3028
STAR’LEY FRANKEL AND KAROL 5. MYSELS
therefore, not probable, although possible, but a more likely explanation should be noted. During its formation, the second-black film is in contact-and presumably in equilibrium-with an exudate, the pressed-out liquid visible in Figure 13c. As shown elsewhere,14 the exudate has a different composition than the bulk liquid, in particular, it is poorer in NaCl. Hence,
it is possible that the properties of the second-black film change with time as it becomes gradually equilibrated with the bulk solution. Acknowledgments. We are indebted to Dr. Malcolm Jones for exploratory flash photographs. This work was supported in part by the National Science Foundation.
The Bursting of Soap Films. 11. Theoretical Considerations] by Stanley Frankel and Karol J. Mysels Los Angeles, California 00056, and Research Department, R. J . Reynolds Tobacco Company, Winston-Salem, North Carolina 27102 (Received December 9,1068)
411 N . Martel,
Several aspects of the bursting of thin liquid films of low viscosity are considered, neglecting air resistance. I t is shown that any significant aureole preceding the rim of the expanding hole is related to large changes of surface tension as the film shrinks and thickens. Methods for calculating these changes from observable quantities are given both for the general case and for self-similar bursting in which all features of the aureole expand linearly with time. The latter represents a limiting case in which the relaxation of surface tension is negligible during bursting. This limiting case is discussed in more detail for unidimensional bursting and several experimentally observed features of the bursting of soap films are accounted for qualitatively. After a thin liquid film such as a soap film develops the tiniest hole, surface tension forces will cause this hole to grow and the film to disappear. This is so rapid a process that the published experimental studies are rather sketchy. A theoretical estimate of the rate at which the hole grows was given by Dupr6 over a century ago.2 This estimate was endorsed by R a ~ l e i g hbut , ~ in 1960, Culick4 showed that it involved a basic error which exaggerated the rate by a factor of d2. The simple premise used by both Dupr6 and Culick was that the process is analogous to the contraction of the surface of a pure liquid in that the surface tension remains constant everywhere. As a corollary, the film must be undisturbed except at the edge of the hole where the collapsed material collects in a toroidal rim. This was in agreement with experimental work available on although a large body of other studies, including Gibbs’ considerations of surface elasticity,ll showed that solutions, and especially thin films of solutions, do not have a constant surface tension during contraction, especially rapid contraction. Recent experimental work of McEntee and Mysels, described in the first paper of this series,12shows that, in fact, film bursting often involves disturbances (the aureole) clearly visible on high-speed photographs, which grow more rapidly than the hole itself, and also that under some conditions the velocity of the rim is much below that T h e Journal of Physical Chemistry
calculated for an undisturbed film. An aureole has been also noted by Liebman, et a1.13 The present paper is concerned with the further development of the theory of bursting in an undisturbed film as an important limiting case, and with the role of (1) Presented in part at the 155th National Meeting of the American Chemical Society, San Francisco, Calif., April 1968. (2) A. DuprB, Ann. Chim. Phys., (4)11, 194 (1867); “Theorie MBcanique de la Chaleur,” Gauthiers-Villars, Paris, 1868. (3) Lord Rayleigh, Proc. Roy. Inst., 13, 261 (1891); Nature, 44, 249 (1891); “Scientific Papers,” Dover, New York, N. Y., Vol. 111, 1964, p 448. (4) F. E. C. Culick, J . A p p l . Phys., 31, 1128 (1960). (5) A. J. DeVries, Rec. Trav. Chim. Pays-Bas, 77, 383 (1958); Rubber Chem. Technol., 31, 1142 (1958). (6) W. E. Ranz, J . A p p l . Phys., 30, 1950 (1959). (7) J. Plateau, “Statique de Liquides,” Vol. 11, Gauthiers-Villars, Paris, 1873, p 317; Bull. Acad. Roy. Belgique, (3)2, 3 (1881). (8) G. van der Mensbrugghe, Ann. SOC.Scientifque Bruzelles, 31, 1 (1897). (9) Reference 8 cites C. Marangoni and P. Stefanelli, Monografia delle Bolle Liquide, Pira, 1873, pp 68-76, which could not be located despite an extensive search and seems different from the article by the same authors, Nuovo Cimento, 7-8, 301 (1872); 9, 236 (1873). (10) M. Kornfels, Z h . Tekh. Fiz., 24, 1520 (1954). (11) J. W. Gibbs, “Collected Papers,” Longmans Green & Co., Inc., New York, N. Y., 1931, p 302. (12) W. R. McEntee and K. J. Mysels, J . Phys. Chem., 73, 3018 (1969). (13) I. Liebman, J. Corry, and H. E. Perlee, Science, 161, 373 (1968).
