Langmuir 1991, 7, 3070-3075
3070
Stability of Interconnected Bubble Networks Robert S . Hansen Ames Laboratory and Department of Chemistry, Iowa State University, Ames, Iowa 50011 Received February 28,1991. In Final Form: May 2, 1991 Two connected bubbles stabilized by films of the same tension in the same external environment can only be in equilibrium if they have the same radius. But where u is the surface tension, A is the external area of a bubble, and E = 2(du/d In A) is the Gibbs elasticity, the equilibrium is only stable if E - u > 0. The fluctuation caused by an infinitesimal transfer of material from one bubble to the other will grow or decrease exponentially in time depending on whether E - u is negative or positive, and models for calculatingthis rate are given. Literature data on the surface tensions of aqueous solutions of the alcohols CH3(CH2),-10H,6 4 n 5 10 are used to model bubble stability in two modes. First, it is supposed that the bubble film is 1 pm thick with ita aqueous phase uniform in alcohol concentration within the film; conditions for *windows of stability” are derived, within which bubbles can be expanded or contracted. Second,the film is assumed to consist of an “oil”layer 6 pm thick with a 1pm thick aqueous layer on either side of it; at equilibrium the alcohol concentration in the ”oil” is assumed K times that in the aqueous phase. All transport is assumed diffusion-limited. With the film initially at rest, the area is subjected to periodic cycling, and both steady periodic and transient responses of the surface tension are calculated. This model is intended to simulatequalitatively the behavior of the collectionof alveoli in human respiration. For suitable parameter choices certain aspects of the latter behavior are rather well simulated; specifically there is a substantial amplitude to the surface tension variation and its phase shift is such that the surface tension is greater during the area expansion, less during contraction, than if it is exactly in phase with the area cycling.
Introduction A spherical bubble of radius r formed by a thin film of a liquid whose surface tension is u in an ambient gas of pressure POhas an internal pressure P given by the Laplace equation
P = Po + 4u/r
(1)
(The factor 4 rather than the familiar 2 reflects the fact that the liquid film has both an external and an internal surface.) If two such bubbles with the same external pressure Po are connected (e.g., by a small tube), they can only be in equilibrium if their radii are the same. But this equilibrium is unstable, for an infinitesimal transfer of gas from one bubble to the other will cause the loser’s radius to decrease and its internal pressure correspondingly to increase, while the gainer’s radius increases and its internal pressure decreases. The process will therefore continue; the loser’s internal pressure will reach a maximum when it is a hemisphere whose radius is that of the tube, and it will continue to lose gas until it is a spherical segment with the same radius as that of the gainer bubble. This evolution is illustrated in Figure 1,and was well discussed in C. V. Boys’s lovely book on soap bubb1es.l For two connected bubbles to be in stable equilibrium, it is necessary for their internal pressures to be equal, but it is also necessary that an increase in r lead to an increase in P; i.e., for stability
As u and r are both positive this is plainly impossible if duldr = 0, as is expected to be the case for a pure liquid. Considering the initial stage of foam development to be a collection of discrete bubbles, it should be noted that if the gas has any solubility in the liquid the diffusion process can play the role of the tube in gas transport from bubble ~~
~~~
(1) Boys, C.V.Soap Bubbles, Their Colors and the Forces that Mold Them; Dover Publications, Inc.: New York, 1959.
Figure 1. Evolution of a volume fluctuation in two connected bubbles with constant surface tension initiallyat equilibrium.If VZis increased by AVz and VI correspondingly decreased, P1 will increase and PZdecrease, so the initial fluctuation will grow. PI will reach a maximum when rl = PT (dotted curve) and transfer will continueuntil rl and rz are again equal (hemisphericalsection on left side figure). Relations among parameters: Pj = PO + 4u/rj, vj = 4/3arF,PO>> 4ufr, for j = 1,2; at t = 0, rj = ri.
