> <latexit sha1_base64="rHZKegjpSzYExDVJRbDCLhblYfc=">AAACI3icbVBNS8MwGE7n15xfVY9egkPw4mhFUQRh6sXjBPcBax1pmm5haVqSdDBK/4sX/4oXD8rw4sH/YtrtoJsvhDw8z/Mm7/t4MaNSWdaXUVpaXlldK69XNja3tnfM3b2WjBKBSRNHLBIdD0nCKCdNRRUjnVgQFHqMtL3hXa63R0RIGvFHNY6JG6I+pwHFSGmqZ145IVIDL4AteA1PoMMQ7zMCHS9ivhyH+kpH2VNauDBi6U2WVRxRmHpm1apZRcFFYM9AFcyq0TMnjh/hJCRcYYak7NpWrNwUCUUxI/rdRJIY4SHqk66GHIVEummxYwaPNOPDIBL6cAUL9ndHikKZD6yd+axyXsvJ/7RuooJLN6U8ThThePpRkDCoIpgHBn0qCFZsrAHCgupZIR4ggbDSsVZ0CPb8yougdVqzz2vWw1m1fjuLowwOwCE4Bja4AHVwDxqgCTB4Bq/gHXwYL8abMTE+p9aSMevZB3/K+P4BCpGklQ==</latexit> V = hvAi <latexit sha1_base64="rHZKegjpSzYExDVJRbDCLhblYfc=">AAACI3icbVBNS8MwGE7n15xfVY9egkPw4mhFUQRh6sXjBPcBax1pmm5haVqSdDBK/4sX/4oXD8rw4sH/YtrtoJsvhDw8z/Mm7/t4MaNSWdaXUVpaXlldK69XNja3tnfM3b2WjBKBSRNHLBIdD0nCKCdNRRUjnVgQFHqMtL3hXa63R0RIGvFHNY6JG6I+pwHFSGmqZ145IVIDL4AteA1PoMMQ7zMCHS9ivhyH+kpH2VNauDBi6U2WVRxRmHpm1apZRcFFYM9AFcyq0TMnjh/hJCRcYYak7NpWrNwUCUUxI/rdRJIY4SHqk66GHIVEummxYwaPNOPDIBL6cAUL9ndHikKZD6yd+axyXsvJ/7RuooJLN6U8ThThePpRkDCoIpgHBn0qCFZsrAHCgupZIR4ggbDSsVZ0CPb8yougdVqzz2vWw1m1fjuLowwOwCE4Bja4AHVwDxqgCTB4Bq/gHXwYL8abMTE+p9aSMevZB3/K+P4BCpGklQ==</latexit> V = hvAi 20 E Anderson, Ann Rev Fluid Mech (1989) E How does this neutral object move? it is not rigid - diffuse clouds of counter-ions move in opposite direction of the charged particle force-free motion from slip: exterior field: driven particles self-phoresis: active particles
70. Main theoretical questions The problem is classical and the motion is governed by Newton’s equations. We then need to know: FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic 21
71. Main theoretical questions ‣ What are the forces and torques acting on the particles due to slip? ‣ How are these modified by the presence of boundaries? ‣ What is the rigid body motion of particles under these forces? ‣ How do we take into account, simultaneously, the many-body character of the hydrodynamic and phoretic interactions? The problem is classical and the motion is governed by Newton’s equations. We then need to know: FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic 21
72. Main theoretical questions ‣ What are the forces and torques acting on the particles due to slip? ‣ How are these modified by the presence of boundaries? ‣ What is the rigid body motion of particles under these forces? ‣ How do we take into account, simultaneously, the many-body character of the hydrodynamic and phoretic interactions? The problem is classical and the motion is governed by Newton’s equations. We then need to know: , where is the normal component of the fluid stress. This is to be obtained from the Stokes equation FH i = ∫ f dSi f = ̂ ρi ⋅ σ FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic 21
73. Main theoretical questions ‣ What are the forces and torques acting on the particles due to slip? ‣ How are these modified by the presence of boundaries? ‣ What is the rigid body motion of particles under these forces? ‣ How do we take into account, simultaneously, the many-body character of the hydrodynamic and phoretic interactions? The problem is classical and the motion is governed by Newton’s equations. We then need to know: , where is the normal component of the fluid stress. This is to be obtained from the Stokes equation FH i = ∫ f dSi f = ̂ ρi ⋅ σ FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic 21 r · v = 0, r · + ⇠ = 0, fluid velocity = pI + ⌘(rv + (rv)T ) r · v = 0, r · + ⇠ = 0, fluid stress Boundary velocity = rigid body motion + active slip active slip boundary condition v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i )
74. Main theoretical questions ‣ What are the forces and torques acting on the particles due to slip? ‣ How are these modified by the presence of boundaries? ‣ What is the rigid body motion of particles under these forces? ‣ How do we take into account, simultaneously, the many-body character of the hydrodynamic and phoretic interactions? The problem is classical and the motion is governed by Newton’s equations. We then need to know: Given the slip, we seek to generalise Stokes law for active particles. f , where is the normal component of the fluid stress. This is to be obtained from the Stokes equation FH i = ∫ f dSi f = ̂ ρi ⋅ σ FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic 21 r · v = 0, r · + ⇠ = 0, fluid velocity = pI + ⌘(rv + (rv)T ) r · v = 0, r · + ⇠ = 0, fluid stress Boundary velocity = rigid body motion + active slip active slip boundary condition v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i )
