Langmuir 1989, 5, 1234-1241
1234
effect is most pronounced when short-tailed surfactants are used. In furture work, we will examine to what extent the introduction of reversible binding between the stickers affects these calculations. Currently, there are no reports in the literature of a systematic study of the effect of surfactant concentration on the association behavior of polymersurfactant aggregates. We hope the predictions presented above will en-
courage experimentalists to explore this behavior.
Acknowledgment. A.C.B. acknowledges financial support from the donors of the Petroleum Research Fund, administered by the American Chemical Society, the NSF through Grant DMR-8718899, and the Union Carbide Corp. We thank Dr. Chris Lantman for several helpful discussions.
Spinning of Partially Engulfed Drops Y. Shaot and T. G. M. van de Ven* Pulp and Paper Research Institute of Canada and Department of Chemistry, McGill University, Montreal, Quebec, Canada H3A 2A7 Received October 14, 1988. I n Final Form: April 24, 1989
The deformation of spinning partially engulfed drops has been studied both theoretically and experimentally. Two drops of different liquids are suspended in a third liquid and brought into contact. Provided that one drop does not engulf the other, the drops will form a partially engulfed pair, the shape of which is determined by the Laplace equation and the Neumann triangle boundary condition at the three-phase contact line. The shapes of quasi-stationary drop pairs were found to be fully consistent with theoretical predictions. Slight discrepancies can be ascribed to small amounts of impurities accumulating at interfaces. By subjecting a drop pair to a rotation about its axis of revolution, it will deform to a degree depending on the relative density differences, the interfacial tensions, the drop volumes, and the angular velocity. It is predicted that the total length of the drop pair increases with increasing angular velocity, while the radius of the contact area decreases. Experiments were in good agreement with these theoretical predictions.
Introduction When three immiscible phases make contact with each other, the equilibrium shapes of the drops or bubbles are governed by the Laplace equation,1,2subject to the condition that at the three-phase contact line (TPL) the interfacial tensions of these three phases form a closed Neumann triangle.3 Using the Laplace equation, Huh and Scriven4calculated the shape of an axisymmetric fluid interface in which a solid object is positioned. Princen et al.23”7 investigated the mechanical equilibrium of solid axisymmetricparticles floating in a horizontal liquidffluid interface. Ivanov et a1.8 measured the equilibrium shape of fluid and solid particles at a fluid interface and calculated the line tension at the TPL. Having found a theoretical solution for the shape of a liquid drop spinning about a horizontal axis, Princen et al.’ improved the method for measuring interfacial tensions with a spinning drop apparatus proposed by V ~ n n e g u t . ~The calculation and measurement of shapes of deformable interfaces are important in many fields related to colloid and interface science, such as petroleum recovery, mineral flotation, etc. The validity of the Neumann triangle has been investigated experimentally with partial success by some authors. In this paper, we propose a method whereby the validity of the Neumann triangle can be tested by studying the shape of a spinning liquid drop pair. The shapes of partially engulfed pairs are also relevant to the concept of line tension, which affects these shapes for small drops. By means of a bispinner,l2l3we can produce partially engulfed drop pairs suspended in a third immiscible phase. When a drop pair spins horizontally about its axis of rotation, ‘Present address: Department of Textile Chemistry, China Textile University, 1882 West Yan An Road, Shanghai, P.R. China.
0743-7463/89/2405-1234$01.50/0
its shape changes to a degree dependent upon the boundary conditions at the TPL. The experimental resulta were compared with those obtained numerically from the Laplace equation. The calculations show that the shape of drop pairs and the radius of the TPL change only by a few percent when the angular velocity was increased to 10 000 rpm. Experiments confirm the general predicted trends, but small deviations were observed which were attributed to the presence of impurities. To investigate these effects further, additional experiments must be performed at higher angular velocities which, because of speed limitations of the bispinner, are presently not possible with our experimental technique.
