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Continuous and discontinuous transitions between two types of capillary bridges on a beaded chain pulled out from a liquid† Filip Dutka,

*ab Zbigniew Rozynek

c

a and Marek Napio ´ rkowski

Capillary bridges can be used for fabricating new materials and structures. Here, we describe theoretically and validate experimentally the mechanism of formation of capillary bridges during a process in which a beaded chain is being pulled out from a liquid with a planar surface. There are two types of capillary bridges present in this system, namely the sphere–planar liquid surface bridge initially formed between the spherical bead leaving the liquid bath and the initially planar liquid surface, and the sphere– sphere capillary bridge formed between neighbouring beads in the part of the chain above the liquid Received 24th February 2017, Accepted 30th May 2017

surface. During the process of pulling the chain out of the liquid, the sphere–planar liquid surface bridge

DOI: 10.1039/c7sm00396j

the chain, this transition can be either continuous or discontinuous. The transition is continuous

transforms into the sphere–sphere bridge. We show that for monodisperse spherical beads comprising when the diameter of the spherical beads is larger than the capillary length. Otherwise, the transition is

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discontinuous, likewise the capillary force acting on the chain.

1 Introduction Capillary bridges play an important role in many physical phenomena, including agglomeration,1–3 mechanical strengthening,4–6 surface adhering,7,8 rheological response,9–11 capillarygripping12,13 and self-assembly.14–16 They are often utilized in materials science, e.g. for fabricating new materials using capillary suspensions,17–19 new one-dimensional structures,20–23 in nanolithography,24–26 microplating,27 for pattern formation,28 and in printed electronics technologies.29 It is thus essential to know their properties and behavior. A substantial effort is directed towards understanding the mechanisms of their formation and shape development,30–33 rupturing34–37 and evaporation.38,39 Capillary bridges exist at solid contacts between spheres,40–43 rods,44–46 plates,47,48 a mix of these,49–53 bodies with other shapes,54–57 and between porous objects.58,59 They are also formed, when a particle is being pulled out from a suspension. Then, a liquid rises to a certain height above the bulk planar level, and forms a concave meniscus. This meniscus is called a capillary bridge between a particle and a planar liquid surface. a

Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa, Poland. E-mail: [email protected]; Tel: +48 2255 32908 b Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warszawa, Poland c Faculty of Physics, Institute of Acoustics, Adam Mickiewicz University, Umultowska ´, Poland 85, 61-614 Poznan † Electronic supplementary information (ESI) available: Two movies and two animations. See DOI: 10.1039/c7sm00396j

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Such capillary bridges can be used for determining the surface tension,60–63 in a way similar to the methods that employ a Wilhelmy plate,64–66 a cylindrical fiber,67–69 or a toroidal ring.70,71 In this article, we analyze the development of capillary bridges formed on a beaded chain being pulled out from a liquid, Fig. 1. We study the formation of a sphere–planar liquid

Fig. 1 (a) Scheme illustrating a beaded chain with spherical particles of radius R being pulled out from a liquid with a planar surface. The contact angle y = 0, i.e. the bridge is tangential to the sphere. (b) The sphere– planar liquid surface capillary bridge is marked by a darker color and the corresponding surface capillary force F sp is directed downwards. (c) The sphere–sphere capillary bridge is marked by a darker color and the corresponding sphere–sphere capillary force F ss between neighbouring particles in the chain acting on the lower sphere is directed upwards.

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surface bridge, Fig. 1b, and its transition into a sphere–sphere capillary bridge, Fig. 1c. This research originates from the observation of the formation of liquid bridges during a novel process of fabricating beaded chains. The assembling of these chains and the accompanying process of pulling them out from a liquid suspension is induced by an electric field acting on the system.72 A needleshaped electrode is dipped into a dispersion of conductive particles. The electric field polarizes the particles which form a chain. As the electrode is being elevated, the particles forming the chain are successively pulled out from the liquid. Fig. 2a shows an example of such a chain composed of hundreds of microparticles (30 mm in radius). Owing to the action of the liquid bridges (clearly resolved in the blow-up of Fig. 2a) the assembled structure remains stable outside the liquid environment even in the absence of the electric field. We observed that the assembly process proceeded unevenly when the beads forming the chain were small (micrometers) and smoothly when the beads had millimeter size, see Fig. 2b and Movies S1 and S2 (ESI†). The way this process proceeded was related to the mechanism of the capillary bridge formation. Here, we provide a theoretical description of that mechanism and show that for monodispersed spherical beads forming a chain, the transition between two types of bridges can be either continuous or discontinuous. The transition is continuous when the diameter of the spheres is larger than the capillary pffiffiffiffiffiffiffiffiffiffiffi length l ¼ g=rl g, where g is the surface tension coefficient, rl the liquid density, and g the gravitational acceleration. Otherwise, the transition is discontinuous, as is the capillary force acting on the chain. Our theoretical predictions were confirmed in experiments. We performed experiments with two beaded chains composed of either large (1 mm and 2 mm radius) or small (25 mm and 30 mm radius) spheres being pulled out from a liquid for which l = 1.45 mm. In order to deal with a permanent chain, we either glued the spheres or assembled them by employing an alternating electric field in the process described in ref. 72. In short,