75. Boundary integral representation of Stokes equations Fluid velocity at any point in the bulk is given in terms of integrals on the surface of the colloids Odqvist 1930, Jackson 1962, Ladyzhenskaia 1969, Pozrikidis 1992 force per unit area (traction) boundary velocity <latexit sha1_base64="Jl+Izs7/1Vhgnr7jO3oOGsUpQXo=">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</latexit> v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi 22
76. Boundary integral representation of Stokes equations Fluid velocity at any point in the bulk is given in terms of integrals on the surface of the colloids Odqvist 1930, Jackson 1962, Ladyzhenskaia 1969, Pozrikidis 1992 force per unit area (traction) boundary velocity <latexit sha1_base64="Jl+Izs7/1Vhgnr7jO3oOGsUpQXo=">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</latexit> v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi potential single layer double layer Laplace equation r2 = 0 <latexit sha1_base64="KO463eF9mPVoLrveTmOrR8F+3Dk=">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</latexit> (r) = Z h ˜ G(r, ri)˜(ri) ˜ K↵(ri, r)ˆ ⇢↵ (ri) i dSi 22
77. Boundary integral representation of Stokes equations Fluid velocity at any point in the bulk is given in terms of integrals on the surface of the colloids Green’s function Stress tensor Odqvist 1930, Jackson 1962, Ladyzhenskaia 1969, Pozrikidis 1992 force per unit area (traction) boundary velocity <latexit sha1_base64="Jl+Izs7/1Vhgnr7jO3oOGsUpQXo=">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</latexit> v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi potential single layer double layer Laplace equation r2 = 0 <latexit sha1_base64="KO463eF9mPVoLrveTmOrR8F+3Dk=">AAACy3icdVFNa9wwEJWdfqTbr2177MV0KWwgWezQklwKITk0UAop7SaBlVnGsnYtIktGGrfdKD72D/bWa39J5Y2hzWY7IPTmzRv0NJNVUliM419BuHHn7r37mw96Dx89fvK0/+z5qdW1YXzMtNTmPAPLpVB8jAIlP68MhzKT/Cy7OGrrZ1+5sUKrL7ioeFrCXImZYICemvZ/08qKIc20zO2i9Jczzda7HSoU0kMxn1AUMufufbOi2XY386kTTbPVqakV8xKa4VrNTif64HMKsir+I9xeMUULQEdNof/2Lb2vN+K9p/nnNulN+4N4FC8jug2SDgxIFyfT/k+aa1aXXCGTYO0kiStMHRgUTPKmR2vLK2AXMOcTDxWU3KZuuYsmeu2ZPJpp44/CaMn+2+GgtK1ZrywBC7taa8l1tUmNs/3UCVXVyBW7fmhWywh11C42yoXhDOXCA2BGeK8RK8AAQ7/+dgjJ6pdvg9PdUfJ2FH96Mzg47MaxSV6SV2RIErJHDsgxOSFjwoLjQAXfgu/hx9CGl+HVtTQMup4X5EaEP/4A6NznwA==</latexit> (r) = Z h ˜ G(r, ri)˜(ri) ˜ K↵(ri, r)ˆ ⇢↵ (ri) i dSi 22
78. Boundary integral representation of Stokes equations Fluid velocity at any point in the bulk is given in terms of integrals on the surface of the colloids Green’s function Stress tensor Odqvist 1930, Jackson 1962, Ladyzhenskaia 1969, Pozrikidis 1992 force per unit area (traction) boundary velocity flow satisfies boundary conditions at non-particle boundaries <latexit sha1_base64="Jl+Izs7/1Vhgnr7jO3oOGsUpQXo=">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</latexit> v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi potential single layer double layer Laplace equation r2 = 0 <latexit sha1_base64="KO463eF9mPVoLrveTmOrR8F+3Dk=">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</latexit> (r) = Z h ˜ G(r, ri)˜(ri) ˜ K↵(ri, r)ˆ ⇢↵ (ri) i dSi 22
79. Boundary integral representation of Stokes equations Fluid velocity at any point in the bulk is given in terms of integrals on the surface of the colloids Green’s function Stress tensor the Green’s function and the Stress tensor satisfy Stokes equation the integral admits analytical solution by Galerkin discretization for smooth boundaries like spheres problem reduced from the bulk three-dimensional flow to the two-dimensional surfaces of the colloids Odqvist 1930, Jackson 1962, Ladyzhenskaia 1969, Pozrikidis 1992 force per unit area (traction) boundary velocity flow satisfies boundary conditions at non-particle boundaries <latexit sha1_base64="Jl+Izs7/1Vhgnr7jO3oOGsUpQXo=">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</latexit> v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi potential single layer double layer Laplace equation r2 = 0 <latexit sha1_base64="KO463eF9mPVoLrveTmOrR8F+3Dk=">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</latexit> (r) = Z h ˜ G(r, ri)˜(ri) ˜ K↵(ri, r)ˆ ⇢↵ (ri) i dSi 22
