Content uploaded by Sergey Karabasov
Author content
All content in this area was uploaded by Sergey Karabasov on Dec 07, 2022
Content may be subject to copyright.
1
The interfacial layer breaker: a violation of Stokes’ law in high-speed
Atomic Force Microscope flows
Fan Li1, Stoyan K. Smoukov1, Ivan Korotkin1,2, Makoto Taiji3, and
Sergey Karabasov1*
1 The School of Engineering and Materials Science, Queen Mary University of London,
Mile End Road, E1 4NS London, United Kingdom
2 Mathematical Sciences, University of Southampton, University Road, SO17 1BJ
Southampton, United Kingdom
3 Laboratory for Computational Molecular Design, Computational Biology Research
Core, RIKEN Quantitative Biology Center (QBiC), 1-6-5 Minatojima Minamimachi,
Chuo-Ku, Kobe, Hyogo, 650-0047 Japan
Abstract
Structured water near surfaces is important in non-classical crystallization,
biomineralization, and restructuring of cellular membranes. In addition to equilibrium
structures, studied by atomic force microscopy (AFM), high-speed AFM (H-S AFM)
can now detect piconewton forces in microseconds. With increasing speeds and
decreasing tip diameters, there is a danger that continuum water models will not hold,
and molecular dynamics (MD) simulations would be needed for accurate predictions.
MD simulations, however, can only evolve over tens of nanoseconds due to memory
and computational efficiency/speed limitations, so new methods are needed to bridge
the gap. Here we report a hybrid, multi-scale simulation method, which can bridge the
size and timescale gaps to existing experiments. Structured water is studied between a
moving silica AFM colloidal tip and a cleaved mica surface. The computational domain
includes 1472766 atoms. To mimic the effect of long-range hydrodynamic forces
occurring in water, when moving the AFM tip at speeds from to 30 m/s, a
5
×
10
―
7
2
hybrid multiscale method with local atomistic resolution is used, which serves as an
effective open-domain boundary condition. The multiscale simulation is thus
equivalent to using a macroscopically large computational domain with equilibrium
boundary conditions. Quantification of the drag force shows breaking of continuum
behaviour. Non-monotonic dependence on both tip speed and distance from the surface
imply breaking of the hydration layer around the moving tip at timescales smaller than
water cluster formation and strong water compressibility effects at the highest speeds.
Introduction
The intricacy of water structure at nanoscale distances from a material surface has
been observed by both experiments and equilibrium molecular dynamic (MD)
simulations [1-6]. The study of this structure is important in non-classical
crystallization [7-9], biomineralization [10], and tribochemical wear [11]. Interfacial
water properties are also pertinent for the restructuring processes of cellular membranes
[12-14], organismal signalling [15]. The advent of advanced High-Speed Atomic Force
Microscopy (H-S AFM) opens new opportunities for measuring the dynamic
piconewton forces, which occur within microseconds and nanometre structural
rearrangements with subsecond resolution [16, 17]. AFM tools are used to detect the
equilibrium structure of water layers, but the dynamic behaviour is still beyond the
experimental reach in the sub-microsecond and nanosecond region typical of structural
rearrangements and bio-catalytic molecular systems [18-20]. While the timestep of
molecular dynamics (MD) simulations is orders of magnitude too small and so is their
system sizes to model such processes, which involve collective dynamics of many water
molecules, the hybrid multi-scale methods, which combine MD with continuum fluid
dynamics simulations [21-24] provide an attractive opportunity for modelling of
realistic-size systems while resolving details of the molecular ordering. Here we report
the detection of water structuring between a silica AFM colloidal tip and a cleaved mica
surface. The quantification of the drag force on the tip as a function of tip-wall distance
and tip velocity shows non-monotonic force, which becomes negative for high-speed
regimes, implying variations not only viscosity but also density. At the highest speeds
3
these are consistent with breaking of the hydration layer around the moving tip and
formation of a layered water structure in the sub-nanometer gap between the tip and the
substrate surface.
Figure 1.A) Schematic representation of one-half of the symmetric
computational domain (26 nm x 19 nm x 15.6 nm). The symmetry plane
corresponds to the top boundary in the z direction. The silica tip is represented by
a sphere ( ), d is a distance between the tangential plane of sphere and
𝑹
=
𝟓 𝒏𝒎
the potassium ions (K) layer of the mica surface. The sphere is suspended by a
massless spring from the top of the box, and the resulting force F on tip is
measured in accordance with Hooke’s law, , where k and Δz are the
𝑭
=
―
𝒌
·
𝚫
𝐳
virtual spring constant and the displacement. The approach speed V of the tip is
varied. B) A cut through the simulation domain showing the decomposition of the
hybrid multiscale simulation domain into several regions. The silica tip and mica
surface are modelled at the full-atom resolution. In addition, a part of the water
volume in a rectangular slab, which covers the mica surface with a 3.7 nm
thickness and the water in a spherical shell, which covers the silica tip with a 2.79
nm thickness are also modelled at all-atom resolution (water in the pure atomistic
MD region is shown in dark blue). Outside of the pure atomistic region, a multi-
scale interface region is placed where the MD particles are subjected to a
4
hydrodynamic force including the Brownian motion. The thickness of the multi-
scale region is selected to include at least a few cut-off radii of van der Waals forces
for best performance (see Supporting Information for further details). The
multi-scale interface and the outer region, where water molecules are fully
dominated by hydrodynamics forces (Buffer) are shown in light blue. The
hydrodynamic forces are consistent with the macroscopic Stokes solution in a
large water volume surrounding the MD domain so that no reflection of pressure
waves occurs at the boundaries of the multiscale domain.
