Langmuir 1992,8, 2962-2965
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Brownian Motion at Liquid-Gas Interfaces. 1. Diffusion Coefficients of Macroparticles at Pure Interfaces B. Radoev,*pt M. Nedjalkov,t and V. Djakovichi Department of Physical Chemistry and Department of Inorganic Chemistry, University of Sofia, 1126 Sofia, Bulgaria Received April 14,1992. In Final Form: October 21,1992 The diffusion coefficient of macroparticles floating on a liquid-gasinterfaceis treated from the viewpoint of the Einstein relation D = kT/B,kT being the Boltzmann factor. The resistancecoefficient B = 6 ~ p a f ( h l a) is presented as a function of the position h of the sphere with radius a with respect to the interface (Figure 1). The hydrodynamic viscous force is obtained using the reciprocal theorem. In the case of a half-immersed sphere (h = a), it is shown that the coefficient f is equal to f(1) = 0.6. Free movement of polymer (melamin) particles on a waterlair interface and in the bulk is observed. The data for the displacements AX are statistically analyzed. The relation ( ( A X ) 2 )vs At is checked for (AX)i with normal distribution. In the cases when it satisfies the the Einstein-Smoluchowski relation ( ( A X ) 2 )= 2DAt at significantlevel, the corresponding diffusion coefficients are estimated. The coefficients in the bulk, D, (bulk) = 0.6 pm2/s,and at the interface, D, (surface) = 0.4 pm2/s,are obtained and compared with the theoretical ones, D(=kTIB) = 0.42-0.24 pm2/s (particle radii a = 0.5-0.9 pm determined by transmission electron microscopy). The validity of the theoretical model and some specific problems connected with the statistical procedures are discussed.
I. Introduction In recent investigations the transport phenomena at fluid interfaces are treated as a result of two types of processes.’ The first one, the so-calledintrinsic interfacial transport processes,by analogy with the three-dimensional linear nonequilibrium thermodynamics,2 is described by two-dimensional fluxes and forces j , = Lax,,where L, represents transport coefficients (e.g. surface viscosity M,, coefficient of surface diffusion D,, etc.), and X , = -V&, the corresponding driving forces, where V, is the surface gradient operator. The other type of processes involves interfacebulk interactionssuch as viscous friction between the interface and the adjacent bulk liquid, diffusion towardlfrom the interface, adsorptionldesorption of surfactants, etc. In reality these two kinds of effects act simultaneously, giving an ambiguous (mixed) nature to the experimentally measured quantities. The development of relevant models describing correctly the Ypure”surface and bulk effects is one of the main applications of the theory. Examples in this respect are the studies of Boussinesq3 and Levich4on the mobility of small bubbles in viscous fluids. In his classical paper5 “Eine neue Bestimmung der Molektildimensionen”, Einstein showed that the diffusion coefficient D for solid spheres, in the dilute limit, could be expressed, in terms of the hydrodynamic resistance of an single sphere, D = kTIB, where B = 6 ~ is the ~ Stokes’ a resistance force coefficient, p is the dynamic viscosity of the fluid, and a is the sphere radius. This relation was later generalized6 to cases with more complicated hydrodynamics, with the appropriate resistance coefficient B. Mass transfer on fluid interfaces is a typical example of such complicated hydrodynamics. Here the diffusing particles are located between two contiguous phases in a t Department of Physical Chemistry. t
Department of Inorganic Chemistry.
(1)Lucaasen-Reynders,E. H.; Lucassen, J. Adu. Colloid Interface Sci. 1969,2, 348. Levich, V. Physicochem. Hydrodymmics; Prentice-Hall: Englewood Cliffs, NJ, 1962; Chapter 8. Sorensen, T. S.In Lecture Notes in Physics 105; Springer-Verlag: Berlin, 1979; p 1. (2) de Groot, S.; Mazur, P. Non-Equilibrium Thermodymmics;NorthHolland Publishing Co.: Amsterdam, 1962; Chapter 4. (3) Boussinesq, I. 1913, 156, 983; Ann. Chim. Phys. 1913,29, 349. (4) Levich, V. G. Zh. Fiz. Khim. 1948,22, 721. (5) Einstein, A. Ann. Phys. 1906, 19, 289.
