Langmuir 1986,2, 248-250
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Equilibrium Conformations of Liquid Drops on Thin Cylinders under Forces of Capillarity. A Theory for the Roll-up Process B. J. Carroll Unilever Research Port Sunlight Laboratory, Bebington, Wirral, Merseyside L63 3 J W , United Kingdom Received January 11, 1985. I n Final Form: September 26, 1985 The way that the conformation of a liquid drop adhering to a circular, cylindrical fiber changes when the contact angle is increased from a low initial value is discussed in terms of the classical Laplace excess pressure in the drop. It is shown that an axially symmetrical conformation becomes metastable when the contact angle exceeds a critical value which is dependent on the fiber radius and the drop volume. Beyond this point an axially unsymmetrical conformation is favored. The transition between conformations is accompanied by a reduction in the drop/fiber contact area over and above that expected from the contact angle increment bringing about the metamorphosis. The transition described corresponds to the classical roll-up process of detergency first described by N. K. Adam.
Introduction When a liquid drop of definite volume is placed on a thin circular cylindrical fiber of definite radius, it adopts a final conformation that is determined solely by the contact angle. When the contact angle is small or zero, the conformation is symmetrical with respect to the cylinder axis, but when the contact angle is sufficiently large (how large is dependent on the drop volume and cylinder radius) a qualitatively different, nonsymmetrical conformation results. The two conformations are illustrated in Figure 1. The existence of these different conformations has been recognized for a considerable time, and Ada" was possibly the first to appreciate the practical significance of the difference for the process of the detergency of oils from fabrics. The areas of contact between the drop and substrate are substantially lower in the unsymmetrical conformation than in the symmetrical and therefore the strength of adhesion of the drop to the fiber is less in the former case than in the latter. The former conformation is also the more susceptible to viscous drag, so that overall, removal is greatly facilitated when the unsymmetrical conformation occurs. Adam's term for the transition between the two conformations-"the roll-up process"-is both descriptive and succinct and has a secure place in the language of surface chemistry. To date, however, no quantitative description of the mechanism of the transition between the two conformations has been published. Two possible reasons for this gap in the literature of capillarity are, first, the lack of experimental data and, second, the mathematical complexities associated with the description of the two conformations, especially the second, highly asymmetrical one. The experimental problem originates in the difficulty of varying the contact angle in a controlled way through a range of values in which the transition occurs and of doing this without otherwise disturbing the system: Such changes in the contact angle are generally induced by the addition of surfactant to the system and the roll-up, if it does occur, usually happens in a time comparable with that needed for the establishment of diffusive and hydrodynamic stasis. These problems can be entirely avoided by studying a system in which the drop volume rather than the contact angle is varied. Such systems are encountered when an oil drop attached to a fiber is immersed in a surfactant solution capable of solubilizing the oil. In this arrangement, which has been used
to make quantitative measurements of the rate of solubilization, the drop volume decreases smoothly over a convenient time interval (of order minutes), while the contact angle remains nearly constant.2 The sequence in Figure 2 illustrates the solubilization of an oil drop attached to a fiber. The transition between conformations (b) apd (c) is of most interest in the present context. In symmetrical conformations it is possible to calculate both the drop volume and the contact angle from linear system parameters3 so that one pair of critical drop volume and contact angle values emerges from this sequence. If the system parameters are varied by changing the constitution of all three phases, a set of such critical values can be compiled and compared with theory. Progress on the theoretical aspect of the problems has been made by applying a perturbation-type analysis to a description of the system developed in recent paper^.^^^ This is fully described in the next section, where it is demonstrated that for a drop of given volume on a circular cylinder of fixed diameter, the axially symmetrical conformation becomes metastable when the contact angle exceeds a certain value and that this critical value is related to the drop volume and to the cylinder radius: increasing the drop volume or decreasing the cylinder radius tends to favor the axisymmetrical conformation.
Theory The drop conformation that is symmetrical with respect to the cylinder axis is characterized by three linear parameters: the cylinder radius xl, the drop median radius out of the plane of the paper x 2 , and the drop length L (Figure 3). Division of the last two parameters by the first gives the two dimensionless quantities n = x2/x1 and = L/xl and reduces the description of the system to two variables. It is shown in ref 3 how from these two quantities the contact angle 0, the drop volume, and the excess internal pressure (AP) can be obtained. The starting point for the present analysis is the expression derived in ref 3 for the Laplace excess pressure:
e
in which R1-'and R2-' are the principal curvatures of the surface at a point and y is the interfacial tension. Ref(2) Carroll, B. J. J. Colloid Interface Sci. 1981,79, 126.
(1)Adam, N. K. J . SOC.Dyers Colour. 1937,53,122
0743-7463/86/2402-0248$01.50/0
(3) Carroll, B. J. J. Colloid Interface Sci. 1976, 57, 488. (4) Carroll, B. J. J . Colloid Interface Sci. 1984, 97, 195.
0 1986 American Chemical Society
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Langmuir, Vol. 2, No. 2, 1986 249
Equilibrium Conformations of Liquid Drops
,------.
- I
I
-
'\
'..___/' Figure 4. hisymmetrical perturbation of the axisymmetrical conformation.
Figure 1. Two principal conformations of a liquid drop on a cylinder (gravity negligible).
L 0
30
60
8
I
Figure 5. Graph of critical values of n VB. critical values of 8.
d
5
Figure 2. Solubilization of an oil drop by an aqueous surfactank showing transition between the two conformations.
