Disjoining Pressure in Partial Wetting on the Nanoscale

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Disjoining pressure in partial wetting at nanoscale Florentina Samoila, and Lucel Sirghi Langmuir, Just Accepted Manuscript • Publication Date (Web): 09 May 2017 Downloaded from http://pubs.acs.org on May 11, 2017

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Disjoining pressure in partial wetting at nanoscale Florentina Samoila and Lucel Sirghi* Iasi Plasma Advanced Research Center (IPARC), Faculty of Physics, “Alexandru Ioan Cuza” University of Iasi, Iasi-700506, Romania

The partial wetting at nanoscale may result in formation of sessile liquid nanodroplets on flat substrates. In this case, the molecular forces generate a strong interaction between nanodroplet interfaces. This interaction is expressed in the mean-field approximation by the disjoining pressure and determines an important deviation from the spherical cap shape of the nanodroplets. This deviation is observed on the atomic force microscopy images of sessile nanodroplets of oleic acid on glass. The disjoining pressure was manipulated by hydroxylation of glass surface. This surface modification generated a strong negative disjoining pressure due to structural forces arisen from orientation of oleic acid molecules with polar head towards substrate. As result, the shape of oleic acid nanodroplets showed large deviations from the spherical cap shape, the liquid-vapour interface tilting angle with respect to the plane substrate having a maximum (herein, considered as the contact angle) at certain distance from the substrate, followed by its decrease to zero at the droplet edge. Integration of augmented Young-Laplace equation, where the dependence of the negative structural disjoining pressure on interface separation distance was assumed to be an exponential decay, yielded height profiles of droplets in good agreement with the experiment.

*

Corresponding author: [email protected]

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1. Introduction Exposure of a solid surface to a certain substance in form of vapour or solute in a liquid solution may result in adsorption of substance molecules in form of continuous thin films (complete wetting) or patches/droplets (partial wetting)1. At macroscopic scale, the partial wetting is usually investigated by contact angle measurements on macroscopic sessile droplets, which have spherical cap shape when gravitational or electric forces are negligible. At nanoscale, formation of nanometer-scale liquid patches or droplets in partial wetting requires direct investigations by microscopy techniques2.Wetting phenomena at nanoscale are ubiquitous in the nature and they may play crucial roles in geologic3 and biologic4 systems. Wetting phenomena at nanoscale are important also in many technological processes such as lubrication and corrosion5,

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, surface decontamination7, fabrication of micro electro

mechanical systems (MEMS)8, and micro-contact printing9. Comparing to macroscale, partial wetting phenomena at nanoscale are more complex due to effects of contact line tension10, enhanced liquid evaporation, surface heterogeneity, and local topography of the surface11. For dynamic wetting phenomena, the liquid height profile at moving contact lines cannot have a wedge and a precursor wetting film should be present12. Moreover, macroscopically thermodynamic parameters as contact angle, superficial tension and contact line tension may have different values at nanoscale13. Therefore, study of nanoscale wetting phenomena has theoretical importance besides the practical one. The experimental investigation of nanoscale wetting phenomena requires the use of special microscopy techniques as interference microscopy, confocal microscopy, environmental scanning electron microscopy14, atomic force microscopy (AFM)15 and scanning force polarization microscopy16. Also, special techniques are necessary for sample preparation in these studies. Small nanodroplets of various liquids on flat substrates were obtained by various techniques as: air spray, electro spray, micropipette dispensing, adsorption from nanoemulsions, and nanoscale AFM tip


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dispensing2. Condensation of saturated vapour on a cooled substrate results also in formation of small liquid nanodroplets17. In this case, the droplet size may vary according to a random fractal distribution18, the droplet radii having predominantly values above a critical value determined by the substrate sub cooling temperature19. The droplet size distribution and area coverage of substrate is important for determining of the transfer coefficient of heat at substrate surface, which in this case is known to be one order of magnitude larger than in the case of uniform liquid films adsorbed on substrate (complete wetting). Formation of nanodroplets with a relatively wide size distribution allows for study of the effect of contact line tension on the contact angle by using Young modified equation20.

