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fluid interface
The Effect of Gravity on the Drainage of a Thin Liquid Film between a Solid Sphere and a Liquid/Fluid Interface P. G. SMITH AND T. G. M. VAN DE VEN Pulp and Paper Research Institute of Canada and Department of Chemistry, McGill University, Montreal Quebec, Canada, H3A 2A 7
Received October 26, 1983; accepted February 20, 1984 A study has been made of the influence of gravitational forces on the thinning of the liquid film which forms as a solid sphere comes to rest on a liquid/fluid interface. It is found that rates of drainage can be dramatically affectedby the ratio of gravity to surface tension forces within the film. At long times a secondaryfilm can possiblybe formed which spreads out radiallyfrom the apex of the sphere. INTRODUCTION Some interesting features of the film thinning process can be readily demonstrated by a simple experiment in which a buoyant polystyrene sphere is allowed to rise through a more dense glycerol medium to a planar glycerol/air surface. The sphere comes to rest at the interface, forming an elevated meniscus which traps a thin film of glycerol. The film drains and eventually ruptures creating a point of three-phase contact. The position of rupture seems always to be asymmetrical, occurring not at the apex of the sphere but at the intersection of the glycerol film and the bulk meniscus. The resulting three-phase line spreads out approximately radially from this point, causing a rotation of the sphere until an equilibrium position is reached. A typical sequence of events is shown in the photographs in Fig. 1. The thinning of these films has been studied by Hartland (1) who formulated a governing partial differential equation describing the thickness of the film as a function of position and time. The equation is of fourth order in the spatial variable and first order with respect to time requiring four boundary conditions and an initial profile for complete specifica-
tion. The solution is complicated by the existence of "moving" or time-dependent boundary conditions at the apex of the film so that characteristics such as thinning rate or instantaneous profile are very much dependent on the history of the system. The governing equation was solved in (1) employing a finite difference scheme using an experimental profile for the initial conditions. Profiles obtained display a fairly flat region over the main portion of the film with a sharp contraction at the film periphery. Although the theory cannot predict rupture points, the film is thinnest at the constriction and is therefore most likely to break at this position. In the same paper, Hartland reported experimental results for the system consisting of a hollow aluminum sphere rising toward the interface between golden syrup and liquid parrafin. In general, theory and experimental resuits were in accord. Jones and Wilson (2) reexamined Hartland's equation and found analytical solutions valid under certain special conditions. In particular the shape of the film was considered to be already established and changes to this shape were assumed to occur only very slowly, i.e., the system was treated as being in quasistatic equilibrium.
456 0021-9797/84 $3.00 Copyright © 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984
EFFECT OF GRAVITY ON THE DRAINAGE OF A THIN LIQUID FILM
457
FIG. 1. Rupture of a thin glycerolfilm around a 2-mm polystyrenesphere at the glycerol/airinterface. Rotation of the sphere was observedas the three-phaseline expanded. Two distinct regions were identified. The first extends over the main portion of the film where the liquid/fluid interface is close to that of the sphere. This we will call the main film region. Here the pressure in the film is approximately hydrostatic with a slight excess due to film curvature. During the initial stages of thinning a balance exists between gravity and surface tension forces and the profile moves from one equilibrium position to another as the film drains. When gravity is unimportant the shape is that of a spherical cap which intersects the solid sphere at the film periphery. The second, much smaller region, which we will call the transition region, is located at the film periphery. Here the thickness of the film is so small that viscous forces are significant. This region forms the intersection between the main body of the film and the exterior meniscus on which the sphere rests. Here the pressure drop between the main film and the bulk liquid just beyond the constriction is balanced by viscous stresses generated by the draining liquid. Expansions of the governing partial differential equation were made in each of these regions and a solution was found using the method of matched asymptotic expansions.
The time and spatial dependencies were kept separate throughout the treatment and, while seemingly simplifying the resultant expressions, this approach leads to complications when discussing the effect of gravity on the thinning film. Jones and Wilson made some attempt to include gravitational effects in the development of film profiles. However, this was limited to a description of small deviations from the zero-gravity case. In this paper we will discuss the influence of gravity on the rate of thinning of such films and discuss the likely development of profiles in films where both surface tension and gravity are important. THEORY
1. General Considerations Consider a solid sphere resting on a deformed liquid/fluid interface as shown schematically in Fig. 2. In accordance with the notation of Jones and Wilson (2), we define polar coordinates r, 0 having their origin at the center of the sphere such that the film thickness H may be expressed as the difference between r at the liquid fluid/film interface and r = R the sphere radius. H = H(O, t) is a Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984
458
SMITH AND VAN DE VEN
I
TRANSITIONREGION
FIG. 2. A buoyant solid sphere resting at a liquid/fluid interface.Two possibleprofilesare shown at short (1) and long (2) thinning times. Thicknessof the film has been greatlyexaggeratedfor clarity.
function of both 0 and time t, and q5 is the value of 0 at the film periphery. The sphere is considered to have come to rest on the interface and the exterior meniscus is assumed to be approximately that due to a floating sphere with three-phase contact angle 0 = 0 °. In this assumption the weight of liquid in the film itself has been neglected. The value of 0 is determined by the shape of the exterior meniscus which itself is governed by the balance of forces on the sphere. The dependence of 4~ on system parameters such as R, surface tension, and relative densities, is not simple but a comprehensive numerical treatment has been presented by Huh and Mason (3) and Rapacchietta and Neumann (4). The governing equation is derived from an expansion of the Navier-Stokes equations in terms of the small parameter 6 = H(0, O)/R with 6 ,~ 1 (lubrication approximation). A convenient measure of the Bond number of the system is ~ = R2Apg/~ where Ap is the difference in densities between the liquid and the fluid, a is the interfacial tension, and g the acceleration due to gravity. Dimensionless quantities used are h = H/H(O, 0) and t' = t / T where T is a time scale to be determined. The governing equation, relating h and 0, is (2) Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984
0(02h
Oh
0-0 ~002 + cot 0 ~ + 2h
) x
+ -~ sin 0 E
- -
h 3 sin O
g(o,t')
[1]
where F = -
£
~ Oh sin OdO. Ot'
The system parameter ~ is given by ~ = 31aR/ ~T63, # being the dynamic viscosity of the liquid film. The first term on the left-hand side of Eq. [1] originates from the pressure gradient due to surface tension (the term in brackets is an expression for the curvature of the liquid film valid for small deviations from a spherical cap). The second term arises from a gravitational component of the pressure gradiem, while the right-hand side of the equation represents the viscous stresses due to the flow of liquid within the film (F is a measure of the dimensionless fluid flux). Jones and Wilson (2) showed that ~ = 0(6) such that in the main film, called the outer region in (2), the equation reduces to first order to
o(o h
ah
0-0 I~002 + cot 0 ~-~ + 2h
) + ~ sin 0 = O.
