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ISSN 0021 3640, JETP Letters, 2012, Vol. 95, No. 11, pp. 560564. © Pleiades Publishing, Inc., 2012. Original Russian Text © M.N. Skryabina, E.V. Lyubin, M.D. Khokhlova, A.A. Fedyanin, 2012, published in Pis'ma v Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki, 2012, Vol. 95, No. 11, pp. 638642.

Probing of Pair Interaction of Magnetic Microparticles with Optical Tweezers¶
M. N. Skryabina, E. V. Lyubin, M. D. Khokhlova, and A. A. Fedyanin
Faculty of Physics, Moscow State University, Moscow, 119991 Russia e mail: fedyanin@nanolab.phys.msu.ru
Received April 18, 2012

Magnetic interaction of paramagnetic Brownian submicron sized particles is studied by optical tweezers technique. Correlation analysis allows one to extract magnetic interaction of two particles 0.4 µm in size, which are optically trapped at the distance of 3 µm one from each other and placed in a static magnetic field of 30 Oe, from the background of their Brownian motion. The magnetic interaction force is estimated to be of approximately 100 fN. Two configurations of the mutual orientation of the magnetic field vector and the line connecting two centers of optical traps are used in the experiment. For field vector orientation paral lel/perpendicular to this line, the magnetic interaction is detected by the cross correlation function increase/decrease in comparison with the absence of magnetic field on the time scales of 1 ms. DOI: 10.1134/S0021364012110094

Magnetic liquids, which are suspensions of mag netic microparticles, have been actively studied in recent decades. These suspensions have numerous practical applications, such as the production of mag netic memory devices [1] and cancer treatment by the hyperthermia method [2]. Single magnetic micropar ticles are used for sensitive monitoring of the proper ties of media [3, 4] and as an instrument allowing one to manipulate single biological cells and macromole cules. As an example, the DNA torsion modulus was measured using magnetic microparticles [5]. The col lective behavior of magnetic microparticles in an external alternating magnetic field was studied in numerous works [6, 7]; however, the problem of pair interaction between magnetic microparticles in sus pension was discussed only in a few papers [4, 8]. Since for these systems magnetic interaction forces on the micron spatial scales are on the order of 100 fN, it is reasonable to use optical tweezers for measuring weak magnetic forces [9, 10]. Optical tweezers are based on the formation of a potential well for transpar ent dielectric microobjects located near the focus of the tightly focused laser beam that allows trapping of microobjects and manipulation of them. Optical twee zers have a lot of applications for studying force inter actions between biological cells [11], measuring elastic characteristics of biological cell membranes and indi vidual macromolecules [12], and studying lumines cent [13] and nonlinear optical [14] properties of sin gle microparticles. This method was also used for studying the behavior of magnetic microparticles in a rotating magnetic field [15] and measuring the mag
¶The

netic moments of trapped particles [8]. However, detection of weak magnetic forces between micropar ticles remains a complicated problem because of Brownian motion. In this paper, a cross correlation function analysis is proposed for detection of the mag netic interaction force between two trapped particles. Previously, this method was used for studying the hydrodynamic interaction between microparticles [16]. Two optically trapped microparticles interact by means of the liquid medium in which they are dipped. Their equations of motion are as follows: d Rn = dt
2

m=1

nm ( R n R m ) [ kr m + F m ( t ) ] ,

(1)

where n, m = 1, 2 are the particle numbers; Rn is the particle radius vector; Fn(t) is the stochastic Brownian force acting on the nth particle; k is the effective stiff ness tensor determining trapping force krn, which appears when the nth particle is displaced by the dis tance rn = Rn R0n from the optical trap center R0n; and nm is the Oseen tensor. Its elements can be writ ten as follows: ^^ I I + RR (2) nn ( R ) = , nm ( R ) = , 8 R where = 3d is the friction coefficient for a spheri cal particle with the diameter d in a liquid medium ^ with the viscosity , I is a 3 в 3 unit matrix, R is the unit vector parallel to the vector connecting particle position points R = R2 R1, and R = R is the dis tance between the particles. Solution of Eqs. (1) gives

article was translated by the authors.

