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Formation of liquid menisci in flexible nanochannels
Journal of Colloid and Interface Science 329 (2009) 133–139
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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Formation of liquid menisci in flexible nanochannels J.W. van Honschoten ∗ , M. Escalante, N.R. Tas, M. Elwenspoek Transducers Science and Technology, MESA + Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 22 July 2008 Accepted 27 September 2008 Available online 7 October 2008
In this paper we analyze the characteristic shape of the liquid meniscus at the fluid air interface in nanochannels of less than 80 nm height capped by a flexible membrane. Because of the induced negative pressure difference between the liquid pressure and the pressure outside, the 0.18 μm thin membrane on top of the channels bends downward. This elastocapillary equilibrium between the surface tension of the wetting liquid and the mechanical forces in the capillary results in a very peculiar shape of the interfacial meniscus, visible from the top through the transparent membrane. For increasing deflection of the membrane, the meniscus is seen to protrude along the channel and its curvature changes from concave to convex in the center. We present an analytical model to describe the meniscus shape in the deformed channel for small membrane deflections. We also show that the protrusion length of the meniscus, which can be measured easily, is an accurate and useful indicator for the membrane deflection. Experimental results on nanochannels filled with ethanol and water are presented and the observed menisci are seen to be in good agreement with the proposed model. © 2008 Elsevier Inc. All rights reserved.
Keywords: Elastocapillarity Nanofluidics Fluid interfaces Nanochannels
1. Introduction When a wetting liquid is brought into contact with a solid surface, a curved fluid–air interface, the so-called meniscus, is developed due to surface tension of the liquid interface. The physical origin and the shape of a meniscus of a wetting liquid in contact with a surface have been a subject of specific interest in capillarity since a long time, starting in particular with the study of Hauksbee [1] on the capillary forces in a liquid in contact with glass plates or tubes, and Maxwell [2]. In 1806, Laplace [3] was the first to deduce an analytic expression for the static shape of the meniscus on a planar wall, a solution that was referred to in 1924 in Bouasse [4], and by Landau [5]. The behavior of a capillary surface in a wedge of prescribed interior corner was addressed in 1969 by Concus and Finn [6], while the static meniscus shape on a vertical fiber in a fluid bath was studied numerically, by White and Tallmadge [7], and analytically [8,9]. While most of the studies on this subject concentrate on a welldefined fluid–air interface (such as a planar wall placed into a liquid reservoir or a fluid imbibing a corner of a capillary) we focus in this paper on the case of a fluid in a deformable confinement with which it interacts, resulting in an elastocapillary equilibrium between the surface tension forces of the wetting fluid and the mechanical forces of the deformed capillary. The deflection of the flexible membranes due to the tensile capillary forces is a promi-
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0021-9797/$ – see front matter doi:10.1016/j.jcis.2008.09.082
©
2008 Elsevier Inc. All rights reserved.
nent effect of elastocapillarity on the nanoscale. In nanochannels, which can be defined as capillaries having at least one dimension on the order of 100 nm or smaller [10–15], capillary action is very prominent due to the large surface-to-volume ratio. The capillary action present in nondeformable nanoscale capillaries and the fluid behavior in these channels were studied systematically by Sobolev et al. [16], by Tas et al. [17] for one-dimensional nanochannels with a height down to 50 nm, and for even smaller cross-sections down to 5 nm by Haneveld et al. [18]. While the phenomenon of elastocapillarity is becoming a subject of increasing interest [19,20], since it can be used in microfabrication technologies [21,22] and can also have damaging effects in microelectromechanical structures [23–28] or in pulmonary airway closure [29], it has not been studied extensively on the nanoscale yet. Capillary action in flexible tubes of nanometer sizes was recently studied numerically [30], especially with respect of the surface-tension-induced buckling of these tubes, while the dynamics of thin films of fluid within small elastic tubes were also studied theoretically by e.g. Halpern and Grotberg [31] and Grotberg and Jensen [32]. As recent work on elastocapillarity in structures of nanodimensions demonstrates [33,34], the shape of the meniscus in these cases is very sensitive to its nanoconfinement and can attain a characteristic shape. To study the meniscus in deforming nanochannels, in this work we investigate hydrophilic nanochannels of about 80 nm with flexible capping layers that are deformed due to the induced negative pressure in the fluid. As a result of the deflection of the membrane, the shape of the meniscus, visible through the thin transparent
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Fig. 1.
layer on top, is substantially modified. As the membrane deflections increase, the fluid protrudes into the center of the channel and the curvature of the meniscus will not be purely concave anymore and becomes even convex in the center. The paper is organized as follows. In Section 2, we describe shortly the fabrication of the hydrophilic nanochannels that are used for the investigation of the development of the meniscus. Then, a description is given of the experiments in which the nanochannels were partially filled with ethanol and water. We describe also the detection of the menisci with an optical microscope, and the method to measure the membrane deflections. In Section 3, a model is presented that describes the shape of the fluid–air interface, starting from the premiss that the sum of the curvatures of the fluid surface in two perpendicular directions is equal to a constant, p /γ (with p the Laplace pressure and γ the liquid surface tension). In Section 4, a comparison between the proposed theoretic model and the experimental observations is given. The model is found to be very appropriate and useful to describe menisci in nanoconfinements. We also obtain the important result that the protrusion length of the meniscus is a very sensitive measure for the deflection of the flexible membranes. This yields a new method to quantify the channel deformation based on observation of this so-called protrusion length. 2. Materials and methods 2.1. Nanochannel fabrication We created arrays of hydrophilic channels of different width in the micrometer range, about 1 cm in length, and height h of 79 nanometer with a capping layer (of total thickness 176 nm) consisting of a laminate of silicon nitride (SiN) and silicon oxide (SiO2 ). In the fabrication process the thickness of the silicon oxide spacer layer, that was grown by dry oxidation at 1100 ◦ C, determined the thickness of the nanochannels. This could be precisely controlled by adjusting the oxidation time. Standard photolithography and 1% HF etching of the silicon oxide layer were carried out to define the nanochannels pattern. The access holes for the fluid supply and the microchannels connecting to the nanochannels were included in a second photolithography process. The channels were covered by a thin membrane consisting of a silicon nitride–silicon oxide composite layer, of thicknesses 97 ± 3 and 79 ± 3 nm, respectively. This layer was transferred by fusion bonding of a second wafer, carrying the plate, to the first wafer, carrying the nanochannel pattern. Before bonding, both wafers were cleaned, first by using a wet chemical HNO3 solution, next in H2 O2 /H2 SO4 (in 1:3 ratio at 130 ◦ C, ‘Piranha’), to assure a hydrophilic surface. The wafer pair was annealed during two hours at 1100 ◦ C in a furnace (N2 ). Finally, the second support wafer was removed by reactive ion etching in a SF6 plasma at room temperature. A more detailed description of the fabrication process is given in [33]. Fig. 1 shows a cross section of the resulting structure. The channel height was measured by means of an ellipsometry measurement of the silicon oxide spacer layer using a Plasmos SD 2002 ellipsometer. The error in the height measurement is estimated to be 3 nm, as a result of the variation in the thickness of the silicon oxide layer on the wafer surface and the precision of the instrument.
