Interaction between Charged Cylinders in Electrolyte Solution

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The Interaction Between Charged Cylinders in Electrolyte Solution. Excluded Volume Effect Bay Huang, Stefano Maset, and Klemen Bohinc J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b05444 • Publication Date (Web): 28 Aug 2017 Downloaded from http://pubs.acs.org on September 2, 2017

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The Journal of Physical Chemistry

The Interaction between Charged Cylinders in Electrolyte Solution. Excluded Volume Eect Beibei Huang,

†Department

∗,†

Stefano Maset,

‡

and Klemen Bohinc

¶

of Experimental Therapeutics, The University of Texas M. D. Anderson Cancer Center, 1901 East Road, Houston, TX 77054, USA

‡Dipartimento ¶Faculty

di Scienze Matematiche Universita' di Trieste 34100 Trieste, Italy

of Health Sciences, University of Ljubljana, 1000 Ljubljana, Slovenia

E-mail: [email protected]

Phone: +7137455746

Abstract Electrostatic interactions govern the physical properties of charged cylindrical structures in electrolyte solutions. Besides the surface charge on the cylinders, another factor inuencing the electrostatic interactions are the mobile ions. The nite size of the mobile ions is included by the excluded volume eect within the lattice statistics, while the electrostatic interactions are considered by means of the mean electrostatic eld. In this paper we consider charged parallel cylinders embedded into an electrolyte solution of mobile monovalent ions. A modied nonlinear Poisson-Boltzmann equation is proposed via variational procedure, and we implement the nite element method to solve it numerically. Excluded volume eect of the system containing two and multiple charged parallel cylinders are taken into account. Numerical results show that the excluded volume eect decreases the concentration of counterion and increases the electrostatic potential near the charged cylinders. The angular distribution of counterion around the particular cylinder is asymmetric. The study of the electrostatic interaction between 1

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two parallel equally charged cylinders reveals that an increase in the free energy is seen when the ionic strength is decreased. The free energy decreases as a function of the cylinders separation distance. On contrary for two oppositely charged cylinders

the free energy increases with increasing cylinder separation distance, while for two cylinders with dierent charged density it shows nonmonotonic variation with the increasing cylinders separation distance.

Introduction The properties of many hard- and soft-matter systems are governed by coulombic interactions among the charged groups in the system. 1 In the past, much attention has been made to charged ionorganic and organic cylindrical structures. Examples are nanotubular structures in dierent phospholipid systems, polyelectrolytes, actin laments, viruses and even bacteria. All these systems can be approximated as charged cylindrical surfaces in contact with an electrolyte solution composed of water molecules and mobile ions. 2 The presence of charged surfaces in an electrolyte solution aects the ion distribution. Counterions are attracted by the charged surface whereas coions are depleted. As a result, a diusive electric double layer is formed. The electric double layer is of fundamental importance in molecular biology, chemistry, and all technologies that employ electrolyte solutions. The interaction between cylindrical macroions plays an important role in their stability. Tuning the balance between attractive and repulsive forces allows to control the thermodynamic properties of assembled system and the response to external conditions. Therefore the interaction between charged cylindrical surfaces embedded in an electrolyte solution was studied. 3 The classical mean-eld theory that describes the electrostatic interactions in electrolyte solutions is the Poisson-Boltzmann (PB) theory. 4,5 This theory considers only coulombic interactions between point-like charges. The correlation between charges and steric eects are not taken into account. Further it is assumed that the charges are embedded in a continuous dielectric medium and the charge on the surfaces is continuously 2


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smeared. Most recently, Tan 42 and Aksimentiev 40,41 dissussed the ion-mediated interaction between DNAs, in which MD simulation were employed to calculate the potentials of mean force (PMFs) between triple-stranded and double-stranded DNAs.

The improvements of Poisson Boltzmann theory to adapt for a variety of specic environment has been studied for years. Antypov et al. 7 studied the eects of adding various local and non-local free energy functionals to the PB free energy functional to include the eects of an ionic hard core. They concluded that the system can be adequately described within the PB approach supplemented with an excluded volume weighted density approach. Borukhov et al. 9 considered the modied Poisson-Boltzmann equation as a special example of the local density approximation, where the excluded volume eect is represented as a homogeneous lattice gas. 11

Another improvement of the PB theory can

be obtained by including direct ion-ion interactions. The uctuation potential 15 due to the self-atmosphere of ion and the ion-ion exclusion volume term were taken into account in the modied Poisson-Boltzmann equation. 1618 The integral equation methods, such as the hypernetted chain (HNC) approximation, have also been carried out. 1923

Another improvement of PB theory focus on the incorporation of ion size, although the treatments for ionic size eects at the level of a local density approximation are known to provide quite poor results. 9 Petris et al. 10 developed Boot-strap(BS) PB theory to calculate the approximate radial distribution functions for a general solution of colloidal particles and point ions.