3029
THEBURSTING OF SOAPFILNS viscous forces in this process; it develops the equations for the general case of disturbed films and their solution for another limiting ideal, that of “unidimensional” bursting from a line source in the absence of significant surface relaxation. The latter case is particularly tractable and experimental evidence indicates that the analysis can account qualitatively for many of the observed phenomena. The model which we shall consider throughout is a film having a uniform initial thickness, 60, surface tension, u0, and density, p . A burst may be initiated at a point and then spread to produce a circular hole to give two-dimensional radial bursting; alternatively it may originate along an (infinite) line and propagate uniformly to both sides of it to give unidimensional bursting. The distance T or 5 , respectively, from the origin is the appropriate principal coordinate. In either case the hole is limited by a rim of varying but large (compared to a0) thickness whose speed, measured say at the center of mass of its crass-section, is u H , the rate of expansion of the hole from its origin. The film beyond the rim may be essentially undisturbed or it may be disturbed in a region, the aureole, which runs ahead of the hole. It may then have a thickness, 6, and surface tension, u , varying with distance and with time, t. If a is only a function of 6 because other changes in u are slow compared to the speed of bursting, we deal with thicknessonly dependent u , This is an important limiting case arising, for example, when the amount of adsorbed surfactant remains constant and surface tension is determined by the area per adsorbed molecule. If desorption occurs, then u will relax at constant 6. We shall show that if bursting were to occur at constant u, i.e., u = uo, there would be no visible aureole unless the viscosity of the liquid is extremely high. Air resistance will be neglected throughout. This last point may be an important limitation in applying the theory to real systems. Speeds in laboratory coordinates are indicated by u and other symbols will be introduced when the coordinate system is changed. Certain factors which are neglected because they do not affect the applicability of the theory to soap films (as contrasted with the effect of air resistance) should be mentioned explicitly. One is slip between the monolayer and the immediately subjacent liquid. This has never been shown to be significant for water. Another is the relative viscous motion of the surface with respect to the central layer of the film. This would correspond to a lagging of the center behind the surface layers as the latter shrink and accelerate. The kinematic viscosity of water is about 0.01 cm2/sec or 1 pu2/psec. Hence, the time for the transfer of momentum across 1 p the distance to the center of a quite thick film-is of the order of 1 psec, i.e., just greater than the blur on a microflash photograph, and decreases with the square of the thickness for thinner films. The third neglected factor is the effect of compressibility of water since the
velocity of sound in water (1.4 mm/psec) is much larger than bursting velocities and the two surfaces of the film remain at atmospheric pressure so that any compressive stress is relieved before affecting the experiment.
1. Undisturbed Film This limiting case can be analyzed completely and the results form the basis of the succeeding discussion of more usual bursting processes. A. Unidimensional Case, Constant Velocity. The simplest case considers a burst propagating from a line origin a t a velocity assumed to be constant. It yields the same velocity as the radial case which will be considered in the following section. Here we deal with a unit length of the rim which moves with a constant velocity, UH, increases in mass by UH80pdtin the time dt, and is acted upon by the surface tension of the two faces of the film, 2u0. Conservation of momentum then requires 2~Odt= (UH6Opdt)UH
(1.1)
Hence UH
= (2Cr0/60p)1’2
(1.2)
This is the result obtained by Culick,* and because of its importance, we call it the Culick velocity, U C , defined by u 2 = 2u0/60p
(1.3)
It may be noted that if conservation of energy were assumed, a factor of ‘/2 would be introduced into the right hand side of eq 1.1and the rim velocity would be increased by 4 2 . This was Dupr@s2result, which is clearly incorrect and implies that bursting is completely frictionless. The surface energy of the film originally occupying the hole of area A is 2Auo. The kinetic energy of the rim ( ‘ / ~ ) ( A ~ O ~=) UAuo C ~ is exactly half as large. Hence, under these assumptions exactly half the surface energy is dissipated, as already noted by Culick. For soap films uo = 30, p = 1, and 6o varies from about to about all in CGS units. Hence, Culick’s velocity is of the order of 8 X lo2 to 8 X loa cm/sec or about 20-200 miles per hour. B. Radial Case. Consider a hole having a radius r and a rim moving with a velocity UE which may vary with r. We assume that the cross-section of the rim is negligible and that it incorporates all the material of the film originally in the area of the hole. Hence, the total mass of the rim is nr260p and for a small sector, de, it is r260pd0/2. The total momentum is held in the rim since the rest of the film is, by hypothesis, undisturbed. The momentum within the small sector considered is r26opu~d0/2. After a very short time interval, dt, the radius of the hole has increased by dr = undt, the rim velocity by duH so that its mass for the small sector is then (r dr)26~pd0/2,neglecting second-order terms in
+
Volume 73, Number 9 September 1960
3030
STA4NLEYFRANKEL AND
dt and de. The force acting upon this sector during that time interval may be observed at a point just beyond the rim, say at r 2dr, where it is equal to 2 ~ 0 ( r 2dr)dB since the film there remains undisturbed according to our assumptions. Conservation of momentum then gives
+
2cro(r
+
+ 2dr)dBdt = (r + dr)260p(uH+ duH)dO/2 - r260p~Hd0/2(1.4)
or, neglecting higher terms and replacing uKdt by dr dUH =
2[(2Uo/p6)
- UH’](~~/TUH) = 2 ( U c 2 - UH2)(dT/rUH)
(1.5)
Hence, the velocity of the rim remains unchanged when it is equal to Culick’s velocity, and if it differs from this value, always tends toward it. Thus, if the rim velocity is constant, it is Culick’s velocity, and if disturbed, it tends to return to it. More specifically, if the disturbed velocity is U H , at radius r’, integration of the last equation gives uH2 - u02 - 4);( ~UH,’ - uc2
For small relative deviations, this simplifies to (1.7) This means that any difference from Culick’s velocity decreases rapidly as a function of the relative increase in the radius of the hole. An important consequence is that any effect of early disturbances due to the first puncturing of the hole will be particularly rapidly eliminated. The Effectof Rim Radius. In the above discussion we have neglected the dimensions of the rim itself and this neglect should be justified. The rim incorporates under present assumptions all the material of the film within the original hole and must have an approximately circular cross-section. If this were exactly circular, the diameter of the rim cross-section would be equal to (r60/2)1”,which for the usual soap film would be less than a tenth to a hundredth of a millimeter when the hole radius reaches 1 cm. Thus, even if the rim crosssection deviates somewhat from circular, its projection on the plane of the film, 2a, is very small in comparison with r and can be neglected in comparison with it. If the calculation leading to eq 1.7 is now repeated, taking into account the finite value of the rim projection 2a and measuring T to the center of this projecbion, the o its momentum mass of the rim becomes (r ~ ) ~ p 6and is correspondingly increased. I n order to be sure that the film is undisturbed, the force acting on the rim may 2dr. Otherwise, the same a be observed a t r reasoning may be followed and since a enters all expressions as a sum with r, it may be neglected leading to the same results.
