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Langmuir 1993,9, 1414-1417
Brownian Motion at Liquid-Gas Interfaces. 2. Effect of Surfactants K. Dimitrov, B. Radoev,; and M. Avramov Department of Physical Chemistry, University of Sofia, 1126 Sofia, Bulgaria Received April 14,1992. In Final Form: February 16,1993
The effect of surfactantson the Brownian motion of macroparticles floating at the liquid-gasinterface is discussed. The drag force of a spherehalf immersedin a liquid containingsolublesurfactants is calculated. It is shown that in the cases of small disturbances in the concentration c (I'OVID) < co (CO, To, initial equilibrium distributions; V , velocity of the sphere; D, bulk diffusion coefficient of the surfactant),the relative decrease of the particle mobility with respect to the mobility at a surfactant-free interface is of the order of (arEIDp)(a= (ar/ac)o,coefficient of surfactant distribution;E = (-I'&y/aI')o, Gibbs elasticity; p, dynamic viscosity of the fluid). The correctness of this result is discussed and compared with some typical data from the literature.
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I. Introduction Surfactants play a significant role in the transfer phenomena at interfaces. The presence of surfactants causesadditional resistance to both normal and tangential deformations of fluid-fluid interfaces due to different dissipative and elastic effects. Many phenomena such as capillary wave damping,'s2 Marangoni instability,3 dynamics and lifetime of foam films? etc. have been successfullyexplained by taking into accountthis complex role of the surfactants. The aim of this paper is to analyze the effect of the surfactants on the drag force of a sphere floating at a flat liquid-gas interface. Since there is no general solution of the hydrodynamic field of a sphere in arbitrary position, here we should consider only a sphere half immersed in the liquid. In a previous paper5 we have shown that the field past a hemisphere translating along a surfactantfree liquid-gas interface coincideswith the classical Stokes field in an unbounded fluid. The resistance coefficient in this case is B = Bapa, rather than the well-known 6apa. IIA. Modeling and Formulation of the Problem The reciprocal theorem6 (see also ref 5) is applied to calculate the additional drag force exerted by Surfactants on a sphere translating along a flat liquid-gas interface
f(vo*P).dS= f(v.P&dS
(11.1)
Here {Vo,po]are the velocities and Viscous Stresses in the Pure System, and b',p)are the velocities and viscous stresses in the presence of surfactants. According to the results obtained in ref 5, eq 11.1 is equivalent to V-(F - FJ = Jpto~,o.Par dY (11e2) where Fo (=3?rpVa)is the drag force in the pure system
* To whom correspondence should be sent. (1) Lucassen-Renders, E. H.; Lucassen,J. Adu. Colloid Interface Sci. 1969,2,348. Levich, V. G. Physicochemical Hydrodynamics; Prentice Hall: Englewood Cliffs, NJ, 1962; Chapter 11. (2) Levich, V. G. Physicochemical Hydrodynamics; Prentice Hall: Englewood Cliffs, NJ, 1962; Chapter 8. (3) Sternling, C. V.; Scriven, L. E. AIChE J. 1959,5,514. Linde, H.; Schwartz, P.; Wilke, H. Lecture Notes in Physics; Sorensen, T . , Ed.; Springer: Berlin, 1979, Vol. 105, p 75. (4) Scheludko,A.Proc. K. Ned. Akad. Wet.,Ser.B 1962,65,87. Radoev, B.; Manev, E.; Ivanov, I. B. Kolloid 2. 1969,243, 1037. (5) Radoev, B.; Nedjalkov, M.; Djakovich, V. Langmoir 1992,8,2962. (6) Perec, J. C. R. Acad. Sci. 1929,188,310. Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics;Prentice Hak Englewood Cliffs, NJ, 1965; Chapter 3, 3.5.
