Thermotaxis of Janus Particles
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Eur. Phys. J. E manuscript No.
(will be inserted by the editor)
Thermotaxis of Janus Particles?
Sven Auschraa,1 , Andreas Bregullab,2 , Klaus Kroyc,1 , Frank Cichosb,2
1 Institute for Theoretical Physics, Leipzig University, 04103 Leipzig, Germany
2 Peter Debye Institute for Soft Matter Physics, Leipzig University, 04103 Leipzig, Germany
arXiv:2103.15165v1 [cond-mat.soft] 28 Mar 2021
Received: date / Accepted: date
Abstract The interactions of autonomous microswim- struggle to locomote through liquid solvents [6, 7], in-
mers play an important role for the formation of col- teractions with boundaries and neighbors and the sens-
lective states of motile active matter. We study them ing of chemical gradients [8] are key features involved
in detail for the common microswimmer-design of two- in the search of food, suitable habitats or mating part-
faced Janus spheres with hemispheres made from differ- ners. Inspired by nature, scientist designed synthetic,
ent materials. Their chemical and physical surface prop- inanimate microswimmers that mimic the characteris-
erties may be tailored to fine-tune their mutual attrac- tics of biological swimmers and are more amenable to
tive, repulsive or aligning behavior. To investigate these a systematic investigation of their interactions. A very
effects systematically, we monitor the dynamics of a sin- popular design exploits self-phoresis [9] for which nu-
gle gold-capped Janus particle in the external tempera- merous experimental and theoretical studies are avail-
ture field created by an optically heated metal nanopar- able [10–12]. Such self-phoretic propulsion relies on the
ticle. We quantify the orientation-dependent repulsion interfacial stresses arising at the particle–fluid interface
and alignment of the Janus particle and explain it in in the self-generated gradient of an appropriate field
terms of a simple theoretical model for the induced ther- (temperature, solute concentration, electrostatic poten-
moosmotic surface fluxes. The model reveals that the tial). On a coarse-grained hydrodynamic level, this ef-
particle’s angular velocity is solely determined by the fect is captured by an effective tangential slip of the
temperature profile on the equator between the Janus fluid along the particle surface [9] that drives the self-
particle’s hemispehres and their phoretic mobility con- propulsion of the swimmer. Accordingly, thermophore-
trast. The distortion of the external temperature field sis [13–15], diffusiophoresis [16–18] and electrophoresis
by their heterogeneous heat conductivity is moreover [19, 20] can deliberately be exploited for (or may in-
shown to break the apparent symmetry of the problem. advertently contribute to) the swimming of Janus par-
ticles [21–29]. The classical Janus-particle design con-
sists of a spherical colloid with hemispheres of distinct
1 Introduction physico-chemical properties, which define a polar sym-
metry axis. Due to the broken symmetry, one expects
Ranging from flocks of birds via schools of fish to colonies the axis to align with an external field gradient [30],
of insects, a distinctive trait displayed by the individ- the direction being determined by the precise surface
ual constituents of motile active matter [1–3] is a unique properties and the chosen solvent [31, 32]. The reori-
capability to adapt to environmental cues [4]. Down to entation of microswimmers in external fields is often
the microbial level where all kinds of “animalcules” [5] referred to as taxis and has been studied for various
phoretic propulsion mechanisms [33–38].
? Contribution to the Topical Issue “Motile Active Matter”, edited What we call an external field can be understood
by Gerhard Gompper, Clemens Bechinger, Roland G. Winkler,
Holger Stark
as a template for the influence of container walls or
a e-mail: sven.auschra@itp.uni-leipzig.de neighboring microswimmers [10, 39–41] that are at the
b e-mail: cichos@uni-leipzig.de core of the rich collective phenomena emerging in ac-
c e-mail: klaus.kroy@uni-leipzig.de tive fluids [42–50]. That microswimmers are constantly
2
exchanging linear and angular momentum with the am- (a) y
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
(b) vs what if ...
bient fluid generally renders their apparent mutual in- they do not fluoresce?
metal particle
ø 1-80 nm
teractions non-reciprocal. Next to the thermodynamic
absorption cross section
u ~!3000 Å2
µ
they heat the environment!
field gradients also the hydrodynamic flow field gener- vs
au
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
what if ...
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
µps
ated by one swimmer at the position of another one (c) vs
they do not fluoresce?
what if ...
n r
↵ = 0.25 µm
x
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
metal particle
ø 1-80 nm
they do not fluoresce?
affects the swimmers’ interactions [10, 51]. Generally, ri absorption cross section
↵ = 1 µm ~!3000 Å2 metal particle
ø 1-80 nm
absorption cross section
u ~!3000 Å2
interactions mediated by hydrodynamic flow fields [52–
they heat the environment!
they heat the environment!
vs
57], optical shadowing [38, 58] and chemical or opti- (d) what if ...
vs
cal patterns [59–61] may have to be considered, and they do not fluoresce?
metal particle
which of these contributions dominate the observed mo- ro ⌦
ø 1-80 nm
absorption cross section
~!3000 Å2
u vs
tion of microswimmers has recently been under debate
they heat the environment!
what if ...
⇡ 1.5 µm
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
they do not fluoresce?
metal particle
ø 1-80 nm
[39, 40, 62].
absorption cross section
~!3000 Å2
they heat the environment!
In the present contribution, we report results from Figure 1 Schematic of the experimental setup and the
an experiment designed to allow for direct measure- phoretic motion. (a) A passive Janus polystyrene (ps) bead
ments of the induced polarization and motion of a single with a thin gold (au) cap (thermophoretic mobilities µps , µau ) is
exposed to the temperature gradient around a laser-heated immo-
passive (i.e., not self-driven) Janus particle in the tem- bilized gold nanoparticle. Particle translation is restricted to the
perature field emanating from a localized heat source sample plane due to the thin liquid film thickness (1.5 µm). The
in its vicinity. In other words, it enables us to single out coordinate frame attached to the particle’s geometric center has
the passive phoretic response of a self-thermophoretic its x-axis aligned with the particle’s symmetry axis and pointing
towards its ps-side, while the z-axis points into the paper plane.
