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Spinning Drop Method Applied to Three-Phase Fluid Equilibria H. M. Princen Strategic Research Center, Paulsboro Technical Center, Mobil Technology Company, Paulsboro, New Jersey 08066 Received March 19, 1999. In Final Form: June 4, 1999 The spinning drop technique is used routinely for measuring single surface or interfacial tensions. We present a simple analysis that extends the method in such a way that three interfacial tensions and three contact angles can be measured simultaneously in a single experiment. In this method, adhering long drops of immiscible fluid phases 1 and 2, surrounded by a heavier immiscible phase 3, are spun about the horizontal tube axis, and the radii of the phase-1 and phase-2 drops, as well as that of the three-phase circle of contact, are measured. The technique may lend itself to the convenient assessment of the potential of a given liquid as an effective antifoaming agent for another liquid.
Introduction The spinning drop technique, in its classical form,1-4 has become one of the most widely accepted absolute methods for measuring surface and interfacial tensions. A horizontal tube is filled with a liquid, and a drop of a less dense, immiscible phase (either a gas or another liquid) is injected. Upon rotation of the tube about its axis, the bubble or drop moves to the axis and increasingly elongates as the angular velocity is raised. At sufficiently high speeds, the drop acquires the shape of a long cylinder with roundedsthough not hemisphericalsends (Figure 1). In this regime, the interfacial tension, γ, is simply given by Vonnegut’s equation:1
γ)
∆Fω2R3 4
(1)
where ∆F is the density difference between the phases, ω is the angular velocity in radians/second, and R is the drop radius in the cylindrical region. The latter is usually measured through a low-power microscope with a micrometer eyepiece and needs to be corrected for the magnification resulting from the curved tube and its contents;5 i.e.,
D ) 2R ) Dapp/n
(2)
where D and Dapp are the real and apparent drop diameters and n is the refractive index of the outer liquid phase (the tube wall can be shown to have no effect). The method has at least four advantages: (1) it does not depend on the wettability of the tube wall; (2) it is suitable for measuring extremely low interfacial tensions; (3) it can handle very viscous systems where other methods may fail; and (4) it can be used for the simultaneous measurement of two tensions, e.g., by injecting an air bubble and an immiscible (1) Vonnegut, B. Rev. Sci. Instrum. 1942, 13, 6. (2) Princen, H. M.; Zia, I.; Mason, S. G. J. Colloid Interface Sci. 1967, 23, 99. (3) Princen, H. M. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley-Interscience: New York, 1969; Vol. 2, pp 1-84. (4) Cayias, J. L.; Schechter, R. S.; Wade, W. H. In Adsorption at Interfaces; Mittal, K. L., Ed.; ACS Symposium Series No. 8; American Chemical Society: Washington, D.C., 1975; p 234. (5) Puig, J. E.; Seeto, Y.; Pesheck, C. V.; Scriven, L. E. Chem. Eng. Commun. 1990, 93, 125.
Figure 1. Long cylindrical spinning drop.
Figure 2. Spinning drop and air bubble (separated).
Figure 3. Spinning drop and air bubble (adhering).
liquid drop in the outer liquid in the same tube (Figure 2). This way, one single experiment yields the surface tension of the outer liquid, as well as the interfacial tension between the two liquids. We have used this procedure routinely and successfully in the past. However, on occasion, the bubble and the drop would come in contact and adhere, with the formation of a liquid-2/air interface between them, as in Figure 3. (The conditions for adherence are addressed in the “Discussion” section below.) As long as at least one of the drop ends is free, i.e., unconfined by the tube ends, this does not affect the measurement of the two tensions, but the phenomenon seemed to be a nuisance, rather than a benefit. However, it turns out that a simple analysis of the adhering doublet permits the additional measurement of the third tension, namely, the surface tension of liquid 2, as well as the three contact angles at the circle of three-phase contact, all in a single experiment! This significantly augments the power of the spinning drop technique. In somewhat related work, Shao and van de Ven6,7 looked instead at small, nearly spherical adhering drops and bubbles, using a “bispinner”8 in which the horizontal tube not only rotates about its longitudinal axis but about a vertical axis as well. This generates an additional centripetal force, acting along the tube axis, which (6) Shao, Y.; van de Ven, T. G. M. Langmuir 1988, 4, 1173. (7) Shao, Y.; van de Ven, T. G. M. Langmuir 1989, 5, 1234. (8) Hollemeyer, S. W.; Mar, A. Alberta Research Council, unpublished results.
