Langmuir 1995,11, 1507- 1510
1507
Brownian Motion at Liquid-Gas Interfaces. 5. Effect of Insoluble Surfactants-Nonstationary Diffusion M. Avramov, K. Dimitrov, and B. Radoev* Department of Physical Chemistry, University of Sofia, 1126 Sofia, Bulgaria Received August 25, 1994@ The nonstationary dynamic of a sphere floating at the liquid-gas interface in the presence of insoluble surfactants is studied. Taking into account the difference between the diffusion (D,) and viscous (Y) coefficients (D$v i a diffusion-limited model for the process is assumed. The additional drag force, hF(t),caused by the surfactant gradients on the moving sphere (Marangoni effect) is analyzed. The numerical result shows two successive periods: an initial stage (AF% 0), followed by an active reaction of the system, with a well-expressed log-increasing resistance force, AF(t) In t. The problem with the unlimited force is discussed from the viewpoint of the Stokes paradox. The analogy with the dynamic behavior of the same system in steady state, is pointed out, where the resistance force proves to be proportional to Reynolds number hF(t) In Re.
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Introduction The presence of surfactants could strongly affect the processes a t fluid interfaces. This influence generally results in blocking the surface tangential motion (Marangoni effect), which in turn leads to a higher energy dissipation in the system. The translation of a sphere a t the liquid-gas interface in the presence of surfactants is a typical case where the Marangoni effect appears. The moving sphere perturbs the equilibrium (homogeneous) surfactant adsorption (compressingin front and stretching behind itself the surfactant), leading thus to an additional force, AF = F - Fo ( F and Fo, drag forces with and without surfactant), proportional to the so-called Marangoni AFIFO Ma, where Ma = Ea/D@, a is the sphere radius, E = (-ray/aT) the Gibbs elasticity, y the surface tension, r the surfactant adsorption, D, the surface diffusion coefficient, and p the dynamic viscosity in the bulk. The behavior of such systems substantially depends on the competition between the transport of surfactant in the bulk and at the surface.2 In the case of insoluble surfactants the mass-transfer is localized at the interface, so some 2D-space peculiarities typical of the process are expected. Thus, for instance, the additional force AF is infinite,3 in contrast to the case of soluble surfactants, where the bulk diffusion ensures a finite additional f ~ r c e . The ~ ? ~inconsistency of the result for insoluble surfactant is proved to be connected with the hydrodynamic model describing the convective transfer in the ~ y s t e m .In ~ this particular case, the infinite force is due to the Stokes flow approximation. In order to evaluate the correct force one should account for a more detailed hydrodynamic model. In ref 3 the Oseen flow was applied, and as a result a finite force depending on the Reynolds number, AFIFO Ma In Re, was obtained (Re = VaIv, Vis the particle velocity, v the fluid kinematic viscosity). The appearance of the Reynolds number in the force expression indicates that besides a viscous stresses' diffusion, the convective momentum transport becomes important for the dynamics. Inertia effects
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* Author to whom correspondence should be sent.
Abstract published in Advance A C S Abstracts, April 1, 1995. (1) Levich, V. G. Physicochemical Hydrodynamics; Prentice-Hall: Engelwood Cliffs, NJ, 1962; Chapter 8. (2) Dimitrov, K.; Radoev, B.; Avramov, M. Langmuir 1993,9,1414. Dimitrov, K.; Avramov, M.; Radoev, B. Comptes Rend. Acad. Bulgare Sci. 2 1994, (3) Dimitrov, K.; Avramov, M.; Radoev, B. Langmuir 1994,10,1596. (4) Mileva, E.; Nikolov, L. J . Colloid Interface Sci. 1993, 161, 63. @
0743-7463/95/2411-1507$09.00/0
z -0
Figure 1. A half-immersed sphere floating at the liquid-gas interface, in presence of an insoluble surfactant, Jd, a diffusion flux; J,, a convective flux.
should be also taken into account during the nonstationary periods. In systems with coupled hydrodynamics and diffusion, the kinetic is controlled by two characteristic times: z D L2/D,, related to the mass transport, and zv L2h,related to the momentum diffusion. Because D, is at least lo3times smaller than v,these two relaxation times are of quite different orders (zD >> zv), so the hydrodynamics could be considered as a quasi-stationary one. Similar considerations were taken into account in the present paper in order to analyze the nonstationary distribution of an insoluble surfactant perturbed by the translating sphere at the liquid-gas interface. The next two sections are devoted to the formulation and solution of the problem, and in the last one, some more general aspects of the Brownian motion at fluid interfaces are discussed.
