A reexamination of the measurement of dynamic surface tensions using the maximum bubble pressure method

A Reexamination of the Measurement of Dynamic Surface Tensions Using the Maximum Bubble Pressure Method P E T E R R. G A R R E T T AND D A V I D R. W A R D Unilever Research Port Sunlight Laboratory, Port Sunlight, Wirral, Merseryside, United Kingdom Received September 14, 1988; accepted December 5, 1988 The m a x i m u m bubble pressure method for measuring dynamic surface tension is reexamined, using high speed cinematography, particularly with respect to the process of bubble formation and the determination of the true surface age. A method for direct measurement of the time of bubble growth is suggested which readily permits routine determination of surface age. Sources of error are discussed and the evaluation of the method using submicellar surfactant solutions is described. © 1989UnileverResearch 1. I N T R O D U C T I O N

The use of the m a x i m u m bubble pressure method for dynamic surface tensions of surface ages < 1 s has been described by Austin et al. (1) and by Kloubek (2, 3). In this method the pressure difference across an extremely fine capillary is increased up to the m a x i m u m Laplace pressure whereupon bubble formation commences. The pressure difference is then more or less maintained at that value throughout the experiment. Unfortunately the details of the process of bubble formation are imperfectly understood. This means that estimation of true surface age using this method is uncertain. We have therefore used high speed cinematography to examine the process in order to improve understanding and facilitate more meaningful interpretation of surface age. Austin et al. ( 1 ) divide the process of bubble formation into two periods. The first is the period immediately after release of a bubble. It is supposed that the new meniscus then formed at the capillary is free of any adsorbed material and therefore has zero surface age. Any surfactant present then adsorbs under quiescent conditions causing a reduction of surface tension. This first period is ended when a bubble begins to rapidly grow from the meniscus. The process of bubble growth then

represents the second period or "dead time" tD. The true surface age is then supposed to be the time interval between consecutive bubbles minus the dead time to. Here we describe a method for direct measurement of tD SO that the apparent surface age calculated from frequency of bubble formation can be readily corrected. 2. EXPERIMENTAL 2.1. APPARATUS

The apparatus is depicted schematically in Fig. 1. A central feature is a glass thermostated cell containing an upwardly orientated precision bore capillary. Partial v a c u u m is applied to the space above the solution in the cell using an aspirator controlled by a precision micrometer valve. Bubble frequency m a y be controlled by simply adjusting the aspirator water outlet rate. The pressure difference is measured using a water m a n o m e t e r read with a cathetometer. As described by Kloubek (2), that difference remains essentially constant if the bubble frequency remains constant. The small amplitude oscillations which accompany bubble formation can be neglected. This is in part a consequence of the relatively small volumes of the bubbles formed from the extremely fine capillaries used here (~<0.01-cm radius).

475

Journal of Colloidand InterfaceScience, Vol. 132,No. 2, October 15, 1989

0021-9797/89 $3.00 Copyright© 1989UnileverResearch All fightsof reproductionin any formreserved. Forpermissionto reproduce,otherthanthrough the CopyrightClearanceCenter,contactAcademic Press,Inc.

476

GARRETT

to and

AND WARD

ELECTRONIC COUNTER

ASPIRATOR MANOMETER T J=

I He/NeLASER I

l 7

PHOTOCELL ~~~TRANSIENT RECORDER OSCILLOSCOPEb/

FIG. 1. S c h e m a t i c d i a g r a m o f m a x i m u m b u b b l e pressure apparatus.

In practice it is difficult to achieve regular bubble flows at frequencies ~<0.5 s -t. This then sets an upper limit to the surfaces ages which can be investigated with this constant pressure technique. A laser beam (2 m W H e / N e type 145 from Spectra Physics Inc.) is positioned so that it passes just above the capillary orifice and illuminates a photocell (from Ealing Beck Limited). Formation of a bubble interrupts the beam which causes a corresponding change in the photocell signal. The resulting signal pulses are counted by an electronic counter (Type FM610 Feedback Instruments Limited). They are also recorded by a transient recorder (Datalab DL902) which enables them to be displayed on an oscilloscope or chart recorder. This facility is particularly important for diagnosis of faults. Irregular bubble flow (caused, for example, by a wettability problem) and bubble coalescence at the capillary tip are both clearly apparent from the recorded traces but are not apparent from the counter readings. It is also possible to use the traces to calculate the time of bubble growth from hemispherical surface to separation. This time, as we discuss Journal of Colloid and Interface Science, Vol. 132, No. 2, October 15, 1989

below, is in certain circumstances of importance in calculation of the true surface age. The surface tension 3" is calculated from the relationship

3'

= ( A P - pgh _ ~ /ro,

[ll

where AP is the pressure difference measured with the manometer, h is the hydrostatic head at the capillary tip, p is the density of the solution, and r0 is the radius of the capillary, h is measured using a cathetometer. During the experiment there may be a tendency for foam to build up in the cell. This has to be removed before it enters the vacuum outlet. The hydrostatic head must then be remeasured. As a check on the validity of the pressure measurements the radius of a capillary determined directly by photomicrography was compared with that calculated from Eq. [ 1] using pressure measurements with distilled water. The respective values were 0.98 + 0.01 × 10 -2 and 0.99 X 10 -2 cm. For high speed cinematography a HYSPEED camera (from J. Hadland Photographic