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the process requires the sum of attractive dipolar force and capillary forces between neighbouring particles Fss to be sufficiently strong to overcome the capillary force Fsp stemming from a sphere–planar liquid surface bridge. Capillary bridges stabilize the chain. This article is structured as follows: in Section 2, we introduce a theoretical model which describes the formation of a sphere– planar liquid surface bridge, calculate the shape of the bridge in the presence of a gravitational field, and the corresponding capillary force. Then, we track the process of pulling the beaded chain out of the liquid. During this process, the liquid meniscus shape undergoes a change which can be continuous or discontinuous, depending on the radius of spheres and the capillary length, which is discussed in Section 3. Both the sphere–liquid surface and sphere–sphere capillary forces are evaluated and discussed in Section 4. We conclude the paper with a summary in Section 5.

2 Liquid bridge between a sphere and a planar liquid surface The initial level of the planar liquid surface is assumed to be at z = 0, see the inset in Fig. 3. The chain is composed of spheres of radius R that touch one another, and are aligned in a line along the z-axis. The system is assumed to have cylindrical symmetry around the chain axis, and the radius of the container is rmax. Upon pulling out the chain, a sphere–planar liquid surface bridge emerges, and, because the volumes of both the liquid and the spheres are fixed, the liquid level decreases from z = 0 to z = zmin o 0. The energy of the sphere–planar liquid surface bridge, as compared to the initial planar configuration of the liquid surface, is a sum of capillary and gravitational terms ð z1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Esp ½rðzÞ ¼ 2p dz grðzÞ 1 þ r0 ðzÞ2 zmin

1 þ rl gz rðzÞ2 r1 ðzÞ2 2

(1)

prmax 2 g þ 2pRðh z1 Þg;

Fig. 2 (a) Image of a beaded chain formed by small spherical beads of radius R = 30 mm pulled out from a liquid with particle suspension. The chain holds together only due to capillary forces acting between neighbouring spheres. (b) The image of two large spheres (R = 1 mm) aligned along the vertical direction along which the chain is being pulled. The upper particle is the beginning of a chain. The lower particle is just being pulled out from a liquid. The liquid forming a capillary sphere–sphere bridge is well-resolved.


Fig. 3 Schematic shape of a sphere–planar liquid surface bridge formation, during the process of pulling out a beaded chain from a liquid bath of size rmax, and the initial flat configuration of the liquid surface (inset). The chain consists of aligned in a line, with spherical beads of radius R touching each other. The shape of the bridge is cylindrically symmetric and described by a function r(z). The maximum level of the liquid meniscus is z1, and the minimum is zmin. Upon increasing the height h, the characteristic angle b decreases. The system is placed in a gravitational field g.

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2.1

with the constant liquid volume constraint DV½rðzÞ ¼ p

ð z1

p dzrðzÞ2 þ ðh z1 Þ2 ð3R h þ z1 Þ 3 zmin

(2)

prmax 2 jzmin j ¼ 0: The liquid–gas surface tension coefficient is denoted as g, rl is the density of the liquid, g = 9.81 m s2 is the gravitational acceleration, and r = r1(z) describes the surface of the spheres already pulled out above the planar liquid surface. The quantity DV is the difference in volume corresponding to the sphere– planar liquid surface bridge configuration, and the one corresponding to the initial flat surface configuration. Note that in both configurations, the volume of the spheres is taken into account. We assume that the difference between the sphere–gas and the sphere–liquid surface tension coefficients is larger than the liquid–gas surface tension coefficient gsg gsl 4 g,

