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A study of the advancing interface. I. Interface shape in liquid—gas systems
A Study of the Advancing Interface I. Interface Shape in Liquid-Gas Systems R I C H A R D L. H O F F M A N Monsanto Polymers and Petrochemicals Company, 730 Worcester Street, Indian Orchard, Massachusetts 01051
Received February 18, 1974; accepted August 23, 1974 The shape of the advancing liquid-air interface has been studied in a glass capillary over
the range in which viscous and interfacial forces are the dominant factors controlling the system. Plotting the apparent contact angle versus the capillary number plus a shift factor, one obtains a curve which correlates all the data. In the systems examined experimentally in this study, the shift factor is determined solely by the static contact angle between the liquid and the solid substrate. To generate the curve, data were first obtained over the full range of apparent contact angles from 0 to 180° with liquids having shift factors which were equal to zero or negligible. Then the validity of the correlation was tested with data on liquids requiring large shift factors and the results were found to agree with the data obtained on the other liquids. INTRODUCTION W h e n analyzing the flow of a liquid as it moves over a solid surface and displaces a gas, one encounters problems which have not y e t been adequately resolved in the literature. One is a complete definition of the shape of the liquid-gas interface as a function of pertinent dimensionless parameters. As this shape determines b o u n d a r y conditions of the flow, it is i m p o r t a n t t h a t the factors which determine the interface shape are well understood. Another problem is encountered when one considers flow near the contact line of the advancing interface with the solid wall. Application of the classical no-slip b o u n d a r y condition in various m a t h models has led to the unlikely result t h a t the shear stress and pressure fields increase without bound as the contact line is approached (1). The problem of interface shape is considered in this article, and in a subsequent article we will discuss the analysis of flow behind one interface shape
where shear stress and pressure fields in the liquid phase do not increase without bound at the contact line. Although interface shapes and contact lines between a solid, liquid, and a fluid have attracted considerable scientific attention (2,3), most of the studies are concerned with static systems and few with d y n a m i c systems. The limited n u m b e r of studies of d y n a m i c systems can be further divided into two distinct categories, those in which the advancing interface is driven with a c o n s t a n t forward velocity over the solid s u b s t r a t e and those in which the interface shape is in a nonequilibrium state and changing as a result of its recent displacement to a new position. Elliott and Riddiford (4) have suggested t h a t studies of driven interfaces should be distinguished from studies of nonequilibrium interfaces b y the terms d y n a m i c studies and relaxation studies, respectively. Under this distinction we will concern ourselves with dynamic studies only. 228
Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
Copyright • 1975 by Academic Press, Inc. All rights of reproduction in any form reserved.
SHAPE OF ADVANCING INTERFACE Considering flow the dynamic contact line, West (5) reasoned that liquid, moving in an air-filled capillary tube, should move forward and spread outward from the center region of flow in the advancing meniscus, and fold inward to the center region of the receding meniscus. Since West's pioneering study, the occurrence of this general flow pattern has been confirmed experimentally in Newtonian and non-Newtonian systems (6-8), and predicted by various theoretical studies (9-14). More complex flow patterns, generally in the air phase, may occur as suggested by the work of Huh and Scriven (1). Details of the flow pattern remain to be worked out, however, for although the velocity fields predicted are reasonable, the theoretical studies always specify stress and pressure fields which increase without bound at the contact line. These problems aside, one expects that a study of interface shape can be conveniently broken down into a study of several different flow regimes. For simplicity we consider dynamic systems in which a liquid advances to displace a gas and in which gravity forces are negligible. Examining the spectrum of velocities from zero to large values with Newtonian fluids at various viscosity levels, we know that four types of forces will come into play; namely viscous forces, inertial forces and interfacial forces at the liquid-gas interface and the liquid-gas-solid juncture. Which set of these forces is important depends upon the system and the flow rate. To compare magnitudes of the first three forces we use the capillary number, Ca = nV/'y, which is the ratio of viscous forces to interfacial forces at the gas-liquid interface and the Weber number, W e = pV2L/'y, which is the ratio of inertial forces to interfacial forces at the gas-liquid interface (15). In these equations ~ is the viscosity of the liquid, V the interface velocity, 7 the surface tension at the gas-liquid interface, p the density of the liquid and L a characteristic dimension of the system. In capillary systems L will be taken as the tube diameter, and for a meniscus ad-
229
vancing between parallel flat plates, L will be taken as the spacing between the plates. Interfacial forces at the liquid-gas-solid juncture must also be considered. The procedure for dealing with the effect of these forces on interface shape in systems where inertial and interfacial forces are dominant has yet to be established, but with systems in which viscous and interfacial forces are dominant, one can use a dimensionless shift factor which is added to the capillary number. For tile systems examined experimentally in this study, this shift factor depends solely upon the static contact angle and the concepts for obtaining the shift factor for these systems are developed in this study. In a review of the literature, one finds that various studies have touched upon a number of flow regimes as shown by Table I. Riddiford et al. (16-18) have studied the first regime in which interfacial forces dominate, and tile studies of Rose and Heins (19) and Hansen and Toong (20, 21) touch upon flow in the second regime where interfacial and viscous forces predominate. But some of the data obtained by Hansen and Toong appear to enter into the regime where interfacial, inertial, and viscous forces all play a role. Being aware of these flow regimes and the possibility of moving from one to another, we find that the work of Hansen and Toong has raised some unanswered questions. For example, Hansen and Toong have reported apparent contact angles and interface shapes as they are correlated by the capillary number. At low values of the capillary number, they find the interface between a heavy paraffin oil and air is a spherical sector, but at high values of the capillary number, for example nV/'g = 0.212, they report deviations from the spherical shape. One finds, however, that when ~ V / 7 = 0.212 for this system, the Weber number is 0.046. From calculations given in Appendix A, we infer that at this high a Weber number inertial forces may also be important. Thus the question arises as to whether the interface is distorted from a
Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
230
R I C H A R D L. H O F F M A N TABLE I RANGES OF THE CAPILLARYAND WEBER NUMBERS COVERED BY VARIOUS LITERATURE STUDIES
Interracial Ref.
Dominant Forces Interfaeial/viscous
Interfacial/ inertial
Interfacial/inertial/ viscous
}"
16, 17, 18
Ca < 10.5 We < 3 X 10.6
19 (Nujol)
1.4 X 10-4 =< Ca =< 7.6 X 10-~ 3 . 2 X 10-8=< W e < ~ 9 X 10-s
19 (oleic acid)
7.9 N 10-5 < Ca < 2.8 X 10-3 1.8 X 10-7 ~ We <~ 3.6 X 10.4
14, 20, 21
1.7 X 10-5 < Ca < 3.9 X 10-* 3.0 X 10-1° < We < 1.5 X 10-~
This stu dy
4 X 10- 5 < C a < = 3 6 5.4 X 10-1I < We < 1.6 M 10.5
spherical shape b y viscous forces, inertial forces, or b y both. I n the s t u d y to be discussed in this paper, inertial forces were m a i n t a i n e d at a negligible level while the interface shape was examined over the range of 4 X l 0 -5 <_ ~ V / 7 < 36. Deviations from a spherical l i q u i d - a i r interface were never observed. Thus we suspect t h a t the distortions observed b y Hansen and Toong were caused b y inertial forces. Studies of the flow regime where only interfacial and inertial forces dominate in capillary or parallel flat plate systems are unknown to this author. Similar remarks apply to any s t u d y designed to examine the combined effects of viscous, inertial, and interfacial forces upon interface shape when all three are important. Two other flow regimes which deserve brief mention in this review are those in which viscous and the combination of viscous and inertial forces are the sole factors controlling interface shape. W h e n viscous forces alone control the interface shape, the a p p a r e n t contact angle will be 180 ° and invariant. F r o m the results of this s t u d y we infer t h a t this occurs when We<< 0.01 and Ca + F(O~) > 10, where F(O,) is the shift factor.
