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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 5, Iss. 3 — Feb. 10, 2010
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Rotational dynamics of optically trapped nanofibers

Antonio Alvaro Ranha Neves, Andrea Camposeo, Stefano Pagliara, Rosalba Saija, Ferdinando Borghese, Paolo Denti, Maria Antonia Iatì, Roberto Cingolani, Onofrio M. Maragò, and Dario Pisignano  »View Author Affiliations


Optics Express, Vol. 18, Issue 2, pp. 822-830 (2010)
http://dx.doi.org/10.1364/OE.18.000822


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Abstract

We report on the experimental evidence of tilted polymer nanofiber rotation, using a highly focused linear polarized Gaussian beam. Torque is controlled by varying trapping power or fiber tilt angle. This suggests an alternative strategy to previously reported approaches for the rotation of nano-objects, to test fundamental theoretical aspects. We compare experimental rotation frequencies to calculations based on T-Matrix formalism, which accurately reproduces measured data, thus providing a comprehensive description of trapping and rotation dynamics of the linear nanostructures.

© 2010 OSA

1. Introduction

Optical forces are currently employed to study a range of chemical, physical and biological problems, by trapping microscale objects and measuring sub pico-Newton forces [1

1. K. Dholakia, P. Reece, and M. Gu, “Optical micromanipulation,” Chem. Soc. Rev. 37(1), 42–55 (2007). [CrossRef]

3

3. A. Jonáš and P. Zemánek, “Light at work: The use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis 29(24), 4813–4851 (2008). [CrossRef]

]. On optically trapped objects, the mechanisms for rotating consist in exploiting the physical properties of the trapping beam, the trapped object or both. A laser beam can carry intrinsic (spin) or extrinsic (orbital) angular momentum, associated to the polarization and to the light beam phase structure, respectively [4

4. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997). [CrossRef] [PubMed]

7

7. H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. 155(1-3), 169–179 (1998). [CrossRef]

]. Either trapping beams with elliptical polarization or with a rotating linear polarization can be exploited to apply a torque to trapped objects [8

8. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23(1), 1–3 (1998). [CrossRef]

12

12. P. Galajda and P. Ormos, “Orientation of flat particles in optical tweezers by linearly polarized light,” Opt. Express 11(5), 446–451 (2003). [CrossRef] [PubMed]

]. Rotation of trapped particles can also be induced by exploiting the phase structure (such as Laguerre-Gaussian or Bessel beams) or by modifying the spatial intensity profile of the trapping focal spot [13

13. E. Santamato, A. Sasso, B. Piccirillo, and A. Vella, “Optical angular momentum transfer to transparent isotropic particles using laser beam carrying zero average angular momentum,” Opt. Express 10(17), 871–878 (2002). [PubMed]

16

16. S. Sato, M. Ishigure, and H. Inaba, “Optical trapping and rotational manipulation of microscopic particles and biological cells using higher-order mode Nd:YAG laser beam,” Electron. Lett. 27(20), 1831–1832 (1991). [CrossRef]

]. The rotatable object can be spherical, exhibiting a birefringence or a slight absorption, or it can have more complex shapes, as in microfabricated propellers by two-photon polymerization [17

17. P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78(2), 249–251 (2001). [CrossRef]

19

19. E. Higurashi, H. Ukita, H. Tanaka, and O. Ohguchi, “Optically induced rotation of anisotropic micro-objects fabricated by surface micromachining,” Appl. Phys. Lett. 64(17), 2209–2210 (1994). [CrossRef]

] or cylinders with inclined faces [20

20. R. C. Gauthier, “Optical levitation and trapping of a micro-optic inclined end-surface cylindrical spinner,” Appl. Opt. 40(12), 1961–1973 (2001). [CrossRef]

].

Present methods used to rotate nano- and micro-objects require the manipulation of the beam profile or polarization, thus being scarcely efficient due to power loss in the process. Moreover, the trapped object needs to be slightly absorptive, birefringent, or specifically microfabricated. These restrictions are here avoided by a strategy to rotate a dielectric cylinder with flat end faces, based on a non-rotating linear polarized Gaussian (TEM00) beam, carrying neither intrinsic nor extrinsic angular momentum. This enables a detailed analysis of the torque acting on fibers, whose experimental results are compared with calculations of optical trapping and rotation of linear nanostructures through a full electromagnetic theory.

