Dynamic contact angles

JOIJRNAL

OF COLLOID

AND

INTERFACE

SCIENCE

9.3, 389-398

(1967)

Dynamic Contact Angles I. The Effect of Impressed Motion G. E. P. ELLIOTT

I

ANn

A. C. RIDDIFORD

Department of Chemistry, University of Southampton, Southampton, England Received August 19, 1966 An apparatus and procedure is described for growing a bubble of one fluid with constant radial velocity between parallel solid plates, so displacing a second fluid. The process can be reversed, so that both advancing and receding angles can be studied as a function of the interfacial velocity. Results are given for the displacement of satm'ated air by water between siliconed glass plates at 22 ° and 42°C., and between polyethylene plates at 22°C. The displacement of a water-saturated hydrocarbon oil between siliconed glass plates by oilsaturated water, and by an aqueous potassium laurate solution, has also been studied at 22°C. In all systems, a definite dependence of the advancing and receding contact angles upon the interracial velocity is found, except at very low speeds, and a provisionM interpretation is given.

INTRODUCTION Of the many reported studies of contact angles, very few have been concerned with the way in which the angle changes when the three-phase line of contact moves, or is caused to move, over the solid surface. The sparse literature dealing with this aspect has been collected and reviewed elsewhere (i). Yet the matter is one of considerable theoretical and technical importance, and deserves systematic investigation. The study of the effects of forcing the threephase line of contact to move over the solid sm'face at different velocities offers, for example, a means for assessing the point at which control by surface forces gives way to viscous control. If, subsequently, the impressed drive is removed, measurements of the contact angle as the system relaxes back to equilibrium provide, in principle, a means for following changes in the extent of adsorption at one or more of the tba-ee interz Present address: Department of Chemistry, Nottingham and District Technical College, Nottingham, England.

faces a n d for following t h e p e n e t r a t i o n of t h e solid b y one or b o t h fluids, should this t a k e place. I n short, d y n a m i c studies b e a r t h e s a m e r e l a t i o n s h i p to static, e q u i l i b r i u m , m e a s u r e m e n t s as does a n y o t h e r k i n e t i c r a t e s t u d y to t h e c o r r e s p o n d i n g s t a t e of equilibrium. A s such, t h e y p r o v i d e i n f o r m a t i o n w h i c h c a n n o t be o b t a i n e d f r o m s t a t i c measurements. I n t h e p r e s e n t p a p e r , we describe a n a p p a r a t u s w h e r e i n a b u b b l e of one fluid can b e caused to g r o w w i t h a c o n s t a n t r a d i a l

velocity between parallel plates of the test solid, so displacing a second fluid. The process can be reversed, so permitting the study of both advancing and receding angles as a function of the interfacial velocity. This has been done for two different solid surfaces. In Part II, we shall describe relaxation studies made on the same systems after re~ moval of the impressed drive (2). A prelimi~ nary note on this work has appeared elsewhere (3). Part III will describe similar studies made with liquids of large molecular

