Langmuir 1986,2,519-524
519
Summary
I/ 40
43
46
48
52
55
Temperature
50
61
04
87
70
L-B films of tetrakis(cumy1phenoxy)phthalocyaninestearyl alcohol mixed monolayers strongly interact with iodine vapor, and it is possible to simultaneously measure the conductivity and gravimetric changes in the film by a novel planar microelectrode surface acoustic wave measurement technique. The conductivity increased by 4 orders of magnitude and a complex formation stoichiometry of two to four iodine atoms per phthalocyanine ring was measured. The identity of the complexed central metal ion has very little effect on either the magnitude of the conductivity increase or the complex stoichiometry. The conductivity measurement is dependent on the multilayer film thickness unless the film is thicker than the planar microelectrode. The quantity of iodine a phthalocyanine film may absorb is dependent on the film morphology while the magnitude of the conductivity increase is nearly independent of the morphology.
PC)
Figure 9. DSC thermograms of (a) a freshly prepared metal-free phthalocyanineatearyl alcohol (k1mol ratio) 183-layerL-B film, (b) a fused (recycled scan) L-B film,and (c) a pure stearyl alcohol
L-B film.
a t 58.5 “C. These thermograms indicated that the morphology of the G B film is more complicated than a simple fused mixture and is irreversibly altered by the 100 “C thermal treatment.
Acknowledgment. We thank Phillip Berg and Russell Jeffries for making the electrode thickness measurement by interferometry and Wanda Carter for obtaining the DSC data. Registry No. CuPcCP, 93530-47-3; ZnPcCP, 93530-48-4; PtPcCP, 93530-49-5;PdPCCP, 93530-50-8;COPCCP,93530-45-1; NiPcCP, 93530-46-2; H2PcCP,93530-40-6; I,, 7553-56-2.
Osmotic Pressure of Foams and Highly Concentrated Emulsions. 1. Theoretical Considerations H.M. Princen Corporate Research-Science Laboratories, Exxon Research and Engineering Co., Clinton Township, Annandale, New Jersey 08801 Received January 22, 1986. In Final Form: April 14, 1986 The concept of osmotic pressure of foams and concentrated emulsions is extended from a previous 2-D model to real systems. The osmotic pressure is proportional to the interfacial tension and inversely proportional to the mean bubble or drop radius, while it varies from zero to infinity as the volume fraction, 4, of the dispersed phase is raised from about 0.74 to unity. Limiting solutions are presented for the ranges of high and low 4. Links are established between the osmotic pressure and other physical properties and phenomena, such as the reduction in vapor pressure above the system; osmotic flow of continuous phase between two different, contacting systems; the gradient in volume fraction in a foam or emulsion in a gravitational or centrifugal field; and the liquid volume in an infinitely tall, equilibrated foam column.
Introduction In our study of the fundamental properties of concentrated fluid/fluid dispersions, Le., foams and emulsions, we have so far dealt with the structure1i2and rheological properties, such as the shear modulus and yield Another property of interest and related to the above is the “osmotic pressure”, II, of these systems. The concept (1) Princen, H.M. J. Colloid Interface Sci. 1979, 71, 55. (2) Princen, H.M.; Aronson, M. P.; Moser, J. C. J. Colloid Interface Sci. 1980, 75, 246. (3) Princen, H. M. J. Colloid Interface Sci. 1983, 91, 160. (4) Princen, H.M. J. Colloid Interface Sci. 1986,105, 150. (5) Princen, H.M.; Kiss, A. D. J. Colloid Interface Sci., in press.
0743-7463/86/2402-0519$01.50/0
was introduced in ref 1 and is akin to the more familiar concept of the osmotic pressure of conventional solutions. In the following paragraph, the most closely related terms for solutions are quoted in parentheses. In the case of foams or concentrated emulsions, in which the volume fraction @ of the dispersed phase exceeds a value of about 0.74, the bubbles or drops (the “solute molecules”)are no longer spherical but are deformed into more or less polyhedral entities. As a result, their surface free energy (“chemical potential”) is increased. If the system communicates with continuous phase via a freely movable, semipermeable membrane, which is permeable t o all the components of the continuous phase but impermeable to the drops or bubbles, the continuous phase 0 1986 American Chemical Society
Princen
520 Langmuir, Vol. 2, No. 4, 1986 and Continuous Phase
(5) ---)
Membmne
When eq 5 is inserted in eq 2, we find Emulsion
Figure 1. Semipermeablemembrane separating a foam or con-
centrated emulsion from continuous phase.
