Znd. Eng. Chem. Res. 1995,34, 3653-3661
3653
Effects of Surfactant on Multiple Stepwise Coalescence of Single Drops at Liquid-Liquid Interfaces Alex D. Nikolov and Darsh T.Wasan* Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Illinois 60616
The mechanisms of multiple stepwise drop-interface coalescence have been investigated for liquid-liquid systems both with and without surfactant. It was observed t h a t both the coalescence time and the number of coalescence steps depend on the initial drop size and the surfactant concentration. Without surfactant, the drop lifetime decreases as the drop size decreases while, in the presence of surfactant, the drop lifetime increases with decreasing drop size. These observations are explained on the basis of thin liquid film stability and film drainage mechanisms, and it is shown that drop lifetime versus size is governed by the thermodynamic stability of the film rather than film drainage and is dependent on film size and capillary pressure. Film stability is analyzed on the basis of both the concepts of mechanical perturbations and nucleation (hole formation). It is concluded t h a t the experimental data for film lifetime versus film size is in qualitative agreement only with the nucleation mechanism. The mechanism of secondary drop formation from the primary drop and its drainage is interpreted by the Rayleigh capillary instability mechanism, and the effect of interfacial tension on the size of the secondary drop is also analyzed.
Introduction A knowledge of basic drop coalescence phenomenon is important in understanding many 1iquiMiquid interfacial processes such as 1iquiMiquid extraction and phase separation, as in chemical engineering operations. The mechanism of drop coalescence may be considered to occur in several successive stages: (1) the drop approaching the interface, beginning of hydrodynamic interaction and dimple formation; (2) drainage of the dimple and formation of a film with uniform thickness; (3) change of the film thickness or film rupture; (4) complete drainage of the droplet into its homophase or partial drainage and formation of secondary drop(s). The experimental observations show that, without surfactant, the oil droplet formed in the water phase coalesces with the oil/water interface by multiple stepwise drop-size transitions. Such a phenomenon of multiple stepwise drop coalescence (also known as a partial coalescence) at a flat interface has been reported earlier by Mahajan (1930), Charles and Mason (1960a), Davies et al. (19701, and, more recently, by Leblanc et al. (1994) and Boulton-Stone (1994). According to our understanding, three parameters depict the partial coalescence process: number of stepwise coalescence, evolution of the drop size ratio (initial drop to secondary drop), and time interval for the occurrence of successive coalescence stages. Using a high-speed camera, Charles and Mason (1960a,b) monitored the partial coalescence. They observed that, after the rupture of the film, the drop begins to deflate and, due to the internal excess pressure, the drop drains first from the bottom and side and forms a cylinder (column). The height of the cylinder remains virtually constant while its radius decreases. When the circumference of the cylinder is less than or equal to its height, it becomes like an unstable jet in which a Rayleigh disturbance is developed, and when the amplitude becomes equal to the radius of the cylinder, it breaks and the undrained liquid forms a secondary drop. The process can be repeated
* Author to
whom correspondence should be addressed.
FAX: 3121567-3003.E-mail:
[email protected].
several times. On the basis of Tomotika’s theory of unstable jets, the initital to secondary drop size ratio and the conditions for formation of a secondary drop were analyzed. Charles and Mason (1960a) did not discuss the evolution of the drop size ratio versus successive stages of coalescence or the time interval for the occurrence of the successive coalescence stages. Recently, Leblanc et al. (1994)experimentally studied and simulated the deformation of the drop after the film ruptured. They explained, quantitatively, why the coalescence stages are finite and how secondary drop formation can be suppressed by adding a high concentration of surfactant. However the model failed to predict the size ratio. Furthermore, they did not analyze the time interval for the occurrence of the successive coalescence stages. Boulton-Stone (1994) analyzed the interfacial instability produced by a bursting gas bubble a t a surface in the presence of surfactant. In order to study this interfacial instability, a numerical model was used to calculate the shape of bursting gas bubbles a t a free surface and the effect of surface dilatational viscosity on the formation of a high-speed liquid jet. In particular, its shape was examined. They found that the surface shear viscosity had little effect on the surface evolution, but they did not consider the role of film stability. The lifetime of such a liquid film is determined by two processes-thinning (film drainage) and rupture. The film-thinning process is governed by the gravity and capillary pressure. In this paper we have explained the stability of a nonplane parallel (curved) film and the effect of surfactant concentrations on the stepwise thinning parameters: number of stepwise coalescence, drop size ratio, and the time interval for the occurrence of the successive coalescence stages. The stepwise dropleuflat interface coalescence process is monitored by a video camera from the top, and the drop and film size are measured. The fildmeniscus contact angle is calculated from the measurement of interference patterns. By using side video recording of the coalescence process, the main stages governing multiple stepwise coalescence are depicted.
