T H E INTERFACE SYMPOSIUM
Bubbles Foam 1
and
SYDNEY ROSS
lateau (12), in a series of brilliant researches, ex-
P piscated the laws of bubble geometry that hold for all assemblies of bubbles in equilibrium, based on the
minimizing of surface area of liquid films, which is the direct result of the tension of liquid surfaces. The laws are : 1. Three, and only three, foam lamellae meet at an edge; the three coplanar angles at which they are inclined to each other are all equal. Hence, each is 120”. 2 . Four, and only four, of these edges meet at a point; the angles at which they meet are equal to 109” 28’ 16” (the tetrahedral angle). Plateau’s laws are the basis of the structures of foamsbut though necessary, they are not sufficient. Incidentally, they have been known for the past hundred years, but the general law discussed in the present paper, under which Plateau’s laws are subsumed, has not, to my knowledge, been stated explicitly heretofore. This general law takes into account the minimizing of the surface required to contain a given quantity of gas at a given temperature and a given average internal pressure. As this pressure also determines the average curvature of the liquid films, it thereby introduces another restricting condition, which explains, among other things, why a single sphere is not the universal answer to foam morphology. The actual morphology of composite bubbles and foams can be extremely complex and varied, yet remains constantly regulated by the general law. A few lines by Goethe, as translated by Whewell, although aimed at a different set of phenomena, admirably express the present situation : 48
INDUSTRIAL A N D ENGINEERING CHEMISTRY
“ A l l the forms resemble, yet none is the same as the other. T h u s the whole of the throng points at a deep-hidden lawPoints at a sacred riddle. Oh! could I to thee, my beloved friend, Whisper the fortunate word by which the riddle is read.”
The nearest approach made by any previous investigator to an expression of the present law is a simple geometric calculation by Tait, in 1867, for the special case of two bubbles uniting to form a single sphere. This contribution (76) appeared so trivial to the editor of Tait’s collected works that he excluded all but one reference to it. Tait himself, however, saw fit to repeat it in a popular book (77) on elementary physics, from which, as it is not readily available, I quote the relevant passage, altering only the symbols to conform to current usage : “When two complete soap-bubbles are made to unite, the tendency of the liquid film is to contract, that of the (compressed) air inside is to expand. I t becomes a curious question to find which of these actually occurs. “Let their radii, when separate, be rl and r2, and let them form, when united, a bubble of radius R. Then, if P be the atmospheric pressure, the original pressures in the bubbles were 4rJ
P -/- - a n d
P
71
while that of the joint bubble is
4u P + x
4rJ +72
New
Genera1
Law
“By Boyle’s Law the densities are as the pressures. Hence, expressing that no air is lost, we have
coalescence of two submerged bubbles, where P signifies the sum of the atmospheric pressure and the hydrostatic head-i.e., the total external pressure. The surface area is that of only the insides of the spheres,
”>
but the original pressures in the bubbles are only
or P(r?
+ - Ra) + 4u(r12 + - R2) = 0
and ( P
r22
r23
r2
If AV be the diminution of the whole volume occupied by the air, AA that of the whole surface of the liquid film, this condition gives at once 3PAV
+ 4uAA = 0
“AS P and u are both essentially positive, this condition shows that AV and AA must have opposite signs. Hence, both tendencies are gratified, the surface, as a whole, shrinks, and the contained air, as a whole, increases in volume, simultaneously. But the work done by the expanding gas is only about two-thirds of that done by the contracting film. “ I t is worthy of notice that, as is easily proved, the air in a soap-bubble of any finite radius would, at atmospheric pressure, fill a sphere of radius greater than before by the constant quantity 4u/3P.” The equation as originally given by Tait has to be altered if we take into account the loss of area on both sides of the liquid films, which alteration would make it read 3PAV
+ 2uAA = 0
(1)
Equation 1 is now able to describe, for example, the
+ ?),so the same equation results.
Tait made no further use of his calculation other than to obtain the following curious theorem of pure mathematics: The cube of the sum of the squares of any series of numbers is always greater than the square of the sum of the cubes of the same numbers. [Suppose several bubbles of radii r l , r2, r3, etc., coalesce into a single bubble of radius R. The surface area of the latter bubble is less than the sum of the surface areas of the original bubbles ; therefore, rI2
+ + + . . . . > R2 r22
r32
The gas expands, however, as a result of the transformation, so the volume of the large bubble is greater than the sum of the volumes of the original bubbles; therefore, r13 r23 r33 < R3
+ + + . ...
