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Rupture energy and wetting behavior of pendular liquid bridges in relation to the spherical agglomeration process
Journal of Colloid and Interface Science 261 (2003) 161–169 www.elsevier.com/locate/jcis
Rupture energy and wetting behavior of pendular liquid bridges in relation to the spherical agglomeration process Damiano Rossetti,a Xavier Pepin,b and Stefaan J.R. Simons a,∗ a Colloid & Surface Engineering Group, Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, UK b Sanofi Synthelabo Research, Pharmaceutical Science Department, 5, rue Georges Bizet, F-91160 Longjumeau, France
Received 7 June 2002; accepted 2 January 2003
Abstract A novel micro force balance (MFB) is used to investigate the rupture energy of a silicon oil liquid bridge formed in water between two glass particles of either the same or dissimilar surface energy. Rupture energies are integrated from force curves and compared with the models proposed by Simons et al. (Chem. Eng. Sci. 49 (1994) 2331) and Pitois et al. (Eur. Phys. J. B 23 (2001) 79). The latter showed slightly better agreement to the experimental data. Glass ballotini (∼100 µm diameter) are either silanized, in order to increase their wettability toward the oil binder, or kept untreated. Results showed how the interaction between the binder and the particle influences the geometry, the capillary pressure, the force, and the rupture energy of the liquid bridge. Higher values of force and liquid bridge energy were measured between particles characterized by higher interaction (silanized–silanized configuration). A thermodynamic approach to the evaluation of the energy stored in a liquid bridge is also proposed. The mechanical work done to stretch apart the liquid bridge is evaluated as the difference of internal and hysteresis energy between the initial and the rupture configuration of the bridge. This approach showed good agreement with the experimental data only for liquid bridges formed between silanized and untreated glass particles. 2003 Elsevier Science (USA). All rights reserved. Keywords: Spherical agglomeration; Pendular bridges; Wetting hysteresis; Capillary pressure; Force; Energy
1. Introduction Spherical agglomeration is an industrial process traditionally used to separate or recover fine solids dispersed in a liquid suspension through the addition of a second immiscible liquid (binder) which presents an affinity to the solids and is capable of forming small liquid bridges that hold the particles together. Under appropriate physicochemical conditions the desired particles can be selectively agglomerated and removed from the slurry [1]. Recent studies have been undertaken in order to apply spherical agglomeration to other industrial sectors, for example, in the deinking of recycled paper [2], in the removal of heavy metals from wastewater [3], and in the agglomeration of either crystals or powders to produce pharmaceutical drugs [4,5]. The last marks a significant departure from the use of spherical agglomeration as merely a separation * Corresponding author.
E-mail address: [email protected] (S.J.R. Simons).
technique to one that can be used to produce highly valued chemicals. In many agglomeration processes, particles exhibiting different surface properties might be processed together. Such discrepancies in surface energy can create problems during agglomeration, as particles can be selectively wet at the expense of others [6]. On the other hand, the study of the strength and rupture energy of individual liquid bridges, in relation to the surface energy differences exhibited by the particles, can help the understanding of the very first stage of agglomeration, since aggregate formation appears as a balance between rupture energy of individual liquid bridges and kinetic energy of particles. A literature review has revealed (see Table 1) that most studies have been focused on the strength of a liquid bridge between relatively large particles (a few mm) suspended in a gaseous medium (usually air) exhibiting the same surface properties and high wettability towards the binder. This approach, which is very valuable for the number of data made available and the good agreement with values predicted by theory, neglects the common size scale of
0021-9797/03/$ – see front matter 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0021-9797(03)00043-2
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Table 1 Referenced works on liquid bridge force measurements between two particles in a gaseous medium (air) Author Mazzone Simons Pitois Willett
Particles–binder Stainless steel–oil (DBT) Silanized glass–silicon oil Ruby–PDMS oil Sapphire–silicon oil
agglomerated particles (< 250 µm) and the important role that contact angle hysteresis plays during the stretch and rupture of a liquid bridge. The role of contact angle hysteresis in relation to the geometry assumed by a liquid bridge, was observed by Pepin et al. [7,8], who studied the behavior of a glycerol liquid bridge formed in air between glass particles of either similar or dissimilar surface energy. During the bridge separation, the liquid was either held strongly or receded from the particle according to the energy exhibited by the particle and, as a consequence, a pinning or a slippage of the contact angle was observed. Similar behavior was also noted in [9] in a different set of experiments involving measurement of contact angle between a modified silica surface and either water or glycerol liquids. As pointed out by Mazzone [10], gravity is also to be taken into account with large binder volumes, although its effect can be masked off by performing experiments in a second liquid of nearly the same density as the binder [11]. Relevant works regarding liquid bridges suspended in a second immiscible can be found in [11,12]. Mason and Clark [11] investigated the strength of an oil liquid bridge formed between relatively large polythene particles (15-mm radii) suspended in water. They noticed that at small separations the force appeared to pass through a maximum, which favorably agreed with values predicted by theory [13], particularly for the larger volumes used (between 10−2 and 10−1 ml). Wolfram and Pinter [12] alternated water and n-hexane as liquid binder and suspending medium and showed how density differences between the two liquids generate instability in the strength and post-rupture binder distribution due to the increase of gravity effects as the liquid bridge volume increases (from 10−2 to 5 × 10−1 ml). They also recognized the importance of the role played by surfaces of different energy (PTFE and glass) in relation to the geometry of, and the energy stored by, a liquid bridge during separation. The rupture energy, W , of a pendular liquid bridge between spherical particles in air can be found in [14] and [15]. Simons et al. [14] modeled the bridge geometry with a toroidal curve and, in the case of perfect wetting conditions, calculated the integral of the quasi-static capillary force with respect to the separation distances. From the approximated solution of the nontrivial integral a simple expression was derived in terms of dimensionless parameters: = W ≈ 3.6V 0.5 . W γ R2
(1)
Contact angle
Particle radius
Reference
∼10◦ < 20◦ ∼10◦ 0◦
1.98 mm < 100 µm 4 mm 2.3 mm
[10] [17] [26] [33]
In (1), γ is the surface tension of the liquid binder and R are = V /R 3 and W is the radius of the spheres, while V the dimensionless volume and rupture energy of the bridge, respectively. Pitois et al. [15] used a circular approximation for the bridge profile and obtained a simplified expression for the capillary force which, when integrated throughout the separation distance, led to the expression for the dimensionless rupture energy without restrictions to the wettability exhibited by the particles 2V 1/3 W = 2π cos θ (1 + θ/2)(1 − A)V (2) + , π )1/3 /π(1 + θ/2)2 and θ is the contact where A = (1 + 2V and V are defined as angle expressed in radians, while W above. In the present paper the strength of a silicon oil liquid bridge, formed between particles of either the same or dissimilar surface energy suspended in water, is measured throughout separation. Force data are compared with values predicted by theory (see Section 4) and fitted with a third order polynomial curve and, by integration with respect to separation distance, the energy of the liquid bridge between two configurations is calculated. The calculated rupture energies are then compared with the approximations proposed by Simons et al. [14] and Pitois et al. [15], the latter showing a slightly better agreement due to the presence of the contact angle parameter. A comparison is also made of the capillary pressure of liquid bridges formed between silanized–silanized and silanized–untreated particles.
