Transport Mechanisms in Capillary Condensation of Water at a Single

Article pubs.acs.org/Langmuir

Transport Mechanisms in Capillary Condensation of Water at a Single-Asperity Nanoscopic Contact Lucel Sirghi* Faculty of Physics, “Al. I. Cuza” University, blvd. Carol I, no. 11, 700506 Romania S Supporting Information *

ABSTRACT: Transport mechanisms involved in capillary condensation of water menisci in nanoscopic gaps between hydrophilic surfaces are investigated theoretically and experimentally by atomic force microscopy (AFM) measurements of capillary force. The measurements showed an instantaneous formation of a water meniscus by coalescence of the water layers adsorbed on the AFM tip and sample surfaces, followed by a time evolution of meniscus toward a stationary state corresponding to thermodynamic equilibrium. This dynamics of the water meniscus is indicated by time evolution of the meniscus force, which increases with the contact time toward its equilibrium value. Two water transport mechanisms competing in this meniscus dynamics are considered: (1) Knudsen diffusion and condensation of water molecules in the nanoscopic gap and (2) adsorption of water molecules on the surface region around the contact and flow of the surface water toward the meniscus. For the case of very hydrophilic surfaces, the dominant role of surface water transportation on the meniscus dynamics is supported by the results of the AFM measurements of capillary force of water menisci formed at sliding tip−sample contacts. These measurements revealed that fast movement of the contact impedes on the formation of menisci at thermodynamic equilibrium because the flow of the surface water is too slow to reach the moving meniscus.

1. INTRODUCTION Water is the substance of which peculiar properties affect every aspect of life on Earth. Water properties are essential in biological1 and geological processes,2 in friction and adhesion3 phenomena, in mechanical resistance of construction materials,4 etc. Recent developments of nanotechnology5 have drawn attention to the nanoscale properties of water.6 Such properties are relevant in studies of powder adhesion,7 water adsorption in porous materials,8 nanolithography,9 nanotribology,3 microelectromechanic systems (MEMS),10 colloidal physics,11 and biochemistry.12 One of the most studied water phenomenon at nanoscale is the water vapor condensation in small gaps between hydrophilic solid surfaces, i.e., the capillary condensation of water.13 The phenomenon is noticeable in atomic force microscopy14 (AFM) and surface force apparatus15 measurements in humid air, as large attractive forces of water menisci formed between probe and sample surfaces occur. Although the phenomenon involves formation of very small water menisci, the common theoretical description of the phenomenon is based on the macroscopic Kelvin theory of thermodynamic equilibrium of curved liquid−vapor interfaces (menisci).16 Unlike the large liquid−vapor interfaces, which in the gravitational field are flat, the capillary liquid−vapor interfaces are curved and their curvatures affect their thermodynamic equilibrium value of vapor pressure. For concave menisci, this is happening because escape of molecules is harder from the curved menisci (due to stronger interaction with neighbor molecules) than from the © 2012 American Chemical Society

flat ones. Thus, concave water menisci may exist in thermodynamic equilibrium with undersaturated vapor. In this case, the thermodynamic equilibrium is expressed by the well-known Kelvin equation,16 which establishes the equilibrium value of meniscus curvature radius, rk. The Kelvin equation expresses the equality between the pressure reduction in the liquid phase, RT/Vm ln(p/ps), and the Laplace pressure, γw/r. Here, R is the ideal gas constant, T, the absolute temperature, Vm, the molar volume of the liquid water, γw, the surface tension of water, r, the meniscus total curvature radius, p, the water vapor pressure in air, and ps, the water saturation pressure (p < ps). Thus, for water vapor in air at room temperature the meniscus curvature radius at thermodynamic equilibrium with vapor is16

r k [nm] =

0.54 ln(p /ps )

(1)

For p < ps, the eq 1 foresees negative values of rk, which at moderate values of air relative humidity (RH = p/ps) are under 1 nm. This is why such water menisci form at nanoscopic values of the separation distance between hydrophilic solid surfaces. The value of rk can be interpreted as the value of the capillary meniscus curvature radius at which the rate of condensation is equal with the rate of evaporation, the water meniscus being Received: July 27, 2011 Revised: December 10, 2011 Published: January 9, 2012 2558

dx.doi.org/10.1021/la202917d | Langmuir 2012, 28, 2558−2566

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stationary. Concave menisci with |r| < |rk| grows because the rate of condensation is larger than the rate of evaporation, while concave menisci with |r| > |rk| evaporates because the rate of condensation is smaller than the rate of evaporation. While the Kelvin theory of capillary condensation describes well the menisci at thermodynamic equilibrium with the vapor,17 it says nothing about meniscus nucleation and evolution toward thermodynamic equilibrium. The dynamics of water menisci after their initial formation has been studied directly by surface force apparatus18 and AFM19,20 and indirectly by time and humidity variation of the avalanche angle of a granular medium.21 The slow dynamics of capillary menisci formed at multiple asperity contacts between microscopically rough surfaces has been explained by the theory of thermal activation of the condensation.22,23 Recently, the AFM was used to study the dynamics of water menisci formation at a multiple asperity contact between a silica microsphere and a flat silica surface by AFM measurements of capillary adhesion force values at different contact time values.24 The measurements showed an increase of the capillary force with the contact time. This behavior of capillary force has been attributed to growth of the water annulus around the microcontact. However, it has been found that the characteristic time of this evolution is several decimal orders larger than that predicted by the diffusion theory of capillary condensation in the microscopic gap. References 19 and 20 have also reported a time evolution toward equilibrium of water menisci at AFM tip−sample contacts much slower than that predicted by the theory of the diffusion kinetics. The purpose of the present work is to investigate theoretically and experimentally the transport mechanisms leading to the formation of water menisci in thermodynamic equilibrium with the water vapors in air at a single-asperity nanocontact. The investigations establish the time scale of formation of stable capillary water menisci and point to the relevant mechanism of water transportation involved in this process. The theoretical investigation proposes the Knudsen diffusion model of capillary condensation in competition with a model that considers surface adsorption of water molecules on the surface region around the contact and flow of the surface water toward the meniscus. The experimental investigation is based on the AFM measurements of time evolution of capillary force between hydrophilic surfaces of a sharp silicon AFM tip and a glass sample.

