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Optical Trapping Forces for Colloids at the Oil-Water Interface Bum Jun Park and Eric M. Furst* Department of Chemical Engineering and Center for Molecular and Engineering Thermodynamics UniVersity of Delaware, Newark, Delaware 19716 ReceiVed August 7, 2008. ReVised Manuscript ReceiVed September 18, 2008 We calculate the optical trapping forces exerted by a single laser beam strongly focused on a dielectric sphere located at a two-dimensional (2D) oil-water interface. The calculated lateral trapping forces, based on the geometrical optics approximation (GOA), agree with experimental measurements of the trapping force. Importantly, the calculations verify that the radiation force exerted on particles perpendicular to the interface is not sufficient to induce capillary interactions between particle pairs, which could be mistaken for particle-particle interactions. Finally, we find that the trapping forces depend on the three-phase contact angle of the particle at the interface.
Introduction Arthur Ashkin observed that suspended colloidal particles can be moved by the radiation pressure of a laser.1 This discovery led to the development of the optical trap, which employs a strongly focused laser beam to hold a particle in a stable position.2 Since then, the single-beam gradient force trap, or optical tweezer, has become a powerful tool for measuring forces that arise in soft matter systems. Such systems encompass diverse areas, such as biology,3-6 colloid physics,7-14 and polymers and biopolymers.15-17 The interactions of particles confined to the interface of two immiscible phases18 has been of recent interest. Since Pickering initially reported the spontaneous adsorption and pinning of colloidal particles at fluid interfaces, which can subsequently be used as stabilizers,19 numerous investigators have studied colloidal interactions, microstructure, and micromechanics of suspensions at 2D interfaces.20-25 Particles are known to attach to the interface * Corresponding author. E-mail:
[email protected]. (1) Ashkin, A. Phys. ReV. Lett. 1970, 24, 156–159. (2) Ashkin, A.; Dziedzic, J. M.; Bjorkholm, J. E.; Chu, S. Opt. Lett. 1986, 11, 288–290. (3) Ashkin, A.; Dziedzic, J. M. Science 1987, 235, 1517–1520. (4) Ashkin, A.; Dziedzic, J. M. Proc. Natl. Acad. Sci. U.S.A. 1989, 86, 7914– 7918. (5) Kuo, S. C.; Sheetz, M. P. Trends Cell Biol. 1992, 2, 116–118. (6) Svoboda, K.; Block, S. M. Annu. ReV. Biophys. 1994, 23, 247–285. (7) Crocker, J. C.; Grier, D. G. Phys. ReV. Lett. 1996, 77, 1897–1900. (8) Lowen, H. J. Phys.: Condens. Matter 2001, 13, R415-R432. (9) Liu, B.; Goree, J.; Nosenko, V.; Boufendi, L. Phys. Plasmas 2003, 10, 9–20. (10) Pantina, J. P.; Furst, E. M. Langmuir 2004, 20, 3940–3946. (11) Pantina, J. P.; Furst, E. M. Phys. ReV. Lett. 2005, 94, 138301–4. (12) Pantina, J. P.; Furst, E. M. Langmuir 2006, 22, 5282–5288. (13) Pantina, J. P.; Furst, E. M. Langmuir 2008, 24, 1141–1146. (14) Dufresne, E. R.; Spalding, G. C.; Dearing, M. T.; Sheets, S. A.; Grier, D. G. ReV. Sci. Instrum. 2001, 72, 1810–1816. (15) Riveline, D.; Wiggins, C. H.; Goldstein, R. E.; Ott, A. Phys. ReV. E 1997, 56, R1330-R1333. (16) Wang, M. D.; Yin, H.; Landick, R.; Gelles, J.; Block, S. M. Biophys. J. 1997, 72, 1335–1346. (17) Bustamante, C.; Bryant, Z.; Smith, S. B. Nature 2003, 421, 423–427. (18) Binks, B. P.; Horozov, T. S. Colloidal particles at liquid interfaces; Cambridge University Press: New York, 2006. (19) Pickering, S. U. Emulsions 1907, 91, 2001–2021. (20) Aveyard, R.; Clint, J. H.; Nees, D.; Paunov, V. N. Langmuir 2000, 16, 1969–1979. (21) Aveyard, R.; Binks, B. P.; Clint, J. H.; Fletcher, P. D. I.; Horozov, T. S.; Neumann, B.; Paunov, V. N.; Annesley, J.; Botchway, S. W.; Nees, D.; Parker, A. W.; Ward, A. D.; Burgess, A. N. Phys. ReV. Lett. 2002, 88, 246102–4. (22) Park, B. J.; Pantina, J. P.; Furst, E. M.; Oettel, M.; Reynaert, S.; Vermant, J. Langmuir 2008, 24, 1686–1694. (23) Reynaert, S.; Moldenaers, P.; Vermant, J. Langmuir 2006, 22, 4936– 4945. (24) Stancik, E. J.; Fuller, G. G. Langmuir 2004, 20, 4805–4808.
