Ultrafast ballistic diffusion of nanotubular water shells in carbon nanotubes. Direct experimental evidence with two-dimensional (\(D-T_2\)) NMR
J. Hassan
G. Diamantopoulos
L. Gkoura
M. Karayanni
S. Alhassan
M. S. Katsiotis
T. Karagiannis
M. Fardis
N. Panopoulos
H. J. Kim
G. Papavassiliou
J. Hassan
G. Diamantopoulos
L. Gkoura
M. Karayanni
S. Alhassan
M. S. Katsiotis
T. Karagiannis
M. Fardis
N. Panopoulos
H.J. Kim
G. Papavassiliou
Water when confined at the nanoscale acquires extraordinary transport properties. And yet there is no direct experimental evidence of these properties at nanoscale resolution. Here, by using 2D NMR diffusion-relaxation (\(D-T_2\)) spectroscopy, we acquire the diffusion profile of water in single and double-walled carbon nanotubes (CNTs). Remarkably, in double-walled CNTs water is shown to split in two axial components with the inner one acquiring astonishing flow properties: high fragility, ultra-high self-diffusion coefficient, and ``rigid'' molecular environment, revealing a nonuniform ballistic mechanism underlying fast diffusion in hydrophobic nanochannels.
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\begin{document}
\preprint{ULTRAFAST BALLISTIC WATER DIFFUSION IN CARBON NANOTUBES.TEX}  
\title{Ultrafast ballistic diffusion of nanotubular water shells in carbon nanotubes. Direct experimental evidence with two-dimensional (\(D-T_2\)) NMR}

\author{J. Hassan}
\email[Corresponding author: ]{jamal.hassan@kustar.ac.ae}
\affiliation{Department of Applied Math and Sciences, Khalifa University, 127788, Abu Dhabi, UAE}
\author{G. Diamantopoulos}
\affiliation{Department of Applied Math and Sciences, Khalifa University, 127788, Abu Dhabi, UAE}
\author{L. Gkoura}
\affiliation{Institute of Nanoscience and Nanotechnology, NCSR Demokritos, 15310 Aghia Paraskevi, Attiki, Greece}
\author{M. Karayanni}
\affiliation{Institute of Nanoscience and Nanotechnology, NCSR Demokritos, 15310 Aghia Paraskevi, Attiki, Greece}
\author{S. Alhassan}
\affiliation{Department of Chemical Engineering, Petroleum Institute, 2533, Abu Dhabi, UAE}
\author{M. S. Katsiotis}
\affiliation{Department of Chemical Engineering, Petroleum Institute, 2533, Abu Dhabi, UAE}
\author{T. Karagiannis}
\affiliation{Department of Chemical Engineering, Petroleum Institute, 2533, Abu Dhabi, UAE}
\author{M. Fardis}
\affiliation{Institute of Nanoscience and Nanotechnology, NCSR Demokritos, 15310 Aghia Paraskevi, Attiki, Greece}
\author{N. Panopoulos}
\affiliation{Institute of Nanoscience and Nanotechnology, NCSR Demokritos, 15310 Aghia Paraskevi, Attiki, Greece}
\author{H. J. Kim}
\affiliation{Nano-Bio Electron Microscopy Research Group, Korea Basic Science Institute, 169-148, \ Daejeon 305-806, Republic of Korea}
\author{G. Papavassiliou}
\email[Corresponding author: ]{g.papavassiliou@inn.demokritos.gr}
\affiliation{Institute of Nanoscience and Nanotechnology, NCSR Demokritos, 15310 Aghia Paraskevi, Attiki, Greece}
%\affiliation{Department of Chemical Engineering, Petroleum Institute, 2533, Abu Dhabi, UAE}
  
\date{\today}

\begin{abstract}
Water when confined at the nanoscale acquires extraordinary transport properties. And yet there is no direct experimental evidence of these properties at nanoscale resolution. Here, by using 2D NMR diffusion-relaxation (\(D-T_2\)) spectroscopy, we acquire the diffusion profile of water in single and double-walled carbon nanotubes (CNTs). Remarkably, in double-walled CNTs water is shown to split in two axial components with the inner one acquiring astonishing flow properties: high fragility, ultra-high self-diffusion coefficient, and ``rigid'' molecular environment, revealing a nonuniform ballistic mechanism underlying fast diffusion in hydrophobic nanochannels.
\end{abstract}

