Dynamical equations for the contact line of an evaporating sessile drop Eliot Fried Department of Mechanical Engineering McGill University
Background • Many developing and advanced technologies rely on processes involving evaporating sessile drops. Γ LV Γ LV cos θ = m · sin θ = m · Ω Ω m t θ m C Γ LS • Basic modes: ◦ Constant contact radius. ◦ Constant contact angle. ◦ Mixed or stick-slip. • For sufficiently small drops, the time scales of evaporation and contact line motion become comparable. • Due to the absence of a reliable theory and the challenges of making measurements at small length and time scales, a com- plete understanding of the mechanisms governing the various modes is not yet available. NIST, January 21, 2011 1/28
Questions 1. What are the evolution equations for the contact line of an evaporating sessile drop? 2. Are dissipative mechanisms important and, if so, under what circumstances is coupling between those mechanisms signifi- cant? Theory for the liquid-vapor interface alone: • E. Fried, A.Q. Shen & M.E. Gurtin, Theory for solvent, mo- mentum, and energy transfer between a surfactant solution and a vapor atmosphere, Phys. Rev. E 73 (2006), 061601. • D.M. Anderson, P. Cermelli, E. Fried, M.E. Gurtin & G.B. Mc- Fadden, General dynamical sharp-interface conditions for phase transformations in viscous heat-conducting fluids, J. Fluid Mech. 501 (2007), 323–370. NIST, January 21, 2011 2/28
Outline • Variational description of the equilibrium of a sessile drop • Discussion and interpretation of the variational results • The Young–Dupr´ e equation • Questions concerning the variational conditions at the contact line • Mechanical balances at the contact line • Configurational forces • Dynamical equations for the contact line • Take-home points NIST, January 21, 2011 3/28
Variational description of the equilibrium of a sessile drop Γ LV Γ LV cos θ = m · sin θ = m · Ω Ω m t θ m C Γ LS • ̺ . . . liquid density • ψ . . . specific Helmholtz free-energy of liquid (relative to vapor) • ϕ . . . specific gravitational potential-energy (grad ϕ = − g ) • ψ LV . . . Helmholtz free-energy density of liquid-vapor interface • ψ LS . . . Helmholtz free-energy density of liquid-solid interface • ψ SV . . . Helmholtz free-energy density of solid-vapor interface • ψ C . . . Helmholtz free-energy density of contact line NIST, January 21, 2011 4/28
Assumptions • The substrate is rigid, impermeable, and chemically inert with respect to both the liquid and the vapor. • The liquid and vapor are in thermodynamic equilibrium. • ψ is constant, as are ψ LV , ψ LS , and ψ SV , and ψ C . Net potential energy � � � � E = Ω ̺ ( ψ + ϕ ) d v + ψ LV d a + ( ψ LS − ψ SV ) d a + C ψ C d s Γ LV Γ LS Variations • δ x satisfying div( δ x ) = 0 ( ∂ Γ LV ) • δ Γ LV Ω • δm LV = ̺ ( ♥ · δ x − δ Γ LV ) ( ∂ C ) m θ • δ C ( ∂ Γ LV ) ⊥ • δ Γ ⊥ LV = cos θ δ C NIST, January 21, 2011 5/28
Variational description of equilibrium To ensure satisfaction of the constraint div( δ x ) = 0, introduce a multiplier field p (ultimately the pressure of the liquid relative to that of the vapor). Then, the system is in equilibrium only if � Ω p div( δ x ) d v = 0 δ E − or, equivalently, only if � 0 = (grad p − ̺ g ) · δ x d v Ω ψ + p � � � � − ( p + ψ LV K LV ) δ Γ LV d a − ̺ + ϕ δm LV d a Γ LV Γ LV � − ( ψ LV cos θ + ψ LS − ψ SV − ψ C κ C ) δ C d s, C where K LV = − div LV ♥ is twice the mean curvature of Γ LV and κ C is the curvature of C . NIST, January 21, 2011 6/28
Necessary conditions for equilibrium Since δ x can be varied arbitrarily subject to the constraint div( δ x ) = 0 on Ω: grad p = ̺ g on Ω Since δ Γ LV and δm LV can be varied arbitrarily and independently on Γ LV : ψ + p p + ψ LV K LV = 0 and ̺ + ϕ = 0 on Γ LV Since δ C can be varied arbitrarily on C : ψ LV cos θ + ψ LS − ψ SV + ψ C κ C = 0 on C NIST, January 21, 2011 7/28
Discussion and interpretation of the necessary conditions • The relations grad p = ̺ g and p + ψ LV K LV = 0 express force balance in bulk and on the liquid-vapor interface, respectively. The latter is the Young–Laplace equation. • For a single-component system in a gravitational field, the spe- cific Gibbs free-energy ψ + p/̺ + ϕ is the driving force, measured per unit mass, per unit area, for evaporation-condensation. The relation ψ + p ̺ + ϕ = 0 expresses the requirement that, in equilibrium, that driving force must vanish. (Ward & Sasges (1999) discuss the ex- perimental significance of the gravitational contribution.) NIST, January 21, 2011 8/28