swimmer to an external temperature field, without hav- The in-plane angle φ and normal angle θ are measured with re-
ing to bother with the autonomous motion that would spect to the x-axis and z-axis, respectively, and technically (yet
result from a direct (laser-)heating of the swimmer it- not with respect to the particle’s polarity) take the role of what
self. Typically such measurements are difficult to con- is conventionally called “azimuthal” and “polar” angles, respec-
tively. The orientation angle of the swimmer relative to the heat
duct since pairwise collisions are rare at low concentra- source is γ. (b–d) Phoretic translational and rotational veloci-
tions and hard to discern among interfering the many- ties u, Ω, arise from slip fluxes with velocities vs induced by the
body effects, at high concentrations. However, by adopt- temperature gradient, chiefly near the particle equator. Arrow
ing the technique of photon nudging [22, 63] we can di- lengths and orientations indicate the magnitude and direction of
the velocities.
rect an individual Janus particle into the vicinity of the
local heat source [23, 24], without imposing potentially
perturbing external fields. This allows us to precisely back across an inner radius ri , followed by a waiting
record the polarization effects and thereby characterize time of 10 rotational diffusion times τr to allow for the
the elusive phoretic repelling and aligning interactions decay of orientational biases and correlations [66, 67].
with good accuracy and good statistics. Our experimen- All data recording and feedback is carried out in a
tal results are substantiated by a theoretical model that custom-made dark field microscopy setup with an in-
addresses the thermophoretic origin of the interactions verse frame rate and exposure time of 5 ms. Further de-
and complements recent calculations of the phoretic in- tails regarding the sample preparation, the experimen-
teractions between two chemically active particles [64] tal setup, and the position and orientation analysis are
and the axis-symmetric interactions between two diffu- contained in Appendix A–Appendix D. The tempera-
siophoretic Janus particles [65]. ture increment ∆T = 12 K of the heated gold nanopar-
ticle relative to the ambient temperature (T0 = 295 K)
is known from a separate measurement using the ne-
2 Experimental Setup matic/isotropic phase transition of a liquid crystal (see
Appendix E). We account for the direct influence of
We experimentally explore the interaction of a 1 µm
the heating laser on the Janus particle and the phoretic
diameter Janus particle with a 50 nm thin gold cap
velocities, as detailed in Appendix F.
with the temperature field generated by an immobilized
250 nm gold nano-particle optically heated by a focused
laser (wavelength 532 nm). To confine the Janus parti- 3 Results and Discussion
cle to the vicinity of the heat source, we employ the
feedback control technique of photon nudging [22, 63] 3.1 Theory
that exploits its autonomous motion to steer it to a cho-
sen target. As illustrated in Fig. 1 (a), the steering is On the hydrodynamic level of description, the temper-
only activated when the Janus particle leaves an outer ature gradient ∇k T ≡ (I − er er )∇T along the surface
radius ro around the heat source until it has migrated of the Janus particle induces a proportionate interfacial
3
creep flow [68, 69], where er denotes the unit vector nor- Motivated by theoretical studies of chemotactic ac-
mal to the particle surface and I the unit matrix. Since tive colloids [31], we further employ the following model
the interfacial flow is localized near the particle sur- for the angular velocity:
face, it is conveniently represented as a slip boundary
condition with slip velocity [9, 26, 30] Ωz (γ) = Ω1 sin γ + Ω2 sin(2γ), (7)
vs (θ, φ) = µ (θ, φ) ∇k T (θ, φ) . (1) introducing the independent parameters Ω1,2 . This is a
natural extension of Ωz ∝ (µps − µau ) sin γ for particles
The particle surface is parametrized in terms of the in- with isotropic heat conductivity [30], to account for the
plane and normal angles φ and θ, as sketched in Fig. 1 material heterogeneities of the Janus sphere. Similar
(a,b). They technically take the role of "azimuthal" and (reflection) methods as presented in [31, 33, 75] might
"polar" angles, respectively, although these notions are be employed to establish a connection between Ω1,2 and
not associated with the particle’s polar symmetry, here. the material and interaction parameters of the Janus
And µ (θ, φ) is a phoretic mobility characterizing the particle and the ambient fluid, but we do not pursue
varying strength of the creep flow due to the distinct this further, here. The crucial feature is that the term
interfacial interactions with the solvent [30]. The result- ∝ sin(2γ) acknowledges the higher periodicity of the ef-
ing translational propulsion velocity u and the angular fect of the two hemipsheres’ distinct heat conductivities
velocity Ω of the Janus particle of radius a are given onto the rotational motion.
by averages over its surface S: [30, 70] The competition between the phoretic alignment of
1 the Janus particle and its orientational dispersion by
I
u=− dS vs , (2) rotational diffusion can be described by the Fokker–
4πa2 S
Planck equation [42, 76]
1 3
I
Ω=− dS × vs . (3)
4πa2 S 2a ∂t f = −R · (Ω − Dr R) f, (8)
Further analysis of Eqs. (2) and (3) becomes pos- for the dynamic probability density f (t, n) to find the
sible by the experimental observation that the Janus particle at time t with an orientation n (relative to
particle is preferentially aligned with the sample plane. the heat source). The rotational operator R ≡ n × ∇n
This effect is presumably mostly due to the hydrody- includes the nabla operator ∇n with respect to the par-
namic flows induced by the heterogeneous heating in ticle’s orientational degrees of freedom, and Dr denotes
the narrow fluid layers between the particle and the the (effective [77, 78]) rotational diffusion coefficient.
glass cover slides [71–73]. For simplicity, the following With the mentioned approximation of a strict in-plane
analysis assumes perfect in-plane alignment, thereby orientation of the particle axis, Eq. (8) greatly simplifies
neglecting weak perturbations due to rotational Brown- [30] to ∂t f = −∂γ J, with the flux
ian motion and the weak bottom-heaviness of the Janus
particle [74]. For any given temperature profile T (φ, θ) J(γ, t) = −Ωz (γ)f (γ, t) − Dr ∂γ f (γ, t). (9)
at the surface of the Janus sphere, the components of
the translational and rotational velocity can then be In the steady state, the flux J is required to vanish
expressed as identically, and for an angular velocity Ωz of the form
Z 2π (7) the orientational distribution reads
1
ux = − dφ µ(φ) cos φ hT sin θiθ (φ)
Ω1 Ω2
πa 0 −1
f (γ) = N exp cos γ + 2
cos γ , (10)
" # Dr Dr
µps − µau
T T
+ + ,
2πa sin θ θ φ= π
2
sin θ θ φ= 3π
2
with a normalization factor1 N .