10.1021/la990333d CCC: $18.00 © 1999 American Chemical Society Published on Web 08/11/1999
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forces, one arrives at the following general differential equation for the interface of a spinning drop:
d(Y sin φ) ) 2 - βY2 Y dY
Figure 4. Spinning three-phase system, type I (“pinched” case; φ2c < 90°).
where Y ) y/b, with φ being the angle of inclination of the interface at a given point, y the radial position of that point, and b the radius of curvature at the drop end (Figure 4). The parameter β is a nondimensional shape-determining parameter, defined by
β≡ Figure 5. Spinning three-phase system, type I (“flaring” case; φ2c > 90°).
facilitates bringing the dispersed entities together, albeit at the cost of increased mechanical complexity. Seeto et al.9 presented an exhaustive review of the various ways three fluid phases can arrange themselves inside a spinning tube with no interface contacting the tube wall. Although their configurations include those discussed in the present work, they did not address the specific problem being considered here.
cos R1 )
γ232 - γ122 - γ132 2γ12γ13
(3)
cos R2 )
γ132 - γ122 - γ232 2γ12γ23
(4)
cos R3 )
γ122 - γ132 - γ232 2γ13γ23
(5)
Of course, R1 + R2 + R3 ) 360°. Single spinning drops and interfaces have been analyzed in detail before.2,3 We shall borrow from these analyses only those elements that are relevant to our present discussion. By balancing the centrifugal and capillary (9) Seeto, Y.; Puig, J. E.; Scriven, L. E.; Davis, H. T. J. Colloid Interface Sci. 1983, 96, 360.
∆Fω2 3 b 2γ
(7)
where ∆F is the density difference across the interface. The value of β varies from 0 for a spherical drop at low speeds to a limiting value of 16/27 for long cylindrical drops at high speeds. Equation 6 may be integrated once to yield
(
sin φ ) Y 1 -
)
βY2 4
(8)
For the long cylindrical drops we consider here, β ) 16/27, and eq 8 simplifies to
Analysis of Spinning Three-Phase Systems Type-I Configurations. Initially, we encountered configurations as depicted in Figures 4 and 5. We shall refer to these as being of type I. (Rather differentsand perhaps more commonstype-II configurations will be discussed later.) The two “inner” or drop phases are phases 1 and 2, and the “outer” phase is phase 3. Their densities are F1, F2, and F3, respectively, while the interfacial tensions between phases i and j are γij (i, j ) 1, 2, 3; i * j). For phases 1 and 2 to be inner phases, it is necessary that F1 < F3 and F2 < F3. All three phases may be liquid. In practice, however, either phase 1 or phase 2 will usually be air. It is assumed that the speed of rotation is sufficiently high (and/or both drops are sufficiently large) so that each drop is long and cylindrical, with an aspect ratio exceeding about 4. The radii of the phase-1 and phase-2 drops are R1 and R2 (R2 < R1), while the radii of curvature at their free drop ends are b13 and b23. The radius of the threephase contact circle is Rc, and the contact angles are R1, R2, and R3 ()φ1c + φ2c). The mechanical balance of the three tensions at the contact line (“Neumann’s triangle”3) leads to
(6)
(
sin φ ) Y 1 -
4 2 Y 27
)
(9)
The maximum drop radius Ymax ) R/b is reached at φ ) 90°, which yields from eq 9
R/b ) 3/2
(10)
For these long drops, in view of eq 10 and β ) 16/27, eq 7 yields eq 1. Equations 1 and 6-10 apply to each of the two drops in Figure 4, using the appropriate subscripts. Specifically, the two drop tensions are obtained immediately from the measured drop radii:
γ13 )
(F3 - F1)ω2R13 4
(11)
γ23 )
(F3 - F2)ω2R23 4
(12)
and
So far, everything is rather conventional. What follows is novel. At the contact circle, y1 ) y2 ) Rc (which is readily measurable) and the interfacial inclinations are φ1 ) φ1c and φ2 ) φ2c (see Figure 4). We may distinguish two cases. In the first case, the contact circle is “pinched” as in Figure 4; i.e., Rc < R2 < R1 and φ2c < 90°. It is important to realize that the shapes of the 1-3 and 2-3 interfaces in this contact region are unaffected by the presence of the other drop and must retrace part of the free “noses” of the respective drops. In the second case, drop 2 flares out to meet drop 1, as in Figure 5 so that R2 < Rc < R1 and φ2c > 90°. The shape of the flared part of drop 2 is given by the only other, less familiar solution to the problem for β ) 16/27 and has been analyzed in detail elsewhere.10,11 In either case, eq 9 applies, but with φ < 90° in a pinched (10) Princen, H. M. J. Colloid Interface Sci. 1995, 169, 241. (11) Princen, H. M.; Vaidya, R. N. J. Colloid Interface Sci. 1995, 174, 68.
