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Capillarity-controlled entrainment of liquid by a thin cylindrical filament moving through an interface
C/w&o/
Engineering Science, 1973, Vol. 28, pp. 23-30.
Pem~n
PESS.
Printed in Great Britain
Capillarity-controlled entrainment of liquid by a thin cylindrical filament moving through an interface B. J. CARROLL and J. LUCASSEN Unilever Research Port Sunlight Laboratory, Port Sunlight, Wirral, Cheshire, England (Received 6 January 1972) Abstract-A theory is presented predicting the thickness of liquid lilms formed in withdrawal experiments with thin cylinders. If gravity can be neglected, an analogy exists with the thinning process of small soap tilms by capillary suction. Based on this analogy a theory has been derived predicting film thickness to be proportional to the square root of the withdrawal speed and of the viscosity and inversely proportional to the square root of the interfacial tension. Withdrawal experiments carried out in an oil-water system at varying speed and inter-facialtension continned this theory. In the presence of surfactant the interface resists extension and tilm thicknesses -under otherwise equal conditions-were found to be about 60 per cent higher than in the absence of surfactant. The importance of surface dilational properties on formation of liquid cylinders and cylindrical films and the relevance to emulsification is discussed qualitatively.
In this original theory and in its later extensions and modifications [2,3] drainage and the establishment of an ultimate film thickness is essentially determined by gravity. Just as in the case of macroscopic soap films pulled out of a solution [4], the expected thickness is found to be proportional to the two-thirds power of the withdrawal speed. While such behaviour is found experimentally in the case of withdrawal of flat plates, the experimental evidence in support of any theory on cylinder withdrawal seems to be very scarce. In this paper we will argue that for film deposition on thin cylinders, gravity cannot be an important factor and that, in fact, the film thickness will be determined by capillary suction only. An alternative theory and experimental material in support of it will be presented. Furthermore, the effect surface dilational and shear properties will have on cylinder withdrawal-not considered hitherto-will be discussed.
INTRODUCTION
A SOLID body withdrawn from a liquid frequently becomes covered by a film of that liquid. An understanding of the factors determining the thickness of deposited films is of importance for such diverse fields as wire coating, deposition of emulsion on photographic films, mutual replacement of fluids in capillaries or in fabrics during detergency, spin drying and the determination of errors in titrations and in viscosity measurements. Theories relating film thickness with withdrawal speed and with surface and bulk properties of the liquid have been developed for flat and for cylindrical surfaces. Most of these theories fall back on a paper by Landau and Levich [ 11. In this paper equations were derived for the shape of the surface above the meniscus- where the entrained liquid can be considered as isolated from the bulk-as well as for the meniscus surface far away from the solid, where the surface shape is practically unaltered by the withdrawal. The condition that these two expressions should match in the transition zone between these two regions then supplied an equation for the film thickness as a function of withdrawal speed, viscosity and density of the liquid and of the gravity constant.
THEORY
At first we suppose that in the systems to be considered the condition of dynamic wetting [5] has been fulfilled. This means that when a solid is withdrawn from a liquid, a continuous film of the liquid is left behind. It does not 23
B. J. CARROLL and J. LUCASSEN
necessarily mean, though, that the contact angle - as measured through the liquid phase - is zero. It is sufficient that the speed with which the three-phase contact line tends to retreat towards the liquid meniscus is smaller than the withdrawal speed. We consider a thin cylinder, initially at rest, passing through a fluid-fluid interface (Fig. 1). The radius of the cylinder, R, is small compared to the capillary length, a,
(where y is the interfacial tension, p1 and p2 are the densities of the lower and upper phase respectively and g is the gravity constant) or in other words the Goucher number[61, Go = R/a is small compared to unity. Under these conditions the interface up to distances of the order a from the cylinder axis will have the shape of a catenoid [7] irrespective of the direction in which gravity acts. This situation, in which pressure changes across the interface are negligible, and Cylinder moves with speed U.
in which its two radii of curvature are of equal magnitude but opposite sign, will be disturbed as soon as the cylinder starts moving through the interface. A pressure difference will be created between point C (Fig. 1) inside the cylindrical film which is being deposited, and point D, where the surface shape can be considered unaltered, of magnitude AP=&.
It is because of this pressure difference, the capillary suction, that the film at C will drain as soon as the movement of the cylinder stops. But also during the withdrawal, drainage because of capillary suction takes place continually. It is easy to see qualitatively that the higher the withdrawal speed, the thicker will be the film when it leaves the meniscus region because the suction then has less chance to cause thinning. This thinning process in the meniscus region should be dimensionally analogous to the process of thinning of microscopic soap films[g] which obeys the Reynolds equation[9]. Assuming that for the present problem the radius of the soap film can be replaced by the height of the meniscus region, which is of the order (R + h) , and the film thickness by the final thickness h of our cylindrical film, we obtain for the Reynolds equation 1 ~=,,(RB+Yj.,)2AP-t
(3)
where B is a drainage factor (4/3 for solid discs) and r) is the liquid viscosity. The time it takes for a liquid element, carried along by the cylinder, to reach the final thickness h is of the order
and substitution finally gives u*r a = Capillary
length
-
R+h
Fig. 1. Entrainment of a liquid film by a thin cylinder.
24
of Eqs. (2) and (4) into Eq. (3)
Capillarity-controlled
entrainment
adjoining aqueous phase. The uppermost phase consisted of petrol ether which served to dissolve the adhering film of the lowermost phase after it has travelled through the aqueous phase in between the two oil phases. Subsequent analysis of the uppermost phase provided the film thickness.
