A Hydrodynamic Mechanism for the Coalescence of Liquid Drops. I. Theory of Coalescence at a Planar Interface Sidney B, Langl and C. R. Wilke Department of Chemical Engineering and Lawreme Radiation Laboratory, University o j California, Berkeley, Calif.
The coalescence of a liquid drop a t a planar interface was studied theoretically. The mechanism of coalescence was found to occur in two parts. In the first part (film thinning) a thin spherical shell of material between the drop and the interface i s slowly squeezed out under the combined action of surface and gravity forces. When the film becomes sufficiently thin, the second part (film rupture) of the mechanism occurs. Because a denser liquid always overlies a less dense liquid a t one of the interfaces of the film, the interface i s inherently unstable with respect to long-wavelength disturbances (Taylor instability). If such a disturbance i s introduced into the proper interface, the disturbance will grow exponentially in time until the film disintegrates, causing coalescence of the drop. A sufficiently intense disturbance of any wavelength can also rupture the film, causing coalescence. A maior source of disturbances i s sonic noise. The effects of drop radius and the physical properties of the liquids are discussed.
T h e iiidustrial physical scientist or engineer is frequently faced with the problem of either separatiiig n liquid dispersion into its coinpoiieiits or making such a dispersion stable for a long peroid of time. =i major factor in the study of these proliems is the rate of coalescence of drops of t'he dispersed phase. Although much iiiformatioii has been obtained on coalescence by studies of t,he rate of separat,ion of a dispersii)ii>a niore complete understandiiig of coalescelice requires : I tiet:iiled aiialj-sis of a simple system. Several workers have studied the coalescence of a drop a t a flat interface, but 110 coinplete theory has been proposed. It was the goal of this work t o provide such :I theory aiid to test' it experimeiitally. Kheii a, drop (phase 1 ) of :I niore dense liquid falls through i~ less dciire fluid (ph 2 ) to the interface between the phase-1 aiitl l)h:ise-2 liquitls, the drop often reim ins a t the interface a period of time before it coalesces wit'h the bulk phase-1 material. This time period has been called the coalescence tinir, drop rest time, or drop lifetime. Some of the earliest, iiivestigations of this pheiionienon were made by Reynolds (1881) aiid Mahajan (1934). Cockbaiii aiid NcRoberts (1953) noted that'the rest' times of drops were not constant, but that. if a sufficient iiuniber of drops was examined! a rcl)roducitjle rest-time distributioii curve could be olitiiiiied. Studies have been carried out on the effect of adclitioii of salt.; (Elton aiid Picknett, 1957; Picknett, 1957) mid eniulsifying agents (Gillespie and Itideal, 1956; Glass, et al., 1970; Hoclgboii and Lee, 1969; Hodgson and JVoods, 1969; Sielsen, et al., 1958; Katanabe, et al., 1958) and theoretical aiid esperinieiit:d iiivestigations of the effect of drop size, temi)erat,ure,electric fields, ionizing radiation, and other vuriul)les have herii performed (Allan, et al., 1961; hllan and l I : ~ s o n ,1961 ; Brown aiid Hansoii, 1967, 1965; Charles, 1959; Charles and l I ~ i s o n 1960:i, , 196011; Chester, 1965; Hartland, Present nddreis, 1)epartmerit of Chemical Engirieering, 1IcGill University, 3Ioiitreal 110, Quebec., Canada. T o whom i~orrespoiidericeshould be sent.
1967; Jeffregs and Hawksley, 1965; MacKay and hIasoii, 1963). It was inimediately recogiiized from the great variety, and occasional lack of coiiristeiicy, iii the theories described in the literature that a h i c unclerstantling of the coalescelice process could be obtained only by study of L: relatively simple tem. Aiccordiiigly,the scope of this investigatioii is governed by the following considerations: coalescence would be at flat interfaces only; only the first stage of coalescence would be examined; t'he liquid systems considered wvonltl iict contain a third component, a restriction that was later relaxed to allow the study of the effect of minute traces of :L coiitaminating material; an esplanatioii of the random nature of coalescence times was of major importance; and the effect of drop size and system propert'ies on droli rest times should be ascert'ained. I n this paper, a theoretical description of the co''I1escence process for a single drop is given. The experirneiital studies and a discussion of the application of t,he theory to the experiniental are presented in another paper (Lang and Wilke, 1971). ;1form for the presentatioii of coalescence data has been :idopted from the literature ( C h c i r l ~1959; ~ Picknett, 1957). plot of the fraction of drops coalesced within a certain time t , 2's. time t ! corresponds t o a statistical tlistrihutioii curve. Accordingly, such a graph will be called :i rest-time distribution curve. The coalescence measurements used i n making such a plot are taken one a t a time. d convention has been adopted for identification of the drop and continuous phases. The drop phase is always called 1, mhet'her it is niore or less dense thnii the continuous phase. The continuous phase is always called phase 2 in the discussion. I n a liquid system name, the first fluid is phase 1 and the second fluid is phase 2 . For esninl)le, in the Jwter-benzene sy:.tem, water is phase 1 (the drop phase) and lmizene is phase 2 (the continuous phase). Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
329
Model Outline
The coalescence of a sirtgle drop was observed to proceed in the following manner. T h e drop was released from the dropping tip and almost imme:diately reached a motionless quasiequilibrium position a t the interface. The drop remained almost spherical and WE1s apparently separated from the remainder of the phase-I liquid by a thin film of phase-2 material. After a period o E time the drop coalesced, generally leaving a smaller daughtt ?r drop at the interface. These observations, the randomne:3s of coalescence times, and several studies on the instabilit y of certain liquid configurations suggested the theoretical model here. The model consists of two successive steps: a film-thinning mechanism and a film-rupture mechanism. Either one of these steps may he rate-determining in a coallescence process. The rate of decrease in thickness of the phase-2 film as a function of the physical properties of the system and the drop size can be calculstted from the thinning-film step. The derivation requires several assumptions concerning the geometry of the drop and interface. A number of models have been proposed in the pas,t for the shape of a drop and the phase4 film as the drosp approaches the interface. The models include a solid eiphere and a sphere with a diskshaped flattened area approaching a rigid interface (Charles, 1959; Picknett, 1957); a fiolid sphere approaching a deformable interface (Hartland, 1968), and a deformable drop approaching a deformahle interface (Burrill and Woods, 1969; Chappelear, 1961; Frankel and Mysels, 1962; Hartland, 1967; Jeffreys and Hawk;sley, 1965; Princen, 1963; Princen and Mason, 1965). Some > h’? and if a time scale is chosen such t h a t tl = 0, then eq 2 becomes the simpler form
where Q(P) =
1
sin2p(1 +
‘/2
cos p sin2 p
+ cos p )
(3) Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
33 1
Figure 4. Exaggerated view of thinning-film Film thicknesses at three values of t are shown
model.