(3)

leading to the contact angle y = 0 which is the case in our experiments with silicon oil and steel spheres.73 There is always a liquid layer coating the solid substrate, and its thickness is proportional to Ca2/3, where Ca denotes the capillary number Ca = mv/g, where m is viscosity of the liquid and v the pulling velocity. For small pulling velocities v B 0.01 mm s1 rendering Ca B 106, this thickness is four orders of magnitude smaller than the diameter of the spheres.68 Therefore, in our analysis we ignore the presence of the coating layer, assuming that the bridge meniscus is tangential to the sphere at z = z1. Obviously, the analysis holds also for gsg gsl = g, when there is no coating layer. For a given height h, the equilibrium profile r(z) = req(z) minimizes the functional Esp[r(z)], eqn (1), under the constant volume constraint 0¼

d Esp ½rðzÞ DpDV½rðzÞ ; drðzÞ rðzÞ¼req ðzÞ

(4)

which gives the equation for the shape of the interface

00

1

req 1 þ req

0

ðzÞ2

1=2

req ðzÞ 1 þ req

0

3=2 ðzÞ2

¼

Dp z 2: g l

For large containers, rmax c l, R, one can consider the limiting case of an infinite system, rmax - N. For such a case, there is no fixed volume constraint, thus, the pressure difference vanishes, Dp = 0. The equilibrium shape of the interface r0(z) fulfills the equation 00

1 r0 ð1 þ r00 ðzÞ2 Þ

1=2

r0 ðzÞ ð1 þ r00 ðzÞ2 Þ

3=2

z ¼ 2: l

(6)

For contact angle y = 0, the boundary conditions at the sphere, z1 = h R(1 + cos b), for a given angle b, Fig. 3, are r0(z1) = R sin b, r0 0 (z1) = cot b.

(7)

One would expect that for small spheres gravity is not significant, and one can ignore the term z/l2 on the rhs of eqn (6). Then, the equilibrium shape would be r0(z) = w cosh(z z0)/w, with w and z0 being two integration constants. Note, that for angles b o p/2, for z = z0, the bridge has minimal width equal to w = r0(z0). The slope of the interface at z = 0 equals r0 0 (0) = sinh z0/w, and because w, z0 4 0 are finite, it cannot be infinite. The condition that the position of the interface has both infinite value and slope at z = 0 cannot be satisfied. Thus, the function r0(z) = w cosh(z z0)/w is not an acceptable equilibrium shape of the bridge for our considerations. It turns out that the term z/l2 in eqn (6) provides that r0 0 (0) - N, and it cannot be neglected. A procedure of numerical integration of eqn (6) is described in Appendix A. It turns out that there exists a particular value of the height h = hmax (maximal height for a given sphere radius), such that for h o hmax there exist two solutions of eqn (6) corresponding to different angles b, Fig. 4. On the other hand, for large heights h 4 hmax, the solution of eqn (6) ceases to exist. We checked that in the case h o hmax when two solutions exist, the solution corresponding to the larger value of angle b always has lower energy than the solution corresponding to the lower value of b, Fig. 4. Hence, the solution for larger angle b describes the equilibrium capillary bridge, while the one with lower b the metastable capillary bridge. For equilibrium solution, the angle b is a decreasing function of h, and for metastable solution, it is an increasing function of h, Fig. 4.

(5)

The expression on the left-hand side of the above equation represents the mean curvature multiplied by factor two, Dp is the Lagrange multiplier, which is the difference in liquid and gas pressures (Laplace pressure). If the hydrostatic pressure corresponding to height z, i.e., zrlg is small compared to the Laplace pressure Dp, then the liquid meniscus forms a constant mean curvature surface,49,74,75 and can be described analytically by elliptic integrals.76,77 In our analysis, we take into account the presence of a gravitational field, and the shape of the bridge can be determined only numerically.

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Equilibrium and metastable states

3 Continuous and discontinuous transitions We note that some of the solutions of eqn (6) are non-physical, because when the presence of the lower, neighbouring sphere is taken into account, then the liquid–gas interface crosses this sphere. In other words, there exists 0 o z o z1, for which r0(z) o r1(z). In order to track the distance between the interface and the neighbouring lower sphere, the parameter qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 ðhÞ ¼ min r0 ðzÞ2 þ ðz ðh 3RÞÞ2 R (8) 0 o z o z1


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Fig. 4 (a) Shapes of equilibrium (full line) and metastable (dashed line) sphere–planar liquid surface bridges. (b) The plot of angle b and the energy of a sphere–planar liquid surface bridge Esp (inset) as a function of height h for equilibrium (full line) and metastable (dashed line) solutions. Sphere radius R = 25 mm and capillary length l = 1.45 mm.