Figure 1. Geometry of a small water meniscus surrounding the contact between a spherical tip of radius Rt and a flat sample. The meniscus contact angle at the tip and sample contact lines is assumed to be zero (θ = 0). The meniscus principal curvature radii are r2 in the horizontal plane and r1 in the vertical plane. In the approximation of small menisci (Rt ≫ r2), α ≅ 0 and the meniscus height h ≅ 2|r1|.

the meniscus height, h, at the contact lines of the meniscus with the sample and tip surfaces as h = 2r1 cos θ, where θ is the contact angle of the meniscus with the sample and tip surfaces, the following approximate relation can be deduced from Pythagoras theorem applied in triangle OAB in Figure 1:

r2 2 ≅ 2R th = 4R tr1 cos θ

(2)

Therefore, for small menisci at thermodynamic equilibrium, the following approximate relations for meniscus principal curvature radii holds:

r1 ≅ r k

(3)

and

r2 ≅ 2 R tr k cos θ

(4)

These relations allow for computation of the capillary force generated by the water meniscus as

γ Fc ≅ w πr2 2 ≅ 4πγw R t cos θ r1

2. THEORETICAL MODELS AND CALCULATIONS This section analyzes the properties of water meniscus at thermodynamic equilibrium and the transport mechanisms of water during meniscus evolution toward its equilibrium configuration. 2.1. Capillary Force of Water Meniscus at Thermodynamic Equilibrium. Figure 1 presents a sketch of the water meniscus surrounding the contact between the hydrophilic surfaces of an AFM tip and a flat sample. The shape of the AFM tip is assumed to be spherical with tip curvature radius Rt, while the meniscus total curvature 1/r is the sum of principal curvatures, here assumed to be 1/r1 in the vertical plan and 1/r2 in the horizontal plan. The meniscus principal radii of curvature are r1 < 0 and r2 > 0 with |r1| ≪ r2. Therefore, the total curvature radius of the meniscus is approximately r1, which means that at the thermodynamic equilibrium r1 ≅ rk. The value of r2 can be determined from simple geometrical considerations in the approximation of small menisci (r2 ≪ Rt). Considering

(5)

The most striking feature of this approximate formula is that it foresees a capillary force value that is independent of the curvature of the meniscus. This result is important in analysis of water transport mechanisms involved in meniscus evolution toward equilibrium. According to diffusion model of capillary condensation,25 during condensation the meniscus curvature radius increases (in absolute value) until it reaches the value rk. Equation 5 foresees no important change in meniscus force associated with this change in meniscus curvature. However, as will be shown in the Experimental Section, the dynamic of the meniscus during its evolution toward equilibrium is associated with an important change of the meniscus force. Rabinovich et al.26 showed that for the case of single asperity contacts between real surfaces, the nanoscale roughness of surfaces has the effect of decreasing of the capillary force at low values of humidity. According to this theory, the capillary force depends on meniscus curvature radius as 2559


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⎛ 1.817 × rms ⎞⎟ Fc ≅ 4πγw R t cos θ⎜1 + ⎝ 2r cos θ ⎠

Article

where |r| < |rk| and p′ < p. Considering that the tip and sample surfaces reflect totally the water molecules, the pressure drop, Δp = p − p′, determines a net inward flux of water molecules

(6)

where the curvature radius of the meniscus is considered negative and rms is the root-mean-square roughness of the sample surface. In the case of perfectly smooth surfaces, an important variation of the capillary force during meniscus evolution toward thermodynamic equilibrium can be related to time evolution of the meniscus contact angle, θ, at the contour lines with the solid body surfaces. Modification of meniscus curvature radius can be accompanied also by a decrease of θ from an initial value to the thermodynamic equilibrium value, which results in an increase of capillary force from a initial value to the equilibrium value. 2.2. Capillary Condensation by Knudsen Diffusion. This section describes the diffusion model of capillary condensation of water vapor in a small gap formed at the contact between a sphere (modeling the AFM tip) and a plan (modeling the flat surface of a sample). It is assumed that a small water meniscus forms instantaneously at the nanoscopic contact by coalescence of the adsorbed water layers on the two surfaces. Then, the meniscus grows through condensation of the water vapor that diffuses in the nanoscopic gap. Since the water molecule mean free path, λw, is much larger than the tip− sample separation distance, the diffusion of water molecules in the nanoscopic gap is controlled by their collisions with the tip and sample surfaces (Knudsen diffusion). The water mean free path is computed by λw = RT/pNAσw, where R is the ideal gas constant, T is the absolute temperature, p is the atmospheric pressure, NA is Avogadro’s number, and σw is the water molecule collision cross section area. In atmospheric air at normal conditions of temperature and pressure, λw is about 50 nm. Figure 2 shows a sketch of growing capillary meniscus due

