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Surface forces between colloidal particles at high
hydrostatic pressure
D. W. Pilat, B. Pouligny, A. Best, T. A. Nick, R. Berger, H.-J. Butt
To cite this version:
D. W. Pilat, B. Pouligny, A. Best, T. A. Nick, R. Berger, et al.. Surface forces between colloidal
particles at high hydrostatic pressure. Physical Review E , American Physical Society (APS), 2016,
93 (022608(1-8)), pp.343 - 354. �10.1103/PhysRevE.93.022608�. �hal-01366149�
Surface forces between colloidal particles at high hydrostatic pressure
D. W. Pilat,
1
B. Pouligny,
2
A. Best,
1
T
. A. Nick,
1
R. Berger,
1,*
and H.-J. Butt
1
1
Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany
2
Centre de Recherche Paul-Pascal, 115 Avenue Schweitzer, 33600 Pessac, France
It was recently suggested that the electrostatic double-layer force between colloidal particles might weaken
at high hydrostatic pressure encountered, for example, in deep seas or during oil recovery. We have addressed
this issue by means of a specially designed optical trapping setup that allowed us to explore the interaction of a
micrometer-sized glass bead and a solid glass wall in water at hydrostatic pressures of up to 1 kbar. The setup
allowed us to measure the distance between bead and wall with a subnanometer resolution. We have determined
the Debye lengths in water for salt concentrations of 0.1 and 1 mM. We found that in the pressure range from 1
bar to 1 kbar the maximum variation of the Debye lengths was <1 nm for both salt concentrations. Furthermore,
the magnitude of the zeta potentials of the glass surfaces in water showed no dependency on pressure.
I. INTRODUCTION
Surf
ace forces determine the stability, structure, and rhe-
ology of suspensions and emulsions, and are often present
in environments with high hydrostatic pressure. Oceans are
one example of an aqueous, high-pressure environment. With
depths of up to 11 km, as in the case of the Mariana Trench, the
hydrostatic pressure in oceans can reach up to 1.1 kbar. One
important maritime system where the stability is governed by
surface forces is colloidal organic carbon, which is “one of
the largest reservoirs of organic carbon on the planet” [
1].
Wells and Goldberg, for example, found that the removal of
colloidal organic carbon from bulk sea water at a depth of up
to 5 km mainly takes place by aggregation into larger colloids
and subsequent sedimentation to the sea floor [2]. From the
structure of colloidal organic carbon aggregates, these authors
concluded that there is a repulsive force between the particles
that must be overcome before aggregates can be formed.
Surface forces at high hydrostatic pressure are also found in
oil recovery. The rheology of water-based drilling muds at
high pressure has been under investigation for more than two
decades [3]. The rheology of suspensions is determined by
interactions between individual particles [4]. To understand
the rheology of suspensions at high pressures, surface forces
between the suspended particles, e.g., charged, micron-sized
clay platelets, need to be investigated.
Various methods have been used to measure surface forces,
such as the surface-force apparatus [5,6], the colloidal probe
technique [7,8], and optical trapping or total internal reflection
microscopy [9–11]. These methods can quantify surface forces
between different materials, under different conditions and at
different temperatures. Measurements of surface forces at a
hydrostatic pressure of up to 170 bars have been reported
by Schurtenberger and Heuberger [12,13], who measured
the f orce between two mica surfaces in carbon dioxide with
an extended surface-force apparatus. They investigated the
critical Casimir effect along the supercritical ridge of carbon
dioxide [13].
*
berger@mpip-mainz.mpg.de
One fundamental surface force in an aqueous electrolyte
is the electrostatic double-layer force. This is of interest
because most surfaces are charged in water, e.g., the sur-
faces of colloidal organic carbon particles and clay platelets
[
1,3,14,15]. The double-layer force determines the stability
of
many aqueous suspensions. Between a sphere and a plane
with the same charge density σ and for distances much larger
than the Debye length λ
D
, the double-layer force can be
approximated by [8,14,16]
F
DL
=
4πλ
D
Rσ
2
ǫ
0
ǫ
exp
−
D
λ
D
= F
0
ex
p
−
D
λ
D
. (1)
Here, R is
the radius of the sphere; ǫ
0
and ǫ are the vacuum
permittivity and the dielectric constant of the medium between
the surfaces, respectively. D is the shortest distance between
the surface of the sphere and the plane. The Debye length can
be calculated by [14]
1
λ
D
=
e
2
ǫ
0
ǫk
B
T
i
c
i
Z
2
i
. (2)
Here, e is
the elementary charge; k
B
and T are the Boltzmann
constant and the temperature in units of K, respectively. c
i
and Z
i
are the concentration and valency of the ith ion sort.