Theory As is well-known, the equilibrium shapes of drops or bubbles are based on the Laplace equation:
AP = P1 - Pz =
.(;+ $)
where P1and P2 are the pressures inside and outside the drop, respectively. l / R 1 and 1 / R 2are the principal curvatures of the interface, and u is the interfacial tension. (1) Princen, H. M.; Zia, I. Y. Z.; Mason, S.G. J. Colloid Interface Sci. 1967, 23, 99. (2) Princen, H. M. Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1969; Vol. 2, p 2. (3) Boruvka, L.; Neumann, A. W. J. Chem. Phys. 1977,66, 5464. (4) Huh, C.;Scriven, L. E. J. Colloid Interface Sci. 1969, 30, 323. (5) Princen, H.M.; Mason, S. G. J. Colloid Sci. 1965, 20, 246. (6) Princen, H.M.; Mason, S. G . J. Colloid Sci. 1965, 20, 156. (7)Huh, C.;Mason, S. G. J. Colloid Interface Sci. 1974, 47, 271. (8) Ivanov, I. B.; Kralchevski, P. A.; Nikolov, A. D. J.Colloid Interface Sci. 1986, 112, 97. (9) Vonnegut, B. Reo. Sci. Instrum. 1942, 13, 6. (10) Fox, W. J. Am. Chem. SOC.1945,67,700. (11) Miller, N. F.J. Phys. Chem. 1941,45, 1025. (12) Hollemever, S. W.: Mar, A. “The Bispinner: A Novel Liquid Micro-manupulator”(to appear). (13) Shao, Y.;van de Ven, T. G. M. Langmuir 1988,4, 1173.
0 1989 American Chemical Society
Langmuir, Vol. 5, No. 5, 1989 1235
Spinning of Partially Engulfed Drops
c glass tube wall
Y.
phase 3
phase 2 I
I
Figure 2. Neumann triangle at the interfaces of a three-phase system. Figure 1. Partially engulfed drop pair rotating about its horizontal axis.
Let us consider two immiscible liquid drops (phases 1 and 2) suspended in a third immiscible liquid (phase 3), brought together to form a partially engulfed drop pair in a glass tube spinning about its horizontal axis, as shown in Figure 1. The densities of phases 1 and 2 were chosen to be smaller than that of the medium. The angular velocity is assumed to be high enough to overcome the buoyancy so that the liquid drop pair will remain on the horizontal axis. Cylindrical coordinates ( x , y ) are chosen, as shown in Figure 1. The origin Oi (i = 1, 2 , 3 ) is located at the axes of a drop phase (depending on which section of the drop is calculated). X is toward the TPL, x o and y o are the semiaxes of a drop, and 9 is the angle between the normal of the interface at ( x , y ) and the negative X-direction. For incompressible liquids, the pressure outside the drop is
where cyij = Cijb,3 with Cij = Apijw2/2aii. Because of the relation sin 9 _ -dx tan 9 = -cos 9 dy eq 7 can be written as dy/dx =
Equation 9 is the fundamental equation for calculating the shape of a spinning drop pair in three-phase systems. The Neumann triangle of interfacial tensions serves as the common boundary condition for the surfaces meeting at the TPL. The three angles 9i (i = 1 , 2 , 3 ) of the Neumann triangle in thermodynamic equilibrium (see Figure 2 ) depend on the interfacial tensions aij (i # j # k = l, 2, 3)14 ajk2
cos 9i = where p j is the density of the outside phase and Pois the pressure outside the drop at y = 0. The pressure inside the drop is (3)
where p i is the density of the liquid drop phase i ( p j > p i ) (i = 1 , 2 ) and bi is the radius of the curvature of the drop surface at the origin Oi. The principal curvatures in eq 1 are defined by
-1= - sin 9 (5) R2 y Substituting eq 2-5 into the Laplace equation (eq l ) ,one obtains d(sin 9 ) + -sin = -9- - 2 dY Y bi
APijw2Y2 20ij
(6)
in which Apij = p j - p i ; aij is the interfacial tension between phases i and j. Integrating eq 6 yields r
- u..2 - uik2 ij
2aijaik
(10)
from which it can be shown that these three angles satisfy the requirement g1 9 2 9 3 = 277 (11) Hence, one can calculate Bi directly from the interfacial tensions of the system. In order to obtain a partially engulfed drop pair, the three spreading coefficients si = g j k - (“ij + b i k ) (12)
+ +
must satisfy the condition14J5 si < 0 (i = 1, 2, 3 )
(13)
Numerical Procedures For a given three-phase system and a given angular velocity, C , in eq 9 is fixed. For liquid drop 1 (see Figure 2), the integration region of eq 9 is divided into three parts: (i) From the origin O1 to point A (xo, yo). A t point A, dyldx = 0. (ii) From point A to point B at the TPL. At point B, the boundary condition that must be satisfied is the Neumann triangle, i.e. (dy/dx)B13= tan (14)
+
where # is the slope angle at B related to the Neumann angles Bi. The symbol Bij refers to the slope at point B associated with the interface between phases i and j . At (14)Torza, S.; Mason, S. G. J. Colloid Interface Sci. 1970, 33, 67. (15)Smith, P. G.; van de Ven, T. G. M. Colloids Surf. 1981,2, 387.
Shao et al.