80. Ritz-Galerkin discretization 23 boundary velocity v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) Mazur and Van Saarloos, Physica A 1982; Hess 2015; RS et al. JSTAT 2015, PRL 2016, JOSS 2020 <latexit sha1_base64="Jl+Izs7/1Vhgnr7jO3oOGsUpQXo=">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</latexit> v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi
81. Ritz-Galerkin discretization 23 <latexit sha1_base64="KPkzMO0bzgStJqDXXaeq6O1g2KA=">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</latexit> vA i Ri + ⇢i = 1 X l=1 V(l) i · Y(l 1)(ˆ ⇢i ), boundary velocity v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) Mazur and Van Saarloos, Physica A 1982; Hess 2015; RS et al. JSTAT 2015, PRL 2016, JOSS 2020 <latexit sha1_base64="Jl+Izs7/1Vhgnr7jO3oOGsUpQXo=">AAAC1nicfVJba9swFJbdXbrslm6PexELgxbaYI+W9WVQuocN9tLRpQ1ExhzLciwqWUaSA8G4Dy1jr/tte9uP2H+YnHiwJmEHhD6d7zvSuSgpBTc2CH55/ta9+w8ebj/qPX7y9Nnz/s6LC6MqTdmIKqH0OAHDBC/YyHIr2LjUDGQi2GVy9aHlL2dMG66Kr3ZeskjCtOAZp2CdK+7/nsU1AVHm0OySRInUzKXbat3svT8gvLCYnPIpnuCPf3UkYXZNvF/fPce82ctcxFK7gcQH+HPHd9dOQcqN2v2VvEgOtiY6V427YBk1++9TbQVReh7zXtwfBMNgYXgdhB0YoM7O4v5PkipaSVZYKsCYSRiUNqpBW04Fa3qkMqwEegVTNnGwAMlMVC/G0uA3zpPiTGm3XCMX3n8japCmzdQpJdjcrHKtcxM3qWx2HNW8KCvLCrp8KKsEtgq3M8Yp14xaMXcAqOYuV0xz0ECt+wltE8LVktfBxdtheDQMvhwOTk67dmyjV+g12kUheodO0Cd0hkaIeufe3Lvxbv2xf+1/878vpb7XxbxEd8z/8QfErunV</latexit> v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi
82. Ritz-Galerkin discretization 23 <latexit sha1_base64="KPkzMO0bzgStJqDXXaeq6O1g2KA=">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</latexit> vA i Ri + ⇢i = 1 X l=1 V(l) i · Y(l 1)(ˆ ⇢i ), boundary velocity v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) Mazur and Van Saarloos, Physica A 1982; Hess 2015; RS et al. JSTAT 2015, PRL 2016, JOSS 2020 Expansion of boundary fields in tensorial spherical harmonics Y(l) - dimensionless, symmetric, irreducible Cartesian tensors of rank l that form a complete, orthogonal basis on the sphere Y(l)(ˆ ⇢) = ( 1)l⇢l+1r(l)⇢ 1 <latexit sha1_base64="f9kVoPJo+AjtI2MEG3lUOl9ykEY=">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</latexit> Y (0) = 1, Y (1) ↵ = ˆ ⇢↵, Y (2) ↵ = 3ˆ ⇢↵ ˆ ⇢ ↵ <latexit sha1_base64="Jl+Izs7/1Vhgnr7jO3oOGsUpQXo=">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</latexit> v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi
83. Ritz-Galerkin discretization 23 <latexit sha1_base64="KPkzMO0bzgStJqDXXaeq6O1g2KA=">AAACmnicbVFbb9MwFHbCZaPcCnvgAR4sKqRWQJVMIPZSaQyJi3gZaO2G6hI5jtNac+zIPplUWflR/BXe+Dc4aZC6jSNZ/vSd79zTUgoLUfQnCG/cvHV7Z/dO7+69+w8e9h89nlldGcanTEttzlJquRSKT0GA5Gel4bRIJT9Nzz80/tMLbqzQ6gTWJV8UdKlELhgFTyX9XyTVMrPrwn/uok6cqH86UlBYMSrd+7omqVgOt0XfW9HLbYqYlW7ZRjyaEFsViZOTuEklVA7rGrcp09zNuhJDOaoJyzT8c/xoydfxqL5UjawobPK3gaNXvaQ/iMZRa/g6iDswQJ0dJ/3fJNOsKrgCJqm18zgqYeGoAcEkr3uksryk7Jwu+dxDRQtuF65dbY1feCbDuTb+KcAtux3haGGbVr2yGcRe9TXk/3zzCvKDhROqrIArtimUVxKDxs2dcCYMZyDXHlBmhO8VsxU1lIG/ZrOE+OrI18Fsfxy/HUff3gwOj7p17KKn6Dkaohi9Q4foMzpGU8SCJ8Ek+Bh8Cp+FR+GX8OtGGgZdzB66ZOHJX5pz0PU=</latexit> vA i Ri + ⇢i = 1 X l=1 V(l) i · Y(l 1)(ˆ ⇢i ), boundary velocity v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) Mazur and Van Saarloos, Physica A 1982; Hess 2015; RS et al. JSTAT 2015, PRL 2016, JOSS 2020 Expansion of boundary fields in tensorial spherical harmonics Y(l) - dimensionless, symmetric, irreducible Cartesian tensors of rank l that form a complete, orthogonal basis on the sphere Y(l)(ˆ ⇢) = ( 1)l⇢l+1r(l)⇢ 1 <latexit sha1_base64="f9kVoPJo+AjtI2MEG3lUOl9ykEY=">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</latexit> Y (0) = 1, Y (1) ↵ = ˆ ⇢↵, Y (2) ↵ = 3ˆ ⇢↵ ˆ ⇢ ↵ <latexit sha1_base64="Jl+Izs7/1Vhgnr7jO3oOGsUpQXo=">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</latexit> v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi
84. Ritz-Galerkin discretization 23 <latexit sha1_base64="KPkzMO0bzgStJqDXXaeq6O1g2KA=">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</latexit> vA i Ri + ⇢i = 1 X l=1 V(l) i · Y(l 1)(ˆ ⇢i ), boundary velocity v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) Mazur and Van Saarloos, Physica A 1982; Hess 2015; RS et al. JSTAT 2015, PRL 2016, JOSS 2020 Expansion of boundary fields in tensorial spherical harmonics Y(l) - dimensionless, symmetric, irreducible Cartesian tensors of rank l that form a complete, orthogonal basis on the sphere Y(l)(ˆ ⇢) = ( 1)l⇢l+1r(l)⇢ 1 <latexit sha1_base64="f9kVoPJo+AjtI2MEG3lUOl9ykEY=">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</latexit> Y (0) = 1, Y (1) ↵ = ˆ ⇢↵, Y (2) ↵ = 3ˆ ⇢↵ ˆ ⇢ ↵ <latexit sha1_base64="Jl+Izs7/1Vhgnr7jO3oOGsUpQXo=">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</latexit> v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi irreducible coefficients <latexit sha1_base64="iQHbbcsmxQA6COVfndimfSPtIt8=">AAACD3icbVC5TgMxFPRyhnAFKGksIlBool0EgjKChjJI5JCyIfI6bxMr3kP2W0S02j+g4VdoKECIlpaOv8E5CkgYydJo5j17PF4shUbb/rYWFpeWV1Zza/n1jc2t7cLObl1HieJQ45GMVNNjGqQIoYYCJTRjBSzwJDS8wdXIb9yD0iIKb3EYQztgvVD4gjM0Uqdw5CI84PieVEE3S92AYd/zaf0uLUlXi17AjrOsUyjaZXsMOk+cKSmSKaqdwpfbjXgSQIhcMq1bjh1jO2UKBZeQ5d1EQ8z4gPWgZWjIAtDtdJwjo4dG6VI/UuaESMfq742UBVoPA89MjtLqWW8k/ue1EvQv2qkI4wQh5JOH/ERSjOioHNoVCjjKoSGMK2GyUt5ninE0FeZNCc7sl+dJ/aTsnJXtm9Ni5XJaR47skwNSIg45JxVyTaqkRjh5JM/klbxZT9aL9W59TEYXrOnOHvkD6/MHLHWdWg==</latexit> V(l )
85. Ritz-Galerkin discretization 23 Fluid flow due to the irreducible mode lσ known body forces and torques FP TP known slip coefficients <latexit sha1_base64="KnWT5FtbfEMMkcOCh8BAgABqYNk=">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</latexit> v(r) = G(1s) · FP + G(2a) · TP + X l =2s ⇧(l ) V(l ) <latexit sha1_base64="KPkzMO0bzgStJqDXXaeq6O1g2KA=">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</latexit> vA i Ri + ⇢i = 1 X l=1 V(l) i · Y(l 1)(ˆ ⇢i ), boundary velocity v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) Mazur and Van Saarloos, Physica A 1982; Hess 2015; RS et al. JSTAT 2015, PRL 2016, JOSS 2020 Expansion of boundary fields in tensorial spherical harmonics Y(l) - dimensionless, symmetric, irreducible Cartesian tensors of rank l that form a complete, orthogonal basis on the sphere Y(l)(ˆ ⇢) = ( 1)l⇢l+1r(l)⇢ 1 <latexit sha1_base64="f9kVoPJo+AjtI2MEG3lUOl9ykEY=">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</latexit> Y (0) = 1, Y (1) ↵ = ˆ ⇢↵, Y (2) ↵ = 3ˆ ⇢↵ ˆ ⇢ ↵ <latexit sha1_base64="Jl+Izs7/1Vhgnr7jO3oOGsUpQXo=">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</latexit> v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi irreducible coefficients <latexit sha1_base64="iQHbbcsmxQA6COVfndimfSPtIt8=">AAACD3icbVC5TgMxFPRyhnAFKGksIlBool0EgjKChjJI5JCyIfI6bxMr3kP2W0S02j+g4VdoKECIlpaOv8E5CkgYydJo5j17PF4shUbb/rYWFpeWV1Zza/n1jc2t7cLObl1HieJQ45GMVNNjGqQIoYYCJTRjBSzwJDS8wdXIb9yD0iIKb3EYQztgvVD4gjM0Uqdw5CI84PieVEE3S92AYd/zaf0uLUlXi17AjrOsUyjaZXsMOk+cKSmSKaqdwpfbjXgSQIhcMq1bjh1jO2UKBZeQ5d1EQ8z4gPWgZWjIAtDtdJwjo4dG6VI/UuaESMfq742UBVoPA89MjtLqWW8k/ue1EvQv2qkI4wQh5JOH/ERSjOioHNoVCjjKoSGMK2GyUt5ninE0FeZNCc7sl+dJ/aTsnJXtm9Ni5XJaR47skwNSIg45JxVyTaqkRjh5JM/klbxZT9aL9W59TEYXrOnOHvkD6/MHLHWdWg==</latexit> V(l )
86. l Symmetric (ls) Antisymmetric (la) Trace (lt) 1 2 3 Fluid flow due to the irreducible mode lσ RS et al. JSTAT 2015, PRL 2016, JOSS 2020 24
87. l Symmetric (ls) Antisymmetric (la) Trace (lt) 1 2 3 SO(3) invariant way to classify active flows. The -th mode of the flow decays as in an unbounded geometry. It has three independent terms: symmetric irreducible gradients of a Green’s function G of Stokes equation its curl and, its Laplacian. l v ∝ r−l ∇l−2(∇ × G) ∇(l−1)G ∇l−3(∇2G) Fluid flow due to the irreducible mode lσ RS et al. JSTAT 2015, PRL 2016, JOSS 2020 24
88. l Symmetric (ls) Antisymmetric (la) Trace (lt) 1 2 3 SO(3) invariant way to classify active flows. The -th mode of the flow decays as in an unbounded geometry. It has three independent terms: symmetric irreducible gradients of a Green’s function G of Stokes equation its curl and, its Laplacian. l v ∝ r−l ∇l−2(∇ × G) ∇(l−1)G ∇l−3(∇2G) Fluid flow due to the irreducible mode lσ RS et al. JSTAT 2015, PRL 2016, JOSS 2020 Charged colloid in a non-conducting fluid 24
