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 • 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 2
Why do fluid ids spread? • Spreading forces – Body • gravity, centrifugal – Surface • thermocapillarity Young-Dupre’ – Contact static contact- • wetting angle • Competition can lead to instabilities! 3
Heating conditions motivated by experiments in Behringer group (Duke University) heat transfer (conductive) substrate heating (non-uniform) NOTE: 2 temperature scales 4
Thermocapil illary forces Marangoni stress (shear) surface tension equation of state substrate heating (non-uniform) heat transfer (conductive) radial gradient axial gradient 5
wetting vs. spreading modeling microscopic effects using macroscopic quantities wetting spreading dynamic contact-line law Young-Dupre equation force imbalance (dynamics): force balance (statics) mobility exponent
Definition sketch choose applied temperature distribution consistent with experiment 7
Solution method incompressible, Newtonian fluid lubrication quasi-static approximation approximation 8
Quasi-static spreading steady droplet shape (small heating) “+ auxiliary conditions” Bond axial Marangoni radial Marangoni centrifugal Imbalance of contact-line forces drive motion Dynamic CL Law response mobility exponent Map the problem to the contact line! 9
Outline of results • 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 10
Isothermal spreading equilibrium shapes spreading law base flow governing equation Ehrhard (1991) gravity dominant surface tension dominant spreading laws Tanner (1979), Chen (1988)
axial gradient vs. radial gradient Smith 95 (JFM)—2D Ehrhard 91 (JFM) thermo- capillary flows droplet shapes 12
Approach to equilibrium axial gradient radial gradient 13
Competition + + vs. equilibrium and Descartes’ rule of signs 14
Bi-stability + force balance bifurcation diagram energy landscape 15
Approach to equilibrium de-coupled CL dimpled drop ridge 16
Centrifugal effects equilibrium equation centrifugal forces can replace/overcome the effect of heat transfer! `slices’ of parameter space 17
Logarithmic temperature profile equilibrium equation lumped parameter with heat transfer is necessary to achieve bi-stability retraction rates are consistent with Mukhopadhyay & Behringer 2009 18
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 Michael Shearer, Karen Daniels, Joshua Dijksman 19
Quasi-static spreading steady droplet shape (small heating) “+ auxiliary conditions” Imbalance of contact-line forces drive motion response mobility exponent Dynamic CL Law static contact-angle Map the problem to the contact line! 20
Evolution equation auxiliary conditions dimensionless numbers capillary scale with σ Bond (surface tension) centrifugal Biot (heat transfer) thermal gradient thermocapillary slip length
Evolution equation auxiliary conditions dimensionless numbers Capillary Bond Centrifugal Slip length Thermocapillary Biot Thermal gradient
Field equations velocity field pressure temperature incompressibility surface tension relationship Stokes flow energy balance
Boundary conditions free surface substrate volume contact-line
Wetting modeling microscopic effects using macroscopic quantities Young-Dupre equation: force balance (statics) wetting ( ) non-wetting ( )
Evolution equation auxiliary conditions dimensionless numbers Capillary Bond Centrifugal Slip length Thermo-capillary Biot Thermal gradient
Spreading unbalanced forces (dynamics) Spreading law spreading wetting (advancing contact angle)
Why do fluid ids spread? centrifugal forces (body) gravity (body) wetting (substrate) thermocapillarity (surface) Young-Dupre’ static contact- angle 28
Competition + + vs. 29
30
axial gradient radial gradient Smith 95 (JFM)—2D Ehrhard 91 (JFM) flows shapes 31
Quasi-static spreading steady droplet shape “+ auxiliary conditions” Imbalance of contact-line forces drive motion response mobility exponent Dynamic CL Law static contact-angle applied temperature gradient Map the problem to the contact line! 32
Approach to equilibrium axial gradient radial gradient mobility exponent 33
Competition axial-cool, radial-in axial-heat, radial-out vs. equilibrium and Descartes’ rule of signs 34
Heating conditions ambient temperature heat transfer (conductive) substrate heating (non-uniform) Marangoni stress (shear) equation of state Experiments by Behringer group (Duke University) 35
Why do fluid ids spread? centrifugal forces (body) gravity (body) wetting (substrate) thermocapillarity (surface) Young-Dupre’ static contact- angle 36
Gravity-driven spreading equilibrium shapes spreading law base flow force balance surface tension dominant gravity dominant power laws
Quasi-static limit ( C 0 ) equilibrium time-dependence in BC Marangoni numbers Map the problem to the contact line!
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
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