Formation, evolution, and rheology of two-dimensional foams in spread monolayers at the air—water interface

Formation, Evolution, and Rheology of Two-Dimensional Foams in Spread Monolayers at the Air-Water Interface JACOB LUCASSEN, 1 SILVI~RE AKAMATSU, AND FRANCIS R O N D E L E Z Laboratoire de Structure et Rdactivit~ aux Interfaces, Universit~ Pierre et Marie Curie (Paris V1), 11 Rue Pierre et Marie Curie, 75231 Paris, Cedex 5, France Received September 5, 1990; accepted December 4, 1990 We discuss the properties of two-dimensional (2-D) foams formed in the gas-liquid coexistence region of spread monolayers on the basis of observations on a surface-active fluorescent dye. Their appearance and general behavior are analogous to those of three-dimensional (3-D) foams. The line tension between the gas and liquid 2-D phase regions plays a role similar to that of the surface tension in conventional foams. It determines the thinning of the strips of liquid monolayer phase which separate the gas cells in the 2-D polygonal surface structure: a lower surface pressure in the corner regions causes the equivalent of the Plateau-border suction. The rate of strip thinning can be slowed down by several orders ifa gradient in line tension opposes monolayer flow. Such a gradient presupposes the existence of the unidimensional equivalent of surface activity, for which there is so far only circumstantial evidence. The line tension also controls cell disproportionation, i,e., the growth of large cells at the expense of smaller ones, and "line activity" can also reduce the rate of this process. Finally, the rearrangement process, in which contact between four strips changes into two three-strip contacts, is based on the minimization of the total line energy. It was observed that this rearrangement actually occurs before a small central cell has disappeared to cause the four-strip contact. This p h e n o m e n o n was shown to be theoretically expected. © 1991AcademicPress,Inc. INTRODUCTION

Rafts of bubbles on the surface of a soap solution were first used by Bragg ( 1 ) in order to simulate the effect of dislocations in a crystalline lattice. Recently, there has been an increased interest in such systems, commonly called two-dimensional (2-D) foams. As the change in their bubble size and bubble size distribution with time is more conveniently observed and easier analyzed than those of conventional three-dimensional (3-D) foams, they are particularly suitable as model systems enabling physicists ( 2 - 6 ) to study order-disorder transitions and to improve the understanding of shape and size changes of crystalline domains in metals. In rheology ( 7 - 9 ) , they have been considered a simplified two~To w h o m correspondence should be addressed at present address: Mathenesselaan 11,2343 HA Oegstgeest, The Netherlands.

dimensional model for the flow of three-dimensional foams. In all these studies, the 2-D foams considered were in reality 3-D foams, either floating as a single layer of bubbles or confined by a geometrical constraint to the thickness of a single bubble. In a novel development in surface chemistry (10-12), it has been shown that two-dimensional foams in a more strict sense of the word can occur in phase coexistence regions of spread monolayers of surface active molecules at the air-water interface. It had been known for a long time (13) that monolayers of compounds such as long-chain fatty acids undergo a sequence of phase transitions with increasing monolayer density, from two-dimensional gas to liquid and solid phases. The coexistence regions of two adjoining phases can extend over a wide range of surface densities. Until recently, little was known about the monolayer structure in these coexistence regions. The ad-

434 0021-9797/91 $3.00 Copyright© 1991by AcademicPress,Inc. All rightsof reproductionin any form reserved.

Journalof Colloidand InterfaceScience,Vol. 144,No. 2, July 1991

SPREAD MONOLAYERS vent of fluorescence microscopy (14), however, has made it possible to study the mechanism of two-dimensional phase transitions and to obtain a clearer picture of the morphology of the resulting composite monolayers. It was found (11 ) that when a gas-liquid two-phase region is approached by expansion of a liquid monolayer, circular (2-D) gas "bubbles" are formed which upon expansion are forced into a polygonal network which has the appearance of a foam. Thus, the "disperse" phase of this two-dimensional foam consists of islands of gaseous phase surrounded by the "continuous" liquid phase. Just as in the case of three-dimensional foams, one can clearly distinguish "films," the rectangular strips of continuous phase which separate two adjoining polygons, and "Plateau borders," where three of such strips meet. In the following, we will use the term strip to indicate the two-dimensional film of a monolayer. The two-dimensional foam bubbles will be called gas

cells. The behavior of the 2-D foams as it can be observed under the microscope is strongly reminiscent of 3-D foams. In some respects, however, there are differences. In the first place, drainage due to gravity obviously does not take place, and consequently the width of the monolayer strips, the size of the gas cells, and the average monolayer density remain uniform. This gives 2-D foams the appearance of stability. Second, their formation does not require agitation as it usually does for their 3D counterparts. Expansion of a "liquid" monolayer to a preset density is sufficient, and this is analogous to foam generation caused by opening a pressurised liquid container, e.g., a beer bottle. In the process of growth, there ought to be another difference compared with c o m m o n 3D foams. For these systems, the gas phase consists predominantly of air. Its diffusion across films limits the rate of the inevitable bubble size disproportionation; in foams where air has been removed deliberately ( 15 ) and where the gas phase only consists of water