0888-588519512634-3653$09.00/00 1995 American Chemical Society
3654 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
VCR
A
IMAGE ANALYZER
light source
Figure 1. Experimental setup for studying the dmplplane interface coalescence. The photograph depicta the size of primary, secondary, and tertiary toluene drops in the presence of m o m NaLS.
Experimental Study Drop Coalescence and Rest Time Measurements. The experiments were confined to the study of single drops at the waterhluene interface plane. The water used was purified by a Milli-Q water system (Millipore), and the toluene was spectroscopic grade (Alltech No. 166451). Observations were made of the system both with and without ionic surfactant sodium dodecyl sulfate (NaLS) BDH, specially purified product No. 4215 at moVL. The watedtoluene interface was purified several times by suction after preequilibration (see Aderangi and Wasan, 1995). A sketch of the drop coalescence experimental setup is shown in Figure 1. The drop coalescence study was conducted in a glass Pyrex cell which consisted of two parts: a 4 cm diameter glass cylinder with an optical plane parallel bottom and a plane parallel glass plate on top. When the coalescence of the water droplet a t the watedoil flat interface is studied, the inner part of the glass cylinder is covered by a very thin strip of Teflon tape. This causes the oiVwater interface to form a slightly convex shaped meniscus which forces the coalescing water droplets to stay in the center part of the interface. The radius of the convex meniscus is much larger than the radius of droplet, so one can assume that the convex meniscus is flat compared with the drop radius (e.g., see the photograph shown in Figure 5e). On the side wall of the glass cell, two syringes are affied as shown in Figure 1. By pushing the piston of the vertically oriented (to the glass cell wall) syringe
with the help of a fine, micrometric screw, a drop with a diameter of about 3 mm is formed on the tip of a tiny metal syringe needle. By pushing the piston of the second syringe (oriented horizontally to the cell wall), this drop is removed from the tip by the water flow and is placed by gravity a t the interface. Measurement of the FilmMeniscus Contact Angle. We used an Epival Interphako microscope in conjunction with a CCD camera and image analyzer to monitor the drophterface coalescence process. This microscope is capable of viewing an oil drop at a flat interface in transmitted as well in reflected light at the same time. A schematic of the experimental arrangement is given in Figure 2. The oil drop was created by the procedure described in the previous paragraph. When the drop approaches the oiVwater interface, the curved film is formed. The interference patterns (known as Newton rings) appear when the system is illuminated from above by monochromatic light. The Newton rings are due to differences in the optical paths, between the light beams, reflected by two film interfaces in the meniscus region. A typical photomicrograph is shown in the figure. By measurement the distance between the bright and dark Newton rings which appear at the point of constructive and destructive interference of the light beams reflected from the meniscus surface, the meniscus profile is obtained. The theoretical profile of the meniscus is obtained by integrating the Laplace equation using the Runge-Kutta method. When the distance between the two meniscus interfaces Ldr) and Lz(r) satisfy the condition for interference, Le., A =
Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 3655 VCR
rvedl I
I
I-
d
m
W Figure 2. Schematic of the experimental setup for contact angle interferometry. The minophotographdepiets the Newton interference patterns obtained from the film meniscus.
R
-
5
E E ,
DROP COALESCENCE
I
U W
t; E
-U0
3
0 1
1 2 3 4 5 6 INITIAL DROP ,SLCONDARY D R E LIFE TIME
_-
LIFE TIME
7
8
9
DROP LIFE TIME [ sec ]
Figure 3. Multiple stepwise coalescence data without surfactant.
iw/2, i = 0,1,2,..., A = 2(L1 - Ldk, and i is odd for the dark fringes and even for the bright ones, k is the water refractive index at the wavelength w = 546 nm. The position of the interference patterns in combination with the Laplace equation is used to obtain information on the profile of the film meniscus interfaces. The microscopic contact angle is, hy definition, subtended between the two extrapolated meniscus surfaces to zero film thickness. The description of this method for calculating the meniscus profile and the contact angle at the
fildmeniscus has been published elsewhere (Dimitrov et al., 1990; Lobo et al., 1990). To calculate the meniscus profile, the interfacial tension needs to be measured. The interfacial tension of water/toluene was measured with a Wilhelmy plate tensiometer (Kruss K 8). All measurements were performed at 20 "C. The interfacial tension for water/ toluene at 20 "C was 36.2f 0.15 mN/m. In the presence of moVL NaLS the measured interfacial tension was 34.1 f 0.1 mN/m.