From the two inequalities, it follows that: (r12
+ + + .. . r22
r32
.>3
> (r13 + rz8 + r 2 + . . .. ) 2
Let Tait’s calculation be the stimulus to search for the relation between AV and AA for other morphological changes that take place spontaneously when VOL. 6 1
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OCTOBER 1 9 6 9
49
two or more bubbles are in close contact. Consider as examples: (A) the coalescence of two unequal bubbles; (13) the partial fusion of two equal bubbles to form a composite bubble (Figure 1); (C) the change in size of two unequal bubbles caused by the transfer of gas from the smaller into the larger bubble, by means of diffusion through the liquid medium between them. I n each case, calculations only slightly more complex than those performed by Tait disclose that Equation 1 still describes the relation between A V and AA. In case (b), the volume and surface area of a composite 27 bubble of two equal parts are: Volume = - rr3; 12 27 Area = - m2. In case (C), let dn moles of gas diffuse
4
from a bubble of initial radius, rl, into a bubble of initial radius, r z . If the gas is ideal, then inside any spherical bubble :
+
16 Hence RTdn = - ordr 4 Pr2dr. The quantity of 3 gas lost by one bubble equals that gained by the other; therefore,
16 3
16 3
- nmldrl f 4nPr12drl = - rrarzdrz f 4~Pr2~drz Following the calculation to the end gives 3PAV
One more calculation should be undertaken. Let v1 spherical bubbles of radius rl, v z of radius r2, va of radius r3, etc., coalesce to form a new distribution of bubble sizes-uiz. Nl bubbles of radius R1, Nz of radius Rz, N 3 of radius R3, etc. The calculation of the whole A V and AA is carried out in the same way as those previously described, and gives the same resultnamely, 3PAV 2oAA = 0. Tait’s equation is, therefore, of far wider applicability than its promulgator ever suggested or suspected. I t describes, for example, the decrease of interfacial area resulting from gas transfer between spherical bubbles submerged in a liquid, whether the mechanism of the transfer be the result of coalescence of bubbles or by diffusion of gas through the liquid medium that separates the bubbles. And as we shall see, cases of partial coalescence, where two or more spheres combine to form a composite bubble (Figure l), are also described by the same equation, since they are included in the general proof to be given below. The success of this equation with free-floating bubbles leads us to try its applicability to foams. Mane-
+
Figure 7. Cross sections of composite bubbles or bubble clusters showing how dzyerences of pressure between bubbles are reflected in the daxerent curvatures of the dividing partitions 50
INDUSTRIAL A N D ENGINEERING CHEMISTRY
+ 2oAA = 0
gold (70) distinguishes between two morphological types of foam; the first, designated kugelschaum (spherefoam), consists of spherical bubbles widely separated from each other ; the second, designated polyederschaum (polyhedron-foam), consists of bubbles that are nearly polyhedral in shape, with thin, plane films of liquid (or at least films of very low curvature) between them. This is a convenient classification because distinctly different spontaneous changes take place in each type of foam. I n kugelschaum there is a hydrodynamical outflow of liquid from the foam, caused by the buoyancy of the bubbles, during which process all of the bubbles expand, and the total area of the liquid surface also expands. Tait's equation does not apply here because the external pressure is not constant, including as it does an additional hydrostatic pressure that changes during the process. But one can readily derive the correct equation for a bubble rising in a fluid. Let a spherical bubble of volume V I and area A1 be submerged a distance 121 from the top of the liquid; on rising within the medium, let its distance from the top be hz, its volume, V Z ,and its area, A%; the relation between these quantities is then
3PAV
(a)
Time
0, magnification (linear)
I
-
16
+ 2uAA $. 3gpA(hV) = 0
Equation 2, like Equation 1, is a special case of the new general law derived below (Equation 8). The hydrodynamic process described by Equation 2 is accompanied by another process which is much slower, but which remains in operation after the first process has virtually ceased, and indeed accelerates as the liquid films become thinner-namely, the diffusion of the contained gas from the smaller to the larger bubbles, and from the larger bubbles a t the top of the foam to the outside atmosphere. The two processes of liquid drainage and gas transference are sufficiently separated in time so that each can be distinguished and its rate measured (15). As for the process of gas transference between spheres, with only negligible changes of hydrostatic pressure, we have already demonstrated that Equation 1 applies. Kugelschaum is not always merely a rapidly-passed transitional stage in the life of a foam. This form can be made to persist for an appreciable time before the polyhedral foam structure is established. Clark (5) and de Vries (7) have published photographic illustrations
AUTHOR Sydney Ross is Professor of Colloid Science in the Department of Chemistry, Rensselaer Polytechnic Institute, Troy, N.Y. 12181. This paper was presented as part of the interfaces I1 Symposium on Chemistry and Physics, Washington, D . C., June 10-12, 1968.