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transition zone7 with finite depth 6 (of the order of some to several tens molecular diameters). For particles of size comparable to the width of the transition zone ( a Ia), there are problems with a continuum treatment of the fluid, the viscosity is no longer constant, etc. For “big” particles (a >> 61, with interface contacting almost completely the homogeneous bulk phases, the transition zone could be approximated as a surface of discontinuit9 with infinitesimal thickness as compared to the particle radius, and the coefficient B is calculated from the viscous forces in the homogeneous phases. In the next section a general expression of the drag force of a big particle floating a t a flat fluid-gas interface is derived. For the special case of a half-immersed sphere, this drag force is computed and then used to estimate the Brownian diffusivity, which is compared with experiments in a later section.
11. Drag Force Problem Following Peresg the reciprocal theorem is applied to obtain the resistance force of a sphere floating along a flat liquid-gas interface
$(v.P’)d S = $(v’.P)d S
(11.1)
where v and P are the velocity and stress tensors of the Stokes’ flow past sphere in unbounded fluid and v’ and P’are the velocity and stress tensors of the investigated flow. The path of integration in (11.1) includes a half-space z I 0 with a spherical concavity located on the boundary ( z = 0) (see Figure 1). As far as Stokes’ flow velocity goes to infinity as v,-r-l, respectively, ( V-P),, r-3,the onlynonzero terms in (11.1)are the integrals over thesphere (r = a, z I 0 ) and over the plane surface z = 0 (around the
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(6)Kubo, R. Rep. Prog. Phys. 1966,29, 255. (7) Ono, S.;Kondo, S.Molecular Theory ofSurface Tension inliquids. In Handbuch der Physik; Springer-Verlag: Berlin, 1960; vol. 10, part 2,
Chapter 3. (8)Brenner, H.; Leal, L. G. J. Colloid Interface Sci. 1982,88, (l), 136. (9) Perec, J. C. R. Hebd. Seances Acad. Sci. 1929,188,310. Happel, J.; Brenner, H. Low Reynolds number hydrodynamics; Prentice-Hall: Englewood-Cliffs, NJ, 1965; Chapter 3, Section 3.5.
0 1992 American Chemical Society
Brownian Motion at Liquid-Gas Interfaces
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sphere), r 1 1 ( I , radius of the three-phase contact solidliquid-gas) #(v*P’) d S =
s s
{ + r=a
)(v.P’) d S
(11.2)
z=c
r>l
The integral J(v.P’)r=a dS over the immersed part of the sphere defines its drag force P: J(V*P’)r=ad S = V F , with V = constant being the translation velocity of the sphere. The other integral, J(v’-P) d S = VF is equal to the Stokes force F corresponding to the immersed part of the sphere area S’: F = 67rpVa(S’/47ra2). For the sake of convenience both forces F and F are compared at the same velocity V. The integrals over the liquidlgas surface ( z = 0, r 1 1) depend on the boundary conditions, which in our model have the form
Px,’ = PyZ, = u,’ = 0, at z = 0, r 1 1
(11.3)
According to (11.3) the tangential motion at fluid-gas interface is free (zero viscous stress tensor components Ps,’), with neglected normal deformations ( v i = 0). Thus the reciprocal theorem (11.1)could be rewritten in a more useful form
V ( F - F, =
J (upz,’- u,‘P,.. - u~’P,,Jdx dy
(11.4)
z=O r2a
