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NANOSYSTEMS: Dweik J., et al. Nanosystems:
PHYSICS, CHEMISTRY, MATHEMATICS Phys. Chem. Math., 2023,14 (4), 428-437.
http://nanojournal.ifmo.ru
Original article DOI 10.17586/2220-8054-2023-14-4-428-437
The Space Charge Model. A new analytical approximation solution of Poisson-Boltzmann equation: the extended homogeneous approximation
Jalal Dweik", Hassan Farhatb, Joumana Younisc Jinan University, Tripoli, Lebanon
°[email protected], [email protected], [email protected] Corresponding author: Jalal Dweik, [email protected]
Abstract The validity of different analytical approximations solution is studied using the classical Poisson-Boltzmann (PB) equation based on a mean-field description of ions as ideal point charges in combination with the assumption of fully overlapped electrical double layers in the membrane pores. The electrical conductivity is calculated by numerical and approximate analytical methods in order to explain the process of ion transport. In this paper, a new analytical approximation named the extended homogeneous approximation (EH) is presented, which provides better results than the homogeneous approximation based on Donnan theory. Also, the results show that the electrical conductivity calculated by the EH, is coherent with the numerical method within specific limits.
Keywords space charge model, Poisson-Boltzmann (PB) equation, electrical conductivity, extended homogeneous (EH) approximation
For citation Dweik J., Farhat H., Younis J. The Space Charge Model. A new analytical approximation solution of Poisson-Boltzmann equation: the extended homogeneous approximation. Nanosystems: Phys. Chem. Math., 2023, 14 (4), 428-437.
1. Introduction
Nanofiltration membranes (NF) are particularly relevant for industrial applications because of their valuable features, such as pore diameters smaller than 2 nm [1-3]. NF has several advantages over other filtration processes, including lower energy consumption, lower operating pressures, and the ability to selectively remove specific components. One of the most interesting applications of these membranes is the separation of ions [4,5]. The separation of ions in NF occurs due to a combination of size exclusion, charge repulsion, and electrostatic attraction. The selectivity of NF membranes can be further enhanced by adjusting the pH of the solution or using different membrane materials with specific surface charges. However, the mechanism of transport through NF membranes is not yet well-established. To shed light on this topic, numerous researchers have utilized different macro and mesoscopic approaches [6-10] to elucidate the physical and chemical mechanisms underlying membrane transport. The space charge (SC) model, which is based on the Poisson-Boltzmann (PB) equation [11], describes ions as ideal point charges and assumes fully overlapped electrical double layers in the membrane pores. This model is well-suited for studying ion transport through charged nanopores membrane. However, the accuracy of the SC model and other approximations, such as Donnan's theory or the homogeneous theory based on the electroneutrality in the pore, is challenged when applied to NF. In this paper, the SC model is applied to study the limit of effectiveness of different areas that provide a physical idea about the behavior of the salt-nanopore system using analytical approximations (homogeneous approximation, good exclusion of co-ions, and plane approximation) and numerical calculations obtained from the PB equation. In this paper, a new analytical approximation solution, named the extended homogeneous approximation (EH) is presented. This approximation allows the researchers to use only one approximation instead of all other approximations mentioned above within specific limits. Finally, the validity of the EH approximation is studied by calculating the electrical conductivity and depends on three main parameters: pore radius, concentration of ions in the bulk, and surface charge density of the pore. These three parameters play a significant role in the transport through the pores. The remainder of this paper is organized as follows. The description of the models is given in section 2. The results and discussion are presented in section 3. The conclusions and suggestions are given in section 4.
2. Space charge model
The space charge model was developed by Osterle and colleagues [12-14]. It is a theoretical framework used to describe the behaviour of charged solutes near charged surfaces, such as nanofiltration membranes. It considers the electrostatic interactions between the charged solutes and the charged membrane surface. This model considers axial and radial variations in pressure, electric potential, and concentration. Assume a simple cylindrical pore of radius rp and length d > rp, filled with an aqueous solution with two ionic species [i = 1 (counter-ion), 2 (co-ion)]. The pore's
© Dweik J., Farhat H., Younis J., 2023
electrical properties are defined by an assumed uniform surface charge density that depends on the ionic concentrations and the pore surface pH (Fig. 1).