Stern 13 introduced

a simple model for the nite size of ions. He assumed that some of the mobile counterions adsorb onto the at surface, thus creating a region that separates the charged surface from the diuse part of the electric double layer. The incorporation of steric eects directly into the diuse ion layer on the basis of a lattice gas model goes back to Bikerman. 66 Freise 24 introduced the excluded volume eect by a pressure-dependent potential, while Wicke and Eigen 25 used a thermodynamic approach. The Poisson-Nernst-Planck (PNP) equation was

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generalized by introducing nite size of ions. 68 Recently, the nite size of particles has been incorporated into the PB theory, based on a lattice statistics model, 2628 by using other functional density approaches 29,31 and by considering the ions and solvent molecules as hard spheres. 35,36 Ionic size eects are discussed through electrostatic properties of an ionic solution with multiple ionic species, 32,34 and ion specic interactions in aqueous solutions have been also studied. 30,33 Also Monte Carlo simulations are widely used in order to describe the nite-sized counterions. 36,37 Boublik-Mansoori-Carnahan-Starling-Leland (BMCSL) accounts the excluded volume interactions based on phenomenological equation of state. 38,39 Lue and coworkers 31 extended BMCSL model for the purpose to describe a hard sphere uid of particles with dierent sizes. The PB equation was soled in dierent geometries. D. Harries solved non-linear PB equation for two charged parallel cylinders. 12 Linearized Poisson-Boltzmann (PB) theory was used to solve the case of low surface charge density. 6 The interaction between two charged surfaces was also studying using counterion condensation theory. Monte Carlo simulations were applied to investigate the distribution of nite size ions around a rodlike polyelectrolytes in aqueous solutions. 4750 Also the interaction between two parallel cylindrical surfaces was considered. 4446 Within the PB theory the repulsive electrostatic interaction was observed. The potential of mean force of a pair of identical rodlike polyions mediated by the condensed counterions was calculetd. 51 In this work we formulate a modied PB model for a system of parallel cylinders embedded in an electrolyte solution. Our model represent a mean-eld approach in the sense that correlations between charges are neglected, however the nite size of ions is included by the lattice model. Minimisation of the free energy of the system through a variational procedure results in a modied Boltzmann distribution for ions. Inserting modied Boltzmann distributions into Poisson's equation yields a partial dierential equation for the electrostatic potential that we solved numerically using Finite Element Method. 61,62 The solver relies on the mesh generator DistMesh. 53

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Modied Poisson Boltzmann Theory

Figure 1: Schematic presentation of two cylinders at a distance h surround by counterions and coions of nite size a. The radius of each cylinder is R. The system we consider is schematically displayed in Figure 1. It consists of several innitely long parallel cylinders of diameter 2R. The distance between the cylinders is h. We identify the plane y = 0 of a Cartesian coordinate system {x, y, z} with the plane which includes the axis of two cylinders. The y axis is oriented parallel with cylinder axis. The cylinders are negatively charged with a surface charge density σ and embedded in an aqueous solution containing positively and negatively charged ions with bulk concentration n0 . The diameter of ions is a. The positively charged ions are attracted by negatively charged surfaces and are called counterions. The negatively charged ions are depleted from charged surfaces 5



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and are denoted as coions. The system is overall charge neutral

Z

[ρf ix (r) + ρ(r)]d3 r = 0 ,

(1)

V

where ρf ix (r) is the xed charge density on the cylinder surfaces and ρ(r) is the volume charge density of counterions and coions that are present at position r. The integration in Eq.1 extends over the volume of the system V . The electrostatic mean-eld free energy of the system, F , measured in units of the thermal energy kB T (here kB is Boltzmann's constant and T is the absolute temperature) can be expressed as

F kB T

Z XZ nω (r) 1 2 3 nω (r) ln = [∇u(r)] d r + d3 r 8πlB V n 0 ω=± V P Z X ns − ω=± nω (r) d3 r . + ns − nω (r) ln ns − 2n0 V ω=±

