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Direct Measurement of Interparticle Forces by the Optical Trapping Technique Tadao Sugimoto,* Tetsuya Takahashi, Hiroyuki Itoh, Shun-ichi Sato, and Atsushi Muramatsu Institute for Advanced Materials Processing, Tohoku University, Katahira 2-1-1, Aobaku, Sendai 980-77, Japan Received May 23, 1997. In Final Form: August 18, 1997X The intrinsic interactive forces between two free colloidal particles suspended in aqueous solutions have been measured as a function of the separation by applying the optical trapping technique to a monodispersed polystyrene latex of a mean diameter 2.13 µm. For this purpose, specific technology for the precise measurement of the surface-to-surface separation and force strength has been developed. The results were in excellent agreement with the prediction of the DLVO theory in a variety of electrolyte concentrations.
Introduction The so-called DLVO theory developed by Derjaguin, Landau,1 Verwey, and Overbeek2 has successfully been used by colloid scientists to explain the dynamic behavior of colloidal particles. However, direct experimental verification of the theory has started rather recently. Particularly, Israelachvili et al.3,4 developed a novel surface force apparatus (SFA) for the direct measurement of planeto-plane interaction between mica plates with high resolution of the orders of 0.1 nm for the interplane distance and 10-8 N for the interactive force and found an excellent agreement with the DLVO theory. Shubin and Kekicheff5 also found similar results in the study of the interplane forces of mica plates in lithium nitrate solutions over wide concentration and pH ranges using the surface forces apparatus (SFA). Derjaguin et al.6-8 developed a new setup for force measurement between crossed cylindrical bodies, where the forces were measured by using a negative feedback in terms of the electric current and the separation was preset by a positional servo system. They measured interactive forces between glass filaments in KCl solutions and obtained satisfactory agreement with the DLVO theory. Also, Parsegian et al.9,10 measured forces between bilayers of dihexadecyldimethylammonium in acetate solutions using the osmotic stress method. Meanwhile, Ducker et al.11 measured the interparticle forces between spherical silica particles of 3.5 µm by setting them on a stage and fixing one to a tip of a cantilever in an atomic force microscope12 (AFM). In a similar manner, * To whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, October 1, 1997. (1) Derjaguin, B. V.; Landau, L. Acta Physichochim. URSS 1941, 14, 633. (2) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (3) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975. (4) Pashley, R. M.; Israelachvili, J. N. J. Colloid Interface Sci. 1984, 97, 446. (5) Shubin, V. E.; Kekicheff, P. J. J. Colloid Interface Sci. 1993, 155, 108. (6) Derjaguin, B. V.; Rabinovich, Ya. I.; Churaev, N. V. Nature 1977, 265, 520. (7) Derjaguin, B. V.; Rabinovich, Ya. I.; Churaev, N. V. Nature 1978, 272, 313. (8) Rabinovich, Ya. I.; Derjaguin, B. V.; Churaev, N. V. Adv. Colloid Interface Sci. 1982, 16, 63. (9) Parsegian, V. A.; Rand, R. P.; Rau, D. C. Methods Enzymol. 1986, 127, 400. (10) Parsegian, V. A.; Rand, R. P.; Fuller, N. L. J. Phys. Chem. 1991, 95, 4777. (11) Ducker, W. A.; Senden, T. I.; Pashley, R. M. Nature 1991, 353, 239. (12) Binnig, E. L.; Quate, C. F.; Gerber, C. Phys. Rev. Lett. 1986, 56, 930.
