Nanoscale Capillary Flows in Alumina: Testing the Limits of Classical

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Nanoscale Capillary Flows in Alumina: Testing the Limits of Classical Theory Wenwen Lei, and David R. McKenzie J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.6b01021 • Publication Date (Web): 23 Jun 2016 Downloaded from http://pubs.acs.org on June 28, 2016

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The Journal of Physical Chemistry Letters

Nanoscale Capillary Flows in Alumina: Testing the Limits of Classical Theory Wenwen Lei and David R. McKenzie School of Physics, University of Sydney, NSW, 2006, Australia

ACS Paragon Plus Environment

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ABSTRACT Anodic aluminium oxide (AAO) membranes have well formed cylindrical channels, as small as 10 nm in diameter, in a close packed hexagonal array. The channels in AAO membranes simulate very small leaks that may be present for example in an aluminum oxide device encapsulation. The 10 nm alumina channel is the smallest that has been studied so far for its moisture flow properties and provides a stringent test of classical capillary theory. We measure the rate at which moisture penetrates channels with diameters in the range 10 nm to 120 nm with moist air present at 1 atm on one side and dry air at the same total pressure on the other. We extend classical theory for water leak rates at high humidities by allowing for variable meniscus curvature at the entrance and show that the extended theory explains why the flow increases greatly when capillary filling occurs and enables the contact angle to be determined. At low humidities our measurements for air filled channels agree well with theory for the interdiffusive flow of water vapor in air. The flow rate of water filled channels is one order less than expected from classical capillary filling theory and is coincidentally equal to the helium flow rate, validating the use of helium leak testing for evaluating moisture flows in aluminum oxide leaks.

KEYWORDS: Moisture flow; AAO membrane; Washburn theory; nanochannel


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The transport of moisture, defined here as water vapor or liquid water, either individually or in combination, through channels of nanometre or micrometre dimensions, is important in many fields. Cell biology, biological and chemical sensing, drug delivery technology, water desalination and encapsulation of devices with electronic components are all dependent on a knowledge of moisture flows. Many electronic devices that underpin information technology, medical technology and energy technology are moisture sensitive and require hermetic encapsulation. Some examples are solar cells, organic light emitting devices (OLEDs) and medical implantable devices (for example, the ‘bionic’ ear and eye). Such devices must be hermetically encapsulated and, although moisture transport rates are critical to their lifetime, at the present time helium is most often used in hermeticity testing of their encapsulations because of the convenience of a helium test. However, the flow rate of moisture through carbon nanotubes is much greater than expected from the flow rate of helium gas

1-3

and there are

examples of helium leak tight membranes that are water permeable 4. It is important therefore to correlate moisture flow rates in a device with helium flow rates in the same device. Recent work on silica capillaries by the Lei et al. 5 shows that, in general, it is not safe to assume that a helium flow rate measurement is an accurate reflection of the penetration rate of moisture without further investigation, since the effects of capillary filling and of the tangential momentum accommodation coefficient (TMAC) 6 increase the rate of moisture flow through small channels in a manner not predictable from a helium flow rate measurement. There is currently a lack of knowledge and experience of the moisture flow rate through channels of the materials commonly used in the manufacture of device encapsulations. Moisture flow rates through silica channels have been studied, but only for channel diameters larger than 200 nm. Molar flow rates that are bounded on the low side by the helium flow rate and bounded on the high side by the Washburn


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flow rate which is at least one order of magnitude larger than the helium flow rate for channels greater than 200 nm but less than 500 nm diameter 5. By the term Washburn flow we refer to flows driven by the Laplace pressure difference across a meniscus that are inhibited by the liquid viscosity. The flow of moisture through cylindrical capillaries smaller than 200 nm has not been studied and is a problem directly relevant to the flow of moisture into and out of an enclosure, but is also relevant to the mechanisms of water extraction from the atmosphere 7. Well formed channels of cylindrical shape with nanometer and micrometer diameters are formed in anodic aluminium oxide (AAO) membranes by electrochemical anodization and are of interest in many applications 8, due to their high surface area (up to 250 m2g-1), high porosity (1011 pores cm-2), highly ordered and monodisperse pores, tunable thickness, excellent chemical, thermal and mechanical stability and biocompatibility 9,10. There are many studies on gas flows through porous media

