Langmuir 1990,6, 1269-1212 suppressed, but the energy migration is efficient. Therefore, the high potential of the anthryl vesicle as an efficient photo-harvesting system was shown. However, the problem of the excimer formation which may work as an energy trap remained unresolved, although the number of the excimer site may not be very large.
1269
Acknowledgment. We thank Dr. H. Hasegawa of the Department of Polymer Chemistry, Kyoto University, for measuring the TEM photographs. M.S. also acknowledges the financial support from the Grant-in-Aid for Scientific Research No. 63470095 from the Ministry of Education, Science, and Culture, Japan.
Effect of a Soap Film on the Catenary P. Mohazzabi,"J. P. McCrickard, and F. Behroozi Department of Physics, University of Wisconsin-Parkside, Kenosha, Wisconsin 53141 Received November 20,1989 The shape of a string, suspended from both ends, under the combined effect of gravity and surface tension is studied. Exact equations of the system are derived which produce results that are in good agreement with the experimental observations. The shape of the string, to a good approximation, turns out to be an arc of a circle.
Introduction The determination of the shape acquired by a uniform chain which is suspended from two points, known as the catenary (from Latin catena for chain), has a long and venerable history.' Galileo believed the shape to be a parabola. However, Huygens showed that the curve is nonalgebraic and therefore not parabolic. In 1691 and again in 1692,Leibnitz published the correct equation of the catenary but did not give a proof. It was Jacob (James) Bernoulli (1654-1705)who furnished the proof establishing that Leibnitz's equation for the catenary was correct. He further studied the shape of strings with variable density and under action of a central force.2 Nowadays, the catenary is usually treated as an example of the application of calculus of variations in modern textbook^.^ We became intrigued with the effect of a soap film on the shape of a hanging string after observing our children at play. They were making large soap bubbles using a meter-long piece of knitting yarn tied to a horizontal rod. Each end of the yarn was tied in a loop which could slide along the rod. With the two ends together, one dipped the rod and yarn into a soap solution and removed it. While the rod was held in a horizontal position, the ends of the yarn were pulled apart, and a film was formed as illustrated in Figure 1. When the rod was jerked rapidly normal to the film, air pressure forced the film to bulge and shed a large bubble. Interesting and amusing as the bubbles were, we were more intrigued by the shape of the string under the combined action of gravity and surface tension. Upon closer examination, we noticed several novel features. In describing these features, it is convenient to refer to Figure (1) See, for example: Rouse Ball, W. W .A Short Account of the History Mathematics; Dover: New York, 1960. (2) Eves, H. Great Moments in Mathematics. The Mathematical Association of America, 1981. ( 3 )Boas,M. L.Mathematical Methods in the Physical Sciences, 2nd ed.; Wiley: New York, 1983. of
0143-146319012406-1269$02.50/0
ili D
Figure 1. String suspended from a rod under the combined action of gravity and surface tension of a soap film. The soap film fills the region enclosed by AOBA.
1. When the ends of the string are pulled apart by a distance 2x0, the lower end of the string remains clumped together for a total length 21, 1 on each side. For small values of 2x0, the angle of the string with the rod, 60, and the length 1 are relatively large. As the ends are pulled apart, 1 and 60 both decrease, until 90 reaches zero. Pulling the ends further apart causes the string to contact the rod as shown in Figure 2. Under this condition, 60 remain zero, but the clumped length and the distance between the two contact points decrease as the ends are separated further. In this paper, we derive the equation of the string by a straightforward application of Newtonian mechanics and 0 1990 American Chemical Society
Mohazzabi et al.
