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THE PHYSICAL CHEMISTRY OF FLOTATION. I

THESIGNIFICANCE OF CONTACT ANGLEIN FLOTATION IAN WILLIAM WARK Department o j Chemistry, University of Melbourne, Melbourne, Australia Received September 17, 1951 CONTENTS

I. Equation to the surface of a stationary bubble of air i; water.. . . . . . . . . . . . . 624 11. Arithmetical solution of the equation for bubbles of different sizes., . . . . . . . 626 111. Equation connecting volume of bubble, angle of contact, and area of airsolid contact.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .627 IV. Arithmetical solutions of the ng: (a) constant values of angle of contact; (b) constant values of area of contact; 628 (c) constant values of bubble volume.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Replacement of solid-water contact by solid-air contact. Tenacity of adhesion. . . . . . . . . . . . . . . . . 633 VI. True and apparent area of contact.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 VII. Difficulties in experimental determinations of angle of contact-hysteresis and its physical nature.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 VIII. The stability of air-mineral attachments. Extension to moving systems. . 638 640 IX. Froths and air-mineral aggregates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. Maximum size of particle which will float.. ............................ 642 INTRODUCTION

Adhesion between a bubble of air and a solid surface is the basis of the flotation process. I n order that a small particle of mineral may become incorporated in the froth at the top of the machine, it is necessary first for it to become attached to (i.e., to adhere to) an air bubble of sufficient buoyancy to carry it upwards. It is the purpose of this communication to investigate some of the physical and chemical principles underlying the adhesion between a bubble of air and a single solid particle and to draw attention to some of the problems awaiting solution before an explanation of the physical nature and behavior of froth systems can be obtained. The froth differs from the simple case in which a single particle is considered, in that each bubble is coated by numerous small particles of mineral instead of merely being attached to one particle. Nevertheless a discussion of the principles involved for a single particle may be of some value in considering the general case. It has been shown (1) in the course of an investigation being carried out in the University of Melbourne for and a t the expense of a group of mining 623

624

IAN WILLIAM WARK

companies' that an air bubble in water will not adhere to a clean surface of any of the common sulfide or gangue minerals or to metals. When, however, a unimolecular film of a xanthate is adsorbed by the mineral, the air of the bubble spreads t o a limited extent over the mineral surface, partly replacing the aqueous phase in so doing. Spreading continues until a definite angle between the air-water interface and the water-mineral interface, 0, is attained which is determined by the well-known relationship :COS

e

=

-

T.8

Taw

T*.

where T,,,T,,,and T,, are the surface tensionsat the air-solid, solid-water, and water-air interfaces, respectively. Different conditions of the mineral surface and different reagents lead t o variations in the contact angle. One of the major objects of this paper is t o investigate the significance of changing contact angle on the tenacity of contact between air and mineral and, through it, on floatability. Incidentally, it should be noted that the molecular mechanism through which adhesion is achieved is unimportant in thermodynamical and mechanical discussions.

FIG.1 I. EQUATION TO T H E SURFACE OF A STATIONARY BUBBLE OF AIR I N WATER

All theories of capillarity are in agreement concerning the difference in pressure on the two sides of a curved interface between two fluids, namely, that this pressure difference a t any point is equal to

1 Broken Hill South Ltd., North Broken Hill Ltd., Zinc Corporation Ltd., Electrolytic Zinc Co. of A/asia Ltd., Mt. Lye11 Mining and Railway CO.,Burma Corporation Ltd.

625

PHYSICAL CHEMISTRY O F FLOTATION. I

where T,, is the surface tension and R and R‘ are the two principal radii of curvature at that point. Assuming this difference in pressure, Bashforth and Adams (2) show how the equation to the surface may be derived. Since this work is not generally available, a modified development is presented here. Let Po and PO’be the pressures on the inner and outer sides, respectively, of the surface at 0 and P Aand PA‘ be the corresponding pressures at any point A . Let be the angle which the normal to the surface at A makes with the axis of revolution, and let 0 be its supplement. The two principal radii of curvature are equal at 0; let each be denoted by b. One of the two principal radii of curvature at A is x/sin +; let the other be denoted by p. It is obvious that

+

PA

- Po

= u,gz

PA’

- Po’

= uzgz

and where c1 and

u2 are

the densities of air and water respectively. Po

-

P,

- Po’

= T,,

PA’ = T,,

2/6

*

Then (3)

( sin 7 6 -I-’p)

(4)

Whence, subtracting equations 3 and 4 and substituting from equations 1 and 2,

or

In the case of a bubble of air in a liquid, u1 Following Bashforth and Adams let 96’

(UI

-

~ 2 )

Twa

u2 is

negative.