cos θ ' = cos θ −

τ 123 , σ 12 ⋅ rc

(1)

where θ is the macroscopic contact angle measured for large liquid droplets (large values of contact line curvature radius, rc), θ’, the nanoscopic value of contact angle for small liquid nanodroplets, τ123 is the contact line tension at vapour (1)-liquid(2)-solid(3) phase separation, and σ12 is the superficial tension of liquid-vapour interface. The concept of contact line tension is nevertheless controversial, the experimentally determined values being scattered in a very large range (10-5 to 10-13 J/m) as compared to the theoretically predicted values (10-11 J/m). On the other hand, the difference between the macro- and nano-scale values of the contact angle can be regarded as caused by the interfacial forces that are very important at the edges of sessile nanoscopic liquid droplets. Pompe and Herminghaus9 used this complementary theoretical approach to study the effects of molecular forces on the edges of nanoscopic droplets of hexaethylene glycol on hydrophobized silicon wafer, aqueous CaCl2 solution on mica and water on silicon wafer. They have analyzed the curvature of AFM topography images of liquid nanodroplets to determine the interaction between liquid-vapour and liquid-solid interfaces. For the hexaethylene glycol nanodroplets on hydrophobized


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silicon substrate, they have found a large deviation from the spherical cap shape, with a drastic change of curvature from convex to concave at proximity of the contact line. Checco and Guenoun21 have analyzed the height profiles of AFM topography images of alkane nanodroplets on hydrophobic silanized silicon substrates and found good fit with spherical cap shape. They also found an important deviation of droplet height profile from the spherical cap at droplet edges, but interpreted this deviation as caused by tip-sample convolution effects in the noncontact AFM images. Following the work of de Gennes and coworkers22, Xu and Salmeron23 derived an approximate relationship that shows how the interfacial forces affect the contact angle values for nanoscale sessile droplets with shapes close to spherical cap. They have found an important decrease of contact angle with the height (below 20 nm) of nanodroplets of glycerol on contaminated mica, fact that was interpreted as an effect of negative potential energy of the interfacial interaction. Same approach was followed by Moldovan et al24 for study of contact angle of sessile nanodrops of glycerol and sulfuric acid on highly oriented pyrolytic graphite and aluminum. They have also found an important decrease of the contact angle with the height of the nanodroplets and, based on this dependence, determined values of the interfacial potential energy. In the present paper, the atomic force microscopy technique is used to analyze the shape and size distribution of sessile nanodroplets of oleic acid formed by vapour condensation on glass substrates. Samples consisting of glass substrates with sessile oleic acid nanodroplets with various sizes are particularly suitable for non contact AFM investigations due to high viscosity and very low evaporation speed of oleic acid25. To highlight the effect of interfacial forces on the shape of nanodroplets, we used substrates consisting in either solvent-cleaned glass or hydroxylated glass. Hydroxylation of the glass surface gives rise to a strong negative structural component of the disjoining pressure because the oleic acid molecules tend to organize with the polar head towards the hydroxylated


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substrate. The negative disjoining pressure changes very much the curvature of the sessile oleic acid droplets, a large departure from the spherical cap shape being detected in the vicinity of droplet edges, where an inflexion point in the droplet height profile is detected. At the inflexion point, the liquid-vapour interface have the maximum tilting angle with respect to the substrate and this maximum value of the interface tilting angle is considered herein as the nanoscopic value of contact angle. A theoretical model based on the augmented YoungLaplace equation in which the negative structural disjoining pressure dependence on the droplet interfacial distance is considered as an exponential decay is used to explain the experimental findings. The augmented Young-Laplace equation in cylindrical geometry with appropriate values of disjoining pressure is numerical integrated to determine height profiles of the nanodroplets that fit well the experiment. The augmented Young-Laplace equation in planar geometry has been numerically integrated by Kuchin et al26 to study the influence of the disjoining pressure on the equilibrium interfacial profile in the transition zone between a wetting thin film and a capillary meniscus. They have proved that the Van der Waals, electrical and structural components of disjoining pressure affect greatly the contact angle and the shape of the liquid-vapor interface in the transition region. Although the geometry is different, the findings of the present study agree well with the theoretical study of Kurchin et al. 2. Theoretical model The effect of molecular forces in a nanometer-thick liquid film on a substrate is approximated in the mean-field theory by the concept of disjoining pressure, which is conceived as the superficial density of force acting between the two parallel interfaces of the film. The disjoining pressure is positive when the molecular forces act to separate the interfaces and negative when they act to approach the two interfaces. The disjoining pressure may result from the action of surface forces of different nature27, as Van der Waals,


