A numerical study of two opposing polyelectrolyte brushes by the self-consistent field theory

Abstract

The self-consistent field theory (SCFT) is employed to numerically study the interaction and interpenetration between two opposing polyelectrolyte (PE) brushes formed by grafting PE chains onto the surfaces of two infinitely long and parallel columns with rectangular-shaped cross-sections immersed in a good solvent. The distribution of counter-ions and the dependence of the brush height on various system parameters are also examined. The numerical result reveals that the brush height scales linearly with the grafting density, the average degree of ionization of PE chains and the chain length. The numerical result suggests that, at a brush separation comparable to the brush height, the two opposing PE brushes shrink in order to avoid mutual interpenetration. The numerical study strongly indicates that the compression and retraction of PE brushes is more dominant than the mutual interpenetration for two apposing PE brushes. It is found that, except in the long chain length and/or small system volume limits, in a salt free solution, only about 50% of the counter-ions are trapped inside the PE brushes. Moreover, the relationship between the present two dimensional PE brushes and one dimensional planar PE brushes in terms of the percentage of the counter-ions trapped inside the PE brushes is discussed.

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I. Introduction

Polyelectrolyte (PE) brushes consist of charged polymer chains densely grafted onto planar or curved surfaces by covalent bonds or physical adsorption,^1,3 which are immersed in a solvent. PE brushes play a critical role in modifying and regulating material surface properties, and are important in chemical, biological and material contexts. In the last few decades, PE brushes have received much attention because of their technological applications in colloidal stabilization,⁴ surface modification,⁵ oil recovery,⁶ lubrication,^7,8 smart materials,⁹ etc. Because of their wide applications in so many fields, PE brushes have been studied intensively by theories,^10–17 simulations,^18–35 and experiments.^36–43

PE brushes share many features with neutral polymer brushes, but qualitatively new properties emerge due to the presence of charged monomers and counter-ions in the brush. From scaling arguments,^10–12 in the so-called osmotic brush regime where almost all the counter-ions are trapped inside the brush, by balancing the chain elasticity with the counter-ion osmotic pressure, the equilibrium brush height is found to be linearly proportional to the chain length, but independent of the grafting density. However, recent experimental as well as simulation works^21,40,42 show a weak increase of the brush height with the grafting density. A so-called non-linear osmotic brush regime has been proposed.⁴²

The interactions between polyelectrolyte brushes which are grafted onto two opposing surfaces have recently attracted much attention in experiments^36–40 and simulations.^23–30 These interactions are closely related to the stabilization of colloids grafted with PE brushes, the lubrication of surfaces by grafting PE brushes, etc. The interactions between polyelectrolyte brushes depend on polymer chain flexibility, the entropic effect of counter-ions, the electrostatic force of the system and the spatial steric factors. Theories and experiments using the surface force apparatus have studied the frictional behavior of PE brushes^8,44–52 and revealed a very low frictional coefficient when sliding one PE brush relative to the other.⁵¹ Molecular simulation techniques such as molecular dynamics (MD) and dissipative particle dynamics (DPD) have been used to study the interpenetration and the frictional behavior between two opposing PE brushes.^{23,25,26,30,32,50} From MD simulations, it is found that PE brushes have a lower frictional coefficient and a lower degree of monomer interpenetration than neutral polymer brushes with the same grafting density and chain length, and the PE brushes can support a much higher normal load than the neutral brushes for the same degree of compression due to the counter-ion osmotic pressure.³⁰ DPD simulation studies show that the frictional coefficient is directly related to the interpenetration depth of polyelectrolyte chains, and a larger interpenetration depth results in a higher frictional coefficient.³²

It should be pointed out that in the studies of the interactions and mutual interpenetration between two PE brushes, virtually all the simulation works thus far focused on infinitely large planar PE brushes. The case of finite sized PE brushes is rarely touched upon. On the other hand, there are quite a few numerical studies of the interactions and mutual interpenetration between two finite sized neutral polymer brushes with different geometries such as spherical brushes, tablet brushes.^2,53–58 In this paper, we employ a continuum self-consistent field theory (SCFT) to numerically study the interaction and mutual interpenetration between two PE brushes grafted on the surfaces of two opposing infinitely long and parallel columns with rectangular cross-sections immersed in a solvent. The SCFT offers computational advantages over molecular simulations. Thus, the dependences of the interaction and mutual interpenetration between the two PE brushes on various system parameters such as the grafting density, the chain length, the charge fraction of PE chains, the ionic strength, and the brush separation can be fully explored. These dependences have rarely been investigated systematically in molecular simulations. Moreover, the amount of counter-ions trapped inside the brushes grafted on the surfaces of two columns can be conveniently analyzed numerically using SCFT, an interesting issue receiving very little attention before. The paper is organized as follows. In Section II, the theory and methods used in the present paper are described. In Section III, results and discussions concerning the brush height, the interaction and mutual penetration between the polyelectrolyte brushes, as well as the amount of counter-ions trapped inside the brushes, are presented. In Section IV, the main results are summarized.

II. Theory, model equations, and numerical methods

In this paper, we consider a system of volume V comprising two rectangle PE brushes permanently grafted on the surfaces of two infinitely long and parallel columns with rectangular cross-sections immersed in solvent with mobile point-like mono-valent cations and anions. A schematic diagram and a cartoon of the cross-section of the system are shown in Fig. 1a and b, respectively.