(2)

The rst term on the right hand side of Eq.2 corresponds to the energy stored in the electrostatic eld, here expressed in terms of the commonly used dimensionless electrostatic potential u = eΦ/kB T , instead of the electrostatic potential Φ. The operator ∇ denotes the gradient of the scalar eld u. The Bjerrum length lB = e2 /(4π0 kB T ) is introduced instead of the dielectric constant of the aqueous solution ( e is the elementary charge and 0 is the permittivity of free space). At room temperature the Bjerrum length in the aqueous solution is 0.7 nm. The second and third terms describe the non-ideal mixing free energy of the counterions, coions and water molecules where nω are corresponding local ion concentrations. The sum runs over counterions ( ω = +1) and cions (ω = −1). Counterions, coions and water molecules in the solution occupy one and only one site of a nite volume. A lattice with an adjustable lattice constant a is introduced. The lattice constant a is a measure for the ionic diameter. Here we assume that both counterions and

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coions have the same size. The number density of lattice sites is

ns =

1 . a3

(3)

In thermal equilibrium, the free energy F = F (n+ , n− ) adopts a minimum with respect to the local concentrations of counterions n+ and coions n− . Upon calculating the rst variation δF (n+ , n− ) and demanding it equals zero, we nd the equilibrium concentration distributions of ions

nω (r) =

e−ωu(r) n0 · w cosh u(r) 1 − w 1 + 1−w

(4)

where w = 2n0 a3 . Inserting the equilibrium distributions n+ (r) and n− (r) into Poisson's equation ∇2 u(r) = −4πlB [ρf ix + ρ(r)]/e, with ρ(r) = e[n+ (r) − n− (r)], leads to a modied PB equation 11,66

∇2 u(r) = ρf ix (r) +

1 sinh u(r) . · w cosh u(r) 1 − w 1 + 1−w

(5)

Eq.5 has to be solved subject to the boundary conditions

∇u(r)

= surf ace

4πlB lD σ , e

(6)

√ where lD = 1/ 8πlB n0 is the Debye length. Far from the cylinders we set potential to 0 for brevity.

u(r) = 0 .

(7)

∞

Solving Modied Poisson Boltzmann Equation We adopt Finite Element method to solve the nonlinear Poisson Boltzmann Equation Eq.5 proposed in previous section with two Neumann boundary conditions

∇u· n

=0 L1

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and

∇u· n where lc =

4πlB lD σ, e

= lc L2

L2 denotes the surface of the cylinders and L1 denotes the far boundary.

The details of procudure solving the Eq.5 via FEM are given in Appendix .

The Interaction Energy Calculation The stability of the system containing multiple cylinders can be evaluated by the calculation of the free energy. From Eq.2 follows

F kT

( X Z

) (Z ) X n uρ ω f ix ω=± = udV + dS + 2 2 Ω Γ i j i j X Z X nω (r) + nω log dV n0 (r) Ωi ω=± i P XZ X ns − ω=± nω (r) nω (r) ln + ns − dV n − 2n s 0 Ω i ω=± i P

(8)

where the concentration of ions is given by Eq.4 and the potential follows from the solution of dierential equation Eq.5 subject to boundary conditions Eq.6 and Eq.7. The integral space Ω is discretized into Ωi . The boundary on the cylinders Γ is discretized into Γi . The interaction energy is dened as ∆F = F − Fref , where Fref corresponds to the free energy of the system without cylinders. As the interaction between mobile ions are ignored,

Fref = 0.

Numerical Results and Discussion Our theoretical model yields the electrostatic potential, ion concentration and charge density proles, between many like-charged cylindrical surfaces embedded in a solution of monovalent

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ions. We also calculate the free energy between two cylinders as a function of their separation distance. In the following, we analyze these properties based on numerical solutions of the nonlinear partial dierential Eq.5 subject to the boundary conditions Eq.6 and Eq.7. In numerical test, we use DistMesh 53 to generate mesh for 2D space. The mesh contains triangular elements which are the pieces on which the polynomials are dened. We rst test the electrostatic potential generated from the two identical cylinders with increasing size of mobile ions a with their distance xed.

(a) a = 0 nm

(b) a = 0.4 nm

(c) a = 0.8 nm

(d) a = 1.2 nm

Figure 2: 3D plots of the absolute value of the electrostatic potential around two parallel cylindrical surfaces for four dierent ion sizes a. The distance between two cylinders is 2 nm. The radius of the cylinder is 1.0nm. All lengths are scaled with respect to the Debye length. The model parameters are lc = 7.90 and n0 = 0.1 mol/l.