1270 Langmuir, Vol. 6, No. 7, 1990 ds E c o s 0 - Tsin 0 = -2u-sin d0 d0
0
(7)
ds dTsin d0 0 + T cos0 = ( 2 ~ ~00- sAg)-d0
(8)
Multiplying eq 7 by -sin 0 and eq 8 by cos 0 and then adding the two resulting equations gives
11 I Figure 2. Same legend as in Figure 1, except that the surface tension is large enough to pull the string up to the rod, making zero contact angles at E and F.
show that the observed features are well explained by the theory. In particular, for a given string of known parameters (length, linear mass density) and a soap solution of known surface tension, the theory predicts the values of length 1 and angle 00 when the end separation is given. To test our theoretical predications, we made several measurements of the angle Oo, length 1, and the end separation under various conditions. In all cases studied, the theory accounts very well for the observed results.
Theory We begin by considering an element As of the string, shown in Figure 1, under the combined action of gravity and surface tension. The element of the force exerted on As by the soap film is 2aAs, where u is the surface tension of the soap solution. The factor of 2 in this expression accounts for the fact that the film consists of two surfaces. Static equilibrium of As requires that the net force on As be zero. Applying E F x = 0 and E F y = 0, respectively, leads to
@) = 0 (1) 2
(
( T - ~ ) s i n ( B - ~ ) + 2 o A s s i n a -T +) -:A
(
3
O = r/2, T = IXg where 1 is half of the total clumped length. Imposing this boundary condition on eq 9, we obtain
Substitution of T from eq 9 into either eq 5 or eq 6 results in a second-order differential equation for s in terms of 0: (11)
This is the equation governing the shape of the string and can be integrated once to give
1-1ds
2u = -2 In lcos e--[ + In C = In C d0 Ag where In C is the integration constant, with C > 0. Noting that ds/dO > 0, this result reduces to
In
C
-ds=
(12) do (cos 0 - 2u/Ag)2 The constant C in this equation can be determined by applying the boundary condition given by eq 10. The result is
c = 2ul/Ag and therefore
sin O +
- -AgAs=O
ds T = (2u-Ag~o~O)d0 It should be noted that the arc length s is chosen such that it increases with 0, Le., ds/dO > 0. Furthermore, at point 0 (Figure l), where clumping starts,we have 0 = r/2, and the tension in the string is equal to the weight of half of the clumped string. This leads to the boundary condition
(2)
where T is the tension and X is the mass per unit length of the string. Noting that a = ~ / -20, rearranging eqs 1 and 2, and dividing each equation by AO, we obtain, respectively
-ds--
a1
do (cos 0 - a)2 where a is a dimensionless constant defined by
(13) (Y
= 261
Xg.
In order to find an equation describing the shape of the string, we first note that (14)
- 2u-As sin 0 (3) A0
If we now take the limit as A0
-
As (2u cos 0 - Ag)- (4) A0
0, we find
d cos 0) = -2a-ds sin 0 -(T d0 d0 d ds -(T sin 0) = (2a cos 0 - Ag)-
d0 d0 Expanding the left-hand sides of eqs 5 and 6 gives
(15)
The negative signs on the right-hand sides of eqs 14 and 15 are due to the fact that s increases with 0. Combining eqs 13-15, we find the two parametric equations a1 cos 0 -dx =do (COso-a)z Q = - a1 sin0 do (cos 0 - a)2 Integration of the first equation depends on whether a is larger, equal to, or less than 1. However, as will be discussed later, the values of (Y are typically much larger
Langmuir, Vol. 6,No. 7, 1990 1271
Effect of a Soap Film on the Catenary than 1. For a
> 1, we obtain
1.6 I
a2sin8 -X = 1 (a2- ~ ) ( C-YCOS 8) 1.2
I
I
I
I
I
0.08
0.10
R
where C, is the integration constant. Similarly, the second equation can be integrated (without any condition on a) to give
where Cy is another integration constant. The integration constants C, and Cy may be evaluated by fixing the origin of the coordinate system. If we choose our coordinate system as shown in Figure 1 with the origin at point 0, where clumping begins, then we have x = y = 0 at 8 = ~ / 2 Applying . these conditions to eqs 16 and 17, we find a 2a a+1 c, = tan-' (a2 - 1)1/2] - 1 (a2- 1)3/2 cy= -1 Equations 16 and 17 with C, and Cygiven as above are the parametric equations of the string down to the point where clumping begins. Elimination of the parameter 8 gives the explicit equation of the string:
+
[
0.00
0.04 xo/L
0.02
0.06 .
