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### kralchevsky_2001

Course: PHYS 7450, Fall 2009
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Opinion Current in Colloid &amp; Interface Science 6 Z2001. 383 401 Capillary forces and structuring in layers of colloid particles Peter A. Kralchevsky U , 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....

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Opinion Current in Colloid & Interface Science 6 Z2001. 383 401 Capillary forces and structuring in layers of colloid particles Peter A. Kralchevsky U , 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. 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 In general, we call capillary forces' interactions between particles, which are mediated by fluid interfaces. The interest in these forces has grown due to their recognised importance for the self-assembly of macroscopic and microscopic ZBrownian. particles and even of protein molecules and viruses w1 3 x. In some cases, the liquid phase forms a capillary bridge between two particles or bodies. Then the capillary force is directed normally to the planes of the contact lines on the particle surfaces ZFig. 1a.. The normal capillary-bridge forces can be attractive or repulsive depending on whether the capillary bridge Abbre iations: 2D, Two-dimensional; 3D, Three-dimensional; AFM, Atomic Force Microscope; PDMS, PolyZdimethylsiloxane. U Corresponding author. Tel.: q359-2-962-5310; fax: q359-2962-5643. E-mail addresses: pk@ ltph.bol.bg Z P.A. Kralchevsky . , nd@ltph.bol.bg ZN.D. Denkov.. is concave or convex. Attractive forces of this type lead to 3D Zthree-dimensional. aggregation and consolidation of bodies built up from particulates. A spontaneous formation of sub-micrometer gas-filled capillary bridges in water seem to be the most probable explanation of the hydrophobic surface force. In other cases, each individual particle causes some perturbation in the shape of a liquid interface or film. The overlap of the perturbations Zmenisci. around two particles gives rise to a lateral capillary force between them ZFig. 1b,c,d,e.. This force could be attractive or repulsive depending on whether the overlapping menisci, formed around the two particles, are similar Zsay, both concave. or dissimilar Zone is concave and the other is convex.. The attractive lateral capillary forces cause 2D Ztwo-dimensional. aggregation and ordering in a rather wide scale of particle sizes: from 1 cm down to 1 nm. Below we first briefly review recent publications on capillary forces. Next we shortly discuss studies in which structuring under the action of lateral capillary 1359-0294r01r\$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 0 2 9 4 Z 0 1 . 0 0 1 0 5 - 4 384 P.A. Kralche sky, N.D. Denko r Current Opinion in Colloid & Interface Science 6 (2001) 383 401 Fig. 1. Types of capillary forces: Za. 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 Zb,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. Zb. In the case of flotation force the deformations are caused by the particle weight and buoyancy. In the case of immersion forces Zc,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 infinite Zc. and finite Zd. menisci, see Eqs. Z5. and Z14.. Ze. The forces between particles of undulated or irregular contact line can be described as interactions between capillary multipoles', in analogy with electrostatics; see Eq. Z15.. forces is reported. Comprehensive reviews on capillary forces and particle structuring can be found in Kralchevsky and Nagayama 2000, 2001 w2 ,3 x. surface tension force exerted around the annulus of the meniscus: Fc s y Z 2 r sin y r 2 Pc . Z0 F F . Z1. 2. Normal (capillary-bridge) force 2.1. Definition, measurements and physical importance Here we summarise the most important information and briefly review recent publications on capillary-bridge forces. A detailed review can be found in Chapter 11 of Kralchevsky and Nagayama 2001 w3 x. The presence of a liquid bridge between two solid surfaces ZFig. 1a. leads to their interaction through a capillary force, Fc , owing to the pressure difference across the curved interface and the action of the Here is the surface Zinterfacial . tension, Pc is the difference between the pressures inside and outside the bridge Zthe capillary pressure., r and are the radial coordinate and the meniscus slope angle corresponding to an arbitrary cross-section of the meniscus. For example, s r2 for a section across the neck of a bridge and then Eq. Z1. can be presented in the form Fc s y2 r 0 Z1 y p ., where r 0 is the radius of the neck and p s Pc r 0r2 is the dimensionless capillary pressure. In general, y - p- q . According to the classification of Plateau, with the increase of p the shape of the capillary bridge becomes, con- P.A. Kralche sky, N.D. Denko r Current Opinion in Colloid & Interface Science 6 (2001) 383 401 385 secutively, concave nodoid Z y - p - 0 . , catenoid Z ps 0., concave unduloid Z0 - p - 1r2., cylinder Z ps 1r2., convex unduloid Z1r2 - p- 1., sphere Z ps 1. and convex nodoid Z1 - p- q .. For p- 1 the capillary-bridge force is attractive Z Fc - 0., whereas for p) 1 it becomes repulsive Z Fc ) 0.; for a bridge with spherical meniscus we have Fc s 0. The effect of capillary bridges is essential for the assessment of the water saturation in soils and the adhesive forces in any moist unconsolidated porous media, for the dispersion of pigments and wetting of powders, for the adhesion of dust and powder to surfaces, for flocculation of particles in three-phase slurries, for liquid-phase sintering of fine metal and polymer particles, for obtaining of films from latex and silica particles, for calculation of the capillary evaporation and condensation in various porous media, for estimation of the retention of water in hydrocarbon reservoirs and for granule consolidation w3 x. The action of capillary-bridge force is often detected in experiments with atomic force microscopy ZAFM. w4 6x. For example, Fujihira et al. w4x used AFM as a friction-force microscope. At higher humidity in the atmosphere they detected a higher friction, which was attributed to the presence of an aqueous bridge due to capillary condensation. AFM was also used to measure the interaction between a small solid spherical particle and a gas bubble attached to a substrate w7 10 ,11x. In fact, after the particle enters the air liquid interface, the bubble plays the role of a gaseous capillary bridge between the particle and the substrate. The measured capillary-bridge force is non-monotonic and depends considerably on the three-phase contact angle w7x. In some experiments a hysteresis of the contact angle was detected; from the measured capillary force one can determine the advancing and receding contact angles on individual particles and to check whether there is hysteresis w8 10 x. Capillary bridges between two fluid phases are found to play an important role in the process of antifoaming by dispersed oil drops w12 14 ,15x. When an oil droplet bridges between the surfaces of an aqueous film, two scenarios of film destruction are proposed: Zi. dewetting of the droplet could cause film rupture; and Zii. the formed oil bridge could have an unstable configuration and the film could break at the centre of the expanding destabilised bridge. The latter mechanism was recorded experimentally with the help of a high-speed video camera w13x and the results were interpreted in terms of the theory of capillarybridge stability w14 x. 2.2. Theoretical calculations of the capillary-bridge force To calculate Fc for a given configuration of the capillary bridge one can use Eq. Z1., along with some appropriate expressions for the meniscus shape. The contact angle, contact radius and the radius of the neck are connected by simple analytical expressions, see equations 11.35 11.38 in Kralchevsky and Nagayama w3 x. The profile, surface area and volume of a bridge can be expressed in terms of elliptic integrals, see Table 11.1 in Kralchevsky and Nagayama w3 x. The elliptic integrals can be computed by means of the stable numerical methods of arithmetic geometric mean', see Chapter 17.6 in Abramowitz and Stegun w16x. Alternatively, the Laplace equation of capillarity can be solved numerically to determine the shape of the bridge and the capillary pressure. For example, in this way Dimitrov et al. w17x estimated the capillary forces between silica particles in amorphous monolayers. Likewise, Aveyard et al. w18x calculated the liquid bridge profile in a study of the effects of line tension and surface forces on the capillary condensation of vapours between two solid surfaces. Various approximate expressions for Fc are available for pendular rings, that is a liquid capillary bridge formed around the point at which a spherical particle of radius R touches a planar surface Zplate.. If the radii of the contact lines are much smaller than R, one can use the formula derived by Orr et al. w19x: Fc f y2 R Z cos 1 q cos 2 . Z2. where 1 and 2 are the contact angles at the surfaces of the particle and the plate. If the radii of the contact lines are not much smaller than R, one can use several alternative expressions for Fc , all of them derived in the framework of the so called toroid' or circle' approximation: the generatrix of the bridge surface is approximated with a circumference, see equations 11.11 11.13 in Kralchevsky and Nagayama w3 x. Using the same approximation, de Lazzer et al. w20x derived analytical expressions for Fc for the cases when the particles are spherical, paraboloidal or conical. The obtained formulas were verified against the respective exact computer solutions. Kolodezhnov et al. w21x derived a set of equations which provide a convenient way to compute the shape of the capillary bridge between two identical spherical particles and to calculate Fc . At the last step numerical solution was used. The accuracy of the circle approximation was verified against the exact solution for various values of the system parameters. In addition, an approximate relationship between the capillary force and the moisture of a powder was derived w21x. Willett et al. w22x obtained closed-form approximated expressions for the capillary-bridge force between equal and unequal spheres as a function of the separation distance and for a given bridge volume and contact angle. These authors developed also a 386 P.A. Kralche sky, N.D. Denko r Current Opinion in Colloid & Interface Science 6 (2001) 383 401 method for measuring the capillary forces arising from microscopic pendular liquid bridges. The experimental force-vs.