On the use of the differential maximum bubble pressure method for determining equilibrium surface tensions of slowly equilibrating liquids or liquids of varying density

On the Use of the Differential Maximum Bubble Pressure Method for Determining Equilibrium Surface Tensions of Slowly Equilibrating Liquids or Liquids of Varying Density The differential m a x i m u m bubble pressure method for determining surface tensions is often used with simplified theoretical equations which allow for simple and rapid measurements. While this approach frequently yields surface tensions accurate and precise enough for practical applications, two cases where accuracy suffers significantly are discussed. One source of error occurs when a term involving density has been neglected even though significant density differences occur between calibration standards and liquid samples. Another source of error enters when equilibrium surface tensions are determined from dynamic measurements in which only one capillary bubbling rate is varied. This occurs when the larger of the capillaries is considered to principally correct for immersion depth and liquid density and its response to the surface tension itself is neglected. The origins and significance of these errors are discussed. © 1989 Academic Press, Inc.

INTRODUCTION

It is still not unusual to see Eq. [1] used with simplifications to one or both terms (4, 6, 8-11 ), for example,

Accurate measurements of equilibrium and dynamic surface tensions are very important for a wide variety of applications. For these, the m a x i m u m bubble pressure method, M B P M ( 1, 2), has some advantages in that the influence of surface impurities is small and a wide variety of liquids can be accommodated, from simple hydrocarbons to molten metals. Historically the method has not been as widely adopted as, for example, the du Nouy ring, Wilhelmy plate, or drop-weight methods because it has been less easy to use and generally less accurate. Recently, there has been a resurgence of interest in its use owing largely to the availability of improved means to conveniently measure small pressures ( 3 - 7 ) . The differential MBPM particularly has been improved and is increasingly used as a replacement for the more traditional methods. The method is very convenient, especially for practical applications where the liquid density may vary frequently and over a significant range. However, it is apparent that several systematic errors are frequently introduced which are the subject of this paper.

pt = (23,O)/r + Apgt.

[2]

Here r, the capillary tip radius, has been substituted for b, assuming a hemispherical bubble. But the actual bubble shape at m a x i m u m internal pressure is the same as that described by Bashforth and A d a m s for sessile and pendant drops. The correct radius factor b can be evaluated from their analysis by successive approximations. The second error lies in the use of the capillary tip immersion depth, t, rather than l. The former is of course easier to measure but this assumption neglects the increasing hydraulic pressure created by the growing bubble. These errors persist despite having been pointed out previously (1, 2). Although for some purposes great accuracy is not needed, the magnitude of the resulting errors is probably still not fully appreciated, especially for the differential MBPM. The differential MBPM involves two capillaries of differing internal radii, but equal immersion depth, thus:

= (23,~/bj + Ap~gz~) -- (23,~/b2 + Ap2gz:). [3]

THEORY W h e n bubbles form at the tip of an immersed capillary the pressure in the tube, p t , is the s u m of pressures due to surface forces, p', and hydrostatic forces, ph: pt = pS + ph = (23,O)/b + Apgl.

[1]

Here b is the radius of curvature at the apex of the bubble, hp is the density difference between gas and liquid, and 1 is the immersion depth to the apex of the bubble. As the bubble grows the internal pressure will go through a maxi m u m value which is uniquely related to the surface tension, and which is readily measured.

Here l = t + z, and the Apgt terms have canceled out. The introduction of errors into the analysis comes about when simplified forms of Eq. [3] are used. Although many researchers construct their own differential MBPM apparatus, for illustration we will consider the commercially available surface tensiometer Sensa-Dyne Model 5000 or 6000 (Chem-Dyne Research Corp., Milwaukee, WI) which is described elsewhere ( 3 ). For the calculations values for glass capillary tip radii and immersion depth of r~ = 0.50 ram, r2 = 2.00 m m , t~ = t2 = 1.50 cm were used. These have been found by the author to yield a practical compromise between accuracy and precision, some other choices will be compared.

534 0021-9797/89 $3.00 Copyright © 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.

Journal of Colloid and Interface Science. Vol. 133, No. 2. December 1989

NOTES RESULTS C a s e 1. L i q u i d s

without Surface

Active Agents

For pure liquids or solutions without surface active agents 3"7 = 3"~ in Eq. [ 3 ], so Ap,,2 = 23,°(1/bl - l/b2) + Apg(zl -- Z2).

[4]

For each capillary we can define ha = p t / ( A p g ) so that h~ = 23"°/(Apgbl) + zl = a2/bl + z,,

[5]

a2 = 23,°l(Aag).

[61

where

Defining Xl = a 2/ha then for any value of r~/a there is a unique value of X,/r~ (a minimum) which is available in tables constructed by Sugden (12). In the differential method we have h,, h2 and Xl, X2, so a 2 = (hi -- h 2 ) / ( 1 / X l - l/X2).

