Sta$s$cal mechanics of hystere$c capillary phenomena: predic$ons of contact angle on rough surfaces and liquid reten$on in unsaturated porous media

Sta$s$cal mechanics of hystere$c capillary phenomena: predic$ons of contact angle on rough surfaces and liquid reten$on in unsaturated porous media Michel Louge h@p://grainflowresearch.mae.cornell.edu/ InterPore 2017, Ro@erdam, May 10, 2017

Ludwig Eduard Boltzmann (February 20, 1844 September 5, 1906) Hysteresis Collec$ve interac$ons Sta$s$cal Mechanics

Sta$s$cal Mechanics of Unsaturated Porous Media

Sta$s$cal Mechanics of Unsaturated Porous Media Jin Xu and Michel Louge, Phys. Rev. E 92 (2015)

Sta$s$cal Mechanics of Unsaturated Porous Media Predict the reten$on curve from pore geometry and surface energies with an equilibrium theory Satura$on (- ) Hysteresis Tension (- ) Jin Xu and Michel Louge, Phys. Rev. E 92 (2015)

Sta$s$cal Mechanics of Unsaturated Porous Media Drainage and imbibi$on are first- order phase transi$ons 1 phase transi$on Satura$on (- ) phase transi$on 0 Tension (- ) Jin Xu and Michel Louge, Phys. Rev. E 92 (2015)

Geometry: pores and necks a p v p Adv. Water Res.(2015) Pot, et al a n Courtesy Valérie Pot and Philippe Baveye Pore volume v p pore surface area a p each pore has several necks with overall cross- sec$on a n

Ising model: the simplest sta$s$cal mechanics σ = 1 pore full of liquid filling state σ σ = +1 pore full of gas Filling state variable

Satura$on filling state σ σ = 1 pore full of liquid σ = +1 pore full of gas Degree of Satura$on S # 0 S = θ 1 ν = 1 σ & % ( $ 2 ' 1 θ = liquid volume fraction ν = solid volume fraction

Two state variables: pore filling and tension σ = 1 pore full of liquid filling state σ σ = +1 pore full of gas mean field σ = f (ψ * ) gas- liquid interfacial energy γ lg rela$ve tension 1/3 ψ * = ψv p γ lg

Energy of a single pore ( ) E = σ 2 ψ v p +γ lg a p cosθ e γ lg a nσ pressure work pore wall energy neck interface energy Young contact angle cosθ e = γ gs γ sl γ lg E * = E γ lg v p 2/3

Dimensionless energy of a single pore ( ) E * = σ 2 ψ * +α cosθ e λσ pressure work pore wall energy neck interface energy Young contact angle θ e α a p v p v p 1/3 λ a n v p v p 1/3 specific pore wall area specific neck cross sec$on

Collec$ve interac$ons distribu$on of α and λ 1 * hysteresis S 2λ * latent energy ~ λ numerical simula$ons 0 of Patrick Richard ψ! *

Haines jumps: latent energy dissipa$on data of Armstrong & Berg, Phys. Rev E 88 (2013) model: mass conserva$on and viscous dissipa$on of latent energy

Sta$s$cal Mechanics of the Triple Gas- Solid- Liquid Contact Line Michel Louge, Phys. Rev. E 95 (2017)

Sta$s$cal mechanics of the contact line Predict advancing and receding contact angles from surface cavity geometry and surface energies with an equilibrium theory advancing contact line data of Shibuishi, et al (1996)

State variables σ = +1 cavity full of gas σ = 1 cavity full of liquid cavity filling Empty state advancing contact line χ =1 χ = 0 σ χ probability of overcoming Full a cavity receding contact line data of Shibuishi, et al (1996)

Geometry of a rough surface: cavi$es and necks v p a 0 a s cavity volume ; cavity opening area ; cavity surface area a p solid surface area several necks of of index (i) and cross sec$on a n,i

Energy of a single surface cavity σ E = γ "# ( 2χ 1)a + a cosθ a σ $ lg 2 0 p e n % E * E a γ 0 lg rela$ve energy α a p a 0 cavity area λ a n a 0 neck cross- sec$on

First- order phase transi$on χ =1 χ = 0 ΔE c * = λ Latent energy

Energy required for line displacement without no surface surface cavi$es cavi$es Tadmor (2004) dg = γ sl da sl +γ lg da lg +γ gs da gs

Energy required for line displacement without no surface surface cavi$es cavi$es Tadmor (2004) dg = γ sl da sl +γ lg da lg +γ gs da gs constant volume spherical cap da gs da sl = 1 da lg da sl = cosθ cosθ e = γ gs γ sl γ lg

Energy required for line displacement without no surface surface cavi$es cavi$es Tadmor (2004) dg = γ sl da sl +γ lg da lg +γ gs da gs constant volume spherical cap da gs da sl = 1 da lg da sl = cosθ cosθ e = γ gs γ sl γ lg poten$al energy change upon incremen$ng da sl dg = γ lg da sl ( ) cosθ cosθ e

With surface cavi$es dg = γ lg da sl ( ) cosθ cosθ a Advance cosθ = ( 1 ε)cosθ ε σ d χ +εδe a e 1 0 checkered solid surface cavity filling latent energy

Advancing vs receding line dg = γ lg da sl ( ) cosθ cosθ a Advance cosθ = ( 1 ε)cosθ ε σ d χ +εδe a e 1 0 different sign dg = γ lg da sl ( ) cosθ cosθ r Recession cosθ = ( 1 ε)cosθ ε σ d χ εδe r e 1 0 latent energy

Six dis$nct regimes Advancing angle Receding angle Each regime allows zero, one or two transi$ons.

Six dis$nct regimes Advancing angle Receding angle Example: regime III No phase transition with advancing line

Six dis$nct regimes Advancing angle Receding angle allowed transi$ons

Six dis$nct regimes Advancing angle Receding angle Cassie- Baxter state (1944)

Six dis$nct regimes Advancing angle Receding angle Metastable states: cavi$es do not fill spontaneously before recession Callies and Quéré, Soi Ma@er (2005)

Hysteresis scale ( ) cosθ r cosθ a Bed of rods hydrophilic solid superhydrophobic geometrical exclusion superhydrophobic hydrophobic solid

Comparison with data λ = 0.34 ± 0.15 α = 5.0 ±1.8 ε = 0.73± 0.09 Shibuichi, et al, J. Phys Chem. (1996) mixtures of water and 1,4 dioxane on surfaces of alkylketene dimer (AKD) and dialkylketone (DAK)

Comparison with data Regime I cosθ r cosθ a = 4λε Lam, et al, Adv. Colloid Interface Sci (2002) FC- 732- coated silicon wafer <100> surface Wenzel state

Ludwig Eduard Boltzmann (February 20, 1844 September 5, 1906) Sta$s$cal Mechanics is a useful framework for analyzing capillary phenomena. Jin Xu and Michel Louge, Phys. Rev. E 92 (2015) Michel Louge, Phys. Rev. E 95 (2017)

Jin Xu Patrick Richard h@p://grainflowresearch.mae.cornell.edu/index.html