Strongly Anisotropic Dielectric Relaxation of Water at the Nanoscale

Letter pubs.acs.org/JPCL

Strongly Anisotropic Dielectric Relaxation of Water at the Nanoscale Cui Zhang,†,∥ François Gygi,‡ and Giulia Galli*,†,§ †

Department of Chemistry, University of California, Davis, California 95616, United States Department of Computer Science, University of California, Davis, California 95616, United States § Department of Physics, University of California, Davis, California 95616, United States ‡

S Supporting Information *

ABSTRACT: We carried out atomistic simulations of water at the nanoscale, and we investigated the dielectric response of the liquid as a function of the distance between hydrophobic confining surfaces. We found that dipolar fluctuations are modified by the presence of surfaces up to strikingly large distances, i.e., tens of nanometers. Fluctuations are suppressed by approximately an order of magnitude in the z direction, perpendicular to the interface, and they are enhanced (up to 50%) in the x−y plane, giving rise to strong anisotropies in the components of the dielectric relaxation. Such anisotropies originate from the very presence of interfaces, and not from the details of the interaction between water and the hydrophobic surfaces. Our findings are consistent with recent terahertz and ultrafast infrared pump−probe spectroscopy experiments and bear important implications for the study of solvation under confinement. SECTION: Physical Processes in Nanomaterials and Nanostructures

W

simulations of ref 22 reported a substantial decrease (∼20% to ∼50%) of ϵ0 of bulk water when confined in a nanocavity with few nanometer diameter. On the contrary, the simulations of ref 23 claimed a larger average value of ϵ0 for the liquid confined between two graphene sheets at separations ranging from ∼1 to 3 nm. We carried out MD simulations of water confined between graphene sheets, which revealed a complex, anisotropic dielectric response of the liquid as a function of the confining distance: dipolar fluctuations are greatly modified by the presence of surfaces up to distances of tens of nanometers, with the appearance of two types of relaxation times, consistent with recent experiments.15 Dipolar fluctuations are suppressed in the z direction, perpendicular to the surfaces, up to an order of magnitude, while they are enhanced in the x−y plane for distances smaller than ∼5 nm. Correspondingly, the Debye relaxation time varies by about 2 orders of magnitude in the z direction. Remarkably, for confinement lengths up to ∼30 nm, the magnitude of the dipolar fluctuations along z is almost independent of surface separation, and depends weakly on the interactions between water and the hydrophobic surfaces, consistent with the experimental results of ref 26. We performed MD simulations27 of bulk liquid water and water confined between two parallel graphene sheets using the SPC/E potential,28 and the water−carbon force field proposed by Werder et al.29 Periodic boundary conditions were applied

ater is ubiquitous and so is its presence in the proximity of surfaces. The dynamics of the liquid at interfaces and under confinement plays a key role in determining the physical properties of inorganic materials1,2 and biomolecules,3−5 including transport properties through pores in minerals and cells, and the dynamics of proteins. However, great uncertainties remain on the dynamics of water and its response to electric fields under confinement, in spite of notable progress in recent years in probing the properties of the liquid at the nanoscale, in various thermodynamic conditions.6−14 Recently, new insight on the dielectric relaxation of water at interfaces and under confinement has been obtained from terahertz spectroscopy measurements15 on hydrated model membranes, and from theory.16−18 Tielrooij et al.15 reported new types of water dynamics in thin interfacial layers, as identified by shorter and longer relaxation times than in the bulk. In addition, the nonperturbative theories of refs 16 and 17 showed complex dielectric profiles of water arising in the proximity of surfaces. Numerous measurements of the dielectric constant ϵ0 of bulk water at ambient conditions have been reported since the end of the 19th century,19 and in the past 20 years the dielectric response of the bulk liquid has been extensively modeled at the atomistic level, using empirical potentials20 and in some cases, first-principle calculations.21 By contrast, measurements of the dielectric tensor of confined water are challenging; in addition, only few microscopic simulations of the dielectric behavior of the confined liquid have appeared in the literature,16,17,22−25 with no systematic study of the liquid dielectric relaxation as a function of the confining distance. Some of the existing studies are controversial; for example, the molecular dynamics (MD) © 2013 American Chemical Society

