Three-Layer Dielectric Models with Spherical Cavities ... - Science Direct

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Procedia Engineering 31 (2012) 1033 – 1038

International Conference on Advances in Computational Modeling and Simulation

Three-Layer Dielectric Models with Spherical Cavities for Reaction Potential Calculations Zhenli Xua* a

Department of Mathematics, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

Abstract In multi-scale modeling of electrostatic interactions, the atomic and the macroscopic regions are usually separated by a dielectric boundary, leading to a sharp dielectric jump. This two-layer model is relatively easy to develop analytical methods to calculate the potential, for example, the spherical harmonic expansion and the image method can be used. In many problems, however, it is suggested to introduce a transition layer between the two dielectric regions. For this purpose, a new three-layer dielectric model is proposed in this paper. The basic principle to construct such a model is to construct a dielectric function such that its square root is the combination of two fundamental solutions of a Poisson-Boltzmann equation. The parameter in this equation can adjust the profile and thus obtain more physical transition of the dielectric permittivity. With this dielectric function, the Poisson equation can be solved analytically by generalizing the Kirkwood series expansion. The result is significant in many fields of physical and biological applications.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Kunming University of Science and Technology Open access under CC BY-NC-ND license. Keywords: Reaction field; Yukawa potential; Electrostatics; Three-layer dielectric models; Spherical harmonics

1. Introduction Continuum electrostatic models have been widely used in a lot of physical and biological fields. In the multi-scale modeling, a spherical geometry of dielectric cavity is often employed to separate the atomic and macroscopic regions as analytical algorithms based on the spherical harmonics expansion and the methods of images can be incorporated instead of numerical solutions of the Poisson equation. Typical

Corresponding author.Tel:+86-13611683459 * E-mail address: [email protected].

1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2012.01.1138

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Zhenli Xu / Procedia Engineering 31 (2012) 1033 – 1038

applications [1-3] include reaction-field models of biomolecular solvation, colloidal suspensions and semiconductor quantum dots. Traditionally, the step-like two-layer dielectric model is used due to the simplicity. This two-layer model treats the inside and outside of the spherical interface with two different dielectric constants, leading to a sharp jump at the boundary. The jump results in unphysical solutions for the reaction-field calculations when a source charge is close to the surface [4]. In colloidal suspensions, it is also known that there is a region near the charged colloidal surface with an intermediate dielectric permittivity due to the restricted motion of water molecules in the Stern layer, and therefore a three-layer model with a transition layer will be more physical. The development of analytical tools for such model is then significant because it can be applied in practical molecular dynamics or Monte Carlo simulations. Usually, the Poisson equation with a space-dependent dielectric function cannot be simply solved by analytical approaches. To tailor a continuous dielectric function for exact solutions of the Poisson equation, Qin et al. [5] proposed a three-dielectric-layer model by constructing a special dielectric permittivity, H (r ) , which connects a low dielectric H i and a higher one H o , through a harmonic interpolation technique to the square root of the permittivity; namely, the resultant H (r ) is harmonic. This technique makes a transformation from the governing equation with varying coefficients to a Poisson equation with constant coefficients, and thus leads to an analytical solution by using the series expansion. This approach is further extended to study self-polarization energies of semiconductor spherical quantum dots and construct analytical models for prolate/oblate geometries by Deng and his collaborator [6,7]. In the present paper, we present a new class of transition functions which also leads to the exact solvability of the Poisson equation, among which the quasiharmonic dielectric model of Qin et al. is a special case. We find if function H (r ) is a linear combination of the sinhc function, sinh(Or)r , and coshc function, cosh(Or)r , then the Poisson equation with such a dielectric function can also have an exact solution. The special case of O 0 gives the quasiharmonic model. Thus this paper provides more freedoms of degree to determine the shape of the dielectric profile with the use of the parameter O , possibly leading to a more physically significant dielectric model. The organization of this paper is as follows. In Section 2, we will give a brief review of three-layer models and then introduce the new model for the Poisson equation. In Section 3, the analytical solution of the model is presented. Example and summary are drawn in Section 4. 2. The dielectric model We study three-layer models under spherical coordinates. A spherical cavity : of radius a with dielectric H i . Outside a larger sphere ( r ! b ! a ), the domain is represented by a continuum medium with a dielectric constant H o . The transition layer between the two spheres is with thickness W b  a , and its dielectric permittivity is a radial-dependent continuous function. A source point charge qs is located at rs . In this setting, the electrostatic potential ) satisfies the following Poisson equation, (1) ’ ˜ H (r )’)(r) 4SqsG (r  rs ), where G is the Dirac delta function. On the two boundaries among the three layers, the continuities of the potentials and the normal displacements require that, )(r ) )(r ) , and w r )(r ) w r )(r ) for r a or b , (2) where r  and r  are, respectively, the inner and outer limits at position r . The solution can be uniquely determined by Eqs. (1) and (2) and the decay property of the potential at the far field. In Eq. (1), the dielectric function H (r ) takes H i for the inner sphere, and H o for the outer layer. Between the two layers, an analytical profile has to be assigned to connect these two values. For an arbitrary function, it is known that, when the charge is not present, a general solution of the potential in the intermediate layer can be represented by,