[2]
EFFECT OF G R A V I T Y ON T H E D R A I N A G E OF A T H I N LIQUID FILM
In the transition region, termed the inner region in (2), we may rescale the variables with respect to ~ such that the fluid flux is of comparable importance with the other terms. Defining e,,~ = h and ~O = 0 - 4~ the governing equation, to first order in ~ becomes ~O 3X 3Z
= A(¢, t3
[3]
where A(~b, t') = F(4~, t')/sin q~ and F(4~, t') is the value of the flux evaluated at 0 = q~. By the continuity of normal stress on the liquid interface we may show that 03,,Y(/003 is proportional to the pressure gradient and so Eq. [3] is of the form of the expression for the flux between two parallel plates. To match the solution valid in the main film 0Y ----'-C O0
as
0~-~
where C is the slope of the profile from the inner film region where 0 = ~. Similarly, to match solutions to the exterior meniscus oEx 002
--'B
as
0~+oo
where B is a constant and, due to our scaling, the product EB is related to the shape of the outer meniscus at the sphere surface. We may rescale the variables as p
= A-IC3,Uff
and
[4]
Jones and Wilson have solved Eq. [4] numerically with the results (2), =
1.2537 as
and
d2p ~ 1.2205 dz 2
z ----~ +c~3.
Thus ~ a A - I C s O 2 as
V = 2~r
h sin OdO
[5]
where V is made dimensionless by the factor H(O, O)R2, so that
__{f0 osin00}
F(~b,t') = 1 d A - - sin ~ sin ~ Ot' 1 dV 27r sin q~ dt'
[61
and thus CS = (Oh~ 5 -
\0-01~
B dV 2~ra sin ~ dt' "
[7]
We see that there is a direct relationship between the incoming slope of the liquid film from the main film region and the fluid flux. This result is implicit though not immediately obvious from Jones and Wilson's treatment because of their separate treatment of space and time variables. In view of the high sensitivity of the fluid flux to the film profile it is probable that even small changes in the film shape due to the action of gravity will have large effects on the rates of thinning of such films. 2. lnitial Stages o f Thinning
such that Eq. [3] may be rewritten
Pmin
sion for the dimensionless liquid volume of the main film region
z = A-1C40
d ° p p 3 = 1. dz 3
459
O~+~
Returning to the governing equation in the main film region and introducing the transformation h* = (6/)~)h, Eq. [2] may be written 02h* 0 Oh* d0-----i- + cot 00 + 2h* = D + cos 0,
[8]
where we have integrated once with respect to 0, D being the integration constant. Note that this transformation excludes the possibility X/6 = 0, but this case is adequately treated by Jones and Wilson (2). The solution to Eq. [8] is h* = D +
( + cos 0 1 n l t a n 0 ] ) Qkl
with a = B A C -5 ~ 0.61.
At this point we may introduce an expres-
1
-~cos01n]sin0l+Ecos0
[9]
Journal of Colloid and Interface Science, Vol. 100, No. 2, August1984
460
SMITH AND VAN DE VEN
where Q and E are also constants of integration. From the two boundary conditions
h*=0 Oh* 00
shows that Eq. [7] can be written as
(
at 0=4~
=0
at
9~ )'
0=0
we may eliminate Q and D, with the result
CUrv "~ R
- -~ cos 0 In 2 cos 2
+ E(cos 0 - cos 40.
Oh*
0---~ = sin 0[~ ln(2 cos2 ~) cos 0 +3(1+cos0)
] E
[11]
and L~
h* sin
OdO= or(1 - cos ~)2
x EE- ln(2 cos= )- 1 [12] where V* is the scaled value of the film volume, i.e., V* = 6/X V. We may combine Eqs. [11] and [12] to eliminate E such that V*=
[Oh*'~ + tk
[13]
where a and/3 are functions of ~bonly, defined
ot =
=
-
-y
3#Ra s R + 3" 1 3'
84.
[16]
Here 3" is the meridional radius of curvature of the exterior meniscus at the surface of the sphere. Since 3" is not a simple function of system parameters (see, e.g., Huh and Scriven (5)), the expression will be left in this form. However, we can estimate that if R ~ 3" and for the following experimental parameters from Hartland's paper; i.e., R = 6.3 × 10-3 m, a = 40 mN m -l, # = 7.80 Pa-sec, with 6 "~ 0.1 and q~ = 65 ° , then T "~ 6 hr. The time scale is highly sensitive to changes in ~b and 6. Returning to Eq. [15] we may write
(~)5 V - -~ #
dV at'
=
[17]
which is a first-order ordinary differential equation, the solution of which is k V = [4(J + t')]-1/4 + ~ 13 [18] where =
1 [Vo-X
1
~(1 -- COS ~)2
sin
and /~
(g + 3")
T=4~raasin4~
j
as
[15]
we may determine the order of the parameter and hence the time scale T:
[101 E incorporates the time dependence of the solution, i.e., E = E(t'). We may also write,
dV ---7" dt
By setting BaS/21ra sin ~b = - 1 , we fix the time scale T and also define the parameter B. Matching ~B to an expression for the curvature of the outer meniscus (2)
= ~ c o s 4 ~ l n 2cos ~
V* = 27r
Ba'
V - -~ ¢~ = 2 r a sin ¢
71"(1
E os0
COS ¢~)2 "l
3
+ cos 4~
"
Rewriting Eq. [13] as
00]~ Journal of Colloid and Interface Science.
