Experimental study of fluctuations
in two out-of-equilibrium systems
using optical traps
Juan Rubén Gómez-Solano
Research Internship Report
Master 2 Sciences de la Matière
Supervisors: Sergio Ciliberto and Artyom Petrosyan
Laboratoire de Physique
Ecole Normale Supérieure de Lyon
49, allée d’Italie
69364 Lyon cedex 07
France
April-July 2008.
Abstract
This work is devoted to the study of displacement fluctuations of micro-sized particles
in two different systems driven out of thermal equilibrium: an aging colloidal glass and a
fluid subject to a vertical temperature gradient below the onset of convection. In the first
part we address the issue of the validity of the fluctuation dissipation theorem (FDT) and
the time evolution of viscoelastic properties during aging of aqueous suspensions of a clay
(laponite RG) in a colloidal glass phase. Given the conflicting results reported in the literature for different experimental techniques, our goal is to check and reconcile them using
simultaneously passive and active microrheology techniques. For this purpose we measure
the thermal fluctuations of micro-sized brownian particles immersed in the colloidal glass
and trapped by optical tweezers. We find that both microrheology techniques lead to compatible results even at low frequencies and no violation of FDT is observed. In the second
part, we attempt to study the influence of non-equilibrium hydrodynamic fluctuations on
the motion of a trapped particle immersed in a fluid layer (ultrapure water) subjected to
a vertical temperature gradient. We estimate the contribution of non-equilibrium fluctuations to the power spectral density of displacement fluctuations of the particle and the
frequency range where it can be measured. Only very preliminary results are available for
this experiment.
Contents
1 FDT in an aging colloidal glass
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 FDT in Laponite suspensions . . . . . . . . . . . . . . .
1.2 Experimental description . . . . . . . . . . . . . . . . . . . . .
1.2.1 Sample preparation . . . . . . . . . . . . . . . . . . . .
1.2.2 Optical tweezers setup and data acquisition . . . . . . .
1.2.3 Creation of multiple traps . . . . . . . . . . . . . . . . .
1.2.4 Calibration of optical tweezers . . . . . . . . . . . . . .
1.2.5 Microrheology techniques . . . . . . . . . . . . . . . . .
1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Viscoelastic properties of the aging colloidal glass . . . .
1.3.2 Properties of non-equilibrium fluctuations during aging
1.3.3 Effective temperatures . . . . . . . . . . . . . . . . . . .
1.3.4 Probability density functions of heat fluctuations . . . .
1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Motion of a Brownian particle in a fluid below the onset of RayleighBénard convection
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Influence of non-equilibrium fluctuations on the motion of a Brownian particle
2.3 Experimental description . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Conclusion and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 1
FDT in an aging colloidal glass
1.1
Introduction
The description of out-of-equilibrium systems is a topic of major interest in physics since
in most systems, energy or matter flows are not negligible but drive them into non-ergodic
states for which equilibrium statistical mechanics is not applicable. The typical examples of
such systems are slowly stirred systems and glasses. Unlike equilibrium statistical mechanics, whose foundations are well established since long time ago, non-equilibrium statistical
mechanics is still being developed.
One attempt to develop a statistical mechanics description of non-equilibrium slowly
evolving systems is to extend some equilibrium concepts to them, namely the so called fluctuation dissipation theorem (FDT). FDT relates the power spectral density of equilibrium
fluctuations x of systems in contact with a thermal bath at temperature T to the response
χ to a weak external perturbation with a prefactor given by T
h|x(ω)|2 i =
4kB T
Im{χ(ω)}.
ω
(1.1)
Despite of the fact that a thermodynamic temperature is not rigorously defined for nonequilibrium systems, a generalization of FDT can be done by means of an effective temperature Tef f (ω, tw ) which describes fluctuations at a time scale 1/ω. It is defined as
Tef f (ω, tw ) =
ωh|x(ω, tw )|2 i
,
4kB Im{χ(ω, tw )}
(1.2)
where tw denotes the age of the system, i.e. the waiting time since the systems left equilibrium. In general, for a non-equilibrium system either the spectral density of fluctuations
and the response function depend on tw and then violations of FDT occur when Tef f (ω, tw )
also depends on it.
Glasses are one of the examples of non-equilibrium systems that have been studied
extensively over the past years. They are disordered systems at microscopic length scales
formed after a quench from an ergodic phase (e.g. an ordinary liquid in equilibrium) to
a non-ergodic phase (e.g. a supercooled liquid). After a quench, a glassy system relaxes
1.1 Introduction
3
Figure 1.1: Chemical structure of laponite. The disc-shaped particles form a house-of-cards
structure in aqueous suspensions.
asymptotically to a new phase in equilibrium with the new temperature T of its surroundings, but the time needed to reach equilibrium may be astronomical. Since the system is
non-ergodic during its relaxation, there is no invariance under time translations and then
its age tw is well defined. The slow time evolution of their physical properties (e.g. viscosity) is known as aging. A number of theoretical, numerical and experimental studies have
shown that two different effective temperatures are found for structural glasses [1], [2] and
spin glasses [3], [4], [5] . One of them is the effective temperature associated to the fast
rattling fluctuations (high frequency modes) and is equal to T . The other effective temperature is the one associated to the slowest structural rearrangements (low frequency modes)
whose value is larger than T and decreases with time to T . Both effective temperatures
have the properties of thermodynamic temperatures in the sense that they correspond to
two different thermalization processes at different time scales [6].
1.1.1
FDT in Laponite suspensions
Regarding the existence of an effective temperature higher than the bath temperature for
structural and spin glasses, several recent works have attempted to look for violations of
FDT in another kind of non-equilibrium systems: colloidal glasses. Unlike structural or
spin glasses, the formation of a colloidal glass does not require a temperature quench but
the packing of colloidal particles at a certain low concentration in water forming a glassy
structure.
Aqueous suspensions of clay laponite have been studied as a prototype of colloidal
glasses. Laponite is a synthetic clay formed by electrostatically charged disc-shaped par0.7− whose dimensions are 1 nm
ticles of chemical formula Na+
0.7 [Si8 Mg5.5 Li0.3 O20 (OH)4 ]
(width) and 25 nm (diameter). When laponite powder is mixed in water, the resulting
suspension undergoes a sol-gel transition in a finite time (e.g. a few hours for 3 wt %), i.e.,
it turns from a viscous liquid phase (sol) into a viscoelastic solid-like phase (gel). During
this aging process, because of electrostatic attraction and repulsion, laponite particles form
a house of cads-like structure (Fig. 1.1).
Several properties of laponite suspensions during aging have been extensively studied,
such as viscoelasticity [7], translational and rotational diffusion [8] and optical susceptibility
1.2 Experimental description
(a)
4
(b)
Figure 1.2: 1.2(a) Time evolution of the effective temperature of an aqueous suspension
of laponite particles (2.4 wt % concentration) during aging measured by using passive microrheology [13]. Tef f increases in time indicating a violation of FDT; 1.2(b) time evolution
of the effective temperature for 2.8 wt % concentration using active microrheology [12]. Unlike Fig. 1.2(b), in this case no violation of FDT is observed for any frequency and any
aging time.
[9] using techniques such as rheology and dynamic light scattering. However, the available
results obtained so far for the issue of the validity of the FDT in this system are contradictory (see Fig. 1.2). Bellon et al [10] reported an effective temperature from dielectric
measurements indicating a strong violation of FDT at low frequencies (f < 40 Hz), whereas
the same group did not observe any violations from mechanical measurements [10]. Abou et
al [11] observed that the effective temperature increases in time from the value of the bath
temperature to a maximum and then it decreases to the bath temperature. Jabbari-Farouji
et al [12] used a combination of passive and active microrheology techniques (see Subsection 1.2.5) without any observation of deviations of the effective temperature from the bath
temperature over several decades in frequency (1 Hz−10 kHz). Greinert et al [13] observed
that the effective temperature increases in time using a passive microrheology technique.
Finally, Jabbari-Farouji et al [14] found again no violation of FDT for the frequency range
of 1 Hz−10 kHz.