This equation predicts the thickness of liquid films deposited on thin cylindrical wires or threads as a function of withdrawal speed, cylinder radius, liquid viscosity and interfacial tension. Because of the dimensional approach we have followed, the value of the constant B is unknown and should be obtained from experiment. The equation for the thinning of microscopic, circular soap films[S] has been derived starting from the assumption that due to the presence of surfactant both film surfaces behave as rigid walls and resist extension. In the case of microscopic wetting films of pure liquids on solid substrates-when the liquid-air surface does not resist extension- it was predicted [ lo] that the thinning speed should be higher by a factor 4. By analogy, in the case of withdrawal by thin cylinders it may also be expected that the constant B in Eq. (5) can be higher by at most a factor 4 when the interface in the meniscus region does not resist extension. We may, therefore, expect that addition of surface active material will in general increase the thickness of deposited films, not only because the surface tension is being reduced (see Eq. 5) but also because small amounts of surfactant will cause the surface to resist extension so that the drainage factor B will be smaller.
(b) Materials The water used was de-ionised and distilled from alkaline permanganate in a nitrogen atmosphere and had a surface tension of 72.7 mNm+ at 20.2”C. BDH spectroscopic grade carbon tetrachloride (yaw = 44.2 mNm-’ at 23°C) was used without further purification, Nujol (commercial) and BDH “Analar” 60-80” petrol ether (yaw = 50 mNm-‘) were passed through an alumina column prior to use. Tetradecyl trimethyl ammonium bromide (TTAB) was recrystallised three times from acetone. Measurements of the interfacial tension, carried out by means of the Wilhelmy-plate method using a micro-roughened PTFE plate, did not show a minimum around the critical micelle concentration (C, = 0.0034 mol 1.-l). The aqueous phase lies above the oil phase in the present system, and it was found necessary to lower the PTFE plate into the aqueous phase through a thin layer of petrol ether to prevent the entrainment of air bubbles on its surface. The petrol ether was sucked off immediately afterwards. The apparent contact angle on the micro-roughened PTFE (measured through the oil) was zero. Filaments of FEP (fluorinated ethylenepropylene copolymer, akin to PTFE) were supplied by Du Pont de Nemours International SA, Geneva.
EXPERIMENTAL
(a) Principle of the technique In order to reduce the effect of gravity in the experiments, it was decided to study withdrawal in a system of two liquids with only slightly differing densities. Withdrawal experiments in two-liquid systems have not been reported hitherto. We may expect, however, that the theoretical considerations of the previous section are applicable without modification as long as the viscous resistance met in the film-forming phase is high as compared to that occurring in the other liquid. Cylindrical monofilaments were made to run upwards through a three-phase system. The bottom phase consisted of a Nujol-carbon tetrachloride mixture slightly heavier than the
(c) Entrainment apparatus The oil/water system was contained in a volumetrically calibrated glass vessel T (Fig. 2). A filament F of radius O-138 mm, wound on a thin-walled stainless steel drum S (dia. 60 mm), was drawn through an oil/water interface OJW by the drum D (dia. 93 mm), which was driven through a gear box by a constant speed electric motor (Rayleigh-Halstrup M 12OC). After 25
B. J. CARROLL
and J. LUCASSEN
pre-equilibrated in a stoppered flask for at least several hours. The apparatus was then run for an interval to ‘sweep’ the lower interface free of adsorbed trace impurities, after which 0, was replaced by fresh petrol ether. The amount of oil entrained was determined by gravimetric analysis of 0, for the involatile Nujol component after running the apparatus for a known time. About 60 m of filament passed through the apparatus in the course of one run. Long, fluid cylinders are inherently unstable [l 11and break up into droplets which, however, are retained on the filament. Trials with a variety of materials (nylon, terylene, steel) showed that a high contact angle (measured in oil) often leads to detachment of the drops when they hit the upper (0,/W) interface. The detached drops then remain in the aqueous phase and thus result in a low estimate of the amount of oil entrained. The FEP filaments used in the present study did not show this disadvantage because the contact angle was either zero or low. Side advantages in the use of FEP include its inert chemical nature and its low rigidity. Also, smooth running of the filament is augmented since the static coefficient of friction is low (O-1 approx) and lower than the dynamic coefficient by a factor of four. (Data from literature published by the manufacturer.)
Fig. 2. Entrainment apparatus (schematic).
