Bo Sun [1], Jiayi Lin [2], Ellis Darby [2], Alexander Y. Grosberg [1]
and David G. Grier [1]
[1] Department of Physics and Center for Soft Matter Research,
New York University, New York, NY 10011
[2] NEST+m 111 Columbia Street, New York, NY 10002
Stochastic heat engines such as thermal ratchets and Brownian motors use non-conservative forces to eke fluxes of energy or probability out of otherwise random thermal fluctuations (1). In virtually all previous reports, such noise-driven machines have relied on a time-dependent force to rectify fluctuations. Here, we demonstrate that time-independent force fields also can create stochastic heat engines, so long as the force has an irrotational component capable of confining the particle and a non-vanishing solenoidal component. The resulting interplay of advection and diffusion gives rise to toroidal probability currents that we refer to as Brownian vortexes. As an illustration, we reinterpret the recently discovered circulation of a colloidal sphere in an optical tweezer (2) in light of this insight and demonstrate that it constitutes a practical realization of a Brownian vortex. One consequence is the prediction, which we confirm both through simulation and experimentally, that a trapped particle's circulation can undergo flux reversal with continuous changes in laser power or temperature. These observations reveal that flux reversal in Brownian vortexes proceeds through a surprising and distinctive two-stage mechanism.
Our discussion focuses on a single particle's
motions through a viscous medium that also acts
as a thermodynamic heat bath at temperature
.
The particle moves under the influence
of a static force field,
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Thermal fluctuations enable the particle to explore the force landscape,
and we further assume that
confines the particle so that its
probability density
does not change with time.
The probability flux,
Both the magnitude and the direction
of
can vary with
position.
The rate and direction of the particle's
circulation therefore depend on the domain over which
the particle can diffuse at temperature
.
Changing the temperature changes this range and therefore
can reverse the sense of the overall circulation.
The possibility of temperature-dependent
flux reversal distinguishes Brownian vortex circulation
from the more familiar interplay of advection and diffusion
in such systems as the
electric current flowing through a battery-powered circuit.
As a concrete example, we consider the motions of a colloidal sphere in a optical tweezer, a single-beam optical gradient force trap created with a strongly focused beam of light (4). Recent three-dimensional particle-tracking measurements (2) have revealed a previously unsuspected toroidal bias in the trapped particle's diffusion, which is depicted schematically in Fig. 1(a). Whereas intensity gradients draw the particle to the beam's focus with a manifestly conservative restoring force, non-conservative radiation pressure biases its fluctuations (2). This cannot constitute an example of a Brownian motor, as was suggested in Ref. (2), because Brownian motors rely on time-dependent forcing to break detailed balance (1). A more detailed examination of the forces acting on the trapped particle instead reveals this system's true nature as a Brownian vortex.
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Figure 2(a) shows streamlines of the force field computed
with Lorenz-Mie theory (5,6) for
an 800 nm diameter polystyrene sphere
trapped by an optical tweezer in water.
This is a fully vectorial treatment for a beam
with vacuum wavelength
of
propagating in the
direction and
brought to a focus by an ideal lens with numerical aperture 1.4.
Streamlines are projected into the
plane
in cylindrical coordinates,
.
Figures 2(b) and (c) show the irrotational and
solenoidal components of the force field, respectively, which
were obtained through the Helmholtz-Hodge decomposition (3).
The particle's thermally driven trajectory
through
was computed
with a Brownian dynamics simulation of the Langevin equation
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Figure 3(a) shows streamlines at a comparatively
large laser power,
, for which the particle is well localized near the optical axis.
Under these conditions, the particle circulates in a single
toroidal vortex, much as was
predicted in Ref. (2) and
portrayed in Fig. 1(a).
The local circulation rate,
,
is uniformly positive.
Reducing the laser power does not change the structure of the
force field, but reduces its overall magnitude.
This is equivalent, therefore, to increasing the effective
temperature.
Doing so increases the range over which the particle can wander
and enables it to populate a second, concentric counter-rotating
vortex, as plotted in Fig. 3(b) for
, and indicated
schematically in Fig. 1(b).
At still lower laser power (or higher temperature), the outer vortex
subsumes the inner vortex, and the probability current circulates
once again in a single toroidal roll, but with its direction
reversed.
Complete flux reversal
is demonstrated in Fig. 3(c) for
.
Such two-stage flux reversal is not observed in
Brownian motors or related temporally-driven stochastic heat engines.
Its origin can be found in the vorticity
of the force field
in Fig. 3.
Although the solenoidal component of the optical force,
,
is directed uniformly upward, its curl,
,
changes direction with distance from the optical axis.
So long as the particle's
probability density is concentrated in regions where
is positive, as is the
case in Fig. 3(a),
the overall circulation of the probability flux also is positive.
When the particle wanders into regions of negative
vorticity, it circulates in the retrograde direction, as shown
in Fig. 3(b).
In both cases, the non-conservative part of the optical force field
redistributes
downstream of the beam's focal point
and diffusion provides the return current.
The single-roll structure reasserts itself in the flux-reversed state
when gradients in
become large enough for diffusion
to outstrip advection along the optical axis.
We also observed flux reversal in Brownian vortex circulation
through experimental observations of colloidal
spheres trapped in optical tweezers.
Our system consists of 1.5
diameter
colloidal silica spheres (Bangs Labs, Lot SS04N/5252)
dispersed in a 50
thick layer of water that is hermetically
sealed between a clean glass slide and a No. 1.5 cover slip.
The sample is mounted on the stage of an inverted light microscope
(Nikon TE 2000U) where it is observed with a
numerical
aperture 1.4 oil immersion objective lens (Nikon Plan Apo).
The same objective lens is used to focus four holographic
optical tweezers (8,10,9,11)
arranged at the corner of a square with 30
sides near the midplane of the sample.
These traps are powered by a single laser (Coherent Verdi 5W,
)
that is imprinted with a computer-generated hologram (10)
by a liquid-crystal spatial light modulator
(Hamamatsu X7665-16) before being projected into the sample.
The trap array is designed so that two of the traps have
nearly the same intensity, the third is slightly brighter, and
the fourth brighter still.
This enables us to seek out intensity-dependent differences
in simultaneously acquired data sets, and thus to avoid artifacts
due to vibrations or other instrumental
fluctuations.
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The trapped spheres' three-dimensional motions are measured
with nanometer resolution
through quantitative analysis (12) of images
obtained with holographic video microscopy
(14,13).
Holographic images are obtained by illuminating the sample
with the collimated beam from
a HeNe laser (Uniphase 10 mW) operating at 632.8 nm.
Light scattered by the particles interferes with
the unscattered portion of the beam in the microscope's
focal plane to create an in-line hologram that is magnified
and recorded by a video camera (NEC TI-324AII) at 30 frames per
second.
Radiation pressure due to the
intensity of the imaging
beam is negligible compared with thermal forces and
forces due to the optical trap and so does not affect
the particles' trajectories.
Both fields of each interlaced holographic video frame
were analyzed with Lorenz-Mie
light scattering theory (5,12)
to measure each sphere's three-dimensional position
, with 3 nm in-plane resolution and
10 nm axial resolution (12)
A total of thirty two-minute-long trajectories were acquired
at constant laser power for the four particles.
At the end of each acquisition period, the trapped particles
were moved automatically out of the field of view to
acquire background holograms and to confirm the system's
stability.
Each particle's trajectories were analyzed with
Eqs. (5) and (6)
to visualize the mean circulation
and the results combined into maps of the mean circulation
for each trap.
In all, more than 100,000 holograms were analyzed for each trap.
Figure 4 shows streamlines of the trajectories
for three of the four particles, the fourth serving as a control
for Fig. 4(c).
These results confirm not only the presence of toroidal circulation
in the particles' motions, but also the appearance of flux
reversal as a function of trap strength.
Each trap-particle combination
is characterized by its apparent (2)
in-plane stiffness,
, which is obtained from
statistical analysis of the particle's
measured in-plane fluctuations (10), stiffer
traps corresponding to higher laser power
and lower effective temperature.
The traps' apparent axial stiffness is a factor of 5 smaller
than their lateral stiffness because axial intensity gradients
are correspondingly weaker (2,15).
The stiffest trap, shown in Fig. 4(a), concentrates
its particle's probability density
closest to the
optical axis and displays a single roll circulating in the positive
sense.
The weaker trap in Fig. 4(b) allows the
trapped particle to wander further afield, where it enters
into concentric counter-rotating rolls, similar to the
simulated results in Fig. 3(b).
The weakest traps, one of which is represented in Fig. 4(c),
both display a single retrograde roll in
,
and thus demonstrate complete flux reversal.
The probability distribution is centered lower in the weaker
traps because of gravity acting on the silica spheres, whose
density exceeds that of the
surrounding water.
This additional conservative force does not directly
contribute to the particles' circulation but does affect
what region of the optical force field the particle occupies
for a given laser power.
Undoubtedly, this influenced the trend in Fig. 4, but
does not change our interpretation of the phenomenon as
two-stage flux reversal in a Brownian vortex.
Instrumental fluctuations cannot account for our observations because all four measurements were performed simultaneously in a static array of optical traps derived from the same laser beam. The particles are sufficiently separated from each other and from the walls of their container that hydrodynamic coupling also is unlikely to have influenced their motion (16). Rather, Brownian vortex circulation, including power-dependent flux reversal, appears to be an inherent aspect of the statistics of colloidal spheres in optical tweezers.
In this study, we have introduced the Brownian vortex as a distinct class of noise-driven machines. Unlike stochastic heat engines driven by time-dependent forces (1), Brownian vortexes arise in static force fields possessing both potential and non-conservative solenoidal components. Because one-dimensional force fields have no solenoidal component, the Brownian vortex has no one-dimensional manifestation. Not any static force field, furthermore, can support a Brownian vortex. For example, a force field lacking a sufficiently strong confining potential cannot establish the requisite probability-conserving steady-state. Still other force fields establish circulating steady states without thermal noise. An example of this is provided by the ring-like optical trap known as an optical vortex (17) that exerts torques on trapped objects (18) through its helical wavefront structure (19). These are deterministic machines rather than stochastic heat engines, and so are not Brownian vortexes.
Although the simulations and experiments presented here focus on colloidal circulation in optical tweezers, the Brownian vortex is a general phenomenon. Seeking its signature in such contexts as biological networks and financial systems, as well as in new mechanical models, should provide opportunities for future research. Further work also is required to elucidate Brownian vortexes' thermodynamic properties, particularly the considerations that determine their thermodynamic efficiency.
We acknowledge fruitful conversations with Yohai Roichman. This work was supported by the MRSEC program of the NSF under grant number DMR-0820341. B.S. acknowledges support from the Kessler Family Foundation.