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region and φ > 90° in a flaring region. Thus, we obtain
sin φ1c )
[
( )]
[
( )]
Rc 4 Rc 1b13 27 b13
axis, the net capillary pressure across the doublet must be 0; i.e.,
2
(13)
γ23 γ12 γ13 + )0 b23 b12 b13
(19)
and
Rc 4 Rc sin φ2c ) 1b23 27 b23
Because of eq 10, this leads to
2
(14) b12 )
Since
β12 ≡
and, similarly,
Rc 3 Rc ) b23 2 R2
γ13/R1 - γ23/R2 > 0
[
( )]
(15)
[
( )]
(16)
3 Rc 1 Rc 2 R1 2 R1
3 Rc 1 Rc 2 R2 2 R2
3
with the understanding that one selects φ2c < 90° when Rc/R1 < Rc/R2 < 1 (pinched case) and φ2c > 90° when Rc/R1 < 1 < Rc/R2 (flaring case). Equations 15 and 16 thus permit the determination of φ1c and φ2c from the measured values of R1, R2, and Rc. Since the external contact angle R3 ) φ1c + φ2c, we can now calculate the third tension, γ12, from eq 5 in the form
γ12 ) (γ132 + γ232 + 2γ13γ23 cos R3)1/2
(17)
Finally, the remaining contact angles, R1 and R2, are then obtained from eqs 3 and 4. Although of no practical consequence in applying the above method, it is interesting to consider the shape of the phase-1/phase-2 interface separating the adhering drops. This interface must have a shape-determining factor, β12, that generally differs from 16/27. For example, for the special case of F1 ) F2, the interface must be a spherical cap; i.e., β12 ) 0. Since we have defined the phases such that F3 > F1, F3 > F2 and R1 > R2, eqs 11 and 12 imply that
F3 - F1 γ13 < F3 - F2 γ23
(22)
which, because of eqs 11, 12, and 18, leads to the condition
3
( )
γ13 F3 - F1 γ23 > > γ23 F3 - F2 γ13
and
0
0) 2γ12 27 (γ /R - γ /R )3 13 1 23 2 (21)
As b12 > 0, we obtain from eq 20
substitution in eqs 13 and 14 leads to
φ2c ) sin-1
(20)
and, therefore,
Rc R1 3 Rc Rc ) ) b13 R1 b13 2 R1
φ1c ) sin-1
γ12 2 (>0) 3 γ13/R1 - γ23/R2
(18)
We may now distinguish four possible type-I cases. (In this context, it does not matter whether the system is pinched or flaring.) Case 1: The Phase-1/Phase-2 Interface Is Curved toward Phase 2 (or R1 + φ1c > 180° as in Figure 4) and F3 > F2 > F1. Since in this case the heavier fluid is located on the convex side of the 1-2 interface, according to the convention adopted in ref 11, this implies positive values of β12 and b12, the radius of curvature in the axis. When we traverse the drops in Figure 4 from left to right on the
2
(23)
Since the density term is greater than unity, this condition can be met only when γ13/γ23 > 1, or
γ13 F3 - F1 > >1 γ23 F3 - F2
(case 1)
(24)
Case 2: The Phase-1/Phase-2 Interface Is Curved toward Phase 2 (or R1 + φ1c > 180° as in Figure 4) and F3 > F1 > F2. In this case, the heavier fluid is located on the concave side of the interface. By convention,11 this implies negative values of β12 and b12. Following the same reasoning as above, we find
b12 )
γ12 2 (1> γ23 F3 - F2
(case 2)
(27)
Case 3: The Phase-1/Phase-2 Interface Is Curved toward Phase 1 (or R1 + φ1c < 180°) and F3 > F2 > F1. The expressions for b12 and β12 turn out to be identical to those for case 1, except that both must now be negative, since the heavier phase is located on the concave side of the interface. As a consequence, we now require that
γ13/R1 - γ23/R2 < 0
(28)
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and
φ2c ) sin-1
Figure 6. Type-II configuration (“pinched” case; φ2c < 90°).