3858 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
5
-
4
E E a 3
DROP COALESCENCE RATIO
+HIIuIL
r
DROP M E
=
R.
-L
i+1
ifl+l
2R 1
I
W
li
I 5 2 P
n
0
SECOND
CT
1
--
2
3
4
FOURTH DROP LIFE TIME
Figure 4. Multiple stepwise coalescence data (with
5
-
6
7
8
9
DROP LIFE TIME -I min 1
mol& NaLS).
Experimental Observations. We first studied the coalescence phenomenon of a toluene droplet a t the waterltoluene homophase without surfactant for a preequilibrated system, and the following phenomenon was observed. The toluene droplet arrives a t the water/ toluene interphase and rests for a second. During the rest time, a thin film forms, which thins and finally ruptures. The size of this arrived droplet is marked as initial drop size in Figure 3. This droplet begins to coalesce by regularly reducing its size in a stepwise manner and its sizes are marked as second, third, fourth, etc., up to six steps of coalescence. In Figure 3, the time reproducibility of the multiple stepwise dropsize coalescence process is shown in the form of a histogram. One can clearly see five coalescence steps before the droplet coalesces completely. For the system without surfactant, the time needed for coalescence to occur decreases by decreasing the drop size. It is important t o note that the reproducibility of drop diameter is better than 85%,when the scattering for the coalescence occurrencetime is much larger as shown in Figure 3, and increases with the decrease of the drop size. In the presence of m o m NaLS surfactant, we observed the same multiple stepwise drop coalescence behavior (see Figure 4), but this time the coalescing drop had to pass through four different sizes before coalescing completely. In the presence of surfactant, two major differences have been observed. First, the coalescence time in the presence of surfactant is an order of magnitude larger than without surfactant (see and compare Figures 3 and 4). Second, the time needed for coalescence to occur increases with decreasing drop size;
this is the opposite of what we observed in the case without surfactant. To understand the mechanism of multiple stepwise coalescence phenomenon, we rideorecorded the process from the side. Figure 5 shows a sequence of microphotographic pictures which depict the multistepwise coalescence process of a single toluene droplet with a flat, water/toluene interface in the presence of m o m NaLS. Figure 5a shows the toluene drop with a diameter of about 0.4 cm resting a t the watedtoluene interface. Figure 5b depicts the drop drainage right &r the film rupture. Under the capillary pressure and buoyancy force, the drop drains and a cylindricallike column is formed. Under the interfacial pertutbation the cylinder breaks and a secondary drop is formed (see Figure 5c.d). The secondary drop rests a t the toluendwater interface for a while (see Figure 5e), and then the second cycle of the stepwise coalescence pmcess starts. On the basis of the above experimental observations for droplet coalescence at an interface, one can distinguish two major phenomena which need to be discussed and explained first, the mechanism of the multiple stepwise coalescence of droplets at the interface and, second, the role of surfactant on the coalescence time versus drop size.
Discussion of the Experimental Results Analyses of the Drop Stepwise Coalescence Process. The observations discussed in the previous paragraph show that, without surfactant, the single toluene droplet did not coalesce completely but, rather,
Ind. Eng. Chem. Res., Vol. 34, No. 10,1995 3657
external meniscus
A
PRIMARY DROP
b
I
B
C
\
1
D
E
Figure 6. Possible mechanism for stepwise drop Coalescence. (A) film ruptures; (B)drop begins tn drain, the flow stretches the drop shape, and a cylinder-like column is formed (C) under the interfa"a1 perturbation the cylinder is broken up; (D)a secondary drop is formed; (E)a secondary drop.
