Density of electric field energy around two surface-charged spheres surrounded by electrolyte II. The smaller sphere is inside the larger one

István P. Sugár; István P. Sugár

doi:10.3934/biophy.2022006

AIMS Biophysics

2022, Volume 9, Issue 1: 61-71. doi: 10.3934/biophy.2022006

Previous Article Next Article

Research article Special Issues

Density of electric field energy around two surface-charged spheres surrounded by electrolyte II. The smaller sphere is inside the larger one

István P. Sugár ^,

Department of Neurology, Icahn School of Medicine at Mount Sinai, New York, NY 10029

Received: 20 January 2022 Revised: 27 February 2022 Accepted: 07 March 2022 Published: 18 March 2022

Based on the generalized version of Newton's Shell Theorem [7] the electric field energy density, u_F around two surface-charged spheres surrounded by electrolyte where the smaller sphere is inside the larger one is analytically calculated. According to the calculations when the surfaces of the spheres are farther from each other than four times of the Debye length the field energy density around and inside the smaller sphere is basically independent from the presence of the larger sphere. The electric field energy density is maximal when the smaller sphere touches the inner surface of the larger sphere and the maximum of u_F is located at the touching point on the outer surface of the larger sphere.

Keywords:

Citation: István P. Sugár. Density of electric field energy around two surface-charged spheres surrounded by electrolyte II. The smaller sphere is inside the larger one[J]. AIMS Biophysics, 2022, 9(1): 61-71. doi: 10.3934/biophy.2022006

Related Papers:

[1]	István P. Sugár . Density of electric field energy around two surface-charged spheres surrounded by electrolyte I. The spheres are separated from each other. AIMS Biophysics, 2022, 9(2): 86-95. doi: 10.3934/biophy.2022008
[2]	István P. Sugár . Electric energies of a charged sphere surrounded by electrolyte. AIMS Biophysics, 2021, 8(2): 157-164. doi: 10.3934/biophy.2021012
[3]	István P. Sugár . A generalization of the Shell Theorem. Electric potential of charged spheres and charged vesicles surrounded by electrolyte. AIMS Biophysics, 2020, 7(2): 76-89. doi: 10.3934/biophy.2020007
[4]	István P. Sugár . A generalization of the Shell Theorem. Electric potential of charged spheres and charged vesicles surrounded by electrolyte. AIMS Biophysics, 2023, 10(1): 23-24. doi: 10.3934/biophy.2023003
[5]	Ganesh Prasad Tiwari, Santosh Adhikari, Hari Prasad Lamichhane, Dinesh Kumar Chaudhary . Natural bond orbital analysis of dication magnesium complexes [Mg(H₂O)₆]²⁺ and [[Mg(H₂O)₆](H₂O)_n]²⁺; n=1-4. AIMS Biophysics, 2023, 10(1): 121-131. doi: 10.3934/biophy.2023009
[6]	David E. Shoup . Diffusion-controlled reaction rates for clusters of binding sites on a cell. AIMS Biophysics, 2016, 3(4): 522-528. doi: 10.3934/biophy.2016.4.522
[7]	Roman Marks . Bubble mediated polymerization of RNA and DNA. AIMS Biophysics, 2022, 9(2): 96-107. doi: 10.3934/biophy.2022009
[8]	Massimo Fioranelli, Maria Grazia Roccia, Aroonkumar Beesham, Dana Flavin, M. Ghaeni, Faissal AZIZ . A model for considering effects of extremely low frequency electromagnetic fields on quail embryonic cells. AIMS Biophysics, 2022, 9(3): 198-207. doi: 10.3934/biophy.2022017
[9]	István P. Sugár, Parkson Lee-Gau Chong . Monte Carlo simulations of the distributions of intra- and extra-vesicular ions and membrane associated charges in hybrid liposomes composed of negatively charged tetraether and zwitterionic diester phospholipids. AIMS Biophysics, 2017, 4(2): 316-336. doi: 10.3934/biophy.2017.2.316
[10]	Arjun Acharya, Madan Khanal, Rajesh Maharjan, Kalpana Gyawali, Bhoj Raj Luitel, Rameshwar Adhikari, Deependra Das Mulmi, Tika Ram Lamichhane, Hari Prasad Lamichhane . Quantum chemical calculations on calcium oxalate and dolichin A and their binding efficacy to lactoferrin: An in silico study using DFT, molecular docking, and molecular dynamics simulations. AIMS Biophysics, 2024, 11(2): 142-165. doi: 10.3934/biophy.2024010

Abstract

1. Introduction

The head groups of membrane lipids have either single charge (e.g. tetraether lipids [1], phosphatidic acid (PA), phosphatidylserine (PS), phosphatidylethanolamine (PE), and phosphatidylinositol (PI)) or electric dipole (e.g. phospholipids, such as dimyristoyl-, dipalmitoyl- and distearoylphosphatidyl choline (DMPC, DPPC and DSPC, respectively)).