Method
Figure 1 (a) shows the schematic representation of the simulated AFM device,
where a silica tip functionalized by hydroxy group on surface with a radius of 5 nm that
corresponds to a mid-range tip size compared to the values reported in the AFM
experiments [25, 26]. The tip is attached by a massless virtual spring and suspended
above mica substrate at a distance d. The simplification of AFM device as a massless
spring attached to a spherical tip is a well-established practice in molecular modelling
for it simplifies comparison of the numerical simulation with the analytical Stokes’
solution in the continuum hydrodynamics theory [16, 27]. In both cases, the drag force
acting on a sphere moving in viscous liquid is considered. Details of the force
comparison between the results of the multi-scale calculations and predictions based on
the Stokes’ law are presented in the following paragraph. Following [28, 29], a detailed
atomistic model of the mica substrate (K[Si3AlO8]Al2O2(OH)2]) has been implemented.
In the model, the Si→Al charge defects are distributed in agreement with the solid state
Nuclear Magnetic Resonance data [30]. The silica tip and mica substrate are fully
immersed in water. For water-mica, water-tip and silica-mica modelling, the recently
developed INTERFACE force field is applied which was shown to provide good results
for interfacial systems [31]. Figure 1 (b) illustrates the partition of simulation domain
in the multiscale model, where the external boundary conditions in the continuum part
of the simulation are prescribed in accordance with the macroscopic Stokes model by
introducing the Fluctuating Hydrodynamics (FH) force in the equations of MD particles.
5
The centre of the domain including the water layers on material surfaces is simulated
at the all-atom resolution, hence all the interactions at all-atom level are accurately
captured as in the conventional MD method. The continuum flow effects are enforced
by the moving tip via the effective boundary conditions using the multiscale method
(see the method details in Supporting Information). Therefore, the input parameters of
the model are the tip-substrate distance, d and the AFM speed, V while the output is the
force on the AFM tip (Fig. 1).
It should be noted that the bulk hydrodynamic flow effect induced by the moving
AFM boundary decays slowly with the distance. Furthermore, the simulation time
needed for averaging out the thermal velocity fluctuations of water atoms (~ 100 m/s),
which amplitude is much larger than the hydrodynamic signal, is macroscopically large
too. Altogether this means that using the standard all-atom MD approach with
equilibrium boundary conditions would require a very large computational domain in
space and time, which is prohibitively expensive. Therefore, the implemented
multiscale method essentially truncates the computational domain, while preserving the
atomistic resolution where it is critical to account for short-range interatomic
interactions.
To calculate the total force exerted on the AFM tip, individual forces in the all-
atom region from the surrounding water atoms and the mica substrate are summed up
and time averaged. The averaging volume comprising the water and mica atoms around
the AFM tip, which defines the force exerted on the AFM tip, is governed by short-
range interatomic interactions between those atoms and the microscopic AFM tip. This
microscopic volume is the effective volume, which corresponds to the statistical
ensemble averaging used in the current simulations. The collective force associated
with the pure hydrodynamic effect is computed as a difference between the total AFM
force with and without the AFM tip motion.
To validate the FH/MD model, the tip velocity is set to zero, and the resulting
force-distance curve solution is compared with the solution obtained from the all-atom
molecular dynamics simulation performed for the same system with periodic boundary
6
conditions. The force is calculated in accordance with Hooke’s law, , where
𝐹
=
―
𝑘
·Δ
z
k and Δz are the virtual spring constant and the displacement in the z-direction. The
distance d in the force-distance curve is defined as the distance between the tangential
plane of sphere and the potassium ions (K) layer of the mica surface. Notably, as shown
in Fig. S1 (a) in Supporting Information, the difference between the FH/MD and the
reference all-atom MD solution is within 0.05 nN. Furthermore, the sensitivity of the
MD solution to the averaging window, 2 ns versus 3 ns is also approximately within
the same 0.05 nN error bar. (See Fig. S1 (b)). It can be noted that this difference could
be further reduced by running the simulations for a longer time to better converge the
models statistically.