Xow by use of either eq 2 or 3 and eq 1, we can determine the film thickness a t any time from a knowledge of the drop radius and the physical properties of the system. Lang and Kilke (1971) describe coalescence measurements on the five systems water-benzene, tributyl phosphatewater, xater-anisole, ethylene glycol-benzene, and waterAroclor 1248. Because these systems exhibit a large range of
Figure 5.
Dimensionless film thickness, H 2 = h’2/a Calculated dimensionless film thickness H2 as a function of time tz for five liquid-liquid systems
Curve la lb 2a 2b 3a 3b 4a 4b 5a 5b
System Hz0-benzene HzO-benzene TBP-H20 TBP-Hz0 HpO-Aroclor HnO-Aroclor HzO-anisole HsO-anisole Ethylene glycol-benzene Ethylene glycol-benzene 1
332
relevant physical properties, the theories developed in this paper are illustrated numerically using the properties of these systems. Plots of film thickness as a function of time for the five systems and several drop sizes are given in Figure 5. Ewers and Sutherland (1952) have stated thtt the thickness of a soap film would have to be less than 50 A for it to be ruptured through random molecular motion. We note from Figure 5 that, with the possible exception of the very smallest drops, the times needed for these films t o decrease in thickness to 50 A are several orders of magnitude greater than the experimental rest times of 1 to 100 sec (Lang and Wilke, 1971). We can, therefore, conclude that a n additional mechanism other than the squeezing out of a thin film by surface and gravity forces is needed to explain drop coalescence. This additional mechanism is proposed in the next section. Electrical Effects and Disjoining Pressure. Several additional factors might modify the results in eq 3 for very small values of h’. T h e electroviscous effect might retard the flow of phase-2 material from within t h e spherical shell and thus diminish the rate of thinning of t h e shell. Calculations b y Elton (1948a, 194813) indicate t h a t this effect is not of great importance until the distance of separation is less than cm. Calculated film thicknesses a t coalescence (Lang and Wilke, 1971) were usually greater than cm for larger drops (greater than 0.1 cm) and thus electroviscosity probably had a negligible effect. However, in the case of Picknett’s experimentation with the wateranisole system, the drop sizes were very small, and calculated viscosities based on Reynolds’ thinning model showed the predicted effect of electroviscosity (Picknett, 1957). Further, Picknett also observed a decrease in drop rest times with
Ind. Eng. Chern. Fundam., Vol. 10, No. 3, 1971
a, cm 0.5 0.1 0.5 0.1 0.5 0.1 0.5 0.1 0.5 0.1
P, deg
47.5 8.8 34.7 6.6 86.0 15.6 13.8 2.7 121.2 26.5
increasing concentration of electrolyte in the aqueous phase. According to Elton’s result, the effective viscosity of an electrolyte solution decreases with increasing concentration of electrolyte. Pickiiett’s coalesceiice data again seem to confirm the presence of an electroviscous effect. When a fluid is squeezed out between two bodies, a resistiiig force can be measured which is not due to the viscosity of the fluid. This force is a measure of the long-range attraction of the molecules of the two bodies on the molecules of the fluid. It is manifested only wheii the fluid layer is very thin. Derjaguiii arid Kussakov (1939a, 1939b) have termed it the disjoiiiiiig pressure and have measured it,s magnitude in a number of systems as a function of t,he film thickiiess. They found a typical value of the disjoining pressure was 500 dyii/cm2 for a film thickness of 10-5 cm and much less for thicker films. Thus, the disjoining pressure would have little effect on drop rest times unless the film thickness was esceedingly small. If t,he diffuse elect,ric double layers adjacent to two iiiterfaces overlap, the interfaces can be expected to repel one another. The magnitude of this repulsion increases with increasing potential drop across each double layer and will increase with decreasing film thickness. llerjaguin and Kussakov (1939b) state that for aqueous films, the effective double-layer thickness is 10-4 to 10-5 cin for pure n-ater and less for electrolyte solutions. This effect is probably of lesser importance in coalescelice measurements in pure systems but may be extremely important when surface-active agents or electrolyt,es are present8.The electric-double-layer effect will be discussed further (Laiig aiid Kilke, 1971). Other Models, Drop Nonsphericity, and Film Nonuniformity. Ainumber of simple uniform filni models are described by Chappelear (1961) and N a c K a y and Mason (1963). 111 all cases, the time required for the film to thin t o a thickness 1 ~ is’ proportional ~ t o h’?-2, a result identical with t’he one given by eq 3. This proportioiialitp is a direct consequence of t h e parabolic velocity prnfilep used in all of these models. However, the dependeiice of the filmthinning rate oii the physical properties of t’he liquids and oil the drop diameter is differeiit in the model propo~edhere. The difference results from the fact that the spherical-zone contact area between the liquid drop and the film is replaced with aii equivaleiit planar area in each of the cases listed by Chappelear and RIacKay aiid Ilason. The model developed here is more realistic because the true spherical geometry is not distorted iii order to solve the hydrodynamics problem. Recent experimental and theoretical studies have showi that drops a t interfaces are oblate spheroidal in shape rather than spherical (Burrill and Il-oods, 1969; Hartland, 1967, 1969; Priiiceii, 1963; Princeii and Masoil, 1965). H o w v e r , the contact area between the drop