Bulk solution (C)
Fig. 1. Description of the model with the important parameters
Suppose that the ionic transport is at the steady state, so that there is no time function in the governing equations. The continuity equation for each ion can be expressed as
V- ji =0, (1)
where ji is the molar flux density of ion i, described by the extended Nernst-Planck equation
F
ji = -DiVci - RTziDiCjV$ + CiU, (2)
where Di is the local diffusion coefficient of the ion i, ci is the local concentration of the species considered, zi is the ion valence, $ is the total local electric potential, F is the Faraday constant.
The flow velocity is obtained by the Navier-Stokes equation. Neglecting all inertial terms ("creeping flow approximation") and the inclusion of electrical forces, the Navier-Stokes equation is expressed by
nV2U = pV$ + Vp, (3)
where p is the local pressure applied to the fluid, n is the dynamic viscosity of the solution, p is the ionic charge density, which is defined by the following formula
2
p = F^ZiCi. (4)
i= l
When the density of the fluid is constant, there is, in addition, the incompressibility condition
VU = 0. (5)
The key of the space charge model is the Poisson-Boltzmann (PB) equation for electric potential and ionic concentration
V2$ = p, (6)
£
V2$ =--1----1----(7)
22 ox2 r or or2
where £ = £0£r is the absolute dielectric constant of the solution, £0 is the dielectric constant in vacuum (£0 = 8.85PF/m), £r is the relative dielectric constant for water (£r = 78.54).
The normal component of the molar flux density and velocity is equal to zero at the surface. Thus the boundary condition for Eq. (6) takes the following form:
= dci
p dr
ji,r
zi F
r=rp + RTCi ~dT
0. (8)
By symmetry, another boundary condition of Eq. (6) which corresponds to the concentration and electric potential at the pore center is fined:
dci dr
d $
= 0, jr-r=0 dr
= 0, (9)
r=0
and so
ji,r
0.
The global surface charge aw is related to the gradient of the potential via the Gaussian equation:
дФ
dr
£
(10)
(11)
If the tangential component of the velocity at the pore surface is zero (no slip), then we can write the boundary condition of the Navier-Stokes equation
1 0. (12)
By symmetry,
dux dr
0.
(13)
The total local electric potential $(x, r) can be divided into two parts:
$(x, r) = ^(r) + <^(x), (14)
where ^(r) is the potential due to the electrical double layer at the surface on the radial direction and the potential y(x) is due to the bulk (outside the pore) along the x axis. Assume that the variation of y(x) along x axis is negligible compared to the variation of the radial component, then Eq. (7) becomes as follows
1 дф д2ф
У2Ф « +
(15)
r dr dr2
Assuming that Eqs. (8) and (10) allow the approximation ji (r) = 0 to be used for any r, then this equation can be integrated between 0 < r < rp to find the "radial equilibrium" approximation which gives one the concentration c^(r) (Boltzmann distribution)
ci(r) « ki(r)viC, (16)
where
ki (r) = exp
iFф(r) RT
(17)
C is the electrolyte concentration equivalent to the concentration of a bulk solution in equilibrium with the pore at position x and vi (the stoichiometric coefficient) represents the number of ions coming from the dissociated electrolytes. In the case of two ionic species i = 1, 2, the condition of electroneutrality is described by vizi + v2z2 = 0 or vi|zi| = v2|z2|:
(Counter-ion)^1 (Co-ion)V2 —» vi(Counter-ion)Z1 + v2(Co-ion)Z2 The index (i = 1) represents the counter ions and the index (i = 2) represents the co-ions.
After substitution of Eqs. (15) and (16) in (6), the Poisson-Boltzmann equation which determines the electric potential inside the pore is obtained. It is presented in dimensionless form [14] by the following relation:
2д
дф
where
dr \ dr
ф = _ ^ф(х>г) > 0, RT
e+V> _ g-"/-
ы
|zi| '
Ac is the Debye length associated with the concentration C given by:
Ac =
RT£
2v1 z2F 2C
C-1/2.