M A X I M U M BUBBLE PRESSURE M E T H O D

Instruments Limited) was used at about 5000 frames s -1 with a macrolens. The film type was high speed video news, tungsten 7250 (Eastman Ektachrome ). All experiments were done with the cell thermostated at 25 °C. Equilibrium surface tensions were measured by the Wilhelmy plate method. 2.2. MATERIALS

Sodium dodecyl sulfate (SDS) was purified from "Analar" grade by HPLC and recrystallized from propanol/water mixtures. This sample did not exhibit a minimum in a plot of surface tension against log (concentration). Sodium fatty acid methyl ester sulfonate ( C 12 H25" CHSO3 N a . C O 2 C H 3 ) was of purity 93 -----5 wt% (by NMR). Major impurities were water and sodium methyl sulfate and no minimum in a plot of surface tension against log(concentration) was found. Dodecyl dimethylamino propane sulfonate ( C12 H25 ( CH3 )2N" (CH2)3803 ) was recrystallized from an acetone/propane-2-ol mixture and found to have a purity of 98% by chromatography. Sodium metaborate was BDH laboratory grade (of nominal specification 98%). Water was distilled from alkaline potassium permanganate and then redistilled. 3. RESULTS A N D DISCUSSION

3.1. BUBBLE FORMATION

3.1.1. Time of Bubble Growth ("Dead Time") For a detailed investigation of the maxim u m bubble pressure method we have selected distilled water and a submicellar SDS solution. In Figs. 2 and 3 we present stills from a high speed cinematographic film illustrating the process of bubble formation as a function of time in distilled water and 7 × 10 -3 M S D S solution, respectively. The interval between successive bubble separation events may be seen to be divisible into two parts. The first part, where the system is quiescent, concerns

477

the presence of a hemispherical meniscus near the top of the capillary. The second part concerns the rapid growth of a bubble. As we have described, Austin et al. ( 1 ) have suggested that the true surface age is the reciprocal of the bubble frequency minus the time for bubble growth which is the so-called dead time. This implies that the air-water surface formed at bubble separation is essentially denuded of surfactant. The dead time tD was estimated from high speed films or directly from the bubble event traces made by the interrupted laser beam (the method of calculation of tD by this method is shown in Fig. 4). Comparison of tD obtained by both methods as a function of bubble frequency is made in Tables I and II for distilled water and 7 × 10-3 M SDS solution, respectively. Aggreement is seen to be reasonable, confirming the suitability of the laser method for measurement of tD. Also shown in Tables I and II are the surface tensions calculated from hydrostatic head and manometer readings. It is clear from these results that the calculated surface tension of distilled water is not a function of bubble frequency within experimental error up to a frequency of 37 s 1. This represents some evidence that the measured pressure is in fact determined by capillarity rather than any other factor (such as viscous and inertial forces) even up to relatively high frequencies. In this we are in agreement with Kloubek (3). In Fig. 5 we plot tD/tB as a function of bubble frequency, where tB is the total time interval between bubbles. The results are for both distilled water and 7 × 10 -3 M SDS solution with a nominal 0.01-em-radius capillary at 25°C. In the same plot we present tD/ta calculated using the empirical expression for tD obtained by Ausin et al. ( 1 ) with a 0.005-cmradius capillary tD = 31.9 -- 0.252f,

[2]

where f ( = 1/ta) is the bubble frequency. If we are to accept the validity of correction of surface age for tD then it is clear from Fig. 5 that the correction is significant at bubble Journal of Colloid and Interface Science, Vol. 132,No. 2, October 15, 1989

478

GARRETT

a

AND WARD

b

e

d

e

f

FIG. 2. High speed movie stills (at 5000 s -1 ) of bubble growth (distilled water, 25 °C, bubble frequency 14.4 s - l ) . (a) t = 0, bubble separation; (b) t = 8.6 ms; (c) t = 20.8 ms; (d) t = 51.5 ms, end of quiescent period and bubble starts to expand, start of dead time; (e) t - 57.6 ms, bubble growth; (f) t = 69.4 ms, nascent bubble separation. External diameter of the capillary tube is 0.49 cm.

frequencies in excess of about 1 s -1. Indeed extrapolation of the curves shown in Fig. 5 would indicate that at a sufficiently high bubble frequency fc we have tB = tD whereupon a hemisphere does not form at the capillary Journal of Colloid and Interface Science, Vol. 132, No. 2, October 15, 1989

and the nature of bubble formation changes. We have in fact observed irregular bubble flow and coalescence of growing bubbles with separated bubbles at very high frequencies. Stills illustrating this phenomenon are shown in Fig.

MAXIMUM BUBBLE PRESSURE METHOD

a

b

e

ti

e

t

479

FIG. 3. High speed movie stills (at 5000 s -l ) of bubble growth (7 x 10 -s MSDS, 25°C, bubble frequency 15.3 s-l). (a) t = 0, bubble separation; (b) t = 7.5 ms; (c) t = 19.6 ms; (d) t = 41.4 ms, end of quiescent period and bubble starts to expand, start of dead time; (e) t = 65.0 ms, nascent bubble separation; (f) t = 65.4 ms, bubble separation and end of dead time. External diameter of the capillary tube is 0.49 cm.

6. K l o u b e k ( 3 ) states t h a t in t h e s e c i r c u m s t a n c e s t h e r e is a s l o w r e d u c t i o n i n f r e q u e n c y a n d a m a r k e d rise in gas v o l u m e t h r o u g h p u t . T h e p r e s s u r e in t h e c a p i l l a r y is t h e n n o l o n g e r related to the surface tension of the liquid.

