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Brownian Motion at Liquid-Gas Interfaces. 4. Experiments with Glass Microspheres in the Presence of Surfactants K. Starchev, M. Avramov, and B. Radoev" Department of Physical Chemistry, University of Sofia, 1126 Sofia, Bulgaria Received June 7, 1994. I n Final Form: February 13, 1 9 9 P Brownian motion (BM) of glass microspheres (-1 pm size) at the water-air interface in the presence of dodecylamine(DDA)is directly measured with the use of an image analyzing method. This method has been tested by BM measurements in the bulk and proved to be accurate. The experimental results, recalculated as particle diffusion coefficients D,, are investigated from the view point of a theory earlier developed by us. Accordingto this theory, D, should be a function of the Marangoni number Ma = E d D p , where E = (-rayfar)is the Gibbs elasticity, a = (aF/&) is the distribution coefficient, r and c are the surfactant (DDA) adsorption and bulk concentration, and D is the surfactant bulk diffusion coefficient. Rather unexpected, D, shows no significant dependence on Marangoni number even for almost saturated states of adsorption. More over, the values of the diffusion coefficients at the surface (D, 0.3 pm2 s-l) and in the bulk (D, 0.2 pm2 s-l) seem to be of one and the same order of magnitude. The discrepancies
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between the theory and experiment are discussed from a more general dynamic and adsorption state point of view.
Introduction Similar is the role of insoluble surfactants6 One of the most powerful tools for studying the diffusion AF/Fo = (ErJD&)[lnR e ...I processes in fluids is Brownian motion. Although modern experimental methods are based on dynamic light scatThe appearance of the Reynolds number Re = Valv ( v = tering, direct observation of Brownian motion first prop l ~the , kinematic viscosity and Q the fluid density) in the posed by Perrinl is still in praxis. Combined with digital Brownian motion dynamics, usually modeled by the Stokes image processing, it has been successfully applied for hydrodynamics is connected with some peculiarities, as investigation of concentrated suspensions,2 in memb r a n e ~and , ~ at liquid-gas interface^.^ In our ~ t u d i e s ~ - ~ shown in ref 6, typical for the two-dimensional perturbations (here the perturbation of the adsorption r). Note we have begun analyzing, both theoretically and experithat all above cited relations are derived for small mentally, the Brownian motion of a microsphere floating perturbations of the surfactant distribution, equivalent at liquid-gas interfaces with special attention paid to the to small Marangoni numbers. But as discussed below, role of the surfactants on the process. The main effect the typical cases in the experimental praxis are just the taken into account here was the so-called Marangoni opposite, Ma >> 1. effect,I equivalent to an additional force AF = F - FO(F The additional drag force A F via the Einstein relation and FOare the drag forces with and without surfactant) D, = kBTlB ( B is the particle friction coefficient, defined acting on the sphere. This force results from perturbation by the Stokes law F = SV, V is the particle velocity) of the equilibrium surfactant distribution when particle corresponds to changes in the particle diffusion coefficient translates on the surface. In the case of soluble surfacas follows: AFIFo= ADJD,, with AD, =D,(O) - D,, where tants, a simple relation between AF and the surfactant D,(O) and D, are the particle diffusion coefficients on pure parameters was carried out5 surfaces and on surfaces with an adsorbed surfactant on AF/Fo = (EdDp)f(crD8/Dr,) it, respectively. These and other effects and problems make the exwhere in the factor ( E d D p ) ,called by some authorss the perimental study of Brownian motion at liquid interfaces Marangoni (Ma) number, E = ( - r a y / X ) is the Gibbs reasonable and interesting. In ref 4 we have described elasticity, a = ( X / & )is the distribution coefficient, r and our experimental setup and have given some preliminary c are the surfactant adsorption and bulk concentration, results of Brownian motion with polymer microspheres D and D, are its bulk and surface diffusion coefficients, (- lpm). The aim ofthis paper is to summarize the results r p is the particle radius, and p is the liquid dynamic from the new experiments with glass micro spheres in the viscosity. The functionflaDslDr,) reflects the competition presence of dodecylamine hydrochloride (DDA) as a between the surface and bulk d i f f ~ s i o n . ~ surfactant and to discuss them from the viewpoint of the Marangoni dynamics effect model.
+
* To whom correspondence should be sent.
