952
S. G.BANKOFF
the pure components, and we do not reach it by calculating expansion from a n activity coefficient. This approach is, however, not altogether unsatisfactory, because free energy, and hence y, can be calculated with fair approximation because of the
Vol. 60
cancelling of uncertainties in entropy and enthalpy, noted above. We thank Dr. Berni J. Alder and Professor R. L. Scott for their criticisms, and the Atomic Energy Commission for its support of this work.
THE CONTORTIONAL ENERGY REQUIREMENT IN THE SPREADIXG OF LARGE DROPS BY S. G. BANKOFF’ Argonne National Laboratory, Lemont, Ill. Received January 16, 1066
The equation of the surface of a semi-infinite sheet of liquid spreading uni-directionally on a solid is deduced. For a particular liquid a t a given temperature, a single curve describes all possible surface configurations, regardless of contact angle. The reversible free energy change for a sheet of liquid climbing a barrier is then calculated. This is shown to be a measure of the energy barrier, and corresponds roughly to Good’s concept of “contortional energy,” which has, however, serious inconsistencies. For semi-infinite sheets on any surface, or for finite sheets on perfectly smooth surfaces, there is no hysteresis of the contact angle upon advancing the barrier into the line of contact, if the surface energies are assumed to be time-independent. For finite sheets on rough surfaces, the hysteresis of the contact angle depends upon the predominant orientation of the surface roughnesses, since Young’s Equation must hold microscopically everywhere. This is partially confirmed by Bartell and Shepard’s observations.
Introduction WenzeP has formulated a n expression which describes the contact angle in the spreading of a liquid drop over a rough surface. However, this expression is strictly valid only for very small drops (which are not appreciably flattened by gravity) resting on a surface whose roughnesses are very minute. Numerous metastable equilibrium states are possible, and for macroscopic roughnesses the energy barrier in passing from metastable to stable equilibrium may become controlling, as indicated by the data of Bartell and S h e ~ a r d . Bikerman14 ~ in discussing the motion of a drop front over a rough surface, has pointed out that the climbing of a drop front over a ridge requires “contortion (Le., extension) of the liquid-air interface,” and consequently an increase in the energy of the system. Good5 has related this contortional energy t o the observed contact angle hysteresis, and suggested that it could, in principle, be calculated for simple barrier configurations. As a preliminary to this calculation, the two-dimensional equation for the surface of a semi-infinite sheet of liquid spreading uni-directionally over a solid surface is derived. A very interesting result is obtained, in that the resulting equilibrium equation contains only one parameter, which is a function only of the liquid properties. Hence, a single equation serves to define the surface contour of a particular liquid a t a particular temperature, regardless of the contact angle, surface properties or surface configuration. This result should have interest for applications other than the one a t hand. The reversible free energy for a sheet of liquid climbing a sharp, plane barrier of arbitrary height and inclination is then calculated. If the horizontal surface and barrier are perfectly smooth, and if (1) Rose Polytechnio Institute, Terre Haute, Ind. (2) R. N. Wenzel, I n $ . Eng. Chem., 86,988 (1938). ( 3 ) J. W. Shepard and F. E. Bartell, THIS JOURNAL, 67,458 (1953). (4) J. J. Bikerman. {bid.,64, 653 (1950). (5) R. J. Good, J. Am. Chem. Soc., 7 4 , 5041 (1952).
time-dependent effects are excluded,6 it is shown that there will be no contact angle hysteresis. This will be true also if the surface is microscopically rough, but the liquid is semi-infinite in extent. For the usual case of a microscopically rough surface and a liquid of finite extent, the contact angle will increase as the volume of liquid is increased until the energy increment exceeds the L‘contortional energy,” when the liquid rapidly climbs the barrier. It is shown that the reversible free energy change is a rough measure of the contortional energy requirement for a drop front to advance over a ridge. These considerations lead to a simple relationship between the advancing and receding contact angles, which was, in fact, observed by Bartell and Shepard.3 Surface of a Semi-infinite Sheet of Liquid.I n order to compute the energy change for a large drop in climbing over a ridge, we must first obtain the equation for the equilibrium free liquid surface. We may consider the drop to be axially symmetrical, and if we .consider a really large drop, the problem reduces itself to a semi-infinite sheet of liquid spreading unidirectionally against a normal, plane barrier. Our first task is then to derive the equation for the free liquid surface under equilibrium conditions. Referring to Fig. 1, the horizontal asymptote to the free liquid surface will be chosen as the x-axis. The location of the z-axis is arbitrary. One principal radius of curvature of the liquid surface is everywhere infinite, and both principal radii of curvature are infinite a t x = a . The Gibbs condition for equilibrium of a curved interface’ reduces to ( h- P1) $72 =
Y 3
(1)
where p g and p1 are the gas and liquid densities, g the local acceleration due to gravity, y the sur(6) The expression “time-dependent effects” is used herein to include adsorption and any other surface altering effcts. (7)J. W. Gibbs. ”Colleated Works,” Vol. I.. Yale Univ. Press, New Haven, Conn., 1948, p. 282.
