JOURNAL OF COLLOID SCIENCE ].7, 39-48 (1962)

M O V I N G INTERFACES AND CONTACT ANGLE R A T E - D E P E N D E N C Y 1

Walter Rose and R. W. Heins

University of Illinois, Urbana, Illinois

Received September 6, 1960; revised May 10, 1961

INTRODUCTION

The porous media, multiphase displacement problem discussed, for ex- ample, at length by Scheidegger (1) shows the importance of being able to describe the spontaneous imbibition and movement of wetting liquid into and through capillary channel systems initially filled with nonwetting fluid(s). An underlying problem, elementary but important, is to find a sensible way to talk about interface motion in simple (e.g., circular) capil- lary tubes under the action of surface, gravity, and/or hydrodynamic forces.

We may suppose that the Navier-Stokes equations apply (1), and, to- gether with the appropriate equations of state and the mass and energy continuity conditions, the relevant equations of motion readily are ob- tained. A difficulty arises, however, in attempting to relate the actions of the surface forces (as necessarily viewed on the molecular scale) to the resultant capillary driving force (meaningful with reference to the actions on fluid"particles" comprised of a statistically large collection of contiguous molecules). The latter must take the form of the pressure gradient vec- tor appearing in the Navier-Stokes equations, which are acting in the vi- cinity of the moving fluid-fluid interface.

The recent papers of Buff (2) and Scriven (3) refer to and cite the per- tinent literature bearing on this tremendously complicated question of what gives rise (in the fundamental sense) to interface shapes and motions. We choose to circumvent much of the analytic difficulty others have considered, however, by describing an experimental method intended approximately to measure the capillary driving force directly.

From our work, we have obtained new evidence of the inherent rate- dependency of the capillary driving force, which feature ultimately may prove useful in checking various fundamental theories of capillary action. A.lso, it is suggested that our data on capillary tube systems have direct application to the development of theories of displacement in porous media of complex internal geometry.

1 This work was supported in par t by funds received from the Chemical Warfare Laboratories, Directorate of Research.

.~9

40 ROSE AND HEINS

THEORY

Since it is our purpose to study rate-dependency, we choose an experi- mental system where the capillary flow will occur under steady-state con- ditions. In this way acceleration effects are avoided, and we may apply directly a Poiseuille-like treatment. For example, if we are looking at the motion of a slug of liquid (of constant length) traveling at constant rate through a circular capillary, this will be achieved by imposing a constant pressure drop to the air across the ends of the tube. Assuming the liquid to be incompressible and neglecting the viscous drag of the air in the tube, we may then write:

oo 0 ,) ,,, v =

where R is the tube radius, L is the liquid slug length, n is the liquid vis- cosity, ~ P is the pressure drop (driving force), ~ is the surface tension of the liquid, and ~A and 62 are the advancing and receding contact angles, re- spectively.

The question now to be examined is whether or not the contact angle can be taken as a constant of the system (i.e., rate-independent) along with the other fixed parameters (i.e., R, L, 7, and ~/). An affirmative answer is im- plicit, for example, in the work of Barter (4) and Brittin (5), but the data of this paper imply that an interaction exists; this means that contact angle must be taken as a rate-dependent variable.

In effect, our procedure rests on the assumption that the curvature of a slowly moving fluid-fluid interface, advancing or receding along a tube of uniform (circular) section, will be a constant curvature surface; hence, the angle of contact will be a direct measure of this curvature as in the case of stationary interfaces under conditions of static equilibrium. With this assumption, the equations of motion of the moving interface follow, as given, for example, by Barrer (4) (who limits attention to cases where the acceleration terms of the Navier-Stokes equations can be dropped) and Brittin (5) (who retains the acceleration terms in an approximate sort of way).

Various investigators already have in fact postulated that the contact angle of an interface in motion will change with rate (4-7), and some data, meager but suggestive, have appeared (8, 9). Additional data are to be found in this paper. These data seem to show that neglect of the rate-de- pendency of contact angle will lead to serious error in predicting capillary flows, for example, by the methods of ]3arrer and Brittin.

Other serious questions can be raised about the adequacy of traditional treatments of capillary rise and capillary movement, even if the ambiguity of how to express the capillary driving force is resolved. This is because of the nonlinearity of the Navier-Stokes equations and the attending problems

MOVING INTERFACES AND CONTACT ANGLE RATE-DEPENDENCY 41

of integration which result. Thus, Schlicting (10) applies boundary layer theory to approximate the increased energy loss at the tube ends (due to velocity profile transition), and likely, this type of a correction should be introduced to show the transition from parabolic to linear velocity profile as the fluid particles approach (and move away from) the region of the moving interface (cf. Wolff's anticipation of this consideration (11)).