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electrostatic, structural, and steric. In the present work, contributions of Van der Waals and structural forces are considered. Therefore, Π( z) = Π m ( z) + Π s ( z) ,

(2)

where Π(z) is the total disjoining pressure in a liquid film with thickness z, Πm(z), the Van der Waals component of disjoining pressure and Πs(z), the structural component of the disjoining pressure. While Πm(z) arises from the molecular dispersion interactions present in every molecular system, Πs(z) arises from the energy required to form a ordered structure in the thin molecular film as compared to the disordered structure in the bulk liquid28. For the experimental model considered in this work, i.e. oleic acid nanodroplets on glass, the contribution of electrostatic forces to the disjoining pressure can be neglected because both the glass surface and the oleic acid molecules have a comparable weak acid character29, 30. Recently, efforts have been made to extend the concept of disjoining pressure to films with nonuniform thickness, i.e. films with sloping interfaces. Based on the minimization of the total free energy of a thin liquid drop on a solid surface, Wu and Wang31 determined an equation for molecular component of the disjoining pressure in nonuniform thin films (with slightly tilted interfaces). Dai et al32 improved the model developed by Wu and Wang and found the following equation for the disjoining pressure, Πm, in nonuniform thin films:

(

)

Π m ( z ) = − A123 ⋅ 4 − 3 z x2 − 3 z xx ⋅ z / 24π z 3 ,

(3)

where A123 is the Hamaker constant for the interaction of media 1(vapour) and 3(substrate) through the medium 2 (adsorbed liquid film), z is the local thickness of the film, zx is the local slope of the film interfaces (first derivative of z with respect to the lateral distance, x) and zxx is the second derivative of z with respect to x. In the present work, the tilting angle of liquidvapour interface near the contour line of sessile nanodroplets on plane solid substrates is


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small (< 15°) so that zx and zxx are very small and their contributions can be neglected. The Hamaker constant can be calculated from the refractive index, dielectric constant, and absorption frequencies of the interaction materials23. Since the values of refractive index and dielectric constant are smaller for oleic acid than for glass33, the Hamaker constant for oleic acid thin films on glass is negative. This determines positive values of the Van der Waals component of disjoining pressure, which explains the adsorption of oleic acid on the clean (not hydroxylated) glass surface. The Hamaker constant for oil film on glass has typical values around 10-21 J,34 which determines positive disjoining pressure values around 106 Pa at a film thickness of 1 nm. The structural component of the disjoining pressure of oleic acid on hydrophilic substrates may arise from the tendency of molecules to orient with their polar heads towards the hydrophilic surface of the substrate. This tendency is enhanced by hydroxylation of the glass surface. In agreement with experimental observations, we consider the structural component of disjoining pressure negative with an exponential decaying dependence35 on the interface distance, z: Π s ( z ) = B ⋅ exp( −α ⋅ z ) ,

(4)

with the constants α and B (B < 0). The structural component of the disjoining pressure is negative and contributes to thinning of oleic acid thin films adsorbed on the hydroxylated glass. For the hydroxylated glass, Πs is considered as the dominant component of the disjoining pressure in oleic acid nanodroplets. For the solvent-cleaned glass, Πs is dominant at relatively very small values of z (few nanometers). The shape of liquid-vapour interface of the nanodroplets is determined by the augmented Laplace equation:

p 2 − p1 = σ 12 ⋅ H − Π ( z ) ,


(5)

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where p2 and p1 are pressure values in liquid and vapour phases, respectively, H is the local curvature of the vapour-liquid interface, σ12, the superficial tension of the vapour-liquid interface, and Π(z), the disjoining pressure given by eqs 2-4. Benet at al [36] showed that for very small values of z (z < 1nm), σ12(z) and Π(z) and are interconnected, the heightdependent superficial tension being proportional to the derivative ∂Π ( z ) / ∂z . However, far from the droplet edge (z > 1nm) the effect can be neglected. In the present approach we consider the constant macroscopic value for σ12. Then, the eq 5 can be written as:

H=

∆p + Π ( z )

σ 12

,

(6)

where ∆p = p2-p1 is the excess pressure in liquid phase with respect to the vapour phase and is considered as being constant all over the liquid-vapour interface. The effect of gravitational force is neglected in eq 6 because the variation of hydrostatic pressure ρgz (ρg is the density of gravitational force in liquid) is negligible at nanoscale. The eq (6) shows that the positive disjoining pressure acts to increase the local curvature of the nanodroplet liquid-vapour interface, while the negative disjoining pressure acts to decrease it. Considering the axis symmetrical geometry of the sessile nanodroplets on a flat substrate (placed at the height z = 0), their shape is completely described by droplet height profile z(r) (Figure 1). In this case, the mean curvature of liquid interface, H, is37:

H=

dϕ sin ϕ , + ds r

(7)

where ϕ is the tilting angle of the interface with respect to horizontal (Or) direction and ds is an element of interface length (Figure 1). Then, the following two equations (see Figure 1) determine the projections of ds on the r and z axis:


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dr = cos ϕ ds

(8)

dz = − sin ϕ ds

(9)

Use of the eqs 6 and 7 in the eqs 8 and 9 yields the following equations determining the droplet profile z(r): cos ϕ  dr  dϕ = ∆p / σ + Π ( z ) / σ − sin ϕ / r  12 12  − sin ϕ  dz =  dϕ ∆p / σ 12 + Π ( z ) / σ 12 − sin ϕ / r

(10)

Let us consider the case when Π(z) is negative, which means that structural component is the dominant component of Π, irrespective the value of z. At relatively large values of z (the top part of the nanodroplet), the effect of disjoining pressure is weak (|Π| σ12/R. For the case of positive disjoining pressure, the repulsive interface forces act to increase the liquid-vapour interface curvature and in this case the curvature of the droplet height profile z(r) is always positive (no inflection point M on the droplet height profile). Therefore, the repulsive disjoining pressure (Π > 0) acts to increase the droplet contact angle while the attractive disjoining pressure (Π < 0) acts to decrease it. This is illustrated by Figure 2, which presents the solutions of augmented Laplace equation for the cases of positive disjoining pressure (Π dominated by Πm described by eq 3) and negative disjoining pressure (Π dominated by Πs described by eq 4). The effect of molecular component of


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disjoining pressure on the droplet shape of is weak and visible only at very small values of z (near the substrate). The effect of structural component of disjoining pressure on the droplet shape is much stronger and the droplet height profile is far from that of a spherical cap. The functions Πm(z) and Πs(z) and Π= Πm(z)+Πs(z) used in integration of the eq 10 are presented in fig 2 b). The position of the infection point on the height profile of the droplet is determined by Π(z0) =-σ12/R. It is worth mentioning here that the eq 10 is not accurate for very small values of z because the mean-field theory of disjoining pressure uses the approximation of a continuum field of molecular forces, which implies coverage of the substrate with at least one molecular layer (thickness around 2.4 nm in the case of a monolayer of oleic acid molecules oriented vertically38). Therefore, the present theoretical model cannot describe accurately the shape of the nanodroplets at their edges, i. e. the contact line. At their edges, the nanodroplet surface may be discontinuous, or it may be connected to a uniform adsorbed film on the substrate. In the latter case, the nanodroplets are thermodynamic equilibrium with a stable adsorbed film of thickness e and eq (5) can be used to determine the relationship between disjoining pressure values at thickness e and zmax (top of nanodroplets)39: a)

b)


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Figure 2 a) Illustration of the effect of disjoining pressure on the shape of sessile droplets on a flat substrate. The shape of the nanodroplet unaffected by the disjoining pressure (Π = 0) corresponds to a spherical cap with the radius of 1000 nm and height of 30 nm. The curvature of droplet liquid-vapour interface increases for Π = Πm > 0 and decreases for Π = Πs < 0. The dependences on interface distance, z, described by eqs 3 and 4 for Π(z) have been assumed. In the latter case, the droplet height profile shows an inflexion point M. b) Plots of Πs(z), Πm(z) and Π =Πs(z)+Πm (z) taken for simulation of the effect of disjoining pressure on the shape of nanodroplets illustrated in a).