0743-7463/93/2409-1414$04.00/0
and F is the force in the presence of surfactants; V and a are the velocity of translation and radius of the sphere, respectively. The path of integration is over the flat liquidgas interface z = 0, p 1 a (see Figure 1). The quantities referred to the interface are denoted with the subscript 8. The stress components P,,", PyZo at pure liquid-gaa interface are zero, while in the presence of surfactants they should counteract the surface tension gradient V,y (Marangoni effect2) P,, = V,y at z = 0 (11.3) Thus the balance (11.2) takes the form V*(F- Po) =
J V , ~ * Vdx, ~ dy
(11.4)
p2.a
Since vo is known the reciprocal theorem permits evaluation of the additional force without solving the hydrodynamical problem. For linear perturbations, the general relation V,y = (dy/ar)V,r could be expressed by the equilibrium values (dy/dr)o and the order of the force AF by the Gibbs' elasticity E = -(rdy/dr)o. As a result, only the surfactant's distribution at the interface is required to evaluate AF, IIB. Surfactant Distribution in the Bulk and at the Interface The constant equilibrium values of the concentration in the bulk and at the liquid-gas interface, when the fluid is at rest, are designated as cg and ro, respectively. The floating sphere generates a hydrodynamic field v which evokes disturbances in the surfactant's distribution both in the bulk and at the liquid-gas interface, c(x,y,z) and r ( x , y ) ,respectively. The following model is proposed for their steady-state distribution~:~J (a)
(b)
v2c = 0,atz < o v,.[(ro+ r ) ~+]D(ac/az)= D,V:r, at z = o
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acldr = 0, at r = a, z I O (11.5) c -0 at r m, r -+Oat p (4 r = LYC at z = 0;(Y = carlac), (e) Equation II.5a defines the concentration distribution in the bulk. The coupling between the hydrodynamics and the mass transfer is localized at the liquid-gaa interface-see relations II.5b and 11.3: D, D,, bulk and (C)
-.
(7) Sorensen, T. Notes Mod. Phys. 1979, 105, 1-10.
0 1993 American Chemical Society
Langmuir, Vol. 9, No. 5,1993 1415
Brownian Motion at Liquid-Gas Interfaces
h
surface diffusion coefficients of the surfactants (not to be confused with the diffusion coefficient of the macroparticle D,,see eq IV.2). Condition 11.5~describes the particle interfaceas noninteractingwith the surfactants (reflecting boundary). Actually this convective diffusion problem is a nonlinear one but, as already mentioned, it could be linearized by standard perturbation methods. The perturbation orders in the series co c + ..., ro + + ...,and vo + VI + are determined by the two boundary conditions (II.5b) and (11.3):
...
+
v,-[(ro+ r + ...I (vs0+ v,, + ...)I + oa(co+ c1 + ...I/& = D,v,2(ro + rl + ...I pa(v,o+ v,,)/az = war), v p 0+ rl + ...I
(11.5b')
(11.3') They are defining some of the first terms in the series:
(a)
(b) (C)
co, ro= constant; v
=0
rova.vso+ Daciaz = CUD,V,~C
(11.6)
Figure 1. A sphere translating in the x direction; the liquid is situated in the region z I0, r 2 a.
(see A.1). The odd terms (a,) and the even terms (b,) are separated for the sake of clarity:
pav,,/az = ca-//wOv,cs
The zeroth-order solution (I1.6a) corresponds to equilibrium concentration in a system at rest. Note that in contrast to the mass transfer in the bulk where V-v = 0 and, correspondingly, the momentum and mass fluxes are not coupled, at fluid-fluid interfaces the convection and diffusion interact. The first-order solution includes the nonzero (but unperturbed) hydrodynamic field vo # 0 and the first perturbation in the concentration c, related to uo by eq II.6b. It depends also on the parameter (CUDslaD) c reflecting the competition between the surface and bulk diffusion. Case c >> 1 is briefly discussed in section IV. The force balance (11.4) is simplified by this first-order approximation to
m
(Further r and p read dimensionless variables r rla; p 2 da.1 The results of the solution are given below. Some used formulae are listed in the Appendix. By substitution of A,, = (-lIn(2n - 1)!!/2"n!, C 3r0V/ 40and e E aJ),/aD eq 111.3 can be rewritten as follows (see A2) 2 x n ( 2 n + l)A,[a,
- e(2n - l)b,-11p-2"
= C(1-
AF = ca-//ac)JvaoqcSd s
-
(111.5)
(11.4')
Equation IL6c defines the second-order perturbation of the velocity u1 dependingon the first-order term c. This analysis indicates that c (VI'oID) C co and u1 V ( d / D p ) C UO, representing the corresponding terms in the concentration series and in the velocity series,respectively. The concentration distribution problem (II.5a) in firstorder approximation is solved in the next section.
