Langmuir 1994,10, 1351-1353
1351
Brownian Motion: A Tool To Determine the Pair Potential between Colloid Particles Kristina Vondermassen, Jorn Bongers, Andreas Mueller, and Heiner Versmold* Institut fiir Physikalische Chemie, R WTH, 0-52060 Aachen, Federal Republic of Germany Received December 13, 1993. I n Final Form: March 29, 1994e Exciting light and neutron scattering as well as microscopic experiments have been performed recently with colloidal systems.l-e Dependingon the nature of the colloid particles and the experimentalconditions, two- or three-dimensionalsupermolecular fluid-,glass-,and crystal-like ordered states have been obtained. Apart from fundamental insight into the structure and dynamics of colloidal systems, such investigations are often suitable to relate theory more directly to experiment than possible for atomic or molecular systems. In spite of the invaluable information obtainable from the above-mentioned experiments, significant lack in understanding the supermolecular structures remains as long as they cannot be related to the interaction between the constituent particles. We propose here to use the particle positions generated by Brownian motion for the determination of effective pair potentials U(r) by microscopy. potential depends on parameters like T, e, and I; Le., U(r) is an effective potential or free energy of a pair. In fact, the potential eq 1 has frequently been used with an adjustable particle charge 2 ., to interpret experimental data on the static structure factor SCQ). No direct experimental method for the determination of the effective pair potential U(r)between spherical particles appears to be available so far. In the present paper we want to demonstrte how effective pair potentials U(r) can be determined experimentally. In 1908 Perrin studied the sedimentation equilibrium of colloid particles in water with a microscope.'g He counted the number of particles at a given height and succeeded to verify a formula on the distribution of particles in the gravitation field, which at that time was just derived by Einsteinz0 and Smoluchowski.zl Perrin determined Avogaro's number N A with an astonishing where K Z = 2e2I/ckT, I = cciziz/2 is the ionic strength, 1 / ~ accuracy. In the present context it is interesting to note that, essentially, Perrin determined the potential of the is the well-known Debye screening length, and 2 ., is the colloid particles in the external gravitational field. More effective particle charge. For colloidal systems the pair recently, ordinary and reflection interference contrast (RIC) microscopies have been used to study the interaction * To whom correspondence should be addressed. of particles with the gravitational field and with an * Abstract published in Aduance ACS Abstracts, May 1, 1994. underlying substrate perpendicular to the field.22 (1) Safran, S.A,, Clark, N. A. Eds. Physics of Complex and SuperIn all these experiments the interaction of colloid molecular Fluids; Wiley: New York, 1987. (2) Pusey, P. N. In Liquids, Freezing and the Glass Transition; particles with systematic external fields was considered. Levesque, D., Hansen, J.-P., Zinn-Justin, J., Eds.; North-Holland: In the present paper we call attention to the fact that this Amsterdam, 1991. method can be generalized to determine the internal (3) Pusey, P. N.; van Megen, W. Nature 1986,320, 340. (4) Clark, N. A.; Hurd, A. J.; Ackerson, B. J. Nature 1979, 281, 57. interaction potential U(r) between colloid particles as well. (5) Ackerson, B. J.; Cleark, N. A. Phys. Reu. Lett. 1981, 46, 123. Due to Brownian motion one finds pairs of particles at (6) Hachisu, S.;Yoshimura, S. Nature 1980,283, 188. various particle distances. The probability to observe a (7) Kruyt, H. R.,Ed. Colloid Science; Elsevier: New York, 1952. (8) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: pair at a particular distance r apart from each other is Oxford, 1987; Vole. I and 11. essentially given by the pair distribution function g(r), (9) Verwey, E.. J. W.; Overbeek, J. Th. G. Theory of the Stability of well-known from liquid-state theories.z3 g(r) is related to Lyophobic Colloids; Elsevier: Amsterdam, 1952. (IO)Derjaguin, B. V.; Churaev, N. V.; Muller, V. M. Surface Forces the potential of mean force W)as Charge-stabilized colloid systems have been discussed in terms of the DLVO potential7-I1for more than 40 years. In contrast to a large number of theoretical treatments, experimental confirmation is scarce: Surface force measurements with crossed cylindrical sheets of mica8J0J1 and atomic force microscopyl2 have partially confirmed the DLVO theory. On the other hand, the DLVO approach has been questioned, both on theoretica11%16and on experimental g r ~ u n d s . ~ ~ J ~ For charged spherical particles of diameter cr the electrostatic part of the potential is often assumed to be of the approximate DLVO form
