the shape of liquid drops approaching a deformable liquid—liquid interface in three-phase systems

The Shape of Liquid Drops Approaching a Deformable Liquid-Liquid Interface in Three-Phase Systems

D. ROBINSON AND S. HARTLAND

Department of Chemical Engineering

The University of Nottingham (Ct. Britain)

(Received: 19 August, 1969; in final form: 22 September, 1969)

ABSTRACT

The shapes of liquid drops of phase 1 immersed in

phase 2 approaching a deformable liquid-liquid inter-

face between immiscible phases 2 and 3 of equal re-

fractive index, have been measured photographically. For pure systems in which the densities of phases

1 and 3 are equal, the drop dimensions do not vary

signtjicantly vvith time and agree with those theoreti-

cally predicted. This is also true when surface active

agents are present only in the bulk interface but not

when they are present in the drop surface. As in two-

phase systems, unsymmetrical drainage of the film

of phase 2 is sometimes observed.

INTRODUCTION

The dimensions of a liquid drop of phase 1, sur- rounded by an immiscible liquid of phase 2, approaching a bulk liquid interface between phases 2 and 3, are shown in Fig. 1. The equilibrium shape

Phase 3

draining film of phase 2

Fig. 1. Dimensions of a liquid drop approaching a deformable liquid-liquid interface.

has been computed by Princen’ as a function of drop size, interfacial tension and density difference, when phases 1 and 3 are identical and the liquid interface extends to infinity (so it eventually becomes plane). Because of the equal densities p, and p3 on each side of the draining film, its overall shape is spherical; the drop dimensions of this pure two-phase system

22

The Chemical Engineering Journal (1) (1970)--Q Elsevier Publishing Company Ltd, England-Printed in Great Britain

have been shown to agree experimentally with those predicted. However, when surface active agents were included in the drop and bulk phase, so that the densities of phases 1 and 3 remained the same, the equilibrium shapes of the drops were seen to deviate from the predicted values.3 As the ages of the drop and bulk phases are not the same, it is likely that the interfacial tensions cl2 and cz3, between phases 1 and 2 and phases 2 and 3, will differ. The drop shape for such a three-phase system with unequal interfacial tensions, but with the densities p 1 and p3 being equal, has been predicted by Princen and Mason.4 However, the experimental drop dimensions when surface active agents are present still do not agree with predicted values. This was explained in terms of the mobility of the interfaces bounding the draining film. *

This paper extends the experimental investigation to a pure three-phase system in which the interfacial tension between the drop and the draining film is different from that between the two bulk phases. However, the system is chosen in such a way that the densities of phases 1 and 3 are equal, so that the overall shape of the draining film is again spherical. It is one purpose of this paper to compare the experimental and theoretical drop shapes for this three- phase system. In some experiments surface active agents are added to either the drop, or to one of the bulk phases, to investigate their effect on drop shape.

The variation in thickness (with time and position) of the draining film, and its subsequent rupture, are also investigated.

EXPERIMENTAL

Three-phase systems were employed consisting of silicone oil MS200 (1000 c.s.),” 50 % aqueous glycerol,

* Silicone Oil MS200 (1000 c.s.) manufactured by Midland Silicones Ltd.

THE SHAPE OF LIQUID DROPS APPROACHING A DEFORMABLE LIQUID-LIQUID INTERFACE 23

and nitromethane or castor oil. The physical properties of these liquids are given in Table 1.

TABLE 1 PHYSICAL PROPERTIES OF LIQUIDS MEASURED

AT 22.5”C.

Physical properties

Castor Oil

aqueous Silicme Nitvo-

glycerol methane

Density (g/cc) 0.961 1.139 0.973 1.132 Refractive index 1.4786 1.4052 1.4050 1.3819

.-,_ - - _,_ Interfacial tension

(dynes/cm) 17.30 30.02 7.34

50% aqueous glycerol has the same refractive index as the silicone oil, and these liquids were always used as phases 2 and 3 so that the drop of phase 1 could be seen without distortion. The density of nitromethane is close to that of 50% aqueous glycerol and higher than that of silicone oil, so drops of nitromethane were allowed to approach the bulk interface from above. The density of castor oil is close to that of silicone oil and less than that of 50% aqueous glycerol, so drops of castor oil were allowed to approach the bulk interface from below.