Fig. 1 (a) A schematic diagram of the rectangle-shaped cross-section of two infinitely long columns under study is displayed. The PE chains are grafted on the surfaces of these two columns which are immersed in a solvent. (b) A cartoon depicting the cross-section of the system including the grafted PE chains and the released counter-ions.

For simplicity, it is assumed that the system is translationally invariant in the direction perpendicular to the cross-section of the two columns (along the axial z direction). Thus, each physical quantity can be viewed as a two-dimensional one on a per unit length in the z direction basis. Accordingly, the 3D position vector r⃑ is reduced to a 2D one, i.e., r⃑ = xi⃑ + yj⃑ (i⃑ and j⃑ are unit vectors in the x, y directions, respectively). The two brushes are symmetric with respect to y-axis, and each brush is symmetric with respect to x-axis of the system (see Fig. 1a). Here, we define L_x, L_y as the system sizes along the x-axis and y-axis, respectively. We denote the distance between the two brushes as D, the length and width of rectangle-shaped cross-sections of the substrates as L₀ and W₀, respectively. The system volume per unit length in z direction is L_x × L_y. The area of the cross-section of each substrate is S = W₀ × L₀. Thus, the free volume of the system excluding the grafting substrates per unit length in z direction is V_f = L_x × L_y − 2S. The periodic boundary condition is applied to the periphery of the system, i.e., along both the x and y directions. In this paper, we use abbreviated notations such as 1D, 2D and 3D brushes to characterize a PE brush formed by grafting PE chains on a substrate with a particular geometry. In this definition, the number 1, 2, or 3 is equal to the subtraction of 3 by the number of infinitely large dimensions of the grafting substrate. Therefore, a PE brush formed by grafting PE chains on an infinitely large two dimensional surface is called 1D brush, i.e., 1 ≡ 3 − 2. The PE brush shown in Fig. 1b with the substrates of finite L₀, W₀ is defined as a 2D brush because the substrates are infinitely long in the z direction, i.e., 2 ≡ 3 − 1.

It is assumed that there are n_p PE chains per unit length along the z direction uniformly grafted onto the surfaces of the two columns. We assume that the solvent molecules and the polymer segments are of the same size and each with a volume ρ⁻¹₀ = a³. Then the grafting density which expresses the average number of grafted chains per unit surface area in the z direction is σ_g = n_p/[4(L₀ + W₀)]. The average volume fraction of the grafted chains is [small phi, Greek, macron]_P = n_PNρ⁻¹₀/[L_x·L_y − 2W₀·L₀] on a per unit length in the z direction basis with N the number of monomers per PE chain.

SCFT for PEs treats the many-chain problem as an effective single flexible charged chain in a mean-field which is to be determined self-consistently. To derive SCFT, the first step is writing down the partition function in the form of standard many-dimensional integral representation over the particle degrees of freedom. Inside the Hamiltonian, the electrostatic contribution in the form ∫dr⃑[ψ(r⃑)[small rho, Greek, circumflex]_e(r⃑) − ε|∇ψ(r⃑)|²/2] with ε, ψ, [small rho, Greek, circumflex]_e respectively denoting the dielectric permittivity of the medium, the electric potential and the total charge density, is included.⁵⁹ Next, by using the delta function transformation, the partition function is transformed into a functional integral representation over the auxiliary field functions. It is not possible to evaluate the functional integral exactly. With the mean-field approximation, the functional integral can be evaluated by the saddle-point technique. The minimization of the free energy functional with respective to the field variables leads to a set of SCF equations. In SCFT which is a mean-field statistical model, the charge density fluctuation and electrostatic correlation are neglected. Thus, it is only applicable to weakly charged PE systems. Also, SCFT models charged polymer chains as thread like, Gaussian chains. So in the present study, the strongly stretched chain regime with the dimensionless grafting density σ_ga² approaching 1 is not accessible.

Within the SCFT, the mean-field free energy in units of k_BT per chain is given in the following compact form:^59,60

In the above equation, ϕ_j with j = P, S, and ± are the dimensionless volume fractions of PE chains, solvent molecules and mobile ions. χ_PS denotes the Flory–Huggins interaction parameter between monomers and solvent molecules and ω_j denotes the conjugate interaction field within the SCFT formalism. ψ(r⃑) denotes the electric potential field in unit of k_BT/e, with k_B the Boltzmann constant, e the elementary charge and T the absolute temperature. The free energy per chain contains the enthalpic and entropic contributions, i.e., F_E = U − TS. And it can be split into various enthalpic and entropic terms: U = U_χ + U_e and S = S_P + S_S + S₊ + S₋ with each term having the following expression: U_χ/k_BT = ∫dr⃑χ_PSNϕ_Pϕ_S/V, U_e/k_BT = ∫dr⃑[−ε|∇ψ|²/2]/V, S_P/k_B = [small phi, Greek, macron]_P ln(Q_P/[small phi, Greek, macron]_P) + ∫dr⃑ω_Pϕ_P/V, S_S/k_B = N[small phi, Greek, macron]_S ln(Q_S/[small phi, Greek, macron]_S) + ∫dr⃑ω_Sϕ_S/V, S₊/k_B = N[small phi, Greek, macron]₊ ln(Q₊/[small phi, Greek, macron]₊), S₋/k_B = N[small phi, Greek, macron]₋ ln(Q₋/[small phi, Greek, macron]₋).⁶⁰ A physical explanation of the negative sign of the electrostatic energy U_e based on a capacitor model has been given by Matsen.⁶¹ The dimensionless dielectric permittivity ε = 6ε₀ε_sk_BT/(ρ₀e²a²), in which ε₀ and ε_s denote the vacuum permittivity and the dielectric constant of the system, respectively. ε = 0.4762 used in this study corresponds to ε_s = 78.0 which is equal to the dielectric constant of water at the room temperature. In this study, χ_PS is set to zero, corresponding to a good solvent condition for the PE chains in water. Such a zero χ_PS is quite reasonable because PE chain backbones contain polar ionizable functional groups.

