Hydrodynamic
effects in soft, heterogeneous films studied by quartz crystal
microbalance (QCM).
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| Heterogeneous vs. Homogenous
interfaces: |
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| An
example of
a homogeneous layer (black-and-white) on a QCM crystal (green gradient)
in liquid (blue). There are many examples of such films, including
polymers on surfaces, lipid bilayers, and confluent cell layers. |
An example of
a laterally heterogeneous layer of surface-adsorbed nanoparticles
(proteins, liposomes, virus or polymer particles...) on a QCM crystal
(green gradient)
in liquid (blue). |
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Shear-acoustic techniques, such as QCM,
are in principle sensitive not only to the amount of material present
on the surface but to its organization at the interface. Extracting
quantitative information about packing density and size of the
surface-adsorbed particles has been difficult however, in part due to a
lack of understanding of the mechanisms behind the QCM response from
laterally heterogeneous films in liquids. In four recent papers
(Johannsmann et al., 2008, Rojas et al. 2008, Tellechea et al., 2009,
Johannsmann et al. 2009), we investigate the behavior of lateraly
heterogeneous films as they are sheared at MHz frequencies in QCM. Specifically, we focus on the
hydrodynamic effects and the mechanisms of energy dissipation in these
films.
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QCM
of homogenous systems
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Interpretation
of acoustic response from homogeneous layers (see the
image above on the left) is relatively straight-forward. It
is based on mechanical or equivalent circuit models. There are two
possibilities:
- The film does not
dissipate energy.
The difference in dissipation between the bare crystal and the crystal
+ film, ΔΓ, is ~ 0. In this case, the frequency shift
due to the film is proportional to the aerial mass density of the film
via the Sauerbrey relationship, Δf/n
= - mfilm * constant.
- The
film does
dissipate energy. A complex elastic modulus G = G'+iG"
is defined, where
the real part represents energy stored, and the imaginary part - energy
dissipated; in general both depend on frequency. The frequency and
bandwidth
shifts then are the real and imaginary parts, respectively, of the
following expression:
Δf + iΔΓ
~ - constant*(mfilm)*(1-ρliqGliq/ρfilmGfilm)
Recall, that mfilm is
in the units of mass per unit
area, and that for a Newtonian liquid, Gliq = iωη, where η is liquid viscosity.
ρ is
density and ω = 2 πf.
Here are several
good references on the subject: Johannsmann,
D. J. Studies of Viscoelasticity
with the QCM, in Piezoelectric
Sensors, Ed. Steinem and Janshoff, Springer Verald 2006;
Johannsmann,
D. J., Appl. Phys. 89 6356 -
6364 (2001), Macromol. Chem. Phys. 200
501-516 (1999); Geelhood S. J., J. Electrochem. Soc. 149 H33-H38 (2002); Reed et al. J.
Appl. Phys. 68 1993 - 2001
(1990). |
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| Hydrodynamic effects in
heterogeneous films: AFM-QCM studies of ferritin adsorption on gold
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| To establish a quantitative relationship
between the number of the surface-adsorbed particles and the frequency
shift Δf measured by QCM, we combined our network
analyzer-based QCM (IQCM,
impedance-based quartz crystal
microbalance) from Resonant Probes, GmbH, with a Nanoscope atomic force microscope
from Veeco. The
combination allowed us to visualize individual molecules of protein
ferritin adsorbed on the surface of gold-coated quartz crystals while
at the same time recording frequency and bandwidth shifts due to
ferritin adsorption.
Crucial to the success of these experiments was the use of freshly
purified, largely monomeric, ferritin.
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Ferritin
is essentially a 12 nm protein ball with a 6 nm iron core. The protein
part
of ferritin, apoferritin, is shown above. One such
monomer consists of 24 subunits.
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Tapping mode AFM images of ferritin adsorbed
on the surface of gold-coated QCM crystals. Images taken at two
different surface coverages are shown. Left image: 2 um x 2 um x 50 nm;
Right image: 3 um x 3 um x 50 nm.
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The
frequency shift measured with the QCM, and the corresponding Sauerbrey
mass, are plotted along the Y-axes. The mass and surface packing
density calculated from AFM images are plotted along the X-axes. Data
obtained with the purified ferritin are shown in red, with non-purified
- in blue.
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The
AFM-QCM comparison clearly demonstrates the non-linear relationship
between the number of the ferritin molecules adsorbed on the surface
and the frequency shift determined by QCM. This non-linearity arises
from hydrodynamic effects:
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2D Finite
Element Method Calculations
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FEM
calculations generate pressure profiles and flow velocities in the
liquid around oscillating 12 nm truncated cylinders that represented
ferritin molecules (image on the left).