Received: May 29, 2013 Accepted: July 11, 2013 Published: July 11, 2013 2477

dx.doi.org/10.1021/jz401108n | J. Phys. Chem. Lett. 2013, 4, 2477−2481

The Journal of Physical Chemistry Letters

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along x, y, and z. We used supercells containing from 3456 to 221184 water molecules. Simulations were carried out in the NVT ensemble30 at 300 K with a time step of 1 fs. Long-range electrostatic interactions were treated by particle mesh Ewald method with a precision of 10−5.31,32 Each simulation was equilibrated for at least 5 ns before collecting statistics for 20− 80 ns. For bulk simulations, in some cases we compared results obtained with the SPC/E potential to those obtained with flexible33 and polarizable34 potentials (see Figure S1). In several cases, we compared results obtained with 3456 and 221184 water molecules (see Figure S2). Simulations of water confined in the z direction were carried out at graphene separations of 0.94, 1.44, 2.31, 4.74, 8.16, 15.96, and 31.56 nm, respectively. The cell length along z was at least 4 times larger than the graphene separation.

S3), which substantially deviates from the sine distribution found in the bulk, corresponding to random orientations. Although notable, the observed structural changes in proximity of the interface are not the most remarkable changes occurring under confinement. The most striking effect was found in the dielectric properties of the system, in particular, in the changes of the molecular dipole relaxation time and in the dipolar fluctuations. The relaxation time τ of the total dipole moment M⃗ in the bulk (τ = 10.51 ± 0.11 ps, see Figure S4) was computed from the decay time of the total dipole autocorrelation function Φ(t) = ⟨M⃗ (0)·M⃗ (t)⟩/⟨M⃗ 2⟩. We obtained a modest overestimate of the Debye relaxation time derived from experiments, 8.24 ± 0.40 ps, by fitting the measured dielectric response using a Debye model.36 Upon confinement, we found that τ decreased by more than 2 orders of magnitude in the direction perpendicular to the surfaces, as shown in Figure 2 (also see

Figure 1. Radial dipole−dipole space correlation function F(r) and its x, y, and z components of (a) liquid water, and water confined between graphene sheets separated by (b) d = 1.44 nm and (c) d = 4.74 nm. Oscillations of F(r) persist up to ∼1.5 nm (also see Figure S1 and Figure S2).

Figure 2. Cartesian components of the relaxation time of water under confinement as a function of surface separation distance d, at 300 K. τα is obtained by averaging the results calculated at 10 ns time intervals over 80 ns trajectories. The standard deviations are reported as the error bars. Orange and purple dashed lines indicate the Debye relaxation time τ of bulk water computed at 300 K and a density of 0.99 g/cm3 and 0.90 g/cm3, respectively.

Table S1; the components of Φ(t) as a function of time and the details of their fit in the case of confined water are reported in Figure S5). On the other hand, the decay time in the x−y plane is remarkably increased by 150% for ∼1 nm surface separation, but resumes a value close to that of the bulk at ∼5 nm. The small deviations from bulk results for distances larger than 5 nm are ascribed to small density variations of the different samples of the confined liquid generated in our simulations, and to the absence of long-range interactions in the z direction. Correspondingly, the amplitude of the fluctuations of the zcomponent of the total dipole moment (Mz) are greatly suppressed up to an order of magnitude, compared to bulk values, for d up to ∼30 nm, while those of the x and y components (Mx and My) are enhanced for d < ∼5 nm. The striking difference between the fluctuations of Mz and (Mx, My) is shown in Figure 3 for d = 8.16 nm, and up to ∼30 nm separations in Figure S6. A quantitative extrapolation of our data is difficult, as simulations could only be performed up to ∼30 nm separation; however, we speculate that extrapolation of the values obtained for τz might yield d of the order of a micrometer. Indeed this is approximately the distance at which the Laplace pressure on our sample (ΔP = 4γ/d, computed using the experimental value of the surface tension of water γ =

We first analyzed the effect of confinement on the structural properties and dipolar orientations of the fluid. Figure 1 shows the radial dipole−dipole correlation F(r) defined as 1 F (r ) = N

