Rotational dynamics of optically trapped polymeric nanofibers

The optical trapping of polymeric nanofibers and the characterization of the rotational dynamics are reported. A strategy to apply a torque to a polymer nanofiber, by tilting the trapped fibers using a symmetrical linear polarized Gaussian beam is demonstrated. Rotation frequencies up to 10 Hz are measured, depending on the trapping power, the fiber length and the tilt angle. A comparison of the experimental rotation frequencies in the different trapping configurations with calculations based on optical trapping and rotation of linear nanostructures through a T-Matrix formalism, accurately reproduce the measured data, providing a comprehensive description of the trapping and rotation dynamics.


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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 [1,2]. In particular, optical trapping of elongated nanoparticles, including nanowires [3] and nanotubes [4], is gaining an increasing interest because of the high shape anisotropy and unique physical properties of these systems. Among linear nanostructures, polymeric nanofibers are novel nanomaterials with many strategic applications ranging from scaffolding for tissue-engineering to integrated photonics [5,6] and electronics [7,8]. However, optical trapping and manipulation of polymer nanofibers has never been reported, despite the understanding of the optical forces and torques acting on these objects as well as their trapping dynamics might open a new range of applications, exploiting the polymeric fibers as local probes or active elements in microrheology [9] and microfluidics [10], and in next generation Photonic Force Microscopy [3]. Furthermore, the nanofibers are characterized by subwavelength diameters and lengths in the range 10-100 m, therefore constituting ideal systems for studying effects occurring in the intermediate regime between the Rayleigh scattering and geometrical optics.
A laser beam can carry intrinsic (spin) or extrinsic (orbital) angular momentum, associated to the polarization and to the light beam phase structure, respectively [11]. Either trapping beams with elliptical polarization or with a rotating linear polarization can be exploited to apply a torque to trapped objects. Rotation in trapped particles can also be induced by exploiting the phase structure or the astigmatism of the trapping laser beam [12 and references therein]. 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 [13] or cylinders with inclined faces [14].
In this work we trap polymeric nanofibers and characterize their rotational dynamics in different trapping configurations by a different method. We employ a strategy to rotate a dielectric cylinder with flat end faces, based on a non-rotating linear polarized Gaussian (TEM 00 ) beam, carrying neither intrinsic nor extrinsic angular momentum. The relevant advantages of this approach is that the trapped object does not need to be slightly absorptive, birefrigent, or specifically microfabricated. In 3 addition, one does not need to manipulate the beam profile or polarization, being in this way more effective. 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.
A detailed treatment of our modelling procedure is given in Ref.s [15,16]. Here we give a short account of calculations of the optical force and torque applied to polymer nanofibers (see Supplemental files for more details) [17]. 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 particles [15]. Once the field is known, the radiation force and torque exerted on a trapped particle is calculated by resorting to momentum conservation for the combined system of field and particle.
Then we use an approach based on the Transition (T-)Matrix formalism [18], to calculate the optical force and torque applied to the trapped object, of arbitrary size and symmetry, which is being modelled as an aggregate of spheres with size below the radiation wavelength. In the specific case of polymer nanofibers we calculate the radiation force ( rad F ) and torque ( rad M ) exerted by the optical tweezers, by modelling the nanostructures as linear chains of spheres with diameter, D , and length, L , equal to the fiber diameter and length, respectively. In particular, the calculations of the torque can be obtained for any orientation of the polymer fiber and for different trapping positions.
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  [19]. Symmetry considerations reduce the relevant hydrodynamics parameters to the translational,   and ||  , and rotational, Rot  , mobilities [4,17]. Specifically when center-of-mass rotation is considered, the rotational mobility is: that is dependent both on the length of the fiber and on the pivot point position. Thus calculating the torque from our electromagnetic theory and using Eq. (1) and Eq. (2) yields the theoretical rotation frequency for the trapped fiber, that we can directly compare to measured experimental values.
Experimentally, our optical trap is custom-built on an inverted microscope (Zeiss Axiovert 40) as shown in Fig. 1(a), and based on a Ti:Sapphire laser (  =800 nm, Coherent). This is strongly focused to a diffraction-limited spot on the objective focal plane, by overfilling the back aperture of an oilimmersion infinity-corrected objective lens (100×/1.3, Zeiss Plan-Neofluar) [20]. Bright field images and videos are recorded by a charge coupled device camera using the same objective lens as the trapping laser. The polymeric nanofibers are fabricated by electrostatic spinning [21,22], 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) [17] and dispersing the fibers (having diameters in the range 200-600 nm) in distilled water after complete solvent evaporation.
The sample cell comprises a poly(dimethylsiloxane) (PDMS) 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.
The dynamics and tracking of the polymer fibers is investigated by means of the back-scattered light from the same laser used to trap the sample. In particular, the applied torque is related to the rotation of the trapped nanofiber, characterized by measuring the time evolution of the backscattered light. Collecting this light by imaging the back focal plane of the microscope objective onto a silicon quadrant detector provides a direct, non-contact method to measure the drag torque. Upon Fourier processing, the particle rotation frequency is recovered by the power spectrum density [23] with high accuracy, and with larger bandwidth and better resolution than frame-by-frame video tracking. In order to characterize the trapped fibers, we first test the alignment between the nanofiber and the polarization direction of the trapping beam for the configuration depicted in Fig. 1 We then analyze the dynamics of tilted fibers by measuring the rotating frequency as a function of the incident optical power. 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. 3(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 values. An increase of the rotation frequency is observed for angles approaching sin =1, an effect that is not related to the increasing tilt, that would cause a decrease of the rotating frequency if the pivot point is unchanged (Fig. 4). We attribute the increase of the rotation frequency for high tilt angles to a progressive shift of the trapping point from the fiber tip towards the fiber center-of-mass.
In fact, our calculation reproduce quite well the experimental results when assuming a progressive shift of the trapping point from the fiber tip towards its center-of-mass mass (up to 4 L
In conclusion we demonstrated optical trapping and manipulation of polymeric nanofibers, introducing the control of rotation over elongated nanostructures by tilting the trapped fiber, which allows to achieve rotation frequencies up to 10 Hz. The measured rotation frequencies in the different trapping configurations are well reproduced by calculations based on a T-Matrix formalism for optical force and torque. The manipulation of this novel class of nanomaterials hold promises for a wealth of applications, such as photonic circuits or microfluidic devices, that can benefit from the controlled manipulation and rotation of the nanofibers, and the assembly of active polymeric fibers in ordered arrays. In particular the control over length and size makes polymer nanofibers ideal probes in next generation Photonic Force Microscopy.

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 [1]. 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 [2] and torque [3,4] 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 [5]. 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 momentum conservation 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 [2,3]:

Hydrodynamics
When dealing with quantitative measurements of optical trapping and rotation on polymer nanofibers, a crucial issue to be considered is the hydrodynamics of the trapped particle. For rigid rod-like structures of length L and diameter D , the viscous drag is described by an anisotropic hydrodynamic mobility tensor [7,8], the components of which depend on the length of the structure ( L ) and on the length-to-diameter ratio p L D  as [7]: where   and ||  are the translational mobilities, transverse and parallel to the main axis respectively, Rot  is the rotational mobility about the center-of-mass,  is the water dynamical viscosity, and i  are end corrections (calculated in [7] as polynomial of   1 ln 2 p  ). Note that when rotation of a linear nanostructure occurs about a point shifted by  from the center-of-mass, the rotational mobility changes as: (