size (4). 389

390

ELLIOTT

AND

RIDDIFORD

and the cam is driven through an extensive gear train by a 1,500 r.p.m., 0.25 h.p. synchronous motor, capable of giving a constant clockwise and anticlockwise torque. The spring-loading on the piston ensures that the 0= / =~ -~,%:o injected liquid can be withdrawn smoothly from between the test plates on reversal of / the drive. The drop profile is illuminated by an elecFro. I. Test plates and plate-holder (exploded, tronic flash (Braun: Hobby Automatic and to scale). E.F.3) and photographed using a traveling microscope, fitted with a 4.75-inch camera EXPERIMENTAL lens and a plate camera attachment. These Apparatus. The apparatus consists of are mounted on an optical bench, standing three basic units, the test surfaces and their on a slate-topped balance bench. Particular care must be taken to minimize holder, the injection system, and the optical vibration. The synchronous motor is system. The test surfaces, in the form of plates mounted above the bench, independently of 10.8 X 8.2 cm, and typically 1-2 mm. in the gear train and injector, the flexible drive coupling being the only point of union. The thickness, are rigidly separated at 1.5 ram. alignment of the motor with the gears and by spacing pieces and by upper and lower pressure plates (Fig. i). The pressure plates the isolation of moving parts by sponge and spacers were made from ~6 inch brass rubber washers are critical features. Materials. Distilled water was redistilled plate. The whole assembly is easily dismantled and reassembled for cleaning pur- before use, and its purity checked by surface poses. Two threaded rods, brazed into the tension measurements. For the work with two liquids, the water was passed through an upper pressure plate, permit the suspension and leveling of the assembly in the optical alumina column which had been eluted continuously for several days before use. This path. water was stored under Bayol, and its purity For work at room temperature, the aschecked by measurement of its interfacial sembly is surrounded by a glass box, 20 X 20 X I0 cm. deep, fitted with a Perspex lid. tension against the hydrocarbon oil. The hydrocarbon oil, Bayol D, was kindly The box serves as a container for the fluid to provided by the Iraq Petroleum Co. Ltd. It be displaced, whether this is the saturated is a light-fraction paraffin mixture said to vapor of the displacing liquid, or a second contain no branched molecules. It was passed liquid. For studies of liquids displacing vapors at higher temperatures, the box is through an alumina column, removing a replaced by an air bath, provided with yellow band from the oil, giving a liquid of windows for the light path. In this case, the surface tension 27.5 ± 0.25 dyne cm.-1 against air at 20°C., and an interfacial tenair bath also encloses the barrel of the hypodermic syringe used to store and inject the sion of 49.5 ± 0.3 dyne era. -1 against water, stable over a period of at least 24 hr. Before displacing fluid. use, the oil was saturated with water for a The needle of the hypodermic syringe minimum of 5 days. During this period, the passes through a 1 inch diameter hole in the center of the upper pressure plate, and then interfacial tension fell to 47.5 ± 0.5 dyne cm.-1, remaining constant thereafter. This through a small hole (No. 70 drill, 0.028 inch diameter) in the center of the upper test decrease is attributed to mutual saturation plate. The plunger for the syringe is actu- of the two phases. Aqueous solutions of potassium laurate ated by a spring-loaded piston bearing were prepared with a highly purified speciagainst a cam surface. The cam form was men of laurie acid (m.p. 42.9-43.5°C., calculated to give a range of radial growth velocities of known values (see Appendix), M.W. 209) kindly supplied by Prof. N. K.

DYNAMIC CONTACT ANGLES Adam. Fresh solutions were prepared for each run, and were checked by measuring the interracial tension against water-saturated Bayol (mean value 46.0 ± 0.5 dyne era. -1). A purified grade of dimethyldiehlorosilane was subjected to trap-to-trap distillation under vacuum, and then diluted with carbon tetrachloride on the line. At first, the solutions were stored on the line under nitrogen. Subsequently, however, little change was found when the solutions were kept in tightly stoppered glass bottles, and this more convenient procedure was then adopted. Polyethylene sheet was supplied by Prof. N. I4. Adam through the eourtesy of I.C.I. (Plastics Division), Ltd. It had been made by the high-pressure process, and was reported to have 21.3 methyls and 0.73 total unsaturation per 1,000 carbon atoms. No plasticizer had been added, and the use of mold lubricants had been avoided by molding between sheets of Melinex (polyethylene terephthalate) film. All other materials were of analytical reagent grade. Preparation of test plates. We used siliconed glass and polyethylene as the two test solids. The glass plates were boiled in carbon tetrachloride for 5 min., in a chromic acid cleaning mixture for 5 min., and finally in distilled water. At this stage they were completely water-wet and were then oven-dried. The dry plates were immersed in a 2.5 % v / v solution of diehlorodimethylsilane in carbon tetrachloride and allowed to stand for 5 rain. with periodic shaking. They were then removed, rinsed with aqueous acetone, and quickly sprayed with distilled water. Finally, they were oven-dried at ll0°C, for 50 min., and allowed to ecol. Except for cutting to size, and drilling the upper plate, the polyethylene sheet was used as supplied. These, and the glass plates, were handled with forceps at all times. Procedure. For the single-liquid runs, the cleaned external container, containing a little water, was placed in position around the hypodermic, the lid closed, and the ap-