strongly.26 The energy required to remove a particle at the interface with radius a on the order of micrometers, Eatt depends on the three-phase contact angle θ and the interfacial tension γ, Eatt ) πa2γ(1 ( cos θ)2. This energy is typically many orders of magnitude greater than thermal energy kBT. Recently, optical traps have been used to measure the interactions of colloids at the air-water27 and oil-water interfaces.20–22 These direct measurements of the interaction forces between particle pairs exhibited long-range repulsion, which is consistent with reported theoretical models of repulsive charge interactions.28-30 It has also been observed that longrange attractive interactions exist, which are possibly due to the capillary interactions caused by the interface deformation.27,31-39 Overall, force measurements are important for understanding the time-dependent and steady-state structures that colloidal particles form at interfaces;22,39 this, in turn, enables the rational design of material parameters, such as the interfacial rheology and stability. Ultimately, the resulting control of colloidal interactions provides opportunities to design new materials and material properties, such as colloidosomes40 or the enhanced stabilization of immiscible fluids.41,42 Two primary concerns arise when trapping particles at a fluid interface compared to trapping in a single phase: First, the (25) Hoekstra, H.; Vermant, J.; Mewis, J.; Fuller, G. G. Langmuir 2003, 19, 9134–9141. (26) Binks, B. P. Curr. Opin. Colloid Interface Sci. 2002, 7, 21–41. (27) Dasgupta, A. R.; Ahlawat, A. S.; Gupta, A. P. K. J. Opt. A: Pure Appl. Opt. 2007, 9, S189-S195. (28) Pieranski, P. Phys. ReV. Lett. 1980, 45, 569–572. (29) Hurd, A. J. J. Phys. A: Math. Gen. 1985, 45, L1055-L1060. (30) Frydel, D.; Dietrich, S.; Oettel, M. Phys. ReV. Lett. 2007, 99, 118302–4. (31) Vassileva, N. D.; vandenEnde, D.; Mugele, F.; Mellema, J. Langmuir 2005, 21, 11190–11200. (32) Golestanian, R.; Goulian, M.; Kardar, M. Phys. ReV. E 1996, 54, 6725– 6734. (33) Wurger, A. Phys. ReV. E 2006, 74, 041402–9. (34) Kralchevsky, P. A.; Nagayama, K. Langmuir 1994, 10, 23–36. (35) Nikolaides, M. G.; Bausch, A. R.; Hsu, M. F.; Dinsmore, A. D.; Brenner, M. P.; Gay, C.; Weitz, D. A. Nature 2002, 420, 299–301. (36) Stamou, D.; Duschl, C.; Johannsmann, D. Phys. ReV. E 2000, 62, 5263– 5272. (37) Oettel, M.; Dominguez, A.; Dietrich, S. Phys. ReV. E 2005, 71, 051401– 16. (38) Dominguez, A.; Oettel, M.; Dietrich, S. J. Phys.: Condens. Matter 2005, 17, S3387-S3392. (39) Park, B. J.; Furst, E. M.; Masschaele, K.; Vermant, J.,in preparation. (40) Dinsmore, A. D.; Hsu, M. F.; Nikolaides, M. G.; Marquez, M.; Bausch, A. R.; Weitz, D. A. Science 2002, 298, 1006–1009. (41) Herzig, E. M.; White, K. A.; Schofield, A. B.; Poon, W. C. K.; Clegg, P. S. Nat. Mater. 2007, 6, 966–971. (42) Clegg, P. S. J. Phys.: Condens. Matter 2008, 20, 113101.