2. Theory

2.1 Radiation force and torque

Light forces are generated by the scattering of electromagnetic fields incident on a particle, hence the quantitative understanding of optical trapping has to rely on the scattering theory of electromagnetic radiation [29

29. F. Borghese, P. Denti, and R. Saija, Scattering from model nonspherical particles (Springer, Berlin, 2007), 2nd ed.

33

33. A. A. R. Neves, A. Fontes, C. L. Cesar, A. Camposeo, R. Cingolani, and D. Pisignano, “Axial optical trapping efficiency through a dielectric interface,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76, 061917 (2007). [CrossRef]

]. The difficulties arising from the use of the full scattering theory are generally overcome by solving the problem in different regimes depending on the size of the scatterer. Moreover the models traditionally used for calculating optical forces are based on approximations which often limit the discussion only to spherical particles. On the contrary, in order to calculate the radiation force [34

34. F. Borghese, P. Denti, R. Saija, and M. A. Iatì, “Optical trapping of nonspherical particles in the T-matrix formalism,” Opt. Express15,11984–11998 (2007); 15, 14618-14618 (2007). http://www.opticsinfobase.org/abstract.cfm?&id=141215 [CrossRef]

] and torque [35

35. F. Borghese, P. Denti, R. Saija, M. A. Iatì, and O. M. Maragò, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100(16), 163903 (2008). [CrossRef] [PubMed]

,36

36. F. Borghese, P. Denti, R. Saija, and M. A. Iatì, “Radiation torque on nonspherical particles in the transition matrix formalism,” Opt. Express 14(20), 9508–9521 (2006). [CrossRef] [PubMed]

] we use the full scattering theory in the framework of the transition matrix (T-matrix) approach. In fact, this approach is quite general as it applies to particles of any shape and refractive index for any choice of the wavelength. Our starting point is the calculation of the field configuration in the focal region of a high numerical aperture (NA) objective lens in absence of any particle, using the procedure originally formulated by Richards and Wolf [37

37. B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]

]. The resulting field is considered as the field incident on the particles, and the radiation force and torque exerted on any particle within the region is calculated by resorting to conservation of linear and angular momentum for the combined system of field and particles. As a result the optical force and torque exerted on a particle turn out to be given by the integrals [34

34. F. Borghese, P. Denti, R. Saija, and M. A. Iatì, “Optical trapping of nonspherical particles in the T-matrix formalism,” Opt. Express15,11984–11998 (2007); 15, 14618-14618 (2007). http://www.opticsinfobase.org/abstract.cfm?&id=141215 [CrossRef]

,36

36. F. Borghese, P. Denti, R. Saija, and M. A. Iatì, “Radiation torque on nonspherical particles in the transition matrix formalism,” Opt. Express 14(20), 9508–9521 (2006). [CrossRef] [PubMed]

]:
FRad=r2Ωr^TMdΩ
(1)
MRad=r3Ωr^TM×r^dΩ
(2)
where the integration is over the full solid angle, r is the radius of a large (possibly infinite) sphere surrounding the particle centre, and TM is the time averaged Maxwell stress tensor:
TM=18πRe[n2EE*+BB*12(n2|E|2+|B|2)I]
(3)
where ⊗ denotes dyadic product, I is the unit dyadic and n is the refractive index of the medium surrounding the particle. When the incident field is a polarized plane wave, the components of the radiation force along the direction of the unit vector v^ξ are given by [34

34. F. Borghese, P. Denti, R. Saija, and M. A. Iatì, “Optical trapping of nonspherical particles in the T-matrix formalism,” Opt. Express15,11984–11998 (2007); 15, 14618-14618 (2007). http://www.opticsinfobase.org/abstract.cfm?&id=141215 [CrossRef]

]:
FRadξ=r216πRe(r^v^ξ)[n2(|ES|2+2EIES)+(|BS|2+2BIBS)]dΩ
(4)
where EI and BI are the incident fields, while ES and BS are the fields scattered by the particle. In turn the radiation torque takes on the form:
MRad=18πr3Ren2r^(EI+ES)(EI+ES)×r^dΩ
(5)
Expanding the incident field in a series of vector spherical harmonics with (known) amplitudes Wlmp, the scattered field can be expanded on the same basis with amplitudes Al'm'p'. The relation between the two amplitudes is given byAl'm'p'=plmSl'm'lmp'pWlmp, where Sl'm'lmp'p is the T-matrix of the particle. In this framework, several kinds of non-spherical particles can be modeled as aggregates of spherical scatterers with size below the radiation wavelength. The elements of the T-matrix are calculated in a given frame of reference through the inversion of the matrix of the linear system obtained by imposing to the fields boundary conditions across each spherical surface [29