435

vapor, the process is extremely rapid. In 2-D foams there does not seem to be an equivalent of air; yet their rate of growth seems quite slow. This can be attributed to another important difference compared with the 3-D foams: Being located in a surface, 2-D foams are not autonomous systems and all equilibration processes which involve either surface diffusion or surface convection should be accompanied by a coupled movement of adjoining bulk liquid. As a last point of difference it should be mentioned that the driving force for all changes occurring in a 2-D foam is not based on surface tension and on the tendency to minimize the total surface area, as it is for 3D foams, but instead on line tension between the 2-D gas and liquid phases and on the tendency of the system to minimize the total length of the two-phase contact line. In this paper we will describe some observations on rearrangements and cellular growth for 2-D foams. We will also investigate some aspects of strip stability and surface flow in strips and the possibility of an effect of line elasticity. Finally, the relationship between microscopic structure and phenomenological surface properties, such as surface pressure and surface viscosity, will be discussed. EXPERIMENTAL All the 2-D foams studied in this paper have been observed in spread monolayers of NBDHDA [ 4- (hexadecylamino)-7-nitrobenz-2oxa-l,3-diazole], a fluorescent dye commercially available from Molecular Probes Inc. (U.S.A.). Its structure is typical of long-chain surfactants, with a hydrophobic 16-carbon chain and a hydrophilic polar head group consisting of the conjugated aromatic rings, the tertiary amino, and the nitro groups. Its purity was found to be higher than 99%, as measured by thin-layer chromatography. It was used without further purification. The monolayers were formed in a Teflon trough filled with ultrapure water (passed through a Millipore, Milli-Q-Organex system) Journal of Colloid and Interface Science, VoL 144, No. 2, July 1991

436

LUCASSEN, AKAMATSU, AND RONDELEZ

acidified to p H 2 by 0.01 mol.liter -1 HC1. Droplets of a chloroform (p.a. Merck) solution of the compound to be spread with a concentration between 10 -4 and 5 × 10 -4 mol. liter -1 were deposited at the air-water interface in 5IA aliquots by means of a Hamilton syringe. The small droplets spread instantaneously. After evaporation of the solvent a monolayer with initial surface densities of around 500 A2 per molecule was obtained. The monolayer could be compressed or expanded using a movable Teflon barrier. The surface pressure was measured and recorded by means of a platinum Wilhelmy plate connected with a sensitive differential transformer (Schenck, Germany). Care was taken that the platinum was always fully wetted by the aqueous solution. A typical isotherm, measured at a temperature of 20 + 0.2°C is shown in Fig. 1. For areas per molecule higher than 66 + I A 2, the surface pressure is found to be less than 0.4 ___0.2 m N m-l; it slowly decreases with increasing area per molecule and is zero within measuring error at 80 A2 per molecule. A wide region of these low surface densities represents the coexistence between 2-D liquid and gas phases, as will be shown later from fluores-

cence microscopy observations. Unfortunately, our technique did not permit sufficiently accurate monitoring of the surface pressure in the liquid-gas coexistence region. At areas per molecule smaller than 66 + 1 A2, there is a rapid increase in surface pressure. This corresponds to the formation of the socalled liquid expanded (LE) phase, often observed in fatty acid monolayers. Below 40 +__1 ~2 per molecule, the surface pressure becomes suddenly independent of the molecular area. At this collapse pressure, three-dimensional crystals begin to form. Fluorescence microscopy observations have been performed using a Polyvar Met (Reichert-Jung, Austria) microscope fitted with an LWD 20X objective. A set of filters permits the excitation of fluorescence at 460 nm and the detection of the emitted light at 525 nm. The images were observed using a SIT vidicon camera (Lhesa Electronique, France) and viewed on a TV monitor. Overall magnification was 37 urn/cm and lateral resolution was 2.5 urn. Figure 2 shows images of the dye monolayer at decreasing surface densities, ranging between 70 and 600 A 2 per molecule. The 2-D liquid phase shows up as white, while

;urface Pressure (raN/m) 25

20

Cl6 H38

I tiN

10

~ N

/

N 0 2

5

0

I

30

40

50

60

70 80 Area per Molecule ( ~, 2)

I

I

90

100

FIG. 1. Surfacepressure vs area curve for the fluorescentdye NBD-HDA. Journal of Colloid and Interface Science, Vol. 144, No. 2, July 1991

110

SPREAD MONOLAYERS

437

FIG. 2. The formation of a 2-D foam by expansion of an NBD-HDA monolayer.