10.1021/la802575k CCC: $40.75 2008 American Chemical Society Published on Web 11/04/2008
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magnitude of the trapping forces must be accurately calibrated in order to measure colloidal interactions at the interface. However, the force depends on the two-phase contact line, which determines whether particles sit predominantly in one of the two phases.22 Second, the radiation pressure exerted on particles could deform the local interface, causing an apparent attraction between particles due to capillary forces.27,31–39 To date, trapping forces at liquid-liquid interfaces have not been examined in detail, despite these potential issues. In this paper, we present analytical calculations of optical trapping forces for colloidal particles at the liquid-liquid interface of two immiscible fluids with different indices of refraction. Our aim is to accurately describe the trapping forces in order to facilitate accurate calibration of the optical traps, and verify that trapping does not give rise to artifacts in colloidal interaction measurements. In order to do this, first, we review the geometric optics approximation used in these calculations. Second, we integrate over all incident rays to calculate the trapping forces and the radiation forces exerted on the sphere confined at the oil-water interface while considering the three-phase contact angle, and then introduce our experimental methods for measuring trapping forces at the liquid-liquid interface. Third, the calculated trapping forces are compared to the experimental results. This work will provide a reference for a comparison with experimental results at 2D interfaces which might include the presence of the additional interaction, such as the capillary force induced by the interface deformation. Trapping Forces for a Single Ray in a Single Phase. In the ray optic regime, where a diameter of an object is much larger than the wavelength of a light (2a . λ0), reflections and refractions of the light induce a momentum change.43,45 The momentum change generates an optical force that moves the object into the laser focus. The resulting force is decomposed into two components, a scattering force in the direction of the light propagation and a gradient force in the direction of the spatial light gradient. In order to trap the object successfully, the gradient force must be larger than the scattering force. To calculate these optical trapping forces analytically, the geometrical optics approximation (GOA) is commonly used as a simple formalism in the ray optic regime. However, the criteria for applying GOA are still ambiguous and require some clarification.43,46-48 Simmons et al.44 measured the trapping force of particles with increasing diameter; in the Rayleigh regime (2a , λ0), the force increases with size, while it becomes independent of particle diameter as the ray optic regime is approached. This suggests that the converging rays partially strike the particle in the former regime and fully strike the particle in the latter regime. Such an observation is consistent with the Ashkin’s calculation43 in which the force scales as (2a)3 in the Rayleigh regime and does not change with the size in the ray optic regime. Although there are no quantitative theories for the intermediate regime (2a ≈ λ0), we realize that the criterion for GOA depends on the relation between the diameter of the particle and the waist size of the beam ω0. Considering the lateral displacement ∆x of the particle, the ray optic regime would be retained if all incident rays hit the particle before escaping from the optical trap. For instance, assuming 2a ) 3 µm, λ0 ) 1064 nm, and 2ω0 ) 0.4 µm, the escaping displacement ∆x* is approximately the halfradius of the particle44 which is ∼0.75 µm. Thus, it is possible
Park and Furst
Figure 1. Geometry of the pathway of the reflected and refracted rays after a single ray with power P hits a dielectric sphere in 3D space. The incident ray is split into a reflected ray of power PR and refracted rays of power PT2Rn.
