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Precisely controlled manipulation of nonadherent single cells is often a pre‐requisite for their detailed investigation. Optical trapping provides a versatile means for positioning cells with submicrometer precision or measuring forces with femto‐Newton resolution. A variant of the technique, called indirect optical trapping, enables single‐cell manipulation with no photodamage and superior spatial control and stability by relying on optically trapped microtools biochemically bound to the cell. High‐resolution 3D lithography enables to prepare such cell manipulators with any predefined shape, greatly extending the number of achievable manipulation tasks. Here, it is presented for the first time a novel family of cell manipulators that are deformable by optical tweezers and rely on their elasticity to hold cells. This provides a more straightforward approach to indirect optical trapping by avoiding biochemical functionalization for cell attachment, and consequently by enabling the manipulated cells to be released at any time. Using the photoresist Ormocomp, the deformations achievable with optical forces in the tens of pN range and present three modes of single‐cell manipulation as examples to showcase the possible applications such soft microrobotic tools can offer are characterized. The applications describe here include cell collection, 3D cell imaging, and spatially and temporally controlled cell–cell interaction.
Optical forces necessary to deform the bending rods and torsion strings. A) Overlaid optical microscopic images of a rod bending test structure in its relaxed and open states; the inset shows the scheme of the open state. The yellow line shows the path of the upper trap that was followed by the upper trapping sphere. The Δx displacement of the stationary sphere, used for force calculation, is also shown. B) The von Mises stress distribution calculated in COMSOL Multiphysics for the bending test structure when a pair of 50 pN opening forces are applied to the ends of the two arms. The 3D model dimensions fit to the 6 mW power and 20 µm s⁻¹ scan speed experimental structure. The stress in the cross section of the elastic rod is shown in the inset. The color bar values are expressed in N m⁻² units. C) The measured optical force acting on each trapping sphere as the function of the measured opening angle. Results are shown for bending rods polymerized with different laser powers (4.5–6 mW as shown in the legend). The gray dashed lines show the linear fit to the averaged curves. D) Overlaid optical microscopic images of structure II in its relaxed and distorted states. The inset shows the overlaid schemes of these states. E) The von Mises stress distribution calculated in COMSOL Multiphysics for the torsion test structure deformed by a force of 20 pN applied to the end of the deflecting arm. The 3D model dimensions fit to the 5.5 mW laser power and 20 µm s⁻¹ scan speed (440 nm) experimental structure. The inset shows the stress distribution inside the flexible part. The color bar values are expressed in N m⁻² units. F) The optical force acting on the trapping sphere of the torsion test structure as the function of the measured torsion angle. Results are shown for torsion strings polymerized with different laser powers (4.5–5.5 mW as shown in the legend). G) The Young's moduli of the nanorods calculated by Equation (1) as the function of laser power (scan speeds 20 µm s⁻¹ for all). D) Overlaid optical microscopic images of structure II in its relaxed and distorted states. The inset shows the overlaid schemes of these states.
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RESEARCH ARTICLE
www.advmat.de
Optically Actuated Soft Microrobot Family for Single-Cell
Manipulation
Gergely T. Iványi, Botond Nemes, Ilona Gróf, Tamás Fekete, Jana Kubacková,
Zoltán Tomori, Gregor Bánó, Gaszton Vizsnyiczai, and Lóránd Kelemen*
Precisely controlled manipulation of nonadherent single cells is often a
pre-requisite for their detailed investigation. Optical trapping provides a
versatile means for positioning cells with submicrometer precision or
measuring forces with femto-Newton resolution. A variant of the technique,
called indirect optical trapping, enables single-cell manipulation with no
photodamage and superior spatial control and stability by relying on optically
trapped microtools biochemically bound to the cell. High-resolution 3D
lithography enables to prepare such cell manipulators with any predefined
shape, greatly extending the number of achievable manipulation tasks. Here,
it is presented for the first time a novel family of cell manipulators that are
deformable by optical tweezers and rely on their elasticity to hold cells. This
provides a more straightforward approach to indirect optical trapping by
avoiding biochemical functionalization for cell attachment, and consequently
by enabling the manipulated cells to be released at any time. Using the
photoresist Ormocomp, the deformations achievable with optical forces in the
tens of pN range and present three modes of single-cell manipulation as
examples to showcase the possible applications such soft microrobotic tools
can offer are characterized. The applications describe here include cell
collection, 3D cell imaging, and spatially and temporally controlled cell–cell
interaction.
1. Introduction
Single-cell investigation methods, such as single-cell genetics,[1]
proteomics,[2]or morphological classification[3]have risen to
G. T. Iványi, B. Nemes, I. Gróf, T. Fekete, G. Vizsnyiczai, L. Kelemen
HUN-REN Biological Research Centre
Szeged Institute of Biophysics
Temesvári krt. 62, Szeged 6726, Hungary
E-mail: kelemen.lorand@brc.hu
G. T. Iványi
Doctoral School of Multidisciplinary Medical Sciences
University of Szeged
Szeged 6720, Hungary
The ORCID identification number(s) for the author(s) of this article
can be found under https://doi.org/10.1002/adma.202401115
© 2024 The Author(s). Advanced Materials published by Wiley-VCH
GmbH. This is an open access article under the terms of the Creative
Commons Attribution-NonCommercial License, which permits use,
distribution and reproduction in any medium, provided the original work
is properly cited and is not used for commercial purposes.