(4)
2π
1 3.2 Experimental Results
Z
uy = − dφ µ(φ) sin φ hT sin θiθ (φ), (5)
πa 0
Figure 2 displays the experimental results for the mag-
3 h i
Ωz = (µ ps − µ au ) hT iθ |φ= 3π − hT i |φ= π
θ , (6) nitude of the phoretic propulsion speed u(γ, r) as a
4πa2 2 2
function of the distance r from and orientation γ to
where we have Rintroduced the average over the normal the heat source. The speed u decays with the squared
π
angle h•iθ ≡ 21 0 dθ sin θ(•). All other velocity compo- h i. √
nents give zero contributions, as the detailed derivation
1N = eB−A e2A D A2+2 √ B − D A−
B
√2B
2 B
B with A ≡
of Eqs. (4)–(6) in Appendix G shows.
2 Rx 2
Ω1 /Dr , B ≡ Ω2 /Dr , and D(x) ≡ e −x
0
y
dy e .
4
(a) (c) (a) (b)
1.8 90° 90°
1
speed |u| [µm/s]
1.6
0.5 1.4
1K 2K 1K 2K
180° 0° 180° 0°
orientation 1.2
0 120
30 150
60 180 1.0
0.1 90
1 2 3 4 0 45 90 135 180
270° 270°
(b) (d)
mean angular velocity z [°/s]
100 20
angular velocity z [°/s]
50 15
(c) (d)
90°
0 10 90°
50 5
0
100 180°
1K 2K
0° 180°
1K 2K
0°
1 2 3 4 0 45 90 135 180
distance to heat source r [µm] orientation angle [°]
Figure 2 Distance- (a,b) and orientation- (c,d) depen-
dence of translational and (mean) angular swim speed. 270°
270°
The Janus particle’s swim speed u (a) and mean angular speed
Ω̄z (averaged over initial orientations γ) (b) both decay like
r−2 (dash-dotted line) in the distance r from the heat source, Figure 3 Numerically determined azimuthal tempera-
as expected from Fourier’s law of heat diffusion. Upper and lower ture variations. The temperature increments hT iθ (φ) − T0 (in
branch in panel (b) correspond to clockwise and counterclockwise Kelvin) along the Janus particle circumference but averaged over
rotation, respectively. The orientational dependence of the swim the normal angle, are depicted for 4 different orientations γ and
speed in panel (c), measured at a distance r = 1.25 µm from a fixed distance r = 1.25 µm between the Janus particle and the
1/2
the heat source, conforms with the theoretical fit u2 2
x + uy source: (a) γ = 0°, (b) γ = 90°, (c) γ = 180°, (d) γ = 270°.
with ux,y obtained from Eqs. (4), (5) using numerically deter-
mined temperature profiles (see Fig. 3). (d). To resolve the
orientation-dependence of the angular velocity Ωz , data in the in-
terval r = 1-4 µm was pooled. The theoretical fit (solid curve) was The translational and rotational speeds depend on
obtained from Eq. (6), again using the numerically determined the orientation γ to the heat source, due to the Janus-
temperature profiles. The least-square fits in c) and d) yield faced particle surface and its heterogeneous mobility
µps = 2.88 µm2 /sK and µau = 1.82 µm2 /sK for the mobilities.
coefficients µ and thermal conductivities κ. We have
The alternative fits shown in panel d) follow from Eq. (7) with
Ω1,2 as independent fit parameters (dashed) and Ωz = Ω0 sin γ therefore also analyzed the particle’s motion as a func-
[30], with Ω0 as fit parameter (dotted), and yield Ω0 = 17.4 °/s, tion of the initial orientation γ. The experimental re-
Ω1 = 17.2 °/s, Ω2 = −3.25 °/s. sults are plotted in Figure 2 (c) and (d). For the trans-
lational phoretic speed u we observe a clear minimum
between γ = 50° and γ = 135°. Local maxima are ob-
reciprocal distance, as expected for an external temper- served when the polymer side is facing the heat source
ature gradient ∇T ∝ 1/r2 consistent with Fourier’s law. (γ = 180°) or pointing away from it (γ = 0°). That the
The maximum speed is u = 2 µm/s at a distance of r = latter orientation displays a smaller speed suggests that
1.25 µm. Closer to the heat source, tracking errors limit the polymer side yields the major contribution.
the acquisition of reliable data. The experiments also In spite of averaging Ωz over the measured distance
provide direct evidence for a thermophoretic rotational range (1 µm–4 µm) the γ-dependent angular velocity ex-
motion of the Janus particle. According to Eq. (6), the hibits some residual scatter. It is still seen to vanish for
boundary temperatures as well as the phoretic mobility γ = 0° and γ = 180° [Fig. 2 (d)], in line with the ex-
coefficients must therefore differ between the gold and pected symmetry of the temperature field around the
polystyrene parts of the particle. Figure 2 (b) shows the axis of the Janus particle. At γ ≈ 90°, we observe a
mean angular velocity Ω̄ for clockwise (+) and counter- maximum angular and minimum translational speed.
