Langmuir 1992,8, 3073-3082
3073
Prediction and Measurement of the Optical Trapping Forces on a Microscopic Dielectric Sphere John Y. Walz' and Dennis C. Prieve Department of Chemical Engineering] Carnegie Mellon University, Pittsburgh] Pennsylvania 15213 Received April 30,1992. I n Finial Form: July 7,1992 A geometric optics model for predicting the axial and radial componente of the radiation force acting on a dielectric sphere is presented. The geometric optics approach is known to be valid for spheres with diameters much larger than the wavelength of incident light. An experimental method of measuring the axial radiation force component using the recent technique of total internal reflectionmicroscopy (TIRM) is then described. The TIRM technique was used to measure the net weight of a single sphere (e.g., 10 pm polystyrene latex) caught in a stable optical trap. The trap was formed by focusing relatively weak beams (typically less than 2 mW output power) from an argon ion laser to a spot size of approximately 11 pm. Forces as small as 5 X lo-*' N were detected. Finally, an experimental procedure for measuring the radial intensity profiie of a laser beam is presented. The technique involves measuring the twodimensional spatial intensity profile of an evanescent wave formed under total internal reflection of the beam.
Introduction
rays can be treated as parallel. This better simulates the conditions under which radiation pressure is used in our The optical trapping of small particles using intense experiments where the upward axial force must remain light beams has been demonstrated on a wide range of constant over a specified propagation distance, requiring systems. In 1970,Ashkin' reported the first successful a beam of low divergence. trapping of micrometer-sizeparticles with radiation forces We then present an experimental method of measuring produced by visible laser light. Polystyrene spheres up to the axial componentof radiation pressure using the recent 2.68 pm diameter suspended in water were trapped at the technique of totalinternal reflection microscopy (TIRM). intersection of two tightly-focused beams from an argon Introduced by Prieve et a1.F TIRM utilizes the scattering ion laser. In 1971,Ashkin and Dziedzic? were ableto leviate of an evanescentwave to accurately measurethe separation 20-pm glass spheres against the force of gravity in both air distance between a single microscopic sphere levitated and vacuum. In 1986,Ashkin et reported the first above a large flat plate. It has been used to measure the single-beam trap in which particles ranging in size from interaction potential energy profile between the sphere =25 nm to 10 pm were trapped both radially and axially and plate!JO as well as the sphere's hindered diffusion by a single, tightly focused beam. Use of these forces to coefficient."J2 trap biological particles was discussed by Ashkin and In 1989,Brown et d.13described the first experimental Dziedzic,' who studied viruses and bacteria, and later by apparatus combining the TIRM and radiation pressure Sat0 et ala6who trapped yeast cells. techniques. Using radiation pressure to push the sphere In this paper, we first present a geometric optics model away from the plate with constant force, Brown and for predicting the axial and radial components of the Staples14were able to measure the hindered mobility of radiation force exerted on a dielectric sphere. The the particle. Since the mobility is a known function of algorithm ia based on the technique presented by van de separation distance, the absolute sphere-plate separation Hulst6for calculating the scattering profile from a sphere distance could be determined. An independent method which is much larger than the wavelength of incident light. for measuring this distance was later presented by Prieve The model simulates a single, focused beam directed and Walz.16 upward against a dielectric sphere, forming an axial trap The approach presented here uses TIRM to measure between the radiation force and the downward gravitathe net weight (gravity minus applied radiation force) of tional force. From a simple linear momentum balance a single microscopic sphere in an aqueous solution which using the incident and scattered rays, both the axial and is being acted on by an upward radiation force. Because radial components of the force are predicted. of the extreme sensitivity of TIRM, forces as small as 5 x lo-" N could be measured. No evidence of significant This approach is similar to that described recently by photophoretic forces arising from uneven heating of the Ashkin7 for calculating the axial and radial forces on a surrounding fluid was observed. dielectric sphere in a single-beam trap. The primary Finally, a method for measuring the radial intensity difference is that our model assumes that the divergence profile of a laser beam is presented. These measurements of the incident beam is small enough so that the incident are significant because of the importance of the shape of Towhomco~~ndenc%ehouldbeaddreesedattheDepartment this profile in forming a stable opticaltrap. Thetechnique ofChemicalEnninwrina.TulaneUnivereity,New Orleana,LA70118.
hhkin, ~ : P h y s Rku. . Lett. 1970, a,isS. Athkin,A;Dziedeic, J. M.Appl. Phys. Lett. 1971,19,283. Ashkin, A.; Dziedzic, J. M.; Bjorkholm, J. E.; Chu, S. Opt. Lett. 1986,II, 288. (4) Athkin,A.; Dziedzic, J. M. Science 1987, 236, 1617. (6) &to, 5.;Ohyumi, M.; Shibata, H.; Inaba, H.; Ogawa, Y. Opt. Lett. 1991, 16, 282. (6) van de H h t , H. C. Light Scattering by Small Particles; Dover: New York, 1975. (7) Aahkin, A. Biophys. J. 1992, 61, 669.