The dimensionless mean-field SCF equations are as follows⁵⁹

1 − ϕ_P(r⃑) − ϕ_S(r⃑) = 0 (2)

ω_P(r⃑) = χ_PSNϕ_S(r⃑) + η(r⃑) (3)

ω_S(r⃑) = χ_PSNϕ_P(r⃑) + η(r⃑) (4)

ω_±(r⃑) = Nν_±ψ(r⃑) (5)

where the Lagrange multiplier η(r⃑) is invoked to enforce the incompressibility condition, α_P denotes the average degree of ionization or the charge fraction of PE chains with the smeared charge distribution, ν_P = −1, ν_± = ±1 are for the charge valences of PE chains and the mobile ions, respectively. [small phi, Greek, macron]_S and [small phi, Greek, macron]_± are respectively the average volume fraction of solvent molecules and dimensionless number densities of mobile ions in the system. Note that [small phi, Greek, macron]₊ = α_P[small phi, Greek, macron]_P + C_s, [small phi, Greek, macron]₋ = C_s with C_s the concentration of the added 1–1 salt. In eqn (1) and (6)–(8), Q_P, Q_± and Q_S are the partition functions of single PE chain, the counter-ions and solvent, respectively, which are defined as follows

It can be seen from eqn (5), (7) and (9) that, the electrostatics is on the Poisson–Boltzmann level. The SCFT equations including the following modified diffusion equation, i.e., eqn (13), are numerically solved on Cartesian grids with the grafting substrates excluded from the computational domain.⁶⁰ The grid space Δx is 0.0255N^1/2a, and the pseudo-time step Δs is in the range of 0.00125–0.005.

In eqn (6), the propagators q_f(r⃑, s) and q_g(r⃑, s) denote the probability distribution function of the end-segment of a polymer chain of length s at r⃑ with one end (q_f(s = 0)) free and the other end (q_g(s = 0)) grafted onto the substrate, respectively. They both satisfy the following modified diffusion equation:

We apply the same boundary conditions and initial conditions for q_f(r⃑, s) and q_g(r⃑, s) as those proposed by Kim and Masten,⁵³ which are suitable for polymer brushes with a fixed uniform grafting density on the surface of the grafting substrate. For the present study, the Dirichlet boundary condition is applied on the surface of the grafting substrates with q_f(r, s) = 0, q_g(r, s) = 0. The initial condition for q_f(r, s) is q_f(r, s = 0) = 1 at all computational grids except the surface grids on the substrates. At the first layer of the computational grids away from the surface of the grafting substrates, q_g(s = 0) equals the inverse of q_f(s = 1) at the same grids, while q_g(s = 0) equals zero at all other grids.

The density profile of the PE brushes is examined in the study. Due to the symmetry, we only consider the left brush which is formed by grafting PE chains on the surface of the left column. Using the propagators of the grafted PE chains, we can obtain the density profile of the left brush as:

The boundary conditions and initial conditions in numerically solving eqn (13) to obtain q_f(r⃑, s) and q_g(r⃑, s) in eqn (14) are the same as those outlined in the preceding paragraph. Note that q_g(r⃑, s) in eqn (14) is only related to the left column.

The monomer density profiles of the PE chains grafted on the left and right surfaces of the left column are

ϕ^LL_P(r⃑) is used to compute the brush height H_B of the left surface of the left brush. The brush height can be calculated through the following

The expressions for other surface brush heights have similar forms. H_R, H_T represent the brush height of the right surface, the top surface of the left brush, respectively. By symmetry, the brush height of the bottom surface of the left brush is the same as that of the top surface. ϕ^LR_P(r⃑) is used to compute the degree of interpenetration D_I between the left and right brushes which is defined as follows

The average concentration of the counter-ions of the system is ϕ^avg₊ = [small phi, Greek, macron]₊ = α_P[small phi, Greek, macron]_P. When the brush height H_R of the right surface of the left brush H_R > D/2, the volume of the brushes is V_brush = 2(H_B + W₀ + D/2)·(2H_T + L₀) − 2(W₀·L₀) and the average concentration of the counter-ions inside the two PE brushes is

If H_R < D/2, the volume of the brushes will be V_brush = 2(H_B + W₀ + H_R)·(2H_T + L₀) − 2(W₀·L₀) and the average concentration of the counter-ions inside the two PE brushes is

