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7 2D-Confinement of Fluids 2D-confinement of fluids, Wetting, and Spreading

7 2D-Confinement of Fluids 2D-confinement of fluids, Wetting, and Spreading

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7.7



2D-Confinement of Fluids, Wetting, and Spreading



347



7.7.1 Phase Transitions Induced by Nanoconfinement

of Liquid Water

The structure and properties of water in nanometerscale confinement are important

in numerous fields including biological self-assembly, fuel-cell technology, nanotribology, etc. (see [7.125]). Molecular dynamics (MD) simulations of water confined

by two parallel hydrophobic walls with a wall–wall separation of 0.8 nm show at

T = 300 K and low water density (ρ = 0.8 g/cm2 ) two peaks in the density ρ (z)

(Fig. 7.33a), whereas at high density (ρ = 1.15 g/cm2 ) a transition to three peaks

occurs. In the latter case, the molecules next to the walls arrange in a crystal-like

structure – templated by the wall – and do not diffuse (Fig. 7.33b). The central slab

is liquid-like (Fig. 7.33c) without long-range order of the molecules, which readily

diffuse. When the water density is decreased to ρ = 1.05 g/cm3 a few vacancy

defects are observed in the crystal-like structure (Fig. 7.33d), which disappears in an

order-to-disorder or “melting” transformation at ρ = 0.8 g/cm3 , where the bilayer

is stable.

The dynamics of nanoconfined water, as studied by atomic force microscopy

(AFM), are found to be orders of magnitude slower than those of bulk water [7.126]

and comparable to the dynamics measured in supercooled water at 170–210 K.



Fig. 7.33 (a) Molecular dynamics simulation of the density profiles ρ (z) of water between

two walls in a distance d = 0.8 nm for densities above (ρ = 1.15, 1.05 g/cm3 ) and below

(ρ = 0.8 g/cm3 ) the phase transition from a trilayer to a bilayer. (b) and (c) Snapshots of the

system at ρ = 1.15 g/cm3 , d = 0.8 nm, and t = 500 ps with a layer close to a wall (b) which is

ordered and a central slab (c) which is disordered. (d–f) Same as (b) for the densities 1.05, 0.8,

and 0.5 g/cm2 , respectively. As the density decreases, first vacancy-like defects are introduced into

the ordered structure next to the wall. Then, an order–disorder structural change occurs. (Reprinted

with permission from [7.125]. © 2009 American Physical Society)



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7 Nanomechanics – Nanophotonics – Nanofluidics



7.7.2 Fluid Flow and Wetting

Steady-state flow of incompressible fluids in a channel width 2 h, driven for

example, by gravity ρg or a pressure gradient dP/dy, can be described by the Navier–

Stokes equations. The solution for the velocity in the direction of flow, y, as a

function of the distance from the wall, z, has a parabolic profile (Fig. 7.34) given by

Uy (z) = (ρg/2η) [δ + h]2 − z2

where η is the viscosity and δ, the slip length, is the distance into the wall at which

the velocity extrapolates to zero. Conventionally, the slip length is assumed to be

zero for wetting fluids but it may be non-zero for non-wetting fluids, which for

very small channels can significantly enhance the fluid flow. Integrating the velocity profile over the cross-sectional area of, e.g., a pipe gives for no-slip boundary

conditions the Hagen–Poiseuille law for the flux through the pipe, yielding a flow

rate proportional to the pressure difference between the tube ends and to the fourth

power of the tube’s internal radius, and inversely proportional to the tube length and

to the viscosity of the fluid.

By taking into account the force exerted by the wall on the fluid, the evolution

of the boundary conditions with the wetting properties can be reproduced [7.128,

7.129]. In particular, a transition from a no-slip boundary condition in a wetting

situation (contact angle θ ∼ 0◦ ) to a partial-slip boundary condition with a slip

length of some tens of nanometers in a non-wetting situation (θ ∼ 120◦ ) is obtained

[7.128].



Fig. 7.34 Parabolic velocity

profile UY (z) for Poiseuille

flow in a capillary of width

2 h. (Reprinted with

permission from [7.127].

