Dynamic interactions of ions with carbon nanotubes in water

NIM B Beam Interactions with Materials & Atoms

Nuclear Instruments and Methods in Physics Research B 256 (2007) 167–171 www.elsevier.com/locate/nimb

Dynamic interactions of ions with carbon nanotubes in water D.J. Mowbray, Z.L. Misˇkovic´ *, F.O. Goodman Department of Applied Mathematics, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 Available online 16 January 2007

Abstract We extend the two dimensional hydrodynamic model, describing the collective electronic response of a single-wall carbon nanotube to include dielectric media surrounding the nanotube. We make use of the dielectric function for water recently developed by Emfietzoglou et al. based on the optical data of Hayashi et al. to model a carbon nanotube in water. Calculations of the stopping force and self-energy (image potential) are given for channeling of fast ions through the nanotube, showing both the radial and velocity dependence. Ó 2006 Elsevier B.V. All rights reserved. PACS: 79.20.Rf; 34.50.Bw; 34.50.Dy Keywords: Carbon nanotubes; Water; Aqueous solution; Stopping force; Self-energy; Image potential

1. Introduction Some of the most intriguing applications of carbon nanotubes (NTs) are expected in the area of biomedical research where they can be used, for example, for targeted drug delivery, or as artificial ion channels in cell membranes. In that context, there has been a recent increase of interest in the interactions of NTs with aqueous environments, especially addressing the hydrophobic properties of carbon NTs and how this affects the transport of water through and around NTs. A number of molecular dynamic (MD) simulations of the interactions between water and carbon NTs have been conducted in recent years [1,2]. Several interesting effects of water on single-wall carbon nanotubes (SWNTs) have been studied experimentally in various contexts, such as NT probes [3], their electrical conductivity [4], NT transistors [5], functionalization of NTs with aqueous solutions [6], NT synthesis in water [7], flexibility of NTs [8] and their alignment in aqueous solution [9].

*

Corresponding author. Tel.: +1 519 888 4567; fax: +1 519 746 4319. E-mail address: [email protected] (Z.L. Misˇkovic´).

0168-583X/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2006.11.108

In the present contribution, we study the effects of water on the dielectric response of carbon NTs to external charged particles moving through NTs under channeling conditions. To that effect, we extend here the previously developed 2-D hydrodynamic model for carbon valence electrons in the NT’s walls [15] to include the electrostatic coupling with dielectric media outside the nanotube. We further improve on the dielectric constant models for water used by Yannouleas et al. [10] and Liu et al. [11] for NTs, by employing a dielectric function for water based on the optical data of Hayashi et al. [12], developed by Emfietzoglou et al. [13,14]. Specifically, we calculate both the stopping force and the image potential of a proton moving paraxially inside a SWNT surrounded by water. Our results can be of possible utility for, e.g. studies of ion transport in biologically relevant environments. Atomic units will be used throughout, unless indicated otherwise. 2. Basic theory We begin by modeling our SWNT system as a 2-D electron gas on a cylinder of radius a with a background dielectric constant nt (which we take equal to 1), and

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encapsulated by an outer medium with dielectric constant w for radial distances r > b. The electron number density per unit area at position ra on the SWNT, given in cylindrical coordinates by ra ={a, u, z}, is assumed to have the form n0 + n1(ra, t), where n0 is the equilibrium number density. For a single fluid model, as given by Wang and Misˇkovic´ [15], the electron gas consists of four valence electrons per carbon atom on the SWNT, so that n0  0.428 a.u., which screens the positive ionic background charge of r+  0.428 a.u. Using va(ra, t) to denote the velocity field of the electron fluid, the linearized Continuity Equation is

Finally, since the total electric potential is defined as the work done to move a test charge at infinity to a position r, U(r, x) must be a continuous function of r. Following the method of Doerr and Yu [18], we consider the total electric potential to be the sum of the screened external perturbing potential, Uext(r, x) and the induced potential on the nanotube and the polarizable boundary Uind(r, x), so that Uðr; xÞ ¼ Uext ðr; xÞ þ Uind ðr; xÞ; ð6Þ 2 and the Poisson equation, r Uðr; xÞ ¼ 4pqðr; xÞ, gives r2 Uext ðr; xÞ ¼ 4pqext ðr; xÞ;