11,12

and some studies on condensation in and evaporation through nanoscale

porous media, for example through compacted powders, oxide xerogels and Vycor glass

13,14

. It

is difficult to compare the experimental results for two phase flows with theoretical predictions due to the diverse channel geometry, including variable cross sectional shape and size and in most cases a complex interconnected structure 15-18. There are several works on evaporation and condensation of Ar toluene

21

19

, organic solvents (perfluoromethylcyclohexane

20

and isopropanol and

) in AAO membranes and a study of water vapor/liquid water interfaces in AAO

membranes with modified surfaces

22

. A study of moisture flows in unmodified AAO

membranes has so far not been reported. Here we address the behavior of flow in the channels of these membranes, as alumina is a material relevant to active implantable medical devices where it used to form feedthroughs 23 and examine for the first time moisture flow rates in well formed channels of diameter as small as 10 nm. We compare the results with an extended theory for


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capillary condensation combined with the interdiffusive flow of water vapor in air. We also compare with the expected flow of helium gas through the same channel.

Figure 1. (a) Four types of moisture leak, combining liquid water and water vapor in a channel. Type a: vapor only flow; Type b: the channel contains a condensed liquid slug; Type c: flow where the inlet side is covered with water; Type d: both inlet and outlet are covered with water.


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(b) A leak type intermediate between a type b and a type c leak in which the meniscus on the inlet side has a radius of curvature defined by the Kelvin equation that is larger than the radius

of the meniscus on the outlet side. (c) Scanning electron microscopy (SEM) images of the four sizes of channel provided by InRedox LLC (USA). (d) An idealized schematic of the AAO membrane provide by InRedox LLC (USA) and its incorporation into the mass loss experiment. Capillary flow had been studied in the 19th century by Young 24 and Laplace 25. Capillary condensation is likely to occur very readily in nanoscale channels, leading to flow phenomena that are driven by the Laplace pressure difference across the meniscus of the condensed liquid. Washburn 26 combined the Laplace pressure across the meniscus with the Poiseuille law and obtained a relation between the length of a liquid column in a capillary and the time. For the AAO membranes studied here, the 10 nm capillary should fill in time less than 1 min according to the classical Washburn theory. In this case, flow will occur by means of evaporation from the outlet meniscus and the capillary will refill as required, driven by Laplace pressure. In our recent paper 5 we classified water leaks into four types: a vapor only leak (type a), a “slug flow” in which there are menisci inside the channel defining a liquid “slug” (type b), a Washburn type leak in which a meniscus forms inside the channel through capillary filling from a reservoir of liquid at the inlet (type c) and a Poiseuille flow between liquid reservoirs (type d). These leak types are reproduced in Figure 1. The flow rates predicted from classical theory have been compared with observations 5,27. For materials other than carbon nanotubes, the classical theory has been found to be satisfactory without invoking large slip lengths for liquid flows or anomalously small tangential momentum accommodation coefficients for water vapor flows.


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Here we are concerned with flows between moist air and dry air, a realistic case for many applications. For such interdiffusive flows, we consider three cases described by Ernst and Hemond 28: Knudsen interdiffusive flow with Kn > 10; “classical” interdiffusive flow with Kn < 0.01; and intermediate interdiffusive flow for Kn between these ranges. When Knudsen interdiffusive flow applies, collisions with the channel walls rather than collisions with the background gas determine the mean free path. Knudsen interdiffusive flow therefore obeys the same equation as ordinary Knudsen flow for an ideal gas with an appropriate mean free path. The molar flow rate for water vapor, MKI, is therefore:

∆ 4 2 = , = 3

(1a)

where is the Knudsen diffusion coefficient, is the radius of the channel, ∆ is pressure

difference across membrane, is the Boltzmann constant, is the temperature, is the channel

length, is the molar mass of water in kg/mol and is the molecular mass.

