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Theories without time-reversal symmetry in soft matter

Theories without time-reversal symmetry in soft matter

C54eed30623e5d69ba809b96832b8d30?s=128

Rajesh Singh

July 20, 2021
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  1. Rajesh Singh Department of Applied Mathematics and Theoretical Physics Theories

    without time-reversal symmetry in soft matter
  2. MSc (Phy), IITK 08/10 - 06/12 PhD, IMSc Chennai 08/12

    - 03/18 Postdoc, University of Cambridge 03/18 - to date 2
  3. MSc (Phy), IITK 08/10 - 06/12 PhD, IMSc Chennai 08/12

    - 03/18 Postdoc, University of Cambridge 03/18 - to date Multi-lane exclusion models (driven particle systems on a lattice): JSTAT 2012 2
  4. MSc (Phy), IITK 08/10 - 06/12 PhD, IMSc Chennai 08/12

    - 03/18 Postdoc, University of Cambridge 03/18 - to date Hydrodynamic interactions (HI) of colloidal spheres with slip: JSTAT 2015, PRL 2016, EJCM 2017, JPC 2018 Multi-lane exclusion models (driven particle systems on a lattice): JSTAT 2012 2
  5. MSc (Phy), IITK 08/10 - 06/12 PhD, IMSc Chennai 08/12

    - 03/18 Postdoc, University of Cambridge 03/18 - to date Phoretic interactions of colloidal spheres with slip: JCP 2019, JOSS 2020 Hydrodynamic interactions (HI) of colloidal spheres with slip: JSTAT 2015, PRL 2016, EJCM 2017, JPC 2018 Multi-lane exclusion models (driven particle systems on a lattice): JSTAT 2012 2
  6. MSc (Phy), IITK 08/10 - 06/12 PhD, IMSc Chennai 08/12

    - 03/18 Postdoc, University of Cambridge 03/18 - to date Phoretic interactions of colloidal spheres with slip: JCP 2019, JOSS 2020 Applications PRE 2017#, FDT of two colloids with HI; JPC C 2018#: electrohydrodynamic fl ows; PNAS 2018#: boundaries control collective dynamics; PRL 2020#: freezing colloids by heating them; PRL 2020: Hamiltonian description of green algae Volvox dance near boundaries due to HI #: direct experimental collaboration Hydrodynamic interactions (HI) of colloidal spheres with slip: JSTAT 2015, PRL 2016, EJCM 2017, JPC 2018 Multi-lane exclusion models (driven particle systems on a lattice): JSTAT 2012 2
  7. MSc (Phy), IITK 08/10 - 06/12 PhD, IMSc Chennai 08/12

    - 03/18 Postdoc, University of Cambridge 03/18 - to date Phoretic interactions of colloidal spheres with slip: JCP 2019, JOSS 2020 Applications PRE 2017#, FDT of two colloids with HI; JPC C 2018#: electrohydrodynamic fl ows; PNAS 2018#: boundaries control collective dynamics; PRL 2020#: freezing colloids by heating them; PRL 2020: Hamiltonian description of green algae Volvox dance near boundaries due to HI #: direct experimental collaboration Hydrodynamic interactions (HI) of colloidal spheres with slip: JSTAT 2015, PRL 2016, EJCM 2017, JPC 2018 Multi-lane exclusion models (driven particle systems on a lattice): JSTAT 2012 A novel self-shearing instability interrupts growth of droplets by splitting them - due to a stress of nonequilibrium origin - leading to a dynamic steady-state: PRL 2019 The fi rst phase- fi eld modelling of self- propulsion of cells using scalar order parameters alone: PRR 2020 Active scalar fi eld theory 2
  8. MSc (Phy), IITK 08/10 - 06/12 PhD, IMSc Chennai 08/12

    - 03/18 Postdoc, University of Cambridge 03/18 - to date Phoretic interactions of colloidal spheres with slip: JCP 2019, JOSS 2020 Applications PRE 2017#, FDT of two colloids with HI; JPC C 2018#: electrohydrodynamic fl ows; PNAS 2018#: boundaries control collective dynamics; PRL 2020#: freezing colloids by heating them; PRL 2020: Hamiltonian description of green algae Volvox dance near boundaries due to HI #: direct experimental collaboration Hydrodynamic interactions (HI) of colloidal spheres with slip: JSTAT 2015, PRL 2016, EJCM 2017, JPC 2018 Multi-lane exclusion models (driven particle systems on a lattice): JSTAT 2012 Bayesian inference of the stationary Gauss- Markov process: SciRep 2017#, PRE 2018; and a non-stationary Gauss-Markov process: arXiv:2005.09625, 2010.11783 A Ritz method for fi nding most probable transition paths between meta-stable states of stochastic dynamical systems: PRR 2020 Stochastic processes A novel self-shearing instability interrupts growth of droplets by splitting them - due to a stress of nonequilibrium origin - leading to a dynamic steady-state: PRL 2019 The fi rst phase- fi eld modelling of self- propulsion of cells using scalar order parameters alone: PRR 2020 Active scalar fi eld theory 2
  9. Plan of the talk Experimental observation of incomplete phase separation

    in active matter Theory of binary fl uid 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: fi eld-theoretic and particle-based theories of active matter
  10. This work has been done with Michael E. Cates I.

    Incomplete phase separation in scalar active matter
  11. micrometer size - non-equilibrium processes on the surface create exterior

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

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

    fl ow and may lead to self-propulsion feeding or fuel => break time-reversal symmetry locally Active particles: special colloids 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 fl uid
  14. 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
  15. 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?
  16. 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 fi eld number of active particles is conserved - use a conserved scalar fi eld
  17. 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 fi eld number of active particles is conserved - use a conserved scalar fi eld start with a theory of phase separation in binary mixtures add minimal terms which break TRS recipe to build an active fi eld theory
  18. 7 How do binary mixtures (such as oil-water) phase separate

    from a homogenous mixed phase? Theory of coarsening kinetics in binary fl uid mixtures Bray, Adv. in Phys. 43, 357 (1994)
  19. 7 How do binary mixtures (such as oil-water) phase separate

    from a homogenous mixed phase? Theory of coarsening kinetics in binary fl uid mixtures Consider a scalar fi eld describing the local composition in a symmetric binary mixture <latexit sha1_base64="eq5CDmOoRMQIIXGe5NNNGnPlUps=">AAACJnicbVDLSsNAFJ3UV62vqEs3wSJUxJKIoptCrRuXFewDmhAm00k7dDIJMxOhhHyNG3/FjYuKiDs/xUkbirYeGDiccy93zvEiSoQ0zS+tsLK6tr5R3Cxtbe/s7un7B20RxhzhFgppyLseFJgShluSSIq7Eccw8CjueKO7zO88YS5IyB7lOMJOAAeM+ARBqSRXr9nRkFTsAMqh5yc8Pa3ZPocosfkwdJPb9HxGGmk6l87mUsnVy2bVnMJYJlZOyiBH09Undj9EcYCZRBQK0bPMSDoJ5JIgitOSHQscQTSCA9xTlMEACyeZxkyNE6X0DT/k6jFpTNXfGwkMhBgHnprM8ohFLxP/83qx9G+chLAolpih2SE/poYMjawzo084RpKOFYGIE/VXAw2hqkmqZrMSrMXIy6R9UbWuqubDZbneyOsogiNwDCrAAtegDu5BE7QAAs/gFUzAu/aivWkf2udstKDlO4fgD7TvH/6VprQ=</latexit> (r) = ⇢A ⇢B ⇢A + ⇢B = 1 = +1 <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> B-phase A-phase Bray, Adv. in Phys. 43, 357 (1994)
  20. 7 How do binary mixtures (such as oil-water) phase separate

    from a homogenous mixed phase? Theory of coarsening kinetics in binary fl uid mixtures Consider a scalar fi eld describing the local composition in a symmetric binary mixture <latexit sha1_base64="eq5CDmOoRMQIIXGe5NNNGnPlUps=">AAACJnicbVDLSsNAFJ3UV62vqEs3wSJUxJKIoptCrRuXFewDmhAm00k7dDIJMxOhhHyNG3/FjYuKiDs/xUkbirYeGDiccy93zvEiSoQ0zS+tsLK6tr5R3Cxtbe/s7un7B20RxhzhFgppyLseFJgShluSSIq7Eccw8CjueKO7zO88YS5IyB7lOMJOAAeM+ARBqSRXr9nRkFTsAMqh5yc8Pa3ZPocosfkwdJPb9HxGGmk6l87mUsnVy2bVnMJYJlZOyiBH09Undj9EcYCZRBQK0bPMSDoJ5JIgitOSHQscQTSCA9xTlMEACyeZxkyNE6X0DT/k6jFpTNXfGwkMhBgHnprM8ohFLxP/83qx9G+chLAolpih2SE/poYMjawzo084RpKOFYGIE/VXAw2hqkmqZrMSrMXIy6R9UbWuqubDZbneyOsogiNwDCrAAtegDu5BE7QAAs/gFUzAu/aivWkf2udstKDlO4fgD7TvH/6VprQ=</latexit> (r) = ⇢A ⇢B ⇢A + ⇢B = 1 = +1 <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> B-phase A-phase The scalar fi eld is conserved: ˙ = r · J Bray, Adv. in Phys. 43, 357 (1994)
  21. 7 How do binary mixtures (such as oil-water) phase separate

