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Shape and effective spring constant of liquid interfaces probed at the nanometer scale: finite size effects

We investigate the shape and mechanical properties of liquid interfaces down to nanometer scale by atomic force microscopy (AFM) and scanning electron microscopy (SEM) combined with in situ micromanipulation techniques. In both cases, the interface is probed with a cylindrical nanofiber with radius R of the order of 25-100 nm. The effective spring constant of the nanomeniscus oscillated around its equilibrium position is determined by static and frequency-modulation (FM) AFM modes. In the case of an unbounded meniscus, we find that the effective spring constant k is proportional to the surface tension {\gamma} of the liquid through k = (0.51 +- 0.06) {\gamma}, regardless of the excitation frequency from quasistatic up to 450 kHz. A model based on the equilibrium shape of the meniscus reproduces well the experimental data. Electron microscopy allowed to visualize the meniscus profile around the fiber with a lateral resolution of the order of 10 nm and confirmed its catenary shape. The influence of a lateral confinement of the interface is also investigated. We showed that the lateral extension L of the meniscus influences the effective spring constant following a logarithmic evolution k ~ 2{\pi}{\gamma}/ln(L/R) deduced from the model. This comprehensive study of liquid interface properties over more than four orders of magnitude in meniscus size, shows that advanced FM-AFM and SEM techniques are promising tools for the investigation of mechanical properties of liquids down to nanometer scale.


INTRODUCTION
Liquid menisci are ubiquitous in many natural and technological processes. For example, they change the physical properties of wet granular media, 1 allow the motion of insects and selfassembly of particles at a liquid interface. [2][3][4] These surface effects become even more pronounced when scaling down in size. 5 The resulting capillary force may be used to manipulate 6 or fabricate [7][8] 3D objects, whereas it may be detrimental in the development of MEMS and NEMS. It also leads to artifacts in atomic force microscopy (AFM) imaging. [9][10] Many studies have been devoted to the understanding of the capillary force exerted by a liquid meniscus bridging two solid surfaces, 11 for many relevant geometries and for either volatile 12 or non-volatile liquids. Recently, the development of AFM allowed to probe capillary forces down to nanometer scale, the versatility of AFM modes giving a large variety of solicitations. Static measurements gave information on the liquid nanomeniscus condensed at the tip-substrate contact 13 while non-conventional probes with various types of micro-or nano-fibers at their extremity provided a model geometry to investigate wetting properties at nanoscale. 14-17 The latter case gives access to liquid surface tension by rod-in-free surface type measurements. [18][19] The AFM cantilever can also be used in a passive way to monitor droplet oscillation modes 20 whereas dynamic AFM modes 9, 21 and thermal noise 22-23 also gave access to dissipation processes at stake in the meniscus.
Since the mechanical properties of interfaces are strongly related to their shape, it is also important to develop methods to visualize the liquid interface down to the nanometer scale. Electron microscopy combined with in situ micromanipulation techniques now provides a powerful way to tackle this issue. Performing mechanical test in a SEM (Scanning Electron Microscope) adds the advantage of visualizing the experiment, but requires samples that can sustain electron irradiation and above all, vacuum. To study liquids, one has to use nanoaquariums, 24 or if manipulation is needed, liquids with a very low vapor pressure (e.g. ionic liquids IL). [25][26] In this context, few studies have considered small perturbations around the equilibrium position which leads to the useful description of the liquid bridge as a spring with an effective spring constant mainly depending on contact angle and size. [27][28] Other situations such as droplets or bubbles, 29-30 contact lines 31 or biological membranes 32 have been considered. Interestingly, in most cases, the effective spring constant depends on the lateral extension of the liquid interface deformation through where is the radius of the probe. However, due to the difficulty to probe a large range in , this logarithmic variation has not been assessed experimentally.
Here, we combine two high resolution microscopy techniques (AFM and SEM) in order to study the static and dynamic behavior of a nanomeniscus created by dipping a nanofiber in a liquid interface ( Fig. 1). The cylindrical geometry was used to probe both the shape and mechanical properties of the liquid interface down to the nanometer scale. We determined the static and dynamic effective 3 spring constant of the meniscus from experiments performed in the frequency modulation AFM mode (FM-AFM) and showed how this value is altered by a lateral confinement of the meniscus.
Interestingly, the use of nanoprobes allowed us to study this effect over many orders of magnitude in lateral size of interfaces. Scanning electron microscopy (SEM) combined with in situ nanomanipulation technique was used to directly visualize the shape of the nanomeniscus with high resolution. A theoretical model allows interpreting the results of the experiments on both the shape and the stiffness of nanomenisci, leading to a comprehensive study of the elasticity of liquid interfaces down to nanometer scale.