clockwise (−) rotation, with the mean over positive To compare the experimental results to our theo-
and negative values of the initial orientation γ taken retical expectations (4)–(7), we require further infor-
separately. The angular velocity also decays with the mation on the angular dependence of the temperature
squared reciprocal distance from the heat source from at the surface of the Janus particle. For this purpose, we
Ω̄ = 100 °/s at short distances. numerically solved the complex heat conduction prob-
5
lem with a commercial PDE solver [79] (Appendix H). (a) 0.8 (b)
The obtained profiles of the mean temperature incre- 0.8 0.7
probability density
ment hT iθ (φ)−T0 along the circumference of the Janus 0.6
particle are displayed in Fig. 3. They reveal that the 0.6
largest temperature difference between the gold (au) 0.5
and polystyrene (ps) side is attained when the polymer 0.4 0.4
is facing the heat source, confirming the experimental 0.3
trend. They also exhibit unequal mean boundary tem- 0.2
peratures hT iθ |φ=3π/2 and hT iθ |φ=π/2 , as required by (d)
0.7 (c)
Eq. (6) for angular motion. 0.60
probability density
The experimental results on the translational and 0.6 0.55
the angular velocity as a function of the orientation an-
gle γ can be compared to the theoretical predictions 0.5 0.50
(4)–(6) while using the numerically calculated surface 0.45
temperature profiles to obtain estimates for the phoretic 0.4
0.40
mobility coefficients pertaining to the different surface 0.3
regions of the Janus particle. A least-square fit of the 0 45 90 135 180 0 45 90 135 180
theoretical prediction (6) for the angular velocity Ωz orientation angle [°] orientation angle [°]
yields our best estimate for µps − µau . Inserting it into Figure 4 Probability density to find the Janus sphere
Eqs. (4) and (5) for the translational velocity compo- pointing at an angle γ to the heat source. The panels show
nents, another least-square fit for the phoretic speed data (symbols) measured for various distances r between particle
and heat source: (a) 1.1 µm, (b) 1.7 µm, (c) 2.3 µm, (d) 2.8 µm.
(u2x +u2y )1/2 eventually yields the optimum values µps = The solid lines are best fits by Eq. (10), with free fit parameters
2.88 µm2 /sK and µau = 1.82 µm2 /sK for the phoretic Ω1,2 /Dr . The dashed lines are fits by f ∝ exp[(Ω0 /Dr ) cos γ ]
mobilities. The theoretical fits are shown in Fig. 2 (c,d) [30] for an angular speed profile Ωz = Ω0 sin γ with Ω0 /Dr as
as solid lines, while the dashed line is a fit of Eq. (7) a free fit parameter. These fits yield for Ω0 /Dr , Ω1 /Dr , Ω2 /Dr
the values (a) 0.596, 0.722, -0.284, (b) 0.493, 0.526, -0.091, (c)
with Ω1,2 as independent fit parameters. It nicely re- 0.355, 0.377, -0.0855, (d) 0.224, 0.228, -0.0248, respectively.
produces the experimental data. In contrast, assuming
the rotational speed to be of the form Ωz ∝ sin θ [30]
(dotted), as for homogeneous heat conductivity, misses if the hemispheres do not have the same heat con-
the experimentally observed asymmetry. ductivities, this will distort the temperature profile in
Besides these dynamical properties, we also assessed the surrounding fluid in an unsymmetric, orientation-
the stationary distribution of the Janus particle’s orien- dependent manner. Secondly, their generally unequal
tation relative to the heat source, at various distances. thermo-osmotic mobility coefficients µ will translate the
Figure 4 verifies that the particle aligns with the exter- resulting surface temperature gradients differently into
nal temperature gradient. In accordance with the pos- phoretic motion. The numerically determined tempera-
itive angular velocities observed for 0 < γ < 180° in ture profiles for our Janus particle, shown Fig. 3, reveal
Fig. 2 (d), we measure a significantly higher probabil- that the presence of the Janus particle indeed distorts
ity to find the particle’s gold cap pointing towards the the external field significantly, and that the difference
heat source than away from it. between the heat conductivities of the two hemispheres
matters. The large thermal conductivity of gold cre-
ates an almost isothermal temperature profile on the
3.3 Discussion gold cap (even if the thin film conductivity is some-
what lower than the bulk thermal conductivity). The
The motion of a colloidal particle in an external tem- resulting temperature distribution is for some orienta-
perature gradient is determined by the thermo-osmotic tions γ reminiscent of the temperature distribution on
surface flows [69] induced by the temperature gradi- the surface of a self-propelled Janus particle. In the lat-
ents along the particle’s surface via its physio-chemical ter case, the metal cap itself is the major light absorber
interactions with the solvent. Knowing both the tem- and thus the heat source creating the surrounding tem-
perature profile and interfacial interaction characteris- perature gradient. In our case the gradient is primarily
tics should thus allow the behavior of our Janus par- caused by the external heat source, but modulated by
ticle in an external temperature field to be explained. the presence of the Janus sphere. Unless the particle’s
Note, however, that the heterogeneous material proper- symmetry axis is perfectly aligned with the heat source
ties of the Janus particle matter in two respects. First, [Fig. 3 (a),(c)], the mean temperature profile is gen-
6
erally asymmetric along the particle’s circumference. 6
(a) (b)
velocity components ux, y [µm/s]
6
Such asymmetric distortions of the temperature field
were not considered in previous theoretical studies [30] 5
speed |u| [µm/s]
4
but matter for the proper interpretion of Eqs. (4)–(6) 4
for the particle’s linear and angular velocities. 2
3
Equation (5) yields the transverse thermophoretic 0 2
velocity, uy , of the particle, i.e., the velocity perpendic-
1
ular to its symmetry axis. Assuming a constant temper- 2
0 45 90 135 180 0 45 90 135 180
ature on the gold hemisphere, the only contribution for orientation angle orientation angle
the transverse velocity uy results from the temperature Figure 5 Orientation dependence of the phoretic propul-
gradients along the polystyrene side — due to the sin φ sion (a) Longitudinal and transverse velocity components ux
term in Eq. (5) — and uy is determined by the mobil- and uy (uy starting at the origin) from Eqs. (4) and (5), re-
ity coefficient µps . The velocity component ux along the spectively. For the phoretic mobilities, three sets of values are
considered: those obtained from the fits in Fig. 2 (b,d), namely
particle’s symmetry axis contains two terms according µps = 2.88 µm2 /sK and µau = 1.82 µm2 /sK (solid lines);
to Eq. (4). The first term yields a propulsion along the µps = 0.5 µm2 /sK and µau = −0.55 µm2 /sK (dashed); µps =
symmetry axis to which both hemispheres contribute −1.82 µm2 /sK and µau = −2.88 µm2 /sK (dotted); (b) the cor-
−1/2
according to the cos φ term. It tends to suppress the responding total propulsion speeds u2 2
x + uy .
details at the au–ps interface, where the temperature
gradients are typically most pronounced. Hence, the
temperature profile in the vicinity of the particle poles and only a lesser local maximum is seen at γ = 0° (au-
and the corresponding mobilities largely determine the side facing the heat source), in Fig. 2 (c).
