Accurate measurement of liquid transport through nanoscale conduits

Nanoscale liquid transport governs the behaviour of a wide range of nanofluidic systems, yet remains poorly characterized and understood due to the enormous hydraulic resistance associated with the nanoconfinement and the resulting minuscule flow rates in such systems. To overcome this problem, here we present a new measurement technique based on capillary flow and a novel hybrid nanochannel design and use it to measure water transport through single 2-D hydrophilic silica nanochannels with heights down to 7 nm. Our results show that silica nanochannels exhibit increased mass flow resistance compared to the classical hydrodynamics prediction. This difference increases with decreasing channel height and reaches 45% in the case of 7 nm nanochannels. This resistance increase is attributed to the formation of a 7-angstrom-thick stagnant hydration layer on the hydrophilic surfaces. By avoiding use of any pressure and flow sensors or any theoretical estimations the hybrid nanochannel scheme enables facile and precise flow measurement through single nanochannels, nanotubes, or nanoporous media and opens the prospect for accurate characterization of both hydrophilic and hydrophobic nanofluidic systems.


I. Mass ow resistance and Washburn equation
Take a rectangular-shaped channel, with width and height of w and h. If the width of the channel is much larger than its height (w h), with the neglect of inertia term the momentum equation governing the incompressible ow in this channel can be written as ( Figure 1b): µ d 2 u dz 2 = dp dX (1) with µ being the uid viscosity and z the direction along the channel height. The slip boundary condition for the lower and upper walls of the channel (z = 0, h) can be expressed as: in which l s is the slip length. Accordingly, the velocity prole and the mass ow rate can be found as: 2µ dp dX (z 2 − hz − hl s ) wherein ρ is the uid density. The slip boundary condition covers the no slip" boundary condition if one simply sets l s = 0. Now, dening hydraulic resistance as r = ∆P/ṁ, the hydraulic resistance of a channel per unit length would be obtained as: Combining equations (4) and (5) yields a relation for liquid meniscus location in the channel (X ) as a function of time, when a pressure dierence of ∆P is applied: ρwh dX dt = ∆P RX X 2 = 2At, A = 1 ρwh ∆P R (6) If the pressure dierence is replaced by the capillary pressure, ∆P = 2σ cos(θ) h , with σ being the surface tension and θ the contact angle, and assuming l s = 0, the following equation, known as the Washburn equation, governing the location of a meniscus as a function of time in capillary llings can be derived: One can use this resistance concept to derive a governing equation describing capillary ow in a hybrid channel (Figures 1b,c). If R * is the resistance per unit length of the test channel with length L * , and the meniscus is at location x in the reference channel, then mass ow rate in the hybrid channel can be written as: Dening η = R/R * , this equation can be written as:

II. Error analysis
Possible detection range of this method along with the expected error can be understood from a comprehensive error analysis. Without loss of generality let's assume the observation channel is the reference channel. Experimental A can be determined from a set of (T i , X i ), i = 1..m, measured in the reference channel when the water is introduce from the reference channel side (Figure 1b), by minimizing error using the following relations: Similarly, η can be found from a set of (t i , x i ), i = 1..n, measured in the hybrid channel when the water is introduced from the test channel side (Figure 1c): Given η = f (t i , x i , A), the temporal error (E t ), the spatial error (E x ), and the error associated with A (E A ) determine the total error as: Calculations showed that the temporal error is insignicant compared to the other terms , and can be safely ignored. Therefore, for the sake of brevity only derivations of E A and E x are presented.