+
+ +
The Journal of Phgsical Chemistrg
KAROL J. AfYSELS
11. Viscous Restraint of the Film Thickening The model of an undisturbed film bordering on a toroidal rim ( i e . , one of circular cross-section) which underlies the above calculations is unrealistic in assuming a completely abrupt transition between undisturbed film and rim. Since viscous forces resist the change of shape of a film element, there must be a transition zone of finite width in which each element of film shrinks and thickens in the course of flowing into the “bulk” liquid of the rim. Although one might be tempted to look to this viscous transition zone for an explanation of the aureole, an order of magnitude calculation suffices to show that the aureole of a soap film cannot be accounted for in this way. The shrinking and thickening of a film element in the transition zone involves a shearing of its liquid content at a rate w , which is its fractional change of area or thickness per unit time. By reason of the viscosity of the liquid, energy is dissipated in it at the rate of qw2 per unit volume. A characteristic time associated with the passage of the liquid through this transition zone will be r = w-l. The dissipated energy will then be of the order of qw26r per unit area. As we have seen above, half the surface energy of the film is dissipated during bursting and if we assume that all of this is dissipated by viscous friction, we obtain VO ‘v q d 6 r
(2.1)
or r‘v-
6 UO/T
For soap solutions in water, the denominator (uo/q) is about 3 X lo3 cm/sec or of the same order as Culick’s velocity of the rim. Hence, the distance covered by the sheared fluid element in this characteristic time is comparable with U C T 5 6. Thus, a film element can thicken against viscous resistances while moving a distance comparable to its thickness, which is of the order of microns or less and therefore below the limit of present experiments. Even if a generous multiple of 6 is allowed for the width of the transition zone, it still would be negligible compared to the observed aureoles which are measured in millimeters. The above argument is based on the assumption that the viscosity of the intralamellar liquid is of the same order as that of water. This is consistent with results of experiments on the thinning14 and on the formation16of films, but can be debated.16 However, the fact that the (14) K.J. Mysels, K. Shinoda, and S. Frankel, “Soap Films, Studies of Their Thinning and a Bibliography,” Pergamon Press, New York, N. Y.,1959,Chapter 111. (15) J. Lyklema, P. C. Scholten, and K. J. Mysels, J. Phys. Chem., 69, 116 (1965). (16) J. Th. G. Overbeek, Discussions Faraday Soc., 42, 12 (1966); B.V. Derjaguin, ibid., 42, 137 (1966); P. C. Scholten, ibid., 42, 136 (1966).
303 1
THEBURSTING OF SOAPFILMS dimensions of the aureole are not strongly dependent on film thickness would imply a remarkable compensation between thickness and viscosity in the film if it were a greatly increased viscosity that is responsible for the aureole. Finally, bursting experiments on films from glycerine-rich solutionsl2 have shown that the width of the aureole dloes not increase with the viscosity of the solution. An independent argument that the observed aureoles are not due to viscous forces is that the width of a transition zone determined by viscous resistance to thickening would be constant rather than increasing in proportion to the time since the beginning of the bursting, as the observed aureoles do (at least approximately). We conclude, therefore, that viscous forces make a negligible contribution to the observed width of the aureole and seek an explanation based on changes in surface tension accompanying very quick changes in film thickness and therefore of its area.