Ca = 0.212 We = 0.046
On the other h a n d when the combination of viscous and inertial forces control interface shape, one expects t h a t these forces will u l t i m a t e l y lead to the e n t r a i n m e n t of a thin film of gas between the liquid and solid which displaces their juncture downstream into the liquid phase. We know of no a t t e m p t to s t u d y this flow regime in capillaries or between flat plates, but the effect has occurred in film coating studies b y D e r y a g i n and Levi (22) and P e r r y (23). As evidenced b y their studies, this e n t r a i n m e n t occurs once the contact angle reaches 180 ° and W e > 0.4 L while 0.02 < Ca < 0.2. F o r these systems the characteristic length, L in cm, can p r o b a b l y be taken as the distance normal to the film over which the bulk of the meniscus curvature is observed. Returning to a consideration of the flow regime in which viscous and interfacial forces are dominant, one finds t h a t the d a t a obtained by Rose and Heins and Hansen and Toong cover only a small portion of this flow regime. Thus, a systematic s t u d y of the full range over which viscous and interfacial forces have a dominant effect on the interface shape is still required. I t is the purpose of this p a p e r to present this s t u d y and to this end we con-
Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
SHAPE OF ADVANCING INTERFACE
231
A-~
~ P OLYCARBONATE BASE ~HEX NUT GLUEDTO DRIVE PLATE YlO. 1. Advancing meniscus apparatus: AFI--advancing fluid interface, PBT--precision bore tube, A--locknuts and Teflon 0-Rings, C--connection to variable speed drive.
sider only the case of a liquid which moves over a solid surface and displaces a gas. From this study we find that the interface shape in this flow regime is directly related to the capillary number plus a shift factor. The need for the shift factor, although previously unused, is established by this study, and severat liquid-solid-gas combinations were tested to demonstrate this fact. EQUIPMENT AND PROCEDURES
The advancing meniscus apparatus used in this study is shown in Fig. 1. Simple in design, the instrument centers about a precision bore glass tube nominally five inches long with an inside diameter of 1.955 mm. A slug of liquid, one to three centimeters long, is drawn into the tube followed by a stainless steel plunger with an outside diameter of roughly 1.880 mm. The glass capillary is then inserted into holes in the Teflon guides which hold the tube snugly. Vee notches in the guides rest on the upper two brass rods shown in Fig. 1. Two more brass rods located three-quarters of an inch directly beneath those shown provide a rigid frame for the apparatus. Driving the tube over the plunger via a screw (1/8-40 NC-2A) connected to a variable speed motor, one obtains an advancing liquid-air interface which is essentially stationary with respect to lab coordinates. For the tests described in this paper the interface velocity varied from roughly 0.00008 to 0.06 cm/sec. The interface
was viewed and photographed through a Bausch and Lomb Stereozoom 7 microscope, Cat. No. 31-27-01, which was tilted so that the optical axis of the lens through which the interface was photographed was normal to the axis of the glass tube. An optical flat, prepared with a microscope slide coverslip and decalin between the slip and the glass tube prevented refraction of the light rays by the external glass-air interface. A correction for the refraction of the interface image by the liquid-glass or air-glass interface inside the tube is made using the refractive index of the glass and the fluid through which the image is transmitted to the glass. See Appendix B. The plunger velocity relative to the tube is obtained by measuring the revolutions per minute of the screw driving the glass tube over the plunger. The interface velocity differs from the plunger velocity by an amount proportional to the backflow of liquid in the space between the tube and the plunger. See Appendix C. The presence of the plunger at one end of the liquid slug will have some effect on the flow therein. Our concern is that its presence has no effect on the flow field near the advancing interface. Some studies in the literature have dealt with this problem and, from the results obtained, one finds that the flow is essentially fully developed at a distance of two pipe diameters from the interface (7, 10, 12, 13). Thus, with a liquid slug 1-3 cm long ahead of
Journal of Colloid and Inlerface Science, Vol. 50, No. 2, February 1975
232
RICHARD L. HOFFMAN
(a) (b)
Fro. 2. Photographs of the advancing liquid-air interface. In each photograph the liquid is left of the interface. Photograph (a) shows a liquid at rest with Oi = 0, but in the other two photographs the velocity is high enough to bow the interface out into the air phase.
(c) the plunger, the plunger is not expected to affect the flow field near the advancing interface. With the inside diameter of the precision bore tube used in these experiments, gravity forces can be neglected, and the radius of
PIPE WALL
I
FIG. 3. Quantities measured on the interface image.
curvature can be assumed constant. Hansen and Toong (14, 20, 21) question this assumption suggesting that the curvature m a y change in a range of 10.5 cm or less from the contact line which the advancing interface makes with the solid. Molecular interactions m a y dominate here and preclude the use of continuum concepts. Although this m a y be true, it does not alter the fact that the major portion of the flow field is determined by the interface shape~'external to this region. Hansen and Toong also presented data which show major distortion of the interface, but as noted in the introduction of this paper, the distortion apparently results from inertial effects. Measurements at various positions on the meniscus profile in our tests always indicated a constant radius of curvature within our measurement capabilities. (Changes in the curvature at tess than 10.5 cm from the contact line, of course, were not detectable.)
Journal of Colloid and Interface Science, Vol. 50, No. 2, F e b r u a r y 1975
233
S H A P E OF A D V A N C I N G I N T E R F A C E
To determine the radius of curvature, photographs of the advancing interface were obtained as shown in Fig. 2. Measurements of xR2, and H as shown on Fig. 3 were then obtained with the aid of a magnifying glass and a rule calibrated in hundredths of an inch. The equation Image radius
-
Actual radius of the circle
xRp RM
=
nl _
_
[X]
na
derived in Appendix B, enables one to obtain RM and then one determines the radius of curvature, Re, of the advancing interface by the equation Ro = { E R ~ 2 +
(Ro- H)231/2}/M, I-2 3
where M is the magnification, and in Eq. I-i] no and ns are the refractive indices of the glass and the fluid between the meniscus and the glass, respectively. Thus the equation for the cosine of the apparent contact angle, 0M, becomes cos 0M = Rp/Rc,
[-3]
where we neglect any changes in R~ which may occur close to the wall. See Fig. 4. To determine the magnification obtained in the pictures used for these calculations, the inside diameter of the glass tube was also measured on one picture from each set of runs. The tube diameter of 1.955 4-0.01 m m was originally obtained by photographing the tube and a calibrated microscope grid under the same magnification. The measurement of H and xR~ is a variable
@k, I
R
FLUID
~
AIR
FIG. 4. Apparent contact angle of an advancing liquid-air interface.
source of error which can generally be minimized by measuring the largest xR~ and associated H possible. In theory this would imply that one should measure H at the position where the interface contacts the wall making RM equal to the pipe radius. In practice, however, the interface image near the wall is lost, due to the critical angle of refraction, and thus the contact line of the interface with the wall is not visible. This being the case, the xRp and H values measured will generally be somewhat less than the theoretical maximum. Just how much less depends on lighting conditions and the refractive indices of the glass and the fluid through which the image is viewed. In calculating OM,the greatest source of error comes from the measurement of H and xRp. With the measurements made in this study we estimate the error was usually of the order of 5% for each value although in a few cases one or both measurements may have been in error by as much as 10%. Five different liquids were used in this study. The pertinent material properties of
T A B L E II 1V][ATERIALPARAMETERS Ot" TEST LIQUIDS Liquid
Viscosity (poise)
Specific gravity 25°C/25°C
Surface tension (dyn/cm)
Refractive I ndex
Static contact angle (degrees)
G.E. Silicone Fluid SF-96 Brookfield Std. Viscosity Fluid Dow C o m i n g 200 Fluid Ashland Chem. Admex 760 Santicizer 405
9.58 988 24,300 1093 112
0.974 0.977 0.98 1.15 1.13
21.3 21.7 21 43.8 43.4
1.405 1.405 1.405 1.471 1.477
0 0 12 69 67
Journal of Colloid and Inie~Jace Science, Vol. 50, No. 2, February 1975
234
RICHARD L. HOFFMAN
the liquids are listed in Table II. All values are at 24°C unless as otherwise noted. Viscosity measurements in cone and plate flow were made with the Weissenberg Rheogoniometer, Model R-16, manufactured by Sangamo Controls, Ltd., Bognor Regis, England. The surface tension of each liquid was obtained with a DuNouy ring tensiometer. Values for the specific gravity of each liquid are those listed by the manufacturer, and the static contact angles given are those measured in the advancing meniscus apparatus built for this study. Under the convention used, a liquid which completely wets the glass will have a static contact angle of zero. Unlike all the other materials, Admex 760 and Santicizer 405 were tested in the capillary after the glass surface had been altered by a silicone treatment used to give a large static contact angle as desired. To obtain this large contact angle, the glass was treated by a procedure outlined in Ref. (24). A 2% solution of a Dow Corning type 200 silicone fluid (200 cs) in tetrachloroethylene was used for the treatment, and after the silicone was applied, the capillary was heated at 100°C for one hour and then baked at 275°C for roughly two and one-half hours. Prior to testing the tube was then rinsed out with reagent grade 2-butanone. All the liquids tested were Newtonian in their viscosity behavior over the applicable range of shear rates with the exception of the Dow Coming 200 fluid. This material which was essentially Newtonian at the lower shear rates became pseudoplastic in the region of higher shear. For our purposes, however, we assumed that the material was Newtonian with a viscosity of 24,300 poise. For all the data taken this amounts to an error of less than 4-10% except for the one point at ~V/'y = 36. Here at the maximum shear rate at the wall the viscosity is roughly 20,000 poise. RESULTS AND DISCUSSION Data obtained on various liquid systems via the procedures outlined above are listed in Table I I I and plotted in Fig. 5 as the apparent
contact angle OM versus nV/~,+ F(O,). A few of the data points listed in Table I I I are omitted from Fig. 5 because of crowding. The term, F(Os), generally a constant for any liquid-solid-gas system, is a shift factor which compensates for the fact that the contact angle resulting from interfacial forces alone will vary with each liquid-solid-gas system considered. The method for determining F(O,) depends upon one of two types of behavior. For some systems one finds that interfacial forces between the solid and the liquid do not change when flow occurs, but with other systems changes such as those reported principally by Riddiford and co-workers can occur (4, 16-18, 25, 26). Let us consider first those systems in which a change does not occur. In such systems the shift factor, F(O,), is determined solely by the static contact angle, 08, and is zero when 0, is zero. Thus, when 08 is zero, the correlation between the apparent contact angle, OM, and nV/~ + F(O,) is easily established because F(O~) is zero and all other parameters can be measured. When the static contact angle is not zero, the same correlation applies if the shift factor F(O~) is used. To determine the shift factor one need only measure the static contact angle, 0~, and then read the value of ~V/'r + F(O~)= O+ F(O~) from Fig. 5 when 0~ is used as 0 u a n d V = 0. Using the concepts outlined above, we established Fig. 5 by first testing the silicone oils for which 08 and thus F(O~) were zero. See Table II. In generating the curve, data from the G. E. and Brookfield silicone fluids were used to establish F(O,) for the Dow Coming silicone fluid which has a static contact angle slightly greater than zero. Over the range tested, however, F(O,) makes a negligible correction to the data obtained for this liquid. Once the curve was established with data on the silicone fluids for which F(O~) was zero or negligible, the data on Admex 760 was used to test the concepts involving the shift factor. Since 08 = 69 ° we find from Fig. 5 that F(O,) = 2.35 N 10-2, and using this shift
Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
SHAPE OF ADVANCING INTERFACE
~gg~gggggggg
235
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Journal of Colloid and Interface Science, Vol. 50, No, 2, February 1975
236
RICHARD L. HOFFMAN ........ I ........ I ........ I ........ I ........ I ........ I ........ 18( 17C
16C 150 ~ 14.0' 130 120
t,
II0
IOC
90 8C 7C 6C 5C 4C 3C 2C l0 I 10"5
......