The above error of the multiscale model solution is much smaller than the
uncertainty of the computed force associated with the thermal fluctuations of water
molecules in the volume corresponding to the tip and the typical averaging time of MD
solutions. On the one hand, the uncertainty in the AFM force measurement due to the
Brownian motion can be estimated as the difference between the forces applied on the
AFM sphere and its mirror image in the upper part of the symmetrical computational
domain (which fluctuates randomly for different AFM distances and speeds in
accordance with random nature of the Brownian force, as shown in Fig. S2 in
Supporting Information). On the other hand, for the case when the AFM tip is
approaching the mica surface slowly and is considered at a distance greater than the
hydration layer from the mica surface, the uncertainty in the force associated with the
Brownian motion can be estimated as a perturbation of Stokes’ law,
𝛿𝐹~6𝜋𝑅𝜇
∙
𝑉𝑎𝑟(
. Here is the dynamic viscosity of bulk water and is the thermal velocity
𝑉
′
)
𝜇
𝑉
′
fluctuation. The variance corresponds to averaging over the Grand-Canonical ensemble
in accordance with the Fluctuating Hydrodynamics theory [21], ,
𝑉𝑎𝑟
(
𝑉
′
)
=
𝑘
𝐵
𝑇
𝜌
0
𝑉
0
where is the characteristic control volume, is the Boltzmann
𝑉
0
=
4𝜋
𝑅
3
/3
𝑘
𝐵
constant, is the temperature, and is the bulk water density at equilibrium
𝑇
𝜌
0
conditions. For the large AFM distance of 0.77 nm and the low speed of ,
5
×
10
―
7
𝑚/𝑠
7
where Stokes’ law should be approximately valid, the theoretical estimate gives
. It can also be noted that the latter uncertainty associated with the
𝛿𝐹
=
0.2
3
4 𝑛𝑁
Brownian motion in the small molecular system considered is in the order of the
positive drag force predicted by the continuum Stokes’ theory at medium-high AFM
speeds ( at 0.3 m/s). This explains why the classical positive
𝐹
𝑆𝑡𝑜𝑘𝑒𝑠
=
0.283 𝑛𝑁
hydrodynamic drag effect is not resolved in the AFM force-distance curve (Fig. S2).
Furthermore, the estimated uncertainty due the Brownian motion is in good
agreement with the numerical simulation result, i.e., one half of the magnitude of the
force difference between the AFM sphere and its mirror image in the same conditions,
(Fig. S2 in Supporting Information). Hence, in the remaining part of the article,
0.3 𝑛𝑁
the numerically computed one half of the absolute difference between the computed
forces on the AFM tip and its mirror image will be used as an estimate of the standard
deviation of the statistical error for each tip speed and mica-tip distance.
Results of additional calculations with the FH/MD method for a test molecular
liquid system are included in Supporting Information (Fig.S5). The test system shares
a few pertinent features with the water flow over the AFM tip, such as flows through
the hybrid FH/MD interface region, whilst allowing a direct comparison with a
reference all-atom MD solution in this case. The comparison shows that the moving
multiscale interface does not lead to noticeable numerical artefacts in computing the
molecular diffusion coefficient.
Results
Properties of water close to the mica surface simulated with the FH/MD
solution are analysed separately in Fig. 2. To reduce the effect of the AFM tip, the
stationary regime is considered where the tip is fixed at a distance from the mica of
, which is much larger than the radius of van der Waals forces. From
𝑑
=
0.77 𝑛𝑚
Fig. 2 (a), it can be noted that the water atoms are absorbed next to the mica cations,
one per vacancy of the mica crystal, in accordance with the experimental and previous
numerical studies [32, 33]. Notably, the periodic structure formed by absorption of
8
water atoms in the mica structure is crucial for the hydration layer formation. For
example, as demonstrated in the atomic force microscope experiments using frequency
modulation [2] and equilibrium all-atom MD simulations [4, 34], the hydration layer is
responsible for the amplitude of the repulsive peak of the AFM force-distance curve at
when the tip velocity is zero (Fig. 4 (b) and Fig.S1 (a)). Further
𝑑
=
0.5
―
0.6 nm
discussion of the force-distance curve can be found in the Supporting Information.
Fig. 2 (b) shows the distribution of the number density of water in the (wall-normal)
z-direction from the mica surface. The density profile is calculated by averaging of the
mass of water molecules contained in each vertical bin between the mica surface and
the maximum height (0.05 nm) of the water column considered in the mica-tip gap over
2 ns. It should be reminded that, as in Fig.2a, the point of interest here is the water
density profile close to the mica and away from the AFM tip. Therefore, to exclude the
effects of non-uniform water confinement between the mica surface and the AFM tip,
the water volume corresponding to the square area of 10 nm x10 nm under the AFM tip
was excluded from the averaging when computing the water density profile on mica.