In view of the conflicting results obtained for different experimental techniques, it is
necessary to compare simultaneously at least two of them in the same colloidal glass sample.
In this spirit we study the time evolution of the effective temperature and viscoelastic
properties of laponite suspensions by using simultaneously passive and active microrheology
techniques by means of optical tweezers manipulation, as explained in the following chapter.
1.2 Experimental description
5
Figure 1.3: Diagram of the sample cell used during the experiment. The diagram is not in
proper scale.
1.2
1.2.1
Experimental description
Sample preparation
Physical properties of Laponite suspensions are very sensitive to the method used during their preparation [19]. Hence, an experimental protocol must be followed in order to
perform reproducible measurements of their aging properties. Laponite RD, the most frequently studied grade and the one studied in this work, is a hygroscopic powder that must
be handled in a controlled dry atmosphere. The powder is mixed with ultrapure water at
a weight concentration of 2.8 % and pH = 10 in order to assure chemical stability of the
samples. At lower pH, the decomposition of clay particles occur. CO2 absorption by water
can modify the pH of the samples and then their aging properties. For these purposes,
the preparation of the samples is done entirely within a glove box filled with circulating
nitrogen. We choose wt 2.8 % since it is known that at this concentration laponite gelation
takes place in few hours after the preparation. The suspension is vigorously stirred by a
magnetic stirrer during 30 minutes. The resulting aqueous suspension is filtered through
a 0.45 µm micropore filter in order to destroy large particle aggregates and obtain a reproducible initial state. The initial aging time (tw = 0) is taken at this step. Immediately
after filtration, a small volume fraction (0.02 ml per 50 ml of sample) of silica microspheres
(diameter = 2 µm) is injected into the suspensions. Microspheres play the role of probes
to study the aging of the colloidal glass, as explained further. The sample is placed in a
ultrasonic bath during 10 minutes to destroy small undesired bubbles that could be present
during the optical tweezers manipulation. The suspension is introduced in a sample cell
consisting on a microscope slide and a coverslip separated by a cylindrical spacer of inner
diameter 15 mm and thickness 2 mm, as shown in Fig. 1.3. The sample cell is sealed with
araldite adhesive in order to avoid evaporation and direct contact with CO2 from air.
1.2 Experimental description
(a)
6
(b)
Figure 1.4: 1.4(a) Optical tweezers setup used during the experiment: 1. Beam from an
infrared laser diode (device not shown), 2. Piezoelectric stage, 3. Microscope objective, 4.
Sample cell, 5. Condenser, 6. Halogen lamp, 7. Dichroic mirror, 8. High speed camera.
The paths and the directions of the trapping beam and the detection beam is are depicted
in red and blue, respectively; 1.4(b) Configuration of three optical traps separated by a
distance D = 9.3 × 10−6 m. The bright spots in the image correspond to three probe
particles of 2 µm diameter trapped by them.
1.2.2
Optical tweezers setup and data acquisition
Optical tweezers rely on the use of laser beams to trap dielectric particles by means of the
effect of radiation pressure [15]. When a micron-sized dielectric particle of refractive index
exceeding that of the surrounding medium, is close to a Gaussian laser beam, the refraction
of light through it results in two effective forces. A gradient force is directed towards the
beam axes (the region of highest light intensity) whereas a scattering force acts in the
direction of the incident beam. If the laser beam is extremely focused (e.g. by means of
a microscope objective), there is an additional gradient force which pushes the particle
towards the laser focus [16]. For sufficiently high intensity gradients, the scattering force is
compensated by the additional gradient force causing free particles to be trapped close to
the beam focus and resulting in a restoring hookean force −kx in the plane perpendicular
to the beam propagation [17] (see Fig. 1.3).
The present experiment is performed in a typical optical tweezers system (Fig. 1.4(a))
at room temperature (22 ± 1◦ C). It consists on a Nd:YAG DPSS 1 laser (Laser Quantum
λ = 1064 nm) whose beam is strongly focused by a microscope oil-immersion objective
(×63, NA = 1.4). Matching optics between the laser source and the objective are needed
in order to direct and produce a collimated laser beam that overfills the objective entrance.
The sample cell is placed on a piezoelectric stage (NanoMax-TS MAX313/M) in order to
1
Diode pumped solid state
1.2 Experimental description
7
have mechanical control in three dimensions of nanometric accuracy during the trapping
process. The sample cell is placed with the glass plate at the top and the coverslip at the
bottom in such a way that the focus of the laser beam is located 20 µm above the coverslip
inner surface (see Fig. 1.3). The whole system is mounted on an optical table in order to
get rid of low frequency mechanical noise.
The light source used for detection is a halogen lamp whose beam is focused in the
sample by a condenser lens. The halogen light passes through the sample and then the
resulting intensity contrast signal is detected by a high speed camera (Mikrotron MC1310)
after emerging from the microscope objective. Once probe particles are trapped properly,
images with a resolution of 240×80 pixels are recorded at a sampling rate of 200 frames per
second during 50 seconds and saved in AVI format every 97 seconds (i.e. 10000 images per
AVI file) for further data processing. Hence, the aging time of the system tw is measured
in multiples of 97 s while the smallest measurable time scale (the time elapsed between
successive positions of the probe) corresponds to 5 ms and the highest accessible frequency
is 100 Hz. The calculation of the baricenter position of the trapped bead on the x − y
plane (perpendicular to the laser beam propagation) is done by means of a MATLAB
image processing program. Then, we obtain a time series (x(t), y(t)) for each coordinate
at each waiting time tw . In the following, x defines the direction along which the position
of the optical trap is oscillated in time, while y corresponds the orthogonal direction. For
simplicity, all our calculations rely on x(t) coordinates only.
1.2.3
Creation of multiple traps
Multiple probe particles are needed to be trapped within the same sample in order to
perform simultaneously passive and active microrheology measurements. Since two trapped
particles are needed for each technique, as explained in Subsection 1.2.5, at least three
different optical traps must be created simultaneously, as shown in Fig. 1.4(b). For passive
measurements, two traps of different stiffness and fixed positions are needed. We label the
weakest trap as ’3’ while the strongest one as ’2’. For active measurements, one needs one
trap in a fixed position in order to measure the power spectral density of displacement
fluctuations. For convenience, we choose trap ’2’ for such purpose. In addition, one needs
an oscillating trap in order to measure the response of the probe to an external forcing. This
trap is labeled as ’1’. The separation distance between adjacent traps is D = 9.3 × 10−6
m, which is sufficiently large to avoid correlations between their motions.
The creation of three optical traps on the x − y plane is implemented by means of two
coupled acousto-optic deflectors (AOD). A high radio frequency voltage signal is sent from
a signal generator to the AOD. Sound waves are created in the crystal inside the AOD,
forming a Bragg diffraction grating which efficiently deflects the incident laser diode beam.
The deflection angle of the output beam (the first order diffracted spot) is proportional to
the high frequency input signal. By varying the frequency of the signal generator, the angle
of the output laser beam is adjusted. By coupling orthogonally in series both AOD, one
can control the angle of the output beam in 2D. In order to create three traps, the laser
beam scans three different positions along the y direction. The visitation frequency for each
position must be large enough in order to avoid the diffusive motion of the particle through
1.2 Experimental description
8
−16
10
−17
−18
10
x
S (f) (m2 / Hz)
10
−19
10
fc = 5.17 Hz
experimental data
Lorentzian fit
−1
10
0
1
10
2
10
10
f (Hz)
Figure 1.5: Power spectral density of displacement fluctuations of a a probe particle kept by
trap ’2’. The red curve is the average over five experimental spectra. The corner frequency
obtained by the Lorentzian fit is fc = 5.17 Hz.
the surrounding medium during the absence of the beam. The oscillation of the position
of trap ’1’ is accomplished by deflecting the laser beam along x with a given waveform
x0 (t) when it scans the position that it would visit in the absence of deflection along x.
The stiffness of each trap is proportional to the time that the laser beam stays in the
corresponding position. We check that by selecting a ratio of 40:40:20 for the visiting time
of traps ’1’, ’2’ and ’3’, respectively we obtain a stiffness ratio of 39.6:39.7:20.7, respectively.