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Zhenli Xu / Procedia Engineering 31 (2012) 1033 – 1038 f

¦ ª«¬ A (r)r

)(r)

n

n

 Bn (r )

n 0

1 º P (cos] ) r n1 » ¼ n

,

(3)

where Pn is the Legendre polynomial, ] is the angle between r and rs , and the coefficients An and Bn are determined by solving two ordinary differential equations [8]. A random choice of the dielectric function is inefficient for computing the coefficients as numerical methods have to be used to solve these two equations, even though it is a very simple profile such as a linear or cosine-like profile. This could be solved by approximating H (r ) through piecewise constants. The solution of the Poisson equation with piecewise constant coefficients can be then obtained analytically or numerically for layered spheres [812]. An alternative is to use the quasiharmonic dielectric model [5] with H (r ) D  E / r 2 , where 1 and 1 / r are two harmonic functions, and the coefficients are determined by the continuity conditions at the interfaces. The harmonicity of H (r ) implies that the Poisson equation (1) can be transformed to a form of constant coefficient and thus the solution can be obtained using a spherical harmonics expansion [13]. In this paper, we consider a novel method to construct analytically solvable three-layer models. The principle is based on the following result [14]. Suppose function ) satisfies the Poisson equation (1). If the square root of H (r ) satisfies the following Poisson-Boltzmann (PB) equation, (4) ’2 H (r )  O2 H (r ) 0, for O ! 0 , then we have, (5)  H (r ) ’2  O2 H (r ))(r) 4SqsG (r  rs ). The PB equation includes two fundamental solutions, exp( rOr ) / r , where the negative exponent one is the screened Coulomb potential which is also called the Yukawa potential. We interpolate these two functions to obtain the dielectric permittivity of the transition layer as:

>



H (r ) §¨D ©

exp(Or ) r

E

@

exp(Or ) · ¸ r ¹

2

(6)

,

which satisfies the continuity conditions H (a) H i and H (b) H o , and thus the coefficients are given by, D

a H i exp(Oa)  b H o exp(Ob) exp(2Oa)  exp(2Ob)

, E

 exp(Oa  Ob)

a H i exp(Ob)  b H o exp(Oa) exp(2Oa)  exp(2Ob)

.

The advantage of using Eq. (6) in comparison with the quasiharmonic model is that parameter O can be varied, and thus the transition profile is adjustable, as is shown in Figure 1. When O tends to zero, Eq. (6) reduces to the quasiharmonic model. 3. Exact solution of the Poisson equation We solve the Poisson equation with the new dielectric function in three cases: the source charge in the interior, transition, and exterior layers, respectively. Case 1: Source charge in the interior layer. Let < H (r ))(r) , and suppose the point charge is located within the interior layer. By Eq. (5) in the intermediate layer, < satisfies the PB equation ’2  O2

)(r )

@

(7)

¦

where in (˜) and kn (˜) are the modified spherical Bessel functions [15] which are defined in terms of the Bessel function J v (˜), v

I v (r )

 1 J v (  1r ), K v (r )

in (r )

I 1 (r ), 2r n  2

S

k n (r )

S I v ( r )  I v ( r ) , 2 sin(vS ) S 2r

K n  1 (r ). 2

The unknown constant coefficients An(1) , Bn(1) , Cn(1) and Dn(1) can be determined by the boundary conditions (2), the orthogonality of Legendre polynomials, together with the expansion of reciprocal distance for the Coulomb potential [15], i.e., we have 1/ | r  rs |

f

¦

n 0

rsn r ( n 1) Pn (cos] ) for rs  r .

