Langmuir 2002, 18, 9751-9756
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Small Particle at a Fluid Interface: Effect of Contact Angle Hysteresis on Force and Work of Detachment Olivier Pitois*,† and Xavier Chateau‡ LPMDI, Universite´ de Marne-la-Valle´ e, baˆ timent Lavoisier, 5 boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Valle´ e Cedex, France, and LMSGC (UMR 113 LCPC-ENPC-CNRS), 2 Alle´ e Kepler, Cite´ Descartes, 77420 Champs-sur-Marne, France Received March 28, 2002. In Final Form: September 10, 2002 This paper deals with the removal of a small sphere initially attached to a liquid interface. The sphere is small enough (0.3-1 mm) for the capillary force to dominate the interaction and large enough for the line tension effect to be negligible. We have measured simultaneously the force and the geometric parameters of the system as a function of the relative (sphere/interface) separation distance during the detachment process, with a high precision. This procedure allows us to quantify the effect of the contact angle hysteresis during the detachment process with respect to the force-path curve. It is shown that the previous work, which assumes a constant receding contact angle, does not describe our experimental data when the hysteresis effect dominates. By analytical integration of the capillary force experienced by the sphere during the detachment process, the first closed-form analytical expression for the detachment work was obtained. Comparison with our experimental data and with the existing numerical calculations showed good agreement. The effect of contact angle hysteresis on the detachment work is also quantified.
1. Introduction The interactions between liquid/liquid or liquid/gas interfaces and solid particles are central in separation processes. In flotation processes, for example, separation is achieved through the selective attachment of particles to gas bubbles. Insofar as the mineral industry is concerned with flotation, a lot of work has been devoted to the understanding and the control of the elementary steps in the process, that is, attachment and detachment of particles to bubbles (see ref 1 for a recent review). In this framework, the problem of a solid particle resting at a fluid interface has been extensively studied, both experimentally and theoretically. From an experimental point of view, force-path diagrams have been measured. Most of this work has been performed with millimetric spheres, and reasonable accordance was found with numerical results.2-4 In the framework of sphere tensiometry, accurate results have been obtained by Hartland et al.5,6 for example. In comparing their results with numerical calculations, the authors developed a procedure to determine simultaneously the surface tension and the contact angle of a system. To obtain accurate force measurements, sphere diameter was several millimeters. Recently, Butt7,8 used atomic force microscope technology to measure the force of detachment of colloidal particles captured by liquid/ gas interfaces. This apparatus has the advantage of enabling accurate measurements with small particles, but the weak spring stiffness does not allow one to obtain the † ‡
LPMDI, Universite´ de Marne-la-Valle´e. LMSGC (UMR 113 LCPC-ENPC-CNRS).
(1) Ralston, J.; Fornasiero, D.; Hayes, R. Int. J. Miner. Process. 1999, 56, 133-164. (2) Nutt, C. W. Chem. Eng. Sci. 1960, 12, 133-141. (3) Hotta, K.; Takeda, K.; Iinoya, K. Powder Technol. 1974, 10, 231242. (4) Schulze, H. J. Int. J. Miner. Process. 1977, 4, 241-259. (5) Gunde, R.; Hartland, S.; Ma¨der, R. J. Colloid Interface. Sci. 1995, 176, 17-30. (6) Zhang, L.; Ren, L.; Hartland, S. J. Colloid Interface Sci. 1996, 180, 493-503. (7) Butt, H.-J. J. Colloid Interface Sci. 1994, 166, 109-117. (8) Preuss, M.; Butt, H.-J. Int. J. Miner. Process. 1999, 56, 99-115.