,,,,,,
10 "4
,
tO'*
, i,,,I,
....
tO -2
,H,
"rlV
T
t
, ,L,i,i
+ F(es)
,
I
i lllltl
,
tO
, ,ill
tO a
FIG. 5. Effect of flow on the apparent contact angle of an advancing liquid-air interface: (e) G.E. Silicone SF-96, F(0.~) = 0; (&) Brookfield Silicone, F(O~) = 0; (q?) Dow Corning Silicone, F(O,~) = 1.5 ;< 10-4; (El) Ashland Chemical Admex 760, F(O,) = 2.35 X 10-2; (+) Santielzer 405, F(O~) = 2.1 X 10-2; the data were taken at 24°C. factor one finds that the data fit the curve established with the silicone oils. Similar results were obtained with Santicizer 405 for which 08 = 67 °. Figure 6 is an expanded plot of a portion of our data along with that obtained by other authors (14, 19, 20). Their data provide us with further tests of the shift factor concept. Static contact angles for the liquids studied by the other authors were estimated by extrapolation to zero velocity in each case. These values were then used to obtain a shift factor for each system by the procedures outlined above. The data of Hansen and Toong agree very well with our data as do the points obtained from their theoretical work. On the other hand, the data of Rose and Heins fall somewhat above the data obtained in this study. The reason for the difference is unknown unless it involves experimental techniques a n d / or incorrect estimates of the static contact angles which were obtained by extrapolating their data to zero velocity. We do find, however, that the apparent contact angle, 0~, Journal of Colloid and Interface Science,
VoL
,50, No.
2, February
becomes progressively less sensitive to the capillary number as values of the capillary number decrease below F(O~). This must be true if the concept of the shift factor is correct. Considering all the data in Figs. 5 and 6 and reasoning inductively, we expect that the curve of Fig. 5 characterizes the shape of any liquid-air interface in motion when only viscous and interfacial forces are important, interfacial forces are not changed by motion, the viscosity behavior is Newtonian and the viscosity of the air is negligible when compared to the liquid. Up to now we have been considering systems in which the interfacial forces between the solid and the liquid do not change when flow occurs. There are some systems, however, where changes do occur. Riddiford and coworkers have done the bulk of the work in this area (4, 16-18, 25, 26). For the liquidsolid-gas systems which they have studied, they find that the apparent contact angle is usually independent of velocity at very low rates. But as the interfacial velocity is in1975
SHAPE OF ADVANCING INTERFACE .....
1
i
"1
I
I'l
li
I
I
1
I
237
I
I
I I I I
I
I
I
I
11
I
I
I Ill
I
I
I
I
I
I I'
120
IO0
~o
~
40
I~l
® 20
I
10-4
I
I
I
I
Ill[
I
10-3
I
I
"lTV + P(O s) 7/
10-1
II
10-1
FIG. 6. Effect of flow on the apparent contact angle of an advancing liquid-air interface: ( I ) G.E. Silicone SF-96, capillary diameter (ID) = 0.1955 crn, 0~ = 0 ° ; (~x) Brookfield Silicone, ID = 0.1955 cm, 0s = 0°; (@) Nujol, Hansen, and Toong (20), ID = 0.2380 cm, 0., = 22°; (®) Nujol, Hansen, and Toong (20), ID = 0.1210 cm, 08 = 36°; (X) Theory values, Hansen, and Toong (14), 0,~ = 21°; (E]) Nujol, Rose, and Heins (19), ID = 0.066 and 0.110 cm, 0~ = 23°; (®) Oleic acid, Rose and Heins (19), ID = 0,066 and 0.110, 0~ = 32°. creased above a critical value, the apparent contact angle changes with velocity until at higher speeds the rate of change diminishes and a limiting value is reached. The limiting value is generally less than 180 ° . To explain their results, Riddiford et al. postulate that molecular orientation occurs at the advancing liquid-solid-gas boundary when interface velocities are low enough, but as velocities are increased the molecules become randomly oriented at the boundary with an attendant change in the interracial forces and thus the apparent contact angle. A new but constant contact angle, 0~x, finally results from this process. To correlate data on systems such as these, one can no longer use the static contact angle, 0,, to determine the shift factor, F (08) because the interracial forces have changed at the advancing contact line. I t should be possible in m a n y cases, however, to handle these systems without difficulty. For example, all the shifts in contact angle observed by Riddi-
ford and co-workers generally occur before viscous forces begin to have an appreciable effect on the interface shape, As a case in point, with the water systems on various solid substrates, the shift is complete when Ca < 3 X 10-6, but with all the static contact angles being >__70° we see from Fig. 5 that viscous forces should have no effect until Ca is the order of 2 X 10-2. This being the case, the correlation of Fig. 5 should still apply if the shift factor F(O~) is determined by using the apparent contact angle 0M~x instead of 0~. An experimental test of this hypothesis has not been made. SUMMARY AND CONCLUSIONS Studying the shape of an advancing interface in a liquid gas system, we find that the apparent contact angle can be correlated as a function of the capillary number plus a shift factor when interracial and viscous forces are the dominant factors controlling the system. When interracial forces between a solid and a
Journal of Colloid and Interface Science, Vol. 50, No. 2, F e b r u a r y 1975
238
RICHARD L. HOFFMAN
1,2
I ,//'--PIPE WALL ........ -7- ............. R-LIQUID ~-GAS -TR~~__~_ __~ _~_ _IZF%-C_ NTEIR E~-_-_'~ " l I
I I I
FIG. 7. Advancing interface region considered in the momentum balance.