Jw =

D k dp kBT dx

(8)

where Dk is the Knudsen diffusion coefficient of water molecules, T is the absolute temperature, and dp/dx is the radial gradient of pressure. The pressure gradient (dp/dx) at the meniscus surface (x = r2) can be approximated considering that the whole drop in water vapor pressure takes place in a sheath, referred hereafter as to the Knudsen sheet, with thickness λw (see Figure 2). This assumption is justified by the fact that λw is large in comparison with the meniscus size (λw ≅ 50 nm in atmospheric air at room temperature) and thus the inward flux of water molecules has little effect on the water molecules density at the entrance in the sheath. Under these conditions, the average value of the pressure vapor gradient in the Knudsen sheet is

p Δp p − p′ = = s {RH − exp[−γw Vm/ RT |r|]} Δx λw λw

(9)

However, the pressure gradient in the Knudsen sheet changes according the continuity equation

Jw (x) S(x) = const

(10)

where Jw(x) is the density of water molecules flux determined by Knudsen diffusion and S(x) is the lateral area of the tip− sample gap at a distance x from the center (see Figure 2). The equation of continuity of the flux of particle in the Knudsen sheath

D k (r2) dp D (x ) d p S(r2) = k S( x ) kBT dr2 kBT dx

(11)

is used to determine the water vapor gradient at the meniscus surface, dp/dr2. Simple geometry considerations for Rt ≫ x gives

⎛ x ⎞3 S(x) = S(r2)⎜ ⎟ ⎝ r2 ⎠

(12)

The Knudsen diffusion coefficient is Figure 2. Sketch of the water transportation mechanism through Knudsen diffusion of water molecules in the nanoscopic gap formed at the contact between a spherical AFM tip and a flat sample. All the calculations were made in the approximation of small menisci (|r1| ≪ r2 ≪ Rt).

Dk =

8RT πμw

(13)

where μw is the water molar mass and h the gap distance. In the case of tip−sample separation distance at meniscus surface, h is approximately r22/(2Rt) and the tip−sample separation distance at the distance x from the center is approximately x2/(2Rt). Therefore,

to Knudsen diffusion of water molecules in the nanoscopic gap between the tip and sample surfaces. It is assumed that, immediately after its formation, the meniscus total curvature radius, r, is not in equilibrium with the water vapor pressure, p, which means that |r| < |rk|, and this is why water molecules condense in the gap leading to growing of |r| until the equilibrium value |rk| is reached. During condensation, the water vapor pressure, p′, and the corresponding density (n′ = p′/ kBT, where kB is the Boltzmann constant) at the meniscus surface is determined by r according to the Kelvin equation

p′ = p0 exp[ − γw Vm/RT |r|]

h 3

⎛ x ⎞2 D k (x) = D k (r2)⎜ ⎟ ⎝ r2 ⎠

(14)

Use of the eqs 11 and 13 in the eq 10 results in

dp dp ⎛ r2 ⎞5 ⎜ ⎟ = dx dr2 ⎝ x ⎠

(15)

Then, considering the eq 15 in computation of the mean value of pressure gradient in the Knudsen sheath

(7) 2560


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Δp 1 = Δr λw

∫r

r2 +λ w ⎛ dp ⎞

2

⎜ ⎟ dr ⎝ dr ⎠

2p VmD k (r2) d|r | = s {RH − exp[−γw Vm/ RT |r|]} R tRT dt

(16)

results in

Δp r dp ≅ 2 Δr 4λ w dr2

where R = NAkB. Using the value r0 = γVm /RT (r0 = 0.54 nm at room temperature) and the classical equation of the Knudsen diffusion coefficient with the value 2|r| for the pore diameter

(17)

1/2 2|r| ⎛ 8RT ⎞ ⎜⎜ ⎟⎟ D k (r2) = 3 ⎝ πμw ⎠

Therefore, the vapor pressure gradient at the meniscus surface is

dp 4λ w p − p′ = dr2 r2 λw 4p = s {RH − exp[ − γw Vm/ RT |r|]} r2

1/2 d|r | 2|r| ⎛ 8RT ⎞ r0 2ps ⎜⎜ ⎟⎟ = [RH − exp( − r0/ dt 3 ⎝ πμw ⎠ γw R t

(18)

|r|)]

(19)

1/2 ps d|r | 4 ⎛ 8r0 ⎞ ⎟⎟ = ⎜⎜ |r|[RH − exp( − r0/ dt 3 ⎝ πρw γw ⎠ R t

|r|)] 1/2

1 4 ⎛ 8r0 ⎞ ⎟⎟ = ⎜⎜ τw 3 ⎝ πρw γw ⎠

2ps D (r ) dN = 8πR t1/2|r|3/2 k 2 kBT R t1/2|r|1/2 dt

Rt

|r | d|r | = [RH − exp( − r0/|r|)] τw dt

which is

(28)

(29)

For ps = 3000 Pa (saturation water vapor pressure at room temperature) and Rt = 30 nm, τw is about 1.7 μs. Numerical integration of eq 29 for r(0) = rk/2 and τw = 1.7 μs gives the solution r(t) plotted in Figure 3. The small value of τw in this case establishes a very fast meniscus formation process, which makes the phenomenon difficult to be investigated by a commercially available AFM setup. However, in practice, a much slower meniscus dynamics was revealed by the pull-off force measurements presented here and by other authors.19,24 This is happening because the present theoretical model neglected the adsorption of water molecules during their collisions with surfaces of the two bodies in contact. If one assumes a coefficient of adsorption a (0 < a < 1) at each collision of molecules with the solid surfaces and considers an average number of collision in the Knudsen sheath N = λw/⟨h⟩, where ⟨h⟩ is the average separation distance between surfaces in the Knudsen sheath, then the flux of water particles at meniscus surface diminishes by a factor of (1 − a)N. This leads to an increase of τw by roughly the same number of times. Thus, in presence of surface adsorption, the characteristic time of meniscus formation is