is introduced. It turns out that for R o l/2, this distance is always positive for equilibrium bridges, while for R 4 l/2, it is zero for certain htr o hmax, Fig. 5. Thus, for large spheres (R 4 l/2) at htr, such that d2(htr) = 0, a continuous transition takes place, i.e. the sphere–planar liquid surface bridge transforms continuously into: (1) the bridge between two neighbouring spheres and (2) the bridge between the lower sphere and the planar liquid surface, Animation S1 (ESI†). In contrast, for small spheres (R o l/2), there is a discontinuous transition (jump-change) between these two configurations. It takes

Fig. 5 The distance d2(h) for equilibrium bridges (full lines) and metastable bridges (dashed lines), for spheres with radius R = 25 mm o l/2 (green) and R = 2 mm 4 l/2 (orange). The capillary length equals l = 1.45 mm.


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Fig. 6 (a) Plot of the height htr, at which the sphere–sphere bridge forms, as a function of the radius of spheres R forming a chain. For R o l/2, the liquid meniscus exhibits a jump and the sphere–sphere bridge forms discontinuously. At the point (R/l, htr/l) = (0.5, 1.63) – marked by a dot – the transition changes its character, and for R 4 l/2, it is continuous. The insets show the projected areas S of spheres and liquid onto a plane parallel to the z axis during the increase of Dz in two experiments, one for R = 25 mm (upper left), and one for R = 2 mm (lower right). The values of parameter R/l corresponding to the insets are marked with the arrows. (b and c) Two frames from Movie S2 (ESI†) for R = 25 mm showing the liquid meniscus jump and discontinuous formation of the sphere–sphere bridge. The elevation difference between these two frames is Dz = 1.3 mm.

place at height htr = hmax, for which the sphere–planar liquid interface bridge ceases to exist, Animation S2 (ESI†). At this height, the sphere–sphere bridge and sphere–planar liquid interface bridge connecting the lower sphere with the planar liquid surface form discontinuously. Thus, the particular value of the radius R = l/2, in the (R,h) space, corresponds to a point, where the line of discontinuous transitions meets the line of continuous transitions, Fig. 6. We note that during the process of pulling the chain out of the liquid, the above transition takes place periodically, with period Dh = 2R. To test quantitatively the existence of both discontinuous and continuous transitions, we measured the projected areas S of spheres and liquid onto a plane parallel to the z axis in two experiments, for R = 25 mm o l/2 and R = 2 mm 4 l/2 (see the insets in Fig. 6, and Movies S1 and S2 (ESI†)). For R = 25 mm, we observed an abrupt decline of the projected area with increasing elevation Dz, which repeats periodically when the sphere–sphere bridge is formed discontinuously. Contrary, for R = 2 mm, the projected area is a smooth function of Dz, since the sphere–sphere transition is continuous.

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vector directed upwards, can be calculated (see Appendix B) on the basis of the surface free energy in eqn (1) Fsp ðhÞ ¼

dEsp ½r0 ðzÞ dh

¼ 2pRg sin2 b rl gz1 pR2 sin2 b ð z1 þ rl g dzpr1 ðzÞ2 :

(9)

0

Fig. 7 The volume of the sphere–sphere bridge Vtr formed during the transitions as a function of the radius of spheres R. The inset shows the capillary force F 2 acting upward on the lower sphere. The dots separate the discontinuous and continuous transitions.

The first term describes the contribution to the total capillary force that acts at the liquid–gas–sphere contact line. The second term is the product of hydrostatic pressure depending on the height of the bridge z1 and the cross-section area of the sphere at z = z1. The third term corresponds to the buoyancy force acting upwards. In the case of the sphere–sphere bridge, one can distinguish two forces,41 F1 = F1ez and F2 = F2ez. Force F1 acts on the upper sphere and is directed downward: F1 ¼ 2pRg sin2 b1 rl gz1 pR2 sin2 b1 ð z1 þ rl g dzpr1 ðzÞ2 :

(10)

h2R -

Force F2 acts on the lower sphere and is directed upward: F2 ¼ 2pRg sin2 b2 þ rl gz2 pR2 sin2 b2 Fig. 8 Two sequences of photos, obtained in the experiments, illustrating the variations of the shape of the sphere–planar liquid surface bridge which take place in the process of pulling the chain of spheres out of the liquid. The upper panel corresponds to the chain vertical velocity v = 0.02 mm s1, and the lower panel corresponds to v = 2 mm s1. In both cases, R = 2 mm 4 l/2 = 0.725 mm. Grey circles with yellow edges were added to mark both the shape and the position of the spheres.

