Unprecedentedly rapid transport of single-file rolling water molecules

Abstract

The realization of rapid and unidirectional single-file water-molecule flow in nanochannels has posed a challenge to date. Here, we report unprecedentedly rapid unidirectional single-file water-molecule flow under a translational terahertz electric field, which is obtained by developing a Debye double-relaxation theory. In addition, we demonstrate that all the single-file molecules undergo both stable translation and rotation, behaving like high-speed train wheels moving along a railway track. Independent molecular dynamics simulations help to confirm these theoretical results. The mechanism involves the resonant relaxation dynamics of H and O atoms. Further, an experimental demonstration is suggested and discussed. This work has implications for the design of high-efficiency nanochannels or smaller nanomachines in the field of nanotechnology, and the findings also aid in the understanding and control of water flow across biological nanochannels in biology-related research.

Graphical abstract

Keywords

water molecules / carbon nanotubes / molecular dynamics / terahertz electric field / electrohydrodynamics / Debye double-relaxation theory

1 Introduction

Owing to the importance of water to human life, this substance has attracted considerable attention from scientists in various fields, including physics, chemistry, biology, nanotechnology, and environmental science. Currently, the study of water flow across nanochannels is an important topic. Its significance is at least two-fold, as this behavior has implications for seawater desalination or water purification by using filtering methods [1–5] and it is also of importance as regards the understanding and control of water flow across water nanochannels in biological membranes [1, 6–10]. In both of these cases, achievement of a fast unidirectional flow of single-file water molecules across nanochannels has posed a significant challenge to date. For this purpose, nanochannels such as carbon nanotubes are often used for water transportation because of their (almost) frictionless interfaces [1, 8, 9, 11–16]. Accordingly, researchers have developed and studied many new techniques for transporting water through nanochannels, e.g., diffusion osmosis, pressure differences, thermal gradients, channel deformation, and radial breathing mode vibrations [17–21]. In particular, electrical methods have been widely used to generate unidirectional water flow. For instance, researchers have designed various types of electric fields using linear gradient fields, rotating fields, or electrostatic charges [9, 10, 22–28].

Because of these efforts, the maximum (linear) transport speed of water molecules in nanochannels has increased significantly (e.g., refs. [1, 7, 14, 21, 22, 25, 27, 29]), which encourages exploration of new approaches to this topic. Moreover, it is well known that the function of charge groups in biological membrane proteins is vital to water permeation through aquaporins [30–34] (biological implication). For instance, asparagine-proline-alanine regions in both aquaporin-1 and glycerol uptake facilitator proteins and mediate the action of a selective water filter [33, 34]. On the other hand, environmental oscillations arising from thermal motion occur when water molecules are transported through nanochannels [21] (physical implication).

In this work, we investigate a model system (see Fig. 1) that is inspired by these biological and physical implications. Hence, we determine that single-file water molecules experience rapid transport, behaving like the wheels of a high-speed train moving along a railway. Therefore, we label this type of transport a water nanotrain. Our investigation is based on a phenomenological Debye double-relaxation theory together with independent molecular dynamics simulations.

2 Debye double-relaxation theory

Let us begin by considering an isolated water molecule at room temperature. It is known that the spontaneous polarizability of the water molecule can be characterized by a permanent dipole moment P. Then, an external electric field E with angular frequency ω can be added to the water molecule. In this case, if E\to 0, p, which denotes the average E-direction projection of P, is sufficiently small, because of the irregular thermal motion of the molecule. Hence, for E\to 0, we obtain p\to 0. Certainly, owing to the competition between thermal motion (which tends to cause irregular molecular motion) and E (which tends to direct the molecule in the E direction), an increasing E causes p to increase up to p=|P|. Thus, p depends on the strength of E, which allows us to regard p as an effectively induced dipole moment. Next, we model the water molecule as a spherical nanoparticle with radius a, and assume that its p can be expressed in the same manner as the induced dipole moment of a spherical dielectric particle [35–39], such that

\begin{array}{}\text{(1)}& p=4\pi {\epsilon }_0{a}^{3}G(\omega )E,\end{array}

where {\epsilon }_0 is the vacuum permittivity and E=|E|. G(\omega ) denotes the ω-dependent dipole factor (also called the Clausius–Mossottii factor, which characterizes the dielectric contrast between the molecule and vacuum), where

\begin{array}{}\text{(2)}& G(\omega )={\displaystyle \frac{\widehat{\epsilon }(\omega )-1}{\widehat{\epsilon }(\omega )+2}}.\end{array}