The boundary conditions at the center and surface of the pore can be presented in dimensionless form [11-14]:
дф dr
where
4 * дФ
= 4a ,
/=i dr
1r
r=0
4 V Am
Am is the Debye length associated with the membrane charge density
A = , 2eRT ^ 1/2
Am
. |zi IF2|Xm|
and R is the constant of the ideal gas and Xm is defined by
Xm
|Xm |
-1/2
2a w
FTP
(18)
(19)
(20)
(21) (22)
(23)
(24)
r
r=r
P
r=r
P
r
z
2
r
p
A
c
r
r
p
ж
0
The membrane charge density |Xm | is expressed in moles per unit volume of the pore. It is proportional to |aw | and inversely proportional to the radius of the pore rp. The average flux density in the pore can be calculated by integrating the local flux using the formula:
rP
- 2 i
ji,x = ~2 ji,xrdr. (25)
rp
0
The average electrolyte flow density is denoted by J, the volume flow density - by Jv and the current density - by I. It should be noted that the volume flow rate Jv is a very good approximation to the barycentric velocity Um [13].
J = ^2 ji, Jv = ux, (26)
i=i
with the current density given by
2
1 = F^ZiJi- (27)
i=l
From Eq. (3), using Eqs. (8) and (13) for ux and component x from Eq.(2) and averaging the fluxes over the pore cross-section by Eq. (27), the following system of transport equations is obtained:
J = -Lii (£) - (£) - Li3 (^) •
J = -L21 (H0) - L22 (t) - L23 (g) , (28)
1 = -L311 -L32{dx)-L33{£)'
where p is the chemical potential given by
and P0 is the effective pressure
with n the osmotic pressure defined according to:
p = RT ln(C ), (29)
Po = P - n, (30)
n = RT (v! + v2 )C. (31)
The term of conductivity is given by
k = L33 = l( d~A , (32)
dx where
L33 = L33 + L33 > (33)
L33 = [diffusive conductivity] + [convective conductivity] such as
kd = L'33 =(CvRZTF2) [Diziki - D2Z2k2] , (34)
Kc = L33 = ((CviziFrp)2 r/ ^ \kig - k2g] . (35)
Using the definition
i
y = 2 yr dr, (36)
0
ki and kig can be calculated with the function which is defined [12,13] by
1 r" 1
g = j — j (ki — k2 ) r'dr'dr''.
(37)
The function defined above can be simplified by using the PB equation and Green's theorem to find
g(r)=2(rp/Xc)-2 U(r) - . (38)
2
The conductivity can therefore be calculated from the electrostatic potential (r ) inside the pore. So, in this case, it is necessary to solve the PB equation in dimensionless form to evaluate V>(f)(V>w = i> (rp). This potential is a function of the dimensionless parameters rp/Xm and rp/Xc, which are proportional, respectively, to |aw |1/2 and Cs1/2
r
^ = 7.485 (zirpaw)1/2 , = 3.288 (^zjCr^2 , (39)
"m "c
with the surface charge density aw in C/m2, the pore radius rp in nm and the bulk concentration C in moles/l.
2.1. Homogeneous analytical approximation (H)
The homogeneous approximation is used to analyze the behaviour of ions inside a cylindrical pore. In a charged pore, ions move through the pore driven by electrostatic forces due to the presence of a fixed charge distribution on the pore surface. The homogeneous approximation assumes that the charge density along the length of the pore is uniformly distributed. By making this approximation, the problem of ion transport in the cylindrical pore can be simplified. This simplification greatly facilitates the mathematical analysis. The homogeneous theory, which assumes a uniform potential, is valid when the pore radius is small and the surface charge density is lower, according to the space charge model [15]. Within the membrane pore, the macroscopic Donnan equilibrium theory can be used in combination with global charge electroneutrality then
0,
i=1
with
where
Ci — kiViC,
ki
exp
Z 7
The value of the potential is given by Eq. (42) which also allows one to write down:
e — kh - k-
|Xm|
|zi |viC
with
e — 4(rpMm)2
(rP/Xcf
(40)
(41)
(42)
(43)
kh = e^0 (44)
and £ is the normalized membrane charge density in dimensionless form given in terms of (rp/Xm) and (rp/Xc) by
(45)
In the case of the symmetric salt (t = 1), the solution of the quadratic equation kh - ekh - 1 = 0 is given by the formula
kh = 1 (e+Ve2+4). (46)
2
2.2. The approximate solutions of the Poisson-Boltzmann equation
2.2.1. Numerical method. The simplified Poisson-Boltzmann Eq. (18) is a non-linear partial differential equation and a fundamental equation used to find the electrostatic potential in porous systems [14]. In this work, a numerical approach uses the finite difference method to approximate solutions in one dimension. The proposed numerical method involves dis-cretizing the spatial coordinate, applying suitable boundary conditions, and iteratively updating the electrostatic potential until convergence.
