731
Bursting of Soap Films
as being formed by the addition of singlet (SN), to singlet S2N2. Analogous processes would then lead to higher polymers.
Acknowledgment. We are grateful to the Data Processing Center of Kyoto University for its generous permission to use the FACOM 230-75 Computer. We owe thanks to Professor Alan G. MacDiarmid and to his outstanding co-workers for informing us about their valuable experimental result, and to the former for improving our English before the submission. We are also grateful to Dr. Hideyuki Konishi for his kind help in performing the MO calculations. This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education of Japan (No. 065101). References and Notes (1) (a) V. V. Walatka, Jr., M. M. Labes, and J. H. Perlstein, Phys. Rev. Left., 31, 1139 (1973); (b) C. Hsu and M. M. Labes, J. Chem. Phys., 61, 4640 (1974). (2) Some recent references are (a) R. L. Greene, P. M. Grant, and G. B. Street, Phys. Rev. Lett., 34, 89 (1975); (b) A. A. Bright, M. J. Cohen, A. F. Garito, A. J. Heeger, C. M. Mikulski, P. J. Russo, and A. G. MacDarmid, ibid., 34, 206 (1975); (c) R. L. Greene, G. B. Street, and L. J. Suter, ibid., 34, 577 (1975); (d) W. D. Gill, R. L. Greene, G. B. Street, and W. A. Little, ibid., 35, 1732 (1975); (e) P. M. Grant, R. L. Greene, and G. 8. Street, ibid., 35, 1743 (1975); (f) L. Ley, ibid., 35, 1796 (1975); (9) P. Mengel, P. M. Grant, W. E. Rudge, B. H. Schlechtman, and D. W. Rice, ibid., 35, 1803 (1975); (h) L. Pintschovius, H. P. Geserich, and W. Moller, Solid State Common., 17, 477 (1975); (i) C. H. Chen, J. Silcox, A. F. Garito, A. J. Heeger, and A. G. MacDiarmid., Phys. Rev. Lett., 36, 525 (1976).
(3) (a) D. E. Parry and J. M. Thomas, J. Phys. C, 8, L$5 (1975); (b) W. E. Rudge and P. M. Grant, Phys. Rev. Left., 35, 1799 (1975); (c) W. I. Friesen, A. J. Berlinsky, B. Bergersen, L. Weiler, and T. M. Rice, J. Phys. C , 8, 3549 (1975); (d) V. T. RaJanand L. M. Falicov, Phys. Rev., B12, 1240 (1975); (e) A. Zunger, J . Chem. Phys., 83, 4854 (1975); (f) H. Kamimura, A. M. Glazer, A. J. Grant, Y. Natsume, M. Schreiber, and A. D. Yoffe, J. Phys. C,9, 291 (1976); (9) A. A. Bright and P. Soven, Solid State Common., 18, 317 (1976). (4) A. G. MacDiarmid, C. M. Mikulski, P. J. Russo, M. S.Saran, A. F. Garito, and A. J. Heeger, J. Chem. Soc., Chem. Commun., 476 (1975); (b)C. M. Mlkulskl, P. J. Russo, M. S.Saran, A. G. MacDlarmM, A. F. Garb, and A. J. Heeger, J. Am. Chem. SOC.,97,6358 (1975). (5) R. H. Baughman, R. R. Chance, and M. J. Cohen, J. Chem. Phys., 64, 1869 (1976). (6) (a) T. Yonezawa, H. Konishi, and H. Kato, Bull Chem. SOC.Jpn., 42, 933 (1969); (b) C. C. J. Roothaan, Rev. Md. Phys., 23, 69 (1951); (c) H. Konlshi, H. Kato, and T. Yonezawa, Theor. Chim. Acta (BerL), 19, 71 (1970); (d) H. Yamabe, H. Kato, and T. Yonezawa, Bulf Chem SOC.Jpn., 44, 22 (1971); (e) H. Yamabe, H. Kato, and T. Yonezawa, ibid, 44, 611 (1971). (7) The two-center Coulomb repulsion integrals are calculated by the Ohno approximation (K. Ohno, Theor. Chim. Acta (Berl.), 2, 219 (1964)), and the one-center exchange integrals are evaluated by the Slater-Condon parameters