O n s e t o f this p h e n o m e n o n t h e r e f o r e sets t h e m i n i m u m accessible s u r f a c e age w i t h this m e t h o d at a b o u t 20 m s (i.e., 20 X 10 -3 s). I n Fig. 7 w e p r e s e n t d e a d t i m e as a f u n c t i o n o f f r e q u e n c y for capillaries o f d i f f e r e n t r a d i u s Journal of Colloid and Interface Science, Vol. 132, No. 2, October 15, 1989

480

GARRETT AND WARD

g

bubble growth commences

separation photocell signal

i

~k

time

FIG. 4. Calculation of dead time from transient recorder traces.

but similar lengths. It is clear that tD is a strong function of capillary radius, which fact probably in part explains the difference between Austin et al. ( 1 ) and the present work. Equation [2] is, however, seen to be independent of surface tension. Indeed Austin et al. ( 1 ) do not even describe the solution which was used for their measurements upon which Eq. [2] was based. That to is a function of the measured dynamic surface tension is, however, implied by our results shown in Fig. 5 and in Tables I and II. In this then we disagree with Austin et al. ( 1 ).

3.1.2. Process o f B u b b l e Growth In order to investigate the nature of bubble growth and separation we made measurements

of bubble volumes from high speed movie stills using a film motion analyzer. The diameter of the bubble was determined at frequent intervals and the volume V calculated, assuming axial symmetry, using the relationship n

V = ~ 7r[(y i - Y i _ l ) ( x i -t- xi_1)21/4,

[31

i=1

where xi is the bubble diameter and Yi is the axial coordinate. We present in Fig. 8 a plot illustrating the increase of bubble volume with time during the dead time period for 7 × 10-3 M S D S with a bubble frequency of 15.3 s -1. A linear plot means that over most of this period the rate of bubble growth is constant, which implies that the overall pressure drop is also essentially

TABLE I Use of Various Methods for Measuring Frequency and Dead Time for Distilled Water Frequency (s-j)

Dead time (ms)

Counter

Transient recorder

Cinematography

Transient recorder

Cinematography

Surface tension (rnN ra i)

2.2 7.5 14.5 18.8 28.0 36.8

2.7 7.1 14.9 18.7 27.6 37.0

-7.3 14.4 18.5 27.8 37.1

20 19 19 19 18 19

-19 19 18 17 19

72.25 72.7 71.9 73.1 72.6 73.1

Journal of Colloid and Interface Science, Vol. 132, No. 2, October 15, 1989

481

MAXIMUM BUBBLE PRESSURE METHOD TABLE II Use of Various Methods for Measuring Frequency and Dead Time for 7 × Frequency (s-1) Counter

Transient recorder

1.3

M SDS Solution

Dead time (ms) Cinematography

Transient recorder

Cinematography

Surface tension (mN m -I)

--5.8 7.4 9.3 -15.3 19.2

26 25 25 25 24 26 24 24 24 24 17

29 -28 26 28 -24 29 -25 18

44.5 45.5 45.5 46.8 46.0 46.8 48.3 48.1 49.3 50.4 52.7

--

4.5 5.9 7.5 9.2 12.4 16.3 19.0 23.1 27.4 46.5

10 -3

4.0 6.2 7.9 8.6 12.5 16.1 19.1 22.8 27.3 46.2

i

27.3 45.8

c o n s t a n t . T h e g e o m e t r y o f t h e b u b b l e is, h o w ever, c o n t i n u o u s l y c h a n g i n g d u r i n g b u b b l e g r o w t h , w h i c h m e a n s t h a t t h e c a p i l l a r y pressure is also c h a n g i n g . T h a t s u c h c h a n g e s in capillary pressure do not influence the overall

p r e s s u r e d r o p i m p l i e s t h a t t h e c a p i l l a r y press u r e in t h e b u b b l e is s m a l l c o m p a r e d to t h e overall pressure d r o p . T h a t inertial a n d viscous forces are r e l a t i v e l y u n i m p o r t a n t in d e t e r m i n ing t h e s h a p e o f t h e b u b b l e c a n b e s h o w n i f

tD/t B

0.7 f

///

//x/

o

0.4--

///

/o/

////I/// /////

0.20" 3

0.1

// /

I

0

10

20 bubble f r e q u e n c y / s

J

I

30

40

-1

FIG. 5. tD/tB as function of bubble frequency. (©) This work/recorder trace, 7 × 10 -3 M SDS, 0.01-cm capillary, 25°C; ( e ) This work/recorder trace, distilled water, 0.01-cm capillary, 25°C; (- - -) empirical equation of Austin et al. ( 1), 0.005-cm capillary. Journal of Colloid and Interface Science, Vol. 132, No. 2, October 15, 1989

482

GARRETT

a

AND

WARD

b

¢

tl

FIG. 6. High speed movie stills of bubble coalescence at extremely high gas flow rates (distilled water, 25°C). (a) t = 0; (b) t = 0.8 ms; (c) t = 1.2 ms; (d) t = 1.4 ms. External diameter of capillary tube is 0.49 em.

we attempt to compare experimental bubble profiles with those calculated assuming a balance of capillary and hydrostatic forces. In these circumstances the bubble shape should be described by the well-known "pendant drop" equation (4, 5)