[14] Vol. 100,No. 2, August1984
Vo is the initial film volume at t' = 0. This may be obtained using the initial condition h(0, 0) = 1. The initial value of E, Eo, is then found from Eq. [10]: 1 16 E ° = 1 - c o s ~ b LX
cos q~ 3
× In(2 cos2 ~ ) + ~1 In
2]
[191
EFFECT OF GRAVITY
ON THE DRAINAGE
and hence V0 = a-(1 - cos 4~)2 × lEo - 1 ln(2 cos2 ~ ) - 1] •
[201
Since we may write Eq. [12] as
V = q~E + q2 where ql = 7r(1 - cos ~b)2 and q2 = - 3 Iln(2 cosZ ~ ) + ~] we may express E = E(t') by ),
qlE = [4(J + t')] -1/4 - q2 + -~ fl where
)k
[21]
-4
For given values of ~,/~ we may find E = E(t'), and by Eq. [10], the development of film profiles. The time variation of the minimum thickness at the constriction (0 = ~) may be found from the expression a~mi n =
1.2537aC 2 B
[221
where C = -(Oh/aO)o is a function o r e = E(t').
3. Long Time Thinning Inspection of Eq. [18] shows that as t' oo, Vbecomes negative for all finite values of q~ between 0 and r, indicating a breakdown in the solution at long times. Exactly when and why the solution breaks down is best seen by considering the physics of the situation. During the drainage process the film thickness will, due to the flattening action of gravity, tend to zero at the sphere apex while still remaining finite in the rest of the main film region. As the film thickness at the apex becomes comparable to that in the transition region, we may expect the formation of a sec-
OF A THIN
LIQUID FILM
461
ond thin film, also of thickness 0(bh) extending axially from the apex of the sphere as the total volume of liquid in the film decreases. In this case the boundary conditions we have used can no longer be valid and we cannot expect the solution we have found to hold after the formation of the new film. The nature of the boundary conditions at the intersection of this secondary film and the main film must be obtained through a rescaling of the equations of motion within the new film together with a similar procedure for matching profiles with the main film. We may thus define a second transition region located at this intersection where O = and ff = ~p(t') (see Fig. 2, profile 2). The governing equation will be similar to Eq. [3] with ~k(t') replacing q~in the expression for the liquid flux. It is reasonable to assume that the solution retains its quadratic dependence as z --- +oo such that in the main film both h and Oh/O0 tend to zero at O = ¢/. In effect, this is equivalent to assuming zero contact angle between the two films. Since the thickness of the secondary film is of the same order as the transition region, one would expect viscous stresses to be present here also, and it is possible that such forces may effect the movement of the intersection (0 = ~) over the solid surface. However, it is likely that the value of the fluid flux associated with the deposition of the secondary film is much smaller than that through the transition region and so at present the effect of viscous stresses on the movement of the intersection will be ignored. We now have three boundary conditions for the main film region, namely h(~)=0,
h(¢)=0,
(0h~ = 0 , \utl/
so that we may determine Q, E, and D directly from Eq. [9] and hence V and dV]dt' from Eqs. [5] and [7], respectively. The rate of expansion of the secondary film d~b/dt' can then be found from
d~b= 1 dV dt' dV/d~ (It'
[231
where d~/dt' is a function of )~/~ and ~b only. Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984
462
SMITH AND VAN DE VEN The m i n i m u m thickness of the film at the constriction has been calculated from Eq. [22] for ~b -- 60 ° and k/~ = 10 (Fig. 3b) and was found to vary from 8.3 × 10 -3 ~Ho at t' = 0 to 7.3 × 10 -4 ~n0 at the limiting profile. At low values of k/~ the film approximates the shape of a spherical cap. At higher values even initial profiles show a marked influence of gravity, and in each case we see an increasing importance of gravity (greater distortion from the spherical shape) as the film thins. This is a necessary feature o f the solutions since a profile at a given value o f ),/6 after some time t' must coincide with an initial profile with a correspondingly higher value of X/6. Figure 4 shows on an expanded scale the development of the main film after the formation of the secondary film. The limiting profile (i.e., ~b = 0) is shown for ~ = 60 ° and k/~ = 10 together with the profiles for ~ = 12, 24, 36 °, and their associated times. As expected, the m o v e m e n t of ~ is fastest at the beginning where the sensitivity to the change in volume is greatest. Thinning becomes very slow as ~b ~ 4 and V--~ 0. This type of development m a y provide an
However, the expressions for Q, E, and D, and hence the derivatives in Eq. [23], are somewhat lengthy, and so it was found more convenient to express both dV/dt' and V in terms of the three constants Q, E, and D and obtain ~ = ~(t') numerically from the approximation AV= V~-V~+~=
dt--7 .At'
[241
where V~ is the volume of the inner region at a particular value of ~ and At' is the time required for the m o v e m e n t of the intersection
from ~ to ~ + A~. Calculations were performed on an Apple II computer taking the value of A~b to be qV200. RESULTS
AND
DISCUSSION
The time evolution of the film profiles from the present theory is shown in Fig. 3 for q~ = 60 ° and various initial values of k/& Profiles are shown at t' = 0, 10 -5, and increasing decades of time including the limiting profile before the formation of the secondary film, where h(0, t') = 0. Also given in Fig. 3 are the times required to reach these limiting shapes.