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Figure 2 shows three dimensional distribution of the electrostatic potential around two charged parallel cylindrical surfaces. Figure 2 a represents calculation for classic PB theory where the excluded volume is not taken into account ( a = 0nm). The potential distributions for dierent lattice constants a are shown in Figures 2 b-d. All calculations were made in the Cube box with dimension 6.0 measured in Debye length. The radius of the cylinder is

1.0nm, and the distance between two surfaces of the cylinders is 2.0nm. The other model parameters are for lc = 7.90 and n0 = 0.1 mol/l. The former corresponds to surface charge density σ = 0.15 C/m2 , i.e. one elementary charge per 1.1 nm2 . In all cases the highest values of the potential is obtained close to the cylindrical surfaces. In the region between both cylinders the potential surface has a form of a saddle. Far from both cylinders the potential converges to zero. The highest absolute values of the potential are reached in the case of the largest lattice constant ( a = 1.2 nm). In this case the potential on the cylinder shows a very asymmetric angular distribution. Figure 3 shows the counterion and coion proles in the plane y −z , x = 0. The counterion concentration proles n+ exhibit a maximal values at the closest vicinity of both cylinders. On contrary the coion concentration prole n− reaches in this region a minimum. The concentration of counterions decreases with increasing distance from the cylinders. Far from both cylinders the concentrations approaches the corresponding bulk value. Larger ions cause more pronounced and broader peak. For lattice constant a = 1.2 nm the width of the peak is around 5 nm, whereas for point-like ions the peak width is around 3.5 nm. Figure 4 shows the counterion and coion proles in the plane x − z , y = 0. The region

1 < |x/lD | < 3 corresponds to the inner side of the cylinders. With increasing ion size the concentration close to the cylinder decreases. The reason is the limited space between both cylinders. For higher surface charge densities and larger ions the concentration of counterions close to the cylinder surface saturates. The closest counterion packing is achieved, all lattice sites near the surface are occupied. For larger ions and higher surface charge densities the thickness of the electric double layer is proportional to the surface charge density and to the

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volume occupied by one ion. In the limit of point-like ions thickness of the electric double layer is given by the Debye length. Our theory distinguishes between occupied and free lattice sites. All free lattice sites are occupied by water with a relative permittivity of 78. The lattice model has a considerable eect on potential and counterion concentration. Contrary, PB theory considers ions as dimensionless and the counterion concentration near a charged surface can be very high. It is also important to note that in PB theory the counterions and coions are assumed to be suspended in a continuous dielectric medium. In our theory the ion-ion interactions other than the hard core interaction are neglected. Nevertheless, in the case of monovalent ions, the ion-ion interaction eects are negligible. However, the ion-ion interaction play an important role for multivalent ions. We also assumed a permittivity of 78 for the surrounding water. Nevertheless, it is known that bound water has a reduced permittivity [34]. In addition, high ion concentrations will reduce the permittivity, especially in the vicinity of the charged surface. Taking these eects into account, e.g. by a position-dependent permittivity, will result in a minor correction of our calculations. It will lead to a slightly higher eld close to the surface and reduce the condensation eect. Although Ref. 63 indicates Debye length related to the the concentration of mobile ions, for simplicity we still treat it a constant. Figure 3 indicates with the size of mobile ions increasing, the concentration of ions along x = 0 increases correspondingly. It is noted that along the curve y = 0 with the size of mobile ions increasing, the concentration of ions on the adjacent of the cylinder's surfaces increases in the rst phase and then decreases, although concentration in the middle increases all the time. Eq. 5 can be used also for the space containing multiple parallel cylinders. Inserting electrostatic potential into Eq.4 follows the distribution of mobile ions in the space containing multiple parallel cylinders (see Figs.5, 6). In principle we can estimate the electrostatic potential barrier for the diusion of the small charged particles by the potential surface.

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(a) a = 0 nm, no volume excluded eect

(b) a = 0.4 nm

(c) a = 0.8 nm

(d) a = 1.2 nm

Figure 3: Concentration proles of counterions n+ and coions n− , in units of mol/l. The proles are shown in the plane: y − z , x = 0. All lengths are given in units of Debye length. The dimensionless surface charge density is lc = 7.90. The distance between two cylinders is 2 nm. The radius of the cylinder is 1.0nm. a = 0 corresponds to the point-like ions without excluded eect. The bulk concentration of ions is 0.1 mol/l.

