Solutions for the electric potential and field distribution in cylindrical core-shell nanoparticles using the image charge method

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1 Solutions for the electric potential and field distribution in cylindrical core-shell nanoparticles using the image charge method Nader Daneshfar, and Nasrin Moradbeigi Citation: AIP Advances 5, 1714 (15); View online: View Table of Contents: Published by the American Institute of Physics Articles you may be interested in Orientation of core-shell nanoparticles in an electric field Applied Physics Letters 91, (7); 1.163/ Robust luminescence of the silicon-vacancy center in diamond at high temperatures AIP Advances 5, (15); 1.163/

2 AIP ADVANCES 5, 1714 (15) Solutions for the electric potential and field distribution in cylindrical core-shell nanoparticles using the image charge method Nader Daneshfar a and Nasrin Moradbeigi Department of Physics, Razi University, Kermanshah, Iran (Received 6 July 15; accepted 7 December 15; published online 15 December 15) This article considers the problem of finding the electrostatic potential that is given in terms of a scalar function called Green function in dielectric cylindrical nanoparticles with core-shell structure using the image charge method. By using this method that allows us to solve differential form of electric potential problem by the Green function, we investigate the distribution of the electric field in the configuration of a cylindrical nanoparticle surrounded by a continuum dielectric medium. By utilizing this well-known method, we obtain exact analytical formulas for the electrostatic potential and the electric field inside the shell, core and surrounding space of nanoparticle that can be applied to analysis of electromagnetic problems, electrostatic interactions in biomolecular simulations and also computer simulations of condensed-matter media. C 15 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3. Unported License. [ I. INTRODUCTION The electrostatics play an important and dominant role in many physical/chemical problems, biological and nanoscale systems. 1 One of the basic problems in electrostatic is to obtain the electrostatic potential and the electric field. A simple and direct calculation of the electrostatic potential and therefore the electric field is called image charge method or method of image charge (IC).,3 IC method is an approach for finding and describing the electric potential and field distributions in structures with special geometries, without specifically solving a differential equation. 4 It is an analytical method for solving some electrostatic problems (specific types of boundary value problems in electrostatics) which can be used for calculating the electric field. 5 Method of IC can be also applied to study different problems such as analysis of electron or ion trajectories, field-induced diffusion in scanning tunneling microscope experiments and field-emission diodes. 6 In particular, image charges play an important role in charge transport through molecules and single-molecule junctions, the electrostatic interactions in computer simulations of biomolecules, the surface tension of electrolyte solutions and the adsorption of polyelectrolytes. 7 9 So far, only a few of researches were investigated the method of image charges to study of cylindrical structures. For example, an analytical expression for the electric potential produced by a point charge located inside and outside of a cylindrical pore studied by Cui. 1 Also, Xu et al. studied reaction fields due to a point charge in a hybrid ion-channel model using the IC method. 11 In this work, the IC method is developed to cylindrical core-shell nanostructures for finding the electrostatic potential, and therefore the electric field in dielectric media. Since the fundamental problem in electrostatics is to find the electric field that plays a significant role in design of electromagnetic devices. It is important to note that a multilayer dielectric structure often play a determinant role for their physical-chemical properties. For example, Qin et al. used the IC Method for a three-dielectric-layer hybrid solvation model of biomolecules. 1 In the present paper, we will obtain the electric potential and field a Electronic address: ndaneshfar@gmail.com, ndaneshfar@razi.ac.ir /15/5(1)/1714/16 5, Author(s) 15