to bubble, and a pure liquid therefore cannot maintain such a collection. This is one reason a pure liquid cannot foam. In later stages of foam development, bubbles are only separated by thin films of liquid and develop into polyhedra with the separating liquid films thinning through Plateau border suction, van der Waals forces, and gravitational drainageS2v3A model for film rupture has been well developed by Vrij4and Vrij and Overbeek5 and is based on fluctuations due to spontaneous capillary waves growing exponentially when the films are sufficiently thin; double-layer repulsion due to surfactants a t the film surfaces is a major factor in resisting the thinning and rupture process. The conditions for stability of an interconnected bubble network can be adequately developed from considerations of the stability of two connected bubbles asjust illustrated. In the following we discuss the conditions under which fluctuations will grow or decay, and how fast this will occur, (2) Adamson, A. W. The Physical ChemistryofSurfaces, 5thed.;John Wiley & Sons, Inc.: New York, 1990; pp 544-50. (3)Mysels, K.J.; Shinoda, K.; Frankel, S. Soap Films - Studies of Their Thinning and a Bibliography; Pergamon: New York, 1959. (4) Vrij, A. Discuss. Faraday SOC. 1966, 42, 23. (5) Vrij, A.; Overbeek, J. Th. G. J. Am. Chem. SOC.1968, 90,3074.
0 1991 American Chemical Society
Langmuir, Vol. 7, No. 12, 1991 3071
Stability of Interconnected Bubble Networks and the conditions under which connected bubbles can be jointly periodically expanded and contracted while maintaining stability.
Theory A. Evolution of a Pressure Fluctuation. Suppose, in the system shown in Figure 1,a fluctuation causes PI to be greater than P2. A treatment by Landau and Lifshitz6is easily adapted to a discussion of its evolution; at constant T
s, and an initial fluctuation in r2 would Then T = 1.2 X grow or decrease tenfold each 2.8 ms. B. Gibbs Elasticity: Some Stability Implications. We now focus attention on the Gibbs elasticity E and consider first the case in which the surface-active agent is only present at the surface. Let r (mol/cm2) be its density there; then conservation requires I'A = constant, d In A = -d In I', and so
The stability condition E - u > 0 hence means in which q is the gas viscosity. We shall be concerned with cases in which u I72 dyn/cm, r 1 100 pm, PO= 1 atm (1.013 X 106dyn/cm2). Hence, 4u/r < 0.0285P0,and except in the PI - PZterm, we can replace PIand PZby PO to sufficient approximation, obtaining
d In u > d In r
which is Gibbs's7 eq 531. Documentation of the function We wish to develop this in powers of Arz to terms in (Ar#. From conservation of volume we have (ri + + (ri + Ar2)3 = 2ri3, whence Ar1 = -Arz(l 2Arz/ri). Using this for j = 1,2 in
+
together with rZ2 = ri2 (1 + Arz/rJz, we obtain from (3a) dr, dAr, -=-dt dt
--
E(I') defined by eq 7a could be accomplished experimentally, e.g. by classical film balance methods or, in fortunate cases, from monolayer equation of state data.8~9 An interesting example of the latter is furnished by a myristic acid monolayer on dilute aqueous HCl. Where A = uo u is the spreading pressure and A the area per molecule in A2, the ?r - A diagrams for this system are discussed a t length by Adam ref 19, pp 58ff, especially pp 59 and 66) and show an extensive liquid expanded range. LangmuirlO was able to represent data in this range with an equation of state; from his parameters the equation of state a t 25 "C is (T
accurate to terms in (Ard2. Now
and as the bubble external area A = 4w2, d In r = l/z(d In A). The quantity E = 2(du/d In A) was introduced by Gibbs7 (see pp 301-2 and eq 6431, who appears to have been the first to address the surface stability problem thermodynamically, and is now called the Gibbs elasticity in his honor. From (4c) it follows that
and using this in (4b), defining u = Arz/ri at t, uo = Arp/ri a t t = 0, we obtain on integration UO
Plainly u diminishes with t if E > u and grows with t if E < u. The relaxation time r for the decay or growth process is given by
To estimate its magnitude, suppose IE - a1 = 10 dyn/cm, ri/rT = 2, q = 1.85 X 10-4 P (air at 25 OC) and 1 = 1 cm. (6) Landau, L. D.;Lifshitz, E. M. Fluid Mechanics; Sykes, J. B., Reid, W. H., Transls.; Pergamon Press: London, 1959. (See problem 6, p 59.) (7)Gibbs, J. W. Thermodynumics; Collected works; Yale University Press: New Haven, CT, 1948; Vol. I. See especially discussion on pp 242-44, eqs 522-531, and pp 301-2.