Π (e) = Π ( zmax ) − σ 12 ⋅ H

(15)

3. Materials and methods Numerical solutions have been found by integration of eq 10, in which a dependences of Π(z) given by eqs 3 and 4 have been considered, using a forth order Runge-Kutta algorithm with variable integration step implemented by the Matlab ode15s function. For the case Π(z) < 0, the eq 10 is stiff in M(z0, r0) and the integration variable ϕ does not have a monotone variation. Therefore, in this case the integration has been made in two parts: a) from ϕ0 = 0 to ϕ > ϕmax = θ, to yield the top part of the droplet profile, and b) from ϕ0’ = θ to

ϕ = 0, to yield the bottom part of the profile (near the substrate). The initial conditions, r(ϕ0), z(ϕ0), and the values of constants in eq 4 have been chosen to obtain the best fit of the numerical solutions with the experimental droplet profiles. During the integration of the top part of droplet height profile, while the algorithm approached the value ϕmax, dϕ/dr 0 and the integration step decreased towards the smallest value allowed by the integration algorithm (4.4⋅10-14), when the integration of the top part of droplet profile stopped yielding the value θ = ϕmax. To obtain the height profile of the liquid-vapour interface in the concave region (near


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the drop edge), the eq 10 are integrated with initial conditions r = r0+∆r, z =z0+∆z at ϕ’0 = θ, where ∆r and ∆z are small variations of r and z, respectively (∆r=2 nm and ∆z =-∆r⋅tanθ). Near the substrate, the eq 10 becomes stiff again because the component Πm(z) ∞. Therefore the integration stops at certain height above the substrate, where the integration step decreased towards the smallest value allowed by the integration algorithm. Figure 2 shows example of solutions obtained for Π = Πs < 0, Π = 0, and Π = Πm > 0. The density of points on the plots shows how the integration algorithm decreases the integration variable step near the inflexion point, M and near the substrate surface. For the experimental part, oleic acid (technical grade) has been procured from Sigma Aldrich. Glass substrates (Cover glasses from Agar Scientific, 13 mm in diameter) were cleaned with acetone, ethanol and deionized water (each step for 15 minutes) in a sonication bath. To change surface properties, the glass substrates cleaned by organic solvents were further treated by exposure for 10 minutes to a d. c. discharge plasma in water vapour at low pressure40. Plasma treatment of glass substrates resulted in further removal of airborne adsorbed contaminant molecules41 and surface hydroxylation, which has been proved by the X-ray photoelectron spectra of the treated glass surfaces42. Finally, the either solvent-cleaned or plasma-hydroxylated glass substrates were placed upside-down on the top of a Petri dish that contained a small volume (2 mL) of oleic acid heated at 80°C. The temperature of substrates has been maintained at lower temperature (60°C) to enhance adsorption and condensation of oleic acid molecules on the glass substrates. After the deposition of oleic acid on the glass substrates, the samples were loaded on a commercial atomic force microscope (XE70 from Park Systems Corp., South Korea) for acquisition of topography images of oleic acid nanodroplets. The topography images were obtained by scanning sample surfaces in true noncontact mode with commercial silicon AFM