CONTORTIONAL ENERGY REQUIREMENT IN SPREADING OF LARGEDROPS
July, 1956
face tension, and R the radius of curvature of the liquid surface. R can be eliminated by ds = - R d a
CY
=
22
5
I
WATER at 25'C
(2)
where s is arc length, and CY is the angle of inclination of the tangent with the horizontal. Equations l and 2 can be readily solved to give - B cos
-z (centimeters)
0.07
953
+ const.
(3)
WATER a t 1 0 0 ° C
0.I
CCI, a t 76.8"C.
0.3
where B =
0.4
- PA7
(PI
When 2 = 0, CY = 0, for a sheet of liquid advancing t o the left and hence eq. 3 becomes B(l
- COS C Y ) = 222
corresponding to convex and concave menisci. This is easily solved, with the substitution
- 2B 22
O I w I 2 n
(6)
giving 5
= B V ~(log tan;
+ 2 cos w, + const.
0.5
-02
0
0.2
(4)
A similar result was obtained by Coghill and Anderson8 in their consideration of the "edge effect." CY can be eliminated from eq. 4 to give
cosw = 1
1
0.5
0.4
1.0
0.6 0.8
x (centimeters).
Fig. 1 .
barrier. For simplification, the roughness element is supposed t o be a plane wall lying parallel to the line of contact. I n practice, it makes a difference whether we consider that the wall and the horizontal surface area are perfectly smooth, or minutely rough, ie., whether the roughnesses have roughnesses. Consider a sheet of liquid advancing to the left against a wall inclined a t an angle cp with the horizontal (Fig. 2). We take the location of the
GAS
I/
J.-
(7)
For z = 0, w may be 0 or 2a,resulting in 12: = - m or These correspond to a semi-infinite sheet adco vancing to the right or t o the left, respectively. If we choose the latter case by limiting w between A and 2a, eq. 7 yields, upon elimination of w
+
~
The signs in front of the square root terms would be reversed for a sheet advancing t o the right. If we choose our coordinate system such that x = 0 when z = 2B'/s, the constant in eq. 8 vanishes identically. A plot of eq. 8 for water at 25 and 100°, and for carbon tetrachloride at its normal boiling point, is given in Fig. 1. It should be noted that, for a particular liquid a t a particular temperature, a single curve describes all possible surface configurations, regardless of contact angle, since B is the only parameter in eq. 8. This result should be of significance for applications other than the one a t hand. For instance, it is expected t o be of value in computing the dimensions of a surface cavity of given shape which will just barely entrap gas as the liquid drop spreads over it. The Climbing of a Barrier by a Sheet of Liquid.Having obtained an expression for the equilibrium free surface of a large sheet of liquid, we now take up the problem of the energy change in climbing a ( 8 ) W. H.Coghill and C. 0.Anderson, U. 8.Bur. Minea Tech. Paper 262, 47 (1923).
Fig. 2.
line of contact as xl, x1 and assume that the liquid extends for a distance L behind the line of contact. We shall assume first of all that the horizontal surface and the barrier are very smooth, so that all surface planes offered to the line of contact are parallel to the direction of the surface itself. Then, by the principle of a microscopic force balance, the contact angle through the liquid, 0, is fixed when the liquid is at rest, and is the same whether the line of contact is resting against the horizontal surface or the wall. Since L and cu1 are fixed by the system chosen, eq. 4 and 8 fix the surface contour, and the volume of liquid a t equilibrium. The free energy in this case is given by F = M h was YiaL YEL' (9) where, for unit width of sheet, s is the arc length of the liquid surface, L' is an arbitrary length of the wall, M the weight of liquid and h is the height of the center of gravity of the liquid above the horizontal surface. The subscripts 1, 2 and 3 refer to the solid, gaseous and liquid phases.