where v = v(r) is the local fluid velocity, R~a is the radius of curvature of the advancing interface and R , is the radius of curvature of the interface when no fluid motion occurs, p is the liquid density and 3' the surface tension. By rearranging Eq. I-All and defining the dimensionless parameters (v/V) =
v0;
(r/R)
=
r0;
and (Rc~/R) = (R~,)o
liquid do not change as flow occurs, the shift factor is determined solely by the static contact angle between the liquid and the solid substrate. Although untested a technique is also suggested to obtain the shift factor for systems in which interfacial forces do change. As a result of this study, data is now available over the complete range of contact angles from 0 to 180 °, and other flow regimes are delineated in which different sets of forces control the interface shape. APPENDIX A: DISTORTION OF THE ADVANCING INTERFACE BY INERTIAL FORCES To estimate the minimum Weber number at which inertial forces begin to affect interface shape, consider a fluid whose static interface bows out into the gas phase with a radius of curvature equal to the pipe radius. (Alternatively one can consider a highly viscous fluid moving with a velocity in excess of that required to give C a + F(O,) > 1 0 2 SO that 0M = 180°.) This being the case, viscous forces have no effect on the interface shape. See Fig. 5. To facilitate the analysis, we place a cylindrical coordinate system in the interface at the center line of the pipe and stipulate that it moves with the interface. See Fig. 7. The arrows left of the plane 1, 2 show the local direction of flow relative to the convected coordinate system. Writing a momentum balance about the fluid in the crosshatched region of Fig. 7, one finds that L R Ov227rrdr = fo R 2ERTa
23"] 2~rrdr,
R~/
EAI-]
one obtains 1 RpV 2 fl
2
v
vo2rodro
Jo = f 0 1 ro 1 - -1
-1
(Rca) o
1 dro,
[A2]
(Res)o
where V is the average fluid velocity relative to a stationary coordinate system and R is the pipe radius. (Rc~)0 = 1 as required by our previous assumptions. Now since inertial forces are balanced by the pressure generated through distortion of the interface, let us presume, as a first approximation, that inertial forces cause a 1% deviation from the radius of curvature of the static interface only in the region r0 _< ½. Thus we have (Roe)o=0.99
for r o < ½
and (R~a)o= 1.0 for ½ < r 0 _ < 1. When these conditions are applied to Eq. I-A2] one finds that 1 RpV 2 fl
2
Jo
7
1
[-A3~
vo2rodro -
792
or upon rearrangement Rp V2 We--
7
[A4]
1
396
L
vo2rodro
for the 1% deviation prescribed. Before inertial forces become important, Vo at the plane 1, 2 should be approximated reasonably well by the parabolic velocity profile found far from
Journal of Colloid and Interlace Science, Vol. 50, No. 2, February'1975
SHAPE OF ADVANCING INTERFACE
~ ~ % i /
239
LINENORMALTO THE PIPEWALL
otS;zfY-. . . . . . . . . . . . . . .
'lRp
"*"
POSITION ON THE > MENISCUS
RADIAL
~ ~ I D E
WALLOF THEGLASSTUBE
FIG. 8. Schematic of light refraction which leads to radial distortion of the liquid-air interface image: The view shown is normal to the axis of the pipe. the advancing interface, given by v0 = 1 -- 2r0~
[A5]
in our convected coordinate system. The justification for this approximation arises from results derived in the second paper of this series. For the purposes of this analysis we assume that the velocity profile is well approximated by Eq. [A5] since inertial forces are just beginning to be important. Turbulence should not be a factor since we can set Ca and thus the fluid viscosity as high as we wish. Thus by substituting Eq. [A5] into Eq. [A4] and integrating one finds that the 1% deviation prescribed for our system occurs when 1
We = - - ~ 0.015. 66
[A6]
Strictly speaking, this result applies only when the interface is bowed out into the gas phase with an apparent contact angle equal to I80 °. With different values of the apparent contact angle, one may expect some change in the We required for a 1% deviation in the radius of curvature. The situation for other apparent contact angles has not been examined, however, because the results should be nominally the same. APPENDIX B: CORRECTIONS FOR THE REFRACTION OF LIGHT A correction must be made for refraction of the interface image by the liquid-glass or
air-glass interface inside the tube. To determine this correction, consider the polymerair interface, i.e., the meniscus, and the inside of the glass tube as shown in Fig. 8. Part of a circle formed by the intersection of the meniscus and a plane normal to the axis of the pipe is shown with radius RM. From Snell's law, one finds that ns sin ¢ = n~ sin 0,
[B1]
where no and n s are the refractive indices of the glass and the fluid between the meniscus and the glass, respectively. Further we know that sin ~ = RM/Rp [B2] and sin 0 = xRp/Rp = x, [B3] where 0_< x < 1 . Combining [B2], and [B3], one finds that
Eqs.
EBI],
R ,, = x - R ~ ( ~ c / n l ) ,
Image radius
xRp
Actual radius of the circle
R.~r
-
no
Eli
which is the equation required to correct for the radial distortion of the interface image. No distortion of the image occurs in the axial direction. Refraction of the light rays by the outer air-glass interface was prevented by an optical flat prepared with a microscope slide cover slip and decalin between the slip and the glass tube. Capillary forces held the fluid in place
Journal of Colloid and Interface Science,
VoL50,No.2,February1975
240
RICHARD L. HOFFMAN
and the refractive index of the decalin matched the refractive index of the glass. APPENDIX C: VELOCITY CORRECTION FOR THE BACKFLOW OF LIQUID
The interface velocity differs from the plunger velocity by an amount proportional to the backflow of the liquid into the space between the plunger and the glass tube. The three factors causing this backflow are simple shearing flow caused by the relative motion of the plunger and glass tube, flow caused by surface tension at the liquid-air interface between the plunger and the glass tube, and flow caused by pressure built up in the slug of liquid ahead of the plunger. One can make calculations to show that the latter effect is negligible and the flow caused by surface tension is negligible at all values of ~V/3' > 2 X 10-~. The correction for backflow due to the relative motions of the plunger and the glass tube is determined as follows. Relative to the glass tube, the plunger has a velocity W?I and, since the gap between the plunger and the tube is small compared to the tube radius, we expect the average velocity of the liquid in this gap to be essentially Vp1/2. Thus the velocity of the advancing liquid air interface, Vr, becomes
Vr
= { VpI~'R~ 2 -
½
× ( v ~ l ~-R~,~ -
v.1 ~R~)}/~R~
or v , = vp~ (1 - ~ [ 1 -
(R~I/R~)~),
EclJ
where Rp1 is the radius of the plunger and Rp is the radius of the glass tube. Over the length of the plunger involved in this flow, the diameter varied from 1.829 to 1.880 mm with the main portion of it being 1.880 ram. Variations in the radius of the precision bore glass tubing should be negligible. Thus from ECI~ one finds that 0.94 < (VI/Vp1) <~0,96 and, for the calculations in this study, we used V1 = 0.96 Vp1. The correction for backflow caused by surface tension at the liquid air interface
between the plunger and the glass tube can be calculated assuming that the liquid wets both the plunger and the tube with a radius of curvature equal to one-half the distance between them. Considering this to be the worst case possible, one finds that the error in V~ incurred by neglecting this effect is less than ten percent when (nV/~)L2 > 2 X 10.4 (inches), where L2 is the distance between the interface in this annular region and the tip of the plunger. For the tests under consideration in this paper, L~ was generally greater than one inch, and the correction for this surface tension effect was ignored at all levels of ~V/7. REFERENCES 1. Htr~I, C., AND SCmVEN, L. E., J. Colloid Interface Sd. 35, 85 (1971). 2. ZISMAN, W. A., Advan. Chem. Ser. 43, 1 (1964). 3. ADA3LSON,A. M., "Physical Chemistry of Surfaces," 2nd ed. Interscience, New York, 1967. 4. ELLIOTT, G. E. P., AND RIDDIFORD,A. C., Recent Progr. Surface Sci. 2, 111 (1964). 5. WEST, G. D., Proc. Roy. Soc., London Set. A 86, 20 (1911). 6. SCHWARTZ,A. M., RADER, C. A., AND HUEY, E., Advan. Chem. Set. 43, 250 (1964). 7. KAt~NIS,A., AVIDMASON, S. G., J. Colloid Interface Sci. 23, 120 (1967). 8. GAUTHIER,V., GOLDSMITH,H. L., AVIDMASON, S. G., Trans. Soc. Rheol. 15, 297 (1971). 9. MO~'FATT,H. K., J. Fluid Mech. 18, 1 (1964). 10. BHATTACI-IARJI,S., AND SAVlC, P., "Proceedings of Heat Transfer Fluid Mechanic Institute," p. 248, 1965. 11. BATAILLE,J., Compt. Rend. Acad. Sci. Paris 262, 843 (1966). 12. ARIBERT, J., Compt. Rend. Acad. Sci. Paris 264, 914 (1967) ; 265, 81 (1967). 13. DIJDA, J. L., AND VRE~ITAS,J. S., J. Fluid Mech. 45, 247 (1971). 14. HANSEN, R. J., AND TOONO, T. Y., J. Colloid Interface Sci. 37, 196 (1971), 15. CATCHPOLE, J. P., AND FULFORD, G., Ind. Eng. Chem. 58, 46 (1966). 16. ELL1OTT, G. E. P., AND RIDDIFORD, A. C., J. Colloid Interface Sci. 23, 389 (1967). 17. LOWE, A. C., AND RIDDIFORD,A. C., J. Chem. Soc. D 387 (1970). 18. ELLIOTT, G. E. P., AND RIDDI:FORD,A. C., Nature 195, 795 (1962).
Journal of Colloid and Interface Science, Vol. 50, No. 2. February 1975
SHAPE OF ADVANCING,~INTERFACE 19. ROSE, W., ANDHEIRS, R. W., Y. Colloid Sci. 17, 39 (1962). 20. HANSEN, R. J., AND TOONG, T. Y., 3. Colloid Interface Sci. 36, 410 (1971). 21. HANSEN, R. J,, Sc. D. Thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 1968. 22. DERYAGIN,B. V., AND LEVI S. ~V[.,"Film Coating Theory," Focal Press, London, 1964.