2. Materials and methods Since spherical agglomerates are usually formed from sub-250-µm particles, glass ballotini of diameter in the range 90–130 µm have been used as the solid particles, while silicon oil 100cSt (VWR International) and water (“Analar” VWR International) have been used as bridging and suspending liquid, respectively. The particles are soda lime glass spheres (Sigmund Lindner Type S), washed in chromosulfuric acid (Acros Chimica) and dried. A sample was soaked for 30 min in a silanizing agent, Repelcote VS (2% solution of dimethyldichlorosilane in octamethylcyclotetrasiloxane), obtained from VWR International. Silanized particles were washed first with toluene (Spectranal, 99.9%
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Fig. 1. Schematic of the MFB set-up used to measure the adhesive force of a liquid binder formed between pairs of particles submerged in a second immiscible liquid.
pure from Aldrich) and then with methanol (VWR International) before final drying. The interfacial tension of the water–oil interface was measured using a Kruss 12 tensiometer with the Wilhelmy plate method and resulted in the value of 37.8 mN/m at 21 ◦ C. Fairbrother and Simons [16,17] have developed a novel device, known as a micro force balance (MFB), in order to study liquid bridges between particles in relation to granulation processes. The MFB allows the investigation of both the geometry and the adhesion force exerted by a liquid bridge between two particles during their separation. The technique has already been used to measure particle–binder interaction in a gaseous (air) medium [8]. In this work it has been modified to allow force measurements between two submerged particles [18]. In brief, the measurement procedure is as follows. A pair of appropriately shaped micropipettes hold the glass ballotini under a water objective in an optically clear dish which is filled with the water. Once a drop of silicon oil is administered on one particle (using a third pipette that is not illustrated in Fig. 1), the two particles are first brought together to form the bridge and then separated until rupture of the bridge occurs. One of the two micropipettes is highly flexible in the direction of separation (see Fig. 1) and, due to the separation movement imposed by the second micropipette, the bend of the flexible pipette deflects proportionally to the adhesive force exerted by the liquid bridge. The deflection of the micropipette is calculated via image analysis as the difference of movement between the undisturbed and the actual position of the particle, as illustrated in Fig. 2. The separation process is recorded with a camera capturing up to 25 frames/s and stills from the video are taken in order to carry out further image analysis. Figure 2 shows the steps to formation and separation of a liquid bridge. In Fig. 2a, the straight pipette onto which a silicon oil droplet has been previously administered is moved toward the flexible pipette (whose spring constant is k). At a certain close distance between the two particles, when the liquid interface first touches the other particle, the flexible pipette is caused to “jump” towards the other pipette to form the liquid bridge. The straight pipette is pushed further to reduce the gap between the two particles (Fig. 2b) and then is driven away until the bridge is ruptured (Fig. 2c). The deflection with respect to the undisturbed
Fig. 2. Schematic of the flexible micropipette deformation during the phases of liquid bridge formation and particle separation.
configuration, e, can be either positive or negative according to the configuration assumed by the bridge (i.e., whether the bridge pushes or pulls the particle together). The flexible micropipette is precalibrated in order to determine its spring constant (usually from 0.1 to around 0.7 µN/µm) and the total force exerted is thus calculated as per Eq. (3) F = ke.
(3)
In a new version of the technique a flexostrip device is used to measure the interaction force with the deflection measured directly by a linear voltage differential transducer [19,20]. Figures 3 and 4 show the rupture sequences of a liquid bridge formed between two silanized particles and a silanized and an untreated particle, respectively. The different geometries assumed by the liquid bridge can be explained in term of the wettability between the particle and the binder. Silanized particles (well wetted particles) hold the binder (silicon oil) more strongly than untreated (poorly wetted particles) and therefore, during separation, the solid– liquid interface remains almost constant on the silanized surface. In contrast, this interface reduces dramatically on untreated particles. The resulting geometries, nodoid (Fig. 3) and unduloid (Fig. 4), can be compared in terms of strength, energy stored, and pressure profile, which will be discussed in more detail in Section 4.
3. Theory 3.1. Wetting hysteresis On the majority of real surfaces, the spreading of a liquid is achieved for an apparent advancing contact angle θad , superior to the receding angle θr obtained when the liquid is retrieved from the wet surface. This hysteresis has received extensive theoretical explanations related to the physical and chemical nature of the surface [21,22]. Practically, the wetting hysteresis of a solid surface is seen at the threephase contact line, which either moves or is pinned at a given position. For a solid surface, pinning of the contact line occurs each time the apparent contact angle θ is inferior
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Fig. 3. Separation sequence of a silicon oil liquid bridge formed in water between two silanized glass particles of radius 63 µm (left) and 58 µm (right), separated at 1 µm/s. The volume of the bridge is ∼91,200 µm3 .
Fig. 4. Separation sequence of a silicon oil liquid bridge formed in water between an untreated glass particle (left) and a silanized one (right) of radius 45 µm and 58 µm, respectively. Separation speed is 1 µm/s, the volume of the bridge is ∼133,000 µm3 .
to θad and superior to θr . Measuring the receding contact angle in circumstances where the binder is mechanically forced to detach from a substrate can qualitatively assess the interaction between the two. The lower the contact angle the higher the interaction energy and vice versa. Advancing and receding contact angles can be measured by a variety of techniques; in the present work they were evaluated via image analysis of the profile assumed by the liquid bridge for a given configuration.
Fig. 5. Geometric parameters describing a liquid bridge.
3.2. Bridge geometry The configuration of a liquid/liquid interface at rest is described by the Young–Laplace equation, which relates the difference in hydrostatic pressure P = Pbr − Pext across the interface to the local mean curvature and the interfacial tension of the interface [23] (Pbr , Pext are the pressures inside the bridge and in the suspending medium, respectively). The Young–Laplace equation can be rewritten in Cartesian coordinates in terms of the analytical expression of the local mean curvature, resulting in [24] P 1 y = − , γow y(1 + y 2 )1/2 (1 + y 2 )3/2
(4)
where y and y are the first and second derivatives of the liquid bridge profile described by the function y(x), while γow refers to the oil/water interface. Equation (4) can be rearranged into a Bernoulli-type differential equation and, following classical integration,
Equation (6) allows the P across the interface to be calculated once the interfacial tension γow and the boundary values of the function describing the bridges are known. The boundary values, yA , yB , (θA + βA ), (θB + βB ), can be determined by analysis of the images captured during the separation sequence. Equation (4) can be solved numerically by the value of P and by imposing the condition that at the extremes P and Q, the function assumes the value yA and yB . In Section 4 the theoretical profile of the bridge is compared with those obtained experimentally for different separation distances. Calculating the revolution integral of the profile y(x) and subtracting the volume of the particleto-binder interface (cap of sphere), the volume of the bridge is eventually determined. 3.3. Bridge strength
is obtained, where C is an integral constant [24]. It is possible to substitute the coordinates of the points P and Q (Fig. 5) into (5) and since yP ,Q = cotg(θA,B + βA,B ), the constant C can be eliminated:
The force exerted by a liquid bridge at rest depends on both the surface tension of the interface and the capillary effects due to the curvature of the bridge. Two different approaches are commonly used to calculate this static force, which lead to slightly different values. In the first case, the force is determined by considerations at the neck of the bridge [25],
yA sin(θA + βA ) − yB sin(θB + βB ) P = . 2γow yA2 − yB2
F n = 2πr1 γow − πr12 P ,
P 2 y = y +C (1 + y 2 )1/2 2γow
(5)
(6)
(7)
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while the second method uses values at the boundary [23] between the particle and the liquid bridge, as indicated in F b = 2πRA sin βA γow sin(βA + θA ) 2 − πRA sin2 βA P ,
(8)