passage through the aqueous phase W the entrained oil film was removed quantitatively in an upper oil phase 0, by a PTFE scrubber P. The oil phases used were (0,) 60/80’ petrol ether, redistilled for the hexane fraction and (9) a Nujol-carbon tetrachloride mixture (164 : 1 w/w) of density 1.06 Mgm+ (1.06 g cmm3) and viscosity 0.0166 kgrn-’ se+ (O-166 poise). The oil phases had been equilibrated with water before use. All experiments were performed in a clean room held at a temperature of 19.5 -t-0*8”C. Both drums D and S ran in ball races. The freerunning drum S was braked by a tissue paper pad pressed against the polished edge of one of the drum flanges. The side-arm of the vessel T was lined with a PTFE tube, through which the filament ran. With this arrangement stray impulses of frictional origin were small and were masked by the large, constant friction due to the pad. The filament could be drawn through the apparatus without jerking for extended intervals of time. Before an experiment the filament was wound without overlapping onto S via a scrubber immersed in petrol ether. All glassware was immersed in chromic acid and was thoroughly rinsed with de-ionised and finally with distilled water prior to use. The phases 0, and W were
RESULTS
AND
DISCUSSION
1. Comparison with theory The density difference between the aqueous phase and the entrained oil phase was 0.06 Mgm+ (0.06 g cmm3). The interfacial tension was varied between 4 and 48 mNm-I (4 and 48 dynes cm-‘) by using different concentrations of TTAB in the aqueous phase. Under those conditions, the Goucher number for a filament of radius O-138 mm varies between 0.01 and O-04, small enough to obey condition (1). The filament speed range covered was between about 2.5 and 50 mm set+. When the cylindrical liquid film has left the meniscus region drainage can be supposed to have virtually halted; gradients in capillary pressure are absent, the effect of gravity is 26
Capillarity-controlled entrainment
negligible and viscous shearing forces exerted by the aqueous phase are unimportant. Under these conditions plug flow may be assumed and the thickness of the entrained film could be calculated from the entrained mass per set, Q, according to
Q = ?r~p{(R+h)~-~~}
(6)
in which p is the oil density and u the filament speed. Film thicknesses varied between 3 and 70 pm. Values for the dimensionless thickness, hl (h + R), obtained under different conditions -by varying speed, interfacial tension and (in one case) the oil viscosity-are plotted in Fig. 3 as a function of the capillary number ugly, in a bilogarithmic graph. From this graph it is immediately obvious that a clear distinction exists between the results obtained for film withdrawal in the absence and in the presence of surface active material. In both cases straight lines with a slope of nearly exactly O-5 gave a very close fit with the experiments. The analogy with the microscopic soap
films obeying the Reynolds equation would suggest that B, the constant in Eq. (5), should not be much different from unity. In fact, values for B, of 2.0 and 043 were found in the absence and presence of surfactant respectively. This confirms the prediction that surfactant should cause the oil-water interface to resist extension, thereby giving rise to higher thicknesses of entrained fihns under otherwise equal conditions. The Levich-Detjaguin theory [2] would predict a slope of 2/3 in Fig. 3 (see dashed line) according to ‘3,
(7)
(It has been tacitly assumed that the term R should be replaced by (R + h) when the assumption h e R, made in Ref. [2], is not always valid.) Although Eq. (7) may hold at higher Goucher numbers, it is evident that the present results do not agree with a 2/3 slope. It can therefore be concluded that film deposition on cylinders in systems where gravity is unimportant is described by Eq. (5). The importance of this conclusion for deposition on thin threads and filaments is obvious. For example, film thicknesses at low capillary numbers can be much higher than predicted from any other theory. Very little experimental work has been reported on cylinder withdrawal at low Goucher numbers. The only data which can be compared with the present results are those of Goucher and Ward [ 121. Represented in Fig. 3 by the inverted triangles, they clearly belong to the class of surfactant-containing systems found in the present study. Although in Goucher and Ward’s experiments no surfactant was present, loci(y) no special care was taken to remove surface active contaminants, as is true of all published Fig. 3. Dependence of dimensionless film thickness on capillary number. Interfacial tension, y, speed u, and viswork on cylinder withdrawal. In the present cosity n have been varied. R = 0.138 mm. Dashed line: study the importance of purification was clearly expected behaviour according to Ref. 121, Drawn lines: demonstrated in preliminary experiments; in expected behaviour, this paper. x y 47.9 mNm-I, speed varied; r) = 0.17 poise. C-y varied; speed 24.3 mm set-‘; r) = using impure (technical) carbon tetrachloride and 0.17 poise, A y 16.5 mNm-I; speed varied; r) = 0.17 poise, Nujol, results were obtained which all lay close 0 y 21.2mNm-1; speed varied; r) =O.l7poise, EJ y 49.0 to the top line in Fig. 3, pointing to the presence mNm-I; speed 9.73 mm set-*; 1) = 204poise. V Data of of surface active impurities. Goucher and Ward [ 121. 27
B. J. CARROLL and J. LUCASSEN
2. Eflect of surface rheological properties It should be stressed here that the action of surface active material at a deforming interface is to spread out more evenly a local deformation (for example an expansion) over the available interfacial area. This is because the local expansion causes interfacial tension gradients and these in turn cause movement of interface and corresponding liquid drag towards the deformation source. For a periodic interfacial compression and expansion of low amplitude the phenomenon of propagation of a deformation has been fully analysed and was shown to be linked with the properties of a longitudinal surface wave [ 13,141. In that case, the effectiveness of propagation increases with increasingly elastic interfacial behaviour and decreases with the frequency of the experiment which can be considered to be a characteristic deformation rate. A quantitative measure of interfacial elasticity is the dilational modulus E = drldln A which gives the interfacial tension change resulting from a defined dilational area change and which increases from zero to a finite value on introducing surfactant into a “clean” interface. Qualitatively the same kind of behaviour is to be expected for continuous interfacial deformation during a withdrawal experiment. At low enough surfactant concentration the dilational modulus will be so small that the interfacial deformation cannot propagate beyond the meniscus region. As a consequence, the necessary expansion of the interface caused by the movement of the filament