= P

Equation 6 then becomes -1 + -sin - -9 x

p

2

Pz

-6’1

(7)

in which fl is negative. If p and $ are expressed2in terms of 2 and z it becomes apparent that the differential equation cannot be exactly solved. Using equation 6, Bashforth and Adams have constructed tables by which corresponding values 2Expression of

p

and 4 in terms of x and z : 1

d2z dx2

- = - A

P

. i\ l +

($11:”’”

626

IAN WILLIAM WARR

. of

2, 9,and z may be approximately obtained for any given values of b and p. p determines the form of the bubble and b its magnitude, but since p depends partly on 6 , the magnitude of the bubble partly determines its form. These tables are extensive enough for the calculation of the shape of the interface between two fluids for which u2 > ul,Le., where p is positive. Where p is negative, however, they are not nearly as comprehensive, and for bubbles of air in water it is only over a limited range that the calculations are possible. It will be apparent that the nature of any solid surface t o which the bubble of air may be attached has no influence on the shape of the bubble, for the nature of the solid cannot affect any of the pressure terms used in deriving the equation to the surface. The bubble fits on to the surface of the solid with a definite angle of contact, and above the plane of contact the shape is the same as that of a bubble of greater depth. Fortunately Bashforth and Adams tabulate the angle 4, which is he supplement of the angle of contact, e, measured across the water, at the line of triple contact. This angle was directly measured in the paper already cited (1).

-X

X

FIQ. 2. SHAPESAND SIZESOF STATIONARY AIR BUBBLES IN WATER (IN

CM.)

11. ARITHMETICAL SOLUTION OF THE EQUATION FOR BUBBLES OF DIFFERENT SIZES

Since fi = g

- Q) a b 2 (01

TW,

and, from Kaye and Laby's Physical and Chemical Constants, g = 980 cm. per sec2for Melbourne, (ul a2) = 0.997 at 2OoC., allowing for the

-

PHYSICAL CHEMISTRY OF FLOTATION. I

627

saturation of the air by water vapor, and T,, = 72.8 dynes per centimeter (Bohr's value), the value of p reduces to - 13.43 b2. Figure 2 shows not only the shapes but the actual sizes in centimeters of a series of bubbles of air in water corresponding to different values of p. These have been calculated from Bashforth and Adams' values of x/b and z / b , and the value of b corresponding to each value of p. There is good agreement between the shapes of calculated curves and photographs of actual bubbles. Worthington (3) states that pendant drops of liquids are unstable with regard to surface oscillations at points where they are reentrant. The writer has not been able to obtain any reasonably stable reentrant surfaces, though if the buoyancy of the bubble is balanced by an upper supporting tube, reentrant surface may persist for some time in the absence of vibrations. 111. EQUATION CONNECTING VOLUME OF BUBBLE, ANGLE OF CONTACT, AND AREA OF AIR-SOLID

CONTACT

Bashforth and Adams develop by two methods a formula for the volume contained by a bubble above any given plane in the bubble. The following analysis is based upon their second derivation. Assume that a mass of water of the same size and shape as that portion of the bubble being considered replaces it in the water. The vertical forces acting on this mass of water, which obviously is in equilibrium, are: (1) Its weight V.az.g acting downwards. (2) The force due to the hydrostatic pressure P i a t the base of the volume, P i .7r.x2 upwards. (3) The resultant, F , (acting downwards) of the hydrostatic pressure over the curved part of the surface. Whence F

+ V U t g = ?rx2PA'

(8)

For the bubble of air, the vertical forces acting on the portion under consideration are: (1) The same force, F , acting downwards. (2) Its weight Vag acting downwards. (3) The surface tension force, 27rxTw,. sin$, acting downwards. (4) The pressure of the air lying under the plane considered, 7rx2 PA. (If the bubble is cut off, not by an imaginary plane but by a solid surface, this pressure term is placed by an equal reaction acting upwards.) Whence F

+ Vulg + 2 ~ x 2 " ~sin . cp = ?rx2PA

(9)

628

IAN WILLIAM WARK

Equations 8, 9, and 4 reduce to the equation of Bashforth and Adams, ViZ.,

or, introducing p,

where, as before, /3 is negative. Since V is always positive, sin 4 / x must always be greater than l / p . Bashforth and Adams have constructed numerical tables for different values of fi and b which enable a numerical solution for V-correct, if necessary, to 1part in 100,000-to be made over a limited range. It would be possible to construct a three-dimensional model to show corresponding values of the three variables, V , 8, and x for any two fluids, assuming T,,, ~1 - uz, and g to be known, i.e., b 2 / p to be fixed. It is, however, easier in the first place to construct sectional diagrams which represent the relationship between any two of the three variables when the third is constant. These sectional diagrams are now considered. IV. ARITHMETICAL SOLUTION O F THE EQUATION OF I11

