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Studies of dynamic contact angles on solids
Studies of Dynamic Contact Angles on Solids A N T H O N Y M. S C H W A R T Z aND S I L V E S T R E B. T E J A D A
The Gillette Company Research Institute, Rockville, Maryland 20850 In forced spreading systems, three different modes of Od-V behavior have been found, each of which predominates in a different velocity range. In the lowest velocity range, with systems involving the low viscosity, low boiling, nonpolar liquid hexane, 0d was found equal to Oeq (the Elllott-Riddiford or Hansen-Miotto mode). In the next higher velocity range, which extends to very low velocities for all other systems studied, the behavior described by Eq. 6 (the Blake-Haynes mode) predominates. At still higher velocities, the behavior described by Eq. [9] (the Friz mode) becomes superposed on the Blake-Haynes mode, and eventually predominates up to the range where 0e approaches 90° and Eq. [9] becomes inapplicable. In the Blake-Haynes mode the major force opposing advance of the liquid front is the solid-liquid interracial viscosity. In the Friz mode it is the bulk viscosity of the liquid. The roughness of solid surfaces has no appreciable effect on the Od-V relationship, provided the physicochemical character of the surfaces is the same and the roughness is random. If the process of roughening alters the physicochemical character the 0d-V behavior of the roughened surface may differ from that of the smooth one. There is no qualitative difference between the Od-V behavior of systems in which Oeq is zero and systems in which Oeqis positive. INTRODUCTION The problem of controlling the flow of liquids in and out of containers at zero gravity depends on certain aspects of capillarity t h a t have relatively little applicability under normal terrestrial gravity, and about which little information has previously been available. One of these is the dynamic contact angle, which at zero gravity becomes an i m p o r t a n t factor in the sloshing of liquids against container walls and in damping the surface disturbances t h a t accompany the sloshing. I n a previous publication (1), the dynamic contact angle behavior (i.e., contact angle vs. velocity of the liquid-solidv a p o r line along the solid surface) of several different liquid-solid-vapor systems was reported. The static equilibrium contact angle of all these systems was zero, and the solid surfaces were smooth b y usual engineering standards. An empirical equation was derived relating dynamic contact angle (0d) to velocity (V), but the coefficients in this equation could not be related to any specific Copyright © 1972 by Academic Press, Inc.
physical properties of the liquid and solid constituting the system. The present work was undertaken with three separate objectives in mind: (i) to study more extensively the contact anglevelocity relationship of selected systems in which the surface was smooth and the static contact angle was either zero or positive and less t h a n 90 °. (ii) to determine the effect of surface roughening on the Oe--V relationship of these systems. (iii) to develop a theoretical basis for relating dynamic contact angle behavior to the physical or chemical properties of the system's components, so t h a t the Od-V relationship of unexplored systems might hopefully be predicted. All systems studied were of the solid-liquid-air type and in all of t h e m the liquid consisted of a single component. BACKGROUND DISCUSSION
Qualitative Considerations If we bring a mass of liquid of arbitrary initial shape into contact with a solid surface
Journal of Colloid and Interface Science, Vol. 38, No. 2, February I972
359
360
SCHWARTZ AND T E J A D A
of arbitrary configuration, the liquid-vapor interface along the three-phase line of contact (the LSV line) will tend to tilt until the equilibrium contact angle 0eq is reached. At the same time the LV interface will tend to adjust itself to a state of constant curvature and minimum area compatible with the constraints of the system. This situation is illustrated by the behavior of a drop of liquid on touching a flat solid surface as shown in Fig. la-c: In Fig. la, the drop has very recently touched the surface and 0 is still almost 180 °. In Fig. lb, some time later, 0 has become smaller; in (c), the system has come to equilibrium, the LSV line has stopped moving and 0 has reached its equilibrium value. The angles G and 0b formed between the LV and SL interfaces while the L S V line is moving along the solid surface are dynamic contact angles, for which we use the general symbol 0d. In Fig. 1, the LV interface is shown at all three stages in its equilibrium constantcurvature minimum-energy shape, that of a spherical zone. This is of course possible if the outward motion of the LSV line is very slow. In general, however, the LV interface
is
will not be in a state of constant curvature while liquid is in motion at the LSV line. Consider, for example, the representation of a wire running through a liquid shown in Fig. 2. In Fig. 2a, the system is at rest. Since we are assuming throughout this discussion that gravity effects are negligible the pressure differential across the LV interface is constant at all points, as is the mean curvature of the interface. The LSV line is at an equilibrium level P above the bulk liquid surface Q, and the contact angle at LSV is at its equilibrium value 0eq (illustrated as equal to zero). In Fig. 2b, we have started to move the filament S downward into the bulk liquid, and can visualize a hypothetical transitory situation in which (assuming no slippage at the SL interface) the LV surface has been distorted as shown to produce a high pressure region at H. The liquid accordingly flows toward the SL interface at the LSV line (Fig. 2b) and prevents the contact angle from increasing indefinitely. A steady state is reached in which liquid removed from the LSV region by the moving SL interface is replenished by liquid forced in by the Laplacian pressure differential. The
II a
II b
1 c
FIG. 1. Stages in self-spreading o f f , d r o p .
f p . . . . . . . . . .
o____S_ S
a
b
c
FIG. 2. Establishment of a dynamic contact angle: forced spreading; (arrow) direction of motion of solid S; ( ..... ) P equilibrium level of LSV boundary; (- -) Q is liquid level base line. Journal of Colloid and Interface Science, Vol. 38, No. 2, February 1972
DYNAMIC CONTACT ANGLES ON SOLIDS
361
plane. Following Y is a region of liquid B sufficiently thick to have complete liquid properties. Following B is the bulk region E of normally behaving liquid. The layer Y, in the range of 1000 A or less thick, has been observed and the region B has also been observed in drops spreading on smooth fiat plates (2). When the system reaches equilibrium the region B disappears, merging into E. It is not known if Y persists or, if so, how far along the solid surface it extends. Experimentally, it is difficult to define a contact angle in the a and b stages of spreading. The layer Y is quite invisible in profile. The layer B supposedly has a thickness in the range of 1-2 ~ and is equally invisible in conventional experimental systems. The "dynamic contact angle" we measure in a system of this type is the angle formed between the solid surface and the projection of the LV interface above B, as shown in Fig. 3b. No detailed picture of forced spreading, analogous to the above picture of self-spreading, has to our knowledge been presented in ~,Lv(COS 0eq - cos 0e). [1] the literature of surface chemistry. Forced The forces which oppose motion of the spreading in the high velocity regime is LSV line are viscous in nature. Viscosity at studied in hydraulics under t h e general the SL interface (for all liquids) and at the heading of "open channel surging." In parLV interface (for multieomponent liquids) ticular the problem "propagation of a surge may, however, differ greatly from ordinary on a dry bed" deals with an LSV line moving bulk viscosity. The effective retarding force across a dry solid surface impelled by presin the thin portions of liquid near the LSV sure. The parameters that are considered, line, particularly in a system of zero 0eq, however, do not include the angle formed may therefore differ from that calculated on between the flat solid surface and the adthe basis of bulk viscosity only. vancing wall of water. In the usual elemenWork by previous investigators (2-5) tary treatment this angle is assumed to be presents the picture of self-spreading illus- 90 ° at the start of the flow and (ideally) to trated in Fig. 3, which shows one profile of a remain constant. In the low velocity region self-spreading drop. In Fig. 3a, shortly after the drop has been touched to the solid, the strong pull of the SV interface has caused a thin layer of liquid Y to move out from the bulk of the liquid. Since the effective pull 1 Is ] of SV cannot extend above the SV plane Is more than a very few molecular diameters b the layer Y is quite thin and is possibly not FIG. 3. Primary (Y) and secondary (B) layers fluid in the dimension normal to the solid in self-spreading. LV interface remains in a slightly distorted configuration, sustained by the viscosity of the liquid and opposed by the LV surface tension, as shown in Fig. 2e, and the contact angle comes to a steady value, 0d, which is greater than O,q. In the system of Fig. 1, the only forces tending to change 0d are molecular forces acting in the LSV line. In Fig. 2, these molecular forces are supplemented by a hydrodynamic force generated by the distorted LV interface, which is dependent on the motion of the filament through the bulk liquid. The two systems accordingly differ in an important and fundamental manner. We shall refer to one as "self-spreading," and the other as "forced spreading." The driving forces in self-spreading can be analyzed in terms of Young's equation; and for a system in which the solid-vapor interracial tension ~'sv, and the solid-liquid interracial tension 7s~, have attained their equilibrium value but 0d > 0eq the force on 1 em of LSV line is potentially equal to:
Journal of Colloid and Interface Science, Vol. 38, No. 2, F e b r u a r y 1972
362
SCHWARTZ AND TEJADA
of forced spreading it can be presumed that surface tension and viscosity might play the same roles they do in self-spreading.
.....