0743-746319212408-3073$03.00/0
(8) Prieve,D. C.; Luo, F.; Lanni, F. Faraday Dbcws. Chem. SOC.1987, 83, 22. (9) Bike, S. G.; Prieve, D. C. Znt. J. Multiphase Flow 1990, 16, 727. (10) Prieve, D. C.; Frej, N. A. Langmuir 1990, 6,396. (11) Prieve, D. C.; Bike, S. G.; Frej, N. A. Faraday Dbcws. Chem. Soc. 1990,90,209. (12) Frej, N. A.; Prieve, D. C. J. Chem. Phys., in prm. (13) Brown, M. A.; Smith,A. L.; Staples, E. J. Longmuir 1989,6,1319. (14) Brown, M. A.; Staples, E. J. Langmuir ISSO, 6, 1260. (16) Prieve, D. C.; Walz, J. Y . Appl. Opt., in p m .
0 1992 American Chemical Society
Walz and %eve
3074 Langmuir, Vol. 8, No. 12,1992
IX
IX
/, -t
'p
/
m
Figure 2. Coordinate system and angles used in the ray optics model. The origin is at the center of the sphere. The incident rays of the radiation pressure beam travel in the +z direction. I
I
t
,,,I, l
t
I
l l l l l ..I,,\
. .. ..................... . . ...............................................
Figure 1. Schematic illustratingthe origin of both the axial and radial force components produced when a TE& model laser beam strikes a large dielectric sphere. The refractive index of the sphere is greater than that of the surrounding fluid. involves measuring the two-dimensional spatial intensity profile of an evanescent wave formed by the laser under the conditions of total internal reflection at an interface. Good agreement between the measured and expected profiles was obtained. Radiation Pressure Background. Maxwell's theory stipulates that a propagating beam of light will possess momentum as well as energy.6 The magnitude of this momentum is given by
Figure 3. Example path of araythrough the sphere. The integer p denotes the number of contacts made with the inner surface
momentum = energylv (1) where Y is the velocity of light in the propagating medium, while its direction is the direction of propagation of the beam.16 When a propagating beam strikes an interface between two optically different media, a reflected and transmitted beam will be produced. The combined momentum of these two beams will differ from that of the incident beam. To satisfy conservation of linear momentum, a force is exerted on the interface. A light beam strikinga sphericalparticlewill be scattered in a well-defined scattering pattern. Because the momentum of the scattered light is different from that of the incident beam, a force will be exerted on the sphere. If the incident beam has a nonuniform intensity profile, such as the common Gaussianprofileproduced by a laser operating in the TE& mode, this force will contain both an axial component (parallelto the axis of propagation) and a radial component (perpendicularto the direction of propagation). Aa illustrated in Figure 1,this radial component results from an axial asymmetry in the scattered intensity profile. The radiation pressure forces, which result from a momentum exchange between the incident beam and particle, must be distinguishedfrom photophoretic forces. The latter arise when absorption of radiant energy by the particle leads to uneven heating of the surrounding fluid. We assume that these photophoretic forces are insignificant relative to the radiation pressure forces. Evidence of this is presented in the discussion section below.
The magnitude and directionof both the axial and radial components of radiation pressure depend on the sizes of the beam and sphere, the wavelength of incident light, and the refractive indices of both the sphere and surrounding medium. In this work, the divergence of the focused beam is assumed small enough so that the incident rays can be treated as parallel. Under these conditione, the directionof the axialforce componentwill alwaysequal the direction of the incident beam. The direction of the radial component,however,whether toward or away from the axis of the beam, depends on the refractive index of the sphere relative to its environment. For the case of a polystyrene sphere in water (-1yatylene > %ater), the force will be toward the axis. The particle can thus be held in a stable radial trap at the axis of the beam. Review of Geometric Optics. Geometric optics is a mathematical technique for calculating the scattering of light by an object. Although approximate in that diffraction effects are ignored, the approach has proven adequate provided the scattering particle is much larger than the wavelength of incident light. For spheres, good agreementwith the rigorousMie theory has been obtained for radius-to-wavelength ratios of at least 3." In geometric optics,the incident beam of light is treated as a bundle of rays, each capable of pursuing an independent path through the sphere. Each incident ray, containing a prescribed amount of energy and linear momentum, produces a set of scattered rays, having
(16)Kerker, M. The Scattering of Light and Other Electromagnetic Radiation; Academic Press: New York, 1969.
(17)Newton, R. G. Scattering Theory of Waues and Particles; Springer-Verlag: New York, 1966.
of the sphere.