Between lipids containing head groups with electric dipole there is short range interaction, i.e. where the two-body potential decays algebraically at large distances with a power equal or larger than the spatial dimension [2]. Theoretical models of lipid membranes usually focus on systems where there is short range lateral interactions between nearest neighbor lipids [3],[4] because it is enough to consider only the interactions between nearest-neighbor lipid molecules. It is much more difficult to model a lipid membrane containing single charged head groups [5]. Between lipids with single charged head groups there is long range interaction, i.e. where the two-body potential decays algebraically at large distances with a power smaller than the spatial dimension [2] and thus modeling this system one has to consider the entire system rather than the interactions between the nearest-neighbor lipids. In order to get closer to the solution of this problem recently we developed a generalized version of Newton's Shell Theorem [6],[7] to calculate the electric potential, V around a surface-charged sphere (of radius R₁) surrounded by electrolyte at a distance Z from the center of the sphere (see also Eqs 9,10 in ref.7):

$V (Z) = \frac{k_{e} \cdot Q_{1} \cdot λ_{D}}{ϵ_{r} \cdot Z \cdot R_{1}} \cdot e^{- \frac{Z}{λ_{D}}} \cdot s i n h (\frac{R_{1}}{λ_{D}}) a t Z > R_{1}$ (1)

$V (Z) = \frac{k_{e} \cdot Q_{1} \cdot λ_{D}}{ϵ_{r} \cdot Z \cdot R_{1}} \cdot e^{- \frac{R_{1}}{λ_{D}}} \cdot s i n h (\frac{Z}{λ_{D}}) a t Z < R_{1}$ (2)

where $k_{e} = {(4 π ϵ_{0})}^{- 1}$ is the Coulomb's constant, λ_D is the Debye length, Q₁ is the total charge of the homogeneously charged surface of the sphere of radius R₁, ϵ_r is the relative static permittivity of the electrolyte. Deriving Eqs 1,2 the general solution of the Screened Poisson Equation was utilized (see Eq 4 in ref.7 or A5 in Appendix 1), an equation that is valid if the electrolyte is electrically neutral [8]. It is important to note that the Screened Poisson Equation (Eq A4) is different from the Poisson-Boltzmann equation (see Eqs A1,A3). The Poisson-Boltzmann equation can be used to calculate the potential energy of an arbitrary, electroneutral, ion solution (i.e. electrolyte). However, for the solution (see Eq A2) one has to know the charge density of the ions in the electrolyte (i.e. the Boltzmann distribution; see Eq A3), which depends on the potential, V, itself. Thus only approximative solution is available (the Debye- Hückel approximation [9]), that is valid when $| z_{i} e V / (k_{B} T) | ≪ 1$ (where e: charge of an electron, z_i: charge number of the i-th type of ion, k_B: Boltzmann constant, T: absolute temperature).

Using the Screened Poisson Equation (Eq A4) one can calculate the potential energy of an electrolyte that contains also external charges. The external charges are embedded into the electrolyte (like the charges of the surface-charged sphere) but not part of the electrolyte itself. For the solution one has to know the charge density of the external charges (see Eq 4 in ref.7 or Eq A5 in Appendix 1), i.e. distribution of the charges on the surface-charged sphere and not the distribution of the ions in the electrolyte. In our case it is assumed that the charges on the surface of the sphere are homogeneously distributed and in this case Eqs 1,2 is the exact solution of the Screened Poisson Equation.

Note that recently by using Eqs 1,2 electric energies have been calculated [10], such as the electric potential energy needed to build up a surface-charged sphere, and the field and polarization energy of the electrolyte inside and around the surface-charged sphere.

In this paper the density of electric field energy is calculated around two surface-charged spheres where the smaller sphere is located inside the larger one and the entire system is embedded in neutral electrolyte. This system is close to a charged vesicle [1] or to a cell [11] where charged lipids are located both on the outer and inner leaflet of the membrane, i.e. two concentric surface-charged spheres. It also models an eukaryote [12] where neutral phospholipids such as sphingomyelin and zwitterionic phosphatidylcholine are located primarily in the outer leaflet of the plasma membrane, and most anionic phospholipids, such as phosphatidic acid (PA), phosphatidylserine (PS), phosphatidylethanolamine (PE), and phosphatidylinositol (PI) are located in the inner leaflet of the plasma membrane (represented by the large surface-charged sphere of our model). Eukaryotes also have a single nucleus enveloped by double layer of lipid membranes which may contain charged lipids too (representing the smaller surface-charged sphere of our model). Note that these two charged spheres of an eukaryote are not necessarily concentric. Finally, our model is generalized for the case when the large surface-charged sphere contains several smaller surface-charged spheres. This system may also model osteoclast cells [12] containing many nuclei.