Equation (1) may be related to the surface potential ψ
0
by
using the linearized Grahame equation σ =
ǫǫ
0
ψ
0
λ
D
[14].
The double-layer force is well understood and characterized
at ambient pressure, but has never directly been measured at
high hydrostatic pressure. The effect of hydrostatic pressure
on the double-layer force between colloidal particles in
hydrothermal water is controversial [17–19]. It has been
reported that the magnitude of the zeta potential of quartz in
an aqueous NaCl solution (concentration 0.01 M)atpH = 4
decreased by a factor of 5 when the hydrostatic pressure
was increased from 1 to 4 bars [
20]. Such a decrease in the
zeta
potential with increasing pressure was also hypothesized
in order to explain the thickening behavior of an Aerosil
silica suspension at a pressure of up to 150 bars [21]. A
decrease in the zeta potential indicates a decrease in the surface
potential ψ
0
and would thus decrease the amplitude F
0
of the
double-layer force. In addition, the dielectric constant of water
ǫ depends weakly on the hydrostatic pressure: It increases from
1
ǫ = 78 at 1 bar to ǫ = 82 at 1 kbar [22](T = 25
◦
C). Thus,
for an increase of the hydrostatic pressure from 1 bar to 1 kbar,
an increase of λ
D
and a change of F
0
by approximately 2.5%
are expected [Eq. (2)].
In this article we report on the measurements of surface
forces in aqueous electrolytes at a hydrostatic pressure of up
to 1 kbar using an optical trapping method. Our aim is to
elucidate the pressure dependency of the double-layer force in
aqueous solutions at room temperature.
II. MATERIALS AND METHOD
Bowman et al. measured the bulk viscosity of water
at a hydrostatic pressure of up to 13 kbars by means of
optically trapping a glass bead, thus demonstrating that optical
trapping at high hydrostatic pressure works effectively [23]. To
achieve force measurements at high hydrostatic pressures, we
developed a long-working-distance optical trapping method
based on a method reported by Nadal et al. [
24]. We performed
force-distance
measurements (force curves) inside an optical
high-pressure cell between a colloidal glass bead and the upper
inner wall of a rectangular glass capillary in aqueous NaCl
solutions [Fig. 1(a)].
A. The optical setup
A long-working-distance (WD) optical trap, an interferom-
eter, and an inverse optical microscope compose the optical
setup [Fig. 1(b)]. The trapping laser (Laser Quantum, Finesse
pure) was a diode-pumped solid-state laser that provided
linearly polarized light at a wavelength of λ = 532 nm with
a Gaussian beam profile at a laser power of 3 W. The
power of the trapping beam was attenuated by combining a
motorized
λ
2
plate and a polarization-sensitive beam-splitter
cube (Workshop of Photonics, Motorized Standard Watt Pilot,
high-power version). The rotation of the
λ
2
plate and thus the
attenuation were controlled by a computer. A second pair
of a fixed
λ
2
plate and a polarization-sensitive beam-splitter
cube split the trapping beam with a power ratio of 1:2. The
trapping beam was thus limited to a maximum power of
1 W. The power of the trapping beam after exiting the fixed
attenuator (P
Laser
) was monitored for all experiments. The
power splitter (also a motorized standard Watt Pilot) split
the trapping laser into two perpendicularly polarized beams
with an adjustable power ratio. The downwards trapping
beam was directed into the condenser (Nikon, 50 × , NA =
0.4, WD = 22 mm) and the upwards trapping beam into the
objective (Nikon, 50 × , NA = 0.6, WD = 11 mm) with both
foci overlapping inside the high-pressure cell (HPDS High
Pressure Cell [
25]). The space between the objective and
the
condenser was >3cm, which was large enough to mount
the high-pressure cell in between. The high-pressure cell was
clamped onto a platform connected to a motorized translation
stage (Vision Lasertechnik), allowing the cell to be moved
relative to the foci of the trapping beams. Furthermore, both
trapping beams could be blocked by computer-controlled
shutters. Measurement of the radii of the beam waists of
both trapping beams inside the high-pressure cell resulted in
w
0
= 2.7 ± 0.4 μm.
