Attraction of Like-Charged Walls with Counterions Only: Exact Results for the 2D Cylinder Geometry

Šamaj, L.

doi:10.1007/s10955-020-02642-9

Cited by 7 publications

(7 citation statements)

References 88 publications

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“…see also [59,60,61]. The counterion density at the contact with the charged line n(0) = 2πσ 2 is in agreement with the contact value theorem (4.6) taken at the coupling constant Γ = 2.…”

Section: The Exactly Solvable Coupling γ =supporting

confidence: 82%

“…Another advantage of the 2D one-component systems is an explicit representation of their partition function and particle densities for the sequence of the coupling constants Γ = 2γ where γ is a positive integer (in the fluid regime), expressing the integer powers of Vandermonde determinant by using either Jack polynomials [53,54] or anticommuting-field theory defined on a one-dimensional (1D) chain of sites [55,56,57,58]. In the case of uniform surface charge densities with counterions only at Γ = 2, the density profile of counterions for the one-wall geometry was derived in [59,60] and the effective interaction between two parallel asymmetrically charged plates at distance d in [61].…”

Section: Introductionmentioning

confidence: 99%

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Electric double layers with modulated surface charge density: Exact 2D results

Šamaj

2022

Preprint

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Electric double layers (EDL) with counterions only, say electrons with the elementary charge −e, in thermal equilibrium at the inverse temperature β are considered. In particular, we study the effect of the surface charge modulation on the particle number density profile and the pressure. The mobile particles are constrained to the surface of a 2D cylinder and immersed in vacuum (no dielectric image charges). An EDL corresponds to the end-circle of the cylinder which carries a (fixed) position-dependent line charge density. The geometries of one single EDL and two EDLs at distance d are considered; the particle density profile is studied for both geometries, the effective interaction of two EDLs is given by the particle pressure on either of the line walls. For any coupling constant Γ ≡ βe 2 = 2 × integer, there exists a mapping of the 2D one-component Coulomb system onto the 1D interacting anticommuting-field theory defined on a chain of sites. Using specific transformation symmetries of anticommuting variables, the contact value theorem is generalized to the EDL with modulated line charge density. For the free-fermion coupling Γ = 2 it is shown that, under certain conditions, the matrix of interaction strengths between anticommuting variables diagonalizes itself which permits one to obtain exact formulas for the particle density profile as well as the pressure. The obtained results confirm the previous indications of weak-coupling and Monte Carlo observations that the surface charge inhomogeneity implies an enhancement of the counterion density at the contact with the charged line and a diminution of the pressure between two parallel lines in comparison with the uniformly charged ones (with the same mean charge densities).

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“…see also [59,60,61]. The counterion density at the contact with the charged line n(0) = 2πσ 2 is in agreement with the contact value theorem (4.6) taken at the coupling constant Γ = 2.…”

Section: The Exactly Solvable Coupling γ =supporting

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Electric double layers with modulated surface charge density: Exact 2D results

Šamaj

2022

Preprint

Self Cite

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show abstract

“…The sum over j simplifies itself in the thermodynamic limit N, W → ∞, with the fixed ratio N/W = σ; in this limit, the wall corresponds to an infinite line and charges interact via the pure 2D logarithmic Coulomb potential. Introducing the continuous variable t = 1 N j + 1 see also [59][60][61]. The counterion density at the contact with the charged line n(0) = 2πσ 2 is in agreement with the contact value theorem (4.6) taken at the coupling constant Γ = 2.…”

Section: The Exactly Solvable Coupling γ =supporting

confidence: 67%

“…Another advantage of the 2D one-component systems is an explicit representation of their partition function and particle densities for the sequence of the coupling constants Γ = 2γ where γ is a positive integer (in the fluid regime), expressing the integer powers of Vandermonde determinant by using either Jack polynomials [53,54] or anticommuting-field theory defined on a one-dimensional (1D) chain of sites [55][56][57][58]. In the case of uniform surface charge densities with counterions only at Γ = 2, the density profile of counterions for the one-wall geometry was derived in [59,60] and the effective interaction between two parallel asymmetrically charged plates at distance d in [61].…”