-distance curves were found to agree excellently with the respective theoretical dependencies calculated by numerical integration of the Laplace equation w22x. 2.3. Nucleation of bridges: capillary condensation and ca itation If the length of a capillary bridge is gradually increased, the bridge becomes unstable and ruptures at a given critical Zmaximum. length. Conversely, if the distance between two approaching parallel hydrophilic plates in humid atmosphere becomes smaller than the maximum length of the stable water bridges, then such bridges can spontaneously appear due to capillary condensation' of vapours. Likewise, if the distance between two parallel hydrophobic plates in water is smaller than the maximum length of the stable vapour-filled bridges, then such bridges can appear owing to a spontaneous cavitation due to fluctuational formation and growth of critical bridges nuclei w3 x. The maximum length, h max , of a stable nodoid-shaped bridge can be estimated by means of the asymptotic expression: h max s 2 cos < Pc < c Z 70 - c - 90 . Z3. see equations 11.76 11.78 in Kralchevsky and Nagayama w3 x, c is the contact angle measured across the bridge phase; for c out of the above interval a more complicated expression for h max is to be used. For a bridge, which is in chemical equilibrium with the ambient mother phase, one has: < Pc < s Py P0 for vapor- filled bridge Z kTrVm . ln Z P0rP . for liquid bridge Z4. Here P is the outer Zusually the atmospheric. pressure, P0 is the equilibrium vapour pressure of the liquid and P is the partial pressure of the vapours in the ambient gas phase, Vm is the volume per molecule in the liquid phase, k is the Boltzmann constant and T is temperature. For example, taking Ps 1 atm, contact angle s 90 y c s 94 , s 72.75 mNrm and P0 s 2337 Pa from Eqs. Z3. and Z4. one calculates h max s 103 nm for a vapour-filled bridge between two parallel hydrophobic plates in water at 20 C. Note that the latter estimate holds for degassed Zdeaerated. water. The presence of dissolved gas much facilitates the cavitation. In such cases the shape of the capillary bridge could be approximated with a cylinder w23x, instead of considering a concave nodoid. Capillary bridging between two glass surfaces in a humid atmosphere was observed by Yaminsky w24x; the formation of a water bridge by capillary condensation was detected as a discontinuity in the force distance dependence. Xiao and Qian w25x investigated, by AFM, the dependence of the capillary-bridge force on humidity. In the case of less hydrophilic surfaces they detected an adhesive force, which is independent of humidity and in agreement with Eq. Z2.. In contrast, between more hydrophilic surfaces Zenhanced capillary condensation. these authors measured a humidity-dependent capillary-bridge force. The latter means that the simplifying assumptions used to derive Eq. Z2. are not satisfied and more complicated theoretical expressions have to be applied to interpret the data w25x. Capillary condensation of water bridges was established not only when the ambient mother phase is a humid atmosphere, but also when this phase is oil w26x and even a bicontinuous microemulsion w27x. With the help of a surface force apparatus, Claesson et al. w26x detected capillary bridging between mica surfaces immersed in triolein, which had been preequilibrated with water. With a similar technique Petrov et al. w27x measured the force between two mica surfaces immersed in a microemulsion ZAOTrdecanerbrine.. A contribution of capillary bridge-forces was detected and the force profile was experimentally obtained on both approach and separation. As already mentioned, the capillary-bridge force is one of the major candidates for explanation of the attractive hydrophobic surface force. Gaseous bridges could appear even if there is no dissolved gas in the water phase. The pressure inside a bridge can be as low as the equilibrium vapour pressure of water owing to the high interfacial curvature of nodoid-shaped bridges, see Eqs. Z3. and Z4. above. Alternatively, the gas bridges could be formed by merging of two bubbles attached to the two opposite approaching surfaces. A number of recent studies w23,26 35x provided evidence in support of the capillary-bridge origin of the long-range hydrophobic surface force. In particular, the observation of steps' in the experimental data was interpreted as an indication for separate acts of bridge nucleation or coalescence of attached bubbles w23x. Both mechanisms are possible; for example the more difficult bridging upon the first contact of two hydrophobic surfaces w32x can be attributed to nucleation, whereas the easier bridging upon the second, third, etc. contacts could be due to the coalescence of residual attached bubbles obtained after destruction of the initial bridge. The experiments by Ederth w33x show that the hydrophobic attraction becomes more pronounced during the process of an experiment, because gases from the air dissolve into the originally P.A. Kralche sky, N.D. Denko r Current Opinion in Colloid & Interface Science 6 (2001) 383 401 387 degassed water, resulting in an easier capillary bridging when the surfaces approach each other. Indications about the existence of nano-bubbles attached to a hydrophobic surface were established by means of AFM experiments w35x. It is still unclear why the gas in such bubbles does not dissolve in the surrounding water in view of their high surface curvature which produces a great internal pressure. 3. Lateral capillary forces 3.1. Theoretical background First we briefly consider the theoretical aspects of the lateral capillary forces, following references by Kralchevsky and Nagayama w2 ,3 x. The applied aspects, related to 2D structuring of particles, are reviewed in subsequent sections. As mentioned in the introduction, the origin of the lateral capillary forces is the overlap of perturbations in the shape of a liquid surface due to the presence of attached particles. The larger the interfacial deformation created by the particles, the stronger the capillary interaction between them. It is known that two similar particles floating on a liquid interface attract each other ZFig. 1b.. This attraction appears because the liquid meniscus deforms in such a way that the gravitational potential energy of the two particles decreases when they approach each other. Hence the origin of this flotation capillary force is the particle weight Zincluding the Archimedes force.. Capillary interaction appears also when the particles Zinstead of being freely floating. are partially immersed Zconfined. in a liquid layer; this is the immersion capillary force ZFig. 1c,d.. The deformation of the liquid surface in this case is related to the wetting properties of the particle surface, i.e. to the position of the contact line and the magnitude of the contact angle, rather than to gravity. The flotation and immersion forces can be attractive or repulsive. For the systems depicted in Fig. 1b,c Zmenisci decaying at infinity., solving the Laplace equation of capillarity for small meniscus slope, 2 s q 2 , in cylindrical coordinates Z r, ., one can determine the interfacial shape around a single particle: Z r . s AK 0 Z qr . Z5. Here is the difference between the mass densities of the two fluids, g is the acceleration due to gravity and is the derivative of the disjoining pressure with respect to the film thickness. Eq. Z5. describes a meniscus which is exponentially decaying at infinity. Furthermore, one can apply the superposition approximation, i.e. assume that the interfacial deformation caused by two particles ZFig. 1b,c. is equal to the sum of the deformations caused by the separate particles in isolation. Then, in view of Eq. Z5., the energy of lateral capillary interaction between the two particles is obtained in the form w2 ,3 ,36x: Wf y2 Q1 Q2 K 0 Z qL . Z7. where L denotes the distance between the centres of the two particles, Q i ' ri sin i Z i s 1,2. are the socalled capillary charges', ri and i are the radii of the contact line and the slope angle at the contact line of the respective particle Zsee Fig. 1b for the notation.. W represents a variation in the gravitational energy in case of flotation force, or in the energy of wetting in case of immersion force. The lateral capillary force is given by the derivative F s yd WrdL, which yields: F f y2 Q1 Q2 qK 1Z qL . , Z rk g L . Z8. ZK 1 modified Bessel function.. The asymptotic form of Eq. Z8. for qL< 1 Z qy1 s 2.7 mm for water.: F s y2 Q1 Q2rL, Z r k g L g qy1 . Z9. where K 0 is the modified Bessel function of the second kind and zeroth order and A is a constant of integration, q2 s 2 looks like a two-dimensional analogue of Coulomb's law for the electric force. This is the reason for calling Q1 and Q2 capillary charges'. Generally speaking, the capillary charge characterises the local deviation of the meniscus shape from planarity at the three-phase contact line. The flotation and immersion capillary forces exhibit similar dependence on the interparticle separation, L, but very different dependencies on the particle radius and the surface tension of the liquid. The different physical origin of these forces results in different magnitudes of the corresponding capillary charges'. In this respect there is an analogy with the electrostatic and gravitational forces, which obey the same power law, but differ in the physical meaning and magnitude of the force constants Zcharges, masses.. In this particular case, when R1 s R 2 s R and r k < L < qy1 , one can derive w2 ,3 x: F A Z R 6r . K 1Z qL . F A R K 1Z qL . 2 gr .r q s Zy Z for thick films . Z for thin films . Z6. for flotation force for immersion force Z 10 . 