complete force-path diagram and its associated work of detachment. From a theoretical point of view, forces acting on the particle during the detachment process have been considered and force-path diagrams were obtained through numerical procedures.4,9,10 In the case of negligible gravity effects, an analytical expression for the interface geometry was proposed.9,11 More recently, O’Brien showed that within these simplified conditions the interface distortion could be linearly related to the capillary force,12 which can be compared to the Hooke’s law obtained theoretically by de Gennes et al.13,14 and experimentally by Nadkarni and Garoff15 for the two-dimensional line pinning problem in the framework of contact angle hysteresis. With regard to flotation, the particle/interface aggregate stability against forces relevant for flotation conditions has been studied. In a simplified treatment, a stability criterion based on the work of detachment of the particle from the interface has been proposed,4,9 and more recently this approach has been found to reasonably describe experimental results obtained within controlled flotation conditions.16 The work of detachment involved in this energy approach is still calculated using numerical procedures,1 assuming a constant receding contact angle during the detachment process. The contact angle hysteresis is widely encountered within partially wetting solid/liquid systems, but to our knowledge, the quantitative study of this effect on force-path diagrams and detachment energy has never been undertaken. In the present paper, we report experimental results for the force and the work of detachment for a small sphere captured by a liquid/gas interface, focusing on contact (9) Scheludko, A.; Toshev, B. V.; Bojadjiev, D. T. J. Chem. Soc., Faraday Trans. 1976, I-72, 2815-2828. (10) Rapacchietta, A. V.; Neumann, A. W. J. Colloid Interface Sci. 1977, 59, 555-567. (11) James, D. J. J. Fluid Mech. 1974, 63, 657-664. (12) O’Brien, S. B. G. J. Colloid Interface Sci. 1996, 183, 51-56. (13) Joanny, J.-F.; de Gennes, P.-G. J. Chem. Phys. 1984, 81, 552562. (14) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827-862. (15) Nadkarni, G. D.; Garoff, S. Europhys. Lett. 1992, 20, 523-528. (16) Crawford, R.; Ralston, J. Int. J. Miner. Process. 1988, 23, 1-24.
10.1021/la020300p CCC: $22.00 © 2002 American Chemical Society Published on Web 11/13/2002
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The work of detachment, that is, the work necessary to remove the particle from the interface, can be easily evaluated from the results of the force computation. The work necessary to move in a quasistatic way the particle from the interface at a position defined by D ) D0 to a position D ) D1 is equal to
Wtot )
Figure 1. Sphere at the liquid-gas interface.
angle hysteresis effect. These data are compared to previous work and also to a new analytical expression for the detachment work. The paper is structured as follows: theoretical expressions are presented in section 2, section 3 presents the experimental setup, and the results are presented and discussed in section 4, before we conclude in section 5. 2. Theory The system studied is presented in Figure 1: a solid sphere is attached to a liquid/air interface. The volumic masses of the three phases are F1, F2, and Fs for the gas, the liquid, and the solid phases, respectively. In the following, the volumic mass of the gas phase will be neglected so that we take F1 ) 0 and F2 ) F. When the system is at rest, the forces acting on the sphere are: The capillary force,
F Bc ) -2πRσ cos φ cos(φ + θ)e bz
bz F Bb ) [(P0 - FgR sin φ)πR2 cos2 φ + FgVim(φ)]e
(2)
where P0 is the liquid pressure at z ) 0 and Vim(φ) is the volume of the particle being immersed into the liquid phase:
Vim(φ) )
3
πR (1 + sin φ)2(2 - sin φ) 3
(3)
The weight force,
1
0
Ft dD
)
R (σ/Fg)1/2
(4)
Then, the total force required to maintain the particle at rest is
e z ) -F Bc - F Bb - F Bw F Bt ) Ftb
(5)
The relative position of the sphere to the interface is defined by the distance D, which is equal to the distance from the planar liquid surface at infinity to the center of the particle:
D ) - lim z(r) rf∞
(6)
where z and r are the cylindrical polar coordinates of the interface. For a given value of D, the value of φ is determined by numerical integration of the YoungLaplace differential equation,4,10 and the Ft ) f(D) curve (force-path diagram) can be constructed, assuming a constant value for the contact angle.