/
6r
Pertinent Prior Studies
Although much careful and important work has been done on dynamic contact angles, relatively little is of direct pertinence to the present study, either because it dealt with multicomponent liquids and selective adsorption during spreading, or because it did not attempt to relate 0a vs. V behavior to physical parameters of the system. Several investigators in recent years have studied the Od-V relationship in self-spreading systems. A group of three recent papers by Schonhorn et al. (6), Van Oene et al. (7), and Yin (8) all deal with the spreading of droplets of molten polymers on smooth surfaces. Cherry and Holmes (9) studied the spread of molten droplets of polyethylene on stainless steel at 150°C, and confirmed the experimental results of previous investigators. Their theoretical treatment is based on the idea that the activation energy for flow is the rate-controlling factor in spreading. Using Eyring's theory of absolute reaction rates they arrive at the equation: d cos 0d dt
-
c. cos
[21
Oeq. a. e -~t,
where c = 7~.vx2y/vVh where V~ is the volume of the moving hydrodynamic unit, y is the dimension of the hydrodynamic unit in the direction of movement and x is the distance between sites. On this basis 7Lv/~ is the controlling factor. They note, however, that with low viscosity liquids surface-liquid in= teractions may become important. Equation [2] can be rearranged (16) to the form: V = "Y~'v "V-h-x~Y ( eOs O~ -
c°s
,
FIG. 4. R o s e - H e i n s s y s t e m : l i q u i d m o v e s f r o m R t o w a r d A u n d e r p r e s s u r e difference P1 - P~ •
spreading. They pushed slugs of liquid through a glass tube by means of air pressure, as sho~al in Fig. 4. Their systems, within the velocity ranges studied, showed an apparently linear relationship between cos 0d and velocity. Elliott and Riddiford (11, 12) studied the forced spreading of water on polyethylene and silieoned glass surfaces. The static advancing contact angles in these systems were 100 and 105 °, respectively. Their results are shown in Figs. 5 and 6. At very low velocities, 0e remained equal to 0eq. At a certain critical velocity the angle started to increase rapidly then leveled off. The authors ascribe this behavior to a molecular relaxation phenomenon occurring at the advancing three-phase boundary line, in accordance with ideas previously proposed by Hansen and Miotto (13). Friz (14) made a theoretical hydrodynamic analysis of a slug of liquid being pushed through a tube [the same system considered by Rose and Heins (10)] and derived an equation for the shape of the liquid front and the apparent contact angle. Friz assumed that the wall of the tube was already wet with a thin layer of liquid (thickness = 5, an important parameter in his derivation), and pictured the dynamic contact angle as an asymptotic limit to the actual LV profile. His equation for the dynamic contact angle is:
[3]
so that a plot of V against (cos 0~q - cos 0d) would be linear. So far as we are aware Rose and Heins (10) made the first study of contact angle vs. liquid velocity under conditions of forced
6d
tan 0e = 3.4 \TLv/
[4]
where 0a is the dynamic contact angle, u0 is the forward velocity of the liquid, ~ is visconsity, and XLv, surface tension. Coney and Masica (15) tested Friz' theory experimentally, using rectangular glass tub:
Journal of Colloid and Interface Science, Vol. 38, No. 2, February 1972
DYNAMIC CONTACT ANGLES ON SOLIDS [
•
,,2
O,(d,~)
Q
363
'D
A
•
,o.r/ l
"
0
S
I0 IS INTFRFRClAL VI:LOClTY In~m mJn-I~
2O
FIG. 5. A d v a n c i n g c o n t a c t angles as a f u n c t i o n of the velocity of an ~ i r / w a t e r interface moving over siliconed glass at 22°C ( 0 ) ; siliconed glass at 42°C (®) ; and polyethylene at 22°C ([E) ; from Ref. (12).
ing which had been prewetted. The experiments were conducted in a weightless environment, and all the liquids used had zero static contact angles on the glass. Velocities ranged from 1.4 to 28 cm/sec; liquid surface tensions from 16.6 to 24.4 dyn/cm; and liquid viscosities from 0.56 to 6.7 cP. They found that the theory adequately predicted the contact angle dependence on interface velocity and liquid properties. Qualitative agreement was obtained between the theoretical and experimental interracial shapes. Blake and Haynes (16), working in a lower velocity regime, derived a very persuasive theoretical treatment of dynamic contact angles. Using the theory of absolute reaction rates, they picture the essential motion to be a sliding of the molecules along the solid surface from the liquid to the vapor side of the LSV line. The work expended in causing the flow is: W = VLv(COS 0o~ -
cos Od)
[5]
and this is used to raise or lower the aetiva-
tion energy for forward or reverse molecular migration along the solid surface. Blake and Haynes thus regard the retarding force as an interfacial viscosity, in contrast to the bulk viscosity used by the other investigators cited. Their final equations are: V = 2Kk sinh [ . ~~Lv (cos 0eq
--
COS
Od
il
[6] ,
V - 2KX~/LV(cos eeq-- cos 0d),
[7]
AnkT
(C0S0eq V = KX exp [r 7Lv ~ 7 -
Is]
cos 0d)J ,
where V is the forward velocity of the liquid, K is number of molecular displacements occurring per unit time per unit length of the LSV line, X is the average distance between sites on the solid surface, An is surface con-
Journal of Colloid and Interface Science,
Vol. 38, No. 2, February 1972
364
SCHWARTZ AND TEJADA
II0
109
0
I08
107
106
lOS
104
103
102
~--
I01
I00 0
I
I INTERFACIAL
I
2
VELOCITY (ram~1~
FIG. 6. An expansion of part of Fig. 5 showing the low velocity regions at 22°C for a siliconed glass surface ( 0 ) and polyethylene (m); from Ref. (12).
centration of sites, and the other symbols have their usual significance. Equation [6] is the master equation, valid over the whole velocity range. From the mathematical properties of the hyperbolic sine function it follows that Eq. [7] becomes approximately valid if the argument of sinh (i.e., the expression in brackets) in Eq. [6] is small. Equation [8] becomes approximately valid if the argument of sinh in Eq. [6] is large. If we designate (cos 8eq - cos 0~) by A cos, plots of either A. cos vs. V or cos 8d vs. V would b e linear irL the range where Eq. [7] was valid, and they would be exponential (linear on semilog paper) in the range where Eq. [8] was valid. Journal of Colloid and Interface Science,
Blake and Haynes obtained experimental data for the system benzene-water-sillconed glass, using cylindrical capillary tubes of 0.2 mm diameter and velocities in the range of zero to 1 cm/sec (600 ram/rain). Their plots of cos 02 vs. velocity on semilog paper were 5near, as shown in Fig. 7, indicating t h a t Eq. [8] was governing in this system. One of the few experimental studies of the effect of surface roughness on the 0¢-V relationship has been made by Bascom et al. (5). They found that surface scratches across the path of flow decrease the spreading rate but scratches parallel to the path increase it. The rate on a polished surface is half way
Vol. 38, No. 2, February 1972
DYNAMIC CONTACT ANGLES ON SOLIDS 0
;
365
d
-0"2
-0,4
COS([) -0"6
-Off
'-3 Io 4
'-2
10
'-,
I0
V
Io
CM SEC -I
FIG. 7. Advancing and receding dynamic contact angles: benzene-water-siliconed glass; ( 0 and [])
benzene advancing toward wager in two slightly different tubes; (Q) water advancing toward benzene; from I~ef. (16). 0.12C
O.IIC OIOC VERTICAL SCRATC 0.09C E0.08C 0.07C i,g ~ O F ~ I Z O N T A L -
o.o6o
SCRATCHES
w 0050
0.o3oI 0.020
0010 0000 0
500
tO00
1500 2000 2500 SPREADINGTIME,SEC.