391

paratus allowed to stand for an hour. The measured relative humidity was then 80 % :I: 5 %, and little change was detected thereafter for periods of up to 8 hr. After an hour, the prepared test plates were fixed in the plate-holder, which in turn was attached to the injection system by the suspension rods. The syringe needle entered the hole in the upper test plate, protruding half way between the plates. The container was then reclosed and allowed to stand for a further 30 rain. During this period, the plates were aligned in the light path, if necessary, by adjusting the suspension rods. A small drop was formed between the plates by drawing the earn over the piston to a previously calibrated mark, and observations were made on the drop diameter to ensure that local saturation of the vapor phase was being maintained. The motor drive to the cam was then engaged to give an advancing drop. The radial growth rate of the interface was checked by observing the image on the camera screen. Provided the run was proceeding smoothly, i.e., provided the drop was growing concentrically, these observed growth rates agreed well with those predicted by the cam equation, and photographs of the interface were then taken at known times. The interface was allowed to advance a little further, and then halted. The motor drive was reversed, causing drop recession. A measurement of the rate of recession was made on the camera screen, again as a cheek, and then photographs were taken of the receding interface. By increasing the period between halting the advancing interface and starting the recession, the time of immersion could be made the same (ca. 45 min.) as increasing recession rates were studied. On an average, one run in five was abandoned because of eccentric drop growth. Receding angles were difficult to characterize at speeds greater than 2 ram. rain.-1, since the interface tended to move in jerks. For the study of the two-liquid systems, the container was filled with water-saturated Bayol. During the immersion of the plates, care had to be taken to ensure that no air bubbles were trapped between them. For a reason which will appear later, measure-

392

ELLIOTT

Jl/ .2

.o

AND

RIDDIFORD

l o

•

109

00

e A (d~g)

I0 8

,o,W

/

,o,

,o~

L2

OA(d'g) lOS

~oo

o

]

s

I

I

~o ~s INTERF;,J~IALVELOCITY Imm mln'J'J

20

~O4

FIG. 2. Advancing contact angles as a function

of the velocity of an air/water interface moving over siliconed glass at 22°C. (--O--), siliconed glass at 42°C. (--®--), and polyethylene at 22°C.

,03 ~o2

(--m--). IO1

ments of the radial growth velocity were sometimes based on observations of the foremost part of the liquid/liquid profile. A number of observations were made on the two-liquid systems b y disconnecting the cam from the motor drive, and impressing a hand drive to the cam. This produced rates estimated as up to 100 ram. min. -1. With the geared injection system, the lowest convenient radial growth velocity was 0.5 mm. rain. -I. In an attempt to study lower velocities, observations were made on drops sliding down inclined plates, using a slight modification of Macdougall and Ockrent's method (5). Photographs were taken at the first discernible motion, and usually 15 rain. after formation of the drop. Contact angles were taken from enlargements of the photographs. The tangents were constructed using a tangentometer which could be read to 15' of arc (6). Provided the angle was in the range 5 0 ° 1 3 0 °, no significant loss of accuracy followed from setting the tangent b y eye. RESULTS The results are displayed in graphical form. Thus the advancing angles for the displacement of air by water between siliconed glass plates at 22 ° and at 42°C. are shown as a function of the interfacial velocity in Fig. 2, together with the results for poly-

,oo 0

I

I INTERFACIAL V E L O C I T Y

I

2 Imm rnirTI/

FIG. 3. An expansion of part of Fig. 2 showing

the low velocity regions at 22°C. for a siliconed glass surface (--0--) and polyethylene (--[E--). ethylene plates at 22°C. Figure 3 is an expanded form, showing the low-velocity region for siliconed glass and polyethylene at 22°C. The corresponding low-velocity regions for the receding angles are shown in Fig. 4, the results at higher recession velocities being too erratic to merit reporting (see later). Figure 5 shows the dependence of 0~ on the interfacial velocity for the displacement of water-saturated Bayol b y Bayolsaturated water, and b y 1.14 X 10-4 M aqueous potassium laurate, between siliconed glass plates at 22°C. Each point represents the mean of at least five, and more usually ten, observations. In the case of the displacement of air b y water, these mean values are assessed as being reliable to 2=1 ° , whereas for the two-liquid systems the corresponding assessment is 2=1.5 ° for 0 < 170 °, and 2=3° for 0 > 170 °. The case of nonwetting (8~ = 180 ° , see Fig. 5) was characterized by a visible layer of the hydrocarbon oil remaining on the solid surface, and so was precisely defined.

393

DYNAMIC CONTACT ANGLES

i

]

~

\

2 4 ) are those obtained from the sliding drop studies. Macdougall and Ockrent (5) found that 0x reached a maximum, constant, value before sliding ensued. It seems reasonable, therefore, to regard the sliding drop values as corresponding to zero interracial velocity. DISCUSSION

°R @'g)

~k

o

~

tNTERFACIAL VELOCITY I mm mln-I~

FIG. 4. Receding contact angles at 22°C. as a function of the velocity of an air/water interface

moving over silieoned glass ( - - Q - - ) and polyethylene (--V]--).