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PROBING OF PAIR INTERACTION OF MAGNETIC MICROPARTICLES

561

the expression for the cross correlation function for vector components of particle positions rn [16]: r 1, i ( 0 ) r 2, i ( t ) = kB T [e 2 ki
t ( 1 + i ) ki /

F

magn

F

magn

e

t ( 1 i ) ki /

] , (3)
1 (a) 2 x

where i = x, y, z, angle brackets denote time averaging, kBT is the product of the Boltzmann constant and absolute temperature, and ki is the effective trap stiff ness in the direction of the i axis. If the line connecting the traps centers is parallel to the x axis, then x = 3d/4L, and y = z = 3d/8L, where L is the distance between the trap centers. The minimum of the cross correlation function is observed at times of about i = /ki. It is equal to kBTi/eki and is induced by hydro dynamic interaction between the particles. Additional interaction between microparticles, for instance magnetic interaction, leads to the displace ment of particles from equilibrium positions in optical traps. This displacement has a magnitude on the order of Fmagn/ki ~ 106 cm. The equipartition theorem for an object in a harmonic potential with stiffness ki can be written as ki r n, i /2 = kBT/2 [10], and the root mean square displacement of particle Brownian motion is estimated to be r n, i ~ 106 cm. There fore, particle displacement from the optical trap cen ter due to Brownian motion is comparable with dis placement under magnetic force action. This compli cates direct measurement of magnetic forces between particles. However, magnetic forces might change the shape of the cross correlation function of Brownian displacement of particles. Indeed, magnetic moments of microparticles lead to either attraction or repulsion of particles. These attractive or repulsive forces change the probability of Brownian displacements of particles from the equilibrium positions in optical traps. In the case of attractive force, the first particle displacement to the right (see Fig. 1a) leads to the increase in the attractive force acting on the second particle. There fore, the probability of the second particle to be dis placed to the left increases, which leads to the decrease in the cross correlation function. In the case of repul sive forces between the particles, the cross correlation function increases. The first particle displacement to the right induces the increase in the repulsive force acting on the second particle, which will more proba bly be shifted to the right (see Fig. 1b). In the present paper, the interaction between para magnetic microparticles is studied by correlation function analysis combined with optical tweezers technique. Interaction is detected by magneto induced changes in the cross correlation function of Brownian displacements of trapped particles. Experimental samples were a water suspension of composite paramagnetic particles with a mean size of 0.4 µm made from silica doped with ferric oxide (III) (Sileks, Russia). The experimental setup of double
JETP LETTERS Vol. 95 No. 11 2012
2 2

F

magn

F

magn

1 (b)

2

x

Fig. 1. Schematic of the influence interaction on cross correlations in particles in optical traps; Fmagn is force; forces of (a) attraction and (b) particles.

of magnetic particle Brownian motion of magnetic interaction repulsion act between

trap optical tweezers with magnetic field application is shown in Fig. 2. Optical traps were formed by tightly focusing the radiation from two infrared CW lasers with the wavelength of 1064 nm into the sample area. The sample suspension was placed between two cover slips separated by 0.15 mm thick gap. The laser beam power in the sample area was approximately 10 mW per trap. The root mean square amplitudes of Brown ian motion displacements of particles were equal to approximately 10 nm, which was sufficient for regis tration of particle displacements by the data acquisi tion system. The system of lenses made it possible to control the positions of optical traps in the sample area by moving the lenses perpendicular to the beam prop agation direction. An external magnetic field was applied to the sample using a system of electromagnets consisting of four independent coils each having a tip pole expansion. A homogeneous magnetic field with different magnetic vector orientations was created by controlling the current through each coil. The field strength was about 30 Oe. For visualization of the experiment, white LED radiation passed through the condenser, illuminated the sample, and then was reg istered using a CCD camera. A typical picture of two trapped microparticles is shown in the inset of Fig. 3. Position sensitive diodes (PSDs) were used for parti cle displacement measurement in real time with preci sion of about 1 nm. Signals from the PSDs were pro

562

SKRYABINA et al.

Fig. 3. Time dependences of normalized cross correlation functions of Brownian displacements for two optically trapped particles: squares are for the absence of magnetic field, circles are for the external magnetic field vector with the magnitude of 30 Oe oriented parallel to the line con necting the two laser traps, and triangles are for the mag netic field vector of the same value and oriented perpen dicular to this line. Inset: microimages of two trapped microparticles. Fig. 2. Schematic of the setup of the double trap optical tweezers: (1, 2) YAG Nd lasers; (3) objective lens; (4)chamber with paramagnetic suspension of microparti cles; (5) condenser lens; (6) system of electromagnets for magnetic field application; (7) LED, (8) position sensitive photodiodes; (9) current boost; (10) analog to digital and digital to analog converters; (11) CCD camera; (13) two magnetic microparticles trapped in two optical traps 12.

for the particle size dispersion and the impact of the particle form, the cross correlation functions were normalized to the dispersion of Brownian displace ments (x2): g(t) = B(t) x1 x
2 2 2

=

x 1 ( 0 ) x 2 ( t ) x1 x
2 2 2

.