EXPERIMENTAL SECTION
The experiment consists in dipping in and withdrawing a nanofiber from the free surface of a liquid bath. Two types of tips were used in AFM experiments, both with a nanocylinder ca. 50 nm in diameter and ca. 500 nm in length at their apex, but with different chemical nature (Silicon probes CDP55 from Team Nanotec and Ag 2 Ga probes from NaugaNeedles). The spring constant of the cantilevers was chosen of the order of , which is much larger than the expected meniscus spring constant while being sufficiently soft to measure forces with good resolution. In our instrument, the nanofiber is inclined by 11° with respect to the vertical. The resulting corrections to the total force coming from the asymmetric deformation of the interface 33 (1,8 %) and from the torque contribution in the AFM measurement 18 (0,4 %) are neglected in the following. Under the e-beam of the SEM, some ionic liquids (generally with the heaviest molecular mass) showed a change from liquid to partially solidified behaviour. The measurements of meniscus shapes were carried out on IL2-2 and IL2-4 that remained liquid even after prolonged exposition.
Possible solidification under the e-beam has been discussed on ILs similar to the ones used here, mainly for TEM 37-39 for doses ranging from 10 +19 e.m -2 to 3.10 +26 e.m -2 at acceleration voltages above 120 kV. We performed our SEM observations at a tension of 2 to 5 kV and at doses ranging from 10 +19 e.m -2 to 10 +21 e.m -2 , which means that we stayed on the lower end of the transmitted energy to keep a liquid behaviour.
For AFM measurements, the liquids were put into small cylindrical vessels with various dimensions.
A 5 mm large and 2 mm deep hole was drilled in a Teflon sample holder. Since the radius of the hole is larger than the capillary length (which is of the order of 2 mm in our experiments), the meniscus can be considered as unbounded (see Section III). Confined liquid interfaces were obtained by milling 100 µm and 10 µm holes in a silicon wafer with a focused ion beam (FIB-SEM Helios NanoLab 600i, FEI). In all cases, the reservoir was filled using a microinjector (Narishige) and the volume of liquid was precisely adjusted in order to get a flat interface pinned on the sharp edges of the reservoir. This is a crucial issue to avoid the formation of a meniscus and the mobility of the contact line at the edge of the reservoir which may strongly modify the mechanical response of the interface. 40 5 The AFM experiments were performed on a PicoForce instrument (Bruker) operated in the frequency-modulation mode (FM-AFM) using a phase-lock-loop device (HF2PLL, Zürich Intruments).
The nanocylinder at the extremity of the tip was dipped in and withdrawn from the liquid bath, with a typical Z range of 1 µm at a velocity of 2 µm.s -1 . The capillary force was first measured in static mode from the deflection of the cantilever. Then, the same Z ramp was applied while oscillating the tip at a constant amplitude of 7 nm and monitoring the frequency shift .
In order to visualize the shape of the meniscus we also performed experiments in a combined