the gas phase regions are black. This is due to a very large difference in the molecular densities between the two phases. As can be seen, when upon expansion of the homogeneous LE phase, the L E - G coexistence region is first entered, circular gas cells begin to appear. These expand much more rapidly than the surrounding LE phase, and when they touch one another they begin to deform out of the circular shape and adopt the appearance of a foam. For equal-sized gas

cells, this transition should occur when the area fraction of the gas phase exceeds the value of 0.9069, corresponding to the close packing of identical circles in a plane (7). More and more pronounced polygonal structures are formed at increasing rarefaction of the monolayer. Typically well-developed foams are observed for areas per molecule ranging between 200 and 1000 A 2. The foams form by simple expansion and no agitation is required. A typical gas cell diameter is 100 t~m, but sizes can Journal of Colloid and Interface Science, Vol. 144, No, 2, July 1991

438

LUCASSEN, AKAMATSU, AND RONDELEZ

range between 10 um and a few hundred micrometers, evidencing the heterogeneity of the foam structure. In practice it is difficult to generate a foam which is homogeneous over the whole trough surface; sometimes even large foam regions can coexist with large liquid or gaseous regions. On the other hand, the width of the liquid strips separating two neighboring gas cells in a foam region appears to be uniform over a large area of observation (several square millimeters). Its value is of the order of a few micrometers. The homogeneity of the 2-D foam can be improved by addition of up to 1 mol% of cholesterol to the NBD-HDA spreading solutions. The number of nucleating gas cells is thereby increased and their size consequently decreases. The size distribution becomes significantly narrower and the foam regions occupy a larger fraction of the total surface (Fig. 3 ). The 2-D foams are extremely long-lived and can be observed for several days after the expansion has been stopped. However, they are undoubtedly metastable as shown by continuous geometrical rearrangements. Figure 4 shows an example where a four-sided gas cell decreases in size, apparently reaches an unsta-

ble geometrical configuration, and suddenly moves and changes into a three-sided cell. The whole process takes about 10 s, which is rapid compared with all other observable changes but slow compared with a similar rearrangement in a 3-D foam. FORCE EQUILIBRIUM AND DRAINAGE IN 2-D FOAMS The strips of liquid phase in a two-dimensional foam are bordered on two sides by dilute (gaseous) phase. A line tension should be attributed to the two edges, and the fact that three strips are always found to meet at 120 ° angles shows that this line tension is the perfect one-dimensional analogue of a surface tension. By the same token, we can formulate a twodimensional version of Laplace's Law, 3-

2xII

R'

where 2xII is the difference between surface pressures on both sides of a line, r is the line tension, and R is the line's radius of curvature. As in the three-dimensional case, the higher pressure--here surface pressure--is found at

FIG. 3. Foam with enhanced monodispersityby addition of cholesterol. Journal of Colloid and Interface Soience, Vol. 144, No. 2, July 1991

[11

SPREAD M O N O L A Y E R S

439

FIG. 4. Rearrangement in a 2-D foam where a four-sided cell shrinks, moves, and changes into a threesided cell before vanishing. (A) 0 s; (B) 16 s; (C) 19 s; (D) 21 s; (E) 23 s; (F) 25 s. Journal of Colloid and lmerface Science, Vol. 144, No. 2, July 199

440

LUCASSEN, AKAMATSU, AND RONDELEZ

the concave side of a curved line, for example, inside a cell of the gas phase. The surface pressure within the two-dimensional Plateau borders (where three strips meet) should therefore be lower than in the strips. This pressure difference should lead to flow toward the border and consequently to strip thinning. In 3-D foams thinning by border suction does occur, but it is very much overshadowed by the action of gravity. The downward flow it causes in the Plateau borders leads to the marginal regeneration which greatly accelerates the rate of film thinning (16). Our experiments show that for 2-D foams, strip thinning is only noticeable during foam formation. In stationary foams, the strip width seems to remain constant and, for one type of monolayer, depends largely on the gas-to-liquid area ratio. This suggests that there is a mechanism by which strip thinning is abruptly slowed down during the foam formation. A one-dimensional disjoining pressure seems an unlikely cause as the sudden slowdown appears to occur over a wide range of strip widths--from 20 down to 1 t~m--and there is no apparent tendency to reach an equilibrium width. A more likely cause is an abrupt change in flow regime as this does occur during drainage of 3-D foam lamellae in the absence of marginal regeneration (17). In that case the films thin initially very quickly according to a plug-flow mechanism; i.e., the film surface flows with the same velocity as the film interior. In that first stage surface tension gradients are generated which then gradually cause a resistance against further tangential movement of the film surfaces, changing them from "mobile" to "rigid." Thus, the modification of the drainage regime from plug flow to laminar flow causes a drastic reduction in the thinning rate. The thinner the film, the smaller the surface tension gradient required to bring the film drainage to a virtual halt by this mechanism. For a similar mechanism in 2-D foams to explain the sudden reduction in the rate of strip thinning, it is necessary to postulate in the first place that there is "line activity," i.e., Journal of Colloid and Interface Science, Vol. 144,No. 2, July 1991

the one-dimensional equivalent of surface activity ( 18 ), and in the second place that gradients in line tension can occur. Such gradients can only be expected when there is more than one surfactant. Even when surface-active materials of the highest purity are used, it can be assumed--in analogy with the 3-D case--that amounts of a second surface-active component required to achieve the desired effect are in general so minute that they escape detection. An estimate of the flow rate through a strip, once its boundary has become rind, can be based on a comparison with the monolayer flow in a surface channel between two reservoirs with slightly differing surface tension. Such a system can be used to measure surface shear viscosity but the coupling between surface and bulk flow leads to viscous dissipation in the bulk, even in the absence of surface viscosity. The surfaces in two reservoirs with surface tensions al and ~r2 are connected by an infinitely deep channel with parallel walls (Fig. 5A). The length and width of the channel are b and a, respectively. The surface density is r and is supposed not to vary significantly in the range of surface tensions considered. For the flux in such a system is found (19). Q = fo ~ rvdy = 0 . 2 7 r ~