Generally, in order to evaluate (11.4) one has to know the behavior of P,,’, v,’, i.e., to solve the complete hydrodynamic problem, but for the special case of a halfimmersed sphere, the right-hand side of (11.4) could be determined from symmetry considerations only. In this case the plane z = 0 (see Figure lb) lies on the axis of symmetry of the Stokes flow (v.P) and, as it is well-known, all terms in the integral vanish, (Pzz= Pyz= V , ) ~ ==O0, and the drag force F becomes simply equal to the half of the total Stokes force F, = 67rpVa
F’ = F,/2 = 37rpVa
b)
(11.5)
A similar symmetry argument can be used to show the validity of (11.5) for any half-immersed body with a plane of symmetry coinciding with the plane of the interface, with F, being the Stokes resistance for this body in an unbounded fluid. From (11.5) it follows that the diffusion coefficient of a hemisphere floating at a liquid-gas interface is D = kT1 37rpa. This relation no longer holds in the presence of surfactants. With surfactants, because of the Marangoni effect,l a gradient of the surface tension would arise, and the boundary condition (11.3) PzS, = 0 would be no longer valid. Note, that the Marangoni effect is equivalent to an additional resistance force.1° The friction coefficientof a sphere floating with its center at an arbitrary distance h above the liquid-gas interface (Figure 1a)can be written in the form B(h) = 67rpaf(h/a) with the factor f(hla) accounting for the hydrodynamic particleinterface interaction. The explicit form of the dependence f(h/a) is not known (due to lack of knowledge of the general solution v’,P’), but numerical values o f f could be given for some characteristic positions h: f(0) = 0.716, pendant sphere,ll Figure IC;f(1) = 0.5, the case of hemisphere considered here, Figure lb; f(2) = 0, sessile (10) Dimitrov, K.; Radoev,B.;Avramov, M. Submitted for publication in Langmuir. (11) Wakiya, S.Coll. Eng. Res. Report (No. 6) 1957, Mar 30. Happel, J.; Brenner, H. Low Reynolds number hydrodynamics; Prentice-Halk Englewood Cliffs, NJ, 1965;Chapter 6, Section 6.3;Chapter 3, Sectin 3.6.
Figure 1. Schematic presentation of the model “macroparticle liquidlgas interface” considered in the text: (a) particle at arbitrary position with respectto the interface (a,radius; 1, threephase contact line); (b)position “hemisphere”(h = 1, drag force F(h/l = 1) = 3upVa; (c) position “pendant sphere”: h = 0, F(h/l = 0 ) = 0.716rpVa; (d) position “sessile sphere” (h = 2a, F(h/l = 2 ) = 0 ) (see text). Table I. Data from the Perrin Paper ‘Mouvement Brownien et realite moleculairen* At-30s At=60s At=90s At=120s ( ( A Z ) ~pm2 ), 50.2 113.5 128 144 45 86.5 140 195 Displacements ( in [pm21 for two series of gomme-gotte particles: radius, a = 0.212 pm; wscosity of the suspension, 0.011; 0.012 P. Each value is averaged from the results obtained for 60 particles.
1 4
T
r
Figure 2. Schematic presentation of the setup for Brownian movement registration on a liquid/gas interface: 1, glaee vessel; 2, microscope; 3, glass cover; 4, laser; 5, glass mirror;6, TV camera; 7, computer; 8, TV monitor.
sphere, Figure Id. The drag force in this last case is zero because the contact area is zero. 111. Experimental Section A classical method for measuring the diffusivity is the direct registration of the Brownian motion. This method is particularly useful for investigations of diffusion at interfaces, where the
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Radoev et al.