Notably, the resulting density profile has a prominent peak in the vicinity of the mica
surface followed by several smaller peaks and deeps. Again, this is in good qualitative
agreement with typical interfacial water structure observed experimentally.
Having validated the force-distance curve of the multiscale model for the
stationary AFM regime, and shown the known water layering near a mica surface, we
investigated the fate of water layering in a dynamic context. We performed a series of
simulations for a range of approach speeds of the AFM tips from
5
×
10
―
7
𝑚/𝑠
(typical speed of the AFM cantilever in experiments) to (hypothetical high-
30 𝑚/𝑠
speed AFM regime).
We find that though layering under the tip close to the mica is disturbed due to the
presence of the moving curved geometry. For the relatively large mica-tip distance,
d=0.77 nm, the higher the tip speed, the more layering could be observed, which is
especially notable at 30 m/s, which corresponds to an oscillation in the density profile
at around 0.6 nm in Fig. 3(a). Fig. 3 (b) shows the variation of water density with height
9
for a different AFM tip-mica separation distance d=0.53 nm and a range of approach
speeds, 0-30 m/s. For such small mica-tip distance, regardless of the tip speed, the
hydration layer cannot fit in the gap due to the proximity of the AFM tip.
Figure 2.A) The first mica layer in contact with the water. The interface is
composed of potassium ions from the mica (blue) and oxygen atoms adsorbed in
vacant potassium cavities in the mica (bigger red Ovac). Oxygen (smaller red Olatt),
silicon (grey) and aluminium (pink) make the rest of the mica lattice. B) Plot of
average number density of water atoms in planar 0.05 nm thick slices immediately
above the mica (and excluding the 10x10 nm square area below the sphere, (white))
as a function of the slice’ height, h in the z-coordinate direction. The potassium
ions’ plane of mica surface is taken as h = 0, and densities are plotted up to a height
of 1.6 nm. The mica-tip distance from the same reference is, = .
𝒅
𝟎.𝟕𝟕 𝒏𝒎
10
Figure 3. Averaged number density of water atoms in a planar slice below the
tip sphere (blue, 10 nm x 10 nm) as a function of the slice’ height, h in the z-
coordinate direction; the blue area in this figure corresponds to the white area in
Figure 2(b). Bin thickness used in the averaging is 0.05 nm. (a) and (b) correspond
to 0.77 nm and 0.53 nm mica-tip distance, d, respectively. Note the distinctly
layered structure of the water density distribution at 30 m/s tip speed in (a).
In turn, the dynamic layering of water near the tip leads to significant, non-intuitive
departures from the expected forces on the AFM tip in accordance with the continuum
hydrodynamics.
Simulation results for various tip/mica distances, h and AFM speeds, V are
summarised in Fig.4 (b) and (c). Fig.4 (a) provides a roadmap to these results in the
form of sketches. On the left, three typical tip/mica configuration regimes are presented:
(1) the equilibrium situation when the tip is moving at a low speed with hydration layers
formed on both the surfaces and (2) and (3) when the tip moving at high speed further
or closer to the mica, so that the hydration layer on the tip does not have time to form.
The second and the third columns show schematics of the force balances, where the
total force includes both the conventional interatomic, van der Waals and electrostatic
forces, which mostly depend on the tip-mica distance, and the collective effect of
moving water molecules (the hydrodynamic force). The second column corresponds to
the classical Stokesian hydrodynamics, which ignore the non-continuum effects such
as the hydration layer, thereby always predicts a positive total repulsion force on the
tip. In accordance with this model, the hydrodynamic drag of a small sphere moving in
a viscous liquid only depends on the liquid viscosity, sphere diameter and speed, but
critically is in the direction opposite the body velocity [35]. The third column shows
the actual balance of the forces in accordance with the simulation FH/MD results. In
accordance with the multiscale simulations, the hydrodynamic force is not only of a
different magnitude but also in the opposite direction compared to the Stokes theory.
The potential role of slip was excluded as a cause, as according to the continuum theory,
11
the drag force reduces by 1.5 times when the hydrophilic nonslip condition on the
spherical tip surface is replaced by the hydrophobic full-slip [36]. And regardless of the
wall boundary condition on a moving sphere down, the Stokes’ drag should always
correspond to a repulsive force. However, the actual simulation results show an
attractive hydrodynamic force. It can be hypothesised that the collective attractive force
may be associated with destroying the hydration layers around the fast-moving
microscopic body, which redistributes pressure around the tip with creating an upstream
rarefaction zone in the water volume by analogy with the ice breaker.
The total forces experienced on the nanometer tip when moving at different speeds
up to 30 m/s are shown with their error bars in Figure 4 (b). This dataset is used to
compute the hydrodynamic interactions by subtracting the well-defined interatomic
(van der Waals and electrostatic) interactions from the total force. We plot the resulting
hydrodynamic forces in Fig. 4 (c), showing they are non-monotonic, large in magnitude,
especially in the highest velocity case, 30 m/s, and opposite to their usual direction in
accordance with the Stokes model. Typical flow regimes (1), (2), and (3) discussed in
Fig. 4(a) are clearly marked on the force distribution curves.