ð7Þ

2

on1 ðra ; tÞ ~ a  va ðra ; tÞ ¼ 0; þ n0 r ot

r Uind ðr; xÞ ¼ 4p½n1 ðra ; xÞdðr  aÞ  rb ðrb ; xÞdðr  bÞ: ð1Þ

ð8Þ

~ a  1 eu o þ ez o is the gradient operator tangent to where r a ou oz the cylindrical shell. The linearized momentum-balance equation is then

Denoting the Fourier Transform in cylindrical coordinates of an arbitrary function A(r, u, z, x) as X Z dk ~ m; k; xÞ; eimu eikz Aðr; ð9Þ Aðr; u; z; xÞ ¼ 2 ð2pÞ m

ova ðra ; tÞ ~ a ~ ¼ ra Uðra ; tÞ  r a n1 ðra ; tÞ ot n0 b~ 2 þ r a  ra n1 ðra ; tÞ  cva ðra ; tÞ; n0

ð2Þ

~ a Uðra ; tÞ is where U is the total electric potential, so that r the tangential force on an electron due to the electric field on the SWNT. The second term arises from the 2-D Thomas–Fermi equilibrium kinetic energy of the electron gas [16,17], and may be considered to be the classical pressure of the electron gas, where a  pn0. The third term arises from the von Weizsa¨cker gradient correction to the equilibrium kinetic energy of the electron gas [17], where b ¼ 14. The last term in Eq. (2) represents the frictional force on an electron due to scattering on the positive charge background, which we take to be vanishingly small c ! 0+. ~  E ¼ 4pq, Gauss’ Law for the electric field gives r where the total charge density q(r, t) is given by qðr; tÞ ¼ rb ðrb ; tÞdðr  bÞ  n1 ðra ; tÞdðr  aÞ þ qext ðr; tÞ; ð3Þ where rb(rb, t) is the polarization charge density on the boundary of the dielectric medium at rb = {b, u, z}, and qext(r, t) is the external perturbing charge density. Integrating Gauss’ Law at r = b, and working with the Fourier transform with respect to time, the total electric potential U(r, x) satisfies   oU oU  ¼ 4prb ðrb ; xÞ: ð4Þ or bþ or b Furthermore, by integrating Gauss’ Law for the electric ~  D ¼ 4pqf , where D  (r, x)E and displacement field, r qf is the total free charge density, the total potential U(r, x) satisfies at r = b   oU oU w ðxÞ   nt ðxÞ  ¼ 0: ð5Þ or bþ or b

the Green’s function in cylindrical coordinates is X Z dk 1 0 0 ¼ eimðuu Þ eikðzz Þ gðr; r0 ; m; kÞ; 2 kr  r0 k ð2pÞ m

ð10Þ

where g(r, r 0 ; m, k) is the radial Green’s function, given by Jackson [19] as gðr; r0 ; m; kÞ ¼ 4pI m ðkr< ÞK m ðkr> Þ;

ð11Þ

where r< = min{r, r 0 }, r> = max{r, r 0 } and Im and Km are modified Bessel’s functions of the first and second kind, respectively. Following the procedure outlined previously [20–24], we find the Fourier transform of the induced potential is given by n1 e ind ¼ ae U ½gðr; a; m; kÞ  bRgðr; b; m; kÞgðb; a; m; kÞ nt  e ext  oU þ bRgðr; b; m; kÞ ; or 

ð12Þ

b

with e n 1 being the Fourier transform of the induced number density on the NT, given by   U ext  e ext ðaÞ þ bgða; b; m; kÞRoe U or  b e ; ð13Þ n 1 ¼ 1 a v þ nt ½gða; a; m; kÞ þ bRag2 ðb; a; m; kÞ where we define R v

w  nt ; 4p½nt  ðnt  w ÞkbI m ðkbÞK 0m ðkbÞ n0 ðk 2 þ m2 =a2 Þ 2

aðk 2 þ m2 =a2 Þ þ bðk 2 þ m2 =a2 Þ  xðx þ icÞ

ð14Þ :