The shape of the liquid bridge between two adjacent spheres that is formed in a continuous way is described by eqn (6). Once this bridge is formed, its volume and shape remain unchanged, and there is no liquid flow along the surface of the spheres.40,41,43 On the other hand, in the case of the sphere– sphere bridge formed discontinuously, we assume that its shape is such that it minimizes the surface free energy. Thus, also in this case, the shape of the bridge is described by eqn (6). The volume of the liquid bridge between two adjacent spheres is determined at transition. It is a function of the radius of the spheres R, and does not depend on h, Fig. 7. In situations in which the velocity of the chain being pulled out from the liquid cannot be neglected, the volume of the bridge depends on the velocity, Fig. 8.

4 Capillary forces During the process of pulling the chain of spheres out from the liquid bath, i.e., for increasing values of parameter h, the shape of the bridge undergoes modifications, which induces modifications of the capillary force acting downward on the chain. The capillary force Fsp(h) = Fsp(h)ez, where ez denotes the unit

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ð h2R þ rl g dzpr1 ðzÞ2 ;

(11)

z2

where b2 is the angle between the vertical direction and the radius directed to the liquid–gas–sphere contact line, which is located at height z2 (see inset in Fig. 7). Both these forces have the same structure as the capillary force acting between the sphere and the planar liquid surface, eqn (9). For R Z l/2, one can check that Fsp(htr) = Fsp(htr 2R) + F1 + F2.

(12)

On the other hand, the sum of the forces for the sphere–sphere bridge equals the weight of the liquid bridge (see Appendix C) F1 + F2 = rlgVtr,

(13)

where Vtr denotes the volume of the sphere–sphere bridge. Finally, we obtain Fsp(htr) = Fsp(htr 2R) rlgVtr.

(14)

Thus, in the case R Z l/2 (and for very small velocities), one can obtain the volume of the sphere–sphere bridge from the capillary force measurements, Fig. 9. In the case of small radii (R o l/2), the weight of the sphere– sphere bridge does not compensate the difference between the sphere–planar liquid interface forces that pop up during the transition Fsp(htr) o Fsp(htr 2R) rlgVtr,

(15)

and one observes discontinuity in the force acting on the chain. For example, for R = 25 mm and l = 1.45 mm, the weight of the


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Fig. 9 The capillary force Fsp acting on the sphere, see Fig. 1, as a function of height h for stable (solid line) and metastable (dashed line) bridges. The black solid line joins the transition point Tr, of height h = htr, with point A, of height htr 2R, where the new period begins. Point B denotes the height, above which, in the system without the presence of the second lower sphere, the bridge cease to exist. The radius of the sphere is R = 2 mm 4 l/2, where the capillary length equals l = 1.45 mm. In the case of the continuous transition, the difference Fsp(htr 2R) Fsp(htr) equals the weight of the liquid bridge.

sphere–sphere bridge rlgVtr/2pRg = 3 104, and it is three orders of magnitude smaller than the difference between the sphere–planar liquid interface forces, Fig. 10. We note, that the absolute value of the capillary force acting upward (see inset in Fig. 7), is smaller than the maximum of the absolute value of sphere–liquid surface bridge |F2| o maxh|Fsp|. Thus, in the case when a chain is formed by assembling spheres from a suspension (as, for example, in the assembly route described in ref. 72), one needs, on top of the capillary force, an additional force of attraction acting between the spheres (e.g., the dipolar force).

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Fig. 11 The experimental setup: (a) a chain of glued steel spheres of radii R = 2 mm is being pulled out from 10cSt silicone oil by a stepper motor with constant velocity. The weight difference is measured by using a microscale during the experiment; (b) close-up on the spheres, silicone oil bath and the camera recording the experiment.

To check the validity of our model, we prepared a chain of spheres of radius R = 2 mm (R 4 l/2, where l = 1.45 mm). The adjacent spheres in the chain were glued together. The chain was very slowly pulled out from a 10cSt silicone oil bath with v = 0.02 mm s1 to ensure quasi-static conditions. A microscale was used as a very precise dynamometer, Fig. 11. For a given height h, the weight on the microscale m(h)g was equal to the weight of the silicone oil reduced by: the weight of the liquid in the sphere–sphere bridges, the absolute value of the sphere–planar liquid surface capillary force |Fsp(h)|, and the buoyancy force. The buoyancy force, during the process of pulling out the spheres, decreases by a factor proportional to the volume of the spheres drawn above the z = 0 level ðh DFb ðhÞ ¼ rl g dzpr1 ðzÞ2 : (16) 0