388 P.A. Kralche sky, N.D. Denko r Current Opinion in Colloid & Interface Science 6 (2001) 383 401 In other words, the flotation force decreases, while the immersion force increases, when the interfacial tension increases. Besides, the flotation force decreases with the decrease of R much more strongly than the immersion force. Thus, the flotation force is negligible for R - 5 y 10 m, whereas the immersion force can be significant even when R s 2 nm. The latter force is one of the main factors causing the self-assembly of small colloidal particles and protein macromolecules confined in thin liquid films or lipid bilayers; see Kralchevsky and Nagayama w2 ,3 x for details. The analogy between the capillary and electrostatic interactions was discussed in more details by Paunov w37x. This analogy can be extended further: in addition to the immersion force between capillary charges' Zmonopoles., one can also examine immersion forces between capillary multipoles', see below. 3.2. Flotation capillary forces Here we briefly describe some specific and recent results for the flotation forces. The capillary charge' for floating particles can be estimated from the expression w3 ,36x 1 Q i f q 2 R 3 Z 2 y 4 Di q 3cos i 6 Z i s 1,2. i y cos 3 i a sliding particle' method for determining the coefficient of surface shear viscosity of surfactant adsorption monolayers from the measured f d . Similar experimental method was applied to measure the yield stress of protein adsorption layers, which exhibit elastic and plastic behaviour w39x. Whitesides et al. w40 42x investigated in detail the self-assembly of mesoscale objects under the action of lateral capillary forces. In the case of heavy Zor light. particles, the flotation capillary force was responsible for the observed interparticle attraction. To estimate the force per unit area of the particle contact line these authors applied the Laplace equation for a meniscus of translational symmetry w40 42x. In some of the studies they used floating hexagonal plates of alternatively changing hydrophobic and hydrophilic edges Zsides. w41 x. In such cases, the immersion force between capillary multipoles, in conjunction with the flotation force, is responsible for the observed self-assembly ZSection 3.4.. 3.3. Immersion force between capillary charges' As already mentioned, in the case of the immersion force the interfacial deformation is related to the wetting properties of the particle surface Zposition and shape of the contact line and magnitude of the contact angle., rather than to particle weight and buoyancy, Fig. 1c. The asymptotic Eq. Z8. can be used to calculate the immersion force only for large interparticle distance L, for which the capillary charges' Q i ' ri sin i are independent of L. However, for shorter distances both ri and i become functions of L, in such cases one can calculate the immersion capillary force using the procedure described in section 7.3.2 of Kralchevsky and Nagayama w3 x. There one can find computational procedures for various configurations: two spherical particles; two vertical cylinders; sphere and cylinder; sphere and vertical wall, etc. In addition, two types of boundary conditions are considered: Zi. fixed contact angle; and Zii. fixed contact line, which affects the magnitude of the immersion capillary force; see section 7.3.4 in Kralchevsky and Nagayama w3 x. In particular, if the position Zelevation. of the contact line is fixed at the particle surface, i.e. zs h c s const., then the energy of capillary interaction between two equal particles of circular contact lines of radius rc ZFig. 1c. is given by the expression w43x: K 1Z qrc . y 1 qr K Z qL . 2 c 0 K 0 Z qrc . q K 0 Z qL . ., Z 11 . where Di s Z i II .rZ I II ., i , I and II are the mass densities of the particle and lower and upper fluid phases, respectively. Eq. Z11. allows one to calculate the capillary charge Q i directly from the particle radius R i and three-phase contact angle i . The convenient asymptotic expressions, Eqs. Z8. and Z11., can be used to calculate the flotation capillary force for qR i < 1 and L ) 4 R i . In all other cases one could apply a more accurate computational procedure, which is described in section 8.1.5 of Kralchevsky and Nagayama w3 x. If a vertical plate is partially immersed in a liquid, a capillary meniscus is formed in a vicinity of the plate Zwall.. The overlap of the latter meniscus with the meniscus around a floating particle gives rise to a capillary force Zparticle wall interaction ., which is described by equation 8.106 in Kralchevsky and Nagayama w3 x. If a particle slides along an inclined meniscus, for small Reynolds numbers the capillary force is completely counterbalanced by the hydrodynamic drag force and then the velocity of particle motion is proportional to the capillary force w38x. The coefficient of proportionality gives the hydrodynamic drag coefficient, f d , which turns out to be dependent on the type of surfactant and the density of its adsorption layer at the interface. Petkov et al. w38x developed W Z L. s 2 qrc h 2 c P.A. Kralche sky, N.D. Denko r Current Opinion in Colloid & Interface Science 6 (2001) 383 401 389 Fig. 2. The hydrophobic thickness of an inclusion Ztransmembrane protein. can be Za. greater or Zb. smaller than the thickness, h, of the non-disturbed phospholipid bilayer. The overlap of the deformations around two similar inclusions gives rise to attraction between them w3 ,43x. The interaction energy can be estimated by means of Eq. Z12., h c is the mismatch between the hydrophobic thicknesses of the inclusion and bilayer. done with micrometer-sized latex spheres encapsulated within the bilamellar membrane of a giant lipid vesicle w49,50 x. These experiments neatly reveal the film deformation caused by the particles and the related attraction between them. For such systems the meniscus profile around a single particle obeys the equation Zfor small meniscus slope.: 1 d d r s Ps const. r dr dr z / Z 13. y K 1Z qrc . K 0 Z qrc . where P is the pressure jump across the meniscus. P is constant if the effect of the gravitational hydrostatic pressure is negligible. The general solution of Eq. Z13. is: Z r . s A q Bln r q Z Pr4 . r 2 Z 14. Z 12 . where q is defined by Eq. Z6. for liquid interfaces or films. Eq. Z12. can also be applied to describe the energy of interaction between two inclusions Zfor instance membrane proteins. in a bilayered lipid membrane. In the latter case q f w4 rZ h .x1r2 , where is the shear elastic modulus in the hydrocarbon-chain zone and h is its thickness ZFig. 2., for details see Kralchevsky et al. w43x and Chapter 10 in Kralchevsky and Nagayama w3 x. Note that the interaction energy, as given by Eq. Z12., is proportional to h 2 , i.e. to the c squared mismatch between the hydrophobic zones of the inclusion and bilayer. This interaction can be one of the reasons for aggregation of membrane proteins in biomembranes w43 45x. The immersion capillary force can also be operative between particles captured in a spherical Zrather than planar. thin liquid film or lipid vesicle w3 ,46x. In this case the capillary charge' characterises the local deviation of the meniscus shape from sphericity Zrather than from planarity. at the contact line. Maenosono et al. w47x examined experimentally the motion of millimetre-sized spheres which are partially immersed in a wetting film ZFig. 1c.. Employing Eq. Z9. to estimate the immersion force, these authors determined the friction coefficient Zrelated to the drag force. from the data for the particle law of motion LZ t ., where t is time. It turned out that the friction with the substrate is negligible and the main hydrodynamic resistance comes from the viscous friction in the liquid film w47x. It is worthwhile noting that Eqs. Z5. Z9. are valid for menisci decaying at infinity ZFig. 1c.. However, it is possible the particle size is much greater than the film thickness, as shown in Fig. 1d. For example, Velikov et al. w48 x observed a strong lateral attraction between latex particles of diameter 2 R f 7 m, entrapped in a foam film of thickness which is at least 100 times smaller. Analogous observations have been where A and B are constants of integration. In other words, Eq. Z13. has no axisymmetric solution which is finite at infinity Z r ., cf. Eqs. Z5. and Z14.. The latter fact implies that the meniscus around a particle must end at a peripheral contact line Zof radius rp ., out of which the film is plane-parallel Z ' 0., see Fig. 1d. Hence, we deal with a finite' meniscus. In this case the overlap of the menisci and the interaction between the particles, begins when they are at a distance L - 2 rp from each other. Such types of interaction is obviously different from that described by Eq. Z7., which for long distances yields WA Z qL.y1 r2 expZyqL.. The problem with the immersion capillary force, F, in the case of finite menisci was examined theoretically in Danov et al. w50 x and in a more detailed article by Danov et al. ZLangmuir 2001, submitted.. Both the theory and experiment show that in the investigated case F corresponds to attraction, whose magnitude, N F NsN F Z L. N has a peculiar non-monotonic behaviour: for short distances N F N increases with L, then N F N has a maximum; and decreases further. Moreover, the capillary interaction exhibits hysteresis: on approach of two particles one has F s 0 for L ) 2 rp , the interaction begins with a jump when the two peripheral contact lines touch each other. In contrast, on separation of the two interacting particles Zand of the two overlapping menisci. one has F / 0 for L ) 2 rp . With further increases in L the configuration of the meniscus becomes unstable and it splits to two separate menisci around the respective individual particles. 