(a) 8 constant. Figure 3 shows the relationship between V and x for a series of different (constant) values of 8. The irregular choice of angles in constructing this figure was necessitated by the incompleteness of the tables upon which the calculations were based. The meaning of the graph is best illustrated by considering for one of the curves (for example that for 90") the shapes of the bubbles corresponding to a number of points on the curve. The numbers 3,4, etc., correspond to those portions of the bubbles of figure 2 above 0 = 90" for values of p of - 0.3, 0.4 etc., respectively. The points marked 4a and 4b correspond t o bubbles, one of which (4b) is merely a continuation of the other to a position where, for the second time, an angle of 90" develops. As explained above, this involves a reentrant surface and thus for flotation the portion of the curve beyond 6a is probably unimportant. Two important curves follow from figure 3. These show respectively (1) maximum volume of bubble for a given value of 8 (figure 4) and (2) maximum value of x for given values of 8 (figure 5). The figures of figure 5 are immediately applicable in flotation, but the maximum values of V given in figure 4 all correspond to bubbles with reentrant surfaces. The maximum values for non-reentrant bubbles are also plotted in the dotted curve; these would be of more interest with regard to flotation.

PHYSICAL CHEMISTRY O F FLOTATION. I

629

ozE I 0.2

0.

I

!

0.3 X

FIQ. 3. RELATIONSHIP BETWEENVOLUMEOF BUBBLEAND RADIUSOF CIRCLEOF CONTACT FOR VARIOUS ANGLESOF CONTACT

Experimental verification. Figure 5 has been verified experimentally, the points marked by a circle being experimental values. This verification

630

IAN WILLIAM WARK

FIQ. 4. MAXIMUMBUBBLEVOLUMESFOR GIVENCONTACT ANGLES

FIG. 5. MAXIMUMVALUESOF RADIUSOF CIRCLEOF CONTACT FOR GIVENCONTACT ANULES 0 Experimental points

63 1

PHYSICAL CHEMISTRY OF FLOTATION. I

is tedious, a large number of bubbles which will just hang to the surface being measured for a series of different xanthates, which give characteristic contact angles. The experimental points were determined by Mr. A. B. Cox without preknowledge of the position of the theoretical curve.

3

I60

0

e FIG.6. RELATIONSHIP BETWEEN VOLUME OF BUBBLEAND ANGLEOF CONTACT FOR VARIOUS VALUESOF RADIUSOF CIRCLEOF CONTACT

(b) x constant. Figure 6 shows the relationship between V and e for a series of different (constant) values of x. The meaning of the curves is

632

FIG. 7. MAXIMUM BUBBLEVOLUMES FOR GIVEN VALUES OF RADIUS OF CIRCLE OF CONTACT

e FIG. 8. RELATIONSHIP BETWEEN RADIUSOF CIRCLE OF CONTACT AND ANGLEOF CONTACT FOR VARIOUS VALUESOF BUBBLEVOLUME

PHYSICAL CHEMISTRY OF FLOTATION. I

633

again best understood by tracing, for one of the curves-e.g., that for x = 0.125 cm.-the shapes of the bubbles corresponding to various points on the curve, Along the portion OA the angle of contact is steadily decreasing to the minimum value, but along AB it increases again as reentrant surfaces appear. From B to C, though the angle is still increasing, the volume is diminishing, owing to increasing flatness of the bubble. Two important curves follow immediately from figure 6. These show, respectively, (1) the minimum value of 8 for given values of x, and (2) the maximum value of V for given values of x (figure 7). It is obvious that the curve showing the minimum values of 8 plotted against x is identical with that of figure 5, which shows the maximum value of x plotted against 8. (c) V constant. Figure 8 shows the relationship between 8 and z for a series of different (constant) values of V . It will be seen that both arms of any of these curves approach the x-axis asymptotically. The points on the lower arm all represent bubbles with reentrant surfaces; at A , however, reentrant surfaces disappear and from A back to the z-axis the bubbles have the simplest form. V. REPLACEMENT O F SOLID-WATER

CONTACT BY SOLID-AIR

CONTACT

Tenacity of adhesion The conditions which determine whether air will displace water a t the surface of a solid follow immediately from thermodynamics. This has been realized by several writers, but others have given incorrect analyses. Restatement may be helpful. For any rearrangement of the surfaces to occur when an air bubble is brought into contact with a submerged mineral surface, it is necessary that the potent,ial energy should decrease. Since, under the conditions of flotation practice, the air-mineral interface cannot be created without the simultaneous destruction of mineral-water and water-air interfaces, the condition for replacement of water by air is evidently3 Tm