In Eq. (2), the complex permittivity \widehat{\epsilon }(\omega ), which describes the effective dielectric response of the water molecule to E, may be formulated in terms of the Debye double-relaxation model [40–43]

\begin{array}{}\text{(3)}& \widehat{\epsilon }(\omega )={\epsilon }_{\mathrm{\infty }}+{\displaystyle \frac{\mathrm{\Delta }{\epsilon }_1}{1+\mathrm{i}\omega {\tau }_1}}+{\displaystyle \frac{\mathrm{\Delta }{\epsilon }_2}{1+\mathrm{i}\omega {\tau }_2}},\end{array}

where \mathrm{i}=\sqrt{-1} and {\epsilon }_{\mathrm{\infty }} is the permittivity at the high-frequency limit. The last two terms on the right-hand side of Eq. (3) describe the two relaxation processes, each arising from one type of atom (namely, O or H) in the molecule. In Eq. (3), \mathrm{\Delta }{\epsilon }_1 and \mathrm{\Delta }{\epsilon }_2 denote the amplitudes of the two relaxation processes with two corresponding relaxation times, {\tau }_1 and {\tau }_2, respectively [44]. For clarity, we set {\tau }_1 (or {\tau }_2) to represent the relaxation time of the O (or H) atom, thus yielding {\tau }_1>{\tau }_2. Clearly, the substitution of E\to 0 into Eq. (1) yields p\to 0, which echoes the preceding requirement. In addition, as the strength of the permanent dipole moment of a water molecule has a known value of 1.85 D, this value can be used to estimate the maximum strength of E. This validates Eq. (1).

As mentioned in the preceding section, the primary aim of this work is to achieve a fast unidirectional water-molecule flow through a nanochannel. For this purpose, we attempt to apply a translational electric field generated by two symmetric nanoelectrode arrays, as shown in Fig. 1(a). The fixed nanoelectrodes align on both sides of the nanochannel, and their charges vary according to a periodic arrangement comprising four time intervals (i.e., t_1\to t_2\to t_3\to t_4\to \mathrm{\dots }). The + and - symbols denote positive and negative electric charges, respectively. The field shown in the figure is defined as being in the positive direction, because each pair of adjacent charges with the same sign (positive or negative) on either side skips in the positive z direction. Note that we utilize a nanoelectrode design that is in accordance with the microelectrode arrangement used in experiments on traveling electric fields. These arrangements are used to transfer dielectric particles (e.g., yeast cells) [45–48]. To proceed, we set the water molecule density to a low value so that the interactions between the molecules can be neglected.

According to the charge arrangement depicted in Fig. 1(a), the E\to 0 generated at the nanochannel central axis (y=0) can be expressed as the summation of three components (E_x, E_y, and E_z along the x-, y-, and z-axes, respectively), with

\begin{array}{}\text{(4)}& E=E_x\widehat{x}+E_y\widehat{y}+E_z\widehat{z}.\end{array}

Here, E_x=0 because of the symmetry of the structure along the x-axis. Further, E_y and E_z are expressed in the following forms in terms of time t [45]

\begin{array}{}\text{(5)}& E_y=A(y=0)\cos \left(\omega t-{\displaystyle \frac{2\pi z}{\lambda }}\right),\end{array}

\begin{array}{}\text{(6)}& E_z={\displaystyle \frac{\lambda }{2\pi }}{A}^{\prime }(y=0)\sin \left(\omega t-{\displaystyle \frac{2\pi z}{\lambda }}\right),\end{array}

where A(y=0)\equiv A(0) is the E strength determined by all the electric charges on the nanoelectrodes, and A^{\prime }(y=0)\equiv A^{\prime }(0) is the y-derivative of A(0). In Eqs. (5)–(6), ω specifically denotes the angular frequency of the charge shifting and λ denotes the wavelength, as indicated in Fig. 1(a).

We are now in a position to derive the force F and torque \Gamma acting upon each water molecule due to E, in order to obtain expressions for both the linear speed v_z and the angular speed \Omega of the water molecules. For simplification, we assume that each water molecule moves along the central axis of the nanochannel. In particular, some narrow nanochannels, such as a (10, 0) carbon nanotube (the radius of which is 0.386 nm), provide a quasi-one-dimensional (1D) inner space for molecular transport. In such confinement, the net effect of the water transport can be neglected, except in the axial direction.