\pacs{88.30.rh, 89.40.Cc, 66.10.cg, 68.35.Fx, 82.56.Fk, 47.56.+r}
\maketitle

Molecular dynamics (MD) simulations have shown that water in very narrow sub-nanometer CNT channels forms an ordered one-dimensional chain, which spontaneously fills the hydrophobic CNT channels \cite{Hummer2001}. The estimated flow rates are comparable to those observed in biological pores such as aquaporin, gramicidin, and bacteriorhodosin \cite{Kong2001,Pomes2002,Lanyi2000}. Other MD simulations have shown that by increasing the CNT diameter nanotube water forms ice nanotubes comprised of a rolled sheet of cubic ice \cite{Koga2001}, while by further increasing the CNT diameter water is organized in a stratified nanotubular structure with a ``chain''-like water component at the center of the CNTs \cite{Bordin2013,Joseph2008,Kolesnikov2004,Striolo2005}. This intriguing molecular arrangement is central in  explaining fast water diffusion in CNTs \cite{Holt2006}, which according to MD simulations takes place either in the form of a single file \cite{Kalra2003}, where water molecules are prevented to surpass each other due to space limitation, or in a ballistic way where clusters of hydrogen bonded water molecules are diffusing very fast in a highly coordinated way \cite{Hummer2001,Striolo2006}. Here, on the basis of $^1$H Nuclear Magnetic Resonance (NMR) diffusion-relaxation (\(D-T_2\)) spectroscopy we acquire experimentally for the first time the diffusion profile of water in single and double-walled CNTs at nanoscale resolution. Experiments unveil a novel stratified ballistic mechanism as the dominant mechanism of ultrafast water diffusion in narrow CNTs.\\
Figure~\ref{Fig1} summarizes the main results of the present work. In single-walled CNTs (SWCNTs) with inner diameter \(d\approx1.2\) nm NMR experiments show the presence of a single nanotube water component with self-diffusion coefficient $D$ comparable to that of free water. Remarkably, in double-walled CNTs (DWCNTs) with inner diameter \(\approx3.5\) nm a second water component is observed with strikingly high $D$-values, approximately four times higher than that of bulk water at 285 K. The intensity of the NMR signals of the two water components correlates excellently with the radial water density distribution in DWCNTs (green line plot in Figure~\ref{Fig1}b) from MD simulations \cite{Bordin2013,Joseph2008}, as shown below. These results are in agreement with theoretical calculations, which predict that in ultra-narrow CNTs nanotube water diffuses in the shape of hollowed nanotubular clusters with a ballistic-type mechanism \cite{Striolo2005,Striolo2006}. The diffusion is observed to enhance further as shown below by decreasing the water content, in agreement with theoretical predictions \cite{Striolo2006}.\\
Figure~\ref{Fig1}c shows the inverse of the measured self-diffusion coefficient, $1/D$ vs. $1000/T$ of nanotube water in both CNTs together with that of bulk water (black circles). This kind of diagrams are very useful in understanding the way that intermolecular interactions propagate in freezing liquids. The blue line is the theoretical simulation of an ideal ``strong'' liquid with $1/D$ exhibiting Arrhenius temperature dependence, $\frac{1}{{D}}\propto \:exp{\left ({\frac{U}{k_BT}} \right)}$. As expected, bulk water deviates significantly from the Arrhenius behavior; within Angels concept of fragility \cite{Speedy1976} bulk water is considered to be fragile \cite{Prielmeier1987,Dehaoui2015}, i.e. it slows down gradually by cooling and freezes rapidly only near the water glass transition temperature $T_g$. The black line is theoretical fit to the empirical formula $\frac{1}{{D}}\propto \frac{1}{{D_0}}{(\frac{T-T_S}{T_S})}^\gamma$ which is widely used in the literature to fit the non-Arrhenius temperature dependence of the water transport properties \cite{Speedy1976}. $T_S$ in the formula is related to the glass transition temperature $T_g$. Data are excellently fitted with \(\gamma\approx2\) and \(T_S\approx215\) K, in agreement with previous reported values \cite{Perakis2011}. In the case of SWCNTs, the $1/D$ values of nanotube water matches excellently with those of bulk water. Most important, in DWCNTs  the central water component (cyan spheres in Figure~\ref{Fig1}b) acquires extremely low $1/D$  values (\(\gamma\approx3\) and \(T_S\approx218\) K), which vary slowly by decreasing temperature, indicating that the water structure of the central water component is robust upon cooling \cite{Bordin2013,Mauro2014}. The robustness of the nanotube water structure is further confirmed in Figure~\ref{Fig1}d, which presents  the $^1$H NMR spin-lattice to spin-spin relaxation \(T_1-T_2\) correlation spectrum of water in DWCNTs at room temperature. Nanotube water, interstitial water, and free bulk water are easily discriminated in the spectrum, according to their \(T_2/T_1\) ratio \cite{SM}. In case of nanotube water, the very low \(T_2/T_1<<1\) ratio is clear evidence that water molecules are in a solid-like than in a liquid-like molecular environment. Evidently, diffusing entities in both CNTs are not single water molecules but nanotubular water clusters diffusing in a ballistic way very fast along the CNTs axis. \\
\indent
\begin{figure}[h]
	\includegraphics[width=0.5\textwidth]{fig1b.png}
\caption{\label{Fig1} Schematic presentation of water diffusion in SWCNTs (a) and DWCNTs (b). The water structure has been sketched by combining \(D-T_2\) NMR experimental data with recent MD simulations \cite{Bordin2013,Joseph2008}. The green plots are sketches of the relevant water density.  (c) The inverse of the experimental self-diffusion coefficient $1/D$ as a function of $1000/T$. The blue line shows the temperature dependence of an ideal ``strong'' liquid following Arrhenius law. The solid black and the dashed cyan lines are theoretical fits as described in the text. (d) $^1$H NMR spin-lattice to spin-spin relaxation ($T_1-T_2$) correlation spectroscopy of water saturated DWCNTs in external magnetic field of $9.4$ Tesla at room temperature.}