1236 Langmuir, Vol. 5, No. 5, 1989
Table I. Main Parameters for Calculating the Shape of a
SDinning Partially Engulfed Droo PaiP
w,
rpm 0
1400 2000
Figure 3. Shapes of partially engulfed drop pairs calculated with the Laplace equation and the Neumann triangle at w = 0 and w = 10000 rpm, respectively.
the same time, this yields the volume V, of part a (the volume of revolution of the curve OIAB about the X-axis) of phase 1 by integration
V = S r y 2 dx
(15)
between x = 0 and x = OIB. For both parts i and ii above, Cij is equal to C13. (iii) From point O2 to B, O2 being chosen as the origin and the X-direction being toward the TPL. For this region, Cii = Clz. The boundary condition is (dy/dx),,, = tan 6 (16) in which 6 is the slope angle at point B of curve OzB. 6 is related to the Neumann angles by 6 = 2 ~ - 8 ~ - $or 6 = r - I ) (17) depending on whether the drop pair has the shape shown in Figure 3 or Figure 2. Equation 15 also gives the volume Vb of part b (enclosed by curve 02B)of phase 1. The total volume of liquid drop 1 (phase 1) is v, = v, v, (18) For liquid drop 2, point O3 is chosen as the origin, and the X-axis is in the direction of the TPL. Equation 9 is integrated (with C, = C23) from point O3 to point D (where dyldx = 0) and next from D to B where (dy/dx)& = tan Q (19)
+
with Q = I ~ + ~ ~ -orI ) f l = I ) - 8 3
(20)
depending on the shape of the drop pair. Integrating eq 15 between x = 0 and x = O,B yields V, and thus the volume of liquid drop 2: v2 = v, - v b (21) For a given system, the contact angles are defined by eq 10, but the slope angle I) is still unknown. Also unknown is the value of bi (the curvature at Oi).We must arbitrarily choose bl and tj and integrate eq 9 to obtain the drop shape (x,y), the volume V,, and the radius r of the TPL (which equals y at point B). For the section of drop 1 between O2 and B, we make an arbitrary choice of b2 to obtain V,‘ and r’. Usually the value of r’is not the same as the value of r obtained from bl. We keep iterating b2 until r’ = r. For this bz value, we also determine vb. According to the Laplace equation, the pressure at points O1 and O3 must be the same. This requireds that 413 - u12 _ + -4Z3
bl
bz
b3
(22)
(16)Torza, S.;Mason, S. G.Kolloid-Z. Z . Polym. 1971,246, 593.
3400 5000
8000 loo00
+, deg 32.5 32.5 32.4 32.2 32.0 31.0 29.5
Cij, Cln = 0
bi,b m m b, = 0.5100 c,; = 0 b2 = 0.4550 C32 = 0 b3 = 0.5512 C13 = 0.06395 bl = 0.5085 C21 = 0.1629 b2 0.4526 C32 = 0.04654 b3 = 0.5498 C13 = 0.1305 bl = 0.5065 Cp1 = 0.3325 b2 = 0.4508 C32 = 0.09500 b3 = 0.5476 C13 0.3712 bl = 0.5013 C21 = 0.9610 b2 = 0.4445 C32 = 0.2145 bs = 0.5423 Cis = 0.8151 b1 = 0.4935 C2l = 2.0183 b2 = 0.4338 C32 = 0.5936 b3 = 0,5341 C13 = 2.0881 bl = 0.4110 C21 = 5.3204 bz 0.4012 C32 = 1.5196 b3 = 0.5118 C13 = 3.2627 bl 0.4505 Cp1 = 8.3131 b2 = 0.3880 C32 = 2.3744 b3 = 0.4898
r, m m 0.4299 0.4299 0.4300 0.4294 0.4294 0.4290 0.4289 0.4289 0.4290 0.4218 0.4218 0.4218 0.4263 0.4263 0.4264 0.4221 0.4221 0.4223 0.4191 0.4191 0.4192
relative error between r: %
0.02 0.05 0.02 0.0
0.02 0.05 0.02
a V l = 0.18 mm3; V 2 = 0.52 mms (for other details see text). is calculated from eq 22. Obtained from (bl,b2)and bB.