241
23. PERRY, ~_. T., Ph.D. Thesis, University of Minnesota, 1967. 24. JOHANSSON, O. K., AND TOROK, J. j., Proc. Inst. Radio Eng. 34, 296 (1964). 20, PHILLIPS, M. C., AND RIDDIFORD,A. C., J. Colloid Interface Sci. 41, 77 (1972). 26. LowE, A. C., AND RIDDIFORD, A. C., Canad. J. Chem. 48, 865 (1970).
Journal of Colloid and InterJace Science. Vol. 50. No. 2. February 1975
Received February 18, 1974; accepted August 23, 1974 The shape of the advancing liquid-air interface has been studied in a glass capillary over
the range in which viscous and interfacial forces are the dominant factors controlling the system. Plotting the apparent contact angle versus the capillary number plus a shift factor, one obtains a curve which correlates all the data. In the systems examined experimentally in this study, the shift factor is determined solely by the static contact angle between the liquid and the solid substrate. To generate the curve, data were first obtained over the full range of apparent contact angles from 0 to 180° with liquids having shift factors which were equal to zero or negligible. Then the validity of the correlation was tested with data on liquids requiring large shift factors and the results were found to agree with the data obtained on the other liquids. INTRODUCTION W h e n analyzing the flow of a liquid as it moves over a solid surface and displaces a gas, one encounters problems which have not y e t been adequately resolved in the literature. One is a complete definition of the shape of the liquid-gas interface as a function of pertinent dimensionless parameters. As this shape determines b o u n d a r y conditions of the flow, it is i m p o r t a n t t h a t the factors which determine the interface shape are well understood. Another problem is encountered when one considers flow near the contact line of the advancing interface with the solid wall. Application of the classical no-slip b o u n d a r y condition in various m a t h models has led to the unlikely result t h a t the shear stress and pressure fields increase without bound as the contact line is approached (1). The problem of interface shape is considered in this article, and in a subsequent article we will discuss the analysis of flow behind one interface shape
where shear stress and pressure fields in the liquid phase do not increase without bound at the contact line. Although interface shapes and contact lines between a solid, liquid, and a fluid have attracted considerable scientific attention (2,3), most of the studies are concerned with static systems and few with d y n a m i c systems. The limited n u m b e r of studies of d y n a m i c systems can be further divided into two distinct categories, those in which the advancing interface is driven with a c o n s t a n t forward velocity over the solid s u b s t r a t e and those in which the interface shape is in a nonequilibrium state and changing as a result of its recent displacement to a new position. Elliott and Riddiford (4) have suggested t h a t studies of driven interfaces should be distinguished from studies of nonequilibrium interfaces b y the terms d y n a m i c studies and relaxation studies, respectively. Under this distinction we will concern ourselves with dynamic studies only. 228
Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
Copyright • 1975 by Academic Press, Inc. All rights of reproduction in any form reserved.
SHAPE OF ADVANCING INTERFACE Considering flow the dynamic contact line, West (5) reasoned that liquid, moving in an air-filled capillary tube, should move forward and spread outward from the center region of flow in the advancing meniscus, and fold inward to the center region of the receding meniscus. Since West's pioneering study, the occurrence of this general flow pattern has been confirmed experimentally in Newtonian and non-Newtonian systems (6-8), and predicted by various theoretical studies (9-14). More complex flow patterns, generally in the air phase, may occur as suggested by the work of Huh and Scriven (1). Details of the flow pattern remain to be worked out, however, for although the velocity fields predicted are reasonable, the theoretical studies always specify stress and pressure fields which increase without bound at the contact line. These problems aside, one expects that a study of interface shape can be conveniently broken down into a study of several different flow regimes. For simplicity we consider dynamic systems in which a liquid advances to displace a gas and in which gravity forces are negligible. Examining the spectrum of velocities from zero to large values with Newtonian fluids at various viscosity levels, we know that four types of forces will come into play; namely viscous forces, inertial forces and interfacial forces at the liquid-gas interface and the liquid-gas-solid juncture. Which set of these forces is important depends upon the system and the flow rate. To compare magnitudes of the first three forces we use the capillary number, Ca = nV/'y, which is the ratio of viscous forces to interfacial forces at the gas-liquid interface and the Weber number, W e = pV2L/'y, which is the ratio of inertial forces to interfacial forces at the gas-liquid interface (15). In these equations ~ is the viscosity of the liquid, V the interface velocity, 7 the surface tension at the gas-liquid interface, p the density of the liquid and L a characteristic dimension of the system. In capillary systems L will be taken as the tube diameter, and for a meniscus ad-
229
vancing between parallel flat plates, L will be taken as the spacing between the plates. Interfacial forces at the liquid-gas-solid juncture must also be considered. The procedure for dealing with the effect of these forces on interface shape in systems where inertial and interfacial forces are dominant has yet to be established, but with systems in which viscous and interfacial forces are dominant, one can use a dimensionless shift factor which is added to the capillary number. For tile systems examined experimentally in this study, this shift factor depends solely upon the static contact angle and the concepts for obtaining the shift factor for these systems are developed in this study. In a review of the literature, one finds that various studies have touched upon a number of flow regimes as shown by Table I. Riddiford et al. (16-18) have studied the first regime in which interfacial forces dominate, and tile studies of Rose and Heins (19) and Hansen and Toong (20, 21) touch upon flow in the second regime where interfacial and viscous forces predominate. But some of the data obtained by Hansen and Toong appear to enter into the regime where interfacial, inertial, and viscous forces all play a role. Being aware of these flow regimes and the possibility of moving from one to another, we find that the work of Hansen and Toong has raised some unanswered questions. For example, Hansen and Toong have reported apparent contact angles and interface shapes as they are correlated by the capillary number. At low values of the capillary number, they find the interface between a heavy paraffin oil and air is a spherical sector, but at high values of the capillary number, for example nV/'g = 0.212, they report deviations from the spherical shape. One finds, however, that when ~ V / 7 = 0.212 for this system, the Weber number is 0.046. From calculations given in Appendix A, we infer that at this high a Weber number inertial forces may also be important. Thus the question arises as to whether the interface is distorted from a
Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
230
R I C H A R D L. H O F F M A N TABLE I RANGES OF THE CAPILLARYAND WEBER NUMBERS COVERED BY VARIOUS LITERATURE STUDIES
Interracial Ref.
Dominant Forces Interfaeial/viscous
Interfacial/ inertial
Interfacial/inertial/ viscous
}"
16, 17, 18
Ca < 10.5 We < 3 X 10.6
19 (Nujol)
1.4 X 10-4 =< Ca =< 7.6 X 10-~ 3 . 2 X 10-8=< W e < ~ 9 X 10-s
19 (oleic acid)
7.9 N 10-5 < Ca < 2.8 X 10-3 1.8 X 10-7 ~ We <~ 3.6 X 10.4
14, 20, 21
1.7 X 10-5 < Ca < 3.9 X 10-* 3.0 X 10-1° < We < 1.5 X 10-~
This stu dy
4 X 10- 5 < C a < = 3 6 5.4 X 10-1I < We < 1.6 M 10.5
spherical shape b y viscous forces, inertial forces, or b y both. I n the s t u d y to be discussed in this paper, inertial forces were m a i n t a i n e d at a negligible level while the interface shape was examined over the range of 4 X l 0 -5 <_ ~ V / 7 < 36. Deviations from a spherical l i q u i d - a i r interface were never observed. Thus we suspect t h a t the distortions observed b y Hansen and Toong were caused b y inertial forces. Studies of the flow regime where only interfacial and inertial forces dominate in capillary or parallel flat plate systems are unknown to this author. Similar remarks apply to any s t u d y designed to examine the combined effects of viscous, inertial, and interfacial forces upon interface shape when all three are important. Two other flow regimes which deserve brief mention in this review are those in which viscous and the combination of viscous and inertial forces are the sole factors controlling interface shape. W h e n viscous forces alone control the interface shape, the a p p a r e n t contact angle will be 180 ° and invariant. F r o m the results of this s t u d y we infer t h a t this occurs when We<< 0.01 and Ca + F(O~) > 10, where F(O,) is the shift factor.