89. l Symmetric (ls) Antisymmetric (la) Trace (lt) 1 2 3 SO(3) invariant way to classify active flows. The -th mode of the flow decays as in an unbounded geometry. It has three independent terms: symmetric irreducible gradients of a Green’s function G of Stokes equation its curl and, its Laplacian. l v ∝ r−l ∇l−2(∇ × G) ∇(l−1)G ∇l−3(∇2G) Fluid flow due to the irreducible mode lσ RS et al. JSTAT 2015, PRL 2016, JOSS 2020 Charged colloid in a non-conducting fluid Charged colloid in a conducting fluid 24
90. 25 Lighthill, CPAM 1952; Blake JFM 1971; RS et al. PRL 2016, JPC 2018 Generalized Stokes laws active slip boundary condition v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) irreducible slip coefficients: <latexit sha1_base64="iQHbbcsmxQA6COVfndimfSPtIt8=">AAACD3icbVC5TgMxFPRyhnAFKGksIlBool0EgjKChjJI5JCyIfI6bxMr3kP2W0S02j+g4VdoKECIlpaOv8E5CkgYydJo5j17PF4shUbb/rYWFpeWV1Zza/n1jc2t7cLObl1HieJQ45GMVNNjGqQIoYYCJTRjBSzwJDS8wdXIb9yD0iIKb3EYQztgvVD4gjM0Uqdw5CI84PieVEE3S92AYd/zaf0uLUlXi17AjrOsUyjaZXsMOk+cKSmSKaqdwpfbjXgSQIhcMq1bjh1jO2UKBZeQ5d1EQ8z4gPWgZWjIAtDtdJwjo4dG6VI/UuaESMfq742UBVoPA89MjtLqWW8k/ue1EvQv2qkI4wQh5JOH/ERSjOioHNoVCjjKoSGMK2GyUt5ninE0FeZNCc7sl+dJ/aTsnJXtm9Ni5XJaR47skwNSIg45JxVyTaqkRjh5JM/klbxZT9aL9W59TEYXrOnOHvkD6/MHLHWdWg==</latexit> V(l )
91. 25 FH i = T T ij ·V i T R ij ·⌦ i 1 X l =1s (T, l ) ij · V(l ) j , TH i = RT ij ·V i RR ij ·⌦ i 1 X l =1s (R, l ) ij · V(l ) j . generalised friction tensors for slip boundary condition Solution of boundary integral equation gives Lighthill, CPAM 1952; Blake JFM 1971; RS et al. PRL 2016, JPC 2018 Generalized Stokes laws active slip boundary condition v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) irreducible slip coefficients: <latexit sha1_base64="iQHbbcsmxQA6COVfndimfSPtIt8=">AAACD3icbVC5TgMxFPRyhnAFKGksIlBool0EgjKChjJI5JCyIfI6bxMr3kP2W0S02j+g4VdoKECIlpaOv8E5CkgYydJo5j17PF4shUbb/rYWFpeWV1Zza/n1jc2t7cLObl1HieJQ45GMVNNjGqQIoYYCJTRjBSzwJDS8wdXIb9yD0iIKb3EYQztgvVD4gjM0Uqdw5CI84PieVEE3S92AYd/zaf0uLUlXi17AjrOsUyjaZXsMOk+cKSmSKaqdwpfbjXgSQIhcMq1bjh1jO2UKBZeQ5d1EQ8z4gPWgZWjIAtDtdJwjo4dG6VI/UuaESMfq742UBVoPA89MjtLqWW8k/ue1EvQv2qkI4wQh5JOH/ERSjOioHNoVCjjKoSGMK2GyUt5ninE0FeZNCc7sl+dJ/aTsnJXtm9Ni5XJaR47skwNSIg45JxVyTaqkRjh5JM/klbxZT9aL9W59TEYXrOnOHvkD6/MHLHWdWg==</latexit> V(l )
92. 25 FH i = T T ij ·V i T R ij ·⌦ i 1 X l =1s (T, l ) ij · V(l ) j , TH i = RT ij ·V i RR ij ·⌦ i 1 X l =1s (R, l ) ij · V(l ) j . generalised friction tensors for slip boundary condition Solution of boundary integral equation gives Lighthill, CPAM 1952; Blake JFM 1971; RS et al. PRL 2016, JPC 2018 Generalized Stokes laws Consistent with the linearity of Stokes flow The generalised friction tensors relate the modes of slip and hydrodynamic forces The expression for the generalised friction tensors is obtained in terms of a Green’s function of the Stokes equation active slip boundary condition v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) irreducible slip coefficients: <latexit sha1_base64="iQHbbcsmxQA6COVfndimfSPtIt8=">AAACD3icbVC5TgMxFPRyhnAFKGksIlBool0EgjKChjJI5JCyIfI6bxMr3kP2W0S02j+g4VdoKECIlpaOv8E5CkgYydJo5j17PF4shUbb/rYWFpeWV1Zza/n1jc2t7cLObl1HieJQ45GMVNNjGqQIoYYCJTRjBSzwJDS8wdXIb9yD0iIKb3EYQztgvVD4gjM0Uqdw5CI84PieVEE3S92AYd/zaf0uLUlXi17AjrOsUyjaZXsMOk+cKSmSKaqdwpfbjXgSQIhcMq1bjh1jO2UKBZeQ5d1EQ8z4gPWgZWjIAtDtdJwjo4dG6VI/UuaESMfq742UBVoPA89MjtLqWW8k/ue1EvQv2qkI4wQh5JOH/ERSjOioHNoVCjjKoSGMK2GyUt5ninE0FeZNCc7sl+dJ/aTsnJXtm9Ni5XJaR47skwNSIg45JxVyTaqkRjh5JM/klbxZT9aL9W59TEYXrOnOHvkD6/MHLHWdWg==</latexit> V(l )