Fig. 2.
2.2. Determination of membrane deflection For an accurate determination of the membrane deflections of the partially filled fluid channels, an atomic force microscope (‘AFM,’ Veeco, Dim. 3100), was used to make images of the nanochannels. The chip surface was scanned by the AFM tip with a lateral resolution of 40 nm, both in the direction perpendicular to and parallel to the arrays of fluid channels. Each channel array consisted of a set of rectangular capillaries of a length of 10 mm and varying width, with the channel widths of interest ranging from 3.4 ± 0.1 to 7.5 ± 0.1 μm in 7 steps. Since the bending of the membrane is strongly dependent on the width of the channel (the wider channels have a lower mechanical stiffness), this set of channel widths permits the investigation of a wide range of membrane deflections and the corresponding meniscus shapes. Prior to the filling, it was verified that the membranes did not noticeably deflect due to the scanning of the tip over the surface. Because of the SiN/SiO2 surface roughness and the presence of small pollutants on this layer, the error in the height measurements amounted to approximately ±2 nm. A droplet of 5 μl liquid was introduced into the access holes by pipetting so that the array of nanochannels could fill by capillary action. We used VLSI Selectipur-ethanol (Merck, 99.8% purity), and water (Milli-Q, specific resistance 18 M cm). When the filling process had stopped and the fluid plugs remained almost constant in time (only slowly decreasing due to evaporation), the membrane bendings were recorded by scanning the AFM tip across the channels. Fig. 2 shows a typical scan along the width of part of one channel array along the channel widths with the AFM. The experiments were performed under constant temperature, measured to
J.W. van Honschoten et al. / Journal of Colloid and Interface Science 329 (2009) 133–139
135
Fig. 3.
be 22.0 ± 0.5 ◦ C and repeated for three different channel arrays under the same conditions and the same volumes of the liquid drops. To verify if the retreat of the menisci due to the continuous evaporation of the fluid in the channels was relatively slow compared with the time of measurement, one can make an estimation for this velocity of retreat. Using a global approach and neglecting radiation effects we approximate the process as a uniform evaporation across the curved interface. We write for the change of the volume V of the liquid per time dV /dt = q A /ρ · H ν , with A the area of the liquid–air interface, q the average heat flux (∼2 kJ/s m2 ), ρ the density of the fluid and H ν the vaporization enthalpy (2.27 × 106 J/kg and 8.41 × 105 J/kg for water and ethanol, respectively) [35–37], and take a meniscus length along the channel of about l = 5 μm. For the characteristic time τ for displacement of the meniscus over its length one finds then: τ = lρ · H ν /q ≈ 5 s and 2 s for water and ethanol, respectively. The time t of scanning one channel with the AFM is t ≈ 0.1 s, so that t /τ ≈ 2 to 5 × 10−2 . Therefore, although evaporation certainly plays a role on time scales of more than seconds, during the measurements the effect is slow and the process can be considered as quasi-static. This was confirmed by the experiments: many scans were done consecutively without showing any measurable difference. The relation between the thus measured membrane deflections of the different channels due to the negative pressure of the liquid in the channel, and their widths is seen in Fig. 3. Also shown in the figure is the relation between the deflection and channel width according to the mechanical model described in the next section (Eqs. (4) and (5)), for a flexural rigidity of the capping layer of κ = 6.1 × 10−11 N m and γ = 0.023 N/m. 2.3. Observation of the menisci Next, the chip was observed under an upright microscope (Leica LM/DM) in bright field mode and again the channels were filled with ethanol and water. A Mitutoyo 20× objective NA = 0.28 was used. As a result of the used amplification, the size of a pixel in the resolution amounted to approximately 20 nm and the edge of a meniscus could be resolved within a few pixels. However, due to physical limitations of the optics, like diffraction and scattering effects, the final resolution of the meniscus determination will be around 200 nm. The experiments with this microscope were performed on the same day and in fresh channel arrays on the chip to avoid any contamination. The process of the filling and after this the evaporation of the liquid from the capillaries, that is, the quasi-static equilibrium we are interested in, was recorded by video imaging with a frame rate of 8 images/s. Fig. 4 shows a typical image from the video capture of the partially filled nanochan-
Fig. 4.
Fig. 5.
nels, in which the menisci at the ethanol–air interface can be clearly distinguished. It was found that the measurement differences between corresponding channels of different channel arrays lie within the errors due to the inaccuracy of the channel height and width, material parameters, and the optical resolution. Subsequently, a MATLAB 7.5.0 software program, equipped with an ‘edge detection’ tool was used to analyze the recorded images of the fluid menisci by reading the grayscale images and interpreting the darkness contrast of the pixels. The points of largest contrast are assumed to give the best representation of the fluid– air interface and are therefore used to compose the shapes of the menisci. Varying the threshold value of this edge detection from 0.4 to 0.6 on the grayscale of the image had no significant effect on the obtained curves, confirming that the contrast was sufficiently sharp. Fig. 5 illustrates the procedure by showing a representative picture of the meniscus in a 5.4 ± 0.1 μm wide channel together with the calculated curve as described below. 3. Analytical modeling Let the fluid have a contact angle θ and surface tension γ . The curvatures of the surface in the x- and y-direction are 1/ R 1 and 1/ R 2 . We can consider quasi-static situations in which the fluid
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nomial function in x with a maximum deflection U 0 at x = w /2 determined by the lateral and tensile forces. Introducing normalized variables ξ = 1 − 2x/ w and U = u /h, we can write the solution of (3) for the normalized deflection at the place of the meniscus as
U (ξ ) = U 0 1 − ξ 2
2
(4)
with U 0 = − pm w 4 f ( K )/16κ h. Fig. 6.