(the “solvent”) tends to pass through the membrane into the dispersion in order to decrease the deformation of the dispersed units and the associated excess surface free energy. The flux of continuous phase can be suppressed by applying a pressure, 11, to the membrane. This is the osmotic pressure of the system; it equals the force exerted by the dispersed phase per unit area of the membrane. As in conventional solutions, II increases with increasing concentration, or 4. In ref 1 we showed that for a monodisperse, 2-D model, II is given by
where u is the interfacial tension, R is the radius of the cylindricaldrops (when undeformed), and 4ois the volume fraction corresponding to close-packed, monodisperse circular cylinders (@o = 0.9069). In real, 3-D systems the dependence of 11 on 4 is expected to be considerably more complicated, but the simple dependence on u / R will be retained. It is the purpose of this study to probe the dependence of II on 4 on the basis of geometrical models and to indicate that detailed knowledge of II(4) enables one to solve a number of interesting practical problems. It will be assumed throughout that the systems, although metastable in a thermodynamic sense, are stable from a practical point of view. Thus, on the time scale of a typical experiment, there is no significant change in drop or bubble size as a result of either coalescence or diffusion (Ostwald ripening). It is furthermore assumed that the contact angle at the perimeter of the thin films of continuous phase is Theory Free-Energy Approach. Consider a closed container in which a volume V of foam or concentrated emulsion is separated from the continuous phase by a movable, semipermeable piston or membrane (Figure 1). Gravity is considered to be absent or negligible, so that 4 is uniform in the fluid dispersion. To maintain equilibrium, a pressure 11 must be applied to the membrane. When the membrane is displaced downward to force an infiitesimal volume dV of continuous phase from the fluid dispersion into the upper layer of continuous phase, we may write for this reversible process: -11 dV = u d S (2) where d S is the increase in interfacial area resulting from the increased deformation of the drops or bubbles; u is considered to be constant. Let VI and V2 be the volumes of the dispersed and continuous phases in the dispersion. Then, V = VI + V, (VI = constant) (3) (4)
where S/Vl is the surface area per unit volume of the dispersed phase. Equation 6 is a general result that applies to polydisperse as well as monodisperse systems. When integrated, it yields
where do is the volume fraction corresponding to closepacked spheres (where n = 0), and Sois the corresponding surface area. For a monodisperse system, +o = 0.7405 and So/ Vl = 3 / R . For typical polydisperse systems, we have experimental evidence that, if anything, r j o is usually smaller (=0.72), which implies somewhat less efficient pa~king.~p*?~ The exact value of 4oshould depend slightly on the details of the size distribution. Also, for a polydisperse system
SO/Vl = 3/R32
(8)
where R32 is the Sauter or surface-volume mean radius of the (spherical) drops. As 4 increases, the drops increasingly deform until, at 4 = 1, all drops are fully developed polyhedra and S S,. For a monodisperse system, it is reasonable to assume5 that in this limit s,/s, = 1.10 (9)
-
independent of the assumed type of polyhedron (Le., rhombic or pentagonal dodecahedron, or tetrakaidecahedron). However, since none of these polyhedra are true minimum-surface bodies (the edges and faces do not always meet at the required angles of 109’28’ and 120°, respectively), the correct value of S,/So may be slightly smaller, as for Kelvin’s “minimal tetrakaidecahedron”.6 On the other hand, for reasons stated b e f ~ r epolydis,~ persity may give rise to slightly larger values. When eq 8 and 9 are inserted in eq 7 for 4 = 1,we find (10)
where
fi is the reduced osmotic pressure, defined n- l - n u/R32
as (11)
+
Although the exact dependence of fI (or S/So) on is unknown, eq 10 may serve as a test of any proposed theory, provided the estimate of eq 9 is not significantly in error. Unfortunately, small deviations in S,/So are greatly magnified in the value of the integral. Finally, it is noted that S,/So plays an important role in determining the shear modulus of these system^.^ Force Approach. Let the semipermeable membrane be perfectly wetted by the continuous phase, so that the (6) Ross, S.Am. J . Phys. 1978, 46, 513.