with the geometric quantities defined as in Fig. 5. In the first step of separation, where a neck is not yet formed, the use of (7) can be misleading and thus the force is better calculated by using (8). Once the neck is formed the two methods can be compared. During separation the dynamic effects due to the viscosity of the liquid bridge must also be accounted for. Between two equisized spheres held by an infinitely extended liquid bridge, the viscous force is given by [26] 1 dD 3 , Fvis = πηR 2 (9) 2 D dt where η is the dynamic viscosity of the liquid, R the radius of the particles, and D the bridge separation distance as indicated in Fig. 5. In the following section, (9) is used to compare the magnitude of the viscous force with the total static force of either (7) or (8). 3.4. Rupture energy of a liquid bridge The rupture energy model proposed by Simons et al. [14] is derived by integration of the total liquid bridge force, which is written as X tan β , F = πγ R(1 + X tan β − X sec β) (10) X sec β − 1 where X = (1 + D/2R) and β is the half-filling angle, equal, for geometric similarity, to that shown in Fig. 5. In the integration of (10) through separation distance D, the half-filling angle β was considered a constant. The approximation, due to the difficulty in relating the angle β to the separation distance D, seems reasonable for particles that exhibit a strong interaction toward the binder where the solid–liquid interface stays almost constant (see Fig. 3). The expression of the dimensionless rupture energy calculated between any two configurations Xmin and Xmax is 2 X 2 tan β cos β − tan β + X sin2 β W = 2π 2 Xmin + tan2 β cos3 β ln(X sec β − 1) (11) Xmax
which, when plotted against the liquid bridge volume, yielded (1). Pitois et al. [15] used a different simplified expression for the total liquid bridge force: −1/2
2V . F = 2πγ R cos θ 1 − 1 + (12) πD 2 R In (12), θ is the contact angle between the particle and the binder. Using the approximation that θ stays constant
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throughout separation, (12) is integrated with respect to the rupture distance D to obtain the rupture energy, which is then rearranged in nondimensional form,
D 1
1/2 2 V , W = 2π D cos θ 1 − 1 + (13) 2 πD D2
is the nondimensionless rupture distance D = where D 1/3 D/R. If D1 = 0 and D2 = (1 + θ/2)V , which is the expression of the dimensionless rupture distance as proposed by Lian [27], Eq. (2) is obtained. In the present work the liquid bridge energy is evaluated between a small, but nonzero, initial separation distance and rupture. Experimental difficulties prevent observation of liquid bridges at zero separation distance and, therefore, some physiochemical parameters of the bridge remain undetermined. Hence, Eqs. (11) and (13) are used instead of (1) and (2). On a thermodynamic basis, a liquid bridge can be considered a closed system that changes its energy by means of the work done by the micropipette, while its mass remains constant. The latter seems a reasonable assumption for two immiscible liquids, such as used in this work. A more important assumption is to neglect losses due to viscous effects. The last statement can be justified by comparing viscous forces with capillary and surface tension forces, as will be addressed in Section 4. The comparison shows that this assumption is legitimate. This is not a general case though, as Pitois et al. detailed in [15,26]. In view of the previous assumptions, the energy stored by a liquid bridge can be evaluated by using the first law of thermodynamics as the variation of internal and hysteresis energy between two different configurations, U = −(P V ) + (γow Abr ) + Eh = −w,
(14)
where U is the internal energy, P and V the capillary pressure and the volume of the bridge, respectively, Abr the area of the bridge delimited by the liquid–liquid interface, Eh the hysteresis energy, and w the external work, which is negative, by convention, when work is done on the system (liquid bridge). Abr can be evaluated once the profile of the liquid bridge is known, as indicated in x=D
y(x) 1 + y 2 (x) dx.
Abr = 2π
(15)
x=0
The hysteresis energy is defined as the difference of work of adhesion [12,22,28] between two liquid bridge configurations and can be written Eh = −γow Aop (cos θ2 − cos θ1 ),
(16)
where Aop is the area of the cap of the sphere at the contact between the oil and the particle, while θ1 and θ2 refer to the contact angles assumed by the bridge on one particle in two different configurations. Hysteresis energy must be calculated for both particles.
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4. Results and discussion 4.1. Liquid bridge strength and geometry Recorded sequences of liquid bridge separation were analyzed via image analysis (Microsoft PowerPoint) in order to determine by (6) the parameter P /γow , which leads to the solution of the Young–Laplace equation (4). Figures 6 and 7 refer to the rupture sequence of the bridge formed between the silanized particles, shown in Fig. 3. It is useful to note that the x-axis of the geometric profiles is the length of the liquid bridge, indicated as h in Fig. 5, and not the separation distance D. Theoretical values of the liquid bridge strength are calculated using both Eqs. (7) and (8) for all the experimental configurations verified with the Young–Laplace equation. In Fig. 7 the arrows reflect the sequence of initial formation, compression, and separation of the liquid bridge. It can be noted that the initial force (D ∼ 30 µm) is lower than the value obtained at the same D after the liquid bridge has been compressed between the two particles. This fact can be ex-
plained by the larger particle to binder interface formed upon compression, which then leads to higher forces in the following phase of separation, as predicted by (8). This behavior was also noticed between silanized and untreated particles (Fig. 8), where the receding phase on the untreated particles only occurred in the late phase of separation, when an unduloid profile is formed (Figs. 4 and 9). It is important to make clear that the configurations shown in Figs. 6b and 9b were the last recorded (with a camera acquiring 25 frames per second) before rupture occurred but are not likely to be the rupture configurations. The rupture of a liquid bridge is a quick process, which involves some complex phenomena, and only a relatively fast camera (i.e., 500 frames/s) can investigate the evolution of the bridge breakage [29]. The effect of the receding interface on poorly wetted particles (untreated glass) plays an important role in the liquid bridge strength and, as a consequence, adhesion forces measured during liquid bridge separation are lower than in the case where both particles present a high wettability
Fig. 8. Force versus separation distance for liquid bridge formed between silanized and untreated particles. Sequence is shown in Fig. 4. Bridge volume ∼133,000 µm3 .
Fig. 6. Geometry profiles of the liquid bridge shown in Fig. 3: (a) D = 7 µm, P = 1455 Pa; (b) D = 68 µm, P = 1540 Pa.
Fig. 7. Force versus separation distance for liquid bridge shown in Fig. 3. Bridge volume is ∼91,200 µm3 .
Fig. 9. Initial and prerupture configurations of the liquid bridge formed between a silanized and an untreated particle (left side of graphs), shown in Fig. 4: (a) D = 12 µm, P = 2268 Pa; (b) D = 75 µm, P = 1758 Pa.
D. Rossetti et al. / Journal of Colloid and Interface Science 261 (2003) 161–169
Fig. 10. Liquid bridge strength versus separation distance in dimensionless form.
toward the binder (silanized particles). This situation clearly appears by comparison of the dimensional force plots shown in Figs. 7 and 8, where the maximum adhesive force between silanized particles (Fig. 7) is almost double that determined between silanized and untreated particles (Fig. 9). Figure 10 shows the comparison of the dimensionless force–separation curves obtained for the two sets of experiments. In the experiments carried out between silanized particles (filled triangles), higher maximum liquid bridge adhesiveness was always measured, while from the plots obtained between particles of different surface energies (open squares) the force curves seem to level off throughout separation. Viscous forces were also analyzed and shown to be negligible for the experimental conditions used. Introducing the capillary number Ca = vη/γow as the ratio between viscous and surface tension effects, with v being the particles separation speed, and using (9) with the input data R = 51 µm (average value for the experiments), η = 0.096 Pa s, D = 10 µm, dD/dt = 1 µm/s results in a value of 1.1 × 10−4 µN for the viscous force results, while Ca is 2.5 × 10−6 . Increasing the separation distance and keeping all the other quantities the same, reduces the viscous force even further. This result agrees with [30,31], in which the stress induced on particle agglomerates deformed at increasing compression speeds is studied and viscous force contribution to the deformation of the wet mass was found to become predominant only when the capillary number Ca was above a threshold of 10−4 . According to (7), the complementary effects of the increase in neck radius rise (r1 in Fig. 5) and reduction of the capillary pressure inside the liquid bridge increase the liquid bridge force. Figure 11 shows the trend of r1 versus the volume of the liquid bridge for all the configurations where the maximum liquid bridge force was observed. The graph illustrates that the neck radius is larger in the set of experiments where the highest forces have been measured (silanized– silanized particles). However, the neck radius is related to the maximum liquid bridge force only when the particle diameter is small enough. Mason and Clark [11] and Bayramli and Van de Ven [32] showed that when the particle diameter is of the order of cm the increase of the liquid bridge volume does not influence the maximum liquid bridge force, while an increase of the neck radius is still observed [11].
167
Fig. 11. Half bridge neck height versus volume of the bridge measured for bridge configurations of maximum strength.