Ca = 0.212 We = 0.046
On the other h a n d when the combination of viscous and inertial forces control interface shape, one expects t h a t these forces will u l t i m a t e l y lead to the e n t r a i n m e n t of a thin film of gas between the liquid and solid which displaces their juncture downstream into the liquid phase. We know of no a t t e m p t to s t u d y this flow regime in capillaries or between flat plates, but the effect has occurred in film coating studies b y D e r y a g i n and Levi (22) and P e r r y (23). As evidenced b y their studies, this e n t r a i n m e n t occurs once the contact angle reaches 180 ° and W e > 0.4 L while 0.02 < Ca < 0.2. F o r these systems the characteristic length, L in cm, can p r o b a b l y be taken as the distance normal to the film over which the bulk of the meniscus curvature is observed. Returning to a consideration of the flow regime in which viscous and interfacial forces are dominant, one finds t h a t the d a t a obtained by Rose and Heins and Hansen and Toong cover only a small portion of this flow regime. Thus, a systematic s t u d y of the full range over which viscous and interfacial forces have a dominant effect on the interface shape is still required. I t is the purpose of this p a p e r to present this s t u d y and to this end we con-
Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
SHAPE OF ADVANCING INTERFACE
231
A-~
~ P OLYCARBONATE BASE ~HEX NUT GLUEDTO DRIVE PLATE YlO. 1. Advancing meniscus apparatus: AFI--advancing fluid interface, PBT--precision bore tube, A--locknuts and Teflon 0-Rings, C--connection to variable speed drive.
sider only the case of a liquid which moves over a solid surface and displaces a gas. From this study we find that the interface shape in this flow regime is directly related to the capillary number plus a shift factor. The need for the shift factor, although previously unused, is established by this study, and severat liquid-solid-gas combinations were tested to demonstrate this fact. EQUIPMENT AND PROCEDURES
The advancing meniscus apparatus used in this study is shown in Fig. 1. Simple in design, the instrument centers about a precision bore glass tube nominally five inches long with an inside diameter of 1.955 mm. A slug of liquid, one to three centimeters long, is drawn into the tube followed by a stainless steel plunger with an outside diameter of roughly 1.880 mm. The glass capillary is then inserted into holes in the Teflon guides which hold the tube snugly. Vee notches in the guides rest on the upper two brass rods shown in Fig. 1. Two more brass rods located three-quarters of an inch directly beneath those shown provide a rigid frame for the apparatus. Driving the tube over the plunger via a screw (1/8-40 NC-2A) connected to a variable speed motor, one obtains an advancing liquid-air interface which is essentially stationary with respect to lab coordinates. For the tests described in this paper the interface velocity varied from roughly 0.00008 to 0.06 cm/sec. The interface
was viewed and photographed through a Bausch and Lomb Stereozoom 7 microscope, Cat. No. 31-27-01, which was tilted so that the optical axis of the lens through which the interface was photographed was normal to the axis of the glass tube. An optical flat, prepared with a microscope slide coverslip and decalin between the slip and the glass tube prevented refraction of the light rays by the external glass-air interface. A correction for the refraction of the interface image by the liquid-glass or air-glass interface inside the tube is made using the refractive index of the glass and the fluid through which the image is transmitted to the glass. See Appendix B. The plunger velocity relative to the tube is obtained by measuring the revolutions per minute of the screw driving the glass tube over the plunger. The interface velocity differs from the plunger velocity by an amount proportional to the backflow of liquid in the space between the tube and the plunger. See Appendix C. The presence of the plunger at one end of the liquid slug will have some effect on the flow therein. Our concern is that its presence has no effect on the flow field near the advancing interface. Some studies in the literature have dealt with this problem and, from the results obtained, one finds that the flow is essentially fully developed at a distance of two pipe diameters from the interface (7, 10, 12, 13). Thus, with a liquid slug 1-3 cm long ahead of
Journal of Colloid and Inlerface Science, Vol. 50, No. 2, February 1975
232
RICHARD L. HOFFMAN
(a) (b)
Fro. 2. Photographs of the advancing liquid-air interface. In each photograph the liquid is left of the interface. Photograph (a) shows a liquid at rest with Oi = 0, but in the other two photographs the velocity is high enough to bow the interface out into the air phase.
(c) the plunger, the plunger is not expected to affect the flow field near the advancing interface. With the inside diameter of the precision bore tube used in these experiments, gravity forces can be neglected, and the radius of
PIPE WALL
I
FIG. 3. Quantities measured on the interface image.
curvature can be assumed constant. Hansen and Toong (14, 20, 21) question this assumption suggesting that the curvature m a y change in a range of 10.5 cm or less from the contact line which the advancing interface makes with the solid. Molecular interactions m a y dominate here and preclude the use of continuum concepts. Although this m a y be true, it does not alter the fact that the major portion of the flow field is determined by the interface shape~'external to this region. Hansen and Toong also presented data which show major distortion of the interface, but as noted in the introduction of this paper, the distortion apparently results from inertial effects. Measurements at various positions on the meniscus profile in our tests always indicated a constant radius of curvature within our measurement capabilities. (Changes in the curvature at tess than 10.5 cm from the contact line, of course, were not detectable.)
Journal of Colloid and Interface Science, Vol. 50, No. 2, F e b r u a r y 1975
233
S H A P E OF A D V A N C I N G I N T E R F A C E
To determine the radius of curvature, photographs of the advancing interface were obtained as shown in Fig. 2. Measurements of xR2, and H as shown on Fig. 3 were then obtained with the aid of a magnifying glass and a rule calibrated in hundredths of an inch. The equation Image radius
-
Actual radius of the circle
xRp RM
=
nl _
_
[X]
na
derived in Appendix B, enables one to obtain RM and then one determines the radius of curvature, Re, of the advancing interface by the equation Ro = { E R ~ 2 +
(Ro- H)231/2}/M, I-2 3
where M is the magnification, and in Eq. I-i] no and ns are the refractive indices of the glass and the fluid between the meniscus and the glass, respectively. Thus the equation for the cosine of the apparent contact angle, 0M, becomes cos 0M = Rp/Rc,
[-3]
where we neglect any changes in R~ which may occur close to the wall. See Fig. 4. To determine the magnification obtained in the pictures used for these calculations, the inside diameter of the glass tube was also measured on one picture from each set of runs. The tube diameter of 1.955 4-0.01 m m was originally obtained by photographing the tube and a calibrated microscope grid under the same magnification. The measurement of H and xR~ is a variable
@k, I
R
FLUID
~
AIR
FIG. 4. Apparent contact angle of an advancing liquid-air interface.
source of error which can generally be minimized by measuring the largest xR~ and associated H possible. In theory this would imply that one should measure H at the position where the interface contacts the wall making RM equal to the pipe radius. In practice, however, the interface image near the wall is lost, due to the critical angle of refraction, and thus the contact line of the interface with the wall is not visible. This being the case, the xRp and H values measured will generally be somewhat less than the theoretical maximum. Just how much less depends on lighting conditions and the refractive indices of the glass and the fluid through which the image is viewed. In calculating OM,the greatest source of error comes from the measurement of H and xRp. With the measurements made in this study we estimate the error was usually of the order of 5% for each value although in a few cases one or both measurements may have been in error by as much as 10%. Five different liquids were used in this study. The pertinent material properties of
T A B L E II 1V][ATERIALPARAMETERS Ot" TEST LIQUIDS Liquid
Viscosity (poise)
Specific gravity 25°C/25°C
Surface tension (dyn/cm)
Refractive I ndex
Static contact angle (degrees)
G.E. Silicone Fluid SF-96 Brookfield Std. Viscosity Fluid Dow C o m i n g 200 Fluid Ashland Chem. Admex 760 Santicizer 405
9.58 988 24,300 1093 112
0.974 0.977 0.98 1.15 1.13
21.3 21.7 21 43.8 43.4
1.405 1.405 1.405 1.471 1.477
0 0 12 69 67
Journal of Colloid and Inie~Jace Science, Vol. 50, No. 2, February 1975
234
RICHARD L. HOFFMAN
the liquids are listed in Table II. All values are at 24°C unless as otherwise noted. Viscosity measurements in cone and plate flow were made with the Weissenberg Rheogoniometer, Model R-16, manufactured by Sangamo Controls, Ltd., Bognor Regis, England. The surface tension of each liquid was obtained with a DuNouy ring tensiometer. Values for the specific gravity of each liquid are those listed by the manufacturer, and the static contact angles given are those measured in the advancing meniscus apparatus built for this study. Under the convention used, a liquid which completely wets the glass will have a static contact angle of zero. Unlike all the other materials, Admex 760 and Santicizer 405 were tested in the capillary after the glass surface had been altered by a silicone treatment used to give a large static contact angle as desired. To obtain this large contact angle, the glass was treated by a procedure outlined in Ref. (24). A 2% solution of a Dow Corning type 200 silicone fluid (200 cs) in tetrachloroethylene was used for the treatment, and after the silicone was applied, the capillary was heated at 100°C for one hour and then baked at 275°C for roughly two and one-half hours. Prior to testing the tube was then rinsed out with reagent grade 2-butanone. All the liquids tested were Newtonian in their viscosity behavior over the applicable range of shear rates with the exception of the Dow Coming 200 fluid. This material which was essentially Newtonian at the lower shear rates became pseudoplastic in the region of higher shear. For our purposes, however, we assumed that the material was Newtonian with a viscosity of 24,300 poise. For all the data taken this amounts to an error of less than 4-10% except for the one point at ~V/'y = 36. Here at the maximum shear rate at the wall the viscosity is roughly 20,000 poise. RESULTS AND DISCUSSION Data obtained on various liquid systems via the procedures outlined above are listed in Table I I I and plotted in Fig. 5 as the apparent