111. The Interpretation of Aureoles in Terms of Surface Tension Changes Assuming that the aureole is due to changes in surface tension, one may ask whether it is possible to determine from a study of the aureole what these changes are. Once such information is available, it should be possible to interpret it further in molecular terms and to obtain information about nonequilibrium surface behavior and about desorption kinetics on a microsecond scale. We will, therefore, examine first the general problem of relating observable quantities to changes in surface tension, although it does not lead to analytical solutions. We will then proceed to a simpler problem which is soluble and provides some insight into the mechanism involved. A . Unidimensional Bumtiny. Since, at present, individual film elements cannot be tagged and followed visually, we are restricted to the observation of lines of equal thickness, 6, which correspond to color fringes (or lines of constant intensity of reflected monochromatic light). In principle, at least, the position of many such lines can be determined a t successive time intervals to give the complete history of the aureole 6(x,t). It is, therefore, also possible to obtain experimentally the rate of advance of the position at which a particular thickness occurs, or, briefly, the velocity of a given thickness ~g
= (dx/dt),
(34
Since film thicknesses change, these constant-thickness lines move with respect to the film material and ug is generally different from the velocity of a film element, u , a t the instant when their positions coincide. To identify film elements it is convenient to introduce the original position of a film element, its distance, X , from the origin, the line along which the film is severed. Thus, X is invariant for a particular film element during
Path of Fi Im Element Path of Conalant Thickness
+-Aureole
ti
Undislurbed Film
Front X
X
Figure 1. Schematic presentation of the bursting of a film in terms of laboratory coordinates 2 and t, and of the original position of a film element in the undisturbed film, X.
I
I
I I
x x+dx
I
x*
Figure 2. Schematic cross-section of a bursting film (omitting the rim) at two successive instants. The dots and lines indicate the positions of the same film element which has moved to the right and also thickened during the process.
the bursting process; in the as yet undisturbed film, x = X . Figure 1 shows some relationships between the x, X , and t coordinates. The velocity of a film element is
u = (dx/dt),
(3.2) Consider a film cross-section a t two successive instants as shown in Figure 2. Conservation of mass requires that the total mass per unit width between some line in the undisturbed film and a given film element remain constant during bursting. Neglecting any density changes, the same applies to volume, and in the crosssection, to its area. Thus
(3.3)
where xo is a point in the undist'urbed film and x the Volume 79, Number 0 September 1960
3032
STANLEY FRANKEL AND KAROL J. MYSELS
coordinate of a film element. Performing the differentiation gives
f
dx -
a(.)($)
X
= 0
(3.4)
and, therefore, the film element velocity is given by”
This equation permits the calculation of the film element velocity at all points and times so that u(x,t) is obtained. Differentiation (which we assume for the moment to be possible) gives the acceleration du/& at any point. This acceleration is due to the gradient of surface tension forces 2bu/dx and is related to it by
bu/dt = (2/pS) (b5/bx) (3.6) As 6(x,t) is already assumed to be known, this equation can be solved at any point and time to give (da/dx) (x,t). Since the surface tension of the undisturbed film is known, it is possible to obtain the value of the surface tension at any point and time by integration
-
U8
= (62u2
- 61u1)/(62 - 61)
(3.8)
I n passing through the shock line, mass flf undergoes a change in velocity of u2 - u1under the influence of the surface tension difference 51 - U Z . Conservation of momentum requires that M(u2
- Ul)
=
-2(u2
- 51)
(399)
Eliminating us between these equations gives 52
-
UI
= -p6182(~2
- ~1)~/2(62 - Si)
(3.10)
Alternatively, if us is the observed quantity, one of the other quantities, for example, UZ,may be eliminated to give p(61/6z)(uS
- ~ 1 ) ~ ( 6-2 Si)
The Journal of Physical Chemistry
2(51
- ~ 2 ) (3.11)
4
Figure 3. T h e quantities describing a shock wave in a bursting film.
This expression is particularly useful in examining a shock a t the outer edge of the aureole which runs into undisturbed film and may be called the frontal shock. In that case 61 = 60, UI = 50,and u1 = 0 so that eq 3.11 reduces to ~(60/&)~s’(62
Shock Waves. The above treatment is valid for continuously varying thicknesses. When a shock wave is present, as has been frequently observed by RIcEntee and RiIysels,12;.e., when the thickness jumps abruptly from one value 61 to another & (Figure 3) with a corresponding jump in velocity from u1 to uz,eq 3.4-3.7 become dubious. To find the required change in surface tension (from 51 to 4,we consider the change of momentum across the shock line in a coordinate system attached to this line which is moving with a velocity us. The mass of the thinner film entering the shock line per unit width and time is A l = (u,- ul)p&, that of thicker film leaving it M = (u,- u2)p82. Since these two masses are equal as indicated
I
- So)
= 2(50
- UZ)
(3.12)