1/P/b +

sin x/b

- 2 + {3y/b,

[4]

where P is the radius of curvature at the point (x, y) and q~ is the slope of the point (x, y). fl is given by -gpb2/'V, where O is the density of the liquid and b is the radius of curvature at the point of intersection of the axial coordinate with the surface of the bubble. In Fig. 9 we compare the shape of a bubble 1 ms before detachment in 7 × 10-3 M SDS solution Journal of ColloM and Interface Science, Vol. 132, No. 2, October 15, 1989

(at f = 15.3 s - I ) , with that calculated using Eq. [4]. We note that solution of this equation can only be obtained using numerical methods which are described in the Appendix. Agreem e n t between theory and experiment is seen to be reasonable. In addition to this evidence we note also that Fainerman (6) has made an order of magnitude estimate of the significance of the viscous resistance of the water to growth of the bubble and finds that it can be neglected for small capillaries ( o f the size we are considering here, i.e., <0.1 c m ) . If then the effect of inertial and viscous forces in the bubble is relatively small then we must follow Kloubek (3) and Fainerman (6) and attribute the pressure difference during the dead time period of bubble growth to viscous resistance as air

483

MAXIMUM BUBBLE PRESSURE METHOD capillary

radius

3.1.3. Volumes and Surface Tensions of Bubbles at Separation

O 0.0051 cm tD/ms

0.0073 cm •

0 . 0 0 9 9 cm

•

0 . 0 1 2 8 cm

o ~

•

¢

20

_

~, ¢,

o-

.

¢,

The absence of a significant contribution from viscous and inertial forces to the process of bubble separation implies that the volume of the separated bubble Vshould be related to the surface tension 3'B of the bubble by the "drop volume" equation (7)

=

¢,

3'B = Vpg/27rrf(r/vii3), _-

¢

I

i

I

10

20

SO

bubble

frequency/s

-1

FIG. 7. Dead time to as function of capillary radius and frequency for 7 X 10 -3 M SDS solution at 25°C (from transient recorder traces).

passes through the capillary. The dead time, as described semiquantitatively by Fainerman (6), should therefore be a function of both capillary length and radius. Direct evidence for this is presented by Kloubek (3) in a comparison of volume flow rate as a function of pressure drop at bubble frequencies >fo with similar measurements made using unimmersed capillaries. We therefore now arrive at a picture of bubble formation. Initially the pressure marginally exceeds the capillary pressure of the hemisphere whereupon the radius increases. As the capillary pressure decreases the gas flow rate through the capillary increases so that the overall pressure difference remains almost constant. At a certain point the changing capillary and hydrostatic forces on the growing bubble may be neglected and the gas flow rate becomes constant. In the case of the example shown in Fig. 8 this transition is seen to occur about 5 ms after the initial growth of the bubble away from a hemisphere. The bubble then continues to expand until a capillary instability develops and separation occurs.

[5]

where f ( r / V 1/3) is an empirical correction factor and r is the radius of the origin of the bubble. If then the surface of the bubble is always completely denuded of surfactant at the point of separation the bubble volume should not be affected by the presence of surfactant. In Fig. 10 we present plots of separated bubble volumes against frequency for distilled water and 7 X 10 -3 M SDS solution. Unfortunately the plots exhibit significant scatter

V/lO-3cm bubble

4.0

separation

occurs

here

3.0

2,0

/

/

/

~ e d e a d time= t D

:~;r,':°' [

" "~0.

I

t 20

10

J 30

t/ms

FIG. 8. Rate of growth of bubble from analysis of high speed movie films (7 X 10 -3 M SDS at bubble frequency 15.4 s -1 and 25°C; ~0.01-cm capillary radius). Journal of Colloid and Interface Science, Vol. 132, No. 2, October 15, 1989

484

GARRETT AND WARD the foregoing argument. It is also consistent with both a constant dead time and overall pressure drop (see Table I). We note, however, that the volumes of the bubbles formed in SDS solution are significantly lower than those formed in distilled water. The foregoing argument implies that the surfaces of such bubbles must therefore have a lower surface tension and be contaminated with surfactant. In such a case the relationship between bubble volume and frequency becomes subtle. Thus the higher the frequency, the higher the overall pressure drop because of the smaller age and therefore higher surface tension of the hemisphere air-water surface. This means that bubble growth will be more rapid but that the volume before separation will be higher because the greater rate of area increase implies higher surface tensions. These opposing effects then lead to dead times which are only weakly dependent upon frequency for surfactant solutions (see Fig. 7). We would, however, expect an increasing bubble volume with increasing frequency in the case of surfactant solution. Unfortunately the scatter of the results shown in Fig. 10 for SDS solution is too large to establish whether this has occurred.

units y/arbitrarY60~__ ~ ' , , . ~

,o

4O

20--

10--

o,

~ 4

8

12

x/arbitrary

l

i

I

16

20

24

units

FIG. 9. Inverted bubble profile ~ 1.0 ms before separation (7 × 10-3MSDS at bubble frequency 15.4s -1 and 25°C). Experimental points are from movie stills; solid line is theoretical from Eq. [4]. (N.B.: 1 arbitrary unit = 0.00375 cm.) (probably mainly due to camera focusing error). Nevertheless it is clear that the bubble volume is essentially constant in the case of distilled water. This is of course consistent with V/IO

-3cm 3

5

•

•

•

• 4

-

0 0

3

-

2

-

1

-

0 0

•

•

0

0 O

0

O 0

•

0

0

I

I

I

I

10

20

30

40

bubble

frequency:

f/S -1

FIG. 10. Effect of surfactant on bubble volume after separation. (©) 7 × distilled water at 25°C. Journal of Colloid and Interface Science, Vol. 132, No. 2, October 15, 1989