11.
c. Figure 5. Sequence of microphotographic pictures depicting the multi-stepwise coalescence process of toluene drop with a flat waterltnluene interface in the presence of mol& NaLS. (a) Toluene drop a t waterlbluene interface; (b)aRer the film ruptures a neck is formed; (e) the drainage flow stretches the drop shape and a cylindrical column is formed, under the interfacial perturbation the cylinder is broken up; (d) a secondary drop is farmed (e) secondary toluene drop a t the waterltoluene interface.
coalesced partially with the flat, watedtoluene interface, leaving behind a smaller, secondary drop which also did not coalesce completely and which also formed a secondary droplet. Without surfactant, we observed up to six
coalescence steps before the initial droplet coalesced completely (see Figure 3). With a low surfactant concentration, we observed a similar, multiple stepwise drop-size coalescence behavior but this time with five coalescence steps (see Figure 4). The experimental observations also showed, clearly, that in the presence of surfactant the coalescence time of the secondary formed droplet increased. The video recorded observations suggest (see the microphotographs shown in Figure 5) that the partial coalescence occurred because of the capillary instability of the neck formed during the drop drainage. After the film rupture (and as a result of film rupture) the drop's liquid begins to drain and a jet is formed because of the action of excess pressure (capillary pressure) and buoyancy force (see Figure 6B). The jet pressure stretches the drop shape and the external drop meniscus turns from the oil phase to the water phase by changing the size of the radius of curvature in the plane of observation (see Figures 6A-C and compare the position of the drop's external meniscus). Thus, Charles and Mason (1960a) analyzed the formation of a secondary drop based on the Ftayleigh instability of a n infinitely long, cylindrical column of an incompressible liquid. Such an infinitely long cylindrical column is unstable to small perturbations and
3658 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
-
2R,=
Table 1. Effect of Drop Radius and Interfacial Tension on Stepwise Coalescence
2Ro
without surfactant
............ i
coalescence step [stagel
drop radius ratio [ryi+d
1 2 3 4 5
4.5 0.25 1.8 1.9 2.3 2.5 2.40
*
with moVL NaLS coalescence drop radius step [stage] ratio [ri~+il 1 3.2 i 0.2 2 3 4
2.2 2.8 3.0 2.8
Rayleigh’s model prediction.
sscMdary drm
Figure 7. Modeling the drop stepwise coalescence. 5.86
breaks up into segments that, under the action of surface tension, form individual, equal drops (Rayleigh, 1899; Chandrasekhar, 1961; Levich, 1962). Charles and Mason (1960a)assumed that, during the drop drainage which follows the rupture of the film, the shape of the primary drop is stretched by the drainage flow driven by drop capillary pressure (resulting from the spherical-like interface) until a cylindrical-like column is formed, as shown in Figure 6B. As a result of this shape transition, the capillary pressure increases because the radius of the column, R, continues to decrease until the wavelength disturbance, 1 > 2nR, grows in amplitude and causes the column to break up (see Figure 6C,D) and form a secondary drop. The sketch in Figure 7 shows the transition of the drop shape to a cylinder with a size ratio of ZO= AopJ 2Ro = RdRo = 4.5 (interfacial tension u = 36.2 mN/m); the value 4.5 is based on Rayleigh’s theory, a t which point the interfacial perturbation breaks up the cylinder and a secondary drop is formed. In the above relation, ZOis the dimensionless parameter at the dominant or a t which the disturbances lead optimum wavelength ,Iopt t o the breakup of the cylinder. On the basis of this relation and assuming that the volume of the secondary drop is equal to that of the column, the volume ratio of the primary to the secondary drop, r;/;+l.was estimated by Charles and Mason (1960b):
where ri is the radius of the drop at the ith step, r;+l is the radius of drop a t the i+lth step and the value ofZo is calculated from Tomokita’s theory of unstable jet as P n is a coefficient Zo = 4.7 a t u = 30 mN/m; p = a t n = 1. The drop ratio rgi+1 = 2.4 and is consistent with the Rayleigh model prediction. If we assume that during the drop drainage the volume of the secondary drop is equal to that of the column in the conditions given by ZO = AoPJ2Ro and further drainage of the undulated cylinder during the breakup time is neglected (the process following the Rayleigh model), then n = 1and we can obtain, ri/,+1= 2.4. This value agrees with the theoretical prediction of Charles and Mason. Our results for rg;+l, both with and without surfactant, are presented in Table 1. In both cases, the values of the drop size ratio versus coalescence steps pass to an unsymmetrical minimum. The minimum for ri/i+1 versus coalescence steps is a t the second drop coalescence step (see the data presented in Table 1). After the minimum, the values of the coalescence ratio slightly increase and, after the third coalescence stage,
.