29. F. Borghese, P. Denti, and R. Saija, Scattering from model nonspherical particles (Springer, Berlin, 2007), 2nd ed.

]. Here we stress that each element of the T-matrix turns out to be independent both on the direction of propagation and on the polarization of the incident field. Thus they do not change when the incident field is a superposition of plane waves with different direction of propagation, i.e. for the description of a focused laser beam in the angular spectrum representation [38

38. L. Novotny, and B. Hecht, Principles of Nano-Optics (Cambridge University Press, New York, 2006).

].

For polymer nanofibers, we calculate the radiation force (Frad) and torque (Mrad) exerted by the optical tweezers, by modeling the nanostructures as linear chains of spheres with diameter, D, and length, L, equal to the fiber diameter and length, respectively. In Fig. 1(a)
Fig. 1 (a) Sketch of the geometric configuration for a fiber trapped and rotated about its center-of-mass (black dot). Rotation occurs in the xy plane. (b) Geometry of the optical trapping of a fiber rotated about a point shifted by ξ towards the edge of the fiber (white dot).
-1(b) we schematize the geometrical configuration of the system. In particular, the calculations of the torque can be obtained for any orientation of the polymer fiber and for different trapping positions.

2.2 Hydrodynamics

When dealing with quantitative comparisons between theory and experiments, a crucial issue to be addressed is the hydrodynamics of the trapped particle. For linear nanostructures (rigid rod-like structures), the viscous drag is described by an anisotropic hydrodynamic mobility tensor, whose components depend on the length of the linear structure (L) and on the length-to-diameter ratio, p=L/D [39

39. S. Broersma, “Viscous force and torque constants for a cylinder,” J. Chem. Phys. 74(12), 6989–6990 (1981). [CrossRef]

]. Symmetry considerations reduce the relevant hydrodynamics parameters to the translational, Γ and Γ, and rotational, ΓRot, mobilities [22

22. O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, and A. C. Ferrari, “Femtonewton force sensing with optically trapped nanotubes,” Nano Lett. 8(10), 3211–3216 (2008). [CrossRef] [PubMed]

], specifically when center-of-mass rotation is considered:
Γ=lnp+δ4πηL,Γ||=lnp+δ||2πηL,ΓRot=3(lnp+δRot)πηL3
(6)
where Γ and Γ|| are the translational mobilities, transverse and parallel to the main axis respectively, ΓRot is the rotational mobility about the center-of-mass, where η is the dynamical viscosity of the surrounding medium, and δRot represents end corrections, calculated as polynomial of (ln2p)1 [39

39. S. Broersma, “Viscous force and torque constants for a cylinder,” J. Chem. Phys. 74(12), 6989–6990 (1981). [CrossRef]

]. On the other hand, when the pivot point of the rotation is shifted by a value ξ from the center-of-mass, we need to change L with L+2ξ in the rotational mobility [Eq. (6)].
ΓRot(L+2ξ)=3(lnp˜+δ˜Rot)πη(L+2ξ)3
(7)
where p˜=(L+2ξ)/D is an effective length-to-diameter ratio and δ˜Rot is an effective end correction that takes into account the shift of the rotation pivot point with respect to the center-of-mass. For a rotating optically trapped polymer nanofiber, the radiation torque Mrad is counterbalanced by the hydrodynamic viscous torque (for the low Reynolds number regime) Mhydro=Ω/ΓRotn^ (n^ is the rotation axis) and the fiber rotates at a constant rotation frequency:
Ω=|Mrad|ΓRot(L+2ξ)
(8)
that is dependent both on the length of the fiber and on the pivot point position. This relation holds when the fiber rotates in a plane orthogonal to the optical axis, i.e. for a situation fulfilled in our experiments where the tilting angle is close to 90° i.e. sinθ≈1. Calculating the torque from our electromagnetic theory and Eq.s (7) and (8) yields the theoretical rotation frequency for the trapped fiber, directly comparable to experimental values.