In this work the density of the electric field energy inside and outside of two surface-charged spheres are calculated at different locations. The density of the electric field energy at a point can be calculated by the following equation [13]:

$u_{F} = \frac{ϵ_{r} ϵ_{0}}{2} \underline{E} \cdot \underline{E}$ (3)

where E is the vector of the electric field strength at the considered point, ϵ₀ is the absolute vacuum permittivity and ϵ_r is the relative permittivity of the electrolyte.

2. Model

Here by using the recently generalized Shell Theorem [7] we calculate the density of electric field energy, u_F produced by two surface-charged spheres (see Figure 1) surrounded outside and inside by electrolyte where the smaller sphere is located inside the larger sphere.

Figure 1. Two surface-charged spheres surrounded outside and inside by electrolyte where the smaller sphere is located inside the larger one.

DownLoad: Full-Size Img PowerPoint

Z: the distance between the centers of the spheres (dashed blue line); R₁ and R_L is the radius of the smaller and larger sphere, respectively; D₁ and D_L is the distance between point P₁ and the center of the smaller and larger sphere, respectively; E₁ and E_L is the field strength created in point P₁ by the smaller and larger surface-charged sphere, respectively.

The origin of the coordinate system (x, y) is attached to the center of the larger sphere and the coordinates of point P₁ are x_p and y_p. The coordinates of the center of the larger and smaller sphere are $(x_{L}, y_{L}) = (0, 0)$ and $(x_{1}, y_{1}) = (- Z, 0)$ , respectively.

In order to calculate the density of electric field energy one has to determine the electric field strength (see Eq 3), i.e. the gradient of the electric potential. The potential produced by the smaller sphere, V₁ at a distance D₁ from its center can be calculated by Eqs 1,2 (or Eqs 9,10 in ref.[7]). The electric field strength created by the smaller sphere at point P₁ (see Figure 1) is:

${\underline{E}}_{1} = - g r a d (V_{1}) = - (\frac{d V_{1}}{d D_{1}} \frac{d D_{1}}{d x_{P}}, \frac{d V_{1}}{d D_{1}} \frac{d D_{1}}{d y_{P}}) = - \frac{d V_{1}}{d D_{1}} (\frac{d \sqrt{{(x_{P} - x_{1})}^{2} + {(y_{P} - y_{1})}^{2}}}{d x_{P}}, \frac{d \sqrt{{(x_{P} - x_{1})}^{2} + {(y_{P} - y_{1})}^{2}}}{d y_{P}}) = - \frac{d V_{1}}{d D_{1}} (\frac{x_{P} - x_{1}}{D_{1}}, \frac{y_{P} - y_{1}}{D_{1}}) = (- \frac{d V_{1}}{d D_{1}} \frac{x_{P} + Z}{D_{1}}, - \frac{d V_{1}}{d D_{1}} \frac{y_{P}}{D_{1}}) = (E_{1 x}, E_{1 y})$ (4)

where

$\frac{d V_{1}}{d D_{1}} = {\begin{matrix} \frac{k_{e} Q_{1} λ_{D}}{ϵ_{r} R_{1}} s i n h (R_{1} / λ_{D}) [- \frac{e^{- \frac{D_{1}}{λ_{D}}}}{D_{1}^{2}} - \frac{e^{- \frac{D_{1}}{λ_{D}}}}{D_{1} λ_{D}}] i f D_{1} > R_{1} \\ \frac{k_{e} Q_{1} λ_{D}}{ϵ_{r} R_{1}} e^{- \frac{R_{1}}{λ_{D}}} [- \frac{s i n h (\frac{D_{1}}{λ_{D}})}{D_{1}^{2}} + \frac{c o s h (\frac{D_{1}}{λ_{D}})}{D_{1} λ_{D}}] i f D_{1} < R_{1} \end{matrix}$ (5)

where λ_D is the Debye length and $Z \leq R_{L} - R_{1}$ .

