Thermal Fluctuation Effects in Magnetophoresis of Superparamagnetic Microbeads

Guntars Kitenbergs, Kristaps Dzilna, Kaspars Ērglis and Andrejs Cēbersa,*

a Laboratory of Soft Materials, Chair of Theoretical Physics, Faculty of Physics and Mathematics, University of Latvia, Zellu 8, LV-1002, Riga, Latvia

*E-mail: aceb@tesla.sal.lv

Abstract. Thermal fluctuation effects in magnetophoresis of commercial superparamagnetic beads are analyzed. By tracking the trajectories of the beads the distribution functions of their displacements in two perpendicular directions are determined. This allows us to determine simultaneously the mean velocity of the beads and their diffusion coefficient. Magnetic moment of a single bead is found by applying the Einstein relation. Distribution of the magnetic moment values are analyzed for two commercial samples of the superparamagnetic beads.

Keywords: Magnetophoresis, thermal fluctuations, magnetic moment, microparticles, nanoparticles. PACS: 05.40.Jc, 87.50.ch, 75.50.Tt, 75.50.Mm.

INTRODUCTION

Determination of the magnetic properties of micron size objects is an important problem in different fields of the physics. Different methods have been proposed. The magnetic moments of magnetotactic bacteria determined by the chains of magnetosomes are measured by tracking reversal of their motion at sudden change of the direction of the applied magnetic field [1,2], from the analysis of their back and forth rotational motion in a rotating magnetic field [3]. Information about the magnetic properties of a flagellated single magnetotactic bacterium may be obtained by analyzing its motion in alternating magnetic fields [3,4]. It should be noted that the methods based on the investigation of the magnetic or optical properties of an ensemble of the magnetic microobjects [5] do not give direct access to the properties of single particles.

Measurement of the magnetic moment of single objects, for example cells, gives information on the amount of the magnetic nanoparticles adsorbed by the cells [6,7]. This allows the prediction of their behavior for the control of tissue regeneration, drug transport under the action of the magnetic field, and other applications [8,9].

Here a new method for the measurement of the magnetic moment of single microobjects based on magnetophoresis in a given non-homogeneous magnetic field under the action of thermal fluctuations is developed. It is illustrated by the measurement of the magnetic moment distribution for two samples of the commercially available superparamagnetic beads (Spherotech and Ademtech Masterbeads). These beads are widely used in biomedicine for purposes of separation [10] and transport [11].

EXPERIMENTAL SETUP

A special magnetophoresis module (see Fig.1) has been created to fit on the Leica DMI3000B microscope. Also a Dantec PIV (Particle Image Velocimetry) system is attached, using a MicroStrobe light source. HiSense camera and Dantec DynamicStudio software are used for the data registration. Camera records 1344×1024 pixel 12-bit image series with 10 Hz and is oriented to match x and y axis directions. As a result particle motion analysis can be done separately for magnetophoresis direction (x-axis) and the perpendicular direction (y-axis). Image scale factor is

291.0=Γ μm/px. A permanent magnet (1 in Fig.1) creates a non-homogenous magnetic field. In the measuring position ( 12=x mm) the magnetic field is 09.0=B T and the magnetic field gradient is 3.4/ =∂∂ xB T/m. These

141

values are considered to be constant in the field of vision. For measurements we use two different superparamagnetic beads – Spherotech SVM-10-10 1.31 μm and Ademtech Masterbeads 500 nm. Sample is introduced between two slides at a distance of 0.028 mm. This makes a cell (3 in Fig.1) where we put a weak concentration particle dilution in TE buffer (few particles per the field of vision).

The PIV system allows us to adjust image exposure parameters to find the best contrast. The camera has high sensitivity, therefore exposure time is less than 5 milliseconds. With DynamicStudio it is possible to record series of pictures with more than 10,000 images.

FIGURE 1. Microscope magnetophoresis module. Orientations of axes are shown.

1 - permanent neodymium magnet, 2 – 40x lens, 3 – sample 10×10×0.028 mm3 cell, 4 – MicroStrobe light source.

