Current Opinion in Colloid & Interface Science 6Ž. 2001 383᎐401 Capillary forces and structuring in layers of colloid particles Peter A. KralchevskyU, Nikolai D. Denkov Laboratory of Chemical Physics and Engineering, Faculty of Chemistry, Uni¨ersity of Sofia, 1 James Bourchier A¨enue, Sofia 1164, Bulgaria Abstract ‘Capillary forces’ are interactions between particles mediated by fluid interfaces. Recent advances in this field have been achieved by experiments and theory on lateral capillary forces, which are due to the overlap of menisci formed around separate particles attached to an interface. In particular, we should mention the cases of ‘finite menisci’ and ‘capillary multipoles’. The capillary-bridge forces were investigated in relation to capillary condensation and cavitation, surface-force measurements and antifoaming by oily drops. The studies on colloidal self-assembly mediated by capillary forces developed in several promising directions. The obtained structures of particles have found numerous applications. ᮊ 2001 Elsevier Science Ltd. All rights reserved. Keywords: Capillary interactions; Lateral capillary forces; Capillary bridges; Colloidal self-assembly; Arrays of particles; Particulate monolayers 1. Introduction is concave or convex. Attractive forces of this type lead to 3DŽ. three-dimensional aggregation and con- In general, we call ‘capillary forces’ interactions solidation of bodies built up from particulates. A between particles, which are mediated by fluid inter- spontaneous formation of sub-micrometer gas-filled faces. The interest in these forces has grown due to capillary bridges in water seem to be the most their recognised importance for the self-assembly of probable explanation of the hydrophobic surface force. macroscopic and microscopicŽ. Brownian particles and In other cases, each individual particle causes some even of protein molecules and virusesw 1᎐3.ⅷ x perturbation in the shape of a liquid interface or film. In some cases, the liquid phase forms a capillary The overlap of the perturbationsŽ. menisci around bridge between two particles or bodies. Then the two particles gives rise to a lateral capillary force capillary force is directed normally to the planes of between themŽ. Fig. 1b,c,d,e . This force could be the contact lines on the particle surfacesŽ. Fig. 1a . attractive or repulsive depending on whether the The normal capillary-bridge forces can be attractive overlapping menisci, formed around the two particles, or repulsive depending on whether the capillary bridge are similarŽ.Ž say, both concave or dissimilar one is concave and the other is convex. The attractive lat- eral capillary forces cause 2DŽ. two-dimensional ag- gregation and ordering in a rather wide scale of ¨ Abbre iations: 2D, Two-dimensional; 3D, Three-dimensional; particle sizes: from 1 cm down to 1 nm. AFM, Atomic Force Microscope; PDMS, PolyŽ. dimethylsiloxane U Corresponding author. Tel.: q359-2-962-5310; fax: q359-2- Below we first briefly review recent publications on 962-5643. capillary forces. Next we shortly discuss studies in E-mail addresses: [email protected]Ž. P.A. Kralchevsky , which structuring under the action of lateral capillary [email protected]Ž. N.D. Denkov . 1359-0294r01r$ - see front matter ᮊ 2001 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 0 2 9 4Ž. 0 1 00105-4 384 P.A. Kralche¨sky, N.D. Denko¨rCurrent Opinion in Colloid & Interface Science 6() 2001 383᎐401 Fig. 1. Types of capillary forces:Ž. a The normal capillary forces can be due to either liquid-in-gas or gas-in-liquid capillary bridges, which lead to particle᎐particle and particle᎐wall interactions, the force is directed normally to the contact line. In the case of lateral capillary forces Ž.b,c,d,e the force is parallel to the contact line. The interaction is due to the overlap of interfacial deformations created by the separate particles.Ž. b In the case of flotation force the deformations are caused by the particle weight and buoyancy. In the case of immersion forces Ž.c,d,e the deformations are related to the wetting properties of the particle surface: position and shape of the contact line; and magnitude of the contact angle. When the deformation around an isolated particle is axisymmetric, we deal with ‘capillary charges’, one can distinguish cases of infiniteŽ. c and finite Ž. d menisci, see Eqs. Ž. 5 and Ž 14 . Ž. e The forces between particles of undulated or irregular contact line can be described as interactions between ‘capillary multipoles’, in analogy with electrostatics; see Eq.Ž. 15 . forces is reported. Comprehensive reviews on capil- surface tension force exerted around the annulus of lary forces and particle structuring can be found in the meniscus: Kralchevsky and Nagayama 2000, 2001w 2ⅷⅷ ,3x . sy␲ ␴ ␪y 2 F␪F␲ FccŽ2r sin rP.