The percentage of the counter-ions inside the two PE brushes is

III. Results and discussion

In this study, the monomer size is fixed as a = 0.7 nm. The length of each object varies from L₀ = 2.0 × 2.887a to L₀ = 20.0 × 2.887a, which corresponds to 2.0R_g to 20.0R_g of a flexible PE chain with N = 50, where is the radius of gyration of a corresponding Gaussian chain. It should be pointed out that other length scales, such as N^3/5a (the coil size for a neutral chain in a good solvent), Na (due to the fact that the brush height is linearly proportional to the chain length as demonstrated in this paper and simple scaling arguments¹³), could be adopted as well. In principle, we are free to choose any reasonable and well defined length scale as long as it does not overrate its impact on the physical interpretation of the results. As a matter of fact, R_g was used to non-dimensionalize the SCFT equations for weakly charged and flexible PE systems by Shi and Wang.⁵⁹ R_g was also chosen as the length scale in their numerical SCFT study of the stimuli-response of charged diblock copolymer brushes by Meng and Wang.²² The width of the grafting substrates is changed from W₀ = 1.0R_g(50) to W₀ = 4.0R_g(50). Except in special circumstances, the width is fixed as W₀ = 2.0R_g(50). The grafting density of PE chains is varied from σ_g = 0.05 to σ_g = 0.4. The chain length is varied from N = 50 to N = 400. The degree of ionization is in the range of α_p = 0.05 to α_p = 0.4. The distance between the two substrates denoted as D is changed from D = 4.0R_g(50) to D = 36.0R_g(50). The concentration of the added salt is adjusted from C_s = 0.0001 to C_s = 0.05. As stated in the preceding section, χ_PS is fixed as zero.

A. The interaction and the degree of interpenetration between the two opposing PE brushes

We have examined the variations of the free energy of the system and the degree of interpenetration of the two PE brushes with the distance of separation between the opposing PE brushes. A typical example is shown in Fig. 2a and c. It can be clearly seen from Fig. 2a that, with decreasing brush separation, the free energy increases, indicating a repulsive interaction between the two opposing PE brushes. The free energy decomposition into various enthalpic and entropic terms corresponding to Fig. 2a is displayed in Fig. 2b. Fig. 2b clearly shows that, the repulsive interaction between the two opposing PE brushes stems from the entropic penalty from the PE chains, i.e., the decrease of S_P/k_B with decreasing PE brush separation D. Also, as expected, the degree of interpenetration between the two brushes increases with decreasing brush separation (see Fig. 2c). For the system corresponding to Fig. 2a, the unperturbed brush height is about 3.76R_g (see Fig. 3a). Here and in the following, the phrase “unperturbed brush height” refers to the brush height defined by eqn (16) at the condition of L_x/2 − W₀ − D/2 ≫ H_B. Please note that the periodic boundary condition is applied along the x direction. Fig. 2a and c reveal that, even in the regime of large brush separation (D ≥ 16R_g), the impact of the brush separation on the free energy and the degree of interpenetration starts to emerge. From the results shown in Fig. 2a and c and subsequent figures (Fig. 3c, 4b, 6b and c and 8a and b and the associated discussion in the main text), it can be concluded that, in the regime where the brush separation is greater than four times the unperturbed brush height, i.e., D > 4H_B, the two PE brushes are well-separated. In the regime of D < 4H_B, the two PE brushes are interacting with each other which results in compression and interpenetration of the two brushes. A very important finding in this paper is that, for two approaching PE brushes, the compression and retraction of PE brushes is more dominant than the mutual interpenetration (see Fig. 6b and c where D_I ≈ 0.16). This is the reason why the numerical value of the degree of interpenetration D_I is not big even when the two brushes are in close contact with D ≈ H_B as shown in Fig. 2c. Because the main focus of this paper is on the dependences of the degree of interpenetration between the two opposing brushes, the brush height and the amount of the counter-ions trapped inside the two brushes on various system parameters, how the interaction in terms of energetics between the opposing brushes responses to system parameters will not be discussed further.

Fig. 2 (a) Plot of the variation of the free energy of the system with the distance of separation D between the two PE brushes. The system parameters are N = 50, α_P = 0.2, σ_g = 0.1, S = W₀ × L₀ = 2.0R_g × 8.0R_g and the system size of L_x × L_y = (76.0R_g) × (44.0R_g). There is no added salt in the system. The relatively small change in the free energy upon the approach of the two PE brushes as shown in the figure is due to the facts that F_E is measured on a per chain in the unit of k_BT basis (see eqn (1)) and a much larger system size is used compared to the grafting substrates. Note that D is in the unit of R_g in the figure. (b) The free energy decomposition into various enthalpic and entropic terms due to the PE chains, counter-ions and the solvent for the systems corresponding to (a). Please refer to the definitions of U_e/k_BT, S₊/k_B, S_P/k_B and S_S/k_B which are in the main text immediately below eqn (1). (c) Plot of the dependence of the degree of interpenetration D_I on the distance between the two PE brushes is displayed. The system parameters are the same as those in (a).

Fig. 3 (a) The brush height H_B is plotted against the average degree of ionization α_P. The system parameters are N = 50, σ_g = 0.1, W₀ = 2.0R_g, L₀ = 8.0R_g, D = 16.0R_g and the system size of (76.0R_g) × (44.0R_g). There is no added salt in the system. Note that the brush height is in the unit of R_g. (b) The dependence of the percentage of trapped counter-ions inside the two PE brushes on the average degree of ionization α_P is displayed. The system parameters are the same as those in (a). (c) The plots of the degree of interpenetration D_I between the two PE brushes against the average degree of ionization α_P at different brush separations. The other system parameters are the same as (a).