+
+
+
S.G. BANKOFF
954
If the liquid were in equilibrium a t the base of the wall (shown as the dotted in line Fig. 2), the free energy in this case would be F’ = M’h’
+
7’23s’
+
+
~13L
(10)
y12-L’
Vol. 60
H e then defines a “contortional energy,” F,, due to the extension of the liquid-air interface in climbing over a ridge such that the drop will cease to change its ‘shapewhen
and the decrement would be AF = F’ - F = (s’ - ~ ) y 2 3 (L‘M’ - E M ) (11) From eq. 1 and 2 it can be shown that
+
This is, on the face of it, open t o a t least two objections. First, bF/dAl3must be negative in order for the drop to spread a t all, and not positive, as implied by eq. 19. Secondly, the energy term, F,, where aland a2are the angles of inclination with the is not dimensionally consistent with the other horizontal of the tangents to the surface a t x = x1 terms, whose units are energy per unit area. The meaning of eq. 19 is therefore not clear, and it and x = z1- L. Similarly seems better t o state that the drop will cease t o tan s t a n 4 spread simply when S‘
-s
= B’lzlog
(tan
tan
4)
(13)
The fact that the arc lengths s and s‘ are segments of the same curve, as pointed out previously, has some interesting consequences. As L approaches infinity, s’ - s approaches a constant value. This may also be ssen fJom eq. 13, where at2+ x2 as L + a,. Also, h’ < h, and M‘ < M , so that ,&M - h’Mr = ( h - h’)M’ + h(M - M’) + m as L --t a. Hence -AF is infinite for a semi-infinite sheet. This means that if a barrier, a t any inclination, $, were advanced into a semi-infinite sheet of liquid, the liquid would immediately begin t o climb the barrier, with no hysteresis of contact angle, even if the surface roughness were such that some variation in the apparent contact angle were possible. We now consider the case of finite L. The gravitational potential energy change is
(14)
where (15)
z’=x+b
since the effect of changing CY a t the line of contact is simply to translate the z-axis. This displacement may be shown to be b = 2B’h (sin
2- sin T )
(16)
el’
From eq. 14 and 15 one can derive
hfM‘ - &M = - b p
[z2 4 B
- z”z 4
Equations 13, 16 and 17, when substituted into eq. 11, constitute a solution for the equilibrium free energy change upon beginning to climb the barrier. This quantity would superficially appear to correspond to the “contortional energy” proposed by several ~ r i t e r s . ~ JHowever, the concept is not simple, and deserves careful .analysis. Good,6 in a treatment valid only for drops small enough to have spherical surfaces, shows that
bA13bF
=
7’23 (COS h s d
- COS eeq) =
Yza A
COS
e (18)
which is, of course, the Gibbs condition. The situation is complicated by the fact that the climbing of the barrier is normally not a reversible process. For the system shown in Fig. 2, the retaining wall a t x1 - L may be considerea to be a piston, which is reversibly retracted until the liquid assumes the dotted line configuration. The piston is then reversibly returned to its original position, the liquid simultaneously climbing the wall. Since the liquid surface is now always more flat than it was initially, the center of gravity will be lower, and the net work of the liquid on the surroundings will be positive. This, however, is not the actual path, which is highly irreversible. If the piston is returned to its original position, while, a t the same time, restraining the liquid from climbing the wall, the work input, which is equal to - AF of eq. 11, is a measure of the departure from equilibrium. AF is therefore a measure of the energy input required to cause the liquid to leave its equilibrium state a t the base of the wall. It is probably higher than the actual energy input, despite the irreversibility of the process. This is because the liquid surface can assume dynamic nonequilibrium shapes which limit the energy changes principally to the region near the line of contact. Returning to our model of a perfectly flat horizontal surface and wall, a small displacement of the piston to the left from its initial position will immediately cause the liquid to climb the barrier until it reaches a new position of equilibrium some distance away. This is because L, a1and M are fixed for a smooth surface, leaving no degrees of freedom. Hence, the equilibrium is unstable for a piston displacement in this direction only. Good5 considers 7 2 3 A cos 0 to be a driving force which determines the ability of the liquid to climb a ridge. In the above case A cos 0 is zero, and yet the smallest displacement causes the liquid to climb a finite distance. It is apparent, therefore, that the displacement of the cosine of the contact angle from the equilibrium value is not adequate, per se, as a measure of the driving force. If we now consider that both the horizontal surface and the wall have numerous minute surface ridges, it is possible for the microscopic force balances to be satisfied with a change in the apparent contact angle for a differential movement of the
.
