Contact Between a Gas Bubble and a Solid Surface and Froth Flotation BY A. SCHELUDKO SL. TSCHALJOWSKA AND A. FABRIKANT University of Sofia Dept. of Chemistry Institute of Physical Chemistry Institute of Mining and Geology Sofia Bulgaria Received 9th April 1970 The process of formation of the contact silica surface/air when pressing a liquid meniscus on to the solid/liquid interface is investigated. The kinetics of expansion of the contact area and the final states characterized by contact angles in both directions (by spreading and withdrawing of the contact) are examined. Preliminary data about the effect of surfactants on the velocity of expansion of the contact and on the contact angle and its hysteresis are presented. The new methods applied in the investigations are described. The results are compared with flotation experiment data.The purpose of this paper is to investigate the contact formed by pressing an air bubble on to a solid surface employing the method and approach used in the investi- gation of microscopic free black fi1ms.l. The data obtained are compared to flotation measurements on the basis that flotation is only possible when the film separating the particle from the gas/liquid interface breaks. ANGLES OF CONTACT A liquid meniscus is forced through a thin-walled circular silica tube of internal radius R = 0.102 cm towards a polished plate of vitreous silica until Newton’s rings appear around the tip of the meniscus i.e. at a distance of several lOOOA from the plate. The pressure upon the meniscus equal to 2Y COS ORIR + hPo9 (1) is maintained strictly constant.Its value is registered photo-electrically on a precise aqueous manometer. In eqn (l) y is the surface tension of the solution p o its density g gravitational acceleration and h the depth of immersion of the tube in the solution. The separating film formed from appropriate solution of dodecylamine hydro- chloride (1.35 x mol/l) in these conditions breaks forming an exposed contact whose radius r increases to y o . This process is observed and recorded in reflected light on a metallographic microscope with a camera. As the silica tube is much narrower than the vessel containing the solution the depth of immersion remains constant so that in all the following treatments the hydrostatic component of the pressure hpog is eliminated. After solving the vaiiational problem for the minimum surface at constant volume of a body of rotation about the axis OZ as described in ref.(l) (4) we obtain for the capillary pressure The profile of the meniscus after formation of contact is shown in fig. 1. Pr = 2y(R cos &-r sin 0,)/(R2 -r2). (2) 112 A . SCHELUDKO SL. TSCHALJOWSKA AND A . FABRIKANT 113 In the process treated here both of the angles OR and 8 form as a result of the recession of the liquid on identical surfaces of vitreous silica. Thus at r = yo we obtain 8R = 8 = O0. At constant external and hydrostatic pressure according to (1) and (2) we obtain tan O0 = ro/R. I (3) FIG. 1.-Profile of a meniscus contacting a plane surface. After establishment of ro we gradually reduce the external pressure. At first the circumference remains fixed. It begins to narrow only after a specific lowering of the pressure APM.This irreversibility (hysteresis) of the process is due to the fact that in order to start moving in the direction of wetting the angle 8 has to increase from e0 to O1. The angle OR also becomes bigger than Oo and two states are now possible. (a) OR increases more slowly than 8 and the shift of the circumference at 8 = 8 takes place at fixed position in the region 8 = 8R.l i.e. at constant H equal to Ho . (b) 8R reaches O1 before Or and the shift of Or corresponds to the condition 8 = 8R = 81. The solution of the problem for the shape of the body shown in fig. 1 for 8 = 0 = Q0 is and for the case (a) H = Ho = const. Ifo = R(l -tan O,) (4) where AE = E(A,,q2) -E(L,,q2) and AF = F(A,,q2) -F(AR,q2) (E and F are elliptic integrals of the second and first kind tabulated e.g.in ref. (5)), 114 CONTACT BETWEEN A BUBBLE AND A SURFACE From these equations the intermediate value OK,1 is calculated and from the expression cos cos 20 2A tan 8 sin 0 tan 28 sin 8 = ____ - - - ~ the angle O1 is obtained. If the latter is bigger than the solution is correct If not we have case (b) 0 = OR = 8 and calculate 01 according to (6) but with &,I = 81. TABLE 1 PH 1' dyn cm-1 ro cm x 102 APM dyn cm-2 8 61 4.6 71 0.82 - 4" 30' - 5.3 71 1.48 - 48 8" 15' -17" 6.6 71 4.28 GOO 22" 45' 51' 8.1 70 7.37 893 35" 50' 52" 11 52 9.42 2315 42" 45' 57" The values of Oo and O1 obtained for various pH regulated by ten-fold diluted universal buffer and controlled by a glass-electrode are shown in table l.