Langmuir, Vol. 2, No. 4, 1986 521
Osmotic Pressure of Foams and Emulsions F=ll
1 (15)
-
A similar calculation for the tetrakaidecahedron (not shown) leads to
Membrane
Film
Drops
Figure 2. Semipermeable membrane and adjacent layer of flattened drops or bubbles. Thickness of film between drops and membrane highly exaggerated. adjacent dispersed drops are flattened against it but remain separated from it by a thin film of continuous phase. The thickness of this film is determined by a balance of the thickness-dependent disjoining pressure, l, in the film and the capillary pressure, II, = aK, where K is the mean curvature of the free drop interface outside the film (Figure 2). As a result, per unit area of the membrane the drops exert a force F which is equal to the osmotic pressure and is given by
where f(4) is the fraction of the membrane area occupied by the thin films. It has been discussed before in connection with wall slip effects of concentrated emulsions in viscometric measurements: As 4 goes from q50 to unity and the drops deform from spheres into polyhedra with increasingly curved edges, it is clear that f(4) varies from zero to unity, while ~ ( 4varies ) from 2/R to infinity between these limits. Hence, I I = O at II-00
R32*
“Low”Volume Fraction. Unless 4 approaches unity, it is extremely difficult to analyze the deformation of the drops and to predict how f and K in eq 12, or S/Vl in eq 6, vary with 4. It is clear, however, that, for reasonably small deformation,f(4) increases rapidly while ~ ( 4remains ) relatively constant, i.e., 4 4 ) = 2/R (17)
In ref 4 we have measured f(4) for typical, polydisperse oil-in-water emulsions and found empirically that, up to 4 = 0.975, 3.20 f(4) = 1 (R, + 7.70)’12 where R, is the volume ratio of the dispersed and continuous phases, Le.,
as 4 - 1
However, as is the case with S/So, we do not know a priori ) with 4 over the whole range of how f(4) and ~ ( 4 vary interest.
-
Limiting Solutions High Volume Fraction (4 1). As the volume fraction approaches unity (we shall neglect, for the time being, the thickness of the films between the drops), the drops become more and more polyhedral. What little continuous phase is left resides in the linear and tetrahedral plateau borders along the sides and at the corners of the polyhedra, respectively. For sufficiently high 4, the volume of the tetrahedral borders becomes negligible compared to that of the linear borders. The latter have constant cross section and are bounded by three circular arcs of radius r . For the rhombic dodecahedron of equivalent-sphere radius R , we have shown2 that
For the regular pentagonal dodecahedron, ~ ( 4was ) found to be about 5% higher.2 It can be shown that for the tetrakaidecahedron (or truncated octahedron), K ( $ ) , is about 1.5% lower than that predicted by eq 13. Thus, again, the result is rather insensitive to the particular choice of the ultimate polyhedron, and we shall use the rhombic dodecahedron as the basic unit. It is shown in the Appendix that, for this geometry, f(4) can also be readily evaluated. The result is
f(4) = [ I - 1.892(1 - 4)’/’]’ so that, from eq 12-14,
which is again very close to eq 15. The above results are valid for monodisperse systems; for polydisperse systems, R is replaced by the mean radius
(4
-
1)
(14)
One difficulty with that particular experimental system was that, to explain certain results for the yield stress, we had to assume that, at least for R, = 6 or 4 = 0.85, the aqueous films between neighboring drops had an equilibrium thickness, h, of about 0.1 pm, which is not negligible compared to the mean drop radius R 3 2 = 8.75 pm. Since we wish to refer all properties to systems with h / 2 R = 0, eq 19 has to be corrected in two ways: (1) Because of the presence of the films, the individual areas of contact of the drops with the membrane are separated more from each other than would be the case if the film thickness were truly negligible. It can be shown that this leads to a small upward correction in f by a factor of (1 h/2R3# = 1.011. (2) More important, because of the finite film thickness, the system has a higher “effective volume fraction” than the actual experimental volume fraction + ( h ) . It can be that the corrected value of $I is related to 4(h)via h 1 -1 -= 1.1054(h)1/3 2R32
+