(translated from Pouerkhnostnye Sily); Plenum: New York, 1987. (11) Israelachvili,J. N. Intermolecular and Surface Forces;Academic Press: London, 1985. (12) Ducker, W.A.; Senden, T. J.; Pashley, R. M. Nature 1991,353, 239. (13) Ltiwen, H.; Madden, P. A,; Hansen, J.-P. Phys. Reu. Lett. 1992, 68,1081. (14) Belloni, L. J. Chem. Phys. 1986,85, 519. (15)Sogami, I.; Ise, N. J.Chem. Phys. 1983,81,6320. Sogami, I. Phys. Lett. 1983,AM, 199. See also: Sogami, I. In Ordering and Organization in Ionic Solutions, Proceedings of the XIX Yamada Conference; Ise, N., Sogami, I., Eds.; World Scientific: Singapore, 1988. (16) The approach of ref 15 has been criticized again by Overbeek,J. T. G. J. Chem. Phys. 1987,87, 4406. (17) Sirota, E. B.; Ou-Yang, H. D.; Sinha, S. H.; Chaikin, P. M.; Axe, J. D.; Fuji, Y. Phys. Reu. Lett. 1989, 62, 1524. (18) Smalley, M. V. Mol. Phys. 1990, 91, 1251. See also: Faraday Discuss. 1990, 90,182-192.
0743-7463/94/2410-1351$04.50/0
For an ordered system \k(r) is not the effective pair potential of interest but the potential of a pair in the presence of the interaction with all neighboring particles (19) Perrin, J. Comput. Rend. 1908,146,967. See also: Perrin, J. Les Atomes; Libraire Felix Alkan: Paris, 1913. (20) Einstein, A. An. Phys. 1906,17,549. (21) van Smoluchowski, M. R. An. Phys. 1906, 21, 756. (22) Prieve, D. C.; Alexander, B. M.; Science 1976,231,1269. Prieve, D. C.; Frej, N. A. Langmuir 1990,6, 369. (23) Hansen, J. P.;McDonald,I. R. Theory of SimpleLiquid;Academic Press: London, 1976.
0 1994 American Chemical Society
Letters
1352 Langmuir, Vol. 10, No. 5, 1994 as background. Many attempts have been undertaken to determine U(r)fromg(r) for ordered atomic and molecular systemsa2*It turns out that usually several U(r)are equally well compatible with a measured g(r). Thus, this route is not particularly successful. By contrast, an isolated pair of particles at infinite dilution will experience the pair interaction only. It can be shown that in this limit the pair distribution function g(r) is directly related to the effective pair potential U(r)as23 /
Ob
Fromg(r) obtained in this limit the effective pair potential can be calculated as U(r)/kT = -ln[g(r)l
(4)
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Figure 1. Experimental (I = 1.5 X 10-8 ( O ) ,1.2 X 10-6(a),and 0.5 x 10-6 (v)mol/L) and ,calculated (-1 pair distribution functions g(r). Details are given in the text. 3, I
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To the best of our knowledge, this method has not been used so far to determine effective pair potentials U(r)in colloid science. The dispersions studied consisted of polystyrene latex particles in water, charge stabilized by sulfate surface groups. The diameter as determined by ligyht scattering and electron microscopy amounted to u = 770 f 50 nm. Due to the strongly acidic sulfate groups, H+ions dissociate I a from the particles, which results in a negative particle 0.5 I 1:s 2 2.5 3 315 4 charge. By adding an electrolyte to the dispersions, the r Ipm particles can be screened and thus the interaction between Figure 2. Experimentaland calculated effective pair potentials the particles varied. U(r)/kT. Symbols are as in Figure 1. In order to obtain different effective pair potentials U(r), -12p. ' dilute suspensions at three different rather low ionic strengths were investigated. From a carefully cleaned stock suspension samples were prepared by adding NaCl which had approximately the ionic strength I = 1.5 X 10-6 ( 0 ) ,1.2 X 10-6 (o), and 0.5 X 10-6 (v)mol/L. The dispersions were kept in a cell with a thin optical flat bottom. Microscopic observation was carried out through the cell bottom with a reversed metallurgial L microscope (METALLOVERT, Leitz, Wetzlar, Germany). i -18 The field of vision was recorded with a video system (HAMAMATSU Model C 2400 and DVS 3000; Panasonic Videorecorder Model AG 7330) interfaced