in the channel slowly evaporates. One can write the curvatures in terms of the function z = f (x, y ) describing the meniscus shape
and use the notation ∇ f ≡ (∂x f , ∂ y f ). Then the Young–Laplace equation becomes [5,38]:
−∇ ·
∇ f
=
1 + (∇ f )2
p (l) − p ( g )
γ
,
(1)
where the left-hand side is the sum of the curvatures 1/ R 1 + 1/ R 2 and p (l) and p ( g ) represent the pressures in the liquid and in the gas, respectively. Equation (1) has to be solved for f , together with the boundary condition that the fluid surface and the channel walls intersect at the contact angle θ . The rectangular channel is covered by an elastic membrane of thickness t, that is pulled downward due to the negative pressure inside the liquid. This downward deflection can be written as u (x, z). The channel is directed along the z-axis, with its height in the y-direction and width in the x-direction. A schematic picture of the geometry is seen in Fig. 6. The deflection u (x, z) obeys the equation of elasticity [39]: 2 −κ 2 u + S x ∂x2 u + 2S xz ∂xz u + S z ∂z2 u 0, membrane–air, = p (l) − p ( g ) , membrane–liquid,
(2)
where = ∂x2 + ∂z2 , κ = Et 3 /12(1 − ν 2 ), E is the effective Young’s modulus and ν the Poisson ratio of the membrane composite. S x and S z denote the tensile stresses in the material in the x- and z-direction respectively, with S xz corresponding to the shear forces in the membrane. Besides u (x, z) should fulfill the boundary conditions, that read u (x, z) = 0 and ∂x u (x, z) = 0 at x = 0 and x = w. Equations (1) and (2) are coupled via the pressure difference p = p (l) − p ( g ) . This makes them quite difficult to solve. Besides, the membrane area on which the pressure acts is defined by the shape of the meniscus, which is still unknown. However, the problem can be significantly simplified if we realize that for the long rectangular channels with characteristic lengths L much larger than the width w and with large tensile stress S x , the z-derivatives of u are much smaller than the x-derivatives. Further, we consider situations in which the length of the meniscus along the channel is smaller than, or in the order of the channel width w, which is the case in our experiments. Then the deflection in the elasticity problem is virtually independent of z and we are left with a quasi one-dimensional problem, the so-called cylindrical bending of a long rectangular plate [35]. It reads
−κ ∂x4 u + S x ∂x2 u =
0,
membrane–air,
p (l) − p ( g ) ,
membrane–liquid,
(3)
of which the solution be found analytically.1 For small deflections we can well approximate the solution u (x) by a fourth-order poly-
1 The bending stiffness for cylindrical bending of a plate of width w under a uniform pressure load p, including the effect of an intrinsic stress in the membrane,
(5)
Here pm is the effective pressure acting downward on the membrane at the place of the meniscus and f ( K ) a dimensionless shape factor that depends on the intrinsic stresses in the membrane.1 U 0 , representing the center deflection at x = w /2, is a positive constant ( p < 0) for points within the plug and its value decreases to zero at distances of several times w from the plug. This also implies that Eqs. (4) and (5) are only valid under the condition that the meniscus does not protrude over lengths along z of many times the width w. The capping membrane and the bottom of the channel are separated by the local height hl (x) = h − u (x) at a given z. Since this distance is much smaller than w, the curvature in the y-direction can be calculated locally as the curvature of the meniscus between two flat plates separated by the distance hl (x): 1 R2
=−
2 cos θ hl (x)
(6)
.
Realizing that the liquid surface crosses the two plates at the contact angle θ we can approximate in good precision the y-dependence of f (x, y ) by a parabolic profile: f (x, y ) = f (x) +
1 4
hl (x) cot θ 1 −
2y
2
hl (x)
.
(7)
If θ is small the wedge-like films near the top and bottom of the channel do not give a significant contribution to the observed contrast, but the central part of the channel, where ∂ y f = 0, does. Therefore, the curvature in the x-direction 1/ R 1 can be calculated at ∂ y f = 0. In this case the observed contrast nearly coincides with f (x), for which we can find a much simpler equation than the original Eqs. (1) and (2). If θ is not small the contrast can be calculated using an appropriate procedure of averaging over y with the help of (7). This situation will be discussed elsewhere. In our quasi one-dimensional approach, when the meniscus length w, the original Eqs. (1), (2) are reduced to the relatively simple one-dimensional equation
−
d dx
df dx
1+
df 2
=
p
γ
+
2 cos θ h(1 − U (ξ ))
.
(8)
dx
Equation (8) includes an unknown function f (x) and the unknown pressure difference p between the liquid and the gas. It has to be solved together with the boundary conditions df /dξ |ξ =±1 = ∓ cot θ that can be used to define p and one of the arbitrary constants. The second integration constant can be left arbitrary because we are interested in the meniscus shape, not in its position. In principle, this constant can be fixed by demanding a
is for example given by Timoshenko and Woinowsky-Krieger [39]. In a simple representation the compliance is given by u 0 ( K , p )/ p = ( w /2)4 f ( K )/κ , with u 0 is the center deflection, κ the flexural rigidity of the composite membrane and f ( K )−1 the stiffening due to the presence of the residual stress, as a function of the dimensionless parameter K = w (Σ σi t i / D )1/2 , with σi and t i the stress and the thickness of layer i, respectively, and f ( K ) = ( K 2 /2 + K / sinh K − K / tanh K )/ K 4 . For the pressure in the fluid we have p = −2γ cos θ/h, and realizing that at the place of the meniscus, only half of the downward forces act, we obtain pm = p /2, and therefore U 0 = γ cos θ w 4 f ( K )/16κ h.
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137
constant volume of the liquid plug, but this is more important for consideration of dynamics of the meniscus than in the quasi-static situation we consider now. The solution of (8) can be presented as w
f (ξ ) =
dξ P (ξ )
1 − P 2 (ξ )
2
where P (ξ ) is defined by P (ξ ) =
wp 2γ
ξ+ √
w cos θ 4hξ1
√
(9)
,
U0
2ξ1
ξ2
arctan
ξ ξ1 + ξ + ln ξ2 ξ1 − ξ
(10)
with ξ12,2 = 1/ U 0 ± 1. The constant p can be determined from the nonlinear equation P (1) = − cos θ . Because ξ1,2 and U 0 depend on p the solution can be found only numerically.
(a)
4. Results and discussion To compare the theoretical predictions for the meniscus shape with the observed menisci, we used the microscopic images of the channels that were interpreted by the contrast detection routine. The thus obtained curves are shown in Fig. 7, represented by the dots. To calculate the curves that follow from the model, we applied Eqs. (9) and (10), using the following values for the material parameters and dimensions: E SiN = (2.60 ± 0.1) × 102 GPa, E SiO = (70 ± 10) GPa, νSiNi = 0.27 ± 0.01, νSiO = 0.17 ± 0.01 [40], t SiNi = (97 ± 3) nm, and t SiO = (79 ± 3) nm for the silicon nitride and the silicon oxide layers in the membrane respectively, and θ = 0◦ , γ = 0.023 N/m (ethanol), γ = 0.072 N/m (water). The membrane stiffening f ( K )−1 ranged from 32.7 for w = 3.4 μm to 65.4 at w = 7.5 μm, based on a tensile stress of 1.09 ± 0.04 GPa for the silicon nitride and a compressive stress of 0.30 ± 0.05 GPa for the silicon oxide [41]. These theoretical predictions are depicted in Fig. 7 as well. Additionally, a detailed picture of the 5.4 μm wide channel is shown in Fig. 5, together with the theoretical prediction. A good agreement between the model curves and the experimental shapes is seen, both qualitatively and quantitatively (that is, there is no scaling factor involved). The proposed model appears therefore to be rather appropriate to describe menisci of fluid–air interfaces in a flexible nanoconfinement. The elastocapillary equilibrium can thus be well described by a relative straightforward model interconnecting the interfacial forces of the wetting liquid and the mechanical forces in the capillary. The presented analysis may therefore also complement to studies of surface-tension-induced buckling of thin elastic tubes such as observed in pulmonary airway closure [30–32]. These studies model the long airways by elastic circular tubes, and show that fluid inertial effects become important especially in the late stages of airway closure, whereas the investigated fluid–mechanical interactions for small mechanical deformations show many resemblances with those described in this paper. For example, the critical width for collapse of these flexible tubes in the low-inertial regime is comparable to our findings. The current problem can be characterized by a characteristic length, determined by the mechanical and surface tension forces, that indicates a typical deflection of the membrane. A force balance of these two forces yields for this characteristic (‘elastocapillary’) deflection δ0 :
δ0 =
w 2
4
f ( K )γ cos θ
κh
.