son. (b)
Time = 15 min, magnification (linear) = 16. Foamed oil-in-water emulConcentration of air = 92.5 vol. yo. Viscosity of liquid = 11 CP
Figure 2. Transformation of the morphology of foam from kugelschaum to polyederschaum on aging [from ref. (7), by consent of the author and permission of the Royal Netherlands Chemical Society ]
r
showing both the frequent occurrence and the longevity of kugelschaum. de Vries shows pictorially the effect of the bulk viscosity of the foaming liquid on the persistence of kugelschaum. Figure 2a is a photomicrograph of a foam produced from an oil-in-water emulsion, prepared by dispersing a mixture of 57 vols of monochlorobenzene and 43 vols of paraffin oil in 100 vols of 0.18M sodium laurate solution. The viscosity of the emulsion was 11 cp a t a shear rate of 900 sec-l. The foam was produced by beating air into the emulsion by means of an electrically-driven household mixer, and it contained 92.5 vol y, of air. Figure 2b shows the foam 15 min after its production. The morphological structure changed fairly quickly from kugelschaum to polyederschaum. Figure 3a is a photomicrograph of a foam produced by the same method from an emulsion of similar composition, but with an oil content of 707, and, as a consequence, a n appreciably higher viscositynamely, 65 cp at a shear rate of 900 sec-'. The air content of the foam was the same as in Figure 2a (92.5 VOL. 6 1
NO. 1 0 O C T O B E R 1 9 6 9
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(a)
(b) sion.
Time = 0, magnification (linear)
3
16
Time = 15 min magnification (linear) = 16. Foamcd oil-in-water cmulConccntration bf air = 92.5 vol. %. Viscosity of liquid = 65 CP
Figure 3. Persistence of the kugelschaum morphology of foam with time [fromref. (71, by consent of the author andpermission of the Royal Netherlands Chemical Society]
Figure 4. ( A ) Two double-walled bubbles form a composite bubble by partial coalescence. ( B ) T w o submerged bubbles cannot form a composite bubble
vol %). The rate of drainage of foam 2 was much less than that in foam 1, so that even 15 min after its production, the morphology was still that of spherical bubbles (Figure 3 b ) , although they were of much larger size because of gas transfer between bubbles during the 15-min period. The kugelschaum morphology is preferred in fire-fighting foams, as the higher concentration of water allows it to absorb more heat before it releases carbon dioxide. The higher density of kugelschaum is also a desirabIe property in foam rubber, in foamed-polymer applications, in shaving foams, in milk shakes, and in whipped cream. Thus, all long-lived foams that are of interest for their industrial application are desired in the kugelschaum form, and formulations are so developed to produce and retain it. I n kugelschaum we need not be concerned about the geometry of composite bubbles because submerged bubbles in the body of the foam cannot come together to create isolated composites. O n close approach, the bubbles either repel one another or the liquid between
them withdraws to allow complete coalescence to a larger sphere. Partial coalescence requires the presence of two interfaces, which are lacking in submerged bubbles, whose tensions are required to complete the Neumann triangle of forces in equilibrium. (See Figures 4A and 4B.) The geometry of kugelschaum is, therefore, that of spheres of different sizes, changing in size chiefly because of gas diffusion from smaller to larger bubbles, and possibly, on occasion, by direct coalescence of bubbles. For both of these processes, we have already demonstrated that Equation 1 is valid. The pressure external to the submerged bubbles is not, however, merely the atmospheric pressure P: T h e hydrostatic pressure in a volume of foam varies with the height. Let the symbol Ph represent the pressure in the liquid medium just outside the inner wall of a bubble in the foam, then
52
INDUSTRIAL AND ENGINEERING C H E M I S T R Y
where P is the pressure of the atmosphere, p is the density of the liquid in the foam, and h is the hydrostatic height