* b3
Hence b3 is determined by bl and b2. Integrating eq 9 yields V, and r”. If the Neumann triangle is satisfied everywhere, r”must be equal to r. The exact agreement between r”and r, which is shown in the following section, confirms that the calculated drop shapes satisfy the Neumann triangle at every point of the TPL. By this procedure, we can calculate the volumes Vl and V2 from the arbitrary choices bl and I) for given values of Cij and uik In a practical situation, the volumes are given and we have to find the corresponding values of bl and I). This can be done by starting with an arbitrary choice of bl and tj and finding better choices by iteration until the new choices result in the chosen volumes VA and VB. To do this, we chose a linear extrapolation scheme. We started with three arbitrary sets of bl and tj, corresponding to three points in the plane V = 0 in an orthogonal Cartesian coordinate system (bl, I), V). For each point, we calculated the correspondingvolume Vl, resulting in three points (bl, I), V,) which forms a plane. The intersection of this plane with the plane V = VA yields a straight line. A similar calculation of the volumes Vz and the intersection of the corresponding plane with the plane V = VB yields a second straight line. The intersection of the projection of these two lines on plane V = 0 yields a new choice (bl, I)). After the worst choice is discarded, this procedure is repeated, resulting in a better choice after each iteration. Usually about five iterations are required to obtain the correct values of bl and I) within 0.05%, which is about the relative error in the fourth-order Runge-Kutte method” used to integrate eq 9 numerically.
Numerical Results We used the procedure described above to calculate the radius r of the TPL and the shape (x,y) of the drop pair at different angular velocities from eq 9. To show the consistencies of the results, listed in Table I are the main parameters calculated for a three-phase system: p1 = 0.917, (17) Franks, R. G. In Mathematical Modeling in Chemical Engineering; Wiley: New York, 1967;p 59. (18)Clunk, J. S.; Goodman,J. F.; Ingram, B. T. Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1971;Vol. 3, p 187.
Langmuir, Vol. 5, No. 5, 1989 1237
Spinning of Partially Engulfed Drops p2 = 0.964, p3 = 1.061 (kg/dm3); u12 = 3.1, ~ 1 = 3 24.2,
~223
= 22.4 (mN/m). Hence, according to eq 10, O1 = 128.5', O2 = 57.7O, and O3 = 173.8O (corresponding to one of our
experimental systems). From Table I, the following can be concluded: (i) The slope angle # of liquid drop 1 at the TPL (see Figure 2) decreases with increasing angular velocity, which is related to the shape change of the drop pair. (ii) The radius r of the TPL decreases with increasing angular velocity, but very slowly, and even for w as high as 10000 rpm r decreases only by 2.5%. (iii) The radii r calculated from (bl,bz) are, within numerical error, the same as those calculated from b3, both at w = 0 and w # 0. The exact agreement between the radii r obtained from (bl,bz)and b3 shows that the calculated shape of the drop pair satisfies the Laplace equation and that the Neumann triangle is satisfied everywhere at TPL. The calculations for other systems gave the same conclusions. As an additional check, we verified the force balance at point B of the TPL (see Figure 2):
where ( l / R l + 1/R2)iB(i = I, 11, 111) is the sum of the principal curvatures at point B of parts a and b of liquid drop 1 and drop 2, respectively. Since d2x/dy2 1
_-
RI
[l
+(d~/dy)~]~/~
(24)
we can obtain the values of Ri from the values of y, dx/dy, and d2x/dy2at point B from the integration results. For the same system listed in Table I, at w = 10000 rpm, we obtained
These values satisfy eq 23 with a relative error of 0.07 % , i.e., of the same order as that of the integration program (0.05%). Thus, the calculation results suggest that the force balance is met everywhere for all angular velocities (including w = 0). This also shows the accuracy of the calculations. By means of this calculation program, we can calculate the shape of the drop pair at different angular velocities a,the change of the radius r of the TPL with w , and the effect of different interfacial tensions and drop pair volumes on the shape changes of the drop-pair. An example of changes in drop shape is given in Figure 3, which shows a partially engulfed drop pair at w = 0 (stationary condition) and w = 10 000 rpm (for the system discussed in Table I). The location of the TPL is taken as a reference. It can be seen that when the pair is spinning its length increases and its shape is flattened. Both drop pairs satisfy the Laplace equation and the Neumann triangle. In this respect, it is worthwhile pointing out that the Neumann triangle can be verified by measuring the shape of nonspinning drop pairs since different boundary