to apply GOA in this example regardless of ∆x, assuming the amount of the rays corresponding to 133°). The interface is located at a distance -a cos θ from the center of the sphere. The distance l between B′ and an arbitrary point along the contact line with the interface is given by l ) ∆x cos β′ ( [(a sin θ)2 - (∆x sin β′)2]1/2 at ∆x > a sin θ and l ) ∆x cos β′ + [(a sin θ)2 - (∆x sin β′)2]1/2 at ∆x e a sin θ where a sin θ is always larger than ∆x sin β′. Thus, the boundary conditions for the third regime become 0 < β′ < π, and R1′ < R′ < π/2 for incident rays from the water phase and Rmin′′ < R′ < R1′ for incident rays from the oil phase. The
Figure 4. (a) Geometry for a strongly focused laser beam. (b) Geometry of the vertical plane (ABCD). (c) Geometry of the incident plane (OBE).
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Figure 5. Views of the bottom (a) and side (b) projections when a single laser beam traps a particle at an the oil-water interface. The particle sits in the oil phase due to the high contact angle (θ g 133°).
value of Rmin′′ is the minimum polar angle for refraction at the oil-water interface and is determined by the refractive indices of the water and oil as well as the objective numerical aperture, Rmin′′ ) (π/2) - sin-1[(n1/n3) sin φmax]. The critical angle that distinguishes incident rays from the water and oil phase R1′ is given by R1′ ) (π/2) - tan-1{-[l/(a cos θ)]}. Using these boundary conditions, the integrals for the Y- and Z-direction forces are,
FY ) 2
∫0π ∫RR
1
min
I(r′, z)(Qg,oil sin γ′ -
n1 b ×b Qs,oil cos γ′) |T T β| dR′ dβ′ + c R 2
∫0π ∫Rπ⁄2 I(r′, z)(Qg,water sin γ′ 1
n1 b ×b Qs,water cos γ′) |T T β| dR′ dβ′ c R FZ ) 2
∫0π ∫RR
1
min
I(r′, z)(Qs,oil sin γ′ -
n1 b ×b Qg,oil cos γ′) |T T β| dR′ dβ′ + c R 2
∫0π ∫Rπ⁄2 I(r′, z)(Qs,water sin γ′ 1
n1 b ×b Qg,water cos γ′) |T T β| dR′ dβ′ (7) c R where FY and FZ indicate the trapping force and the radiation force, parallel and perpendicular to the interface, respectively. However, it is difficult to integrate these equations directly since the boundary conditions include the variable R1′. Instead, we use the change of coordinates β′ ) u and R′ ) (1 - V)g1(u) + Vg2(u), where g1(u) ) Rmin′′ and g2(u) ) R1′ for the first integration of FY and FZ, and g1(u) ) R1′ and g2(u) ) π/2 for the second integration. The new integration limits are then 0 < β′ < π and 0 < R′ < 1. Also, (g2(u) - g1(u)), which represents the Jacobian ∂(β′, R′)/∂(u, V), is multiplied to the right-hand-side of eq 7. The general equation for this is given by,
∫0π ∫gg f(R′, β′) dR′ dβ′ ) ∫0π ∫01 f((1 - V)g1(u) + 2
1
Vg2(u), u)(g2(u) - g1(u)) dV du (8) Finally, for a circularly polarized beam, the trapping forces are averaged over the forces, generated from two orthogonally polarized beams.43,48 This concludes our discussion of the methods used to calculate the trapping and scattering forces generated by a focused laser beam on a particle at the oil-water interface. Before reporting
Figure 6. Dimensionless efficiencies Qs (open squares) and -Qg (open circles) of the scattering force and the gradient force calculated by eq 9 in the aqueous phase. Solid lines indicate the values of Q obtained from eq 3.
the calculations, we first review the experimental materials and methods used to compare our results.