	\end{figure}
\begin{figure}[t]
	\includegraphics[width=0.5\textwidth]{fig2b.png}
	\caption{\label{Fig2} Contour plots of $^{1}$H NMR \(D-T_2\) spectra of water in SWCNT (left panel) and DWCNT (right panel) at selected temperatures. Two main components are seen at each temperature, corresponding to nanotube water confined in the CNTs (low $T_2$ values) and interstitial and bulk water external to the CNTs (high $T_2$ values). Plots at the left side of the contour plots are the relevant $D$-projections.}
	\end{figure}
\indent
2D (\(D-T_2\)) NMR experiments were performed in the stray field of a superconductive magnet with a constant linear magnetic field gradient \(G=34.7\) Tesla/m. The conventional 1D NMR method for measuring water diffusion in a constant linear magnetic field gradient is to monitor the $^{1}$H NMR spin-echo decay and then fitting the obtained decay curve according to relation $M=M_0\:exp{\left( { - \frac{{2\tau }}{{T_2 }} - \frac{2}{3}D\gamma ^2 G^2 \tau ^3 } \right)}$ where $\gamma=26.7522$ ($10^{7}$ rad/sT) is the gyromagnetic ratio for $^{1}$H, and $D$ the water self-diffusion coefficient ($D=2.3\times10^{-9}$ m$^{2}$/s for bulk water at room temperature). In case of non-uniform diffusion processes with multiple $D$ values, diffusion is expressed with a distribution function $f(D)$, which is obtained by implementing an appropriate ``inversion algorithm'' \cite{Mitchell2012} as explained in the Supplemental Material \cite{SM}. Furthermore, in case of water diffusion in a bundle of randomly oriented CNTs, $f(D)$ curve acquires a characteristic asymmetric shape with a tail towards low $D$ values, which is the fingerprint of 1D restricted diffusion. It is noticed that $^{1}$H NMR studies of water diffusion inside CNTs have been already reported, presenting mainly effective $D$ values \cite{Das2010,Liu2014}. However, it is difficult with 1D NMR diffusion experiments to disentangle water components with relatively close $D$ coefficients, as those in nanotubular channels.\\
\indent
To overcome this obstacle, we take benefit of the fact that nanotube water acquires very short relaxation times. In general, liquids confined in porous structures relax faster than bulk liquids, due to fast exchange between slow relaxing bulk liquid and fast relaxing adsorbed liquid (\(T_{2,bulk}>>T_{2,surf}\)). In such case, the mean relaxation rate is expressed as $\frac{1}{{T_2 }}\propto k\frac{S}{V}$ where $k = T_{2,surf}^{-1}$ \cite{Zimmerman1957}. However, in hydrophobic CNTs water is not in contact with the CNT walls and the short values indicate solid-like environment at molecular scale rather than fast exchange between bulk and adsorbed water molecules. This assumption is confirmed by the very low \(T_2/T_1<<1\) ratio, as shown in the  $^{1}$H NMR $T_1-T_2$ spectrum of Figure~\ref{Fig1}d. The 2D \(D-T_2\) results of water in SWCNT and DWCNT are shown in Figure~\ref{Fig2} at selected temperatures. Plots were acquired by inversion of the corresponding spin-echo decay curves with a non-negative Tikhonov regularization algorithm \cite{Mitchell2012}. Since short $T_2$ values might create significant distortions in the $D-T_2$ spectra, the validity of the inversion has been confirmed by comparing experimental with simulated spectra, as shown in the Supplemental Material. Two main peaks are observed, corresponding to nanotube water (short $T_2$ component at $\approx0.5$ ms) and external water (long $T_2$ component at $\approx10-12$ ms); the former becomes more distinct at temperatures lower than 285 K. Further discussion on the $T_2$ analysis is presented in the Supplemental Material \cite{SM}. In both CNTs, external water attains $D$ value $\approx2.5\times10^{-9}$ m$^{2}$/s at room temperature, which is very close to that of bulk water. Upon lowering temperature water mobility decreases and the peak shifts to lower $D$ values. At the same time, the intensity of the NMR water signal decreases rapidly. We notice that part of the signal persists even at temperatures lower than the typical freezing point of bulk water ($\approx273$ K). Therefore, this signal cannot be assigned to ``bulk'' water but rather to interstitial water restricted in the space between CNT bundles with molecules possessing liquid-like mobility.\\
\indent
The $D$ projections shown at the left side of the contour plots have a contribution from both signals; the diffusion peak from water inside the CNTs is distorted or even masked by the signal from the external water.
\begin{figure}[t]
	\includegraphics[width=0.5\textwidth]{fig3b.png}
	\caption{\label{Fig3}  $D$-projections of nanotube water in (a) SWCNTs and (b) DWCNTs. The insets show the selected areas of projection. For better visualization spectra are normalized to one.The red line at 275 K in SWCNTs is simulation of water diffusion in a bundle of randomly oriented CNTs \cite{SM}. The dark cyan line is the relevant simulation of unrestricted bulk water diffusion. In DWCNTs nanotube water shows stratified fast diffusion. The red colored line corresponds to shell water as in SWCNTs, while the cyan colored line to an axial fast diffusing water component.}
	\end{figure}
\indent
In view of this fact, the $D$ projections of nanotube water were calculated separately for both SWCNTs and DWCNTs in the way presented in Figures~\ref{Fig3} and~\ref{Fig4}. In the case of SWCNTs at temperatures below 285 K, $D$ projections exhibit the characteristic low-$D$ tail (Figure~\ref{Fig3}a), encountered with 1D diffusion in randomly oriented CNTs \cite{Callaghan1979}. This is manifested with the theoretical red line (details are presented in the Supplemental Material \cite{SM}), which simulates accurately the experimental spectrum in the left panel of Figure~\ref{Fig3} at temperature of 275 K.  Most important, the restricted diffusion in SWCNTs acquires $D$ values close to that of free water, as presented in Figure~\ref{Fig1}c, in agreement with MD calculations \cite{Alexiadis2008}. \\