[7]

535

Eq. [9 ] to generate apparent values of 7 °. The errors range from about +0.5 m N / m (hexane) to - 4 5 m N / m (mercury) using the simple calibration method. The general dependence of the errors on liquid density for the calibration curve Eel. [9] is shown for several contours of constant absolute surface tension in Fig. 1. The origin and extent of this kind of error can be seen from Fig. 2. Here the absolute surface tension--differential maximum pressure relationship is shown for contours of liquid density, calculated by the Sugden method. It is apparent that the "constant" n in Eq. [8 ] is actually a function of liquid density, as derived above. When calibrating liquids have different density the calibration curve crosses the contours of Fig. 2. This makes the calibration curve nonlinear. The accuracy of measured surface tensions for unknown liquids then depends on both the density and surface tension of the unknowns and how close they are to the standards used. The density dependence is shown in the equations proposed by Schrrdinger and others (14-16), which for the differential MBPM and dropping higher terms is Apl.2 = 23" °(( 1/rl ) -- ( 1/r2)) + (2Aag/3)(r~ -- r2).

To calculate 3' o one first calculates ha and h2 knowing p] and p~. Next, one assumes AT, = rl, 22 ~ r2 and calculates a 2 and r~/a, rz/a. Sugden's table then gives X~/ rl, X2/r2 hence Xl, Xz. From Eq. [7] a new value for a 2 is generated. By successive approximations an accurate value of a 2 is obtained which yields 3, ° from Eq. [6]. Volkov and Volyak ( 13 ) have tabulated solutions to this procedure, but their tables are not accurate enough for the present comparisons, A common method ( 3, 6-9) of calibrating and operating the differential MBPM involves simplifying Eq. [4] to the linear equation Apl.2 = m3"° + lt.

[8]

Equation [8] only holds to the extent that the relative sizes and shapes of droplets issuing from the two capillaries are independent of surface tension and density (e.g., n = Aog(zj - z2) = constant ). Calibration is achieved using two standard liquids and solving Eq. [ 8 ] for m and n. This technique is very convenient and may be accurate enough for certain practical purposes, but it can lead to a significant error, as the following example shows. A typical calibration with water and methanol, using literature values of surface tension and density at 20.0°C, together with the capillary dimensions given earlier, yields the calibration equation ApL2 = 2.9703,0 -- 8.57 Pa.

[9]

A set of sample liquids was chosen to span a range of densities from 0.66 to 13.6 g/ml and surface tensions from 18.4 to 465, maximum differential pressures, Ap, were calculated from accepted surface tension values using the Sugden method. The values for Ap were substituted into

[10]

Equation [10] is of the same form as Eq. [8] but shows that n is not constant if the liquids have different densities. This is not normally appreciated because one of the reasons for using the two capillaries at equal immersion depth is supposedly to remove the need to consider (measure) the liquid densities. Examples can be found in the literature cited above where either the single capillary or differential MBPM has been used without accounting for all the density terms. Again, the absolute value of the measurement error increases with increasing divergence between the densities of the sample liquids and those of the calibration liquids. Equation [8] is a reasonable approximation only if the calibration liquids and all the sample liquids have about the same densities. To get surface tensions accurate to within 0.2 m N / m requires that all liquids have the same density to within about 0.06 g/ml. Other capillary radii combinations introduce different degrees of the same dependence. Using rl = 0.25 m m and rE = 2.00 m m would improve the density tolerance to about 0.12 g] ml. Using r~ = 1.00 m m and r2 = 4.00 mm, on the other hand, would degrade the density tolerance to about 0.02 g/ml. Thus from Eq. [9 ] the errors for, e.g., glycerol and mercury would be - 4 and - 119 m N / m, respectively. If liquids of varying density are of interest then one must either use more exact equations in the calculations and include the liquid densities or use pairs of standards for calibrating that are matched in density to the samples of interest. It should also be remembered that the calibration liquids should be chosen to span a narrow range o f surface tension within which the test liquids fall. This is because in the differential MBPM, r2 is fairly large at 2 or Journal of Colloidand InterfaceScience, Vol. 133.No. 2, December1989

536

NOTES 2 /. 1.5

E z E

30 rnN/m

70

0.5 o

,~

-0.5

'"

-1.5 -2 -2.5

I

l

0.6

I

018

I

1

1.2

1.4

Liquid Density (g/mL)

FIG. 1. Surface tension error contours for constant absolute surface tension liquids of varying density.

4 m m for which even the full Schr6dinger equation is not accurate to 0.2 m N / m (14). These considerations are important because the differential MBPM is c o m m o n l y used

for practical liquids whose densities and surface tensions m a y span a large range, e.g., from aqueous solutions and hydrocarbon liquids to polymer melts, to molten metals.