12
Figure 4. A) Schematic of the sphere approaching the structured water layer
above the mica plane; the thickness of the interfacial water layers depends on the
mica-tip gap and the tip speed and comparison of the total tip force predicted by
classical hydrodynamic theory and FH/MD. For , at small speeds
𝒅
=
𝟎.𝟕𝟕 𝒏𝒎
(denoted by Num. 1), a hydration layer occurs on the tip surface. At high speed
(denoted by Num. 2), the hydration layer on the AFM tip breaks down because of
water molecules depleting from it to bulk thereby resulting in occurrence of
layered water structure, which corresponds to the single hydration layer on the
mica surface, while the hydration layer on the mica stays the same. As the tip
approaches the mica at high speed even closer (denoted by Num. 3) both the
hydration layer on the tip and the mica are broken down because of the
confinement. Centre and right: comparison of the total force based on the
predicted hydrodynamic force from Stokes’ law and the numerical simulation
result.
B) Total force on the AFM tip as a function of distance from the mica surface
for velocity 0, 5 × 10-7, 1, and 30 m/s. The error bar is the absolute difference of
total force between the AFM tip and its symmetrical image on the top. The total
force includes van der Waals (VdW) and electrostatic (El’cs) force from water and
mica substrate for the zero tip velocity. For other velocities, the total force also
includes hydrodynamic force because of the incorporated flow. C) The effective
Hydrodynamic drag force on the sphere, i.e. the difference between the computed
13
AFM forces in the stationary regime and the force in the flow regime measured
for the same mica-tip distance.
The answer to the puzzling behaviour of the hydrodynamic force lies in the
violation of the continuum theory due to breaking of the hydration layer on the moving
tip surface. When the tip-mica distance d is in the range of a small integer number of
hydration layers, the forces can no longer be described by the continuum theory. Further,
the formation of such hydration layer structures is associated with the collective
behaviour of water clusters, which require a small but finite time to form, in the order
of 10-11 s according to [37]. At the same time, the tip approaching the mica surface at a
speed of 30 m/s passes through the characteristic length scale of water structure, such
as the equilibrium intermolecular distance between the oxygen atoms, 0.31 nm in about
10-11 s. This shows that, for the very high approach speeds, there is little time for the
hydration layers to form on the AFM tip. This observation is consistent with the
decreased number density around the tip, as shown in Fig. S3 in the Supporting
Information. Furthermore, there is another important physical time scale for this
problem, which is related to water compressibility. As the water squeezed between the
mica surface and the approaching tip is pushed and displaced to the tip sides, the
propagating pressure waves need to cover the distance equal at least to the tip diameter
(10 nm). Assuming the pressure waves propagate in water at normal speed of sound
conditions (1500 m/s) the time for a single wave to traverse the tip is also similar 0.67
x 10-11 s. This means the water compressed under the approaching AFM tip at a speed
of 30 m/s does not have time to settle to the equilibrium pressure state, which would
require many passing-through cycles of the pressure wave. Altogether, this suggests
that, depending on the tip speed and size, the water in the sub-nano-meter gap between
the approaching tip and the mica can be highly compressible, thereby potentially
leading to the formation of strong non-divergent flows and rarefaction under the tip. In
turn, this generates a negative pressure zone below the spherical tip, which cannot be
explained by the classical Stokes model that assumes water incompressibility. The
compressibility of water is another potential cause for the oscillation in the water
14
density at the 30 m/s tip speed, as shown earlier in Fig. 3 (a). In accordance with the
numerical solution, the characteristic size of this oscillation, which occurs close to the
AFM tip surface, is about 0.2 nm. By combining this space length with the tip speed
(30m/s), one obtains the corresponding time scale of the oscillation, 0.67 10-11 s, which
is remarkably close to the water compressibility time scale discussed above.
Conclusions
In conclusion, we demonstrate a hybrid continuum - molecular dynamics (FH/MD)
method capable of simulating 1472766 atoms, in a 7700 nm3 box size system for 2 to 3
ns with MD-like precision, while the outer continuum part of the model, which drives
the continuum bulk flow effects is many orders of magnitude larger. Hence, the key
feature of the multiscale solution used here is that it allows bridging the gap in time
scales between the fast atoms/ thermal fluctuations and the slow continuum/ AFM
motion.
Notably, the cantilever structure above the AFM tip in the multiscale simulation
was approximated by a massless spring model consistently with the molecular
dynamics literature [16, 27]. This is a reasonable approximation because it is equivalent
to focusing on the localized flow effects around the microscopic tip, which exhibit the
non-Stokesian behaviour due to the breach of the continuity assumption at high-speed
AFM regimes. These effects are treated as a perturbation to the continuum bulk flow
imposed away from the moving AFM tip. At the same time, the size of the full AFM
system does not influence the simulation time significantly, as any increases beyond
the immediate vicinity of the mica and the tip are treated only by continuum mechanics.