FIG. 1. (a) Surface-force measurement inside a high-pressure
cell: The high-pressure cell allowed light to be transmitted through
two sapphire windows (window thickness = 2.2 mm). The cell was
filled with water that could be pressurized by a piston screw pump (not
shown). A sealed rectangular glass capillary filled with the sample
solution and the glass beads was placed inside the cell. A single
glass bead was pushed against the upper wall of the glass capillary
by a moderately focused trapping beam (green). The distance D was
determined by measuring the interference between the reflection of
a second laser beam (red) off the surface of the bead and the wall.
(b) Optical trapping and interferometry setup for measurements inside
a high-pressure cell: The components were mounted on a vertical
optical breadboard. Thick green line: beam path of the trapping laser.
Thin red line: Beam path of interference laser. Yellow: Light path of
optical microscope. The drawing of the high-pressure cell (HP Cell)
is based on the high-pressure cell described by Hartmann et al. [
25]
and
a technical drawing provided by
Andreas Zumbach (RECORD
Maschinenbau GmbH, K
¨
onigsee, Germany). APD: Avalanche
photodiode.
2
In order to measure the bead-glass distance, we used an
ultra-low noise laser-diode module (Coherent, ULN, 5 mW,
λ = 635 nm). The high-power dielectric mirrors (CVI Melles
Griot) reflected the light of the trapping laser and allowed the
light of the interference laser to be (partially) transmitted. The
interference laser was aligned on top of the trapping beam.
The rays reflected off the surfaces of the bead and the wall
interfered [Fig. 1(a) ], and were guided into an avalanche pho-
todiode (Thorlabs, APD130A/M) via a nonpolarizing beam-
splitter cube, a notch filter (Thorlabs,wavelength of center =
633 nm, FWHM = 25 nm), and an optical fiber (Thorlabs,
M69L01, core diameter = 300 μm,NA = 0.39). The notch
filter was slightly tilted, shifting the reflected light to larger
wavelengths, thus allowing a small part of the interference
signal to enter the microscope camera. A similar optical
setup for the same interferometry technique was published
recently by Suraniti et al. [
26]. The illumination, condenser,
objective, tube lens, and camera formed an inverse optical
microscope. Monitoring the reflected interference pattern at
635 nm together with the microscope image of a trapped
bead allowed alignment of the interference laser. The optical
fiber allowed for collection of the isolated on-axis interference
signal. The intensity of the interference signal measured with
the APD was recorded with a sampling rate of 50 kHz, thus
fulfilling the Nyquist criterion as the fastest recorded frequency
was of the order of 5 kHz.
B. Sample preparation
In order to realize defined aqueous sample solutions
subject to high hydrostatic pressure, an 8-mm-long rectangular
glass capillary (VitroCom, VitroTubes 5010-050, borosilicate
rectangular glass capillary, nominal inner dimensions of cross
section: 0.1 mm × 1 mm) was used as a capsule inside the
high-pressure cell [Fig. 1(a)]. Two aqueous solutions of NaCl
were prepared with concentrations of c
0
= 0.10 ± 0.01 mM
and c
0
= 1.0 ± 0.1mM. These low NaCl concentrations were
chosen in order to avoid aggregation of particles or attachment
to the walls. The Debye lengths were larger than 8 nm in
both concentrations of NaCl. Due to this slow decay of the
double-layer force, changes in the force of the order of 1
pN corresponded to changes in distance that could be well
resolved. The solutions were prepared with ultrapure water
provided by a Sartorius Arium 611 VF purification system
(resistivity of 18.2 M cm).
The glass beads were supplied as a dry powder (DUKE
Scientific, 9000 Series, 8-µm-diameter borosilicate glass mi-
crospheres, Fig. 2). They were dispersed in ultrapure water and
sonicated for at least 15 min. We mixed 1 ml of NaCl solution
and 50 µl of the bead dispersion before filling the capillary,
thus diluting the sample solutions to a salt concentration of
c
0
= 95 ± 10 μM and c
0
= 0.95 ± 0.1mM, respectively
(concentration of beads <0.1 v/v%).ThepH values of
the samples were approximately pH = 4–4.5 (measured with
MERCK pH-indicator strips Acilit). The capillary was filled
with the sample solution by capillary rise, sealed by dipping
the ends of the capillary into a pressure-transmitting sealing
paste (Bayer, high-viscosity Baysilone paste), and placed
in the middle of the high-pressure cell. Consequently, any
exchange between the aqueous sample solution and the
FIG. 2. Scanning electron microscopy images of the beads used
here, shown at two different magnifications. Beads were supplied
as a dry powder. For scanning electron microscopy imaging a small
amount of the dry powder was poured onto a Si wafer coated with
20 nm of Pt. A 3-nm-thick layer of Pt was then sputtered onto the
sample. (b) Closeup of (a).
pressure-transmitting water was avoided. The high-pressure
cell was then filled with water, closed, and connected to the
high-pressure periphery consisting mainly of a piston screw
pump (HiP, High Pressure Equipment Company) to pressurize
the water with a maximum pressure of 1 kbar.