Similarly, the electric field strength created by the large sphere at point P₁ (i.e. at a distance D_L from its center; see Figure 1) is:

$\begin{matrix} {\underline{E}}_{L} = - g r a d (V_{L}) = - (\frac{d V_{L}}{d D_{L}} \frac{d D_{L}}{d x_{P}}, \frac{d V_{L}}{d D_{L}} \frac{d D_{L}}{d y_{P}}) \\ = - \frac{d V_{L}}{d D_{L}} (\frac{d \sqrt{{(x_{P} - x_{L})}^{2} + {(y_{P} - y_{L})}^{2}}}{d x_{P}}, \frac{d \sqrt{{(x_{P} - x_{L})}^{2} + {(y_{P} - y_{L})}^{2}}}{d y_{P}}) \\ = - \frac{d V_{L}}{d D_{L}} (\frac{x_{P} - x_{L}}{D_{L}}, \frac{y_{P} - y_{L}}{D_{L}}) = (- \frac{d V_{L}}{d D_{L}} \frac{x_{P}}{D_{L}}, - \frac{d V_{L}}{d D_{L}} \frac{y_{P}}{D_{L}}) = (E_{L x}, E_{L y}) \end{matrix}$ (6)

where one can construct $\frac{d V_{L}}{d D_{L}}$ from Eq 5 by changing D₁ to D_L, R₁ to R_L and Q₁ to Q_L.

$\begin{matrix} u_{F} (x_{P}, y_{P}) = \frac{ϵ_{r} ϵ_{0}}{2} \underline{E} \cdot \underline{E} = \frac{ϵ_{r} ϵ_{0}}{2} ({\underline{E}}_{1} + {\underline{E}}_{L}) \cdot ({\underline{E}}_{1} + {\underline{E}}_{L}) \\ = \frac{ϵ_{r} ϵ_{0}}{2} ([E_{1 x} + E_{L x}], [E_{1 y} + E_{L y}]) \cdot ([E_{1 x} + E_{L x}], [E_{1 y} + E_{L y}]) \\ = \frac{ϵ_{r} ϵ_{0}}{2} ({[E_{1 x} + E_{L x}]}^{2} + {[E_{1 y} + E_{L y}]}^{2}) \end{matrix}$ (7)

3. Results

Here by using Eq 7 the density of the electric field energy, u_F, is calculated around two surface- charged spheres (where the smaller sphere is located inside the larger sphere) surrounded in- and outside by electrolyte. The radius of the larger and smaller sphere is: $R_{L} = 10^{- 6} m$ and $R_{1} = 0.2 R_{L}$ , respectively. The surface charge density of the homogeneously charged spheres is ρ_s = −0.266 C/m² (the surface charge density of the PLFE lipid vesicles [1]). The total charge of the larger and the smaller sphere is, $Q_{2} = ρ_{s} 4 π R_{2}^{2} = - 3.3427 \cdot 10^{- 12} C$ and $Q_{1} = ρ_{s} 4 π R_{1}^{2} = - 1.337 \cdot 10^{- 13} C$ , respectively. This system is axially symmetric, where the symmetry axis is the straight line connecting the centers of the spheres. The center of the attached coordinate system is at the center of large sphere and the x axis is defined by the symmetry axis. Because of the axial symmetry of the system it is enough to calculate u_F along straight lines parallel to the symmetry axis (see Figure 2), where the same y_P coordinate belongs to each straight line. The surface-charged spheres are surrounded by electrolyte containing monovalent ions. The considered electrolyte ion concentrations (of the positive ion) are: 0.00001, 0.001 and 0.1 mol/m³ and the respective Debye lengths are: 3.05 · 10⁻⁶, 3.05 · 10⁻⁷ and 3.05 · 10⁻⁸ m (see Table 1 in ref.7), and the relative permittivity of the electrolyte is ϵ_r = 78.

Figure 2. Locations of a small surface-charged sphere inside a large surface-charged sphere.

DownLoad: Full-Size Img PowerPoint

Inside a large surface-charged sphere of radius $R_{L} (= 10^{- 6} m)$ a small surface-charged sphere of radius $R_{1} (= 0.2 R_{L})$ is located. The electric field energy density, u_F is calculated at the three different locations of the small sphere, i.e. in Figure 3, Figure 4 and Figure 5 the center of the small sphere is at $Z = 0.8 R_{L}$ (see small orange circle), $Z = 0.5 R_{L}$ (see small purple circle) and Z = 0 (see small dark red circle at the center of the large circle), respectively. In Figures 3–5 the electric field energy densities are calculated along the five horizontal (dashed red, green, blue, grey, black) lines.

Figure 3. Density of the electric field energy around two surface-charged spheres.

DownLoad: Full-Size Img PowerPoint

$Z = 0.8 R_{L}$ .