Experimental Section Materials and Methods. Materials. The optical trapping forces are measured at an oil-water interface to compare with the calculated forces. Ultrapurified water (>18.2 M Ω · cm) is used to form a subphase, while the superphase is n-decane (Acros Organics, 99+%) which has been passed through aluminum oxide column (Acros Chemical, acidic activated, particle size 100-500 µm) to remove polar components. We use surfactant-free, charge-stabilized polystyrene (PS) particles with a surface charge 9.1 µC/cm2 and diameter 2a ) 3.1 ( 0.2 µm (Interfacial Dynamics Corporation). The particle solution is diluted to 2 × 10- 2 w/w%, and 5 µL of six parts diluted particle solution and four parts isopropyl alcohol (Sigma Aldrich) are dispersed as a monolayer at the oil-water interface.22,23 A specially designed fluid cell22 consisting of a glass outer cylinder and an inner cylinder made of aluminum is placed on the microscope stage. The outer cylinder is attached to a 40 mm circular no. 1.5 coverglass using a fast curing UV epoxy (Norland Products, NOA 81). A Teflon ring is inserted into the bottom of the inner cylinder in order to pin the contact line of an oil-water interface. The height of the inner cylinder is controlled by a micrometer to achieve a hydrostatic equilibrium and a flat interface. All glassware is cleaned using a Plasma cleaner (Harrick Plasma, PDC 32-G) immediately before constructing the cell to achieve good wetting conditions for the water. The laser tweezer apparatus is constructed around an inverted microscope (Zeiss Axiovert 200).10 The laser is a 4-W CW Nd: YAG laser (λ0 ) 1064 nm, Coherent Compass 1064-400M). The laser beam passes through a pair of perpendicular acousto-optic deflectors (AODs; AA Opto-electronics AA.DTS.XY-400). Using the AOD, the beam angle incident in the back aperture of the objective
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Figure 7. Dimensionless efficiencies Qs and -Qg of the scattering force and the gradient force. (a) Solid and dashed lines are Qs and -Qg calculated from eq 9 in the aqueous phase, respectively. Open and filled symbols show Qs and -Qg at the oil-water interface in the orders of the reflection at the particle-water interface and the next sequential refractions at the particle-oil or particle-water interfaces (i.e., WOWO, WOWW, WOOO, and WOOW). (b) Long-dashed and solid lines, and short-dashed and dotted lines indicate Qs and -Qg for WOWO and OOOO, respectively. Open and filled symbols are Qs and -Qg when the incident ray (0 < θ3 < 70°) is from the oil phase (i.e., OOWO, OOWW, and OOOW).
measured previously by both pendant drop22–24 and gel trapping22,54 with a small amount of sodium dodecylsulphate (SDS, Sigma Aldrich 98%) in the water phase. The optical trap rigidity κt is then given by the slop in a plot of FS versus ∆x.
Results and Discussion Trapping Forces for a Single Ray in a Single Phase. We first verify our calculations by computing the dimensionless efficiencies for a particle immersed in a single phase (water). Figure 6 shows a comparison of the dimensionless trap force Q calculated from the full equation (eq 3) and eq 2 at n ) 2, given by, Figure 8. Maximum incident angle as a function of the displacement ∆x of a particle with a ) 1.55 µm.
is controlled by a Labview computer program. This enables optical trapping over an area of approximately 100 × 100 µm2 in the focal plane of the microscope and gives us the ability to generate multiple time-shared traps. A water immersion objective (63× NA 1.2, Zeiss C-Apochromat) with a working distance of approximately 200-300 µm is used. A second objective (20×, Zeiss Achroplan) with a low numerical aperture (NA 0.45) is used to calibrate the laser power. Calibration of Laser Powers. The actual laser power delivered to the sphere is determined by the calibration of two objectives with low and high NA,45,50 Phigh ) Plow[Thigh(λ0)]/[Tlow(λ0)]. Plow is directly measured by the optical power meter (Newport, 1815-c) in front of the low-NA objective, and Thigh(λ0) and Tlow(λ0) are the transmittances of the low- and high-NA objectives which are measured by a dualobjective technique (i.e., T63× = 30%, and T20× = 80% for λ0 ) 1064 nm).51,52 Drag Calibration of Trapping Forces. To calibrate the optical trap strength, a particle at the oil-water interface is held and subjected to drag forces by translating the microscope stage at several constant velocities between U ) 7 and 30 µm/s.10,22 The particle is displaced ∆x from its equilibrium position in the optical trap in response to the Stokes drag force, FS ) 6πaηeffU, where the effective viscosity is approximated as an average based on the exposed surface area of the particle to each liquid phase,22,53 ηeff ) [ηoil(1 - cos θ) + ηwater(1 + cos θ)]/2. The three-phase contact angle θ has been (49) Optics Guide 4; Melles Griot: Irvine, CA, 1988. (50) Neuman, K. C.; Chadd, E. H.; Liou, G. F.; Bergman, K.; Block, S. M. Biophys. J. 1999, 77, 2856–2863. (51) Misawa, H.; Koshioka, M.; Sasaki, K.; Kitamura, N.; Masuhara, H. J. Appl. Phys. 1991, 70, 3829–3836. (52) Viana, N. B.; Rocha, M. S.; Mesquita, O. N.; Mazolli, A.; Maia Neto, P. A. Appl. Opt. 2006, 45, 4263–4269. (53) Danov, K.; Aust, R.; Durst, F.; Lange, U. J. Colloid Interface Sci. 1995, 175, 36–45.