In some experiments the 50% aqueous glycerol solution contained polyoxyethylene sorbitan mono- oleate which is a water-soluble non-ionic surfactant known as ‘Tween 80.‘” In further experiments the castor oil contained sorbitan mono-oleate, an oil soluble surface active agent known as ‘Sorbester P 17’.” It is thus possible to study the effect of surface active agent on the bulk interface alone, the drop interface alone and on both together.

The three-phase systems studied are summarised in Table 2. Systems A and B are pure systems whereas

TABLE 2 SUMMARY OF THREE-PHASE SYSTEMS

STUDIED ______

System Drop phase 1 Bulk phase 2 Bulk phase 3 ___-_____

A Castor oil 50 % Aqueous Silicone oil glycerol

B Nitromethane Silicone oil 50 “/d Aqueous glycerol

C Castor oil + 50 % Aqueous Silicone oil 10-Z volume glycerol fraction ‘Sor- bester P 17’

D Castor oil 50 % Aqueous Silicone oil glycerol + lo-3 volume fraction ‘Tween 80’

E Nitromethane Silicone oil 50 % Aqueous glycerol + 10-3 volume fraction ‘Tween 80’

* ‘Sorbester P 17’ and ‘Tween 80’ manufactured by Howards of Ilford Ltd.

systems C, D and E contain surfactant in one of the phases. The presence of ‘Sorhester P 17’ in the castor

oil drops of system C affects the tension in the drop interface but not in the bulk interface, however

‘Tween 80’ in phase 3 of system E only affects the tension at the bulk interface. The presence of ‘Tween 80’ in phase 2 of system D affects both the tension at the bulk interface and at the drop interface.

The interfacial tensions were measured by the pendant drop method (Andreas et al. 5, and are given in Table 1 and Fig. 2; this method allows the variation of interfacial tension with time to be followed when surface active agents are present.

Fig. 2. Variation of interfacial tension with time for systems containing surface active reagent: (a) interfacial tension between castor oil plus 10 -2 volume fraction ‘Sorbester P 17’ and 507; aqueous glycerol; (b) interfacial tension between castor 011 and 50% aqueous glycerol plus 10 -3 volume fraction ‘Tween 80’; (c) mterfacial tension between silicone oil and 50% aqueous glycerol plus 10-s volume fraction ‘Tween 80’.

The immiscible bulk phases were placed in a 5 cm cube perspex cell and allowed to equilibrate for one hour when one of the bulk phases contained surface active agent. Thisallows theinterfacialtension to approach its minimum equilibrium value. 0.1, 0.2, 0.3, 0.4 and 0.5 ml drops of nitromethane or castor oil were allowed to approach the bulk interface and were photographed as previously described2*3,6 at different times throughout the drainage period.

RESULTS AND DISCUSSION

Typical photographs of the three-phase systems studied are shown in Figs. 3 to 7. The negatives were back-projected and the magnified drop dimensions x’~, ztc and cp’, shown in Fig. 1 measured. The ratio x’,/z’, and the value of cp’, are independent of magnifi- cation, and so may be compared directly with the theoretical values of x,/z, and cpc predicted by

24 D. ROBINSON AND S. HARTLAND

TABLE 3

DROP DIMENSIONS OF SYSTEM A

~12 = 17.30 dynes/cm; E = 0.365

023 = 30.02 dynes/cm; z/c12 = 3.178

_________

Drop volume Drainage (ml) time (set)

Experimental Theoretical rdc12

XCIZC w XCIZC PC

0.1 0.1 0.1 0.1

0.2 0.2

0.2 0.2 0.2

0.3 0.3 0.3

0.4 04 0.4

0.5 0.5

7

1’: 35

15’ :i 27

:i 27

0.52

0.909 0.51 0.52 0.51

1.151 0.65 0.67

0.68 1.151 0.67

0.65

0.77 1.319 0.80

0.79

0.88 1.452 @86

0.88

1.557 0.95 0.95

$1 20” 0.52 20”

20.5”

0 is 0.67 26”

26” 0

% 0.67 26”

28” 3

:s 0.78 28”

0

% 0.86 31” 34”

35” 35” 0.94 33”

Princen and Mason. Comparison between these values is used as the criterion of agreement between the experimental and theoretical drop shapes. The magnified film thickness 6’ was also measured at intervals of a few degrees from the vertical axis of the drop. Knowledge of the magnified value of x’, enables the magnified radius of the spherical cap R’ =

x’,/sin cpc to be obtained, and hence the dimension- less film thickness 6/R. This is plotted against the angular distance from the vertical axis of the drop in Figs. 8 to 12.