220 210 2O0 190 n v

.8

180

1.0 1.2

170 3

13.

_e nn

160

150 140

130 120

-5 B 0

110 100 9O 80 7O 6O 50 4O

20

'

' 40 Surfoce

' Tension

6~ 0

(mN/m)

FIG. 2. Pressure-surface tension contours for constant density liquids of varying surface tension. Journal of Colloid and Interface Science, Vol. 133, No, 2, December 1989

80

NOTES

537

Case 2. Liquids Containing Surface Active Agents For pure liquids and quickly equilibrating solutions one can measure differential m a x i m u m bubble pressures and use Eq. [4], or for constant density liquids Eq. [8]. However, when solutions equilibrate slowly, dynamic surface tensions are exhibited. The differential MBPM can be used to study these by determining apparent surface tensions as a function of bubble frequency. The longer the bubble period the longer the a m o u n t of equilibration is allowed between the growing bubbles and the liquid of interest. Two problems arise. First, with growing bubbles, new surface area is constantly being created and the surface tension measured for a given bubble period will not, in general, exactly equal the surface tension measured by a dynamic method in which no new surface area is created for the same a m o u n t of equilibration time (e.g., dynamic Wilhelmy plate or pendant drop). Further, since one capillary is larger than the other, the rates of increase of surface area are not equal at each capillary for any given bubble rate. This would be most significant at fast bubbling rates (where other factors become important as well) and correction factors have been discussed by Kloubek (17-19). We will consider a simpler case in which apparent dynamic surface tensions are measured in order to obtain the equilibrium surface tension of a solution. At bubble periods longer than 1 s the effective surface age can be taken to be 10 to 20% less than the bubble period (18) so that at long enough bubble periods good agreement should eventually be obtained between measured and equilibrium surface tensions. A second error arises during the c o m m o n practice of varying the smaller capillary bubble rate while either not controlling the larger capillary bubble rate or holding it constant. This procedure is experimentally convenient and is usually justified on the grounds that the pressure contribution from the larger capillary is small and serves mainly to correct for the hydrostatic pressure ( 1, 3). Although it is true that the larger capillary contributes a smaller pressure, significant errors can still be introduced. The conditions of Eq, [4] are m e t for a given equilibration time only if both capillaries are bubbling with the same period and if that period is sufficiently large (i.e., the same dynamic surface tension operates at each). If this is not true, as when the two capillaries bubble at different rates, then it is Eq. [ 3 ] that m u s t be used. We can calculate the error involved in determining the equilibrium surface tension by varying only one capillary bubble period as follows. Method 1. The calculation is illustrated using a sample isolated from a commercial oil sands hot water flotation process stream (7), containing anionic surfactants, simple electrolyte, and some colloidal solids. The dynamic surface tension behavior shown in Fig. 3 was determined by the differential M B P M keeping both capillaries always at the same bubble period, which was varied from 1 to 35 s.

........

O'H2 0

g, ,r. 60

55

J

10~

~

2~0

~

3~0

Bubble Time

J

4-~0

(s)

FIG. 3. Dynamic surface tension behavior of an aqueous surfactant-containing solution from an industrial process stream. After (7).

U n d e r these conditions, and at the longer bubble periods, Eq. [4] holds. Figure 3 shows dynamic surface tensions of 3" ° = 60.8 m N / m at a bubble period of 30 s and -y ° = 69.3 m N / m at a bubble period of 1.0 s. The equilibrium surface tension is very nearly reached at 60.8 m N / m . For bubble periods of 30 s (both capillaries) the Sugden method yields Apl,2 = 171.0 Pa; for periods set to 1.0 s, Ap~.2 = 197.6 Pa. Substituting into Eq. [8] yields Ap = (3.13 Pa- m / m N ) ' y ° - 19.2 Pa for these measurements (all at constant density). Method 2. If now capillary 1 is set for a bubble period of 30 s while capillary 2 is set for 1.0 s, the Sugden method yields: Capillary 1

Capillary 2

a~ = 2v~'/(A.og) = 12.4 m m 2 rl/al = 0.142

azz = 23~/(Apg) = 14.2 m m z rz/az = 0.531

Xi/rl

X2/rz =

from Sugden's table: = 0.987 hi = 25.1 m m p] = 246.0 Pa

0.831 hz = 8.54 m m p [ = 83.6 Pa

therefore: Apl,2 = 162.4 Pa. Using the calibration parameters from Method 1, this method would yield a surface tension of 58.0 m N / m for Journal of Colloid and Interface Science, V ol. 133, No. 2, December 1989

538

NOTES

the liquid. Comparison with the real equilibrium surface tension of 60.8 m N / m shows that Method 2 introduced an error o f - 2 . 8 mN/m. This result goes against intuition. With one capillary bubbling faster than the other one might have expected the apparent surface tension to have been intermediate between 60.8 and 69.3 mN/m. The theoretical calculation shows, however, that maximum bubble pressure increases with surface tension by a larger factor at the smaller capillary than at the larger capillary. To have achieved an accuracy of 0.2 m N / m would have required that the larger capillary produce a bubble period of at least 21 s. Thus in dealing with unknown liquids, a better technique would be either to make both capillaries bubble with the same periods, increased until constant values of surface tension are obtained ( 7 ), or to hold the larger capillary at a constant but very long period while varying the smaller capillary period until constant values of surface tension are obtained (19). In both cases periods longer than 60 s may well be required to approach equilibrium values. The measurement of truly dynamic surface tensions with the differential maximum bubble pressure technique is more complicated since the two capillaries will produce bubbles with different rates of increase of surface area ( 1719). This source of error does not normally cancel but must be added to those discussed above. CONCLUSION For liquids of varying density, and where dynamic surface tension behavior is of interest, the use of the differential MBPM with simplified theoretical equations can lead to significant errors. The density related errors are pervasive because it is so often incorrectly assumed that immersing both capillaries to equal depths cancels out all the effects of liquid density. A different kind of error occurs in determining the equilibrium surface tensions of dynamic systems when the same bubble formation periods are not used at each capillary. This follows directly from the theory but since the larger capillary requires a lower pressure to form bubbles it is sometimes assumed that it acts only to cancel one of the hydrostatic pressure terms. Furthermore, the error is in the direction opposite to what might intuitively be assumed. These systematic errors can easily go undetected, or be attributed to other causes. The density effects become important whenever there is significant density variation, such as for solutions of high ionic strength, organic liquids, or for molten metals. The effects of how the bubble periods are controlled on apparent equilibrium surface tensions are almost always important whenever there is significant dynamic surface tension character.

Journalof Colloidand Inte(faceScience,Vol, 133,No. 2, December1989

ACKNOWLEDGMENTS This work was done in response to discussions held with many colleagues who use the differential maximum bubble pressure technique. Particularly helpful discussions with E. Eddy Isaacs (Alberta Research Council) and Victor P. Janule (Chem-Dyne Research Corp.) are gratefully acknowledged. REFERENCES 1. Harkins, W D., in "Physical Methods of Organic Chemistry" (A. Weissberger, Ed.), Vol. 1, Part 1, p. 757. Interscience, New York, 1959. 2. Padday, J. F., in "Surface and Colloid Science" (E. Matijevic, Ed.), Vol. 1, p. 101. Wiley-lnterscience, New York, 1969. 3. Janule, V. P., Klus, J. P., Gibbons, E., and Brodsky, E., Proc. World Congr. Chem. Eng. 2nd 5, 499 (1981). 4. Mysels, K. J., Langmuir 2, 428 (1986). 5. Miller, T. E., and Meyer, W. C., Amer. Lab. 16, 91 (1984). 6. Woolfrey, S. G., Banzon, G. M., and Groves, M. J., J. Colloid Interface Sci. 112, 583 (1986). 7. Schramm, L. L., Smith, R. G., and Stone, J. A., Colloids Surf. i l , 247 (1984). 8. Brian. B. W.. and Chen, J. C.. AIChE J. 33, 316 (1987). 9. Mitra, P. K., and Roy, T. K., Indian Foundry .L 33, 15 (1987). 10. Goldmann, L. S., and Krall, B., Rev. Sci. Instrum. 47, 324 (1976). 11. Huang, D. D., Nikolov, A., and Wasan, D. T., Langmuir 2, 672 (1986). 12. Sugden, S., J. Chem. Soc. 121, 858 (1922). 13. Volkov, B. N., and Volyak, L. D., Russ. J. Phys. Chem. 46, 598 (1972). 14. Kisil', I. S., Mal'ko, A. G., and Dranchuk, M. M., Russ. J. Phys. Chem. 55, 177 ( 1981 ). 15. Sugden, S., J. Chem. Soc. 125, 27 (1924). 16. Cuny, K. H., and Wolf, K. L., Ann. Phys, 17, 57 (1956). 17. Kloubek, J., Tenside 5, 317 (1968). 18. Kloubek, J., J. Colloid Interface Sci. 41, 1 (1972). 19. Kloubek, J., J. Colloid Interface Sci. 41, 7 (1972). LAURIER L. SCHRAMM

Petroleum Recover), Institute 3512 33rd Street N.W. Calgary, Alberta, Canada, T2L Z46 Received February 17, 1989; accepted March 15. 1989