Classical interdiffusive flow applies when the mean free path is much smaller than the diameter of the channel so that we can neglect the influence of the walls on the interdiffusive process. We write the molar flow rate of water vapor for classical interdiffusive flow, MCI, in the special case of molecules consisting of rigid elastic spheres as: ∆ 3 ( ) 1 1 = , = ( + ) , 2 8 = ( + )/2

(1b)

where is the “classical” interdiffusive flow coefficient, is the mean molecular diameter of the interdiffusing gases, and are molecular diameters of the two components

respectively and is the total pressure.


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For intermediate interdiffusive flow the theory has been described by Ernst and Hemond 28 and Remick and Geankopolis 29. We write the water vapor molar flow rate, MII, as: =

∆ 1 1 , = 1/( + )

(1c)

where is the coefficient for the interdiffusive flow at intermediate Kn. Remick and

Geankopolis 29 compared this theory with experiment and found it worked well for molecules

consisting of the test gases they used, N2 and He. Ernst and Hemond 28 found the equation worked well for single atoms but somewhat overestimated flow for polyatomic molecules for

which internal degrees of freedom make collisions less elastic. At high total pressures, in Eq.

(1c) approaches and at low pressures it approaches . When air is present with no total pressure gradient, the flow of water vapor is described by interdiffusion theory (Eqs. 1(a-c)).

Helium is an important gas for leak testing and its use for this purpose is well established. The equation of Cha and McCoy 30 for ideal gas flow in a cylindrical channel is convenient for describing the helium flow rate that would be measured in a helium leak test, where one side of the leak is at vacuum and the other side has one atmosphere of helium present. This equation gives a single analytic expression for molar flow rate across all flow regimes and requires only two fitting parameters, the Maxwell TMAC 31, (α, 0 (specular collision) ≤ α ≤ 1(diffusive collision)) and an empirical parameter (c0, 5 ≤ c0 ≤ 7) 30, as we have recently demonstrated for nitrogen 32:

∆ 8 2 − ) √ 1 # = $ % & + , 2 8 *+ 3√ )

1 1 × .1 − /0 *+ 12+ℎ*+ 41 − 256 $ %7 − 9 2 − ) 64 5 2(1 + ) 3 *+)

(2)


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where 5 is the function 4*+/√, 6 is the first order Bessel function of the first kind. In the

definition of Maxwell’s TMAC, it was proposed that there are two possible scattering processes for a molecule moving in a channel and colliding with the wall: reflection and adsorption/desorption. Reflection preserves momentum in the direction of travel () = 0), while

adsorption/desorption destroys it () = 1). For helium, we used a value of /0 of 5.0 and a TMAC of 1.0. Here we use the Maxwell definition of TMAC, bearing in mind that other more realistic definitions can be used at the cost of some complexity 32. For water vapor flows we also assumed a TMAC of 1.0. We now refine the definition of a type b and type c leaks by allowing an intermediate leak type where the liquid that extends somewhat outside the entrance of the capillary and is terminated there by a meniscus with a variable radius of curvature, as shown in Figure 1. When the radii of curvature of the two menisci are the same, it is a type b leak. When the radius of the meniscus at the inlet is infinite, it is a type c leak. The meniscus curvature at the inlet side is calculated from the water vapor partial pressure at the entrance using the Kelvin equation to obtain the pressure (P1) above the meniscus: = : ;

<

= >:? @AB CD(EF ⁄EGH )JG

(3)

where K is surface tension of water, L is the density of water, M is the water contact angle, Ps is the saturation pressure at room temperature. From Eq. (3), the radius of the meniscus on the inlet of the channel is obtained: =

2K/NOM L P+( : ⁄ )