(8)
When the condition , 1 is satisfied, one can show that the interaction is definitely dominated by the capillary force, allowing the buoyancy and the particle weight to be neglected so that Ft ≈ Fc. Note that this approximation is fully justified in the case of floated particles. Besides, it was established that in the asymptotic limit f 0, the interface distortion h is evaluated by the approximate expression9,11
h ) R cos(φ) cos(θ + φ) ×
(
ln
)
4 - γ (9) (cos(φ) + cos(φ) sin(θ + φ))
where λ ) 0.57721... is the Euler constant and h is related to the relative position D:
h ) R sin(φ) - D
4 F Bw ) - πFsgR3b ez 3
(7)
The value of D0 corresponds to the equilibrium configuration of the system when no external force is applied, that is, Ft ) 0. The value of D1 depends on the way the system is perturbed. If the particle is moved from its initial position by means of an imposed displacement, the system configurations are stable until its extreme position value is reached. On the contrary, if a force is imposed to move the particle, D1 equals the value of D corresponding to the extreme value of the force. This latter case is obviously relevant for the situation encountered within flotation process conditions,4 and only this condition will be considered in the following. In the case of small particles, simplifications can be put in this treatment. The relevant parameter to consider is the Bond number , defined as the ratio of the sphere radius to the interface capillary length:
(1)
where the unit vector b ez points vertically upward, R is the sphere radius, σ is the liquid/gas surface tension, the angle φ defines the position of the contact line on the spherical particle (filling angle), and θ is the solid/liquid contact angle. The buoyancy force, applied by the liquid phase,
∫DD
(10)
The parametric eqs 1 and 9 can be used to plot the force acting on the sphere as a function of the interface distortion, which can be expressed as
Ft ≈ k(φ,θ,) × h
(11)
where k(φ,θ,) is a stiffness coefficient. It can be shown that eq 11 can be used to describe the sphere/interface interaction as soon as the particle diameter does not exceed a few hundred micrometers. As a consequence, theoretical values for the force will be plotted in the following using this approach (eq 11 completed by eq 10 will be used to plot force-path curves), instead of using the complete numerical treatment.4,9,10 Quite recently, O’Brien12 points out the approximate linear dependence of the capillary force on the interface distortion, that is, k(φ,θ,) = k(θ,), and compared eq 11 to the Hooke’s law established by de Gennes in the two-dimensional case:
F ) kh
(12)
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Figure 2. Schema of the apparatus used for force-path diagram measurement. The picture is an enlargement of the meniscus formed around the sphere (the sphere diameter is 500 µm).
where k is an effective elasticity constant. Note that in this latter case, the work necessary to detach the interface from its pinning point is simply expressed as
1 W ) kh12 2
(13)
where h1 is the extreme interface distortion. Taking into account the action of the capillary force only, one can calculate the detachment work:
Wc )
∫
c
D1 c D0
-Fc dD
(14)
where Fc(Dc0) ) 0 and Dc1 is the value of the displacement corresponding to the detachment of the sphere from the interface into the gas phase. As the filling angle must belong to the subset [-π/2,π/2], the initial configuration of the system is reached when φ ) φc0 ) π/2 - θ. For an applied force condition, it is easily shown from eq 1 that the last stable configuration of the system is defined by φc1 ) -θ/2. Then eq 14 can be written
Wc )
-θ/2 dh - Fc(R cos(φ) - ) dφ ∫π/2-θ dφ
(15)
Substituting eqs 1 and 9 into eq 15, the work of detachment was computed. After some rearrangements, the following analytical expression was obtained:
{(
θ 1 θ2 θ Wc ) 2πσR2 1 - sin 7 sin2 + 8 sin + 3 + 4 2 2 2 4 1 θ cos4 ln (16) -γ 2 2 cos θ/2(1 + sin θ/2)
(
)(
)
)}