\begin{figure*} [t]
\includegraphics[width=\textwidth,height=6cm]{fig4b.png}
\caption{\label{Fig4} (a) The $D$-profile of water in DWCNTs at RT for three samples with different water content. For better visualization the three spectra are rescaled accordingly. The blue line is the $D$-profile of the fully hydrated SWCNTs. (b) The $^{1}$H NMR $D-T_2$ spectrum of  DWCNTs with intermediate water content  (black line in the left panel). }
 \end{figure*}

\indent
In DWCNTs, the $D$ distribution of water inside the CNTs attains a more intriguing picture. At $T=285$ K, diffusion of nanotube water is observed to be resolved in two components with peaks centered at $D\approx2.58\times10^{-9}$ m$^{2}$/s, and $9.60\times10^{-9}$ m$^{2}$/s (Figure~\ref{Fig3}b). This is in agreement with MD simulations \cite{Alexiadis2008}, which show that by increasing the CNTs diameter, a stratified arrangement of water molecules takes place \cite{Bordin2013,Joseph2008,Barati2011}. The split of water in two components is explainable by considering repulsive Coulomb forces \cite{Head1993} between next neighboring oxygen atoms, which become important when water is squeezed into narrow hydrophobic CNTs. In view of these facts water inside DWCNTs is considered to be organized in two axial water components as shown in Figure~\ref{Fig1}b, with the inner one exhibiting astonishing diffusion enhancement, approximately four times higher than that of bulk water at 285 K. Apparently, the lower $D$ value of the outermost axial water component is due to ``friction'', i.e. interaction of water molecules with defects and oxygen moieties, which unavoidably are present on the CNT walls. Most important, fast diffusion shows up in a rigid molecular environment as witnessed by the very low $^1$H NMR  \(T_2/T_1<<1\) ratio presented in Figure~\ref{Fig1}d. A similar short  \(T_2/T_1\) ratio is observed in SWCNTs. This is evidence that diffusion in narrow CNTs takes place with a ballistic-type mechanism \cite{Hummer2001,Striolo2006} instead of a single-file mechanism \cite{Kalra2003,Das2010}. \\	
\indent

Remarkably, by reducing the water content fast diffusion phenomena become more prominent  in agreement with MD simulations  \cite{Striolo2006}. Figure~\ref{Fig4}a shows the $D$-distribution profile of water at room temperature in three DWCNTs samples with different water content. By reducing water a very broad distribution of $D$ unveils, which  according to Figure~\ref{Fig4}b  decomposes into a very broad fast diffusing component belonging to the nanotube water and a second one belonging to external water. 
\\	
\indent
In conclusion, herein we report the first experimental measurement of fast water diffusion in single and double-walled CNTs at nanoscale resolution with a 2D $^{1}$H NMR method. It is noticed that the time span of the presented NMR diffusion measurements is $\approx1-2$ ms, i.e.\ two-to-three orders of magnitude longer than in MD calculations. This allows assessment even of ultra-slow motions of macromolecules, such as those through biological membranes. Combining the NMR experimental results with existing MD calculations, we conclude that fast diffusion in DWCNTs takes place as random stratified axial motion of water shells with enhanced diffusion at the center, a discovery of significant importance in the efforts to understand fluidic properties through extremely narrow pores and membranes \cite{Garcia2012,Geng2014}, especially of biological interest \cite{Fornasiero2008}.
\section{ACKNOWLEDGEMENTS}
This work was supported by Khalifa University Fund (210065). G.P. and H.J.K. would like also to express their gratitude to IRSES FP7 project Nanomag (295190). 
  
\raggedright
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supplemCNTs.tex
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\preprint{SUPPLEMENTAL INFORMATION: ULTRAFAST BALLISTIC DIFFUSION .....}  
\title{Ultrafast ballistic diffusion of nanotubular water shells in carbon nanotubes. Direct experimental evidence with two-dimensional (\(D-T_2\)) NMR}

\author{J. Hassan}
\email[Corresponding author: ]{jamal.hassan@kustar.ac.ae}
\affiliation{Department of Applied Math and Sciences, Khalifa University, 127788, Abu Dhabi, UAE}
\author{G. Diamantopoulos}
\affiliation{Department of Applied Math and Sciences, Khalifa University, 127788, Abu Dhabi, UAE}
\author{L. Gkoura}
\affiliation{Institute of Nanoscience and Nanotechnology, NCSR Demokritos, 15310 Aghia Paraskevi, Attiki, Greece}
\author{M. Karayanni}
\affiliation{Institute of Nanoscience and Nanotechnology, NCSR Demokritos, 15310 Aghia Paraskevi, Attiki, Greece}
\author{S. Alhassan}
\affiliation{Department of Chemical Engineering, Petroleum Institute, 2533, Abu Dhabi, UAE}
\author{M. S. Katsiotis}
\affiliation{Department of Chemical Engineering, Petroleum Institute, 2533, Abu Dhabi, UAE}
\author{T. Karagiannis}
\affiliation{Department of Chemical Engineering, Petroleum Institute, 2533, Abu Dhabi, UAE}
\author{M. Fardis}
\affiliation{Institute of Nanoscience and Nanotechnology, NCSR Demokritos, 15310 Aghia Paraskevi, Attiki, Greece}
\author{N. Panopoulos}
\affiliation{Institute of Nanoscience and Nanotechnology, NCSR Demokritos, 15310 Aghia Paraskevi, Attiki, Greece}
\author{H.J. Kim}
\affiliation{Nano-Bio Electron Microscopy Research Group, Korea Basic Science Institute, 169-148, \ Daejeon 305-806, Republic of Korea}
\author{G. Papavassiliou}
\email[Corresponding author: ]{g.papavassiliou@inn.demokritos.gr}
\affiliation{Institute of Nanoscience and Nanotechnology, NCSR Demokritos, 15310 Aghia Paraskevi, Attiki, Greece}

\pacs{88.30.rh, 89.40.Cc, 66.10.cg, 68.35.Fx, 82.56.Fk, 47.56.+r}
\keywords{carbon nanotubes, water transportation, self-diffusion in liquids (mass diffusion), diffusion, multidimensional NMR, flows through porous media}