1.2.4
Calibration of optical tweezers
In the present work, the calibration of optical tweezers, i.e. the determination of the effective stiffness k associated to the hookean force exerted by the trap on brownian particles,
is based on the passive technique described in [17]. We prepare a calibration sample by
introducing a small fraction of silica beads (diameter 2R = 2 µm) in a simple viscous fluid
(glycerol wt 40% in water) with known dynamic viscosity η within a sample cell at thermal
equilibrium (temperature T ). The overdamped motion of a probe particle is well described
by the Langevin equation
p
(1.3)
γ ẋ + kx = 2kB T γξ(t),
where γ = 6πηR and ξ(t) is a delta-correlated stochastic noise term of mean 0 and variance
1 associated to the fast random collisions of the particles of the fluid with the probe. The
power spectral density of the fluctuations described by (1.3) is a Lorentzian
Sx (f ) = h|x(f )|2 i =
kB T
2
γπ (fc2 +
f 2)
,
(1.4)
where the corner frequency fc (defined by Sx (fc ) = Sx (0)/2) is given by
fc =
k
.
2πγ
(1.5)
1.2 Experimental description
9
Fig. 1.5 shows the typical experimental power spectral density of displacement fluctuations
for a particle kept by a fixed trap. Note that under the present experimental conditions we
are able to resolve frequencies as low as of order 10−1 Hz. Eq. (1.5) allows to determine
the value of k by fitting the power spectral density into a Lorentzian curve and the value of
the corresponding corner frequency leads to the determination of the stiffness of the optical
trap
k = 12π 2 ηRfc .
(1.6)
In this way, we obtained the following values of the stiffness of each optical trap
k1 = 7.12 pN/µm,
k2 = 7.15 pN/µm,
k1 = 3.73 pN/µm,
respectively. Note that trap ’1’ must be held in a fixed position during calibration since the
power spectral density of the corresponding trapped probe must be measured at equilibrium
(no external forcing acting on it).
1.2.5
Microrheology techniques
The determination of effective temperature, viscosity and elasticity of laponite samples
during aging is done using microrheology. It consists on the measurement of the motion
of probe particles embedded in the colloidal glass and manipulated by the optical tweezers
system. This is the approach followed by [12], [13] and [14] leading to conflicting results
between passive and active measurements.
There are two different kinds of microrheology techniques depending on the manipulation of the probes by the optical trap: passive and active.
Passive microrheology
In passive microrheology (PMR), an optical trap acts as a passive element keeping a probe
particle in a fixed position only (like traps ’2’ and ’3’ in Fig. 1.4(b)). One is interested in
the measurement of the displacement fluctuations x(t) around the fixed position in order
to obtain information about the properties of the medium.
In the present work, PMR measurements of effective temperature and elasticity of the
colloidal glass are based on the method proposed by Greinert et al [13] This method is based
on a generalization of the equipartition relation to glassy systems argued by Berthier and
Barrat [18]. Since a laponite suspension becomes viscoelastic as it ages, a probe particle
immersed in it is subject to an additional force Fe = −ke x due to the elasticity of the
colloidal glass, where ke denotes the effective elastic stiffness. Then, by introducing an
effective temperature Tef f by means of the generalized equipartition relation
1
1
(k + ke )hδx2 i = kB Tef f ,
2
2
(1.7)
1.2 Experimental description
10
where hδx2 i is the ensemble variance of x and is a priori different from the time variance
since the system is non-ergodic. In an aging system, the quantities of interest ke and Tef f
can be determined from the measurement of the displacement variances hδx2 i2 and hδx2 i3
of probe particles ’2’ and ’3’ in Fig. Tef f and ke are given by
Tef f (tw ) =
ke (tw ) =
(k3 − k2 )hδx2 i2 hδx2 i3
,
hδx2 i2 − hδx2 i3
(1.8)
k3 hδx2 i3 − k2 hδx2 i2
.
hδx2 i2 − hδx2 i3
(1.9)
hδx2 i2 and hδx2 i3 are calculated as the arithmetic means of (x − hxi)2 for the 10000 data
points using a moving time window of 3 s at aging time tw , for each one of particles 2
and 3, respectively. The time window of 3 s is chosen sufficiently short to assure that the
viscoelasticity of the colloidal glass remains almost constant. However, one must be aware
that this time window is not sufficiently long to take into account the role of low frequency
fluctuations at late stages during aging, and variances could be largely underestimated in
this regime, as found by [22], leading to a bad measurement of Tef f . In the present work
we use such time window in order to compare our results with those found in [13]. Longer
time windows were also checked without observing any major difference in the late-time
behavior of Tef f with respect to that corresponding to 3 s, due to large data variability
associated to PMR, as discussed in Subsection 1.3.3
Active microrheology
Active microrheology (AMR) involves the active manipulation of probe particles by an external force exerted by optical tweezers. In our case, we apply an oscillatory force f0 (t, ω) at
certain frequencies ω by means of the spatial oscillation x0 (t, ω) of trap ’1’. One determines
the linear response function χ(t), defined by means of the convolution
Z t
χ(t − t0 )f0 (t0 , ω)dt0 ,
x(t) =
−∞
but in practice, one measures directly its Fourier transform
χ(ω) =
x(ω)
,
f0 (ω)
(1.10)
in order to resolve the viscoelastic properties of aging colloidal glasses.
As mentioned before, there are a priori two times scales in the glassy system. The
displacements fluctuations evolve in the fast time scale t (x = x(t)) while viscoelastic
properties evolve in a slow aging time tw , (γ = γ(tw , ω), ke = ke (tw , ω)). The effective
hookean force on the particle due the relative displacement x(t) − x0 (t, ω) with respect to
the laser beam focus is now −k(x(t) − x0 (t, ω)) while the force due the evolving elasticity
of the medium is −ke (tw , ω)x(t). Hence, the motion of the probe particle is described by
the following Langevin equation
q
γ(tw , ω)ẋ + k(x − x0 (t, ω)) + ke (tw , ω)x = 2kB Tef f (tw , ω)γ(tw , ω)ξ(t).
(1.11)
1.3 Results
11
In order to derive a relation between the linear response function and the frequencydependent viscoelastic properties of the colloidal glass, Eq. (1.11) must be recasted with
no stochastic term, as
γ(tw , ω)ẋ + (k + ke (tw , ω))x = f0 (t, ω),
(1.12)
where f0 (t, ω) = kx0 (t, ω) is the active force term. The Fourier transform of Eq. (1.12)
is computed over a time window of 10 s (such that γ and ke are constant) and movingaveraged over 50 seconds. It leads to the following expression for the inverse of the response
function at given frequency and aging time
1
χ(tw , ω)
= iωγ(tw , ω) + (k + ke (tw , ω)),
(1.13)
Therefore, by measuring directly the mechanical response at a given frequency of the particle motion to the applied external force, it is possible to resolve both the relative viscosity
and the elasticity of the colloidal glass during aging by means of the expressions
1
ωγ(tw , ω)
= Im{
},
k
kχ(tw , ω)
(1.14)
ke (tw , ω)
1
= Re{
} − 1.
(1.15)
k
kχ(tw , ω)
In our case, the position of trap ’1’ is oscillated in time along the x direction at three
different frequencies ω = ωi , i = 1, 2, 3 simultaneously according to
x0 (t, ω) = A(sin(ω1 t) + sin(ω2 t) + sin(ω3 t)),
(1.16)
where f1 = ω1 /2π = 0.3 Hz, f2 = ω2 /2π = 0.5 Hz, f3 = ω3 /2π = 1.0 Hz, and A = 9.2×10−7
m. Higher frequency sinusoidal oscillations (ω/2π = 2.0, 4.0, 8.0 Hz) were also checked in
order to compare our results at low frequencies to higher ones.
AMR allows to determine directly the effective temperature of the colloidal glass at
given frequency ω and aging time tw by means of the expression (1.2), as suggested by [12].