elements move towards the meniscus; the ensuing liquid drag is the basic cause for the increased thickness of deposited films in the presence of surfactant. A similar change in behaviour can be easily demonstrated in the familiar process of sucking off the surface of a Langmuir through by means of a small pipette. As long as surfactant is present, dust particles in the surface far away from the pipette tip will move. When surfactants have been removed this motion stops. In order to find exactly where the transition from non-elastic to elastic behaviour occurs, a quantitative analysis of the system would be necessary. This does not seem feasible for the case of a continuous withdrawal because of the complex geometry and the intractable relationship between the local values of the dilational modulus, the rate of interfacial deformation and the previous history of the interface (which is stored in gradients in surfactant concentration below the surface). For thin soap films it has been calculated [ 151 that the transition from non-elastic to elastic behaviour takes place over a rather small range of (low) e-values. This would correspond to a narrow range of surfactant concentrations. A similarly sharp transition may be expected for cylinder withdrawal, explaining why so few points lie in between the drawn lines of Fig. 3. Some measurements made at low surface pressure seem to point to a change in behaviour. It is doubtful, however, whether the present experimental technique will permit a full investigation of the transition region. As mentioned above, for microscopic soap films the constant B decreases by a factor 4 when one of the surfaces changes from mobile to rigid. One may wonder why in the case of cylinder withdrawal for a similar transition only a factor 2.5 is found. It should be stressed here that the analogy between film thinning and cylinder withdrawal is not complete. For the small soap film with two rigid surfaces these surfaces can only move towards each other. This ensures maximal velocity gradients in the liquid and a maximal increase in viscous resistance as
$!= 2r(R+h)u will all be concentrated in the meniscus and the relative rate of expansion of interfacial elements there will be high. At higher surfactant concentration, the required area increase will be distributed far beyond the meniscus region and the relative area increase for each single element will be the smaller the larger is the available area. A consequence of this “elastic” interfacial behaviour is that remote interfacial 28
Capillarity-controlled entrainment
compared to the case where one of the surfaces is mobile. For cylinder withdrawal, however, even if the relative area increase of a surface element entering the meniscus region is negligible due to a high dilational modulus, the geometry of the system forces the element to move upwards with a speed smaller than that of the filament. It gradually picks up speed and only on leaving the meniscus will it have the same speed as the filament. As compared with the non-elastic interface the velocity gradients in the liquid and the ensuing viscous resistance may well show a smaller increase than is found for microscopic soap films. In this context it is interesting to note that while interfacial elements entering the meniscus may undergo a negligible extension, they should always undergo a very large shearing deformation. A high surface shear modulus, as is often reported for protein and polymer solutions, could therefore have a dramatic effect on the conditions in the entrance region and the resulting film thickness. Surface shear moduli in the present systems are probably small, however.
surface behaviour, however, continuous transport of interface and adhering liquid towards the meniscus region will slow down the thinning process there and postpone the moment of rupture. The pulling of liquid cylinders out of bulk liquid or large droplets is not only the basis of some fibre spinning processes but can also often be considered as the initial stage of emulsification. In order to understand the effect of surfactants on these processes it will be essential to know how they affect the conditions in the meniscus region. Another aspect which supported and free liquid cylinders have in common is their inherent instability which eventually leads to the formation of a row of droplets. This droplet formation can for free cylinders be considered as yet another important aspect of the emulsification process. From the cylindrical films formed in the above-mentioned withdrawal experiments droplets were always formed at some distance above the meniscus. The mechanism of their formation will be discussed in a subsequent paper.
CONCLUSIONS NOTATION
The qualitative picture put forward in this paper is that the formation of thin cylindrical films can be regarded as the result of a not fully completed process of capillary suction and also that increased film thicknesses will be found when liquid drag is caused by elastic interfacial behaviour. This picture should apply just as well to the formation of free, unsupported liquid cylinders out of a cylindrical meniscus. The important difference is in the absence of a solid backbone which prevents unlimited thinning for the supported cylindrical films. Because of this the difference between non-elastic and elastic interfacial behaviour will be relatively more important for the free cylinders. For nonelastic behaviour only the liquid in the meniscus region will be pulled out, a process which is finished very rapidly. In the case of elastic
A ii
Chf Go
g h L R t u
element of surface capillary length = d27~ (p1 - pz) g constant defined by Eq. (3) critical micelle concentration goucher number = R/a acceleration due to gravity entrained film thickness pressure rate of mass entrainment filament radius time filament speed
Greek symbols y interfacial tension n viscosity coefficient p density
REFERENCES [l] LANDAU L. D. and LEVICH V. G., Acta phys. Chim. USSR 1942 17 41. [21 DERJAGUIN B. V. and LEVI S. M., Film Coafing Theory, p. 46. Focal Press, New York 1964.
29
B. J. CARROLL
and J. LUCASSEN
[3] TALLMADGE J. A. and GUTFINGER CH., Ind. Engng Chem. (ht. Edn.) 1967 59 19. K. J. and COX M. C., J. ColloidSci. 1962 17 136. is] CHUN HUH and SCRIVEN L. E., .i. Colloid Znterf. Sci. 197 135 85. [6] TALLMADGE J. A. and GUTFINGER CH., A.Z.Ch.E. Jl1965 10 774. 171 See e.g. LANDAU L. D. and LIFSHITZ E. M., Fluid Mechanics, p. 234. Pergamon Press, Oxford 1959. 181 SHELUDKO A. and MANEV E., Tmns. Faraduv Sot. 1968 64 1123. _ [9] REYNOLDS O., Phil. Trans. 1866 177 157. [lo] SHELUDKO A. and PLATIKANOV D., Kolloidzeitschrif 1961175 150. [ll] LordRAYLEIGH,Proc.R.Soc. 18792971. [12] GOUCHER F. S. and WARD H., Phil. Mug. 1922 44 (6) 1002. [ 131 LUCASSEN J., Trans. Faraday Sot. 196867 222 1. [14] LUCASSEN J. and TEMPEL M. V.D., to be published J. Colloid Znte$. Sci. [IS] VRIJ A., HESSELINK F. TH., LUCASSEN J. and TEMPEL M. V.D., Proc. Konink. Ned. Akad. Wefenschuppen 1970 B73 124.
