^{1}

### Abstract

The continuum self-consistent field (SCF) theory is applied to the study of the adsorption of flexible polyelectrolytes (PEs) onto the surfaces of two parallel and infinitely long charged columns, taking into account the short-range monomer-surface non-Coulombic interaction. Due to the complex interplay between the electrostatic and surface interactions, very interesting PE adsorption behaviors in terms of the degree of charge compensation and the bridging chain conformation are found from the numerical solution of the SCF equations. The screening-enhanced salt effect and the permanent adsorption of PEs, irrespectively of the salt concentration, emerge in the presence of the monomer-surface non-electrostatic interaction. The numerical results reveal that, for relatively weak monomer-surface interactions, the degree of charge compensation decreases with increasing monomer-surface interaction. Numerical result shows that, for the strength of monomer-surface interaction above the desorption-adsorption critical value and in a salt-free solution, the total amount of the adsorbed PE chains is linearly proportional to the surface charge density in the high PE charge fraction regime.

The author thanks the financial supports from the National Natural Science Foundation of China (NSFC projects 21074062), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. C.T. acknowledges the supports from K. C. Wong Magna and Hu Lan Funds at Ningbo University.

I. INTRODUCTION

II. THEORY, MODEL EQUATIONS, AND NUMERICAL METHODS

III. RESULTS AND DISCUSSIONS

A. The interaction between the two charged objects

B. Dependences of the degree of charge compensation and the total amount of the bridging chain conformation on the strength of the short-range monomer-surface interaction and the salt concentration

C. Dependences of the degree of charge compensation, the total adsorbed amount, and the total amount of the bridging chain conformation on the charge fraction of PE chains

D. Dependences of the degree of charge compensation, the total adsorbed amount, and the total amount of the bridging chain conformation on the surface charge density

IV. SUMMARY AND CONCLUSIONS

### Key Topics

- Surface charge
- 42.0
- Adsorption
- 34.0
- Double layers
- 27.0
- Polymers
- 24.0
- Electrostatics
- 18.0

## Figures

A schematic representation of the cross-sections of the system under study. The two charged columns are immersed in a polyelectrolyte solution.

A schematic representation of the cross-sections of the system under study. The two charged columns are immersed in a polyelectrolyte solution.

(a) Effect of the salt on the degree of charge compensation at different strengths of monomer-surface non-Coulombic interaction for a specified system parameters of α_{ P } = 0.2, σ*a* ^{2} = 0.04762, *H* = 4 × 2.887*a*, and a system size of (20.0 × 2.887*a*) × (19.0 × 2.887*a*). In the figure, the legends B-I correspond to *u* _{0} = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.7, 1.0, respectively. (b) Effect of the strength of the monomer-surface non-Coulombic interaction on the degree of charge compensation at different salt concentrations for the same system as in Figure 2(a). In the inset, the boundary layer thickness is plotted against the strength of the monomer-surface non-Coulombic interaction at three different salt concentrations. (c) The logarithmic (base 10) plot of the total amount of the bridging chain conformation against the strength of the monomer-surface non-Coulombic interaction at different salt concentrations for the same system as in Figure 2(a). Please note that, in this figure and the subsequent figures, Σ is in the unit of *Rg* ^{2}.

(a) Effect of the salt on the degree of charge compensation at different strengths of monomer-surface non-Coulombic interaction for a specified system parameters of α_{ P } = 0.2, σ*a* ^{2} = 0.04762, *H* = 4 × 2.887*a*, and a system size of (20.0 × 2.887*a*) × (19.0 × 2.887*a*). In the figure, the legends B-I correspond to *u* _{0} = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.7, 1.0, respectively. (b) Effect of the strength of the monomer-surface non-Coulombic interaction on the degree of charge compensation at different salt concentrations for the same system as in Figure 2(a). In the inset, the boundary layer thickness is plotted against the strength of the monomer-surface non-Coulombic interaction at three different salt concentrations. (c) The logarithmic (base 10) plot of the total amount of the bridging chain conformation against the strength of the monomer-surface non-Coulombic interaction at different salt concentrations for the same system as in Figure 2(a). Please note that, in this figure and the subsequent figures, Σ is in the unit of *Rg* ^{2}.