[75 170 ~ < 0A(,g)

,6s

Isc

!

Iss; Isc s]

I 1o

1 15

[NTENFACIAL VELOCITY (rammlgI)

Fro. 5. Advancing

contact angles at 22°C. as a

function of the interracial velocity for :Bayolsaturated water ( - - Q - - ) , and 1.14 ;4 10.4 M aqueous potassium laurate (--I2]--), displacing watersaturated Bayol between silieoned glass plates.

The interracial velocities are judged to be reliable to within ~=0.05 mm. rain. -I over the range 0-3.0 ram. rain. -1, ±0.1 ram. rain. -~ over the range 3.0-6.0 mm. rain. -1, and -4-0.5 mm. min. -1 for speeds higher than this. The values shown at zero velocity (Figs.

Advancing angles. There can be no doubt that 0~ is a function of the interracial velocity in the systems studied. The results are reasonably reproducible, particularly when one bears in mind the comparatively large area of solid surface used, and that a fresh pair of test plates were used for each speed. This suggests that the velocity effect cannot be attributed to chance contamination. For the single liquid systems, 0x appears to be sensibly independent of the interracial velocity in the range 0-1 mm. rain. -1. Above 1 ram. rain. -1, 0~ at first increases linearly with the velocity, but at higher speeds the rate of change diminishes until finally a limiting value is reached. For the two-liquid systems, the behavior is similar, except that there does not appear to be a region at low velocities in which 0x is independent of the growth rate. From Fig. 5, it will be seen that the upper limiting value for 0~ is 180 ° in the case of the two-liquid systems, and it must be pointed out that it is difficult to correlate an advancing angle of 180 ° with any particular velocity. The rate of advance of the foremost part of the liquid/liquid interface does not truly reflect the relative motion between the advancing liquid and the solid surface when the line of thi'ee-phase contact is stationary. The velocity effect found in these systems bears a striking resemblance to the dependence found by Ablett (7) for water displacing air over a paraffin wax surface in a study which has received far less attention than it merits. Again, the absence of a velocity effect below about 1 mm. rain. -~ for the singleliquid systems supports X/arnold and Mason's view that there is little or no change in 0~ at low interfacial velocities (8). Both these aspects of the relationship between 0x and the interracial velocity demand

394

ELLIOTT AND I~IDDIFORD

a kinetic interpretation, and we have suggested (3) that Hansen and Miotto's views (9) should form a starting point. On this approach, v, the interracial velocity, is to be compared with v~, the "natural" displacement velocity, where vn is given by the ratio of the peripheral thickness 1 to r, the relaxation time of the most slowly relaxing molecule at the periphery. For the single-liquid systems, this leads to the view that, if the air/water interface is caused to move sufficiently slowly over the solid surface, the water molecules in the periphery have ample time to orient themselves against the solid surface, such that no displacement from the equilibrium state is observable; accordingly, 0x is independent of v. As v is increased, however, a point is reached at which it is no longer negligible in comparison with v.. If one supposes, and reasonably, that at this stage the tension of water against its saturated vapor has still its equilibrium value, this must lead to an increase in VsL, the specific free surface energy of the solid/water interface, and hence to an increase in the observed contact

angle. Experimentally, this increase in 0x is observed when the interracial velocity exceeds ca. 1 ram. min. -~ (Fig. 3), so we m a y take this value as an underestimate for v~. Then, with I of molecular dimensions, r is less than 10.5 see., as compared with r0 = 10-I° see., the relaxation time for bulk water. Since r and r0 must be related through the equation r = r0 exp ( E / R T ) , E, an estimate of the energy barrier for the adsorption of water molecules on the solid surface, must be less than 7 kcal. mole-l; this is not um'easonable. This view is, however, predicated on the assumption that, during advance, water does not penetrate the solid surface--a point which is discussed in the following, and with respect to relaxation phenomena (2, 3). As the interracial velocity is increased beyond 1 mm. min. -I, the degree of disorientation of the water molecules against the solid surface, immediately behind the advancing interface, must increase; and it is tempting to suppose that the upper limiting