3000
3500 4000
FIG. 8. Effect of directional scratches on distance versus time for 10% pristane in distilled squalene on vertical steel plates; from Ref. (5). between the two, as shown in Fig. 8. On this basis we should expect that the rate of spreading over a randomly rough surface would be the same as over a polished surface. I n summary, all workers except Friz regard ~/Lv(A cos) as the sole driving force tending to advance the LSV line. I n Friz' treatment, the movement is a surge, with the driving force an exterior pressure. All workers except Friz and Blake-Haynes regard the bulk viscosity ~ of the liquid as the sole resistance force. Friz' resistance force is
a function of both ~ and "YLv. Blake and Haynes' resistance force can be regarded as a solid-liquid interracial viscosity which is dependent on the properties of the SL and SV interfaces ~s well as those of the bulk liquid. The r~w data of the experiments hereinafter reported (19) are in the form of tables of 0d VS. V. For purposes of theorizing, the data may be plotted in three different ways: 1. As A cos vs. V: This is referred to as a cosine plot. If it is linear over any substantial
Journal of Colloid and Interface Science,
Vol. 38, No. 2, February 1972
366
SCHWARTZ AND TEJADA
range, it might indicate a relationship of the type shown in Eqs. [3] or [7]. If the cosine plot is curved, and has the general form of sinh function, the numerical values of the coefficients in Eq. [6] can be estimated, and from them values of K, ~, and An can be derived. These derived values can then be compared for magnitude with physically reasonable values of K, ~, and An. 2. As a plot of A cos vs. log V: This is referred to as a Haynes plot. If it is linear it would indicate a Blake-Haynes mechanism in which the argument of the sinh function is large, as in Eq. [8]. 3. As a plot of log tan 0d VS. log V/~LvV: This is referred to as a Friz plot. If it is linear it would suggest a Friz (14) mechanism, as described b y Eq. [4]. Equation [4] requires that such a plot have a slope of 0.33 and it is also obvious that the equation cannot be valid for values of Odapproaching 90 °. The mathematical form of the relationship, however, suggests the physical situation postulated by Friz, i.e., an outside pressure as the driving force, with viscosity as one resistance force and a continuously increasing back pressure against the advancing liquid front as a second resistance force. Mere linearity of a Friz plot would not neces-
sarily indicate that a Friz type of mechanism was veritably applicable. The same data would also give a linear Haynes plot for values of Od between about 38 and 73 °. Within this range, the relationship between the cosine of the angle and the log tangent of the angle is very nearly linear. EXPERIMENTAL METHODS The relationship between 0d and V was studied for the 10 liquid-solid systems designated in Table I by either X Y or Y. This distinguishes them from the previously studied systems (1) which are designated by X. T h e five systems designated b y Y are those for which the static contact angle ~eq is greater than zero but less than 90 ° . In these systems new data were developed for the solid in both the smooth state and at three different degrees of roughness. The five systems designated X Y had previously been studied in the smooth state only and these prior smooth state data were used in the present analysis. New data were developed for these systems at three different degrees of roughness. All the new data extended into a much lower velocity range than the old. The properties of the liquids used, and the values of the
TABLE I SYSTEMS STUDIEDa Solid Liquid
Di (2-ethylhexyl)sebacate (DOS) Diethylphthalate (DEP) Benzyl alcohol I-Iexadeeane Isopropanol Absolute ethanol Alpha bromonaphthalene Freon TF Octane Hexane
Stainless steel
Titanium
Aluminum
X X XY X XY
X
X X X
X
X
Nylon
PMMA b
Teflon
Y
XY
Water Methylene iodide
X X X
X
X X X X Y X
X X
XY Y Y
XY
Y
X = smooth solid only, data from Ref. (1); XY = data for smooth solid from Ref. (1); data for roughened solid from present study. Y = data for both smooth and roughened solid from present study. b Poly(methyl methaerylate). Journal of Colloid and Interface Science, Vo]. 38, No. 2, February 1972
DYNAMIC CONTACT ANGLES ON SOLIDS TABLE
367
II
PROPERTIES OF TEST LIQUIDS
Liquid
Viscosity (cP)
Surface tension (dyn/cm)
Static contact angle on smooth surfacea n/'rLV
Di (2-ethylhexyl) sebacate
25.0
31.2
0.808
(DOS) Diethyl phthalate (DEP) Benzyl alcohol Hexadecane Isopropanol Absolute alcohol Alpha bromonaphthalene Freon TF Octane Hexane Water Methylene iodide
10.10 5.80 3.32 2.22 1.20 2.3 0.694 0.542 0.326 1.00 0.50
37.5 39.0 27.6 21.7 22.75 44.6 19.0 21.8 18.4 72.0 50.7
0.270 0.1485 0.1205 0.1025 0.0529 0.0516 0.0365 0.0249 0.0180 0.0139 0.00985
Stainless Titanium Aluml- Nylon PMMA Teflon steel num
61 0 0 0 0 0
0 0 0 0
0
0
0 0 0
0 0 0 0 16 0
0 0
26 0
0 70 41
a C o n t a c t a n g l e v a l u e s m e a s u r e d e x p e r i m e n t a l l y ; t h e s e cheek l i t e r a t u r e v a l u e s (17, 18).
static contact angles of all the systems, are given in Table II. The apparatus, cleaning and manipulative procedures, and methods of measuring both 0d and V were the same as described previously (i). The apparatus was very slightly modified to enable velocities as low as 0.0002 era/see to be measured. The solids were used in the form of cylindrical filaments 20 mils in diameter. The liquids were the purest grade available commercially. The precision of the measurements of 0d was about 4-1 °. Each value of 0d was obtained by averaging at least I0 individual measurements. The range of these sets of i0 or more seldom exceeded 5 ° .
Preparation of Roughened Specimens Roughened filaments were prepared by sandblasting. The filament from a supply spool was passed upward through the sandblasting box, receiving the blast from three nozzles placed symmetrically around the filament. The filaments were examined microscopically to check that the roughening was complete and uniform. Usually one pass was sufficient to accomplish this result. The three different degrees of roughness were obtained by using three different sizes of carborundum grit; Nos. 120, 180, and 240. The
appearance of the specimens roughened by any one grit varied very little with the material, i.e., the No. 120 roughened steel, nylon, PMMA, and Teflon all appeared to have about the same roughness, and all were rougher than the No. 180 specimens. Figure 9 shows the appearance of the smooth and roughened stainless steel wires under the scanning electron microscope at 180X magnification. Direct measurements of roughness could not be made on the filaments. To estimate the roughness flat specimens of each material were sandblasted in the same manner as the filaments, and their appearance under the microscope was checked against that of the correspondingly roughened filaments. The "center line average deviation" roughness of flat specimens of the P M M A , nylon, and stainless steel was measured by a Taylor-I-Iobson "Talysurf" machine fitted with an automatic integrator. The roughness of the No. 120 specimens was 26-35 mieroinehes; No. 180 specimens, 20-23 mieroinehes; and No. 240 specimens, 11-15 mieroinehes. The smooth specimens had a roughness near the resolution limit of the instrument, about 2 microinehes. The traces were quite uniform with regard to direction along the specimen, indicating well-randomized roughening. Precise meas-
Journal of Colloid and Interface Science, Vol. 38, No. 2, February 1972
368
SCHWARTZ AND TEJADA
FIG. 9. Smooth and roughened stainless steel wires; scanning electron microscope 180×. (a) Smooth; (b) No. 240 roughened; (e) No. 180 roughened; and (d) No. 120 roughened. urements could not be obtained on the Teflon due to its softness. DISCUSSION OF RESULTS T a b u l a t e d numerical data for the 10 different solid-liquid combinations studied are presented in a concurrent report (19). For purposes of analysis, the data were arranged and diagramed in the form cosine plots,
Haynes plots, and Friz plots. The systems designated X in Table I gave what appeared to be approximate straight lines in all three methods of plotting. This was due, however, to the fact that the data were confined to the relatively high velocity ranges. Those systems (designated Y) for which data in the low velocity range were also available gave quite a different picture. Aside from the
Journal of Colloid and Interface Science, Voi. 38, No. 2, F e b r u a r y 1972
D Y N A M I C CONTACT ANGLES ON SOLIDS
369 O
I I
I
t j ! ! I
30
(9 ®
/o
[
I I ! / 1 I I I I O I
lb u w,
t ! I t I !
v
~20
/o 1 I l l ! g9 /
o
I I !
/
®/ I 1 I I
a/ !
10
/ (9
!
[
/
.e Q
//
9t /
......
~>/ ,e
e7° - .1
I .3
.2
I .4
! .5
! .6
/~cos
FIG. 10. Smooth Teflon-octane system: cosine plot; points are experimental; (..... ) sinh function based on computed values of parameters; V = 0.36 sinh 13.6 A cos. nylon-hexane and stainless steel-hexane systems, which are discussed below-, they showed the following typical behavior: The cosine plots were curved, and had the general form of an exponential or hyperbolic sine curve. A typieM plot is shown in Fig. 10. A t t e m p t s to find coefficients which would fit Eq. [6] to these curves were reasonably successful so far as the lower velocity ranges
were concerned. At higher velocities the coefficients had to be changed markedly, indicating t h a t Eq. [6] was not valid over the whole velocity range. The shape of the curves indicated t h a t Eqs. [7] or [3] could not be valid, at least in the low velocity range. The Friz plots all departed from linearity, as expected, as 0d approached 90 °. Below this
Journal of Colloid and Interface Science, V o l . 38, N o . 2, F e b r u a r y
1972
370
SCHWAI~TZ AND TEJADA b /
f
.i0.0
,o, ~"d/ /
0
,I.0
Q
0
0~
a
~'0~
~
O
!
.01
I
0.i
1.0
~v
~'Lv
Fro. 11. Friz plots: (a) smooth Teflon-octane; and (b) No. 180 grit roughened stainless steel-hexadeeane. region the typical Friz plot had the general form of two intersecting straight lines, as shown in Fig. lla, or an upward trending curve (Fig. 11b) which might be interpreted as two linear regions with a gradual rather than a sharp intersection. Comparisons of one O¢-V function with another is most conveniently made by means of the Haynes plots. These typically consisted of two essentially linear regions intersecting more or less sharply as shown in Figs. 12 and 13, the slope (AA cos/A log V) of the high velocity leg (fast side slope) being greater than that of the low velocity leg (slow side slope). These slopes, and the velocity Vo at which the transition between the two occurs, are the key fencures of the Haynes plot in the regions pertinent to this study. They are given for the stainless steel systems in Table III.
The similarity between the Friz and Haynes plots in certain velocity ranges is shown by comparing Figs. l l a and 14. This graphic behavior suggested that at least two mechanisms were governing the 6~-V relationship, one predominating at lower and the other at higher velocities. If this were the case, and the two mechanisms were the Blake-Haynes (according to Eq. [8]) and the Friz, we would expect the Blake-Haynes to govern at lower velocities and the Friz at higher velocities. To test this hypothesis the fast side slopes and the slow side slopes of the Haynes plots were used separately to evaluate An and k, using the Blake-Haynes method (16) which starts with Eq. [8]. The values of X calculated from fast side slopes are of the order of 5-10 A, which is considered small and less in accord with current theories of the liquid state than the values of 15-3~)
Journal of Colloid and Interface Science, Vol. 38, No. 2, February 1972
DYNAMIC CONTACT ANGLES ON SOLIDS 1.
371
? i
.9 .8 z
k
a
/t1
.? jd o
.3 .2
.01 '
O. i
l.()
100
Veloc~y (era/see)
FIG. 12. Haynes plots; No. 240 grit roughened solid-liquid systems: (a) stainless steel-hexadeeane, 0~q = 0; (b) PMMA-hexane, 0ca = 0; (e) nylon-~-bromonaphthalene, 0eq = 16; (d) Teflon-octane, 0~q = 26; and (e) Teflon-DOS, 0~q = 61. ].
d
b
/'
//
""
l/ja
.7
uc
.6 .5 ,,f 7
.4
7
//77
d
.3
c .2 .]