Optical !happing Forces on a Dielectric Sphere
predictable energy and momentum. In the absence of absorption, the energy contained in all of the scattered rays must equal the total incident energy. The totallinear momentum of the scattered rays, however, differs from that of the incident rays, resulting in a net force on the sphere. Determining this force requires calculating the mmentum of both the incident and scattered rays. The algorithm used to track the direction and energy of rays through the sphere is based on that presented by van de In this model, a nonabsorbing sphere of radius a and refractive index n3 rests in a homogeneous medium of refractive index n2. A Cartesian coordinate system with origin at the center of the sphere is utilized (seeFigure 2a). The incident rays of the radiation pressure beam approach the sphere in the +z direction. The point of contact, P, of the incident ray with the sphere is characterized by an azimuthal angle, 4 (0 I4 I2 ~ and ) a contact angle r (0 IT Id 2 ) . 4 = 0 or T describes the x-z plane while 4 = 7r/2 describes the y-z plane. If P’ is the projection of P onto the x-y plane, then r is the angle 2 normal between OP’ and OP. Thus r = ~ / indicates incidence at the center of the sphere while r = 0 describes a grazing ray (tangential to surface). Note that as defined, r is the complement of the angle of incidence, which is usually measured from the normal to the surface at the point of incidence. Consider a ray of intensity IO incident on the sphere at some position P, defined by the angles r and 4. The power (energyhime) of this ray, ni,, will be given by
n,
= Ioa2cos r sin r d r dt$ez (2) where ezis the unit vector in the positive z direction. After p internal reflections (see Figure 31, a scattered ray will emerge. The direction of this ray can be described by the azimuthal angle 4 and a scattering angle B (0 IB IT ) . B describes the angle which the scattered ray makes with the direction of the incident rays, in this case the +z axis (see Figure 2b). Because of the symmetry of the sphere, the incident and scattered ray will lie in the same plane containing the origin; thus both are described by the same azimuthal angle 4. If we let f equal the angle between the internal rays and the sphere (Figure 3), then from Snell’s law we can write n2 cos r = n3 cos f (3) All emerging rays will make the angle T with the surface. It can be showns that the scattering angle B defined in the interval 0 IB IT can be found from the relationship 27 - 2 p f = j27r + qB (4) where j is an integer and q = +1 or -1. The amplitude of the scattered ray will equal that of the incident ray multiplied by the cumulative Fresnel coefficient. The reflection coefficients for the perpendicular cfi) and parallel cfi) polarization states will be given by n2 sin T - n3 sin f n3 sin r - n2 sin f f 1 = n2 sin r n3 sin f f 2 = n3 sin r n2 sin f (5) Let ti represent the product of the Fresnel coefficients for all interfaces encountered by the scattered ray prior to emerging from the sphere. Applying the Fresnel coefficients at each interface givess
+
+
Langmuir, Vol. 8, No. 12, 1992 3076
In the current model, the direction of the electric field of the incident light can be specified as parallel to either the x axis (4 = 0) or y axis (4 = 7r/2). At any point of contact with the sphere, we can divide the incident electric field into perpendicular and parallel components and calculate the intensity of each component independently. Let Z@,r$) be the fractional energy of the incident ray contained in the scattered ray. For the case in which the electric field of the incident ray is parallel to the x axis, the azimuthalangle 4 equalsthe anglebetween the incident plane (which must contain the center of the sphere) and the plane of polarization (parallel to the x axis). Thus
Z@,4) = tl@)sin24 + t2@) cos24 (7) Likewise, for the case in which the electric field is parallel to the y axis, we can write Z@,4) = tl@)cos24 + t2@) sin24 (8) The energyof the scattered ray emergingfrom the sphere will contain both axial (e,) and radial (e,) components. The rate of energy of the scattered ray is then m dp,4)I0a2 cos r sin d r d4[cos Be, + sin Be,] (9) Summing the energies of all scattered rays produced from a single incident ray gives m
nout= x E @ , 4 ) I o a 2cos T sin r d r d+[cos Be, + sin h,] P-0
(10)
the magnitude of which equals the power of the incident ray (eq 2). Gaussian Intensity Profile. Equation 10 gives the total scattered energy produced by a single ray of intensity IO striking the sphere at incident angles 4 and 7. If the incident beam has a Gaussian intensity profile, such as that produced from a TEMw mode laser, IO will depend upon the position of the incident ray relative to the center of the beam. The intensity of a Gaussian laser beam at radial distance r from the center is given byl8
where Pis the power of the laser and w ois the characteristic beam radius (commonly referred to as the waist radius). For the simple case in which the sphere is located at the axis of the beam, the radial distance r can be written in terms of the sphere radius, a, and the incident angle T as r = a cos r (12) For a more general expression, we need to consider the case where the two centers are offset by a radial distance 6r. A schematic of the situation is shown in Figure 4. The origin of the coordinate system remains at the center of ) ~distance from the sphere. For any contact point ( 4 , ~the the centerline of the beam to this contact point will be given by r = [(6r+ a cos T sin 4)2+ (a cos r cos 4)231/2 (13) For the case of 6r = 0, eq 13 reduces to eq 12. Equations 2 and 10 can now be rewritten as
Il, =
%exp( >rw;/~
=w0
This equation predicts that the beam’s profile varies linearly with distance at large distances. The divergence of the beam, OD, is defined as the angle between the beam’s profile and center axis in this linear region (see Figure 5). Thus
Sincethe divergenceangles are typicallyvery small (order 1 mrad), tan (OD) can be approximated by OD, giving x/rwO (26) A plot of the predicted divergence (eq 23) of a 5.4-pm beam (Xo = 514.5 nm) in water is shown in Figure 6. Also shown is the linear, asymptotic divergence predicted with eq 24. The characteristic length rwo2/X for this system is approximately 240Nm. As seen, the beam’s profile, which is flat close to the waist, becomes essentially linear after propagation distances greater than lo00 pm. Predicted Radiation Forces. Shown in Figure 7 are the predicted axial and radial components (calculated using eqs 16,18,and 19) acting on a 10-pm polystyrene sphere in water. The incident beam (Ao = 514.5 nm) has been focused to a waist radius of 5.4 pm and has a total power of 1mW. Results for the electric field, E, polarized parallel to the IC axis (solid) and y axis (dashed) are presented. The graph plots radiation force, as a fraction of gravitational force in water, versus the center-to-center distance between the sphere and beam, as a fraction of beam radius. Thus a dimensionless force of 1means the sphere has zero net weight (radiation force equals gravity), OD
%
Center-to-Center
Dittance/Beam
Waist Radius
Figure 7. Predicted axial and radial components of radiation pressure force acting on a 10-pm polystyrene sphere in water. The incident beam (b= 514.5 nm)contains 1 mW of power and has been focused to a waist radius of 5.4 pm.