3.4. Immersion forces between capillary multipoles' The weight of a floating micrometer-sized particle is too small to create any surface deformation. However, surface deformations could appear if the contact 390 P.A. Kralche sky, N.D. Denko r Current Opinion in Colloid & Interface Science 6 (2001) 383 401 line on the particle surface is irregular Zsay, undulated as in Fig. 1e., rather than a perfect circumference. In this case, instead of Eq. Z5., the meniscus shape around a single particle is described by the expression: Zr, .s ms1 K m Z qr .Z A m cos m q Bm sin m . Z 15 . This equation is the respective solution of the linearized Laplace equation of capillarity for small meniscus slope, 2 s q 2 , in cylindrical coordinates Z r, .. Here A m and Bm are constants of integration. For qr < 1 one has K mZ qr . A Z qr .ym and then Eq. Z15. reduces to a multipole expansion Za 2D analogue of Eq. Z15. in electrostatics .. The terms with m s 1, 2, 3,... correspond to dipole', quadrupole', hexapole', etc. In fact, such multipoles were experimentally realised by Bowden et al. w40,41 x by the appropriate hydrophobization or hydrophilization of the sides of floating hexagonal plates. From a theoretical viewpoint, the capillary force between particles of irregular or undulated contact line is a kind of immersion force insofar as it is related to the particle wettability, rather than to the particle weight. Eq. Z5. is the zeroth-order term of the expansion in Eq. Z15.. For m G 1 the capillary force can cause not only translation, but also rotation of the particles. Theoretical description of this capillary force was recently given by Stamou et al. w51 x for rough colloidal spheres. These authors note that for freely floating particles the capillary force will spontaneously rotate each particle around a horizontal axis to annihilate the capillary dipole moment. Therefore, the term with m s 1 in Eq. Z15. has to be skipped and the leading multipole order in the capillary force between such two particles is the quadrupole quadrupole interaction Z m s 2.; the respective interaction energy is w51 x: W Z L . s y12 h2 cos Z 2 c A q2 B . rc4 L4 Z m s 2. Z 16 . Here h c is amplitude of the undulation of the contact line, whose average radius is rc . The angles A and B are subtended between the diagonals of the respective quadrupoles and the line connecting the centres of the two particles. For two particles in contact Z Lrrc s 2. and optimal orientation, cosZ2 A q 2 B . s 1, one obtains Ws yZ3r4. h2 . Thus, c for interfacial tension s 35 mNrm the interaction energy W becomes greater than the thermal energy kT for undulation amplitude h c ) 2.2 A. This result is really astonishing: even a minimal roughness of the contact line could be sufficient to give rise to a significant capillary attraction, which may produce 2D aggregation of colloidal particles attached to a fluid interface, also see Lucassen w52x and Chapter 12 in Kralchevsky and Nagayama w3 x. However, in the angstrom scale the fluid interfaces are not smooth: they are corrugated by thermally excited fluctuation capillary waves, whose amplitude is typically 3 6 A. Hence, one can expect that the effect of the contactline undulations will become significant when their amplitude is greater than the background stochastic noise, that is for nanometre and larger amplitudes w3 x. Note that both Eqs. Z12. and Z16. give WA h 2 . c With respect to the force, F s ydZ W .rdL, for the immersion force between two capillary charges' Zmonopoles. we have F A 1rL, whereas in the case of two quadrupoles the force is F A 1rL5. In other words, for qr < 1, the range of action of the capillary force decreases with the increase of the multipole order, as it could be expected. Consequently, if capillary charges Zi.e. multipoles of the lowest order. are present, as a rule they dominate the capillary interaction. Another difference between the cases of interacting capillary charges' and multipoles' is that the interaction is isotropic for charges, whereas for multipoles Z m G 2. its sign and magnitude depend on the particle mutual orientation. For that reason, as sketched in Fig. 3a,e, the immersion force between quadrupoles Z m s 2. will tend to organise them in a square lattice, rather than in a hexagonal one. Particles hexapoles Z m s 3. can be arranged into a hexagonal lattice with voids ZFig. 3c., or without voids ZFig. 3b,f.. In fact, such lattices Zhexagonal with voids and square. have been observed in the experiments of Bowden et al. w40,41 x. The angular dependence of the immersion force between multipoles leads to the conclusion that this force could hardly produce formation of 2D crystals from a mixture of particles with different multipole orders m. In principle, it is possible this capillary interaction is able to induce a phase separation' of the mixture into ordered domains of particles with the same m. Another possibility, suggested in reference w51 x, is that the particles could form simple linear aggregates. When two or more modes contribute with a considerable weight to the multipole expansion, Eq. Z15., the capillary force is expected to be unable to cause two-dimensional crystallisation of the respective particles, because of their inadequate relative orientations w51 x. It is worthwhile noting that Lucassen w52x was the first who developed a quantitative theory of the capillary forces between particles with an undulated contact line. He considered rough-edged' cubic particles P.A. Kralche sky, N.D. Denko r Current Opinion in Colloid & Interface Science 6 (2001) 383 401 391 edge of the cubic particles is equal to . With s 40 mNrm, s 1 m, h c s 100 nm and h c s 20 nm Eq. Z17. yields Edil f 124 mNrm and Esh f 8 mNrm. The latter values of the surface elasticities are comparable with those for protein adsorption layers w53x. However, the predictions of Eq. Z17. have not yet been verified experimentally. 4. Self-assembly of particles under the action of capillary forces 4.1. Assembly of spherical particles in wetting films The process of formation of well ordered 2D particle arrays in evaporating wetting films was known for many years w54 59x. Regular arrays of latex particles were often fabricated by evaporation of drops of suspensions on solid substrates to prepare samples for optical studies w54x or for modelling the process of paint drying w55x. In the 1970s the mica spreading technique' was introduced for preparation of 2D crystals of viruses and proteins w56,57x, which were suitable for structural analysis by electron microscopy. Yoshimura et al. w58x, developed a mercury spreading technique and obtained 2D crystals from a dozen of protein w3 ,59x. The mechanism and the governing forces of the 2D particle assembly in evaporating wetting films were clarified about a decade ago w60,61x. A two-stage process was observed: Z1. nucleus formation, under the action of attractive capillary immersion forces; and Z2. crystal growth, through convective particle flux caused by the water evaporation from the already ordered array Zsee Fig. 4a.. It was shown that one can produce ordered mono- and multilayers of particles by appropriate control of the shape of the liquid film surface and of the water evaporation rate w60x. The term convective assembly' was introduced w59x to identify this mechanism of particle ordering under the action of capillary immersion force and hydrodynamic drag force. During the last few years, the method of convective assembly has found a wide application, see Section 4.6. Ordered arrays from various nano- and micronsized particles were obtained w62 72 ,73,74 , 75,76 ,77 81x by the method described in Denkov et al. w60,61x or its modifications. Several research groups suggested new versions of equipment or procedures Zbased on the same principles and governing forces., which were aimed at improving the control or at simplifying the procedures for particle array formation. Thus, Dimitrov and Nagayama w65x developed a set-up which resembles in construction the dip-coating apparatus: by appropriate control of the rates of water evaporation and substrate withdrawal, they suc- Fig. 3. The 2D arrays formed by capillary quadrupoles Z m s 2. and hexapoles Z m s 3. w41 ,51 x; the signs q' and y'' denote, respectively, positive and negative capillary charges', i.e. convex and concave local deviations of the meniscus shape from planarity at the contact line. Za. Quadrupoles of square shape form tetragonal close-packed array, irrespectively of the location of the capillary charges Zon the sides or on the corners of the square.. Zb. Plates with a hexagonal shape form a close-packed array only if the charges are located on the corners, whereas Zc. porous Zopened. hexagonal array is formed when the charges are located on the hexagon sides. Zd. Quadrupoles having the shape of hexagons form linear aggregates w51 x. Ze. Quadrupoles having circular shape will form square array, Zf. circular hexapoles can form close-packed hexagonal array. Note that the aggregate structure depends on both the distribution of the capillary charges and the geometrical shape of the elementary cell. with a sinusoidal contact line. The calculated capillary force exhibits a minimum: it is attractive at long distances and repulsive at short separations. Any interfacial deformation, either by dilatation or by shear, will take the particles out of their equilibrium positions Zat the minimum. and will, therefore, be resisted. As a consequence, the particulate monolayer will exhibit dilatational and shear elastic properties w52x. The respective surface dilatational and shear elastic moduli can be estimated by means of the expressions w3 x: Edil f 2 2 3 h3 c , hc Esh f 2 2 h2 c 2 Z 17. where is the wavelength of the undulations, h c is their average amplitude and h c is the standard deviation of the amplitude from its mean value. It is assumed that h c < h c and that the length of the 392 P.A. Kralche sky, N.D. Denko r Current Opinion in Colloid & Interface Science 6 (2001) 383 401 Fig. 4. Ordering of spherical particles in wetting films on substrates: Za. Main driving forces in the convective assembly process: F is a lateral capillary immersion force, between particles captured in the thin liquid film. Fd is a hydrodynamic force, which drags the particles suspended in the thicker layers towards the thinner regions. Fd is caused by a hydrodynamic flux, JW , which compensates the water evaporated from thinner regions w60,61x. Zb. Ordering of particles on patterned