Fig. 12. Capillary pressure versus separation distance for two silanized (a) and untreated–silanized (b) particles.
4.2. Capillary pressure of a liquid bridge Capillary pressure, which is determined by (6), is plotted in Fig. 12 for the two sets of experiments. When both particles are silanized (Fig. 12a), the pressure profile seems to follow a parabolic curve, which decreases its concavity at increasing bridge volume. With particles of different surface energy (Fig. 12b), the parabolic behavior exhibited by the smaller volumes becomes more asymmetric as the volume increases. A mutual comparison of the two profiles is presented in Fig. 13, which shows higher values for the untreated–silanized particle configuration. The lower strength of the untreated–silanized particle configuration can therefore be explained by both the presence of a thinner bridge neck and higher capillary pressures that work against the bridge strength. 4.3. Energy stored by a liquid bridge The higher strength measured between silanized particles is reflected by higher energy values W stored during separation (Fig. 14). Experimental force diagrams were fitted
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Fig. 13. Normalized pressure versus separation distance.
Fig. 14. Experimental values of liquid bridge energy.
with a third-order polynomial curve and integrated through separation distance in order to calculate the energy W . The higher energy stored in liquid bridges is beneficial to counterbalance the kinetic effects which leads to particle separation and therefore, during mixing, particles with higher surface energy are more likely to exist in an agglomerated form. A further effect that hinders the agglomeration of poorly wetted particles is the amount of binder volume left on the particle after separation. Due to the receding of the particle–binder interface, in fact, a very small amount of binder remains available on low-surface-energy particles to form other liquid bridges, which ultimately disadvantages agglomeration. Figure 15 shows a comparison between the normalized experimental energy data and the values calculated from the models proposed by Simons et al. and Pitois et al. per Eqs. (11) and (13), respectively. Data are plotted against the binder-to-particles volume fraction, Vbr /Vs , and are presented in Table 2. The two models give reasonably good agreement with the experimental data considering the magnitude of the energies and volumes involved and that calculations are made from data acquired by image analysis, which will incur some degree of error. For the untreated– silanized configuration, where the Simons et al. assumption of perfect wetting is not valid, the Pitois model shows slightly better agreement due to the presence of the contact angle parameter. The values used for the contact angle in the two sets of experiments, which are the lower contact angles measured on the particles just before rupture, are shown in Table 2. All the quantities required for the calculation of (11) and (13) are determined via image analysis. Finally, the thermodynamic approach to calculating the liquid bridge energy between untreated and silanized parti-
Fig. 15. Comparison of experimental liquid bridge energy with the models proposed by Simons and Pitois: (a) two silanized particles, (b) untreated–silanized particles. Table 2 Data presented in Fig. 15 V /R 3
exp W
Simons W
%err
Pitois W
%err
Silanized–silanized particles, receding contact angle 35◦ ± 3.7 0.32 0.34 1.25 0.15 0.40 0.25 0.04
1.70 1.36 2.52 0.93 1.73 1.30 0.40
1.29 0.52 1.39 0.90 1.51 1.25 0.44
−24.0% −61.7% −45.0% −3.5% −13.0% −3.7% 8.2%
1.39 0.61 1.40 0.87 1.50 1.21 0.38
−18.5% −54.9% −44.6% −6.0% −13.2% −6.9% −5.6%
Untreated–silanized particles, receding contact angle 52◦ ± 6.2 1.02 1.51 0.84 −44.6% 0.95 −37.3% 0.40 0.97 1.19 23.1% 1.16 20.3% 1.04 1.53 1.22 −20.6% 1.18 −23.0% 0.38 0.97 1.55 60.5% 0.99 2.9% 0.05 0.18 0.16 −10.0% 0.17 −6.6% 0.40 1.00 0.81 −19.2% 0.80 −19.7%
cles using (14) and (16) (the latter applied to both particles) is shown in Fig. 16. In each experiment the thermodynamic energy is evaluated between the initial and final configurations (indicated by the subscripts 1 and 2 in (16)) used to calculate the experimental liquid bridge energy by means of force plot integration. The data shown in Fig. 16 seem to confirm that during a quasi-static separation the energy stored by a liquid bridge can also be calculated using a thermodynamic approach in terms of variation of capillary pressure, increase of interfacial bridge area, and modification of the three-phase contact line due to variations of solid–liquid interfacial area and contact angle formed between the liquid meniscus and the particle. However, good agreement with the experimental data was not found for silanized particles except in a few cases, which might be explained by capil-
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References
Fig. 16. Experimental liquid bridge energy compared with values obtained by thermodynamic analysis for the separation of silanized–untreated particles. The fitting line shows a nonlinear trend (dotted line) at low liquid bridge volumes. A linear trend (continuous line) approximates the liquid bridge energy when the bridge volume is increased.
lary pressure variations that cannot be accounted for near the bridge rupture, due to the camera limitations. The thermodynamic approach to the energy stored by a liquid bridge seems to support the important effect that hysteresis energy can play in surface chemistry phenomena. The experimental data for silanized and untreated particles (already presented in Fig. 14 on a logarithmic scale) show that the work done to separate the particles increases with the volume of the liquid bridge, although not linearly at low bridge volumes (dotted line). The trend becomes linear (continuous line) when the bridge volume is increased and the equation of the fitting line is given in Fig. 16. 5. Conclusion The energy stored by a liquid bridge formed between micrometric-size glass particles of either similar or dissimilar surface energy has been investigated. The results show how liquid bridges formed between particles exhibiting high interaction with a liquid binder present higher strength and store more energy than configurations where one particle shows lower interaction than the other. It was also noticed that the particle wettability highly influences both the geometry and the capillary pressure of the liquid bridge. These results will be of interest in agglomeration processes where particles of different surface energy are formulated together, in order to explain which particles are likely to agglomerate and which may segregate. Acknowledgment This work is supported by the Engineering and Physical Sciences Research Council (EPSRC) of the UK, under Grant GR/L77720.