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(b) a = 0.4 nm

(c) a = 0.8 nm

(d) a = 1.2 nm

Figure 4: Concentration proles of counterions n+ and coions n− , in units of mol/l. The proles are shown in the plane: x − z , y = 0. All length are given in units of Debye length. The dimensionless surface charge density is lc = 7.90. The distance between two cylinders is 2 nm. The radius of the cylinder is 1.0nm. a = 0 corresponds to the point-like ions without excluded eect. The bulk concentration of ions is 0.1 mol/l.

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The local space lled with ions attracted by cylinders will create a dry environment in the vicinity of charged cylinders. The electrostatic potential is inuenced by the distance between cylinders. The decreasing distance between cylinders leads to the electrostatic potential increase. This corresponds to the increase of concentration of coions and counterions. Thus Figure 5 shows a situation where the distance between each pair of cylinders reduced to 1.0 nm, the triangular zone between 3 cylinders lling full of mobile ions and becomes a dry region. The ion distribution is also inuenced by the ion size a. Figure 6 shows the situation where the size of ions reach to a certain scale, the dry zone between 4 cylinders with xed position appears. Without excluded volume eect this phenomenon does not exist. Both tests are carried out in the Cube box with dimensionless length 6.0 (use Debye length as unit). The radius of cylinder are still 1.0 nm and the dimensionless surface charge density

lc = 7.90. Finally, we consider the interaction free energy of the system composed of two cylinders in electrolyte solution of monovalent ions with nite size. Figure 7 shows the free energy as a function of the distance between two like-charged cylinders for three dierent bulk concentrations and two lattice side parameters. In all cases the free energy decreases with increasing separation distance between the cylinders.

For point-like ions

an increase in

the free energy is seen when the ionic strength is decreased. The increasing parameter a leads to the larger values of the free energy.

For larger ions, the free energy decrease

when the ionic strength is decreased.

The force between the charged cylinders can

be evaluated by the the negative slope of the free energy curve. For ions of nite size the increasing bulk concentration n0 increases the critical distance at which the force drops to zero (see Figure 7b). This eect is not observed for point-like ions (see Figure 7a).

These results are in agreement with previous studies for two parallel like-charged surfaces embedded in an electrolyte of monovalent ions. 69 Within PB theory the repulsive interaction between like charged planar surfaces was calculated. Repulsive interaction was obtained also for two parallel cylinders embedded in 14


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(a) h = 2.0 nm in overall perspective

(b) h = 2.0 nm, in top-down perspective

(c) h = 3.0 nm, in overall perspective

(d) h = 3.0 nm, in top-down perspective

Figure 5: The distribution of mobile ions in the box containing 3 cylinders, two of which are with dierent distance h. The values in the plot indicate the sum of the concentration of the ions and counter ions at r.

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(a) a = 0 nm, no excluded volume eect

(b) a = 0.5 nm

(c) a = 1.0 nm

(d) a = 1.2 nm

Figure 6: The distribution of ions and Pcounter ions, the value in plots indicate the sum of concentration of ions and counter ions n± (r). The distance between each two neighbouring cylinders is h = 2.0 nm.

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a solution of monovalent point-like ions. 12 We continue with two oppositely charged cylinders. The free energy as a function of the separation distance D is shown in Figure 8. Only attraction between two cylinders is observed. If the cylinders are charged with dierent surface charge densities then the transition from the attraction to repulsion is obtained (Figure 8b).


(b) a = 0.4 nm

Figure 7: Interaction energy between two like-charged cylinders as a function of the separation distance D in Debye length units for three dierent bulk concentrations n0 . a) point like ions and b) ions with excluded volume. Model parameter lc = 7.90. Then the distribution of mobile ions around the two cylinders is taken into account, we adopted the bicylindrical coordinates proposed in Ref. 12 We solved the equations via Newton-Raphson (NR) iteration and collocation with bicubic Hermite basis functions(NRBH solver). We compare the concentration of coions n− and counter ions n+ from two solvers, as illustrated in Figure 9.