Let us define u(1) ( An(1) , Bn(1) , Cn(1) , Dn(1) )T . Then for n 0,1,", we have a system Mu (1) f (1) with §  Hi an ¨ ¨ ¨ 0 ¨ ¨  2nH 3 / 2 a n 1 i ¨ ¨ 0 ¨ ©

M

qs rsn

and f (1)

Hi an2

in (O a )

k n ( Oa )

in ( Ob )

k n (Ob )

2OH i in ' (Oa)  H ' (a)in (Oa ) 2OH oin ' (Ob)  H ' (b)in (Ob)

0 

2OH i k n ' (Oa)  H ' (a)k n (Oa ) 2OH o k n ' (Ob)  H ' (a)k n (Ob)

Ho b n1

0 2( n 1)H o3 / 2 b n 2

· ¸ ¸ ¸ ¸, ¸ ¸ ¸ ¸ ¹

(8)

a,0,2(n  1)H i ,0 T . The solution of the linear system is given in Eq. (12) in Appendix.

Case 2: Source charge in the transition layer. When the position of the source is in the transition layer, the general solution for the potential in the three layers reads: ­ f ( 2) n ° n 0 An r Pn (cos] ), 0 d r d a, ° f qs exp(O |r  rs |) °  1 Bn( 2)in (Or )  Cn( 2) k n (Or ) Pn (cos] ), a d r d b, ® |r  rs | n 0 (r ) ( r )H ( rs ) H H ° ° f D ( 2) r  ( n 1) Pn (cos] ), r t b, ° ¯ n 0 n

¦

)(r )

¦ >

@

(9)

¦

where the Yukawa potential can be expanded in terms of the Legendre polynomials [15], ­ f ( 4 n  2) O in (Ors )k n (Or ) Pn (cos] ), for rs d r , ° ° n 0 S ® f ( 4 n  2) O ° in (Or )k n (Ors ) Pn (cos] ), otherwise, ° ¯ n 0 S The coefficients u(2) ( An(2) , Bn(2) , Cn(2) , Dn(2) )T are determined by solving system Mu (2) coefficient matrix M is the same as expression (8) and, exp( O |r  rs |) |r  rs |

f ( 2)



¦ ¦

( 4n  2)Oq s

S H (rs )

kn (Ors )in (Oa),

f (2) , where, the

in (Ors )kn (Ob), kn (Ors )m32, in (Ors )m43 T .

The solution of the linear system is similarly given in Eq. (12) in Appendix. Case 3: Source charge in the outer layer. When rs ! b , the general solution for the potential in the three layers is expressed by,

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Zhenli Xu / Procedia Engineering 31 (2012) 1033 – 1038

­ f (3) n ° n 0 An r Pn (cos] ), 0 d r d a, ° f ° (10) Bn(3)in (Or )  Cn(3) k n (Or ) Pn (cos] ), a d r d b, )(r ) ® 1 n 0 (r ) H ° ° qs 1 f Dn(3) r  ( n 1) Pn (cos ] ), r t b, ° H |r  r |  n 0 s ¯ o where we need to use the expansion of reciprocal distance for r  rs to expand the Coulomb potential. Similarly, the coefficients u(3) ( An(3) , Bn(3) , Cn(3) , Dn(3) )T are determined by solving Mu (3) f (3) , where, f (3) qs bn 1H o1 / 2rs(n 1) 0, b,0,2nH o T . The solution of the linear system is again given in Eq. (12).

¦

¦ > ¦

@

4. Example and summary To illustrate the solution property of the proposed model, we consider a toy example. The interior spherical cavity is with radius a =1 ϳ, the interior and exterior dielectrics are H i 1 and H o 2 , the thickness of the transition layer takes 0.2 and 1 ϳ, and the parameter O takes 0.2, 0.5 and 0.8. We calculate the self-polarization energy Vs (r) of a unit point charge, say, qs 1e and rs r , with the energy defined by Vs (r) e)rf (r) / 2 , and, ­ f (1) n ° n 0 An r , 0 d r d a, ° f ° 1 Bn( 2)in (Or )  Cn( 2) k n (Or ) , a d r d b, ® n 0 (r ) H ° ° f D (3) r  ( n 1) Pn (cos] ), r t b. ° ¯ n 0 n

¦

) rf (r )

¦ >

@

(11)

¦

The results of the self-polarization energy are plotted in Figure 2, which demonstrates two ways to reduce the singularity. One way is to increase the size of the transition layer. The other way is to vary the value of parameter O . As is illustrated in Figure 2(b), the minimum value of the self-polarization energy changes from -347 to -208 kcal/mol when increasing O from 0.2 to 0.8. And thus we conclude the new model is available to improve the treatment method of the boundary singularity if picking a suitable O .

Figure 2. Self-polarization energy of a unit point charge as a function of r with (a) W

0.2 and (b) 1 ϳ, respectively.

