Supplementary Information
Yajie Lei and Yongsheng Leng
Simulation Methods
1. Liquid-vapor molecular dynamics (LVMD) simulations
Liquid-vapor molecular dynamics (LVMD) simulations have been performed in
previous studies. These include capillary wave, surface tension and other interesting
phenomena at water liquid-vapor interfaces
1-5
. In our LVMD simulation, the simulation
system consists of a liquid droplet of argon and two face-centered cubic (f.c.c.) crystal
walls with (111) surfaces. For liquid argon, force parameters in simple Lennard-Jones (LJ)
atomic potential are given in the literature ( = 0.2381 kcal/mol and  = 0.3405 nm) 6.
Each confining wall is composed of a central wall that has the same interaction strength
as the argon-argon interaction (i.e., wf = ff = , where w and f stand for “wall” and
“fluid”, respectively), and two side walls on which the wall-fluid interaction is decreased
to ¼  to keep argon molecules in this region in a liquid phase. Without the loss of
generality, we assume wf = and the f.c.c. wall lattice constant a = 0.5405 nm, giving
the first neighbor distance of f.c.c. wall particle as a/(2)1/2 = 0.3822 nm. This value is in
fact the argon LJ equilibrium distance6, i.e., (2)1/6 . Thus, the ratio of the wall particle
radius to that of argon equals 1.0 (This commensurate wall geometry is slightly different
from those in early studies7, 8 based on wall-fluid equal-density parameters). We find that
different wf and wall lattice constant a yield qualitatively similar force profile (see
Discussion 1), though the critical layer number, nc, below which the liquid-to-solid phase
transition is different. A standard cutoff distance of 2.5 is used in simulations. The sizes
of MD simulation box along the x-, y-, and z-directions are 76.59, 11.42, and 3.71 nm,
respectively. The central wall of interest has a length of 10.7 nm along the x-direction.
The simulation system contains 4319 liquid argon atoms and 8823 solid wall atoms. The
Nose-Hoover thermostat 9, 10 is used to control the temperature at 85K, corresponding to a
liquid state of argon.
2. Determination of vapor pressure in LVMD simulation
1
In our previous study11, we showed that when a liquid droplet confined between two
solid walls becomes quite elongated during squeeze out, artificial oscillatory waves in the
lateral direction arise if the conventional pressure-control technique is used. This problem
comes from the scaling of MD particle coordinates and simulation box lengths in the
elongated direction. By introducing vapor phases around the liquid phase, the lateral
pressure can be maintained to a very low vapor pressure without any coordinate scaling
that leads to artificial oscillatory waves.
Figure S1 shows a snapshot of the equilibrium configuration of the molecular system
at zero normal force. The separation between the two surfaces stabilizes around 4.67 nm.
The liquid film is slightly squeezed out, while one or two layers of fluid molecules
crystallize near central wall surfaces, consistent with early results in MD simulations 8.
This typical one- or two-layer of liquid molecules crystallized near wall surface are also
seen for different wall structure (not shown in the figure).
FIG. S1. Snapshot of the equilibrium configuration of an argon droplet confined between two
solid walls. Liquid and vapor phases are clearly seen in the confined region. The lateral
dimension of confining wall is 76.59  3.71 nm. Argon molecules are in blue, the central and side
wall atoms are in red and grey.
The vapor pressures at different layer thickness are calculated based on a simple
thermodynamic definition of pressure, which is given by 12
pg  f B 2 mkT
(S1)
where fB is the frequency or the number of particles going through a unit cross-sectional
area per unit time, and m, k, and T are the mass of argon (m = 6.632×10-23g), the
Boltzmann constant, and temperature, respectively. Equilibrium MD runs for different
layers are preformed to calculate fB in a time span of 0.5 ns.
2
Under extreme confinement, when a large portion of argon droplet is squeezed out
during normal compression, the far-end argon molecules have to be removed from the
simulation system to maintain a proper liquid-vapor interface. We find that the removal
of these argon molecules does not influence the static solvation pressures and subsequent
layer transitions. Figure S2 shows the variation of vapor pressure versus wall separation.
The mean value around 15 atm is a vanishingly low pressure compared with the normal
repulsive pressures of solid phase, indicating that the layered solid phase is a strongly
anisotropic material under nanoconfinement.
FIG. S2. Variation of vapor pressure versus wall distance. The numbers in the figure indicate
different layers of confined film.