(11)
When δ0 approaches h, the meniscus will protrude over large distances along the channel, the membrane may collapse (i.e., stick to the channel bottom) and nonlinear effects may become important. Setting δ0 equal to h, we find a characteristic width w c
(b)
(c)
(d) Fig. 7.
at which the deflection becomes important; one obtains w c = 2h1/2 (κ / f ( K )γ cos θ)1/4 . To analyze the protrusion of the meniscus along the channel, one can define the protrusion length, l p , as the distance between the front of the meniscus and its attachment point at the wall of the channel. The observed dependence, as measured by means of
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with a wetting liquid, ethanol. It was observed that the shape of the meniscus at the liquid–air interface is very sensitive to its nanoconfinement and that it can attain a specific equilibrium shape characterized by both a convex and a concave curvature. We presented a model to describe this elastocapillary equilibrium and the meniscus shape for relative small deflections of the channel capping layers, and found a good agreement with the experimental observations. It was also found that the so-called protrusion length of the meniscus in the flexible nanoconfinement is a sensitive and accurate measure for the membrane deflection. This allows for an accurate and convenient determination for the deflection of thin membranes in nanochannels.
Fig. 8.
the AFM, is shown in Fig. 8. The protrusion length l p has been normalized by w c . The solid line represents the theoretical prediction following from the model. As is clearly seen, the protrusion length of the meniscus is a sensitive measure for the deflection of the membrane. This provides us with a new method to quantify the deflection of flexible sheets: the observation of the protrusion length l p is a very convenient and accurate procedure to determine the deflection of the channel membrane. As mentioned above, the value of the pressure in these cases can only be found numerically from Eqs. (9) and (10). The thus determined p attains a value of −6.7 × 105 Pa at l p / w = 1. The reader will note that Fig. 8 has been plotted for deflections for a limited region of U 0 only. The reason for this is related with the following. The function f according to Eq. (8) is only defined if P (ξ ) < 1 for all −1 < ξ < 1. For increasing deflections U 0 , however, P can become larger than 1 and the solution of (8) does not exist anymore. It corresponds to the situation that df /dx → ∞. This singularity is encountered at δ0 approaching 0.11, where the infinite steepness of f then occurs at ξ = ±0.44 and which corresponds for the current material parameters with a channel width of w = 6.45 μm. The experimental observations of very far protruded menisci with almost infinite steepness, as illustrated by the upper channels in Fig. 4, seem to confirm this. It was also observed that beyond this critical point, the fluid plugs can even detach from the wall and the meniscus shapes change drastically. The origin and consequences of the singularity will be discussed in a future paper on elastocapillarity. In the limit of a stiff membrane, U 0 → 0, it is easy to find the expected result for normal menisci p = 2 cos θ(1/ w + 1/h). In this case P (ξ ) = −ξ cos θ and we reproduce from Eq. (9) the normal meniscus shape: f (x) = ( w /2 cos θ)(1 − 1 − ξ 2 cos2 θ ). To illustrate, finally, the general applicability of the presented model we applied the described approach to the results obtained in [33]. In that work, the authors analyzed water plugs in hydrophilic nanochannels that were subject to significant negative pressure due to the tensile capillary forces. We performed the edge detection routine and curve fitting procedure to the optical micrograph of the fluid meniscus presented in that paper. We thus found a normalized protrusion length of that meniscus of l p / w c = 0.52 ± 0.06, corresponding to a membrane deflection of 9.7 ± 0.8 nm, and a negative pressure in the liquid of −13.8 ± 8 bar. This corresponds quite well with the measured values given there: a deflection of 11 ± 4 nm and a negative pressure of −17 ± 10 bar, respectively. It is also close to the theoretically expected value of −12 bar. 5. Summary In conclusion, we analyzed theoretically and experimentally the fluid menisci in elastically deformed nanochannels that were filled
Acknowledgment It is a pleasure to thank Dr. V.B. Svetovoy for his mathematical support and valuable comments. References [1] F.R.S. Hauksbee, Philosophical Transactions (1683–1775), vol. 26, The Royal Society, London, 1708–1709, pp. 258–266. [2] J.C. Maxwell, Capillary Action, in: Encycl. Britt., ninth ed., Samuel L. Hall, New York, 1875. [3] P.S. Laplace, Celest. Mech. IV (1806). [4] H. Bouasse, Capillarité: Phénomènes Superficiels, Delagrave, Paris, 1924. [5] L.D. Landau, E.M. Lifschitz, Course of Theoretical Physics, vol. 6, Fluid Mechanics, second ed., Butterworth/Heineman, Oxford, 1971. [6] P. Concus, R. Finn, Proc. Natl. Acad. Sci. 63 (1969) 292–299. [7] D.A. White, J.A. Tallmadge, J. Fluid Mech. Digital Arch. 23 (1965) 325–335. [8] D.F. James, J. Fluid Mech. 63 (1974) 657–664. [9] L.L. Lo, J. Fluid Mech. 132 (1983) 65–78. [10] M.G. Stern, M.W. Geis, J.E. Curtin, J. Vac. Sci. Technol. B 15 (1997) 2887. [11] J. Han, H.G. Craighead, Science 288 (2000) 1026. [12] M. Foquet, J. Korlach, W. Zipfel, W.W. Webb, H.G. Craighead, Anal. Chem. 74 (2002) 1415. [13] H. Cao, Z. Yu, J. Wang, J.O. Tegenfeldt, R.H. Austin, E. Chen, W. Wu, S.Y. Chou, Appl. Phys. Lett. 81 (2002) 174. [14] N.R. Tas, J.W. Berenschot, P. Mela, H.V. Jansen, M. Elwenspoek, A. Van den Berg, Nano Lett. 2 (2002) 1031. [15] J. Haneveld, H.V. Jansen, J.W. Berenschot, N.R. Tas, M. Elwenspoek, J. Micromech. Microeng. 13 (2003) S62. [16] V.D. Sobolev, N.V. Churaev, M.G. Velarde, Z.M. Zorin, J. Colloid Interface Sci. 222 (2000) 51. [17] N.R. Tas, J. Haneveld, H.V. Jansen, M. Elwenspoek, A. Van den Berg, Appl. Phys. Lett. 85 (2004) 15. [18] J. Haneveld, N.R. Tas, N. Brunets, H.V. Jansen, M. Elwenspoek, J. Appl. Phys. 104 (2008) 014309. [19] H.-Y. Kim, L. Mahadevan, J. Fluid Mech. 548 (2006) 141–150. [20] J. Bico, B. Roman, L. Moulin, A. Boudaoud, Nature 432 (2004) 690. [21] C. Py, P. Reverdy, L. Doppler, J. Bico, B. Roman, C.N. Baroud, Phys. Rev. Lett. 98 (2007) 156103. [22] R.R.A. Syms, E.M. Yeatman, V.M. Bright, G.M. Whitesides, J. Microel. Mech. Syst. 12 (2003) 4. [23] T. Tanaka, M. Morigami, N. Atoda, Jpn. J. Appl. Phys. 32 (1993) 6059. [24] C.H. Mastrangelo, C.H. Hsu, J. Microelectromech. Syst. 2 (1993) 33–43. [25] C.H. Mastrangelo, C.H. Hsu, J. Microelectromech. Syst. 2 (1993) 44–55. [26] N.R. Tas, T. Sonnenberg, H.V. Jansen, R. Legtenberg, M. Elwenspoek, J. Micromech. Microeng. 