DOI: 10.1002/adma.202401115
the forefront of biological research in the
last decade. These methods require physical
handling of individual cells on macroscopic
scales (for example, using fluorescence-
activated cell sorting (FACS)) or with
microfluidic systems.[4]Micropipettes[5]
and microtraps[6]are tools for holding
cells in specific positions, whereas single
cells transportation and rotation can be
carried out in a controlled manner among
others, with active movable microtools,
such as substrate-attached[7]and unteth-
ered microgrippers,[8]with electrophoretic
systems using high-frequency electric
fields[9]or with optothermal traps operat-
ing with localized laser heating.[10]Optical
tweezers (OT) have also been shown to
be very effective for direct[11]and indirect
manipulation[12]for arranging, observing,
and diagnosing single cells. Indirect opti-
cal manipulation prevents the cells from
photodamage, and because the trapped
intermediate object has a higher refractive
index, it allows for more precise manipu-
lation. Optically driven cell manipulating
microtools have long been reported in the
form of simple spheres[13]or more complex, tailor-made
structures.[12b,c,14]These tools are characterized as noninva-
sive and nontethered, unlike micropipettes, or microgrippers,
therefore no external physical wiring is necessary for their
J. Kubacková, Z. Tomori
Department of Biophysics
Institute of Experimental Physics SAS
Watsonova 47, Košice 04001, Slovakia
G. Bánó
Department of Biophysics
Faculty of Science
P. J. Šafárik University in Košice
Jesenná 5, Košice 04154, Slovakia
G. Vizsnyiczai
Department of Biotechnology
University of Szeged
Szeged 6720, Hungary
Adv. Mater. 2024,36, 2401115 2401115 (1 of 12) © 2024 The Author(s). Advanced Materials published by Wiley-VCH GmbH
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operation; it allows their use in any segment of a microfluidic
channel that can be reached by the focusing objective. Although
micropipettes[15]can apply many orders of magnitude larger
forces on the cells than optical traps, they have enormous size
and have to be physically introduced from outside of the sample
volume. Shape-changing gripping microtools were developed in
substrate-tethered and nontethered versions to collect or move
cells around but they operate either through external wiring or
tubing[7]or by changing the pH or temperature of the whole
environment[8a,b]limiting their applicability.
Indirect optical manipulation offers precise actuation of the
trapped cells with fluctuation around 100 nm due to the achiev-
able optical forces in the range from tens to hundreds of
pikonewtons. When complex microstructures are used for cell
manipulation,[12b,c,16]they can approximately be as small as the
cells (10–30 μm). Probably the most difficult controlled actua-
tion task is when the cell is rotated to an arbitrary orientation
and is held there as steadily as possible during, while the ex-
periment is carried out. Dielectrophoretic actuation have been
reported to rotate the cells at high speed for 3D imaging, but
it can be done only at given locations in the sample space de-
termined by the electrode arrangement, and it lacks the capa-
bility of holding the cell steadily in a desired orientation.[9]Re-
cently, opto-thermal cell manipulation schemes were published
taking advantage of localized heating-generated force fields.[10]
The method requires much lower optical power to trap the sensi-
tive cells than optical tweezers, but in return, the trapping forces
are also an order of magnitude lower. One can move, collect
or sort particles even of molecular size with the method, but
these manipulations are mostly restricted to the 2D plane where
the thermal effect is induced with limited possibilities to move
to the third dimension. The rotation this method offers is also
continuous, lacking the capability for steadily holding a cell at
any fixed position. In contrast, with indirect optical manipula-
tion using a properly designed intermediate tool, cells can be
turned to any desired orientation and held there as long as nec-
essary with very low fluctuation.[12c]The major drawback of this
method is that the cells need to be attached to the microtools
usually with biochemical means and they cannot be released
afterward.
The current technological status of multiphoton
polymerization[17]allows for developing optically actuated
microtools in the direction of mobile microstructures, which
are deformable with optical trap forces. Deformable struc-
tures, often referred to as soft robots, have been reported in
many sizes ranging from several millimeters down to tens of
micrometers,[8c,18]and are usually built up as multimaterial
systems.[8a,19]When such structures are made of only one elastic
material, the most straightforward mode of operation is based
on recovering their shape after deformation, similar to ordinary
lab tweezers.[8c]In single-cell applications, such microtools can
be associated with cells by locking on to them via elastic forces
generated during structure deformation rather than chemical or
biochemical attachment. Elastic deformations also make such
associations reversible: the cells could easily be released after
performing the required manipulation task. The control over
the microtools dimensions and mechanical properties, namely
their Young’s modulus is critical for their operation: both must
be kept low in order to deform the structures with forces typical
of optical manipulations. The smallest achievable feature size
with multiphoton polymerization is around 100 nm, and the
bulk Young’s modulus of most common photopolymers is in
the 1–5 GPa range (SU8: 4 GPa, IP-L: 3.9 GPa, IP-S: 5.1 GPa).[20 ]
However, it has been observed that the elastic moduli of poly-
mer wires with submicrometer cross-section are reduced in a
material-dependent manner when compared to the bulk value.
For photoresist SCR 500 the shear modulus decreased from
150 MPa to the range of 0.39–0.77 MPa for nanowires with
160–240 nm average thickness.[21]The reduction was attributed
to both a lower degree of crosslinking and a relatively larger
surface area with dangling polymer chains.[22]The decrease is
less dramatic for one of the most successful commercial resins,
IP-DIP: for the as-polymerized 240–440 nm wide nanowires,
the Young’s modulus is 0.8–2.4 GPa compared to 4.5 GPa
for the factory bulk value.[23]Our previous experiments with
OT-assisted deformation revealed that 200 nm wide nanowires
made of the photopolymer Ormocomp have a Young’s modulus
as low as 3.5 MPa, that is 300 times lower than the 1 GPa
bulk value.[24]Also, nanowires of this material were recently
measured to have a Young’s modulus as low as 50 MPa.[25 ]The
high elasticity of Ormocomp nanowires was previously utilized
to prepare microfluidic flow velocity measuring device[26]and to
study single cardiac cell contraction.[27]Its biocompatibility[27,28 ]
makes it a suitable material for preparing soft robots for live cell
applications.
In this paper, we show that with the careful design of geome-
try and fabrication parameters, two-photon polymerization (TPP)
can be used to make self-supported, mobile microtools out of
the elastic photopolymer Ormocomp that can be deformed with
OT. Deformability is achieved by incorporating either a bend-
able nanorod or a torsion nanostring as the key elastic element
into the microtool design. These nanorod and nanostring com-
ponents are the simplest elastic structures possible, being much
more compact and occupying much less space than those intro-
duced earlier in the form of a spiral spring[29]or a helical cross
springs.[30]To demonstrate the diversity of potential applications,
we present a family of such elastic microtools developed for spe-
cific single-cell manipulation tasks. The family includes three
members: the first is a cell transporter designed to enclose and
move single cells in a microfluidic chamber with no force ap-
plied to the cell for its holding. The second microtool is a cell
tweezer that is able to hold and spatially manipulate cells with
high stability and precision, the capability of which is demon-
strated by applying it in multiview microscopy imaging. Finally,
the third presented microtool allows for inducing cell-to-cell in-
teractions with precise spatial and temporal control. We show the
potential of these microtools by performing proof-of-concept ma-
nipulation experiments performed on nonadherent mammalian
cells.
2. Results
The deformability of the presented microtools is made possi-
ble by the insertion of either bending rods or a torsional string
into the structure (Figure 1). The bending moment or torque re-
quired for their deformation is determined by their geometry and
material characteristics (Equations (1)and(2)). For bending two
nanorods that span between two rigid rods by applying force at
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Figure 1. Geometrical model of the deformation of microstructures with
A) two nanorods and B) a nanostring as the elastic element (colored red).
their ends as shown in Figure 1A, the bending moment Mfor
small distortions is
MFL=E𝜋t3hΘ
l(1)
where Fis the force applied at the two ends, Lis the distance
between the bending rods and the trapping spheres, Eis the
Young’s modulus, tis half the thickness of the bar, his half of
its height, lis its length, and Θis the half angle of the two rigid
rods;[31]here, for a rod of ellipsoid cross section t3h𝜋/4 =Iis
called the second moment of inertia, and the bending radius Ris
expressed as l/(2Θ).
In the case of a two-end-fixed torsional string (Figure 1B), the
torque necessary to twist it at the middle by angle Φis
TFa=ΦG𝜋r4
l(2)
where Fis the tangential force, ais the arm length, ris the ra-
dius of the string, lis the half of its length, and Gis the torsional
modulus (G=E/(2*(1+𝜈)), where𝜈is the Poisson’s ratio, which
for most plastics is 0.35 <𝜈<0.45).[32,33 ]
2.1. Optical Deformation of the Nanowires
In both cases of elastic elements, their elasticity and the moment
or torque exerted by the optical trap’s force determine the achiev-
able deformation. Since the available trapping force is limited,
the exertable torque can only be increased by the length of the
lever arm, which, in turn, will be limited by a practical constraint
of avoiding excessively large structures. On the other hand, in-
creasing deformation by making the elastic elements “softer” is
undesirable, as it would risk the collapse of the structures during
their development and washing. Thus, it is logical to optimize the
geometrical parameters of the elastic elements in Equations (1)
and (2) to match with the available torque. In order to keep our
design’s size practical, we use a fixed lengthlfor the flexible ele-
ment and choose to optimize its thickness parameters (t, h,orr)
by tuning the fabrication laser power.
We design all structures with the same trapping sphere-
handles with a diameter of 4.5 ±0.2 μm, that when trapped
with the maximum power of our laser tweezer (225 mW per trap
at the objective’s entrance pupil), it produces a trap stiffness of
30 ±1pNμm1. The lever arm lengths were chosen to be be-
tween 30 and 50 μm in the various microtools.
2.1.1. Bending Rod
We test the applicability of the bending rods with an opening”
test structure that has a lever arm length of 35 μmandabend-
ing rod prepared with 20 μms
1scan speed and 4.5–6 mW
laser power (Figure 2A). COMSOL simulation showed that stress
builds only in the elastic part of the structure during deformation
(Figure 2B). It also predicted the amount of deformation: as an
example, the structure with bending rod dimensions of 400 nm
thickness and 1.15 μm height (achievable with fabrication param-
eters 6 mW power and 20 μms
1scan speed, Figure S2, Support-
ing Information) can be bent by 10°with 50 pN force. This value
fits well with the measured ones, as shown next. The experiments
showed that the force to open these structures increased linearly
with the opening angle at all laser powers for small angles (up to
15°for 5 mW, 8°for 5.5 mW, and 4°for 6 mW, Figure 2B).
Not surprisingly, the force required to open the structures by the
same amount drastically increases with the polymerization laser
power: for instance, for 10°opening 2.4, 14.5, 22, and 34 pN is
needed in the 4.5, 5, 5.5, and 6 mW cases, respectively; 30°open-
ing can be realized only with the rods made with 4.5 and 5 mW.
This is the combined consequence of the increase of rod thick-
ness and height with the laser power, from 0.3 to 0.4 μm and from
0.5 to 1.15 μm, respectively (Figure S1, Supporting Information),
and of the increase of the Young’s modulus, as explained in the
next paragraph. Based on these curves, the parameter couple of
5mWand20μms
1demonstrated a well-balanced performance
suitable for practical use across all the presented structures. This
combination allowed easy opening to 30°with a force of 45 pN,
ensuring mechanical stability to prevent damage or collapse dur-
ing the development and collection process.
According to Equation (1), the dependencies of force versus
opening angle, along with the nanostring dimensions (Figure
S1, Supporting Information), provide a way to directly obtain the
Young’s modulus for the bending bars at each polymerization pa-
rameter. Rearranging Equation (1) reveals that, the Young’s mod-
ulus depends on the F/Θratio, which was obtained through a lin-
ear fit to the low-angle linear regime of the averaged Fversus Θ
curves (dashed gray line in Figure 2B). The ratio, as provided by
the slope of the fit, increased from 17.6 pN/rad (4.5 mW) to 276.8
pN/rad (6 mW). From these values, the Young’s modulus was de-
termined to be between 2.6 and 7.2 MPa, as shown in Figure 2C.
It corresponds nicely with the 3.5 MPa value, we determined ear-
lier with a completely different method for nanorods prepared
with 6 mW laser power and 20 μms
1scan speed.[24]The ori-
gin of the Young’s modulus dependence on the polymerization
parameters is beyond the scope of this paper.
2.1.2. Torsion String
The torsion string test experiments were conducted using a
surface-mounted test structures bearing a 30 μm long lever arm
and a 13 μm long torsion string fabricated with 20 μms
1scan
speed and 4.5–5.5 mW laser power (Figure 2D,E). Here, the sim-
ulations also show that the majority of the stress builds up in the
nanorod while the rest of the structure is practically unaffected
(Figure 2E). The force required for the same amount of distor-
tion increased with the polymerization laser power once again:
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Figure 2. Optical forces necessary to deform the bending rods and torsion strings. A) Overlaid optical microscopic images of a rod bending test structure
in its relaxed and open states; the inset shows the scheme of the open state. The yellow line shows the path of the upper trap that was followed by the
upper trapping sphere. The Δxdisplacement of the stationary sphere, used for force calculation, is also shown. B) The von Mises stress distribution
calculated in COMSOL Multiphysics for the bending test structure when a pair of 50 pN opening forces are applied to the ends of the two arms. The 3D
model dimensions fit to the 6 mW power and 20 μms
1scan speed experimental structure. The stress in the cross section of the elastic rod is shown
in the inset. The color bar values are expressed in N m2units. C) The measured optical force acting on each trapping sphere as the function of the
measured opening angle. Results are shown for bending rods polymerized with different laser powers (4.5–6 mW as shown in the legend). The gray
dashed lines show the linear fit to the averaged curves. D) Overlaid optical microscopic images of structure II in its relaxed and distorted states. The
inset shows the overlaid schemes of these states. E) The von Mises stress distribution calculated in COMSOL Multiphysics for the torsion test structure
deformed by a force of 20 pN applied to the end of the deflecting arm. The 3D model dimensions fit to the 5.5 mW laser power and 20 μms
1scan
speed (440 nm) experimental structure. The inset shows the stress distribution inside the flexible part. The color bar values are expressed in N m2
units. F) The optical force acting on the trapping sphere of the torsion test structure as the function of the measured torsion angle. Results are shown
for torsion strings polymerized with different laser powers (4.5–5.5 mW as shown in the legend). G) The Young’s moduli of the nanorods calculated by
Equation (1) as the function of laser power (scan speeds 20 μms
1for all). D) Overlaid optical microscopic images of structure II in its relaxed and
distorted states. The inset shows the overlaid schemes of these states.
strings made with 5.5 mW can be twisted up to 15°–20°, while
those made with 4.5 mW to more than 90°. The initial linear parts
of the force versus angle plots have slopes of 3.17 pN deg1for
5.5 mW, 1.57 pN deg1for 5 mW and only 0.71 pN deg1for
4.5 mW. When normalized with the rod length, a more general
value is resulted describing the torsion string itself: it is 95.1,
47.1, and 21.3 pN deg1μm1for the strings polymerized with
5.5, 5, and 4.5 mW laser power, respectively. When a structure
based on the torsion string mounts a cell to the substrate (see
Section 2.4 below), one can easily calculate the force that keeps
the cell in place by measuring the twisting angle relative to equi-
librium and the lever arm’s length between the string and the cell
surface.
2.2. Collecting Cells: Cell Transporter
The purpose of the cell transporter structure (structure III, see
the Experimental Section) is to selectively collect single, nonad-
herent cells in a microfluidic environment and deposit them to
a predefined location without applying significant force on them
(Figure 3A). The wireframe design of the cages (Figure 3C)was
intended to reduce the viscous drag on the structure during trans-
portation. As a result, using the maximum trapping laser power,
the structures can be translated at speeds up to 300 μms
1in the
direction parallel to their main axis and 200 μms
1perpendicular
to it. The two halves of the 30 μm diameter cages could be easily
opened to larger than the typical cell diameter (12–18 μm), allow-
ing the transporter to enclose and pick up cells without difficulty.
The scheme of the process of cell collection is shown in
Figure 3A and snapshots of the actual experiment in Figure 3B
(see also Movie S1, Supporting Information). The target cell is se-
lected visually from the ensemble of cells settled on the glass sub-
strate. The cell transporter, trapped at three points, approaches it
by moving the microscope stage. Then the OT opens up the trans-
porter structure, forming a gap somewhat larger than the cell size
between the two cages. Moving the stage, the cell is maneuvered
in between the cages which then are closed (“collect”). After this
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Figure 3. Single-cell collection with optically actuated microstructure. A) The scheme of the cell collection procedure. B) Brightfield microscopic images
of the major steps of the collection procedure (Movie S1, Supporting Information). The red arrows indicate the direction of the optical trap’s movement
for opening up the structure. The defocused image of the cell container’s rim is visible during transport. C) The electron micrograph of the cell mover
with the elastic rods overlaid with red. The inset shows the height of the bending rod (0.83 μm) and the thickness of the vertical rod (0.63 μm) that
connects the two bending rods and supports the trapping ellipsoid. D) Electron micrograph of the cell container.
point, the traps serve only to move the structure and the cages are
kept closed around the cell merely by elastic forces. Next, while
holding the structure with the OT, the cell transporter is lifted
from the substrate by moving the focusing objective (“lift”). The
structure is then transported toward the targeted container at a
height that is higher than the height of its fence (Figure 3D)by
moving the sample stage (“transport”). The noticeable displace-
ment of the cell inside the cage shows that the structure does not
squeeze the cell at all, which is the main advantage of cell trans-
lation with this structure. After maneuvering the cell transporter
over the container’s fence, it is lowered slightly below the height
of the fence. The cell is released here by opening the cell trans-
porter (“release”) with the OT. At this point, the cell slowly rolls
out of the cage and falls to the bottom of the container. One cell
collection cycle between two consecutive cell collection steps lasts
about 3 min.
The compactness of this structure is limited by a minimum
distance between the trapping spheres and the cell containing
cage. When it is too small, during cell release, the radiation pres-
sure of the strongly focused trapping beam pushes the falling cell
up toward the spheres eventually blocking the beam and causing
the OT to release the structure; a minimum of 20 μm cage-to-
sphere distance was therefore necessary. For similar reasons the
cell transporter structure may not be moved lower than 30 μm
above the substrate during cell release as the radiation pressure of
the trapping beam would push the already settled cells toward the
cell transporter’s spheres. Considering maneuvering, it is crucial
that the vicinity of the target cell is not crowded with other cells
during the collection step, as they may interact with the OT.
2.3. Imaging Cells: Cell Tweezers
For applications where it is crucial to hold a cell with minimal
fluctuations (for instance imaging), we introduce the elastic cell
tweezers structure (Figure 4A,B). We previously demonstrated
the benefit of indirectly trapping a single cell with a polymer mi-
crotool for precise 3D imaging.[12c]In the procedure introduced
here, the tweezer tool’s elasticity, rather than a chemical or bio-
chemical attachment, keeps the cell attached to the structure.
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Figure 4. Cell tweezers structure and its application for multiview microscopic imaging. A) Schematic view of the nonadherent cell held with the cell
tweezers structure. B) Electron micrograph of the cell tweezer structure; the flexible rods are highlighted with red, the cell holding pins with green.C)
Brightfield microscopy snapshots of the cell collection procedure. The yellow stars mark the position of the OTs. D) Fluctuation of a cell measured in
a stationary position held with an optically trapped tweezers structure. E) Brightfield-fluorescence composite images of a fluorescent nanobead-labeled
cell held with the tweezers structure in two different orientations reached by rotating the microstructure by 90°with the OT. F) Maximum intensity
projection images of aligned image stacks recorded on fluorescent bead-decorated cells originating from four different orientations and that of the fused
image stack. G) Normalized intensity traces observed along the Zand Xaxes on the bead marked with a yellow arrow on panel (F). The more than three
times reduction of the image width along the Zaxis demonstrates the resolution enhancement.
Another natural advantage of the nonpermanent attachment is
that the cell can be released after inspection (see Movie S2,Sup-
porting Information). We demonstrate the capabilities of the new
cell tweezers microtool with 3D imaging using multiview mi-
croscopy. The imaging task naturally demands that the microtool
does not distort the image of the cell. Consequently, it was con-
structed with a minimum number of elements, each as thin as
possible. The negligible residual distortion is illustrated in Figure
S2 (Supporting Information). Fluorescent nanobeads attached to
the cells were used to assess the imaging quality.
Picking up a cell with the tweezers structure (Figure 4C)was
similar to the process shown for the cell transporter tool. First,
a cell with enough nanobeads on it was selected using fluores-
cent observation. Then, in brightfield mode, the trapped tweezers
structure approached the cell, it was opened up, the cell was ma-
neuvered in between the small pins of the microtool, and finally
the OT closed them onto the cell. At this stage the pins touch
the cell’s surface, and the microtool’s elasticity alone can hold
them together. We occasionally had to move the trapping spheres
closer to one another with the OT to hold the cells more firmly to
reduce their fluctuation. For precise imaging, the tweezed cell’s
spatial stability must be good enough not to cause blur, which
we quantified by holding a cell stationary and observing its posi-
tion (Figure 4D). Approximately 5000 frames were recorded, and
a correlation algorithm was used to determine the cell’s move-
ment. The widths at the half of the maximum of the cell’s posi-
tion probability were 24.75 and 23.9 nm along the axes Xand Y,
respectively, both being small enough to allow for high-resolution
imaging even with long exposure times. The force each holding
pin exerts on the cell surface can be estimated from the depen-
dence of the optical force on the opening angle of the structures
(Figure 2B), the length of the rigid rods and the distance of the cell
from the bending rods. Considering the various angles between
the two rigid rods when cells of different diameter are entrapped
between them (from 3°to 11.5°), we estimated that each pin ex-
erts a force between 4 and 15 pN on the cell; the higher value is
expected for the larger cells (for details see Figure S3, Supporting
Information).
Multiview microscopic imaging requires independent Z-stack
recordings of the cell in multiple orientations that are achieved
by rotations around an axis perpendicular to the optical axis.
Figure 4E depicts such a case in which the cell is rotated around
the Xaxis by 90°to a new orientation by actuating the optically
trapped cell tweezers. In our case, the Z-stack image series were
recorded in orientations 45°from one another. The processing
of the independent Z-stack image series was carried out as pre-
viously described.[12c]Shortly, an iterative 3D deconvolution step
with 40 iterations (Richardson–Lucy algorithm) was performed
first, followed by a registration process of the image stacks with a
direct search algorithm, aligning the different orientations to the
first, 0°orientation. The first four images in Figure 4F show the
result of the deconvolution and alignment steps; it displays the
significant elongation of the bead images as the result of the in-
herent anisotropic resolution of the fluorescent microscope. The
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Figure 5. Cell–cell interaction with indirectly trapped cells in the vertical direction. A) Schematic draw of the interaction: the lower cell is substrate-
attached, while the upper one is optically maneuvered. B) Electron micrograph of the cell-retaining structure; the flexible string is highlighted with red.
C) Brightfield image of a substrate-mounted cell. D) Stages of the cell–cell interaction procedure; for explanation, see main text. For “tilted touch,” the
structure was tilted by 15°. E) Cell–cell interaction experiment with fluorescently labeled cells observed with a combination of fluorescent imaging and a
weak background illumination to visualize the cell manipulating microstructures. The red arrow points to a nanobead selected on the surface-mounted
cell, while the green arrow points to one attached to the cell held with the tweezers structure above the mounted one. F) Position distribution of a
substrate-mounted cell, similar to that shown in panel (C), due to its residual fluctuation after it being pressed against the two vertical bars.
final fusion process was carried out by the pixelwise multiplica-
tion of the aligned image stacks, yielding a single 3D stack of
isotropic optical resolution (Figure 4F “Fused”). The resolution
enhancement is demonstrated qualitatively by the nonelongated
images of the nanobeads and quantitatively in Figure 4G by plot-
ting the intensity traces over a selected bead along the Zaxis from
the as-recorded stack and from the fused one. The width of the
intensity peak is reduced from about 2.1–0.6 μm.
2.4. Cell–Cell-Interaction Using a Pair of Structures
We present two modes for realizing cell-cell interactions, one
with axial and one with lateral direction of approach (Figures 5A,
and 6A). Both approaches rely on a pair of OT-actuated cell ma-
nipulator structures: a surface-mounted cell-retaining microtool,
and a mobile microtool. The cell-retaining microtool is used to
anchor one cell to the substrate in a fixed position, while the
second one to maneuver and push another cell against the first
cell either from the side or from above. Because precise spatio-
temporal control of these cells is critical, they must be held with
minimal fluctuation.
The cell-retaining microtool operates on the torsion force risen
in a distorted string (Figure 5B,C). A tiltable rod is attached to this
string with one end equipped with a sphere for the OT, while the
other end presses the cell against a substrate-mounted counter-
part. To allow for the lateral and axial approaches, two types of
counterparts were designed, each leaving the respective surfaces
of the cell freely accessible. In the axial approach, the counter-
part is simply a pair of slightly tilted columns, while in the lateral
approach it is a ring with a plane perpendicular to the substrate
glass. Cell capturing (Movie S3, Supporting Information) begins
with tilting the rod with the OT against the string’s torsion resis-
tance, followed by moving the cell in between the other end of the
rod and the counterpart with another trap. This second trap di-
rectly illuminates the cell for a few seconds, however such a short
exposure is unlikely to cause damage to the cell.[34]When the trap
is turned off, the rod’s end pushes the cell against the counter-
part. This cell-retaining force is controlled by the polymerization
parameters of the string as shown in Figure 2E,andFigure5F
depicts cell fluctuation as held by the retaining structure. It was
found to be 40 and 30 nm along the Xand Yaxes, respectively;
the smaller fluctuation along the Yaxis can be explained by the
cell being pressed against the counterpart along this axis.
In the vertical approach the mobile microtool was the cell
tweezers structure described in the previous section. Fluorescent
bead-decorated cells were used to better track the axial position of
the overlapping cells. At the beginning of the process (Figure 5D;
and Movie S4, Supporting Information), the upper surface of the
substrate-mounted cell and the bottom surface of the optically
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Figure 6. Cell–cell interaction with cells translated in the lateral direction. A) Arrangement of the cells held by the two types of structures. B) Brightfield
image of the ring of the cell holding structure VI. C) Brightfield optical microscopic image of the cell–cell interaction with lateral movement. On the left,
the two cells are separated; the stage moves the substrate-mounted cell in the direction of the thick white arrow. On the right, the two cells come into
contact, and the optically manipulated one is slightly pushed out by the other one. The distance between the accommodation ring and the retaining pin
is highlighted in purple.
actuated cell were brought into a common focal plane (“co-focus”)
by moving the focusing objective (“lift”) and the optical traps ver-
tically. Next, the lower cell was transferred underneath the up-
per one by moving the microscope stage (“approach”). For the
interaction, the optically actuated cell was lowered by the focus-
ing objective to touch the lower cell (“touch”). This step pushed
the mobile cell slightly upwards as evidenced from the additional
defocusing of the trapping ellipsoid in Figure 5D. The two cells
can even slide over each other at this position by moving the stage
sideways. This kind of interaction is evidenced by the slight lat-
eral displacement of the upper cell. The ability to hold the upper
cell in various orientations (Figure 5D,E, “tilted touch”) allows to
change its contact surface presented to the other cell. The vertical
approach allows for the entire interacting membrane regions to
be observed in the image plane.
The lateral approach was conceived with the advantage in mind
that the cell images do not overlap, so the effects of the interac-
tion can be observed undisturbed in both cells. For this approach,
a special structure was designed (Figure 6A,B) that is holding the
cell with the elastic force of a bent rod that pushes the cell against
a ring. To grab a cell, the elastic rod is pulled away from the ring
by the OT, allowing a cell to be maneuvered into the space be-
tween the two (see Movie S5, Supporting Information). Finally,
the trap is turned off, the elastic rod relaxes, and the cell is cap-
tured. Once the cell is mounted, the structure is maneuvered by
the OT holding the four trapping ellipsoids.
To start the interaction, the movable cell is held steadily with
the OTs, and the substrate-mounted cell is pushed against it
by slowly translating the microscope stage (see Figure 6C;and
Movie S6, Supporting Information). During this, the circumfer-
ence of the cells are usually brought into a common focus, but
their mutual position along the optical axis can also be varied
to some extent. It was frequently observed that one of the cells
pushes the other slightly out from its accommodation ring; in
Figure 6C,2μm of such movement is seen. In these cases, the
elastic element held the cell in place and returned it to its origi-
nal position after separation. Structure VI is designed in such a
way that mechanical forces exerted during the interaction can be
measured with it, however, it was beyond the scope of the present
paper. A possible scenario is that the two cells bind to each other
after making a controlled contact. The structure’s ring and the
four trapping ellipsoids are arranged, so that if the two attached
cells have to be separated, a large pulling force can be exerted
without allowing the mobile cell to slip out of the structure due
to the ring’s diameter being significantly smaller than that of the
cell. In this case, while the substrate-mounted cell is pulled away
by moving the microscope stage, the position of the optically held
structure can be used to measure the force between them at the
moment of their detachment.[35]
3. Conclusion
Deformable microtools prepared with multiphoton polymeriza-
tion and actuated with optical tweezers were introduced in proof-
of-concept single-cell manipulation experiments. The key com-
ponents of the microtools are elastic nanorods that can be bent
or twisted, acting as hinges for the rigid parts of the structures.
The prepared structures can be deformed with optical forces of no
more than 60 pN when the fabrication parameters are carefully
balanced, resulting in elastic rod cross sections of a few hundred
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Figure 7. Schematic representations of the elastic, deformable microstructures used in the experiments. See the main text for a more detailed description.
Structures I and II were used for the bending and torsion tests, while structures III-VII for cell manipulation. The focused beam of the optical tweezers
is illustrated with structure II as it is trapping its trapping sphere.
nanometers. At the same time, these structures proved to be rea-
sonably stable in terms of shape integrity under the load of the
moving liquid during the fabrication procedure. A fundamental
feature of the microtools is that the microtool-cell association is
not permanent, meaning that when the required task is finished,
the structures can eventually release the cells. First, the micro-
tools were used to select, capture, and collect cells to a predefined
location inside a microfluidic system, using 3D translation, with-
out exerting considerable force on the cells. Second, fluorescence
multiview imaging with firmly captured cells actuated with 6°of
freedom and with better than 50 nm precision was demonstrated.
Finally, cell–cell interaction was demonstrated in two directions
of approach, vertical and lateral, with mounting one of the cells
to the bottom surface of the microfluidic channel and maneu-
vering the other one to it for touching with precise temporal and
spatial control. The completed tasks demonstrated that multipho-
ton polymerization combined with optical manipulation can pre-
pare and actuate tools for complex soft-robotics applications in
the field of single-cell research.
4. Experimental Section
Microstructure Polymerization:The elastic cell manipulator structures
were fabricated by two-photon polymerization (TPP) using a custom-made
setup.[24]The essential elements of the setup are an ultrashort-pulsed
fiber laser (𝜆=785 nm, pulse length =100 fs, C-Fiber A, Menlo Sys-
tems, Germany) as light source, a high NA objective (40x oil, 1.3 NA, C
Plan-Apochromat, Zeiss, Germany) that focuses the beam into the Ormo-
comp (Microresist GmbH, Germany) photopolymer, a 3-axis piezo stage
(P-124 731.8L and P-721.10, Physik Instrumente GmbH, Germany) for
nanometric translation of the sample relative to the focus and an acousto-
optical modulator to change the power of the polymerizing laser beam
(APE 130 101, APE GmbH, Germany). The TPP system was controlled by
a Labview program. The Ormocomp microstructures were polymerized
onto standard glass cover slides (Menzel Gläser, Germany). A plastic ring
of 2 mm height and 10 mm diameter was glued to the center of the cleaned
slides with Norland optical adhesives (NOA81, Thorlabs Inc, USA) and a
1μL droplet of Ormocomp was drop-casted at its center. The plastic ring
ensures that the polymerized microstructures remain in liquid through-
out the development and sample preparation procedure preventing their
collapse. After the polymerization, the sample was rinsed in Ormocomp’s
developer solution (Ormodev, Microresist GmbH, Germany) for 10 min
twice and then transferred to ethanol for another 2 times 10 min rinse.
During the second bath in the developer, the entire sample was illumi-
nated with a microscope mercury lamp (HBO50) to facilitate postpoly-
merization.
First, test lines were polymerized both in lateral (referred to as rods)
and axial direction (strings). The parameter matrix for the rods ranged
from 4 to 9 mW laser power and from 1 to 100 μms
1scan speed; for the
strings these ranges were 4– 7 mW and 3–100 μms
1.The10μm long rods
were polymerized 7 μm above the substrate between supporting blocks,
while the 11 μm long strings were polymerized inside a thick vertical frame
(see Figure S1D, Supporting Information). The test lines were critical point
dried (CPD) for scanning electron microscopy (SEM) imaging. Their cross-
section dimensions were determined with the ImageJ software.
Structures for Elasticity Test:All the deformable microstructures
(Figure 7) consisted of three main regions: i) deformable rods or strings
(colored as red), ii) spheres that interact with the optical trap (designed
with 4 μm diameter, colored as gray) and iii) rigid, nondeformable ele-
ments (colored as blue).
For the optical deformability tests, two different types of structures
were designed. The first type tested the bending elastic rods (Figure 7,
structure I). These structures consisted of two 35 μm long rigid rods,
connected at one end by two 10 μm long parallel bending rods; the two
rods were needed for structural stability and in-plane only motion. The
scan speed for the polymerization of the bending rods was 20 μms
1,
while the laser power was varied from 4.5 to 6 mW. A trapping sphere
was polymerized at the other end of each rigid rod. The rods were bent,
therefore the structure was opened simply by moving one of the spheres
by translating the corresponding optical trap away from the other (yellow
line in Figure 2A). During translation, two parameters were monitored
by video microscopy: the increasing angle between the rigid rods (2Θon
Figure 1A) and the position of the stationary sphere. As the translated
sphere moves, the elastic stress risen due to the bending nanorods
applies a force on both trapping spheres, which is balanced by the
trapping force. The two spheres are held by optical forces of the same
amplitude but opposite directions (forces Fon Figure 1A). In the resting
state before the opening of the structure, the center of the stationary
sphere assumes the position of the nontranslating trapping focus. During
opening however, the elastic force pulls this sphere out of the resting
position with a measurable Δxdisplacement. The optical force Frequired
for the opening was obtained by multiplying this displacement by the k
trap stiffness of the sphere (measured separately) using the expression
F=−k·Δx.
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The second type tested the torsion of the vertical strings. This struc-
ture (Figure 7, structure II) consisted of a rigid frame to the torsion string,
the string itself and a horizontal rigid rod attached perpendicularly to the
center of the string. The 13 μm long string was prepared with 20 μms
1
scan speed using 4.5, 5, and 5.5 mW laser power. The 3 μm long thick
middle section of the string was needed to attach the rigid rod to it. The
trapping sphere was prepared at the end of the 30 μm long rod. The string
was twisted by moving the sphere with the optical trap along the circumfer-
ence of a circle with a radius of 30 μm (yellow line in Figure 2D)in1°steps
(0.53 μm steps along the circumference). In the resting position, again, the
sphere’s center coincides with the trapping focus position. During twist-
ing, as the trapping focus moves through predetermined positions, it car-
ries the sphere with it, which distorts the torsion string via the rigid rod.
As a result, the stress risen in the string displaces the trapping sphere’s
center from the trapping focus position. The force required to distort the
string was calculated from this Δxdisplacement, and from the separately
measured ktrap stiffness of the sphere.
Cell Manipulator Structures:Three microstructures were prepared for
three distinct cell manipulation tasks. The elastic parts (bending rods and
torsion strings) in all the structures were polymerized with 5 mW laser
power and 20 μms
1scan speed, the trapping spheres with 8 mW and
32 μms
1and the rigid parts with 8 mW and varying scan speed.
The first structure, the cell transporter (Figure 7, structure III), consists
of two half sphere-shaped cages with a diameter of 30 μm each. The cages
are connected by two bending rods and can encapsulate a nonadherent
cell of about 12–18 μm without squeezing them. The cages are made up
of thin lines to reduce viscous drag. A trapping sphere is attached to each
cage with a 20 μm long rigid rod. An extra ellipsoid was added to the struc-
ture to hold it steadily when lifted from the surface with three optical traps.
A cell container structure of 200 μm diameter was also polymerized where
the cell transporter could collect the selected cells. It was large enough to
accept the entire cell transporter structure and its 50 μm height and the
slightly inward-tilted wall ensured that the cells could not migrate passively
over it.
The cell tweezers structure (Figure 7, structure IV) is based on the rod
bending test structure but it is completed with cell holding pins. Its pur-
pose is to attach a cell to itself to prevent the cell from moving relative to
the structure. As a result, the cell precisely follows every movement made
with the tool. There are altogether six 5 μm long pins (three on each rigid
rods) to anchor the cell firmly to the trapped microstructure. The structure
is opened similarly to test structure I with two traps, and an extra ellipsoid
was also added for the 3D manipulation using three traps.
The cell-retaining structure (Figure 7, structure V) is based on the tor-
sion string test structure but has been modified at three points. First, the
long rigid rod extends to both sides of the torsion string. At one end, the
trapping sphere is attached, while the other end is designed for holding the
cells. Second, the string-holding frame structure is rotated to form a 70°
angle with the long rigid rod. Finally, two slightly tilted substrate-mounted
bars are added. These bars provide support against which the free end of
the rigid rod pushes the cell.
The cell holding structure type VI is designed for cell-to-cell interaction
via lateral movement. The central ring part of this structure accommodates
the cell which is pushed against the ring by a short rod completed with a
trapping sphere. The short rod is attached to the end of the elastic part of
the structure, a bending rod and can be moved with the optical tweezers to
create space for the cell. The elasticity of the bending rod pushes the cell
against the ring after the trap is turned off. The ring diameter allows the
cell to protrude on the other side presenting a surface for interaction with
another cell. The four symmetrically positioned trapping ellipsoids allow
for complete optical manipulation, especially for pushing and pulling the
cell. Structure VII is the counterpart of structure VI and can mount a cell
to the substrate by using a ring to accommodate the cell.
The development of the structures shown in Figure 7is finished dif-
ferently than that of the test lines. After the ethanol rinse, the structures
were transferred to deionized water via the following water-ethanol mix-
ture series: i) 100% ethanol, ii) 66% ethanol, 34% water, iii) 33% ethanol,
67% water, iv) 100% water; the sample was rinsed in each mixture for
1 min. After this, the deionized water was replaced with 1% w/w BSA in
1X PBS. For the opening and torsion test experiments, no cells were used.
The rod bending test structures, the cell transporter, and the cell tweezer
structures were released mechanically from the substrate with a glass mi-
cropipette. The cells were of the type K562 (LGC Standards, UK, cat no.
CCL-243),[12c]fixed with 4% formaldehyde and kept in 1X PBS for short-
term storage at 4 °C. Whenever cells were used, they were added in the
form of 2 μL suspension to the 200 μL BSA solution covering the struc-
tures previously released from the substrate. After adding the cells, a cover
slide was placed over the sample contained within the plastic ring to elim-
inate the free liquid surface.
Optical Tweezers Setup:The optical tweezers (OT) setup used for the
present experiments was introduced earlier in detail.[12c]A wide-field fluo-
rescence microscope (Nikon Eclipse Ti) combined with a holographic op-
tical tweezers setup was used for observing and actuating the structures.
The trapping laser beam (THFL-1P400-COL50, BKtel Photonics, France)
was manipulated by a spatial light modulator (Pluto NIR, Holoeye, Ger-
many) to create and move multiple trapping focal spots in the sample
chamber. The beams were focused into the sample by an Olympus mi-
croscope objective (UPlanSApo, 60x, water immersion, NA 1.2). The posi-
tions of the trapping focal spots were controlled by a home-made software.
The microstructures and the cells were observed in bright field (GS3-U3-
23S6M CCD camera, Point Grey Research Inc., Canada) and in fluorescent
(ORCA Flash 4.0v3 CMOS camera, Hamamatsu, Japan) mode. The fluo-
rescent light source was a metal halide lamp (Lumen 200S, Prior Scientific,
Inc., USA). The positions of the trapping spheres were determined with
correlation-based image analysis routines written in MATLAB.
The process of recording and analyzing multiview images was de-
scribed in detail earlier.[12c]Shortly, Z-stack image series of the fluorescent
bead-decorated and optical trap-actuated cells were recorded at predefined
orientations, resulting in five 3D recordings of the bead images. The image
analysis started with the 3D deconvolution of the stacks with a recorded
point-spread function of the microscope. Then, the stacks were aligned to
each other, using a direct search algorithm. In the final step, the aligned
stacks were fused by the pixelwise multiplication of the aligned stacks.
The calculation of the trap stiffness for the microstructures’ trapping
spheres was based on bright-field microscopy recordings of the fluctuation
of trapped, nontethered spheres using 0.5 ms exposure time. The stiffness
(k) was calculated from the variance (𝜎) of the sphere position data using
k=kBT/𝜎,2 where kBis the Boltzmann constant and Tis the absolute
temperature.[36]
Numerical Simulations:Numerical simulations were run in COMSOL
Multiphysics. The goal was to see the stress distribution within the mi-
crostructures under typical experimental load conditions. Furthermore,
cells held by the microstructures were modeled to determine the extent of
their deformation and the stress induced in them. The numerical model for
the microstructures used Young’s moduli of 7.2 MPa for the bending rods,
6 MPa for the torsion strings, and 1 GPa for the rigid elements and line
dimensions that were obtained by scanning electron microscopy (Figure
S1, Supporting Information). The model for cell deformation used a 15 μm
diameter spherical cell with a 10 μm diameter nucleus. The Young modu-
lus of K562 cells was previously measured to be 40[37]or 400 Pa.[38 ]In the
present simulations, 100 and 200 Pa Young’s moduli for the cell and nu-
cleus, respectively, were chosen. The cell was pushed against a solid ring
by a 1 μm diameter bead with a total force of 10 pN. This configuration
replicates the experimental conditions of structures VI and VII.
Supporting Information
Supporting Information is available from the Wiley Online Library or from
the author.
Acknowledgements
G.V. was supported by the János Bolyai Research Scholarship of the
Hungarian Academy of Sciences (No. BO/00290/21/11) and by the
ÚNKP-23-5-SZTE-717 New National Excellence Program of the Ministry
Adv. Mater. 2024,36, 2401115 2401115 (10 of 12) © 2024 The Author(s). Advanced Materials published by Wiley-VCH GmbH
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for Innovation and Technology from the source of The National Research,
Development and Innovation Fund. This work was supported by the Na-
tional Research, Development and Innovation Office, Hungary, under
Grant No. F.K. 138520. This work was also supported by the Slovak Re-
search and Development Agency, Grant No. APVV-21-0333, and by the
grant agency of the Ministry of Education, Science, Research and Sports
of the Slovak Republic, Grant No. VEGA 2/0101/22. The authors are grate-
fully acknowledge the contribution of prof. Mária Deli in the revision pro-
cess. [Correction added on June 10, 2024, after first online publication:
Affiliations were corrected.]
Conflict of Interest
The authors declare no conflict of interest.
Data Availability Statement
The data that support the findings of this study are available from the
corresponding author upon reasonable request.
Keywords
3D microfabrication, deformable microstructures, elastic photopolymers,
optical trap, single-cell manipulation, soft robotics
Received: January 22, 2024
Revised: May 17, 2024
Published online: June 5, 2024
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... Inspired by many of these applications we simultaneously optimize the design of the metalens and the particle for different objectives; such as, attracting and repelling the particle, or trapping the particle in a target point in space. The derivation of the proposed framework paves the way for further optimization of more complex optomechanical systems, such as the design of optically actuated mechanical devices [30], the design of novel active particles beyond self-propelling particles [31], novel optically-driven particle paths [32][33][34], or many-body systems [31,35,36]. Note that the base code developed for this framework is open-source and is readily available at: www.topopt.dtu.dk and https://github.com/bmdaj/MST_TopOpt. ...
... Following the motivation behind Equation 30, we can compare the radiation and the force experienced by an infinite interface under normal plane-wave illumination to our optimized design. A perfectly reflecting (|r| 2 = 1) interface would experience a radiation pressure P rad = ε 0 E 2 0 = 8.85 pN/µm 2 . ...
... In this case, the incoming plane-wave will be totally reflected, resulting in maximal momentum exchange between the plane-wave and the interface in the simulation domain. We calculate the force for a perfectly reflecting surface across the width of the simulation domain by using Equation 30, and find a total upward force of ⟨F y ⟩ = 37.54 pN/µm, which is ∼ 4 times larger than the force obtained for our optimized design, and ∼ 13 times larger than the baseline. In the following sections we will show how (nearly) all energy/momentum that is available in the simulation domain may be harnessed by the use of a metalens structure, hereby approaching the reference value for small particles relative to the model domain width. ...
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