* The values of y are obtained by the Wilhelmy method using a mica plate.The correc- tions for incomplete wetting of the plate are made the manner recommended in ref. (7). EXPANSION OF THE CONTACT The process of expansion of the contact r ( t ) is either recorded photographically Following when it is fast or visually read on an eye-piece micrometer when slow. ref. (2) the velocity of expansion is represented as drldt = a&. Here (7) 8 = ~(COS 0 - cos 80) (8) is the tangent to the solid-surface force acting at time t from the beginning of the process and a is the instantaneous mobility (the reciprocal coefficient of friction). Both co-factors correspond to a unit length of the circumference of wetting. As At is a function of Y through the non-equilibrium angle 8 = Q, in order to obtain the values of a for different Y the slopes dr/dt (obtained by graphical differentiation of the curves r ( t ) ) ought to be divided by Zit.The latter is easy to calculated with the aid of eqn (2) At = y[Jl-(r cos 0,/R)2-cos 8,] (9) if we assume that the angle 8 = O0 remains constant during expansion at constant pressure. * The details of the method its theory and the additional measurements ofy and floatability will be published separately. FIG. 2.-Photographs of the contact expansion frequency 12 framesis ( a ) pH 5.3 at 100 xniagnifica- tion ; (b) and ( c ) pH 8.1 at 40 x and 100 x . The appearance of the contact is marked by an asterisk. To face page 11 5.1 A . SCHELUDKO S L . TSCHALJOWSKA A N D A . FABRIKANT 115 At low pH the expansion is very rapid as is seen from the photographs in fig.2 4 and the final state yo is established roughly within a minute. On fig. 3 curve 1 the corresponding values of a(r) are given. Within the limits of scatter they are the radius of contact x 10' cm FIG. 3.-Contact mobility as a function of the contact radius. and 0 pH 5.3). Curve 1 for low pH (0 pH 4.6 Curve 2 for pH 11 (the first point 0 is obtained from -rm) ; curve 3 for pH 8.1. same at pH = 4.6 and 5.3 and of the order of cm2 s-' dyn-l. Apparently the friction here is determined by the viscosity in the volume of the liquid which is almost independent of pH. The volume mechanism of the dissipation of energy in this case is corroborated by the fact that a here coincides with a = 5 x obtained in ref. (2) for expansion of a black film of radius 7 x In that paper the volume mechanism of friction was proved by the independence of a on the film tension at given r over a wide range of values of the film tension.At high pH the situation is completely different. The time of establishment of ro is of the order of several hours. The dependence a(r) at pH = 11 (fig. 3 curve 2) shows that a in this case first reaches values which are four orders lower i.e. the friction is 10 thousand times greater than at low pH. As the volume viscosity cannot rise significantly with increase of pH the velocity of expansion is determined by surface or peripheral friction. It is interesting as well that a now decreases extremely steeply with the expansion so that the data are represented on a log scale. This effect is completely rheological-the friction increases dramatically when the deforming force diminish.The process of surmounting the rheological obstructions by big initial forces is clearly demonstrated at intermediate values of p H ~ 8 . On the photographs in fig. 2b and c it is seen that the circumference of wetting first tears and a starlike expan- sion takes place its velocity being of the same order as at low pH. With decreasing cm. I16 CONTACT BETWEEN A BUBBLE AND A SURFACE a, a regular circular circumference forms gradually (fig. 2b) which expands very slowly the values of a being close to those at pH 1 1 (fig. 3 curve 3). COMPARISON WITH FLOTATION As a basis of this comparison the curve in fig. 4 is used. The latter is obtained with quartz particles (radius R,z2 x cm) in a laboratory flotation cell under the same conditions (solution washing of the quartz) as in the measurements of contact angles.Although the shape of such curves depends on the construction and 6 7 a 9 10 PH FIG. 4.