In this paper, therefore, we are suggesting that simple capillary flows must be understood as background for the more complicated study of the capillarity of porous media systems. On the other hand, the classic theories of simple capillary flows, as summarized in the work of Barrer (4) and Brittin (5), are incomplete in not representing complete analytic solutions of the Navier-Stokes equations (for physically real initial and boundary conditions) and in not identifying fully the origin and the action of the capillary driving force.

At this time, we are not prepared to resolve all the difficulties of analysis. Instead, we postulate that contact angle is a measure of interfacial curva- ture, and therefore a measure of the capillary driving force for systems of

FIG. 1. Receding and advanc ing contac t angle observat ions for the NujoI-air-glass system, at 22°C. The numbers refer to the sequence, showing exposures t a k e n at a ra te of 32 frames per second. Pic ture 4 shows the in terface at rest, and the hor izonta l (ink) l ine drawn in each frame marks this posi t ion of rest ; hence the dis tance be tween the line and the menisci of the o ther pictures is a rough measurement of veloci ty and accelerat ion (deceleration) of in terface movement .

42 ROSE AND HEINS

simple geometry . By observing the r a t e -dependency of contac t angle, we

proceed to the ex ten t of ob t a in ing empir ica l ly de te rmined corrections to be appl ied to the simplified analyses of Barrer and Br i t t in .

EXPERIME~TTAL METHOD AND RESULTS

Of the several procedures for measur ing contac t angle ra te -dependency

described in the l i tera ture , the Ab le t t (8) me thod (where the solid is made to move th rough an otherwise u n d i s t u r b e d l iquid- l iquid interface) and the

TABLE I Temperature = 22°C., Nujol viscosity = 1.05 poise, Nujol surface tension = 30.1

dynes/cm., Receding contact ~ngle = 0 °.

Tube radius R Pressure, AP Velocity, Slug length Advancing (cm.) (dynes/cm3) V (cm./sec.) L (cm.) contact angle,

Oa (deg.)

0.033 249 0.0147 1.07 30.8 0.033 498 0.0275 1.05 34.1 0.033 747 0.0366 1.01 40.1 0.033 996 0.0586 0.94 40.2 0.033 1245 0.0928 0.98 41.8

0.033 1245 0.1012 0.78 50.7 0.033 1494 0.1014 1.81 49.6 0.033 2441 0.2138 0.61 62.4 0.033 3113 0.2181 1.00 64.7 0.033 249 0.0116 1.03 28.0

0.033 249 0.0129 1.03 30.9 0.033 498 0.0180 1.03 34.1 0.033 742 0.0223 1.03 39.2 0.033 1003 0.0472 1.03 40.9 0.033 373 0.0124 2.86 27.3

0.033 622 0.0231 2.86 30.6 0.033 871 0.0281 2.86 33.3 0.033 1120 0.0405 2.86 33.9 0.033 1245 0.0479 2.86 34.3 0.033 124 0.0050 2.86 24.0

0.033 186 0.0050 2.86 25.3 0.033 311 0.0149 2.86 27.8 0.033 87 0.0041 2.86 18.3 0.033 435 0.0149 2.86 28.5 0.055 124 0.0101 2.54 22.8

0.055 236 0.0202 2.54 26.0 0.055 361 0.0314 2.54 32.2 0.055 548 0.0481 2.54 36.8 0.055 896 0.0886 2.54 44.6

MOVING INTERFACES AND CONTACT ANGLE RATE-DEPENDENCY 43

TABLE I (Continued) Temperature -- 22°C., oleic acid viscosity = 0.256 poise, oleic acid surfacv tension

-- 32.5 dynes/cm., Receding contact angle = 0 °.

Tube radius R Pressure, AP Velocity, Slug length Advancing contact angle, (cm.) (dynes/cm?) V (cm/sec.) L (em.) 0,~ (deg.)

0.05 249 0. 050 1.44 38.3

0.05 373 0.0857 1.33 43.7 0.05 498 0.1091 1.17 47.3 0.05 622 0.188 1.01 47.2 0.05 747 0.172 1.32 50.4 0.05 846 0.2595 1.07 48.8

0.05 996 0.359 1.14 58.3 0.05 124 0.0106 1.10 35.9 0.033 498 0.0826 0.76 38.2 0.033 249 0.0316 1.16 32.3 0.033 373 0.0441 1.10 32.0

0.033 498 0.0407 0.93 35.2 0,033 622 0.0650 0.94 37.7 0.033 622 0.1566 0.63 41.1 0.033 498 0.0936 0.84 40.4 0.033 747 0.160 0.74 41.5 0.033 124 0.010 0.88 30.2

Yarnold and Mason (9) method (wherein the interface is caused to move relative to a stationary solid) were found to be unsuitable for the present work. For example, we were unable to reproduce results in the a t tempt to compare equivalent systems by these older methods.