INTERFACE
We first studied the case of an unbounded interface using the large 5 mm container. The nanomeniscus formed around the tip was probed mechanically using the AFM operated in static or dynamic modes as described below: -For static measurements, the capillary force is computed from the z cantilever deflection as . An example of curve obtained using a 60 nm in diameter nanocylinder dipped in the free interface of IL2-2 ionic liquid is reported on Fig. 2(a). The interpretation of such curves has already been discussed in several papers. 14-17 The hysteresis between advancing and receding curves is similar to what is obtained macroscopically with a Wilhelmy balance technique. It results from the presence of a large number of defects at the surface of the tip, yet too small to be resolved by SEM.
The advancing and receding contact angles are estimated from the mean values of the two flat parts of the curves using the capillary force . 41 In the case of Fig. 2 when the direction of motion of the tip is reversed (1), due to hysteresis, the contact line remains pinned until the receding contact angle is reached; just before the meniscus snaps off (2), when the nanomeniscus is pinned at the tip apex. 15 The slopes of the F(z) measured on these two portions gives the static spring constant with an accuracy of 20 % (including the uncertainty on the determination of the cantilever spring constant which is of order 7 %).

6
-The same configuration was investigated using the frequency modulation AFM (FM-AFM) mode. 42 In this mode, the cantilever is vibrated at its resonance frequency using a phase lock loop (PLL) device. A PID modulates the excitation signal in order to maintain constant tip oscillation amplitude. The interaction of the tip with the liquid leads to a frequency shift where is the resonant frequency in air and to a change in excitation amplitude . The advantage of this mode compared to the standard Amplitude Modulation (AM-AFM) mode used in air, is that it allows to measure independently the conservative (through ) and the dissipative (through ) contributions of the interaction. In this paper, we concentrate on the conservative contribution that shows up as the frequency shift due to the presence of the meniscus.
The system can be modeled as a harmonic oscillator with a frequency in air where is the effective mass of the cantilever. A shift in resonance frequency may therefore have two origins: (i) a change of spring constant: indeed, the meniscus acts as a spring with stiffness k dyn pulling on the tip which leads to an effective spring constant ; (ii) a change in the mass of the system: the oscillation of the nanofiber induces a velocity field in the liquid. The mass of this viscous layer around the fiber then adds to the system. The curve reported on Fig. 2 can therefore be interpreted as follows: the sudden positive frequency shift when the meniscus is created is attributed to the meniscus stiffness while the negative slope for positive Z is an added mass effect. Since the added mass effect is negligible at the creation of the meniscus, the associated frequency shift reads . 42 The dynamic meniscus spring constant can therefore be determined using with an uncertainty of order 12 %. The same procedure can be performed using a higher oscillation mode of the cantilever (see Supplementary Information). The frequency of the second mode is given by , 43 whereas its effective spring constant reads . Note that, since the thickness and therefore the mass of the viscous layer is a function of the liquid viscosity, 44 the negative slope for positive Z is also a function of the liquid viscosity as observed in Fig. 2b.
It is important to note that the pinning of the contact line during the tip oscillation is crucial to guarantee that the meniscus is deformed following the tip motion. This is justified by the fact that the oscillation amplitude (7 nm for all reported data) is much smaller than the distance required to switch from advancing to receding contact angles, observed when the tip motion is reversed [of the order of 100 nm in Fig. 2(a)]. It is also validated by the fact that the frequency shift is independent of the oscillation amplitude up to 21 nm (see Supplementary Information). As discussed above, the use of non-volatile liquid was also dictated by this constraint. These conditions guarantee that the spring constant is measured on a pinned meniscus oscillated at a frequency of the order of 75 kHz for the fundamental mode and 450 kHz for the second mode . 7 The simple cylindrical geometry combined with the use of non-volatile liquids therefore gives a model system with well controlled boundary conditions for FM-AFM measurements. This allows to extract quantitative measurements, whereas in the study of Jai et al 45 evaporation and conical tips were leading to poorly defined menisci. The experimental results are plotted in Fig. 3. We gathered on the same graph all the spring constant values measured in the unbounded meniscus configuration, in static and dynamic modes, at both and frequencies for the latter and with three tips of similar diameter (60 nm) but different chemical nature, two of them in silicon and one in Ag 2 Ga alloy.
The spring constant values were plotted as a function of the liquid surface tension. Within the uncertainty of the measure, all data points fall on a same curve, indicating that there is no effect of the frequency of excitation on the response of the liquid interface (up to 450 kHz). This is a strong indication that FM-AFM allows to decouple the effects of interfacial and bulk properties of the liquid which both contribute to the response of the system. Note that this holds for the relatively large viscosities of the liquid used in this study (ranging from 36 mPa.s to 500 mPa.s).
In a first approximation, all data can be fitted by a linear relationship that shows that the spring constant is proportional to the surface tension of the liquid. In the following, we normalize the spring constant values by the surface tension . We find, for the whole series of liquids investigated, that . The spring constant of an unbounded liquid meniscus is therefore approximately half of the surface tension.