© 2007 Nature Publishing

Group)



7.7.3 Superhydrophobic Surfaces

Non-wettability or superhydrophobicity is a well-known effect in many natural

materials, such as the “lotus effect” of lotus leaves (see Sect. 11.9). Artificial



7.7



2D-Confinement of Fluids, Wetting, and Spreading



349



Fig. 7.35 (a) Scanning electron micrograph (SEM) of a superhydrophobic Sb2 O3 film. (b–d)

Optical images of water drops of different sizes (17–22 μl) on the as-synthesized Sb2 O3 film (b),

the water droplet profile (4 μl) on the Sb2 O3 film (c) and a sliding water drop (4 μl) on the surface

tilted by less than 5◦ (d). (Reprinted with permission from [7.130]. © 2008 Institute of Physics)



superhydrophobicity has been demonstrated by antimony oxide (Sb2 O3 )

micro-nanoscale hierarchical surface structures [7.130]. For the preparation of

the nanocrystalline Sb2 O3 surface coating a solution of hexadecylamine (HDA,

CH3 (CH2 )15 NH2 ) is added to the Sb(OCH(CH3 )2 )3 precursor yielding cilium-like

nanostructures 10–30 nm in diameter and 5–30 nm in length with interspaces of

5–15 nm (see Fig. 7.35a). Water drops on the surface (Fig. 7.35) show superhydrophobic behavior with a contact angle of about 159◦ ± 2◦ (Fig. 7.35c). Moreover,

the water drop is hardly able to stick to the surface, as indicated by a small sliding

angle (less than 5◦ ), allowing water drops to roll off quite easily (Fig. 7.35d). By xray photoelectron spectroscopy (XPS) a substantial surface contribution of carbon

is detected. From this it is concluded that the presence of alkyl chains contributes to

the hydrophobicity of the microscale–nanoscale Sb2 O3 structures [7.130].



7.7.4 Liquid Spreading Under Nanoscale Confinement

The macroscopic spreading of a liquid droplet on a solid surface proceeds via the

formation of a very thin “precursor film” ahead of the droplet’s contact line (see

[7.131]) where the length of the precursor is predicted by de Gennes [7.132] to be

Lp = a S γ Ca−1



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7 Nanomechanics – Nanophotonics – Nanofluidics



where S = γSv − γSL − γ is the spreading parameter, with γ the liquid surface tension and γ SV and γ SL the solid–vapor and solid-liquid interfacial energies,

respectively. The capillary number Ca = η V/γ depends on the viscosity η and the

spreading velocity V of the liquid, a is a molecular length scale. The film thickness h is expected to decrease as the inverse of the distance from the apparent

droplet contact line until the film is “truncated” at a thickness h0 = a 3γ 2S

significantly larger than a molecular size. The truncation region corresponds to the

microscopic contact line. The investigation of these spreading processes is important

for emerging technologies aiming at the formation and manipulation of smaller and

smaller amounts of liquid.

The spreading of the non-volatile liquid squalone (C30 H62 ) along wettable

nanostripes embedded in a non-wettable (CH3 terminated) surface of an octadecyltrichlorosilane (OTS) monolayer on a Si wafer is studied by atomic force

microscopy (AFM) [7.131] (see Fig. 7.36). The maximum liquid thickness turns

out to depend on the stripe width (Fig. 7.36b). The film is thinner (by about 0.4 nm)



Fig. 7.36 (a) Temporal evolution of a transverse cross-sectional AFM profile of liquid nanostripes

(lateral scale bar 50 s, vertical scale bar 400 nm). (b) Height of the liquid versus time on the narrow

(150 nm wide) and wide (650 nm) stripe, as extracted from (a). (c) The length Lp of the precursor

film versus the inverse of the spreading velocity V0 . From the linear fit, the spreading parameter

S = 3 × 10−4 N/m can be estimated. (Reprinted with permission from [7.131]. © 2009 American

Physical Society)



7.8



Fast Transport of Liquids and Gases Through Carbon Nanotubes



351



on the narrow line (150 nm) compared to the wider line (650 nm). This is the effect

of the nanoscale lateral confinement on the fluid morphology. The pinning of the

contact line along the boundaries of the wettable stripe imposes a curvature to the

liquid’s free surface. Consequently for sufficiently narrow stripes, the film height

is expected to decrease with the stripe width [7.133]. From the measurements carried out at various velocities V0 , it is found that Lp scales as 1/V0 (Fig. 7.36c), as

predicted by de Gennes’ model above.