T,, (4),but no such cases are known. VI. TRUE AND APPARENT AREA OF CONTACT

When a submerged bubble of air is pressed against a solid surface it, sometimes fails to spread uniformly. Portions of the surface are in true contact with the air, but others may be fouled or for some other reason the air makes no true contact with them. The area of true contact will not then be identical with the apparent area of contact, and in estimating the tenacity of contact from the equation, W = T,, (1 - cos 8) ergs per unit area, the true rather than the apparent area of contact must be employed. It is important that the relationship between these two areas should be known. Some experiments a t a cerussite surface using a bubble of carbon tetrachloride throw light on this question. If the surface be freshly prepared in the manner described in the third paper of this series, the carbon tetrachloride does not spread immediately over the whole surface. Contact first occurs a t a series of small isolated areas whose extent gradually increases. The water is gradually forced out between them and ultimately, save for a few irregular patches, there is a complete disc of contact between the carbon tetrachloride and the mineral. A few patches of the surface remain wetted by thin layers of water. These observations, 4 Since the surface of the bubble is curved a t the point of contact, and since, further, there is a rearrangement of the air-water interface after disruption, small corrections are necessary. The curvature introduces a “pressure” correction; that it is neligible for fairly large bubbles is proved by the absence of variation in contact angle with size of air bubble, and also follows from Lyons’ work on floating lenses (J. Chem. SOC.1930, 623).

PHYSICAL CHEMISTRY OF FLOTATION. I

635

which extend over only a fraction of a minute, are possible because of the high refractive index of the carbon tetrachloride. The line of vision should be at right angles to an illuminating beam of light and inclined at 20" to the horizontal. If, however, the surface of the mineral is oiled, or is coated by what is called in flotation an organic collecting agent, the carbon tetrachloride spreads outwards very quickly, and no surface inclusions of the aqueous phase remain. It may be concluded that where the surface is in a receptive condition for air-contact, the true and apparent areas of contact are identical. This conclusion is supported by the uniform manner in which a bubble of air leaves the surface if it is pulled away. Ostwald ( 5 ) has formulated an ingenious theory in which he claims that only a unimolecular ring of the organic collecting agent is necessary to ensure attachment of the bubble to the particle. His main evidence in favor of this theory is the very doubtful statement that sufficient collector is not present t o form a unimolecular film over the whole surface. The above expression for the tenacity of sticking proves, however, that such a ring contact would be so unstable as to collapse with the slightest displacement unless the moving air-water-mineral boundary carried with it the ring of collector, i.e., that the attraction between mineral and collector was so small as t o permit free movement of collector molecules over the surface. This it is not, for if a mineral which has had contact with a xanthate solution for a minute be removed and washed in several changes of water, it still retains over its whole surface the power of attachment to an air bubble. Recent measurements of surface tension (6) indicate moreover that there is little adsorption of xanthate in the air-water interface. A contaminated patch in the center of the air-mineral contact will not, of course, influence the stability of the contact for small displacements, but it will cause a rapid unopposed contraction of the area of contact the moment the displacement of the bubble wall reaches it. I n the experiments described in the paper of Wark and Cox (1) and in those of the following papers, there has been little or no fouling of the surface. When one is approaching a region of non-sticking, however, as for example when alkali is added to a xanthate solution in contact with galena, the surface does change in such a way that the true area of contact is apparently greatly diminished. Even on a partly fouled surface, at points where sticking is possible, the full angle of contact may be developed. VII. DIFFICULTIES I N THE EXPERIMENTAL DETERMINATION OF ANGLE OF CONTACT-HYSTERESIS

Sulman (7), Ablett ( 8 ) , and Langmuir (9) have shown that the angle of contact varies according to whether the line of contact between air, water,

I

636

IAN WILLIAM WAHK

and solid is advancing or receding. The angle 0 ” is greater when the water-solid contact is being replaced by an air-solid contact ‘than the angle 0’ when the reverse is occurring. The difference Of’-# is termed the “hysteresis.” Sulman has shown that the variation lies within certain well-defined limits, which he claimed were dependent on the mineral. Ablett also showed that the limits were clearly defined on a paraffin surface and, furthermore, that the mean of 8” and 0’ was very close to the equilibrium value, 8 ; also that the difference between 8” and 8’ was a function of the rate of motion, and was constant over a range from 0.4 to 4 mm. per second. The angle of contact of a bubble in equilibrium on a solid surface is (subject to certain minor corrections) given by the expression