of the foam above the point where the external pressure is Ph. I n polyederschaum, no significant spontaneous diffusion of gas takes place between the foam cells, as the films separating them are almost planar, thus indicating that the pressures inside contiguous cells are nearly equal. As was first pointed out by Plateau, however, a powerful capillary suction is exerted upon the liquid within the films, as a result of the pronounced curvature of the Gibbs angles (also known as the Plateau borders)--i.e., the “lines” in the foam structure where three films meet and the “points” where four lines meet. The stability of polyederschaum is due to the loss of fluidity in the lamellae, which would otherwise recede and rupture in response to the capillary suction. The loss of fluidity in its turn may result from more than one set of conditions: it may arise from surface plasticity, which extends to a sufficient distance into the underlying liquid, so that the effects of opposite surfaces of a thin liquid film overlap; it may result from the presence of a gelating polymer or associated complex in the bulk liquid that thickens the whole liquid medium; or it may be the result of electrostatic repulsion between two like-charged surfaces of the film, the electric force being strong enough to overcome the mechanical forces of capillarity and hydrodynamic drainage (9, 7 7). A number of adjacent polyhedral foam cells, separated by lamellae of very small curvature, may exist in the interior of the foam, but the films that separate the foam from the external atmosphere are curved, which means that the gas inside the foam is at a higher pressure than the atmosphere. The polyhedral-foam structure, in spite of its appearance to a cursory glance (Figure 2 b ) , and despite its name, does not consist of real polyhedra. During the process of hydrodynamic drainage, the result of which is to bring the original spherical bubbles close to one another, the original differences in pressure between bubbles of different size are not equalized, and although rupture of films has increased the size of the foam cells, the remaining statistical distribution of sizes is evidence that differences in pressure between them still exist. Two polyhedral cells of the same size would certainly have a liquid film of zero curvature established between them, but the greater the disparity of size between two contiguous cells, the greater the curvature of the septum between them. Examination of Figure 2 b shows slight curvature of the septum between foam cells, the more pronounced where the difference in size is greater. Because the gas in each polyhedral foam cell is at a pressure (p - P) above that of the atmosphere, there is an imaginary sphere of radius r that contains gas at the same pressure as that in the cell, where
(3) Let us call this imaginary sphere the “sphere of equiv-
alent curvature.” Form any simple cases of composite bubbles, the sphere of equivalent curvature is clearly the circumsphere of the foam cell-that is, the sphere of minimum volume sufficient to contain the foam cell. Perhaps the two spheres are always the same for any polyhedral-foam cell; I am inclined to think so, but that is a theorem still to be proved. Fortunately, the result I wish to demonstrate does not depend upon identifying the sphere of equivalent curvature with any geometric entity of the structure. It is sufficient to be able to assert that the volume of the polyhedral-foam cell, V, can be expressed as some unknown fraction of the volume of the sphere of equivalent curvature, so that we may write V = ar3, where a is a constant and r is the radius of the sphere of equivalent curvature. Similarly the area, A , of each polyhedral-foam cell can be expressed as some other unknown constant, 6 , times the square of the radius of the sphere of equivalent curvature: A = br2. Let the area, A , correspond to the share of the total interfacial area of the foam that can properly be traced to the individual cell-Le., it will be the inside surface of the cell if each wall is shared by another cell, but, for walls on the outermost surface of the foam, both outer and inner faces will be counted in making up the value of A . I shall now prove that the ratio of the constants, a / b , is the same for any foam cell. The gas inside every element of a foam structure, whether it be contained inside a single isolated bubble or inside a polyhedral-foam cell, does not expand spontaneously, because the energy that would be thus liberated would be completely used up in creating new surface of liquid film; nor do the containing walls of a foam element spontaneously contract, because the energy liberated in this way would be used in compressing the gas. I n all such elements, therefore, the condition for stability is given by the equation
(p
- P)dV
udA
(4)
where A is as defined a t the end of the preceding paragraph. The volume and surface area of any foam element is given by the equations
V = ara and A = br2 (5) where r is the radius of the sphere of equivalent curvature. Substituting Equations 3 and 5 into Equation 4, gives 6a = b.