3. Method

3.1 Realization of nanofibers

The polymeric nanofibers are fabricated by electrostatic spinning (ES) [40

40. D. Li and Y. Xia, “Electrospinning of Nanofibers: Reinventing the Wheel?” Adv. Mater. 16(14), 1151–1170 (2004). [CrossRef]

,41

41. D. H. Reneker and I. Chun, “Nanometre diameter fibres of polymer, produced by electrospinning,” Nanotechnology 7(3), 216–223 (1996). [CrossRef]

], exploiting a high electrostatic field (~0.9 kV cm−1) to stretch a jet of polymer solution. Our samples are made by spinning a formic acid solution of poly(methylmethacrylate) (PMMA) with concentration of 26% (w/w). Due to the formic acid high conductivity and dielectric constant, the realization of uniform and beads-free fibers with a narrow size dispersion and diameters in the sub-micrometer range can be obtained. In a typical ES process, a 0.5 ml of PMMA solution is loaded into a 1.0 mL plastic syringe tipped with a 19-gauge stainless steel needle. The positive lead from a high voltage supply (XRM30P, Gamma High Voltage Research Inc., Ormond Beach, FL) is connected to the metal needle applying a bias of 9 kV. The solution is injected at the end of the needle at a constant rate of 10 µL/min by a syringe pump (33 Dual Syringe Pump, Harvard Apparatus Inc., Holliston, MA), which prevents dripping at the end of the metallic capillary. Fibers are collected as non-woven mat on an aluminum collector negatively biased at −2 kV and placed at a distance of 12 cm from the needle. All the ES experiments are performed at room temperature with air humidity about 40%. Finally, fibers are mechanically removed from the collector and stored in a vial containing distilled water. To allow the subsequent separation and fragmentation, the suspension of fibers in distilled water is sonicated for 1 hour at 25 °C before the trapping experiments.

3.2 Optical tweezers

Experimentally, our optical trap is custom-built on an inverted microscope (Zeiss Axiovert 40) as shown in Fig. 2(a)
Fig. 2 (a) Scheme of the optical tweezers set-up with detection using backscattered light. The arrows indicate the light paths. (b)-(e): Optical rotation of the polymer fiber, schematized (top, Media 1), and imaged (bottom). The fiber main axis is tilted by θ from the optical axis, the trapping point is shifted by ξ from the center-of-mass (CM). Scale bar 2.5 µm.
, and based on a Ti:Sapphire laser (λ = 800 nm, Coherent). This is strongly focused to a diffraction-limited spot on the objective focal plane, by overfilling the back aperture of an oil-immersion infinity-corrected objective lens (100 × /1.3, Zeiss Plan-Neofluar) [42

42. A. A. R. Neves, A. Fontes, L. Y. Pozzo, A. A. de Thomaz, E. Chillce, E. Rodriguez, L. C. Barbosa, and C. L. Cesar, “Electromagnetic forces for an arbitrary optical trapping of a spherical dielectric,” Opt. Express 14(26), 13101–13106 (2006). [CrossRef] [PubMed]

]. Bright field images and videos are recorded by a charge coupled device camera using the same objective lens as the trapping laser. The dispersed polymer nanofibers are placed in a sample cell comprises a poly(dimethylsiloxane) chamber in conformal contact with a glass cover slip, thus defining a 100 µL volume of the water suspension of fibers. The cover slip is mounted on a piezoelectric stage, allowing travelling over 300 µm along each axis with nanometric spatial resolution.

4. Results and discussion

Figures 2(b)-(e) highlight the stages of fiber trapping and rotation. The particle is picked up from glass with low optical power [< 50 mW, Fig. 2(b)], and taken to a distance from the cover slip slightly larger than the fiber length. Once in this position the fiber stands upright, aligning its longitudinal axis to that of the optical axis of the laser beam [Fig. 2(c)]. We then increase the trapping power (100-400 mW) to confine the fiber in a stiff trap (Media 2). In this configuration we can translate the beam and the fiber along all the three axes. By approaching the cover slip to the fiber using the piezo-stage, the glass surface is led again in contact with the bottom tip of the fiber, which starts to tilt by an angle θ [Fig. 2(d)]. Above a critical angle depending on the trapping power, the fiber begins to rotate at constant rate (Ω) (Media 1). For some samples, we can also finally lower the piezo-stage, thus leaving the fiber not in contact with the glass surface while continuing to rotate in its tilted configuration [Fig. 2(e)]. This rotation mechanism is different from previously observed complex oscillations of tin oxide nanowires, attributed to the particle asymmetric cross-section shape [44

44. P. J. Pauzauskie, A. Radenovic, E. Trepagnier, H. Shroff, P. Yang, and J. Liphardt, “Optical trapping and integration of semiconductor nanowire assemblies in water,” Nat. Mater. 5(2), 97–101 (2006). [CrossRef] [PubMed]

].