6 (1996) 385–397. [27] R. Maboudian, R.T. Howe, J. Vac. Sci. Technol. B 15 (1) (1997) 1–20. [28] C.Y. Hui, A. Jagota, Y.Y. Lin, E.J. Kramer, Langmuir 18 (2002) 1394–1407. [29] M. Heil, J.P. White, J. Fluid Mech. 462 (2002) 79–109. [30] A.L. Hazel, M. Heil, Proc. R. Soc. A 461 (2058) (2005) 1847–1868. [31] D. Halpern, J.B. Grotberg, J. Fluid Mech. 244 (1991) 615–632. [32] J.B. Grotberg, O.E. Jensen, Annu. Rev. Fluid Mech. 36 (2004) 121–147. [33] N.R. Tas, P. Mela, T. Kramer, J.W. Berenschot, A. Van den Berg, Nano Lett. 3 (2003) 1537. [34] J.W. van Honschoten, M. Escalante, N.R. Tas, H.V. Jansen, M. Elwenspoek, J. Appl. Phys. 101 (2007) 094310. [35] D.S. Golovko, P. Bonanno, S. Lorenzoni, F. Stefani, R. Raiteri, E. Bonaccurso, J. Micromech. Eng. 18 (2008) 095026. [36] R.J. Hunter, Foundations of Colloid Science, second ed., Oxford Univ. Press, New York, 2007. [37] R.G. Picknett, R. Bexon, J. Colloid Interface Sci. 61 (1977) 336–350.
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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Formation of liquid menisci in flexible nanochannels J.W. van Honschoten ∗ , M. Escalante, N.R. Tas, M. Elwenspoek Transducers Science and Technology, MESA + Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 22 July 2008 Accepted 27 September 2008 Available online 7 October 2008
In this paper we analyze the characteristic shape of the liquid meniscus at the fluid air interface in nanochannels of less than 80 nm height capped by a flexible membrane. Because of the induced negative pressure difference between the liquid pressure and the pressure outside, the 0.18 μm thin membrane on top of the channels bends downward. This elastocapillary equilibrium between the surface tension of the wetting liquid and the mechanical forces in the capillary results in a very peculiar shape of the interfacial meniscus, visible from the top through the transparent membrane. For increasing deflection of the membrane, the meniscus is seen to protrude along the channel and its curvature changes from concave to convex in the center. We present an analytical model to describe the meniscus shape in the deformed channel for small membrane deflections. We also show that the protrusion length of the meniscus, which can be measured easily, is an accurate and useful indicator for the membrane deflection. Experimental results on nanochannels filled with ethanol and water are presented and the observed menisci are seen to be in good agreement with the proposed model. © 2008 Elsevier Inc. All rights reserved.
Keywords: Elastocapillarity Nanofluidics Fluid interfaces Nanochannels
1. Introduction When a wetting liquid is brought into contact with a solid surface, a curved fluid–air interface, the so-called meniscus, is developed due to surface tension of the liquid interface. The physical origin and the shape of a meniscus of a wetting liquid in contact with a surface have been a subject of specific interest in capillarity since a long time, starting in particular with the study of Hauksbee [1] on the capillary forces in a liquid in contact with glass plates or tubes, and Maxwell [2]. In 1806, Laplace [3] was the first to deduce an analytic expression for the static shape of the meniscus on a planar wall, a solution that was referred to in 1924 in Bouasse [4], and by Landau [5]. The behavior of a capillary surface in a wedge of prescribed interior corner was addressed in 1969 by Concus and Finn [6], while the static meniscus shape on a vertical fiber in a fluid bath was studied numerically, by White and Tallmadge [7], and analytically [8,9]. While most of the studies on this subject concentrate on a welldefined fluid–air interface (such as a planar wall placed into a liquid reservoir or a fluid imbibing a corner of a capillary) we focus in this paper on the case of a fluid in a deformable confinement with which it interacts, resulting in an elastocapillary equilibrium between the surface tension forces of the wetting fluid and the mechanical forces of the deformed capillary. The deflection of the flexible membranes due to the tensile capillary forces is a promi-
*
Corresponding author. Fax: +31 53 4893343. E-mail address: [email protected] (J.W. van Honschoten).
0021-9797/$ – see front matter doi:10.1016/j.jcis.2008.09.082
©
2008 Elsevier Inc. All rights reserved.
nent effect of elastocapillarity on the nanoscale. In nanochannels, which can be defined as capillaries having at least one dimension on the order of 100 nm or smaller [10–15], capillary action is very prominent due to the large surface-to-volume ratio. The capillary action present in nondeformable nanoscale capillaries and the fluid behavior in these channels were studied systematically by Sobolev et al. [16], by Tas et al. [17] for one-dimensional nanochannels with a height down to 50 nm, and for even smaller cross-sections down to 5 nm by Haneveld et al. [18]. While the phenomenon of elastocapillarity is becoming a subject of increasing interest [19,20], since it can be used in microfabrication technologies [21,22] and can also have damaging effects in microelectromechanical structures [23–28] or in pulmonary airway closure [29], it has not been studied extensively on the nanoscale yet. Capillary action in flexible tubes of nanometer sizes was recently studied numerically [30], especially with respect of the surface-tension-induced buckling of these tubes, while the dynamics of thin films of fluid within small elastic tubes were also studied theoretically by e.g. Halpern and Grotberg [31] and Grotberg and Jensen [32]. As recent work on elastocapillarity in structures of nanodimensions demonstrates [33,34], the shape of the meniscus in these cases is very sensitive to its nanoconfinement and can attain a characteristic shape. To study the meniscus in deforming nanochannels, in this work we investigate hydrophilic nanochannels of about 80 nm with flexible capping layers that are deformed due to the induced negative pressure in the fluid. As a result of the deflection of the membrane, the shape of the meniscus, visible through the thin transparent
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Fig. 1.