and the film still retains the shape of a spherical zone. 13ecauee of the nonlinearity of the differential equations describing the drop shape in these models, it is not iiossible to determine the radius or area of the spherical zone analytically. Thus no aiialj-tic nieans exists for determining the rate of thinning of the film beneath a noiispherical drop. However, because the contact area i.: spherical, it s e e m reasonable that our eq 1 aiid 3 should be good approximations to the true situatioii. It has also been established both experimeiitally and theoretically that the phase-:! film is, in general, nonuniform i i i thickness and often has a “dimpled” forii: with tlie thiiine.st region located a t a distance from its vert:cal axis (Burrill and T o o d s , 1969; Frankel and l I y s & ~1962; Hartland, 1967 ; Princeii, 1963; Priiiceii aiid Mason, 1965). It is extremely difficult to compute t’herate of film thiniiiiig in this geometry,
although some calculations by Frankel and LIysels (1962) indicate that the rate of thiiining of the thinnest region is approximately the same as the rate of approach of a sphere with a flattened area to a plane (MacKay and Mason, 1963). Reasonable agreement was found between experimental average film thicknesses measured by Hartland (1967) and values calculated by use of eq 1 and 3, despite the fact that the film was severely dimpled. The experimental thicknesses were usually less thaii the calculated values. The discrepancy resulted because bhe drop was tilted, enabling tlie film to drain inore rapidly. The possibility of drop tilt was first suggested by Jeffreys aiid Hawksley (1965). Tilt, which can only occur with nonspherical drops, is probably dependent’ on the previous history of t’he drop and is not likely to be amenable to theoretical treatment. 111 conclusion, the model proposed here provides a good approximation to the true thinning rate even t,hough the geometry is idealized. The model is useful iii the interpretation of experimental data (Lang aiid Kilke, 1971). Film Instability
The mechanism that causes the rupture of the phase-2 film, aiid t,hus coalescence, is discussed in this section. R e assume that a disturbance having a n angular frequency, q, generat,es a siiiusoidal wave of waveleiigth, X, and wave number, k = 2?r/X, in the interface between the bulk phase-1 and phase-2 fluids. This wave is propagated into the lower interface of the phase-2 film. By a mechanism discussed below, a wave will also be formed in t’heupper iiiterface of the phase-2 film. If the amplitudes of either or both waves become equal to the film thickness in any region the film will be ruptured, creating a “hole” through which the drop material can flow into the bulk phase-1 material causing drop coalesceiice. Two types of film rupture are consider~dhere. The first type is due to a Taylor instability, originally discussed by Taylor (1950) for a n ideal fluid and checked experimentally by Lewis (1950). Dellman aiid Peiiiiirigtoii (1954) extended the theory to iiiclude the effects of surface tension and viscosity. The secoiid type of film rupture coiisiders the effect of large disturbances on a film. Taylor Instability. Inviscid Solution. T h e Taylor instability will now be used t o explain why t h e thin film of phase-2 liquid described above disintegrates i n much less time t h a t t,hat calculated from the drop-approach model alone. T h e analysis is based on the following major assumptions. (a) T h e fluids behave in a lionviscous nianner and their st,ate of motion can be described by a velocity potential. T h e inriscid restriction will later be relaxed. T h e fluids are incompressible. (b) T h e film of phase-2 liquid is assumed planar aiid horizontal. This restriction is necessary in order that a n aiialytic solution can be found. The problem can be solved numerically without this assumption but no major changes result. (c) For simplicity, wave propagation in a single dimension is considered. As shown by Lamb (1945) in related wave propagation problems, this assumption is not’ a very restrictive one, and relaxation of it would have an insignificant effect on the results. (d) Init,ially, both t’he upper and lower interfaces are assumed infinite iii ext’eiit. As a result, the drop radius does not’ appear in the derivat’ion. The effect of fiiiite extent’ of the upper interface is considered in a later subsection. (e) Only iiifiiiitesimal disturbances are considered. This restriction is caused by the mathematical difficulties of treating a more general, noiiliiiear case. (f) *Ill velocities are very small. -4ssume that a layer of fluid density, p l , and thickness, 2h, Ind. Eng. Chem. Fundom., Vol. 10, No. 3, 1971
333
f
The velocity potentials in the three phases may be related to one another with the equation of motion (Lamb, 1945). Because of assumptions (a) and (f), the equation becomes
P_ - - a@ p dt - gz
P
p3
Figure 6. Sketch of geometry of flat-film model with Taylor instability. Waves shown correspond to eq 4 at
(7)
Let us first look at the case of no interfacial tension. Then the dynamic boundary condition requires t h a t there be no discontinuity in pressure across the interfaces. If eq 7 is written for each phase and the pressures are equated at the interfaces, we can derive a n expression for the ratio of the amplitudes of the waves at the upper and lower interfaces, y = a/p.Assuming for simplicity that p1 = p 3 (which is the physical situation in coalescence), the following equation for the growth constant, q, results
t = O