2.2.2. Analytical solution. Additionally, the article explores analytical approximations solution for specific scenarios with simplified charge distributions and boundary conditions, providing insights into the behavior of simple systems (Good Co-Exclusion Approximation, Plane approximation, and Debye-HUuckel approximation). By comparing the numerical and analytical approaches, researchers gain valuable tools to efficiently analyze charged systems in radial direction and better understand their electrostatic behaviour in porous system. However, the validity of the analytical approximations depends on the value of the dimensional electrostatic potential V> (r ). Therefore, the ranges of validity of different types of analytical approximations vary in function of the parameters (rp/Xm, rp/Xc).
2.2.3. Good Co-Exclusion Approximation (GCE). The concept of "Co-Exclusion" arises from the fact that ions of the same charge have a strong tendency to repel each other, leading to an excluded volume effect. The "Good Co-Exclusion Approximation" is an approach used to simplify the PBE solution by incorporating this excluded volume effect, particularly, for ionic solutions with concentrated charges. In the Good Co-Exclusion Approximation, the ions are treated as hard spheres, and their excluded volume effects are taken into account by modifying the boundary conditions in the PBE. Instead of considering the actual physical boundaries of the ions, the boundary conditions are adjusted to reflect the presence of neighboring ions and their excluded volumes. This approximation assumes that the ions cannot penetrate each other's excluded volumes and considers their spatial arrangement to account for the effect of ionic crowding. By using the Good Co-Exclusion Approximation, the PBE can be simplified, and the computational cost can be significantly reduced while still providing reasonably accurate results for systems with concentrated ionic solutions. However, it is important to note that this approximation is most applicable to systems with relatively low ion concentration. It is valid for high potential values f > 1, because the second term on the right of Eq. (18) becomes negligible. In this limit, the solution of PB equation [13,14] between 0 < r < 1 is given by
0(r) = 2ln ( — r„
+ f (r a*),
with
f (T a * ) = ln
16a * (1 + a * )
(1 + a * - a * r2)2_
(47)
(48)
By combining Eq. (22) and Eq. (47), the GEC approximation is valid when f (0) > 1 (minimum value of f at the origin)
-f I < 2.426
Ac
Am
4+1 ^
-1/2
(49)
Staying within the range of validity of the GEC approximation, when the value of a * becomes small, the function appearing in Eq. (47) can be simplified. In this case, the homogeneous approximation for the electrostatic potential is valid in the whole range 0 < r < 1 with
|Xm 1
The homogeneous (rp/Ac) < 1.21 (rp/Am).
pV'C?) _
GEC approximation is valid when
(50)
16a *
(rpAc )2
(rp/Am) < 2 and inequality (49) simplifies to
2.2.4. Plane and Debye-Huckel approximation. Solving the Poisson-Boltzmann equation directly in three dimensions can be challenging, especially for complex systems. In some cases, it is possible to simplify the equation and make approximations to make the problem more tractable. One such approximation is the plane approximation or the Debye-Huckel (DH) approximation. The plane approximation assumes that the charge distribution is confined to a single plane, and the system's behaviour is effectively two-dimensional. This approximation is valid for situations where the charged species are confined to a surface. Solving this simplified one-dimensional Poisson-Boltzmann equation can provide valuable insights into the electrostatic behaviour near the charged plane, making it easier to study systems like charged membranes. The DH approximation is a method to simplify the PB equation under specific conditions related to dilute solutions and weak electrostatic interactions between ions. For a solution containing a symmetric salt at a concentration C, the nonlinear PB equation can be solved analytically, where the potential f is different from zero only at the pore surface. In this case one has:
1 d20 r dr dr2
(51)
Apply the change of variable r = 1 - t, and consider a uniform potential f w at the surface, the variation of the potential becomes strong near the surface of the pore, as opposed to inside the pore where its value is close to zero. In this approximation [14], the electrostatic potential at a distance t from the surface satisfies the plane approximation of the PB equation.