Assuming that h was constant over the limited range of 4 considered here, we have corrected eq 19 accordingly and plotted the results in Figure 3. The shift in 4 amounts to about 0.014. Figure 4 shows the two limiting solutions for fi, as calculated according to eq 16 and 18, up to their intersection at 4 = 0.925. The lower limit represents a lower bound, since ~ ( 4must ) increase moderately with 4 also. At intermediate values of 4 there will be a smooth transition from one limit to the other, such that eq 10 is obeyed. One has surprisingly little freedom in drawing a
522 Langmuir, Vol. 2, No. 4, 1986
Princen
-
Form
or
Emulsion
= 9.p 0.74
Z=O0.70
0.75
0.80
-
0.85
Continuous Phrw
$
Figure 3. Fractional area of membrane ”contacted* by adjacent drops as function of c$.
f
4 1
3
Figure 4. Reduced osmotic pressure R(qj). The dashed lines, intersecting at the vertical mark, are limiting solutions for high and low volume fraction, respedively. I is trial curve that conforms to SJS, = 1.10; 11 is the same for &/So = 1.083. curve that meets all three requirements. One such approximate curve (I) is indicated in Figure 4. Curve 11, which clings to the limiting solutions much more closely, corresponds to the case where S,/So = 1.083 so that the value of the integral in eq 10 is 0.25, instead of 0.30.
Problems Involving Osmotic Pressure One can think of a variety of problems that may be approached by using the concept of osmotic pressure. Some of these are the following. Vapor Pressure of Continuous Phase. It was concluded in ref 1that the vapor pressure, pv, of the continuous phase above a foam or concentrated emulsion is reduced below that of the bulk continuous phase, (pJ0, according to pv =
(pV)oe-*v/RT
(22)
where P is the molar volume of the continuous phase, R is the gas constant, and Tis the absolute temperature. For example, for an oil-in-water emulsion with u = 5 dyn/cm, R32= 5 pm, and I#J = 0.99, we calculate from eq 22 II = 3.9 X lo4 dyn/cm2, so that pV/(pv),,= 0.99997, a very small effect indeed. For the effect to become appreciable, R32 must be much smaller or I#J must approach unity much more closely. Emulsions or Foams in Contact. When two concentrated fluid dispersions have different osmotic pressures and are brought into contact, either directly or via a freely
Figure 5. Ruilibrated foam or emulsion column in gravitational
field.
movable semipermeable membrane, the (common) continuous phase will move from the system with the lower to that with the higher II, until the osmotic pressures are equalized. (Even without a membrane separating the systems, the droplets are not expected to “mix”,since their relative positions are essentially locked in because of the high degree of crowding.) A rather trivial example is that of two emulsions that differ only in 4. Continuous phase will move from the more dilute emulsion (low II) to the more concentrated one (high II) until the two emulsions have become identical in all respects. From the initial volumes and volume fractions, it is straightforward to calculate the final, uniform volume fraction. No knowledge of II(I#J) is required. More interesting is the case where the contacting emulsions differ in mean drop size but have identical volume fractions and interfacial tensions. The emulsion with the smaller RB2has the higher II, so that continuous phase is sucked into it. At equilibrium, II must be equalized in both emulsions, which implies not only that their volumes change but that their final volume fractions are, in fact, different. To predict the final situation, II(I#J) must be known, along with the initial volumes, volume fraction, and mean drop sizes. Emulsions and Foams in an External Field. When a column of foam or emulsion is placed in a gravitational or centrifugal field, continuous phase drains out of the column and collects at the bottom (or at the top, if the dispersed phase is the heavier phase, as in typical waterin-oil emulsions). At equilibrium, drainage ceases and a gradient in I#J is set up along the field direction, z (Figure 5). Knowledge of $ ( z ) is important, for example, in describing the process of foam drainage (e.g., ref 7-9). Let us assume that the drops or bubbles are sufficiently small, such that the capillary pressure inside a (spherical) drop far exceeds the change in hydrostatic pressure across one drop diameter: 2a/R
>> 2R(Ap)g
or
where Ap is the density difference between the phases and g is the acceleration due to gravity. Then, the drops at z = 0 are essentially spherical (“kugelschaum”),so that d(O) (7) Leonard, R. A.; Lemlich, R. MChE J. 1966,11, 18. (8) Kann, K. B.Colloid J. (USSR)(Engl. Trawl.) 1978,40,714,927; 1979, 41, 364, 569, 574. (9) Krotov, V. V. Colloid J.(USSR)(Engl. Trawl.) 1980,42,903,912; 1981, 43, 33, 233.