3. Experimental Section 3.1. Setup. We constructed an apparatus allowing the measurement of complete force-path diagrams, that is, F ) f(D) curves, through classical tensiometry techniques. The scheme of the apparatus is presented in Figure 2. The particle is fixed under the beam of an electromagnetic (counter-reaction) scale, allowing the force to be measured with a resolution of 0.1 µN without any appreciable displacement of the sphere. A liquid bath can be moved up and down along the vertical axis at a constant speed: 5 µm/s. The dimensions of the bath have been chosen so that the position of the horizontal plane containing the interface is not modified by the particle immersion. Images of the particle region can be captured by means of a lens and a camera connected to a computer. 3.2. Measurement Procedure. The particle is first partially immersed inside the liquid phase, and the time t0 corresponding to the particle/interface contact detection is then used as a reference time. The relative position of the particle to the interface during the immersion, D, is thus calculated from the time and
Figure 3. Force-path diagram for a 300 µm diameter sphere in contact with pure water. Experimental attachment and detachment curves are presented. Theoretical curves are calculated from eq 11 assuming either a constant contact angle (30° or 65°) or a contact angle variation (between 65° and 30°). the interface velocity v by means of the relation
D ) R - (t - t0)v
(17)
The maximum error for D values has been estimated to be lower than 5 µm. Then the movement is reversed and the particle is pulled out from the liquid bath, until detachment occurs. In the same time, the force is continuously recorded, as well as images of the particle and the distorted interface. Some image processing then allows the measurement of the filling angle φ and the contact angle θ, with a precision of about 2°. The interface distortion is then determined using eqs 10 and 17. 3.3. Material. The particles used are polished ruby spheres of radius 150, 250, 350, and 500 µm, mechanically held on stainless steel rods. Sphere roughness is in the order of 1 µm. The spheres were cleaned before each measurement in a ultrasonic bath using acetone and then absolute ethanol, followed by chromic-sulfuric acid. Then they were rinsed with ultrapure water and dried. The same cleaning procedure was used for the glass vessel containing the liquid. The liquid used is ultrapure water (ultrapure water system Milli-Q plus), and the associated surface tension is close to 72 mN/m (in the following, the parameter σ will be set to this value). Experiments were performed at room temperature (20 °C).
4. Results and Discussion Force-path diagrams have been obtained for attachment and detachment processes. Typical F ) f(D) curves are presented in Figure 3 for a 300 µm diameter ruby sphere. F represents the forces acting on the sphere except its own weight. For this system, the Bond number is close to 5 × 10-2 and the ratio of the buoyancy forces over the capillary force is less than 1% (for the largest sphere this ratio reaches 5%). Consequently, the force F and the capillary force Fc will not be differentiated in the following, although the measured value of the filling angle allows us to deduce Fc from the force measurements. The lower curve in Figure 3 corresponds to the attachment process, and the first nonzero force value is associated with the meniscus formation. As the sphere is further pushed toward the liquid bath, the attractive force is decreased and becomes repulsive (negative values). The movement is reversed when the center of the sphere is located below the plane of the undisturbed interface (D < 0). The sphere is then pulled out from the bath, and the force rapidly increases until a maximum value is reached. Note that the meniscus remains stable until a separation distance greater than the sphere diameter is reached. The presence of two distinct curves in the force-path diagram indicates
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Figure 5. Capillary force as a function of the interface distortion for a 300 µm diameter sphere in contact with pure water.
Figure 4. Contact angle and filling angle are presented as a function of the interface distortion for the attachment process (a) and the detachment process (b).