Conclusion In this work we made an eort to explore the eect of excluded volume of mobile ions in the electrostatic interaction between parallel charged (ionic) cylinders embedded into an 17



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(a)

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(b)

Figure 8: Interaction energy dierence of the system containing double cylinders. The surfaces of cylinders are with (a) dimensionless opposite charge density lc1 = 6.32, lc2 = −6.32; (b) lc1 = 8.43, lc2 = −2.11, with dierent size of mobile ions a. a = 0 corresponds to point-like ions. All lengths are given in units of Debye length. Mobile ions concentration in both cases are n0 = 0.1 mol/l electrolyte solution. Equations have been derived for the electrostatic potential in the region outside of the cylinders . The resulting modied Poisson-Boltzmann equation was solved numerically via nite element method (FEM) with a merit of easily introducing in new objects by adding extra boundary conditions. We implemented the solver for solving the modied nonlinear PB equation through FEM incorporated with the mesh tool DistMesh. Finally, the free energy of the system was calculated. Numerical results indicate the decrease of counterion concentration proles close to the charged cylinders if the excluded volume of ions is applied. The interaction energy between two like-charged cylinders reveals repulsive interaction independently on the ion size and bulk concentration of monovalent ions in the solution. The interaction energy as a function of the distance between two oppositely charged parallel cylinders indicates nonmonotonic behavior along with the increasing distance when the excluded volume of ions is taken into account. We expect that our present results could contribute to better understanding of

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Figure 9: Comparison of the solution for the distribution of ions and counter ions along the curve y = 0 between x = −1.0 and x = 1.0, via nite element solver (FEM solver) and Newton-Raphson bicubic Hermite method (BRBH solver), and ions size is a = 0.8 nm

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interaction between DNA 70 and other cylinder type macromolecules. It is a good starting point to study how common condensing agents such as polyamines inuence condensation of cylindrical macroions 71,72 .

Appendix Finite Element method is employed to solve the nonlinear Poisson Boltzmann Equation Eq.5 with corresponding boundary conditions proposed in previous section. According to Galerkin's method, solving Eq.5 is to nd solution u ∈ H 1 (Ω) such that for all v ∈ H 1 (Ω)

Z

Z (∇ · (∇u)) v =

Ω

Ω

v sinh u 1 − w + w cosh u

(9)

according to Green's Theorem we use integration by parts for the left side

Z

Z (∇ · (∇u)) v = −

Ω

Z ∇u · ∇v +

lc v

Ω

L2

and then we obtain the weak form of the Eq.5

Z

Z ∇u · ∇v +

Ω

Ω

v sinh u − 1 − w + w cosh u

Z lc v = 0

(10)

L2

It is noted that L2 is composed by two boundaries corresponding to the two surfaces of cylinders, and they are natural boundary conditions which appear inside the formulation and can be easily expanded if new cylinders are added. The continuous Galerkin method is adopted to discritize u by Eq.11

u = u1 φ1 + u2 φ2 + . . . + uN φN

(11)

where N equals the number of mesh points, since there are only Neumann boundaries. φi 20


Page 21 of 31

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are basis functions, and u is decomposed as a unique linear combination of the elements of the basis. Substitute v by trial function φi and insert Eq.11 into Eq.10 we obtain the discrete form Eq.12

X

Z

Z ∇φj · ∇φi +

uj Ω

j

Ω

P Z φi sinh j uj φj P − lc φi = 0. 1 − w + w cosh j uj φj L2

(12)

For solving this nonlinear Eq.12 we dene function Fi in terms of u = (uj )

Fi (u) =

X

Z

Z ∇φj · ∇φi +

uj Ω

j

Ω

P Z φi sinh j uj φj P lc φi − 1 − w + w cosh j uj φj L2

(13)

And in its corresponding Jacobian Matrix the (i, j) entry reads

∂Fi (u) = ∂uj

P cosh j uj φj (1 − w) + w P ∇φj · ∇φi + φi φj (1 − w + w cosh j uj φj )2 Ω Ω

Z

Z

(14)

We adopt P1 Linear Elements. Supports of basis functions φj and φi have intersecting area

Ω(i,j) that substitutes Ω in the above integrals. It means that the value of entries (i, j) in the Jacobian matrix is non-zero only when node i and j are adjacent. Hence, the Jacobian matrix is a sparse band and diagonal matrix.

Acknowledgement The authors are grateful to Prof. Daniel Harries, Prof. Benzhuo Lu, Prof. Vladimir Baulin and Prof. Shuxing Zhang. The authors gratefully acknowledges nancial help from China NFS grant no.11001257 for computer equipment.

21



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Interaction between Charged Cylinders in Electrolyte Solution

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