-Flotation recovery of quartz as a function of pli. regime of the flotation cell the shape of the curve in fig. 4 is sufficiently typical (see also ref (8)) to permit a comparison with the contact investigation. The decrease of floatability at high pH although the contact angles retain their high values is due to the sharp deceleration of the contact expansion. Hence our interpretation is based on the kinetics of contact expansion. According to the theory of attachment of particles to bubbles,g* lo a spherical particle under the action of a detaching force G breaks away from the bubble when the chord of the contacted part becomes equal to or smaller than 2r at * r,, = JGR,/2ny Therefore it is necessary that the contact shall succeed in expanding at least to We simplify the problem this size in order to retain the attachment or flotation.by neglecting the inertial forces and take G = G = 4xR;(P - Po)& (11) * In order to simplify the calculation the hysteresis of the contact angle [lo] is not taken into account here and later. A. SCHELUDKO SL. TSCHALJOWSKA AND A. FABRIKANT 117 where p is the density of the particles we obtain r,=2 x cm at p-po = 2 and y (see table 1) in the range 71 dyn cm-1 (rm = 1 . 7 ~ cm) to 52 dyn cm-1 (v = 2 x cm). At such small r,, the chord and the diameter of contact are almost identical and the time z for development of the contact is very short. In these conditions the integration of (7) is simple as the initial (r-0) values y(1-cos 0,) for & and a.for a can be used so that From fig. 3 curve 1 we obtain at low pH a. z 7 x and from this z = 3.3 x s for pH = 5.3 (y = 71 O0 = 8" 15'). Since at pH = 5.3 the recovery is some 10 % (fig. 4) then if we assume that it is roughly inversely proportional to the time of contact we find the latter to be of the order of a few milliseconds in good agreement with other data.ll This result confirms such an intrepretation and allows a possible estimate of a for Y = 2 x at the other end of the flotation curve at pH 1 1. Here the recovery is of the same order of magnitude but the direct determina- tion of the initial value of a is very difficult because of the steep change of a with r. Substituting in (12) z = 3.3 x y = 52 and O0 = 42" 45' according to table 1 we obtain a = 4.4 x at pH = 11 the first point of curve 2 fig.3. This result indicates that friction in the periphery of the contact which hampers the recovery at high pH is for small r = r much less than that measured for larger contacts. This first point confirms the sharp decrease of a and the rheological character of the peripheral friction. The latter is apparently closely connected with the hysteresis of the contact angle. In conclusion the present paper shows that the main part of the time of contact in this investigation of froth flotation is not the time of thinning of the separating film to its r ~ p t u r e ~ but the time of expansion of the contact after rupture to the size necessary for attachment. In other conditions e.g.for very small particles the time z strongly decreases (eqn. (12)) and it is possible that the thinning of the separating film becomes the controlling process. Possibly this is the cause of the decrease in floatability of small particles. for r = 2 x A. Scheludko B. Radoev and T. Kolarov Trans. Faraday SOC. 1968 64 2213. T. Kolarov A. Scheludko and D. Exerowa Trans. Faraday SOC. 1968,64 2864. A. Scheludko Kolloid-Z. 1963 191 52; A. Scheludko S1. Tschaljowska A. Fabrikant H. Schulze and B. Radoev Freiberger Forschung. in press. D. Exerowa I. Ivanov and A. Scheludko Ann. Universitb Sofia 1961/62 56 157. V. Beliakov R. Kravtzova and M. Rappoport Tables of Elliptic Integrals (Russ.) (Moscow 1962) vol. 1. H. Britton and R. Robinson J . Chern. SOC. 1931 1456. ' A. W. Neumann and W. Tanner Tenside 1967 4 220. R. Dean and P. Ambrose U.S. Bur. Mines. Bull. 1944,449 ; A. Gaudin Flotation (New York 1957) chap. 10 36 ; A. Gaudin and D. Fuerstenau Min. Eng. 1955 66 ; M. Eigeles and M. Volova 8th Znt. Mineral Processing Congr. (Leningrad 1968) S 12 p. 1. C . W. Nutt Chem. Eng. Sci. 1960 12 133. l o A. Scheludko B. Radoev and A. Fabrikant Ann. Unioersitb Sofia in press. I ' V. Klasscn and V. Mokrousov An Introduction f o the Theory of Flotation (Russ.) (Moscow 1959) pp. 139-144 ; H. Kirchberg and E. Topfer 7th Int. Min. Process Cong. (New York) p. 157.