Ž0 . Ž.1 2. Normal() capillary-bridge force ␴ Here is the surfaceŽ. interfacial tension, Pc is the difference between the pressures inside and outside 2.1. Definition, measurements and physical importance the bridgeŽ. the capillary pressure , r and ␪ are the radial coordinate and the meniscus slope angle corre- Here we summarise the most important informa- sponding to an arbitrary cross-section of the menis- tion and briefly review recent publications on capil- cus. For example, ␪s␲r2 for a section across the lary-bridge forces. A detailed review can be found in neck of a bridge and then Eq.Ž. 1 can be presented in w ⅷ x sy ␲␴ y Chapter 11 of Kralchevsky and Nagayama 2001 3 . the form Fc 2 r00Ž.1 p , where r is the radius s r ␴ The presence of a liquid bridge between two solid of the neck and p Prc 0 2 is the dimensionless surfacesŽ. Fig. 1a leads to their interaction through a capillary pressure. In general, yϱ-p-qϱ. Accord- capillary force, Fc, owing to the pressure difference ing to the classification of Plateau, with the increase across the curved interface and the action of the of p the shape of the capillary bridge becomes, con- P.A. Kralche¨sky, N.D. Denko¨rCurrent Opinion in Colloid & Interface Science 6() 2001 383᎐401 385 secutively, concave nodoid Ž.y ϱ - p - 0, capillary bridge one can use Eq.Ž. 1 , along with some catenoid Ž.ps0 , concave unduloid Ž 0-p-1r2, . appropriate expressions for the meniscus shape. The cylinder Ž.ps1r2 , convex unduloid Ž 1r2-p-1, .contact angle, contact radius and the radius of the sphere Ž.ps1 and convex nodoid Ž 1-p-qϱ .. For neck are connected by simple analytical expressions, - - ᎐ p 1 the capillary-bridge force is attractive Ž.Fc 0, see equations 11.35 11.38 in Kralchevsky and Na- ) ) w ⅷ x whereas for p 1 it becomes repulsive Ž.Fc 0 ; for a gayama 3 . The profile, surface area and volume of s bridge with spherical meniscus we have Fc 0. a bridge can be expressed in terms of elliptic inte- The effect of capillary bridges is essential for the grals, see Table 11.1 in Kralchevsky and Nagayama assessment of the water saturation in soils and the w3ⅷ x . The elliptic integrals can be computed by means adhesive forces in any moist unconsolidated porous of the stable numerical methods of ‘arithmetic᎐geo- media, for the dispersion of pigments and wetting of metric mean’, see Chapter 17.6 in Abramowitz and powders, for the adhesion of dust and powder to Stegunwx 16 . Alternatively, the Laplace equation of surfaces, for flocculation of particles in three-phase capillarity can be solved numerically to determine the slurries, for liquid-phase sintering of fine metal and shape of the bridge and the capillary pressure. For polymer particles, for obtaining of films from latex example, in this way Dimitrov et al.wx 17 estimated the and silica particles, for calculation of the capillary capillary forces between silica particles in amorphous evaporation and condensation in various porous me- monolayers. Likewise, Aveyard et al.wx 18 calculated dia, for estimation of the retention of water in hydro- the liquid bridge profile in a study of the effects of carbon reservoirs and for granule consolidationw 3ⅷ x . line tension and surface forces on the capillary con- The action of capillary-bridge force is often de- densation of vapours between two solid surfaces. tected in experiments with atomic force microscopy Various approximate expressions for Fc are avail- Ž.AFMwx 4᎐6 . For example, Fujihira et al. wx 4 used able for pendular rings, that is a liquid capillary AFM as a friction-force microscope. At higher humid- bridge formed around the point at which a spherical ity in the atmosphere they detected a higher friction, particle of radius R touches a planar surfaceŽ. plate . which was attributed to the presence of an aqueous If the radii of the contact lines are much smaller than bridge due to capillary condensation. R, one can use the formula derived by Orr et al.wx 19 : AFM was also used to measure the interaction fy ␲␴ ␪ q ␪ between a small solid spherical particle and a gas Fc 2 RŽ.cos 12cos Ž.2 bubble attached to a substratew 7᎐10ⅷ ,11x . In fact, ᎐ ␪ ␪ after the particle enters the air liquid interface, the where 12and are the contact angles at the surfaces bubble plays the role of a gaseous capillary bridge of the particle and the plate. If the radii of the between the particle and the substrate. The measured contact lines are not much smaller than R, one can capillary-bridge force is non-monotonic and depends use several alternative expressions for Fc, all of them considerably on the three-phase contact anglewx 7 . In derived in the framework of the so called ‘toroid’ or some experiments a hysteresis of the contact angle ‘circle’ approximation: the generatrix of the bridge was detected; from the measured capillary force one surface is approximated with a circumference, see can determine the advancing and receding contact equations 11.11᎐11.13 in Kralchevsky and Nagayama angles on individual particles and to check whether w3ⅷ x .