Osmotic Pressure of Foams and Emulsions
Langmuir, Vol. 2, No. 4, 1986 523
i1 .M 1 1
0.95
0.90
0
0.85
0.80 0.75 1
I
I
1
1
2
3
4
z Figure 6. Volume fraction 4 w.reduced height 2, as derived from curves I and I1 in Figure 4.
- -
= m0 = 0.74. As z a,the drops assume the shape of fully developed polyhedra and 6 1 ("polyederschaum"). Now imagine a horizontal surface of unit area placed in the column at height 2. The upward force exerted on the bottom of this surface by the dispersed phase equals the local osmotic pressure and is given by the combined buoyant force of all the drops below the surface, i.e.,
(24)
or
If there is a vertical gradient in R32 as well, its local value must be used in eq 25. If the column is subjected to a centrifugal field, g is replaced by u2L, where L is the distance to the center of rotation. Equations 24 and 25 provide the link between n($)and +(z). Clearly, for fi to be a unique function of 4, the latter must be a unique function of ApgRS2z/uor
where u = (a/Apg)1/2is the capillary constant, and R32Z a2 Therefore, the simplest form of eq 25 is
and, conversely,
Hence, once either fi(4) or m(Z) is firmly established, the other may be derived from eq 27 or 26, respectively. For example, the approximate curves for fi(r#~)in Figure 4 lead to the corresponding functions &(E) shown in Figure 6. In reality,an equilibrated foam or emulsion column does not necessarily rest on a layer of continuous phase but may extend upward from a lower region where 4 > & In that case, the +(z) plot of Figure 4 is realized only from that particular volume fraction upward, while the z = 0 level is located at a corresponding distance below that of the bottom of the column. Liquid Content in an Infinitely Tall, Equilibrated Foam Column. This problem was poaed by Krotov? who arrived at a range of answers, depending on the assump-
Figure 7. (a) Foam or emulsion column resting on continuous phase in a U-tube. (b) T w o arms connected via semipermeable plug.
tions made. Of course, once d(2) is f m l y established, the correct answer is obtained from
V / A = Jm(l - 4) dz = (a2/R32)xm(1- 4) dz' 0
(28)
where V is the volume of continuous phase in an infinitely tall column of cross-sectional area A. The answer is finite and is represented by the area above either of the curves in Figure 6. For these curves, when continued to 4 = 1 with our limiting solution, i.e., eq 15 combined with eq 27 and 28, we find
V / A = O . ~ & I ~ / R (curve ~ ~ I)
(294
or
V / A = 0 . 2 4 ~ ~ / R ~(curve ~ 11) (29b) Both values are considerably in excess of the maximum value of Krotov's proposed range of 1 u2 v 1 a2 - $, and ll > 0 there. However, this upsets the equilibrium m(z) in the right-hand arm, so that the continuous phase will drain down to the bottom, while more continuous phase enters through the plug. This sets up a continuous clockwise circulation which may be used for the perpetual generation of energy. What is wrong with this scheme? Concluding Remarks Speculation on m(z) of foams have been advanced before by Kann: Krotov,Band Pertsov et d.l0 They use the local "aeration ratio" (10)Pertaov, A. V.; Chemin, V. N.; Chistyakov, B. E.; Shchukin, E.