Edser (10) concludes that if, owing to hysteresis, 0 alters, T,,must have altered. (It is not stated why T,, and not T,, has been assumed to alter.) This would be true if the bubble were still in equilibrium, but under such conditions it is not. If the vessel in which the mineral rests is shaken lightly, e.g., by tapping the table upon which it stands, the angle of contact reverts to the equilibrium value e, Hundreds of tests have supported this contention. Apart from any experimental verification, however, it is obvious that the stress applied at a point of the bubble can have no effect on the surface energies of the distant interfaces mineral-water and mineral-air. Some secondary force must assist in the maintenance of apparent equilibrium, if variations in T,,, T,,,etc., are inadmissable. A frictional force alone could be responsible. To make the argument specific, let it be assumed that the stress a t the base has been caused by an upward extension of the bubble which has the effect of narrowing it along its length. When the top of the bubble is raised, an attempt is made to increase the volume of the bubble. This reduces the pressure of the bubble, and thereby creates an excess external pressure which tends to push in the bubble wall. Yet some force, resulting in the hysteresis, prevents the base of the bubble from contracting, though in equalizing the pressure the bubble wall alters in shape and a new angle of contact develops. Only a horizontal opposing force could be effective in preventing motion along the mineral surface under the stress of this pressure. This horizontal force is thus responsible for the new value of the angle. Furthermore this force must disappear when true equilibrium is reestablished, for the original angle is again obtained. Such a force, which acts parallel to the direction of possible motion and disappears when stable equilibrium is reached, is

PHYSICAL CHEMISTRY OF FLOTATION. I

637

customarily styled a frictional force. Hysteresis of contact angle is thus a manifestation of f r i c t i ~ n . ~ Adam and Jessop (11) have already suggested that hysteresis is due to friction, but their paper has not received the attention it deserves. This is perhaps due to the fact that, beyond the statement that the degree of hysteresis is less at smooth than at rough surfaces, they give no reason for this interpretation. Our experience confirms this statement. It is by no means certain that the laws of friction at solid surfaces may be applied when considering solid-fluid contacts, but, assuming that they may, it follows that the frictional force opposing motion of the bubble over the surface is proportional to our L of equation 16. Assuming that hysteresis is due to friction, Adams and Jessop develop a formula for 8, namely, COS

e

= 1/2 (COSeH

+

COS

e’)

(14)

In deriving this expression the authors equate terms of unequal dimensions, but their final expression is correct. If H dynes per centimeter of air-water-mineral boundary be the frictional force it may be shown that H = 1/2 T,, (COSe‘

- COS err)

(15)

Since 8 * - 8’ may be as high as 60’, the frictional force may be of an order of magnitude as high as one-quarter that of the surface tension. Ablett’s results for paraffin wax are accurate enough for testing equation 14. He found that err = 11309’

and

e’

= 96’20’

Whence 1/2 (COSerr

+

COS

e’)

= 0.2517

and

eOiElo. = 104035’ Ablett’s 0, as measured, was 104’34’. Ablett wrongly expected 1/2 (e” 48’) to equal 8. The mean value is 104’44’ which differs from the deter6 The difference between interfacial energy (or tension) and friction is as follows: Interfacial energy changes become manifest with destruction and creation of interfaces; the frictional force and energy consumed thereby are associated with displacements of surfaces, not primarily with changes in their extent.

638

IAN WILLIAM WARK

mined value of 8 by more than the experimental error. A better check of the theory could be obtained when using smaller values of 8, for which the difference between mean and cosine mean is greater. VIII. THE STABILITY O F AIR-MINERAL

ATTACHMENTS

There has been some misconception with regard to the influence of hysteresis on the stability of air-mineral aggregates. Though there may be some justification for Sulman’s claim that hysteresis imparts a greater range of stability to a floated particle, there can be none for Edser’s statement that, “NO particle could float stably but for the possibility of variation of the contact angle, for if this were constant, a slight tilt would inevitably cause the particle to sink.” Sulman later (b) states, “But for hysteresis a mineralized bubble could only have a brief existence.” Were these statements correct the measurement of contact angle would lose some of the significance which we have given to it. It would need to be supported by an estimation of the hysteresis. It devolves upon us, therefore, t o explain why it is considered that these statements are incorrect. It has been proved above that when the air-mineral contact is formed there is a decrease in free energy of the system amounting to T (1 - cos e) ergs per of contact. Since (1 cos e) is always positive, there is invariably a decrease of free energy on the sticking of the bubble. This amounts to saying that the process occurs spontaneously. To liberate the air bubble from the surface would require the expenditure of an equal amount of work. This implies that the equilibrium is a stable one with regard to attempts at disruption. While contact between mineral and air is being established, and the air is spreading over the surface of the mineral from a small nuclear point of attachment, frictional forces would tend to retard the spreading. There would thus be a tendency to prevent the true equilibrium angle and area of contact being reached, so that the maximum tenacity of sticking between air and mineral would develop rather slowly. Nevertheless, because of the violent agitation in the flotation boxes, it is probable that the true contact angle would develop within a reasonable time. The initial effect of hysteresis is therefore to prevent maximum attachment between air and mineral, which is the primary step in flotation. The effects of hysteresis on the stability of an attachment already established must also be considered. It is obvious that, as the contact angle can be raised slightly by hysteresis, the stability of contact under a transient stress can be increased-momentarily a t least. Sulman’s claim that hysteresis imparts a greater range of stability is thus justified in this case. An analysis of the effects of all possible types of stress which may be applied to a bubble in contact with a mineral surface leaves one in doubt whether hysteresis ever imparts much greater stability to the attachment.