7a '2

t2l

where the y-coordinate is in the surface perpendicular to the channel wall, and ~ is the substrate viscosity. For a strip of a 2-D foam in contact with a Plateau border, the surface pressure difference of ~-/R causes flow from the center of the strip to its edge (Fig. 5B). Therefore, the length of the channel should be taken as half the strip length. Furthermore, the result of the suction is not a flow through a channel with constant width, but a thinning of the channel, in this case the strip. From the supposition of a virtually constant surface coverage, I', we obtain da o.sb -~ = -a~,

[3]

441

SPREAD M O N O L A Y E R S

A

(

o1

I

>

<

b

B

/ aa

0-1

>

b

/

FIG. 5. Comparison between monolayer flow in (A) a channel viscometer and (B) a strip in a 2-D foam.

where ~ is the monolayer flow velocity in the b-direction, averaged over the width of the channel. Combining [ 1], [ 2 ], and [ 3 ] we find for the rate of strip thinning da

a3 T

- ~ - = 0.54 ~b2--~ .

[41

Because of our lack of knowledge of the values for the line tension in 2-D foams, only a rough guess can be made of the expected rate of thinning. Taking a typical value of 10-2 for a/b, of 10-2 p for the subphase viscosity 7/, of 10-3 cm for R, the radius of curvature of the Plateau border, and of 10-6 dyn for the line tension, we obtain for the characteristic rate d In a/dt ~ 5 X 10 -6 s -l. This corresponds to a half-value time of strip thinning of the order of l0 to 100 h. This is very slow indeed. The high resistance against strip drainage could also be deduced from observations on isolated strips where rupture had taken place. While the radial flow in ruptured soap films is a very rapid process, in the 2-D foams we observed strips which were detached at one end, with the rest apparently frozen in the shape it had before rupture.

Equation [4] is analogous to Reynolds' equation (20) for the thinning of 3-D diskshaped films with rigid walls. It is well-known that genuine Reynolds drainage for soap films is very slow. In films with a high shear modulus, marginal regeneration is hampered by a high resistance against the lateral movement of film elements which it requires. Reynolds drainage is the only possibility and it results in a much slower rate of film thinning. The rate of thinning for 3-D films with completely mobile surfaces is larger (21 ) than that predicted by Reynolds' equation by a factor of(b/a) 2. When we assume the same applies to the 2-D strips, the thinning rate can be expected to be 10 4 times faster in the initial stage when the strip edges are still mobile. It should be stressed that in deriving Eq. [4 ], surface viscosity has been ignored and all the resistance against flow is caused by viscous friction in the liquid below the surface. Any surface viscosity would reduce the rate of thinning even further (22). MEC H A N IS M OF C E L L U L A R G R O W T H IN T W O - D I M E N S I O N A L FOAMS

In 3-D foam there is no space-filling arrangement of polyhedra with fiat faces which Journal of Colloid and Interface Science, Vol. 144,No. 2, July 1991

442

LUCASSEN, AKAMATSU, AND RONDELEZ

satisfies simultaneously the requirements, prescribed by mechanical equilibrium, of an angle of 120 ° between meeting films and an angle of 109 ° 28' between meeting Plateau borders (23). This means that in a 3-D foam there will always be curved surfaces and consequently there is always diffusion of gas from smaller to larger bubbles. No matter how well the films in a foam are protected against drainage and rupture, the foam will always experience disproportionation by diffusion and will ultimately disappear. For 2-D foams the situation is different in principle. A plane-filling arrangement of hexagons obeys the required meeting angle of 120 ° between the strips, while leaving them uncurved. Consequently, the diffusion between gas cells is absent. An array of hexagons in the absence of other polygons should therefore be indefinitely stable. This still should hold true if the hexagonal cells are not equalsized. As can be seen by comparing the sets of hexagons A, B, and C in Fig. 6, the total length of lines remains equal when the size of the central hexagon is increased or diminished. Experiments with quasi-2-D foams (3-D foams enclosed between two parallel glass plates) confirm that disproportionation does start at faults--usually consisting of pentagons or heptagons--in an originally hexagonal array of gas cells (4). Polygons with fewer than six sides will diminish in area by diffusion. This is because the mechanical requirement of a 120 ° meeting angle at the corners will force the sides to become curved outward which results in a higher surface pressure inside the cell. By the same token, polygons with more

than six sides will tend to increase in size. The relationship between the area change and the number of sides has been expressed quantitatively in Von Neumann's law (24). Von Neumann's paper is frequently referred to, but not readily accessible. We will derive this law for the simple case of regular polygons. For the more general derivation, the reader is referred to Von Neumann's original work. Let the number of sides of the polygon be n. The corner angle is then