first term in Eq. (4). The second term, which only de- From our fits in Fig. 2 we obtained µps > µau > 0.
pends on the boundary values of the (weighted) mean The first condition ensures the correct sign for the an-
temperature at the au-ps interface and the mobility step gular velocity according to Eq. (6) and Fig. 2(d), and is
µps − µau , is of opposite sign and thus reduces the total in agreement with previous findings for thermo-osmotic
propulsion velocity. (It disappears if µps ≈ µau .) interfacial flows [69]. The step in the phoretic mobility
at the particle equator determines the magnitude and
Figure 5 illustrates the orientation dependence of sign of the angular velocity [30]. While different abso-
the phoretic velocity compoents ux and uy obtained lute values can lead to the same step height µps − µau ,
from Eqs. (4) and (5). The longitudinal component ux the γ−dependence of the translational velocity also con-
(along the particle’s symmetry axis) is positive or neg- straints these absolute values. This is illustrated by the
ative depending on whether the ps-hemisphere faces dashed and dotted lines in Fig. 5, representing other
away from or towards the heat source. Its smooth sign combinations of phoretic mobilities, including negative
change at γ ≈ 90° simply reflects the fact that the signs (µps > 0, µau < 0 or µau < 0 < µps ). Such choices
interaction is overall repulsive. Notice, however, that would result in a quantitative and qualitative mismatch
the higher thermal conductivity of the gold cap creates between theory and data. They also serve to demon-
a surface temperature contribution mimicking that for strate that the motion of the particle is very sensitive
an optically heated Janus swimmer. The ensuing (self-) to these values, for a given temperature profile.
propulsion along the x direction shifts the zero cross- According to Eq. (6), the angular velocity compo-
ing slightly from 90°. This thermophoretic "swimmer- nent Ωz only depends on the equatorial interfacial val-
contribution" to the propulsion is not generally parallel ues at φ = π/2 and 3π/2 of the average temperature
to the direction of the external temperature gradient, hT iθ and the jump in the mobility coefficients. In other
unless it is perfectly aligned to the heat source, thereby words, the details of the temperature profile on both
causing subtle deviations from predictions for parti- sides of the particle are irrelevant for the rotational mo-
cles with isotropic heat conductivity [30]. The trans- tion as long as the two boundary temperatures and the
verse velocity component uy naturally vanishes if the two mobility coefficients differ appreciably, but rota-
particle axis is aligned or anti-aligned with the heat tional motion will cease if either pair coincides. Hence,
source (γ = 0° and γ = 180°). It attains a maximum irrespective of the negligible temperature gradient on
at γ ≈ 120°, when the polystyrene hemisphere is ori- the gold side, the rotational velocity is sensitive to the
ented somewhat towards the heat source, which allows thermo-osmotic mobility coefficient µau , which can thus
for the maximum lateral surface temperature gradients. confidently be inferred from the measurement. Com-
For the same reason, the maximum propulsion speed u pared to the substantial thermal-conductivity contrast,
is attained for γ = 180° (ps-side facing the heat source) the role of mass anisotropy, which can lead to similar
7
polarization effects [80–82], plays presumably a negligi-
ble role in our experiments, as the thin gold cap makes
the Janus particle only slightly bottom heavy.
4 Conclusions
To summarize, we have investigated the interaction of
a single gold-capped Janus particle with the inhomo-
geneous temperature field emanating from an immobi-
lized gold nanoparticle. The setup allows for a precise
and well-controlled study of thermophoretic interpar-
ticle interactions that dominate in dilute suspensions
of thermophoretic microswimmers. To our knowledege,
Figure 6 Raster electron microscopy image of the prepared
this is the first time, the repulsion of the Janus par- Janus Particles.
ticle from the heat source and its thermophoretically
induced angular velocity have quantitatively been mea-
sured. An interesting consequence of the induced an- Acknowledgements: We acknowledge financial sup-
gular motion is an emerging polarization of the Janus port by the Deutsche Forschungsgemeinschaft via SPP
particle in the thermal field, which should generalize 1726 "Microswimmers" (KR 3381/6-1, KR 3381/6-2 and
to any type of Janus swimmer in a motility gradient. CI 33/16-1, CI 33/16-2).
In our case, it means that the metal cap preferentially
points towards the heat source.
In combination with numerically determined surface
temperature profiles for various particle-heat source ori-
entations, the standard hydrodynamic model for col-
Author Contribution Statement
loidal phoretic motion was found to nicely reproduce
our experimental data. Theory and observation corrob-
AB conducted the experiments and analyzed the data.
orate that the rotational motion hinges on two neces-
FC carried out the numerical calculations. SA carried
sary conditions: (i) the phoretic mobilities of the Janus
out analytical calculations. KK, FC and SA drafted the
hemispheres must be distinct and (ii) the values of the
manuscript.
driving field (in our cases the temperature) must differ
accross the equator — irrespective of its behavior in be-
tween. In return, we could therefore infer the phoretic
mobilities from the observed rotational and transla- Appendix A: Preparation of the Janus particles
tional motion in an external field gradient. We found
them to be positive for both polystyrene and gold. The Janus particles have been prepared on standard
As an interesting detail, we found that the distinct microscopy glass cover slips, which have been treated
heat conductivities moreover break the naively expected in an oxygen plasma. A solution of polystyrene beads
symmetry of the particle’s translational and rotational (R = 0.25 µm; Microparticles GmbH) is deposited on
speeds as a function of the orientation, and, accordingly, these cover slips in a spin coater at 8000 rpm. The parti-
of the resulting polarization of the Janus sphere with cle concentration of the bead solution has been adjusted
respect to the heat source. The observed asymmetries such that the particles do not form a closed packed
are quantitatively explained by the high heat conduc- monolayer but settle as rather isolated particles. This
tivity of gold, which renders the metal cap virtually reduces the number of aggregates formed during the
isothermal. This induces a robust translational motion gold layer depostion. The samples have been further
that mimicks the self-propulsion of a Janus swimmer in covered with a 5 nm chromium and a 50 nm gold film
its self-generated temperature gradient, along it sym- by evaporation in a vacuum chamber. The chromium
metry axis. Since phoresis generally involves gradients layer has been added to make sure that the gold layer
in some (typically long-ranged) thermodynamic fields, adheres to the glass slide when removing the Janus par-
our principal results should also apply to similar setups ticles from the glass substrate by sonification. Fig. 6
involving other types of phoretic mechanisms. displays a REM image of the prepared particles.
