Langmuir 1993,9, 3724-3727
3724
The Aging of a Liquid Drop V. A. Kuzt Instituto de Fisica de Liquidos y Sistemas Biol6gicos, IFLYSIB (U.N.L.P.-CONICET-CIC), C.C. 565, 1900 La Plata, Argentina Received October 23, 1 9 9 P The aging of a liquid drop is made evident when any surface property, eg. surface tension, surface density, or surface concentration, ia modified in the come of time. In the present study, the diffusion of solute inside the drop will be considered the only agent which produces these changes. The surface mass conservation law is derived in spherical geometry. This law links the time variation of the surface mass density with the gradient of the solute concentration in the drop. The solute concentration inside the drop is found by solving the diffusion equation; in the gas phase, the solute concentration ie assumed to be constant. In the dilute case, the Surface density changes with time following an integrodifferential equation of Volterra type. The solution of this equation, obtained as a Fourier series, together with a two-dimension equation of state, makes it poesible to find the surface tension variation with time. The present result is compared with those obtained for a plane interface in refs 1 and 3. While the latter worh predict a diminution proportional to t1/2 and t , respectively,the present model indicates an exponential decay of surface tension with time. Also the theory is tested with experimental dynamic measurement obtained by the pendant-drop technique and with data gathered from the oscillating jet method. 1. Introduction
When a fresh interface is formed from a surfactant solution, interfacial tension is found to decrease with time as surfactant diffuses to the interface and adsorbs. Diffusion,adsorption, and desorption of soluble surfactant are the mechanism that may cause surface tension to vary with both position and time. Ward and Tordai made the first general model for a diffusion controlled adsorption. Papers detailing new solutions and, under special conditions, deriving analytical or power-series solutions also considered this prob1em.a While all the above works were done for a plane interface, these results may not always be applied for curved surfaces, in particular when the radii of curvature are of the order of the interfacial layer thickness. This thickness is, to an order of magnitude, (Dt)lf2. It depends on the diffusion constant D and on the surface aging time t. Sort time, diffusion constant of the order of D z le7or 10-8 cm2/s (small polymer or protein system) and small drops radii could represent a delicatesituation for the applicationof the standard (plane interface) theories. This circumstance gives rise to exploring the problem in the adequate and natural geometry at the expense of handling a mathematically complicated system. In order to avoid the same difficulties of this geometry, it will be assumed that the substantial change occurs only in the radial direction. Then the surface mass conservation law, which links surface and bulk properties of the system, is derived. The solute concentration of the drop is found by solving the diffusion equation with a point source somewhere inside the sphere 0 Ir Ia. At r = a, the concentration and its derivative are obtained as a Fourier series. The substitution of this result into the surface mass conservation law leads to an integrodifferential equation of Volterra type. By supposing a linear relation between the bulk concentration and the surface mass density, this integrodifferential equation was solved by using Laplace transform. Then by means of a simple + Member of the Consejo Nacional de Inveetigacionea Cientlficae y TBcnicae (CONICET) of Argentina. Abstract published in Advance ACS Abstracts, October 1,1993. (1)Ward, A. F. H.;Tordai, L. J. J. Chem. Phys. 1946,485,63. (2)Haneen, R. 5.J. Phys. Chem. 1960,64,637. (3)Kuz, V. A. Phys Rev. A Gen. Phys. 1991,44,(12)8414. (4)Varoqui, R.;Pefferkorn, E. J. Colloid Znterface Sci. 1986,109,520.
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two-dimension equation of state, which links surface tension and surface density, it was found how surface tension diminishes with time. 2. The Mass Balance Surface Equation
In this section the surface mass balance equation for a drop is derived. Surface properties, in the Gibbs version of the surface thermodynamics: result from considering the interface as a mathematical surface having a constant tension as its most notable property. Another approach is to consider the interface as a two-dimensional fluid with singular properties.6J Also it is possible to view it, going back to the van der Waals ideas, as a finite regions9 represented by a real interphase confiied between two fluids, where each point of this interface will be characterized by a continuous function with continuous derivatives. The position of the interface will be somewhere between r - c < a < r + c, 2c being the width of the interface. Then, by following the latter model of i n t e r f a ~ e ~the v~ surface mass equation is obtained by integrating once the corresponding three-dimension bulk equation. It is supposed that the concentration of the system is such that no convective motion of the drop's fluid occurs. There is no Marangoni instability due to solute concentration neither coming from the bulk nor produced at the Surface of the drop.1° Only diffusion of solute from the inside liquid toward the surface of the drop occurs. The general three-dimensionalequation to be integrated reads
where C = pJp is the mass fraction of solute, p , and p are (5)Fbwlinaon, J. S.;Widon, B. Molecular Theory of Cupillarity; Clarendon Press: Oxford, 1982. (6)Scriven, L. E.;Starling,C. V. J. Fluid Mech. lSSa,I9,321. (7)Bedeaux, D.;Albano, A. M.; Mazur, P. Physical A 1976,82A,438. (8) Goodrich, F. C . Proc. R. Soc. London, Ser. A 1981,374,341. (9)Garam, A. N.;Kuz, V.A; Vila, M.A. J. Colloid Interface Sci. 1987, 119.49. (10) Chifu, E.;Stan, I.; Fmta, Z.; Gavrila, E.J. Colloid Interface Sei. 1985,93,140.