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probes (CSG 30 from NT-MDT, Russia) driven to oscillation at a frequency slightly above the resonance frequency of the cantilever with a low free amplitude (about 6 nm). The set point amplitude has been fixed to about 3 nm and the feedback gain was set to maximum possible in order to obtain best images of the nanodroplets. AFM images (not shown) of bare glass substrates taken before and after plasma treatment showed a small decrease of the RMS roughness from 0.24 nm to 0.21 nm as result of plasma cleaning and hydroxylation. These values of glass surface roughness were much smaller than the characteristic lengths (height, curvature and contour radii) of oleic acid nanodroplets, thus indicating that the substrate roughness may have a negligible effect on the nanodroplet shapes. Figure 3 a) shows a typical topography image of oleic acid nanodroplets formed by condensation on hydroxylated glass during 30 minutes. The original AFM images have been levelled to the planar surface of the glass substrate (height = 0) by a homemade Matlab processing and analyzing software. The image shows a good coverage of substrate with droplets with more or less symmetric shapes (circular contour lines). Formation of nanodroplets with asymmetric shapes can be attributed to the heterogeneity at nanoscale of the plasma treated glass surface. Droplet height profiles, z(r), in a vertical plane crossing the axis of symmetric nanodroplets were selected, processed for removing tip convolution effects by inverse tip image method43 and then analyzed as shown in Figure 3 b). The tip image has been obtained by scanning sharp edges of a standard silicon grid (TGG1 from NT-MDT Spectrum Instruments). It has been found that tip convolution had little effects on the shape of the scanned oleic acid nanodroplets. After deconvolution, the droplet height profiles were fitted with the circular shape to determine the curvature radius of the spherical cap, R, the radius of the contour line, rc, the droplet height, h, and the contact angle, θ, as indicated in Figure 3 b). Then, the position of inflexion point M on the droplet height profile is identified to determine z0, r0 and interface maximum slopping angle, θ = ϕmax.


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Figure 3 a) Topography image (5 × 5 µm2) of oleic acid nanodroplets formed on hydroxylated glass during 30 minutes. b) Typical height profile, z(r), of a nanodroplet (along the segment AB on Figure 3 a) showing the spherical cap symmetry. The left side of the profile has been mirrored to be compared to the right side of the profile.

4. Shape of the oleic acid nanodroplets on glass substrates. Effect of the disjoining pressure The shape of oleic acid nanodroplets on glass can be approximated with spherical cap, at list for the upper part of nanodroplets, where disjoining pressure is much smaller than the Laplace pressure. Figure 4 shows a comparison between height profiles of two oleic acid nanodroplets with closed values of curvature radius, R, and height, h, but with noticeable different shapes due to a difference in disjoining pressure values. This difference in disjoining pressure was determined by surface hydroxylation of the glass substrate (Figure 4 b) as compared with the surface of solvent-cleaned glass substrate (Figure 4 a). For the oleic acid nanodroplets on solvent-cleaned glass the disjoining pressure is weak and its effect on the nanodroplet shape is hardly visible at the nanodroplet edge. As result, the shape of nanodroplet is well approximated to a spherical cap with a contact angle determined by the droplet curvature radius and height.


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Figure 4 Effect of disjoining pressure on the shape of oleic acid nanodroplets on a) non hydroxylated glass and on b) hydroxylated glass. The size of the analyzed nanodroplets is roughly the same (contour radius rc ≈ 275nm and h ≈ 30 nm). The values of curvature radius, R, at the droplet apex are slightly different. The experimental z(r) profile of the droplets is compared with spherical cap shape (negligible effect of disjoining pressure) and solutions of augmented Laplace equation. The level z0 separating the region dominated by the excess Laplace pressure ∆p = σ/R to that dominated by the disjoining pressure Π(z) is figured on the graphs.

The critical height of the droplet liquid-vapour interface (at which Π = -σ12/R) has a small value in this case (z0 ≈ 2.5 nm). For the sessile nanodroplets on hydroxylated glass, the effect of disjoining pressure on the droplet shape is much more important. The top part of the droplet (dominated by the Laplace pressure) can be approximated to spherical shape, but with a curvature radius slightly larger than that determined by the liquid-vapour excess pressure. On this part, the effect of the disjoining pressure is noticed as a decrease of droplet curvature (or increase of curvature radius). In this case, the critical height of droplet liquid-vapour interface has a much larger value (z0 ≈ 12 nm). Towards the edge (z < z0) the height profile is concave because in this region the negative disjoining pressure is larger than the Laplace pressure. Figure 5 presents the plots of the total disjoining pressure (Π= Πm + Πs) dependence on the liquid height as they were determined by the best fit of the augmented Young-Laplace solutions to the experimental height profiles presented in fig 4. The main difference between


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the values of Π for oleic acid on the hydroxylated and not hydroxylated glass substrates is given by the interaction length of the structural component of the disjoining pressure, which was 8 nm for the hydroxylated glass and 2.5 nm for non-hydroxylated glass. Therefore, hydroxylation of the glass surface restricted the random orientation of oleic acid molecules within a thickness of few molecular monolayers.

