^{1,a)}, Jayendran C. Rasaiah

^{1,b)}and Gerhard Hummer

^{2,b)}

### Abstract

We used molecular dynamics simulations to investigate the thermodynamics of filling of a (6,6) open carbon nanotube (diameter *D* = 0.806 nm) solvated in TIP3P water over a temperature range from 280 K to 320 K at atmospheric pressure. In simulations of tubes with slightly weakened carbon-water attractive interactions, we observed multiple filling and emptying events. From the water occupancy statistics, we directly obtained the free energy of filling, and from its temperature dependence the entropy of filling. We found a negative entropy of about −1.3 *k* _{ B } per molecule for filling the nanotube with a hydrogen-bonded single-file chain of water molecules. The entropy of filling is nearly independent of the strength of the attractive carbon-water interactions over the range studied. In contrast, the energy of transfer depends strongly on the carbon-water attraction strength. These results are in good agreement with entropies of about −0.5 *k* _{ B } per water molecule obtained from grand-canonical Monte Carlo calculations of water in quasi-infinite tubes in vacuum under periodic boundary conditions. Overall, for realistic carbon-water interactions we expect that at ambient conditions filling of a (6,6) carbon nanotube open to a water reservoir is driven by a favorable decrease in energy, and opposed by a small loss of waterentropy.

The authors thank Dr. S. Vaitheeswaran and Dr. H. Yin for discussions. J.C.R. thanks the National Science Foundation for support under Grant No. CHE 05489187. A.W. and J.C.R. thank the University of Maine Supercomputing Cluster for generous allocations of computing time and resources and Dr. John Koskie and Dr. Steve Cousins for their assistance. G.H. is supported by the Intramural Research Program of the National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health. This study utilized the high-performance computational capabilities of the Biowulf PC/Linux cluster at the National Institutes of Health, Bethesda, MD (http://biowulf.nih.gov).

I. INTRODUCTION

II. METHODS

A. MD simulations of open nanotubes

B. Thermodynamics of water transfer

C. Infinite periodic nanotubes

III. RESULTS AND DISCUSSION

A. Dependence of water occupancy on nanotube-water interactions and temperature

B. Entropy of transfer

C. Energy of transfer

IV. CONCLUSIONS

### Key Topics

- Entropy
- 71.0
- Carbon nanotubes
- 38.0
- Water energy interactions
- 25.0
- Nanotubes
- 24.0
- Energy transfer
- 23.0

##### B82B1/00

## Figures

Simulation system. (a) Slab view of the nanotube filled with water and surrounded by water reservoir. (b) Close-up of the single file of H-bonded water molecules inside the nanotube.

Simulation system. (a) Slab view of the nanotube filled with water and surrounded by water reservoir. (b) Close-up of the single file of H-bonded water molecules inside the nanotube.

Free energies of transfer of water into a (6,6) nanotube open to a reservoir as a function of the occupancy number *N* for temperatures ranging from 280 K to 320 K for *λ* = 0.752 and 0.785.

Free energies of transfer of water into a (6,6) nanotube open to a reservoir as a function of the occupancy number *N* for temperatures ranging from 280 K to 320 K for *λ* = 0.752 and 0.785.

Sensitivity of water occupancy in the open (6,6) nanotubes to temperature. The left panels show the water occupancy *N* as a function of time from MD simulations at *T* = 280, 300, and 320 K (bottom to top) with nanotube-water attractive interactions scaled by *λ* = 0.785. The right panels show the corresponding normalized occupancy histograms *P*(*N*). Dashed vertical lines indicate the probability *P*(*N* = 5) of the filled state at the lowest and highest temperature simulated.

Sensitivity of water occupancy in the open (6,6) nanotubes to temperature. The left panels show the water occupancy *N* as a function of time from MD simulations at *T* = 280, 300, and 320 K (bottom to top) with nanotube-water attractive interactions scaled by *λ* = 0.785. The right panels show the corresponding normalized occupancy histograms *P*(*N*). Dashed vertical lines indicate the probability *P*(*N* = 5) of the filled state at the lowest and highest temperature simulated.

Transfer free energy per water molecule in units of *k* _{ B } *T* as a function of the inverse temperature for open (6,6) nanotubes immersed in TIP3P water. The energy of transfer (Δ*U* _{ N } */N*) and entropy of transfer (−Δ*S* _{ N } * /Nk* _{ B }) are obtained from the slope and intercept, respectively, of lines fitted to the MD data obtained with the Andersen thermostat (middle, bottom) and weak-coupling thermostat (top). Results are shown for *λ* = 0.752 (top), *λ* = 0.785 (middle), and *λ* = 0.8 (bottom) between 280 and 320 K (filled symbols: Andersen^{26} thermostat; open symbols: weak-coupling^{25} thermostat).

Transfer free energy per water molecule in units of *k* _{ B } *T* as a function of the inverse temperature for open (6,6) nanotubes immersed in TIP3P water. The energy of transfer (Δ*U* _{ N } */N*) and entropy of transfer (−Δ*S* _{ N } * /Nk* _{ B }) are obtained from the slope and intercept, respectively, of lines fitted to the MD data obtained with the Andersen thermostat (middle, bottom) and weak-coupling thermostat (top). Results are shown for *λ* = 0.752 (top), *λ* = 0.785 (middle), and *λ* = 0.8 (bottom) between 280 and 320 K (filled symbols: Andersen^{26} thermostat; open symbols: weak-coupling^{25} thermostat).