while a dimensionless distance of 1 means the center-tocenter distance equals WO. Several observations from the graph are worth noting. First, because of the symmetry of the beam, the radial force is zero at the center. Second, the maximum radial force can actually exceed the maximum axial force. This is extremelyuseful in our experimentssince the axial force must be kept low enough to prevent lifting the particle. Third, even at power levels as low as 1 mW, both the axial and radial components can be greater than the sphere’s weight in the absenceof radiation pressure. This illustrates the usefulness of the radiation pressure technique for manipulating microscopic particles in any direction. Finally, the forces are not substantially different for the two polarizations. Thus, regardless of the direction of any displacement between the center of the sphere and beam axis (Le., whether parallel or perpendicular to the incident electric field), both the axial and radial forces will be approximately equal. Total Internal Reflection Microscopy Description of Technique. The technique of total internal reflection microscopy (TIRM) has been described by Prieve and co-workers.e12 In TIRM, a laser beam is made incident on a glasslwater or micalwater interface at a supercritical angle of incidence, forming an evanescent wave in the water (see Figure 8). A dielectric particle, such as a polystyrenesphere, levitated above the interface by electrostatic repulsion scatters the evanescent wave. Since the intensity of the wave decays exponentiallyaway from the interface, the intensity of the scattering light measured over some solid cone will depend upon the separation distance between the sphere and solid surface. Prieve and Walzl* showed that for microscopic (7-30 pm) polystyrene spheres in an aqueous medium, this scattered intensity will vary exponentiallywith separation distance (from large h down to contact) with the same decay constant which characterizes the evanescent wave. Thus
where Isca(h)is the scattered intensity at separation
Walz and Prieve
3078 Langmuir, Vol. 8, No. 12,1992
fX
I
\ I ?
I
I
Water, n, Glass or Mica, n,
/I\
5" I
-&I: .4k+
I
HNL
Incident Ray
I
#
I
Figure 8. Scattering of an evanescentwave by a dielectric sphere in total internal reflection microscopy (TIRM).
distance h, 10is the scattered intensity at h = 0, k2 is the wavenumber in the fluid (k2 = 27rnz/b), nl and n2 are the refractive indices of the plate and fluid, respectively, and Bi is the incident angle. Measuring the scattering intensity provides a sensitive and instantaneous measure of the separation distance. In order to determine the absolute separation distance using eq 27, it is necessary to measure the scattering intensity from the sphere at a known position h. This is accomplished by increasing the ionic strength of the solution, which effectively screens the double-layer repulsive forcesand allows the sphere to settle into a primary potential energy minimum very close to the surface (h = 0). Since this requires sacrificing the particle, it is the last step of the experimental procedure. Measurement of Colloidal Forces with TIRM. TIRM has proven to be an effective tool for measuring the forces of interaction between single microscopic particles in equilibrium above a flat planar i n t e r f a ~ e . ~InJ ~a typical experiment, a 10-pm polystyrene sphere is levitated by electrostatic repulsion above a glass or mica surface in an aqueous medium. Due to Brownian motion, the sphere will diffuse around an equilibrium separation distance where the net force acting on it is zero. By monitoring the separation distances sampled by the sphere for a statistically long period of time, the probability density p ( h ) can be deduced by constructing a histogram of elevations, h. This can then be convertedto a potential energy profile using Boltzmann's equation (29)
where p ( h ) is the probability of finding the sphere at separation distance h, A is a normalization constant to ensure that all probabilities sum to 1,$(h)is the potential energy of the sphere located at h, k g is Boltzmann's constant, and T is absolute temperature. The net force acting on the sphere at any separation distance h will simply be the slope of the potential energy profile at that point. The three major contributions to the sphere's potential energy will be gravitational, electrostatic repulsion, and van der Waals attraction. Thus
AIL
BE
Figure 9. A simple schematicof the combined TIRM/radiation pressure experimental setup: AIL, argon ion laser;HNL,heliumneon h e r ; BE, beam expander; RHP, right angle prism; FL, focusing lens; PR, prism; SYP, syringe pump; FLT, filter; FW, fluid well; OBJ,microscope objective; INF, interference filter; PMT, photomultiplier tube. The arrow indicate the degrees of freedom of movement with a particular component.