C. Experimental procedure
Using optical microscopy, we selected a single bead for
the experiment, trapped it at the lower wall of the capillary,
and separated it from the other beads by moving the cell
laterally. Then the bead was pushed upwards to the upper inner
wall of the capillary, where the force curves were measured
[Fig. 1(a)]. It is known that the radius, the roughness, and
the
surface charge density of a bead have an influence on
the electrostatic double-layer force [Eq. (1)] [27]. Thus, in
one experimental series the same bead was used to record
all force curves at a given salt concentration. This procedure
ensured comparability of the force curves recorded at different
pressures.
Since the radiation pressure force acting on the bead (F
RP
)
is proportional to P
Laser
[
11], a force curve was recorded by
stepwise
increasing P
Laser
. The force driving the bead against
the wall is given by
F
bead
=−F
RP
+ F
G
− F
B
. (3)
3
Here, F
G
and F
B
are gravity and bouyancy, which were
constant during the experimental procedure. Forces F
RP
and
F
B
act in the opposite direction to F
DL
, thus the negative
sign. At each step of P
Laser
(single measurement) the bead
was pushed against the wall by the trapping beam [Fig. 1(a)].
Starting a few micrometers away from the wall, the bead moved
against the wall. F
bead
was determined from analysis of the
bead’s trajectory far from the wall (D>7λ
D
).Closetothe
wall, the bead reached an equilibrium distance D
eq
, where
the force pushing the bead against the wall was in equilibrium
with the repulsive double-layer force, i.e., F
bead
=−F
DL
(D
eq
).
Evaluation of the bead’s thermal motion yielded the value
of D
eq
[24]. The attenuation of the trapping beam and the
data acquisition were computer controlled. Thus, a force
curve consisting of 10–20 force-distance pairs was recorded
automatically within 1–3 min.
D. Data analysis
The distance between bead and wall was determined by
evaluating the reflected interference signal collected by the
avalanche photodiode (I
APD
). The intensity-distance relation-
ship of the interference signal is given by [28]
2I
APD
= I
max
+ I
min
−
(
I
max
− I
min
)
cos
4πn
1
λ
D +
.
(4)
The refractive index of water n
1
at hydrostatic pressure up to
P = 1 kbar was previously determined by Schiebener et al.
[29]. I
max
and I
min
are the intensities of a maximum and
a minimum of the interference signal, and is the phase
of the cosine function upon contact of bead and wall. For
consecutive extrema in the measured interference signal, the
corresponding difference of D is D =
λ
4n
1
≈ 119 nm. The
trajectory of the bead while moving towards the wall could
thus be deduced by determining the temporal positions of
the extrema in the interference signal (Fig. 3, blue empty
squares). After the minimum at approximately 1.12 s in Fig. 3
the reflected interference signal was strictly monotonic in D;
thus Eq. (4) could be inverted in order to determine D (Fig. 3,
blue line).
At distances D between 0.2 and 3 µm (Fig. 3,leftof
1.13 s), the double-layer force was negligible because for
the here used salt concentrations D>7λ
D
. When the bead
moved through the liquid, F
bead
was counteracted by the
hydrodynamic force F
H
, given by
F
H
=−6πRvη
1 +
R
D
. (5)
Here, η is
the dynamic viscosity of the liquid, and v = dD/dt
and R are the velocity and radius of the bead, respectively.
Wall effects due to draining the liquid between the bead and
the wall are considered by using the factor (1 + R/D)[14],
which
is an approximation of the only numerically computable
exact Brenner factor [30]. Owing to the moderate focusing of
the trapping beam, the width of the beam increased by less
than 0.3% at a distance of 3 µm from the beamwaist. Thus F
RP
and consequently F
bead
were considered to be constant during
one single measurement.
FIG. 3. Example of a bead pushed against the wall by the trapping
beam with P
Laser
= 124 mW. Black line and left axis: Intensity of
the reflected interference signal I
APD
between bead and wall versus
time. Empty blue squares, blue line, and right axis: Trajectory of the
bead. D is the absolute distance of the bead and the upper inner wall
of the glass capillary [Eq. (4)]. At times >1.13
s the bead showed
thermal motion around the distance D
eq
where F
bead
=−F
DL
(D
eq
).