B. The dependences of the brush height, the degree of interpenetration and the amount of counter-ions trapped inside the brushes on system parameters

1. The effect of the average degree of ionization of PE brushes. It can be clearly seen from Fig. 3a that, in a salt-free solution, the brush height scales linearly with the average degree of ionization of PE chains grafted on column-like objects with finite-sized rectangular cross-sections, i.e., H_B ∝ α_P. Such a linear scaling is different from the power law scaling with exponent of 1/2 for infinitely large planar (one dimensional, 1D) PE brushes in the so-called osmotic brush regime predicted by simple scaling arguments.¹³ As shown in Fig. 3b, only about half of the counter-ions released from PE chains is trapped inside the two brushes, which is in contrast to the nearly 100% trapping of counter-ions inside 1D planar osmotic PE brushes. In the present two-dimensional PE brushes as opposed to the 1D osmotic PE brushes, counter-ions have a higher dimensional space to explore, resulting in an appreciable amount of counter-ions outside the PE brushes. Also, as shown in Fig. 3b, the percentage of the counter-ions trapped inside the brushes has a weak dependence on the average degree of ionization of PE chains. It is found from the numerical study that the separation between two PE brushes D has a negligible influence on the total amount of counter-ions trapped inside the two PE brushes.

The effect of the average degree of ionization of PE brushes on the degree of interpenetration between the two PE brushes is displayed in Fig. 3c. At relatively wide brush separation (D = 8.0R_g, D = 16.0R_g) which is larger than the unperturbed brush height (see Fig. 3a), the degree of interpenetration increases slightly with α_P due to the increase of the brush height with α_P. The relative increase in D_I with α_P is rather small compared with the increase of brush height with α_P (from ∼3 to ∼5R_g). On the other hand, at a brush separation comparable with the unperturbed brush height, the degree of interpenetration is virtually insensitive to the change of α_P. These results clearly demonstrate that the compression and retraction of PE brushes is a dominant feature for two approaching PE brushes. It is interesting to note that MD simulation results have also revealed the similar feature that the PE brushes shrinks as they approach each other trying to avoid interpenetration.^23,30 It should be pointed out that in MD simulations of PE brushes, the polymer chains are fully charged, i.e., α_P = 1, well beyond the range of SCFT calculations.

2. The effect of the grafting density of PE chains. Fig. 4a shows that the brush height scales linearly with respect to the grafting density σ_g. This linear scaling is in sharp contrast to the independence of the brush height on the grafting density for 1D planar PE brushes based on simple scaling arguments.¹³ However, the linear relationship is consistent with that between the brush height and the grafting density obtained from the MD simulation by Csajka et al.¹⁹ and DPD simulation by Ibergay et al.³⁴ Moreover, the linear scaling shown in Fig. 4a is in agreement of the scaling relationship between the brush height and the grafting density proposed for the so-called non-linear osmotic brush.^14,34,42 Please bear in mind that the non-linear osmotic brush regime corresponds to highly stretched and strongly charged limit of PE brushes. As shown in the inset, the percentage of trapped counter-ions inside the brushes is nearly independent of the grafting density. Again, similar to what is shown in Fig. 3b, about half of the counter-ions is trapped inside the PE brushes.

Fig. 4 (a) The brush height H_B is plotted against the grafting density at α_P = 0.2. Other system parameters are the same as those in Fig. 3a. There is no added salt in the system. The inset shows the dependence of the percentage of trapped counter-ions inside the two PE brushes on the grafting density. (b) The plots of the degree of interpenetration D_I between the two PE brushes against the grafting density σ_g at different brush separations. The other system parameters are the same as (a).

As shown in Fig. 4b, at relatively wide brush separation beyond the unperturbed brush height, the degree of interpenetration increases slightly with increasing grafting density due to the increase of brush height with σ_g. Quite interestingly, when the brush separation (D = 4.0R_g) is comparable with the brush height (see Fig. 4a), the degree of interpenetration between the two brushes slightly decreases with increasing grafting density, presumably due to the unfavorable electrostatic repulsion between the two PE brushes. Once more, similar to the feature observed in Fig. 3c and 4b unambiguously shows that the compression and retraction of PE brushes is a dominant feature for two approaching PE brushes.