values found for 0A correspond to the situation where the interface is advancing so rapidly that the disorientation is complete. Since this occurs at ca. 8 mm. min. -1 for siliconed glass, and at 12 mm. min. -1 for polyethylene (Fig. 2), one would then take these as overestimates for v~, i.e., r > 10-6 see., with E > 5 keal. mole -~. Again, this appears not unreasonable. Since the silicone layer presents a dosepacked methyl surface (10) and behaves essentially as a paraffinie surface (11), it is not surprising that the advancing angles for water are generally higher than those on polyethylene, save at the highest velocities. The specific free surface energy of polyethylene is certainly greater than that of a paraffinic surface (12), reflecting the exposure of methylene groups; and one might have expected the low-velocity region, where 0A is constant, would have extended to higher velocities than in the ease of silieoned glass, which is not observed (Fig. 3). One would also expect that the upper limiting value for 0x would be reached at somewhat higher speeds, which is found (Fig. 2). The possibility that penetration of water into the polyethylene surface occurs at low interfacial velocities (2) may, however, account for the results shown in Fig. 3. Penetration certainly occurs during static measurements (13). Interpretation of the results shown in Fig. 5 for the displacement of Bayol by water is more difficult. For one thing, it is not certain that the advancing water phase will scour all the hydrocarbon oil from the solid surface when 0x < 180°; it may well leave an adsorbed layer behind. Nor is it certain that the velocity effect is attributable to the slow relaxation of water molecules on the solid surface behind the advancing front; it could be due to slow relaxation of water and/or hydrocarbon molecules in the liquid/liquid interface, or to effects at both these interfaces. On the other hand, the apparent absence of a region at low velocities where 0x is sensibly constant must mean that the natural displacement velocity is smaller in this ease. I t would need to be smaller than the value found for the single-liquid systems only by a factor of two for the constant

DYNAMIC CONTACT ANGLES region to be unobservable with our apparatus. Relaxation studies (2) indicate that 0~ does, in fact, become constant at sufficiently low velocities. Receding angles. Because of the possibility of penetration of the solid surfaees by water, receding angles were determined after a constant immersion time of 45 rain. The results for two systems at interfacial velocities ranging between 0 and 2 mm. min. -1 are shown in Fig. 4. Comparing these results with the advaneing values (Fig. 3), it is apparent that hysteresis is present in both systems, whether one compares the constant, low-velocity, values for 0A and OR or the values obtained by extrapolating the range where the contact angles vary linearly with velocity back to zero speed. T h a t the receding angles are more closely similar for these two surfaces than are the advancing angles suggests t h a t penetration of water into these surfaces is an important factor governing hysteresis in these particular systems. Irregular penetration is thought to be the reason for the difficulty we experienced in getting sufficiently reproducible results for interfaeial velocities above 2 mm. min.-L Very few of the bubbles withdrew concentrically, but although the results are too erratic to warrant reporting in detail, they do suggest that a lower limiting value for OR is reached at interfaeial velocities higher than ca. 10 mm. rain. -~, again tending to confirm Ablett's observations (7). We have suggested (3) that the absence of hysteresis in this system, and his ability to study receding angles at speeds above 2 mm. rain. -1, could be explained if complete penetration had occurred on his paraffin wax surface; but his results would equally well support the view that no penetration had taken place, and this is perhaps more likely when one considers t h a t his zero velocity value was 104.7 ° . At low velocities, a region in which OR is constant is to be expected, and the onset of the velocity effect is now to be interpreted in terms of a displacement of 3'sv, the specific free surface energy of the solid/vapor interface, further away from its equilibrium value. As the interface recedes, a disoriented layer of water molecules is left behind, the

