Capillary forces on colloids at fluid interfaces S. Dietrich Max - PowerPoint PPT Presentation

Capillary forces on colloids at fluid interfaces S. Dietrich Max Planck Institute for Intelligent Systems, Stuttgart, Germany and Inst. for Theoretical and Applied Physics, University of Stuttgart, Germany collaborators: J. Bleibel 1 , 4 , A. Dom´ ınguez 2 , J. Guzowski 3 , M. Oettel 4 , M. Tasinkevych 1 1 MPI-IS, Stuttgart, Germany 2 F´ ısica Te´ orica, Universidad de Sevilla, Spain 3 Dept. of Mechanical Engeneering, Princeton University, USA 4 Inst. for Applied Physics, University of T¨ ubingen, Germany 1

introduction colloids (nm...μm) trapped at fluid interfaces: two-dimensional structures • basic research on 2d systems • well-defined cluster shapes, pattern formation (e.g., Kosterlitz-Thouless transition) • potential build-up of 3d structures on a solid water / air ( ca 2 μ m ) water / air ( Ag, − : 0.5 μ m ) melting Zahn, Lenke and Maret, PRL 82 , 2721 (1999) R.P. Sear et al ., PRE 59 , R6255 (2004). colloid assembly controlled by effective interactions 2

colloids on planar water / air interfaces Ghezzi and Earnshaw, J.Phys.: Cond.Matt. 9 , L517 (1997) 100 μ m 50 μ m 100 μ m 50 μ m 3

monolayers at fluid interfaces glass spheres spheres air–water at air–oil ( R ≈ 2 µm ) ( R ≈ 24 µm ) Zahn et al. PRL 90 (2003) 155506 Aubry & Singh, PRE 77 (2008) 056302 micropost ellipsoids and rods oil–water oil–water (— : 21 µm ) (— : 100 µm ) Loudet et al. PRL 94 (2005) 018301 Cavallaro et al. PNAS 27 (2011) 20923 4

capillary forces • deformation of interface relative to reference plane u ( r ) given pressure normal to the interface Π( r ) • interface in mechanical equilibrium for given Π( r ) |∇ u | ≪ 1 • approximation: small deviations from flat interface: (very good for realistic conditions) local vertical mechanical balance: ∇ 2 u = 1 γ ( − Π) + u λ 2 Young-Laplace equation λ = capillary length ( ∼ mm ) γ = surface tension in–plane mechanical balance: �� � � F S || = = − S dA ( − Π) ∇ || u ∂ S dℓ n γ || S capillary force on region S n n : normal to e z −∇ || u ( x, y ) and normal to tangent of ∂ S M¨ uller, Deserno, Guven, EPL 69 (2005) Dom´ ınguez, Oettel, S.D., JCP 128 (2008) 5

capillary forces on single particle � ∇ 2 u = 1 γ ( − Π) + u F S || = − S dA ( − Π) ∇ || u λ 2 gravitational or electrostatic analogy interfacial deformation u ↔ gravitational potential � capillary length λ = γ/ (∆ ̺g ) ↔ “screening” length density of vertical force Π ↔ − mass density vertical force f (capillary monopole) ↔ − mass particle–interface contact line ↔ generation of multipolar moments air water ⇒ effective capillary interaction screened 2D gravity 6

two colloids: capillary monopoles Sf monopole f d F − F V cap capillary monopole: f = vertical force 1 2 3 d/λ 0 F = − V ′ cap ( d ) capillary force: ∼ d − 1 / 2 e − d/λ effective potential: � d � V cap ( d ) = − f 2 2 πγ K 0 ∼ ln d λ λ  plasma parameter   (number of interacting neighbors) mean interparticle separation ℓ ∼ 10 − 100 µ m ( λ/ℓ ) 2 ∼ 10 2 − 10 4 ≫ 1 ⇒   capillary length λ ≈ 1 mm ⇒ long–ranged λ , f , γ , R , ℓ easily tunable in experiments Kralchevsky & Nagayama, Adv. Colloid Interface Sci. (2000) Oettel & S.D., Langmuir (2008) 7

several colloids 2   K 0    ≈ 2   ln    f r f r single particle = capillary monopole = f  r   u = = mass in a 2d world buoyancy, electric fields ... “gravitational” potential between 2 π γ ln ( d 2 λ ) V ( d )≈− f u ( d )≈ − f two colloids cut off at capillary length λ 3  V ~ R 6 f ~ R gravity: colloids at water-air R = 10 μ m → V ∼ 1 k B T interfaces − 6 k B T R = 1 μ m → V ∼ 10 8

capillary multipoles • two arbitrary capillary “charge distributions” ⇒ multipoles q (1) and q (2) = at distance d l k symmetry axis of multipoles • capillary potential: � exp( il Φ 1 + ik Φ 2 ) Φ c lk q (1) q (2) Φ U cap = γ 2 1 l k d | l | + | k | d l,k ∈ Z \ 0 two freely floating ellipsoids: per- manent capillary quadrupoles U cap ∼ ∆ u 2 max d − 4 theory: exp: confirmed for tip-tip side–side: ∼ d − 3 . 1 Loudet, PRL 97 (2006) 9

small charged colloids – induced multipoles r 0, ref Δ h u ( r ) f total vertical force A r 0 ϵ= f / 2 π γ r 0, ref γ/ 2 ∫ A d 2 r [(∇ r u ) 2 + u 2 /λ 2 ] F = change of surface area and restoring force − ∫ A d 2 r Π( r ) u ( r )− f Δ h „pulling“ the meniscus and „pushing“ the colloid + γ/( 2r 0 ) ∫ ∂ A dl ( u −Δ h ) 2 + O (ϵ 3 ) change in colloid surface energy • need model for Π ( r ) form renormalized electrostatics • minimize with respect to u( r ) and h • V att = F ( d ) - F ( d→∞ ) , V rep : direct interaction of the particles 10 Dominguez, Oettel, S.D., PRE (2005), J. Chem. Phys. (2007)