As the electric force F acting on an electric dipole is known to be F=(p\cdot \nabla )E, the z-directed force component F_z for the water molecule is given by

\begin{array}{}\text{(7)}& F_z=-{\displaystyle \frac{4{\pi }^2{\epsilon }_0{a}^3}{\lambda }}A{(0)}^2\text{Im}[G(\omega )]\widehat{z},\end{array}

where Im[G(\omega )] () denotes the imaginary part of G(\omega ). Clearly, the presence of F_z helps to increase the z-directed v_z of the water molecule.

When linear transportation of the water molecule along the z-axis within the nanochannel begins, under the influence of F_z, the interfacial friction between the water and the carbon nanotube induces a drag force F_f that acts on the water in the -z direction. This tends to decrease the linear speed of the molecule. Because of the linear relationship between F_f and v_z [49], we make a phenomenological assumption that these terms are proportional to each other. Then, we express this assumption in Stokes’ form: F_f=-6\pi {\eta }_{l}a{v}_z\widehat{z}, where {\eta }_l is a coefficient analogous to the effective linear viscosity. When the system reaches equilibrium, i.e., F_z+F_f=0, v_z reaches a constant value. This is defined as

\begin{array}{}\text{(8)}& v_z=-{\displaystyle \frac{2\pi {\epsilon }_0{a}^2}{3{\eta }_l\lambda }}A{(0)}^2\text{Im}[G(\omega )].\end{array}

Clearly, v_z is proportional to the imaginary part of G(\omega ).

On the other hand, it is easy to determine that the phase of p [Eq. (1)] lags behind that of the electric field E [Eq. (4)], thus yielding a non-zero angle between p and E. This angle induces a \Gamma on the water molecule, \Gamma =p\times E\equiv {\Gamma }_x\widehat{x}+{\Gamma }_y\widehat{y}+{\Gamma }_z\widehat{z}, which causes the molecule to rotate. Here, both {\Gamma }_y and {\Gamma }_z are roughly zero at the central axis (y=0), and

\begin{array}{}\text{(9)}& {\Gamma }_x=-{\epsilon }_0\lambda a^3{A}^{\prime }(a)A(0)\text{Im}[G(\omega )].\end{array}

Hence, we consider the {\Gamma }_x-induced rotations on the yz-plane only, i.e., the rotation of the φ depicted in Fig. 1(b). Here, φ denotes the angle between the z-axis and N, which is the projection of the P of the water molecule on the yz-plane. Similar to the above analysis of F_f, we assume that the magnitude of the drag torque {\Gamma }_d acting on the water molecule is proportional to \Omega, with {\Gamma }_d=-8\pi {\eta }_r{a}^3\Omega, also in Stokes’ form. Here, {\eta }_r is a coefficient analogous to the effective rotational viscosity. Like F_f, {\Gamma }_d also arises from the interfacial friction between the water and the nanotube. In balance between {\Gamma }_x and {\Gamma }_d is achieved, \Omega also reaches a constant (similar to the constant v_z discussed above), such that

\begin{array}{}\text{(10)}& \Omega =-{\displaystyle \frac{{\epsilon }_0\lambda }{8\pi {\eta }_r}}{A}^{\prime }(a)A(0)\text{Im}[G(\omega )].\end{array}

Further, similar to v_z [Eq. (8)], \Omega [Eq. (10)] is proportional to \text{Im}[G(\omega )].

In Fig. 2, \text{Im}[G(\omega )] is plotted for various {\tau }_1, {\tau }_2, \mathrm{\Delta }{\epsilon }_1, and \mathrm{\Delta }{\epsilon }_2, considering Eq. (8). Hence, it is apparent from this figure that v_z always exhibits two peaks regardless of the values of the latter four terms. From Eq. (3), we conclude that the two peaks originate from the two relaxation processes of the H and O atoms in the water molecule, respectively. Figure 2 also implies that a typical ω exists at which v_z reaches a maximum, namely, v_{z,max}. Thus, the parameters that determine the behavior of the two relaxation processes (namely, {\tau }_1, {\tau }_2, \mathrm{\Delta }{\epsilon }_1, and \mathrm{\Delta }{\epsilon }_2) have evident effects on the position and magnitude of v_{z,max}. In particular, {\tau }_2 has a more prominent influence on the position of v_{z,max} relative to the ω axis than {\tau }_1. Likewise, \mathrm{\Delta }{\epsilon }_2 influences the magnitude of v_{z,max} more strongly than \mathrm{\Delta }{\epsilon }_1. These results also hold for \Omega as, similar to v_z, \Omega is proportional to \text{Im}[G(\omega )] in accordance with Eq. (10). That is, the typical ω yielding v_z=v_{z,max} also causes simultaneous maximum \Omega ({\Omega }_{max}). This means that, when the molecule is transported along the z-axis at v_{z,max}, it also rotates within the yz-plane at a speed of {\Omega }_{max}. Clearly, in this situation, all the single-file molecules behave like the wheels of a high-speed train rolling along a railway and, thus, we can refer to this type of transport as a water nanotrain. In fact, according to Fig. 2, this water nanotrain behavior holds over the entire frequency range.