1 0 -3

M SDS at 25°C; (O)

M A X I M U M BUBBLE PRESSURE M E T H O D

485

We may in fact use either the bubble profile that surface tension rises to a high value which, shown in Fig. 9 or the separated bubble vol- of course, need not necessarily be equal to that umes given in Fig. 10 to calculate the surface of distilled water. Thus the characteristic time tensions of bubbles at the point of separation. for diffusion in a surfactant solution is of the In the case of the bubble profile shown in Fig. order F2/2C2D, where I" is the surface excess, 9 we find a surface tension of about 52 m N C is the concentration, and D may be the m-1 using the pendant drop tables of Stauffer m o n o m e r or micelle diffusion coefficient (5). This value concerns the bubble only 1 ms whichever is the most appropriate. This means before separation and should therefore com- that if 0 >> 2C2D/p2 then the surface tension pare with a value calculated from separated must increase dramatically. If, however, this bubble volumes. inequality is not satisfied at some stage during Calculation of the surface tension from the bubble growth then the surface tension will separated bubble volumes shown in Fig. 10 decrease. For 7 × 10 -3 M SDS solutions we was done using the drop volume equation [ 5 ] find 2C2D/F 2 to be of the order 102 s-1.1 Toand the tables of Wilkinson (7). Values of 74 ward the point of bubble separation, however, + 3 m N m - l were obtained for distilled water. the measured dead times shown in Table II A surface tension of 55 m N m -1 was found indicate values of 0 of order 20-30 s -1. Adin the case of the bubble in SDS solution for sorption should therefore be occurring and which application of the pendant drop equa- surface tension decreasing at the point of bubtion [4] to the shape before separation gave ble separation. This we have found. 52 m N m -~. In view of the experimental scatActual separation of a bubble occurs exter observed with these measurements of sep- tremely rapidly. The neck at the base of the arated bubble volumes these results must be bubble breaks in <0.2 ms--whereupon the considered reasonable and represent further residual meniscus adopts a hemispherical evidence that inertial and viscous forces are shape (see Figs. 2 and 3 ). The pressure at the relatively unimportant in determining the na- point of rupture appeared to be slightly less ture of the growth and separation of bubbles than the maximum capillary pressure because under these circumstances. the hemispherical meniscus showed a tenOrder of magnitude theoretical calculations dency to move a small distance into the capcan also be used to suggest that bubbles in 7 illary. This was particularly marked in distilled × 10-3 M SDS solution are contaminated with water. Presumably the pressure then relatively surfactant at the point of separation. Thus ex- slowly builds up until it just exceeds the sum amination of Fig. 8 reveals that over most of of the hydrostatic and capillary pressures the dead time the volume rate of bubble (which has been declining due to adsorption growth is constant. In the case of a spherical during this period) whereupon the meniscus bubble undergoing a constant volume growth moves to begin the whole cycle again. These rate we can write for the relative rate of area considerations imply that small pressure flucincrease (8) tuations occur during the experiment. Examination of the manometer with a cathetod l n A / d t = 0 = 2/3t. [6] meter reveals that these fluctuations are, howwhere t is the time from commencement of ever, barely perceptible (i.e., <0.005 cm -= 0.02 bubble growth. Clearly d In A / d t has a max- m N m - l ) under normal operating conditions imum value at the start of the process. Here (i.e., frequencies >0.5 s -1 ). In principle, howof course the bubble is not a complete sphere ever, if a sufficiently sensitive pressure meaand d V / d t is not constant (see Fig. 8) so Eq. suring device is used it should be possible to [6 ] does not precisely apply and we do not measure bubble frequencies by monitoring find d l n A / d t = ~ at t = 0. It is during this stage of maximal rate of relative area increase For relevant data for this calculation see Section 3.3 Journal of Colloid and Interface Science, Vol. 132,No. 2, October 15, 1989

486

GARRETT AND WARD

these changes. Miller and Meyer (9) in fact describe an apparatus which has that capability (albeit for a larger capillary, larger bubble volumes, and therefore probably larger pressure fluctuations than found here). 3.2. POSSIBLE ERRORS IN MEASUREMENT OF SURFACE TENSION D U E TO EXPANSION OF THE A I R - W A T E R SURFACE

Implicit in the use of Eq. [ 1] for calculation of surface tension is the assumption that the radius of the meniscus at the point of maxim u m Laplace pressure is identical to that of the capillary, i.e., the meniscus forms a hemisphere. In order for the maximum Laplace pressure Pc to correspond to the capillary pressure at the point immediately preceding expansion of the meniscus from hemispherical shape we must have --~-]

~<

0

(at r --~ r0),

[71

where r is the radius of the air-water surface at any time t and r0 is the radius of the capillary. When adsorbed surfactant is present both 3" and r will change with time as the hemisphere grows into a bubble. This means that we require >~ 3'

(at r --~ ro)

[8]

If condition [7] is to be satisfied. Unfortunately we cannot estimate dr/dt at r --~ ro from the high speed movie stills with meaningful precision. We cannot therefore estimate with any degree of certainty the conditions under which condition [7] will be violated, giving rise to high erroneous values of 3". It is, however, possible to show that if dr/ dt is sufficiently high so that surfactant does not have time to diffuse to the expanding surface then in general condition [7] will be violated. Thus for a surfactant exhibiting Langmuir-Szyskowski adsorption behavior we find

(d3")_ -~

RTI'~

(r ~ - r )