D k (r2)ps dN = 16π|r| {RH dt kBT (21)

The rate of increase of the number of water molecules in the meniscus can be also expressed as a function of the meniscus volume, V, as

(22)

where Vm is the molar volume of liquid water and V is expressed by the following approximate equation

V ≅ 4πR t|r|2

ps

establishes the value of the characteristic time of meniscus formation, τw. Then, eq 28 is written as

{RH − exp[ − γw Vm/ RT |r|]}

N dV dN = A Vm dt dt N dV d|r | = A Vm d|r| dt N d|r | = A 8πR t|r| Vm dt

(27)

The expression

(20)

where r2 ≅ 2(Rt|r|)1/2. Thus, the rate of increase of the number of water molecules in the liquid phase is

− exp[ − γw Vm/RT |r|]}

(26)

Using μw = ρwVm (ρw is density of liquid water) in the eq 26 results in the following equation:

Simple geometry considerations (see Figure 1) in the approximation of small menisci (r2 ≪ Rt) formed between perfectly wetting tip and sample surfaces (null water contact angles) give

S(r2) = 2πr2(2|r|) = 8πR t1/2|r|3/2

(25)

in the eq 24, one obtains

The variation of the number of water molecules in the condensed phase is

dN = Jw (r2) S(r2) dt

(24)

(23)

Equating the two expressions of dN/dt, gives the following firstorder nonlinear differential equation for the time evolution of the meniscus curvature radius during capillary condensation

τw ′ = 2561

1/2 3 ⎛ πρw γw ⎞ ⎟ ⎜ (1 − a)λ w / ⟨h⟩ 4 ⎝ 8r0 ⎠

1

(30)


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Figure 4. Sketch of the capillary water meniscus at the contact between completed wetting surfaces of AFM tip and sample. The thickness of the water layer on sample surface is h. The value h0 corresponds to the thickness of water layer on flat surface, far from the meniscus, at thermodynamic equilibrium with the water vapor at the pressure p. Variation of h nearby the meniscus contact line with the sample determines the radial gradient of disjoining pressure and the speed vr of the radial flow of water.

Figure 3. Time dependence of meniscus radius as numerical solutions of eq 29 for Rt = 30 nm, RH = 0.5, ps = 3000 Pa, and r(0) = rk/2. The solution neglecting molecule adsorption on tip and sample surfaces is computed for τw = 1.7 μs, while the solution assuming an adsorption coefficient a = 0.5 and an average number of collisions in the Knudsen sheath N = 10 is computed for τw = 1.7 ms.

The surface adsorption of molecules can increase very much the value of τw. Thus, considering a = 0.5, λw = 50 nm, and ⟨h⟩ = 5 nm gives τ′w ≅ 103τw = 1.7 ms. A numerical solution, r(t), for this value of τw is also plotted in Figure 3. In this case, the growing of meniscus curvature radius toward the equilibrium value rk is much slower, the equilibrium being reached in about 10 ms. As will be pointed out in the Experimental Section, this time scale of formation of stable water menisci is close to the experimental value in the case of water menisci formed at the contact between a hydrophilic AFM tip and sample surfaces. The formation of the initial water meniscus by coalescence of the water layers adsorbed on the two surfaces leads to a local depletion of the adsorbed water layers, which increases the surface adsorption of water in the Knudsen sheath. This can result in a much longer time of growing of the meniscus through Knudsen diffusion, in which case the water transportation in the layers of adsorbed water on the two surfaces in contact is the phenomenon that establishes the meniscus growth rate. 2.3. Capillary Condensation through Surface Adsorption and Transport of Water Molecules. In the followings it is assumed that, after the initial formation through coalescence of the water layers adsorbed on the two surfaces, the meniscus grows until the thermodynamic equilibrium through surface adsorption and transportation of water molecules. This meniscus dynamic is determined by the flow of water toward the meniscus in the surrounding layer of adsorbed water on the sample surface (Figure 4). This is determined by the radial gradient of the disjoining pressure, Π, in the surface water layer surrounding the meniscus. In this region, the meniscus curvature is determined by the sum of disjoining, Π, and the Laplace, γw/r, pressures. For nanoscopic menisci at equilibrium with vapor, the following equation holds27

γw r

+ Π(h) =

RT ln(p /p s) Vm

layer on the sample surface in a region situated far from the water meniscus (r = ∞):

Π(h0) =

RT ln(p /p s) Vm

(32)

To determine the radial gradient of disjoining pressure at the meniscus contact line with the sample, the knowledge of the dependence of the disjoining pressure of water on water film thickness is necessary. In the following, the dependence

Π(h) =

A h

(33)

proposed by Pashley28,29 for thin water layers (h < 40 nm) on glass or silica is considered. The value of the constant A (A < 0) can be expressed as a function of the thermodynamic equilibrium value of the thickness of adsorbed water layer on the flat surface, h0. Thus, according to the eq 31

γ A RT = w = ln(p /p s) h0 rk Vm

(34)

Therefore, the radial gradient of disjoining pressure is

dΠ dΠ dh = dr dh dr

(35)

where

(31)