................
a = lo[dyn/cm] u = 30 [ dyn / cm 1
9=o.l32[g/cma ]
4.82 4.70
.................... ..............
4.46 0.W 0.01 0.02 0.03 0.04 0.05 0.08 0.01 0.W 0.09 0.11
DROP MUDIUS I CM J
Figure 8. Effect of viscosity, interfacial tension, and drop radius on the dimensionless parameter Z,.
for the case without surfactant, reach a value which correlates well with the Rayleigh model (see the value calculated by eq 2 a t n = 1). To analyze the effect of viscosity, interfacial tension, and drop size on the drop ratio, we used the relation derived by Weber (1931) for the dimensionless parameter, ZO:
where p is viscosity, u is interfacial tension, and e is density. The effect of viscosity, interfacial tension, and drop size on the dimensionless parameter, ZO,is graphically presented in Figure 8. One can see that both ZOand r,/,+1(see eq 1) will decrease with a decrease in the viscosity and ZOand r,/;+lwill increase with decreasing interfacial tension and drop size. This is what we really observed (see the results presented in Table 1). One notes that, for the initial drop, the value of the drop coalescence ratio is highest and does not follow the prediction of the Rayleigh model. Charles and Mason (1960a) observed a similar phenomenon, but they did not analyze it. A possible explanation for this observation could be given based on the role of capillary pressure and buoyancy force on the drop drainage and necking-down process of the column. There is a “race” between the drop drainage and necking-down process, and if the capillary pressure is low (large droplet or low interfacial tension) the hydrostatic pressure, AOpt Aeg, (large droplet), is relatively high (due to the buoyancy force)and will cause a smaller size column to form during the drop drainage (see Figure 7). Thus, the coalescence process will lead to the higher drop radius ratio, r,/,+l. During the multiple stepwise coalescence process, the drop size decreases and the capillary pressure increases proportionally to
Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 3659
(R, - R,+l)/(R,Ri+l).For the smallest droplets, with a radius of about 3 x 10-3cm, the capillary pressure is 2 x lo4 dyn/cm2. Thus, the increase in the capillary pressure leads to a decrease of the drop drainage time and, thus, the time required for disturbances to attain critical amplitude, Ro,may not be enough. As a result, one can expect that, at a certain drop size and depending on the capillary pressure and drop fluid viscosity, the drop will coalesce completely into the bulk phase. More research needs to be done, especially by accurately monitoring of the coalescence process of the last (smallest) drop. Effect of Surfactant on the Stepwise Coalescence Process. It has been observed that with a low concentration of surfactant such as mol/L NaLS, the multiple stepwise coalescence process versus time has a different trend than without surfactant (see and compare Figures 3 and 4). At low surfactant concentrations, the drop step coalescence time (which includes drop rest time, drop drainage time, and neck break up time) increases by decreasing the drop size. The experimental observation (see the photomicrograph presented in Figure 5) shows that both the drop drainage and neck breakup occur in less than a second. When an oil drop gently arrives onto an oiywater interface, it rests on the interface and a spherical water film is formed. To estimate the drainage time, one uses the spherical-planar approach proposed by Charles and Mason (1960b). They used the Stefan (1874) and Reynolds (1886) approach that considers a sphere of radius R approaching a n unbounded plane. To render the equation tractable for the approaching sphere, a parabola of the same radius of curvature a t the apex has been substituted. The weakness in this approach is the assumption that the pressure acting on both film surfaces is equal. It is important to note that, for the spherical film, the Stefan-Reynolds approach may not be as simple t o apply as Charles and Mason (1960b) pointed out because of the pressure gradient normal to the film surfaces. However for the final equation they obtained (3) (4)