-

PHYSICAL CHEMISTRY O F FLOTATION. I

639

The bubble and mineral must part company by relative motion in a direction at right angles to the surface and the force of friction, being parallel to the surface, cannot be effective in preventing such motion. On the other hand, friction does oppose easy motion of a bubble over the surface of a solid or of a solid over the surface of a bubble, but this is probably disadvantageous. Some experiments with diphenyl ether CBH50C6H6 confirm these views. The density of this substance is 1.07 and it melts at 28°C. If the melt be allowed to cool on a glass plate, very large smooth crystals form. On the smooth surface there is practically no hysteresis effect, yet nothing known to us floats so readily. It is true that the floatability of large plates of the compound by very small bubbles is due to the very small difference between its density and that of water, but the ability of the aggregate to withstand large stresses is, we suspect, due to the absence of hysteresis and the consequent flexibility which enables adjustment of the partners of the aggregate to meet any external stress.

Extension to moving systems Even when air-mineral attachment is possible, there are certain other conditions to be fulfilled in order that a particle may float. The capillary force of attraction between air and mineral must obviously be greater than their tendency to part. The capillary force of attraction, L, corrected for the hydrostatic pressure difference has been shown to be given by the expression, L

= ~TxT,. sin

c+

e - zx2Tw.

or, since 0 is the supplement of 4,

+

3

The tendency to part may be evaluated from considerations of the dynamics of the motion of the air-mineral aggregate. Let the. tension between mineral and air be E. Let ul, UZ, and a3 be the densities of air, flotation liquor, and solid respectively, and let VI and V z be the volumes of air and mineral in the small aggregate under consideration. Let the acceleration of the system be f upwards. Considering the motions of the bubble and particle separately;

-E

=

YIUI~

- U,g(ua - UJ

=

vz~8.f

Uig(oz

-

UI)

and E

THE JOURNAL OF PHYBICAL CEEMIBTRY, YOL. XXXVII, NO. 5

640

IAN WILLIAM WARK

M hence

and

If UI may be neglected, E = vlgaz, which is equal to the upward thrust due to the buoyancy of the air. Three cases arise: (1) The bubble and mineral both sink iff is negative, that is, if v mf

VZU3

> (VI 4- V Z ) U Z

and L

>E

(2) The bubble and mineral both rise iff is positive, Le., if mu1

+

v2u3




E

(3) The bubble and mineral part company if L < E Yet another contingency may arise. It is possible that disruption of the bubble itself may occur in preference to separation of the air-mineral contact. Let S be the area of this contact. If SIbe the area of the bubble at the level where disruption occurs, the work done against the cohesive forces is 2S1T,,. If

-

2S1T,, < S Twa(l-

COS

e)

the bubble will therefore break in preference to separation of air and mineral. Practically, this means that the bubble should have a reentrant surface and in general is obtained only with bubbles whose contact angle exceeds 90". This explains why only bubbles whose contact angle is is .above 90" leave a residual small bubble if they are forced by increasing size to leave the surface. IX. FROTHS AND AIR-MINERAL

AGGREGATES

Regarding the size of the air bubble as infinite, the principles of the preceding sections are directly applicable to film flotation, but film flotation is unimportant nowadays. A frothing agent is invariably added to promote the formation of a relatively stable froth. A large surface for collecting the minerals is thereby provided. The physical principles underlying the formation of a froth have been clearly defined by Edser (lo), but there is much confusion in the subse-