lies between two semi-infinite (in the z direction) layers of fluid, the upper fluid with a density p i ; the lower one a density p 3 . At a time t = 0 the interfaces are perturbed in a n arbitrary manner. The form of the interfaces can be expressed at a n y instant later, not violating assumption (e), by 2 decomposition of the disturbances into a Fourier series. Because only linear operations mill be performed, we can solve this problem in terms of a single Fourier component and then generalize the result. Therefore, let the surfaces be represented by the standing wave equations Th
=
(Yeqt
COS
kX
+h
From this result, we see t h a t there are not just two but four growth constants. The two imaginary values of q correspond to waves traveling in the positive and negative x directions, the negative real value corresponds to damping, and the positive real value represents instability. Instability, as determined by the presence of a real, positive growth constant exists regardless of the sign of the density difference term. The situation can be clarified by a n examination of the equations when h is very large; then we find y =
-
P1
- sinh 2hk
-
kdpi
P2
- ”)
Pd
sinh 2hk
- cosh 2hk (9)
and
and 7-h =
-peqt
COS kX
-h
(4)
Heie, the amplitudes corresponding to a wave number k are (Y and -p. The growth constant q is the quantity of interest. If q is real and positive, the disturbance will grow in time; if it is real and negative, the disturbance will be damped out; and if it is imaginary, the disturbance will remain the same in magnitude, but the surface will oscillate in a periodic manner. If q is imaginary, it is the angular frequency of the waves and q = C,k, where C, is the phase velocity of the waves. The geometry is depicted schematically in Figure 6. The goal of this computation is as follons. If one or both of the waves grow exponentially in time, they mill quickly achieve such an amplitude that the film will be ruptured. It is not necessary for the film to become thinner only under the action of ordinary surface and gravity forces, because a new mechanism for film rupture has been considered. The motion of each layer of fluid must obey the equation of continuity, which, b y assumption (a), is expressed by Laplace’s equation (Milne-Thomson, 1955).
where @ is the velocity potential. The velocity potentials in the three phases may be found from eq 5 , the two boundary conditions that the velocity potentials of the two semi-infinite masses must vanish a t i a , and the four boundary conditions contained in the kinematic condition
334 Ind.
Eng. Chem. Fundam., Vol. 10, No.
3, 1971
Suppose p 1 is greater than p 2 ; from physical intuition we would expect waves a t the upper interface to be unstable (because a denser fluid overlies a less dense one) and a t the lower one to be stable. Substitute the positive value of q 2 from eq 10 into eq 9. We then find that y =
- (2
E+
1) sinh 2hk
- cosh 2hk
(11)
which has a very large absolute value. Thus, the unstable wave is of much greater initial amplitude a t the upper interface than a t the lower, as expected. If we substitute the negative value of q2 into eq 9, we find y = sinh 2hk
- cosh 2hk
(12)
which approaches zero as h becomes large. This indicates t h a t the stable wave exists predominantly a t the lower interface, again confirming our intuition. Suppose now that h is very small. Then y (see Lang, 1962a, 1962b) approaches -1, and we see that both the stable and the unstable waves must have equal initial amplitude. Before continuing, we can summarize two important conclusions. First, in a system containing two free interfaces, two wave systems exist corresponding to a single wave number. If the interfaces are far apart, the wave systems act independently of one another, each in a separate interface. If the two interfaces are close together, however, both wave systems exist in each interface, the relative initial amplitudes of the waves in the upper and lower interfaces depending upon y. Second, if a denser phase overlies a less dense phase in the layered system, the system will always be in-
:.I
where
RZ =
[2g(p1 - PZ)(PI sinh 2hk
e = g(pl -
2
d486S q1.874
I b
0.5
1.5
1.0
-
PZ)
-
(15) uk2
+
C
0
pZ)
-
$ = -g(p1
- PZ cosh 2hk)I2 + 40$pz2
20
02
dz
When p1 > p 2 , a selection of before the term gives the amplitude ratio for the stable wave (which is predominant a t the lower interface) and - gives the amplitude ratio for the unstable wave (which predominates a t the upper interface). We have reached an additional interesting conclusion, in showing that the presence of a nonzero interfacial tension has placed an upper limit upon the value of the wave number k t h a t will cause a n instability to develop. I t is obvious that the real positive value of y must reach a maximum between k = 0 and k = [ ( g / u ) ( p l - PZ)]~". Graphs of q us. k for three different film thicknesses and the five systems studied by Lang and Wilke (1971) are presented in Figure 7. Values of y for the water-benzene system are shown in Figure 8. Note that q increases as 2h increases, and that the y corresponding to the unstable wave increases as 2h increases. Let us now apply this discussion to the coalescence of a liquid drop. For an instability to develop, disturbances create waves in the interface between the bulk phases. These waves will be stable because here pz is above. This wave will be propagated through the phase-2 film to the upper (or drop) interface. The efficiency of propagation y decreases as 2hk becomes smaller; i.e., for a given value of film thickness,
1 I
0.I
Wave number kkm-') Figure 7. Growth factor as a function of film thickness and wave number for five liquid-liquid systems. The three film thicknesses are: curve a, 10-1 cm; curve b, cm; curve c, 10-5 cm. The minimum wave numbers which can affect drops of radii 0.5 and 1.0 cm are shown. The radius of the smallest drop which is subject to a Taylor instability is given for each system