First of all, one needs to synchronize the input forcing signal x0 (t, ω) with the response
of the trapped bead x(t). The Fourier transform of the response function is determined
by using Eq. (1.10) for a probe particle driven by trap ’1’: in practice we divide the
power spectral density of the output signal |x(ω)|2 by the corresponding transfer function
x∗ (ω)f0 (ω). On the other hand, one needs to determine the power spectral density of the
displacement fluctuations in the absence of external forcing but for the same value of the
trapping stiffness. For this reason, traps ’1’ and ’2’ are created with the same stiffness. We
measure the power spectral density of fluctuations of a particle kept by trap ’2’.
1.3
1.3.1
Results
Viscoelastic properties of the aging colloidal glass
We first present separately the results of the time evolution of viscosity and elasticity of
the colloidal glass during aging. The time evolution of the dimensionless quantity ωγ/k1 ,
1.3 Results
12
linearly proportional to the dynamic viscosity η of the colloidal glass, is shown in Fig. 1.6(a).
As expected, it increases continuously as the system ages. On the other hand, the evolution
of the stiffness ke is qualitatively different, as shown in Fig. 1.6(b). For tw < 300 min, ke ≈
0, revealing an entirely viscous nature of the laponite suspension, while for tw > 300 min it
becomes viscoelastic, with ke increasing dramatically in aging time from 0 to 10 times the
stiffness of the second trap k2 = 7.15 pN/µm. Both PMR and AMR lead to consistent and
complementary results and we notice that AMR measurements are more accurate, leading
to a very small dispersion of data around the mean trend. Instead, PMR measurements
become very sensitive to the inverse of the difference hδx2 i2 −hδx2 i3 as tw increases, leading
to increasingly large data dispersion for tw > 350 min. In order to compare our AMR results
with previous rheological measurements, in Fig. 1.6(c) we plot the time evolution of the
modulus of the complex viscosity, given by |η ∗ | = (γ 2 + k 2 /ω 2 )1/2 /6πR. We observe that
|η ∗ | increases almost exponentially two orders of magnitude during the first 500 minutes of
aging. The behavior of |η ∗ | is in good agreement with previous rheological measurements
[21].
1.3.2
Properties of non-equilibrium fluctuations during aging
The probability density functions of displacement fluctuations around their mean positions
δx computed for particles ’2’ and ’3’ over the time window of 3 s, are Gaussian, as shown
in Fig. 1.7(a). Their variances hδx2 i2 and hδx2 i3 , evolve in aging time, as shown in Fig.
1.7(b). Two regimes can be identified: for tw . 300 min, both variances are quite constant,
while for tw & 300 min they decrease up to one order of magnitude at tw = 500 min.
The transition between one regime to the other is abrupt and occurs at tw ≈ 300 min
≡ tg . This time corresponds to the transition from a purely viscous liquid-like phase to
the formation of a viscoelastic glassy phase associated to the house of cards structure, as
shown previously. It is noticeable that the transition point from the plateau to the decaying
curve of the variances depends slightly on the value of the stiffness of the optical trap. We
observe that the transition occurs first (around tw = 250 min) for the weakest trap (’3’)
than for the strongest one (’2’) (around tw = 300 min), revealing that the motion of a
probe particle is sensitive to the relative strength of the optical trap with respect to the
elasticity of the colloidal glass close to the gelation point. In other words, the displacement
fluctuations of a particle trapped by a weak optical tweezer are more easily constrained by
the increasingly elastic medium.
Fig. 1.8(a) shows the time evolution of the power spectral density for each frequency
studied ω = ωi , i = 1, 2, 3. Changes of the value of the power spectral density during
aging are due to the change of viscoelastic properties of the colloidal glass. The time
behavior of χ(ω, tw ) is completely different from that of ergodic liquids at equilibrium
whose power spectral density is constant in time. The nontrivial shape of |x(ω, tw )|2 has a
maximum which depends on the value of the corresponding frequency. Fig. 1.8(b) shows
the time evolution of the imaginary part of the Fourier transform of the response function
at each frequency ω = ωi , i = 1, 2, 3, calculated by means of Eq. (1.10). We observe the
same behavior in time for each frequency, indicating that |x(ω, tw )|2 and Im{χ(ω, tw )} are
related by a proportionality constant during aging, satisfying a generalized FDT relation
1.3 Results
13
1
ωγ/k
8
10
6
10
PMR
AMR (f = 0.3 Hz)
AMR (f = 0.5 Hz)
AMR (f = 1.0 Hz)
f = 0.3 Hz
f = 0.5 Hz
f = 1.0 Hz
0
10
|η*| (Pa s)
10
15
f = 0.3 Hz
f = 0.5 Hz
f = 1.0 Hz
klap / k
12
5
4
−1
10
−2
0
0
10
0
2
−3
100
200
300
400
500
100
200
300
tw (min)
tw (min)
(a)
(b)
400
10
0
100
200
300
400
500
tw (min)
(c)
Figure 1.6: 1.6(a) Time evolution of viscosity of the colloidal glass obtained by means of
AMR; 1.6(b) time evolution of elasticity of the colloidal glass obtained either by PMR and
AMR; 1.6(c) time evolution of the modulus of the complex viscosity
(1.2), as explained in the following. Note that a bad measurement of the response function
(e.g. an unsynchronized measurement of the external force f0 and the response of the
particle x) can lead to a wrong determination of the effective temperature during aging
and consequently an apparent violation of FDT could be observed, as probably occurred
in [11].
1.3.3
Effective temperatures
Effective temperatures obtained by means of both PMR and AMR are shown in Fig. 1.9(a).
In the case of PMR, for tw < 200 min the effective temperature is very close to the bath
temperature T = 295 K with very few data dispersion around it. This is due to the fact
that at this aging stage the aqueous laponite suspension behaves as a viscous liquid leading
to small and constant data dispersion of hδx2 i. For 300 min < tw < 400 min, we observe an
apparent increase of Tef f (tw ), indicating a possible violation of FDT. However, we argue
that this is not an actual physical effect but an artifact of the PMR method, as pointed
out before by [22]. By regarding the facts that the decay onset of hδx2 i occurs first for the
2 −1 −1
weakest trap and that Tef f ∝ (hδx2 i−1
2 −hδx i3 ) , we conclude that PMR always leads to
an apparent increase of the effective temperature of the colloidal glass close to the gelation
point when using optical traps of very different stiffness. PMR would lead to no apparent
increase of Tef f only in the unpractical limit k2 ≈ k3 . Nevertheless, in this limit PMR
measurements are useless because they lead to extremely inaccurate and fluctuating values
of Tef f and ke . In addition, by monitoring the effective temperature for longer times as the
colloidal glass becomes more and more viscoelastic, we observe an increasing variability of
Tef f even leading to negative values of Tef f due to the fact that the dispersions of hδx2 i2
and hδx2 i3 are comparable to hδx2 i2 − hδx2 i3 .
We can also check that data smoothing can be an artifact in PMR leading to an apparent
dramatic increase of the effective temperature at late aging times. By convoluting the data
points of the variances hδx2 i2 and hδx2 i3 with rectangular time windows of 16 min, we
calculate the corresponding simple moving averages at each aging time tw in order to
1.3 Results
14
−15
1.8
x 10
2
<δ x >2
1.6
<δ x2>
t = 30 min
3
w
tw = 95 min
−1
10
pdf
1.2
<δ x >3 (s)
2
2
1
2
w
tw = 500 min
<δ x > (m )
t = 460 min
−2
10
<δ x2> (s)
2
tw = 175 min
tw = 340 min
1.4
0.8
0.6
−3
10
0.4
0.2
t
g
0
0
−4
10
−1.5
−1
−0.5
0
δ x (m)
0.5
1
1.5
100
200
300
400
500
t (min)
−7
w
x 10
(a)
(b)
Figure 1.7: 1.7(a) PDFs of δx = x − hxi at different waiting times tw for particle ’2’; 1.7(b)
Time evolution of displacement variances for particles ’2’ and ’3’. The thin dashed line
indicates the approximate aging time when gelation begins to take place. The label ’(s)’
corresponds to the curves smoothed over a moving time window of 16 min.