Hence, measurement of the speed of the outer edge and of the thickness immediately before and behind it is sufficient to give the surface tension drop occurring across it. Returning now to eq 3.7 we can combine it with eq 3.11 obtained previously to solve, in principle, the problem of obtaining the surface tension of the bursting film from a knowledge of the shape of the aureole whether it contains shock waves or not. Time Dependence. It is likely that the area changes and, therefore, thickness changes will be an important variable determining changes in surface tension. Hence, a(6,t) will be a more significant function than u(x,t). Since S(z,t), the shape of the aureole, is assumed to be known, the required transformation is, in principle, trivial. It must be noted that this procedure requires only the knowledge of the shape of the aureole S(z,t), but involves two differentiations and two integrations of this information, so that quite accurate data may be required to give meaningful results. Furthermore, in general, the surface tension may be expected to depend not only on the momentary state of the film, namely its thickness, but also on its past history, which determines the extent to which it has approached equilibrium. This history will depend on x or t in a definite way for any given film, but will vary with the rate of bursting, and, there(17) An alternative expression which sometimes may be more accessible experimentally is obtained by integrating along the ordinate instead of the abscissa. This yields in the unidimensional case
3033
THEBURSTING OF SOAPFILMS fore, with the thickness, even for films of a given composition. Because of this double dependence it should be possible to study the effect of this past history over a considerable range of effects. After generalizing these results to the practically important radial case, we shall consider the simple case in which surface tension is dependent only on the thickness of the film and is not otherwise dependent on its past history. B. Radial Case. We assume that the bursting process is symmetrical about the origin. The general reasoning of the unidimensional case can then be applied directly, replacing x by r and X by R, with the modification that the volume of the film between r and r dr is proportional not only to its thickness, 6, but also to its distance from the origin, r. If we denote the original position of a film element by R, conservation of volume gives
+
(3.13)
and the velocity of a film element is given by
extent that the real film departs from this limiting behavior, we are dealing with relaxation phenomena of time scales comparable with bursting times due, e.g., to the solubility of the monolayer and therefore with the kinetics of its desorption or with frictional effects in the surrounding medium. The treatment is based on the method of characteristics which is discussed in some textbooks of gas dynamics.‘* The compression of the monolayer is in our case due to the shrinkage of the film surface as a film element thickens and this shrinkage can be conveniently expressed in terms of a parameter, a,which can be defined by any of the following ratios: of the present area, A , of a film element to its original area, Ao; of its initial thickness, 6,,,to its present one, 6; of its present width, dx, to its original one, dX, where X is again the original coordinate of the film element for a unidimensional burst a =
A/&
= 60/6 =
(44 Our first purpose is to describe the bursting process assuming that the function (4.2) describing the variation of surface tension with shrinkage is known. Since a varies between 0 and 1, this function may be represented as a line of the kind shown in Figure 4. It has the value uo at a = 1 and its derivative a t that point is positive. The derivative at that point cannot be less than half of Gibb’s film elasticity modulus E fJ
in strict analogy with eq 3.3 and 3,5.17 The remaining treatment of the unidimensional case can be adapted directly to the radial one since the radial character of the burst does not affect the reasoning. In particular, although the shock line becomes a circle, we can restrict the consideration of the thicknesses and forces involved to such a narrow band on both sides of it that the changes with r become negligible. The argument is analogous to that used in deriving (section I-B) Culick’s velocity for the undisturbed radial case. We shall return to the interpretation of experimental data after a more general discussion of the dynamics of bursting when u is a function of 6 only.
IV. Unidimensional Film Bursting When Surface Tension is Thickness-Only Dependent We now consider the mechanism of film bursting for the ideal limiting case of unidimensional bursting with surface tension dependent only on the thickening of the film as it shrinks, and not on the past history or on the rate of thickening. These simplifying assumptions permit much more explicit solutions of the basic equations and thus more insight into the mechanism involved. The assumption that surface tension depends only on the thickness of the film and therefore only on its shrinkage is equivalent to saying that the film surface behaves during bursting in the way an insoluble monolayer is supposed to behave in a Langmuir trough. The insoluble monolayer is assumed to be in equilibrium. Our shrinking film is in a quasi-equilibrium. To the
(bx/bX),
E
= $(a)
2d~’/[(dA)/A],=l = 2 ( d ~ ’ / d a ) , = l (4.3)
The prime is used to emphasize that Gibbs’ definition relates to a true equilibrium state for each thickness of the film, whereas the states represented in Figure 4 may be only quasi-equilibrium ones. From our knowledge of monolayers we can expect that the surface tension continues to decrease as the film area decreases. Otherwise, there seem to be no a priori limitations on the shape of the curve. The present position, x, of a film element is a function of its original position, X,and of time, t
dX,O (4.4) The partial derivatives of 4 are, respectively, the
x
=
shrinkage and the velocity
bx/bX =
a
ax/& = u
(4.5)
Cross-differentiation shows that ba/bt
=
bu/bX
(4.6)
(18) E.g., V. L. Streeter, “Fluid Mechanics,” 4th ed, MoGraw-Hill Book Co., Inc., New York, N. Y . , 1966.
Volume 73, Number 9 September 1969
3034
STANLEY FRANKEL AND KAROL J. MYSELS
In view of eq 4.6 and 4.7, respectively, we can replace these two equations by
in which w and u appear completely symmetrically. By subtracting the second from the first one and rearranging, we obtain
a
0
I
Figure 4. The relation of a quasi-equilibrium variation of surface tension u with shrinkage Q! to Gibbs’ film elasticity and to the parameter A = U,.