Water molecules form a locally ordered superstructure on TiO2 anatase (101), a

most efficient photocatalyst for the dissociation of water [7.134].



7.8 Fast Transport of Liquids and Gases Through

Carbon Nanotubes

Molecularly smooth hydrophobic graphitic walls and nanoscale inner diameters

of carbon nanotubes (CNTs) give rise to ultraefficient transport of water and gas,

by orders of magnitude faster than through other pores of similar size. The water

transport mechanism has a similarity to the transport mechanisms of biological

membrane ion channels. The combination of transport efficiency and selectivity

makes CNT membranes a promising technology for future water desalination, water

purification, nanofiltration, and gas separation applications [7.135].



7.8.1 Limits of Continuum Hydrodynamics at the Nanoscale

Advances in nanoscience, which have pushed the dimensions of synthetic objects

closer to the molecular scale, pose a fundamental question: does the continuumbased description of fluid flow, governed by the Navier–Stokes equations, still work

in a situation where the reduction in size causes the variables to vary appreciably over the molecular length. The size of a water droplet that still can be treated

as a continuum object with less than 1% statistical fluctuation of a property can

be estimated to contain 104 water molecules with a droplet diameter of ∼ 27

molecules, or ∼ 6.5 nm. According to molecular dynamics (MD) simulations (see

[7.135]), Navier–Stokes theory predictions are approached by channel widths of ten

molecular diameters. Flow through CNTs with lateral dimensions of less than 2 nm

evidently falls in the field of non-continuum fluidics, or nanofluidics.



7.8.2 Water Transport in CNTs

It is surprising that hydrophilic liquids, especially water, enter and fill very narrow

and hydrophobic CNTs. MD simulations (see [7.135]) confirm this behavior and

demonstrate that water molecules confined in such a small space, form a single-file



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7 Nanomechanics – Nanophotonics – Nanofluidics



Fig. 7.37 (a) Structure of the hydrogen-bonded water chain inside a carbon nanotube (CNT)

[7.136]. (b) High-resolution transmission electron micrograph (HRTEM) of the cross-section of

a double-walled carbon nanotube (DWNT) in a Si3 Nx membrane [7.137]. (Reprinted with permission from [7.136] (a) and [7.137] (b). © 2001 Nature Publishing Group (a) and © 2006 AAAS (b))



configuration (Fig. 7.37a) that is unseen in bulk water. Such 1D hydrogen-bonded

structures are highly reminiscent of the water wires observed in the biological

aquaporin water-transporting channels (see Sect. 11.5) with also a hydrophobic

lining inside (see [7.135]). In CNTs, the friction at the channel walls is so low

that the water transport is no longer governed by Hagen–Poiseuille flow, but

nanotube entrance and exit. The calculated water transport rates approach 5.8

water molecules per nanosecond per nanotube, similar to the transport rates in

aquaporins (see [7.135]), yet – due to the different dimensions of the two systems – one cannot imply that the same mechanism is responsible for CNTs

and aquaporins.

The experimentally determined water transport rates in sub-2 nm double wall

carbon nanotube (DWNT, Fig. 7.37b) membranes reveal a flow enhancement that

is at least 2–3 orders of magnitude faster than no-slip, hydrodynamic flow calculated using the Hagen–Poiseuille equation whereas the calculated slip length for

sub-2 nm CNTs is as large as hundreds of nanometers, which is almost three orders

of magnitude larger than the pore size [7.135]. The measured water flux compares

well with that predicted by MD simulations (see [7.135]).

MD simulations further show that water infiltration into and defiltration out of

CNTs is influenced by gas molecules [7.138].



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