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Formulation of the Problem In the present study, the model of a translating sphere half-immersed at the flat liquid-gas interface is ~ s e d ~ - ~ (see Figure 1).The mass (surfactant)transfer in this case obeys the surface balance?
+
where r = To rl,(Toand rlbeing the initial (equilibrium) adsorption and its perturbation, respectively), and v is the tangential hydrodynamic velocity a t the interface (V is the tangential nabla operator). Zero perturbation and reflecting solid (sphere) interface are supposed respectively as initial and boundary conditions of eq 1: (5) Radoev, B.; Nedjalkov, M.; Djakovich, V. Langmuir 1992,8,2962.
1995 American Chemical Society
Auramou et al.
1508 Langmuir, Vol. 11, No. 5, 1995
r,(e,yl,t=O) = 0
(ar1/a&
=
o
(2a,b)
It should be noted that the boundary condition (2b) corresponding to an indifferent (impenetrable) solid surface is only one of the reasonable adsorption models. The model of constant adsorption, Tl(e=a) = 0, for instance, is another realistic case, but as will be demonstrated in the last section, the additional forces (AF)of these two limits are quite similar. In addition, a t the liquid-gas interface, a dynamic boundary condition (the formal expression of the Marangoni effect) should be satisfied. For the case of a flat (nondeformable) interface, this boundary condition is
p z s = p(av/az) = -vy = (-ay/ar)vr at z = o
(3)
Eqs 1 and 3 clearly show the coupling between mass and velocity distribution and generally state a nonlinear problem. For small adsorption perturbations, rl < To, equivalent to a small Peclet number (Pe = Va/D, < l),eq 1 could be linearized and takes the f ~ r m ~ - ~
ar1 + ToV;v
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at
= D,V2T1
The solution of eq 1' depends on a hydrodynamic field generated by a floating sphere, with given viscous stresses and velocity distributions at the fluid interface (see the boundary condition, eq 3). In the cases of a small Marangoni number (Ma < l),the asymptotic analysis shows that the velocity could be expanded in a series of Ma powers:2v = vo Mavl+ Ma2vz ...,with the leading term vo corresponding to a free interface (without surfactant, i.e. satisfying the condition (&/~Z),,O = 0). For our model, a half-immersed sphere, vo coincides in fact with the well-known Stokes flow for an unbound fluid,5 and eq 1' is reduced to a simple diffusion problem (in dimensionless form):
+
AF/Fo = Ma&alvo4T dS
(4')
The gradient VT is a function of the surface velocity, v (coupled in the general case with the T distribution, see eq 3),but for small Marangoni numbers, the leading term T depends only on the unperturbed velocity, VO, and thus facilitates the additional resistance force estimation, avoiding solution of the hydrodynamics. The usefulness of eq 4' depends on the form of T(e,t). In our case, the nonstationary solution is a very complicated expression (see eq 7), and only after a rather sophisticated technique (see Appendix) does one succeed to obtain a result convenient for numerical calculations (eq 8').