In summary, we have developed a new three-layer dielectric model for spherical cavities for the treatment of the singularity due to the dielectric jump. The Poisson equation with the dielectric function is analytically solvable by using the spherical harmonics expansion. This model is useful in applications where charges could be close to the dielectric boundary. Appendix A: Solution of the linear matrix We present the solution of the 4u 4 linear system derived in Section 3. Let vector f

( f1, f 2 , f3 , f 4 )T

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Zhenli Xu / Procedia Engineering 31 (2012) 1033 – 1038

and matrix M (mij ) 4u4 with identities m14, m21, m34 and m41 0 as shown in Eq. (8). Let, T1 m12m23  m13m22, T2 m13m42  m12m43, T3 m22m33  m23m32, T4 m32m43  m33m42, and constant D m31m44T1  m24m31T2  m11m44T3  m11m24T4 . Then, the solution of system Mu f is given by, u

§ m44T3  m24T4 · § m44 (m13m32  m12m33 ) · § m24T2  m44T1 · ¨ ¸ ¨ ¸ ¨ ¸ ¨ m31(m23m44  m24m43) ¸ f1 ¨ m44 (m11m33  m13m31) ¸ f 2 ¨ m11(m24m43  m23m44 ) ¸ f 3   ¨ m (m m  m m ) ¸ ¨ ¸ ¨ m (m m  m m ) ¸ 22 44 ¸ D ¨ m44 ( m12m31  m11m32 ) ¸ D 24 42 ¸ D ¨ 31 24 42 ¨ 11 22 44 ¨ m (m m  m m ) ¸ ¨ m (m m  m m ) ¸ ¨m T  m T ¸ 22 43 ¹ 23 42 ¹ 11 4 © 11 23 42 © 31 22 43 © 31 2 ¹ m m m  m m ( ) § 24 12 33 13 32 · ¨ ¸ ¨ m24 (m13m31  m11m33 ) ¸ f 4 ¨ ¸ . ¨ m24 (m11m32  m12m31) ¸ D ¨m T  m T ¸ 11 3 © 31 1 ¹

(12)

Acknowledgements The author acknowledges the financial support from the Chinese Ministry of Education (Grant No. NCET-09-0556) and NSFC (Grant No. 11026057). References [1] F. Fogolari, A. Brigo, H. Molinari, The Poisson-Boltzmann equation for biomolecular electrostatics: a tool for structural biology, J. Mol. Biol. 15 (2002) 377-392. [2] A. Okur, C. Simmerling, Hybrid explicit/implicit solvation methods, Annu. Rep. Comput. Chem. 2 (2006) 97-109. [3] R. H. French et al., Long range interactions in nanoscale science, Rev. Mod. Phys. 82 (2) (2010) 1887-1944. [4] Y. Lin et al., An image-based reaction field method for electrostatic interactions in molecular dynamics simulations of aqueous solutions, J. Chem. Phys. 131 (2009) 154103. [5] P. Qin, Z. Xu, W. Cai, D. Jacobs, Image charge methods for a three-dielectric-layer hybrid solvation model of biomolecules, Commun. Comput. Phys. 6 (2009) 955-977. [6] S. Deng, A robust numerical method for self-polarization energy of spherical quantum dots with finite confinement barriers, Comput. Phys. Commun. 181 (2010) 787-799. [7] C. Xue, S. Deng, Three-dielectric-layer hybrid solvation model with spheroidal cavities in biomolecular simulations, Phys. Rev. E 81 (2010) 016701. [8] A. Sihvola, I. V. Lindell, Polarizability and effective permittivity of layered and continuously inhomegeneous dielectric spheres, J. Electrom. Waves Appl. 3 (1989) 37-60. [9] I. V. Lindell, M. E. Ermutlu, A. H. Sihvola, Electrostatic image theory for layered dielectric sphere, IEE Proc.-H 139 (1992) 186-192. [10] P. G. Bolcatto, C. R. Proetto, Self-polarization energies of semiconductor quantum dots with finite confinement barriers, Phys. Stat. Sol. 220 (2000) 191-194. [11] P. G. Bolcatto, C. R. Proetto, Partially confined excitons in semiconductor nanocrystals with a finite size dielectric interface, J. Phys. Condens. Matter 13 (2001) 319-334. [12] J. L. Movilla, J. Planelles, Image charges in spherical quantum dots with an off-centered impurity: algorithm and numerical results, Comput. Phys. Commun. 170 (2005) 144-152. [13] J. G. Kirkwood, Theory of solutions of molecules containing widely separated charges with special applications to awitterions, J. Chem. Phys. 2 (1934) 351-361. [14] D. L. Clements, Fundamental solutions for second order linear elliptic partial differential equations, Comput. Mech. 22 (1998) 26-31. [15] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1964.