γ h0 dΠ A =− 2 =− w 2 dh h r kh

(36)

dh = tan θ dr

(37)

and θ is the microscopic tilting angle of the adsorbed water layer at the region of contact with the meniscus. The velocity of the water flow in water layer at contact lines with the meniscus is determined by the radial gradient of disjoining pressure according the Navier−Stokes equation

where h is the thickness of the water layer on the sample surface. Inside the meniscus, h is large, Π(h) ≅ 0, and eq 31 reduces to the Kelvin equation. On the other hand, eq 31 establishes the equilibrium thickness, h0, of the adsorbed water 2562


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this region (surface water layer with equilibrium thickness, h0, does not adsorb water molecules anymore). Increase of the thickness of the depleted water layer toward h0 results in less water adsorption.

d2vr

1 dp = 2 η dz (38) ws dr where ηws is viscosity of surface water. Considering the movement of water molecules in the adsorbed film with no shear force at the water−air interface (dvr /dz = 0 at z = h) and no slip (vr = 0 at z = 0) at the water−solid interface, the following solution of eq 38 is found vr(z) =

1 dΠ (hz − z 2/2) ηws dr

3. EXPERIMENTAL SECTION 3.1. Materials and Methods. The AFM force spectroscopy measurements were performed by a commercial AFM apparatus (Nova from NT-MDT) with silicon AFM probes (NSG11 from NTMDT). Before measurements, the surface of the AFM probes were cleaned and hydrophilized by plasma treatment33 in low-pressure air plasma for 10 min. The force constant of the AFM probes was determined by the thermal noise method,34 and the curvature radius of the AFM tip was determined by analysis of high-resolution images of the sharp edges of a standard grating probe (TGG1 from NT-MDT). The AFM samples, consisting of smooth glass microscope slides from J. Melvin Freed Brand, were cleaned by treatments in UV/ozone (30 min) and ethanol (10 min). The hydrophilicity of the AFM probes and samples surfaces were characterized by measurements of water contact angle values with a goniometer (Digidrop from GBX Instruments Co.) for small sessile droplets of deionized water. The measurements revealed very low water contact angles (less of 5°) of the AFM probe and sample surfaces after their cleaning. In a first series of force spectroscopy measurements, the water meniscus pull-off force was measured as a function of the tip−sample contact time. The measurements were performed in air at a temperature and relative humidity around 23 °C and 50%, respectively, with a plasma-cleaned AFM probe with a force constant of 8.8 N/m and tip curvature of about 30 nm. The time resolution of the force spectroscopy measurement was determined by the characteristic time of the mechanical response of the AFM cantilever (2.5 μs in the present experiment) and the time resolution of force curve data (0.2 ms in the present experiment). The air temperature and humidity were monitored by a precision hygrometer (HM 34 from Vaisala). The tip−sample contact time was modified either by an increase of the sample approaching and retracting speed or by an increase of the probe displacement after the tip−sample contact. To avoid formation of water droplets on sample surface,35 any single pulloff force measurement was performed at a different place on the sample surface with no tip−sample contact between the measurements, i.e., between the single pull-off measurements the AFM tip was kept at a certain separation distance from the sample surface. This assures the same initial state of tip−sample contact. Figure 5 shows three force curves acquired at three values of tip− sample contact time. The contact time is determined as the time spent by the tip in contact with the sample surface, i.e., the time laps between the moment of tip snap-in on the sample surface (indicated by the letter A in Figure 5) and the moment the meniscus starts elongating (indicated by letter B in Figure 5). The pull-off force was measured as the maximum value of the meniscus attractive force (measured at the moment B). The plots in Figure 5 show an increase of the pull-off force with the increase of the contact time. In this experiment, the contact time has been increased by keeping the approaching and retracting speed constant (200 nm/s) and by increasing the probe displacement after the AFM tip−sample contact. This procedure assured a constant and negligible value of the hydrodynamic viscous force,36 but had the effect of increasing of the repulsive force with the increase of the contact time. This may lead to an increase of the tip− sample contact area and, in case of presence of hysteretic adhesive forces, to a corresponding increase of the adhesive force. One of the most used models of contact adhesion, the Johnson, Kendall, and Roberts (JKR) model,37 considers the pull-off force as being generated by short-range attractive forces on the tip sample contact area as

(39)

where h is the thickness of the water film and z the distance from the water−solid interface. The average radial velocity of water molecules, ⟨vr⟩, in the adsorbed water layer is

⟨vr⟩ =

1 h

∫0

h

vr(h) dz =

h2 dΠ 3ηws dr

(40)

Use of eqs 35−37 in eq 40 results in the following expression of the average value of the radial velocity of water flow:

⟨vr⟩ = =

γw h0 3ηws |r k|

tan θ (41)

A realistic value of ⟨vr⟩ can be found if realistic values for the parameters h0, ηws, and θ are used in eq 41. It is known that the viscosity of water nanoconfined between hydrophilic surfaces can be much larger than the bulk value.30,31 At present there are conflicting results reported in the literature on the water viscosity of nanoconfined water. Computer simulation and AFM measurements have shown that ηws can reach a value of 35 Pa·s on water layers thinner then 0.5 nm on hydrophilic surfaces,31 while interfacial force microscope measurements32 have indicated a value of 1.5 kPa·s. Also, there is a lack of information on the microscopic contact angle at the meniscus transition region with the adsorbed water layer. These uncertainties prevent an exact evaluation of ⟨vr⟩. However, a rough estimation based on chosen value of ηws = 35 Pa·s, tan θ = 0.01, h0 = 0.3 nm, and rk = −0.78 nm (RH = 50%) gives a value of 2.6 μm/s for ⟨vr⟩. Considering that the meniscus volume is V = 4πRt, the time of meniscus formation is