At this stage of the discussion, the reader may question the accuracy of the water nanotrain concept, as we have made certain assumptions in order to develop this phenomenological theory. Therefore, so as to strengthen the validity of our theoretical findings, we next employ independent molecular dynamics simulations that are free from these assumptions.

3 Molecular dynamics simulations: Comparison with theory

The molecular dynamics simulations described here are performed using the Gromacs 3.3.1 [50] simulation system, which produces results such as those shown in Fig. 1(a). The nanochannel, which is a z-directed (10,0) single-walled carbon nanotube, has 0.386-nm radius and 9.590-nm length. All the carbon atoms are immobilized. To echo the above theoretical setting, we apply a low-density condition to the water molecules such that the initial state corresponds to scattered water molecules within the nanotube. In fact, the low-density condition eliminates interaction between the water dipoles, which would affect the responses of certain water molecules to the external fields. Thus, the directions of the water dipoles can be better mediated by the electric fields under the low-density condition. The water model used here is known as the transferable intermolecular potential three-point model [51]. Further, all atoms considered in this study obey the Lennard-Jones interaction, the parameters of which are taken from Ref. [8]. Each ion is positioned at y=\pm 0.728 nm [see Fig. 1(a)] and \lambda =1.6nm [as denoted in Fig. 1(a)]. We also set each ion to carry an electric charge of 5e or -5e in our simulations. The simulation box has dimensions of 4.016\times 4.616\times 9.701nm³, which is established using a periodic boundary condition in all three dimensions. We perform the simulations in a canonical ensemble at 300 K with a 2-fs time step.

Figure 3 displays the simulation (dots) and theory (lines) results. Note that we calculate \Omega from \Omega =|(1/t){\int }_0^t\varphi (t^{\prime })d{t}^{\prime }|, where \varphi (t^{\prime }) is the time-dependent angle between the z-axis and N. For the theory results (lines), we adopt {\epsilon }_{\mathrm{\infty }}=1.77, which is an approximate value to experimentally measured results [37]. Finally, {\tau }_1=0.21 ps and {\tau }_2=0.015 ps. Note that {\tau }_1 is close to the experimental values yielded by terahertz (THz) spectroscopy studies [41, 42], and that the {\tau }_2 we employ has the same order of magnitude as the water relaxation result yielded by the simulations [25]. The details of the comparison between simulation and theory are given below.

In Fig. 3(a), the simulated v_z displays two peaks at 0.625 and 12.5 THz, which are in accordance with the theoretical predictions. However, in Fig. 3(b), the simulated \Omega exhibits only one evident peak (located at an ω value close to 12.5 THz). The disappearance of the 0.625-THz peak in the simulation results for \Omega implies that the water-molecule rotation is significantly affected by Brownian motion. From a practical perspective, Brownian motion (which was not considered in the above phenomenological theory) affects \Omega more significantly for an ω value of 0.625 THz than 12.5 THz. The source of this behavior may be related to the frequency of the water-molecule spontaneous oscillation [52, 53]. However, further investigation of this topic is required. In this sense, for a field with 12.5-THz frequency, changes to the field-directed orientation of a water molecule occur so rapidly that the thermal motion cannot follow. Hence, the effect of the thermal motion degrades at 12.5 THz (\gg 1THz). This can explain why a peak located in the vicinity of 12.5 THz can be obtained through our simulations [see Fig. 3(b)], in agreement with the peak obtained at the same position in the theoretical results. Additionally, as regards v_z, the random forces of the thermal fluctuation are symmetrical in the z direction and the effect of the random force is cancelled. As a result, the thermal motion has negligible effects on the v_z obtained from the simulations in Fig. 3(a), thus yielding two peaks similar to those predicted by theory.

In general, Fig. 3 exhibits agreement between the theory and simulations, with this agreement being obtained by correction for the presence of the thermal fluctuation. However, the theory and simulations do not always agree in terms of \Omega. In other words, the water nanotrain concept proposed based on the above phenomenological theory can be disturbed by such random forces. Therefore, we analyze our simulation data carefully in order to identify the specific frequency region in which the water nanotrain concept is viable.