DATA PROCESSING AND ANALYSIS

Software in MatLab was developed to process image series. At first the recorded image is read and its preprocessing is carried out. The preprocessing increases image contrast. Then particle positions are found. These steps are done for each image separately to save memory resources. Using detected positions particle tracking is done. For these steps special functions developed in [12] are used. Due to the random character of the process only long trajectories (more than 200 images) are selected. Maximum trajectory length depends on both technical limitations (maximal camera speed – 12Hz) and particle velocity (particle magnetic moment and size, magnetic field). Finally average velocities from particle coordinates x and y are calculated.

An example of the tracked particle trajectory sample by image composition is illustrated in Fig. 2.

FIGURE 2. Trajectory of a single Spherotech 1.31 μm particle. The particle velocity is calculated from its coordinate data.

142

Theoretical Model

The theoretical model is based on the theory of the Brownian motion for a single magnetic particle under the action of the constant Kelvin force )0,/(),( xBmBmfff yx ∂∂=∇==

( m is the magnetic moment of the particle,

xB ∂∂ / is the x component of the magnetic field gradient) [8,13]. The probability density );,( tyxP ∆∆∆ of the particle displacements x∆ and y∆ along x and y axes during the time interval t∆ reads

∆

∆−

∆∆−∆

−∆

=∆∆∆−

tDy

tDtfx

tDtyxP x

4exp

4)(

exp4

1);,(221α

π. (1)

Here D is the diffusion coefficient, α is the hydrodynamic drag coefficient.

The relation (1) shows that the motions along x and y axes are independent and the following relation for the densities of the probability distribution )( , yxvP of the average velocity along x and y axes may be obtained

∆−−

∆=

−

Dtfv

DtvP yxyx

yx 4)(

exp21)(

2,

1,

,

απ

. (2)

The relation (2) shows that although the average velocity of the bead along the magnetic field gradient is

determined by the Kelvin force Bm∇ nevertheless some particles due to the thermal fluctuation effects move faster and some slower than by expected mean velocity Bmv ∇= −1

0 α . Tracking the particle motion, obtaining its statistics and matching it by the normal distribution for the given time interval t∆ gives the diffusion coefficient of the particle and its mean velocity 0v . Since the diffusion coefficient and the hydrodynamic drag coefficient are connected by the Einstein relation α/TkD B= this allows us to measure the magnetic moment of the single bead for the given magnetic field. It is important to notice that this approach is valid independent of how strong the hydrodynamic interaction of the particle with the walls of the cell used in the measurements.

FIGURE 3. Average velocity histograms of a single particle (Fig.2) in x and y axis directions for 5.2=∆t s. Probability density (2) for the parameters D , 0v which are found by fit, are showed by red lines.

Velocity Analysis

By particle tracking the coordinates of the bead are found for different time moments and average velocities v at different observation time t∆ intervals are calculated. For each t∆ a pair of average velocity histograms is constructed (see Fig.3). Parameters D and 0v and their statistical errors (95% confidence level) are found from

143

calculated average velocity data for each trajectory using MatLab normfit function, which tests if the data distribution follows the normal distribution.

To obtain more precise parameter values, we calculate and fit average velocity distributions for different t∆ . To eliminate other effects including particle sedimentation we choose only those particles, for which the determined values of the parameters D and 0v do not depend on the observation time t∆ (see Fig 4.). As a result more precise diffusion coefficient and mean velocity values are obtained, which allow us to calculate magnetic moment of individual particles.

FIGURE 4. Parameter D and 0v values in dependence on observation time t∆ for one particle. Calculated mean values are shown. Each point represents data from a histogram (see Fig. 3)

RESULTS AND DISCUSSION

Using the described method of data analysis we have found the values of the magnetic moments for 20 different particles of both types of beads. For comparison the data of independent magnetic measurements are used: the magnetic moment of the Spherotech beads is estimated according to [14] and for the Ademtech beads the density and size given by supplier together with a magnetization curve (taken by M. M. Maiorov, IPUL) are used. Obtained data and the results of the measurements are shown in Fig. 5. Although the characterized particle ensemble is small nevertheless a noticeable dispersion is detected and demonstrated for both types of particles. The magnetic moment values of the same type of particles can differ by more than two times, therefore having very different resulting attraction to an external magnetic field.