Figure 3. Graph of 00 as a function of x o f L obtained from eq 21. At x o / L = 0, the value of Bo is A f 2. At Bo = 0, the value of x o / L is l / a . a sin 80 -
CY - COS 80
L
2 (a2-
a+l
tan-' 2a
[ (a2- 1)1/2])'(
sin Bo -
a - a - cos 80
tan-' [(a+ 1) tan (s0/2)]
(a2- 1)'/2
+
(a2- 1)1/2
2a
a+l
(a2 - 1)1/2
A graph of this equation, in fact, correctly predicts the observed shape of the string in the unclumped region, i.e., region OB in Figure 1. We now set out to manipulate our equations so that we can actually calculate some measurable quantities for comparison. In a typical experiment, the length of the string and the separation of the two supporting points (A and B in Figure 1) are chosen arbitrarily. The clumped length and the angle 60 at the supporting points can then be measured. We first integrate eq 13, yielding S,L-'ds = alJy2 d8 (a- COS e)2
If a,X O , and L are known, eq 21 can be solved numerically for 80, and then 1 can be calculated from eq 19 or eq 20. The values of 80 and 1 can then be compared with the measured quantities. Before turning to the measurements, however, we would like to point out that eqs 19 and 21 qualitatively predict the observed behavior of the string as the two ends are pulled apart. When the ends are together, xo = 0, and eqs 19 and 21 predict that I = L and 80 = ~ / 2 that ; is, the total length of the string is clumped and hangs straight down. As the ends are pulled apart, x o increases, and eqs 19 and 21 indicate that 1 and do both decrease. When xo reaches the critical value xo = L/a, the solution to eq 21 is Bo = 0, while eq 20 predicts that the value of 1 is given by
where L is half of the total length of the string. Carrying out the integrations, we find
Also, since xo denotes the value of x at 0 = do, we find from eq 16 that
2
(a2 - 1)'/2
tan-'
[
+ 1) tan (8,/2)
(a
(a2- 1)'/2
Dividing eq 20 by eq 19, we find
I+
For xo > L / a , eq 21 has no physically meaningful solution obtained for 80. Figure 3 shows 80 as a function of X O / L from eq 21. As can be seen, the values of x o / L are limited between zero and 1/a. This last prediction agrees with the observed behavior. Recall that as the ends A and B are pulled apart, eventually the string begins to adhere to the rod as shown in Figure 2. According to our results, this should first occur when xo = L / a . As the ends are pulled still further apart, a greater length adheres to the rod, effectively decreasing the hanging length. Our results are consistent with this provided that we interpret 2L as the actual hanging length of the string (EDF in Figure 2), rather than its total length (ADB), and t h a t 2x0 is interpreted as the distance between the contact points (EF) rather than the distance between its ends (AB). According
1272 Langmuir, Vol. 6, No. 7, 1990
to this interpretation, as the ends are pulled further apart the effective hanging length decreases, and since x o = L/a, xo decreases as well. This results in the length 1 decreasing according to eq 22. Consequently, L, X O , and 1 all decrease proportionately after 00 reaches zero.
Experimental Section To check our calculations, we used a length of braided nylon string, a glass rod, and a 1 % detergent solution in water. By use of a standard Cahn balance, the surface tension of the detergent solution was measured to be 24 dyn/cm. The linear mass density of the nylon string was determined to be 0.446 g/m when wet. With these physical parameters, the dimensionless parameter a turns out to be very nearly 11. The experiment consisted of measuring carefully the length of the various segments of the string. The angles were measured by projecting the shape of the string on a screen to achieve better accuracy. Figures 1 and 2 depict the shape of the string under two different load conditions. In Figure 1, the string hangs under ita own weight when the total length is large. In this case, the effective total length of the string can be considerably increased by hanging a small weight to the end of the clumped length (point D). In Figure 2, where the total length is shorter, the surface tension pulls the string up to the rod, forming zero contact angles. In this case, however, as mentioned earlier, 2x0 is the distance between the two zero-angle contact points (E and F) and 2L is the length of the string which is hanging between these two contact points. The result of our measurements for the two cases considered is as follows:
Mohazzabi et al.