Dotted red line: $y_{P} = 0.75 R_{1}$ ; green line: $y_{P} = R_{1}$ ; blue line: $y_{P} = 0.7 R_{L}$ ; grey line: $y_{P} = R_{L}$ ; black line: $y_{P} = 1.01 R_{L}$ . The concentration of the monovalent positive (or negative) ion in the electrolyte is: A) $C = 0.1 m o l / m^{3}$ ; B) $C = 0.001 m o l / m^{3}$ ; C) $C = 0.00001 m o l / m^{3}$ .

The connection point between the large sphere and the small sphere (represented by orange circle in Figure 2) is at $x_{P} = - R_{L}$ and $y_{P} = 0$ . At this point there is no electrolyte and Eqs 4–7 are not applicable. Similar situations take place when the horizontal line crosses the circles in Figure 2. The x_p coordinates of these cross sections, $x_{p}^{c}$ can be calculated by:

$x_{p}^{c} = \pm \sqrt{R_{L}^{2} - y_{p}^{2}} (crossing the large circle, i . e . y_{P} < R_{L})$ (8)

and

$x_{p}^{c} = - Z \pm \sqrt{R_{1}^{2} - y_{p}^{2}} (crossing the small circle located at - R_{L} < x_{1} < 0 and y_{P} < R_{1})$ (9)

$x_{p}^{c} = Z \pm \sqrt{R_{1}^{2} - y_{p}^{2}} (crossing the small circle located at 0 < x_{1} < R_{L} and y_{P} < R_{1})$ (10)

where x₁ is the x coordinate of the center of the small sphere.

Figure 4. Density of the electric field energy around two surface-charged spheres.

DownLoad: Full-Size Img PowerPoint

$Z = 0.5 R_{L}$

Figure 5. Density of the electric field energy around two surface-charged spheres.

DownLoad: Full-Size Img PowerPoint

$Z = 0$

Note in Figures 3–5 the sharp maxima of the density of the electric field energy appear where the horizontal line at the respective y_P crosses the charged sphere(s). These crossing points, $x_{p}^{c}$ 's, can be calculated by Eqs 8–10. In the case of $y_{P} = 0.75 R_{1}$ the values of the crossing points are listed at the first column of Table 1.

It is also important to note that $| {\underline{E}}_{1} | ~ Q_{1}$ and $| {\underline{E}}_{L} | ~ Q_{L}$ (see Eqs 4–6) and thus in the case of total surface charges $a \cdot Q_{1}$ and $a \cdot Q_{L}$ (where $0 < a < 1$ is a constant) the electric field energy density will be a² times of the above calculated $u_{F} (x_{P}, y_{P})$ values (see Eq 7).

4. Discussion

In this work the solution of the screened Poisson equation ([7] and Eq A5 in Appendix 1) is used to calculate the field energy density around two surface-charged spheres where the small sphere is located inside the large sphere. This solution is not restricted to small potentials (<< 25 mV) like in the case of the Debye-Hückel approximation of the Poisson-Boltzmann equation [9] where the superposition principle is not applicable either. This is an important advantage because the measured absolute value of the Zeta potentials of the cells are usually higher than 25 mV (e.g. –57.89 ± 22.63 mV on ARO cells, –40.41 ± 5.10 mV on C32TG cells, −46.99 ± 18.71 mV on RT4 cells, –40.13 ± 9.28 mV on TK cells, and −43.03 ± 5.52 mV on UM-UC-14 cells [14].

The considered two spheres (with homogeneously charged surfaces) electrically interact. If the lateral movement of the charges on the spheres would not be restricted the interaction of the smaller sphere (located inside the larger sphere) with the larger sphere would result in inhomogeneous distribution of the surface charges on both spheres. However, the free lateral diffusion of proteins and lipids are usually restricted in biological membranes not only by direct collisions with structures where immobile proteins are crowded, but also by electrostatic deflection, hydrophobic mismatches, and other mechanisms [15].

The density of the electric field energy depends on the electric field strength (Eq 3), i.e. the gradient of the electric potential (Eqs 4,6). In the case of a single surface-charged sphere surrounded by electrolyte with low ion concentration the potential inside the sphere is close to constant (see red curve in Figure 3A in ref. [7]) and thus the absolute value of the electric field strength is close to zero. On the other hand, outside the sphere the absolute value of the potential and also the electric field strength decrease with increasing distance from the surface of the sphere (see red curve in Figure 3A in ref. [7]). At higher electrolyte ion concentration, because of the increased screening effect, the absolute value of the potential and also the electric field strength decrease faster with increasing distance from the surface of the sphere. In this case inside the sphere toward its center the absolute value of the potential and the electric field strength also decrease (see curves in Figure 3A,B in ref. [7]).