395

degree of disorientation being a function of the interfacial velocity. If the solid is impermeable, and if there is no equilibrium film pressure, these molecules must desorb into the vapor phase; and desorption must still occur even if there is a finite film pressure at equilibrium, but in this case it will be the excess molecules which leave the surface. On the other hand, if the solid is permeable, they must re-orient themselves with respect to the water molecules in the underlying surface and with respect to the vapor phase. The difficulties connected with the study of receding angles are particularly acute in the ease of the two-liquid systems. In very few cases was it found possible to cause the water drop to withdraw concentrically. These served to show that OR decreases from 114 =t= 1.5 ° at 0.5 mm. min. -~ to 101.8 -41.5 ° at 2.6 mm. rain. -~, i.e., there is again a velocity effect. When the growth or withdrawal was eccentric, both for the singleand two-liquid systems, it was usually noted that, at some point on the periphery, the contact angle against the upper test surface differed from that against the lower. As has been stated, the receding ease was always more prone to this difficulty than the advancing situation. These difficulties are well illustrated by the following observations. A water drop was grown in Bayol at a sufficiently high velocity to give 0x = 180 °. When the drive was stopped and then reversed, the interface began to recede initially with 0R = 180 ° (Fig. 6A). After a time, the interface caught on to one of the surfaces, usually the lower, and the angle began to fall rapidly. Attempts to measure ORat this stage gave values in the range 158°-168 ° (Fig. 6B). When the interface caught in this way, no oil layer could be seen between the water profile and the bottom plate, although it was still discernible along the upper plate, on which OR was still 180 °. Finally, the interface also caught on the upper plate (Fig. 6C) and then, usually by a series of jerks, adoped the form shown in Fig. 6D. This was followed by inversion, giving OR < 90 °, as sketched in Fig. 6E. If the cam was now stopped, the interface would rapidly assume a configuration for which the contact angle was greater

396

ELLIOTT AND RIDDIFORD

__C OR= "E,

c

0 R = 160 ° or 180 ° D

0 R:

IBO °

01~- 1 0 0 ° or ISO ° E

l od

0 R= 8o

°

FIG. 6. Receding meniscus of water in siliconed glass/Bayol/water system, showing spasmodic movement and inversion.

than 90 °, i.e., it would invert again. We estimated that, during recession, when the interface jerked back suddenly, the interfacial velocity exceeeded 20 ram. rain. -1. Moving the cam rapidly by hand gave recession rates of the order 100 mm. rain. -1, and values for 0R as low as 70 °. Effect of a surface-active agent. In order to see whether similar effects could be observed in the presence of a surfactant, the advancing angles for the displacement of watersaturated Bayol by aqueous 1.14 X 10-4 M potassium laurate solution were measured and are compared with the pure two-liquid system in Fig. 5. I t will be seen that 0~ is reduced by 10 ° at the lowest velocities, and that the trend toward the upper limiting value of 180 ° is slower, being reached at ca. 8 mm. min. -~. A few measurements indicated that the effect of the surfaetant was to reduce 0R, at low velocity, by 14 °, i.e., hysteresis was increased. The velocity effect is still present, and, since the concentration of surfaetant was chosen such that the tension of the newly formed liquid/liquid interface reaches its equilibrium value quickly (14), the effect is to be traced to the solid/solution interface. Further discussion of these preliminary observations would appear to be unwar-

ranted at this stage. I t should be noted, however, that we found the interfacial tension of the aqueous solution against water-saturated Bayol consistently fell from its original value of 46 to a final value of 45 dyne era. -1. In view of the absence of a corresponding fall with the pure liquids, this is ascribed to some transfer of potassium laurate between the mutually saturated phases. Effect of temperature. The meager data concerning the temperature coefficient of contact angles have been reviewed elsewhere (1), and reference to more recent studies has been made in other communications (11, 15). These studies were concerned with static measurements on single-liquid systems and, surprisingly, most of them suggest that the temperature coefficient is zero, within the limits of experimental error. This means that the temperature coefficient of the difference ~/sv -- VsL must be proportional 2 to the temperature coefficient of the liquid surface tension. I t has been suggested (15) that this arises when there is no equilibrium film pressure at the solid/vapor interface, and it has been argued that the system siliconed glass/water/air is a partieular example (11). No dynamic study appears to have been reported, and the results shown in Fig. 2 are therefore of interest. There is no difference at any interfacial velocity, within the experimental scatter, when the temperature is raised through 20 ° . This implies not only the absence of a film pressure but also a very weak interaction energy between water and the unpenetrated silicone surface, as discussed. Other points. It is clear that our discussion has been based upon the further assumption that the influence of gravity may be neglected. We chose a gap spacing of 1.5 ram. in the light of Mack's demonstration (16) that the effect of gravity on droplets of the order 0.8 ram. in height is small. This seems reasonable inasmuch as there was no detectable difference in contact angle against the upper and lower plates for fully satisfactory runs. Where a difference appeared, as in Fig. 6, this could be attributed to dif2 And not equal, view (1).

as wrongly

stated

in our re-

DYNAMIC CONTACT ANGLES ferences in the state of the two experimental sm'faces. On the whole, siliconed glass forms a very satisfactory test surface. It exhibits excellent specular reflection and good reproducibility, and has been used by us in a number of other studies (2, 4, i0, Ii, 15).