On the other hand, a direct (photographic) measurement of the contact angles characterizing interfaces in motion has proved feasible, and has given consistent and reproducible results, as described below. In addition, since the measurements are made in particular capillary systems of interest (i.e., circular capillary tubes of small bore), kinematic similitude is assured, and a direct connection between the contact angle data and the associated capillary driving force can be stated.

We have used 20-cm. lengths of precision bore Pyrex tubing (Fischer and Porter), and we photograph the movement of a liquid slug forced through the tube by gas displacement at constant pressure. The optical system in- eludes a camera (Path6 Webo model M) in series with a microscope, giving pictures of the moving interface as illustrated in Fig. 1. In general, the camera was operated at speeds of 8 to 32 frames per second, depending on rate-of-interface movement, and images as appearing on the 16-ram. film strip represented five- to tenfold magnifications.

44 ROSE AND HEINS

1.0

0.8

0.6

0 0 .4

0.2

)

°O o

®

] i ! w 0 0.04 o.oe o.,~, o.=6 0 2 0 o.~4

Velocity ( c m s / s e c )

80

6 0

20 ~

FIG. 2. The Nujol contac t angle da ta (Table I) p lo t ted versus the ra te of in ter face movement . (The s t ra igh t l ine curves represent a leas t -squares fit.)

Table I provides a summary of representative data we have obtained for oleic acid and Nujol systems. Experiments were Mso undertaken with glycerol, but they are not reported here as we question the adequacy of our temperature control in this case (meaning that viscosity was not known precisely).

Our procedure was to clean the glass tubes between successive runs by flushing with a liquid solvent, degreasing with ethylene dichloride, immersing in a dichromate-sulfuric acid solution, then rinsing with distilled water and drying with acetone and anhydrous ether. A liquid slug was then introduced and its length was measured. Slug motion was then initiated by setting the gas drive pressure at some appropriate vMue. Velocity of move- ment was calculated from the displacements observed on the film strip, and advancing and receding contact angles were measured by the method of Bartell and Merrill (12). 2 Since the tube radius was known, and since the viscosity and surface tension values were independently determined, all the parameters of Eq. [1] were thus obtained.

2 Each value of t~A in Table I is the average of six separate de te rmina t ions .

MOVING INTERFACES AND CONTACT ANGLE RATE-DEPENDENCY 4 5

09

"o 0.7

(.3

0

Ct)

30 .~

® Olelc Acid

A Nuj01

0

7"0

®

®.

O L

o t

O O A "A

A

0.5 _ _ (3.5 0°5 0.7 0.9

Cos 0 A (measured)

FIG. 3. M e a s u r e d va l ue s of a d v a n c i n g c o n t a c t ang le (Tab le I) v e r s u s v a l u e s cal- c u l a t e d b y Eq . [1].

As suggested by Fig. 1, the receding contact angle was always zero (or near-zero) for all rates of oleic acid and Nujol slug movement. In contrast, the advancing angles were seen to increase as the interface velocity in- creased, as Fig. 2 shows, in a way qualitatively consistent with the original work of Ablett (8). To be remembered in connection with what Fig. 2 shows is the assertion of Templeton (13) that the advancing contact angle for Nuiol-air-glass systems is always zero.

Figure 3 shows that the data of Table I are consistent with the meaning of Eq. [1], which is to say that the slug length to tube radius ratio (usually greater than 100) was large enough for the flow to be in the Poiseuille regime. Before this equation can be used to predict flow velocities, how- ever, the data of Table I demonstrate that the contact angle rate-depend- ency must be either independently measured or derived from theoretical considerations.

A dimensional analysis has prompted us to see if the data of Table I can be correlated in terms of the Reynold's number, RVp/~?, the friction factor, R 2 A P/L nV, the Leverett number, ,y/R AP, and the slug length to pore

46 ROSE AND HEINS

I.C

®. ° . 7

O ' ® ® Cos 0 A O

0.6

® 0.5

0.4

A m'

A

Z~

A A

A

,s.

A

O Oleic Acid

A Nujol

0 . ~ I | I , |

o o .~ 0.0.4 o.~ o.~ o;o o',2 o.,4. o.,~ o.~ o.~o

FIG. 4. Measured values of advancing contact angle (Tab]e I) versus the dimen- sionless group, VnL/vR.

radius ratio, L/R; however, as illustrated by Fig. 4, no way was found to interconnect the behavior of the oleic acid and Nujol systems.