(32)
where G is the gravitational force, Frpis the upward radiation force,g is the gravitational constant, and ppand pf are the densities of the particle and fluid, respectively. For separation distances several times larger than the Debye length, the double-layer potential between a spherical particle and a flat plate can be expressed using linear superpositionand the Derjaguin approximation asm
&(h) = B exp(-Kh) (valid for a >> h >> K-') (33) B = l&a( k,T 2 tanh tanh 4kgT 4kgT (valid for a 1:l electrolyte) (34) where e is the unit charge, K - ~ the Debye length, Q the dielectric constant of water, and 91 and 9 2 the Stern potentials of the sphere and plate, respectively. In low ionic strength solutions (e.g., I1 mM NaCl), the equilibrium separation distance of a 10-pm sphere is typically large enough that van der Waalsattractive forces are insignificant?' We can therefore calculate the potential energy of the sphere as
--)
(*)
(*)
4(h) = (G - F,)h + B exp(-Kh) (35) This equation describesthe potential energyprofde near a secondary potential energy minimum formed by a balance between electrostatic and gravitational forces. If the sphere could be pushed closer to the surface, the attractive van der Waals force would become dominant and a much deeper primary potential well would be formed very close to the surface (contact being prevented by steric repulsion). Experimental Section
(30)
Apparatus. To incorporate radiation pressure, several modifications to the TIRM apparatus have been made (see Figure 9). A 5-mW helium-neon laser (XO = 632.8 nm)is used to create the evanescentwave in the fluid. The M-mW argon ion laser (XO = 514.5 nm)is used to form the radiation pressure beam. The
The gravitational potential will equal the net weight of the particle times ita separation distance above the plate. Thus
(20) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (21) Prieve, D. C.; Alexander, B. M. Langmuir 1987,3,788.
4 4 ) = 9&
+ &(h) + 4Jvdw(h)
Langmuir, Vol. 8, No. 12, 1992 3079
Optical !happing Forces on a Dielectric Sphere beam passes through a 4X beam expander and is then guided up through the center of the microscope objective with right angle prisms mounted on precision translation stages. (Asexplained above, expanding the beam first allows focusing to a smaller spot size with the focusing lens.) A 40 mm focal length precision achromatic doublet lens placed under the prism focusesthe beam to a spot size of approximately 11pm. This lens is mounted on a three-dimensional precision translation system for accurate alignment and focusing. The fluid well, which rests on top of the prism stage, has been redesigned asa flow cell to allow the fluid medium to be exchanged without disrupting the cell. A syringe pump (Sage Instruments, Model 355, Cambridge, MA) is used to pump fluid into the cell. The bottom of the flow cell has been designed to allow different materials to be used as the bottom of the well where the particle interactions occur. Both BK-7 glass and mica have been used. When data are collected, a laser interference filter is placed above the microscope objective. This fiiter allows only the light scattered from the helium neon laser to pass into the photomultiplier tube. No significant leakage of the argon ion beam through the fiiter has been detected. A final note should be added concerning eye protection. Since the focused radiation pressure beam passes directly into the eyepiece of the microscope objective, accidentally looking at this beam without some type of protection could result in serious eye injury. To protect against this, a dual fiiter holder was placed above the objective and fitted with a stop to impede its removal. This ensures that some type of filter, either neutral density or interference, is always in place to block the beam. Procedure. Described below is the typical procedure used to measure the potential profiles of a 10-pm polystyrene sphere above a mica sheet. Two solutions (typically 0.5 and 100 mM) are prepared using deionized ultrafiitered water. After salt is added, the solutions are again fiitered through a 0.22-pm fiiter (Millipore Corp., Bedford, MA) to remove any insoluble impurities. Monodisperse polystyrene particles of mean diameter 9.87 f 0.206 pm were purchased from Duke Scientific Corp. (Palo Alto, CA) as an aqueous dispersion of 0.32% solids. T w o or three drops of this dispersion are added to approximately 50 mL of the low ionic strength solution. This solution is then placed in an ultrasonic bath for approximately 1min to disperse the particles and dislodge any air bubbles. The Plexiglas flow cell is cleaned with soap and water and allowed to dry. A thin (0.00025-0.00030 in.) mice sheet (AshevilleSchoonmaker Mica, Newport News, VA) is cut and washed with deionized ultrafiltered water and allowed to dry. The mica sheet is glued to the bottom of the flow cell using silicone rubber and allowed to dry overnight. At the start of the experiment, the prism and optical glassware are cleaned with spectrographic grade acetone. The prism is leveled using a sealed glass vial level. The helium-neon laser is then aligned by back-reflecting from the prism to the mirror to ensure that normal incidence has been attained. One or two drops of immersion oil ( n =~1.515) are placed on top of the prism. The mica flow cell is blown with air to remove dust and then slowly placed on the oil. Care is taken to remove any air bubbles trapped beneath the mica. The inlet and outlet tubing are connected to the flow cell and the entire assembly, including the inlet syringe, is flushed with the low ionic strength solution. The flow cell is then emptied and several milliliters of the particle solution added. The microscope objective, which has been cleaned to remove dust, is then lowered into the solution and the open space between the objective and well of the flow cell covered with Parafilm to alleviate dust contamination and evaporation. The particles are then allowed to settle for approximately 30 min. After settling, an isolated sphere is centered in the microscope's field-of-view. An aperture on the microscope photometry system is set to define a rectangular Sampling region, typically twice the diameter of the sphere, from which scattered light is collected by the photomultiplier tube. The horizontal position of the evanescent wave on the mica is adjusted with the two translation stages used to control the helium-neon beam until a maximum scattering intensity is obtained. This ensures that the center of the Gaussian beam is
0
-
striking the micdwater interface just below the particle. The radiation pressure beam is aligned by placing a neutral density filter (density = 4,transmittance = lW) above the objective and viewing the focused beam through the eyepiece of the microecope. By use of the precision translation stages on both the argon ion beamandfocusinglens, the position, size,andshapeofthefocused spot can be controlled. The beam is considered focused when the spot size is a minimum. The f i i t measurement taken is the background reading. This is obtained by moving the prism with the translation stages until the particle is displaced out of the sampling region and recording the scattered intensity. The particle is then moved back into the region and the radiation pressure beam applied to it to prevent lateral diffusion. The intensity of the beam is kept low enough so that the upward radiation force does not exceed the downward gravitational force. Scattering intensity measurements are then recorded at five or six radiation pressure levels. Typically 50 000 readings at 10-msintervals are recorded at each power level. The background reading is rechecked for consistency. The syringe pump is then fiied with the high ionic strength solution and the well slowly flushed out. The high ionic strength solution screensthe double-layer repulsion and allows the particle to settle to the surface. The flow must be kept low enough so that the particle is not forced out of the radial trap formed by the radiation pressure beam. Since the flow cell is not rigidly sealed at the top, another syringe pump or aspirator is used to withdraw fluid from the cell during flushing. Once flushing is complete, a scattering measurement is taken. This will be the 10value needed in eq 27. If the intensity is still fluctuating, indicating that the particle has not settled, a small amount of additional salt is added and the intensity rechecked. The salt concentration is always kept low enough so that the refractive index of the solution is not significantly affectad.