solid surfaces. The lateral capillary immersion force, F, acting between particles localised in a given domain, focuses these particles towards the domain center. The conventional method of convective assembly leads to formation of Zc. closepacked mono- and multilayers of particles, whereas the assembly on patterned substrates can be used to fabricate Zd. small particle clusters or Ze. regular arrays of particles with desired lattice shape and separation w85 x. In Qin et al. w88 x the particles are formed in situ in the microdroplets sitting over the hydrophilic domains. ceeded in obtaining centimetre sized, homogeneous in thickness, ordered arrays of latex particles. Other versions of the method utilise a deposition of the suspension with a plate Zplaying the role of a brush. or through an extruder w3 ,59,68x. In other studies w82x, the suspension spreading has been facilitated by applying the spin-coating technique Zrotating substrate ., which is widely used in the polymer industry for casting polymer films. The quality of the final ordered array can be improved if appropriate agitation by sonication is applied during the film drying process w67x. A relatively simple procedure was introduced by Micheletto et al. w66x: a drop of suspension was placed on a glass plate, which was then tilted at an appropriate angle. The array formation starts at the upper edge of the drop and proceeds downwards. Jiang et al. w72 x proposed an efficient procedure for the formation of large single-crystal colloidal multilayers of silica particles by means of controlled evaporation of the disperse medium and the ensuing action capillary forces. Experiments with latex particles on mercury substrates confirmed the important role of the immersion capillary forces for the convective assembly process and showed that the electrical potential of mercury can be used as another parameter for control of the particle substrate interaction and thereby, of the ordering process w3 ,59x. One interesting observation in these studies was the size separation of the particles, when mixtures of particles of various sizes were used w1,77x: the larger particles always collected in the centre of the close-packed hexagonal cluster, surrounded by smaller particles. The reason for this segregation is that the larger particles are pressed by the film surface and start attracting each other by capillary immersion forces before the smaller particles. It was demonstrated w78x that another liquid, perfluorinated oil, can be used as a sub-phase for convective particle assembly. The advantages of the liquid substrates Zmolecular smoothness and mobility of their surfaces . were discussed in relation to the quality of the final ordered arrays. The optical-microscope observations w78x showed that the process is governed by the same forces as in the case of solid substrate: the immersion capillary and hydrodynamic drag forces. Several studies w48 ,79 81x showed that the convective assembly method is operative even for particles, which are captured in free-standing Zfoam. films. Therefore, the solid or liquid substrates are not a necessary component of the assembly process, because the liquid film itself acts as a 2D matrix for particle packing. Following this approach Denkov et al. w79 81x developed a new procedure for the preparation of vitrified aqueous films, which contain monoor multi-layers of particles, suitable for electron cryomicroscopy w83x. The method was applied to nanometre sized latex particles and to monodisperse vesicles made of a lipid protein mixture. 4.2. Self-assembly of particles on patterned substrates or in capillary networks During the last years, several new approaches to the particle self-assembly into complex 2D structures have emerged. One way to fabricate such structures is to use a solid substrate, whose surface is chemically patterned, e.g. by the microcontact printing method w84x, so that different domains of desired shape and size are formed. Aizenberg et al. w85 x showed that micro-patterned substrates, bearing cationic and anionic regions, can be utilised. The particles preferentially stick to the domains of the opposite electric charge. Next, the substrate is rinsed with water, thus removing the non-attached particles. After the rinsing, one observes that the attached particles are con- P.A. Kralche sky, N.D. Denko r Current Opinion in Colloid & Interface Science 6 (2001) 383 401 393 tained in residual water droplets ZFig. 4b.. Upon evaporation of water, each droplet compresses the particles towards the domain centre leaving an ordered cluster after drying is complete. By this procedure, very well ordered 2D particle arrays of desired symmetry and lattice constant Zwhich could be much larger than the particle diameter. were produced w85 x. Furthermore, small clusters of 5 to 7 particles were assembled in the nodes of a 2D periodic lattice ZFig. 4d.. It was demonstrated that other types of particle surface interactions can be used to induce a deposition of particles onto patterned substrates w86,87x. One can conclude that this method for the formation of complex structures has a big potential for further development and applications. An interesting modification of this technique was suggested by Qin et al. w88 x and Zhong et al. w89x who used micro-droplets, formed on the hydrophilic domains of the patterned substrate Zsimilar to those described above w85 x., as micro-crystallisation or micro-chemical reactors, in which the particles were formed in situ, during the drying process. Thus, well ordered arrays of micro- and nano-particles were formed with lattice geometry and spacing, which replicated the initial pattern. In this technique, the particle size and lattice constant can be conveniently controlled by varying the domain size and the concentration of the initial suspension. There are experimental indications that the capillary forces play some role in the deposition of micelles on solid surfaces w90,91x. Massay et al. w90x succeeded to orient specially designed core shell micelles of a cylindrical shape along nano-sized grooves, created by electron beam lithography and reactive ion etching on the surface of a silicon wafer. Thus, nanoscopic lines of approximately 3 nm in height and 15 nm in width were formed, which followed the contours of the lithographic grooves. A different approach to the formation of complex 2D and 3D structures of ordered micro-spheres in pre-formed templates was proposed by Kim et al. w92 x. First, a PDMS stamp is produced, whose surface has a relief of micro-channels of the desired shape and connectivity. This stamp is placed over a solid support and the formed network of microcapillaries is set in contact with a suspension of latex particles. The capillaries spontaneously suck-in suspension and close-packed 2D and 3D arrays of particles are found to fill-up the micro-channels under appropriate conditions. It should be noted that the ordering of the latex particles in these experiments is not driven by capillary forces. The authors showed w92 x that the particles pack in the micro-capillaries mainly under the action of a hydrodynamic drag force, created by the water evaporation at the capillary exits, which resembles the analogous process in the convective assembly method w60,61x. The term MIMIC Zmicro-moulding in capillaries . is used to name this technique w92 x. The latter can be applied to a variety of liquids, which solidify Zor which contain a substance able to solidify. upon the appropriate treatment: polymerisation, curing, crystallisation, etc. Thus, a solid micro-pattern of complex structure can be formed from various materials in the capillaries. The removal of the PDMS stamp results in a patterned substrate. A further chemical or physical treatment can be used to detach the pattern from the support and to obtain free-standing structures. Formations of polyurethane Zfree-standing or attached to SirSiO2 substrate., various salts ZKH 2 PO4 , CuSO4 , K 3 FeZCN. 6 and others., silica and metals were fabricated by means of the MIMIC procedure w92 x. A rich variety of patterns can be produced using this relatively simple method. Following a similar idea, Park et al. w93 x obtained large crystalline assemblies of particles by injecting latex suspension, under pressure, in a narrow slit between two planar surfaces. Micro-channels were made in one of the side-walls of the slit, so that the fluid was able to flow through these channels under the action of the applied pressure. The channels were smaller in size than the diameter of the latex particles, therefore the latter remained captured in the slit. By optimising the applied pressure, water evaporation rate and intensity of sonication, perfectly ordered crystalline assemblies of various thickness were fabricated. Capillary immersion forces could be involved in the observed formation of close-packed arrays of nanoparticles in condensed alkylamine films w94,95x. The role of the meniscus, mediating the capillary interaction, can be played by amphiphilic bilayers, which probably form in the structured alkylamine films. A comprehensive review on various techniques unconventional for fabricating complex nanostructures was recently published by Xia et al. w96x. 4.3. Self-assembly of non-spherical objects into 2D arrays In the framework of the so-called meso-scale selfassembly ZMESA. project, Whitesides et al. w40,41 , 97 102 x developed a new procedure for the fabrication of complex 2D structures from millimetre and sub-millimetre sized plates floating on the interface between water and perfluorodecalin. The plates were made of PDMS moulds of desired shape Zsquare, hexagon, hexagonal rings, or more complex. and their side walls were selectively rendered hydrophobic or hydrophilic. By addition of aluminium oxide to the PDMS mould, the authors were able to vary the mass 394 P.A. Kralche sky, N.D. Denko r Current Opinion in Colloid & Interface Science 6 (2001) 383 401 density of the plates between that of water Z1 grcm3 . and perfluorodecalin Z1.9 grcm3 .. Depending on the mass of the plates and on the hydrophobicityrhydrophilicity of their side walls, lateral