3 1714- N. Daneshfar and N. Moradbeigi AIP Advances 5, 1714 (15) distribution at different regions of a dielectric core-shell nanoparticle with cylindrical geometry by using the IC method. II. THEORY AND FORMULATION Let us apply method of IC to a cylindrical core-shell nanoparticle with a dissimilar dielectric constant than the surrounding medium. Consider the configuration shown in Fig. 1 that a dielectric core-shell nanoparticle in the cylindrical geometry with radii a and b (b > a), located in a medium (a continuum dielectric) with permittivity ε 3. Three media are homogeneous and isotropic. In our model, the cylindrical core-shell structure is infinitely long. This means that the cylindrical nanoparticle length has to be assumed much larger than its radius. Note also that nanotubes are cylinders with lengths typically on the order of microns and diameters of about a nanometer. A point charge q placed inside the shell at the position ŕ = ( ρ, ϕ, ź) from the center while r = (ρ, ϕ, z) is the observation point. The problem is to find the electrostatic potential and the electric field at any point in three regions ρ < a, a < ρ < b and ρ > b, because the IC method is very useful tool to calculate the electrostatic potentials and the electric fields in many problems with special symmetry. A charged particle in a medium induces an image charge. In a core-shell system, the presence of a second interface increases the complexity of the problem than a particle with radius R. So, it is important to calculate the electrostatic potential produced by the charged particle and the image charge. It should be noted that the physical idea behind this method is the solution for the potential. The method of image charges has a more important application in potential boundary value problems in electrostatics, magnetostatic and elsewhere. The most general formulation for the electrostatic potential is given in terms of a scalar function called the Green function which is closely related to the potential due to a charge and its image. The method of images can be used for finding the electrostatic potential and the electric field due to a point charge inside a cylindrical shell. In fact, a charged particle in one medium induces an image charge in the other and therefore the electrostatic potential in the physical region is the sum of potentials of the point charge and its image. However, the electrostatic potential is given by the solution to Poisson s equation. 13 Thus, mathematically, the starting point is the equation for the Green function as introduced by Jackson 14 FIG. 1. Model for a core-shell nanocylinder with the inner and outer radii a and b immersed in a medium with the dielectric constant ε 3, while the dielectric constants of core and shell are and ε, respectively. Also, ŕ is the position of a point charge, q.

4 N. Daneshfar and N. Moradbeigi AIP Advances 5, 1714 (15) G(r,r ) = 4πδ(r r ). (1) From a physical point of view, a Green function produces a response at an observation point r due to a unit point source at r. To finding the Green function for potential problems in cylindrical geometry, we can write the delta function, δ(r r ), in cylindrical coordinates. Thus, we obtain G(r,r ) = 4π ρ δ(ρ ρ )δ(ϕ ϕ )δ(z z ), () Here, owing to the cylindrical geometry of nanoparticle, the Green function can be expressed in terms of orthonormal functions 14 : G(r,r ) = 1 dke im(φ φ ) cos[k(z z )]g π m (ρ, ρ ), (3) m= It should be noted that the φ and z delta functions has been written in terms of orthonormal functions. Substituting Eq. (3) into Eq. (1) and using the Laplacian in cylindrical coordinates, we obtain an equation for the radial Green function 1 ρ ρ (ρ g m ρ ) ( m ρ + k )g m = 4π ρ δ(ρ ρ ). (4) For ρ ρ, the radial Green function converted to a homogeneous radial equation that the solutions of this equation are called modified Bessel functions. 14 g m (k, ρ, ρ ) = A 1 I m (k ρ < ) + B 1 K m (k ρ < ) (ρ < ρ ) A I m (k ρ > ) + B K m (k ρ > ) (ρ > ρ ), (5) where the coefficients A 1, A, B 1 and B are functions of ρ to be determined by the proper boundary conditions and must in fact be linear combinations of modified Bessel functions K m (ρ ) and I m (ρ ). Also, ρ < (ρ > ) is the smaller (larger) of ρ and ρ. However, after some algebraic calculations and using the boundary conditions which are derived from the full set of Maxwell equations, we can obtain the electric potential (Green function) inside the core, shell and the environment medium in terms of real functions. Without going into the full details (owing to lengthy mathematical algebraic calculations), the Green function in all regions can be obtained. Therefore, the Green function inside the core due to the image charge which is given by G core (r,r ) = 4 π 1 1 M Σ (k) I (k ρ )K (k ρ) + dk cos[k(z z )] Σ Σ m(k)i m (k ρ )K m (k ρ) cos[m(φ φ )]. (6) In the shell, a < ρ < b, the Green function is sum of contributions of the actual and image charge G shell (r,r ) = G 1 (r,r ) + G (r,r ), (7) where G 1 (r,r ) is the potential produced by the actual charge and has the form G 1 (r,r ) = 4 πε 1 1 M q 1 I (k ρ > )K (k ρ < ) + dk cos[k(z z )] I m (k ρ > )K m (k ρ < ) cos[m(φ φ )], (8) and the potential produced by the image charge has the form G (r,r ) = 4 πε 1 1 M Θ (k) dk cos[k(z z )] K (ka) I (ka) I (k ρ > )I (k ρ < )