4πR tr k 2 R 1/2|r k|3/2 = t 2πr2h0⟨vr⟩ h0⟨vr⟩ (42) With h0 = 0.3 nm, Rt = 30 nm, rk = 0.78 nm, and ⟨vr⟩ = 2600 nm/s, a value of 4.8 ms is computed for τ. Therefore, the time scale for water meniscus formation through surface adsorption and flow of water is larger than the time scale of water meniscus formation through diffusion and condensation of water from the vapor phase under conditions of no surface adsorption of water molecules. As will be pointed out in the Experimental Section, this value of the time required for formation of stable water menisci is closed to the experimentally found value in the case of water menisci formed at the contact between a hydrophilic AFM tip and sample surfaces. Finely, one notices that it is the radial flow of water that keeps the water layer in the meniscus surrounding region depleted and this determines the rate of water adsorption in this region. The rate of adsorption of water molecules in the depleted region of the adsorbed water layer surrounding the meniscus is controlled by the thickness of the adsorbed layer in τ=

FJKR =

3 πR γa 2

(43)

where R is the tip radius and γa is the work of adhesion of tip and sample surfaces. Therefore, according to the JKR model, the tip− sample contact pull-off force is independent of the contact pressure 2563


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experiment, and the increase of the pull-off force versus the contact time can be attributed to formation and evolution toward equilibrium of the water meniscus at the tip−sample contact. In a second series of force spectroscopy experiments, the dependence of the meniscus pull-off force on the sliding speed of the tip−sample contact has been investigated. To do this, the sample was moved laterally, forth and back, with a frequency of 5 Hz and a speed controlled by the amplitude of the movement (Figure 6). Then,

Figure 6. Movements of the AFM tip and sample during force spectroscopy measurements of pull-off force dependence on the sliding speed of the AFM tip−sample contact. force spectroscopy measurements were performed on the moving sample by keeping a constant contact time (around 0.4 s) and a constant pushing (repulsive) force on the tip−sample contact. 3.2. Results and Discussion. The dependence of the meniscus breakup force on the contact time found in the pull-off force measurements is represented in Figure 7. It is noticed that the

Figure 5. Force curves taken in air (RH = 50%, t = 23 °C) for three values of contact time: 5 ms (a), 10 ms (b), and 40 ms (c). The contact time has been measured between moments indicated by A and B on the plots.

Figure 7. Dependence of the meniscus pull-off force on the tip− sample contact time. The continuous line is a fit of the experimental data with a relaxation-type formula F(t) = F0 + ΔF[1 − exp(−t/τ)], where F0 = 15 nN, ΔF = 15 nN, and τ = 10 ms.

and contact time. Computation of FJKR pull-off force in the case of the present experiments for Rt = 30 nm and γa = 10 mJ/m2 for hydrophilic silica surfaces in water gives a value of about 1.42 nN. This value is much smaller than the values of the pull-off force observed in Figure 5. The ion electrostatic force and hydrodynamic forces have also negligible contributions to the tip−sample pull-off force measured in the present experiment. This has been proved by the results (see the Supporting Information) of the pull-off force measurements performed in deionized water (pH = 5.7) with the same AFM tip and sample at constant approaching and retracting speed (200 nm/s) for various values of contact time and maximum impinging force. These measurements showed, irrespective of the contact time value, negligible adhesive force between hydrophilic surfaces of the AFM tip and glass sample in pure water. This indicates that the capillary force is the main component of the pull-off force measured in the present

meniscus breakup force increases by about 2 times by the increase of the contact time from a few milliseconds to about 40 ms. The continuous line on the graph represents the best fit of the experimental data with a relaxation-type time dependence, where the relaxation time, τ, is interpreted as the characteristic time required for meniscus to reach the thermodynamic equilibrium. For the data presented in Figure 7, τ = 10 ms. The increase of force with the contact time is associated with the meniscus growing from its initial size determined by the amount of water on the two surfaces before the contact to the size of the meniscus at thermodynamic equilibrium. However, as has been pointed out in the previous section, in the case of perfectly smooth surfaces, this large increase of capillary force with the contact time cannot be related to the increase of the meniscus curvature radius. 2564


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slow to reach the moving meniscus. Therefore, at large vs the water meniscus forms by coalescence of the adsorbed water layers on the tip and sample surfaces and cannot reach the thermodynamic equilibrium because the flow of the surface water toward the meniscus is impeded by the fast change of the contact position on the sample surface.

This is happening because the meniscus capillary force increases with the size of the meniscus and decreases with the increase of the absolute value of the meniscus curvature radius. The two effects compensate each other, so that the capillary force does not change very much if the water meniscus reaches its thermodynamic equilibrium by decrease of its curvature. This is indicated by eq 5, which in the approximation of small menisci (r2 ≪ Rt) foresees a capillary force value that is independent of the meniscus curvature. This means that the thermodynamic equilibrium of the meniscus may be reached not only by variation of meniscus curvature, but also by variation of other meniscus parameters. Equation 5 suggests that the thermodynamic equilibrium of the meniscus may be reached by variation of the contact angle, θ, at the meniscus contact lines with the tip or sample. In this case, the meniscus evolves toward equilibrium by changing the contact angle θ at its contour lines with the surfaces in contact. On the other hand, in the case of single-asperity contacts between surfaces with finite roughness at nanoscale, an important variation of capillary force with the meniscus curvature radius is foreseen by the theory of Rabinovich et al.26 For the case of glass surface used in the present experiments, rms = 0.13 nm. Under these conditions, according the eq 6, an increase of absolute value of r from 0.18 nm (Kelvin radius at RH = 5%) to 0.78 nm (Kelvin radius at RH = 50%) results in an increase of capillary force by a factor of 2. In a second set of force spectroscopy measurements, the pull-off force required to breakup the water meniscus formed at a sliding contact between the AFM tip and sample was measured. The dependence of the pull-off force on the lateral speed, vs, of the sliding contact is represented by the plot in Figure 8. At low values of vs the