• Eliminating p between the relations for the liquid-vapor inter- face yields a combined balance ̺ ( ψ + ϕ ) = ψ LV K LV that can be imposed in place of the balance ψ + p/̺ + ϕ = 0. This combined balance is reminiscent of the Gibbs–Thomson condition arising in models of solidification. • When ψ C is negligible, the relation ψ LV cos θ + ψ LS − ψ SV − ψ C κ C = 0 reduces to an equation first derived by Gibbs (1878). Boruvka & Newmann (1977) provide a substantially broader generaliza- tion that allows the substrate to be deformable and accounts for dependence of the various interfacial free-energy densities on suitable strain measures. NIST, January 21, 2011 9/28
The Young equation Ω − ( σ SV ) m ( σ LS ) m σ LV cos θ + σ LS − σ SV = 0 θ ( σ LV ) Questions ◦ How, if at all, are the variational condition and the Young equation related? ◦ How can a contribution from line tension (or line energy) be incorporated in the Young equation? ◦ What are the forms of the vertical and tangential components of the vectorial force balance that encompasses the Young equation? NIST, January 21, 2011 10/28
Controversy surrounding these and other related questions is on- going. See: • R. Finn, Contact angle in capillarity, Phys. Fluids 18 (2006), 047102. • I. Lunati, Young’s law and the effects of interfacial energy on the pressure at the solid-fluid interface, Phys. Fluids 19 (2007), 118105. • R. Finn, Comments related to my paper “The contact angle in capillarity”, Phys. Fluids 20 (2008), 107104. • Y.D. Shikhmurzaev, On Young’s (1805) equation and Finn’s (2006) ‘counterexample’, Phys. Lett. A 372 (2008), 704–707. NIST, January 21, 2011 11/28
Mechanical balances at the contact line Suppose that: • The liquid-vapor, liquid-solid, and solid-vapor interfaces Γ LV , Γ LS , and Γ SV are endowed with (symmetric and tangential) Cauchy stresses ❚ LV , ❚ LS , and ❚ SV . • The contact line C is endowed with Cauchy line stress τ C and a line force r C e , where e denotes the upward unit normal on the substrate. • The interfaces and the contact line are not sufficiently massy to warrant the inclusion of interfacial or contact-line inertia. The linear- and angular-momentum balances for a segment L of C are then given by � � � � L ( ❚ LS − ❚ SV ) m d s − L ❚ LV t d s + τ C � ∂ L + L r C e d s = 0 � NIST, January 21, 2011 12/28
and � � L ( x − o ) × ( ❚ LS − ❚ SV ) m d s − ( x − o ) × ❚ LV t d s L � � + ( x − o ) × τ C � ∂ L + L ( x − o ) × r C e d s = 0 . � • Since L is arbitrary, these balances localize to ( ❚ LS − ❚ SV ) m − ❚ LV t + ∂ τ C ∂s + r C e = 0 , where s denotes arclength along C , and t ⊗ τ C = τ C ⊗ t . • The latter balances implies that τ C is tangential to C : τ C = τ C t . NIST, January 21, 2011 13/28
• Anderson, Cermelli, Fried, Gurtin & McFadden establish the representation ❚ LV = ψ LV ( 1 − ♥ ⊗ ♥ ) − ❙ LV for the Cauchy stress on the liquid-vapor interface, where ❙ LV = ❙ ⊤ LV is a viscous extra stress. • For a rigid substrate, the results of Anderson, Cermelli, Fried, Gurtin & McFadden can be adapted to yield ❚ LS − ❚ SV = ( ψ LS − ψ SV )( 1 − e ⊗ e ) . • The force r C e is a reaction, to the constraint of substrate rigid- ity, that is determined by the vertical component of the linear- momentum balance for C by r C = ψ LV sin θ − e · ❙ LV t . NIST, January 21, 2011 14/28
• In view of the Frenet relation ∂ t /∂s = κ C m , the linear-momentum balance for C has normal and tangential components ψ LV cos θ − m · ❙ LV t + ψ LS − ψ SV + τ C κ C = 0 and t · ❙ LV t = ∂τ C ∂s . • The first of the above equations does not coincide with the variational condition ψ LV cos θ + ψ LS − ψ SV + ψ C κ C = 0 unless m · ❙ LV t = 0 and τ C = ψ C . • If ψ C = τ C = 0, then the variational condition and the Young equation coincide. NIST, January 21, 2011 15/28
Observations This approach resolves: • Issues concerning balances normal to the substrate and tan- gential to the contact line. • The connection between the interfacial tensions and the inter- facial energies: σ LV = ψ LV + tr ❙ LV Additional questions • Why does the variational approach yield two equilibrium con- ditions on the liquid-vapor interface but only a single condition at the contact line? • Does τ C = ψ C in equilibrium? • Does a single balance suffice away from equilibrium? NIST, January 21, 2011 16/28
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