S0743-7463(97)00541-6 CCC: $14.00
Meagher13 measured the interactive forces between a silica particle and silica plate, and Li et al.14 measured the interparticle forces between polystyrene particles of 2 µm. They found an agreement with the trend of the DLVO formula at some concentrations of electrolyte and determined parameters such as the Debye lengths of the electric double layer and the charge densities of the particles on assumption of the DLVO theory by curve fitting. However, it does not necessarily mean there is agreement between the experiments and the theory, because the agreement is presumed beforehand in this procedure. In addition, such a way of measurement may not completely be free from some problems, such as the possibility of additional effects of long-range interactive forces from the cantilever, stage, and neighboring particles on the stage on the measurement of the interaction between one pair of particles, the influence of the indefinite position of a particle glued to the cantilever on the force evaluation, etc. Hence, if possible, direct measurement of the intrinsic interparticle forces with a statistically ample number of pairs of free colloidal particles may be more desirable. On the other hand, the optical trapping technique with a focused laser beam, developed by Ashkin et al.,15 has become an important tool for the precise manipulation of micrometer-size particles such as biological cells,16 polystyrene latex spheres,17 etc. Brown et al.18 measured the interactive forces between a polystyrene particle (∼10 µm) and a quartz glass wall by combining the total internal reflection microscopy with the optical trapping technique with a single laser beam. But, the absolute separation between them was not determined. Grier et al.19,20 studied the influence of a glass wall on the interaction between small polystyrene particles of 0.652 µm close to the wall using the double laser beam technique21 and found some long-distance attractive interaction between the particles 3-5 µm apart when they were close to the wall (∼2.5 µm), while there was only a repulsive force when they were away from the wall (∼9.5 µm). This interesting result (13) Meagher, L. J. Colloid Interface Sci. 1992, 152, 293. (14) Li, Y. Q.; Tao, N. J.; Pan, J.; Garcia, A. A.; Lindsay, S. M. Langmuir 1993, 9, 637. (15) Ashkin, A.; Dziedzic, J. M.; Bjorkholm, J. E.; Chu, S. Opt. Lett. 1986, 11, 288. (16) Ashkin, A.; Dziedzic, J. M.; Yamane, T. Nature 1987, 330, 769. (17) Sato, S.; Ohyumi, M.; Shibata, H.; Inaba, H.; Ogawa, Y. Opt. Lett. 1991, 16, 282. (18) Brown, M. A.; Smith, A. L.; Staples, E. J. Langmuir 1989, 5, 1319. (19) Croker, J. C.; Grier, D. G. Phys. Rev. Lett. 1994, 73, 352. (20) Larsen, A. E.; Grier, D. G. Nature 1997, 385, 230. (21) Simon, A.; Libchaber, A. Phys. Rev. Lett. 1992, 68, 3375.
© 1997 American Chemical Society
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may suggest the possibility of considerable influence of the cantilever, stage, and/or neighboring particles on the stage on the subtle measurement of the interactive forces of micrometer-size particles in an AFM. However, since they estimated the surface-to-surface separations of particles from optical microscopy on the indefinite fringeto-fringe distances and evaluated the interparticle forces from the average of the broadly scattered migration distances of each particle after cutting the trapping beam, this method is not suitable for the measurement of interparticle forces as a function of the separation less than a few microns, as dealt with by the DLVO theory. Moreover, since they analyzed their results by the conventional curve fitting to the DLVO formula, the experimental result as well as the DLVO theory may not be assessed from this procedure. As a consequence, one may find that the intrinsic interparticle forces at separations less than a few microns have never successfully been measured, and thus experimental verification of the DLVO theory for colloidal particles has not been achieved as yet. In the meantime, we have continued to develop the technology of the optical trapping with a single or double laser beam.17,22,23 The objective of this paper is to report the results of our study on the fascinating issue to determine the intrinsic interparticle forces as a function of the separation.
Figure 1. Distribution of electrophoretic mobility of the polystyrene particles in the inset synthesized for this study.