Thus, the mass difference Dm(Dh) = m(h) m(htr 2R), where Dh = h (htr 2R), equals DmðDhÞ ¼

1 Fsp ðhÞ Fsp ðhtr 2RÞ g ðh rl dzpr1 ðzÞ2 :

(17)

htr 2R

After one period, using eqn (14), we obtain 4 Dmð2RÞ ¼ rl Vtr rl pR3 : 3

(18)

The first term is the mass of one sphere–sphere liquid bridge and the second term is the mass of liquid displaced by one sphere. The comparison of experimental data and the theoretical curve is shown in Fig. 12. We note that the agreement is particularly satisfactory for Dh/2R o 0.5. Fig. 10 Capillary force in the sphere–planar liquid surface bridge Fsp as a function of height h for equilibrium (solid line) and metastable (dashed line) bridges. The black solid line joins the transition point Tr, of height h = htr, with point A, of height htr 2R, where the new period begins. The weight of the sphere–sphere bridge rlgVtr/2pRg = 3 104 does not compensate the difference Fsp(htr 2R) Fsp(htr), and one observes a jump in the force acting on the chain. The radius of the sphere is R = 25 mm o l/2, where the capillary length equals l = 1.45 mm.


5 Summary We described theoretically and validated experimentally the mechanism of capillary bridge formation on a beaded chain pulled out from a liquid. Two types of capillary bridges come into play. The first type is the bridge connecting the sphere with a planar liquid surface, and in the process of pulling out the

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Fig. 12 Comparison of theoretical predictions and experimental measurements of the mass difference Dm during the process of pulling out R = 2 mm steel spheres at height Dh = h (htr 2R) from a 10cSt silicone oil bath. For Dh = 2R, the absolute value of the mass difference equals the sum of mass of one sphere–sphere liquid bridge, and the mass of liquid displaced by one sphere, Dm(2R) = 33.27 mg. Measurements were averaged over four spheres, and the capillary tension was taken to be g = 19.7 mN m1, oil density rl = 950 kg m3 (provided by Dow Corning, USA), and the maximum radius of the system in numerical calculations to rmax = 10 mm.

chain of spheres from the liquid, this bridge appears first. Then, when the next sphere is pulled out from the liquid, this bridge transforms into the bridge between the adjacent spheres and the bridge connecting the lower sphere with the surface of the liquid. We showed that this process can be continuous or discontinuous depending on the ratio of the sphere radius and the capillary length R/l. For R/l Z 2, it is continuous, and for R/l o 2, it is discontinuous. In our experiments on spheres being pulled out from the silicon oil, we tested quantitatively the existence of discontinuous and continuous transitions by measuring the spheres and liquid projected areas during the increase of elevation. The shape of the meniscus of the bridge is given by the solution of eqn (6), in which the mean curvature on its lhs depends on the local height of the meniscus. There are two solutions of this equation, and the one corresponding to the larger value of b always has smaller surface free energy. The metastable bridge corresponding to the smaller angle b could be observed in the reverse experiment, when a sphere is pushed towards a flat liquid surface. Besides the shape of the bridges, the accompanying capillary forces acting in the system were also calculated. It turns out that the constant capillary force binding two adjacent spheres is not sufficiently strong to prevent the breaking of the chain in the process of pulling it out from the liquid. It is smaller than the maximal value of the sphere–planar liquid surface capillary force acting downwards. This maximal value is attained at a height smaller than the transition height. Thus, an additional attractive force acting between the adjacent spheres is needed to provide the stability of the chain. This can be the dipolar force;72 in the reported experiment, the spheres were simply glued together. We compared our theoretical predictions with experimental data corresponding to the case R 4 l/2. The plot of the theoretically predicted capillary force fits well with the experimental data, in particular for larger R/l values.