1 1
herently unstable. The growth constant of the instability increases without limit as the wave number increases. Let us now see how these results are modified by the inclusion of interfacial tension. The Young-Laplace equation (Adamson, 1960) shows that the difference in pressure on the two sides of a n interface is not zero, but is equal to the interfacial tension times the sum of the reciprocals of the radii of curvature of the interface. Xoting assumption (e) and again letting pl = p 3 , we can derive a n expression for p2 q2 =
- (uk3pl + uk3p2coth 2hk) p12
+ PZ' +
2 ~ 1 coth ~ 2
i
6
2hk
(13)
where R1
=
- I)] + k2[g2(pl - P Z ) ~ ( P ~ '
ka[u'p22(Coth22hk
+ + 2p1pz coth 2hk)l ~ 2 '
It can be easily shown from eq 13 that positive values of q2 (regions of instability) only exist when the wave number obeys the expression k
i [ g h - PZ)/Crl1'2
=
kr
(14)
Thus, we have derived an upper limit k T on the value of wave numbers t h a t will permit instability t o develop. The limit is based only on physical properties, not geometrical ones. The result is the same as that found by Bellman and Pennington (1984) for a single interface. The following expression can be derived for y
-3-
I
0
0.4
I
I
0.8 1.2 Wave number k (crn-l)
1
I.6
3
Figure 8. Amplitude ratio corresponding to the unstable disturbance as a function of wave number and film thickness, Physical properties used correspond to the waterbenzene system Ind. Eng. Chem. Fundarn., Vol. 10, No. 3, 1971
335
exponential term and viscous damping acts as a negative exponential, the value of the growth constant with viscous damping is given by Q o = q + T
-0 a, In
where T is the viscous damping constant. The method consists of equating the average rate of dissipation of energy due t o viscosity to the rate of change of the energy contained in the progressive surface waves. The amplitude of the waves then decreases proportionally t o the term exp[rt]. If we assume t h a t the motion is irrotational, the average rate of dissipation of energy is (Lamb, 1945)
-0.02-
Y
)I L
0
c 0
.I-
-0.03-
en
c ._ 0
where u = -V@ and b/bn represents differentiation with respect to the outward normal. The integration is performed over the bounding surfaces of the fluid mass and the bar indicates a time average. The total energy (both potential and kinetic) contained in the progressive waves is (Lamb, 1945)
E
0 -0
In
a
-0.04 -
0 In 0
.-
> -0.05-
I I
-o*06/
d.2 d.4
d.6 0.8 IlO 112 1l.4 Wave n u m b e r , k (ern-')
Il6
If we equate the average rate of dissipation of energy to the rate of change of the wave energy
118'
Figure 9. Viscous-damping factor as a function of wave number for various film thicknesses. Physical properties correspond to water-benzene system
we find that
the efficiency is loivest for a large wavelength (small k ) , and for a given wavelength, the efficiency decreases as the film becomes thinner. The rate a t which instability grows depends upoii both h and L, as can be seen in Figure 7 . Equations 13 and 15 will also be found useful in determining frequency of the disturbance as a function of wave -2k2(!y2 T
=
+
$2
f 2 sinh2 2hk ~~~
Pl(Y2
$1
[(-y2
= $3.
+ l)(sinh 4hk + 4-y sinh 2hk)l
+ 1) + 2 PZX k [(-y2 + l)(sinh 4hk + 47 sinh 2hk)l
number and the amplitude ratios in regions outside the instabilit'y limit of eq 14. Effect of Viscosity. T h e exact solution for t h e effect of viscosity oii the stability of a free surface entails the solution of the Navier-Stokes equations. Bellman and Peniiingtoii (1954) solved t h e problem for a single free surface and found that, it was only possible to find a n upper bound for t8hevalues of 9 2 without resorting to lengthy numerical techniques. They found that viscosit'y had no effect on t'he stahilit'y criterion but t,hat it did decrease the value of q 2 . A method has beeii developed by Stokes for computing bhe rate of viscous dampiiig of surface waves (Lamb, 1945). The method involves the very strong assumptions that the flow be irrotatioiial and that velocities be very small a t points distant from the interface. If, however, the viscosities of the two phases are sinal1 aiid of the same order of magnitude, t,his method produces results very similar to those obtained by means of the Yavier--Stokes equations, even though the assumptions are partially violated. Several situations have been examined to illustrate this point (Lang, 1962a, 1962b). Because an unstable wave grows in amplit,ude as a positive 336
The appropriate differentiations and integrations can be potentials. Then, assuming t h a t performed on the velocity the contributions of E , aiid E, in the interfaces are additive, the damping factor is found as in eq 20 where p l = p3 and
Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
A graph of values of T for various film thicknesses is given in Figure 9 using data for the water-benzene system. For low viscosity fluids, the correction to the growth factor is quite small. Even in the ethylene glycol-benzene system, in which one phase was relatively viscous (ethylene glycol viscosity, 16.0 cP), the correction to the maximum value of q was only 9%. Effect of Drop Radius. I t is now necessary to reconsider assumptions (b) and (d). It is apparent that, if t h e wavelength of the disturbance in t h e fluid is much smaller t h a n some characteristic dimension of t h e drop, the interfaces will behave as if infinite in extent and also as if planar. However, if t h e wavelength of the disturbance is much greater t h a n some characteristic dimension of the drop, me would expect the drop to be unaffected by the disturbance. Thus, as the wavelength diminishes in comparison t o the droll dimension, t h e conditions of assumptions (b) and (d) are met more closely. We now calculate the masimum wavelength which can affect a drop interface. Let us assume that the natural frequencies of vibration of the interfaces of the film of phase-2