Abstract published in Advance A C S Abstracts, April 15, 1995. (1)Perrin, J.Ann. Chim. Phys. (Paris)1909,18(8), 5. (2)Schaertl, W.; Sillescu, H. J.Colloid Interface Sci. 1993,155,313. Melo, (3)Gaub, H.; Sackmann, E.Biophys. J.1984,45,725.Vaz, W.; E.; Thompson, T. Biophys. J. 1989,56,869. (4)Radoev, B.; Nedjalkov, M.; Djakovich,V. Langmuir 1992,8,2962. ( 5 ) Dimitrov, K.;Radoev, B.; Avramov, M. Langmuzr 1993,9,1414. Avramov, M.; Radoev, B. Langmuir 1994,10,1596. (6)Dimitrov, K.; (7)Levich, V. G. Physicochemical Hydrodynamics; Prentice-Hall: Engelwood Cliffs, NJ, 1962;Chapter 8. (8)Schwartz, P. Dr. Sci. Thesis, Berlin, 1985;Chapter 3,3.1 @
Materials and Methods Apparatus. As mentioned above,the experimentalsetup was first described in ref 4. In its present version there are some improvements of quality images. The particles were illuminated from the bottom side of the cell by a Hg lamp and the scattered light is observed using a dark field condenser. The glass cell was covered by a standard microscope slide, in order to eliminate the evaporation and the convection from the gas phase. The cell is
0743-746319512411-1731$09.00/0 0 1995 American Chemical Society
1732 Langmuir, Vol. 11,No.5, 1995
Starchev et al.
Table 1 1 2 3 4 5 6
7.0 7.5 8.4 9.5 10.5 11.0
0.7 x 0.9 x 10-10 6.5 x 36.5 x 78.6 x 94.9 x
1.19 x 1.49 x 5.10 x 4.01 x 2.45 x 2.12 x
71.0 68.0 62.2 55.5
106 lo6 lo6 lo6 lo6 106
placed inside a second vessel for additional isolation, and thus the nonthermic motions have been substantially reduced. The pictures were stored on a videotape recorder and further processed by IBM PC computer via a frame grabber board (see Image Analyzing). Materials and Sample Preparation. Spherical glass particles (Duke Sci. Corp., Catalog No. 145)with size 1.6 f 0.3 pm have been used. The particles have been washed in bichromic mixture and many times rinsed in double distilled water. As the glass microspheresare with specificgravity 2.42g/cm3,they could be attached on a liquid-gas interface by capillary forces only, i.e. the particles should be hydrophobized. The hydrophobization was carried out with dodecylamine (DDA), known as a good collector for glass and quartzg The degree of hydrophobization is characterized usually by three-phase contact (TPC) angles. For the system quartz-DDA water solution-air, the cited values of TPC angles (for pH interval 5-11)11 are in the range of 2040°,good enough for the adhesion of l-pm particles on the liquidair interface. Because of the small particle sizes in our case, the actual TPC angles could not be experimentally measured, but it was definitely established whether the particles were captured or not on the interface. By focusing the optical system on the liquid-gas interface, the hydrophilic particles disappear after a while (due to sedimentation) but the hydrophobized ones (those attached at the interface) remain focused. DDA is known as a strong surfactant as well. According to ref 10,in the pH-interval 5-11, DDA is a base, C12H25NH2 H2O = C12H25NH3++ OH-, with pK = 4.4(K= 4 x lo+). From the experimental data of y(pH) and the dissociation equilibrium it follows that the neutral amine RNH2 is in this case the more surface active form (see Table 1, and ref 10). Its concentration cs could be recalculated from the analytical concentration co = M of DDA uiu the relation cs = c&+ 1014-pK-pH). In 1.1 x such cases of mixture of ionic surfactants (here F W H 2 and the exact evaluation of the adsorption characteristics ri, WlaCi, Xi/&are rather complicated.12 On the other hand, the presence in the system of a component with a higher surface activity permits neglecting, in a rough approximation, the contribution of the rest of the components to the elasticity (E). Therefore, for the estimation ofE we limitedourselveswithin the Gibbsequation for the neutral form only, rs= -(c$RT)(ay/aC,). Moreover, as seen from Figure 4, there is no significant dependence of the diffusioncoefficient on the Murungoni number, i.e. no correlation with the surfactant activity (Mu aE/Dp Ts(ay/&s). The possible corrections of the Mu values therefore will only shift the data series for given pH value along the Mu axis but will not affect the distribution of the experimental D, values (see Figure 4). In any case to avoid such ambiguities the study of Mu effect should be carried out with a simpler (nonionic) surfactant. Brownian motion measurements in the bulk were performed using latex particles of diameters 1.12, 1.7, and 2.54 pm (see Figure 3). The suspensions were diluted enough to neglect the hydrodynamic interactions between the particles in the bulk. Image Analyzing. The digitalization procedure was performed by a video frame grabber (256x 256 pixels with 256 gray shade levels). One of the problems here was the localization of the real coordinates of the particle, as the observations were in a scattered light, and the images (spots) of particle were several times larger than their real dimensions. The tests have proved