h
0
1.0
c
d
0.5
0 0
20 °
40 °
60 °
20 °
40 °
60 °
o
FIG. 3. Film profilesh vs Ofor q~ = 60° and various initial values of the ratio k/t~ indicating the effect of gravity. (a)),/~ = 1, (b) k/fi = 10, (c) k/t~ = 20, (d) k/8 = 50. Profiles are given for t' = 0, 10-5, 10-4, and increasing decades of time until a limiting profile is reached when h(0, F) = 0. The time to reach this profile is (a) 15,510 T, (b) 1.54 T, (c) 0.09 T, (d) 1.98 × 10 -3 T, where Tis the time scale (see text). Journal ufColloid and Interface Science, Vol. 100, No. 2, August 1984
EFFECT OF GRAVITY ON THE DRAINAGE OF A THIN LIQUID FILM
463
0.1
h
o.o5
o 0
20 °
40 °
60 °
0 FIG. 4.
Developmento f
the main film region after the formation o f the secondary film, ~ = 60 °,
~/~ = 10.
explanation for some of Hartland's experimental profiles, particularly at long times. For example, his profile for t = 72,000 sec shows a flat film extending from 0 = 0 to 45 o followed by a small hump just before the contraction point, indicative of a well-formed secondary film. It is not clear, however, that all films will follow this course, and it is highly likely that some will rupture (presumably at the constriction) before they arrive at this stage. The effect of gravity on the rate of drainage is best seen by considering a numerical example. We will use again the experimental parameters of Hartland (1), with ~, = 5.2 and the approximation R "~ 3'. (This will not be exactly true, but in comparison to the other terms in Eq. [ 16], an order of magnitude approximation is sufficient.) For 6 = 0.08 and X/6 ~ 63 (1), the limiting profile is reached after "~ 11 sec. At lower values of X (the ratio of gravity to surface tension forces) drainage times are dramatically increased such that if X = 12, 1, 0.5--and all other variables remain the same--the limiting profile is reached after ~ 7 6 sec, ~ 2 3 rain, and "-'40 hr, respectively. This comparison is, of course, somewhat restricted since in an experimental situation, lowering X by reducing either R or Ap, will also result in a change in 4) which from Eq. [16] can be seen to be very important. No attempt is made here to incorporate changes
in q~ into our estimates for thinning times, because of the complex way 4) is related to the other parameters. However, the general trend is clear; even small changes in the ratio of gravity to surface tension forces can have profound effects on the rates of thinning. CONCLUDING REMARKS
The treatment presented here describes the behavior of thinning films when both surface tension and gravity forces are important. The analysis of Jones and Wilson, developed for X/6 ~ 1 must have substantial limitations in experimental situations since we also have the condition ~ ~ 1 which imposes severe restrictions on the value of X. (In particular these two conditions can never be fulfilled for Hartland's system where ~, = 5.2) Although we have lifted some restrictions on the value of ~/6, we should note a further complication which may arise in the later stages of thinning when X/~ is large. Under these circumstances the action of gravity may cause fairly large deviations from the spherical cap shape such that the curvature of the meniscus can n o longer be represented by the expression used in Eq. [1 ]. If this effect is not prominent before the limiting profile is reached it may become important as the secondary film develops. Another limitation of Journal of Colloid and Interface Science, VoL 100, No. 2, August 1984
464
SMITH AND VAN DE VEN
the present theory is the quasi-static nature of the solutions, i.e., no information has been obtained about the development of profiles from arbitrary initial conditions. Such solutions can only be found from a numerical treatment of Eq. [1], but then we still have difficulty in defining what initial profile should be used. A comparison of our results with numerical solutions is at present not possible since no comprehensive numerical treatment of Eq. [1] is available. The only definitive treatment would be to consider movement of the sphere from large distances toward an initially undisturbed liquid/fluid interface. A recent paper by Leal and Lee (6) has considered the transport of a sphere toward and eventually through such an interface under conditions of constant force on the sphere or constant sphere velocity. Their analysis concentrates on the dynamics of what we have termed the exterior meniscus and, although not specifically designed to treat liquid film profiles, their determination of the initial deformation of the liquid/fluid interface could provide the basis for a more general treatment of the film drainage problem. APPENDIX: NOMENCLATURE a A B C D E
constant "~ 0.61 function of mass flux defined in Eq. [6] coefficient o f a 2 x / a O 2 as O ~ ~ , found by matching to exterior meniscus slope of film at 0 = constant of integration, Eq. [8] time-dependent function defined in Eq.
F g H h
liquid flux Eq. [1] acceleration due to gravity film thickness dimensionless film thickness
h*
scaled dimensionless film thickness h* = (~/X)h
[9]
h = H/H(O, O)
Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984
,,~ J O p R r t t' T
thickness in the inner region CX = h volume function (V0 - X#/~)-4, Eq. [18] origin of polar coordinates parameter defined in Eq. [4] radius of sphere polar coordinate time dimensionless time t' = t / T time scale V / V * volume of liquid film based on h/h* z parameter defined in Eq. [4] Greek Letters
a, B functions of ~b defined in Eq. [13] meridional radius of curvature of exterior meniscus lubrication parameter ~ = H(0, O)/R system parameter, Eq. [ 1] 0 polar coordinate O scaled polar coordinate EO = 0 ~, Bond number ~, = R2A#g/tr # film viscosity p film density, Ap liquid/fluid density difference cr surface tension 4) value of 0 at film periphery ~b intersection of secondary film and main film region ACKNOWLEDGMENTS The authors thank Dr. R. G. Cox and Dr. S. G. Mason for valuable discussion and advice. REFERENCES 1. Hartland, S., Chem. Eng. Sci. 24, 987 (1969). 2. Jones, A. F., and Wilson, S. D. R., J. Fluid Mech. 87, 263 (1978). 3. Huh, C., and Mason, S. G., J. Colloid Interface Sci. 47, 271 (1974). 4. Rapacchietta,A. V., and Neumann, A. W., J. Colloid Interface Sci. 59, 555 (1977). 5. Huh, C., and Scriven, L. E., J. Colloid Interface Sci. 30, 323 (1969). 6. Leal, L. G., and Lee, S. H., Adv. Colloid Interface Sci. 17, 61 (1982).