Section: Introductionmentioning

confidence: 99%

Electric double layers with modulated surface charge density: exact 2D results

Samaj¹

2022

J. Phys. A: Math. Theor.

Self Cite

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Electric double layers (EDL) with counterions only, say electrons with the elementary charge −e, in thermal equilibrium at the inverse temperature β are considered. In particular, we study the eﬀect of the surface charge modulation on the particle number density proﬁle and the pressure. The mobile particles are constrained to the surface of a 2D cylinder and immersed in vacuum (no dielectric image charges). An EDL corresponds to the end-circle of the cylinder which carries a (ﬁxed) position-dependent line charge density. The geometries of one single EDL and two EDLs at distance d are considered; the particle density proﬁle is studied for both geometries, the eﬀective interaction of two EDLs is given by the particle pressure on either of the line walls. For any coupling constant Γ ≡ βe2= 2 × integer, there exists a mapping of the 2D one-component Coulomb system onto the 1D interacting anticommuting-ﬁeld theory deﬁned on a chain of sites. Using speciﬁc transformation symmetries of anticommuting variables, the contact value theorem is generalized to the EDL with modulated line charge density. For the free-fermion coupling Γ = 2 it is shown that, under certain conditions, the matrix of interaction strengths between anticommuting variables diagonalizes itself which permits one to obtain exact formulas for the particle density proﬁle as well as the pressure. The obtained results conﬁrm the previous indications of weak-coupling and Monte Carlo observations that the surface charge inhomogeneity implies an enhancement of the counterion density at the contact with the charged line and a diminution of the pressure between two parallel lines in comparison with the uniformly charged ones (with the same mean charge densities).

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“…It is well known that charged colloids (i.e., macroions) have typically a low relative dielectric constant (

≈ 2−5) which is much smaller than that of the surrounding solvent (e.g., for water

≈ 80). In most of the simulation and theoretical works [ 12 , 13 , 14 , 15 , 16 , 17 , 18 ], it is generally assumed that the electrolyte solution and electrode have the same relative dielectric constant.…”

Section: Introductionmentioning

confidence: 99%

On Capacitance and Energy Storage of Supercapacitor with Dielectric Constant Discontinuity

Zhou

2022

Nanomaterials

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The classical density functional theory (CDFT) is applied to investigate influences of electrode dielectric constant on specific differential capacitance Cd and specific energy storage E of a cylindrical electrode pore electrical double layer. Throughout all calculations the electrode dielectric constant varies from 5, corresponding to a dielectric electrode, to εwr = 108 corresponding to a metal electrode. Main findings are summarized as below. (i): By using a far smaller value of the solution relative dielectric constant εr = 10, which matches with the reality of extremely narrow tube, one discloses that a rather high saturation voltage is needed to attain the saturation energy storage in the ultra-small pore. (ii): Use of a realistic low εr = 10 value brings two obvious effects. First, influence of bulk electrolyte concentration on the Cd is rather small except when the electrode potential is around the zero charge potential; influence on the E curve is almost unobservable. Second, there remain the Cd and E enhancing effects caused by counter-ion valency rise, but strength of the effects reduces greatly with dropping of the εr value; in contrast, the Cd and E reducing effects coming from the counter-ion size enhancing remain significant enough for the low εr value. (iii) A large value of electrode relative dielectric constant εrw always reduces both the capacitance and energy storage; moreover, the effect of the εrw value gets eventually unobservable for small enough pore when the εrw value is beyond the scope corresponding to dielectric electrode. It is analyzed that the above effects take their rise in the repulsion and attraction on the counter-ions and co-ions caused by the electrode bound charges and a strengthened inter-counterion electrostatic repulsion originated in the low εr value.

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Attraction of Like-Charged Walls with Counterions Only: Exact Results for the 2D Cylinder Geometry

Cited by 7 publications

References 88 publications

Electric double layers with modulated surface charge density: Exact 2D results

Electric double layers with modulated surface charge density: Exact 2D results

Electric double layers with modulated surface charge density: exact 2D results

On Capacitance and Energy Storage of Supercapacitor with Dielectric Constant Discontinuity

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