\date{\today}

\maketitle

In this supplementary information, we provide additional details on certain aspects of the study reported in the manuscript.\\
\indent 
The following issues are discussed:\\
\indent  
1. Materials and Methods.\\
\indent 
2. $^{1}$H NMR 2D diffusion - spin-spin relaxation time (\(D-T_{2}\)) method in permanent linear magnetic field gradient.\\
\indent 
3. $^{1}$H NMR diffusion study of water in randomly oriented nanochannels.\\
\indent 
4.  $^{1}$H NMR relaxation analysis of water in CNTs. \\
\indent
5.  Comparison between Simulated and Experimental Diffusion Measurements. \\
\indent

\section{1. MATERIALS AND METHODS}
Purified and open ended carbon nanotubes (SWCNT and DWCNT) were purchased from SES research, USA. The outer diameter of the CNTs used in this work was $\approx2$ nm for the SWCNT and $\approx5$ nm for the DWCNT. Both kinds of CNTs had average length $\approx1$ $\mu m$. \

Morphological analysis of the carbon nanotubes (CNTs) was performed with Transmission Electron Microscopy (TEM) using a FEI Tecnai G20 microscope with a 0.11 nm point to point resolution, operated at 200 kV. For this analysis, approximately 1 mg of each sample was dispersed in 20 ml of high purity cyclohexane (Merck, 99.9 \%) via sonication ($\approx15$ s); a drop of each suspension was deposited on copper grids (400 mesh) covered with thin amorphous carbon film (lacey carbon). To avoid contamination, CNT samples were inserted in the TEM machine immediately following preparation. Bright field images were collected at several magnifications in order to observe structure and size homogeneity. Figure~\ref{FigS1} shows representative TEM images of the single walled (SWCNT) and double walled (DWCNT) samples used in this study.

\begin{figure}[b]
	\includegraphics[width=0.8\textwidth]{FigS1b.png}
	\caption{\label{FigS1} TEM images for the SWCNT (left) and DWCNT (right) used in this study. The outer diameters of the SWCNT and DWCNT samples were  $\approx2$ nm and  $\approx5$ nm, respectively.}
	\end{figure}

$^{1}$H NMR two-dimensional diffusion-relaxation (2D \(D-T_{2}\)) measurements were performed in the stray field of a 4.7 T Bruker superconductive magnet providing a 34.7 T/m linear magnetic field gradient at $^{1}$H NMR frequency of 101.324 MHz. The experiments were carried out by using a pulse sequence with more than 5000 pulses (details are presented below). The sample temperature was controlled by an ITC5 temperature controller in a flow type Oxford cryostat. The accuracy of the temperature was $\approx0.1$ K. An equilibrium time period of about thirty minutes was allowed at each temperature before collecting data.\\
\indent
NMR data were analyzed using a 2D non-negative Tikhonov regularization inversion algorithm code developed by the authors, as discussed below.\\
In order to prepare fully water saturated samples, CNT powder was immersed in deionized water inside NMR quartz tubes. Samples were then let to come to equilibrium at room temperature for a couple of hours. In order to force water to enter the tubes (with air extrusion), the NMR tubes with the sample were connected to a vacuum system and air was carefully pumped out until water immersed in the powder CNTs sample. The NMR tubes were then left at ambient conditions for different time intervals to control the water content in the CNTs.  Each sample was then flame sealed under vacuum. The external water and nanotube water content in the samples was verified with NMR by examining the $T_2$ distribution. The same sample preparation procedure was followed for both CNT types used in this study.


\section{2. $^{1}$H NMR 2D DIFFUSION-SPIN-SPIN RELAXATION TIME (\(D-T_{2}\)) METHOD IN PERMANENT LINEAR MAGNETIC FIELD GRADIENT}
Two-dimensional $^1$H NMR \(D-T_{2}\) was implemented, in order to study water diffusion in single and double walled CNTs.  
Measurements were performed on a broadband home-built NMR spectrometer operating in the frequency range 5 MHz - 1 GHz. The probe was placed in the fringe field of a 4.7 T Bruker magnet with a linear magnetic field gradient G=34.7 T/m at $^{1}$H NMR frequency of 101.324 MHz \cite{Kimmich1991,Leu2005}. The advantage of using a permanent magnetic field gradient instead of a pulsed one, is that (i) G in the fringe field of a superconductive magnet is very strong, allowing the measurement of very slow diffusion processes, (ii) the time between pulses can be set to very short values, due to the absence of pulsed gradient stabilization effects, allowing measurements of very short $T_{2}$ values, and (iii) the very high G value allows reliable measurement of diffusion in \(D-T_2\) experiments with very low $T_2$ values.\\
\indent
The \(D-T_{2}\) pulse sequence is shown in the upper panel of Figure~\ref{FigS2}. It consists of a \(\pi/2-t_D-\pi\) (Hahn echo) pulse sequence, followed by a train of 5000 $\pi$ pulses, generating a Carr-Purcell-Meiboom-Gill (CPMG) spin-echo decay (SED) train \cite{Eurydice2015,Eurydice2014}. The $\pi/2$ pulse duration was set to $4\mu$s and the CPMG interpulse time distance ``$t$'' to $30\mu$s. \(D-T_{2}\) data were acquired with a series of 1D experiments. Data in the ``direct'' ($T_{2}$) dimension, were collected in a single shot, by recording simultaneously the intensity of all CPMG echoes. Data in the ``indirect'' dimension (D) were accumulated in successive experiments by incrementing $t_{D}$ \cite{Eurydice2015,Eurydice2014,Mitchell2012}. Fifty experiments were performed with $t_{D}$ values between 30 $\mu$s to 1500 $\mu$s in logarithmic scale.\\
\indent
The lower panel in Figure~\ref{FigS2} demonstrates SEDs for three different $t_{D}$ values. The inset is the signal intensity variation of the first spin-echo as a function of $t_{D}$. The solid line in the inset is fit to the relation \cite{Kimmich1991,Leu2005,Eurydice2015,Eurydice2014}:
\begin{equation} 
M=M_0 exp(-\frac{2t_D}{T_2}-\frac{2}{3}\gamma^2 DG^2 t_D ^3)\label{equation}
\end{equation}
\begin{figure}[t]
	\includegraphics[width=0.8\textwidth]{FigS2b.png}
	\caption{\label{FigS2}(Upper panel) The \(D-T_{2}\) NMR pulse sequence. Experiments were operated in a constant linear magnetic field gradient $G=34.7$ T/m. The signal intensity of the first spin-echo depends on  $t_{D}$ (see inset of the lower panel) according to relation 1. The CPMG spin-echo decay, following the first echo, provides information on $T_{2}$. The ``diffusion'' axis was scanned with fifty different $t_{D}$ values ranging from 20 $\mu$s to 1000 $\mu$s. (Lower panel) CPMG spin-echo decay for the SWCNTs at room temperature, acquired at three different $t_{D}$ values; 50, 133 and 236 $\mu$s. The inset is the decay of the first echo, together with the fit by relation 1.}
	\end{figure}