In this work two surface-charged spheres (with the same surface charge density) are considered where the smaller sphere is located inside the larger sphere. The above mentioned electric properties of a single surface-charged sphere remain the same for the smaller sphere (located inside a larger sphere) if the surfaces of the spheres are far enough from each other (farther than 4 λ_D), i.e. the absolute value of the potential decreases close to zero between the surfaces of the two spheres. However, when part of the surfaces of the two spheres are close enough to each other one sphere contributes to the potential and electric field strength around the other sphere. The electric field energy density is particularly high at the place where the surfaces of the two spheres touch each other. This maximal electric field energy density is very close to the outer surface of the larger sphere. Thus one can detect at the outer surface of the erythrocyte when the nucleus is getting close.

The electric field energy density has maximum when the horizontal line crosses the circles in Figure 2. The x_p coordinates of these cross sections, $x_{p}^{c}$ can be calculated by Eqs 8–10. When the x axis of the coordinate system is the horizontal line (i.e. $y_{P} = 0$ ) the electric field energy density is particularly high at the place where the surfaces of the two spheres touch each other (see orange circles in Figure 2) at close to zero electrolyte ion concentration (i.e. $C = 0.00001 m o l / m^{3}$ ):

$\begin{matrix} u_{F} (x_{P} = - R_{L +}, y_{P} = 0; - x_{1} = Z = 0.8 R_{L} = R_{L} - R_{1}) \\ = \frac{ϵ_{r} ϵ_{0}}{2} (\cdot {[E_{1 x} + E_{L x}]}^{2} + {[E_{1 y} + E_{L y}]}^{2}) \\ = \frac{ϵ_{r} ϵ_{0}}{2} ({[{(- \frac{d V_{1}}{d D_{1}})}_{D_{1} = R_{1 +}} \frac{x_{p} - x_{1}}{R_{1 +}} + {(- \frac{d V_{L}}{d D_{L}})}_{D_{L} = R_{L +}} \frac{x_{p}}{R_{L +}}]}^{2} + {[0 + 0]}^{2}) \\ ≅ \frac{ϵ_{r} ϵ_{0}}{2} ({[- \frac{k_{e} Q_{1}}{ϵ_{r} R_{1 +}^{2}} - \frac{k_{e} Q_{L}}{ϵ_{r} R_{L +}^{2}}]}^{2}) = \frac{2 ρ_{s}^{2}}{ϵ_{r} ϵ_{0}} = 1.4798 \cdot 10^{8} J m^{- 3} \end{matrix}$ (11)

This maximal electric field energy density is very close to the outer surface of the larger sphere ( $| x_{P} | = R_{L +} \geq R_{L}$ ) on the x axis of the coordinate system. The x axis also crosses the small and large spheres at $x_{p} = - 3 R_{1}$ and $x_{p} = R_{L}$ , respectively. Very close to these coss sections, at the outer side of the spheres, the field energy density is only a quarter of the above maximal value.

In general the first maximum of u_F (see the left maximum in Figures 3–5) is getting smaller when the center of the small sphere approaches the center of the large sphere. This is the case because the interaction between the spheres is reducing when the average distance between the surfaces of the two spheres is increasing.

In the case of horizontal lines where $y_{P} > 0 E_{1 y} + E_{L y}$ contributes also to u_F. This contribution is particularly high by E_1y when $x_{p} ≅ - Z$ or by E_Ly when $x_{p} ≅ 0$ relative to the contribution by E_1x and E_Lx, respectively.

When the location of the center of the small and large sphere is identical (i.e. $Z = 0$ ) then because of the additional symmetry $u_{F} (x_{P}, y_{P}) = u_{F} (- x_{p}, y_{P})$ at any value of y_p (see Figure 5).

When $y_{p} > R_{1}$ the horizontal line crosses only the surface of the large sphere at two points and these are symmetric crossing points (where the y axis is the symmetry axis). The distance of the left crossing point from the y axis is similar to the distance of the right crossing point from the y axis (see Eq 8). Because of this symmetry if the small sphere only slightly affect the field strength along the horizontal line then $u_{F} (x_{P}, y_{P}) ≅ u_{F} (- x_{p}, y_{P})$ at any location of the small sphere along the x axis (see blue, black and grey lines in Figures 3–5).