w
-p ) sinh(-i/>). Ac
After integration, the solution of this equation can be found using the boundary condition at the pore surface:
" 1 + u(t) "
0 (i) = 2 ln
1 - u(t)_
where
u = tanh(h)e
— (rp/Ac)t
h = ^arsinh
(rp/Am) 2 (rp/Ac)
(52)
(53)
(54)
2
r
p
2
At the limit, for a low potential and dilute solution, the solution takes the form of the plane DH approximation.
0©
(fp/An
(rp/Ac)
)2 -e
(rp/Ac)t
(55)
We observe that the plane approximation is valid when rp/Ac > 11. It's important to note that the plane approximation is valid only when the charged particles are distributed primarily along the x-axis and when the distance between the charged particles and the plane is much larger than the characteristic size of the charged particles themselves. In cases where the system deviates significantly from these conditions, the plane approximation may not be accurate.
3. Results and discussion
3.1. Extended homogeneous approximation (EH)
EH is a new analytical approximation solution based on the Donnan theory (homogeneous approximation in the pore). It can be studied by adding a small radial variation 0^f) < 1. The potential can be expressed as 0(f) = 0o + a01 (f), an approximate analytical solution to Eq. (18) with a development parameter, a, which depends on |aw | and takes the value 1 at the end of the calculation. The expansion of the PB equation to the first order of a according to the boundary condition given by:
(56)
2V2(0 1) - W0 1 = R,
d0 i df
dfi df
r=o
with
W =
(A:)2 [kh+], r =(A;)2 [kh -]
For a given value of 0o, the analytical solution to Eq. (73) is as follows
f (fp/Am) 01 (f) = —-
Io(mf )
_ I1(m) _
-2
o/An
(57)
(58)
(59)
(60)
with
m = (W/2)1/2 , (61)
10 and I1 are the modified Bessel functions of order 0 and 1, respectively.
When the value of 0 0 is zero, we fall on the Debye-Hukel approximation, for which the approximation is valid if
0 i(1) < 1 then
An
Io(V ) I1(V )
with
V
1+TA1/2 / fp
2
A;
(62)
(63)
The EH approximation can be represented in the plane (rp/Am, rp/Ac) by the inequality 01 (1) < 1. If this radial function tends to 0, it moves to the domain of validity of the homogeneous approximation (rp/Am < 2). For the GEC zone, the extended homogeneous approximation is valid for rp/Am < 2.4, while outside the GEC zone, the EH zone includes and extends the homogeneous and the Debye-Hukel (DH) zones. Fig. 2 summaries the validity domains of different analytical approximations solution [16].
3.2. Electrical conductivity
The electrical conductivity in charged nanopores is determined by various factors, including the size of the nanopore, the surface charge density (Table 1), the concentration and type of ions presented in the pore (Table 2), and the properties of the surrounding medium. However, the behaviour of electrical conductivity in charged nanopores can be complex and dependent on many factors, and may require detailed computational studies to fully understand. The conductivity is computed by numerical approach using equation (33) and compared to the conductivity calculated by homogeneous and extended homogeneous analytical approximations. Using the EH approximation, the conductivity k can be calculated analytically and is given by:
k = LH + ¿L33, (64)
where the homogeneous conductivity is given by
L33
(CV1NF2/RT) (D1Mkh + D2MV) + ((CV1Z1F)2 LP) £2-
(65)
0
2
f
m
2
Valid Homogeneous Approximation
(8) j (7)
(1) # « 1 ; (2) ■ % / 1 / / l3) / X 1 / (4) -GEC -DH -EH
— 1 $ >> l] (6)
D 1 2 3 Ie. 4 s e
Fig. 2. Representation of the validity domains of different analytical solution approximations [16]. Representation of different zones: (1) and (2) - Validity of the Homogeneous and Debye-HUckel (DH) zone; (1), (2) and (3) - Validity of EH zone; (5) and (6) - Validity of the Homogeneous zone and inhomogeneous GEC; (4) - Intermediate zone; (7) - Validity of the plane approximation; (8) - Validity of the plane Debye-Huckel approximation. The blue curve delimits the area of EH and the dashed line delimits the H area.