contact angle OM versus nV/~,+ F(O,). A few of the data points listed in Table I I I are omitted from Fig. 5 because of crowding. The term, F(Os), generally a constant for any liquid-solid-gas system, is a shift factor which compensates for the fact that the contact angle resulting from interfacial forces alone will vary with each liquid-solid-gas system considered. The method for determining F(O,) depends upon one of two types of behavior. For some systems one finds that interfacial forces between the solid and the liquid do not change when flow occurs, but with other systems changes such as those reported principally by Riddiford and co-workers can occur (4, 16-18, 25, 26). Let us consider first those systems in which a change does not occur. In such systems the shift factor, F(O,), is determined solely by the static contact angle, 08, and is zero when 0, is zero. Thus, when 08 is zero, the correlation between the apparent contact angle, OM, and nV/~ + F(O,) is easily established because F(O~) is zero and all other parameters can be measured. When the static contact angle is not zero, the same correlation applies if the shift factor F(O~) is used. To determine the shift factor one need only measure the static contact angle, 0~, and then read the value of ~V/'r + F(O~)= O+ F(O~) from Fig. 5 when 0~ is used as 0 u a n d V = 0. Using the concepts outlined above, we established Fig. 5 by first testing the silicone oils for which 08 and thus F(O~) were zero. See Table II. In generating the curve, data from the G. E. and Brookfield silicone fluids were used to establish F(O,) for the Dow Coming silicone fluid which has a static contact angle slightly greater than zero. Over the range tested, however, F(O,) makes a negligible correction to the data obtained for this liquid. Once the curve was established with data on the silicone fluids for which F(O~) was zero or negligible, the data on Admex 760 was used to test the concepts involving the shift factor. Since 08 = 69 ° we find from Fig. 5 that F(O,) = 2.35 N 10-2, and using this shift
Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
SHAPE OF ADVANCING INTERFACE
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Journal of Colloid and Interface Science, Vol. 50, No, 2, February 1975
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, ,ill
tO a
FIG. 5. Effect of flow on the apparent contact angle of an advancing liquid-air interface: (e) G.E. Silicone SF-96, F(0.~) = 0; (&) Brookfield Silicone, F(O~) = 0; (q?) Dow Corning Silicone, F(O,~) = 1.5 ;< 10-4; (El) Ashland Chemical Admex 760, F(O,) = 2.35 X 10-2; (+) Santielzer 405, F(O~) = 2.1 X 10-2; the data were taken at 24°C. factor one finds that the data fit the curve established with the silicone oils. Similar results were obtained with Santicizer 405 for which 08 = 67 °. Figure 6 is an expanded plot of a portion of our data along with that obtained by other authors (14, 19, 20). Their data provide us with further tests of the shift factor concept. Static contact angles for the liquids studied by the other authors were estimated by extrapolation to zero velocity in each case. These values were then used to obtain a shift factor for each system by the procedures outlined above. The data of Hansen and Toong agree very well with our data as do the points obtained from their theoretical work. On the other hand, the data of Rose and Heins fall somewhat above the data obtained in this study. The reason for the difference is unknown unless it involves experimental techniques a n d / or incorrect estimates of the static contact angles which were obtained by extrapolating their data to zero velocity. We do find, however, that the apparent contact angle, 0~, Journal of Colloid and Interface Science,
VoL
,50, No.
2, February
becomes progressively less sensitive to the capillary number as values of the capillary number decrease below F(O~). This must be true if the concept of the shift factor is correct. Considering all the data in Figs. 5 and 6 and reasoning inductively, we expect that the curve of Fig. 5 characterizes the shape of any liquid-air interface in motion when only viscous and interfacial forces are important, interfacial forces are not changed by motion, the viscosity behavior is Newtonian and the viscosity of the air is negligible when compared to the liquid. Up to now we have been considering systems in which the interfacial forces between the solid and the liquid do not change when flow occurs. There are some systems, however, where changes do occur. Riddiford and coworkers have done the bulk of the work in this area (4, 16-18, 25, 26). For the liquidsolid-gas systems which they have studied, they find that the apparent contact angle is usually independent of velocity at very low rates. But as the interfacial velocity is in1975
SHAPE OF ADVANCING INTERFACE .....
1
i
"1
I
I'l
li
I
I
1
I
237
I
I
I I I I
I
I
I
I
11
I
I
I Ill
I
I
I
I
I
I I'
120
IO0
~o
~
40
I~l
® 20
I
10-4
I
I
I
I
Ill[
I
10-3
I
I
"lTV + P(O s) 7/
10-1
II
10-1
FIG. 6. Effect of flow on the apparent contact angle of an advancing liquid-air interface: ( I ) G.E. Silicone SF-96, capillary diameter (ID) = 0.1955 crn, 0~ = 0 ° ; (~x) Brookfield Silicone, ID = 0.1955 cm, 0s = 0°; (@) Nujol, Hansen, and Toong (20), ID = 0.2380 cm, 0., = 22°; (®) Nujol, Hansen, and Toong (20), ID = 0.1210 cm, 08 = 36°; (X) Theory values, Hansen, and Toong (14), 0,~ = 21°; (E]) Nujol, Rose, and Heins (19), ID = 0.066 and 0.110 cm, 0~ = 23°; (®) Oleic acid, Rose and Heins (19), ID = 0,066 and 0.110, 0~ = 32°. creased above a critical value, the apparent contact angle changes with velocity until at higher speeds the rate of change diminishes and a limiting value is reached. The limiting value is generally less than 180 ° . To explain their results, Riddiford et al. postulate that molecular orientation occurs at the advancing liquid-solid-gas boundary when interface velocities are low enough, but as velocities are increased the molecules become randomly oriented at the boundary with an attendant change in the interracial forces and thus the apparent contact angle. A new but constant contact angle, 0~x, finally results from this process. To correlate data on systems such as these, one can no longer use the static contact angle, 0,, to determine the shift factor, F (08) because the interracial forces have changed at the advancing contact line. I t should be possible in m a n y cases, however, to handle these systems without difficulty. For example, all the shifts in contact angle observed by Riddi-
ford and co-workers generally occur before viscous forces begin to have an appreciable effect on the interface shape, As a case in point, with the water systems on various solid substrates, the shift is complete when Ca < 3 X 10-6, but with all the static contact angles being >__70° we see from Fig. 5 that viscous forces should have no effect until Ca is the order of 2 X 10-2. This being the case, the correlation of Fig. 5 should still apply if the shift factor F(O~) is determined by using the apparent contact angle 0M~x instead of 0~. An experimental test of this hypothesis has not been made. SUMMARY AND CONCLUSIONS Studying the shape of an advancing interface in a liquid gas system, we find that the apparent contact angle can be correlated as a function of the capillary number plus a shift factor when interracial and viscous forces are the dominant factors controlling the system. When interracial forces between a solid and a
Journal of Colloid and Interface Science, Vol. 50, No. 2, F e b r u a r y 1975
238
RICHARD L. HOFFMAN
1,2
I ,//'--PIPE WALL ........ -7- ............. R-LIQUID ~-GAS -TR~~__~_ __~ _~_ _IZF%-C_ NTEIR E~-_-_'~ " l I
I I I
FIG. 7. Advancing interface region considered in the momentum balance.
where v = v(r) is the local fluid velocity, R~a is the radius of curvature of the advancing interface and R , is the radius of curvature of the interface when no fluid motion occurs, p is the liquid density and 3' the surface tension. By rearranging Eq. I-All and defining the dimensionless parameters (v/V) =
v0;
(r/R)
=
r0;
and (Rc~/R) = (R~,)o
liquid do not change as flow occurs, the shift factor is determined solely by the static contact angle between the liquid and the solid substrate. Although untested a technique is also suggested to obtain the shift factor for systems in which interfacial forces do change. As a result of this study, data is now available over the complete range of contact angles from 0 to 180 °, and other flow regimes are delineated in which different sets of forces control the interface shape. APPENDIX A: DISTORTION OF THE ADVANCING INTERFACE BY INERTIAL FORCES To estimate the minimum Weber number at which inertial forces begin to affect interface shape, consider a fluid whose static interface bows out into the gas phase with a radius of curvature equal to the pipe radius. (Alternatively one can consider a highly viscous fluid moving with a velocity in excess of that required to give C a + F(O,) > 1 0 2 SO that 0M = 180°.) This being the case, viscous forces have no effect on the interface shape. See Fig. 5. To facilitate the analysis, we place a cylindrical coordinate system in the interface at the center line of the pipe and stipulate that it moves with the interface. See Fig. 7. The arrows left of the plane 1, 2 show the local direction of flow relative to the convected coordinate system. Writing a momentum balance about the fluid in the crosshatched region of Fig. 7, one finds that L R Ov227rrdr = fo R 2ERTa
23"] 2~rrdr,
R~/
EAI-]
one obtains 1 RpV 2 fl
2
v
vo2rodro
Jo = f 0 1 ro 1 - -1
-1
(Rca) o
1 dro,
[A2]
(Res)o
where V is the average fluid velocity relative to a stationary coordinate system and R is the pipe radius. (Rc~)0 = 1 as required by our previous assumptions. Now since inertial forces are balanced by the pressure generated through distortion of the interface, let us presume, as a first approximation, that inertial forces cause a 1% deviation from the radius of curvature of the static interface only in the region r0 _< ½. Thus we have (Roe)o=0.99
for r o < ½
and (R~a)o= 1.0 for ½ < r 0 _ < 1. When these conditions are applied to Eq. I-A2] one finds that 1 RpV 2 fl
2
Jo
7
1
[-A3~
vo2rodro -
792
or upon rearrangement Rp V2 We--
7
[A4]
1
396
L
vo2rodro
for the 1% deviation prescribed. Before inertial forces become important, Vo at the plane 1, 2 should be approximated reasonably well by the parabolic velocity profile found far from
Journal of Colloid and Interlace Science, Vol. 50, No. 2, February'1975
SHAPE OF ADVANCING INTERFACE
~ ~ % i /
239
LINENORMALTO THE PIPEWALL
otS;zfY-. . . . . . . . . . . . . . .