It is important to note that, for spherical film at equilibrium thickness, the disjoining pressure, z1= a/&, and thus the film-thinning driving force, is half of the drop capillary pressure. After integration, the time required for the film to change its thickness from hi t o hz was obtained by Charles Mason (1960b) from the following equation: 6npRd2 h l tIg = -In F h2 By using the above equation, we estimate that drainage time (the time film takes t o reach equilibrium thickness) is of the order of less than 1/100 s, for the film forms between a drop with a radius of about 50 pm and a flat interface a t an interfacial tension of 25 mN/m. The equilibrium film thickness is assumed to be 20 nm (it is known that plane parallel film ruptures at such a film thickness) and the contact angle, @MI, to be 3.2” (this value has been experimentally measured,
see discussion below). However, eq 5 fails to explain the experimental observation of the drop step coalescence time in the presence of surfactant as shown in Figure 4 and it also cannot predict the trend of the drop lifetime. By analyzing this equation, one can conclude that film drainage in the presence of low surfactant is not the rate-limiting step of drop coalescence. Thus, it seems that the limiting stage for the drop coalescence process is the thermodynamic stability of the film. For many years the topic of film stability has been the subject of long discussions in the literature: Frenkel (1955),de Vries (19581, Scheludko (19621, Derjaguin and Gutop (1962), Derjaguin and Prokhorov (1981), Vrij (19661, Jeffeys and Hawksley (19651, Hodgson and Woods (19691, Ivanov at al. (19701, Jain et a1 (19761, Zapryanov et al. (19831, Malhotra and Wasan (1988), and Kralchevsky et al. (1990). Nevertheless, this problem has not yet been solved. Until now, one could summarize the two major concepts concerning film stability that have been accepted as thermodynamic and kinetic. It is often assumed that film rupture is due to the growth of instabilities whose origin may be thermal fluctuation or mechanical perturbations. Vries (1958) postulated that a film can rupture if a hole (created by thermal or mechanical fluctuations) forms spontaneously in the film. The activation energy required to form such a “hole” is approximately d c r 2 (her is film thickness at which film ruptures and u is surface tension) and if the critical film thickness (the thickness at which film ruptures) is greater than 10 nm (such as in the case of very low and low surfactant concentrations), oh,2 >> KT and therefore, holes cannot be formed. Later, de Vries (1966) modified his “hole” mechanism by saying that the fluctuations can corrugate a deformable interface and, if the film disjoining pressure isotherm contains a part where dwdh > 0, then the amplitude of the wave could lead to the local film thickness where the van der Waals interaction force is predominant. In this way, the film becomes thermodynamicallyunstable and ruptures. This idea was later explored by Vrij (1966), Scheludko (19671, Jain and Ruckenstein (19791, etc. Vrij (1966) was the first to account for the fact that the growth of surface disturbances occurs in the thinning film and developed a theoretical expression for the film critical thickness. He has demonstrated, theoretically, that the film is unstable (with respect to fluctuations) when the wavelengths are larger than a critical wavelength:
a,
=
22a [- m]
lf2
The film ruptures when the A,, reaches an amplitude of hl2. Vrij identified the value of the critical thickness, h,,, by minimizing the sum of the drainage and rupture times. He derived the following expression for the critical film thickness of rupture:
h,,
tzr
* 0.268 -
(7)
where A is the Hamaker constant and f is a numerical coefficient approximately equal to 6. The mean time of film rupture, r, is given by the following expression:
3860 Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 Table 2. Experimental Measurements and Theoretical Calculations for the Film Lifetime theoretical calculation expt drop coalescence step
Rd
(Cm)
0.065 0.030 0.0105
/ * I ,
rf (cm)
0 (deg)
(dyn/cm2)
her (A)
t (SI
A= hCr(A)
0.0036 0.0015 0.0005
2.7 2.9 3.2
384.5 833.5 2381
213 149 93
1.1 0.18 0.017
412 287 180
Pd2
THEORY EXPERIMENTAL DATA
.L 10
20
50
40
FILM RADIUS [ m ] Figure 9. Comparison of theoretical calculation of film lifetime with experimental measurements.
