Some experimental observations of the surface elasticity of surfactant solutions

Some Experimental Observations of the Surface Elasticity of Surfactant Solutions F. W. P I E R S O N AND S T E P H E N W H I T A K E R Department of Chemical Engineering, University of California, Davis, California 95616 Received February 17, 1977; accepted June 16, 1977 Previous experimental data concerning the propagation of waves in ripple tanks and the onset of waves in falling liquid films are reexamined in order to obtain new values of the compositional surface elasticity. Values of the elasticity from these dynamic experiments are compared with thermodynamic values. The results, especially those from the ripple tank experiments, are encouraging. INTRODUCTION

There are numerous multiphase flow phenomena which are dramatically influenced by the presence of surface active agents (1, 6, 11-13). Often this influence is the result of the dependence of the surface tension on the surface concentration which leads to an elastic-like behavior of the surface. This effect shows up in the tangential stress condition at a phase interface in terms of the thermodynamic quantity, y(Ocr/Oy). Here o- represents the surface tension (dyn/cm) and I/is the surface concentration (moles/cm2). For small disturbances, the stresses resulting from variations in the surface concentration of an insoluble surfactant are identical to those that would result if the surface were treated as an elastic membrane (3, 23). Because of this it is reasonable to think of y(O~/Oy) as a compositional elasticity and denote it by = -~(O~rlO~,).

[1]

Since the surface concentration, y, can be influenced by mass transfer between the surface and the substrate, in addition to the deformation of the surface, the behavior of solutions of soluble surfactants are quite complicated. When the surfactant is treated as in-

soluble, the analysis of wave p h e n o m e n a is greatly simplified provided one deals with small or linearized disturbances. For example, an analysis of the stability of vertical falling liquid films (23, 24) for an insoluble surfactant indicates that a critical Reynolds number exists and is related to the elasticity by f

1if3 11/5

NRe,e = 1.1 ---~--j~]

.

[23

Here g is the magnitude of the gravity vector, v is the kinematic viscosity, and p is the fluid density. The Reynolds number used here is defined as N a e = uoh/v,

[3]

where u0 is the surface velocity and h is the film thickness. The onset of waves on a falling liquid film of surfactant solution does indeed give rise to a p h e n o m e n a suggestive o f a critical Reynolds number. Several investigators (5, 18-21) have reported critical Reynolds numbers by observing the wave inception line as a function of Reynolds number leading to plots such as those shown in Figs. 1 and 2. Although one can estimate critical Reynolds numbers from the curves shown in Figs. 1 and 2, theoretical analyses for pure liquids (4, 16) and soluble surfactant

129 0021-9797/78/0631-0129502.00/0 Journal of Colloid and Interface Science, Vol. 63, No. 1, January 1978

Copyright © 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.

130

PIERSON AND WHITAKER I

I

I

I

f

( ~ Distilled Water 50

(~l

(~) 3 x 10-4 Molar Hexanoic Acid ( ~ 87 x 10-4 Molar Hexanoic Acid

"-7

==

Q

40

~

(~) 3 x 10-3 Molar Hexanoic Acid

30 uJ 20

10

0

I

I

I

I

I

5

10

50

100

500

1,000

NRe

FIa. 1. Wave inception line as a function of Reynolds number for hexanoic acid solutions. solutions (23) indicate an absence of a true critical Reynolds number. What in fact happens with surfactant solutions is that a point is reached where there is a sudden decrease in the growth rate, and a small lowering of the Reynolds n u m b e r leads to the absence of observable waves. The column used in this study was 2 m long and the solid lines in Figs. 1 and 2 represent points at which w a v e s were observed. For smaller Reynolds numbers no w a v e s could be detected visually o v e r the entire length of the column, and the broken 60

,

lines in Figs. 1 and 2 simply represent a guess as to an appropriate Reynolds number to use with Eq. [2] in order to estimate e. Following the suggestion of L u c a s s e n and H a n s e n (8) we refer to values of the elasticity determined from an insoluble surfactant model as apparent elasticities and designate them as eaop. L u c a s s e n and H a n s e n have determined values of eapv from ripple tank data for both hexanoic and heptanoic acid solutions. Their values for hexanoic acid, and values obtained from the w a v e inception line plots

I

l

I

r

i

@ 50

@ Distilled Water

I

O

1 x 10-5 Molar Heptanoic Acid

I

v~

= -'

40

(~)

o!

1 x 10-4 Molar HeptanrJic Acid \

\ \

@

(~) l x 10.3 Molar Heptanoic Acid

10

o

,

I

,

I

,

5

10

50

100

500

1,000

NRe

FIG. 2. Wave inception line as a function of Reynolds number for heptanoic acid solutions. Journal of Colloid and Interface Science, Vol. 63, No. 1, January 1978

SURFACE ELASTICITY OF SURFACTANT SOLUTIONS 10.0 0.0--

I

I

I

I

i

I

I

I

I

I

I

I

I

131 I

5.0--

o

4,0--

0 0

0

o

%

2.0--

0

O

0

Q

O ~

1.0--

~ o.00.6-0,4--

O Ripple Tank Data ([.ucassen & Hansen) O Critical Reynolds Number Data

0.2--

0.1 10-4

I 2

I 4

I I I 6 8 10-3

I 2

I 4

I 6

I I 8 10-2

I 2

I 4

I 6

10-i

c, moles/liter

FIG. 3. Apparent surface elasticity as a function of concentration for hexanoic acid solutions.

given in Fig. 1 are shown in Fig. 3. There we see that the ripple tank values of eapp are slightly above those obtained from the falling liquid film data, but surprisingly close to the latter values. It is important to note that the ripple tank data were obtained at a frequency of 200 Hz, while the frequencies observed for the falling liquid film were on the order of 20 Hz. At lower frequencies mass transfer from the substrate to the surface tends to reduce the surface concentration gradients and thus the surface stresses; this in effect reduces the apparent elasticity and we are relieved to see the falling liquid film values fall beneath the ripple tank values in Fig. 3. In Fig. 4 we have shown similar data for heptanoic acid, and there the descrepancy between the two sets of data is quite noticeable especially at the higher concentrations where mass transfer effects are more important owing to the change in the slope of the isotherm. From the work of Hedge and Slattery (7) or Lucassen and Hansen we know that a key parameter in assessing the effect of mass transfer is the quantity

Here ~ is the diffusivity, o~is the frequency, and dy/dc is the slope of the isotherm. At high concentrations dy/dc--0 0 and the effect of mass transfer becomes more important leading to the decreased values of Eaop shown in Fig. 4. Although the mass transfer process in a falling liquid film is more complex, the parameter, ~, does ap. . . .

1

. . . .

I

'

40 20

E

6

g

,

8 1.0 0.8 0.6

• Ripple Tank Data (Lucassen& Hansen)

0.4

OCritical Reynolds Number Data

0.2

. . . .

0.1 10-4

2

4

I 6 0 10-3

. . . . 2

4

I 6 0 10-2

, 2

C, moles/liter =

( dy/-1(~/2oJ)1/2.

[4]

FIG. 4. Apparent surface elasticity as a function of concentration for heptanoic acid solutions. Journal of CoUoid and Interface Science, Vol. 63, N o . 1, Ja n u a ry 1978

132

PIERSON AND WHITAKER 10-1

disturbance theory and experimental observations can be profitably compared. Since theory and experiment contain no mutual observable, it was decided to use the wave number of the fastest growing wave, o~m,as a parameter which could be related to the critical Reynolds number. The wave number is given by

10-2

am

10-3 Otm =

10-4

1'0

1'~0

1000

NRe

Fic. 5. Wave n u m b e r as a function of Reynolds number and elasticity number: K = 10 4.

pear with the frequency replaced by uo/k. Here u0 is the surface velocity and k is the wave length. Although e~pp, being a process parameter, has little value for the general analysis of surfactant solutions, it does provide an upperbound for any mechanical surface elasticity that these solutions may exhibit. In order to extract values of the true compositional surface elasticity, e, from wave inception line data one must perform an extensive set of calculations involving the stability of the film. These calculations involve a numerical solution of the Orr-Sommerfeld equation and they produce values for the growth rate, wave velocity and wave length. Comparison of the observed wave length and wave velocity with theoretical calculations is relatively straightforward and our results in this area are available elsewhere (16, 17). Comparing these parameters with wave inception line data, or in particularly with the pseudo-critical Reynolds number, presents something of a problem. Atherton and Homsy (2) have commented on the difficulty of analyzing the wave inception line problem for pure liquids; however, in our case the surfactants give rise to a rather dramatic phenomenon and there is some hope that linear Journal o f Colloid and Interface Science, Vol. 63, N o . 1, J a n u a r y 1978

2~h/km,

[5]

and calculated values of o~mare shown as a function of Reynolds number in Figs, 5 and 6. The wave number equivalent to a 2 m wave length, o~2, is also shown in these figures. The surface tension number, N~, and Schmidt number, Nsc, are defined as

N~ = o'(2/p3v4g) zla,

[6]

Nsc = vl@,

[7]

and the values used are representative of surfactant solutions of hexanoic and heptanoic acid. Carrying out calculations that matched the individual properties of the several solutions studied would have required excessive computer time. Since the comparison between theory and experiment is qualitative, extensive calculations were considered to be unnecessary. The elasticity number, NE~, is given by NEI = 10-i

E(2/p3p4g) 1/3, I

I

11o

i 100

[8]

am

1o-2

1°-4

1N

NRe FIc, 6. Wave n u m b e r as a function of Reynolds number and elasticity number: K = 10 2.

133

SURFACE ELASTICITY OF SURFACTANT SOLUTIONS

and K is the dimensionless slope o f the isotherm given by

K=L(dT/ h \ dc

lg

. . . . . .i . .

[9]

200

The curves shown in Figs. 5 and 6 are similar to the growth rate curves for systems of this type, i.e., there is a rapid decrease in O~mfor a range o f Reynolds number depending on NE1 and K. Deciding upon a point on the calculated curves that is equivalent to the pseudo-critical Reynolds number obtained from Figs. 1 and 2 is rather arbitrary. Since the length o f our experimental falling liquid film was 2 m we chose that length as the critical wave length and determined NR~.~ as the intersection of the curves for am and oe2. Since no waves longer than about ! cm, i.e., am ~ 0.05, were ever observed, we should be overestimating the elasticity required to produce a given critical Reynolds number. This means that our value of • determined from wave inception line data will probably be too large. Values o f • are much easier to determine from the ripple tank data of Lucassen and Hansen. T h e y presented an analysis for soluble surfactants, but were reluctant to

100 80

I

lOOO 800 60O 400

--

I I I I I I I I I I I 0 Lucassen & Ftansen,.~ - I x10-I cnl2/ see • Leeassen & HarJsen,~ 4.3xiO-6cm2/sec 0 Critical ReynoldNumberOata,.~5 ] xlO-Scai2/sec

I

o

no/"

o

•

20

6 4 2i I

~

4

6 8 10-3 2

I

c,

/

1Q

OO /

6.~8"~

0

0 CriticM Reynolds Number Data,.~: I x 10-5 cra2/sec

/

• Hedles & $1attery,

4.1D 2.~

10-4

~

2

.~ :7.9x 10-7 era'?./see

4

6 010-3

2

4

6 8 10-2

2

c, moles / liter

FIG. 8. Comparison of thermodynamic and dynamic values of • for heptanoic acid solutions.

use it because it required the knowledge o f the slope o f the isotherm, dT/dc. We have been less cautious with their data and have used their soluble surfactant analysis and the isotherm, c = {a(TIT~)/[1

-

(T/T~)]}10-2bwy~, [10]

to extract values o f e from the published values o f Eapp. These are compared with values of • calculated from Eqs. [10] and [11] in Figs. 7 and 8.

+ b(T/T~)2].

¢o

I 1.0 10-4 2

~

O /

20O

40

ThermodynamicData

o- - o-° = 2.3RTT°°[Iog (1 - T/T°°)

/

Tberm0oy~micData

'

• Lucassen & Ha~en..~= 5.3 x 10-5 cm2/sec 400

}

I

0 I_ucassen & Hansen,.~= 1 x 10- 5 cm2/sec

I

I II

J

4 6 8 10-2 2

I

I

4 6

10-1

moles/ Inet

Fin. 7. Comparison of thermodynamic and dynamic values of • for hexanoic acid solutions.

[11]

The values of a, b, and T ~ were determined (14, 15) from surface tension bulk concentration measurements and are listed in Table I. In Fig. 7 we have shown values o f e calculated from the work of Lucassen and Hansen using two different values of the molecular diffusivity. The value of 4.3 x 10 -~ cm2/sec was suggested by Lucassen and Hansen, while the value of 1 x 10-5 cmZ/sec was that used throughout all of our own analysis of falling liquid films and is consistent with the choice o f Schmidt Journal of Colloid and Interface Science, Vo|. 63, No. l, January 1978

134

PIERSON AND WHITAKER TABLE I Values of the Parameters a, b, and y~ a

"F¸~

Substance

(g-rnoles/cm 3)

b

(g-moles/cmz)

Hexanoic acid Heptanoic acid

8.6 × 10-6

0.30

5.7 × 10-1°

2.6 x 10 6

0.51

5.5 x 10-1°

namic property of surfactant solutions. Ripple tank data are in good agreement with the thermodynamic values while wave inception line observations provide only qualitative agreement. ACKNOWLEDGMENT

number given by Ns¢ = 103. The WilkeChang (25) correlation gives a value of 7.2 × 10-6 cm2/sec for hexanoic acid so the values used here are quite reasonable. As one would expect, use of a smaller diffusivity gives rise to a smaller predicted value of •. The agreement between the work of Lucassen and Hansen and values of e calculated from Eqs. [10] and [11] is surprisingly good, while the results obtained from the critical Reynolds number data are too large by a factor of 20. Comparison of Fig. 7 with Fig. 3 indicates good agreement between • and •app for concentrations less than 10-3 moles/liter, while at a concentration of 10-1 moles/liter eaop is too low by a factor of 500. In Fig. 8 we again see good agreement between the ripple tank data of Lucassen and Hansen, while the critical Reynolds number data give values of • which are generally too large. In Fig. 8 we have also included a single value determined by Hedge and Slattery (7) who independently analyzed the ripple tank data of Mann (9, 10). The agreement between these separate investigations along with the qualitative agreement with the falling liquid film experiments suggests that Eqs. [10] and [11], along with the parameters given in Table I, can be used with some confidence to predict compositional elasticities for hexanoic and heptanoic acid solutions. CONCLUSIONS

Dynamic experiments have been used to obtain values of the compositional surface elasticity, •, which is in fact a thermodyJournal of Colloid and Interface Science, Vol. 63, No. 1, January 1978

This work was supported by National Science Foundation Grant ENG74-13037. NOMENCLATURE

a b c g h R T u0 a a2 3' Y~ • eapp k u P oJ o°°

adsorption isotherm parameter, g-moles/cm3 adsorption isotherm parameter bulk concentration, g-moles/liter molecular diffusivity, cm2/sec magnitude of the gravity vector, cm/sec 2 film thickness, cm gas constant, cal/g-mole °K absolute temperature, °K surface velocity, cm/sec 2rrh/h, wave number wave number corresponding to a two meter wave length surface concentration, g-moles/cm2 adsorption isotherm parameter, g-moles/cm2 compositional surface elasticity, dyn/cm apparent compositional surface elasticity, dyn/cm wave length, cm kinematic viscosity, cm2/sec fluid density, g/cm3 frequency, Hz surface tension, dyn/cm surface tension of water, dyn/cm

Dimensionless groups N~

o'(2/pSu4g ) v3, surface tension

NRe N~z

uoh/u, Reynolds number • (2/p3u4g) 1/3, elasticity number h-l(dy/dc), dimensionless slope of

number

K

adsorption isotherm

( dy/ dc )-l( @/2oo) 112, dimensionless mass transfer parameter

SURFACE ELASTICITY OF SURFACTANT SOLUTIONS REFERENCES 1. Anshus, B. E., and Acrivos, A., Chem. Eng. Sci. 22, 389 (1967). 2. Atherton, R. W., and Homsy, G. M., Chem. Eng. J. 6, 273 (1973). 3. Benjamin, T. B., Arch. Mech. Stos 16, 615 (1964). 4. Benjamin, T. B., J. Fluid Mech. 2, 554 (1957). 5. Cerro, R. L., and Whitaker, S., J. Colloid Interface Sci. 37, 33 (1971). 6. Francis, R. C., and Berg, J. C., Chem. Eng. Sci. 22, 685 (1967). 7. Hedge, M. G., and Slattery, J. C., J. Colloid Inter.face Sci. 35, 183 (1971). 8. Lucassen, J., and Hansen, R. S., J. Colloid Interface Sci. 23, 319 (1967). 9. Mann, J. A., Jr., Ph.D. thesis, Iowa State Univ., Ames, Iowa, 1962. 10. Mann, J. A., Jr., and Hansen, R. S., J. Colloid Interface Sci. 18, 805 (1963). 11. Mansfield, W. W., Chem. Eng. Sci. 29, 1593 (1974). 12. Narasimhan, T. V., and Davis, E. J., I&EC Fund. Quart. 11,490 (1972). 13. Palmer, H. J., and Berg, J. C., J. Fluid Mech. 51, 385 (1972).

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14. Pierson, F. W., and Whitaker, S., J. Colloid Interface Sci. 54, 203 (1976). 15. Pierson, F. W., and Whitaker, S., J. Colloid Interface Sci. 54, 219 (1976). 16. Pierson, F. W., and Whitaker, S., I&EC Fund. Quart. 16, 401 (1977). 17. Pierson, F. W., Ph.D. thesis, Dept. Chem. Engr., Univ. of California, Davis, Calif., 1974. 18. Portalski, S., and Clegg, A. J., Chem. Eng, Sci. 27, 1257 (1972). 19. Stainthorp, F. P., and Alien, J. M., Trans. Inst. Chem. Eng. 42, 185 (1965). 20. Strobel, W. J., and Whitaker, S., AIChE J. 15, 527 (1969). 21. Tailby, S. R., and Portalski, S., Trans. Inst. Chem. Eng. 38, 324 (1962). 22. Vogtl~inder, J. G., and Meijboom, F. W., Chem. Eng. Sci. 29, 949 (1974). 23. Whitaker, S., I&EC Fund. Quart. 3, 132 (1964). 24. Whitaker, S., and Jones, L. O., AIChEJ. 12, 421 (1966). 25. Wilke, C. R., and Chang, P. R., AIChE J. L 264 (1955).

Journal of Colloidand Interface Science, Vol.63, No. 1, January 1978