Error associated with A: E A can be expressed as: in which both ∂η ∂A and δA must be determined. δA can be written as: From equation (9) we get Given the very small contribution of the temporal error (| ∂A ∂T i δT | | ∂A ∂X i δX|), δA can be written as: Here, δX is the spatial resolution (of the microscope), and τ is the time interval between two consecutive frames. The total number of data points is calculated as: m = T max /τ = L 2 /2Aτ , with L being length of the reference channel. Moreover, one can write T i = iτ , which yields: Here, for the sake of simplicity we have assumed m 1. Having found δA, next we need to nd a relation for ∂η/∂A to plug into equation (12). Equation (10) for ∂η/∂A yields: Deriving equations for asymptotic cases of η 1 and η 1 and then combining the equations yields relations that can accurately reproduce the error throughout the entire parametric space. In case of η 1: x i = Aη L * t i and ∂η/∂A = −η/A. Hence: In case of η 1: , yielding the following relation for E A : Combining equations (14) and (15) yields: Spatial error (E x ): A similar approach is adopted for determining the spatial error. E x can be expressed as: ∂η ∂x can be found from equation (10): Again, E x can be analytically found for asymptotic cases of η 1, and η 1: with ζ being the time interval between two consecutive frames and δx being the spatial resolution of the microscope (which may or may not be the same as τ and δX). Combining the two equations above yields: Our full numerical solution of the error (with no estimation about η) indicated that equations (11), (16) and (20) can very accurately estimate the error. According to these relations, at small values of η, the major source of error is the term associated with c 1 which is proportional to and is independent of η and L * . At large values of η, however, the terms associated with c 2 and c 4 are dominant and the error is proportional to √ Aζδx LL * η which grows large with increasing η, and also may be reduced by choosing a longer test channel (L * ). Numerical value of the error for τ = ζ = 10 ms (100 fps), δx = δX =1 micron, L=350 micron and L * =50 micron, for water owing in a reference channel with h=30 nm, suggests that error at low η range is negligible (Figure 1d). At large values of η, however, error can be large and in order to reduce the error, a higher frame rate as well as a longer test channel must be considered. For example, measuring η up to 10 4 with only 20% error is possible, if the length of test channel is increased up to 2 mm and a high speed camera with a frame rate as high as 10 4 is utilized. The spatial and temporal resolutions can be further improved by utilizing techniques such as Field Eect Transistors (FET) 1 along the test/reference conduits and cross-channel current measurements using E-beam dened metal electrodes on two sides of the channel.
It is worth noting that both η and A maybe derived from a single experiment without any need to do the reference channel experiment, i.e., by introducing water from test channel side and using equation (8) along with a least square tting method. However, further error analysis showed that results obtained from this approach are not as accurate as the two-step approach. In particular, at small values of η (η 1), x = Aη L * t which suggests that in such a case only A × η can be found from a single experiment.

III. Decrease in channel height after anodic bonding
Although the height of nanochannels before bonding have been very accurately measured several times with AFM to ensure the consistency between dierent measurements, height of the nanochannels after bonding may not be the same as before bonding. It's known that applying too large of a voltage during anodic bonding may cause a deection equal to the height of nanochannels, in which case nanochannels collapse. This deection is a function of applied voltage, channels' width and the thickness of oxide layer.
2 In our case, however, we observe that increasing the voltage more than a certain value rst decreases the height of channels without channels collapsing, and by continuing to increase the applied voltage nally nanochannels collapse. In order to understand this phenomenon and nd a bonding recipe that ensures minimal change in the channel heights we used a hybrid channel with 16.2 nm test channel. First we bonded the silicon chip with glass by applying 250 Volts at 400 o C and performed the capillary lling experiment, after oxygen plasma. Next, we applied 300 V to the same chip and re-bonded the chip (at the same temperature) and did the capillary experiment. We continued this experiment with 400 and 450 volts too. The results are presented in Figure S1. At 250 Volts, the channels are not completely bonded and a large variation in lling speed from one channel to another is observed. ( Figure S1a) Imperfect bonding at this relatively low voltage gives rise to the measured values of actual resistance to be smaller than theoretical values for some channels in Figure S1c, shown by dashed ellipses. In addition complex ow pattern between channels caused irregularities in the measured x-t curves for this chip which gave rise to large tting errors and yielded some very large resistance values, too.
As the voltage increases to 350V lling speed of dierent channels become consistent.