(4)

From the Laplace pressure, the pressure difference Q across the meniscus on the inlet of the channel is given by:


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Q = RS −

2K L P+( : ⁄ ) = RS − /NOM

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(5)

where RS is the pressure of the atmosphere. The radius of the meniscus on the outlet side of the channel is related to the radius of the channel and to the contact angle. The pressure

difference across this meniscus is: Q = RS −

2K 2K/NOM = RS −

(6)

2K/NOM L P+( : ⁄ ) − /NOM

(7)

Thus, the pressure difference driving the flow is: ∆ = Q − Q =

and the corresponding molar flow rate is obtained from the Poiseuille law: T =

U L U L 2K/NOM L P+( : ⁄ X ) ∆ = [ − ] 8V 8V /NOM

(8)

where V is the viscosity of water. The theory of Eq. (8) applies when the inlet humidity is sufficient to form condensation while for lower inlet humidities there is no condensation and the interdiffusive theory for water vapor in air applies. We will refer Eq. (8) to as extended capillary filling theory and encompasses slug flow (type b) at low inlet humidities, passing to Washburn flow (type c) at large inlet humidities. For a flow exiting into dry air from a short capillary as in our case, the evaporation rate from a meniscus when located at the exit would be larger than the refilling rate of the capillary 5. The flow rate is therefore limited by the result of Eq. (8) for the nearly filled capillary. A balance will be maintained between the refilling rate and the evaporation rate (impeded by a short length of vapor flow) from the meniscus located inside the capillary. The column of liquid is not accelerated, avoiding the need for the correction to the Washburn filling rate introduced by Bosanquet 33 and discussed further by Das et al. 34 and Das and Mitra 35.


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Figure 2. A comparison of the molar flow rate calculated from our extended capillary filling theory (Eq. (8)) for water flow combined with the interdiffusion theory (Eqs. 1(a-c)) for water vapor flow as a function of humidity for four channel diameters. The points are our experimental values for single channels derived from measurements on AAO membranes. The channel length is the membrane thickness of 0.05mm. The inset shows that a contact angle of above 75o can be determined from the fit to experimental data of the extended capillary filling theory. AAO membranes of thickness 0.05 mm were sourced from InRedox LLC (USA) and measurements were made using the mass loss method, in which the humidity of the air at the inlet was controlled by the use of water-glycerol mixtures to adjust the partial pressure of water vapor above the liquid surface 36. This method uses glass vials (volume of 2 ml and inner diameter of 6 mm on the top, Thermo Scientific, Australia) charged with ~1.8 ml of de-ionized water or a mixture of de-ionized water and glycerol (pharmaceutical grade). A glass slide was used as a support structure for the AAO membranes by drilling a hole (6 mm diameter) through


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the centre. AAO membranes were then mounted above the hole and sealed with epoxy resin (Selleys, Australia). These samples were then sealed with epoxy resin onto the vials containing the liquid charge as shown in Figure 1 (d). After curing at room temperature overnight, the vials were weighed on a microbalance to an accuracy of 100 µg. By measuring a control sample (only with glass lid without hole), the effect of any mass loss from the epoxy resin and the presence of leaks in the seal could be accounted for. For assessing flow rates with air at the outlet, containers were placed in a desiccator containing silica gel desiccant and weighed every 2 days. The results for the molar flow rate of moisture as a function of the relative humidity at the inlet and for four different diameters are shown in Figure 3. The results are compared to the theory that assumes interdiffusive flow (Eqs. 1(a)-1(c)) until capillary filling occurs, when a sharp rise occurs as the flow is dominated by a Washburn type filling and subsequent flow (Eq. (8)). The humidity at which the rise occurs is sensitive to the contact angle. A contact angle of somewhat above 75o for the alumina wall gives a good fit to the capillary filling portion of the data and is in agreement with other measurements of the water contact angle of alumina 37.