5 N. Daneshfar and N. Moradbeigi AIP Advances 5, 1714 (15) Θ (k) Θ m (k) K m(ka) I m (ka) I m(k ρ > )I m (k ρ < ) cos[m(φ φ )] I (kb) K (kb) K (k ρ < )K (k ρ > ) Θ m (k) I m(kb) K m (kb) K m(k ρ < )K m (k ρ > ) cos[m(φ φ )] + Θ (k) K (ka) I (kb) I (ka) K (kb) I (k ρ < )K (k ρ > ) + MΘ m (k)i m (k ρ < )K m (k ρ > ) cos[m(φ φ )]. (9) In the outside space of nanoparticle, the Green function that is finite at infinity can be written G out (r,r ) = 4 πε M Λ (k) I (k ρ)k (k ρ ) + dk cos[k(z z )] Λ m (k)i m (k ρ)k m (k ρ ) cos[m(φ φ )] (1) where M = I m(kb)k m (ka) I m (ka)k m (kb), while I m and K m are the modified Bessel functions of the first and second kind. Note that the Green s function, G(r,r ) = 1 r r, is the potential due to a charge located at r. The IC method can be exploited very conveniently for this system. We can obtain the electric potential inside the core (ρ < a), inside the shell (a < ρ < b) and outside the nanoparticle (ρ > b). It should be noted that the potential due to a point charge can be found as a sum of two potentials: i) due to the charge itself, ii) due to an image charge. On this basis, we can determine the electric potential in each region using φ = q ε r r. However, by applying the continuity of the electric potential G core (r,r ) ρ=a = G shell (r,r ) ρ=a and G shell (r,r ) ρ=b = G out (r,r ) ρ=b and the electric displacement continuity at each interface, Gcore (r,r ) ρ ρ=a = Gshell (r,r ) ρ ρ=a and Gshell (r,r ) ρ ρ=b = Gout (r,r ) ρ ρ=b yields the total electric potential in each region. A. Derivation of the electric potential 1. The electric potential in the region ρ < a The electrostatic potential inside the core due to the image charge is φ 1 (r,r ) = 4 π 1 1 M Σ (k) I (k ρ )K (k ρ) + dk cos[k(z z )] Σ m (k)i m (k ρ )K m (k ρ) cos[m(φ φ )]. (11). The electric potential in the region a < ρ < b The electrostatic potential inside the shell due to the image and actual charge is φ (r,r ) = 4 πε 1 1 M dk cos[k(z z )] q 1 I (k ρ > )K (k ρ < ) + q Θ (k) I m (k ρ > )K m (k ρ < ) cos[m(φ φ )] K (ka) I (ka) I (k ρ > )I (k ρ < )

6 N. Daneshfar and N. Moradbeigi AIP Advances 5, 1714 (15) Θ (k) Θ m (k) K m(ka) I m (ka) I m(k ρ > )I m (k ρ < ) cos[m(φ φ )] I (kb) K (kb) K (k ρ < )K (k ρ > ) Θ m (k) I m(kb) K m (kb) K m(k ρ < )K m (k ρ > ) cos[m(φ φ )] + Θ (k) K (ka) I (kb) I (ka) K (kb) I (k ρ < )K (k ρ > ) + MΘ m (k)i m (k ρ < )K m (k ρ > ) cos[m(φ φ )], (1) where the first and second terms are for the actual charge in the shell while other terms emerges from the image charge. 3. The electric potential in the region ρ > b The electrostatic potential in the surrounding space is finite at infinity and becomes φ 3 (r,r ) = 4 πε M Λ (k) I (k ρ)k (k ρ ) + dk cos[k(z z )] Λ m (k)i m (k ρ)k m (k ρ ) cos[m(φ φ )], (13) where Σ m (k), Θ m (k) and Λ m (k) are unknown coefficients and determined by the boundary conditions. The electrostatic potentials given by Eqs. (11)-(13) will be determined by applying the proper boundary conditions. On the other hand, the electrostatic potential must satisfy the boundary conditions. Thus, by utilizing the continuity of the electric potential and the electric displacement at the boundary ρ = a, we find: and q Θ m(k) = Σ m(k), m =,1,,... (14) ε ε Σ m (k) = q K m(ka)i m(ka) I m (ka)k m(ka) Θ m (k) g m (k) I m(kb)k m (k ρ ) K m (kb)i m (k ρ ) f m (k) Θ m (k) + I m(kb)k m (ka)k m (k ρ )I m(ka) K m (kb)i m (ka)i m (k ρ )K m(ka) c m (k) Θ m (k), (15) where I m and K m denotes the derivative of the modified Bessel functions with respect to the argument. Thus, Σ m (k) and Θ m (k) are obtained by using Eqs. (14) and (15) 1 g m f m + c m Σ m (k) = ε 1 g ε m ε f ε m 1 ε + c q, (16) m 1 Θ m (k) = 1 ε ε 1 g ε m ε f ε m 1 ε + c q. (17) m 1 Also, by applying the continuity of the electric potential and the electric displacement at the boundary ρ = b, we find:

7 N. Daneshfar and N. Moradbeigi AIP Advances 5, 1714 (15) and Λ m (k) = q K m(ka)i m (k ρ ) I m (ka)k m (k ρ Θ m (k) ) a m (k) I m(kb)k m(kb) I m(kb)k m (kb) b m (k) q Θ m(k) = Λ m(k), (18) ε ε ε 3 Θ m (k) + I m(kb)k m (ka)k m(kb)i m (k ρ ) K m (kb)i m (ka)i m(kb)k m (k ρ ) d m (k) Using Eqs. (18) and (19), we can obtain the following relation for Λ m (k) Θ m (k). (19) 1 a m b m + d m Λ m (k) = ε 1 a ε m ε b ε m 3 ε + d q. () m 3 ε 3 B. Derivation of the electric field Here, we give a brief layout of the derivation of electric field at different regions. An electrostatic field can be described by a scalar potential. The electrostatic field is then given by the negative gradient of the potential as E = φ, so that we can find the electric field in each region of a core-shell cylindrical system. 1. The electric field for ρ < a For ρ < a E 1 (r) = 4 1 ( E1 (ρ)ρ + E 1 (φ)φ + E 1 (z)z ), (1) π 1 M where the components of electric field are E 1 (ρ) = dk cos[k(z z Σ (k) )] I (k ρ )kk (k ρ) + Σ m (k)i m (k ρ )kk m(k ρ) cos[m(φ φ )], () E 1 (φ) = 1 dk cos[k(z z )] ρ Σ m (k)i m (k ρ )K m (k ρ)( m) sin[m(φ φ )], (3) E 1 (z) = + dk( k) sin[k(z z Σ (k) )] I (k ρ )K (k ρ) Σ m (k)i m (k ρ )K m (k ρ) cos[m(φ φ )]. (4) To calculate the electric field at the region a < ρ < b, we consider two regions a < ρ < ρ and ρ < ρ < b.. The electric field for a < ρ < ρ The electric field in the region a < ρ < ρ is E (r) = 4 1 ( E (ρ)ρ + E (φ)φ + E (z)z ), (5) πε 1 M where the components of electric field for this case can be written