( )
γ13 F3 - F1 γ23 > < γ23 F3 - F2 γ13
2
(29)
As the density term is greater than unity, these two inequalities cannot simultaneously be met. Therefore, it appears that case 3 is not a viable configuration. Case 4: The Phase-1/Phase-2 Interface Is Curved toward Phase 1 (or R1 + φ1c < 180°) and F3 > F1 > F2. The expressions for b12 and β12 turn out to be identical to those for case 2, except that both must now be positive, and we arrive at the following condition:
1>
γ13 F3 - F1 > γ23 F3 - F2
(case 4)
(30)
γ23 ) (γ122 + γ132 + 2γ12γ13 cos R1)1/2
γ13 )
(F3 - F1)ω2R13 4
(31)
and
(F1 - F2)ω2R23 γ12 ) 4
(32)
As before, the angles φ1c and φ2c are obtained from
φ1c ) sin-1
[
( )]
3 Rc 1 Rc 2 R1 2 R1
3
(33)
(34)
(35)
Finally, the remaining contact angles, R2 and R3, are then obtained from eqs 4 and 5. As in type-I configurations, it is interesting to consider the shape of the third interface, which in this case is the phase-2/phase-3 interface at the left-hand side of Figures 6 and 7. Again, the value of its shape-determining factor will generally differ from 16/27. Since in this case we have defined the phases such that F3 > F1 > F2 and R1 > R2, eqs 31 and 32 imply that
0
0) 3 γ12/R2 + γ13/R1
(37)
2 2 4 (F3 - F2)ω γ23 (>0) 27 (γ /R + γ /R )3
(38)
b23 ) Type-II Configurations. By using water as the external phase (phase 3), air as one of the internal phases (phase 2), and some simple organic liquid (e.g., benzene, dodecane, or hexadecane) as the other internal phase (phase 1), we observed different equilibrium configurations. Upon making contact with the air bubble, the organic liquid would surround the air bubble over most of its length but would leave the bubble ends free, i.e., in direct contact with the external water phase. Increasing the volume of the organic droplet ultimately caused one (but not both) of the bubble ends to be engulfed, leading to a situation as in Figure 6. This is the equivalent of the pinched case for type-I configurations (cf. Figure 4), while Figure 7 shows the equivalent of the flaring case (cf. Figure 5). For either configuration to be stable, it is obvious that F2 must be smaller than F1 and R2 < R1. With the dimensions and angles as defined in Figure 6, the analysis proceeds as before, with only minor modifications. Two of the tensions can be calculated directly from
3
again with the understanding that one selects φ2c < 90° when Rc/R1 < Rc/R2 < 1 (pinched case) and φ2c > 90° when Rc/R1 < 1 < Rc/R2 (flaring case). Since contact angle R1 ) φ2c - φ1c, we can now calculate the third tension, γ23, from eq 3 in the form
Figure 7. Type-II configuration (“flaring” case; φ2c > 90°).