Experimental Results Typical Potential Energy Profile. A typical potential energy profile measured using TIRM is shown in Figure 10. These data were taken on a l0-pm polystyrene sphere levitated above a mica sheet in a 0.5 m M NaCl solution. There are two distinct regions of this profile. At smaller separation distances,the interaction is dominated by the strong electrostatic repulsion and the potential decays exponentiallywith distance. At large separationdistances, the electrostaticrepulsion becomes insignificant and the
Walz and %eve
3080 Langmuir, Vol. 8, No. 12,1992 0.30 Expected VOlUC
-
0.25
I
0.20
a
-.cm ;0.15 -
-
0 0 0
=
0.10
0.05
0.00 0.0 Separatlon Dialonce, nm
Figure 11. Potential energy profiles from a 10-pm polystyrene sphere in a 0.5 mM NaCl solution taken at three different levels of radiation pressure. The power levels refer to the measured output power of the argon ion laser. The profiles have been offset by 1 keT for ease of viewing. potential energy profile becomes linear, reflecting the constant force of gravity. The slope of this linear portion will equal the net weight of the particle. In the absence of radiation pressure, this can easily be calculated from the known diameter and density of the sphere. In previous work without radiation pressure, gravitational forces corresponding to masses as small as 10 pg have been measured to within a root mean square error of 6%.lo A comparison of these data with the theoretical profile (eq 35) is also shown. For these calculations, G - F, = 0.19 pN (equal to the slope of the linear region of the measured data), K - ~= 13.58nm (predicted from the known salt concentration), and B = 3.23 X lo-'' J (found by matching the equilibrium separation distance). As shown by Prieve and Frej,lothe shape of the predicted profile is a function of only the Debye length, K-', and net weight (G-F,). ItisindependentofB. Thus thegoodagreement in the shape of the profile in the region where doublelayer forces dominate (smallest separation distances) was obtained without any adjustable parameters. B can be determined, however, by matching the equilibrium separation distance, h,. (Since the surface potentials of the sphere and mica *I and q 2 , are not known, B cannot be predicted independently.) Measurement of Axial Radiation Force. The axial component of the radiation force is measured with TIRM by measuring the net weight (G - Frp)of a suspended sphere at various laser output levels. For example, shown in Figure 11are three potential energy profiles measured on a 10-pmpolystyrene sphere levitated abovea mica sheet in a 0.5 mM NaCl solution. (The minima in the curves have been intentionallyoffset by 1kBTfor ease of viewing.) The three profiles were taken at three different power levels of the radiation pressure laser-0.88,1.34, and 2.00 mW. The theoretical curves were obtained in the manner described for Figure 10. As seen, increasing the upwarddirected radiation pressure results in a lower slope of the gravity-dominated region of the curve, indicating a lower net weight of the sphere. Shown in Figure 12 is a plot of this net particle weight, found by calculating the slope of the linear portion of the potential energy profiles versus laser output power for six
0.5
1 .o
1.5
2.0
5
Laser Output Power, mW
Figure 12. Net weight of a 1O-pm polystyrene sphere asa function of output power of the radiation pressure h e r . The expected value is the weight calculated using a sphere diameter of 9.87 h 0.206 pm and a density of 1.055 g/cm3.