Spring constant of an unbounded liquid interface
In order to interpret these results, we have developed a model to compute the spring constant of a liquid interface. We consider a meniscus pinned on a fiber with a diameter dipped in an unbounded liquid bath [ Fig. 1 and Fig. 4(a)]. The meniscus spring constant extracted from the capillary force F exerted by the meniscus reads (1) where H is the altitude of the contact line with respect to the flat part of the meniscus i.e. the height of the meniscus (see Fig. 1). The shape of the liquid interface is solution of the equation (2) which is obtained by a balance, at each point of the interface, of the hydrostatic pressure with the Laplace pressure imposed by the curvature of the meniscus. is the inverse of the capillary length defined above. It is interesting to note that, in contrast to the 2D meniscus case, 44 Eq. (2)    Note that the characteristic time for equilibration of a liquid meniscus can be estimated as where A is the amplitude of deformation and is the capillary velocity of the liquid. In the experimental conditions, we find which is much larger than the FM-AFM excitation frequency used here. Even in dynamic mode, the meniscus can therefore be considered at equilibrium at each instant, consistently with the observation that .

Spring constant of a laterally confined liquid interface
In this section, we consider the case of a liquid container with a radius smaller than the capillary length as schematized in Fig. 4(b). The shape of the interface is still given by Eq. (2) The height of the meniscus of extension deduced assuming reads (6) Combining with Eq. (1), the expression of the reduced spring constant is obtained: The spring constant of the liquid interface strongly depends on its lateral extension. In a first approximation, this evolution can be described by keeping only the logarithmic term in Eq. (7), leading to an approximate formula: This logarithmic evolution is similar to the one calculated on bubbles or droplets 29, 50 or contact lines 31 and is characteristic of liquid behaviour.

COMPARISON WITH EXPERIMENTS
In this section, we report experiments performed by SEM and FM-AFM in order to assess the conclusions of the model described in the theoretical model section.

SEM observation of nanomenisci
The model presented above is based on a meniscus profile described by a catenary curve [Eq. (5)].
We verified this hypothesis by directly visualizing the shape of the meniscus in a SEM. The experimental procedure involving advanced in situ micromanipulation techniques has been described in Section II. Since the nanofibers used in the AFM experiments could not be attached to the SEM micromanipulation tungsten tip, we performed these experiments with homemade carbon nanocones (carbon nanocones terminated by a nanotube 34 ).
The motion of the tip was slow enough (about 500 nm/s) to allow SEM monitoring of the process.
The motion was stopped a few seconds in order to acquire good resolution images. An example of SEM micrograph of nanomeniscus around a 180 nm diameter carbon fiber is reported in Fig. 5(a).
The profile of the meniscus is observed over several microns with a lateral resolution on the order of 10 nm. Since the unperturbed interface profile is not flat due to the use of droplets, we subtracted this latter shape from the measured profile in order to get the influence of the probe on the interface. An example of such a profile extracted with a MATLAB program is reported in black in