The acceleration of a film element is given by
Here, A is an important parameter defined by A = [(2/p&)du/da]*”
1
= laAda
(4.9)
According to this definition dw - = -A da
(4.10)
w=Ofora=l
(4.11)
and
Furthermore, since, according to eq 4.8 and 4.9, A and w depend only on a The Journal of Physical ChembtTU
w =
(4.8)
and depending, under our assumptions, only on the shrinkage, a. It has the dimensions of a velocity in the X,t system, i.e., length of original film per unit time. The physical meaning of a velocity in this system is the rate at which the original film is undergoing a process. Thus, e.g., UH is the velocity at which original film is consumed by the advancing rim and U , the velocity at which the original film passes through the state of having shrunk to a. We will show shortly that the value of U , is A. We may note that A is related to a in a simple way since its square is, except for the constant factor 2/pSo, the slope of the u vs. a plot as shown in Figure 4. We now introduce w , another function of a and also dimensionally a velocity (but in the x,t system) and defined by .(a)
This relation means that a certain combination of derivatives of the difference between w and u is zero, Le., that this difference is unchanging along certain paths. This combination of derivatives defines a path in the X,t plane whose slope, dX/dt = -A, so that the path is a curve extending from high values of X , i e . , undisturbed film, into the bursting part and to the rim. However, according to its definition, u is zero in the undisturbed film and so is w according eq 4.11. Hence these two quantities are equal there, and by eq 4.14 stay equal along any such line. Since the line has an arbitrary origin, it can be made to pass through any point of the X,t space and we obtain the result
u for all X and t
(4.15)
Since w is by definition a function of only the shrinkage, a, this relation means that u,the velocity of a film element, is determined by its shrinkage only and can be obtained from a knowledge of the .(a) function by eq 4.8 and 4.9 which give
u, =
Ada = L1[(2/p8,)du/da]’” d a
(4.16)
To obtain the meaning of A, we now add the two equations (4.13) and rearrange to obtain (k+A&)(w+u)
= O
(4.17)
Since we know that w equals u everywhere, this relation shows that there is a path in the X,t space along which both w and u are constant. Since they have equal values and w is a function of a only, this must be, therefore, a path of constant shrinkage. This path is a line whose slope, dX/dt = A, is therefore a constant. Thus, A is indeed equal to the velocity U , of a line of constant shrinkage, in terms of original film position, X , as we have anticipated earlier. Hence, in the aureole of this type of system, each thickness has a constant velocity and the aureole expands linearly with time. The bursting may be called self-similar. Equation 4.14 shows that a certain quantity propa-
3035
THEBURSTING OF SOAPFILMS I
-d 6
2
y
v, -
0.5
a
I
“C
I
Figure 5. (a) Four simple hypothetical shapes for the u--a! curve. (b) Corresponding slopes of van[ (2/p&)du/da] ’/z. ( 0 ) Corresponding shapes of u, (eq 4.16). (d) Detail of (c). Figure 6 . The distribution of thickness and of velocities in aureoles corresponding to the systems of Figure 5 .
gates to the left in X , t with speed A. Equation 4.17 describes the rightward propagation of another quantity at the same spced. Thus, A plays a role similar to that of the speed of sound (depending on density) in aerodynamics. 2o I n graphical terms we have already seen that Ua2 varies with the slope of the u us. a plot as shown in Figure 4. Kow, if the values of U , thus obtained are plotted against a , the velocity of the film element, u = w, is given according to the definition of w (eq 4.9) by the area under this curve extending from the right side limit to the a corresponding to the thickness of this film element. These relations are illustrated for several simple cases in Figure 5. The coordinates in this figure are made dimensionless by expressing surface tension in terms of that of the undisturbed film, go, and all the velocities in terms of Culick’s velocity, UC. A and U , are thus related to u and to the slope of the u us. a plot, but U , is not an observable quantity since it is in the X , t system. It can be related, however, to the observable u,,which is measured in the x,l laboratory coordinates by U, =
aU,
+ u = aU, +
l1
U,da
(4.18)
The first term in these expressions represents the contraction from dX to dx and the second takes into account the movement of the film in laboratory coordi-
nates. It may be seen that at the outer edge of the aureole where a = 1, the second term vanishes and u, = bra;whereas, as we shall see in more detail later, at the rim where a tends to zero this first term vanishes, u, = U , and the film elements have the rim velocity which is given by the integral over the whole range of a’s. We can now construct a profile of the aureole since eq 4.18 gives a relation between a and u, and the latter is proportional to the position of the film element which has undergone a given thickening, p = 1/a. This is shown in Figure 6 along with the distribution of velocities u through the aureole for the systems of Figure 5 . Shock Waves. Since U , varies with the slope of the u vs. a line, there may be regions in which it may be less for thinner films than for thicker ones as may be noted in Figure 4. This happens when the line is concave downwards and means that the thicker film would overtake the thinner one so that the latter disappears and an abrupt change of thickness, a “shock wave,” is created. Under these conditions the above analysis breaks down and we must supplement it by a determination of shock velocity and shock extent ( i e . , its change of thickness). (19) Ya. B. Zel’dovich and Yu. P. Railer, “Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena,” 1-1-1 1, Academic Press, New York, N. Y., 1966. (20) H. Lamb, “Hydrodynamics,” Dover, New York, N. Y., 1945, Chapter X.