141 MYSELS
30
Engineering Science, 1973, Vol. 28, pp. 23-30.
Pem~n
PESS.
Printed in Great Britain
Capillarity-controlled entrainment of liquid by a thin cylindrical filament moving through an interface B. J. CARROLL and J. LUCASSEN Unilever Research Port Sunlight Laboratory, Port Sunlight, Wirral, Cheshire, England (Received 6 January 1972) Abstract-A theory is presented predicting the thickness of liquid lilms formed in withdrawal experiments with thin cylinders. If gravity can be neglected, an analogy exists with the thinning process of small soap tilms by capillary suction. Based on this analogy a theory has been derived predicting film thickness to be proportional to the square root of the withdrawal speed and of the viscosity and inversely proportional to the square root of the interfacial tension. Withdrawal experiments carried out in an oil-water system at varying speed and inter-facialtension continned this theory. In the presence of surfactant the interface resists extension and tilm thicknesses -under otherwise equal conditions-were found to be about 60 per cent higher than in the absence of surfactant. The importance of surface dilational properties on formation of liquid cylinders and cylindrical films and the relevance to emulsification is discussed qualitatively.
In this original theory and in its later extensions and modifications [2,3] drainage and the establishment of an ultimate film thickness is essentially determined by gravity. Just as in the case of macroscopic soap films pulled out of a solution [4], the expected thickness is found to be proportional to the two-thirds power of the withdrawal speed. While such behaviour is found experimentally in the case of withdrawal of flat plates, the experimental evidence in support of any theory on cylinder withdrawal seems to be very scarce. In this paper we will argue that for film deposition on thin cylinders, gravity cannot be an important factor and that, in fact, the film thickness will be determined by capillary suction only. An alternative theory and experimental material in support of it will be presented. Furthermore, the effect surface dilational and shear properties will have on cylinder withdrawal-not considered hitherto-will be discussed.
INTRODUCTION
A SOLID body withdrawn from a liquid frequently becomes covered by a film of that liquid. An understanding of the factors determining the thickness of deposited films is of importance for such diverse fields as wire coating, deposition of emulsion on photographic films, mutual replacement of fluids in capillaries or in fabrics during detergency, spin drying and the determination of errors in titrations and in viscosity measurements. Theories relating film thickness with withdrawal speed and with surface and bulk properties of the liquid have been developed for flat and for cylindrical surfaces. Most of these theories fall back on a paper by Landau and Levich [ 11. In this paper equations were derived for the shape of the surface above the meniscus- where the entrained liquid can be considered as isolated from the bulk-as well as for the meniscus surface far away from the solid, where the surface shape is practically unaltered by the withdrawal. The condition that these two expressions should match in the transition zone between these two regions then supplied an equation for the film thickness as a function of withdrawal speed, viscosity and density of the liquid and of the gravity constant.
THEORY
At first we suppose that in the systems to be considered the condition of dynamic wetting [5] has been fulfilled. This means that when a solid is withdrawn from a liquid, a continuous film of the liquid is left behind. It does not 23
B. J. CARROLL and J. LUCASSEN
necessarily mean, though, that the contact angle - as measured through the liquid phase - is zero. It is sufficient that the speed with which the three-phase contact line tends to retreat towards the liquid meniscus is smaller than the withdrawal speed. We consider a thin cylinder, initially at rest, passing through a fluid-fluid interface (Fig. 1). The radius of the cylinder, R, is small compared to the capillary length, a,
(where y is the interfacial tension, p1 and p2 are the densities of the lower and upper phase respectively and g is the gravity constant) or in other words the Goucher number[61, Go = R/a is small compared to unity. Under these conditions the interface up to distances of the order a from the cylinder axis will have the shape of a catenoid [7] irrespective of the direction in which gravity acts. This situation, in which pressure changes across the interface are negligible, and Cylinder moves with speed U.
in which its two radii of curvature are of equal magnitude but opposite sign, will be disturbed as soon as the cylinder starts moving through the interface. A pressure difference will be created between point C (Fig. 1) inside the cylindrical film which is being deposited, and point D, where the surface shape can be considered unaltered, of magnitude AP=&.
It is because of this pressure difference, the capillary suction, that the film at C will drain as soon as the movement of the cylinder stops. But also during the withdrawal, drainage because of capillary suction takes place continually. It is easy to see qualitatively that the higher the withdrawal speed, the thicker will be the film when it leaves the meniscus region because the suction then has less chance to cause thinning. This thinning process in the meniscus region should be dimensionally analogous to the process of thinning of microscopic soap films[g] which obeys the Reynolds equation[9]. Assuming that for the present problem the radius of the soap film can be replaced by the height of the meniscus region, which is of the order (R + h) , and the film thickness by the final thickness h of our cylindrical film, we obtain for the Reynolds equation 1 ~=,,(RB+Yj.,)2AP-t
(3)
where B is a drainage factor (4/3 for solid discs) and r) is the liquid viscosity. The time it takes for a liquid element, carried along by the cylinder, to reach the final thickness h is of the order
and substitution finally gives u*r a = Capillary
length
-
R+h
Fig. 1. Entrainment of a liquid film by a thin cylinder.
24
of Eqs. (2) and (4) into Eq. (3)
Capillarity-controlled
entrainment
adjoining aqueous phase. The uppermost phase consisted of petrol ether which served to dissolve the adhering film of the lowermost phase after it has travelled through the aqueous phase in between the two oil phases. Subsequent analysis of the uppermost phase provided the film thickness.