and simple relations between a n and b n are obtained a. = C; 3a, - C = 3 4 ; a, = e(2n - l)b,-l (n > 1) (111.6)
The boundary condition over the sphere r = 1,eq II.5c,
N
IIIA. Analytical Solution Cartesian [x,y,z),cylindrical {p,z,cp),and spherical {r,O,cp) coordinates will be used for the solution of the problem (Figure 1). The unperturbed Stokes' field vo (n, unit vector)
+
vo = (a/4r)[(3 a2/r2)V- 3(1- a2/r2)n(V-n)] (111.1)
gives for the term I'o(V,.vo) the following expression: ro(v,-vo)= -(3rov~/4p2)(i - a2/p2)COS (C (111.2) Obviously the angular dependence of c(p,z,cp) = c(p,z) cos cp follows from (111.2), and the boundary condition (11.6b) at z = 0 takes the form = - ( ~ / ~ ) ~ , V C Z ( I -+UpD(ac/aa ~/~~) (111.3) The symmetry of the problem makes easier the presentation of c(p,z) by c(r,B) in a spherical harmonic series
-
P - ~ ) a,
n=1
is sin 0 i-cose
+
a, -
+ ~)P~,,(cose) +
,=,
xb,2(n
+ l)P1,,+,(cOs 8) = 0 (111.7)
n=O
Substitution of a, from (111.6) in (111.7), with P12(cos0) = -3 sin 0 cos 8, leads to ( b n = b,/C) m
sin 0 3)p12,+2(C0Se11 = 3 sin e COS e - -(111.8) 1- COSe
In this form eq 11.4' with v, from (111.1) gives for the additional force A F = (7ri2) a C(a-//ac)F(c)
- IIC
Multiplication of both sides of eq 111.8 by P12,+1(cos B) and integration over 19(7r/2,7r)gives b,, taking into account
1416 Langmuir, Vol. 9, No. 5, 1993 2.4@
Dimitrov et al. with a series of finite systemswith dimensions N from 20 to 650 equations. The following simple run for FNvs N at fixed t value has been observed
'j
F N ( E = const,N) = N/(A
+ BN)
(111.13)
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This finding allowed us to obtain an estimation of F(E) as a limit of FN(N -,e) from (111.131, thus obviating labor-consumingcalculations of (111.10) for largeNvalues. The calculated F(e)dependence is plotted in Figure 2.
o'80 0.40
8 I O - * IO-'
1
lo2
lo3
lo4
19!410
Figure 2. Computed values of F(t)vs In (e) from eqs 111.10-11, 13.
that the boundary condition (eq 111.7) is defied over the hemisphere. After some integrations (see A3) the final result is obtained (X, = A,[(2n + 1)(2n + 3)I2b,; a, = 2 - [3/4(n + l)(n + 2)]; Bn E 1- [9/2(n + 2)(2n - l)]) m
F(E)= 1-
X,[(2n
$
+ 3)An+ll-2+
anx,
n=0(2n+ 1)(2n + 3)' (4n + 3)€Xk
(111.10)
-
2(k + n + 2)(2k - 2n + 1) (4n + 3)8, (111.11) (2n + l)(n + 1)
Equation 111.11 is an infinite linear system for X,, its numerical solution being the object of the next section. This complicated form of the solution is due to the nontrivial boundary condition (111.3). The observant reader will notice that the presence of two diffusion coefficients (D,DB)leads to derivatives in the condition (111.3) of the same order as in eq II.5a. Therefore in the solution (111.4) both a, and b, are nonzero. The presence of both odd and even spherical functions strongly complicates the procedure described above (because the Legendre polynomials are only partially orthogonal in the interval r/2, n e e A.3). In the case when t is zero (negligible surface diffusion) the solution has a simpler form 4n+3 F=l-Z A;anSn n=o4(n + 1)3
(111.12)
with a finite sum, F(t = 0) = 2.2. IIIB. Numerical Analysis
The main calculation problem is connected with the infinite system (111.11). Here it has been approximated (8) Radoev,B.;Manev,E.; Ivanov,I. Kolloid Z. 1969,243,1037. Radoev, B.; Dimitrov, D.; Ivanov, 1. Colloid Polym. Sci. 1974,252,50. Vaseilieff, Chr.; Manev, E.; Ivanov, I. VI. Int. Tagung uber Crenzflachenuktiue Stoffe; Bad Stuer, Abhand. Akad. Wise. DDR, Abtl. Math. Naturwiss. Technik; Akademie Verlag: Berlin, 1987; Vol. 1, p 465.