0.25
−16
x 10
<|x(ω,tw)|2> (m2 / Hz)
8
7
ω / 2π = 0.3 Hz
ω / 2π = 0.5 Hz
ω / 2π = 1.0 Hz
0.2
k Im χ(ω,tw) / ω (s)
9
6
5
4
3
ω / 2π = 0.3 Hz
ω / 2π = 0.5 Hz
ω / 2π = 1.0 Hz
0.15
0.1
0.05
2
1
0
0
100
200
300
400
500
0
0
100
200
300
tw (min)
tw (min)
(a)
(b)
400
500
Figure 1.8: 1.8(a) Time evolution of the power spectral densities of displacement fluctuations of particle ’2’ for the three frequencies studied; 1.8(b) Time evolution of the imaginary
part of the Fourier transform of the response functions of particle ’1’.
1.3 Results
15
8
7
6
2
4
Teff (f) / T
T
eff
/T
5
2.5
PMR
AMR (f = 0.3 Hz)
AMR (f = 0.5 Hz)
AMR (f = 1.0 Hz)
PMR (smoothed)
3
2
1.5
1
1
0.5
0
−1
0
100
200
300
400
0
500
0
1
10
tw (min)
10
f (Hz)
(a)
(b)
Figure 1.9: 1.9(a) Time evolution of effective temperatures of the colloidal glass during
aging obtained either by PMR and AMR; 1.9(b) Aging time average of the effective temperature obtained by AMR for different frequencies.
obtain smooth curves, as shown in Fig. 1.7(b). However, this smoothing method results in
overestimated values of the variances for tw > tg and consequently, the corresponding value
of the effective temperature is also overestimated showing a steep increase as tw increases,
as shown in Fig. 1.9(a). This is very similar to the behavior of Tef f reported by [13] (see
Fig. 1.2(a)) and we assert that such increase is not an actual violation of FDT but the
result of data smoothing.
The AMR results for the effective temperature at different frequencies are shown in Fig.
1.9(a). Unlike PMR, we verify that there is no actual systematic increase of Tef f as the
colloidal glass ages. The effective temperatures recorded by the probe particles subject to
the non-equilibrium fluctuations in the colloidal glass at a given time scale 1/ω are equal
to the bath temperature during aging even for the slowest modes studied (∼ 1/f1 ≈ 3 s).
Unlike PMR measurements, we find that the variability of the data is constant in aging
time, which implies that AMR is a more reliable method even during gelation. In order to
check if Tef f (ω, tw ) = T within experimental accuracy, by taking into account the constant
behavior of Tef f and of the dispersion around its mean value we can compute the aging
time average of Tef f (ω, tw ) for tw ∈ [tiw = 15 min, tfw = 500 min]
Tef f (ω) =
1
tfw − tiw
Z
tfw
tiw
Tef f (ω, t0w )dt0w ,
(1.17)
for every frequency ω. Fig. 1.9(b) shows the results of Tef f (ω) with their respective error
bars corresponding to the standard deviations of the data sets. For comparision, we also
determined the effective temperature of the colloidal glass under the same experimental
conditions at higher frequencies (f = 2.0, 4.0 and 8.0 Hz) without observing any deviation
of the effective temperatures from the bath temperature within experimental accuracy
1.3 Results
16
τ = 50 ms
τ = 250 ms
τ = 500 ms
τ=1s
τ = 2.5 s
τ=5s
−1
10
−1
tw = 340 min
10
tw = 435 min
tw = 470 min
pdf
pdf
−2
−2
10
10
tw = 500 min
−3
10
−3
10
−4
−6
−4
−2
0
Q’τ / kB T
(a)
2
4
6
10
0
1
2
3
4
5
|Q’|τ / kB T
(b)
Figure 1.10: 1.10(a) PDFs of Q0τ at tw = 95 min for different values of τ ; 1.10(b) PDFs
of |Q0τ | for τ = 5 s at different aging times tw after the onset of gelation. Blue points
correspond to Q0τ > 0 while red ones to Q0τ < 0.
(Fig. 1.9(b)). We conclude that there is no violation of FDT for this aging non-equilibrium
system with its effective temperatures equal to the bath temperature, regardless of the
measured time scale.
1.3.4
Probability density functions of heat fluctuations
Finally, we present the results for the probability density functions (PDF) of the heat
transfer between the colloidal glass and the surroundings. Since non-equilibrium fluctuating
forces due to the collisions of the colloidal glass particles with a micron-sized probe particle
do work Wτ (of ensemble average hWτ i = 0) on it in a given time interval τ , a fraction
of Wτ must be dissipated in the form of heat Qτ . Qτ is a stochastic variable which may
be either positive (heat received by the colloidal glass) or negative (heat transferred to the
surroundings). In an ergodic system in equilibrium with a thermal bath at temperature T ,
the mean heat transfer must vanish hQτ i = 0 in the absence of any external forcing in order
to satisfy the first law of thermodynamics. However, in a system out of equilibrium this
situation is not necessarily fulfilled due to non-ergodicity. A mean heat transfer hQτ i > 0
would be a signature of an effective temperature of the colloidal glass Tef f (ω, tw ) > T
during a time lag τ ≈ 2π/ω. Hence, by investigating possible asymmetries in the PDF of
Qτ for different time lags and at different aging times, one could find possible violations of
FDT in the aging colloidal glass.
In order to calculate the heat transfer during a time lag τ , Eq. (1.11) (with x0 (t) = 0
since we are interested in the situation without any external forcing) is multiplied by ẋ
and integrated from tw to tw + τ (with τ ¿ tw ), leading to an extension of the first law
of thermodynamics for a probe particle subject to non-equilibrium thermal fluctuations of
1.4 Conclusion
17
the colloidal glass
where
∆Uτ (tw ) = Qτ (tw ),
(1.18)
1
2π
∆Uτ (tw ) = (k + ke (tw , ))(x(tw + τ )2 − x(tw )2 ),
2
τ
(1.19)
and
2π
Qτ (tw ) = −γ(tw , )
τ
Z
tw +τ
r
0 2
0
ẋ(t ) dt +
tw
2kB Tef f (tw ,
2π
2π
)γ(tw , )
τ
τ
Z
tw +τ
ξ(t0 )ẋ(t0 )dt0 .
tw
(1.20)
In practice we determine the PDF of the stochastic variable Q0τ ≡ (k/(k + ke (tw , 2π/τ ))Qτ
by means of Eqs. (1.18) and (1.19) for particle ’2’. At every aging time tw , we obtain
a data set of {Q0τ (tw )}τ of 10000−int(τ /200 s) points (τ measured in seconds). Then we
compute the corresponding PDF at every tw . The results are shown in Fig. 1.10. PDFs of
Q0τ are exponentially decaying and symmetric with respect to the maximum value located
at Q0τ = 0. At a given time tw , for different values of τ we check that there is no asymmetry
in PDFs, which implies that there is no net mean heat flux taking place at any time scale
τ , as shown in Fig. 1.10(a). In addition, Fig. 1.10(b) shows that even during the gelation
regime (tw > tg ) no such asymmetries can be detected. The absence of any asymmetry
in the PDF of Qτ confirms the absence of any effective temperature of the colloidal glass
higher than the bath temperature even for fluctuating modes taking place at time scales
as slow as τ = 5 s.
1.4
Conclusion
The use of multiple traps allows us to check simultaneously two microrheology techniques
that had led to conflicting results in the past. We conclude that there is no conflict between
them but the PMR technique may lead to artifacts since it becomes very inaccurate as the
system becomes very viscoelastic. We find that passive measurements are consistent with
the idea of no mean heat transfer between the aging colloidal glass and the surrounding
environment even for long time scales, for which possible violations of FDT could be expected. This result confirms that the effective temperature of the colloidal glass and the
environment is always the same during aging, even during the aging time when the PMR
artifact show an apparent increase of the effective temperature. At the same time, we
check by means of AMR at very low frequencies that there is no actual violation of FDT
in the aging colloidal glass. An adventage of the AMR technique is that it also allows to
resolve the slowly evolving viscosity and elasticity of the colloidal glass in order to observe
its transition from a purely viscous fluid to a viscoelastic one. All these results show that
FDT is actually a very robust property even in a glassy system.