3. Effects of the chain length of PE chains. As shown in Fig. 5a and its inset, the brush height calculated from the grafted PE chains on the left side of the left substrate increases linearly with the chain length for relatively short PE chains, i.e., N ≤ 100. Such a linear scaling relationship is consistent with the prediction for osmotic PE brushes based on simple scaling arguments.¹³ With a further increase in the chain length, the brush height starts to deviate from the linear scaling law, and eventually levels off in the long chain length limit. Such a deviation from the linear scaling relationship is due to the effect of compression and mutual interpenetration of the opposing brushes introduced by the periodic boundary condition along the x direction of the system. The separation between the two brushes formed by grafting PE chains on the left side of the left substrate and the right side of the right substrate is 96a(2 × (L_x/2 − W₀ − D/2)) due to the periodic boundary condition along the x direction of the system. It can be observed from Fig. 5a that, at N = 150, the brush separation of 96a as calculated above is about 4 times the brush height. At such a brush separation, the two brushes start to interact with each other. It can be seen from Fig. 5b that, the degree of interpenetration increases rapidly with the chain length for N > 100, but seems to saturate in the long chain length limit. It should be pointed out that, even though the grafting surfaces in the present study are finite, which means that there is a reduced entropic penalty for the polymer chains, such an effect does not alter the linear scaling law between the brush height and the chain length. This is because even in the unconfined space, the size of PE chains is still a linear function of the chain length for the Gaussian chain like conformation adopted in SCFT. The free energy of a charged Gaussian chain iswhere R_e is the end-to-end distance of the PE chain, l_B denotes the Bjerrum length. In the above equation, the first and the second terms on the right hand side denote the chain conformational and electrostatic contributions to the free energy, respectively. Minimizing the above equation with respect to R_e leads to R_e ∼ N. It is interesting to compare the scaling law between the brush height and the chain length for PE brushes with that of neutral polymer brushes grafted on finite sized surfaces. By performing dynamic light scattering (DLS) experiments on neutral polymer chains grafted on spherical particles, Ohno, et al.⁶² and Dukes, et al.⁶³ found that the brush height scales respectively as H_B ∼ N^4/5 for shorter chains corresponding to the so-called concentrated polymer brush (CPB) regime, and H_B ∼ N^3/5 for long enough chains corresponding to semi-dilute polymer brush (SDPB) regime. The transition from H_B ∼ N^4/5 to H_B ∼ N^3/5 occurs at a crossover distance r_c. The scaling of H_B ∼ N^3/5 uniquely reflects the increased free space available to the chain segments further away from a curved grafting surface. Using small-angle neutron scattering (SANS) and SCFT calculation to study neutral polymer chains grafted on nano-particles, Hore et al. confirmed the above brush height scaling laws in CPB and SDPB regimes.⁶⁴ As is well-known that, the end-to-end distance of a neutral polymer chain scales as R_e ∼ N^1/2, R_e ∼ N^3/5 in theta and good solvent conditions, respectively. Note that in Hore's experiments, the neutral polymer brushes are imbedded in homo-polymer melts which correspond to a theta solvent condition.

Fig. 5 (a) Plot of the brush height H_B against the chain length of PE chains. Note that the monomer size a instead of R_g(N) is used as the length scale of H_B because N is changing. The system parameters are α_p = 0.1, σ_g = 0.1, D = 96.0a, W₀ = 6.0a, L₀ = 24.0a, And the system size is (204.0a) × (120.0a). This is no added salt in the system. In the inset, the plot of the brush height against the chain length in the range of 30 ≤ N ≤ 100 is displayed. (b) The degree of interpenetration is plotted against the chain length N. The system parameters are the same as those in (a). (c) Plots of the average concentration of the counter-ions inside the two PE brushes (red-colored curve with circles) and in the system (black-colored line with squares) against the chain length N. The system parameters are the same as those in (a). There is no added salt in the system. In the inset, the plot of corresponding percentage of the counter-ions inside the two PE brushes against the chain length N is displayed. (d) The 2D contour plot of the monomer density of two PE brushes with N = 150. Note that the x-axis, y-axis are in a unit of R_g(150), respectively. (e) The 2D contour plot of the counter-ion concentration at N = 150. In (d) and (e), the other system parameters are the same as those in (a).

The average concentration and the number percentage of the counter-ions inside the PE brushes show a non-trivial dependence on the chain length as shown in Fig. 5c and its inset. There is a minimum in both the average concentration and the percentage plots at an intermediate chain length. It can be imagined that, in the long chain length limit, the PE brushes will extend over the whole system space. Thus, in such a limit, the curve of the average concentration of the counter-ions trapped inside the PE brushes shown in the main figure will eventually merge with the straight line corresponding to the average concentration of the counter-ions in the system, and the percentage of counter-ions trapped inside the PE brushes will eventually approach 100%. Note that the percentage of counter-ions trapped inside the PE brushes is again around 50% in the parameter range of N studied.

Typical examples of the 2D contour plots of the monomer density and the counter-ion concentration are shown in Fig. 5d and e, respectively. The contours, to a large extent, exhibit an elliptic shape. Nevertheless, it is expected that, the assumption of a rectangular shape of the contours of the monomer density and the counter-ion concentration in eqn (16)–(20) should not alter the main results and conclusions in this paper in any significant way. It can be seen from Fig. 5d and e that, the extension of counter-ions is wider than that of monomers, indicating an appreciable amount of counter-ions outside the PE brushes.

4. The effect of the dimension of the substrates along the y direction. The brush height is found to increase with the dimension of the grafting substrates in the y direction, as shown in Fig. 6a. Such an increase is due to the geometric and confining effect of grafted PE chains. The PE chains in the interior of brushes are confined by nearby chains more severely than those near the corners of the columns. Thus, these inner PE chains will be stretched more strongly than those near the corners. As L₀ increases, the fraction of grafted PE chains in the interior of the brushes increases also. Therefore, the brush height defined in eqn (16) increases with increasing dimension of the grafting substrates in the y direction. Moreover, it can be seen from the inset of Fig. 6a, the degree of interpenetration between the two brushes somewhat follows the same trend as the brush height with the size of the cross-sections of the grafting substrates in the y direction. With the further increase of the y dimension of the grafting substrates, L₀ approaches the y dimension of the system L_y. Then the 2D PE brushes in the present study crossovers to 1D planar PE brushes because the periodic boundary condition is applied in the y direction in our model. In the limit of L₀ = L_y, seamless slabs along the y direction will be formed, leading to perfect 1D PE brushes. In the model building, the top/bottom surfaces of the substrates are grafted with PE chains, so the limit of L₀ = L_y can not be realized in the present model. Nevertheless, at the largest value of L₀ = 20R_g with L_y = 24R_g in Fig. 6a, the gap spacing between neighboring slab-shaped grafting substrates aligned along the y direction is only 4R_g(2 × (L_y − L₀)/2 = 4). As shown in Fig. 6b, at L₀ = 20R_g, the edge effect in the vicinity of the gaps is small, and somewhat close to perfect 1D PE brush behaviors could be expected.