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Figure 3 Theoretical and our experimental results for molar flow rate for AAO membranes with 10, 20, 40 and 120 nm pore diameters. The areal density of the channels in the membranes was provided by the manufacturer. Theoretical lines are He flow (―), interdiffusion theory (---), meniscus evaporation into air (---), Poiseuille law (―) with two pressure difference Ps and 1 atm and Washburn flow (---) with two contact angles (0o and 75o). Our experimental results are 10, 20, 40 and 120 nm ().All the empty symbols are derived from literature measurements of capillary filling rates: quartz capillaries of 45-180 nm diameter, Sobolev et al. (◇ ◇) 38, porous Vycor of 6.8 and 9.8 nm pore diameter ,Gruener et al. (◁ ◁) 16 .

The maximum measured flow rates at 298 K for each membrane operating between saturated and dry air are plotted in Figure 3. The pore densities for the AAO membranes of 10, 20, 40 and 120 nm pore diameters are (1.6 ± 0.5) × 1011, (5.8 ± 1) × 1010, (1 ± 0.5) × 1010 and (9.8 ± 1) × 108


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pores cm-2. We observe time dependence in the flow rates over times of the order of weeks. The maximum flow rates occur at the beginning of the test and decrease with time. Time dependent effects have been previously observed only over short times for filling of silica 38-44 and silicon capillaries 15,40,45,46. The measurements of Figure 3 are for the highest flow rates and the time dependences are discussed further in the Supporting Information. A possible mechanism for the time dependence is the blocking of the capillaries over time from the deposition of dissolved material at the exit. This was suggested by the observation that heating of the membrane to more than 100°C is required to restore the initial flow rate. The interdiffusion theory shown in Figure 3 as a green dotted line is the molar flow rate of water vapor at saturation pressure in air at 1 atm diffusing through an air filled channel into air dry air at 1 atm. To check that the flow rates at high inlet humidities were not sensitive to the liquid configuration at the inlet we compared the flows for the upright and inverted cases. In the inverted case, the inlet is definitely covered by liquid, while in the upright case, it may not be. The molar flow rates of pure water for upward vials and downward vials were very similar, confirming that the same capillary filling conditionsare achieved at high humidities in the upright case without forced direct contact with liquid water. While results for flow rates of moisture through small AAO channels are not available in the literature, there are measurements of the filling rate of small capillaries of silica 38 and of the capillary filling rate of porous Vycor glass 16

. The silica and Vycor capillary filling rate measurements were converted to an equivalent

moisture flow rate for presentation in Figure 3 and show similarities with our results for AAO membranes. It is evident that the maximum observed flow rates in all the AAO channels is approximately one order of magnitude less than the predicted Washburn flow rate for a contact angle of 75o shown


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in Figure 3. The flow is not limited by the rate of evaporation of the outlet meniscus, as shown in the work of Birdi et al. 47 and as shown in Figure 3, the meniscus evaporation rate is much larger than the observed flow rates for these channel sizes. Reasons for the capillary flow rates to fall short of the Washburn prediction have been discussed by Oyarzua et al. 43 and relate to electroviscous effects, nanobubbles, strong liquid-surface interactions and the dynamic contact angle. Note also that we observe an enhancement factor of 460 over liquid Poiseuille flow in the same channels for a pressure difference of 1atm. This large factor is a consequence of the very large Laplace pressure that greatly exceeds 1 atm. The helium flow rate for the same channels, shown as the solid black line in Figure 3 lies above the line for the interdiffusive flow of water vapor by approximately two orders of magnitude for these channel sizes. There is a near coincidence of the helium flow rate with our maximum moisture flow rates that is unexpected but is a result of the one order of magnitude deficit in the moisture flow rate relative to the expected Washburn flow rate discussed above. In this work, we measured molar moisture flow rates through ordered nanopores in AAO membranes subjected to a gradient in the partial pressure of water vapor between moist air and dry air, both at 1 atm pressure. These are the smallest well- defined channels of alumina so far measured for their moisture conductance properties. The measured flow rates are in good agreement with our extended classical theory of flow except that the maximum flow rates are approximately one order of magnitude less than expected. The measurements are consistent with rough walled channels with a contact angle of 75o. Liquid flows occur without slip. There is a chance coincidence with the expected flow rates of helium gas through the same channels that gives credibility to the use of helium leak testing for leak channels in alumina, enabling rapid and convenient testing of maximum moisture flow rates in alumina based encapsulations.