This is because: (1) majority of the atomistic forces are short ranged (a few nanometers)
and (2) the important length scales of the problem such as the AFM tip radius and the
distance between the AFM tip and the substrate wall are 5 nanometer and a sub-
nanometer, respectively. Therefore, in static conditions, the entire AFM system could
be modelled at all atom resolution within a modest MD box at equilibrium boundary
conditions. What cannot be modelled by pure MD methods is the slow motion of the
AFM tip, which generates substantial bulk-flow hydrodynamic effect on the boundary
15
conditions of the water atoms surrounding the AFM tip. Because this hydrodynamic
effect is much slower than thermal fluctuations, one would require a prohibitively long
time averaging/ MD simulation times to separate this collective effect from thermal
noise. At the same time, the simulation of the bulk-flow effects using a continuum fluid
dynamics method is many orders of magnitude faster than the equivalent all-atom MD
model, and which speed-up is utilised in the suggested multiscale model. For a 5 nm
sphere, 0.5 - 0.77 nm from a mica surface, moving towards it at 1 - 30 m/s, we show
that the continuum Stokes’ law expression for drag force fails spectacularly. Rather
than a drag force opposite the direction of motion, the effective hydrodynamic force is
in the direction of the motion, and can be significantly larger than the respective
electrostatic and van der Waals contributions. The collective force of water molecules
deviates from the continuum, where the time to rearrange molecules during the high-
speed AFM tip motion falls below the time of formation of water clusters and the time
for the water density around the tip to equilibrate. With decrease in tip sizes and higher
frequencies current high-speed AFM (HS-AFM) experiments are starting to approach
velocities considered here. FH/MD simulations would be critical for interpreting the
force components once the operation velocity in future advanced H-S AFM
experiments reaches 1 m/s.
The current simulation results can be viewed as a high-speed extension of recent
experimental observations from [38], which revealed that the atomic-scale features
observed in force-distance curves in AFM experiments reflect the inhomogeneous
perturbation introduced by the tip and the substrate in the liquid.
Supporting Information
FH/MD method and its validation for the force-distance AFM curve; Validation of the
FH/MD method for a test molecular system with moving the atomistic/continuum
interface.
Acknowledgements
16
The work of F.L. was supported by the China Scholarship Council (CSC). I.A.K.
and S.A.K. gratefully acknowledge the funding under the European Commission Marie
Skłodowska-Curie Individual Fellowship Grant No. H2020-MSCA-IF-2015-700276
(HIPPOGRIFFE). S.S. gratefully acknowledge EPSRC fellowship EP/R028915/1. The
work was also supported by European Commission in the framework of the RISE
program, Grant No. H2020-MSCA-RISE-2018-824022-ATM2BT.
Corresponding author: s.karabasov@qmul.ac.uk
References
(1) Umeda, K.; Zivanovic, L.; Kobayashi, K.; Ritala, J.; Kominami, H.; Spijker,
P.; Foster, A. S.; Yamada, H. Atomic-resolution three-dimensional hydration structures
on a heterogeneously charged surface. Nat. Commun 2017, 8 (1), 2111. DOI:
10.1038/s41467-017-01896-4.
(2) Fukuma, T.; Ueda, Y.; Yoshioka, S.; Asakawa, H. Atomic-Scale Distribution
of Water Molecules at the Mica-Water Interface Visualized by Three-Dimensional
Scanning Force Microscopy. Phys. Rev. Lett. 2010, 104 (1), 016101. DOI:
10.1103/PhysRevLett.104.016101.
(3) Argyris, D.; Ashby, P. D.; Striolo, A. Structure and Orientation of Interfacial
Water Determine Atomic Force Microscopy Results: Insights from Molecular
Dynamics Simulations. ACS Nano 2011, 5 (3), 2215-2223. DOI: 10.1021/nn103454m.
(4) Tsukada, M.; Watanabe, N.; Harada, M.; Tagami, K. Theoretical simulation of
noncontact atomic force microscopy in liquids. J. Vac. Sci. Technol. B 2010, 28 (3),
C4C1-C4C4. DOI: 10.1116/1.3430541.
(5) Watkins, M.; Reischl, B. A simple approximation for forces exerted on an
AFM tip in liquid. J. Chem. Phys. 2013, 138 (15), 154703. DOI: 10.1063/1.4800770.
(6) Fukuma, T.; Reischl, B.; Kobayashi, N.; Spijker, P.; Canova, F. F.; Miyazawa,
K.; Foster, A. S. Mechanism of atomic force microscopy imaging of three-dimensional
hydration structures at a solid-liquid interface. Phys. Rev. B 2015, 92 (15), 155412. DOI:
10.1103/PhysRevB.92.155412.