( Figure S1b and S1c) By further increasing the applied voltage to 400V, the curve maintains its shape and only shifts upward (the blue and black curves in Figure S1c in Figure S1 allows us to conclude that the hybrid channel scheme is a reliable tool to measure very small dierences in the channel heights.
After the experiments the glass layer of the bonded chip was dissolved in HF and again 300 nm oxide was grown on the Si chip. Channel heights in this chip was measured using AFM and perfect agreement with initial measurement was observed. This indicated that any deformation as a result of bonding occurs to the glass and not to the silicon chip.
It can be shown that glass deection cannot be more than a few nanometers and for such a small deection, plastic deformation is not expected. Instead, we hypothesize that conformation of the glass to the round edges of the silicon chip, as shown in Figure S2, reveals that the speed of meniscus is much faster than predicted by Washburn equation (Figures S3a and 3b). Addressing this phase in capillary lling which seems to be mainly ruled by the corner ows and lm ows is out of scope of this paper; nevertheless, to obtain A actual one has to consider the entrance eect and hence in this work only data of menisci located in the range of 250 micron to 500 micron from the entrance of the refer-ence channels were used to t the theoretical curves. It's worthwhile mentioning that this phenomenon was a lot less pronounced in capillary lling of shallower reference channels with h = 50nm. To nd the correction factor C, A actual is compared with the theoretical A estimated by assuming ∆P = 2σ cos(θ)( 1 h + 1 w ), θ = 0, µ = 1 mPa.s, σ=0.07 N/m, w=3 micron, along with the height of the reference channel for each channel set. As already discussed, A has contributions from both resistance term as well as the pressure term.
However, given the relatively large height of the reference channels and also because the electroviscosity eect cannot be more than a few percent, 46 negligible deviation of the hydraulic resistance from theory is expected for the reference channels (as demonstrated in Figure 4b) and the correction factor C can be mainly attributed to the capillary pressure. In fact, surface quality of the nanochannels, i.e., roughness and hydrophilicity, can be the major role players and in particular hydrophilicity of the surfaces can be slightly dierent from chip to chip and also may vary depending on the preparation of the chips, yielding dierent contact angles. In fact distorted menisci and/or menisci with dierent contact angles have been observed in our experiments reecting the interplay between the capillary force and the viscous forces ( Figure S3c). The contact angles shown in Figure   S3c are of course the in-plane contact angles (the top view of the channel), and the capillary pressure by formation of a curved meniscus in this plane cannot be more than 3% of the total capillary pressure (h/w ≈ 3%); however, one can expect the same behavior in the meniscus shape to be observed along the channel height which is the dominating term. quality of the channels deteriorate over time. The correction factor C for the reference channels of such a chip reached the values of up to 2 over time ( Figure S4a). Therefore, for the purpose of this paper only the results of the experiments with fresh chips (right after Piranha cleaning, bonding with glass, and applying oxygen plasma) were used.
However study of the old unused chips enabled us to better evaluate the performance of the hybrid nanochannels scheme, and also to get some insight about mechanism of the formation of bubbles in the nanochannels. When water is introduced from the test channel side of an old unused chip, some air pockets were trapped in several spots along the channel.
1012 These bubbles had not been observed for most of the fresh chips of similar dimensions, or the results were discarded otherwise ( Figure S4b). Repeating the experiment several times in a few consecutive days indicated that air pockets are always consistently trapped in the same spots ( Figures S4c,d) and indicated that at those spots silica surface is relatively hydrophobic. On the other hand, when water was introduced from the reference channel side no air was trapped in those spots ( Figure S4f ), indicating that in addition to the local hydrophobicity of the surface acting as the weak points along the channel, the hydraulic resistance behind the meniscus is another factor giving rise to entrapment of air in the channel. In fact, when the meniscus reaches a hydrophobic site it is momentarily distorted. This distortion along with increase in the contact angle give rise to a reduced capillary pressure. Now if the resistance behind the meniscus is small enough to allow the reduced capillary pressure drive the water through the entire channel, no air would be trapped; otherwise, water ows from the corners and forms another meniscus downstream, leaving some air behind. Finally the air pocket forms a bubble because of the liquid pressure. (Figures S4g,h,i) Corner ows even in case of fresh chips with no hydrophobic sites can be observed. Here we experimentally observed that choice of an excessively long test channel by imposing a huge resistance to the reference channel causes the corner ows to become the dominant mode of liquid transport ( Figure S5b).