+ 11.2)(A - 16.5) = 411
(84
The liquid expanded film represented by this equation covers the approximate range 29.5 IA I 52; transitions occur as A < 29.5 (to an intermediate film) and as A > 52 (to a gaseous film) (ref 2, pp 131-41). From (8a) with uo = 72 dyn/cm it follows that
E-a=
411(3A - 16.5)
(A - 16.5)'
- 83.2
(8b)
This is negative for A > 38.7, increases to 23.1 dyn/cm at A = 35 and to 82.6 dyn/cm at A = 30. As A is reduced below the A = 29.5 transition area E - u becomes negative. Hence, the region 29.5 IA I38.7 is a "window of stability" for interconnected bubbles stabilized by myristic acid; if one were to vary their radii periodically, one would need to stay within this range so at most their areas could vary by a factor 1.31 and their radii by a factor 1.15. In the more common wateraoluble surfactant-stabilized bubbles such as Boys's1 soap bubbles, we have two Burfactant-rich films separated by an aqueous layer approximately 1 pm in thickness in which the surfactant is appreciably soluble. Reduction of surfactant surface density on area expansion is partially compensated by replenishment from the liquid film; transfer in the reverse direction occurs on contraction. Papers by van den Tempel, Lucassen, and Lucassen-Reyndersl' and by Lucassen (8) Gaines, G. L. Jr. Insoluble Monolayers at Liquid-Gas Interfaces; Interscience Publishers: New York, 1966. (9) Adam, N. K. The Physics and Chemistry of Surfaces; Dover Publications: New York, 1968. (10) Langmuir, I. J. Chem. Phys. 1933,1, 756. (11) van den Tempel, M.; Lucassen, J.; Lucassen-Reynders, E. H. J . Phys. Chem. 1966,69,1798.
3072 Langmuir, Vol. 7, No. 12, 1991
Hansen
and Lucassen-Reynders12are particularly germane to this problem. In expansion or contraction of the film, it is presumed that film volume Ah (A = area, h = film thickness) is conserved. It is also assumed that total surfactant is conserved; in the simplest case it is presumed that surfactant concentration c (mol/cm3) is uniform through the aqueous layer, as if diffusion were fast on the expansion-contraction time scale or the aqueous layer were well stirred. Hence, Ahc + 2Ar is conserved, and therefore
dlne --=---dlnA
1 d8 2rmd6 8d1nA-hdc+2rmd8=
Using (loa), we recover the two forms
-d In A = d In h = (hdc + 2dr)(2r)-'
(9) In the simplest case, it is also assumed that r remains in equilibrium with c according to the Frumkin or regular localized monolayer m0de1.l~ In this model, where 8 = F/Fm, rmbeing the surfactant monolayer saturation density, a , the surfactant activity in the liquid layer, B and a parameters
d In A
r,RT
= -r,RT[ln
(1- e)
+ a8']
although many situations permit the assumption of Henry's law for the surfactant activity, so that ita activity is proportional to its concentration and the proportionality constant can be incorporated in the parameter B, many surfactants form micelles extensively above a critical micelle concentration (cmc) and their activities vary little (often treated as if not at all) with total concentration above the cmc. Hence, so long as the total concentration in the aqueous film exceedsthe cmc, the surfactant activity, and hence 8 and r , will remain nearly constant. On film expansion, surfactant to maintain 8 constant will be furnished by dissociation of micelles at nearly constant activity so long as the micelles exist; the reverse of this process will occur on contraction. The micelles in the liquid layer hence effectively "buffer" 8 and r , keeping them essentially constant on film expansion and contraction. In this case E = 0 (strictly, very small) and connected bubbles will not be stable. Second, if a > 2 there will be an activity a t which two surface phases coexist; in the model described by eqs 10, if the surface coverage in one phase is 01, that in the other is (1 - 81) (ref 13, section 1008). Some representative sets of the triplet (a,B1, Ba) are (4.69, 0.01, 0.0092), (4.05, 0.02, 0.0174), (3.27, 0.05, 0.0380),(2.75,0.10,0.0639), and (2.20,0.25,0.111). While these two phases coexist a t a given temperature, r and the surfactant activity in the liquid film are fixed and will remain fixed on expansion or contraction, with the relative amounts of the two phases changing as needed to keep surfactant activity in the liquid film constant. In this case the two surface phases effectively "buffer" a and a so long as they coexist, E = 0 during their coexistence, and the connected bubble system will be unstable. In the Henry's law case, we can replace a by c in eq 10a and develop an expression for E as follows: da dlne d In 0 d In A Using (lob) and (91, we have
-2-
(12) Lucassen, J.; Lucassen-Reynders, E. H. J. Colloid Interface Sci. 1967,25,496.