capillary forces of different sign and magnitude appear between the neighbouring sides of two approaching plates: some of the sides repel, while others attract each other. As a result, the plates acquire an optimal mutual orientation, which minimises the overall free energy of the system. A shape selective, lock-and-key mechanism is imposed in this way. Taking into account the millimetre size of the plates and the different wettability of their side walls, one can realise that the lateral capillary forces in these systems presents a combination of the flotation force, driven by gravity ZFig. 1b., with the immersion force between the capillary multipoles' ZFig. 1e.. By using a mild agitation Zoscillatory rotation of the container with controlled amplitude and frequency., the authors were able to balance the capillary interaction with the disrupting hydrodynamic shear force, so that well ordered arrays were generated. Whitesides et al. w40,41 ,97 102 x were able to produce in this way a large variety of complex structures: 2D closepacked or porous Zopen. arrays; linear oligomer- and polymer-like aggregates from similar or complementary in shape Zlock-and-key. plates; closed rings of given size and shape; polymer-like branched aggregates; and many others ZFig. 3a,b,c,d.. The observed shape-selective recognition and self-assembly were qualitatively explained by the local deformation of the meniscus around each side of the plates, which gives rise to lateral capillary forces. A far reaching analogy with the receptor ligand interactions in chemistry was drawn and the term capillary bonds' was introduced to designate this type of directed capillary interaction. One can envisage a virtually unlimited variety of hierarchical structures that can be obtained by the MESA approach. The attempts to reduce the characteristic size of the plates down to a micrometer size resulted in less ordered arrays, in which the density of defects was higher w101x. This result was attributed to two main reasons: Zi. the reduced magnitude of the buoyancy force, which affects the deformation of the fluid interface and thereby the flotation force; and Zii. the reduced magnitude of the hydrodynamic shear forces, imposed by the oscillatory motion, for the smaller particles. The second problem could be probably be overcome by using some other means of system agitation Ze.g. sonication.. The first problem, however, recalls the necessity for a more rigorous scaling of the magnitude of capillary forces acting between such shaped objects having neither rotational nor translational symmetry; a corresponding theory is still missing. The MESA approach was successfully applied for fabrication of complex 3D structures as well. By using the surface of a suspended emulsion drop Zwater-in-oil or oil-in-water. as a template, a self-assembled, viruslike shell was formed from gold hexagonal rings with size of approximately 100 m w103 x. To prevent the shell disassembly upon drying, the authors deposited a thin silver layer over the ordered hexagonal rings by using a micro-electrode. The electro-deposition welded the rings into a spherical assembly, which was sufficiently robust to withstand the strong capillary forces upon drying and to remain intact in air. This is one of the most complex tailored procedures, involving a stage of particle self-assembly under the action of lateral capillary forces, which has so far been realised. A simpler version of this technique was first suggested by Velev et al. w104,105x, who ordered micrometer sized latex beads on the surface of emulsion drops. 4.4. Formation of balls and rings of structured spherical particles Recently Velev et al. w106 x developed a new procedure for assembly of micro- and nano-particles into free-standing, millimetre-sized ordered structures of various shapes. Aqueous droplets of colloidal suspension were placed on the surface of an inert liquid substrate Zperfluoromethyldecalin.. The slow evaporation of water led to shrinking of the aqueous drops and to a gradual concentration of the suspended particles. Eventually, a compact 3D colloid crystal was formed. The shape and size of the final assembly can be controlled by various factors, such as the size of the initial drop, particle concentration and presence of surfactants. The latter affect the interfacial tensions Zand thereby the shape of the drop. and the appearance of hydrodynamic fluxes, as a result of the surfactant-driven Marangoni effect during evaporation. By appropriate control of the conditions, Velev et al. w106 x obtained spherical, discoidal, dimpled and toroidal Zring-shaped. assemblies. Anisotropic assemblies were obtained by using mixtures of magnetic q non-magnetic or plastic q gold particles. This method seems rather versatile and can be applied to a wide variety of inorganic and plastic particles and their mixtures. Rings of particles, deposited on solid substrates, are often observed after drying of suspension drops on solid substrates w60,107 109x. The optical observations and the theoretical models have shown that the prevailing force, which determines the particle transport and arrangement in these systems, is the hydrodynamic drag force, caused by the liquid evaporation w60,61x. The ring formation in these experiments is explained by the following scenario w60,107 109x: P.A. Kralche sky, N.D. Denko r Current Opinion in Colloid & Interface Science 6 (2001) 383 401 395 firstly, some of the particles irreversibly stick to the substrate at the three-phase contact line Zdrop periphery. under the action of normal capillary force. The stuck particles pin the contact line so that the process proceeds at a fixed contact radius. The liquid evaporation from the edge is compensated by liquid flow from the interior, which drags the suspended particles towards the drop periphery ZFig. 5a.. As a result, most of the particles get accumulated in the region of the contact line and are compressed there by the liquid meniscus at the final stage of drying. The ring size in these experiments is determined by the initial radius of the contact line. Therefore, this method can be further developed by depositing suspension drops on well defined lattice domains on patterned substrates, similar to those described in the Section 4.3 w84,85 ,86 88 ,89x. In this way, well ordered arrays of rings of given size could be fabricated in a controlled manner. Ohara et al. w110,111x studied the formation of submicrometer rings upon drying of wetting films containing very small nanoparticles Z3 5 nm in diameter.. A characteristic feature of this system is that the particles are so small, that various surface forces become operative in the wetting film of thickness comparable to the particle diameter. These surface forces may induce film destabilisation and hole formation. Therefore, the authors Ohara and Gelbart, and Ohara et al. w110,111x have hypothesised that the particulate rings in their experiments are formed as a result of the nucleation and expansion of holes in the wetting film at the final stage of its drying. The rim of the expanding hole drags the particles away and arranges them into annular rings ZFig. 5b.. The effects of solvent evaporation and surface forces on the conditions for hole nucleation were studied w110x. 4.5. The 2D clusters and foam-like formations of particles on a fluid interface Several studies w51 ,112 115 ,116 ,117,118x reported the formation of 2D clusters and foam-like structures from micrometer-sized spherical particles on the air water interface. The interparticle spacing in these arrays was about several particle diameters. The structuring was observed after spreading a drop of suspension over the surface of an aqueous subphase. In some of these studies, the results can be explained either by pure electrostatic repulsion between the particles w114,115 x, or by an interplay of the common flotation capillary force ZFig. 1b. with the van der Waals and electrostatic forces w112,113x. Other observations w51 ,116 ,117,118x, however, required the authors to invoke a very long-range attraction between the particles. This could be neither the van der Waals force Zwhich was negligible at such separations. nor the flotation capillary force, because the particles were too small and the respective interaction energy was well below the thermal energy, kT. It is worthwhile noting that in these experiments the disperse medium of the suspension, used for spreading of the latex particles, was a methanolrwater mixture Z9r1.. Stamou et al. w51 x suggested that the governing, long-range attractive force in the latter experiments is created by nano-scopic irregularities of the three-phase contact line, i.e. by immersion force between capillary multipoles' ZFig. 1e. and developed a respective theoretical model. One should note, however, that the observed evolution from 2D foam to 2D clusters in these experiments resembles very much the processes observed when an aqueous suspension of latex particles was spread over liquid perfluorodecalin sub-phase w78x. Observations in reflected light of the particle assembly process on perfluorodecalin proved that the latex particles were captured in thin aqueous films w78x and the assembly process was driven by the immersion force between capillary charges', which is sufficiently strong and long-ranged to explain the observed phenomena. Therefore, it is worthwhile verifying whether some oily film Zsay from dissolved styrene oligomers. is not generated on the surface of the aqueous sub-phase when spreading the suspension of latex particles in methanol. 4.6. Applications of 2D particle arrays Here we briefly overview the main areas, in which the 2D colloidal arrays, assembled with the help of capillary forces, find application. Fig. 5. Two possible ways for fabrication of circular particulate rings on substrates: Za. by evaporating drops of suspensions; in this case the particle transport is driven by a hydrodynamic drag force, Fd . Zb. By hole formation and subsequent expansion, which leads to lateral capillary immersion force at the rim periphery. The simultaneous formation of many holes in the layer leads to formation of 2D foam-like assembly. 