that the contact angle is not the same during attachment and detachment processes. This is confirmed by following the values of the contact angle, which have been drawn in Figure 4. It can be seen in Figure 4a that the contact angle remains approximately constant during the attachment process. The values of the filling angle measured in the same time are presented together. It appears that the filling angle increases as the contact angle remains unaltered, which suggests that the contact line is freely sliding over the solid surface. These measurements allow us to determine the advancing contact angle, whose value is close to θa ) 65°. These experimental data are correctly described by the theoretical values assuming a constant contact angle equal to 65°. The behavior of the contact line during the detachment process appears to be very different (see Figure 4b). During the first moments of the receding displacement, the value of the contact angle is strongly decreased while the filling angle remains unchanged. While the contact angle reaches its lowest value and then remains constant, the filling angle begins to decrease, until detachment occurs. This behavior is characteristic of a pinning and then sliding motion of the contact line. The contact line remains pinned at the same position until a critical value for the contact angle is reached. This latter value, which is close to 30°, defines the receding contact angle, θr. This transition in the values of the contact angle, observed as the bath movement is reversed, has been previously observed during dynamical contact angle measurements. Lam et al.,17 for example, reported similar behavior for the contact line of growing
and shrinking liquid drops on planar surfaces. In this careful experiment, the contact line is forced to advance and then to recede in a controlled manner. The pinning of the contact line was observed, resulting in a decrease in the contact angle values, from the advancing value to the receding one. Note that the energy barrier against receding of the three-phase contact line can be high enough to induce stick-slip behavior or complete pinning of the contact line (no receding motion). Although contact angle hysteresis has been studied extensively in the past several decades, its origins are not completely understood. However, the presence of contact angle hysteresis is generally attributed to surface roughness18-20 and/or surface heterogeneity21-23 inherent in every natural solid surface. For the problem under consideration in this paper, the main consequence of contact angle hysteresis is that the theoretical data, calculated assuming a constant value for the receding contact angle equal to 30°, do not fit the whole experimental curve. The discrepancy can be clearly observed for the small forces region in Figure 3, showing the error made in assuming a constant contact angle value. To closely describe the experimental results, a theoretical curve can be obtained from a new procedure, simulating the transition in contact angle values. The initial value of θ is set equal to θa, and then the capillary force is calculated per eqs 10 and 11 for values of θ in the range θa - θr while the filling angle is kept equal to its initial value. When θ reaches the receding contact angle value, this parameter is kept constant while the filling angle evolves until detachment. The resulting curve is plotted in Figure 3 and shows that this procedure enables us to accurately describe our experimental data. The validity of eq 11 has been checked. The measured force is plotted as a function of the interface distortion in Figure 5, as well as values provided by eq 11 assuming a constant contact angle equal to 30°. It can be seen that agreement is excellent for both attachment and detachment experimental curves. This is surprising insofar as the contact angle value corresponding to the attachment curve is twice the value assumed in the theoretical (17) Lam, C. N. C.; Wu, R.; Li, D.; Hair, M. L.; Neumann, A. W. Adv. Colloid Interface Sci. 2002, 96, 169-191. (18) Bartell, F. E.; Shepard, J. W. J. Phys. Chem. 1953, 57, 211. (19) Johnson, R. E., Jr.; Dettre, R. H. Adv. Chem. Ser. 1964, 43, 112. (20) Oliver, J. F.; Huh, C.; Mason, S. G. Colloids Surf. 1980, 1, 79. (21) Johnson, R. E., Jr.; Dettre, R. H. J. Phys. Chem. 1964, 68, 1744. (22) Neumann, A. W.; Good, R. J. J. Colloid Interface Sci. 1972, 38, 341. (23) Decker, E. L.; Garoff, S. Langmuir 1997, 13, 6321.
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Figure 6. Theoretical capillary force as a function of the interface distortion, plotted for several values of the contact angle.
Figure 8. Experimental and theoretical detachment work plotted as a function of sphere diameter.
agreement for the smallest spheres (R ) 150, 250 µm) and only a reasonable one for the largest spheres, suggesting the validity of eq 18 to be restricted to sphere diameters lower than 500 µm. These results support the proposition of O’Brien12 and justify the use of eq 12 in describing the interaction of a small sphere captured at a fluid interface. In this regard, eq 18 completes this approach. The work associated with the interface distortion during the detachment process can be evaluated from the F ) f(h) curve. As the maximum for the force is found for φ ) -θ/2, the corresponding maximum interface distortion is
h1 =
2πRσ cos2(θ/2) k
(19)
and the associated distortion work can be calculated:13,14 Figure 7. Experimental capillary force divided by the “effective elasticity constant” k given by eq 12 as a function of the interface distortion.