8
Appendix B: Sample preparation
dark field illumination
The samples consist of two glass cover slips, which were lamp
rinsed with acetone, ethanol and deionized water, and
treated with an oxygen plasma. They have been fur-
ther coated with Pluronic F-127 (Sigma Aldrich) in
a 5 % aqueous solution for a few hours. The Pluronic
piezo stage, sample, detection
is adsorbed to the glass surface and residual Pluronic laser steering
has been removed by rinsing the coated slides with D
deionized water. A mixture of Janus particles and R =
125 nm gold colloids (British Biocell) was then deposited AOD
between the two slides and sealed with polydimethyl-
siloxane (PDMS) to prevent evaporation of the solu- 532nm EM CCD
tion. The typical thickness of the liquid layer between
the glass slides has been adjusted to be on the order of
the diameter of the Janus particle (≈ 1 µm) to prevent ADC LabVIEW
motion in vertical direction. feedback control
Figure 7 Experimental setup used for the experiments. See text
Appendix C: Experimental setup for explanations and additional details.
The experimental setup consists of 2 parts: the heating
and the illumination part. For the heating part a com-
mon laser source at a wavelength of 532 nm was used. 1.75
(a) (b) 1.0 mW 1.5 mW
This beam was first enlarged by a beam expander to
fully illuminate an acousto-optic-deflector (AOD). This 1.70
AOD is utilized to freely steer the focused beam within 4 µm
refractive index
1.65
the sample. The optical path is arranged such that the
2.0 mW 2.5 mW
beam waist is approximately 500 nm. This beam is then 1.60
focused by an oil immersion objective lens (Olympus
100x NA 0.5-1.3) into the sample.
1.55 3.0 mW 3.5 mW
The illumination of the sample is realized by an oil 1.50
immersion dark field condenser (Olympus NA 1.2-1.4). 15 20 25 30 35 40
Temperature [°C]
The scattered white light is collected by the objective
and imaged on the CCD-camera (Andor iXon). For the
spatial position of the sample a piezo-scanner was used
(Physik Instrumente, PI). Figure 8 (a) The refractive index of 5CB in the nematic phase
(< 35 °C) and the isotropic phase (> 35 °C)[83] (b) Example
images of the growing isotropic bubble with heating power.
Appendix D: Particle tracking
center is determined. The x− and y−coordinates of the
To determine the position of the Janus particles a bi- geometric center are fitted with fx = ax x + bx and
nary image at a threshold above the background noise fy = ay x+by , respectively. From both fits, the in-plane
was taken. The particle with the larger visual area was orientation can be determined by γ = arctan (ax /ay ).
identified as the Janus particle the smaller one as the
gold heat source. The geometric centers of the visual im-
ages was identified with the particle position. For small Appendix E: Measurering the temperature
distances (< 1 µm) beween the Janus particle and the profile on the surface of the heated gold
gold particle, the determination of the position of both nanoparticle
particles fails. In this case the data is disregarded.
The image of he Janus particle is further analyzed with For the estimate of the temperature increment ∆TAu
multiple binary images that are obtained by limiting at the surface of the gold nanoparticle heat source an
the maximum image intensity to thresholds between additional experiment has been performed. In this ex-
10 % and 80 %. For each binary image the geometric periment the solvent was replaced by a liquid crystal
9
molten bubble scatters the incident white light and ap-
pears as a bright ring in the dark field microscope. Ex-
200 50
individual particles
linear fit
ample images are displayed in Fig. 8(b) for different
incident heating powers. The black circle indicates the
150
40 estimated bubble size. Its knowledge allows the surface
temperature of the gold colloid in the liquid crystal to
be estimate by:
T5CB [K]
30
TH2O [K]
100
20 rph (35◦ − T0 )
∆TAu = . (E.3)
R
50
10
Since the heat equation is linear, the estimate for the
temperature increment in water is determined by its
0 0
thermal conductivity κH2 O = 0.6 W/(mK) relative to
0.0 1.0 2.0 3.0
that of the isotropic liquid crystal, κ = 0.15 W/(mK).
Heating Power [mW]
The result is displayed in Fig. 9 where the approximate
Figure 9 Surface temperature increment on the gold col- temperature increment in water is displayed in addi-
loid (relative to the ambient temperature of the solvent) for a tion to the estimated temperature increment in 5CB
liquid-crystalline solvent and water, respectively. Symbols repre- in dependence on the heating power. From the linear
sent the experiment. The dashed line is a linear fit to the data to
extract the temperature increment per heating power. The values
fit a temperature increment per heating power of ≈
in water have been calculated from the known thermal conduc- 14 K/(mW) can be obtained.
tivities of the liquid crystal and water.
(5CB). Its nematic-to-isotropic melting transition upon
Appendix F: Influence of the laser heating on
heating beyond Tph = 35 °C [83, 84] was employed as a
the Janus particle
temperature sensor. The colloidal heat source generates
a radial temperature field
The focused laser beam (beam-waist ≈ 500 nm) used
∆TAu (r) R Pabs for the heating of the immobile gold colloid may also
T (r) = T0 + = T0 + (E.1) heat the Janus particle directly. To quantify this effect,
r 4πκr
the experiment was repeated with and without the im-
with κ being the thermal conductivity of the medium, mobile gold colloid with identical focus position. The
Pabs the absorbed power, proportional to the incident influence of the laser beam can be estimated by cal-
light power Pinc , T0 = 22 °C the ambient temperature, culating the particle velocity u and the radial velocity
and R = 125 nm the radius of the gold colloid. vR . The particle velocity is obtained by projecting the
Whenever the temperature exceeds the phase transition translational step ∆si = ri−1 − ri onto the particle
temperature T (r) > Tph the nematic order melts. Since orientation nio and then performing the ensemble av-
the molecular temperature field T (r) varies locally, the erage u = ∆si · nio i /∆t divided by the experimental
phase transition is confined to the vicinity of the heat timescale ∆t being the exposure time of the camera.
source. Due to the radially symmetric shape of the tem- Fig. 10 (a) displays the absolute value of u for 3 different
perature profile, an isotropic bubble forms around the orientations of the particle relative to the heat source
gold colloid if ∆TAu + T0 > Tph . The size rph of the φ. The particle velocity is always positive as a result
bubble scales linearly with ∆TAu and therefore with of the direct laser illumination, and quickly diminishes
the heating power Pinc : over a length scale comparable to the laser beam width.