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In our theoretical model, we assumed zero contact angle. Allowing for a non-zero y value can lead to the stabilization of the metastable bridges corresponding to the y = 0 case. For y 4 0, the transition from the sphere–planar liquid surface bridge into the sphere–sphere bridge will occur for smaller values of h, and the particular value of R dividing discontinuous and continuous transitions will decrease below l/2. Such a change in stability would cause the phase diagram and the formation of liquid bridges scenarios to be more complicated. One can also investigate the situation when the spheres do not touch each other, with a fixed non-zero distance between them. In this case, for y = 0, the distance between the meniscus interface and the neighbouring lower sphere will increase, see Fig. 5, and the particular value of R dividing discontinuous and continuous transitions will increase above l/2. Thus, investigating the cases with non-zero contact angles, and a non-zero distance between spheres opens a new perspective for finding interesting effects. This is left for further analysis. Summarizing, we have explained theoretically the mechanisms of capillary bridge formation in the process of pulling the chain of spheres out of a liquid bath. We predicted the existence of continuous and discontinuous transitions between two types of bridges. The transition is continuous when the diameter of the spheres is larger than the capillary length. In the opposite case, the sphere–sphere bridge forms discontinuously, and there is a jump in the capillary force acting on the chain. Thus, the capillary length sets the lower limit for the diameter of the spheres, for which the beaded chain formation out from a liquid dispersion is a smooth process.

Appendix A

Procedure of determining the shape of liquid bridges

In order to determine the shapes of liquid bridges, we solve eqn (6) numerically using the shooting method.51,78 An essential ingredient of this method is checking whether the boundary condition r0(z = 0) = N is fulfilled for the probe function at hand. In our numerical calculations, we check this condition using a large but finite parameter denoted as rN. We also introduce another auxiliary quantity zN. It is defined as the minimal value of z in the range [0, h R(1 + cos b)], for which r0(zN) r rN. Such a defined quantity zN depends on the angle b, i.e., we have zN(b). Note that if r0(0) o rN, we adjust the angle b to increase the value of r0. The equilibrium profile corresponds to zN = 0. The function zN(b) is non-differentiable (has a kink), Fig. 13. Depending on the value of h, the equation zN(b) = 0 can have zero, one, or two solutions, at which points the function zN(b) is non-differentiable. Every such point corresponds to a solution of eqn (6). For h 4 hmax, we have no solution, for h = hmax – one solution, for 2R o h o hmax – two solutions, and for 0 o h o 2R – one solution. While two solutions of eqn (6) are possible, the equilibrium shape of the bridge corresponds to the one with smaller surface energy. We checked that the solution with larger b always has smaller surface energy. To check how the choice of rN influences the results, we performed calculations of b(h) for different rN values, Fig. 14.


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The sphere–liquid planar surface capillary force Fsp is then given by

Fsp 1 dEsp ½r0 ðzÞ ¼ 2pg 2pg dh ! 1 dEsp ½rðzÞ dr0 ðzÞ @Esp ½r0 ðzÞ þ ¼ 2pg drðzÞ r¼r0 ðzÞ dh @h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 ¼ R þ dz r0 ðzÞ 1 þ r0 ðzÞ R ð

i dr ðzÞ z 1 r ðzÞ 1 l2 dh ð qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ R þ dz r0 ðzÞ 1 þ r00 ðzÞ2 R

dðz z1 Þ jr00 ðz1 Þ r10 ðz1 Þj

þ

Fig. 13 Function zN(b) for different heights: (a) h = 1 mm, (b) h = 2 mm, (c) h = 2.5 mm, (d) h = 3 mm, (e) h = 3.06 mm, and (f) h = 3.5 mm. Radius of a sphere is R = 1 mm, and capillary length l = 1.45 mm. For h r 2R, function zN(b) = 0 between zero and a certain value of b, and then increases up to b = 1801. In these calculations, we took rN = 106R.

dðz z1 Þ jr00 ðz1 Þ r10 ðz1 Þj

idr ðzÞ z 1 r1 ðzÞYðr0 ðzÞ r1 ðzÞÞYðzÞ 2 l dz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð r0 ðz1 Þ 1 þ r00 ðz1 Þ2 R d r1 ðzÞ2 1 z1 ¼ R þ cot b þ 2 dzz 0 0 2l 0 dz jr0 ðz1 Þ r1 ðz1 Þj þ

We note that starting from rN 4 4l, the results do not change significantly, so in the following calculations we will take rN to be in the vicinity of 10l.