8 N. Daneshfar and N. Moradbeigi AIP Advances 5, 1714 (15) E (ρ) = + q dk cos[k(z z )] q 1 I (k ρ )kk (k ρ) I m (k ρ )kk m(k ρ) cos[m(φ φ )] Θ (k) K (ka) I (ka) I (k ρ )ki (k ρ) Θ m (k) K m(ka) I m (ka) I m(k ρ )ki m(k ρ) cos[m(φ φ )] Θ (k) I (kb) K (kb) K (k ρ )kk (k ρ) Θ m (k) I m(kb) K m (kb) K m(k ρ )kk m(k ρ) cos[m(φ φ )] + Θ (k) K (ka) I (kb) I (ka) K (kb) ki (k ρ)k (k ρ ) + Θ m (k) K m(ka) I m (kb) I m (ka) K m (kb) ki m(k ρ)k m (k ρ ) cos[m(φ φ )], (6) E (φ) = 1 dk cos[k(z z )] ρ q I m (k ρ )K m (k ρ)( m) sin[m(φ φ )] + Θ m (k) K m(ka) I m (ka) I m(k ρ )I m (k ρ)m sin[m(φ φ )] + Θ m (k) I m(kb) K m (kb) K m(k ρ )K m (k ρ)m sin[m(φ φ )] Θ m (k) K m(ka) I m (kb) I m (ka) K m (kb) I m(k ρ)k m (k ρ )m sin[m(φ φ )], (7) E (z) = dk( k) sin[k(z z )] qi (k ρ )K (k ρ) + q I m (k ρ )K m (k ρ) cos[m(φ φ )] Θ (k) K (ka) I (ka) I (k ρ )I (k ρ) Θ m (k) K m(ka) I m (ka) I m(k ρ )I m (k ρ) cos[m(φ φ )] Θ (k) I (kb) K (kb) K (k ρ )K (k ρ) Θ m (k) I m(kb) K m (kb) K m(k ρ )K m (k ρ) cos[m(φ φ )] + Θ (k) + K (ka) I (ka) Θ m (k) K m(ka) I m (ka) 3. The electric field for ρ < ρ < b The electric field in the region ρ < ρ < b is I (kb) K (kb) I (k ρ)k (k ρ ) I m (kb) K m (kb) I m(k ρ)k m (k ρ ) cos[m(φ φ )]. (8) E 3 (r) = 4 1 ( E3 (ρ)ρ + E 3 (φ)φ + E 3 (z)z ), (9) πε 1 M

9 N. Daneshfar and N. Moradbeigi AIP Advances 5, 1714 (15) E 3 (ρ) = + q dk cos[k(z z )] q 1 ki (k ρ)k (k ρ ) ki m(k ρ)k m (k ρ ) cos[m(φ φ )] Θ (k) K (ka) I (ka) ki (k ρ)i (k ρ ) Θ m (k) K m(ka) I m (ka) I m(k ρ)ki m (k ρ ) cos[m(φ φ )] Θ (k) I (kb) K (kb) kk (k ρ)k (k ρ ) k Θ m (k) I m(kb) K m (kb) K m(k ρ)k m (k ρ ) cos[m(φ φ )] + Θ (k) K (ka) I (kb) I (ka) K (kb) I (k ρ )kk (k ρ) + k Θ m (k) K m(ka) I m (kb) I m (ka) K m (kb) I m(k ρ )K m(k ρ) cos[m(φ φ )], (3) E 3 (φ) = 1 dk cos[k(z z )] ρ q I m (k ρ)k m (k ρ )m sin[m(φ φ )] + Θ m (k) K m(ka) I m (ka) I m(k ρ)i m (k ρ )m sin[m(φ φ )] + Θ m (k) I m(kb) K m (kb) K m(k ρ)k m (k ρ )m sin[m(φ φ )] Θ m (k) K m(ka) I m (kb) I m (ka) K m (kb) I m(k ρ )K m (k ρ)m sin[m(φ φ )], (31) E 3 (z) = dk sin[k(z z )]k qi (k ρ)k (k ρ ) + q I m (k ρ)k m (k ρ ) cos[m(φ φ )] Θ (k) K (ka) I (ka) I (k ρ)i (k ρ ) Θ m (k) K m(ka) I m (ka) I m(k ρ)i m (k ρ ) cos[m(φ φ )] Θ (k) I (kb) K (kb) K (k ρ)k (k ρ ) Θ m (k) I m(kb) K m (kb) K m(k ρ)k m (k ρ ) cos[m(φ φ )] + Θ (k) + 4. The electric field for ρ > b K (ka) I (ka) Θ m (k) K m(ka) I m (ka) The electric field in the region ρ > b is I (kb) K (kb) I (k ρ )K (k ρ) I m (kb) K m (kb) I m(k ρ )K m (k ρ) cos[m(φ φ )]. (3) E 4 (r) = 4 1 ( E4 (ρ)ρ + E 4 (φ)φ + E 4 (z)z ), (33) πε 3 1 M