Figure 1. Explanation of the image recognition procedure. A simple amplitude filter based on the dispersion of the signal is used to eliminate the electronic noise. The nonfiltered amplitudes are considered as a scattered light spot of the tracked particle and a weighted mean is used to evaluate its coordinates. the first momentum of the intensity as a quite satisfactory definition of the particle coordinates (X,, Yp)
+
Here X are the coordinates of a pixel with intensity I(X). The summation region C (see Figure 1) is defined according the
m+),condition I > 3(12 - 7")y". This in fact is an amplitude filter-
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(9)Schulze,H.J.Physico-ChemicalElementaryP m s s e s in Flotation; Elsiever: Amsterdam, 1984;Chapter 4,4.2.2. (10)Petrov, J.; Panaiotov, I.; Tchaljovska, S.; Scheludko, A. Ann. Uniu. Sofia (Fac.Chim.) 1971, 66, 191. (11)Radoev, B.;Alexandrova, L.; Tchaljovska, S. Int. J. Mineral Processing 1990,28, 127. Schulze, H.J.; Chalyovska, S.; Scheludko, A.; et al. Freiberger Forschungsh. A. 1977,568, 11. (12)Eriksson, J. C.; Ljunggren, S. Colloids Surf.1989,38, 179.
ing procedure which considers the signals smaller than 3(12 - PI'/"as an electronic noise. Another way is the frequency filtering but this procedure is many more times expensive and does not lead to considerably better results compared to the described above simple amplitude filter. The aim of the tracking procedure here performed was to get information about the particles diffusion coefficient D,, using the classical Brownian motion law:
(AX-
z2 =WPAt
The averaging was performed on a single particle trajectory, for different At, by the following procedure. For At = 1 s, for example, the AX-set contains all differences (Xj+l- Xj), in the trajectory, for AT = 2 s, all combinations (Xj+z- Xj), etc. (see Figure 2). Thus obtained values of the mean square displace-
z2
ments presented in coordinates (AX vs At determine the coefficient D, using a linear regression. For long time intervals (At > 15-20 s) a nonlinearity was observed (seeFigure 21,probably due to long distance interactions with the other particles at the surface, but more complicated nonstationary effects are not excluded. The Brownian motion must be a fully stochastic process so the displacements M j must be normally distributed. In several series we have tested the distribution of AX and for the greatest part of the cases the Kolmogorov test was ~atisfied.~
Results and Discussion The described here method for direct observation of Brownian motion was tested by bulk diffusion coefficient
Brownian Motion at Liquid-Gas Interfaces
Langmuir, Vol. 11,No. 5, 1995 1733
o o o o 5 -
A 2-AX
. I o
'
I'
IW'I
0 O
0
0'
4
I
:b
U 2 .
O
0
0 0
0
a4
5,
d 1
"8 @)
0.2
6' '
,'
0 8 n
-
Figure 2. Sample dependence fl-fl vs At. In some cases a deviation from the linear dependence occurs. In the calculation procedure for D, only the points situated close around the line are taken into account.
I
A
.
Figure 4. D, vs Ma (-surface activityE ) for particles floating at the liquid gas interface. Linear regression: intercept 0.30 f 0.05 @m2s-l) slope (1.22 x f (1.48 x @m2s-l dyn-l) (i.e., the slope confidence interval includes the zero). The numbers at the top of the different series correspondto the numbers in Table 1.
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0
could be the values of Ma number realized in ovr experiments. They are much greater than 1 (Ma lo6, see Figure 4), while the cited above theof16 is valid for Ma < 1. Generally,for the usual surfactants, the condition Ma 1 could be reached at extremely low bulk surfactant concentrations. This is connected with the small values of the product Dp dyn, so that the condition Ma 1 is equivalent to concentrations where a73 = T(ay/aC) (dyn). It should be noted that the two cases Ma >< 1 correspond to two different controlled transfer processes. When Ma 1,the processes are diffusion limited: while in the other case Ma > 1, where our systems hold, the transport is controlled by the momentum rate transfer (i.e., by convection).14 These considerations show, that the limit Ma > 1 is more important, at least from an experimental point of view, and deserves special analysis.