Journal of Colloid and Interface Science,

dI" dt

'

[9]

Vol.132,No. 2, October15, 1989

where I" oo is the saturation adsorption, R is the gas constant, and T is the temperature. If no surfactant enters or leaves the surface during expansion we must also find

dP dt

P dA

[10]

:

where A is the surface area of the meniscus. Simple geometrical considerations enable us to deduce that

dt - 27r 2r + ~ -

r~ +

-~,

[111

which implies that dA/dt --~ oe as r --~ to. Therefore if we combine Eqs. [9]-[11] with condition [ 8 ] we find that condition [ 7 ] cannot be satisfied. For dilute surfactant solutions where diffusion is slow we would therefore expect condition [7] to be violated, yielding errors in measurement of 3'. Consideration of the nature of the bubble forming process suggests that the problem may occur regardless of the bubble frequency. Thus examination of Table II reveals that the time of bubble formation (dead time) is only weakly dependent upon bubble frequency. Examination of Fig. 10 reveals that the same is true of the separated bubble volume. Indeed in the case of higher bubble frequencies surface tension at r = r0 will in general be greater than for lower bubble frequencies and therefore concentration gradients will be steeper, affording more rapid transport to the expanding air-water surface. This will tend to make violation of condition [7] less likely as the frequency increases. In conclusion then condition [ 7 ] may be violated. If present this problem will occur over the whole frequency range, if anything becoming worse the lower the frequency. Appearance of errors will therefore be revealed if surface tensions obtained by extrapolation to 1 / t = f = 0 are higher than the corresponding equilibrium measurements obtained by, say, the Wilhelmy plate method.

487

M A X I M U M BUBBLE PRESSURE M E T H O D

3.3.

EVALUATION OF THE METHOD USING SUBMICELLAR SOLUTIONS

As we have seen a hemispherical meniscus is formed by the separating bubble in <0.2 ms. The separating bubble is not, however, necessarily uncontaminated with adsorbed surfactant. Whether the newly formed hemisphere is also contaminated with adsorbed surfactant can only be established by inference. Thus if we calculate the theoretical dependence of surface tension upon time for a surfactant solution and compare with experiment then agreement should be obtained by making a correction for "dead time" if the hemisphere is essentially uncontaminated. Such comparison should also afford indication of the extent to which condition [ 7 ] is violated. In contrast to micellar solutions, theoretical description of the kinetics of adsorption from submicellar solution is fairly well understood provided diffusion is the rate determining process. We therefore selected a range of submicellar solutions for evaluation of the method. Joos and Rillaerts (8) (and also Fainerman (6)) propose solutions to a convective diffusion equation for monomeric surfactant solutions which purport to describe adsorption with the maximum bubble pressure method. These approaches assume, however, that a planar surface is initially formed after bubble separation which subsequently undergoes a gradual change in radius up to the point of maximum capillary pressure as gas flows into the capillary at a constant volume rate. Examination of Figs. 2 and 3 reveals that this assumption is probably incorrect in that a hemispherical surface forms almost instantaneously upon separation of the bubble. During the time interval up to attainment of maxim u m capillary pressure the meniscus appears to have a constant hemispherical profile. In this we are in agreement with Bendure (10) (who considered the maximum bubble pressure method for high surface ages of 1-100 s). We must therefore select a solution to the diffusion equation which concerns adsorption to an initially clean surface of constant area.

Hansen ( 11 ) has shown that for the case of a single component monomeric nonionic (or anionic surfactant in the presence of excess electrolyte) surfactant solution we may write

R T ~ 2 [ 1__1_11/2 "g=%+~\TrDtJ "

[12]

In the case o f a monomeric anionic surfactant solution in the absence of electrolyte, Eq. [ 12 ] should be modified so that 2RTT2 (_}__11'/2 u = Ue + - - - - - C - \ ~ D t ]

[13]

provided l/1/t is small. Ye is the equilibrium surface tension. Equations [ 12 ] and [ 13 ] have the advantage of yielding 3' as an explicit function of V l / t and require only knowledge of D, %, and F (which may be obtained from equilibrium measurements of d Y e / d In C at the relevant concentration). On the other hand, rigorous solution of the relevant diffusion equation (of Ward and Tordai (12)) requires detailed knowledge of the surface equation of state for the surfactant. Such solutions do, however, indicate that Eq. [ 12 ] is reasonably accurate for values of 7 - "Yeup to about 5 m N m -1 with deviations becoming more marked for low surfactant concentrations ( 11 ). In Fig. 11 we compare the predictions of Eq. [ 13 ] with experiment for submicellar SDS solutions of various concentrations. Times were calculated using the dead time correction so that t = 1 / f - to, where to was directly obtained from transient recorder traces. Values for r were found from the adsorption equation of Lucassen-Reynders (13) for SDS, C/a =

1 --r-/-r~

exp[-2Er(r/r~)2]'

[14] where F °° = 7.64 × 1 0 - m M c m -2, a = 5.73 × 10 .6 M c m -3, and/-t = 1.36. Values for % were taken from Elworthy and Mysels (14). A diffusion coefficient of 6 × 10 .6 cm 2 s -~, obtained from bulk phase measurements, was taken from Kamenka et al. ( 15 ). Agreement between theory and experiment

JournalofColloidandInterfaceScience,Vol. 132,No. 2, October 15, 1989

488

GARRETT AND WARD

Y/ mNm -1 70 3.3xlO-3M

f

60

l

)