liquid are the same as those of a membrane having the shape of a zone of a sphere. Then the lowest frequency which could affect the drop would correspond to the fundamental frequency of the spherical-zone membrane. Love (1888) showed t h a t the fundamental frequency could be found from the smallest eigenvalue of the equation P,(cos p )
=
0
(21)
where the wave number, k , is given by
k
=
[Y(Y
=
=
Er
=
.[LA + (& [l
CY
cos kz)']
432.7 ~
+ EO + E, = 2(E, + E,)
dz - A]
=
'/4aa2k2Aper wavelength
where a is the drop diameter (cm) and I-, is the zone angle (degrees), deviates by a maximum of oiily 37&from the exact solution when p is as large as 90". All wave numbers equal t o or greater than k M shown in eq 22 would be expected to have a n effect 011 the interfaces of the phase-2 film. We now have both a lower limit (eq 22) and a n upper limit (eq 14) t o the wave numbers that can cause a Taylor instability. The lower wave number limits for several drop sizes and the minimum drop size for which a Taylor instability is possible are also shown in Figure 7. Large Amplitude Wave Instability. The Taylor instabili t y describes t h e mechanism b y which a n infinitesimal disturbance of t h e proper wavelength can grow t o an amplitude large enough t o rupture t h e phase-2 film. However, t h e Taylor instability is restricted t o a definite wave number range. A very intense disturbance having a n y wave number can generate waves sufficiently large in amplitude to directly rupture t h e phase-2 film. I n this section, calculations are made of t h e energy required to establish a large amplitude wave system. Only one exact solution to the equations of ideal fluid motion for arbitrarily large waves is known-Gerstner's trochoidal wave (Milne-Thomson, 1955). However, the trochoidal wave analysis is limited t o a single fluid surface above which is a vapor of negligible density. Therefore, the following analysis will be confined to sinusoidal waves. If the wave amplitude is much smaller than the wavelength, the sinusoidal wave will be a good approximation to the exact trochoidal wave. First, let us calculate the energy contained in a wave system at a single interface. The total energy of a progressive wave is made up of three contributions: the kinetic energy, the potential energy due t o gravity, and the potential energy due t o surface forces. The kinetic energy is also equal to the potential energy (Lamb, 1945). 'rhus we have
E
E,
+ 1)1"2
Here P,(cos p ) is the Legendre function of noniuteger order Y and p is the zone angle shown in Figure 2. Values of k as a function of p have been found numerically. By observing t h a t a spherical zone membrane can be approximated by a flat circular membrane when p is small, it is possible t o find a n asymptotic solution t o the problem of finding the smallest eigenvalue of eq 21. The asymptotic solution k,
respectively. Here p L and p3 represent the densities of the lower and upper fluids, respectively, a is the amplitude of the waves, and the energies are given per square centimeter of surface. The potential energy due to surface forces is found by multiplying the interfacial tension by the increase in interfacial area due to the waves
=
2Er
(23)
The kinetic energy and potential energy due to gravity are given by
(26)
or '/4aa2k2per unit area. From eq 23, 25, and 26, the total energy is then
E
= l/za'[g(pL
-
PU)
+ bk21
(27)
A stationary or standing wave would have the same total energy (Lamb, 1945). According to eq 12, in a two-interface system which is stable with respect to a Taylor instability, there are two systems of waves having different angular frequencies corresponding to a single wave number. Each wave system consists of a large amplit'ude wave a t one interface and a small amplitude wave a t the other interface. The two amplitudes are related according to eq 15. R e designate the wave having its largest amplitude e a t the upper interfa subscript Cr, and the wave m with its largest amplitude wave a t the lower interface by a subscript L . Equation 27 can be written for each of the two waves of each system. The sum of the energies of the two waves gives the energy for the system. The energies contained in t'he systems I - and L , respectively, are Eo and EL, and
.
EL =
'/d3L2[g(pi
- pz)(l
- YL')
+ gk'(1 + Y L ~ ) I
(29)
Here the amplitudes of the major wave a t the upper interface and the major wave a t the lower interface are au a n d OL, respectively. The amplitude ratios, yu and y ~are , calculated from eq 15, using - and respectively, for the signs of the 4%term. Let us assume t h a t the total energy propagated into a film is equally distributed between the two wave systems, i.e., EU = EL. I3y some algebraic manipulation of eq 28 and 29, it can be shown t h a t the largest wave that can be formed in a two-interface system will have an amplitude a ~if ,the wave number is greater than k~ (expressed by eq 14). It is now possible to calculate from eq 28 the total wave energy required so t h a t the largest of the four waves has a magnitude equal t o the thickness of the phase-2 film, t h a t is, the energy necessary t o rupture the film. The total energy necessary for film rupture as function of wave number and film thickness for the five systems discussed by Lang and Wilke (1971) is shown in Figure 10.