Volume 7 3 , h’umber 0 September 1969
STANLEY FRANKEL AND KAROL J. MYSELS
q -
I a
I
a
I
I
cx
I
64
a
Figure 7. The convex u-a plots. Solid lines show the convex plot and dashed lines any part of the original u-a plot which has been replaced by straight lines corresponding to shock waves.
Let U sbe the shock velocity in the X,t coordinates. The mass of film over which it passes (per unit width and time) is pSoUs. If we designate the quantities immediately in front of the shock by 1 and those immediately behind it by 2, the change in velocity at the shock is ~1
- uz =
Usal
-
U s ~= z Us(al -
az)
(4.19)
Conservation of momentum gives
Ue2 = (2/PSO)(Ul
- UZ>/(a!l
-
012)
(4.20)
which differs from the definition of A = U , (eq 4.8) by having a difference quotient instead of a derivative. Thus, the shock velocity is related to a chord connecting two points on the u us. a line in the same way as U , is to the tangent a t a point. It can be seen (Figure 7) that the actual shock corresponds to the chord subtending the largest possible part of the curve because otherwise it would either be overrunning slower moving film thicknesses ahead or being overrun by that behind it. The number of shocks actually present depends, of course, on the shape of the u us. a curve. Of special interest are conditions near a = 1 and a = 0. If the curve has the shape shown in Figure 7(b), the The Journal of Physical Chemistry
tangent to the main portion which passes through the (uo,l) coordinates of the original film represents a frontal shock preceeding the aureole, u1 is then uo and a1 = 1. Behind the rim, Le., within the hole, the film does not exist so that its “surface tension’’ may be regarded as zero. Since no material moves out of the rim, there is no outgoing momentum and the rim behaves like a shock wave leading to an infinitely thick film, i e . , 0 1 ~ = 0. Hence, the rim may be represented as a shock defined by a line passing through the origin and tangent to the curve. Thus, the behavior of the film may be described by the original considerations provided that all portions of the u us. a curve which can be subtended by tangents at two points, or tangents passing through the origin or through the coordinates of the undisturbed film, be replaced by these lines. We may call such a curve a convex u us. a! plot (Note that it is convex downward). It is this plot which must be used in eq 4.8 and 4.9 to obtain velocities for this ideal case as shown in Figure 5 . If the u us. 01 curve lies entirely above the diagonal line connecting the coordinates of the undisturbed film to the origin (Figure 7(d)), the convex plot becomes this straight line and only one shock wave, namely the rim itself, is present. The rim shock moves into undisturbed film and we are dealing with the situation discussed in section I. The velocity of this shock wave is the same in terms of the original film as in laboratory coordinates so that
us=
(2uo/pGo)1’a = us = uc
(4.21)
and, as would be expected, we find that the rim has Culick’s velocity. The existence of an aureole means that the u us. a! curve dips below the diagonal line connecting the values for the undisturbed film with the origin. Rim velocity is then given according to eq 4.16 by
S, 1
UH =
U,da
(4.22)
with U , taken along the convex plot of the u us. a curve. Thus, U H is the average of U , over all a values. It can be shown that U H is never larger than Culick’s velocity for the ideal system under discussion. For this purpose, let us call f(a)the fractional deviation of U, from its average value so that =
uH(1 + f ( a ) )
(4.23)
Let us also note that according to eq 4.8, pSoUU2/2is the slope of the convex u us. a plot. Integration along this line gives 1
uo = ( ~ 6 , / 2 ) sUU2da 0
Combining these two equations, we obtain
(4.24)
3037
THEBURSTING OF SOAP FILMS
2u0/pSO =
1
l 6 d x = tuo& p(p)dp = tu060Z(p)
l l u a ‘ ( l + f ( ~ ~ ) ) ~=d a
The first of these integrals is equal to unity; the second vanishes, since it is proportional to the total deviation from the average; and the third is necessarily 2 0. We obtain, therefore
+ J f2(a)da)
where I ( p ) stands for the definite integral of P from p = 0 (or, what is equivalent, from the rim) to p . For large p , I = p . Experimentally, I may be obtainable more easily by integrating from p to a point po in the undisturbed film (5.4)
1
UH’ = ( 2 n o / p 6 0 ) / ( 1
a
(4.26)
which shows that the maximum value of U H is obtained when the deviations vanish. It is then equal to U C ; furthermore, u, E U H ; and the convex u us. a plot is a straight line and we deal with bursting into undisturbed film. It may be noted that fractional deviations from this maximum rim velocity depend on the average square of deviations of u, from UC, hence, when these latter deviations are small, the difference between the rim speed and Culick’s speed is veyy small.