396 P.A. Kralche sky, N.D. Denko r Current Opinion in Colloid & Interface Science 6 (2001) 383 401 The ordered 2D arrays of monodisperse latex particles attracted the attention of researchers a long time ago due to their interesting optical properties w54,62,72 ,73,74 ,119,120 ,121 123x. When the particle size and spacing are comparable to the wavelength of the illuminating light, a rich variety of optical phenomena, caused by the light diffraction and interference can be observed. Since the lattice constant and the refractive index of the particles can be varied in wide ranges, the optical properties of the arrays can be finely tuned. Therefore, the 2D colloid crystals are studied in the literature as optical elements, such as diffraction gratings, interference filters, antireflection coatings, micro-lenses, etc. w54,62,72 ,119x. Along with these more conventional applications, recent studies were directed to study the application of particle arrays as basic photonic elements and as photonic band-gap crystals, i.e. as periodic dielectric structures, in which the photons behave in a manner similar to that of electrons in semiconductors w73,74 ,120 , 121 123x. During the last several years rapid progress was achieved in developing procedures for fabrication of regular, highly porous structures from various materials by using 2D and 3D colloid crystals as templates w124 127 ,128 ,129,130 ,131,132x. Such porous structures have been obtained from inorganic oxides, polymers, glassy carbon, semiconductors and metals. These structures are of great interest due to their unique optical properties and possible applications in catalysis, including the photo- and electro-catalysis. A comprehensive overview on this subject can be found in Holland et al., Jiang et al. and Velev and Kaler w127 ,128 ,130 x. Ordered 2D particle monolayers were successfully applied as lithographic masks for fabrication of regular nano-structures on silicon substrates by etching or vacuum deposition of metal w75,76 ,82,133x. For this purpose, Burmeister et al. w76 x further developed the convective assembly method by including several additional steps: The pre-formed dry 2D array of latex particles was reinforced by vacuum deposition of metal or by thermal annealing. These procedures led to shrinking of the openings between the latex beads, without completely closing them. Afterwards, the glass substrate was slowly dipped into a water bath and the strengthened particle monolayer floated off onto the water surface. If a metal deposition is used, the substrate remains covered with a regular array of nanometal dots, which replicate the voids between the latex particles in the initial array. However, the floating particle array can be transferred from the water surface onto another solid substrate for further experiments. In this way, a free standing, transportable lithographic mask was produced. This technique has been termed natural' w133x, nanosphere' w82x or col- loid monolayer lithography' w75,76 x. The use of patterned substrates, allows one to obtain 2D crystal symmetries, which are different from the simple close-packed hexagonal array w76 x. The process of particle ordering in wetting films is related to paint coatings. Recent studies showed w134 136 x that the paint layer formation from aqueous dispersions resembles in some aspects the convective assembly process. In many systems Zespecially in those containing surfactants . the drying of the paint layer is not uniform but occurs through a moving zone, which separates the wet Zthicker. from the dry Zthinner . films this compaction' zone moves in the plane of the layer so that the dry regions expand with time at the expense of the wet regions. A hydrodynamic force Zcaused by the water evaporation. drags the suspended particles from the wet regions towards the dry regions, just as in the convective assembly method w60,61x. This hydrodynamic flux redistributes also the electrolytes and surfactants, so that the final composition of the dry layer is non-uniform in the plane of the layer w134,135x. The degree of particle ordering in the final layer depends primarily on the processes taking place in the compaction zone, where the capillary forces press the particles against each other and against the substrate. The capillary bridge forces play a dominant role also in the process of particle deformation Zand to some extent, of particle fusion. in the latex layers w134 136 ,137,138x. Ordered monolayers of particles are used also in some biological studies. Thus, Miyaki et al. w139x studied the interaction of neutrophil-type biological cells with particle arrays as a function of the particle size. They found that the contact of the cells with the particle array activated the cell to an extent, which strongly depended on the particle size. Thus, the particle arrays serve as convenient micro-patterned surfaces for studying the cell adhesion. The wide ranges of available particle sizes and compositions, along with the possibility to graft various bio-active molecules onto the particle surface, imply that these studies will expand in the future. 5. Summary and conclusions The review of publications from the last 2 3 years shows that there is a permanent interest in the various kinds of capillary forces, which often play an essential role in fundamental and applied studies. The capillary bridge forces ZFig. 1a. are investigated in relation to AFM experiments w4 6,8 10 ,11,32x, studies on capillary condensation and measurements by the surface-force apparatus w18,24 27x, capillary cavitation and long-range hydrophobic surface force w23,28 35 x, antifoaming action of oily drops P.A. Kralche sky, N.D. Denko r Current Opinion in Colloid & Interface Science 6 (2001) 383 401 397 w13,14 ,15x. The flotation lateral capillary force ZFig. 1b. is employed in surface rheological measurements w38x and for self-assembly of floating meso-scale objects w40,41 ,97 102 x. The common immersion capillary force ZFig. 1c. is reported to take part in the 2D aggregation and ordering of micrometer-sized and sub-micrometer particles confined in wetting and free-standing liquid films w54 72 ,73,74 ,75,76 , 77 82x. Systematic presentation of the theory of the aforementioned types of capillary forces ZFig. 1a c. was recently published w2 ,3 x. New theoretical developments are devoted to the description of the immersion force in the case of finite menisci ZFig. 1d. w50 x and the forces between capillary multipoles' ZFig. 1e. w51 x. The latter two themes mark directions for future research, both theoretical and experimental. The studies on self-assembly of colloidal particles, mediated by the action of capillary forces, develops along the following major directions: improving the procedures for particle assembly in wetting films w72 x or in shrinking drops w106 x, structuring on patterned substrates or in capillary networks w85 ,86 88 ,89,90,92 x, and self-assembly of floating non-spherical objects into complex 2D arrays w41 ,102 x. The obtained ordered structures of particles have found miscellaneous applications. Acknowledgements The authors are grateful to Dr Theodor Gurkov, Ms Petia Vlahovska, Mr Krassimir Velikov, Mr Stanislav Kotsev, Ms Denitza Lambreva and Mr Alexander Zdravkov, who helped to collect the literature sources, as well as to Ms Mariana Paraskova for preparing the figures. References and recommended reading of special interest of outstanding interest w1x Yamaki M, Higo J, Nagayama K. Size-dependent separation of colloidal particles in two-dimensional convective self-assembly. Langmuir 1995;11:2975 2978. w2x Kralchevsky PA, Nagayama K. Capillary interactions between particles bound to interfaces, liquid films and biomembranes. Adv. Colloid Interface Sci. 2000;85:145 192. An overview, comparison and discussion of recent results, both theoretical and experimental, about lateral capillary forces. w3x Kralchevsky PA, Nagayama K. Particles at fluid interfaces and membranes: attachment of colloid particles and proteins to interfaces and formation of two-dimensional arrays. Amsterdam: Elsevier, 2001. Chapters 7 9: theory and experiment on lateral capillary forces. Chapter 10: interactions between inclusions in lipid membranes. Chapter 11: capillary bridges and capillary-bridge forces. Chapter 12: capillary forces between particles of irregular contact line. Chapter 13: two-dimensional crystallisation of particulates and proteins. Chapter 14: effect of oil drops and particulates on the stability of foams Zantifoaming.. w4x Fujihira M, Aoki D, Okabe Y, Takano H, Hokari H. Effect of capillary force on friction force microscopy: a scanning hydrophilicity microscope. Chemistry Letters, 1996:499 500. ZJapan. w5x Behrend OP, Oulevey F, Gourdon D et al. Intermittent contact: tapping or hammering? Applied Physics A 1998;66:S219 S221. w6x Suzuki H, Mashiko S. Adhesive force mapping of frictiontransferred PTFE film surface. Applied Physics A 1998;66:S1271 S1274. w7x Fielden ML, Hayes RA, Ralston J. Surface and capillary forces affecting air bubble particle interactions in aqueous electrolyte. Langmuir 1996;12:3721 3727. w8x Preuss M, Butt H-J. Direct measurement of particle-bubble interactions in aqueous electrolyte: dependence on surfactant. Langmuir 1998;14:3164 3174. w9x Preuss M, Butt H-J. Measuring the contact angle of individual colloidal particles. J. Colloid Interface Sci. 1998; 208:468 477. w10x Ecke S, Preuss M, Butt H-J. Microsphere tensiometry to measure advancing and receding contact angles on individual particles. J. Adhesion Sci. Technol. 1999;13:1181 1191. Silanized silica spheres, 4.1 m in diameter, were used. The distance to which a sphere, attached to the AFM cantilever, jumps into its equilibrium position at the air-liquid interface of a drop or an air bubble was measured. From these distances the contact angles were calculated. No hysteresis was measured with microspheres, whereas a considerable hysteresis was established with similarly prepared planar silica surfaces. w11x Preuss M, Butt H-J. Direct measurement of forces between particles and bubbles. Int. J. Miner. Process. 