¼ R sin2 b þ

z1 r1 ðz1 Þ2 1 2 2l l2 2

ð z1

dzr1 ðzÞ2 ;

0

(20) B Calculation of the capillary force The functional in eqn (1) can be rewritten, using theHeaviside step function Y(z) and Dirac delta function d(z), in the following form Esp ½rðzÞ 2pg ð qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ dz rðzÞ 1 þ r0 ðzÞ2

where r0(z) is the equilibrium shape of the sphere–planar liquid surface bridge interface. In the above calculation we used the fact that the function r1(z) depends on the parameter h only through the difference z h, i.e., r1(z) = rs(z h), where rs(z) is a function describing submerged spheres (h = 0). To calculate qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 r0 ðz1 Þ r10 ðz1 Þ, we assumed that r0 ðz1 Þ 1 þ r00 ðz1 Þ2 R r00 (z1) = cot(b + y), and then took the limit y - 0.

i z 2 2 R Y rðzÞ r rðzÞ r ðzÞ ð ðzÞ Þ þ R YðzÞYðh zÞ 1 1 2l2 ð

dzdðzÞrðzÞ2 2:

þ

C

Volume of the bridge

To determine the volume of the sphere–sphere bridge, one can integrate eqn (6) once to obtain 0

(19)

0

00

d r0 ðzÞ r0 ðzÞ r0 ðzÞr0 ðzÞr0 ðzÞ ¼ 1=2 3=2 dz ð1 þ r00 ðzÞ2 Þ1=2 ð1 þ r00 ðzÞ2 Þ ð1 þ r00 ðzÞ2 Þ

(21)

z d ¼ 2 ðr0 ðzÞÞ2 ; 2l dz and hence z1 z ð r0 ðzÞ zr0 ðzÞ2 1 1 z1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ þ dzr0 ðzÞ2 ; 2l2 z2 2l2 z2 1 þ r00 ðzÞ2 z

(22)

2

which gives ð z1 dzp r0 ðzÞ2 r1 ðzÞ2 Vtr ¼ z2

¼ 2pl2 R sin2 b1 sin2 b2

Fig. 14 Dependence of b on h for equilibrium (upper branch) and metastable (lower branch) solutions for different choices of rN: (a) rN = l, (b) rN = 2l, (c) rN = 3l, (d) rN = 4l, 5l, 6l, 7l, 8l, 9l, and 10l. The inset shows a close-up of transition regions. In all the calculations, R = l/2.


þ pR2 z1 sin2 b1 z2 sin2 b2

ð z1

dzpr1 ðzÞ2

(23)

z2

¼

1 ðF1 þ F2 Þ: rl g

Soft Matter, 2017, 13, 4698--4708 | 4705

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The total capillary force acting on two adjacent spheres is equal to the weight of the liquid bridge between them.41

Acknowledgements F. D. was supported by the Foundation for Polish Science, Poland, within the project Homing Plus/2012-6/3, co-financed by the European Regional Development Fund. Z. R. acknowledges financial support from the National Science Centre, Poland, through the OPUS programme (2015/19/B/ST3/03055) and the Foundation for Polish Science, through the Homing Plus programme (2013-7/13). The authors are grateful to Dr Jan Guzowski for numerous comments and suggestions.

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Electronic Supplementary Material (ESI) for Soft Matter. This journal is © The Royal Society of Chemistry 2017

Continuous and discontinuous transitions between two types of capillary bridges on a beaded chain pulled out from a liquid. Filip Dutka, Zbigniew Rozynek, and Marek Napiórkowski

Supplementary Movie Captions

Supplementary Movie 1. The process of pulling the chain of spheres with radius R = 2 mm out of the 10 cSt silicone oil with the vertical velocity v = 0.01 mm/s. The sphere-planar liquid surface bridge transforms into the sphere-sphere bridge continuously. The chain was assembled by gluing the spheres together. Section of the video frame (700px x 620px) was used to measure the projected area S of the spheres and liquid onto a plane parallel to the z axis which was then used in making the inset in Fig. 6.

Supplementary Movie 2. The process of pulling the chain of spheres with radius R = 25 µm out of the 10 cSt silicone oil with the vertical velocity v = 0.01 mm/s. The sphere-planar liquid surface bridge transforms into the sphere-sphere bridge discontinuously. The chain was assembled by employing the alternating electric field. Section of the video frame (300px x 250px) was used to measure the projected area S of the spheres and liquid onto a plane parallel to the z axis which was then used in making the inset in Fig. 6.

Supplementary Animation 1. The process of pulling the chain of spheres with radius R = 2 mm > λ /2, where the capillary length λ = 1.45 mm. The sphere-planar liquid surface bridge transforms into the sphere-sphere bridge continuously. This animation is based on our theoretical analysis and corresponds to Supplementary Movie 1.

Supplementary Animation 2. The process of pulling the chain of spheres with radius R = 25 µm < λ /2, where the capillary length λ = 1.45 mm. The sphere-planar liquid surface bridge transforms into the sphere-sphere bridge discontinuously. This animation is based on our theoretical analysis and corresponds to Supplementary Movie 2.