In the presence of a broad distribution of diffusion processes and relaxation times, the \(D-T_{2}\) signal intensity decays according to the relation \cite{Eurydice2015,Eurydice2014}:
\begin{equation} 
\frac{g(t_D,t)}{g(0,0)}=\int\int^{+\infty}_{-\infty}(e^{-\frac{DG^2\gamma^2 (t_D)^3}{12}} e^{-\frac{t_D}{T_2}})e^{-\frac{t}{T_2}}f(D,T_2)\;dD\;dT_2\label{equation}
\end{equation}
where $f(D,T_2)$ is the distribution function of $D$ and $T_2$.\\
For the determination of $f(D,T_2)$ a change of variables was applied \cite{Eurydice2015,Eurydice2014} $t_t=t+t_D$. This allows separation of the $D$ and $T_2$ axis, resulting in the following expression:
\begin{equation} 
\frac{g(t_D,t)}{g(0,0)}=\int\int^{+\infty}_{-\infty}(e^{-\frac{DG^2\gamma^2 (t_D)^3}{12}})(e^{-\frac{t_t}{T_2}})f(D,T_2)\;dD\;dT_2\label{equation}
\end{equation}
The penalty of this transformation is that a small number of echoes at the beginning of the data array is not taken into consideration in the 2D inversion. In this study, the first 5 echoes out of 5000 echoes were excluded, which accounts that NMR signals with $T_2$ component shorter than $\approx0.2$ ms become invisible.\\
\indent
\(D-T_{2}\) spectra (i.e.\ the  $f(D,T_2)$ distribution function) were obtained using a 2D non-negative Tikhonov regularization algorithm \cite{Mitchell2012,Day2011} developed in Matlab. In this method 2D data are lexicographically transformed in a single column and subsequently inverted by applying a 1D Tikhonov inversion. The inverted data are presented as 1D vector, where the 2D spectrum is encrypted. Unwrapping this again in two dimensions provides the \(D-T_{2}\) spectrum.\\
\indent
\section{3. $^1$H NMR DIFFUSION STUDY OF WATER IN RANDOMLY ORIENTED NANOCHANNELS}
In case of diffusion in a bundle of randomly oriented one-dimensional (1D) nanochannels, relation 1 describing unrestricted 3D diffusion does not hold anymore, and it should be modified in order to take into account the 1D properties of the diffusion and the random orientation of the CNTs in respect to the external magnetic field gradient $G$. This issue was addressed by Callaghan \cite{Callaghan1979}, who demonstrated that water diffusion in endosperm tissue of wheat grains, when simulated with diffusion in 1D randomly oriented array of capillaries, averaged over all solid angles, gives spin-echo attenuation $R_{1D}$ according to relation:
\begin{equation}
R_{1D}=\int^{1}_{0}e^{-kDx^2}\;dx\label{equation}\\
\end{equation}
For diffusion in the stray field gradient $G$ of the magnet, $k$ is given by \(k=\frac{2}{3}\gamma^2 G^2 t_D^3\). The difference in the spin-echo decay between unrestricted 3D diffusion and restricted 1D diffusion is excellently demonstarted in Figure~\ref{FigS3}, which shows the decay of the first CPMG echo   as a function of  \(t_D^3\) for external water (white circles) and nanotube water (yellow circles) in the case of partially water-saturated DWCNTs at room temperature, in comparison with the theoretical SED curves acquired by relations 1 for bulk water diffusion (blue line) and 4 for 1D restricted diffusion (black line), as shown in the inset of Figure~\ref{FigS3}. While the spin-echo attenuation for unrestricted 3D diffusion decreases ``linearly'',  the spin-echo intensity of 1D diffusion decreases with a successively declining slope. It is noticed that the inversion of the theoretical 1D spin-echo decay curve gives rise to an asymmetric distribution of $D$ with a long tail towards the low $D$ values (Figure~\ref{FigS4}). This is the fingerprint of uniform 1D restricted diffusion in a set of randomly oriented nanochannels. Clearly, SEDs of nanotube water in DWCNTs (i) reveal the profile of a 1D restricted diffusion and (ii) at short times a very fast diffusion is observed as demonstrated by the very steep initial slope, demonstrating the presence of a very fast diffusing nanotube water component. 