93. 25 FH i = T T ij ·V i T R ij ·⌦ i 1 X l =1s (T, l ) ij · V(l ) j , TH i = RT ij ·V i RR ij ·⌦ i 1 X l =1s (R, l ) ij · V(l ) j . generalised friction tensors for slip boundary condition Solution of boundary integral equation gives Lighthill, CPAM 1952; Blake JFM 1971; RS et al. PRL 2016, JPC 2018 Generalized Stokes laws Consistent with the linearity of Stokes flow The generalised friction tensors relate the modes of slip and hydrodynamic forces The expression for the generalised friction tensors is obtained in terms of a Green’s function of the Stokes equation We use the above in Newton’s laws to obtain the rigid body motion FH i + FP i + ˆ F i = 0, TH i + TP i + ˆ T i = 0. Body Brownian Hydrodynamic active slip boundary condition v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) irreducible slip coefficients: <latexit sha1_base64="iQHbbcsmxQA6COVfndimfSPtIt8=">AAACD3icbVC5TgMxFPRyhnAFKGksIlBool0EgjKChjJI5JCyIfI6bxMr3kP2W0S02j+g4VdoKECIlpaOv8E5CkgYydJo5j17PF4shUbb/rYWFpeWV1Zza/n1jc2t7cLObl1HieJQ45GMVNNjGqQIoYYCJTRjBSzwJDS8wdXIb9yD0iIKb3EYQztgvVD4gjM0Uqdw5CI84PieVEE3S92AYd/zaf0uLUlXi17AjrOsUyjaZXsMOk+cKSmSKaqdwpfbjXgSQIhcMq1bjh1jO2UKBZeQ5d1EQ8z4gPWgZWjIAtDtdJwjo4dG6VI/UuaESMfq742UBVoPA89MjtLqWW8k/ue1EvQv2qkI4wQh5JOH/ERSjOioHNoVCjjKoSGMK2GyUt5ninE0FeZNCc7sl+dJ/aTsnJXtm9Ni5XJaR47skwNSIg45JxVyTaqkRjh5JM/klbxZT9aL9W59TEYXrOnOHvkD6/MHLHWdWg==</latexit> V(l )
94. 26 Active Brownian motion: theory FH i + FP i + ˆ F i = 0, TH i + Body Brownian Hydrodynamic T T ij ·Vj T R ij · ⌦j + FP i + ˆ Fi 1 X l =1s (T, l ) ij · V(l ) j = 0, RT ij ·Vj RR ij · ⌦j + TP i + ˆ Ti 1 X l =1s (R, l ) ij · V(l ) j = 0. Brownian Body Mazur and Van Saarloos, Physica A 1982; Ladd, JCP 1988; RS et al. JSTAT 2015, PRL 2016, JPC 2018
95. 26 Active Brownian motion: theory FH i + FP i + ˆ F i = 0, TH i + Body Brownian Hydrodynamic invert for rigid body motion T T ij ·Vj T R ij · ⌦j + FP i + ˆ Fi 1 X l =1s (T, l ) ij · V(l ) j = 0, RT ij ·Vj RR ij · ⌦j + TP i + ˆ Ti 1 X l =1s (R, l ) ij · V(l ) j = 0. Brownian Body Mazur and Van Saarloos, Physica A 1982; Ladd, JCP 1988; RS et al. JSTAT 2015, PRL 2016, JPC 2018
96. 26 Active Brownian motion: theory FH i + FP i + ˆ F i = 0, TH i + Body Brownian Hydrodynamic Propulsion tensors new tensors that relate modes of slip to rigid body motion Mobility matrices well-known for passive suspension Vi = µT T ij · FP j + µT R ij · TP j + q 2kBTµT T ij · ⌘T j + q 2kBTµT R ij · ⇣R j + 1 X l =2s ⇡(T, l ) ij · V(l ) j + VA i ⌦i = µRT ij · FP j + µRR ij · TP j | {z } Passive + q 2kBTµRT ij · ⇣T j + q 2kBTµRR ij · ⌘R j | {z } Brownian + 1 X l =2s ⇡(R, l ) ij · V(l ) j + ⌦A i | {z } Active there are infinitely many propulsion tensors in contrast to the four mobility matrices White noises the correlated noise does not depend on propulsion tensors invert for rigid body motion T T ij ·Vj T R ij · ⌦j + FP i + ˆ Fi 1 X l =1s (T, l ) ij · V(l ) j = 0, RT ij ·Vj RR ij · ⌦j + TP i + ˆ Ti 1 X l =1s (R, l ) ij · V(l ) j = 0. Brownian Body Mazur and Van Saarloos, Physica A 1982; Ladd, JCP 1988; RS et al. JSTAT 2015, PRL 2016, JPC 2018
97. 26 Active Brownian motion: theory FH i + FP i + ˆ F i = 0, TH i + Body Brownian Hydrodynamic Propulsion tensors new tensors that relate modes of slip to rigid body motion Mobility matrices well-known for passive suspension Vi = µT T ij · FP j + µT R ij · TP j + q 2kBTµT T ij · ⌘T j + q 2kBTµT R ij · ⇣R j + 1 X l =2s ⇡(T, l ) ij · V(l ) j + VA i ⌦i = µRT ij · FP j + µRR ij · TP j | {z } Passive + q 2kBTµRT ij · ⇣T j + q 2kBTµRR ij · ⌘R j | {z } Brownian + 1 X l =2s ⇡(R, l ) ij · V(l ) j + ⌦A i | {z } Active there are infinitely many propulsion tensors in contrast to the four mobility matrices White noises the correlated noise does not depend on propulsion tensors invert for rigid body motion T T ij ·Vj T R ij · ⌦j + FP i + ˆ Fi 1 X l =1s (T, l ) ij · V(l ) j = 0, RT ij ·Vj RR ij · ⌦j + TP i + ˆ Ti 1 X l =1s (R, l ) ij · V(l ) j = 0. Brownian Body Mazur and Van Saarloos, Physica A 1982; Ladd, JCP 1988; RS et al. JSTAT 2015, PRL 2016, JPC 2018
98. Known: chemical surface flux at the particle boundaries jA = D⇢ · rc <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> Desired: rigid body motion of the particles 27 RS et al. JPC 2018, JCP 2019, JOSS 2020 What about phoretic interactions?
99. Known: chemical surface flux at the particle boundaries jA = D⇢ · rc <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> Desired: rigid body motion of the particles 27 RS et al. JPC 2018, JCP 2019, JOSS 2020 What about phoretic interactions?