N

∑ i=1

1 ni(r )

ni(r )

∑→̂μ⎯ i ·→̂μ⎯ j (1)

j=1

where ni(r) is the number of molecules within distance r and (r μ⎯ and →̂ μ⎯ are the normalized dipole + δr) from molecule i; →̂ i

j

moments of molecule i and j, respectively. We found that F(r) is inhomogeneous in liquid water at ambient conditions over a region of radius ≃1.5 nm, with enhanced oscillations as the temperature is lowered, e.g., from 300 to 220 K. Under confinement, we observed the appearance of anisotropies in F(r) for short surface separation lengths (d), which persist up to about 5 nm: dipolar spacial correlations are less structured (more homogeneous) along z compared to the x−y plane, as shown in Figure 1b. These anisotropies are related to the preference of molecules to lie in the x−y plane close to the interface and are consistent with the results of ab initio simulations on smaller cells.35 Such preference is evident in the angular distribution of molecular dipole moments (see Figure 2478



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Figure 3. Left panel: snapshot of water confined between graphene sheets at a separation distance d = 8.16 nm. We defined three different regions: an interfacial region including water molecule within 4 Å to the interfaces, corresponding to distances over which the density of the liquid is different from that of bulk water; a central region (Δd = 5.7 nm) where the density of the liquid is the same as in the bulk throughout 80 ns simulations; and an intermediate region with small density variations between the interfacial and central regions. Oxygen atoms are depicted as blue, pink, and red spheres in the three regions, respectively. (a−c) Cartesian components Mx (black), My (red) and Mz (blue) of the total dipole moment of the confined liquid in the three regions: (a) interfacial; (b) intermediate; and (c) central, normalized by the square root of the number of molecules in each region.

72 mN/m) equals the atmospheric pressure. It is also at ∼1 μm that the average net dipole moment on a single water molecule (⟨M⟩/N) in a bulk sample reaches values close to zero ( 5 nm such modifications are merely due to the presence of interfaces. From the Cartesian components of the total dipole moment, we evaluated the components of the dielectric tensor in the x−y plane using conducting boundary conditions;37 these are given by εαα = 1 +

4π (⟨MαMα⟩ − ⟨Mα⟩⟨Mα⟩) VkBT

where α = x or y; V is the volume of the system; kB and T are the Boltzmann constant and temperature, respectively; and Mα is the α component of the total dipole moment M⃗ . Figure 4

Figure 4. Parallel components of the dielectric tensor of confined water as a function of surface separation distance d, at 300 K. Orange and purple dashed lines indicate the dielectric constant ϵ0 of liquid water computed at 300 K and a density of 0.99 g/cm3 and 0.90 g/cm3, respectively. Note that with the SPC/E potential at 0.99 g/cm3 and 300 K, ϵ0 = 71.53 ± 0.14, compared with the experimental value of 78.

shows that the components of the dielectric tensor parallel to the surface (ϵxx and ϵyy) exhibit substantial variations with respect to their bulk values, and they increase by about 50% for a surface separation of ∼1 nm (see Table S2 for details). The ϵxx and ϵyy values decrease as a function of surface separation and become similar to those of the bulk for d ≥ ∼5 nm. The small deviations from bulk results of ϵxx and ϵyy for distances larger than 5 nm are ascribed to two effects: small density variations of the different samples of the confined liquid generated in our simulations, and the effect of cutting out longranged dipole correlations in the z direction.

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Figure 5. (a) Density distributions and (b) parallel components ϵ∥ (= ϵxx = ϵyy) of the dielectric tensor of water confined between surfaces at a distance d = 4.74 nm. The liquid was represented by a slab with 3456 H2O; we used Lennard-Jones oxygen−carbon interaction parameters of σCO = 3.19 Å, ϵCO = 0.392 kJ/mol (black lines) proposed in ref 29 and σ′CO = (1/2)σCO, ϵ′CO = (1/2)ϵCO (red lines). Cartesian components Mx (black), My (red), and Mz (blue) of the total dipole moment of the confined liquid are shown in panels (c) and (d) for simulations carried out with σCO, ϵCO and σ′CO, ϵ′CO, respectively.