ð15Þ

3. Results and discussion We model a single-walled carbon nanotube in water using the Drude-type dielectric function for water devel-

D.J. Mowbray et al. / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 167–171

oped by Emfietzoglou et al. [13,14], w(x) = e1(x) + ie2(x), where " # ioniz X fj ðE2j  x2 Þ 2 e1 ðxÞ ¼ 1 þ Ep 2 2 ðE2j  x2 Þ þ ðcj xÞ j ( ) excit X fj ðE2j  x2 Þ½ðE2j  x2 Þ2 þ 3ðcj xÞ2  2 þ Ep ; 2 2 2 ½ðE2j  x2 Þ þ ðcj xÞ  j ð16Þ " # iX oniz fj cj x e2 ðxÞ ¼ E2p H ðx  EC ; DÞ 2 2 2 ðEj  x2 Þ þ ðcj xÞ j ( ) excit X 2f j c3j x3 þ E2p ; ð17Þ ½ðE2j  E2 Þ2 þ ðcj xÞ2 2 j with fj, cj and Ej being the oscillator strength, damping energy and transition energy coefficients, respectively, given in Table 1 based on the optical data of Hayashi et al. [12]. The ionization energy cut-off at EC = 7 eV in water is ensured by the function H(x  EC, D) defined by H(x, D) = 1/(1 + ex/D). A sharp cut-off, as employed by Emfietzoglou et al., is obtained by letting D ! 0+, a smoothed cut-off, as employed by Dingfelder et al. [25], is obtained by taking D  0.64 eV, and a dielectric function without an ionization energy cut-off is obtained by letting D/x ! 0+. Moreover, since carbon NTs are known to be hydrophobic [9], we model the encapsulating water by a dielectric boundary at both b = a, and b = a + Dr, using ˚ , as found from molecular dynamic simulation Dr  3.4 A by Moulin et al. [2]. The stopping force on an ion of charge Q traveling paraxially to the SWNT with speed v at r0(t) = {r0, u0, vt}, is the force opposing the ion’s motion. Its magnitude equals the usual stopping power, defined by S  Q oUozind jr¼r0 . This quantity describes, in our model, the energy loss of a channeled particle per unit path length due to the collective electron excitations on the NT wall, which are modified by the electrostatic coupling with the polarization of the surrounding water molecules giving rise to the effective surface charge rb on the water boundary at r = b. The self-energy, or image potential, of the ion is defined by

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Eself  Q2 Uind jr¼r0 . This quantity gives rise to the conservative image force which deflects the channeled particle towards the NT wall. Although we only calculate this potential inside the NT, our model can be used straightforwardly to evaluate the image potential outside the NT, thus providing an opportunity to analyze the toroidal electron image states between the NT and the surface of water in aqueous solutions, if any [27]. ˚ and selfFigs. 1 and 2 show the stopping force in eV/A energy in eV respectively, versus speed v in a.u. of an ion traveling paraxially at r0 = 0, for a dielectric boundary sep˚ , calculated using a sharp cut-off, aration of Dr = 0 and 3.4 A smooth cut-off and no cut-off of the ionization energy at EC = 7 eV. One can notice from these figures that both the stopping and the self-energy at the axis of the NT are rather insensitive to the presence of water for low ion speeds, as opposed to the high-speed region. More interestingly, there appear structures in both the stopping and self-energy curves related to the ionization threshold at EC in the imaginary part of the dielectric function. These structures are more pronounced for sharper transition regions D of the cut-off at EC,

Table 1 The Drude model parameters for the IXS optical data [14,12] Transition, j Excitations 1 (A1B1) 2 (B1A1) 3 (Ryd A + B) 4 (Ryd C + D) 5 (Diffuse bands) Ionizations 6 7 8 9 10 a

(1b1) (3a1) (1b2) (2a1) (K shell)a

Ej [eV]

cj [eV]

8.10 10.10 12.10 13.51 14.41

1.90 1.95 2.94 5.06 2.64

0.0045 0.0046 0.0030 0.0190 0.0050

16.30 17.25 28.00 42.00 450.00

14.00 10.91 27.38 28.68 360.00

0.2300 0.1600 0.1890 0.2095 0.3143

fj

Based on the NIST K-shell photoelectric cross-section database [26].