Solution After Laplace transformation of eq l", taking into account the zero initial condition (see eq 2a), one gets
where the tilde denotes Laplace images of the functions, the prime denotes the derivatives over the variable y = e&, and s is the Laplace variable. The solution of eq 5 with the boundary condition (2b) is
+
with dimensionless variables e =@/a,t = tD$a2,vo = vdv, and V = aV. This particular form of eq 1" is obtained by the use of Stokes flow in the convective term of eq l', V,VO = e-2(e-2 - 1)cos (the right hand side of eq l"),and with the substitution Tl(e,p,t) = Perl(&) cos p. The solution of eq 1"is given in the next section. Here the evaluation of the viscous force, a very important characteristic of the Brownian motion, will be briefly outlined. The problem with the viscous (drag) force in the presence of surfactants could be substantially advanced by use of the so-called reciprocal theorem:2,6
Here 11(y), Kl(y) are modified Bessel functions (general solutions of the homogeneous equation),Ll(y)is a modified Strouve function (a partial solution of the nonhomogeneous equation), and K ( S ) is a function determined from the boundary condition (2b):
The inverse transformation of T(Q,s)is carried out by using the theorem for inverse Laplace transformation (see the Appendix). The final result for the distribution T(Q,t) is
where J and Yare Bessel functions, His a Strouve function,
(4)
Here again, AF = F - Fo is the additional resistance due to the Marangoni effect [Fo= F(T=O) = 3nVpa is the drag force of a half-immersed sphere at the pure liquid-gas interface5], vo is the Stokes velocity, and pzsis the stress distribution at the liquid-gas interface in the presence of surfactants. The integration path in eq 4 is over the liquid-gas interface (8 z a). The benefit of the reciprocal theorem becomes evident in its combination with the stress boundary condition (eq 3). Replacing pzsby VT, eq 4 takes the form (6)Peres, J. Compt. Rend. 1929,188, 310.
In the solution (7) the term [exp(-tx2) - l]G(e,x) satisfies the homogeneous part of eq 1" with initial and boundary conditions (2a,b). The evolution of the additional force AF(t) could be obtained from the result for T ( e , t ) in eq 4', and after a trivial integration over yl (0, 227) one gets \
Brownian Motion at Liquid-Gas Interfaces
Fo
Langmuir, Vol. 11, No. 5, 1995 1509
f0
= Mafit) =
{
+
+
MaJw de 2(1 - 1 / 3 ~ ~(l/e ) ~ 1/3e3)r}( 8 )
2i
i l
1.8
The integration with respect to could be performed directly (note, that the g-dependence of the brackets is localized in G(e,x),eq 7), so we come to an expression of the time-dependence of AF At), amenable further to a numerical evaluation:
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1.2
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0.8
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0.4 L
t 0
Results and Discussion The computed results offit) are given in Figure 2. They should be treated as a dynamic response of the system to a stepwise particle velocity (Heaviside function). The graphic clearly shows an initial period (AF 0), followed by an active reaction of increasing resistance force, with a well-expressed logarithmic branch, AFnonst In t. The unlimited increase of the force is a contradictory result, but it becomes plausible when comparing it with the steady state of the same system. In general, the distribution r(@) strongly depends on the hydrodynamicfield v (see eq 1'1, and for classical Stokes flow (VO)the stationary force AFstokes tends to infinity,2 i.e. formally confirming the asymptotic nature of eq 8' for long times. The analysis shows2 that this divergence is connected with the incorrectness of the Stokes model far from the sphere. Using the Oseen flow as a more precise approximation, the additional stationary drag force, AFoswn, proves to be finite, but proportional to the logarithm of the Reynolds number, so all cases discussed here have a uniform presentation:
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(3 = Ma('/4
+ In e);
= Ma(lI4
Stokes
+ In r + ...);
r = (eCRe)-l;
(2oseen
C = 0.5572, the Euler constant (9)
The formal analogy between the log relations in stationary and nonstationary systems reflects similar dynamic situations. The disturbances raised by the surfactant elasticity (Marangoni effect) could not be counterbalanced only by the viscous (dissipative) forces. This is connected with the well-known peculiarities of 2D-perturbations (here, the surface distribution of the insoluble surfactant), and as a result, inertia forces (the nonstationary term in eq 1 and/or the Oseen term in the hydrodynamics2)become important. In the case of soluble surfactants, where all perturbations are spread in 3Dspace, such paradox does not appear, and the inertia plays only a second-order role. It is interesting that the classical Stokes paradox is quite similar to the problem discussed here. It concerns the unlimited drag force for a steady translation of a cylinder (2D-flow)in unbound viscous fluid [in contrast to the finite drag force of a sphere (3D-flow)]. Here, it would be instructive to cite Stokes himself about this p r ~ b l e m"The : ~ pressure of the cylinder on the fluid continually tends to increase the quantity of fluid which it carries with it, while the friction of the fluid at a distance (7) Stokes, G. G. Trans. Camb. Phil. SOC.1850, 9 (II), 8.