An analysis of the above equations leads to the conclusion that the critical film thickness of rupture will decrease with the decrease of the film radius and with increasing capillary pressure. To verify film stability mechanisms, experimental information about film radius and capillary pressure is needed. Such information for film radius, drop radius, and fildmeniscus contact angle has been obtained by us by using reflected light interferometry and transmitted light microscopy. In the case of film with a small contact angle (smaller than 4” and film thickness > 20 nm), the film radius is not visually distinct and cannot be accurately measured directly. In this case, while viewing the drop from the top in reflected light, we observed four to five interference pattermNewton rings in the three-phase contact region. The film radius, contact angle, and capillary pressure were calculated by using measurements of the radii of the common interference rings. The calculation procedure is described in the paper published by Lobo et al. (1990). The results for the drop radius, film radius, contact angle, and capillary pressure are presented in Table 2. The calculated data for film rupture thickness, h,,, and film rupture time, t, for two values of Hamaker constant and erg are also presented in Table 2. For the first coalescence stage, the data are not presented because of the technical difficulty in simultaneously monitoringthe drop radius and Newton rings. One notes that the results for the film radius and contact angle for the first and fifth drop size transitions are not presented. To measure the initial drop size, one needs low magnification, while for Newton ring distance
A=
erg
erg 5 (8)
t (s)
28 4.6 0.44
20 50 170
measurements a higher magnification is needed and, for that reason, the procedure for the calculation of rf and 0 requires certain simultaneous information about both parameters. The results for the fifth drop coalescence transition are also not presented because, in the case of the smallest droplets, the measurements of the Newton rings were not very certain. Nevertheless, the presented results for film life time versus drop size show, clearly, that the theory of the capillary instability fails to explain, qualitatively, the drop size coalescence effect versus time in the presence of surfactant (see Figure 4). One can expect however that, without surfactant, the theory of capillary instability could predict the coalescence time versus drop size, even quantitatively (see Figure 3). Derjaguin and Gutop (1962) proposed another mechanism concerning the stability of very thin films. They suggest a nucleation (hole) mechanism for the rupture of very thin, bilayer films. One can see that such an idea originated with Vries concept of hole formation in the thinning film, which we have already discussed. This is analgous to the theory of cavitation-in-liquid developed by Frenkel(1955). Nevertheless, Derjaguin and Gutop (1962) and Derjaguin and Prokhorov (1981) considered nucleation stability for the Newton film (bilayer film). They considered it is as a quasi-crystalline carrier of a two-dimensional lattice gas of defects (vacancies) which condensed and formed holes. Recently, Prokhorov and Derjaguin (1988) developed a generalized theory for the film stability combiningboth concepts, Le., the mechanical and vacancy mechanisms. They assumed that the fluctuations can lead to the formation of very small holes in the film which grow to a critical size causing the film to rupture. The critical size of the hole depends on work done by the film tension y and the work for formation of the perimeter of the hole with K being the line tension of the two-dimensional nucleus. The work for hole formation has a maximum at hole radius rcr = K / Y . The film lifetime, z, of a spherical film of radius rf formed after droplet interactions with the flat interface of radius is given by
Here p* is the two-dimensional viscosity of the film and r is the number of molecules per unit area of the film. A comparison of experimental measurements and theoretical calculation of the film lifetime of a stepwise coalescing toluene drop a t the plane waterloil interface versus film radius is presented in Figure 9. The fitting parameter is the line tension, and the best fit has been obtained by choosing 4 x dyn for the line tension. It is noted that eq 9 correctly predicts the effect of film size and capillary pressure versus film mean lifetime. Nevertheless, such agreement should be accepted only qualitatively. For example, the radius of the critical hole is found to be about 8 nm and the work for the formation of the critical hole is 2 x 103kT. Thus, the effect of line tension improved the probability of critical
Ind.Eng. Chem. Res., Vol. 34,No. 10,1995 3661
hole formation. Unfortunately, a definitive method for direct measurement of the line tension with values of about dyn has not been developed yet and, for that reason, further study is needed to improve our understanding of the mechanism of film stability.
Conclusions We have studied the mechanisms for the multiple stepwise drop coalescence with its homophase. The mechanism of secondary drop formation from the coalescence of a primary drop a t a 1iquidAiquid interface quantitatively is interpreted by Rayleigh’s theory of capillary instability. An interpretation is given for the observed minima of the drop ratio versus coalescence step based on the role of capillary pressure and bouyancy force. We have shown that drop lifetime versus size is governed by film stability rather than film drainage. The film lifetime is dependent on film size and capillary pressure. It has been found that, without surfactant, the drop lifetime decreases as the drop size decreases while, in the presence of a low concentration of surfactant (NaLS), the drop lifetime increases with decreasing drop size. Film stability is analyzed on the basis of the concept of the nucleation mechanism proposed by Prokhorov and Derjaguin (1988). The experimental measurements are in quantitative agreement with the theoretical prediction of the nucleation mechanism. Acknowledgment This work was supported in part by the National Science Foundation and by the US. Department of Energy.
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Received for review March 15, 1995 Revised manuscript received August 8, 1995 Accepted August 9,1995@
IE950177F Abstract published in Advance ACS Abstracts, September 15,1995. @