In the case of $0 < y_{p} < R_{1}$ the horizontal line crosses twice the large and twice the small sphere. In the case of the dotted red lines in Figures 3–5 $y_{p} = 0.75 R_{1}$ and each curve has four maxima. The hight of each maxima depends on the square of the field strength at the respective crossing point (see Eq 3), which is related to the x and y components of the field strengths created by the small sphere (E_1x, E_1y) and by the large sphere (E_Lx, E_Ly) (see Eq 7). In order to find out the reason of the hight of each maximum of the curves shown in Figures 3B–5B (i.e. at $C = 0.001 m o l / m^{3}$ ) in Table 1 these x and y components of the field strengths are listed.

Table 1. Values of the x and y components of the electric field strength at the cross sections between a horizontal line (at $y_{p} = 0.75 \cdot R_{1}$ ) and two spheres of radii R₁ and R_L.

Cross #	x_P [m]	u_F [J]	E_1x [V/m]	E_Lx [V/m]	E_1y [V/m]	E_Ly [V/m]
			Z = 0.8 R_L
1	−9.9·10⁻⁷	7.4·10⁷	1.8·10⁸	2.47·10⁸	−1.42·10⁸	−3.74·10⁷
2	−9.4·10⁻⁷	2.2·10⁷	2.29·10⁸	−1.16·10⁸	−2.45·10⁸	1.86·10⁷
3	−6.6·10⁻⁷	4.65·10⁷	−2.29·10⁸	−5.45·10⁷	−2.45·10⁸	1.24·10⁷
4	9.9·10⁻⁷	2.15·10⁷	−9.7·10⁴	−2.47·10⁸	−8.2·10³	−3.74·10⁷
			Z = 0.5 R_L
1	−9.9·10⁻⁷	2.73·10⁷	3·10⁷	2.47·10⁸	−9.22·10⁶	−3.74·10⁷
2	−6.4·10⁻⁷	2.97·10⁷	2.29·10⁸	−5.16·10⁷	−2.45·10⁸	1.2·10⁷
3	−3.6·10⁻⁷	4.1·10⁷	−2.29·10⁸	−2.2·10⁷	−2.45·10⁸	9.16·10⁶
4	9.9·10⁻⁷	2.15·10⁷	−3.2·10⁵	−2.47·10⁸	−3.22·10⁴	−3.74·10⁷
			Z = 0.0 R_L
1	−9.9·10⁻⁷	2.2·10⁷	2.6·10⁶	2.47·10⁸	−3.97·10⁵	−3.74·10⁷
2	−1.4·10⁻⁷	3.63·10⁷	2.29·10⁸	−7.6·10⁶	−2.45·10⁸	8.17·10⁶
3	1.4·10⁻⁷	3.63·10⁷	−2.29·10⁸	7.6·10⁶	−2.45·10⁸	8.17·10⁶
4	9.9·10⁻⁷	2.2·10⁷	−2.6·10⁶	−2.47·10⁸	−3.97·10⁵	−3.74·10⁷

| Show Table

DownLoad: CSV

For example in the case of $Z = 0.5 R_{L}$ (thus $x_{1} = - Z = - 5 \cdot 10^{- 7} m$ ) the reason that the mimum at cross section 3 is higher than at cross section 2 is that at cross section 3 both E_1x(3) and E_Lx(3) are negative while at cross section 2 E_1x(2) is positive and E_Lx(2) is negative. Because of this at cross section 3 ${[E_{1 x} + E_{L x}]}^{2}$ much larger than at cross section 2. Actually because of the symmetry $| E_{1 x} (3) | = | E_{1 x} (2) |$ but $s i g n E_{1 x} (3) \neq s i g n E_{1 x} (2)$ because $s i g n (\frac{x_{p} (3) - x_{1}}{D_{1}}) \neq s i g n (\frac{x_{p} (2) - x_{1}}{D_{1}})$ (see Eq 4).

As an other example in the case of $Z = 0.0 R_{L}$ the maximum at cross section 3 is higher than at cross section 4. The reason is that $| E_{L y} (4) | \cdot | E_{L x} (4) |$ while $| E_{1 y} (3) | ≅ | E_{1 x} (3) | ≅ | E_{L x} (4) |$ . Note that $| E_{L y} (4) | \cdot | E_{L x} (4) |$ because the direction of ${\underline{E}}_{L} (4)$ is close to the direction of the x axis.