PHYSICAL CHEMISTRY OF FLOTATION. I

64 1

quent literature of flotation. They will be summarized here in so far as this is necessary for the development of certain deductions. For froth formation it is necessary that the soluble frother should be used in such a concentration that there is a finite (positive or negative) rate of change of surface tension with respect t o concentration. Only when this condition is satisfied is there adsorption of the frother in the air-liquid interface, but the amount of adsorption is dependent on -C.- dTwa where C is the dC concentration, and T,, is the static surface tension. The dynamic surface tension differs less from that of pure water than does the static. Edser demonstrates that any sudden strain applied to the surface will displace the adsorption equilibrium in such a manner that the surface tension is raised; in extreme cases the dynamic value may be reached. The value of the restorative force is therefore dependent upon the difference between the static and dynamic values of the surface tension. Certain observations which do not seem to have been recorded elsewhere, though they must surely be familiar to the operators of flotation plants, may be explained along similar lines. There is a correlation between the size of bubble in and stability of the froth and the concentration of the frother. Starting with very dilute solutions the stablest bubbles are large; as the concentration of the frother is increased, the average size of the relatively stable bubbles becomes smaller, until ultimately (12), no stable froth can be obtained when the solution becomes saturated. I n the most dilute solutions there is probably insufficient adsorption of the frother a t the surface to prevent coalescence between the bubbles or to exert much stabilizing influence on those that are formed; the bubbles are large and ephemeral. With more of the frother present coalescence is inhibited, for with each coalescence there is a decrease in total area of film surface and a consequent additional increase in the concentration of the frother in the new bubble. Relatively stable bubbles form and the size of these does not vary over very wide limits. Some coalescence still occurs, resulting in larger bubbles which collapse on rising to the surface. The reason for the empirical rule of Gaudin, Haynes, and Haas is not clear, but it must be connected with the appearance of a film of oil as a discrete phase.

A i r m i n e r a l aggregates Bartsch (13) has investigated the influence of insoluble oils, of colloidal particles, of gangue and sulfide minerals and of soluble salts on the solubility of the froth produced by a soluble frother. This paper is of outstanding importance with respect to stability of froth systems. It is of an empirical nature, however, and some consideration of the theoretical basis is desirable.

642

IAN WILLIAM WARK

Some of the conclusions of the preceding sections may be applied in the study of air-mineral aggregates. It follows, for instance, from the fundamental equation for contact angle, that the contact angles a t each of the mineral particles armoring a bubble are uninfluenced by the presence of the other particles.6 The shape of the bubble, however, would be determined by the particles. The particles collected by a bubble on its way to the surface slide downwards until they receive lateral support from the more or less continuous film of particles a t the bottom. As the bubble becomes mineralized, each particle becomes more and more nearly surrounded by others, but each particle makes contact with both air and water. When the bubble reaches the froth proper, it meets a shower of particles from collapsed bubbles sliding downwards between the bubbles, and many of these are captured. Those bubbles which become most completely covered are the most stable. If they are not collected when they reach the surface, they, too, collapse. The rate of collapse may, however, be exceedingly slow. The slow rate of collapse is probably due to prevention of rapid draining by the solid particles. At the top of the froth, much of the mineral liberated as the bubbles collapse is supported by those particles which are still securely bound to the froth. Lower down, the bubbles are separated by columns of liquid, some of which are thin, and it is a t the surfaces between the water columns and the air bubbles that the particles ride. Some particles may effect contact with two neighboring bubbles of air, but whether this would increase or decrease the stability of the mineralized froth systems, it is difficult to decide. X. MAXIMUM SIZE OF PARTICLE WHICH WILL FLOAT

Edser (10) shows that a very large particle can be floated by “skin flotation” a t an air-water interface. A disc of large radius is supported almost entirely by the hydrostatic pressure of the water, “the surface tension serving merely to prevent the liquid from flowing over the disc.” Such a disc must be thin, however, and Edser shows how its thickness may be calculated. Gaudin, Groh, and Henderson (14) have attempted to calculate the maximum size of a galena particle floatable by skin flotation. They have, however, neglected the term due to hydrostatic pressure differences. They show that the surface tension forces are large enough to float a cube of galena with an edge length of 2 mm., but they do not demonstrate that sufficient water is displaced to provide sufficient buoyancy, which is of course an essential for flotation. Figure 9 of Edser’s paper suggests that the buoyancy would be insufficient. 6 If water drains away the particle may ultimately be supported by other particles and not by capillary forces; there would be no angle of contact in such a case.