Results for each of the systems listed in Table 2 are summarised in Tables 3 to 7, and discussed separately below.

Fig. 3. Photograph of 0.2 ml castor oil drop in 50% aqueous glycerol approaching an interface with silicone oil after 15 set

drainage time.

System A

The experimental and theoretical drop dimensions for castor oil drops rising through 50 ‘A aqueous glycerol towards a silicone oil interface are shown in Table 3. The experimental ratio of x,/z, agrees with that predicted by Princen and Mason4 for all drop volumes investigated, but the experimental value of p= only agrees with those predicted for 0.1, 0.2 and 0.3 ml castor oil drops, the experimental values of cpc for 0.4 and 0.5 ml drops being about 7% greater than predicted. These larger drops deform the bulk interface more than the smaller drops, and in addi- tion are closer to the cell walls, so that the interface may not become plane within the limited size of cell used. Table 3 also shows that both x,/z, and q,c remain constant as the drainage time increases.

Profiles of the draining film are shown in Fig. 8 and are symmetrical about the vertical axis of the drop, with the film being thinnest at the edge and thickest at the centre.

System B The experimental and theoretical drop dimensions

for nitromethane drops falling through silicone oil towards a 50”/ aqueous glycerol interface are shown in Table 4. As with the pure system A, the experimental values of x,/z, agree with the theoretical values, and the angle cpc also agrees even for large drops.


Fig. 4a

Fig. 4b

Fig. 4c Fig. 4. Photographs of 0.5 ml nitromethane drop in silicone oil approaching an interface with 50% aqueous glycerol at different drainage times: (a) 5 set drainage time, (b) 15 set

drainage time, (c) 20 set drainage time.

Fig. 5. Photograph of 0.3 ml castor oil drop plus 10-2 volume fraction ‘Sorbester P 17’ in 50% aqueous glycerol approaching an interface with silicone oil after 47 set drainage

time.

Fig. 6b

Fig. 6c

Fig. 6d

Fig. 6. Photographs of 0.5 ml castor oil drop in 50% aqueous glycerol plus 10-s volume fraction ‘Tween 80’ approaching an Interface with silicone oil at different times: (a) 20 set drainage

time, (b) 125 set, (c) 127 set, (d) 150 sec.


Fig. 7a

Fig. 7b

“0 4 -

3 - x

n 2 -

Ia I -

0 I I I

-30 -25 -20-15 -10 -5 0 5 IO I5 20 25 30

Fig. 7c Angular distance from vertlca: axis of drop (degrees)

Fig. 7. Photographs of 0.4 ml nitromethane drop in silicone oil approaching an interface with 50% aqueous glycerol plus 10-s volume fraction ‘Tween 80’ at different drainage times: (a) 12 set drainage time, (b) 19 set drainage time, (c) 24 set

drainage time.

Fig. 9. Profiles of the draining film beneath a 0.5 ml nitromethane drop in silicone oil MS200 approaching an interface with 50% aqueous glycerol. The bulk interface has been

arbitrarily straightened out.

Profiles of the draining film are shown in Fig. 9

and the film is seen to be initially symmetrical about the vertical axis of the drop, being thinnest at the edge and having a secondary minimum in the centre.

Because of the circulation patterns within the drop,6 the film thickens at the centre and then unsymmetrical drainage occurs, as demonstrated in Fig. 4.

IO

9

8

I 15 set

0’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ -30-25-20-15-10 -5 0 5 IO 15 20 25 30 35

Angular distance from vertical axis of drop (degrees)

Fig. 8. Profiles of the drainage film beneath a 0.2 ml castor oil drop in 50% aqueous glycerol approaching an interface with silicone oil MS200. The bulk interface has been arbitrarily

straightened out.