In this case, we measured D
eq
= 176.4 ± 0.9nmandF
bead
= 19.6 ±
0.3 pN. The error of the absolute force value was 22% and arose
mainly from the error in determining the bead radii R = 0.5 μm
due to the resolution of the microscope. The red line is a fit on the
interval [0.2 µm, 3 µm] with Eq. (5). We notice an envelope of the
cosine signal in the measured intensity signal that is not accounted
for by Eq. (4). This envelope is due to a lensing effect of the bead.
W
ith our setup it is possible to measure relative distances
˜
D = D + D
0
, where D
0
is an offset. Thus, the absolute
distance D is not known apriori. For the calculation we defined
˜
D := 0 being the position corresponding to the last extremum
of the measured intensity in a single measurement (Fig. 3,
approximately at 1.12 s). Solving Eq. (5) and substituting
D =
˜
D − D
0
yielded the implicit solution for the bead’s
trajectory,
t − t
0
=−
6πη
F
H
[R(
˜
D − D
0
) + R
2
ln(
˜
D − D
0
)], (6)
which was used to fit the trajectory of the bead during
laser-driven sedimentation starting from a bead-wall distance
of 3 µm (Fig. 3, red line). Values for the viscosity of
water were taken from the NIST Chemistry WebBook [31].
From this model we derived F
bead
as F
H
=−F
bead
, and D
0
.
For the measurement shown in Fig. 3 we obtained F
bead
=
19.6 ± 0.3 pN and D
0
=−216 ± 5nm.D
0
is identical for
all single measurements belonging to one force curve. For
the force curve recorded at ambient pressure and a NaCl
concentration of 0.1 mM in water (Fig. 4), we found a mean
value of D
0
=−202 ± 15 nm, where the uncertainty is the
standard deviation.
The refractive indices of the glass beads and the glass
wall were higher than the refractive index of water. Thus,
the phase of the cosine function in Eq. (4) is defined and
becomes φ = π [32]. Consequently, D
0
can only assume
values that are multiples of −
λ
2n
and we can, therefore,
conclude that the absolute distance
between the bead and wall
was D =
˜
D +
λ
2n
≈
˜
D + 238 nm. We attribute the difference
between the measured value of D
0
(i.e., −202 ± 15 nm) and
4
![FIG. 1. (a) Surface-force measurement inside a high-pressure cell: The high-pressure cell allowed light to be transmitted through two sapphire windows (window thickness = 2.2mm). The cell was filledwithwater that could be pressurized by a piston screw pump (not shown). A sealed rectangular glass capillary filled with the sample solution and the glass beads was placed inside the cell. A single glass bead was pushed against the upper wall of the glass capillary by a moderately focused trapping beam (green). The distance D was determined by measuring the interference between the reflection of a second laser beam (red) off the surface of the bead and the wall. (b)Optical trapping and interferometry setup formeasurements inside a high-pressure cell: The components were mounted on a vertical optical breadboard. Thick green line: beam path of the trapping laser. Thin red line: Beam path of interference laser. Yellow: Light path of optical microscope. The drawing of the high-pressure cell (HP Cell) is based on the high-pressure cell described by Hartmann et al. [25] and a technical drawing provided by Andreas Zumbach (RECORD Maschinenbau GmbH, Königsee, Germany). APD: Avalanche photodiode.](/figures/fig-1-a-surface-force-measurement-inside-a-high-pressure-1kpr2u89.webp)
![FIG. 3. Example of a bead pushed against the wall by the trapping beam with PLaser = 124mW. Black line and left axis: Intensity of the reflected interference signal IAPD between bead and wall versus time. Empty blue squares, blue line, and right axis: Trajectory of the bead. D is the absolute distance of the bead and the upper inner wall of the glass capillary [Eq. (4)]. At times >1.13 s the bead showed thermal motion around the distance Deq where Fbead = −FDL(Deq). In this case, we measuredDeq = 176.4± 0.9 nm and Fbead = 19.6± 0.3 pN. The error of the absolute force value was 22% and arose mainly from the error in determining the bead radii R = 0.5μm due to the resolution of the microscope. The red line is a fit on the interval [0.2 µm, 3 µm] with Eq. (5). We notice an envelope of the cosine signal in the measured intensity signal that is not accounted for by Eq. (4). This envelope is due to a lensing effect of the bead.](/figures/fig-3-example-of-a-bead-pushed-against-the-wall-by-the-2up65kc5.webp)