\begin{figure}[h]
	\includegraphics[width=0.9\textwidth]{FigS3b.png}
	\caption{\label{FigS3} Decay of the first Spin Echo in the CPMG echo train of external (white circles) and Nanotube water (yellow circles) in DWCNTs as a function of \(t_D^3\). The inset shows the theoretical $^{1}$H NMR Spin Echo decays of free water molecules (blue line), and water molecules diffusing in one-dimensional randomly oriented nanochannels (black line), where $k$ in the x-axis is equal to \(k=\frac{2}{3}\gamma^2 G^2 t_D^3\).  Clearly, Nanotube water exhibits 1D restricted diffusion, whereas external water follows relation S1, which is manifestation of unrestricted diffusion.}
	\end{figure}

\begin{figure}[h]
	\includegraphics[width=0.6\textwidth]{FigS4b.png}
	\caption{\label{FigS4}Self-diffusion coefficient $D$ distribution simulations acquired by implementing a 1D Tikhonov regularization algorithm on the theoretical $^{1}$H NMR Spin Echo decays of Figure~\ref{FigS3}.}
	\end{figure}


\section{4.  $^{1}$H NMR RELAXATION ANALYSIS OF WATER IN CNTs}
In order to distinguish among the various water groups in terms of their dynamics, $^{1}$H NMR spin-lattice relaxation $T_1$ and spin-spin relaxation $T_2$ measurements were performed in magnetic field 9.4 T (Larmor frequency 400 MHz) and in the stray field of a 4.7 T superconductive magnet at Larmor frequency 101.324 MHz. The main panel in Figure~\ref{FigS5} shows the spin echo decay obtained with a CPMG pulse sequence in field 9.4 T, on a saturated sample of DWCNTs at room temperature.  The yellow line is the theoretical fit by implementing a 1D nonnegative Tikhonov regularization algorithm. Gray circles in the inset demonstrate the inverted $T_2$ distribution. Water is shown to be grouped in two main $T_2$ peaks; a low intensity narrow peak with $T_2=1.2$ sec, corresponding to $T_2$ values of bulk water, and a second broad line with a peak at 10 msec, which corresponds to interstitial water and water inside the carbon nanotubes. This assignment is confirmed with $T_1$ measurements (cyan circles), obtained with an inversion recovery \(\pi-t-\pi/2\) pulse sequence. According to Figure~\ref{FigS5} $T_1$ shows two peaks: a peak with $T_1$ value coinciding with the long $T_2$ component (as expected for bulk water), and the second one with $T_1$ value sufficiently higher than the short $T_2$ component, as expected for interstitial and nanotube water molecules.\\

\begin{figure}[h]
	\includegraphics[width=0.9\textwidth]{FigS5b.png}
	\caption{\label{FigS5}(Main panel) $^{1}$H NMR CPMG spin echo decay of water in a saturated sample of DWCNT at room temperature, in a magnetic field 9.4 T (400 MHz). The yellow line is theoretical fit by applying the 1D Tikhonov regularization inversion. The inset presents the relevant $^{1}$H NMR $T_2$ distribution (gray circles), together with the $^{1}$H NMR spin-lattice relaxation time $T_1$ distribution (cyan circles) and the $^{1}$H NMR $T_2$ distribution of the same sample in the stray field of a 4.7 T (101.324 MHz), green circles.}
	\end{figure}
\indent
Most notable, the $T_2$ projection of the \(D-T_{2}\) plot at 295 K (green circles), coincides with the $T_2$ distribution of nanotube and interstitial water at 400 MHz (black circles). This is direct evidence that water in the \(D-T_{2}\) measurements is not bulk water but primarily interstitial water (long $T_2$ component) and nanotube water (short $T_2$ component), as discussed in the article.    

\begin{figure}[h]
	\includegraphics[width=0.8\textwidth]{FigS6b.png}
	\caption{\label{FigS6}$^{1}$H NMR $T_1-T_2$ correlation spectroscopy of a water saturated DWCNT at room temperature in the external magnetic field of 9.4 T. The inset shows the corresponding $^{1}$H NMR spectrum at repetition times TR=0.5 sec and 5 sec, respectively.}
	\end{figure}
\indent

To further enlighten the different  water groups probed in our NMR experiments,  $^{1}$H NMR $T_1-T_2$ correlation spectroscopy was performed at room temperature on a water saturated DWCNT sample. The results are presented in Figure~\ref{FigS6}. This type of spectroscopy provides information about the ``liquidity'' of the local molecular environment: in general, molecules in unconstraint liquid environment are characterized by \(T_{1}\approx T_{2}\), whereas molecules in rigid (solid) environment show \(T_2<<T_1\). In this context, bulk water with \(T_{1}\approx T_{2}\), interstitial water with T$_{1}$ moderately higher than $T_2$, and solid-like nanotube water with \(T_2<<T_1\) are clearly visible in Figure~\ref{FigS6}. The inset shows the relevant $^{1}$H NMR spectra at 400 MHz and at two different repetition times. Two $^1$H NMR peaks are observed, which are splitted by \(\approx 2 \) kHz; a high frequency peak with long $T_1$ as concluded by the rapid decrease of the signal intensity at short repetition time TR, which is assigned to bulk water, and a low frequency peak with fast relaxation (shorter $T_1$), which is assigned to interstitial and nanotube water.
\indent

\begin{figure}[h]
	\includegraphics[width=1\textwidth]{FigS7b.png}
	\caption{\label{FigS7}$T_2$ projections of the \(D-T_{2}\) spectra of water in (a) SWCNTs and (b) DWCNTs. For clarity, all $T_2$ projections have been normalized.  Fits (dashed lines) were performed with inversion of a stretched exponential spin echo decay and a log-norm distribution of $T_2$s . The short $T_2$ component corresponds to the nanotube water and the long $T_2$ component to the interstitial water. By decreasing temperature strong wipe out of the NMR signal from the interstitial water is observed.}
	\end{figure}	
\indent