or
( )]
[
3 Rc 1 Rc 2 R2 2 R2
and
β23 )
12
2
13
1
Experimental Section We used a University of Texas (Model 500) spinning drop tensiometer. As a quantitative example, we report on a type-I (case-1) system that was just slightly pinched. (Some closely related systems were of the slightly flaring variety.) The outer phase 3 was UCON 75-H-1400 (ex Union Carbide), a linear random copolymer of ethylene oxide and propylene oxide with 75% by weight of oxyethylene and 25% by weight of oxypropylene groups. Phase 1 was air, and Phase 2 was SHF-63, a poly(Rolefin) (PAO) fluid ex Mobil Chemical Company. The UCON and PAO fluids were preequilibrated for 24 h. The density and refractive index of the equilibrated UCON fluid were F3 ) 1.0849 g/cm3 and n3 ) 1.4638, respectively. The density of the equilibrated PAO was F2 ) 0.8205 g/cm3, while for air we take F1 ) 0. Table 1 shows all measured and derived quantities, the latter in the order in which they were obtained using the analysis presented above. The tube speed is given in terms of the period T in milliseconds/revolution, as indicated on the instrument. The appropriate form of eq 1 then is
( )
∆F Dapp γ ) 1.234 × 106 2 n T
3
(39)
where ∆F is the density difference in grams/cubic centimeter and Dapp is the apparent diameter in centimeters. Not indicated is β12. By using eq 21, we obtained an average value of β12 ) 0.596, which is only slightly greater than the “magic” number 16/27 ()0.592 592...). It is seen that repeat measurements at a given speed are quite consistent but that there are some differences in the results when
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Table 1. Measured and Derived Quantities for the UCON/PAO/Air System (n3 ) 1.4638; G3 - G2 ) 0.2644 g/cm3; G3 - G1 ) 1.0849 g/cm3) measured
derived
T, ms/rev
(D2)app, cm
(D1)app, cm
(Dc)app, cm
Rc/R2
Rc/R1
6.21 6.21 9.83 9.83 13.21 13.21 13.21 13.21 14.61 14.61 av
0.1372 0.1371 0.1876 0.1874 0.2267 0.2269 0.2268 0.2267 0.2436 0.2434
0.1473 0.1471 0.2003 0.2006 0.2436 0.2434 0.2435 0.2436 0.2614 0.2616
0.1346 0.1344 0.1860 0.1860 0.2186 0.2185 0.2186 0.2185 0.2389 0.2388
0.9810 0.9805 0.9915 0.9925 0.9643 0.9630 0.9638 0.9638 0.9807 0.9811
0.9138 0.9139 0.9286 0.9272 0.8974 0.8977 0.8977 0.8970 0.9139 0.9128
γ23, mN/m
γ13, mN/m
φ2c, deg
φ1c, deg
R3, deg
γ12, mN/m
R1, deg
R2, deg
6.97 6.95 7.11 7.08 6.95 6.96 6.95 6.95 7.04 7.03 7.00
35.37 35.23 35.50 35.66 35.36 35.27 35.31 35.36 35.72 35.80 35.46
88.13 88.07 89.15 89.26 86.47 86.35 86.43 86.43 88.09 88.13 87.65
81.56 81.57 83.00 82.86 79.98 80.01 80.01 79.94 81.57 81.47 81.20
169.69 169.64 172.15 172.12 166.45 166.36 166.45 166.37 169.67 169.60 168.85
28.55 28.42 28.47 28.66 28.65 28.55 28.60 28.66 28.81 28.92 28.63
177.50 177.48 178.05 178.06 176.75 176.70 176.73 176.73 177.49 177.49 177.30
12.82 12.88 9.80 9.82 16.80 16.94 16.82 16.90 12.85 12.91 13.86
Figure 8. Engulfment of drop 2 by drop 1 (a) and of drop 1 by drop 2 (b). the speed is changed. The reasons for this are not totally clear. Some nonequilibrium effects may be involved. As indicated above, we observed type-II behavior with a number of other systems, but these were not analyzed in detail for two reasons. First, in these systems, angle R3 (see Figure 6) tended to be so close to 180° that the exact location of the threephase contact line was difficult to pin down. Second, the outline of the (phase-2) air bubble was not always visible because of refractive effects associated with the surrounding phase-1 droplet. In those cases where that outline was clearly visible, γ12 could be obtained from the apparent diameter of the phase-2 bubble by using eq 39 but with n ) n1 (not n3) as the magnification factor, as pointed out by Puig et al.12
Discussion In practice, establishing contact between the two drops in a spinning drop tensiometer may require some manipulation, such as temporary tilting of the instrument. However, even then, it will be impossible to establish adhesion if the spreading coefficient S3(12) of phase 3 with respect to phases 1 and 2 is positive, i.e., if
S3(12) ) γ12 - γ23 - γ13 > 0
(40)
Even when S3(12) < 0, adhesion may not occur if the film of phase 3 separating phases 1 and 2ssometimes referred to as an asymmetric or “pseudoemulsion” filmsis metastable due to positive disjoining forces. When such disjoining forces are absent, contact should lead to adhesion when S3(12) < 0. Once contact is established, the ultimate configuration will be one of those sketched in Figures 4-7, provided that each of the three tensions is smaller than the sum of the other two. If, on the other hand, γ23 > γ12 + γ13 or S1(23) > 0, drop 1 will engulf drop 2 as in Figure 8a, while drop 2 will “engulf” drop 1 if γ13 > γ12 + γ23 or S2(13) > 0, in which case a thin film of phase (12) Puig, J. E.; Seeto, Y.; Pesheck, C. V.; Scriven, L. E. J. Colloid Interface Sci. 1992, 148, 459.