Qs ) 1 - R cos(π + 2θ1) + T2[cos R + R cos(R + β) + R2 cos(R + 2β)] (9) Qg ) -R sin(π + 2θ1) + T2[sin R + R sin(R + β) + R2 sin(R + 2β)] The values of Qs and Qg obtained using eq 9 show a good agreement with the full equation (which is identical to the Ashkin’s results43), indicating that the contributions of the first reflection and the next three refraction terms are sufficient to determine the trapping force for a single ray. Thus, only four terms will be used for the calculation at 2D interfaces (see the bottom tables in Figures 2 and 3). Trapping Forces for a Single Ray at the Oil-Water Interface. Figure 7a shows the dimensionless efficiency Q calculated for a single ray in the geometry a in which the incident ray enters from the water phase (see Figure 2). Solid and dashed lines indicate the dimensionless efficiency Qs of the scattering force and -Qg of the gradient force in the aqueous phase, and the open and filled symbols show Qs and -Qg at the oil-water interface, when the first reflection to the medium or the first refraction through the particle occurs at the particle-water interface (W), and the second reflection inside the particle or the second refraction at the particle-oil interface (O). The subsequential rays are reflected at the particle-oil (O) or particle-water interfaces (W) (i.e., WOWO, WOWW, WOOO, and WOOW). Regardless of the refraction order after the second refraction, Qs and -Qg do not change. Also, Qs decreases more than -Qg, compared to particles in a single aqueous phase. That is, -Qg at the interface is always larger than Qs regardless of the incident angle. Even though it is difficult to quantify the magnitude of the total Z-direction forces (which must be integrated for all (54) Paunov, V. N. Langmuir 2003, 19, 7970–7976.
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Figure 9. Comparison of the calculated lateral forces using eqs 7 and 8, and the drag experiments for different laser powers in both the water phase (a) and the oil-water interface (b) containing 0.1 mM SDS in water (θ ) 140°).