Table 11. Varinncer 2,in pm*, va the Time Period (At), in a, for Dirplacementr Ax Satirfying Certain Statirtical Conditionr (See Section IV.Statirtical Analyrir) (a) Brownian Motion in the Bulk Averaged from Nine Different Particles; Regression Coefficients in Equation 9 = a + b(At): a = 0.2; b = 1.2 At=3s At=6s At=9s At = 12s 3.7 7.2 11.2 14.5
9,pm2
At = 15 s 17.8
(b)Brownian Motion at the Air/Water Interface from Five Different Particlea with Regression Coefficients a = 0.1 and b = 0.8 9,lrm2
At=15s 14.4
At1306 26.0
At-456 32.5
At=60s 45.5
standard dynamiclight scattering method" is not suitable, since there is not enough scattered light. In his famous paper "Mouvement Brownien et realite moleculaire", Perrin1*in 1906 measured the displacement (Az) of gomme gotte particles with diameter 2a 0.5 pm and via the relation ( (Az)2)/At = (RTI 3 7 p a ) l N has ~ estimated the the Avogadro number NA.The data reported there, NA = (6.8-7.3) X 1029 showing a remarkable coincidence with the recent value NA= 6.02 X 1029, prove the correctness in the interpretation of the experiment as a whole, but no systematic analysis of the character of the registered motion has been carried out. Moreover, the data on the time reported in the same paperl12show some dependence of ( discrepancy with the theoretical linear Einstein-Smoluchowski = 2DAt (see Table I). relation It should be noted that the deviations of the NA values determined by Perrin are not statistically distributed around the actual value since they are higher. The observed deviations both of the time dependence ( ( A z ) 2 )and of the NAvalues from the theoretical ones are most probably due to uncontrollable noises, but they could reflect physical effects such as inhomogeneities in the velocity field intrinsic relaxation times,18etc. In all these cases additional statistics should be carried out (see section IV). In order to obtain information on the stochastical character of the motion, we performed experiments with single particles in a specially designed setup (see the sketch in Figure 2). Macroparticles (melamin polymer, diameter -1 pm, density 1.06 g/cmS)were deposited by a glass fiber on a slightly concave meniscusin the measuring cell (l),Figure 2. Specialpreliminary cleaning of the liquid interface should be performed since otherwise the particle motion would be practically immobilized. This is an indication of the important role of surfactants in the transport at the surfaces.1° The particles were illuminated by a laser beam (Ne-He laser, X = 0.628 pm (4),ultramicroscopy method). The scattered light was observed by a biological microscope with objective lens F = 6.3 and free work distance of 1 mm, combined with a photo objective lens PK 16 and TV camera (6). The obtained video signal (magnification X2230) was directed simultaneously to a PC (7) and a video monitor (8). The pattern of the individual particles was traced on the monitor at constant times intervals At, and via an image analysis program the coordinates of the points (Xi,Yi)were obtained. The Brownian motion in the bulk was investigated by replacing the glass vessel 1 (in Figure 2) with a standard BCvker camera (for erythrocyte counting) filled up with a dilute suspension of the same polymer particles.
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IV. Statistical Analysis Data series of consecutive points (Xi(ti),Yi(ti)j,i L= 1, ...,n,n = 80-120 observed for the individual particles a t the surface, and in the bulk, are used for the statistical calculations. The conditions characterizing the normal random processes, (1) the hypothesis for normal distribution of AX and A Y using the w2 criterion,15(2)the independence (12) Perrin, J. Ann. Chim. Phys. (Paris) 1909, 18 (8), 5. (13) Mann, Adin J., Jr. Langmuir 1986,l (l),10. Radoev, B.; Teekov, R. Adv. Colloid Interface Sci. 1990,33, ( 2 4 ,79.
(14) Puaey,P.N.;Tough,R.J.A.InDy~micLightScattering;Pecora, R., Ed.; Plenum Prees: New York, 1986, Chapter 2. (15) Balehev, L. N.; Smirnov, N. V. Tables of moth. statistics; Mir: Moscow,1983.
Ate758 58.5
At=90s 68.8
At=105s 81.4
At-120s 94.4
At=135s 107.4
of AX vs AY, and (3)the homogeneity of the variances ux2 ( = ( ( A X ) z ) ) ,uy2, are analyzed separately, since the experimentally observed movements could be a result of different factors (hydrodynamic convections, mechanical noises, etc.). The mean values (AX),( A Y ) of the drift and the variances uz2, uy2 are determined for series satisfying conditions 1-3 at a significance level of 0.05. A random process is Brownian motion if the displacementa are normally distributed, and the variance of this distribution increases linearly with time. That is the EinsteinSmoluchomki relation must hold. The Einstein-Smoluchowski relation u2 = 2DAt is checked by plotting the variances u2vs (At). The particles are observed to drift at nearly constant drift. The data are sampled a t time intervals, with length chosen according to the magnitude of the drift (Tables 11). The particle diffusion coefficient D, is estimated from the slope of u2 vs (At).