2 will envelop drop 1, as in Figure 8b. Of course, the two phase-2 lobes in Figure 8b need not be of equal length; in fact, more often than not, only one lobe is observed. It is clear that this version of the spinning drop technique may be well suited to assess the potential effectiveness of one liquid as an antifoaming agent for another liquid. Depending on the relative densities, either phase 3 or one of the drop phases may play the role of the potential antifoam. Even simple visual observation of the threephase contact region indicates whether either the relevant spreading coefficient, S, or bridging coefficient, B, is indeed positive, as is believed to be required for antifoam effectiveness.13-15 In addition, the method has the distinct advantage that the three fluids are in intimate contact and at equilibrium so that the measured values of the three tensions are much more reliable than if they were measured separatelyseach with its own errorsin the three binary systems. The present procedure will thus give a much more reliable estimate of the sign and magnitude of the spreading and/or bridging coefficients. We have analyzed various types of adhering doublets and derived a necessary, though probably insufficient, condition for the existence of each. We do not pretend to have solved all aspects of the problem completely. Specifically, it is not yet clear what exact functional relationship(s) between the three densities and three interfacial tensions determines whether a given system assumes a type-I or a type-II configuration (or whether both configurations may in fact be realized). This problem awaits further analysis. However, it must be emphasized that such detail is not necessary for the successful application of the method. Finally, the analysis assumes that Neumann’s triangle is the appropriate condition for equilibrium at the threephase contact line. A finite line tension would cause some deviation. In fact, the method could be used in principle to evaluate the line tension from such deviation and its dependence on Rc, which may be varied by varying the speed. Since reasonable estimates16 of the line tension (13) In this context, the spreading and bridging coefficients are defined as S ≡ γL - γA - γA/L and B ≡ γL2 + γA/L2 - γA2, respectively, where γL is the surface tension of the liquid to be defoamed, γA is the surface tension of the potential liquid antifoam, and γA/L is the interfacial tension between the two. The sign of B may also be ascertained from the contact angle measured through the liquid to be defoamed (R1, R2, or R3 depending on which of the phases plays this role). If that angle is greater than 90°, then B > 0; otherwise, B < 0. For more details, see refs 14 and 15. (14) Garrett, P. R. J. Colloid Interface Sci. 1980, 76, 587. (15) Garrett, P. R. In Defoaming: Theory and Industrial Applications; Garrett, P. R., Ed.; Surfactant Science Series 45; Marcel Dekker: New York, 1993; Chapter 1, p 1. (16) Aveyard, R.; Clint, J. H.; Nees, D.; Paunov, V. Colloids Surf. A 1999, 146, 95.
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are all in the range of 10-10-10-12 N, however, we believe that its effect can be safely ignored in any practical, i.e., macroscopic, system being considered here.
for the experimental work, and J. Lucassen, R. N. Vaidya, A. Jackson, and R. B. Saeger for their valuable suggestions and comments.
Acknowledgment. I thank Mobil Technology Company for permission to publish this paper, R. K. DeMura
LA990333D