layer on top, is substantially modified. As the membrane deflections increase, the fluid protrudes into the center of the channel and the curvature of the meniscus will not be purely concave anymore and becomes even convex in the center. The paper is organized as follows. In Section 2, we describe shortly the fabrication of the hydrophilic nanochannels that are used for the investigation of the development of the meniscus. Then, a description is given of the experiments in which the nanochannels were partially filled with ethanol and water. We describe also the detection of the menisci with an optical microscope, and the method to measure the membrane deflections. In Section 3, a model is presented that describes the shape of the fluid–air interface, starting from the premiss that the sum of the curvatures of the fluid surface in two perpendicular directions is equal to a constant, p /γ (with p the Laplace pressure and γ the liquid surface tension). In Section 4, a comparison between the proposed theoretic model and the experimental observations is given. The model is found to be very appropriate and useful to describe menisci in nanoconfinements. We also obtain the important result that the protrusion length of the meniscus is a very sensitive measure for the deflection of the flexible membranes. This yields a new method to quantify the channel deformation based on observation of this so-called protrusion length. 2. Materials and methods 2.1. Nanochannel fabrication We created arrays of hydrophilic channels of different width in the micrometer range, about 1 cm in length, and height h of 79 nanometer with a capping layer (of total thickness 176 nm) consisting of a laminate of silicon nitride (SiN) and silicon oxide (SiO2 ). In the fabrication process the thickness of the silicon oxide spacer layer, that was grown by dry oxidation at 1100 ◦ C, determined the thickness of the nanochannels. This could be precisely controlled by adjusting the oxidation time. Standard photolithography and 1% HF etching of the silicon oxide layer were carried out to define the nanochannels pattern. The access holes for the fluid supply and the microchannels connecting to the nanochannels were included in a second photolithography process. The channels were covered by a thin membrane consisting of a silicon nitride–silicon oxide composite layer, of thicknesses 97 ± 3 and 79 ± 3 nm, respectively. This layer was transferred by fusion bonding of a second wafer, carrying the plate, to the first wafer, carrying the nanochannel pattern. Before bonding, both wafers were cleaned, first by using a wet chemical HNO3 solution, next in H2 O2 /H2 SO4 (in 1:3 ratio at 130 ◦ C, ‘Piranha’), to assure a hydrophilic surface. The wafer pair was annealed during two hours at 1100 ◦ C in a furnace (N2 ). Finally, the second support wafer was removed by reactive ion etching in a SF6 plasma at room temperature. A more detailed description of the fabrication process is given in [33]. Fig. 1 shows a cross section of the resulting structure. The channel height was measured by means of an ellipsometry measurement of the silicon oxide spacer layer using a Plasmos SD 2002 ellipsometer. The error in the height measurement is estimated to be 3 nm, as a result of the variation in the thickness of the silicon oxide layer on the wafer surface and the precision of the instrument.
Fig. 2.
2.2. Determination of membrane deflection For an accurate determination of the membrane deflections of the partially filled fluid channels, an atomic force microscope (‘AFM,’ Veeco, Dim. 3100), was used to make images of the nanochannels. The chip surface was scanned by the AFM tip with a lateral resolution of 40 nm, both in the direction perpendicular to and parallel to the arrays of fluid channels. Each channel array consisted of a set of rectangular capillaries of a length of 10 mm and varying width, with the channel widths of interest ranging from 3.4 ± 0.1 to 7.5 ± 0.1 μm in 7 steps. Since the bending of the membrane is strongly dependent on the width of the channel (the wider channels have a lower mechanical stiffness), this set of channel widths permits the investigation of a wide range of membrane deflections and the corresponding meniscus shapes. Prior to the filling, it was verified that the membranes did not noticeably deflect due to the scanning of the tip over the surface. Because of the SiN/SiO2 surface roughness and the presence of small pollutants on this layer, the error in the height measurements amounted to approximately ±2 nm. A droplet of 5 μl liquid was introduced into the access holes by pipetting so that the array of nanochannels could fill by capillary action. We used VLSI Selectipur-ethanol (Merck, 99.8% purity), and water (Milli-Q, specific resistance 18 M cm). When the filling process had stopped and the fluid plugs remained almost constant in time (only slowly decreasing due to evaporation), the membrane bendings were recorded by scanning the AFM tip across the channels. Fig. 2 shows a typical scan along the width of part of one channel array along the channel widths with the AFM. The experiments were performed under constant temperature, measured to
J.W. van Honschoten et al. / Journal of Colloid and Interface Science 329 (2009) 133–139
135
Fig. 3.
be 22.0 ± 0.5 ◦ C and repeated for three different channel arrays under the same conditions and the same volumes of the liquid drops. To verify if the retreat of the menisci due to the continuous evaporation of the fluid in the channels was relatively slow compared with the time of measurement, one can make an estimation for this velocity of retreat. Using a global approach and neglecting radiation effects we approximate the process as a uniform evaporation across the curved interface. We write for the change of the volume V of the liquid per time dV /dt = q A /ρ · H ν , with A the area of the liquid–air interface, q the average heat flux (∼2 kJ/s m2 ), ρ the density of the fluid and H ν the vaporization enthalpy (2.27 × 106 J/kg and 8.41 × 105 J/kg for water and ethanol, respectively) [35–37], and take a meniscus length along the channel of about l = 5 μm. For the characteristic time τ for displacement of the meniscus over its length one finds then: τ = lρ · H ν /q ≈ 5 s and 2 s for water and ethanol, respectively. The time t of scanning one channel with the AFM is t ≈ 0.1 s, so that t /τ ≈ 2 to 5 × 10−2 . Therefore, although evaporation certainly plays a role on time scales of more than seconds, during the measurements the effect is slow and the process can be considered as quasi-static. This was confirmed by the experiments: many scans were done consecutively without showing any measurable difference. The relation between the thus measured membrane deflections of the different channels due to the negative pressure of the liquid in the channel, and their widths is seen in Fig. 3. Also shown in the figure is the relation between the deflection and channel width according to the mechanical model described in the next section (Eqs. (4) and (5)), for a flexural rigidity of the capping layer of κ = 6.1 × 10−11 N m and γ = 0.023 N/m. 2.3. Observation of the menisci Next, the chip was observed under an upright microscope (Leica LM/DM) in bright field mode and again the channels were filled with ethanol and water. A Mitutoyo 20× objective NA = 0.28 was used. As a result of the used amplification, the size of a pixel in the resolution amounted to approximately 20 nm and the edge of a meniscus could be resolved within a few pixels. However, due to physical limitations of the optics, like diffraction and scattering effects, the final resolution of the meniscus determination will be around 200 nm. The experiments with this microscope were performed on the same day and in fresh channel arrays on the chip to avoid any contamination. The process of the filling and after this the evaporation of the liquid from the capillaries, that is, the quasi-static equilibrium we are interested in, was recorded by video imaging with a frame rate of 8 images/s. Fig. 4 shows a typical image from the video capture of the partially filled nanochan-
Fig. 4.