the surface. These results clearly indicate that the properties of the confined liquid may not be described by an average dielectric constant, which is often adopted in complex large scale simulations of, e.g., biological systems. Our findings show that the definition of average dielectric properties of confined water may lead to an unphysical description of the liquid in the proximity of surfaces. Finally, in one case we tested the dependence of our results on the force fields, and we carried out simulations with a polarizable force field34 for d = 1.47 nm. In this case, we found a decrease of the molecular dipole moment in close proximity to the graphene surfaces, consistent with previous ab initio results.35 However, our main findings were again unaffected, and the suppression of the fluctuations of Mz was found to be the same, quantitatively, as in the case of the SPC/E force field. In summary, we investigated the water dielectric response as a function of the distance between confining, hydrophobic surfaces; our study brought several new insights into the properties of confined water: (i) We showed that at a confining distance of about 5 nm, the dielectric constant of water attains the same values as in the bulk, in the directions parallel to the confining surfaces. On the other hand, the anisotropy in the direction perpendicular to the graphene sheets persists for much larger distances, of at least tens of nanometers. (ii) We found a substantial enhancement (up to 50%) in the values of the dielectric constant in the directions parallel to the confining surfaces, for distances smaller than 5 nm. (iii) We showed that in the case of hydrophobic surfaces the dielectric anisotropy of the confined liquid stems from the confinement (i.e., the very presence of interfaces) and not from the details of the water− surface interaction. This result is consistent with the findings of recent experiments26 on water confined by reverse micelle, showing that the dynamics of the fluid is mostly governed by the presence of interfaces. Finally, we reported specific values for the Debye relaxation times as a function of the confining distance, whose knowledge is expected to be of interest for the interpretation of future experimental studies and for modeling confined water in complex environments.

In order to examine possible dependence of the results reported above on the technical details of our simulations, we repeated several of our calculations by increasing the number of molecules in the sample at fixed distance and density, for example, from 3456 to 31104 water molecules at d = 8.16 nm (see Figure S9). In addition, we compared results obtained with a Nosé−Hoover thermostat and with the one proposed in ref 30; we changed the temperature of the graphene sheets from 0 to 300 K, and we varied the cutoff radii chosen for the Ewald sums and the van der Waals interaction. We also compared results obtained by varying the vacuum region in the transverse direction of our MD cell. In all cases, the findings reported above were unaffected. Furthermore, we investigated the influence of the water− surface interactions. We repeated some of our simulations with van der Waals interaction parameters arbitrarily reduced by a factor of 2. Such a big reduction greatly influences the density and the components of the dielectric tensor parallel to the surfaces (ϵ// = ϵxx = ϵyy), as shown in Figure 5a,b, respectively. However, the qualitative behavior reported in Figure 4 persists, and the suppression of fluctuations along z is clearly present (see Figure 5c,d), indicating that the effect found here is robust against the details of the interaction between water and the hydrophobic surfaces, and it originates from the very presence of interfaces. To further confirm that the effect stems from the presence of interfaces, we carried out a simulation of a water droplet (3456 molecules) confined between graphene sheets, representing water confined in all three directions. As expected, suppression of the dipolar fluctuations was found in all directions (see Figure S10). It is interesting to note that ϵ∥ is always enhanced in the vicinity of the surface, within a region of ∼5 Å, even in the case of weak water−surface interactions, where one observes an overall decrease of the density in the proximity of the interface. We emphasize that such behavior does not mean that the dielectric constant of water increases in the proximity of the interface, as the ϵ∥ enhancement is accompanied by a strong decrease of the relaxation time in the direction perpendicular to 2480



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ASSOCIATED CONTENT

S Supporting Information *

Additional details of our simulations of bulk water and water confined within graphene sheets. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Address ∥

Department of Chemistry, Princeton University, Princeton, NJ 08544 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by DOE Grant DE-SC0005180. We thank M. Parrinello and D. Pan for useful discussions. REFERENCES

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Strongly Anisotropic Dielectric Relaxation of Water at the Nanoscale

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