∆TAu R The radial velocity vR = ∆si · eiR i /∆t being the en-
rph = ∼ ∆TAu ∼ Pinc (E.2) semble average of the scalar product of ∆si and the
(35◦ − T0 )
unit vector in radial direction eiR divided by the exper-
In the experiments, the size of the isotropic bub- imental timescale ∆t decays on similar length scales as
ble rph as the function of the incident power Pinc is the particle velocity. Even though the influence of the
of interest. Its observation in the dark field setup, (see direct laser heating on the Janus particle diminishes
Fig. 8(a), [83]) exploits the refractive-index change upon rather quickly with increasing distance, its influence is
melting. Similar to a colloidal particle with a refrac- still noticeable, and was therefore subtracted for the
tive index deviating from the surrounding material the velocities presented in the main text.
10
(a)
20
averaged speed for γ (b) 20 where |S| = 4πa2 is the area of the particle surface S.
15 It is experimentally observed that the particle pref-
particle velocity [µm/s]
15 Gaussian intensity profile
erentially aligns horizontally with the close-by cover
radial velocity [µm/s]
10
10
5 slides. This observation enters our theory through the
5
0
assumption that the swimmer rotates only about the z-
0
-5
axis, i.e., perpendicular to the observation plane. This
-5
implies that the swimmer also translates only in the x-y
-10
-10 plane. Once the swimmer’s z-axis remains invariant, the
0 1 2 3 4 0 1 2 3 4
radial distance to heat source [µm] radial distance to heat source [µm]
surface-temperature profile consequently always obeys
the (approximate) symmetry
Figure 10 (a) Particle speed u and (b) radial speed vR in de-
pendence on radial distance to the heat source for three orienta-
T (φ, π/2 − α) = T (φ, π/2 + α), α ∈ [0, π/2] (G.11)
tions γ of the particle relative to the heat source.
in the normal angle, in accord with the heterogeneous
Appendix G: Derivation of the phoretic material composition of the Janus sphere. The local
velocities phoretic mobility µ may likewise be expressed as
Appendix G.1: The setup µps for 0 ≤ φ ≤ φpa
µ = µ(φ) ≡ µau for φpa < φ ≤ φap , (G.12)
The considered setup of a Janus particle exposed to
µps for φap < φ < 2π
an external heat source, and conventions used in the
following derivations, are summarized in Fig. 1. The where µps and µau are the constant phoretic mobili-
induced slip velocity vs at the surface of a Janus sphere ties corresponding to the polystyrene and gold part of
of radius a is given by [see Eq. (1)] the swimmer, respectively, and φpa and φap denote the
angles pertaining to the equator between the distinct
vs (θ, φ) = µ(θ, φ)∇k T (θ, φ), (G.4)
surface materials.
where the in-plane angle φ and normal angle θ are em-
ployed to parametrize the particle surface (rather than
conventional polar coordinates adjusted to the particle Appendix G.2: Rotation
symmetry). In the above equation, µ(φ, θ) is the ther-
mophoretic mobility and We start with the term r̂ × vs inside the integral on the
∂θ T ∂φ T r.h.s. of Eq. (G.10). Plugging in Eqs. (G.4) and (G.5),
∇k T ≡ θ̂ + φ̂ (G.5) and using r̂ × θ̂ = φ̂ and r̂ × φ̂ = −θ̂, one obtains
a a sin θ
denotes the tangential part of the temperature gradient
µ(φ) ∂φ T
at the particle surface, expressed in terms of spherical r̂ × vs = (∂θ T )φ̂ − θ̂ (G.13)
a sin θ
coordinates. As they are constantly used in the follow-
ing derivations, we note the corresponding unit vectors: − sin φ
µ(φ)
> = (∂θ T ) cos φ (G.14)
r̂ ≡ cos φ sin θ, sin φ sin θ, cos θ , (G.6) a
0
>
θ̂ ≡ cos φ cos θ, sin φ cos θ, − sin θ , (G.7)
− cos φ cos θ
µ(φ)(∂φ T )
> + − sin φ cos θ . (G.15)
φ̂ ≡ − sin φ, cos φ, 0 . (G.8) a sin θ
sin θ
The translational and rotational phoretic velocities, u
and Ω, follow from vs as [see Eqs. (2) and (3)] The symmetry relation (G.11) implies
Z2π Zπ ∂θ T (φ, π/2 − α) = −∂θ T (φ, π/2 + α), (G.16)
1 1
I
u=− dS vs = − dφ dθ sin θ vs , (G.9)
|S| 4π ∂φ T (φ, π/2 − α) = ∂φ T (φ, π/2 + α), (G.17)
S 0 0
3 1 for α ∈ [0, π/2]. Hence, when calculating the surface
I
Ω=− dS × vs average of Eq. (G.13), the x and y-components vanish:
2a |S|
S
Z2π Zπ Zπ Zπ
3
=− dφ dθ sin θ r̂ × vs , (G.10) dθ sin θ ∂θ T = dθ sin θ cos θ ∂φ T = 0. (G.18)
8πa
0 0
0 011
Via Eq. (G.10), the remaining z-component is given by by virtue of the symmetry relation (G.16). We now
Z 2π Z π decompose the remaining x and y components of the
3 sin θ
Ωz = − dφ µ(φ) dθ ∂φ T (θ, φ) sin θ translational velocity into u(θ) + u(φ) , corresponding
8πa 0 0 a sin θ to the contributions ∂θ T and ∂φ T of the temperature
3
Z 2π gradient (G.5), respectively. We furthermore apply inte-
=− dφ µ(φ)∂φ hT i (φ), (G.19) gration by parts to get rid of the temperature gradients
4πa2 0
and deal with the bare temperature profiles instead.
where we introduced the mean (θ−averaged) tempera-
ture hT iθ (φ) via
∂θ -part
Rπ
dθ • (φ, θ) sin θ 1 π
Z
h•iθ (φ) ≡ 0 R π = dθ • (φ, θ) sin θ. Using Eq. (G.9), the θ-derivative of the temperature
0
dθ sin θ 2 0
gradient (G.5) gives
(G.20)
1 µ(φ)
I
In contrast to Eqs. (4)–(6) in the main text, we omit cos φ cos θ
u(θ) ≡ − dS (∂θ T )
the subscript θ in the averaging notation h•i throughout |S| a sin φ cos θ
S
the rest of this section for the sake of brevity. Using the
mobility profile (G.12) and 2π-periodicity of µ and T Z2π Zπ
1 cos φ
in the angle φ, the angular velocity simplifies to =− dφ µ dθ sin θ cos θ ∂θ T.