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ASSOCIATED CONTENT Supporting Information The Supporting Information describes condensation of AAO nanopores, time dependent phenomenon and roughness of the channel wall. This material is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] Author Contributions W. L. and D. M. developed the new Washburn theory. W. L. designed the mass loss method, measured weight loss of the AAO membranes and did the theoretical calculations for all the theories used in this paper. The manuscript was written through contributions of both authors. ACKNOWLEDGMENTS The authors acknowledge the provision of SEM pictures of the AAO membranes provided by the manufacturer, InRedox LLC. (USA). REFERENCES (1) Majumder, M.; Chopra, N.; Andrews, R.; Hinds, B. J. Enhanced Flow in Carbon Nanotubes. Nature 2005, 438, 44. (2) Majumder, M.; Chopra, N.; J.Hinds, B. Mass Transport through Carbon Nanotube Membranes in Three Different Regimes: Ionic Diffusion and Gas And Liquid Flow. ACS Nano 2011, 5, 3867-3877.


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(3) Holt, J. K.; Park, H. G.; Wang, Y.; Stadermann, M.; Artyukhin, A. B.; Grigoropoulos, C. P.; Noy, A.; Bakajin, O. Fast Mass Transport through Sub-2-Nanometer Carbon Nanotubes. Science 2006, 312, 1034-1037. (4) Nair, R. R.; Wu, H. A.; Jayaram, P. N.; Grigorieva, I. V.; Geim, A. K. Unimpeded Permeation of Water through Helium-Leak-Tight Graphene-Based Membranes. Science 2012, 335, 442-444. (5) Lei, W.; Fong, N.; Yin, Y.; Svehla, M. J.; McKenzie, D. R. Science of Water Leaks: Validated Theory for Moisture Flow in Microchannels and Nanochannels. Langmuir 2015, 31, 11740-11747. (6) Lei, W.; McKenzie, D. R. Enhanced Water Vapor Flow in Silica Microchannels: The Effect of Adsorbed Water on Tangential Momentum Accommodation. J. Phys. Chem. C 2015, 119, 22072-22079. (7) Shannon, M. A.; Bohn, P. W.; Elimelech, M.; Georgiadis, J. G.; Marinas, B. J.; Mayes, A. M. Science and Technology for Water Purification in the Coming Decades. Nature 2008, 452, 301-310. (8) Aramesh, M.; Cervenka, J.: Nanomedicine; ONE CENTRAL Press: Manchester, UK, 2014. (9) Brüggemann, D. Nanoporous Aluminium Oxide Membranes as Cell Interfaces. J. Nanomater. 2013, 2013, 1-18. (10) La Flamme, K. E.; Popat, K. C.; Leoni, L.; Markiewicz, E.; La Tempa, T. J.; Roman, B. B.; Grimes, C. A.; Desai, T. A. Biocompatibility of Nanoporous Alumina Membranes for Immunoisolation. Biomaterials 2007, 28, 2638-2645. (11) Pant, L. M.; Mitra, S. K.; Secanell, M. Absolute Permeability and Knudsen Diffusivity Measurements in PEMFC Gas Diffusion Layers and Micro Porous Layers. J. Power Sources 2012, 206, 153-160. (12) Carrigy, N. B.; Pant, L. M.; Mitra, S. K.; Secanell, M. Knudsen Diffusivity and Permeability of PEMFC Microporous Coated Gas Diffusion Layers for Different Polytetrafluoroethylene Loadings. J. Electrochem. Soc. 2013, 160, F81-F89. (13) Gregg, S. J.; Sing, K. S. W.: Adsorption, Surface Area and Porosity; Academic Press: London, UK, 1982. (14) Valiullin, R.; Naumov, S.; Galvosas, P.; Karger, J.; Woo, H. J.; Porcheron, F.; Monson, P. A. Exploration of Molecular Dynamics during Transient Sorption of Fluids in Mesoporous Materials. Nature 2006, 443, 965-968. (15) Acquaroli, L. N.; Urteaga, R.; Berli, C. L.; Koropecki, R. R. Capillary Filling in Nanostructured Porous Silicon. Langmuir 2011, 27, 2067-2072. (16) Gruener, S.; Hofmann, T.; Wallacher, D.; Kityk, A. V.; Huber, P. Capillary Rise of Water in Hydrophilic Nanopores. Phys. Rev. E 2009, 79, 067301. (17) M., K.; M., J.; A., S. Application of Large Pore MCM-41 Molecular Sieves to Improve Pore Size Analysis Using Nitrogen Adsorption Measurements. Langmuir 1997, 13, 6267-6273. (18) Neimark, A. V.; Ravikovitch, P. I.; Vishnyakov, A. Adsorption Hysteresis in Nanopores. Phys. Rev. E 2000, 62, R1493. (19) Bruschi, L.; Fois, G.; Mistura, G.; Sklarek, K.; Hillebrand, R.; Steinhart, M.; Gősele, U. Adsorption Hysteresis in Self-Ordered Nanoporous Alumina. Langmuir 2008, 24, 10936-10941.