Particles Since uniform polystyrene latices commercially available were of very broad distributions in surface charge density and since well-defined particles were indispensable for our purpose, monodispersed polystyrene (PS) particles were synthesized by soap-free emulsion polymerization24 in a 300 cm3 aqueous emulsion containing 0.87 mol dm-3 styrene monomers, 1.0 × 10-3 mol dm-3 K2S2O8, and 1.0 × 10-2 mol dm-3 NaCl at 70 °C under a mild agitation with a Teflon impeller for 24 h. A part of the resulting particles was used as seeds to be grown furthermore under the same conditions but at 65 °C. The PS latex was repeatedly washed with doubly distilled water and completely deionized with cation- and anionexchange resins. The styrene monomers were purified by distillation twice before use. The mean diameter of the so obtained polystyrene particles was 2.13 µm with a standard deviation of 0.06 µm, and the standard deviation of the electrophoretic mobility of each particle from the average was 3.3%, as shown in Figure 1 (cf. 22% for a commercially available standard PS latex). The diameter of a particle was chosen to be sufficiently larger than the beam diameter at the focus of the objective of the microscope (∼1 µm). The Hamaker constant of the PS particles in the water-polystyrene system was 5.0 × 10-21 J, as determined from the critical coagulation concentration of the electrolyte NaClO4,25 which was reasonably in consistant with the data in the literature.26 Apparatus Figure 2 shows a block diagram of the used apparatus. The Nd:YAG laser beam of 30 mW in a doughnut mode (1.064 µm in wavelength) was split into two polarized beams by polarizing beam splitter A. One of them was reflected by rotatable mirror B, whose angle was regulated (22) Sato, S.; Harada, Y.; Waseda, Y.; Sugimoto, T. Rev. Laser Eng. 1996, 24, 1193. (23) Sato, S.; Inaba, H. Opt. Quantum Electron. 1996, 28, 1. (24) Ottewill, R. H.; Shaw, J. N. Kolloid Z. Z. Polylm. 1967, 215, 161. (25) Sihite, T.; Sasaki, H.; Muramatsu, A.; Usui, S. Shigen to Sozai 1991, 107, 215. (26) Visser, J. Adv. Colloid Interface Sci. 1972, 3, 331.
Figure 2. Block diagram of the apparatus.
with a micrometer, to control the interbeam distance, while the other was reflected by fixed mirror C and passed through the ND filter to control the beam power within the resolution of 10-13 N. The two beams finally reached the sample through a water-immersion objective in the microscope. An aperture ring was attached to the objective to adjust the numerical aperture to 1.09 in order to limit the convergence angle within 55.4° for completely separate irradiation of each particle even when the two particles were attached together. On the other hand, in order to set the focal positions of the two beams exactly at the same level, the angle of the fixed mirror C was previously adjusted using the knife-edge method.27 Measurements and Results The density of the aqueous media fixed at pH 7.5 containing different concentrations of NaClO4 was adjusted to that of the PS particles equal to 1.053 at 25 °C by mixing D2O with H2O to cancel the effect of gravity. In order to avoid the effects of the walls of the sample cell and the other neighboring particles, the focal position to the objective was set sufficiently away from the cell bottom (>> 10 µm) and the concentration of the PS particles was (27) Arnaud, J. A.; Hubbard, W. M.; Mandeville, G. D.; de la Claviere, B.; Franke, E. A.; Franke, J. M. Appl. Opt. 1971, 10, 2775.
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Figure 4. Log plot of the experimental repulsive components extracted from the data of the total forces in Figure 3, in comparison with the solid lines for the theoretical repulsive forces (FR). Figure 3. Interparticle forces as a function of the surfaceto-surface separation between two polystyrene particles at different electrolyte concentrations: (a) 1.15 × 10-4, (b) 1.00 × 10-3, and (c) 5.00 × 10-3 mol dm-3. Experimental results are shown by symbols (0, 4, O), while the solid curves are calculated ones from the DLVO theory.