4. CONCLUSION This work investigated and discussed the transport mechanisms involved in formation of water menisci at a single-asperity nanoscopic contact between hydrophilic surfaces. On the basis of the experimental evidence collected by atomic force microscopy measurements of meniscus capillary force, it is considered that the water menisci forms instantaneously at the nanoscopic contacts by coalescence of the surface water layers, and this is followed by growth of the menisci toward their thermodynamic equilibrium configuration. Two water transport mechanisms involved in the water meniscus dynamics were considered: (1) Knudsen diffusion and condensation of water molecules in the nanoscopic gap and (2) adsorption of water molecules on the surface region around the contact and flow of the surface water toward the meniscus. The first mechanism assumed that the meniscus evolve toward its thermodynamic equilibrium configuration by change of its curvature radius. The initial meniscus has a smaller curvature radius (in absolute value) and because of this the vapor condenses in the nanoscopic gap, leading to meniscus growth and an increase of its curvature radius toward the thermodynamic equilibrium value, which is the Kelvin radius. The diffusion of water vapor in the nanoscopic gap is considered to be of Knudsen type because of frequent collisions of water molecules with the surfaces forming the nanoscopic gap. If the adsorption of molecules on surfaces is neglected, this theoretical model foresees a value of the time required for the meniscus to reach the thermodynamic equilibrium that is much shorter than that indicated by the experiments. However, the water molecule adsorption on the surface region around the single-asperity nanospic contact, where the adsorbed water layers are depleted, cannot be neglected. Water adsorption on the surfaces around the contact has the effect of diminishing the diffusion flux of water molecules, a fact that results in an increase of the meniscus growing time by a few decimal orders of magnitude. In this latter case, the time scale of the meniscus evolution is close to that observed in experiments. However, for nanoscopic contacts between ideally smooth surfaces, the diffusion model cannot explain properly the experimentally observed evolution of the capillary force during meniscus evolution toward thermodynamic equilibrium. For single-asperity contacts between ideally smooth surfaces, the meniscus capillary force has a weak dependence on the meniscus curvature, and the large variation of the meniscus capillary force found in experiments is not justified by the change of the meniscus curvature. In this case, the capillary force variation during the meniscus evolution toward thermodynamic equilibrium can be attributed to a change of the meniscus contact angles with the surfaces in contact. The important variation of capillary force during evolution of the meniscus toward equilibrium can be explained also by the effect of surface roughness. Real surfaces have a finite roughness at nanoscale and this gives a strong dependence of the capillary force on the meniscus curvature, in which case the hypothesis of variation of meniscus contact angle is not necessary. Because of the lack of information on the microscopic contact angle of the water menisci, it is impossible to decide which of the two effects is responsible for the large

Figure 8. Dependence of the meniscus pull-off (capillary) force on the contact sliding speed. pull-off force have values close to those measured for water menisci in thermodynamic equilibrium; i.e., water menisci formed at static AFM tip−sample contacts for large contact time values. While vs increases, the pull-off force required to breakup the water meniscus decreases, which means that the relative lateral movement of the AFM tip against the sample impeded the formation of menisci at thermodynamic equilibrium with water vapor and surface adsorbed layers. This is happening because the position of the tip−sample contact and water collection on the sample surface is changing with the speed vs. The lateral movement of the sample alters the meniscus symmetry due to shear stress and/or contact line hysteresis and this may contribute to the decreasing effect of vs on the meniscus pull-off force. However, this effect cannot explain the large decrease of pull-off force and why this decrease reaches saturation. Indeed, the plot in Figure 8 shows that at values of vs larger than a critical value, which in this case is about 3 μm/s, the meniscus capillary force does not decrease by a further increase of vs and remains constant. This critical value of vs is on the same order of magnitude with the mean value of radial velocity of water flow on the surface (⟨vr⟩ expressed by eq 41). This is explained by the fact that at vs > ⟨vr⟩ the radial flow of the surface water is too 2565