˚ , plotted versus speed v in a.u. on an ion Fig. 1. The stopping force in eV/A traveling paraxially at r0 = 0 due to a single-walled (11,9) carbon nanotube ˚ alone (     ), and surrounded by water with of radius a = 6.89 A ˚ , calculated using a boundary separation Dr = jb  aj = 0 and 3.4 A dielectric function with a sharp cut-off (—–), no cut-off (– –) and a smoothed cut-off ( ).

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Fig. 2. The self-energy in eV, plotted versus speed v in a.u. of an ion traveling paraxially at r0 = 0 due to a single-walled (11,9) carbon nanotube ˚ alone (     ), and surrounded by water with of radius a = 6.89 A ˚ , calculated using a boundary separation Dr = jb  aj = 0 and 3.4 A dielectric function with a sharp cut-off (—–), no cut-off (– –) and a smoothed cut-off ( ).

and their positions depend on the separation Dr between the NT and the water boundary. Given the strong dependence of our results on the parameters D and Dr, it is desirable to physically motivate the values to be used in modeling the effects of water on the NT’s dielectric response. ˚ and self-energy Fig. 3 shows the stopping force in eV/A in eV, versus speed v in a.u. for an ion traveling paraxially ˚ , and 5 A ˚ , for a dielectric boundary separation at r0 = 0, 3 A ˚ , calculated using a smooth cut-off of the ionof Dr  3.4 A ization energy at EC = 7 eV. Fig. 4 shows the stopping ˚ and self-energy in eV, versus radial distance force in eV/A ˚ r0 in A for an ion traveling paraxially with speed v = 0, 3 a.u., 5 a.u., 7 a.u. and 9 a.u., for a dielectric boundary ˚ , calculated using a smooth cutseparation of Dr  3.4 A off of the ionization energy at EC = 7 eV. These figures confirm that the above conclusions on the effects of water on stopping and image force are also true when the ion gets close to the NT wall. As shown in Figs. 1–4, the main effect of the surrounding water is to reduce the magnitude of both the stopping

˚ and self-energy in eV, plotted versus Fig. 3. The stopping force in eV/A ˚ , and 5 A ˚ , due speed v in a.u. for an ion traveling paraxially at r0 = 0, 3 A ˚ alone to a single-walled (11,9) carbon nanotube of radius a = 6.89 A ˚ ( (     ), and in water with boundary at b = 10.29 A ).

force and self-energy for ions channeled at high speeds. This can be explained by considering the resonance condition, x = kv, for electronic excitations in the system by the ion moving paraxially at speed v, as follows. At low frequencies (say, x  5 eV), the dielectric function of water may be approximated as w  1.9, so that the surrounding water ‘‘screens’’ the electrostatic interaction between the nanotube electrons. This suppresses the nanotube’s plasmon frequency at long wavelengths, thus reducing the high-speed magnitudes of the stopping force and selfenergy. On the other hand, at high frequencies (say, x > 17 eV), the real part of the dielectric function of water tends to one, so that, at shorter wavelengths, the nanotube’s plasmon frequency is unchanged by the presence of water. This leaves both the stopping force and selfenergy for ions channeled at low speeds largely unaffected by the surrounding water. In future work, we plan to refine and generalize our modeling of the dielectric properties of water by using the proper Poisson–Boltzmann theory for the formation of a double electric layer in an electrolyte in the vicinity of the NT’s wall.

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˚ and self-energy in eV, plotted versus Fig. 4. The stopping force in eV/A ˚ for an ion traveling paraxially with speed v = 0, 3 a.u., position r0 in A 5 a.u., 7 a.u. and 9 a.u., due to a single-walled (11,9) carbon nanotube of ˚ alone (     ), and in water with boundary at radius a = 6.89 A ˚ ( b = 10.29 A ).