.01
'o1:
b'
::o Velocity @m/sac)
FIG. 13. Haynes plots; nylon-methylene iodide system: (a) smooth; (b) No. 240 grit roughened; (e) No. 180 grit roughened; and (d) No. 120 grit roughened. A calculated from slow side slopes. This supports the hypothesis that in the low velocity region the B l a k e - H a y n e s mechanism predominates. Further support comes when K
in Eq. [8] is evaluated, from X and tile intereept of the slow side leg with the A cos = 0 axis. Substituting the values of K, An, and X into Eq. [6] and plotting the curve, it is
Journal of Colloid and Interface
Science,
VoI. 38, No. 2, February 1972
372
SCHWARTZ AND TEJADA TABLE III ]:~OUGttENED STAINLESS STEEL SYSTEMS
No. Smooth 120
Hexadecane Slope slow side Slope fast side V~ (cm/sec) Lowest V measured (cm/sec)
0.531
Alcohol Slope slow side Slope fast side V~ (cm/sec) Lowest V measured
0.54
180
0'140 84
0 80
240
0.051
1.7
1.3 0.02
0.06
2
0.0~ /
0.040.19
0.27
0.02 0.2~
1.0
12 0.03
0.02
(cm/see)
Hexane Slope slow side Slope fast side Vc (cm/sec) Lowest V measured (cm/sec)
0.30
i
O~
/
D.23 0.7 0.35 0 38
10,0
.18 .5 0.06
12 0.25
a This is an Elliott-Riddiford type system (11, 12). 1,01
.7
o/
<1 .6
.5 ,4
.2
! i
i.
F~G. 14. Smooth
10. Velocity (cm/sec)
Teflon-octane; Haynes plot.
Journal of Colloid and Interface Science, Vol. 38, No. 2, February 1972
i00
DYNAMIC CONTACT ANGLES ON SOLIDS
373
t. .9
.8 .I jb 0
.5
/l/f/f
.4
C
/J
/jr/
~J ,2
,]. ] .0 Velocity
10 (cm/sec)
'
100
Fro. 15. Haynes plot; nylon-hexane systems; A cos = 0 at low velocities: (a) smooth; (b) No. 240 grit roughened; (c) No. 180 grit roughened; and (d) No. 120 grit roughened. found to superpose the experimental curve of A cos vs. V remarkably well in the low velocity region. The experimental curve in this region is thus shown to be a sinh curve, in conformity with the Blake-Haynes theory. This is illustrated in Fig. 10. In the higher velocity region, corresponding to the fast side legs of both the Haynes plots and the Friz plots, the data can be well represented by an equation having the form of Eq. [4], i.e., tan
Od ~
(~V ~ b a \.y~/ .
[9]
This represents a surging motion in which the bulk fluidity 1/7 rather than the interfaeial fluidity KX is the governing parameter. Since the liquid surges onto a dry bed, and since the flow is superposed on a flow of the Blake-Haynes type the parameters a and b should differ from those of Eq. [4], and the scaling factor should be some unascertained combination of KX, 7, and "/~v rather than
~/7~v-
The nylon-hexane system (Fig. 15) and the stainless stecl-hexane system (Table III) showed the low velocity behavior described by Elliott and Riddiford, i.e., Cd remained equal to 0ca. Above the critical velocity at which 0e began to exceed 0eq the Haynes plot remained linear, showing no sign of a Friz type surge within the velocity limits of the experiment. This is ascribed to the low viscosity of the liquid. In forced spreading the general form of the Od-V relationship over the range from 0d = 0.q to 0e --* 90 ° may accordingly be described as follows: At the lowest velocities there is a region in which 0a -- 0e~. This region was observed experimentally in the present study only with systems in which hexane was the liquid (although there is a suggestion of similar behavior in the octane-No. 180 grit roughened Teflon system). Theoretically, according to the views of Hansen and Miotto (13) such a region should exist at sufficiently low velocities for all liquids.
Journal of Colloid and Interface Science, Vol. 38, No. 9, February 1972
374
SCHWARTZ AND TEJADA
c /d 1/
.9
.
,8
.7
cc.6 c .5
•4
,3
.2
a/ .1
r •0 [
0 .]
Velocity
i
i
1 .0
]0
(cm/sec)
Fie. 16. Effect of roughness on the 0d-V relstionship ;stainless steel-hexadecsne systems : (a)smooth; (b) No. 240 grit roughened; (c) iNTo.180 grit roughened; snd (d) No. 120 grit roughened.
Above this "Hansen-Miotto region" is a region in which the behavior described by Blake and Haynes, Eq. 6, predominates. In this region, the major driving force is measured by 7Lv" A cos and the major retarding force is the solid-liquid interfacial viscosity measured by 1/KX. This parameter is a property of the solid-liquid pair, rather than of the liquid alone or the solid alone. The Blak'e-Haynes behavior is, of course, superposed on whatever Hansen-Miotto behavior may exist. Above a relatively narrow critical velocity range a surging mode of flow of the type described by Friz (Eqs. [4] and [9]) becomes superposed on the Blake-Haynes mode, and becomes increasingly predominant as the velocity is further increased.
Effect of Magnitude of the Static Contact Angle Whether the static equilibrium contact angle is zero or positive appears to have no effect on the form of the 0~vs. V relationship. Figure 12 shows Haynes plots of three posiJournal of Colloid and Inte~rface Science, V o l . 38, N o . 2, F e b r u a r y
tire angle systems and two representative zero angle systems, all at the same degree of roughening, No. 240 grit. Ordering the systems with regard to Oeqand slope, it is evident that no correlation exists. This result is not unexpected since there is no theoretical reason why any functional relationship should exist between Oeq and the form of the Od-V curves.
Effect of Roughening and Roughness The roughened surfaces used in the present studies were produced by blasting with grit of three different sizes but of the same chemical composition. The blasting destroys the old surface of the substrate and exposes a new surface which may, or may not, differ in physicochemicsl properties from the old one. Presumably the three new surfaces formed by the three different sizes of grit will have identical physicochemical properties. If we wish to ascertain the effect of roughness (i.e., of surface geometry alone) on the 0x-V relationship, comparisons should 1972
DYNAMIC CONTACT ANGLES ON SOLIDS be made of the three roughened surfaces with one another. If, on the other hand, we wish to ascertain the effect of roughening (i.e., of changing the original surface b y abrading and disrupting it), comparison should be made between the original smooth surface
and any of the three roughened surfaces. As shown in Table III, in each of the three stainless steel systems the three roughened specimens show no significant differences among themselves, but they all appear to differ from the corresponding smooth specimen in fast side slope, the only p a r a m e t e r evaluated in the smooth systems. This would suggest t h a t blasting changed the physicochemical character of the stainless steel surface, but t h a t degree of roughness has little if any effect on the 0d vs. V relationship. Haynes plots of the four stainless steelhexadecane systems are shown in Fig. 16. ACKNOWLED GMENT The work oH which this paper is based was performed under Contract NAS3-11522 with NASA Lewis Research Center, to whom we are grateful for permission to publish. We wish especially to acknowledge the invaluable help and constructive criticism of Mr. Donald A. Petrash and Mr. E. P. Symons of NASA Lewis Research Center throughout the investigation. REFERENCES 1. ELLISON, A. H., AND TEgADA, S. B., N A S A Contract. Rep. CR 72441, 1968.
375
2. HARDY,W. B., Collect. Pap. 624 (1936). Cambridge University Press. 3. HARDY,W. B., Collect. Pap. 7tl (1936). Cambridge University Press. 4. BANGt-IAM, D. H., AND SAWERIS, Z., Trans. Faraday Soc. 34, 554 (1938). 5. BxSCOM, W. D., COTTINGTON, R. L., AND SINGLETERRY, C. R., Advan. Chem. Ser. 43,
355. American Chemical Society, Washington, D.C. (1964). 6. SCHONHORN, I-I., FRIscIt, H. L., AND KWEI, T. W., J. Appl. Phys. 37, 4967 (1966). 7. VAN OENE, H., CHANG, Y. F., AND NEWMAN, S., J. Adhes. 1, 54 (1969). 8. YIN, T. P., J. Phys. Chem. 73, 2413 (1969). 9. CHERRY,B. W., AND HOLMES, C. M., J. Colloid Interface Sci. 29, 174 (1969). 10. ROSE, W., AND I-IEINS, R. W., J. Colloid Sci.
17, 39 (1962). 11. ELLIOT, G. E. P., AND RIDDIFORD, A. C.,
Nature n4843, 795 (1962). 12. ELLIOT, G. E. P., AND RIDDIFORD, A. C., J. Colloid Interface Sci. 23, 389 (1967). 13. HANSEN, R. S., AND MIOTTO, M., J. Miner. Chem. Soc. 79, 1765 (1957). 14. FRIZ, G., Z. Angew. Phys. 19(4), 374 (1965). 15. CONEY, T. A., AND MASICA, W. J., N A S A Tech. Note TND-5115 (1969). 16. BLAKE, T. D., AND HAYNES, J. M., J. Colloid Interface Sci. 30, 421 (1969). 17. ELLISON,A. H., AND ZISM~XN,W. A., J. Phys. Chem. 58, 503 (1954). 18. FORT,T., JR., Advan. Chem. Ser. 43, 302. American Chemical Society, Washington, D.C. (1964). 19. SCHWARTZ,A. M., AND TEJADA, S. B., _NASA Contract. Rep. CR 72728, 1970.