This information is then used to calculate the stress/speed ratio at
the interface, which yields the frequency and bandwidth shifts by the
small load approximation. FEM calculations are described in great
detail in Johannsmann
et al. 2008. |
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In the
figure above, frequency and bandwidth shifts observed experimentally
with purified ferritin (large symbols), plotted as a function of
surface coverage determined from AFM images, are compared with the
results of the 2D finite element method (FEM) calculations (lines).
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There
is very good agreement between the data and the results of the
simulations - including the small dissipation shifts observed for this
nearly-Sauerbrey system. This demonstrates, that taking into account
hydrodynamic effects at the laterally heterogeneous interface, it is
possible to reproduce the experimentally observed non-linear
relationship between the frequency shifts and feritin surface coverage.
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Experimental
details of the AFM-QCM setup are given in Rojas et
al. 2008, while
the detailes of the FEM calculations and comparison with the purified
ferritin data are given in Johannsmann
et al. 2008.
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Hydrodynamic effects in
heterogeneous films: The ΔΓ/Δf ratio
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In the case of thin,
homogeneous films, the ratio of bandwidth shift to the frequency
shift can be easily calculated from the model described above. To first order, it turns out to be
independent of the adsorbed mass, as follows:
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where
J' is the elastic compliance of the film. (As we remark in the paper,
a higher-order expansion predicts an increase in the ratio with
film thickness. See Tellechea
et all., 2009). In the plots shown below, we demonstrate that when
nano-sized particles adsorb to the surface, the ratio decreases with
the magnitude of the frequency shift but incrases with the particle
size:
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| 2D
Finite Element Method Calculations |
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Multiphysics
Module of the COMSOL software package was used to calculate the
pressure profiles and flow velocities in the liquid around oscilating
hemicylinders (image above). These were used to calculate stress at the
interface. Calculations were performed in two directions: with the flow
paralel and perpendicular to the main cylinder axes.
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Calculated
ΔΓ/Δf
vs Δf /n plot (5th overtone). Forces acting on the crystal surface were
calculated by finite element method (FEM, see the panels on the right).
The small load approximation was then used to convert these forces into
frequency and bandwidth shifts. The small load approximation states
that the frequency/bandwidth shifts are proportional to the
stress-to-speed ration at the interface.
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| In the
plot above, CPMV stands for cow pea mosaic virus, and extruded DPPC
liposomes of two sizes were used. Scroll down to see diagrams and
cryoTEM images of the liposomes. |
| Comparison
between the experimental ΔΓ/Δf
vs Δf /n plot
(above, left) and the one obtained from the FEM calculations clearly
demonstrate that the negative slope of the plot can be explained by
hydrodynamic considerations. Questions remain as to the linearity of
the ΔΓ/Δf
vs Δf n plot. Keep in mind, that in the case of
acoustically homogeneous, thin films, we expect ΔΓ/Δf
vs Δf /n plot
to be independent of Δf /n. There
are examples of such plots in the literature: e.g., Tsortos et al. Biophys J. 94, 2706, 2008. See also the supplementary
information for the Tellechea et al. 2009 paper. |
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| Model-independent
analysis of QCM data on colloidal particle adsorption |
| In many cases, we would
like to know the absolute height of the surface-adsorbed nanoparticles
(proteins, liposomes, viruses...), or at least how it changes as the
particles adsorb to the surface. We recently
observed, that we can gain information about sizes of the
surface-adsorbed particles by examining the ΔΓ/Δf
vs Δf /n
plot on many overtones. Such a plot is shown below on the left. |
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CPMV,
cow pea mosaic virus: ~ 28 nm
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DPPC liposomes
~ 80
nm
~ 120 m
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Converting the
value of Δf /n at the intercept of the ΔΓ/Δf with the X-axis (blue arrowheads on the
left) into a Sauerbrey mass yields, assuming a density of ~ 1 g/cm3,
the following heights: for the 28 nm CPMV particles: 29 ± 0.8
nm. For 83 nm DPPC liposomes: 81 ± 6 nm. For 114 ± 8 nm DPPC liposomes: 111 ± 11 nm. Z-averaged sizes are quoted for the
liposomes.
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Cryo-TEM images,
shown on the right, of DPPC liposomes (stiff; left image) and DOPC
liposomes (soft, middle and right images) adsorbed on
silica particles (dark spheres) indeed reveal that their shapes are
different: DPPC liposomes are quasi-hexagonal, while DOPC liposomes are
dome-shaped and deformed. Both sets of liposomes are approximately of
the same size, and the silica particles are 300 nm in diameter.
The magnifications of the images are slightly different.
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DPPC
liposomes
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DOPC liposomes
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CryoTEM work was performed by Edurne Tellechea and Marta Gallego at the Electron Microscopy platform at CIC bioGUNE, CPMV experiments
were performed by Ralf Richter using a Q-Sense system,
and CPMV itself was
prepared by Nicole Steinmetz. FEM calculations were performed by Diethelm Johannsmann.
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| Energy dissipation
in films of adsorbed nanospheres studied by QCM |
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