Chapter 2
Motion of a Brownian particle in a
fluid below the onset of
Rayleigh-Bénard convection
2.1
Introduction
The second work developed during this Research Internship concerns the influence of nonequilibrium hydrodynamic fluctuations in a fluid below the onset of Rayleigh-Bénard convection on the motion of a micro-sized particle immersed in the fluid and trapped by optical
tweezers. Our goal is to investigate experimentally whether non-equilibrium hydrodynamic
fluctuations are locally felt or not by the brownian particle by detecting modifications of
its power spectral density of fluctuations, which is Lorentzian at equilibrium.
The statistical properties of non-equilibrium hydrodynamic fluctuations have been studied during the past two decades and are very well established [23]-[29]. Unlike spatially
short-ranged fluctuations in thermal equilibrium fluids, fluctuations in non-equilibrium
states are long-ranged, typically encompassing the spatial size of the system close enough to
a critical point of instability. A theoretical approach to this problem is based on fluctuating
hydrodynamics. The ordinary deterministic equations describing the hydrodynamics of the
system are transformed into a set of stochastic partial differential equations by assuming
that the dissipative fluxes contain an average part and a stochastic part. By applying a local
extension of FDT to non-equilibrium states, one obtains a set of linearized coupled Langevin
equations where dissipative fluxes play the role of random-noise terms. A consequence of
the local extension of FDT to fluctuating hydrodynamics of non-equilibrium states is that
the noise correlation functions become dependent on the position in the fluid which leads
to long-ranged correlations [26]-[29], as verified in several scattering experiments [23]-[25],
[29].
The non-equilibrium situation under study in this work is the typical Rayleigh-Bénard
system. A fluid layer is confined between two horizontal plates acting as thermal baths
at different temperatures and separated by a distance d much smaller than the horizontal
size L of the layer. When the temperature T1 of the lower thermal bath is larger than
2.2 Influence of non-equilibrium fluctuations on the motion of a Brownian
particle
19
the temperature T2 of the upper bath (T2 < T1 ), an important dimensionless parameter
controls the dynamics of the fluid: the Rayleigh number. It is defined as
Ra =
αgd3 ∆T
,
DT ν
(2.1)
where ∆T = T1 − T2 , g is the gravitational acceleration, α the isobaric thermal expansion
coefficient, DT the thermal diffusivity and ν the kinematic viscosity of the fluid. There
exists a critical value Rac of Ra which establish the onset of convective motion of the fluid
(Rayleigh-Bénard convection). When Ra < Rac , the fluid remains in a quiescent state
and the heat is transferred from the lower to the upper plate by thermal conduction (heat
diffusion). For Ra ≥ Rac the system becomes unstable against perturbations around a
certain critical wavenumber (qc = 3.1163/d) and a convective motion in the form of rolls
appear. In the case of the rigid boundary conditions used in the present experiment, it
is possible to show analitically that the onset of Rayleigh-Bénard convection corresponds
to Rac = 1708. In the following, we are interested in non-equilibrium fluctuations in the
quiescent state only (Ra < Rac ). It is important to remark that in the present work we
look for a local effect of non-equilibrium fluctuations which could differ noticeably from the
integral effect (over all length scales within the Rayleigh-Bénard cell) observed in scattering
experiments [23]-[25], [29].
2.2
Influence of non-equilibrium fluctuations on the motion
of a Brownian particle
In order to study the influence of non-equilibrium hydrodynamic fluctuations below the
convective instability on the motion of a Brownian particle trapped by optical tweezers, we
first estimate the order of magnitude of their amplitudes to figure out their extent in the
power spectrum of displacement fluctuations of the particle. For this purpose, we consider
temperature, δT (~r, t) density δρ(~r, r) and velocity δ~v (~r, t) fluctuations around the mean
quiescent state of the fluid (T (~r), ρ(~r), ~v (~r)), given by
T (~r) = T1 +
∆T
z,
d
α∆T
z),
d
~v (~r) = ~0,
ρ(~r) = ρ1 (1 +
(2.2)
(2.3)
(2.4)
where ρ1 is the value of the density of the fluid at the bottom of the layer and z is the
vertical coordinate in the direction of the temperature gradient. By using the Boussinesq
approximation and linearizing the full hydrodynamic equations around the conductive solution (2.2)−(2.4), one obtains the fluctuating Boussinesq equations for δT and δw (the z
component of δ~v )
∂
∂ 2 δT
∂ 2 δT
(∇2 δw) = ν∇2 (∇2 δw) + αg(
+
) + F1 ,
2
∂t
∂x
∂y 2
(2.5)
2.2 Influence of non-equilibrium fluctuations on the motion of a Brownian
particle
20
∂δT
∆T
= DT ∇2 δT −
δw + F2 ,
(2.6)
∂t
d
where F1 and F2 represent the contribution of rapidly varing short-ranged fluctuations
of the Landau stress tensor and the random heat flow, respectively. A rough estimate
of the amplitudes of non-equilibrium temperature and density fluctuations can be done
by means of the expresions given in [27] and [29] for the mean variance of temperature
fluctuations in the case of shadowgraph experiments. After solving Eqs. (2.5) and (2.6) in
Fourier frequency and wavevector space by means of a first-order Galerkin approximation,
it is found that close and below the onset of Rayleigh-Bénard convection, the variance of
temperature fluctuations (averaged along the thickness of the fluid layer) is given by [27]
Z
1 d
(∆Tc )2 1
σ2
1
2
√ ,
hδT i ≡
dzh|δT |2 i = kB T
(2.7)
d 0
ρdν 2 4ξ˜02 Rc σ + 0.515 −²
where ξ˜02 = 0.062, σ = ν/DT is the Prandtl number, ² = Ra/Rac − 1 is the dimensionless
Rayleigh number, and ∆Tc is the critical temperature difference between lower and upper
plates, corresponding to Rac for given values of α, d, DT and ν. Furthermore, the variance
of density fluctuations is simply given by
hδρ2 i = α2 ρ2 hδT 2 i.
(2.8)
We now estimate the order of magnitude of their amplitudes in the case of water and
investigate their possible influence on the motion of a Brownian particle trapped by a
focused laser beam. For this purpose, we consider a particle of mass m and radius R
trapped by optical tweezers of stiffness k in a fixed position. By taking into account that
close enough to the instability (² → 0− ), the spatial variation of hydrodynamic fluctuations
occurs in a length scale of order 1/qc ∼ d much larger than R, and assuming that FDT is
valid in the surrounding fluid, then the motion of the particle in the plane perpendicular
to the temperature gradient is described by the following Langevin equation
(m +
p
2πR3
(ρ + δρ))ẍ + γ(ẋ − δv) + kx = 2kB (T + δT )γξ(t),
3
(2.9)
where hξ(t)i = 0 and hξ(t)ξ(t0 )i = δ(t − t0 ) The fist term in the left-hand side of Eq.
3
(2.9) takes into account the mass 2πR
3 (ρ + δρ) of the surrounding fluid, while the second
term corresponds to Stokes’ law applied to the particle moving with a velocity ẋ − δv with
respect to the fluid in the presence of a fluctuation δv. The right-hand side term is the
usual stochastic force acting on the brownian particle due to the fast random collision of
the molecules of the fluid at a local temperature T + δT and it results from the application
of the local FDT around the Brownian particle. For the typical values of the parameters
of the system used in the experiment (α ≈ 2 × 10−4 K−1 , DT ≈ 10−7 m2 s−1 , ν ≈ 10−6
m2 s−1 , d = 2 × 10−3 m, ρ ≈ 103 kgm−3 , R = 10−6 m, ∆T ∼ 10 K), we find by using Eqs.
(2.7) and (2.8) that the ratio between the amplitude of density fluctuations and the mean
density of the fluid is
q
hδρ2 i
ρ
∼
10−7
.