Received October 26, 1983; accepted February 20, 1984 A study has been made of the influence of gravitational forces on the thinning of the liquid film which forms as a solid sphere comes to rest on a liquid/fluid interface. It is found that rates of drainage can be dramatically affectedby the ratio of gravity to surface tension forces within the film. At long times a secondaryfilm can possiblybe formed which spreads out radiallyfrom the apex of the sphere. INTRODUCTION Some interesting features of the film thinning process can be readily demonstrated by a simple experiment in which a buoyant polystyrene sphere is allowed to rise through a more dense glycerol medium to a planar glycerol/air surface. The sphere comes to rest at the interface, forming an elevated meniscus which traps a thin film of glycerol. The film drains and eventually ruptures creating a point of three-phase contact. The position of rupture seems always to be asymmetrical, occurring not at the apex of the sphere but at the intersection of the glycerol film and the bulk meniscus. The resulting three-phase line spreads out approximately radially from this point, causing a rotation of the sphere until an equilibrium position is reached. A typical sequence of events is shown in the photographs in Fig. 1. The thinning of these films has been studied by Hartland (1) who formulated a governing partial differential equation describing the thickness of the film as a function of position and time. The equation is of fourth order in the spatial variable and first order with respect to time requiring four boundary conditions and an initial profile for complete specifica-
tion. The solution is complicated by the existence of "moving" or time-dependent boundary conditions at the apex of the film so that characteristics such as thinning rate or instantaneous profile are very much dependent on the history of the system. The governing equation was solved in (1) employing a finite difference scheme using an experimental profile for the initial conditions. Profiles obtained display a fairly flat region over the main portion of the film with a sharp contraction at the film periphery. Although the theory cannot predict rupture points, the film is thinnest at the constriction and is therefore most likely to break at this position. In the same paper, Hartland reported experimental results for the system consisting of a hollow aluminum sphere rising toward the interface between golden syrup and liquid parrafin. In general, theory and experimental resuits were in accord. Jones and Wilson (2) reexamined Hartland's equation and found analytical solutions valid under certain special conditions. In particular the shape of the film was considered to be already established and changes to this shape were assumed to occur only very slowly, i.e., the system was treated as being in quasistatic equilibrium.
456 0021-9797/84 $3.00 Copyright © 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984
EFFECT OF GRAVITY ON THE DRAINAGE OF A THIN LIQUID FILM
457
FIG. 1. Rupture of a thin glycerolfilm around a 2-mm polystyrenesphere at the glycerol/airinterface. Rotation of the sphere was observedas the three-phaseline expanded. Two distinct regions were identified. The first extends over the main portion of the film where the liquid/fluid interface is close to that of the sphere. This we will call the main film region. Here the pressure in the film is approximately hydrostatic with a slight excess due to film curvature. During the initial stages of thinning a balance exists between gravity and surface tension forces and the profile moves from one equilibrium position to another as the film drains. When gravity is unimportant the shape is that of a spherical cap which intersects the solid sphere at the film periphery. The second, much smaller region, which we will call the transition region, is located at the film periphery. Here the thickness of the film is so small that viscous forces are significant. This region forms the intersection between the main body of the film and the exterior meniscus on which the sphere rests. Here the pressure drop between the main film and the bulk liquid just beyond the constriction is balanced by viscous stresses generated by the draining liquid. Expansions of the governing partial differential equation were made in each of these regions and a solution was found using the method of matched asymptotic expansions.
The time and spatial dependencies were kept separate throughout the treatment and, while seemingly simplifying the resultant expressions, this approach leads to complications when discussing the effect of gravity on the thinning film. Jones and Wilson made some attempt to include gravitational effects in the development of film profiles. However, this was limited to a description of small deviations from the zero-gravity case. In this paper we will discuss the influence of gravity on the rate of thinning of such films and discuss the likely development of profiles in films where both surface tension and gravity are important. THEORY
1. General Considerations Consider a solid sphere resting on a deformed liquid/fluid interface as shown schematically in Fig. 2. In accordance with the notation of Jones and Wilson (2), we define polar coordinates r, 0 having their origin at the center of the sphere such that the film thickness H may be expressed as the difference between r at the liquid fluid/film interface and r = R the sphere radius. H = H(O, t) is a Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984
458
SMITH AND VAN DE VEN
I
TRANSITIONREGION
FIG. 2. A buoyant solid sphere resting at a liquid/fluid interface.Two possibleprofilesare shown at short (1) and long (2) thinning times. Thicknessof the film has been greatlyexaggeratedfor clarity.
function of both 0 and time t, and q5 is the value of 0 at the film periphery. The sphere is considered to have come to rest on the interface and the exterior meniscus is assumed to be approximately that due to a floating sphere with three-phase contact angle 0 = 0 °. In this assumption the weight of liquid in the film itself has been neglected. The value of 0 is determined by the shape of the exterior meniscus which itself is governed by the balance of forces on the sphere. The dependence of 4~ on system parameters such as R, surface tension, and relative densities, is not simple but a comprehensive numerical treatment has been presented by Huh and Mason (3) and Rapacchietta and Neumann (4). The governing equation is derived from an expansion of the Navier-Stokes equations in terms of the small parameter 6 = H(0, O)/R with 6 ,~ 1 (lubrication approximation). A convenient measure of the Bond number of the system is ~ = R2Apg/~ where Ap is the difference in densities between the liquid and the fluid, a is the interfacial tension, and g the acceleration due to gravity. Dimensionless quantities used are h = H/H(O, 0) and t' = t / T where T is a time scale to be determined. The governing equation, relating h and 0, is (2) Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984
0(02h
Oh
0-0 ~002 + cot 0 ~ + 2h
) x
+ -~ sin 0 E
- -
h 3 sin O
g(o,t')
[1]
where F = -
£
~ Oh sin OdO. Ot'
The system parameter ~ is given by ~ = 31aR/ ~T63, # being the dynamic viscosity of the liquid film. The first term on the left-hand side of Eq. [1] originates from the pressure gradient due to surface tension (the term in brackets is an expression for the curvature of the liquid film valid for small deviations from a spherical cap). The second term arises from a gravitational component of the pressure gradiem, while the right-hand side of the equation represents the viscous stresses due to the flow of liquid within the film (F is a measure of the dimensionless fluid flux). Jones and Wilson (2) showed that ~ = 0(6) such that in the main film, called the outer region in (2), the equation reduces to first order to
o(o h
ah
0-0 I~002 + cot 0 ~-~ + 2h
) + ~ sin 0 = O.