IV. Discussion
The order of the additional drag force A F is defined by the following combination of physical parameters, ?raV&/D (see the notations introduced before eq (IILS)), but the dimensionless quantity (AF)/F = (&/8Dp)F(t) gives the real measure of the importance of this effect. Surfactants always affect the hydrodynamics of liquidgas interfaces by this factor (&/Dp), when the mass transfer is controlled by the diffusion in the bulk (see for instance the rate of thinning of soap films stabilized by soluble surfactants8 or the influence of bulk-to-surface diffusion interchange on dynamic tension excess on the liquid-gas interfaceg). It should be noted that for a broad range of surfactants this factor is greater than 1;e.g. sodium dodecyl sulfate, CH&H2)110S03Na, in the concentration range 0.1 mM Ic I1mM gives8&/Dp lo4;capric acid, c = 0.001 mM, evesg&/Dp 5 X lo6,etc. These values are in accord w t h observations5J0confirming that surfactants could strongly block fluid interfaces. It should be noted that our solution here could not be applied for the cases of &/Dp > 1without additional analyses. The system reacts against the disturbances c and I' by diffusion both in the bulk and at the interface. The computed function F(c)in Figure 2 shows the competition between these two kinds of transport expressed by the parameter t d $ a D . The resistance F decreases with the rise of e from F(0) = 2.2 to F(-) = 1. The limit t does not describe the case of insoluble surfactants (Le. D 0), as it could be formally expected from the general statement of the problem, eq 11.5. This case will be analyzed in more detail in the next paper of this series. We note here only that the inconsistency of the Stokes solution leads to an infinite drag force. The use of a more correct Oseen velocity results
-
-
-- -
AF = rEa(3Va/4D,)(ln Re + 1.04 + Re/4 + ...I (IV.l) Comparison between AF = (3~/8)Va(aE/D)F(t) (see eq III.9) and A F from (IV.l) clearly illustrates the nonequivalence of the two kinds of mass transfer. In the case of soluble surfactants the surface diffusion cannot prevail t over bulk diffusion, even in the cases when D$D -. The weaker effect of the surface diffusion is most obvious in the cases of insoluble surfactants; due to the slower decay of the two-dimensional disturbances, higher approximations in the Stokes field should be taken into account, and the Reynolds number appears in the resistance force. The knowledge of the dissipative force F = BV leads directlyto the correspondingvalue of the particle diffusion
--
(9) Panaiotov, I.; Dimitrov, D.; Ivanova, M. J . Colloid Interface Sci. 1979, 69 (2), 318. (10) Guastalla,J. Thesis, University of Paris,1948;Teissie, J.;Tocanne, J.-F.; Baudras, A. Eur. J. Biochem. 1978,83, 77.
Brownian Motion at Liquid-Gas Interfaces
Langmuir, Vol. 9, No. 5,1993 1417
coefficient viaEinstein relationD, = kT/B (kT, Boltzmann factor). Thus the expressions for AF derived here lead to the determination of the relative change of the diffusion coefficients ADp D,(O) - D,(r) (a)
mp,m
=( a r ~ / ~ m c )
(IV.2)
(b) hDdD,(r) = (Ea/D&[ln Re + ...I for soluble and insoluble surfactants, respectively. The influence of the surfactantson the mobility, i.e., on the diffusion coefficients of macroparticles predicted in (IV.21, can be easily checked by direct Brownian motion measurements. By an appropriate choice of the surfactants, a very strong dependence of D, on the surfactant's concentration can be expected.
In order to determine (dC/dz),,o in eq 111.3, d/dz = r-l dld cos 8 is presented at z = 0. The next equality was additionally used d[~12,+l(~)l/~lz:zl,,o = 0;d[P1,,(r)1/dxl,,, = 2n(2n + 1)A, (8.766.2) A3. Upon transition from eq 111.8 to eq III.11 multiple integration by parte is carried out, using the definition (Al) of the associatedLegendre functionsand the following known integrals:
JO1[~,(r)l2 dx = n(n + 1)/(2n + 1) (7.112.1)
Appendix The numbers in parentheses of the formulascited below are taken from ref 11. Al. P', are associated Lengendre functions: 8) = d[P,(cos @]/de A2. At z = 0 (i.e. cos 8 = 0) one obtains P~,(COS
P',,(O) = 0; P12n+1(0) = -(an
(8.810)
+ UA,; P2,+l(0)= 0; P2,(0) = A, (8.756.1)
(11) Gradshteyn, I. S.;Ryzhik, I. M. Tables of Integrals, Series, and Products; Nauka: Moscow, 1971.