Figure~\ref{FigS7} presents the $T_2$ projections of \(D-T_{2}\) spectra at selected temperatures for both SWCNT and DWCNT. Each projection consists mainly of two peaks; a short $T_2$ peak corresponding to the nanotube water, i.e.\ water in the interior of the CNTs, and a long $T_2$ component corresponding mainly to the interstitial water.

Most astonishing, in the double-walled CNTs the short $T_2$ water component shows higher diffusion coefficient $D$ values in comparison to the long $T_2$ water component, as clearly seen in Figure 4b of the manuscript. A similar surprising result was observed in pulsed gradient diffusion experiments of water in hydrating collagen II \cite{Knauss1996}, where the water fraction with the low diffusion was shown to have longer $T_2$ and hence higher local mobility, in comparison to the water fraction with the high diffusion, which was shown to have shorter $T_2$ and hence reduced local mobility.

\begin{figure}[h]
	\includegraphics[width=0.8\textwidth]{FigS8b.png}
	\caption{\label{FigS8}Temperature dependence of water-$T_2$ inside a) SWCNT and b) DWCNT samples.}
	\end{figure}
\indent

Figure~\ref{FigS8} demonstrates the temperature dependence of the $T_2$ peak positions for both the SWCNT and DWCNT samples. Upon lowering temperature, the $T_2$ value of the nanotube water in the SWCNT (green circles in Figure~\ref{FigS8}a shifts to lower values, from $\approx2$ ms at room temperature to $\approx$0.2 ms at 265 K. In the DWCNTs $T_2$ of the nanotube water (green circles in Figure~\ref{FigS8}b) remains almost constant at about 0.5 ms. In the case of interstitial water (yellow circles), an apparent local minimum at 285 K is observed for both samples, which might be related to the change of hydrophobicity, as discussed in a recent NMR publication \cite{Wang2008}. More work is needed to examine this issue.
 
\section{5.  COMPARISON BETWEEN SIMULATED AND EXPERIMENTAL DIFFUSION MEASUREMENTS}
Short $T_2$ values might strongly distort the self-diffusion coefficient D-distribution \cite{Kausik2016}. This is so as short $T_2$ might enhance the spin echo decay at long interpulse time delays, giving rise to illusive high D-values (tails in the direction of the high D-values). 
\indent

\begin{figure}[h]
	\includegraphics[width=0.8\textwidth]{FigS9b.png}
	\caption{\label{FigS9}Comparison of the self-diffusion coefficient $D$ distribution of water in (i) fully hydrated DWCNTs at 285K (black line in the upper pannel) and (ii) partially saturated DWCNTs at 293K (black line in the lower pannel), with simulated data (blue line), as obtained from \(D-T_2\) experiments. It is noticed that simulated data acquire similar $T_2$ distribution with the experimental data but only one $D$-value, $D=1.8$ x $10$$^{-9}$ m$^2$/sec.}
	\end{figure}
\indent
 In order to examine the reliability of the inversion algorithm used in this work at very short $T_2$ values, we have compared the $D$-profiles of experimental and simulated \(D-T_2\) data, by using simulated data with $T_2$ values similar to the experimental ones, but only a single diffusion coefficient $D$ value. All other used inversion parameters were exactly the same for both simulated and experimental data. Figure ~\ref{FigS9} presents the results of this comparison for the cases of (i) fully hydrated DWCNTs at 285K and (ii) partially saturated DWCNTs at $293$K. Despite the very short $T_2$ value, the simulated data are shown to acquire only a slight broadening of the $D$-distribution, which is an order of magnitude narrower than that of the experimental spectra. This provides straightforward evidence that the short $T_2$ values in the very strong permanent magnetic field gradient $G=34.7$ T/m used in this work do not create any significant artifact in the \(D-T_2\) spectra.   The proposed method acquires reliable information even in the case where the external water masks almost completely the $D$-projection of the nanotube water, as shown in Figure ~\ref{FigS10}. 
\indent
\begin{figure}[h]
	\includegraphics[width=0.8\textwidth]{FigS10b.png}
	\caption{\label{FigS10} The $D$-projection of water in fully water saturated DWCNTs at room temperature,  as obtained from the relevant \(D-T_2\) spectrum. The blue (red) lines corespond to the individual $D$-projections of the external (nanotube) water with the high (short) $T_2$ values.}
	\end{figure}
\indent

Furthermore, It is noticed that the validity of our experimental analysis is in agreement with recent results by Kausik et al. \cite{Kausik2016}, on the sensitivity of \(D-T_{2}\) measurements. The authors introduced two parameters \( \alpha \) and C (equation 3 and 5 in their article), which measure the relative strength of diffusion/relaxation and what they call diffusion contrast. According to their analysis the inversion of the $D-T_2$ data is reliable if \(\alpha \geq 1\)  and C has values between $0- 1$, with the $1$ being the ideal value. In our study, for the short \(D-T_{2}\) component \( \alpha = 1.3 \)  and C has intermediate values reaching a maximum value of $0.7$. 
\raggedright	
\bibliographystyle{unsrt}
\bibliography{suppref}

\end{document}

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