Consider now a perturbation of the drop from ita equilibrium conformation involving a small movement of the two three-phase contact lines, as shown in Figure 4. A t a constant drop volume, the effect will be to increase one curvature (R;') and to decrease the other (R;') 80 that a net pressure change
a(@) = r(6R;'
+ 6R;')
occurs in the vicinity of point A. If the perturbation of the equilibrium position of the two contact lines is such as to produce a change (6x4 of the drop median radius (6x2 > 0). then it is possible to write
(4) L
Figure 3. System parameters for the axisymmetrieel conformation. erence to Figure 3 shows that there are three different positions on the drop surface where one of the principal radii of curvature is obvious from inspection. These positions are labeled A-C. At A, the radius out of the plane of the paper is clearly equal to xz. at B, which is a point of inflexion, the radius in the plane of the paper is necessarily infinite (zero curvature), and a t C, the radius out of the plane of the paper is xl sec 8. By use of eq 1, it is obviously possible to write down an expression for the second radius of curvature in each case and doing this for point A the following equations for the reduced principal curvatures in and normal to the plane of the paper are obtained:
Rz-' = x , / R z = l / n
(3) For n > 1, both these reduced curvatures are of the same sign, and Rl-' < or > R2-' for n > or < sec 8 (8 < 90").
6R2-'
dR2-' u
-6n dn
(5)
The negative sign in eq 4 originates in the fact that while dR,-l/dn is negative for the unperturbed, axially symmetrical drop (cf. eq 2, in the present perturbation R1-' increaseswith increase in n. The approximation is made that the change in R;I in going from the unperturbed to perturbed states is equal to minus the change in this quantity going between two unperturbed states characterized by the same change in the parameter n. The two differential coefficients are readily evaluated and the exaression for the Dressure change becomes
The magnitude of this pressure change depends upon that of the expression in parentheses and it is easily shown that it changes sign when n and 8 satisfy the equation
2n3 cos 8 - 3n2 + 1 = 0 (7) The locus of points entisfying eq 5 in the (n, 0) plane is given in Figure 5. For points above this line NAF') < 0; for points below, NAP) > 0. Consider now a perturbation
250 Langmuir, Vol. 2, No. 2, 1986
Carroll
n l
O
I
-___--Figure 6. Perturbed &symmetrical conformation as a precursor of rolling up.
of the two contact lines of the type shown in Figure 6, where in the upper part of the figure the two contact lines move together and in the lower part they move apart. In the initial stages of such a perturbation, nearly equal areas of both the solid/oil and solidlwater interfaces are created and destroyed, with a net zero contribution to the free energy change from each interface. Changes in the Laplace pressure distribution in the system then solely control the response of the system to the perturbation. Except when &(Up) = 0, there will in general be a pressure difference set up between the upper and lower points of the drop, originating in the movement together of the contact lines in the upper part. If 6(U)> 0, there will be a flow of liquid from the upper to the lower part of the drop. This flow will once again decrease R;’ and since Ri’ is also less than its initial unperturbed value, the flow is sustained. The regions in Figure 5 separated by the continuous line therefore represent sets of values of (n,0 ) for stable (above the line) and metastable (below the line) conformations that possess axial symmetry. Comparison with Data and Discussion. In Figure 7 are plotted the “critical coordinates” measured on liquid drops from a wide range of solubilization experiments. Also plotted is the curve given by eq 7. The critical coordinates plotted were necessarily measured on the penultimate photograph in a sequence ending in a roll-up movement, so that the measured value of n is expected to be higher than the “true” critical value. The otherwise diffuse nature of the data can be attributed to external factors (notably variability in the background building vibration levels, which may play a part in initiating the movement of the three-phase contact lines) and to internal factors, including deviations of the fiber geometry from that of a smooth cylinder (cf. ref 4). Both these factors would tend to shift a given point to the left, toward low 0 values. Bearing in mind that all accessible experimental points must (because they correspond to precritical drops)
0
25
0
50
75
Figure 7. Critical n vs. critical 8 (Figure 5) with experimental points plotted.
in any case lie to the left of the theoretical line and given the likely directions of the sources of experimental bias listed above, reasonable agreement with theory is apparent. This suggests that the approximations necessarily made in developing the equations in the previous section are reasonable. In conclusion, it seems worthwhile to comment on the use of the term “roll-up” in the literature. Although it was originally applied by Adam to the movement of a liquid drop off a cylinder as the contact angle is increased, the term has also appeared recently to describe the movement of a drop on a plane surface under the same circumstances. To the present author it seems preferable to reserve the term roll-up for the former process only. Quite apart from historical considerations, it is possible to demarcate clearly between the two cases. In the case of the plane surface, when the contact angle is increased, the contact area between drop and substrate decreases. In the case of a cylindrical substrate, an increase in contact angle brings about an analogous change in the phase contact areas, but if the angle is near the critical value before it is further increased, the transition between conformations that occurs results in a further change in this contact area and, in addition, the profile of the drop-in cylinder system becomes radically different. The process that occurs on a cylinder is clearly differentiable from the simpler behavior observed on a plane substrate and therefore merits its individual designation.
Acknowledgment. Some of the experimental work was performed by M. Carve11 and P. Doyle. P. Doyle also assisted in the compilation of the experimental data from photographic records.