397 B

(i

~-

ACKNOWLEDGMENT

2r

P

)

t

FIG. 7. The trapped bubble

This work forms part of a study financed by the Iraq Petroleum Co., Ltd. REFERENCES i. ELLIOTT, G. E. P., AND RIDDIFO~D, A. C., Recent Progr. Surface Sci. 2,111 (1964). 2. ELLIOTT, G. E. P., AND RIDDIFORD, A. C.,

To be published. 3. ELLIOTT, G. E. P., AND RIDDIFORD, Nature 195, 795 (1962). 4. PHILLIPS, M. C., AND RIDDIFORD,

L

A. C.,

A. C., To

be published.

.'J'-;:Y=:q ,.... A

5. i'ViAcDOUGALL, G., AND OCKRENT, C.~ Proc. 6.

7. 8. 9. 10. Ii. 12. 13. 14.

Roy. Soe. (London) A180, 151 (1942). LIVINGSTON, R., "Techniques of Organic Chemistry," Vol. 8, p. 182. Interscience, New York, 1953. ABL~TT, R., Phil. Mag. 46, 244 (1923). YARNOLD, G. D., AND MASON, B. J., Proc. Phys. Soe. (London) B62, 125 (1949). HANSEN, R. S., AND MIOTTO, IVl., J. Am. Chem. Soc. 79, 1765 (1957). PHILLIPS, M. C., AND RIDDIFORD, A. C., Proc. 4th Intern. Congr. Surface Activity, in press. PHILLIPS, M. C., AND RIDDIFOI%D, A. C., J. Colloid Interracial Sci. 22, 149 (1966). ZISMAN,W. A., Advan. Chem. Set. 43, 1 (1964). ADAM,N. K., ANDELLIOTT, G. E. P., J. Chem. Soc. 1962, 2206. WA~D, A. F. H., in "Surface Chemistry Supplement to Research," p. 56. Butterworths, London, 1940.

15. PHILLIPS, M . C., AND RIDDIFOI%D, A. C. N a -

205, 1005 (1965). 16. MACK, G. L., J. Phys Chem. 40, 159 (1936). ture

APPENDIX

4

) ......

FI~. 8. The cam and hypodermle driving a piston of length l into a cylinder of cross-sectional area A (Fig. 8). We require to find a function f such t h a t ÷ = eonst, when ~ = const., the dots denoting differentiation with respect to time. The volume of the bubble V m a y be expressed b y V = Vo + A ( R -

Re),

[A1]

[A2]

where V0 is the initial volume of the bubble when R = R0 and ¢ = 0. Alternatively, from Fig. 7, V = ~rr2h + 2~rar,

Consider a bubble, of contact radius r, growing between parallel plates of separation h (Fig. 7). The bubble is to be caused to grow with a constant contact radial velocity b y injection of fluid at the mid-point B. The injection system is to be formed b y a cam of equation R = f(~)

(

[A3]

where 2a is the total area of the bubble profile diminished b y 2rh. I t is to be expected t h a t a will v a r y with ~ b u t will be constant for a given ~. I n order of magnitude, it is given b y a N

7rh~

[A41

398

ELLIOTT AND RIDDIFORD

From Eq. [A3],

(a ~

v

r = - h + _~ + ~/

Intega'ating Eq. [A9] between R and R0 (~ and 0) gives the required function

yI.

[A5]

¢~

R = Ro +

and from Eqs. [A2] and [A3], =- A!~ = 2~rrh÷ + 2~ra~;

[A6]

2~

Voh)"2"

[A10]

Now dR

Introducing Eqs. [A5] and [A7] into [A6] gives 27r. ÷ ( _~)1/~ d-~ = A - ~ a ' + .

For computing the cam, we chose to work with ~ in degrees, ~ in revolutions per minute, ~ in rain. see.-1, R, R0 and h in millimeters, a and A in square millimeters, and Vo in cubic millimeters. Then Eq. [A10] becomes

dR

[AS]

Substituting for V from Eq. [A2], we obtain dR d~,

2~r A

~{ Vo h (o a: + --~r + AhR -

-

A h Ro)

[A9]

"~ .

R = Ro + 3-6-x

2

[All]

We t o o k R - Ro = 50ram. f o r ~ = 55 °, h = 1.55 ram., A = 7r6.52/4, and ro 8 mm., whence, from Eq. [A4], ~/~bN 1.