estimated by Hinze and Jaff6 (J. Hinze and H. H. Jaff6, J. Chem. Phys., 38, 1834 (1963)). (8) 90.42 and 89.58’ for the S-N-S and the N-S-N angles, respectively. (9) 1.651 and 1.657 A alternatively. (10) D. R. Salahub and R. P. Messmer, J. Chem. Phys., 64, 2039 (1976). (11) D. 9. Adams, A. J. Banister, D. T. Clark, and D. Kilcast, Int. J . Sulfur Chem., 1, 143 (1971). (12) J. Bragin and M. V. Evans, J . Chem. Phys., 51, 268 (1969). (13) For example, see D. S. McClure, J. Chem. Phys., 20, 682 (1952). (14) M. Blume and R. E. Watson, Proc. R . SOC.(London), Ser. A , 271, 565 (1963). (15) For example, see T. Yonezawa, H. Kat6, and H. Kato, Theor. Chim. Acta (Berl.), 13, 125 (1969). (16) Unpublished observations, A. G. MacDiarmid and M. S. Saran.
The Bursting of Soap Films. 8. Rim Velocity in Radial Bursting Karol J. Myseis” General Atomic Company, P.O. Box 81608, San Diego, California 92138
and B. R. Vijayendran Celanese Research Company, P.O. Box 1001, Summit, New Jersey (Received September 17, 1976) Publication costs assisted by The General Atomic Company
The theory of the propagation of the bursting of a soap film from a linear origin shows that in the absence of surface relaxation the rim velocity cannot exceed the characteristic Culick‘s velocity of a rim without aureole. Experiments on radial bursting propagating from a point source indicated that higher velocities are observed. Numerical analysis of radial bursting shows that under some conditions Culick’s velocity may be indeed exceeded but only by a few percent. The study of the bursting of soap films can provide valuable information about both the equation of state of adsorbed monolayers at surface ressures and concentrations not otherwise accessible3! and also about the kinetics of desorption on the submillisecond time scalea4 The value of any interpretation of the experimental results depends, of course, on the accuracy of the experimental measurements and the validity of the theory used in their interpretation. An effort has been made therefore to analyze5p6a number of anomalies that have been encountered in these studies, and the present paper is the last in this series. The theory of unidimensional bursting propagating from a line defect in a uniform sheet can be fully developed under the assumption that the surface tension is only dependent on surface shrinkage, Le., that desorption is negligible during the short time involved.8 This as-
sumption is used throughout the present paper and, based on the work of ref 3, is believed to be applicable to most of the experimental studies available. The theory shows that the velocity of the rim of the hole cannot exceed a characteristic velocity u, called Culick’s velocity u,
= (2ao/p60)”2
(1)
where u is the surface tension, p the density, 6 the thickness of the film, and the subscript 0 indicates the condition of the undisturbed film. Culick’s velocity is obtained when the film beyond the growing hole is completely undisturbed. When, as is in fact always the case, a region of shrinking film, the “aureole”, precedes the hole, then it has been shown’ that rim velocity has to be smaller than 4.