PHYSICAL CHEMISTRY OF FLOTATION. I

643

If applied t o flotation by a single bubble, their method is therefore equivalent to using the expression V,(Q -

ul)g =

2 ~ 2 Tsin , ~ e~

in place of our equation 10.' Whenf = 0, VI(U2

-

01)

=

Vdu3

-

Ud

and from these equations V , can be calculated. However, using the principles of the preceding sections, allowance can be made for the differences in hydrostatic pressure, and thus a closer approximation to maximum floatable size of particle by a submerged bubble may be obtained. It should be remembered, however, that the calculation is valid only for a particle possessing a large flat surface. This surface must be of sufficient extent for the bubble of maximum volume for a given contact angle to fit on the surface without touching the edges of the particle. The maximum volume of air for a given contact angle may then be determined from figure 3, and also the corresponding area of contact. The values apply strictly only for stationary systems, but they would also be applicable to the limiting case of a particle just so big that the bubble could carry it to the surface with an infinitesimally small acceleration. Then if crl may be neglected it follows, since f = 0, that VlUZ

=

VAu3

- uz)

(19)

To take a specific case, let us calculate the size of the largest particle of galena which could be floated for an angle of contact of 90". From figure 3 the maximum value of z is 0.25 cm., i.e., the diameter of the circle of contact with the bubble must be 0.5 cm. and its volume, read from figure 3, is 0.06 ~ m (The . ~ bigger volumes of figure 3 correspond to bubbles with reentrant surfaces, which are of no interest in flotation.) A bubble of this size could, by equation 19, float a particle of galena (density 7.5) of volume 0.06/'(7.5 - 1) cmn3i.e., 0.0092 ~111.~The thickness of this particle would be not greater than 0.005 cm. For an angle of contact of 125O, the maximum value of z is 0.46 and the corresponding volume of air is 0.15 This would float a particle of galena 0.023 ~ m If, . ~however, still keeping t9 a t 125" we again make z = 0.25, V becomes 0.015 ~ mand . the ~ biggest particle which can be floated is but one-quarter the thickness that can be floated for the smaller angle of 90". This is because the bubble corresponding to the higher angle is the flatter. It is apparent, therefore, in the special case of the flotation of This approximation is similar to that formerly in use for the estimation of surface tension by the drop weight method, but now rendered unnecessary by the use of certain tables based upon the equation of Bashforth and Adams.

644

IAN WILLIAM WARK

a single particle by a single bubble that the higher of two possible contact angles may not lead to the best flotation. It might be wondered whether the observed low collecting power of the higher xanthates may be due, in part, to the high angles of contact produced by them, but it must be borne in mind that amyl xanthate, which is a good collector, leads to an angle of contact not far short of the maximum value for xanthates. In practical flotation there are several other factors which influence the maximum floatable size of particle. Firstly, the air-water interface may touch the edges of the particle. The contact angle might then differ slightly from that at a plane surface and in any case it no longer determines the slope of the air-water interface with the vertical. Larger bubbles may therefore be attached for a given contact angle, and very small cubes of galena may be floated because of this factor alone. Secondly, contact with more than one face of the particle may be possible. Thirdly, when the bubble is armored by a large number of small particles, it may, for a given base of contact and a given contact angle with one particle, have a much larger volume than is possible if that particle alone were attached to it. It is impossible to evaluate mathematically the significance of these factors and therefore the maximum floatable size of particle in a flotation machine cannot be exactly evaluated. Gaudin, Groh, and Henderson state that the coarsest galena particles on which reliable flotation was obtained in machines is about 0.4mm. diameter. The writer wishes to express his thanks to Mr. A. B. Cox and Professor T. M. Cherry for their help in the preparation of this paper. REFERENCES (1) WARKAND Cox: Am. Inst. Mining Met. Engrs., Tech. Pub. 461. (2) BASHFORTH AND ADAMS:Capillary Action. Cambridge (1883). Proc. Roy. SOC.London 32, 362 (1881). (3) WORTHINGTON: (4) BARTELLAND OSTERHOF: Colloid Symposium Monograph V, 113 (1928). (5) OSTWALD:Ko1loid:Z. 68, 179 (1932). (6) DE WILLET AL.: J. Am. Chem. SOC.64,444, 455 (1932). (7) SULMAN: Trans. Inst. Mining Met. 29, 44 (1919), and Third Empire Min. and Met. Congress, South Africa, 1930. (8) ABLETT:Phil. Mag. 46, 244 (1932). (9) LANGMUIR: Trans. Faraday SOC. (Pt. 111) 16, 62-74 (1920). (10) EDSER:Brit. Assocn. Repts. on Colloid Chemistry 4, 263 (1922). (11) ADAMAND JESSOP:J. Chem. SOC.1926,1865. (12) GAUDIN,HAYNES,AND HAAS:Flotation Fundamentals, Part 4, p. 19. University of Utah. (13) BARTSCH: Kolloidchem. Beihefte 20, 1 (1924). (14) GAUDIN,GROH,AND HENDERSON: Am. Inst. Mining Met. Engrs., Tech. Pub. 414 (1931).