(a) The double logarithmic plot of the boundary layer thickness against the charge fraction of PE chains at a surface charge density of σ*a* ^{2} = 0.04762 and *u* _{0} = 0 for the same system as in Figure 2(a). There is no added salt in the system. In the inset, the double logarithmic plot of the boundary layer thickness against the surface charge density at a PE chain charge fraction of α_{ P } = 0.2 and *u* _{0} = 0 is displayed. (b) The plots of the degree of charge compensation against the charge fraction of PE chains at different monomer-surface non-Coulombic interactions for the same system as in Figure 2(a) with a surface charge density of σ*a* ^{2} = 0.04762. There is no added salt in the system. In the inset, the total amounts of the adsorbed PE chains are plotted against the charge fraction of PE chains at different monomer-surface non-Coulombic interactions. The legends in the inset are the same as in the figure. Please note that, for all the data points at different α_{ P } and *u* _{0} in the inset, the total amount of the adsorbed chains is normalized by that at α_{ P } = 0.05 and *u* _{0} = 0.

(a) The double logarithmic plot of the boundary layer thickness against the charge fraction of PE chains at a surface charge density of σ*a* ^{2} = 0.04762 and *u* _{0} = 0 for the same system as in Figure 2(a). There is no added salt in the system. In the inset, the double logarithmic plot of the boundary layer thickness against the surface charge density at a PE chain charge fraction of α_{ P } = 0.2 and *u* _{0} = 0 is displayed. (b) The plots of the degree of charge compensation against the charge fraction of PE chains at different monomer-surface non-Coulombic interactions for the same system as in Figure 2(a) with a surface charge density of σ*a* ^{2} = 0.04762. There is no added salt in the system. In the inset, the total amounts of the adsorbed PE chains are plotted against the charge fraction of PE chains at different monomer-surface non-Coulombic interactions. The legends in the inset are the same as in the figure. Please note that, for all the data points at different α_{ P } and *u* _{0} in the inset, the total amount of the adsorbed chains is normalized by that at α_{ P } = 0.05 and *u* _{0} = 0.

(a) The plots of the degree of charge compensation against the surface charge density at different monomer-surface non-Coulombic interaction strengths for the same system as in Figure 2(a) with a PE charge fraction of α_{ P } = 0.2. There is no added salt in the system. In the figure, the legends B-E correspond to *u* _{0} = 0.0, 0.3, 0.5, 1.0, respectively. In the inset, the total amounts of the adsorbed PE chains are plotted against the surface charge density at different strengths of the monomer-surface non-Coulombic interaction. The legends in the inset are the same as in the figure. Please note that in the inset, for all the data points at different *u* _{0} and σ*a* ^{2}, the total amount of the adsorbed PE chains is normalized by that at a surface charge density of σ*a* ^{2} = 0.02381 and *u* _{0} = 0.0. (b) The plots of the total amounts of the adsorbed PE chains against the surface charge density at different monomer-surface non-Coulombic interaction strengths (the lower and the upper curves correspond to *u* _{0} = 0.3 and *u* _{0} = 1.0, respectively) for the same system as in Figure 4(a) with a PE charge fraction of α_{ P } = 0.4. In the lower right inset, the total amounts of the adsorbed PE chains are plotted as a function of the surface charge density at different monomer-surface non-Coulombic interaction strengths for the same system as in Figure 4(a) with a PE charge fraction of α_{ P } = 0.05. The legends in the lower right inset are the same as in the figure. Please note that in both the figure and the lower right inset, the total amount of the adsorbed PE chains corresponding to different data points is normalized by the total adsorbed amount at a surface charge density of σ*a* ^{2} = 0.02381 and *u* _{0} = 0.3. In the upper left inset, the monomer density distributions along the central horizontal line (x-axis) at different PE charge fractions (α_{ p } = 0.05, 0.2, 0.4) at the same monomer-surface non-Coulombic interaction of *u* _{0} = 1.0 and the surface charge density of σ*a* ^{2} = 0.02381 for the same system as in Figure 4(a) are displayed. In the upper left inset, the curves with the peak value in a decreasing order correspond to increasing PE chain charge fraction. (c) The plot of the total amount of the bridging chain conformation against the surface charge density for the same system as in Figure 4(a) at *u* _{0} = 0.0. In the inset, the total amounts of the bridging chain conformation are plotted against the surface charge density at different monomer-surface non-Coulombic interaction strengths. The legends C-E in the inset are the same as in Figure 4(a).