Water nanotrain motion is found to occur at ω values of 0.625–25THz (i.e., the shaded region in Fig. 3). In this frequency region, the water molecules exhibit a combination of linear propagation and unidirectional rotation, rather than the random flip-flop reorientation typically exhibited by water in carbon nanotubes [54, 55]. As an example, please see the “movie-1.avi” movie file in the supplementary material, which shows an example of the water nanotrain motion in the 12.5 THz field. In this case, when the linear transport of the water reaches a maximum v_{z,max} (see Fig. 3), the water molecules also undergo the most regular rotation. In particular, we find that v_{z,max} reaches a value of 1080m/s in this case, which is higher than all the values reported in the literature (e.g., Refs. [1, 7, 14, 21, 22, 25, 27, 29]). Furthermore, when the water molecules undergo the train-wheel-like motion within certain translational electric fields, both v_z and \Omega are almost constant. The linear propagation and rotation of the water molecules are almost uniform. This coincides with our phenomenological assumptions in the theoretical analysis, i.e., that F_f in the z direction is closely related to v_z and \Omega. In fact, the relation between the interfacial friction and water velocity is more complex when the water velocity is relatively high [56].

In addition, we analyze ω values lower than 0.625THz, finding that the relevant motion is quite different to the train-wheel-like motion discussed above. As an example, please see the “movie-2.avi” movie file in the supplementary material, which was recorded at 0.568 THz. Such ω values correspond to a phase-propagation motion, because of the restriction of the phase velocities of the translational z-directed E [25]. In addition, we also investigate those ω larger than 25THz, and observe that the corresponding motion is also distinct from the train-wheel-like motion. For instance, see the “movie-3.avi” movie file in the supplementary material, which was recorded at 62.5 THz. At such ω, no (average linear) transportation occurs, because the frequencies are too large for the water molecules to follow the translation of the z-directed electric field.

Figure 4(a) shows the simulation results for a series of higher-density systems that were obtained by adding more water molecules to our systems. This figure indicates that a higher water-molecule density corresponds to smaller v_z. A higher density leads to smaller gaps between the water molecules, so that the resultant stronger water electrostatic interactions (i.e., hydrogen bonds and Coulomb potentials) also act on the water dipole orientations [52]. As the coupling between the water and the field is disturbed, the influence of the transport on the water declines. Figure 4(b) shows the simulation results for a series of systems in which the nanoelectrodes have a different number of elementary charges at 2.5 THz. Therefore, this simulation examines the effect of electric fields of various magnitudes on the transport. It is apparent that v_z increases with increased E magnitude, as expected from Eq. (8).

4 Conclusion and discussion

By developing the phenomenological Debye double-relaxation theory and performing independent molecular dynamics simulations, we have revealed an unprecedentedly fast unidirectional flow of single-file water molecules within a nanochannel in a model system. This approach was inspired by both biological (the function of charge groups in biological membrane proteins is vital to water permeation through aquaporins) and physical implications (environmental oscillations arising from thermalmotion occur when water molecules are transported through nanochannels). Interestingly, we have found that the water molecules exhibit a train-wheel-like motion, i.e., a combination of both translational and rotational motion; we have labeled this behavior a water nanotrain. The underlying mechanism originates from the resonant relaxation dynamics of the H and O atoms.

The flow speed obtained in this work can be further increased by applying stronger fields in an appropriate manner. Also, the water-train water molecules (the angular speed of which can be controlled) may be potential substitutes for fullerene wheels in smaller nanocar designs [57]. A future experimental demonstration of this work could be performed using a carbon-nanotube nanochannel or other types of nanochannels or nanopores (which may even have a larger radius to allow a greater number of water molecules to be transported side-by-side). Besides water molecules, the present investigation could also be extended to treat other polar nanoparticles, such as proteins or DNA segments [58]. Furthermore, as molecular polarizability plays a crucial role in transportation, as revealed in this work, our method has an advantage in that selective transportation is achieved. In other words, the separation of polar molecules from non-polar molecules, or polar molecules from other molecules with different polar properties, can be achieved. To summarize, the findings of this work have implications for nanotechnology, as regards the design of high-efficiency nanochannels or smaller nanodevices such as nanocars. These findings are also helpful for understanding and controlling water flow across biological nanochannels in biology-related research.

Movie captions

All the movies are based on the simulation results. More details are presented in the article.

Movie 1. Train-wheel-like motion for water molecules at 12.5THz.

Movie 2. Phase-propagation motion for water molec-ules at 0.568THz.

Movie 3. No linear transport occurs at 62.5THz. Note that the water molecules are still affected by the applied field and experience regular oscillation.

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