output power levels. The data clearly agree well with the expected linear relationship. From the slope of this line, -0.117 pN/mW, we can determine the axial force applied to the sphere at any output laser power. It should be emphasized that due to losses in the optical train (e.g., absorption, scattering, reflection), the rate of energy actually strikingthe sphere could be considerablylessthan that produced by the laser. For example, using the results of Figure 7 (and assuming the sphere remains at the axie of the beam), 1mW of incident power should produce an axial force of 0.290 pN. Comparingthis with the measured value of 0.117 pN per milliwatt of laser output power yields an efficiency of approximately 40 5%. Since these optical losses are difficult to quantify, determiningthe axial force by measuring the net weight of the sphere is much more accurate than predicting the force using the laser output power and estimated losses. The intercept of the linear regression in Figure 12 is 0.289 f 0.011 pN. By comparison, the predicted weight of the 9.87 f 0.206-pm polystyrene sphere in water (pp = 1.055 g/cm3) is 0.271 f 0.017 pN, a difference of 6.6%. Measurement of Gaussian Intensity Profile of Laser Beam. The ability to hold a particle on the axis of the beam results from the nonuniform radial profile of the beam's intensity. In the geometricoptics calculations described above, a perfect Gaussian profile is assumed. Because of the importance of this profile in forming a stable optical trap, we have devised a method to experimentally measured it using TIRM. A schematic is shown in Figure 13. The argon ion laser, which is normally used as the radiation pressure laser, is used to form the evanescent wave. (Radiation pressure is not required for this procedure.) The path of the beam is arranged so that ita point of contact with the glass or micdfluid interface can be accurately controlled with a mirror and right-angle prism mounted on precision translation stages. Each of these stages provides a resolution of 1 pm. A 10-pmpolystyrene particle above a glass or mica slide in a high ionic strength (100mM NaC1) solution is centered in the microscope aperture. At this ionic strength, electrostatic repulsive forces are effectively screened and the sphere becomes trapped in a primary minimum very
Langmuir, Vol. 8, No. 12,1982 3081
Optical Trapping Forces on a Dielectric Sphere
mh
-- +
I/
Figure 13. An illustration of the experimental technique used to measure the intensity profile of a h e r beam. The circular beam, which strikes the front face of the prism at normal incidence, forms an ellipse upon reflection from the upper face. The double-headed arrows indicate the possible directions of movement of the beam.
.'
0.2 -2
t
,
e
2
0
-1
Radial D l s p l a e w " t .
mm
Figure 14. Radial intensity profile measured on an evanescent wave produced at a ghs/water interface with an incident angle of 6 8 O . The long and short axes refer to the two primary axes of the elliptical-shaped evanescent field.
close to the surface. This can be verified by the absence of a fluctuations in the scattered intensity, indicating no vertical movement. By manipulation of the translation stages, the center of the evanescent field can be moved relative to the sphere. This changes the intensity of the field striking the particle, which results in an equivalent change in the scattered intensity. The intensity profile of the beam is found by systematically scanning the beam across the sphere in discrete steps and measuring the scattering intensity at each position. Since the incident beam strikes theglasslwater interface at a 68O angle of incidence, the evanescent field formed by the circular laser beam will be elliptical. The ratio of the two axes of the ellipse will be a function of the angle of incidence. Specifically long axis =
short axis cos ei
(36)
where 8i is the angle of incidence and the short axis is the diameter of the circular incident beam. Each of the two translation stages is arranged such that its direction of travel is parallel to one of the axes of the ellipse. The results of these measurements are shown graphically in Figure 14. Intensity (relative to the maximum value) is plotted versus displacement for scans along both axes. The symbols are measured values while the lines are a Gaussian regression of the data. The regression model used is
where I(r) is the intensity at displacement r, I,, is the maximum intensity at the beam's center, and w is the waist radius of the beam. The regression program calculates the beam radius which provides the best fit to the measured data. Ae seen,the measured profile show very good agreement with the predicted Gaussian shape. The two beam radii calculated were 1.865 f 0.014 mm for the long axis and 0.701f 0.003mm for the short axis. Estimating the angle of incidence with these values using eq 36 gives 67.9 f 0.3O,which agrees well with the independently measured prism angle of 68O. Discussion Effect of Brownian Movement on Radiation Force. A critical assumption in our experimental procedure is that the vertical radiation pressure force on the sphere does not change as the sphere diffuses vertically around the equilibrium point. This requires that the beam radius remain constant over the sphere's full range of vertical movement. Using eq 24,we can calculate the minimum waist size to ensure that the spreading of the beam can be ignored. (Note that this is a worst-case scenario, since it utilizes the linear region of the beam where the spreading is greatest.) For example, the maximum amount of vertical movement typically observed in our TIRM experiments is 500 nm. If we desire a maximum of 0.19% change in the beam radius over this distance (which corresponds to a 0.2 !% change in the applied force),then the minimum waist radius is approximately0.3pm. This is well below the 6.4 pm focused waist radius of our current experimentalsetup; thus beam divergence should not be a problem. In addition to the effects of vertical fluctuations, the magnitude of the axial force also depends on the distance between the beam axisand the center of the sphere (Figure 7). It is thus possible that lateral diffusion of the particle in the radial plane can lead to fluctuations in this force. To determine if this is a problem, it is first necessary to calculate the radial potential profile, created by the radial force, in which the sphere is trapped. For anyradial center-to-center displacement, r, the radial potential, ,&t can be found as
4r(r) =
S,'F(+)d~
(38)
whereF(r) is the radial force shown in Figure 7. With this equation, the graph shown in Figure 15 was created. Plotted here is the radial potential energy of the sphere, in units Of kBT, versus the dimensionless center-to-center distance used in Figure 7. Also plotted is the magnitude of the axial force relative to its value at a center-to-center distance of zero. (The electricfield was assumed polarized parallel to the x axis for these calculations.) Assuming the particle will remain within 7kBTof the center (a 99.9!% probability),then from Figure 15,the maximum deviation in the axial force is predicted to be lese than0.4 9%. Lateral fluctuations around the equilibrium position should therefore have a negligible effect on the axial radiation force. Photophoreeis Effects. A common problem encountered in radiation pressure work is the development of photophoretic forces. The surface of a spherical polystyrene particle in water acts as a crude focusing lens, concentrating the incident radiation at various spots in the interior of the sphere. This can result in localized "hot spots", which leads to uneven heating of the sur-
Walz and R i e v e
3082 Langmuir, Vol. 8, No.12, 1992 ‘0
..