Entropy (top) and energy (bottom) of transfer per water molecule in infinite periodic tubes with *λ* = 1 and different repeat lengths *L* and diameters *D* as a function of the average spacing per particle Δ*z = L/N* along the tube axis at 300 K. Wide and narrow pores of diameters 0.835 nm and 0.777 nm were obtained by scaling the carbon-carbon bond lengths of the original pore (*D* = 0.806 nm) from 0.14 nm to 0.145 and 0.135 nm, respectively. The vertical line indicates the equilibrium spacing of Δ*z* ≈ 0.26 nm in the open nanotube.^{1} Lines are linear and quadratic fits to the entropy and energy, respectively. The bottom plot shows Δ*U* _{ N }/*N* (symbols with error bars: from Monte Carlo simulations), and Δ*A* _{ N }/*N* (magenta line: obtained by combining the fits to the energy and entropy; symbols: from Eqs. (2) and (4)) for the original pore diameter (*D* = 0.806 nm).

Entropy (top) and energy (bottom) of transfer per water molecule in infinite periodic tubes with *λ* = 1 and different repeat lengths *L* and diameters *D* as a function of the average spacing per particle Δ*z = L/N* along the tube axis at 300 K. Wide and narrow pores of diameters 0.835 nm and 0.777 nm were obtained by scaling the carbon-carbon bond lengths of the original pore (*D* = 0.806 nm) from 0.14 nm to 0.145 and 0.135 nm, respectively. The vertical line indicates the equilibrium spacing of Δ*z* ≈ 0.26 nm in the open nanotube.^{1} Lines are linear and quadratic fits to the entropy and energy, respectively. The bottom plot shows Δ*U* _{ N }/*N* (symbols with error bars: from Monte Carlo simulations), and Δ*A* _{ N }/*N* (magenta line: obtained by combining the fits to the energy and entropy; symbols: from Eqs. (2) and (4)) for the original pore diameter (*D* = 0.806 nm).

Comparison of transfer free energies Δ*A* _{ N }–Δ*A* _{5} for different values of *λ* calculated directly from the logarithm of the occupancy probabilities (open symbols: 280 K; filled symbols: 320 K), and from global fits to the free energies and free energy derivatives with respect to *λ* (upper and lower lines for *N*<5 are at 280 and 320 K, respectively). In the fits, the entropy and enthalpy were assumed to be constant and quadratic polynomials in *λ*, respectively.

Comparison of transfer free energies Δ*A* _{ N }–Δ*A* _{5} for different values of *λ* calculated directly from the logarithm of the occupancy probabilities (open symbols: 280 K; filled symbols: 320 K), and from global fits to the free energies and free energy derivatives with respect to *λ* (upper and lower lines for *N*<5 are at 280 and 320 K, respectively). In the fits, the entropy and enthalpy were assumed to be constant and quadratic polynomials in *λ*, respectively.

Thermodynamic driving force and entropy-enthalpy compensation for water filling of the (6,6) nanotube at 300 K. Free energy (black triangles), energy (blue circles), and entropy of transfer (red squares) as a function of the occupancy for *λ* = 0.752 (open symbols) and 0.785 (filled symbols).

Thermodynamic driving force and entropy-enthalpy compensation for water filling of the (6,6) nanotube at 300 K. Free energy (black triangles), energy (blue circles), and entropy of transfer (red squares) as a function of the occupancy for *λ* = 0.752 (open symbols) and 0.785 (filled symbols).

## Tables

Lennard-Jones parameters of the *λ*-dependent carbon-water interactions.

Lennard-Jones parameters of the *λ*-dependent carbon-water interactions.

Thermodynamics of transferring TIP3P water into an open and solvated nanotube at *T* = 300 K in units of kJ/mol for *λ* = 0.752 and 0.785. Results for the completely filled *N* = 5 state are in italics. The first and second line for each *N* list the results obtained from fits of the temperature dependence; the third line lists the results obtained directly from differences of the system enthalpies and free energies. Results in line one for each *N* are for weak-coupling thermostat simulations,^{25} and in lines two and three for Andersen^{26} thermostat simulations. Numbers in parentheses indicate estimated statistical errors in the last digits.

Thermodynamics of transferring TIP3P water into an open and solvated nanotube at *T* = 300 K in units of kJ/mol for *λ* = 0.752 and 0.785. Results for the completely filled *N* = 5 state are in italics. The first and second line for each *N* list the results obtained from fits of the temperature dependence; the third line lists the results obtained directly from differences of the system enthalpies and free energies. Results in line one for each *N* are for weak-coupling thermostat simulations,^{25} and in lines two and three for Andersen^{26} thermostat simulations. Numbers in parentheses indicate estimated statistical errors in the last digits.

Thermodynamic properties of bulk TIP3P water and real water near ambient conditions from MD simulations in an *NVT* ensemble. The excess entropies (last column) are per particle. Numbers in parentheses indicate estimated statistical errors in the last digits.

Thermodynamic properties of bulk TIP3P water and real water near ambient conditions from MD simulations in an *NVT* ensemble. The excess entropies (last column) are per particle. Numbers in parentheses indicate estimated statistical errors in the last digits.

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