(−²)1/4
2.2 Influence of non-equilibrium fluctuations on the motion of a Brownian
particle
21
Hence, the contribution of inertial fluctuating effects to the power spectral density h|x(ω)|2 i
of the displament fluctuations of the trapped particle are undetectable
in practice, since
p
even the mean inertial term is only detectable at frequencies ω & k/m ∼ 105 rad/s for
a typical optical trap of stiffness k ∼ 10−5 Nm−1 . Therefore, for the range of measurable
frequencies in the present experiment (0.1 − 10000 Hz), one can easily neglect the inertial
contribution in Eq. (2.9) to h|x(ω)|2 i. Moreover, the relative amplitude of temperature
fluctuations with respect to the mean temperature of the surrounding fluid is
q
hδT 2 i
10−5
∼
,
T
(−²)1/4
which is also undetectable under our experimental conditions. Hence, for practical purposes, the motion of the trapped brownian particle is well described by the following
Langevin equation if one assumes that the amplitude of velocity fluctuations are not negligible
p
γ(ẋ − δv) + kx = 2kB T γξ(t).
(2.10)
By Fourier transforming Eq. (2.10) and ensemble-averaging, one readly obtains the expression of the power spectral density of displacement fluctuations of the Brownian particle
h|x(ω)|2 i =
4kB T γ + γ 2 h|δv(ω)|2 i
= h|x(ω)|2 iE + h|x(ω)|2 iN E .
k2 + γ 2 ω2
(2.11)
Eq. (2.10) shows that velocity fluctuations of the non-equilibrium fluid actually modify the
form of the power spectral density of fluctuations of the particle in the fluid at equilibrium
(∆T = 0), h|x(ω)|2 iE = 4kB T γ/(k 2 + γ 2 ω 2 ), with a non-equilibrium contribution given by
h|x(ω)|2 iN E = γ 2 h|δv(ω)|2 i/(k 2 + γ 2 ω 2 ). The estimation of such non-equilibrium contribution is more delicate than those of temperature and density fluctuations since there is no
analytical expression in the literature for the correlation function of velocity fluctuations.
Non-equilibrium velocity fluctuations are expected to be detectable at low frequencies only
due to the effect of critical slowing down close to the bifurcation. In fact, it has been
demonstrated that below the onset of convection there exist two non-equilibrium hydrodynamic modes: a slower heat-like mode of decay rate Γ− and a faster viscous mode of decay
rate Γ+ [27], [29]. Γ± = Γ± (q) are very complicated wavenumber-dependent functions
given explicitly in [29]. Taking into account that close to the instability the characteristic
wavenumber is the critical one q = qc , we can estimate the order of magnitude of such
decay rates for the typical values of the parameters of the experimental system (for ∆T =
10 K)
DT
Γ+ ∼ 500 2 ∼ 10 rad s−1 ,
d
Γ− ∼ 0.002Γ+ ∼ 0.02 rad s−1 .
Thus, it is expected that close enough to the onset of convection, non-equilibrium hydrodynamic fluctuations will lead to an increase of the noise level of the power spectral density
of displacement fluctuations of the micro-sized particle with respect to the spectrum at
equilibrium in the low frequency side f . max{Γ+ , Γ− }/(2π) = Γ+ /(2π) ≈ 1.5 Hz of
2.3 Experimental description
22
Figure 2.1: Diagram of the sample cell and the temperature control chamber used during
the experiment. The ultrapure water layer is depicted in blue while the circulating water
in the cooling chamber is in gray dashed lines. The diagram is not in proper scale.
the spectrum. An exact calculation of the non-equilibrium contribution h|x(ω)|2 iN E for
the system under study requires the knowledge of the power spectral density of velocity
fluctuations, which is out of the scope of the present work. We attempt to determine
experimentally h|x(ω)|2 iN E instead, by noting that it must diverge (probably as (−²)−1/4
like in the case of the variance of temperature fluctuations (2.7)) due to the proximity
of the bifurcation. Hence, our goal is to take the system close enough to criticality and
detect an increase in the Lorenztian power spectral density of displacement fluctuations of
the probe particle at low frequencies. Note that the system must not be taken extremely
close to criticality (² → 0− ) since in this case Γ± would be much smaller than the values
estimated previously due to the critical slowing down of the system, which would be out
of the detectable frequency range of our experiment.
2.3
2.3.1
Experimental description
Sample preparation
Ultrapure water is the fluid used in this experiment. Since we attempt to measure just
the influence of non-equilibrium hydrodynamic fluctuations on the motion of a trapped
micro-sized bead, we must reduce the presence of any other source of perturbations (e.g.
the presence of dust nearby, a high concentration of microzed beads, air bubbles, etc.). For
this purpose, the water sample is filtered through a 0.22 µm micropore filter. A very small
fraction of silica micro-sized particles (diameter = 2 µm) is injected in the water sample.
Then, we dilute it in ultrapure water several times in order to reduce the concentration of
micro-sized particles as much as possible. Once small bubbles are destroyed by means of a
ultrasonic bath during 10 minutes, the sample is injected into a cylindrical cell similar to
the one used for laponite (see Fig. 2.1) with a separation of d = 2 mm between the inner
2.3 Experimental description
23
Figure 2.2: Diagram of the PSD detection system.
surfaces of the microscope slide and the coverslip.
2.3.2
Experimental setup
Trapping system
In order to keep the micro-sized particle in a fixed mean position, we use an optical tweezer
setup like the one described in the first part of this work (Fig. 1.4(a)) with some slight
modifications. A laser diode (Lumics λ = 845 nm) is used as the trapping beam. The
sample cell is placed on the piezoelectric stage with the microscope slide at the top and
the coverslip at the bottom. Once the particle is trapped, the focus of the trapping laser
beam is adjusted 25 µm above the coverslip inner surface (see Fig. 2.1).
Detection system
Unlike the detection system used in the aging experiment, in this case we choose one
based on a position sensing scheme (Fig. 2.2). In addition to the trapping laser, a second
laser is used for detection. We use a HeNe laser whose Gaussian beam is focused by the
microscope objective very close to the focus of the trap. The trapped silica bead of radius
R = 1 µm acts as a microlens of focal distance f1 = 0.6 µm. Then, when its center is
located exactly at 1.6 µm from the focus of the HeNe beam, the rays refracted by the bead
emerge parallel to the +z direction. The lens array shown in Fig. 2.2 allows to obtain a
final laser spot (at a distance f2 +4f3 from the center of the condenser) of constant intensity
profile independently of the fluctuations of the trapped bead along the z direction. This
spot is detected by a duolateral position sensing diode [PSD (DL100-7PCBA3)] of square
active area of 10 mm × 10 mm which measures the relative displacement of the laser spot
centroid with respect to its centroid. The optical system is aligned in such a way that the
laser spot centroid is located exactly in the PSD centroid in the absence of any refracting
silica bead. In the presence of a bead, fluctuations around its the mean position in the
x − y plane deflect the resulting laser spot from the centroid of the PSD. When the laser
spot impinges on the photodiode, currents are created and collected by electrodes located
2.4 Preliminary results
24
at opposite edges. The currents collected at each edge are converted into voltages (X1 , X2
along x and Y1 , Y2 along y) that are processed in order to provide the difference outputs
X1 − X2 , Y1 − Y2 and the sum outputs X1 + X2 , Y1 + Y2 . The differences are then externally
normalized by the sums in order to obtain the coordinates (X = (X1 − X2 )/(X1 + X2 ),
Y = (Y1 − Y2 )/(Y1 + Y2 )) of the laser spot centroid. For simplicity, we record the X
coordinate only. The position x(t) of the trapped bead with respect to its mean position is
proportional to the position X of the laser spot centroid and the proportionality constant
can be determined by calibration of the apparatus. The signal X is filtered by a low-pass
filter (Stanford SR640) with a cutoff of 4 kHz in order to get rid of high frequency modes,
which are useless in this experiment. Once filtered, a time series {x(t)} of 1000000 data
points are recorded at a scan rate of 8192 Hz for a given value of ∆T . Several spectra are
recorded in order to have good statistics when averaging to obtain h|x(ω)|2 i.