2e - -[I 1-8
(13) Fowler, R. H.; Guggenheim, E. A. Statistical Thermodynamics; The University Press: Cambridge, U.K., 1956;Chapter X, pp 429-35.
8)'
e-2ao[ 1- 2a8(1- e)]
- 2ae(1 -e)] x 2rme(1 hc - e) [ I - 2a8(1-8)]
(lob)
Two special situations deserve comment a t this point. First,
E
2BI',(l-
and therefore
--E T
(
I t should be plain that eqs 11pertain only to single-surfacephase regions; under this circumstance 1- 2a8(1- e) 1 0, (d In r/de) I0, and -(d In 8/d In A) 5 1. Lucassen and Lucassen-Reyndersl' give illustrative plots of r / F,RT vs BandE/2I',RTvsr/rmRTforh =Oandh=2BI',(their Figures 1, 2, and 5, respectively, for various values of a (their H,/RT). The stability condition E - u > 0 takes a form particularly simple to analyze if cy = 0. In this case E-u=r,RT
{~ B[c
l+-
2Br,
+
+
(1 B c ) ~ ] - ~
+
I
In (1 Bc) - uo (12a) Hommelen14 has reported the dependence of surface tensions of aqueous solutions of the primary normal alcohols, CnH2,,+1OH, and their solubilities in water, for 6 I n I10a t 20 "C. With c in mol/cm3 and co the saturation value of c, a model qualitatively in accord with his results a t the simplified level implicit in the choice a = 0 in eqs 10and 12employsthe followingparameter choices: 10IOF, = 6.0 mol/cm2, uo = 72.8 dyn/cm, B = 77.9 X 4" cm3/mol, co = 0.246 X 4-" mol/cm3. We choose h = 1pm = lo-' cm, and therefore have the following parameter group values: I',RT = 14.62 dyn/cm, uo/rmRT = 4.98, h/2BFm= 1070 X 4-", Bco = 19.16 (so Bc = 19.16c/c0). Rearranging eq 12 with these values, we have
E - a32(~)[1+-(1 ---=38. F,RT
CO
1070 4"
+19.1&)2]-1+ CO
The group (E - u)/I',RT is negative, corresponding to instability, for all c/co 5 1if n = 6 or 7. (eq 12b obviously does not apply if C/CO > 1; an attempt to reach such a concentration would result in formation of the alcoholrich phase, and in the well-stirred liquid-phase model, this would fix u independent of A so long as this phase were present, and so E would be zero.) For n 1 8 (E u)/r,RT is positive, corresponding to stability, from a lower value of C/CO of about 0.1 (0.116, 0.105, 0.102 for n = 8 , 9 , 10, respectively) up to C/CO = 1; its values at C/CO (14) Hommelen, J. R. J. Colloid Sci. 1959, 14, 385.