Acknowledgements This work was supported by the Natural Sciences and Engineering Research Council of Canada. ZLM also acknowledges support by the Premier’s Research Excellence Award. References [1] Y. Bushuev, S. Davletbaeva, F.F. Muguet, Sensors 5 (2005) 139. [2] F. Moulin, M. Devel, S. Picaud, Phys. Rev. B 71 (2005) 165401. [3] K. Moloni, M.R. Buss, R.P. Andres, Ultramicroscopy 80 (1999) 237.

171

[4] Z. Zahab, L. Spina, P. Poncharal, C. Marlie`re, Phys. Rev. B 62 (2000) 10000. [5] W. Kim, A. Javey, O. Vermesh, Q. Wang, Y. Li, H. Dai, Nano Lett. 3 (2003) 193. [6] D.M. Guldi, G.M.A. Rahman, N. Jux, D. Balbinot, U. Hartnagel, N. Tagmatarchis, M. Prato, J. Am. Chem. Soc. 127 (2005) 9830. [7] D.N. Futaba, K. Hata, T. Yamada, K. Mizuno, M. Yumura, S. Iijima, Phys. Rev. Lett. 95 (2005) 056104. [8] S. Andreev, D. Reichman, G. Hummer, J. Chem. Phys. 123 (2005) 194502. [9] L. Zhu, Y. Xiu, D.W. Hess, C.-P. Wong, Nano Lett. 6 (2005) 2641. [10] C. Yannouleas, E.N. Bogachek, U. Landman, Phys. Rev. B 53 (1996) 10225. [11] X. Liu, J.L. Spencer, A.B. Kaiser, W.M. Arnol, Curr. Appl. Phys. 4 (2004) 125. [12] H. Hayashi, N. Watanabe, Y. Udagawa, C.-C. Kao, Proc. Natl. Acad. Sci. USA 97 (2000) 6264. [13] D. Emfietzoglou, H. Nikjoo, Radiat. Res. 163 (2005) 98. [14] D. Emfietzoglou, F.A. Cucinotta, H. Nikjoo, Radiat. Res. 164 (2005) 202. [15] Y.-N. Wang, Z.L. Misˇkovic´, Phys. Rev. A 69 (2004) 022901. [16] S. Lundqvist, N.H. March (Eds.), Theory of the Inhomogeneous Electron Gas, Plenum Press, New York, 1983. [17] R.G. Parr, W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989. [18] T.P. Doerr, Y.-K. Yu, Am. J. Phys. 72 (2004) 190. [19] J.D. Jackson, Classical Electrodynamics, third ed., John Wiley & Sons, Inc., New York, 1999. [20] D.J. Mowbray, Z.L. Misˇkovic´, F.O. Goodman, Phys. Rev. B 74 (2006) 195435. [21] D.J. Mowbray, Z.L. Misˇkovic´, F.O. Goodman, Y.-N. Wang, Phys. Lett. A 329 (2004) 94. [22] D.J. Mowbray, Z.L. Misˇkovic´, F.O. Goodman, Y.-N. Wang, Phys. Rev. B 70 (2004) 195418. [23] D.J. Mowbray, S. Chung, Z.L. Misˇkovic´, F.O. Goodman, Y.-N. Wang, Nucl. Instr. and Meth. B 230 (2005) 142. [24] D.J. Mowbray, S. Chung, Z.L. Misˇkovic´, Carbon Nanotubes, in: V. Popov, P. Lambin (Eds.), Springer, Dordrecht, 2006, p. 177. [25] M. Dingfelder, D. Hantke, M. Inokuti, H.G. Partzke, Radiat. Phys. Chem. 53 (1999) 1. [26] C.T. Chantler, K. Olsen, R.A. Dragoset, A.R. Kishore, S.A. Kotochigova, D.S. Zucker, X-Ray Form Factor, Attenuation and Scattering Tables, National Institute of Standards and Technology, Gaithersburg, MD, 2005. [27] B.E. Granger, P. Kral, H.R. Sadeghpour, M. Shapiro, Phys. Rev. Lett. 89 (2002) 135506; M. Zamkov, N. Woody, B. Shan, H.S. Chakraborty, Z. Chang, U. Thumm, P. Richard, Phys. Rev. Lett. 93 (2004) 156803.