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(20) Alvine, K. J.; Shpyrko, O. G.; Pershan, P. S.; Shin, K.; Russell, T. P. Capillary Filling of Anodized Alumina Nanopore Arrays. Phys. Rev. Lett. 2006, 97, 175503. (21) Casanova, F.; Chiang, C. E.; Li, C. P.; Roshchin, I. V.; Ruminski, A. M.; Sailor, M. J.; Schuller, I. K. Gas Adsorption and Capillary Condensation in Nanoporous Alumina Films. Nanotechnology 2008, 19, 315709. (22) Lee, J.; Laoui, T.; Karnik, R. Nanofluidic Transport Governed by the Liquid/Vapour Interface. Nat. Nanotechnol. 2014, 9, 317-323. (23) Lu, H.; Svehla, M. J.; Skalsky, M.; Kong, C.; Sorrell, C. C. Pt-Al2O3 Interfacial Bonding in Implantable Hermetic Feedthroughs: Morphology and Orientation. J. Biomed. Mater. Res. B: Appl. Biomater. 2012, 100, 817-824. (24) Young, T. An Essay on the Cohesion of Fluids. Phil. Trans. R. Soc. Lond. 1805, 95, 65-87. (25) Laplace, P. S.: Traité de Mécanique Céleste; Duprat: Paris, France, 1799. (26) Washburn, E. W. The Dynamics of Capillary Flow. Phys. Rev. 1921, 17, 273-283. (27) Lei, W.; Rigozzi, M. K.; McKenzie, D. R. The physics of confined flow and its application to water leaks, water permeation and water nanoflows: a review. Rep. Prog. Phys. 2016, 79, 025901. (28) Ernst, M. J.; Hemond, H. F. Multicomponent Vapor Transport Model for Viscous, Transitional, and Molecular Flow. J. Vac. Sci. Technol. A 1995, 13, 2962-2971. (29) Remick, R. R.; Geankoplis, C. J. Binary Diffusion of Gases in Capillaries in the Trasition Region between Knudsen and Molecular Diffusion. Ind. Eng. Chem. Fundam. 1973, 12, 214-220. (30) Cha, C. Y.; McCoy, B. J. Application of Third-Order Constitutive Relations to Poiseuille Flow of A Rarefied Gas. J. Chem. Phys. 1971, 54, 4373-4383. (31) Maxwell, J. C. On Stresses in Rarified Gases Arising from Inequalities of Temperature. Phil. Trans. R. Soc. Lond. 1879, 170, 231-256. (32) Lei, W.; McKenzie, D. R. Revisiting Maxwell’s Tangential Momentum Accommodation Coefficient: A Study of Nitrogen Flow in A Silica Microtube Across All Flow Regimes. Ann. Phys. 2014, 351, 828-836. (33) Bosanquet, M. A. On the Flow of Liquids into Capillary Tubes. Philos. Mag. 1923, 6, 525-531. (34) Das, S.; Waghmare, P. R.; Mitra, S. K. Early Regimes of Capillary Filling. Phys. Rev. E 2012, 86, 067301. (35) Das, S.; Mitra, S. K. Different Regimes in Vertical Capillary Filling. Phys. Rev. E 2013, 87, 063005. (36) Glycerine Producers' Association: Physical Properties of Glycerine and Its Solutions; Glycerine Producers' Association, New York, 1963. (37) Redón, R.; Vázquez-Olmos, A.; Mata-Zamora, M. E.; Ordóñez-Medrano, A.; Rivera-Torres, F.; Saniger, J. M. Contact Angle Studies on Anodic Porous Alumina. Rev. Adv. Mater. Sci 2006, 11, 79-87. (38) Sobolev, V. D.; Churaev, N. V.; Velarde, M. G.; Zorin, Z. M. Surface Tension and Dynamic Contact Angle of Water in Thin Quartz Capillaries. J. Colloid Interface Sci. 2000, 222, 51-54. (39) Thamdrup, L. H. j.; Persson, F.; Bruus, H.; Kristensen, A.; Flyvbjerg, H. Experimental Investigation of Bubble Formation during Capillary Filling of SiO2 Nanoslits. App. Phys. Lett. 2007, 91, 163505.