Figure 5 Dependence on the height, z, of the total disjoining pressure as resulted from the best fit of the solutions of augmented Young-Laplace equation to the experimental height profiles of oleic acid nanodroplets on hydroxylated and not hydroxylated glass (shown in figure 4).

The effect of disjoining pressure on the droplet shape in its top part is more pronounced in droplets with small height. Figure 6 shows the experimental and theoretical height profiles of two different sessile nanodroplets on hydroxylated glass. The droplets have comparable curvature radii, but noticeable different height values. Theoretical droplet height profile


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matching the experimental height profile for the droplet with small height (h = 12 nm) showed a much larger curvature radius (R = 1300 nm) than that determined by the liquidvapour excess pressure (R = 900 nm) because in this case the negative disjoining pressure is large and acts to reduce the droplet curvature. However, the effect of disjoining on the droplet curvature radius on its top part is negligible for the nanodroplet with large height (h = 39 nm) where the droplet curvature on its top is slightly increased (R = 1050 nm) as compared to the curvature radius determined by the excess pressure (R = 1000 nm).

Figure 6 Effect of disjoining pressure on the shape of oleic acid nanodroplets on hydroxylated glass for a) a nanodroplet with small height (h = 13 nm), and b) a nanodroplet with large height (h = 39 nm). The values of curvature radius, R, at the droplet apex are slightly different. The experimental z(r) profile of the droplets is compared with spherical cap shape (negligible effect of disjoining pressure) and solutions of augmented Young-Laplace equation. The level z0 separating the region dominated by the Laplace pressure and that dominated by the disjoining pressure Π(z) is figured on the graphs. The parameter, R, taken in the theoretical model is the droplet curvature radius determined solely by the liquid-vapour excess pressure (∆p = 2σ/R).


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5. Statistic analysis of oleic acid nanodroplets formed by condensation on cooled glass surfaces exposed to oleic acid vapour. As described in the previous section, the shape of sessile oleic acid nanodroplets on glass substrates changed as result of plasma hydroxylation of the substrates. Overall, the hydroxylation of glass surface had a big impact on the surface coverage (number of droplets on unit area) and the size of nanodroplets. As a rule, for the same exposure conditions, the number of sessile nanodroplets was much larger on plasma hydroxylated glass than on solvent-cleaned glass. However, the size of nanodroplets on solvent-cleaned glass was slightly larger. Figure 7 shows topography images of oleic acid droplets deposited by condensation of oleic acid vapour for 10 minutes on solvent-cleaned glass (part A) and on plasma-hydroxilated glass (part B). On average, about 6 nanodroplets/µm2 formed on hydroxylated glass surface, while only 1 nanodroplet/µm2 formed on the solvent-cleaned glass substrate. The size of droplets, especially their height, was also different on the two glass substrates. The plot in Figure 8 shows the distributions of droplet height and curvature radius values for the nanodroplets found on an area of 25 square microns of glass substrates. While for the hydroxylated glass there was found a number of about 200 droplets, for the solvent-cleaned glass, the number of droplets was about 30.


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Figure 7 Topography images of sessile oleic acid nanodroplets formed by condensation on solvent cleaned glass (part A) and on plasma hydroxylated glass (part B)

The droplet height and curvature radius values are not independent parameters, the droplet height having a linear dependence on curvature radius, the slope of this dependence being determined by the value of the droplet contact angle, which was around 10° for the nanodroplets formed on hydroxylated glass and around 11.5° for the nanodroplets formed on solvent-cleaned glass .

Figure 8 Dependence of droplet height on the curvature radius for nanodroplets formed by condensation (10 minutes) on plasma hydroxylated glass (back) and solvent-cleaned glass (red).