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V. Results and Discussion The experimental data for nine particles (At 3 e), measured in the bulk, and for five particles (At 15 a), measured at the interface, are summarized in parta a and b of Table 11, respectively. The series Xi(ti) is divided into sectors,with an evaluated, almost constant drift. The results obtained by this method, shown in Table 11, fit quite well to a linear dependence u2(At) = a + bAt. The deviation of the constant term a from zero was found to be statistically insignificant at the 96 % confidence level. Note,that the nonstationary parta of the drifta would give additional, erroneous values of the variances, leading to deviations of the u2(At) from the theoretical linear dependence, with a zero intercept. The values of the diffusion coefficients, calculated from the slopes b (=2D) are D,(bulk) = 0.6 pm2 s-l and D,(surface) = 0.4 pm2 s-l, respectively. Transmission electron microscopy investigations show that the polymer particles are nearly spherical in shape, with radii predominantly in the interval 0.64.9 pm. The values of the bulk diffusion coefficienta for particles with these dimensions, in a fluid with 7 1cP, a t T 300 K, are to be D ( = k T / 6 ~ 7 a ) 0.42-0.24 pm2 8-1.
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Two possible effects for the values of D,(surface) 0.4 pm2 s-l, a value very close to those of the bulk diffusion coefficients, are the depth of immersion of the sphere in the liquid h (seeFigure 1)and the action of the surfactants. As shown in the second part of this study,1° due to the Marangoni effect surfactants could reduce drastically (up to lo"timea) the mobility of translating spheres at constant depth h. At the same time, the capillary forces (the threephase contact angle) controlling h also depend on the presence of surfactants, so that the discussion of the D,(surface) values would need more theoretical as well as experimental investigations. A factor possibly contributing to the measured D, in the bulk being higher than the theoretical prediction is a
Langmuir, Vol. 8, No. 12, 1992 2966
Brownian Motion at Liquid-Gas Interfaces selection bias toward observing particles with smaller than averaged size. The measurements of Brownian displacements in the bulk were taken from the upper part of the Btirker camera, where, because of sedimentation, there tend to be smaller particles. The nonuniformdistribution of the viscous stresses on the interface of the sphere should lead to ita rotation. From the minimum energy dissipation theorem (Helmholtz'l), it follows that the accounting for the cross effects should decrease the resistance. Actually, for the case cited in the drag force problem section, translation of a nonrotating pendant sphere (Figure IC), the coefficientf(0) = 0.716 is higher thanthat with rotation, f(0) = 0.694." However, for hemispheres floating on flat interfaces the rotation should not influence the drag force. According to eq 11.4 the integral in the right-hand side is zero regardless of the type of field v', and the resistance coefficient should always be equal to f(1) = 0.5.
The assumption of a flat liquid-gas interface is another approximation of the model. The real shape of the interface is determined by the normal stress boundary condition, pzr = pr + pgt, where pZE= -p + 2p(auZ/az)ie the viscous stress tensor component, pr the capillary pressure, pgt the hydrostatic pressure, and {((n,y) the deviation of the interface from ita flat form. Supposing small deformations, one could analyze the order of the terms in the normal stress boundary condition and thus to estimate the order of the deformation amplitude { as follows: p z z Vpla;P-/ N a 9 tla (Vp/y)(l + pga2/r). For particles with radius a 1 pm, particle velocity (root mean square velocity) V (kT/m)l/Z 0.1 cm s-', interfacial tension y 50 dyn cm-', p 1g cm-3, and the order of the amplitude of deformations is much smaller than the particle radius, {/a lo+.
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