1999;56:99 115. w12x Aveyard R, Clint JH. Liquid lenses at fluidrfluid interfaces. J. Chem. Soc. Faraday Trans. 1997;93:1397 1403. w13x Denkov ND, Cooper P, Martin J-Y. Mechanisms of action of mixed solid-liquid antifoams. 1. Dynamics of foam film rupture. Langmuir 1999;15:8514 8529. w14x Denkov ND. Mechanisms of action of mixed solid-liquid antifoams. 2. Stability of oil bridges in foam films. Langmuir 1999;15:8530 8542. A new, bridging-stretching', mechanism of foam destruction by oil drops was proposed based on systematic experimental investigations. The small oil bridges in foam films are stable, whereas the larger bridges are unstable Zfew milliseconds lifetime.. Initially stable small bridges could be later transformed into unstable ones due to the thinning of the foam films and the transfer of pre-spread oil. w15x Denkov ND, Marinova KG, Christova C, Hadjiiski A, Cooper P. Mechanisms of action of mixed solid-liquid antifoams: 3. Exhaustion and reactivation. Langmuir 2000;16:2515 2528. w16x Abramowitz M, Stegun IA. Handbook of mathematical functions. New York: Dover, 1965. w17x Dimitrov AS, Miwa T, Nagayama K. A comparison between the optical properties of amorphous and crystalline monolayers of silica particles. Langmuir 1999;15:5257 5264. w18x Aveyard R, Clint JH, Paunov VN, Nees D. Capillary condensation of vapours between two solid surfaces: effects of line tension and surface forces. Phys. Chem. Chem. Phys. 1999;1:155 163. w19x Orr FM, Scriven LE, Rivas AP. Pendular rings between solids: meniscus properties and capillary force. J. Fluid Mech. 1975;67:723 742. w20x de Lazzer A, Dreyer M, Rath HJ. Particle surface capillary forces. Langmuir 1999;15:4551 4559. 398 P.A. Kralche sky, N.D. Denko r Current Opinion in Colloid & Interface Science 6 (2001) 383 401 forces: synthesis using the capillary bond. J. Am. Chem. Soc. 1999;121:5373 5391. The interactions between floating hexagonal plates, 5.4 mm in diameter, were experimentally examined. The faces of the plates were functionalized to be hydrophilic or hydrophobic; in this way various types of capillary multipoles' were realized. The strength and the directionality of the interactions can be tailored by manipulating the geometric size and the pattern of hydrophobization. The type of 2D lattice of the assemblies was investigated as a function of the sort of particles. w42x Grzybowski BA, Bowden N, Arias F, Yang H, Whitesides GM. Modeling menisci and capillary forces from the millimeter to the micrometer size range. J. Phys. Chem. B 2001;105:404 412. w43x Kralchevsky PA, Paunov VN, Denkov ND, Nagayama K. Stresses in lipid membranes and interactions between inclusions. J. Chem. Soc. Faraday Trans. 1995;91:3415 3432. w44x Gil T, Ipsen JH, Mouritsen OG, Sabra MC, Sperotto MM, Zuckermann MJ. Theoretical analysis of protein organization in lipid membranes. Biochim. Biophys. Acta 1998; 1376:245 266. w45x Mansfield SL, Gotch AJ, Harms GS, Johnson CK, Larive CK. Complementary analysis of peptide aggregation by NMR and time-resolved laser spectrometry. J. Phys. Chem. B 1999;103:2262 2269. w46x Kralchevsky PA, Paunov VN, Nagayama K. Lateral capillary interaction between particles protruding from a spherical liquid layer. J. Fluid Mech. 1995;299:105 132. w47x Maenosono S, Dushkin CD, Yamaguchi Y. Direct measurement of the viscous force between two spherical particles trapped in a thin wetting film. Colloid Polym. Sci. 1999;277:993 996. w48x Velikov KP, Durst F, Velev OD. Direct observation of the dynamics of latex particles confined inside thinning water air films. Langmuir 1998;14:1148 1155. Dynamics of micrometer-sized latex particles confined in thinning foam films was investigated. The behavior of the particles depend on several experimental factors, incl. the type of surfactant. In some experiments the captured particles tended to form small 2D aggregates; the latter were observed to attract each other from distances, which could reach up to 100 m. This attraction was attributed to the lateral immersion force. At higher particle concentration the formation of 2D-foam' structure was observed. w49x Dietrich C, Angelova M, Pouligny B. Adhesion of latex spheres to giant phospholipid vesicles: statics and dynamics. J. Phys. II Fr. 1997;7:1651 1682. w50x Danov KD, Pouligny B, Angelova MI, Kralchevsky PA. Strong capillary attraction between spherical inclusions in a multilayered lipid membrane. In: Iwasawa Y, Oyama N, Kunieda H, editors. Amsterdam: Elsevier, 2001. Strong attraction has been experimentally observed between two spherical latex particles, which are included in the membrane of a giant spherical phospholipid vesicle. This attraction was interpreted as a lateral capillary force resulting from the overlap of the menisci formed around each of the two particles. The theoretical results about the force are in line with the experimentally observed trends. w51x Stamou D, Duschl C, Johannsmann D. Long-range attraction between colloidal spheres at the air-water interface: the consequence of an irregular meniscus. Phys. Rev. E 2000;62:5263 5272. Theory of the interactions between floating spherical particles with undulated contact line Zcapillary multipoles'. is developed. A convenient analytical expression is obtained for the energy of interaction between two capillary quadrupoles. Frustrations of the formation of some types of 2D lattices due to inadequate particle w21x Kolodezhnov VN, Magomedov GO, Mal'tsev GP. Refined determination of shape for the free surface of the liquid region in analysis of capillary interaction of powder particles. Colloid J. Russ. 2000;62:443 450. w22x Willett CD, Adams MJ, Johnson SA, Seville JPK. Capillary bridges between two spherical bodies. Langmuir 2000; 16:9396 9405. w23x Attard P. Thermodynamic analysis of bridging bubbles and a quantitative comparison with the measured hydrophobic attraction. Langmuir 2000;16:4455 4466. w24x Yaminsky VV. Long range attraction in water vapor. Capillary forces relevant to polywater'. Langmuir 1997;13:2 7. w25x Xiao X, Quian L. Investigation of humidity-dependent capillary force. Langmuir 2000;16:8153 8158. w26x Claesson PM, Dedinaite A, Bergenstahl B, Campbell B, Christenson H. Interaction between hydrophobic mica surfaces in triolein: triolein surface orientation, solvation forces, and capillary condensation. Langmuir 1997;13:1682 1688. w27x Petrov P, Olsson U, Wennerstrom H. Surface forces in bicontinuous microemulsions: water capillary condensation and lamellae formation. Langmuir 1997;13:3331 3337. w28x Carambassis A, Jonker LC, Attard P, Rutland MW. Forces measured between hydrophobic surfaces due to a submicroscopic bridging bubble. Phys. Rev. Lett. 1998;80: 5357 5360. w29x Considine RF, Hayes RA, Horn RG. Forces measured between latex spheres in aqueous electrolyte: Non-DLVO behavior and sensitivity to dissolved gas. Langmuir 1999;15:1657 1659. w30x Considine RF, Drummond CJ. Long-range force of attraction between solvophobic surfaces in water and organic liquids containing dissolved air. Langmuir 2000;16:631 635. w31x Mahnke J, Stearnes J, Hayes RA, Fornasiero D, Ralston J. The influence of dissolved gas on the interactions between surfaces of different hydrophobicity in aqueous media. Part I. Measurement of interaction forces. Phys. Chem. Chem. Phys. 1999;1:2793 2798. w32x Yakubov GE, Butt H-J, Vinogradova O. Interaction forces between hydrophobic surfaces. Attractive jump as an indication of formation of stable' submicrocavities. J. Phys. Chem. B 2000;104:3407 3410. w33x Ederth T. Substrate and solution effects on the long-range hydrophobic' interactions between hydrophobized gold surfaces. J. Phys Chem. B 2000;104:9704 9712. w34x Ishida N, Sakamoto M, Miyahara M, Higashitani K. Attraction between hydrophobic surfaces with and without gas phase. Langmuir 2000;16:5681 5687. w35x Ishida N, Inoue T, Miyahara M, Higashitani K. Nano bubbles on a hydrophobic surface in water observed by tapping-mode atomic force microscopy. Langmuir 2000;16:6377 6380. w36x Chan DYC, Henry JD, White LR. The interaction of colloidal particles collected at fluid interfaces. J. Colloid Interface Sci. 1981;79:410 418. w37x Paunov VN. On the analogy between lateral capillary interactions and electrostatic interactions in colloid systems. Langmuir 1998;14:5088 5097. w38x Petkov JT, Danov KD, Denkov ND, Aust R, Durst F. Precise method for measuring the shear surface viscosity of surfactant monolayers. Langmuir 1996;12:2650 2653. w39x Petkov JT, Gurkov TD, Campbell BE, Measurement of the yield stress of gel-like protein layers on liquid surfaces by means of an attached particle. Langmuir 2001, 17, in press. w40x Bowden N, Terfort A, Carbeck J, Whitesides GM. Self-assembly of mesoscale objects into ordered two-dimensional arrays. Science 1997;276:233 235. w41x Bowden N, Choi IS, Grzybowski BA, Whitesides GM. Mesoscale self-assembly of hexagonal plates using lateral capillary P.A. Kralche sky, N.D. Denko r Current Opinion in Colloid & Interface Science 6 (2001) 383 401 orientations are discussed. The theoretical predictions are compared with experimental data for structuring of latex particles spread over the surface of water. w52x Lucassen J. Capillary forces between solid particles in fluid interfaces. Colloids Surf. 1992;65:131 137. w53x Pe...

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Air Force Academy - COMP - 2300
NORTH CAROLINA CENTRAL UNIVERSITY DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE COMP 2300, Fall 2008 SYLLABUS DISCRETE STRUCTURES FOR COMPUTING MWF 1:00 1:50 PM, room 2226 Science Complex Advancing Teaching, Scholarship, and Service through Diversi
Air Force Academy - COMP - 2300
COMP 2300 Discrete Structures for Computing, Fall 2008October 27, 2008Quiz #4 (Solution)Consider the recursively defined sequence a1 = 1, an = an-1 + 2n - 1, n Z, n 2 (a) Compute the first five terms of the sequence (3 pts.) a1 = 1 a2 = a1 + 2
Air Force Academy - COMP - 2300
COMP 2300 Discrete Structures for Computing, Fall 2008November 12, 2008Quiz #5 (Solution)Compare each two of the three given functions, using the appropriate asymptotic notation (5 pts. each): f (x) = x3 + x2 , Solution: According to the theorem