On the other hand, experimental measurements on two-dimensional, radial bursting originating from a point The Journal of Physlcal Chemistry, Vol. 81 No. 8 , 1977 ~
732
Karol J. Mysels and B. R. Vljayendran
defect consistently showed6i10that when windage effects, i.e., the slowing down by the drag of the surrounding atmosphere, were eliminated or reduced, the rim velocity was slightly higher than u,. As reported, the difference was within the limit of experimental errors but the fact that it was observed under very different conditions using two very different techniques made it credible. The theory of radial bursting cannot be developed analytically to the same extent as that of unidimensional bursting so that no theorem about maximum rim velocity is available in this experimentally interesting case. Thus there is no real contradiction between theory and experiment but rather a disturbing area of ignorance. Numerical exploration of radial bursting seems to be the only way of clarifying this problem and the present paper is devoted to such an approach. That rim velocity has to be equal to u,when the aureole is nonexistent is a consequence of conservation of momentum. The existence of the aureole can, however, cause deviations. In the absence of desorption, with surface tension independent of time, there is no natural time constant for the system and velocities of all features, e.g., of any thickness, are constant.8 It is convenient then to express velocities and positions in terms of Culick's velocity u,. Thus
P = (r/t)/% (2) is the reduced velocity and position of a feature located a t a radial distance r at time t. The aureole can then be characterized by the reduced thickness, or thickening, 0 = 6/a0, as a function of p. It is said to be self-similar. Within the visible aureole the velocity of all features must increase with their reduced position. Otherwise the slower feature is overtaken by the faster one and a shock wave, Le., an abrupt change of thickness, occurs.8 The details of the thickness change within such a shock wave do not affect the calculations based on momentum changes, as long as the total width over which they take place is smaller than the uncertainty of the position of the shock wave. Furthermore, the same resulta apply to radial phenomena as long as this width is negligible compared to the radius of the shock. Hence the rim of the hole in radial bursting can be considered as a shock wave and on this basis it can be shown2that the limiting thickening 01 = 6 l / a 0 satisfies the relation
Pz = 4 I Z 2 / P l Z b l / d
(3)
where I is the reduced amount of material per angular unit between the point considered and the center. Thus, Il is the reduced amount of material in the rim per radian. I is conveniently obtained8 by difference from the material within the visible film and aureole from a point po in the undisturbed film and the point considered, i.e.
I = Po2/2 - s;,PP dP
(4)
The limiting surface tension can be obtained8 by integration of the local relation given by applying Newton's second law to a film element:
(5) which is eq 5.22 of ref 8 written in reduced units. Thus, if the shape of the aureole, Le., 0as a function p, is known, one can calculate the value of u by obtaining the value of I from eq 4 and the corresponding value of duldp from eq 5 at shccessive points and integrating the latter from p o where Q = uo to p. Once the criterion of eq 3 is The Journal of Physical Chemistry, Vol. 81, No. 8 , 1977
1,
!
'
I
pm
P
I
1.2
P,
Figure 1. The locus of radial rim velocities for a flat aureole of varying thickness and fixed frontal velocity p,. The locus reaches its maximum, pm,for the aureole thlckness shown. Dashed lines refer to the unidimensional case in which rim velocity decreases monotonically with thlckness.
Ip
Figure 2. The locus of radial rim velocities for a linear aureole of varying slope and fixed frontal velocity pt. The rim velocity p,of the aureole shown is less than the maximum, pm,of the locus. Dashed lines refer to the comparable unidlmensional case.
satisfied, the rim has been reached and thus its velocity obtained. Such a calculation can be made for experimentally obtained aureole profiles or for theoretical ones. The latter permits the varying of parameters to determine the factors affecting the rim velocity. Most of the results to be reported assumed a monomial aureole, i.e.
P
1 + 4 P f - PI"
(6) where pf is the frontal outer edge of the aureole, a determines its steepness, and n ita curvature. The simplest case of a flat aureole (Figure 1)corresponds to n = 0. This is also the only one for which the rim velocity could be obtained by solving through successive approximations a closed expression." For higher values of n including the linear aureole (Figure 2) numerical integration of eq 5 was required with step by step testing of eq 3. The results are summarized in Figures 1-3. In all cases the rim velocity tended, as expected, to u,as a approached zero, i.e., as the aureole vanished. In addition, the generalization that in the unidimensional case rim velocity cannot exceed u, was confirmed as shown by the dotted lines in Figures 1and 2. In the radial case, however, for small values of a, the rim velocity always exceeded u,. At a certain value of a, a,, rim velocity was maximum and for higher values of a it decreased. Thus there was a maximum rim velocity pmfor each value of pf. This maximum depended on pf,again tending to unity, i.e., to u,,as pf approached unity, i.e., as the aureole disappeared. =
Bursting of Soap Films
t
733
I
Figure 3. The locus of pm,the maximum radial rim velocity for a given p,, as a function of the p , values for several values of the exponent for monomial aureoles defined by eq 6. The dots indicate the location of the maximum of each line. Note that the ordinate covers only 2.5 % and that the maxima increase only slightly between n = 2 and 4.