'lRp
"*"
POSITION ON THE > MENISCUS
RADIAL
~ ~ I D E
WALLOF THEGLASSTUBE
FIG. 8. Schematic of light refraction which leads to radial distortion of the liquid-air interface image: The view shown is normal to the axis of the pipe. the advancing interface, given by v0 = 1 -- 2r0~
[A5]
in our convected coordinate system. The justification for this approximation arises from results derived in the second paper of this series. For the purposes of this analysis we assume that the velocity profile is well approximated by Eq. [A5] since inertial forces are just beginning to be important. Turbulence should not be a factor since we can set Ca and thus the fluid viscosity as high as we wish. Thus by substituting Eq. [A5] into Eq. [A4] and integrating one finds that the 1% deviation prescribed for our system occurs when 1
We = - - ~ 0.015. 66
[A6]
Strictly speaking, this result applies only when the interface is bowed out into the gas phase with an apparent contact angle equal to I80 °. With different values of the apparent contact angle, one may expect some change in the We required for a 1% deviation in the radius of curvature. The situation for other apparent contact angles has not been examined, however, because the results should be nominally the same. APPENDIX B: CORRECTIONS FOR THE REFRACTION OF LIGHT A correction must be made for refraction of the interface image by the liquid-glass or
air-glass interface inside the tube. To determine this correction, consider the polymerair interface, i.e., the meniscus, and the inside of the glass tube as shown in Fig. 8. Part of a circle formed by the intersection of the meniscus and a plane normal to the axis of the pipe is shown with radius RM. From Snell's law, one finds that ns sin ¢ = n~ sin 0,
[B1]
where no and n s are the refractive indices of the glass and the fluid between the meniscus and the glass, respectively. Further we know that sin ~ = RM/Rp [B2] and sin 0 = xRp/Rp = x, [B3] where 0_< x < 1 . Combining [B2], and [B3], one finds that
Eqs.
EBI],
R ,, = x - R ~ ( ~ c / n l ) ,
Image radius
xRp
Actual radius of the circle
R.~r
-
no
Eli
which is the equation required to correct for the radial distortion of the interface image. No distortion of the image occurs in the axial direction. Refraction of the light rays by the outer air-glass interface was prevented by an optical flat prepared with a microscope slide cover slip and decalin between the slip and the glass tube. Capillary forces held the fluid in place
Journal of Colloid and Interface Science,
VoL50,No.2,February1975
240
RICHARD L. HOFFMAN
and the refractive index of the decalin matched the refractive index of the glass. APPENDIX C: VELOCITY CORRECTION FOR THE BACKFLOW OF LIQUID
The interface velocity differs from the plunger velocity by an amount proportional to the backflow of the liquid into the space between the plunger and the glass tube. The three factors causing this backflow are simple shearing flow caused by the relative motion of the plunger and glass tube, flow caused by surface tension at the liquid-air interface between the plunger and the glass tube, and flow caused by pressure built up in the slug of liquid ahead of the plunger. One can make calculations to show that the latter effect is negligible and the flow caused by surface tension is negligible at all values of ~V/3' > 2 X 10-~. The correction for backflow due to the relative motions of the plunger and the glass tube is determined as follows. Relative to the glass tube, the plunger has a velocity W?I and, since the gap between the plunger and the tube is small compared to the tube radius, we expect the average velocity of the liquid in this gap to be essentially Vp1/2. Thus the velocity of the advancing liquid air interface, Vr, becomes
Vr
= { VpI~'R~ 2 -
½
× ( v ~ l ~-R~,~ -
v.1 ~R~)}/~R~
or v , = vp~ (1 - ~ [ 1 -
(R~I/R~)~),
EclJ
where Rp1 is the radius of the plunger and Rp is the radius of the glass tube. Over the length of the plunger involved in this flow, the diameter varied from 1.829 to 1.880 mm with the main portion of it being 1.880 ram. Variations in the radius of the precision bore glass tubing should be negligible. Thus from ECI~ one finds that 0.94 < (VI/Vp1) <~0,96 and, for the calculations in this study, we used V1 = 0.96 Vp1. The correction for backflow caused by surface tension at the liquid air interface
between the plunger and the glass tube can be calculated assuming that the liquid wets both the plunger and the tube with a radius of curvature equal to one-half the distance between them. Considering this to be the worst case possible, one finds that the error in V~ incurred by neglecting this effect is less than ten percent when (nV/~)L2 > 2 X 10.4 (inches), where L2 is the distance between the interface in this annular region and the tip of the plunger. For the tests under consideration in this paper, L~ was generally greater than one inch, and the correction for this surface tension effect was ignored at all levels of ~V/7. REFERENCES 1. Htr~I, C., AND SCmVEN, L. E., J. Colloid Interface Sd. 35, 85 (1971). 2. ZISMAN, W. A., Advan. Chem. Ser. 43, 1 (1964). 3. ADA3LSON,A. M., "Physical Chemistry of Surfaces," 2nd ed. Interscience, New York, 1967. 4. ELLIOTT, G. E. P., AND RIDDIFORD,A. C., Recent Progr. Surface Sci. 2, 111 (1964). 5. WEST, G. D., Proc. Roy. Soc., London Set. A 86, 20 (1911). 6. SCHWARTZ,A. M., RADER, C. A., AND HUEY, E., Advan. Chem. Set. 43, 250 (1964). 7. KAt~NIS,A., AVIDMASON, S. G., J. Colloid Interface Sci. 23, 120 (1967). 8. GAUTHIER,V., GOLDSMITH,H. L., AVIDMASON, S. G., Trans. Soc. Rheol. 15, 297 (1971). 9. MO~'FATT,H. K., J. Fluid Mech. 18, 1 (1964). 10. BHATTACI-IARJI,S., AND SAVlC, P., "Proceedings of Heat Transfer Fluid Mechanic Institute," p. 248, 1965. 11. BATAILLE,J., Compt. Rend. Acad. Sci. Paris 262, 843 (1966). 12. ARIBERT, J., Compt. Rend. Acad. Sci. Paris 264, 914 (1967) ; 265, 81 (1967). 13. DIJDA, J. L., AND VRE~ITAS,J. S., J. Fluid Mech. 45, 247 (1971). 14. HANSEN, R. J., AND TOONO, T. Y., J. Colloid Interface Sci. 37, 196 (1971), 15. CATCHPOLE, J. P., AND FULFORD, G., Ind. Eng. Chem. 58, 46 (1966). 16. ELL1OTT, G. E. P., AND RIDDIFORD, A. C., J. Colloid Interface Sci. 23, 389 (1967). 17. LOWE, A. C., AND RIDDIFORD,A. C., J. Chem. Soc. D 387 (1970). 18. ELLIOTT, G. E. P., AND RIDDI:FORD,A. C., Nature 195, 795 (1962).
Journal of Colloid and Interface Science, Vol. 50, No. 2. February 1975
SHAPE OF ADVANCING,~INTERFACE 19. ROSE, W., ANDHEIRS, R. W., Y. Colloid Sci. 17, 39 (1962). 20. HANSEN, R. J., AND TOONG, T. Y., 3. Colloid Interface Sci. 36, 410 (1971). 21. HANSEN, R. J,, Sc. D. Thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 1968. 22. DERYAGIN,B. V., AND LEVI S. ~V[.,"Film Coating Theory," Focal Press, London, 1964.
241
23. PERRY, ~_. T., Ph.D. Thesis, University of Minnesota, 1967. 24. JOHANSSON, O. K., AND TOROK, J. j., Proc. Inst. Radio Eng. 34, 296 (1964). 20, PHILLIPS, M. C., AND RIDDIFORD,A. C., J. Colloid Interface Sci. 41, 77 (1972). 26. LowE, A. C., AND RIDDIFORD, A. C., Canad. J. Chem. 48, 865 (1970).
Journal of Colloid and InterJace Science. Vol. 50. No. 2. February 1975