[1] C.I. House, C.J. Veal, in: R.A. Williams (Ed.), Colloid and Surface Engineering: Application in the Process Industries, Butterworth– Heinemann, London, 1992, p. 189. [2] J. Grant, T. Blain, Pulp Pap. Can. 98 (1) (1997) 51. [3] U. Broeckel, in: Proc. 7th Int. Symp. Agglom., Albi, France, 29–31 May, 2001, Vol. 2, pp. 647–653. [4] Y. Kawashima, F. Cui, H. Takeuchi, T. Niwa, T. Hino, K. Kiuchi, Pharm. Res. 12 (7) (1995) 1040. [5] A.H.L. Chow, M.W.M. Leung, Drug. Dev. Ind. Pharm. 22 (4) (1996) 357. [6] S.A. Schildecrout, J. Pharm. Pharm. Sci. 36 (8) (1984) 502. [7] X. Pepin, D. Rossetti, S.J.R. Simons, J. Colloid Interface Sci. 232 (2000) 298. [8] X. Pepin, D. Rossetti, S.M. Iveson, S.J.R. Simons, J. Colloid Interface Sci. 232 (2000) 289. [9] T.E. Yen, R.S. Chahal, T. Salman, Can. Metall. Quart. 12 (3) (1973) 231. [10] D.N. Mazzone, G. Tardos, R. Pfeffer, J. Colloid Interface Sci. 113 (2) (1986) 544. [11] G. Mason, C.G. Clark, Chem. Eng. Sci. 20 (1965) 859. [12] M. Wolfram, J. Pinter, Acta Chim. Hung. 100 (1979) 433. [13] R.A. Fisher, J. Agr. Sci. 16 (1926) 492. [14] S.J.R. Simons, J.P.K. Seville, M.J. Adams, Chem. Eng. Sci. 49 (14) (1994) 2331. [15] O. Pitois, P. Moucheront, X. Chateau, Eur. Phys. J. B 23 (1) (2001) 79. [16] R.J. Fairbrother, S.J.R. Simons, Part. Part. Syst. Charact. 15 (1) (1998) 16. [17] S.J.R. Simons, R.J. Fairbrother, Powder Technol. 110 (1–2) (2000) 44. [18] D. Rossetti, S.J.R. Simons, in: Proc. 7th Int. Symp. Agglom., Albi, France, 29–31 May, 2001, Vol. 1, pp. 137–145, and Powder Technol., in press. [19] M.T. Spyridopoulos, S.J.R. Simons, in: United Engineering Foundation, Proc. Froth Flotation/Dissolved Air Flotation: Bridging the Gap, Lake Tahoe, California, 20–25 May, 2001, Colloids Surf. A, in press. [20] F. Pratola, S.J.R. Simons, A.J. Jones, Trans. IChemE A 80 (2003) 441. [21] J.F. Joanny, P.G. De Gennes, J. Chem. Phys. 81 (1) (1984) 553. [22] C.W. Extrand, J. Colloid Interface Sci. 207 (1998) 11. [23] F.M. Orr, L.E. Scriven, P. Rivas, J. Fluid Mech. 67 (4) (1975) 723. [24] F.R.E. De Bisschop, W.J.L. Rigole, J. Colloid Interface Sci. 88 (1) (1982) 117. [25] R.H. Heady, J.W. Cahn, Metall. Trans. 1 (1970) 185. [26] O. Pitois, P. Moucheront, X. Chateau, J. Colloid Interface Sci. 231 (1) (2000) 26. [27] G. Lian, C. Thornton, M.J. Adams, J. Colloid Interface Sci. 161 (1993) 138. [28] D. Quere, M.J. Azzopardi, L. Delattre, Langmuir 14 (8) (1998) 2213. [29] J.F. Padday, J. Fluid Mech. 352 (1997) 177. [30] S.M. Iveson, N.W. Page, J.D. Litster, in: Proc. 7th Int. Symp. Agglom., Albi, France, 29–31 May, 2001, Vol. 2, pp. 541–547. [31] S.J.R. Simons, X. Pepin, D. Rossetti, Int. J. Miner. Process. (2003), to be published. [32] E. Bayramli, T.G.M. Van de Ven, J. Colloid Interface Sci. 116 (2) (1987) 503. [33] C.D. Willett, M.J. Adams, S.A. Johnson, J.P.K. Seville, Langmuir 16 (24) (2000) 9396.
Rupture energy and wetting behavior of pendular liquid bridges in relation to the spherical agglomeration process Damiano Rossetti,a Xavier Pepin,b and Stefaan J.R. Simons a,∗ a Colloid & Surface Engineering Group, Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, UK b Sanofi Synthelabo Research, Pharmaceutical Science Department, 5, rue Georges Bizet, F-91160 Longjumeau, France
Received 7 June 2002; accepted 2 January 2003
Abstract A novel micro force balance (MFB) is used to investigate the rupture energy of a silicon oil liquid bridge formed in water between two glass particles of either the same or dissimilar surface energy. Rupture energies are integrated from force curves and compared with the models proposed by Simons et al. (Chem. Eng. Sci. 49 (1994) 2331) and Pitois et al. (Eur. Phys. J. B 23 (2001) 79). The latter showed slightly better agreement to the experimental data. Glass ballotini (∼100 µm diameter) are either silanized, in order to increase their wettability toward the oil binder, or kept untreated. Results showed how the interaction between the binder and the particle influences the geometry, the capillary pressure, the force, and the rupture energy of the liquid bridge. Higher values of force and liquid bridge energy were measured between particles characterized by higher interaction (silanized–silanized configuration). A thermodynamic approach to the evaluation of the energy stored in a liquid bridge is also proposed. The mechanical work done to stretch apart the liquid bridge is evaluated as the difference of internal and hysteresis energy between the initial and the rupture configuration of the bridge. This approach showed good agreement with the experimental data only for liquid bridges formed between silanized and untreated glass particles. 2003 Elsevier Science (USA). All rights reserved. Keywords: Spherical agglomeration; Pendular bridges; Wetting hysteresis; Capillary pressure; Force; Energy
1. Introduction Spherical agglomeration is an industrial process traditionally used to separate or recover fine solids dispersed in a liquid suspension through the addition of a second immiscible liquid (binder) which presents an affinity to the solids and is capable of forming small liquid bridges that hold the particles together. Under appropriate physicochemical conditions the desired particles can be selectively agglomerated and removed from the slurry [1]. Recent studies have been undertaken in order to apply spherical agglomeration to other industrial sectors, for example, in the deinking of recycled paper [2], in the removal of heavy metals from wastewater [3], and in the agglomeration of either crystals or powders to produce pharmaceutical drugs [4,5]. The last marks a significant departure from the use of spherical agglomeration as merely a separation * Corresponding author.
E-mail address: [email protected] (S.J.R. Simons).
technique to one that can be used to produce highly valued chemicals. In many agglomeration processes, particles exhibiting different surface properties might be processed together. Such discrepancies in surface energy can create problems during agglomeration, as particles can be selectively wet at the expense of others [6]. On the other hand, the study of the strength and rupture energy of individual liquid bridges, in relation to the surface energy differences exhibited by the particles, can help the understanding of the very first stage of agglomeration, since aggregate formation appears as a balance between rupture energy of individual liquid bridges and kinetic energy of particles. A literature review has revealed (see Table 1) that most studies have been focused on the strength of a liquid bridge between relatively large particles (a few mm) suspended in a gaseous medium (usually air) exhibiting the same surface properties and high wettability towards the binder. This approach, which is very valuable for the number of data made available and the good agreement with values predicted by theory, neglects the common size scale of
0021-9797/03/$ – see front matter 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0021-9797(03)00043-2
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Table 1 Referenced works on liquid bridge force measurements between two particles in a gaseous medium (air) Author Mazzone Simons Pitois Willett
Particles–binder Stainless steel–oil (DBT) Silanized glass–silicon oil Ruby–PDMS oil Sapphire–silicon oil
agglomerated particles (< 250 µm) and the important role that contact angle hysteresis plays during the stretch and rupture of a liquid bridge. The role of contact angle hysteresis in relation to the geometry assumed by a liquid bridge, was observed by Pepin et al. [7,8], who studied the behavior of a glycerol liquid bridge formed in air between glass particles of either similar or dissimilar surface energy. During the bridge separation, the liquid was either held strongly or receded from the particle according to the energy exhibited by the particle and, as a consequence, a pinning or a slippage of the contact angle was observed. Similar behavior was also noted in [9] in a different set of experiments involving measurement of contact angle between a modified silica surface and either water or glycerol liquids. As pointed out by Mazzone [10], gravity is also to be taken into account with large binder volumes, although its effect can be masked off by performing experiments in a second liquid of nearly the same density as the binder [11]. Relevant works regarding liquid bridges suspended in a second immiscible can be found in [11,12]. Mason and Clark [11] investigated the strength of an oil liquid bridge formed between relatively large polythene particles (15-mm radii) suspended in water. They noticed that at small separations the force appeared to pass through a maximum, which favorably agreed with values predicted by theory [13], particularly for the larger volumes used (between 10−2 and 10−1 ml). Wolfram and Pinter [12] alternated water and n-hexane as liquid binder and suspending medium and showed how density differences between the two liquids generate instability in the strength and post-rupture binder distribution due to the increase of gravity effects as the liquid bridge volume increases (from 10−2 to 5 × 10−1 ml). They also recognized the importance of the role played by surfaces of different energy (PTFE and glass) in relation to the geometry of, and the energy stored by, a liquid bridge during separation. The rupture energy, W , of a pendular liquid bridge between spherical particles in air can be found in [14] and [15]. Simons et al. [14] modeled the bridge geometry with a toroidal curve and, in the case of perfect wetting conditions, calculated the integral of the quasi-static capillary force with respect to the separation distances. From the approximated solution of the nontrivial integral a simple expression was derived in terms of dimensionless parameters: = W ≈ 3.6V 0.5 . W γ R2
(1)
Contact angle
Particle radius
Reference
∼10◦ < 20◦ ∼10◦ 0◦
1.98 mm < 100 µm 4 mm 2.3 mm
[10] [17] [26] [33]
In (1), γ is the surface tension of the liquid binder and R are = V /R 3 and W is the radius of the spheres, while V the dimensionless volume and rupture energy of the bridge, respectively. Pitois et al. [15] used a circular approximation for the bridge profile and obtained a simplified expression for the capillary force which, when integrated throughout the separation distance, led to the expression for the dimensionless rupture energy without restrictions to the wettability exhibited by the particles 2V 1/3 W = 2π cos θ (1 + θ/2)(1 − A)V (2) + , π )1/3 /π(1 + θ/2)2 and θ is the contact where A = (1 + 2V and V are defined as angle expressed in radians, while W above. In the present paper the strength of a silicon oil liquid bridge, formed between particles of either the same or dissimilar surface energy suspended in water, is measured throughout separation. Force data are compared with values predicted by theory (see Section 4) and fitted with a third order polynomial curve and, by integration with respect to separation distance, the energy of the liquid bridge between two configurations is calculated. The calculated rupture energies are then compared with the approximations proposed by Simons et al. [14] and Pitois et al. [15], the latter showing a slightly better agreement due to the presence of the contact angle parameter. A comparison is also made of the capillary pressure of liquid bridges formed between silanized–silanized and silanized–untreated particles.