I t also tended to unity as pf became large as shown in Figure 3. Hence, there was a maximum of pmfor every value of n. These maxima increased somewhat as n increased from 0 to 2 but after that the rise was very small. The maximum shifted to higher pf values as n increased. Thus, clearly the velocity of the rim of a radial aureole can exceed u,. The magnitude of the difference is, however, rather small. The largest difference obtained in our calculations was less than 2.5% and from the trend of the values it seems unlikely that any other theoretical shape would give significantly higher ones. Neither the exponential shape nor a number of binomial ones have shown any marked increase. It should be noted that the shape of the aureole is determined by the variation of surface tension with shrinkage of the film as may be seen from eq 5. Hence, a rim velocity approaching the maximum value would require a special shape of the surface tension vs. shrinkage curve to give the necessary shape to the aureole. On a purely statistical basis, most real systems should give values below this.
A previously published experimental aureole, that of solution A, Figure 1 of ref 3, was fitted to the best significant polynomial expression and its rim velocity calculated for a number of assumed pf values around the experimentally determined one. (The experimental uncertainty is larger in measuring absolute p values needed here than in the relative values involved in determining the aureole shape.) Rim velocities exceeding Culick’s value were found with a maximum of 1 . 0 2 1 at ~ ~the experimental pf value of 1.50. Thus, clearly theory and experiment both show that in radial self-similar bursting the rim velocity can exceed Culick’s velocity u, but only by a small amount. The higher values encountered in the present stud are not far from those reported by McEntee and MyselsIT who found for the thickest films an average excess of about 8% in u2, Le., about 2.8% in u over Culick’s velocity. On the other hand, it is likely that any larger excesses, such as those seen in Figure 1 of ref 6, are likely to be due to experimental errors. Acknowledgment. This work was begun and initial results obtained while the authors were members of the Research Department of R. J. Reynolds Industries in Winston-Salem, N.C. References and Notes (1) Presented in part before the Dlvisbn of Callold and Surface Chemistry at the 161st National Meeting of the American Chemical Society, Los Angeles, Calif., March 1971. (2) G. Frens, K. J. Mysels, and B. R. Vijayendran, Spec. Discuss. Farshy Soc., 1, 12 (1970). (3) B. R. Vijayendran, J . Phys. Chem., 79, 2501 (1975). (4) A. T. Florence and 0.Frens, J . Phys. Chem., 76, 3024 (1972). (5) K. J. Mysels and A. T. Florence, J . Phys. Chem., 78, 234 (1974). (6) K. J. Mysels and B. R. ViJayendran, J. phys. Chem., 77, 1692 (1973). (For a different point of view, see ref 7.) (7) G. Frens, J . Phys. Chem., 78, 1949 (1974). (8) S. Frankel and K. J. Mysels, J . Phys. Chem., 73, 3028 (1969). (9) F. E. C. Culick, J . Appl. Phys., 31, 1128 (1960). (10) W. R. McEntee and K. J. Mysels, J. Phys. Chem., 73, 3018 (1969). (1 1) This expression Is
a Z p 4t p2[P(l.5a- 0.5 + a lnp/pf)- a ] t 0.5P2= 0 where p represents pI and P
Po(l - a ) .
The Journal of Physlcal Chemlstty, Vol. 81, No. 8, 1977