This equation predicts the thickness of liquid films deposited on thin cylindrical wires or threads as a function of withdrawal speed, cylinder radius, liquid viscosity and interfacial tension. Because of the dimensional approach we have followed, the value of the constant B is unknown and should be obtained from experiment. The equation for the thinning of microscopic, circular soap films[S] has been derived starting from the assumption that due to the presence of surfactant both film surfaces behave as rigid walls and resist extension. In the case of microscopic wetting films of pure liquids on solid substrates-when the liquid-air surface does not resist extension- it was predicted [ lo] that the thinning speed should be higher by a factor 4. By analogy, in the case of withdrawal by thin cylinders it may also be expected that the constant B in Eq. (5) can be higher by at most a factor 4 when the interface in the meniscus region does not resist extension. We may, therefore, expect that addition of surface active material will in general increase the thickness of deposited films, not only because the surface tension is being reduced (see Eq. 5) but also because small amounts of surfactant will cause the surface to resist extension so that the drainage factor B will be smaller.
(b) Materials The water used was de-ionised and distilled from alkaline permanganate in a nitrogen atmosphere and had a surface tension of 72.7 mNm+ at 20.2”C. BDH spectroscopic grade carbon tetrachloride (yaw = 44.2 mNm-’ at 23°C) was used without further purification, Nujol (commercial) and BDH “Analar” 60-80” petrol ether (yaw = 50 mNm-‘) were passed through an alumina column prior to use. Tetradecyl trimethyl ammonium bromide (TTAB) was recrystallised three times from acetone. Measurements of the interfacial tension, carried out by means of the Wilhelmy-plate method using a micro-roughened PTFE plate, did not show a minimum around the critical micelle concentration (C, = 0.0034 mol 1.-l). The aqueous phase lies above the oil phase in the present system, and it was found necessary to lower the PTFE plate into the aqueous phase through a thin layer of petrol ether to prevent the entrainment of air bubbles on its surface. The petrol ether was sucked off immediately afterwards. The apparent contact angle on the micro-roughened PTFE (measured through the oil) was zero. Filaments of FEP (fluorinated ethylenepropylene copolymer, akin to PTFE) were supplied by Du Pont de Nemours International SA, Geneva.
EXPERIMENTAL
(a) Principle of the technique In order to reduce the effect of gravity in the experiments, it was decided to study withdrawal in a system of two liquids with only slightly differing densities. Withdrawal experiments in two-liquid systems have not been reported hitherto. We may expect, however, that the theoretical considerations of the previous section are applicable without modification as long as the viscous resistance met in the film-forming phase is high as compared to that occurring in the other liquid. Cylindrical monofilaments were made to run upwards through a three-phase system. The bottom phase consisted of a Nujol-carbon tetrachloride mixture slightly heavier than the
(c) Entrainment apparatus The oil/water system was contained in a volumetrically calibrated glass vessel T (Fig. 2). A filament F of radius O-138 mm, wound on a thin-walled stainless steel drum S (dia. 60 mm), was drawn through an oil/water interface OJW by the drum D (dia. 93 mm), which was driven through a gear box by a constant speed electric motor (Rayleigh-Halstrup M 12OC). After 25
B. J. CARROLL
and J. LUCASSEN
pre-equilibrated in a stoppered flask for at least several hours. The apparatus was then run for an interval to ‘sweep’ the lower interface free of adsorbed trace impurities, after which 0, was replaced by fresh petrol ether. The amount of oil entrained was determined by gravimetric analysis of 0, for the involatile Nujol component after running the apparatus for a known time. About 60 m of filament passed through the apparatus in the course of one run. Long, fluid cylinders are inherently unstable [l 11and break up into droplets which, however, are retained on the filament. Trials with a variety of materials (nylon, terylene, steel) showed that a high contact angle (measured in oil) often leads to detachment of the drops when they hit the upper (0,/W) interface. The detached drops then remain in the aqueous phase and thus result in a low estimate of the amount of oil entrained. The FEP filaments used in the present study did not show this disadvantage because the contact angle was either zero or low. Side advantages in the use of FEP include its inert chemical nature and its low rigidity. Also, smooth running of the filament is augmented since the static coefficient of friction is low (O-1 approx) and lower than the dynamic coefficient by a factor of four. (Data from literature published by the manufacturer.)
Fig. 2. Entrainment apparatus (schematic).