capillary potential total effective potential V = V rep + V att V att position of minimum: κ d > 10 − 1 r 0, ref ≤ 1: κ κ − 1 additional length scale conditions for appearance of minimum: • κ R ~ 1 (Debye-Hueckel screening length) • → ϵ= f /( 2 π γ r 0 )≥ 0.5 colloidal charge density > 1.... 5 e / nm 2 (rather large) 11 Oettel, Dominguez, S.D., JPCM 17 (2005)

effective interactions of colloids on nematic films with surfactant added 2R ≈ 7  m h ≈ 60  m 2R ≈ 1 μ m h ≈ 60  m h ≈ 7 − 10  m 2R ≈ 7  m 2R ≈ 1  m h ≈ 60 μ m cluster formation: capillary attraction vs. elastic repulsion I. I. Smalyukh et al., Phys. Rev. Lett. 93, 117801 (2004) 5 V el ( d ≫ R ,h →∞)∝( R d ) , quadrupolar repulsion V men ( d ,h )∝ ( h ) log ( R ) + ∣ const ∣ ( d ) 6 5 R d R , M. Oettel, A. Dominguez, M. Tasinkevych and S. Dietrich, Eur. Phys. J. E (2009)

capillary interactions on sessile droplets fixed (+) and probe particles Guzowski,Tasinkevych, Dietrich, Eur. Phys. J. E (2010); Soft Matter (2011)

interface and colloid fluctuations interface fluctuation (colloid and contact line fixed) contact line pinned contact line pinned colloids are fixed colloid fluctuations: vertical position orientation contact line position mean field meniscus contact line pinned contact line pinned colloids free to tilt Lehle, Oettel, PRE 75 (2007) 14

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collective dynamics driven by capillary attraction: cosmology in the petri dish fluid of capillary monopoles self–gravitating fluid long range, screened long range ∇ 2 U − U λ 2 = − f ∇ 2 Φ = 4 πGm̺ γ̺ particle conservation particle conservation ∂̺ ∂̺ ∂t = −∇ · ( ̺ v ) ∂t = −∇ · ( ̺ v ) overdamped (Stokesian) dynamics inertial (Newtonian) dynamics � ∂ v � ̺ v Γ = −∇ p + f̺ ∇ U ̺ ∂t + ( v · ∇ ) v = −∇ p − m̺ ∇ Φ f : capillary monopole ̺ : particle number density ̺ : colloid number density p : pressure p : pressure λ : capillary length Γ : mobility ∇ = ∇ || 16

mean-field diffusion equation (ensemble averaged) F ∂ρ δ F ∂ t =−∇⋅ (ρ v )=Γ ∇ [ρ∇ δρ( r )] (DDFT) ρ 0 ( r ) Γ ∇⋅ [∇ p (ρ)− f ρ∇ U ] A = . “expanding” flow: “collapsing” flow: repulsive 2d pressure attractive capillarity coarse-grained (averaged) interface deformation: 2 U  r − 1 2 U  r =− f U [ρ( r ) ,f , λ] ∇   r   screened Poisson equation 17

attractive energy of a colloidal cluster λ 2 λ 2 L capillary energy per particle 8 γ × { L ( 1 + 2ln λ L ) 2 e cap = 1 2 f N ∑ i < j V ( r ij )≈−ρ L 2 λ 2 L λ 2 λ 2 L L e short energy per particles from repulsions: p [ρ( r )] due to thermal motion ( ), colloidal hard cores, charges, ... Dominguez, Oettel, S.D., PRE 82, 011402 (2010) Pergamenshchik, PRE 85, 021403 (2012) 18

cluster stability critical system size = Jeans' length e short ~ k B T For , a classic result is recovered: f √ 8 γ e cap = e short → L J ≃ 1 J. H. Jeans, ρ e short “The Stability of a Spherical Nebula", Philosophical Transactions of the Royal Society homogeneous of London A 199,1 (1902) distribution stable for any size L λ 2 2 e cap ∼ L e cap ∼λ L homogeneous distribution stable for L < LJ system collapses until new equilibrium is reached 19

linear stability analysis  r ,t = 0   r ,t  mean field diffusion equation: U  r ,t = U 0  U  r ,t  t / k     k ,t ~ e Fourier transform and linear stability analysis: exponential collapse stable characteristic scales: Jeans' length f √ γ p' (ρ 0 ) K = 1 1 ρ 0 Jeans' time stable γ T T = 2 ρ 0 Γ f λ K ≤ 1 : all modes stable 20

experimental realization of collapse conditions: 1 1 √ l ρ 0 ≪λ , R < K < λ initial density with reduced mean interparticle separation q = 1 R √ l ρ 0 particle radius example: charged colloids at air-water interface 21

f √ Jeans' length γ p' (ρ 0 ) K = 1 1 1 = ρ 0 K ( R,q ) f exp.: due to external electric field induced dipoles dipole-dipole int. p (ρ 0 ) from MC q = 1 R √ l ρ 0 suitable size range for q = 30 λ K = 1 R K = 1 Aubry, Singh, Janjua, Nudurupati, PNAS 105, 3711 (2008) Dominguez, Oettel, S.D., PRE 82, 011402 (2010) 22