[2]
EFFECT OF G R A V I T Y ON T H E D R A I N A G E OF A T H I N LIQUID FILM
In the transition region, termed the inner region in (2), we may rescale the variables with respect to ~ such that the fluid flux is of comparable importance with the other terms. Defining e,,~ = h and ~O = 0 - 4~ the governing equation, to first order in ~ becomes ~O 3X 3Z
= A(¢, t3
[3]
where A(~b, t') = F(4~, t')/sin q~ and F(4~, t') is the value of the flux evaluated at 0 = q~. By the continuity of normal stress on the liquid interface we may show that 03,,Y(/003 is proportional to the pressure gradient and so Eq. [3] is of the form of the expression for the flux between two parallel plates. To match the solution valid in the main film 0Y ----'-C O0
as
0~-~
where C is the slope of the profile from the inner film region where 0 = ~. Similarly, to match solutions to the exterior meniscus oEx 002
--'B
as
0~+oo
where B is a constant and, due to our scaling, the product EB is related to the shape of the outer meniscus at the sphere surface. We may rescale the variables as p
= A-IC3,Uff
and
[4]
Jones and Wilson have solved Eq. [4] numerically with the results (2), =
1.2537 as
and
d2p ~ 1.2205 dz 2
z ----~ +c~3.
Thus ~ a A - I C s O 2 as
V = 2~r
h sin OdO
[5]
where V is made dimensionless by the factor H(O, O)R2, so that
__{f0 osin00}
F(~b,t') = 1 d A - - sin ~ sin ~ Ot' 1 dV 27r sin q~ dt'
[61
and thus CS = (Oh~ 5 -
\0-01~
B dV 2~ra sin ~ dt' "
[7]
We see that there is a direct relationship between the incoming slope of the liquid film from the main film region and the fluid flux. This result is implicit though not immediately obvious from Jones and Wilson's treatment because of their separate treatment of space and time variables. In view of the high sensitivity of the fluid flux to the film profile it is probable that even small changes in the film shape due to the action of gravity will have large effects on the rates of thinning of such films. 2. lnitial Stages o f Thinning
such that Eq. [3] may be rewritten
Pmin
sion for the dimensionless liquid volume of the main film region
z = A-1C40
d ° p p 3 = 1. dz 3
459
O~+~
Returning to the governing equation in the main film region and introducing the transformation h* = (6/)~)h, Eq. [2] may be written 02h* 0 Oh* d0-----i- + cot 00 + 2h* = D + cos 0,
[8]
where we have integrated once with respect to 0, D being the integration constant. Note that this transformation excludes the possibility X/6 = 0, but this case is adequately treated by Jones and Wilson (2). The solution to Eq. [8] is h* = D +
( + cos 0 1 n l t a n 0 ] ) Qkl
with a = B A C -5 ~ 0.61.
At this point we may introduce an expres-
1
-~cos01n]sin0l+Ecos0
[9]
Journal of Colloid and Interface Science, Vol. 100, No. 2, August1984
460
SMITH AND VAN DE VEN
where Q and E are also constants of integration. From the two boundary conditions
h*=0 Oh* 00
shows that Eq. [7] can be written as
(
at 0=4~
=0
at
9~ )'
0=0
we may eliminate Q and D, with the result
CUrv "~ R
- -~ cos 0 In 2 cos 2
+ E(cos 0 - cos 40.
Oh*
0---~ = sin 0[~ ln(2 cos2 ~) cos 0 +3(1+cos0)
] E
[11]
and L~
h* sin
OdO= or(1 - cos ~)2
x EE- ln(2 cos= )- 1 [12] where V* is the scaled value of the film volume, i.e., V* = 6/X V. We may combine Eqs. [11] and [12] to eliminate E such that V*=
[Oh*'~ + tk
[13]
where a and/3 are functions of ~bonly, defined
ot =
=
-
-y
3#Ra s R + 3" 1 3'
84.
[16]
Here 3" is the meridional radius of curvature of the exterior meniscus at the surface of the sphere. Since 3" is not a simple function of system parameters (see, e.g., Huh and Scriven (5)), the expression will be left in this form. However, we can estimate that if R ~ 3" and for the following experimental parameters from Hartland's paper; i.e., R = 6.3 × 10-3 m, a = 40 mN m -l, # = 7.80 Pa-sec, with 6 "~ 0.1 and q~ = 65 ° , then T "~ 6 hr. The time scale is highly sensitive to changes in ~b and 6. Returning to Eq. [15] we may write
(~)5 V - -~ #
dV at'
=
[17]
which is a first-order ordinary differential equation, the solution of which is k V = [4(J + t')]-1/4 + ~ 13 [18] where =
1 [Vo-X
1
~(1 -- COS ~)2
sin
and /~
(g + 3")
T=4~raasin4~
j
as
[15]
we may determine the order of the parameter and hence the time scale T:
[101 E incorporates the time dependence of the solution, i.e., E = E(t'). We may also write,
dV ---7" dt
By setting BaS/21ra sin ~b = - 1 , we fix the time scale T and also define the parameter B. Matching ~B to an expression for the curvature of the outer meniscus (2)
= ~ c o s 4 ~ l n 2cos ~
V* = 27r
Ba'
V - -~ ¢~ = 2 r a sin ¢
71"(1
E os0
COS ¢~)2 "l
3
+ cos 4~
"
Rewriting Eq. [13] as
00]~ Journal of Colloid and Interface Science.