passage through the aqueous phase W the entrained oil film was removed quantitatively in an upper oil phase 0, by a PTFE scrubber P. The oil phases used were (0,) 60/80’ petrol ether, redistilled for the hexane fraction and (9) a Nujol-carbon tetrachloride mixture (164 : 1 w/w) of density 1.06 Mgm+ (1.06 g cmm3) and viscosity 0.0166 kgrn-’ se+ (O-166 poise). The oil phases had been equilibrated with water before use. All experiments were performed in a clean room held at a temperature of 19.5 -t-0*8”C. Both drums D and S ran in ball races. The freerunning drum S was braked by a tissue paper pad pressed against the polished edge of one of the drum flanges. The side-arm of the vessel T was lined with a PTFE tube, through which the filament ran. With this arrangement stray impulses of frictional origin were small and were masked by the large, constant friction due to the pad. The filament could be drawn through the apparatus without jerking for extended intervals of time. Before an experiment the filament was wound without overlapping onto S via a scrubber immersed in petrol ether. All glassware was immersed in chromic acid and was thoroughly rinsed with de-ionised and finally with distilled water prior to use. The phases 0, and W were
RESULTS
AND
DISCUSSION
1. Comparison with theory The density difference between the aqueous phase and the entrained oil phase was 0.06 Mgm+ (0.06 g cmm3). The interfacial tension was varied between 4 and 48 mNm-I (4 and 48 dynes cm-‘) by using different concentrations of TTAB in the aqueous phase. Under those conditions, the Goucher number for a filament of radius O-138 mm varies between 0.01 and O-04, small enough to obey condition (1). The filament speed range covered was between about 2.5 and 50 mm set+. When the cylindrical liquid film has left the meniscus region drainage can be supposed to have virtually halted; gradients in capillary pressure are absent, the effect of gravity is 26
Capillarity-controlled entrainment
negligible and viscous shearing forces exerted by the aqueous phase are unimportant. Under these conditions plug flow may be assumed and the thickness of the entrained film could be calculated from the entrained mass per set, Q, according to
Q = ?r~p{(R+h)~-~~}
(6)
in which p is the oil density and u the filament speed. Film thicknesses varied between 3 and 70 pm. Values for the dimensionless thickness, hl (h + R), obtained under different conditions -by varying speed, interfacial tension and (in one case) the oil viscosity-are plotted in Fig. 3 as a function of the capillary number ugly, in a bilogarithmic graph. From this graph it is immediately obvious that a clear distinction exists between the results obtained for film withdrawal in the absence and in the presence of surface active material. In both cases straight lines with a slope of nearly exactly O-5 gave a very close fit with the experiments. The analogy with the microscopic soap
films obeying the Reynolds equation would suggest that B, the constant in Eq. (5), should not be much different from unity. In fact, values for B, of 2.0 and 043 were found in the absence and presence of surfactant respectively. This confirms the prediction that surfactant should cause the oil-water interface to resist extension, thereby giving rise to higher thicknesses of entrained fihns under otherwise equal conditions. The Levich-Detjaguin theory [2] would predict a slope of 2/3 in Fig. 3 (see dashed line) according to ‘3,
(7)
(It has been tacitly assumed that the term R should be replaced by (R + h) when the assumption h e R, made in Ref. [2], is not always valid.) Although Eq. (7) may hold at higher Goucher numbers, it is evident that the present results do not agree with a 2/3 slope. It can therefore be concluded that film deposition on cylinders in systems where gravity is unimportant is described by Eq. (5). The importance of this conclusion for deposition on thin threads and filaments is obvious. For example, film thicknesses at low capillary numbers can be much higher than predicted from any other theory. Very little experimental work has been reported on cylinder withdrawal at low Goucher numbers. The only data which can be compared with the present results are those of Goucher and Ward [ 121. Represented in Fig. 3 by the inverted triangles, they clearly belong to the class of surfactant-containing systems found in the present study. Although in Goucher and Ward’s experiments no surfactant was present, loci(y) no special care was taken to remove surface active contaminants, as is true of all published Fig. 3. Dependence of dimensionless film thickness on capillary number. Interfacial tension, y, speed u, and viswork on cylinder withdrawal. In the present cosity n have been varied. R = 0.138 mm. Dashed line: study the importance of purification was clearly expected behaviour according to Ref. 121, Drawn lines: demonstrated in preliminary experiments; in expected behaviour, this paper. x y 47.9 mNm-I, speed varied; r) = 0.17 poise. C-y varied; speed 24.3 mm set-‘; r) = using impure (technical) carbon tetrachloride and 0.17 poise, A y 16.5 mNm-I; speed varied; r) = 0.17 poise, Nujol, results were obtained which all lay close 0 y 21.2mNm-1; speed varied; r) =O.l7poise, EJ y 49.0 to the top line in Fig. 3, pointing to the presence mNm-I; speed 9.73 mm set-*; 1) = 204poise. V Data of of surface active impurities. Goucher and Ward [ 121. 27
B. J. CARROLL and J. LUCASSEN
2. Eflect of surface rheological properties It should be stressed here that the action of surface active material at a deforming interface is to spread out more evenly a local deformation (for example an expansion) over the available interfacial area. This is because the local expansion causes interfacial tension gradients and these in turn cause movement of interface and corresponding liquid drag towards the deformation source. For a periodic interfacial compression and expansion of low amplitude the phenomenon of propagation of a deformation has been fully analysed and was shown to be linked with the properties of a longitudinal surface wave [ 13,141. In that case, the effectiveness of propagation increases with increasingly elastic interfacial behaviour and decreases with the frequency of the experiment which can be considered to be a characteristic deformation rate. A quantitative measure of interfacial elasticity is the dilational modulus E = drldln A which gives the interfacial tension change resulting from a defined dilational area change and which increases from zero to a finite value on introducing surfactant into a “clean” interface. Qualitatively the same kind of behaviour is to be expected for continuous interfacial deformation during a withdrawal experiment. At low enough surfactant concentration the dilational modulus will be so small that the interfacial deformation cannot propagate beyond the meniscus region. As a consequence, the necessary expansion of the interface caused by the movement of the filament