~
r ...................................
10-10
lo4
I
yLLlyl
io4
...."
.........
"
. . . . .
... .................. _
" _
"
Illll.Y
lo4
io4
; "
.......... ioo
.yll.l
/ .
-,-,-
10'
Ill.l.Y
.
io4
t
Figure 2. Numerical values of the drag force vs t, scaled with a2/D,. The straight line represents the solution with the
boundary condition (ar/a@>,=1= 0 (eq 2b); the solution with boundary condition T1(@=a>= 0 is given as a dashed line. The magnification illustrates the differences in the initial periods of these two solutions.
from the cylinder continually tends to diminish it. In the case of a sphere, these two causes eventually counteract each other, and the motion becomes uniform. But in the case of a cylinder, the decrease due to the friction of the surrounding fluid and the quantity carried increases indefinitely as the cylinder moves on". Finite drag force for a cylinder is obtained, if the inertia terms are partially taken into accounteither after the manner of Oseens (with a In Re depending force), or in the frame of the nonstationary Stokes equations (with a log time) increasing resistanceg). The above discussed peculiarities reflect the flow far from the perturbation source and should not substantially depend on the boundary conditions at the floating sphere. This could be illustrated by solving the problem (eqs 1-3) with boundary condition T1(Q=l) = 0, instead of eq 2b, (X/ag),=, = 0. As it can be seen at Figure 2, the AF(t)values for long times in both cases are quite similar, the log-branches are parallel. Some differences are observed in the initial periods. The reflecting solid-liquid interface (eq 2b) leads to a nonmonotonous fct), while the force evolution for a constant adsorption is without extrema. The logarithmic dependencies of the force on the time and on the Reynolds number could lead to interesting effects in the Brownian motion at the surfaces, in membra ne^,^ etc. Thus, for instance, the question what should the Brownian kinetics look like a t liquid-gas interfaces in the presence of strong surfactants (at high Marangoni number) is not a trivial one. It could not be answered in the frame of the classical Langevin equation, a t least because of the nonlinear (with respect to the velocity, V) resistance force, F = VB(V)(see eq 9), and its study needs a special nonlinear analysis. lo
Acknowledgment. These investigations are partially supported by the Bulgarian National Science Foundation, project X-202. Appendix The inverse transformation of the Laplace solution for the adsorption perturbation r(Q,t) is a nontrivial problem (8) Lamb, H. Hydrodynamics; Dover, 1954; Chapter 9. Van Dyke, M. Perturbation Methods in Fluid Mechanics; Academic Press: New York, 1964. (9) Saffman, P. G.; Delbriick,Proc. Natl. Acad. Sci., U.SA.Biophys. 1975, 72 (No. 8), 3111. (10) Mazur, P.; Bedeaux, D. Langmuir 1992,8, 2947. (11) Abramovitz,M.; Stegun,I. Handbook ofMathematicalFunctions; NBS: 1964.
Auramou et al.
1510 Langmuir, Vol. 11, No. 5, 1995
,r
m Im
The last two integrals can be evaluated analytically:
-JC_esgg(s) 1 2in
Figure 3. Integration region for the inverse Laplace transformation.
ds = lim g(s) = 0 S-00
Taking into account that K, I , and L can be expressed by Y ,J , and H over the imaginary axis, a series offormulae (see the list at the end of the Appendix) is easily obtained. This way the final result for the inverse image of (eq 7) reads:
and needs special explanations. Part of the transformation can be done using standard formulae:12 Relations between I,K, L and J, Y, H
L,
The main problem is the calculation of the inverse image of g(e,s). Because s is a complex variable, and g(e,s)is a function of &, the integration path should exclude the branch points = 0 (see Figure 3).13Using the substitutions s = x2einwhen the integral is taken over L+ and s = x2e-is when the integral is taken over L-, Yl@(~,s)) gets the form:
(12) Erdelyi, A,; Batteman, H. Tables of Integral Transforms; McGraw-Hill: New York, 1954. (13) Carslaw, H.; Jeager, J. Conduction ofHeat in Solids; Oxford University Press: New York, 1947.
s =x2ein
&=ix
ds=-&dx