D i s c u s s i o n

It is realized, of course, that the shape of fluid-fluid interfaces in motion is determined by a host of complex physicoehemical interactions between components and phases, and by presently unspecified hydrodynamic effects. To treat these is clearly beyond the scope of this paper except to mention some of the considerations which lead one to expect that contact angle will be rate-dependent. Thus, it may be noted that if either or both of the fluid phases imbibing (and draining) into (and out of) capillary channels contain surfactants and/or disassociated ions, then the surface energy of M1 fluid-solid and fluid-fluid interfaces will depend upon to what extent the interfaces are saturated with respect to the concentration of surfactants and/or ions adsorbed at the respective surfaces (14). A high rate of inter- face movement can be imagined, therefore, especiMly when a capillary channel of varying cross-sectional area is considered, where there is not sufficient time for the fluid-solid and/or fluid-fluid interfaces to become saturated in the sense that an equilibrium condition of surface activity has been attained. A lower rate of interface movement correspondingly can be

MOVING INTERFACES AND CONTACT ANGLE RATE-DEPENDENCY z~7

imagined where there is a condition of equilibrium in surface activity. A variation in contact angle with rate, therefore, is to be expected.

As for the hydrodynamic effects, we follow the reasoning of Barter (4) that the high velocity of the fluid particles in the central pore space (as compared to the low velocity near the channel walls) must lead to a rate- dependent flattening of moving interfaces, causing an increase of advancing contact angle with increasing rate and a corresponding decrease in receding contact angle. We may also reason that an increase in advancing contact angle with rate is a simple expression of Le Chatelier's rule that a system placed under stress moves such as to oppose the applied stress. Thus, a moving fluid-fluid interface can be made to move faster by supplying pres- sure energy. If this is supplied in the direction of causing the wetting fluid to imbibe faster into a capillary channel, a flattening of the interface is to be expected, meaning the advancing contact angle is increased, as (by Eq. [1]) this is in the direction of diminishing the capillary driving force. The eor- responding argument for the drainage ease indieates, aeeording to the rule of Le Chatelier, that the receding contact angle will decrease with increasing rate of interface movement.

CONCLUSION

We have described a simple and effective method for observing the eurva- ture of fluid-fluid interfaces in motion in capillary tubes. Our data show that this curvature, as measured by the contact angle, is rate-dependent at least for oleic acid and Nujol advancing into an air-filled glass capillary. The range of values observed, moreover, is sufficiently large so that neglect of this effect can lead to large error in predicting capillary motions. The indicated linearity of cos 0 versus rate (Fig. 2) confirms the espeetat]ons recently reported by Siegel (15).

ACKNOWLEDGMENTS

This paper is an extension of the Master 's thesis of G. R. Ramagopal (University of Illinois, 1960). The drawings and most of the calculations are the work of B. Singh.

REFERENCES

i. SCHEIDEGGER, A. E., "The Physics of Flow through Porous Media." Macmillan, New York, 1960.

2. BUFF, F. P., "The Theory of Capillarity", Handbuch der Physik, Chapter X (Structure of Liquids), pp. 281-304, Springer-Verlag (1960).

3. SCRIVEN, L. E., "Dynamics of a fluid interface. Equation of motion for Newtonian surface fluids." Chem. Eng. Sci. 2, 98-108 (1960).

4. B•RRER, II,. M., "F lu id flow in porous media ." Discussions Faraday Soc. No. 3 61-72 (1948).

5. BRITTIN, W. E., "Liquid rise in a capillary tube ," J. Appl. Phys. 17, 37-44 (1946). 6. LEGRAND, E. J., aND RENS~, W. A., " D a t a on rate of capillary r ise ." J. Appl.

Phys. 16, 843-846 (1945).

48 ROSE AND HEINS

7. YARNOLD, G. D., AND MASON, B. J., "A theory of the angle of contact." Proc. Phys. Soc. (London) B62, 121-125 (1949).

8. ABLETT, R., "An investigation of the angle of contact between water and wax." Phil.'Mag. 46, 244-256 (1923).

9. YARNOLD, G. D., AND MASON, B. J., "The angle of contact between water and wax." Proc. Phys. Soc. (London) B62, 125-128 (1949).

10. SCHLICTING, H., "Boundary Layer Theory." McGraw-Hill, New York, 1960. 11. WOLFF, H. C., "Steady motion of viscous liquids in capillary tubes." Trans.

Wisconsin Acad. Sci. 12, 550-553 (1899). 12. BARTELL, F. E. AND MERRILL, E. J. , "Determination of adhesion tension of

liquids against solids. A microscopic method for the measurement of inter- facial contact angles." J. Phys. Chem. 36, 1178-1190 (1932).

13. TEMPI~ETON, C. C., "A study of displacements in microscopic capillaries." Trans. A.I .M.E. , v. 201, pp. 162-168, (1954).

14. EI~LIOTT, T. A., AND LEESE, L., "Dynamic contact angles. I. Change in air- solution-solid contact angles with t ime." J. Chem. Soc. 1957, 22-30.

15. SIEGEL, ROBERT, "Transient capillary rise in reduced and zero-gravity fields." J. Applied Mechanics 83, 165-170 (1961).