incident rays) using this calculation for a single ray, the radiation force perpendicular to the interface is intuitively less than what would be expected for a particle in the water phase. While this would normally lead to enhanced stability in the Z-axis, the implication here is that the radiation pressure is weaker, reducing the possibility of the optical traps causing a deformation of the interface. This will be discussed in detail below. Figure 7b shows the comparison of Q as a function of the incident angle calculated for a single ray for the geometries a and b, in which the latter is the case that the transmitted ray from the oil-water interface strikes the particle at the particle-oil interface. This will occur when θ > 90° (see Figure 3). At the particle-oil interface (O), the refracted ray is separated into the refracted and reflected rays at the particle-oil interface (O). The next successive rays are again refracted and reflected at particle-oil (O) or particle-water (W) interfaces. In this figure, open symbols and filled symbols indicating Qs and -Qg for several possible pathways (i.e., OOWO, OOWW, and OOOW) in the geometry are identical to each other in the range of θ < 70°, suggesting that they are independent of the order of the refractions after the second refraction when the incident angle θ3 is smaller than ∼70°. Above this angle, the refractive angle outside of the particle is larger than 90°, causing a total internal reflection inside the particle. The short-dashed and dotted lines show the Qs and -Qg in the case of the order of OOOO which is same to the immersed particle in the oil phase. Similarly, below 70° of the incident angle, -Qg is always larger than Qs. The values of Qs and -Qg in geometry b are smaller than those calculated for geometry a (long-dashed and solid lines, respectively). These differences will be considered below for the integration over all rays to calculate the total trapping force. Comparison of Analytical Calculations with Experimental Lateral Trapping Forces. We now integrate over all incident rays to calculate the total trapping force at the interface. As discussed earlier, typical values of the three-phase contact angle determine the limits of the integration over all incident rays to the third regime, a sin θ e L - ∆x. In Figure 7, we show the trapping forces at the interface do not change with the reflection order inside the particle and only depend on the originating medium of the incident ray (i.e., water or oil). Thus, in order to calculate the trapping forces in the third regime, each arbitrary equation in bottom tables in Figures 2 and 3 is chosen for the cases of the water- and oil-incidence, respectively. Moreover, it is necessary to verify whether total internal reflection occurs inside the particle, which happens for θ3 g 70° in geometry b. As shown in Figure 8, the maximum incident angle is calculated as a function of the displacement ∆x. Considering the particle radius of 1.55 µm, it is about 65° at ∆x ) 1.4 µm. This suggests
that total reflection does not occur over the majority of displacements. The calculated lateral trapping forces -FY using eqs 7 and 8 are shown in Figure 9. The calculated forces exhibit good agreement with the experimental drag calibration results for different laser powers in both the water phase (Figure 9a), and the oil-water interface containing 0.1 mM SDS in water (θ ) 140°, Figure 9b). The slightly larger deviation between the calculated and experimental forces in the water phase compared to those for particles at the oil-water interface may be due to out-of-plane displacement of the particle during dragging.43 The pinning of the particle at the interface reduces the displacement of the particle in the Z-direction due to the high energetic penalty of deforming the interface. The degree of agreement between the calculations and experimental results could be considered surprising, since the analytical calculation is based on the geometric optics approximation in the ray optic regime, which assumes the particle diameter is much larger than the wavelength of the light, 2πa/λ0 g 100.48 Nonetheless, the experimental conditions, in which a ) 1.55 µm and λ0 is 1064 nm, is obviously intermediate between the ray optic and Rayleigh regimes. Radiation Forces. Figure 10a and b show calculations of the radiation force acting in the Z-direction in the pure aqueous phase and at the interface (θ ) 140°) as the laser power is varied. When the laser focus is on the center of the particle (the lateral trapping force in this case is zero due to the symmetry), the radiation forces increase linearly with the laser power, and the force for a particle completely immersed in the aqueous phase is larger than that for a particle at the interface (Figure 10c) as predicted in earlier for single ray calculations. In typical drag experiments at the pure oil-water interface (θ ) 133°), P is 13 mW and the trap stiffness κt is about 1 pN/µm (6 time-shared traps). The calculated radiation force at P ) 13 mW is (0.78/6) pN ) 0.13 pN, which corresponds to the same order of magnitude of the thermal energy for a particle held by the trap with κt ) (1/6) pN/µm at the interface. Therefore, this force is too small to change the three-phase contact angle θ of a particle pinned at the interface; that is, the amount of energy 1 × 106 kBT is needed to change θ from 133° (virgin system) to 134°, and to deform the interface. This result is significant, because a strong force that pushes the particle perpendicular to the interface may cause the interface to deform, which would induce an unexpected force (i.e., capillary force27,31–38 in the pair interactions22,39). More quantitatively, the capillary force FCap caused by the interface deformation due to the vertical force Fv perpendicular to the interface is given by FCap ∼ -FRεF(a , r), where FR is the repulsive force between two particles with a separation r, and
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Figure 10. Radiation forces (Z-direction) as a function of the displacement ∆x in the aqueous phase (a) and at the interfaces (θ ) 140°) (b) with varying the laser power. (c) Radiation forces as a function of the laser power at ∆x ) 0 in the aqueous phase (solid line) and at the interfaces (dashed line), respectively.