(7) Gerrard, N.; Gattinoni, C.; McBride, F.; Michaelides, A.; Hodgson, A. Strain
Relief during Ice Growth on a Hexagonal Template. J. Am. Chem. Soc. 2019, 141 (21),
8599-8607. DOI: 10.1021/jacs.9b03311.
(8) Michaelides, A.; Morgenstern, K. Ice nanoclusters at hydrophobic
metal surfaces. Nat. Mater. 2007, 6 (8), 597-601. DOI: 10.1038/nmat1940.
(9) Wu, S.; He, Z.; Zang, J.; Jin, S.; Wang, Z.; Wang, J.; Yao, Y.; Wang, J.
Heterogeneous ice nucleation correlates with bulk-like interfacial water. Sci. Adv. 2019,
5 (4), eaat9825. DOI: 10.1126/sciadv.aat9825.
(10) Lu, H.; Huang, Y.-C.; Hunger, J.; Gebauer, D.; Cölfen, H.; Bonn, M. Role of
Water in CaCO3 Biomineralization. J. Am. Chem. Soc. 2021, 143 (4), 1758-1762. DOI:
10.1021/jacs.0c11976.
17
(11) Wang, X.; Kim, S. H.; Chen, C.; Chen, L.; He, H.; Qian, L. Humidity
Dependence of Tribochemical Wear of Monocrystalline Silicon. ACS Appl. Mater.
Interfaces 2015, 7 (27), 14785-14792. DOI: 10.1021/acsami.5b03043.
(12) Mondal, J. A.; Nihonyanagi, S.; Yamaguchi, S.; Tahara, T. Structure and
Orientation of Water at Charged Lipid Monolayer/Water Interfaces Probed by
Heterodyne-Detected Vibrational Sum Frequency Generation Spectroscopy. J. Am.
Chem. Soc. 2010, 132 (31), 10656-10657. DOI: 10.1021/ja104327t.
(13) Moilanen, D. E.; Spry, D. B.; Fayer, M. D. Water Dynamics and Proton
Transfer in Nafion Fuel Cell Membranes. Langmuir 2008, 24 (8), 3690-3698. DOI:
10.1021/la703358a.
(14) Zhao, Q.; Majsztrik, P.; Benziger, J. Diffusion and Interfacial Transport of
Water in Nafion. J. Phys. Chem. B 2011, 115 (12), 2717-2727. DOI:
10.1021/jp1112125.
(15) Khesbak, H.; Savchuk, O.; Tsushima, S.; Fahmy, K. The Role of Water H-
Bond Imbalances in B-DNA Substate Transitions and Peptide Recognition Revealed
by Time-Resolved FTIR Spectroscopy. J. Am. Chem. Soc. 2011, 133 (15), 5834-5842.
DOI: 10.1021/ja108863v.
(16) Rico, F.; Russek, A.; González, L.; Grubmüller, H.; Scheuring, S.
Heterogeneous and rate-dependent streptavidin–biotin unbinding revealed by high-
speed force spectroscopy and atomistic simulations. Proc. Natl. Acad. Sci. 2019, 116
(14), 6594-6601. DOI: 10.1073/pnas.1816909116.
(17) Valotteau, C.; Sumbul, F.; Rico, F. High-speed force spectroscopy:
microsecond force measurements using ultrashort cantilevers. Biophys. Rev. 2019, 11
(5), 689-699. DOI: 10.1007/s12551-019-00585-4.
(18) Banerjee, D.; Pal, S. K. Direct observation of essential DNA dynamics:
melting and reformation of the DNA minor groove. J. Phys. Chem. B 2007, 111 (36),
10833-10838.
(19) Andreatta, D.; Pérez Lustres, J. L.; Kovalenko, S. A.; Ernsting, N. P.; Murphy,
C. J.; Coleman, R. S.; Berg, M. A. Power-law solvation dynamics in DNA over six
decades in time. J. Am. Chem. Soc. 2005, 127 (20), 7270–7271.
(20) Brauns, E. B.; Madaras, M. L.; Coleman, R. S.; Murphy, C. J.; Berg, M. A.
Complex local dynamics in DNA on the picosecond and nanosecond time scales. Phys.
Rev. Lett. 2002, 88 (15), 158101.
(21) Korotkin, I.; Karabasov, S.; Nerukh, D.; Markesteijn, A.; Scukins, A.;
Farafonov, V.; Pavlov, E. A hybrid molecular dynamics/fluctuating hydrodynamics
method for modelling liquids at multiple scales in space and time. J. Chem. Phys. 2015,
143 (1), 014110.
(22) Markesteijn, A.; Karabasov, S.; Scukins, A.; Nerukh, D.; Glotov, V.;
Goloviznin, V. Concurrent multiscale modelling of atomistic and hydrodynamic
processes in liquids. Philos. Trans. Royal Soc. 2014, 372 (2021). DOI:
10.1098/rsta.2013.0379.