FIGURE 5. Individual particle magnetic moment comparison with estimated values. Left – Spherotech beads, right – Ademtech beads.

144

In our opinion magnetic moment, and accordingly magnetic property dispersion should be strictly characterized. Typical data include only particle size distribution, and measurement made by dynamic light scattering (DLS), which does not directly characterize particle magnetic properties.

A huge problem in this experimental setup is particle sedimentation due to gravity. Due to this the particles not only leave the microscope focus, but also change the mean velocity by approaching the sample walls (following Faxen’s law), resulting in higher dispersion of the average velocity distribution. Until now this problem is solved by adjusting the focus to the middle of the sample.

Particle sedimentation speed has been determined for the Spherotech beads and is 8.1≈Sv μm/min. This results in maximal experiment length limitation around ten minutes. If this effect is eliminated (proved by analysis of magnetophoresis motion in a more viscous medium), the obtained results are more precise and without influence of the sedimentation.

CONCLUSION

A new method for the measurement of the magnetic moment of single superparamagnetic beads is proposed. It is based on the tracking of the particle motion in non-homogeneous magnetic field under the action of thermal fluctuations. Obtained statistics due to the Einstein relation allows one to obtain the hydrodynamic drag coefficient and the magnetic moment of a single superparamagnetic bead. This is demonstrated by the measurements of the distribution of magnetic moments for the two types of the commercially available superparamagnetic beads.

ACKNOWLEDGMENTS

This work is supported by the grants of University of Latvia ESS2009/86 and Z-6191-100. We are thankful to M. M. Maiorov for the magnetization measurements.

REFERENCES

1. F. S. Nogueira, H. G. P. Lins de Barros, Eur. Biophys. J. 24, 13-21 (1995) 2. D. M. S. Esquivel, and H. G. P. Lins de Barros, J. Exp. Biol. 121, 153-163 (1986) 3. K. Ērglis, Qi Wen, V. Ose, A. Zeltins, A. Sharipo, P. A. Janmey, and A. Cēbers, Biophys. J. 93, 1402-1412 (2007) 4. H. Lins de Barros, and D. Acota-Avalos, JMMM 320:e215-e217 (2008) 5. P. C. Logofatu, I. Ardelean, D. Apostol, I. Iordacxhe, M. Bojan, Cr. Moisescu, and B. Ionita, J. Appl. Phys. 103, 094911

(2008) 6. C. Wilhelm, F. Gazeau, and J. - C. Bacri., Eur. Biophys. J. 31, 118-125 (2002) 7. C. Wilhelm, F. Gazeau, J. Roger, J. N. Pons, and J. - C. Bacri. Langmuir 18, 8148-8155 (2002) 8. Q. A. Pankhurst, J. Connoly, S. K. Jones, and J. Dobson, J.Phys. : Appl. Phys. D 36, R167-R181 (2003) 9. Q. A. Pankhurst, N. K. T. Thanh, S. K. Jones, and J. Dobson, J.Phys.:Appl. Phys. D 42, 224001 (2009) 10. K. E. McCloskey, J. J. Chalmers, M. Zborowski, Anal. Chem. 75, 6868-6874 (2003) 11. B. Babic, R. Ghai, and K. Dmitrov., Appl. Phys. Lett. 92, 053901-053901-3 (2008) 12. D. Blair and E. Dufresne. Matlab particle tracking. http://physics.georgetown.edu/matlab/, December 2008. 13. E. Blums, A. Cebers, and M. M. Maiorov, Magnetic Liquids, Berlin - New York: de Gruyter, 1997. 14. J. Schotter. “Development of a magnetoresistive biosensor for the detection of biomolecules”, PhD thesis, University of

Bielefeld, 2004.

145

Copyright of AIP Conference Proceedings is the property of American Institute of Physics and its content may

not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written

permission. However, users may print, download, or email articles for individual use.