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(7) Restagno, F.; Bocquet, L.; Crassous, J.; Charlaix, E. Colloids Surf. A 2002, 206, 69. (8) Striolo, A.; Chialvo, A. A.; Cummings, P. T.; Gubbins, K. E. Langmuir 2003, 19 (20), 8583−8591. (9) Rezhok, S; Piner, R.; Mirkin, C. A. J. Phys. Chem. B 2003, 107, 751. (10) Schenge, M.; Li, X.; Schaefer, J. A. Tribol. Lett. 1999, 6, 215− 220. (11) de Lazzer, A.; Dreyer, M.; Rath, H. J. Langmuir 1999, 15, 4551− 4559. (12) Shao, Z. F.; Mou, J.; Czajkowsky, D. M.; Yang, J.; Yuan, J. Y. Adv. Phys. 1996, 45, 1−86. (13) Davis, H. T. Statistical Mechanics of Phases, Interfaces, and Thin Films; VCH: New York, 1996. (14) Sirghi, L; Szoszkiewich, R.; Riedo, E. Langmuir 2006, 22, 1093. (15) Matsuoka, H.; Fukui, S. Langmuir 2002, 18, 6796. (16) Israelachivili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1992. (17) Maeda, N.; Israelachivili, J. N.; Kohonen, M. M. PHAS 2003, 100, 803. (18) Crassous, J.; Charlaix, E. Phys. Rev. Let. 1997, 78, 2425. (19) Wei, Z.; Zhao, Y.-P. J. Phys. D: Appl. Phys. 2007, 40, 4368. (20) Xu, L.; Lio, A.; Hu, J.; Ogletree, D. F.; Salmeron, M. J. Phys. Chem. B 1998, 102, 540. (21) Restagno, F.; Bocquet, L.; Crassous, J.; Charlaix, E. Colloids Surf. A 2002, 206, 69. (22) Gnecco, E.; Riedo, E; Bennewitz, R.; Meyer, E.; Brune, H. Mater. Res. Soc. Symp. Proc. 2004, 790, P.1.3.1. (23) Szoszkiewicz, R.; Riedo, E. Phys. Rev. Let. 2005, 95, 135502. (24) Rabinovich, Y. I.; Singh, A.; Hahn, M.; Brown, S.; Moudgil, B. Langmuir 2011, 27, 13514. (25) Kohonen, M. M.; Maeda, N.; Christenson, H. K. Phys. Rev. Lett. 1999, 82, 4667. (26) Rabinovich, Y. I.; Adler, J. J.; Esayanur, M. S.; Ata, A.; Singh, R. K.; Moudgil, B. Adv. Colloid Interface Sci. 2002, 96, 213. (27) Aveyard, R.; Clint, J. H.; Paunov, V. N.; Nees, D. Phys. Chem. Chem. Phys. 1999, 1, 155. (28) Pashley, R. M. J. Colloid Interface Sci. 1980, 78, 246. (29) Shanahan, M. E. R. Langmuir 2001, 17, 8229. (30) Goertz, M. P.; Moore, N. W. Prog. Surf. Sci. 2010, 85, 347−397. (31) Li, T.-D.; Gao, J.; Szoszkiewicz, R.; Landman, U.; Riedo, E. Phys. Rev. B 2007, 75, 115415−1. (32) Feibelman, P. J. Langmuir 2004, 20, 1239−1244. (33) Sirghi, L.; Kylian, O.; Gilliland, D.; Ceccone, G.; Rossi, F. J. Phys. Chem. B 2006, 110, 25975. (34) Burnham, N. A.; Chen, X.; Hodges, C. S.; Matei, G. A.; Thoreson, E. J.; Roberts, C. J.; Davies, M. C.; Tendler, S. J. B. Nanotechnology 2003, 14, 1. (35) Xu, L.; Lio, A.; Hu, J.; Ogletree, D. F.; Salmeron, M. J. Phys. Chem. B 1998, 102, 540. (36) Cai, S; Bhushan, B. Nanotechnology 2007, 18, 465704. (37) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London A 1971, 324, 301.

variation of the capillary force observed experimentally during the water meniscus formation. The second water transport mechanism investigated in the paper, i.e., water adsorption on the surface region around the contact and flow of surface water toward the growing meniscus, is supported by the AFM force spectroscopy measurements of capillary force of water menisci formed at sliding nanoscopic contacts. These measurements revealed that the movement of the tip−sample contact impeded the formation of water menisci at thermodynamic equilibrium. At large sliding velocity values, the flow of the surface water toward the meniscus is too slow to reach the moving meniscus, which in this case is formed by coalescence of the adsorbed water layers on the tip and sample surfaces and cannot reach the thermodynamic equilibrium. Both water transport mechanisms can contribute to the time evolution of water menisci formed by capillary condensation at single-asperity contacts between hydrophilic surfaces. The time scale foreseen by the Knudsen diffusion mechanism in the absence of surface adsorption is much shorter than that observed experimentally. However, adsorption of molecules on the very hydrophilic surfaces that forms the nanoscopic gap diminishes significantly the diffusion flux of water molecules and results in an increase of the time scale by a few orders of magnitude. In the case of strong surface adsorption, the water adsorption on the surface region around the contact and flow of surface water toward the growing meniscus may be the relevant water transport mechanism. This latter mechanism is supported by the AFM force spectroscopy measurements of capillary force of water menisci formed at sliding nanoscopic contacts between very hydrophilic surfaces.

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ASSOCIATED CONTENT

S Supporting Information *

Plots of force curves taken in deionized water (pH = 5.7, t = 23 °C) for three values of contact time, 5 ms (a), 15 ms (b), and 20 ms (c), recorded at constant approaching and retracting speed (200 nm/s) by increasing the probe displacement after the AFM tip−sample contact. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

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ACKNOWLEDGMENTS This work was supported by CNCSIS, IDEI Research Program of Romanian Research, Development and Integration National Plan II, Grant no. 267/2011.

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REFERENCES

(1) Lynden-Bell, R. M.; Moris, S. C.; Borrow, J. D.; Finney, J. L.; Harper, C. L., Jr. (Eds.) Water and Life: The Unique Properties of H2O; CRC Press: Boca Raton, FL, 2010. (2) Ingebritsen, S. E.; Sanford, W. E.; Neuzil, C. E. Groundwater in Geologic Processes, 2nd ed.; Cambridge University Press: Cambridge, 2006. (3) Bushan, B. Nanotechnology and Nanomechanics. An Introduction, 2nd ed.; Sringer-Verlag: Berlin, 2008; p 356. (4) Gudmundsson, K. Nordic J. Building Phys. 2003, 3, 1. (5) Bushan, B. Handbook of Nanotechnology, 2nd ed.; Sringer-Verlag: Berlin, 2007. (6) Striolo, A. Theor. Comput. Chem. 2007, 18, 245. 2566


Transport Mechanisms in Capillary Condensation of Water at a Single

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