2. Materials and methods Since spherical agglomerates are usually formed from sub-250-µm particles, glass ballotini of diameter in the range 90–130 µm have been used as the solid particles, while silicon oil 100cSt (VWR International) and water (“Analar” VWR International) have been used as bridging and suspending liquid, respectively. The particles are soda lime glass spheres (Sigmund Lindner Type S), washed in chromosulfuric acid (Acros Chimica) and dried. A sample was soaked for 30 min in a silanizing agent, Repelcote VS (2% solution of dimethyldichlorosilane in octamethylcyclotetrasiloxane), obtained from VWR International. Silanized particles were washed first with toluene (Spectranal, 99.9%
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Fig. 1. Schematic of the MFB set-up used to measure the adhesive force of a liquid binder formed between pairs of particles submerged in a second immiscible liquid.
pure from Aldrich) and then with methanol (VWR International) before final drying. The interfacial tension of the water–oil interface was measured using a Kruss 12 tensiometer with the Wilhelmy plate method and resulted in the value of 37.8 mN/m at 21 ◦ C. Fairbrother and Simons [16,17] have developed a novel device, known as a micro force balance (MFB), in order to study liquid bridges between particles in relation to granulation processes. The MFB allows the investigation of both the geometry and the adhesion force exerted by a liquid bridge between two particles during their separation. The technique has already been used to measure particle–binder interaction in a gaseous (air) medium [8]. In this work it has been modified to allow force measurements between two submerged particles [18]. In brief, the measurement procedure is as follows. A pair of appropriately shaped micropipettes hold the glass ballotini under a water objective in an optically clear dish which is filled with the water. Once a drop of silicon oil is administered on one particle (using a third pipette that is not illustrated in Fig. 1), the two particles are first brought together to form the bridge and then separated until rupture of the bridge occurs. One of the two micropipettes is highly flexible in the direction of separation (see Fig. 1) and, due to the separation movement imposed by the second micropipette, the bend of the flexible pipette deflects proportionally to the adhesive force exerted by the liquid bridge. The deflection of the micropipette is calculated via image analysis as the difference of movement between the undisturbed and the actual position of the particle, as illustrated in Fig. 2. The separation process is recorded with a camera capturing up to 25 frames/s and stills from the video are taken in order to carry out further image analysis. Figure 2 shows the steps to formation and separation of a liquid bridge. In Fig. 2a, the straight pipette onto which a silicon oil droplet has been previously administered is moved toward the flexible pipette (whose spring constant is k). At a certain close distance between the two particles, when the liquid interface first touches the other particle, the flexible pipette is caused to “jump” towards the other pipette to form the liquid bridge. The straight pipette is pushed further to reduce the gap between the two particles (Fig. 2b) and then is driven away until the bridge is ruptured (Fig. 2c). The deflection with respect to the undisturbed
Fig. 2. Schematic of the flexible micropipette deformation during the phases of liquid bridge formation and particle separation.
configuration, e, can be either positive or negative according to the configuration assumed by the bridge (i.e., whether the bridge pushes or pulls the particle together). The flexible micropipette is precalibrated in order to determine its spring constant (usually from 0.1 to around 0.7 µN/µm) and the total force exerted is thus calculated as per Eq. (3) F = ke.
(3)
In a new version of the technique a flexostrip device is used to measure the interaction force with the deflection measured directly by a linear voltage differential transducer [19,20]. Figures 3 and 4 show the rupture sequences of a liquid bridge formed between two silanized particles and a silanized and an untreated particle, respectively. The different geometries assumed by the liquid bridge can be explained in term of the wettability between the particle and the binder. Silanized particles (well wetted particles) hold the binder (silicon oil) more strongly than untreated (poorly wetted particles) and therefore, during separation, the solid– liquid interface remains almost constant on the silanized surface. In contrast, this interface reduces dramatically on untreated particles. The resulting geometries, nodoid (Fig. 3) and unduloid (Fig. 4), can be compared in terms of strength, energy stored, and pressure profile, which will be discussed in more detail in Section 4.
3. Theory 3.1. Wetting hysteresis On the majority of real surfaces, the spreading of a liquid is achieved for an apparent advancing contact angle θad , superior to the receding angle θr obtained when the liquid is retrieved from the wet surface. This hysteresis has received extensive theoretical explanations related to the physical and chemical nature of the surface [21,22]. Practically, the wetting hysteresis of a solid surface is seen at the threephase contact line, which either moves or is pinned at a given position. For a solid surface, pinning of the contact line occurs each time the apparent contact angle θ is inferior
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Fig. 3. Separation sequence of a silicon oil liquid bridge formed in water between two silanized glass particles of radius 63 µm (left) and 58 µm (right), separated at 1 µm/s. The volume of the bridge is ∼91,200 µm3 .
Fig. 4. Separation sequence of a silicon oil liquid bridge formed in water between an untreated glass particle (left) and a silanized one (right) of radius 45 µm and 58 µm, respectively. Separation speed is 1 µm/s, the volume of the bridge is ∼133,000 µm3 .
to θad and superior to θr . Measuring the receding contact angle in circumstances where the binder is mechanically forced to detach from a substrate can qualitatively assess the interaction between the two. The lower the contact angle the higher the interaction energy and vice versa. Advancing and receding contact angles can be measured by a variety of techniques; in the present work they were evaluated via image analysis of the profile assumed by the liquid bridge for a given configuration.
Fig. 5. Geometric parameters describing a liquid bridge.