[14] Vol. 100,No. 2, August1984
Vo is the initial film volume at t' = 0. This may be obtained using the initial condition h(0, 0) = 1. The initial value of E, Eo, is then found from Eq. [10]: 1 16 E ° = 1 - c o s ~ b LX
cos q~ 3
× In(2 cos2 ~ ) + ~1 In
2]
[191
EFFECT OF GRAVITY
ON THE DRAINAGE
and hence V0 = a-(1 - cos 4~)2 × lEo - 1 ln(2 cos2 ~ ) - 1] •
[201
Since we may write Eq. [12] as
V = q~E + q2 where ql = 7r(1 - cos ~b)2 and q2 = - 3 Iln(2 cosZ ~ ) + ~] we may express E = E(t') by ),
qlE = [4(J + t')] -1/4 - q2 + -~ fl where
)k
[21]
-4
For given values of ~,/~ we may find E = E(t'), and by Eq. [10], the development of film profiles. The time variation of the minimum thickness at the constriction (0 = ~) may be found from the expression a~mi n =
1.2537aC 2 B
[221
where C = -(Oh/aO)o is a function o r e = E(t').
3. Long Time Thinning Inspection of Eq. [18] shows that as t' oo, Vbecomes negative for all finite values of q~ between 0 and r, indicating a breakdown in the solution at long times. Exactly when and why the solution breaks down is best seen by considering the physics of the situation. During the drainage process the film thickness will, due to the flattening action of gravity, tend to zero at the sphere apex while still remaining finite in the rest of the main film region. As the film thickness at the apex becomes comparable to that in the transition region, we may expect the formation of a sec-
OF A THIN
LIQUID FILM
461
ond thin film, also of thickness 0(bh) extending axially from the apex of the sphere as the total volume of liquid in the film decreases. In this case the boundary conditions we have used can no longer be valid and we cannot expect the solution we have found to hold after the formation of the new film. The nature of the boundary conditions at the intersection of this secondary film and the main film must be obtained through a rescaling of the equations of motion within the new film together with a similar procedure for matching profiles with the main film. We may thus define a second transition region located at this intersection where O = and ff = ~p(t') (see Fig. 2, profile 2). The governing equation will be similar to Eq. [3] with ~k(t') replacing q~in the expression for the liquid flux. It is reasonable to assume that the solution retains its quadratic dependence as z --- +oo such that in the main film both h and Oh/O0 tend to zero at O = ¢/. In effect, this is equivalent to assuming zero contact angle between the two films. Since the thickness of the secondary film is of the same order as the transition region, one would expect viscous stresses to be present here also, and it is possible that such forces may effect the movement of the intersection (0 = ~) over the solid surface. However, it is likely that the value of the fluid flux associated with the deposition of the secondary film is much smaller than that through the transition region and so at present the effect of viscous stresses on the movement of the intersection will be ignored. We now have three boundary conditions for the main film region, namely h(~)=0,
h(¢)=0,
(0h~ = 0 , \utl/
so that we may determine Q, E, and D directly from Eq. [9] and hence V and dV]dt' from Eqs. [5] and [7], respectively. The rate of expansion of the secondary film d~b/dt' can then be found from
d~b= 1 dV dt' dV/d~ (It'
[231
where d~/dt' is a function of )~/~ and ~b only. Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984
462
SMITH AND VAN DE VEN The m i n i m u m thickness of the film at the constriction has been calculated from Eq. [22] for ~b -- 60 ° and k/~ = 10 (Fig. 3b) and was found to vary from 8.3 × 10 -3 ~Ho at t' = 0 to 7.3 × 10 -4 ~n0 at the limiting profile. At low values of k/~ the film approximates the shape of a spherical cap. At higher values even initial profiles show a marked influence of gravity, and in each case we see an increasing importance of gravity (greater distortion from the spherical shape) as the film thins. This is a necessary feature o f the solutions since a profile at a given value o f ),/6 after some time t' must coincide with an initial profile with a correspondingly higher value of X/6. Figure 4 shows on an expanded scale the development of the main film after the formation of the secondary film. The limiting profile (i.e., ~b = 0) is shown for ~ = 60 ° and k/~ = 10 together with the profiles for ~ = 12, 24, 36 °, and their associated times. As expected, the m o v e m e n t of ~ is fastest at the beginning where the sensitivity to the change in volume is greatest. Thinning becomes very slow as ~b ~ 4 and V--~ 0. This type of development m a y provide an
However, the expressions for Q, E, and D, and hence the derivatives in Eq. [23], are somewhat lengthy, and so it was found more convenient to express both dV/dt' and V in terms of the three constants Q, E, and D and obtain ~ = ~(t') numerically from the approximation AV= V~-V~+~=
dt--7 .At'
[241
where V~ is the volume of the inner region at a particular value of ~ and At' is the time required for the m o v e m e n t of the intersection
from ~ to ~ + A~. Calculations were performed on an Apple II computer taking the value of A~b to be qV200. RESULTS
AND
DISCUSSION
The time evolution of the film profiles from the present theory is shown in Fig. 3 for q~ = 60 ° and various initial values of k/& Profiles are shown at t' = 0, 10 -5, and increasing decades of time including the limiting profile before the formation of the secondary film, where h(0, t') = 0. Also given in Fig. 3 are the times required to reach these limiting shapes.