We are now equipped to demonstrate that Tait’s equation expresses a general law. Let the total volume of a foam structure consisting of v elements be V, and let its total surface area be A . Then
L‘
=
a1r13
A =
b1r12
+ a ~ r 2+~ + . . . . = + b z r 2 + bar2 + . . .bir,. . . =
V
airt3 ( 6 )
~ 3 r 3 ~
V
btr:
(7)
The symbols r1, r 2 , . . . r t . , .represent the radii of the spheres of equivalent curvature of each foam element, respectively. The total quantity of gas inside the foam VOL 61
NO. 1 0 O C T O B E R 1 9 6 9
53
Y
4u
pi= P+-
re
The evaluation of the quantity of gas (assumed to be an ideal gas) in the foam is made as follows:
nR T = p -
+ 2 g.- A (8A) V V Equation 8A, another form of Equation 8, makes it more apparent that in a foam the interfacial area per unit volume is uniquely determined (a) by the average excess pressure of the gas above that of the atmosphere and (b) by the surface tension of the liquid. p"'
is given by C ( p t V t ) / R T where
=
Applications of the Equation of State
2
v
PV
4- - B -
PV
2 + -ju*A
3
bp?
In this development, we have made use of the result 6ai = b,, proved above for any foam element. Now consider the whole foam structure described above to undergo a spontaneous change such that the gas expands and the liquid films contract. The change may come about either by internal diffusion of gas from one cell to another or by rupture of films and consequent coalescence of cells. Let a new structure result by either mechanism or by a blend of both, assuming that all of the gas originally in the foam remains inside. From Equation 8 it follows immediately that 3P.AV
+ 2u.AA = 0
(see Eq. 1)
where AV and A A are the changes in volume and internal area, respectively, from one structure of the foam to the other. The foregoing proof reveals that Tait's equation, although of general applicability to spherical and composite bubbles of all degrees of complexity of structure, is still only a special case of the more general Equation 8, which may properly be called the equation of state of a foam: PV
2 +uA = nRT 3
(see Eq. 8)
The equation of state relates the foam variables of moles of gas, volume of gas, temperature, and area of liquidgas interface, and also describes the influence of the atmospheric pressure and the surface tension of the liquid. For some purposes it is convenient to use the concept of the average pressure of the gas within a body of foam. Let the average pressure, p"', be defined as
P"'
=
5 @tW2(Vi)
and it can then be readily deduced that 54
INDUSTRIAL A N D ENGINEERING CHEMISTRY
The experimental verification of the equation of state by means of Equation 1 requires a foam so stable that it can undergo internal changes without losing any of the contained gas. Actually, this requirement is impossible to meet, as a certain quantity of gas is always lost by diffusion from the outermost layer of bubbles in contact with the atmosphere. If, however, the outermost liquid films do not rupture, the loss of air can be kept to the irreducible minimum. Some foams of interest for their applications would be satisfactory-for example, whipped cream, shaving cream, and fire-fighting foam. The specific interfacial area of such stable foams can be measured by photography. The technique of the photographic method is described by various authors, in particular Clark (5) and de Vries (7), who have also published photographic illustrations of both kugelschaum and polyederschaum. A more rapid and convenient method to determine the specific surface is to measure the light absorbed due to scattering by the foam films, making use of a relation, derived by Clark and Blackman (6),between the loss of light on transmission and the specific surface of the foam :
A
= :(k
- 1)
where k is a proportionality constant, 10 is the intensity of the incident light, and I is the intensity of the light after transmission through the foam. The expansion of the gas in the foam can be determined indirectly by measuring the change of the foam density as the foam ages. The relative change in V is much less pronounced than the change in A , but densities can be measured with high precision. A relatively large volume of foam, possibly 3 or 4 I., is required. A container of appropriate size is filled with the foam and weighed at periodic intervals, after each incremental volume of foam created by the expansion of the gas has been carefully swept off the top. The volume of gas in 1 cm3of foam is given by :
v = 1 - (D/p) (9) where D is the density of the foam, and p is the density of the liquid. Both A and V are to be measured on the same foam as a function of time in order to obtain the data required to test the application of Tait's equation to foam. From Equation 9, d V = -dD/p
which, substituted in the differential form of Equation 1, gives
3 5 d D - 2adA
= 0
P
or
Equation 10 is useful in providing for an experimental test of Equation 1 in situations too co_mplex to be treated by geometry-situations where both gas diffusion and coalescence of bubbles may be taking place as the foam decays. If Equation 1 holds for the dynamics of foam decay, the plot of specific surface area us. foam density will be linear. The test of the validity of the equation lies not only in the linearity of the plot, but in the numerical value of the slope, which should equal (2up/3P). g cm-2, This numerical value is of the order of which shows how much less pronounced is the expansion of the gas compared to the shrinking of the liquid films. The value of P that ought to be used is the average pressure external to the bubbles, in which the hydrostatic head is averaged as its mid-point value: Le.,
In practice, the second term on the right-hand side of Equation 11 is very small compared with the atmospheric pressure, so that PaV = P will usually be a satisfactory approximation. Equation 10 is predicated on the supposition that the decay of the foam is an isothermal process. The expansion of the gas results in cooling, which would be maximum for an adiabatic process. The expansion that actually takes place during the decay of a stable foam is very slow; moreover, the gas is subdivided into small volumes, each surrounded by liquid films which serve as a heat source. The expansion would therefore tend to be isothermal, although some irreversibility could remain, resulting in cooling; but this cooling would be only a small fraction of what would have resulted from an adiabatic expansion. Simultaneously, the contraction of the liquid films is accompanied by the release of heat (78)) which is absorbed by the liquid and by the gas. In 1 1. of foam with an internal surface of 300 cm2/cm3, the heat released by the total disappearance of the liquid films is about 2 cal, which, especially as it is released slowly, would have negligible effect on the temperature of the liquid; that which is absorbed by the gas would help toward ensuring that the gas expansion is isothermal. The total expansion of the gas on the decay of the foam is about 20 cm3 per 1. of the initial foam, and the maximum possible cooling for an adiabatic expansion would be about 4OC. The gas expansion occurs, however, under conditions far more favorable for an isothermal process than for an adiabatic process. The reasoned conclusion is then that the decay of a stable foam is practically isothermal.