Figure 11. Dependence of the trapping force on the three-phase contact angle as a function of the displacement ∆x at P ) 10mW. (a) Lateral trapping force, -FY. (b) Radiation force, FZ.
εF ) -Fv/(2πγOWa sin θ) is the ratio of the interfacial tension γOW and Fv acting on the particle by other sources (i.e., gravitational, electrostatic, hydrostatic, etc.).37,38 Thus, the capillary force becomes negligible in the range of |εF| , 1. For instance, when a particle with a ) 1.55 µm at a virgin oil-water interface (θ ) 133° and γOW = 50 mN/m) is trapped, the vertical force Fv caused by the radiation force FZ is about 0.1∼0.2 pN at 0 < ∆x < 1 µm, and its corresponding value of |εF| is 2.8 × 10-7-5.6 × 10-7, satisfying the above condition (|εF| , 1). This confirms that the radiation force in a typical tweezer experiment is not sufficient to create significant capillary interactions between particles; it is negligible compared to the repulsive force between particles. Dependence of Trapping Forces on the Three-Phase Contact Angle. The wetting properties of particles at the oil-water interface depend on the amount of a surfactant in each phase. For example, the three-phase contact angle changes from 133° to 164° for SDS concentrations ranging from 0 to 1 mM in the aqueous subphase.22 Such changes in the contact angle will lead to change the boundary conditions of the integrals in
eqs 7 and 8, and the ultimate trapping forces. In Figure 11, the trapping forces are calculated while varying the contact angle. The values of θ ) 0° and 180° mean that a particle is immersed in the water and oil phase, respectively. At θ ) 90°, all rays enter from the water phase, and for θ > 90°, some incident rays are from the water phase and others from the oil phase. For θ ) 110°, ∆x is only considered in the range between 0 to 1 µm, since the second regime in which 0 e L - ∆x e a sin θ becomes considerable for ∆x > 1 µm. As shown in Figure 11a, the lateral trapping forces for the particle immersed in the water and oil phases are the largest and smallest, respectively. This means that the lateral force increases as the portion of the incident rays from the water phase increases. Similar results are obtained from the calculation of radiation forces shown in Figure 11b.
Conclusions In this work, we calculated the trapping forces on a dielectric particle for a single laser beam at the oil-water interface based
13392 Langmuir, Vol. 24, No. 23, 2008
on Ashkin’s calculation in 3D space.43 We summarize the six key results of this work: First, we found that the contribution of the first reflected ray and three successive refracted rays at particlemedia interfaces are sufficient to accurately determine the trapping force for a single ray. Second, it was shown that the trapping force for a single ray at the oil-water interface decreases compared to that in the aqueous phase due to the difference of the refractive index of the second medium. Moreover, the most important factor for determining the trapping force for a single ray at the interface is the phase the incident ray strikes the particle. It is interesting to note that the trapping force is independent from the order of the second and third refractions to the media. Third, the calculated scattering force for a single ray at the interface decreases significantly compared to the gradient force, unlike the ratio of these forces for a particle immersed in a single (water) phase. This provides the insight that the axial radiation pressure
Park and Furst
at the oil-water interface is reduced, and this further decreases the possibility that the radiation pressure in the Z-direction will lead to induced capillary interactions between particles. Fourth, we found that the lateral trapping forces integrated over all rays are consistent with the experimental results, even though the system used in the calculations and experiments is intermediate to the ray optic and Rayleigh regimes. Fifth, we confirmed that the radiation force is too small to cause the significant capillary interactions. Finally, the calculated lateral trapping force and radiation force depend on the three-phase contact angle, and change in the range from the maxima in the aqueous phase to the minima in the oil phase. Acknowledgment. We gratefully acknowledge financial support from the National Science Foundation (NSF CBET-0553656). LA802575K