Langmuir, Vol. 7,No. 12, 1991 3073
Stability of Interconnected Bubble Networks = 1are 3.0, 12.4, and 25.1 for n = 8, 9, and 10, and for n = 8 and 9 there are intermediate maximum values of 4.2 and 12.6 at c/co = 0.48 and 0.66. The range of c/co over which E - u is positive can be used to calculate the factor f by which a film can be expanded in area. Suppose, for example, that c / c o = 1 initially, and we seek the largest fthat leaves the film stable for the purpose of maintaining interconnected bubbles. Conservation of volume requires Ah = A&,, so if A = fA0, h = ho/f. Conservation of surfactant then requires 21'mho-1Bco(l co = 2f7mho-'Bc(l + Bc)+ C, SO with BCO= 19.16, f = (1 + 19.16c/co)(20.16c/co)-1[l+ 0.5261hocorm-l(l-c/co)]. With 104ho= 1.0 cm, 10"rm = 6.0 mol/cm2, co = 0.246 X 4-n mol/cm3, and the previously given lower stability limits, the maximum values off are 1.78, 1.53, and 1.46 for n = 8,9, and 10. Note that the term preceding the brackets is the ratio of initial to final values of 0 for these expansions, and the term containing hoc0 decreases rapidly as n increases. With decreasing solubility, the aqueous solution plays a steadily decreasing role in replenishing adsorbed surfactant as the film expands. C. HumanRespiration: SomeCharacteristicParameters. Effective transport of oxygen to the blood in human respiration depends on its transport across a large surface area (about 70 m2) of the walls of approximately 3 X los small sacs called alveoli, which are interconnected through a hierarchy of tubes of progressively decreasing radii (trachea, bronchi, and bronchioles). If we model the alveoli as a collection of interconnected spherical bubbles of equal radii, their radii must be about 136 pm and the total volume about 3.18 L. Normal respiration is about 0.5 L/breath at a frequency of 15/min; plainly both volume and frequency would be greater under vigorous exertion. The alveolar walls are in fact "bumpy", varying in thickness to accommodate capillaries. They are covered by a wetting film containing a number of surface-active compounds of which dipalmitoylphosphatidylcholine(DPPC) is particularly important. Surprisingly, the elastic properties of the wall contribute a relatively small fraction to the work of breathing; 3/4ths,to7/&hs of this work can apparently be accounted for through surface properties of the wetting film. Abnormally low supplies of surface-active agents in this film are associated witha serious, often fatal condition called respiratory distress syndrome. Very readable surveys of respiratory physiology,including the importance of wetting-film surface properties, have been given by ComWest,16 and Hills.17 Hills,17 in his chapter 6, gives an extensive discussion of alveolar models, including strengths and weaknesses of each (including the bubble model). The marked hysteresis in n-A plots for DPPC monolayers, dependent on frequency and amplitude of cycling, is especially interesting and is discussed in all three surveys; Hills's treatment is particularly extensive including discussionsof controversiesboth in measurement techniques and data interpretation. D. Volume Cycling of Interconnected Bubbles: A Qualitative Respiration Simulation. We now investigate a surface film model such that an interconnected bubble network based on it might simulate some aspects of respiration mechanics. We consider a planar film occupying the space -l/zhz 5 z 5 l/2h2 consisting of 3 components (1,an oil; 2, water; 3, a surfactant), 2 phases
+
+
(15) Comroe, J. H., Jr. Physiology of Respiration, 2nd ed.; Yearbook Medical Publishers, Inc.: Chicago, 1974 (especially Chapters 1, 2, and
IO).
(16) West, J. B. I n Respiratory Physiology - T h e Essentials, 3rd ed.; Williams, Wilkins: Baltimore, 1985 (especially Chapters 1, 2, and 7). (17) Hills, B. A. T h e Biology of Surfactant; Cambridge University Press: Cambridge, U.K., 1988 (especially Chapters 5-7).