July, 19%
HYDRATION OF COLLOIDAL SILICA PARTICLES
piston to the right. This will continue until the line of contact, by small movements, is no longer able to find a sufficient number of planes of proper orientation to provide the apparent contact angle demanded by the relationships of eq. 4 and 8. At that point, the equilibrium becomes unstable, and the liquid climbs the wall. The contact angle hysteresis is determined, therefore, not by the inclination or height of the barrier (within reasonable limits), but by the predominant orientation of the surface roughnesses. If the roughnesses consist of a series of ridges whose sides are inclined to the horizontal by an angle, 9, we would expect ea = e., 'p (21) This relationship was in fact, observed by Bartell and Shepard3for liquids of high surface tension spreading over blocks of paraffin whose surfaces had been ruled to give asperities of known inclination. Eliminating runs in which large amounts of air were obviously entrapped, it was found that the height of the asperities had little effect, and that eq. 21 was obeyed, if 8eq. is taken as 8,, instead of (ea 8,)/2, as is usually done, for want of more precise knowledge. The situation is complicated here by the fact that their surfaces were ruled in two directions, so that spreading within the valleys, as climbing over the ridges, was an important mode of advance and recession. For this reason, presumably, their data for liquids of low surface energy do not follow eq. 21 so well (Table I).
+
+
IN
AQUEOUS SOLUTION
955
TABLE I ADVANCING AND RECEDING:CONTACTANGLES I N THE SPREADING: OF LIQUIDDROPS OVER MACHINED PARAFFIN SURFACES (DATA,OF SHEPARD AND BAR TELL^) Liquid Water 3 M CaClz Glycerol Ethylene glycol Methyl cellosolve Methanol
y, Smooth dynes/ surface om. 8s or
f a
-
30' or
72 110' 9Q0 129O 99' 84.5 119 109 136 114 63.2 97 90 115 92 47.5 81 74 93 65 30 62 42 76 12 22.6 42 27 58 0
% = 45' + = 60o'r or
142O 149 127 102 82 50
94" 115 83 49 0 0
Be
160' 96' 174 125 145 68 120 17 93 0 0 0
Conclusions (1) A semi-infinite sheet of liquid would be expected t o display no time-independent hysteresis of contact angle, ie., hysteresis attributable to surface roughness. ( 2 ) A method is given for calculating the reversible free energy change of a finite sheet of liquid upon climbing a barrier of known height and inclination. This corresponds roughly to the "contortional energy,'' and hence gives a measure of the contact angle hysteresis to be expected on advancing over a surface whose roughnesses have random inclinations and orientations. If, however, the roughnesses have a predominant inclination, this will tend to determine the contact angle hysteresis, in accordance with eq. 21.
DEGREE OF HYDRATION OF' PARTICLES OF COLLOIDAL SILICA I N AQUEOUS SOLUTION BY R. K. ILER AND R. L. DALTON Grasselli Chemicals Department, Experimental Station, E. I . du Pont de Nemours and Company, Inc. Received January 16, 1966
The viscosity of colloidal dispersions of silica has been studied, with the objective of determining the amount of water bound to the surface of the discrete, spherical amorphous particles. By applying the Mooney equation for the viscosity of a dispersion of spheres in a liquid medium, i t is calculated from the viscosity data that there is a monomolecular layer of water molecules immobilized, probably through hydrogen bonding, at the hydroxylated surface of the silica particles.
The degree to which the molecules in water are immobilized a t the surface of a highly polar solid apparently varies with the nature of the surface. For example, Bertil Jacobson' concluded that near the surface of molecules of sodium desoxyribonucleate, water molecules are oriented to form an immobilized lattice-oriented layer similar to ice. On the other hand, Vand2 has concluded that the viscosity of an aqueous solution of sucrose is consistent with the assumption that there is approximately a single layer of water molecules hydrogen bonded to the hydroxyl groups of the sugar molecule. It was of interest to determine whether the surface of amorphous silica immobilizes or otherwise binds water molecules. The viscosity of colloidal dispersions of amorphous silica has therefore been studied with the objective of determining the amount of water bound to the surface. ,
(1) B. Jacobson, J . Am. Chem. Soc., 77,2919 (1955). ( 2 ) V. Vand, TEIIE JOURNAL,52, 314 (1948).
Vand3 and Mooney4 have developed equations for a dispersion of spheres in a liquid, relating the relative viscosity of the dispersion to the volume fraction of the dispersed phase. The dispersed phase includes the volume of the solid particle plus any part of the medium ivhich is bound to the surface of the particles. With spheres larger than 100 mp diameter, the presence of a single molecular layer of solvent bound to the surface of the particle would have little effect on the volume fraction of the dispersed phase. However, in the case of spherical particles smaller than 10 or 20 mp in diameter, one or two molecular layers of solvent attached to the surface would appreciably increase the volume of the dispersed phase. This would have a marked influence on the viscosity. In the case of particles 5 mp in diameter, for example, the presence of two molecular layers of water, having a thickness of the (3) V. Vand, ibid., 52, 277. 300 (1948). (4) M. Mooney, J . CoZZoid Sci., 6 162 (1951).
,