    from a homogenous mixed phase? Theory of coarsening kinetics in binary fl uid mixtures Consider a scalar fi eld describing the local composition in a symmetric binary mixture <latexit sha1_base64="eq5CDmOoRMQIIXGe5NNNGnPlUps=">AAACJnicbVDLSsNAFJ3UV62vqEs3wSJUxJKIoptCrRuXFewDmhAm00k7dDIJMxOhhHyNG3/FjYuKiDs/xUkbirYeGDiccy93zvEiSoQ0zS+tsLK6tr5R3Cxtbe/s7un7B20RxhzhFgppyLseFJgShluSSIq7Eccw8CjueKO7zO88YS5IyB7lOMJOAAeM+ARBqSRXr9nRkFTsAMqh5yc8Pa3ZPocosfkwdJPb9HxGGmk6l87mUsnVy2bVnMJYJlZOyiBH09Undj9EcYCZRBQK0bPMSDoJ5JIgitOSHQscQTSCA9xTlMEACyeZxkyNE6X0DT/k6jFpTNXfGwkMhBgHnprM8ohFLxP/83qx9G+chLAolpih2SE/poYMjawzo084RpKOFYGIE/VXAw2hqkmqZrMSrMXIy6R9UbWuqubDZbneyOsogiNwDCrAAtegDu5BE7QAAs/gFUzAu/aivWkf2udstKDlO4fgD7TvH/6VprQ=</latexit> (r) = ⇢A ⇢B ⇢A + ⇢B = 1 = +1 <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> B-phase A-phase Gaussian white noise mechanical part by solving for fl ow di ff usive part from a chemical potential µ The current has three parts: <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">AAACQ3icbVDLSsNAFJ3UV62vqks3waIIaklE0Y0g6kLEhYJ9YBPKzWTaDk4mcWZSKCH/5sYfcOcPuHGhiFvBSc2iPi4MczjnHu69x4sYlcqynozC2PjE5FRxujQzOze/UF5cqsswFpjUcMhC0fRAEkY5qSmqGGlGgkDgMdLwbk8yvdEnQtKQX6tBRNwAupx2KAalqXb5xvFC5stBoL/kPF0/3B4lHA4eg9QJ4s1Rup86UY9uOvJOqGTnNP1hudDDfUi3Su1yxapawzL/AjsHFZTXZbv86PghjgPCFWYgZcu2IuUmIBTFjKQlJ5YkAnwLXdLSkENApJsMM0jNNc34ZicU+nFlDtlRRwKBzFbUnQGonvytZeR/WitWnQM3oTyKFeH4e1AnZqYKzSxQ06eCYMUGGgAWVO9q4h4IwErHnoVg/z75L6jvVO29qnW1Wzk6zuMoohW0ijaQjfbRETpDl6iGMLpHz+gVvRkPxovxbnx8txaM3LOMfpTx+QWT1rOh</latexit> J = rµ + v + p 2D⇤, The scalar fi eld is conserved: ˙ = r · J Bray, Adv. in Phys. 43, 357 (1994)
  22. φ4 fi eld theory of passive phase separation ˙ =

    r · J Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I no fl ow the fl uid fl ow acts to minimise the deformation of the droplet 8 Hohenberg & Halperin 1977; Bray, Adv in Phys 1994; Kendon et al, JFM 2001; Cates and Tjhung, JFM 2018
  23. φ4 fi eld theory of passive phase separation ˙ =

    r · J Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">AAACQ3icbVDLSsNAFJ3UV62vqks3waIIaklE0Y0g6kLEhYJ9YBPKzWTaDk4mcWZSKCH/5sYfcOcPuHGhiFvBSc2iPi4MczjnHu69x4sYlcqynozC2PjE5FRxujQzOze/UF5cqsswFpjUcMhC0fRAEkY5qSmqGGlGgkDgMdLwbk8yvdEnQtKQX6tBRNwAupx2KAalqXb5xvFC5stBoL/kPF0/3B4lHA4eg9QJ4s1Rup86UY9uOvJOqGTnNP1hudDDfUi3Su1yxapawzL/AjsHFZTXZbv86PghjgPCFWYgZcu2IuUmIBTFjKQlJ5YkAnwLXdLSkENApJsMM0jNNc34ZicU+nFlDtlRRwKBzFbUnQGonvytZeR/WitWnQM3oTyKFeH4e1AnZqYKzSxQ06eCYMUGGgAWVO9q4h4IwErHnoVg/z75L6jvVO29qnW1Wzk6zuMoohW0ijaQjfbRETpDl6iGMLpHz+gVvRkPxovxbnx8txaM3LOMfpTx+QWT1rOh</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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 fl ow the fl uid fl ow 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:
  24. φ4 fi eld theory of passive phase separation ˙ =

    r · J Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I no fl ow the fl uid fl ow acts to minimise the deformation of the droplet 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:
  25. φ4 fi eld theory of passive phase separation ˙ =

    r · J Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I no fl ow the fl uid fl ow acts to minimise the deformation of the droplet 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
  26. φ4 fi eld theory of passive phase separation ˙ =

    r · J Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I no fl ow the fl uid fl ow acts to minimise the deformation of the droplet 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
  27. φ4 fi eld theory of passive phase separation ˙ =

    r · J Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I no fl ow the fl uid fl ow acts to minimise the deformation of the droplet 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
  28. φ4 fi eld theory of passive phase separation ˙ =

    r · J Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I no fl ow the fl uid fl ow acts to minimise the deformation of the droplet 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?
  29. 9 Model H: Ostwald ripening Chemical potential raised at curved

    interface μs <latexit sha1_base64="ZAGPOvulsxZljt/eAzG3uVV8924=">AAACC3icbVDLSsNAFJ3UV62vqEs3Q4tQNyURRTdC0Y3LWuwDmhAm00k7dGYSZiZCCd278VfcuFDErT/gzr9x2mahrQcuHM65l3vvCRNGlXacb6uwsrq2vlHcLG1t7+zu2fsHbRWnEpMWjlksuyFShFFBWppqRrqJJIiHjHTC0c3U7zwQqWgs7vU4IT5HA0EjipE2UmCXPZ4Gqto8gVfQiyTCmTdAnKNJ5iVDGoSwOQnsilNzZoDLxM1JBeRoBPaX149xyonQmCGleq6TaD9DUlPMyKTkpYokCI/QgPQMFYgT5WezXybw2Ch9GMXSlNBwpv6eyBBXasxD08mRHqpFbyr+5/VSHV36GRVJqonA80VRyqCO4TQY2KeSYM3GhiAsqbkV4iEyiWgTX8mE4C6+vEzapzX3vObcnVXq13kcRXAEyqAKXHAB6uAWNEALYPAInsEreLOerBfr3fqYtxasfOYQ/IH1+QPRLJpK</latexit> µs(R) = bR <latexit sha1_base64="ypZaUtqe6AylfSkUkDkDZ9KWAX8=">AAACGXicbVDLSgMxFM34rPU16tJNsAgVocyIohuh6MZlLfYBnTLcSTNtaDIzJBmhDv0NN/6KGxeKuNSVf2P6WGjrgcDhnHO5uSdIOFPacb6thcWl5ZXV3Fp+fWNza9ve2a2rOJWE1kjMY9kMQFHOIlrTTHPaTCQFEXDaCPrXI79xT6VicXSnBwltC+hGLGQEtJF82/FE6qti9QhfYi+UQDKvC0KA7wwzL+kxP8DVIT7Go5j3QDX4dsEpOWPgeeJOSQFNUfHtT68Tk1TQSBMOSrVcJ9HtDKRmhNNh3ksVTYD0oUtbhkYgqGpn48uG+NAoHRzG0rxI47H6eyIDodRABCYpQPfUrDcS//NaqQ4v2hmLklTTiEwWhSnHOsajmnCHSUo0HxgCRDLzV0x6YPrRpsy8KcGdPXme1E9K7lnJuT0tlK+mdeTQPjpAReSic1RGN6iCaoigR/SMXtGb9WS9WO/WxyS6YE1n9tAfWF8/rKiffQ==</latexit> µs(R) = 0 bR + µ⇣ Bray, Adv. in Phys. 43, 357 (1994) Widely separated droplets of varying sizes
  30. 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 fi nite system μ∞ = μs Chemical potential raised at curved interface μs <latexit sha1_base64="ZAGPOvulsxZljt/eAzG3uVV8924=">AAACC3icbVDLSsNAFJ3UV62vqEs3Q4tQNyURRTdC0Y3LWuwDmhAm00k7dGYSZiZCCd278VfcuFDErT/gzr9x2mahrQcuHM65l3vvCRNGlXacb6uwsrq2vlHcLG1t7+zu2fsHbRWnEpMWjlksuyFShFFBWppqRrqJJIiHjHTC0c3U7zwQqWgs7vU4IT5HA0EjipE2UmCXPZ4Gqto8gVfQiyTCmTdAnKNJ5iVDGoSwOQnsilNzZoDLxM1JBeRoBPaX149xyonQmCGleq6TaD9DUlPMyKTkpYokCI/QgPQMFYgT5WezXybw2Ch9GMXSlNBwpv6eyBBXasxD08mRHqpFbyr+5/VSHV36GRVJqonA80VRyqCO4TQY2KeSYM3GhiAsqbkV4iEyiWgTX8mE4C6+vEzapzX3vObcnVXq13kcRXAEyqAKXHAB6uAWNEALYPAInsEreLOerBfr3fqYtxasfOYQ/IH1+QPRLJpK</latexit> µs(R) = bR <latexit sha1_base64="ypZaUtqe6AylfSkUkDkDZ9KWAX8=">AAACGXicbVDLSgMxFM34rPU16tJNsAgVocyIohuh6MZlLfYBnTLcSTNtaDIzJBmhDv0NN/6KGxeKuNSVf2P6WGjrgcDhnHO5uSdIOFPacb6thcWl5ZXV3Fp+fWNza9ve2a2rOJWE1kjMY9kMQFHOIlrTTHPaTCQFEXDaCPrXI79xT6VicXSnBwltC+hGLGQEtJF82/FE6qti9QhfYi+UQDKvC0KA7wwzL+kxP8DVIT7Go5j3QDX4dsEpOWPgeeJOSQFNUfHtT68Tk1TQSBMOSrVcJ9HtDKRmhNNh3ksVTYD0oUtbhkYgqGpn48uG+NAoHRzG0rxI47H6eyIDodRABCYpQPfUrDcS//NaqQ4v2hmLklTTiEwWhSnHOsajmnCHSUo0HxgCRDLzV0x6YPrRpsy8KcGdPXme1E9K7lnJuT0tlK+mdeTQPjpAReSic1RGN6iCaoigR/SMXtGb9WS9WO/WxyS6YE1n9tAfWF8/rKiffQ==</latexit> µs(R) = 0 bR + µ⇣ Bray, Adv. in Phys. 43, 357 (1994) Widely separated droplets of varying sizes
  31. 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 fi nite system μ∞ = μs Chemical potential raised at curved interface μs <latexit sha1_base64="ZAGPOvulsxZljt/eAzG3uVV8924=">AAACC3icbVDLSsNAFJ3UV62vqEs3Q4tQNyURRTdC0Y3LWuwDmhAm00k7dGYSZiZCCd278VfcuFDErT/gzr9x2mahrQcuHM65l3vvCRNGlXacb6uwsrq2vlHcLG1t7+zu2fsHbRWnEpMWjlksuyFShFFBWppqRrqJJIiHjHTC0c3U7zwQqWgs7vU4IT5HA0EjipE2UmCXPZ4Gqto8gVfQiyTCmTdAnKNJ5iVDGoSwOQnsilNzZoDLxM1JBeRoBPaX149xyonQmCGleq6TaD9DUlPMyKTkpYokCI/QgPQMFYgT5WezXybw2Ch9GMXSlNBwpv6eyBBXasxD08mRHqpFbyr+5/VSHV36GRVJqonA80VRyqCO4TQY2KeSYM3GhiAsqbkV4iEyiWgTX8mE4C6+vEzapzX3vObcnVXq13kcRXAEyqAKXHAB6uAWNEALYPAInsEreLOerBfr3fqYtxasfOYQ/IH1+QPRLJpK</latexit> µs(R) = bR <latexit sha1_base64="ypZaUtqe6AylfSkUkDkDZ9KWAX8=">AAACGXicbVDLSgMxFM34rPU16tJNsAgVocyIohuh6MZlLfYBnTLcSTNtaDIzJBmhDv0NN/6KGxeKuNSVf2P6WGjrgcDhnHO5uSdIOFPacb6thcWl5ZXV3Fp+fWNza9ve2a2rOJWE1kjMY9kMQFHOIlrTTHPaTCQFEXDaCPrXI79xT6VicXSnBwltC+hGLGQEtJF82/FE6qti9QhfYi+UQDKvC0KA7wwzL+kxP8DVIT7Go5j3QDX4dsEpOWPgeeJOSQFNUfHtT68Tk1TQSBMOSrVcJ9HtDKRmhNNh3ksVTYD0oUtbhkYgqGpn48uG+NAoHRzG0rxI47H6eyIDodRABCYpQPfUrDcS//NaqQ4v2hmLklTTiEwWhSnHOsajmnCHSUo0HxgCRDLzV0x6YPrRpsy8KcGdPXme1E9K7lnJuT0tlK+mdeTQPjpAReSic1RGN6iCaoigR/SMXtGb9WS9WO/WxyS6YE1n9tAfWF8/rKiffQ==</latexit> µs(R) = 0 bR + µ⇣ <latexit sha1_base64="LRaCBugJGvpb+fI3gKGqA12rQiQ=">AAACR3icbZBLSwMxFIUz9V1fVZdugkWogmVGFN0IohtxpcWq0BlLJs20oclMSO4IZZh/58atO/+CGxeKuDRtZ+HrQuDwnVzuvSdUghtw3WenNDE5NT0zO1eeX1hcWq6srF6bJNWUNWkiEn0bEsMEj1kTOAh2qzQjMhTsJuyfDv2be6YNT+IrGCgWSNKNecQpAYvalTu/kwBu4CPsR5rQ7BzXGlt5tuurHm+HOfaVThQkhet3iZQkzxrWECyC1hh7ltxt12Arxzsj4jV8zbs9CNqVqlt3R4X/Cq8QVVTURbvyZBeiqWQxUEGMaXmugiAjGjgVLC/7qWGK0D7pspaVMZHMBNkohxxvWtLBUaLtiwGP6PeOjEhjBtKetSkJ9Mxvbwj/81opRIdBxmOVAovpeFCUCmyDGYaKO1wzCmJgBaGa210x7REbBNjoyzYE7/fJf8X1bt3br7uXe9XjkyKOWbSONlANeegAHaMzdIGaiKIH9ILe0Lvz6Lw6H87n+GvJKXrW0I8qOV97irEl</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
  32. 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 fi nite system μ∞ = μs Chemical potential raised at curved interface μs <latexit sha1_base64="ZAGPOvulsxZljt/eAzG3uVV8924=">AAACC3icbVDLSsNAFJ3UV62vqEs3Q4tQNyURRTdC0Y3LWuwDmhAm00k7dGYSZiZCCd278VfcuFDErT/gzr9x2mahrQcuHM65l3vvCRNGlXacb6uwsrq2vlHcLG1t7+zu2fsHbRWnEpMWjlksuyFShFFBWppqRrqJJIiHjHTC0c3U7zwQqWgs7vU4IT5HA0EjipE2UmCXPZ4Gqto8gVfQiyTCmTdAnKNJ5iVDGoSwOQnsilNzZoDLxM1JBeRoBPaX149xyonQmCGleq6TaD9DUlPMyKTkpYokCI/QgPQMFYgT5WezXybw2Ch9GMXSlNBwpv6eyBBXasxD08mRHqpFbyr+5/VSHV36GRVJqonA80VRyqCO4TQY2KeSYM3GhiAsqbkV4iEyiWgTX8mE4C6+vEzapzX3vObcnVXq13kcRXAEyqAKXHAB6uAWNEALYPAInsEreLOerBfr3fqYtxasfOYQ/IH1+QPRLJpK</latexit> µs(R) = bR <latexit sha1_base64="ypZaUtqe6AylfSkUkDkDZ9KWAX8=">AAACGXicbVDLSgMxFM34rPU16tJNsAgVocyIohuh6MZlLfYBnTLcSTNtaDIzJBmhDv0NN/6KGxeKuNSVf2P6WGjrgcDhnHO5uSdIOFPacb6thcWl5ZXV3Fp+fWNza9ve2a2rOJWE1kjMY9kMQFHOIlrTTHPaTCQFEXDaCPrXI79xT6VicXSnBwltC+hGLGQEtJF82/FE6qti9QhfYi+UQDKvC0KA7wwzL+kxP8DVIT7Go5j3QDX4dsEpOWPgeeJOSQFNUfHtT68Tk1TQSBMOSrVcJ9HtDKRmhNNh3ksVTYD0oUtbhkYgqGpn48uG+NAoHRzG0rxI47H6eyIDodRABCYpQPfUrDcS//NaqQ4v2hmLklTTiEwWhSnHOsajmnCHSUo0HxgCRDLzV0x6YPrRpsy8KcGdPXme1E9K7lnJuT0tlK+mdeTQPjpAReSic1RGN6iCaoigR/SMXtGb9WS9WO/WxyS6YE1n9tAfWF8/rKiffQ==</latexit> µs(R) = 0 bR + µ⇣ <latexit sha1_base64="LRaCBugJGvpb+fI3gKGqA12rQiQ=">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</latexit> ˙ R = J(R) 2 b / R  1 R⇤(t) 1 R For a system with many droplets, is set by mean radius μ∞ R*(t) <latexit sha1_base64="ypZaUtqe6AylfSkUkDkDZ9KWAX8=">AAACGXicbVDLSgMxFM34rPU16tJNsAgVocyIohuh6MZlLfYBnTLcSTNtaDIzJBmhDv0NN/6KGxeKuNSVf2P6WGjrgcDhnHO5uSdIOFPacb6thcWl5ZXV3Fp+fWNza9ve2a2rOJWE1kjMY9kMQFHOIlrTTHPaTCQFEXDaCPrXI79xT6VicXSnBwltC+hGLGQEtJF82/FE6qti9QhfYi+UQDKvC0KA7wwzL+kxP8DVIT7Go5j3QDX4dsEpOWPgeeJOSQFNUfHtT68Tk1TQSBMOSrVcJ9HtDKRmhNNh3ksVTYD0oUtbhkYgqGpn48uG+NAoHRzG0rxI47H6eyIDodRABCYpQPfUrDcS//NaqQ4v2hmLklTTiEwWhSnHOsajmnCHSUo0HxgCRDLzV0x6YPrRpsy8KcGdPXme1E9K7lnJuT0tlK+mdeTQPjpAReSic1RGN6iCaoigR/SMXtGb9WS9WO/WxyS6YE1n9tAfWF8/rKiffQ==</latexit> µs(R) = 0 bR + µ⇣ smaller drops shrink and bigger drops grow material is transported from small drops to large ones by di ff usion <latexit sha1_base64="2btw/Rj7G+MrzrPMRvSLgaSzP9c=">AAAB83icbVBNSwMxEM3Wr1q/qh69BIvgqeyKoseiF49VrC10l5JNZ9vQbBKSrFCW/g0vHhTx6p/x5r8xbfegrQ8GHu/NMDMvVpwZ6/vfXmlldW19o7xZ2dre2d2r7h88GplpCi0qudSdmBjgTEDLMsuhozSQNObQjkc3U7/9BNowKR7sWEGUkoFgCaPEOim87+UhKMO4FJNetebX/RnwMgkKUkMFmr3qV9iXNEtBWMqJMd3AVzbKibaMcphUwsyAInREBtB1VJAUTJTPbp7gE6f0cSK1K2HxTP09kZPUmHEau86U2KFZ9Kbif143s8lVlDOhMguCzhclGcdW4mkAuM80UMvHjhCqmbsV0yHRhFoXU8WFECy+vEwez+rBRd2/O681ros4yugIHaNTFKBL1EC3qIlaiCKFntErevMy78V79z7mrSWvmDlEf+B9/gB1LZH1</latexit> R✏ <latexit sha1_base64="F+eCqzv3OsZFbVxCRluB+ci6TEc=">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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
  33. 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⇤,
  34. 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 fl uid fl ow even in absence of external forces extensile/contractile dipolar fl ow along the swimming axis Ramaswamy, Annu. Rev. Condens. Matter Phys. 2010 ˙ = r · J Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤,
  35. φ4 fi eld theory of active phase separation RS et

    al. PRL 2019, PRR 2020; Cates and Tjhung JFM 2018; Tiribocchi et al. PRL 2015 Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I ˙ = r · J Active model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, 11
  36. φ4 fi eld theory of active phase separation RS et

    al. PRL 2019, PRR 2020; Cates and Tjhung JFM 2018; Tiribocchi et al. PRL 2015 ⌃A = (˜  )S <latexit sha1_base64="AMPxuO9iYBYmiqIs71LmxRklFT4=">AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow=</latexit> <latexit sha1_base64="AMPxuO9iYBYmiqIs71LmxRklFT4=">AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow=</latexit> <latexit sha1_base64="AMPxuO9iYBYmiqIs71LmxRklFT4=">AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow=</latexit> <latexit sha1_base64="AMPxuO9iYBYmiqIs71LmxRklFT4=">AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow=</latexit> mechanical activity parameter positive (extensile microswimmers) negative (contractile microswimmers) minimal term breaking TRS in mechanical sector ˜ κ Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I ˙ = r · J Active model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, 11
  37. φ4 fi eld theory of active phase separation RS et

    al. PRL 2019, PRR 2020; Cates and Tjhung JFM 2018; Tiribocchi et al. PRL 2015 ⌃A = (˜  )S <latexit sha1_base64="AMPxuO9iYBYmiqIs71LmxRklFT4=">AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow=</latexit> <latexit sha1_base64="AMPxuO9iYBYmiqIs71LmxRklFT4=">AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow=</latexit> <latexit sha1_base64="AMPxuO9iYBYmiqIs71LmxRklFT4=">AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow=</latexit> <latexit sha1_base64="AMPxuO9iYBYmiqIs71LmxRklFT4=">AAACJnicbVDLSsNAFJ3UV62vqks3wSLURUsigm4KVTcuK7UPaGK5mUzaoZMHMxOhhHyNG3/FjYuKiDs/xUnbRW09MNzDOfcy9x4nYlRIw/jWcmvrG5tb+e3Czu7e/kHx8Kgtwphj0sIhC3nXAUEYDUhLUslIN+IEfIeRjjO6y/zOM+GChsGjHEfE9mEQUI9ikErqF2uWEzJXjH1VEqtJBz6kT8lNWquULUmZS6wRRBFUZuV8sbuZFvrFklE1ptBXiTknJTRHo1+cWG6IY58EEjMQomcakbQT4JJiRtKCFQsSAR7BgPQUDcAnwk6mZ6b6mVJc3Qu5eoHUp+riRAK+yHZTnT7IoVj2MvE/rxdL79pOaBDFkgR49pEXM12GepaZ7lJOsGRjRQBzqnbV8RA4YKmSzUIwl09eJe2LqmlUzYfLUv12HkcenaBTVEYmukJ1dI8aqIUwekFvaII+tFftXfvUvmatOW0+c4z+QPv5BcsSpow=</latexit> mechanical activity parameter positive (extensile microswimmers) negative (contractile microswimmers) minimal term breaking TRS in mechanical sector ˜ κ Contractile along the normal to the interface ˜ κ < 0 Extensile along the normal to the interface ˜ κ > 0 ⌃A ⇠ (r )(r ) ⌃A ⇠ (r )(r ) = 1 = +1 <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="zIIe8PRT1KE/tx4MElAV6rssooU=">AAACA3icbVDLSgMxFM34rPU16k43wSIIQpkRQTdC0Y3LCvYBnaFk0rQNzWSG5I5YhgE3/oobF4q49Sfc+Tem01lo64HA4Zx7c5ITxIJrcJxva2FxaXlltbRWXt/Y3Nq2d3abOkoUZQ0aiUi1A6KZ4JI1gINg7VgxEgaCtYLR9cRv3TOleSTvYBwzPyQDyfucEjBS1973gD1Afk8aCEJHWerFQ3554mZdu+JUnRx4nrgFqaAC9a795fUimoRMAhVE647rxOCnRAGngmVlL9EsNhFkwDqGShIy7ad5doaPjNLD/UiZIwHn6u+NlIRaj8PATIYEhnrWm4j/eZ0E+hd+ymWcAJN0GtRPBIYITwrBPa4YBTE2hFDFzVsxHRJFKJjayqYEd/bL86R5WnWdqnt7VqldFXWU0AE6RMfIReeohm5QHTUQRY/oGb2iN+vJerHerY/p6IJV7OyhP7A+fwCNXZgS</latexit> <latexit sha1_base64="XJx+pKEirN30bSLrBUI49M+QrtA=">AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK</latexit> v = ˜  0/ Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I ˙ = r · J Active model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, 11
  38. Incomplete phase separation ˜ κ < 0 Complete phase separation

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

    ˜ κ > 0 Active model H: nucleation and growth 12 -1 1 RS and Cates PRL (2019)
  40. Active model H: self-shearing instability (SSI) A spontaneous stretching motion

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

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

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

    larger droplets by stretching them. Ostwald ripening: smaller drops shrink and bigger drops grow. The result is a dynamic steady-state maintained by SSI. Self-shearing instability Small drops disappear Big drops grow in size Steady-state dynamics A spontaneous stretching motion of the interface A contractile active stress ˜ κ < 0 ⟹ γv < 0 SSI Ostwald ripening RS and Cates PRL (2019) The spontaneous stretching motion results in SSI => splitting the droplet 13
  44. 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
  45. Length scale determined by activity SSI Ostwald ripening RS and

    Cates PRL (2019) From the mechanical tension and fl uid viscosity , we can construct just one quantity with the dimensions of velocity γv η Vv = v/⌘ ˙ = r · J Active Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I <latexit sha1_base64="XJx+pKEirN30bSLrBUI49M+QrtA=">AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK</latexit> v = ˜  0/ 14
  46. Length scale determined by activity SSI Ostwald ripening RS and

    Cates PRL (2019) From the mechanical tension and fl uid viscosity , we can construct just one quantity with the dimensions of velocity γv η Vv = v/⌘ Ostwald process gives another rate V ( ¯ R) = ˙ ¯ R / M / 2 B ¯ R2 ˙ = r · J Active Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I <latexit sha1_base64="XJx+pKEirN30bSLrBUI49M+QrtA=">AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK</latexit> v = ˜  0/ 14
  47. Length scale determined by activity SSI Ostwald ripening RS and

    Cates PRL (2019) From the mechanical tension and fl uid viscosity , we can construct just one quantity with the dimensions of velocity γv η Vv = v/⌘ Ostwald process gives another rate V ( ¯ R) = ˙ ¯ R / M / 2 B ¯ R2 ¯ R / ✓ v 2 B ⌘M ◆ 1/2 ⇠ v 1/2 Balancing the two rates, we get scaling for droplet size ˙ = r · J Active Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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 <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
  48. Length scale determined by activity SSI Ostwald ripening RS and

    Cates PRL (2019) From the mechanical tension and fl uid viscosity , we can construct just one quantity with the dimensions of velocity γv η Vv = v/⌘ Ostwald process gives another rate V ( ¯ R) = ˙ ¯ R / M / 2 B ¯ R2 ¯ R / ✓ v 2 B ⌘M ◆ 1/2 ⇠ v 1/2 Balancing the two rates, we get scaling for droplet size ˙ = r · J Active Model H <latexit sha1_base64="Oil5AofUfIHx4lKM2Tcc2UzhaL0=">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</latexit> J = rµ + v + p 2D⇤, Stokes equation <latexit sha1_base64="m1HA2FWWD8E8IPh8sY0cte2/nck=">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</latexit> rp + ⌘r2v = r · ⌃E r · ⌃A ⌃E = S, ⌃A = (˜  )S, S ⌘ (r )(r ) 1 d |r |2I <latexit sha1_base64="XJx+pKEirN30bSLrBUI49M+QrtA=">AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK</latexit> v = ˜  0/ 14
  49. 15 Activity in di ff usive sector Cates and Tjhung

    JFM 2018; Tjhung et al PRX 2018 ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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> 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⇤,
  50. 15 Activity in di ff usive sector Cates and Tjhung

    JFM 2018; Tjhung et al PRX 2018 ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> J = Mrµ + J⇣ + p 2DM⇤, J⇣ = ⇣(r2 )r <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, <latexit sha1_base64="iU+JbNDhU90L9SncVWcfxeqtL+Y=">AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT</latexit> J⇣ = rµ⇣ + r ⇥ A
  51. 15 Activity in di ff usive sector Cates and Tjhung

    JFM 2018; Tjhung et al PRX 2018 ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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=">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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=">AAACmXicbVFta9swEJa9ty57y1rYl34xCxsd3YIdBuuHFbpXSlmhY0tbiNJwluVGVJZd6VyWCf2n/ZZ927+Z7AS2Nj0Q9/Dcne65u7SSwmAc/wnCGzdv3b6zcrdz7/6Dh4+6j1cPTVlrxoeslKU+TsFwKRQfokDJjyvNoUglP0rPPjTxowuujSjVd5xVfFzAqRK5YICemnR/0QJwmuZ2zz3ffrVP01JmZlZ4Z6mCVIKjRb35L2li6U+O4DapOddoBx/33aWaL751Bu4lPa8hsxT5D2xFWs0zZ6Plj7ZbtzFvdmIHjlZT8eI6HZ53rjPp9uJ+3Fq0DJIF6JGFHUy6v2lWsrrgCpkEY0ZJXOHYgkbBJHcdWhteATuDUz7yUEHBzdi2ol30zDNZlJfaP4VRy/5fYaEwjUyf2UxmrsYa8rrYqMZ8a2yFqmrkis0b5bWMsIyaM0WZ0JyhnHkATAuvNWJT0MDQH7NZQnJ15GVwOOgncT/5+rq3836xjhWyTp6SDZKQN2SH7JIDMiQseBK8DT4Fn8P18F24G+7NU8NgUbNGLln47S+iRtBQ</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, e ff ective nonlocal chemical potential <latexit sha1_base64="iU+JbNDhU90L9SncVWcfxeqtL+Y=">AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT</latexit> J⇣ = rµ⇣ + r ⇥ A
  52. 15 Activity in di ff usive sector Cates and Tjhung

    JFM 2018; Tjhung et al PRX 2018 ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> J = Mrµ + J⇣ + p 2DM⇤, J⇣ = ⇣(r2 )r <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">AAACmXicbVFta9swEJa9ty57y1rYl34xCxsd3YIdBuuHFbpXSlmhY0tbiNJwluVGVJZd6VyWCf2n/ZZ927+Z7AS2Nj0Q9/Dcne65u7SSwmAc/wnCGzdv3b6zcrdz7/6Dh4+6j1cPTVlrxoeslKU+TsFwKRQfokDJjyvNoUglP0rPPjTxowuujSjVd5xVfFzAqRK5YICemnR/0QJwmuZ2zz3ffrVP01JmZlZ4Z6mCVIKjRb35L2li6U+O4DapOddoBx/33aWaL751Bu4lPa8hsxT5D2xFWs0zZ6Plj7ZbtzFvdmIHjlZT8eI6HZ53rjPp9uJ+3Fq0DJIF6JGFHUy6v2lWsrrgCpkEY0ZJXOHYgkbBJHcdWhteATuDUz7yUEHBzdi2ol30zDNZlJfaP4VRy/5fYaEwjUyf2UxmrsYa8rrYqMZ8a2yFqmrkis0b5bWMsIyaM0WZ0JyhnHkATAuvNWJT0MDQH7NZQnJ15GVwOOgncT/5+rq3836xjhWyTp6SDZKQN2SH7JIDMiQseBK8DT4Fn8P18F24G+7NU8NgUbNGLln47S+iRtBQ</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, e ff ective nonlocal chemical potential <latexit sha1_base64="XbSTBBHNMy7G1Rbi0kv+i7vyfAM=">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</latexit> µs(R) = 0 bR + µ + µ⇣ Chemical potential raised at curved interface: μs <latexit sha1_base64="iU+JbNDhU90L9SncVWcfxeqtL+Y=">AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT</latexit> J⇣ = rµ⇣ + r ⇥ A
  53. 15 Activity in di ff usive sector Cates and Tjhung

    JFM 2018; Tjhung et al PRX 2018 The active contributions can reverse the sign of chemical fl ux and thus the sign of e ff ective tension => reverse Ostwald ripening ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> J = Mrµ + J⇣ + p 2DM⇤, J⇣ = ⇣(r2 )r <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, e ff ective nonlocal chemical potential <latexit sha1_base64="XbSTBBHNMy7G1Rbi0kv+i7vyfAM=">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</latexit> µs(R) = 0 bR + µ + µ⇣ Chemical potential raised at curved interface: μs <latexit sha1_base64="iU+JbNDhU90L9SncVWcfxeqtL+Y=">AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT</latexit> J⇣ = rµ⇣ + r ⇥ A
  54. 15 Activity in di ff usive sector Cates and Tjhung

    JFM 2018; Tjhung et al PRX 2018 The active contributions can reverse the sign of chemical fl ux and thus the sign of e ff ective tension => reverse Ostwald ripening e ff ective tension in di ff usive sector <latexit sha1_base64="KmlrZqHgOXPGYRlr1GxkuG4hVyQ=">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</latexit> = (⇣S0 2 S1) ⇣ 2 ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit 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=">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⇤, e ff ective nonlocal chemical potential <latexit sha1_base64="XbSTBBHNMy7G1Rbi0kv+i7vyfAM=">AAACLHicbVBbS8MwGE29O29VH30JDkERRiuKvgjiXnycw11gHeVrlm7BpC1JKszSH+SLf0UQHxziq7/DdBvi7UDgcM75knwnSDhT2nFG1szs3PzC4tJyaWV1bX3D3txqqjiVhDZIzGPZDkBRziLa0Exz2k4kBRFw2gpuq4XfuqNSsTi60cOEdgX0IxYyAtpIvl31ROqr/foBPsdeKIFkXh+EAN/JMy8ZMD/A9Rwf4iKWedxc3IMcfwn3VEPu22Wn4oyB/xJ3SspoippvP3u9mKSCRppwUKrjOonuZiA1I5zmJS9VNAFyC33aMTQCQVU3Gy+b4z2j9HAYS3Mijcfq94kMhFJDEZikAD1Qv71C/M/rpDo862YsSlJNIzJ5KEw51jEumsM9JinRfGgIEMnMXzEZgKlMm35LpgT398p/SfOo4p5UnOvj8sXltI4ltIN20T5y0Sm6QFeohhqIoAf0hF7RyHq0Xqw3630SnbGmM9voB6yPT0orpxA=</latexit> µs(R) = 0 bR + µ + µ⇣ Chemical potential raised at curved interface: μs <latexit sha1_base64="iU+JbNDhU90L9SncVWcfxeqtL+Y=">AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT</latexit> J⇣ = rµ⇣ + r ⇥ A
  55. 15 Activity in di ff usive sector Cates and Tjhung

    JFM 2018; Tjhung et al PRX 2018 The active contributions can reverse the sign of chemical fl ux and thus the sign of e ff ective tension => reverse Ostwald ripening e ff ective tension in di ff usive sector <latexit sha1_base64="KmlrZqHgOXPGYRlr1GxkuG4hVyQ=">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</latexit> = (⇣S0 2 S1) ⇣ 2 ˙ = r · J Active Model H µ = µE + µ , µE = F , µ = |r |2. <latexit sha1_base64="83uUWeWjVVtldWY2V1Cg5tFT0i4=">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</latexit> <latexit 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=">AAACmXicbVFta9swEJa9ty57y1rYl34xCxsd3YIdBuuHFbpXSlmhY0tbiNJwluVGVJZd6VyWCf2n/ZZ927+Z7AS2Nj0Q9/Dcne65u7SSwmAc/wnCGzdv3b6zcrdz7/6Dh4+6j1cPTVlrxoeslKU+TsFwKRQfokDJjyvNoUglP0rPPjTxowuujSjVd5xVfFzAqRK5YICemnR/0QJwmuZ2zz3ffrVP01JmZlZ4Z6mCVIKjRb35L2li6U+O4DapOddoBx/33aWaL751Bu4lPa8hsxT5D2xFWs0zZ6Plj7ZbtzFvdmIHjlZT8eI6HZ53rjPp9uJ+3Fq0DJIF6JGFHUy6v2lWsrrgCpkEY0ZJXOHYgkbBJHcdWhteATuDUz7yUEHBzdi2ol30zDNZlJfaP4VRy/5fYaEwjUyf2UxmrsYa8rrYqMZ8a2yFqmrkis0b5bWMsIyaM0WZ0JyhnHkATAuvNWJT0MDQH7NZQnJ15GVwOOgncT/5+rq3836xjhWyTp6SDZKQN2SH7JIDMiQseBK8DT4Fn8P18F24G+7NU8NgUbNGLln47S+iRtBQ</latexit> <latexit sha1_base64="Vda/T+EABB9thuaXfu3OmtQQw48=">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</latexit> <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, e ff ective nonlocal chemical potential <latexit sha1_base64="XbSTBBHNMy7G1Rbi0kv+i7vyfAM=">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</latexit> µs(R) = 0 bR + µ + µ⇣ Chemical potential raised at curved interface: μs <latexit sha1_base64="iU+JbNDhU90L9SncVWcfxeqtL+Y=">AAACKnicbVDLSgNBEJyNrxhfUY9eBoMgiGFXFL0IPi7iKYJRIRtC72TWDJmdXWZ6hbjke7z4K15yUIJXP8RJjKKJBQ1FVTfdXUEihUHX7Tu5qemZ2bn8fGFhcWl5pbi6dmPiVDNeZbGM9V0AhkuheBUFSn6XaA5RIPlt0D4f+LcPXBsRq2vsJLwewb0SoWCAVmoUT/0IsBWE9LKR+Y8coUuP6a6vIJBA/Sj9UXfot4gi4oZ+z502iiW37A5BJ4k3IiUyQqVR7PnNmKURV8gkGFPz3ATrGWgUTPJuwU8NT4C14Z7XLFVg19Wz4atdumWVJg1jbUshHaq/JzKIjOlEge0cHGjGvYH4n1dLMTyqZ0IlKXLFvhaFqaQY00FutCk0Zyg7lgDTwt5KWQs0MLTpFmwI3vjLk+Rmr+wdlN2r/dLJ2SiOPNkgm2SbeOSQnJALUiFVwsgTeSGv5M15dnpO33n/as05o5l18gfOxyfpOqZT</latexit> J⇣ = rµ⇣ + r ⇥ A Forward Ostwald ripening γϕ > 0 Reverse Ostwald ripening γϕ < 0
  56. 16 Steady-states in the plane of e ff ective tensions

    ˙ = r · J Active Model H <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, <latexit sha1_base64="hHrH1Qh7oykcidiyawuqJauU8m4=">AAACC3icbZC7TsMwFIadcivlVmBkiVohMZUEgWCsYGEsEr1IbRQ57mlr1U4i+6Siirqz8CosDCDEyguw8Ta4bQZoOZKlT/9/jn38B7HgGh3n28qtrK6tb+Q3C1vbO7t7xf2Dho4SxaDOIhGpVkA1CB5CHTkKaMUKqAwENIPhzdRvjkBpHoX3OI7Bk7Qf8h5nFI3kF0sdhAec3ZMq6E7STp9KSf3RaQbOxC+WnYozK3sZ3AzKJKuaX/zqdCOWSAiRCap123Vi9FKqkDMBk0In0RBTNqR9aBsMqQTtpbMdJvaxUbp2L1LmhGjP1N8TKZVaj2VgOiXFgV70puJ/XjvB3pWX8jBOEEI2f6iXCBsjexqM3eUKGIqxAcoUN7vabEAVZWjiK5gQ3MUvL0PjrOJeVJy783L1OosjT45IiZwQl1ySKrklNVInjDySZ/JK3qwn68V6tz7mrTkrmzkkf8r6/AEt45u/</latexit> v/ 0 <latexit sha1_base64="IOOGtSA4k3m0H3LcmZZVWyB+xtw=">AAACEHicbVC7TsMwFHV4lvIKMLJEVAimkiAQjBUsjEWiD6mpKse9ba3aSWTfIKoon8DCr7AwgBArIxt/g9tmgJYjWTo+597r6xPEgmt03W9rYXFpeWW1sFZc39jc2rZ3dus6ShSDGotEpJoB1SB4CDXkKKAZK6AyENAIhtdjv3EPSvMovMNRDG1J+yHvcUbRSB37yEd4wMmcVEE3S/0+lZJ2Uj8e8Owkv7lZxy65ZXcCZ554OSmRHNWO/eV3I5ZICJEJqnXLc2Nsp1QhZwKyop9oiCkb0j60DA2pBN1OJ4tkzqFRuk4vUuaE6EzU3x0plVqPZGAqJcWBnvXG4n9eK8HeZTvlYZwghGz6UC8RDkbOOB2nyxUwFCNDKFPc7OqwAVWUocmwaELwZr88T+qnZe+87N6elSpXeRwFsk8OyDHxyAWpkBtSJTXCyCN5Jq/kzXqyXqx362NaumDlPXvkD6zPH2t6nhA=</latexit> / 0 <latexit sha1_base64="IOOGtSA4k3m0H3LcmZZVWyB+xtw=">AAACEHicbVC7TsMwFHV4lvIKMLJEVAimkiAQjBUsjEWiD6mpKse9ba3aSWTfIKoon8DCr7AwgBArIxt/g9tmgJYjWTo+597r6xPEgmt03W9rYXFpeWW1sFZc39jc2rZ3dus6ShSDGotEpJoB1SB4CDXkKKAZK6AyENAIhtdjv3EPSvMovMNRDG1J+yHvcUbRSB37yEd4wMmcVEE3S/0+lZJ2Uj8e8Owkv7lZxy65ZXcCZ554OSmRHNWO/eV3I5ZICJEJqnXLc2Nsp1QhZwKyop9oiCkb0j60DA2pBN1OJ4tkzqFRuk4vUuaE6EzU3x0plVqPZGAqJcWBnvXG4n9eK8HeZTvlYZwghGz6UC8RDkbOOB2nyxUwFCNDKFPc7OqwAVWUocmwaELwZr88T+qnZe+87N6elSpXeRwFsk8OyDHxyAWpkBtSJTXCyCN5Jq/kzXqyXqx362NaumDlPXvkD6zPH2t6nhA=</latexit> / 0 RS and Cates PRL (2019)
  57. 16 Steady-states in the plane of e ff ective tensions

    ˙ = r · J Active Model H <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, di ff usive sector mechanical sector <latexit sha1_base64="XJx+pKEirN30bSLrBUI49M+QrtA=">AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK</latexit> v = ˜  0/ <latexit sha1_base64="KmlrZqHgOXPGYRlr1GxkuG4hVyQ=">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</latexit> = (⇣S0 2 S1) ⇣ 2 models H (always +ve) <latexit sha1_base64="GxJ7cDOVOO9yyHn3LIfDMaLR0hw=">AAACEXicbVDLSsNAFJ34rPVVdelmsAjdWJOqWBdC0Y3LCvYBTRpuptN26EwSZyZCCfkFN/6KGxeKuHXnzr8xfSy09cCFwzn3cu89XsiZ0qb5bSwsLi2vrGbWsusbm1vbuZ3dugoiSWiNBDyQTQ8U5cynNc00p81QUhAepw1vcD3yGw9UKhb4d3oYUkdAz2ddRkCnkpsr2D0QAtzYTC5tdS91fFS2BxCGgKEdnyTHF147LiVJ1s3lzaI5Bp4n1pTk0RRVN/dldwISCeprwkGplmWG2olBakY4TbJ2pGgIZAA92kqpD4IqJx5/lODDVOngbiDT8jUeq78nYhBKDYWXdgrQfTXrjcT/vFaku2UnZn4YaeqTyaJuxLEO8Cge3GGSEs2HKQEiWXorJn2QQHQa4igEa/bleVIvFa2zonl7mq9cTePIoH10gArIQueogm5QFdUQQY/oGb2iN+PJeDHejY9J64IxndlDf2B8/gBgh5y0</latexit> 0 = p 8a3/9b2 <latexit sha1_base64="hHrH1Qh7oykcidiyawuqJauU8m4=">AAACC3icbZC7TsMwFIadcivlVmBkiVohMZUEgWCsYGEsEr1IbRQ57mlr1U4i+6Siirqz8CosDCDEyguw8Ta4bQZoOZKlT/9/jn38B7HgGh3n28qtrK6tb+Q3C1vbO7t7xf2Dho4SxaDOIhGpVkA1CB5CHTkKaMUKqAwENIPhzdRvjkBpHoX3OI7Bk7Qf8h5nFI3kF0sdhAec3ZMq6E7STp9KSf3RaQbOxC+WnYozK3sZ3AzKJKuaX/zqdCOWSAiRCap123Vi9FKqkDMBk0In0RBTNqR9aBsMqQTtpbMdJvaxUbp2L1LmhGjP1N8TKZVaj2VgOiXFgV70puJ/XjvB3pWX8jBOEEI2f6iXCBsjexqM3eUKGIqxAcoUN7vabEAVZWjiK5gQ3MUvL0PjrOJeVJy783L1OosjT45IiZwQl1ySKrklNVInjDySZ/JK3qwn68V6tz7mrTkrmzkkf8r6/AEt45u/</latexit> v/ 0 <latexit sha1_base64="IOOGtSA4k3m0H3LcmZZVWyB+xtw=">AAACEHicbVC7TsMwFHV4lvIKMLJEVAimkiAQjBUsjEWiD6mpKse9ba3aSWTfIKoon8DCr7AwgBArIxt/g9tmgJYjWTo+597r6xPEgmt03W9rYXFpeWW1sFZc39jc2rZ3dus6ShSDGotEpJoB1SB4CDXkKKAZK6AyENAIhtdjv3EPSvMovMNRDG1J+yHvcUbRSB37yEd4wMmcVEE3S/0+lZJ2Uj8e8Owkv7lZxy65ZXcCZ554OSmRHNWO/eV3I5ZICJEJqnXLc2Nsp1QhZwKyop9oiCkb0j60DA2pBN1OJ4tkzqFRuk4vUuaE6EzU3x0plVqPZGAqJcWBnvXG4n9eK8HeZTvlYZwghGz6UC8RDkbOOB2nyxUwFCNDKFPc7OqwAVWUocmwaELwZr88T+qnZe+87N6elSpXeRwFsk8OyDHxyAWpkBtSJTXCyCN5Jq/kzXqyXqx362NaumDlPXvkD6zPH2t6nhA=</latexit> / 0 <latexit sha1_base64="IOOGtSA4k3m0H3LcmZZVWyB+xtw=">AAACEHicbVC7TsMwFHV4lvIKMLJEVAimkiAQjBUsjEWiD6mpKse9ba3aSWTfIKoon8DCr7AwgBArIxt/g9tmgJYjWTo+597r6xPEgmt03W9rYXFpeWW1sFZc39jc2rZ3dus6ShSDGotEpJoB1SB4CDXkKKAZK6AyENAIhtdjv3EPSvMovMNRDG1J+yHvcUbRSB37yEd4wMmcVEE3S/0+lZJ2Uj8e8Owkv7lZxy65ZXcCZ554OSmRHNWO/eV3I5ZICJEJqnXLc2Nsp1QhZwKyop9oiCkb0j60DA2pBN1OJ4tkzqFRuk4vUuaE6EzU3x0plVqPZGAqJcWBnvXG4n9eK8HeZTvlYZwghGz6UC8RDkbOOB2nyxUwFCNDKFPc7OqwAVWUocmwaELwZr88T+qnZe+87N6elSpXeRwFsk8OyDHxyAWpkBtSJTXCyCN5Jq/kzXqyXqx362NaumDlPXvkD6zPH2t6nhA=</latexit> / 0 RS and Cates PRL (2019)
  58. 16 Steady-states in the plane of e ff ective tensions

    ˙ = r · J Active Model H <latexit sha1_base64="uGFc44TDdM2NdroI1WBUc8vzZCk=">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</latexit> J = rµ + J⇣ + p 2D⇤, di ff usive sector mechanical sector <latexit sha1_base64="XJx+pKEirN30bSLrBUI49M+QrtA=">AAACKXicbVBNS8NAEN34bf2qevQSLIKnmoiiF6HoxWMFq0JTymQzrUt3k7A7EcuSv+PFv+JFQVGv/hHTD8GvBwOP92Z2Z16YSmHI896cicmp6ZnZufnSwuLS8kp5de3CJJnm2OCJTPRVCAaliLFBgiRepRpBhRIvw97JwL+8QW1EEp9TP8WWgm4sOoIDFVK7XAsIb2n4jtUY5TboglLQtjf5UUBCRmiDHqQp5F+G9fJ850vMS+1yxat6Q7h/iT8mFTZGvV1+CqKEZwpj4hKMafpeSi0LmgSXmJeCzGAKvAddbBY0BoWmZYcb5u5WoURuJ9FFxeQO1e8TFpQxfRUWnQro2vz2BuJ/XjOjzmHLijjNCGM++qiTSZcSdxCbGwmNnGS/IMC1KHZ1+TVo4FSEOwjB/33yX3KxW/X3q97ZXqV2PI5jjm2wTbbNfHbAauyU1VmDcXbHHtgze3HunUfn1XkftU4445l19gPOxyeuqalK</latexit> v = ˜  0/ <latexit sha1_base64="KmlrZqHgOXPGYRlr1GxkuG4hVyQ=">AAACU3icbVHPa9RAFJ6kVetqda1HL4OLUA9dkqLoRSh68dii2xZ2lvAyedkddyYZZl7ENeR/FMGD/4gXD3ayXcG2Phj4+L7385vcauUpSX5G8db2rdt3du4O7t3fffBw+Gjv1NeNkziRta7deQ4etapwQoo0nluHYHKNZ/nyXa+ffUbnVV19pJXFmYF5pUolgQKVDT8Jwi+07tM6LLpWzMEYyIRdKP6Gi9KBbMUSrIV98RUJuDBACwm6/dBlCT/gh0KHccVVIX0eOvXpB3/lrsuGo2ScrIPfBOkGjNgmjrPhd1HUsjFYkdTg/TRNLM1acKSkxm4gGo8W5BLmOA2wAoN+1q5v6fizwBS8rF14FfE1+29FC8b7lclDZr+3v6715P+0aUPl61mrKtsQVvJyUNloTjXvDeaFcihJrwIA6VTYlcsFBBspfMMgmJBeP/kmOD0cpy/HycmL0dHbjR077Al7yvZZyl6xI/aeHbMJk+wb+8X+RCz6Ef2O43j7MjWONjWP2ZWIdy8AVfa0tQ==</latexit> = (⇣S0 2 S1) ⇣ 2 models H (always +ve) <latexit sha1_base64="GxJ7cDOVOO9yyHn3LIfDMaLR0hw=">AAACEXicbVDLSsNAFJ34rPVVdelmsAjdWJOqWBdC0Y3LCvYBTRpuptN26EwSZyZCCfkFN/6KGxeKuHXnzr8xfSy09cCFwzn3cu89XsiZ0qb5bSwsLi2vrGbWsusbm1vbuZ3dugoiSWiNBDyQTQ8U5cynNc00p81QUhAepw1vcD3yGw9UKhb4d3oYUkdAz2ddRkCnkpsr2D0QAtzYTC5tdS91fFS2BxCGgKEdnyTHF147LiVJ1s3lzaI5Bp4n1pTk0RRVN/dldwISCeprwkGplmWG2olBakY4TbJ2pGgIZAA92kqpD4IqJx5/lODDVOngbiDT8jUeq78nYhBKDYWXdgrQfTXrjcT/vFaku2UnZn4YaeqTyaJuxLEO8Cge3GGSEs2HKQEiWXorJn2QQHQa4igEa/bleVIvFa2zonl7mq9cTePIoH10gArIQueogm5QFdUQQY/oGb2iN+PJeDHejY9J64IxndlDf2B8/gBgh5y0</latexit> 0 = p 8a3/9b2 <latexit sha1_base64="hHrH1Qh7oykcidiyawuqJauU8m4=">AAACC3icbZC7TsMwFIadcivlVmBkiVohMZUEgWCsYGEsEr1IbRQ57mlr1U4i+6Siirqz8CosDCDEyguw8Ta4bQZoOZKlT/9/jn38B7HgGh3n28qtrK6tb+Q3C1vbO7t7xf2Dho4SxaDOIhGpVkA1CB5CHTkKaMUKqAwENIPhzdRvjkBpHoX3OI7Bk7Qf8h5nFI3kF0sdhAec3ZMq6E7STp9KSf3RaQbOxC+WnYozK3sZ3AzKJKuaX/zqdCOWSAiRCap123Vi9FKqkDMBk0In0RBTNqR9aBsMqQTtpbMdJvaxUbp2L1LmhGjP1N8TKZVaj2VgOiXFgV70puJ/XjvB3pWX8jBOEEI2f6iXCBsjexqM3eUKGIqxAcoUN7vabEAVZWjiK5gQ3MUvL0PjrOJeVJy783L1OosjT45IiZwQl1ySKrklNVInjDySZ/JK3qwn68V6tz7mrTkrmzkkf8r6/AEt45u/</latexit> v/ 0 <latexit sha1_base64="IOOGtSA4k3m0H3LcmZZVWyB+xtw=">AAACEHicbVC7TsMwFHV4lvIKMLJEVAimkiAQjBUsjEWiD6mpKse9ba3aSWTfIKoon8DCr7AwgBArIxt/g9tmgJYjWTo+597r6xPEgmt03W9rYXFpeWW1sFZc39jc2rZ3dus6ShSDGotEpJoB1SB4CDXkKKAZK6AyENAIhtdjv3EPSvMovMNRDG1J+yHvcUbRSB37yEd4wMmcVEE3S/0+lZJ2Uj8e8Owkv7lZxy65ZXcCZ554OSmRHNWO/eV3I5ZICJEJqnXLc2Nsp1QhZwKyop9oiCkb0j60DA2pBN1OJ4tkzqFRuk4vUuaE6EzU3x0plVqPZGAqJcWBnvXG4n9eK8HeZTvlYZwghGz6UC8RDkbOOB2nyxUwFCNDKFPc7OqwAVWUocmwaELwZr88T+qnZe+87N6elSpXeRwFsk8OyDHxyAWpkBtSJTXCyCN5Jq/kzXqyXqx362NaumDlPXvkD6zPH2t6nhA=</latexit> / 0 <latexit sha1_base64="IOOGtSA4k3m0H3LcmZZVWyB+xtw=">AAACEHicbVC7TsMwFHV4lvIKMLJEVAimkiAQjBUsjEWiD6mpKse9ba3aSWTfIKoon8DCr7AwgBArIxt/g9tmgJYjWTo+597r6xPEgmt03W9rYXFpeWW1sFZc39jc2rZ3dus6ShSDGotEpJoB1SB4CDXkKKAZK6AyENAIhtdjv3EPSvMovMNRDG1J+yHvcUbRSB37yEd4wMmcVEE3S/0+lZJ2Uj8e8Owkv7lZxy65ZXcCZ554OSmRHNWO/eV3I5ZICJEJqnXLc2Nsp1QhZwKyop9oiCkb0j60DA2pBN1OJ4tkzqFRuk4vUuaE6EzU3x0plVqPZGAqJcWBnvXG4n9eK8HeZTvlYZwghGz6UC8RDkbOOB2nyxUwFCNDKFPc7OqwAVWUocmwaELwZr88T+qnZe+87N6elSpXeRwFsk8OyDHxyAWpkBtSJTXCyCN5Jq/kzXqyXqx362NaumDlPXvkD6zPH2t6nhA=</latexit> / 0 To summarise, incomplete phase separation in active scalar fi eld theories is a generic consequence of an e ff ective interfacial tension - mechanical (causing fl ow) or di ff usive (causing Ostwald ripening) - becoming negative. This is achieved by adding non-local terms, which break TRS, to equilibrium model H RS and Cates PRL (2019)
  59. 17 II. Phoresis and Stokesian hydrodynamics of active particles Michael

    E. Cates Ronojoy Adhikari
  60. 18 Nonequilibrium processes - in a thin layer on the

    surface drive exterior fl uid fl ow - diverse mechanisms. The resulting fl uid 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 fl uid How to study active matter systems in absence of time- reversal symmetry for particle-level dynamics?
  61. 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
  62. 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
  63. 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 fl uid fl ow boundary condition on the surface of a no-slip sphere, with velocity and angular velocity , is given as: Vi Ωi v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) is the radius vector of the -th particle. ρi i
  64. 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 fl uid fl ow boundary condition on the surface of a no-slip sphere, with velocity and angular velocity , is given as: Vi Ωi v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) is the radius vector of the -th particle. ρi i
  65. 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 fl uid fl ow boundary condition on the surface of a no-slip sphere, with velocity and angular velocity , is given as: Vi Ωi v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i ) is the radius vector of the -th particle. ρi i
  66. What is slip? Charged colloid in a non-conducting fl uid

    Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric fi eld <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
  67. What is slip? Charged colloid in a non-conducting fl uid

    V = qE 6⇡⌘b size-dependent velocity: Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric fi eld <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
  68. What is slip? Charged colloid in a non-conducting fl uid

    V = qE 6⇡⌘b size-dependent velocity: exterior fl ow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric fi eld <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
  69. What is slip? Charged colloid in a non-conducting fl uid

    V = qE 6⇡⌘b size-dependent velocity: exterior fl ow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric fi eld <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 fl uid 20 E
  70. What is slip? Charged colloid in a non-conducting fl uid

    V = qE 6⇡⌘b size-dependent velocity: exterior fl ow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric fi eld <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 fl ow decays as 1/r3 colloidal charge is balanced by a di ff used 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 fl uid 20 E
  71. What is slip? Charged colloid in a non-conducting fl uid

    V = qE 6⇡⌘b size-dependent velocity: exterior fl ow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric fi eld <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 fl ow decays as 1/r3 colloidal charge is balanced by a di ff used 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 fl uid 20 E Anderson, Ann Rev Fluid Mech (1989) E
  72. What is slip? Charged colloid in a non-conducting fl uid

    V = qE 6⇡⌘b size-dependent velocity: exterior fl ow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric fi eld <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 fl ow decays as 1/r3 colloidal charge is balanced by a di ff used 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 fl uid 20 E Anderson, Ann Rev Fluid Mech (1989) E How does this neutral object move?
  73. What is slip? Charged colloid in a non-conducting fl uid

    V = qE 6⇡⌘b size-dependent velocity: exterior fl ow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric fi eld <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 fl ow decays as 1/r3 colloidal charge is balanced by a di ff used 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 fl uid 20 E Anderson, Ann Rev Fluid Mech (1989) E How does this neutral object move? it is not rigid - di ff use clouds of counter-ions move in opposite direction of the charged particle
  74. What is slip? Charged colloid in a non-conducting fl uid

    V = qE 6⇡⌘b size-dependent velocity: exterior fl ow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric fi eld <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 fl ow decays as 1/r3 colloidal charge is balanced by a di ff used 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 fl uid Slip fl ow <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 - di ff use clouds of counter-ions move in opposite direction of the charged particle
  75. What is slip? Charged colloid in a non-conducting fl uid

    V = qE 6⇡⌘b size-dependent velocity: exterior fl ow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric fi eld <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 fl ow decays as 1/r3 colloidal charge is balanced by a di ff used 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 fl ow. The resulting fl uid stress may cause self-propulsion Charged colloid in a conducting fl uid Slip fl ow <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=">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="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 - di ff use clouds of counter-ions move in opposite direction of the charged particle
  76. What is slip? Charged colloid in a non-conducting fl uid

    V = qE 6⇡⌘b size-dependent velocity: exterior fl ow decays as 1/r Helmholtz (1879); Smoluchowski (1903); Anderson, Ann Rev Fluid Mech (1989) Force on a sphere in an electric fi eld <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 fl ow decays as 1/r3 colloidal charge is balanced by a di ff used 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 fl ow. The resulting fl uid stress may cause self-propulsion Charged colloid in a conducting fl uid Slip fl ow <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 - di ff use clouds of counter-ions move in opposite direction of the charged particle force-free motion from slip: 
 exterior fi eld: driven particles 
 self-phoresis: active particles
  77. 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
  78. Main theoretical questions ‣ What are the forces and torques

    acting on the particles due to slip? ‣ How are these modi fi ed 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
  79. Main theoretical questions ‣ What are the forces and torques

    acting on the particles due to slip? ‣ How are these modi fi ed 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 fl uid 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
  80. Main theoretical questions ‣ What are the forces and torques

    acting on the particles due to slip? ‣ How are these modi fi ed 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 fl uid 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, fl uid velocity = pI + ⌘(rv + (rv)T ) r · v = 0, r · + ⇠ = 0, fl uid stress Boundary velocity = rigid body motion + active slip active slip boundary condition v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i )
  81. Main theoretical questions ‣ What are the forces and torques

    acting on the particles due to slip? ‣ How are these modi fi ed 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 fl uid 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, fl uid velocity = pI + ⌘(rv + (rv)T ) r · v = 0, r · + ⇠ = 0, fl uid stress Boundary velocity = rigid body motion + active slip active slip boundary condition v(ri) = Vi + ⌦i ⇥ ⇢i + vA i (⇢i )
  82. 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
  83. 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
  84. 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=">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</latexit> (r) = Z h ˜ G(r, ri)˜(ri) ˜ K↵(ri, r)ˆ ⇢↵ (ri) i dSi 22
  85. 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 fl ow satis fi es 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
  86. 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 fl ow to the two-dimensional surfaces of the colloids Odqvist 1930, Jackson 1962, Ladyzhenskaia 1969, Pozrikidis 1992 force per unit area (traction) boundary velocity fl ow satis fi es 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
  87. 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
  88. 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=">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</latexit> v↵(r) = Z h G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ ⇢ v (ri) i dSi
  89. 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 fi elds 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
  90. 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 fi elds 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
  91. 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 fi elds 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 coe ffi cients <latexit sha1_base64="iQHbbcsmxQA6COVfndimfSPtIt8=">AAACD3icbVC5TgMxFPRyhnAFKGksIlBool0EgjKChjJI5JCyIfI6bxMr3kP2W0S02j+g4VdoKECIlpaOv8E5CkgYydJo5j17PF4shUbb/rYWFpeWV1Zza/n1jc2t7cLObl1HieJQ45GMVNNjGqQIoYYCJTRjBSzwJDS8wdXIb9yD0iIKb3EYQztgvVD4gjM0Uqdw5CI84PieVEE3S92AYd/zaf0uLUlXi17AjrOsUyjaZXsMOk+cKSmSKaqdwpfbjXgSQIhcMq1bjh1jO2UKBZeQ5d1EQ8z4gPWgZWjIAtDtdJwjo4dG6VI/UuaESMfq742UBVoPA89MjtLqWW8k/ue1EvQv2qkI4wQh5JOH/ERSjOioHNoVCjjKoSGMK2GyUt5ninE0FeZNCc7sl+dJ/aTsnJXtm9Ni5XJaR47skwNSIg45JxVyTaqkRjh5JM/klbxZT9aL9W59TEYXrOnOHvkD6/MHLHWdWg==</latexit> V(l )
  92. Ritz-Galerkin discretization 23 Fluid fl ow due to the irreducible

    mode lσ known body forces and torques FP TP known slip coe ffi cients <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 fi elds 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 coe ffi cients <latexit sha1_base64="iQHbbcsmxQA6COVfndimfSPtIt8=">AAACD3icbVC5TgMxFPRyhnAFKGksIlBool0EgjKChjJI5JCyIfI6bxMr3kP2W0S02j+g4VdoKECIlpaOv8E5CkgYydJo5j17PF4shUbb/rYWFpeWV1Zza/n1jc2t7cLObl1HieJQ45GMVNNjGqQIoYYCJTRjBSzwJDS8wdXIb9yD0iIKb3EYQztgvVD4gjM0Uqdw5CI84PieVEE3S92AYd/zaf0uLUlXi17AjrOsUyjaZXsMOk+cKSmSKaqdwpfbjXgSQIhcMq1bjh1jO2UKBZeQ5d1EQ8z4gPWgZWjIAtDtdJwjo4dG6VI/UuaESMfq742UBVoPA89MjtLqWW8k/ue1EvQv2qkI4wQh5JOH/ERSjOioHNoVCjjKoSGMK2GyUt5ninE0FeZNCc7sl+dJ/aTsnJXtm9Ni5XJaR47skwNSIg45JxVyTaqkRjh5JM/klbxZT9aL9W59TEYXrOnOHvkD6/MHLHWdWg==</latexit> V(l )
  93. l Symmetric (ls) Antisymmetric (la) Trace (lt) 1 2 3

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

    SO(3) invariant way to classify active fl ows. The -th mode of the fl ow 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 fl ow due to the irreducible mode lσ RS et al. JSTAT 2015, PRL 2016, JOSS 2020 24
  95. l Symmetric (ls) Antisymmetric (la) Trace (lt) 1 2 3

    SO(3) invariant way to classify active fl ows. The -th mode of the fl ow 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 fl ow due to the irreducible mode lσ RS et al. JSTAT 2015, PRL 2016, JOSS 2020 Charged colloid in a non-conducting fl uid 24
  96. l Symmetric (ls) Antisymmetric (la) Trace (lt) 1 2 3

    SO(3) invariant way to classify active fl ows. The -th mode of the fl ow 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 fl ow due to the irreducible mode lσ RS et al. JSTAT 2015, PRL 2016, JOSS 2020 Charged colloid in a non-conducting fl uid Charged colloid in a conducting fl uid 24
  97. 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 coe ff i cients: <latexit sha1_base64="iQHbbcsmxQA6COVfndimfSPtIt8=">AAACD3icbVC5TgMxFPRyhnAFKGksIlBool0EgjKChjJI5JCyIfI6bxMr3kP2W0S02j+g4VdoKECIlpaOv8E5CkgYydJo5j17PF4shUbb/rYWFpeWV1Zza/n1jc2t7cLObl1HieJQ45GMVNNjGqQIoYYCJTRjBSzwJDS8wdXIb9yD0iIKb3EYQztgvVD4gjM0Uqdw5CI84PieVEE3S92AYd/zaf0uLUlXi17AjrOsUyjaZXsMOk+cKSmSKaqdwpfbjXgSQIhcMq1bjh1jO2UKBZeQ5d1EQ8z4gPWgZWjIAtDtdJwjo4dG6VI/UuaESMfq742UBVoPA89MjtLqWW8k/ue1EvQv2qkI4wQh5JOH/ERSjOioHNoVCjjKoSGMK2GyUt5ninE0FeZNCc7sl+dJ/aTsnJXtm9Ni5XJaR47skwNSIg45JxVyTaqkRjh5JM/klbxZT9aL9W59TEYXrOnOHvkD6/MHLHWdWg==</latexit> V(l )
  98. 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 coe ff i cients: <latexit sha1_base64="iQHbbcsmxQA6COVfndimfSPtIt8=">AAACD3icbVC5TgMxFPRyhnAFKGksIlBool0EgjKChjJI5JCyIfI6bxMr3kP2W0S02j+g4VdoKECIlpaOv8E5CkgYydJo5j17PF4shUbb/rYWFpeWV1Zza/n1jc2t7cLObl1HieJQ45GMVNNjGqQIoYYCJTRjBSzwJDS8wdXIb9yD0iIKb3EYQztgvVD4gjM0Uqdw5CI84PieVEE3S92AYd/zaf0uLUlXi17AjrOsUyjaZXsMOk+cKSmSKaqdwpfbjXgSQIhcMq1bjh1jO2UKBZeQ5d1EQ8z4gPWgZWjIAtDtdJwjo4dG6VI/UuaESMfq742UBVoPA89MjtLqWW8k/ue1EvQv2qkI4wQh5JOH/ERSjOioHNoVCjjKoSGMK2GyUt5ninE0FeZNCc7sl+dJ/aTsnJXtm9Ni5XJaR47skwNSIg45JxVyTaqkRjh5JM/klbxZT9aL9W59TEYXrOnOHvkD6/MHLHWdWg==</latexit> V(l )
  99. 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 fl ow 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 coe ff i cients: <latexit sha1_base64="iQHbbcsmxQA6COVfndimfSPtIt8=">AAACD3icbVC5TgMxFPRyhnAFKGksIlBool0EgjKChjJI5JCyIfI6bxMr3kP2W0S02j+g4VdoKECIlpaOv8E5CkgYydJo5j17PF4shUbb/rYWFpeWV1Zza/n1jc2t7cLObl1HieJQ45GMVNNjGqQIoYYCJTRjBSzwJDS8wdXIb9yD0iIKb3EYQztgvVD4gjM0Uqdw5CI84PieVEE3S92AYd/zaf0uLUlXi17AjrOsUyjaZXsMOk+cKSmSKaqdwpfbjXgSQIhcMq1bjh1jO2UKBZeQ5d1EQ8z4gPWgZWjIAtDtdJwjo4dG6VI/UuaESMfq742UBVoPA89MjtLqWW8k/ue1EvQv2qkI4wQh5JOH/ERSjOioHNoVCjjKoSGMK2GyUt5ninE0FeZNCc7sl+dJ/aTsnJXtm9Ni5XJaR47skwNSIg45JxVyTaqkRjh5JM/klbxZT9aL9W59TEYXrOnOHvkD6/MHLHWdWg==</latexit> V(l )
  100. 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 fl ow 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 coe ff i cients: <latexit sha1_base64="iQHbbcsmxQA6COVfndimfSPtIt8=">AAACD3icbVC5TgMxFPRyhnAFKGksIlBool0EgjKChjJI5JCyIfI6bxMr3kP2W0S02j+g4VdoKECIlpaOv8E5CkgYydJo5j17PF4shUbb/rYWFpeWV1Zza/n1jc2t7cLObl1HieJQ45GMVNNjGqQIoYYCJTRjBSzwJDS8wdXIb9yD0iIKb3EYQztgvVD4gjM0Uqdw5CI84PieVEE3S92AYd/zaf0uLUlXi17AjrOsUyjaZXsMOk+cKSmSKaqdwpfbjXgSQIhcMq1bjh1jO2UKBZeQ5d1EQ8z4gPWgZWjIAtDtdJwjo4dG6VI/UuaESMfq742UBVoPA89MjtLqWW8k/ue1EvQv2qkI4wQh5JOH/ERSjOioHNoVCjjKoSGMK2GyUt5ninE0FeZNCc7sl+dJ/aTsnJXtm9Ni5XJaR47skwNSIg45JxVyTaqkRjh5JM/klbxZT9aL9W59TEYXrOnOHvkD6/MHLHWdWg==</latexit> V(l )
  101. 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
  102. 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
  103. 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 matrice s 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 in fi nitely 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
  104. 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 matrice s 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 in fi nitely 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
  105. Known: chemical surface fl ux 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?
  106. Known: chemical surface fl ux 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?
  107. Known: chemical surface fl ux 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?
  108. Known: chemical surface fl ux 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?
  109. 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 potentia l ‣ Problem reduced to the choice of a Green’s function and surface fl ux (or slip ) ‣ The slip is fi xed by choosing leading modes which match experimental dat a ‣ The boundary conditions in the bulk fl ow is implemented by choosing appropriate Green’s functions. No need to simulate the fl uid explicitly, just like in Coulomb's law for evaluating electrostatic interactions 28 28
  110. 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 potentia l ‣ Problem reduced to the choice of a Green’s function and surface fl ux (or slip ) ‣ The slip is fi xed by choosing leading modes which match experimental dat a ‣ The boundary conditions in the bulk fl ow is implemented by choosing appropriate Green’s functions. No need to simulate the fl uid explicitly, just like in Coulomb's law for evaluating electrostatic interactions Experimental fl ow Theoretical fl ow slip expansion truncated to l=3. Thutupalli, Geyer, RS, Adhikari, and Stone, PNAS 2018 28 28
  111. 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 potentia l ‣ Problem reduced to the choice of a Green’s function and surface fl ux (or slip ) ‣ The slip is fi xed by choosing leading modes which match experimental dat a ‣ The boundary conditions in the bulk fl ow is implemented by choosing appropriate Green’s functions. No need to simulate the fl uid explicitly, just like in Coulomb's law for evaluating electrostatic interactions GitHub.com/rajeshrinet/PyStokes [20K downloads] Experimental fl ow Theoretical fl ow slip expansion truncated to l=3. Thutupalli, Geyer, RS, Adhikari, and Stone, PNAS 2018 28 28
  112. 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 potentia l ‣ Problem reduced to the choice of a Green’s function and surface fl ux (or slip ) ‣ The slip is fi xed by choosing leading modes which match experimental dat a ‣ The boundary conditions in the bulk fl ow is implemented by choosing appropriate Green’s functions. No need to simulate the fl uid explicitly, just like in Coulomb's law for evaluating electrostatic interactions GitHub.com/rajeshrinet/PyStokes [20K downloads] Experimental fl ow Theoretical fl ow 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
  113. 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 potentia l ‣ Problem reduced to the choice of a Green’s function and surface fl ux (or slip ) ‣ The slip is fi xed by choosing leading modes which match experimental dat a ‣ The boundary conditions in the bulk fl ow is implemented by choosing appropriate Green’s functions. No need to simulate the fl uid explicitly, just like in Coulomb's law for evaluating electrostatic interactions GitHub.com/rajeshrinet/PyStokes [20K downloads] Experimental fl ow Theoretical fl ow 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
  114. Colloids tethered to an interface - free to move in

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

    the plane Caciagli, RS et al. PRL 2020 D Joshi Erika Eiser A Caciagli 29
  116. 30 Optically trap one of the colloids and study the

    optofluidic interactions Caciagli, RS et al. PRL 2020
  117. 30 Optically trap one of the colloids and study the

    optofluidic interactions Caciagli, RS et al. PRL 2020
  118. 30 Optically trap one of the colloids and study the

    optofluidic interactions Caciagli, RS et al. PRL 2020
  119. 31 Water Oil Puzzle: what causes motion into the hot

    region? Caciagli, RS et al. PRL 2020
  120. 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
  121. 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 fl ow 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
  122. 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 fl ow 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
  123. 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 fl ow 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
  124. 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 fl ow 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 opto fl uidic force can written as the gradient of a potential. FH Flow-induced attraction FH FH 0 1 Caciagli, RS et al. PRL 2020
  125. 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 fl ow 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 opto fl uidic 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 opto fl uidic 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.
  126. Summary: fi eld-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 e ff ective equilibrium description 32 Field-theoretic Particle-based theories of active matter Particle-based Phoresis and Stokesian hydrodynamics without resolving fl uid degrees of freedom