• 5.gx10-3M

t

5or [ ~ e m~

-

7x10-3M

I 401

__t 0

5

I

I

lO

15

%/~'i- / 8 - 1 / 2

FIG. 11. Dynamic surface tensions 3'o of submicellar SDS solutions at 25°C. Comparison between diffusion theory (Eq. [13]) and experiment. Solid lines are from [ 13]; boxes are literature values of equilibrium surface tensions (17). is seen to lie mostly within 2 m N m -~. Evidence of greater systematic deviation for the most dilute solution (3 X 10 -3 M ) may possibly reflect failure of Eq. [13] at high values of 1V~. We should note, however, that our measurements were made at 25°C and those of Elworthy and Mysels (14) at 20°C, which would imply that values of-y~ substituted into Eq. [13] should be up to about 0,5 m N m -I lower if account is taken of the probable temperature dependence of %. This factor would produce slightly better agreement for the most dilute solutions and slightly worse agreement for the other solutions. We should recognize that use of the square of the adsorption in Eq. [13 ] places exaggerated reliance upon the reliability of the empirical adsorption equation [ 14] for SDS. Thus the experimental values of F used in deriving Eq. [ 14 ] were obtained by differentiating the Elworthy and Mysels (14) plots of 7e against log(concentration), which procedure must introduce uncertainties. Other sources of uncertainty involve capillary radius calibration Journal of Colloid and Interface Science, Vol, 132, No. 2, October 15, 1989

error, manometer reading error, violation of condition [ 7 ] (leading to high values of 3' at 1 / t --~ 0), and even errors in the original measurement of ~/e by Elworthy and Mysels (14). In view of these factors the observed agreement must be considered reasonable especially when we note that random errors in measurements of dynamic surface tension by other methods are typically of the order of + 1 m N m-1 (8, 16, 17). We made similar comparisons between Eq. [12] and experiment for a fatty acid ester sulfonate (C12H25- CHSO3Na- CO2CH3) at 1.5 X 10 -3 M in excess electrolyte solution (7 X 10-3 MNaBO2). Unfortunately in this case the diffusion coefficient for the surfactant m o n o m e r is not known. Such diffusion coefficients are, however, only weakly dependent upon molecular structure--measured values being in the region of 2-6 × 10 -6 cm e s- 1 ( 15, 17-19). These values are consistent with the empirical relationship of Longsworth (20) concerning diffusion coefficients and molecular volumes. We therefore somewhat arbitrarily selected a diffusion coefficient of 3 X 10 -6 cm 2 s -1 for the fatty acid ester sulfonate, noting that it is the square root of this quantity which appears in Eq. [ 12 ], which in turn implies limited sensitivity to the exact value selected. Our plots of 3% against log(concentration) for this material were not sufficiently accurate to permit better than rough estimates to be made of F. This was found to lie between 4.4 × 10 -1° and 6.5 X 10 -1° M cm -2. Comparison between Eq. [ 12 ] and experiment for the dodecyl fatty acid sulfonate is shown in Fig. 12a for both extreme values of F. Agreement is seen to be less satisfactory than for the SDS solutions if the low value for P is used. It seems probable then that a value near 6.5 X 10-1° M cm -2 is more reasonable, in which case agreement between Eq. [ 12 ] and experiment becomes at least as good as that found for SDS with Eq. [ 13 ]. Finally, we made comparison between the prediction of Eq. [ 12 ] and experiment using a solution of a zwitterionic surfactant, DDPS

MAXIMUM BUBBLE PRESSURE METHOD

489

The gradient of the experimental value of 3' against 1 ~ is apparently steeper than the theoretical value which would imply either a lower diffusion coefficient than that selected (i.e., ~ 1 × 10 -6 c m 2 S - I ) , breakdown of Eq. . . e e ~ " [12] at comparatively low values of V l / t , or • / r = 4 4x10 lOMcm -2 o errors in P. 1"5x10-3M We conclude that measurement of the time dependence of 3"D by the maximum bubble C12H25 CHSO3Na C02CH3 pressure method appears to yield values which in 7x10 3M NaBO2 are in reasonable agreement with theoretical I I 0 5 10 expectation. Since all times in this study were Y/mNm -1 V ~ / $ 1/2 subjected to the dead time correction then this 60 correction appears to be justified. Failure to apply it, we should note, results in steeper slopes of 3, against 1V~ and therefore accentuated discrepancy between experiment and theory. The available evidence then suggests s o •~• / that the surface formed upon bubble separation is essentially uncontaminated with surfactant. C12 H25 (CH312N ( CH312S03 ( D D P S ) On the whole, agreement between theory I I and experiment is sufficiently good as ~ --* 40-0 5 10 0 to imply that violation of condition [ 7 ] has V~/s 1/2 not been observed. Nevertheless we feel that FIG. 12. Dynamic surfacetensions of submicellar sur- although the errors due to this cause are clearly factant solutions at 25°C, Comparison betweendiffusion small in the systems investigated here, comtheory (Eq. [12]) and experiment. Solid lines are from parison between 3" at V1/t or 1/t -+ 0 and % Eq. [ 12]; boxesare equilibriumsurfacetensionsusingthe measured by some independent method Wilhelmy plate method. should always be made. Y/mNrn -1 60 --

r ~ 6.5X 10"10 Mcm 2

401

(C12H25(CH3)eN(CH2)3SO3), at2 N 1 0 - 3 M in distilled water. Again the diffusion coefficient is not known for this compound. The diffusion coefficients for the corresponding tetradecyl and hexadecyl homologues have, however, been inferred by Lucassen (19) from longitudinal wave measurements. Thus values for these compounds apparently lie in the region of 3.5-4.0 X 10 -6 cm 2 s -1 and show no obvious chain length dependence. We therefore feel justified in selecting a value of 3.5 × 10 -6 cm 2 s -1 for DDPS. Values o f f (=3.03 X 10 -l° M cm -2) and % were found from results of Lucassen and Giles (21 ). In Fig. 12b we compare experiments with the predictions of Eq. [ 12 ] for DDPS. Agreement is seen to lie within about 3 m N m - k