Journal of Colloidand Interface Science, Vol.38, No. 2, FebruaryI972
The Gillette Company Research Institute, Rockville, Maryland 20850 In forced spreading systems, three different modes of Od-V behavior have been found, each of which predominates in a different velocity range. In the lowest velocity range, with systems involving the low viscosity, low boiling, nonpolar liquid hexane, 0d was found equal to Oeq (the Elllott-Riddiford or Hansen-Miotto mode). In the next higher velocity range, which extends to very low velocities for all other systems studied, the behavior described by Eq. 6 (the Blake-Haynes mode) predominates. At still higher velocities, the behavior described by Eq. [9] (the Friz mode) becomes superposed on the Blake-Haynes mode, and eventually predominates up to the range where 0e approaches 90° and Eq. [9] becomes inapplicable. In the Blake-Haynes mode the major force opposing advance of the liquid front is the solid-liquid interracial viscosity. In the Friz mode it is the bulk viscosity of the liquid. The roughness of solid surfaces has no appreciable effect on the Od-V relationship, provided the physicochemical character of the surfaces is the same and the roughness is random. If the process of roughening alters the physicochemical character the 0d-V behavior of the roughened surface may differ from that of the smooth one. There is no qualitative difference between the Od-V behavior of systems in which Oeq is zero and systems in which Oeqis positive. INTRODUCTION The problem of controlling the flow of liquids in and out of containers at zero gravity depends on certain aspects of capillarity t h a t have relatively little applicability under normal terrestrial gravity, and about which little information has previously been available. One of these is the dynamic contact angle, which at zero gravity becomes an i m p o r t a n t factor in the sloshing of liquids against container walls and in damping the surface disturbances t h a t accompany the sloshing. I n a previous publication (1), the dynamic contact angle behavior (i.e., contact angle vs. velocity of the liquid-solidv a p o r line along the solid surface) of several different liquid-solid-vapor systems was reported. The static equilibrium contact angle of all these systems was zero, and the solid surfaces were smooth b y usual engineering standards. An empirical equation was derived relating dynamic contact angle (0d) to velocity (V), but the coefficients in this equation could not be related to any specific Copyright © 1972 by Academic Press, Inc.
physical properties of the liquid and solid constituting the system. The present work was undertaken with three separate objectives in mind: (i) to study more extensively the contact anglevelocity relationship of selected systems in which the surface was smooth and the static contact angle was either zero or positive and less t h a n 90 °. (ii) to determine the effect of surface roughening on the Oe--V relationship of these systems. (iii) to develop a theoretical basis for relating dynamic contact angle behavior to the physical or chemical properties of the system's components, so t h a t the Od-V relationship of unexplored systems might hopefully be predicted. All systems studied were of the solid-liquid-air type and in all of t h e m the liquid consisted of a single component. BACKGROUND DISCUSSION
Qualitative Considerations If we bring a mass of liquid of arbitrary initial shape into contact with a solid surface
Journal of Colloid and Interface Science, Vol. 38, No. 2, February I972
359
360
SCHWARTZ AND T E J A D A
of arbitrary configuration, the liquid-vapor interface along the three-phase line of contact (the LSV line) will tend to tilt until the equilibrium contact angle 0eq is reached. At the same time the LV interface will tend to adjust itself to a state of constant curvature and minimum area compatible with the constraints of the system. This situation is illustrated by the behavior of a drop of liquid on touching a flat solid surface as shown in Fig. la-c: In Fig. la, the drop has very recently touched the surface and 0 is still almost 180 °. In Fig. lb, some time later, 0 has become smaller; in (c), the system has come to equilibrium, the LSV line has stopped moving and 0 has reached its equilibrium value. The angles G and 0b formed between the LV and SL interfaces while the L S V line is moving along the solid surface are dynamic contact angles, for which we use the general symbol 0d. In Fig. 1, the LV interface is shown at all three stages in its equilibrium constantcurvature minimum-energy shape, that of a spherical zone. This is of course possible if the outward motion of the LSV line is very slow. In general, however, the LV interface
is
will not be in a state of constant curvature while liquid is in motion at the LSV line. Consider, for example, the representation of a wire running through a liquid shown in Fig. 2. In Fig. 2a, the system is at rest. Since we are assuming throughout this discussion that gravity effects are negligible the pressure differential across the LV interface is constant at all points, as is the mean curvature of the interface. The LSV line is at an equilibrium level P above the bulk liquid surface Q, and the contact angle at LSV is at its equilibrium value 0eq (illustrated as equal to zero). In Fig. 2b, we have started to move the filament S downward into the bulk liquid, and can visualize a hypothetical transitory situation in which (assuming no slippage at the SL interface) the LV surface has been distorted as shown to produce a high pressure region at H. The liquid accordingly flows toward the SL interface at the LSV line (Fig. 2b) and prevents the contact angle from increasing indefinitely. A steady state is reached in which liquid removed from the LSV region by the moving SL interface is replenished by liquid forced in by the Laplacian pressure differential. The
II a
II b
1 c
FIG. 1. Stages in self-spreading o f f , d r o p .
f p . . . . . . . . . .
o____S_ S
a
b
c
FIG. 2. Establishment of a dynamic contact angle: forced spreading; (arrow) direction of motion of solid S; ( ..... ) P equilibrium level of LSV boundary; (- -) Q is liquid level base line. Journal of Colloid and Interface Science, Vol. 38, No. 2, February 1972
DYNAMIC CONTACT ANGLES ON SOLIDS
361
plane. Following Y is a region of liquid B sufficiently thick to have complete liquid properties. Following B is the bulk region E of normally behaving liquid. The layer Y, in the range of 1000 A or less thick, has been observed and the region B has also been observed in drops spreading on smooth fiat plates (2). When the system reaches equilibrium the region B disappears, merging into E. It is not known if Y persists or, if so, how far along the solid surface it extends. Experimentally, it is difficult to define a contact angle in the a and b stages of spreading. The layer Y is quite invisible in profile. The layer B supposedly has a thickness in the range of 1-2 ~ and is equally invisible in conventional experimental systems. The "dynamic contact angle" we measure in a system of this type is the angle formed between the solid surface and the projection of the LV interface above B, as shown in Fig. 3b. No detailed picture of forced spreading, analogous to the above picture of self-spreading, has to our knowledge been presented in ~,Lv(COS 0eq - cos 0e). [1] the literature of surface chemistry. Forced The forces which oppose motion of the spreading in the high velocity regime is LSV line are viscous in nature. Viscosity at studied in hydraulics under t h e general the SL interface (for all liquids) and at the heading of "open channel surging." In parLV interface (for multieomponent liquids) ticular the problem "propagation of a surge may, however, differ greatly from ordinary on a dry bed" deals with an LSV line moving bulk viscosity. The effective retarding force across a dry solid surface impelled by presin the thin portions of liquid near the LSV sure. The parameters that are considered, line, particularly in a system of zero 0eq, however, do not include the angle formed may therefore differ from that calculated on between the flat solid surface and the adthe basis of bulk viscosity only. vancing wall of water. In the usual elemenWork by previous investigators (2-5) tary treatment this angle is assumed to be presents the picture of self-spreading illus- 90 ° at the start of the flow and (ideally) to trated in Fig. 3, which shows one profile of a remain constant. In the low velocity region self-spreading drop. In Fig. 3a, shortly after the drop has been touched to the solid, the strong pull of the SV interface has caused a thin layer of liquid Y to move out from the bulk of the liquid. Since the effective pull 1 Is ] of SV cannot extend above the SV plane Is more than a very few molecular diameters b the layer Y is quite thin and is possibly not FIG. 3. Primary (Y) and secondary (B) layers fluid in the dimension normal to the solid in self-spreading. LV interface remains in a slightly distorted configuration, sustained by the viscosity of the liquid and opposed by the LV surface tension, as shown in Fig. 2e, and the contact angle comes to a steady value, 0d, which is greater than O,q. In the system of Fig. 1, the only forces tending to change 0d are molecular forces acting in the LSV line. In Fig. 2, these molecular forces are supplemented by a hydrodynamic force generated by the distorted LV interface, which is dependent on the motion of the filament through the bulk liquid. The two systems accordingly differ in an important and fundamental manner. We shall refer to one as "self-spreading," and the other as "forced spreading." The driving forces in self-spreading can be analyzed in terms of Young's equation; and for a system in which the solid-vapor interracial tension ~'sv, and the solid-liquid interracial tension 7s~, have attained their equilibrium value but 0d > 0eq the force on 1 em of LSV line is potentially equal to:
Journal of Colloid and Interface Science, Vol. 38, No. 2, F e b r u a r y 1972
362
SCHWARTZ AND TEJADA
of forced spreading it can be presumed that surface tension and viscosity might play the same roles they do in self-spreading.
.....