100. Known: chemical surface flux at the particle boundaries jA = D⇢ · rc <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> Desired: rigid body motion of the particles 27 Hydrodynamic interactions Exterior Fluid Flow Phoretic interactions Many-Body Slip Chemical Surface Flux RS et al. JPC 2018, JCP 2019, JOSS 2020 What about phoretic interactions?
101. Known: chemical surface flux at the particle boundaries jA = D⇢ · rc <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> <latexit sha1_base64="SGSU0cST+ShgkVmul3CrdpmMehQ=">AAACI3icbVDLSgMxFM34rPVVdekmWAQ3lhkRFEGoj4XLCrYWOrXcyaRtbCYZkoxQhv6LG3/FjQuluHHhv5hpu6jWAyGHc+7l3nuCmDNtXPfLmZtfWFxazq3kV9fWNzYLW9s1LRNFaJVILlU9AE05E7RqmOG0HisKUcDpfdC7yvz7J6o0k+LO9GPajKAjWJsRMFZqFc4eH1I/AtMlwPHF4Pzw2g8kD3U/sp+vutInoTRTWuoLCDgMSL5VKLoldwQ8S7wJKaIJKq3C0A8lSSIqDOGgdcNzY9NMQRlGOB3k/UTTGEgPOrRhqYCI6mY6unGA960S4rZU9gmDR+p0RwqRzja0ldk5+q+Xif95jcS0T5spE3FiqCDjQe2EYyNxFhgOmaLE8L4lQBSzu2LSBQXE2FizELy/J8+S2lHJc0ve7XGxfDmJI4d20R46QB46QWV0gyqoigh6Rq/oHX04L86bM3Q+x6VzzqRnB/2C8/0Dqp6lhA==</latexit> Desired: rigid body motion of the particles 27 Hydrodynamic interactions Exterior Fluid Flow Phoretic interactions Many-Body Slip Chemical Surface Flux RS et al. JPC 2018, JCP 2019, JOSS 2020 What about phoretic interactions?
102. RS et al. JCP 2019, JOSS 2020 Active Brownian motion: simulations ˙ Ri = Vi, ˙ pi = ⌦i ⇥ pi . ‣ The positions and orientations of the colloids are updated as ‣ The Steric repulsion is modelled using the truncated Lennard-Jones potential ‣ Problem reduced to the choice of a Green’s function and surface flux (or slip) ‣ The slip is fixed by choosing leading modes which match experimental data ‣ The boundary conditions in the bulk flow is implemented by choosing appropriate Green’s functions. No need to simulate the fluid explicitly, just like in Coulomb's law for evaluating electrostatic interactions 28 28
103. RS et al. JCP 2019, JOSS 2020 Active Brownian motion: simulations ˙ Ri = Vi, ˙ pi = ⌦i ⇥ pi . ‣ The positions and orientations of the colloids are updated as ‣ The Steric repulsion is modelled using the truncated Lennard-Jones potential ‣ Problem reduced to the choice of a Green’s function and surface flux (or slip) ‣ The slip is fixed by choosing leading modes which match experimental data ‣ The boundary conditions in the bulk flow is implemented by choosing appropriate Green’s functions. No need to simulate the fluid explicitly, just like in Coulomb's law for evaluating electrostatic interactions Experimental flow Theoretical flow slip expansion truncated to l=3. Thutupalli, Geyer, RS, Adhikari, and Stone, PNAS 2018 28 28
104. RS et al. JCP 2019, JOSS 2020 Active Brownian motion: simulations ˙ Ri = Vi, ˙ pi = ⌦i ⇥ pi . ‣ The positions and orientations of the colloids are updated as ‣ The Steric repulsion is modelled using the truncated Lennard-Jones potential ‣ Problem reduced to the choice of a Green’s function and surface flux (or slip) ‣ The slip is fixed by choosing leading modes which match experimental data ‣ The boundary conditions in the bulk flow is implemented by choosing appropriate Green’s functions. No need to simulate the fluid explicitly, just like in Coulomb's law for evaluating electrostatic interactions GitHub.com/rajeshrinet/PyStokes [20K downloads] Experimental flow Theoretical flow slip expansion truncated to l=3. Thutupalli, Geyer, RS, Adhikari, and Stone, PNAS 2018 28 28
105. RS et al. JCP 2019, JOSS 2020 Active Brownian motion: simulations ˙ Ri = Vi, ˙ pi = ⌦i ⇥ pi . ‣ The positions and orientations of the colloids are updated as ‣ The Steric repulsion is modelled using the truncated Lennard-Jones potential ‣ Problem reduced to the choice of a Green’s function and surface flux (or slip) ‣ The slip is fixed by choosing leading modes which match experimental data ‣ The boundary conditions in the bulk flow is implemented by choosing appropriate Green’s functions. No need to simulate the fluid explicitly, just like in Coulomb's law for evaluating electrostatic interactions GitHub.com/rajeshrinet/PyStokes [20K downloads] Experimental flow Theoretical flow slip expansion truncated to l=3. Thutupalli, Geyer, RS, Adhikari, and Stone, PNAS 2018 28 PNAS 2018 - with the labs of Dr. Thutupalli (Bangalore) and Prof. Stone (Princeton) J Phy Chem C 2018 - with the lab of Prof. T Pradeep at IIT Madras Sci Rep and Phy Rev E 2017 - with the lab of Prof. Banerjee at IISER Kolkata PRL 2020 - with the lab of Prof. Eiser at Cavendish Laboratory, University of Cambridge PRL 2020: Hamiltonian description of green algae Volvox dance near boundaries due to HI 28
106. RS et al. JCP 2019, JOSS 2020 Active Brownian motion: simulations ˙ Ri = Vi, ˙ pi = ⌦i ⇥ pi . ‣ The positions and orientations of the colloids are updated as ‣ The Steric repulsion is modelled using the truncated Lennard-Jones potential ‣ Problem reduced to the choice of a Green’s function and surface flux (or slip) ‣ The slip is fixed by choosing leading modes which match experimental data ‣ The boundary conditions in the bulk flow is implemented by choosing appropriate Green’s functions. No need to simulate the fluid explicitly, just like in Coulomb's law for evaluating electrostatic interactions GitHub.com/rajeshrinet/PyStokes [20K downloads] Experimental flow Theoretical flow slip expansion truncated to l=3. Thutupalli, Geyer, RS, Adhikari, and Stone, PNAS 2018 28 PNAS 2018 - with the labs of Dr. Thutupalli (Bangalore) and Prof. Stone (Princeton) J Phy Chem C 2018 - with the lab of Prof. T Pradeep at IIT Madras Sci Rep and Phy Rev E 2017 - with the lab of Prof. Banerjee at IISER Kolkata PRL 2020 - with the lab of Prof. Eiser at Cavendish Laboratory, University of Cambridge PRL 2020: Hamiltonian description of green algae Volvox dance near boundaries due to HI 28