The aging of foam is a phenomenon by means of which the generality of Equation 1 can be tested; it also provides an opportunity for its application. Liquids that are able to sustain a stable foam are characterized in terms of the relative ease of foam formation using a given mode of producing foam, and by the stability of the foam once formed. The relative ease of foam formation, or foamability, depends on viscosity, surface tension, and perhaps other physical properties of the solution; it may be measured by the specific surface of interfacial film per unit volume of foam. A more fundamental measure of foamability would be the net increase in potential energy acquired by the gas plus liquid when it is converted into foam. Three sources of potential energy of foam reside in the extension of the liquid surface, the compression of the gas, and the hydrostatic potential of the fluid inside the foam lamellae. The latter source of energy is soon spent and is, besides, no more than a casual concomitant. Equation 1 provides us with a way to approximate the potential energy in the remaining two sources. The measurement needed could be a determination of the interfacial area per unit volume of foam using the photographic method or the method of light transmission. Alternatively, the measurement could be a determination of the compression of the gas in the foam. The degree of compression could be measured by isolating a fixed volume of the original foam in a sealed container, waiting for the eventual complete separation of the gas, and then measuring the increase of pressure above that of the atmosphere. The time of waiting could be reduced by injecting a small quantity of a nonvolatile antifoaming agent. The former measurement gives AA; the latter, after calculation, gives AV. The potential energy of foam is measured by the work done as the foam decays. The actual work done as the bubbles expand would be close to that of a theoretical isothermal reversible gas expansion ; indeed, this process affords what is perhaps the closest practical approximation to the ideal unattainable expansion. For such a process
-dU
=
FdV
- adA
(12)
where U is the potential energy and p is the average pressure of gas inside the foam. Equation 1 enables us to calculate the potential energy approximately, the approximation being the substitution of P,,AV for the reversible work of expansion. Therefore -AU = PavAV
- uAA
(13)
and
A U = -u.AA 5 3 Substituting boundary conditions gives
VOL. 6 1
NO. 1 0 O C T O B E R 1 9 6 9
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The gain of the potential energy of the foam, therefore, parallels exactly the formation of A cm2 of interface per cm3 of foam. The value of U for a given mode of producing foam measures the foamability of the solution. If AV is determined, the foamability can be calculated by Equation 9 :
where Pa, is defined by Equation 5 . The increase of potential energy that accompanies the conversion of a given amount of gas into bubbles is shown graphically in Figure 5 , calculated according to Equation 14, for some simple foam elements. The greater the degree of dispersion of the gas, the more potential energy is acquired by the foam. A fundamental measure of the stability of foam would be the rate of dissipation of this potential energy. This suggestion has the merit of expressing both foamability and foam stability by the same unit, rather than by arbitrary units. Among suggested measures of foam stability are the average lifetime of bubbles determined singly (8); the average lifetime of a bubble in a dynamic foam (2); the average time of unit volume of liquid in a foam ( 4 ) ; the average lifetime of unit volume of gas in a foam ( 4 ) ; the rate of disappearance of the interfacial area (5); the rate of disappearance of bubbles (7); as well as numerous arbitrary measures that depend on the design of special apparatus (74). Some authors have derived rate equations for the decay of foam, with the intention of using the rate constant as a measure of foam stability. Now, just as the potential energy of foam resides in two sources, so two different mechanisms are responsible for the dissipation of that energy as the foam decays-namely, the diffusion of the gas through the liquid films, and the rupture of the films themselves. If the films collapse readily, as thick films are sometimes able to do, then the foam will be relatively short-lived, because this mechanism is the more rapid mode of the two to dissipate the potential energy. O n the other hand, if the