+,
Effect and Analysis of Disturbances
The disturbances discussed above can arise in a number of ways, i.e., through fluctuating fluid pressures caused b y sonic and subsonic noise, through fluid motion caused by thermal or velocity gradients, through interfacial turbulence caused b y chemical potential gradients, or even by fluid motion caused by molecular-scale disturbances. I n general, the most important disturbance is t h a t caused b y soilic nuise. Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
337
phases because of the difference in momenta of the sound beams on opposite sides of the interface. Surface waves can be imposed upon the interface and the pressure will do work on the fluid particles composing these waves. The waves will then propagate along the interface away from the point of incidence of the sound beam. The rate a t which energy is carried away by the waves is determined by the group velocity of the waves. By such a n argument, we could write a n energy balance around the point of incidence of the sound beam which would enable us to compute the wave amplitudes. A knowledge of the intensity of the sound, its angle of incidence, its frequency, and the physical properties of the two liquids is needed. Sufficient information for our purposes can be obtained without a detailed solution of the problem, however. We first examine the relationship between the frequency of the disturbance and the length of the waves a t the interface. Equations relating frequency (v), wavelength (A), wave number ( k ) , and phase velocity (C,) to the liquid physical properties are given by Lamb (1945)
N
E
- Ethylene glycol-Benzene
4P
0, v
c
N ,I
ln ln 0, 1 0 .-
f
L
0
-E ._ L ? c
cl
2 0
c
h
P
W
y 2 = -
w
1 (2T)Z
0
40
20
60
80
100
Wave number, k=2n/X(cm-~) Figure 10. Total energy per unit area necessary for film rupture as a function of wave number and phase-2 film thickness for five liquid-liquid systems
The following description gives a possible way in which a disturbance might arise through a sonic noise. Suppose a plane wave of sound impinges upon the interface between two liquid phases (Landau and Lifshitz, 1959). A portion of the sound beam will be reflected back into the first phase and another portion will be refracted because of the change in acoustic impedance between the media and will pass on into the second phase. A pressure will be exerted on the interface between the
-
LP1+
P1 PZ pz
gk
+
P1
+ PZ k j l
A graph of wave number as a function of frequency is given in Figure 11. Each frequency produces a wave having a wave number either less than g(pl - p ~ ) / or v greater. (This parameter is the same as the Taylor instability criterion in eq 14.) The waves with small wave numbers (long wavelengths) are called gravity waves; their behavior is principally a function of the density difference between the phases. The second type of wave is called a capillary wave and its properties depend upon the interfacial tension. Thus a n y disturbance creates one of two types of waves, one type of which can initiate a Taylor instability film rupture (provided that its wave number is greater than the limit described by eq 22), and the other which can cause a large amplitude wave instability (provided that its energy is sufficiently great). A relationship between the sound pressure and the amplitude of the waves generated a t the interface between the bulk liquid phases will now be derived. The energy of a progressive wave is given by eq 27. Energy is carried away from
100
T^ E
s x
1
IO
C 0
B
s
1 I
100
IO
Frequency U (cycles sac-')
Figure 11.
338 Ind.
Frequency as a function of wave number for waves traveling at a liquid-liquid interface
Eng. Chem. Fundam., Vol. 10, No. 3, 1971
(30)
the point of incidence of the sonic disturbance a t the group velocity (Lamb, 1945), which is equal to a constant times the phase velocity given in eq 30. The constant lies between '/z and 3/2, depending upon the wavelength. The energy available for producing the surface waves will be proportional to the energy of the sound wave which, in turn, is proportional to the square of the sound pressure (Landau a n d Lifshitz, 1959). We express the proportionality between the energy available for production of surface waves and the rate of transmission of energy along the surface. After some rearrangement we find
where P,,, is the root-mean-square sound pressure in dyn/ cm2. We can now state several qualitative conclusions. The amplitude of the waves produced is proportional to the sound pressure. If the frequency of the sound is low and the density difference between phases is small, the waves will be of large amplitude (gravity waves). If the surface tension is low and the sound is in the low sonic range, the amplitude of the waves will also be large (capillary waves). These conclusions are in agreement with the results obtained in the large amplitude wave analysis.
Table 1. Effect of System Properties on M o d e l Parameters
Parameter
tz
9 (Growth constant in Taylor instability model)
1
IT
U
a PZ
AP U
h k AP
(Absolute value of amplitude ratio, Taylor instability model)
I
U
h k PlJ P2
(Absolute value of visous damping ratio, Taylor instability model)
PI, PZ
h k AP
alJ
(Magnitude of large amplitude wave) a
Summary of Model
The model developed in this chapter will now be briefly summarized. A liquid drop falls off the dropping tip and almost instantaneously reaches a quasi-equilibrium position a t the interface. T h e force of gravity and the surface forces attempting t o restore the planar shape of the interface gradually squeeze the film of phase 2 out from beneath the drop against the viscous forces. If the film becomes much thinner than lop4 cm, electroviscous, disjoining-pressure, and doublelayer repulsion forces also help t o retard the thinning of the film. It can be seen t h a t the drop may not coalesce in tens of thousands of seconds, if film thinning is the only important effect. However, the upper surface of the phase-2 film is inherently unstable with respect to long-wavelength disturbances. It can also be ruptured b y almost a n y sufficiently intense disturbance, especially as it attenuates. As the film becomes thinner, the efficiency with which long-wavelengt'h disturbances can be propagated into the upper film surface decreases slight,ly. These low-frequency disturbances may be beats generated by interference between higher frequency sources, for example. A wave perturbed b y low-frequency disturbances will grow exponentially in time a t the upper