Q 1993 American Chemical Society
The Aging of a Liquid Drop
Langmuir, Vol. 9, No. 12,1993 3725
the solute density and the total density (massper unit volume). D is the diffusion coefficient. By integrating this equation we have
ac I(I+' + 2Dp E = Dp at a--t
;:
s
a+-t
a--(
1ac D ac -(-) r dr d r - 7 p cot 9 -1ao
I(I+' + 2Dp
s
a3 n=l
a
na
2
2
1
naD(-l)n~oteDn'X/a'C (A) dX (7) Now, with the help of eq 6, the second term of the righthand side of eq 3 and can be evaluated and
where Si(na) is the sine integral Si(na) =
nrsin x dx x
The substitution of eqs 7 and 8 into eq 3 gives
ac dr -(-) r ar
a+L1
a--(
ar
2,~== - Ce-DnP't/a'n cos n a
a+-t-
where r = J:zps(r) dr, the surface density or surface concentration of the solute (mol/cm2)a t the interface. With : 1 is indicated the difference between the respective quantity evaluated at each side of the interface. There is no evaporation. The total density of the system p remains constant during time. In the following, for the sake of simplicity, it is assumed that the concentration C depends only on r, then eq 2 reduces to
ar at = Dp
ac
+a
(3)
ar
n=-
- = - 2 ( ~ ~ / C~e 3- D)n z P t / a 2 (na) cos n a X this equation links surface and bulk properties of the system. The evolution in time of the surface density depends on the bulk concentration gradient, evaluated at each side of the drop's surface.
3. Aging of a Drop As soon as solute inside the drop begins to diffuse toward the surface, a process of aging start to happen. The irreversible thermosolute cross effect is disregarded. The drop is in "thermal equilibrium" with ita vapor. In this situation the diffusion of solute, in the radial direction, is found by solving eq 1. The solution for a given initial concentration f(r) somewhere in the sphere is given byll 2 n-C = - Ce-Dn'*t/dar n-1
nar sin -{s:rf'(r)' a!
nar' sin -dr' -
at
n=l
a
na
2
2
- sin -- naD(-l)D ~eDn'T2X/a'C(X) dX
[ -- Si(na)1
2(Dp/a3)~e~Dn'"~a'(na2) n-1 - nr a
X
na
- sin -- naD(-l)n~eDn~'"~a'C(X)dX
2 2 The above equation can be written as follows
ar
- + 2(D2pa3/a3)(-1)"~ot'mC(X) X at
-u n=l
a!
where C(h) is the concentration at r = a and C(r=O,t) = 0 and C(r,t) is the solution of the diffusion equation. The concentration depends on the coordinate rand on the time t (radial flow). In the case that the initial solute concentrationisgivenbyf(r') = 6(r'-a/2)Q, whereQisaconstant, eq 4 becomes
1
naD(-l)n~eDn'*'X~a'C(h) dX ( 5 ) This equation gives the evolution of the solute in the liquid drop for a given initial source point concentration of solute. From eq 5, the gradient of C is
and at the (L-V) surface is (11) Carelaw, H. S.; Jaeger, J. C. Conduction of Heat in Solids; Clarendon Press: Oxford, 1947; Chapter IX.
n=na (a2Dp/a2)Q~e-Dn'IPt/a'(n)2Si(na) sin - (10) n=l 2 We have here a general equation which links, for a given source point, surface arid bulk concentration. In order to solve this equation, it is necessary to write an explicit relation between the surface concentration r a n d the bulk concentration C. At this point it could be argued that the standard Langmuir equation12
used on a plane surface, can be also applied to the present geometry. As far as I know there is no direct discussion of this point in the literature. It will be interesting to know if curvature could play here some role as it does in relating pressure difference and surface tension (Laplace equation). Ad hoc corrections for the application of planar time-dependent interfacial tension theories are usually done for interpreting experiments dealing with growing drops or bubbles.13-16 But let us assume, in the present situation, that the drop's radius remains unchanged. For every dilute solutions of un-ionized solute the following relation holds12 (12) Davies, J. T.; Rideal, E. K. Interfacial Phenomena; Academic Press: New York, 1963. (13) Miller, R. Colloids Surf. 1990, 46, 75-83. (14) Garret,P. T.;Ward, D. R. J. ColloidInterface Sci. 1989,132,476. (15) Mysels, K.J.; Stafford, R. E. Colloids Surf. 1990,51, 105-114.
3726 Langmuir, Vol. 9,No. 12,1993
Kuz
r(t)= 6-'c(r=a,t) (12) where 6 is an arbitrary constant. This "adsorption isotherm" represents an "ideal isotherm". By substitution of this relation into eq 9, it becomes
ar -+ 2(D2p?r3/a3)6xI'(X)X
-
' 1 3
at
n-or
h
Y
a linear differential equation. In first approximation this equation reads
(?r2Dp/a2)QSi(?r) e-D*zt/u*(14) This is an approximate version of eq 13. rl indicates the order of the approximation. By use of the Laplace transform,16the solution of eq 14 is r,(t) = F'B, sinh(8t) (15) where a = Da2/a2,A1 = 2(D2p?r3/a3)6Si(?r),B1 = apQSi(?r), and o2 = A1 + (a/2I2. This equation gives the time dependence of the surfactant adsorption to the liquidvapor drops' interface. We present below some important approximated theoretical results found for planar interfaces. From the Ward and Tordai1p4 solution, the first term is
r(t)= ~ C O ( D ~ / T ) ' / ~ (16) where C" and D are the bulk surfactant concentration and the bulk phase diffusion constant. The Varoqui and Pefferkorn asymptotic solution4 (0 < t