Spring constant of unbounded meniscus
In order to make a quantitative comparison of the experimental values of with the model, we first considered the case of unbounded meniscus reported in Section III. In order to take into account the effect of the contact angle, we plotted in Fig. 6, for each experiment, the value as a function of the contact angle deduced from the measured capillary force. Note that for clarity, we used as the average of advancing and receding contact angles. The results are compared with the theoretical model [Eq. (7)] using and .  (see Supplementary Information). Interestingly, the angle of the liquid interface with 14 respect to the vertical can be deduced, for each point of the curve, using . We can therefore, with a single experiment, measure a whole curve as reported in Fig. 8 in blue dots.
In order to increase the range of contact angles above the maximum experimentally available value (30°), we also reported on Fig. 8, in blue dashed line, the modelisation of the capillary force. 54 We observe that the dimensionless spring constant , varies from 0 to a value which could be as large as and is limited experimentally to . These values are significantly larger than the values obtained for an unbounded meniscus also reported in red in Fig. 8. The curve from the NADIS experiment is well described by Eq. (7) using the value deduced from the fit of the force curve, as plotted in blue solid line in Fig. 8. This demonstrates that the description of the effective meniscus stiffness provided by Eq. (7) remains valid down to very small values and can therefore be applied to nanomenici. Effect of lateral extension of the interface. The four configurations studied experimentally provide a method to study the elasticity of liquid interfaces, with meniscus size ranging from millimetric down to nanometric scale. In terms of the relevant parameter of the model, this covers more than four orders of magnitude, from to . We plotted in Fig. 9, the dimensionless spring constant as a function of . Note that in the case of NADIS, we considered the value extrapolated using the modelization for a contact angle of 50° which is the mean value of the contact angles obtained in the nanofiber experiments. These results show that the confinement leads to a hardening of the interface by a factor that may be as large as 20 compared to a free unbounded liquid interface. The results can be directly compared to the theoretical model using Eq. (7) using a contact angle of 50°. This is plotted in Fig. 9 in solid line together with the approximate expression proposed in Eq.
(8). This simplified expression which slightly overestimates the spring constant provides a good approximation. The deviation remains smaller than 10 % for and, as expected, increases significantly for smaller values. Figure 9 shows that both curves reproduce fairly well the experimental data over about 5 orders of magnitudes in values.
In a fist approximation, the hardening of a liquid interface due to its lateral confinement is inversely proportional to the logarithmic type ( ). This is similar to other objects involving liquids interface such as droplets or bubbles 29 or contact line. 31 However, this logarithmic dependence was not verified experimentally. This hardening is qualitatively similar to the one observed with lipid membranes 55 or to the case a solid beam with both ends clamped whose bending spring constant increases when decreasing its length. 56 However, in the latter case the variation of effective spring constant ( for unstressed beam) is more pronounced than for liquids systems.

CONCLUSIONS
The nanomeniscus created by dipping nanofibers in a liquid interface was investigated using advanced microscopy techniques. We showed that the dynamic FM-AFM mode gives a simple method to measure the dynamic spring constant of the meniscus. With an unbounded interface, the reduced effective spring constant was found to be proportional to the liquid surface tension following . We extended this study to the case of interfaces with micron scale lateral dimensions and used non-conventional liquid nanodispensing technique to reach nanometer scale. A strong influence of the meniscus size was evidenced with a 20-fold increase of the spring constant compared to the unbounded interface. A theoretical model based on the equilibrium shape of the interface leads to the expression of the effective spring constant which depends on the liquid contact angle and the ratio of the interface to probe dimensions .
Neglecting the influence of contact angle, a simple expression was derived and provides an approximate solution within 10%. The four orders of magnitude range of experimental values available in our experiment, are described by this theoretical model with good agreement. This is, to our knowledge, the first experimental evidence of the logarithmic dependence of the effective spring constant, which was also predicted for other liquid systems.
Electron microscopy combined with in situ micromanipulation techniques was also used to visualize the shape of the (ionic) liquid interface around the nanoprobe with a 10 nm lateral resolution. The

FM-AFM: measurements at the second mode
In order to study the influence of oscillation frequency and to reach larger solicitation frequencies, we repeated the FM-AFM experiments, locking the PLL on the second oscillation mode of the cantilever with a frequency . 2 The results reported on Fig. S2 show similar trends to the ones observed for the fundamental mode i.e a positive jump resulting from the meniscus formation followed by a decrease of the frequency shift due to added mass effects. The meniscus spring constant is derived from where is the effective spring constant of the cantilever corresponding to the second mode which is found to verify . 2 Note that the curve obtained with the IL2-10 liquid is significantly different for the ones reported in Fig. S2 and is not reported. Excitation and deflection curves also evidence a drastic change in the response of this liquid. Since this liquid is the more viscous one ( ), this may come from the large dissipation associated with the oscillation of this liquid at high frequency and reveals the limits of our procedure.