calculation. To clarify this point, the dependence of the contact angle on the theoretical F ) f(h) curve has been drawn in Figure 6 for several values of the contact angle (15° < θ < 90°). The change in the contact angle appears to modify the maximum force value only. The set of theoretical F ) f(h) curves could thus be approximately described by a single linear curve, whose slope depends on the Bond number only, that is, k(φ,θ,) = k(), and whose extreme value, for a fixed value, depends on the contact angle only. Equations 9 and 11 suggest that the mean slope could be reasonably described by the analytical form
k() )
2πσ A + B ln
(18)
where A and B are two constants. An optimization procedure was performed to obtain numerical values for the constants A and B. Theoretical curves from eqs 1 and 9 were fitted with eq 11 completed by eq 18, within a large set for θ and values, by doing a least-squares fit. Numerical values are A ) 0.6285 and B ) -0.96646. A F ) k()h curve obtained using eq 18 associated with the constants A and B is plotted in Figure 5 for the system described above. It can be seen that experimental results are correctly described using this approach. A comparison is also presented in Figure 7 for several sphere diameters. Each experimental value is divided by the corresponding value of k, calculated using eq 18. Figure 7 shows a good
Wd =
∫0h
1
kh dh
(20)
Using eqs 18 and 19, Wd is expressed as follows:
Wd ≈
1 [2πRσ cos2(θ/2)]2 ) 2k πR2σ cos4(θ/2)[A + B ln ] (21)
Obviously, the quantity Wd does not correspond to the work of detachment, because the distortion process is associated with the sliding motion of the contact line occurring in the same time. However, we observed that eq 21 could be easily modified in order to provide values for the detachment work that are very close to the theoretical one. In this regard, the constant A can be set to a value A′ close to 2.45, producing errors less than 5% of the theoretical values for contact angles in the range 0-90° and sphere radii less than 100 µm. The work of detachment is now determined by numerical integration of the experimental F ) f(D) curves, and the results are compared to data provided by eqs 16 and 21, as well as data obtained following the conventional numerical scheme.4 Comparison is presented in Figure 8. First, it can be seen that experimental results are correctly described by theoretical curves, even for the largest spheres and despite the variation of contact angle occurring in the small forces region. Equations 16 and 21 provide values very close to numerical ones, which makes these expressions useful for evaluating the work of detachment. We now quantify the influence of contact angle hysteresis on the detachment work. This latter quantity is calculated by numerical integration of theoretical F ) f(D)
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lower than 5% are observed for moderate receding contact angle and hysteresis values. For the experimental system considered here, the error is about 3%, which is in accordance with results presented in Figure 8. 5. Conclusions
Figure 9. Relative variation of the detachment work as a function of contact angle hysteresis (with reference to the nohysteresis case) for several receding contact angle values. Calculations are performed for the experimental system considered in the present paper and for a sphere diameter equal to 150 µm.
curves obtained through the procedure described above, which enables the contact angle value to evolve between the advancing contact angle value and the receding one. For a given hysteresis value, the variation of the detachment work ∆W is calculated with reference to the zero hysteresis case W. These calculations are performed for the parameters corresponding to the experimental system and for a sphere radius equal to 150 µm. The quantity ∆W/W is plotted in Figure 9 as a function of the contact angle hysteresis (in the range 0-60°) considered for several values for the receding contact angle (in the range 0-60°). It can be seen that the overestimation made in neglecting the contact angle variation increases as the contact angle hysteresis increases and that the error is all the more significant when the receding contact angle value is high. Even if errors equal to 20% are quite conceivable, errors
Dedicated experiments allowed us to quantify the effect of contact line pinning and the associated contact angle hysteresis on the force and the work of removal of a small sphere captured at a water/air interface. The measurement of force-path curves with a high precision showed that neglecting contact angle hysteresis during the detachment process could induce non-negligible deviations. On the other hand, taking this effect into account within the theoretical treatment allowed us to obtain close agreement with experimental data. The analytical integration of the capillary force experienced by the sphere during the detachment from the interface has been performed, providing a simple closedform expression for evaluating the detachment work. As the values provided by this expression were found to be very close to experimental data and to values obtained from the conventional numerical scheme, we think this latter expression to be useful for applications involving spheres of diameter lower than 500 µm. With regard to the detachment work, the effect of contact angle hysteresis was found to remain moderate as soon as the receding contact angle value is not too high (0-40°). Finally, the theoretical work initiated by O’Brien has been completed. First, the approximate linear relation between the capillary force and the interface distortion has been experimentally verified. It has been shown that the slope of the F-h curve does not depend on the contact angle, enabling this latter relation to be determined by a single parameter, for which a simple expression has been proposed. LA020300P