Fig. 5.
nels, in which the menisci at the ethanol–air interface can be clearly distinguished. It was found that the measurement differences between corresponding channels of different channel arrays lie within the errors due to the inaccuracy of the channel height and width, material parameters, and the optical resolution. Subsequently, a MATLAB 7.5.0 software program, equipped with an ‘edge detection’ tool was used to analyze the recorded images of the fluid menisci by reading the grayscale images and interpreting the darkness contrast of the pixels. The points of largest contrast are assumed to give the best representation of the fluid– air interface and are therefore used to compose the shapes of the menisci. Varying the threshold value of this edge detection from 0.4 to 0.6 on the grayscale of the image had no significant effect on the obtained curves, confirming that the contrast was sufficiently sharp. Fig. 5 illustrates the procedure by showing a representative picture of the meniscus in a 5.4 ± 0.1 μm wide channel together with the calculated curve as described below. 3. Analytical modeling Let the fluid have a contact angle θ and surface tension γ . The curvatures of the surface in the x- and y-direction are 1/ R 1 and 1/ R 2 . We can consider quasi-static situations in which the fluid
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nomial function in x with a maximum deflection U 0 at x = w /2 determined by the lateral and tensile forces. Introducing normalized variables ξ = 1 − 2x/ w and U = u /h, we can write the solution of (3) for the normalized deflection at the place of the meniscus as
U (ξ ) = U 0 1 − ξ 2
2
(4)
with U 0 = − pm w 4 f ( K )/16κ h. Fig. 6.
in the channel slowly evaporates. One can write the curvatures in terms of the function z = f (x, y ) describing the meniscus shape
and use the notation ∇ f ≡ (∂x f , ∂ y f ). Then the Young–Laplace equation becomes [5,38]:
−∇ ·
∇ f
=
1 + (∇ f )2
p (l) − p ( g )
γ
,
(1)
where the left-hand side is the sum of the curvatures 1/ R 1 + 1/ R 2 and p (l) and p ( g ) represent the pressures in the liquid and in the gas, respectively. Equation (1) has to be solved for f , together with the boundary condition that the fluid surface and the channel walls intersect at the contact angle θ . The rectangular channel is covered by an elastic membrane of thickness t, that is pulled downward due to the negative pressure inside the liquid. This downward deflection can be written as u (x, z). The channel is directed along the z-axis, with its height in the y-direction and width in the x-direction. A schematic picture of the geometry is seen in Fig. 6. The deflection u (x, z) obeys the equation of elasticity [39]: 2 −κ 2 u + S x ∂x2 u + 2S xz ∂xz u + S z ∂z2 u 0, membrane–air, = p (l) − p ( g ) , membrane–liquid,
(2)
where = ∂x2 + ∂z2 , κ = Et 3 /12(1 − ν 2 ), E is the effective Young’s modulus and ν the Poisson ratio of the membrane composite. S x and S z denote the tensile stresses in the material in the x- and z-direction respectively, with S xz corresponding to the shear forces in the membrane. Besides u (x, z) should fulfill the boundary conditions, that read u (x, z) = 0 and ∂x u (x, z) = 0 at x = 0 and x = w. Equations (1) and (2) are coupled via the pressure difference p = p (l) − p ( g ) . This makes them quite difficult to solve. Besides, the membrane area on which the pressure acts is defined by the shape of the meniscus, which is still unknown. However, the problem can be significantly simplified if we realize that for the long rectangular channels with characteristic lengths L much larger than the width w and with large tensile stress S x , the z-derivatives of u are much smaller than the x-derivatives. Further, we consider situations in which the length of the meniscus along the channel is smaller than, or in the order of the channel width w, which is the case in our experiments. Then the deflection in the elasticity problem is virtually independent of z and we are left with a quasi one-dimensional problem, the so-called cylindrical bending of a long rectangular plate [35]. It reads
−κ ∂x4 u + S x ∂x2 u =
0,
membrane–air,
p (l) − p ( g ) ,
membrane–liquid,
(3)
of which the solution be found analytically.1 For small deflections we can well approximate the solution u (x) by a fourth-order poly-
1 The bending stiffness for cylindrical bending of a plate of width w under a uniform pressure load p, including the effect of an intrinsic stress in the membrane,
(5)
Here pm is the effective pressure acting downward on the membrane at the place of the meniscus and f ( K ) a dimensionless shape factor that depends on the intrinsic stresses in the membrane.1 U 0 , representing the center deflection at x = w /2, is a positive constant ( p < 0) for points within the plug and its value decreases to zero at distances of several times w from the plug. This also implies that Eqs. (4) and (5) are only valid under the condition that the meniscus does not protrude over lengths along z of many times the width w. The capping membrane and the bottom of the channel are separated by the local height hl (x) = h − u (x) at a given z. Since this distance is much smaller than w, the curvature in the y-direction can be calculated locally as the curvature of the meniscus between two flat plates separated by the distance hl (x): 1 R2
=−
2 cos θ hl (x)
(6)
.
Realizing that the liquid surface crosses the two plates at the contact angle θ we can approximate in good precision the y-dependence of f (x, y ) by a parabolic profile: f (x, y ) = f (x) +
1 4
hl (x) cot θ 1 −
2y
2
hl (x)
.
(7)
If θ is small the wedge-like films near the top and bottom of the channel do not give a significant contribution to the observed contrast, but the central part of the channel, where ∂ y f = 0, does. Therefore, the curvature in the x-direction 1/ R 1 can be calculated at ∂ y f = 0. In this case the observed contrast nearly coincides with f (x), for which we can find a much simpler equation than the original Eqs. (1) and (2). If θ is not small the contrast can be calculated using an appropriate procedure of averaging over y with the help of (7). This situation will be discussed elsewhere. In our quasi one-dimensional approach, when the meniscus length w, the original Eqs. (1), (2) are reduced to the relatively simple one-dimensional equation
−
d dx
df dx
1+
df 2
=
p
γ
+
2 cos θ h(1 − U (ξ ))
.
(8)
dx
Equation (8) includes an unknown function f (x) and the unknown pressure difference p between the liquid and the gas. It has to be solved together with the boundary conditions df /dξ |ξ =±1 = ∓ cot θ that can be used to define p and one of the arbitrary constants. The second integration constant can be left arbitrary because we are interested in the meniscus shape, not in its position. In principle, this constant can be fixed by demanding a
is for example given by Timoshenko and Woinowsky-Krieger [39]. In a simple representation the compliance is given by u 0 ( K , p )/ p = ( w /2)4 f ( K )/κ , with u 0 is the center deflection, κ the flexural rigidity of the composite membrane and f ( K )−1 the stiffening due to the presence of the residual stress, as a function of the dimensionless parameter K = w (Σ σi t i / D )1/2 , with σi and t i the stress and the thickness of layer i, respectively, and f ( K ) = ( K 2 /2 + K / sinh K − K / tanh K )/ K 4 . For the pressure in the fluid we have p = −2γ cos θ/h, and realizing that at the place of the meniscus, only half of the downward forces act, we obtain pm = p /2, and therefore U 0 = γ cos θ w 4 f ( K )/16κ h.