Fig. 6 (a) The plot of the brush height H_B with the cross-sectional size of the grafting substrates along the y direction. The size of the substrates along the x direction is fixed at W₀ = 2.0R_g. The system parameters are α_P = 0.2, σ_g = 0.1, N = 50, D = 4.0R_g, L_x × L_y = (36.0R_g × 24.0R_g), and there is no added salt in the system. In the inset, the degree of interpenetration is plotted against the dimension of the grafting substrates in the y direction. (b) The 2D contour plot of the monomer density of two PE brushes at L₀ = 20.0R_g. The other system parameters are the same as (a). The x and y axes are in the unit of R_g. (c) The plot of the monomer density profile along the central horizontal line of (b) (at y = 0). (d) The percentage of trapped counter-ions inside the brushes is plotted against the dimension of the substrates in the y direction. The system parameters are the same as (a).

As pointed out in subsection A, the monomer density profile shown in Fig. 6b and the corresponding profile along the x axis shown in Fig. 6c together with the corresponding degree of interpenetration D_I ≈ 0.16 clearly demonstrate that the compression and retraction of PE brushes is more dominant than the mutual interpenetration for two apposing PE brushes. Data analysis of the monomer density profile of Fig. 6c shows that only about 34% of the total amount of monomers is located within 2R_g(D/2) from the grafting substrate for the unperturbed PE brush. Whereas for the two apposing PE brushes, the percentage of the total amount of monomers located within D/2(2R_g) from the grafting substrate is about 84% ((1 − D_I) × 100%, see eqn (17)). Such a significant increase from 34% to 84% results from a strong compression and retraction of grafted PE chains. Because the grafting substrates in our model, i.e., the columns, are assumed to be infinitely long in the z direction, 3D PE brushes, e.g., spherical PE brushes, can not be attained in the present model.

As the size of the grafting substrates along the y direction increases, more PE chains are grafted on the substrates, thus the average counter-ion concentration in the system increases also when the system size is fixed (note that the free volume of the system decreases slightly with increasing L₀). Numerical result also reveals that the average counter-ion concentration inside the PE brushes increases with L₀ (data not shown in Fig. 6d). With the increase of the size of the grafting substrates along the y direction, Fig. 6d clearly shows that the percentage of counter-ions trapped inside the PE brushes also increases and eventually approaches the value corresponding to 1D planar PE brushes. For 1D PE brushes, our numerical study suggests that there is still an appreciable amount of counter-ions outside the brushes.

5. The effect of the free volume of the system. The brush height is found to increase slightly with increasing free volume, and quickly approaches a constant (see Fig. 7a). Please note that the first data point in Fig. 7a corresponds to a system size in the x direction of L_x = 36R_g, such that the separation between the two brushes formed by grafting PE chains on the left side of the left substrate and the right side of the right substrate is 16R_g(2 × (L_x/2 − W₀ − D/2)). Thus the brush separation is about 4 times the unperturbed brush height (∼3.77R_g, see Fig. 7a), leading to a smaller brush height as indicated by the first data point in Fig. 7a. On the other hand, although the system dimension shown in Fig. 6a is the same as that corresponding to the first data point in Fig. 7a, due to its smaller D which is 4.0R_g, the left side of the left substrate and the right side of the right substrate are well-separated. Therefore, in Fig. 6a, the brush height measured from the left side of the left substrate differs little from that obtained by using a much larger system dimension of 76R_g × 44R_g.

Fig. 7 (a) The plot of the brush height H_B with the free volume of the system V_f. Note that the system volume per unit length in the z direction is in unit ofR²_g. The system parameters are α_P = 0.2, σ_g = 0.1, N = 50, D = 16.0R_g, S = W₀ × L₀ = 2.0R_g × 8.0R_g, L_x × L_y = 36R_g × 24R_g, 40R_g × 24R_g, 44R_g × 24R_g, 56R_g × 24R_g, 48R_g × 32R_g, 56R_g × 36R_g, 76R_g × 44R_g. There is no added salt in the system. (b) The percentage of trapped counter-ions inside the brushes is plotted against the free volume of the system. The system parameters are the same as those in (a). Note that the data in (a) and (b) are not affected by the way the system volume is increased, i.e., increasing L_x, increasing L_y, or increasing L_x and L_y simultaneously.

The numerical result reveals that the percentage of counter-ions trapped inside the brushes decreases slightly with the increase of the free volume and eventually approaches a constant (see Fig. 7b). The eventual level off of the percentage of the counter-ions trapped inside the brushes in the large system size limit reflects the competition and balance between the strong electrostatic attraction of the charges on the grafted chains with the counter-ions and the entropic effect of the counter-ions. Based on the discussion in the preceding subsection, it can be expected that, for 1D PE brushes, the percentage of the counter-ions trapped inside the brushes will also approach a constant in the large system size limit. Furthermore, it can be anticipated that, in the small free volume limit, the PE brushes will extend over the whole system. Therefore, 100% of the counter-ions will be trapped inside the brushes in such a limit.