We then analyze the dynamics of tilted fibers by measuring the rotating frequency as a function of the incident optical power (Media 4). A time series of the quadrant photodetector signal over 20 s is used to determine the rotation frequency, from the frequency peak of the power spectrum [Fig. 4(b)]. Since only one sharp and symmetrical peak is detected along with its harmonics in the power spectral density, we conclude that the observed frequency is that of a continuous rotation of the fiber without nutation. We find that the rotation frequency increases linearly with the trapping power [linear fits through the origin shown in Fig. 4(c)], which rules out the occurrence of nonlinear effects in the investigated experimental range. Moreover, Ω decreases upon increasing the fiber length, as expected from Eq. (8).

5. Conclusions

Acknowledgments

This work was partially supported by the Italian Minister of University and Research through the FIRB programs RBIN045NMB and RBIP06SH3W, and by the Apulia Regional Strategic Project PS_144. The authors gratefully acknowledge R. Stabile for the SEM images.

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P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78(2), 249–251 (2001). [CrossRef]

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34.

F. Borghese, P. Denti, R. Saija, and M. A. Iatì, “Optical trapping of nonspherical particles in the T-matrix formalism,” Opt. Express15,11984–11998 (2007); 15, 14618-14618 (2007). http://www.opticsinfobase.org/abstract.cfm?&id=141215 [CrossRef]

35.

F. Borghese, P. Denti, R. Saija, M. A. Iatì, and O. M. Maragò, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100(16), 163903 (2008). [CrossRef] [PubMed]

36.

F. Borghese, P. Denti, R. Saija, and M. A. Iatì, “Radiation torque on nonspherical particles in the transition matrix formalism,” Opt. Express 14(20), 9508–9521 (2006). [CrossRef] [PubMed]

37.

B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]

38.

L. Novotny, and B. Hecht, Principles of Nano-Optics (Cambridge University Press, New York, 2006).

39.

S. Broersma, “Viscous force and torque constants for a cylinder,” J. Chem. Phys. 74(12), 6989–6990 (1981). [CrossRef]

40.

D. Li and Y. Xia, “Electrospinning of Nanofibers: Reinventing the Wheel?” Adv. Mater. 16(14), 1151–1170 (2004). [CrossRef]

41.

D. H. Reneker and I. Chun, “Nanometre diameter fibres of polymer, produced by electrospinning,” Nanotechnology 7(3), 216–223 (1996). [CrossRef]

42.

A. A. R. Neves, A. Fontes, L. Y. Pozzo, A. A. de Thomaz, E. Chillce, E. Rodriguez, L. C. Barbosa, and C. L. Cesar, “Electromagnetic forces for an arbitrary optical trapping of a spherical dielectric,” Opt. Express 14(26), 13101–13106 (2006). [CrossRef] [PubMed]

43.

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75(3), 594–612 (2004). [CrossRef]

44.

P. J. Pauzauskie, A. Radenovic, E. Trepagnier, H. Shroff, P. Yang, and J. Liphardt, “Optical trapping and integration of semiconductor nanowire assemblies in water,” Nat. Mater. 5(2), 97–101 (2006). [CrossRef] [PubMed]

45.

F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Radiation torque exerted on a spheroid: Analytical solution,” Phys. Rev. A 78(1), 013843 (2008). [CrossRef]

OCIS Codes
(160.5470) Materials : Polymers
(170.0180) Medical optics and biotechnology : Microscopy
(170.4520) Medical optics and biotechnology : Optical confinement and manipulation
(160.4236) Materials : Nanomaterials
(350.4855) Other areas of optics : Optical tweezers or optical manipulation
(290.5825) Scattering : Scattering theory

ToC Category:
Optical Trapping and Manipulation

History
Original Manuscript: September 11, 2009
Revised Manuscript: October 19, 2009
Manuscript Accepted: October 28, 2009
Published: January 6, 2010

Virtual Issues
Vol. 5, Iss. 3 Virtual Journal for Biomedical Optics

Citation
Antonio Alvaro Ranha Neves, Andrea Camposeo, Stefano Pagliara, Rosalba Saija, Ferdinando Borghese, Paolo Denti, Maria Antonia Iatì, Roberto Cingolani, Onofrio M. Maragò, and Dario Pisignano, "Rotational dynamics of optically trapped nanofibers," Opt. Express 18, 822-830 (2010)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-2-822


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