(23) Korotkin, I. A.; Karabasov, S. A. A generalised Landau-Lifshitz fluctuating
hydrodynamics model for concurrent simulations of liquids at atomistic and continuum
resolution. J. Chem. Phys. 2018, 149 (24), 244101. DOI: 10.1063/1.5058804.
18
(24) Li, F.; Korotkin, I.; Farafonov, V.; Karabasov, S. A. Lateral migration of
peptides in transversely sheared flows in water: An atomistic-scale-resolving
simulation. J. Mol. Liq. 2021, 337, 116111. DOI:
https://doi.org/10.1016/j.molliq.2021.116111.
(25) Heath, G. R.; Scheuring, S. High-speed AFM height spectroscopy reveals µs-
dynamics of unlabeled biomolecules. Nat. Commun 2018, 9 (1), 4983. DOI:
10.1038/s41467-018-07512-3.
(26) Uchihashi, T.; Kodera, N.; Ando, T. Guide to video recording of structure
dynamics and dynamic processes of proteins by high-speed atomic force microscopy.
Nat. Protoc. 2012, 7 (6), 1193-1206. DOI: 10.1038/nprot.2012.047.
(27) Hu, X.; Nanney, W.; Umeda, K.; Ye, T.; Martini, A. Combined Experimental
and Simulation Study of Amplitude Modulation Atomic Force Microscopy
Measurements of Self-Assembled Monolayers in Water. Langmuir 2018, 34 (33),
9627-9633. DOI: 10.1021/acs.langmuir.8b01609.
(28) Heinz, H.; Koerner, H.; Anderson, K. L.; Vaia, R. A.; Farmer, B. L. Force
Field for Mica-Type Silicates and Dynamics of Octadecylammonium Chains Grafted
to Montmorillonite. Chem. Mater. 2005, 17 (23), 5658-5669. DOI:
10.1021/cm0509328.
(29) Heinz, H.; Vaia, R. A.; Farmer, B. L. Interaction energy and surface
reconstruction between sheets of layered silicates. J. Chem. Phys. 2006, 124 (22),
224713. DOI: 10.1063/1.2202330.
(30) Heinz, H.; Suter, U. W. Surface Structure of Organoclays. Angew. Chem. Int.
Ed. 2004, 43 (17), 2239-2243. DOI: 10.1002/anie.200352747.
(31) Heinz, H.; Lin, T.-J.; Kishore Mishra, R.; Emami, F. S. Thermodynamically
Consistent Force Fields for the Assembly of Inorganic, Organic, and Biological
Nanostructures: The INTERFACE Force Field. Langmuir 2013, 29 (6), 1754-1765.
DOI: 10.1021/la3038846.
(32) Martin-Jimenez, D.; Garcia, R. Identification of Single Adsorbed Cations on
Mica–Liquid Interfaces by 3D Force Microscopy. J. Phys. Chem. Lett. 2017, 8 (23),
5707-5711. DOI: 10.1021/acs.jpclett.7b02671.
(33) Park, S.-H.; Sposito, G. Structure of Water Adsorbed on a Mica Surface. Phys.
Rev. Lett. 2002, 89 (8), 085501. DOI: 10.1103/PhysRevLett.89.085501.
(34) Kobayashi, K.; Liang, Y.; Amano, K.-i.; Murata, S.; Matsuoka, T.; Takahashi,
S.; Nishi, N.; Sakka, T. Molecular Dynamics Simulation of Atomic Force Microscopy
at the Water–Muscovite Interface: Hydration Layer Structure and Force Analysis.
Langmuir 2016, 32 (15), 3608-3616. DOI: 10.1021/acs.langmuir.5b04277.
(35) Brenner, H. The slow motion of a sphere through a viscous fluid towards a
plane surface. Chem. Eng. Sci. 1961, 16 (3), 242-251. DOI:
https://doi.org/10.1016/0009-2509(61)80035-3.
(36) Luo, H.; Pozrikidis, C. Effect of surface slip on Stokes flow past a spherical
particle in infinite fluid and near a plane wall. J. Eng. Math. 2008, 62 (1), 1-21. DOI:
10.1007/s10665-007-9170-6.
(37) O. L. Lange, L. K., E.-D. Schulze. Water and Plant Life Problems and
Modern Approaches; Springer, 1976.
19
(38) Hernández-Muñoz, J.; Uhlig, M. R.; Benaglia, S.; Chacón, E.; Tarazona, P.;
Garcia, R. Subnanometer Interfacial Forces in Three-Dimensional Atomic Force
Microscopy: Water and Octane near a Mica Surface. J. Phys. Chem. C 2020, 124 (48),
26296-26303. DOI: 10.1021/acs.jpcc.0c08092.
TOC Graphic