107. Colloids tethered to an interface - free to move in the plane Caciagli, RS et al. PRL 2020 D Joshi Erika Eiser A Caciagli 29
108. Colloids tethered to an interface - free to move in the plane Caciagli, RS et al. PRL 2020 D Joshi Erika Eiser A Caciagli 29
109. 30 Optically trap one of the colloids and study the optofluidic interactions Caciagli, RS et al. PRL 2020
110. 30 Optically trap one of the colloids and study the optofluidic interactions Caciagli, RS et al. PRL 2020
111. 30 Optically trap one of the colloids and study the optofluidic interactions Caciagli, RS et al. PRL 2020
112. 31 Water Oil Puzzle: what causes motion into the hot region? Caciagli, RS et al. PRL 2020
113. 31 Water Oil <latexit sha1_base64="egp3jPbevT+u1Hbj6Umu6KE9ihI=">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</latexit> FT = µT µ? rT1 1 Thermophoresis into the interface Puzzle: what causes motion into the hot region? Caciagli, RS et al. PRL 2020
114. 31 Water Oil FP 1 <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> Monopolar flow once the colloid is stalled <latexit sha1_base64="egp3jPbevT+u1Hbj6Umu6KE9ihI=">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</latexit> FT = µT µ? rT1 1 Thermophoresis into the interface Puzzle: what causes motion into the hot region? Caciagli, RS et al. PRL 2020
115. 31 Water Oil FP 1 <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> Monopolar flow once the colloid is stalled <latexit sha1_base64="egp3jPbevT+u1Hbj6Umu6KE9ihI=">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</latexit> FT = µT µ? rT1 1 Thermophoresis into the interface Puzzle: what causes motion into the hot region? Flow-induced attraction FH FH 0 1 Caciagli, RS et al. PRL 2020
116. 31 Water Oil FP 1 <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> Monopolar flow once the colloid is stalled <latexit sha1_base64="egp3jPbevT+u1Hbj6Umu6KE9ihI=">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</latexit> FT = µT µ? rT1 1 Thermophoresis into the interface Puzzle: what causes motion into the hot region? Flow-induced attraction FH FH 0 1 Caciagli, RS et al. PRL 2020
117. 31 Water Oil FP 1 <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> Monopolar flow once the colloid is stalled <latexit sha1_base64="egp3jPbevT+u1Hbj6Umu6KE9ihI=">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</latexit> FT = µT µ? rT1 1 Thermophoresis into the interface Puzzle: what causes motion into the hot region? The optofluidic force can written as the gradient of a potential. FH Flow-induced attraction FH FH 0 1 Caciagli, RS et al. PRL 2020
118. 31 Water Oil FP 1 <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> <latexit sha1_base64="DBROXI55FdFnyJi/ukcMpAI2ilw=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiQi6LIoiMsK9gFtDJPppB06mYSZSaGE/IkbF4q49U/c+TdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nrep2bWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3BZ+d0qlYrF41LOEehEeCRYygrWRfNseRFiPgzC7y333KWvlvl13Gs4caJW4JalDiZZvfw2GMUkjKjThWKm+6yTay7DUjHCa1wapogkmEzyifUMFjqjysnnyHJ0ZZYjCWJonNJqrvzcyHCk1iwIzWeRUy14h/uf1Ux1eexkTSaqpIItDYcqRjlFRAxoySYnmM0MwkcxkRWSMJSbalFUzJbjLX14lnYuG6zTch8t686asowoncArn4MIVNOEeWtAGAlN4hld4szLrxXq3PhajFavcOYY/sD5/AH5fk40=</latexit> Monopolar flow once the colloid is stalled <latexit sha1_base64="egp3jPbevT+u1Hbj6Umu6KE9ihI=">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</latexit> FT = µT µ? rT1 1 Thermophoresis into the interface Puzzle: what causes motion into the hot region? The optofluidic force can written as the gradient of a potential. FH Flow-induced attraction FH FH 0 1 Caciagli, RS et al. PRL 2020 An attractive optofluidic potential centred about the hot region <latexit sha1_base64="jIlCeInyuCr6GnNciEj6veDMO2s=">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</latexit> (r) = µT 4⇡⌘µ?µk  1 1 + h r⇤ + 2 1 + h3 r⇤3 @zT1.
119. Summary: field-theoretic and particle-based theories of active matter Incomplete phase separation in scalar active matter A self-shearing instability (SSI), due to active contractile stress, interrupts growth of droplets by splitting them. The result is a dynamic steady-state maintained by the self- shearing instability and Ostwald ripening. SSI Ostwald ripening Analytical and numerics-friendly formalism to study phoresis and Stokesian hydrodynamics of colloids with surface slip. The non-equilibrium steady state due to heating (freezing by heating) admits an effective equilibrium description 32 Field-theoretic Particle-based theories of active matter Particle-based Phoresis and Stokesian hydrodynamics without resolving fluid degrees of freedom