But since 0 I0 Ix/2, then 8/2 < 1. Ignoring the second term in eq 16, which is of the order of I/$, we find
x=--
sin B 1 a - cos e + c* and y is given by eq 17. Fixing the origin as before, we find C, = 1/a and Cy = -1. Therefore
E=---1
sin8 a-cos6 2=- cos e 1 a - cos e If we further neglect cos B compared to a,we find 1
a
r
1 7 = $1 - sin 6)
(23)
Equations 23 and 24 are the parametric equations of a circle. Elimination of the parameter 6' gives (x - ;)2
+ y2 =
(y
(25)
which is a circle of radius l / a , centered at ( l / a ,0). It is interesting to note that using eq 25 alone and some straightforward geometrical considerations we obtain
case 1 (Figure 1):
Bo = 53O, L = 212 cm, xo = 4.2 cm, 1 = 198 cm case 2 (Figure 2):
x0
Bo = 0, L = 28 cm, ro = 3.15 cm, 1 = 23 cm Discussion In the first experiment, when the distance between the two support points and the total length of the string were chosen arbitrarily, we used the values of xo and L in eq 21 and solved numerically for 00. We found 00 = 51.8'. Then either eq 19 or 20 gave 1 = 199 cm. These compare with the measured values of 00 = 53' and 1 = 198 cm, with errors of 2 7; and 0.572 , respectively. In the second experiment, where the contact angle 60 was zero, we chose 00 and L, half of the total length, as input for calculations. With these data, eqs 19-21 gave xo = 2.55 cm and 1 = 24.1 cm, These compare with the measured values of xo = 3.15 cm and I = 23 cm, with errors of 19 % and 5 7% , respectively. As the results indicate, the theoretical equations developed in this work successfully predict the correct behavior of the catenary influenced by a soap film. The larger discrepancy between theory and the second experiment may be attributed to the fact that the string used was not perfectly flexible. Residual normal forces could influence the string, especially in the second experiment, where higher curvatures were involved. It is interesting to note that since the values of a are typically much larger than 1, we may approximate the results for a >> 1. First we note that in eq 16 we find
(
CY
I(~+ 2 I--~~)=~L
(26)
+ -x2 - 00) = L ( l -sin e,)
(27)
For the first experiment with L = 212 cm and xo = 4.2 cm, eqs 26 and 27 yield do = 50.2' and 1 = 199 cm. The differences between these values and those obtained from the exact equations are 3.1 7% and zero, respectively. For the second experiment with 00 = 0, eqs 26 and 27 yield xo = 2.23 cm and 1 = 24.5 cm. These are within 12.5% and 1.7 7% ,respectively, of the corresponding exact solutions. Finally, we would like to point out that if we set u = 0 in eq 11, then instead of eq 12 we get -ds =--
c
de cos2% Note that since we are measuring s from point B in Figure 1, in this case ds/de < 0. Combining this equation with eqs 14 and 15, integrating the resulting equations, and eliminating 0 give the equation of the catenary
(q)
( Y - c,7 = cosh )
where C1and CZ are the integration constants.
Acknowledgment. The assistance of Professor Bernard Abraham in measuring the surface tension of our soap solutions is gratefully acknowledged. n = 3, 126925-25-5; Registry No. H[Ca~Na~-3Nb~03,-1] H[CazNa,-3NbnO~,-1] n = 4,126925-26-6; H[CazNa,-3NbnO3,1]n = 5, 126925-27-7;M360, 86904-09-8; M600, 77110-54-4.