V. The Interpretation of Self-Similar Results I n the previous section we have shown that when surface tension is “thickness-only dependent,” lines of equal thickness propagate at constant speeds which decrease from the outer edge of the aureole to the rim of the hole. Thus, as the aureole expands with time, it remains self-similar, all of its dimensions increasing proportionately to the time since the hole originated. This similarity of early and late stages of the aureole results from the lack of a time constant for any relaxation process in the theory so that there is no criterion for defining the meaning of “early” or “late.” For the same reason radial bursting will be self-similar when the surface tension is “thickness-only dependent.” We shall now consider the problem of obtaining the relation between surface tension and thickness for such selfsimilar bursts, both unidimensional and radial. Unidimensional Self-Similar Bwsting. A convenient coordinate is the velocity relative to some reference velocity uo p
= x/tuo
(5.1)
The choice of uo is arbitrary and depends on experimental or computational convenience. It may, for example, be set equal to uc or U H depending on circumstances. The origin corresponds to p = 0, and a t sufficiently large p the film is undisturbed. The film thickness can be expressed in terms of the thickening, P, which is the reciprocal of the a employed previously 6 =60Pb) (5.2) Thus, P is unity in undisturbed film and increases as the rim is approached. For p values within the hole, we will regard p as having the value zero. The volume of liquid per unit width to the left of a point, x, is obtained by integration of the film thickness
(5.3)
The speed, u,of a film element is the rate of change of x, keeping expression 5.3 constant. Hence uoSoZ(p)
+ tuoao(dl/dp) (-x/t2uo) + utuoSo(dZ/dp) (l/tuo) = 0
(5.5)
or
+ Px/t)/P
u = (-uJ
=
PUO
- UOZ/@
(5.6)
For large p, u becomes zero as expected. The acceleration of a film element can be expressed as follows ($+u$)u=(-;+u;)&
P du =
qu- puo) du
- =
-
dP
X
This acceleration must be equal to (2/PS) (bn/bx) = (2/PSOP) (dUldP) (PlX)
(5.8)
Equating these two expressions gives da/dp = -p6ouoZ(du/dp)/2 =
- (P60UOI/2) ( h l P ’ ) (dD/dP)
(5.9)
so that du/dp
=
(p6ouo2/2)(I/P)’(d@/dp)
(5.10)
This expression can be integrated numerically to provide u as a function of p . Since 6 is also known as a function of p , this gives then u as a function of 6 or a. The above result can be generalized to the radial burst as shown below, but for the unidimensional system further interpretations are possible. Thus, if @ is used as the independent variable, eq 5.10 can be rewritten = (p6ouo2/2)(Z/p)’d@
(5.11)
which can be numerically integrated to give directly u as a function of p and therefore of a or 6. The value of the important parameter A = U , can be introduced using eq 4.18 to give A = (u, - u ) / a
(PUO- u ) / a
(5.12)
In view of eq 5.6, this can be simplified to A = uoI(p)
(5.13)
Radial Proportional Bursting. The analysis follows vo‘olume78,Number 9
September 1969
3038
STAWLEY FRAWKEL AND KAROL J. MYSELS
the same pattern as the unidimensional case with replacing x. Thus, p is defined by p = r/tuo
T
(5.14)
and P retains its definition. The volume of liquid may be evaluated within a sector of one radian (to eliminate numerical factors) by lr8dr
=
(tuo)2So[ppP(p)dp 0
=
(tu0)~60I(p)(5.15)
I is now the integral of pP from the origin t o p and can be evaluated also by PVO
I
= po2/2
- Jp-
(5.16)
PP(PNP
The velocity of a film element is again obtained by taking the time derivative with the volume of expression 5.15 unchanged so that
21
+ t2(dI/dp)(-r/t2uo) + u(t2dI/dp)(l/tuo) = 0
(5.17)
Hence u = [(pPr/tuo) - 2Il/(PP/UO) =
puo
- ( ~ I ? ~ P P(5.18) )
The acceleration of a film element is then
- puo)du -= r
dP
du --2 1 ~ 0rP dP
(5.19)
da
-=
dp
( P S O I U O[1 ~ /P ) (21/pzP2)(P
+ pdP/dp) I
(5.22)
which can again be integrated numerically but is considerably more complicated and does not lend itself to simplification or further analysis along the lines employed in the unidimensional case.
Conclusion The above discussion shows that the existence of large aureoles preceding the advancing rim of a bursting film points to large changes of surface tension as the film shrinks. These changes can be obtained from experimentally accessible quantities by procedures described in sections 111 and V. The treatment of the limiting case of unidimensional bursting with the neglect of any relaxations of surface tension as given in section IV cannot be readily generalized to radial bursting nor does it take into consideration the effects of air resistance. NeverthelesR, it explains a number of experimental observations and permits the drawing of some qualitative conclusions. Thus it shows that: (A) Very large changes of surface tension with shrinkage must be present since a linear decrease, or one that is less rapid for initial than for final stages, would lead only to a rim of negligible width and an undisturbed film ahead of it (Figure 7(d)). (B) Only certain parts of the relation between surface tension and shrinkage can be accessible experimentally, whereas others-the concave ones-are unrealized, being “hidden” in shock waves (Figure 7). This explains the existence of such shock waves and the constant speed of any that fronts the aureole. (C) The rim speed is relatively insensitive to the existence of, and changes in, the aureole unless the latter is highly developed, because of first-order compensating effect of deviations (eq 4.26 and Figure 5). On the other hand, an aureole which runs far ahead of the hole can be associated with a considerably reduced speed. It should be noted, however, that the demonstration of the existence of a maximum rim velocity (eq 4.26) has not been generalized to the radial case.
Acknowledgment. We are indebted to Dr. G. Frens for helpful comments.
The Journal of Physical Chemistry