elements move towards the meniscus; the ensuing liquid drag is the basic cause for the increased thickness of deposited films in the presence of surfactant. A similar change in behaviour can be easily demonstrated in the familiar process of sucking off the surface of a Langmuir through by means of a small pipette. As long as surfactant is present, dust particles in the surface far away from the pipette tip will move. When surfactants have been removed this motion stops. In order to find exactly where the transition from non-elastic to elastic behaviour occurs, a quantitative analysis of the system would be necessary. This does not seem feasible for the case of a continuous withdrawal because of the complex geometry and the intractable relationship between the local values of the dilational modulus, the rate of interfacial deformation and the previous history of the interface (which is stored in gradients in surfactant concentration below the surface). For thin soap films it has been calculated [ 151 that the transition from non-elastic to elastic behaviour takes place over a rather small range of (low) e-values. This would correspond to a narrow range of surfactant concentrations. A similarly sharp transition may be expected for cylinder withdrawal, explaining why so few points lie in between the drawn lines of Fig. 3. Some measurements made at low surface pressure seem to point to a change in behaviour. It is doubtful, however, whether the present experimental technique will permit a full investigation of the transition region. As mentioned above, for microscopic soap films the constant B decreases by a factor 4 when one of the surfaces changes from mobile to rigid. One may wonder why in the case of cylinder withdrawal for a similar transition only a factor 2.5 is found. It should be stressed here that the analogy between film thinning and cylinder withdrawal is not complete. For the small soap film with two rigid surfaces these surfaces can only move towards each other. This ensures maximal velocity gradients in the liquid and a maximal increase in viscous resistance as
$!= 2r(R+h)u will all be concentrated in the meniscus and the relative rate of expansion of interfacial elements there will be high. At higher surfactant concentration, the required area increase will be distributed far beyond the meniscus region and the relative area increase for each single element will be the smaller the larger is the available area. A consequence of this “elastic” interfacial behaviour is that remote interfacial 28
Capillarity-controlled entrainment
compared to the case where one of the surfaces is mobile. For cylinder withdrawal, however, even if the relative area increase of a surface element entering the meniscus region is negligible due to a high dilational modulus, the geometry of the system forces the element to move upwards with a speed smaller than that of the filament. It gradually picks up speed and only on leaving the meniscus will it have the same speed as the filament. As compared with the non-elastic interface the velocity gradients in the liquid and the ensuing viscous resistance may well show a smaller increase than is found for microscopic soap films. In this context it is interesting to note that while interfacial elements entering the meniscus may undergo a negligible extension, they should always undergo a very large shearing deformation. A high surface shear modulus, as is often reported for protein and polymer solutions, could therefore have a dramatic effect on the conditions in the entrance region and the resulting film thickness. Surface shear moduli in the present systems are probably small, however.
surface behaviour, however, continuous transport of interface and adhering liquid towards the meniscus region will slow down the thinning process there and postpone the moment of rupture. The pulling of liquid cylinders out of bulk liquid or large droplets is not only the basis of some fibre spinning processes but can also often be considered as the initial stage of emulsification. In order to understand the effect of surfactants on these processes it will be essential to know how they affect the conditions in the meniscus region. Another aspect which supported and free liquid cylinders have in common is their inherent instability which eventually leads to the formation of a row of droplets. This droplet formation can for free cylinders be considered as yet another important aspect of the emulsification process. From the cylindrical films formed in the above-mentioned withdrawal experiments droplets were always formed at some distance above the meniscus. The mechanism of their formation will be discussed in a subsequent paper.
CONCLUSIONS NOTATION
The qualitative picture put forward in this paper is that the formation of thin cylindrical films can be regarded as the result of a not fully completed process of capillary suction and also that increased film thicknesses will be found when liquid drag is caused by elastic interfacial behaviour. This picture should apply just as well to the formation of free, unsupported liquid cylinders out of a cylindrical meniscus. The important difference is in the absence of a solid backbone which prevents unlimited thinning for the supported cylindrical films. Because of this the difference between non-elastic and elastic interfacial behaviour will be relatively more important for the free cylinders. For nonelastic behaviour only the liquid in the meniscus region will be pulled out, a process which is finished very rapidly. In the case of elastic
A ii
Chf Go
g h L R t u
element of surface capillary length = d27~ (p1 - pz) g constant defined by Eq. (3) critical micelle concentration goucher number = R/a acceleration due to gravity entrained film thickness pressure rate of mass entrainment filament radius time filament speed
Greek symbols y interfacial tension n viscosity coefficient p density
REFERENCES [l] LANDAU L. D. and LEVICH V. G., Acta phys. Chim. USSR 1942 17 41. [21 DERJAGUIN B. V. and LEVI S. M., Film Coafing Theory, p. 46. Focal Press, New York 1964.
29
B. J. CARROLL
and J. LUCASSEN
[3] TALLMADGE J. A. and GUTFINGER CH., Ind. Engng Chem. (ht. Edn.) 1967 59 19. K. J. and COX M. C., J. ColloidSci. 1962 17 136. is] CHUN HUH and SCRIVEN L. E., .i. Colloid Znterf. Sci. 197 135 85. [6] TALLMADGE J. A. and GUTFINGER CH., A.Z.Ch.E. Jl1965 10 774. 171 See e.g. LANDAU L. D. and LIFSHITZ E. M., Fluid Mechanics, p. 234. Pergamon Press, Oxford 1959. 181 SHELUDKO A. and MANEV E., Tmns. Faraduv Sot. 1968 64 1123. _ [9] REYNOLDS O., Phil. Trans. 1866 177 157. [lo] SHELUDKO A. and PLATIKANOV D., Kolloidzeitschrif 1961175 150. [ll] LordRAYLEIGH,Proc.R.Soc. 18792971. [12] GOUCHER F. S. and WARD H., Phil. Mug. 1922 44 (6) 1002. [ 131 LUCASSEN J., Trans. Faraday Sot. 196867 222 1. [14] LUCASSEN J. and TEMPEL M. V.D., to be published J. Colloid Znte$. Sci. [IS] VRIJ A., HESSELINK F. TH., LUCASSEN J. and TEMPEL M. V.D., Proc. Konink. Ned. Akad. Wefenschuppen 1970 B73 124.
141 MYSELS
30