For all values of n except 6, a will differ from 120 °, and in order to satisfy the requirement of force equilibrium, the polygon sides should become circular arcs with a base angle of

Figure 8A shows as an example the configuration for four-sided symmetry (n = 4), for which ~ equals 7r/12, or 15 °. When the radius of the circumscribed circle equals Rc, the radius of curvature of the arcs is found to be Rcsin ( zr / n ) sin 6

R -

[71

The rate of area change is proportional to the difference in surface pressure between the outside and inside of the gas cell,

2xrc

r R

r sin 6 RcsinOr/n) ,

[8]

with the length of its circumference L =

A

B

c

2 Rcn6 sin(~-/n) sin

[9]

and with a permeability coefficient t~. This results in yon Neumann's law:

FIG. 6. Arrayof hexagonalcellsof varyingpolydispersity.

The total line length is equal for configurations (A), (B), and (C). Journal of Colloid and Interface Science, Vol. 144,No. 2, July 1991

~--~LAxa-=27rr~ ~-

1 .

[10]

SPREAD MONOLAYERS

As the rate of area change is constant, the rate of change in the linear dimension of a gas cell should be inversely proportional to its size. For cells with fewer than six sides this means a dramatic increase in the rate of change just before they disappear. This accelerated decrease in size is shown by the last three photographs (D, E, and F) of Fig. 4. The time interval between the successive photographs is 2 s, while the change in size between E and F is more pronounced than that between D and E. There is another reason why the process of disappearance of small gas cells appears to be self-accelerating. For a cell with sides of unequal initial length, the area decrease will result in the loss of a side, an increased curvature, and an even larger rate of area change. The protection against disproportionation for 2-D foams with hexagonal gas cells of equal or nearly equal size--as predicted by Eq. [10]--is absent when the foam is in the process of being generated and the polygon structure has not yet been reached, i.e., when the area fraction of the disperse phase is lower than 0.9069. A dispersion of circular gas cells (the 2-D equivalent of a Kugelschaum) cannot be stable against disproportionation if there is only the slightest distribution of curvatures. Even when this distribution is initially very narrow, it cannot be prevented from continually widening. Thus, only if dense hexagonal packing can be obtained sufficiently quickly so that the size distribution of the circular cells is still narrow enough to allow formation of the required hexagonal array, disproportionation can be halted. In that case the curvature distribution for these circular cells is replaced by just two curvatures for the hexagons, the zero curvature for the straight strips, and one uniform single curvature for all the Plateau borders, determined by the phase ratio and the film width. Whether or not such a situation can be achieved will for each system depend on the rates of nucleation, foam cell growth, and imposed expansion.

443

REARRANGEMENTS OF FOUR-STRIP CONTACTS

When during the coarsening process of 3D foams small bubbles disappear, contacts between four, rather than three, films are occasionally formed. This gives rise to a mechanically unstable situation which is resolved by a sudden rearrangement resulting in two threefilm contacts. The same process occurs in twodimensional foams, as shown in Fig. 4. Like all changes in foam, the rearrangement can be ascribed to the tendency to reduce the system's free energy, which is accomplished by minimizing the total line energy and therefore the total strip length. This is schematically illustrated in Fig. 7. The four-strip contact in Fig. 7A gives a total length of 2,L]/2 = 2.83 L, while in Fig. 7B, with the two three-strip contacts, this has been reduced to (1 + ]/3)*L = 2.73L. In general, four-strip contacts originate because of the disappearance of a central gas cell as shown in Fig. 4 and as schematically depicted in Fig. 8. As long as this cell is present, the formation of direct four-strip contact is prevented and mechanical stability seems ensured. Yet, it is always observed that rearrangement occurs before the central gas cell has completely disappeared and that during the process the remainder of it moves toward one of the new three-strip contacts. Hereby

A

FIG. 7. A four-strip contact (A) is unstable with respect to two three-strip contacts (B) as the total strip length decreases in a transformation from (A) to (B).