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Page 19 of 19

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(40) Han, A.; Mondin, G.; Hegelbach, N. G.; de Rooij, N. F.; Staufer, U. Filling Kinetics of Liquids in Nanochannels as Narrow as 27nm by Capillary Force. J Colloid Interface Sci. 2006, 293, 151-157. (41) Tsukahara, T.; Maeda, T.; Hibara, A.; Mawatari, K.; Kitamori, T. Direct Measurements of the Saturated Vapor Pressure of Water Confined in Extended Nanospaces Using Capillary Evaporation Phenomena. RSC Advances 2012, 2, 3184-3186. (42) Fisher, L. R.; Lark, P. D. An Experimental Study of the Washburn Equation for Liquid Flow in Very Fine Capillaries. J. Colloid Interface Sci. 1979, 69, 486-492. (43) Oyarzua, E.; Walther, J. H.; Mejia, A.; Zambrano, H. A. Early Regimes of Water Capillary Flow in Slit Silica Nanochannels. Phys. Chem. Chem. Phys. 2015, 17, 14731-14739. (44) Tas, N. R.; Haneveld, J.; Jansen, H. V.; Elwenspoek, M.; van den Berg, A. Capillary Filling Speed of Water in Nanochannels. Appl. Phys. Lett. 2004, 85, 3274. (45) Haneveld, J.; Tas, N. R.; Brunets, N.; Jansen, H. V.; Elwenspoek, M. Capillary Filling of Sub-10nm Nanochannels. J. Appl. Phys. 2008, 104, 014309. (46) Phan, V. N.; Nguyen, N. T.; Yang, C.; Joseph, P.; Djeghlaf, L.; Bourrier, D.; Gue, A. M. Capillary Filling in Closed End Nanochannels. Langmuir 2010, 26, 13251-13255. (47) Birdi, K. S.; Vu, D. T. A Study of the Evaporation Rates of Small Water Drops Placed on A Solid Surface. J. Phys. Chem. 1989, 93, 3702-3703.


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Nanoscale Capillary Flows in Alumina: Testing the Limits of Classical

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