(I, an oil phase, simulating the alveolar wall, occupying the space -l/Zhl C z C l/zhl; 11, an aqueous phase, simulating the wetting films, occupying the spaces -l/ 2h2 5 z I-l/zhl and '/2hl Iz 5 l/2h2). The space outside the film is assumed to be occupied by an inert gas; its solubility in the film is assumed negligible as are the vapor pressures of the film components, CI(z,t) and CII(Z,~) are the concentrations of surfactant in phases I and 11, and rowand rwgare its surface densities at the oil-aqueous and aqueous-gas interfaces. We suppose CI and CII symmetric about z = 0 and at equilibrium at their interfaces with a distribution coefficient K; i.e., C~('/zhl,t) = KC11(l/2hl,t),with row in equilibrium with C1(l/2hl,t) and in equilibrium with C ~ ~ ( l / z h ~ ,We t). CII(l/zhl,t) and rwg imagine the film initially at rest and at equilibrium with cI = cII= c;~,row= r:w,rwg= r:g, A = A ~ hl, = hy, hp = h:. The solubilities of oil in the aqueous phase and of water in the oil phase are assumed negligible. DI and DII are surfactant diffusion coefficients in phases I and 11. Transport in this model is assumed solely limited by diffusion. At t = 0 we initiate a periodic variation in A, so that A = A & + b sin ut) with b small compared to 1,and consider its consequences to terms first order in b. Conservation of volumes of phases I and I1 then requires hi = hP(1 - b sin ut) to this order, and conservation of surfactant requires
c,
A [ c1'2 dz C +~ I ~ ~ ~dz2+ C row I+ II',]
=
AO{'/2[hjCI + (4- hl)C~Il+ r8,
+ I'tgI
Let AQ = Q - 80,Q = CI, CII, row, or rWg. Then to terms first order in b
-b(r:w+ rig)sin ut
(13a)
The standard diffusion equations with initial conditions d2ACi aACi =ACi(z,O) = 0 i = I or I1 (13b) az2 at can be solved subject to the symmetry conditions ACi(-z,t) = ACi(z,t),the condition for surfactant distribution equilibrium at the ow boundary ACI(1/2h:,t) = KACII ('1 ,h;,t),andtherequirementsfortheequilibriumbetween absorbed layers and contiguous aqueous phase at the ow and wg boundaries ArOw = (dr,,/dC~~)AC~~(l/zh:,t), AFwg = (d~wg/dCII)ACII( 1/2h:,t) (thederivativesbeingevaluated at CII = Ct1), and eq 13a. The solution, obtained in the present case by Laplace transform techniques, is
Di-
ACi(z,t) = -bu(I':,
+ rpg)Cf,(Pj,z)e'j'/F'(Pj)i = 1,II (14)
in which
dC11
cosh 1/2qIIh2+ -(sinh 1 1/2qIIh2- sinh 1/2911h1)]+ QII ,
li sinh 'lpqIhl cosh l/zqIIhl 91
theoj are the roots of F(s) = 0, qI = (s/D1)lI2,q11= ( s / D I I ) ' / ~ F'(Pj) = dF/ds evaluated at s = Pj
3074 Langmuir, Vol. 7,No. 12,1991
Hansen
fI(Dj,z) = K cosh (Djh;2/4D#2 cosh [(Pj/DI)"2zl (14b)
fII(pj,z) = cosh [j3,h; 2/4DI)1/2 cosh [(Pj/DII)1/2zl (14~) The two roots s = *io of (s2+ w 2 ) = 0 yield two terms in eq 14,obviouslycomplex conjugates, describing the steady periodic behavior of ACi(z,t);the remaining roots give rise to terms in the transient whose decay with time describes the approach to steady periodic behavior. We now simplify these results and specialize parameters to a model intended to simulate some aspects of human respiration. As previously stated,the alveolar wall-wetting film interface (simulated here by the ow interface, appears to contribute little to the work of respiration compared to the wg interface. We, therefore, neglect surface properties of the ow interface, setting I?:, and dI',,/dCII both equal to zero in eq 14. Reasonable estimates can be made for some of the parameters in the model: breathing frequency 15 cycles/min or w = 1.571 rad/s, wall (oil film) thickness ho = 6 pm = 6 X cm, wall plus wetting films thickness cm, diffusion coefficient for surfactant in h,h = 8 X water DII = 5 X lo+ cm2/s (about the value for sucrose in water at 25 OC). There is little guidance for selection of the diffusion coefficient for surfactant in the alveolar wall or its distribution coefficient between wall and aqueous phase. Assuming D inversely proportional to solvent viscosity and supposing wall transport properties perhaps somewhat similar to those of olive oil, we choose DI = 5 X cm2/sec. The surfactant (here modeled as a long-chain alcohol) is expected to strongly favor the oil phase, i.e. K >> 1. We base the surface properties of the wg interface on the model of Hommelen's14results for the surface tensions of aqueous solutions of the alcohols CH3(CH2)n-10H,6 In I10, reflected in the parameters used in eq 12b. Hence, where c is now the initial concentration of the surfactant in the aqueous phase (formerly C;J and co its saturation concentration we have I':g = rm(19.16c/co) (1 + 19.16c/co)-' and dr,,/dCII = 19.16rm/co(l 19.16c/c0)-~with 10IOrm= 6.0 mol/cm2, r m R T= 14.62 dyn/cm, and co = 0.246 ( 4 9 mol/cm3. We suppose the initial value of 0 is 0.9, so that 19.16c/co = 0 ( l - e)-' = 9; the spreading pressure will be expected to cycle about r m R TIn 10 = 33.7 dyn/cm, and u correspondingly about 72 - ?r = 38.3 dyn/cm. Equation 14 can be specialized to z = 1/2h; (the wg interface) and the Gibbs adsorption theorem used to set Au = - I'RTACII(1/2h:, t)/C$. It is convenient to define the parameter groups u = 1/2q:Ih; = ( S ~ ; ~ / ~ D and ~ J ' / ~= h~co(19.16c/co)2~ (38.32rJ' = '/gh;(dI'w /dCII)-'; with the parameters given log 6 = 5.933 - 0.6d21n, so = 0.816 (i.e., about 1) if n = 10, is greater than 1 if n < 10 and less than 1 if n > 10. There results