limited to 50 mg dm-3 (average interparticle separation = 55 µm). Also, the sample dispersion was previously purged of CO2 by N2 gas. Though the weak laser beams caused no thermal convection, an infrared filter was used to prevent a likely effect of the lighting to the sample. The relationship between the trapping force and the beam power was predetermined at 25 ( 1 °C by the aid of the Stokes equation from the critical beam power at the moment of the start of the displacement of a particle trapped by a gradually attenuating beam against the frictional force of the used mixed medium in horizontal motion with a sample cell on a moving stage driven by a linear motor at different constant rates.17 The interparticle force was determined at 25 ( 1 °C by detecting the critical beam power at the moment of the displacement of a particle trapped by an attenuating beam, whereas the other particle of the pair was fixed by another beam at the full power. This measurement was repeated with an identical particle pair at different interparticle distances without releasing a counterpart. The surfaceto-surface separation between the two particles of a pair was accurately measured within the resolution of 1 nm from the difference between the central distances calibrated from the VTR images of the two particles at the trapped positions and in the state attached together by addition of a concentrated electrolyte after one series of measurement at different separations for this pair. By this procedure, the errors from the direct measurement of the fringe-to-fringe distance with the blurred optical images of the particle fringes could be avoided. The measurement was repeated for ten pairs at each electrolyte concentration in order to minimize the possible statistical errors. When it was difficult to detect the critical trapping force at each interparticle separation, as found in the region of a sharp increase of the repulsive force at the electrolyte concentrations of 1.00 × 10-3 and 5.00 × 10-3 mol dm-3, the interparticle separation was gradually changed at each constant beam power for three pairs. The results for electrolyte concentrations [(a) 1.15 × 10-4, (b) 1.00 × 10-3, and (c) 5.00 × 10-3 mol dm-3] are shown in Figure 3, where the experimental results are indicated by symbols (0, 4, O), while the solid curves are
those calculated from the DLVO theory for a pair of colloidal spheres28 with the Hamaker constant A ) 5.0 × 10-21 J and ζ potentials used for the Stern potentials (ψδ),29 as obtained from electrophoresis of the PS particles in the dispersions at 25 ( 1 °C with the Henry formula:30 (a) -35.5, (b) -28.5, (c) -22.0 mV at the respective electrolyte concentrations. For the calculation of the theoretical curves, the following equations of force as derivatives of the corresponding energies were used:
FR )
( )
zFψδ 64πamRT exp(-κh) tanh2 κ 4RT FA )
-32Aa6 3h (2a + h)3 (4a + h)2 2
where FR and FA are the repulsive force for κa >> 1 and the attractive force, respectively (total force F ) FR + FA), a is the particle radius, h is the separation, m is the molarity of the symmetric electrolyte of z-z type, z ) 1, F is the Faraday constant, and κ is the Debye-Hu¨ckel parameter. Obviously, the experimental results including the weak attraction range are in excellent agreement with the theory, substantiating the latter. Figure 4 shows the log plots of the repulsive components (FR) against separation, where the symbols indicate the experimental repulsive forces obtained by subtracting the theoretical FA from the data of the total forces in Figure 3 and the solid lines are the theoretical ones. From the slopes of the experimental linear lines determined by the method of least squares, one obtains the Debye lengths, 1/κ, as (a) 26.2 (28.4) nm, (b) 9.5 (9.62) nm, and (c) 4.6 (4.30) nm, where the values in the parentheses are theoretical ones. Lastly, if the solid density of a particle is too high or too low to be adjusted to the medium density, the powers of the two beams should be changed equally in order to keep the vertical positions of the two particles at the same level. Also, when we measure much smaller interparticle separations in the range of much higher ionic strengths, some automatic recording system for the change of interparticle distance may be helpful for determining the precise separations at critical trapping forces. LA970541A (28) Overbeek, J. T. G. Adv. Colloid Interface Sci. 1982, 16, 17. (29) Hunter, R. J. Zeta Potential in Colloid Science; Principles and Applications; Academic Press: London, 1981; p 241. (30) Henry, D. C. Proc. R. Soc. London 1931, 133, 106.