(4)

cessed by an analog to digital converter with a sam pling rate of 20 kHz [10]. In the experiment, two magnetic microparticles were optically trapped at the distance of 10 µm above the lower coverslip of the chamber. The distance L between the particles was fixed at 3 µm. First, the sta tistics of Brownian displacements of the particles were recorded for 100 s without an external magnetic field, and the cross correlation functions of the displace ments were determined. In this case, the magnetic moments of particles and their magnetic interaction forces were negligible and the cross correlation func tions were similar to those of dielectric beads given by Eq. (3). Then cross correlation functions of Brownian displacements were measured in parallel and perpen dicular configurations of external dc magnetic field application with the magnitude of the field strength vector H equal to 30 Oe. The magnetic field vector in the parallel configuration was directed along the line connecting the trap centers, i.e., parallel to the x axis. In the perpendicular configuration, it was perpendic ular to the x axis, as shown in Fig. 1. The cross correlation functions of the particle dis placements along the x axis, B(t) = x 1 ( 0 ) x 2 ( t ) , were analyzed since the projections of the magnetic inter action forces on the x axis were maximal. To account

Typical normalized cross correlation functions are shown in Fig. 3. The cross correlation curves exhibit a time delayed anticorrelation with a pronounced min imum at 0.3 ms induced by hydrodynamic interaction between the particles. The depth of the minimum is about 0.04 when the magnetic field is absent. This allows one to give estimates of the stiffness of optical traps and the diameter of particles which appear to be kx 2 в 102 dyn/cm and d 4 в 105 cm, respectively. A decrease in the cross correlation functions is observed when the magnetic field vector orientation is parallel to the x axis, meaning that the anticorrelation becomes more pronounced. When the magnetic field vector is perpendicular to the x axis, an increase in the cross correlation functions is observed, indicating the appearance of correlation of the particle displace ments. In the presence of an external magnetic field, mag netic moments were induced in the particles. In case of magnetic field vector orientation parallel or perpen dicular to the x axis, the magnetic particles were ori ented so that their magnetic moments were codirec tional with each other and the external magnetic field (Fig. 4). In the dipole approximation, the x axis pro
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563

jections of magnetic forces for the magnetic vector parallel to the x axis can be written as follows [4, 8]: F
||, 1

= F

||, 2

= 6M . 4 R

2

(5)
Fig. 4. Schematic diagram of mutual orientation of mag netic moments and forces. The magnetic field vector is (a) parallel and (b) perpendicular to the line connecting the optical trap centers; Fmagn is the magnetic interaction force, H is the external magnetic field vector, and R is the vector connecting the positions of particles. Magnetic moments are shown by gray arrows.

In the case of magnetic vector orientation perpendic ular to this axis, the magnetic force projections are as follows: F
, 1

= F

, 2

=

3M , 4 R

2

(6)

where M is the value of the microparticle magnetic moment. If particle magnetic moments are perpendic ular to the line connecting the trap positions, the repulsive forces appear between the particles. In the case of orientation of the magnetic moments parallel to the line connecting the trap positions, the attractive forces take place. For the same magnetic moment val ues, the attractive force magnitude is double the repul sive force value. With regard to the magnetic interac tion forces, the equations of particle motion can be written as follows: d Rn = dt
2

influences the form of the cross correlation function of displacements: B
magn

(t) =

kB T e 2

t ( 1 + x ) kx /

k

e

t ( 1 x ) k' / x

x

k 'x

,

(9)

m=1

nm ( R n R m )
magn, m

(7)