Journal of Colloid and Interface Science, Vol. 144, No. 2, July 1991

444

LUCASSEN, AKAMATSU, AND RONDELEZ

the cell changes its symmetry from fourfold to threefold. This spontaneous process suggests that even in the presence of a central gas cell, rearrangement is accompanied by a decrease in free energy. In order to verify this, we must calculate the total strip length for both configurations shown in Fig. 8. We postulate that the part of the system in which the changes occur is enclosed in an imaginary diamond-shaped box with edge length a. Both in the left- and in the righthand-side situation we impose the condition that the strips should meet at 120 ° angles. The lengths r and s characterize the cell size in the fourfold and in the threefold symmetry,

respectively. A relationship between these quantities can be found if we assume that the cell area stays constant during the rearrangement,

Al = r 2 +

r21[sin2(Tr/12) /12

t a n ( ; / 1 2 ) }[11]

3s 2 { 1 An = -~- 2 cos(rr/6)

1]} 121

r

+ [sin2(Tr/6)

tan(Tr/6)

'

and All a r e the areas before and after the change, respectively. w h e r e A1

[131

A1 = AII

R -= s = r

~i

7r/12 1 1 + sinZ(Tr/12) tan(a-/12) 3 7r/2 3 cos(Tr/6) + 4 sin2(Tr/6) 4 tan(Tr/6)

The total length of the strips in the two cases are LI and L n , respectively:

E1 = a~{V1 + c o s , + V1 - c o s , ) + 4r

7r/12 sin(Tr/12)

[151

a Zll = .

sin(2~-/3)

+

f] -7-~ rr/6 ~Stsint,, / 6)

Journal of Colloid and Interface Science,

ergetically more favorable. Table I shows the value of (r/a)crit as a function of the shape of the enclosing box. Rearrangements hardly ever conform to the idealized shapes used in our calculations. Only the experiments shown in Fig. 4 come reasonably close to the case of e = 90 °. Taking for r and a the average of the four sides of the configuration, we find (r/a)c~t ~ 0.19. Considering the rather low measuring accuracy for strip lengths, this is in satisfactory agreement with the predicted value. Two other observa-

1 }. [161 2 cos(It/6)

Now we can determine the difference between L n and LI a s a function of the size of the central gas cell. For decreasing cell size, the difference gradually diminishes and changes sign at a critical size, (r/a)c,it, well before the cell has vanished. At that point the system with the three-sided cell becomes enVol. 144,No. 2, July 1991

[141

TABLE i (degrees) 30 22 14 6 2

~(degrees)

(r/a),~t

90 68.1 45.5 21.2 7.6

0.194 0.183 0.149 0.084 0.034

445

SPREAD MONOLAYERS

A

B

FlG. 8. Transformation of a four-sided cell (A) into a three-sided cell (B). See text.

tions for systems with slightly curved strips gave r/a values of 0.18 and 0.21. EFFECT OF "LINE ELASTICITY"

As mentioned under Experimental, it was found that the addition of small quantities of cholesterol to the fluorescent dye monolayer leads to a foam with a more uniform gas cell size and a higher stability (see Fig. 3 ). We will discuss a possible explanation for this observation. For three-dimensional foams, the disproportionation of gas bubbles is governed by Laplace's law which gives the hydrostatic pressure difference between the inside and outside of a bubble: 2oR

~p = --.

[17]

Two bubbles of equal size will have equal gas pressures but the equilibrium between them is labile, and any small fluctuation in size will cause a self-enlarging pressure difference. This pressure difference results in gas diffusion through the liquid phase from small to large bubbles and thus in disproportionation. The rate of this process will normally increase with a growing disparity in size except when there are changes in o- overcompensating those in R. Thus, if a decrease in surface tension of a shrinking bubble and a corresponding increase in that of an expanding one outweigh the changes in the bubble radii, disproportionation will cease. For an initially monodisperse bubble dispersion this occurs when the elastic modulus exceeds half the surface ten-

sion, as was first formulated by Gibbs (25): do- - > ed - d In A

2

[181

The two-dimensional analogue of this condition is easily derived, dr 0d- - > -r, dln L

[19]

where 0d is the elastic modulus of a surface phase boundary, which is defined as the change in line tension resulting from a fractional increase in line length. At present there is no known method for measuring line tension in two-dimensional foams, let alone for measuring possible changes due to strip extension. We have already indicated above that the drainage behavior of the strips in the foam suggests that there are likely to be gradients in line tension and therefore a finite line elasticity. Boys (26), as long ago as 1912, proposed a simple picture to visualize the origin of line tension for the abrupt transition between black and colored areas in soap films. He assumed that there are steep banks between areas of different thickness and that the line tension is due to "an increased surface equal in amount to the heights of these banks." Such banks could also exist at the phase boundaries in a monolayer, and the line tension should then be related to an effective surface tension multiplied by the bank height. Cholesterol could have a considerable "line activity," as it is likely to be sterieally incomJournal of Colloid andlnterface Science, Vol. i44, No. 2, July 1991