Finally, the analytical equation, Eq 7, for the calculation of the electric field energy density of two surface-charged spheres (the smaller sphere located inside the larger sphere), can be generalized for the case when N small surface-charged spheres are located inside the large sphere (see Appendix 2). Also when the radius of the smaller sphere approaches zero the total surface charge of the smaller sphere, Q₁ approaches zero too and consequently the electric field strength of the smaller sphere, E₁ approaches zero. Thus, based on Eq 7 one can calculate the field energy density around a single charged sphere by:

$\begin{matrix} u_{F} (x_{P}, y_{P}) = \frac{ϵ_{r} ϵ_{0}}{2} \underline{E} \cdot \underline{E} = \frac{ϵ_{r} ϵ_{0}}{2} ({\underline{E}}_{L}) \cdot ({\underline{E}}_{L}) \\ = \frac{ϵ_{r} ϵ_{0}}{2} (E_{L x}, E_{L y}) \cdot (E_{L x}, E_{L y}) \\ = \frac{ϵ_{r} ϵ_{0}}{2} ({[E_{L x}]}^{2} + {[E_{L y}]}^{2}) \end{matrix}$

5. Conclusions

Acknowledgments

The author is very thankful for Chinmoy Kumar Ghose.

Conflict of interest

The author declares no conflict of interest.

References

[1]	Chong PLG (2010) Archaebacterial bipolar tetraether lipids: Physico-chemical and membrane properties. Chem Phys Lipids 163: 253-265. https://doi.org/10.1016/j.chemphyslip.2009.12.006
[2]	Ewald PP (1921) Die berechnung optischer und elektrostatischer gitterpotentiale. Ann Phys-berlin 369: 253-287. https://doi.org/10.1002/andp.19213690304
[3]	Sugár IP, Thompson TE, Biltonen RL (1999) Monte carlo simulation of two-component bilayers: DMPC/DSPC mixtures. Biophys J 76: 2099-2110. https://doi.org/10.1016/S0006-3495(99)77366-2
[4]	Almeida PFF (2009) Thermodynamics of lipid interactions in complex bilayers. BBA-Biomembranes 1788: 72-85. https://doi.org/10.1016/j.bbamem.2008.08.007
[5]	Bohinc K, Špadina M, Reščič J, et al. (2022) Influence of charge lipid head group structures on electric double layer properties. J Chem Theory Comput 18: 448-460. https://doi.org/10.1021/acs.jctc.1c00800
[6]	Newton I (1999) The Principia: Mathematical Principles of Natural Philosophy. Berkeley: University of California Press 590.
[7]	Sugár IP (2020) A generalization of the shell theorem. Electric potential of charged spheres and charged vesicles surrounded by electrolyte. AIMS Biophys 7: 76-89. https://doi.org/10.3934/biophy.2020007
[8]	Fetter AL, Walecka JD (2003) Theoretical Mechanics of Particles and Continua. New York: Dover Publications 307-310.
[9]	Holtzer AM (1954) The collected papers of Peter JW Debye. Interscience, New York-London, 1954. xxi+ 700 pp., $9.50. J Polym Sci 13: 548. https://doi.org/10.1002/pol.1954.120137203
[10]	Sugár IP (2021) Electric energies of a charged sphere surrounded by electrolyte. AIMS Biophys 8: 157-164. https://doi.org/10.3934/biophy.2021012
[11]	Ma Y, Poole K, Goyette J, et al. (2017) Introducing membrane charge and membrane potential to T cell signaling. Front Immunol 8: 1513. https://doi.org/10.3389/fimmu.2017.01513
[12]	Bar-Shavit Z (2007) The osteoclast: a multinucleated, hematopoietic-origin, bone-resorbing osteoimmune cell. J Cell Biochem 102: 1130-1139. https://doi.org/10.1002/jcb.21553
[13]	Griffiths DJ (2005) Introduction to electrodynamics. AM J Phys 73: 574. https://doi.org/10.1119/1.4766311
[14]	Nishino M, Matsuzaki I, Musangile FY, et al. (2020) Measurement and visualization of cell membrane surface charge in fixed cultured cells related with cell morphology. PLoS One 15: e0236373. https://doi.org/10.1371/journal.pone.0236373
[15]	Trimble WS, Grinstein S (2015) Barriers to free diffusion of proteins and lipids in plasma membrane. J Cell Biol 208: 259-271. https://doi.org/10.1083/jcb.201410071

biophy-09-01-006-s001.pdf

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Biophysics

1.0 2.2

Metrics

Article views(1977) PDF downloads(56) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(5) / Tables(1)

AIMS Biophysics

Density of electric field energy around two surface-charged spheres surrounded by electrolyte II. The smaller sphere is inside the larger one

Related Papers:

Abstract

1. Introduction

2. Model

3. Results

4. Discussion

5. Conclusions

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Biophysics

Density of electric field energy around two surface-charged spheres surrounded by electrolyte II. The smaller sphere is inside the larger one

Related Papers:

Abstract

1. Introduction

2. Model

3. Results

4. Discussion

5. Conclusions

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog