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Spreading and bi-stability of droplets driven by thermocapillary and - - PowerPoint PPT Presentation

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Spreading and bi-stability of droplets driven by thermocapillary and centrifugal forces Joshua Bostwick North Carolina State University Workshop on Surfactant Driven Thin Film Flows Fields Institute, Toronto, ON 2/22/2012 1 Outline

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Spreading and bi-stability of droplets driven by thermocapillary and centrifugal forces

Joshua Bostwick North Carolina State University Workshop on Surfactant Driven Thin Film Flows Fields Institute, Toronto, ON 2/22/2012

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  • Definition sketch

– Spreading mechanisms

  • thermocapillarity
  • wetting vs. spreading
  • Quasi-static spreading

– axial vs. radial thermal gradients

  • flows, interface shapes and spreading rates

– bi-stability

  • competition

– effect of applied temperature profile

  • linear vs. logarithmic heating

Outline

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  • Spreading forces

– Body

  • gravity, centrifugal

– Surface

  • thermocapillarity

– Contact

  • wetting
  • Competition can lead to instabilities!

Why do fluid ids spread?

Young-Dupre’ static contact- angle

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Heating conditions

heat transfer (conductive) substrate heating (non-uniform)

motivated by experiments in Behringer group (Duke University)

NOTE: 2 temperature scales

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Thermocapil illary forces

surface tension equation of state substrate heating (non-uniform) Marangoni stress (shear) heat transfer (conductive)

axial gradient radial gradient

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wetting vs. spreading

Young-Dupre equation

force balance (statics)

modeling microscopic effects using macroscopic quantities dynamic contact-line law

force imbalance (dynamics): mobility exponent

wetting spreading

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Definition sketch

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choose applied temperature distribution consistent with experiment

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Solution method

incompressible, Newtonian fluid lubrication approximation quasi-static approximation

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Quasi-static spreading

steady droplet shape (small heating)

Map the problem to the contact line!

Imbalance of contact-line forces drive motion

“+ auxiliary conditions” mobility exponent response

Dynamic CL Law

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Bond centrifugal radial Marangoni axial Marangoni

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  • large parameter space

– equilibrium, flow fields and path to equilibrium

  • review isothermal spreading
  • linear temperature distribution

– small heating – isorotational spreading

  • axial vs. radial thermal gradients
  • competition and bi-stability

– centrifugal effects

  • logarithmic temperature distribution

– compare retraction laws to experiment

Outline of results

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Isothermal spreading

spreading laws equilibrium shapes spreading law base flow surface tension dominant gravity dominant governing equation Tanner (1979), Chen (1988) Ehrhard (1991)

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axial gradient radial gradient

droplet shapes thermo- capillary flows Ehrhard 91 (JFM) Smith 95 (JFM)—2D

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vs.

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Approach to equilibrium

axial gradient radial gradient

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Competition

equilibrium and Descartes’ rule of signs vs.

+ +

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Bi-stability

bifurcation diagram force balance energy landscape

+

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Approach to equilibrium

drop

de-coupled CL dimpled ridge

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Centrifugal effects

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equilibrium equation centrifugal forces can replace/overcome the effect of heat transfer! `slices’ of parameter space

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Logarithmic temperature profile

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lumped parameter equilibrium equation with

retraction rates are consistent with Mukhopadhyay & Behringer 2009

heat transfer is necessary to achieve bi-stability

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Concluding remarks

  • bi-stability <---> competition
  • centrifugal forces can enlarge regions of bi-stability

– thermal conditions may be relaxed – more control

  • map regions of indefinite spreading
  • generalized to other heating conditions

Acknowledgement: NSF FRG Grant # DMS-0968258

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Michael Shearer, Karen Daniels, Joshua Dijksman

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SLIDE 20

Quasi-static spreading

steady droplet shape (small heating)

Map the problem to the contact line!

Imbalance of contact-line forces drive motion

“+ auxiliary conditions” mobility exponent response static contact-angle

Dynamic CL Law

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Evolution equation

auxiliary conditions dimensionless numbers capillary Bond centrifugal slip length thermocapillary Biot (heat transfer) thermal gradient

scale with σ

(surface tension)

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Evolution equation

auxiliary conditions dimensionless numbers Capillary Bond Centrifugal Slip length Thermocapillary Biot Thermal gradient

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Field equations

surface tension relationship energy balance velocity field pressure temperature incompressibility Stokes flow

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Boundary conditions

substrate volume free surface contact-line

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Wetting

Young-Dupre equation:

force balance (statics) non-wetting ( ) wetting ( )

modeling microscopic effects using macroscopic quantities

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Evolution equation

auxiliary conditions dimensionless numbers Capillary Bond Centrifugal Slip length Thermo-capillary Biot Thermal gradient

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Spreading

Spreading law unbalanced forces (dynamics) wetting (advancing contact angle) spreading

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Why do fluid ids spread?

centrifugal forces (body) thermocapillarity (surface) gravity (body) wetting (substrate) Young-Dupre’ static contact- angle

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Competition

vs.

+ +

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SLIDE 30

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SLIDE 31

axial gradient radial gradient

shapes flows Ehrhard 91 (JFM) Smith 95 (JFM)—2D

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SLIDE 32

Quasi-static spreading

steady droplet shape

Map the problem to the contact line!

Imbalance of contact-line forces drive motion applied temperature gradient

“+ auxiliary conditions” mobility exponent response static contact-angle

Dynamic CL Law

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SLIDE 33

Approach to equilibrium

axial gradient radial gradient

mobility exponent

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Competition

equilibrium and Descartes’ rule of signs axial-cool, radial-in axial-heat, radial-out vs.

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Heating conditions

ambient temperature equation of state heat transfer (conductive) substrate heating (non-uniform) Marangoni stress (shear) Experiments by Behringer group (Duke University)

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Why do fluid ids spread?

centrifugal forces (body) thermocapillarity (surface) gravity (body) wetting (substrate) Young-Dupre’ static contact- angle

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Gravity-driven spreading

power laws equilibrium shapes spreading law base flow force balance surface tension dominant gravity dominant

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Quasi-static limit (C 0 )

Marangoni numbers equilibrium time-dependence in BC

Map the problem to the contact line!

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Results

  • Large parameter space
  • Unforced spreading (base-flow)

– power laws

  • Spreading by thermal-gradients (forced)

– axial vs. radial gradients

  • similarities, mechanisms and power laws

– equilibrium, stability and bifurcation

  • surface chemistry (wetting)

– bi-stability

  • competition