9 -

8 - 5 set

7 -

6 -

5 -

TABLE 4

DROP DIMENSIONS OF SYSTEM B

cri 2 = 7.34 dynes/cm; E = 0.196

07.3 = 30.02 dynes/cm; v’ct a = 4.609

Drop volume (ml)

Drainage time (set)

Experimental Theoretical MC12

xclzc w xc/zc (PC

0.1 40 1.318 0.69 Z.5” C 0.69 15; 0.2 30 1.668 0.90 0.91 18.5” 0.3 20 1.913 I .ot 20.5” 1.07 0.5 15 2.258 1.21 22” 1.20

2:o 0


System C The experimental drop dimensions for castor oil

drops plus lo-’ volume fraction ‘Sorbester P 17’

rising through 50% aqueous glycerol towards a silicone oil interface are shown in Table 5a. The

TABLE 5 DROP DIMENSIONS OF SYSTEM C

(a) Experimental

Drop volume Drainage time

(ml) (set)

0.1 0.1 0.1

0.2 0.2 0.2

0.3 0.3 0.3

0.4 0.4

0.5 0.5 9:

0.58 0.61 0.61

0.76 27.5” 0.77 29.5” 0.78 29.5”

0.84 0.91 0.89

0.99 1 .oo

1.05 1.05

g50 24

31.5”

;‘,r,o

36.5” 37”

(6) Theoretical

012 = 17.30 dynes/cm; E = 0.365

02s = 30.02 dynes/cm; 2/ct2 = 3.178

xclzc 9c

____ ____ 0.52

0.67

% 0

0.78 28”

0.86 0.94 ;;:

Drop volume (ml)

0.1 0.909

0.2 1.151 0.3 1.319

;I: 1.452 1.557

___- __-___

(c) Theoretical

(~12 = 14.50 dynes/cm; E = 0.326

(~23 = 30.02 dynes/cm; l/cts = 3.469

0.1 0.992 0.56

0.2 1.256 0.72

ii0 0

0.3 1.440 0.84 28”

0.4 1.585 0.93 0.5 1.700 1 .Ol ;y:

variation of interfacial tension, oi2, with time is shown in Fig. 2, and is seen to vary from 17.30 dynes/cm initially to 14.50 dynes/cm after 390 seconds. The theoretical dimensions when c1 2 is equal to 17.30 dynes/cm are shown in Table 5b and those for 14.50 dynes/cm in Table 5c.

The experimental values of x,/z, are up to 13 % greater than predicted in Table 5b and up to 7 % greater than predicted in Table 5c. The experimental values of cpC are up to 20% greater than predicted.

By reducing the value of 0, 2 to 12.70 dynes/cm the

predicted values of x,/z, can be made to agree with the experimental values, but this increases the diver-

gence between the values of cpC. This discrepancy between the experimental and

predicted drop dimensions may be explained by the circulation within the drop induced by its approach to the interface, and by drainage in the film. 3 This circulation sweeps surface active molecules out of the film and concentrates them in the free surface of the drop. The interfacial tension between the drop and the draining film is thus greater than that for the free surface of the drop. The drop will then sink further into the bulk interface than expected, and its free surface will be flatter accounting for the high experimental values of x,/z, and cpC. No agreement between the experimental values and the theoretical values of Princen and Mason4 can be anticipated because of the non-uniform distribution of interfacial tension around the drop.

The draining film profiles are shown in Fig. 10, and are similar to those for system A. The film is symmetrical about the vertical axis of the drop, being thinnest at the edge and thickest in the centre.

8- 7- 6-

mg 5- x 4-

$ 3-

2- I-

7sec

12 set

11 1 1 I I I I I I I 1 I I

o-3O-25-2O-15 -10 -5 0 5 IO 15 20 25 30


Fig. 10. Profiles of the draining film beneath a 0.2 ml castor oil drop containing 10-Z volume fraction ‘Sorbester P 17’ in 50% aqueous glycerol approaching an interface with silicone oil MS200. The bulk interface has been arbitrarily straightened

out.

System D The experimental drop dimensions for castor oil

drops rising through 50% aqueous glycerol plus lo- 3 volume fraction ‘Tween 80’ towards a silicone oil interface are shown in Table 6a. In contrast to the behaviour of system A the experimental values of x,/z, and cpC increase with drainage time.

The variation of interfacial tensions, cp 1 2 and qpz 3, with time is shown in Fig. 2. The interfacial tension C, 2 varies from 17.30 dynes/cm initially to 5.93 dynes/cm after 70 min and the interfacial tension cz3 varies from 30.02 dynes/cm initially to 8.00 dynes/cm after 60 min. In the experiments the interface was aged for 60 min before a drop was released,


TABLE 6 DROP DIMENSIONS OF SYSTEM D

(a) Experimental

Drop volume Drainage time (ml) (set)

;:: 9: 0.90 0.94

0.2 I 1.22 0.2 55 1.26 0.2 140 1.29

0.3 1.46 0.3

25: 1.58

0.4 7 1.55

0.5 10 1.66 0.5 115 1.74

0

iZ.5’ 46”

55.5” 56”

(b) Theoretical

012 = 17.30 dynes/cm; E = 0.684 crzs = 8.00 dynes/cm; v’cts = 3.178

Drop volume (ml)

;:: 0.909 1.151 0.88 0.70 47” 0

0.3 1.319 1.00 z:o

0.4 1.452 1.09 0.5 1.557 1.17 2::

(c) Theoretical

CJ , 2 = 5.93 dynes/cm ; E = 0.426 02s = 8.00 dynes/cm; dct 2 = 5.425

Drop volume (ml)

8::. 1.551 1.964 0.95 1.24 ;;:

;:; 2.251 2.419 1.61 1.45 49” 51” 0.5 2.658 1.74 53”

so cz3 should approach its minimum value. It is not so easy to predict the interfacial tension of the drop, as the initial decrease in interfacial tension with time is very rapid. Therefore, the theoretical drop dimensions for all four possible combinations of the limiting conditions are considered. Those corresponding to the minimum value of oz3, and the maximum and minimum values of cri 2, are shown in Tables 6b and 6c.

Poor agreement is observed between the theoretical drop dimensions shown in Table 6b, obtained from the maximum value of oi2, and the experimental results. However, at long drainage times there is reasonable agreement with the dimensions obtained from the minimum value of c 1 2 shown in Table 6c. This is unexpected, since the assumed value of 0 1 2 would only be obtained after ageing the drop for 70 min, and the drop is released immediately after its formation.

The values of 40~ corresponding to the maximum value of oz3 are always about 50% less than the experimental values for both limiting values of or2. The ratios of x,/z, corresponding to the maximum value of c 1 2 are also about 50 % less than the experimental values, but the ratios of x,/z, corresponding to the minimum value are only about 10 % less.

The draining film profiles illustrated in Fig. 11

IO set

20 set

OLI”““““” -60-5040-30-20-10 0 10 20 30 40 50 60


Fig. 11. Profiles of the draining film beneath a 0.5 ml castor oil drop in 50% aqueous glycerol containing 10-s volume fraction ‘Tween 80’ approaching an interface with silicone oil MS200. The bulk interface has been arbitrarily straightened

out.

show that unsymmetrical drainage occurs resulting in rapid thinning. As demonstrated in Fig. 6, rupture occurs in two places on the periphery where the film is very thin; the two receding edges move across the drop, and when they meet small drops of phase 2 are formed from the draining film. After rupture of the film the drop is wetted by two different liquids, and its shape becomes distorted. Usually rupture only occurs at one position on the periphery, and Mar7 has also observed central rupture.

System E The experimental drop dimensions for nitro-

methane drops descending through silicone oil towards an interface with 50% aqueous glycerol plus lo- ’ volume fraction ‘Tween 80,’ are shown in Table 7a. In this case the surfactant is in the lower bulk phase on the opposite side of the interface to the draining film, and since the bulk interface has been

TABLE I

DROP DIMENSIONS OF SYSTEM E

(a) Experimental

Drop volume (ml)

Drainage time (set)

XCIZC Bc

0.1 0.1

1:: 0.83 39.5” 0.83 38.5”

0.2 ;; 1.07 ;::

jZf50 105 1.39 1.23 49.5”

0.5 95 1.41 53.5”


TABLE ‘J-continued

(6) Theoretical

a12 = I.34 dynes/cm; E = 0.478 crzs = 8.00 dynes/cm; 4~12 = 4578

Drop volume (ml)

0.1 1.309 0.83 0.2 1.657 1900

1.06 0.3 i.23

;;I

50” 8:; 2.092 2.243 1.46 1.36 :zO 0

aged for 60 min and there is little induced circulation within the bulk phase, the interfacial tensions

g12 anda,, should remain constant at 7.34 dynes/cm and 8.00 dynes/cm respectively. The drop dimensions predicted by these conditions are given in Table 7b, and do in fact agree with the experimental measure- ments within the expected error.

The draining film profiles in Fig. 12 show that the film is initially symmetrical about the vertical axis of the drop, being thinnest at the edge and thickest at the centre. However, unsymmetrical drainage occurs and causes localised thinning.

II -

10 -

9-

e-

7-

6-

-50 -40 -30 -20 -10 0 10 20 30 40 50 60

Angular d!stance from verllcal axis of drop ( degrees)

Fig. 12. Profiles of the draining film beneath a 0.4 ml nitromethane drop in silicone oil MS200 approaching an interface with 50% aqueous glycerol containing 10-3 volume fraction ‘Tween 80’. The bulk interface has been arbitrarily straightened

out.

CONCLUSIONS

The shape of a liquid drop of phase 1 approaching a deformable liquid-liquid interface through phase 2 agrees with the predicted values for a pure three- phase system. Agreement is also observed when the surface active agent is included in phase 3 and the bulk interface is aged for some time. However, when surface active agent is included in either phases 1 or 2, the drop dimensions do not agree with those predicted. This discrepancy is explained in terms of the non-uniform interfacial tension in the drop surface, induced by drainage in the film and circulation within the drop.

Unsymmetrical drainage occurs beneath the liquid

nitromethane drops descending through silicone oil

and approaching a 50% aqueous glycerol interface, with or without surfactant. Unsymmetrical drainage

also occurs above the castor oil drops when surfactant is included in the 50% aqueous glycerol. How- ever, symmetrical drainage occurs when castor oil drops with or without surfactant approach a pure deformable interface.

ACKNOWLEDGEMENT

The authors would like to thank Professor W. Smith for his encouragement, and the United

Kingdom Science Research Council for a generous grant.

NOMENCLATURE

Cl2 (Pi -P&/fl12

g acceleration due to gravity

R radius of curvature of surface of draining film

x, radial dimension measured from vertical axis

of drop to edge of draining film

2, vertical dimension measured from top of free surface of drop to edge of draining film

Greek symbols

6 film thickness

& g12/@12 + az3>

P1,2,3 density of phase 1, 2 or 3

O12,23 interfacial tensions between phases 1 and 2

or between phases 2 and 3

cpc inclination of edge of draining film to horizontal axis

Superscript

’ denotes a magnified dimension

REFERENCES

1. PRINCEN, H. M., J. Colloid Sci., 1963 18 178. 2. HARTLAND, S., Trans. Ins&. Chem. Engrs., 1967 45 T97. 3. HARTLAND, S., Trans. Instn. Chem. Engrs., 1968 46 T275. 4. PRJNCEN, H. M., AND MASON, S. G., J. Colloid Sci., 1965

20 246. 5. ANDREAS,J.M.,HAUSER,E.A.,ANDTUCKER,W.B., J. Phys.

Chem., 1938 42 1001. 6. HARTLAND, S., Chem. Eng. Sci., 1969 24 611. 7, MAR, A., AND MASON, S. G., Kolloid-Zeitschrift, 1968 224

161.


RESUME

On determine par photographie la forme des gouttes

liguides d’une phase (1) dispersde dans une phase (2), se

rapprochant d’une interface liquide-liquide entre des phases (2) et (3) non miscibles de m&me indice de

refraction. Pour des systbmes ‘puts dont Ies phases (1) et (3)

ont meme densite’, les dimensions des gouttes n’e’voluent

pas de maniere sign$cative et sont en accord avec celles prevues par la the’orie. Ceci est Pgalement vrai

quand des agents tensio-act@ sontpresents d l’interface

entre les phases 2 et 3, mais non quand ils sont presents a la surface des gouttes. Comme dans un systeme a

deux phases, on observe parfois un entrainement dis-

symetrique du film de la phase (2).

ZUSAMMENFASSUNG

Es werden die Formen von Fliissigkeitstropfen der

Phase 1 photographisch untersucht, die in Phase 2

eingetaucht, sich einer verformbaren Fliissigkeits- Fliissigkeits GrenzJache zwischen nichtmischbaren

Phasen 2 und 3 von gleichem Brechungsindex nahern. Bei reinen Systemen, bei dene die Dichten der

Phasen 1 und 3 gleich sind, ist keine signtjikante k;zderung in Abhtingigkeit von der Zeit festzustellen.

Die Ergebnisse stimmen mit den theoretisch voraus-

gesagten iiberein. Dies ist ebenfalls erfiillt bei Anwesen- heit oberJIiichenaktiver Substanzen in der 2-3 Grenz-

flache jedoch nicht wenn diese in der Tropfenobetfldche

vorkommen. Wie in zweiphasigen Systemen wird manchmal ein unsymmetrisches AbJieDen des Films

der Phase 2 beobachtet.

the shape of liquid drops approaching a deformable liquid—liquid interface in three-phase systems

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