FM-AFM: effect of oscillation amplitude
The experiments reported in the article were performed using small oscillation amplitude in order to guarantee that the contact line remains pinned on the tip during the oscillation. In order to verify this, we reproduced the experiments with increasing amplitude (Fig.   S3). We observe that the oscillation amplitude has no effect for oscillation amplitudes smaller than . For larger values, the frequency shift (and therefore the spring constant) starts to decrease. This is due to the fact that the oscillation amplitude becomes larger than the distance required to switch from advancing to receding contact angle. The contact line does not remain 23 anymore pinned on the tip during the whole oscillation. The resulting motion of the contact line leads to a decrease of the effective meniscus stiffness. Figure S3. Relative frequency shift measured on IL2-2 at the fundamental oscillation mode ( ) as a function of oscillation amplitude .

Shape of liquid interface and meniscus height
The shape of the liquid interface is solution of the equation (1) which is obtained by a balance, at each point of the interface, of the hydrostatic pressure with the Laplace pressure imposed by the curvature of the meniscus. is the inverse of the capillary length defined above.

Unbounded meniscus
In the case of an unbounded meniscus, the boundary conditions read and . An approximate solution of Eq. (1) based on the asymptotic matching technique in the case of a cylindrical fiber was first proposed by Poisson 3 and derived independently by Derjaguin 4 and James. 5 The solution close to the fiber , where gravity [right hand side in Eq. (1)] can be neglected, leads to a catenary profile whereas the solution far from the fiber , obtained for vanishing slopes is a Bessel function. 6 Matching these two approximate solutions leads to: where is the Euler constant. These two solutions are plotted on Fig. S4 together with the numerical solution of Eq. (1) for two different values of the ratio , namely and which corresponds to fibers with diameter 400 µm and 60 nm, respectively. In the case of a nanoprobe such as the one used in the experiment of the article, the solution is a very good approximation up to very large distances compared to the fiber radius (up to ), consistent with the fact that gravity can be neglected for such small size probes [see inset Fig. S4(b)]. inset: same as Fig. S4(b) with a logarithmic scale for .
In all cases, the height of the meniscus on the fiber reads: The subscript refers to unbounded interfaces with infinite extension. Note that, for fibers with small diameter ( ), the Euler constant can be neglected as shown in ref 7 .

Laterally confined liquid interface
In the case of a liquid container with a radius smaller than the capillary length, the shape of the interface is still given by Eq. (1) but the boundary condition far from the fiber becomes , assuming a pinning of the liquid on the sharp edges of the container. Given the nanometric size of our probes, as shown in Fig. S4, the interface profile can be approximated by a catenary solution . The constant is determined using as boundary condition , leading to the interface profile 25 (5) The height of the meniscus of extension is deduced as and, assuming reads (6)

Capillary forces in liquid nanodispensing
An example of force curve measured during the retraction of a NADIS tip 8 from a surface is reported in Fig. S5. It was obtained with a tip with an aperture of diameter 280 nm. A comprehensive study of the capillary forces experienced by the tip during the deposition process was reported in ref 9 . It showed that, for nanochannel sizes larger than 200 nm, the curves can be fitted considering a constant pressure constraint. In this case, the liquid flow in the channel is sufficient to maintain an equilibrium pressure between the reservoir and the meniscus, at all time.
The only adjustable parameter in the model is the value i.e. the radius of the wetted part of the tip, since is known experimentally as the radius of the droplet left on the surface after the meniscus breakage. In the case reported Fig. S5, we find and leading to . This value was used in the article to calculate the associated spring constant.