/
6r
Pertinent Prior Studies
Although much careful and important work has been done on dynamic contact angles, relatively little is of direct pertinence to the present study, either because it dealt with multicomponent liquids and selective adsorption during spreading, or because it did not attempt to relate 0a vs. V behavior to physical parameters of the system. Several investigators in recent years have studied the Od-V relationship in self-spreading systems. A group of three recent papers by Schonhorn et al. (6), Van Oene et al. (7), and Yin (8) all deal with the spreading of droplets of molten polymers on smooth surfaces. Cherry and Holmes (9) studied the spread of molten droplets of polyethylene on stainless steel at 150°C, and confirmed the experimental results of previous investigators. Their theoretical treatment is based on the idea that the activation energy for flow is the rate-controlling factor in spreading. Using Eyring's theory of absolute reaction rates they arrive at the equation: d cos 0d dt
-
c. cos
[21
Oeq. a. e -~t,
where c = 7~.vx2y/vVh where V~ is the volume of the moving hydrodynamic unit, y is the dimension of the hydrodynamic unit in the direction of movement and x is the distance between sites. On this basis 7Lv/~ is the controlling factor. They note, however, that with low viscosity liquids surface-liquid in= teractions may become important. Equation [2] can be rearranged (16) to the form: V = "Y~'v "V-h-x~Y ( eOs O~ -
c°s
,
FIG. 4. R o s e - H e i n s s y s t e m : l i q u i d m o v e s f r o m R t o w a r d A u n d e r p r e s s u r e difference P1 - P~ •
spreading. They pushed slugs of liquid through a glass tube by means of air pressure, as sho~al in Fig. 4. Their systems, within the velocity ranges studied, showed an apparently linear relationship between cos 0d and velocity. Elliott and Riddiford (11, 12) studied the forced spreading of water on polyethylene and silieoned glass surfaces. The static advancing contact angles in these systems were 100 and 105 °, respectively. Their results are shown in Figs. 5 and 6. At very low velocities, 0e remained equal to 0eq. At a certain critical velocity the angle started to increase rapidly then leveled off. The authors ascribe this behavior to a molecular relaxation phenomenon occurring at the advancing three-phase boundary line, in accordance with ideas previously proposed by Hansen and Miotto (13). Friz (14) made a theoretical hydrodynamic analysis of a slug of liquid being pushed through a tube [the same system considered by Rose and Heins (10)] and derived an equation for the shape of the liquid front and the apparent contact angle. Friz assumed that the wall of the tube was already wet with a thin layer of liquid (thickness = 5, an important parameter in his derivation), and pictured the dynamic contact angle as an asymptotic limit to the actual LV profile. His equation for the dynamic contact angle is:
[3]
so that a plot of V against (cos 0~q - cos 0d) would be linear. So far as we are aware Rose and Heins (10) made the first study of contact angle vs. liquid velocity under conditions of forced
6d
tan 0e = 3.4 \TLv/
[4]
where 0a is the dynamic contact angle, u0 is the forward velocity of the liquid, ~ is visconsity, and XLv, surface tension. Coney and Masica (15) tested Friz' theory experimentally, using rectangular glass tub:
Journal of Colloid and Interface Science, Vol. 38, No. 2, February 1972
DYNAMIC CONTACT ANGLES ON SOLIDS [
•
,,2
O,(d,~)
Q
363
'D
A
•
,o.r/ l
"
0
S
I0 IS INTFRFRClAL VI:LOClTY In~m mJn-I~
2O
FIG. 5. A d v a n c i n g c o n t a c t angles as a f u n c t i o n of the velocity of an ~ i r / w a t e r interface moving over siliconed glass at 22°C ( 0 ) ; siliconed glass at 42°C (®) ; and polyethylene at 22°C ([E) ; from Ref. (12).
ing which had been prewetted. The experiments were conducted in a weightless environment, and all the liquids used had zero static contact angles on the glass. Velocities ranged from 1.4 to 28 cm/sec; liquid surface tensions from 16.6 to 24.4 dyn/cm; and liquid viscosities from 0.56 to 6.7 cP. They found that the theory adequately predicted the contact angle dependence on interface velocity and liquid properties. Qualitative agreement was obtained between the theoretical and experimental interracial shapes. Blake and Haynes (16), working in a lower velocity regime, derived a very persuasive theoretical treatment of dynamic contact angles. Using the theory of absolute reaction rates, they picture the essential motion to be a sliding of the molecules along the solid surface from the liquid to the vapor side of the LSV line. The work expended in causing the flow is: W = VLv(COS 0o~ -
cos Od)
[5]
and this is used to raise or lower the aetiva-
tion energy for forward or reverse molecular migration along the solid surface. Blake and Haynes thus regard the retarding force as an interfacial viscosity, in contrast to the bulk viscosity used by the other investigators cited. Their final equations are: V = 2Kk sinh [ . ~~Lv (cos 0eq
--
COS
Od
il
[6] ,
V - 2KX~/LV(cos eeq-- cos 0d),
[7]
AnkT
(C0S0eq V = KX exp [r 7Lv ~ 7 -
Is]
cos 0d)J ,
where V is the forward velocity of the liquid, K is number of molecular displacements occurring per unit time per unit length of the LSV line, X is the average distance between sites on the solid surface, An is surface con-
Journal of Colloid and Interface Science,
Vol. 38, No. 2, February 1972
364
SCHWARTZ AND TEJADA
II0
109
0
I08
107
106
lOS
104
103
102
~--
I01
I00 0
I
I INTERFACIAL
I
2
VELOCITY (ram~1~
FIG. 6. An expansion of part of Fig. 5 showing the low velocity regions at 22°C for a siliconed glass surface ( 0 ) and polyethylene (m); from Ref. (12).
centration of sites, and the other symbols have their usual significance. Equation [6] is the master equation, valid over the whole velocity range. From the mathematical properties of the hyperbolic sine function it follows that Eq. [7] becomes approximately valid if the argument of sinh (i.e., the expression in brackets) in Eq. [6] is small. Equation [8] becomes approximately valid if the argument of sinh in Eq. [6] is large. If we designate (cos 8eq - cos 0~) by A cos, plots of either A. cos vs. V or cos 8d vs. V would b e linear irL the range where Eq. [7] was valid, and they would be exponential (linear on semilog paper) in the range where Eq. [8] was valid. Journal of Colloid and Interface Science,
Blake and Haynes obtained experimental data for the system benzene-water-sillconed glass, using cylindrical capillary tubes of 0.2 mm diameter and velocities in the range of zero to 1 cm/sec (600 ram/rain). Their plots of cos 02 vs. velocity on semilog paper were 5near, as shown in Fig. 7, indicating t h a t Eq. [8] was governing in this system. One of the few experimental studies of the effect of surface roughness on the 0¢-V relationship has been made by Bascom et al. (5). They found that surface scratches across the path of flow decrease the spreading rate but scratches parallel to the path increase it. The rate on a polished surface is half way
Vol. 38, No. 2, February 1972
DYNAMIC CONTACT ANGLES ON SOLIDS 0
;
365
d
-0"2
-0,4
COS([) -0"6
-Off
'-3 Io 4
'-2
10
'-,
I0
V
Io
CM SEC -I
FIG. 7. Advancing and receding dynamic contact angles: benzene-water-siliconed glass; ( 0 and [])
benzene advancing toward wager in two slightly different tubes; (Q) water advancing toward benzene; from I~ef. (16). 0.12C
O.IIC OIOC VERTICAL SCRATC 0.09C E0.08C 0.07C i,g ~ O F ~ I Z O N T A L -
o.o6o
SCRATCHES
w 0050
0.o3oI 0.020
0010 0000 0
500
tO00
1500 2000 2500 SPREADINGTIME,SEC.