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137
constant volume of the liquid plug, but this is more important for consideration of dynamics of the meniscus than in the quasi-static situation we consider now. The solution of (8) can be presented as w
f (ξ ) =
dξ P (ξ )
1 − P 2 (ξ )
2
where P (ξ ) is defined by P (ξ ) =
wp 2γ
ξ+ √
w cos θ 4hξ1
√
(9)
,
U0
2ξ1
ξ2
arctan
ξ ξ1 + ξ + ln ξ2 ξ1 − ξ
(10)
with ξ12,2 = 1/ U 0 ± 1. The constant p can be determined from the nonlinear equation P (1) = − cos θ . Because ξ1,2 and U 0 depend on p the solution can be found only numerically.
(a)
4. Results and discussion To compare the theoretical predictions for the meniscus shape with the observed menisci, we used the microscopic images of the channels that were interpreted by the contrast detection routine. The thus obtained curves are shown in Fig. 7, represented by the dots. To calculate the curves that follow from the model, we applied Eqs. (9) and (10), using the following values for the material parameters and dimensions: E SiN = (2.60 ± 0.1) × 102 GPa, E SiO = (70 ± 10) GPa, νSiNi = 0.27 ± 0.01, νSiO = 0.17 ± 0.01 [40], t SiNi = (97 ± 3) nm, and t SiO = (79 ± 3) nm for the silicon nitride and the silicon oxide layers in the membrane respectively, and θ = 0◦ , γ = 0.023 N/m (ethanol), γ = 0.072 N/m (water). The membrane stiffening f ( K )−1 ranged from 32.7 for w = 3.4 μm to 65.4 at w = 7.5 μm, based on a tensile stress of 1.09 ± 0.04 GPa for the silicon nitride and a compressive stress of 0.30 ± 0.05 GPa for the silicon oxide [41]. These theoretical predictions are depicted in Fig. 7 as well. Additionally, a detailed picture of the 5.4 μm wide channel is shown in Fig. 5, together with the theoretical prediction. A good agreement between the model curves and the experimental shapes is seen, both qualitatively and quantitatively (that is, there is no scaling factor involved). The proposed model appears therefore to be rather appropriate to describe menisci of fluid–air interfaces in a flexible nanoconfinement. The elastocapillary equilibrium can thus be well described by a relative straightforward model interconnecting the interfacial forces of the wetting liquid and the mechanical forces in the capillary. The presented analysis may therefore also complement to studies of surface-tension-induced buckling of thin elastic tubes such as observed in pulmonary airway closure [30–32]. These studies model the long airways by elastic circular tubes, and show that fluid inertial effects become important especially in the late stages of airway closure, whereas the investigated fluid–mechanical interactions for small mechanical deformations show many resemblances with those described in this paper. For example, the critical width for collapse of these flexible tubes in the low-inertial regime is comparable to our findings. The current problem can be characterized by a characteristic length, determined by the mechanical and surface tension forces, that indicates a typical deflection of the membrane. A force balance of these two forces yields for this characteristic (‘elastocapillary’) deflection δ0 :
δ0 =
w 2
4
f ( K )γ cos θ
κh
.
(11)
When δ0 approaches h, the meniscus will protrude over large distances along the channel, the membrane may collapse (i.e., stick to the channel bottom) and nonlinear effects may become important. Setting δ0 equal to h, we find a characteristic width w c
(b)
(c)
(d) Fig. 7.
at which the deflection becomes important; one obtains w c = 2h1/2 (κ / f ( K )γ cos θ)1/4 . To analyze the protrusion of the meniscus along the channel, one can define the protrusion length, l p , as the distance between the front of the meniscus and its attachment point at the wall of the channel. The observed dependence, as measured by means of
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with a wetting liquid, ethanol. It was observed that the shape of the meniscus at the liquid–air interface is very sensitive to its nanoconfinement and that it can attain a specific equilibrium shape characterized by both a convex and a concave curvature. We presented a model to describe this elastocapillary equilibrium and the meniscus shape for relative small deflections of the channel capping layers, and found a good agreement with the experimental observations. It was also found that the so-called protrusion length of the meniscus in the flexible nanoconfinement is a sensitive and accurate measure for the membrane deflection. This allows for an accurate and convenient determination for the deflection of thin membranes in nanochannels.
Fig. 8.
the AFM, is shown in Fig. 8. The protrusion length l p has been normalized by w c . The solid line represents the theoretical prediction following from the model. As is clearly seen, the protrusion length of the meniscus is a sensitive measure for the deflection of the membrane. This provides us with a new method to quantify the deflection of flexible sheets: the observation of the protrusion length l p is a very convenient and accurate procedure to determine the deflection of the channel membrane. As mentioned above, the value of the pressure in these cases can only be found numerically from Eqs. (9) and (10). The thus determined p attains a value of −6.7 × 105 Pa at l p / w = 1. The reader will note that Fig. 8 has been plotted for deflections for a limited region of U 0 only. The reason for this is related with the following. The function f according to Eq. (8) is only defined if P (ξ ) < 1 for all −1 < ξ < 1. For increasing deflections U 0 , however, P can become larger than 1 and the solution of (8) does not exist anymore. It corresponds to the situation that df /dx → ∞. This singularity is encountered at δ0 approaching 0.11, where the infinite steepness of f then occurs at ξ = ±0.44 and which corresponds for the current material parameters with a channel width of w = 6.45 μm. The experimental observations of very far protruded menisci with almost infinite steepness, as illustrated by the upper channels in Fig. 4, seem to confirm this. It was also observed that beyond this critical point, the fluid plugs can even detach from the wall and the meniscus shapes change drastically. The origin and consequences of the singularity will be discussed in a future paper on elastocapillarity. In the limit of a stiff membrane, U 0 → 0, it is easy to find the expected result for normal menisci p = 2 cos θ(1/ w + 1/h). In this case P (ξ ) = −ξ cos θ and we reproduce from Eq. (9) the normal meniscus shape: f (x) = ( w /2 cos θ)(1 − 1 − ξ 2 cos2 θ ). To illustrate, finally, the general applicability of the presented model we applied the described approach to the results obtained in [33]. In that work, the authors analyzed water plugs in hydrophilic nanochannels that were subject to significant negative pressure due to the tensile capillary forces. We performed the edge detection routine and curve fitting procedure to the optical micrograph of the fluid meniscus presented in that paper. We thus found a normalized protrusion length of that meniscus of l p / w c = 0.52 ± 0.06, corresponding to a membrane deflection of 9.7 ± 0.8 nm, and a negative pressure in the liquid of −13.8 ± 8 bar. This corresponds quite well with the measured values given there: a deflection of 11 ± 4 nm and a negative pressure of −17 ± 10 bar, respectively. It is also close to the theoretically expected value of −12 bar. 5. Summary In conclusion, we analyzed theoretically and experimentally the fluid menisci in elastically deformed nanochannels that were filled
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