+
Au = 131.6wb~e@/F'(Dj)dyn/cm
(15a)
j
where now
+
F(s) = (s2 + w2)(1 (ucosh u)-' x [O.lOK[ tanh 7.5 u cosh 0.75 u + [(sinh u - sinh 0.75u)l) (15b) This results on dividing both numerator and denominator of the basic Laplace transform for ACII('/2h;,t) by [(dI',,/dc) cosh '/,qIh!cosh 1/2qIIhi];while the form of
(15b) differs from that of (14a) the roots @j(for the same parameters) are unaffected. This form is especially convenient for discussing the roots s = f iw giving rise to the steady periodic response, because quantities such as u-l sinh u can be expressed in rapidly convergent power series; e.g., for u = (iwh;2/4DII)'i2
The factor 131.6 dyn/cm arises from RTl%l(dI'/dc)-' = rmRT(19.16c/co). The roots of F ( s ) = 0 other than s = *iw are those corresponding to F(s)(s2+ w2)-' = 0 and involve values of u that are pure imaginaries. For these it is desirable to convert F(s)(s2+ from a function of u to a function of u, where u = iu and s = 4DIIv2/hi2. The hyperbolic functions of u are thus converted to trigonometric functions of u ; e.g., cosh u = cos u, u-' sinh u = u-' sin u, u-tanh u = u-l tan u. All terms are invariant on replacement of u by -u, in F(s) (s2 + so if u = r is a root, so is u = -r; but both correspond to the same value of s, so it is only necessary to consider the positive roots. There results Aa = 132b
sin(wt + 4)
+
-
(G2 H2)1/2 m
2 ~ T j ( c 0rj)[(l s + T;)R(rj)]-' exp(-wtT;') j=3
I
(16a)
in which Tj = (wh~2/4DI,rz), G = 1+ (0.4002K + 0.2499)(, H = 0.3157 K t , tan 4 = k/G,the rj are the roots in u of F(s)(s2+ w2)-' = 0 (r3 being the lowest), and
R(u) = cos u - u sin u + O.lOKt(7.5 sec27 . 5 ~cos 0 . 7 5 ~0.75 tan 7 . 5 ~ sin 0.75~)+ [(cos u - 0.75 cos 0.75~)(16b) The first term within the braces of eq 16a represents the steady periodic response to the area cycling;the summation that follows it, the transient, decaying in time to zero as the steady response is approached. Some observations follow quickly from eq 16a. As the parameters 5 0, 6 0, (G2+ P)1/2 1, and the steady periodic response is exactly in phase with the area cycling, the adsorbed surfactant behaves in this respect as if it were in an insoluble monolayer. If K[ >> 1 and 5 5 1, (G2 112)1/2 0.51Kt; the steady periodic response is inversely proportional to K t and can become vanishingly small. The surface tension is then "buffered" to constancy by the bathing aqueous solution, and the system cannot stabilize interconnected bubbles. If K t >> 1, 4 tan-l(0.3157/0.4002) = 38.3" independent of K f . In the transient, for parameters the given, Tj = 0.0502rj-2 and wt/Tj = 31.25rj2t. Inspection of the functionof u resulting from F(s)(s2+ w2)-' in eq 15b shows that the aggregate of terms other than that multiplied by K t is positive for u < '/2?r, and therefore the first three roots r3, r4, and r5 must lie in the first three ranges of u in which tan 7 . 5 ~is negative. For K[ = 1, = 0.01 these are 0.2552, 0.6496, and 1.0640, and the first three terms following the summation symbol in eq 16a are (71.4e-2.10f+ 3.8e-13,2t+ 0.47e-31.3t)X For fixed K t = 1 these terms would change very little if 5 were increased from 0.01 to 0.10, The transient with these parameter choices is hence rather small even initially; its decay is quite rapid compared to
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Langmuir, Vol. 7, No. 12,1991 3075