Temperature control system
A temperaute control system is needed in order to apply a temperature gradient of magnitude ∆T /d in the +z direction across the ultrapure water layer, below the onset of Rayleigh
Bénard convection. An objective heating ring is placed around the metal jacket of the oilimmersion microscope objective (×63) in order to keep it at a constant high temperature
T1 . The highest allowed temperature without damaging the objective is T1 = 37◦ C. Then,
the metal jacket plays the role of a high temperature reservoir, keeping the temperature of
the lower plate of the sample cell (the coverslip) at a constant value. On the other hand,
a low temperature reservoir is implemented by means of a small chamber with circulating
water placed above the upper plate (microscope slide) of the sample cell (see Fig. 2.1). The
cooling chamber consists on a metallic ring-shaped canal for water circulation encompassed
between two heat-conducting sapphire windows. Water enters the chamber and flows (in a
laminar regime) along the canal absorbing the heat transferred from the sample cell by the
lower sapphire plate. Heat is evacuated when the water flow leaves the chamber, keeping
the microscope slide at a constant temperature T2 . The lowest temperature that we can
be reached by the circulating water is T2 = 15◦ C without the effect of water condensation
from air vapor on the optical surfaces. The inner surface of the canal is conic-shaped in
order to allow the whole deflected laser beams (both trapping and detecting beams) to pass
through the upper sapphire window, as shown in Fig. 2.1. The space between the inner
surface of the canal and both saphire windows is filled with immersion oil to avoid a large
mismatch of the optical path of the laser beams when passing through different optical
media in the chamber.
The critical value of the temperature difference that must be applied between both
reservoirs is calculated by means of Eq. (2.1). For the values of the parameters of the
system used in the experiment, we obtain ∆Tc = 11 K. Then, we must choose ∆T < 11 K
in order to take the ultrapure water layer below the onset of convection.
2.4 Preliminary results
25
−3
10
−3
10
−4
10
−4
<|x(f)|2> (V2 / Hz)
<|x(f)|2> (V2 / Hz)
10
−5
10
−5
10
−6
10
−6
10
experimental
Lorentzian fit
experimental
Lorentzian fit
−7
−7
10
−1
10
0
10
1
2
10
10
3
10
f (Hz)
(a)
4
10
10
−1
10
0
10
1
2
10
10
3
10
4
10
f (Hz)
(b)
Figure 2.3: 2.3(a) Power spectral density of displacement fluctuations of the trapped particle in the fluid at equilibrium (25◦ C with no heating and no cooling); 2.3(b) power
spectral density in the fluid at equilibrium (29◦ C with heating and cooling). In both cases
the plotted curves are averages over 10 spectra.
2.4
Preliminary results
No reliable results are obtained for the effect of non-equilibrium velocity fluctuations in the
spectra of the trapped Brownian particle, given several technical difficulties encountered
during the experiment, as discussed in the following.
First of all, we measure the power spectral density of displacements when the whole
system is in equilibrium (∆T = 0) at room temperature (25◦ C). The power spectral density
is Lorentzian (Fig. 2.3(a)), as expected at equilibrium. In order to test the performance of
the temperature control system inside the water layer, we increase both the temperature
of the objective heating ring and the circulating water in the upper chamber at T = 35◦
C. In this case, the system is also at equilibrium (∆T = 0) and the spectrum must be
Lorentzian, but we should expect an increase of the value of the corner frequency fc ∝ 1/η
and the low frequency level of the spectrum ∝ T /η since the value of the viscosity of water
η decreases. The expected relative magnitudes of such quantities at T = 35◦ C with respect
to those at T = 25◦ C should be 1.24 for the corner frequency and 1.28 for the spectrum
level. Figure 2.3(b) shows the power spectral density at T = 35◦ , which is Lorentzian.
However, the values of the relative magnitudes obtained from the Lorentzian fit are 1.07
and 1.10, respectively. By looking for the temperature which corresponds to such ratios,
we find the actual temperature of the water layer is 29◦ instead of 35◦ C. Then, we observe
that there are large heat losses from the heating ring to the coverslip of the sample cell
and from the circulating water source to the the microscope slide and there is not enough
temperature control even at equilibrium. Thus, under this experimental conditions it is
very difficult to take the system close enough and below the critical temperature difference
∆Tc = 11 K where the amplitude of non-equilibrium velocity fluctuations is extremely
enhanced. However, we could not improve the temperature control system due to lack of
time.
2.4 Preliminary results
26
−2
10
−3
10
−3
10
−4
<|x(f)|2> (V2 / Hz)
<|x(f)|2> (V2 / Hz)
10
−4
10
−5
10
−5
10
−6
10
−6
10
experimental
Lorentzian fit
−7
10
−7
−1
10
0
10
1
2
10
10
f (Hz)
(a)
3
10
4
10
10
−1
10
0
10
1
2
10
10
3
10
4
10
f (Hz)
(b)
Figure 2.4: 2.4(a) Power spectral density of displacement fluctuations of the trapped particle in the presence of a temperature gradient with interacting small particles; 2.4(a) power
spectral density in the presence of a temperature gradient. The plotted curve is the average over five spectra with neither any apparent interaction with small particles nor any
apparent horizontal convection.
Another major problem encountered during the application of the vertical temperature
gradient is the presence of a uniform horizontal water flow around the trapped particle even
below the theoretical critical temperature difference ∆Tc = 11 K. This flow is thought to
be due to non-zero horizontal temperature gradients in the microscope slide, leading to a
uniform horizontal convective motion of the fluid close to the microscope slide. The force
of this convective flow is strong enough to drag free Brownian particles close to the trapped
one and then strongly perturbing its motion. Even in the absence of free beads close to
the trapped one, very small particles that cannot be filtered by the 0.22 µm micropore
filter get close to the trapped bead and interact with it, modifying enormously the low
frequency side (f < 2 Hz) of the spectrum, as shown in Fig. 2.4(a). Unfortunatly, this is
the frequency range where non-equilibrium hydrodynamic fluctuations are expected to take
place and their effect on the motion of the trapped bead is easily hidden by any influence of
neighboring small particles. By keeping only the time series with no apparent interactions
of small particles and no apparent horizontal convective flow, we obtain the power spectral
density curve shown in Fig. 2.4(b). Unlike spectra at equilibrium (Figs. 2.3(a) and 2.3(b)),
in this case it is possible to observe a slight increase of the spectrum level for f < 1 Hz, which
suggests the effect of slow non-equilibrium hydrodynamic fluctuations due to the vertical
temperature gradient across the fluid layer, as expected from the calculations carried out in
Section 2.2. However, given the lack of knowledge of the actual temperature gradient and
the possible existence of additional perturbing sources, this increase could be an artifact
and not the actual effect of non-equilibrium velocity fluctuations of water.
2.5 Conclusion and perspectives
2.5
27
Conclusion and perspectives
We have roughly estimated the possible effects of non-equilibrium hydrodynamic fluctuations on the motion a micro-sized particle immersed in water and trapped by optical
tweezers. Unlike scattering experiments performed in the past, in this case we are interested in a local effect. We find that only velocity fluctuations could be non-negligible and
modify the lower frequency side of the power spectral density of displacement fluctuations.
We must be aware that the results obtained here are very preliminary and they are not
completely reliable given the several sources of low frequency noise observed during the
measurements. Then, in order to enhance the effect, we must look for an optimal value of
the Rayleigh number corresponding to a good compromise between the critical slowing down
and the amplitude of fluctuations below the onset of instability. An analytical calculation
of the expression for the power spectral density of velocity fluctuations in the x − y plane
is also needed in order to know the actual contribution of non-equilibrium fluctuations to
the power spectral density of the probe particle. We need to improve in the future the
temperature control system in order to take the system close enough to and below the
onset of convection and be able to resolve the contribution of non-equilibrium fluctuations.
One should also take into account the presence of possible additional hydrodynamic effects
since the probe particle was trapped at 25 µm above the inner surface of the lower plate
where viscosity could damp the non-equilibrium effect. One possible direction is the use of
video microscopy in order to study the motion of multiple free micro-sized particles in a
very thin Rayleigh-Bénard cell.
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