interface until its amplitude is sufficient t o rupture the film and cause coalescence. An intense disturbance might rupture the film even though its wavelength is too short t o promote a n unstable situation. The shape of the drop rest-time distribution curve can now' be explained qualitatively. Unt'il the phase-2 film is sufficiently thin, no disturbance will be intense enough either to rupture the film directly or be propagated through to the upper surface of the film (if the disturbance has a sufficiently long wavelength to cause Taylor instability but is short enough t o affect the drop). This sets t'he minimum time of coalescence. When the film is thicker, the disturbances cause waves that grow much more rapidly (Figure lo), thus causing the median drop rest time t o be lower t h a n bhe average of the lowest and highest rest' times. The general shape of the distribution curve is due to the random arrival of disturbances. Table I contains a summary of the way in which the various
AP
(Time required for film to reach thickness of h'* in film-thinning model)
IY
Property
U
Eo Prma
(Amplitude of wave produced b y disturbance)
AP U
k
Sign of d(parameters)/ d(property)
Positive Negative Positive Positive Positive Negative Positive q = y(k) exhibits a maximum Positive Positive Positive Positive Negative Positive Positive Positive n'egative Positive Positive Negative Negative (Positive for gravity) (Negative for capillary waves)
parameters developed in the coalescence model vary with physical properties. The variation of film thickness h' with physical properties was determined from eq 3. It will be noted that, t o a good approximation, J ( p ) is inversely proportional t o & ( p ) . The behavior of 92, y, 7 , a C p ,and 01 were determined from eq 13, 15, 20, 28, and 31, respectively. We see t h a t to promote rapid coalescence by means of the Taylor instability, we require large values of y2, a , and y ( , a small value of 7 and a value of IC within the instability region described by eq 14 and 22. Small values of pi and ~2 and a large value of P,,, will assist in satisfying the h', a, and requirements, but we note t h a t , wit'h regard to A p and u , the requirements of the thinning model and the Taylor instability model are different. For rapid coalescence, a small A p and a large u are required in the thinning model, but the opposite is true in the Taylor instability problem. This paradox gives a theoretical explanation for the experimental difficult'ies in predicting the physical property dependence of drop-rest times. I n the case of t'he large amplitude wave instabilities, a rapidly decreasing value of h' and a large value of a are required for rapid coalescence. Again, a small value of 112 a n d a large value of P,,, will assist in the satisfaction of the h' and CY requirements, but where the thinning niodel requires a large value of u , the rapid rupture of the film is promoted b y a small value of U . A low value of A p will promote rapid coalescence by decreasing the value of h' for a given t , but it will decrease t'he value of auz.The validity of the model described here is examined in light of experimental coalescence data by Lang and Wilke (1971).
I 1,
1
IT I
Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
339
z
Acknowledgment
The authors are grateful for the helpful suggestions of Theodore Vermeulen and Peter TT’. AT. John. S. B. L. expresses his thanks to the University of California and to the National Science Foundation for financial help in the form of fellowships. This work mas partially performed under the auspices of the 17.S. Atomic Energy Commission.
c,
ds dv E
E, El,
E, E D
E, g
H h’ h J (PI k kM IZT
n
P
P P,,, Pu(c0s p ) Q Pu
Q (PI S
t W
X
Y 2
radius of sphere in film-thinning and spherical shell models, em = phase velocity of waves, cin/sec = differential surface element, em2 = differential volume element, em3 = wave total energy per unit wave length by unit width, ergs/cm2 = wave gravitational potential energy per unit wavelength by unit width, ergs//cm2 = wave kinetic energy per unit wavelength b y unit n-idt,h, ergs/cm2 = wave surface potential energy per unit wavelength by unit width, ergs/cm2 = average rate of energy dissipation due t o viscosity per unit wave length by unit IT-idt’h,ergs/cm2 sec = average energy content of surface waves per unit wavelengt’h by unit width, ergs/cm2 = acceleration due to gravity, cm/sec2 = dimensionless film thickness, H = h’/u = thickness of spherical shell, cni = half thickness of flat film, cm = functioii of p defined b y eq 2 = wave number, equals 2 ~ : / h , = minimuni wave number n-hich can affect, drop (eq 2 2 ) , em-‘ = wave number for Taylor instability criterion (ea 14), em-’ = o u h a r d normal to a surface = zone angle defined in Figure 2, degrees or radians = pressure, dyn/cm2 = root-mean-square sound pressure amplitude, d yn / c m2 = Legendre function of noninteger order v and argument cos p = growth constant in stability analysis when q is real; q = C,k when q is imaginary, see-’ = growth constant considering viscous damping, sec-’ = trigonometric function of p defined by eq 1 = dimension defined in Figure 2, em = time, see = dimension defined in Figure 2 , ern = Cartesian coordinate = Cartesian coordinate = Cartesian coordinate
i 0 V2
performed, em2 viscous damping constant, see-‘ velocity potential, crn2/sec defined by ic. = - g ( p l - P Z ) - ah2 nabla or del, vector operator Laplacian operator
be
SUBSCRIPTS I 2
=
wave system having wave predominant at upper or lower interface, respectively
=
GREEKLETTERS cy = amplitude of a wave a t the upper interface of tiflat film, em 3! = amplitude of a wave at the lolver interface of a flat film, ern = amplitude ratio, equals 01 p Y = elevation of upper and lower surfaces of a 11 flat film, cm e = defined by e = g(pi - p 2 ) - ak2 x = wavelength, cm = absolute viscosity of phases 1 and 2, re1 1 , P2 spectively, P V = frequency, HZ P 1 , P ? , p3, A p = densities of phases 1, 2 , and 3 and density difference, respectively, g/cm3 U = surface or interfacial tension, dyn/cm B = summation operator
__
340 Ind.
= = = = =
7
r,
Nomenclature
a
= surface area over which integration is to
Eng. Chem. Fundam., Vol. 10,
No. 3, 1971
literature Cited
Adamson, 4.W.,“Physical Chemistry of Surfaces,” Interscience Publishers, Inc., New York, N. Y., 1960, pp 4-6. Allan, R. S., Charles, G. E., Mason, S. G., J . Colloid Sei. 16, 150 11 961 ) \----,
Allan, R . S., Mason, S. G., Trans. Faraday SOC.57, 2027 (1961). Bellman, I