3.2. Bridge geometry The configuration of a liquid/liquid interface at rest is described by the Young–Laplace equation, which relates the difference in hydrostatic pressure P = Pbr − Pext across the interface to the local mean curvature and the interfacial tension of the interface [23] (Pbr , Pext are the pressures inside the bridge and in the suspending medium, respectively). The Young–Laplace equation can be rewritten in Cartesian coordinates in terms of the analytical expression of the local mean curvature, resulting in [24] P 1 y = − , γow y(1 + y 2 )1/2 (1 + y 2 )3/2
(4)
where y and y are the first and second derivatives of the liquid bridge profile described by the function y(x), while γow refers to the oil/water interface. Equation (4) can be rearranged into a Bernoulli-type differential equation and, following classical integration,
Equation (6) allows the P across the interface to be calculated once the interfacial tension γow and the boundary values of the function describing the bridges are known. The boundary values, yA , yB , (θA + βA ), (θB + βB ), can be determined by analysis of the images captured during the separation sequence. Equation (4) can be solved numerically by the value of P and by imposing the condition that at the extremes P and Q, the function assumes the value yA and yB . In Section 4 the theoretical profile of the bridge is compared with those obtained experimentally for different separation distances. Calculating the revolution integral of the profile y(x) and subtracting the volume of the particleto-binder interface (cap of sphere), the volume of the bridge is eventually determined. 3.3. Bridge strength
is obtained, where C is an integral constant [24]. It is possible to substitute the coordinates of the points P and Q (Fig. 5) into (5) and since yP ,Q = cotg(θA,B + βA,B ), the constant C can be eliminated:
The force exerted by a liquid bridge at rest depends on both the surface tension of the interface and the capillary effects due to the curvature of the bridge. Two different approaches are commonly used to calculate this static force, which lead to slightly different values. In the first case, the force is determined by considerations at the neck of the bridge [25],
yA sin(θA + βA ) − yB sin(θB + βB ) P = . 2γow yA2 − yB2
F n = 2πr1 γow − πr12 P ,
P 2 y = y +C (1 + y 2 )1/2 2γow
(5)
(6)
(7)
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while the second method uses values at the boundary [23] between the particle and the liquid bridge, as indicated in F b = 2πRA sin βA γow sin(βA + θA ) 2 − πRA sin2 βA P ,
(8)
with the geometric quantities defined as in Fig. 5. In the first step of separation, where a neck is not yet formed, the use of (7) can be misleading and thus the force is better calculated by using (8). Once the neck is formed the two methods can be compared. During separation the dynamic effects due to the viscosity of the liquid bridge must also be accounted for. Between two equisized spheres held by an infinitely extended liquid bridge, the viscous force is given by [26] 1 dD 3 , Fvis = πηR 2 (9) 2 D dt where η is the dynamic viscosity of the liquid, R the radius of the particles, and D the bridge separation distance as indicated in Fig. 5. In the following section, (9) is used to compare the magnitude of the viscous force with the total static force of either (7) or (8). 3.4. Rupture energy of a liquid bridge The rupture energy model proposed by Simons et al. [14] is derived by integration of the total liquid bridge force, which is written as X tan β , F = πγ R(1 + X tan β − X sec β) (10) X sec β − 1 where X = (1 + D/2R) and β is the half-filling angle, equal, for geometric similarity, to that shown in Fig. 5. In the integration of (10) through separation distance D, the half-filling angle β was considered a constant. The approximation, due to the difficulty in relating the angle β to the separation distance D, seems reasonable for particles that exhibit a strong interaction toward the binder where the solid–liquid interface stays almost constant (see Fig. 3). The expression of the dimensionless rupture energy calculated between any two configurations Xmin and Xmax is 2 X 2 tan β cos β − tan β + X sin2 β W = 2π 2 Xmin + tan2 β cos3 β ln(X sec β − 1) (11) Xmax
which, when plotted against the liquid bridge volume, yielded (1). Pitois et al. [15] used a different simplified expression for the total liquid bridge force: −1/2
2V . F = 2πγ R cos θ 1 − 1 + (12) πD 2 R In (12), θ is the contact angle between the particle and the binder. Using the approximation that θ stays constant
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throughout separation, (12) is integrated with respect to the rupture distance D to obtain the rupture energy, which is then rearranged in nondimensional form,
D 1
1/2 2 V , W = 2π D cos θ 1 − 1 + (13) 2 πD D2
is the nondimensionless rupture distance D = where D 1/3 D/R. If D1 = 0 and D2 = (1 + θ/2)V , which is the expression of the dimensionless rupture distance as proposed by Lian [27], Eq. (2) is obtained. In the present work the liquid bridge energy is evaluated between a small, but nonzero, initial separation distance and rupture. Experimental difficulties prevent observation of liquid bridges at zero separation distance and, therefore, some physiochemical parameters of the bridge remain undetermined. Hence, Eqs. (11) and (13) are used instead of (1) and (2). On a thermodynamic basis, a liquid bridge can be considered a closed system that changes its energy by means of the work done by the micropipette, while its mass remains constant. The latter seems a reasonable assumption for two immiscible liquids, such as used in this work. A more important assumption is to neglect losses due to viscous effects. The last statement can be justified by comparing viscous forces with capillary and surface tension forces, as will be addressed in Section 4. The comparison shows that this assumption is legitimate. This is not a general case though, as Pitois et al. detailed in [15,26]. In view of the previous assumptions, the energy stored by a liquid bridge can be evaluated by using the first law of thermodynamics as the variation of internal and hysteresis energy between two different configurations, U = −(P V ) + (γow Abr ) + Eh = −w,
(14)
where U is the internal energy, P and V the capillary pressure and the volume of the bridge, respectively, Abr the area of the bridge delimited by the liquid–liquid interface, Eh the hysteresis energy, and w the external work, which is negative, by convention, when work is done on the system (liquid bridge). Abr can be evaluated once the profile of the liquid bridge is known, as indicated in x=D
y(x) 1 + y 2 (x) dx.
Abr = 2π
(15)
x=0
The hysteresis energy is defined as the difference of work of adhesion [12,22,28] between two liquid bridge configurations and can be written Eh = −γow Aop (cos θ2 − cos θ1 ),
(16)
where Aop is the area of the cap of the sphere at the contact between the oil and the particle, while θ1 and θ2 refer to the contact angles assumed by the bridge on one particle in two different configurations. Hysteresis energy must be calculated for both particles.
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4. Results and discussion 4.1. Liquid bridge strength and geometry Recorded sequences of liquid bridge separation were analyzed via image analysis (Microsoft PowerPoint) in order to determine by (6) the parameter P /γow , which leads to the solution of the Young–Laplace equation (4). Figures 6 and 7 refer to the rupture sequence of the bridge formed between the silanized particles, shown in Fig. 3. It is useful to note that the x-axis of the geometric profiles is the length of the liquid bridge, indicated as h in Fig. 5, and not the separation distance D. Theoretical values of the liquid bridge strength are calculated using both Eqs. (7) and (8) for all the experimental configurations verified with the Young–Laplace equation. In Fig. 7 the arrows reflect the sequence of initial formation, compression, and separation of the liquid bridge. It can be noted that the initial force (D ∼ 30 µm) is lower than the value obtained at the same D after the liquid bridge has been compressed between the two particles. This fact can be ex-
plained by the larger particle to binder interface formed upon compression, which then leads to higher forces in the following phase of separation, as predicted by (8). This behavior was also noticed between silanized and untreated particles (Fig. 8), where the receding phase on the untreated particles only occurred in the late phase of separation, when an unduloid profile is formed (Figs. 4 and 9). It is important to make clear that the configurations shown in Figs. 6b and 9b were the last recorded (with a camera acquiring 25 frames per second) before rupture occurred but are not likely to be the rupture configurations. The rupture of a liquid bridge is a quick process, which involves some complex phenomena, and only a relatively fast camera (i.e., 500 frames/s) can investigate the evolution of the bridge breakage [29]. The effect of the receding interface on poorly wetted particles (untreated glass) plays an important role in the liquid bridge strength and, as a consequence, adhesion forces measured during liquid bridge separation are lower than in the case where both particles present a high wettability
Fig. 8. Force versus separation distance for liquid bridge formed between silanized and untreated particles. Sequence is shown in Fig. 4. Bridge volume ∼133,000 µm3 .
Fig. 6. Geometry profiles of the liquid bridge shown in Fig. 3: (a) D = 7 µm, P = 1455 Pa; (b) D = 68 µm, P = 1540 Pa.
Fig. 7. Force versus separation distance for liquid bridge shown in Fig. 3. Bridge volume is ∼91,200 µm3 .
Fig. 9. Initial and prerupture configurations of the liquid bridge formed between a silanized and an untreated particle (left side of graphs), shown in Fig. 4: (a) D = 12 µm, P = 2268 Pa; (b) D = 75 µm, P = 1758 Pa.