films have a pronounced superficial plasticity, they do not break readily and gas diffusion then assumes the major role in dissipating the potential energy of the foam. As the gas-diffusion process process is relatively slow, the foam is correspondingly stable. But even in this case, the diffusion could not occur if the films were too rigid to contract. Equation 1 requires that the two processes occur simultaneously, if one is prevented, so is the other; if one is promoted, so is the other. de Vries (7) has derived a theoretical expression for the rate of diffusion of gas out of a small bubble in a foam, which causes the bubble to shrink and ultimately to disappear. The rate a t which the bubble shrinks is given by the equation
56
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Figure 5. Relative potential-energy levels for the subdivision of a given amount of gas into a number of simple or composite double-walled bubbles
where ro is the initial value of the bubble radius at t = 0 ; r is the bubble radius after a time interval t; D is the diffusion constant; S is the solubility of the gas per ml of liquid a t a pressure of 1 dyne/cm2; and 8 is the thickness of the liquid film between bubbles. In general, each large bubble in a foam is surrounded by a number of smaller ones. de Vries assumes that the radius of the larger bubble is many times larger than that of any of the small bubbles (e.g., 20 times as large). This assumption is the basis of the derivation. I t follows, therefore, that the loss of interfacial area experienced by the small bubbles as they shrink is much greater than the area of interface gained by the large bubble as it expands. Equation 16, therefore, gives as a good approximation that
RT DSu 9
A0 - A = 1Ga-*P
e
t
= k,t
From Equation 1 it also follows that:
32 RT DSu2 V - V0 -- -a-.-*t 3 P2 8
= k,t
Equations 17 and 18 refer to foams that decay by gas diffusion only. Less stable foams decay more rapidly by immediate rupture of thick liquid films and coalescence of adjacent bubbles. Clark (5),by means of direct photographic measurements of the interfacial area of certain foams, found that his results could be described by the first-order rate equation:
A
= AOe-OLt
(19)
where A . is the specific area of interface at t = 0 ; A is the specific interfacial area after a time interval t; and cr is the rate constant. There is no evidence, how-
ever, that Clark’s foams were decaying by the sole mechanism of film collapse; they were, in fact, firefighting foams, stabilized by a mixture of soap and hydrolyzed blood; and the data themselves could be as well described by Equation 17 as by Equation 19. We should look at much less stable foams than these to find a decay that will depend primarily on the mechanism of film collapse. For many such less-stable foams, whose decay has been measured by observing the rate at which liquid is released, an empirical description of behavior is given by the equation
V = Voe-at (20) where V is the volume of liquid in the foam a t time t; Vo is the volume of liquid in the foam when t = 0 ; and a is a constant dependent on temperature and on the nature of the liquid from which the foam is formed. Equation 20 has been found to hold for such widely diverse substances as beer ( 3 ) , saponin solutions (7), and solutions of lauryl sulfonic acid (73). The liquid that a t first escapes from the foam stems from hydrodynamic drainage, but Ross (73) has shown that it is only towards the completion of this process that Equation 20 applies. Thereafter the liquid exudate is derived from the collapse of the foam lamellae. The rate of liquid efflux is, therefore, dependent on the rate of film collapse, and Equation 20 is an indirect, but nevertheless faithful, reflection of the rate at which the area of the liquid surface is disappearing. That it should be an exponential function means that the rate of bubble coalescence depends only on the amount of interface present at any time. If Equation 19 describes the rate of contraction of the interfacial area, the simultaneous expansion of the contained gas would, by Equation 1, be described by
-