3000
3500 4000
FIG. 8. Effect of directional scratches on distance versus time for 10% pristane in distilled squalene on vertical steel plates; from Ref. (5). between the two, as shown in Fig. 8. On this basis we should expect that the rate of spreading over a randomly rough surface would be the same as over a polished surface. I n summary, all workers except Friz regard ~/Lv(A cos) as the sole driving force tending to advance the LSV line. I n Friz' treatment, the movement is a surge, with the driving force an exterior pressure. All workers except Friz and Blake-Haynes regard the bulk viscosity ~ of the liquid as the sole resistance force. Friz' resistance force is
a function of both ~ and "YLv. Blake and Haynes' resistance force can be regarded as a solid-liquid interracial viscosity which is dependent on the properties of the SL and SV interfaces ~s well as those of the bulk liquid. The r~w data of the experiments hereinafter reported (19) are in the form of tables of 0d VS. V. For purposes of theorizing, the data may be plotted in three different ways: 1. As A cos vs. V: This is referred to as a cosine plot. If it is linear over any substantial
Journal of Colloid and Interface Science,
Vol. 38, No. 2, February 1972
366
SCHWARTZ AND TEJADA
range, it might indicate a relationship of the type shown in Eqs. [3] or [7]. If the cosine plot is curved, and has the general form of sinh function, the numerical values of the coefficients in Eq. [6] can be estimated, and from them values of K, ~, and An can be derived. These derived values can then be compared for magnitude with physically reasonable values of K, ~, and An. 2. As a plot of A cos vs. log V: This is referred to as a Haynes plot. If it is linear it would indicate a Blake-Haynes mechanism in which the argument of the sinh function is large, as in Eq. [8]. 3. As a plot of log tan 0d VS. log V/~LvV: This is referred to as a Friz plot. If it is linear it would suggest a Friz (14) mechanism, as described b y Eq. [4]. Equation [4] requires that such a plot have a slope of 0.33 and it is also obvious that the equation cannot be valid for values of Odapproaching 90 °. The mathematical form of the relationship, however, suggests the physical situation postulated by Friz, i.e., an outside pressure as the driving force, with viscosity as one resistance force and a continuously increasing back pressure against the advancing liquid front as a second resistance force. Mere linearity of a Friz plot would not neces-
sarily indicate that a Friz type of mechanism was veritably applicable. The same data would also give a linear Haynes plot for values of Od between about 38 and 73 °. Within this range, the relationship between the cosine of the angle and the log tangent of the angle is very nearly linear. EXPERIMENTAL METHODS The relationship between 0d and V was studied for the 10 liquid-solid systems designated in Table I by either X Y or Y. This distinguishes them from the previously studied systems (1) which are designated by X. T h e five systems designated b y Y are those for which the static contact angle ~eq is greater than zero but less than 90 ° . In these systems new data were developed for the solid in both the smooth state and at three different degrees of roughness. The five systems designated X Y had previously been studied in the smooth state only and these prior smooth state data were used in the present analysis. New data were developed for these systems at three different degrees of roughness. All the new data extended into a much lower velocity range than the old. The properties of the liquids used, and the values of the
TABLE I SYSTEMS STUDIEDa Solid Liquid
Di (2-ethylhexyl)sebacate (DOS) Diethylphthalate (DEP) Benzyl alcohol I-Iexadeeane Isopropanol Absolute ethanol Alpha bromonaphthalene Freon TF Octane Hexane
Stainless steel
Titanium
Aluminum
X X XY X XY
X
X X X
X
X
Nylon
PMMA b
Teflon
Y
XY
Water Methylene iodide
X X X
X
X X X X Y X
X X
XY Y Y
XY
Y
X = smooth solid only, data from Ref. (1); XY = data for smooth solid from Ref. (1); data for roughened solid from present study. Y = data for both smooth and roughened solid from present study. b Poly(methyl methaerylate). Journal of Colloid and Interface Science, Vo]. 38, No. 2, February 1972
DYNAMIC CONTACT ANGLES ON SOLIDS TABLE
367
II
PROPERTIES OF TEST LIQUIDS
Liquid
Viscosity (cP)
Surface tension (dyn/cm)
Static contact angle on smooth surfacea n/'rLV
Di (2-ethylhexyl) sebacate
25.0
31.2
0.808
(DOS) Diethyl phthalate (DEP) Benzyl alcohol Hexadecane Isopropanol Absolute alcohol Alpha bromonaphthalene Freon TF Octane Hexane Water Methylene iodide
10.10 5.80 3.32 2.22 1.20 2.3 0.694 0.542 0.326 1.00 0.50
37.5 39.0 27.6 21.7 22.75 44.6 19.0 21.8 18.4 72.0 50.7
0.270 0.1485 0.1205 0.1025 0.0529 0.0516 0.0365 0.0249 0.0180 0.0139 0.00985
Stainless Titanium Aluml- Nylon PMMA Teflon steel num
61 0 0 0 0 0
0 0 0 0
0
0
0 0 0
0 0 0 0 16 0
0 0
26 0
0 70 41
a C o n t a c t a n g l e v a l u e s m e a s u r e d e x p e r i m e n t a l l y ; t h e s e cheek l i t e r a t u r e v a l u e s (17, 18).
static contact angles of all the systems, are given in Table II. The apparatus, cleaning and manipulative procedures, and methods of measuring both 0d and V were the same as described previously (i). The apparatus was very slightly modified to enable velocities as low as 0.0002 era/see to be measured. The solids were used in the form of cylindrical filaments 20 mils in diameter. The liquids were the purest grade available commercially. The precision of the measurements of 0d was about 4-1 °. Each value of 0d was obtained by averaging at least I0 individual measurements. The range of these sets of i0 or more seldom exceeded 5 ° .
Preparation of Roughened Specimens Roughened filaments were prepared by sandblasting. The filament from a supply spool was passed upward through the sandblasting box, receiving the blast from three nozzles placed symmetrically around the filament. The filaments were examined microscopically to check that the roughening was complete and uniform. Usually one pass was sufficient to accomplish this result. The three different degrees of roughness were obtained by using three different sizes of carborundum grit; Nos. 120, 180, and 240. The
appearance of the specimens roughened by any one grit varied very little with the material, i.e., the No. 120 roughened steel, nylon, PMMA, and Teflon all appeared to have about the same roughness, and all were rougher than the No. 180 specimens. Figure 9 shows the appearance of the smooth and roughened stainless steel wires under the scanning electron microscope at 180X magnification. Direct measurements of roughness could not be made on the filaments. To estimate the roughness flat specimens of each material were sandblasted in the same manner as the filaments, and their appearance under the microscope was checked against that of the correspondingly roughened filaments. The "center line average deviation" roughness of flat specimens of the P M M A , nylon, and stainless steel was measured by a Taylor-I-Iobson "Talysurf" machine fitted with an automatic integrator. The roughness of the No. 120 specimens was 26-35 mieroinehes; No. 180 specimens, 20-23 mieroinehes; and No. 240 specimens, 11-15 mieroinehes. The smooth specimens had a roughness near the resolution limit of the instrument, about 2 microinehes. The traces were quite uniform with regard to direction along the specimen, indicating well-randomized roughening. Precise meas-
Journal of Colloid and Interface Science, Vol. 38, No. 2, February 1972
368
SCHWARTZ AND TEJADA
FIG. 9. Smooth and roughened stainless steel wires; scanning electron microscope 180×. (a) Smooth; (b) No. 240 roughened; (e) No. 180 roughened; and (d) No. 120 roughened. urements could not be obtained on the Teflon due to its softness. DISCUSSION OF RESULTS T a b u l a t e d numerical data for the 10 different solid-liquid combinations studied are presented in a concurrent report (19). For purposes of analysis, the data were arranged and diagramed in the form cosine plots,
Haynes plots, and Friz plots. The systems designated X in Table I gave what appeared to be approximate straight lines in all three methods of plotting. This was due, however, to the fact that the data were confined to the relatively high velocity ranges. Those systems (designated Y) for which data in the low velocity range were also available gave quite a different picture. Aside from the
Journal of Colloid and Interface Science, Voi. 38, No. 2, F e b r u a r y 1972
D Y N A M I C CONTACT ANGLES ON SOLIDS
369 O
I I
I
t j ! ! I
30
(9 ®
/o
[
I I ! / 1 I I I I O I
lb u w,
t ! I t I !
v
~20
/o 1 I l l ! g9 /
o
I I !
/
®/ I 1 I I
a/ !
10
/ (9
!
[
/
.e Q
//
9t /
......
~>/ ,e
e7° - .1
I .3
.2
I .4
! .5
! .6
/~cos
FIG. 10. Smooth Teflon-octane system: cosine plot; points are experimental; (..... ) sinh function based on computed values of parameters; V = 0.36 sinh 13.6 A cos. nylon-hexane and stainless steel-hexane systems, which are discussed below-, they showed the following typical behavior: The cosine plots were curved, and had the general form of an exponential or hyperbolic sine curve. A typieM plot is shown in Fig. 10. A t t e m p t s to find coefficients which would fit Eq. [6] to these curves were reasonably successful so far as the lower velocity ranges
were concerned. At higher velocities the coefficients had to be changed markedly, indicating t h a t Eq. [6] was not valid over the whole velocity range. The shape of the curves indicated t h a t Eqs. [7] or [3] could not be valid, at least in the low velocity range. The Friz plots all departed from linearity, as expected, as 0d approached 90 °. Below this
Journal of Colloid and Interface Science, V o l . 38, N o . 2, F e b r u a r y
1972
370
SCHWAI~TZ AND TEJADA b /
f
.i0.0
,o, ~"d/ /
0
,I.0
Q
0
0~
a
~'0~
~
O
!
.01
I
0.i
1.0
~v
~'Lv
Fro. 11. Friz plots: (a) smooth Teflon-octane; and (b) No. 180 grit roughened stainless steel-hexadeeane. region the typical Friz plot had the general form of two intersecting straight lines, as shown in Fig. lla, or an upward trending curve (Fig. 11b) which might be interpreted as two linear regions with a gradual rather than a sharp intersection. Comparisons of one O¢-V function with another is most conveniently made by means of the Haynes plots. These typically consisted of two essentially linear regions intersecting more or less sharply as shown in Figs. 12 and 13, the slope (AA cos/A log V) of the high velocity leg (fast side slope) being greater than that of the low velocity leg (slow side slope). These slopes, and the velocity Vo at which the transition between the two occurs, are the key fencures of the Haynes plot in the regions pertinent to this study. They are given for the stainless steel systems in Table III.
The similarity between the Friz and Haynes plots in certain velocity ranges is shown by comparing Figs. l l a and 14. This graphic behavior suggested that at least two mechanisms were governing the 6~-V relationship, one predominating at lower and the other at higher velocities. If this were the case, and the two mechanisms were the Blake-Haynes (according to Eq. [8]) and the Friz, we would expect the Blake-Haynes to govern at lower velocities and the Friz at higher velocities. To test this hypothesis the fast side slopes and the slow side slopes of the Haynes plots were used separately to evaluate An and k, using the Blake-Haynes method (16) which starts with Eq. [8]. The values of X calculated from fast side slopes are of the order of 5-10 A, which is considered small and less in accord with current theories of the liquid state than the values of 15-3~)
Journal of Colloid and Interface Science, Vol. 38, No. 2, February 1972
DYNAMIC CONTACT ANGLES ON SOLIDS 1.
371
? i
.9 .8 z
k
a
/t1
.? jd o
.3 .2
.01 '
O. i
l.()
100
Veloc~y (era/see)
FIG. 12. Haynes plots; No. 240 grit roughened solid-liquid systems: (a) stainless steel-hexadeeane, 0~q = 0; (b) PMMA-hexane, 0ca = 0; (e) nylon-~-bromonaphthalene, 0eq = 16; (d) Teflon-octane, 0~q = 26; and (e) Teflon-DOS, 0~q = 61. ].
d
b
/'
//
""
l/ja
.7
uc
.6 .5 ,,f 7
.4
7
//77
d
.3
c .2 .]