в [ kr m + F m ( t ) + F

],

where Fmagn, n is the magnetic interaction force acting on the nth particle. Neglecting the changes in the dis tance between the particles induced by their Brownian motion along the y and z axes, we consider that the dis tance between the particles can be written as R = L + x2 x1, where x1 and x2 are the x axis projections of Brownian displacements of particles. Therefore, the impact of magnetic forces on the cross correlation functions can be estimated. The Brownian displace ments of particles are on the order of tens of nanome ters; i.e., they are much less than the distance between the particles that is on the order of several microns. Therefore, magnetic forces (5) and (6) can be expanded in the small parameter /L (x2 x1)/L (first two terms of the expansions are retained): F F
||, 1

where k 'x = kx 48M2/L5 if the external magnetic field vector is parallel to the x axis and k 'x = kx + 24M2/L5 if it is perpendicular to the x axis. In the presence of a magnetic field, the dispersion of the Brownian dis placements is as follows: x2 = kBT(kx + k 'x )/2kx k 'x . The largest magneto induced changes in the cross correlation function are observed for t = 0. Consider the expression for the difference g(0) = gmagn(0) g(0) revealing the magneto induced changes in the normalized cross correlation functions. For parallel and perpendicular magnetic field configurations, it can be written as follows: g || ( 0 ) = 24 M , 5 2 L k x 24 M
2 2

(10)

12 M . g ( 0 ) = 5 2 L k x + 12 M According to expressions (10), the normalized cross correlation function change is negative for the attrac tive forces and positive for the repulsive forces, and g || ( 0 ) > g ( 0 ) , which is in agreement with the experimental results. Cross correlation function changes can be estimated as g ( 0 ) 0.01 and the magnetic moment value is approximately M 5 в 10 11 erg/G (see Fig. 3). Magnetic interaction forces are estimated using Eqs. (5) and (6). The values of the forces are approximately equal to 108 dyn, i.e., on the order of 100 fN. In conclusion, the presence of magnetic interac tion considerably changes the statistical properties of the Brownian motion of particles. Namely, the cross correlation function of Brownian displacements of two optically trapped paramagnetic particles changes considerably in the presence of an external magnetic

= F = F

||, 2

6 M 24 M , 4 5 L L
2 2

2

2

(8)

, 1

, 2

3 M 12 M 4 + . 5 L L

Each of Eqs. (8) consists of two components. The first component is independent of Brownian displace ments and proportional to M2/L4. This force compo nent displaces the particles from the equilibrium posi tions in the optical trap. The second component depends on the Brownian displacements and therefore
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SKRYABINA et al. 4. L. E. Helseth, J. Phys. D 40, 3030 (2007). 5. K. C. Neuman and A. Nagy, Nature Meth. 5, 491 (2008). 6. S. Melle, O. G. Calderon, M. A. Rubio, and G. G. Ful ler, J. Non Newtonian Fluid Mech. 102, 135 (2002). 7. S. Melle and J. E. Martin, J. Phys. Chem. 118, 9875 (2003). 8. L. E. Helseth, Opt. Commun. 276, 277 (2007). 9. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, Opt. Lett. 11, 288 (1986). 10. K. C. Neuman and S. M. Block, Rev. Sci. Instum. 75, 2787 (2004). 11. M. D. Khokhlova, E. V. Lyubin, A. G. Zhdanov, et al., J. Biomed. Opt. 17, 025001 (2012). 12. C. Cecconi, E. A. Shank, S. Marqusee, and C. Busta mante, Methods Mol. Biol. 749, 255 (2011). 13. A. Zhdanov, M. P. Kreuzer, S. Rao, et al., Opt. Lett. 33, 2749 (2008). 14. X. Vidal, A. Fedyanin, A. Molinos Gomez, et al., Opt. Lett. 33, 699 (2008). 15. G. Romano, L. Sacconi, M. Capitanio, and F. S. Pavo ne, Opt. Commun. 215, 323 (2003). 16. S. C. Meiners and S. R. Quake, Phys. Rev. Lett. 82, 277 (1999).

field. The changes in the cross correlation functions are dependent on the values and directions of mag netic interaction forces. Therefore, the suggested method allows detecting weak magnetic forces on the order of femtonewtons, whereas the direct measure ment of these forces is complicated because of the presence of stochastic Brownian motion. Such parti cle interaction forces considerably affect various pro cesses in magnetic suspensions, such as magnetic par ticle aggregation, magnetic fluid flow, and magnetic field induced structure formation. This work was performed using CKP facilities and supported by the Russian Foundation of Basic Research and the Ministry of Education and Science of the Russian Federation. We thank A. Zhigulin for providing the experimental samples. REFERENCES
1. R. Palm and V. Korenivski, New J. Phys. 11, 1 (2009). 2. V. M. Laurent, S. Henon, E. Planus, et al., J. Biomech. Eng. 124, 408 (2002). 3. B. H. McNaughton, K. A. Kehbein, J. N. Anker, and R. Kopelman, J. Phys. Chem. 110, 18958 (2006).

JETP LETTERS

Vol. 95

No. 11

2012