000
1.000
002
004
Cintir-to-C.nt.r
006 Ddant./B.em
008
0 10
0 12
W d % t Rodiur
Figure 15. Radial potential energy profiie calculated using the radial force results from Figure 7. (Theelectric field is assumed to be parallel to the x axis.) Also shown is the axial force relative to its maximum value at a center-to-centerdistance of zero.
rounding fluid and the developmentof convectivecurrents. These currents can generate motion of the sphere which can appear to arise from a radiation pressure force. Since these photophoretic forces could result in unsteady movement of the particle and thus alter the measured potential energy profile, it is important they be insignificant relative to the actual radiation pressure forces. There are a number of indications that no significant photophoresis is occurring in our experiments. First, no anomalies in the particle motion or in the measured potential energy profiles were observed at the higher radiation pressure levels. Second, no time-dependent changes in the measured potential energy profiles was observed, suggesting a lack of any significant heating of the fluid. Third, in similar experiments with polystyrene spheres, Brown et al.13 noticed a change in the far-field scattering pattern after prolonged exposure to highintensity argon ion beams. No such changeswere observed in our experiments, even after several hours of exposure. Finally, similar experiments performed by Ashkin’ utilizing polystyrene spheres in water showed no photophoresis effects. Uses of Radiation Pressure. The ability to manipulate and position a single, microscopic particle makes radiation pressure a powerful tool for studying colloidal interactions near interfaces. For example, combiningthis technique with TIRM reduces distortions in the measured potential energy profiles caused by lateral movement of the particle. To obtain representative probability distributions, the separation distance between particle and interface must be monitored for a long period of time so that the particle can sample ita potential energy levels many times. Without radiation pressure this is difficult, since the particle will wander laterally due to Brownian motion. This results in the particle interacting with different areas of the surface (with possibly different surface potentials) or moving out of the sampling region. With radiation pressure, the particle can be held at a fixed lateral position indefinitely. Furthermore, the radial force component is strong enough to hold a given particle at a fixed position while the surroundingfluid is replaced,permitting a carefulstudy of the interaction forces as a function of solution ionic strength. Another useful experiment would be to position a single particle at different positions along the surface and measure the potential energy profile at each point.
This would allow measuringthe spatialvariation in surface potential of the plate. In addition to potential energy profiles, radiation pressure provides the capacity for measuring the mobility of a particle near an interface. As demonstrated by Brown and Staples,14this can be done by applying a step change in the upward force large enough to lift the particle. Measuring the rate of rise allows calculatingthe hindered mobility as a function of separation distance. Similar experiments could be performed on particles coated with an adsorbed layer of polymer to determine how much additional resistance is produced. If a spherical particle is held in a fixed position against a steady shear flow, the particle will rotate with constant angular velocity. Any deviations in the surface potential of the sphere will result in a regular, periodic vertical movement which should be discernible from the random Brownian motion. Measuring this movement would thus provide some indication of the spatial variation in surface potential of the sphere. Finally, radiation pressure should allow interaction potentials to be measured over a wider range of separation distances. With the current apparatus, in which the radiation pressure is applied from below, the equilibrium separation distance is controlled primarily by the ionic strength of the solution. (Recall that the radiation force must be kept below the gravitational force to avoid lifting the particle.) Ignoring van der Waals attraction, this distance is typically 7-10debye lengths. Raising the ionic strength suppresses the double-layer repulsion and thus allows the particle to settle closer, yet also increases the likelihood that the particle will escape the secondary potential energy barrier to become trapped in a primary potential energyminimum and lost to further experiments. One way around this problem is to apply the radiation force from above and push the particle closer to the flat plate. Since there would no longer be a concern about lifting the sphere, much higher radiation forces could be used. Interactions at smaller separation distances could then be measured at a constant ionicstrength. In addition to reducing the chance of the particle sticking, by eliminating the need to use different ionic strengths, problems caused by changesin either the surfacepotential or van der Waals attraction with ionic strength would be avoided. Future Work The technique of geometricoptics providesaconvenient method for calculating the axial and radial components of the radiation force exerted on a sphere. Although the model currently assumes parallel incident rays, simulating a divergent beam should not require significant modifications. As described by Ashkin,’ in such a model both the intensity and direction of an incidentray would depend on ita radial position relative to the center of the beam. This would allow predicting the axial and radial forces in a single-beam, or gradient, trap. TIRM also provides the unique capability of accurately measuring the axial component of the radiation force. A technique for measuring the radial component was describedby Sato et al? This involvesmeasuringthe viscous drag force required to pull a sphere out of ita radial potential trap. Performing similar experiments with our experimentalsetup should also be possible and would thus allow accurately determining both the axial and radial force components. Registry No. Polystyrene, 9003-53-6.