10 N. Daneshfar and N. Moradbeigi AIP Advances 5, 1714 (15) where the components of electric field for this case can be written E 4 (ρ) = dk cos[k(z z )] Λ (k) ki (k ρ)k (k ρ ) + Λ m (k)ki m(k ρ)k m (k ρ ) cos[m(φ φ )], (34) E 4 (φ) = 1 dk cos[k(z z )] ρ Λ m (k)i m (k ρ)k m (k ρ )( m) sin[m(φ φ )], (35) E 4 (z) = dk( k) sin[k(z z )] Λ (k) I (k ρ)k (k ρ ) + Λ m (k)i m (k ρ)k m (k ρ ) cos[m(φ φ )]. (36) III. NUMERICAL RESULTS AND DISCUSSION In this section, we first discuss the limiting behavior of Σ m (k) and Θ m (k) as k using the asymptotic expression for the modified Bessel functions because the behavior of potentials is closely related to them. 15 In fact, to gain insight into the summation in the electrostatic potential equations, we will investigate the behavior of intermediate functions because the electrostatic potential depends on the intermediate functions that into the series expansions. In other words, the fields in each dielectric region are expressed in terms of these functions. Σ m (k) lim = lim k q k = Θ m (k) lim = lim k q k = 1 g m (k) f m (k) + c m (k) 1 g (k) ε ε f m (k) ε 1 ε + c m (k) ε 1 ( b ρ ) m 1 + ε ( b ρ ) m ( ε ε ). (37) 1 1 ε 1 g m (k) ε f m (k) ε + c m (k) ε 1 ε 1 + ε ( b ρ ) m ( ε ). (38) When b, the relations in Ref. 1 for a cylindrical pore can be obtained: Σ m (k) lim = k q Θ m (k) lim k q 1 + ε. (39) = 1 ε 1 + ε. (4) where is in agreement with obtained result in Ref. 1. For m =, the limiting behavior of Σ (k), Θ (k) and Λ (k) as k tends to zero (or infinity) can be written Σ (k) lim = lim k q k Θ (k) lim = lim k q k 1 g (k) f (k) + c (k) 1 g (k) ε ε f (k) ε 1 ε + c (k) ε 1 = 1, (41) 1 ε 1 g (k) ε f (k) ε + c (k) ε = 1 ε, (4)

11 N. Daneshfar and N. Moradbeigi AIP Advances 5, 1714 (15) Λ (k) 1 a b + d lim = lim ε k q k 1 a ε ε b ε 3 ε + d =. (43) 3 ε 3 Figures -4 show the variation of the functions Σ m (k), Θ m (k) and Λ m (k) versus variable kb by varying the dielectric constant of core, shell and the surrounding medium. The numerical results are calculated for m =,1,,3,4. The position of charge is b 3 while b = a. Since molecules are very small and typically just several nanometers in size, we consider a core-shell nanoparticle with a = 5 nm and b = 1 nm. Noted that nanomaterials have sizes ranging from about one nanometer up to several hundred nanometers, comparable to many biological molecules such as antibodies, proteins FIG.. The k dependence of function Σ m (k) by varying the dielectric constant of core and shell for m=, 1,, 3, and 4.The dielectric constants of core and shell are =.5 and 3, ε = 1.5 and.5, respectively.

12 N. Daneshfar and N. Moradbeigi AIP Advances 5, 1714 (15) FIG. 3. The k dependence of function Θ m (k) by varying the dielectric constant of core and shell for m=, 1,, 3, and 4. The dielectric constants of core and shell are =.5 and 3, ε = 1.5 and.5, respectively. and etc. 16 It can be seen that most of the variation and shift of Σ m (k) and Θ m (k) occurs for small m values and there is an overlap when m increases (Figs and 3), while this behavior is different for Λ m (k) (Fig. 4). Also, Σ m (k) and Θ m (k) are affected by the dielectric constants of core and shell, in which Λ m (k) is affected by the dielectric constants of shell and environment medium. The results clearly show the influence of the dielectric on the electric potential. Figures 5-7 explains the electric field behavior in the core, shell and the surrounding dielectric of particle because electric fields can create much stronger forces on particles in solution than other forces. In Fig. 5 we show the electric field distribution in each region of nanoparticle as a function

