1. Capillary force between particles: Force mediated by a fluid interface Interactions between Particles with an Undulated Contact Line at a Fluid Interface: Capillary Multipoles of Arbitrary Order Paper Submitted to: Journal of Colloid and Interface Science by Krassimir Danov, Peter Kralchevsky, Boris Naydenov and Günter Brenn

2. Kinds of Capillary Forces KEY: The lateral capillary forces are due to the overlap of the menisci formed around the separate particles attached to a fluid interface.

3. Origin of the Lateral Capillary Force F (1) = F (2) = F F = F (  ) + F (p) F()F() F (p) (force due to surface tension) (force due to pressure) Presence of second particle breaks the axial symmetry  nonzero integral force

4. (a) Brownian motion - thick layer (2R > h)(b) Aggregation after particle protrusion (2R < h) Capillary Immersion Forces and 2D Crystallization Two consecutive stages of 2D array formation (latex particles 1.7  m in diameter) evaporation

5. Convective Self-Assembly of Colloids and Proteins Role of Water Evaporation: Drives convective flow from the suspension toward array. N. Denkov et al., Nature 361 (1993) 26. Protein ferritin, 2R = 12 nm  Latex beads, 2R = 1.7  m 

6. Meniscus Overlap around Particles of Undulated Contact Line Rough-edged particles have undulated contact line when attached to a fluid interface. Copolymer latex particles (PS/HEMA): Cardoso et al. Langmuir 15 (1999) 4447.

7. Capillary Multipoles Meniscus around a particle of undulated contact line: H(r,  ) = (A m cos m  + B m sin m  ) Analogy with electrostatics: m = 0 – “capillary charges” m = 1 – “capillary dipoles” m = 2 – “capillary quadrupoles” m = 3 – “capillary hexapoles”.................................................. The capillary force spontaneously rotates a floating particle to annihilate its dipole moment (m = 1)  The leading multipole orders are the charges and quadrupoles.

8. The signs “+” and “  ” = local deviations of the contact line from planarity. (a) Initial state. (b) After rotation of the particles at angles  A and  B =  A. Asymptotic formula, Stamou et al. Phys. Rev. E 62 (2000) 5263: For typical parameter values  W becomes greater than the thermal energy kT for undulation amplitude H > 3 Å. (Even minimal roughness  capillary force). See also: P. Kralchevsky, N. Denkov, K. Danov, Langmuir 17 (2001) 7694–7705. Interaction between “Capillary Quadrupoles”

9. 2D arrays: capillary quadrupoles (m = 2) and hexapoles (m = 3) Quadrupoles form square arrayHexapoles form hexagonal array Bowden, Whitesides et al., J. Am. Chem. Soc. 121 (1999) 5373-5391. A.B.D. Brown, C.G. Smith, et al., Phys. Rev. E 62 (2000) 951-960: curved disks, having one hydrophilic and one hydrophobic side.

10. Generalization of the Theory to Arbitrary Capillary Multipoles The Laplace equation is solved in bipolar coordinates ( ,  ) Boundary conditions:

11. Expression for the Interaction Energy W(L,  A,  B ) The meniscus shape, z =  ( ,  ), is found by solving the Laplace equation. Next, the interaction energy (the surface excess surface energy) is calculated: Results: where S A (L), S B (L), and G (L) are known functions, given by infinite series; The dependence of W(L) on  A and  B is given explicitly by the cosine.

12. Example: Hexapoles Energy of interaction between two capillary hexapoles (m A = m B = 3) for two identical particles: H A = H B = H; r A = r B = r c. The different curves correspond to different phase angle , denoted in the figure; the dashed line is the asymptotic expression for large L and  = 0. For 5  <  < 25 , the dependence  W(L) has a minimum. For 0   < 5  the interaction is attraction at all distances. The energy of capillary interaction is very large compared to the thermal energy kT, even for undulations of 10 nm amplitude. The asymptotic formula becomes sufficiently accurate for L/(2r c ) > 1.5.

13. Asymptotic expressions for  W(L) for some values of m A and m B Type of interaction ( m A, m B ) Interaction Energy  W(L) charge – quadrupole (0, 2) charge – multipole (0, m B ) quadrupole – quadrupole (2, 2) quadrupole – hexapole (2, 3) hexapole – hexapole (3, 3) multipole – multipole (m A, m B )

14. Rheology of Particulate Monolayers (response of a “quadrupole” monolayer to deformations) Shear deformation  Yield Stress: (small effect) Shear Elasticity: (considerable effect) E S /  *  169 H/r c = 0.1 and  = 70 mN/m E S  16.1 mN/m Predictions which have to be checked experimentally !

15. Rheology of Particulate Monolayers (response of a “hexapole” monolayer to deformations) Shear deformation  Shear Elasticity: (considerable effect) H/r c = 0.1 and  = 70 mN/m E S  44.8 mN/m Predictions that have to be checked experimentally !

16. Conclusions 1. As a generalization of previous studies, here, we derive expressions for the interaction between two capillary multipoles of arbitrary order. 2. Simpler asymptotic expressions for the interaction energy at not-too- short interparticle distances are also derived. 3. Depending on , the interaction is either monotonic attraction, or it is attraction at long distances but repulsion at short distances. 4. Typically, for H  5 nm, the interaction energy is much greater than the thermal energy kT. 5. Owing to the angular dependence of the interaction energy, the adsorption monolayer of capillary multipoles has a considerable shear elasticity, and should behave as a 2D elastic solid, rather than 2D fluid. 6. The forces between capillary multipoles could influence many phenomena with particles, particle monolayers and arrays at fluid interfaces, but experimentally, these effects are insufficiently explored.