Table 11. Tendency To Engulf or Separate by a Drop Pair in Systems with Different Interfacial Tensions w , rpm $, deg V,: mm3 Xm," mm AXOT, % System A" 3400 5000 8000 10000
57.0 57.0 57.0 57.0 56.5
0.0093 0.0090 0.0088 0.0083 0.0081
0 3400 5000 8000 loo00
32.5 32.2 32.0 31.0 29.5
0.0737 0.0737 0.0738 0.0746 0.0769
0
0.0521 0.0518 0.0515 0.0511 0.0505
-0.6 -1.2 -1.9 -3.1
0.204 0.207 0.209 0.218 0.229
+1.5 +2.5 r+6.9 +12.3
System Bb
"e, = 160.30, e2 = 21.70. be1 = 135.00, e2 = 50.00; for both SYStems VI = 0.18 and V2 = 0.52 mm3. V , = the volume of part a of liquid drop 1. dXoT= the length of part of a liquid drop 1 in the X-direction (see Figure 2). conditions at the TPL would lead to different shapes. However, as it is difficult to find a system in which all phases have the same density, a drop pair will usually sediment or float, making it more difficult to observe. Furthermore, if only its length instead of the whole shape were measured, it is possible, in principle, that different boundary conditions at the TPL could lead to the same length. Spinning the drop pair has the advantage of keeping the pair at a proper position and, at the same time, reducing the systematic error in the measurements (see the following section). Moreover, observing the variations in shape with varying angular velocity and comparing the results with shapes calculated with the Neumann triangle boundary condition provide an unambiguous test of its validity. Calculations for different systems with different interfacial tensions show that drop pairs can have different tendencies to engulf or separate with increasing angular velocity. Table I1 shows the results for systems with the same densities as the examples discussed in Table I but with different interfacial tensions (which can be realized by adding surfactants to the system). It can be seen that for the lower interfacial tension system (system B) both V, and X m increase with increasing angular velocity while the slope # decreases. This suggests that the drop pair has a tendency to separate. For the high interfacial tension system (system A), # is almost unchanged, but V, and Xm decrease with increasing angular velocity. Hence this system shows a tendency toward engulfing. Changes in the shape of drop pairs as a function of angular velocity for various systems with different interfacial tensions are shown in Figure 4, where L is the total length of the drop pair along the x-axis, 1 the length of liquid drop 2, and r the radius of the TPL (cf. Figure 1). The subscript "0" refers to these values when w = 0. From Figure 4b it can be seen that with increasing angular velocity drop 2 elongates along the spinning axis. For given volumes of the drops, the lower the interfacial tension the more the liquid drop elongates. This can be concluded directly from the governing equations. We know that the shape of a drop is related to the parameter aij (see eq 7). For a given angular velocity w and a given density difference Apij, the value of aij is high when uij is low. From this, it follows that a drop in a lower interfacial tension system changes its shape more easily. Figure 4c shows the change in the total length of the drop pair, which follows the same trend as changes in 1. For curve 1, L / L ois somewhat smaller than l / l o because this system tends toward engulfment, a process that decreases L.
1238 Langmuir, Vol. 5, No. 5, 1989
Shao et al. 1.00 0.99
-rro 0.98 0.97
-
1.04 1.oo
1.00
0.96 1.00 1 1.08 1.06 1.04 1.02
wfj
To
-
L
io
1.08 1.06 1.04 1.02 1.00
0 1 2 3 4 5 6 7 8 9 1 0 Angular Velocity (1O3rpm1 Figure 4. Shape of drop pairs as a function of the angular velocity for various interfacial tensions. Curve 1: u12= 3.1, u13 = 32.6, ~ 2= 3 29.7. Curve 2: uI2 = 3.1, ~ 1 = 3 27.2, ~ 2 = 3 25.1. Curve 3: u12 = 3.1, 613 = 24.2, ~ 2 =3 22.4. Vl = 0.18 and V2 = 0.52 mm3.
0 1 2 3 4 5 6 7 8 9 1 0 Angular Velocity C103rpm) Figure 6. Shape of drop pairs BS a function of the angular velocity for various volume ratios (Vl/V2) indicated in the figure, V = Vl + V , = 0.7 mm3; interfacial tensions as for curve 3 in Figure 4.
-rr o 098 -rr0 0.98
-
0.97
-
0.96 1.12
-
a 1
-
097 - a 096-, 112 l
l
l
l
l
l
-1 /
-
,
1
1
1
108
LO
104 1.00
-
1.12
Lo 1.08
-L
1.04
Lo 1.04 l.08
1.00
1.00
0
0 1 2 3 4 5 6 7 8 9 1 0 Angular Velocity (103rpm)
Figure 5. Shape of drop pairs as a function of the angular velocity. Curve A: V = Vl V2 = 0.7 mm3. Curve B: V = Vl V2 = 1.09 mm3. V , / V 2= 1.59; interfacial tensions as for curve 3 in Figure 4.
+
+
The change in the radius r of the TPL is opposite changes in length. Figure 4a shows that the highest interfacial tension system (curve 1) has the largest rate of change in r (in contrast to the smallest change in 1 and L). This is also due to the fact that this system has a tendency toward total engulfment. The changes in radius r of the TPL are more complicated because of the opposite tendencies of separation and coalescence and because of the constraints put on it by the Neumann triangle boundary condition. Since the change in shape of a liquid drop is related to the parameter ai,(=Cijbt),for given values of Apij, ai,,and w the larger bi (or the larger the volume) the larger the change in drop shape. Figure 5 shows the changes in the lengths 1 and L and radius r for systems with different volumes but with the same volume ratio. It can be seen that the system with the largest volume has the largest
1
2 3 4 5 6 7 8 9 10 Angular Velocity (103rpm)
Figure 7. Shape of drop pairs as a function of the angular velocity for different parameters Cij Curve C1: C13 = 377.2, C12= 961.0, Czs = 274.5 at w = 3400 rpm. Curve C2: C13 = 754.4, Clz = 1922, Cz3 = 549.0 a t w = 3400 rpm. Volumes as in Figure 4.
change in shape with increasing angular velocity. Figure 6 shows the changes in drop pair shapes for systems with a given total volume but various volume ratios. This figure indicatesthat when the volume of liquid drop 1 is almost the same as that of liquid drop 2 the change in shape of the drop pair is greater than for pairs in which the volume of liquid drop 1 is either larger or smaller than that of liquid drop 2. Changes in interfacial tensions cause a change in the Neumann triangle boundary condition as well as in the parameter Cij. It is possible to change Cij without changing the boundary condition by changing the density of the drops (Le., changing Ap;,). Figure 7 shows the comparison of two systems with different C, values. It can be seen that the larger the value of Cijthe larger the changes in shape. Experimental Section Bispinner Technique. A bispinner is an instrument designed to manipulate small particles, drops, or bubbles.12J3 It consists
Spinning of Partially Engulfed Drops
Langmuir, Vol. 5, No.5, 1989 1239
Table 111. Exwrlmsntal TbreaPbw Systems. wtem
I I1 111
P
b3
glycerol + 0.2 g/L aqueoun SDS solution (k3w/w) glycerol + 0.5 g/L aqueous SDS solution (k3w/w) glycerol + 0.75 g/L aqueous SDS solution (1:3 w/w)
eta
01.
3.1 3.1 3.1
32.6 27.2 24.2
8, 29.7 25.1 22.4
8% 21.7 50.0 57.7
160.3 135.0 128.5
e, 178.0 175.0 173.8
SI
Sa
Sa
-6.0 -5.2 -4.9
-0.2 -1.0 -1.3
-59.2 -49.2 -43.5
*Interfacialtension8 810 measured hy the pendant dmp method (mN/m). ejcalculated from eq 10 (degrees).
theory. meter mm exptl:mm bt 0.5100 0.0.0061 para-
*
4 Figure 8. Schemati- of a partiaUy engulfed drop pair rotating in the glasa tube of the hispinner prior to and after eontak of a glass tube which rotates about two perpendicular axes (see Figure 8). We denote the rate of rotation about the vuticle axis by ov and the one abwt the horizontal axis by q,.The horizontal rotation maintains the dispersoids on the horizontal axis. thus avoiding huoyaney effects. while the vertical rotation pushes two droplets against each other or pulls them apart (depending on the density difference between the droplets and the medium). In our experiments,the densities of the liquid drop are smaller than that of the medium. After two drops of liquid phases 1 and 2 are injected,q,is gradually increased to keep the drops on the spinnii aria. and, subsequently, q is i n d to muse the drops to move toward each other. Under the correct conditions, this will result in a partially engulfed drop pair. In this study, we only investigated the relationship between the shape of the drop pair and the angular velocity about the horizontal axia (w).Therefore,when two drop hecame a partiauy engulfed pair, ov was gradually decreased to zero, while wH was varied in the range 18(t3400 rpm. For lower values of q,,it was not pmible to maintain a drop pair on the tube axis. Photographs of spinning drop pairs were taken with a Nikon F camera with a Zek mom lens, using a flash light source (Strobex System 236). Sin= 0~ = 0 during the experiments, is written as w in what follows. When the shapes of the drop pair were measured,corrections for the refraction of light at the curved glass interface were made." Exwrimental Systems. In order to obtain partidy engulfed three-phase systems, we must choose systems which have spreading coefficientsSi satisfying the condition Si < 0 (i = 1, 2,3). We used silimneoil "IF 5CS (Dow Coming pl = 917 kg/m? BS phase 1, caster oil (Fisher Scientific Co., p2 = 964 kg/ms) as phase 2. and a glyceml-wntermixture with d i u m dodecyl sulfate (SDS)surfactant as phase 3 (p3 = 1061 kg/m3). We were unable to find a three-phasesystem with Si < 0 without surfactants. The interfacial tensions. contactangles. and spreading coefficients are Listed in Table 111. For system Ill, we used two different volumes System III(1) has volumes of 0.18 and 0.52 m3and system III(2) of 0.67 and 0.42 mm3 for liquid drops 1 and 2, respectively.
c
r
0.5512 0.4299
-
0.5Wo
OaOM
0.43!213+ 0.0046
meter
from interfad tension," deg
fmmmhape ofdrop pair:dq
8,
1285 + 0.6
131.2
para-
(140.1)
* 0.8
4
67.7
&
(46.8) 173.8 0.2 (174.1)
* 0.6
* 0.6 175.7 * 0.2 65.2
.At u 18Orpm. bj aleuLtsd I"m d d u s s o f r and h by vsins eq 28 bCdculntdfrom interfad tension values (using eq 10) taLen immediately after mntact; in parentheses are valugl from tenaions o b s e d 24 h later. eCalculatsdfrom e, = 180" - 01 - 8; 0, = 180. - 101 - 1. where con 01 = r/b,, nm B * r/b* and EOS 7
-
r/ba.
Results and Discussion Figure 9 shows photographa of system III(1) (cf. Table 111) at o = 180 rpm and at o = 3400 rpm, identical with the system shown in Figure 3. It should be noted that the drop pair deformation appears exaggerated due to the refraction of liiht at the glass tube. From Figure 9, we can see that r, V,, and.Vb vary with w, although the changes are rather small. In Table IV, the experimental results obtained at o = 180 rpm are compared with the theoretical values for a nonspinning drop (o = 0 ) for system III(1). From the theory, it can be concluded that the differences in shape between drop pairs spinning at o = 0 and 180 rpm are too small to measure. When o = 0 rpm, all shapes are sections of spheres. It is helpful to make an analysis of the experimental error. The experimental values of bi were calculated from the measured values of the height of the spherical cap hi and the radius r of the TPL by using the geometrical relation
bi = r-
dt + 1 2di
where di = hi/r. Since all distances are obtained by multiplying the measured values on a negative by a known calibration factor, the error in r and hi depends on this calibration factor, but the error in didoes not. The calibration factor is obtained by photographinga solid sphere
Figure 9. Photographs of drop pairs at different angular velocities.
w = 180 and 3400
rpm
1240 Langrnuir, Vol. 5, No. 5, 1989
'8c1
Shao et al.
' I
System 1
W System II (0.5 1
Figure 10. Photographs of partially engulfed three-phase systems in different surfactant concentrations,w = 180 rpm. The number undernenth each photograph represents the SDS concentration in g/L.
of known diameter 1.00 i 0.01 mm; i.e., it contains a systematic error of about 1%. The sizes of the images on the negative can be estimated with a random error of about 0.25% in rand 0.15% in hi, resulting in an error of about 0.3% in de It follows from eq 26 that the error in bi is mainly due to the error in the calibration constant and is about 1%. The observed values of bi and r are, within experimental error, in agreement with the theoretical values. In contrast, errors in r/bi are much smaller since, like di, they do not depend on the calibration constant. It can be estimated from eq 26 that the error in r/bi is about 0.2%. resulting in errors in Si, as indicated in Table IV. It can be seen that 8, and 0, are slghtly different from the values obtained from eq 11 (i.e., from the Neumann triangle), in which case errors in 0, are determined by the uncertainties in the values of uii. The observed discrepancy is evidently due to impurities present in the liquids used. This conclusion was drawn from measurements of interfacial tension performed 24 h after contact between two liquids was made. Little variation was found in u,,, but u13and uzI each decreased by about 1 mN m-'. Most likely, since glycerol and water were of high purity, the silicone and castor oils contained surface-active contaminants which accumulated at the oil/water interface but not a t the oil/oil interface. Conkc$ angles Oi calculatd from the interfacial tensions of the saturated systems are included (in parentheses) in Table IV. It can be seen that 0, was hardly affected, but 8, and e, increased and decreased with time, respectively. The observed differences in values of Si obtained from interfacial tensions and drop pair shapes are most likely due to the fact that, in the experiments, a certain time elapsed (about 30 min) between formation of the interface and photographing'the spinning drop pair. During this time interval, O1 increased and 8, decreased somewhat. I t can be concluded therefore that the shape of partially engulfed drop pairs is fully consistent with that
calculated from the Laplace equation with the Neumann triangle boundary condition. J3ecause of the presence of surfactants, small Marangoni effects could operate in our systems. However, if present, they play an insignificant role, as can be concluded from the agreement between theory and experiment. Figure 10 shows photographs of systems I, 11, III(1).and III(2) a t w = 180 rpm. I t can be seen that the contact angles e, change with surfactant concentration and interfacial tension. The change in & seems more apparent. We checked the theoretical results with experiments in which different surfactant concentrations were used. In Table I1 we showed that the lower interfacial tension system III(1) (the same as system B) has a tendency to separate, but the higher interfacial tension system I (as system A) has tendency to engulf. The experimental results for the difference in length X, at w = 0 and w = 3400 rpm are AX, = -0.9% for system I and AX, = +2.1% for system III(1) compared to the theoretical values of -0.6% and +1.5%. Although there is some error between experimental and theoretical results, they do show the predicted tendency to separate for system III(1) and to engulf for system I. In Figure 11, the theoretical results for the drop pair shapes (solid lines) are compared with the experimental ones (symbols)when the system is spinnii. The distances r, I, and L are compared with the values ro, Io, and Lo obtained for a slow spinning drop (w = 180 rpm), in which case the deviations from a stationary drop are too small to measure. The random errors in r, I, and L are estimated as 0.25%, 0.2%, and 0.15%. respectively, resulting in errors in r/ro, [ / I o , and L / L o of 0.35%. 0.3%. and 0.2%. respectively. It is of interest to note that these errors are much smaller than the total errors in r, 1, or L , which are about 1% because the ratios are independent of the calibration
Langmuir, Vol. 5, No. 5, 1989 1241
Spinning of Partially Engulfed Drops 1.012 I
-rr0
-L 1.008 LO
loo6 1.004
-1
I
I
I
I
I
t
0.996
1,
L 1.02
c
-ror 0.992 0 . 7 M
I
0
1 2 3 4 Angular Velocity C1O3rpm1
5
Figure 11. Comparison of the experimental and theoretical results for changes in the shape of drop pairs as a function of angular velocity for various SDS concentrations. Numbers on the curves represent the SDS concentration in g/L. Symbols are experimental results, and solid curves are calculated theoretically.
constant used in the magnification. This shows one of the advantages of the bispinner: by comparing shapes at different angular velocities, no absolute length calibration is necessary. It can be seen from Figure 11that although variations in drop shape are small (only a few percent), the experimental technique is sufficiently accurate to pick up these variations. Within experimental error, all relative deformations are in agreement with the predictions of the theory. The experimental results also show that the length of the drop pair in the system with a lower surfactant concentration changes more slowly than in the system with a higher concentration, but the radius r of the TPL changes differently due to the different tendency toward engulfment or separation. It is of interest to note that, in agreement with theory, the variations in r are more complex than those in L (or 1). The length of the drop pair increases with SDS concentration at a given angular velocity, since lower interfacial tensions cause drops to deform more readily (Figure 12a). However, the radius r of the TPL goes through a maximum value with increasing surfactant concentration (Figure 12b), reflecting the trends toward separation or engulfment.
Conclusions By means of a bispinner, a technique for manipulating
I I 0.984 I 1 I 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 SDS cgi11
Figure 12. Variations in the shape of a drop pair by varying the SDS concentration at constant angular velocity (w = 3400 rpm): (a) variations in total length; (b) variations in TPL radius.
small drops, one can produce a partially engulfed drop pair in a three-phase system and measure its changes in shape when it is spun about its horizontal axis. The drop formation depends on the boundary condition at the threephase contact line (TPL). The shape of the drop pair can be calculated by numerical integration of the Laplace equation using the Neumann triangle as the boundary condition at the TPL. The results show that a spinning drop pair in a low interfacial tension system changes its shape more than when the interfacial tension is high. Moreover, a drop pair in a low interfacial tensioli system has a tendency to separate, while for systems with a high interfacial tension there is a tendency toward engulfment. The radius r of the TPL decreases with increasing angular velocity, while the length of the drop pair increases. The change of r depends on the tendency toward separation or engulfment of the drop pair. Drop pairs with a large volume change their shape more readily than those with a small volume. When the volume of liquid drop 1of the drop pair is close to that of liquid drop 2, the changes in shape are more prominent than when the volume of liquid drop 1differs from that of liquid drop 2. With the bispinner, the results can be checked experimentally. The results prove the validity of the Neumann triangle in three-phase systems, in both stationary and rotating states. Small discrepancies are attributed to the impurities present in the liquids used and possibly to small Marangoni effects.