h
0
1.0
c
d
0.5
0 0
20 °
40 °
60 °
20 °
40 °
60 °
o
FIG. 3. Film profilesh vs Ofor q~ = 60° and various initial values of the ratio k/t~ indicating the effect of gravity. (a)),/~ = 1, (b) k/fi = 10, (c) k/t~ = 20, (d) k/8 = 50. Profiles are given for t' = 0, 10-5, 10-4, and increasing decades of time until a limiting profile is reached when h(0, F) = 0. The time to reach this profile is (a) 15,510 T, (b) 1.54 T, (c) 0.09 T, (d) 1.98 × 10 -3 T, where Tis the time scale (see text). Journal ufColloid and Interface Science, Vol. 100, No. 2, August 1984
EFFECT OF GRAVITY ON THE DRAINAGE OF A THIN LIQUID FILM
463
0.1
h
o.o5
o 0
20 °
40 °
60 °
0 FIG. 4.
Developmento f
the main film region after the formation o f the secondary film, ~ = 60 °,
~/~ = 10.
explanation for some of Hartland's experimental profiles, particularly at long times. For example, his profile for t = 72,000 sec shows a flat film extending from 0 = 0 to 45 o followed by a small hump just before the contraction point, indicative of a well-formed secondary film. It is not clear, however, that all films will follow this course, and it is highly likely that some will rupture (presumably at the constriction) before they arrive at this stage. The effect of gravity on the rate of drainage is best seen by considering a numerical example. We will use again the experimental parameters of Hartland (1), with ~, = 5.2 and the approximation R "~ 3'. (This will not be exactly true, but in comparison to the other terms in Eq. [ 16], an order of magnitude approximation is sufficient.) For 6 = 0.08 and X/6 ~ 63 (1), the limiting profile is reached after "~ 11 sec. At lower values of X (the ratio of gravity to surface tension forces) drainage times are dramatically increased such that if X = 12, 1, 0.5--and all other variables remain the same--the limiting profile is reached after ~ 7 6 sec, ~ 2 3 rain, and "-'40 hr, respectively. This comparison is, of course, somewhat restricted since in an experimental situation, lowering X by reducing either R or Ap, will also result in a change in 4) which from Eq. [16] can be seen to be very important. No attempt is made here to incorporate changes
in q~ into our estimates for thinning times, because of the complex way 4) is related to the other parameters. However, the general trend is clear; even small changes in the ratio of gravity to surface tension forces can have profound effects on the rates of thinning. CONCLUDING REMARKS
The treatment presented here describes the behavior of thinning films when both surface tension and gravity forces are important. The analysis of Jones and Wilson, developed for X/6 ~ 1 must have substantial limitations in experimental situations since we also have the condition ~ ~ 1 which imposes severe restrictions on the value of X. (In particular these two conditions can never be fulfilled for Hartland's system where ~, = 5.2) Although we have lifted some restrictions on the value of ~/6, we should note a further complication which may arise in the later stages of thinning when X/~ is large. Under these circumstances the action of gravity may cause fairly large deviations from the spherical cap shape such that the curvature of the meniscus can n o longer be represented by the expression used in Eq. [1 ]. If this effect is not prominent before the limiting profile is reached it may become important as the secondary film develops. Another limitation of Journal of Colloid and Interface Science, VoL 100, No. 2, August 1984
464
SMITH AND VAN DE VEN
the present theory is the quasi-static nature of the solutions, i.e., no information has been obtained about the development of profiles from arbitrary initial conditions. Such solutions can only be found from a numerical treatment of Eq. [1], but then we still have difficulty in defining what initial profile should be used. A comparison of our results with numerical solutions is at present not possible since no comprehensive numerical treatment of Eq. [1] is available. The only definitive treatment would be to consider movement of the sphere from large distances toward an initially undisturbed liquid/fluid interface. A recent paper by Leal and Lee (6) has considered the transport of a sphere toward and eventually through such an interface under conditions of constant force on the sphere or constant sphere velocity. Their analysis concentrates on the dynamics of what we have termed the exterior meniscus and, although not specifically designed to treat liquid film profiles, their determination of the initial deformation of the liquid/fluid interface could provide the basis for a more general treatment of the film drainage problem. APPENDIX: NOMENCLATURE a A B C D E
constant "~ 0.61 function of mass flux defined in Eq. [6] coefficient o f a 2 x / a O 2 as O ~ ~ , found by matching to exterior meniscus slope of film at 0 = constant of integration, Eq. [8] time-dependent function defined in Eq.
F g H h
liquid flux Eq. [1] acceleration due to gravity film thickness dimensionless film thickness
h*
scaled dimensionless film thickness h* = (~/X)h
[9]
h = H/H(O, O)
Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984
,,~ J O p R r t t' T
thickness in the inner region CX = h volume function (V0 - X#/~)-4, Eq. [18] origin of polar coordinates parameter defined in Eq. [4] radius of sphere polar coordinate time dimensionless time t' = t / T time scale V / V * volume of liquid film based on h/h* z parameter defined in Eq. [4] Greek Letters
a, B functions of ~b defined in Eq. [13] meridional radius of curvature of exterior meniscus lubrication parameter ~ = H(0, O)/R system parameter, Eq. [ 1] 0 polar coordinate O scaled polar coordinate EO = 0 ~, Bond number ~, = R2A#g/tr # film viscosity p film density, Ap liquid/fluid density difference cr surface tension 4) value of 0 at film periphery ~b intersection of secondary film and main film region ACKNOWLEDGMENTS The authors thank Dr. R. G. Cox and Dr. S. G. Mason for valuable discussion and advice. REFERENCES 1. Hartland, S., Chem. Eng. Sci. 24, 987 (1969). 2. Jones, A. F., and Wilson, S. D. R., J. Fluid Mech. 87, 263 (1978). 3. Huh, C., and Mason, S. G., J. Colloid Interface Sci. 47, 271 (1974). 4. Rapacchietta,A. V., and Neumann, A. W., J. Colloid Interface Sci. 59, 555 (1977). 5. Huh, C., and Scriven, L. E., J. Colloid Interface Sci. 30, 323 (1969). 6. Leal, L. G., and Lee, S. H., Adv. Colloid Interface Sci. 17, 61 (1982).