Discussion
1. Comparison between results using different wf and wall lattice constant a
In order to see how the wall-fluid geometric parameter wf and wall lattice constant a
influence the force oscillation and phase transition, we further investigate wf = 0.887
and a = 0.408 nm (the same geometric parameters used in previous studies7), while
keeping the wall-fluid interaction wf =. Note that in this case, the first neighbor distance
of f.c.c. wall particle is reduced to a/(2)1/2 = 0.288 nm, corresponding to a very high
number density of wall ( = 0.0588 Å-3) (the number density of liquid argon at 85 K is
0.02 Å-3). The ratio of the wall particle radius to that of argon is also decreased to ~ 0.75.
Figures S3 shows force oscillations and structure factor variations for different geometric
parameters. The force profiles are qualitatively similar (Fig. S3 A). However, the critical
3
layer number, nc, at which the liquid-to-solid phase transition is different: we find that
this transition happens at nc = 7 for commensurate contact, and nc = 5 for
incommensurate contact. This is also shown in Fig. S3 B the very low structure factors at
larger distances (n = 6 and 7) for the incommensurate case. Figure S4 shows the
snapshots of the solid phases of the commensurate and incommensurate systems at n = 5
layers at the force minima (negative pressure). The figure clearly shows that the liquid-tosolid phase transition is confinement-induced, rather than pressure- or surface-crystallineinduced, although the commensurate contact usually promotes the earlier transition.
A
B
FIG. S3. Comparisons of (A) force oscillations and (B) structure factor variations for different
wall-fluid geometric parameters (wf) and wall lattice constants (a). Number n indicates different
layers of confined film.
4
A
B
FIG. S4. The 5-layered solid phases of (A) commensurate system at D=19.1 Å and p =  0.5 MPa,
and (B) incommensurate system at D = 18.7A and p =  16.9 MPa.
2. Force profiles using different spring force constants
In order to understand how the spring force constant influence the force profiles, we
further carried out normal approach and retraction simulations using additional two
spring constants. The results from very soft (ky = 7.5 N/m) and very hard (ky = 1500 N/m)
springs, as well as that from ky = 150 N/m, are shown in Fig. S5. The figure clearly shows
that the unstable regions explored by different springs are quite different. Soft spring (ky =
7.5 N/m) yields a slightly early jump (a*  b*), while the force minima are all repulsive,
which can be calculated as Qk = Pk – ky, where Pk and Qk are the k-th force peak and
valley, respectively, and  is one monolayer thickness of the film (roughly 0.357 nm,
which is a little larger than the argon LJ parameter ). It turns out that the spring force
constant (ky) plays a dominant role in determining the depths of force valleys during one
swipe of approach. As shown in Fig. S5, the force profile measured by very hard spring
(ky = 1500 N/m) during normal approach almost touches down the maximum adhesion
forces obtained from retraction simulation with ky = 150 N/m. This indicates that very
hard springs can access more force regions in which the spring force constants are greater
than the force profile gradient F/D. This result is consistent with early analysis in
surface force measurements
13
. However, the force signal using hard spring are much
5
noisy than that of soft spring. In other words, the soft spring gives smoother force profile
with higher force resolution.
The force profile in retraction using the soft spring also has an early jump (the blue
line corresponding to c*  d* or 3  4 transition in Fig. S5). Adhesion force is decreased
by ky during this transition. MD animation analysis shows that this unstable transition
proceeds so quickly that liquid film has no time to fully reorganize into a solid phase. The
transition is simply from a three-layer solid phase to a four-layer liquid phase, in contrast
to the case of ky = 150 N/m, in which repulsive force was obtained in solid phase (Fig. 3B
and 3D).
FIG. S5. Comparison of the force profiles measured by different springs. The different spring
force constants give qualitatively the same force maxima. The force profile measured by the soft
spring has higher force resolution but less accessible force regions due to unstable jumps, while
the force profile measured by the hard spring has lower force resolution but larger accessible
force regions. Force segments ab and cd correspond to ky = 150 N/m unstable transitions during
normal approach and retraction, while a*b* and c*d* correspond to the case of ky = 7.5 N/m. The
numbers 2 - 7 indicate different layers of confined film.
3. Structures of the solid phases at low and high pressures before the n  n - 1 layer
transition
When the solid phase of confined film is formed, further compression of the film
before n  n - 1 layer transition results in more compact ordered phase. This is shown in
Fig. S6 for n = 6 commensurate system, in which the in-plane number-density
distribution of argon increases as the pressure is increasing. The less ordered and more
6
ordered configurations shown in Fig. S7 correspond to the solid dots in the approach
force profile in Fig. 1B.
FIG. S6. The in-plane number density distribution of argon increases with the pressure before the
n  n - 1 layer transition.
7
FIG. S7. Molecular configurations of solidified films at different layer thickness. Less ordered
structures under low pressures and more ordered structures under high pressures are shown in
panels a and b, respectively.
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