The hydraulic permeability of the membrane is given by the following formula
L0p = r2p/&n
and the correction term of the homogeneous approximation of order 1 is given by
(Cv^Frpf e2
ÔL33
2n
1+1 1o(m) if 1o(m)
4 mIi(m) 2 y Ii (m)
(66)
(67)
Table 1. The Parameter rp/Xm, the surface charge density and membrane charge density Xm for rp = 2 nm
rp/Xm aw (C/m2) Xm (mol/l)
0.18 0.0003 0.0031
4 0.15 1.55
6.4 0.37 3.83
8 0.58 6.01
10.4 0.97 10.05
2
Table 2. Parameter rp/Xc and concentration in the bulk for rp = 2 nm
rp/Xc C (M)
0.2 10-3
0.6 10-1
6.5 1
11.4 3
When £ tends to 0, then the correction term ¿L33 tends to 0. In this case, the conductivity in the homogeneous zone is attained. Fig. 3 shows that at low surface charge densities, the electrical conductivity is dominated by the properties of the electrolyte solution, and the conductivity decreases as the size of the nanopore decreases due to increased surface effects. At higher surface charge densities, the electrical conductivity may increase due to the formation of ion transport channels or ion hopping mechanisms. At low surface charge density, Fig. 3 shows that the conductivity in a charged pore is proportional to the ionic concentration of the counter ions compensated by the volume charge density of the membrane (contribution of the electrical double layer). It turns out that the SC model predicts very well the experimentally observed conductivity in cylindrical nanopores of radius rp > 3 nm [16]. This indicates that the model captures the relevant physical phenomena and properties of the nanopores accurately. In this context, the homogeneous approximation is coherent with the numerical method up to rp/Am « 3, while the EH remains coherent untill to rp/Am « 6. Indeed, the EH approximations can be valuable analytical methods to calculate physical quantities of certain systems, including membranes for low and high surface charge density aw . The issue of the validity of this model for small pore radius rp < 2 nm remains an open question.
Homogeneous
Numeric
EH
Fig. 3. Conductivity k as a function of parameter rp/Am in a nanopore of radius rp = 2 nm containing KCl salt. The conductivity is calculated by numerical, Homogeneous and extended homogeneous approximation
4. Conclusion and perspective
The aim of this paper was to highlight the influence of physical parameters on the ionic transfer, especially the concentration of the bulk and the surface charge density aw. Different analytical approximations based on the space charge (SC) model (Homogeneous, Good Co-Ion Exclusion and Planar approximation) is described in this paper. A new analytical approximation named the extended homogeneous approximation is presented. The behavior of the salt-porous system depends mainly on the surface charge density and the concentration in the bulk. The calculation of the conductivity by numerical and approximate analytical methods shows that the Extended Homogeneous approximation (EH) is coherent with the numerical method up to a limit value of rp/Am « 6, corresponding to 0.37 value of surface charge density. Then, without going through a heavy numerical method, the EH approximations can be used as simple analytical methods to calculate the physical quantities of the membrane such as electrical conductivity and flow potential within specific limits, help the development of new applications in water treatment, food processing, and the recovery of valuable materials from industrial waste streams.
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Submitted 4 July 2023; revised 25 July 2023; accepted 28 July 2023
Information about the authors:
Jalal Dweik - Jinan university, Tripoli Lebanon - Main Campus: Zaytoun Abi-Samra, P. O. Box: 818; ORCID 0009-004-2988-0918; [email protected]
Hassan Farhat - Jinan university, Tripoli Lebanon - Main Campus: Zaytoun Abi-Samra, P. O. Box: 818; ORCID 0009-0005-4696-9786; [email protected]
Joumana Younis - Jinan university, Tripoli Lebanon - Main Campus: Zaytoun Abi-Samra, P. O. Box: 818; ORCID 0000-0003-0576-4264; [email protected]
Conflict of interest: the authors declare no conflict of interest.
