Theory of Optical Chromatography - Analytical Chemistry (ACS

---

Anal. Chem. 1997, 69, 2701-2710

Theory of Optical Chromatography Takashi Kaneta, Yasunori Ishidzu, Naoki Mishima, and Totaro Imasaka*

Department of Chemical Science and Technology, Faculty of Engineering, Kyushu University, Hakozaki, Fukuoka 812, Japan

To evaluate the performance of optical chromatography, a number of equations are theoretically derived using a ray-optics model. These mathematical formalisms are experimentally verified by determining the relationship between the velocity of motion of a polystyrene bead with respect to the intensity of an applied radiation force under the condition where there exists no applied fluid flow. The force is confirmed to be at a maximum at the focal point and to decrease with increasing distance from this position. The radiation force is verified to be proportional to the square of the particle size when the particle diameter is much smaller than the beam diameter. In addition, the radiation force is ascertained to be proportional to the laser power. These results are in excellent agreement with the proposed theoretical model, which is based on ray optics. Furthermore, by analogy with conventional chromatography, fundamental parameters such as retention distance, selectivity, theoretical plate number, and resolution are calculated, and optimum conditions for chromatographic separation are discussed. Based on the results obtained, the dynamic range can be extended by increasing laser power and decreasing flow rate. Peak broadening is primarily caused by variations in laser power and flow rate of the medium for large particles (>1 µm). It is possible, in theory, to distinguish particles whose diameters differ by less than 1% for particles with a diameter larger than 1 µm. Three sizes of polystyrene beads are well separated at a flow rate of 20 µm s-1 and a laser power of 700 mW. This technique is also applied to the separation of human erythrocytes. Two fractions, one consisting of cells ranging from 1.5 to 2.4 µm in diameter and another consisting of cells ranging from 3.5 to 5.7 µm in diameter, are observed. Optical chromatography is useful for separation and size measurement of particles and biological cells. Some years ago, Ashkin reported the development of a new optical technique for the control and manipulation of a microsphere using laser radiation pressure.1 This technique is called laser trapping, laser manipulation, optical tweezers, or laser levitation. Many scientists have applied this technique to polymer particles1,2 and even biological cells.3,4 Recently, this technique was used in analytical chemistry for the characterization of an (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.; Yamane, T. Nature 1987, 330, 769-771. (4) Wright, W. H.; Sonek, G. J.; Tadir, Y.; Berns, M. W. IEEE J. Quantum Electron 1990, 26, 2148-2157. S0003-2700(97)00079-6 CCC: $14.00

© 1997 American Chemical Society

ion-exchange resin.5 The theory for the evaluation of this method is based on a ray-optics model for large particles6,7 and on an electromagnetic model for small particles.7-10 We recently proposed a new optical technique for the separation of particles using laser radiation pressure, which we referred to as optical chromatography.11 This technique is based on the equilibrium between two forces: one is the optical pressure induced by a laser radiation force and the other is the resistance force induced by a flowing medium. In this approach, the laser beam is focused into a capillary filled with water which is counterflowing hydrodynamically to the propagation direction of the laser beam. The optical pressure produced by the laser accelerates the particle in a direction opposite to the liquid flow. The particle stops at a point where the radiation force is equal to the force induced by the medium flow. If two types of particles are suspended in the medium, they equilibrate at different positions, depending on the size and the refractive index of the particle. A more detailed description of the mechanism is described in our previous paper.11 Optical chromatography has great potential value as a new modern separation technique. Previously, we demonstrated the separation of polystyrene beads of different diameters. However, the theoretical basis of this technique has not yet been elucidated. To evaluate this technique, it is necessary to develop a theory so that the advantages and the unavoidable limitations are clarified. In this study, a theory of optical chromatography is developed which is based on a ray-optics model. Many mathematical equations are derived for the evaluation of optical chromatography. The radiation force applied to a particle and its equilibrated position are calculated as functions of the particle radius and refractive index. The radiation force applied to a particle is measured using a video camera and is compared with the theoretical value calculated using the equations derived. The experimental results are in good agreement with the theoretical values for particles ranging in diameter from 1 to 10 µm. In analogy with conventional chromatography, many fundamental parameters, such as retention distance (rather than retention time), selectivity, theoretical plate number, and resolution are calculated, and the optimum parameters for separation of particles are derived by computer simulation. The factors that affect these (5) Kim, H.; Hayashi, M.; Nakatani, K.; Kitamura, N.; Sasaki, K.; Hotta, J.; Masuhara, H. Anal. Chem. 1996, 68, 409-414. (6) Ashkin, A. Biophys. J. 1992, 61, 569-582. (7) Wright, W. H.; Sonek, G. J.; Berns, M. W. Appl. Phys. Lett. 1993, 63, 715717. (8) Irvine, W. M. J. Opt. Soc. Am. 1965, 55, 16-21. (9) Kim, J. S.; Lee, S. S. J. Opt. Soc. Am. 1983, 73, 303-312. (10) Barton, J.; Alexander, D.; Schaub, S. J. Appl. Phys. 1989, 66, 4594-4602. (11) Imasaka, T.; Kawabata, Y.; Kaneta, T.; Ishidzu, Y. Anal. Chem. 1995, 67, 1763-1765.

Analytical Chemistry, Vol. 69, No. 14, July 15, 1997 2701

distribution. Thus, the light intensity decreases exponentially as the distance from the beam axis increases. The intensity of the light irradiating at an incidence angle of θ is expressed as a function of the distance, r, and the particle radius, a, by the following equation:

( )

I ) I0 exp

-2r2 ω2

(1)

Figure 1. Ray-optics model applied to optical chromatography. The parameters Fa1 and Fa2 are the radiation forces generated by refractions of beam a, and Fb1 and Fb2 are the radiation forces generated by refractions of beam b. The parameters Fat and Fbt are the sum of Fa1 and Fa2, and the sum of Fb1 and Fb2, respectively. The radiation forces generated by the reflections are not shown, in order to simplify the figure.

where I0 is the intensity at the beam center and ω is the beam radius. Since r ) a sin θ, eq 2 can be derived from eq 1.

parameters are also discussed in quantitative terms. More importantly, this technique is applied to the separation of three types of polystyrene beads as well as human erythrocytes and is useful for the evaluation of particle and biological cell size.

The power of the incident beam on a small area of the particle, dP, is expressed by

THEORY Model. First, we assume that a single-transverse-mode laser beam with a Gaussian intensity distribution irradiates a single sphere whose refractive index and radius are n and a, respectively. The radiation force in optical chromatography can be calculated mathematically for this system in a manner similar to that for the ray-optics model developed for the laser trapping technique described by Ashkin6 and for the optical levitation technique described by Roosen and Imbert.12 There are significant differences between optical chromatography and the aforementioned techniques. The most notable differences are the presence of a liquid flow in optical chromatography and the incident angle of the optical wave. Also, the laser beam is tightly focused in laser trapping but loosely focused in optical chromatography. Figure 1 shows an optical wave traveling toward the particle. We assume that all the rays are traveling parallel to the direction of propagation of the sphere since a lens with a long focal length is used. This assumption is valid in most cases in optical chromatography, but care should be exercised when the particle radius becomes close to or larger than the beam radius, i.e., a g ω. In this case, the particle is trapped at the focal point when the flow rate is sufficiently small, i.e., laser trapping occurs. Two types of radiation forces are applied to the particle: one is the radial force, called a gradient force, and the other is the axial force, called a scattering force. The gradient force component attracts the particle to the center line of the laser beam axis, and ultimately, all the particles are aligned to the beam center. As a result of this, the particle is depicted at the central line of the laser beam in Figure 1. On the other hand, the scattering force component accelerates the particle in the propagation direction of the laser beam. These forces are generated by a momentum change produced by the light at the point of refraction and reflection at the surface of the bead. Under a medium flow, the particle drifts to a point where the radiation force is identical to the force induced by the medium flow, resulting in a separation of particles as a function of size. As described above, the laser beam is assumed to be operated in a single-transverse mode and to have a Gaussian intensity

Substituting r ) a sin θ and dr ) a cos θ dθ in eq 3, dP can be rewritten as

(12) Roosen, G.; Imbert, C. Phys. Lett. 1976, 59A, 6-8.

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Analytical Chemistry, Vol. 69, No. 14, July 15, 1997

(

I ) I0 exp

)

-2a2 sin2 θ ω2

(2)

dP ) 2πr drI

dP ) 2πa2 sin θ cos θ I0 exp

(

) πa2 sin 2θ I0 exp

(3)

(

)

-2a2 sin2 θ dθ ω2

)

-2a2 sin2 θ dθ ω2

(4)

Taking into account only the scattering force, the conversion efficiency from an optical radiation to the pressure applied to a particle is calculated using the Fresnel reflection and transmission coefficients, R and T, respectively, as follows:

[

Q(θ) ) 1/2 1 + R cos 2θ -

]

T2{cos(2θ - 2φ) + R cos 2θ} 1 + R2 + 2R cos 2φ

(5) The parameter Q(θ) is a dimensionless factor. The values of R and T are also expressed as functions of θ and φ:

{

R ) 1/ 2

{

T ) 1/2

sin2(θ - φ)

sin2(θ + φ)

}

tan2(θ - φ) +

(6)

tan2(θ + φ)

}

sin 2θ sin φ sin 2θ sin φ + 2 2 sin (θ + φ) sin (θ + φ) cos2(θ - φ)

(7)

where φ is a refraction angle specified in Figure 1 and R + T ) 1. The value of φ is determined using the refractive indexes of the medium, n1, and the sphere, n2, and the following relationship is given by Snell’s law:

n1 sin θ ) n2 sin φ

(8)

The radiation force at the incident angle of θ, F(θ), is related to Q(θ) as follows:

n1P F(θ) ) Q(θ) c

(9)

Figure 2. Schematic diagram of the experimental apparatus.

where P is the incident power and c is the velocity of light. Using eqs 4-7, the radiation force generated on dP is represented by

n1 dP c

dF(θ) ) Q(θ)

(

)

n1 a -2a2 sin2 θ dθ ) Q(θ) π sin 2θ I0 exp c ω ω2

(10)

Therefore, the total scattering force applied to the particle is calculated by integrating eq 10.

F) )

∫

π/2

0

dF

n1 πI a2 c 0

∫

{ (ωa ) sin θ}Q(θ) sin 2θ dθ (11)

π/2

2

exp -2

0

2

Equation 11 can be rewritten by using the total laser power, P ) πω2I0 /2:

F)

2n1P a c ω

( )∫ 2

π/2

{ (ωa ) sin θ}Q(θ) sin 2θ dθ 2

exp -2

0

2

(12)

When (a/ω)2 , 1, the first term in the integral can be approximated to 1. Equation 12 can then be simplified to eq 13.

F)

)

2n1P a c ω

( )∫ 2

π/2

0

sin 2θ Q(θ) dθ

2n1P a 2 Q* c ω

(13)

∫

(14)

Q* )

()

π/2

0

sin 2θ Q(θ) dθ

The parameter ω is a function of the distance from the beam waist. We can then calculate the magnitude of the radiation force at any point on the axis of the laser beam from eq 13. EXPERIMENTAL SECTION Apparatus. A block diagram of the experimental apparatus constructed in this study is shown in Figure 2. A quartz cell is mounted on a stage which can be translated three-dimensionally and rotated two-dimensionally in order to align the channel axis of the cell to the propagation direction of the laser beam and to adjust the position of the beam waist correctly inside the cell. A capillary tube (GL Science Inc., Tokyo, Japan; 200 µm i.d., 375

µm o.d.) is inserted coaxially into the channel of the cell. The distal end of the capillary is immersed into a buffer solution. The outlet of the cell is connected to another capillary, the end of which is immersed into a reservoir containing water. Four types of polystyrene latex beads with different diameters (refractive index, 1.59; diameter, 0.997, 3.03, 5.85, and 10.53 µm) were obtained from Polyscience Inc. (Warrington, PA). These particles were dispersed in water (refractive index, 1.33; viscosity, 1 × 10-3 Pa s) and introduced from the inlet side of the capillary to the quartz cell using a siphon method; the inlet side of the capillary was dipped into the sample solution, which was raised above the outlet side of the solution in order to allow the sample to flow. An argon ion laser (BeamLok 2060-7S, multiline mode, 454.5514.5 nm, Spectra-Physics, Mountain View, CA; or Model GLG3200, single-line mode, 514.5 nm, Nippon Electric Co., Tokyo, Japan) is focused by a lens (focal length, 50 mm) into the capillary. The diameter of the beam waist is calculated to be 4.67 µm using eqs 19 and 20 below. The position of the beam waist is positioned 200-500 µm into the capillary. The velocity of the particles is monitored by a video camera connected to a microscope (Model Scopeman MS 603, magnification 210× or 400×, Moritex, Tokyo, Japan) equipped with a video tape recorder (Model SVO 260, Sony, Tokyo, Japan). A sharp cutoff filter (dielectric filter, cutoff wavelength 500 nm, Sigma Koki, Saitama, Japan) is placed between the quartz cell and the video camera to isolate the scattered light of the laser beam. The acceleration of individual particles can be visually observed on the monitor of the video recorder. The velocity of the particle is calculated by measuring the time period required for the particle to pass through a specified distance in the capillary. The radiation force was measured after stopping the medium flow. The force was calculated from the above velocity data on the basis of Stokes’s law using eq 16 below. A typical flow rate was 20 µm s-1 in this study. A similar study was also undertaken using biological cells, prepared by the following procedure. Sample. A sample of human erythrocyte cells was prepared as follows: 8.0 mL of a human blood sample was diluted to 20 mL with 0.9% NaCl solution. After centrifugation, the solution was allowed to stand for 1 h at room temperature (25 °C), in order to facilitate separation of the human blood into plasma and erythrocytes. Then, 8 mL of the lower portion of the solution, containing erythrocytes, was removed using a Pasteur pipet and diluted to 300 mL with 0.9% NaCl solution. This sample solution was diluted and used for the measurements. RESULTS AND DISCUSSION Dependence of Radiation Force on Particle Size. To verify the derived equations, the dependence of the radiation force on the particle size was investigated by measuring the velocity of the particle in a nonflowing medium. The capillary tube and the quartz cell were filled with a sample solution containing polystyrene beads (1, 3, or 6 µm), and the laser beam was focused into the capillary. As described in the theoretical section, the radiation force is dependent on the particle radius divided by the beam radius (a/ω). The beam radius, ω, is a function of the distance from the focal point and increases proportionally when the particle is located sufficiently far away from the focal point. In Figure 3, the radiation forces observed for these particles are plotted as a function of the distance from the beam waist. The solid lines are obtained by theoretical calculation (see eq 13), while the symbols represent experimental data points. The experimental values Analytical Chemistry, Vol. 69, No. 14, July 15, 1997

2703

Figure 3. Relationship between radiation force and distance from beam waist. Particle diameter: (1) 1, (2) 3, and (3) 6 µm. Laser, BeamLok 2060-7S. The laser power is assumed to be 0.45 W. The solid curve is calculated by assuming Q ) 0.129. The measurement is carried out without flow in the sample solution.

Figure 4. Relationship between particle diameter and maximum radiation force obtained at the beam waist. The parameters used are the same as those in Figure 2. The open circles are experimental values. The broken lines are calculated by assuming that the particle size is much smaller than the diameter of the laser beam. The beam waist radius is assumed to be (1) ω0 ) 4.67 and (2) 7.62 µm. The former value is obtained by assuming the TEM00 mode of the laser and the latter value by fitting to the experimental data. The solid curve (3) is calculated theoretically for particles including a > ω0, assuming that ω0 ) 7.62 µm.

agree quite well with the theoretical values. The dependence of the radiation force on the distance from the beam waist is explained by a change in the beam diameter along the beam axis. In other words, the intensity of the laser beam per unit area changes as a function of the distance from the beam waist, and it changes the radiation force applied to the particle aligned along the beam axis. As shown in Figure 3, the radiation force is maximum at the beam waist and decreases with increasing distance from the waist. The relationship between the particle diameter and the maximum radiation force observed at the beam waist is shown in Figure 4. When the particle diameter is much smaller than the beam diameter, the maximum radiation force is proportional to the square of the particle size. This result is consistent with the theory (see eq 13) of optical chromatography. The transverse mode of the laser is assumed to be TEM00, and the beam radius at the waist can be calculated to be 4.67 µm from the laser beam radius at the lens and the focal length of the lens. Obviously, the theoretical line (1) in Figure 4 does not entirely fit the experimental data. The data agree quite well with the theoretical values 2704 Analytical Chemistry, Vol. 69, No. 14, July 15, 1997

Figure 5. Relationship between laser power and radiation force applied to polystyrene beads. Particle diameter, 6 µm. Laser, BeamLok 20060-7S: (1) 0.5, (2) 1, and (3) 2 W. The solid curves are calculated assuming Q ) 0.129.

shown by line 2 in Figure 4 when the laser beam radius is assumed to be 7.62 µm. These data suggest that the laser beam is not correctly focused as expected from theory; i.e., the laser does not have a TEM00 mode. The deviation from a singletransverse mode may be attributed to the poor beam quality of the laser itself and to aberrations in the focusing lens used in this experiment. Thus, care should be exercised with respect to degradation of the beam quality during beam propagation, since this will introduce serious disagreement between the radiation forces obtained theoretically and those obtained experimentally. It is interesting to note that the observed data deviate from a straight line when the particle diameter approaches the beam diameter. This is also predicted by the theory (eq 12), as shown in curve 3 of Figure 4, since the cross section of the particle is sufficiently large to receive nearly all of the laser beam and the radiation force becomes saturated above 10 µm. For collimated light with a flat intensity profile, the radiation force reaches a maximum value at a ) ω. It should be noted that, in the experiments, the maximum radiation force is not obtained at a ) ω, because the intensity profile of the incident beam is Gaussian rather than flat. This is caused by the dependence of the radiation force on the incident angle of the light. The term Q(θ) in eq 9 is at a maximum when the incident angle of the light is near 70°. The total radiation pressure acting on a particle can be found by integrating the product of Q(θ) and the light intensity (eq 11). Thus, for a Gaussian profile beam, the maximum radiation pressure is shifted to particles with radii smaller than the beam radius (a < ω). When the particle radius is larger than the beam radius (a > ω), the radiation force decreases and eventually becomes independent of the particle size. This is because the particle receives all of the light and the laser beam is incident perpendicular to the surface of the particle. The momentum change is not as efficient as that in the case where a ) ω. The relationship between laser power and radiation pressure is shown in Figure 5. Only particles with a diameter of 6 µm are employed in the experiment, and the laser power is changed. As shown in this figure, radiation force is proportional to laser power. Similar results are also obtained when particles with different diameters are used, and the correlation coefficients obtained are summarized in Table 1. As can be seen from the results in Table 1, the observed conversion coefficient agrees well with the theoretical value (Q ) 0.129) calculated from eq 14, when the beam radius is assumed to be 7.62 µm.

Table 1. Relationship between Laser Power and Maximum Radiation Force Applied to a Particle at the Beam Waista particle size (µm)

linear equation

correl coeff, r

Q1

Q2

0.997 3.03 5.85 10.53

Fmax ) 5.45P Fmax ) 55.6P Fmax ) 160P Fmax ) 299P

0.992 0.996 0.972 0.992

0.054 0.060 0.046 0.027

0.144 0.159 0.122 0.071

a F max is the maximum radiation force (in pN) observed at beam waist. P is the laser power (in W). Laser, GLG3200. The laser power was adjusted to 0.26, 0.32, 0.37, 0.42, 0.48, and 0.53 W in the experiments. The conversion coefficients Q1 and Q2 were calculated by assuming ω0 ) 4.67 µm and ω0 ) 7.62 µm, respectively.

Retention Behavior. The most important parameter in optical chromatography is the position of the particle equilibrated by the balance between the radiation force and the force induced by a medium flow. As shown in Figures 3 and 5, the radiation force can be calculated as a function of distance from the beam waist. Resistance force, induced by a medium flow, can be calculated from Stokes’s law. Thus, the position of the particle is readily calculated, permitting an estimation of the radius of the particle from the experimental data without requiring calibration using standard microspheres. In optical chromatography, the total force applied to a particle, Ft, is given by

(15)

Ft ) F - Ff

Figure 6. Relationship between retention distance and particle diameter. The retention distance is calculated using the following parameters: P ) 1 W; λ ) 488 nm; n ) 1.59; ω0 ) 10 µm; v ) 50 µm s-1; and η ) 8.9 × 10-4 Pa s.

distance from the focal point, defined as the retention distance, and the confocal distance, respectively, and λ is the wavelength of the laser. Substituting eq 19 for eq 18, the relationship between the particle size and the retention distance is obtained by

2n1P a2 Q* ) 6πηav c ω 2{1 + (z/z )2} 0 c

Substituting eq 20 into eq 21 and rearranging the equation, the retention distance is calculated as

z) where the negative sign of Ff indicates that the propagation direction of a light is opposite to the direction of a medium flow. According to Stokes’s law, the resistance force applied to the particle moving at a velocity v is given by

Ff ) 6πηav

(16)

where η is the viscosity of a medium. Substituting eqs 13 and 16 for eq 15, the total force can be calculated as follows:

Ft )

2n1P a 2 Q* - 6πηav c ω

()

(17)

When the particle is stopped at an equilibrated position, Ft ) 0:

2n1P a 2 Q* ) 6πηav c ω

()

(18)

The beam radius, ω, is calculated by eq 19, when a singletransverse-mode laser beam with a Gaussian intensity distribution is focused by a lens.

{ ( )}

ω2 ) ω02 1 +

zc )

πω02 λ

z zc

2

(19)

(20)

where ω0 is the beam radius at the focal point, z and zc are the

(21)

x

πω02 λ

n1PQa

3πηvcω02

-1

(22)

The calibration curve of the retention distance calculated from eq 22 for a particle with diameter d() 2a) is shown in Figure 6. The retention distance increases with increasing particle diameter, since the larger particle receives a larger quantity of light and is pushed away from the beam waist by a larger radiation force. Thus, optical chromatography can be used for particle separation. In fact, the retention distance is proportional to the square root of the particle diameter in the middle of the range (1-6 µm). Figure 6 provides additional information concerning dynamic range in optical chromatography. The retention distance decreases steeply around 0.2 µm. The minimum value of the particle diameter, dmin, is determined as the size of the particle that passes through the focal point, at which the radiation force is maximum. The minimum diameter is obtained by substituting z ) 0 in eq 22:

dmin ) 2amin )

3πηvcω02 n1PQ*

(23)

Consequently, dmin is determined by the parameter vω02/P. It should be noted that the ray-optics model cannot be applied to small particles whose diameters are comparable to or less than the wavelength of the laser.8-10 The experimental results obtained in the present study indicate that the ray-optics model is valid for particles larger than 1 µm. Wright and co-workers reported that the radiation force should be calculated using an electromagnetic model for a 1-µm particle when a laser emitting at 1064 nm is Analytical Chemistry, Vol. 69, No. 14, July 15, 1997

2705

used for optical levitation.7 Thus, we may have to use an electromagnetic model, instead of the ray-optics model, when the particle size becomes close to the wavelength of the laser. Consequently, our results calculated for particles smaller than 1 µm may be valid only when a UV laser is used. There is also a maximum particle diameter, dmax, which can be determined by optical chromatography. This limitation can be attributed to the fact that the radiation force is no longer dependent on the particle size when the particle radius, a, is much larger than the beam radius, ω, as described above. Under such conditions, the laser beam is irradiated on the particle perpendicularly at the center of the surface, so that the radiation force becomes

F)

2n1P Q(0) c

(24)

where Q(0) is the efficiency parameter at θ ) 0. Thus, dmax is given by the following equation:

dmax ) 2amax )

2n1PQ(0) 3πηvc

(25)

The parameters n1, Q(0), π, η, and c are constants, and therefore dmax is dependent on P/v but is independent of ω. The dynamic range of optical chromatography, defined as dmax/dmin, is calculated from eqs 23 and 25 as follows:

(

)

dmax 2 n1P 2 ) /9 Q*Q(0) dmin πηvcω02

Analytical Chemistry, Vol. 69, No. 14, July 15, 1997

Selectivity. Selectivity in optical chromatography is defined as the slope in Figures 5 and 6, and is written as

S)

| | ∂ log z ∂ log a

(27)

Assuming that the retention distance, z, is sufficiently long in comparison with the confocal distance, zc, eq 19 can be rewritten as

ω) (26)

The dynamic range of the retention distance, which is calculated by assuming reasonable parameters (listed in Figure 6 caption), extends 3 orders of magnitude, as shown in Figure 6. It is interesting to note that two positions are possible for a particle whose diameter is larger than dmax. This is due to the fact that such a particle can be equilibrated at two positions: one far from the beam waist (regular separation mode) and the other rather near to the beam waist, where the particle diameter is much larger than the beam diameter, thus giving a larger efficiency in light collection but a smaller efficiency in momentum change of the light. If the particle diameter could be continuously decreased, the particle would pass through a beam waist by a liquid flow when d ) dmax. Equation 26 also indicates that the dynamic range can be expanded by using a high-power laser and a low flow rate and by focusing the laser beam tightly. The effects of these parameters are calculated and are shown in Figure 7. Only curves below dmax are drawn in these figures. The dynamic range can be expanded to both sides of the particle diameter by increasing the laser power and by decreasing the flow rate. It is noteworthy that the dynamic range extends more than 4 orders of magnitude under optimized conditions. The range can be expanded only to a smaller side by a tight beam focus, since the higher range is restricted by dmax. However, it should be noted that the separation of the particles becomes smaller, due to the large gradient of the radiation force, when the laser beam is tightly focused. At this condition, collision of the particles, which is neglected in this theory, would become a serious problem. 2706

Figure 7. Relationship between retention distance and particle diameter. P/v ) (1) 106, (2) 105.5, (3) 105, (4) 104.5, and (5) 104 W m s-1. The parameters not specified are the same as those described in Figure 6.

ω0 λ z) z zc πω0

(28)

Substituting eq 28 for eq 18, the relationship between particle size and retention distance can be simplified to

z)

x

ω0 λ

n1πPQ* a 3ηvc

(29)

By calculating logarithms for both the right-hand and left-hand terms, eq 29 can be expressed as

log z ) 1/2 log a + 1/2 log

ω0 n1πPQ* + log 3ηvc λ

(30)

Figure 8 shows the relationship between selectivity and particle diameter, which is calculated from eqs 27 and 30. The selectivity is 0.5 at midrange, which corresponds to a particle diameter of 10-6-10-4 m. The selectivity increases steeply for particles with smaller diameters and gradually for particles with larger diameters as the laser power decreases and the flow rate increases. In other words, a flat (constant selectivity) region can be expanded by increasing the laser power and by decreasing the flow rate. On the other hand, selectivity increases with an increase in beam waist size. A high selectivity can be ascribed to a small gradient of the optical density near the beam waist, indicating that the distance between the particles with slightly different diameters increases at this region. Selectivity gradually degrades for particles with larger diameters, since the radiation force saturates and becomes independent of the particle size, as shown in Figure 4. The flat region can be extended to smaller diameters by focusing the laser beam tightly.

Since the displacement, ζ2 - ζ1, is small, eq 35 can be written as

[

C(ζ2) ) C(ζ1) exp -

]

κ(ζ2 - ζ1)2 2kT

(36)

Equation 36 indicates that the concentration distribution can be written by using a normal error function containing the following standard deviation:

σtherm ) xkT/κ Figure 8. Relationship between selectivity and particle diameter. The parameters used for computer simulation are the same as those described in Figure 7.

Theoretical Plate Number. Based on analogy with chromatography, a theoretical plate number is calculated for performance evaluation of optical chromatography. Assuming that the variation of the signal intensity in optical chromatography is caused by thermal fluctuation, i.e., Brownian motion, of the particle and by instrument fluctuation, induced by variations in laser power and flow rate, the total fluctuation of the signal, namely the standard deviation, σtot, is represented by

σtot2

2

) σtherm + σinst

2

C(z1)

[

) exp -

]

U(z2) - U(z1) kT

(32)

Here, U(z1) and U(z2) are the potentials applied to the particles located at positions z1 and z2, respectively, k is the Boltzmann’s constant, and T is the temperature. To simplify the discussion, the particle position, z, is normalized by the confocal distance, zc:

C(ζ2) C(ζ1)

[

) exp -

]

U(ζ2) - U(ζ1) kT

(33)

where ζ ) z/zc. When a particle is moved from ζ1 to ζ2, e.g., by Brownian motion, the particle tends to come back to the equilibrated position, ζ1, assuming that the particle travels only in one dimension. Thus, the particle behaves like a mass connected to a spring. The potential energy is given by

U(ζ2) - U(ζ1) ) 1/2κ(ζ22 - ζ12)

(34)

where κ is a constant, corresponding to the strength of a spring. Substituting this relationship for eq 29, the concentration distribution can be simplified as follows:

[

C(ζ2) ) C(ζ1) exp -

The variation of the signal induced by instrumental change may be caused by fluctuations in the laser power, the flow rate, etc. The particle position, ζ, is given by the following equation after rearranging eq 21:

1 + ζ2 )

2n1 a2 P Q* 6πηac ω 2 v

(38)

0

A drift in laser power is independent of the flow rate, so that ∆ζ is represented by

∆ζ )

∂ζ ∂ζ ∆P + ∆v ∂P ∂v

(39)

(31)

where σtherm and σinst are the standard deviations of the thermal and instrumental fluctuations, respectively. Thermal fluctuation induced by Brownian motion is represented by

C(z2)

(37)

]

κ(ζ22 - ζ12) 2kT

(35)

Thus, the standard deviation of the signal as a result of instrumental change is given by

σinst )

1 ζ2 + 1 ∆P ∆v 2 ζ P v

(

)

(40)

Consequently, the total standard deviation is represented by

σtot2 )

(

(

))

kT 1 ζ2 + 1 ∆P ∆v + κ 2 ζ P v

2

(41)

By analogy with conventional chromatography, the theoretical plate number in optical chromatography is defined as follows:

N)

ζ2 σtot(ζ)2

(42)

The theoretical plate number calculated as a function of the particle diameter assuming different parameters for P/v is shown in Figure 9. In this calculation, instrument noise is assumed to be 0.05%, which can be achieved by using a semiconductor laser whose output power is regulated by a feedback control and by using a high-precision pump for a liquid flow. The theoretical plate number increases with an increase in the parameter, P/v, so that a high-power laser and a low flow rate are preferential for efficient particle separation, as shown in Figure 9. Needless to say, the theoretical plate number can be substantially improved by reducing fluctuations in instrument conditions. Such fluctuations cause band broadening for the separation of particles larger than 1 µm. On the other hand, the theoretical plate number is largely reduced as a result of Brownian motion for particles smaller than 1 µm. In addition, fluctuation occurs from a Analytical Chemistry, Vol. 69, No. 14, July 15, 1997

2707

Figure 9. Relationship between theoretical plate number and particle diameter. The parameters used for computer simulation are the same as those described in Figure 7.

Figure 10. Relationship between resolution and particle diameter. The parameters used for computer simulation are the same as those described in Figure 7.

resonance in the radiation pressure for such small particles.7-10 This effect has been observed experimentally by Ashkin and Dziedzic as a change in the force necessary for levitation of a droplet as it evaporates.13,14 Thus, the radiation force fluctuates as the particle size approaches the wavelength of the laser. With this in mind, the theoretical plate number may decrease more rapidly for small particles (