Stability of Interconnected Bubble Networks the cycling period. While this result is based on a model in which a film initially at rest at t = 0 suddenly is subjected to a periodic area variation so that its area A = A d 1 + b sin ut), comparable transient decay behavior is to be expected if the film, having reached steady periodic behavior with the area variation parameters b and w, had these changed to b’ and w’. This aspect of the model simulates a transition in respiration parameters such as might accompany a change from rest to exercise. To simulate the steady periodic response in respiration it is clear that the product K[ cannot be large compared to one-else the amplitude of ACTwill be negligible, and an interconnected bubble system based on such a system, unstable-nor small compared to one-else the surfactant might as well behave as an insoluble monolayer, and this does not appear to be the case in respiration. Hence for proper respiration simulation, we must have K [ 1, log K [ = 5.933 - 0.6021n + log K = 0. We expect K >> 1; for K = 10 or 100, one would need n = 11 to 12 or 13 to achieve K [ 1. For K[ = 1and assuming K >> 1, use of G = 1.4002 and H = 0.3157 in eq 16a leads to a steady periodic response AgSp= 91.7b[sin (ut + 4)] dyn/cm with 4 = 12.7O. For b = 0.1 corresponding to a 10% amplitude in area variation there would hence be a 9 dyn/cm amplitude in surface tension variation (Le., 18 dyn/cm between maximum and minimum). The phase shift 4 is such that the surface tension is higher during expansion over most of the expansion, lower during compression than would be the case if 4 were zero so that surface tension and area variations were exactly in phase. The model with K [ = 1hence reasonably simulates the magnitude and phase of steady periodic response of surface tension to a periodic variation in area and the transient associated with attainment of steady response, which appear to be characteristic of human respiration. The CHS(CH2),-1OH alcohols that might be expected to satisfy K[ 1depend of course on the estimate of K , but values of n in the range 11to 14appear to be reasonable estimates. These involve what appear to be quite reasonable extrapolations from Hommelen’s14data, which covered 6 I n 5 10. E. Some Caveats. 1. The conservation equations for film volume and surfactant given prior to eq 9 neglect volume and surfactant contents of Plateau borders. In soap films supported on wire frames and at film junctions in polyhedral foams, these importantly affect film drainage rates. Such phenomena have been discussed at length by
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Mysels, Shinoda, and Franke13 and some rate estimates have been given by Hansen and Derderian.18 The border characteristics depend both on film and support; if, as appears to be the case in Figure 1, the film meets the support undistorted and with a 90’ contact angle, there would appear to be no effect on the conservation equations. If the film spreads on the support, a border suction would result. 2. Evaporation of the solvent (e.g., water) has been neglected. If the gas phase is supposed saturated with solvent vapor, solvent transport to or from the film should be small. 3. The conservation equations and the transport problem under Theory, section D, are set up for a planar film, whereas the bubbles used to simulate the alveoli are supposed spherical. As the total film thickness is assumed 8 pm and the bubble radius 136 pm, if the latter is taken to the film center, a variation in outer film area will be 3 5% greater, in inner film area 3% less, than the mean area variation. Since the last is supposed small in first order compared to total area, these differences can be neglected in the present case; Markin and Volkovlghave presented a general approach to the treatment of spherical interfaces.
Acknowledgment. J. Adin Mann, Jr., and J. Lucassen s h q e d my first investigations of capillary waves and associated dynamical surface properties, and I have benefitted from many subsequent discussions of these and related topics with them. K. J. Mysels, both through his published work and many discussions, has greatly increased my understanding of soap films. Years of stimulating interactions with A. W. Adamson on surface chemistry in general, including dynamical surface properties, are gratefully acknowledged. R. J. Engen, Professor of Veterinary Physiology and Pharmacology at Iowa State University, first introduced me to the critical role of the lung surfactant in respiration and the unusual character of 7r-A plots of this material. The Ames Laboratory-DOE is operated for the U.S. Department of Energy by Iowa State University under Contract No. W-7405-Eng-82.This research was supported by the Office of Basic Energy Sciences, Materials Sciences Division. (18) Hansen, R. S.;Derderian, E. J. In Foams; Akers, R. J., Ed.; Academic Press: New York, 1976; pp 1-16. (19) Markin, V. S.;Volkov, A. G. J. Colloid Interface Sci. 1990,135, 553.