to a H P Vectra 386/25 PC. Particle recognition and calculation of the 1. pair distribution function g(r) was carried out with the PC by applying methods also used in standard computer simulation programs.2s Details will be given elsewhere.26 the glass surface were carried out. No influence of this Microscopic experiments on concentrated ordered coldistance on g(r) was observed. loidal systems have been reported p r e v i o ~ s l y .We ~ ~ ~ ~ ~ ~Figure ~ 1 shows the pair distribution functions g(r) for emphasize again that for concentrated systems g(r) is not the three samples. A t small r an excluded volume is visible, related in a simple manner to the effective pair potential which is largest for the dispersion with the lowest ionic U(r). Our experiments, therefore, were carried out with strength due to the longest range of the repulsive interachighly dilute dispersions. In preliminary experiments first tion. We want to call attention to the fact that at large the concentration for which three-body interactions are r the value 1 is approached by the g(r) without maxima negligible was determined by diluting the system until no or oscillations. further change of g(r) was observed. In order to obtain Next, the effective pair potential U(r)was determined g(r) at the required low particle concentration with a by applying eq 4. Results are shown in Figure 2. A t small reasonable signal to noise ratio, it was necessary to evaluate r there is a steep decrease of the U(r)lkT curves for the more than 20 000 video pictures for eachg(r). Further, in three ionic strenghts. As expected the potential is shorter order to make sure that the glass surface of the cell did ranged the higher the ionic strength. It is interesting to not influence the result, measurements with the focal plane note that within experimental accuracy the U(r)lkTcurves of the microscope at distances between 2 and 6 pm from smoothly approach zero at large r; i.e., there is no indication of an attractive interaction for the dispersions investigated. (24)Dupuy, J., Dianoux, A. J., Eds. Microscopic Dynamics and The screened Coulomb potential eq 1has widely been Structure of Liquids; Plenum: New York, 1978. considered most approapriate for charge-stabilized col(25) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1987. loidal systems. Next, we investigate whether this could (26) Bongers, B.; Mueller, A.; Versmold, H. T o be published. be born out of our data. If eq 1 is correct, a plot of ln(27)Murray, C. A,; Van Winkle, D. H. Phys. Rev. Lett. 1987,58,1200. ( r U ( r ) / k Tversus ) r - u should result in a straight line with (28) Ise, N.;Matsuoka, H.; Ito, K.; Yoshida, H. Faraday Discuss. 1990, the slope - K . For the different samples such plots are 90,153.
-'b
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Letters shown in Figure 3. Since the logarithm of a negative argument is not defined, corresponding data points are missing in Figure 3. Within the experimental accuracy the data are well represented by straight lines. The K values obtained from the slopes (K = 3.9 X lo6 (0),3.5X lo6(13, and 2.4 X 106 (v)m-1) are in fair agreement with our previously given estimate estimates of the ionic strength of the dispersions. Finally, the K values can be used to calculate U ( r ) / k T and g(r). The solidly drawn curves in Figures 1 and 2 were obtained with an effective particle charge of 190 e-. Although there is some scatter of the experimental data, a reasonable agreement is obtained. One reason that the data become less accurate at small r is that kTis no longer dominant and fewer instances are available for sampling.
Langmuir, Vol. 10, No. 5, 1994 1353 Concerning the question whether there is a purely electrostatic attractive interaction between charged spheri~ ~ present J ~ investigation does not cal particles or n ~ t , the indicate such an attraction. However, we do not claim that this first microscopic investigation can already give a fiial answer. Further experiments under various conditions such as variation of the particle size, particle charge, surfactants, different counterions, etc. are necessary and possible to obtain more conclusive results. The experimental method presented in this paper appears particularly suited for such investigations.
Acknowledgment. Financial support of the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie is gratefully acknowledged.