13 N. Daneshfar and N. Moradbeigi AIP Advances 5, 1714 (15) FIG. 4. The k dependence of function Λ m (k) by varying the dielectric core and shell for m=, 1,, 3, and 4. The dielectric constants of shell and embedding medium are ε =.5 and 3, ε 3 = 1.5 and.5, respectively. of ρ for a source charge at (ρ = b 3, φ =, z = ) with =, ε = 1.5 and ε 3 = 1 in disconnected graphs. The electric filed at the core increases as ρ increases while the electric field in the outside of nanoparticle decreases as shown in Figs. 5(a) and 5(d), respectively. On the other hand, there is a monotonic increase and a uniform decrease in the behavior of the electric field in the core and the surrounding medium of nanopartice. The electric field is singular and increases rapidly as the field position approaches the source charge as shown in Fig. 5(b). Furthermore, the electric field decreases rapidly and thus the electric field becomes weaker as the field position recedes the source charge as shown in Figs. 5(c). As a result, the field is stronger in the shell than in the core and surrounding medium. Also, in Fig. 6 we show a comparison of the electric field at all regions summing over the

14 N. Daneshfar and N. Moradbeigi AIP Advances 5, 1714 (15) FIG. 5. The electric filed as a function of ρ for a source charge at (ρ = b 3, φ =, z = ) with =, ε = 1.5 and ε 3 = 1. (a) ρ < a; (b) a < ρ < ρ ; (c) ρ < ρ < b; (d) ρ > b. first 4, 6, 8, and 1 terms for a source charge at (ρ = b 3, φ =, z = ) along the line (φ =,z = ) when =, ε = 1.5 and ε 3 = 1. Figure 6(a) clearly shows that the difference between the first 8 and 1 terms is small than other parts of this figure. It is important to note that fast convergence can be achieved with only a few terms and good numerical accuracy is obtained. 1 In Fig. 7(a) we have shown the normalized electric field in all regions in a graph over the domain for ρ on [,b] for

15 N. Daneshfar and N. Moradbeigi AIP Advances 5, 1714 (15) FIG. 6. The electric filed as a function of ρ for a source charge at (ρ = b 3, φ =, z = ) with ε = 1.5 and ε 3 = 1 with summing over the first 4, 6, 8, and 1 terms. comparison as shown in Fig. 5 in disconnected plots. In addition, Fig. 7(b) shows the electric field from b = nm to 8b = 8 nm where b is the radius of the outer cylinder or shell. The electric field decreases when ρ increases from 1 to 8 nm. It is clear from the behavior of the radial part of the electric field that the filed drops off to asymptotically large b. The electric field therefore decreases

16 N. Daneshfar and N. Moradbeigi AIP Advances 5, 1714 (15) FIG. 7. The electric filed as a function of ρ for a source charge at (ρ = b 3, φ =, z = ) along the line (φ =,z = ) with =, ε = 1.5 and ε 3 = 1. outside the cylinder at the region away from the electric charge when the radius of shell increases. In other words, the electric field is reduced with increasing ρ. IV. CONCLUDING REMARKS In concluding, we have developed the IC method to study the electrostatic potential and the electric field for cylindrical nanoparticles with core-shell geometry. The key and important findings contains analytical expressions for the electrostatic fields and potentials at different regions of the system. Also, we have investigated the behavior of the electric filed at each region, because the calculation of the electric field is important and needed in a field-emission device. It was found that the electric field decreases with increasing distance from the source charge. Finally, we believe the IC method can be applied to study many different problems and these results can be useful for biomolecule simulations and biomedical science. ACKNOWLEDGEMENTS We would like to thank Dr M. Tabrizi for fruitful discussions. 1 B. Honig and A. Nicholls, Science 68, 1144 (1995). D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice-Hall, Inc., New Jersey, 1999). 3 H. J. W. Müller-Kirsten, Electrodynamics: An introduction including quantum effects (World Scientific Publishing, Singapore, 4). 4 R. Messina, J. Chem. Phys. 117, 116 (). 5 H. Cheng and L. Greengard, SIAM J. Appl. Math. 58, 1 (1998). 6 G. Mesa, E. Dobado-Fuentes, and J. J. Saenz, J. Appl. Phys. 79, 39 (1996). 7 R. Wang and Z-G. Wang, J. Chem. Phys. 139, 147 (13).

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