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0.6
0.8
1
1.2
1.4
5' Ip"
1.6
1.8
2
1
Figure 3. Results D, vs rP-l for a Brownian particle in the volume used for accuracy test of the proposed experimental equipment. The slope of this line 0.18 f 0.04 (u3 s-l) (at confidence limit 0.9)is very close to the theoretical value ( k ~ T l 6nv)= 0.21 (um3s-l).
measurements of latex particles. The experimental data presented in coordinates difhsion coefficients (D,) vs inverse particles sizes r,-l are shown in Figure 3. The confidence limits for the regression line ( f 2 standard deviations) are given with a dashed line. The agreement with the theoretical relation Db = (k~T/6np)r,-lis good; the experimental slope equal to 0.18 f 0.04 Cum3 s-l) is very close to the theoretical value ( k ~ T / 6 n p=) 0.21 Cum3 s-l),so we have considered this technique accurate enough for Brownian motion measurements a t the surface. The experimental data about Brownian motion of glass microspheres (r, = 0.8 f 0.15 pm) at the interface surfactant water solution-air (for six concentrations of DDA) are plotted in coordinates D , vs. Ma (Figure 4). The data for D, could be presented as a function of the surfactant concentration cs or as a function of the pH values, but we preferred Ma as a coordinate in order to compare the experimental results with the the01-y:~D,(O)/D,(I')- 1 = Maf (aDJrp)(see Introduction). The impression from Figure 4, confirmed by the linear regression analysis, is that there is no significant dependence of the diffusion coefficient on the Marangoni number, i.e., no correlation with the surfactant activity (Ma d / D p T(@/aC,). It should be noted that the factor f l a D J r p ) is a weak function of its argument and could not compensate the linear dependence of Ma.5J3One reason for this discrepancy between a theory and experiment
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Concluding Remarks Besides the discussed above discrepancy with the low surfactant activity model (Ma l),the experimental data in Figure 4 provoke some additional speculations. The values of the particle diffusion coefficients at the surface (D, 0.3 pm2 s-l), for instance, are of the same order of magnitude as the diffusion coefficients in the bulk (Dp(rp = 0.85pm) 0.19pm2s-l, see Figure 3. In the same time, the estimated surface concentrations r in our case vary more than 10 times, including dilute gaseous (at pH = 7) as well as condensed (pH = 11)states of adsorption. Note, that the diffusion coefficient (proportional to the particle mobility B-l) at totally packed interfaces15should tend to zero.16 As a matter of fact, in our experiments, almost depressed diffusion was observed only in the cases of strong uncontrollable contamination. The difference between the dispersions of the diffusion coefficients a t the surface = 0.6, Figure 4) and in the bulk ([(ADp)23'2/% = 0.11,Figure 3) is another feature of the experiment we would like to discuss briefly.
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(13) Dimitrov, IC;Avramov, M.; Radoev, B. C. R. A c d . Bulg. Sci. 1994,47 (2), 57. (14) Panaiotov,I.; Dimitrov, D.; Ivanova,M. J. Colloid Interface Sci. 1979,69,318. Panaiotov,I.; Sanfeld, A.; Bois, A.; Baret, J. F. J . Colloid Interface Sci. 1983, 96, 315. ( 15) Sobotka, H.MonomolecularLayers; American Association for the Advancement of Science: Washington, DC, 1954; Chapter 3. (16) Guastalla, J. Thesis, University of Paris, 1948.
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1734 Langmuir, Vol. 11, No. 5, 1995
An additional source of dispersions at the surface could be the difference in the immersion depth of the particles in the liquid, but as shown in Ref 4,the hydrodynamic resistance coefficient B is a weak function on the particle depth of immergion. A possible explanation of the large values of the surface diffusion coefficients and their dispersions could be the heterogeneity in the adsorption r. The traditional view point is that the surfactant distribution at the interfaces is homogeneous, i.e. no local phase transitions, clusters, etc. exist. However, in some cases (phospholypide monolayers) heterogeneous structures are directly observed by
fluorescence microscopy.l7 The possibility of Brownian motion as a detector of nonhomogeneities at liquid interfaces needs careful analysis and will be a subject of further experiments.
Acknowledgment. These investigations are partially supported by the Science Foundation of the University of Sofia, Project 233. LA940447J (17)Mohwald, H.Annu. Rev. Phys. Chem. 1990,41, 441.