6. The effect of the added salt. Fig. 8a shows the effect of the added salt ions on the unperturbed brush height. As expected, with increasing salt concentration, due to the electrostatic screening effect from the salt ions, the brush height decreases. At high enough salt concentration, the PE brushes behaves like neutral brushes, and the brush height is no longer dependent on the added salt. For the system shown in Fig. 8a, the Debye screening length is reduced from about 4.0a at C_S = 0 to 1.0a at C_S = 0.04, where a is the monomer size with a = 0.7 nm. For the salt-fee systems studied in the preceding sections, the Debye screening length due to the counter-ions released form the PE chains is in the range of 2.0–5.0a, which is one to two orders of magnitude smaller than the system size. As shown in Fig. 8b, the degree of interpenetration between the two opposing PE brushes exhibits a very weak dependence on the salt concentration. This again demonstrates that, for two opposing PE brushes at a separation comparable with the unperturbed brush height, due to the strong electrostatic repulsion between the two brushes at low salt concentrations, PE chains retract to try to avoid mutual interpenetration. So consistent with the result in subsection B4, the compression and retraction of PE brushes is more dominant than the mutual interpenetration for two apposing PE brushes.

Fig. 8 (a) The plot of the brush height H_B with the added salt concentration is displayed. The system parameters are α_p = 0.2, σ_g = 0.1, N = 50, D = 4.0R_g, W₀ × L₀ = 2.0R_g × 8.0R_g and the system size of (76.0R_g) × (44.0R_g). (b) The degree of interpenetration is plotted against the salt concentration. The system parameters are the same as those in (a).

IV. Summary and conclusions

In this paper, the self-consistent field theory is employed to numerically study the interaction, interpenetration between two opposing 2D polyelectrolyte (PE) brushes immersed in a solvent. The dependences of the brush height and the percentage of the counter-ions trapped inside the PE brushes on various system parameters are also investigated.

It is found that the brush height scales linearly with both the average degree of ionization of PE chains and the grafting density. Such scaling relationships deviate from the predictions of simple scaling arguments for osmotic PE brushes, but are consistent with MD and DPD simulations. In particular, the linear scaling relationship between the brush height and the grafting density is in agreement with the prediction for non-linear osmotic PE brushes. When the two PE brushes are farther apart and there is virtually no interpenetration, the brush height also scales with the chain length in a linear fashion, which is consistent with the scaling arguments. Due to the geometric and confining effect of grafted PE chains, the brush height of 2D PE brushes increases with increasing one dimensional size of the substrates, and eventually approach the brush height of 1D PE brushes. Numerical study indicates that the brush height increases slightly with increasing free volume of the system, and quickly levels off. As expected, with the addition of mono-valent salt ions, due to the electrostatic screening, the brush height decreases and approaches that of the neutral brushes.

The numerical study shows different dependences of the degree of interpenetration on system parameters at different brushes separations. When the brush separation is relatively large compared with the brush height, the degree of interpenetration between the two opposing PE brushes is found to increase with increasing average degree ionization of PE chains, the grafting density, the chain length and one dimensional size of the grafting substrates. Numerical results reveal that, at a brush separation comparable to the brush height, the degree of interpenetration does not increase further with increasing average degree of ionization of PE chains and grafting density. The saturation of the degree of interpenetration with these system parameters indicates that PE brushes shrink in order to reduce unfavorable electrostatic repulsion between the two PE brushes. Such a saturation of the degree of interpenetration has also been observed in molecular dynamics simulations. The numerical SCFT study in this paper convincingly shows that the compression and retraction of PE brushes is more dominant than the mutual interpenetration for two apposing PE brushes.

The numerical results show that, in a salt-free solution, the percentage of counter-ions trapped inside the brushes exhibits relatively weak dependence on the average degree of ionization of PE chains, and is nearly independent of the grafting density. Furthermore, the numerical study suggests that, except in the long chain length and/or small system volume limits, in a salt free solution, only about 50% of the counter-ions is trapped inside the PE brushes. This percentage around fifty is irrespective of the system size, underlining the competition and balance of the electrostatic attraction of the counter-ions to the brushes, which favors the trapping of the counter-ions inside the brush, and the entropic effect of the counter-ions, which favors the escape of the counter-ions from the brushes. By increasing the size of the grafting substrates in the lateral direction, the 2D PE brushes in the current model can crossover to 1D PE brushes. It can be inferred from the crossover from 2D to 1D PE brushes that, except in the long chain length and/or small system size limits, only about 60% of the counter-ions is trapped inside the 1D PE brushes in a salt-free solution. It is anticipated that, for 3D PE brushes such as PE chains grafted on a spherical surface, due to the stronger entropic effect of counter-ions in a higher dimensional space, in the large system size limit, the percentage of counter-ions trapped inside the brushes will approach zero. Such a study of 3D PE brushes is currently underway.

Acknowledgements

The authors thank the financial supports from the National Natural Science Foundation of China (NSFC projects 21074062, 21374052, 61320106014, 11174163), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Sate Education Ministry. C. T. acknowledges the support from K. C. Wong Magna at Ningbo University.

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