Explaining Ultra-High Water Flow Rates Observed in Carbon Nanotube Membranes

Explaining ultra-high water flow rates observed in carbon nanotube membranes

Davide Mattia,^1* Francesco Calabr˜²

 ¹Department of Chemical Engineering, University of Bath, UK

2 DIEI, Universitˆ di Cassino, Italy

d.mattia@bath.ac.uk, www.bath.ac.uk/nanotech

Liquid flow through nanoporous media has been studied in disciplines as diverse as membrane science, soil permeability and cell physiology. In all these fields, though, the emphasis has been on the macroscopic outcome, often neglecting the effects of intermolecular forces and liquid-pore wall physicochemical interactions on liquid behaviour. The availability of carbon nanotubes (CNTs) as a platform for mass transport studies at the nanoscale has generated a large number of experimental and molecular dynamics studies which have shown unexpected fluid behaviours compared to the macroscale. The most striking one has been the observation of several orders of magnitude higher liquid flow velocities compared to the no-slip Haagen-Poiseuille equation. 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These experimental results have generated an intense debate about the underlying physical phenomena and whether such flow can be explained within the continuum model on which traditional fluid mechanics is based or whether there is a size threshold below which it no longer applies (Figure 1). These results also hold great promise for filtration applications in general and water desalination in particular. ADDIN EN.CITE <EndNote><Cite><Author>Lee</Author><Year>2011</Year><RecNum>1294</RecNum><DisplayText><style Article">17</ref-type><contributors><authors><author>Lee, Kah Peng</author><author>Arnot, Tom C.</author><author>Mattia, D.</author></authors></contributors><titles><title>A review of reverse osmosis membrane materials for desalination--Development to date and future potential</title><secondary-title>Journal of Membrane Science</secondary-title></titles><periodical><full-title>Journal of Membrane Science</full-title><abbr-1>J. Membr. Sci.</abbr-1><abbr-2>J Membr Sci</abbr-2></periodical><pages>1-22</pages><volume>370</volume><number>1-2</number><keywords><keyword>Desalination</keyword><keyword>Reverse osmosis (RO)</keyword><keyword>Membrane material</keyword><keyword>Membrane performance</keyword><keyword>Nano-materials</keyword></keywords><dates><year>2011</year></dates><isbn>0376-7388</isbn><call-num>1294</call-num><urls><related-urls><url>http://www.sciencedirect.com/science/article/B6TGK-51SPS08-2/2/6f8473a0951f30842ad2aace7de13864</url></related-urls></urls><electronic-resource-num>DOI: 10.1016/j.memsci.2010.12.036</electronic-resource-num></record></Cite></EndNote>³

   

Figure 1. Water plugs in 40-60 nm and 2-5 nm diameter carbon nanotubes, respectively, showing a transition from a continuum to a sub-continuum behaviour. 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The micrograph on the left has been obtained using an environmental SEM by preferential condensation of water inside a CNT. The micrograph on the right was obtained using high resolution TEM and shows the coexistence of water liquid and vapor in a capped CNT.

The attempts made so far to explain these phenomena have focused on the slip length at the wall and its possible dependencies on a wide range of parameters, from contact angle to roughness, viscosity and shear rate. All these solutions are somewhat unsatisfying when slip values in the order of tens of micrometers for membranes with CNTs with diameters of only few nanometers are obtained. Moreover, no model has yet been capable of explicitly relating the flow enhancement with solid-liquid molecular interactions occurring at the channel walls.

Here a model capable of explaining the observed dependence of the flow enhancement on the membrane channels' characteristic dimensions and surface wettability is proposed. The model can accurately predict the effect of each component for membrane channels and nanotubes of different sizes, wall surface chemistry and structure. Comparison with the experimental and modelling data available shows good agreement (Figure 2). ADDIN EN.CITE <EndNote><Cite><Author>Mattia</Author><Year>2012</Year><RecNum>1472</RecNum><DisplayText><style db-id="zsta22rsnd5zpfesz2opddwwttf2sxfsfx09">1472</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Mattia, D.</author><author>Calabr˜, F.</author></authors></contributors><titles><title>Explaining high flow rate of water in carbon nanotubes via solid–liquid molecular interactions</title><secondary-title>Microfluidics and Nanofluidics</secondary-title></titles><periodical><full-title>Microfluidics and Nanofluidics</full-title><abbr-1>Microfluid. Nanofluid.</abbr-1></periodical><pages>1-6</pages><keywords><keyword>work of adhesion</keyword><keyword>surface diffusion</keyword><keyword>slip</keyword><keyword>solid-liquid interaction</keyword></keywords><dates><year>2012</year></dates><publisher>Springer Berlin / Heidelberg</publisher><isbn>1613-4982</isbn><call-num>1472</call-num><urls><related-urls><url>http://dx.doi.org/10.1007/s10404-012-0949-z</url></related-urls></urls><electronic-resource-num>10.1007/s10404-012-0949-z</electronic-resource-num></record></Cite></EndNote>⁶ The proposed model is based on a two-phase model for water flow in cylindrical channels with reduced viscosity at the wall, , and a non-zero velocity at the wall which is solely a function of solid-liquid interactions (ex. graphitic carbon and water) and the applied pressure. By solving the Haagen-Poiseuille equation with this velocity at the wall term, the following expression for the fluid flow enhancement, , at the wall is obtained:

            

where  and  are the tube/channel/pore length and radius, respectively;  is the thickness of the annular region and  is the liquid bulk density;  is the surface diffusion term arising from the chemical potential generated by the pressure gradient, ADDIN EN.CITE <EndNote><Cite><Author>Ruckenstein</Author><Year>1983</Year><RecNum>1314</RecNum><DisplayText><style no-slip boundary condition of hydrodynamics</title><secondary-title>Journal of Colloid and Interface Science</secondary-title></titles><periodical><full-title>Journal of Colloid and Interface Science</full-title><abbr-1>J. Colloid Interface Sci.</abbr-1><abbr-2>J Colloid Interface Sci</abbr-2></periodical><pages>488-491</pages><volume>96</volume><number>2</number><keywords><keyword>slip</keyword><keyword>surface diffusion</keyword></keywords><dates><year>1983</year></dates><isbn>0021-9797</isbn><call-num>1314</call-num><urls><related-urls><url>http://www.sciencedirect.com/science/article/B6WHR-4CT7DM2-NX/2/a2dbcea1f67474c1626147d2f0f2fb1c</url></related-urls></urls><electronic-resource-num>10.1016/0021-9797(83)90050-4</electronic-resource-num></record></Cite></EndNote>⁷ and  is the work of adhesion which represents the strength of the solid-liquid interaction at the tube/channel/pore wall. The two latter terms can be measured independently and at the macroscale for any solid-liquid couple.

Two recent publications have offered further validation of the proposed model: First, a molecular dynamics paper of water flow in CNTs of different lengths has found a linear dependence of the flow enhancement with the former. ADDIN EN.CITE <EndNote><Cite><Author>Nicholls</Author><Year>2011</Year><RecNum>1358</RecNum><DisplayText><style Jason</author></authors></contributors><titles><title>Water transport through (7,7) carbon nanotubes of different lengths using molecular dynamics</title><secondary-title>Microfluidics and Nanofluidics</secondary-title></titles><periodical><full-title>Microfluidics and Nanofluidics</full-title><abbr-1>Microfluid. Nanofluid.</abbr-1></periodical><pages>257-264</pages><volume>12</volume><number>1-4</number><keywords><keyword>Physics and Astronomy</keyword></keywords><dates><year>2011</year></dates><publisher>Springer Berlin / Heidelberg</publisher><isbn>1613-4982</isbn><call-num>1358</call-num><urls><related-urls><url>http://dx.doi.org/10.1007/s10404-011-0869-3</url></related-urls></urls><electronic-resource-num>10.1007/s10404-011-0869-3</electronic-resource-num></record></Cite></EndNote>⁸

Second, water flow measurements in hydrophilic alumina nanochannels with diameters ranging from 20 nm to 100 nm have shown to be well represented by the above equation. For the smallest channels tested, the water permeability is more than double than the theoretical prediction using the Haagen-Poiseuille equation. Though such an enhancement is significantly smaller than what has been measured in hydrophobic carbon nanotubes, it clearly shows that flow enhancement and water slippage occur on hydrophilic surfaces as well, contrary to existing theoretical models (Figure 2, red empty circles ○).  ADDIN EN.CITE <EndNote><Cite><Author>Lee</Author><Year>2012</Year><RecNum>1486</RecNum><DisplayText><style Peng</author><author>Leese, Hannah</author><author>Mattia, Davide</author></authors></contributors><titles><title>Water flow enhancement in hydrophilic nanochannels</title><secondary-title>Nanoscale</secondary-title></titles><periodical><full-title>Nanoscale</full-title></periodical><keywords><keyword>slip</keyword><keyword>flow enhancement</keyword><keyword>Hagen-poisuille</keyword><keyword>AAO</keyword></keywords><dates><year>2012</year></dates><publisher>The Royal Society of Chemistry</publisher><isbn>2040-3364</isbn><call-num>1486</call-num><urls><related-urls><url>http://dx.doi.org/10.1039/C2NR30098B</url></related-urls></urls></record></Cite></EndNote>⁹

Finally, the above equation has been successfully used to model molecular dynamics simulations of water flow in 2nm diameter silicon carbide (SiCNTs, yellow diamonds ♦) and boron nitride (BNNTs, blue squares ■) nanotubes.¹⁰

Figure 2. Normalized flow enhancement as a function of nanotube radius (dashed curve). Literature: ○ - (Thomas and McGaughey, Nano Lett, 2009); ● - (Holt et al., Science, 2006); × - (Thomas et al., Int J Therm Sci, 2010); ▲ - (Majumder et al., Nature, 2005); ♦ - (Du et al., Langmuir, 2011); ■ - (Whitby and Quirke, Nano Lett, 2007). Literature values (open symbols for MD and full symbols for experimental) are reported as a range between minimum and maximum possible values.  ○ Enhancement values for water flow in alumina nanopores; ♦ SiCNTs;  ■ BNNTs; Adapted from ref.s  ADDIN EN.CITE  ADDIN EN.CITE.DATA 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