(V - V ) = (P Vo)e-”$ (21) where 7 is the volume of the gas at atmospheric pressure, P o is the volume of the gas in the foam at t = 0, and V is the volume of the gas in the foam after a time t. An exponential rate of film contraction and of gas expansion means that the total potential energy of the foam is also decreasing exponentially with time. This result suggests some obvious mechanical analogies. For example, the hydrostatic head of a tank containing liquid that is emptying through an opening at the bottom decays exponentially, because the rate of efflux is at all times proportional to the hydrostatic head; similarly, the potential energy of a compressed gas escaping through a “leak” declines exponentially. In all such cases the rate of decay of the potential energy is proportional to the potential energy that still remains. These analogies imply that the decay of foam is also essentially the dissipation of its potential energy, which is stored up in the compressed gas and the extended liquid surface, at a rate determined by the nature of the physical means available to promote the occurrence-in this case, the readiness of the liquid films either to allow gas diffusion or to rupture. The two different means affect the rate
function differently. The exponential rate of film contraction corresponds to the mechanical analogies instanced above. The gas diffusion is a more complex mechanism whose rate depends on the area of interface and on the gas compression, both referring to the smaller bubbles. It is also dependent on the distribution of bubble sizes in the foam. I n a foam of bubbles of uniform size, there would at first be no interbubble diffusion at all except from the outermost layer to the atmosphere, and then in a lesser degree to the bubbles next underneath; the over-all rate of diffusion would be small. In a foam of bubbles of a wide distribution of sizes, the internal diffusion would be much more rapid. In our mechanical analogy it is as if the orifice grew wider as the hydrostatic head decreased. That two different rate functions should exist for the two different mechanisms of foam decay is an advantage to the investigator, who could thereby detect the operating mechanism by his measurements of foam collapse. A relatively simple apparatus could be designed, based on the preceding considerations, to measure both foamability and foam stability. A container is fitted with a sensitive pressure gauge that can detect the difference of a few centimeters of water between the inside and outside of the container, using for example a mechanicalinput transducer of appropriate range of response. The container would also have a device, which could be operated from the outside, for agitating the foamy liquid-e.g., a perforated paddle that could be moved up and down. A small quantity of a foamy liquid is placed in the container and converted into foam by external application. The inclusion of some of the air in the foam causes the pressure inside the vessel to drop. The extent of the pressure reduction measures the efficiency of the foam formation. As the foam decays, the original pressure is restored. A recording potentiometer attached to the transducer would thus record the foamability of the liquid, the efficiency of the mechanism for foam conversion, and the foam stability. REFERENCES (1) Arbuzov, K. N. and
Grebenshchikov, B. N., J . Phys. Chem. (USSR), io, 32 (1937). (2) Bikerman, J. J., Trans. Faraday Soc., 34, 634 (1938). (3) Blom, J. and Prip, P., Wochchr. Brau., 59, 1 1 (1936). (4) Brady, A. P. and Ross, S., J.Amer. Chem. Soc., 66, 1348 (1944). (5) .Clarh, N. 0 “A Study of Mechanically Produced Foam for Combating P e w 4 Flrea, D. S. i:R., Chemistry Research, Special Report No. 6, H.M.S.C. London, 1947. (6) Clark, N. 0. and Blackman, M., Trans. Faraday Soc., 44, 7 (1948). (7) de Vries, A. J., Rcc. Tim. Chim., 77, 81 (1958). (8) Hardy, Sir W., J.Chem. Soc., 1925, 127, 1207; Talmud, D. and Suchowolskaja, S., Z . Physrk. Chem., A154,277 (1931). (9) Kitchener J. A. “Recent Progr. in Surface Sci. ” J. F Danielli K G A Pankhurst, C. kiddiford, Eds, Academic Pres;, New‘York, 1464, ’Voi. 1; Chap. 2. (10) berg, Manegold, 1953. E., “Schaum”, Chemie und Technik Verlagsgesellschaft, Heidel-
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(1 1 ) Mysels, K. J. et al. in Discusstons of tha Faraday SOC.No. 42 (1 966). (12) Plateau, J. A. F.,“Statique der Liquids,” Paris, 1873, Vol. 1, Chap. 5. (13) Ross, S., J. Phys. Chem., 47, 266 (1943). (14) Ross, S.,Znd. Eng. Chem. (Anal. Edition), 15, 329 (1943). (15) Ross, S. and Cutillas, M. J., J . Phys. Chem., 59, 863 (1955). (16) Tait, P. G., Proc. Roy. Soc. Edinburgh, 6 , 292 (1867-8). (17) Tait, P. G.,“Propertiea of Matter,” A. and C. Black, Edinburgh, 1885, Chap. 12. (18) Thomson, W. (LSfd Kelvin) Phil Mag. 1859 ( 4 ) 17, 61; “Mathematical and Physlcal Papers, Cambridie Univ., 1911, Vbl. 5,’pp 55-6.
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