.01
'o1:
b'
::o Velocity @m/sac)
FIG. 13. Haynes plots; nylon-methylene iodide system: (a) smooth; (b) No. 240 grit roughened; (e) No. 180 grit roughened; and (d) No. 120 grit roughened. A calculated from slow side slopes. This supports the hypothesis that in the low velocity region the B l a k e - H a y n e s mechanism predominates. Further support comes when K
in Eq. [8] is evaluated, from X and tile intereept of the slow side leg with the A cos = 0 axis. Substituting the values of K, An, and X into Eq. [6] and plotting the curve, it is
Journal of Colloid and Interface
Science,
VoI. 38, No. 2, February 1972
372
SCHWARTZ AND TEJADA TABLE III ]:~OUGttENED STAINLESS STEEL SYSTEMS
No. Smooth 120
Hexadecane Slope slow side Slope fast side V~ (cm/sec) Lowest V measured (cm/sec)
0.531
Alcohol Slope slow side Slope fast side V~ (cm/sec) Lowest V measured
0.54
180
0'140 84
0 80
240
0.051
1.7
1.3 0.02
0.06
2
0.0~ /
0.040.19
0.27
0.02 0.2~
1.0
12 0.03
0.02
(cm/see)
Hexane Slope slow side Slope fast side Vc (cm/sec) Lowest V measured (cm/sec)
0.30
i
O~
/
D.23 0.7 0.35 0 38
10,0
.18 .5 0.06
12 0.25
a This is an Elliott-Riddiford type system (11, 12). 1,01
.7
o/
<1 .6
.5 ,4
.2
! i
i.
F~G. 14. Smooth
10. Velocity (cm/sec)
Teflon-octane; Haynes plot.
Journal of Colloid and Interface Science, Vol. 38, No. 2, February 1972
i00
DYNAMIC CONTACT ANGLES ON SOLIDS
373
t. .9
.8 .I jb 0
.5
/l/f/f
.4
C
/J
/jr/
~J ,2
,]. ] .0 Velocity
10 (cm/sec)
'
100
Fro. 15. Haynes plot; nylon-hexane systems; A cos = 0 at low velocities: (a) smooth; (b) No. 240 grit roughened; (c) No. 180 grit roughened; and (d) No. 120 grit roughened. found to superpose the experimental curve of A cos vs. V remarkably well in the low velocity region. The experimental curve in this region is thus shown to be a sinh curve, in conformity with the Blake-Haynes theory. This is illustrated in Fig. 10. In the higher velocity region, corresponding to the fast side legs of both the Haynes plots and the Friz plots, the data can be well represented by an equation having the form of Eq. [4], i.e., tan
Od ~
(~V ~ b a \.y~/ .
[9]
This represents a surging motion in which the bulk fluidity 1/7 rather than the interfaeial fluidity KX is the governing parameter. Since the liquid surges onto a dry bed, and since the flow is superposed on a flow of the Blake-Haynes type the parameters a and b should differ from those of Eq. [4], and the scaling factor should be some unascertained combination of KX, 7, and "/~v rather than
~/7~v-
The nylon-hexane system (Fig. 15) and the stainless stecl-hexane system (Table III) showed the low velocity behavior described by Elliott and Riddiford, i.e., Cd remained equal to 0ca. Above the critical velocity at which 0e began to exceed 0eq the Haynes plot remained linear, showing no sign of a Friz type surge within the velocity limits of the experiment. This is ascribed to the low viscosity of the liquid. In forced spreading the general form of the Od-V relationship over the range from 0d = 0.q to 0e --* 90 ° may accordingly be described as follows: At the lowest velocities there is a region in which 0a -- 0e~. This region was observed experimentally in the present study only with systems in which hexane was the liquid (although there is a suggestion of similar behavior in the octane-No. 180 grit roughened Teflon system). Theoretically, according to the views of Hansen and Miotto (13) such a region should exist at sufficiently low velocities for all liquids.
Journal of Colloid and Interface Science, Vol. 38, No. 9, February 1972
374
SCHWARTZ AND TEJADA
c /d 1/
.9
.
,8
.7
cc.6 c .5
•4
,3
.2
a/ .1
r •0 [
0 .]
Velocity
i
i
1 .0
]0
(cm/sec)
Fie. 16. Effect of roughness on the 0d-V relstionship ;stainless steel-hexadecsne systems : (a)smooth; (b) No. 240 grit roughened; (c) iNTo.180 grit roughened; snd (d) No. 120 grit roughened.
Above this "Hansen-Miotto region" is a region in which the behavior described by Blake and Haynes, Eq. 6, predominates. In this region, the major driving force is measured by 7Lv" A cos and the major retarding force is the solid-liquid interfacial viscosity measured by 1/KX. This parameter is a property of the solid-liquid pair, rather than of the liquid alone or the solid alone. The Blak'e-Haynes behavior is, of course, superposed on whatever Hansen-Miotto behavior may exist. Above a relatively narrow critical velocity range a surging mode of flow of the type described by Friz (Eqs. [4] and [9]) becomes superposed on the Blake-Haynes mode, and becomes increasingly predominant as the velocity is further increased.
Effect of Magnitude of the Static Contact Angle Whether the static equilibrium contact angle is zero or positive appears to have no effect on the form of the 0~vs. V relationship. Figure 12 shows Haynes plots of three posiJournal of Colloid and Inte~rface Science, V o l . 38, N o . 2, F e b r u a r y
tire angle systems and two representative zero angle systems, all at the same degree of roughening, No. 240 grit. Ordering the systems with regard to Oeqand slope, it is evident that no correlation exists. This result is not unexpected since there is no theoretical reason why any functional relationship should exist between Oeq and the form of the Od-V curves.
Effect of Roughening and Roughness The roughened surfaces used in the present studies were produced by blasting with grit of three different sizes but of the same chemical composition. The blasting destroys the old surface of the substrate and exposes a new surface which may, or may not, differ in physicochemicsl properties from the old one. Presumably the three new surfaces formed by the three different sizes of grit will have identical physicochemical properties. If we wish to ascertain the effect of roughness (i.e., of surface geometry alone) on the 0x-V relationship, comparisons should 1972
DYNAMIC CONTACT ANGLES ON SOLIDS be made of the three roughened surfaces with one another. If, on the other hand, we wish to ascertain the effect of roughening (i.e., of changing the original surface b y abrading and disrupting it), comparison should be made between the original smooth surface
and any of the three roughened surfaces. As shown in Table III, in each of the three stainless steel systems the three roughened specimens show no significant differences among themselves, but they all appear to differ from the corresponding smooth specimen in fast side slope, the only p a r a m e t e r evaluated in the smooth systems. This would suggest t h a t blasting changed the physicochemical character of the stainless steel surface, but t h a t degree of roughness has little if any effect on the 0d vs. V relationship. Haynes plots of the four stainless steelhexadecane systems are shown in Fig. 16. ACKNOWLED GMENT The work oH which this paper is based was performed under Contract NAS3-11522 with NASA Lewis Research Center, to whom we are grateful for permission to publish. We wish especially to acknowledge the invaluable help and constructive criticism of Mr. Donald A. Petrash and Mr. E. P. Symons of NASA Lewis Research Center throughout the investigation. REFERENCES 1. ELLISON, A. H., AND TEgADA, S. B., N A S A Contract. Rep. CR 72441, 1968.
375
2. HARDY,W. B., Collect. Pap. 624 (1936). Cambridge University Press. 3. HARDY,W. B., Collect. Pap. 7tl (1936). Cambridge University Press. 4. BANGt-IAM, D. H., AND SAWERIS, Z., Trans. Faraday Soc. 34, 554 (1938). 5. BxSCOM, W. D., COTTINGTON, R. L., AND SINGLETERRY, C. R., Advan. Chem. Ser. 43,
355. American Chemical Society, Washington, D.C. (1964). 6. SCHONHORN, I-I., FRIscIt, H. L., AND KWEI, T. W., J. Appl. Phys. 37, 4967 (1966). 7. VAN OENE, H., CHANG, Y. F., AND NEWMAN, S., J. Adhes. 1, 54 (1969). 8. YIN, T. P., J. Phys. Chem. 73, 2413 (1969). 9. CHERRY,B. W., AND HOLMES, C. M., J. Colloid Interface Sci. 29, 174 (1969). 10. ROSE, W., AND I-IEINS, R. W., J. Colloid Sci.
17, 39 (1962). 11. ELLIOT, G. E. P., AND RIDDIFORD, A. C.,
Nature n4843, 795 (1962). 12. ELLIOT, G. E. P., AND RIDDIFORD, A. C., J. Colloid Interface Sci. 23, 389 (1967). 13. HANSEN, R. S., AND MIOTTO, M., J. Miner. Chem. Soc. 79, 1765 (1957). 14. FRIZ, G., Z. Angew. Phys. 19(4), 374 (1965). 15. CONEY, T. A., AND MASICA, W. J., N A S A Tech. Note TND-5115 (1969). 16. BLAKE, T. D., AND HAYNES, J. M., J. Colloid Interface Sci. 30, 421 (1969). 17. ELLISON,A. H., AND ZISM~XN,W. A., J. Phys. Chem. 58, 503 (1954). 18. FORT,T., JR., Advan. Chem. Ser. 43, 302. American Chemical Society, Washington, D.C. (1964). 19. SCHWARTZ,A. M., AND TEJADA, S. B., _NASA Contract. Rep. CR 72728, 1970.
Journal of Colloidand Interface Science, Vol.38, No. 2, FebruaryI972