D.Dokl. Akad. Nauk SSSR
1978,238,1395.
524 Langmuir, Vol. 2, No. 4,1986
Princen I
A
I
to characterize the system. On the basis of a polyhedral model, Karma arrives at
K = (c'z
+ C)2
(31) where c' and C are purported to be constants, but their values are not clearly defined. Kroto@ interprets Kann's work as indicating that C = 1.96, which follows from putting z = 0 in eq 31 and @(O) = 0.74 in eq 30; the constant c' is still undetermined, however. Pertsov et al.1° conclude that C = 1.63 and c' = 1.63R/a2,while Krotov: on the basis of improved geometrical arguments, corrects these values to C = 1.74 and c' = 1.74R/a2. Consistent with this, Krotov predicts that for large z:
K = ( ~ ' 2 )=~ 3 . 0 3 ( R z / ~ ~=) 3.03Z2 ~
(32)
(b)
(8)
Figure 8. (a) Rhombic dodecahedron which circumscribes each
drop in a 3-D, monodisperse system. AB is a threefold symmetry
axis which runs perpendicular to the wall. (b) Film between drop and solid wall at high 4 (shaded area). Larger hexagon is projected drop outline.
Our own analysis, embodied in eq 15 or 16, coupled with eq 27, when applied to values of 4 extremely close to unity, leads to
dodecahedron but it will have rounded edges, whose radii of curvature are r. As shown in eq 15 of ref 2
or
It is reasonable to assume that the droplets are arranged in layers in hexagonal packing parallel to the confining wall. This can be accomplished only when one of each droplet's threefold symmetry axes, such as AB in Figure 8a, is normal to the wall. Then, the projection of each droplet onto the wall (i.e., its available cross-sectional area) is a regular hexagon whose side length, c, is equal to the width of one of the rhombi (Figure 8a). It can be shown from the shape of the rhombus that
Z=fi=
0.5759 (1 - 4)'/2
-
K = 3.02Z2
(33b)
depending on whether the rhombic dodecahedron or the tetrakaidecahedron is used as the basic polyhedron. In either case, the result is in very close agreement with Krotov's. However, the utility of eq 32 and 33 is restricted to large 2 or K . Although an equation of the type of eq 31 may be better capable of describing the system, it can do so only over a limited range of 2 or 4, since c' and C cannot strictly be pure constants. Our analysis above extends the description to the region of low 4 and into the transition region but there is a limit to how far one can go along this route. Instead, we are initiating careful experimental work that is aimed at the accurate measurement of n($)and/or 4(2) over the full range of 4. Two approaches are being considered: (i) the direct measurement of ll of well-characterized systems in a "conventional" osmometer and/or (ii) the measurement of 4 ( z ) in an equilibrated emulsion column. Experience to date suggests that the latter is the more convenient and fruitful approach.
Appendix Fractional Area f(4) for 3-D, Monodisperse System As 4 1. Each face of the dodecahedron'l is a rhombus with a major diagonal a and a minor diagonal, b, such that a / b = 2ll2 (Figure &I). At high volume fraction, the shape of each drop away from the wall approaches that of the
-
(11) Lissant, K.J. J. Colloid Interface Sci. 1966, 22, 462.
c = 0.5773~
(-4-2) Each dodecahedron immediately adjacent to the wall will be flattened against it; i.e., three of its rhombic surfaces disappear and are replaced by a single hexagon of side length c parallel to the wall. Because of the rounded edges of the drop, the flat film between the drop and the wall occupies a smaller hexagon whose sides are moved inward over a distance r (Figure 8b). The fraction, f , of the wall area occupied by the films is then simply the ratio of the areas of the smaller and larger hexagons. It is readily shown that the length of the sides of the smaller hexagon is
e = c - 2r tan 30 = c - 2r/3'l2
64-3)
so that
Substituting for c and r according to eq A-2 and A-1, we find
f = [l - 1.892(1 - $)'/2]2
($
-+
1)
(A-5)