446

LUCASSEN, AKAMATSU, AND RONDELEZ

patible with the main component in the monolayer and is thus forced out into the onedimensional phase boundaries. It would therefore be enriched in the "banks" and in this way it could impart similar elastic properties to a line as it would to a surface. As the surface elastic modulus for cholesterol can be extremely high (27), it seems quite likely that cholesterol will have the effect of making the 2-D foam obey Eq. [19], reduce disproportionation, and thereby exert a stabilizing influence. Weis and McConnell (28, 29) have suggested that a similar effect of cholesterol on line tension could explain the behavior in a 2-D solid-fluid coexistence region for phospholipids. They observed that increasing concentrations of cholesterol in the phospholipid resulted in longer and thinner crystalline domains. THE RELATION BETWEEN SURFACE STRUCTURE AND PHENOMENOLOGICAL SURFACE PROPERTIES Disperse structures such as two-dimensional foams confer to a surface the character of a composite material and will therefore affect its macroscopic mechanical and rheological properties. Their three-dimensional counterparts, foams as well as concentrated emulsions, are non-Newtonian fluids with a considerably higher viscosity than that of the component Newtonian fluids. This is due to the domination of the rheological behavior by interfaces. A similar situation prevails in two-dimensional foams, where lines take the place of interfaces. Both interfacial shear and dilational moduli can be expected to be considerably larger than they are for the component phases, especially when a small cell size causes a large line length per unit area and a correspondingly high resistance against deformation. Princen (7) approached the shear rheology of 3-D foams with a two-dimensional model, but he considered "such a system unrealistic." As real 2-D foams are now known to exist, Princen's conclusions with regard to two-dimensional shear modulus and yield stress are Journal of Colloid and Interface Science, Vol. 144, No. 2, July 1991

directly applicable to the foams discussed in the present paper. The main message is that both shear modulus and yield stress are expected to be proportional to the line (or strip) tension and inversely proportional to the gas cell dimension. It should be emphasized that especially the surface shear rheology is totally dominated by the line tension; for the separate 2-D gas and liquid phases, shear modulus and yield stress are usually vanishingly small. A combination of microscopical observation of the geometry o f a 2-D foam with Princen's theory could provide in principle a useful method for deriving line tension from the measured surface shear modulus. An analysis of the dilational surface properties could also give information on the line tension. Derjaguin (30) was the first to obtain an expression for the compressibility (or inverse dilational modulus) of a 3-D foam. His derivation was based on free-energy considerations. Variations in total surface area and in the state of compression of the gas phase were taken into account and the liquid phase was considered incompressible. Schwartz and Princen (31 ) as well as Ross and Morrison (32, 33) extended and refined Derjaguin's treatment. The only major difference in the case of the two-dimensional foam is that not only the constituent 2-D gas, but also the liquid phase, has a finite compressibility. Details of the derivation of the modulus for such a composite surface will be given elsewhere (34). A structured two-phase system in a surface will affect not only the dynamic dilational and shear properties but also the static surface pressure. According to the phase rule, this surface pressure should at full thermodynamic equilibrium be invariant in a coexistence region between two adjoining phases. Princen (35, 36) did show, however, that an osmotic pressure should be attributed to concentrated dispersions, which has its origin in the resistance against deformation of dispersed droplets or bubbles. The smaller the particles are, the larger this resistance is. When Princen's argument is extended to a 2-D foam, it leads to the conclusion that the observed surface

447

SPREAD MONOLAYERS

pressure lies above its equilibrium value. When the foam eventually disappears, either by strip rupture or by gas cell growth through disproportionation, this osmotic pressure vanishes and the surface pressure adopts its thermodynamic value. If the monolayer material is genuinely pure, there is no mechanism for long-term stabilization of a 2-D foam and therefore this should lead to a fast approach of the equilibrium surface pressure. For a highly purified monolayer, it has actually been observed (37) that in a phase coexistence region a constant surface pressure was obtained more rapidly than usual. CONCLUDING REMARKS

We have discussed various striking properties of 2-D foams in monolayers spread at the air-water surface, such as the effect of line tension on the mechanism of cellular growth, the sudden reduction of the rate of strip thinning, and the existence of a critical size for the rearrangement of a four-sided to a three-sided gas cell. It should be stressed that systems with the topology and the characteristic rheological properties of two-dimensional foams are not limited to gas-liquid coexistence regions in monolayers. Lrsche and Mrhwald (38), for example, observed in a coexistence region between liquid-expanded and liquid-condensed phases of a phospholipid monolayer a structure which appears indistinguishable from a 2-D foam (Fig. 4 of Ref. (38)). However, an important difference is that the foam-like structure is formed by compression rather than by expansion as in this case the disperse phase is the most condensed one. Systems such as that observed in Ref. (38) can be considered the 2-D analogs of concentrated (or high internal phase ratio) emulsions. It can also be expected that systems with the structure of a 2-D foam or concentrated emulsion are not necessarily confined to phase coexistence regions but could also be formed from two monolayers with a limited miscibility. An important property of such systems