(a) The plots of the degree of charge compensation against the surface charge density at different monomer-surface non-Coulombic interaction strengths for the same system as in Figure 2(a) with a PE charge fraction of α_{ P } = 0.2. There is no added salt in the system. In the figure, the legends B-E correspond to *u* _{0} = 0.0, 0.3, 0.5, 1.0, respectively. In the inset, the total amounts of the adsorbed PE chains are plotted against the surface charge density at different strengths of the monomer-surface non-Coulombic interaction. The legends in the inset are the same as in the figure. Please note that in the inset, for all the data points at different *u* _{0} and σ*a* ^{2}, the total amount of the adsorbed PE chains is normalized by that at a surface charge density of σ*a* ^{2} = 0.02381 and *u* _{0} = 0.0. (b) The plots of the total amounts of the adsorbed PE chains against the surface charge density at different monomer-surface non-Coulombic interaction strengths (the lower and the upper curves correspond to *u* _{0} = 0.3 and *u* _{0} = 1.0, respectively) for the same system as in Figure 4(a) with a PE charge fraction of α_{ P } = 0.4. In the lower right inset, the total amounts of the adsorbed PE chains are plotted as a function of the surface charge density at different monomer-surface non-Coulombic interaction strengths for the same system as in Figure 4(a) with a PE charge fraction of α_{ P } = 0.05. The legends in the lower right inset are the same as in the figure. Please note that in both the figure and the lower right inset, the total amount of the adsorbed PE chains corresponding to different data points is normalized by the total adsorbed amount at a surface charge density of σ*a* ^{2} = 0.02381 and *u* _{0} = 0.3. In the upper left inset, the monomer density distributions along the central horizontal line (x-axis) at different PE charge fractions (α_{ p } = 0.05, 0.2, 0.4) at the same monomer-surface non-Coulombic interaction of *u* _{0} = 1.0 and the surface charge density of σ*a* ^{2} = 0.02381 for the same system as in Figure 4(a) are displayed. In the upper left inset, the curves with the peak value in a decreasing order correspond to increasing PE chain charge fraction. (c) The plot of the total amount of the bridging chain conformation against the surface charge density for the same system as in Figure 4(a) at *u* _{0} = 0.0. In the inset, the total amounts of the bridging chain conformation are plotted against the surface charge density at different monomer-surface non-Coulombic interaction strengths. The legends C-E in the inset are the same as in Figure 4(a).

## Tables

The various enthalpic and entropic contributions to the free energy per chain of a polyelectrolyte salt-fee solution with immersed two charged objects at different gap spacings. The system parameters are: *N* = 50, χ_{ PS } = 0.0, , *u* _{0} = 0.5,α_{ P } = 0.05, σ*a* ^{2} = 0.04762, *L* _{ o } = 2.887*a*, and the system size of (12.0 × 2.887*a*) × (5.0 × 2.887*a*). In solving the self-consistent field equations, the size of the time step is 0.002, and the mesh size is 0.015625 *Rg*. In the table, *U* _{ sr } / *k* _{ B } *T*, *U* _{ e } / *k* _{ B } *T*, *S* _{ P } / *k* _{ B }, *S* _{ S } / *k* _{ B }, *S* _{+} / *k* _{ B }, *S* _{−} / *k* _{ B }denote, respectively, the short-range monomer-surface enthalpic contribution, the electrostatic part, the entropic contributions of polymer chains, solvent, counter-ions, and co-ions. Please refer to Ref. 30 for the definitions of the enthalpic and entropic terms.

The various enthalpic and entropic contributions to the free energy per chain of a polyelectrolyte salt-fee solution with immersed two charged objects at different gap spacings. The system parameters are: *N* = 50, χ_{ PS } = 0.0, , *u* _{0} = 0.5,α_{ P } = 0.05, σ*a* ^{2} = 0.04762, *L* _{ o } = 2.887*a*, and the system size of (12.0 × 2.887*a*) × (5.0 × 2.887*a*). In solving the self-consistent field equations, the size of the time step is 0.002, and the mesh size is 0.015625 *Rg*. In the table, *U* _{ sr } / *k* _{ B } *T*, *U* _{ e } / *k* _{ B } *T*, *S* _{ P } / *k* _{ B }, *S* _{ S } / *k* _{ B }, *S* _{+} / *k* _{ B }, *S* _{−} / *k* _{ B }denote, respectively, the short-range monomer-surface enthalpic contribution, the electrostatic part, the entropic contributions of polymer chains, solvent, counter-ions, and co-ions. Please refer to Ref. 30 for the definitions of the enthalpic and entropic terms.

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