4πa sin φ
0 0
Zφpa
3 (G.28)
Ωz = − µps dφ ∂φ hT i (φ) (G.21)
4πa2
0 Applying integration by parts and using
Zφap Z2π !
+ µau dφ ∂φ hT i (φ) + µps dφ ∂φ hT i (φ) ∂θ (sin θ cos θ) = cos2 θ − sin2 θ,
φpa φap
the θ-integral in Eq. (G.28) can be written as
(G.22)
3 Zπ Zπ
=− µps [hT i (φpa ) − hT i (φap )] (G.23) cos θ
4πa2 dθ sin θ cos θ ∂θ T = dθ sin θ sin θ − T
tan θ
+ µau [hT i (φap ) − hT i (φpa )] (G.24) 0 0
3 cos θ
= (µps − µau ) [hT i (φap ) − hT i (φpa )] . (G.25) =2 sin θ − T ,
4πa2 tan θ
(G.29)
The final experession yields Eq. (6) upon identifying
φpa = π/2 and φap = 3π/2 for a half-coated Janus with the θ-average as defined in Eq. (G.20). The θ-
sphere. contribution to the velocity thus reads
Z2π
Appendix G.3: Translation
(θ) dφ µ(φ) cos θ cos φ
u =− sin θ − T (φ) .
2πa tan θ sin φ
Plugging Eq. (G.5) in to Eq. (G.4), and using the ex- 0
pessions for the unit vectors (G.6)–(G.8), the local slip (G.30)
velocity at the particle surface reads
∂φ -part
cos φ cos θ − sin φ
(∂θ T )µ(φ) (∂φ T )µ(φ)
vs = sin φ cos θ + cos φ .
Analogously, the φ-derivative of the temperature gradi-
a a sin θ
− sin θ 0 ent (G.5) contributes
(G.26)
Z2π Zπ
Calculating the surface average of Eq. (G.26), one finds
1 − sin φ
u (φ)
≡− dφ µ(φ) dθ (∂φ T ). (G.31)
that its z-component vanishes, because 4πa cos φ
0 0
Zπ
dθ sin2 θ ∂θ T = 0, (G.27) The φ-derivative appearing on the r.h.s. of the above
0
equation can be pulled out of the first integral. The12
remaining θ-integration of the bare temperature profile
can be expressed as
Zπ
T
dθ T (φ, θ) = 2 ≡ 2Te(φ), (G.32)
sin θ
0
with the θ-average h•i as defined in Eq. (G.20). With ↵ = 1what
µmif ...
↵ = 0.25 µm
the profile (G.12) of the local phoretic mobility µ(φ), they do not fluoresce?
20 µm
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
one finds metal particle
ø 1-80 nm
Au PS
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
absorption cross section
Z 2π ~!3000 Å2
(φ) dφ µ(φ) sin φ
u = ∂φ Te(φ)
− cos φ
they heat the environment!
0 2πa
1.25 µm
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
Zφpa
µps sin φ
= dφ ∂φ Te
2πa − cos φ
0
Zφap
µau sin φ
+ dφ ∂φ Te
2πa − cos φ
φpa Figure 11 Geometry of the setup used for the numerical tem-
perature calculations.
Z2π
µps sin φ
+ dφ ∂φ Te (G.33)
2πa − cos φ
φap (G.36)
φpa φpa
where we replaced Te(φ) by hT / sin θi [see Eq. (G.32)]
µps sin φ µau sin φ
= Te − Te and used the identity 1 + sin2 θ − cos2 θ = 2 sin2 θ to
2πa − cos φ φap
2πa − cos φ φap
Z 2π arrive at the term hT sin θi appearing inside the integral
dφ µ(φ) cos φ in the last line of Eq. (G.36). Setting φpa = π/2 and
− Te (G.34)
0 2πa sin φ φap = 3π/2 renders Eqs. (4) and (5).
µps − µau
= ×
2πa
Appendix H: Finite-element simulation of the
sin(φpa ) sin(φap )
Te(φpa ) − Te(φap ) temperature field
− cos(φpa ) − cos(φap )
Z2π
To calculate the surface temperature of the Janus parti-
dφ µ(φ) cos φ e
− T, (G.35)
2πa sin φ cle at different orientations, we use the COMSOL Multi-
0
physics® software [79] to employ a finite-element solver
where we applied integration by parts from (G.33) to for the considered heat conduction problem sketched in
(G.34), and exploited 2π-symmetry. Fig. 11.
The Janus particle is realized as a polystyrene parti-
Combining both contributions cle of 1 µm diameter with a gold cap which is tapered to
the edges and has a maximum thickness of 50 nm. The
Adding the results (G.30) and (G.35) for u(θ) and u(φ) ,
heat source is a gold sphere of 250 nm diameter placed
one finally arrives at
at 1.25 µm distance from the Janus particle center. Both
µps − µau particles are placed in a box of an edge length of 20 µm.
ux
= ×
uy 2πa The gold nanoparticle is heated with a heat source den-
" sity of 1 × 1015 W m−3 . Other parameters used for the
sin(φpa ) T
(φpa ) numerical calculations are listed in Table 1.
− cos(φpa ) sin θ
#
sin(φap ) T
− (φap ) References
− cos(φap ) sin θ
Z2π 1. Sriram Ramaswamy. The mechanics and statistics
dφ µ(φ) cos φ
−2 hT sin θi (φ), of active matter. Annual Review of Condensed Mat-
πa sin φ ter Physics, 1(1):323–345, Aug 2010.
0You can also read