would be that the range of surface pressures at which a foam structure occurs can be very wide. The rheology could be manipulated by varying the degree of dispersion and the phase ratio and this provides a flexibility which could make them ideal candidates for introducing strength in some biological surfaces. It would be interesting to investigate whether the lung surfactants (39), the deficiency of which causes the respiratory distress syndrome in newborn babies, give the required mechanical properties to alveolar surfaces by forming a structure similar to that of a 2-D foam. A treatment of the syndrome is based (40) on injection of suspensions of dipalmitoyl phosphatidyl choline in the lungs. One could imagine a mechanism by which the spreading of small particles of phospholipid on a surface already covered with protein gives rise to a composite surface with the topology and the mechanical characteristics of a 2-D foam. ACKNOWLEDGMENTS The financial support by Elf Aquitaine is gratefully acknowledged. One of us (J.L.) thanks Dr. C. Jablon (D.R.D.I/EIf) for arranging his sabbatical stay at the University of Paris VI. We thank Dr. E. H. Lucassen-Reynders for many critical remarks. REFERENCES 1. Bragg, W. L., and Nye, J. F., Proc. R. Soc. London A 190, 474 (1947). 2. Aboav, D. A., Metallography 13, 43 (1980). 3. Weaire, D., and Rivier, N., Contemp. Phys. 25, 59 (1984). 4. Beenakker, C. W. J., Phys. Rev. Lett. 57, 2454 (1986). 5. Stavans, J., and Glazier, J. A., Phys. Rev. Lett. 62, 1318 (1989). 6. Flare, F., Sci. News 136, 72 (1989). 7. Princen, H. M., J. Colloid Interface Sci. 91, 160 (1983). 8. Kraynik, A. M., Annu. Rev. Fluid Mech. 20, 325 (1988). 9. Reinelt, D. A., and Kraynik, A. M., J. Fluid Mech. 215, 431 (1990). 10. Moore, B., Knobler, C. M., Broseta, D., and Rondelez, F., ~L Chem. Soc. Faraday Trans. 2 82, 1753 (1986). 11. Rondelez, F., Baret, J. F., and Suresh, K. A., in "Physico-Chernical Hydrodynamics," Vol. 174, p. Journal of Colloidand InterfaceScience, Vol. 144,No. 2, July 1991

448

LUCASSEN, AKAMATSU, AND RONDELEZ

857. NATO ASI Series, Physics. Plenum, New York, 1988. 12a.Stine, K. J., Rauseo, S. A., Moore, B. G., Wise, J. A., and Knobler, C. M., Phys. Rev. A 41, 6884 (1990). 12b.Berge, B., Simon, A. J., and Libchaber, A., Phys. Rev. A 41, 6893 (1990). 13. Adam, N. K., "The Physics and Chemistry of Surfaces," p. 43. Oxford Univ. Press, London/New York, 1941. 14. Lrsche, M., and M~Shwald, H., Rev. Sci. Instrum. 55, 1968 (1984). 15. Schwarz, H. W., Rec. Tray. Chim. Pays Bas 84, 771 (1964). 16. Mysels, K. J., Shinoda, K., and Frankel, S., "Soap Films." Pergamon, London/New York/Paris, 1959. 17. Lucassen, J., in "Surfactant Science Series," Vol. 11, p. 217. Dekker, New York, 1981. 18. Mysels, K. J., Langmuir 4, 1305 (1988). 19. Landau, L., and Lifchitz, E., "Mrcanique des Fluides," p. 304. Editions Mir, Moscow, 1971. 20. Reynolds, O., Philos. Trans. R. Soc. London 177, 157 (1886). 21. Vrij, A., Hesselink, F. Th., Lucassen, J., and van den Tempel, M., Proc. K. Ned. Akad. Wet. B73, 124 (1970). 22. Hansen, R. S., J. Phys. Chem. 63, 637 (1959). 23. Princen, H. M., and Levinson, P., J. Colloidlnterface Sci. 120, 172 (1987).

Journal of Colloid and Interface Science, Vol. 144, No. 2, July 199 l

24. Von Neumann, J., in "Metal Interfaces" (C. J. Herring, Ed.), p. 108. American Society for Metals, Cleveland, Ohio, 1952. 25. Gibbs, J. W., "The Scientific Papers," Vol. 1, p. 244. Dover, New York, 1981. 26. Boys, C. V., Proc. R. Soc. London A 87, 340 ( 1912). 27. Lucassen, J., Trans. Faraday Soc. 64, 2230 (1967). 28. Weis, R. M., and McConnell, H. M., Nature 310, 47 (1984). 29. Weis, R. M., and McConnell, H. M., J. Phys. Chem. 89, 4453 (1985). 30. Derjaguin, B., KolloidZ. 64, 1 (1933). 31. Schwartz, L. W., and Prineen, H. M., J. Colloidlnterface Sci. 118, 201 ( 1987 ). 32. Ross, S., Ind. Eng. Chem. 61, 48 (1969). 33. Morrison, I. D., and Ross, S., Z ColloidlnterfaceSci. 95, 97 (1983). 34. Lucassen, J., "Dilational Surface Properties of Composite Surfaces," submitted for publication. 35. Princen, H. M., Langmuir 2, 519 (1986). 36. Princen, H. M., Langmuir4, 164 (1988). 37. Pallas, N. R., and Pethica, B. A., J. Chem. Soc. Faraday Trans. 1, 83, 585 (1987). 38. Lrsche, M., and Mrhwald, H., Eur. Biophys. J. 11, 35 (1984), 39. Bangham, A. D., Lung 165, 17 (1987). 40. Notter, R. H., and Shapiro, D. L., Clin. Perinatol. 14, 433 (1987).