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Fig. 10. Liquid bridge strength versus separation distance in dimensionless form.
toward the binder (silanized particles). This situation clearly appears by comparison of the dimensional force plots shown in Figs. 7 and 8, where the maximum adhesive force between silanized particles (Fig. 7) is almost double that determined between silanized and untreated particles (Fig. 9). Figure 10 shows the comparison of the dimensionless force–separation curves obtained for the two sets of experiments. In the experiments carried out between silanized particles (filled triangles), higher maximum liquid bridge adhesiveness was always measured, while from the plots obtained between particles of different surface energies (open squares) the force curves seem to level off throughout separation. Viscous forces were also analyzed and shown to be negligible for the experimental conditions used. Introducing the capillary number Ca = vη/γow as the ratio between viscous and surface tension effects, with v being the particles separation speed, and using (9) with the input data R = 51 µm (average value for the experiments), η = 0.096 Pa s, D = 10 µm, dD/dt = 1 µm/s results in a value of 1.1 × 10−4 µN for the viscous force results, while Ca is 2.5 × 10−6 . Increasing the separation distance and keeping all the other quantities the same, reduces the viscous force even further. This result agrees with [30,31], in which the stress induced on particle agglomerates deformed at increasing compression speeds is studied and viscous force contribution to the deformation of the wet mass was found to become predominant only when the capillary number Ca was above a threshold of 10−4 . According to (7), the complementary effects of the increase in neck radius rise (r1 in Fig. 5) and reduction of the capillary pressure inside the liquid bridge increase the liquid bridge force. Figure 11 shows the trend of r1 versus the volume of the liquid bridge for all the configurations where the maximum liquid bridge force was observed. The graph illustrates that the neck radius is larger in the set of experiments where the highest forces have been measured (silanized– silanized particles). However, the neck radius is related to the maximum liquid bridge force only when the particle diameter is small enough. Mason and Clark [11] and Bayramli and Van de Ven [32] showed that when the particle diameter is of the order of cm the increase of the liquid bridge volume does not influence the maximum liquid bridge force, while an increase of the neck radius is still observed [11].
167
Fig. 11. Half bridge neck height versus volume of the bridge measured for bridge configurations of maximum strength.
Fig. 12. Capillary pressure versus separation distance for two silanized (a) and untreated–silanized (b) particles.
4.2. Capillary pressure of a liquid bridge Capillary pressure, which is determined by (6), is plotted in Fig. 12 for the two sets of experiments. When both particles are silanized (Fig. 12a), the pressure profile seems to follow a parabolic curve, which decreases its concavity at increasing bridge volume. With particles of different surface energy (Fig. 12b), the parabolic behavior exhibited by the smaller volumes becomes more asymmetric as the volume increases. A mutual comparison of the two profiles is presented in Fig. 13, which shows higher values for the untreated–silanized particle configuration. The lower strength of the untreated–silanized particle configuration can therefore be explained by both the presence of a thinner bridge neck and higher capillary pressures that work against the bridge strength. 4.3. Energy stored by a liquid bridge The higher strength measured between silanized particles is reflected by higher energy values W stored during separation (Fig. 14). Experimental force diagrams were fitted
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Fig. 13. Normalized pressure versus separation distance.
Fig. 14. Experimental values of liquid bridge energy.
with a third-order polynomial curve and integrated through separation distance in order to calculate the energy W . The higher energy stored in liquid bridges is beneficial to counterbalance the kinetic effects which leads to particle separation and therefore, during mixing, particles with higher surface energy are more likely to exist in an agglomerated form. A further effect that hinders the agglomeration of poorly wetted particles is the amount of binder volume left on the particle after separation. Due to the receding of the particle–binder interface, in fact, a very small amount of binder remains available on low-surface-energy particles to form other liquid bridges, which ultimately disadvantages agglomeration. Figure 15 shows a comparison between the normalized experimental energy data and the values calculated from the models proposed by Simons et al. and Pitois et al. per Eqs. (11) and (13), respectively. Data are plotted against the binder-to-particles volume fraction, Vbr /Vs , and are presented in Table 2. The two models give reasonably good agreement with the experimental data considering the magnitude of the energies and volumes involved and that calculations are made from data acquired by image analysis, which will incur some degree of error. For the untreated– silanized configuration, where the Simons et al. assumption of perfect wetting is not valid, the Pitois model shows slightly better agreement due to the presence of the contact angle parameter. The values used for the contact angle in the two sets of experiments, which are the lower contact angles measured on the particles just before rupture, are shown in Table 2. All the quantities required for the calculation of (11) and (13) are determined via image analysis. Finally, the thermodynamic approach to calculating the liquid bridge energy between untreated and silanized parti-
Fig. 15. Comparison of experimental liquid bridge energy with the models proposed by Simons and Pitois: (a) two silanized particles, (b) untreated–silanized particles. Table 2 Data presented in Fig. 15 V /R 3
exp W
Simons W
%err
Pitois W
%err
Silanized–silanized particles, receding contact angle 35◦ ± 3.7 0.32 0.34 1.25 0.15 0.40 0.25 0.04
1.70 1.36 2.52 0.93 1.73 1.30 0.40
1.29 0.52 1.39 0.90 1.51 1.25 0.44
−24.0% −61.7% −45.0% −3.5% −13.0% −3.7% 8.2%
1.39 0.61 1.40 0.87 1.50 1.21 0.38
−18.5% −54.9% −44.6% −6.0% −13.2% −6.9% −5.6%
Untreated–silanized particles, receding contact angle 52◦ ± 6.2 1.02 1.51 0.84 −44.6% 0.95 −37.3% 0.40 0.97 1.19 23.1% 1.16 20.3% 1.04 1.53 1.22 −20.6% 1.18 −23.0% 0.38 0.97 1.55 60.5% 0.99 2.9% 0.05 0.18 0.16 −10.0% 0.17 −6.6% 0.40 1.00 0.81 −19.2% 0.80 −19.7%
cles using (14) and (16) (the latter applied to both particles) is shown in Fig. 16. In each experiment the thermodynamic energy is evaluated between the initial and final configurations (indicated by the subscripts 1 and 2 in (16)) used to calculate the experimental liquid bridge energy by means of force plot integration. The data shown in Fig. 16 seem to confirm that during a quasi-static separation the energy stored by a liquid bridge can also be calculated using a thermodynamic approach in terms of variation of capillary pressure, increase of interfacial bridge area, and modification of the three-phase contact line due to variations of solid–liquid interfacial area and contact angle formed between the liquid meniscus and the particle. However, good agreement with the experimental data was not found for silanized particles except in a few cases, which might be explained by capil-
D. Rossetti et al. / Journal of Colloid and Interface Science 261 (2003) 161–169
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References
Fig. 16. Experimental liquid bridge energy compared with values obtained by thermodynamic analysis for the separation of silanized–untreated particles. The fitting line shows a nonlinear trend (dotted line) at low liquid bridge volumes. A linear trend (continuous line) approximates the liquid bridge energy when the bridge volume is increased.
lary pressure variations that cannot be accounted for near the bridge rupture, due to the camera limitations. The thermodynamic approach to the energy stored by a liquid bridge seems to support the important effect that hysteresis energy can play in surface chemistry phenomena. The experimental data for silanized and untreated particles (already presented in Fig. 14 on a logarithmic scale) show that the work done to separate the particles increases with the volume of the liquid bridge, although not linearly at low bridge volumes (dotted line). The trend becomes linear (continuous line) when the bridge volume is increased and the equation of the fitting line is given in Fig. 16. 5. Conclusion The energy stored by a liquid bridge formed between micrometric-size glass particles of either similar or dissimilar surface energy has been investigated. The results show how liquid bridges formed between particles exhibiting high interaction with a liquid binder present higher strength and store more energy than configurations where one particle shows lower interaction than the other. It was also noticed that the particle wettability highly influences both the geometry and the capillary pressure of the liquid bridge. These results will be of interest in agglomeration processes where particles of different surface energy are formulated together, in order to explain which particles are likely to agglomerate and which may segregate. Acknowledgment This work is supported by the Engineering and Physical Sciences Research Council (EPSRC) of the UK, under Grant GR/L77720.
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