4. SUMMARY We have examined in some detail the nature of the process of bubble formation in the maximum bubble pressure method used at constant pressure. The balance of evidence favors interpretation of surface age after the manner of Austin et al. ( 1 ), where a "dead time" correction is applied. Thus if we assume that the meniscus formed upon separation of a bubble is essentially uncontaminated with surfactant then we find reasonable agreement between measured 3" as a function of V1/t and diffusion theory. We have drawn particular attention to possible errors due to an increase of capillary pressure as the meniscus starts to expand beyond hemispherical shape (i.e., d P c / d t > 0). Journal of Colloid and Interface Science, Vol. 132, No. 2, October 15, 1989

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GARRETT AND WARD

Although we find that any errors from this source for the systems reported here are small it seems clear that 3' at 1/ t ~ 0 should preferably be compared with equilibrium surface tension measured by some independent method (such as the Wilhelmy plate method). We find no evidence for inertial or viscous effects which would be expected to cause systemic errors. Indeed apart from possible problems due to dPc/dt > 0 the method would appear to have some merit. Thus it is suitable for surface ages in the range 20-200 ms. The method can be readily thermostated and we have regularly used it at temperatures of up to 75°C. Only small samples of solution are required. A disadvantage of the method as described here is a requirement that the solution transmit laser light with reasonable efficiency. In principle, however, monitoring of pressure fluctuations using a sensitive transducer could be used to avoid this limitation although this may present problems with respect to accurate estimation of the dead time. Another limitation is the requirement that the capillary be water-wet. This means that the method as described here probably cannot be used with cationic surfactants. APPENDIX: CALCULATION OF BUBBLE SHAPES

We express the pendant drop equation (4) in a reduced form so that 1 sin ~b + - - 2 +/337, P 92

[All

where/5 = P/b, Y = x/b, and 37 = y/b. Solutions of Eq. [All require knowledge of/3 which may be found from the ratio S -- De/ Ds using literature tables (4, 5). Here De is the equatorial diameter of the bubble and Ds is the diameter at y = De. In order to permit accurate interpolation, literature tables (4, 5 ) of 13 as a function of S were fitted to spliced polynomials of the form J = ~ A i Si-1, i=l

[A2]

Journalof Colloidand InterfaceScience,Vol.132,No. 2, October15, 1989

where Ai are constants determined by linear regression. The size parameter b may be obtained from the relationship b = De(/3/H) °'5,

[A3]

where H can be accurately found from S using existing literature tables. The method of Winkel (22) may now be used to obtain solutions of Eq. [ A 1]. ACKNOWLEDGMENTS We acknowledge the assistance of J. Mander, who did much of the experimental measurement. We also acknowledge helpful comments from Dr. P. Joos. REFERENCES 1. Austin, M., Bright, B. B., and Simpson, E. A., J. Colloid Interface Sci. 23, 108 ( 1967 ). 2. Kloubek, J., Tenside 5 ( 11 / 12), 317 ( 1968 ). 3. Kloubek, J., J. Colloid Interface Sci. 41, 1-22 (1972). 4. Fordham, S., Proc. R. Soc, London, Ser. A, 194, 1 (1948). 5. Stauffer, C, E., J. Phys. Chem. 69, 1933 (1965). 6. Fainerman, V. B., Kolloidn. Zh. 41(1), 111-116 (1979). 7. Wilkinson, M. C., J. Colloid Interface Sci. 40, 14 (1972). 8. Joos, P., and Rillaerts, E., J, Colloid Interface Sci. 79, 96 (1981). 9. Miller, T. E., and Meyer, W. C., International Laboratory, April, 28 (1984). 10. Bendure, R. L., J. Colloid Interface Sci. 35, 238 (1971). 11. Hansen, R. S., J. Phys. Chem. 64, 637 (1960). 12. Ward, A. F. H., and Tordai, L., J. Chem. Phys. 14, 453 (1946). 13. Lucassen-Reynders, E. H., unpublished results. 14. Elworthy, P. H., and Mysels, K. J., J. Colloid Interface Sci. 21, 331 (1966). 15. Kamenka, N., Lindman, B., and Brun, B., Colloid Polym. Sci. 252, 144 (1974). 16. Joos, P., Bleys, G., and Petre, G., J. Chim. Phys. 79(4), 387 (1982). 17. van den Bogaert, R., and Joos, P., J. Phys. Chem. 84, 190 (1980). 18. Lucassen, J., and van den Tempel, M., Chem. Eng. Sci. 27, 1283 (1972). 19. Lucassen, J., Faraday Discuss. Chem. Soc. 59, 76 (1975). 20. Longsworth, L. G., J. Amer. Chem. Soc. 75, 5705 (1953). 21. Lucassen, J., and Giles, D., unpublished results. 22. Winkel, D., J. Phys. Chem. 69, 348 (1965).