For the nanodroplets formed on the hydroxylated glass, an important deviation from the linear dependence and a saturation of droplet height values were noticed in the region of large curvature radii (> 1000 nm). However, the height and size of nanodroplets increased by the increase of the deposition time (exposure time of substrates to oleic acid vapour) towards 20 and 30 minutes (plots not shown). The contact angle of the nanodroplets, which was


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measured as the maximum tilting slope of the droplet height profile, showed a much larger variation for the nanodroplets formed on the hydroxylated glass than for the nanodroplets formed on solvent-cleaned glass. Figure 9 shows the distribution of contact angle values for the two substrates. The distributions were fitted to Gauss distribution to yield the most probable (mean) and standard deviation values. Thus, for the nanodroplets formed on hydroxylated glass θ = 7.75°±1.25°, while for the nanodroplets formed on solvent-cleaned glass θ = 10.75° ± 0.6°. These values are much smaller than the macroscopic contact angles measured on macroscopic sessile droplets on the two substrates, i.e. θ = 33.8° ± 1.2° for the plasma-hydroxylated glass and θ = 28.6° ± 4.4° for the solvent-cleaned glass. We believe that this large difference between macroscopic and nanoscopic values of contact angle is an effect of interfacial forces (disjoining pressure) important at nanoscale, but negligible at macroscale.

Figure 9 Distributions of contact angle values for the nanodroplets formed by condensation on hydroxylated glass (black) and, respectively, on solvent cleaned glass (red). Exposure time of cooled


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substrates to oleic acid vapour was 10 minutes. The distributions were fitted to Gaussian distributions to yield θ = 7.75±1.25 ° for droplets deposited on hydroxylated glass and θ = 10.75 ± 0.6 ° for droplets deposited on solvent cleaned glass.

The height z0 of the inflexion point on the droplet height profiles is much larger for the nanodroplets on the hydroxylated glass than for the droplets on solvent-cleaned glass. For the droplets on hydroxylated glass a relatively large variation of z0 has been observed because z0 increases with the increase of droplet height, fact illustrated by fig 6. Figure 10 presents the histograms with value distributions of z0 for oleic acid nanodroplets on hydroxylated and solvent-cleaned glass substrates. The values of z0 on the solvent-cleaned glass are much smaller due to a much smaller effect of the structural disjoining pressure on this substrate.


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Figure 10 Value distributions of z0 for the nanodroplets formed by condensation on hydroxylated glass (black) and, respectively, on solvent cleaned glass (red). Solid lines are shown to guide the eye.

6. Conclusion Condensation of oleic acid vapour on glass substrates resulted in deposition of sessile oleic acid nanodroplets. The atomic force microscopy technique was used to analyze the shape and size of the sessile oleic acid nanodroplets formed on glass substrates cleaned by organic solvents and then hydroxilated by a low-pressure discharge plasma treatment. For the oleic acid nanodroplets formed on plasma-hydroxylated glass an important deviation of the droplet shape from the spherical cap shape has been observed. This deviation has been attributed to the molecular forces determining an important disjoining pressure at the nanodroplet interfaces with the substrate and vapour, respectively. Hydroxylation of glass surface gave rise of a strong negative disjoining pressure due to structural forces arisen from orientation of oleic acid molecules with polar head towards substrate. As result, the local curvature of liquid-vapour interface of the acid nanodroplets on the hydroxylated glass showed large variations, especially in the vicinity of the droplet edges. Analysis of droplet height profiles showed occurrence of an inflexion point where the profile curvature changes from convex on the top part of the droplet to concave on the vicinity of the droplet edge. The tilting angle of the liquid-vapour interface with respect to the substrate has a maximum value at the profile inflexion point, maximum that is considered as the nanoscopic value of liquidsolid contact angle. For the oleic acid sessile droplets on glass substrates, the nanoscopic values of contact angles were much smaller than their macroscopic values, the difference being larger for the hydroxylated glass substrates as compared to the solvent-cleaned glass substrates. A theoretical model explaining these experimental findings is proposed. The


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model is based on the augmented Young-Laplace equation, in which the liquid excess pressure is considered as equal to the sum of Laplace and disjoining pressures. Thus, for the case of negative disjoining pressure, the Laplace pressure and, consequently, the liquidvapour interface curvature are decreased. For the case of positive disjoining pressure, the Laplace pressure and liquid-vapour interface curvature are increased. For the case of sessile oleic acid nanodroplets on hydroxylated glass, the assumption of an exponential decay dependence of the negative structural disjoining pressure on the interface separation distance resulted in numerical solutions of augmented Young-Laplace equation in good agreement with the experiment.

Acknowledgement 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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Disjoining Pressure in Partial Wetting on the Nanoscale

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