Since ow at the sharp corners moves quite faster than the bulk ow 1416 and can easily ll the nanochannelsas narrow as 3 micronwith a dierent speed, they are considered a major problem is accurate measurement of the capillary ows. The hybrid nanochannel design, however, gives us the latitude to eliminate or greatly reduce the corner ows by adjusting the total hydraulic resistance through right choice of test channel length (L * ) and adjusting the driving capillary pressure by right choice of the reference channel height (h).
We've been able to take advantage of the chips with deteriorated surface properties to better evaluate the hybrid nanochannel scheme and it's capability in decoupling the hydraulic resistance from the driving pressure. Using an old unused hybrid chip (a few months after bonding the chip) with h * =28 nm experiments were performed, after applying Oxygen Plasma. By introducing water from the reference channel side the correction factor C was measured to be C = 1.84 +0.29 −0.16 , about 50% larger than the fresh chips ( Figure 4c). When water is introduced from test channel side it advances for about ∼50 micron before any bubble is formed. We used this part of the data to nd α for this channel height. Results showed that while the value of A in the reference channel has signicantly decreased, the which can cause a deformation in the channels at the location of meniscus, giving rise to an increased capillary pressure and also increased hydraulic resistance which is known to boost the lling rate.
19 This error, however, can be avoided using our method as data is only collected when meniscus moves in the reference channels of large height. Needless to say, in case of our measurements due to large thickness of the glass and silicon wafers (500 micron) and small width of the channels (3 micron) elastic deformation can be safely neglected, but if the nanochannels lack strong mechanical supports, this deformation can be signicant. Another advantage of this method is that since only the ratio of the uid properties in the test and reference channels determines η (and value of A is experimentally found), use of this method to a large extent eliminates the errors associated with the temperature dependence of uid properties. This method other than characterization of the hydrophilic nanochannels can be used to characterize 1-D nanotubes and nanoporous media, even if they are hydrophobic. In fact characterization of hydrophobic conduits/nanoporous materials which don't allow for spontaneous lling can be done through their integration with a hydrophilic reference channel. Similar approach that has been explained would be used for characterization of such hydrophobic-hydrophilic hybrid channels, except that for the experiment that starts from the test channel side (hydrophobic side) we have to provide some external pressure to drive the water through the hydrophobic test channel until water enters the hydrophilic reference channel. The external pressure can be immediately removed once water enters the hydrophilic part, or it can be maintained to further drive water through the hydrophilic reference channel. In the second case, when introducing water from the hydrophilic channel side for the second measurement, the same external pressure also needs to be applied. In case of CNTs, it has been previously shown that water spontaneously lls the CNT and applying extra pressure is not needed.
1 The real challenges of CNT ow characterization include integration of CNTs in a hybrid setting and tracking the location of meniscus as a function of time. We are currently working on solving these two challenges and will report our results in another paper.
The realm of validity of the hybrid nanochannels scheme is the validity of Washburn equation, which if for any reason violated, the method may not be applied. In addition, if the liquid of interest is non-evaporating, or cannot be removed from the hybrid channels after the rst experiment, this method fails to work. Finally, rate dependence of the dynamic contact angle can introduce error to the results of this method. In our method, the velocity of meniscus can be widely dierent between the two experiments, i.e., the lling experiment that starts from the test channel side and another one that starts from the reference channel side, which means the driving capillary pressure can be dierent between the two experiments. However, in our measurements since the capillary numbers in all cases are very small (C a = uµ/σ < 10 −4 with u being the velocity), variations in cos(θ) due to dierent lling rates is no more than 1%, 22 consistent with previous contact angle measurements in nanochannels.